LMFDB - Embedded newform 507.2.b.d.337.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(337,507)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.2.b.d.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.56155i q^{2} +1.00000 q^{3} -4.56155 q^{4} +0.561553i q^{5} +2.56155i q^{6} +3.56155i q^{7} -6.56155i q^{8} +1.00000 q^{9} -1.43845 q^{10} +2.00000i q^{11} -4.56155 q^{12} -9.12311 q^{14} +0.561553i q^{15} +7.68466 q^{16} -2.56155 q^{17} +2.56155i q^{18} -1.12311i q^{19} -2.56155i q^{20} +3.56155i q^{21} -5.12311 q^{22} -2.00000 q^{23} -6.56155i q^{24} +4.68466 q^{25} +1.00000 q^{27} -16.2462i q^{28} -5.68466 q^{29} -1.43845 q^{30} -1.56155i q^{31} +6.56155i q^{32} +2.00000i q^{33} -6.56155i q^{34} -2.00000 q^{35} -4.56155 q^{36} -3.43845i q^{37} +2.87689 q^{38} +3.68466 q^{40} +2.56155i q^{41} -9.12311 q^{42} -0.438447 q^{43} -9.12311i q^{44} +0.561553i q^{45} -5.12311i q^{46} +8.24621i q^{47} +7.68466 q^{48} -5.68466 q^{49} +12.0000i q^{50} -2.56155 q^{51} +11.6847 q^{53} +2.56155i q^{54} -1.12311 q^{55} +23.3693 q^{56} -1.12311i q^{57} -14.5616i q^{58} +11.1231i q^{59} -2.56155i q^{60} +12.1231 q^{61} +4.00000 q^{62} +3.56155i q^{63} -1.43845 q^{64} -5.12311 q^{66} +0.438447i q^{67} +11.6847 q^{68} -2.00000 q^{69} -5.12311i q^{70} +14.0000i q^{71} -6.56155i q^{72} +1.87689i q^{73} +8.80776 q^{74} +4.68466 q^{75} +5.12311i q^{76} -7.12311 q^{77} +9.56155 q^{79} +4.31534i q^{80} +1.00000 q^{81} -6.56155 q^{82} -9.12311i q^{83} -16.2462i q^{84} -1.43845i q^{85} -1.12311i q^{86} -5.68466 q^{87} +13.1231 q^{88} -13.1231i q^{89} -1.43845 q^{90} +9.12311 q^{92} -1.56155i q^{93} -21.1231 q^{94} +0.630683 q^{95} +6.56155i q^{96} -4.43845i q^{97} -14.5616i q^{98} +2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 10 q^{4} + 4 q^{9} - 14 q^{10} - 10 q^{12} - 20 q^{14} + 6 q^{16} - 2 q^{17} - 4 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{27} + 2 q^{29} - 14 q^{30} - 8 q^{35} - 10 q^{36} + 28 q^{38} - 10 q^{40}+ \cdots + 52 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 10 * q^4 + 4 * q^9 - 14 * q^10 - 10 * q^12 - 20 * q^14 + 6 * q^16 - 2 * q^17 - 4 * q^22 - 8 * q^23 - 6 * q^25 + 4 * q^27 + 2 * q^29 - 14 * q^30 - 8 * q^35 - 10 * q^36 + 28 * q^38 - 10 * q^40 - 20 * q^42 - 10 * q^43 + 6 * q^48 + 2 * q^49 - 2 * q^51 + 22 * q^53 + 12 * q^55 + 44 * q^56 + 32 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^66 + 22 * q^68 - 8 * q^69 - 6 * q^74 - 6 * q^75 - 12 * q^77 + 30 * q^79 + 4 * q^81 - 18 * q^82 + 2 * q^87 + 36 * q^88 - 14 * q^90 + 20 * q^92 - 68 * q^94 + 52 * 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 $

$2$	2.56155i	1.81129i	0.424035	+	0.905646i	$0.360613\pi$
$2$	2.56155i	1.81129i	−0.424035	+	0.905646i	$0.639387\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−4.56155	−2.28078
$5$	0.561553i	0.251134i	0.992085	+	0.125567i	$0.0400750\pi$
$5$	0.561553i	0.251134i	−0.992085	+	0.125567i	$0.959925\pi$
$6$	2.56155i	1.04575i
$7$	3.56155i	1.34614i	0.739579	+	0.673070i	$0.235025\pi$
$7$	3.56155i	1.34614i	−0.739579	+	0.673070i	$0.764975\pi$
$8$	− 6.56155i	− 2.31986i
$9$	1.00000	0.333333
$10$	−1.43845	−0.454877
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	−4.56155	−1.31681
$13$	0	0
$13$	0	0
$14$	−9.12311	−2.43825
$15$	0.561553i	0.144992i
$16$	7.68466	1.92116
$17$	−2.56155	−0.621268	−0.310634	−	0.950530i	$-0.600541\pi$
$17$	−2.56155	−0.621268	−0.310634	+	0.950530i	$0.600541\pi$
$18$	2.56155i	0.603764i
$19$	− 1.12311i	− 0.257658i	−0.991667	−	0.128829i	$-0.958878\pi$
$19$	− 1.12311i	− 0.257658i	0.991667	−	0.128829i	$-0.0411218\pi$
$20$	− 2.56155i	− 0.572781i
$21$	3.56155i	0.777195i
$22$	−5.12311	−1.09225
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	− 6.56155i	− 1.33937i
$25$	4.68466	0.936932
$26$	0	0
$27$	1.00000	0.192450
$28$	− 16.2462i	− 3.07025i
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	−1.43845	−0.262623
$31$	− 1.56155i	− 0.280463i	−0.990119	−	0.140232i	$-0.955215\pi$
$31$	− 1.56155i	− 0.280463i	0.990119	−	0.140232i	$-0.0447847\pi$
$32$	6.56155i	1.15993i
$33$	2.00000i	0.348155i
$34$	− 6.56155i	− 1.12530i
$35$	−2.00000	−0.338062
$36$	−4.56155	−0.760259
$37$	− 3.43845i	− 0.565277i	−0.959226	−	0.282639i	$-0.908790\pi$
$37$	− 3.43845i	− 0.565277i	0.959226	−	0.282639i	$-0.0912097\pi$
$38$	2.87689	0.466694
$39$	0	0
$40$	3.68466	0.582596
$41$	2.56155i	0.400047i	0.979791	+	0.200024i	$0.0641019\pi$
$41$	2.56155i	0.400047i	−0.979791	+	0.200024i	$0.935898\pi$
$42$	−9.12311	−1.40773
$43$	−0.438447	−0.0668626	−0.0334313	−	0.999441i	$-0.510643\pi$
$43$	−0.438447	−0.0668626	−0.0334313	+	0.999441i	$0.510643\pi$
$44$	− 9.12311i	− 1.37536i
$45$	0.561553i	0.0837114i
$46$	− 5.12311i	− 0.755361i
$47$	8.24621i	1.20283i	0.798935	+	0.601417i	$0.205397\pi$
$47$	8.24621i	1.20283i	−0.798935	+	0.601417i	$0.794603\pi$
$48$	7.68466	1.10918
$49$	−5.68466	−0.812094
$50$	12.0000i	1.69706i
$51$	−2.56155	−0.358689
$52$	0	0
$53$	11.6847	1.60501	0.802506	−	0.596645i	$-0.203500\pi$
$53$	11.6847	1.60501	0.802506	+	0.596645i	$0.203500\pi$
$54$	2.56155i	0.348583i
$55$	−1.12311	−0.151440
$56$	23.3693	3.12286
$57$	− 1.12311i	− 0.148759i
$58$	− 14.5616i	− 1.91203i
$59$	11.1231i	1.44811i	0.689745	+	0.724053i	$0.257723\pi$
$59$	11.1231i	1.44811i	−0.689745	+	0.724053i	$0.742277\pi$
$60$	− 2.56155i	− 0.330695i
$61$	12.1231	1.55220	0.776102	−	0.630607i	$-0.217194\pi$
$61$	12.1231	1.55220	0.776102	+	0.630607i	$0.217194\pi$
$62$	4.00000	0.508001
$63$	3.56155i	0.448713i
$64$	−1.43845	−0.179806
$65$	0	0
$66$	−5.12311	−0.630611
$67$	0.438447i	0.0535648i	0.999641	+	0.0267824i	$0.00852613\pi$
$67$	0.438447i	0.0535648i	−0.999641	+	0.0267824i	$0.991474\pi$
$68$	11.6847	1.41697
$69$	−2.00000	−0.240772
$70$	− 5.12311i	− 0.612328i
$71$	14.0000i	1.66149i	0.556650	+	0.830747i	$0.312086\pi$
$71$	14.0000i	1.66149i	−0.556650	+	0.830747i	$0.687914\pi$
$72$	− 6.56155i	− 0.773286i
$73$	1.87689i	0.219674i	0.993950	+	0.109837i	$0.0350329\pi$
$73$	1.87689i	0.219674i	−0.993950	+	0.109837i	$0.964967\pi$
$74$	8.80776	1.02388
$75$	4.68466	0.540938
$76$	5.12311i	0.587661i
$77$	−7.12311	−0.811753
$78$	0	0
$79$	9.56155	1.07576	0.537879	−	0.843022i	$-0.319226\pi$
$79$	9.56155	1.07576	0.537879	+	0.843022i	$0.319226\pi$
$80$	4.31534i	0.482470i
$81$	1.00000	0.111111
$82$	−6.56155	−0.724602
$83$	− 9.12311i	− 1.00139i	−0.865624	−	0.500695i	$-0.833078\pi$
$83$	− 9.12311i	− 1.00139i	0.865624	−	0.500695i	$-0.166922\pi$
$84$	− 16.2462i	− 1.77261i
$85$	− 1.43845i	− 0.156022i
$86$	− 1.12311i	− 0.121108i
$87$	−5.68466	−0.609459
$88$	13.1231	1.39893
$89$	− 13.1231i	− 1.39105i	−0.718504	−	0.695523i	$-0.755173\pi$
$89$	− 13.1231i	− 1.39105i	0.718504	−	0.695523i	$-0.244827\pi$
$90$	−1.43845	−0.151626
$91$	0	0
$92$	9.12311	0.951150
$93$	− 1.56155i	− 0.161925i
$94$	−21.1231	−2.17868
$95$	0.630683	0.0647067
$96$	6.56155i	0.669686i
$97$	− 4.43845i	− 0.450656i	−0.974283	−	0.225328i	$-0.927655\pi$
$97$	− 4.43845i	− 0.450656i	0.974283	−	0.225328i	$-0.0723454\pi$
$98$	− 14.5616i	− 1.47094i
$99$	2.00000i	0.201008i
$100$	−21.3693	−2.13693
$101$	3.43845	0.342138	0.171069	−	0.985259i	$-0.445278\pi$
$101$	3.43845	0.342138	0.171069	+	0.985259i	$0.445278\pi$
$102$	− 6.56155i	− 0.649691i
$103$	7.56155	0.745062	0.372531	−	0.928020i	$-0.378490\pi$
$103$	7.56155	0.745062	0.372531	+	0.928020i	$0.378490\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	29.9309i	2.90714i
$107$	8.24621	0.797191	0.398596	−	0.917127i	$-0.369498\pi$
$107$	8.24621	0.797191	0.398596	+	0.917127i	$0.369498\pi$
$108$	−4.56155	−0.438936
$109$	17.8078i	1.70567i	0.522177	+	0.852837i	$0.325120\pi$
$109$	17.8078i	1.70567i	−0.522177	+	0.852837i	$0.674880\pi$
$110$	− 2.87689i	− 0.274301i
$111$	− 3.43845i	− 0.326363i
$112$	27.3693i	2.58616i
$113$	−14.8078	−1.39300	−0.696499	−	0.717558i	$-0.745260\pi$
$113$	−14.8078	−1.39300	−0.696499	+	0.717558i	$0.745260\pi$
$114$	2.87689	0.269446
$115$	− 1.12311i	− 0.104730i
$116$	25.9309	2.40762
$117$	0	0
$118$	−28.4924	−2.62294
$119$	− 9.12311i	− 0.836314i
$120$	3.68466	0.336362
$121$	7.00000	0.636364
$122$	31.0540i	2.81149i
$123$	2.56155i	0.230967i
$124$	7.12311i	0.639674i
$125$	5.43845i	0.486430i
$126$	−9.12311	−0.812751
$127$	−9.56155	−0.848451	−0.424225	−	0.905557i	$-0.639454\pi$
$127$	−9.56155	−0.848451	−0.424225	+	0.905557i	$0.639454\pi$
$128$	9.43845i	0.834249i
$129$	−0.438447	−0.0386031
$130$	0	0
$131$	−17.3693	−1.51756	−0.758782	−	0.651345i	$-0.774205\pi$
$131$	−17.3693	−1.51756	−0.758782	+	0.651345i	$0.774205\pi$
$132$	− 9.12311i	− 0.794064i
$133$	4.00000	0.346844
$134$	−1.12311	−0.0970215
$135$	0.561553i	0.0483308i
$136$	16.8078i	1.44125i
$137$	1.43845i	0.122895i	0.998110	+	0.0614474i	$0.0195717\pi$
$137$	1.43845i	0.122895i	−0.998110	+	0.0614474i	$0.980428\pi$
$138$	− 5.12311i	− 0.436108i
$139$	10.9309	0.927144	0.463572	−	0.886059i	$-0.346567\pi$
$139$	10.9309	0.927144	0.463572	+	0.886059i	$0.346567\pi$
$140$	9.12311	0.771043
$141$	8.24621i	0.694456i
$142$	−35.8617	−3.00945
$143$	0	0
$144$	7.68466	0.640388
$145$	− 3.19224i	− 0.265101i
$146$	−4.80776	−0.397893
$147$	−5.68466	−0.468863
$148$	15.6847i	1.28927i
$149$	− 6.56155i	− 0.537543i	−0.963204	−	0.268772i	$-0.913382\pi$
$149$	− 6.56155i	− 0.537543i	0.963204	−	0.268772i	$-0.0866177\pi$
$150$	12.0000i	0.979796i
$151$	− 15.3693i	− 1.25074i	−0.780329	−	0.625369i	$-0.784949\pi$
$151$	− 15.3693i	− 1.25074i	0.780329	−	0.625369i	$-0.215051\pi$
$152$	−7.36932	−0.597731
$153$	−2.56155	−0.207089
$154$	− 18.2462i	− 1.47032i
$155$	0.876894	0.0704339
$156$	0	0
$157$	−4.36932	−0.348709	−0.174355	−	0.984683i	$-0.555784\pi$
$157$	−4.36932	−0.348709	−0.174355	+	0.984683i	$0.555784\pi$
$158$	24.4924i	1.94851i
$159$	11.6847	0.926654
$160$	−3.68466	−0.291298
$161$	− 7.12311i	− 0.561379i
$162$	2.56155i	0.201255i
$163$	− 15.8078i	− 1.23816i	−0.785328	−	0.619080i	$-0.787506\pi$
$163$	− 15.8078i	− 1.23816i	0.785328	−	0.619080i	$-0.212494\pi$
$164$	− 11.6847i	− 0.912419i
$165$	−1.12311	−0.0874337
$166$	23.3693	1.81381
$167$	6.24621i	0.483346i	0.970358	+	0.241673i	$0.0776962\pi$
$167$	6.24621i	0.483346i	−0.970358	+	0.241673i	$0.922304\pi$
$168$	23.3693	1.80298
$169$	0	0
$170$	3.68466	0.282600
$171$	− 1.12311i	− 0.0858860i
$172$	2.00000	0.152499
$173$	−3.75379	−0.285395	−0.142698	−	0.989766i	$-0.545578\pi$
$173$	−3.75379	−0.285395	−0.142698	+	0.989766i	$0.545578\pi$
$174$	− 14.5616i	− 1.10391i
$175$	16.6847i	1.26124i
$176$	15.3693i	1.15851i
$177$	11.1231i	0.836064i
$178$	33.6155	2.51959
$179$	13.1231	0.980867	0.490433	−	0.871479i	$-0.336838\pi$
$179$	13.1231	0.980867	0.490433	+	0.871479i	$0.336838\pi$
$180$	− 2.56155i	− 0.190927i
$181$	−9.68466	−0.719855	−0.359927	−	0.932980i	$-0.617199\pi$
$181$	−9.68466	−0.719855	−0.359927	+	0.932980i	$0.617199\pi$
$182$	0	0
$183$	12.1231	0.896166
$184$	13.1231i	0.967448i
$185$	1.93087	0.141960
$186$	4.00000	0.293294
$187$	− 5.12311i	− 0.374639i
$188$	− 37.6155i	− 2.74339i
$189$	3.56155i	0.259065i
$190$	1.61553i	0.117203i
$191$	−0.876894	−0.0634499	−0.0317249	−	0.999497i	$-0.510100\pi$
$191$	−0.876894	−0.0634499	−0.0317249	+	0.999497i	$0.510100\pi$
$192$	−1.43845	−0.103811
$193$	− 19.4924i	− 1.40310i	−0.712623	−	0.701548i	$-0.752493\pi$
$193$	− 19.4924i	− 1.40310i	0.712623	−	0.701548i	$-0.247507\pi$
$194$	11.3693	0.816269
$195$	0	0
$196$	25.9309	1.85220
$197$	− 11.3693i	− 0.810030i	−0.914310	−	0.405015i	$-0.867266\pi$
$197$	− 11.3693i	− 0.810030i	0.914310	−	0.405015i	$-0.132734\pi$
$198$	−5.12311	−0.364083
$199$	23.1771	1.64298	0.821490	−	0.570223i	$-0.193143\pi$
$199$	23.1771	1.64298	0.821490	+	0.570223i	$0.193143\pi$
$200$	− 30.7386i	− 2.17355i
$201$	0.438447i	0.0309257i
$202$	8.80776i	0.619712i
$203$	− 20.2462i	− 1.42101i
$204$	11.6847	0.818090
$205$	−1.43845	−0.100466
$206$	19.3693i	1.34952i
$207$	−2.00000	−0.139010
$208$	0	0
$209$	2.24621	0.155374
$210$	− 5.12311i	− 0.353528i
$211$	7.31534	0.503609	0.251804	−	0.967778i	$-0.418976\pi$
$211$	7.31534	0.503609	0.251804	+	0.967778i	$0.418976\pi$
$212$	−53.3002	−3.66067
$213$	14.0000i	0.959264i
$214$	21.1231i	1.44395i
$215$	− 0.246211i	− 0.0167915i
$216$	− 6.56155i	− 0.446457i
$217$	5.56155	0.377543
$218$	−45.6155	−3.08947
$219$	1.87689i	0.126829i
$220$	5.12311	0.345400
$221$	0	0
$222$	8.80776	0.591138
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	−23.3693	−1.56143
$225$	4.68466	0.312311
$226$	− 37.9309i	− 2.52312i
$227$	− 1.12311i	− 0.0745431i	−0.999305	−	0.0372716i	$-0.988133\pi$
$227$	− 1.12311i	− 0.0745431i	0.999305	−	0.0372716i	$-0.0118667\pi$
$228$	5.12311i	0.339286i
$229$	− 0.246211i	− 0.0162701i	−0.999967	−	0.00813505i	$-0.997411\pi$
$229$	− 0.246211i	− 0.0162701i	0.999967	−	0.00813505i	$-0.00258949\pi$
$230$	2.87689	0.189697
$231$	−7.12311	−0.468666
$232$	37.3002i	2.44888i
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	−4.63068	−0.302072
$236$	− 50.7386i	− 3.30280i
$237$	9.56155	0.621090
$238$	23.3693	1.51481
$239$	− 0.630683i	− 0.0407955i	−0.999792	−	0.0203977i	$-0.993507\pi$
$239$	− 0.630683i	− 0.0407955i	0.999792	−	0.0203977i	$-0.00649326\pi$
$240$	4.31534i	0.278554i
$241$	− 2.80776i	− 0.180864i	−0.995903	−	0.0904320i	$-0.971175\pi$
$241$	− 2.80776i	− 0.180864i	0.995903	−	0.0904320i	$-0.0288248\pi$
$242$	17.9309i	1.15264i
$243$	1.00000	0.0641500
$244$	−55.3002	−3.54023
$245$	− 3.19224i	− 0.203944i
$246$	−6.56155	−0.418349
$247$	0	0
$248$	−10.2462	−0.650635
$249$	− 9.12311i	− 0.578153i
$250$	−13.9309	−0.881066
$251$	30.7386	1.94021	0.970103	−	0.242695i	$-0.0780314\pi$
$251$	30.7386	1.94021	0.970103	+	0.242695i	$0.0780314\pi$
$252$	− 16.2462i	− 1.02342i
$253$	− 4.00000i	− 0.251478i
$254$	− 24.4924i	− 1.53679i
$255$	− 1.43845i	− 0.0900791i
$256$	−27.0540	−1.69087
$257$	16.1771	1.00910	0.504549	−	0.863383i	$-0.331659\pi$
$257$	16.1771	1.00910	0.504549	+	0.863383i	$0.331659\pi$
$258$	− 1.12311i	− 0.0699215i
$259$	12.2462	0.760943
$260$	0	0
$261$	−5.68466	−0.351872
$262$	− 44.4924i	− 2.74875i
$263$	−15.3693	−0.947713	−0.473856	−	0.880602i	$-0.657138\pi$
$263$	−15.3693	−0.947713	−0.473856	+	0.880602i	$0.657138\pi$
$264$	13.1231	0.807671
$265$	6.56155i	0.403073i
$266$	10.2462i	0.628236i
$267$	− 13.1231i	− 0.803121i
$268$	− 2.00000i	− 0.122169i
$269$	3.36932	0.205431	0.102715	−	0.994711i	$-0.467247\pi$
$269$	3.36932	0.205431	0.102715	+	0.994711i	$0.467247\pi$
$270$	−1.43845	−0.0875411
$271$	1.06913i	0.0649450i	0.999473	+	0.0324725i	$0.0103381\pi$
$271$	1.06913i	0.0649450i	−0.999473	+	0.0324725i	$0.989662\pi$
$272$	−19.6847	−1.19356
$273$	0	0
$274$	−3.68466	−0.222598
$275$	9.36932i	0.564991i
$276$	9.12311	0.549146
$277$	−17.6847	−1.06257	−0.531284	−	0.847194i	$-0.678290\pi$
$277$	−17.6847	−1.06257	−0.531284	+	0.847194i	$0.678290\pi$
$278$	28.0000i	1.67933i
$279$	− 1.56155i	− 0.0934877i
$280$	13.1231i	0.784256i
$281$	2.80776i	0.167497i	0.996487	+	0.0837486i	$0.0266893\pi$
$281$	2.80776i	0.167497i	−0.996487	+	0.0837486i	$0.973311\pi$
$282$	−21.1231	−1.25786
$283$	1.31534	0.0781889	0.0390945	−	0.999236i	$-0.487553\pi$
$283$	1.31534	0.0781889	0.0390945	+	0.999236i	$0.487553\pi$
$284$	− 63.8617i	− 3.78950i
$285$	0.630683	0.0373584
$286$	0	0
$287$	−9.12311	−0.538520
$288$	6.56155i	0.386643i
$289$	−10.4384	−0.614026
$290$	8.17708	0.480175
$291$	− 4.43845i	− 0.260186i
$292$	− 8.56155i	− 0.501027i
$293$	− 24.5616i	− 1.43490i	−0.696609	−	0.717451i	$-0.745309\pi$
$293$	− 24.5616i	− 1.43490i	0.696609	−	0.717451i	$-0.254691\pi$
$294$	− 14.5616i	− 0.849247i
$295$	−6.24621	−0.363668
$296$	−22.5616	−1.31136
$297$	2.00000i	0.116052i
$298$	16.8078	0.973648
$299$	0	0
$300$	−21.3693	−1.23376
$301$	− 1.56155i	− 0.0900064i
$302$	39.3693	2.26545
$303$	3.43845	0.197534
$304$	− 8.63068i	− 0.495004i
$305$	6.80776i	0.389811i
$306$	− 6.56155i	− 0.375099i
$307$	− 10.1922i	− 0.581702i	−0.956768	−	0.290851i	$-0.906062\pi$
$307$	− 10.1922i	− 0.581702i	0.956768	−	0.290851i	$-0.0939383\pi$
$308$	32.4924	1.85143
$309$	7.56155	0.430162
$310$	2.24621i	0.127576i
$311$	10.8769	0.616772	0.308386	−	0.951261i	$-0.400211\pi$
$311$	10.8769	0.616772	0.308386	+	0.951261i	$0.400211\pi$
$312$	0	0
$313$	−1.31534	−0.0743475	−0.0371738	−	0.999309i	$-0.511835\pi$
$313$	−1.31534	−0.0743475	−0.0371738	+	0.999309i	$0.511835\pi$
$314$	− 11.1922i	− 0.631614i
$315$	−2.00000	−0.112687
$316$	−43.6155	−2.45357
$317$	− 23.0540i	− 1.29484i	−0.762133	−	0.647420i	$-0.775848\pi$
$317$	− 23.0540i	− 1.29484i	0.762133	−	0.647420i	$-0.224152\pi$
$318$	29.9309i	1.67844i
$319$	− 11.3693i	− 0.636560i
$320$	− 0.807764i	− 0.0451554i
$321$	8.24621	0.460259
$322$	18.2462	1.01682
$323$	2.87689i	0.160075i
$324$	−4.56155	−0.253420
$325$	0	0
$326$	40.4924	2.24267
$327$	17.8078i	0.984772i
$328$	16.8078	0.928054
$329$	−29.3693	−1.61918
$330$	− 2.87689i	− 0.158368i
$331$	− 23.8078i	− 1.30859i	−0.756238	−	0.654297i	$-0.772965\pi$
$331$	− 23.8078i	− 1.30859i	0.756238	−	0.654297i	$-0.227035\pi$
$332$	41.6155i	2.28395i
$333$	− 3.43845i	− 0.188426i
$334$	−16.0000	−0.875481
$335$	−0.246211	−0.0134520
$336$	27.3693i	1.49312i
$337$	−2.12311	−0.115653	−0.0578265	−	0.998327i	$-0.518417\pi$
$337$	−2.12311	−0.115653	−0.0578265	+	0.998327i	$0.518417\pi$
$338$	0	0
$339$	−14.8078	−0.804247
$340$	6.56155i	0.355850i
$341$	3.12311	0.169126
$342$	2.87689	0.155565
$343$	4.68466i	0.252948i
$344$	2.87689i	0.155112i
$345$	− 1.12311i	− 0.0604660i
$346$	− 9.61553i	− 0.516934i
$347$	−13.6155	−0.730920	−0.365460	−	0.930827i	$-0.619088\pi$
$347$	−13.6155	−0.730920	−0.365460	+	0.930827i	$0.619088\pi$
$348$	25.9309	1.39004
$349$	13.8078i	0.739113i	0.929208	+	0.369556i	$0.120490\pi$
$349$	13.8078i	0.739113i	−0.929208	+	0.369556i	$0.879510\pi$
$350$	−42.7386	−2.28448
$351$	0	0
$352$	−13.1231	−0.699464
$353$	17.6847i	0.941259i	0.882331	+	0.470630i	$0.155973\pi$
$353$	17.6847i	0.941259i	−0.882331	+	0.470630i	$0.844027\pi$
$354$	−28.4924	−1.51436
$355$	−7.86174	−0.417258
$356$	59.8617i	3.17267i
$357$	− 9.12311i	− 0.482846i
$358$	33.6155i	1.77664i
$359$	− 15.3693i	− 0.811162i	−0.914059	−	0.405581i	$-0.867069\pi$
$359$	− 15.3693i	− 0.811162i	0.914059	−	0.405581i	$-0.132931\pi$
$360$	3.68466	0.194199
$361$	17.7386	0.933612
$362$	− 24.8078i	− 1.30387i
$363$	7.00000	0.367405
$364$	0	0
$365$	−1.05398	−0.0551676
$366$	31.0540i	1.62322i
$367$	20.0540	1.04681	0.523404	−	0.852084i	$-0.324662\pi$
$367$	20.0540	1.04681	0.523404	+	0.852084i	$0.324662\pi$
$368$	−15.3693	−0.801181
$369$	2.56155i	0.133349i
$370$	4.94602i	0.257132i
$371$	41.6155i	2.16057i
$372$	7.12311i	0.369316i
$373$	3.63068	0.187990	0.0939948	−	0.995573i	$-0.470036\pi$
$373$	3.63068	0.187990	0.0939948	+	0.995573i	$0.470036\pi$
$374$	13.1231	0.678580
$375$	5.43845i	0.280840i
$376$	54.1080	2.79040
$377$	0	0
$378$	−9.12311	−0.469242
$379$	11.3153i	0.581230i	0.956840	+	0.290615i	$0.0938599\pi$
$379$	11.3153i	0.581230i	−0.956840	+	0.290615i	$0.906140\pi$
$380$	−2.87689	−0.147582
$381$	−9.56155	−0.489853
$382$	− 2.24621i	− 0.114926i
$383$	26.7386i	1.36628i	0.730287	+	0.683140i	$0.239386\pi$
$383$	26.7386i	1.36628i	−0.730287	+	0.683140i	$0.760614\pi$
$384$	9.43845i	0.481654i
$385$	− 4.00000i	− 0.203859i
$386$	49.9309	2.54141
$387$	−0.438447	−0.0222875
$388$	20.2462i	1.02785i
$389$	3.05398	0.154843	0.0774213	−	0.996998i	$-0.475331\pi$
$389$	3.05398	0.154843	0.0774213	+	0.996998i	$0.475331\pi$
$390$	0	0
$391$	5.12311	0.259087
$392$	37.3002i	1.88394i
$393$	−17.3693	−0.876166
$394$	29.1231	1.46720
$395$	5.36932i	0.270160i
$396$	− 9.12311i	− 0.458453i
$397$	12.0540i	0.604972i	0.953154	+	0.302486i	$0.0978165\pi$
$397$	12.0540i	0.604972i	−0.953154	+	0.302486i	$0.902184\pi$
$398$	59.3693i	2.97591i
$399$	4.00000	0.200250
$400$	36.0000	1.80000
$401$	− 18.5616i	− 0.926920i	−0.886118	−	0.463460i	$-0.846608\pi$
$401$	− 18.5616i	− 0.926920i	0.886118	−	0.463460i	$-0.153392\pi$
$402$	−1.12311	−0.0560154
$403$	0	0
$404$	−15.6847	−0.780341
$405$	0.561553i	0.0279038i
$406$	51.8617	2.57385
$407$	6.87689	0.340875
$408$	16.8078i	0.832108i
$409$	− 18.3693i	− 0.908304i	−0.890924	−	0.454152i	$-0.849942\pi$
$409$	− 18.3693i	− 0.908304i	0.890924	−	0.454152i	$-0.150058\pi$
$410$	− 3.68466i	− 0.181972i
$411$	1.43845i	0.0709534i
$412$	−34.4924	−1.69932
$413$	−39.6155	−1.94935
$414$	− 5.12311i	− 0.251787i
$415$	5.12311	0.251483
$416$	0	0
$417$	10.9309	0.535287
$418$	5.75379i	0.281427i
$419$	−17.7538	−0.867329	−0.433665	−	0.901074i	$-0.642780\pi$
$419$	−17.7538	−0.867329	−0.433665	+	0.901074i	$0.642780\pi$
$420$	9.12311	0.445162
$421$	14.7538i	0.719056i	0.933134	+	0.359528i	$0.117062\pi$
$421$	14.7538i	0.719056i	−0.933134	+	0.359528i	$0.882938\pi$
$422$	18.7386i	0.912182i
$423$	8.24621i	0.400945i
$424$	− 76.6695i	− 3.72340i
$425$	−12.0000	−0.582086
$426$	−35.8617	−1.73751
$427$	43.1771i	2.08949i
$428$	−37.6155	−1.81822
$429$	0	0
$430$	0.630683	0.0304142
$431$	− 2.87689i	− 0.138575i	−0.997597	−	0.0692876i	$-0.977927\pi$
$431$	− 2.87689i	− 0.138575i	0.997597	−	0.0692876i	$-0.0220726\pi$
$432$	7.68466	0.369728
$433$	−25.2462	−1.21326	−0.606628	−	0.794986i	$-0.707478\pi$
$433$	−25.2462	−1.21326	−0.606628	+	0.794986i	$0.707478\pi$
$434$	14.2462i	0.683840i
$435$	− 3.19224i	− 0.153056i
$436$	− 81.2311i	− 3.89026i
$437$	2.24621i	0.107451i
$438$	−4.80776	−0.229724
$439$	1.31534	0.0627778	0.0313889	−	0.999507i	$-0.490007\pi$
$439$	1.31534	0.0627778	0.0313889	+	0.999507i	$0.490007\pi$
$440$	7.36932i	0.351318i
$441$	−5.68466	−0.270698
$442$	0	0
$443$	14.7386	0.700254	0.350127	−	0.936702i	$-0.386138\pi$
$443$	14.7386	0.700254	0.350127	+	0.936702i	$0.386138\pi$
$444$	15.6847i	0.744361i
$445$	7.36932	0.349339
$446$	−20.4924	−0.970344
$447$	− 6.56155i	− 0.310351i
$448$	− 5.12311i	− 0.242044i
$449$	− 8.24621i	− 0.389163i	−0.980886	−	0.194581i	$-0.937665\pi$
$449$	− 8.24621i	− 0.389163i	0.980886	−	0.194581i	$-0.0623348\pi$
$450$	12.0000i	0.565685i
$451$	−5.12311	−0.241238
$452$	67.5464	3.17712
$453$	− 15.3693i	− 0.722113i
$454$	2.87689	0.135019
$455$	0	0
$456$	−7.36932	−0.345100
$457$	− 28.6155i	− 1.33858i	−0.743002	−	0.669289i	$-0.766599\pi$
$457$	− 28.6155i	− 1.33858i	0.743002	−	0.669289i	$-0.233401\pi$
$458$	0.630683	0.0294699
$459$	−2.56155	−0.119563
$460$	5.12311i	0.238866i
$461$	− 36.8078i	− 1.71431i	−0.515060	−	0.857154i	$-0.672230\pi$
$461$	− 36.8078i	− 1.71431i	0.515060	−	0.857154i	$-0.327770\pi$
$462$	− 18.2462i	− 0.848891i
$463$	26.6847i	1.24014i	0.784546	+	0.620071i	$0.212896\pi$
$463$	26.6847i	1.24014i	−0.784546	+	0.620071i	$0.787104\pi$
$464$	−43.6847	−2.02801
$465$	0.876894	0.0406650
$466$	− 66.6004i	− 3.08520i
$467$	26.0000	1.20314	0.601568	−	0.798821i	$-0.294543\pi$
$467$	26.0000	1.20314	0.601568	+	0.798821i	$0.294543\pi$
$468$	0	0
$469$	−1.56155	−0.0721058
$470$	− 11.8617i	− 0.547141i
$471$	−4.36932	−0.201327
$472$	72.9848	3.35940
$473$	− 0.876894i	− 0.0403196i
$474$	24.4924i	1.12497i
$475$	− 5.26137i	− 0.241408i
$476$	41.6155i	1.90744i
$477$	11.6847	0.535004
$478$	1.61553	0.0738925
$479$	6.24621i	0.285397i	0.989766	+	0.142698i	$0.0455779\pi$
$479$	6.24621i	0.285397i	−0.989766	+	0.142698i	$0.954422\pi$
$480$	−3.68466	−0.168181
$481$	0	0
$482$	7.19224	0.327597
$483$	− 7.12311i	− 0.324113i
$484$	−31.9309	−1.45140
$485$	2.49242	0.113175
$486$	2.56155i	0.116194i
$487$	− 1.12311i	− 0.0508928i	−0.999676	−	0.0254464i	$-0.991899\pi$
$487$	− 1.12311i	− 0.0508928i	0.999676	−	0.0254464i	$-0.00810071\pi$
$488$	− 79.5464i	− 3.60090i
$489$	− 15.8078i	− 0.714852i
$490$	8.17708	0.369403
$491$	19.7538	0.891476	0.445738	−	0.895163i	$-0.352941\pi$
$491$	19.7538	0.891476	0.445738	+	0.895163i	$0.352941\pi$
$492$	− 11.6847i	− 0.526785i
$493$	14.5616	0.655819
$494$	0	0
$495$	−1.12311	−0.0504798
$496$	− 12.0000i	− 0.538816i
$497$	−49.8617	−2.23660
$498$	23.3693	1.04720
$499$	− 28.4924i	− 1.27550i	−0.770245	−	0.637748i	$-0.779866\pi$
$499$	− 28.4924i	− 1.27550i	0.770245	−	0.637748i	$-0.220134\pi$
$500$	− 24.8078i	− 1.10944i
$501$	6.24621i	0.279060i
$502$	78.7386i	3.51428i
$503$	−11.7538	−0.524076	−0.262038	−	0.965058i	$-0.584395\pi$
$503$	−11.7538	−0.524076	−0.262038	+	0.965058i	$0.584395\pi$
$504$	23.3693	1.04095
$505$	1.93087i	0.0859226i
$506$	10.2462	0.455500
$507$	0	0
$508$	43.6155	1.93513
$509$	6.80776i	0.301749i	0.988553	+	0.150874i	$0.0482089\pi$
$509$	6.80776i	0.301749i	−0.988553	+	0.150874i	$0.951791\pi$
$510$	3.68466	0.163159
$511$	−6.68466	−0.295712
$512$	− 50.4233i	− 2.22842i
$513$	− 1.12311i	− 0.0495863i
$514$	41.4384i	1.82777i
$515$	4.24621i	0.187110i
$516$	2.00000	0.0880451
$517$	−16.4924	−0.725336
$518$	31.3693i	1.37829i
$519$	−3.75379	−0.164773
$520$	0	0
$521$	−37.9309	−1.66178	−0.830891	−	0.556436i	$-0.812169\pi$
$521$	−37.9309	−1.66178	−0.830891	+	0.556436i	$0.812169\pi$
$522$	− 14.5616i	− 0.637342i
$523$	−23.8617	−1.04340	−0.521701	−	0.853129i	$-0.674702\pi$
$523$	−23.8617	−1.04340	−0.521701	+	0.853129i	$0.674702\pi$
$524$	79.2311	3.46122
$525$	16.6847i	0.728178i
$526$	− 39.3693i	− 1.71658i
$527$	4.00000i	0.174243i
$528$	15.3693i	0.668864i
$529$	−19.0000	−0.826087
$530$	−16.8078	−0.730083
$531$	11.1231i	0.482702i
$532$	−18.2462	−0.791074
$533$	0	0
$534$	33.6155	1.45469
$535$	4.63068i	0.200202i
$536$	2.87689	0.124263
$537$	13.1231	0.566304
$538$	8.63068i	0.372095i
$539$	− 11.3693i	− 0.489711i
$540$	− 2.56155i	− 0.110232i
$541$	29.7386i	1.27856i	0.768972	+	0.639282i	$0.220768\pi$
$541$	29.7386i	1.27856i	−0.768972	+	0.639282i	$0.779232\pi$
$542$	−2.73863	−0.117634
$543$	−9.68466	−0.415608
$544$	− 16.8078i	− 0.720627i
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−24.9309	−1.06597	−0.532984	−	0.846126i	$-0.678929\pi$
$547$	−24.9309	−1.06597	−0.532984	+	0.846126i	$0.678929\pi$
$548$	− 6.56155i	− 0.280296i
$549$	12.1231	0.517402
$550$	−24.0000	−1.02336
$551$	6.38447i	0.271988i
$552$	13.1231i	0.558556i
$553$	34.0540i	1.44812i
$554$	− 45.3002i	− 1.92462i
$555$	1.93087	0.0819609
$556$	−49.8617	−2.11461
$557$	14.0691i	0.596128i	0.954546	+	0.298064i	$0.0963409\pi$
$557$	14.0691i	0.596128i	−0.954546	+	0.298064i	$0.903659\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	−15.3693	−0.649472
$561$	− 5.12311i	− 0.216298i
$562$	−7.19224	−0.303386
$563$	−1.36932	−0.0577098	−0.0288549	−	0.999584i	$-0.509186\pi$
$563$	−1.36932	−0.0577098	−0.0288549	+	0.999584i	$0.509186\pi$
$564$	− 37.6155i	− 1.58390i
$565$	− 8.31534i	− 0.349829i
$566$	3.36932i	0.141623i
$567$	3.56155i	0.149571i
$568$	91.8617	3.85443
$569$	−40.7386	−1.70785	−0.853926	−	0.520394i	$-0.825785\pi$
$569$	−40.7386	−1.70785	−0.853926	+	0.520394i	$0.825785\pi$
$570$	1.61553i	0.0676670i
$571$	−19.3693	−0.810581	−0.405290	−	0.914188i	$-0.632830\pi$
$571$	−19.3693	−0.810581	−0.405290	+	0.914188i	$0.632830\pi$
$572$	0	0
$573$	−0.876894	−0.0366328
$574$	− 23.3693i	− 0.975416i
$575$	−9.36932	−0.390728
$576$	−1.43845	−0.0599353
$577$	− 29.6847i	− 1.23579i	−0.786261	−	0.617894i	$-0.787986\pi$
$577$	− 29.6847i	− 1.23579i	0.786261	−	0.617894i	$-0.212014\pi$
$578$	− 26.7386i	− 1.11218i
$579$	− 19.4924i	− 0.810077i
$580$	14.5616i	0.604636i
$581$	32.4924	1.34801
$582$	11.3693	0.471273
$583$	23.3693i	0.967858i
$584$	12.3153	0.509612
$585$	0	0
$586$	62.9157	2.59902
$587$	14.6307i	0.603873i	0.953328	+	0.301936i	$0.0976330\pi$
$587$	14.6307i	0.603873i	−0.953328	+	0.301936i	$0.902367\pi$
$588$	25.9309	1.06937
$589$	−1.75379	−0.0722636
$590$	− 16.0000i	− 0.658710i
$591$	− 11.3693i	− 0.467671i
$592$	− 26.4233i	− 1.08599i
$593$	− 44.4233i	− 1.82425i	−0.409917	−	0.912123i	$-0.634442\pi$
$593$	− 44.4233i	− 1.82425i	0.409917	−	0.912123i	$-0.365558\pi$
$594$	−5.12311	−0.210204
$595$	5.12311	0.210027
$596$	29.9309i	1.22602i
$597$	23.1771	0.948575
$598$	0	0
$599$	−0.384472	−0.0157091	−0.00785455	−	0.999969i	$-0.502500\pi$
$599$	−0.384472	−0.0157091	−0.00785455	+	0.999969i	$0.502500\pi$
$600$	− 30.7386i	− 1.25490i
$601$	−35.9309	−1.46565	−0.732825	−	0.680417i	$-0.761799\pi$
$601$	−35.9309	−1.46565	−0.732825	+	0.680417i	$0.761799\pi$
$602$	4.00000	0.163028
$603$	0.438447i	0.0178549i
$604$	70.1080i	2.85265i
$605$	3.93087i	0.159813i
$606$	8.80776i	0.357791i
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	7.36932	0.298865
$609$	− 20.2462i	− 0.820418i
$610$	−17.4384	−0.706062
$611$	0	0
$612$	11.6847	0.472324
$613$	22.8617i	0.923377i	0.887042	+	0.461688i	$0.152756\pi$
$613$	22.8617i	0.923377i	−0.887042	+	0.461688i	$0.847244\pi$
$614$	26.1080	1.05363
$615$	−1.43845	−0.0580038
$616$	46.7386i	1.88315i
$617$	− 10.8078i	− 0.435104i	−0.976049	−	0.217552i	$-0.930193\pi$
$617$	− 10.8078i	− 0.435104i	0.976049	−	0.217552i	$-0.0698072\pi$
$618$	19.3693i	0.779148i
$619$	24.3002i	0.976707i	0.872646	+	0.488353i	$0.162402\pi$
$619$	24.3002i	0.976707i	−0.872646	+	0.488353i	$0.837598\pi$
$620$	−4.00000	−0.160644
$621$	−2.00000	−0.0802572
$622$	27.8617i	1.11715i
$623$	46.7386	1.87254
$624$	0	0
$625$	20.3693	0.814773
$626$	− 3.36932i	− 0.134665i
$627$	2.24621	0.0897050
$628$	19.9309	0.795328
$629$	8.80776i	0.351189i
$630$	− 5.12311i	− 0.204109i
$631$	14.4384i	0.574786i	0.957813	+	0.287393i	$0.0927885\pi$
$631$	14.4384i	0.574786i	−0.957813	+	0.287393i	$0.907212\pi$
$632$	− 62.7386i	− 2.49561i
$633$	7.31534	0.290759
$634$	59.0540	2.34533
$635$	− 5.36932i	− 0.213075i
$636$	−53.3002	−2.11349
$637$	0	0
$638$	29.1231	1.15299
$639$	14.0000i	0.553831i
$640$	−5.30019	−0.209508
$641$	−26.1771	−1.03393	−0.516966	−	0.856006i	$-0.672939\pi$
$641$	−26.1771	−1.03393	−0.516966	+	0.856006i	$0.672939\pi$
$642$	21.1231i	0.833662i
$643$	38.5464i	1.52012i	0.649852	+	0.760061i	$0.274831\pi$
$643$	38.5464i	1.52012i	−0.649852	+	0.760061i	$0.725169\pi$
$644$	32.4924i	1.28038i
$645$	− 0.246211i	− 0.00969456i
$646$	−7.36932	−0.289942
$647$	47.6155	1.87196	0.935980	−	0.352054i	$-0.114517\pi$
$647$	47.6155	1.87196	0.935980	+	0.352054i	$0.114517\pi$
$648$	− 6.56155i	− 0.257762i
$649$	−22.2462	−0.873240
$650$	0	0
$651$	5.56155	0.217974
$652$	72.1080i	2.82397i
$653$	14.8769	0.582178	0.291089	−	0.956696i	$-0.405982\pi$
$653$	14.8769	0.582178	0.291089	+	0.956696i	$0.405982\pi$
$654$	−45.6155	−1.78371
$655$	− 9.75379i	− 0.381112i
$656$	19.6847i	0.768557i
$657$	1.87689i	0.0732246i
$658$	− 75.2311i	− 2.93281i
$659$	14.2462	0.554954	0.277477	−	0.960732i	$-0.410502\pi$
$659$	14.2462	0.554954	0.277477	+	0.960732i	$0.410502\pi$
$660$	5.12311	0.199417
$661$	− 30.3693i	− 1.18123i	−0.806954	−	0.590615i	$-0.798885\pi$
$661$	− 30.3693i	− 1.18123i	0.806954	−	0.590615i	$-0.201115\pi$
$662$	60.9848	2.37024
$663$	0	0
$664$	−59.8617	−2.32309
$665$	2.24621i	0.0871043i
$666$	8.80776	0.341294
$667$	11.3693	0.440222
$668$	− 28.4924i	− 1.10240i
$669$	8.00000i	0.309298i
$670$	− 0.630683i	− 0.0243654i
$671$	24.2462i	0.936015i
$672$	−23.3693	−0.901491
$673$	6.75379	0.260339	0.130170	−	0.991492i	$-0.458448\pi$
$673$	6.75379	0.260339	0.130170	+	0.991492i	$0.458448\pi$
$674$	− 5.43845i	− 0.209481i
$675$	4.68466	0.180313
$676$	0	0
$677$	25.6155	0.984485	0.492242	−	0.870458i	$-0.336177\pi$
$677$	25.6155	0.984485	0.492242	+	0.870458i	$0.336177\pi$
$678$	− 37.9309i	− 1.45673i
$679$	15.8078	0.606646
$680$	−9.43845	−0.361948
$681$	− 1.12311i	− 0.0430375i
$682$	8.00000i	0.306336i
$683$	− 36.1080i	− 1.38163i	−0.723030	−	0.690816i	$-0.757251\pi$
$683$	− 36.1080i	− 1.38163i	0.723030	−	0.690816i	$-0.242749\pi$
$684$	5.12311i	0.195887i
$685$	−0.807764	−0.0308631
$686$	−12.0000	−0.458162
$687$	− 0.246211i	− 0.00939355i
$688$	−3.36932	−0.128454
$689$	0	0
$690$	2.87689	0.109521
$691$	2.30019i	0.0875032i	0.999042	+	0.0437516i	$0.0139310\pi$
$691$	2.30019i	0.0875032i	−0.999042	+	0.0437516i	$0.986069\pi$
$692$	17.1231	0.650923
$693$	−7.12311	−0.270584
$694$	− 34.8769i	− 1.32391i
$695$	6.13826i	0.232837i
$696$	37.3002i	1.41386i
$697$	− 6.56155i	− 0.248537i
$698$	−35.3693	−1.33875
$699$	−26.0000	−0.983410
$700$	− 76.1080i	− 2.87661i
$701$	−19.3693	−0.731569	−0.365785	−	0.930700i	$-0.619199\pi$
$701$	−19.3693	−0.731569	−0.365785	+	0.930700i	$0.619199\pi$
$702$	0	0
$703$	−3.86174	−0.145648
$704$	− 2.87689i	− 0.108427i
$705$	−4.63068	−0.174402
$706$	−45.3002	−1.70490
$707$	12.2462i	0.460566i
$708$	− 50.7386i	− 1.90687i
$709$	− 25.4924i	− 0.957388i	−0.877982	−	0.478694i	$-0.841110\pi$
$709$	− 25.4924i	− 0.957388i	0.877982	−	0.478694i	$-0.158890\pi$
$710$	− 20.1383i	− 0.755775i
$711$	9.56155	0.358586
$712$	−86.1080	−3.22703
$713$	3.12311i	0.116961i
$714$	23.3693	0.874575
$715$	0	0
$716$	−59.8617	−2.23714
$717$	− 0.630683i	− 0.0235533i
$718$	39.3693	1.46925
$719$	−1.36932	−0.0510669	−0.0255335	−	0.999674i	$-0.508128\pi$
$719$	−1.36932	−0.0510669	−0.0255335	+	0.999674i	$0.508128\pi$
$720$	4.31534i	0.160823i
$721$	26.9309i	1.00296i
$722$	45.4384i	1.69104i
$723$	− 2.80776i	− 0.104422i
$724$	44.1771	1.64183
$725$	−26.6307	−0.989039
$726$	17.9309i	0.665477i
$727$	−39.6695	−1.47126	−0.735630	−	0.677383i	$-0.763114\pi$
$727$	−39.6695	−1.47126	−0.735630	+	0.677383i	$0.763114\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 2.69981i	− 0.0999246i
$731$	1.12311	0.0415396
$732$	−55.3002	−2.04395
$733$	53.4924i	1.97579i	0.155131	+	0.987894i	$0.450420\pi$
$733$	53.4924i	1.97579i	−0.155131	+	0.987894i	$0.549580\pi$
$734$	51.3693i	1.89608i
$735$	− 3.19224i	− 0.117747i
$736$	− 13.1231i	− 0.483724i
$737$	−0.876894	−0.0323008
$738$	−6.56155	−0.241534
$739$	6.24621i	0.229771i	0.993379	+	0.114885i	$0.0366501\pi$
$739$	6.24621i	0.229771i	−0.993379	+	0.114885i	$0.963350\pi$
$740$	−8.80776	−0.323780
$741$	0	0
$742$	−106.600	−3.91342
$743$	− 37.3693i	− 1.37095i	−0.728097	−	0.685474i	$-0.759595\pi$
$743$	− 37.3693i	− 1.37095i	0.728097	−	0.685474i	$-0.240405\pi$
$744$	−10.2462	−0.375644
$745$	3.68466	0.134995
$746$	9.30019i	0.340504i
$747$	− 9.12311i	− 0.333797i
$748$	23.3693i	0.854467i
$749$	29.3693i	1.07313i
$750$	−13.9309	−0.508683
$751$	30.1080	1.09865	0.549327	−	0.835607i	$-0.314884\pi$
$751$	30.1080	1.09865	0.549327	+	0.835607i	$0.314884\pi$
$752$	63.3693i	2.31084i
$753$	30.7386	1.12018
$754$	0	0
$755$	8.63068	0.314103
$756$	− 16.2462i	− 0.590869i
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	−28.9848	−1.05278
$759$	− 4.00000i	− 0.145191i
$760$	− 4.13826i	− 0.150110i
$761$	15.3693i	0.557137i	0.960416	+	0.278569i	$0.0898600\pi$
$761$	15.3693i	0.557137i	−0.960416	+	0.278569i	$0.910140\pi$
$762$	− 24.4924i	− 0.887267i
$763$	−63.4233	−2.29608
$764$	4.00000	0.144715
$765$	− 1.43845i	− 0.0520072i
$766$	−68.4924	−2.47473
$767$	0	0
$768$	−27.0540	−0.976226
$769$	− 18.0000i	− 0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$	− 18.0000i	− 0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$	10.2462	0.369248
$771$	16.1771	0.582603
$772$	88.9157i	3.20015i
$773$	− 7.75379i	− 0.278884i	−0.990230	−	0.139442i	$-0.955469\pi$
$773$	− 7.75379i	− 0.278884i	0.990230	−	0.139442i	$-0.0445309\pi$
$774$	− 1.12311i	− 0.0403692i
$775$	− 7.31534i	− 0.262775i
$776$	−29.1231	−1.04546
$777$	12.2462	0.439330
$778$	7.82292i	0.280465i
$779$	2.87689	0.103075
$780$	0	0
$781$	−28.0000	−1.00192
$782$	13.1231i	0.469281i
$783$	−5.68466	−0.203153
$784$	−43.6847	−1.56017
$785$	− 2.45360i	− 0.0875728i
$786$	− 44.4924i	− 1.58699i
$787$	1.17708i	0.0419584i	0.999780	+	0.0209792i	$0.00667838\pi$
$787$	1.17708i	0.0419584i	−0.999780	+	0.0209792i	$0.993322\pi$
$788$	51.8617i	1.84750i
$789$	−15.3693	−0.547162
$790$	−13.7538	−0.489338
$791$	− 52.7386i	− 1.87517i
$792$	13.1231	0.466309
$793$	0	0
$794$	−30.8769	−1.09578
$795$	6.56155i	0.232714i
$796$	−105.723	−3.74727
$797$	41.6155	1.47410	0.737049	−	0.675840i	$-0.236219\pi$
$797$	41.6155	1.47410	0.737049	+	0.675840i	$0.236219\pi$
$798$	10.2462i	0.362712i
$799$	− 21.1231i	− 0.747282i
$800$	30.7386i	1.08677i
$801$	− 13.1231i	− 0.463682i
$802$	47.5464	1.67892
$803$	−3.75379	−0.132468
$804$	− 2.00000i	− 0.0705346i
$805$	4.00000	0.140981
$806$	0	0
$807$	3.36932	0.118606
$808$	− 22.5616i	− 0.793713i
$809$	37.3002	1.31140	0.655702	−	0.755019i	$-0.272373\pi$
$809$	37.3002	1.31140	0.655702	+	0.755019i	$0.272373\pi$
$810$	−1.43845	−0.0505419
$811$	− 1.56155i	− 0.0548335i	−0.999624	−	0.0274168i	$-0.991272\pi$
$811$	− 1.56155i	− 0.0548335i	0.999624	−	0.0274168i	$-0.00872812\pi$
$812$	92.3542i	3.24100i
$813$	1.06913i	0.0374960i
$814$	17.6155i	0.617424i
$815$	8.87689	0.310944
$816$	−19.6847	−0.689101
$817$	0.492423i	0.0172277i
$818$	47.0540	1.64520
$819$	0	0
$820$	6.56155	0.229139
$821$	− 26.4924i	− 0.924592i	−0.886726	−	0.462296i	$-0.847026\pi$
$821$	− 26.4924i	− 0.924592i	0.886726	−	0.462296i	$-0.152974\pi$
$822$	−3.68466	−0.128517
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	− 49.6155i	− 1.72844i
$825$	9.36932i	0.326198i
$826$	− 101.477i	− 3.53085i
$827$	− 34.7386i	− 1.20798i	−0.796992	−	0.603990i	$-0.793577\pi$
$827$	− 34.7386i	− 1.20798i	0.796992	−	0.603990i	$-0.206423\pi$
$828$	9.12311	0.317050
$829$	19.4924	0.677000	0.338500	−	0.940966i	$-0.390081\pi$
$829$	19.4924	0.677000	0.338500	+	0.940966i	$0.390081\pi$
$830$	13.1231i	0.455510i
$831$	−17.6847	−0.613474
$832$	0	0
$833$	14.5616	0.504528
$834$	28.0000i	0.969561i
$835$	−3.50758	−0.121385
$836$	−10.2462	−0.354373
$837$	− 1.56155i	− 0.0539752i
$838$	− 45.4773i	− 1.57099i
$839$	19.6155i	0.677203i	0.940930	+	0.338602i	$0.109954\pi$
$839$	19.6155i	0.677203i	−0.940930	+	0.338602i	$0.890046\pi$
$840$	13.1231i	0.452790i
$841$	3.31534	0.114322
$842$	−37.7926	−1.30242
$843$	2.80776i	0.0967045i
$844$	−33.3693	−1.14862
$845$	0	0
$846$	−21.1231	−0.726227
$847$	24.9309i	0.856635i
$848$	89.7926	3.08349
$849$	1.31534	0.0451424
$850$	− 30.7386i	− 1.05433i
$851$	6.87689i	0.235737i
$852$	− 63.8617i	− 2.18787i
$853$	6.12311i	0.209651i	0.994491	+	0.104826i	$0.0334284\pi$
$853$	6.12311i	0.209651i	−0.994491	+	0.104826i	$0.966572\pi$
$854$	−110.600	−3.78467
$855$	0.630683	0.0215689
$856$	− 54.1080i	− 1.84937i
$857$	−31.4384	−1.07392	−0.536958	−	0.843609i	$-0.680427\pi$
$857$	−31.4384	−1.07392	−0.536958	+	0.843609i	$0.680427\pi$
$858$	0	0
$859$	20.4384	0.697351	0.348675	−	0.937244i	$-0.386632\pi$
$859$	20.4384	0.697351	0.348675	+	0.937244i	$0.386632\pi$
$860$	1.12311i	0.0382976i
$861$	−9.12311	−0.310915
$862$	7.36932	0.251000
$863$	− 2.49242i	− 0.0848430i	−0.999100	−	0.0424215i	$-0.986493\pi$
$863$	− 2.49242i	− 0.0848430i	0.999100	−	0.0424215i	$-0.0135072\pi$
$864$	6.56155i	0.223229i
$865$	− 2.10795i	− 0.0716725i
$866$	− 64.6695i	− 2.19756i
$867$	−10.4384	−0.354508
$868$	−25.3693	−0.861091
$869$	19.1231i	0.648707i
$870$	8.17708	0.277229
$871$	0	0
$872$	116.847	3.95692
$873$	− 4.43845i	− 0.150219i
$874$	−5.75379	−0.194625
$875$	−19.3693	−0.654802
$876$	− 8.56155i	− 0.289268i
$877$	19.4384i	0.656390i	0.944610	+	0.328195i	$0.106440\pi$
$877$	19.4384i	0.656390i	−0.944610	+	0.328195i	$0.893560\pi$
$878$	3.36932i	0.113709i
$879$	− 24.5616i	− 0.828441i
$880$	−8.63068	−0.290940
$881$	37.9309	1.27792	0.638962	−	0.769239i	$-0.279364\pi$
$881$	37.9309	1.27792	0.638962	+	0.769239i	$0.279364\pi$
$882$	− 14.5616i	− 0.490313i
$883$	11.8078	0.397363	0.198681	−	0.980064i	$-0.436334\pi$
$883$	11.8078	0.397363	0.198681	+	0.980064i	$0.436334\pi$
$884$	0	0
$885$	−6.24621	−0.209964
$886$	37.7538i	1.26836i
$887$	49.3693	1.65766	0.828830	−	0.559501i	$-0.189007\pi$
$887$	49.3693	1.65766	0.828830	+	0.559501i	$0.189007\pi$
$888$	−22.5616	−0.757116
$889$	− 34.0540i	− 1.14213i
$890$	18.8769i	0.632755i
$891$	2.00000i	0.0670025i
$892$	− 36.4924i	− 1.22186i
$893$	9.26137	0.309920
$894$	16.8078	0.562136
$895$	7.36932i	0.246329i
$896$	−33.6155	−1.12302
$897$	0	0
$898$	21.1231	0.704887
$899$	8.87689i	0.296061i
$900$	−21.3693	−0.712311
$901$	−29.9309	−0.997142
$902$	− 13.1231i	− 0.436952i
$903$	− 1.56155i	− 0.0519652i
$904$	97.1619i	3.23156i
$905$	− 5.43845i	− 0.180780i
$906$	39.3693	1.30796
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	5.12311i	0.170016i
$909$	3.43845	0.114046
$910$	0	0
$911$	−10.7386	−0.355787	−0.177893	−	0.984050i	$-0.556928\pi$
$911$	−10.7386	−0.355787	−0.177893	+	0.984050i	$0.556928\pi$
$912$	− 8.63068i	− 0.285790i
$913$	18.2462	0.603861
$914$	73.3002	2.42455
$915$	6.80776i	0.225058i
$916$	1.12311i	0.0371085i
$917$	− 61.8617i	− 2.04285i
$918$	− 6.56155i	− 0.216564i
$919$	44.4924	1.46767	0.733835	−	0.679328i	$-0.237729\pi$
$919$	44.4924	1.46767	0.733835	+	0.679328i	$0.237729\pi$
$920$	−7.36932	−0.242959
$921$	− 10.1922i	− 0.335846i
$922$	94.2850	3.10511
$923$	0	0
$924$	32.4924	1.06892
$925$	− 16.1080i	− 0.529626i
$926$	−68.3542	−2.24626
$927$	7.56155	0.248354
$928$	− 37.3002i	− 1.22444i
$929$	12.8078i	0.420209i	0.977679	+	0.210105i	$0.0673804\pi$
$929$	12.8078i	0.420209i	−0.977679	+	0.210105i	$0.932620\pi$
$930$	2.24621i	0.0736562i
$931$	6.38447i	0.209243i
$932$	118.600	3.88488
$933$	10.8769	0.356094
$934$	66.6004i	2.17923i
$935$	2.87689	0.0940845
$936$	0	0
$937$	−3.43845	−0.112329	−0.0561646	−	0.998422i	$-0.517887\pi$
$937$	−3.43845	−0.112329	−0.0561646	+	0.998422i	$0.517887\pi$
$938$	− 4.00000i	− 0.130605i
$939$	−1.31534	−0.0429245
$940$	21.1231	0.688960
$941$	− 2.49242i	− 0.0812507i	−0.999174	−	0.0406253i	$-0.987065\pi$
$941$	− 2.49242i	− 0.0812507i	0.999174	−	0.0406253i	$-0.0129350\pi$
$942$	− 11.1922i	− 0.364663i
$943$	− 5.12311i	− 0.166831i
$944$	85.4773i	2.78205i
$945$	−2.00000	−0.0650600
$946$	2.24621	0.0730306
$947$	− 10.7386i	− 0.348959i	−0.984661	−	0.174479i	$-0.944176\pi$
$947$	− 10.7386i	− 0.348959i	0.984661	−	0.174479i	$-0.0558242\pi$
$948$	−43.6155	−1.41657
$949$	0	0
$950$	13.4773	0.437260
$951$	− 23.0540i	− 0.747576i
$952$	−59.8617	−1.94013
$953$	34.9848	1.13327	0.566635	−	0.823969i	$-0.308245\pi$
$953$	34.9848	1.13327	0.566635	+	0.823969i	$0.308245\pi$
$954$	29.9309i	0.969048i
$955$	− 0.492423i	− 0.0159344i
$956$	2.87689i	0.0930454i
$957$	− 11.3693i	− 0.367518i
$958$	−16.0000	−0.516937
$959$	−5.12311	−0.165434
$960$	− 0.807764i	− 0.0260705i
$961$	28.5616	0.921340
$962$	0	0
$963$	8.24621	0.265730
$964$	12.8078i	0.412510i
$965$	10.9460	0.352365
$966$	18.2462	0.587062
$967$	− 9.12311i	− 0.293379i	−0.989183	−	0.146690i	$-0.953138\pi$
$967$	− 9.12311i	− 0.293379i	0.989183	−	0.146690i	$-0.0468619\pi$
$968$	− 45.9309i	− 1.47627i
$969$	2.87689i	0.0924192i
$970$	6.38447i	0.204993i
$971$	52.9848	1.70036	0.850182	−	0.526488i	$-0.176492\pi$
$971$	52.9848	1.70036	0.850182	+	0.526488i	$0.176492\pi$
$972$	−4.56155	−0.146312
$973$	38.9309i	1.24807i
$974$	2.87689	0.0921816
$975$	0	0
$976$	93.1619	2.98204
$977$	15.8229i	0.506220i	0.967438	+	0.253110i	$0.0814535\pi$
$977$	15.8229i	0.506220i	−0.967438	+	0.253110i	$0.918547\pi$
$978$	40.4924	1.29480
$979$	26.2462	0.838833
$980$	14.5616i	0.465152i
$981$	17.8078i	0.568558i
$982$	50.6004i	1.61472i
$983$	27.6155i	0.880799i	0.897802	+	0.440399i	$0.145163\pi$
$983$	27.6155i	0.880799i	−0.897802	+	0.440399i	$0.854837\pi$
$984$	16.8078	0.535812
$985$	6.38447	0.203426
$986$	37.3002i	1.18788i
$987$	−29.3693	−0.934836
$988$	0	0
$989$	0.876894	0.0278836
$990$	− 2.87689i	− 0.0914337i
$991$	40.3542	1.28189	0.640946	−	0.767586i	$-0.278542\pi$
$991$	40.3542	1.28189	0.640946	+	0.767586i	$0.278542\pi$
$992$	10.2462	0.325318
$993$	− 23.8078i	− 0.755517i
$994$	− 127.723i	− 4.05114i
$995$	13.0152i	0.412608i
$996$	41.6155i	1.31864i
$997$	20.6155	0.652900	0.326450	−	0.945214i	$-0.394147\pi$
$997$	20.6155	0.652900	0.326450	+	0.945214i	$0.394147\pi$
$998$	72.9848	2.31029
$999$	− 3.43845i	− 0.108788i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.b.d.337.4		4
3.2	odd	2		1521.2.b.h.1351.1		4
13.2	odd	12		39.2.e.b.22.1	yes	4
13.3	even	3		507.2.j.g.316.1		8
13.4	even	6		507.2.j.g.361.1		8
13.5	odd	4		507.2.a.g.1.2		2
13.6	odd	12		39.2.e.b.16.1	✓	4
13.7	odd	12		507.2.e.g.484.2		4
13.8	odd	4		507.2.a.d.1.1		2
13.9	even	3		507.2.j.g.361.4		8
13.10	even	6		507.2.j.g.316.4		8
13.11	odd	12		507.2.e.g.22.2		4
13.12	even	2	inner	507.2.b.d.337.1		4
39.2	even	12		117.2.g.c.100.2		4
39.5	even	4		1521.2.a.g.1.1		2
39.8	even	4		1521.2.a.m.1.2		2
39.32	even	12		117.2.g.c.55.2		4
39.38	odd	2		1521.2.b.h.1351.4		4
52.15	even	12		624.2.q.h.529.2		4
52.19	even	12		624.2.q.h.289.2		4
52.31	even	4		8112.2.a.bk.1.2		2
52.47	even	4		8112.2.a.bo.1.1		2
65.2	even	12		975.2.bb.i.724.1		8
65.19	odd	12		975.2.i.k.601.2		4
65.28	even	12		975.2.bb.i.724.4		8
65.32	even	12		975.2.bb.i.874.4		8
65.54	odd	12		975.2.i.k.451.2		4
65.58	even	12		975.2.bb.i.874.1		8
156.71	odd	12		1872.2.t.r.289.1		4
156.119	odd	12		1872.2.t.r.1153.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.b.16.1	✓	4	13.6	odd	12
39.2.e.b.22.1	yes	4	13.2	odd	12
117.2.g.c.55.2		4	39.32	even	12
117.2.g.c.100.2		4	39.2	even	12
507.2.a.d.1.1		2	13.8	odd	4
507.2.a.g.1.2		2	13.5	odd	4
507.2.b.d.337.1		4	13.12	even	2	inner
507.2.b.d.337.4		4	1.1	even	1	trivial
507.2.e.g.22.2		4	13.11	odd	12
507.2.e.g.484.2		4	13.7	odd	12
507.2.j.g.316.1		8	13.3	even	3
507.2.j.g.316.4		8	13.10	even	6
507.2.j.g.361.1		8	13.4	even	6
507.2.j.g.361.4		8	13.9	even	3
624.2.q.h.289.2		4	52.19	even	12
624.2.q.h.529.2		4	52.15	even	12
975.2.i.k.451.2		4	65.54	odd	12
975.2.i.k.601.2		4	65.19	odd	12
975.2.bb.i.724.1		8	65.2	even	12
975.2.bb.i.724.4		8	65.28	even	12
975.2.bb.i.874.1		8	65.58	even	12
975.2.bb.i.874.4		8	65.32	even	12
1521.2.a.g.1.1		2	39.5	even	4
1521.2.a.m.1.2		2	39.8	even	4
1521.2.b.h.1351.1		4	3.2	odd	2
1521.2.b.h.1351.4		4	39.38	odd	2
1872.2.t.r.289.1		4	156.71	odd	12
1872.2.t.r.1153.1		4	156.119	odd	12
8112.2.a.bk.1.2		2	52.31	even	4
8112.2.a.bo.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.56155i q^{2} +1.00000 q^{3} -4.56155 q^{4} +0.561553i q^{5} +2.56155i q^{6} +3.56155i q^{7} -6.56155i q^{8} +1.00000 q^{9} -1.43845 q^{10} +2.00000i q^{11} -4.56155 q^{12} -9.12311 q^{14} +0.561553i q^{15} +7.68466 q^{16} -2.56155 q^{17} +2.56155i q^{18} -1.12311i q^{19} -2.56155i q^{20} +3.56155i q^{21} -5.12311 q^{22} -2.00000 q^{23} -6.56155i q^{24} +4.68466 q^{25} +1.00000 q^{27} -16.2462i q^{28} -5.68466 q^{29} -1.43845 q^{30} -1.56155i q^{31} +6.56155i q^{32} +2.00000i q^{33} -6.56155i q^{34} -2.00000 q^{35} -4.56155 q^{36} -3.43845i q^{37} +2.87689 q^{38} +3.68466 q^{40} +2.56155i q^{41} -9.12311 q^{42} -0.438447 q^{43} -9.12311i q^{44} +0.561553i q^{45} -5.12311i q^{46} +8.24621i q^{47} +7.68466 q^{48} -5.68466 q^{49} +12.0000i q^{50} -2.56155 q^{51} +11.6847 q^{53} +2.56155i q^{54} -1.12311 q^{55} +23.3693 q^{56} -1.12311i q^{57} -14.5616i q^{58} +11.1231i q^{59} -2.56155i q^{60} +12.1231 q^{61} +4.00000 q^{62} +3.56155i q^{63} -1.43845 q^{64} -5.12311 q^{66} +0.438447i q^{67} +11.6847 q^{68} -2.00000 q^{69} -5.12311i q^{70} +14.0000i q^{71} -6.56155i q^{72} +1.87689i q^{73} +8.80776 q^{74} +4.68466 q^{75} +5.12311i q^{76} -7.12311 q^{77} +9.56155 q^{79} +4.31534i q^{80} +1.00000 q^{81} -6.56155 q^{82} -9.12311i q^{83} -16.2462i q^{84} -1.43845i q^{85} -1.12311i q^{86} -5.68466 q^{87} +13.1231 q^{88} -13.1231i q^{89} -1.43845 q^{90} +9.12311 q^{92} -1.56155i q^{93} -21.1231 q^{94} +0.630683 q^{95} +6.56155i q^{96} -4.43845i q^{97} -14.5616i q^{98} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 10 q^{4} + 4 q^{9} - 14 q^{10} - 10 q^{12} - 20 q^{14} + 6 q^{16} - 2 q^{17} - 4 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{27} + 2 q^{29} - 14 q^{30} - 8 q^{35} - 10 q^{36} + 28 q^{38} - 10 q^{40}+ \cdots + 52 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 10 * q^4 + 4 * q^9 - 14 * q^10 - 10 * q^12 - 20 * q^14 + 6 * q^16 - 2 * q^17 - 4 * q^22 - 8 * q^23 - 6 * q^25 + 4 * q^27 + 2 * q^29 - 14 * q^30 - 8 * q^35 - 10 * q^36 + 28 * q^38 - 10 * q^40 - 20 * q^42 - 10 * q^43 + 6 * q^48 + 2 * q^49 - 2 * q^51 + 22 * q^53 + 12 * q^55 + 44 * q^56 + 32 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^66 + 22 * q^68 - 8 * q^69 - 6 * q^74 - 6 * q^75 - 12 * q^77 + 30 * q^79 + 4 * q^81 - 18 * q^82 + 2 * q^87 + 36 * q^88 - 14 * q^90 + 20 * q^92 - 68 * q^94 + 52 * q^95

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