LMFDB - Embedded newform 6480.2.a.bx.1.3

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:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.571993$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} +5.24482 q^{7} -2.67282 q^{11} +3.81681 q^{13} +3.52884 q^{17} +4.67282 q^{19} +4.95684 q^{23} +1.00000 q^{25} -1.85601 q^{29} +8.67282 q^{31} +5.24482 q^{35} -2.67282 q^{37} -3.67282 q^{41} -3.52884 q^{43} -9.26329 q^{47} +20.5081 q^{49} -2.85601 q^{53} -2.67282 q^{55} -4.20166 q^{59} -7.96080 q^{61} +3.81681 q^{65} +0.859966 q^{67} +15.1625 q^{71} +6.28797 q^{73} -14.0185 q^{77} -5.63362 q^{79} +3.89917 q^{83} +3.52884 q^{85} -11.0000 q^{89} +20.0185 q^{91} +4.67282 q^{95} -3.83528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 5 q^{7} + 2 q^{11} + 2 q^{17} + 4 q^{19} + 7 q^{23} + 3 q^{25} - 7 q^{29} + 16 q^{31} + 5 q^{35} + 2 q^{37} - q^{41} - 2 q^{43} + 13 q^{47} + 10 q^{49} - 10 q^{53} + 2 q^{55} + 6 q^{59}+ \cdots + 30 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 + 5 * q^7 + 2 * q^11 + 2 * q^17 + 4 * q^19 + 7 * q^23 + 3 * q^25 - 7 * q^29 + 16 * q^31 + 5 * q^35 + 2 * q^37 - q^41 - 2 * q^43 + 13 * q^47 + 10 * q^49 - 10 * q^53 + 2 * q^55 + 6 * q^59 - 11 * q^61 - q^67 + 14 * q^71 + 16 * q^73 - 12 * q^77 + 6 * q^79 + 21 * q^83 + 2 * q^85 - 33 * q^89 + 30 * q^91 + 4 * q^95 + 30 * 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
$7$	5.24482	1.98235	0.991177	−	0.132543i	$-0.0423143\pi$
$7$	5.24482	1.98235	0.991177	+	0.132543i	$0.0423143\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.67282	−0.805887	−0.402943	−	0.915225i	$-0.632013\pi$
$11$	−2.67282	−0.805887	−0.402943	+	0.915225i	$0.632013\pi$
$12$	0	0
$13$	3.81681	1.05859	0.529296	−	0.848437i	$-0.322456\pi$
$13$	3.81681	1.05859	0.529296	+	0.848437i	$0.322456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.52884	0.855869	0.427934	−	0.903810i	$-0.359241\pi$
$17$	3.52884	0.855869	0.427934	+	0.903810i	$0.359241\pi$
$18$	0	0
$19$	4.67282	1.07202	0.536010	−	0.844212i	$-0.319931\pi$
$19$	4.67282	1.07202	0.536010	+	0.844212i	$0.319931\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.95684	1.03357	0.516787	−	0.856114i	$-0.327128\pi$
$23$	4.95684	1.03357	0.516787	+	0.856114i	$0.327128\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.85601	−0.344653	−0.172327	−	0.985040i	$-0.555128\pi$
$29$	−1.85601	−0.344653	−0.172327	+	0.985040i	$0.555128\pi$
$30$	0	0
$31$	8.67282	1.55769	0.778843	−	0.627219i	$-0.215807\pi$
$31$	8.67282	1.55769	0.778843	+	0.627219i	$0.215807\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.24482	0.886536
$36$	0	0
$37$	−2.67282	−0.439410	−0.219705	−	0.975566i	$-0.570509\pi$
$37$	−2.67282	−0.439410	−0.219705	+	0.975566i	$0.570509\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.67282	−0.573599	−0.286799	−	0.957991i	$-0.592591\pi$
$41$	−3.67282	−0.573599	−0.286799	+	0.957991i	$0.592591\pi$
$42$	0	0
$43$	−3.52884	−0.538143	−0.269071	−	0.963120i	$-0.586717\pi$
$43$	−3.52884	−0.538143	−0.269071	+	0.963120i	$0.586717\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.26329	−1.35119	−0.675595	−	0.737273i	$-0.736113\pi$
$47$	−9.26329	−1.35119	−0.675595	+	0.737273i	$0.736113\pi$
$48$	0	0
$49$	20.5081	2.92973
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.85601	−0.392304	−0.196152	−	0.980574i	$-0.562845\pi$
$53$	−2.85601	−0.392304	−0.196152	+	0.980574i	$0.562845\pi$
$54$	0	0
$55$	−2.67282	−0.360403
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.20166	−0.547010	−0.273505	−	0.961871i	$-0.588183\pi$
$59$	−4.20166	−0.547010	−0.273505	+	0.961871i	$0.588183\pi$
$60$	0	0
$61$	−7.96080	−1.01928	−0.509638	−	0.860389i	$-0.670221\pi$
$61$	−7.96080	−1.01928	−0.509638	+	0.860389i	$0.670221\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.81681	0.473417
$66$	0	0
$67$	0.859966	0.105062	0.0525308	−	0.998619i	$-0.483271\pi$
$67$	0.859966	0.105062	0.0525308	+	0.998619i	$0.483271\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.1625	1.79945	0.899726	−	0.436454i	$-0.143766\pi$
$71$	15.1625	1.79945	0.899726	+	0.436454i	$0.143766\pi$
$72$	0	0
$73$	6.28797	0.735952	0.367976	−	0.929835i	$-0.380051\pi$
$73$	6.28797	0.735952	0.367976	+	0.929835i	$0.380051\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.0185	−1.59755
$78$	0	0
$79$	−5.63362	−0.633832	−0.316916	−	0.948454i	$-0.602647\pi$
$79$	−5.63362	−0.633832	−0.316916	+	0.948454i	$0.602647\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.89917	0.427989	0.213995	−	0.976835i	$-0.431352\pi$
$83$	3.89917	0.427989	0.213995	+	0.976835i	$0.431352\pi$
$84$	0	0
$85$	3.52884	0.382756
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	0	0
$91$	20.0185	2.09851
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.67282	0.479422
$96$	0	0
$97$	−3.83528	−0.389414	−0.194707	−	0.980861i	$-0.562376\pi$
$97$	−3.83528	−0.389414	−0.194707	+	0.980861i	$0.562376\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.4896	−1.04376	−0.521879	−	0.853020i	$-0.674769\pi$
$101$	−10.4896	−1.04376	−0.521879	+	0.853020i	$0.674769\pi$
$102$	0	0
$103$	−4.10478	−0.404456	−0.202228	−	0.979338i	$-0.564818\pi$
$103$	−4.10478	−0.404456	−0.202228	+	0.979338i	$0.564818\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.7345	1.90780	0.953901	−	0.300122i	$-0.0970275\pi$
$107$	19.7345	1.90780	0.953901	+	0.300122i	$0.0970275\pi$
$108$	0	0
$109$	−9.50811	−0.910711	−0.455356	−	0.890310i	$-0.650488\pi$
$109$	−9.50811	−0.910711	−0.455356	+	0.890310i	$0.650488\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.43196	−0.887284	−0.443642	−	0.896204i	$-0.646314\pi$
$113$	−9.43196	−0.887284	−0.443642	+	0.896204i	$0.646314\pi$
$114$	0	0
$115$	4.95684	0.462228
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18.5081	1.69664
$120$	0	0
$121$	−3.85601	−0.350547
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.6089	−1.65128	−0.825638	−	0.564200i	$-0.809185\pi$
$127$	−18.6089	−1.65128	−0.825638	+	0.564200i	$0.809185\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.3456	1.34076	0.670378	−	0.742020i	$-0.266132\pi$
$131$	15.3456	1.34076	0.670378	+	0.742020i	$0.266132\pi$
$132$	0	0
$133$	24.5081	2.12512
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.489634	0.0418323	0.0209161	−	0.999781i	$-0.493342\pi$
$137$	0.489634	0.0418323	0.0209161	+	0.999781i	$0.493342\pi$
$138$	0	0
$139$	10.6913	0.906824	0.453412	−	0.891301i	$-0.350207\pi$
$139$	10.6913	0.906824	0.453412	+	0.891301i	$0.350207\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−10.2017	−0.853106
$144$	0	0
$145$	−1.85601	−0.154134
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.87448	0.317410	0.158705	−	0.987326i	$-0.449268\pi$
$149$	3.87448	0.317410	0.158705	+	0.987326i	$0.449268\pi$
$150$	0	0
$151$	14.6728	1.19406	0.597029	−	0.802220i	$-0.296348\pi$
$151$	14.6728	1.19406	0.597029	+	0.802220i	$0.296348\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.67282	0.696618
$156$	0	0
$157$	−13.1809	−1.05195	−0.525976	−	0.850499i	$-0.676300\pi$
$157$	−13.1809	−1.05195	−0.525976	+	0.850499i	$0.676300\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	25.9977	2.04891
$162$	0	0
$163$	−20.8745	−1.63502	−0.817508	−	0.575917i	$-0.804645\pi$
$163$	−20.8745	−1.63502	−0.817508	+	0.575917i	$0.804645\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.00395288	−0.000305883 0	−0.000152941	−	1.00000i	$-0.500049\pi$
$167$	−0.00395288	−0.000305883 0	−0.000152941		1.00000i	$0.500049\pi$
$168$	0	0
$169$	1.56804	0.120618
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.6129	−1.56717	−0.783584	−	0.621285i	$-0.786611\pi$
$173$	−20.6129	−1.56717	−0.783584	+	0.621285i	$0.786611\pi$
$174$	0	0
$175$	5.24482	0.396471
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.10478	−0.306806	−0.153403	−	0.988164i	$-0.549023\pi$
$179$	−4.10478	−0.306806	−0.153403	+	0.988164i	$0.549023\pi$
$180$	0	0
$181$	−4.43196	−0.329425	−0.164712	−	0.986342i	$-0.552670\pi$
$181$	−4.43196	−0.329425	−0.164712	+	0.986342i	$0.552670\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.67282	−0.196510
$186$	0	0
$187$	−9.43196	−0.689733
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.1625	1.24183	0.620916	−	0.783877i	$-0.286761\pi$
$191$	17.1625	1.24183	0.620916	+	0.783877i	$0.286761\pi$
$192$	0	0
$193$	5.89522	0.424347	0.212173	−	0.977232i	$-0.431946\pi$
$193$	5.89522	0.424347	0.212173	+	0.977232i	$0.431946\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.8353	−1.55570	−0.777850	−	0.628450i	$-0.783690\pi$
$197$	−21.8353	−1.55570	−0.777850	+	0.628450i	$0.783690\pi$
$198$	0	0
$199$	13.2488	0.939180	0.469590	−	0.882885i	$-0.344402\pi$
$199$	13.2488	0.939180	0.469590	+	0.882885i	$0.344402\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.73445	−0.683225
$204$	0	0
$205$	−3.67282	−0.256521
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−12.4896	−0.863926
$210$	0	0
$211$	−0.384851	−0.0264942	−0.0132471	−	0.999912i	$-0.504217\pi$
$211$	−0.384851	−0.0264942	−0.0132471	+	0.999912i	$0.504217\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.52884	−0.240665
$216$	0	0
$217$	45.4874	3.08788
$218$	0	0
$219$	0	0
$220$	0	0
$221$	13.4689	0.906016
$222$	0	0
$223$	11.4280	0.765276	0.382638	−	0.923898i	$-0.375016\pi$
$223$	11.4280	0.765276	0.382638	+	0.923898i	$0.375016\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.9401	−1.19072	−0.595362	−	0.803458i	$-0.702991\pi$
$227$	−17.9401	−1.19072	−0.595362	+	0.803458i	$0.702991\pi$
$228$	0	0
$229$	−4.91369	−0.324706	−0.162353	−	0.986733i	$-0.551908\pi$
$229$	−4.91369	−0.324706	−0.162353	+	0.986733i	$0.551908\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.59442	0.300990	0.150495	−	0.988611i	$-0.451913\pi$
$233$	4.59442	0.300990	0.150495	+	0.988611i	$0.451913\pi$
$234$	0	0
$235$	−9.26329	−0.604270
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.2386	−1.69723	−0.848617	−	0.529008i	$-0.822564\pi$
$239$	−26.2386	−1.69723	−0.848617	+	0.529008i	$0.822564\pi$
$240$	0	0
$241$	−0.895217	−0.0576660	−0.0288330	−	0.999584i	$-0.509179\pi$
$241$	−0.895217	−0.0576660	−0.0288330	+	0.999584i	$0.509179\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.5081	1.31021
$246$	0	0
$247$	17.8353	1.13483
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.10478	−0.132853	−0.0664264	−	0.997791i	$-0.521160\pi$
$251$	−2.10478	−0.132853	−0.0664264	+	0.997791i	$0.521160\pi$
$252$	0	0
$253$	−13.2488	−0.832943
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.69129	−0.292635	−0.146317	−	0.989238i	$-0.546742\pi$
$257$	−4.69129	−0.292635	−0.146317	+	0.989238i	$0.546742\pi$
$258$	0	0
$259$	−14.0185	−0.871065
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.81681	−0.605330	−0.302665	−	0.953097i	$-0.597876\pi$
$263$	−9.81681	−0.605330	−0.302665	+	0.953097i	$0.597876\pi$
$264$	0	0
$265$	−2.85601	−0.175444
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.4504	1.24688	0.623442	−	0.781869i	$-0.285734\pi$
$269$	20.4504	1.24688	0.623442	+	0.781869i	$0.285734\pi$
$270$	0	0
$271$	5.03920	0.306110	0.153055	−	0.988218i	$-0.451089\pi$
$271$	5.03920	0.306110	0.153055	+	0.988218i	$0.451089\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.67282	−0.161177
$276$	0	0
$277$	1.04711	0.0629147	0.0314573	−	0.999505i	$-0.489985\pi$
$277$	1.04711	0.0629147	0.0314573	+	0.999505i	$0.489985\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.75914	0.104941	0.0524706	−	0.998622i	$-0.483290\pi$
$281$	1.75914	0.104941	0.0524706	+	0.998622i	$0.483290\pi$
$282$	0	0
$283$	−2.87844	−0.171105	−0.0855527	−	0.996334i	$-0.527266\pi$
$283$	−2.87844	−0.171105	−0.0855527	+	0.996334i	$0.527266\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.2633	−1.13708
$288$	0	0
$289$	−4.54731	−0.267489
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.9114	1.45534	0.727671	−	0.685927i	$-0.240603\pi$
$293$	24.9114	1.45534	0.727671	+	0.685927i	$0.240603\pi$
$294$	0	0
$295$	−4.20166	−0.244630
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.9193	1.09413
$300$	0	0
$301$	−18.5081	−1.06679
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−7.96080	−0.455834
$306$	0	0
$307$	1.32322	0.0755203	0.0377602	−	0.999287i	$-0.487978\pi$
$307$	1.32322	0.0755203	0.0377602	+	0.999287i	$0.487978\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.95289	−0.280853	−0.140426	−	0.990091i	$-0.544847\pi$
$311$	−4.95289	−0.280853	−0.140426	+	0.990091i	$0.544847\pi$
$312$	0	0
$313$	9.61515	0.543480	0.271740	−	0.962371i	$-0.412401\pi$
$313$	9.61515	0.543480	0.271740	+	0.962371i	$0.412401\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.3064	−0.691199	−0.345599	−	0.938382i	$-0.612324\pi$
$317$	−12.3064	−0.691199	−0.345599	+	0.938382i	$0.612324\pi$
$318$	0	0
$319$	4.96080	0.277751
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.4896	0.917508
$324$	0	0
$325$	3.81681	0.211719
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−48.5843	−2.67854
$330$	0	0
$331$	−1.54731	−0.0850478	−0.0425239	−	0.999095i	$-0.513540\pi$
$331$	−1.54731	−0.0850478	−0.0425239	+	0.999095i	$0.513540\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.859966	0.0469850
$336$	0	0
$337$	32.3170	1.76042	0.880210	−	0.474585i	$-0.157402\pi$
$337$	32.3170	1.76042	0.880210	+	0.474585i	$0.157402\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−23.1809	−1.25532
$342$	0	0
$343$	70.8475	3.82541
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.0841	1.34658	0.673291	−	0.739377i	$-0.264880\pi$
$347$	25.0841	1.34658	0.673291	+	0.739377i	$0.264880\pi$
$348$	0	0
$349$	20.7490	1.11067	0.555333	−	0.831628i	$-0.312591\pi$
$349$	20.7490	1.11067	0.555333	+	0.831628i	$0.312591\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.1153	−0.964183	−0.482091	−	0.876121i	$-0.660123\pi$
$353$	−18.1153	−0.964183	−0.482091	+	0.876121i	$0.660123\pi$
$354$	0	0
$355$	15.1625	0.804740
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.3720	−1.33908	−0.669542	−	0.742774i	$-0.733510\pi$
$359$	−25.3720	−1.33908	−0.669542	+	0.742774i	$0.733510\pi$
$360$	0	0
$361$	2.83528	0.149225
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.28797	0.329128
$366$	0	0
$367$	−6.18319	−0.322760	−0.161380	−	0.986892i	$-0.551594\pi$
$367$	−6.18319	−0.322760	−0.161380	+	0.986892i	$0.551594\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.9793	−0.777685
$372$	0	0
$373$	−16.8145	−0.870624	−0.435312	−	0.900280i	$-0.643362\pi$
$373$	−16.8145	−0.870624	−0.435312	+	0.900280i	$0.643362\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.08405	−0.364847
$378$	0	0
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.79608	0.347263	0.173632	−	0.984811i	$-0.444450\pi$
$383$	6.79608	0.347263	0.173632	+	0.984811i	$0.444450\pi$
$384$	0	0
$385$	−14.0185	−0.714447
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8.61515	−0.436805	−0.218403	−	0.975859i	$-0.570085\pi$
$389$	−8.61515	−0.436805	−0.218403	+	0.975859i	$0.570085\pi$
$390$	0	0
$391$	17.4919	0.884603
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.63362	−0.283458
$396$	0	0
$397$	22.4033	1.12439	0.562195	−	0.827005i	$-0.309957\pi$
$397$	22.4033	1.12439	0.562195	+	0.827005i	$0.309957\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.9137	0.694816	0.347408	−	0.937714i	$-0.387062\pi$
$401$	13.9137	0.694816	0.347408	+	0.937714i	$0.387062\pi$
$402$	0	0
$403$	33.1025	1.64895
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.14399	0.354114
$408$	0	0
$409$	30.6129	1.51371	0.756855	−	0.653583i	$-0.226735\pi$
$409$	30.6129	1.51371	0.756855	+	0.653583i	$0.226735\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−22.0369	−1.08437
$414$	0	0
$415$	3.89917	0.191403
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.7305	0.670779	0.335389	−	0.942080i	$-0.391132\pi$
$419$	13.7305	0.670779	0.335389	+	0.942080i	$0.391132\pi$
$420$	0	0
$421$	15.9216	0.775971	0.387985	−	0.921665i	$-0.373171\pi$
$421$	15.9216	0.775971	0.387985	+	0.921665i	$0.373171\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.52884	0.171174
$426$	0	0
$427$	−41.7529	−2.02057
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.2672	1.21708	0.608540	−	0.793523i	$-0.291755\pi$
$431$	25.2672	1.21708	0.608540	+	0.793523i	$0.291755\pi$
$432$	0	0
$433$	9.24086	0.444088	0.222044	−	0.975037i	$-0.428727\pi$
$433$	9.24086	0.444088	0.222044	+	0.975037i	$0.428727\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.1625	1.10801
$438$	0	0
$439$	17.4320	0.831982	0.415991	−	0.909369i	$-0.363435\pi$
$439$	17.4320	0.831982	0.415991	+	0.909369i	$0.363435\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.62967	0.172451	0.0862254	−	0.996276i	$-0.472519\pi$
$443$	3.62967	0.172451	0.0862254	+	0.996276i	$0.472519\pi$
$444$	0	0
$445$	−11.0000	−0.521450
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−39.6785	−1.87254	−0.936271	−	0.351278i	$-0.885747\pi$
$449$	−39.6785	−1.87254	−0.936271	+	0.351278i	$0.885747\pi$
$450$	0	0
$451$	9.81681	0.462256
$452$	0	0
$453$	0	0
$454$	0	0
$455$	20.0185	0.938480
$456$	0	0
$457$	5.69356	0.266333	0.133167	−	0.991094i	$-0.457485\pi$
$457$	5.69356	0.266333	0.133167	+	0.991094i	$0.457485\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.3170	0.806534	0.403267	−	0.915082i	$-0.367875\pi$
$461$	17.3170	0.806534	0.403267	+	0.915082i	$0.367875\pi$
$462$	0	0
$463$	−25.7674	−1.19751	−0.598757	−	0.800931i	$-0.704339\pi$
$463$	−25.7674	−1.19751	−0.598757	+	0.800931i	$0.704339\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.04711	0.0484544	0.0242272	−	0.999706i	$-0.492287\pi$
$467$	1.04711	0.0484544	0.0242272	+	0.999706i	$0.492287\pi$
$468$	0	0
$469$	4.51037	0.208269
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.43196	0.433682
$474$	0	0
$475$	4.67282	0.214404
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−22.5759	−1.03152	−0.515761	−	0.856733i	$-0.672491\pi$
$479$	−22.5759	−1.03152	−0.515761	+	0.856733i	$0.672491\pi$
$480$	0	0
$481$	−10.2017	−0.465156
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.83528	−0.174151
$486$	0	0
$487$	−0.923855	−0.0418638	−0.0209319	−	0.999781i	$-0.506663\pi$
$487$	−0.923855	−0.0418638	−0.0209319	+	0.999781i	$0.506663\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.08631	−0.364930	−0.182465	−	0.983212i	$-0.558408\pi$
$491$	−8.08631	−0.364930	−0.182465	+	0.983212i	$0.558408\pi$
$492$	0	0
$493$	−6.54957	−0.294978
$494$	0	0
$495$	0	0
$496$	0	0
$497$	79.5243	3.56715
$498$	0	0
$499$	12.6834	0.567786	0.283893	−	0.958856i	$-0.408374\pi$
$499$	12.6834	0.567786	0.283893	+	0.958856i	$0.408374\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.93837	−0.220191	−0.110096	−	0.993921i	$-0.535116\pi$
$503$	−4.93837	−0.220191	−0.110096	+	0.993921i	$0.535116\pi$
$504$	0	0
$505$	−10.4896	−0.466783
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.2096	0.674152	0.337076	−	0.941477i	$-0.390562\pi$
$509$	15.2096	0.674152	0.337076	+	0.941477i	$0.390562\pi$
$510$	0	0
$511$	32.9793	1.45892
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.10478	−0.180878
$516$	0	0
$517$	24.7591	1.08891
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.1809	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$521$	22.1809	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$522$	0	0
$523$	9.46495	0.413873	0.206937	−	0.978354i	$-0.433651\pi$
$523$	9.46495	0.413873	0.206937	+	0.978354i	$0.433651\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	30.6050	1.33317
$528$	0	0
$529$	1.57030	0.0682740
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14.0185	−0.607207
$534$	0	0
$535$	19.7345	0.853195
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−54.8145	−2.36103
$540$	0	0
$541$	−2.42405	−0.104218	−0.0521091	−	0.998641i	$-0.516594\pi$
$541$	−2.42405	−0.104218	−0.0521091	+	0.998641i	$0.516594\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.50811	−0.407282
$546$	0	0
$547$	15.2818	0.653401	0.326700	−	0.945128i	$-0.394063\pi$
$547$	15.2818	0.653401	0.326700	+	0.945128i	$0.394063\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.67282	−0.369475
$552$	0	0
$553$	−29.5473	−1.25648
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.9793	1.65160	0.825802	−	0.563960i	$-0.190723\pi$
$557$	38.9793	1.65160	0.825802	+	0.563960i	$0.190723\pi$
$558$	0	0
$559$	−13.4689	−0.569674
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.60894	0.194243	0.0971217	−	0.995273i	$-0.469036\pi$
$563$	4.60894	0.194243	0.0971217	+	0.995273i	$0.469036\pi$
$564$	0	0
$565$	−9.43196	−0.396806
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.3826	0.477183	0.238591	−	0.971120i	$-0.423314\pi$
$569$	11.3826	0.477183	0.238591	+	0.971120i	$0.423314\pi$
$570$	0	0
$571$	−45.2258	−1.89264	−0.946320	−	0.323231i	$-0.895231\pi$
$571$	−45.2258	−1.89264	−0.946320	+	0.323231i	$0.895231\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.95684	0.206715
$576$	0	0
$577$	5.14399	0.214147	0.107073	−	0.994251i	$-0.465852\pi$
$577$	5.14399	0.214147	0.107073	+	0.994251i	$0.465852\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	20.4504	0.848427
$582$	0	0
$583$	7.63362	0.316152
$584$	0	0
$585$	0	0
$586$	0	0
$587$	43.2818	1.78643	0.893215	−	0.449630i	$-0.148444\pi$
$587$	43.2818	1.78643	0.893215	+	0.449630i	$0.148444\pi$
$588$	0	0
$589$	40.5266	1.66987
$590$	0	0
$591$	0	0
$592$	0	0
$593$	33.3641	1.37010	0.685050	−	0.728496i	$-0.259780\pi$
$593$	33.3641	1.37010	0.685050	+	0.728496i	$0.259780\pi$
$594$	0	0
$595$	18.5081	0.758758
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−40.1153	−1.63907	−0.819534	−	0.573030i	$-0.805768\pi$
$599$	−40.1153	−1.63907	−0.819534	+	0.573030i	$0.805768\pi$
$600$	0	0
$601$	−5.96306	−0.243238	−0.121619	−	0.992577i	$-0.538809\pi$
$601$	−5.96306	−0.243238	−0.121619	+	0.992577i	$0.538809\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.85601	−0.156769
$606$	0	0
$607$	15.1400	0.614515	0.307258	−	0.951626i	$-0.400589\pi$
$607$	15.1400	0.614515	0.307258	+	0.951626i	$0.400589\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−35.3562	−1.43036
$612$	0	0
$613$	28.5081	1.15143	0.575716	−	0.817650i	$-0.304723\pi$
$613$	28.5081	1.15143	0.575716	+	0.817650i	$0.304723\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.4320	0.701784	0.350892	−	0.936416i	$-0.385878\pi$
$617$	17.4320	0.701784	0.350892	+	0.936416i	$0.385878\pi$
$618$	0	0
$619$	−29.6521	−1.19182	−0.595909	−	0.803052i	$-0.703208\pi$
$619$	−29.6521	−1.19182	−0.595909	+	0.803052i	$0.703208\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−57.6930	−2.31142
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.43196	−0.376077
$630$	0	0
$631$	40.1233	1.59728	0.798641	−	0.601808i	$-0.205553\pi$
$631$	40.1233	1.59728	0.798641	+	0.601808i	$0.205553\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18.6089	−0.738473
$636$	0	0
$637$	78.2755	3.10139
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.41349	0.174322	0.0871612	−	0.996194i	$-0.472220\pi$
$641$	4.41349	0.174322	0.0871612	+	0.996194i	$0.472220\pi$
$642$	0	0
$643$	−19.8313	−0.782071	−0.391036	−	0.920376i	$-0.627883\pi$
$643$	−19.8313	−0.782071	−0.391036	+	0.920376i	$0.627883\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.6873	1.67821	0.839106	−	0.543967i	$-0.183079\pi$
$647$	42.6873	1.67821	0.839106	+	0.543967i	$0.183079\pi$
$648$	0	0
$649$	11.2303	0.440828
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.3641	−1.46217	−0.731085	−	0.682286i	$-0.760986\pi$
$653$	−37.3641	−1.46217	−0.731085	+	0.682286i	$0.760986\pi$
$654$	0	0
$655$	15.3456	0.599604
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.43196	0.211599	0.105800	−	0.994387i	$-0.466260\pi$
$659$	5.43196	0.211599	0.105800	+	0.994387i	$0.466260\pi$
$660$	0	0
$661$	17.0946	0.664904	0.332452	−	0.943120i	$-0.392124\pi$
$661$	17.0946	0.664904	0.332452	+	0.943120i	$0.392124\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	24.5081	0.950384
$666$	0	0
$667$	−9.19997	−0.356224
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21.2778	0.821421
$672$	0	0
$673$	−39.4011	−1.51880	−0.759400	−	0.650624i	$-0.774507\pi$
$673$	−39.4011	−1.51880	−0.759400	+	0.650624i	$0.774507\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.2695	0.471555	0.235778	−	0.971807i	$-0.424236\pi$
$677$	12.2695	0.471555	0.235778	+	0.971807i	$0.424236\pi$
$678$	0	0
$679$	−20.1153	−0.771956
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.8952	−0.455158	−0.227579	−	0.973760i	$-0.573081\pi$
$683$	−11.8952	−0.455158	−0.227579	+	0.973760i	$0.573081\pi$
$684$	0	0
$685$	0.489634	0.0187080
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.9009	−0.415290
$690$	0	0
$691$	5.26724	0.200375	0.100188	−	0.994969i	$-0.468056\pi$
$691$	5.26724	0.200375	0.100188	+	0.994969i	$0.468056\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.6913	0.405544
$696$	0	0
$697$	−12.9608	−0.490925
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22.1131	−0.835200	−0.417600	−	0.908631i	$-0.637129\pi$
$701$	−22.1131	−0.835200	−0.417600	+	0.908631i	$0.637129\pi$
$702$	0	0
$703$	−12.4896	−0.471055
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−55.0162	−2.06910
$708$	0	0
$709$	17.4112	0.653892	0.326946	−	0.945043i	$-0.393980\pi$
$709$	17.4112	0.653892	0.326946	+	0.945043i	$0.393980\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	42.9898	1.60998
$714$	0	0
$715$	−10.2017	−0.381520
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19.4689	0.726068	0.363034	−	0.931776i	$-0.381741\pi$
$719$	19.4689	0.726068	0.363034	+	0.931776i	$0.381741\pi$
$720$	0	0
$721$	−21.5288	−0.801776
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.85601	−0.0689306
$726$	0	0
$727$	23.9255	0.887349	0.443675	−	0.896188i	$-0.353675\pi$
$727$	23.9255	0.887349	0.443675	+	0.896188i	$0.353675\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.4527	−0.460579
$732$	0	0
$733$	−4.11535	−0.152004	−0.0760019	−	0.997108i	$-0.524216\pi$
$733$	−4.11535	−0.152004	−0.0760019	+	0.997108i	$0.524216\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.29854	−0.0846678
$738$	0	0
$739$	−16.8145	−0.618533	−0.309267	−	0.950975i	$-0.600084\pi$
$739$	−16.8145	−0.618533	−0.309267	+	0.950975i	$0.600084\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−27.3602	−1.00375	−0.501874	−	0.864941i	$-0.667356\pi$
$743$	−27.3602	−1.00375	−0.501874	+	0.864941i	$0.667356\pi$
$744$	0	0
$745$	3.87448	0.141950
$746$	0	0
$747$	0	0
$748$	0	0
$749$	103.504	3.78194
$750$	0	0
$751$	−4.47116	−0.163155	−0.0815775	−	0.996667i	$-0.525996\pi$
$751$	−4.47116	−0.163155	−0.0815775	+	0.996667i	$0.525996\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.6728	0.533999
$756$	0	0
$757$	−17.7753	−0.646056	−0.323028	−	0.946389i	$-0.604701\pi$
$757$	−17.7753	−0.646056	−0.323028	+	0.946389i	$0.604701\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−40.3905	−1.46415	−0.732077	−	0.681222i	$-0.761449\pi$
$761$	−40.3905	−1.46415	−0.732077	+	0.681222i	$0.761449\pi$
$762$	0	0
$763$	−49.8683	−1.80535
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.0369	−0.579060
$768$	0	0
$769$	−39.6992	−1.43159	−0.715795	−	0.698311i	$-0.753935\pi$
$769$	−39.6992	−1.43159	−0.715795	+	0.698311i	$0.753935\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.97927	0.322962	0.161481	−	0.986876i	$-0.448373\pi$
$773$	8.97927	0.322962	0.161481	+	0.986876i	$0.448373\pi$
$774$	0	0
$775$	8.67282	0.311537
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17.1625	−0.614909
$780$	0	0
$781$	−40.5266	−1.45015
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.1809	−0.470448
$786$	0	0
$787$	−35.8952	−1.27953	−0.639763	−	0.768572i	$-0.720968\pi$
$787$	−35.8952	−1.27953	−0.639763	+	0.768572i	$0.720968\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−49.4689	−1.75891
$792$	0	0
$793$	−30.3849	−1.07900
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.8458	1.34057	0.670284	−	0.742104i	$-0.266172\pi$
$797$	37.8458	1.34057	0.670284	+	0.742104i	$0.266172\pi$
$798$	0	0
$799$	−32.6886	−1.15644
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16.8066	−0.593094
$804$	0	0
$805$	25.9977	0.916300
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−27.8722	−0.979935	−0.489968	−	0.871741i	$-0.662991\pi$
$809$	−27.8722	−0.979935	−0.489968	+	0.871741i	$0.662991\pi$
$810$	0	0
$811$	−48.3249	−1.69692	−0.848459	−	0.529262i	$-0.822469\pi$
$811$	−48.3249	−1.69692	−0.848459	+	0.529262i	$0.822469\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.8745	−0.731201
$816$	0	0
$817$	−16.4896	−0.576899
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.7098	−1.03688	−0.518439	−	0.855115i	$-0.673487\pi$
$821$	−29.7098	−1.03688	−0.518439	+	0.855115i	$0.673487\pi$
$822$	0	0
$823$	30.2056	1.05290	0.526451	−	0.850206i	$-0.323522\pi$
$823$	30.2056	1.05290	0.526451	+	0.850206i	$0.323522\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.14003	0.0396429	0.0198214	−	0.999804i	$-0.493690\pi$
$827$	1.14003	0.0396429	0.0198214	+	0.999804i	$0.493690\pi$
$828$	0	0
$829$	−17.9608	−0.623804	−0.311902	−	0.950114i	$-0.600966\pi$
$829$	−17.9608	−0.623804	−0.311902	+	0.950114i	$0.600966\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	72.3698	2.50746
$834$	0	0
$835$	−0.00395288	−0.000136795 0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.21183	−0.0418369	−0.0209185	−	0.999781i	$-0.506659\pi$
$839$	−1.21183	−0.0418369	−0.0209185	+	0.999781i	$0.506659\pi$
$840$	0	0
$841$	−25.5552	−0.881214
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.56804	0.0539422
$846$	0	0
$847$	−20.2241	−0.694908
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13.2488	−0.454162
$852$	0	0
$853$	35.3641	1.21084	0.605422	−	0.795905i	$-0.293004\pi$
$853$	35.3641	1.21084	0.605422	+	0.795905i	$0.293004\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.8123	−1.42828	−0.714140	−	0.700003i	$-0.753182\pi$
$857$	−41.8123	−1.42828	−0.714140	+	0.700003i	$0.753182\pi$
$858$	0	0
$859$	11.7753	0.401770	0.200885	−	0.979615i	$-0.435618\pi$
$859$	11.7753	0.401770	0.200885	+	0.979615i	$0.435618\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.1193	0.889111	0.444556	−	0.895751i	$-0.353362\pi$
$863$	26.1193	0.889111	0.444556	+	0.895751i	$0.353362\pi$
$864$	0	0
$865$	−20.6129	−0.700859
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.0577	0.510797
$870$	0	0
$871$	3.28233	0.111217
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.24482	0.177307
$876$	0	0
$877$	−52.6235	−1.77697	−0.888484	−	0.458908i	$-0.848241\pi$
$877$	−52.6235	−1.77697	−0.888484	+	0.458908i	$0.848241\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.8089	−0.431543	−0.215771	−	0.976444i	$-0.569227\pi$
$881$	−12.8089	−0.431543	−0.215771	+	0.976444i	$0.569227\pi$
$882$	0	0
$883$	−21.7529	−0.732044	−0.366022	−	0.930606i	$-0.619281\pi$
$883$	−21.7529	−0.732044	−0.366022	+	0.930606i	$0.619281\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.0471	1.10961	0.554807	−	0.831979i	$-0.312792\pi$
$887$	33.0471	1.10961	0.554807	+	0.831979i	$0.312792\pi$
$888$	0	0
$889$	−97.6005	−3.27341
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−43.2857	−1.44850
$894$	0	0
$895$	−4.10478	−0.137208
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.0969	−0.536861
$900$	0	0
$901$	−10.0784	−0.335760
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.43196	−0.147323
$906$	0	0
$907$	−49.1849	−1.63316	−0.816579	−	0.577234i	$-0.804132\pi$
$907$	−49.1849	−1.63316	−0.816579	+	0.577234i	$0.804132\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.45269	0.213787	0.106894	−	0.994270i	$-0.465910\pi$
$911$	6.45269	0.213787	0.106894	+	0.994270i	$0.465910\pi$
$912$	0	0
$913$	−10.4218	−0.344911
$914$	0	0
$915$	0	0
$916$	0	0
$917$	80.4851	2.65785
$918$	0	0
$919$	19.5552	0.645067	0.322533	−	0.946558i	$-0.395466\pi$
$919$	19.5552	0.645067	0.322533	+	0.946558i	$0.395466\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	57.8722	1.90489
$924$	0	0
$925$	−2.67282	−0.0878819
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−51.1730	−1.67893	−0.839466	−	0.543412i	$-0.817132\pi$
$929$	−51.1730	−1.67893	−0.839466	+	0.543412i	$0.817132\pi$
$930$	0	0
$931$	95.8308	3.14073
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.43196	−0.308458
$936$	0	0
$937$	−37.6627	−1.23039	−0.615193	−	0.788377i	$-0.710922\pi$
$937$	−37.6627	−1.23039	−0.615193	+	0.788377i	$0.710922\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25.8145	−0.841530	−0.420765	−	0.907170i	$-0.638238\pi$
$941$	−25.8145	−0.841530	−0.420765	+	0.907170i	$0.638238\pi$
$942$	0	0
$943$	−18.2056	−0.592856
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.83923	0.0597671	0.0298835	−	0.999553i	$-0.490486\pi$
$947$	1.83923	0.0597671	0.0298835	+	0.999553i	$0.490486\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	39.7674	1.28819	0.644097	−	0.764944i	$-0.277233\pi$
$953$	39.7674	1.28819	0.644097	+	0.764944i	$0.277233\pi$
$954$	0	0
$955$	17.1625	0.555364
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.56804	0.0829263
$960$	0	0
$961$	44.2179	1.42638
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.89522	0.189774
$966$	0	0
$967$	−9.70541	−0.312105	−0.156053	−	0.987749i	$-0.549877\pi$
$967$	−9.70541	−0.312105	−0.156053	+	0.987749i	$0.549877\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−13.8247	−0.443656	−0.221828	−	0.975086i	$-0.571202\pi$
$971$	−13.8247	−0.443656	−0.221828	+	0.975086i	$0.571202\pi$
$972$	0	0
$973$	56.0739	1.79765
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35.9770	1.15101	0.575503	−	0.817800i	$-0.304806\pi$
$977$	35.9770	1.15101	0.575503	+	0.817800i	$0.304806\pi$
$978$	0	0
$979$	29.4011	0.939662
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.5490	0.719201	0.359601	−	0.933106i	$-0.382913\pi$
$983$	22.5490	0.719201	0.359601	+	0.933106i	$0.382913\pi$
$984$	0	0
$985$	−21.8353	−0.695730
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−17.4919	−0.556210
$990$	0	0
$991$	13.8767	0.440809	0.220405	−	0.975409i	$-0.429262\pi$
$991$	13.8767	0.440809	0.220405	+	0.975409i	$0.429262\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13.2488	0.420014
$996$	0	0
$997$	35.5058	1.12448	0.562241	−	0.826974i	$-0.309939\pi$
$997$	35.5058	1.12448	0.562241	+	0.826974i	$0.309939\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.bx.1.3		3
3.2	odd	2		6480.2.a.bu.1.3		3
4.3	odd	2		3240.2.a.r.1.1		3
9.2	odd	6		2160.2.q.j.1441.1		6
9.4	even	3		720.2.q.j.241.1		6
9.5	odd	6		2160.2.q.j.721.1		6
9.7	even	3		720.2.q.j.481.1		6
12.11	even	2		3240.2.a.q.1.1		3
36.7	odd	6		360.2.q.d.121.3	✓	6
36.11	even	6		1080.2.q.d.361.3		6
36.23	even	6		1080.2.q.d.721.3		6
36.31	odd	6		360.2.q.d.241.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.d.121.3	✓	6	36.7	odd	6
360.2.q.d.241.3	yes	6	36.31	odd	6
720.2.q.j.241.1		6	9.4	even	3
720.2.q.j.481.1		6	9.7	even	3
1080.2.q.d.361.3		6	36.11	even	6
1080.2.q.d.721.3		6	36.23	even	6
2160.2.q.j.721.1		6	9.5	odd	6
2160.2.q.j.1441.1		6	9.2	odd	6
3240.2.a.q.1.1		3	12.11	even	2
3240.2.a.r.1.1		3	4.3	odd	2
6480.2.a.bu.1.3		3	3.2	odd	2
6480.2.a.bx.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +5.24482 q^{7} -2.67282 q^{11} +3.81681 q^{13} +3.52884 q^{17} +4.67282 q^{19} +4.95684 q^{23} +1.00000 q^{25} -1.85601 q^{29} +8.67282 q^{31} +5.24482 q^{35} -2.67282 q^{37} -3.67282 q^{41} -3.52884 q^{43} -9.26329 q^{47} +20.5081 q^{49} -2.85601 q^{53} -2.67282 q^{55} -4.20166 q^{59} -7.96080 q^{61} +3.81681 q^{65} +0.859966 q^{67} +15.1625 q^{71} +6.28797 q^{73} -14.0185 q^{77} -5.63362 q^{79} +3.89917 q^{83} +3.52884 q^{85} -11.0000 q^{89} +20.0185 q^{91} +4.67282 q^{95} -3.83528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 5 q^{7} + 2 q^{11} + 2 q^{17} + 4 q^{19} + 7 q^{23} + 3 q^{25} - 7 q^{29} + 16 q^{31} + 5 q^{35} + 2 q^{37} - q^{41} - 2 q^{43} + 13 q^{47} + 10 q^{49} - 10 q^{53} + 2 q^{55} + 6 q^{59}+ \cdots + 30 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 + 5 * q^7 + 2 * q^11 + 2 * q^17 + 4 * q^19 + 7 * q^23 + 3 * q^25 - 7 * q^29 + 16 * q^31 + 5 * q^35 + 2 * q^37 - q^41 - 2 * q^43 + 13 * q^47 + 10 * q^49 - 10 * q^53 + 2 * q^55 + 6 * q^59 - 11 * q^61 - q^67 + 14 * q^71 + 16 * q^73 - 12 * q^77 + 6 * q^79 + 21 * q^83 + 2 * q^85 - 33 * q^89 + 30 * q^91 + 4 * q^95 + 30 * q^97

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