LMFDB - Embedded newform 7616.2.a.bt.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7616,2,Mod(1,7616)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7616.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,2,0,0,0,-5,0,11,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.453749.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 $ x^5 - 2x^4 - 5x^3 + 10*x^2 + x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 119)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.609440$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82084 q^{3} -2.51889 q^{5} -1.00000 q^{7} +4.95716 q^{9} +5.25717 q^{11} +4.20342 q^{13} -7.10539 q^{15} +1.00000 q^{17} -0.781120 q^{19} -2.82084 q^{21} +2.42281 q^{23} +1.34480 q^{25} +5.52084 q^{27} -3.42230 q^{29} +6.94170 q^{31} +14.8296 q^{33} +2.51889 q^{35} +7.02282 q^{37} +11.8572 q^{39} +1.83631 q^{41} -8.63974 q^{43} -12.4865 q^{45} -1.34285 q^{47} +1.00000 q^{49} +2.82084 q^{51} +1.95716 q^{53} -13.2422 q^{55} -2.20342 q^{57} -2.43776 q^{59} -6.16369 q^{61} -4.95716 q^{63} -10.5879 q^{65} -1.53435 q^{67} +6.83436 q^{69} -1.65664 q^{71} -14.3197 q^{73} +3.79346 q^{75} -5.25717 q^{77} +12.6262 q^{79} +0.701939 q^{81} +11.8451 q^{83} -2.51889 q^{85} -9.65376 q^{87} +4.49202 q^{89} -4.20342 q^{91} +19.5814 q^{93} +1.96755 q^{95} +12.9267 q^{97} +26.0606 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 5 q^{7} + 11 q^{9} + 2 q^{11} - 2 q^{13} + 8 q^{15} + 5 q^{17} - 6 q^{19} - 2 q^{21} - 10 q^{23} + 21 q^{25} + 26 q^{27} + 8 q^{29} + 6 q^{33} - 8 q^{37} + 14 q^{39} + 18 q^{41} - 8 q^{43}+ \cdots + 62 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 - 5 * q^7 + 11 * q^9 + 2 * q^11 - 2 * q^13 + 8 * q^15 + 5 * q^17 - 6 * q^19 - 2 * q^21 - 10 * q^23 + 21 * q^25 + 26 * q^27 + 8 * q^29 + 6 * q^33 - 8 * q^37 + 14 * q^39 + 18 * q^41 - 8 * q^43 - 10 * q^47 + 5 * q^49 + 2 * q^51 - 4 * q^53 - 24 * q^55 + 12 * q^57 - 8 * q^59 - 22 * q^61 - 11 * q^63 - 30 * q^65 - 16 * q^67 + 32 * q^69 - 2 * q^71 + 10 * q^73 + 14 * q^75 - 2 * q^77 + 18 * q^79 + 25 * q^81 + 12 * q^83 - 26 * q^87 + 20 * q^89 + 2 * q^91 + 28 * q^93 - 22 * q^95 + 12 * q^97 + 62 * q^99

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
$3$	2.82084	1.62861	0.814307	−	0.580434i	$-0.197117\pi$
$3$	2.82084	1.62861	0.814307	+	0.580434i	$0.197117\pi$
$4$	0	0
$5$	−2.51889	−1.12648	−0.563241	−	0.826293i	$-0.690446\pi$
$5$	−2.51889	−1.12648	−0.563241	+	0.826293i	$0.690446\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.95716	1.65239
$10$	0	0
$11$	5.25717	1.58510	0.792548	−	0.609810i	$-0.208754\pi$
$11$	5.25717	1.58510	0.792548	+	0.609810i	$0.208754\pi$
$12$	0	0
$13$	4.20342	1.16582	0.582909	−	0.812537i	$-0.301914\pi$
$13$	4.20342	1.16582	0.582909	+	0.812537i	$0.301914\pi$
$14$	0	0
$15$	−7.10539	−1.83460
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−0.781120	−0.179201	−0.0896006	−	0.995978i	$-0.528559\pi$
$19$	−0.781120	−0.179201	−0.0896006	+	0.995978i	$0.528559\pi$
$20$	0	0
$21$	−2.82084	−0.615559
$22$	0	0
$23$	2.42281	0.505190	0.252595	−	0.967572i	$-0.418716\pi$
$23$	2.42281	0.505190	0.252595	+	0.967572i	$0.418716\pi$
$24$	0	0
$25$	1.34480	0.268960
$26$	0	0
$27$	5.52084	1.06249
$28$	0	0
$29$	−3.42230	−0.635505	−0.317752	−	0.948174i	$-0.602928\pi$
$29$	−3.42230	−0.635505	−0.317752	+	0.948174i	$0.602928\pi$
$30$	0	0
$31$	6.94170	1.24677	0.623383	−	0.781917i	$-0.285758\pi$
$31$	6.94170	1.24677	0.623383	+	0.781917i	$0.285758\pi$
$32$	0	0
$33$	14.8296	2.58151
$34$	0	0
$35$	2.51889	0.425770
$36$	0	0
$37$	7.02282	1.15455	0.577273	−	0.816552i	$-0.304117\pi$
$37$	7.02282	1.15455	0.577273	+	0.816552i	$0.304117\pi$
$38$	0	0
$39$	11.8572	1.89867
$40$	0	0
$41$	1.83631	0.286783	0.143391	−	0.989666i	$-0.454199\pi$
$41$	1.83631	0.286783	0.143391	+	0.989666i	$0.454199\pi$
$42$	0	0
$43$	−8.63974	−1.31755	−0.658774	−	0.752341i	$-0.728925\pi$
$43$	−8.63974	−1.31755	−0.658774	+	0.752341i	$0.728925\pi$
$44$	0	0
$45$	−12.4865	−1.86138
$46$	0	0
$47$	−1.34285	−0.195875	−0.0979374	−	0.995193i	$-0.531225\pi$
$47$	−1.34285	−0.195875	−0.0979374	+	0.995193i	$0.531225\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.82084	0.394997
$52$	0	0
$53$	1.95716	0.268836	0.134418	−	0.990925i	$-0.457083\pi$
$53$	1.95716	0.268836	0.134418	+	0.990925i	$0.457083\pi$
$54$	0	0
$55$	−13.2422	−1.78558
$56$	0	0
$57$	−2.20342	−0.291850
$58$	0	0
$59$	−2.43776	−0.317369	−0.158685	−	0.987329i	$-0.550725\pi$
$59$	−2.43776	−0.317369	−0.158685	+	0.987329i	$0.550725\pi$
$60$	0	0
$61$	−6.16369	−0.789180	−0.394590	−	0.918857i	$-0.629113\pi$
$61$	−6.16369	−0.789180	−0.394590	+	0.918857i	$0.629113\pi$
$62$	0	0
$63$	−4.95716	−0.624543
$64$	0	0
$65$	−10.5879	−1.31327
$66$	0	0
$67$	−1.53435	−0.187451	−0.0937254	−	0.995598i	$-0.529878\pi$
$67$	−1.53435	−0.187451	−0.0937254	+	0.995598i	$0.529878\pi$
$68$	0	0
$69$	6.83436	0.822760
$70$	0	0
$71$	−1.65664	−0.196607	−0.0983035	−	0.995156i	$-0.531342\pi$
$71$	−1.65664	−0.196607	−0.0983035	+	0.995156i	$0.531342\pi$
$72$	0	0
$73$	−14.3197	−1.67600	−0.837998	−	0.545674i	$-0.816274\pi$
$73$	−14.3197	−1.67600	−0.837998	+	0.545674i	$0.816274\pi$
$74$	0	0
$75$	3.79346	0.438031
$76$	0	0
$77$	−5.25717	−0.599110
$78$	0	0
$79$	12.6262	1.42056	0.710281	−	0.703919i	$-0.248568\pi$
$79$	12.6262	1.42056	0.710281	+	0.703919i	$0.248568\pi$
$80$	0	0
$81$	0.701939	0.0779932
$82$	0	0
$83$	11.8451	1.30017	0.650085	−	0.759862i	$-0.274733\pi$
$83$	11.8451	1.30017	0.650085	+	0.759862i	$0.274733\pi$
$84$	0	0
$85$	−2.51889	−0.273212
$86$	0	0
$87$	−9.65376	−1.03499
$88$	0	0
$89$	4.49202	0.476153	0.238076	−	0.971246i	$-0.423483\pi$
$89$	4.49202	0.476153	0.238076	+	0.971246i	$0.423483\pi$
$90$	0	0
$91$	−4.20342	−0.440638
$92$	0	0
$93$	19.5814	2.03050
$94$	0	0
$95$	1.96755	0.201867
$96$	0	0
$97$	12.9267	1.31251	0.656256	−	0.754538i	$-0.272139\pi$
$97$	12.9267	1.31251	0.656256	+	0.754538i	$0.272139\pi$
$98$	0	0
$99$	26.0606	2.61919
$100$	0	0
$101$	1.82279	0.181374	0.0906872	−	0.995879i	$-0.471094\pi$
$101$	1.82279	0.181374	0.0906872	+	0.995879i	$0.471094\pi$
$102$	0	0
$103$	1.40734	0.138670	0.0693349	−	0.997593i	$-0.477912\pi$
$103$	1.40734	0.138670	0.0693349	+	0.997593i	$0.477912\pi$
$104$	0	0
$105$	7.10539	0.693415
$106$	0	0
$107$	16.0074	1.54749	0.773745	−	0.633497i	$-0.218381\pi$
$107$	16.0074	1.54749	0.773745	+	0.633497i	$0.218381\pi$
$108$	0	0
$109$	−16.5143	−1.58179	−0.790893	−	0.611954i	$-0.790384\pi$
$109$	−16.5143	−1.58179	−0.790893	+	0.611954i	$0.790384\pi$
$110$	0	0
$111$	19.8103	1.88031
$112$	0	0
$113$	−1.61548	−0.151971	−0.0759857	−	0.997109i	$-0.524210\pi$
$113$	−1.61548	−0.151971	−0.0759857	+	0.997109i	$0.524210\pi$
$114$	0	0
$115$	−6.10278	−0.569087
$116$	0	0
$117$	20.8370	1.92638
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	16.6378	1.51253
$122$	0	0
$123$	5.17993	0.467059
$124$	0	0
$125$	9.20705	0.823503
$126$	0	0
$127$	11.4177	1.01316	0.506580	−	0.862193i	$-0.330909\pi$
$127$	11.4177	1.01316	0.506580	+	0.862193i	$0.330909\pi$
$128$	0	0
$129$	−24.3714	−2.14578
$130$	0	0
$131$	9.96907	0.871002	0.435501	−	0.900188i	$-0.356571\pi$
$131$	9.96907	0.871002	0.435501	+	0.900188i	$0.356571\pi$
$132$	0	0
$133$	0.781120	0.0677317
$134$	0	0
$135$	−13.9064	−1.19687
$136$	0	0
$137$	−19.3454	−1.65279	−0.826395	−	0.563091i	$-0.809612\pi$
$137$	−19.3454	−1.65279	−0.826395	+	0.563091i	$0.809612\pi$
$138$	0	0
$139$	6.31075	0.535271	0.267636	−	0.963520i	$-0.413758\pi$
$139$	6.31075	0.535271	0.267636	+	0.963520i	$0.413758\pi$
$140$	0	0
$141$	−3.78797	−0.319005
$142$	0	0
$143$	22.0981	1.84793
$144$	0	0
$145$	8.62038	0.715884
$146$	0	0
$147$	2.82084	0.232659
$148$	0	0
$149$	−7.73465	−0.633647	−0.316824	−	0.948484i	$-0.602616\pi$
$149$	−7.73465	−0.633647	−0.316824	+	0.948484i	$0.602616\pi$
$150$	0	0
$151$	13.4088	1.09119	0.545596	−	0.838049i	$-0.316303\pi$
$151$	13.4088	1.09119	0.545596	+	0.838049i	$0.316303\pi$
$152$	0	0
$153$	4.95716	0.400762
$154$	0	0
$155$	−17.4854	−1.40446
$156$	0	0
$157$	12.3806	0.988082	0.494041	−	0.869439i	$-0.335519\pi$
$157$	12.3806	0.988082	0.494041	+	0.869439i	$0.335519\pi$
$158$	0	0
$159$	5.52084	0.437831
$160$	0	0
$161$	−2.42281	−0.190944
$162$	0	0
$163$	−10.7487	−0.841901	−0.420951	−	0.907084i	$-0.638303\pi$
$163$	−10.7487	−0.841901	−0.420951	+	0.907084i	$0.638303\pi$
$164$	0	0
$165$	−37.3542	−2.90802
$166$	0	0
$167$	−10.4586	−0.809314	−0.404657	−	0.914469i	$-0.632609\pi$
$167$	−10.4586	−0.809314	−0.404657	+	0.914469i	$0.632609\pi$
$168$	0	0
$169$	4.66872	0.359132
$170$	0	0
$171$	−3.87214	−0.296110
$172$	0	0
$173$	−3.83241	−0.291373	−0.145686	−	0.989331i	$-0.546539\pi$
$173$	−3.83241	−0.291373	−0.145686	+	0.989331i	$0.546539\pi$
$174$	0	0
$175$	−1.34480	−0.101657
$176$	0	0
$177$	−6.87654	−0.516872
$178$	0	0
$179$	17.5918	1.31488	0.657438	−	0.753509i	$-0.271640\pi$
$179$	17.5918	1.31488	0.657438	+	0.753509i	$0.271640\pi$
$180$	0	0
$181$	16.9212	1.25774	0.628870	−	0.777510i	$-0.283518\pi$
$181$	16.9212	1.25774	0.628870	+	0.777510i	$0.283518\pi$
$182$	0	0
$183$	−17.3868	−1.28527
$184$	0	0
$185$	−17.6897	−1.30057
$186$	0	0
$187$	5.25717	0.384442
$188$	0	0
$189$	−5.52084	−0.401582
$190$	0	0
$191$	−4.97998	−0.360339	−0.180169	−	0.983636i	$-0.557665\pi$
$191$	−4.97998	−0.360339	−0.180169	+	0.983636i	$0.557665\pi$
$192$	0	0
$193$	18.1725	1.30808	0.654042	−	0.756458i	$-0.273072\pi$
$193$	18.1725	1.30808	0.654042	+	0.756458i	$0.273072\pi$
$194$	0	0
$195$	−29.8669	−2.13881
$196$	0	0
$197$	−4.33864	−0.309116	−0.154558	−	0.987984i	$-0.549395\pi$
$197$	−4.33864	−0.309116	−0.154558	+	0.987984i	$0.549395\pi$
$198$	0	0
$199$	−16.8964	−1.19775	−0.598877	−	0.800841i	$-0.704386\pi$
$199$	−16.8964	−1.19775	−0.598877	+	0.800841i	$0.704386\pi$
$200$	0	0
$201$	−4.32816	−0.305285
$202$	0	0
$203$	3.42230	0.240198
$204$	0	0
$205$	−4.62545	−0.323055
$206$	0	0
$207$	12.0102	0.834769
$208$	0	0
$209$	−4.10648	−0.284051
$210$	0	0
$211$	23.7938	1.63803	0.819017	−	0.573770i	$-0.194520\pi$
$211$	23.7938	1.63803	0.819017	+	0.573770i	$0.194520\pi$
$212$	0	0
$213$	−4.67312	−0.320197
$214$	0	0
$215$	21.7625	1.48419
$216$	0	0
$217$	−6.94170	−0.471233
$218$	0	0
$219$	−40.3937	−2.72955
$220$	0	0
$221$	4.20342	0.282752
$222$	0	0
$223$	−1.14333	−0.0765629	−0.0382814	−	0.999267i	$-0.512188\pi$
$223$	−1.14333	−0.0765629	−0.0382814	+	0.999267i	$0.512188\pi$
$224$	0	0
$225$	6.66637	0.444425
$226$	0	0
$227$	−13.4560	−0.893108	−0.446554	−	0.894757i	$-0.647349\pi$
$227$	−13.4560	−0.893108	−0.446554	+	0.894757i	$0.647349\pi$
$228$	0	0
$229$	5.86006	0.387243	0.193622	−	0.981076i	$-0.437977\pi$
$229$	5.86006	0.387243	0.193622	+	0.981076i	$0.437977\pi$
$230$	0	0
$231$	−14.8296	−0.975719
$232$	0	0
$233$	2.41891	0.158468	0.0792341	−	0.996856i	$-0.474753\pi$
$233$	2.41891	0.158468	0.0792341	+	0.996856i	$0.474753\pi$
$234$	0	0
$235$	3.38249	0.220649
$236$	0	0
$237$	35.6166	2.31355
$238$	0	0
$239$	−8.20587	−0.530794	−0.265397	−	0.964139i	$-0.585503\pi$
$239$	−8.20587	−0.530794	−0.265397	+	0.964139i	$0.585503\pi$
$240$	0	0
$241$	26.1567	1.68490	0.842450	−	0.538775i	$-0.181113\pi$
$241$	26.1567	1.68490	0.842450	+	0.538775i	$0.181113\pi$
$242$	0	0
$243$	−14.5824	−0.935464
$244$	0	0
$245$	−2.51889	−0.160926
$246$	0	0
$247$	−3.28337	−0.208916
$248$	0	0
$249$	33.4132	2.11747
$250$	0	0
$251$	−7.98792	−0.504193	−0.252097	−	0.967702i	$-0.581120\pi$
$251$	−7.98792	−0.504193	−0.252097	+	0.967702i	$0.581120\pi$
$252$	0	0
$253$	12.7371	0.800774
$254$	0	0
$255$	−7.10539	−0.444957
$256$	0	0
$257$	9.29833	0.580014	0.290007	−	0.957025i	$-0.406342\pi$
$257$	9.29833	0.580014	0.290007	+	0.957025i	$0.406342\pi$
$258$	0	0
$259$	−7.02282	−0.436377
$260$	0	0
$261$	−16.9649	−1.05010
$262$	0	0
$263$	3.59317	0.221564	0.110782	−	0.993845i	$-0.464664\pi$
$263$	3.59317	0.221564	0.110782	+	0.993845i	$0.464664\pi$
$264$	0	0
$265$	−4.92986	−0.302839
$266$	0	0
$267$	12.6713	0.775470
$268$	0	0
$269$	−6.51331	−0.397124	−0.198562	−	0.980088i	$-0.563627\pi$
$269$	−6.51331	−0.397124	−0.198562	+	0.980088i	$0.563627\pi$
$270$	0	0
$271$	1.29935	0.0789296	0.0394648	−	0.999221i	$-0.487435\pi$
$271$	1.29935	0.0789296	0.0394648	+	0.999221i	$0.487435\pi$
$272$	0	0
$273$	−11.8572	−0.717629
$274$	0	0
$275$	7.06982	0.426326
$276$	0	0
$277$	12.0668	0.725026	0.362513	−	0.931979i	$-0.381919\pi$
$277$	12.0668	0.725026	0.362513	+	0.931979i	$0.381919\pi$
$278$	0	0
$279$	34.4111	2.06014
$280$	0	0
$281$	1.77411	0.105834	0.0529172	−	0.998599i	$-0.483148\pi$
$281$	1.77411	0.105834	0.0529172	+	0.998599i	$0.483148\pi$
$282$	0	0
$283$	−10.8859	−0.647101	−0.323550	−	0.946211i	$-0.604876\pi$
$283$	−10.8859	−0.647101	−0.323550	+	0.946211i	$0.604876\pi$
$284$	0	0
$285$	5.55016	0.328763
$286$	0	0
$287$	−1.83631	−0.108394
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	36.4643	2.13758
$292$	0	0
$293$	−23.7828	−1.38940	−0.694702	−	0.719298i	$-0.744464\pi$
$293$	−23.7828	−1.38940	−0.694702	+	0.719298i	$0.744464\pi$
$294$	0	0
$295$	6.14044	0.357511
$296$	0	0
$297$	29.0239	1.68414
$298$	0	0
$299$	10.1841	0.588960
$300$	0	0
$301$	8.63974	0.497986
$302$	0	0
$303$	5.14181	0.295389
$304$	0	0
$305$	15.5257	0.888996
$306$	0	0
$307$	−17.4011	−0.993134	−0.496567	−	0.867999i	$-0.665406\pi$
$307$	−17.4011	−0.993134	−0.496567	+	0.867999i	$0.665406\pi$
$308$	0	0
$309$	3.96990	0.225840
$310$	0	0
$311$	−14.1485	−0.802288	−0.401144	−	0.916015i	$-0.631387\pi$
$311$	−14.1485	−0.802288	−0.401144	+	0.916015i	$0.631387\pi$
$312$	0	0
$313$	4.04918	0.228873	0.114437	−	0.993431i	$-0.463494\pi$
$313$	4.04918	0.228873	0.114437	+	0.993431i	$0.463494\pi$
$314$	0	0
$315$	12.4865	0.703536
$316$	0	0
$317$	32.6583	1.83427	0.917136	−	0.398575i	$-0.130495\pi$
$317$	32.6583	1.83427	0.917136	+	0.398575i	$0.130495\pi$
$318$	0	0
$319$	−17.9916	−1.00734
$320$	0	0
$321$	45.1543	2.52026
$322$	0	0
$323$	−0.781120	−0.0434627
$324$	0	0
$325$	5.65275	0.313558
$326$	0	0
$327$	−46.5843	−2.57612
$328$	0	0
$329$	1.34285	0.0740338
$330$	0	0
$331$	−14.3272	−0.787496	−0.393748	−	0.919219i	$-0.628822\pi$
$331$	−14.3272	−0.787496	−0.393748	+	0.919219i	$0.628822\pi$
$332$	0	0
$333$	34.8132	1.90775
$334$	0	0
$335$	3.86486	0.211160
$336$	0	0
$337$	21.9938	1.19808	0.599038	−	0.800720i	$-0.295550\pi$
$337$	21.9938	1.19808	0.599038	+	0.800720i	$0.295550\pi$
$338$	0	0
$339$	−4.55701	−0.247503
$340$	0	0
$341$	36.4936	1.97624
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−17.2150	−0.926824
$346$	0	0
$347$	18.2108	0.977606	0.488803	−	0.872394i	$-0.337434\pi$
$347$	18.2108	0.977606	0.488803	+	0.872394i	$0.337434\pi$
$348$	0	0
$349$	−13.6621	−0.731313	−0.365657	−	0.930750i	$-0.619156\pi$
$349$	−13.6621	−0.731313	−0.365657	+	0.930750i	$0.619156\pi$
$350$	0	0
$351$	23.2064	1.23866
$352$	0	0
$353$	−7.82913	−0.416703	−0.208351	−	0.978054i	$-0.566810\pi$
$353$	−7.82913	−0.416703	−0.208351	+	0.978054i	$0.566810\pi$
$354$	0	0
$355$	4.17289	0.221474
$356$	0	0
$357$	−2.82084	−0.149295
$358$	0	0
$359$	36.9947	1.95251	0.976253	−	0.216634i	$-0.0695079\pi$
$359$	36.9947	1.95251	0.976253	+	0.216634i	$0.0695079\pi$
$360$	0	0
$361$	−18.3899	−0.967887
$362$	0	0
$363$	46.9326	2.46332
$364$	0	0
$365$	36.0698	1.88798
$366$	0	0
$367$	−31.6345	−1.65131	−0.825654	−	0.564177i	$-0.809194\pi$
$367$	−31.6345	−1.65131	−0.825654	+	0.564177i	$0.809194\pi$
$368$	0	0
$369$	9.10286	0.473876
$370$	0	0
$371$	−1.95716	−0.101611
$372$	0	0
$373$	28.3894	1.46995	0.734974	−	0.678095i	$-0.237194\pi$
$373$	28.3894	1.46995	0.734974	+	0.678095i	$0.237194\pi$
$374$	0	0
$375$	25.9716	1.34117
$376$	0	0
$377$	−14.3853	−0.740883
$378$	0	0
$379$	21.6022	1.10963	0.554814	−	0.831974i	$-0.312789\pi$
$379$	21.6022	1.10963	0.554814	+	0.831974i	$0.312789\pi$
$380$	0	0
$381$	32.2077	1.65005
$382$	0	0
$383$	17.4402	0.891153	0.445577	−	0.895244i	$-0.352999\pi$
$383$	17.4402	0.891153	0.445577	+	0.895244i	$0.352999\pi$
$384$	0	0
$385$	13.2422	0.674886
$386$	0	0
$387$	−42.8286	−2.17710
$388$	0	0
$389$	−30.6552	−1.55428	−0.777141	−	0.629326i	$-0.783331\pi$
$389$	−30.6552	−1.55428	−0.777141	+	0.629326i	$0.783331\pi$
$390$	0	0
$391$	2.42281	0.122527
$392$	0	0
$393$	28.1212	1.41853
$394$	0	0
$395$	−31.8040	−1.60024
$396$	0	0
$397$	−11.0023	−0.552189	−0.276095	−	0.961130i	$-0.589040\pi$
$397$	−11.0023	−0.552189	−0.276095	+	0.961130i	$0.589040\pi$
$398$	0	0
$399$	2.20342	0.110309
$400$	0	0
$401$	3.98115	0.198809	0.0994046	−	0.995047i	$-0.468306\pi$
$401$	3.98115	0.198809	0.0994046	+	0.995047i	$0.468306\pi$
$402$	0	0
$403$	29.1788	1.45350
$404$	0	0
$405$	−1.76811	−0.0878579
$406$	0	0
$407$	36.9201	1.83006
$408$	0	0
$409$	21.0687	1.04178	0.520890	−	0.853624i	$-0.325600\pi$
$409$	21.0687	1.04178	0.520890	+	0.853624i	$0.325600\pi$
$410$	0	0
$411$	−54.5704	−2.69176
$412$	0	0
$413$	2.43776	0.119954
$414$	0	0
$415$	−29.8365	−1.46462
$416$	0	0
$417$	17.8016	0.871750
$418$	0	0
$419$	−15.8730	−0.775446	−0.387723	−	0.921776i	$-0.626738\pi$
$419$	−15.8730	−0.775446	−0.387723	+	0.921776i	$0.626738\pi$
$420$	0	0
$421$	−12.7787	−0.622794	−0.311397	−	0.950280i	$-0.600797\pi$
$421$	−12.7787	−0.622794	−0.311397	+	0.950280i	$0.600797\pi$
$422$	0	0
$423$	−6.65672	−0.323661
$424$	0	0
$425$	1.34480	0.0652323
$426$	0	0
$427$	6.16369	0.298282
$428$	0	0
$429$	62.3352	3.00957
$430$	0	0
$431$	−39.1793	−1.88720	−0.943601	−	0.331084i	$-0.892586\pi$
$431$	−39.1793	−1.88720	−0.943601	+	0.331084i	$0.892586\pi$
$432$	0	0
$433$	−16.2650	−0.781648	−0.390824	−	0.920465i	$-0.627810\pi$
$433$	−16.2650	−0.781648	−0.390824	+	0.920465i	$0.627810\pi$
$434$	0	0
$435$	24.3168	1.16590
$436$	0	0
$437$	−1.89250	−0.0905307
$438$	0	0
$439$	−1.97235	−0.0941354	−0.0470677	−	0.998892i	$-0.514988\pi$
$439$	−1.97235	−0.0941354	−0.0470677	+	0.998892i	$0.514988\pi$
$440$	0	0
$441$	4.95716	0.236055
$442$	0	0
$443$	−34.1280	−1.62147	−0.810734	−	0.585414i	$-0.800932\pi$
$443$	−34.1280	−1.62147	−0.810734	+	0.585414i	$0.800932\pi$
$444$	0	0
$445$	−11.3149	−0.536377
$446$	0	0
$447$	−21.8182	−1.03197
$448$	0	0
$449$	−6.09912	−0.287835	−0.143918	−	0.989590i	$-0.545970\pi$
$449$	−6.09912	−0.287835	−0.143918	+	0.989590i	$0.545970\pi$
$450$	0	0
$451$	9.65376	0.454578
$452$	0	0
$453$	37.8241	1.77713
$454$	0	0
$455$	10.5879	0.496370
$456$	0	0
$457$	−14.9650	−0.700034	−0.350017	−	0.936743i	$-0.613824\pi$
$457$	−14.9650	−0.700034	−0.350017	+	0.936743i	$0.613824\pi$
$458$	0	0
$459$	5.52084	0.257691
$460$	0	0
$461$	25.9096	1.20673	0.603365	−	0.797465i	$-0.293826\pi$
$461$	25.9096	1.20673	0.603365	+	0.797465i	$0.293826\pi$
$462$	0	0
$463$	19.7323	0.917037	0.458518	−	0.888685i	$-0.348380\pi$
$463$	19.7323	0.917037	0.458518	+	0.888685i	$0.348380\pi$
$464$	0	0
$465$	−49.3234	−2.28732
$466$	0	0
$467$	10.7158	0.495868	0.247934	−	0.968777i	$-0.420248\pi$
$467$	10.7158	0.495868	0.247934	+	0.968777i	$0.420248\pi$
$468$	0	0
$469$	1.53435	0.0708497
$470$	0	0
$471$	34.9238	1.60920
$472$	0	0
$473$	−45.4205	−2.08844
$474$	0	0
$475$	−1.05045	−0.0481979
$476$	0	0
$477$	9.70194	0.444221
$478$	0	0
$479$	6.99221	0.319482	0.159741	−	0.987159i	$-0.448934\pi$
$479$	6.99221	0.319482	0.159741	+	0.987159i	$0.448934\pi$
$480$	0	0
$481$	29.5199	1.34599
$482$	0	0
$483$	−6.83436	−0.310974
$484$	0	0
$485$	−32.5610	−1.47852
$486$	0	0
$487$	11.2113	0.508032	0.254016	−	0.967200i	$-0.418248\pi$
$487$	11.2113	0.508032	0.254016	+	0.967200i	$0.418248\pi$
$488$	0	0
$489$	−30.3203	−1.37113
$490$	0	0
$491$	−26.1467	−1.17999	−0.589993	−	0.807408i	$-0.700870\pi$
$491$	−26.1467	−1.17999	−0.589993	+	0.807408i	$0.700870\pi$
$492$	0	0
$493$	−3.42230	−0.154133
$494$	0	0
$495$	−65.6437	−2.95047
$496$	0	0
$497$	1.65664	0.0743105
$498$	0	0
$499$	0.887794	0.0397431	0.0198716	−	0.999803i	$-0.493674\pi$
$499$	0.887794	0.0397431	0.0198716	+	0.999803i	$0.493674\pi$
$500$	0	0
$501$	−29.5022	−1.31806
$502$	0	0
$503$	23.2662	1.03739	0.518694	−	0.854960i	$-0.326418\pi$
$503$	23.2662	1.03739	0.518694	+	0.854960i	$0.326418\pi$
$504$	0	0
$505$	−4.59141	−0.204315
$506$	0	0
$507$	13.1697	0.584888
$508$	0	0
$509$	1.45906	0.0646718	0.0323359	−	0.999477i	$-0.489705\pi$
$509$	1.45906	0.0646718	0.0323359	+	0.999477i	$0.489705\pi$
$510$	0	0
$511$	14.3197	0.633467
$512$	0	0
$513$	−4.31244	−0.190399
$514$	0	0
$515$	−3.54494	−0.156209
$516$	0	0
$517$	−7.05959	−0.310480
$518$	0	0
$519$	−10.8106	−0.474534
$520$	0	0
$521$	−42.0865	−1.84384	−0.921921	−	0.387377i	$-0.873381\pi$
$521$	−42.0865	−1.84384	−0.921921	+	0.387377i	$0.873381\pi$
$522$	0	0
$523$	−17.2674	−0.755051	−0.377525	−	0.925999i	$-0.623225\pi$
$523$	−17.2674	−0.755051	−0.377525	+	0.925999i	$0.623225\pi$
$524$	0	0
$525$	−3.79346	−0.165560
$526$	0	0
$527$	6.94170	0.302385
$528$	0	0
$529$	−17.1300	−0.744783
$530$	0	0
$531$	−12.0844	−0.524417
$532$	0	0
$533$	7.71876	0.334337
$534$	0	0
$535$	−40.3208	−1.74322
$536$	0	0
$537$	49.6238	2.14143
$538$	0	0
$539$	5.25717	0.226442
$540$	0	0
$541$	24.7411	1.06370	0.531851	−	0.846838i	$-0.321497\pi$
$541$	24.7411	1.06370	0.531851	+	0.846838i	$0.321497\pi$
$542$	0	0
$543$	47.7320	2.04837
$544$	0	0
$545$	41.5978	1.78185
$546$	0	0
$547$	−4.66104	−0.199292	−0.0996459	−	0.995023i	$-0.531771\pi$
$547$	−4.66104	−0.199292	−0.0996459	+	0.995023i	$0.531771\pi$
$548$	0	0
$549$	−30.5544	−1.30403
$550$	0	0
$551$	2.67323	0.113883
$552$	0	0
$553$	−12.6262	−0.536922
$554$	0	0
$555$	−49.8999	−2.11813
$556$	0	0
$557$	13.1120	0.555573	0.277787	−	0.960643i	$-0.410399\pi$
$557$	13.1120	0.555573	0.277787	+	0.960643i	$0.410399\pi$
$558$	0	0
$559$	−36.3164	−1.53602
$560$	0	0
$561$	14.8296	0.626108
$562$	0	0
$563$	27.6650	1.16594	0.582971	−	0.812493i	$-0.301890\pi$
$563$	27.6650	1.16594	0.582971	+	0.812493i	$0.301890\pi$
$564$	0	0
$565$	4.06921	0.171193
$566$	0	0
$567$	−0.701939	−0.0294787
$568$	0	0
$569$	10.8911	0.456578	0.228289	−	0.973593i	$-0.426687\pi$
$569$	10.8911	0.456578	0.228289	+	0.973593i	$0.426687\pi$
$570$	0	0
$571$	−40.3446	−1.68837	−0.844184	−	0.536053i	$-0.819915\pi$
$571$	−40.3446	−1.68837	−0.844184	+	0.536053i	$0.819915\pi$
$572$	0	0
$573$	−14.0477	−0.586853
$574$	0	0
$575$	3.25819	0.135876
$576$	0	0
$577$	30.0190	1.24971	0.624855	−	0.780741i	$-0.285158\pi$
$577$	30.0190	1.24971	0.624855	+	0.780741i	$0.285158\pi$
$578$	0	0
$579$	51.2618	2.13037
$580$	0	0
$581$	−11.8451	−0.491418
$582$	0	0
$583$	10.2891	0.426131
$584$	0	0
$585$	−52.4861	−2.17003
$586$	0	0
$587$	−38.3982	−1.58486	−0.792432	−	0.609960i	$-0.791185\pi$
$587$	−38.3982	−1.58486	−0.792432	+	0.609960i	$0.791185\pi$
$588$	0	0
$589$	−5.42230	−0.223422
$590$	0	0
$591$	−12.2386	−0.503430
$592$	0	0
$593$	5.71631	0.234741	0.117370	−	0.993088i	$-0.462554\pi$
$593$	5.71631	0.234741	0.117370	+	0.993088i	$0.462554\pi$
$594$	0	0
$595$	2.51889	0.103264
$596$	0	0
$597$	−47.6621	−1.95068
$598$	0	0
$599$	−29.9765	−1.22481	−0.612404	−	0.790545i	$-0.709798\pi$
$599$	−29.9765	−1.22481	−0.612404	+	0.790545i	$0.709798\pi$
$600$	0	0
$601$	28.2586	1.15269	0.576346	−	0.817206i	$-0.304478\pi$
$601$	28.2586	1.15269	0.576346	+	0.817206i	$0.304478\pi$
$602$	0	0
$603$	−7.60602	−0.309741
$604$	0	0
$605$	−41.9087	−1.70383
$606$	0	0
$607$	−43.1772	−1.75251	−0.876253	−	0.481851i	$-0.839965\pi$
$607$	−43.1772	−1.75251	−0.876253	+	0.481851i	$0.839965\pi$
$608$	0	0
$609$	9.65376	0.391190
$610$	0	0
$611$	−5.64456	−0.228355
$612$	0	0
$613$	12.1412	0.490380	0.245190	−	0.969475i	$-0.421150\pi$
$613$	12.1412	0.490380	0.245190	+	0.969475i	$0.421150\pi$
$614$	0	0
$615$	−13.0477	−0.526133
$616$	0	0
$617$	23.0826	0.929270	0.464635	−	0.885502i	$-0.346185\pi$
$617$	23.0826	0.929270	0.464635	+	0.885502i	$0.346185\pi$
$618$	0	0
$619$	−7.86057	−0.315943	−0.157972	−	0.987444i	$-0.550495\pi$
$619$	−7.86057	−0.315943	−0.157972	+	0.987444i	$0.550495\pi$
$620$	0	0
$621$	13.3759	0.536757
$622$	0	0
$623$	−4.49202	−0.179969
$624$	0	0
$625$	−29.9155	−1.19662
$626$	0	0
$627$	−11.5837	−0.462610
$628$	0	0
$629$	7.02282	0.280018
$630$	0	0
$631$	35.8212	1.42602	0.713010	−	0.701154i	$-0.247331\pi$
$631$	35.8212	1.42602	0.713010	+	0.701154i	$0.247331\pi$
$632$	0	0
$633$	67.1186	2.66773
$634$	0	0
$635$	−28.7600	−1.14131
$636$	0	0
$637$	4.20342	0.166545
$638$	0	0
$639$	−8.21222	−0.324871
$640$	0	0
$641$	−5.24743	−0.207261	−0.103631	−	0.994616i	$-0.533046\pi$
$641$	−5.24743	−0.207261	−0.103631	+	0.994616i	$0.533046\pi$
$642$	0	0
$643$	−32.1828	−1.26917	−0.634583	−	0.772855i	$-0.718828\pi$
$643$	−32.1828	−1.26917	−0.634583	+	0.772855i	$0.718828\pi$
$644$	0	0
$645$	61.3887	2.41718
$646$	0	0
$647$	−16.6171	−0.653286	−0.326643	−	0.945148i	$-0.605917\pi$
$647$	−16.6171	−0.653286	−0.326643	+	0.945148i	$0.605917\pi$
$648$	0	0
$649$	−12.8157	−0.503061
$650$	0	0
$651$	−19.5814	−0.767457
$652$	0	0
$653$	−0.159913	−0.00625786	−0.00312893	−	0.999995i	$-0.500996\pi$
$653$	−0.159913	−0.00625786	−0.00312893	+	0.999995i	$0.500996\pi$
$654$	0	0
$655$	−25.1110	−0.981167
$656$	0	0
$657$	−70.9851	−2.76939
$658$	0	0
$659$	−32.4256	−1.26312	−0.631561	−	0.775326i	$-0.717586\pi$
$659$	−32.4256	−1.26312	−0.631561	+	0.775326i	$0.717586\pi$
$660$	0	0
$661$	31.9068	1.24103	0.620516	−	0.784194i	$-0.286923\pi$
$661$	31.9068	1.24103	0.620516	+	0.784194i	$0.286923\pi$
$662$	0	0
$663$	11.8572	0.460495
$664$	0	0
$665$	−1.96755	−0.0762985
$666$	0	0
$667$	−8.29157	−0.321051
$668$	0	0
$669$	−3.22515	−0.124691
$670$	0	0
$671$	−32.4036	−1.25093
$672$	0	0
$673$	31.4641	1.21285	0.606425	−	0.795141i	$-0.292603\pi$
$673$	31.4641	1.21285	0.606425	+	0.795141i	$0.292603\pi$
$674$	0	0
$675$	7.42441	0.285766
$676$	0	0
$677$	−24.6433	−0.947120	−0.473560	−	0.880762i	$-0.657031\pi$
$677$	−24.6433	−0.947120	−0.473560	+	0.880762i	$0.657031\pi$
$678$	0	0
$679$	−12.9267	−0.496083
$680$	0	0
$681$	−37.9573	−1.45453
$682$	0	0
$683$	24.5793	0.940502	0.470251	−	0.882533i	$-0.344163\pi$
$683$	24.5793	0.940502	0.470251	+	0.882533i	$0.344163\pi$
$684$	0	0
$685$	48.7289	1.86184
$686$	0	0
$687$	16.5303	0.630670
$688$	0	0
$689$	8.22675	0.313414
$690$	0	0
$691$	−30.5847	−1.16350	−0.581749	−	0.813368i	$-0.697632\pi$
$691$	−30.5847	−1.16350	−0.581749	+	0.813368i	$0.697632\pi$
$692$	0	0
$693$	−26.0606	−0.989960
$694$	0	0
$695$	−15.8961	−0.602973
$696$	0	0
$697$	1.83631	0.0695550
$698$	0	0
$699$	6.82337	0.258084
$700$	0	0
$701$	39.8182	1.50391	0.751956	−	0.659213i	$-0.229111\pi$
$701$	39.8182	1.50391	0.751956	+	0.659213i	$0.229111\pi$
$702$	0	0
$703$	−5.48567	−0.206896
$704$	0	0
$705$	9.54148	0.359353
$706$	0	0
$707$	−1.82279	−0.0685531
$708$	0	0
$709$	1.22040	0.0458331	0.0229166	−	0.999737i	$-0.492705\pi$
$709$	1.22040	0.0458331	0.0229166	+	0.999737i	$0.492705\pi$
$710$	0	0
$711$	62.5902	2.34732
$712$	0	0
$713$	16.8184	0.629854
$714$	0	0
$715$	−55.6625	−2.08166
$716$	0	0
$717$	−23.1475	−0.864459
$718$	0	0
$719$	−14.9740	−0.558435	−0.279218	−	0.960228i	$-0.590075\pi$
$719$	−14.9740	−0.558435	−0.279218	+	0.960228i	$0.590075\pi$
$720$	0	0
$721$	−1.40734	−0.0524122
$722$	0	0
$723$	73.7839	2.74405
$724$	0	0
$725$	−4.60230	−0.170925
$726$	0	0
$727$	−24.6970	−0.915959	−0.457980	−	0.888963i	$-0.651427\pi$
$727$	−24.6970	−0.915959	−0.457980	+	0.888963i	$0.651427\pi$
$728$	0	0
$729$	−43.2406	−1.60150
$730$	0	0
$731$	−8.63974	−0.319552
$732$	0	0
$733$	5.36434	0.198136	0.0990682	−	0.995081i	$-0.468414\pi$
$733$	5.36434	0.198136	0.0990682	+	0.995081i	$0.468414\pi$
$734$	0	0
$735$	−7.10539	−0.262086
$736$	0	0
$737$	−8.06634	−0.297127
$738$	0	0
$739$	−20.4094	−0.750772	−0.375386	−	0.926869i	$-0.622490\pi$
$739$	−20.4094	−0.750772	−0.375386	+	0.926869i	$0.622490\pi$
$740$	0	0
$741$	−9.26188	−0.340244
$742$	0	0
$743$	−16.9780	−0.622861	−0.311431	−	0.950269i	$-0.600808\pi$
$743$	−16.9780	−0.622861	−0.311431	+	0.950269i	$0.600808\pi$
$744$	0	0
$745$	19.4827	0.713791
$746$	0	0
$747$	58.7180	2.14838
$748$	0	0
$749$	−16.0074	−0.584896
$750$	0	0
$751$	−8.99611	−0.328273	−0.164136	−	0.986438i	$-0.552484\pi$
$751$	−8.99611	−0.328273	−0.164136	+	0.986438i	$0.552484\pi$
$752$	0	0
$753$	−22.5327	−0.821136
$754$	0	0
$755$	−33.7752	−1.22921
$756$	0	0
$757$	−8.75655	−0.318262	−0.159131	−	0.987257i	$-0.550869\pi$
$757$	−8.75655	−0.318262	−0.159131	+	0.987257i	$0.550869\pi$
$758$	0	0
$759$	35.9294	1.30415
$760$	0	0
$761$	−51.5596	−1.86903	−0.934517	−	0.355920i	$-0.884168\pi$
$761$	−51.5596	−1.86903	−0.934517	+	0.355920i	$0.884168\pi$
$762$	0	0
$763$	16.5143	0.597859
$764$	0	0
$765$	−12.4865	−0.451451
$766$	0	0
$767$	−10.2469	−0.369995
$768$	0	0
$769$	1.84214	0.0664292	0.0332146	−	0.999448i	$-0.489426\pi$
$769$	1.84214	0.0664292	0.0332146	+	0.999448i	$0.489426\pi$
$770$	0	0
$771$	26.2291	0.944619
$772$	0	0
$773$	−54.9320	−1.97577	−0.987884	−	0.155196i	$-0.950399\pi$
$773$	−54.9320	−1.97577	−0.987884	+	0.155196i	$0.950399\pi$
$774$	0	0
$775$	9.33518	0.335329
$776$	0	0
$777$	−19.8103	−0.710690
$778$	0	0
$779$	−1.43438	−0.0513918
$780$	0	0
$781$	−8.70923	−0.311641
$782$	0	0
$783$	−18.8939	−0.675214
$784$	0	0
$785$	−31.1854	−1.11306
$786$	0	0
$787$	5.34624	0.190573	0.0952865	−	0.995450i	$-0.469623\pi$
$787$	5.34624	0.190573	0.0952865	+	0.995450i	$0.469623\pi$
$788$	0	0
$789$	10.1358	0.360842
$790$	0	0
$791$	1.61548	0.0574398
$792$	0	0
$793$	−25.9086	−0.920041
$794$	0	0
$795$	−13.9064	−0.493208
$796$	0	0
$797$	−47.0419	−1.66631	−0.833156	−	0.553039i	$-0.813468\pi$
$797$	−47.0419	−1.66631	−0.833156	+	0.553039i	$0.813468\pi$
$798$	0	0
$799$	−1.34285	−0.0475066
$800$	0	0
$801$	22.2676	0.786788
$802$	0	0
$803$	−75.2811	−2.65661
$804$	0	0
$805$	6.10278	0.215095
$806$	0	0
$807$	−18.3730	−0.646761
$808$	0	0
$809$	27.2918	0.959527	0.479763	−	0.877398i	$-0.340722\pi$
$809$	27.2918	0.959527	0.479763	+	0.877398i	$0.340722\pi$
$810$	0	0
$811$	31.6841	1.11258	0.556289	−	0.830989i	$-0.312225\pi$
$811$	31.6841	1.11258	0.556289	+	0.830989i	$0.312225\pi$
$812$	0	0
$813$	3.66525	0.128546
$814$	0	0
$815$	27.0747	0.948386
$816$	0	0
$817$	6.74867	0.236106
$818$	0	0
$819$	−20.8370	−0.728104
$820$	0	0
$821$	−13.2126	−0.461124	−0.230562	−	0.973058i	$-0.574057\pi$
$821$	−13.2126	−0.461124	−0.230562	+	0.973058i	$0.574057\pi$
$822$	0	0
$823$	8.52446	0.297144	0.148572	−	0.988902i	$-0.452532\pi$
$823$	8.52446	0.297144	0.148572	+	0.988902i	$0.452532\pi$
$824$	0	0
$825$	19.9429	0.694322
$826$	0	0
$827$	−2.47994	−0.0862360	−0.0431180	−	0.999070i	$-0.513729\pi$
$827$	−2.47994	−0.0862360	−0.0431180	+	0.999070i	$0.513729\pi$
$828$	0	0
$829$	−47.4995	−1.64972	−0.824862	−	0.565333i	$-0.808748\pi$
$829$	−47.4995	−1.64972	−0.824862	+	0.565333i	$0.808748\pi$
$830$	0	0
$831$	34.0387	1.18079
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	26.3441	0.911676
$836$	0	0
$837$	38.3240	1.32467
$838$	0	0
$839$	−32.9427	−1.13731	−0.568653	−	0.822577i	$-0.692535\pi$
$839$	−32.9427	−1.13731	−0.568653	+	0.822577i	$0.692535\pi$
$840$	0	0
$841$	−17.2879	−0.596134
$842$	0	0
$843$	5.00448	0.172363
$844$	0	0
$845$	−11.7600	−0.404556
$846$	0	0
$847$	−16.6378	−0.571681
$848$	0	0
$849$	−30.7075	−1.05388
$850$	0	0
$851$	17.0149	0.583265
$852$	0	0
$853$	−47.5645	−1.62858	−0.814288	−	0.580460i	$-0.802873\pi$
$853$	−47.5645	−1.62858	−0.814288	+	0.580460i	$0.802873\pi$
$854$	0	0
$855$	9.75348	0.333562
$856$	0	0
$857$	−34.0918	−1.16455	−0.582277	−	0.812991i	$-0.697838\pi$
$857$	−34.0918	−1.16455	−0.582277	+	0.812991i	$0.697838\pi$
$858$	0	0
$859$	−13.1715	−0.449405	−0.224703	−	0.974427i	$-0.572141\pi$
$859$	−13.1715	−0.449405	−0.224703	+	0.974427i	$0.572141\pi$
$860$	0	0
$861$	−5.17993	−0.176532
$862$	0	0
$863$	−5.24872	−0.178668	−0.0893342	−	0.996002i	$-0.528474\pi$
$863$	−5.24872	−0.178668	−0.0893342	+	0.996002i	$0.528474\pi$
$864$	0	0
$865$	9.65342	0.328226
$866$	0	0
$867$	2.82084	0.0958009
$868$	0	0
$869$	66.3782	2.25172
$870$	0	0
$871$	−6.44952	−0.218534
$872$	0	0
$873$	64.0799	2.16878
$874$	0	0
$875$	−9.20705	−0.311255
$876$	0	0
$877$	−45.7500	−1.54487	−0.772434	−	0.635096i	$-0.780961\pi$
$877$	−45.7500	−1.54487	−0.772434	+	0.635096i	$0.780961\pi$
$878$	0	0
$879$	−67.0874	−2.26280
$880$	0	0
$881$	0.719858	0.0242526	0.0121263	−	0.999926i	$-0.496140\pi$
$881$	0.719858	0.0242526	0.0121263	+	0.999926i	$0.496140\pi$
$882$	0	0
$883$	36.3191	1.22224	0.611118	−	0.791539i	$-0.290720\pi$
$883$	36.3191	1.22224	0.611118	+	0.791539i	$0.290720\pi$
$884$	0	0
$885$	17.3212	0.582247
$886$	0	0
$887$	−12.7805	−0.429126	−0.214563	−	0.976710i	$-0.568833\pi$
$887$	−12.7805	−0.429126	−0.214563	+	0.976710i	$0.568833\pi$
$888$	0	0
$889$	−11.4177	−0.382939
$890$	0	0
$891$	3.69021	0.123627
$892$	0	0
$893$	1.04893	0.0351010
$894$	0	0
$895$	−44.3119	−1.48118
$896$	0	0
$897$	28.7277	0.959189
$898$	0	0
$899$	−23.7565	−0.792325
$900$	0	0
$901$	1.95716	0.0652024
$902$	0	0
$903$	24.3714	0.811028
$904$	0	0
$905$	−42.6225	−1.41682
$906$	0	0
$907$	−27.1387	−0.901126	−0.450563	−	0.892745i	$-0.648777\pi$
$907$	−27.1387	−0.901126	−0.450563	+	0.892745i	$0.648777\pi$
$908$	0	0
$909$	9.03586	0.299701
$910$	0	0
$911$	34.9889	1.15923	0.579616	−	0.814890i	$-0.303202\pi$
$911$	34.9889	1.15923	0.579616	+	0.814890i	$0.303202\pi$
$912$	0	0
$913$	62.2717	2.06089
$914$	0	0
$915$	43.7954	1.44783
$916$	0	0
$917$	−9.96907	−0.329208
$918$	0	0
$919$	−19.0980	−0.629984	−0.314992	−	0.949094i	$-0.602002\pi$
$919$	−19.0980	−0.629984	−0.314992	+	0.949094i	$0.602002\pi$
$920$	0	0
$921$	−49.0858	−1.61743
$922$	0	0
$923$	−6.96355	−0.229208
$924$	0	0
$925$	9.44428	0.310526
$926$	0	0
$927$	6.97643	0.229136
$928$	0	0
$929$	38.5323	1.26420	0.632102	−	0.774885i	$-0.282193\pi$
$929$	38.5323	1.26420	0.632102	+	0.774885i	$0.282193\pi$
$930$	0	0
$931$	−0.781120	−0.0256002
$932$	0	0
$933$	−39.9107	−1.30662
$934$	0	0
$935$	−13.2422	−0.433067
$936$	0	0
$937$	40.9170	1.33670	0.668350	−	0.743847i	$-0.267001\pi$
$937$	40.9170	1.33670	0.668350	+	0.743847i	$0.267001\pi$
$938$	0	0
$939$	11.4221	0.372747
$940$	0	0
$941$	−14.9176	−0.486298	−0.243149	−	0.969989i	$-0.578180\pi$
$941$	−14.9176	−0.486298	−0.243149	+	0.969989i	$0.578180\pi$
$942$	0	0
$943$	4.44901	0.144880
$944$	0	0
$945$	13.9064	0.452374
$946$	0	0
$947$	54.3712	1.76683	0.883413	−	0.468595i	$-0.155240\pi$
$947$	54.3712	1.76683	0.883413	+	0.468595i	$0.155240\pi$
$948$	0	0
$949$	−60.1917	−1.95391
$950$	0	0
$951$	92.1239	2.98732
$952$	0	0
$953$	−33.1263	−1.07306	−0.536532	−	0.843880i	$-0.680266\pi$
$953$	−33.1263	−1.07306	−0.536532	+	0.843880i	$0.680266\pi$
$954$	0	0
$955$	12.5440	0.405915
$956$	0	0
$957$	−50.7514	−1.64056
$958$	0	0
$959$	19.3454	0.624696
$960$	0	0
$961$	17.1871	0.554424
$962$	0	0
$963$	79.3510	2.55705
$964$	0	0
$965$	−45.7745	−1.47353
$966$	0	0
$967$	−54.0452	−1.73798	−0.868988	−	0.494834i	$-0.835229\pi$
$967$	−54.0452	−1.73798	−0.868988	+	0.494834i	$0.835229\pi$
$968$	0	0
$969$	−2.20342	−0.0707840
$970$	0	0
$971$	40.0719	1.28597	0.642984	−	0.765880i	$-0.277696\pi$
$971$	40.0719	1.28597	0.642984	+	0.765880i	$0.277696\pi$
$972$	0	0
$973$	−6.31075	−0.202313
$974$	0	0
$975$	15.9455	0.510665
$976$	0	0
$977$	37.2405	1.19143	0.595715	−	0.803196i	$-0.296869\pi$
$977$	37.2405	1.19143	0.595715	+	0.803196i	$0.296869\pi$
$978$	0	0
$979$	23.6153	0.754748
$980$	0	0
$981$	−81.8641	−2.61372
$982$	0	0
$983$	12.0729	0.385067	0.192534	−	0.981290i	$-0.438330\pi$
$983$	12.0729	0.385067	0.192534	+	0.981290i	$0.438330\pi$
$984$	0	0
$985$	10.9286	0.348213
$986$	0	0
$987$	3.78797	0.120572
$988$	0	0
$989$	−20.9324	−0.665612
$990$	0	0
$991$	−6.66778	−0.211809	−0.105905	−	0.994376i	$-0.533774\pi$
$991$	−6.66778	−0.211809	−0.105905	+	0.994376i	$0.533774\pi$
$992$	0	0
$993$	−40.4149	−1.28253
$994$	0	0
$995$	42.5601	1.34925
$996$	0	0
$997$	−13.2321	−0.419064	−0.209532	−	0.977802i	$-0.567194\pi$
$997$	−13.2321	−0.419064	−0.209532	+	0.977802i	$0.567194\pi$
$998$	0	0
$999$	38.7719	1.22669

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bt.1.4		5
4.3	odd	2		7616.2.a.bq.1.2		5
8.3	odd	2		1904.2.a.t.1.4		5
8.5	even	2		119.2.a.b.1.5	✓	5
24.5	odd	2		1071.2.a.m.1.1		5
40.29	even	2		2975.2.a.m.1.1		5
56.5	odd	6		833.2.e.h.18.1		10
56.13	odd	2		833.2.a.g.1.5		5
56.37	even	6		833.2.e.i.18.1		10
56.45	odd	6		833.2.e.h.324.1		10
56.53	even	6		833.2.e.i.324.1		10
136.101	even	2		2023.2.a.j.1.5		5
168.125	even	2		7497.2.a.br.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.2.a.b.1.5	✓	5	8.5	even	2
833.2.a.g.1.5		5	56.13	odd	2
833.2.e.h.18.1		10	56.5	odd	6
833.2.e.h.324.1		10	56.45	odd	6
833.2.e.i.18.1		10	56.37	even	6
833.2.e.i.324.1		10	56.53	even	6
1071.2.a.m.1.1		5	24.5	odd	2
1904.2.a.t.1.4		5	8.3	odd	2
2023.2.a.j.1.5		5	136.101	even	2
2975.2.a.m.1.1		5	40.29	even	2
7497.2.a.br.1.1		5	168.125	even	2
7616.2.a.bq.1.2		5	4.3	odd	2
7616.2.a.bt.1.4		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82084 q^{3} -2.51889 q^{5} -1.00000 q^{7} +4.95716 q^{9} +5.25717 q^{11} +4.20342 q^{13} -7.10539 q^{15} +1.00000 q^{17} -0.781120 q^{19} -2.82084 q^{21} +2.42281 q^{23} +1.34480 q^{25} +5.52084 q^{27} -3.42230 q^{29} +6.94170 q^{31} +14.8296 q^{33} +2.51889 q^{35} +7.02282 q^{37} +11.8572 q^{39} +1.83631 q^{41} -8.63974 q^{43} -12.4865 q^{45} -1.34285 q^{47} +1.00000 q^{49} +2.82084 q^{51} +1.95716 q^{53} -13.2422 q^{55} -2.20342 q^{57} -2.43776 q^{59} -6.16369 q^{61} -4.95716 q^{63} -10.5879 q^{65} -1.53435 q^{67} +6.83436 q^{69} -1.65664 q^{71} -14.3197 q^{73} +3.79346 q^{75} -5.25717 q^{77} +12.6262 q^{79} +0.701939 q^{81} +11.8451 q^{83} -2.51889 q^{85} -9.65376 q^{87} +4.49202 q^{89} -4.20342 q^{91} +19.5814 q^{93} +1.96755 q^{95} +12.9267 q^{97} +26.0606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 5 q^{7} + 11 q^{9} + 2 q^{11} - 2 q^{13} + 8 q^{15} + 5 q^{17} - 6 q^{19} - 2 q^{21} - 10 q^{23} + 21 q^{25} + 26 q^{27} + 8 q^{29} + 6 q^{33} - 8 q^{37} + 14 q^{39} + 18 q^{41} - 8 q^{43}+ \cdots + 62 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 - 5 * q^7 + 11 * q^9 + 2 * q^11 - 2 * q^13 + 8 * q^15 + 5 * q^17 - 6 * q^19 - 2 * q^21 - 10 * q^23 + 21 * q^25 + 26 * q^27 + 8 * q^29 + 6 * q^33 - 8 * q^37 + 14 * q^39 + 18 * q^41 - 8 * q^43 - 10 * q^47 + 5 * q^49 + 2 * q^51 - 4 * q^53 - 24 * q^55 + 12 * q^57 - 8 * q^59 - 22 * q^61 - 11 * q^63 - 30 * q^65 - 16 * q^67 + 32 * q^69 - 2 * q^71 + 10 * q^73 + 14 * q^75 - 2 * q^77 + 18 * q^79 + 25 * q^81 + 12 * q^83 - 26 * q^87 + 20 * q^89 + 2 * q^91 + 28 * q^93 - 22 * q^95 + 12 * q^97 + 62 * q^99

Introduction

L-functions

Modular forms

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