LMFDB - Embedded newform 2912.2.c.b.1457.6

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 = [38] 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:	$38$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.6
Character	$\chi$	$=$	2912.1457
Dual form			2912.2.c.b.1457.33

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40918i q^{3} -0.687773i q^{5} -1.00000 q^{7} -2.80413 q^{9} -3.21615i q^{11} -1.00000i q^{13} -1.65697 q^{15} +6.73233 q^{17} -4.13763i q^{19} +2.40918i q^{21} +7.04171 q^{23} +4.52697 q^{25} -0.471879i q^{27} -4.45687i q^{29} +8.29157 q^{31} -7.74826 q^{33} +0.687773i q^{35} +7.82265i q^{37} -2.40918 q^{39} -4.81888 q^{41} -12.7017i q^{43} +1.92861i q^{45} -7.90753 q^{47} +1.00000 q^{49} -16.2194i q^{51} +10.0261i q^{53} -2.21198 q^{55} -9.96828 q^{57} -3.97386i q^{59} -2.07834i q^{61} +2.80413 q^{63} -0.687773 q^{65} +11.4528i q^{67} -16.9647i q^{69} -11.8606 q^{71} -16.8777 q^{73} -10.9063i q^{75} +3.21615i q^{77} -0.900413 q^{79} -9.54924 q^{81} -2.06372i q^{83} -4.63031i q^{85} -10.7374 q^{87} +11.8949 q^{89} +1.00000i q^{91} -19.9759i q^{93} -2.84575 q^{95} -8.07789 q^{97} +9.01850i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 38 q^{7} - 46 q^{9} + 8 q^{15} + 20 q^{17} - 12 q^{23} - 50 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 38 q^{49} + 8 q^{57} + 46 q^{63} + 20 q^{65} + 12 q^{79} + 62 q^{81} + 56 q^{87}+O(q^{100}) $ 38 * q - 38 * q^7 - 46 * q^9 + 8 * q^15 + 20 * q^17 - 12 * q^23 - 50 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 38 * q^49 + 8 * q^57 + 46 * q^63 + 20 * q^65 + 12 * q^79 + 62 * 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
$3$	− 2.40918i	− 1.39094i	−0.718556	−	0.695469i	$-0.755196\pi$
$3$	− 2.40918i	− 1.39094i	0.718556	−	0.695469i	$-0.244804\pi$
$4$	0	0
$5$	− 0.687773i	− 0.307581i	−0.988103	−	0.153791i	$-0.950852\pi$
$5$	− 0.687773i	− 0.307581i	0.988103	−	0.153791i	$-0.0491481\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.80413	−0.934711
$10$	0	0
$11$	− 3.21615i	− 0.969704i	−0.874596	−	0.484852i	$-0.838873\pi$
$11$	− 3.21615i	− 0.969704i	0.874596	−	0.484852i	$-0.161127\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−1.65697	−0.427827
$16$	0	0
$17$	6.73233	1.63283	0.816415	−	0.577466i	$-0.195958\pi$
$17$	6.73233	1.63283	0.816415	+	0.577466i	$0.195958\pi$
$18$	0	0
$19$	− 4.13763i	− 0.949237i	−0.880192	−	0.474618i	$-0.842586\pi$
$19$	− 4.13763i	− 0.949237i	0.880192	−	0.474618i	$-0.157414\pi$
$20$	0	0
$21$	2.40918i	0.525725i
$22$	0	0
$23$	7.04171	1.46830	0.734149	−	0.678988i	$-0.237581\pi$
$23$	7.04171	1.46830	0.734149	+	0.678988i	$0.237581\pi$
$24$	0	0
$25$	4.52697	0.905394
$26$	0	0
$27$	− 0.471879i	− 0.0908132i
$28$	0	0
$29$	− 4.45687i	− 0.827620i	−0.910363	−	0.413810i	$-0.864198\pi$
$29$	− 4.45687i	− 0.827620i	0.910363	−	0.413810i	$-0.135802\pi$
$30$	0	0
$31$	8.29157	1.48921	0.744605	−	0.667505i	$-0.232638\pi$
$31$	8.29157	1.48921	0.744605	+	0.667505i	$0.232638\pi$
$32$	0	0
$33$	−7.74826	−1.34880
$34$	0	0
$35$	0.687773i	0.116255i
$36$	0	0
$37$	7.82265i	1.28604i	0.765851	+	0.643018i	$0.222318\pi$
$37$	7.82265i	1.28604i	−0.765851	+	0.643018i	$0.777682\pi$
$38$	0	0
$39$	−2.40918	−0.385777
$40$	0	0
$41$	−4.81888	−0.752583	−0.376291	−	0.926501i	$-0.622801\pi$
$41$	−4.81888	−0.752583	−0.376291	+	0.926501i	$0.622801\pi$
$42$	0	0
$43$	− 12.7017i	− 1.93699i	−0.249032	−	0.968495i	$-0.580113\pi$
$43$	− 12.7017i	− 1.93699i	0.249032	−	0.968495i	$-0.419887\pi$
$44$	0	0
$45$	1.92861i	0.287500i
$46$	0	0
$47$	−7.90753	−1.15343	−0.576716	−	0.816945i	$-0.695666\pi$
$47$	−7.90753	−1.15343	−0.576716	+	0.816945i	$0.695666\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 16.2194i	− 2.27117i
$52$	0	0
$53$	10.0261i	1.37719i	0.725147	+	0.688594i	$0.241772\pi$
$53$	10.0261i	1.37719i	−0.725147	+	0.688594i	$0.758228\pi$
$54$	0	0
$55$	−2.21198	−0.298263
$56$	0	0
$57$	−9.96828	−1.32033
$58$	0	0
$59$	− 3.97386i	− 0.517352i	−0.965964	−	0.258676i	$-0.916714\pi$
$59$	− 3.97386i	− 0.517352i	0.965964	−	0.258676i	$-0.0832862\pi$
$60$	0	0
$61$	− 2.07834i	− 0.266104i	−0.991109	−	0.133052i	$-0.957522\pi$
$61$	− 2.07834i	− 0.266104i	0.991109	−	0.133052i	$-0.0424777\pi$
$62$	0	0
$63$	2.80413	0.353287
$64$	0	0
$65$	−0.687773	−0.0853077
$66$	0	0
$67$	11.4528i	1.39918i	0.714543	+	0.699591i	$0.246635\pi$
$67$	11.4528i	1.39918i	−0.714543	+	0.699591i	$0.753365\pi$
$68$	0	0
$69$	− 16.9647i	− 2.04231i
$70$	0	0
$71$	−11.8606	−1.40759	−0.703797	−	0.710401i	$-0.748514\pi$
$71$	−11.8606	−1.40759	−0.703797	+	0.710401i	$0.748514\pi$
$72$	0	0
$73$	−16.8777	−1.97538	−0.987691	−	0.156420i	$-0.950005\pi$
$73$	−16.8777	−1.97538	−0.987691	+	0.156420i	$0.950005\pi$
$74$	0	0
$75$	− 10.9063i	− 1.25935i
$76$	0	0
$77$	3.21615i	0.366514i
$78$	0	0
$79$	−0.900413	−0.101304	−0.0506522	−	0.998716i	$-0.516130\pi$
$79$	−0.900413	−0.101304	−0.0506522	+	0.998716i	$0.516130\pi$
$80$	0	0
$81$	−9.54924	−1.06103
$82$	0	0
$83$	− 2.06372i	− 0.226522i	−0.993565	−	0.113261i	$-0.963870\pi$
$83$	− 2.06372i	− 0.226522i	0.993565	−	0.113261i	$-0.0361297\pi$
$84$	0	0
$85$	− 4.63031i	− 0.502228i
$86$	0	0
$87$	−10.7374	−1.15117
$88$	0	0
$89$	11.8949	1.26085	0.630427	−	0.776249i	$-0.282880\pi$
$89$	11.8949	1.26085	0.630427	+	0.776249i	$0.282880\pi$
$90$	0	0
$91$	1.00000i	0.104828i
$92$	0	0
$93$	− 19.9759i	− 2.07140i
$94$	0	0
$95$	−2.84575	−0.291968
$96$	0	0
$97$	−8.07789	−0.820185	−0.410093	−	0.912044i	$-0.634504\pi$
$97$	−8.07789	−0.820185	−0.410093	+	0.912044i	$0.634504\pi$
$98$	0	0
$99$	9.01850i	0.906393i
$100$	0	0
$101$	7.62035i	0.758254i	0.925345	+	0.379127i	$0.123776\pi$
$101$	7.62035i	0.758254i	−0.925345	+	0.379127i	$0.876224\pi$
$102$	0	0
$103$	5.04500	0.497098	0.248549	−	0.968619i	$-0.420046\pi$
$103$	5.04500	0.497098	0.248549	+	0.968619i	$0.420046\pi$
$104$	0	0
$105$	1.65697	0.161703
$106$	0	0
$107$	16.1219i	1.55856i	0.626675	+	0.779281i	$0.284415\pi$
$107$	16.1219i	1.55856i	−0.626675	+	0.779281i	$0.715585\pi$
$108$	0	0
$109$	− 11.1382i	− 1.06685i	−0.845848	−	0.533424i	$-0.820905\pi$
$109$	− 11.1382i	− 1.06685i	0.845848	−	0.533424i	$-0.179095\pi$
$110$	0	0
$111$	18.8462	1.78880
$112$	0	0
$113$	19.2899	1.81464	0.907320	−	0.420442i	$-0.138125\pi$
$113$	19.2899	1.81464	0.907320	+	0.420442i	$0.138125\pi$
$114$	0	0
$115$	− 4.84309i	− 0.451621i
$116$	0	0
$117$	2.80413i	0.259242i
$118$	0	0
$119$	−6.73233	−0.617152
$120$	0	0
$121$	0.656411	0.0596738
$122$	0	0
$123$	11.6095i	1.04680i
$124$	0	0
$125$	− 6.55239i	− 0.586063i
$126$	0	0
$127$	−12.8222	−1.13778	−0.568892	−	0.822412i	$-0.692628\pi$
$127$	−12.8222	−1.13778	−0.568892	+	0.822412i	$0.692628\pi$
$128$	0	0
$129$	−30.6006	−2.69424
$130$	0	0
$131$	− 3.98575i	− 0.348237i	−0.984725	−	0.174118i	$-0.944292\pi$
$131$	− 3.98575i	− 0.348237i	0.984725	−	0.174118i	$-0.0557075\pi$
$132$	0	0
$133$	4.13763i	0.358778i
$134$	0	0
$135$	−0.324546	−0.0279324
$136$	0	0
$137$	5.92401	0.506122	0.253061	−	0.967450i	$-0.418563\pi$
$137$	5.92401	0.506122	0.253061	+	0.967450i	$0.418563\pi$
$138$	0	0
$139$	3.12806i	0.265318i	0.991162	+	0.132659i	$0.0423516\pi$
$139$	3.12806i	0.265318i	−0.991162	+	0.132659i	$0.957648\pi$
$140$	0	0
$141$	19.0506i	1.60435i
$142$	0	0
$143$	−3.21615	−0.268948
$144$	0	0
$145$	−3.06531	−0.254560
$146$	0	0
$147$	− 2.40918i	− 0.198706i
$148$	0	0
$149$	2.20901i	0.180969i	0.995898	+	0.0904846i	$0.0288416\pi$
$149$	2.20901i	0.180969i	−0.995898	+	0.0904846i	$0.971158\pi$
$150$	0	0
$151$	4.94770	0.402638	0.201319	−	0.979526i	$-0.435477\pi$
$151$	4.94770	0.402638	0.201319	+	0.979526i	$0.435477\pi$
$152$	0	0
$153$	−18.8783	−1.52622
$154$	0	0
$155$	− 5.70272i	− 0.458053i
$156$	0	0
$157$	− 20.8723i	− 1.66579i	−0.553430	−	0.832895i	$-0.686681\pi$
$157$	− 20.8723i	− 1.66579i	0.553430	−	0.832895i	$-0.313319\pi$
$158$	0	0
$159$	24.1546	1.91558
$160$	0	0
$161$	−7.04171	−0.554964
$162$	0	0
$163$	11.1250i	0.871374i	0.900098	+	0.435687i	$0.143495\pi$
$163$	11.1250i	0.871374i	−0.900098	+	0.435687i	$0.856505\pi$
$164$	0	0
$165$	5.32904i	0.414865i
$166$	0	0
$167$	5.73008	0.443407	0.221703	−	0.975114i	$-0.428838\pi$
$167$	5.73008	0.443407	0.221703	+	0.975114i	$0.428838\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	11.6025i	0.887262i
$172$	0	0
$173$	− 8.86006i	− 0.673618i	−0.941573	−	0.336809i	$-0.890652\pi$
$173$	− 8.86006i	− 0.673618i	0.941573	−	0.336809i	$-0.109348\pi$
$174$	0	0
$175$	−4.52697	−0.342207
$176$	0	0
$177$	−9.57373	−0.719605
$178$	0	0
$179$	5.50445i	0.411422i	0.978613	+	0.205711i	$0.0659507\pi$
$179$	5.50445i	0.411422i	−0.978613	+	0.205711i	$0.934049\pi$
$180$	0	0
$181$	14.5090i	1.07845i	0.842163	+	0.539223i	$0.181282\pi$
$181$	14.5090i	1.07845i	−0.842163	+	0.539223i	$0.818718\pi$
$182$	0	0
$183$	−5.00708	−0.370134
$184$	0	0
$185$	5.38021	0.395561
$186$	0	0
$187$	− 21.6522i	− 1.58336i
$188$	0	0
$189$	0.471879i	0.0343242i
$190$	0	0
$191$	−17.4734	−1.26433	−0.632165	−	0.774834i	$-0.717834\pi$
$191$	−17.4734	−1.26433	−0.632165	+	0.774834i	$0.717834\pi$
$192$	0	0
$193$	5.20134	0.374401	0.187200	−	0.982322i	$-0.440059\pi$
$193$	5.20134	0.374401	0.187200	+	0.982322i	$0.440059\pi$
$194$	0	0
$195$	1.65697i	0.118658i
$196$	0	0
$197$	− 5.95897i	− 0.424559i	−0.977209	−	0.212279i	$-0.931911\pi$
$197$	− 5.95897i	− 0.424559i	0.977209	−	0.212279i	$-0.0680887\pi$
$198$	0	0
$199$	−12.1075	−0.858276	−0.429138	−	0.903239i	$-0.641183\pi$
$199$	−12.1075	−0.858276	−0.429138	+	0.903239i	$0.641183\pi$
$200$	0	0
$201$	27.5918	1.94618
$202$	0	0
$203$	4.45687i	0.312811i
$204$	0	0
$205$	3.31429i	0.231480i
$206$	0	0
$207$	−19.7459	−1.37243
$208$	0	0
$209$	−13.3072	−0.920479
$210$	0	0
$211$	− 9.71594i	− 0.668873i	−0.942418	−	0.334437i	$-0.891454\pi$
$211$	− 9.71594i	− 0.668873i	0.942418	−	0.334437i	$-0.108546\pi$
$212$	0	0
$213$	28.5743i	1.95788i
$214$	0	0
$215$	−8.73588	−0.595782
$216$	0	0
$217$	−8.29157	−0.562869
$218$	0	0
$219$	40.6613i	2.74763i
$220$	0	0
$221$	− 6.73233i	− 0.452866i
$222$	0	0
$223$	19.4803	1.30450	0.652249	−	0.758005i	$-0.273826\pi$
$223$	19.4803	1.30450	0.652249	+	0.758005i	$0.273826\pi$
$224$	0	0
$225$	−12.6942	−0.846281
$226$	0	0
$227$	− 5.75489i	− 0.381966i	−0.981593	−	0.190983i	$-0.938833\pi$
$227$	− 5.75489i	− 0.381966i	0.981593	−	0.190983i	$-0.0611675\pi$
$228$	0	0
$229$	4.51943i	0.298652i	0.988788	+	0.149326i	$0.0477104\pi$
$229$	4.51943i	0.298652i	−0.988788	+	0.149326i	$0.952290\pi$
$230$	0	0
$231$	7.74826	0.509798
$232$	0	0
$233$	2.03939	0.133605	0.0668025	−	0.997766i	$-0.478720\pi$
$233$	2.03939	0.133605	0.0668025	+	0.997766i	$0.478720\pi$
$234$	0	0
$235$	5.43858i	0.354774i
$236$	0	0
$237$	2.16925i	0.140908i
$238$	0	0
$239$	−8.01616	−0.518522	−0.259261	−	0.965807i	$-0.583479\pi$
$239$	−8.01616	−0.518522	−0.259261	+	0.965807i	$0.583479\pi$
$240$	0	0
$241$	3.51113	0.226172	0.113086	−	0.993585i	$-0.463927\pi$
$241$	3.51113	0.226172	0.113086	+	0.993585i	$0.463927\pi$
$242$	0	0
$243$	21.5902i	1.38501i
$244$	0	0
$245$	− 0.687773i	− 0.0439402i
$246$	0	0
$247$	−4.13763	−0.263271
$248$	0	0
$249$	−4.97186	−0.315079
$250$	0	0
$251$	12.4594i	0.786430i	0.919447	+	0.393215i	$0.128637\pi$
$251$	12.4594i	0.786430i	−0.919447	+	0.393215i	$0.871363\pi$
$252$	0	0
$253$	− 22.6472i	− 1.42381i
$254$	0	0
$255$	−11.1552	−0.698568
$256$	0	0
$257$	1.98847	0.124037	0.0620186	−	0.998075i	$-0.480246\pi$
$257$	1.98847	0.124037	0.0620186	+	0.998075i	$0.480246\pi$
$258$	0	0
$259$	− 7.82265i	− 0.486076i
$260$	0	0
$261$	12.4977i	0.773586i
$262$	0	0
$263$	−25.0187	−1.54272	−0.771360	−	0.636399i	$-0.780423\pi$
$263$	−25.0187	−1.54272	−0.771360	+	0.636399i	$0.780423\pi$
$264$	0	0
$265$	6.89566	0.423597
$266$	0	0
$267$	− 28.6569i	− 1.75377i
$268$	0	0
$269$	16.1113i	0.982326i	0.871068	+	0.491163i	$0.163428\pi$
$269$	16.1113i	0.982326i	−0.871068	+	0.491163i	$0.836572\pi$
$270$	0	0
$271$	16.8822	1.02552	0.512759	−	0.858532i	$-0.328623\pi$
$271$	16.8822	1.02552	0.512759	+	0.858532i	$0.328623\pi$
$272$	0	0
$273$	2.40918	0.145810
$274$	0	0
$275$	− 14.5594i	− 0.877964i
$276$	0	0
$277$	6.69730i	0.402402i	0.979550	+	0.201201i	$0.0644844\pi$
$277$	6.69730i	0.402402i	−0.979550	+	0.201201i	$0.935516\pi$
$278$	0	0
$279$	−23.2507	−1.39198
$280$	0	0
$281$	−22.9392	−1.36844	−0.684218	−	0.729278i	$-0.739856\pi$
$281$	−22.9392	−1.36844	−0.684218	+	0.729278i	$0.739856\pi$
$282$	0	0
$283$	− 1.57480i	− 0.0936120i	−0.998904	−	0.0468060i	$-0.985096\pi$
$283$	− 1.57480i	− 0.0936120i	0.998904	−	0.0468060i	$-0.0149043\pi$
$284$	0	0
$285$	6.85591i	0.406109i
$286$	0	0
$287$	4.81888	0.284450
$288$	0	0
$289$	28.3243	1.66613
$290$	0	0
$291$	19.4611i	1.14083i
$292$	0	0
$293$	− 27.8609i	− 1.62765i	−0.581111	−	0.813825i	$-0.697382\pi$
$293$	− 27.8609i	− 1.62765i	0.581111	−	0.813825i	$-0.302618\pi$
$294$	0	0
$295$	−2.73311	−0.159128
$296$	0	0
$297$	−1.51763	−0.0880619
$298$	0	0
$299$	− 7.04171i	− 0.407233i
$300$	0	0
$301$	12.7017i	0.732114i
$302$	0	0
$303$	18.3588	1.05468
$304$	0	0
$305$	−1.42942	−0.0818485
$306$	0	0
$307$	21.6337i	1.23470i	0.786689	+	0.617349i	$0.211793\pi$
$307$	21.6337i	1.23470i	−0.786689	+	0.617349i	$0.788207\pi$
$308$	0	0
$309$	− 12.1543i	− 0.691433i
$310$	0	0
$311$	−12.6974	−0.720006	−0.360003	−	0.932951i	$-0.617224\pi$
$311$	−12.6974	−0.720006	−0.360003	+	0.932951i	$0.617224\pi$
$312$	0	0
$313$	23.8266	1.34676	0.673380	−	0.739297i	$-0.264842\pi$
$313$	23.8266	1.34676	0.673380	+	0.739297i	$0.264842\pi$
$314$	0	0
$315$	− 1.92861i	− 0.108665i
$316$	0	0
$317$	0.958688i	0.0538453i	0.999638	+	0.0269226i	$0.00857078\pi$
$317$	0.958688i	0.0538453i	−0.999638	+	0.0269226i	$0.991429\pi$
$318$	0	0
$319$	−14.3339	−0.802547
$320$	0	0
$321$	38.8405	2.16786
$322$	0	0
$323$	− 27.8559i	− 1.54994i
$324$	0	0
$325$	− 4.52697i	− 0.251111i
$326$	0	0
$327$	−26.8339	−1.48392
$328$	0	0
$329$	7.90753	0.435956
$330$	0	0
$331$	6.17550i	0.339436i	0.985493	+	0.169718i	$0.0542857\pi$
$331$	6.17550i	0.339436i	−0.985493	+	0.169718i	$0.945714\pi$
$332$	0	0
$333$	− 21.9358i	− 1.20207i
$334$	0	0
$335$	7.87693	0.430362
$336$	0	0
$337$	−6.59480	−0.359242	−0.179621	−	0.983736i	$-0.557487\pi$
$337$	−6.59480	−0.359242	−0.179621	+	0.983736i	$0.557487\pi$
$338$	0	0
$339$	− 46.4727i	− 2.52405i
$340$	0	0
$341$	− 26.6669i	− 1.44409i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−11.6679	−0.628177
$346$	0	0
$347$	10.6595i	0.572233i	0.958195	+	0.286117i	$0.0923645\pi$
$347$	10.6595i	0.572233i	−0.958195	+	0.286117i	$0.907636\pi$
$348$	0	0
$349$	20.7465i	1.11053i	0.831673	+	0.555266i	$0.187384\pi$
$349$	20.7465i	1.11053i	−0.831673	+	0.555266i	$0.812616\pi$
$350$	0	0
$351$	−0.471879	−0.0251871
$352$	0	0
$353$	5.86613	0.312223	0.156111	−	0.987739i	$-0.450104\pi$
$353$	5.86613	0.312223	0.156111	+	0.987739i	$0.450104\pi$
$354$	0	0
$355$	8.15740i	0.432950i
$356$	0	0
$357$	16.2194i	0.858420i
$358$	0	0
$359$	−32.1557	−1.69711	−0.848556	−	0.529105i	$-0.822528\pi$
$359$	−32.1557	−1.69711	−0.848556	+	0.529105i	$0.822528\pi$
$360$	0	0
$361$	1.88004	0.0989492
$362$	0	0
$363$	− 1.58141i	− 0.0830025i
$364$	0	0
$365$	11.6080i	0.607590i
$366$	0	0
$367$	0.00587334	0.000306586 0	0.000153293	−	1.00000i	$-0.499951\pi$
$367$	0.00587334	0.000306586 0	0.000153293		1.00000i	$0.499951\pi$
$368$	0	0
$369$	13.5128	0.703447
$370$	0	0
$371$	− 10.0261i	− 0.520528i
$372$	0	0
$373$	− 22.2332i	− 1.15119i	−0.817734	−	0.575596i	$-0.804770\pi$
$373$	− 22.2332i	− 1.15119i	0.817734	−	0.575596i	$-0.195230\pi$
$374$	0	0
$375$	−15.7859	−0.815178
$376$	0	0
$377$	−4.45687	−0.229541
$378$	0	0
$379$	4.20029i	0.215754i	0.994164	+	0.107877i	$0.0344053\pi$
$379$	4.20029i	0.215754i	−0.994164	+	0.107877i	$0.965595\pi$
$380$	0	0
$381$	30.8909i	1.58259i
$382$	0	0
$383$	−3.25784	−0.166468	−0.0832339	−	0.996530i	$-0.526525\pi$
$383$	−3.25784	−0.166468	−0.0832339	+	0.996530i	$0.526525\pi$
$384$	0	0
$385$	2.21198	0.112733
$386$	0	0
$387$	35.6172i	1.81053i
$388$	0	0
$389$	20.0798i	1.01809i	0.860741	+	0.509043i	$0.170001\pi$
$389$	20.0798i	1.01809i	−0.860741	+	0.509043i	$0.829999\pi$
$390$	0	0
$391$	47.4071	2.39748
$392$	0	0
$393$	−9.60238	−0.484376
$394$	0	0
$395$	0.619279i	0.0311593i
$396$	0	0
$397$	− 23.0645i	− 1.15758i	−0.815478	−	0.578788i	$-0.803526\pi$
$397$	− 23.0645i	− 1.15758i	0.815478	−	0.578788i	$-0.196474\pi$
$398$	0	0
$399$	9.96828	0.499038
$400$	0	0
$401$	6.25670	0.312445	0.156222	−	0.987722i	$-0.450068\pi$
$401$	6.25670	0.312445	0.156222	+	0.987722i	$0.450068\pi$
$402$	0	0
$403$	− 8.29157i	− 0.413033i
$404$	0	0
$405$	6.56770i	0.326352i
$406$	0	0
$407$	25.1588	1.24707
$408$	0	0
$409$	20.2811	1.00284	0.501418	−	0.865205i	$-0.332812\pi$
$409$	20.2811	1.00284	0.501418	+	0.865205i	$0.332812\pi$
$410$	0	0
$411$	− 14.2720i	− 0.703985i
$412$	0	0
$413$	3.97386i	0.195541i
$414$	0	0
$415$	−1.41937	−0.0696740
$416$	0	0
$417$	7.53604	0.369042
$418$	0	0
$419$	− 31.6138i	− 1.54444i	−0.635357	−	0.772219i	$-0.719147\pi$
$419$	− 31.6138i	− 1.54444i	0.635357	−	0.772219i	$-0.280853\pi$
$420$	0	0
$421$	1.28516i	0.0626348i	0.999509	+	0.0313174i	$0.00997027\pi$
$421$	1.28516i	0.0626348i	−0.999509	+	0.0313174i	$0.990030\pi$
$422$	0	0
$423$	22.1738	1.07812
$424$	0	0
$425$	30.4771	1.47835
$426$	0	0
$427$	2.07834i	0.100578i
$428$	0	0
$429$	7.74826i	0.374090i
$430$	0	0
$431$	−13.9878	−0.673770	−0.336885	−	0.941546i	$-0.609373\pi$
$431$	−13.9878	−0.673770	−0.336885	+	0.941546i	$0.609373\pi$
$432$	0	0
$433$	−12.4382	−0.597740	−0.298870	−	0.954294i	$-0.596610\pi$
$433$	−12.4382	−0.597740	−0.298870	+	0.954294i	$0.596610\pi$
$434$	0	0
$435$	7.38488i	0.354078i
$436$	0	0
$437$	− 29.1360i	− 1.39376i
$438$	0	0
$439$	29.1671	1.39207	0.696034	−	0.718008i	$-0.254946\pi$
$439$	29.1671	1.39207	0.696034	+	0.718008i	$0.254946\pi$
$440$	0	0
$441$	−2.80413	−0.133530
$442$	0	0
$443$	7.38931i	0.351077i	0.984473	+	0.175538i	$0.0561666\pi$
$443$	7.38931i	0.351077i	−0.984473	+	0.175538i	$0.943833\pi$
$444$	0	0
$445$	− 8.18097i	− 0.387815i
$446$	0	0
$447$	5.32190	0.251717
$448$	0	0
$449$	−18.4074	−0.868700	−0.434350	−	0.900744i	$-0.643022\pi$
$449$	−18.4074	−0.868700	−0.434350	+	0.900744i	$0.643022\pi$
$450$	0	0
$451$	15.4982i	0.729783i
$452$	0	0
$453$	− 11.9199i	− 0.560045i
$454$	0	0
$455$	0.687773	0.0322433
$456$	0	0
$457$	21.1297	0.988405	0.494202	−	0.869347i	$-0.335460\pi$
$457$	21.1297	0.988405	0.494202	+	0.869347i	$0.335460\pi$
$458$	0	0
$459$	− 3.17685i	− 0.148283i
$460$	0	0
$461$	− 10.0573i	− 0.468415i	−0.972187	−	0.234207i	$-0.924751\pi$
$461$	− 10.0573i	− 0.468415i	0.972187	−	0.234207i	$-0.0752494\pi$
$462$	0	0
$463$	35.0391	1.62840	0.814202	−	0.580582i	$-0.197175\pi$
$463$	35.0391	1.62840	0.814202	+	0.580582i	$0.197175\pi$
$464$	0	0
$465$	−13.7389	−0.637124
$466$	0	0
$467$	− 22.2151i	− 1.02799i	−0.857792	−	0.513996i	$-0.828165\pi$
$467$	− 22.2151i	− 1.02799i	0.857792	−	0.513996i	$-0.171835\pi$
$468$	0	0
$469$	− 11.4528i	− 0.528841i
$470$	0	0
$471$	−50.2851	−2.31701
$472$	0	0
$473$	−40.8505	−1.87831
$474$	0	0
$475$	− 18.7309i	− 0.859433i
$476$	0	0
$477$	− 28.1145i	− 1.28727i
$478$	0	0
$479$	26.9888	1.23315	0.616574	−	0.787297i	$-0.288520\pi$
$479$	26.9888	1.23315	0.616574	+	0.787297i	$0.288520\pi$
$480$	0	0
$481$	7.82265	0.356682
$482$	0	0
$483$	16.9647i	0.771922i
$484$	0	0
$485$	5.55575i	0.252274i
$486$	0	0
$487$	−4.49681	−0.203770	−0.101885	−	0.994796i	$-0.532487\pi$
$487$	−4.49681	−0.203770	−0.101885	+	0.994796i	$0.532487\pi$
$488$	0	0
$489$	26.8020	1.21203
$490$	0	0
$491$	41.3602i	1.86656i	0.359152	+	0.933279i	$0.383066\pi$
$491$	41.3602i	1.86656i	−0.359152	+	0.933279i	$0.616934\pi$
$492$	0	0
$493$	− 30.0051i	− 1.35136i
$494$	0	0
$495$	6.20268	0.278790
$496$	0	0
$497$	11.8606	0.532021
$498$	0	0
$499$	− 2.02432i	− 0.0906210i	−0.998973	−	0.0453105i	$-0.985572\pi$
$499$	− 2.02432i	− 0.0906210i	0.998973	−	0.0453105i	$-0.0144277\pi$
$500$	0	0
$501$	− 13.8048i	− 0.616752i
$502$	0	0
$503$	13.6836	0.610123	0.305062	−	0.952333i	$-0.401323\pi$
$503$	13.6836	0.610123	0.305062	+	0.952333i	$0.401323\pi$
$504$	0	0
$505$	5.24107	0.233225
$506$	0	0
$507$	2.40918i	0.106995i
$508$	0	0
$509$	− 15.2121i	− 0.674265i	−0.941457	−	0.337133i	$-0.890543\pi$
$509$	− 15.2121i	− 0.674265i	0.941457	−	0.337133i	$-0.109457\pi$
$510$	0	0
$511$	16.8777	0.746624
$512$	0	0
$513$	−1.95246	−0.0862032
$514$	0	0
$515$	− 3.46981i	− 0.152898i
$516$	0	0
$517$	25.4318i	1.11849i
$518$	0	0
$519$	−21.3455	−0.936961
$520$	0	0
$521$	26.9487	1.18064	0.590322	−	0.807168i	$-0.299001\pi$
$521$	26.9487	1.18064	0.590322	+	0.807168i	$0.299001\pi$
$522$	0	0
$523$	0.627284i	0.0274292i	0.999906	+	0.0137146i	$0.00436563\pi$
$523$	0.627284i	0.0274292i	−0.999906	+	0.0137146i	$0.995634\pi$
$524$	0	0
$525$	10.9063i	0.475989i
$526$	0	0
$527$	55.8216	2.43163
$528$	0	0
$529$	26.5857	1.15590
$530$	0	0
$531$	11.1432i	0.483575i
$532$	0	0
$533$	4.81888i	0.208729i
$534$	0	0
$535$	11.0882	0.479384
$536$	0	0
$537$	13.2612	0.572263
$538$	0	0
$539$	− 3.21615i	− 0.138529i
$540$	0	0
$541$	7.91671i	0.340366i	0.985412	+	0.170183i	$0.0544359\pi$
$541$	7.91671i	0.340366i	−0.985412	+	0.170183i	$0.945564\pi$
$542$	0	0
$543$	34.9547	1.50005
$544$	0	0
$545$	−7.66056	−0.328142
$546$	0	0
$547$	29.4627i	1.25973i	0.776704	+	0.629866i	$0.216890\pi$
$547$	29.4627i	1.25973i	−0.776704	+	0.629866i	$0.783110\pi$
$548$	0	0
$549$	5.82793i	0.248730i
$550$	0	0
$551$	−18.4409	−0.785608
$552$	0	0
$553$	0.900413	0.0382894
$554$	0	0
$555$	− 12.9619i	− 0.550201i
$556$	0	0
$557$	− 6.15890i	− 0.260961i	−0.991451	−	0.130480i	$-0.958348\pi$
$557$	− 6.15890i	− 0.260961i	0.991451	−	0.130480i	$-0.0416520\pi$
$558$	0	0
$559$	−12.7017	−0.537224
$560$	0	0
$561$	−52.1639	−2.20236
$562$	0	0
$563$	27.4626i	1.15741i	0.815536	+	0.578706i	$0.196442\pi$
$563$	27.4626i	1.15741i	−0.815536	+	0.578706i	$0.803558\pi$
$564$	0	0
$565$	− 13.2671i	− 0.558149i
$566$	0	0
$567$	9.54924	0.401030
$568$	0	0
$569$	−25.1770	−1.05547	−0.527737	−	0.849408i	$-0.676959\pi$
$569$	−25.1770	−1.05547	−0.527737	+	0.849408i	$0.676959\pi$
$570$	0	0
$571$	− 34.0974i	− 1.42693i	−0.700689	−	0.713467i	$-0.747124\pi$
$571$	− 34.0974i	− 1.42693i	0.700689	−	0.713467i	$-0.252876\pi$
$572$	0	0
$573$	42.0965i	1.75861i
$574$	0	0
$575$	31.8776	1.32939
$576$	0	0
$577$	−34.3580	−1.43034	−0.715170	−	0.698950i	$-0.753651\pi$
$577$	−34.3580	−1.43034	−0.715170	+	0.698950i	$0.753651\pi$
$578$	0	0
$579$	− 12.5310i	− 0.520769i
$580$	0	0
$581$	2.06372i	0.0856174i
$582$	0	0
$583$	32.2453	1.33547
$584$	0	0
$585$	1.92861	0.0797380
$586$	0	0
$587$	4.39822i	0.181534i	0.995872	+	0.0907669i	$0.0289318\pi$
$587$	4.39822i	0.181534i	−0.995872	+	0.0907669i	$0.971068\pi$
$588$	0	0
$589$	− 34.3074i	− 1.41361i
$590$	0	0
$591$	−14.3562	−0.590536
$592$	0	0
$593$	−1.50359	−0.0617451	−0.0308725	−	0.999523i	$-0.509829\pi$
$593$	−1.50359	−0.0617451	−0.0308725	+	0.999523i	$0.509829\pi$
$594$	0	0
$595$	4.63031i	0.189824i
$596$	0	0
$597$	29.1691i	1.19381i
$598$	0	0
$599$	−13.7500	−0.561809	−0.280904	−	0.959736i	$-0.590634\pi$
$599$	−13.7500	−0.561809	−0.280904	+	0.959736i	$0.590634\pi$
$600$	0	0
$601$	7.31520	0.298393	0.149197	−	0.988808i	$-0.452331\pi$
$601$	7.31520	0.298393	0.149197	+	0.988808i	$0.452331\pi$
$602$	0	0
$603$	− 32.1152i	− 1.30783i
$604$	0	0
$605$	− 0.451462i	− 0.0183545i
$606$	0	0
$607$	19.5840	0.794889	0.397445	−	0.917626i	$-0.369897\pi$
$607$	19.5840	0.794889	0.397445	+	0.917626i	$0.369897\pi$
$608$	0	0
$609$	10.7374	0.435101
$610$	0	0
$611$	7.90753i	0.319904i
$612$	0	0
$613$	44.5486i	1.79930i	0.436610	+	0.899651i	$0.356179\pi$
$613$	44.5486i	1.79930i	−0.436610	+	0.899651i	$0.643821\pi$
$614$	0	0
$615$	7.98472	0.321975
$616$	0	0
$617$	−17.9533	−0.722772	−0.361386	−	0.932416i	$-0.617696\pi$
$617$	−17.9533	−0.722772	−0.361386	+	0.932416i	$0.617696\pi$
$618$	0	0
$619$	14.7130i	0.591365i	0.955286	+	0.295683i	$0.0955471\pi$
$619$	14.7130i	0.591365i	−0.955286	+	0.295683i	$0.904453\pi$
$620$	0	0
$621$	− 3.32284i	− 0.133341i
$622$	0	0
$623$	−11.8949	−0.476558
$624$	0	0
$625$	18.1283	0.725132
$626$	0	0
$627$	32.0594i	1.28033i
$628$	0	0
$629$	52.6647i	2.09988i
$630$	0	0
$631$	−17.9131	−0.713110	−0.356555	−	0.934274i	$-0.616049\pi$
$631$	−17.9131	−0.713110	−0.356555	+	0.934274i	$0.616049\pi$
$632$	0	0
$633$	−23.4074	−0.930362
$634$	0	0
$635$	8.81874i	0.349961i
$636$	0	0
$637$	− 1.00000i	− 0.0396214i
$638$	0	0
$639$	33.2587	1.31569
$640$	0	0
$641$	46.0544	1.81904	0.909520	−	0.415660i	$-0.136449\pi$
$641$	46.0544	1.81904	0.909520	+	0.415660i	$0.136449\pi$
$642$	0	0
$643$	− 20.1393i	− 0.794215i	−0.917772	−	0.397107i	$-0.870014\pi$
$643$	− 20.1393i	− 0.794215i	0.917772	−	0.397107i	$-0.129986\pi$
$644$	0	0
$645$	21.0463i	0.828696i
$646$	0	0
$647$	−3.30439	−0.129909	−0.0649544	−	0.997888i	$-0.520690\pi$
$647$	−3.30439	−0.129909	−0.0649544	+	0.997888i	$0.520690\pi$
$648$	0	0
$649$	−12.7805	−0.501679
$650$	0	0
$651$	19.9759i	0.782916i
$652$	0	0
$653$	− 13.9169i	− 0.544610i	−0.962211	−	0.272305i	$-0.912214\pi$
$653$	− 13.9169i	− 0.544610i	0.962211	−	0.272305i	$-0.0877860\pi$
$654$	0	0
$655$	−2.74129	−0.107111
$656$	0	0
$657$	47.3272	1.84641
$658$	0	0
$659$	− 23.5824i	− 0.918638i	−0.888271	−	0.459319i	$-0.848093\pi$
$659$	− 23.5824i	− 0.918638i	0.888271	−	0.459319i	$-0.151907\pi$
$660$	0	0
$661$	47.2373i	1.83732i	0.395050	+	0.918660i	$0.370727\pi$
$661$	47.2373i	1.83732i	−0.395050	+	0.918660i	$0.629273\pi$
$662$	0	0
$663$	−16.2194	−0.629908
$664$	0	0
$665$	2.84575	0.110353
$666$	0	0
$667$	− 31.3840i	− 1.21519i
$668$	0	0
$669$	− 46.9315i	− 1.81448i
$670$	0	0
$671$	−6.68423	−0.258042
$672$	0	0
$673$	−8.68063	−0.334614	−0.167307	−	0.985905i	$-0.553507\pi$
$673$	−8.68063	−0.334614	−0.167307	+	0.985905i	$0.553507\pi$
$674$	0	0
$675$	− 2.13618i	− 0.0822217i
$676$	0	0
$677$	− 9.39355i	− 0.361024i	−0.983573	−	0.180512i	$-0.942225\pi$
$677$	− 9.39355i	− 0.361024i	0.983573	−	0.180512i	$-0.0577754\pi$
$678$	0	0
$679$	8.07789	0.310001
$680$	0	0
$681$	−13.8646	−0.531291
$682$	0	0
$683$	− 7.64309i	− 0.292455i	−0.989251	−	0.146227i	$-0.953287\pi$
$683$	− 7.64309i	− 0.292455i	0.989251	−	0.146227i	$-0.0467131\pi$
$684$	0	0
$685$	− 4.07437i	− 0.155674i
$686$	0	0
$687$	10.8881	0.415407
$688$	0	0
$689$	10.0261	0.381963
$690$	0	0
$691$	38.5172i	1.46526i	0.680626	+	0.732631i	$0.261708\pi$
$691$	38.5172i	1.46526i	−0.680626	+	0.732631i	$0.738292\pi$
$692$	0	0
$693$	− 9.01850i	− 0.342584i
$694$	0	0
$695$	2.15139	0.0816070
$696$	0	0
$697$	−32.4423	−1.22884
$698$	0	0
$699$	− 4.91326i	− 0.185836i
$700$	0	0
$701$	− 18.4756i	− 0.697814i	−0.937157	−	0.348907i	$-0.886553\pi$
$701$	− 18.4756i	− 0.697814i	0.937157	−	0.348907i	$-0.113447\pi$
$702$	0	0
$703$	32.3672	1.22075
$704$	0	0
$705$	13.1025	0.493469
$706$	0	0
$707$	− 7.62035i	− 0.286593i
$708$	0	0
$709$	2.94207i	0.110492i	0.998473	+	0.0552459i	$0.0175943\pi$
$709$	2.94207i	0.110492i	−0.998473	+	0.0552459i	$0.982406\pi$
$710$	0	0
$711$	2.52488	0.0946903
$712$	0	0
$713$	58.3868	2.18660
$714$	0	0
$715$	2.21198i	0.0827232i
$716$	0	0
$717$	19.3123i	0.721233i
$718$	0	0
$719$	4.53648	0.169182	0.0845911	−	0.996416i	$-0.473042\pi$
$719$	4.53648	0.169182	0.0845911	+	0.996416i	$0.473042\pi$
$720$	0	0
$721$	−5.04500	−0.187885
$722$	0	0
$723$	− 8.45892i	− 0.314591i
$724$	0	0
$725$	− 20.1761i	− 0.749322i
$726$	0	0
$727$	0.575488	0.0213437	0.0106718	−	0.999943i	$-0.496603\pi$
$727$	0.575488	0.0213437	0.0106718	+	0.999943i	$0.496603\pi$
$728$	0	0
$729$	23.3668	0.865437
$730$	0	0
$731$	− 85.5120i	− 3.16278i
$732$	0	0
$733$	4.34048i	0.160319i	0.996782	+	0.0801597i	$0.0255430\pi$
$733$	4.34048i	0.160319i	−0.996782	+	0.0801597i	$0.974457\pi$
$734$	0	0
$735$	−1.65697	−0.0611181
$736$	0	0
$737$	36.8339	1.35679
$738$	0	0
$739$	5.99047i	0.220363i	0.993911	+	0.110181i	$0.0351432\pi$
$739$	5.99047i	0.220363i	−0.993911	+	0.110181i	$0.964857\pi$
$740$	0	0
$741$	9.96828i	0.366194i
$742$	0	0
$743$	−14.2196	−0.521665	−0.260833	−	0.965384i	$-0.583997\pi$
$743$	−14.2196	−0.521665	−0.260833	+	0.965384i	$0.583997\pi$
$744$	0	0
$745$	1.51930	0.0556628
$746$	0	0
$747$	5.78694i	0.211733i
$748$	0	0
$749$	− 16.1219i	− 0.589081i
$750$	0	0
$751$	−4.72032	−0.172247	−0.0861234	−	0.996284i	$-0.527448\pi$
$751$	−4.72032	−0.172247	−0.0861234	+	0.996284i	$0.527448\pi$
$752$	0	0
$753$	30.0169	1.09388
$754$	0	0
$755$	− 3.40289i	− 0.123844i
$756$	0	0
$757$	34.5160i	1.25451i	0.778816	+	0.627253i	$0.215821\pi$
$757$	34.5160i	1.25451i	−0.778816	+	0.627253i	$0.784179\pi$
$758$	0	0
$759$	−54.5610	−1.98044
$760$	0	0
$761$	13.0455	0.472900	0.236450	−	0.971644i	$-0.424016\pi$
$761$	13.0455	0.472900	0.236450	+	0.971644i	$0.424016\pi$
$762$	0	0
$763$	11.1382i	0.403231i
$764$	0	0
$765$	12.9840i	0.469438i
$766$	0	0
$767$	−3.97386	−0.143488
$768$	0	0
$769$	15.5443	0.560541	0.280271	−	0.959921i	$-0.409576\pi$
$769$	15.5443	0.560541	0.280271	+	0.959921i	$0.409576\pi$
$770$	0	0
$771$	− 4.79057i	− 0.172528i
$772$	0	0
$773$	− 36.8531i	− 1.32551i	−0.748834	−	0.662757i	$-0.769386\pi$
$773$	− 36.8531i	− 1.32551i	0.748834	−	0.662757i	$-0.230614\pi$
$774$	0	0
$775$	37.5357	1.34832
$776$	0	0
$777$	−18.8462	−0.676102
$778$	0	0
$779$	19.9387i	0.714379i
$780$	0	0
$781$	38.1454i	1.36495i
$782$	0	0
$783$	−2.10311	−0.0751588
$784$	0	0
$785$	−14.3554	−0.512366
$786$	0	0
$787$	19.8020i	0.705866i	0.935649	+	0.352933i	$0.114816\pi$
$787$	19.8020i	0.705866i	−0.935649	+	0.352933i	$0.885184\pi$
$788$	0	0
$789$	60.2745i	2.14583i
$790$	0	0
$791$	−19.2899	−0.685869
$792$	0	0
$793$	−2.07834	−0.0738039
$794$	0	0
$795$	− 16.6129i	− 0.589198i
$796$	0	0
$797$	− 14.0956i	− 0.499292i	−0.968337	−	0.249646i	$-0.919686\pi$
$797$	− 14.0956i	− 0.499292i	0.968337	−	0.249646i	$-0.0803143\pi$
$798$	0	0
$799$	−53.2361	−1.88336
$800$	0	0
$801$	−33.3548	−1.17853
$802$	0	0
$803$	54.2810i	1.91554i
$804$	0	0
$805$	4.84309i	0.170697i
$806$	0	0
$807$	38.8151	1.36636
$808$	0	0
$809$	−13.6176	−0.478770	−0.239385	−	0.970925i	$-0.576946\pi$
$809$	−13.6176	−0.478770	−0.239385	+	0.970925i	$0.576946\pi$
$810$	0	0
$811$	18.9691i	0.666095i	0.942910	+	0.333047i	$0.108077\pi$
$811$	18.9691i	0.666095i	−0.942910	+	0.333047i	$0.891923\pi$
$812$	0	0
$813$	− 40.6721i	− 1.42643i
$814$	0	0
$815$	7.65144	0.268018
$816$	0	0
$817$	−52.5549	−1.83866
$818$	0	0
$819$	− 2.80413i	− 0.0979843i
$820$	0	0
$821$	− 6.42394i	− 0.224197i	−0.993697	−	0.112099i	$-0.964243\pi$
$821$	− 6.42394i	− 0.224197i	0.993697	−	0.112099i	$-0.0357572\pi$
$822$	0	0
$823$	9.99062	0.348251	0.174125	−	0.984723i	$-0.444290\pi$
$823$	9.99062	0.348251	0.174125	+	0.984723i	$0.444290\pi$
$824$	0	0
$825$	−35.0761	−1.22119
$826$	0	0
$827$	− 36.5643i	− 1.27147i	−0.771909	−	0.635733i	$-0.780698\pi$
$827$	− 36.5643i	− 1.27147i	0.771909	−	0.635733i	$-0.219302\pi$
$828$	0	0
$829$	− 45.0922i	− 1.56612i	−0.621948	−	0.783059i	$-0.713658\pi$
$829$	− 45.0922i	− 1.56612i	0.621948	−	0.783059i	$-0.286342\pi$
$830$	0	0
$831$	16.1350	0.559716
$832$	0	0
$833$	6.73233	0.233261
$834$	0	0
$835$	− 3.94099i	− 0.136384i
$836$	0	0
$837$	− 3.91262i	− 0.135240i
$838$	0	0
$839$	−16.0522	−0.554184	−0.277092	−	0.960843i	$-0.589371\pi$
$839$	−16.0522	−0.554184	−0.277092	+	0.960843i	$0.589371\pi$
$840$	0	0
$841$	9.13630	0.315045
$842$	0	0
$843$	55.2645i	1.90341i
$844$	0	0
$845$	0.687773i	0.0236601i
$846$	0	0
$847$	−0.656411	−0.0225546
$848$	0	0
$849$	−3.79396	−0.130209
$850$	0	0
$851$	55.0848i	1.88828i
$852$	0	0
$853$	− 32.9867i	− 1.12944i	−0.825282	−	0.564721i	$-0.808984\pi$
$853$	− 32.9867i	− 1.12944i	0.825282	−	0.564721i	$-0.191016\pi$
$854$	0	0
$855$	7.97985	0.272905
$856$	0	0
$857$	8.97007	0.306412	0.153206	−	0.988194i	$-0.451040\pi$
$857$	8.97007	0.306412	0.153206	+	0.988194i	$0.451040\pi$
$858$	0	0
$859$	− 7.46952i	− 0.254857i	−0.991848	−	0.127428i	$-0.959328\pi$
$859$	− 7.46952i	− 0.254857i	0.991848	−	0.127428i	$-0.0406723\pi$
$860$	0	0
$861$	− 11.6095i	− 0.395652i
$862$	0	0
$863$	4.42663	0.150684	0.0753421	−	0.997158i	$-0.475995\pi$
$863$	4.42663	0.150684	0.0753421	+	0.997158i	$0.475995\pi$
$864$	0	0
$865$	−6.09371	−0.207192
$866$	0	0
$867$	− 68.2382i	− 2.31749i
$868$	0	0
$869$	2.89586i	0.0982352i
$870$	0	0
$871$	11.4528	0.388064
$872$	0	0
$873$	22.6515	0.766636
$874$	0	0
$875$	6.55239i	0.221511i
$876$	0	0
$877$	27.9884i	0.945100i	0.881304	+	0.472550i	$0.156666\pi$
$877$	27.9884i	0.945100i	−0.881304	+	0.472550i	$0.843334\pi$
$878$	0	0
$879$	−67.1218	−2.26396
$880$	0	0
$881$	−13.3354	−0.449280	−0.224640	−	0.974442i	$-0.572121\pi$
$881$	−13.3354	−0.449280	−0.224640	+	0.974442i	$0.572121\pi$
$882$	0	0
$883$	2.41618i	0.0813110i	0.999173	+	0.0406555i	$0.0129446\pi$
$883$	2.41618i	0.0813110i	−0.999173	+	0.0406555i	$0.987055\pi$
$884$	0	0
$885$	6.58455i	0.221337i
$886$	0	0
$887$	40.2075	1.35004	0.675018	−	0.737801i	$-0.264136\pi$
$887$	40.2075	1.35004	0.675018	+	0.737801i	$0.264136\pi$
$888$	0	0
$889$	12.8222	0.430042
$890$	0	0
$891$	30.7117i	1.02888i
$892$	0	0
$893$	32.7184i	1.09488i
$894$	0	0
$895$	3.78581	0.126546
$896$	0	0
$897$	−16.9647	−0.566436
$898$	0	0
$899$	− 36.9545i	− 1.23250i
$900$	0	0
$901$	67.4989i	2.24871i
$902$	0	0
$903$	30.6006	1.01833
$904$	0	0
$905$	9.97889	0.331710
$906$	0	0
$907$	27.7295i	0.920742i	0.887727	+	0.460371i	$0.152284\pi$
$907$	27.7295i	0.920742i	−0.887727	+	0.460371i	$0.847716\pi$
$908$	0	0
$909$	− 21.3685i	− 0.708748i
$910$	0	0
$911$	−14.4713	−0.479456	−0.239728	−	0.970840i	$-0.577058\pi$
$911$	−14.4713	−0.479456	−0.239728	+	0.970840i	$0.577058\pi$
$912$	0	0
$913$	−6.63721	−0.219660
$914$	0	0
$915$	3.44373i	0.113846i
$916$	0	0
$917$	3.98575i	0.131621i
$918$	0	0
$919$	31.3587	1.03443	0.517214	−	0.855856i	$-0.326969\pi$
$919$	31.3587	1.03443	0.517214	+	0.855856i	$0.326969\pi$
$920$	0	0
$921$	52.1193	1.71739
$922$	0	0
$923$	11.8606i	0.390397i
$924$	0	0
$925$	35.4129i	1.16437i
$926$	0	0
$927$	−14.1468	−0.464643
$928$	0	0
$929$	−48.9015	−1.60441	−0.802203	−	0.597052i	$-0.796339\pi$
$929$	−48.9015	−1.60441	−0.802203	+	0.597052i	$0.796339\pi$
$930$	0	0
$931$	− 4.13763i	− 0.135605i
$932$	0	0
$933$	30.5904i	1.00148i
$934$	0	0
$935$	−14.8918	−0.487013
$936$	0	0
$937$	16.8418	0.550198	0.275099	−	0.961416i	$-0.411289\pi$
$937$	16.8418	0.550198	0.275099	+	0.961416i	$0.411289\pi$
$938$	0	0
$939$	− 57.4025i	− 1.87326i
$940$	0	0
$941$	25.4756i	0.830480i	0.909712	+	0.415240i	$0.136302\pi$
$941$	25.4756i	0.830480i	−0.909712	+	0.415240i	$0.863698\pi$
$942$	0	0
$943$	−33.9332	−1.10502
$944$	0	0
$945$	0.324546	0.0105575
$946$	0	0
$947$	48.9591i	1.59096i	0.605981	+	0.795479i	$0.292781\pi$
$947$	48.9591i	1.59096i	−0.605981	+	0.795479i	$0.707219\pi$
$948$	0	0
$949$	16.8777i	0.547872i
$950$	0	0
$951$	2.30965	0.0748955
$952$	0	0
$953$	−53.9832	−1.74869	−0.874344	−	0.485307i	$-0.838708\pi$
$953$	−53.9832	−1.74869	−0.874344	+	0.485307i	$0.838708\pi$
$954$	0	0
$955$	12.0177i	0.388884i
$956$	0	0
$957$	34.5330i	1.11629i
$958$	0	0
$959$	−5.92401	−0.191296
$960$	0	0
$961$	37.7502	1.21775
$962$	0	0
$963$	− 45.2079i	− 1.45680i
$964$	0	0
$965$	− 3.57734i	− 0.115159i
$966$	0	0
$967$	4.69531	0.150991	0.0754955	−	0.997146i	$-0.475946\pi$
$967$	4.69531	0.150991	0.0754955	+	0.997146i	$0.475946\pi$
$968$	0	0
$969$	−67.1097	−2.15588
$970$	0	0
$971$	43.6293i	1.40013i	0.714079	+	0.700065i	$0.246846\pi$
$971$	43.6293i	1.40013i	−0.714079	+	0.700065i	$0.753154\pi$
$972$	0	0
$973$	− 3.12806i	− 0.100281i
$974$	0	0
$975$	−10.9063	−0.349280
$976$	0	0
$977$	−24.1270	−0.771892	−0.385946	−	0.922521i	$-0.626125\pi$
$977$	−24.1270	−0.771892	−0.385946	+	0.922521i	$0.626125\pi$
$978$	0	0
$979$	− 38.2556i	− 1.22266i
$980$	0	0
$981$	31.2330i	0.997195i
$982$	0	0
$983$	53.7433	1.71415	0.857073	−	0.515196i	$-0.172281\pi$
$983$	53.7433	1.71415	0.857073	+	0.515196i	$0.172281\pi$
$984$	0	0
$985$	−4.09842	−0.130586
$986$	0	0
$987$	− 19.0506i	− 0.606388i
$988$	0	0
$989$	− 89.4416i	− 2.84408i
$990$	0	0
$991$	43.0179	1.36651	0.683255	−	0.730180i	$-0.260564\pi$
$991$	43.0179	1.36651	0.683255	+	0.730180i	$0.260564\pi$
$992$	0	0
$993$	14.8779	0.472135
$994$	0	0
$995$	8.32719i	0.263990i
$996$	0	0
$997$	47.1795i	1.49419i	0.664717	+	0.747095i	$0.268552\pi$
$997$	47.1795i	1.49419i	−0.664717	+	0.747095i	$0.731448\pi$
$998$	0	0
$999$	3.69135	0.116789

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.c.b.1457.6		38
4.3	odd	2		728.2.c.b.365.15	✓	38
8.3	odd	2		728.2.c.b.365.16	yes	38
8.5	even	2	inner	2912.2.c.b.1457.33		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.c.b.365.15	✓	38	4.3	odd	2
728.2.c.b.365.16	yes	38	8.3	odd	2
2912.2.c.b.1457.6		38	1.1	even	1	trivial
2912.2.c.b.1457.33		38	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)

\(f(q)\)	\(=\)	\(q-2.40918i q^{3} -0.687773i q^{5} -1.00000 q^{7} -2.80413 q^{9} -3.21615i q^{11} -1.00000i q^{13} -1.65697 q^{15} +6.73233 q^{17} -4.13763i q^{19} +2.40918i q^{21} +7.04171 q^{23} +4.52697 q^{25} -0.471879i q^{27} -4.45687i q^{29} +8.29157 q^{31} -7.74826 q^{33} +0.687773i q^{35} +7.82265i q^{37} -2.40918 q^{39} -4.81888 q^{41} -12.7017i q^{43} +1.92861i q^{45} -7.90753 q^{47} +1.00000 q^{49} -16.2194i q^{51} +10.0261i q^{53} -2.21198 q^{55} -9.96828 q^{57} -3.97386i q^{59} -2.07834i q^{61} +2.80413 q^{63} -0.687773 q^{65} +11.4528i q^{67} -16.9647i q^{69} -11.8606 q^{71} -16.8777 q^{73} -10.9063i q^{75} +3.21615i q^{77} -0.900413 q^{79} -9.54924 q^{81} -2.06372i q^{83} -4.63031i q^{85} -10.7374 q^{87} +11.8949 q^{89} +1.00000i q^{91} -19.9759i q^{93} -2.84575 q^{95} -8.07789 q^{97} +9.01850i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 38 q^{7} - 46 q^{9} + 8 q^{15} + 20 q^{17} - 12 q^{23} - 50 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 38 q^{49} + 8 q^{57} + 46 q^{63} + 20 q^{65} + 12 q^{79} + 62 q^{81} + 56 q^{87}+O(q^{100}) \) 38 * q - 38 * q^7 - 46 * q^9 + 8 * q^15 + 20 * q^17 - 12 * q^23 - 50 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 38 * q^49 + 8 * q^57 + 46 * q^63 + 20 * q^65 + 12 * q^79 + 62 * q^81 + 56 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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