LMFDB - Embedded newform 690.3.k.b.553.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,3,Mod(277,690)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("690.277");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	690.k (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.8011382409$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			553.19
Character	$\chi$	$=$	690.553
Dual form			690.3.k.b.277.19

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(0.815565 + 4.93304i) q^{5} -2.44949 q^{6} +(5.39766 - 5.39766i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(0.815565 + 4.93304i) q^{5} -2.44949 q^{6} +(5.39766 - 5.39766i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +(-5.74860 - 4.11747i) q^{10} -16.8789 q^{11} +(2.44949 - 2.44949i) q^{12} +(-15.3246 - 15.3246i) q^{13} +10.7953i q^{14} +(-5.04285 + 7.04057i) q^{15} -4.00000 q^{16} +(20.2469 - 20.2469i) q^{17} +(-3.00000 - 3.00000i) q^{18} -27.2666i q^{19} +(9.86607 - 1.63113i) q^{20} +13.2215 q^{21} +(16.8789 - 16.8789i) q^{22} +(-3.39116 - 3.39116i) q^{23} +4.89898i q^{24} +(-23.6697 + 8.04642i) q^{25} +30.6493 q^{26} +(-3.67423 + 3.67423i) q^{27} +(-10.7953 - 10.7953i) q^{28} +1.86234i q^{29} +(-1.99772 - 12.0834i) q^{30} -56.4076 q^{31} +(4.00000 - 4.00000i) q^{32} +(-20.6724 - 20.6724i) q^{33} +40.4937i q^{34} +(31.0290 + 22.2247i) q^{35} +6.00000 q^{36} +(14.7883 - 14.7883i) q^{37} +(27.2666 + 27.2666i) q^{38} -37.5375i q^{39} +(-8.23494 + 11.4972i) q^{40} -31.1689 q^{41} +(-13.2215 + 13.2215i) q^{42} +(3.71233 + 3.71233i) q^{43} +33.7579i q^{44} +(-14.7991 + 2.44669i) q^{45} +6.78233 q^{46} +(-52.2797 + 52.2797i) q^{47} +(-4.89898 - 4.89898i) q^{48} -9.26957i q^{49} +(15.6233 - 31.7161i) q^{50} +49.5945 q^{51} +(-30.6493 + 30.6493i) q^{52} +(-29.1368 - 29.1368i) q^{53} -7.34847i q^{54} +(-13.7659 - 83.2644i) q^{55} +21.5907 q^{56} +(33.3947 - 33.3947i) q^{57} +(-1.86234 - 1.86234i) q^{58} -44.5688i q^{59} +(14.0811 + 10.0857i) q^{60} +97.7394 q^{61} +(56.4076 - 56.4076i) q^{62} +(16.1930 + 16.1930i) q^{63} +8.00000i q^{64} +(63.0987 - 88.0952i) q^{65} +41.3448 q^{66} +(29.7413 - 29.7413i) q^{67} +(-40.4937 - 40.4937i) q^{68} -8.30662i q^{69} +(-53.2538 + 8.80429i) q^{70} +112.964 q^{71} +(-6.00000 + 6.00000i) q^{72} +(-69.1242 - 69.1242i) q^{73} +29.5767i q^{74} +(-38.8442 - 19.1345i) q^{75} -54.5333 q^{76} +(-91.1068 + 91.1068i) q^{77} +(37.5375 + 37.5375i) q^{78} +17.0900i q^{79} +(-3.26226 - 19.7321i) q^{80} -9.00000 q^{81} +(31.1689 - 31.1689i) q^{82} +(70.4417 + 70.4417i) q^{83} -26.4430i q^{84} +(116.391 + 83.3659i) q^{85} -7.42467 q^{86} +(-2.28089 + 2.28089i) q^{87} +(-33.7579 - 33.7579i) q^{88} -103.307i q^{89} +(12.3524 - 17.2458i) q^{90} -165.434 q^{91} +(-6.78233 + 6.78233i) q^{92} +(-69.0850 - 69.0850i) q^{93} -104.559i q^{94} +(134.507 - 22.2377i) q^{95} +9.79796 q^{96} +(-78.1167 + 78.1167i) q^{97} +(9.26957 + 9.26957i) q^{98} -50.6368i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{2} - 8 q^{5} - 8 q^{7} + 96 q^{8}+O(q^{10}) $ 48 * q - 48 * q^2 - 8 * q^5 - 8 * q^7 + 96 * q^8	$ 48 q - 48 q^{2} - 8 q^{5} - 8 q^{7} + 96 q^{8} + 8 q^{10} - 32 q^{11} - 24 q^{13} + 24 q^{15} - 192 q^{16} + 72 q^{17} - 144 q^{18} + 32 q^{22} + 24 q^{25} + 48 q^{26} + 16 q^{28} - 24 q^{30} + 24 q^{31} + 192 q^{32} - 24 q^{33} + 288 q^{36} - 128 q^{37} - 16 q^{38} - 16 q^{40} - 40 q^{41} + 48 q^{43} - 136 q^{47} - 80 q^{50} - 48 q^{52} + 144 q^{53} - 144 q^{55} - 32 q^{56} + 96 q^{57} + 8 q^{58} + 128 q^{61} - 24 q^{62} - 24 q^{63} + 184 q^{65} + 48 q^{66} - 144 q^{68} + 40 q^{70} - 40 q^{71} - 288 q^{72} + 40 q^{73} - 72 q^{75} + 32 q^{76} - 104 q^{77} + 96 q^{78} + 32 q^{80} - 432 q^{81} + 40 q^{82} - 88 q^{85} - 96 q^{86} + 120 q^{87} - 64 q^{88} + 24 q^{90} + 144 q^{91} - 96 q^{93} + 312 q^{95} + 480 q^{97} + 584 q^{98}+O(q^{100}) $ 48 * q - 48 * q^2 - 8 * q^5 - 8 * q^7 + 96 * q^8 + 8 * q^10 - 32 * q^11 - 24 * q^13 + 24 * q^15 - 192 * q^16 + 72 * q^17 - 144 * q^18 + 32 * q^22 + 24 * q^25 + 48 * q^26 + 16 * q^28 - 24 * q^30 + 24 * q^31 + 192 * q^32 - 24 * q^33 + 288 * q^36 - 128 * q^37 - 16 * q^38 - 16 * q^40 - 40 * q^41 + 48 * q^43 - 136 * q^47 - 80 * q^50 - 48 * q^52 + 144 * q^53 - 144 * q^55 - 32 * q^56 + 96 * q^57 + 8 * q^58 + 128 * q^61 - 24 * q^62 - 24 * q^63 + 184 * q^65 + 48 * q^66 - 144 * q^68 + 40 * q^70 - 40 * q^71 - 288 * q^72 + 40 * q^73 - 72 * q^75 + 32 * q^76 - 104 * q^77 + 96 * q^78 + 32 * q^80 - 432 * q^81 + 40 * q^82 - 88 * q^85 - 96 * q^86 + 120 * q^87 - 64 * q^88 + 24 * q^90 + 144 * q^91 - 96 * q^93 + 312 * q^95 + 480 * q^97 + 584 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$4$		−	2.00000i		−	0.500000i
$5$	0.815565	+	4.93304i	0.163113	+	0.986607i
$5$	0.815565	+	4.93304i	0.163113	+	0.986607i
$6$	−2.44949			−0.408248
$7$	5.39766	−	5.39766i	0.771095	−	0.771095i	−0.207203	−	0.978298i	$-0.566436\pi$
$7$	5.39766	−	5.39766i	0.771095	−	0.771095i	0.978298	+	0.207203i	$0.0664361\pi$
$8$	2.00000	+	2.00000i	0.250000	+	0.250000i
$9$			3.00000i			0.333333i
$10$	−5.74860	−	4.11747i	−0.574860	−	0.411747i
$11$	−16.8789			−1.53445			−0.767224	−	0.641379i	$-0.778363\pi$
$11$	−16.8789			−1.53445			−0.767224	+	0.641379i	$0.778363\pi$
$12$	2.44949	−	2.44949i	0.204124	−	0.204124i
$13$	−15.3246	−	15.3246i	−1.17882	−	1.17882i	−0.980046	−	0.198772i	$-0.936305\pi$
$13$	−15.3246	−	15.3246i	−1.17882	−	1.17882i	−0.198772	−	0.980046i	$-0.563695\pi$
$14$			10.7953i			0.771095i
$15$	−5.04285	+	7.04057i	−0.336190	+	0.469371i
$16$	−4.00000			−0.250000
$17$	20.2469	−	20.2469i	1.19099	−	1.19099i	0.214203	−	0.976789i	$-0.431285\pi$
$17$	20.2469	−	20.2469i	1.19099	−	1.19099i	0.976789	−	0.214203i	$-0.0687153\pi$
$18$	−3.00000	−	3.00000i	−0.166667	−	0.166667i
$19$		−	27.2666i		−	1.43509i	−0.696514	−	0.717543i	$-0.745267\pi$
$19$		−	27.2666i		−	1.43509i	0.696514	−	0.717543i	$-0.254733\pi$
$20$	9.86607	−	1.63113i	0.493304	−	0.0815565i
$21$	13.2215			0.629596
$22$	16.8789	−	16.8789i	0.767224	−	0.767224i
$23$	−3.39116	−	3.39116i	−0.147442	−	0.147442i
$23$	−3.39116	−	3.39116i	−0.147442	−	0.147442i
$24$			4.89898i			0.204124i
$25$	−23.6697	+	8.04642i	−0.946788	+	0.321857i
$26$	30.6493			1.17882
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$	−10.7953	−	10.7953i	−0.385547	−	0.385547i
$29$			1.86234i			0.0642186i	0.999484	+	0.0321093i	$0.0102225\pi$
$29$			1.86234i			0.0642186i	−0.999484	+	0.0321093i	$0.989778\pi$
$30$	−1.99772	−	12.0834i	−0.0665906	−	0.402781i
$31$	−56.4076			−1.81960			−0.909801	−	0.415046i	$-0.863766\pi$
$31$	−56.4076			−1.81960			−0.909801	+	0.415046i	$0.863766\pi$
$32$	4.00000	−	4.00000i	0.125000	−	0.125000i
$33$	−20.6724	−	20.6724i	−0.626436	−	0.626436i
$34$			40.4937i			1.19099i
$35$	31.0290	+	22.2247i	0.886544	+	0.634992i
$36$	6.00000			0.166667
$37$	14.7883	−	14.7883i	0.399685	−	0.399685i	−0.478437	−	0.878122i	$-0.658797\pi$
$37$	14.7883	−	14.7883i	0.399685	−	0.399685i	0.878122	+	0.478437i	$0.158797\pi$
$38$	27.2666	+	27.2666i	0.717543	+	0.717543i
$39$		−	37.5375i		−	0.962501i
$40$	−8.23494	+	11.4972i	−0.205874	+	0.287430i
$41$	−31.1689			−0.760216			−0.380108	−	0.924942i	$-0.624113\pi$
$41$	−31.1689			−0.760216			−0.380108	+	0.924942i	$0.624113\pi$
$42$	−13.2215	+	13.2215i	−0.314798	+	0.314798i
$43$	3.71233	+	3.71233i	0.0863334	+	0.0863334i	0.748955	−	0.662621i	$-0.230556\pi$
$43$	3.71233	+	3.71233i	0.0863334	+	0.0863334i	−0.662621	+	0.748955i	$0.730556\pi$
$44$			33.7579i			0.767224i
$45$	−14.7991	+	2.44669i	−0.328869	+	0.0543710i
$46$	6.78233			0.147442
$47$	−52.2797	+	52.2797i	−1.11233	+	1.11233i	−0.119500	+	0.992834i	$0.538129\pi$
$47$	−52.2797	+	52.2797i	−1.11233	+	1.11233i	−0.992834	+	0.119500i	$0.961871\pi$
$48$	−4.89898	−	4.89898i	−0.102062	−	0.102062i
$49$		−	9.26957i		−	0.189175i
$50$	15.6233	−	31.7161i	0.312466	−	0.634323i
$51$	49.5945			0.972441
$52$	−30.6493	+	30.6493i	−0.589409	+	0.589409i
$53$	−29.1368	−	29.1368i	−0.549752	−	0.549752i	0.376617	−	0.926369i	$-0.377087\pi$
$53$	−29.1368	−	29.1368i	−0.549752	−	0.549752i	−0.926369	+	0.376617i	$0.877087\pi$
$54$		−	7.34847i		−	0.136083i
$55$	−13.7659	−	83.2644i	−0.250288	−	1.51390i
$56$	21.5907			0.385547
$57$	33.3947	−	33.3947i	0.585871	−	0.585871i
$58$	−1.86234	−	1.86234i	−0.0321093	−	0.0321093i
$59$		−	44.5688i		−	0.755404i	−0.925927	−	0.377702i	$-0.876714\pi$
$59$		−	44.5688i		−	0.755404i	0.925927	−	0.377702i	$-0.123286\pi$
$60$	14.0811	+	10.0857i	0.234686	+	0.168095i
$61$	97.7394			1.60228			0.801142	−	0.598474i	$-0.204226\pi$
$61$	97.7394			1.60228			0.801142	+	0.598474i	$0.204226\pi$
$62$	56.4076	−	56.4076i	0.909801	−	0.909801i
$63$	16.1930	+	16.1930i	0.257032	+	0.257032i
$64$			8.00000i			0.125000i
$65$	63.0987	−	88.0952i	0.970750	−	1.35531i
$66$	41.3448			0.626436
$67$	29.7413	−	29.7413i	0.443900	−	0.443900i	−0.449420	−	0.893321i	$-0.648369\pi$
$67$	29.7413	−	29.7413i	0.443900	−	0.443900i	0.893321	+	0.449420i	$0.148369\pi$
$68$	−40.4937	−	40.4937i	−0.595496	−	0.595496i
$69$		−	8.30662i		−	0.120386i
$70$	−53.2538	+	8.80429i	−0.760768	+	0.125776i
$71$	112.964			1.59104			0.795520	−	0.605928i	$-0.207198\pi$
$71$	112.964			1.59104			0.795520	+	0.605928i	$0.207198\pi$
$72$	−6.00000	+	6.00000i	−0.0833333	+	0.0833333i
$73$	−69.1242	−	69.1242i	−0.946907	−	0.946907i	0.0517525	−	0.998660i	$-0.483519\pi$
$73$	−69.1242	−	69.1242i	−0.946907	−	0.946907i	−0.998660	+	0.0517525i	$0.983519\pi$
$74$			29.5767i			0.399685i
$75$	−38.8442	−	19.1345i	−0.517922	−	0.255127i
$76$	−54.5333			−0.717543
$77$	−91.1068	+	91.1068i	−1.18321	+	1.18321i
$78$	37.5375	+	37.5375i	0.481250	+	0.481250i
$79$			17.0900i			0.216329i	0.994133	+	0.108164i	$0.0344973\pi$
$79$			17.0900i			0.216329i	−0.994133	+	0.108164i	$0.965503\pi$
$80$	−3.26226	−	19.7321i	−0.0407782	−	0.246652i
$81$	−9.00000			−0.111111
$82$	31.1689	−	31.1689i	0.380108	−	0.380108i
$83$	70.4417	+	70.4417i	0.848695	+	0.848695i	0.989970	−	0.141275i	$-0.0451203\pi$
$83$	70.4417	+	70.4417i	0.848695	+	0.848695i	−0.141275	+	0.989970i	$0.545120\pi$
$84$		−	26.4430i		−	0.314798i
$85$	116.391	+	83.3659i	1.36931	+	0.980775i
$86$	−7.42467			−0.0863334
$87$	−2.28089	+	2.28089i	−0.0262171	+	0.0262171i
$88$	−33.7579	−	33.7579i	−0.383612	−	0.383612i
$89$		−	103.307i		−	1.16075i	−0.814349	−	0.580376i	$-0.802906\pi$
$89$		−	103.307i		−	1.16075i	0.814349	−	0.580376i	$-0.197094\pi$
$90$	12.3524	−	17.2458i	0.137249	−	0.191620i
$91$	−165.434			−1.81796
$92$	−6.78233	+	6.78233i	−0.0737210	+	0.0737210i
$93$	−69.0850	−	69.0850i	−0.742849	−	0.742849i
$94$		−	104.559i		−	1.11233i
$95$	134.507	−	22.2377i	1.41587	−	0.234081i
$96$	9.79796			0.102062
$97$	−78.1167	+	78.1167i	−0.805326	+	0.805326i	−0.983922	−	0.178596i	$-0.942844\pi$
$97$	−78.1167	+	78.1167i	−0.805326	+	0.805326i	0.178596	+	0.983922i	$0.442844\pi$
$98$	9.26957	+	9.26957i	0.0945875	+	0.0945875i
$99$		−	50.6368i		−	0.511483i
$100$	16.0928	+	47.3394i	0.160928	+	0.473394i
$101$	−35.5554			−0.352034			−0.176017	−	0.984387i	$-0.556321\pi$
$101$	−35.5554			−0.352034			−0.176017	+	0.984387i	$0.556321\pi$
$102$	−49.5945	+	49.5945i	−0.486220	+	0.486220i
$103$	93.4693	+	93.4693i	0.907469	+	0.907469i	0.996067	−	0.0885985i	$-0.0282388\pi$
$103$	93.4693	+	93.4693i	0.907469	+	0.907469i	−0.0885985	+	0.996067i	$0.528239\pi$
$104$		−	61.2985i		−	0.589409i
$105$	10.7830	+	65.2223i	0.102695	+	0.621164i
$106$	58.2737			0.549752
$107$	−72.7144	+	72.7144i	−0.679574	+	0.679574i	−0.959904	−	0.280330i	$-0.909556\pi$
$107$	−72.7144	+	72.7144i	−0.679574	+	0.679574i	0.280330	+	0.959904i	$0.409556\pi$
$108$	7.34847	+	7.34847i	0.0680414	+	0.0680414i
$109$		−	98.3783i		−	0.902553i	−0.892384	−	0.451277i	$-0.850969\pi$
$109$		−	98.3783i		−	0.902553i	0.892384	−	0.451277i	$-0.149031\pi$
$110$	97.0302	+	69.4985i	0.882093	+	0.631805i
$111$	36.2239			0.326341
$112$	−21.5907	+	21.5907i	−0.192774	+	0.192774i
$113$	−3.23925	−	3.23925i	−0.0286660	−	0.0286660i	0.692629	−	0.721294i	$-0.256453\pi$
$113$	−3.23925	−	3.23925i	−0.0286660	−	0.0286660i	−0.721294	+	0.692629i	$0.756453\pi$
$114$			66.7893i			0.585871i
$115$	13.9630	−	19.4945i	0.121418	−	0.169517i
$116$	3.72468			0.0321093
$117$	45.9739	−	45.9739i	0.392939	−	0.392939i
$118$	44.5688	+	44.5688i	0.377702	+	0.377702i
$119$		−	218.572i		−	1.83674i
$120$	−24.1668	+	3.99544i	−0.201390	+	0.0332953i
$121$	163.898			1.35453
$122$	−97.7394	+	97.7394i	−0.801142	+	0.801142i
$123$	−38.1739	−	38.1739i	−0.310357	−	0.310357i
$124$			112.815i			0.909801i
$125$	−58.9975	−	110.201i	−0.471980	−	0.881609i
$126$	−32.3860			−0.257032
$127$	−21.7786	+	21.7786i	−0.171485	+	0.171485i	−0.787632	−	0.616147i	$-0.788693\pi$
$127$	−21.7786	+	21.7786i	−0.171485	+	0.171485i	0.616147	+	0.787632i	$0.288693\pi$
$128$	−8.00000	−	8.00000i	−0.0625000	−	0.0625000i
$129$			9.09333i			0.0704909i
$130$	24.9965	+	151.194i	0.192280	+	1.16303i
$131$	240.543			1.83620			0.918101	−	0.396346i	$-0.129722\pi$
$131$	240.543			1.83620			0.918101	+	0.396346i	$0.129722\pi$
$132$	−41.3448	+	41.3448i	−0.313218	+	0.313218i
$133$	−147.176	−	147.176i	−1.10659	−	1.10659i
$134$			59.4827i			0.443900i
$135$	−21.1217	−	15.1286i	−0.156457	−	0.112063i
$136$	80.9875			0.595496
$137$	50.7479	−	50.7479i	0.370423	−	0.370423i	−0.497208	−	0.867631i	$-0.665642\pi$
$137$	50.7479	−	50.7479i	0.370423	−	0.370423i	0.867631	+	0.497208i	$0.165642\pi$
$138$	8.30662	+	8.30662i	0.0601929	+	0.0601929i
$139$		−	76.8919i		−	0.553179i	−0.960988	−	0.276589i	$-0.910796\pi$
$139$		−	76.8919i		−	0.553179i	0.960988	−	0.276589i	$-0.0892042\pi$
$140$	44.4495	−	62.0581i	0.317496	−	0.443272i
$141$	−128.059			−0.908217
$142$	−112.964	+	112.964i	−0.795520	+	0.795520i
$143$	258.663	+	258.663i	1.80883	+	1.80883i
$144$		−	12.0000i		−	0.0833333i
$145$	−9.18699	+	1.51886i	−0.0633585	+	0.0104749i
$146$	138.248			0.946907
$147$	11.3529	−	11.3529i	0.0772303	−	0.0772303i
$148$	−29.5767	−	29.5767i	−0.199842	−	0.199842i
$149$		−	211.594i		−	1.42009i	−0.704155	−	0.710046i	$-0.748674\pi$
$149$		−	211.594i		−	1.42009i	0.704155	−	0.710046i	$-0.251326\pi$
$150$	57.9787	−	19.7096i	0.386525	−	0.131398i
$151$	−264.463			−1.75141			−0.875705	−	0.482846i	$-0.839603\pi$
$151$	−264.463			−1.75141			−0.875705	+	0.482846i	$0.839603\pi$
$152$	54.5333	−	54.5333i	0.358771	−	0.358771i
$153$	60.7406	+	60.7406i	0.396997	+	0.396997i
$154$		−	182.214i		−	1.18321i
$155$	−46.0041	−	278.261i	−0.296801	−	1.79523i
$156$	−75.0750			−0.481250
$157$	66.1849	−	66.1849i	0.421560	−	0.421560i	−0.464181	−	0.885741i	$-0.653651\pi$
$157$	66.1849	−	66.1849i	0.421560	−	0.421560i	0.885741	+	0.464181i	$0.153651\pi$
$158$	−17.0900	−	17.0900i	−0.108164	−	0.108164i
$159$		−	71.3704i		−	0.448870i
$160$	22.9944	+	16.4699i	0.143715	+	0.102937i
$161$	−36.6087			−0.227384
$162$	9.00000	−	9.00000i	0.0555556	−	0.0555556i
$163$	−139.340	−	139.340i	−0.854848	−	0.854848i	0.135878	−	0.990726i	$-0.456615\pi$
$163$	−139.340	−	139.340i	−0.854848	−	0.854848i	−0.990726	+	0.135878i	$0.956615\pi$
$164$			62.3377i			0.380108i
$165$	85.1179	−	118.837i	0.515866	−	0.720226i
$166$	−140.883			−0.848695
$167$	−130.406	+	130.406i	−0.780877	+	0.780877i	−0.979979	−	0.199102i	$-0.936197\pi$
$167$	−130.406	+	130.406i	−0.780877	+	0.780877i	0.199102	+	0.979979i	$0.436197\pi$
$168$	26.4430	+	26.4430i	0.157399	+	0.157399i
$169$			300.689i			1.77922i
$170$	−199.757	+	33.0253i	−1.17504	+	0.194266i
$171$	81.7999			0.478362
$172$	7.42467	−	7.42467i	0.0431667	−	0.0431667i
$173$	5.87862	+	5.87862i	0.0339805	+	0.0339805i	0.723893	−	0.689912i	$-0.242351\pi$
$173$	5.87862	+	5.87862i	0.0339805	+	0.0339805i	−0.689912	+	0.723893i	$0.742351\pi$
$174$		−	4.56178i		−	0.0262171i
$175$	−84.3293	+	171.193i	−0.481881	+	0.978246i
$176$	67.5157			0.383612
$177$	54.5854	−	54.5854i	0.308392	−	0.308392i
$178$	103.307	+	103.307i	0.580376	+	0.580376i
$179$			149.359i			0.834409i	0.908813	+	0.417205i	$0.136990\pi$
$179$			149.359i			0.834409i	−0.908813	+	0.417205i	$0.863010\pi$
$180$	4.89339	+	29.5982i	0.0271855	+	0.164435i
$181$	−43.4397			−0.239998			−0.119999	−	0.992774i	$-0.538289\pi$
$181$	−43.4397			−0.239998			−0.119999	+	0.992774i	$0.538289\pi$
$182$	165.434	−	165.434i	0.908980	−	0.908980i
$183$	119.706	+	119.706i	0.654130	+	0.654130i
$184$		−	13.5647i		−	0.0737210i
$185$	85.0123	+	60.8906i	0.459526	+	0.329138i
$186$	138.170			0.742849
$187$	−341.745	+	341.745i	−1.82752	+	1.82752i
$188$	104.559	+	104.559i	0.556167	+	0.556167i
$189$			39.6646i			0.209865i
$190$	−112.270	+	156.745i	−0.590892	+	0.824974i
$191$	−157.052			−0.822261			−0.411131	−	0.911576i	$-0.634866\pi$
$191$	−157.052			−0.822261			−0.411131	+	0.911576i	$0.634866\pi$
$192$	−9.79796	+	9.79796i	−0.0510310	+	0.0510310i
$193$	30.1699	+	30.1699i	0.156320	+	0.156320i	0.780934	−	0.624614i	$-0.214743\pi$
$193$	30.1699	+	30.1699i	0.156320	+	0.156320i	−0.624614	+	0.780934i	$0.714743\pi$
$194$		−	156.233i		−	0.805326i
$195$	185.174	−	30.6143i	0.949610	−	0.156996i
$196$	−18.5391			−0.0945875
$197$	129.404	−	129.404i	0.656875	−	0.656875i	−0.297764	−	0.954639i	$-0.596241\pi$
$197$	129.404	−	129.404i	0.656875	−	0.656875i	0.954639	+	0.297764i	$0.0962409\pi$
$198$	50.6368	+	50.6368i	0.255741	+	0.255741i
$199$			38.6175i			0.194058i	0.995282	+	0.0970290i	$0.0309339\pi$
$199$			38.6175i			0.194058i	−0.995282	+	0.0970290i	$0.969066\pi$
$200$	−63.4323	−	31.2466i	−0.317161	−	0.156233i
$201$	72.8511			0.362443
$202$	35.5554	−	35.5554i	0.176017	−	0.176017i
$203$	10.0523	+	10.0523i	0.0495186	+	0.0495186i
$204$		−	99.1890i		−	0.486220i
$205$	−25.4202	−	153.757i	−0.124001	−	0.750035i
$206$	−186.939			−0.907469
$207$	10.1735	−	10.1735i	0.0491473	−	0.0491473i
$208$	61.2985	+	61.2985i	0.294704	+	0.294704i
$209$			460.231i			2.20206i
$210$	−76.0053	−	54.4393i	−0.361930	−	0.259235i
$211$	229.400			1.08721			0.543603	−	0.839343i	$-0.317060\pi$
$211$	229.400			1.08721			0.543603	+	0.839343i	$0.317060\pi$
$212$	−58.2737	+	58.2737i	−0.274876	+	0.274876i
$213$	138.352	+	138.352i	0.649539	+	0.649539i
$214$		−	145.429i		−	0.679574i
$215$	−15.2854	+	21.3407i	−0.0710950	+	0.0992592i
$216$	−14.6969			−0.0680414
$217$	−304.469	+	304.469i	−1.40309	+	1.40309i
$218$	98.3783	+	98.3783i	0.451277	+	0.451277i
$219$		−	169.319i		−	0.773147i
$220$	−166.529	+	27.5317i	−0.756949	+	0.125144i
$221$	−620.551			−2.80792
$222$	−36.2239	+	36.2239i	−0.163171	+	0.163171i
$223$	−118.733	−	118.733i	−0.532433	−	0.532433i	0.388863	−	0.921296i	$-0.372868\pi$
$223$	−118.733	−	118.733i	−0.532433	−	0.532433i	−0.921296	+	0.388863i	$0.872868\pi$
$224$		−	43.1813i		−	0.192774i
$225$	−24.1393	−	71.0091i	−0.107286	−	0.315596i
$226$	6.47851			0.0286660
$227$	−140.299	+	140.299i	−0.618059	+	0.618059i	−0.945033	−	0.326974i	$-0.893971\pi$
$227$	−140.299	+	140.299i	−0.618059	+	0.618059i	0.326974	+	0.945033i	$0.393971\pi$
$228$	−66.7893	−	66.7893i	−0.292936	−	0.292936i
$229$			208.203i			0.909183i	0.890700	+	0.454591i	$0.150215\pi$
$229$			208.203i			0.909183i	−0.890700	+	0.454591i	$0.849785\pi$
$230$	5.53143	+	33.4575i	0.0240497	+	0.145467i
$231$	−223.165			−0.966083
$232$	−3.72468	+	3.72468i	−0.0160547	+	0.0160547i
$233$	237.245	+	237.245i	1.01822	+	1.01822i	0.999831	+	0.0183890i	$0.00585372\pi$
$233$	237.245	+	237.245i	1.01822	+	1.01822i	0.0183890	+	0.999831i	$0.494146\pi$
$234$			91.9478i			0.392939i
$235$	−300.535	−	215.260i	−1.27887	−	0.916001i
$236$	−89.1376			−0.377702
$237$	−20.9309	+	20.9309i	−0.0883159	+	0.0883159i
$238$	218.572	+	218.572i	0.918368	+	0.918368i
$239$		−	101.498i		−	0.424678i	−0.977196	−	0.212339i	$-0.931892\pi$
$239$		−	101.498i		−	0.424678i	0.977196	−	0.212339i	$-0.0681081\pi$
$240$	20.1714	−	28.1623i	0.0840475	−	0.117343i
$241$	78.4578			0.325551			0.162775	−	0.986663i	$-0.447955\pi$
$241$	78.4578			0.325551			0.162775	+	0.986663i	$0.447955\pi$
$242$	−163.898	+	163.898i	−0.677265	+	0.677265i
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$		−	195.479i		−	0.801142i
$245$	45.7271	−	7.55994i	0.186641	−	0.0308569i
$246$	76.3478			0.310357
$247$	−417.851	+	417.851i	−1.69170	+	1.69170i
$248$	−112.815	−	112.815i	−0.454900	−	0.454900i
$249$			172.546i			0.692957i
$250$	169.199	+	51.2037i	0.676795	+	0.204815i
$251$	−480.688			−1.91509			−0.957545	−	0.288282i	$-0.906916\pi$
$251$	−480.688			−1.91509			−0.957545	+	0.288282i	$0.906916\pi$
$252$	32.3860	−	32.3860i	0.128516	−	0.128516i
$253$	57.2392	+	57.2392i	0.226242	+	0.226242i
$254$		−	43.5572i		−	0.171485i
$255$	40.4475	+	244.651i	0.158618	+	0.959417i
$256$	16.0000			0.0625000
$257$	−128.059	+	128.059i	−0.498284	+	0.498284i	−0.910903	−	0.412619i	$-0.864614\pi$
$257$	−128.059	+	128.059i	−0.498284	+	0.498284i	0.412619	+	0.910903i	$0.364614\pi$
$258$	−9.09333	−	9.09333i	−0.0352454	−	0.0352454i
$259$		−	159.645i		−	0.616390i
$260$	−176.190	−	126.197i	−0.677655	−	0.485375i
$261$	−5.58702			−0.0214062
$262$	−240.543	+	240.543i	−0.918101	+	0.918101i
$263$	−109.271	−	109.271i	−0.415480	−	0.415480i	0.468163	−	0.883642i	$-0.344916\pi$
$263$	−109.271	−	109.271i	−0.415480	−	0.415480i	−0.883642	+	0.468163i	$0.844916\pi$
$264$		−	82.6895i		−	0.313218i
$265$	119.970	−	167.496i	0.452717	−	0.632061i
$266$	294.352			1.10659
$267$	126.525	−	126.525i	0.473875	−	0.473875i
$268$	−59.4827	−	59.4827i	−0.221950	−	0.221950i
$269$			262.095i			0.974330i	0.873310	+	0.487165i	$0.161969\pi$
$269$			262.095i			0.974330i	−0.873310	+	0.487165i	$0.838031\pi$
$270$	36.2503	−	5.99315i	0.134260	−	0.0221969i
$271$	6.42234			0.0236987			0.0118493	−	0.999930i	$-0.496228\pi$
$271$	6.42234			0.0236987			0.0118493	+	0.999930i	$0.496228\pi$
$272$	−80.9875	+	80.9875i	−0.297748	+	0.297748i
$273$	−202.615	−	202.615i	−0.742179	−	0.742179i
$274$			101.496i			0.370423i
$275$	399.519	−	135.815i	1.45280	−	0.493873i
$276$	−16.6132			−0.0601929
$277$	52.1441	−	52.1441i	0.188246	−	0.188246i	−0.606692	−	0.794937i	$-0.707504\pi$
$277$	52.1441	−	52.1441i	0.188246	−	0.188246i	0.794937	+	0.606692i	$0.207504\pi$
$278$	76.8919	+	76.8919i	0.276589	+	0.276589i
$279$		−	169.223i		−	0.606534i
$280$	17.6086	+	106.508i	0.0628878	+	0.380384i
$281$	295.578			1.05188			0.525939	−	0.850522i	$-0.323714\pi$
$281$	295.578			1.05188			0.525939	+	0.850522i	$0.323714\pi$
$282$	128.059	−	128.059i	0.454109	−	0.454109i
$283$	−264.543	−	264.543i	−0.934782	−	0.934782i	0.0632181	−	0.998000i	$-0.479864\pi$
$283$	−264.543	−	264.543i	−0.934782	−	0.934782i	−0.998000	+	0.0632181i	$0.979864\pi$
$284$		−	225.928i		−	0.795520i
$285$	191.973	+	137.502i	0.673588	+	0.482462i
$286$	−517.327			−1.80883
$287$	−168.239	+	168.239i	−0.586199	+	0.586199i
$288$	12.0000	+	12.0000i	0.0416667	+	0.0416667i
$289$		−	530.871i		−	1.83692i
$290$	7.66813	−	10.7058i	0.0264418	−	0.0369167i
$291$	−191.346			−0.657546
$292$	−138.248	+	138.248i	−0.473454	+	0.473454i
$293$	167.891	+	167.891i	0.573007	+	0.573007i	0.932968	−	0.359960i	$-0.117210\pi$
$293$	167.891	+	167.891i	0.573007	+	0.573007i	−0.359960	+	0.932968i	$0.617210\pi$
$294$			22.7057i			0.0772303i
$295$	219.860	−	36.3488i	0.745287	−	0.123216i
$296$	59.1533			0.199842
$297$	62.0171	−	62.0171i	0.208812	−	0.208812i
$298$	211.594	+	211.594i	0.710046	+	0.710046i
$299$			103.937i			0.347614i
$300$	−38.2691	+	77.6883i	−0.127564	+	0.258961i
$301$	40.0759			0.133142
$302$	264.463	−	264.463i	0.875705	−	0.875705i
$303$	−43.5463	−	43.5463i	−0.143717	−	0.143717i
$304$			109.067i			0.358771i
$305$	79.7128	+	482.152i	0.261353	+	1.58083i
$306$	−121.481			−0.396997
$307$	61.9635	−	61.9635i	0.201835	−	0.201835i	−0.598951	−	0.800786i	$-0.704415\pi$
$307$	61.9635	−	61.9635i	0.201835	−	0.201835i	0.800786	+	0.598951i	$0.204415\pi$
$308$	182.214	+	182.214i	0.591603	+	0.591603i
$309$			228.952i			0.740945i
$310$	324.265	+	232.257i	1.04602	+	0.749216i
$311$	−23.6111			−0.0759198			−0.0379599	−	0.999279i	$-0.512086\pi$
$311$	−23.6111			−0.0759198			−0.0379599	+	0.999279i	$0.512086\pi$
$312$	75.0750	−	75.0750i	0.240625	−	0.240625i
$313$	5.59915	+	5.59915i	0.0178887	+	0.0178887i	0.715995	−	0.698106i	$-0.245974\pi$
$313$	5.59915	+	5.59915i	0.0178887	+	0.0178887i	−0.698106	+	0.715995i	$0.745974\pi$
$314$			132.370i			0.421560i
$315$	−66.6742	+	93.0871i	−0.211664	+	0.295515i
$316$	34.1800			0.108164
$317$	191.306	−	191.306i	0.603490	−	0.603490i	−0.337747	−	0.941237i	$-0.609665\pi$
$317$	191.306	−	191.306i	0.603490	−	0.603490i	0.941237	+	0.337747i	$0.109665\pi$
$318$	71.3704	+	71.3704i	0.224435	+	0.224435i
$319$		−	31.4343i		−	0.0985401i
$320$	−39.4643	+	6.52452i	−0.123326	+	0.0203891i
$321$	−178.113			−0.554870
$322$	36.6087	−	36.6087i	0.113692	−	0.113692i
$323$	−552.064	−	552.064i	−1.70918	−	1.70918i
$324$			18.0000i			0.0555556i
$325$	486.038	+	239.421i	1.49550	+	0.736680i
$326$	278.680			0.854848
$327$	120.488	−	120.488i	0.368466	−	0.368466i
$328$	−62.3377	−	62.3377i	−0.190054	−	0.190054i
$329$			564.377i			1.71543i
$330$	33.7193	+	203.955i	0.102180	+	0.618046i
$331$	141.318			0.426944			0.213472	−	0.976949i	$-0.431523\pi$
$331$	141.318			0.426944			0.213472	+	0.976949i	$0.431523\pi$
$332$	140.883	−	140.883i	0.424348	−	0.424348i
$333$	44.3650	+	44.3650i	0.133228	+	0.133228i
$334$		−	260.813i		−	0.780877i
$335$	170.971	+	122.459i	0.510361	+	0.365549i
$336$	−52.8861			−0.157399
$337$	68.1580	−	68.1580i	0.202249	−	0.202249i	−0.598714	−	0.800963i	$-0.704321\pi$
$337$	68.1580	−	68.1580i	0.202249	−	0.202249i	0.800963	+	0.598714i	$0.204321\pi$
$338$	−300.689	−	300.689i	−0.889611	−	0.889611i
$339$		−	7.93452i		−	0.0234057i
$340$	166.732	−	232.782i	0.490388	−	0.684654i
$341$	952.100			2.79208
$342$	−81.7999	+	81.7999i	−0.239181	+	0.239181i
$343$	214.452	+	214.452i	0.625223	+	0.625223i
$344$			14.8493i			0.0431667i
$345$	40.9769	−	6.77459i	0.118774	−	0.0196365i
$346$	−11.7572			−0.0339805
$347$	−331.456	+	331.456i	−0.955204	+	0.955204i	−0.999039	−	0.0438349i	$-0.986042\pi$
$347$	−331.456	+	331.456i	−0.955204	+	0.955204i	0.0438349	+	0.999039i	$0.486042\pi$
$348$	4.56178	+	4.56178i	0.0131086	+	0.0131086i
$349$		−	341.273i		−	0.977859i	−0.872323	−	0.488930i	$-0.837388\pi$
$349$		−	341.273i		−	0.977859i	0.872323	−	0.488930i	$-0.162612\pi$
$350$	−86.8638	−	255.522i	−0.248182	−	0.730064i
$351$	112.613			0.320834
$352$	−67.5157	+	67.5157i	−0.191806	+	0.191806i
$353$	173.856	+	173.856i	0.492509	+	0.492509i	0.909096	−	0.416587i	$-0.136774\pi$
$353$	173.856	+	173.856i	0.492509	+	0.492509i	−0.416587	+	0.909096i	$0.636774\pi$
$354$			109.171i			0.308392i
$355$	92.1293	+	557.254i	0.259519	+	1.56973i
$356$	−206.614			−0.580376
$357$	267.694	−	267.694i	0.749844	−	0.749844i
$358$	−149.359	−	149.359i	−0.417205	−	0.417205i
$359$		−	97.4524i		−	0.271455i	−0.990746	−	0.135728i	$-0.956663\pi$
$359$		−	97.4524i		−	0.271455i	0.990746	−	0.135728i	$-0.0433372\pi$
$360$	−34.4916	−	24.7048i	−0.0958100	−	0.0686245i
$361$	−382.469			−1.05947
$362$	43.4397	−	43.4397i	0.119999	−	0.119999i
$363$	200.733	+	200.733i	0.552985	+	0.552985i
$364$			330.869i			0.908980i
$365$	284.617	−	397.368i	0.779773	−	1.08868i
$366$	−239.412			−0.654130
$367$	323.497	−	323.497i	0.881462	−	0.881462i	−0.112221	−	0.993683i	$-0.535796\pi$
$367$	323.497	−	323.497i	0.881462	−	0.881462i	0.993683	+	0.112221i	$0.0357964\pi$
$368$	13.5647	+	13.5647i	0.0368605	+	0.0368605i
$369$		−	93.5066i		−	0.253405i
$370$	−145.903	+	24.1217i	−0.394332	+	0.0651938i
$371$	−314.542			−0.847821
$372$	−138.170	+	138.170i	−0.371425	+	0.371425i
$373$	70.3440	+	70.3440i	0.188590	+	0.188590i	0.795086	−	0.606496i	$-0.207426\pi$
$373$	70.3440	+	70.3440i	0.188590	+	0.188590i	−0.606496	+	0.795086i	$0.707426\pi$
$374$		−	683.491i		−	1.82752i
$375$	62.7114	−	207.225i	0.167231	−	0.552600i
$376$	−209.119			−0.556167
$377$	28.5397	−	28.5397i	0.0757020	−	0.0757020i
$378$	−39.6646	−	39.6646i	−0.104933	−	0.104933i
$379$		−	339.778i		−	0.896513i	−0.893905	−	0.448256i	$-0.852045\pi$
$379$		−	339.778i		−	0.896513i	0.893905	−	0.448256i	$-0.147955\pi$
$380$	−44.4754	−	269.015i	−0.117041	−	0.707933i
$381$	−53.3464			−0.140017
$382$	157.052	−	157.052i	0.411131	−	0.411131i
$383$	−6.13355	−	6.13355i	−0.0160145	−	0.0160145i	0.699054	−	0.715069i	$-0.253605\pi$
$383$	−6.13355	−	6.13355i	−0.0160145	−	0.0160145i	−0.715069	+	0.699054i	$0.753605\pi$
$384$		−	19.5959i		−	0.0510310i
$385$	−523.737	−	375.130i	−1.36035	−	0.974363i
$386$	−60.3397			−0.156320
$387$	−11.1370	+	11.1370i	−0.0287778	+	0.0287778i
$388$	156.233	+	156.233i	0.402663	+	0.402663i
$389$		−	316.963i		−	0.814814i	−0.913247	−	0.407407i	$-0.866433\pi$
$389$		−	316.963i		−	0.814814i	0.913247	−	0.407407i	$-0.133567\pi$
$390$	−154.560	+	215.788i	−0.396307	+	0.553303i
$391$	−137.321			−0.351204
$392$	18.5391	−	18.5391i	0.0472937	−	0.0472937i
$393$	294.603	+	294.603i	0.749627	+	0.749627i
$394$			258.809i			0.656875i
$395$	−84.3056	+	13.9380i	−0.213432	+	0.0352861i
$396$	−101.274			−0.255741
$397$	73.1005	−	73.1005i	0.184132	−	0.184132i	−0.609021	−	0.793154i	$-0.708438\pi$
$397$	73.1005	−	73.1005i	0.184132	−	0.184132i	0.793154	+	0.609021i	$0.208438\pi$
$398$	−38.6175	−	38.6175i	−0.0970290	−	0.0970290i
$399$		−	360.506i		−	0.903525i
$400$	94.6788	−	32.1857i	0.236697	−	0.0804642i
$401$	−313.533			−0.781878			−0.390939	−	0.920417i	$-0.627850\pi$
$401$	−313.533			−0.781878			−0.390939	+	0.920417i	$0.627850\pi$
$402$	−72.8511	+	72.8511i	−0.181222	+	0.181222i
$403$	864.426	+	864.426i	2.14498	+	2.14498i
$404$			71.1108i			0.176017i
$405$	−7.34008	−	44.3973i	−0.0181237	−	0.109623i
$406$	−20.1046			−0.0495186
$407$	−249.611	+	249.611i	−0.613295	+	0.613295i
$408$	99.1890	+	99.1890i	0.243110	+	0.243110i
$409$		−	543.161i		−	1.32802i	−0.747723	−	0.664011i	$-0.768853\pi$
$409$		−	543.161i		−	1.32802i	0.747723	−	0.664011i	$-0.231147\pi$
$410$	179.177	+	128.337i	0.437018	+	0.313017i
$411$	124.306			0.302449
$412$	186.939	−	186.939i	0.453734	−	0.453734i
$413$	−240.568	−	240.568i	−0.582488	−	0.582488i
$414$			20.3470i			0.0491473i
$415$	−290.042	+	404.941i	−0.698896	+	0.975762i
$416$	−122.597			−0.294704
$417$	94.1729	−	94.1729i	0.225834	−	0.225834i
$418$	−460.231	−	460.231i	−1.10103	−	1.10103i
$419$			42.3838i			0.101155i	0.998720	+	0.0505774i	$0.0161061\pi$
$419$			42.3838i			0.101155i	−0.998720	+	0.0505774i	$0.983894\pi$
$420$	130.445	−	21.5660i	0.310582	−	0.0513477i
$421$	571.283			1.35697			0.678484	−	0.734616i	$-0.262637\pi$
$421$	571.283			1.35697			0.678484	+	0.734616i	$0.262637\pi$
$422$	−229.400	+	229.400i	−0.543603	+	0.543603i
$423$	−156.839	−	156.839i	−0.370778	−	0.370778i
$424$		−	116.547i		−	0.274876i
$425$	−316.323	+	642.152i	−0.744288	+	1.51095i
$426$	−276.704			−0.649539
$427$	527.564	−	527.564i	1.23551	−	1.23551i
$428$	145.429	+	145.429i	0.339787	+	0.339787i
$429$			633.593i			1.47691i
$430$	−6.05530	−	36.6262i	−0.0140821	−	0.0851771i
$431$	−500.121			−1.16037			−0.580187	−	0.814483i	$-0.697020\pi$
$431$	−500.121			−1.16037			−0.580187	+	0.814483i	$0.697020\pi$
$432$	14.6969	−	14.6969i	0.0340207	−	0.0340207i
$433$	−29.3072	−	29.3072i	−0.0676841	−	0.0676841i	0.672454	−	0.740139i	$-0.265240\pi$
$433$	−29.3072	−	29.3072i	−0.0676841	−	0.0676841i	−0.740139	+	0.672454i	$0.765240\pi$
$434$		−	608.939i		−	1.40309i
$435$	−13.1119	−	9.39150i	−0.0301424	−	0.0215897i
$436$	−196.757			−0.451277
$437$	−92.4656	+	92.4656i	−0.211592	+	0.211592i
$438$	169.319	+	169.319i	0.386573	+	0.386573i
$439$			210.861i			0.480321i	0.970733	+	0.240160i	$0.0772000\pi$
$439$			210.861i			0.480321i	−0.970733	+	0.240160i	$0.922800\pi$
$440$	138.997	−	194.060i	0.315902	−	0.441047i
$441$	27.8087			0.0630583
$442$	620.551	−	620.551i	1.40396	−	1.40396i
$443$	−235.554	−	235.554i	−0.531725	−	0.531725i	0.389361	−	0.921085i	$-0.372696\pi$
$443$	−235.554	−	235.554i	−0.531725	−	0.531725i	−0.921085	+	0.389361i	$0.872696\pi$
$444$		−	72.4478i		−	0.163171i
$445$	509.617	−	84.2534i	1.14521	−	0.189334i
$446$	237.465			0.532433
$447$	259.148	−	259.148i	0.579750	−	0.579750i
$448$	43.1813	+	43.1813i	0.0963869	+	0.0963869i
$449$		−	845.610i		−	1.88332i	−0.336568	−	0.941659i	$-0.609266\pi$
$449$		−	845.610i		−	1.88332i	0.336568	−	0.941659i	$-0.390734\pi$
$450$	95.1484	+	46.8699i	0.211441	+	0.104155i
$451$	526.097			1.16651
$452$	−6.47851	+	6.47851i	−0.0143330	+	0.0143330i
$453$	−323.900	−	323.900i	−0.715010	−	0.715010i
$454$		−	280.599i		−	0.618059i
$455$	−134.923	−	816.094i	−0.296533	−	1.79361i
$456$	133.579			0.292936
$457$	−358.568	+	358.568i	−0.784612	+	0.784612i	−0.980605	−	0.195993i	$-0.937207\pi$
$457$	−358.568	+	358.568i	−0.784612	+	0.784612i	0.195993	+	0.980605i	$0.437207\pi$
$458$	−208.203	−	208.203i	−0.454591	−	0.454591i
$459$			148.783i			0.324147i
$460$	−38.9889	−	27.9261i	−0.0847585	−	0.0607088i
$461$	−119.174			−0.258511			−0.129256	−	0.991611i	$-0.541259\pi$
$461$	−119.174			−0.258511			−0.129256	+	0.991611i	$0.541259\pi$
$462$	223.165	−	223.165i	0.483041	−	0.483041i
$463$	−385.179	−	385.179i	−0.831921	−	0.831921i	0.155859	−	0.987779i	$-0.450186\pi$
$463$	−385.179	−	385.179i	−0.831921	−	0.831921i	−0.987779	+	0.155859i	$0.950186\pi$
$464$		−	7.44936i		−	0.0160547i
$465$	284.455	−	397.142i	0.611732	−	0.854069i
$466$	−474.490			−1.01822
$467$	−61.8544	+	61.8544i	−0.132451	+	0.132451i	−0.770224	−	0.637773i	$-0.779856\pi$
$467$	−61.8544	+	61.8544i	−0.132451	+	0.132451i	0.637773	+	0.770224i	$0.279856\pi$
$468$	−91.9478	−	91.9478i	−0.196470	−	0.196470i
$469$		−	321.067i		−	0.684579i
$470$	515.795	−	85.2750i	1.09744	−	0.181436i
$471$	162.119			0.344202
$472$	89.1376	−	89.1376i	0.188851	−	0.188851i
$473$	−62.6602	−	62.6602i	−0.132474	−	0.132474i
$474$		−	41.8618i		−	0.0883159i
$475$	219.399	+	645.393i	0.461892	+	1.35872i
$476$	−437.143			−0.918368
$477$	87.4105	−	87.4105i	0.183251	−	0.183251i
$478$	101.498	+	101.498i	0.212339	+	0.212339i
$479$			161.028i			0.336175i	0.985772	+	0.168087i	$0.0537590\pi$
$479$			161.028i			0.336175i	−0.985772	+	0.168087i	$0.946241\pi$
$480$	7.99087	+	48.3337i	0.0166476	+	0.100695i
$481$	−453.252			−0.942311
$482$	−78.4578	+	78.4578i	−0.162775	+	0.162775i
$483$	−44.8364	−	44.8364i	−0.0928289	−	0.0928289i
$484$		−	327.796i		−	0.677265i
$485$	−449.062	−	321.643i	−0.925900	−	0.663182i
$486$	22.0454			0.0453609
$487$	−568.581	+	568.581i	−1.16752	+	1.16752i	−0.184727	+	0.982790i	$0.559140\pi$
$487$	−568.581	+	568.581i	−1.16752	+	1.16752i	−0.982790	+	0.184727i	$0.940860\pi$
$488$	195.479	+	195.479i	0.400571	+	0.400571i
$489$		−	341.312i		−	0.697980i
$490$	−38.1672	+	53.2871i	−0.0778922	+	0.108749i
$491$	417.266			0.849830			0.424915	−	0.905233i	$-0.360304\pi$
$491$	417.266			0.849830			0.424915	+	0.905233i	$0.360304\pi$
$492$	−76.3478	+	76.3478i	−0.155178	+	0.155178i
$493$	37.7065	+	37.7065i	0.0764838	+	0.0764838i
$494$		−	835.702i		−	1.69170i
$495$	249.793	−	41.2976i	0.504633	−	0.0834295i
$496$	225.631			0.454900
$497$	609.741	−	609.741i	1.22684	−	1.22684i
$498$	−172.546	−	172.546i	−0.346478	−	0.346478i
$499$			569.095i			1.14047i	0.821481	+	0.570236i	$0.193148\pi$
$499$			569.095i			1.14047i	−0.821481	+	0.570236i	$0.806852\pi$
$500$	−220.402	+	117.995i	−0.440805	+	0.235990i
$501$	−319.429			−0.637583
$502$	480.688	−	480.688i	0.957545	−	0.957545i
$503$	575.380	+	575.380i	1.14390	+	1.14390i	0.987731	+	0.156167i	$0.0499137\pi$
$503$	575.380	+	575.380i	1.14390	+	1.14390i	0.156167	+	0.987731i	$0.450086\pi$
$504$			64.7720i			0.128516i
$505$	−28.9977	−	175.396i	−0.0574212	−	0.347319i
$506$	−114.478			−0.226242
$507$	−368.267	+	368.267i	−0.726364	+	0.726364i
$508$	43.5572	+	43.5572i	0.0857425	+	0.0857425i
$509$			108.397i			0.212960i	0.994315	+	0.106480i	$0.0339580\pi$
$509$			108.397i			0.212960i	−0.994315	+	0.106480i	$0.966042\pi$
$510$	−285.099	−	204.204i	−0.559018	−	0.400400i
$511$	−746.219			−1.46031
$512$	−16.0000	+	16.0000i	−0.0312500	+	0.0312500i
$513$	100.184	+	100.184i	0.195290	+	0.195290i
$514$		−	256.118i		−	0.498284i
$515$	−384.857	+	537.318i	−0.747296	+	1.04334i
$516$	18.1867			0.0352454
$517$	882.425	−	882.425i	1.70682	−	1.70682i
$518$	159.645	+	159.645i	0.308195	+	0.308195i
$519$			14.3996i			0.0277449i
$520$	302.388	−	49.9929i	0.581515	−	0.0961402i
$521$	375.872			0.721443			0.360721	−	0.932674i	$-0.382531\pi$
$521$	375.872			0.721443			0.360721	+	0.932674i	$0.382531\pi$
$522$	5.58702	−	5.58702i	0.0107031	−	0.0107031i
$523$	−502.480	−	502.480i	−0.960764	−	0.960764i	0.0384946	−	0.999259i	$-0.487744\pi$
$523$	−502.480	−	502.480i	−0.960764	−	0.960764i	−0.999259	+	0.0384946i	$0.987744\pi$
$524$		−	481.085i		−	0.918101i
$525$	−312.950	+	106.386i	−0.596095	+	0.202640i
$526$	218.542			0.415480
$527$	−1142.08	+	1142.08i	−2.16713	+	2.16713i
$528$	82.6895	+	82.6895i	0.156609	+	0.156609i
$529$			23.0000i			0.0434783i
$530$	47.5260	+	287.466i	0.0896716	+	0.542389i
$531$	133.706			0.251801
$532$	−294.352	+	294.352i	−0.553294	+	0.553294i
$533$	477.651	+	477.651i	0.896156	+	0.896156i
$534$			253.049i			0.473875i
$535$	−418.006	−	299.400i	−0.781320	−	0.559625i
$536$	118.965			0.221950
$537$	−182.927	+	182.927i	−0.340646	+	0.340646i
$538$	−262.095	−	262.095i	−0.487165	−	0.487165i
$539$			156.460i			0.290279i
$540$	−30.2571	+	42.2434i	−0.0560317	+	0.0782286i
$541$	657.157			1.21471			0.607354	−	0.794431i	$-0.292231\pi$
$541$	657.157			1.21471			0.607354	+	0.794431i	$0.292231\pi$
$542$	−6.42234	+	6.42234i	−0.0118493	+	0.0118493i
$543$	−53.2026	−	53.2026i	−0.0979789	−	0.0979789i
$544$		−	161.975i		−	0.297748i
$545$	485.304	−	80.2339i	0.890466	−	0.147218i
$546$	405.230			0.742179
$547$	426.043	−	426.043i	0.778872	−	0.778872i	−0.200767	−	0.979639i	$-0.564344\pi$
$547$	426.043	−	426.043i	0.778872	−	0.778872i	0.979639	+	0.200767i	$0.0643435\pi$
$548$	−101.496	−	101.496i	−0.185211	−	0.185211i
$549$			293.218i			0.534095i
$550$	−263.704	+	535.334i	−0.479462	+	0.973335i
$551$	50.7797			0.0921592
$552$	16.6132	−	16.6132i	0.0300965	−	0.0300965i
$553$	92.2460	+	92.2460i	0.166810	+	0.166810i
$554$			104.288i			0.188246i
$555$	29.5429	+	178.694i	0.0532305	+	0.321971i
$556$	−153.784			−0.276589
$557$	161.428	−	161.428i	0.289816	−	0.289816i	−0.547191	−	0.837008i	$-0.684303\pi$
$557$	161.428	−	161.428i	0.289816	−	0.289816i	0.837008	+	0.547191i	$0.184303\pi$
$558$	169.223	+	169.223i	0.303267	+	0.303267i
$559$		−	113.780i		−	0.203543i
$560$	−124.116	−	88.8989i	−0.221636	−	0.158748i
$561$	−837.102			−1.49216
$562$	−295.578	+	295.578i	−0.525939	+	0.525939i
$563$	3.66442	+	3.66442i	0.00650874	+	0.00650874i	0.710354	−	0.703845i	$-0.248535\pi$
$563$	3.66442	+	3.66442i	0.00650874	+	0.00650874i	−0.703845	+	0.710354i	$0.748535\pi$
$564$			256.117i			0.454109i
$565$	13.3375	−	18.6212i	0.0236063	−	0.0329578i
$566$	529.086			0.934782
$567$	−48.5790	+	48.5790i	−0.0856772	+	0.0856772i
$568$	225.928	+	225.928i	0.397760	+	0.397760i
$569$		−	323.369i		−	0.568312i	−0.958778	−	0.284156i	$-0.908287\pi$
$569$		−	323.369i		−	0.568312i	0.958778	−	0.284156i	$-0.0917133\pi$
$570$	−329.474	+	54.4710i	−0.578025	+	0.0955632i
$571$	634.022			1.11037			0.555185	−	0.831727i	$-0.312647\pi$
$571$	634.022			1.11037			0.555185	+	0.831727i	$0.312647\pi$
$572$	517.327	−	517.327i	0.904417	−	0.904417i
$573$	−192.349	−	192.349i	−0.335687	−	0.335687i
$574$		−	336.478i		−	0.586199i
$575$	107.555	+	52.9811i	0.187052	+	0.0921411i
$576$	−24.0000			−0.0416667
$577$	385.309	−	385.309i	0.667781	−	0.667781i	−0.289421	−	0.957202i	$-0.593463\pi$
$577$	385.309	−	385.309i	0.667781	−	0.667781i	0.957202	+	0.289421i	$0.0934629\pi$
$578$	530.871	+	530.871i	0.918462	+	0.918462i
$579$			73.9007i			0.127635i
$580$	3.03772	+	18.3740i	0.00523744	+	0.0316793i
$581$	760.441			1.30885
$582$	191.346	−	191.346i	0.328773	−	0.328773i
$583$	491.799	+	491.799i	0.843565	+	0.843565i
$584$		−	276.497i		−	0.473454i
$585$	264.286	+	189.296i	0.451770	+	0.323583i
$586$	−335.782			−0.573007
$587$	−151.774	+	151.774i	−0.258558	+	0.258558i	−0.824467	−	0.565909i	$-0.808525\pi$
$587$	−151.774	+	151.774i	−0.258558	+	0.258558i	0.565909	+	0.824467i	$0.308525\pi$
$588$	−22.7057	−	22.7057i	−0.0386152	−	0.0386152i
$589$			1538.05i			2.61128i
$590$	−183.511	+	256.208i	−0.311035	+	0.434252i
$591$	316.975			0.536336
$592$	−59.1533	+	59.1533i	−0.0999212	+	0.0999212i
$593$	−380.622	−	380.622i	−0.641858	−	0.641858i	0.309154	−	0.951012i	$-0.399954\pi$
$593$	−380.622	−	380.622i	−0.641858	−	0.641858i	−0.951012	+	0.309154i	$0.899954\pi$
$594$			124.034i			0.208812i
$595$	1078.22	−	178.259i	1.81214	−	0.299595i
$596$	−423.188			−0.710046
$597$	−47.2966	+	47.2966i	−0.0792238	+	0.0792238i
$598$	−103.937	−	103.937i	−0.173807	−	0.173807i
$599$			267.371i			0.446362i	0.974777	+	0.223181i	$0.0716442\pi$
$599$			267.371i			0.446362i	−0.974777	+	0.223181i	$0.928356\pi$
$600$	−39.4193	−	115.957i	−0.0656988	−	0.193262i
$601$	−733.969			−1.22125			−0.610623	−	0.791921i	$-0.709081\pi$
$601$	−733.969			−1.22125			−0.610623	+	0.791921i	$0.709081\pi$
$602$	−40.0759	+	40.0759i	−0.0665712	+	0.0665712i
$603$	89.2240	+	89.2240i	0.147967	+	0.147967i
$604$			528.926i			0.875705i
$605$	133.670	+	808.516i	0.220941	+	1.33639i
$606$	87.0926			0.143717
$607$	472.725	−	472.725i	0.778789	−	0.778789i	−0.200836	−	0.979625i	$-0.564366\pi$
$607$	472.725	−	472.725i	0.778789	−	0.778789i	0.979625	+	0.200836i	$0.0643658\pi$
$608$	−109.067	−	109.067i	−0.179386	−	0.179386i
$609$			24.6230i			0.0404318i
$610$	−561.865	−	402.439i	−0.921090	−	0.659736i
$611$	1602.33			2.62248
$612$	121.481	−	121.481i	0.198499	−	0.198499i
$613$	57.4176	+	57.4176i	0.0936666	+	0.0936666i	0.752387	−	0.658721i	$-0.228902\pi$
$613$	57.4176	+	57.4176i	0.0936666	+	0.0936666i	−0.658721	+	0.752387i	$0.728902\pi$
$614$			123.927i			0.201835i
$615$	157.180	−	219.447i	0.255577	−	0.356824i
$616$	−364.427			−0.591603
$617$	−316.532	+	316.532i	−0.513018	+	0.513018i	−0.915450	−	0.402432i	$-0.868165\pi$
$617$	−316.532	+	316.532i	−0.513018	+	0.513018i	0.402432	+	0.915450i	$0.368165\pi$
$618$	−228.952	−	228.952i	−0.370473	−	0.370473i
$619$		−	621.040i		−	1.00330i	−0.865072	−	0.501648i	$-0.832727\pi$
$619$		−	621.040i		−	1.00330i	0.865072	−	0.501648i	$-0.167273\pi$
$620$	−556.522	+	92.0082i	−0.897616	+	0.148400i
$621$	24.9199			0.0401286
$622$	23.6111	−	23.6111i	0.0379599	−	0.0379599i
$623$	−557.616	−	557.616i	−0.895049	−	0.895049i
$624$			150.150i			0.240625i
$625$	495.510	−	380.913i	0.792816	−	0.609461i
$626$	−11.1983			−0.0178887
$627$	−563.666	+	563.666i	−0.898989	+	0.898989i
$628$	−132.370	−	132.370i	−0.210780	−	0.210780i
$629$		−	598.835i		−	0.952043i
$630$	−26.4129	−	159.761i	−0.0419252	−	0.253589i
$631$	903.819			1.43236			0.716180	−	0.697916i	$-0.245889\pi$
$631$	903.819			1.43236			0.716180	+	0.697916i	$0.245889\pi$
$632$	−34.1800	+	34.1800i	−0.0540822	+	0.0540822i
$633$	280.957	+	280.957i	0.443850	+	0.443850i
$634$			382.613i			0.603490i
$635$	−125.196	−	89.6728i	−0.197160	−	0.141217i
$636$	−142.741			−0.224435
$637$	−142.053	+	142.053i	−0.223003	+	0.223003i
$638$	31.4343	+	31.4343i	0.0492701	+	0.0492701i
$639$			338.891i			0.530346i
$640$	32.9398	−	45.9888i	0.0514684	−	0.0718575i
$641$	72.2655			0.112739			0.0563694	−	0.998410i	$-0.482048\pi$
$641$	72.2655			0.112739			0.0563694	+	0.998410i	$0.482048\pi$
$642$	178.113	−	178.113i	0.277435	−	0.277435i
$643$	−824.283	−	824.283i	−1.28193	−	1.28193i	−0.939567	−	0.342366i	$-0.888772\pi$
$643$	−824.283	−	824.283i	−1.28193	−	1.28193i	−0.342366	−	0.939567i	$-0.611228\pi$
$644$			73.2175i			0.113692i
$645$	−44.8577	+	7.41620i	−0.0695468	+	0.0114980i
$646$	1104.13			1.70918
$647$	−704.621	+	704.621i	−1.08906	+	1.08906i	−0.0934333	+	0.995626i	$0.529784\pi$
$647$	−704.621	+	704.621i	−1.08906	+	1.08906i	−0.995626	+	0.0934333i	$0.970216\pi$
$648$	−18.0000	−	18.0000i	−0.0277778	−	0.0277778i
$649$			752.274i			1.15913i
$650$	−725.459	+	246.617i	−1.11609	+	0.379411i
$651$	−745.795			−1.14561
$652$	−278.680	+	278.680i	−0.427424	+	0.427424i
$653$	304.501	+	304.501i	0.466311	+	0.466311i	0.900717	−	0.434406i	$-0.143042\pi$
$653$	304.501	+	304.501i	0.466311	+	0.466311i	−0.434406	+	0.900717i	$0.643042\pi$
$654$			240.977i			0.368466i
$655$	196.178	+	1186.61i	0.299508	+	1.81161i
$656$	124.675			0.190054
$657$	207.373	−	207.373i	0.315636	−	0.315636i
$658$	−564.377	−	564.377i	−0.857715	−	0.857715i
$659$		−	138.577i		−	0.210284i	−0.994457	−	0.105142i	$-0.966470\pi$
$659$		−	138.577i		−	0.210284i	0.994457	−	0.105142i	$-0.0335297\pi$
$660$	−237.675	−	170.236i	−0.360113	−	0.257933i
$661$	533.039			0.806413			0.403206	−	0.915109i	$-0.367896\pi$
$661$	533.039			0.806413			0.403206	+	0.915109i	$0.367896\pi$
$662$	−141.318	+	141.318i	−0.213472	+	0.213472i
$663$	−760.017	−	760.017i	−1.14633	−	1.14633i
$664$			281.767i			0.424348i
$665$	605.994	−	846.057i	0.911268	−	1.27227i
$666$	−88.7300			−0.133228
$667$	6.31550	−	6.31550i	0.00946852	−	0.00946852i
$668$	260.813	+	260.813i	0.390438	+	0.390438i
$669$		−	290.834i		−	0.434730i
$670$	−293.430	+	48.5120i	−0.437955	+	0.0724059i
$671$	−1649.74			−2.45862
$672$	52.8861	−	52.8861i	0.0786996	−	0.0786996i
$673$	−624.129	−	624.129i	−0.927384	−	0.927384i	0.0701527	−	0.997536i	$-0.477651\pi$
$673$	−624.129	−	624.129i	−0.927384	−	0.927384i	−0.997536	+	0.0701527i	$0.977651\pi$
$674$			136.316i			0.202249i
$675$	57.4036	−	116.533i	0.0850424	−	0.172641i
$676$	601.377			0.889611
$677$	−40.6078	+	40.6078i	−0.0599820	+	0.0599820i	−0.736461	−	0.676479i	$-0.763505\pi$
$677$	−40.6078	+	40.6078i	−0.0599820	+	0.0599820i	0.676479	+	0.736461i	$0.263505\pi$
$678$	7.93452	+	7.93452i	0.0117028	+	0.0117028i
$679$			843.295i			1.24197i
$680$	66.0505	+	399.514i	0.0971331	+	0.587521i
$681$	−343.662			−0.504643
$682$	−952.100	+	952.100i	−1.39604	+	1.39604i
$683$	−196.064	−	196.064i	−0.287064	−	0.287064i	0.548854	−	0.835918i	$-0.315064\pi$
$683$	−196.064	−	196.064i	−0.287064	−	0.287064i	−0.835918	+	0.548854i	$0.815064\pi$
$684$		−	163.600i		−	0.239181i
$685$	291.729	+	208.953i	0.425882	+	0.305041i
$686$	−428.903			−0.625223
$687$	−254.995	+	254.995i	−0.371172	+	0.371172i
$688$	−14.8493	−	14.8493i	−0.0215833	−	0.0215833i
$689$			893.022i			1.29611i
$690$	−34.2023	+	47.7515i	−0.0495685	+	0.0692050i
$691$	−153.130			−0.221606			−0.110803	−	0.993842i	$-0.535342\pi$
$691$	−153.130			−0.221606			−0.110803	+	0.993842i	$0.535342\pi$
$692$	11.7572	−	11.7572i	0.0169902	−	0.0169902i
$693$	−273.320	−	273.320i	−0.394402	−	0.394402i
$694$		−	662.911i		−	0.955204i
$695$	379.310	−	62.7103i	0.545770	−	0.0902307i
$696$	−9.12356			−0.0131086
$697$	−631.072	+	631.072i	−0.905411	+	0.905411i
$698$	341.273	+	341.273i	0.488930	+	0.488930i
$699$			581.130i			0.831373i
$700$	342.386	+	168.659i	0.489123	+	0.240941i
$701$	−1249.93			−1.78307			−0.891534	−	0.452953i	$-0.850370\pi$
$701$	−1249.93			−1.78307			−0.891534	+	0.452953i	$0.850370\pi$
$702$	−112.613	+	112.613i	−0.160417	+	0.160417i
$703$	−403.228	−	403.228i	−0.573582	−	0.573582i
$704$		−	135.031i		−	0.191806i
$705$	−104.440	−	631.718i	−0.148142	−	0.896054i
$706$	−347.711			−0.492509
$707$	−191.916	+	191.916i	−0.271451	+	0.271451i
$708$	−109.171	−	109.171i	−0.154196	−	0.154196i
$709$		−	499.790i		−	0.704923i	−0.935826	−	0.352461i	$-0.885345\pi$
$709$		−	499.790i		−	0.704923i	0.935826	−	0.352461i	$-0.114655\pi$
$710$	−649.384	−	465.125i	−0.914625	−	0.655106i
$711$	−51.2700			−0.0721097
$712$	206.614	−	206.614i	0.290188	−	0.290188i
$713$	191.288	+	191.288i	0.268286	+	0.268286i
$714$			535.389i			0.749844i
$715$	−1065.04	+	1486.95i	−1.48956	+	2.07965i
$716$	298.719			0.417205
$717$	124.309	−	124.309i	0.173374	−	0.173374i
$718$	97.4524	+	97.4524i	0.135728	+	0.135728i
$719$		−	1307.45i		−	1.81843i	−0.416330	−	0.909213i	$-0.636684\pi$
$719$		−	1307.45i		−	1.81843i	0.416330	−	0.909213i	$-0.363316\pi$
$720$	59.1964	−	9.78678i	0.0822173	−	0.0135927i
$721$	1009.03			1.39949
$722$	382.469	−	382.469i	0.529735	−	0.529735i
$723$	96.0907	+	96.0907i	0.132906	+	0.132906i
$724$			86.8794i			0.119999i
$725$	−14.9852	−	44.0810i	−0.0206692	−	0.0608014i
$726$	−401.467			−0.552985
$727$	399.575	−	399.575i	0.549622	−	0.549622i	−0.376709	−	0.926332i	$-0.622944\pi$
$727$	399.575	−	399.575i	0.549622	−	0.549622i	0.926332	+	0.376709i	$0.122944\pi$
$728$	−330.869	−	330.869i	−0.454490	−	0.454490i
$729$		−	27.0000i		−	0.0370370i
$730$	112.751	+	681.985i	0.154453	+	0.934226i
$731$	150.326			0.205645
$732$	239.412	−	239.412i	0.327065	−	0.327065i
$733$	222.787	+	222.787i	0.303939	+	0.303939i	0.842553	−	0.538614i	$-0.181052\pi$
$733$	222.787	+	222.787i	0.303939	+	0.303939i	−0.538614	+	0.842553i	$0.681052\pi$
$734$			646.993i			0.881462i
$735$	65.2631	+	46.7451i	0.0887933	+	0.0635987i
$736$	−27.1293			−0.0368605
$737$	−502.002	+	502.002i	−0.681142	+	0.681142i
$738$	93.5066	+	93.5066i	0.126703	+	0.126703i
$739$		−	1354.80i		−	1.83329i	−0.399703	−	0.916645i	$-0.630887\pi$
$739$		−	1354.80i		−	1.83329i	0.399703	−	0.916645i	$-0.369113\pi$
$740$	121.781	−	170.025i	0.164569	−	0.229763i
$741$	−1023.52			−1.38127
$742$	314.542	−	314.542i	0.423911	−	0.423911i
$743$	55.0211	+	55.0211i	0.0740526	+	0.0740526i	0.743163	−	0.669110i	$-0.233325\pi$
$743$	55.0211	+	55.0211i	0.0740526	+	0.0740526i	−0.669110	+	0.743163i	$0.733325\pi$
$744$		−	276.340i		−	0.371425i
$745$	1043.80	−	172.568i	1.40107	−	0.231636i
$746$	−140.688			−0.188590
$747$	−211.325	+	211.325i	−0.282898	+	0.282898i
$748$	683.491	+	683.491i	0.913758	+	0.913758i
$749$			784.976i			1.04803i
$750$	144.514	+	269.937i	0.192685	+	0.359915i
$751$	577.871			0.769469			0.384734	−	0.923027i	$-0.374293\pi$
$751$	577.871			0.769469			0.384734	+	0.923027i	$0.374293\pi$
$752$	209.119	−	209.119i	0.278084	−	0.278084i
$753$	−588.720	−	588.720i	−0.781833	−	0.781833i
$754$			57.0793i			0.0757020i
$755$	−215.687	−	1304.61i	−0.285678	−	1.72795i
$756$	79.3291			0.104933
$757$	95.5246	−	95.5246i	0.126188	−	0.126188i	−0.641192	−	0.767380i	$-0.721560\pi$
$757$	95.5246	−	95.5246i	0.126188	−	0.126188i	0.767380	+	0.641192i	$0.221560\pi$
$758$	339.778	+	339.778i	0.448256	+	0.448256i
$759$			140.207i			0.184726i
$760$	313.490	+	224.539i	0.412487	+	0.295446i
$761$	953.861			1.25343			0.626715	−	0.779248i	$-0.284399\pi$
$761$	953.861			1.25343			0.626715	+	0.779248i	$0.284399\pi$
$762$	53.3464	−	53.3464i	0.0700085	−	0.0700085i
$763$	−531.013	−	531.013i	−0.695954	−	0.695954i
$764$			314.104i			0.411131i
$765$	−250.098	+	349.173i	−0.326925	+	0.456436i
$766$	12.2671			0.0160145
$767$	−683.001	+	683.001i	−0.890483	+	0.890483i
$768$	19.5959	+	19.5959i	0.0255155	+	0.0255155i
$769$		−	1519.77i		−	1.97630i	−0.153492	−	0.988150i	$-0.549052\pi$
$769$		−	1519.77i		−	1.97630i	0.153492	−	0.988150i	$-0.450948\pi$
$770$	898.866	−	148.607i	1.16736	−	0.192996i
$771$	−313.679			−0.406847
$772$	60.3397	−	60.3397i	0.0781602	−	0.0781602i
$773$	−19.2189	−	19.2189i	−0.0248627	−	0.0248627i	0.694566	−	0.719429i	$-0.255596\pi$
$773$	−19.2189	−	19.2189i	−0.0248627	−	0.0248627i	−0.719429	+	0.694566i	$0.755596\pi$
$774$		−	22.2740i		−	0.0287778i
$775$	1335.15	−	453.880i	1.72278	−	0.585651i
$776$	−312.467			−0.402663
$777$	195.524	−	195.524i	0.251640	−	0.251640i
$778$	316.963	+	316.963i	0.407407	+	0.407407i
$779$			849.870i			1.09098i
$780$	−61.2286	−	370.348i	−0.0784982	−	0.474805i
$781$	−1906.71			−2.44137
$782$	137.321	−	137.321i	0.175602	−	0.175602i
$783$	−6.84267	−	6.84267i	−0.00873905	−	0.00873905i
$784$			37.0783i			0.0472937i
$785$	380.471	+	272.514i	0.484676	+	0.347152i
$786$	−589.206			−0.749627
$787$	−99.8621	+	99.8621i	−0.126890	+	0.126890i	−0.767700	−	0.640810i	$-0.778599\pi$
$787$	−99.8621	+	99.8621i	−0.126890	+	0.126890i	0.640810	+	0.767700i	$0.278599\pi$
$788$	−258.809	−	258.809i	−0.328438	−	0.328438i
$789$		−	267.659i		−	0.339238i
$790$	70.3676	−	98.2436i	0.0890729	−	0.124359i
$791$	−34.9688			−0.0442084
$792$	101.274	−	101.274i	0.127871	−	0.127871i
$793$	−1497.82	−	1497.82i	−1.88880	−	1.88880i
$794$			146.201i			0.184132i
$795$	352.073	−	58.2072i	0.442859	−	0.0732166i
$796$	77.2351			0.0970290
$797$	−501.918	+	501.918i	−0.629759	+	0.629759i	−0.948007	−	0.318248i	$-0.896906\pi$
$797$	−501.918	+	501.918i	−0.629759	+	0.629759i	0.318248	+	0.948007i	$0.396906\pi$
$798$	360.506	+	360.506i	0.451762	+	0.451762i
$799$			2117.00i			2.64956i
$800$	−62.4931	+	126.865i	−0.0781164	+	0.158581i
$801$	309.921			0.386917
$802$	313.533	−	313.533i	0.390939	−	0.390939i
$803$	1166.74	+	1166.74i	1.45298	+	1.45298i
$804$		−	145.702i		−	0.181222i
$805$	−29.8568	−	180.592i	−0.0370892	−	0.224338i
$806$	−1728.85			−2.14498
$807$	−320.999	+	320.999i	−0.397769	+	0.397769i
$808$	−71.1108	−	71.1108i	−0.0880084	−	0.0880084i
$809$			166.576i			0.205903i	0.994686	+	0.102952i	$0.0328287\pi$
$809$			166.576i			0.205903i	−0.994686	+	0.102952i	$0.967171\pi$
$810$	51.7374	+	37.0572i	0.0638734	+	0.0457497i
$811$	−169.755			−0.209316			−0.104658	−	0.994508i	$-0.533375\pi$
$811$	−169.755			−0.209316			−0.104658	+	0.994508i	$0.533375\pi$
$812$	20.1046	−	20.1046i	0.0247593	−	0.0247593i
$813$	7.86573	+	7.86573i	0.00967495	+	0.00967495i
$814$		−	499.222i		−	0.613295i
$815$	573.729	−	801.011i	0.703962	−	0.982836i
$816$	−198.378			−0.243110
$817$	101.223	−	101.223i	0.123896	−	0.123896i
$818$	543.161	+	543.161i	0.664011	+	0.664011i
$819$		−	496.303i		−	0.605987i
$820$	−307.514	+	50.8405i	−0.375017	+	0.0620006i
$821$	1166.02			1.42024			0.710119	−	0.704081i	$-0.248641\pi$
$821$	1166.02			1.42024			0.710119	+	0.704081i	$0.248641\pi$
$822$	−124.306	+	124.306i	−0.151224	+	0.151224i
$823$	−5.81705	−	5.81705i	−0.00706810	−	0.00706810i	0.703564	−	0.710632i	$-0.251591\pi$
$823$	−5.81705	−	5.81705i	−0.00706810	−	0.00706810i	−0.710632	+	0.703564i	$0.751591\pi$
$824$			373.877i			0.453734i
$825$	655.648	+	322.970i	0.794725	+	0.391479i
$826$	481.135			0.582488
$827$	653.216	−	653.216i	0.789863	−	0.789863i	−0.191609	−	0.981471i	$-0.561370\pi$
$827$	653.216	−	653.216i	0.789863	−	0.789863i	0.981471	+	0.191609i	$0.0613705\pi$
$828$	−20.3470	−	20.3470i	−0.0245737	−	0.0245737i
$829$			1010.10i			1.21846i	0.792995	+	0.609228i	$0.208521\pi$
$829$			1010.10i			1.21846i	−0.792995	+	0.609228i	$0.791479\pi$
$830$	−114.900	−	694.983i	−0.138433	−	0.837329i
$831$	127.726			0.153702
$832$	122.597	−	122.597i	0.147352	−	0.147352i
$833$	−187.680	−	187.680i	−0.225306	−	0.225306i
$834$			188.346i			0.225834i
$835$	−749.654	−	536.945i	−0.897790	−	0.643047i
$836$	920.463			1.10103
$837$	207.255	−	207.255i	0.247616	−	0.247616i
$838$	−42.3838	−	42.3838i	−0.0505774	−	0.0505774i
$839$			1408.48i			1.67876i	0.543542	+	0.839382i	$0.317083\pi$
$839$			1408.48i			1.67876i	−0.543542	+	0.839382i	$0.682917\pi$
$840$	−108.879	+	152.011i	−0.129617	+	0.180965i
$841$	837.532			0.995876
$842$	−571.283	+	571.283i	−0.678484	+	0.678484i
$843$	362.007	+	362.007i	0.429427	+	0.429427i
$844$		−	458.801i		−	0.543603i
$845$	−1483.31	+	245.231i	−1.75539	+	0.290214i
$846$	313.678			0.370778
$847$	884.667	−	884.667i	1.04447	−	1.04447i
$848$	116.547	+	116.547i	0.137438	+	0.137438i
$849$		−	647.996i		−	0.763246i
$850$	−325.830	−	958.475i	−0.383329	−	1.12762i
$851$	−100.299			−0.117861
$852$	276.704	−	276.704i	0.324769	−	0.324769i
$853$	236.049	+	236.049i	0.276728	+	0.276728i	0.831801	−	0.555073i	$-0.187310\pi$
$853$	236.049	+	236.049i	0.276728	+	0.276728i	−0.555073	+	0.831801i	$0.687310\pi$
$854$			1055.13i			1.23551i
$855$	66.7131	+	403.522i	0.0780270	+	0.471955i
$856$	−290.858			−0.339787
$857$	−983.436	+	983.436i	−1.14753	+	1.14753i	−0.160496	+	0.987036i	$0.551309\pi$
$857$	−983.436	+	983.436i	−1.14753	+	1.14753i	−0.987036	+	0.160496i	$0.948691\pi$
$858$	−633.593	−	633.593i	−0.738454	−	0.738454i
$859$		−	1103.77i		−	1.28495i	−0.766309	−	0.642473i	$-0.777909\pi$
$859$		−	1103.77i		−	1.28495i	0.766309	−	0.642473i	$-0.222091\pi$
$860$	42.6815	+	30.5709i	0.0496296	+	0.0355475i
$861$	−412.100			−0.478629
$862$	500.121	−	500.121i	0.580187	−	0.580187i
$863$	847.091	+	847.091i	0.981566	+	0.981566i	0.999833	−	0.0182674i	$-0.00581501\pi$
$863$	847.091	+	847.091i	0.981566	+	0.981566i	−0.0182674	+	0.999833i	$0.505815\pi$
$864$			29.3939i			0.0340207i
$865$	−24.2051	+	33.7939i	−0.0279827	+	0.0390680i
$866$	58.6145			0.0676841
$867$	650.181	−	650.181i	0.749921	−	0.749921i
$868$	608.939	+	608.939i	0.701543	+	0.701543i
$869$		−	288.461i		−	0.331946i
$870$	22.5034	−	3.72043i	0.0258660	−	0.00427635i
$871$	−911.550			−1.04656
$872$	196.757	−	196.757i	0.225638	−	0.225638i
$873$	−234.350	−	234.350i	−0.268442	−	0.268442i
$874$		−	184.931i		−	0.211592i
$875$	−913.278	−	276.380i	−1.04375	−	0.315863i
$876$	−338.638			−0.386573
$877$	989.465	−	989.465i	1.12824	−	1.12824i	0.137775	−	0.990464i	$-0.456005\pi$
$877$	989.465	−	989.465i	1.12824	−	1.12824i	0.990464	−	0.137775i	$-0.0439950\pi$
$878$	−210.861	−	210.861i	−0.240160	−	0.240160i
$879$			411.248i			0.467858i
$880$	55.0634	+	333.057i	0.0625721	+	0.378474i
$881$	122.780			0.139364			0.0696820	−	0.997569i	$-0.477802\pi$
$881$	122.780			0.139364			0.0696820	+	0.997569i	$0.477802\pi$
$882$	−27.8087	+	27.8087i	−0.0315292	+	0.0315292i
$883$	−87.6720	−	87.6720i	−0.0992888	−	0.0992888i	0.655718	−	0.755006i	$-0.272366\pi$
$883$	−87.6720	−	87.6720i	−0.0992888	−	0.0992888i	−0.755006	+	0.655718i	$0.772366\pi$
$884$			1241.10i			1.40396i
$885$	313.790	+	224.754i	0.354565	+	0.253959i
$886$	471.108			0.531725
$887$	−42.3305	+	42.3305i	−0.0477232	+	0.0477232i	−0.730566	−	0.682842i	$-0.760744\pi$
$887$	−42.3305	+	42.3305i	−0.0477232	+	0.0477232i	0.682842	+	0.730566i	$0.260744\pi$
$888$	72.4478	+	72.4478i	0.0815853	+	0.0815853i
$889$			235.107i			0.264462i
$890$	−425.363	+	593.870i	−0.477936	+	0.667270i
$891$	151.910			0.170494
$892$	−237.465	+	237.465i	−0.266216	+	0.266216i
$893$	1425.49	+	1425.49i	1.59629	+	1.59629i
$894$			518.297i			0.579750i
$895$	−736.795	+	121.812i	−0.823234	+	0.136103i
$896$	−86.3626			−0.0963869
$897$	−127.296	+	127.296i	−0.141913	+	0.141913i
$898$	845.610	+	845.610i	0.941659	+	0.941659i
$899$		−	105.050i		−	0.116852i
$900$	−142.018	+	48.2785i	−0.157798	+	0.0536428i
$901$	−1179.86			−1.30950
$902$	−526.097	+	526.097i	−0.583256	+	0.583256i
$903$	49.0827	+	49.0827i	0.0543552	+	0.0543552i
$904$		−	12.9570i		−	0.0143330i
$905$	−35.4279	−	214.290i	−0.0391468	−	0.236784i
$906$	647.799			0.715010
$907$	−134.722	+	134.722i	−0.148536	+	0.148536i	−0.777464	−	0.628928i	$-0.783494\pi$
$907$	−134.722	+	134.722i	−0.148536	+	0.148536i	0.628928	+	0.777464i	$0.283494\pi$
$908$	280.599	+	280.599i	0.309029	+	0.309029i
$909$		−	106.666i		−	0.117345i
$910$	951.017	+	681.172i	1.04507	+	0.748540i
$911$	−1299.86			−1.42685			−0.713424	−	0.700733i	$-0.752856\pi$
$911$	−1299.86			−1.42685			−0.713424	+	0.700733i	$0.752856\pi$
$912$	−133.579	+	133.579i	−0.146468	+	0.146468i
$913$	−1188.98	−	1188.98i	−1.30228	−	1.30228i
$914$		−	717.136i		−	0.784612i
$915$	−492.885	+	688.141i	−0.538672	+	0.752067i
$916$	416.406			0.454591
$917$	1298.37	−	1298.37i	1.41589	−	1.41589i
$918$	−148.783	−	148.783i	−0.162073	−	0.162073i
$919$			939.091i			1.02186i	0.859622	+	0.510931i	$0.170699\pi$
$919$			939.091i			1.02186i	−0.859622	+	0.510931i	$0.829301\pi$
$920$	66.9150	−	11.0629i	0.0727337	−	0.0120248i
$921$	151.779			0.164798
$922$	119.174	−	119.174i	0.129256	−	0.129256i
$923$	−1731.13	−	1731.13i	−1.87554	−	1.87554i
$924$			446.330i			0.483041i
$925$	−231.042	+	469.029i	−0.249776	+	0.507058i
$926$	770.358			0.831921
$927$	−280.408	+	280.408i	−0.302490	+	0.302490i
$928$	7.44936	+	7.44936i	0.00802733	+	0.00802733i
$929$			121.777i			0.131084i	0.997850	+	0.0655418i	$0.0208776\pi$
$929$			121.777i			0.131084i	−0.997850	+	0.0655418i	$0.979122\pi$
$930$	112.687	+	681.597i	0.121168	+	0.732900i
$931$	−252.750			−0.271482
$932$	474.490	−	474.490i	0.509110	−	0.509110i
$933$	−28.9175	−	28.9175i	−0.0309941	−	0.0309941i
$934$		−	123.709i		−	0.132451i
$935$	−1964.56	−	1407.13i	−2.10113	−	1.50495i
$936$	183.896			0.196470
$937$	363.534	−	363.534i	0.387976	−	0.387976i	−0.485989	−	0.873965i	$-0.661540\pi$
$937$	363.534	−	363.534i	0.387976	−	0.387976i	0.873965	+	0.485989i	$0.161540\pi$
$938$	321.067	+	321.067i	0.342289	+	0.342289i
$939$			13.7151i			0.0146060i
$940$	−430.520	+	601.070i	−0.458001	+	0.639437i
$941$	1043.15			1.10855			0.554277	−	0.832332i	$-0.312995\pi$
$941$	1043.15			1.10855			0.554277	+	0.832332i	$0.312995\pi$
$942$	−162.119	+	162.119i	−0.172101	+	0.172101i
$943$	105.699	+	105.699i	0.112088	+	0.112088i
$944$			178.275i			0.188851i
$945$	−195.667	+	32.3490i	−0.207055	+	0.0342318i
$946$	125.320			0.132474
$947$	265.716	−	265.716i	0.280587	−	0.280587i	−0.552756	−	0.833343i	$-0.686424\pi$
$947$	265.716	−	265.716i	0.280587	−	0.280587i	0.833343	+	0.552756i	$0.186424\pi$
$948$	41.8618	+	41.8618i	0.0441580	+	0.0441580i
$949$			2118.61i			2.23246i
$950$	−864.792	−	425.994i	−0.910307	−	0.448415i
$951$	468.603			0.492747
$952$	437.143	−	437.143i	0.459184	−	0.459184i
$953$	−58.5107	−	58.5107i	−0.0613963	−	0.0613963i	0.675742	−	0.737138i	$-0.263823\pi$
$953$	−58.5107	−	58.5107i	−0.0613963	−	0.0613963i	−0.737138	+	0.675742i	$0.763823\pi$
$954$			174.821i			0.183251i
$955$	−128.086	−	774.743i	−0.134121	−	0.811249i
$956$	−202.996			−0.212339
$957$	38.4990	−	38.4990i	0.0402288	−	0.0402288i
$958$	−161.028	−	161.028i	−0.168087	−	0.168087i
$959$		−	547.840i		−	0.571262i
$960$	−56.3246	−	40.3428i	−0.0586714	−	0.0420238i
$961$	2220.82			2.31095
$962$	453.252	−	453.252i	0.471155	−	0.471155i
$963$	−218.143	−	218.143i	−0.226525	−	0.226525i
$964$		−	156.916i		−	0.162775i
$965$	−124.224	+	173.434i	−0.128729	+	0.179725i
$966$	89.6727			0.0928289
$967$	−1009.54	+	1009.54i	−1.04399	+	1.04399i	−0.0450054	+	0.998987i	$0.514331\pi$
$967$	−1009.54	+	1009.54i	−1.04399	+	1.04399i	−0.998987	+	0.0450054i	$0.985669\pi$
$968$	327.796	+	327.796i	0.338633	+	0.338633i
$969$		−	1352.27i		−	1.39554i
$970$	770.705	−	127.418i	0.794541	−	0.131359i
$971$	595.331			0.613111			0.306555	−	0.951853i	$-0.400824\pi$
$971$	595.331			0.613111			0.306555	+	0.951853i	$0.400824\pi$
$972$	−22.0454	+	22.0454i	−0.0226805	+	0.0226805i
$973$	−415.037	−	415.037i	−0.426553	−	0.426553i
$974$		−	1137.16i		−	1.16752i
$975$	302.043	+	888.502i	0.309787	+	0.911284i
$976$	−390.958			−0.400571
$977$	−1126.89	+	1126.89i	−1.15342	+	1.15342i	−0.167562	+	0.985862i	$0.553589\pi$
$977$	−1126.89	+	1126.89i	−1.15342	+	1.15342i	−0.985862	+	0.167562i	$0.946411\pi$
$978$	341.312	+	341.312i	0.348990	+	0.348990i
$979$			1743.71i			1.78111i
$980$	−15.1199	−	91.4543i	−0.0154284	−	0.0933207i
$981$	295.135			0.300851
$982$	−417.266	+	417.266i	−0.424915	+	0.424915i
$983$	542.434	+	542.434i	0.551815	+	0.551815i	0.926964	−	0.375150i	$-0.122409\pi$
$983$	542.434	+	542.434i	0.551815	+	0.551815i	−0.375150	+	0.926964i	$0.622409\pi$
$984$		−	152.696i		−	0.155178i
$985$	743.895	+	532.819i	0.755223	+	0.540933i
$986$	−75.4131			−0.0764838
$987$	−691.217	+	691.217i	−0.700322	+	0.700322i
$988$	835.702	+	835.702i	0.845852	+	0.845852i
$989$		−	25.1783i		−	0.0254583i
$990$	−208.496	+	291.091i	−0.210602	+	0.294031i
$991$	770.639			0.777638			0.388819	−	0.921314i	$-0.372883\pi$
$991$	770.639			0.777638			0.388819	+	0.921314i	$0.372883\pi$
$992$	−225.631	+	225.631i	−0.227450	+	0.227450i
$993$	173.079	+	173.079i	0.174299	+	0.174299i
$994$			1219.48i			1.22684i
$995$	−190.502	+	31.4951i	−0.191459	+	0.0316534i
$996$	345.092			0.346478
$997$	−221.179	+	221.179i	−0.221844	+	0.221844i	−0.809275	−	0.587430i	$-0.800140\pi$
$997$	−221.179	+	221.179i	−0.221844	+	0.221844i	0.587430	+	0.809275i	$0.300140\pi$
$998$	−569.095	−	569.095i	−0.570236	−	0.570236i
$999$			108.672i			0.108780i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.k.b.553.19	yes	48
5.2	odd	4	inner	690.3.k.b.277.19	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.k.b.277.19	✓	48	5.2	odd	4	inner
690.3.k.b.553.19	yes	48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	690.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(0.815565 + 4.93304i) q^{5} -2.44949 q^{6} +(5.39766 - 5.39766i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(0.815565 + 4.93304i) q^{5} -2.44949 q^{6} +(5.39766 - 5.39766i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} +(-5.74860 - 4.11747i) q^{10} -16.8789 q^{11} +(2.44949 - 2.44949i) q^{12} +(-15.3246 - 15.3246i) q^{13} +10.7953i q^{14} +(-5.04285 + 7.04057i) q^{15} -4.00000 q^{16} +(20.2469 - 20.2469i) q^{17} +(-3.00000 - 3.00000i) q^{18} -27.2666i q^{19} +(9.86607 - 1.63113i) q^{20} +13.2215 q^{21} +(16.8789 - 16.8789i) q^{22} +(-3.39116 - 3.39116i) q^{23} +4.89898i q^{24} +(-23.6697 + 8.04642i) q^{25} +30.6493 q^{26} +(-3.67423 + 3.67423i) q^{27} +(-10.7953 - 10.7953i) q^{28} +1.86234i q^{29} +(-1.99772 - 12.0834i) q^{30} -56.4076 q^{31} +(4.00000 - 4.00000i) q^{32} +(-20.6724 - 20.6724i) q^{33} +40.4937i q^{34} +(31.0290 + 22.2247i) q^{35} +6.00000 q^{36} +(14.7883 - 14.7883i) q^{37} +(27.2666 + 27.2666i) q^{38} -37.5375i q^{39} +(-8.23494 + 11.4972i) q^{40} -31.1689 q^{41} +(-13.2215 + 13.2215i) q^{42} +(3.71233 + 3.71233i) q^{43} +33.7579i q^{44} +(-14.7991 + 2.44669i) q^{45} +6.78233 q^{46} +(-52.2797 + 52.2797i) q^{47} +(-4.89898 - 4.89898i) q^{48} -9.26957i q^{49} +(15.6233 - 31.7161i) q^{50} +49.5945 q^{51} +(-30.6493 + 30.6493i) q^{52} +(-29.1368 - 29.1368i) q^{53} -7.34847i q^{54} +(-13.7659 - 83.2644i) q^{55} +21.5907 q^{56} +(33.3947 - 33.3947i) q^{57} +(-1.86234 - 1.86234i) q^{58} -44.5688i q^{59} +(14.0811 + 10.0857i) q^{60} +97.7394 q^{61} +(56.4076 - 56.4076i) q^{62} +(16.1930 + 16.1930i) q^{63} +8.00000i q^{64} +(63.0987 - 88.0952i) q^{65} +41.3448 q^{66} +(29.7413 - 29.7413i) q^{67} +(-40.4937 - 40.4937i) q^{68} -8.30662i q^{69} +(-53.2538 + 8.80429i) q^{70} +112.964 q^{71} +(-6.00000 + 6.00000i) q^{72} +(-69.1242 - 69.1242i) q^{73} +29.5767i q^{74} +(-38.8442 - 19.1345i) q^{75} -54.5333 q^{76} +(-91.1068 + 91.1068i) q^{77} +(37.5375 + 37.5375i) q^{78} +17.0900i q^{79} +(-3.26226 - 19.7321i) q^{80} -9.00000 q^{81} +(31.1689 - 31.1689i) q^{82} +(70.4417 + 70.4417i) q^{83} -26.4430i q^{84} +(116.391 + 83.3659i) q^{85} -7.42467 q^{86} +(-2.28089 + 2.28089i) q^{87} +(-33.7579 - 33.7579i) q^{88} -103.307i q^{89} +(12.3524 - 17.2458i) q^{90} -165.434 q^{91} +(-6.78233 + 6.78233i) q^{92} +(-69.0850 - 69.0850i) q^{93} -104.559i q^{94} +(134.507 - 22.2377i) q^{95} +9.79796 q^{96} +(-78.1167 + 78.1167i) q^{97} +(9.26957 + 9.26957i) q^{98} -50.6368i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{2} - 8 q^{5} - 8 q^{7} + 96 q^{8}+O(q^{10}) \) 48 * q - 48 * q^2 - 8 * q^5 - 8 * q^7 + 96 * q^8	\( 48 q - 48 q^{2} - 8 q^{5} - 8 q^{7} + 96 q^{8} + 8 q^{10} - 32 q^{11} - 24 q^{13} + 24 q^{15} - 192 q^{16} + 72 q^{17} - 144 q^{18} + 32 q^{22} + 24 q^{25} + 48 q^{26} + 16 q^{28} - 24 q^{30} + 24 q^{31} + 192 q^{32} - 24 q^{33} + 288 q^{36} - 128 q^{37} - 16 q^{38} - 16 q^{40} - 40 q^{41} + 48 q^{43} - 136 q^{47} - 80 q^{50} - 48 q^{52} + 144 q^{53} - 144 q^{55} - 32 q^{56} + 96 q^{57} + 8 q^{58} + 128 q^{61} - 24 q^{62} - 24 q^{63} + 184 q^{65} + 48 q^{66} - 144 q^{68} + 40 q^{70} - 40 q^{71} - 288 q^{72} + 40 q^{73} - 72 q^{75} + 32 q^{76} - 104 q^{77} + 96 q^{78} + 32 q^{80} - 432 q^{81} + 40 q^{82} - 88 q^{85} - 96 q^{86} + 120 q^{87} - 64 q^{88} + 24 q^{90} + 144 q^{91} - 96 q^{93} + 312 q^{95} + 480 q^{97} + 584 q^{98}+O(q^{100}) \) 48 * q - 48 * q^2 - 8 * q^5 - 8 * q^7 + 96 * q^8 + 8 * q^10 - 32 * q^11 - 24 * q^13 + 24 * q^15 - 192 * q^16 + 72 * q^17 - 144 * q^18 + 32 * q^22 + 24 * q^25 + 48 * q^26 + 16 * q^28 - 24 * q^30 + 24 * q^31 + 192 * q^32 - 24 * q^33 + 288 * q^36 - 128 * q^37 - 16 * q^38 - 16 * q^40 - 40 * q^41 + 48 * q^43 - 136 * q^47 - 80 * q^50 - 48 * q^52 + 144 * q^53 - 144 * q^55 - 32 * q^56 + 96 * q^57 + 8 * q^58 + 128 * q^61 - 24 * q^62 - 24 * q^63 + 184 * q^65 + 48 * q^66 - 144 * q^68 + 40 * q^70 - 40 * q^71 - 288 * q^72 + 40 * q^73 - 72 * q^75 + 32 * q^76 - 104 * q^77 + 96 * q^78 + 32 * q^80 - 432 * q^81 + 40 * q^82 - 88 * q^85 - 96 * q^86 + 120 * q^87 - 64 * q^88 + 24 * q^90 + 144 * q^91 - 96 * q^93 + 312 * q^95 + 480 * q^97 + 584 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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