LMFDB - Embedded newform 1734.2.b.l.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1734,2,Mod(577,1734)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1734.577"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1734, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1734 = 2 \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1734.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,8,0,8,0,0,0,8,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$13.8460597105$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 102)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1734.577
Dual form			1734.2.b.l.577.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.26197i q^{5} -1.00000i q^{6} -2.94495i q^{7} +1.00000 q^{8} -1.00000 q^{9} -1.26197i q^{10} -3.29769i q^{11} -1.00000i q^{12} -3.36370 q^{13} -2.94495i q^{14} -1.26197 q^{15} +1.00000 q^{16} -1.00000 q^{18} -7.44155 q^{19} -1.26197i q^{20} -2.94495 q^{21} -3.29769i q^{22} +1.41421i q^{23} -1.00000i q^{24} +3.40743 q^{25} -3.36370 q^{26} +1.00000i q^{27} -2.94495i q^{28} +0.522726i q^{29} -1.26197 q^{30} +7.10973i q^{31} +1.00000 q^{32} -3.29769 q^{33} -3.71644 q^{35} -1.00000 q^{36} -0.792706i q^{37} -7.44155 q^{38} +3.36370i q^{39} -1.26197i q^{40} -4.06306i q^{41} -2.94495 q^{42} -0.867091 q^{43} -3.29769i q^{44} +1.26197i q^{45} +1.41421i q^{46} -10.3086 q^{47} -1.00000i q^{48} -1.67271 q^{49} +3.40743 q^{50} -3.36370 q^{52} +9.98868 q^{53} +1.00000i q^{54} -4.16160 q^{55} -2.94495i q^{56} +7.44155i q^{57} +0.522726i q^{58} +7.31543 q^{59} -1.26197 q^{60} -10.7035i q^{61} +7.10973i q^{62} +2.94495i q^{63} +1.00000 q^{64} +4.24489i q^{65} -3.29769 q^{66} +13.5349 q^{67} +1.41421 q^{69} -3.71644 q^{70} +7.38144i q^{71} -1.00000 q^{72} -5.23880i q^{73} -0.792706i q^{74} -3.40743i q^{75} -7.44155 q^{76} -9.71153 q^{77} +3.36370i q^{78} -10.9449i q^{79} -1.26197i q^{80} +1.00000 q^{81} -4.06306i q^{82} -12.3633 q^{83} -2.94495 q^{84} -0.867091 q^{86} +0.522726 q^{87} -3.29769i q^{88} -17.3975 q^{89} +1.26197i q^{90} +9.90591i q^{91} +1.41421i q^{92} +7.10973 q^{93} -10.3086 q^{94} +9.39104i q^{95} -1.00000i q^{96} -6.22466i q^{97} -1.67271 q^{98} +3.29769i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} - 8 q^{9} + 16 q^{15} + 8 q^{16} - 8 q^{18} - 16 q^{19} - 24 q^{25} + 16 q^{30} + 8 q^{32} - 16 q^{33} - 16 q^{35} - 8 q^{36} - 16 q^{38} - 32 q^{47} - 24 q^{49} - 24 q^{50}+ \cdots - 24 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 - 8 * q^9 + 16 * q^15 + 8 * q^16 - 8 * q^18 - 16 * q^19 - 24 * q^25 + 16 * q^30 + 8 * q^32 - 16 * q^33 - 16 * q^35 - 8 * q^36 - 16 * q^38 - 32 * q^47 - 24 * q^49 - 24 * q^50 + 32 * q^53 + 48 * q^59 + 16 * q^60 + 8 * q^64 - 16 * q^66 + 16 * q^67 - 16 * q^70 - 8 * q^72 - 16 * q^76 + 32 * q^77 + 8 * q^81 + 16 * q^83 + 32 * q^87 - 32 * q^89 + 16 * q^93 - 32 * q^94 - 24 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times$.

$n$	$1157$	$1159$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.00000	0.500000
$5$	− 1.26197i	− 0.564371i	−0.959360	−	0.282186i	$-0.908941\pi$
$5$	− 1.26197i	− 0.564371i	0.959360	−	0.282186i	$-0.0910594\pi$
$6$	− 1.00000i	− 0.408248i
$7$	− 2.94495i	− 1.11309i	−0.830819	−	0.556543i	$-0.812128\pi$
$7$	− 2.94495i	− 1.11309i	0.830819	−	0.556543i	$-0.187872\pi$
$8$	1.00000	0.353553
$9$	−1.00000	−0.333333
$10$	− 1.26197i	− 0.399071i
$11$	− 3.29769i	− 0.994292i	−0.867667	−	0.497146i	$-0.834381\pi$
$11$	− 3.29769i	− 0.994292i	0.867667	−	0.497146i	$-0.165619\pi$
$12$	− 1.00000i	− 0.288675i
$13$	−3.36370	−0.932922	−0.466461	−	0.884542i	$-0.654471\pi$
$13$	−3.36370	−0.932922	−0.466461	+	0.884542i	$0.654471\pi$
$14$	− 2.94495i	− 0.787070i
$15$	−1.26197	−0.325840
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	−7.44155	−1.70721	−0.853605	−	0.520921i	$-0.825588\pi$
$19$	−7.44155	−1.70721	−0.853605	+	0.520921i	$0.825588\pi$
$20$	− 1.26197i	− 0.282186i
$21$	−2.94495	−0.642640
$22$	− 3.29769i	− 0.703071i
$23$	1.41421i	0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$	1.41421i	0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$	− 1.00000i	− 0.204124i
$25$	3.40743	0.681485
$26$	−3.36370	−0.659675
$27$	1.00000i	0.192450i
$28$	− 2.94495i	− 0.556543i
$29$	0.522726i	0.0970678i	0.998822	+	0.0485339i	$0.0154549\pi$
$29$	0.522726i	0.0970678i	−0.998822	+	0.0485339i	$0.984545\pi$
$30$	−1.26197	−0.230404
$31$	7.10973i	1.27695i	0.769644	+	0.638473i	$0.220434\pi$
$31$	7.10973i	1.27695i	−0.769644	+	0.638473i	$0.779566\pi$
$32$	1.00000	0.176777
$33$	−3.29769	−0.574055
$34$	0	0
$35$	−3.71644	−0.628194
$36$	−1.00000	−0.166667
$37$	− 0.792706i	− 0.130320i	−0.997875	−	0.0651601i	$-0.979244\pi$
$37$	− 0.792706i	− 0.130320i	0.997875	−	0.0651601i	$-0.0207558\pi$
$38$	−7.44155	−1.20718
$39$	3.36370i	0.538623i
$40$	− 1.26197i	− 0.199535i
$41$	− 4.06306i	− 0.634543i	−0.948335	−	0.317272i	$-0.897233\pi$
$41$	− 4.06306i	− 0.634543i	0.948335	−	0.317272i	$-0.102767\pi$
$42$	−2.94495	−0.454415
$43$	−0.867091	−0.132230	−0.0661151	−	0.997812i	$-0.521060\pi$
$43$	−0.867091	−0.132230	−0.0661151	+	0.997812i	$0.521060\pi$
$44$	− 3.29769i	− 0.497146i
$45$	1.26197i	0.188124i
$46$	1.41421i	0.208514i
$47$	−10.3086	−1.50367	−0.751835	−	0.659351i	$-0.770831\pi$
$47$	−10.3086	−1.50367	−0.751835	+	0.659351i	$0.770831\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−1.67271	−0.238959
$50$	3.40743	0.481883
$51$	0	0
$52$	−3.36370	−0.466461
$53$	9.98868	1.37205	0.686025	−	0.727578i	$-0.259354\pi$
$53$	9.98868	1.37205	0.686025	+	0.727578i	$0.259354\pi$
$54$	1.00000i	0.136083i
$55$	−4.16160	−0.561150
$56$	− 2.94495i	− 0.393535i
$57$	7.44155i	0.985658i
$58$	0.522726i	0.0686373i
$59$	7.31543	0.952388	0.476194	−	0.879340i	$-0.342016\pi$
$59$	7.31543	0.952388	0.476194	+	0.879340i	$0.342016\pi$
$60$	−1.26197	−0.162920
$61$	− 10.7035i	− 1.37045i	−0.728333	−	0.685223i	$-0.759705\pi$
$61$	− 10.7035i	− 1.37045i	0.728333	−	0.685223i	$-0.240295\pi$
$62$	7.10973i	0.902937i
$63$	2.94495i	0.371028i
$64$	1.00000	0.125000
$65$	4.24489i	0.526514i
$66$	−3.29769	−0.405918
$67$	13.5349	1.65355	0.826775	−	0.562532i	$-0.190173\pi$
$67$	13.5349	1.65355	0.826775	+	0.562532i	$0.190173\pi$
$68$	0	0
$69$	1.41421	0.170251
$70$	−3.71644	−0.444200
$71$	7.38144i	0.876015i	0.898971	+	0.438008i	$0.144316\pi$
$71$	7.38144i	0.876015i	−0.898971	+	0.438008i	$0.855684\pi$
$72$	−1.00000	−0.117851
$73$	− 5.23880i	− 0.613155i	−0.951846	−	0.306577i	$-0.900816\pi$
$73$	− 5.23880i	− 0.613155i	0.951846	−	0.306577i	$-0.0991838\pi$
$74$	− 0.792706i	− 0.0921502i
$75$	− 3.40743i	− 0.393456i
$76$	−7.44155	−0.853605
$77$	−9.71153	−1.10673
$78$	3.36370i	0.380864i
$79$	− 10.9449i	− 1.23140i	−0.787980	−	0.615701i	$-0.788873\pi$
$79$	− 10.9449i	− 1.23140i	0.787980	−	0.615701i	$-0.211127\pi$
$80$	− 1.26197i	− 0.141093i
$81$	1.00000	0.111111
$82$	− 4.06306i	− 0.448690i
$83$	−12.3633	−1.35705	−0.678526	−	0.734577i	$-0.737381\pi$
$83$	−12.3633	−1.35705	−0.678526	+	0.734577i	$0.737381\pi$
$84$	−2.94495	−0.321320
$85$	0	0
$86$	−0.867091	−0.0935008
$87$	0.522726	0.0560421
$88$	− 3.29769i	− 0.351535i
$89$	−17.3975	−1.84413	−0.922063	−	0.387040i	$-0.873498\pi$
$89$	−17.3975	−1.84413	−0.922063	+	0.387040i	$0.873498\pi$
$90$	1.26197i	0.133024i
$91$	9.90591i	1.03842i
$92$	1.41421i	0.147442i
$93$	7.10973	0.737245
$94$	−10.3086	−1.06326
$95$	9.39104i	0.963500i
$96$	− 1.00000i	− 0.102062i
$97$	− 6.22466i	− 0.632018i	−0.948756	−	0.316009i	$-0.897657\pi$
$97$	− 6.22466i	− 0.632018i	0.948756	−	0.316009i	$-0.102343\pi$
$98$	−1.67271	−0.168970
$99$	3.29769i	0.331431i
$100$	3.40743	0.340743
$101$	9.88764	0.983857	0.491929	−	0.870636i	$-0.336292\pi$
$101$	9.88764	0.983857	0.491929	+	0.870636i	$0.336292\pi$
$102$	0	0
$103$	6.54168	0.644571	0.322286	−	0.946642i	$-0.395549\pi$
$103$	6.54168	0.644571	0.322286	+	0.946642i	$0.395549\pi$
$104$	−3.36370	−0.329838
$105$	3.71644i	0.362688i
$106$	9.98868	0.970186
$107$	− 3.49207i	− 0.337591i	−0.985651	−	0.168796i	$-0.946012\pi$
$107$	− 3.49207i	− 0.337591i	0.985651	−	0.168796i	$-0.0539878\pi$
$108$	1.00000i	0.0962250i
$109$	− 15.8751i	− 1.52056i	−0.649596	−	0.760279i	$-0.725062\pi$
$109$	− 15.8751i	− 1.52056i	0.649596	−	0.760279i	$-0.274938\pi$
$110$	−4.16160	−0.396793
$111$	−0.792706	−0.0752404
$112$	− 2.94495i	− 0.278271i
$113$	− 19.3827i	− 1.82337i	−0.410893	−	0.911683i	$-0.634783\pi$
$113$	− 19.3827i	− 1.82337i	0.410893	−	0.911683i	$-0.365217\pi$
$114$	7.44155i	0.696965i
$115$	1.78470	0.166424
$116$	0.522726i	0.0485339i
$117$	3.36370	0.310974
$118$	7.31543	0.673440
$119$	0	0
$120$	−1.26197	−0.115202
$121$	0.125218	0.0113835
$122$	− 10.7035i	− 0.969052i
$123$	−4.06306	−0.366354
$124$	7.10973i	0.638473i
$125$	− 10.6099i	− 0.948982i
$126$	2.94495i	0.262357i
$127$	−17.7830	−1.57798	−0.788992	−	0.614404i	$-0.789397\pi$
$127$	−17.7830	−1.57798	−0.788992	+	0.614404i	$0.789397\pi$
$128$	1.00000	0.0883883
$129$	0.867091i	0.0763431i
$130$	4.24489i	0.372302i
$131$	15.2127i	1.32914i	0.747227	+	0.664569i	$0.231385\pi$
$131$	15.2127i	1.32914i	−0.747227	+	0.664569i	$0.768615\pi$
$132$	−3.29769	−0.287027
$133$	21.9150i	1.90027i
$134$	13.5349	1.16924
$135$	1.26197	0.108613
$136$	0	0
$137$	3.61766	0.309078	0.154539	−	0.987987i	$-0.450611\pi$
$137$	3.61766	0.309078	0.154539	+	0.987987i	$0.450611\pi$
$138$	1.41421	0.120386
$139$	− 20.2446i	− 1.71712i	−0.512713	−	0.858560i	$-0.671359\pi$
$139$	− 20.2446i	− 1.71712i	0.512713	−	0.858560i	$-0.328641\pi$
$140$	−3.71644	−0.314097
$141$	10.3086i	0.868144i
$142$	7.38144i	0.619436i
$143$	11.0924i	0.927596i
$144$	−1.00000	−0.0833333
$145$	0.659666	0.0547823
$146$	− 5.23880i	− 0.433566i
$147$	1.67271i	0.137963i
$148$	− 0.792706i	− 0.0651601i
$149$	7.73418	0.633609	0.316804	−	0.948491i	$-0.397390\pi$
$149$	7.73418	0.633609	0.316804	+	0.948491i	$0.397390\pi$
$150$	− 3.40743i	− 0.278215i
$151$	−1.02280	−0.0832346	−0.0416173	−	0.999134i	$-0.513251\pi$
$151$	−1.02280	−0.0832346	−0.0416173	+	0.999134i	$0.513251\pi$
$152$	−7.44155	−0.603590
$153$	0	0
$154$	−9.71153	−0.782578
$155$	8.97229	0.720671
$156$	3.36370i	0.269311i
$157$	23.7502	1.89547	0.947736	−	0.319055i	$-0.103366\pi$
$157$	23.7502	1.89547	0.947736	+	0.319055i	$0.103366\pi$
$158$	− 10.9449i	− 0.870733i
$159$	− 9.98868i	− 0.792154i
$160$	− 1.26197i	− 0.0997677i
$161$	4.16478	0.328231
$162$	1.00000	0.0785674
$163$	3.22037i	0.252239i	0.992015	+	0.126119i	$0.0402522\pi$
$163$	3.22037i	0.252239i	−0.992015	+	0.126119i	$0.959748\pi$
$164$	− 4.06306i	− 0.317272i
$165$	4.16160i	0.323980i
$166$	−12.3633	−0.959580
$167$	4.68231i	0.362328i	0.983453	+	0.181164i	$0.0579865\pi$
$167$	4.68231i	0.362328i	−0.983453	+	0.181164i	$0.942013\pi$
$168$	−2.94495	−0.227208
$169$	−1.68554	−0.129657
$170$	0	0
$171$	7.44155	0.569070
$172$	−0.867091	−0.0661151
$173$	17.4428i	1.32615i	0.748553	+	0.663075i	$0.230749\pi$
$173$	17.4428i	1.32615i	−0.748553	+	0.663075i	$0.769251\pi$
$174$	0.522726	0.0396278
$175$	− 10.0347i	− 0.758551i
$176$	− 3.29769i	− 0.248573i
$177$	− 7.31543i	− 0.549862i
$178$	−17.3975	−1.30399
$179$	8.76606	0.655206	0.327603	−	0.944815i	$-0.393759\pi$
$179$	8.76606	0.655206	0.327603	+	0.944815i	$0.393759\pi$
$180$	1.26197i	0.0940619i
$181$	3.78273i	0.281168i	0.990069	+	0.140584i	$0.0448981\pi$
$181$	3.78273i	0.281168i	−0.990069	+	0.140584i	$0.955102\pi$
$182$	9.90591i	0.734275i
$183$	−10.7035	−0.791227
$184$	1.41421i	0.104257i
$185$	−1.00037	−0.0735489
$186$	7.10973	0.521311
$187$	0	0
$188$	−10.3086	−0.751835
$189$	2.94495	0.214213
$190$	9.39104i	0.681297i
$191$	−0.371014	−0.0268456	−0.0134228	−	0.999910i	$-0.504273\pi$
$191$	−0.371014	−0.0268456	−0.0134228	+	0.999910i	$0.504273\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 1.04026i	− 0.0748793i	−0.999299	−	0.0374397i	$-0.988080\pi$
$193$	− 1.04026i	− 0.0748793i	0.999299	−	0.0374397i	$-0.0119202\pi$
$194$	− 6.22466i	− 0.446904i
$195$	4.24489	0.303983
$196$	−1.67271	−0.119480
$197$	5.29573i	0.377305i	0.982044	+	0.188652i	$0.0604120\pi$
$197$	5.29573i	0.377305i	−0.982044	+	0.188652i	$0.939588\pi$
$198$	3.29769i	0.234357i
$199$	− 21.2738i	− 1.50806i	−0.656841	−	0.754029i	$-0.728108\pi$
$199$	− 21.2738i	− 1.50806i	0.656841	−	0.754029i	$-0.271892\pi$
$200$	3.40743	0.240941
$201$	− 13.5349i	− 0.954678i
$202$	9.88764	0.695692
$203$	1.53940	0.108045
$204$	0	0
$205$	−5.12747	−0.358118
$206$	6.54168	0.455781
$207$	− 1.41421i	− 0.0982946i
$208$	−3.36370	−0.233230
$209$	24.5400i	1.69746i
$210$	3.71644i	0.256459i
$211$	− 12.9900i	− 0.894270i	−0.894466	−	0.447135i	$-0.852444\pi$
$211$	− 12.9900i	− 0.894270i	0.894466	−	0.447135i	$-0.147556\pi$
$212$	9.98868	0.686025
$213$	7.38144	0.505768
$214$	− 3.49207i	− 0.238713i
$215$	1.09425i	0.0746269i
$216$	1.00000i	0.0680414i
$217$	20.9378	1.42135
$218$	− 15.8751i	− 1.07520i
$219$	−5.23880	−0.354005
$220$	−4.16160	−0.280575
$221$	0	0
$222$	−0.792706	−0.0532030
$223$	3.88573	0.260208	0.130104	−	0.991500i	$-0.458469\pi$
$223$	3.88573	0.260208	0.130104	+	0.991500i	$0.458469\pi$
$224$	− 2.94495i	− 0.196768i
$225$	−3.40743	−0.227162
$226$	− 19.3827i	− 1.28932i
$227$	− 4.38875i	− 0.291292i	−0.989337	−	0.145646i	$-0.953474\pi$
$227$	− 4.38875i	− 0.291292i	0.989337	−	0.145646i	$-0.0465260\pi$
$228$	7.44155i	0.492829i
$229$	3.42063	0.226041	0.113021	−	0.993593i	$-0.463947\pi$
$229$	3.42063	0.226041	0.113021	+	0.993593i	$0.463947\pi$
$230$	1.78470	0.117680
$231$	9.71153i	0.638972i
$232$	0.522726i	0.0343187i
$233$	− 0.893211i	− 0.0585162i	−0.999572	−	0.0292581i	$-0.990686\pi$
$233$	− 0.893211i	− 0.0585162i	0.999572	−	0.0292581i	$-0.00931448\pi$
$234$	3.36370	0.219892
$235$	13.0092i	0.848628i
$236$	7.31543	0.476194
$237$	−10.9449	−0.710951
$238$	0	0
$239$	0.389157	0.0251725	0.0125863	−	0.999921i	$-0.495994\pi$
$239$	0.389157	0.0251725	0.0125863	+	0.999921i	$0.495994\pi$
$240$	−1.26197	−0.0814600
$241$	15.7554i	1.01489i	0.861683	+	0.507447i	$0.169411\pi$
$241$	15.7554i	1.01489i	−0.861683	+	0.507447i	$0.830589\pi$
$242$	0.125218	0.00804934
$243$	− 1.00000i	− 0.0641500i
$244$	− 10.7035i	− 0.685223i
$245$	2.11092i	0.134862i
$246$	−4.06306	−0.259051
$247$	25.0311	1.59269
$248$	7.10973i	0.451468i
$249$	12.3633i	0.783494i
$250$	− 10.6099i	− 0.671032i
$251$	15.2423	0.962083	0.481042	−	0.876698i	$-0.340259\pi$
$251$	15.2423	0.962083	0.481042	+	0.876698i	$0.340259\pi$
$252$	2.94495i	0.185514i
$253$	4.66364	0.293201
$254$	−17.7830	−1.11580
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.05696	0.440201	0.220101	−	0.975477i	$-0.429361\pi$
$257$	7.05696	0.440201	0.220101	+	0.975477i	$0.429361\pi$
$258$	0.867091i	0.0539827i
$259$	−2.33448	−0.145057
$260$	4.24489i	0.263257i
$261$	− 0.522726i	− 0.0323559i
$262$	15.2127i	0.939842i
$263$	−5.53808	−0.341493	−0.170746	−	0.985315i	$-0.554618\pi$
$263$	−5.53808	−0.341493	−0.170746	+	0.985315i	$0.554618\pi$
$264$	−3.29769	−0.202959
$265$	− 12.6054i	− 0.774346i
$266$	21.9150i	1.34369i
$267$	17.3975i	1.06471i
$268$	13.5349	0.826775
$269$	− 4.31477i	− 0.263076i	−0.991311	−	0.131538i	$-0.958008\pi$
$269$	− 4.31477i	− 0.263076i	0.991311	−	0.131538i	$-0.0419916\pi$
$270$	1.26197	0.0768012
$271$	−4.45569	−0.270664	−0.135332	−	0.990800i	$-0.543210\pi$
$271$	−4.45569	−0.270664	−0.135332	+	0.990800i	$0.543210\pi$
$272$	0	0
$273$	9.90591	0.599533
$274$	3.61766	0.218551
$275$	− 11.2366i	− 0.677595i
$276$	1.41421	0.0851257
$277$	14.0382i	0.843471i	0.906719	+	0.421736i	$0.138579\pi$
$277$	14.0382i	0.843471i	−0.906719	+	0.421736i	$0.861421\pi$
$278$	− 20.2446i	− 1.21419i
$279$	− 7.10973i	− 0.425649i
$280$	−3.71644	−0.222100
$281$	−8.67913	−0.517753	−0.258877	−	0.965910i	$-0.583352\pi$
$281$	−8.67913	−0.517753	−0.258877	+	0.965910i	$0.583352\pi$
$282$	10.3086i	0.613871i
$283$	14.3623i	0.853753i	0.904310	+	0.426876i	$0.140386\pi$
$283$	14.3623i	0.853753i	−0.904310	+	0.426876i	$0.859614\pi$
$284$	7.38144i	0.438008i
$285$	9.39104	0.556277
$286$	11.0924i	0.655910i
$287$	−11.9655	−0.706301
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	0.659666	0.0387369
$291$	−6.22466	−0.364896
$292$	− 5.23880i	− 0.306577i
$293$	22.0243	1.28668	0.643338	−	0.765582i	$-0.277549\pi$
$293$	22.0243	1.28668	0.643338	+	0.765582i	$0.277549\pi$
$294$	1.67271i	0.0975547i
$295$	− 9.23188i	− 0.537501i
$296$	− 0.792706i	− 0.0460751i
$297$	3.29769	0.191352
$298$	7.73418	0.448029
$299$	− 4.75699i	− 0.275104i
$300$	− 3.40743i	− 0.196728i
$301$	2.55354i	0.147183i
$302$	−1.02280	−0.0588557
$303$	− 9.88764i	− 0.568030i
$304$	−7.44155	−0.426802
$305$	−13.5076	−0.773440
$306$	0	0
$307$	−7.13291	−0.407097	−0.203548	−	0.979065i	$-0.565247\pi$
$307$	−7.13291	−0.407097	−0.203548	+	0.979065i	$0.565247\pi$
$308$	−9.71153	−0.553366
$309$	− 6.54168i	− 0.372143i
$310$	8.97229	0.509592
$311$	19.1212i	1.08427i	0.840293	+	0.542133i	$0.182383\pi$
$311$	19.1212i	1.08427i	−0.840293	+	0.542133i	$0.817617\pi$
$312$	3.36370i	0.190432i
$313$	13.0225i	0.736076i	0.929811	+	0.368038i	$0.119970\pi$
$313$	13.0225i	0.736076i	−0.929811	+	0.368038i	$0.880030\pi$
$314$	23.7502	1.34030
$315$	3.71644	0.209398
$316$	− 10.9449i	− 0.615701i
$317$	− 4.55648i	− 0.255917i	−0.991779	−	0.127959i	$-0.959157\pi$
$317$	− 4.55648i	− 0.255917i	0.991779	−	0.127959i	$-0.0408425\pi$
$318$	− 9.98868i	− 0.560137i
$319$	1.72379	0.0965137
$320$	− 1.26197i	− 0.0705464i
$321$	−3.49207	−0.194908
$322$	4.16478	0.232094
$323$	0	0
$324$	1.00000	0.0555556
$325$	−11.4615	−0.635772
$326$	3.22037i	0.178360i
$327$	−15.8751	−0.877895
$328$	− 4.06306i	− 0.224345i
$329$	30.3584i	1.67371i
$330$	4.16160i	0.229088i
$331$	18.7047	1.02811	0.514053	−	0.857758i	$-0.328144\pi$
$331$	18.7047	1.02811	0.514053	+	0.857758i	$0.328144\pi$
$332$	−12.3633	−0.678526
$333$	0.792706i	0.0434400i
$334$	4.68231i	0.256205i
$335$	− 17.0807i	− 0.933217i
$336$	−2.94495	−0.160660
$337$	− 23.8756i	− 1.30059i	−0.759683	−	0.650294i	$-0.774646\pi$
$337$	− 23.8756i	− 1.30059i	0.759683	−	0.650294i	$-0.225354\pi$
$338$	−1.68554	−0.0916815
$339$	−19.3827	−1.05272
$340$	0	0
$341$	23.4457	1.26966
$342$	7.44155	0.402393
$343$	− 15.6886i	− 0.847103i
$344$	−0.867091	−0.0467504
$345$	− 1.78470i	− 0.0960850i
$346$	17.4428i	0.937729i
$347$	24.9127i	1.33738i	0.743540	+	0.668692i	$0.233145\pi$
$347$	24.9127i	1.33738i	−0.743540	+	0.668692i	$0.766855\pi$
$348$	0.522726	0.0280211
$349$	19.5194	1.04485	0.522425	−	0.852685i	$-0.325027\pi$
$349$	19.5194	1.04485	0.522425	+	0.852685i	$0.325027\pi$
$350$	− 10.0347i	− 0.536377i
$351$	− 3.36370i	− 0.179541i
$352$	− 3.29769i	− 0.175768i
$353$	−25.7115	−1.36849	−0.684243	−	0.729254i	$-0.739867\pi$
$353$	−25.7115	−1.36849	−0.684243	+	0.729254i	$0.739867\pi$
$354$	− 7.31543i	− 0.388811i
$355$	9.31517	0.494398
$356$	−17.3975	−0.922063
$357$	0	0
$358$	8.76606	0.463301
$359$	−19.2482	−1.01588	−0.507939	−	0.861393i	$-0.669593\pi$
$359$	−19.2482	−1.01588	−0.507939	+	0.861393i	$0.669593\pi$
$360$	1.26197i	0.0665118i
$361$	36.3767	1.91456
$362$	3.78273i	0.198816i
$363$	− 0.125218i	− 0.00657225i
$364$	9.90591i	0.519211i
$365$	−6.61122	−0.346047
$366$	−10.7035	−0.559482
$367$	− 5.08708i	− 0.265544i	−0.991147	−	0.132772i	$-0.957612\pi$
$367$	− 5.08708i	− 0.265544i	0.991147	−	0.132772i	$-0.0423877\pi$
$368$	1.41421i	0.0737210i
$369$	4.06306i	0.211514i
$370$	−1.00037	−0.0520070
$371$	− 29.4161i	− 1.52721i
$372$	7.10973	0.368622
$373$	21.9719	1.13766	0.568831	−	0.822454i	$-0.307396\pi$
$373$	21.9719	1.13766	0.568831	+	0.822454i	$0.307396\pi$
$374$	0	0
$375$	−10.6099	−0.547895
$376$	−10.3086	−0.531628
$377$	− 1.75829i	− 0.0905567i
$378$	2.94495	0.151472
$379$	− 10.0798i	− 0.517763i	−0.965909	−	0.258881i	$-0.916646\pi$
$379$	− 10.0798i	− 0.517763i	0.965909	−	0.258881i	$-0.0833539\pi$
$380$	9.39104i	0.481750i
$381$	17.7830i	0.911049i
$382$	−0.371014	−0.0189827
$383$	27.9738	1.42940	0.714698	−	0.699433i	$-0.246564\pi$
$383$	27.9738	1.42940	0.714698	+	0.699433i	$0.246564\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	12.2557i	0.624608i
$386$	− 1.04026i	− 0.0529477i
$387$	0.867091	0.0440767
$388$	− 6.22466i	− 0.316009i
$389$	12.3355	0.625432	0.312716	−	0.949847i	$-0.398761\pi$
$389$	12.3355	0.625432	0.312716	+	0.949847i	$0.398761\pi$
$390$	4.24489	0.214949
$391$	0	0
$392$	−1.67271	−0.0844848
$393$	15.2127	0.767378
$394$	5.29573i	0.266795i
$395$	−13.8122	−0.694968
$396$	3.29769i	0.165715i
$397$	− 7.10626i	− 0.356653i	−0.983971	−	0.178326i	$-0.942932\pi$
$397$	− 7.10626i	− 0.356653i	0.983971	−	0.178326i	$-0.0570683\pi$
$398$	− 21.2738i	− 1.06636i
$399$	21.9150	1.09712
$400$	3.40743	0.170371
$401$	29.6691i	1.48160i	0.671725	+	0.740801i	$0.265554\pi$
$401$	29.6691i	1.48160i	−0.671725	+	0.740801i	$0.734446\pi$
$402$	− 13.5349i	− 0.675059i
$403$	− 23.9150i	− 1.19129i
$404$	9.88764	0.491929
$405$	− 1.26197i	− 0.0627079i
$406$	1.53940	0.0763992
$407$	−2.61410	−0.129576
$408$	0	0
$409$	14.8922	0.736371	0.368185	−	0.929752i	$-0.379979\pi$
$409$	14.8922	0.736371	0.368185	+	0.929752i	$0.379979\pi$
$410$	−5.12747	−0.253228
$411$	− 3.61766i	− 0.178446i
$412$	6.54168	0.322286
$413$	− 21.5436i	− 1.06009i
$414$	− 1.41421i	− 0.0695048i
$415$	15.6022i	0.765881i
$416$	−3.36370	−0.164919
$417$	−20.2446	−0.991380
$418$	24.5400i	1.20029i
$419$	26.2549i	1.28264i	0.767274	+	0.641319i	$0.221612\pi$
$419$	26.2549i	1.28264i	−0.767274	+	0.641319i	$0.778388\pi$
$420$	3.71644i	0.181344i
$421$	16.6683	0.812365	0.406182	−	0.913792i	$-0.366860\pi$
$421$	16.6683	0.812365	0.406182	+	0.913792i	$0.366860\pi$
$422$	− 12.9900i	− 0.632345i
$423$	10.3086	0.501223
$424$	9.98868	0.485093
$425$	0	0
$426$	7.38144	0.357632
$427$	−31.5213	−1.52542
$428$	− 3.49207i	− 0.168796i
$429$	11.0924	0.535548
$430$	1.09425i	0.0527692i
$431$	− 4.86443i	− 0.234312i	−0.993114	−	0.117156i	$-0.962622\pi$
$431$	− 4.86443i	− 0.234312i	0.993114	−	0.117156i	$-0.0373777\pi$
$432$	1.00000i	0.0481125i
$433$	−2.20663	−0.106044	−0.0530220	−	0.998593i	$-0.516885\pi$
$433$	−2.20663	−0.106044	−0.0530220	+	0.998593i	$0.516885\pi$
$434$	20.9378	1.00505
$435$	− 0.659666i	− 0.0316286i
$436$	− 15.8751i	− 0.760279i
$437$	− 10.5239i	− 0.503429i
$438$	−5.23880	−0.250319
$439$	9.89180i	0.472110i	0.971740	+	0.236055i	$0.0758546\pi$
$439$	9.89180i	0.472110i	−0.971740	+	0.236055i	$0.924145\pi$
$440$	−4.16160	−0.198396
$441$	1.67271	0.0796531
$442$	0	0
$443$	5.50565	0.261581	0.130791	−	0.991410i	$-0.458248\pi$
$443$	5.50565	0.261581	0.130791	+	0.991410i	$0.458248\pi$
$444$	−0.792706	−0.0376202
$445$	21.9551i	1.04077i
$446$	3.88573	0.183995
$447$	− 7.73418i	− 0.365814i
$448$	− 2.94495i	− 0.139136i
$449$	33.2572i	1.56951i	0.619808	+	0.784753i	$0.287210\pi$
$449$	33.2572i	1.56951i	−0.619808	+	0.784753i	$0.712790\pi$
$450$	−3.40743	−0.160628
$451$	−13.3987	−0.630921
$452$	− 19.3827i	− 0.911683i
$453$	1.02280i	0.0480555i
$454$	− 4.38875i	− 0.205974i
$455$	12.5010	0.586055
$456$	7.44155i	0.348483i
$457$	17.4812	0.817736	0.408868	−	0.912594i	$-0.365924\pi$
$457$	17.4812	0.817736	0.408868	+	0.912594i	$0.365924\pi$
$458$	3.42063	0.159835
$459$	0	0
$460$	1.78470	0.0832120
$461$	14.5385	0.677123	0.338562	−	0.940944i	$-0.390060\pi$
$461$	14.5385	0.677123	0.338562	+	0.940944i	$0.390060\pi$
$462$	9.71153i	0.451821i
$463$	−1.47703	−0.0686435	−0.0343217	−	0.999411i	$-0.510927\pi$
$463$	−1.47703	−0.0686435	−0.0343217	+	0.999411i	$0.510927\pi$
$464$	0.522726i	0.0242670i
$465$	− 8.97229i	− 0.416080i
$466$	− 0.893211i	− 0.0413772i
$467$	−8.68873	−0.402066	−0.201033	−	0.979584i	$-0.564430\pi$
$467$	−8.68873	−0.402066	−0.201033	+	0.979584i	$0.564430\pi$
$468$	3.36370	0.155487
$469$	− 39.8596i	− 1.84054i
$470$	13.0092i	0.600071i
$471$	− 23.7502i	− 1.09435i
$472$	7.31543	0.336720
$473$	2.85940i	0.131475i
$474$	−10.9449	−0.502718
$475$	−25.3565	−1.16344
$476$	0	0
$477$	−9.98868	−0.457350
$478$	0.389157	0.0177996
$479$	10.9463i	0.500151i	0.968226	+	0.250075i	$0.0804554\pi$
$479$	10.9463i	0.500151i	−0.968226	+	0.250075i	$0.919545\pi$
$480$	−1.26197	−0.0576009
$481$	2.66642i	0.121578i
$482$	15.7554i	0.717638i
$483$	− 4.16478i	− 0.189504i
$484$	0.125218	0.00569174
$485$	−7.85535	−0.356693
$486$	− 1.00000i	− 0.0453609i
$487$	8.75751i	0.396841i	0.980117	+	0.198420i	$0.0635811\pi$
$487$	8.75751i	0.396841i	−0.980117	+	0.198420i	$0.936419\pi$
$488$	− 10.7035i	− 0.484526i
$489$	3.22037	0.145630
$490$	2.11092i	0.0953616i
$491$	−30.4170	−1.37270	−0.686351	−	0.727271i	$-0.740788\pi$
$491$	−30.4170	−1.37270	−0.686351	+	0.727271i	$0.740788\pi$
$492$	−4.06306	−0.183177
$493$	0	0
$494$	25.0311	1.12620
$495$	4.16160	0.187050
$496$	7.10973i	0.319236i
$497$	21.7379	0.975080
$498$	12.3633i	0.554014i
$499$	− 22.3333i	− 0.999777i	−0.866090	−	0.499889i	$-0.833374\pi$
$499$	− 22.3333i	− 0.999777i	0.866090	−	0.499889i	$-0.166626\pi$
$500$	− 10.6099i	− 0.474491i
$501$	4.68231	0.209190
$502$	15.2423	0.680296
$503$	42.3520i	1.88838i	0.329398	+	0.944191i	$0.393154\pi$
$503$	42.3520i	1.88838i	−0.329398	+	0.944191i	$0.606846\pi$
$504$	2.94495i	0.131178i
$505$	− 12.4779i	− 0.555261i
$506$	4.66364	0.207324
$507$	1.68554i	0.0748576i
$508$	−17.7830	−0.788992
$509$	−9.03882	−0.400639	−0.200319	−	0.979731i	$-0.564198\pi$
$509$	−9.03882	−0.400639	−0.200319	+	0.979731i	$0.564198\pi$
$510$	0	0
$511$	−15.4280	−0.682493
$512$	1.00000	0.0441942
$513$	− 7.44155i	− 0.328553i
$514$	7.05696	0.311269
$515$	− 8.25543i	− 0.363778i
$516$	0.867091i	0.0381716i
$517$	33.9947i	1.49509i
$518$	−2.33448	−0.102571
$519$	17.4428	0.765653
$520$	4.24489i	0.186151i
$521$	− 33.4356i	− 1.46484i	−0.680853	−	0.732420i	$-0.738391\pi$
$521$	− 33.4356i	− 1.46484i	0.680853	−	0.732420i	$-0.261609\pi$
$522$	− 0.522726i	− 0.0228791i
$523$	−23.9751	−1.04836	−0.524180	−	0.851608i	$-0.675628\pi$
$523$	−23.9751	−1.04836	−0.524180	+	0.851608i	$0.675628\pi$
$524$	15.2127i	0.664569i
$525$	−10.0347	−0.437950
$526$	−5.53808	−0.241472
$527$	0	0
$528$	−3.29769	−0.143514
$529$	21.0000	0.913043
$530$	− 12.6054i	− 0.547545i
$531$	−7.31543	−0.317463
$532$	21.9150i	0.950135i
$533$	13.6669i	0.591979i
$534$	17.3975i	0.752861i
$535$	−4.40690	−0.190527
$536$	13.5349	0.584618
$537$	− 8.76606i	− 0.378283i
$538$	− 4.31477i	− 0.186023i
$539$	5.51610i	0.237595i
$540$	1.26197	0.0543067
$541$	− 17.5884i	− 0.756182i	−0.925768	−	0.378091i	$-0.876581\pi$
$541$	− 17.5884i	− 0.756182i	0.925768	−	0.378091i	$-0.123419\pi$
$542$	−4.45569	−0.191388
$543$	3.78273	0.162333
$544$	0	0
$545$	−20.0339	−0.858160
$546$	9.90591	0.423934
$547$	4.79884i	0.205183i	0.994724	+	0.102592i	$0.0327135\pi$
$547$	4.79884i	0.205183i	−0.994724	+	0.102592i	$0.967286\pi$
$548$	3.61766	0.154539
$549$	10.7035i	0.456815i
$550$	− 11.2366i	− 0.479132i
$551$	− 3.88989i	− 0.165715i
$552$	1.41421	0.0601929
$553$	−32.2323	−1.37066
$554$	14.0382i	0.596424i
$555$	1.00037i	0.0424635i
$556$	− 20.2446i	− 0.858560i
$557$	−27.0366	−1.14558	−0.572788	−	0.819704i	$-0.694138\pi$
$557$	−27.0366	−1.14558	−0.572788	+	0.819704i	$0.694138\pi$
$558$	− 7.10973i	− 0.300979i
$559$	2.91663	0.123360
$560$	−3.71644	−0.157048
$561$	0	0
$562$	−8.67913	−0.366107
$563$	−38.3143	−1.61475	−0.807377	−	0.590036i	$-0.799114\pi$
$563$	−38.3143	−1.61475	−0.807377	+	0.590036i	$0.799114\pi$
$564$	10.3086i	0.434072i
$565$	−24.4604	−1.02906
$566$	14.3623i	0.603694i
$567$	− 2.94495i	− 0.123676i
$568$	7.38144i	0.309718i
$569$	−18.6210	−0.780633	−0.390317	−	0.920681i	$-0.627634\pi$
$569$	−18.6210	−0.780633	−0.390317	+	0.920681i	$0.627634\pi$
$570$	9.39104	0.393347
$571$	− 3.86349i	− 0.161682i	−0.996727	−	0.0808410i	$-0.974239\pi$
$571$	− 3.86349i	− 0.161682i	0.996727	−	0.0808410i	$-0.0257606\pi$
$572$	11.0924i	0.463798i
$573$	0.371014i	0.0154993i
$574$	−11.9655	−0.499430
$575$	4.81883i	0.200959i
$576$	−1.00000	−0.0416667
$577$	−25.4772	−1.06063	−0.530315	−	0.847801i	$-0.677926\pi$
$577$	−25.4772	−1.06063	−0.530315	+	0.847801i	$0.677926\pi$
$578$	0	0
$579$	−1.04026	−0.0432316
$580$	0.659666	0.0273911
$581$	36.4093i	1.51051i
$582$	−6.22466	−0.258020
$583$	− 32.9396i	− 1.36422i
$584$	− 5.23880i	− 0.216783i
$585$	− 4.24489i	− 0.175505i
$586$	22.0243	0.909817
$587$	0.531636	0.0219430	0.0109715	−	0.999940i	$-0.496508\pi$
$587$	0.531636	0.0219430	0.0109715	+	0.999940i	$0.496508\pi$
$588$	1.67271i	0.0689816i
$589$	− 52.9074i	− 2.18001i
$590$	− 9.23188i	− 0.380070i
$591$	5.29573	0.217837
$592$	− 0.792706i	− 0.0325800i
$593$	12.1266	0.497982	0.248991	−	0.968506i	$-0.419901\pi$
$593$	12.1266	0.497982	0.248991	+	0.968506i	$0.419901\pi$
$594$	3.29769	0.135306
$595$	0	0
$596$	7.73418	0.316804
$597$	−21.2738	−0.870677
$598$	− 4.75699i	− 0.194528i
$599$	14.6476	0.598486	0.299243	−	0.954177i	$-0.403266\pi$
$599$	14.6476	0.598486	0.299243	+	0.954177i	$0.403266\pi$
$600$	− 3.40743i	− 0.139108i
$601$	26.2961i	1.07264i	0.844015	+	0.536320i	$0.180186\pi$
$601$	26.2961i	1.07264i	−0.844015	+	0.536320i	$0.819814\pi$
$602$	2.55354i	0.104074i
$603$	−13.5349	−0.551184
$604$	−1.02280	−0.0416173
$605$	− 0.158022i	− 0.00642451i
$606$	− 9.88764i	− 0.401658i
$607$	24.3315i	0.987584i	0.869580	+	0.493792i	$0.164390\pi$
$607$	24.3315i	0.987584i	−0.869580	+	0.493792i	$0.835610\pi$
$608$	−7.44155	−0.301795
$609$	− 1.53940i	− 0.0623797i
$610$	−13.5076	−0.546905
$611$	34.6752	1.40281
$612$	0	0
$613$	−8.21114	−0.331645	−0.165822	−	0.986156i	$-0.553028\pi$
$613$	−8.21114	−0.331645	−0.165822	+	0.986156i	$0.553028\pi$
$614$	−7.13291	−0.287861
$615$	5.12747i	0.206760i
$616$	−9.71153	−0.391289
$617$	44.5518i	1.79359i	0.442450	+	0.896793i	$0.354109\pi$
$617$	44.5518i	1.79359i	−0.442450	+	0.896793i	$0.645891\pi$
$618$	− 6.54168i	− 0.263145i
$619$	− 21.8112i	− 0.876668i	−0.898812	−	0.438334i	$-0.855569\pi$
$619$	− 21.8112i	− 0.876668i	0.898812	−	0.438334i	$-0.144431\pi$
$620$	8.97229	0.360336
$621$	−1.41421	−0.0567504
$622$	19.1212i	0.766692i
$623$	51.2346i	2.05267i
$624$	3.36370i	0.134656i
$625$	3.64767	0.145907
$626$	13.0225i	0.520484i
$627$	24.5400	0.980032
$628$	23.7502	0.947736
$629$	0	0
$630$	3.71644	0.148067
$631$	41.7021	1.66014	0.830068	−	0.557663i	$-0.188302\pi$
$631$	41.7021	1.66014	0.830068	+	0.557663i	$0.188302\pi$
$632$	− 10.9449i	− 0.435367i
$633$	−12.9900	−0.516307
$634$	− 4.55648i	− 0.180961i
$635$	22.4416i	0.890569i
$636$	− 9.98868i	− 0.396077i
$637$	5.62650	0.222930
$638$	1.72379	0.0682455
$639$	− 7.38144i	− 0.292005i
$640$	− 1.26197i	− 0.0498838i
$641$	13.7067i	0.541381i	0.962666	+	0.270691i	$0.0872521\pi$
$641$	13.7067i	0.541381i	−0.962666	+	0.270691i	$0.912748\pi$
$642$	−3.49207	−0.137821
$643$	− 21.8390i	− 0.861247i	−0.902532	−	0.430624i	$-0.858294\pi$
$643$	− 21.8390i	− 0.861247i	0.902532	−	0.430624i	$-0.141706\pi$
$644$	4.16478	0.164115
$645$	1.09425	0.0430859
$646$	0	0
$647$	−21.2760	−0.836447	−0.418223	−	0.908344i	$-0.637347\pi$
$647$	−21.2760	−0.836447	−0.418223	+	0.908344i	$0.637347\pi$
$648$	1.00000	0.0392837
$649$	− 24.1241i	− 0.946952i
$650$	−11.4615	−0.449559
$651$	− 20.9378i	− 0.820616i
$652$	3.22037i	0.126119i
$653$	− 33.1843i	− 1.29860i	−0.760531	−	0.649301i	$-0.775062\pi$
$653$	− 33.1843i	− 1.29860i	0.760531	−	0.649301i	$-0.224938\pi$
$654$	−15.8751	−0.620765
$655$	19.1980	0.750127
$656$	− 4.06306i	− 0.158636i
$657$	5.23880i	0.204385i
$658$	30.3584i	1.18349i
$659$	−0.649089	−0.0252849	−0.0126425	−	0.999920i	$-0.504024\pi$
$659$	−0.649089	−0.0252849	−0.0126425	+	0.999920i	$0.504024\pi$
$660$	4.16160i	0.161990i
$661$	30.8996	1.20185	0.600927	−	0.799304i	$-0.294798\pi$
$661$	30.8996	1.20185	0.600927	+	0.799304i	$0.294798\pi$
$662$	18.7047	0.726981
$663$	0	0
$664$	−12.3633	−0.479790
$665$	27.6561	1.07246
$666$	0.792706i	0.0307167i
$667$	−0.739246	−0.0286237
$668$	4.68231i	0.181164i
$669$	− 3.88573i	− 0.150231i
$670$	− 17.0807i	− 0.659884i
$671$	−35.2969	−1.36262
$672$	−2.94495	−0.113604
$673$	30.3508i	1.16994i	0.811056	+	0.584968i	$0.198893\pi$
$673$	30.3508i	1.16994i	−0.811056	+	0.584968i	$0.801107\pi$
$674$	− 23.8756i	− 0.919655i
$675$	3.40743i	0.131152i
$676$	−1.68554	−0.0648286
$677$	− 6.53702i	− 0.251238i	−0.992079	−	0.125619i	$-0.959908\pi$
$677$	− 6.53702i	− 0.251238i	0.992079	−	0.125619i	$-0.0400917\pi$
$678$	−19.3827	−0.744386
$679$	−18.3313	−0.703490
$680$	0	0
$681$	−4.38875	−0.168177
$682$	23.4457	0.897783
$683$	10.6469i	0.407392i	0.979034	+	0.203696i	$0.0652953\pi$
$683$	10.6469i	0.407392i	−0.979034	+	0.203696i	$0.934705\pi$
$684$	7.44155	0.284535
$685$	− 4.56539i	− 0.174435i
$686$	− 15.6886i	− 0.598993i
$687$	− 3.42063i	− 0.130505i
$688$	−0.867091	−0.0330575
$689$	−33.5989	−1.28002
$690$	− 1.78470i	− 0.0679423i
$691$	19.7603i	0.751718i	0.926677	+	0.375859i	$0.122652\pi$
$691$	19.7603i	0.751718i	−0.926677	+	0.375859i	$0.877348\pi$
$692$	17.4428i	0.663075i
$693$	9.71153	0.368911
$694$	24.9127i	0.945673i
$695$	−25.5481	−0.969093
$696$	0.522726	0.0198139
$697$	0	0
$698$	19.5194	0.738821
$699$	−0.893211	−0.0337844
$700$	− 10.0347i	− 0.379276i
$701$	39.7169	1.50009	0.750043	−	0.661390i	$-0.230033\pi$
$701$	39.7169	1.50009	0.750043	+	0.661390i	$0.230033\pi$
$702$	− 3.36370i	− 0.126955i
$703$	5.89897i	0.222484i
$704$	− 3.29769i	− 0.124286i
$705$	13.0092	0.489956
$706$	−25.7115	−0.967666
$707$	− 29.1186i	− 1.09512i
$708$	− 7.31543i	− 0.274931i
$709$	44.9863i	1.68950i	0.535163	+	0.844749i	$0.320250\pi$
$709$	44.9863i	1.68950i	−0.535163	+	0.844749i	$0.679750\pi$
$710$	9.31517	0.349592
$711$	10.9449i	0.410468i
$712$	−17.3975	−0.651997
$713$	−10.0547	−0.376551
$714$	0	0
$715$	13.9984	0.523509
$716$	8.76606	0.327603
$717$	− 0.389157i	− 0.0145334i
$718$	−19.2482	−0.718335
$719$	− 34.9648i	− 1.30397i	−0.758233	−	0.651983i	$-0.773937\pi$
$719$	− 34.9648i	− 1.30397i	0.758233	−	0.651983i	$-0.226063\pi$
$720$	1.26197i	0.0470309i
$721$	− 19.2649i	− 0.717463i
$722$	36.3767	1.35380
$723$	15.7554	0.585949
$724$	3.78273i	0.140584i
$725$	1.78115i	0.0661503i
$726$	− 0.125218i	− 0.00464729i
$727$	−9.34543	−0.346603	−0.173301	−	0.984869i	$-0.555443\pi$
$727$	−9.34543	−0.346603	−0.173301	+	0.984869i	$0.555443\pi$
$728$	9.90591i	0.367137i
$729$	−1.00000	−0.0370370
$730$	−6.61122	−0.244692
$731$	0	0
$732$	−10.7035	−0.395614
$733$	−35.0025	−1.29285	−0.646423	−	0.762979i	$-0.723736\pi$
$733$	−35.0025	−1.29285	−0.646423	+	0.762979i	$0.723736\pi$
$734$	− 5.08708i	− 0.187768i
$735$	2.11092	0.0778625
$736$	1.41421i	0.0521286i
$737$	− 44.6339i	− 1.64411i
$738$	4.06306i	0.149563i
$739$	−20.3466	−0.748460	−0.374230	−	0.927336i	$-0.622093\pi$
$739$	−20.3466	−0.748460	−0.374230	+	0.927336i	$0.622093\pi$
$740$	−1.00037	−0.0367745
$741$	− 25.0311i	− 0.919541i
$742$	− 29.4161i	− 1.07990i
$743$	19.1359i	0.702027i	0.936370	+	0.351014i	$0.114163\pi$
$743$	19.1359i	0.702027i	−0.936370	+	0.351014i	$0.885837\pi$
$744$	7.10973	0.260655
$745$	− 9.76033i	− 0.357591i
$746$	21.9719	0.804449
$747$	12.3633	0.452350
$748$	0	0
$749$	−10.2840	−0.375768
$750$	−10.6099	−0.387420
$751$	− 42.2934i	− 1.54331i	−0.636043	−	0.771654i	$-0.719430\pi$
$751$	− 42.2934i	− 1.54331i	0.636043	−	0.771654i	$-0.280570\pi$
$752$	−10.3086	−0.375918
$753$	− 15.2423i	− 0.555459i
$754$	− 1.75829i	− 0.0640332i
$755$	1.29075i	0.0469752i
$756$	2.94495	0.107107
$757$	−46.3076	−1.68308	−0.841538	−	0.540197i	$-0.818350\pi$
$757$	−46.3076	−1.68308	−0.841538	+	0.540197i	$0.818350\pi$
$758$	− 10.0798i	− 0.366114i
$759$	− 4.66364i	− 0.169280i
$760$	9.39104i	0.340649i
$761$	−2.15386	−0.0780775	−0.0390388	−	0.999238i	$-0.512430\pi$
$761$	−2.15386	−0.0780775	−0.0390388	+	0.999238i	$0.512430\pi$
$762$	17.7830i	0.644209i
$763$	−46.7513	−1.69251
$764$	−0.371014	−0.0134228
$765$	0	0
$766$	27.9738	1.01074
$767$	−24.6069	−0.888504
$768$	− 1.00000i	− 0.0360844i
$769$	−22.8143	−0.822705	−0.411353	−	0.911476i	$-0.634944\pi$
$769$	−22.8143	−0.822705	−0.411353	+	0.911476i	$0.634944\pi$
$770$	12.2557i	0.441664i
$771$	− 7.05696i	− 0.254150i
$772$	− 1.04026i	− 0.0374397i
$773$	−14.4131	−0.518403	−0.259201	−	0.965823i	$-0.583459\pi$
$773$	−14.4131	−0.518403	−0.259201	+	0.965823i	$0.583459\pi$
$774$	0.867091	0.0311669
$775$	24.2259i	0.870219i
$776$	− 6.22466i	− 0.223452i
$777$	2.33448i	0.0837489i
$778$	12.3355	0.442247
$779$	30.2355i	1.08330i
$780$	4.24489	0.151992
$781$	24.3417	0.871015
$782$	0	0
$783$	−0.522726	−0.0186807
$784$	−1.67271	−0.0597398
$785$	− 29.9721i	− 1.06975i
$786$	15.2127	0.542618
$787$	− 1.65367i	− 0.0589469i	−0.999566	−	0.0294735i	$-0.990617\pi$
$787$	− 1.65367i	− 0.0589469i	0.999566	−	0.0294735i	$-0.00938305\pi$
$788$	5.29573i	0.188652i
$789$	5.53808i	0.197161i
$790$	−13.8122	−0.491417
$791$	−57.0809	−2.02956
$792$	3.29769i	0.117178i
$793$	36.0034i	1.27852i
$794$	− 7.10626i	− 0.252192i
$795$	−12.6054	−0.447069
$796$	− 21.2738i	− 0.754029i
$797$	37.4699	1.32725	0.663625	−	0.748065i	$-0.269017\pi$
$797$	37.4699	1.32725	0.663625	+	0.748065i	$0.269017\pi$
$798$	21.9150	0.775782
$799$	0	0
$800$	3.40743	0.120471
$801$	17.3975	0.614709
$802$	29.6691i	1.04765i
$803$	−17.2759	−0.609655
$804$	− 13.5349i	− 0.477339i
$805$	− 5.25584i	− 0.185244i
$806$	− 23.9150i	− 0.842369i
$807$	−4.31477	−0.151887
$808$	9.88764	0.347846
$809$	− 51.2959i	− 1.80347i	−0.432291	−	0.901734i	$-0.642295\pi$
$809$	− 51.2959i	− 1.80347i	0.432291	−	0.901734i	$-0.357705\pi$
$810$	− 1.26197i	− 0.0443412i
$811$	1.19528i	0.0419719i	0.999780	+	0.0209860i	$0.00668053\pi$
$811$	1.19528i	0.0419719i	−0.999780	+	0.0209860i	$0.993319\pi$
$812$	1.53940	0.0540224
$813$	4.45569i	0.156268i
$814$	−2.61410	−0.0916242
$815$	4.06401	0.142356
$816$	0	0
$817$	6.45250	0.225745
$818$	14.8922	0.520693
$819$	− 9.90591i	− 0.346140i
$820$	−5.12747	−0.179059
$821$	− 14.3412i	− 0.500511i	−0.968180	−	0.250255i	$-0.919485\pi$
$821$	− 14.3412i	− 0.500511i	0.968180	−	0.250255i	$-0.0805146\pi$
$822$	− 3.61766i	− 0.126180i
$823$	− 45.4114i	− 1.58294i	−0.611207	−	0.791471i	$-0.709316\pi$
$823$	− 45.4114i	− 1.58294i	0.611207	−	0.791471i	$-0.290684\pi$
$824$	6.54168	0.227890
$825$	−11.2366	−0.391210
$826$	− 21.5436i	− 0.749596i
$827$	− 23.4799i	− 0.816475i	−0.912876	−	0.408238i	$-0.866143\pi$
$827$	− 23.4799i	− 0.816475i	0.912876	−	0.408238i	$-0.133857\pi$
$828$	− 1.41421i	− 0.0491473i
$829$	−4.28183	−0.148714	−0.0743571	−	0.997232i	$-0.523690\pi$
$829$	−4.28183	−0.148714	−0.0743571	+	0.997232i	$0.523690\pi$
$830$	15.6022i	0.541559i
$831$	14.0382	0.486978
$832$	−3.36370	−0.116615
$833$	0	0
$834$	−20.2446	−0.701011
$835$	5.90895	0.204488
$836$	24.5400i	0.848732i
$837$	−7.10973	−0.245748
$838$	26.2549i	0.906962i
$839$	29.6465i	1.02351i	0.859131	+	0.511756i	$0.171005\pi$
$839$	29.6465i	1.02351i	−0.859131	+	0.511756i	$0.828995\pi$
$840$	3.71644i	0.128229i
$841$	28.7268	0.990578
$842$	16.6683	0.574429
$843$	8.67913i	0.298925i
$844$	− 12.9900i	− 0.447135i
$845$	2.12711i	0.0731748i
$846$	10.3086	0.354418
$847$	− 0.368761i	− 0.0126708i
$848$	9.98868	0.343013
$849$	14.3623	0.492914
$850$	0	0
$851$	1.12106	0.0384293
$852$	7.38144	0.252884
$853$	− 1.16274i	− 0.0398116i	−0.999802	−	0.0199058i	$-0.993663\pi$
$853$	− 1.16274i	− 0.0398116i	0.999802	−	0.0199058i	$-0.00633663\pi$
$854$	−31.5213	−1.07864
$855$	− 9.39104i	− 0.321167i
$856$	− 3.49207i	− 0.119356i
$857$	− 38.8166i	− 1.32595i	−0.748642	−	0.662975i	$-0.769294\pi$
$857$	− 38.8166i	− 1.32595i	0.748642	−	0.662975i	$-0.230706\pi$
$858$	11.0924	0.378690
$859$	−11.0615	−0.377412	−0.188706	−	0.982034i	$-0.560429\pi$
$859$	−11.0615	−0.377412	−0.188706	+	0.982034i	$0.560429\pi$
$860$	1.09425i	0.0373134i
$861$	11.9655i	0.407783i
$862$	− 4.86443i	− 0.165683i
$863$	8.01331	0.272776	0.136388	−	0.990655i	$-0.456451\pi$
$863$	8.01331	0.272776	0.136388	+	0.990655i	$0.456451\pi$
$864$	1.00000i	0.0340207i
$865$	22.0123	0.748441
$866$	−2.20663	−0.0749845
$867$	0	0
$868$	20.9378	0.710675
$869$	−36.0931	−1.22437
$870$	− 0.659666i	− 0.0223648i
$871$	−45.5273	−1.54263
$872$	− 15.8751i	− 0.537599i
$873$	6.22466i	0.210673i
$874$	− 10.5239i	− 0.355978i
$875$	−31.2457	−1.05630
$876$	−5.23880	−0.177002
$877$	− 14.8632i	− 0.501894i	−0.968001	−	0.250947i	$-0.919258\pi$
$877$	− 14.8632i	− 0.501894i	0.968001	−	0.250947i	$-0.0807419\pi$
$878$	9.89180i	0.333832i
$879$	− 22.0243i	− 0.742863i
$880$	−4.16160	−0.140287
$881$	9.72089i	0.327505i	0.986501	+	0.163753i	$0.0523599\pi$
$881$	9.72089i	0.327505i	−0.986501	+	0.163753i	$0.947640\pi$
$882$	1.67271	0.0563232
$883$	21.2137	0.713896	0.356948	−	0.934124i	$-0.383817\pi$
$883$	21.2137	0.713896	0.356948	+	0.934124i	$0.383817\pi$
$884$	0	0
$885$	−9.23188	−0.310326
$886$	5.50565	0.184966
$887$	− 1.24492i	− 0.0418005i	−0.999782	−	0.0209002i	$-0.993347\pi$
$887$	− 1.24492i	− 0.0418005i	0.999782	−	0.0209002i	$-0.00665323\pi$
$888$	−0.792706	−0.0266015
$889$	52.3699i	1.75643i
$890$	21.9551i	0.735937i
$891$	− 3.29769i	− 0.110477i
$892$	3.88573	0.130104
$893$	76.7123	2.56708
$894$	− 7.73418i	− 0.258670i
$895$	− 11.0625i	− 0.369779i
$896$	− 2.94495i	− 0.0983838i
$897$	−4.75699	−0.158831
$898$	33.2572i	1.10981i
$899$	−3.71644	−0.123950
$900$	−3.40743	−0.113581
$901$	0	0
$902$	−13.3987	−0.446129
$903$	2.55354	0.0849764
$904$	− 19.3827i	− 0.644658i
$905$	4.77370	0.158683
$906$	1.02280i	0.0339804i
$907$	− 16.2330i	− 0.539009i	−0.962999	−	0.269505i	$-0.913140\pi$
$907$	− 16.2330i	− 0.539009i	0.962999	−	0.269505i	$-0.0868600\pi$
$908$	− 4.38875i	− 0.145646i
$909$	−9.88764	−0.327952
$910$	12.5010	0.414404
$911$	− 19.2058i	− 0.636316i	−0.948038	−	0.318158i	$-0.896936\pi$
$911$	− 19.2058i	− 0.636316i	0.948038	−	0.318158i	$-0.103064\pi$
$912$	7.44155i	0.246414i
$913$	40.7704i	1.34930i
$914$	17.4812	0.578226
$915$	13.5076i	0.446546i
$916$	3.42063	0.113021
$917$	44.8005	1.47944
$918$	0	0
$919$	−35.7959	−1.18080	−0.590398	−	0.807112i	$-0.701029\pi$
$919$	−35.7959	−1.18080	−0.590398	+	0.807112i	$0.701029\pi$
$920$	1.78470	0.0588398
$921$	7.13291i	0.235037i
$922$	14.5385	0.478799
$923$	− 24.8289i	− 0.817254i
$924$	9.71153i	0.319486i
$925$	− 2.70109i	− 0.0888112i
$926$	−1.47703	−0.0485383
$927$	−6.54168	−0.214857
$928$	0.522726i	0.0171593i
$929$	− 38.9490i	− 1.27788i	−0.769258	−	0.638938i	$-0.779374\pi$
$929$	− 38.9490i	− 1.27788i	0.769258	−	0.638938i	$-0.220626\pi$
$930$	− 8.97229i	− 0.294213i
$931$	12.4476	0.407953
$932$	− 0.893211i	− 0.0292581i
$933$	19.1212	0.626001
$934$	−8.68873	−0.284304
$935$	0	0
$936$	3.36370	0.109946
$937$	49.5538	1.61885	0.809426	−	0.587222i	$-0.199778\pi$
$937$	49.5538	1.61885	0.809426	+	0.587222i	$0.199778\pi$
$938$	− 39.8596i	− 1.30146i
$939$	13.0225	0.424974
$940$	13.0092i	0.424314i
$941$	− 3.44941i	− 0.112447i	−0.998418	−	0.0562237i	$-0.982094\pi$
$941$	− 3.44941i	− 0.112447i	0.998418	−	0.0562237i	$-0.0179060\pi$
$942$	− 23.7502i	− 0.773823i
$943$	5.74603	0.187117
$944$	7.31543	0.238097
$945$	− 3.71644i	− 0.120896i
$946$	2.85940i	0.0929671i
$947$	21.5300i	0.699631i	0.936819	+	0.349815i	$0.113756\pi$
$947$	21.5300i	0.699631i	−0.936819	+	0.349815i	$0.886244\pi$
$948$	−10.9449	−0.355475
$949$	17.6217i	0.572025i
$950$	−25.3565	−0.822675
$951$	−4.55648	−0.147754
$952$	0	0
$953$	−40.4171	−1.30924	−0.654619	−	0.755959i	$-0.727171\pi$
$953$	−40.4171	−1.30924	−0.654619	+	0.755959i	$0.727171\pi$
$954$	−9.98868	−0.323395
$955$	0.468209i	0.0151509i
$956$	0.389157	0.0125863
$957$	− 1.72379i	− 0.0557222i
$958$	10.9463i	0.353660i
$959$	− 10.6538i	− 0.344030i
$960$	−1.26197	−0.0407300
$961$	−19.5483	−0.630590
$962$	2.66642i	0.0859690i
$963$	3.49207i	0.112530i
$964$	15.7554i	0.507447i
$965$	−1.31278	−0.0422597
$966$	− 4.16478i	− 0.134000i
$967$	19.1408	0.615528	0.307764	−	0.951463i	$-0.400419\pi$
$967$	19.1408	0.615528	0.307764	+	0.951463i	$0.400419\pi$
$968$	0.125218	0.00402467
$969$	0	0
$970$	−7.85535	−0.252220
$971$	52.0327	1.66981	0.834905	−	0.550394i	$-0.185522\pi$
$971$	52.0327	1.66981	0.834905	+	0.550394i	$0.185522\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	−59.6191	−1.91130
$974$	8.75751i	0.280609i
$975$	11.4615i	0.367063i
$976$	− 10.7035i	− 0.342612i
$977$	−50.2515	−1.60769	−0.803845	−	0.594839i	$-0.797216\pi$
$977$	−50.2515	−1.60769	−0.803845	+	0.594839i	$0.797216\pi$
$978$	3.22037	0.102976
$979$	57.3715i	1.83360i
$980$	2.11092i	0.0674309i
$981$	15.8751i	0.506853i
$982$	−30.4170	−0.970646
$983$	36.8882i	1.17655i	0.808661	+	0.588275i	$0.200193\pi$
$983$	36.8882i	1.17655i	−0.808661	+	0.588275i	$0.799807\pi$
$984$	−4.06306	−0.129526
$985$	6.68306	0.212940
$986$	0	0
$987$	30.3584	0.966319
$988$	25.0311	0.796346
$989$	− 1.22625i	− 0.0389925i
$990$	4.16160	0.132264
$991$	− 0.650857i	− 0.0206751i	−0.999947	−	0.0103376i	$-0.996709\pi$
$991$	− 0.650857i	− 0.0206751i	0.999947	−	0.0103376i	$-0.00329061\pi$
$992$	7.10973i	0.225734i
$993$	− 18.7047i	− 0.593577i
$994$	21.7379	0.689486
$995$	−26.8469	−0.851104
$996$	12.3633i	0.391747i
$997$	49.6033i	1.57095i	0.618892	+	0.785476i	$0.287582\pi$
$997$	49.6033i	1.57095i	−0.618892	+	0.785476i	$0.712418\pi$
$998$	− 22.3333i	− 0.706949i
$999$	0.792706	0.0250801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1734.2.b.l.577.1		8
17.2	even	8		1734.2.f.l.829.2		8
17.3	odd	16		102.2.h.b.25.2	✓	8
17.4	even	4		1734.2.a.t.1.1		4
17.8	even	8		1734.2.f.l.1483.2		8
17.9	even	8		1734.2.f.k.1483.3		8
17.11	odd	16		102.2.h.b.49.2	yes	8
17.13	even	4		1734.2.a.u.1.4		4
17.15	even	8		1734.2.f.k.829.3		8
17.16	even	2	inner	1734.2.b.l.577.8		8
51.11	even	16		306.2.l.e.253.2		8
51.20	even	16		306.2.l.e.127.2		8
51.38	odd	4		5202.2.a.bu.1.4		4
51.47	odd	4		5202.2.a.bx.1.1		4
68.3	even	16		816.2.bq.c.433.1		8
68.11	even	16		816.2.bq.c.49.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.h.b.25.2	✓	8	17.3	odd	16
102.2.h.b.49.2	yes	8	17.11	odd	16
306.2.l.e.127.2		8	51.20	even	16
306.2.l.e.253.2		8	51.11	even	16
816.2.bq.c.49.1		8	68.11	even	16
816.2.bq.c.433.1		8	68.3	even	16
1734.2.a.t.1.1		4	17.4	even	4
1734.2.a.u.1.4		4	17.13	even	4
1734.2.b.l.577.1		8	1.1	even	1	trivial
1734.2.b.l.577.8		8	17.16	even	2	inner
1734.2.f.k.829.3		8	17.15	even	8
1734.2.f.k.1483.3		8	17.9	even	8
1734.2.f.l.829.2		8	17.2	even	8
1734.2.f.l.1483.2		8	17.8	even	8
5202.2.a.bu.1.4		4	51.38	odd	4
5202.2.a.bx.1.1		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:	\( N \)	\(=\)	\( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1734.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.26197i q^{5} -1.00000i q^{6} -2.94495i q^{7} +1.00000 q^{8} -1.00000 q^{9} -1.26197i q^{10} -3.29769i q^{11} -1.00000i q^{12} -3.36370 q^{13} -2.94495i q^{14} -1.26197 q^{15} +1.00000 q^{16} -1.00000 q^{18} -7.44155 q^{19} -1.26197i q^{20} -2.94495 q^{21} -3.29769i q^{22} +1.41421i q^{23} -1.00000i q^{24} +3.40743 q^{25} -3.36370 q^{26} +1.00000i q^{27} -2.94495i q^{28} +0.522726i q^{29} -1.26197 q^{30} +7.10973i q^{31} +1.00000 q^{32} -3.29769 q^{33} -3.71644 q^{35} -1.00000 q^{36} -0.792706i q^{37} -7.44155 q^{38} +3.36370i q^{39} -1.26197i q^{40} -4.06306i q^{41} -2.94495 q^{42} -0.867091 q^{43} -3.29769i q^{44} +1.26197i q^{45} +1.41421i q^{46} -10.3086 q^{47} -1.00000i q^{48} -1.67271 q^{49} +3.40743 q^{50} -3.36370 q^{52} +9.98868 q^{53} +1.00000i q^{54} -4.16160 q^{55} -2.94495i q^{56} +7.44155i q^{57} +0.522726i q^{58} +7.31543 q^{59} -1.26197 q^{60} -10.7035i q^{61} +7.10973i q^{62} +2.94495i q^{63} +1.00000 q^{64} +4.24489i q^{65} -3.29769 q^{66} +13.5349 q^{67} +1.41421 q^{69} -3.71644 q^{70} +7.38144i q^{71} -1.00000 q^{72} -5.23880i q^{73} -0.792706i q^{74} -3.40743i q^{75} -7.44155 q^{76} -9.71153 q^{77} +3.36370i q^{78} -10.9449i q^{79} -1.26197i q^{80} +1.00000 q^{81} -4.06306i q^{82} -12.3633 q^{83} -2.94495 q^{84} -0.867091 q^{86} +0.522726 q^{87} -3.29769i q^{88} -17.3975 q^{89} +1.26197i q^{90} +9.90591i q^{91} +1.41421i q^{92} +7.10973 q^{93} -10.3086 q^{94} +9.39104i q^{95} -1.00000i q^{96} -6.22466i q^{97} -1.67271 q^{98} +3.29769i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} - 8 q^{9} + 16 q^{15} + 8 q^{16} - 8 q^{18} - 16 q^{19} - 24 q^{25} + 16 q^{30} + 8 q^{32} - 16 q^{33} - 16 q^{35} - 8 q^{36} - 16 q^{38} - 32 q^{47} - 24 q^{49} - 24 q^{50}+ \cdots - 24 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 8 * q^8 - 8 * q^9 + 16 * q^15 + 8 * q^16 - 8 * q^18 - 16 * q^19 - 24 * q^25 + 16 * q^30 + 8 * q^32 - 16 * q^33 - 16 * q^35 - 8 * q^36 - 16 * q^38 - 32 * q^47 - 24 * q^49 - 24 * q^50 + 32 * q^53 + 48 * q^59 + 16 * q^60 + 8 * q^64 - 16 * q^66 + 16 * q^67 - 16 * q^70 - 8 * q^72 - 16 * q^76 + 32 * q^77 + 8 * q^81 + 16 * q^83 + 32 * q^87 - 32 * q^89 + 16 * q^93 - 32 * q^94 - 24 * q^98

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