LMFDB - Embedded newform 6480.2.a.cb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$51.7430605098$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29268.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 5x + 16 $ x^4 - x^3 - 9x^2 + 5x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.25548$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +4.57301 q^{7} +4.59208 q^{11} -3.67925 q^{13} -2.55395 q^{17} -4.76643 q^{19} +2.57301 q^{23} +1.00000 q^{25} +1.91282 q^{29} -3.46677 q^{31} +4.57301 q^{35} +5.46677 q^{37} +6.64113 q^{41} +6.55395 q^{43} -10.0398 q^{47} +13.9125 q^{49} +4.03813 q^{53} +4.59208 q^{55} -12.2713 q^{59} +9.59208 q^{61} -3.67925 q^{65} +11.3394 q^{67} +1.67925 q^{71} +3.12530 q^{73} +20.9996 q^{77} +13.2823 q^{79} -10.7855 q^{83} -2.55395 q^{85} +4.04905 q^{89} -16.8253 q^{91} -4.76643 q^{95} +17.3792 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + q^{7} - q^{11} + 4 q^{13} + 5 q^{17} - q^{19} - 7 q^{23} + 4 q^{25} + 7 q^{29} + 2 q^{31} + q^{35} + 6 q^{37} + 12 q^{41} + 11 q^{43} - 7 q^{47} + 3 q^{49} + 12 q^{53} - q^{55} - 11 q^{59}+ \cdots + q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + q^7 - q^11 + 4 * q^13 + 5 * q^17 - q^19 - 7 * q^23 + 4 * q^25 + 7 * q^29 + 2 * q^31 + q^35 + 6 * q^37 + 12 * q^41 + 11 * q^43 - 7 * q^47 + 3 * q^49 + 12 * q^53 - q^55 - 11 * q^59 + 19 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 + 9 * q^73 + 32 * q^77 + 24 * q^79 - 23 * q^83 + 5 * q^85 + 21 * q^89 - 14 * q^91 - q^95 + q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.57301	1.72844	0.864218	−	0.503117i	$-0.167813\pi$
$7$	4.57301	1.72844	0.864218	+	0.503117i	$0.167813\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.59208	1.38456	0.692282	−	0.721627i	$-0.256606\pi$
$11$	4.59208	1.38456	0.692282	+	0.721627i	$0.256606\pi$
$12$	0	0
$13$	−3.67925	−1.02044	−0.510221	−	0.860043i	$-0.670436\pi$
$13$	−3.67925	−1.02044	−0.510221	+	0.860043i	$0.670436\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.55395	−0.619424	−0.309712	−	0.950830i	$-0.600233\pi$
$17$	−2.55395	−0.619424	−0.309712	+	0.950830i	$0.600233\pi$
$18$	0	0
$19$	−4.76643	−1.09349	−0.546747	−	0.837298i	$-0.684134\pi$
$19$	−4.76643	−1.09349	−0.546747	+	0.837298i	$0.684134\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.57301	0.536511	0.268255	−	0.963348i	$-0.413553\pi$
$23$	2.57301	0.536511	0.268255	+	0.963348i	$0.413553\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.91282	0.355202	0.177601	−	0.984103i	$-0.443166\pi$
$29$	1.91282	0.355202	0.177601	+	0.984103i	$0.443166\pi$
$30$	0	0
$31$	−3.46677	−0.622651	−0.311326	−	0.950303i	$-0.600773\pi$
$31$	−3.46677	−0.622651	−0.311326	+	0.950303i	$0.600773\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.57301	0.772980
$36$	0	0
$37$	5.46677	0.898732	0.449366	−	0.893348i	$-0.351650\pi$
$37$	5.46677	0.898732	0.449366	+	0.893348i	$0.351650\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.64113	1.03717	0.518585	−	0.855026i	$-0.326459\pi$
$41$	6.64113	1.03717	0.518585	+	0.855026i	$0.326459\pi$
$42$	0	0
$43$	6.55395	0.999468	0.499734	−	0.866179i	$-0.333431\pi$
$43$	6.55395	0.999468	0.499734	+	0.866179i	$0.333431\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.0398	−1.46445	−0.732227	−	0.681061i	$-0.761519\pi$
$47$	−10.0398	−1.46445	−0.732227	+	0.681061i	$0.761519\pi$
$48$	0	0
$49$	13.9125	1.98749
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.03813	0.554679	0.277340	−	0.960772i	$-0.410547\pi$
$53$	4.03813	0.554679	0.277340	+	0.960772i	$0.410547\pi$
$54$	0	0
$55$	4.59208	0.619196
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.2713	−1.59759	−0.798796	−	0.601602i	$-0.794529\pi$
$59$	−12.2713	−1.59759	−0.798796	+	0.601602i	$0.794529\pi$
$60$	0	0
$61$	9.59208	1.22814	0.614070	−	0.789252i	$-0.289531\pi$
$61$	9.59208	1.22814	0.614070	+	0.789252i	$0.289531\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.67925	−0.456355
$66$	0	0
$67$	11.3394	1.38533	0.692667	−	0.721258i	$-0.256436\pi$
$67$	11.3394	1.38533	0.692667	+	0.721258i	$0.256436\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.67925	0.199291	0.0996454	−	0.995023i	$-0.468229\pi$
$71$	1.67925	0.199291	0.0996454	+	0.995023i	$0.468229\pi$
$72$	0	0
$73$	3.12530	0.365789	0.182895	−	0.983133i	$-0.441453\pi$
$73$	3.12530	0.365789	0.182895	+	0.983133i	$0.441453\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	20.9996	2.39313
$78$	0	0
$79$	13.2823	1.49437	0.747185	−	0.664616i	$-0.231405\pi$
$79$	13.2823	1.49437	0.747185	+	0.664616i	$0.231405\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.7855	−1.18386	−0.591931	−	0.805989i	$-0.701634\pi$
$83$	−10.7855	−1.18386	−0.591931	+	0.805989i	$0.701634\pi$
$84$	0	0
$85$	−2.55395	−0.277015
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.04905	0.429198	0.214599	−	0.976702i	$-0.431155\pi$
$89$	4.04905	0.429198	0.214599	+	0.976702i	$0.431155\pi$
$90$	0	0
$91$	−16.8253	−1.76377
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.76643	−0.489026
$96$	0	0
$97$	17.3792	1.76459	0.882297	−	0.470693i	$-0.155996\pi$
$97$	17.3792	1.76459	0.882297	+	0.470693i	$0.155996\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.1460	1.30808	0.654039	−	0.756461i	$-0.273073\pi$
$101$	13.1460	1.30808	0.654039	+	0.756461i	$0.273073\pi$
$102$	0	0
$103$	11.5049	1.13361	0.566806	−	0.823851i	$-0.308179\pi$
$103$	11.5049	1.13361	0.566806	+	0.823851i	$0.308179\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.52396	−0.534022	−0.267011	−	0.963694i	$-0.586036\pi$
$107$	−5.52396	−0.534022	−0.267011	+	0.963694i	$0.586036\pi$
$108$	0	0
$109$	−16.7381	−1.60322	−0.801610	−	0.597847i	$-0.796023\pi$
$109$	−16.7381	−1.60322	−0.801610	+	0.597847i	$0.796023\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.78752	0.356300	0.178150	−	0.984003i	$-0.442989\pi$
$113$	3.78752	0.356300	0.178150	+	0.984003i	$0.442989\pi$
$114$	0	0
$115$	2.57301	0.239935
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.6793	−1.07064
$120$	0	0
$121$	10.0872	0.917016
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.1444	0.988903	0.494451	−	0.869205i	$-0.335369\pi$
$127$	11.1444	0.988903	0.494451	+	0.869205i	$0.335369\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.8249	1.90685	0.953426	−	0.301627i	$-0.0975297\pi$
$131$	21.8249	1.90685	0.953426	+	0.301627i	$0.0975297\pi$
$132$	0	0
$133$	−21.7970	−1.89004
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.44569	−0.379821	−0.189910	−	0.981801i	$-0.560820\pi$
$137$	−4.44569	−0.379821	−0.189910	+	0.981801i	$0.560820\pi$
$138$	0	0
$139$	−11.1668	−0.947152	−0.473576	−	0.880753i	$-0.657037\pi$
$139$	−11.1668	−0.947152	−0.473576	+	0.880753i	$0.657037\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.8954	−1.41287
$144$	0	0
$145$	1.91282	0.158851
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.0868	−1.07211	−0.536057	−	0.844182i	$-0.680087\pi$
$149$	−13.0868	−1.07211	−0.536057	+	0.844182i	$0.680087\pi$
$150$	0	0
$151$	0.358872	0.0292046	0.0146023	−	0.999893i	$-0.495352\pi$
$151$	0.358872	0.0292046	0.0146023	+	0.999893i	$0.495352\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.46677	−0.278458
$156$	0	0
$157$	−13.0698	−1.04308	−0.521541	−	0.853226i	$-0.674643\pi$
$157$	−13.0698	−1.04308	−0.521541	+	0.853226i	$0.674643\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	11.7664	0.927325
$162$	0	0
$163$	18.7872	1.47152	0.735762	−	0.677240i	$-0.236824\pi$
$163$	18.7872	1.47152	0.735762	+	0.677240i	$0.236824\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.2523	−1.10287	−0.551437	−	0.834217i	$-0.685920\pi$
$167$	−14.2523	−1.10287	−0.551437	+	0.834217i	$0.685920\pi$
$168$	0	0
$169$	0.536913	0.0413010
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.18416	−0.698258	−0.349129	−	0.937075i	$-0.613523\pi$
$173$	−9.18416	−0.698258	−0.349129	+	0.937075i	$0.613523\pi$
$174$	0	0
$175$	4.57301	0.345687
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.7872	−1.40422	−0.702109	−	0.712069i	$-0.747758\pi$
$179$	−18.7872	−1.40422	−0.702109	+	0.712069i	$0.747758\pi$
$180$	0	0
$181$	11.9128	0.885473	0.442737	−	0.896652i	$-0.354008\pi$
$181$	11.9128	0.885473	0.442737	+	0.896652i	$0.354008\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.46677	0.401925
$186$	0	0
$187$	−11.7279	−0.857632
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.74571	−0.343387	−0.171694	−	0.985150i	$-0.554924\pi$
$191$	−4.74571	−0.343387	−0.171694	+	0.985150i	$0.554924\pi$
$192$	0	0
$193$	8.55395	0.615727	0.307863	−	0.951431i	$-0.400386\pi$
$193$	8.55395	0.615727	0.307863	+	0.951431i	$0.400386\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.2539	−0.873056	−0.436528	−	0.899691i	$-0.643792\pi$
$197$	−12.2539	−0.873056	−0.436528	+	0.899691i	$0.643792\pi$
$198$	0	0
$199$	−1.71738	−0.121742	−0.0608710	−	0.998146i	$-0.519388\pi$
$199$	−1.71738	−0.121742	−0.0608710	+	0.998146i	$0.519388\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.74737	0.613945
$204$	0	0
$205$	6.64113	0.463836
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−21.8878	−1.51401
$210$	0	0
$211$	−2.53323	−0.174394	−0.0871972	−	0.996191i	$-0.527791\pi$
$211$	−2.53323	−0.174394	−0.0871972	+	0.996191i	$0.527791\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.55395	0.446976
$216$	0	0
$217$	−15.8536	−1.07621
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.39664	0.632086
$222$	0	0
$223$	−22.3318	−1.49545	−0.747725	−	0.664008i	$-0.768854\pi$
$223$	−22.3318	−1.49545	−0.747725	+	0.664008i	$0.768854\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.4177	0.691449	0.345724	−	0.938336i	$-0.387633\pi$
$227$	10.4177	0.691449	0.345724	+	0.938336i	$0.387633\pi$
$228$	0	0
$229$	−17.5634	−1.16062	−0.580311	−	0.814395i	$-0.697069\pi$
$229$	−17.5634	−1.16062	−0.580311	+	0.814395i	$0.697069\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.2328	1.25999	0.629993	−	0.776601i	$-0.283058\pi$
$233$	19.2328	1.25999	0.629993	+	0.776601i	$0.283058\pi$
$234$	0	0
$235$	−10.0398	−0.654924
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.5045	0.679482	0.339741	−	0.940519i	$-0.389661\pi$
$239$	10.5045	0.679482	0.339741	+	0.940519i	$0.389661\pi$
$240$	0	0
$241$	9.32075	0.600402	0.300201	−	0.953876i	$-0.402946\pi$
$241$	9.32075	0.600402	0.300201	+	0.953876i	$0.402946\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	13.9125	0.888834
$246$	0	0
$247$	17.5369	1.11585
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.9125	−1.25686	−0.628432	−	0.777865i	$-0.716303\pi$
$251$	−19.9125	−1.25686	−0.628432	+	0.777865i	$0.716303\pi$
$252$	0	0
$253$	11.8155	0.742833
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.4174	−0.712195	−0.356098	−	0.934449i	$-0.615893\pi$
$257$	−11.4174	−0.712195	−0.356098	+	0.934449i	$0.615893\pi$
$258$	0	0
$259$	24.9996	1.55340
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.57135	−0.281882	−0.140941	−	0.990018i	$-0.545013\pi$
$263$	−4.57135	−0.281882	−0.140941	+	0.990018i	$0.545013\pi$
$264$	0	0
$265$	4.03813	0.248060
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.5954	1.07281	0.536405	−	0.843961i	$-0.319782\pi$
$269$	17.5954	1.07281	0.536405	+	0.843961i	$0.319782\pi$
$270$	0	0
$271$	9.64113	0.585657	0.292828	−	0.956165i	$-0.405404\pi$
$271$	9.64113	0.585657	0.292828	+	0.956165i	$0.405404\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.59208	0.276913
$276$	0	0
$277$	14.1042	0.847440	0.423720	−	0.905793i	$-0.360724\pi$
$277$	14.1042	0.847440	0.423720	+	0.905793i	$0.360724\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.45585	0.206159	0.103079	−	0.994673i	$-0.467130\pi$
$281$	3.45585	0.206159	0.103079	+	0.994673i	$0.467130\pi$
$282$	0	0
$283$	−24.1821	−1.43748	−0.718739	−	0.695280i	$-0.755280\pi$
$283$	−24.1821	−1.43748	−0.718739	+	0.695280i	$0.755280\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.3700	1.79268
$288$	0	0
$289$	−10.4773	−0.616314
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.25429	−0.306959	−0.153480	−	0.988152i	$-0.549048\pi$
$293$	−5.25429	−0.306959	−0.153480	+	0.988152i	$0.549048\pi$
$294$	0	0
$295$	−12.2713	−0.714465
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.46677	−0.547478
$300$	0	0
$301$	29.9713	1.72752
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.59208	0.549241
$306$	0	0
$307$	−1.31148	−0.0748503	−0.0374252	−	0.999299i	$-0.511916\pi$
$307$	−1.31148	−0.0748503	−0.0374252	+	0.999299i	$0.511916\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.86341	−0.502598	−0.251299	−	0.967910i	$-0.580858\pi$
$311$	−8.86341	−0.502598	−0.251299	+	0.967910i	$0.580858\pi$
$312$	0	0
$313$	2.24337	0.126803	0.0634014	−	0.997988i	$-0.479805\pi$
$313$	2.24337	0.126803	0.0634014	+	0.997988i	$0.479805\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.5747	1.15559	0.577794	−	0.816182i	$-0.303914\pi$
$317$	20.5747	1.15559	0.577794	+	0.816182i	$0.303914\pi$
$318$	0	0
$319$	8.78383	0.491800
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.1732	0.677337
$324$	0	0
$325$	−3.67925	−0.204088
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−45.9121	−2.53122
$330$	0	0
$331$	−21.7875	−1.19755	−0.598775	−	0.800917i	$-0.704346\pi$
$331$	−21.7875	−1.19755	−0.598775	+	0.800917i	$0.704346\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.3394	0.619540
$336$	0	0
$337$	12.4457	0.677960	0.338980	−	0.940794i	$-0.389918\pi$
$337$	12.4457	0.677960	0.338980	+	0.940794i	$0.389918\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15.9197	−0.862100
$342$	0	0
$343$	31.6108	1.70682
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.1947	−0.923061	−0.461530	−	0.887124i	$-0.652699\pi$
$347$	−17.1947	−0.923061	−0.461530	+	0.887124i	$0.652699\pi$
$348$	0	0
$349$	9.48381	0.507657	0.253828	−	0.967249i	$-0.418310\pi$
$349$	9.48381	0.507657	0.253828	+	0.967249i	$0.418310\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.4174	0.714134	0.357067	−	0.934079i	$-0.383777\pi$
$353$	13.4174	0.714134	0.357067	+	0.934079i	$0.383777\pi$
$354$	0	0
$355$	1.67925	0.0891256
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12.3970	−0.654289	−0.327144	−	0.944974i	$-0.606086\pi$
$359$	−12.3970	−0.654289	−0.327144	+	0.944974i	$0.606086\pi$
$360$	0	0
$361$	3.71887	0.195730
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.12530	0.163586
$366$	0	0
$367$	20.5714	1.07382	0.536908	−	0.843641i	$-0.319592\pi$
$367$	20.5714	1.07382	0.536908	+	0.843641i	$0.319592\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.4664	0.958728
$372$	0	0
$373$	−4.46309	−0.231090	−0.115545	−	0.993302i	$-0.536861\pi$
$373$	−4.46309	−0.231090	−0.115545	+	0.993302i	$0.536861\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.03776	−0.362463
$378$	0	0
$379$	8.77660	0.450823	0.225412	−	0.974264i	$-0.427627\pi$
$379$	8.77660	0.450823	0.225412	+	0.974264i	$0.427627\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.21212	0.164132	0.0820658	−	0.996627i	$-0.473848\pi$
$383$	3.21212	0.164132	0.0820658	+	0.996627i	$0.473848\pi$
$384$	0	0
$385$	20.9996	1.07024
$386$	0	0
$387$	0	0
$388$	0	0
$389$	25.5914	1.29753	0.648766	−	0.760988i	$-0.275285\pi$
$389$	25.5914	1.29753	0.648766	+	0.760988i	$0.275285\pi$
$390$	0	0
$391$	−6.57135	−0.332328
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.2823	0.668303
$396$	0	0
$397$	14.3902	0.722221	0.361111	−	0.932523i	$-0.382398\pi$
$397$	14.3902	0.722221	0.361111	+	0.932523i	$0.382398\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.1384	1.30529	0.652645	−	0.757663i	$-0.273659\pi$
$401$	26.1384	1.30529	0.652645	+	0.757663i	$0.273659\pi$
$402$	0	0
$403$	12.7551	0.635379
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.1039	1.24435
$408$	0	0
$409$	6.58191	0.325455	0.162727	−	0.986671i	$-0.447971\pi$
$409$	6.58191	0.325455	0.162727	+	0.986671i	$0.447971\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−56.1170	−2.76134
$414$	0	0
$415$	−10.7855	−0.529439
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.7490	−0.622831	−0.311415	−	0.950274i	$-0.600803\pi$
$419$	−12.7490	−0.622831	−0.311415	+	0.950274i	$0.600803\pi$
$420$	0	0
$421$	−1.28226	−0.0624933	−0.0312467	−	0.999512i	$-0.509948\pi$
$421$	−1.28226	−0.0624933	−0.0312467	+	0.999512i	$0.509948\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.55395	−0.123885
$426$	0	0
$427$	43.8647	2.12276
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.89137	−0.331946	−0.165973	−	0.986130i	$-0.553076\pi$
$431$	−6.89137	−0.331946	−0.165973	+	0.986130i	$0.553076\pi$
$432$	0	0
$433$	30.4551	1.46358	0.731790	−	0.681530i	$-0.238685\pi$
$433$	30.4551	1.46358	0.731790	+	0.681530i	$0.238685\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.2641	−0.586671
$438$	0	0
$439$	29.7309	1.41898	0.709488	−	0.704717i	$-0.248926\pi$
$439$	29.7309	1.41898	0.709488	+	0.704717i	$0.248926\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.32964	0.300730	0.150365	−	0.988631i	$-0.451955\pi$
$443$	6.32964	0.300730	0.150365	+	0.988631i	$0.451955\pi$
$444$	0	0
$445$	4.04905	0.191943
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.8456	1.21973	0.609866	−	0.792505i	$-0.291223\pi$
$449$	25.8456	1.21973	0.609866	+	0.792505i	$0.291223\pi$
$450$	0	0
$451$	30.4966	1.43603
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−16.8253	−0.788781
$456$	0	0
$457$	−35.3091	−1.65169	−0.825845	−	0.563897i	$-0.809301\pi$
$457$	−35.3091	−1.65169	−0.825845	+	0.563897i	$0.809301\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.11883	0.145258	0.0726291	−	0.997359i	$-0.476861\pi$
$461$	3.11883	0.145258	0.0726291	+	0.997359i	$0.476861\pi$
$462$	0	0
$463$	3.17472	0.147542	0.0737708	−	0.997275i	$-0.476497\pi$
$463$	3.17472	0.147542	0.0737708	+	0.997275i	$0.476497\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.2808	−0.938482	−0.469241	−	0.883070i	$-0.655472\pi$
$467$	−20.2808	−0.938482	−0.469241	+	0.883070i	$0.655472\pi$
$468$	0	0
$469$	51.8554	2.39446
$470$	0	0
$471$	0	0
$472$	0	0
$473$	30.0963	1.38383
$474$	0	0
$475$	−4.76643	−0.218699
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.0763	0.460396	0.230198	−	0.973144i	$-0.426063\pi$
$479$	10.0763	0.460396	0.230198	+	0.973144i	$0.426063\pi$
$480$	0	0
$481$	−20.1137	−0.917104
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.3792	0.789150
$486$	0	0
$487$	28.3589	1.28506	0.642532	−	0.766259i	$-0.277884\pi$
$487$	28.3589	1.28506	0.642532	+	0.766259i	$0.277884\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.9121	−1.48530	−0.742651	−	0.669679i	$-0.766432\pi$
$491$	−32.9121	−1.48530	−0.742651	+	0.669679i	$0.766432\pi$
$492$	0	0
$493$	−4.88526	−0.220021
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.67925	0.344462
$498$	0	0
$499$	−5.01092	−0.224320	−0.112160	−	0.993690i	$-0.535777\pi$
$499$	−5.01092	−0.224320	−0.112160	+	0.993690i	$0.535777\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−42.1916	−1.88123	−0.940615	−	0.339477i	$-0.889750\pi$
$503$	−42.1916	−1.88123	−0.940615	+	0.339477i	$0.889750\pi$
$504$	0	0
$505$	13.1460	0.584991
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.2332	−0.763848	−0.381924	−	0.924194i	$-0.624738\pi$
$509$	−17.2332	−0.763848	−0.381924	+	0.924194i	$0.624738\pi$
$510$	0	0
$511$	14.2921	0.632243
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.5049	0.506967
$516$	0	0
$517$	−46.1035	−2.02763
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.42901	−0.237849	−0.118925	−	0.992903i	$-0.537945\pi$
$521$	−5.42901	−0.237849	−0.118925	+	0.992903i	$0.537945\pi$
$522$	0	0
$523$	−6.68128	−0.292152	−0.146076	−	0.989273i	$-0.546664\pi$
$523$	−6.68128	−0.292152	−0.146076	+	0.989273i	$0.546664\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.85397	0.385685
$528$	0	0
$529$	−16.3796	−0.712156
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−24.4344	−1.05837
$534$	0	0
$535$	−5.52396	−0.238822
$536$	0	0
$537$	0	0
$538$	0	0
$539$	63.8871	2.75181
$540$	0	0
$541$	35.3087	1.51804	0.759020	−	0.651067i	$-0.225678\pi$
$541$	35.3087	1.51804	0.759020	+	0.651067i	$0.225678\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.7381	−0.716982
$546$	0	0
$547$	−11.5621	−0.494359	−0.247180	−	0.968970i	$-0.579504\pi$
$547$	−11.5621	−0.494359	−0.247180	+	0.968970i	$0.579504\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.11734	−0.388412
$552$	0	0
$553$	60.7399	2.58293
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.0415	−0.764441	−0.382220	−	0.924071i	$-0.624840\pi$
$557$	−18.0415	−0.764441	−0.382220	+	0.924071i	$0.624840\pi$
$558$	0	0
$559$	−24.1137	−1.01990
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.56173	−0.107964	−0.0539820	−	0.998542i	$-0.517191\pi$
$563$	−2.56173	−0.107964	−0.0539820	+	0.998542i	$0.517191\pi$
$564$	0	0
$565$	3.78752	0.159342
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.7003	−0.867804	−0.433902	−	0.900960i	$-0.642864\pi$
$569$	−20.7003	−0.867804	−0.433902	+	0.900960i	$0.642864\pi$
$570$	0	0
$571$	7.16675	0.299919	0.149960	−	0.988692i	$-0.452086\pi$
$571$	7.16675	0.299919	0.149960	+	0.988692i	$0.452086\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.57301	0.107302
$576$	0	0
$577$	−27.2463	−1.13428	−0.567140	−	0.823622i	$-0.691950\pi$
$577$	−27.2463	−1.13428	−0.567140	+	0.823622i	$0.691950\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−49.3222	−2.04623
$582$	0	0
$583$	18.5434	0.767989
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.4949	−0.680818	−0.340409	−	0.940277i	$-0.610566\pi$
$587$	−16.4949	−0.680818	−0.340409	+	0.940277i	$0.610566\pi$
$588$	0	0
$589$	16.5241	0.680865
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.2819	−0.586487	−0.293244	−	0.956038i	$-0.594735\pi$
$593$	−14.2819	−0.586487	−0.293244	+	0.956038i	$0.594735\pi$
$594$	0	0
$595$	−11.6793	−0.478803
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22.3902	−0.914837	−0.457419	−	0.889252i	$-0.651226\pi$
$599$	−22.3902	−0.914837	−0.457419	+	0.889252i	$0.651226\pi$
$600$	0	0
$601$	−12.1003	−0.493582	−0.246791	−	0.969069i	$-0.579376\pi$
$601$	−12.1003	−0.493582	−0.246791	+	0.969069i	$0.579376\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.0872	0.410102
$606$	0	0
$607$	−20.3318	−0.825244	−0.412622	−	0.910902i	$-0.635387\pi$
$607$	−20.3318	−0.825244	−0.412622	+	0.910902i	$0.635387\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	36.9389	1.49439
$612$	0	0
$613$	24.8853	1.00511	0.502553	−	0.864546i	$-0.332394\pi$
$613$	24.8853	1.00511	0.502553	+	0.864546i	$0.332394\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.3988	1.82769	0.913844	−	0.406065i	$-0.133099\pi$
$617$	45.3988	1.82769	0.913844	+	0.406065i	$0.133099\pi$
$618$	0	0
$619$	−19.4275	−0.780858	−0.390429	−	0.920633i	$-0.627673\pi$
$619$	−19.4275	−0.780858	−0.390429	+	0.920633i	$0.627673\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	18.5164	0.741842
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.9619	−0.556696
$630$	0	0
$631$	−7.78752	−0.310016	−0.155008	−	0.987913i	$-0.549540\pi$
$631$	−7.78752	−0.310016	−0.155008	+	0.987913i	$0.549540\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.1444	0.442251
$636$	0	0
$637$	−51.1875	−2.02812
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.3963	−1.16108	−0.580541	−	0.814231i	$-0.697159\pi$
$641$	−29.3963	−1.16108	−0.580541	+	0.814231i	$0.697159\pi$
$642$	0	0
$643$	−9.30132	−0.366808	−0.183404	−	0.983038i	$-0.558712\pi$
$643$	−9.30132	−0.366808	−0.183404	+	0.983038i	$0.558712\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.4647	1.59083	0.795417	−	0.606063i	$-0.207252\pi$
$647$	40.4647	1.59083	0.795417	+	0.606063i	$0.207252\pi$
$648$	0	0
$649$	−56.3509	−2.21197
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.6509	−0.573335	−0.286668	−	0.958030i	$-0.592548\pi$
$653$	−14.6509	−0.573335	−0.286668	+	0.958030i	$0.592548\pi$
$654$	0	0
$655$	21.8249	0.852770
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.02760	0.195847	0.0979237	−	0.995194i	$-0.468780\pi$
$659$	5.02760	0.195847	0.0979237	+	0.995194i	$0.468780\pi$
$660$	0	0
$661$	31.8256	1.23787	0.618937	−	0.785441i	$-0.287564\pi$
$661$	31.8256	1.23787	0.618937	+	0.785441i	$0.287564\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−21.7970	−0.845250
$666$	0	0
$667$	4.92172	0.190570
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.0476	1.70044
$672$	0	0
$673$	10.9015	0.420223	0.210112	−	0.977677i	$-0.432617\pi$
$673$	10.9015	0.420223	0.210112	+	0.977677i	$0.432617\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.0094	−1.07649	−0.538245	−	0.842788i	$-0.680913\pi$
$677$	−28.0094	−1.07649	−0.538245	+	0.842788i	$0.680913\pi$
$678$	0	0
$679$	79.4755	3.04999
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.19544	0.275326	0.137663	−	0.990479i	$-0.456041\pi$
$683$	7.19544	0.275326	0.137663	+	0.990479i	$0.456041\pi$
$684$	0	0
$685$	−4.44569	−0.169861
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.8573	−0.566018
$690$	0	0
$691$	−22.5427	−0.857563	−0.428782	−	0.903408i	$-0.641057\pi$
$691$	−22.5427	−0.857563	−0.428782	+	0.903408i	$0.641057\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.1668	−0.423579
$696$	0	0
$697$	−16.9611	−0.642448
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19.7098	−0.744428	−0.372214	−	0.928147i	$-0.621401\pi$
$701$	−19.7098	−0.744428	−0.372214	+	0.928147i	$0.621401\pi$
$702$	0	0
$703$	−26.0570	−0.982758
$704$	0	0
$705$	0	0
$706$	0	0
$707$	60.1170	2.26093
$708$	0	0
$709$	24.1286	0.906170	0.453085	−	0.891467i	$-0.350323\pi$
$709$	24.1286	0.906170	0.453085	+	0.891467i	$0.350323\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.92006	−0.334059
$714$	0	0
$715$	−16.8954	−0.631853
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.79732	−0.328085	−0.164042	−	0.986453i	$-0.552453\pi$
$719$	−8.79732	−0.328085	−0.164042	+	0.986453i	$0.552453\pi$
$720$	0	0
$721$	52.6121	1.95938
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.91282	0.0710405
$726$	0	0
$727$	5.24247	0.194432	0.0972162	−	0.995263i	$-0.469006\pi$
$727$	5.24247	0.194432	0.0972162	+	0.995263i	$0.469006\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16.7385	−0.619095
$732$	0	0
$733$	−34.1177	−1.26017	−0.630083	−	0.776528i	$-0.716979\pi$
$733$	−34.1177	−1.26017	−0.630083	+	0.776528i	$0.716979\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	52.0716	1.91808
$738$	0	0
$739$	5.53174	0.203488	0.101744	−	0.994811i	$-0.467558\pi$
$739$	5.53174	0.203488	0.101744	+	0.994811i	$0.467558\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.5447	0.570279	0.285140	−	0.958486i	$-0.407960\pi$
$743$	15.5447	0.570279	0.285140	+	0.958486i	$0.407960\pi$
$744$	0	0
$745$	−13.0868	−0.479464
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−25.2612	−0.923023
$750$	0	0
$751$	−22.4944	−0.820831	−0.410416	−	0.911899i	$-0.634616\pi$
$751$	−22.4944	−0.820831	−0.410416	+	0.911899i	$0.634616\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.358872	0.0130607
$756$	0	0
$757$	−14.5714	−0.529605	−0.264802	−	0.964303i	$-0.585307\pi$
$757$	−14.5714	−0.529605	−0.264802	+	0.964303i	$0.585307\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.8147	−0.500783	−0.250392	−	0.968145i	$-0.580559\pi$
$761$	−13.8147	−0.500783	−0.250392	+	0.968145i	$0.580559\pi$
$762$	0	0
$763$	−76.5436	−2.77106
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.1494	1.63025
$768$	0	0
$769$	−0.128627	−0.00463841	−0.00231921	−	0.999997i	$-0.500738\pi$
$769$	−0.128627	−0.00463841	−0.00231921	+	0.999997i	$0.500738\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−44.5427	−1.60209	−0.801044	−	0.598605i	$-0.795722\pi$
$773$	−44.5427	−1.60209	−0.801044	+	0.598605i	$0.795722\pi$
$774$	0	0
$775$	−3.46677	−0.124530
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−31.6545	−1.13414
$780$	0	0
$781$	7.71127	0.275931
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.0698	−0.466480
$786$	0	0
$787$	−32.7864	−1.16871	−0.584355	−	0.811498i	$-0.698652\pi$
$787$	−32.7864	−1.16871	−0.584355	+	0.811498i	$0.698652\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17.3204	0.615842
$792$	0	0
$793$	−35.2917	−1.25324
$794$	0	0
$795$	0	0
$796$	0	0
$797$	53.8765	1.90840	0.954202	−	0.299162i	$-0.0967070\pi$
$797$	53.8765	1.90840	0.954202	+	0.299162i	$0.0967070\pi$
$798$	0	0
$799$	25.6411	0.907118
$800$	0	0
$801$	0	0
$802$	0	0
$803$	14.3516	0.506458
$804$	0	0
$805$	11.7664	0.414712
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−44.3890	−1.56064	−0.780318	−	0.625383i	$-0.784943\pi$
$809$	−44.3890	−1.56064	−0.780318	+	0.625383i	$0.784943\pi$
$810$	0	0
$811$	−23.6339	−0.829898	−0.414949	−	0.909845i	$-0.636201\pi$
$811$	−23.6339	−0.829898	−0.414949	+	0.909845i	$0.636201\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.7872	0.658085
$816$	0	0
$817$	−31.2390	−1.09291
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.81841	−0.237964	−0.118982	−	0.992896i	$-0.537963\pi$
$821$	−6.81841	−0.237964	−0.118982	+	0.992896i	$0.537963\pi$
$822$	0	0
$823$	42.3311	1.47557	0.737785	−	0.675036i	$-0.235872\pi$
$823$	42.3311	1.47557	0.737785	+	0.675036i	$0.235872\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.3700	1.05607	0.528034	−	0.849223i	$-0.322929\pi$
$827$	30.3700	1.05607	0.528034	+	0.849223i	$0.322929\pi$
$828$	0	0
$829$	−45.8420	−1.59216	−0.796078	−	0.605193i	$-0.793096\pi$
$829$	−45.8420	−1.59216	−0.796078	+	0.605193i	$0.793096\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−35.5317	−1.23110
$834$	0	0
$835$	−14.2523	−0.493220
$836$	0	0
$837$	0	0
$838$	0	0
$839$	11.2502	0.388402	0.194201	−	0.980962i	$-0.437789\pi$
$839$	11.2502	0.388402	0.194201	+	0.980962i	$0.437789\pi$
$840$	0	0
$841$	−25.3411	−0.873831
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.536913	0.0184704
$846$	0	0
$847$	46.1288	1.58500
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.0661	0.482179
$852$	0	0
$853$	5.34834	0.183124	0.0915619	−	0.995799i	$-0.470814\pi$
$853$	5.34834	0.183124	0.0915619	+	0.995799i	$0.470814\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	51.9291	1.77387	0.886933	−	0.461899i	$-0.152832\pi$
$857$	51.9291	1.77387	0.886933	+	0.461899i	$0.152832\pi$
$858$	0	0
$859$	−29.4235	−1.00392	−0.501958	−	0.864892i	$-0.667387\pi$
$859$	−29.4235	−1.00392	−0.501958	+	0.864892i	$0.667387\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.9635	−0.543405	−0.271703	−	0.962381i	$-0.587587\pi$
$863$	−15.9635	−0.543405	−0.271703	+	0.962381i	$0.587587\pi$
$864$	0	0
$865$	−9.18416	−0.312271
$866$	0	0
$867$	0	0
$868$	0	0
$869$	60.9931	2.06905
$870$	0	0
$871$	−41.7207	−1.41365
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.57301	0.154596
$876$	0	0
$877$	5.42865	0.183312	0.0916562	−	0.995791i	$-0.470784\pi$
$877$	5.42865	0.183312	0.0916562	+	0.995791i	$0.470784\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.2676	0.345926	0.172963	−	0.984928i	$-0.444666\pi$
$881$	10.2676	0.345926	0.172963	+	0.984928i	$0.444666\pi$
$882$	0	0
$883$	−12.6184	−0.424642	−0.212321	−	0.977200i	$-0.568102\pi$
$883$	−12.6184	−0.424642	−0.212321	+	0.977200i	$0.568102\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.6030	1.26259	0.631293	−	0.775545i	$-0.282525\pi$
$887$	37.6030	1.26259	0.631293	+	0.775545i	$0.282525\pi$
$888$	0	0
$889$	50.9633	1.70926
$890$	0	0
$891$	0	0
$892$	0	0
$893$	47.8540	1.60137
$894$	0	0
$895$	−18.7872	−0.627985
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.63133	−0.221167
$900$	0	0
$901$	−10.3132	−0.343582
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.9128	0.395996
$906$	0	0
$907$	28.3111	0.940055	0.470028	−	0.882652i	$-0.344244\pi$
$907$	28.3111	0.940055	0.470028	+	0.882652i	$0.344244\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7.61317	0.252235	0.126118	−	0.992015i	$-0.459748\pi$
$911$	7.61317	0.252235	0.126118	+	0.992015i	$0.459748\pi$
$912$	0	0
$913$	−49.5278	−1.63913
$914$	0	0
$915$	0	0
$916$	0	0
$917$	99.8057	3.29587
$918$	0	0
$919$	41.2182	1.35966	0.679832	−	0.733368i	$-0.262053\pi$
$919$	41.2182	1.35966	0.679832	+	0.733368i	$0.262053\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.17840	−0.203365
$924$	0	0
$925$	5.46677	0.179746
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.2256	1.15571	0.577857	−	0.816138i	$-0.303889\pi$
$929$	35.2256	1.15571	0.577857	+	0.816138i	$0.303889\pi$
$930$	0	0
$931$	−66.3128	−2.17331
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−11.7279	−0.383545
$936$	0	0
$937$	−15.7113	−0.513265	−0.256632	−	0.966509i	$-0.582613\pi$
$937$	−15.7113	−0.513265	−0.256632	+	0.966509i	$0.582613\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−40.5188	−1.32087	−0.660437	−	0.750881i	$-0.729629\pi$
$941$	−40.5188	−1.32087	−0.660437	+	0.750881i	$0.729629\pi$
$942$	0	0
$943$	17.0877	0.556453
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.0531	−0.846613	−0.423307	−	0.905986i	$-0.639131\pi$
$947$	−26.0531	−0.846613	−0.423307	+	0.905986i	$0.639131\pi$
$948$	0	0
$949$	−11.4988	−0.373266
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.24264	0.267005	0.133503	−	0.991048i	$-0.457378\pi$
$953$	8.24264	0.267005	0.133503	+	0.991048i	$0.457378\pi$
$954$	0	0
$955$	−4.74571	−0.153567
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.3302	−0.656496
$960$	0	0
$961$	−18.9815	−0.612306
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.55395	0.275361
$966$	0	0
$967$	−12.5164	−0.402499	−0.201250	−	0.979540i	$-0.564500\pi$
$967$	−12.5164	−0.402499	−0.201250	+	0.979540i	$0.564500\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−34.5180	−1.10774	−0.553868	−	0.832604i	$-0.686849\pi$
$971$	−34.5180	−1.10774	−0.553868	+	0.832604i	$0.686849\pi$
$972$	0	0
$973$	−51.0657	−1.63709
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−52.6328	−1.68387	−0.841936	−	0.539577i	$-0.818584\pi$
$977$	−52.6328	−1.68387	−0.841936	+	0.539577i	$0.818584\pi$
$978$	0	0
$979$	18.5936	0.594253
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−35.8164	−1.14237	−0.571183	−	0.820823i	$-0.693515\pi$
$983$	−35.8164	−1.14237	−0.571183	+	0.820823i	$0.693515\pi$
$984$	0	0
$985$	−12.2539	−0.390443
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.8634	0.536225
$990$	0	0
$991$	−42.7203	−1.35706	−0.678528	−	0.734574i	$-0.737382\pi$
$991$	−42.7203	−1.35706	−0.678528	+	0.734574i	$0.737382\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.71738	−0.0544446
$996$	0	0
$997$	46.9143	1.48579	0.742895	−	0.669408i	$-0.233452\pi$
$997$	46.9143	1.48579	0.742895	+	0.669408i	$0.233452\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.cb.1.4		4
3.2	odd	2		6480.2.a.bz.1.4		4
4.3	odd	2		3240.2.a.u.1.1		4
9.2	odd	6		2160.2.q.l.1441.1		8
9.4	even	3		720.2.q.l.241.3		8
9.5	odd	6		2160.2.q.l.721.1		8
9.7	even	3		720.2.q.l.481.3		8
12.11	even	2		3240.2.a.s.1.1		4
36.7	odd	6		360.2.q.e.121.2	✓	8
36.11	even	6		1080.2.q.e.361.4		8
36.23	even	6		1080.2.q.e.721.4		8
36.31	odd	6		360.2.q.e.241.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.e.121.2	✓	8	36.7	odd	6
360.2.q.e.241.2	yes	8	36.31	odd	6
720.2.q.l.241.3		8	9.4	even	3
720.2.q.l.481.3		8	9.7	even	3
1080.2.q.e.361.4		8	36.11	even	6
1080.2.q.e.721.4		8	36.23	even	6
2160.2.q.l.721.1		8	9.5	odd	6
2160.2.q.l.1441.1		8	9.2	odd	6
3240.2.a.s.1.1		4	12.11	even	2
3240.2.a.u.1.1		4	4.3	odd	2
6480.2.a.bz.1.4		4	3.2	odd	2
6480.2.a.cb.1.4		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +4.57301 q^{7} +4.59208 q^{11} -3.67925 q^{13} -2.55395 q^{17} -4.76643 q^{19} +2.57301 q^{23} +1.00000 q^{25} +1.91282 q^{29} -3.46677 q^{31} +4.57301 q^{35} +5.46677 q^{37} +6.64113 q^{41} +6.55395 q^{43} -10.0398 q^{47} +13.9125 q^{49} +4.03813 q^{53} +4.59208 q^{55} -12.2713 q^{59} +9.59208 q^{61} -3.67925 q^{65} +11.3394 q^{67} +1.67925 q^{71} +3.12530 q^{73} +20.9996 q^{77} +13.2823 q^{79} -10.7855 q^{83} -2.55395 q^{85} +4.04905 q^{89} -16.8253 q^{91} -4.76643 q^{95} +17.3792 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + q^{7} - q^{11} + 4 q^{13} + 5 q^{17} - q^{19} - 7 q^{23} + 4 q^{25} + 7 q^{29} + 2 q^{31} + q^{35} + 6 q^{37} + 12 q^{41} + 11 q^{43} - 7 q^{47} + 3 q^{49} + 12 q^{53} - q^{55} - 11 q^{59}+ \cdots + q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + q^7 - q^11 + 4 * q^13 + 5 * q^17 - q^19 - 7 * q^23 + 4 * q^25 + 7 * q^29 + 2 * q^31 + q^35 + 6 * q^37 + 12 * q^41 + 11 * q^43 - 7 * q^47 + 3 * q^49 + 12 * q^53 - q^55 - 11 * q^59 + 19 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 + 9 * q^73 + 32 * q^77 + 24 * q^79 - 23 * q^83 + 5 * q^85 + 21 * q^89 - 14 * q^91 - q^95 + q^97

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