LMFDB - Embedded newform 1045.2.b.b.419.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(419,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1045.419");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.b (of order $2$, degree $1$, 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:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 19x^{14} + 144x^{12} + 552x^{10} + 1119x^{8} + 1146x^{6} + 524x^{4} + 83x^{2} + 4 $ x^16 + 19x^14 + 144x^12 + 552x^10 + 1119x^8 + 1146x^6 + 524x^4 + 83*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			419.8
Root			$-0.301154i$ of defining polynomial
Character	$\chi$	$=$	1045.419
Dual form			1045.2.b.b.419.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.301154i q^{2} +1.85377i q^{3} +1.90931 q^{4} +(0.637551 + 2.14325i) q^{5} +0.558272 q^{6} +1.94668i q^{7} -1.17730i q^{8} -0.436480 q^{9} +O(q^{10})$	\(q-0.301154i q^{2} +1.85377i q^{3} +1.90931 q^{4} +(0.637551 + 2.14325i) q^{5} +0.558272 q^{6} +1.94668i q^{7} -1.17730i q^{8} -0.436480 q^{9} +(0.645450 - 0.192001i) q^{10} -1.00000 q^{11} +3.53942i q^{12} +3.75832i q^{13} +0.586251 q^{14} +(-3.97311 + 1.18188i) q^{15} +3.46406 q^{16} -2.16098i q^{17} +0.131448i q^{18} -1.00000 q^{19} +(1.21728 + 4.09212i) q^{20} -3.60871 q^{21} +0.301154i q^{22} -4.03747i q^{23} +2.18246 q^{24} +(-4.18706 + 2.73286i) q^{25} +1.13183 q^{26} +4.75219i q^{27} +3.71681i q^{28} +2.18930 q^{29} +(0.355927 + 1.19652i) q^{30} +3.21529 q^{31} -3.39783i q^{32} -1.85377i q^{33} -0.650787 q^{34} +(-4.17223 + 1.24111i) q^{35} -0.833374 q^{36} -5.64900i q^{37} +0.301154i q^{38} -6.96707 q^{39} +(2.52326 - 0.750591i) q^{40} -3.52926 q^{41} +1.08678i q^{42} -2.74094i q^{43} -1.90931 q^{44} +(-0.278278 - 0.935486i) q^{45} -1.21590 q^{46} -1.87707i q^{47} +6.42159i q^{48} +3.21044 q^{49} +(0.823014 + 1.26095i) q^{50} +4.00596 q^{51} +7.17578i q^{52} -4.59639i q^{53} +1.43114 q^{54} +(-0.637551 - 2.14325i) q^{55} +2.29184 q^{56} -1.85377i q^{57} -0.659317i q^{58} -3.10100 q^{59} +(-7.58588 + 2.25656i) q^{60} -10.0971 q^{61} -0.968298i q^{62} -0.849687i q^{63} +5.90485 q^{64} +(-8.05502 + 2.39612i) q^{65} -0.558272 q^{66} +12.0622i q^{67} -4.12596i q^{68} +7.48457 q^{69} +(0.373765 + 1.25648i) q^{70} +6.42317 q^{71} +0.513870i q^{72} -10.7452i q^{73} -1.70122 q^{74} +(-5.06611 - 7.76186i) q^{75} -1.90931 q^{76} -1.94668i q^{77} +2.09816i q^{78} -10.8068 q^{79} +(2.20851 + 7.42436i) q^{80} -10.1189 q^{81} +1.06285i q^{82} -8.12591i q^{83} -6.89012 q^{84} +(4.63152 - 1.37773i) q^{85} -0.825447 q^{86} +4.05847i q^{87} +1.17730i q^{88} +11.1151 q^{89} +(-0.281726 + 0.0838046i) q^{90} -7.31624 q^{91} -7.70877i q^{92} +5.96042i q^{93} -0.565288 q^{94} +(-0.637551 - 2.14325i) q^{95} +6.29880 q^{96} +11.4492i q^{97} -0.966837i q^{98} +0.436480 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) $ 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9	$ 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99}+O(q^{100}) $ 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9 + 10 * q^10 - 16 * q^11 + 4 * q^14 + 3 * q^15 - 18 * q^16 - 16 * q^19 - 2 * q^20 - 10 * q^21 + 10 * q^24 - 7 * q^25 - 24 * q^26 + 2 * q^29 + 4 * q^30 - 32 * q^31 - 16 * q^34 - 18 * q^35 + 18 * q^36 + 40 * q^39 - 28 * q^40 + 6 * q^41 + 6 * q^44 + 16 * q^45 + 38 * q^49 - 30 * q^50 - 16 * q^51 + 18 * q^54 - 3 * q^55 + 12 * q^56 + 24 * q^59 - 20 * q^60 - 42 * q^61 + 62 * q^64 - 20 * q^65 + 8 * q^66 + 30 * q^69 - 18 * q^70 - 46 * q^71 - 2 * q^74 - 25 * q^75 + 6 * q^76 + 74 * q^79 - 22 * q^80 - 56 * q^81 + 34 * q^84 - 18 * q^85 + 8 * q^86 + 14 * q^89 - 4 * q^90 - 24 * q^91 + 64 * q^94 - 3 * q^95 + 54 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times$.

$n$	$496$	$761$	$837$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	0.301154i		−	0.212948i	−0.994315	−	0.106474i	$-0.966044\pi$
$2$		−	0.301154i		−	0.212948i	0.994315	−	0.106474i	$-0.0339561\pi$
$3$			1.85377i			1.07028i	0.844764	+	0.535139i	$0.179741\pi$
$3$			1.85377i			1.07028i	−0.844764	+	0.535139i	$0.820259\pi$
$4$	1.90931			0.954653
$5$	0.637551	+	2.14325i	0.285121	+	0.958491i
$5$	0.637551	+	2.14325i	0.285121	+	0.958491i
$6$	0.558272			0.227914
$7$			1.94668i			0.735776i	0.929870	+	0.367888i	$0.119919\pi$
$7$			1.94668i			0.735776i	−0.929870	+	0.367888i	$0.880081\pi$
$8$		−	1.17730i		−	0.416240i
$9$	−0.436480			−0.145493
$10$	0.645450	−	0.192001i	0.204109	−	0.0607161i
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$			3.53942i			1.02174i
$13$			3.75832i			1.04237i	0.853444	+	0.521185i	$0.174510\pi$
$13$			3.75832i			1.04237i	−0.853444	+	0.521185i	$0.825490\pi$
$14$	0.586251			0.156682
$15$	−3.97311	+	1.18188i	−1.02585	+	0.305159i
$16$	3.46406			0.866015
$17$		−	2.16098i		−	0.524114i	−0.965052	−	0.262057i	$-0.915599\pi$
$17$		−	2.16098i		−	0.524114i	0.965052	−	0.262057i	$-0.0844008\pi$
$18$			0.131448i			0.0309825i
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$	1.21728	+	4.09212i	0.272192	+	0.915027i
$21$	−3.60871			−0.787484
$22$			0.301154i			0.0642063i
$23$		−	4.03747i		−	0.841872i	−0.907090	−	0.420936i	$-0.861702\pi$
$23$		−	4.03747i		−	0.841872i	0.907090	−	0.420936i	$-0.138298\pi$
$24$	2.18246			0.445492
$25$	−4.18706	+	2.73286i	−0.837412	+	0.546573i
$26$	1.13183			0.221971
$27$			4.75219i			0.914559i
$28$			3.71681i			0.702411i
$29$	2.18930			0.406543			0.203271	−	0.979122i	$-0.434843\pi$
$29$	2.18930			0.406543			0.203271	+	0.979122i	$0.434843\pi$
$30$	0.355927	+	1.19652i	0.0649830	+	0.218453i
$31$	3.21529			0.577483			0.288741	−	0.957407i	$-0.406763\pi$
$31$	3.21529			0.577483			0.288741	+	0.957407i	$0.406763\pi$
$32$		−	3.39783i		−	0.600656i
$33$		−	1.85377i		−	0.322701i
$34$	−0.650787			−0.111609
$35$	−4.17223	+	1.24111i	−0.705235	+	0.209785i
$36$	−0.833374			−0.138896
$37$		−	5.64900i		−	0.928690i	−0.885654	−	0.464345i	$-0.846290\pi$
$37$		−	5.64900i		−	0.928690i	0.885654	−	0.464345i	$-0.153710\pi$
$38$			0.301154i			0.0488537i
$39$	−6.96707			−1.11562
$40$	2.52326	−	0.750591i	0.398962	−	0.118679i
$41$	−3.52926			−0.551177			−0.275589	−	0.961276i	$-0.588873\pi$
$41$	−3.52926			−0.551177			−0.275589	+	0.961276i	$0.588873\pi$
$42$			1.08678i			0.167693i
$43$		−	2.74094i		−	0.417990i	−0.977917	−	0.208995i	$-0.932981\pi$
$43$		−	2.74094i		−	0.417990i	0.977917	−	0.208995i	$-0.0670192\pi$
$44$	−1.90931			−0.287839
$45$	−0.278278	−	0.935486i	−0.0414832	−	0.139454i
$46$	−1.21590			−0.179275
$47$		−	1.87707i		−	0.273799i	−0.990585	−	0.136899i	$-0.956286\pi$
$47$		−	1.87707i		−	0.273799i	0.990585	−	0.136899i	$-0.0437137\pi$
$48$			6.42159i			0.926877i
$49$	3.21044			0.458634
$50$	0.823014	+	1.26095i	0.116392	+	0.178325i
$51$	4.00596			0.560947
$52$			7.17578i			0.995101i
$53$		−	4.59639i		−	0.631362i	−0.948865	−	0.315681i	$-0.897767\pi$
$53$		−	4.59639i		−	0.631362i	0.948865	−	0.315681i	$-0.102233\pi$
$54$	1.43114			0.194754
$55$	−0.637551	−	2.14325i	−0.0859673	−	0.288996i
$56$	2.29184			0.306259
$57$		−	1.85377i		−	0.245538i
$58$		−	0.659317i		−	0.0865726i
$59$	−3.10100			−0.403715			−0.201858	−	0.979415i	$-0.564698\pi$
$59$	−3.10100			−0.403715			−0.201858	+	0.979415i	$0.564698\pi$
$60$	−7.58588	+	2.25656i	−0.979332	+	0.291321i
$61$	−10.0971			−1.29281			−0.646403	−	0.762996i	$-0.723728\pi$
$61$	−10.0971			−1.29281			−0.646403	+	0.762996i	$0.723728\pi$
$62$		−	0.968298i		−	0.122974i
$63$		−	0.849687i		−	0.107050i
$64$	5.90485			0.738107
$65$	−8.05502	+	2.39612i	−0.999102	+	0.297202i
$66$	−0.558272			−0.0687186
$67$			12.0622i			1.47363i	0.676096	+	0.736814i	$0.263671\pi$
$67$			12.0622i			1.47363i	−0.676096	+	0.736814i	$0.736329\pi$
$68$		−	4.12596i		−	0.500347i
$69$	7.48457			0.901036
$70$	0.373765	+	1.25648i	0.0446734	+	0.150179i
$71$	6.42317			0.762291			0.381145	−	0.924515i	$-0.375530\pi$
$71$	6.42317			0.762291			0.381145	+	0.924515i	$0.375530\pi$
$72$			0.513870i			0.0605601i
$73$		−	10.7452i		−	1.25763i	−0.777556	−	0.628814i	$-0.783541\pi$
$73$		−	10.7452i		−	1.25763i	0.777556	−	0.628814i	$-0.216459\pi$
$74$	−1.70122			−0.197763
$75$	−5.06611	−	7.76186i	−0.584984	−	0.896263i
$76$	−1.90931			−0.219012
$77$		−	1.94668i		−	0.221845i
$78$			2.09816i			0.237570i
$79$	−10.8068			−1.21586			−0.607932	−	0.793989i	$-0.708001\pi$
$79$	−10.8068			−1.21586			−0.607932	+	0.793989i	$0.708001\pi$
$80$	2.20851	+	7.42436i	0.246919	+	0.830068i
$81$	−10.1189			−1.12432
$82$			1.06285i			0.117372i
$83$		−	8.12591i		−	0.891934i	−0.895049	−	0.445967i	$-0.852860\pi$
$83$		−	8.12591i		−	0.891934i	0.895049	−	0.445967i	$-0.147140\pi$
$84$	−6.89012			−0.751774
$85$	4.63152	−	1.37773i	0.502358	−	0.149436i
$86$	−0.825447			−0.0890103
$87$			4.05847i			0.435114i
$88$			1.17730i			0.125501i
$89$	11.1151			1.17820			0.589101	−	0.808060i	$-0.299482\pi$
$89$	11.1151			1.17820			0.589101	+	0.808060i	$0.299482\pi$
$90$	−0.281726	+	0.0838046i	−0.0296965	+	0.00883378i
$91$	−7.31624			−0.766950
$92$		−	7.70877i		−	0.803695i
$93$			5.96042i			0.618066i
$94$	−0.565288			−0.0583050
$95$	−0.637551	−	2.14325i	−0.0654113	−	0.219893i
$96$	6.29880			0.642869
$97$			11.4492i			1.16249i	0.813728	+	0.581246i	$0.197435\pi$
$97$			11.4492i			1.16249i	−0.813728	+	0.581246i	$0.802565\pi$
$98$		−	0.966837i		−	0.0976653i
$99$	0.436480			0.0438679
$100$	−7.99438	+	5.21787i	−0.799438	+	0.521787i
$101$	6.86482			0.683075			0.341537	−	0.939868i	$-0.389052\pi$
$101$	6.86482			0.683075			0.341537	+	0.939868i	$0.389052\pi$
$102$		−	1.20641i		−	0.119453i
$103$			13.7258i			1.35244i	0.736699	+	0.676220i	$0.236383\pi$
$103$			13.7258i			1.35244i	−0.736699	+	0.676220i	$0.763617\pi$
$104$	4.42468			0.433876
$105$	−2.30073	−	7.73437i	−0.224529	−	0.754797i
$106$	−1.38422			−0.134447
$107$			14.8545i			1.43603i	0.696025	+	0.718017i	$0.254950\pi$
$107$			14.8545i			1.43603i	−0.696025	+	0.718017i	$0.745050\pi$
$108$			9.07338i			0.873087i
$109$	7.13713			0.683613			0.341806	−	0.939770i	$-0.388961\pi$
$109$	7.13713			0.683613			0.341806	+	0.939770i	$0.388961\pi$
$110$	−0.645450	+	0.192001i	−0.0615412	+	0.0183066i
$111$	10.4720			0.993956
$112$			6.74342i			0.637193i
$113$			1.22524i			0.115261i	0.998338	+	0.0576303i	$0.0183545\pi$
$113$			1.22524i			0.115261i	−0.998338	+	0.0576303i	$0.981646\pi$
$114$	−0.558272			−0.0522870
$115$	8.65332	−	2.57409i	0.806927	−	0.240036i
$116$	4.18004			0.388107
$117$		−	1.64043i		−	0.151658i
$118$			0.933879i			0.0859705i
$119$	4.20673			0.385630
$120$	1.39143	+	4.67756i	0.127019	+	0.427000i
$121$	1.00000			0.0909091
$122$			3.04080i			0.275301i
$123$		−	6.54245i		−	0.589913i
$124$	6.13897			0.551296
$125$	−8.52668	−	7.23158i	−0.762649	−	0.646812i
$126$	−0.255887			−0.0227962
$127$		−	1.78182i		−	0.158111i	−0.996870	−	0.0790554i	$-0.974810\pi$
$127$		−	1.78182i		−	0.158111i	0.996870	−	0.0790554i	$-0.0251904\pi$
$128$		−	8.57392i		−	0.757835i
$129$	5.08109			0.447365
$130$	0.721601	+	2.42580i	0.0632886	+	0.212757i
$131$	18.4204			1.60939			0.804697	−	0.593685i	$-0.202328\pi$
$131$	18.4204			1.60939			0.804697	+	0.593685i	$0.202328\pi$
$132$		−	3.53942i		−	0.308067i
$133$		−	1.94668i		−	0.168799i
$134$	3.63257			0.313807
$135$	−10.1851	+	3.02976i	−0.876597	+	0.260760i
$136$	−2.54413			−0.218157
$137$		−	13.8926i		−	1.18693i	−0.804861	−	0.593463i	$-0.797760\pi$
$137$		−	13.8926i		−	1.18693i	0.804861	−	0.593463i	$-0.202240\pi$
$138$		−	2.25401i		−	0.191874i
$139$	4.35165			0.369102			0.184551	−	0.982823i	$-0.440917\pi$
$139$	4.35165			0.369102			0.184551	+	0.982823i	$0.440917\pi$
$140$	−7.96606	+	2.36965i	−0.673255	+	0.200272i
$141$	3.47967			0.293041
$142$		−	1.93437i		−	0.162328i
$143$		−	3.75832i		−	0.314286i
$144$	−1.51199			−0.125999
$145$	1.39579	+	4.69222i	0.115914	+	0.389668i
$146$	−3.23596			−0.267810
$147$			5.95142i			0.490865i
$148$		−	10.7857i		−	0.886577i
$149$	11.6917			0.957818			0.478909	−	0.877864i	$-0.341032\pi$
$149$	11.6917			0.957818			0.478909	+	0.877864i	$0.341032\pi$
$150$	−2.33752	+	1.52568i	−0.190858	+	0.124571i
$151$	−23.0439			−1.87529			−0.937644	−	0.347596i	$-0.886998\pi$
$151$	−23.0439			−1.87529			−0.937644	+	0.347596i	$0.886998\pi$
$152$			1.17730i			0.0954920i
$153$			0.943222i			0.0762550i
$154$	−0.586251			−0.0472415
$155$	2.04991	+	6.89117i	0.164653	+	0.553512i
$156$	−13.3023			−1.06503
$157$			12.2578i			0.978277i	0.872206	+	0.489138i	$0.162689\pi$
$157$			12.2578i			0.978277i	−0.872206	+	0.489138i	$0.837311\pi$
$158$			3.25452i			0.258916i
$159$	8.52066			0.675732
$160$	7.28240	−	2.16629i	0.575724	−	0.171260i
$161$	7.85967			0.619429
$162$			3.04736i			0.239423i
$163$		−	8.00570i		−	0.627055i	−0.949579	−	0.313527i	$-0.898489\pi$
$163$		−	8.00570i		−	0.627055i	0.949579	−	0.313527i	$-0.101511\pi$
$164$	−6.73843			−0.526183
$165$	3.97311	−	1.18188i	0.309306	−	0.0920089i
$166$	−2.44715			−0.189936
$167$			5.00463i			0.387270i	0.981074	+	0.193635i	$0.0620278\pi$
$167$			5.00463i			0.387270i	−0.981074	+	0.193635i	$0.937972\pi$
$168$			4.24855i			0.327782i
$169$	−1.12494			−0.0865341
$170$	−0.414910	−	1.39480i	−0.0318221	−	0.106976i
$171$	0.436480			0.0333784
$172$		−	5.23330i		−	0.399035i
$173$		−	7.15833i		−	0.544237i	−0.962264	−	0.272119i	$-0.912276\pi$
$173$		−	7.15833i		−	0.544237i	0.962264	−	0.272119i	$-0.0877244\pi$
$174$	1.22223			0.0926567
$175$	−5.32001	−	8.15086i	−0.402155	−	0.616147i
$176$	−3.46406			−0.261113
$177$		−	5.74855i		−	0.432087i
$178$		−	3.34737i		−	0.250896i
$179$	4.12217			0.308106			0.154053	−	0.988063i	$-0.450767\pi$
$179$	4.12217			0.308106			0.154053	+	0.988063i	$0.450767\pi$
$180$	−0.531318	−	1.78613i	−0.0396021	−	0.133130i
$181$	17.5288			1.30291			0.651454	−	0.758688i	$-0.274159\pi$
$181$	17.5288			1.30291			0.651454	+	0.758688i	$0.274159\pi$
$182$			2.20332i			0.163321i
$183$		−	18.7178i		−	1.38366i
$184$	−4.75334			−0.350421
$185$	12.1072	−	3.60152i	0.890141	−	0.264789i
$186$	1.79501			0.131616
$187$			2.16098i			0.158026i
$188$		−	3.58390i		−	0.261383i
$189$	−9.25099			−0.672911
$190$	−0.645450	+	0.192001i	−0.0468258	+	0.0139292i
$191$	−27.5080			−1.99041			−0.995205	−	0.0978155i	$-0.968814\pi$
$191$	−27.5080			−1.99041			−0.995205	+	0.0978155i	$0.968814\pi$
$192$			10.9463i			0.789979i
$193$		−	13.1923i		−	0.949603i	−0.880093	−	0.474802i	$-0.842520\pi$
$193$		−	13.1923i		−	0.949603i	0.880093	−	0.474802i	$-0.157480\pi$
$194$	3.44798			0.247551
$195$	−4.44186	−	14.9322i	−0.318088	−	1.06932i
$196$	6.12970			0.437836
$197$		−	11.4532i		−	0.816006i	−0.912981	−	0.408003i	$-0.866225\pi$
$197$		−	11.4532i		−	0.816006i	0.912981	−	0.408003i	$-0.133775\pi$
$198$		−	0.131448i		−	0.00934159i
$199$	19.7562			1.40048			0.700240	−	0.713907i	$-0.253076\pi$
$199$	19.7562			1.40048			0.700240	+	0.713907i	$0.253076\pi$
$200$	3.21741	+	4.92944i	0.227505	+	0.348564i
$201$	−22.3605			−1.57719
$202$		−	2.06737i		−	0.145460i
$203$			4.26187i			0.299124i
$204$	7.64861			0.535510
$205$	−2.25008	−	7.56409i	−0.157152	−	0.528299i
$206$	4.13358			0.288000
$207$			1.76228i			0.122487i
$208$			13.0190i			0.902708i
$209$	1.00000			0.0691714
$210$	−2.32924	+	0.692876i	−0.160733	+	0.0478130i
$211$	−22.4594			−1.54617			−0.773085	−	0.634303i	$-0.781287\pi$
$211$	−22.4594			−1.54617			−0.773085	+	0.634303i	$0.781287\pi$
$212$		−	8.77591i		−	0.602732i
$213$			11.9071i			0.815862i
$214$	4.47349			0.305801
$215$	5.87453	−	1.74749i	0.400640	−	0.119178i
$216$	5.59477			0.380676
$217$			6.25914i			0.424898i
$218$		−	2.14938i		−	0.145574i
$219$	19.9191			1.34601
$220$	−1.21728	−	4.09212i	−0.0820690	−	0.275891i
$221$	8.12163			0.546320
$222$		−	3.15368i		−	0.211661i
$223$		−	10.3113i		−	0.690499i	−0.938511	−	0.345249i	$-0.887794\pi$
$223$		−	10.3113i		−	0.690499i	0.938511	−	0.345249i	$-0.112206\pi$
$224$	6.61448			0.441949
$225$	1.82757	−	1.19284i	0.121838	−	0.0795226i
$226$	0.368986			0.0245446
$227$		−	19.9756i		−	1.32583i	−0.748696	−	0.662914i	$-0.769320\pi$
$227$		−	19.9756i		−	1.32583i	0.748696	−	0.662914i	$-0.230680\pi$
$228$		−	3.53942i		−	0.234404i
$229$	15.7027			1.03767			0.518833	−	0.854876i	$-0.326367\pi$
$229$	15.7027			1.03767			0.518833	+	0.854876i	$0.326367\pi$
$230$	−0.775200	−	2.60599i	−0.0511152	−	0.171834i
$231$	3.60871			0.237435
$232$		−	2.57747i		−	0.169219i
$233$		−	21.1823i		−	1.38770i	−0.720120	−	0.693850i	$-0.755913\pi$
$233$		−	21.1823i		−	1.38770i	0.720120	−	0.693850i	$-0.244087\pi$
$234$	−0.494022			−0.0322953
$235$	4.02304	−	1.19673i	0.262434	−	0.0780659i
$236$	−5.92075			−0.385408
$237$		−	20.0334i		−	1.30131i
$238$		−	1.26687i		−	0.0821193i
$239$	0.429031			0.0277517			0.0138759	−	0.999904i	$-0.495583\pi$
$239$	0.429031			0.0277517			0.0138759	+	0.999904i	$0.495583\pi$
$240$	−13.7631	+	4.09409i	−0.888403	+	0.264272i
$241$	11.9657			0.770781			0.385391	−	0.922753i	$-0.374067\pi$
$241$	11.9657			0.770781			0.385391	+	0.922753i	$0.374067\pi$
$242$		−	0.301154i		−	0.0193589i
$243$		−	4.50164i		−	0.288780i
$244$	−19.2785			−1.23418
$245$	2.04682	+	6.88077i	0.130766	+	0.439596i
$246$	−1.97029			−0.125621
$247$		−	3.75832i		−	0.239136i
$248$		−	3.78537i		−	0.240371i
$249$	15.0636			0.954617
$250$	−2.17782	+	2.56785i	−0.137738	+	0.162405i
$251$	7.13854			0.450581			0.225290	−	0.974292i	$-0.427667\pi$
$251$	7.13854			0.450581			0.225290	+	0.974292i	$0.427667\pi$
$252$		−	1.62231i		−	0.102196i
$253$			4.03747i			0.253834i
$254$	−0.536602			−0.0336694
$255$	2.55400	+	8.58579i	0.159938	+	0.537663i
$256$	9.22763			0.576727
$257$		−	6.86772i		−	0.428397i	−0.976790	−	0.214198i	$-0.931286\pi$
$257$		−	6.86772i		−	0.428397i	0.976790	−	0.214198i	$-0.0687139\pi$
$258$		−	1.53019i		−	0.0952657i
$259$	10.9968			0.683308
$260$	−15.3795	+	4.57492i	−0.953796	+	0.283725i
$261$	−0.955585			−0.0591492
$262$		−	5.54737i		−	0.342718i
$263$			0.473180i			0.0291775i	0.999894	+	0.0145888i	$0.00464391\pi$
$263$			0.473180i			0.0291775i	−0.999894	+	0.0145888i	$0.995356\pi$
$264$	−2.18246			−0.134321
$265$	9.85121	−	2.93043i	0.605155	−	0.180015i
$266$	−0.586251			−0.0359454
$267$			20.6049i			1.26100i
$268$			23.0304i			1.40680i
$269$	−2.26859			−0.138318			−0.0691591	−	0.997606i	$-0.522032\pi$
$269$	−2.26859			−0.138318			−0.0691591	+	0.997606i	$0.522032\pi$
$270$	0.912426	+	3.06730i	0.0555285	+	0.186670i
$271$	−13.6384			−0.828474			−0.414237	−	0.910169i	$-0.635952\pi$
$271$	−13.6384			−0.828474			−0.414237	+	0.910169i	$0.635952\pi$
$272$		−	7.48575i		−	0.453891i
$273$		−	13.5627i		−	0.820850i
$274$	−4.18382			−0.252754
$275$	4.18706	−	2.73286i	0.252489	−	0.164798i
$276$	14.2903			0.860177
$277$		−	1.64693i		−	0.0989544i	−0.998775	−	0.0494772i	$-0.984244\pi$
$277$		−	1.64693i		−	0.0989544i	0.998775	−	0.0494772i	$-0.0157555\pi$
$278$		−	1.31052i		−	0.0785997i
$279$	−1.40341			−0.0840198
$280$	1.46116	+	4.91198i	0.0873211	+	0.293547i
$281$	16.7390			0.998565			0.499282	−	0.866439i	$-0.333597\pi$
$281$	16.7390			0.998565			0.499282	+	0.866439i	$0.333597\pi$
$282$		−	1.04792i		−	0.0624025i
$283$			9.91741i			0.589529i	0.955570	+	0.294764i	$0.0952412\pi$
$283$			9.91741i			0.589529i	−0.955570	+	0.294764i	$0.904759\pi$
$284$	12.2638			0.727723
$285$	3.97311	−	1.18188i	0.235346	−	0.0700082i
$286$	−1.13183			−0.0669267
$287$		−	6.87034i		−	0.405543i
$288$			1.48308i			0.0873915i
$289$	12.3302			0.725305
$290$	1.41308	−	0.420348i	0.0829791	−	0.0246837i
$291$	−21.2243			−1.24419
$292$		−	20.5158i		−	1.20060i
$293$			14.1780i			0.828289i	0.910211	+	0.414144i	$0.135919\pi$
$293$			14.1780i			0.828289i	−0.910211	+	0.414144i	$0.864081\pi$
$294$	1.79230			0.104529
$295$	−1.97704	−	6.64622i	−0.115108	−	0.386958i
$296$	−6.65059			−0.386558
$297$		−	4.75219i		−	0.275750i
$298$		−	3.52100i		−	0.203966i
$299$	15.1741			0.877541
$300$	−9.67276	−	14.8198i	−0.558457	−	0.855620i
$301$	5.33574			0.307547
$302$			6.93978i			0.399340i
$303$			12.7258i			0.731079i
$304$	−3.46406			−0.198678
$305$	−6.43744	−	21.6407i	−0.368607	−	1.23914i
$306$	0.284056			0.0162384
$307$		−	7.49870i		−	0.427974i	−0.976837	−	0.213987i	$-0.931355\pi$
$307$		−	7.49870i		−	0.427974i	0.976837	−	0.213987i	$-0.0686450\pi$
$308$		−	3.71681i		−	0.211785i
$309$	−25.4445			−1.44749
$310$	2.07531	−	0.617339i	0.117869	−	0.0350625i
$311$	−22.6904			−1.28666			−0.643328	−	0.765591i	$-0.722447\pi$
$311$	−22.6904			−1.28666			−0.643328	+	0.765591i	$0.722447\pi$
$312$			8.20236i			0.464367i
$313$			16.3070i			0.921723i	0.887472	+	0.460862i	$0.152460\pi$
$313$			16.3070i			0.921723i	−0.887472	+	0.460862i	$0.847540\pi$
$314$	3.69148			0.208322
$315$	1.82109	−	0.541718i	0.102607	−	0.0305224i
$316$	−20.6336			−1.16073
$317$		−	26.5837i		−	1.49309i	−0.665336	−	0.746544i	$-0.731712\pi$
$317$		−	26.5837i		−	1.49309i	0.665336	−	0.746544i	$-0.268288\pi$
$318$		−	2.56603i		−	0.143896i
$319$	−2.18930			−0.122577
$320$	3.76464	+	12.6556i	0.210450	+	0.707469i
$321$	−27.5368			−1.53696
$322$		−	2.36697i		−	0.131906i
$323$			2.16098i			0.120240i
$324$	−19.3201			−1.07334
$325$	−10.2710	−	15.7363i	−0.569731	−	0.872892i
$326$	−2.41095			−0.133530
$327$			13.2306i			0.731655i
$328$			4.15501i			0.229422i
$329$	3.65406			0.201455
$330$	−0.355927	−	1.19652i	−0.0195931	−	0.0658662i
$331$	12.6060			0.692887			0.346444	−	0.938071i	$-0.387389\pi$
$331$	12.6060			0.692887			0.346444	+	0.938071i	$0.387389\pi$
$332$		−	15.5148i		−	0.851487i
$333$			2.46567i			0.135118i
$334$	1.50717			0.0824685
$335$	−25.8523	+	7.69024i	−1.41246	+	0.420163i
$336$	−12.5008			−0.681974
$337$		−	6.18984i		−	0.337182i	−0.985686	−	0.168591i	$-0.946078\pi$
$337$		−	6.18984i		−	0.337182i	0.985686	−	0.168591i	$-0.0539217\pi$
$338$			0.338782i			0.0184273i
$339$	−2.27131			−0.123361
$340$	8.84298	−	2.63051i	0.479578	−	0.142660i
$341$	−3.21529			−0.174118
$342$		−	0.131448i		−	0.00710788i
$343$			19.8765i			1.07323i
$344$	−3.22693			−0.173984
$345$	4.77179	+	16.0413i	0.256905	+	0.863635i
$346$	−2.15576			−0.115894
$347$		−	12.5724i		−	0.674921i	−0.941340	−	0.337460i	$-0.890432\pi$
$347$		−	12.5724i		−	0.674921i	0.941340	−	0.337460i	$-0.109568\pi$
$348$			7.74886i			0.415382i
$349$	−17.0337			−0.911793			−0.455897	−	0.890033i	$-0.650681\pi$
$349$	−17.0337			−0.911793			−0.455897	+	0.890033i	$0.650681\pi$
$350$	−2.45467	+	1.60214i	−0.131208	+	0.0856382i
$351$	−17.8602			−0.953308
$352$			3.39783i			0.181105i
$353$			8.51838i			0.453387i	0.973966	+	0.226694i	$0.0727916\pi$
$353$			8.51838i			0.453387i	−0.973966	+	0.226694i	$0.927208\pi$
$354$	−1.73120			−0.0920123
$355$	4.09510	+	13.7665i	0.217345	+	0.730649i
$356$	21.2222			1.12477
$357$			7.79833i			0.412731i
$358$		−	1.24141i		−	0.0656106i
$359$	6.43010			0.339367			0.169684	−	0.985499i	$-0.445725\pi$
$359$	6.43010			0.339367			0.169684	+	0.985499i	$0.445725\pi$
$360$	−1.10135	+	0.327618i	−0.0580464	+	0.0172670i
$361$	1.00000			0.0526316
$362$		−	5.27889i		−	0.277452i
$363$			1.85377i			0.0972979i
$364$	−13.9689			−0.732172
$365$	23.0296	−	6.85059i	1.20543	−	0.358576i
$366$	−5.63696			−0.294648
$367$		−	29.9633i		−	1.56407i	−0.623233	−	0.782037i	$-0.714181\pi$
$367$		−	29.9633i		−	1.56407i	0.623233	−	0.782037i	$-0.285819\pi$
$368$		−	13.9861i		−	0.729074i
$369$	1.54045			0.0801926
$370$	−1.08461	−	3.64615i	−0.0563864	−	0.189554i
$371$	8.94769			0.464541
$372$			11.3803i			0.590039i
$373$			34.6611i			1.79469i	0.441334	+	0.897343i	$0.354505\pi$
$373$			34.6611i			1.79469i	−0.441334	+	0.897343i	$0.645495\pi$
$374$	0.650787			0.0336514
$375$	13.4057	−	15.8065i	0.692269	−	0.816246i
$376$	−2.20988			−0.113966
$377$			8.22808i			0.423768i
$378$			2.78598i			0.143295i
$379$	−1.82041			−0.0935083			−0.0467542	−	0.998906i	$-0.514888\pi$
$379$	−1.82041			−0.0935083			−0.0467542	+	0.998906i	$0.514888\pi$
$380$	−1.21728	−	4.09212i	−0.0624451	−	0.209922i
$381$	3.30309			0.169222
$382$			8.28415i			0.423854i
$383$		−	17.0353i		−	0.870463i	−0.900319	−	0.435232i	$-0.856666\pi$
$383$		−	17.0353i		−	0.870463i	0.900319	−	0.435232i	$-0.143334\pi$
$384$	15.8941			0.811094
$385$	4.17223	−	1.24111i	0.212636	−	0.0632527i
$386$	−3.97292			−0.202216
$387$			1.19637i			0.0608147i
$388$			21.8601i			1.10978i
$389$	21.7778			1.10418			0.552090	−	0.833784i	$-0.313830\pi$
$389$	21.7778			1.10418			0.552090	+	0.833784i	$0.313830\pi$
$390$	−4.49689	+	1.33769i	−0.227709	+	0.0677363i
$391$	−8.72488			−0.441236
$392$		−	3.77966i		−	0.190902i
$393$			34.1472i			1.72250i
$394$	−3.44918			−0.173767
$395$	−6.88990	−	23.1618i	−0.346669	−	1.16539i
$396$	0.833374			0.0418786
$397$			8.81816i			0.442571i	0.975209	+	0.221285i	$0.0710252\pi$
$397$			8.81816i			0.442571i	−0.975209	+	0.221285i	$0.928975\pi$
$398$		−	5.94967i		−	0.298230i
$399$	3.60871			0.180661
$400$	−14.5042	+	9.46681i	−0.725211	+	0.473340i
$401$	−11.0380			−0.551210			−0.275605	−	0.961271i	$-0.588878\pi$
$401$	−11.0380			−0.551210			−0.275605	+	0.961271i	$0.588878\pi$
$402$			6.73397i			0.335860i
$403$			12.0841i			0.601950i
$404$	13.1070			0.652099
$405$	−6.45133	−	21.6874i	−0.320569	−	1.07766i
$406$	1.28348			0.0636980
$407$			5.64900i			0.280011i
$408$		−	4.71624i		−	0.233489i
$409$	25.9583			1.28355			0.641777	−	0.766891i	$-0.278197\pi$
$409$	25.9583			1.28355			0.641777	+	0.766891i	$0.278197\pi$
$410$	−2.27796	+	0.677621i	−0.112500	+	0.0334653i
$411$	25.7538			1.27034
$412$			26.2067i			1.29111i
$413$		−	6.03665i		−	0.297044i
$414$	0.530717			0.0260833
$415$	17.4159	−	5.18068i	0.854911	−	0.254309i
$416$	12.7701			0.626106
$417$			8.06698i			0.395042i
$418$		−	0.301154i		−	0.0147299i
$419$	−8.14823			−0.398067			−0.199034	−	0.979993i	$-0.563780\pi$
$419$	−8.14823			−0.398067			−0.199034	+	0.979993i	$0.563780\pi$
$420$	−4.39280	−	14.7673i	−0.214347	−	0.720569i
$421$	−14.7467			−0.718711			−0.359356	−	0.933201i	$-0.617003\pi$
$421$	−14.7467			−0.718711			−0.359356	+	0.933201i	$0.617003\pi$
$422$			6.76375i			0.329254i
$423$			0.819304i			0.0398359i
$424$	−5.41134			−0.262798
$425$	5.90565	+	9.04813i	0.286466	+	0.438899i
$426$	3.58588			0.173736
$427$		−	19.6559i		−	0.951216i
$428$			28.3617i			1.37092i
$429$	6.96707			0.336373
$430$	−0.526264	−	1.76914i	−0.0253787	−	0.0853156i
$431$	−28.0178			−1.34957			−0.674785	−	0.738014i	$-0.735764\pi$
$431$	−28.0178			−1.34957			−0.674785	+	0.738014i	$0.735764\pi$
$432$			16.4619i			0.792022i
$433$			2.14430i			0.103049i	0.998672	+	0.0515243i	$0.0164080\pi$
$433$			2.14430i			0.103049i	−0.998672	+	0.0515243i	$0.983592\pi$
$434$	1.88497			0.0904813
$435$	−8.69832	+	2.58748i	−0.417053	+	0.124060i
$436$	13.6270			0.652613
$437$			4.03747i			0.193139i
$438$		−	5.99873i		−	0.286631i
$439$	−4.98343			−0.237846			−0.118923	−	0.992903i	$-0.537944\pi$
$439$	−4.98343			−0.237846			−0.118923	+	0.992903i	$0.537944\pi$
$440$	−2.52326	+	0.750591i	−0.120292	+	0.0357830i
$441$	−1.40129			−0.0667281
$442$		−	2.44586i		−	0.116338i
$443$			10.3754i			0.492952i	0.969149	+	0.246476i	$0.0792726\pi$
$443$			10.3754i			0.492952i	−0.969149	+	0.246476i	$0.920727\pi$
$444$	19.9942			0.948883
$445$	7.08646	+	23.8225i	0.335930	+	1.12930i
$446$	−3.10531			−0.147041
$447$			21.6737i			1.02513i
$448$			11.4949i			0.543081i
$449$	−35.9683			−1.69745			−0.848725	−	0.528835i	$-0.822629\pi$
$449$	−35.9683			−1.69745			−0.848725	+	0.528835i	$0.822629\pi$
$450$	−0.359229	−	0.550380i	−0.0169342	−	0.0259451i
$451$	3.52926			0.166186
$452$			2.33935i			0.110034i
$453$		−	42.7183i		−	2.00708i
$454$	−6.01574			−0.282333
$455$	−4.66447	−	15.6805i	−0.218674	−	0.735115i
$456$	−2.18246			−0.102203
$457$		−	27.3842i		−	1.28098i	−0.767966	−	0.640490i	$-0.778731\pi$
$457$		−	27.3842i		−	1.28098i	0.767966	−	0.640490i	$-0.221269\pi$
$458$		−	4.72895i		−	0.220969i
$459$	10.2694			0.479333
$460$	16.5218	−	4.91473i	0.770335	−	0.229151i
$461$	−9.58866			−0.446589			−0.223294	−	0.974751i	$-0.571681\pi$
$461$	−9.58866			−0.446589			−0.223294	+	0.974751i	$0.571681\pi$
$462$		−	1.08678i		−	0.0505615i
$463$			27.4154i			1.27410i	0.770822	+	0.637050i	$0.219846\pi$
$463$			27.4154i			1.27410i	−0.770822	+	0.637050i	$0.780154\pi$
$464$	7.58387			0.352072
$465$	−12.7747	+	3.80007i	−0.592411	+	0.176224i
$466$	−6.37915			−0.295508
$467$			30.6386i			1.41778i	0.705317	+	0.708892i	$0.250805\pi$
$467$			30.6386i			1.41778i	−0.705317	+	0.708892i	$0.749195\pi$
$468$		−	3.13208i		−	0.144781i
$469$	−23.4812			−1.08426
$470$	−0.360400	−	1.21155i	−0.0166240	−	0.0558849i
$471$	−22.7231			−1.04703
$472$			3.65082i			0.168043i
$473$			2.74094i			0.126029i
$474$	−6.03315			−0.277112
$475$	4.18706	−	2.73286i	0.192115	−	0.125392i
$476$	8.03193			0.368143
$477$			2.00623i			0.0918589i
$478$		−	0.129205i		−	0.00590968i
$479$	−5.55489			−0.253809			−0.126905	−	0.991915i	$-0.540504\pi$
$479$	−5.55489			−0.253809			−0.126905	+	0.991915i	$0.540504\pi$
$480$	4.01581	+	13.4999i	0.183296	+	0.616184i
$481$	21.2307			0.968038
$482$		−	3.60354i		−	0.164137i
$483$			14.5701i			0.662961i
$484$	1.90931			0.0867866
$485$	−24.5386	+	7.29946i	−1.11424	+	0.331451i
$486$	−1.35569			−0.0614953
$487$			15.4839i			0.701642i	0.936442	+	0.350821i	$0.114098\pi$
$487$			15.4839i			0.701642i	−0.936442	+	0.350821i	$0.885902\pi$
$488$			11.8874i			0.538118i
$489$	14.8408			0.671122
$490$	2.07217	−	0.616407i	0.0936113	−	0.0278464i
$491$	−25.8915			−1.16847			−0.584233	−	0.811586i	$-0.698605\pi$
$491$	−25.8915			−1.16847			−0.584233	+	0.811586i	$0.698605\pi$
$492$		−	12.4915i		−	0.563162i
$493$		−	4.73103i		−	0.213075i
$494$	−1.13183			−0.0509236
$495$	0.278278	+	0.935486i	0.0125077	+	0.0420470i
$496$	11.1380			0.500109
$497$			12.5039i			0.560875i
$498$		−	4.53647i		−	0.203284i
$499$	33.9769			1.52102			0.760508	−	0.649329i	$-0.224950\pi$
$499$	33.9769			1.52102			0.760508	+	0.649329i	$0.224950\pi$
$500$	−16.2800	−	13.8073i	−0.728065	−	0.617481i
$501$	−9.27746			−0.414486
$502$		−	2.14980i		−	0.0959504i
$503$			11.0065i			0.490756i	0.969428	+	0.245378i	$0.0789120\pi$
$503$			11.0065i			0.490756i	−0.969428	+	0.245378i	$0.921088\pi$
$504$	−1.00034			−0.0445587
$505$	4.37667	+	14.7130i	0.194759	+	0.654721i
$506$	1.21590			0.0540535
$507$		−	2.08539i		−	0.0926155i
$508$		−	3.40204i		−	0.150941i
$509$	20.0893			0.890443			0.445221	−	0.895420i	$-0.353125\pi$
$509$	20.0893			0.890443			0.445221	+	0.895420i	$0.353125\pi$
$510$	2.58565	−	0.769149i	0.114494	−	0.0340585i
$511$	20.9174			0.925332
$512$		−	19.9268i		−	0.880648i
$513$		−	4.75219i		−	0.209814i
$514$	−2.06824			−0.0912264
$515$	−29.4178	+	8.75088i	−1.29630	+	0.385610i
$516$	9.70136			0.427079
$517$			1.87707i			0.0825535i
$518$		−	3.31173i		−	0.145509i
$519$	13.2699			0.582485
$520$	2.82096	+	9.48321i	0.123707	+	0.415866i
$521$	−5.13542			−0.224987			−0.112493	−	0.993652i	$-0.535884\pi$
$521$	−5.13542			−0.224987			−0.112493	+	0.993652i	$0.535884\pi$
$522$			0.287779i			0.0125957i
$523$		−	11.4021i		−	0.498578i	−0.968429	−	0.249289i	$-0.919803\pi$
$523$		−	11.4021i		−	0.498578i	0.968429	−	0.249289i	$-0.0801968\pi$
$524$	35.1701			1.53641
$525$	15.1099	−	9.86210i	0.659448	−	0.430417i
$526$	0.142500			0.00621330
$527$		−	6.94816i		−	0.302667i
$528$		−	6.42159i		−	0.279464i
$529$	6.69880			0.291252
$530$	−0.882511	−	2.96674i	−0.0383338	−	0.128867i
$531$	1.35352			0.0587379
$532$		−	3.71681i		−	0.161144i
$533$		−	13.2641i		−	0.574530i
$534$	6.20527			0.268528
$535$	−31.8369	+	9.47047i	−1.37643	+	0.409444i
$536$	14.2008			0.613383
$537$			7.64158i			0.329758i
$538$			0.683195i			0.0294546i
$539$	−3.21044			−0.138283
$540$	−19.4465	+	5.78474i	−0.836846	+	0.248936i
$541$	31.1731			1.34024			0.670118	−	0.742254i	$-0.266243\pi$
$541$	31.1731			1.34024			0.670118	+	0.742254i	$0.266243\pi$
$542$			4.10726i			0.176422i
$543$			32.4945i			1.39447i
$544$	−7.34262			−0.314812
$545$	4.55028	+	15.2967i	0.194913	+	0.655237i
$546$	−4.08445			−0.174798
$547$			36.3618i			1.55472i	0.629056	+	0.777360i	$0.283442\pi$
$547$			36.3618i			1.55472i	−0.629056	+	0.777360i	$0.716558\pi$
$548$		−	26.5252i		−	1.13310i
$549$	4.40720			0.188095
$550$	−0.823014	−	1.26095i	−0.0350934	−	0.0537671i
$551$	−2.18930			−0.0932673
$552$		−	8.81161i		−	0.375047i
$553$		−	21.0374i		−	0.894603i
$554$	−0.495980			−0.0210722
$555$	6.67641	+	22.4441i	0.283398	+	0.952698i
$556$	8.30863			0.352364
$557$			0.847753i			0.0359204i	0.999839	+	0.0179602i	$0.00571722\pi$
$557$			0.847753i			0.0359204i	−0.999839	+	0.0179602i	$0.994283\pi$
$558$			0.422642i			0.0178919i
$559$	10.3013			0.435700
$560$	−14.4528	+	4.29927i	−0.610744	+	0.181677i
$561$	−4.00596			−0.169132
$562$		−	5.04102i		−	0.212643i
$563$			39.5533i			1.66697i	0.552541	+	0.833486i	$0.313658\pi$
$563$			39.5533i			1.66697i	−0.552541	+	0.833486i	$0.686342\pi$
$564$	6.64375			0.279752
$565$	−2.62599	+	0.781151i	−0.110476	+	0.0328633i
$566$	2.98667			0.125539
$567$		−	19.6983i		−	0.827251i
$568$		−	7.56203i		−	0.317296i
$569$	−18.3932			−0.771082			−0.385541	−	0.922691i	$-0.625985\pi$
$569$	−18.3932			−0.771082			−0.385541	+	0.922691i	$0.625985\pi$
$570$	−0.355927	−	1.19652i	−0.0149081	−	0.0501166i
$571$	33.4442			1.39959			0.699797	−	0.714341i	$-0.253274\pi$
$571$	33.4442			1.39959			0.699797	+	0.714341i	$0.253274\pi$
$572$		−	7.17578i		−	0.300034i
$573$		−	50.9936i		−	2.13029i
$574$	−2.06903			−0.0863597
$575$	11.0339	+	16.9051i	0.460144	+	0.704993i
$576$	−2.57735			−0.107390
$577$		−	32.8041i		−	1.36565i	−0.730581	−	0.682826i	$-0.760751\pi$
$577$		−	32.8041i		−	1.36565i	0.730581	−	0.682826i	$-0.239249\pi$
$578$		−	3.71329i		−	0.154452i
$579$	24.4556			1.01634
$580$	2.66499	+	8.95889i	0.110658	+	0.371998i
$581$	15.8185			0.656264
$582$			6.39178i			0.264948i
$583$			4.59639i			0.190363i
$584$	−12.6503			−0.523475
$585$	3.51585	−	1.04586i	0.145363	−	0.0432409i
$586$	4.26977			0.176383
$587$			26.2752i			1.08449i	0.840220	+	0.542246i	$0.182426\pi$
$587$			26.2752i			1.08449i	−0.840220	+	0.542246i	$0.817574\pi$
$588$			11.3631i			0.468606i
$589$	−3.21529			−0.132484
$590$	−2.00154	+	0.595395i	−0.0824020	+	0.0245120i
$591$	21.2316			0.873353
$592$		−	19.5685i		−	0.804260i
$593$			19.5147i			0.801371i	0.916216	+	0.400686i	$0.131228\pi$
$593$			19.5147i			0.801371i	−0.916216	+	0.400686i	$0.868772\pi$
$594$	−1.43114			−0.0587205
$595$	2.68200	+	9.01608i	0.109951	+	0.369623i
$596$	22.3230			0.914384
$597$			36.6236i			1.49890i
$598$		−	4.56975i		−	0.186871i
$599$	−15.9690			−0.652476			−0.326238	−	0.945288i	$-0.605781\pi$
$599$	−15.9690			−0.652476			−0.326238	+	0.945288i	$0.605781\pi$
$600$	−9.13807	+	5.96436i	−0.373060	+	0.243494i
$601$	−18.0313			−0.735510			−0.367755	−	0.929923i	$-0.619874\pi$
$601$	−18.0313			−0.735510			−0.367755	+	0.929923i	$0.619874\pi$
$602$		−	1.60688i		−	0.0654916i
$603$		−	5.26489i		−	0.214403i
$604$	−43.9979			−1.79025
$605$	0.637551	+	2.14325i	0.0259201	+	0.0871356i
$606$	3.83244			0.155682
$607$			9.02312i			0.366237i	0.983091	+	0.183119i	$0.0586192\pi$
$607$			9.02312i			0.366237i	−0.983091	+	0.183119i	$0.941381\pi$
$608$			3.39783i			0.137800i
$609$	−7.90054			−0.320146
$610$	−6.51720	+	1.93866i	−0.263874	+	0.0784942i
$611$	7.05463			0.285400
$612$			1.80090i			0.0727971i
$613$			35.9969i			1.45390i	0.686690	+	0.726950i	$0.259063\pi$
$613$			35.9969i			1.45390i	−0.686690	+	0.726950i	$0.740937\pi$
$614$	−2.25827			−0.0911362
$615$	14.0221	−	4.17114i	0.565426	−	0.168197i
$616$	−2.29184			−0.0923407
$617$		−	44.8507i		−	1.80562i	−0.430037	−	0.902811i	$-0.641499\pi$
$617$		−	44.8507i		−	1.80562i	0.430037	−	0.902811i	$-0.358501\pi$
$618$			7.66272i			0.308240i
$619$	−36.6485			−1.47303			−0.736514	−	0.676422i	$-0.763530\pi$
$619$	−36.6485			−1.47303			−0.736514	+	0.676422i	$0.763530\pi$
$620$	3.91390	+	13.1574i	0.157186	+	0.528412i
$621$	19.1868			0.769941
$622$			6.83331i			0.273991i
$623$			21.6376i			0.866892i
$624$	−24.1344			−0.966148
$625$	10.0629	−	22.8853i	0.402517	−	0.915413i
$626$	4.91091			0.196279
$627$			1.85377i			0.0740326i
$628$			23.4038i			0.933915i
$629$	−12.2074			−0.486739
$630$	−0.163141	−	0.548430i	−0.00649968	−	0.0218500i
$631$	−39.4249			−1.56948			−0.784739	−	0.619826i	$-0.787203\pi$
$631$	−39.4249			−1.56948			−0.784739	+	0.619826i	$0.787203\pi$
$632$			12.7229i			0.506091i
$633$		−	41.6347i		−	1.65483i
$634$	−8.00579			−0.317950
$635$	3.81888	−	1.13600i	0.151548	−	0.0450807i
$636$	16.2686			0.645090
$637$			12.0658i			0.478066i
$638$			0.659317i			0.0261026i
$639$	−2.80359			−0.110908
$640$	18.3761	−	5.46631i	0.726378	−	0.216075i
$641$	−6.91478			−0.273117			−0.136559	−	0.990632i	$-0.543604\pi$
$641$	−6.91478			−0.273117			−0.136559	+	0.990632i	$0.543604\pi$
$642$			8.29283i			0.327292i
$643$		−	38.8033i		−	1.53025i	−0.643879	−	0.765127i	$-0.722676\pi$
$643$		−	38.8033i		−	1.53025i	0.643879	−	0.765127i	$-0.277324\pi$
$644$	15.0065			0.591340
$645$	3.23945	+	10.8901i	0.127553	+	0.428796i
$646$	0.650787			0.0256049
$647$		−	18.4537i		−	0.725490i	−0.931888	−	0.362745i	$-0.881840\pi$
$647$		−	18.4537i		−	0.725490i	0.931888	−	0.362745i	$-0.118160\pi$
$648$			11.9131i			0.467989i
$649$	3.10100			0.121725
$650$	−4.73905	+	3.09315i	−0.185881	+	0.121323i
$651$	−11.6030			−0.454758
$652$		−	15.2853i		−	0.598620i
$653$		−	12.6249i		−	0.494052i	−0.969009	−	0.247026i	$-0.920547\pi$
$653$		−	12.6249i		−	0.494052i	0.969009	−	0.247026i	$-0.0794533\pi$
$654$	3.98446			0.155805
$655$	11.7439	+	39.4795i	0.458873	+	1.54259i
$656$	−12.2256			−0.477328
$657$			4.69005i			0.182976i
$658$		−	1.10044i		−	0.0428994i
$659$	−37.4031			−1.45702			−0.728509	−	0.685036i	$-0.759786\pi$
$659$	−37.4031			−1.45702			−0.728509	+	0.685036i	$0.759786\pi$
$660$	7.58588	−	2.25656i	0.295280	−	0.0878365i
$661$	−12.6089			−0.490431			−0.245215	−	0.969469i	$-0.578859\pi$
$661$	−12.6089			−0.490431			−0.245215	+	0.969469i	$0.578859\pi$
$662$		−	3.79634i		−	0.147549i
$663$			15.0557i			0.584714i
$664$	−9.56666			−0.371259
$665$	4.17223	−	1.24111i	0.161792	−	0.0481281i
$666$	0.742549			0.0287732
$667$		−	8.83924i		−	0.342257i
$668$			9.55538i			0.369709i
$669$	19.1149			0.739025
$670$	2.31595	+	7.78552i	0.0894729	+	0.300781i
$671$	10.0971			0.389796
$672$			12.2618i			0.473008i
$673$			6.19541i			0.238815i	0.992845	+	0.119408i	$0.0380996\pi$
$673$			6.19541i			0.238815i	−0.992845	+	0.119408i	$0.961900\pi$
$674$	−1.86410			−0.0718023
$675$	−12.9871	−	19.8977i	−0.499873	−	0.765862i
$676$	−2.14786			−0.0826101
$677$			26.5020i			1.01855i	0.860603	+	0.509276i	$0.170087\pi$
$677$			26.5020i			1.01855i	−0.860603	+	0.509276i	$0.829913\pi$
$678$			0.684016i			0.0262695i
$679$	−22.2880			−0.855334
$680$	−1.62201	−	5.45270i	−0.0622012	−	0.209102i
$681$	37.0303			1.41900
$682$			0.968298i			0.0370780i
$683$		−	44.3957i		−	1.69876i	−0.527786	−	0.849378i	$-0.676978\pi$
$683$		−	44.3957i		−	1.69876i	0.527786	−	0.849378i	$-0.323022\pi$
$684$	0.833374			0.0318648
$685$	29.7754	−	8.85724i	1.13766	−	0.338418i
$686$	5.98588			0.228542
$687$			29.1093i			1.11059i
$688$		−	9.49480i		−	0.361986i
$689$	17.2747			0.658113
$690$	4.83091	−	1.43705i	0.183910	−	0.0547074i
$691$	−15.0873			−0.573948			−0.286974	−	0.957938i	$-0.592649\pi$
$691$	−15.0873			−0.573948			−0.286974	+	0.957938i	$0.592649\pi$
$692$		−	13.6674i		−	0.519558i
$693$			0.849687i			0.0322769i
$694$	−3.78623			−0.143723
$695$	2.77440	+	9.32668i	0.105239	+	0.353781i
$696$	4.77805			0.181112
$697$			7.62664i			0.288880i
$698$			5.12978i			0.194165i
$699$	39.2672			1.48522
$700$	−10.1575	−	15.5625i	−0.383919	−	0.588207i
$701$	8.66946			0.327441			0.163720	−	0.986507i	$-0.447650\pi$
$701$	8.66946			0.327441			0.163720	+	0.986507i	$0.447650\pi$
$702$			5.37869i			0.203005i
$703$			5.64900i			0.213056i
$704$	−5.90485			−0.222548
$705$	2.21846	+	7.45780i	0.0835522	+	0.280877i
$706$	2.56535			0.0965481
$707$			13.3636i			0.502590i
$708$		−	10.9757i		−	0.412494i
$709$	−42.7639			−1.60603			−0.803015	−	0.595958i	$-0.796772\pi$
$709$	−42.7639			−1.60603			−0.803015	+	0.595958i	$0.796772\pi$
$710$	4.14584	−	1.23326i	0.155590	−	0.0462833i
$711$	4.71696			0.176900
$712$		−	13.0859i		−	0.490415i
$713$		−	12.9816i		−	0.486166i
$714$	2.34850			0.0878904
$715$	8.05502	−	2.39612i	0.301241	−	0.0896097i
$716$	7.87049			0.294134
$717$			0.795327i			0.0297020i
$718$		−	1.93645i		−	0.0722677i
$719$	−27.6655			−1.03175			−0.515875	−	0.856664i	$-0.672533\pi$
$719$	−27.6655			−1.03175			−0.515875	+	0.856664i	$0.672533\pi$
$720$	−0.963972	−	3.24058i	−0.0359251	−	0.120769i
$721$	−26.7197			−0.995093
$722$		−	0.301154i		−	0.0112078i
$723$			22.1818i			0.824950i
$724$	33.4679			1.24383
$725$	−9.16673	+	5.98306i	−0.340444	+	0.222205i
$726$	0.558272			0.0207194
$727$			20.9256i			0.776088i	0.921641	+	0.388044i	$0.126849\pi$
$727$			20.9256i			0.776088i	−0.921641	+	0.388044i	$0.873151\pi$
$728$			8.61344i			0.319235i
$729$	−22.0118			−0.815250
$730$	−2.06309	−	6.93547i	−0.0763582	−	0.256693i
$731$	−5.92311			−0.219074
$732$		−	35.7381i		−	1.32092i
$733$		−	27.8171i		−	1.02745i	−0.857956	−	0.513723i	$-0.828266\pi$
$733$		−	27.8171i		−	1.02745i	0.857956	−	0.513723i	$-0.171734\pi$
$734$	−9.02359			−0.333067
$735$	−12.7554	+	3.79433i	−0.470490	+	0.139956i
$736$	−13.7186			−0.505676
$737$		−	12.0622i		−	0.444316i
$738$		−	0.463913i		−	0.0170769i
$739$	−9.45503			−0.347809			−0.173904	−	0.984763i	$-0.555638\pi$
$739$	−9.45503			−0.347809			−0.173904	+	0.984763i	$0.555638\pi$
$740$	23.1164	−	6.87641i	0.849776	−	0.252782i
$741$	6.96707			0.255942
$742$		−	2.69464i		−	0.0989232i
$743$			5.01586i			0.184014i	0.995758	+	0.0920070i	$0.0293282\pi$
$743$			5.01586i			0.184014i	−0.995758	+	0.0920070i	$0.970672\pi$
$744$	7.01723			0.257264
$745$	7.45403	+	25.0582i	0.273094	+	0.918061i
$746$	10.4384			0.382175
$747$			3.54679i			0.129770i
$748$			4.12596i			0.150860i
$749$	−28.9169			−1.05660
$750$	−4.76021	−	4.03719i	−0.173818	−	0.147417i
$751$	−42.5515			−1.55273			−0.776363	−	0.630286i	$-0.782938\pi$
$751$	−42.5515			−1.55273			−0.776363	+	0.630286i	$0.782938\pi$
$752$		−	6.50229i		−	0.237114i
$753$			13.2332i			0.482246i
$754$	2.47792			0.0902406
$755$	−14.6917	−	49.3890i	−0.534685	−	1.79745i
$756$	−17.6630			−0.642396
$757$			6.77125i			0.246105i	0.992400	+	0.123053i	$0.0392684\pi$
$757$			6.77125i			0.246105i	−0.992400	+	0.123053i	$0.960732\pi$
$758$			0.548225i			0.0199124i
$759$	−7.48457			−0.271673
$760$	−2.52326	+	0.750591i	−0.0915283	+	0.0272268i
$761$	31.5134			1.14236			0.571180	−	0.820825i	$-0.306486\pi$
$761$	31.5134			1.14236			0.571180	+	0.820825i	$0.306486\pi$
$762$		−	0.994739i		−	0.0360356i
$763$			13.8937i			0.502986i
$764$	−52.5212			−1.90015
$765$	−2.02156	+	0.601352i	−0.0730898	+	0.0217419i
$766$	−5.13025			−0.185364
$767$		−	11.6545i		−	0.420821i
$768$			17.1059i			0.617258i
$769$	−10.0557			−0.362617			−0.181308	−	0.983426i	$-0.558033\pi$
$769$	−10.0557			−0.362617			−0.181308	+	0.983426i	$0.558033\pi$
$770$	−0.373765	−	1.25648i	−0.0134695	−	0.0452805i
$771$	12.7312			0.458503
$772$		−	25.1882i		−	0.906541i
$773$		−	4.14250i		−	0.148995i	−0.997221	−	0.0744977i	$-0.976265\pi$
$773$		−	4.14250i		−	0.148995i	0.997221	−	0.0744977i	$-0.0237353\pi$
$774$	0.360291			0.0129504
$775$	−13.4626	+	8.78694i	−0.483591	+	0.315636i
$776$	13.4792			0.483876
$777$			20.3856i			0.731329i
$778$		−	6.55849i		−	0.235133i
$779$	3.52926			0.126449
$780$	−8.48087	−	28.5101i	−0.303664	−	1.02083i
$781$	−6.42317			−0.229839
$782$			2.62754i			0.0939605i
$783$			10.4040i			0.371807i
$784$	11.1211			0.397184
$785$	−26.2715	+	7.81495i	−0.937670	+	0.278928i
$786$	10.2836			0.366803
$787$		−	6.39855i		−	0.228084i	−0.993476	−	0.114042i	$-0.963620\pi$
$787$		−	6.39855i		−	0.228084i	0.993476	−	0.114042i	$-0.0363798\pi$
$788$		−	21.8677i		−	0.779003i
$789$	−0.877168			−0.0312280
$790$	−6.97527	+	2.07492i	−0.248169	+	0.0738225i
$791$	−2.38515			−0.0848060
$792$		−	0.513870i		−	0.0182596i
$793$		−	37.9483i		−	1.34758i
$794$	2.65563			0.0942447
$795$	5.43235	+	18.2619i	0.192666	+	0.647684i
$796$	37.7207			1.33697
$797$			31.6848i			1.12233i	0.827703	+	0.561166i	$0.189647\pi$
$797$			31.6848i			1.12233i	−0.827703	+	0.561166i	$0.810353\pi$
$798$		−	1.08678i		−	0.0384715i
$799$	−4.05631			−0.143502
$800$	9.28579	+	14.2269i	0.328302	+	0.502997i
$801$	−4.85153			−0.171420
$802$			3.32413i			0.117379i
$803$			10.7452i			0.379189i
$804$	−42.6931			−1.50567
$805$	5.01094	+	16.8453i	0.176612	+	0.593717i
$806$	3.63917			0.128184
$807$		−	4.20545i		−	0.148039i
$808$		−	8.08198i		−	0.284323i
$809$	−34.9827			−1.22993			−0.614963	−	0.788556i	$-0.710829\pi$
$809$	−34.9827			−1.22993			−0.614963	+	0.788556i	$0.710829\pi$
$810$	−6.53126	+	1.94285i	−0.229485	+	0.0682646i
$811$	−27.4687			−0.964556			−0.482278	−	0.876018i	$-0.660190\pi$
$811$	−27.4687			−0.964556			−0.482278	+	0.876018i	$0.660190\pi$
$812$			8.13721i			0.285560i
$813$		−	25.2825i		−	0.886697i
$814$	1.70122			0.0596278
$815$	17.1582	−	5.10404i	0.601027	−	0.178787i
$816$	13.8769			0.485789
$817$			2.74094i			0.0958935i
$818$		−	7.81745i		−	0.273331i
$819$	3.19339			0.111586
$820$	−4.29609	−	14.4422i	−0.150026	−	0.504342i
$821$	17.9999			0.628200			0.314100	−	0.949390i	$-0.398297\pi$
$821$	17.9999			0.628200			0.314100	+	0.949390i	$0.398297\pi$
$822$		−	7.75586i		−	0.270517i
$823$		−	8.49995i		−	0.296290i	−0.988966	−	0.148145i	$-0.952670\pi$
$823$		−	8.49995i		−	0.296290i	0.988966	−	0.148145i	$-0.0473302\pi$
$824$	16.1594			0.562940
$825$	5.06611	+	7.76186i	0.176379	+	0.270233i
$826$	−1.81796			−0.0632550
$827$			10.1919i			0.354406i	0.984174	+	0.177203i	$0.0567049\pi$
$827$			10.1919i			0.354406i	−0.984174	+	0.177203i	$0.943295\pi$
$828$			3.36472i			0.116932i
$829$	4.33629			0.150605			0.0753027	−	0.997161i	$-0.476008\pi$
$829$	4.33629			0.150605			0.0753027	+	0.997161i	$0.476008\pi$
$830$	−1.56018	−	5.24486i	−0.0541547	−	0.182052i
$831$	3.05304			0.105909
$832$			22.1923i			0.769380i
$833$		−	6.93768i		−	0.240376i
$834$	2.42941			0.0841234
$835$	−10.7262	+	3.19071i	−0.371195	+	0.110419i
$836$	1.90931			0.0660347
$837$			15.2796i			0.528142i
$838$			2.45388i			0.0847677i
$839$	42.6097			1.47105			0.735525	−	0.677498i	$-0.236936\pi$
$839$	42.6097			1.47105			0.735525	+	0.677498i	$0.236936\pi$
$840$	−9.10570	+	2.70866i	−0.314177	+	0.0934578i
$841$	−24.2070			−0.834723
$842$			4.44104i			0.153048i
$843$			31.0303i			1.06874i
$844$	−42.8819			−1.47606
$845$	−0.717208	−	2.41104i	−0.0246727	−	0.0829422i
$846$	0.246737			0.00848299
$847$			1.94668i			0.0668887i
$848$		−	15.9222i		−	0.546769i
$849$	−18.3846			−0.630959
$850$	2.72488	−	1.77851i	0.0934628	−	0.0610025i
$851$	−22.8077			−0.781838
$852$			22.7343i			0.778865i
$853$		−	26.8673i		−	0.919917i	−0.887940	−	0.459959i	$-0.847864\pi$
$853$		−	26.8673i		−	0.919917i	0.887940	−	0.459959i	$-0.152136\pi$
$854$	−5.91946			−0.202560
$855$	0.278278	+	0.935486i	0.00951691	+	0.0319930i
$856$	17.4882			0.597735
$857$		−	56.6619i		−	1.93553i	−0.251846	−	0.967767i	$-0.581038\pi$
$857$		−	56.6619i		−	1.93553i	0.251846	−	0.967767i	$-0.418962\pi$
$858$		−	2.09816i		−	0.0716301i
$859$	−51.0188			−1.74074			−0.870370	−	0.492398i	$-0.836120\pi$
$859$	−51.0188			−1.74074			−0.870370	+	0.492398i	$0.836120\pi$
$860$	11.2163	−	3.33649i	0.382472	−	0.113774i
$861$	12.7361			0.434043
$862$			8.43768i			0.287389i
$863$		−	33.1838i		−	1.12959i	−0.825231	−	0.564795i	$-0.808955\pi$
$863$		−	33.1838i		−	1.12959i	0.825231	−	0.564795i	$-0.191045\pi$
$864$	16.1471			0.549336
$865$	15.3421	−	4.56380i	0.521647	−	0.155174i
$866$	0.645766			0.0219440
$867$			22.8574i			0.776277i
$868$			11.9506i			0.405630i
$869$	10.8068			0.366597
$870$	0.779231	+	2.61954i	0.0264184	+	0.0888106i
$871$	−45.3334			−1.53606
$872$		−	8.40257i		−	0.284547i
$873$		−	4.99736i		−	0.169135i
$874$	1.21590			0.0411285
$875$	14.0776	−	16.5987i	0.475909	−	0.561139i
$876$	38.0317			1.28497
$877$		−	43.4040i		−	1.46565i	−0.680418	−	0.732825i	$-0.738202\pi$
$877$		−	43.4040i		−	1.46565i	0.680418	−	0.732825i	$-0.261798\pi$
$878$			1.50078i			0.0506490i
$879$	−26.2828			−0.886499
$880$	−2.20851	−	7.42436i	−0.0744490	−	0.250275i
$881$	−34.1080			−1.14913			−0.574565	−	0.818459i	$-0.694829\pi$
$881$	−34.1080			−1.14913			−0.574565	+	0.818459i	$0.694829\pi$
$882$			0.422005i			0.0142096i
$883$		−	31.9448i		−	1.07503i	−0.843255	−	0.537514i	$-0.819364\pi$
$883$		−	31.9448i		−	1.07503i	0.843255	−	0.537514i	$-0.180636\pi$
$884$	15.5067			0.521546
$885$	12.3206	−	3.66499i	0.414152	−	0.123197i
$886$	3.12461			0.104973
$887$			15.6331i			0.524907i	0.964945	+	0.262453i	$0.0845316\pi$
$887$			15.6331i			0.524907i	−0.964945	+	0.262453i	$0.915468\pi$
$888$		−	12.3287i		−	0.413724i
$889$	3.46863			0.116334
$890$	7.17426	−	2.13412i	0.240482	−	0.0715358i
$891$	10.1189			0.338997
$892$		−	19.6875i		−	0.659187i
$893$			1.87707i			0.0628138i
$894$	6.52713			0.218300
$895$	2.62809	+	8.83485i	0.0878475	+	0.295317i
$896$	16.6907			0.557597
$897$			28.1294i			0.939212i
$898$			10.8320i			0.361469i
$899$	7.03923			0.234771
$900$	3.48938	−	2.27750i	0.116313	−	0.0759165i
$901$	−9.93268			−0.330905
$902$		−	1.06285i		−	0.0353891i
$903$			9.89126i			0.329161i
$904$	1.44248			0.0479761
$905$	11.1755	+	37.5687i	0.371487	+	1.24883i
$906$	−12.8648			−0.427404
$907$		−	15.2429i		−	0.506132i	−0.967449	−	0.253066i	$-0.918561\pi$
$907$		−	15.2429i		−	0.506132i	0.967449	−	0.253066i	$-0.0814390\pi$
$908$		−	38.1395i		−	1.26571i
$909$	−2.99635			−0.0993828
$910$	−4.72226	+	1.40473i	−0.156542	+	0.0465662i
$911$	10.7020			0.354574			0.177287	−	0.984159i	$-0.443268\pi$
$911$	10.7020			0.354574			0.177287	+	0.984159i	$0.443268\pi$
$912$		−	6.42159i		−	0.212640i
$913$			8.12591i			0.268928i
$914$	−8.24688			−0.272783
$915$	40.1170	−	11.9336i	1.32623	−	0.394511i
$916$	29.9813			0.990611
$917$			35.8586i			1.18415i
$918$		−	3.09266i		−	0.102073i
$919$	59.1251			1.95036			0.975178	−	0.221421i	$-0.0710694\pi$
$919$	59.1251			1.95036			0.975178	+	0.221421i	$0.0710694\pi$
$920$	−3.03049	−	10.1876i	−0.0999124	−	0.335875i
$921$	13.9009			0.458050
$922$			2.88767i			0.0951003i
$923$			24.1403i			0.794588i
$924$	6.89012			0.226668
$925$	15.4379	+	23.6527i	0.507597	+	0.777696i
$926$	8.25626			0.271318
$927$		−	5.99102i		−	0.196771i
$928$		−	7.43886i		−	0.244193i
$929$	−41.4480			−1.35986			−0.679932	−	0.733276i	$-0.737991\pi$
$929$	−41.4480			−1.35986			−0.679932	+	0.733276i	$0.737991\pi$
$930$	1.14441	+	3.84715i	0.0375266	+	0.126153i
$931$	−3.21044			−0.105218
$932$		−	40.4435i		−	1.32477i
$933$		−	42.0629i		−	1.37708i
$934$	9.22694			0.301915
$935$	−4.63152	+	1.37773i	−0.151467	+	0.0450566i
$936$	−1.93128			−0.0631260
$937$		−	2.64414i		−	0.0863803i	−0.999067	−	0.0431901i	$-0.986248\pi$
$937$		−	2.64414i		−	0.0863803i	0.999067	−	0.0431901i	$-0.0137521\pi$
$938$			7.07146i			0.230891i
$939$	−30.2294			−0.986500
$940$	7.68121	−	2.28492i	0.250533	−	0.0745259i
$941$	−55.5911			−1.81222			−0.906109	−	0.423045i	$-0.860961\pi$
$941$	−55.5911			−1.81222			−0.906109	+	0.423045i	$0.860961\pi$
$942$			6.84317i			0.222963i
$943$			14.2493i			0.464021i
$944$	−10.7420			−0.349624
$945$	−5.89798	−	19.8272i	−0.191861	−	0.644979i
$946$	0.825447			0.0268376
$947$		−	30.2967i		−	0.984509i	−0.870451	−	0.492255i	$-0.836173\pi$
$947$		−	30.2967i		−	0.984509i	0.870451	−	0.492255i	$-0.163827\pi$
$948$		−	38.2500i		−	1.24230i
$949$	40.3838			1.31091
$950$	−0.823014	−	1.26095i	−0.0267021	−	0.0409106i
$951$	49.2801			1.59802
$952$		−	4.95260i		−	0.160515i
$953$		−	0.456421i		−	0.0147849i	−0.999973	−	0.00739247i	$-0.997647\pi$
$953$		−	0.456421i		−	0.0147849i	0.999973	−	0.00739247i	$-0.00235312\pi$
$954$	0.604185			0.0195612
$955$	−17.5377	−	58.9566i	−0.567508	−	1.90779i
$956$	0.819152			0.0264933
$957$		−	4.05847i		−	0.131192i
$958$			1.67288i			0.0540482i
$959$	27.0445			0.873312
$960$	−23.4606	+	6.97880i	−0.757188	+	0.225240i
$961$	−20.6619			−0.666514
$962$		−	6.39373i		−	0.206142i
$963$		−	6.48367i		−	0.208933i
$964$	22.8463			0.735829
$965$	28.2744	−	8.41076i	0.910186	−	0.270752i
$966$	4.38784			0.141176
$967$			34.6037i			1.11278i	0.830922	+	0.556389i	$0.187814\pi$
$967$			34.6037i			1.11278i	−0.830922	+	0.556389i	$0.812186\pi$
$968$		−	1.17730i		−	0.0378400i
$969$	−4.00596			−0.128690
$970$	2.19826	+	7.38990i	0.0705820	+	0.237275i
$971$	−20.0747			−0.644228			−0.322114	−	0.946701i	$-0.604393\pi$
$971$	−20.0747			−0.644228			−0.322114	+	0.946701i	$0.604393\pi$
$972$		−	8.59501i		−	0.275685i
$973$			8.47127i			0.271576i
$974$	4.66304			0.149414
$975$	29.1715	−	19.0401i	0.934237	−	0.609770i
$976$	−34.9771			−1.11959
$977$			8.02308i			0.256681i	0.991730	+	0.128341i	$0.0409650\pi$
$977$			8.02308i			0.256681i	−0.991730	+	0.128341i	$0.959035\pi$
$978$		−	4.46936i		−	0.142914i
$979$	−11.1151			−0.355241
$980$	3.90800	+	13.1375i	0.124836	+	0.419662i
$981$	−3.11521			−0.0994611
$982$			7.79733i			0.248823i
$983$		−	10.1782i		−	0.324633i	−0.986739	−	0.162317i	$-0.948103\pi$
$983$		−	10.1782i		−	0.324633i	0.986739	−	0.162317i	$-0.0518966\pi$
$984$	−7.70245			−0.245545
$985$	24.5471	−	7.30199i	0.782135	−	0.232661i
$986$	−1.42477			−0.0453739
$987$			6.77380i			0.215612i
$988$		−	7.17578i		−	0.228292i
$989$	−11.0665			−0.351894
$990$	0.281726	−	0.0838046i	0.00895383	−	0.00266349i
$991$	25.7709			0.818641			0.409321	−	0.912391i	$-0.365766\pi$
$991$	25.7709			0.818641			0.409321	+	0.912391i	$0.365766\pi$
$992$		−	10.9250i		−	0.346869i
$993$			23.3686i			0.741581i
$994$	3.76559			0.119437
$995$	12.5956	+	42.3425i	0.399307	+	1.34235i
$996$	28.7610			0.911328
$997$		−	33.3107i		−	1.05496i	−0.849567	−	0.527481i	$-0.823137\pi$
$997$		−	33.3107i		−	1.05496i	0.849567	−	0.527481i	$-0.176863\pi$
$998$		−	10.2323i		−	0.323898i
$999$	26.8451			0.849342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.b.b.419.8	✓	16
5.2	odd	4		5225.2.a.z.1.9		16
5.3	odd	4		5225.2.a.z.1.8		16
5.4	even	2	inner	1045.2.b.b.419.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.b.419.8	✓	16	1.1	even	1	trivial
1045.2.b.b.419.9	yes	16	5.4	even	2	inner
5225.2.a.z.1.8		16	5.3	odd	4
5225.2.a.z.1.9		16	5.2	odd	4

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.301154i q^{2} +1.85377i q^{3} +1.90931 q^{4} +(0.637551 + 2.14325i) q^{5} +0.558272 q^{6} +1.94668i q^{7} -1.17730i q^{8} -0.436480 q^{9} +O(q^{10})\)	\(q-0.301154i q^{2} +1.85377i q^{3} +1.90931 q^{4} +(0.637551 + 2.14325i) q^{5} +0.558272 q^{6} +1.94668i q^{7} -1.17730i q^{8} -0.436480 q^{9} +(0.645450 - 0.192001i) q^{10} -1.00000 q^{11} +3.53942i q^{12} +3.75832i q^{13} +0.586251 q^{14} +(-3.97311 + 1.18188i) q^{15} +3.46406 q^{16} -2.16098i q^{17} +0.131448i q^{18} -1.00000 q^{19} +(1.21728 + 4.09212i) q^{20} -3.60871 q^{21} +0.301154i q^{22} -4.03747i q^{23} +2.18246 q^{24} +(-4.18706 + 2.73286i) q^{25} +1.13183 q^{26} +4.75219i q^{27} +3.71681i q^{28} +2.18930 q^{29} +(0.355927 + 1.19652i) q^{30} +3.21529 q^{31} -3.39783i q^{32} -1.85377i q^{33} -0.650787 q^{34} +(-4.17223 + 1.24111i) q^{35} -0.833374 q^{36} -5.64900i q^{37} +0.301154i q^{38} -6.96707 q^{39} +(2.52326 - 0.750591i) q^{40} -3.52926 q^{41} +1.08678i q^{42} -2.74094i q^{43} -1.90931 q^{44} +(-0.278278 - 0.935486i) q^{45} -1.21590 q^{46} -1.87707i q^{47} +6.42159i q^{48} +3.21044 q^{49} +(0.823014 + 1.26095i) q^{50} +4.00596 q^{51} +7.17578i q^{52} -4.59639i q^{53} +1.43114 q^{54} +(-0.637551 - 2.14325i) q^{55} +2.29184 q^{56} -1.85377i q^{57} -0.659317i q^{58} -3.10100 q^{59} +(-7.58588 + 2.25656i) q^{60} -10.0971 q^{61} -0.968298i q^{62} -0.849687i q^{63} +5.90485 q^{64} +(-8.05502 + 2.39612i) q^{65} -0.558272 q^{66} +12.0622i q^{67} -4.12596i q^{68} +7.48457 q^{69} +(0.373765 + 1.25648i) q^{70} +6.42317 q^{71} +0.513870i q^{72} -10.7452i q^{73} -1.70122 q^{74} +(-5.06611 - 7.76186i) q^{75} -1.90931 q^{76} -1.94668i q^{77} +2.09816i q^{78} -10.8068 q^{79} +(2.20851 + 7.42436i) q^{80} -10.1189 q^{81} +1.06285i q^{82} -8.12591i q^{83} -6.89012 q^{84} +(4.63152 - 1.37773i) q^{85} -0.825447 q^{86} +4.05847i q^{87} +1.17730i q^{88} +11.1151 q^{89} +(-0.281726 + 0.0838046i) q^{90} -7.31624 q^{91} -7.70877i q^{92} +5.96042i q^{93} -0.565288 q^{94} +(-0.637551 - 2.14325i) q^{95} +6.29880 q^{96} +11.4492i q^{97} -0.966837i q^{98} +0.436480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) \) 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9	\( 16 q - 6 q^{4} + 3 q^{5} - 8 q^{6} - 8 q^{9} + 10 q^{10} - 16 q^{11} + 4 q^{14} + 3 q^{15} - 18 q^{16} - 16 q^{19} - 2 q^{20} - 10 q^{21} + 10 q^{24} - 7 q^{25} - 24 q^{26} + 2 q^{29} + 4 q^{30} - 32 q^{31} - 16 q^{34} - 18 q^{35} + 18 q^{36} + 40 q^{39} - 28 q^{40} + 6 q^{41} + 6 q^{44} + 16 q^{45} + 38 q^{49} - 30 q^{50} - 16 q^{51} + 18 q^{54} - 3 q^{55} + 12 q^{56} + 24 q^{59} - 20 q^{60} - 42 q^{61} + 62 q^{64} - 20 q^{65} + 8 q^{66} + 30 q^{69} - 18 q^{70} - 46 q^{71} - 2 q^{74} - 25 q^{75} + 6 q^{76} + 74 q^{79} - 22 q^{80} - 56 q^{81} + 34 q^{84} - 18 q^{85} + 8 q^{86} + 14 q^{89} - 4 q^{90} - 24 q^{91} + 64 q^{94} - 3 q^{95} + 54 q^{96} + 8 q^{99}+O(q^{100}) \) 16 * q - 6 * q^4 + 3 * q^5 - 8 * q^6 - 8 * q^9 + 10 * q^10 - 16 * q^11 + 4 * q^14 + 3 * q^15 - 18 * q^16 - 16 * q^19 - 2 * q^20 - 10 * q^21 + 10 * q^24 - 7 * q^25 - 24 * q^26 + 2 * q^29 + 4 * q^30 - 32 * q^31 - 16 * q^34 - 18 * q^35 + 18 * q^36 + 40 * q^39 - 28 * q^40 + 6 * q^41 + 6 * q^44 + 16 * q^45 + 38 * q^49 - 30 * q^50 - 16 * q^51 + 18 * q^54 - 3 * q^55 + 12 * q^56 + 24 * q^59 - 20 * q^60 - 42 * q^61 + 62 * q^64 - 20 * q^65 + 8 * q^66 + 30 * q^69 - 18 * q^70 - 46 * q^71 - 2 * q^74 - 25 * q^75 + 6 * q^76 + 74 * q^79 - 22 * q^80 - 56 * q^81 + 34 * q^84 - 18 * q^85 + 8 * q^86 + 14 * q^89 - 4 * q^90 - 24 * q^91 + 64 * q^94 - 3 * q^95 + 54 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more