LMFDB - Embedded newform 867.2.d.f.577.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [867,2,Mod(577,867)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(867, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("867.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.836829184.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 $ x^8 + 14x^6 + 61x^4 + 84*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.4
Root			$-2.63640i$ of defining polynomial
Character	$\chi$	$=$	867.577
Dual form			867.2.d.f.577.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.222191 q^{2} +1.00000i q^{3} -1.95063 q^{4} -0.636405i q^{5} -0.222191i q^{6} +1.72844i q^{7} +0.877796 q^{8} -1.00000 q^{9} +0.141404i q^{10} -4.95063i q^{11} -1.95063i q^{12} +2.50625 q^{13} -0.384044i q^{14} +0.636405 q^{15} +3.70622 q^{16} +0.222191 q^{18} +0.950631 q^{19} +1.24139i q^{20} -1.72844 q^{21} +1.09999i q^{22} +2.12220i q^{23} +0.877796i q^{24} +4.59499 q^{25} -0.556867 q^{26} -1.00000i q^{27} -3.37155i q^{28} +9.58704i q^{29} -0.141404 q^{30} +5.27281i q^{31} -2.57908 q^{32} +4.95063 q^{33} +1.09999 q^{35} +1.95063 q^{36} +8.48705i q^{37} -0.211222 q^{38} +2.50625i q^{39} -0.558634i q^{40} -6.92171i q^{41} +0.384044 q^{42} -7.15061 q^{43} +9.65685i q^{44} +0.636405i q^{45} -0.471535i q^{46} +8.10124 q^{47} +3.70622i q^{48} +4.01250 q^{49} -1.02097 q^{50} -4.88877 q^{52} +6.44438 q^{53} +0.222191i q^{54} -3.15061 q^{55} +1.51722i q^{56} +0.950631i q^{57} -2.13016i q^{58} +10.1125 q^{59} -1.24139 q^{60} -2.96983i q^{61} -1.17157i q^{62} -1.72844i q^{63} -6.83940 q^{64} -1.59499i q^{65} -1.09999 q^{66} +7.70129 q^{67} -2.12220 q^{69} -0.244408 q^{70} -0.384044i q^{71} -0.877796 q^{72} +6.51420i q^{73} -1.88575i q^{74} +4.59499i q^{75} -1.85433 q^{76} +8.55687 q^{77} -0.556867i q^{78} +2.37280i q^{79} -2.35866i q^{80} +1.00000 q^{81} +1.53794i q^{82} +13.3581 q^{83} +3.37155 q^{84} +1.58880 q^{86} -9.58704 q^{87} -4.34564i q^{88} -1.98875 q^{89} -0.141404i q^{90} +4.33190i q^{91} -4.13964i q^{92} -5.27281 q^{93} -1.80002 q^{94} -0.604986i q^{95} -2.57908i q^{96} +3.04142i q^{97} -0.891542 q^{98} +4.95063i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 12 q^{4} + 12 q^{8} - 8 q^{9} + 4 q^{13} - 12 q^{15} + 12 q^{16} - 4 q^{18} - 20 q^{19} + 8 q^{21} - 4 q^{25} + 40 q^{26} - 24 q^{30} + 28 q^{32} + 12 q^{33} + 8 q^{35} - 12 q^{36} - 24 q^{38}+ \cdots + 76 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 - 8 * q^9 + 4 * q^13 - 12 * q^15 + 12 * q^16 - 4 * q^18 - 20 * q^19 + 8 * q^21 - 4 * q^25 + 40 * q^26 - 24 * q^30 + 28 * q^32 + 12 * q^33 + 8 * q^35 - 12 * q^36 - 24 * q^38 - 8 * q^42 - 28 * q^43 + 8 * q^47 - 60 * q^50 - 16 * q^52 + 40 * q^53 + 4 * q^55 + 48 * q^59 - 32 * q^60 - 4 * q^64 - 8 * q^66 + 8 * q^67 - 12 * q^69 + 8 * q^70 - 12 * q^72 - 80 * q^76 + 24 * q^77 + 8 * q^81 + 8 * q^83 + 48 * q^84 - 8 * q^86 - 32 * q^87 + 8 * q^89 - 8 * q^93 - 16 * q^94 + 76 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/867\mathbb{Z}\right)^\times$.

$n$	$290$	$292$
$\chi(n)$	$1$	$-1$

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 $

$2$	−0.222191	−0.157113	−0.0785565	−	0.996910i	$-0.525031\pi$
$2$	−0.222191	−0.157113	−0.0785565	+	0.996910i	$0.525031\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.95063	−0.975315
$5$	− 0.636405i	− 0.284609i	−0.989823	−	0.142304i	$-0.954549\pi$
$5$	− 0.636405i	− 0.284609i	0.989823	−	0.142304i	$-0.0454512\pi$
$6$	− 0.222191i	− 0.0907092i
$7$	1.72844i	0.653289i	0.945147	+	0.326644i	$0.105918\pi$
$7$	1.72844i	0.653289i	−0.945147	+	0.326644i	$0.894082\pi$
$8$	0.877796	0.310348
$9$	−1.00000	−0.333333
$10$	0.141404i	0.0447158i
$11$	− 4.95063i	− 1.49267i	−0.665570	−	0.746336i	$-0.731811\pi$
$11$	− 4.95063i	− 1.49267i	0.665570	−	0.746336i	$-0.268189\pi$
$12$	− 1.95063i	− 0.563099i
$13$	2.50625	0.695108	0.347554	−	0.937660i	$-0.387012\pi$
$13$	2.50625	0.695108	0.347554	+	0.937660i	$0.387012\pi$
$14$	− 0.384044i	− 0.102640i
$15$	0.636405	0.164319
$16$	3.70622	0.926556
$17$	0	0
$17$	0	0
$18$	0.222191	0.0523710
$19$	0.950631	0.218090	0.109045	−	0.994037i	$-0.465221\pi$
$19$	0.950631	0.218090	0.109045	+	0.994037i	$0.465221\pi$
$20$	1.24139i	0.277584i
$21$	−1.72844	−0.377176
$22$	1.09999i	0.234518i
$23$	2.12220i	0.442510i	0.975216	+	0.221255i	$0.0710153\pi$
$23$	2.12220i	0.442510i	−0.975216	+	0.221255i	$0.928985\pi$
$24$	0.877796i	0.179179i
$25$	4.59499	0.918998
$26$	−0.556867	−0.109211
$27$	− 1.00000i	− 0.192450i
$28$	− 3.37155i	− 0.637163i
$29$	9.58704i	1.78027i	0.455699	+	0.890134i	$0.349389\pi$
$29$	9.58704i	1.78027i	−0.455699	+	0.890134i	$0.650611\pi$
$30$	−0.141404	−0.0258167
$31$	5.27281i	0.947025i	0.880787	+	0.473512i	$0.157014\pi$
$31$	5.27281i	0.947025i	−0.880787	+	0.473512i	$0.842986\pi$
$32$	−2.57908	−0.455922
$33$	4.95063	0.861794
$34$	0	0
$35$	1.09999	0.185932
$36$	1.95063	0.325105
$37$	8.48705i	1.39526i	0.716457	+	0.697631i	$0.245763\pi$
$37$	8.48705i	1.39526i	−0.716457	+	0.697631i	$0.754237\pi$
$38$	−0.211222	−0.0342647
$39$	2.50625i	0.401321i
$40$	− 0.558634i	− 0.0883278i
$41$	− 6.92171i	− 1.08099i	−0.841347	−	0.540495i	$-0.818237\pi$
$41$	− 6.92171i	− 1.08099i	0.841347	−	0.540495i	$-0.181763\pi$
$42$	0.384044	0.0592593
$43$	−7.15061	−1.09046	−0.545229	−	0.838287i	$-0.683557\pi$
$43$	−7.15061	−1.09046	−0.545229	+	0.838287i	$0.683557\pi$
$44$	9.65685i	1.45583i
$45$	0.636405i	0.0948696i
$46$	− 0.471535i	− 0.0695241i
$47$	8.10124	1.18169	0.590843	−	0.806786i	$-0.298795\pi$
$47$	8.10124	1.18169	0.590843	+	0.806786i	$0.298795\pi$
$48$	3.70622i	0.534947i
$49$	4.01250	0.573214
$50$	−1.02097	−0.144387
$51$	0	0
$52$	−4.88877	−0.677950
$53$	6.44438	0.885204	0.442602	−	0.896718i	$-0.354055\pi$
$53$	6.44438	0.885204	0.442602	+	0.896718i	$0.354055\pi$
$54$	0.222191i	0.0302364i
$55$	−3.15061	−0.424828
$56$	1.51722i	0.202747i
$57$	0.950631i	0.125914i
$58$	− 2.13016i	− 0.279703i
$59$	10.1125	1.31653	0.658267	−	0.752785i	$-0.271290\pi$
$59$	10.1125	1.31653	0.658267	+	0.752785i	$0.271290\pi$
$60$	−1.24139	−0.160263
$61$	− 2.96983i	− 0.380248i	−0.981760	−	0.190124i	$-0.939111\pi$
$61$	− 2.96983i	− 0.380248i	0.981760	−	0.190124i	$-0.0608890\pi$
$62$	− 1.17157i	− 0.148790i
$63$	− 1.72844i	− 0.217763i
$64$	−6.83940	−0.854925
$65$	− 1.59499i	− 0.197834i
$66$	−1.09999	−0.135399
$67$	7.70129	0.940862	0.470431	−	0.882437i	$-0.344098\pi$
$67$	7.70129	0.940862	0.470431	+	0.882437i	$0.344098\pi$
$68$	0	0
$69$	−2.12220	−0.255483
$70$	−0.244408	−0.0292123
$71$	− 0.384044i	− 0.0455777i	−0.999740	−	0.0227888i	$-0.992745\pi$
$71$	− 0.384044i	− 0.0455777i	0.999740	−	0.0227888i	$-0.00725454\pi$
$72$	−0.877796	−0.103449
$73$	6.51420i	0.762430i	0.924487	+	0.381215i	$0.124494\pi$
$73$	6.51420i	0.762430i	−0.924487	+	0.381215i	$0.875506\pi$
$74$	− 1.88575i	− 0.219214i
$75$	4.59499i	0.530584i
$76$	−1.85433	−0.212706
$77$	8.55687	0.975145
$78$	− 0.556867i	− 0.0630527i
$79$	2.37280i	0.266961i	0.991051	+	0.133480i	$0.0426153\pi$
$79$	2.37280i	0.266961i	−0.991051	+	0.133480i	$0.957385\pi$
$80$	− 2.35866i	− 0.263706i
$81$	1.00000	0.111111
$82$	1.53794i	0.169838i
$83$	13.3581	1.46625	0.733123	−	0.680096i	$-0.238062\pi$
$83$	13.3581	1.46625	0.733123	+	0.680096i	$0.238062\pi$
$84$	3.37155	0.367866
$85$	0	0
$86$	1.58880	0.171325
$87$	−9.58704	−1.02784
$88$	− 4.34564i	− 0.463247i
$89$	−1.98875	−0.210807	−0.105404	−	0.994430i	$-0.533613\pi$
$89$	−1.98875	−0.210807	−0.105404	+	0.994430i	$0.533613\pi$
$90$	− 0.141404i	− 0.0149053i
$91$	4.33190i	0.454106i
$92$	− 4.13964i	− 0.431587i
$93$	−5.27281	−0.546765
$94$	−1.80002	−0.185658
$95$	− 0.604986i	− 0.0620703i
$96$	− 2.57908i	− 0.263227i
$97$	3.04142i	0.308809i	0.988008	+	0.154405i	$0.0493459\pi$
$97$	3.04142i	0.308809i	−0.988008	+	0.154405i	$0.950654\pi$
$98$	−0.891542	−0.0900594
$99$	4.95063i	0.497557i
$100$	−8.96313	−0.896313
$101$	−18.5456	−1.84536	−0.922679	−	0.385569i	$-0.874005\pi$
$101$	−18.5456	−1.84536	−0.922679	+	0.385569i	$0.874005\pi$
$102$	0	0
$103$	3.39501	0.334521	0.167260	−	0.985913i	$-0.446508\pi$
$103$	3.39501	0.334521	0.167260	+	0.985913i	$0.446508\pi$
$104$	2.19998	0.215725
$105$	1.09999i	0.107348i
$106$	−1.43189	−0.139077
$107$	− 13.1531i	− 1.27156i	−0.771871	−	0.635779i	$-0.780679\pi$
$107$	− 13.1531i	− 1.27156i	0.771871	−	0.635779i	$-0.219321\pi$
$108$	1.95063i	0.187700i
$109$	17.3896i	1.66562i	0.553561	+	0.832809i	$0.313269\pi$
$109$	17.3896i	1.66562i	−0.553561	+	0.832809i	$0.686731\pi$
$110$	0.700037	0.0667460
$111$	−8.48705	−0.805555
$112$	6.40598i	0.605309i
$113$	− 0.776042i	− 0.0730039i	−0.999334	−	0.0365019i	$-0.988378\pi$
$113$	− 0.776042i	− 0.0730039i	0.999334	−	0.0365019i	$-0.0116215\pi$
$114$	− 0.211222i	− 0.0197828i
$115$	1.35058	0.125942
$116$	− 18.7008i	− 1.73632i
$117$	−2.50625	−0.231703
$118$	−2.24691	−0.206845
$119$	0	0
$120$	0.558634	0.0509961
$121$	−13.5087	−1.22807
$122$	0.659871i	0.0597419i
$123$	6.92171	0.624110
$124$	− 10.2853i	− 0.923648i
$125$	− 6.10630i	− 0.546164i
$126$	0.384044i	0.0342134i
$127$	−10.8519	−0.962950	−0.481475	−	0.876460i	$-0.659899\pi$
$127$	−10.8519	−0.962950	−0.481475	+	0.876460i	$0.659899\pi$
$128$	6.67782	0.590242
$129$	− 7.15061i	− 0.629576i
$130$	0.354393i	0.0310823i
$131$	0.706223i	0.0617030i	0.999524	+	0.0308515i	$0.00982190\pi$
$131$	0.706223i	0.0617030i	−0.999524	+	0.0308515i	$0.990178\pi$
$132$	−9.65685	−0.840521
$133$	1.64311i	0.142476i
$134$	−1.71116	−0.147822
$135$	−0.636405	−0.0547730
$136$	0	0
$137$	12.3125	1.05192	0.525962	−	0.850508i	$-0.323705\pi$
$137$	12.3125	1.05192	0.525962	+	0.850508i	$0.323705\pi$
$138$	0.471535	0.0401398
$139$	− 13.3444i	− 1.13186i	−0.824454	−	0.565928i	$-0.808518\pi$
$139$	− 13.3444i	− 1.13186i	0.824454	−	0.565928i	$-0.191482\pi$
$140$	−2.14567	−0.181342
$141$	8.10124i	0.682247i
$142$	0.0853313i	0.00716085i
$143$	− 12.4075i	− 1.03757i
$144$	−3.70622	−0.308852
$145$	6.10124	0.506680
$146$	− 1.44740i	− 0.119788i
$147$	4.01250i	0.330945i
$148$	− 16.5551i	− 1.36082i
$149$	0.356892	0.0292377	0.0146189	−	0.999893i	$-0.495347\pi$
$149$	0.356892	0.0292377	0.0146189	+	0.999893i	$0.495347\pi$
$150$	− 1.02097i	− 0.0833616i
$151$	−9.13317	−0.743247	−0.371624	−	0.928384i	$-0.621199\pi$
$151$	−9.13317	−0.743247	−0.371624	+	0.928384i	$0.621199\pi$
$152$	0.834460	0.0676837
$153$	0	0
$154$	−1.90126	−0.153208
$155$	3.35564	0.269532
$156$	− 4.88877i	− 0.391414i
$157$	17.5975	1.40443	0.702216	−	0.711964i	$-0.252194\pi$
$157$	17.5975	1.40443	0.702216	+	0.711964i	$0.252194\pi$
$158$	− 0.527215i	− 0.0419430i
$159$	6.44438i	0.511073i
$160$	1.64134i	0.129759i
$161$	−3.66810	−0.289087
$162$	−0.222191	−0.0174570
$163$	− 16.5901i	− 1.29943i	−0.760177	−	0.649717i	$-0.774888\pi$
$163$	− 16.5901i	− 1.29943i	0.760177	−	0.649717i	$-0.225112\pi$
$164$	13.5017i	1.05431i
$165$	− 3.15061i	− 0.245274i
$166$	−2.96806	−0.230366
$167$	− 14.2234i	− 1.10064i	−0.834953	−	0.550321i	$-0.814505\pi$
$167$	− 14.2234i	− 1.10064i	0.834953	−	0.550321i	$-0.185495\pi$
$168$	−1.51722	−0.117056
$169$	−6.71872	−0.516825
$170$	0	0
$171$	−0.950631	−0.0726966
$172$	13.9482	1.06354
$173$	− 1.89331i	− 0.143946i	−0.997407	−	0.0719728i	$-0.977071\pi$
$173$	− 1.89331i	− 0.143946i	0.997407	−	0.0719728i	$-0.0229295\pi$
$174$	2.13016	0.161487
$175$	7.94216i	0.600371i
$176$	− 18.3481i	− 1.38304i
$177$	10.1125i	0.760101i
$178$	0.441884	0.0331206
$179$	22.2469	1.66281	0.831406	−	0.555666i	$-0.187536\pi$
$179$	22.2469	1.66281	0.831406	+	0.555666i	$0.187536\pi$
$180$	− 1.24139i	− 0.0925278i
$181$	11.6142i	0.863276i	0.902047	+	0.431638i	$0.142064\pi$
$181$	11.6142i	0.863276i	−0.902047	+	0.431638i	$0.857936\pi$
$182$	− 0.962511i	− 0.0713460i
$183$	2.96983	0.219536
$184$	1.86286i	0.137332i
$185$	5.40120	0.397104
$186$	1.17157	0.0859039
$187$	0	0
$188$	−15.8025	−1.15252
$189$	1.72844	0.125725
$190$	0.134423i	0.00975205i
$191$	6.63311	0.479955	0.239978	−	0.970778i	$-0.422860\pi$
$191$	6.63311	0.479955	0.239978	+	0.970778i	$0.422860\pi$
$192$	− 6.83940i	− 0.493591i
$193$	− 2.91769i	− 0.210020i	−0.994471	−	0.105010i	$-0.966513\pi$
$193$	− 2.91769i	− 0.210020i	0.994471	−	0.105010i	$-0.0334874\pi$
$194$	− 0.675776i	− 0.0485179i
$195$	1.59499	0.114220
$196$	−7.82690	−0.559064
$197$	− 4.53517i	− 0.323117i	−0.986863	−	0.161559i	$-0.948348\pi$
$197$	− 4.53517i	− 0.323117i	0.986863	−	0.161559i	$-0.0516521\pi$
$198$	− 1.09999i	− 0.0781727i
$199$	2.46029i	0.174405i	0.996191	+	0.0872026i	$0.0277928\pi$
$199$	2.46029i	0.174405i	−0.996191	+	0.0872026i	$0.972207\pi$
$200$	4.03346	0.285209
$201$	7.70129i	0.543207i
$202$	4.12068	0.289930
$203$	−16.5706	−1.16303
$204$	0	0
$205$	−4.40501	−0.307659
$206$	−0.754343	−0.0525576
$207$	− 2.12220i	− 0.147503i
$208$	9.28872	0.644057
$209$	− 4.70622i	− 0.325536i
$210$	− 0.244408i	− 0.0168657i
$211$	− 19.9038i	− 1.37023i	−0.728434	−	0.685116i	$-0.759752\pi$
$211$	− 19.9038i	− 1.37023i	0.728434	−	0.685116i	$-0.240248\pi$
$212$	−12.5706	−0.863353
$213$	0.384044	0.0263143
$214$	2.92251i	0.199778i
$215$	4.55068i	0.310354i
$216$	− 0.877796i	− 0.0597265i
$217$	−9.11373	−0.618681
$218$	− 3.86381i	− 0.261690i
$219$	−6.51420	−0.440189
$220$	6.14567	0.414341
$221$	0	0
$222$	1.88575	0.126563
$223$	−14.3631	−0.961823	−0.480911	−	0.876769i	$-0.659694\pi$
$223$	−14.3631	−0.961823	−0.480911	+	0.876769i	$0.659694\pi$
$224$	− 4.45779i	− 0.297849i
$225$	−4.59499	−0.306333
$226$	0.172430i	0.0114699i
$227$	10.8744i	0.721758i	0.932613	+	0.360879i	$0.117523\pi$
$227$	10.8744i	0.721758i	−0.932613	+	0.360879i	$0.882477\pi$
$228$	− 1.85433i	− 0.122806i
$229$	−18.9900	−1.25489	−0.627447	−	0.778659i	$-0.715900\pi$
$229$	−18.9900	−1.25489	−0.627447	+	0.778659i	$0.715900\pi$
$230$	−0.300087	−0.0197872
$231$	8.55687i	0.563000i
$232$	8.41546i	0.552502i
$233$	7.08079i	0.463878i	0.972730	+	0.231939i	$0.0745069\pi$
$233$	7.08079i	0.463878i	−0.972730	+	0.231939i	$0.925493\pi$
$234$	0.556867	0.0364035
$235$	− 5.15567i	− 0.336319i
$236$	−19.7257	−1.28404
$237$	−2.37280	−0.154130
$238$	0	0
$239$	−5.28872	−0.342099	−0.171049	−	0.985262i	$-0.554716\pi$
$239$	−5.28872	−0.342099	−0.171049	+	0.985262i	$0.554716\pi$
$240$	2.35866	0.152251
$241$	5.79826i	0.373499i	0.982408	+	0.186749i	$0.0597952\pi$
$241$	5.79826i	0.373499i	−0.982408	+	0.186749i	$0.940205\pi$
$242$	3.00153	0.192945
$243$	1.00000i	0.0641500i
$244$	5.79304i	0.370862i
$245$	− 2.55357i	− 0.163142i
$246$	−1.53794	−0.0980558
$247$	2.38252	0.151596
$248$	4.62845i	0.293907i
$249$	13.3581i	0.846538i
$250$	1.35677i	0.0858095i
$251$	−8.33190	−0.525905	−0.262952	−	0.964809i	$-0.584696\pi$
$251$	−8.33190	−0.525905	−0.262952	+	0.964809i	$0.584696\pi$
$252$	3.37155i	0.212388i
$253$	10.5062	0.660522
$254$	2.41120	0.151292
$255$	0	0
$256$	12.1950	0.762190
$257$	4.25565	0.265460	0.132730	−	0.991152i	$-0.457626\pi$
$257$	4.25565	0.265460	0.132730	+	0.991152i	$0.457626\pi$
$258$	1.58880i	0.0989146i
$259$	−14.6694	−0.911509
$260$	3.11123i	0.192951i
$261$	− 9.58704i	− 0.593423i
$262$	− 0.156917i	− 0.00969435i
$263$	9.09999	0.561129	0.280565	−	0.959835i	$-0.409478\pi$
$263$	9.09999	0.561129	0.280565	+	0.959835i	$0.409478\pi$
$264$	4.34564	0.267456
$265$	− 4.10124i	− 0.251937i
$266$	− 0.365084i	− 0.0223848i
$267$	− 1.98875i	− 0.121710i
$268$	−15.0224	−0.917637
$269$	21.3820i	1.30368i	0.758355	+	0.651842i	$0.226003\pi$
$269$	21.3820i	1.30368i	−0.758355	+	0.651842i	$0.773997\pi$
$270$	0.141404	0.00860555
$271$	−19.3213	−1.17368	−0.586842	−	0.809702i	$-0.699629\pi$
$271$	−19.3213	−1.17368	−0.586842	+	0.809702i	$0.699629\pi$
$272$	0	0
$273$	−4.33190	−0.262178
$274$	−2.73572	−0.165271
$275$	− 22.7481i	− 1.37176i
$276$	4.13964	0.249177
$277$	3.34013i	0.200689i	0.994953	+	0.100344i	$0.0319945\pi$
$277$	3.34013i	0.200689i	−0.994953	+	0.100344i	$0.968006\pi$
$278$	2.96501i	0.177829i
$279$	− 5.27281i	− 0.315675i
$280$	0.965565	0.0577035
$281$	−4.67754	−0.279039	−0.139519	−	0.990219i	$-0.544556\pi$
$281$	−4.67754	−0.279039	−0.139519	+	0.990219i	$0.544556\pi$
$282$	− 1.80002i	− 0.107190i
$283$	− 10.0582i	− 0.597897i	−0.954269	−	0.298948i	$-0.903364\pi$
$283$	− 10.0582i	− 0.597897i	0.954269	−	0.298948i	$-0.0966358\pi$
$284$	0.749129i	0.0444526i
$285$	0.604986	0.0358363
$286$	2.75684i	0.163015i
$287$	11.9638	0.706198
$288$	2.57908	0.151974
$289$	0	0
$290$	−1.35564	−0.0796061
$291$	−3.04142	−0.178291
$292$	− 12.7068i	− 0.743609i
$293$	5.76934	0.337048	0.168524	−	0.985698i	$-0.446100\pi$
$293$	5.76934	0.337048	0.168524	+	0.985698i	$0.446100\pi$
$294$	− 0.891542i	− 0.0519958i
$295$	− 6.43563i	− 0.374697i
$296$	7.44990i	0.433017i
$297$	−4.95063	−0.287265
$298$	−0.0792983	−0.00459363
$299$	5.31877i	0.307592i
$300$	− 8.96313i	− 0.517486i
$301$	− 12.3594i	− 0.712383i
$302$	2.02931	0.116774
$303$	− 18.5456i	− 1.06542i
$304$	3.52325	0.202072
$305$	−1.89001	−0.108222
$306$	0	0
$307$	−13.2375	−0.755502	−0.377751	−	0.925907i	$-0.623302\pi$
$307$	−13.2375	−0.755502	−0.377751	+	0.925907i	$0.623302\pi$
$308$	−16.6913	−0.951074
$309$	3.39501i	0.193136i
$310$	−0.745595	−0.0423469
$311$	− 1.29530i	− 0.0734499i	−0.999325	−	0.0367250i	$-0.988307\pi$
$311$	− 1.29530i	− 0.0734499i	0.999325	−	0.0367250i	$-0.0116926\pi$
$312$	2.19998i	0.124549i
$313$	19.1007i	1.07964i	0.841782	+	0.539818i	$0.181507\pi$
$313$	19.1007i	1.07964i	−0.841782	+	0.539818i	$0.818493\pi$
$314$	−3.91001	−0.220655
$315$	−1.09999	−0.0619773
$316$	− 4.62845i	− 0.260371i
$317$	− 15.1043i	− 0.848339i	−0.905583	−	0.424170i	$-0.860566\pi$
$317$	− 15.1043i	− 0.848339i	0.905583	−	0.424170i	$-0.139434\pi$
$318$	− 1.43189i	− 0.0802962i
$319$	47.4619	2.65735
$320$	4.35263i	0.243319i
$321$	13.1531	0.734135
$322$	0.815020	0.0454193
$323$	0	0
$324$	−1.95063	−0.108368
$325$	11.5162	0.638803
$326$	3.68617i	0.204158i
$327$	−17.3896	−0.961645
$328$	− 6.07585i	− 0.335483i
$329$	14.0025i	0.771983i
$330$	0.700037i	0.0385358i
$331$	22.6769	1.24644	0.623218	−	0.782048i	$-0.285825\pi$
$331$	22.6769	1.24644	0.623218	+	0.782048i	$0.285825\pi$
$332$	−26.0568	−1.43005
$333$	− 8.48705i	− 0.465087i
$334$	3.16033i	0.172925i
$335$	− 4.90114i	− 0.267778i
$336$	−6.40598	−0.349475
$337$	− 1.61669i	− 0.0880666i	−0.999030	−	0.0440333i	$-0.985979\pi$
$337$	− 1.61669i	− 0.0880666i	0.999030	−	0.0440333i	$-0.0140208\pi$
$338$	1.49284	0.0811999
$339$	0.776042	0.0421488
$340$	0	0
$341$	26.1037	1.41360
$342$	0.211222	0.0114216
$343$	19.0344i	1.02776i
$344$	−6.27677	−0.338421
$345$	1.35058i	0.0727128i
$346$	0.420677i	0.0226157i
$347$	15.3606i	0.824602i	0.911048	+	0.412301i	$0.135275\pi$
$347$	15.3606i	0.824602i	−0.911048	+	0.412301i	$0.864725\pi$
$348$	18.7008	1.00247
$349$	15.1087	0.808749	0.404374	−	0.914594i	$-0.367489\pi$
$349$	15.1087	0.808749	0.404374	+	0.914594i	$0.367489\pi$
$350$	− 1.76468i	− 0.0943261i
$351$	− 2.50625i	− 0.133774i
$352$	12.7681i	0.680541i
$353$	−31.2482	−1.66317	−0.831586	−	0.555396i	$-0.812567\pi$
$353$	−31.2482	−1.66317	−0.831586	+	0.555396i	$0.812567\pi$
$354$	− 2.24691i	− 0.119422i
$355$	−0.244408	−0.0129718
$356$	3.87932	0.205604
$357$	0	0
$358$	−4.94307	−0.261249
$359$	−2.72503	−0.143822	−0.0719108	−	0.997411i	$-0.522910\pi$
$359$	−2.72503	−0.143822	−0.0719108	+	0.997411i	$0.522910\pi$
$360$	0.558634i	0.0294426i
$361$	−18.0963	−0.952437
$362$	− 2.58057i	− 0.135632i
$363$	− 13.5087i	− 0.709025i
$364$	− 8.44994i	− 0.442897i
$365$	4.14567	0.216994
$366$	−0.659871	−0.0344920
$367$	18.4515i	0.963163i	0.876401	+	0.481581i	$0.159937\pi$
$367$	18.4515i	0.963163i	−0.876401	+	0.481581i	$0.840063\pi$
$368$	7.86536i	0.410010i
$369$	6.92171i	0.360330i
$370$	−1.20010	−0.0623902
$371$	11.1387i	0.578294i
$372$	10.2853	0.533268
$373$	24.4049	1.26364	0.631820	−	0.775115i	$-0.282308\pi$
$373$	24.4049	1.26364	0.631820	+	0.775115i	$0.282308\pi$
$374$	0	0
$375$	6.10630	0.315328
$376$	7.11123	0.366734
$377$	24.0275i	1.23748i
$378$	−0.384044	−0.0197531
$379$	− 5.89057i	− 0.302578i	−0.988490	−	0.151289i	$-0.951658\pi$
$379$	− 5.89057i	− 0.302578i	0.988490	−	0.151289i	$-0.0483425\pi$
$380$	1.18010i	0.0605381i
$381$	− 10.8519i	− 0.555959i
$382$	−1.47382	−0.0754072
$383$	−30.7756	−1.57256	−0.786279	−	0.617871i	$-0.787995\pi$
$383$	−30.7756	−1.57256	−0.786279	+	0.617871i	$0.787995\pi$
$384$	6.67782i	0.340776i
$385$	− 5.44563i	− 0.277535i
$386$	0.648284i	0.0329968i
$387$	7.15061	0.363486
$388$	− 5.93268i	− 0.301186i
$389$	−19.5244	−0.989925	−0.494963	−	0.868914i	$-0.664818\pi$
$389$	−19.5244	−0.989925	−0.494963	+	0.868914i	$0.664818\pi$
$390$	−0.354393	−0.0179454
$391$	0	0
$392$	3.52215	0.177896
$393$	−0.706223	−0.0356243
$394$	1.00768i	0.0507659i
$395$	1.51006	0.0759794
$396$	− 9.65685i	− 0.485275i
$397$	0.904279i	0.0453844i	0.999742	+	0.0226922i	$0.00722378\pi$
$397$	0.904279i	0.0453844i	−0.999742	+	0.0226922i	$0.992776\pi$
$398$	− 0.546655i	− 0.0274013i
$399$	−1.64311	−0.0822583
$400$	17.0301	0.851503
$401$	− 1.26486i	− 0.0631639i	−0.999501	−	0.0315820i	$-0.989945\pi$
$401$	− 1.26486i	− 0.0631639i	0.999501	−	0.0315820i	$-0.0100545\pi$
$402$	− 1.71116i	− 0.0853449i
$403$	13.2150i	0.658285i
$404$	36.1757	1.79981
$405$	− 0.636405i	− 0.0316232i
$406$	3.68185	0.182727
$407$	42.0162	2.08267
$408$	0	0
$409$	28.6350	1.41591	0.707954	−	0.706258i	$-0.249618\pi$
$409$	28.6350	1.41591	0.707954	+	0.706258i	$0.249618\pi$
$410$	0.978756	0.0483373
$411$	12.3125i	0.607329i
$412$	−6.62242	−0.326263
$413$	17.4788i	0.860076i
$414$	0.471535i	0.0231747i
$415$	− 8.50119i	− 0.417307i
$416$	−6.46382	−0.316915
$417$	13.3444	0.653478
$418$	1.04568i	0.0511460i
$419$	− 23.9581i	− 1.17043i	−0.810879	−	0.585214i	$-0.801010\pi$
$419$	− 23.9581i	− 1.17043i	0.810879	−	0.585214i	$-0.198990\pi$
$420$	− 2.14567i	− 0.104698i
$421$	−12.5312	−0.610735	−0.305368	−	0.952235i	$-0.598779\pi$
$421$	−12.5312	−0.610735	−0.305368	+	0.952235i	$0.598779\pi$
$422$	4.42244i	0.215281i
$423$	−8.10124	−0.393896
$424$	5.65685	0.274721
$425$	0	0
$426$	−0.0853313	−0.00413432
$427$	5.13317	0.248412
$428$	25.6569i	1.24017i
$429$	12.4075	0.599040
$430$	− 1.01112i	− 0.0487606i
$431$	17.6728i	0.851267i	0.904896	+	0.425633i	$0.139949\pi$
$431$	17.6728i	0.851267i	−0.904896	+	0.425633i	$0.860051\pi$
$432$	− 3.70622i	− 0.178316i
$433$	14.0076	0.673160	0.336580	−	0.941655i	$-0.390730\pi$
$433$	14.0076	0.673160	0.336580	+	0.941655i	$0.390730\pi$
$434$	2.02499	0.0972028
$435$	6.10124i	0.292532i
$436$	− 33.9206i	− 1.62450i
$437$	2.01743i	0.0965069i
$438$	1.44740	0.0691594
$439$	− 11.5740i	− 0.552398i	−0.961100	−	0.276199i	$-0.910925\pi$
$439$	− 11.5740i	− 0.552398i	0.961100	−	0.276199i	$-0.0890749\pi$
$440$	−2.76559	−0.131844
$441$	−4.01250	−0.191071
$442$	0	0
$443$	−35.6174	−1.69223	−0.846117	−	0.532997i	$-0.821066\pi$
$443$	−35.6174	−1.69223	−0.846117	+	0.532997i	$0.821066\pi$
$444$	16.5551	0.785670
$445$	1.26565i	0.0599977i
$446$	3.19135	0.151115
$447$	0.356892i	0.0168804i
$448$	− 11.8215i	− 0.558513i
$449$	22.1461i	1.04514i	0.852597	+	0.522569i	$0.175026\pi$
$449$	22.1461i	1.04514i	−0.852597	+	0.522569i	$0.824974\pi$
$450$	1.02097	0.0481288
$451$	−34.2668	−1.61356
$452$	1.51377i	0.0712018i
$453$	− 9.13317i	− 0.429114i
$454$	− 2.41619i	− 0.113398i
$455$	2.75684	0.129243
$456$	0.834460i	0.0390772i
$457$	−9.90632	−0.463398	−0.231699	−	0.972787i	$-0.574428\pi$
$457$	−9.90632	−0.463398	−0.231699	+	0.972787i	$0.574428\pi$
$458$	4.21941	0.197160
$459$	0	0
$460$	−2.63449	−0.122834
$461$	9.62367	0.448219	0.224109	−	0.974564i	$-0.428053\pi$
$461$	9.62367	0.448219	0.224109	+	0.974564i	$0.428053\pi$
$462$	− 1.90126i	− 0.0884547i
$463$	15.3013	0.711113	0.355557	−	0.934655i	$-0.384291\pi$
$463$	15.3013	0.711113	0.355557	+	0.934655i	$0.384291\pi$
$464$	35.5317i	1.64952i
$465$	3.35564i	0.155614i
$466$	− 1.57329i	− 0.0728812i
$467$	22.3150	1.03261	0.516307	−	0.856404i	$-0.327307\pi$
$467$	22.3150	1.03261	0.516307	+	0.856404i	$0.327307\pi$
$468$	4.88877	0.225983
$469$	13.3112i	0.614655i
$470$	1.14554i	0.0528400i
$471$	17.5975i	0.810849i
$472$	8.87670	0.408583
$473$	35.4000i	1.62769i
$474$	0.527215	0.0242158
$475$	4.36814	0.200424
$476$	0	0
$477$	−6.44438	−0.295068
$478$	1.17511	0.0537481
$479$	− 15.5571i	− 0.710824i	−0.934710	−	0.355412i	$-0.884341\pi$
$479$	− 15.5571i	− 0.710824i	0.934710	−	0.355412i	$-0.115659\pi$
$480$	−1.64134	−0.0749166
$481$	21.2707i	0.969858i
$482$	− 1.28832i	− 0.0586815i
$483$	− 3.66810i	− 0.166904i
$484$	26.3506	1.19775
$485$	1.93557	0.0878898
$486$	− 0.222191i	− 0.0100788i
$487$	0.0296516i	0.00134364i	1.00000		0.000671822i	$0.000213848\pi$
$487$	0.0296516i	0.00134364i	−1.00000		0.000671822i	$0.999786\pi$
$488$	− 2.60691i	− 0.118009i
$489$	16.5901	0.750228
$490$	0.567382i	0.0256317i
$491$	−6.66935	−0.300984	−0.150492	−	0.988611i	$-0.548086\pi$
$491$	−6.66935	−0.300984	−0.150492	+	0.988611i	$0.548086\pi$
$492$	−13.5017	−0.608704
$493$	0	0
$494$	−0.529375	−0.0238177
$495$	3.15061	0.141609
$496$	19.5422i	0.877471i
$497$	0.663798	0.0297754
$498$	− 2.96806i	− 0.133002i
$499$	18.6469i	0.834748i	0.908735	+	0.417374i	$0.137049\pi$
$499$	18.6469i	0.834748i	−0.908735	+	0.417374i	$0.862951\pi$
$500$	11.9111i	0.532682i
$501$	14.2234	0.635456
$502$	1.85128	0.0826265
$503$	− 25.7396i	− 1.14767i	−0.818970	−	0.573837i	$-0.805454\pi$
$503$	− 25.7396i	− 1.14767i	0.818970	−	0.573837i	$-0.194546\pi$
$504$	− 1.51722i	− 0.0675822i
$505$	11.8025i	0.525205i
$506$	−2.33440	−0.103777
$507$	− 6.71872i	− 0.298389i
$508$	21.1680	0.939180
$509$	41.1950	1.82594	0.912968	−	0.408032i	$-0.133785\pi$
$509$	41.1950	1.82594	0.912968	+	0.408032i	$0.133785\pi$
$510$	0	0
$511$	−11.2594	−0.498087
$512$	−16.0653	−0.709992
$513$	− 0.950631i	− 0.0419714i
$514$	−0.945570	−0.0417073
$515$	− 2.16060i	− 0.0952076i
$516$	13.9482i	0.614035i
$517$	− 40.1062i	− 1.76387i
$518$	3.25940	0.143210
$519$	1.89331	0.0831070
$520$	− 1.40007i	− 0.0613973i
$521$	− 8.98205i	− 0.393511i	−0.980453	−	0.196755i	$-0.936960\pi$
$521$	− 8.98205i	− 0.393511i	0.980453	−	0.196755i	$-0.0630405\pi$
$522$	2.13016i	0.0932344i
$523$	−5.38995	−0.235686	−0.117843	−	0.993032i	$-0.537598\pi$
$523$	−5.38995	−0.235686	−0.117843	+	0.993032i	$0.537598\pi$
$524$	− 1.37758i	− 0.0601799i
$525$	−7.94216	−0.346624
$526$	−2.02194	−0.0881607
$527$	0	0
$528$	18.3481	0.798500
$529$	18.4963	0.804185
$530$	0.911259i	0.0395826i
$531$	−10.1125	−0.438844
$532$	− 3.20510i	− 0.138959i
$533$	− 17.3475i	− 0.751405i
$534$	0.441884i	0.0191222i
$535$	−8.37070	−0.361897
$536$	6.76016	0.291994
$537$	22.2469i	0.960025i
$538$	− 4.75090i	− 0.204826i
$539$	− 19.8644i	− 0.855620i
$540$	1.24139	0.0534210
$541$	1.22765i	0.0527806i	0.999652	+	0.0263903i	$0.00840127\pi$
$541$	1.22765i	0.0527806i	−0.999652	+	0.0263903i	$0.991599\pi$
$542$	4.29302	0.184401
$543$	−11.6142	−0.498413
$544$	0	0
$545$	11.0668	0.474050
$546$	0.962511	0.0411916
$547$	13.5274i	0.578391i	0.957270	+	0.289196i	$0.0933878\pi$
$547$	13.5274i	0.578391i	−0.957270	+	0.289196i	$0.906612\pi$
$548$	−24.0171	−1.02596
$549$	2.96983i	0.126749i
$550$	5.05443i	0.215522i
$551$	9.11373i	0.388258i
$552$	−1.86286	−0.0792887
$553$	−4.10124	−0.174402
$554$	− 0.742148i	− 0.0315308i
$555$	5.40120i	0.229268i
$556$	26.0300i	1.10392i
$557$	−18.3599	−0.777936	−0.388968	−	0.921251i	$-0.627168\pi$
$557$	−18.3599	−0.777936	−0.388968	+	0.921251i	$0.627168\pi$
$558$	1.17157i	0.0495966i
$559$	−17.9212	−0.757986
$560$	4.07680	0.172276
$561$	0	0
$562$	1.03931	0.0438406
$563$	−3.33315	−0.140475	−0.0702377	−	0.997530i	$-0.522376\pi$
$563$	−3.33315	−0.140475	−0.0702377	+	0.997530i	$0.522376\pi$
$564$	− 15.8025i	− 0.665406i
$565$	−0.493877	−0.0207775
$566$	2.23484i	0.0939374i
$567$	1.72844i	0.0725876i
$568$	− 0.337113i	− 0.0141449i
$569$	−29.4369	−1.23406	−0.617029	−	0.786940i	$-0.711664\pi$
$569$	−29.4369	−1.23406	−0.617029	+	0.786940i	$0.711664\pi$
$570$	−0.134423	−0.00563035
$571$	25.8275i	1.08085i	0.841393	+	0.540424i	$0.181736\pi$
$571$	25.8275i	1.08085i	−0.841393	+	0.540424i	$0.818264\pi$
$572$	24.2025i	1.01196i
$573$	6.63311i	0.277102i
$574$	−2.65824	−0.110953
$575$	9.75150i	0.406666i
$576$	6.83940	0.284975
$577$	−22.1850	−0.923575	−0.461788	−	0.886990i	$-0.652792\pi$
$577$	−22.1850	−0.923575	−0.461788	+	0.886990i	$0.652792\pi$
$578$	0	0
$579$	2.91769	0.121255
$580$	−11.9013	−0.494173
$581$	23.0887i	0.957882i
$582$	0.675776	0.0280118
$583$	− 31.9038i	− 1.32132i
$584$	5.71814i	0.236618i
$585$	1.59499i	0.0659447i
$586$	−1.28190	−0.0529547
$587$	−32.1482	−1.32690	−0.663448	−	0.748222i	$-0.730908\pi$
$587$	−32.1482	−1.32690	−0.663448	+	0.748222i	$0.730908\pi$
$588$	− 7.82690i	− 0.322776i
$589$	5.01250i	0.206536i
$590$	1.42994i	0.0588698i
$591$	4.53517	0.186552
$592$	31.4549i	1.29279i
$593$	29.3406	1.20488	0.602438	−	0.798166i	$-0.294196\pi$
$593$	29.3406	1.20488	0.602438	+	0.798166i	$0.294196\pi$
$594$	1.09999	0.0451330
$595$	0	0
$596$	−0.696164	−0.0285160
$597$	−2.46029	−0.100693
$598$	− 1.18178i	− 0.0483268i
$599$	−36.2632	−1.48167	−0.740836	−	0.671686i	$-0.765570\pi$
$599$	−36.2632	−1.48167	−0.740836	+	0.671686i	$0.765570\pi$
$600$	4.03346i	0.164665i
$601$	− 45.7383i	− 1.86570i	−0.360262	−	0.932851i	$-0.617313\pi$
$601$	− 45.7383i	− 1.86570i	0.360262	−	0.932851i	$-0.382687\pi$
$602$	2.74615i	0.111925i
$603$	−7.70129	−0.313621
$604$	17.8155	0.724900
$605$	8.59703i	0.349519i
$606$	4.12068i	0.167391i
$607$	− 33.6640i	− 1.36638i	−0.730241	−	0.683190i	$-0.760592\pi$
$607$	− 33.6640i	− 1.36638i	0.730241	−	0.683190i	$-0.239408\pi$
$608$	−2.45176	−0.0994318
$609$	− 16.5706i	− 0.671475i
$610$	0.419945	0.0170031
$611$	20.3037	0.821400
$612$	0	0
$613$	−5.89120	−0.237943	−0.118972	−	0.992898i	$-0.537960\pi$
$613$	−5.89120	−0.237943	−0.118972	+	0.992898i	$0.537960\pi$
$614$	2.94125	0.118699
$615$	− 4.40501i	− 0.177627i
$616$	7.51118	0.302634
$617$	− 35.2179i	− 1.41782i	−0.705300	−	0.708909i	$-0.749188\pi$
$617$	− 35.2179i	− 1.41782i	0.705300	−	0.708909i	$-0.250812\pi$
$618$	− 0.754343i	− 0.0303441i
$619$	32.8288i	1.31950i	0.751485	+	0.659750i	$0.229338\pi$
$619$	32.8288i	1.31950i	−0.751485	+	0.659750i	$0.770662\pi$
$620$	−6.54562	−0.262878
$621$	2.12220	0.0851611
$622$	0.287805i	0.0115399i
$623$	− 3.43744i	− 0.137718i
$624$	9.28872i	0.371846i
$625$	19.0889	0.763555
$626$	− 4.24402i	− 0.169625i
$627$	4.70622	0.187948
$628$	−34.3262	−1.36976
$629$	0	0
$630$	0.244408	0.00973744
$631$	−10.7057	−0.426186	−0.213093	−	0.977032i	$-0.568354\pi$
$631$	−10.7057	−0.426186	−0.213093	+	0.977032i	$0.568354\pi$
$632$	2.08283i	0.0828506i
$633$	19.9038	0.791103
$634$	3.35603i	0.133285i
$635$	6.90620i	0.274064i
$636$	− 12.5706i	− 0.498457i
$637$	10.0563	0.398446
$638$	−10.5456	−0.417505
$639$	0.384044i	0.0151926i
$640$	− 4.24980i	− 0.167988i
$641$	− 36.9437i	− 1.45919i	−0.683881	−	0.729593i	$-0.739709\pi$
$641$	− 36.9437i	− 1.45919i	0.683881	−	0.729593i	$-0.260291\pi$
$642$	−2.92251	−0.115342
$643$	− 39.3137i	− 1.55038i	−0.631727	−	0.775191i	$-0.717654\pi$
$643$	− 39.3137i	− 1.55038i	0.631727	−	0.775191i	$-0.282346\pi$
$644$	7.15511	0.281951
$645$	−4.55068	−0.179183
$646$	0	0
$647$	−14.8806	−0.585016	−0.292508	−	0.956263i	$-0.594490\pi$
$647$	−14.8806	−0.585016	−0.292508	+	0.956263i	$0.594490\pi$
$648$	0.877796	0.0344831
$649$	− 50.0632i	− 1.96515i
$650$	−2.55880	−0.100364
$651$	− 9.11373i	− 0.357195i
$652$	32.3611i	1.26736i
$653$	− 24.2908i	− 0.950571i	−0.879832	−	0.475285i	$-0.842345\pi$
$653$	− 24.2908i	− 0.950571i	0.879832	−	0.475285i	$-0.157655\pi$
$654$	3.86381	0.151087
$655$	0.449444	0.0175612
$656$	− 25.6534i	− 1.00160i
$657$	− 6.51420i	− 0.254143i
$658$	− 3.11123i	− 0.121289i
$659$	−9.14192	−0.356119	−0.178059	−	0.984020i	$-0.556982\pi$
$659$	−9.14192	−0.356119	−0.178059	+	0.984020i	$0.556982\pi$
$660$	6.14567i	0.239220i
$661$	16.7951	0.653253	0.326627	−	0.945153i	$-0.394088\pi$
$661$	16.7951	0.653253	0.326627	+	0.945153i	$0.394088\pi$
$662$	−5.03861	−0.195831
$663$	0	0
$664$	11.7257	0.455046
$665$	1.04568	0.0405498
$666$	1.88575i	0.0730713i
$667$	−20.3456	−0.787787
$668$	27.7447i	1.07347i
$669$	− 14.3631i	− 0.555309i
$670$	1.08899i	0.0420714i
$671$	−14.7025	−0.567585
$672$	4.45779	0.171963
$673$	− 18.6313i	− 0.718186i	−0.933302	−	0.359093i	$-0.883086\pi$
$673$	− 18.6313i	− 0.718186i	0.933302	−	0.359093i	$-0.116914\pi$
$674$	0.359214i	0.0138364i
$675$	− 4.59499i	− 0.176861i
$676$	13.1057	0.504067
$677$	10.8010i	0.415117i	0.978223	+	0.207559i	$0.0665518\pi$
$677$	10.8010i	0.415117i	−0.978223	+	0.207559i	$0.933448\pi$
$678$	−0.172430	−0.00662212
$679$	−5.25690	−0.201741
$680$	0	0
$681$	−10.8744	−0.416707
$682$	−5.80002	−0.222094
$683$	− 22.2199i	− 0.850221i	−0.905141	−	0.425111i	$-0.860235\pi$
$683$	− 22.2199i	− 0.850221i	0.905141	−	0.425111i	$-0.139765\pi$
$684$	1.85433	0.0709021
$685$	− 7.83571i	− 0.299387i
$686$	− 4.22929i	− 0.161475i
$687$	− 18.9900i	− 0.724514i
$688$	−26.5017	−1.01037
$689$	16.1512	0.615313
$690$	− 0.300087i	− 0.0114241i
$691$	38.5413i	1.46618i	0.680132	+	0.733090i	$0.261923\pi$
$691$	38.5413i	1.46618i	−0.680132	+	0.733090i	$0.738077\pi$
$692$	3.69315i	0.140392i
$693$	−8.55687	−0.325048
$694$	− 3.41300i	− 0.129556i
$695$	−8.49244	−0.322137
$696$	−8.41546	−0.318987
$697$	0	0
$698$	−3.35702	−0.127065
$699$	−7.08079	−0.267820
$700$	− 15.4922i	− 0.585551i
$701$	18.0843	0.683034	0.341517	−	0.939876i	$-0.389059\pi$
$701$	18.0843	0.683034	0.341517	+	0.939876i	$0.389059\pi$
$702$	0.556867i	0.0210176i
$703$	8.06805i	0.304292i
$704$	33.8593i	1.27612i
$705$	5.15567	0.194174
$706$	6.94307	0.261306
$707$	− 32.0550i	− 1.20555i
$708$	− 19.7257i	− 0.741338i
$709$	52.2614i	1.96272i	0.192185	+	0.981359i	$0.438443\pi$
$709$	52.2614i	1.96272i	−0.192185	+	0.981359i	$0.561557\pi$
$710$	0.0543053	0.00203804
$711$	− 2.37280i	− 0.0889869i
$712$	−1.74572	−0.0654236
$713$	−11.1900	−0.419068
$714$	0	0
$715$	−7.89620	−0.295301
$716$	−43.3955	−1.62177
$717$	− 5.28872i	− 0.197511i
$718$	0.605478	0.0225962
$719$	2.25831i	0.0842206i	0.999113	+	0.0421103i	$0.0134081\pi$
$719$	2.25831i	0.0842206i	−0.999113	+	0.0421103i	$0.986592\pi$
$720$	2.35866i	0.0879020i
$721$	5.86808i	0.218539i
$722$	4.02084	0.149640
$723$	−5.79826	−0.215640
$724$	− 22.6550i	− 0.841966i
$725$	44.0523i	1.63606i
$726$	3.00153i	0.111397i
$727$	−39.5601	−1.46720	−0.733601	−	0.679581i	$-0.762162\pi$
$727$	−39.5601	−1.46720	−0.733601	+	0.679581i	$0.762162\pi$
$728$	3.80252i	0.140931i
$729$	−1.00000	−0.0370370
$730$	−0.921132	−0.0340926
$731$	0	0
$732$	−5.79304	−0.214117
$733$	13.5137	0.499139	0.249570	−	0.968357i	$-0.419711\pi$
$733$	13.5137	0.499139	0.249570	+	0.968357i	$0.419711\pi$
$734$	− 4.09977i	− 0.151325i
$735$	2.55357	0.0941899
$736$	− 5.47334i	− 0.201750i
$737$	− 38.1262i	− 1.40440i
$738$	− 1.53794i	− 0.0566125i
$739$	10.7282	0.394642	0.197321	−	0.980339i	$-0.436776\pi$
$739$	10.7282	0.394642	0.197321	+	0.980339i	$0.436776\pi$
$740$	−10.5357	−0.387302
$741$	2.38252i	0.0875240i
$742$	− 2.47493i	− 0.0908575i
$743$	− 43.9921i	− 1.61392i	−0.590609	−	0.806958i	$-0.701113\pi$
$743$	− 43.9921i	− 1.61392i	0.590609	−	0.806958i	$-0.298887\pi$
$744$	−4.62845	−0.169687
$745$	− 0.227128i	− 0.00832131i
$746$	−5.42257	−0.198534
$747$	−13.3581	−0.488749
$748$	0	0
$749$	22.7343	0.830695
$750$	−1.35677	−0.0495421
$751$	− 12.5047i	− 0.456304i	−0.973626	−	0.228152i	$-0.926732\pi$
$751$	− 12.5047i	− 0.456304i	0.973626	−	0.228152i	$-0.0732683\pi$
$752$	30.0250	1.09490
$753$	− 8.33190i	− 0.303631i
$754$	− 5.33870i	− 0.194424i
$755$	5.81240i	0.211535i
$756$	−3.37155	−0.122622
$757$	35.8474	1.30290	0.651449	−	0.758693i	$-0.274162\pi$
$757$	35.8474	1.30290	0.651449	+	0.758693i	$0.274162\pi$
$758$	1.30883i	0.0475390i
$759$	10.5062i	0.381353i
$760$	− 0.531055i	− 0.0192634i
$761$	−23.4468	−0.849944	−0.424972	−	0.905206i	$-0.639716\pi$
$761$	−23.4468	−0.849944	−0.424972	+	0.905206i	$0.639716\pi$
$762$	2.41120i	0.0873485i
$763$	−30.0568	−1.08813
$764$	−12.9388	−0.468108
$765$	0	0
$766$	6.83807	0.247070
$767$	25.3444	0.915133
$768$	12.1950i	0.440051i
$769$	22.8938	0.825573	0.412786	−	0.910828i	$-0.364556\pi$
$769$	22.8938	0.825573	0.412786	+	0.910828i	$0.364556\pi$
$770$	1.20997i	0.0436044i
$771$	4.25565i	0.153264i
$772$	5.69133i	0.204835i
$773$	−34.1287	−1.22753	−0.613763	−	0.789491i	$-0.710345\pi$
$773$	−34.1287	−1.22753	−0.613763	+	0.789491i	$0.710345\pi$
$774$	−1.58880	−0.0571083
$775$	24.2285i	0.870313i
$776$	2.66974i	0.0958382i
$777$	− 14.6694i	− 0.526260i
$778$	4.33815	0.155530
$779$	− 6.57999i	− 0.235753i
$780$	−3.11123	−0.111400
$781$	−1.90126	−0.0680325
$782$	0	0
$783$	9.58704	0.342613
$784$	14.8712	0.531115
$785$	− 11.1991i	− 0.399714i
$786$	0.156917	0.00559703
$787$	13.1730i	0.469568i	0.972048	+	0.234784i	$0.0754383\pi$
$787$	13.1730i	0.469568i	−0.972048	+	0.234784i	$0.924562\pi$
$788$	8.84644i	0.315141i
$789$	9.09999i	0.323968i
$790$	−0.335522	−0.0119373
$791$	1.34134	0.0476926
$792$	4.34564i	0.154416i
$793$	− 7.44313i	− 0.264313i
$794$	− 0.200923i	− 0.00713049i
$795$	4.10124	0.145456
$796$	− 4.79911i	− 0.170100i
$797$	−36.3506	−1.28761	−0.643803	−	0.765191i	$-0.722644\pi$
$797$	−36.3506	−1.28761	−0.643803	+	0.765191i	$0.722644\pi$
$798$	0.365084	0.0129239
$799$	0	0
$800$	−11.8509	−0.418991
$801$	1.98875	0.0702691
$802$	0.281040i	0.00992388i
$803$	32.2494	1.13806
$804$	− 15.0224i	− 0.529798i
$805$	2.33440i	0.0822767i
$806$	− 2.93625i	− 0.103425i
$807$	−21.3820	−0.752682
$808$	−16.2793	−0.572703
$809$	45.1267i	1.58657i	0.608851	+	0.793285i	$0.291631\pi$
$809$	45.1267i	1.58657i	−0.608851	+	0.793285i	$0.708369\pi$
$810$	0.141404i	0.00496842i
$811$	− 3.32815i	− 0.116867i	−0.998291	−	0.0584336i	$-0.981389\pi$
$811$	− 3.32815i	− 0.116867i	0.998291	−	0.0584336i	$-0.0186106\pi$
$812$	32.3232	1.13432
$813$	− 19.3213i	− 0.677626i
$814$	−9.33565	−0.327214
$815$	−10.5580	−0.369830
$816$	0	0
$817$	−6.79759	−0.237817
$818$	−6.36244	−0.222458
$819$	− 4.33190i	− 0.151369i
$820$	8.59255	0.300065
$821$	− 7.46983i	− 0.260699i	−0.991468	−	0.130349i	$-0.958390\pi$
$821$	− 7.46983i	− 0.260699i	0.991468	−	0.130349i	$-0.0416099\pi$
$822$	− 2.73572i	− 0.0954193i
$823$	− 54.1834i	− 1.88871i	−0.328923	−	0.944357i	$-0.606686\pi$
$823$	− 54.1834i	− 1.88871i	0.328923	−	0.944357i	$-0.393314\pi$
$824$	2.98013	0.103818
$825$	22.7481	0.791987
$826$	− 3.88364i	− 0.135129i
$827$	4.95063i	0.172150i	0.996289	+	0.0860752i	$0.0274325\pi$
$827$	4.95063i	0.172150i	−0.996289	+	0.0860752i	$0.972567\pi$
$828$	4.13964i	0.143862i
$829$	−0.0762440	−0.00264806	−0.00132403	−	0.999999i	$-0.500421\pi$
$829$	−0.0762440	−0.00264806	−0.00132403	+	0.999999i	$0.500421\pi$
$830$	1.88889i	0.0655643i
$831$	−3.34013	−0.115868
$832$	−17.1412	−0.594265
$833$	0	0
$834$	−2.96501	−0.102670
$835$	−9.05187	−0.313253
$836$	9.18010i	0.317501i
$837$	5.27281	0.182255
$838$	5.32328i	0.183890i
$839$	36.4479i	1.25832i	0.777276	+	0.629160i	$0.216601\pi$
$839$	36.4479i	1.25832i	−0.777276	+	0.629160i	$0.783399\pi$
$840$	0.965565i	0.0333151i
$841$	−62.9113	−2.16935
$842$	2.78433	0.0959545
$843$	− 4.67754i	− 0.161103i
$844$	38.8249i	1.33641i
$845$	4.27583i	0.147093i
$846$	1.80002	0.0618861
$847$	− 23.3491i	− 0.802283i
$848$	23.8843	0.820191
$849$	10.0582	0.345196
$850$	0	0
$851$	−18.0112	−0.617418
$852$	−0.749129	−0.0256647
$853$	39.9684i	1.36849i	0.729252	+	0.684245i	$0.239868\pi$
$853$	39.9684i	1.36849i	−0.729252	+	0.684245i	$0.760132\pi$
$854$	−1.14055	−0.0390287
$855$	0.604986i	0.0206901i
$856$	− 11.5457i	− 0.394625i
$857$	2.95962i	0.101099i	0.998722	+	0.0505493i	$0.0160972\pi$
$857$	2.95962i	0.101099i	−0.998722	+	0.0505493i	$0.983903\pi$
$858$	−2.75684	−0.0941170
$859$	22.4494	0.765963	0.382981	−	0.923756i	$-0.374897\pi$
$859$	22.4494	0.765963	0.382981	+	0.923756i	$0.374897\pi$
$860$	− 8.87670i	− 0.302693i
$861$	11.9638i	0.407724i
$862$	− 3.92673i	− 0.133745i
$863$	−27.8981	−0.949661	−0.474831	−	0.880077i	$-0.657491\pi$
$863$	−27.8981	−0.949661	−0.474831	+	0.880077i	$0.657491\pi$
$864$	2.57908i	0.0877422i
$865$	−1.20491	−0.0409682
$866$	−3.11236	−0.105762
$867$	0	0
$868$	17.7775	0.603409
$869$	11.7468	0.398484
$870$	− 1.35564i	− 0.0459606i
$871$	19.3013	0.654001
$872$	15.2645i	0.516921i
$873$	− 3.04142i	− 0.102936i
$874$	− 0.448256i	− 0.0151625i
$875$	10.5544	0.356803
$876$	12.7068	0.429323
$877$	− 20.6961i	− 0.698858i	−0.936963	−	0.349429i	$-0.886376\pi$
$877$	− 20.6961i	− 0.698858i	0.936963	−	0.349429i	$-0.113624\pi$
$878$	2.57165i	0.0867889i
$879$	5.76934i	0.194595i
$880$	−11.6768	−0.393627
$881$	− 22.2399i	− 0.749282i	−0.927170	−	0.374641i	$-0.877766\pi$
$881$	− 22.2399i	− 0.749282i	0.927170	−	0.374641i	$-0.122234\pi$
$882$	0.891542	0.0300198
$883$	4.23253	0.142436	0.0712180	−	0.997461i	$-0.477311\pi$
$883$	4.23253	0.142436	0.0712180	+	0.997461i	$0.477311\pi$
$884$	0	0
$885$	6.43563	0.216331
$886$	7.91388	0.265872
$887$	− 36.3047i	− 1.21899i	−0.792789	−	0.609496i	$-0.791372\pi$
$887$	− 36.3047i	− 1.21899i	0.792789	−	0.609496i	$-0.208628\pi$
$888$	−7.44990	−0.250002
$889$	− 18.7568i	− 0.629084i
$890$	− 0.281217i	− 0.00942642i
$891$	− 4.95063i	− 0.165852i
$892$	28.0171	0.938081
$893$	7.70129	0.257714
$894$	− 0.0792983i	− 0.00265213i
$895$	− 14.1580i	− 0.473251i
$896$	11.5422i	0.385598i
$897$	−5.31877	−0.177589
$898$	− 4.92066i	− 0.164205i
$899$	−50.5506	−1.68596
$900$	8.96313	0.298771
$901$	0	0
$902$	7.61380	0.253512
$903$	12.3594	0.411295
$904$	− 0.681206i	− 0.0226566i
$905$	7.39133	0.245696
$906$	2.02931i	0.0674194i
$907$	− 24.3718i	− 0.809251i	−0.914482	−	0.404626i	$-0.867402\pi$
$907$	− 24.3718i	− 0.809251i	0.914482	−	0.404626i	$-0.132598\pi$
$908$	− 21.2119i	− 0.703942i
$909$	18.5456	0.615119
$910$	−0.612546	−0.0203057
$911$	− 26.9178i	− 0.891826i	−0.895076	−	0.445913i	$-0.852879\pi$
$911$	− 26.9178i	− 0.891826i	0.895076	−	0.445913i	$-0.147121\pi$
$912$	3.52325i	0.116666i
$913$	− 66.1312i	− 2.18862i
$914$	2.20110	0.0728059
$915$	− 1.89001i	− 0.0624820i
$916$	37.0425	1.22392
$917$	−1.22066	−0.0403099
$918$	0	0
$919$	−23.8000	−0.785088	−0.392544	−	0.919733i	$-0.628405\pi$
$919$	−23.8000	−0.785088	−0.392544	+	0.919733i	$0.628405\pi$
$920$	1.18553	0.0390859
$921$	− 13.2375i	− 0.436189i
$922$	−2.13830	−0.0704210
$923$	− 0.962511i	− 0.0316814i
$924$	− 16.6913i	− 0.549103i
$925$	38.9979i	1.28224i
$926$	−3.39983	−0.111725
$927$	−3.39501	−0.111507
$928$	− 24.7258i	− 0.811663i
$929$	5.82742i	0.191191i	0.995420	+	0.0955957i	$0.0304756\pi$
$929$	5.82742i	0.191191i	−0.995420	+	0.0955957i	$0.969524\pi$
$930$	− 0.745595i	− 0.0244490i
$931$	3.81440	0.125012
$932$	− 13.8120i	− 0.452427i
$933$	1.29530	0.0424063
$934$	−4.95819	−0.162237
$935$	0	0
$936$	−2.19998	−0.0719084
$937$	−55.4424	−1.81123	−0.905613	−	0.424106i	$-0.860589\pi$
$937$	−55.4424	−1.81123	−0.905613	+	0.424106i	$0.860589\pi$
$938$	− 2.95764i	− 0.0965702i
$939$	−19.1007	−0.623328
$940$	10.0568i	0.328017i
$941$	21.1696i	0.690109i	0.938583	+	0.345054i	$0.112140\pi$
$941$	21.1696i	0.690109i	−0.938583	+	0.345054i	$0.887860\pi$
$942$	− 3.91001i	− 0.127395i
$943$	14.6893	0.478349
$944$	37.4791	1.21984
$945$	− 1.09999i	− 0.0357826i
$946$	− 7.86558i	− 0.255732i
$947$	37.5831i	1.22129i	0.791906	+	0.610643i	$0.209089\pi$
$947$	37.5831i	1.22129i	−0.791906	+	0.610643i	$0.790911\pi$
$948$	4.62845	0.150325
$949$	16.3262i	0.529971i
$950$	−0.970563	−0.0314892
$951$	15.1043	0.489789
$952$	0	0
$953$	21.2794	0.689307	0.344654	−	0.938730i	$-0.387996\pi$
$953$	21.2794	0.689307	0.344654	+	0.938730i	$0.387996\pi$
$954$	1.43189	0.0463590
$955$	− 4.22134i	− 0.136599i
$956$	10.3163	0.333654
$957$	47.4619i	1.53422i
$958$	3.45666i	0.111680i
$959$	21.2813i	0.687210i
$960$	−4.35263	−0.140480
$961$	3.19748	0.103144
$962$	− 4.72616i	− 0.152377i
$963$	13.1531i	0.423853i
$964$	− 11.3103i	− 0.364279i
$965$	−1.85683	−0.0597734
$966$	0.815020i	0.0262229i
$967$	26.2320	0.843563	0.421782	−	0.906697i	$-0.361405\pi$
$967$	26.2320	0.843563	0.421782	+	0.906697i	$0.361405\pi$
$968$	−11.8579	−0.381128
$969$	0	0
$970$	−0.430067	−0.0138086
$971$	20.8243	0.668285	0.334142	−	0.942523i	$-0.391553\pi$
$971$	20.8243	0.668285	0.334142	+	0.942523i	$0.391553\pi$
$972$	− 1.95063i	− 0.0625665i
$973$	23.0650	0.739429
$974$	− 0.00658834i	0 0.000211104i
$975$	11.5162i	0.368813i
$976$	− 11.0069i	− 0.352321i
$977$	−16.2900	−0.521162	−0.260581	−	0.965452i	$-0.583914\pi$
$977$	−16.2900	−0.521162	−0.260581	+	0.965452i	$0.583914\pi$
$978$	−3.68617	−0.117871
$979$	9.84558i	0.314666i
$980$	4.98108i	0.159115i
$981$	− 17.3896i	− 0.555206i
$982$	1.48187	0.0472884
$983$	0.931671i	0.0297157i	0.999890	+	0.0148578i	$0.00472957\pi$
$983$	0.931671i	0.0297157i	−0.999890	+	0.0148578i	$0.995270\pi$
$984$	6.07585	0.193691
$985$	−2.88620	−0.0919621
$986$	0	0
$987$	−14.0025	−0.445704
$988$	−4.64741	−0.147854
$989$	− 15.1750i	− 0.482538i
$990$	−0.700037	−0.0222487
$991$	− 21.6972i	− 0.689234i	−0.938743	−	0.344617i	$-0.888009\pi$
$991$	− 21.6972i	− 0.689234i	0.938743	−	0.344617i	$-0.111991\pi$
$992$	− 13.5990i	− 0.431769i
$993$	22.6769i	0.719630i
$994$	−0.147490	−0.00467810
$995$	1.56574	0.0496373
$996$	− 26.0568i	− 0.825641i
$997$	3.22709i	0.102203i	0.998693	+	0.0511015i	$0.0162732\pi$
$997$	3.22709i	0.102203i	−0.998693	+	0.0511015i	$0.983727\pi$
$998$	− 4.14317i	− 0.131150i
$999$	8.48705	0.268518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.d.f.577.4		8
17.2	even	8		867.2.e.g.829.2		8
17.3	odd	16		867.2.h.k.688.1		16
17.4	even	4		867.2.a.l.1.3		4
17.5	odd	16		867.2.h.i.757.3		16
17.6	odd	16		867.2.h.k.712.2		16
17.7	odd	16		867.2.h.i.733.4		16
17.8	even	8		867.2.e.g.616.3		8
17.9	even	8		51.2.e.a.4.3	✓	8
17.10	odd	16		867.2.h.i.733.3		16
17.11	odd	16		867.2.h.k.712.1		16
17.12	odd	16		867.2.h.i.757.4		16
17.13	even	4		867.2.a.k.1.3		4
17.14	odd	16		867.2.h.k.688.2		16
17.15	even	8		51.2.e.a.13.2	yes	8
17.16	even	2	inner	867.2.d.f.577.3		8
51.26	odd	8		153.2.f.b.55.2		8
51.32	odd	8		153.2.f.b.64.3		8
51.38	odd	4		2601.2.a.be.1.2		4
51.47	odd	4		2601.2.a.bf.1.2		4
68.15	odd	8		816.2.bd.e.625.3		8
68.43	odd	8		816.2.bd.e.769.3		8
204.83	even	8		2448.2.be.x.1441.3		8
204.179	even	8		2448.2.be.x.1585.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.e.a.4.3	✓	8	17.9	even	8
51.2.e.a.13.2	yes	8	17.15	even	8
153.2.f.b.55.2		8	51.26	odd	8
153.2.f.b.64.3		8	51.32	odd	8
816.2.bd.e.625.3		8	68.15	odd	8
816.2.bd.e.769.3		8	68.43	odd	8
867.2.a.k.1.3		4	17.13	even	4
867.2.a.l.1.3		4	17.4	even	4
867.2.d.f.577.3		8	17.16	even	2	inner
867.2.d.f.577.4		8	1.1	even	1	trivial
867.2.e.g.616.3		8	17.8	even	8
867.2.e.g.829.2		8	17.2	even	8
867.2.h.i.733.3		16	17.10	odd	16
867.2.h.i.733.4		16	17.7	odd	16
867.2.h.i.757.3		16	17.5	odd	16
867.2.h.i.757.4		16	17.12	odd	16
867.2.h.k.688.1		16	17.3	odd	16
867.2.h.k.688.2		16	17.14	odd	16
867.2.h.k.712.1		16	17.11	odd	16
867.2.h.k.712.2		16	17.6	odd	16
2448.2.be.x.1441.3		8	204.83	even	8
2448.2.be.x.1585.3		8	204.179	even	8
2601.2.a.be.1.2		4	51.38	odd	4
2601.2.a.bf.1.2		4	51.47	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.222191 q^{2} +1.00000i q^{3} -1.95063 q^{4} -0.636405i q^{5} -0.222191i q^{6} +1.72844i q^{7} +0.877796 q^{8} -1.00000 q^{9} +0.141404i q^{10} -4.95063i q^{11} -1.95063i q^{12} +2.50625 q^{13} -0.384044i q^{14} +0.636405 q^{15} +3.70622 q^{16} +0.222191 q^{18} +0.950631 q^{19} +1.24139i q^{20} -1.72844 q^{21} +1.09999i q^{22} +2.12220i q^{23} +0.877796i q^{24} +4.59499 q^{25} -0.556867 q^{26} -1.00000i q^{27} -3.37155i q^{28} +9.58704i q^{29} -0.141404 q^{30} +5.27281i q^{31} -2.57908 q^{32} +4.95063 q^{33} +1.09999 q^{35} +1.95063 q^{36} +8.48705i q^{37} -0.211222 q^{38} +2.50625i q^{39} -0.558634i q^{40} -6.92171i q^{41} +0.384044 q^{42} -7.15061 q^{43} +9.65685i q^{44} +0.636405i q^{45} -0.471535i q^{46} +8.10124 q^{47} +3.70622i q^{48} +4.01250 q^{49} -1.02097 q^{50} -4.88877 q^{52} +6.44438 q^{53} +0.222191i q^{54} -3.15061 q^{55} +1.51722i q^{56} +0.950631i q^{57} -2.13016i q^{58} +10.1125 q^{59} -1.24139 q^{60} -2.96983i q^{61} -1.17157i q^{62} -1.72844i q^{63} -6.83940 q^{64} -1.59499i q^{65} -1.09999 q^{66} +7.70129 q^{67} -2.12220 q^{69} -0.244408 q^{70} -0.384044i q^{71} -0.877796 q^{72} +6.51420i q^{73} -1.88575i q^{74} +4.59499i q^{75} -1.85433 q^{76} +8.55687 q^{77} -0.556867i q^{78} +2.37280i q^{79} -2.35866i q^{80} +1.00000 q^{81} +1.53794i q^{82} +13.3581 q^{83} +3.37155 q^{84} +1.58880 q^{86} -9.58704 q^{87} -4.34564i q^{88} -1.98875 q^{89} -0.141404i q^{90} +4.33190i q^{91} -4.13964i q^{92} -5.27281 q^{93} -1.80002 q^{94} -0.604986i q^{95} -2.57908i q^{96} +3.04142i q^{97} -0.891542 q^{98} +4.95063i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 12 q^{4} + 12 q^{8} - 8 q^{9} + 4 q^{13} - 12 q^{15} + 12 q^{16} - 4 q^{18} - 20 q^{19} + 8 q^{21} - 4 q^{25} + 40 q^{26} - 24 q^{30} + 28 q^{32} + 12 q^{33} + 8 q^{35} - 12 q^{36} - 24 q^{38}+ \cdots + 76 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 - 8 * q^9 + 4 * q^13 - 12 * q^15 + 12 * q^16 - 4 * q^18 - 20 * q^19 + 8 * q^21 - 4 * q^25 + 40 * q^26 - 24 * q^30 + 28 * q^32 + 12 * q^33 + 8 * q^35 - 12 * q^36 - 24 * q^38 - 8 * q^42 - 28 * q^43 + 8 * q^47 - 60 * q^50 - 16 * q^52 + 40 * q^53 + 4 * q^55 + 48 * q^59 - 32 * q^60 - 4 * q^64 - 8 * q^66 + 8 * q^67 - 12 * q^69 + 8 * q^70 - 12 * q^72 - 80 * q^76 + 24 * q^77 + 8 * q^81 + 8 * q^83 + 48 * q^84 - 8 * q^86 - 32 * q^87 + 8 * q^89 - 8 * q^93 - 16 * q^94 + 76 * q^98

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