LMFDB - Embedded newform 2912.2.c.a.1457.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(1457,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.1457"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.16
Character	$\chi$	$=$	2912.1457
Dual form			2912.2.c.a.1457.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.275879i q^{3} +3.61620i q^{5} +1.00000 q^{7} +2.92389 q^{9} -5.52552i q^{11} +1.00000i q^{13} +0.997636 q^{15} +3.52124 q^{17} +1.10679i q^{19} -0.275879i q^{21} +7.91353 q^{23} -8.07691 q^{25} -1.63428i q^{27} -10.0364i q^{29} +5.12736 q^{31} -1.52438 q^{33} +3.61620i q^{35} -3.62244i q^{37} +0.275879 q^{39} +1.03995 q^{41} -8.73661i q^{43} +10.5734i q^{45} -13.4664 q^{47} +1.00000 q^{49} -0.971439i q^{51} +2.23708i q^{53} +19.9814 q^{55} +0.305341 q^{57} +4.78728i q^{59} -0.861749i q^{61} +2.92389 q^{63} -3.61620 q^{65} -4.23522i q^{67} -2.18318i q^{69} -0.866345 q^{71} -5.70186 q^{73} +2.22825i q^{75} -5.52552i q^{77} +7.29555 q^{79} +8.32081 q^{81} -7.27650i q^{83} +12.7335i q^{85} -2.76883 q^{87} +7.66208 q^{89} +1.00000i q^{91} -1.41453i q^{93} -4.00238 q^{95} +12.7857 q^{97} -16.1560i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) $ 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 0.275879i	− 0.159279i	−0.996824	−	0.0796395i	$-0.974623\pi$
$3$	− 0.275879i	− 0.159279i	0.996824	−	0.0796395i	$-0.0253769\pi$
$4$	0	0
$5$	3.61620i	1.61721i	0.588349	+	0.808607i	$0.299778\pi$
$5$	3.61620i	1.61721i	−0.588349	+	0.808607i	$0.700222\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.92389	0.974630
$10$	0	0
$11$	− 5.52552i	− 1.66601i	−0.553269	−	0.833003i	$-0.686620\pi$
$11$	− 5.52552i	− 1.66601i	0.553269	−	0.833003i	$-0.313380\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0.997636	0.257588
$16$	0	0
$17$	3.52124	0.854027	0.427014	−	0.904245i	$-0.359566\pi$
$17$	3.52124	0.854027	0.427014	+	0.904245i	$0.359566\pi$
$18$	0	0
$19$	1.10679i	0.253915i	0.991908	+	0.126958i	$0.0405212\pi$
$19$	1.10679i	0.253915i	−0.991908	+	0.126958i	$0.959479\pi$
$20$	0	0
$21$	− 0.275879i	− 0.0602018i
$22$	0	0
$23$	7.91353	1.65009	0.825043	−	0.565070i	$-0.191151\pi$
$23$	7.91353	1.65009	0.825043	+	0.565070i	$0.191151\pi$
$24$	0	0
$25$	−8.07691	−1.61538
$26$	0	0
$27$	− 1.63428i	− 0.314517i
$28$	0	0
$29$	− 10.0364i	− 1.86371i	−0.362833	−	0.931854i	$-0.618191\pi$
$29$	− 10.0364i	− 1.86371i	0.362833	−	0.931854i	$-0.381809\pi$
$30$	0	0
$31$	5.12736	0.920901	0.460450	−	0.887685i	$-0.347688\pi$
$31$	5.12736	0.920901	0.460450	+	0.887685i	$0.347688\pi$
$32$	0	0
$33$	−1.52438	−0.265360
$34$	0	0
$35$	3.61620i	0.611249i
$36$	0	0
$37$	− 3.62244i	− 0.595526i	−0.954640	−	0.297763i	$-0.903760\pi$
$37$	− 3.62244i	− 0.595526i	0.954640	−	0.297763i	$-0.0962405\pi$
$38$	0	0
$39$	0.275879	0.0441761
$40$	0	0
$41$	1.03995	0.162413	0.0812064	−	0.996697i	$-0.474123\pi$
$41$	1.03995	0.162413	0.0812064	+	0.996697i	$0.474123\pi$
$42$	0	0
$43$	− 8.73661i	− 1.33232i	−0.745809	−	0.666160i	$-0.767937\pi$
$43$	− 8.73661i	− 1.33232i	0.745809	−	0.666160i	$-0.232063\pi$
$44$	0	0
$45$	10.5734i	1.57619i
$46$	0	0
$47$	−13.4664	−1.96427	−0.982136	−	0.188171i	$-0.939744\pi$
$47$	−13.4664	−1.96427	−0.982136	+	0.188171i	$0.939744\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 0.971439i	− 0.136029i
$52$	0	0
$53$	2.23708i	0.307286i	0.988126	+	0.153643i	$0.0491006\pi$
$53$	2.23708i	0.307286i	−0.988126	+	0.153643i	$0.950899\pi$
$54$	0	0
$55$	19.9814	2.69429
$56$	0	0
$57$	0.305341	0.0404434
$58$	0	0
$59$	4.78728i	0.623251i	0.950205	+	0.311626i	$0.100873\pi$
$59$	4.78728i	0.623251i	−0.950205	+	0.311626i	$0.899127\pi$
$60$	0	0
$61$	− 0.861749i	− 0.110336i	−0.998477	−	0.0551678i	$-0.982431\pi$
$61$	− 0.861749i	− 0.110336i	0.998477	−	0.0551678i	$-0.0175694\pi$
$62$	0	0
$63$	2.92389	0.368376
$64$	0	0
$65$	−3.61620	−0.448534
$66$	0	0
$67$	− 4.23522i	− 0.517414i	−0.965956	−	0.258707i	$-0.916704\pi$
$67$	− 4.23522i	− 0.517414i	0.965956	−	0.258707i	$-0.0832965\pi$
$68$	0	0
$69$	− 2.18318i	− 0.262824i
$70$	0	0
$71$	−0.866345	−0.102816	−0.0514081	−	0.998678i	$-0.516371\pi$
$71$	−0.866345	−0.102816	−0.0514081	+	0.998678i	$0.516371\pi$
$72$	0	0
$73$	−5.70186	−0.667352	−0.333676	−	0.942688i	$-0.608289\pi$
$73$	−5.70186	−0.667352	−0.333676	+	0.942688i	$0.608289\pi$
$74$	0	0
$75$	2.22825i	0.257297i
$76$	0	0
$77$	− 5.52552i	− 0.629691i
$78$	0	0
$79$	7.29555	0.820814	0.410407	−	0.911902i	$-0.365387\pi$
$79$	7.29555	0.820814	0.410407	+	0.911902i	$0.365387\pi$
$80$	0	0
$81$	8.32081	0.924534
$82$	0	0
$83$	− 7.27650i	− 0.798700i	−0.916799	−	0.399350i	$-0.869236\pi$
$83$	− 7.27650i	− 0.798700i	0.916799	−	0.399350i	$-0.130764\pi$
$84$	0	0
$85$	12.7335i	1.38115i
$86$	0	0
$87$	−2.76883	−0.296850
$88$	0	0
$89$	7.66208	0.812178	0.406089	−	0.913833i	$-0.366892\pi$
$89$	7.66208	0.812178	0.406089	+	0.913833i	$0.366892\pi$
$90$	0	0
$91$	1.00000i	0.104828i
$92$	0	0
$93$	− 1.41453i	− 0.146680i
$94$	0	0
$95$	−4.00238	−0.410635
$96$	0	0
$97$	12.7857	1.29819	0.649097	−	0.760706i	$-0.275147\pi$
$97$	12.7857	1.29819	0.649097	+	0.760706i	$0.275147\pi$
$98$	0	0
$99$	− 16.1560i	− 1.62374i
$100$	0	0
$101$	9.46158i	0.941462i	0.882277	+	0.470731i	$0.156010\pi$
$101$	9.46158i	0.941462i	−0.882277	+	0.470731i	$0.843990\pi$
$102$	0	0
$103$	−0.503322	−0.0495938	−0.0247969	−	0.999693i	$-0.507894\pi$
$103$	−0.503322	−0.0495938	−0.0247969	+	0.999693i	$0.507894\pi$
$104$	0	0
$105$	0.997636	0.0973593
$106$	0	0
$107$	3.29814i	0.318843i	0.987211	+	0.159422i	$0.0509629\pi$
$107$	3.29814i	0.318843i	−0.987211	+	0.159422i	$0.949037\pi$
$108$	0	0
$109$	17.1603i	1.64366i	0.569735	+	0.821829i	$0.307046\pi$
$109$	17.1603i	1.64366i	−0.569735	+	0.821829i	$0.692954\pi$
$110$	0	0
$111$	−0.999358	−0.0948548
$112$	0	0
$113$	4.38117	0.412146	0.206073	−	0.978537i	$-0.433932\pi$
$113$	4.38117	0.412146	0.206073	+	0.978537i	$0.433932\pi$
$114$	0	0
$115$	28.6169i	2.66854i
$116$	0	0
$117$	2.92389i	0.270314i
$118$	0	0
$119$	3.52124	0.322792
$120$	0	0
$121$	−19.5313	−1.77558
$122$	0	0
$123$	− 0.286901i	− 0.0258689i
$124$	0	0
$125$	− 11.1267i	− 0.995204i
$126$	0	0
$127$	0.951245	0.0844094	0.0422047	−	0.999109i	$-0.486562\pi$
$127$	0.951245	0.0844094	0.0422047	+	0.999109i	$0.486562\pi$
$128$	0	0
$129$	−2.41025	−0.212211
$130$	0	0
$131$	6.18131i	0.540063i	0.962851	+	0.270032i	$0.0870342\pi$
$131$	6.18131i	0.540063i	−0.962851	+	0.270032i	$0.912966\pi$
$132$	0	0
$133$	1.10679i	0.0959709i
$134$	0	0
$135$	5.90988	0.508642
$136$	0	0
$137$	2.69492	0.230242	0.115121	−	0.993351i	$-0.463274\pi$
$137$	2.69492	0.230242	0.115121	+	0.993351i	$0.463274\pi$
$138$	0	0
$139$	10.0486i	0.852314i	0.904649	+	0.426157i	$0.140133\pi$
$139$	10.0486i	0.852314i	−0.904649	+	0.426157i	$0.859867\pi$
$140$	0	0
$141$	3.71510i	0.312868i
$142$	0	0
$143$	5.52552	0.462067
$144$	0	0
$145$	36.2936	3.01402
$146$	0	0
$147$	− 0.275879i	− 0.0227542i
$148$	0	0
$149$	11.6165i	0.951664i	0.879536	+	0.475832i	$0.157853\pi$
$149$	11.6165i	0.951664i	−0.879536	+	0.475832i	$0.842147\pi$
$150$	0	0
$151$	−4.16990	−0.339342	−0.169671	−	0.985501i	$-0.554270\pi$
$151$	−4.16990	−0.339342	−0.169671	+	0.985501i	$0.554270\pi$
$152$	0	0
$153$	10.2957	0.832361
$154$	0	0
$155$	18.5416i	1.48929i
$156$	0	0
$157$	19.3608i	1.54516i	0.634918	+	0.772580i	$0.281034\pi$
$157$	19.3608i	1.54516i	−0.634918	+	0.772580i	$0.718966\pi$
$158$	0	0
$159$	0.617164	0.0489443
$160$	0	0
$161$	7.91353	0.623674
$162$	0	0
$163$	− 10.0263i	− 0.785320i	−0.919684	−	0.392660i	$-0.871555\pi$
$163$	− 10.0263i	− 0.785320i	0.919684	−	0.392660i	$-0.128445\pi$
$164$	0	0
$165$	− 5.51245i	− 0.429144i
$166$	0	0
$167$	−15.3016	−1.18407	−0.592037	−	0.805911i	$-0.701676\pi$
$167$	−15.3016	−1.18407	−0.592037	+	0.805911i	$0.701676\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	3.23613i	0.247473i
$172$	0	0
$173$	12.8175i	0.974498i	0.873263	+	0.487249i	$0.162000\pi$
$173$	12.8175i	0.974498i	−0.873263	+	0.487249i	$0.838000\pi$
$174$	0	0
$175$	−8.07691	−0.610557
$176$	0	0
$177$	1.32071	0.0992709
$178$	0	0
$179$	7.47709i	0.558864i	0.960165	+	0.279432i	$0.0901462\pi$
$179$	7.47709i	0.558864i	−0.960165	+	0.279432i	$0.909854\pi$
$180$	0	0
$181$	− 22.4833i	− 1.67117i	−0.549360	−	0.835585i	$-0.685129\pi$
$181$	− 22.4833i	− 1.67117i	0.549360	−	0.835585i	$-0.314871\pi$
$182$	0	0
$183$	−0.237739	−0.0175742
$184$	0	0
$185$	13.0995	0.963093
$186$	0	0
$187$	− 19.4567i	− 1.42281i
$188$	0	0
$189$	− 1.63428i	− 0.118876i
$190$	0	0
$191$	11.6789	0.845057	0.422528	−	0.906350i	$-0.361143\pi$
$191$	11.6789	0.845057	0.422528	+	0.906350i	$0.361143\pi$
$192$	0	0
$193$	−11.6391	−0.837800	−0.418900	−	0.908032i	$-0.637584\pi$
$193$	−11.6391	−0.837800	−0.418900	+	0.908032i	$0.637584\pi$
$194$	0	0
$195$	0.997636i	0.0714422i
$196$	0	0
$197$	− 5.59686i	− 0.398760i	−0.979922	−	0.199380i	$-0.936107\pi$
$197$	− 5.59686i	− 0.398760i	0.979922	−	0.199380i	$-0.0638927\pi$
$198$	0	0
$199$	−20.2542	−1.43578	−0.717890	−	0.696156i	$-0.754892\pi$
$199$	−20.2542	−1.43578	−0.717890	+	0.696156i	$0.754892\pi$
$200$	0	0
$201$	−1.16841	−0.0824133
$202$	0	0
$203$	− 10.0364i	− 0.704415i
$204$	0	0
$205$	3.76066i	0.262656i
$206$	0	0
$207$	23.1383	1.60822
$208$	0	0
$209$	6.11559	0.423024
$210$	0	0
$211$	− 12.3571i	− 0.850695i	−0.905030	−	0.425347i	$-0.860152\pi$
$211$	− 12.3571i	− 0.850695i	0.905030	−	0.425347i	$-0.139848\pi$
$212$	0	0
$213$	0.239007i	0.0163765i
$214$	0	0
$215$	31.5933	2.15465
$216$	0	0
$217$	5.12736	0.348068
$218$	0	0
$219$	1.57303i	0.106295i
$220$	0	0
$221$	3.52124i	0.236865i
$222$	0	0
$223$	−14.7408	−0.987120	−0.493560	−	0.869712i	$-0.664305\pi$
$223$	−14.7408	−0.987120	−0.493560	+	0.869712i	$0.664305\pi$
$224$	0	0
$225$	−23.6160	−1.57440
$226$	0	0
$227$	− 0.323771i	− 0.0214894i	−0.999942	−	0.0107447i	$-0.996580\pi$
$227$	− 0.323771i	− 0.0214894i	0.999942	−	0.0107447i	$-0.00342021\pi$
$228$	0	0
$229$	22.2035i	1.46725i	0.679556	+	0.733624i	$0.262173\pi$
$229$	22.2035i	1.46725i	−0.679556	+	0.733624i	$0.737827\pi$
$230$	0	0
$231$	−1.52438	−0.100297
$232$	0	0
$233$	17.2053	1.12716	0.563579	−	0.826062i	$-0.309424\pi$
$233$	17.2053	1.12716	0.563579	+	0.826062i	$0.309424\pi$
$234$	0	0
$235$	− 48.6971i	− 3.17665i
$236$	0	0
$237$	− 2.01269i	− 0.130738i
$238$	0	0
$239$	14.0545	0.909112	0.454556	−	0.890718i	$-0.349798\pi$
$239$	14.0545	0.909112	0.454556	+	0.890718i	$0.349798\pi$
$240$	0	0
$241$	14.8607	0.957262	0.478631	−	0.878016i	$-0.341133\pi$
$241$	14.8607	0.957262	0.478631	+	0.878016i	$0.341133\pi$
$242$	0	0
$243$	− 7.19838i	− 0.461776i
$244$	0	0
$245$	3.61620i	0.231031i
$246$	0	0
$247$	−1.10679	−0.0704234
$248$	0	0
$249$	−2.00744	−0.127216
$250$	0	0
$251$	25.9434i	1.63753i	0.574128	+	0.818766i	$0.305341\pi$
$251$	25.9434i	1.63753i	−0.574128	+	0.818766i	$0.694659\pi$
$252$	0	0
$253$	− 43.7263i	− 2.74905i
$254$	0	0
$255$	3.51292	0.219988
$256$	0	0
$257$	18.1892	1.13461	0.567306	−	0.823507i	$-0.307985\pi$
$257$	18.1892	1.13461	0.567306	+	0.823507i	$0.307985\pi$
$258$	0	0
$259$	− 3.62244i	− 0.225088i
$260$	0	0
$261$	− 29.3453i	− 1.81643i
$262$	0	0
$263$	26.5683	1.63827	0.819135	−	0.573601i	$-0.194454\pi$
$263$	26.5683	1.63827	0.819135	+	0.573601i	$0.194454\pi$
$264$	0	0
$265$	−8.08972	−0.496948
$266$	0	0
$267$	− 2.11381i	− 0.129363i
$268$	0	0
$269$	− 25.0750i	− 1.52885i	−0.644713	−	0.764425i	$-0.723023\pi$
$269$	− 25.0750i	− 1.52885i	0.644713	−	0.764425i	$-0.276977\pi$
$270$	0	0
$271$	−2.65065	−0.161016	−0.0805079	−	0.996754i	$-0.525654\pi$
$271$	−2.65065	−0.161016	−0.0805079	+	0.996754i	$0.525654\pi$
$272$	0	0
$273$	0.275879	0.0166970
$274$	0	0
$275$	44.6291i	2.69123i
$276$	0	0
$277$	− 18.7189i	− 1.12471i	−0.826895	−	0.562356i	$-0.809895\pi$
$277$	− 18.7189i	− 1.12471i	0.826895	−	0.562356i	$-0.190105\pi$
$278$	0	0
$279$	14.9918	0.897538
$280$	0	0
$281$	−14.3291	−0.854803	−0.427402	−	0.904062i	$-0.640571\pi$
$281$	−14.3291	−0.854803	−0.427402	+	0.904062i	$0.640571\pi$
$282$	0	0
$283$	− 14.4237i	− 0.857400i	−0.903447	−	0.428700i	$-0.858972\pi$
$283$	− 14.4237i	− 0.857400i	0.903447	−	0.428700i	$-0.141028\pi$
$284$	0	0
$285$	1.10417i	0.0654056i
$286$	0	0
$287$	1.03995	0.0613862
$288$	0	0
$289$	−4.60084	−0.270637
$290$	0	0
$291$	− 3.52732i	− 0.206775i
$292$	0	0
$293$	− 4.97250i	− 0.290497i	−0.989395	−	0.145248i	$-0.953602\pi$
$293$	− 4.97250i	− 0.290497i	0.989395	−	0.145248i	$-0.0463981\pi$
$294$	0	0
$295$	−17.3118	−1.00793
$296$	0	0
$297$	−9.03024	−0.523988
$298$	0	0
$299$	7.91353i	0.457651i
$300$	0	0
$301$	− 8.73661i	− 0.503570i
$302$	0	0
$303$	2.61026	0.149955
$304$	0	0
$305$	3.11626	0.178436
$306$	0	0
$307$	2.53252i	0.144539i	0.997385	+	0.0722693i	$0.0230241\pi$
$307$	2.53252i	0.144539i	−0.997385	+	0.0722693i	$0.976976\pi$
$308$	0	0
$309$	0.138856i	0.00789925i
$310$	0	0
$311$	−21.4305	−1.21521	−0.607607	−	0.794238i	$-0.707871\pi$
$311$	−21.4305	−1.21521	−0.607607	+	0.794238i	$0.707871\pi$
$312$	0	0
$313$	−31.4457	−1.77742	−0.888708	−	0.458473i	$-0.848397\pi$
$313$	−31.4457	−1.77742	−0.888708	+	0.458473i	$0.848397\pi$
$314$	0	0
$315$	10.5734i	0.595742i
$316$	0	0
$317$	20.0491i	1.12607i	0.826433	+	0.563036i	$0.190367\pi$
$317$	20.0491i	1.12607i	−0.826433	+	0.563036i	$0.809633\pi$
$318$	0	0
$319$	−55.4562	−3.10495
$320$	0	0
$321$	0.909889	0.0507851
$322$	0	0
$323$	3.89728i	0.216850i
$324$	0	0
$325$	− 8.07691i	− 0.448026i
$326$	0	0
$327$	4.73417	0.261800
$328$	0	0
$329$	−13.4664	−0.742425
$330$	0	0
$331$	25.2112i	1.38573i	0.721065	+	0.692867i	$0.243653\pi$
$331$	25.2112i	1.38573i	−0.721065	+	0.692867i	$0.756347\pi$
$332$	0	0
$333$	− 10.5916i	− 0.580417i
$334$	0	0
$335$	15.3154	0.836770
$336$	0	0
$337$	−16.0856	−0.876236	−0.438118	−	0.898918i	$-0.644355\pi$
$337$	−16.0856	−0.876236	−0.438118	+	0.898918i	$0.644355\pi$
$338$	0	0
$339$	− 1.20868i	− 0.0656463i
$340$	0	0
$341$	− 28.3313i	− 1.53423i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	7.89482	0.425043
$346$	0	0
$347$	− 5.29694i	− 0.284355i	−0.989841	−	0.142177i	$-0.954590\pi$
$347$	− 5.29694i	− 0.284355i	0.989841	−	0.142177i	$-0.0454104\pi$
$348$	0	0
$349$	− 19.0084i	− 1.01749i	−0.860916	−	0.508747i	$-0.830109\pi$
$349$	− 19.0084i	− 1.01749i	0.860916	−	0.508747i	$-0.169891\pi$
$350$	0	0
$351$	1.63428	0.0872314
$352$	0	0
$353$	2.67801	0.142536	0.0712680	−	0.997457i	$-0.477295\pi$
$353$	2.67801	0.142536	0.0712680	+	0.997457i	$0.477295\pi$
$354$	0	0
$355$	− 3.13288i	− 0.166276i
$356$	0	0
$357$	− 0.971439i	− 0.0514140i
$358$	0	0
$359$	−22.7147	−1.19884	−0.599418	−	0.800436i	$-0.704601\pi$
$359$	−22.7147	−1.19884	−0.599418	+	0.800436i	$0.704601\pi$
$360$	0	0
$361$	17.7750	0.935527
$362$	0	0
$363$	5.38829i	0.282812i
$364$	0	0
$365$	− 20.6191i	− 1.07925i
$366$	0	0
$367$	3.18895	0.166462	0.0832310	−	0.996530i	$-0.473476\pi$
$367$	3.18895	0.166462	0.0832310	+	0.996530i	$0.473476\pi$
$368$	0	0
$369$	3.04070	0.158292
$370$	0	0
$371$	2.23708i	0.116143i
$372$	0	0
$373$	22.8764i	1.18450i	0.805756	+	0.592248i	$0.201760\pi$
$373$	22.8764i	1.18450i	−0.805756	+	0.592248i	$0.798240\pi$
$374$	0	0
$375$	−3.06963	−0.158515
$376$	0	0
$377$	10.0364	0.516900
$378$	0	0
$379$	14.0537i	0.721891i	0.932587	+	0.360946i	$0.117546\pi$
$379$	14.0537i	0.721891i	−0.932587	+	0.360946i	$0.882454\pi$
$380$	0	0
$381$	− 0.262429i	− 0.0134446i
$382$	0	0
$383$	5.29338	0.270479	0.135239	−	0.990813i	$-0.456820\pi$
$383$	5.29338	0.270479	0.135239	+	0.990813i	$0.456820\pi$
$384$	0	0
$385$	19.9814	1.01835
$386$	0	0
$387$	− 25.5449i	− 1.29852i
$388$	0	0
$389$	21.1015i	1.06989i	0.844887	+	0.534945i	$0.179668\pi$
$389$	21.1015i	1.06989i	−0.844887	+	0.534945i	$0.820332\pi$
$390$	0	0
$391$	27.8655	1.40922
$392$	0	0
$393$	1.70530	0.0860208
$394$	0	0
$395$	26.3822i	1.32743i
$396$	0	0
$397$	− 28.3687i	− 1.42378i	−0.702289	−	0.711892i	$-0.747838\pi$
$397$	− 28.3687i	− 1.42378i	0.702289	−	0.711892i	$-0.252162\pi$
$398$	0	0
$399$	0.305341	0.0152862
$400$	0	0
$401$	−29.0290	−1.44964	−0.724819	−	0.688940i	$-0.758077\pi$
$401$	−29.0290	−1.44964	−0.724819	+	0.688940i	$0.758077\pi$
$402$	0	0
$403$	5.12736i	0.255412i
$404$	0	0
$405$	30.0897i	1.49517i
$406$	0	0
$407$	−20.0159	−0.992150
$408$	0	0
$409$	−34.4641	−1.70414	−0.852069	−	0.523429i	$-0.824652\pi$
$409$	−34.4641	−1.70414	−0.852069	+	0.523429i	$0.824652\pi$
$410$	0	0
$411$	− 0.743472i	− 0.0366728i
$412$	0	0
$413$	4.78728i	0.235567i
$414$	0	0
$415$	26.3133	1.29167
$416$	0	0
$417$	2.77221	0.135756
$418$	0	0
$419$	− 4.27902i	− 0.209044i	−0.994523	−	0.104522i	$-0.966669\pi$
$419$	− 4.27902i	− 0.209044i	0.994523	−	0.104522i	$-0.0333312\pi$
$420$	0	0
$421$	7.01406i	0.341844i	0.985285	+	0.170922i	$0.0546747\pi$
$421$	7.01406i	0.341844i	−0.985285	+	0.170922i	$0.945325\pi$
$422$	0	0
$423$	−39.3742	−1.91444
$424$	0	0
$425$	−28.4408	−1.37958
$426$	0	0
$427$	− 0.861749i	− 0.0417029i
$428$	0	0
$429$	− 1.52438i	− 0.0735976i
$430$	0	0
$431$	29.6150	1.42651	0.713253	−	0.700906i	$-0.247221\pi$
$431$	29.6150	1.42651	0.713253	+	0.700906i	$0.247221\pi$
$432$	0	0
$433$	11.8181	0.567939	0.283970	−	0.958833i	$-0.408348\pi$
$433$	11.8181	0.567939	0.283970	+	0.958833i	$0.408348\pi$
$434$	0	0
$435$	− 10.0126i	− 0.480070i
$436$	0	0
$437$	8.75862i	0.418982i
$438$	0	0
$439$	−15.2077	−0.725822	−0.362911	−	0.931824i	$-0.618217\pi$
$439$	−15.2077	−0.725822	−0.362911	+	0.931824i	$0.618217\pi$
$440$	0	0
$441$	2.92389	0.139233
$442$	0	0
$443$	14.9404i	0.709842i	0.934896	+	0.354921i	$0.115492\pi$
$443$	14.9404i	0.709842i	−0.934896	+	0.354921i	$0.884508\pi$
$444$	0	0
$445$	27.7076i	1.31347i
$446$	0	0
$447$	3.20477	0.151580
$448$	0	0
$449$	29.9678	1.41427	0.707133	−	0.707081i	$-0.249988\pi$
$449$	29.9678	1.41427	0.707133	+	0.707081i	$0.249988\pi$
$450$	0	0
$451$	− 5.74625i	− 0.270581i
$452$	0	0
$453$	1.15039i	0.0540500i
$454$	0	0
$455$	−3.61620	−0.169530
$456$	0	0
$457$	11.0522	0.517001	0.258501	−	0.966011i	$-0.416772\pi$
$457$	11.0522	0.517001	0.258501	+	0.966011i	$0.416772\pi$
$458$	0	0
$459$	− 5.75470i	− 0.268606i
$460$	0	0
$461$	− 41.6333i	− 1.93906i	−0.244977	−	0.969529i	$-0.578780\pi$
$461$	− 41.6333i	− 1.93906i	0.244977	−	0.969529i	$-0.421220\pi$
$462$	0	0
$463$	−36.3379	−1.68877	−0.844384	−	0.535739i	$-0.820033\pi$
$463$	−36.3379	−1.68877	−0.844384	+	0.535739i	$0.820033\pi$
$464$	0	0
$465$	5.11524	0.237213
$466$	0	0
$467$	8.62570i	0.399150i	0.979883	+	0.199575i	$0.0639561\pi$
$467$	8.62570i	0.399150i	−0.979883	+	0.199575i	$0.936044\pi$
$468$	0	0
$469$	− 4.23522i	− 0.195564i
$470$	0	0
$471$	5.34124	0.246112
$472$	0	0
$473$	−48.2743	−2.21965
$474$	0	0
$475$	− 8.93944i	− 0.410170i
$476$	0	0
$477$	6.54097i	0.299490i
$478$	0	0
$479$	−14.0769	−0.643191	−0.321595	−	0.946877i	$-0.604219\pi$
$479$	−14.0769	−0.643191	−0.321595	+	0.946877i	$0.604219\pi$
$480$	0	0
$481$	3.62244	0.165169
$482$	0	0
$483$	− 2.18318i	− 0.0993382i
$484$	0	0
$485$	46.2358i	2.09946i
$486$	0	0
$487$	−30.9096	−1.40065	−0.700324	−	0.713826i	$-0.746961\pi$
$487$	−30.9096	−1.40065	−0.700324	+	0.713826i	$0.746961\pi$
$488$	0	0
$489$	−2.76605	−0.125085
$490$	0	0
$491$	30.4626i	1.37476i	0.726300	+	0.687378i	$0.241238\pi$
$491$	30.4626i	1.37476i	−0.726300	+	0.687378i	$0.758762\pi$
$492$	0	0
$493$	− 35.3405i	− 1.59166i
$494$	0	0
$495$	58.4234	2.62593
$496$	0	0
$497$	−0.866345	−0.0388609
$498$	0	0
$499$	9.18401i	0.411133i	0.978643	+	0.205566i	$0.0659036\pi$
$499$	9.18401i	0.411133i	−0.978643	+	0.205566i	$0.934096\pi$
$500$	0	0
$501$	4.22140i	0.188598i
$502$	0	0
$503$	−3.80294	−0.169565	−0.0847824	−	0.996399i	$-0.527020\pi$
$503$	−3.80294	−0.169565	−0.0847824	+	0.996399i	$0.527020\pi$
$504$	0	0
$505$	−34.2150	−1.52255
$506$	0	0
$507$	0.275879i	0.0122522i
$508$	0	0
$509$	28.6024i	1.26778i	0.773423	+	0.633890i	$0.218543\pi$
$509$	28.6024i	1.26778i	−0.773423	+	0.633890i	$0.781457\pi$
$510$	0	0
$511$	−5.70186	−0.252235
$512$	0	0
$513$	1.80880	0.0798607
$514$	0	0
$515$	− 1.82011i	− 0.0802038i
$516$	0	0
$517$	74.4087i	3.27249i
$518$	0	0
$519$	3.53609	0.155217
$520$	0	0
$521$	20.2308	0.886327	0.443164	−	0.896441i	$-0.353856\pi$
$521$	20.2308	0.886327	0.443164	+	0.896441i	$0.353856\pi$
$522$	0	0
$523$	− 25.8242i	− 1.12921i	−0.825360	−	0.564607i	$-0.809028\pi$
$523$	− 25.8242i	− 1.12921i	0.825360	−	0.564607i	$-0.190972\pi$
$524$	0	0
$525$	2.22825i	0.0972489i
$526$	0	0
$527$	18.0547	0.786475
$528$	0	0
$529$	39.6240	1.72278
$530$	0	0
$531$	13.9975i	0.607439i
$532$	0	0
$533$	1.03995i	0.0450452i
$534$	0	0
$535$	−11.9267	−0.515638
$536$	0	0
$537$	2.06278	0.0890154
$538$	0	0
$539$	− 5.52552i	− 0.238001i
$540$	0	0
$541$	− 7.96843i	− 0.342589i	−0.985220	−	0.171295i	$-0.945205\pi$
$541$	− 7.96843i	− 0.342589i	0.985220	−	0.171295i	$-0.0547950\pi$
$542$	0	0
$543$	−6.20268	−0.266183
$544$	0	0
$545$	−62.0550	−2.65815
$546$	0	0
$547$	14.9906i	0.640953i	0.947256	+	0.320477i	$0.103843\pi$
$547$	14.9906i	0.640953i	−0.947256	+	0.320477i	$0.896157\pi$
$548$	0	0
$549$	− 2.51966i	− 0.107536i
$550$	0	0
$551$	11.1082	0.473224
$552$	0	0
$553$	7.29555	0.310238
$554$	0	0
$555$	− 3.61388i	− 0.153401i
$556$	0	0
$557$	4.35114i	0.184364i	0.995742	+	0.0921819i	$0.0293841\pi$
$557$	4.35114i	0.184364i	−0.995742	+	0.0921819i	$0.970616\pi$
$558$	0	0
$559$	8.73661	0.369519
$560$	0	0
$561$	−5.36770	−0.226625
$562$	0	0
$563$	− 19.3697i	− 0.816334i	−0.912907	−	0.408167i	$-0.866168\pi$
$563$	− 19.3697i	− 0.816334i	0.912907	−	0.408167i	$-0.133832\pi$
$564$	0	0
$565$	15.8432i	0.666529i
$566$	0	0
$567$	8.32081	0.349441
$568$	0	0
$569$	37.1682	1.55817	0.779087	−	0.626916i	$-0.215683\pi$
$569$	37.1682	1.55817	0.779087	+	0.626916i	$0.215683\pi$
$570$	0	0
$571$	− 21.5095i	− 0.900143i	−0.892992	−	0.450072i	$-0.851398\pi$
$571$	− 21.5095i	− 0.900143i	0.892992	−	0.450072i	$-0.148602\pi$
$572$	0	0
$573$	− 3.22197i	− 0.134600i
$574$	0	0
$575$	−63.9169	−2.66552
$576$	0	0
$577$	−15.0168	−0.625159	−0.312579	−	0.949892i	$-0.601193\pi$
$577$	−15.0168	−0.625159	−0.312579	+	0.949892i	$0.601193\pi$
$578$	0	0
$579$	3.21099i	0.133444i
$580$	0	0
$581$	− 7.27650i	− 0.301880i
$582$	0	0
$583$	12.3610	0.511941
$584$	0	0
$585$	−10.5734	−0.437155
$586$	0	0
$587$	− 16.8666i	− 0.696161i	−0.937465	−	0.348080i	$-0.886834\pi$
$587$	− 16.8666i	− 0.696161i	0.937465	−	0.348080i	$-0.113166\pi$
$588$	0	0
$589$	5.67491i	0.233831i
$590$	0	0
$591$	−1.54406	−0.0635141
$592$	0	0
$593$	−24.9466	−1.02443	−0.512217	−	0.858856i	$-0.671176\pi$
$593$	−24.9466	−1.02443	−0.512217	+	0.858856i	$0.671176\pi$
$594$	0	0
$595$	12.7335i	0.522024i
$596$	0	0
$597$	5.58771i	0.228690i
$598$	0	0
$599$	−18.5685	−0.758687	−0.379343	−	0.925256i	$-0.623850\pi$
$599$	−18.5685	−0.758687	−0.379343	+	0.925256i	$0.623850\pi$
$600$	0	0
$601$	−1.28859	−0.0525625	−0.0262813	−	0.999655i	$-0.508367\pi$
$601$	−1.28859	−0.0525625	−0.0262813	+	0.999655i	$0.508367\pi$
$602$	0	0
$603$	− 12.3833i	− 0.504288i
$604$	0	0
$605$	− 70.6292i	− 2.87149i
$606$	0	0
$607$	−4.75102	−0.192838	−0.0964190	−	0.995341i	$-0.530739\pi$
$607$	−4.75102	−0.192838	−0.0964190	+	0.995341i	$0.530739\pi$
$608$	0	0
$609$	−2.76883	−0.112199
$610$	0	0
$611$	− 13.4664i	− 0.544791i
$612$	0	0
$613$	10.0018i	0.403967i	0.979389	+	0.201984i	$0.0647388\pi$
$613$	10.0018i	0.403967i	−0.979389	+	0.201984i	$0.935261\pi$
$614$	0	0
$615$	1.03749	0.0418356
$616$	0	0
$617$	−13.2245	−0.532396	−0.266198	−	0.963918i	$-0.585768\pi$
$617$	−13.2245	−0.532396	−0.266198	+	0.963918i	$0.585768\pi$
$618$	0	0
$619$	23.6609i	0.951011i	0.879713	+	0.475506i	$0.157735\pi$
$619$	23.6609i	0.951011i	−0.879713	+	0.475506i	$0.842265\pi$
$620$	0	0
$621$	− 12.9329i	− 0.518980i
$622$	0	0
$623$	7.66208	0.306975
$624$	0	0
$625$	−0.148100	−0.00592399
$626$	0	0
$627$	− 1.68716i	− 0.0673789i
$628$	0	0
$629$	− 12.7555i	− 0.508595i
$630$	0	0
$631$	−14.7571	−0.587469	−0.293735	−	0.955887i	$-0.594898\pi$
$631$	−14.7571	−0.587469	−0.293735	+	0.955887i	$0.594898\pi$
$632$	0	0
$633$	−3.40906	−0.135498
$634$	0	0
$635$	3.43989i	0.136508i
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	0	0
$639$	−2.53310	−0.100208
$640$	0	0
$641$	−13.7361	−0.542542	−0.271271	−	0.962503i	$-0.587444\pi$
$641$	−13.7361	−0.542542	−0.271271	+	0.962503i	$0.587444\pi$
$642$	0	0
$643$	25.9351i	1.02278i	0.859349	+	0.511390i	$0.170869\pi$
$643$	25.9351i	1.02278i	−0.859349	+	0.511390i	$0.829131\pi$
$644$	0	0
$645$	− 8.71595i	− 0.343190i
$646$	0	0
$647$	−12.2027	−0.479737	−0.239868	−	0.970805i	$-0.577104\pi$
$647$	−12.2027	−0.479737	−0.239868	+	0.970805i	$0.577104\pi$
$648$	0	0
$649$	26.4522	1.03834
$650$	0	0
$651$	− 1.41453i	− 0.0554399i
$652$	0	0
$653$	− 11.5584i	− 0.452316i	−0.974091	−	0.226158i	$-0.927383\pi$
$653$	− 11.5584i	− 0.452316i	0.974091	−	0.226158i	$-0.0726166\pi$
$654$	0	0
$655$	−22.3528	−0.873398
$656$	0	0
$657$	−16.6716	−0.650422
$658$	0	0
$659$	30.8839i	1.20307i	0.798848	+	0.601533i	$0.205443\pi$
$659$	30.8839i	1.20307i	−0.798848	+	0.601533i	$0.794557\pi$
$660$	0	0
$661$	29.5707i	1.15017i	0.818094	+	0.575084i	$0.195031\pi$
$661$	29.5707i	1.15017i	−0.818094	+	0.575084i	$0.804969\pi$
$662$	0	0
$663$	0.971439	0.0377276
$664$	0	0
$665$	−4.00238	−0.155205
$666$	0	0
$667$	− 79.4232i	− 3.07528i
$668$	0	0
$669$	4.06670i	0.157228i
$670$	0	0
$671$	−4.76161	−0.183820
$672$	0	0
$673$	−40.3299	−1.55460	−0.777301	−	0.629128i	$-0.783412\pi$
$673$	−40.3299	−1.55460	−0.777301	+	0.629128i	$0.783412\pi$
$674$	0	0
$675$	13.1999i	0.508065i
$676$	0	0
$677$	29.4826i	1.13311i	0.824025	+	0.566553i	$0.191723\pi$
$677$	29.4826i	1.13311i	−0.824025	+	0.566553i	$0.808277\pi$
$678$	0	0
$679$	12.7857	0.490671
$680$	0	0
$681$	−0.0893217	−0.00342282
$682$	0	0
$683$	− 6.90366i	− 0.264161i	−0.991239	−	0.132081i	$-0.957834\pi$
$683$	− 6.90366i	− 0.264161i	0.991239	−	0.132081i	$-0.0421658\pi$
$684$	0	0
$685$	9.74535i	0.372351i
$686$	0	0
$687$	6.12549	0.233702
$688$	0	0
$689$	−2.23708	−0.0852258
$690$	0	0
$691$	− 48.0500i	− 1.82791i	−0.405816	−	0.913955i	$-0.633012\pi$
$691$	− 48.0500i	− 1.82791i	0.405816	−	0.913955i	$-0.366988\pi$
$692$	0	0
$693$	− 16.1560i	− 0.613716i
$694$	0	0
$695$	−36.3379	−1.37837
$696$	0	0
$697$	3.66191	0.138705
$698$	0	0
$699$	− 4.74659i	− 0.179533i
$700$	0	0
$701$	1.98674i	0.0750381i	0.999296	+	0.0375190i	$0.0119455\pi$
$701$	1.98674i	0.0750381i	−0.999296	+	0.0375190i	$0.988055\pi$
$702$	0	0
$703$	4.00928	0.151213
$704$	0	0
$705$	−13.4345	−0.505974
$706$	0	0
$707$	9.46158i	0.355839i
$708$	0	0
$709$	− 16.3537i	− 0.614175i	−0.951681	−	0.307087i	$-0.900646\pi$
$709$	− 16.3537i	− 0.614175i	0.951681	−	0.307087i	$-0.0993544\pi$
$710$	0	0
$711$	21.3314	0.799990
$712$	0	0
$713$	40.5755	1.51957
$714$	0	0
$715$	19.9814i	0.747261i
$716$	0	0
$717$	− 3.87736i	− 0.144803i
$718$	0	0
$719$	42.8271	1.59718	0.798590	−	0.601875i	$-0.205579\pi$
$719$	42.8271	1.59718	0.798590	+	0.601875i	$0.205579\pi$
$720$	0	0
$721$	−0.503322	−0.0187447
$722$	0	0
$723$	− 4.09976i	− 0.152472i
$724$	0	0
$725$	81.0629i	3.01060i
$726$	0	0
$727$	−13.1739	−0.488594	−0.244297	−	0.969700i	$-0.578557\pi$
$727$	−13.1739	−0.488594	−0.244297	+	0.969700i	$0.578557\pi$
$728$	0	0
$729$	22.9765	0.850983
$730$	0	0
$731$	− 30.7637i	− 1.13784i
$732$	0	0
$733$	6.15717i	0.227420i	0.993514	+	0.113710i	$0.0362735\pi$
$733$	6.15717i	0.227420i	−0.993514	+	0.113710i	$0.963726\pi$
$734$	0	0
$735$	0.997636	0.0367983
$736$	0	0
$737$	−23.4018	−0.862015
$738$	0	0
$739$	10.2396i	0.376669i	0.982105	+	0.188334i	$0.0603088\pi$
$739$	10.2396i	0.376669i	−0.982105	+	0.188334i	$0.939691\pi$
$740$	0	0
$741$	0.305341i	0.0112170i
$742$	0	0
$743$	−19.4710	−0.714321	−0.357161	−	0.934043i	$-0.616255\pi$
$743$	−19.4710	−0.714321	−0.357161	+	0.934043i	$0.616255\pi$
$744$	0	0
$745$	−42.0077	−1.53904
$746$	0	0
$747$	− 21.2757i	− 0.778437i
$748$	0	0
$749$	3.29814i	0.120511i
$750$	0	0
$751$	−22.0975	−0.806350	−0.403175	−	0.915123i	$-0.632093\pi$
$751$	−22.0975	−0.806350	−0.403175	+	0.915123i	$0.632093\pi$
$752$	0	0
$753$	7.15725	0.260825
$754$	0	0
$755$	− 15.0792i	− 0.548788i
$756$	0	0
$757$	1.15259i	0.0418916i	0.999781	+	0.0209458i	$0.00666775\pi$
$757$	1.15259i	0.0418916i	−0.999781	+	0.0209458i	$0.993332\pi$
$758$	0	0
$759$	−12.0632	−0.437866
$760$	0	0
$761$	52.0968	1.88851	0.944253	−	0.329220i	$-0.106786\pi$
$761$	52.0968	1.88851	0.944253	+	0.329220i	$0.106786\pi$
$762$	0	0
$763$	17.1603i	0.621244i
$764$	0	0
$765$	37.2314i	1.34611i
$766$	0	0
$767$	−4.78728	−0.172859
$768$	0	0
$769$	−3.53413	−0.127444	−0.0637220	−	0.997968i	$-0.520297\pi$
$769$	−3.53413	−0.127444	−0.0637220	+	0.997968i	$0.520297\pi$
$770$	0	0
$771$	− 5.01804i	− 0.180720i
$772$	0	0
$773$	32.9280i	1.18434i	0.805814	+	0.592168i	$0.201728\pi$
$773$	32.9280i	1.18434i	−0.805814	+	0.592168i	$0.798272\pi$
$774$	0	0
$775$	−41.4132	−1.48761
$776$	0	0
$777$	−0.999358	−0.0358518
$778$	0	0
$779$	1.15101i	0.0412390i
$780$	0	0
$781$	4.78700i	0.171292i
$782$	0	0
$783$	−16.4022	−0.586168
$784$	0	0
$785$	−70.0125	−2.49885
$786$	0	0
$787$	− 17.5105i	− 0.624182i	−0.950052	−	0.312091i	$-0.898971\pi$
$787$	− 17.5105i	− 0.624182i	0.950052	−	0.312091i	$-0.101029\pi$
$788$	0	0
$789$	− 7.32964i	− 0.260942i
$790$	0	0
$791$	4.38117	0.155777
$792$	0	0
$793$	0.861749	0.0306016
$794$	0	0
$795$	2.23179i	0.0791534i
$796$	0	0
$797$	− 33.4031i	− 1.18320i	−0.806232	−	0.591600i	$-0.798497\pi$
$797$	− 33.4031i	− 1.18320i	0.806232	−	0.591600i	$-0.201503\pi$
$798$	0	0
$799$	−47.4184	−1.67754
$800$	0	0
$801$	22.4031	0.791574
$802$	0	0
$803$	31.5057i	1.11181i
$804$	0	0
$805$	28.6169i	1.00861i
$806$	0	0
$807$	−6.91768	−0.243514
$808$	0	0
$809$	11.3668	0.399636	0.199818	−	0.979833i	$-0.435965\pi$
$809$	11.3668	0.399636	0.199818	+	0.979833i	$0.435965\pi$
$810$	0	0
$811$	32.5379i	1.14256i	0.820756	+	0.571279i	$0.193553\pi$
$811$	32.5379i	1.14256i	−0.820756	+	0.571279i	$0.806447\pi$
$812$	0	0
$813$	0.731261i	0.0256465i
$814$	0	0
$815$	36.2571	1.27003
$816$	0	0
$817$	9.66959	0.338296
$818$	0	0
$819$	2.92389i	0.102169i
$820$	0	0
$821$	− 34.1067i	− 1.19033i	−0.803603	−	0.595166i	$-0.797086\pi$
$821$	− 34.1067i	− 1.19033i	0.803603	−	0.595166i	$-0.202914\pi$
$822$	0	0
$823$	35.0649	1.22229	0.611143	−	0.791520i	$-0.290710\pi$
$823$	35.0649	1.22229	0.611143	+	0.791520i	$0.290710\pi$
$824$	0	0
$825$	12.3122	0.428657
$826$	0	0
$827$	22.8625i	0.795006i	0.917601	+	0.397503i	$0.130123\pi$
$827$	22.8625i	0.795006i	−0.917601	+	0.397503i	$0.869877\pi$
$828$	0	0
$829$	− 45.9793i	− 1.59693i	−0.602042	−	0.798464i	$-0.705646\pi$
$829$	− 45.9793i	− 1.59693i	0.602042	−	0.798464i	$-0.294354\pi$
$830$	0	0
$831$	−5.16417	−0.179143
$832$	0	0
$833$	3.52124	0.122004
$834$	0	0
$835$	− 55.3337i	− 1.91490i
$836$	0	0
$837$	− 8.37954i	− 0.289639i
$838$	0	0
$839$	23.4652	0.810108	0.405054	−	0.914293i	$-0.367253\pi$
$839$	23.4652	0.810108	0.405054	+	0.914293i	$0.367253\pi$
$840$	0	0
$841$	−71.7288	−2.47341
$842$	0	0
$843$	3.95311i	0.136152i
$844$	0	0
$845$	− 3.61620i	− 0.124401i
$846$	0	0
$847$	−19.5313	−0.671104
$848$	0	0
$849$	−3.97920	−0.136566
$850$	0	0
$851$	− 28.6663i	− 0.982669i
$852$	0	0
$853$	1.79276i	0.0613831i	0.999529	+	0.0306915i	$0.00977095\pi$
$853$	1.79276i	0.0613831i	−0.999529	+	0.0306915i	$0.990229\pi$
$854$	0	0
$855$	−11.7025	−0.400217
$856$	0	0
$857$	−22.0714	−0.753943	−0.376972	−	0.926225i	$-0.623035\pi$
$857$	−22.0714	−0.753943	−0.376972	+	0.926225i	$0.623035\pi$
$858$	0	0
$859$	− 4.52672i	− 0.154450i	−0.997014	−	0.0772248i	$-0.975394\pi$
$859$	− 4.52672i	− 0.154450i	0.997014	−	0.0772248i	$-0.0246059\pi$
$860$	0	0
$861$	− 0.286901i	− 0.00977754i
$862$	0	0
$863$	−51.4286	−1.75065	−0.875325	−	0.483534i	$-0.839353\pi$
$863$	−51.4286	−1.75065	−0.875325	+	0.483534i	$0.839353\pi$
$864$	0	0
$865$	−46.3507	−1.57597
$866$	0	0
$867$	1.26928i	0.0431069i
$868$	0	0
$869$	− 40.3117i	− 1.36748i
$870$	0	0
$871$	4.23522	0.143505
$872$	0	0
$873$	37.3841	1.26526
$874$	0	0
$875$	− 11.1267i	− 0.376152i
$876$	0	0
$877$	− 38.9877i	− 1.31652i	−0.752791	−	0.658260i	$-0.771293\pi$
$877$	− 38.9877i	− 1.31652i	0.752791	−	0.658260i	$-0.228707\pi$
$878$	0	0
$879$	−1.37181	−0.0462700
$880$	0	0
$881$	36.2082	1.21989	0.609943	−	0.792445i	$-0.291192\pi$
$881$	36.2082	1.21989	0.609943	+	0.792445i	$0.291192\pi$
$882$	0	0
$883$	− 49.4027i	− 1.66253i	−0.555875	−	0.831266i	$-0.687617\pi$
$883$	− 49.4027i	− 1.66253i	0.555875	−	0.831266i	$-0.312383\pi$
$884$	0	0
$885$	4.77596i	0.160542i
$886$	0	0
$887$	−27.1660	−0.912146	−0.456073	−	0.889942i	$-0.650744\pi$
$887$	−27.1660	−0.912146	−0.456073	+	0.889942i	$0.650744\pi$
$888$	0	0
$889$	0.951245	0.0319037
$890$	0	0
$891$	− 45.9768i	− 1.54028i
$892$	0	0
$893$	− 14.9044i	− 0.498758i
$894$	0	0
$895$	−27.0387	−0.903803
$896$	0	0
$897$	2.18318	0.0728943
$898$	0	0
$899$	− 51.4601i	− 1.71629i
$900$	0	0
$901$	7.87730i	0.262431i
$902$	0	0
$903$	−2.41025	−0.0802081
$904$	0	0
$905$	81.3041	2.70264
$906$	0	0
$907$	22.3716i	0.742838i	0.928465	+	0.371419i	$0.121129\pi$
$907$	22.3716i	0.742838i	−0.928465	+	0.371419i	$0.878871\pi$
$908$	0	0
$909$	27.6646i	0.917577i
$910$	0	0
$911$	−9.55929	−0.316713	−0.158357	−	0.987382i	$-0.550620\pi$
$911$	−9.55929	−0.316713	−0.158357	+	0.987382i	$0.550620\pi$
$912$	0	0
$913$	−40.2064	−1.33064
$914$	0	0
$915$	− 0.859711i	− 0.0284212i
$916$	0	0
$917$	6.18131i	0.204125i
$918$	0	0
$919$	60.0820	1.98192	0.990961	−	0.134148i	$-0.0428298\pi$
$919$	60.0820	1.98192	0.990961	+	0.134148i	$0.0428298\pi$
$920$	0	0
$921$	0.698671	0.0230220
$922$	0	0
$923$	− 0.866345i	− 0.0285161i
$924$	0	0
$925$	29.2581i	0.962002i
$926$	0	0
$927$	−1.47166	−0.0483356
$928$	0	0
$929$	10.2401	0.335966	0.167983	−	0.985790i	$-0.446275\pi$
$929$	10.2401	0.335966	0.167983	+	0.985790i	$0.446275\pi$
$930$	0	0
$931$	1.10679i	0.0362736i
$932$	0	0
$933$	5.91225i	0.193558i
$934$	0	0
$935$	70.3593	2.30100
$936$	0	0
$937$	34.9995	1.14338	0.571692	−	0.820469i	$-0.306287\pi$
$937$	34.9995	1.14338	0.571692	+	0.820469i	$0.306287\pi$
$938$	0	0
$939$	8.67523i	0.283105i
$940$	0	0
$941$	− 20.3362i	− 0.662941i	−0.943466	−	0.331471i	$-0.892455\pi$
$941$	− 20.3362i	− 0.662941i	0.943466	−	0.331471i	$-0.107545\pi$
$942$	0	0
$943$	8.22967	0.267995
$944$	0	0
$945$	5.90988	0.192249
$946$	0	0
$947$	19.1433i	0.622073i	0.950398	+	0.311036i	$0.100676\pi$
$947$	19.1433i	0.622073i	−0.950398	+	0.311036i	$0.899324\pi$
$948$	0	0
$949$	− 5.70186i	− 0.185090i
$950$	0	0
$951$	5.53114	0.179360
$952$	0	0
$953$	41.8842	1.35676	0.678382	−	0.734710i	$-0.262682\pi$
$953$	41.8842	1.35676	0.678382	+	0.734710i	$0.262682\pi$
$954$	0	0
$955$	42.2333i	1.36664i
$956$	0	0
$957$	15.2992i	0.494553i
$958$	0	0
$959$	2.69492	0.0870234
$960$	0	0
$961$	−4.71019	−0.151941
$962$	0	0
$963$	9.64340i	0.310754i
$964$	0	0
$965$	− 42.0893i	− 1.35490i
$966$	0	0
$967$	48.0006	1.54360	0.771798	−	0.635868i	$-0.219358\pi$
$967$	48.0006	1.54360	0.771798	+	0.635868i	$0.219358\pi$
$968$	0	0
$969$	1.07518	0.0345397
$970$	0	0
$971$	− 2.14847i	− 0.0689477i	−0.999406	−	0.0344739i	$-0.989024\pi$
$971$	− 2.14847i	− 0.0689477i	0.999406	−	0.0344739i	$-0.0109755\pi$
$972$	0	0
$973$	10.0486i	0.322145i
$974$	0	0
$975$	−2.22825	−0.0713612
$976$	0	0
$977$	8.19821	0.262284	0.131142	−	0.991364i	$-0.458136\pi$
$977$	8.19821	0.262284	0.131142	+	0.991364i	$0.458136\pi$
$978$	0	0
$979$	− 42.3369i	− 1.35309i
$980$	0	0
$981$	50.1748i	1.60196i
$982$	0	0
$983$	−7.16077	−0.228393	−0.114196	−	0.993458i	$-0.536429\pi$
$983$	−7.16077	−0.228393	−0.114196	+	0.993458i	$0.536429\pi$
$984$	0	0
$985$	20.2394	0.644880
$986$	0	0
$987$	3.71510i	0.118253i
$988$	0	0
$989$	− 69.1374i	− 2.19844i
$990$	0	0
$991$	−22.3913	−0.711284	−0.355642	−	0.934622i	$-0.615738\pi$
$991$	−22.3913	−0.711284	−0.355642	+	0.934622i	$0.615738\pi$
$992$	0	0
$993$	6.95526	0.220719
$994$	0	0
$995$	− 73.2432i	− 2.32196i
$996$	0	0
$997$	− 57.3479i	− 1.81623i	−0.418725	−	0.908113i	$-0.637523\pi$
$997$	− 57.3479i	− 1.81623i	0.418725	−	0.908113i	$-0.362477\pi$
$998$	0	0
$999$	−5.92008	−0.187303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.c.a.1457.16		34
4.3	odd	2		728.2.c.a.365.18	yes	34
8.3	odd	2		728.2.c.a.365.17	✓	34
8.5	even	2	inner	2912.2.c.a.1457.19		34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.c.a.365.17	✓	34	8.3	odd	2
728.2.c.a.365.18	yes	34	4.3	odd	2
2912.2.c.a.1457.16		34	1.1	even	1	trivial
2912.2.c.a.1457.19		34	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.275879i q^{3} +3.61620i q^{5} +1.00000 q^{7} +2.92389 q^{9} -5.52552i q^{11} +1.00000i q^{13} +0.997636 q^{15} +3.52124 q^{17} +1.10679i q^{19} -0.275879i q^{21} +7.91353 q^{23} -8.07691 q^{25} -1.63428i q^{27} -10.0364i q^{29} +5.12736 q^{31} -1.52438 q^{33} +3.61620i q^{35} -3.62244i q^{37} +0.275879 q^{39} +1.03995 q^{41} -8.73661i q^{43} +10.5734i q^{45} -13.4664 q^{47} +1.00000 q^{49} -0.971439i q^{51} +2.23708i q^{53} +19.9814 q^{55} +0.305341 q^{57} +4.78728i q^{59} -0.861749i q^{61} +2.92389 q^{63} -3.61620 q^{65} -4.23522i q^{67} -2.18318i q^{69} -0.866345 q^{71} -5.70186 q^{73} +2.22825i q^{75} -5.52552i q^{77} +7.29555 q^{79} +8.32081 q^{81} -7.27650i q^{83} +12.7335i q^{85} -2.76883 q^{87} +7.66208 q^{89} +1.00000i q^{91} -1.41453i q^{93} -4.00238 q^{95} +12.7857 q^{97} -16.1560i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) \) 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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