LMFDB - Embedded newform 507.2.b.e.337.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [507,2,Mod(337,507)] 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("507.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4,-4,0,0,0,0,4,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.2.b.e.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.414214i q^{2} +1.00000 q^{3} +1.82843 q^{4} -2.82843i q^{5} +0.414214i q^{6} -2.82843i q^{7} +1.58579i q^{8} +1.00000 q^{9} +1.17157 q^{10} +2.00000i q^{11} +1.82843 q^{12} +1.17157 q^{14} -2.82843i q^{15} +3.00000 q^{16} -7.65685 q^{17} +0.414214i q^{18} -2.82843i q^{19} -5.17157i q^{20} -2.82843i q^{21} -0.828427 q^{22} +4.00000 q^{23} +1.58579i q^{24} -3.00000 q^{25} +1.00000 q^{27} -5.17157i q^{28} +2.00000 q^{29} +1.17157 q^{30} -1.17157i q^{31} +4.41421i q^{32} +2.00000i q^{33} -3.17157i q^{34} -8.00000 q^{35} +1.82843 q^{36} +7.65685i q^{37} +1.17157 q^{38} +4.48528 q^{40} +5.17157i q^{41} +1.17157 q^{42} +1.65685 q^{43} +3.65685i q^{44} -2.82843i q^{45} +1.65685i q^{46} +11.6569i q^{47} +3.00000 q^{48} -1.00000 q^{49} -1.24264i q^{50} -7.65685 q^{51} -2.00000 q^{53} +0.414214i q^{54} +5.65685 q^{55} +4.48528 q^{56} -2.82843i q^{57} +0.828427i q^{58} -7.65685i q^{59} -5.17157i q^{60} +13.3137 q^{61} +0.485281 q^{62} -2.82843i q^{63} +4.17157 q^{64} -0.828427 q^{66} +6.82843i q^{67} -14.0000 q^{68} +4.00000 q^{69} -3.31371i q^{70} +2.00000i q^{71} +1.58579i q^{72} -0.343146i q^{73} -3.17157 q^{74} -3.00000 q^{75} -5.17157i q^{76} +5.65685 q^{77} -11.3137 q^{79} -8.48528i q^{80} +1.00000 q^{81} -2.14214 q^{82} +3.65685i q^{83} -5.17157i q^{84} +21.6569i q^{85} +0.686292i q^{86} +2.00000 q^{87} -3.17157 q^{88} -14.8284i q^{89} +1.17157 q^{90} +7.31371 q^{92} -1.17157i q^{93} -4.82843 q^{94} -8.00000 q^{95} +4.41421i q^{96} +3.65685i q^{97} -0.414214i q^{98} +2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 16 q^{10} - 4 q^{12} + 16 q^{14} + 12 q^{16} - 8 q^{17} + 8 q^{22} + 16 q^{23} - 12 q^{25} + 4 q^{27} + 8 q^{29} + 16 q^{30} - 32 q^{35} - 4 q^{36} + 16 q^{38} - 16 q^{40}+ \cdots - 32 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 + 16 * q^10 - 4 * q^12 + 16 * q^14 + 12 * q^16 - 8 * q^17 + 8 * q^22 + 16 * q^23 - 12 * q^25 + 4 * q^27 + 8 * q^29 + 16 * q^30 - 32 * q^35 - 4 * q^36 + 16 * q^38 - 16 * q^40 + 16 * q^42 - 16 * q^43 + 12 * q^48 - 4 * q^49 - 8 * q^51 - 8 * q^53 - 16 * q^56 + 8 * q^61 - 32 * q^62 + 28 * q^64 + 8 * q^66 - 56 * q^68 + 16 * q^69 - 24 * q^74 - 12 * q^75 + 4 * q^81 + 48 * q^82 + 8 * q^87 - 24 * q^88 + 16 * q^90 - 16 * q^92 - 8 * q^94 - 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.414214i	0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$	0.414214i	0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.82843	0.914214
$5$	− 2.82843i	− 1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	− 2.82843i	− 1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$	0.414214i	0.169102i
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	1.58579i	0.560660i
$9$	1.00000	0.333333
$10$	1.17157	0.370484
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	1.82843	0.527821
$13$	0	0
$13$	0	0
$14$	1.17157	0.313116
$15$	− 2.82843i	− 0.730297i
$16$	3.00000	0.750000
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0.414214i	0.0976311i
$19$	− 2.82843i	− 0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$	− 2.82843i	− 0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$	− 5.17157i	− 1.15640i
$21$	− 2.82843i	− 0.617213i
$22$	−0.828427	−0.176621
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.58579i	0.323697i
$25$	−3.00000	−0.600000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 5.17157i	− 0.977335i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	1.17157	0.213899
$31$	− 1.17157i	− 0.210421i	−0.994450	−	0.105210i	$-0.966448\pi$
$31$	− 1.17157i	− 0.210421i	0.994450	−	0.105210i	$-0.0335516\pi$
$32$	4.41421i	0.780330i
$33$	2.00000i	0.348155i
$34$	− 3.17157i	− 0.543920i
$35$	−8.00000	−1.35225
$36$	1.82843	0.304738
$37$	7.65685i	1.25878i	0.777090	+	0.629390i	$0.216695\pi$
$37$	7.65685i	1.25878i	−0.777090	+	0.629390i	$0.783305\pi$
$38$	1.17157	0.190054
$39$	0	0
$40$	4.48528	0.709185
$41$	5.17157i	0.807664i	0.914833	+	0.403832i	$0.132322\pi$
$41$	5.17157i	0.807664i	−0.914833	+	0.403832i	$0.867678\pi$
$42$	1.17157	0.180778
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	3.65685i	0.551292i
$45$	− 2.82843i	− 0.421637i
$46$	1.65685i	0.244290i
$47$	11.6569i	1.70033i	0.526519	+	0.850163i	$0.323497\pi$
$47$	11.6569i	1.70033i	−0.526519	+	0.850163i	$0.676503\pi$
$48$	3.00000	0.433013
$49$	−1.00000	−0.142857
$50$	− 1.24264i	− 0.175736i
$51$	−7.65685	−1.07217
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0.414214i	0.0563673i
$55$	5.65685	0.762770
$56$	4.48528	0.599371
$57$	− 2.82843i	− 0.374634i
$58$	0.828427i	0.108778i
$59$	− 7.65685i	− 0.996838i	−0.866936	−	0.498419i	$-0.833914\pi$
$59$	− 7.65685i	− 0.996838i	0.866936	−	0.498419i	$-0.166086\pi$
$60$	− 5.17157i	− 0.667647i
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	0.485281	0.0616308
$63$	− 2.82843i	− 0.356348i
$64$	4.17157	0.521447
$65$	0	0
$66$	−0.828427	−0.101972
$67$	6.82843i	0.834225i	0.908855	+	0.417113i	$0.136958\pi$
$67$	6.82843i	0.834225i	−0.908855	+	0.417113i	$0.863042\pi$
$68$	−14.0000	−1.69775
$69$	4.00000	0.481543
$70$	− 3.31371i	− 0.396064i
$71$	2.00000i	0.237356i	0.992933	+	0.118678i	$0.0378657\pi$
$71$	2.00000i	0.237356i	−0.992933	+	0.118678i	$0.962134\pi$
$72$	1.58579i	0.186887i
$73$	− 0.343146i	− 0.0401622i	−0.999798	−	0.0200811i	$-0.993608\pi$
$73$	− 0.343146i	− 0.0401622i	0.999798	−	0.0200811i	$-0.00639244\pi$
$74$	−3.17157	−0.368688
$75$	−3.00000	−0.346410
$76$	− 5.17157i	− 0.593220i
$77$	5.65685	0.644658
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	− 8.48528i	− 0.948683i
$81$	1.00000	0.111111
$82$	−2.14214	−0.236559
$83$	3.65685i	0.401392i	0.979654	+	0.200696i	$0.0643203\pi$
$83$	3.65685i	0.401392i	−0.979654	+	0.200696i	$0.935680\pi$
$84$	− 5.17157i	− 0.564265i
$85$	21.6569i	2.34902i
$86$	0.686292i	0.0740047i
$87$	2.00000	0.214423
$88$	−3.17157	−0.338091
$89$	− 14.8284i	− 1.57181i	−0.618347	−	0.785905i	$-0.712197\pi$
$89$	− 14.8284i	− 1.57181i	0.618347	−	0.785905i	$-0.287803\pi$
$90$	1.17157	0.123495
$91$	0	0
$92$	7.31371	0.762507
$93$	− 1.17157i	− 0.121486i
$94$	−4.82843	−0.498014
$95$	−8.00000	−0.820783
$96$	4.41421i	0.450524i
$97$	3.65685i	0.371297i	0.982616	+	0.185649i	$0.0594386\pi$
$97$	3.65685i	0.371297i	−0.982616	+	0.185649i	$0.940561\pi$
$98$	− 0.414214i	− 0.0418419i
$99$	2.00000i	0.201008i
$100$	−5.48528	−0.548528
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	− 3.17157i	− 0.314033i
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	− 0.828427i	− 0.0804640i
$107$	−11.3137	−1.09374	−0.546869	−	0.837218i	$-0.684180\pi$
$107$	−11.3137	−1.09374	−0.546869	+	0.837218i	$0.684180\pi$
$108$	1.82843	0.175940
$109$	5.31371i	0.508961i	0.967078	+	0.254480i	$0.0819045\pi$
$109$	5.31371i	0.508961i	−0.967078	+	0.254480i	$0.918096\pi$
$110$	2.34315i	0.223410i
$111$	7.65685i	0.726756i
$112$	− 8.48528i	− 0.801784i
$113$	−5.31371	−0.499872	−0.249936	−	0.968262i	$-0.580410\pi$
$113$	−5.31371	−0.499872	−0.249936	+	0.968262i	$0.580410\pi$
$114$	1.17157	0.109728
$115$	− 11.3137i	− 1.05501i
$116$	3.65685	0.339530
$117$	0	0
$118$	3.17157	0.291967
$119$	21.6569i	1.98528i
$120$	4.48528	0.409448
$121$	7.00000	0.636364
$122$	5.51472i	0.499279i
$123$	5.17157i	0.466305i
$124$	− 2.14214i	− 0.192369i
$125$	− 5.65685i	− 0.505964i
$126$	1.17157	0.104372
$127$	−5.65685	−0.501965	−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685	−0.501965	−0.250982	+	0.967992i	$0.580754\pi$
$128$	10.5563i	0.933058i
$129$	1.65685	0.145878
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	3.65685i	0.318288i
$133$	−8.00000	−0.693688
$134$	−2.82843	−0.244339
$135$	− 2.82843i	− 0.243432i
$136$	− 12.1421i	− 1.04118i
$137$	10.8284i	0.925135i	0.886584	+	0.462567i	$0.153072\pi$
$137$	10.8284i	0.925135i	−0.886584	+	0.462567i	$0.846928\pi$
$138$	1.65685i	0.141041i
$139$	−7.31371	−0.620341	−0.310170	−	0.950681i	$-0.600386\pi$
$139$	−7.31371	−0.620341	−0.310170	+	0.950681i	$0.600386\pi$
$140$	−14.6274	−1.23624
$141$	11.6569i	0.981684i
$142$	−0.828427	−0.0695201
$143$	0	0
$144$	3.00000	0.250000
$145$	− 5.65685i	− 0.469776i
$146$	0.142136	0.0117632
$147$	−1.00000	−0.0824786
$148$	14.0000i	1.15079i
$149$	− 9.17157i	− 0.751365i	−0.926749	−	0.375682i	$-0.877408\pi$
$149$	− 9.17157i	− 0.751365i	0.926749	−	0.375682i	$-0.122592\pi$
$150$	− 1.24264i	− 0.101461i
$151$	3.51472i	0.286024i	0.989721	+	0.143012i	$0.0456787\pi$
$151$	3.51472i	0.286024i	−0.989721	+	0.143012i	$0.954321\pi$
$152$	4.48528	0.363804
$153$	−7.65685	−0.619020
$154$	2.34315i	0.188816i
$155$	−3.31371	−0.266163
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	− 4.68629i	− 0.372821i
$159$	−2.00000	−0.158610
$160$	12.4853	0.987048
$161$	− 11.3137i	− 0.891645i
$162$	0.414214i	0.0325437i
$163$	− 18.8284i	− 1.47476i	−0.675480	−	0.737378i	$-0.736064\pi$
$163$	− 18.8284i	− 1.47476i	0.675480	−	0.737378i	$-0.263936\pi$
$164$	9.45584i	0.738377i
$165$	5.65685	0.440386
$166$	−1.51472	−0.117565
$167$	3.65685i	0.282976i	0.989940	+	0.141488i	$0.0451886\pi$
$167$	3.65685i	0.282976i	−0.989940	+	0.141488i	$0.954811\pi$
$168$	4.48528	0.346047
$169$	0	0
$170$	−8.97056	−0.688011
$171$	− 2.82843i	− 0.216295i
$172$	3.02944	0.230992
$173$	11.6569	0.886254	0.443127	−	0.896459i	$-0.353869\pi$
$173$	11.6569	0.886254	0.443127	+	0.896459i	$0.353869\pi$
$174$	0.828427i	0.0628029i
$175$	8.48528i	0.641427i
$176$	6.00000i	0.452267i
$177$	− 7.65685i	− 0.575524i
$178$	6.14214	0.460373
$179$	23.3137	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$179$	23.3137	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$180$	− 5.17157i	− 0.385466i
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	13.3137	0.984178
$184$	6.34315i	0.467623i
$185$	21.6569	1.59224
$186$	0.485281	0.0355826
$187$	− 15.3137i	− 1.11985i
$188$	21.3137i	1.55446i
$189$	− 2.82843i	− 0.205738i
$190$	− 3.31371i	− 0.240402i
$191$	3.31371	0.239772	0.119886	−	0.992788i	$-0.461747\pi$
$191$	3.31371	0.239772	0.119886	+	0.992788i	$0.461747\pi$
$192$	4.17157	0.301057
$193$	− 5.31371i	− 0.382489i	−0.981542	−	0.191245i	$-0.938748\pi$
$193$	− 5.31371i	− 0.382489i	0.981542	−	0.191245i	$-0.0612524\pi$
$194$	−1.51472	−0.108750
$195$	0	0
$196$	−1.82843	−0.130602
$197$	0.485281i	0.0345749i	0.999851	+	0.0172874i	$0.00550304\pi$
$197$	0.485281i	0.0345749i	−0.999851	+	0.0172874i	$0.994497\pi$
$198$	−0.828427	−0.0588738
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	− 4.75736i	− 0.336396i
$201$	6.82843i	0.481640i
$202$	− 3.17157i	− 0.223151i
$203$	− 5.65685i	− 0.397033i
$204$	−14.0000	−0.980196
$205$	14.6274	1.02162
$206$	− 0.970563i	− 0.0676223i
$207$	4.00000	0.278019
$208$	0	0
$209$	5.65685	0.391293
$210$	− 3.31371i	− 0.228668i
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−3.65685	−0.251154
$213$	2.00000i	0.137038i
$214$	− 4.68629i	− 0.320348i
$215$	− 4.68629i	− 0.319602i
$216$	1.58579i	0.107899i
$217$	−3.31371	−0.224949
$218$	−2.20101	−0.149071
$219$	− 0.343146i	− 0.0231876i
$220$	10.3431	0.697335
$221$	0	0
$222$	−3.17157	−0.212862
$223$	− 12.4853i	− 0.836076i	−0.908429	−	0.418038i	$-0.862718\pi$
$223$	− 12.4853i	− 0.836076i	0.908429	−	0.418038i	$-0.137282\pi$
$224$	12.4853	0.834208
$225$	−3.00000	−0.200000
$226$	− 2.20101i	− 0.146409i
$227$	− 17.3137i	− 1.14915i	−0.818452	−	0.574576i	$-0.805167\pi$
$227$	− 17.3137i	− 1.14915i	0.818452	−	0.574576i	$-0.194833\pi$
$228$	− 5.17157i	− 0.342496i
$229$	1.31371i	0.0868123i	0.999058	+	0.0434062i	$0.0138209\pi$
$229$	1.31371i	0.0868123i	−0.999058	+	0.0434062i	$0.986179\pi$
$230$	4.68629	0.309005
$231$	5.65685	0.372194
$232$	3.17157i	0.208224i
$233$	−6.97056	−0.456657	−0.228328	−	0.973584i	$-0.573326\pi$
$233$	−6.97056	−0.456657	−0.228328	+	0.973584i	$0.573326\pi$
$234$	0	0
$235$	32.9706	2.15076
$236$	− 14.0000i	− 0.911322i
$237$	−11.3137	−0.734904
$238$	−8.97056	−0.581475
$239$	2.00000i	0.129369i	0.997906	+	0.0646846i	$0.0206041\pi$
$239$	2.00000i	0.129369i	−0.997906	+	0.0646846i	$0.979396\pi$
$240$	− 8.48528i	− 0.547723i
$241$	− 0.343146i	− 0.0221040i	−0.999939	−	0.0110520i	$-0.996482\pi$
$241$	− 0.343146i	− 0.0221040i	0.999939	−	0.0110520i	$-0.00351803\pi$
$242$	2.89949i	0.186387i
$243$	1.00000	0.0641500
$244$	24.3431	1.55841
$245$	2.82843i	0.180702i
$246$	−2.14214	−0.136578
$247$	0	0
$248$	1.85786	0.117975
$249$	3.65685i	0.231744i
$250$	2.34315	0.148194
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	− 5.17157i	− 0.325778i
$253$	8.00000i	0.502956i
$254$	− 2.34315i	− 0.147022i
$255$	21.6569i	1.35620i
$256$	3.97056	0.248160
$257$	4.34315	0.270918	0.135459	−	0.990783i	$-0.456749\pi$
$257$	4.34315	0.270918	0.135459	+	0.990783i	$0.456749\pi$
$258$	0.686292i	0.0427266i
$259$	21.6569	1.34569
$260$	0	0
$261$	2.00000	0.123797
$262$	− 3.31371i	− 0.204722i
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	−3.17157	−0.195197
$265$	5.65685i	0.347498i
$266$	− 3.31371i	− 0.203177i
$267$	− 14.8284i	− 0.907485i
$268$	12.4853i	0.762660i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	1.17157	0.0712997
$271$	27.7990i	1.68867i	0.535817	+	0.844334i	$0.320004\pi$
$271$	27.7990i	1.68867i	−0.535817	+	0.844334i	$0.679996\pi$
$272$	−22.9706	−1.39279
$273$	0	0
$274$	−4.48528	−0.270966
$275$	− 6.00000i	− 0.361814i
$276$	7.31371	0.440234
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	− 3.02944i	− 0.181694i
$279$	− 1.17157i	− 0.0701402i
$280$	− 12.6863i	− 0.758151i
$281$	− 21.1716i	− 1.26299i	−0.775380	−	0.631495i	$-0.782442\pi$
$281$	− 21.1716i	− 1.26299i	0.775380	−	0.631495i	$-0.217558\pi$
$282$	−4.82843	−0.287529
$283$	−28.9706	−1.72212	−0.861061	−	0.508502i	$-0.830199\pi$
$283$	−28.9706	−1.72212	−0.861061	+	0.508502i	$0.830199\pi$
$284$	3.65685i	0.216994i
$285$	−8.00000	−0.473879
$286$	0	0
$287$	14.6274	0.863429
$288$	4.41421i	0.260110i
$289$	41.6274	2.44867
$290$	2.34315	0.137594
$291$	3.65685i	0.214369i
$292$	− 0.627417i	− 0.0367168i
$293$	− 2.14214i	− 0.125145i	−0.998040	−	0.0625724i	$-0.980070\pi$
$293$	− 2.14214i	− 0.125145i	0.998040	−	0.0625724i	$-0.0199304\pi$
$294$	− 0.414214i	− 0.0241574i
$295$	−21.6569	−1.26091
$296$	−12.1421	−0.705747
$297$	2.00000i	0.116052i
$298$	3.79899	0.220070
$299$	0	0
$300$	−5.48528	−0.316693
$301$	− 4.68629i	− 0.270113i
$302$	−1.45584	−0.0837744
$303$	−7.65685	−0.439875
$304$	− 8.48528i	− 0.486664i
$305$	− 37.6569i	− 2.15623i
$306$	− 3.17157i	− 0.181307i
$307$	22.8284i	1.30289i	0.758697	+	0.651444i	$0.225836\pi$
$307$	22.8284i	1.30289i	−0.758697	+	0.651444i	$0.774164\pi$
$308$	10.3431	0.589355
$309$	−2.34315	−0.133297
$310$	− 1.37258i	− 0.0779575i
$311$	10.6274	0.602626	0.301313	−	0.953525i	$-0.402575\pi$
$311$	10.6274	0.602626	0.301313	+	0.953525i	$0.402575\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	− 4.14214i	− 0.233754i
$315$	−8.00000	−0.450749
$316$	−20.6863	−1.16369
$317$	8.48528i	0.476581i	0.971194	+	0.238290i	$0.0765870\pi$
$317$	8.48528i	0.476581i	−0.971194	+	0.238290i	$0.923413\pi$
$318$	− 0.828427i	− 0.0464559i
$319$	4.00000i	0.223957i
$320$	− 11.7990i	− 0.659584i
$321$	−11.3137	−0.631470
$322$	4.68629	0.261157
$323$	21.6569i	1.20502i
$324$	1.82843	0.101579
$325$	0	0
$326$	7.79899	0.431946
$327$	5.31371i	0.293849i
$328$	−8.20101	−0.452825
$329$	32.9706	1.81773
$330$	2.34315i	0.128986i
$331$	26.1421i	1.43690i	0.695578	+	0.718451i	$0.255148\pi$
$331$	26.1421i	1.43690i	−0.695578	+	0.718451i	$0.744852\pi$
$332$	6.68629i	0.366958i
$333$	7.65685i	0.419593i
$334$	−1.51472	−0.0828817
$335$	19.3137	1.05522
$336$	− 8.48528i	− 0.462910i
$337$	−9.31371	−0.507350	−0.253675	−	0.967290i	$-0.581639\pi$
$337$	−9.31371	−0.507350	−0.253675	+	0.967290i	$0.581639\pi$
$338$	0	0
$339$	−5.31371	−0.288601
$340$	39.5980i	2.14750i
$341$	2.34315	0.126888
$342$	1.17157	0.0633514
$343$	− 16.9706i	− 0.916324i
$344$	2.62742i	0.141661i
$345$	− 11.3137i	− 0.609110i
$346$	4.82843i	0.259578i
$347$	−8.68629	−0.466305	−0.233152	−	0.972440i	$-0.574904\pi$
$347$	−8.68629	−0.466305	−0.233152	+	0.972440i	$0.574904\pi$
$348$	3.65685	0.196028
$349$	− 3.65685i	− 0.195747i	−0.995199	−	0.0978735i	$-0.968796\pi$
$349$	− 3.65685i	− 0.195747i	0.995199	−	0.0978735i	$-0.0312040\pi$
$350$	−3.51472	−0.187870
$351$	0	0
$352$	−8.82843	−0.470557
$353$	− 33.4558i	− 1.78067i	−0.455301	−	0.890337i	$-0.650468\pi$
$353$	− 33.4558i	− 1.78067i	0.455301	−	0.890337i	$-0.349532\pi$
$354$	3.17157	0.168567
$355$	5.65685	0.300235
$356$	− 27.1127i	− 1.43697i
$357$	21.6569i	1.14620i
$358$	9.65685i	0.510381i
$359$	− 34.9706i	− 1.84568i	−0.385189	−	0.922838i	$-0.625864\pi$
$359$	− 34.9706i	− 1.84568i	0.385189	−	0.922838i	$-0.374136\pi$
$360$	4.48528	0.236395
$361$	11.0000	0.578947
$362$	− 5.79899i	− 0.304788i
$363$	7.00000	0.367405
$364$	0	0
$365$	−0.970563	−0.0508016
$366$	5.51472i	0.288259i
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	12.0000	0.625543
$369$	5.17157i	0.269221i
$370$	8.97056i	0.466357i
$371$	5.65685i	0.293689i
$372$	− 2.14214i	− 0.111065i
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	6.34315	0.327996
$375$	− 5.65685i	− 0.292119i
$376$	−18.4853	−0.953306
$377$	0	0
$378$	1.17157	0.0602592
$379$	0.485281i	0.0249272i	0.999922	+	0.0124636i	$0.00396739\pi$
$379$	0.485281i	0.0249272i	−0.999922	+	0.0124636i	$0.996033\pi$
$380$	−14.6274	−0.750371
$381$	−5.65685	−0.289809
$382$	1.37258i	0.0702275i
$383$	30.9706i	1.58252i	0.611479	+	0.791261i	$0.290575\pi$
$383$	30.9706i	1.58252i	−0.611479	+	0.791261i	$0.709425\pi$
$384$	10.5563i	0.538701i
$385$	− 16.0000i	− 0.815436i
$386$	2.20101	0.112028
$387$	1.65685	0.0842226
$388$	6.68629i	0.339445i
$389$	26.9706	1.36746	0.683731	−	0.729734i	$-0.260356\pi$
$389$	26.9706	1.36746	0.683731	+	0.729734i	$0.260356\pi$
$390$	0	0
$391$	−30.6274	−1.54890
$392$	− 1.58579i	− 0.0800943i
$393$	−8.00000	−0.403547
$394$	−0.201010	−0.0101267
$395$	32.0000i	1.61009i
$396$	3.65685i	0.183764i
$397$	− 30.9706i	− 1.55437i	−0.629273	−	0.777184i	$-0.716647\pi$
$397$	− 30.9706i	− 1.55437i	0.629273	−	0.777184i	$-0.283353\pi$
$398$	− 8.97056i	− 0.449654i
$399$	−8.00000	−0.400501
$400$	−9.00000	−0.450000
$401$	− 26.1421i	− 1.30548i	−0.757584	−	0.652738i	$-0.773620\pi$
$401$	− 26.1421i	− 1.30548i	0.757584	−	0.652738i	$-0.226380\pi$
$402$	−2.82843	−0.141069
$403$	0	0
$404$	−14.0000	−0.696526
$405$	− 2.82843i	− 0.140546i
$406$	2.34315	0.116288
$407$	−15.3137	−0.759072
$408$	− 12.1421i	− 0.601125i
$409$	− 34.9706i	− 1.72918i	−0.502475	−	0.864592i	$-0.667577\pi$
$409$	− 34.9706i	− 1.72918i	0.502475	−	0.864592i	$-0.332423\pi$
$410$	6.05887i	0.299226i
$411$	10.8284i	0.534127i
$412$	−4.28427	−0.211071
$413$	−21.6569	−1.06566
$414$	1.65685i	0.0814299i
$415$	10.3431	0.507725
$416$	0	0
$417$	−7.31371	−0.358154
$418$	2.34315i	0.114607i
$419$	14.6274	0.714596	0.357298	−	0.933990i	$-0.383698\pi$
$419$	14.6274	0.714596	0.357298	+	0.933990i	$0.383698\pi$
$420$	−14.6274	−0.713745
$421$	37.3137i	1.81856i	0.416186	+	0.909279i	$0.363366\pi$
$421$	37.3137i	1.81856i	−0.416186	+	0.909279i	$0.636634\pi$
$422$	− 4.97056i	− 0.241963i
$423$	11.6569i	0.566776i
$424$	− 3.17157i	− 0.154025i
$425$	22.9706	1.11424
$426$	−0.828427	−0.0401374
$427$	− 37.6569i	− 1.82234i
$428$	−20.6863	−0.999910
$429$	0	0
$430$	1.94113	0.0936094
$431$	8.34315i	0.401875i	0.979604	+	0.200938i	$0.0643988\pi$
$431$	8.34315i	0.401875i	−0.979604	+	0.200938i	$0.935601\pi$
$432$	3.00000	0.144338
$433$	21.3137	1.02427	0.512136	−	0.858905i	$-0.328854\pi$
$433$	21.3137	1.02427	0.512136	+	0.858905i	$0.328854\pi$
$434$	− 1.37258i	− 0.0658861i
$435$	− 5.65685i	− 0.271225i
$436$	9.71573i	0.465299i
$437$	− 11.3137i	− 0.541208i
$438$	0.142136	0.00679150
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	8.97056i	0.427655i
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	−25.9411	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$443$	−25.9411	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$444$	14.0000i	0.664411i
$445$	−41.9411	−1.98820
$446$	5.17157	0.244881
$447$	− 9.17157i	− 0.433801i
$448$	− 11.7990i	− 0.557450i
$449$	31.7990i	1.50069i	0.661048	+	0.750344i	$0.270112\pi$
$449$	31.7990i	1.50069i	−0.661048	+	0.750344i	$0.729888\pi$
$450$	− 1.24264i	− 0.0585786i
$451$	−10.3431	−0.487040
$452$	−9.71573	−0.456989
$453$	3.51472i	0.165136i
$454$	7.17157	0.336579
$455$	0	0
$456$	4.48528	0.210043
$457$	− 7.65685i	− 0.358173i	−0.983833	−	0.179086i	$-0.942686\pi$
$457$	− 7.65685i	− 0.358173i	0.983833	−	0.179086i	$-0.0573141\pi$
$458$	−0.544156	−0.0254267
$459$	−7.65685	−0.357391
$460$	− 20.6863i	− 0.964503i
$461$	5.17157i	0.240864i	0.992722	+	0.120432i	$0.0384280\pi$
$461$	5.17157i	0.240864i	−0.992722	+	0.120432i	$0.961572\pi$
$462$	2.34315i	0.109013i
$463$	24.4853i	1.13793i	0.822363	+	0.568964i	$0.192656\pi$
$463$	24.4853i	1.13793i	−0.822363	+	0.568964i	$0.807344\pi$
$464$	6.00000	0.278543
$465$	−3.31371	−0.153670
$466$	− 2.88730i	− 0.133752i
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	19.3137	0.891824
$470$	13.6569i	0.629944i
$471$	−10.0000	−0.460776
$472$	12.1421	0.558887
$473$	3.31371i	0.152364i
$474$	− 4.68629i	− 0.215248i
$475$	8.48528i	0.389331i
$476$	39.5980i	1.81497i
$477$	−2.00000	−0.0915737
$478$	−0.828427	−0.0378914
$479$	− 25.3137i	− 1.15661i	−0.815820	−	0.578306i	$-0.803714\pi$
$479$	− 25.3137i	− 1.15661i	0.815820	−	0.578306i	$-0.196286\pi$
$480$	12.4853	0.569873
$481$	0	0
$482$	0.142136	0.00647410
$483$	− 11.3137i	− 0.514792i
$484$	12.7990	0.581772
$485$	10.3431	0.469658
$486$	0.414214i	0.0187891i
$487$	− 7.79899i	− 0.353406i	−0.984264	−	0.176703i	$-0.943457\pi$
$487$	− 7.79899i	− 0.353406i	0.984264	−	0.176703i	$-0.0565432\pi$
$488$	21.1127i	0.955727i
$489$	− 18.8284i	− 0.851451i
$490$	−1.17157	−0.0529263
$491$	−30.6274	−1.38220	−0.691098	−	0.722761i	$-0.742873\pi$
$491$	−30.6274	−1.38220	−0.691098	+	0.722761i	$0.742873\pi$
$492$	9.45584i	0.426302i
$493$	−15.3137	−0.689695
$494$	0	0
$495$	5.65685	0.254257
$496$	− 3.51472i	− 0.157816i
$497$	5.65685	0.253745
$498$	−1.51472	−0.0678762
$499$	26.1421i	1.17028i	0.810931	+	0.585141i	$0.198961\pi$
$499$	26.1421i	1.17028i	−0.810931	+	0.585141i	$0.801039\pi$
$500$	− 10.3431i	− 0.462560i
$501$	3.65685i	0.163376i
$502$	0	0
$503$	−7.31371	−0.326102	−0.163051	−	0.986618i	$-0.552134\pi$
$503$	−7.31371	−0.326102	−0.163051	+	0.986618i	$0.552134\pi$
$504$	4.48528	0.199790
$505$	21.6569i	0.963717i
$506$	−3.31371	−0.147312
$507$	0	0
$508$	−10.3431	−0.458903
$509$	11.7990i	0.522981i	0.965206	+	0.261491i	$0.0842140\pi$
$509$	11.7990i	0.522981i	−0.965206	+	0.261491i	$0.915786\pi$
$510$	−8.97056	−0.397223
$511$	−0.970563	−0.0429352
$512$	22.7574i	1.00574i
$513$	− 2.82843i	− 0.124878i
$514$	1.79899i	0.0793500i
$515$	6.62742i	0.292039i
$516$	3.02944	0.133364
$517$	−23.3137	−1.02534
$518$	8.97056i	0.394144i
$519$	11.6569	0.511679
$520$	0	0
$521$	25.3137	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$521$	25.3137	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$522$	0.828427i	0.0362593i
$523$	−15.3137	−0.669622	−0.334811	−	0.942285i	$-0.608672\pi$
$523$	−15.3137	−0.669622	−0.334811	+	0.942285i	$0.608672\pi$
$524$	−14.6274	−0.639002
$525$	8.48528i	0.370328i
$526$	4.97056i	0.216727i
$527$	8.97056i	0.390764i
$528$	6.00000i	0.261116i
$529$	−7.00000	−0.304348
$530$	−2.34315	−0.101780
$531$	− 7.65685i	− 0.332279i
$532$	−14.6274	−0.634179
$533$	0	0
$534$	6.14214	0.265796
$535$	32.0000i	1.38348i
$536$	−10.8284	−0.467717
$537$	23.3137	1.00606
$538$	7.45584i	0.321444i
$539$	− 2.00000i	− 0.0861461i
$540$	− 5.17157i	− 0.222549i
$541$	− 10.0000i	− 0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$	− 10.0000i	− 0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$	−11.5147	−0.494600
$543$	−14.0000	−0.600798
$544$	− 33.7990i	− 1.44912i
$545$	15.0294	0.643790
$546$	0	0
$547$	23.3137	0.996822	0.498411	−	0.866941i	$-0.333917\pi$
$547$	23.3137	0.996822	0.498411	+	0.866941i	$0.333917\pi$
$548$	19.7990i	0.845771i
$549$	13.3137	0.568215
$550$	2.48528	0.105973
$551$	− 5.65685i	− 0.240990i
$552$	6.34315i	0.269982i
$553$	32.0000i	1.36078i
$554$	0.828427i	0.0351965i
$555$	21.6569	0.919282
$556$	−13.3726	−0.567124
$557$	7.79899i	0.330454i	0.986256	+	0.165227i	$0.0528356\pi$
$557$	7.79899i	0.330454i	−0.986256	+	0.165227i	$0.947164\pi$
$558$	0.485281	0.0205436
$559$	0	0
$560$	−24.0000	−1.01419
$561$	− 15.3137i	− 0.646545i
$562$	8.76955	0.369921
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	21.3137i	0.897469i
$565$	15.0294i	0.632293i
$566$	− 12.0000i	− 0.504398i
$567$	− 2.82843i	− 0.118783i
$568$	−3.17157	−0.133076
$569$	42.9706	1.80142	0.900710	−	0.434421i	$-0.143047\pi$
$569$	42.9706	1.80142	0.900710	+	0.434421i	$0.143047\pi$
$570$	− 3.31371i	− 0.138796i
$571$	12.9706	0.542801	0.271401	−	0.962466i	$-0.412513\pi$
$571$	12.9706	0.542801	0.271401	+	0.962466i	$0.412513\pi$
$572$	0	0
$573$	3.31371	0.138432
$574$	6.05887i	0.252893i
$575$	−12.0000	−0.500435
$576$	4.17157	0.173816
$577$	− 31.9411i	− 1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$577$	− 31.9411i	− 1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$578$	17.2426i	0.717199i
$579$	− 5.31371i	− 0.220830i
$580$	− 10.3431i	− 0.429476i
$581$	10.3431	0.429106
$582$	−1.51472	−0.0627871
$583$	− 4.00000i	− 0.165663i
$584$	0.544156	0.0225173
$585$	0	0
$586$	0.887302	0.0366541
$587$	− 10.9706i	− 0.452804i	−0.974034	−	0.226402i	$-0.927304\pi$
$587$	− 10.9706i	− 0.452804i	0.974034	−	0.226402i	$-0.0726962\pi$
$588$	−1.82843	−0.0754031
$589$	−3.31371	−0.136539
$590$	− 8.97056i	− 0.369312i
$591$	0.485281i	0.0199618i
$592$	22.9706i	0.944084i
$593$	20.4853i	0.841230i	0.907239	+	0.420615i	$0.138186\pi$
$593$	20.4853i	0.841230i	−0.907239	+	0.420615i	$0.861814\pi$
$594$	−0.828427	−0.0339908
$595$	61.2548	2.51120
$596$	− 16.7696i	− 0.686908i
$597$	−21.6569	−0.886356
$598$	0	0
$599$	−23.3137	−0.952572	−0.476286	−	0.879290i	$-0.658017\pi$
$599$	−23.3137	−0.952572	−0.476286	+	0.879290i	$0.658017\pi$
$600$	− 4.75736i	− 0.194218i
$601$	−0.627417	−0.0255929	−0.0127964	−	0.999918i	$-0.504073\pi$
$601$	−0.627417	−0.0255929	−0.0127964	+	0.999918i	$0.504073\pi$
$602$	1.94113	0.0791144
$603$	6.82843i	0.278075i
$604$	6.42641i	0.261487i
$605$	− 19.7990i	− 0.804943i
$606$	− 3.17157i	− 0.128836i
$607$	41.9411	1.70234	0.851169	−	0.524892i	$-0.175894\pi$
$607$	41.9411	1.70234	0.851169	+	0.524892i	$0.175894\pi$
$608$	12.4853	0.506345
$609$	− 5.65685i	− 0.229227i
$610$	15.5980	0.631544
$611$	0	0
$612$	−14.0000	−0.565916
$613$	− 47.6569i	− 1.92484i	−0.271561	−	0.962421i	$-0.587540\pi$
$613$	− 47.6569i	− 1.92484i	0.271561	−	0.962421i	$-0.412460\pi$
$614$	−9.45584	−0.381607
$615$	14.6274	0.589834
$616$	8.97056i	0.361434i
$617$	− 34.8284i	− 1.40214i	−0.713093	−	0.701070i	$-0.752706\pi$
$617$	− 34.8284i	− 1.40214i	0.713093	−	0.701070i	$-0.247294\pi$
$618$	− 0.970563i	− 0.0390418i
$619$	− 23.7990i	− 0.956562i	−0.878207	−	0.478281i	$-0.841260\pi$
$619$	− 23.7990i	− 0.956562i	0.878207	−	0.478281i	$-0.158740\pi$
$620$	−6.05887	−0.243330
$621$	4.00000	0.160514
$622$	4.40202i	0.176505i
$623$	−41.9411	−1.68034
$624$	0	0
$625$	−31.0000	−1.24000
$626$	2.48528i	0.0993318i
$627$	5.65685	0.225913
$628$	−18.2843	−0.729622
$629$	− 58.6274i	− 2.33763i
$630$	− 3.31371i	− 0.132021i
$631$	− 43.1127i	− 1.71629i	−0.513408	−	0.858145i	$-0.671617\pi$
$631$	− 43.1127i	− 1.71629i	0.513408	−	0.858145i	$-0.328383\pi$
$632$	− 17.9411i	− 0.713660i
$633$	−12.0000	−0.476957
$634$	−3.51472	−0.139587
$635$	16.0000i	0.634941i
$636$	−3.65685	−0.145004
$637$	0	0
$638$	−1.65685	−0.0655955
$639$	2.00000i	0.0791188i
$640$	29.8579	1.18024
$641$	−30.2843	−1.19616	−0.598078	−	0.801438i	$-0.704069\pi$
$641$	−30.2843	−1.19616	−0.598078	+	0.801438i	$0.704069\pi$
$642$	− 4.68629i	− 0.184953i
$643$	22.8284i	0.900265i	0.892962	+	0.450133i	$0.148623\pi$
$643$	22.8284i	0.900265i	−0.892962	+	0.450133i	$0.851377\pi$
$644$	− 20.6863i	− 0.815154i
$645$	− 4.68629i	− 0.184523i
$646$	−8.97056	−0.352942
$647$	−11.3137	−0.444788	−0.222394	−	0.974957i	$-0.571387\pi$
$647$	−11.3137	−0.444788	−0.222394	+	0.974957i	$0.571387\pi$
$648$	1.58579i	0.0622956i
$649$	15.3137	0.601116
$650$	0	0
$651$	−3.31371	−0.129874
$652$	− 34.4264i	− 1.34824i
$653$	−25.3137	−0.990602	−0.495301	−	0.868721i	$-0.664942\pi$
$653$	−25.3137	−0.990602	−0.495301	+	0.868721i	$0.664942\pi$
$654$	−2.20101	−0.0860663
$655$	22.6274i	0.884126i
$656$	15.5147i	0.605748i
$657$	− 0.343146i	− 0.0133874i
$658$	13.6569i	0.532400i
$659$	−47.3137	−1.84308	−0.921540	−	0.388283i	$-0.873068\pi$
$659$	−47.3137	−1.84308	−0.921540	+	0.388283i	$0.873068\pi$
$660$	10.3431	0.402606
$661$	34.9706i	1.36020i	0.733121	+	0.680099i	$0.238063\pi$
$661$	34.9706i	1.36020i	−0.733121	+	0.680099i	$0.761937\pi$
$662$	−10.8284	−0.420859
$663$	0	0
$664$	−5.79899	−0.225044
$665$	22.6274i	0.877454i
$666$	−3.17157	−0.122896
$667$	8.00000	0.309761
$668$	6.68629i	0.258700i
$669$	− 12.4853i	− 0.482709i
$670$	8.00000i	0.309067i
$671$	26.6274i	1.02794i
$672$	12.4853	0.481630
$673$	−16.6274	−0.640940	−0.320470	−	0.947259i	$-0.603841\pi$
$673$	−16.6274	−0.640940	−0.320470	+	0.947259i	$0.603841\pi$
$674$	− 3.85786i	− 0.148599i
$675$	−3.00000	−0.115470
$676$	0	0
$677$	26.6863	1.02564	0.512819	−	0.858497i	$-0.328601\pi$
$677$	26.6863	1.02564	0.512819	+	0.858497i	$0.328601\pi$
$678$	− 2.20101i	− 0.0845293i
$679$	10.3431	0.396934
$680$	−34.3431	−1.31700
$681$	− 17.3137i	− 0.663463i
$682$	0.970563i	0.0371648i
$683$	− 47.9411i	− 1.83442i	−0.398408	−	0.917208i	$-0.630437\pi$
$683$	− 47.9411i	− 1.83442i	0.398408	−	0.917208i	$-0.369563\pi$
$684$	− 5.17157i	− 0.197740i
$685$	30.6274	1.17021
$686$	7.02944	0.268385
$687$	1.31371i	0.0501211i
$688$	4.97056	0.189501
$689$	0	0
$690$	4.68629	0.178404
$691$	− 5.85786i	− 0.222844i	−0.993773	−	0.111422i	$-0.964460\pi$
$691$	− 5.85786i	− 0.222844i	0.993773	−	0.111422i	$-0.0355405\pi$
$692$	21.3137	0.810226
$693$	5.65685	0.214886
$694$	− 3.59798i	− 0.136577i
$695$	20.6863i	0.784676i
$696$	3.17157i	0.120218i
$697$	− 39.5980i	− 1.49988i
$698$	1.51472	0.0573329
$699$	−6.97056	−0.263651
$700$	15.5147i	0.586401i
$701$	−5.02944	−0.189959	−0.0949796	−	0.995479i	$-0.530279\pi$
$701$	−5.02944	−0.189959	−0.0949796	+	0.995479i	$0.530279\pi$
$702$	0	0
$703$	21.6569	0.816804
$704$	8.34315i	0.314444i
$705$	32.9706	1.24174
$706$	13.8579	0.521548
$707$	21.6569i	0.814490i
$708$	− 14.0000i	− 0.526152i
$709$	4.62742i	0.173786i	0.996218	+	0.0868931i	$0.0276939\pi$
$709$	4.62742i	0.173786i	−0.996218	+	0.0868931i	$0.972306\pi$
$710$	2.34315i	0.0879367i
$711$	−11.3137	−0.424297
$712$	23.5147	0.881251
$713$	− 4.68629i	− 0.175503i
$714$	−8.97056	−0.335715
$715$	0	0
$716$	42.6274	1.59306
$717$	2.00000i	0.0746914i
$718$	14.4853	0.540586
$719$	29.9411	1.11662	0.558308	−	0.829634i	$-0.311451\pi$
$719$	29.9411	1.11662	0.558308	+	0.829634i	$0.311451\pi$
$720$	− 8.48528i	− 0.316228i
$721$	6.62742i	0.246818i
$722$	4.55635i	0.169570i
$723$	− 0.343146i	− 0.0127617i
$724$	−25.5980	−0.951341
$725$	−6.00000	−0.222834
$726$	2.89949i	0.107610i
$727$	10.3431	0.383606	0.191803	−	0.981433i	$-0.438567\pi$
$727$	10.3431	0.383606	0.191803	+	0.981433i	$0.438567\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 0.402020i	− 0.0148794i
$731$	−12.6863	−0.469219
$732$	24.3431	0.899749
$733$	− 36.6274i	− 1.35286i	−0.736505	−	0.676432i	$-0.763525\pi$
$733$	− 36.6274i	− 1.35286i	0.736505	−	0.676432i	$-0.236475\pi$
$734$	− 9.94113i	− 0.366934i
$735$	2.82843i	0.104328i
$736$	17.6569i	0.650840i
$737$	−13.6569	−0.503057
$738$	−2.14214	−0.0788531
$739$	18.1421i	0.667369i	0.942685	+	0.333685i	$0.108292\pi$
$739$	18.1421i	0.667369i	−0.942685	+	0.333685i	$0.891708\pi$
$740$	39.5980	1.45565
$741$	0	0
$742$	−2.34315	−0.0860196
$743$	2.00000i	0.0733729i	0.999327	+	0.0366864i	$0.0116803\pi$
$743$	2.00000i	0.0733729i	−0.999327	+	0.0366864i	$0.988320\pi$
$744$	1.85786	0.0681126
$745$	−25.9411	−0.950409
$746$	4.14214i	0.151654i
$747$	3.65685i	0.133797i
$748$	− 28.0000i	− 1.02378i
$749$	32.0000i	1.16925i
$750$	2.34315	0.0855596
$751$	0.970563	0.0354163	0.0177082	−	0.999843i	$-0.494363\pi$
$751$	0.970563	0.0354163	0.0177082	+	0.999843i	$0.494363\pi$
$752$	34.9706i	1.27525i
$753$	0	0
$754$	0	0
$755$	9.94113	0.361795
$756$	− 5.17157i	− 0.188088i
$757$	51.9411	1.88783	0.943916	−	0.330185i	$-0.107111\pi$
$757$	51.9411	1.88783	0.943916	+	0.330185i	$0.107111\pi$
$758$	−0.201010	−0.00730102
$759$	8.00000i	0.290382i
$760$	− 12.6863i	− 0.460180i
$761$	− 32.4853i	− 1.17759i	−0.808282	−	0.588795i	$-0.799602\pi$
$761$	− 32.4853i	− 1.17759i	0.808282	−	0.588795i	$-0.200398\pi$
$762$	− 2.34315i	− 0.0848832i
$763$	15.0294	0.544102
$764$	6.05887	0.219202
$765$	21.6569i	0.783005i
$766$	−12.8284	−0.463510
$767$	0	0
$768$	3.97056	0.143275
$769$	42.0000i	1.51456i	0.653091	+	0.757279i	$0.273472\pi$
$769$	42.0000i	1.51456i	−0.653091	+	0.757279i	$0.726528\pi$
$770$	6.62742	0.238836
$771$	4.34315	0.156415
$772$	− 9.71573i	− 0.349677i
$773$	34.1421i	1.22801i	0.789303	+	0.614004i	$0.210442\pi$
$773$	34.1421i	1.22801i	−0.789303	+	0.614004i	$0.789558\pi$
$774$	0.686292i	0.0246682i
$775$	3.51472i	0.126252i
$776$	−5.79899	−0.208172
$777$	21.6569	0.776935
$778$	11.1716i	0.400520i
$779$	14.6274	0.524082
$780$	0	0
$781$	−4.00000	−0.143131
$782$	− 12.6863i	− 0.453661i
$783$	2.00000	0.0714742
$784$	−3.00000	−0.107143
$785$	28.2843i	1.00951i
$786$	− 3.31371i	− 0.118196i
$787$	40.7696i	1.45328i	0.687020	+	0.726639i	$0.258919\pi$
$787$	40.7696i	1.45328i	−0.687020	+	0.726639i	$0.741081\pi$
$788$	0.887302i	0.0316088i
$789$	12.0000	0.427211
$790$	−13.2548	−0.471586
$791$	15.0294i	0.534385i
$792$	−3.17157	−0.112697
$793$	0	0
$794$	12.8284	0.455264
$795$	5.65685i	0.200628i
$796$	−39.5980	−1.40351
$797$	−24.3431	−0.862278	−0.431139	−	0.902285i	$-0.641888\pi$
$797$	−24.3431	−0.862278	−0.431139	+	0.902285i	$0.641888\pi$
$798$	− 3.31371i	− 0.117304i
$799$	− 89.2548i	− 3.15761i
$800$	− 13.2426i	− 0.468198i
$801$	− 14.8284i	− 0.523937i
$802$	10.8284	0.382365
$803$	0.686292	0.0242187
$804$	12.4853i	0.440322i
$805$	−32.0000	−1.12785
$806$	0	0
$807$	18.0000	0.633630
$808$	− 12.1421i	− 0.427159i
$809$	18.6863	0.656975	0.328488	−	0.944508i	$-0.393461\pi$
$809$	18.6863	0.656975	0.328488	+	0.944508i	$0.393461\pi$
$810$	1.17157	0.0411649
$811$	− 30.1421i	− 1.05843i	−0.848487	−	0.529217i	$-0.822486\pi$
$811$	− 30.1421i	− 1.05843i	0.848487	−	0.529217i	$-0.177514\pi$
$812$	− 10.3431i	− 0.362973i
$813$	27.7990i	0.974953i
$814$	− 6.34315i	− 0.222327i
$815$	−53.2548	−1.86544
$816$	−22.9706	−0.804131
$817$	− 4.68629i	− 0.163953i
$818$	14.4853	0.506466
$819$	0	0
$820$	26.7452	0.933982
$821$	− 23.7990i	− 0.830590i	−0.909687	−	0.415295i	$-0.863678\pi$
$821$	− 23.7990i	− 0.830590i	0.909687	−	0.415295i	$-0.136322\pi$
$822$	−4.48528	−0.156442
$823$	−15.0294	−0.523893	−0.261947	−	0.965082i	$-0.584364\pi$
$823$	−15.0294	−0.523893	−0.261947	+	0.965082i	$0.584364\pi$
$824$	− 3.71573i	− 0.129444i
$825$	− 6.00000i	− 0.208893i
$826$	− 8.97056i	− 0.312126i
$827$	26.0000i	0.904109i	0.891990	+	0.452054i	$0.149309\pi$
$827$	26.0000i	0.904109i	−0.891990	+	0.452054i	$0.850691\pi$
$828$	7.31371	0.254169
$829$	17.3137	0.601330	0.300665	−	0.953730i	$-0.402791\pi$
$829$	17.3137	0.601330	0.300665	+	0.953730i	$0.402791\pi$
$830$	4.28427i	0.148709i
$831$	2.00000	0.0693792
$832$	0	0
$833$	7.65685	0.265294
$834$	− 3.02944i	− 0.104901i
$835$	10.3431	0.357939
$836$	10.3431	0.357725
$837$	− 1.17157i	− 0.0404955i
$838$	6.05887i	0.209300i
$839$	− 43.2548i	− 1.49332i	−0.665204	−	0.746661i	$-0.731656\pi$
$839$	− 43.2548i	− 1.49332i	0.665204	−	0.746661i	$-0.268344\pi$
$840$	− 12.6863i	− 0.437719i
$841$	−25.0000	−0.862069
$842$	−15.4558	−0.532644
$843$	− 21.1716i	− 0.729188i
$844$	−21.9411	−0.755245
$845$	0	0
$846$	−4.82843	−0.166005
$847$	− 19.7990i	− 0.680301i
$848$	−6.00000	−0.206041
$849$	−28.9706	−0.994267
$850$	9.51472i	0.326352i
$851$	30.6274i	1.04989i
$852$	3.65685i	0.125282i
$853$	− 3.65685i	− 0.125208i	−0.998038	−	0.0626042i	$-0.980059\pi$
$853$	− 3.65685i	− 0.125208i	0.998038	−	0.0626042i	$-0.0199406\pi$
$854$	15.5980	0.533752
$855$	−8.00000	−0.273594
$856$	− 17.9411i	− 0.613215i
$857$	49.5980	1.69423	0.847117	−	0.531406i	$-0.178336\pi$
$857$	49.5980	1.69423	0.847117	+	0.531406i	$0.178336\pi$
$858$	0	0
$859$	−0.686292	−0.0234160	−0.0117080	−	0.999931i	$-0.503727\pi$
$859$	−0.686292	−0.0234160	−0.0117080	+	0.999931i	$0.503727\pi$
$860$	− 8.56854i	− 0.292185i
$861$	14.6274	0.498501
$862$	−3.45584	−0.117707
$863$	− 28.3431i	− 0.964812i	−0.875948	−	0.482406i	$-0.839763\pi$
$863$	− 28.3431i	− 0.964812i	0.875948	−	0.482406i	$-0.160237\pi$
$864$	4.41421i	0.150175i
$865$	− 32.9706i	− 1.12103i
$866$	8.82843i	0.300002i
$867$	41.6274	1.41374
$868$	−6.05887	−0.205652
$869$	− 22.6274i	− 0.767583i
$870$	2.34315	0.0794401
$871$	0	0
$872$	−8.42641	−0.285354
$873$	3.65685i	0.123766i
$874$	4.68629	0.158516
$875$	−16.0000	−0.540899
$876$	− 0.627417i	− 0.0211985i
$877$	42.2843i	1.42784i	0.700228	+	0.713919i	$0.253082\pi$
$877$	42.2843i	1.42784i	−0.700228	+	0.713919i	$0.746918\pi$
$878$	− 7.02944i	− 0.237232i
$879$	− 2.14214i	− 0.0722524i
$880$	16.9706	0.572078
$881$	25.5980	0.862418	0.431209	−	0.902252i	$-0.358087\pi$
$881$	25.5980	0.862418	0.431209	+	0.902252i	$0.358087\pi$
$882$	− 0.414214i	− 0.0139473i
$883$	−27.5980	−0.928746	−0.464373	−	0.885640i	$-0.653720\pi$
$883$	−27.5980	−0.928746	−0.464373	+	0.885640i	$0.653720\pi$
$884$	0	0
$885$	−21.6569	−0.727987
$886$	− 10.7452i	− 0.360991i
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	−12.1421	−0.407463
$889$	16.0000i	0.536623i
$890$	− 17.3726i	− 0.582330i
$891$	2.00000i	0.0670025i
$892$	− 22.8284i	− 0.764352i
$893$	32.9706	1.10332
$894$	3.79899	0.127057
$895$	− 65.9411i	− 2.20417i
$896$	29.8579	0.997481
$897$	0	0
$898$	−13.1716	−0.439541
$899$	− 2.34315i	− 0.0781483i
$900$	−5.48528	−0.182843
$901$	15.3137	0.510174
$902$	− 4.28427i	− 0.142651i
$903$	− 4.68629i	− 0.155950i
$904$	− 8.42641i	− 0.280258i
$905$	39.5980i	1.31628i
$906$	−1.45584	−0.0483672
$907$	12.9706	0.430680	0.215340	−	0.976539i	$-0.430914\pi$
$907$	12.9706	0.430680	0.215340	+	0.976539i	$0.430914\pi$
$908$	− 31.6569i	− 1.05057i
$909$	−7.65685	−0.253962
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	− 8.48528i	− 0.280976i
$913$	−7.31371	−0.242048
$914$	3.17157	0.104906
$915$	− 37.6569i	− 1.24490i
$916$	2.40202i	0.0793650i
$917$	22.6274i	0.747223i
$918$	− 3.17157i	− 0.104678i
$919$	3.31371	0.109309	0.0546546	−	0.998505i	$-0.482594\pi$
$919$	3.31371	0.109309	0.0546546	+	0.998505i	$0.482594\pi$
$920$	17.9411	0.591501
$921$	22.8284i	0.752222i
$922$	−2.14214	−0.0705475
$923$	0	0
$924$	10.3431	0.340265
$925$	− 22.9706i	− 0.755267i
$926$	−10.1421	−0.333291
$927$	−2.34315	−0.0769590
$928$	8.82843i	0.289807i
$929$	11.7990i	0.387112i	0.981089	+	0.193556i	$0.0620022\pi$
$929$	11.7990i	0.387112i	−0.981089	+	0.193556i	$0.937998\pi$
$930$	− 1.37258i	− 0.0450088i
$931$	2.82843i	0.0926980i
$932$	−12.7452	−0.417482
$933$	10.6274	0.347926
$934$	3.31371i	0.108428i
$935$	−43.3137	−1.41651
$936$	0	0
$937$	−21.3137	−0.696289	−0.348144	−	0.937441i	$-0.613188\pi$
$937$	−21.3137	−0.696289	−0.348144	+	0.937441i	$0.613188\pi$
$938$	8.00000i	0.261209i
$939$	6.00000	0.195803
$940$	60.2843	1.96626
$941$	34.1421i	1.11300i	0.830847	+	0.556501i	$0.187856\pi$
$941$	34.1421i	1.11300i	−0.830847	+	0.556501i	$0.812144\pi$
$942$	− 4.14214i	− 0.134958i
$943$	20.6863i	0.673638i
$944$	− 22.9706i	− 0.747628i
$945$	−8.00000	−0.260240
$946$	−1.37258	−0.0446265
$947$	21.0294i	0.683365i	0.939815	+	0.341682i	$0.110997\pi$
$947$	21.0294i	0.683365i	−0.939815	+	0.341682i	$0.889003\pi$
$948$	−20.6863	−0.671860
$949$	0	0
$950$	−3.51472	−0.114033
$951$	8.48528i	0.275154i
$952$	−34.3431	−1.11307
$953$	−40.3431	−1.30684	−0.653421	−	0.756994i	$-0.726667\pi$
$953$	−40.3431	−1.30684	−0.653421	+	0.756994i	$0.726667\pi$
$954$	− 0.828427i	− 0.0268213i
$955$	− 9.37258i	− 0.303290i
$956$	3.65685i	0.118271i
$957$	4.00000i	0.129302i
$958$	10.4853	0.338764
$959$	30.6274	0.989011
$960$	− 11.7990i	− 0.380811i
$961$	29.6274	0.955723
$962$	0	0
$963$	−11.3137	−0.364579
$964$	− 0.627417i	− 0.0202077i
$965$	−15.0294	−0.483815
$966$	4.68629	0.150779
$967$	18.1421i	0.583412i	0.956508	+	0.291706i	$0.0942228\pi$
$967$	18.1421i	0.583412i	−0.956508	+	0.291706i	$0.905777\pi$
$968$	11.1005i	0.356784i
$969$	21.6569i	0.695718i
$970$	4.28427i	0.137560i
$971$	−15.3137	−0.491440	−0.245720	−	0.969341i	$-0.579024\pi$
$971$	−15.3137	−0.491440	−0.245720	+	0.969341i	$0.579024\pi$
$972$	1.82843	0.0586468
$973$	20.6863i	0.663172i
$974$	3.23045	0.103510
$975$	0	0
$976$	39.9411	1.27848
$977$	42.1421i	1.34825i	0.738619	+	0.674123i	$0.235478\pi$
$977$	42.1421i	1.34825i	−0.738619	+	0.674123i	$0.764522\pi$
$978$	7.79899	0.249384
$979$	29.6569	0.947837
$980$	5.17157i	0.165200i
$981$	5.31371i	0.169654i
$982$	− 12.6863i	− 0.404836i
$983$	− 25.3137i	− 0.807382i	−0.914895	−	0.403691i	$-0.867727\pi$
$983$	− 25.3137i	− 0.807382i	0.914895	−	0.403691i	$-0.132273\pi$
$984$	−8.20101	−0.261439
$985$	1.37258	0.0437341
$986$	− 6.34315i	− 0.202007i
$987$	32.9706	1.04946
$988$	0	0
$989$	6.62742	0.210740
$990$	2.34315i	0.0744701i
$991$	4.68629	0.148865	0.0744325	−	0.997226i	$-0.476285\pi$
$991$	4.68629	0.148865	0.0744325	+	0.997226i	$0.476285\pi$
$992$	5.17157	0.164198
$993$	26.1421i	0.829596i
$994$	2.34315i	0.0743201i
$995$	61.2548i	1.94191i
$996$	6.68629i	0.211863i
$997$	−39.2548	−1.24321	−0.621607	−	0.783330i	$-0.713520\pi$
$997$	−39.2548	−1.24321	−0.621607	+	0.783330i	$0.713520\pi$
$998$	−10.8284	−0.342768
$999$	7.65685i	0.242252i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.b.e.337.3		4
3.2	odd	2		1521.2.b.j.1351.2		4
13.2	odd	12		507.2.e.h.22.1		4
13.3	even	3		507.2.j.f.316.2		8
13.4	even	6		507.2.j.f.361.2		8
13.5	odd	4		39.2.a.b.1.2	✓	2
13.6	odd	12		507.2.e.h.484.1		4
13.7	odd	12		507.2.e.d.484.2		4
13.8	odd	4		507.2.a.h.1.1		2
13.9	even	3		507.2.j.f.361.3		8
13.10	even	6		507.2.j.f.316.3		8
13.11	odd	12		507.2.e.d.22.2		4
13.12	even	2	inner	507.2.b.e.337.2		4
39.5	even	4		117.2.a.c.1.1		2
39.8	even	4		1521.2.a.f.1.2		2
39.38	odd	2		1521.2.b.j.1351.3		4
52.31	even	4		624.2.a.k.1.1		2
52.47	even	4		8112.2.a.bm.1.2		2
65.18	even	4		975.2.c.h.274.2		4
65.44	odd	4		975.2.a.l.1.1		2
65.57	even	4		975.2.c.h.274.3		4
91.83	even	4		1911.2.a.h.1.2		2
104.5	odd	4		2496.2.a.bf.1.2		2
104.83	even	4		2496.2.a.bi.1.2		2
117.5	even	12		1053.2.e.e.703.2		4
117.31	odd	12		1053.2.e.m.703.1		4
117.70	odd	12		1053.2.e.m.352.1		4
117.83	even	12		1053.2.e.e.352.2		4
143.109	even	4		4719.2.a.p.1.1		2
156.83	odd	4		1872.2.a.w.1.2		2
195.44	even	4		2925.2.a.v.1.2		2
195.83	odd	4		2925.2.c.u.2224.3		4
195.122	odd	4		2925.2.c.u.2224.2		4
273.83	odd	4		5733.2.a.u.1.1		2
312.5	even	4		7488.2.a.cl.1.1		2
312.83	odd	4		7488.2.a.co.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.2	✓	2	13.5	odd	4
117.2.a.c.1.1		2	39.5	even	4
507.2.a.h.1.1		2	13.8	odd	4
507.2.b.e.337.2		4	13.12	even	2	inner
507.2.b.e.337.3		4	1.1	even	1	trivial
507.2.e.d.22.2		4	13.11	odd	12
507.2.e.d.484.2		4	13.7	odd	12
507.2.e.h.22.1		4	13.2	odd	12
507.2.e.h.484.1		4	13.6	odd	12
507.2.j.f.316.2		8	13.3	even	3
507.2.j.f.316.3		8	13.10	even	6
507.2.j.f.361.2		8	13.4	even	6
507.2.j.f.361.3		8	13.9	even	3
624.2.a.k.1.1		2	52.31	even	4
975.2.a.l.1.1		2	65.44	odd	4
975.2.c.h.274.2		4	65.18	even	4
975.2.c.h.274.3		4	65.57	even	4
1053.2.e.e.352.2		4	117.83	even	12
1053.2.e.e.703.2		4	117.5	even	12
1053.2.e.m.352.1		4	117.70	odd	12
1053.2.e.m.703.1		4	117.31	odd	12
1521.2.a.f.1.2		2	39.8	even	4
1521.2.b.j.1351.2		4	3.2	odd	2
1521.2.b.j.1351.3		4	39.38	odd	2
1872.2.a.w.1.2		2	156.83	odd	4
1911.2.a.h.1.2		2	91.83	even	4
2496.2.a.bf.1.2		2	104.5	odd	4
2496.2.a.bi.1.2		2	104.83	even	4
2925.2.a.v.1.2		2	195.44	even	4
2925.2.c.u.2224.2		4	195.122	odd	4
2925.2.c.u.2224.3		4	195.83	odd	4
4719.2.a.p.1.1		2	143.109	even	4
5733.2.a.u.1.1		2	273.83	odd	4
7488.2.a.cl.1.1		2	312.5	even	4
7488.2.a.co.1.1		2	312.83	odd	4
8112.2.a.bm.1.2		2	52.47	even	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +1.00000 q^{3} +1.82843 q^{4} -2.82843i q^{5} +0.414214i q^{6} -2.82843i q^{7} +1.58579i q^{8} +1.00000 q^{9} +1.17157 q^{10} +2.00000i q^{11} +1.82843 q^{12} +1.17157 q^{14} -2.82843i q^{15} +3.00000 q^{16} -7.65685 q^{17} +0.414214i q^{18} -2.82843i q^{19} -5.17157i q^{20} -2.82843i q^{21} -0.828427 q^{22} +4.00000 q^{23} +1.58579i q^{24} -3.00000 q^{25} +1.00000 q^{27} -5.17157i q^{28} +2.00000 q^{29} +1.17157 q^{30} -1.17157i q^{31} +4.41421i q^{32} +2.00000i q^{33} -3.17157i q^{34} -8.00000 q^{35} +1.82843 q^{36} +7.65685i q^{37} +1.17157 q^{38} +4.48528 q^{40} +5.17157i q^{41} +1.17157 q^{42} +1.65685 q^{43} +3.65685i q^{44} -2.82843i q^{45} +1.65685i q^{46} +11.6569i q^{47} +3.00000 q^{48} -1.00000 q^{49} -1.24264i q^{50} -7.65685 q^{51} -2.00000 q^{53} +0.414214i q^{54} +5.65685 q^{55} +4.48528 q^{56} -2.82843i q^{57} +0.828427i q^{58} -7.65685i q^{59} -5.17157i q^{60} +13.3137 q^{61} +0.485281 q^{62} -2.82843i q^{63} +4.17157 q^{64} -0.828427 q^{66} +6.82843i q^{67} -14.0000 q^{68} +4.00000 q^{69} -3.31371i q^{70} +2.00000i q^{71} +1.58579i q^{72} -0.343146i q^{73} -3.17157 q^{74} -3.00000 q^{75} -5.17157i q^{76} +5.65685 q^{77} -11.3137 q^{79} -8.48528i q^{80} +1.00000 q^{81} -2.14214 q^{82} +3.65685i q^{83} -5.17157i q^{84} +21.6569i q^{85} +0.686292i q^{86} +2.00000 q^{87} -3.17157 q^{88} -14.8284i q^{89} +1.17157 q^{90} +7.31371 q^{92} -1.17157i q^{93} -4.82843 q^{94} -8.00000 q^{95} +4.41421i q^{96} +3.65685i q^{97} -0.414214i q^{98} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 16 q^{10} - 4 q^{12} + 16 q^{14} + 12 q^{16} - 8 q^{17} + 8 q^{22} + 16 q^{23} - 12 q^{25} + 4 q^{27} + 8 q^{29} + 16 q^{30} - 32 q^{35} - 4 q^{36} + 16 q^{38} - 16 q^{40}+ \cdots - 32 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 + 16 * q^10 - 4 * q^12 + 16 * q^14 + 12 * q^16 - 8 * q^17 + 8 * q^22 + 16 * q^23 - 12 * q^25 + 4 * q^27 + 8 * q^29 + 16 * q^30 - 32 * q^35 - 4 * q^36 + 16 * q^38 - 16 * q^40 + 16 * q^42 - 16 * q^43 + 12 * q^48 - 4 * q^49 - 8 * q^51 - 8 * q^53 - 16 * q^56 + 8 * q^61 - 32 * q^62 + 28 * q^64 + 8 * q^66 - 56 * q^68 + 16 * q^69 - 24 * q^74 - 12 * q^75 + 4 * q^81 + 48 * q^82 + 8 * q^87 - 24 * q^88 + 16 * q^90 - 16 * q^92 - 8 * q^94 - 32 * q^95

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