LMFDB - Embedded newform 676.4.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [676,4,Mod(1,676)] 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("676.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(676, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	676.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8852911639$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 114x^{6} - 224x^{5} + 3123x^{4} + 10080x^{3} - 7598x^{2} - 46368x - 33663 $ x^8 - 114x^6 - 224x^5 + 3123x^4 + 10080x^3 - 7598x^2 - 46368x - 33663
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30615$ of defining polynomial
Character	$\chi$	$=$	676.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.78841 q^{3} +2.49640 q^{5} -17.9693 q^{7} +68.8129 q^{9} -57.2769 q^{11} -24.4358 q^{15} -17.5058 q^{17} -22.1225 q^{19} +175.891 q^{21} -131.193 q^{23} -118.768 q^{25} -409.282 q^{27} +74.2527 q^{29} -110.463 q^{31} +560.649 q^{33} -44.8587 q^{35} -241.452 q^{37} -426.704 q^{41} -208.296 q^{43} +171.785 q^{45} +158.624 q^{47} -20.1031 q^{49} +171.354 q^{51} +47.9103 q^{53} -142.986 q^{55} +216.544 q^{57} -441.432 q^{59} +869.398 q^{61} -1236.52 q^{63} +801.045 q^{67} +1284.17 q^{69} +893.624 q^{71} -413.180 q^{73} +1162.55 q^{75} +1029.23 q^{77} -246.527 q^{79} +2148.27 q^{81} +333.857 q^{83} -43.7016 q^{85} -726.816 q^{87} +379.176 q^{89} +1081.26 q^{93} -55.2266 q^{95} -941.263 q^{97} -3941.39 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 140 q^{9} + 176 q^{17} - 40 q^{23} + 84 q^{25} - 432 q^{27} + 968 q^{29} - 80 q^{35} + 1008 q^{43} + 1844 q^{49} + 1808 q^{51} - 1164 q^{53} + 2256 q^{55} + 2448 q^{61} + 3476 q^{69} + 2896 q^{75}+ \cdots + 4800 q^{95}+O(q^{100}) $ 8 * q + 140 * q^9 + 176 * q^17 - 40 * q^23 + 84 * q^25 - 432 * q^27 + 968 * q^29 - 80 * q^35 + 1008 * q^43 + 1844 * q^49 + 1808 * q^51 - 1164 * q^53 + 2256 * q^55 + 2448 * q^61 + 3476 * q^69 + 2896 * q^75 + 3972 * q^77 - 3968 * q^79 + 8264 * q^81 + 3320 * q^87 + 4800 * q^95

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
$3$	−9.78841	−1.88378	−0.941890	−	0.335922i	$-0.890952\pi$
$3$	−9.78841	−1.88378	−0.941890	+	0.335922i	$0.890952\pi$
$4$	0	0
$5$	2.49640	0.223285	0.111643	−	0.993748i	$-0.464389\pi$
$5$	2.49640	0.223285	0.111643	+	0.993748i	$0.464389\pi$
$6$	0	0
$7$	−17.9693	−0.970253	−0.485126	−	0.874444i	$-0.661226\pi$
$7$	−17.9693	−0.970253	−0.485126	+	0.874444i	$0.661226\pi$
$8$	0	0
$9$	68.8129	2.54863
$10$	0	0
$11$	−57.2769	−1.56997	−0.784983	−	0.619517i	$-0.787328\pi$
$11$	−57.2769	−1.56997	−0.784983	+	0.619517i	$0.787328\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−24.4358	−0.420620
$16$	0	0
$17$	−17.5058	−0.249752	−0.124876	−	0.992172i	$-0.539853\pi$
$17$	−17.5058	−0.249752	−0.124876	+	0.992172i	$0.539853\pi$
$18$	0	0
$19$	−22.1225	−0.267118	−0.133559	−	0.991041i	$-0.542641\pi$
$19$	−22.1225	−0.267118	−0.133559	+	0.991041i	$0.542641\pi$
$20$	0	0
$21$	175.891	1.82774
$22$	0	0
$23$	−131.193	−1.18937	−0.594686	−	0.803958i	$-0.702724\pi$
$23$	−131.193	−1.18937	−0.594686	+	0.803958i	$0.702724\pi$
$24$	0	0
$25$	−118.768	−0.950144
$26$	0	0
$27$	−409.282	−2.91727
$28$	0	0
$29$	74.2527	0.475462	0.237731	−	0.971331i	$-0.423596\pi$
$29$	74.2527	0.475462	0.237731	+	0.971331i	$0.423596\pi$
$30$	0	0
$31$	−110.463	−0.639994	−0.319997	−	0.947419i	$-0.603682\pi$
$31$	−110.463	−0.639994	−0.319997	+	0.947419i	$0.603682\pi$
$32$	0	0
$33$	560.649	2.95747
$34$	0	0
$35$	−44.8587	−0.216643
$36$	0	0
$37$	−241.452	−1.07282	−0.536412	−	0.843956i	$-0.680221\pi$
$37$	−241.452	−1.07282	−0.536412	+	0.843956i	$0.680221\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−426.704	−1.62537	−0.812683	−	0.582706i	$-0.801994\pi$
$41$	−426.704	−1.62537	−0.812683	+	0.582706i	$0.801994\pi$
$42$	0	0
$43$	−208.296	−0.738718	−0.369359	−	0.929287i	$-0.620423\pi$
$43$	−208.296	−0.738718	−0.369359	+	0.929287i	$0.620423\pi$
$44$	0	0
$45$	171.785	0.569070
$46$	0	0
$47$	158.624	0.492292	0.246146	−	0.969233i	$-0.420836\pi$
$47$	158.624	0.492292	0.246146	+	0.969233i	$0.420836\pi$
$48$	0	0
$49$	−20.1031	−0.0586096
$50$	0	0
$51$	171.354	0.470478
$52$	0	0
$53$	47.9103	0.124170	0.0620848	−	0.998071i	$-0.480225\pi$
$53$	47.9103	0.124170	0.0620848	+	0.998071i	$0.480225\pi$
$54$	0	0
$55$	−142.986	−0.350550
$56$	0	0
$57$	216.544	0.503191
$58$	0	0
$59$	−441.432	−0.974060	−0.487030	−	0.873385i	$-0.661920\pi$
$59$	−441.432	−0.974060	−0.487030	+	0.873385i	$0.661920\pi$
$60$	0	0
$61$	869.398	1.82484	0.912419	−	0.409258i	$-0.134213\pi$
$61$	869.398	1.82484	0.912419	+	0.409258i	$0.134213\pi$
$62$	0	0
$63$	−1236.52	−2.47281
$64$	0	0
$65$	0	0
$66$	0	0
$67$	801.045	1.46065	0.730323	−	0.683102i	$-0.239370\pi$
$67$	801.045	1.46065	0.730323	+	0.683102i	$0.239370\pi$
$68$	0	0
$69$	1284.17	2.24052
$70$	0	0
$71$	893.624	1.49371	0.746857	−	0.664985i	$-0.231562\pi$
$71$	893.624	1.49371	0.746857	+	0.664985i	$0.231562\pi$
$72$	0	0
$73$	−413.180	−0.662452	−0.331226	−	0.943551i	$-0.607462\pi$
$73$	−413.180	−0.662452	−0.331226	+	0.943551i	$0.607462\pi$
$74$	0	0
$75$	1162.55	1.78986
$76$	0	0
$77$	1029.23	1.52326
$78$	0	0
$79$	−246.527	−0.351094	−0.175547	−	0.984471i	$-0.556169\pi$
$79$	−246.527	−0.351094	−0.175547	+	0.984471i	$0.556169\pi$
$80$	0	0
$81$	2148.27	2.94687
$82$	0	0
$83$	333.857	0.441513	0.220756	−	0.975329i	$-0.429147\pi$
$83$	333.857	0.441513	0.220756	+	0.975329i	$0.429147\pi$
$84$	0	0
$85$	−43.7016	−0.0557660
$86$	0	0
$87$	−726.816	−0.895665
$88$	0	0
$89$	379.176	0.451602	0.225801	−	0.974174i	$-0.427500\pi$
$89$	379.176	0.451602	0.225801	+	0.974174i	$0.427500\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1081.26	1.20561
$94$	0	0
$95$	−55.2266	−0.0596435
$96$	0	0
$97$	−941.263	−0.985266	−0.492633	−	0.870237i	$-0.663965\pi$
$97$	−941.263	−0.985266	−0.492633	+	0.870237i	$0.663965\pi$
$98$	0	0
$99$	−3941.39	−4.00126
$100$	0	0
$101$	385.949	0.380232	0.190116	−	0.981762i	$-0.439114\pi$
$101$	385.949	0.380232	0.190116	+	0.981762i	$0.439114\pi$
$102$	0	0
$103$	−395.674	−0.378514	−0.189257	−	0.981928i	$-0.560608\pi$
$103$	−395.674	−0.378514	−0.189257	+	0.981928i	$0.560608\pi$
$104$	0	0
$105$	439.095	0.408108
$106$	0	0
$107$	1228.14	1.10961	0.554807	−	0.831979i	$-0.312792\pi$
$107$	1228.14	1.10961	0.554807	+	0.831979i	$0.312792\pi$
$108$	0	0
$109$	542.176	0.476431	0.238216	−	0.971212i	$-0.423438\pi$
$109$	542.176	0.476431	0.238216	+	0.971212i	$0.423438\pi$
$110$	0	0
$111$	2363.43	2.02096
$112$	0	0
$113$	−677.411	−0.563942	−0.281971	−	0.959423i	$-0.590988\pi$
$113$	−677.411	−0.563942	−0.281971	+	0.959423i	$0.590988\pi$
$114$	0	0
$115$	−327.510	−0.265569
$116$	0	0
$117$	0	0
$118$	0	0
$119$	314.568	0.242323
$120$	0	0
$121$	1949.64	1.46479
$122$	0	0
$123$	4176.75	3.06183
$124$	0	0
$125$	−608.543	−0.435438
$126$	0	0
$127$	1271.86	0.888656	0.444328	−	0.895864i	$-0.353442\pi$
$127$	1271.86	0.888656	0.444328	+	0.895864i	$0.353442\pi$
$128$	0	0
$129$	2038.89	1.39158
$130$	0	0
$131$	1249.24	0.833179	0.416589	−	0.909095i	$-0.363225\pi$
$131$	1249.24	0.833179	0.416589	+	0.909095i	$0.363225\pi$
$132$	0	0
$133$	397.526	0.259172
$134$	0	0
$135$	−1021.73	−0.651383
$136$	0	0
$137$	−1742.91	−1.08691	−0.543454	−	0.839439i	$-0.682884\pi$
$137$	−1742.91	−1.08691	−0.543454	+	0.839439i	$0.682884\pi$
$138$	0	0
$139$	988.883	0.603424	0.301712	−	0.953399i	$-0.402442\pi$
$139$	988.883	0.603424	0.301712	+	0.953399i	$0.402442\pi$
$140$	0	0
$141$	−1552.68	−0.927369
$142$	0	0
$143$	0	0
$144$	0	0
$145$	185.365	0.106164
$146$	0	0
$147$	196.777	0.110408
$148$	0	0
$149$	−1696.58	−0.932815	−0.466408	−	0.884570i	$-0.654452\pi$
$149$	−1696.58	−0.932815	−0.466408	+	0.884570i	$0.654452\pi$
$150$	0	0
$151$	2603.11	1.40290	0.701449	−	0.712719i	$-0.252537\pi$
$151$	2603.11	1.40290	0.701449	+	0.712719i	$0.252537\pi$
$152$	0	0
$153$	−1204.63	−0.636525
$154$	0	0
$155$	−275.761	−0.142901
$156$	0	0
$157$	−178.988	−0.0909859	−0.0454930	−	0.998965i	$-0.514486\pi$
$157$	−178.988	−0.0909859	−0.0454930	+	0.998965i	$0.514486\pi$
$158$	0	0
$159$	−468.966	−0.233908
$160$	0	0
$161$	2357.44	1.15399
$162$	0	0
$163$	−2080.21	−0.999599	−0.499799	−	0.866141i	$-0.666593\pi$
$163$	−2080.21	−0.999599	−0.499799	+	0.866141i	$0.666593\pi$
$164$	0	0
$165$	1399.61	0.660359
$166$	0	0
$167$	−3039.90	−1.40859	−0.704295	−	0.709907i	$-0.748737\pi$
$167$	−3039.90	−1.40859	−0.704295	+	0.709907i	$0.748737\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−1522.31	−0.680784
$172$	0	0
$173$	51.7355	0.0227363	0.0113681	−	0.999935i	$-0.496381\pi$
$173$	51.7355	0.0227363	0.0113681	+	0.999935i	$0.496381\pi$
$174$	0	0
$175$	2134.18	0.921880
$176$	0	0
$177$	4320.92	1.83492
$178$	0	0
$179$	628.894	0.262602	0.131301	−	0.991343i	$-0.458085\pi$
$179$	628.894	0.262602	0.131301	+	0.991343i	$0.458085\pi$
$180$	0	0
$181$	1661.09	0.682145	0.341072	−	0.940037i	$-0.389210\pi$
$181$	1661.09	0.682145	0.341072	+	0.940037i	$0.389210\pi$
$182$	0	0
$183$	−8510.03	−3.43759
$184$	0	0
$185$	−602.762	−0.239546
$186$	0	0
$187$	1002.68	0.392103
$188$	0	0
$189$	7354.52	2.83049
$190$	0	0
$191$	1917.94	0.726581	0.363291	−	0.931676i	$-0.381653\pi$
$191$	1917.94	0.726581	0.363291	+	0.931676i	$0.381653\pi$
$192$	0	0
$193$	−231.092	−0.0861883	−0.0430942	−	0.999071i	$-0.513722\pi$
$193$	−231.092	−0.0861883	−0.0430942	+	0.999071i	$0.513722\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1630.42	−0.589656	−0.294828	−	0.955550i	$-0.595262\pi$
$197$	−1630.42	−0.589656	−0.294828	+	0.955550i	$0.595262\pi$
$198$	0	0
$199$	−3065.57	−1.09202	−0.546011	−	0.837778i	$-0.683854\pi$
$199$	−3065.57	−1.09202	−0.546011	+	0.837778i	$0.683854\pi$
$200$	0	0
$201$	−7840.96	−2.75154
$202$	0	0
$203$	−1334.27	−0.461318
$204$	0	0
$205$	−1065.23	−0.362920
$206$	0	0
$207$	−9027.75	−3.03127
$208$	0	0
$209$	1267.11	0.419366
$210$	0	0
$211$	−3449.79	−1.12556	−0.562780	−	0.826606i	$-0.690268\pi$
$211$	−3449.79	−1.12556	−0.562780	+	0.826606i	$0.690268\pi$
$212$	0	0
$213$	−8747.16	−2.81383
$214$	0	0
$215$	−519.991	−0.164945
$216$	0	0
$217$	1984.95	0.620956
$218$	0	0
$219$	4044.37	1.24791
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2524.48	−0.758080	−0.379040	−	0.925380i	$-0.623746\pi$
$223$	−2524.48	−0.758080	−0.379040	+	0.925380i	$0.623746\pi$
$224$	0	0
$225$	−8172.77	−2.42156
$226$	0	0
$227$	−1153.06	−0.337142	−0.168571	−	0.985689i	$-0.553915\pi$
$227$	−1153.06	−0.337142	−0.168571	+	0.985689i	$0.553915\pi$
$228$	0	0
$229$	3857.53	1.11316	0.556578	−	0.830795i	$-0.312114\pi$
$229$	3857.53	1.11316	0.556578	+	0.830795i	$0.312114\pi$
$230$	0	0
$231$	−10074.5	−2.86949
$232$	0	0
$233$	−1018.55	−0.286385	−0.143192	−	0.989695i	$-0.545737\pi$
$233$	−1018.55	−0.286385	−0.143192	+	0.989695i	$0.545737\pi$
$234$	0	0
$235$	395.990	0.109921
$236$	0	0
$237$	2413.11	0.661384
$238$	0	0
$239$	1263.24	0.341893	0.170947	−	0.985280i	$-0.445317\pi$
$239$	1263.24	0.341893	0.170947	+	0.985280i	$0.445317\pi$
$240$	0	0
$241$	−2067.94	−0.552729	−0.276365	−	0.961053i	$-0.589130\pi$
$241$	−2067.94	−0.552729	−0.276365	+	0.961053i	$0.589130\pi$
$242$	0	0
$243$	−9977.52	−2.63398
$244$	0	0
$245$	−50.1855	−0.0130867
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3267.93	−0.831713
$250$	0	0
$251$	5126.38	1.28914	0.644570	−	0.764545i	$-0.277036\pi$
$251$	5126.38	1.28914	0.644570	+	0.764545i	$0.277036\pi$
$252$	0	0
$253$	7514.31	1.86727
$254$	0	0
$255$	427.769	0.105051
$256$	0	0
$257$	3961.52	0.961528	0.480764	−	0.876850i	$-0.340359\pi$
$257$	3961.52	0.961528	0.480764	+	0.876850i	$0.340359\pi$
$258$	0	0
$259$	4338.73	1.04091
$260$	0	0
$261$	5109.55	1.21177
$262$	0	0
$263$	−7104.63	−1.66574	−0.832871	−	0.553468i	$-0.813304\pi$
$263$	−7104.63	−1.66574	−0.832871	+	0.553468i	$0.813304\pi$
$264$	0	0
$265$	119.604	0.0277252
$266$	0	0
$267$	−3711.53	−0.850718
$268$	0	0
$269$	5613.88	1.27243	0.636216	−	0.771511i	$-0.280499\pi$
$269$	5613.88	1.27243	0.636216	+	0.771511i	$0.280499\pi$
$270$	0	0
$271$	−441.205	−0.0988978	−0.0494489	−	0.998777i	$-0.515746\pi$
$271$	−441.205	−0.0988978	−0.0494489	+	0.998777i	$0.515746\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6802.66	1.49169
$276$	0	0
$277$	4676.87	1.01446	0.507231	−	0.861810i	$-0.330669\pi$
$277$	4676.87	1.01446	0.507231	+	0.861810i	$0.330669\pi$
$278$	0	0
$279$	−7601.31	−1.63110
$280$	0	0
$281$	1253.34	0.266079	0.133040	−	0.991111i	$-0.457526\pi$
$281$	1253.34	0.266079	0.133040	+	0.991111i	$0.457526\pi$
$282$	0	0
$283$	4172.15	0.876356	0.438178	−	0.898888i	$-0.355624\pi$
$283$	4172.15	0.876356	0.438178	+	0.898888i	$0.355624\pi$
$284$	0	0
$285$	540.580	0.112355
$286$	0	0
$287$	7667.59	1.57702
$288$	0	0
$289$	−4606.55	−0.937624
$290$	0	0
$291$	9213.47	1.85602
$292$	0	0
$293$	−9779.07	−1.94983	−0.974913	−	0.222585i	$-0.928550\pi$
$293$	−9779.07	−1.94983	−0.974913	+	0.222585i	$0.928550\pi$
$294$	0	0
$295$	−1101.99	−0.217493
$296$	0	0
$297$	23442.4	4.58002
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3742.94	0.716743
$302$	0	0
$303$	−3777.83	−0.716273
$304$	0	0
$305$	2170.37	0.407459
$306$	0	0
$307$	1641.63	0.305187	0.152594	−	0.988289i	$-0.451237\pi$
$307$	1641.63	0.305187	0.152594	+	0.988289i	$0.451237\pi$
$308$	0	0
$309$	3873.02	0.713037
$310$	0	0
$311$	5900.48	1.07584	0.537919	−	0.842996i	$-0.319211\pi$
$311$	5900.48	1.07584	0.537919	+	0.842996i	$0.319211\pi$
$312$	0	0
$313$	5268.10	0.951344	0.475672	−	0.879623i	$-0.342205\pi$
$313$	5268.10	0.951344	0.475672	+	0.879623i	$0.342205\pi$
$314$	0	0
$315$	−3086.86	−0.552142
$316$	0	0
$317$	20.4536	0.00362395	0.00181197	−	0.999998i	$-0.499423\pi$
$317$	20.4536	0.00362395	0.00181197	+	0.999998i	$0.499423\pi$
$318$	0	0
$319$	−4252.96	−0.746459
$320$	0	0
$321$	−12021.5	−2.09027
$322$	0	0
$323$	387.272	0.0667133
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5307.03	−0.897491
$328$	0	0
$329$	−2850.37	−0.477647
$330$	0	0
$331$	−6302.42	−1.04656	−0.523282	−	0.852160i	$-0.675292\pi$
$331$	−6302.42	−1.04656	−0.523282	+	0.852160i	$0.675292\pi$
$332$	0	0
$333$	−16615.0	−2.73423
$334$	0	0
$335$	1999.73	0.326141
$336$	0	0
$337$	−3060.14	−0.494648	−0.247324	−	0.968933i	$-0.579551\pi$
$337$	−3060.14	−0.494648	−0.247324	+	0.968933i	$0.579551\pi$
$338$	0	0
$339$	6630.78	1.06234
$340$	0	0
$341$	6326.99	1.00477
$342$	0	0
$343$	6524.72	1.02712
$344$	0	0
$345$	3205.80	0.500274
$346$	0	0
$347$	−203.618	−0.0315008	−0.0157504	−	0.999876i	$-0.505014\pi$
$347$	−203.618	−0.0315008	−0.0157504	+	0.999876i	$0.505014\pi$
$348$	0	0
$349$	−1505.61	−0.230927	−0.115464	−	0.993312i	$-0.536835\pi$
$349$	−1505.61	−0.230927	−0.115464	+	0.993312i	$0.536835\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11133.4	1.67867	0.839337	−	0.543612i	$-0.182944\pi$
$353$	11133.4	1.67867	0.839337	+	0.543612i	$0.182944\pi$
$354$	0	0
$355$	2230.85	0.333524
$356$	0	0
$357$	−3079.12	−0.456483
$358$	0	0
$359$	−4176.37	−0.613985	−0.306992	−	0.951712i	$-0.599323\pi$
$359$	−4176.37	−0.613985	−0.306992	+	0.951712i	$0.599323\pi$
$360$	0	0
$361$	−6369.60	−0.928648
$362$	0	0
$363$	−19083.9	−2.75935
$364$	0	0
$365$	−1031.46	−0.147916
$366$	0	0
$367$	4480.80	0.637319	0.318659	−	0.947869i	$-0.396767\pi$
$367$	4480.80	0.637319	0.318659	+	0.947869i	$0.396767\pi$
$368$	0	0
$369$	−29362.8	−4.14245
$370$	0	0
$371$	−860.917	−0.120476
$372$	0	0
$373$	−363.859	−0.0505092	−0.0252546	−	0.999681i	$-0.508040\pi$
$373$	−363.859	−0.0505092	−0.0252546	+	0.999681i	$0.508040\pi$
$374$	0	0
$375$	5956.67	0.820270
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−9755.44	−1.32217	−0.661086	−	0.750310i	$-0.729904\pi$
$379$	−9755.44	−1.32217	−0.661086	+	0.750310i	$0.729904\pi$
$380$	0	0
$381$	−12449.5	−1.67403
$382$	0	0
$383$	5252.34	0.700737	0.350368	−	0.936612i	$-0.386056\pi$
$383$	5252.34	0.700737	0.350368	+	0.936612i	$0.386056\pi$
$384$	0	0
$385$	2569.37	0.340122
$386$	0	0
$387$	−14333.5	−1.88272
$388$	0	0
$389$	11266.0	1.46841	0.734205	−	0.678928i	$-0.237555\pi$
$389$	11266.0	1.46841	0.734205	+	0.678928i	$0.237555\pi$
$390$	0	0
$391$	2296.64	0.297048
$392$	0	0
$393$	−12228.1	−1.56953
$394$	0	0
$395$	−615.431	−0.0783941
$396$	0	0
$397$	−3001.18	−0.379407	−0.189704	−	0.981841i	$-0.560753\pi$
$397$	−3001.18	−0.379407	−0.189704	+	0.981841i	$0.560753\pi$
$398$	0	0
$399$	−3891.14	−0.488223
$400$	0	0
$401$	7998.63	0.996091	0.498045	−	0.867151i	$-0.334051\pi$
$401$	7998.63	0.996091	0.498045	+	0.867151i	$0.334051\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	5362.95	0.657992
$406$	0	0
$407$	13829.6	1.68430
$408$	0	0
$409$	−10632.1	−1.28539	−0.642694	−	0.766123i	$-0.722183\pi$
$409$	−10632.1	−1.28539	−0.642694	+	0.766123i	$0.722183\pi$
$410$	0	0
$411$	17060.3	2.04750
$412$	0	0
$413$	7932.24	0.945085
$414$	0	0
$415$	833.442	0.0985833
$416$	0	0
$417$	−9679.59	−1.13672
$418$	0	0
$419$	−329.538	−0.0384224	−0.0192112	−	0.999815i	$-0.506115\pi$
$419$	−329.538	−0.0384224	−0.0192112	+	0.999815i	$0.506115\pi$
$420$	0	0
$421$	10594.7	1.22649	0.613247	−	0.789891i	$-0.289863\pi$
$421$	10594.7	1.22649	0.613247	+	0.789891i	$0.289863\pi$
$422$	0	0
$423$	10915.4	1.25467
$424$	0	0
$425$	2079.13	0.237301
$426$	0	0
$427$	−15622.5	−1.77055
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12753.4	−1.42531	−0.712655	−	0.701514i	$-0.752508\pi$
$431$	−12753.4	−1.42531	−0.712655	+	0.701514i	$0.752508\pi$
$432$	0	0
$433$	−14009.8	−1.55490	−0.777448	−	0.628947i	$-0.783486\pi$
$433$	−14009.8	−1.55490	−0.777448	+	0.628947i	$0.783486\pi$
$434$	0	0
$435$	−1814.43	−0.199989
$436$	0	0
$437$	2902.30	0.317703
$438$	0	0
$439$	7296.19	0.793230	0.396615	−	0.917985i	$-0.370185\pi$
$439$	7296.19	0.793230	0.396615	+	0.917985i	$0.370185\pi$
$440$	0	0
$441$	−1383.35	−0.149374
$442$	0	0
$443$	−6098.72	−0.654084	−0.327042	−	0.945010i	$-0.606052\pi$
$443$	−6098.72	−0.654084	−0.327042	+	0.945010i	$0.606052\pi$
$444$	0	0
$445$	946.575	0.100836
$446$	0	0
$447$	16606.8	1.75722
$448$	0	0
$449$	−10846.2	−1.14001	−0.570006	−	0.821640i	$-0.693059\pi$
$449$	−10846.2	−1.14001	−0.570006	+	0.821640i	$0.693059\pi$
$450$	0	0
$451$	24440.3	2.55177
$452$	0	0
$453$	−25480.3	−2.64275
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5353.85	0.548014	0.274007	−	0.961728i	$-0.411651\pi$
$457$	5353.85	0.548014	0.274007	+	0.961728i	$0.411651\pi$
$458$	0	0
$459$	7164.82	0.728595
$460$	0	0
$461$	15097.3	1.52527	0.762637	−	0.646827i	$-0.223904\pi$
$461$	15097.3	1.52527	0.762637	+	0.646827i	$0.223904\pi$
$462$	0	0
$463$	9543.47	0.957932	0.478966	−	0.877833i	$-0.341012\pi$
$463$	9543.47	0.957932	0.478966	+	0.877833i	$0.341012\pi$
$464$	0	0
$465$	2699.26	0.269194
$466$	0	0
$467$	4319.08	0.427972	0.213986	−	0.976837i	$-0.431355\pi$
$467$	4319.08	0.427972	0.213986	+	0.976837i	$0.431355\pi$
$468$	0	0
$469$	−14394.3	−1.41720
$470$	0	0
$471$	1752.01	0.171397
$472$	0	0
$473$	11930.6	1.15976
$474$	0	0
$475$	2627.44	0.253800
$476$	0	0
$477$	3296.85	0.316462
$478$	0	0
$479$	13244.2	1.26335	0.631673	−	0.775235i	$-0.282368\pi$
$479$	13244.2	1.26335	0.631673	+	0.775235i	$0.282368\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−23075.6	−2.17387
$484$	0	0
$485$	−2349.77	−0.219995
$486$	0	0
$487$	−619.602	−0.0576526	−0.0288263	−	0.999584i	$-0.509177\pi$
$487$	−619.602	−0.0576526	−0.0288263	+	0.999584i	$0.509177\pi$
$488$	0	0
$489$	20361.9	1.88302
$490$	0	0
$491$	−5462.14	−0.502042	−0.251021	−	0.967982i	$-0.580766\pi$
$491$	−5462.14	−0.502042	−0.251021	+	0.967982i	$0.580766\pi$
$492$	0	0
$493$	−1299.86	−0.118748
$494$	0	0
$495$	−9839.30	−0.893421
$496$	0	0
$497$	−16057.8	−1.44928
$498$	0	0
$499$	−16005.2	−1.43585	−0.717926	−	0.696119i	$-0.754908\pi$
$499$	−16005.2	−1.43585	−0.717926	+	0.696119i	$0.754908\pi$
$500$	0	0
$501$	29755.8	2.65347
$502$	0	0
$503$	−10055.8	−0.891386	−0.445693	−	0.895186i	$-0.647043\pi$
$503$	−10055.8	−0.891386	−0.445693	+	0.895186i	$0.647043\pi$
$504$	0	0
$505$	963.485	0.0849001
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14572.4	1.26898	0.634490	−	0.772931i	$-0.281210\pi$
$509$	14572.4	1.26898	0.634490	+	0.772931i	$0.281210\pi$
$510$	0	0
$511$	7424.56	0.642746
$512$	0	0
$513$	9054.32	0.779255
$514$	0	0
$515$	−987.763	−0.0845166
$516$	0	0
$517$	−9085.49	−0.772881
$518$	0	0
$519$	−506.408	−0.0428301
$520$	0	0
$521$	14409.9	1.21172	0.605862	−	0.795570i	$-0.292828\pi$
$521$	14409.9	1.21172	0.605862	+	0.795570i	$0.292828\pi$
$522$	0	0
$523$	−4248.32	−0.355193	−0.177597	−	0.984103i	$-0.556832\pi$
$523$	−4248.32	−0.355193	−0.177597	+	0.984103i	$0.556832\pi$
$524$	0	0
$525$	−20890.2	−1.73662
$526$	0	0
$527$	1933.75	0.159840
$528$	0	0
$529$	5044.52	0.414607
$530$	0	0
$531$	−30376.2	−2.48252
$532$	0	0
$533$	0	0
$534$	0	0
$535$	3065.93	0.247761
$536$	0	0
$537$	−6155.87	−0.494684
$538$	0	0
$539$	1151.44	0.0920151
$540$	0	0
$541$	−14369.4	−1.14194	−0.570971	−	0.820970i	$-0.693433\pi$
$541$	−14369.4	−1.14194	−0.570971	+	0.820970i	$0.693433\pi$
$542$	0	0
$543$	−16259.5	−1.28501
$544$	0	0
$545$	1353.49	0.106380
$546$	0	0
$547$	−18950.8	−1.48131	−0.740657	−	0.671884i	$-0.765485\pi$
$547$	−18950.8	−1.48131	−0.740657	+	0.671884i	$0.765485\pi$
$548$	0	0
$549$	59825.8	4.65083
$550$	0	0
$551$	−1642.65	−0.127004
$552$	0	0
$553$	4429.92	0.340650
$554$	0	0
$555$	5900.08	0.451251
$556$	0	0
$557$	−11852.8	−0.901654	−0.450827	−	0.892611i	$-0.648871\pi$
$557$	−11852.8	−0.901654	−0.450827	+	0.892611i	$0.648871\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−9814.63	−0.738635
$562$	0	0
$563$	11585.3	0.867251	0.433625	−	0.901093i	$-0.357234\pi$
$563$	11585.3	0.867251	0.433625	+	0.901093i	$0.357234\pi$
$564$	0	0
$565$	−1691.09	−0.125920
$566$	0	0
$567$	−38603.0	−2.85921
$568$	0	0
$569$	9242.34	0.680948	0.340474	−	0.940254i	$-0.389413\pi$
$569$	9242.34	0.680948	0.340474	+	0.940254i	$0.389413\pi$
$570$	0	0
$571$	−20877.1	−1.53009	−0.765043	−	0.643979i	$-0.777282\pi$
$571$	−20877.1	−1.53009	−0.765043	+	0.643979i	$0.777282\pi$
$572$	0	0
$573$	−18773.5	−1.36872
$574$	0	0
$575$	15581.5	1.13007
$576$	0	0
$577$	−19966.8	−1.44060	−0.720302	−	0.693661i	$-0.755997\pi$
$577$	−19966.8	−1.44060	−0.720302	+	0.693661i	$0.755997\pi$
$578$	0	0
$579$	2262.02	0.162360
$580$	0	0
$581$	−5999.19	−0.428379
$582$	0	0
$583$	−2744.15	−0.194942
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19411.9	1.36493	0.682466	−	0.730917i	$-0.260907\pi$
$587$	19411.9	1.36493	0.682466	+	0.730917i	$0.260907\pi$
$588$	0	0
$589$	2443.72	0.170954
$590$	0	0
$591$	15959.2	1.11078
$592$	0	0
$593$	−19259.0	−1.33368	−0.666841	−	0.745200i	$-0.732354\pi$
$593$	−19259.0	−1.33368	−0.666841	+	0.745200i	$0.732354\pi$
$594$	0	0
$595$	785.289	0.0541071
$596$	0	0
$597$	30007.1	2.05713
$598$	0	0
$599$	27804.6	1.89660	0.948301	−	0.317372i	$-0.102800\pi$
$599$	27804.6	1.89660	0.948301	+	0.317372i	$0.102800\pi$
$600$	0	0
$601$	−9927.10	−0.673769	−0.336884	−	0.941546i	$-0.609373\pi$
$601$	−9927.10	−0.673769	−0.336884	+	0.941546i	$0.609373\pi$
$602$	0	0
$603$	55122.3	3.72264
$604$	0	0
$605$	4867.09	0.327066
$606$	0	0
$607$	−14402.0	−0.963032	−0.481516	−	0.876437i	$-0.659914\pi$
$607$	−14402.0	−0.963032	−0.481516	+	0.876437i	$0.659914\pi$
$608$	0	0
$609$	13060.4	0.869021
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−14778.2	−0.973713	−0.486856	−	0.873482i	$-0.661856\pi$
$613$	−14778.2	−0.973713	−0.486856	+	0.873482i	$0.661856\pi$
$614$	0	0
$615$	10426.9	0.683661
$616$	0	0
$617$	−17161.9	−1.11979	−0.559897	−	0.828562i	$-0.689159\pi$
$617$	−17161.9	−1.11979	−0.559897	+	0.828562i	$0.689159\pi$
$618$	0	0
$619$	7592.09	0.492975	0.246488	−	0.969146i	$-0.420724\pi$
$619$	7592.09	0.492975	0.246488	+	0.969146i	$0.420724\pi$
$620$	0	0
$621$	53694.8	3.46972
$622$	0	0
$623$	−6813.53	−0.438168
$624$	0	0
$625$	13326.8	0.852917
$626$	0	0
$627$	−12402.9	−0.789993
$628$	0	0
$629$	4226.82	0.267940
$630$	0	0
$631$	−6757.72	−0.426340	−0.213170	−	0.977015i	$-0.568379\pi$
$631$	−6757.72	−0.426340	−0.213170	+	0.977015i	$0.568379\pi$
$632$	0	0
$633$	33767.9	2.12031
$634$	0	0
$635$	3175.08	0.198424
$636$	0	0
$637$	0	0
$638$	0	0
$639$	61492.9	3.80692
$640$	0	0
$641$	14252.7	0.878235	0.439117	−	0.898430i	$-0.355291\pi$
$641$	14252.7	0.878235	0.439117	+	0.898430i	$0.355291\pi$
$642$	0	0
$643$	12887.1	0.790383	0.395192	−	0.918599i	$-0.370678\pi$
$643$	12887.1	0.790383	0.395192	+	0.918599i	$0.370678\pi$
$644$	0	0
$645$	5089.89	0.310720
$646$	0	0
$647$	28352.4	1.72280	0.861398	−	0.507931i	$-0.169589\pi$
$647$	28352.4	1.72280	0.861398	+	0.507931i	$0.169589\pi$
$648$	0	0
$649$	25283.8	1.52924
$650$	0	0
$651$	−19429.5	−1.16974
$652$	0	0
$653$	24016.2	1.43925	0.719623	−	0.694365i	$-0.244315\pi$
$653$	24016.2	1.43925	0.719623	+	0.694365i	$0.244315\pi$
$654$	0	0
$655$	3118.60	0.186036
$656$	0	0
$657$	−28432.1	−1.68834
$658$	0	0
$659$	−366.040	−0.0216371	−0.0108186	−	0.999941i	$-0.503444\pi$
$659$	−366.040	−0.0216371	−0.0108186	+	0.999941i	$0.503444\pi$
$660$	0	0
$661$	−7308.10	−0.430033	−0.215017	−	0.976610i	$-0.568981\pi$
$661$	−7308.10	−0.430033	−0.215017	+	0.976610i	$0.568981\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	992.385	0.0578692
$666$	0	0
$667$	−9741.42	−0.565501
$668$	0	0
$669$	24710.6	1.42806
$670$	0	0
$671$	−49796.4	−2.86493
$672$	0	0
$673$	−13894.0	−0.795803	−0.397901	−	0.917428i	$-0.630261\pi$
$673$	−13894.0	−0.795803	−0.397901	+	0.917428i	$0.630261\pi$
$674$	0	0
$675$	48609.6	2.77183
$676$	0	0
$677$	23804.6	1.35138	0.675691	−	0.737185i	$-0.263845\pi$
$677$	23804.6	1.35138	0.675691	+	0.737185i	$0.263845\pi$
$678$	0	0
$679$	16913.9	0.955957
$680$	0	0
$681$	11286.6	0.635102
$682$	0	0
$683$	−20866.6	−1.16902	−0.584508	−	0.811388i	$-0.698712\pi$
$683$	−20866.6	−1.16902	−0.584508	+	0.811388i	$0.698712\pi$
$684$	0	0
$685$	−4350.99	−0.242690
$686$	0	0
$687$	−37759.1	−2.09694
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−5142.24	−0.283097	−0.141549	−	0.989931i	$-0.545208\pi$
$691$	−5142.24	−0.283097	−0.141549	+	0.989931i	$0.545208\pi$
$692$	0	0
$693$	70824.1	3.88223
$694$	0	0
$695$	2468.65	0.134736
$696$	0	0
$697$	7469.81	0.405939
$698$	0	0
$699$	9970.02	0.539486
$700$	0	0
$701$	−8584.99	−0.462554	−0.231277	−	0.972888i	$-0.574290\pi$
$701$	−8584.99	−0.462554	−0.231277	+	0.972888i	$0.574290\pi$
$702$	0	0
$703$	5341.52	0.286571
$704$	0	0
$705$	−3876.11	−0.207068
$706$	0	0
$707$	−6935.25	−0.368921
$708$	0	0
$709$	21568.8	1.14250	0.571252	−	0.820775i	$-0.306458\pi$
$709$	21568.8	1.14250	0.571252	+	0.820775i	$0.306458\pi$
$710$	0	0
$711$	−16964.2	−0.894808
$712$	0	0
$713$	14492.0	0.761191
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12365.1	−0.644051
$718$	0	0
$719$	−32635.3	−1.69276	−0.846379	−	0.532581i	$-0.821222\pi$
$719$	−32635.3	−1.69276	−0.846379	+	0.532581i	$0.821222\pi$
$720$	0	0
$721$	7110.00	0.367254
$722$	0	0
$723$	20241.8	1.04122
$724$	0	0
$725$	−8818.85	−0.451757
$726$	0	0
$727$	−6636.77	−0.338575	−0.169288	−	0.985567i	$-0.554147\pi$
$727$	−6636.77	−0.338575	−0.169288	+	0.985567i	$0.554147\pi$
$728$	0	0
$729$	39660.8	2.01498
$730$	0	0
$731$	3646.40	0.184496
$732$	0	0
$733$	18221.6	0.918188	0.459094	−	0.888388i	$-0.348174\pi$
$733$	18221.6	0.918188	0.459094	+	0.888388i	$0.348174\pi$
$734$	0	0
$735$	491.236	0.0246524
$736$	0	0
$737$	−45881.4	−2.29316
$738$	0	0
$739$	25459.9	1.26733	0.633665	−	0.773607i	$-0.281550\pi$
$739$	25459.9	1.26733	0.633665	+	0.773607i	$0.281550\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22559.2	−1.11388	−0.556942	−	0.830552i	$-0.688025\pi$
$743$	−22559.2	−1.11388	−0.556942	+	0.830552i	$0.688025\pi$
$744$	0	0
$745$	−4235.35	−0.208284
$746$	0	0
$747$	22973.7	1.12525
$748$	0	0
$749$	−22068.9	−1.07661
$750$	0	0
$751$	−21808.6	−1.05966	−0.529830	−	0.848104i	$-0.677744\pi$
$751$	−21808.6	−1.05966	−0.529830	+	0.848104i	$0.677744\pi$
$752$	0	0
$753$	−50179.1	−2.42846
$754$	0	0
$755$	6498.40	0.313246
$756$	0	0
$757$	−12803.7	−0.614742	−0.307371	−	0.951590i	$-0.599449\pi$
$757$	−12803.7	−0.614742	−0.307371	+	0.951590i	$0.599449\pi$
$758$	0	0
$759$	−73553.1	−3.51753
$760$	0	0
$761$	4154.40	0.197893	0.0989467	−	0.995093i	$-0.468453\pi$
$761$	4154.40	0.197893	0.0989467	+	0.995093i	$0.468453\pi$
$762$	0	0
$763$	−9742.53	−0.462259
$764$	0	0
$765$	−3007.24	−0.142127
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−4122.77	−0.193330	−0.0966652	−	0.995317i	$-0.530818\pi$
$769$	−4122.77	−0.193330	−0.0966652	+	0.995317i	$0.530818\pi$
$770$	0	0
$771$	−38776.9	−1.81131
$772$	0	0
$773$	−34102.7	−1.58679	−0.793394	−	0.608708i	$-0.791688\pi$
$773$	−34102.7	−1.58679	−0.793394	+	0.608708i	$0.791688\pi$
$774$	0	0
$775$	13119.5	0.608086
$776$	0	0
$777$	−42469.3	−1.96085
$778$	0	0
$779$	9439.74	0.434164
$780$	0	0
$781$	−51184.0	−2.34508
$782$	0	0
$783$	−30390.3	−1.38705
$784$	0	0
$785$	−446.826	−0.0203158
$786$	0	0
$787$	42573.1	1.92829	0.964147	−	0.265368i	$-0.0854935\pi$
$787$	42573.1	1.92829	0.964147	+	0.265368i	$0.0854935\pi$
$788$	0	0
$789$	69543.0	3.13789
$790$	0	0
$791$	12172.6	0.547166
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1170.73	−0.0522282
$796$	0	0
$797$	−6054.75	−0.269097	−0.134549	−	0.990907i	$-0.542958\pi$
$797$	−6054.75	−0.269097	−0.134549	+	0.990907i	$0.542958\pi$
$798$	0	0
$799$	−2776.85	−0.122951
$800$	0	0
$801$	26092.2	1.15096
$802$	0	0
$803$	23665.6	1.04003
$804$	0	0
$805$	5885.13	0.257669
$806$	0	0
$807$	−54950.9	−2.39698
$808$	0	0
$809$	37902.7	1.64720	0.823601	−	0.567169i	$-0.191962\pi$
$809$	37902.7	1.64720	0.823601	+	0.567169i	$0.191962\pi$
$810$	0	0
$811$	−2418.43	−0.104714	−0.0523568	−	0.998628i	$-0.516673\pi$
$811$	−2418.43	−0.104714	−0.0523568	+	0.998628i	$0.516673\pi$
$812$	0	0
$813$	4318.69	0.186302
$814$	0	0
$815$	−5193.04	−0.223196
$816$	0	0
$817$	4608.02	0.197325
$818$	0	0
$819$	0	0
$820$	0	0
$821$	40835.6	1.73590	0.867950	−	0.496652i	$-0.165438\pi$
$821$	40835.6	1.73590	0.867950	+	0.496652i	$0.165438\pi$
$822$	0	0
$823$	1699.36	0.0719755	0.0359877	−	0.999352i	$-0.488542\pi$
$823$	1699.36	0.0719755	0.0359877	+	0.999352i	$0.488542\pi$
$824$	0	0
$825$	−66587.2	−2.81002
$826$	0	0
$827$	−28148.2	−1.18357	−0.591784	−	0.806097i	$-0.701576\pi$
$827$	−28148.2	−1.18357	−0.591784	+	0.806097i	$0.701576\pi$
$828$	0	0
$829$	1779.51	0.0745535	0.0372768	−	0.999305i	$-0.488132\pi$
$829$	1779.51	0.0745535	0.0372768	+	0.999305i	$0.488132\pi$
$830$	0	0
$831$	−45779.2	−1.91102
$832$	0	0
$833$	351.922	0.0146379
$834$	0	0
$835$	−7588.82	−0.314517
$836$	0	0
$837$	45210.6	1.86704
$838$	0	0
$839$	27406.2	1.12773	0.563865	−	0.825867i	$-0.309314\pi$
$839$	27406.2	1.12773	0.563865	+	0.825867i	$0.309314\pi$
$840$	0	0
$841$	−18875.5	−0.773936
$842$	0	0
$843$	−12268.2	−0.501234
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−35033.7	−1.42122
$848$	0	0
$849$	−40838.7	−1.65086
$850$	0	0
$851$	31676.8	1.27599
$852$	0	0
$853$	49349.2	1.98087	0.990437	−	0.137964i	$-0.0440557\pi$
$853$	49349.2	1.98087	0.990437	+	0.137964i	$0.0440557\pi$
$854$	0	0
$855$	−3800.30	−0.152009
$856$	0	0
$857$	−657.558	−0.0262098	−0.0131049	−	0.999914i	$-0.504172\pi$
$857$	−657.558	−0.0262098	−0.0131049	+	0.999914i	$0.504172\pi$
$858$	0	0
$859$	2866.78	0.113869	0.0569343	−	0.998378i	$-0.481867\pi$
$859$	2866.78	0.113869	0.0569343	+	0.998378i	$0.481867\pi$
$860$	0	0
$861$	−75053.5	−2.97075
$862$	0	0
$863$	18625.8	0.734683	0.367341	−	0.930086i	$-0.380268\pi$
$863$	18625.8	0.734683	0.367341	+	0.930086i	$0.380268\pi$
$864$	0	0
$865$	129.153	0.00507667
$866$	0	0
$867$	45090.7	1.76628
$868$	0	0
$869$	14120.3	0.551206
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−64771.1	−2.51107
$874$	0	0
$875$	10935.1	0.422485
$876$	0	0
$877$	−19727.9	−0.759596	−0.379798	−	0.925069i	$-0.624006\pi$
$877$	−19727.9	−0.759596	−0.379798	+	0.925069i	$0.624006\pi$
$878$	0	0
$879$	95721.5	3.67304
$880$	0	0
$881$	−50424.4	−1.92831	−0.964156	−	0.265337i	$-0.914517\pi$
$881$	−50424.4	−1.92831	−0.964156	+	0.265337i	$0.914517\pi$
$882$	0	0
$883$	−27255.8	−1.03877	−0.519384	−	0.854541i	$-0.673838\pi$
$883$	−27255.8	−1.03877	−0.519384	+	0.854541i	$0.673838\pi$
$884$	0	0
$885$	10786.8	0.409709
$886$	0	0
$887$	11738.1	0.444336	0.222168	−	0.975008i	$-0.428687\pi$
$887$	11738.1	0.444336	0.222168	+	0.975008i	$0.428687\pi$
$888$	0	0
$889$	−22854.5	−0.862221
$890$	0	0
$891$	−123046.	−4.62649
$892$	0	0
$893$	−3509.16	−0.131500
$894$	0	0
$895$	1569.97	0.0586351
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8202.21	−0.304292
$900$	0	0
$901$	−838.710	−0.0310117
$902$	0	0
$903$	−36637.5	−1.35019
$904$	0	0
$905$	4146.76	0.152313
$906$	0	0
$907$	15434.7	0.565051	0.282526	−	0.959260i	$-0.408828\pi$
$907$	15434.7	0.565051	0.282526	+	0.959260i	$0.408828\pi$
$908$	0	0
$909$	26558.3	0.969069
$910$	0	0
$911$	−24957.2	−0.907649	−0.453825	−	0.891091i	$-0.649941\pi$
$911$	−24957.2	−0.907649	−0.453825	+	0.891091i	$0.649941\pi$
$912$	0	0
$913$	−19122.3	−0.693160
$914$	0	0
$915$	−21244.5	−0.767563
$916$	0	0
$917$	−22448.0	−0.808394
$918$	0	0
$919$	16582.3	0.595213	0.297606	−	0.954689i	$-0.403812\pi$
$919$	16582.3	0.595213	0.297606	+	0.954689i	$0.403812\pi$
$920$	0	0
$921$	−16068.9	−0.574906
$922$	0	0
$923$	0	0
$924$	0	0
$925$	28676.8	1.01934
$926$	0	0
$927$	−27227.5	−0.964691
$928$	0	0
$929$	5910.56	0.208740	0.104370	−	0.994539i	$-0.466717\pi$
$929$	5910.56	0.208740	0.104370	+	0.994539i	$0.466717\pi$
$930$	0	0
$931$	444.730	0.0156557
$932$	0	0
$933$	−57756.3	−2.02664
$934$	0	0
$935$	2503.09	0.0875507
$936$	0	0
$937$	−7798.93	−0.271910	−0.135955	−	0.990715i	$-0.543410\pi$
$937$	−7798.93	−0.271910	−0.135955	+	0.990715i	$0.543410\pi$
$938$	0	0
$939$	−51566.3	−1.79212
$940$	0	0
$941$	−9738.22	−0.337361	−0.168681	−	0.985671i	$-0.553951\pi$
$941$	−9738.22	−0.337361	−0.168681	+	0.985671i	$0.553951\pi$
$942$	0	0
$943$	55980.5	1.93316
$944$	0	0
$945$	18359.9	0.632006
$946$	0	0
$947$	13774.3	0.472656	0.236328	−	0.971673i	$-0.424056\pi$
$947$	13774.3	0.472656	0.236328	+	0.971673i	$0.424056\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−200.209	−0.00682672
$952$	0	0
$953$	30604.5	1.04027	0.520135	−	0.854084i	$-0.325882\pi$
$953$	30604.5	1.04027	0.520135	+	0.854084i	$0.325882\pi$
$954$	0	0
$955$	4787.94	0.162235
$956$	0	0
$957$	41629.7	1.40616
$958$	0	0
$959$	31318.8	1.05458
$960$	0	0
$961$	−17588.8	−0.590408
$962$	0	0
$963$	84511.9	2.82799
$964$	0	0
$965$	−576.898	−0.0192446
$966$	0	0
$967$	−48790.3	−1.62253	−0.811266	−	0.584677i	$-0.801221\pi$
$967$	−48790.3	−1.62253	−0.811266	+	0.584677i	$0.801221\pi$
$968$	0	0
$969$	−3790.78	−0.125673
$970$	0	0
$971$	−33077.6	−1.09321	−0.546607	−	0.837389i	$-0.684081\pi$
$971$	−33077.6	−1.09321	−0.546607	+	0.837389i	$0.684081\pi$
$972$	0	0
$973$	−17769.6	−0.585474
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13620.6	0.446020	0.223010	−	0.974816i	$-0.428412\pi$
$977$	13620.6	0.446020	0.223010	+	0.974816i	$0.428412\pi$
$978$	0	0
$979$	−21718.0	−0.708999
$980$	0	0
$981$	37308.7	1.21425
$982$	0	0
$983$	−23886.5	−0.775037	−0.387519	−	0.921862i	$-0.626668\pi$
$983$	−23886.5	−0.775037	−0.387519	+	0.921862i	$0.626668\pi$
$984$	0	0
$985$	−4070.17	−0.131661
$986$	0	0
$987$	27900.6	0.899783
$988$	0	0
$989$	27326.9	0.878611
$990$	0	0
$991$	−3180.18	−0.101939	−0.0509695	−	0.998700i	$-0.516231\pi$
$991$	−3180.18	−0.101939	−0.0509695	+	0.998700i	$0.516231\pi$
$992$	0	0
$993$	61690.7	1.97150
$994$	0	0
$995$	−7652.90	−0.243832
$996$	0	0
$997$	−53027.4	−1.68445	−0.842223	−	0.539129i	$-0.818754\pi$
$997$	−53027.4	−1.68445	−0.842223	+	0.539129i	$0.818754\pi$
$998$	0	0
$999$	98822.0	3.12972

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.4.a.g.1.2		8
13.2	odd	12		52.4.h.a.17.4	✓	8
13.3	even	3		676.4.e.h.529.8		16
13.4	even	6		676.4.e.h.653.7		16
13.5	odd	4		676.4.d.d.337.1		8
13.6	odd	12		676.4.h.e.361.4		8
13.7	odd	12		52.4.h.a.49.4	yes	8
13.8	odd	4		676.4.d.d.337.2		8
13.9	even	3		676.4.e.h.653.8		16
13.10	even	6		676.4.e.h.529.7		16
13.11	odd	12		676.4.h.e.485.4		8
13.12	even	2	inner	676.4.a.g.1.1		8
39.2	even	12		468.4.t.g.433.2		8
39.20	even	12		468.4.t.g.361.3		8
52.7	even	12		208.4.w.c.49.1		8
52.15	even	12		208.4.w.c.17.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.4.h.a.17.4	✓	8	13.2	odd	12
52.4.h.a.49.4	yes	8	13.7	odd	12
208.4.w.c.17.1		8	52.15	even	12
208.4.w.c.49.1		8	52.7	even	12
468.4.t.g.361.3		8	39.20	even	12
468.4.t.g.433.2		8	39.2	even	12
676.4.a.g.1.1		8	13.12	even	2	inner
676.4.a.g.1.2		8	1.1	even	1	trivial
676.4.d.d.337.1		8	13.5	odd	4
676.4.d.d.337.2		8	13.8	odd	4
676.4.e.h.529.7		16	13.10	even	6
676.4.e.h.529.8		16	13.3	even	3
676.4.e.h.653.7		16	13.4	even	6
676.4.e.h.653.8		16	13.9	even	3
676.4.h.e.361.4		8	13.6	odd	12
676.4.h.e.485.4		8	13.11	odd	12

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:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	676.a (trivial)

\(f(q)\)	\(=\)	\(q-9.78841 q^{3} +2.49640 q^{5} -17.9693 q^{7} +68.8129 q^{9} -57.2769 q^{11} -24.4358 q^{15} -17.5058 q^{17} -22.1225 q^{19} +175.891 q^{21} -131.193 q^{23} -118.768 q^{25} -409.282 q^{27} +74.2527 q^{29} -110.463 q^{31} +560.649 q^{33} -44.8587 q^{35} -241.452 q^{37} -426.704 q^{41} -208.296 q^{43} +171.785 q^{45} +158.624 q^{47} -20.1031 q^{49} +171.354 q^{51} +47.9103 q^{53} -142.986 q^{55} +216.544 q^{57} -441.432 q^{59} +869.398 q^{61} -1236.52 q^{63} +801.045 q^{67} +1284.17 q^{69} +893.624 q^{71} -413.180 q^{73} +1162.55 q^{75} +1029.23 q^{77} -246.527 q^{79} +2148.27 q^{81} +333.857 q^{83} -43.7016 q^{85} -726.816 q^{87} +379.176 q^{89} +1081.26 q^{93} -55.2266 q^{95} -941.263 q^{97} -3941.39 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 140 q^{9} + 176 q^{17} - 40 q^{23} + 84 q^{25} - 432 q^{27} + 968 q^{29} - 80 q^{35} + 1008 q^{43} + 1844 q^{49} + 1808 q^{51} - 1164 q^{53} + 2256 q^{55} + 2448 q^{61} + 3476 q^{69} + 2896 q^{75}+ \cdots + 4800 q^{95}+O(q^{100}) \) 8 * q + 140 * q^9 + 176 * q^17 - 40 * q^23 + 84 * q^25 - 432 * q^27 + 968 * q^29 - 80 * q^35 + 1008 * q^43 + 1844 * q^49 + 1808 * q^51 - 1164 * q^53 + 2256 * q^55 + 2448 * q^61 + 3476 * q^69 + 2896 * q^75 + 3972 * q^77 - 3968 * q^79 + 8264 * q^81 + 3320 * q^87 + 4800 * q^95

Introduction

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