LMFDB - Embedded newform 108.5.k.a.41.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(5,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("108.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.k (of order $18$, degree $6$, 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:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			41.3
Character	$\chi$	$=$	108.41
Dual form			108.5.k.a.29.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.90497 + 5.77247i) q^{3} +(11.0592 - 13.1798i) q^{5} +(7.77764 + 2.83083i) q^{7} +(14.3573 - 79.7174i) q^{9} +O(q^{10})$	\(q+(-6.90497 + 5.77247i) q^{3} +(11.0592 - 13.1798i) q^{5} +(7.77764 + 2.83083i) q^{7} +(14.3573 - 79.7174i) q^{9} +(44.3220 + 52.8209i) q^{11} +(-4.19703 + 23.8025i) q^{13} +(-0.283269 + 154.845i) q^{15} +(213.549 + 123.293i) q^{17} +(175.684 + 304.294i) q^{19} +(-70.0452 + 25.3494i) q^{21} +(275.524 + 756.995i) q^{23} +(57.1279 + 323.988i) q^{25} +(361.030 + 633.323i) q^{27} +(-348.016 + 61.3646i) q^{29} +(634.917 - 231.091i) q^{31} +(-610.949 - 108.880i) q^{33} +(123.324 - 71.2013i) q^{35} +(870.013 - 1506.91i) q^{37} +(-108.419 - 188.583i) q^{39} +(629.967 + 111.080i) q^{41} +(1149.85 - 964.841i) q^{43} +(-891.882 - 1070.84i) q^{45} +(-649.053 + 1783.26i) q^{47} +(-1786.79 - 1499.30i) q^{49} +(-2186.26 + 381.374i) q^{51} +1728.31i q^{53} +1186.34 q^{55} +(-2969.62 - 1087.01i) q^{57} +(-442.288 + 527.098i) q^{59} +(995.692 + 362.402i) q^{61} +(337.332 - 579.370i) q^{63} +(267.298 + 318.553i) q^{65} +(1019.70 - 5783.02i) q^{67} +(-6272.21 - 3636.58i) q^{69} +(3889.70 + 2245.72i) q^{71} +(-725.679 - 1256.91i) q^{73} +(-2264.68 - 1907.36i) q^{75} +(195.194 + 536.290i) q^{77} +(-173.955 - 986.547i) q^{79} +(-6148.74 - 2289.05i) q^{81} +(-1712.04 + 301.879i) q^{83} +(3986.66 - 1451.03i) q^{85} +(2048.81 - 2432.63i) q^{87} +(-1131.82 + 653.456i) q^{89} +(-100.024 + 173.246i) q^{91} +(-3050.12 + 5260.71i) q^{93} +(5953.47 + 1049.76i) q^{95} +(10494.3 - 8805.79i) q^{97} +(4847.09 - 2774.87i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{17}{18}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−6.90497	+	5.77247i	−0.767219	+	0.641385i
$3$	−6.90497	+	5.77247i	−0.767219	+	0.641385i
$4$		0			0
$5$	11.0592	−	13.1798i	0.442368	−	0.527193i	−0.498080	−	0.867131i	$-0.665962\pi$
$5$	11.0592	−	13.1798i	0.442368	−	0.527193i	0.940448	+	0.339938i	$0.110406\pi$
$6$		0			0
$7$	7.77764	+	2.83083i	0.158727	+	0.0577720i	0.420162	−	0.907449i	$-0.361973\pi$
$7$	7.77764	+	2.83083i	0.158727	+	0.0577720i	−0.261434	+	0.965221i	$0.584196\pi$
$8$		0			0
$9$	14.3573	−	79.7174i	0.177250	−	0.984166i
$10$		0			0
$11$	44.3220	+	52.8209i	0.366298	+	0.436537i	0.917440	−	0.397875i	$-0.130252\pi$
$11$	44.3220	+	52.8209i	0.366298	+	0.436537i	−0.551142	+	0.834411i	$0.685808\pi$
$12$		0			0
$13$	−4.19703	+	23.8025i	−0.0248345	+	0.140843i	−0.994704	−	0.102782i	$-0.967226\pi$
$13$	−4.19703	+	23.8025i	−0.0248345	+	0.140843i	0.969869	+	0.243625i	$0.0783367\pi$
$14$		0			0
$15$	−0.283269	+	154.845i	−0.00125898	+	0.688201i
$16$		0			0
$17$	213.549	+	123.293i	0.738925	+	0.426619i	0.821678	−	0.569951i	$-0.193038\pi$
$17$	213.549	+	123.293i	0.738925	+	0.426619i	−0.0827531	+	0.996570i	$0.526371\pi$
$18$		0			0
$19$	175.684	+	304.294i	0.486660	+	0.842920i	0.999882	−	0.0153357i	$-0.00488168\pi$
$19$	175.684	+	304.294i	0.486660	+	0.842920i	−0.513222	+	0.858256i	$0.671548\pi$
$20$		0			0
$21$	−70.0452	+	25.3494i	−0.158833	+	0.0574815i
$22$		0			0
$23$	275.524	+	756.995i	0.520839	+	1.43099i	0.869589	+	0.493777i	$0.164384\pi$
$23$	275.524	+	756.995i	0.520839	+	1.43099i	−0.348750	+	0.937216i	$0.613394\pi$
$24$		0			0
$25$	57.1279	+	323.988i	0.0914047	+	0.518382i
$26$		0			0
$27$	361.030	+	633.323i	0.495240	+	0.868756i
$28$		0			0
$29$	−348.016	+	61.3646i	−0.413812	+	0.0729662i	−0.376678	−	0.926344i	$-0.622934\pi$
$29$	−348.016	+	61.3646i	−0.413812	+	0.0729662i	−0.0371335	+	0.999310i	$0.511823\pi$
$30$		0			0
$31$	634.917	−	231.091i	0.660684	−	0.240469i	0.0101521	−	0.999948i	$-0.496768\pi$
$31$	634.917	−	231.091i	0.660684	−	0.240469i	0.650531	+	0.759479i	$0.274546\pi$
$32$		0			0
$33$	−610.949	−	108.880i	−0.561019	−	0.0999813i
$34$		0			0
$35$	123.324	−	71.2013i	0.100673	−	0.0581235i
$36$		0			0
$37$	870.013	−	1506.91i	0.635510	−	1.10073i	−0.350897	−	0.936414i	$-0.614123\pi$
$37$	870.013	−	1506.91i	0.635510	−	1.10073i	0.986407	−	0.164321i	$-0.0525432\pi$
$38$		0			0
$39$	−108.419	−	188.583i	−0.0712814	−	0.123986i
$40$		0			0
$41$	629.967	+	111.080i	0.374757	+	0.0660798i	0.357855	−	0.933777i	$-0.383508\pi$
$41$	629.967	+	111.080i	0.374757	+	0.0660798i	0.0169027	+	0.999857i	$0.494619\pi$
$42$		0			0
$43$	1149.85	−	964.841i	0.621878	−	0.521818i	−0.276515	−	0.961010i	$-0.589180\pi$
$43$	1149.85	−	964.841i	0.621878	−	0.521818i	0.898393	+	0.439192i	$0.144735\pi$
$44$		0			0
$45$	−891.882	−	1070.84i	−0.440436	−	0.528808i
$46$		0			0
$47$	−649.053	+	1783.26i	−0.293822	+	0.807270i	0.701676	+	0.712496i	$0.252435\pi$
$47$	−649.053	+	1783.26i	−0.293822	+	0.807270i	−0.995499	+	0.0947744i	$0.969787\pi$
$48$		0			0
$49$	−1786.79	−	1499.30i	−0.744188	−	0.624448i
$50$		0			0
$51$	−2186.26	+	381.374i	−0.840545	+	0.146626i
$52$		0			0
$53$			1728.31i			0.615278i	0.951503	+	0.307639i	$0.0995388\pi$
$53$			1728.31i			0.615278i	−0.951503	+	0.307639i	$0.900461\pi$
$54$		0			0
$55$	1186.34			0.392177
$56$		0			0
$57$	−2969.62	−	1087.01i	−0.914011	−	0.334568i
$58$		0			0
$59$	−442.288	+	527.098i	−0.127058	+	0.151421i	−0.825823	−	0.563930i	$-0.809289\pi$
$59$	−442.288	+	527.098i	−0.127058	+	0.151421i	0.698765	+	0.715351i	$0.253733\pi$
$60$		0			0
$61$	995.692	+	362.402i	0.267587	+	0.0973938i	0.472329	−	0.881422i	$-0.343413\pi$
$61$	995.692	+	362.402i	0.267587	+	0.0973938i	−0.204742	+	0.978816i	$0.565636\pi$
$62$		0			0
$63$	337.332	−	579.370i	0.0849917	−	0.145974i
$64$		0			0
$65$	267.298	+	318.553i	0.0632657	+	0.0753971i
$66$		0			0
$67$	1019.70	−	5783.02i	0.227156	−	1.28826i	−0.631365	−	0.775486i	$-0.717505\pi$
$67$	1019.70	−	5783.02i	0.227156	−	1.28826i	0.858521	−	0.512778i	$-0.171384\pi$
$68$		0			0
$69$	−6272.21	−	3636.58i	−1.31741	−	0.763826i
$70$		0			0
$71$	3889.70	+	2245.72i	0.771612	+	0.445491i	0.833450	−	0.552596i	$-0.186363\pi$
$71$	3889.70	+	2245.72i	0.771612	+	0.445491i	−0.0618371	+	0.998086i	$0.519696\pi$
$72$		0			0
$73$	−725.679	−	1256.91i	−0.136176	−	0.235863i	0.789870	−	0.613274i	$-0.210148\pi$
$73$	−725.679	−	1256.91i	−0.136176	−	0.235863i	−0.926046	+	0.377411i	$0.876814\pi$
$74$		0			0
$75$	−2264.68	−	1907.36i	−0.402610	−	0.339087i
$76$		0			0
$77$	195.194	+	536.290i	0.0329219	+	0.0904520i
$78$		0			0
$79$	−173.955	−	986.547i	−0.0278729	−	0.158075i	0.967695	−	0.252126i	$-0.0811296\pi$
$79$	−173.955	−	986.547i	−0.0278729	−	0.158075i	−0.995567	+	0.0940505i	$0.970018\pi$
$80$		0			0
$81$	−6148.74	−	2289.05i	−0.937165	−	0.348887i
$82$		0			0
$83$	−1712.04	+	301.879i	−0.248518	+	0.0438204i	−0.296519	−	0.955027i	$-0.595826\pi$
$83$	−1712.04	+	301.879i	−0.248518	+	0.0438204i	0.0480014	+	0.998847i	$0.484715\pi$
$84$		0			0
$85$	3986.66	−	1451.03i	0.551787	−	0.200834i
$86$		0			0
$87$	2048.81	−	2432.63i	0.270685	−	0.321394i
$88$		0			0
$89$	−1131.82	+	653.456i	−0.142888	+	0.0824966i	−0.569740	−	0.821825i	$-0.692956\pi$
$89$	−1131.82	+	653.456i	−0.142888	+	0.0824966i	0.426851	+	0.904322i	$0.359623\pi$
$90$		0			0
$91$	−100.024	+	173.246i	−0.0120787	+	0.0209210i
$92$		0			0
$93$	−3050.12	+	5260.71i	−0.352656	+	0.608245i
$94$		0			0
$95$	5953.47	+	1049.76i	0.659664	+	0.116317i
$96$		0			0
$97$	10494.3	−	8805.79i	1.11535	−	0.935890i	0.116990	−	0.993133i	$-0.462676\pi$
$97$	10494.3	−	8805.79i	1.11535	−	0.935890i	0.998360	+	0.0572434i	$0.0182311\pi$
$98$		0			0
$99$	4847.09	−	2774.87i	0.494551	−	0.283122i
$100$		0			0
$101$	−3163.83	+	8692.56i	−0.310149	+	0.852129i	0.682476	+	0.730908i	$0.260903\pi$
$101$	−3163.83	+	8692.56i	−0.310149	+	0.852129i	−0.992626	+	0.121221i	$0.961319\pi$
$102$		0			0
$103$	3788.01	+	3178.52i	0.357057	+	0.299606i	0.803616	−	0.595148i	$-0.202906\pi$
$103$	3788.01	+	3178.52i	0.357057	+	0.299606i	−0.446560	+	0.894754i	$0.647351\pi$
$104$		0			0
$105$	−440.543	+	1203.53i	−0.0399586	+	0.109163i
$106$		0			0
$107$			15110.5i			1.31981i	0.751350	+	0.659904i	$0.229403\pi$
$107$			15110.5i			1.31981i	−0.751350	+	0.659904i	$0.770597\pi$
$108$		0			0
$109$	−15250.4			−1.28360			−0.641798	−	0.766873i	$-0.721811\pi$
$109$	−15250.4			−1.28360			−0.641798	+	0.766873i	$0.721811\pi$
$110$		0			0
$111$	2691.15	+	15427.3i	0.218420	+	1.25211i
$112$		0			0
$113$	1774.06	−	2114.24i	0.138935	−	0.165576i	−0.692090	−	0.721811i	$-0.743310\pi$
$113$	1774.06	−	2114.24i	0.138935	−	0.165576i	0.831025	+	0.556235i	$0.187754\pi$
$114$		0			0
$115$	13024.1	+	4740.40i	0.984811	+	0.358442i
$116$		0			0
$117$	1837.22	+	676.316i	0.134211	+	0.0494058i
$118$		0			0
$119$	1311.89	+	1563.45i	0.0926410	+	0.110405i
$120$		0			0
$121$	1716.77	−	9736.31i	0.117258	−	0.665003i
$122$		0			0
$123$	−4991.11	+	2869.46i	−0.329904	+	0.189666i
$124$		0			0
$125$	14214.4	+	8206.69i	0.909722	+	0.525228i
$126$		0			0
$127$	13768.7	+	23848.0i	0.853659	+	1.47858i	0.877883	+	0.478875i	$0.158955\pi$
$127$	13768.7	+	23848.0i	0.853659	+	1.47858i	−0.0242240	+	0.999707i	$0.507711\pi$
$128$		0			0
$129$	−2370.19	+	13299.7i	−0.142431	+	0.799212i
$130$		0			0
$131$	−8139.98	−	22364.4i	−0.474330	−	1.30321i	−0.914241	−	0.405170i	$-0.867212\pi$
$131$	−8139.98	−	22364.4i	−0.474330	−	1.30321i	0.439911	−	0.898041i	$-0.355010\pi$
$132$		0			0
$133$	505.004	+	2864.02i	0.0285491	+	0.161910i
$134$		0			0
$135$	12339.8	+	2245.73i	0.677080	+	0.123223i
$136$		0			0
$137$	−9749.91	+	1719.17i	−0.519469	+	0.0915964i	−0.427234	−	0.904141i	$-0.640512\pi$
$137$	−9749.91	+	1719.17i	−0.519469	+	0.0915964i	−0.0922346	+	0.995737i	$0.529401\pi$
$138$		0			0
$139$	−5846.34	+	2127.89i	−0.302590	+	0.110134i	−0.488853	−	0.872366i	$-0.662585\pi$
$139$	−5846.34	+	2127.89i	−0.302590	+	0.110134i	0.186263	+	0.982500i	$0.440362\pi$
$140$		0			0
$141$	−5812.11	−	16060.0i	−0.292345	−	0.807806i
$142$		0			0
$143$	−1443.29	+	833.286i	−0.0705801	+	0.0407495i
$144$		0			0
$145$	−3040.00	+	5265.43i	−0.144590	+	0.250437i
$146$		0			0
$147$	20992.4	+	38.4030i	0.971466	+	0.00177717i
$148$		0			0
$149$	722.838	+	127.456i	0.0325588	+	0.00574100i	0.189904	−	0.981803i	$-0.439182\pi$
$149$	722.838	+	127.456i	0.0325588	+	0.00574100i	−0.157345	+	0.987544i	$0.550293\pi$
$150$		0			0
$151$	−12154.9	+	10199.2i	−0.533086	+	0.447312i	−0.869165	−	0.494521i	$-0.835343\pi$
$151$	−12154.9	+	10199.2i	−0.533086	+	0.447312i	0.336080	+	0.941834i	$0.390899\pi$
$152$		0			0
$153$	12894.6	−	15253.5i	0.550838	−	0.651607i
$154$		0			0
$155$	3975.93	−	10923.8i	0.165491	−	0.454684i
$156$		0			0
$157$	−26195.4	−	21980.5i	−1.06274	−	0.891741i	−0.0683613	−	0.997661i	$-0.521777\pi$
$157$	−26195.4	−	21980.5i	−1.06274	−	0.891741i	−0.994375	+	0.105919i	$0.966222\pi$
$158$		0			0
$159$	−9976.64	−	11934.0i	−0.394630	−	0.472053i
$160$		0			0
$161$			6667.59i			0.257227i
$162$		0			0
$163$	−45253.0			−1.70322			−0.851612	−	0.524172i	$-0.824375\pi$
$163$	−45253.0			−1.70322			−0.851612	+	0.524172i	$0.824375\pi$
$164$		0			0
$165$	−8191.62	+	6848.09i	−0.300886	+	0.251537i
$166$		0			0
$167$	31781.8	−	37876.1i	1.13958	−	1.35810i	0.215230	−	0.976563i	$-0.430950\pi$
$167$	31781.8	−	37876.1i	1.13958	−	1.35810i	0.924353	−	0.381539i	$-0.124606\pi$
$168$		0			0
$169$	26289.6	+	9568.64i	0.920473	+	0.335025i
$170$		0			0
$171$	26779.9	−	9636.27i	0.915834	−	0.329547i
$172$		0			0
$173$	−30120.6	−	35896.3i	−1.00640	−	1.19938i	−0.979850	−	0.199736i	$-0.935992\pi$
$173$	−30120.6	−	35896.3i	−1.00640	−	1.19938i	−0.0265522	−	0.999647i	$-0.508453\pi$
$174$		0			0
$175$	−472.836	+	2681.58i	−0.0154395	+	0.0875619i
$176$		0			0
$177$	11.3287	−	6192.69i	0.000361605	−	0.197666i
$178$		0			0
$179$	−12684.8	−	7323.59i	−0.395894	−	0.228569i	0.288817	−	0.957384i	$-0.406738\pi$
$179$	−12684.8	−	7323.59i	−0.395894	−	0.228569i	−0.684711	+	0.728815i	$0.740071\pi$
$180$		0			0
$181$	−3216.57	−	5571.27i	−0.0981830	−	0.170058i	0.812750	−	0.582613i	$-0.197970\pi$
$181$	−3216.57	−	5571.27i	−0.0981830	−	0.170058i	−0.910933	+	0.412555i	$0.864636\pi$
$182$		0			0
$183$	−8967.18	+	3245.22i	−0.267765	+	0.0969041i
$184$		0			0
$185$	−10239.1	−	28131.8i	−0.299171	−	0.821965i
$186$		0			0
$187$	2952.50	+	16744.5i	0.0844320	+	0.478837i
$188$		0			0
$189$	1015.13	+	5947.77i	0.0284182	+	0.166506i
$190$		0			0
$191$	−61078.9	+	10769.9i	−1.67427	+	0.295218i	−0.928594	−	0.371096i	$-0.878982\pi$
$191$	−61078.9	+	10769.9i	−1.67427	+	0.295218i	−0.745671	+	0.666314i	$0.767871\pi$
$192$		0			0
$193$	−9223.47	+	3357.07i	−0.247617	+	0.0901251i	−0.462847	−	0.886438i	$-0.653172\pi$
$193$	−9223.47	+	3357.07i	−0.247617	+	0.0901251i	0.215230	+	0.976563i	$0.430950\pi$
$194$		0			0
$195$	−3684.52	−	656.632i	−0.0968972	−	0.0172684i
$196$		0			0
$197$	30005.4	−	17323.6i	0.773156	−	0.446382i	−0.0608435	−	0.998147i	$-0.519379\pi$
$197$	30005.4	−	17323.6i	0.773156	−	0.446382i	0.833999	+	0.551766i	$0.186046\pi$
$198$		0			0
$199$	−35471.9	+	61439.1i	−0.895733	+	1.55145i	−0.0628372	+	0.998024i	$0.520015\pi$
$199$	−35471.9	+	61439.1i	−0.895733	+	1.55145i	−0.832895	+	0.553431i	$0.813318\pi$
$200$		0			0
$201$	26341.3	+	45817.8i	0.651995	+	1.13407i
$202$		0			0
$203$	−2880.45	−	507.902i	−0.0698987	−	0.0123250i
$204$		0			0
$205$	8430.94	−	7074.40i	0.200617	−	0.168338i
$206$		0			0
$207$	64301.5	−	11095.7i	1.50065	−	0.258948i
$208$		0			0
$209$	−8286.42	+	22766.7i	−0.189703	+	0.521205i
$210$		0			0
$211$	−48385.5	−	40600.2i	−1.08680	−	0.911934i	−0.0903334	−	0.995912i	$-0.528793\pi$
$211$	−48385.5	−	40600.2i	−1.08680	−	0.911934i	−0.996468	+	0.0839772i	$0.973238\pi$
$212$		0			0
$213$	−39821.6	+	6946.53i	−0.877727	+	0.153112i
$214$		0			0
$215$		−	25825.2i		−	0.558685i
$216$		0			0
$217$	5592.33			0.118761
$218$		0			0
$219$	12266.3	+	4489.99i	0.255755	+	0.0936176i
$220$		0			0
$221$	−3830.96	+	4565.55i	−0.0784373	+	0.0934779i
$222$		0			0
$223$	14199.5	+	5168.20i	0.285538	+	0.103927i	0.480819	−	0.876820i	$-0.340340\pi$
$223$	14199.5	+	5168.20i	0.285538	+	0.103927i	−0.195281	+	0.980747i	$0.562562\pi$
$224$		0			0
$225$	26647.7	+	97.4976i	0.526375	+	0.00192588i
$226$		0			0
$227$	−40600.0	−	48385.2i	−0.787905	−	0.938989i	0.211356	−	0.977409i	$-0.432212\pi$
$227$	−40600.0	−	48385.2i	−0.787905	−	0.938989i	−0.999262	+	0.0384200i	$0.987768\pi$
$228$		0			0
$229$	11889.2	−	67427.2i	0.226716	−	1.28577i	−0.632660	−	0.774430i	$-0.718037\pi$
$229$	11889.2	−	67427.2i	0.226716	−	1.28577i	0.859377	−	0.511343i	$-0.170852\pi$
$230$		0			0
$231$	−4443.52	−	2576.32i	−0.0832729	−	0.0482809i
$232$		0			0
$233$	81880.6	+	47273.8i	1.50824	+	0.870780i	0.999954	+	0.00958931i	$0.00305242\pi$
$233$	81880.6	+	47273.8i	1.50824	+	0.870780i	0.508282	+	0.861191i	$0.330281\pi$
$234$		0			0
$235$	16325.0	+	28275.8i	0.295610	+	0.512011i
$236$		0			0
$237$	6895.96	+	5807.93i	0.122772	+	0.103401i
$238$		0			0
$239$	−33160.0	−	91106.3i	−0.580522	−	1.59497i	−0.787291	−	0.616581i	$-0.788517\pi$
$239$	−33160.0	−	91106.3i	−0.580522	−	1.59497i	0.206769	−	0.978390i	$-0.433705\pi$
$240$		0			0
$241$	11435.1	+	64851.9i	0.196883	+	1.11658i	0.909712	+	0.415239i	$0.136302\pi$
$241$	11435.1	+	64851.9i	0.196883	+	1.11658i	−0.712830	+	0.701337i	$0.752587\pi$
$242$		0			0
$243$	55670.3	−	19687.6i	0.942782	−	0.333411i
$244$		0			0
$245$	−39521.0	+	6968.62i	−0.658409	+	0.116095i
$246$		0			0
$247$	−7980.33	+	2904.60i	−0.130806	+	0.0476094i
$248$		0			0
$249$	10079.0	−	11967.2i	0.162562	−	0.193016i
$250$		0			0
$251$	−21936.7	+	12665.2i	−0.348196	+	0.201031i	−0.663891	−	0.747830i	$-0.731096\pi$
$251$	−21936.7	+	12665.2i	−0.348196	+	0.201031i	0.315694	+	0.948861i	$0.397763\pi$
$252$		0			0
$253$	−27773.4	+	48105.0i	−0.433899	+	0.751534i
$254$		0			0
$255$	−19151.8	+	33032.2i	−0.294530	+	0.507992i
$256$		0			0
$257$	124203.	+	21900.3i	1.88046	+	0.331576i	0.991884	−	0.127148i	$-0.0405823\pi$
$257$	124203.	+	21900.3i	1.88046	+	0.331576i	0.888579	+	0.458724i	$0.151693\pi$
$258$		0			0
$259$	11032.4	−	9257.31i	0.164464	−	0.138002i
$260$		0			0
$261$	−104.728	+	28624.0i	−0.00153738	+	0.420193i
$262$		0			0
$263$	895.943	−	2461.58i	0.0129530	−	0.0355880i	−0.933048	−	0.359753i	$-0.882861\pi$
$263$	895.943	−	2461.58i	0.0129530	−	0.0355880i	0.946001	+	0.324165i	$0.105083\pi$
$264$		0			0
$265$	22778.9	+	19113.8i	0.324370	+	0.272179i
$266$		0			0
$267$	4043.12	−	11045.5i	0.0567146	−	0.154939i
$268$		0			0
$269$			101027.i			1.39616i	0.716021	+	0.698078i	$0.245961\pi$
$269$			101027.i			1.39616i	−0.716021	+	0.698078i	$0.754039\pi$
$270$		0			0
$271$	−29168.0			−0.397162			−0.198581	−	0.980084i	$-0.563633\pi$
$271$	−29168.0			−0.397162			−0.198581	+	0.980084i	$0.563633\pi$
$272$		0			0
$273$	−309.397	−	1773.65i	−0.00415137	−	0.0237981i
$274$		0			0
$275$	−14581.3	+	17377.4i	−0.192811	+	0.229783i
$276$		0			0
$277$	79476.1	+	28926.9i	1.03580	+	0.377001i	0.803287	−	0.595592i	$-0.203082\pi$
$277$	79476.1	+	28926.9i	1.03580	+	0.377001i	0.232515	+	0.972593i	$0.425305\pi$
$278$		0			0
$279$	−9306.30	−	53931.8i	−0.119555	−	0.692845i
$280$		0			0
$281$	83010.4	+	98928.0i	1.05128	+	1.25287i	0.966553	+	0.256468i	$0.0825587\pi$
$281$	83010.4	+	98928.0i	1.05128	+	1.25287i	0.0847312	+	0.996404i	$0.472997\pi$
$282$		0			0
$283$	−3459.59	+	19620.3i	−0.0431968	+	0.244981i	−0.998759	−	0.0498092i	$-0.984139\pi$
$283$	−3459.59	+	19620.3i	−0.0431968	+	0.244981i	0.955562	+	0.294790i	$0.0952498\pi$
$284$		0			0
$285$	−47168.2	+	27117.7i	−0.580711	+	0.333859i
$286$		0			0
$287$	4585.21	+	2647.27i	0.0556667	+	0.0321392i
$288$		0			0
$289$	−11358.3	−	19673.1i	−0.135993	−	0.235547i
$290$		0			0
$291$	−21631.9	+	121382.i	−0.255452	+	1.43340i
$292$		0			0
$293$	−14958.8	−	41099.0i	−0.174245	−	0.478735i	0.821572	−	0.570105i	$-0.193098\pi$
$293$	−14958.8	−	41099.0i	−0.174245	−	0.478735i	−0.995817	+	0.0913700i	$0.970875\pi$
$294$		0			0
$295$	2055.72	+	11658.6i	0.0236222	+	0.133968i
$296$		0			0
$297$	−17451.2	+	47140.1i	−0.197839	+	0.534414i
$298$		0			0
$299$	−19174.8	+	3381.03i	−0.214481	+	0.0378187i
$300$		0			0
$301$	11674.4	−	4249.15i	0.128855	−	0.0468996i
$302$		0			0
$303$	−28331.3	−	78285.0i	−0.308590	−	0.852695i
$304$		0			0
$305$	15787.9	−	9115.17i	0.169717	−	0.0979863i
$306$		0			0
$307$	3347.53	−	5798.10i	0.0355180	−	0.0615189i	−0.847720	−	0.530444i	$-0.822025\pi$
$307$	3347.53	−	5798.10i	0.0355180	−	0.0615189i	0.883238	+	0.468925i	$0.155359\pi$
$308$		0			0
$309$	−44504.0	−	81.4145i	−0.466104	−	0.000852677i
$310$		0			0
$311$	26470.7	+	4667.50i	0.273681	+	0.0482574i	0.308804	−	0.951126i	$-0.400071\pi$
$311$	26470.7	+	4667.50i	0.273681	+	0.0482574i	−0.0351230	+	0.999383i	$0.511182\pi$
$312$		0			0
$313$	119813.	−	100535.i	1.22296	−	1.02619i	0.224300	−	0.974520i	$-0.427991\pi$
$313$	119813.	−	100535.i	1.22296	−	1.02619i	0.998664	−	0.0516685i	$-0.0164539\pi$
$314$		0			0
$315$	−3905.38	−	10853.3i	−0.0393589	−	0.109381i
$316$		0			0
$317$	−19067.4	+	52387.3i	−0.189746	+	0.521323i	−0.997690	−	0.0679366i	$-0.978358\pi$
$317$	−19067.4	+	52387.3i	−0.189746	+	0.521323i	0.807943	+	0.589260i	$0.200581\pi$
$318$		0			0
$319$	−18666.1	−	15662.7i	−0.183431	−	0.153917i
$320$		0			0
$321$	−87224.8	−	104337.i	−0.846505	−	1.01258i
$322$		0			0
$323$			86642.5i			0.830473i
$324$		0			0
$325$	−7951.52			−0.0752806
$326$		0			0
$327$	105304.	−	88032.5i	0.984800	−	0.823280i
$328$		0			0
$329$	−10096.2	+	12032.2i	−0.0932752	+	0.111161i
$330$		0			0
$331$	−37852.5	−	13777.2i	−0.345493	−	0.125749i	0.163445	−	0.986552i	$-0.447739\pi$
$331$	−37852.5	−	13777.2i	−0.345493	−	0.125749i	−0.508938	+	0.860803i	$0.669962\pi$
$332$		0			0
$333$	−107636.	−	90990.2i	−0.970661	−	0.820552i
$334$		0			0
$335$	−64942.1	−	77395.0i	−0.578678	−	0.689641i
$336$		0			0
$337$	36939.7	−	209496.i	0.325262	−	1.84465i	−0.182563	−	0.983194i	$-0.558439\pi$
$337$	36939.7	−	209496.i	0.325262	−	1.84465i	0.507826	−	0.861460i	$-0.330449\pi$
$338$		0			0
$339$	−45.4407	+	24839.5i	−0.000395408	+	0.216144i
$340$		0			0
$341$	40347.2	+	23294.5i	0.346980	+	0.200329i
$342$		0			0
$343$	−19589.1	−	33929.3i	−0.166504	−	0.288394i
$344$		0			0
$345$	−117295.	+	42449.1i	−0.985465	+	0.356640i
$346$		0			0
$347$	−12450.0	−	34206.2i	−0.103398	−	0.284083i	0.877196	−	0.480132i	$-0.159411\pi$
$347$	−12450.0	−	34206.2i	−0.103398	−	0.284083i	−0.980594	+	0.196049i	$0.937189\pi$
$348$		0			0
$349$	−29968.9	−	169962.i	−0.246048	−	1.39541i	−0.818045	−	0.575154i	$-0.804942\pi$
$349$	−29968.9	−	169962.i	−0.246048	−	1.39541i	0.571997	−	0.820256i	$-0.306169\pi$
$350$		0			0
$351$	−16590.0	+	5935.35i	−0.134658	+	0.0481761i
$352$		0			0
$353$	10299.5	−	1816.08i	0.0826543	−	0.0145742i	−0.132168	−	0.991227i	$-0.542194\pi$
$353$	10299.5	−	1816.08i	0.0826543	−	0.0145742i	0.214822	+	0.976653i	$0.431083\pi$
$354$		0			0
$355$	72615.1	−	26429.7i	0.576196	−	0.209718i
$356$		0			0
$357$	−18083.5	−	3222.73i	−0.141888	−	0.0252864i
$358$		0			0
$359$	−65698.6	+	37931.1i	−0.509762	+	0.294311i	−0.732736	−	0.680513i	$-0.761757\pi$
$359$	−65698.6	+	37931.1i	−0.509762	+	0.294311i	0.222974	+	0.974824i	$0.428423\pi$
$360$		0			0
$361$	3430.55	−	5941.89i	0.0263238	−	0.0455942i
$362$		0			0
$363$	44348.2	+	77139.0i	0.336560	+	0.585410i
$364$		0			0
$365$	−24591.3	−	4336.11i	−0.184585	−	0.0325473i
$366$		0			0
$367$	22828.4	−	19155.3i	0.169490	−	0.142219i	−0.554097	−	0.832452i	$-0.686936\pi$
$367$	22828.4	−	19155.3i	0.169490	−	0.142219i	0.723587	+	0.690233i	$0.242492\pi$
$368$		0			0
$369$	17899.6	−	48624.6i	0.131459	−	0.357111i
$370$		0			0
$371$	−4892.56	+	13442.2i	−0.0355458	+	0.0976614i
$372$		0			0
$373$	5382.98	+	4516.86i	0.0386906	+	0.0324653i	0.661928	−	0.749568i	$-0.269738\pi$
$373$	5382.98	+	4516.86i	0.0386906	+	0.0324653i	−0.623237	+	0.782033i	$0.714183\pi$
$374$		0			0
$375$	−145523.	+	25385.2i	−1.03483	+	0.180517i
$376$		0			0
$377$		−	8541.21i		−	0.0600948i
$378$		0			0
$379$	145934.			1.01596			0.507980	−	0.861369i	$-0.330392\pi$
$379$	145934.			1.01596			0.507980	+	0.861369i	$0.330392\pi$
$380$		0			0
$381$	−232734.	−	85190.8i	−1.60328	−	0.586871i
$382$		0			0
$383$	−96300.0	+	114766.i	−0.656491	+	0.782375i	−0.986878	−	0.161470i	$-0.948376\pi$
$383$	−96300.0	+	114766.i	−0.656491	+	0.782375i	0.330387	+	0.943846i	$0.392821\pi$
$384$		0			0
$385$	9226.89	+	3358.31i	0.0622492	+	0.0226569i
$386$		0			0
$387$	−60405.9	−	105516.i	−0.403327	−	0.704523i
$388$		0			0
$389$	63115.2	+	75217.7i	0.417095	+	0.497074i	0.933153	−	0.359479i	$-0.117046\pi$
$389$	63115.2	+	75217.7i	0.417095	+	0.497074i	−0.516058	+	0.856553i	$0.672601\pi$
$390$		0			0
$391$	−34494.1	+	195626.i	−0.225627	+	1.27960i
$392$		0			0
$393$	185304.	+	107438.i	1.19978	+	0.695620i
$394$		0			0
$395$	−14926.3	−	8617.71i	−0.0956662	−	0.0552329i
$396$		0			0
$397$	−89376.0	−	154804.i	−0.567074	−	0.982201i	−0.996853	−	0.0792673i	$-0.974742\pi$
$397$	−89376.0	−	154804.i	−0.567074	−	0.982201i	0.429779	−	0.902934i	$-0.358591\pi$
$398$		0			0
$399$	−20019.5	−	16860.9i	−0.125750	−	0.105909i
$400$		0			0
$401$	−15740.3	−	43246.2i	−0.0978871	−	0.268943i	0.881078	−	0.472971i	$-0.156819\pi$
$401$	−15740.3	−	43246.2i	−0.0978871	−	0.268943i	−0.978965	+	0.204029i	$0.934596\pi$
$402$		0			0
$403$	2835.78	+	16082.5i	0.0174608	+	0.0990249i
$404$		0			0
$405$	−98169.3	+	55724.3i	−0.598502	+	0.339730i
$406$		0			0
$407$	118157.	−	20834.2i	0.713297	−	0.125773i
$408$		0			0
$409$	99480.0	−	36207.8i	0.594688	−	0.216449i	−0.0271021	−	0.999633i	$-0.508628\pi$
$409$	99480.0	−	36207.8i	0.594688	−	0.216449i	0.621790	+	0.783184i	$0.286406\pi$
$410$		0			0
$411$	57399.0	−	68151.9i	0.339798	−	0.403454i
$412$		0			0
$413$	−4932.08	+	2847.54i	−0.0289155	+	0.0166943i
$414$		0			0
$415$	−14955.1	+	25902.9i	−0.0868345	+	0.150402i
$416$		0			0
$417$	28085.6	−	48440.9i	0.161515	−	0.278573i
$418$		0			0
$419$	342282.	+	60353.5i	1.94965	+	0.343775i	0.999488	+	0.0320086i	$0.0101904\pi$
$419$	342282.	+	60353.5i	1.94965	+	0.343775i	0.950159	+	0.311767i	$0.100921\pi$
$420$		0			0
$421$	−124187.	+	104205.i	−0.700668	+	0.587930i	−0.921964	−	0.387277i	$-0.873416\pi$
$421$	−124187.	+	104205.i	−0.700668	+	0.587930i	0.221296	+	0.975207i	$0.428971\pi$
$422$		0			0
$423$	132838.	+	77343.6i	0.742408	+	0.432259i
$424$		0			0
$425$	−27745.8	+	76231.0i	−0.153610	+	0.422040i
$426$		0			0
$427$	6718.23	+	5637.27i	0.0368468	+	0.0309181i
$428$		0			0
$429$	5155.78	−	14085.2i	0.0280143	−	0.0765328i
$430$		0			0
$431$		−	68099.8i		−	0.366599i	−0.983057	−	0.183300i	$-0.941322\pi$
$431$		−	68099.8i		−	0.366599i	0.983057	−	0.183300i	$-0.0586778\pi$
$432$		0			0
$433$	115590.			0.616515			0.308257	−	0.951303i	$-0.400254\pi$
$433$	115590.			0.616515			0.308257	+	0.951303i	$0.400254\pi$
$434$		0			0
$435$	−9403.42	−	53905.9i	−0.0496944	−	0.284877i
$436$		0			0
$437$	−181944.	+	216832.i	−0.952741	+	1.13543i
$438$		0			0
$439$	−419.769	−	152.783i	−0.00217812	−	0.000792770i	0.340931	−	0.940088i	$-0.389258\pi$
$439$	−419.769	−	152.783i	−0.00217812	−	0.000792770i	−0.343109	+	0.939296i	$0.611480\pi$
$440$		0			0
$441$	−145174.	+	120913.i	−0.746467	+	0.621721i
$442$		0			0
$443$	20484.1	+	24412.0i	0.104378	+	0.124393i	0.815704	−	0.578469i	$-0.196350\pi$
$443$	20484.1	+	24412.0i	0.104378	+	0.124393i	−0.711326	+	0.702862i	$0.751905\pi$
$444$		0			0
$445$	−3904.56	+	22143.9i	−0.0197175	+	0.111824i
$446$		0			0
$447$	−5726.91	+	3292.48i	−0.0286619	+	0.0164781i
$448$		0			0
$449$	−6386.03	−	3686.98i	−0.0316766	−	0.0182885i	0.484078	−	0.875025i	$-0.339155\pi$
$449$	−6386.03	−	3686.98i	−0.0316766	−	0.0182885i	−0.515755	+	0.856736i	$0.672488\pi$
$450$		0			0
$451$	22054.1	+	38198.8i	0.108427	+	0.187800i
$452$		0			0
$453$	25054.8	−	140589.i	0.122094	−	0.685100i
$454$		0			0
$455$	1177.18	+	3234.26i	0.00568615	+	0.0156226i
$456$		0			0
$457$	16663.8	+	94505.3i	0.0797889	+	0.452505i	0.998360	+	0.0572493i	$0.0182330\pi$
$457$	16663.8	+	94505.3i	0.0797889	+	0.452505i	−0.918571	+	0.395256i	$0.870656\pi$
$458$		0			0
$459$	−986.543	+	179758.i	−0.00468264	+	0.853225i
$460$		0			0
$461$	7688.93	−	1355.77i	0.0361796	−	0.00637944i	−0.155529	−	0.987831i	$-0.549708\pi$
$461$	7688.93	−	1355.77i	0.0361796	−	0.00637944i	0.191709	+	0.981452i	$0.438597\pi$
$462$		0			0
$463$	−140479.	+	51130.3i	−0.655315	+	0.238515i	−0.648212	−	0.761460i	$-0.724483\pi$
$463$	−140479.	+	51130.3i	−0.655315	+	0.238515i	−0.00710280	+	0.999975i	$0.502261\pi$
$464$		0			0
$465$	35603.4	+	98379.2i	0.164659	+	0.454986i
$466$		0			0
$467$	174259.	−	100609.i	0.799029	−	0.461319i	−0.0441027	−	0.999027i	$-0.514043\pi$
$467$	174259.	−	100609.i	0.799029	−	0.461319i	0.843131	+	0.537708i	$0.180710\pi$
$468$		0			0
$469$	24301.6	−	42091.6i	0.110481	−	0.191359i
$470$		0			0
$471$	307760.	+	563.008i	1.38730	+	0.00253789i
$472$		0			0
$473$	101928.	+	17972.6i	0.455585	+	0.0803319i
$474$		0			0
$475$	−88551.3	+	74303.4i	−0.392471	+	0.329322i
$476$		0			0
$477$	137777.	+	24813.9i	0.605535	+	0.109058i
$478$		0			0
$479$	78432.5	−	215492.i	0.341842	−	0.939203i	−0.643018	−	0.765851i	$-0.722318\pi$
$479$	78432.5	−	215492.i	0.341842	−	0.939203i	0.984860	−	0.173352i	$-0.0554597\pi$
$480$		0			0
$481$	32216.7	+	27033.0i	0.139249	+	0.116844i
$482$		0			0
$483$	−38488.5	−	46039.5i	−0.164982	−	0.197350i
$484$		0			0
$485$		−	235698.i		−	1.00201i
$486$		0			0
$487$	225927.			0.952600			0.476300	−	0.879283i	$-0.341978\pi$
$487$	225927.			0.952600			0.476300	+	0.879283i	$0.341978\pi$
$488$		0			0
$489$	312470.	−	261221.i	1.30675	−	1.09242i
$490$		0			0
$491$	−259155.	+	308848.i	−1.07497	+	1.28110i	−0.117341	+	0.993092i	$0.537437\pi$
$491$	−259155.	+	308848.i	−1.07497	+	1.28110i	−0.957628	+	0.288007i	$0.907008\pi$
$492$		0			0
$493$	−81884.4	−	29803.5i	−0.336905	−	0.122623i
$494$		0			0
$495$	17032.5	−	94571.7i	0.0695135	−	0.385967i
$496$		0			0
$497$	23895.4	+	28477.5i	0.0967391	+	0.115289i
$498$		0			0
$499$	−52018.8	+	295013.i	−0.208910	+	1.18479i	0.682257	+	0.731112i	$0.260999\pi$
$499$	−52018.8	+	295013.i	−0.208910	+	1.18479i	−0.891167	+	0.453675i	$0.850113\pi$
$500$		0			0
$501$	−814.058	+	444993.i	−0.00324325	+	1.77287i
$502$		0			0
$503$	−275570.	−	159100.i	−1.08917	−	0.628833i	−0.155815	−	0.987786i	$-0.549800\pi$
$503$	−275570.	−	159100.i	−1.08917	−	0.628833i	−0.933356	+	0.358953i	$0.883134\pi$
$504$		0			0
$505$	79577.0	+	137831.i	0.312036	+	0.540463i
$506$		0			0
$507$	−236764.	+	85684.8i	−0.921084	+	0.333340i
$508$		0			0
$509$	−149917.	−	411893.i	−0.578648	−	1.58982i	−0.790461	−	0.612512i	$-0.790159\pi$
$509$	−149917.	−	411893.i	−0.578648	−	1.58982i	0.211814	−	0.977310i	$-0.432063\pi$
$510$		0			0
$511$	−2085.96	−	11830.1i	−0.00798850	−	0.0453050i
$512$		0			0
$513$	−129289.	+	221124.i	−0.491279	+	0.840237i
$514$		0			0
$515$	83784.7	−	14773.5i	0.315901	−	0.0557018i
$516$		0			0
$517$	−122961.	+	44754.1i	−0.460029	+	0.167437i
$518$		0			0
$519$	415192.	+	73993.0i	1.54140	+	0.274698i
$520$		0			0
$521$	−76314.7	+	44060.3i	−0.281146	+	0.162320i	−0.633942	−	0.773380i	$-0.718564\pi$
$521$	−76314.7	+	44060.3i	−0.281146	+	0.162320i	0.352796	+	0.935700i	$0.385231\pi$
$522$		0			0
$523$	205191.	−	355401.i	0.750162	−	1.29932i	−0.197581	−	0.980286i	$-0.563309\pi$
$523$	205191.	−	355401.i	0.750162	−	1.29932i	0.947744	−	0.319033i	$-0.103358\pi$
$524$		0			0
$525$	−12214.4	−	21245.7i	−0.0443154	−	0.0770819i
$526$		0			0
$527$	164078.	+	28931.4i	0.590785	+	0.104171i
$528$		0			0
$529$	−282757.	+	237262.i	−1.01042	+	0.847845i
$530$		0			0
$531$	35668.9	+	42825.7i	0.126503	+	0.151885i
$532$		0			0
$533$	−5287.98	+	14528.6i	−0.0186138	+	0.0511411i
$534$		0			0
$535$	199154.	+	167110.i	0.695794	+	0.583840i
$536$		0			0
$537$	129864.	−	22653.6i	0.450338	−	0.0785576i
$538$		0			0
$539$		−	160832.i		−	0.553599i
$540$		0			0
$541$	−492919.			−1.68415			−0.842075	−	0.539361i	$-0.818666\pi$
$541$	−492919.			−1.68415			−0.842075	+	0.539361i	$0.818666\pi$
$542$		0			0
$543$	54370.3	+	19901.9i	0.184400	+	0.0674986i
$544$		0			0
$545$	−168657.	+	200998.i	−0.567821	+	0.676703i
$546$		0			0
$547$	−371903.	−	135361.i	−1.24295	−	0.452398i	−0.364938	−	0.931032i	$-0.618910\pi$
$547$	−371903.	−	135361.i	−1.24295	−	0.452398i	−0.878014	+	0.478634i	$0.841132\pi$
$548$		0			0
$549$	43185.2	−	74170.9i	0.143282	−	0.246087i
$550$		0			0
$551$	−79813.8	−	95118.4i	−0.262890	−	0.313301i
$552$		0			0
$553$	1439.79	−	8165.44i	0.00470813	−	0.0267011i
$554$		0			0
$555$	233091.	+	135144.i	0.756726	+	0.438744i
$556$		0			0
$557$	−359117.	−	207336.i	−1.15751	−	0.668291i	−0.206806	−	0.978382i	$-0.566307\pi$
$557$	−359117.	−	207336.i	−1.15751	−	0.668291i	−0.950707	+	0.310091i	$0.899640\pi$
$558$		0			0
$559$	18139.7	+	31418.9i	0.0580505	+	0.100546i
$560$		0			0
$561$	−117044.	−	98576.9i	−0.371897	−	0.313220i
$562$		0			0
$563$	−16738.4	−	45988.4i	−0.0528077	−	0.145088i	0.910484	−	0.413543i	$-0.135709\pi$
$563$	−16738.4	−	45988.4i	−0.0528077	−	0.145088i	−0.963292	+	0.268455i	$0.913487\pi$
$564$		0			0
$565$	−8245.69	−	46763.6i	−0.0258303	−	0.146491i
$566$		0			0
$567$	−41342.8	−	35209.4i	−0.128598	−	0.109520i
$568$		0			0
$569$	415714.	−	73301.6i	1.28401	−	0.226406i	0.510331	−	0.859978i	$-0.329523\pi$
$569$	415714.	−	73301.6i	1.28401	−	0.226406i	0.773684	+	0.633572i	$0.218412\pi$
$570$		0			0
$571$	−238847.	+	86933.2i	−0.732567	+	0.266633i	−0.681251	−	0.732050i	$-0.738564\pi$
$571$	−238847.	+	86933.2i	−0.732567	+	0.266633i	−0.0513161	+	0.998682i	$0.516342\pi$
$572$		0			0
$573$	359579.	−	426941.i	1.09518	−	1.30035i
$574$		0			0
$575$	−229518.	+	132512.i	−0.694193	+	0.400793i
$576$		0			0
$577$	72698.7	−	125918.i	0.218361	−	0.378212i	−0.735946	−	0.677040i	$-0.763262\pi$
$577$	72698.7	−	125918.i	0.218361	−	0.378212i	0.954307	+	0.298828i	$0.0965957\pi$
$578$		0			0
$579$	44309.2	−	76422.6i	0.132171	−	0.227963i
$580$		0			0
$581$	−14170.2	−	2498.59i	−0.0419782	−	0.00740189i
$582$		0			0
$583$	−91291.2	+	76602.4i	−0.268591	+	0.225375i
$584$		0			0
$585$	29231.9	−	16734.7i	0.0854171	−	0.0488998i
$586$		0			0
$587$	201839.	−	554549.i	0.585773	−	1.60940i	−0.192378	−	0.981321i	$-0.561620\pi$
$587$	201839.	−	554549.i	0.585773	−	1.60940i	0.778151	−	0.628077i	$-0.216158\pi$
$588$		0			0
$589$	181865.	+	152602.i	0.524225	+	0.439877i
$590$		0			0
$591$	−107186.	+	292824.i	−0.306877	+	0.838363i
$592$		0			0
$593$		−	632321.i		−	1.79816i	−0.437786	−	0.899079i	$-0.644237\pi$
$593$		−	632321.i		−	1.79816i	0.437786	−	0.899079i	$-0.355763\pi$
$594$		0			0
$595$	35114.4			0.0991863
$596$		0			0
$597$	−109723.	−	628996.i	−0.307857	−	1.76481i
$598$		0			0
$599$	−258279.	+	307805.i	−0.719840	+	0.857872i	−0.994615	−	0.103636i	$-0.966952\pi$
$599$	−258279.	+	307805.i	−0.719840	+	0.857872i	0.274775	+	0.961509i	$0.411397\pi$
$600$		0			0
$601$	−338819.	−	123320.i	−0.938034	−	0.341417i	−0.172645	−	0.984984i	$-0.555231\pi$
$601$	−338819.	−	123320.i	−0.938034	−	0.341417i	−0.765389	+	0.643568i	$0.777454\pi$
$602$		0			0
$603$	−446367.	−	164316.i	−1.22760	−	0.451904i
$604$		0			0
$605$	−109337.	−	130302.i	−0.298714	−	0.355993i
$606$		0			0
$607$	−63002.8	+	357306.i	−0.170995	+	0.969758i	0.771672	+	0.636021i	$0.219421\pi$
$607$	−63002.8	+	357306.i	−0.170995	+	0.969758i	−0.942666	+	0.333737i	$0.891690\pi$
$608$		0			0
$609$	22821.3	−	13120.3i	0.0615327	−	0.0353760i
$610$		0			0
$611$	−39722.0	−	22933.5i	−0.106402	−	0.0614311i
$612$		0			0
$613$	51114.3	+	88532.6i	0.136026	+	0.235604i	0.925989	−	0.377551i	$-0.123234\pi$
$613$	51114.3	+	88532.6i	0.136026	+	0.235604i	−0.789963	+	0.613155i	$0.789900\pi$
$614$		0			0
$615$	−17378.7	+	97515.9i	−0.0459480	+	0.257825i
$616$		0			0
$617$	−174957.	−	480690.i	−0.459580	−	1.26268i	−0.925799	−	0.378015i	$-0.876607\pi$
$617$	−174957.	−	480690.i	−0.459580	−	1.26268i	0.466220	−	0.884669i	$-0.345616\pi$
$618$		0			0
$619$	−92539.8	−	524819.i	−0.241517	−	1.36971i	−0.828445	−	0.560070i	$-0.810774\pi$
$619$	−92539.8	−	524819.i	−0.241517	−	1.36971i	0.586929	−	0.809639i	$-0.300337\pi$
$620$		0			0
$621$	−379950.	+	447793.i	−0.985244	+	1.16117i
$622$		0			0
$623$	−10652.7	+	1878.36i	−0.0274463	+	0.00483952i
$624$		0			0
$625$	72146.1	−	26259.0i	0.184694	−	0.0672231i
$626$		0			0
$627$	−74202.8	−	205037.i	−0.188749	−	0.521551i
$628$		0			0
$629$	371581.	−	214533.i	0.939188	−	0.542241i
$630$		0			0
$631$	54673.1	−	94696.5i	0.137314	−	0.237835i	−0.789165	−	0.614181i	$-0.789486\pi$
$631$	54673.1	−	94696.5i	0.137314	−	0.237835i	0.926479	+	0.376346i	$0.122820\pi$
$632$		0			0
$633$	568464.	+	1039.93i	1.41872	+	0.00259536i
$634$		0			0
$635$	466583.	+	82271.2i	1.15713	+	0.204033i
$636$		0			0
$637$	43186.3	−	36237.6i	0.106431	−	0.0893061i
$638$		0			0
$639$	234868.	−	277834.i	0.575205	−	0.680431i
$640$		0			0
$641$	−56474.2	+	155162.i	−0.137447	+	0.377631i	−0.989251	−	0.146229i	$-0.953286\pi$
$641$	−56474.2	+	155162.i	−0.137447	+	0.377631i	0.851804	+	0.523860i	$0.175509\pi$
$642$		0			0
$643$	174635.	+	146536.i	0.422386	+	0.354424i	0.829070	−	0.559145i	$-0.188871\pi$
$643$	174635.	+	146536.i	0.422386	+	0.354424i	−0.406684	+	0.913569i	$0.633315\pi$
$644$		0			0
$645$	149075.	+	178322.i	0.358332	+	0.428634i
$646$		0			0
$647$			167904.i			0.401101i	0.979683	+	0.200550i	$0.0642730\pi$
$647$			167904.i			0.401101i	−0.979683	+	0.200550i	$0.935727\pi$
$648$		0			0
$649$	−47444.9			−0.112642
$650$		0			0
$651$	−38614.9	+	32281.6i	−0.0911156	+	0.0761715i
$652$		0			0
$653$	−381636.	+	454816.i	−0.894999	+	1.06662i	0.102415	+	0.994742i	$0.467343\pi$
$653$	−381636.	+	454816.i	−0.894999	+	1.06662i	−0.997414	+	0.0718765i	$0.977101\pi$
$654$		0			0
$655$	−384780.	−	140049.i	−0.896872	−	0.326435i
$656$		0			0
$657$	−110617.	+	39803.5i	−0.256265	+	0.0922125i
$658$		0			0
$659$	346989.	+	413526.i	0.798997	+	0.952207i	0.999623	−	0.0274604i	$-0.00874203\pi$
$659$	346989.	+	413526.i	0.798997	+	0.952207i	−0.200626	+	0.979668i	$0.564298\pi$
$660$		0			0
$661$	−63023.7	+	357425.i	−0.144245	+	0.818054i	0.823725	+	0.566989i	$0.191892\pi$
$661$	−63023.7	+	357425.i	−0.144245	+	0.818054i	−0.967970	+	0.251065i	$0.919219\pi$
$662$		0			0
$663$	98.1259	−	53639.1i	0.000223232	−	0.122027i
$664$		0			0
$665$	43332.2	+	25017.9i	0.0979869	+	0.0565728i
$666$		0			0
$667$	−142339.	−	246539.i	−0.319943	−	0.554158i
$668$		0			0
$669$	−127880.	+	46279.9i	−0.285727	+	0.103405i
$670$		0			0
$671$	24988.7	+	68655.8i	0.0555006	+	0.152487i
$672$		0			0
$673$	139117.	+	788974.i	0.307151	+	1.74194i	0.613205	+	0.789924i	$0.289880\pi$
$673$	139117.	+	788974.i	0.307151	+	1.74194i	−0.306054	+	0.952014i	$0.599009\pi$
$674$		0			0
$675$	−184565.	+	153150.i	−0.405080	+	0.336132i
$676$		0			0
$677$	318018.	−	56075.2i	0.693865	−	0.122347i	0.184416	−	0.982848i	$-0.440961\pi$
$677$	318018.	−	56075.2i	0.693865	−	0.122347i	0.509448	+	0.860501i	$0.329849\pi$
$678$		0			0
$679$	106549.	−	38780.6i	0.231105	−	0.0841152i
$680$		0			0
$681$	559643.	+	99736.2i	1.20675	+	0.215060i
$682$		0			0
$683$	−638191.	+	368460.i	−1.36807	+	0.789858i	−0.990682	−	0.136196i	$-0.956512\pi$
$683$	−638191.	+	368460.i	−1.36807	+	0.789858i	−0.377392	+	0.926054i	$0.623179\pi$
$684$		0			0
$685$	−85167.7	+	147515.i	−0.181507	+	0.314380i
$686$		0			0
$687$	307126.	+	534213.i	0.650734	+	1.13188i
$688$		0			0
$689$	−41138.3	−	7253.79i	−0.0866578	−	0.0152801i
$690$		0			0
$691$	320796.	−	269180.i	0.671852	−	0.563751i	−0.241761	−	0.970336i	$-0.577725\pi$
$691$	320796.	−	269180.i	0.671852	−	0.563751i	0.913613	+	0.406585i	$0.133281\pi$
$692$		0			0
$693$	45554.1	−	7860.68i	0.0948552	−	0.0163679i
$694$		0			0
$695$	−36610.5	+	100587.i	−0.0757942	+	0.208243i
$696$		0			0
$697$	120834.	+	101392.i	0.248727	+	0.208707i
$698$		0			0
$699$	−838270.	+	146229.i	−1.71565	+	0.299281i
$700$		0			0
$701$			396026.i			0.805911i	0.915220	+	0.402956i	$0.132017\pi$
$701$			396026.i			0.805911i	−0.915220	+	0.402956i	$0.867983\pi$
$702$		0			0
$703$	611390.			1.23711
$704$		0			0
$705$	−275945.	−	101008.i	−0.555194	−	0.203225i
$706$		0			0
$707$	−49214.3	+	58651.3i	−0.0984584	+	0.117338i
$708$		0			0
$709$	471516.	+	171618.i	0.938002	+	0.341405i	0.765376	−	0.643583i	$-0.222553\pi$
$709$	471516.	+	171618.i	0.938002	+	0.341405i	0.172625	+	0.984988i	$0.444775\pi$
$710$		0			0
$711$	−81142.5	−	296.881i	−0.160513	−	0.000587277i
$712$		0			0
$713$	349869.	+	416958.i	0.688219	+	0.820188i
$714$		0			0
$715$	−4979.09	+	28237.8i	−0.00973953	+	0.0552356i
$716$		0			0
$717$	754877.	+	437672.i	1.46838	+	0.851354i
$718$		0			0
$719$	−397408.	−	229444.i	−0.768739	−	0.443832i	0.0636853	−	0.997970i	$-0.479715\pi$
$719$	−397408.	−	229444.i	−0.768739	−	0.443832i	−0.832425	+	0.554138i	$0.813048\pi$
$720$		0			0
$721$	20464.0	+	35444.6i	0.0393658	+	0.0681836i
$722$		0			0
$723$	−453314.	−	381791.i	−0.867207	−	0.730381i
$724$		0			0
$725$	−39762.8	−	109247.i	−0.0756487	−	0.207843i
$726$		0			0
$727$	−103524.	−	587115.i	−0.195872	−	1.11085i	−0.911171	−	0.412028i	$-0.864821\pi$
$727$	−103524.	−	587115.i	−0.195872	−	1.11085i	0.715299	−	0.698819i	$-0.246291\pi$
$728$		0			0
$729$	−270756.	+	457297.i	−0.509475	+	0.860485i
$730$		0			0
$731$	364508.	−	64272.6i	0.682139	−	0.120279i
$732$		0			0
$733$	321783.	−	117119.i	0.598901	−	0.217982i	−0.0247390	−	0.999694i	$-0.507875\pi$
$733$	321783.	−	117119.i	0.598901	−	0.217982i	0.623640	+	0.781712i	$0.285653\pi$
$734$		0			0
$735$	232665.	−	276252.i	0.430682	−	0.511364i
$736$		0			0
$737$	350660.	−	202453.i	0.645581	−	0.372726i
$738$		0			0
$739$	467974.	−	810555.i	0.856905	−	1.48420i	−0.0179608	−	0.999839i	$-0.505717\pi$
$739$	467974.	−	810555.i	0.856905	−	1.48420i	0.874866	−	0.484365i	$-0.160949\pi$
$740$		0			0
$741$	38337.2	−	66122.3i	0.0698207	−	0.120424i
$742$		0			0
$743$	630777.	+	111223.i	1.14261	+	0.201473i	0.712747	−	0.701421i	$-0.247451\pi$
$743$	630777.	+	111223.i	1.14261	+	0.201473i	0.429864	+	0.902894i	$0.358562\pi$
$744$		0			0
$745$	9673.85	−	8117.32i	0.0174296	−	0.0146252i
$746$		0			0
$747$	−515.203	+	140814.i	−0.000923288	+	0.252350i
$748$		0			0
$749$	−42775.2	+	117524.i	−0.0762480	+	0.209490i
$750$		0			0
$751$	−280016.	−	234962.i	−0.496482	−	0.416598i	0.359861	−	0.933006i	$-0.382824\pi$
$751$	−280016.	−	234962.i	−0.496482	−	0.416598i	−0.856342	+	0.516408i	$0.827269\pi$
$752$		0			0
$753$	78363.2	−	214082.i	0.138204	−	0.377563i
$754$		0			0
$755$			272994.i			0.478915i
$756$		0			0
$757$	−24716.3			−0.0431312			−0.0215656	−	0.999767i	$-0.506865\pi$
$757$	−24716.3			−0.0431312			−0.0215656	+	0.999767i	$0.506865\pi$
$758$		0			0
$759$	−85909.7	−	492485.i	−0.149128	−	0.854888i
$760$		0			0
$761$	−492900.	+	587416.i	−0.851118	+	1.01432i	0.148559	+	0.988904i	$0.452537\pi$
$761$	−492900.	+	587416.i	−0.851118	+	1.01432i	−0.999677	+	0.0254195i	$0.991908\pi$
$762$		0			0
$763$	−118612.	−	43171.3i	−0.203742	−	0.0741560i
$764$		0			0
$765$	−58434.5	−	338639.i	−0.0998497	−	0.578648i
$766$		0			0
$767$	−10690.0	−	12739.8i	−0.0181713	−	0.0216557i
$768$		0			0
$769$	116072.	−	658279.i	0.196280	−	1.11316i	−0.714304	−	0.699836i	$-0.753257\pi$
$769$	116072.	−	658279.i	0.196280	−	1.11316i	0.910584	−	0.413324i	$-0.135632\pi$
$770$		0			0
$771$	−984034.	+	565735.i	−1.65539	+	0.951709i
$772$		0			0
$773$	542163.	+	313018.i	0.907342	+	0.523854i	0.879575	−	0.475760i	$-0.157827\pi$
$773$	542163.	+	313018.i	0.907342	+	0.523854i	0.0277668	+	0.999614i	$0.491160\pi$
$774$		0			0
$775$	111142.	+	192504.i	0.185044	+	0.320506i
$776$		0			0
$777$	−22741.1	+	127606.i	−0.0376678	+	0.211363i
$778$		0			0
$779$	76874.3	+	211210.i	0.126679	+	0.348049i
$780$		0			0
$781$	53778.4	+	304992.i	0.0881669	+	0.500019i
$782$		0			0
$783$	−164508.	−	198252.i	−0.268326	−	0.323366i
$784$		0			0
$785$	−579399.	+	102164.i	−0.940240	+	0.165790i
$786$		0			0
$787$	590795.	−	215032.i	0.953866	−	0.347179i	0.182239	−	0.983254i	$-0.441665\pi$
$787$	590795.	−	215032.i	0.953866	−	0.347179i	0.771627	+	0.636075i	$0.219443\pi$
$788$		0			0
$789$	8022.95	+	22169.0i	0.0128878	+	0.0356116i
$790$		0			0
$791$	19783.1	−	11421.8i	0.0316185	−	0.0182549i
$792$		0			0
$793$	−12805.0	+	22179.0i	−0.0203627	+	0.0352692i
$794$		0			0
$795$	−267621.	−	489.579i	−0.423434	−	0.000774619i
$796$		0			0
$797$	423476.	+	74670.3i	0.666672	+	0.117552i	0.496734	−	0.867903i	$-0.334532\pi$
$797$	423476.	+	74670.3i	0.666672	+	0.117552i	0.169938	+	0.985455i	$0.445643\pi$
$798$		0			0
$799$	−358468.	+	300790.i	−0.561509	+	0.471162i
$800$		0			0
$801$	35842.0	+	99607.5i	0.0558634	+	0.155248i
$802$		0			0
$803$	34227.8	−	94040.0i	0.0530820	−	0.145842i
$804$		0			0
$805$	87877.7	+	73738.2i	0.135609	+	0.113789i
$806$		0			0
$807$	−583177.	−	697591.i	−0.895474	−	1.07116i
$808$		0			0
$809$		−	256726.i		−	0.392258i	−0.980578	−	0.196129i	$-0.937163\pi$
$809$		−	256726.i		−	0.392258i	0.980578	−	0.196129i	$-0.0628372\pi$
$810$		0			0
$811$	126232.			0.191924			0.0959618	−	0.995385i	$-0.469407\pi$
$811$	126232.			0.191924			0.0959618	+	0.995385i	$0.469407\pi$
$812$		0			0
$813$	201404.	−	168371.i	0.304711	−	0.254734i
$814$		0			0
$815$	−500461.	+	596426.i	−0.753451	+	0.897928i
$816$		0			0
$817$	495606.	+	180386.i	0.742494	+	0.270246i
$818$		0			0
$819$	12374.7	+	10461.0i	0.0184487	+	0.0155957i
$820$		0			0
$821$	−222431.	−	265083.i	−0.329996	−	0.393274i	0.575379	−	0.817887i	$-0.304855\pi$
$821$	−222431.	−	265083.i	−0.329996	−	0.393274i	−0.905375	+	0.424613i	$0.860410\pi$
$822$		0			0
$823$	−11974.4	+	67910.2i	−0.0176789	+	0.100262i	−0.992370	−	0.123292i	$-0.960655\pi$
$823$	−11974.4	+	67910.2i	−0.0176789	+	0.100262i	0.974692	+	0.223554i	$0.0717659\pi$
$824$		0			0
$825$	373.486	−	204161.i	0.000548740	−	0.299961i
$826$		0			0
$827$	622211.	+	359234.i	0.909760	+	0.525250i	0.880354	−	0.474317i	$-0.157305\pi$
$827$	622211.	+	359234.i	0.909760	+	0.525250i	0.0294062	+	0.999568i	$0.490638\pi$
$828$		0			0
$829$	−45069.7	−	78063.0i	−0.0655806	−	0.113589i	0.831371	−	0.555718i	$-0.187557\pi$
$829$	−45069.7	−	78063.0i	−0.0655806	−	0.113589i	−0.896951	+	0.442129i	$0.854223\pi$
$830$		0			0
$831$	−715760.	+	259033.i	−1.03649	+	0.375106i
$832$		0			0
$833$	−196716.	−	540473.i	−0.283498	−	0.778905i
$834$		0			0
$835$	−147719.	−	837758.i	−0.211868	−	1.20156i
$836$		0			0
$837$	375579.	+	318677.i	0.536106	+	0.454883i
$838$		0			0
$839$	−514240.	+	90674.4i	−0.730536	+	0.128813i	−0.526530	−	0.850156i	$-0.676507\pi$
$839$	−514240.	+	90674.4i	−0.730536	+	0.128813i	−0.204006	+	0.978970i	$0.565396\pi$
$840$		0			0
$841$	−547277.	+	199193.i	−0.773776	+	0.281632i
$842$		0			0
$843$	−1.14424e6	−	203920.i	−1.61014	−	0.286949i
$844$		0			0
$845$	416855.	−	240671.i	0.583810	−	0.337063i
$846$		0			0
$847$	40914.3	−	70865.6i	0.0570306	−	0.0987799i
$848$		0			0
$849$	−89369.1	−	155448.i	−0.123986	−	0.215660i
$850$		0			0
$851$	1.38043e6	+	243407.i	1.90614	+	0.336104i
$852$		0			0
$853$	359641.	−	301774.i	0.494277	−	0.414748i	−0.361279	−	0.932458i	$-0.617660\pi$
$853$	359641.	−	301774.i	0.494277	−	0.414748i	0.855556	+	0.517710i	$0.173215\pi$
$854$		0			0
$855$	169159.	−	459524.i	0.231400	−	0.628602i
$856$		0			0
$857$	−340030.	+	934226.i	−0.462974	+	1.27201i	0.460265	+	0.887782i	$0.347754\pi$
$857$	−340030.	+	934226.i	−0.462974	+	1.27201i	−0.923238	+	0.384228i	$0.874468\pi$
$858$		0			0
$859$	−609585.	−	511502.i	−0.826128	−	0.693204i	0.128270	−	0.991739i	$-0.459058\pi$
$859$	−609585.	−	511502.i	−0.826128	−	0.693204i	−0.954399	+	0.298535i	$0.903502\pi$
$860$		0			0
$861$	−46942.0	+	8188.63i	−0.0633221	+	0.0110460i
$862$		0			0
$863$			260591.i			0.349894i	0.984578	+	0.174947i	$0.0559755\pi$
$863$			260591.i			0.349894i	−0.984578	+	0.174947i	$0.944025\pi$
$864$		0			0
$865$	−806217.			−1.07751
$866$		0			0
$867$	191991.	+	70276.9i	0.255412	+	0.0934920i
$868$		0			0
$869$	44400.3	−	52914.2i	0.0587958	−	0.0700701i
$870$		0			0
$871$	133371.	+	48543.0i	0.175802	+	0.0639868i
$872$		0			0
$873$	−551305.	−	963008.i	−0.723375	−	1.26358i
$874$		0			0
$875$	87322.7	+	104067.i	0.114054	+	0.135924i
$876$		0			0
$877$	108616.	−	615990.i	0.141219	−	0.800893i	−0.829106	−	0.559091i	$-0.811150\pi$
$877$	108616.	−	615990.i	0.141219	−	0.800893i	0.970325	−	0.241803i	$-0.0777386\pi$
$878$		0			0
$879$	340532.	+	197438.i	0.440738	+	0.255536i
$880$		0			0
$881$	−491616.	−	283835.i	−0.633394	−	0.365690i	0.148671	−	0.988887i	$-0.452500\pi$
$881$	−491616.	−	283835.i	−0.633394	−	0.365690i	−0.782065	+	0.623196i	$0.785834\pi$
$882$		0			0
$883$	154303.	+	267260.i	0.197903	+	0.342778i	0.947848	−	0.318722i	$-0.103254\pi$
$883$	154303.	+	267260.i	0.197903	+	0.342778i	−0.749945	+	0.661500i	$0.769920\pi$
$884$		0			0
$885$	−81493.3	−	68635.4i	−0.104048	−	0.0876318i
$886$		0			0
$887$	−324902.	−	892660.i	−0.412957	−	1.13459i	−0.955610	−	0.294633i	$-0.904803\pi$
$887$	−324902.	−	892660.i	−0.412957	−	1.13459i	0.542653	−	0.839957i	$-0.317420\pi$
$888$		0			0
$889$	39578.0	+	224458.i	0.0500784	+	0.284009i
$890$		0			0
$891$	−151615.	−	426237.i	−0.190979	−	0.536903i
$892$		0			0
$893$	−656664.	+	115788.i	−0.823456	+	0.145197i
$894$		0			0
$895$	−236808.	+	86190.9i	−0.295631	+	0.107601i
$896$		0			0
$897$	112884.	−	134032.i	0.140297	−	0.166580i
$898$		0			0
$899$	−206780.	+	119385.i	−0.255853	+	0.147717i
$900$		0			0
$901$	−213089.	+	369081.i	−0.262489	+	0.454644i
$902$		0			0
$903$	−56083.6	+	96730.5i	−0.0687797	+	0.118628i
$904$		0			0
$905$	−109001.	−	19219.8i	−0.133086	−	0.0234667i
$906$		0			0
$907$	452630.	−	379801.i	0.550210	−	0.461681i	−0.324802	−	0.945782i	$-0.605298\pi$
$907$	452630.	−	379801.i	0.550210	−	0.461681i	0.875012	+	0.484101i	$0.160853\pi$
$908$		0			0
$909$	647525.	+	377014.i	0.783662	+	0.456278i
$910$		0			0
$911$	219349.	−	602656.i	0.264301	−	0.726160i	−0.734565	−	0.678539i	$-0.762614\pi$
$911$	219349.	−	602656.i	0.264301	−	0.726160i	0.998865	−	0.0476217i	$-0.0151642\pi$
$912$		0			0
$913$	−91826.6	−	77051.7i	−0.110161	−	0.0924359i
$914$		0			0
$915$	−56398.3	+	154075.i	−0.0673633	+	0.184031i
$916$		0			0
$917$		−	196985.i		−	0.234258i
$918$		0			0
$919$	715806.			0.847548			0.423774	−	0.905768i	$-0.360705\pi$
$919$	715806.			0.847548			0.423774	+	0.905768i	$0.360705\pi$
$920$		0			0
$921$	10354.7	+	59359.2i	0.0122073	+	0.0699792i
$922$		0			0
$923$	−69779.0	+	83159.3i	−0.0819070	+	0.0976130i
$924$		0			0
$925$	537922.	+	195788.i	0.628689	+	0.228824i
$926$		0			0
$927$	307769.	−	256336.i	0.358151	−	0.298298i
$928$		0			0
$929$	−495286.	−	590259.i	−0.573885	−	0.683929i	0.398539	−	0.917152i	$-0.369517\pi$
$929$	−495286.	−	590259.i	−0.573885	−	0.683929i	−0.972423	+	0.233222i	$0.925073\pi$
$930$		0			0
$931$	142316.	−	807114.i	0.164193	−	0.931184i
$932$		0			0
$933$	−209723.	+	120572.i	−0.240925	+	0.138511i
$934$		0			0
$935$	253341.	+	146267.i	0.289790	+	0.167310i
$936$		0			0
$937$	−536614.	−	929442.i	−0.611199	−	1.05863i	−0.991039	−	0.133575i	$-0.957354\pi$
$937$	−536614.	−	929442.i	−0.611199	−	1.05863i	0.379840	−	0.925052i	$-0.375979\pi$
$938$		0			0
$939$	−246969.	+	1.38580e6i	−0.280099	+	1.57170i
$940$		0			0
$941$	−416989.	−	1.14567e6i	−0.470918	−	1.29384i	−0.917015	−	0.398852i	$-0.869409\pi$
$941$	−416989.	−	1.14567e6i	−0.470918	−	1.29384i	0.446097	−	0.894985i	$-0.352814\pi$
$942$		0			0
$943$	89483.7	+	507487.i	0.100628	+	0.570692i
$944$		0			0
$945$	89617.1	+	52398.3i	0.100352	+	0.0586751i
$946$		0			0
$947$	509671.	−	89868.8i	0.568316	−	0.100209i	0.117896	−	0.993026i	$-0.462385\pi$
$947$	509671.	−	89868.8i	0.568316	−	0.100209i	0.450421	+	0.892817i	$0.351274\pi$
$948$		0			0
$949$	32963.4	−	11997.7i	0.0366016	−	0.0133219i
$950$		0			0
$951$	−170744.	−	471799.i	−0.188792	−	0.521670i
$952$		0			0
$953$	−347753.	+	200775.i	−0.382900	+	0.221067i	−0.679079	−	0.734065i	$-0.737621\pi$
$953$	−347753.	+	200775.i	−0.382900	+	0.221067i	0.296179	+	0.955132i	$0.404287\pi$
$954$		0			0
$955$	−533538.	+	924115.i	−0.585004	+	1.01326i
$956$		0			0
$957$	219301.	+	401.184i	0.239451	+	0.000438046i
$958$		0			0
$959$	−80698.0	−	14229.2i	−0.0877456	−	0.0154719i
$960$		0			0
$961$	−357742.	+	300181.i	−0.387367	+	0.325040i
$962$		0			0
$963$	1.20457e6	+	216945.i	1.29891	+	0.233936i
$964$		0			0
$965$	−57758.5	+	158690.i	−0.0620242	+	0.170410i
$966$		0			0
$967$	340877.	+	286030.i	0.364540	+	0.305885i	0.806597	−	0.591102i	$-0.201307\pi$
$967$	340877.	+	286030.i	0.364540	+	0.305885i	−0.442057	+	0.896987i	$0.645751\pi$
$968$		0			0
$969$	−500141.	−	598264.i	−0.532653	−	0.637155i
$970$		0			0
$971$			1.11262e6i			1.18007i	0.807377	+	0.590036i	$0.200887\pi$
$971$			1.11262e6i			1.18007i	−0.807377	+	0.590036i	$0.799113\pi$
$972$		0			0
$973$	−51494.4			−0.0543919
$974$		0			0
$975$	54905.0	−	45899.9i	0.0577567	−	0.0482839i
$976$		0			0
$977$	771833.	−	919835.i	0.808601	−	0.963653i	−0.191239	−	0.981544i	$-0.561250\pi$
$977$	771833.	−	919835.i	0.808601	−	0.963653i	0.999840	+	0.0178902i	$0.00569492\pi$
$978$		0			0
$979$	−84680.6	−	30821.2i	−0.0883525	−	0.0321577i
$980$		0			0
$981$	−218954.	+	1.21572e6i	−0.227518	+	1.26327i
$982$		0			0
$983$	−570006.	−	679307.i	−0.589892	−	0.703006i	0.385693	−	0.922627i	$-0.373962\pi$
$983$	−570006.	−	679307.i	−0.589892	−	0.703006i	−0.975585	+	0.219621i	$0.929518\pi$
$984$		0			0
$985$	103513.	−	587051.i	0.106690	−	0.605067i
$986$		0			0
$987$	258.604	−	141362.i	0.000265461	−	0.145110i
$988$		0			0
$989$	1.04719e6	+	604596.i	1.07061	+	0.618120i
$990$		0			0
$991$	123747.	+	214336.i	0.126005	+	0.218247i	0.922125	−	0.386891i	$-0.126451\pi$
$991$	123747.	+	214336.i	0.126005	+	0.218247i	−0.796120	+	0.605138i	$0.793118\pi$
$992$		0			0
$993$	340899.	−	123371.i	0.345722	−	0.125117i
$994$		0			0
$995$	417467.	+	1.14698e6i	0.421673	+	1.15854i
$996$		0			0
$997$	126217.	+	715814.i	0.126978	+	0.720129i	0.980113	+	0.198439i	$0.0635871\pi$
$997$	126217.	+	715814.i	0.126978	+	0.720129i	−0.853135	+	0.521690i	$0.825302\pi$
$998$		0			0
$999$	1.26846e6	+	6961.52i	1.27100	+	0.00697546i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.41.3	yes	72
3.2	odd	2		324.5.k.a.233.4		72
27.2	odd	18	inner	108.5.k.a.29.3	✓	72
27.25	even	9		324.5.k.a.89.4		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.3	✓	72	27.2	odd	18	inner
108.5.k.a.41.3	yes	72	1.1	even	1	trivial
324.5.k.a.89.4		72	27.25	even	9
324.5.k.a.233.4		72	3.2	odd	2

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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-6.90497 + 5.77247i) q^{3} +(11.0592 - 13.1798i) q^{5} +(7.77764 + 2.83083i) q^{7} +(14.3573 - 79.7174i) q^{9} +O(q^{10})\)	\(q+(-6.90497 + 5.77247i) q^{3} +(11.0592 - 13.1798i) q^{5} +(7.77764 + 2.83083i) q^{7} +(14.3573 - 79.7174i) q^{9} +(44.3220 + 52.8209i) q^{11} +(-4.19703 + 23.8025i) q^{13} +(-0.283269 + 154.845i) q^{15} +(213.549 + 123.293i) q^{17} +(175.684 + 304.294i) q^{19} +(-70.0452 + 25.3494i) q^{21} +(275.524 + 756.995i) q^{23} +(57.1279 + 323.988i) q^{25} +(361.030 + 633.323i) q^{27} +(-348.016 + 61.3646i) q^{29} +(634.917 - 231.091i) q^{31} +(-610.949 - 108.880i) q^{33} +(123.324 - 71.2013i) q^{35} +(870.013 - 1506.91i) q^{37} +(-108.419 - 188.583i) q^{39} +(629.967 + 111.080i) q^{41} +(1149.85 - 964.841i) q^{43} +(-891.882 - 1070.84i) q^{45} +(-649.053 + 1783.26i) q^{47} +(-1786.79 - 1499.30i) q^{49} +(-2186.26 + 381.374i) q^{51} +1728.31i q^{53} +1186.34 q^{55} +(-2969.62 - 1087.01i) q^{57} +(-442.288 + 527.098i) q^{59} +(995.692 + 362.402i) q^{61} +(337.332 - 579.370i) q^{63} +(267.298 + 318.553i) q^{65} +(1019.70 - 5783.02i) q^{67} +(-6272.21 - 3636.58i) q^{69} +(3889.70 + 2245.72i) q^{71} +(-725.679 - 1256.91i) q^{73} +(-2264.68 - 1907.36i) q^{75} +(195.194 + 536.290i) q^{77} +(-173.955 - 986.547i) q^{79} +(-6148.74 - 2289.05i) q^{81} +(-1712.04 + 301.879i) q^{83} +(3986.66 - 1451.03i) q^{85} +(2048.81 - 2432.63i) q^{87} +(-1131.82 + 653.456i) q^{89} +(-100.024 + 173.246i) q^{91} +(-3050.12 + 5260.71i) q^{93} +(5953.47 + 1049.76i) q^{95} +(10494.3 - 8805.79i) q^{97} +(4847.09 - 2774.87i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 72 * q + 9 * q^5 - 102 * q^9	\( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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