LMFDB - Embedded newform 6480.2.a.by.1.1

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:	4.4.62352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 11x^{2} + 12x + 24 $ x^4 - 2x^3 - 11x^2 + 12*x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3240)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.16910$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -3.90115 q^{7} -6.09520 q^{11} -2.02494 q^{13} +2.46199 q^{17} -3.33820 q^{19} +0.830899 q^{23} +1.00000 q^{25} -2.63109 q^{29} -1.83090 q^{31} +3.90115 q^{35} +5.58789 q^{37} -6.65604 q^{41} -11.0520 q^{43} +0.166992 q^{47} +8.21899 q^{49} -12.0234 q^{53} +6.09520 q^{55} -1.73205 q^{59} -13.3163 q^{61} +2.02494 q^{65} -13.2622 q^{67} +10.8829 q^{71} +1.87621 q^{73} +23.7783 q^{77} -7.46410 q^{79} +6.80019 q^{83} -2.46199 q^{85} -12.1493 q^{89} +7.89961 q^{91} +3.33820 q^{95} -0.663907 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 2 q^{7} + 4 q^{11} + 2 q^{17} + 10 q^{23} + 4 q^{25} + 4 q^{29} - 14 q^{31} + 2 q^{35} + 14 q^{37} - 4 q^{41} - 22 q^{43} + 10 q^{49} + 8 q^{53} - 4 q^{55} + 4 q^{61} - 24 q^{67} + 28 q^{71}+ \cdots - 10 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 2 * q^7 + 4 * q^11 + 2 * q^17 + 10 * q^23 + 4 * q^25 + 4 * q^29 - 14 * q^31 + 2 * q^35 + 14 * q^37 - 4 * q^41 - 22 * q^43 + 10 * q^49 + 8 * q^53 - 4 * q^55 + 4 * q^61 - 24 * q^67 + 28 * q^71 + 2 * q^73 + 30 * q^77 - 16 * q^79 + 6 * q^83 - 2 * q^85 + 8 * q^89 - 30 * q^91 - 10 * 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
$7$	−3.90115	−1.47450	−0.737248	−	0.675622i	$-0.763875\pi$
$7$	−3.90115	−1.47450	−0.737248	+	0.675622i	$0.763875\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.09520	−1.83777	−0.918885	−	0.394525i	$-0.870909\pi$
$11$	−6.09520	−1.83777	−0.918885	+	0.394525i	$0.870909\pi$
$12$	0	0
$13$	−2.02494	−0.561618	−0.280809	−	0.959764i	$-0.590603\pi$
$13$	−2.02494	−0.561618	−0.280809	+	0.959764i	$0.590603\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.46199	0.597121	0.298560	−	0.954391i	$-0.403494\pi$
$17$	2.46199	0.597121	0.298560	+	0.954391i	$0.403494\pi$
$18$	0	0
$19$	−3.33820	−0.765836	−0.382918	−	0.923782i	$-0.625081\pi$
$19$	−3.33820	−0.765836	−0.382918	+	0.923782i	$0.625081\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.830899	0.173254	0.0866272	−	0.996241i	$-0.472391\pi$
$23$	0.830899	0.173254	0.0866272	+	0.996241i	$0.472391\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.63109	−0.488582	−0.244291	−	0.969702i	$-0.578555\pi$
$29$	−2.63109	−0.488582	−0.244291	+	0.969702i	$0.578555\pi$
$30$	0	0
$31$	−1.83090	−0.328839	−0.164420	−	0.986390i	$-0.552575\pi$
$31$	−1.83090	−0.328839	−0.164420	+	0.986390i	$0.552575\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.90115	0.659415
$36$	0	0
$37$	5.58789	0.918644	0.459322	−	0.888270i	$-0.348092\pi$
$37$	5.58789	0.918644	0.459322	+	0.888270i	$0.348092\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.65604	−1.03950	−0.519749	−	0.854319i	$-0.673975\pi$
$41$	−6.65604	−1.03950	−0.519749	+	0.854319i	$0.673975\pi$
$42$	0	0
$43$	−11.0520	−1.68541	−0.842707	−	0.538373i	$-0.819039\pi$
$43$	−11.0520	−1.68541	−0.842707	+	0.538373i	$0.819039\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.166992	0.0243583	0.0121792	−	0.999926i	$-0.496123\pi$
$47$	0.166992	0.0243583	0.0121792	+	0.999926i	$0.496123\pi$
$48$	0	0
$49$	8.21899	1.17414
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.0234	−1.65154	−0.825770	−	0.564006i	$-0.809259\pi$
$53$	−12.0234	−1.65154	−0.825770	+	0.564006i	$0.809259\pi$
$54$	0	0
$55$	6.09520	0.821876
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.73205	−0.225494	−0.112747	−	0.993624i	$-0.535965\pi$
$59$	−1.73205	−0.225494	−0.112747	+	0.993624i	$0.535965\pi$
$60$	0	0
$61$	−13.3163	−1.70498	−0.852488	−	0.522747i	$-0.824907\pi$
$61$	−13.3163	−1.70498	−0.852488	+	0.522747i	$0.824907\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.02494	0.251163
$66$	0	0
$67$	−13.2622	−1.62023	−0.810117	−	0.586268i	$-0.800596\pi$
$67$	−13.2622	−1.62023	−0.810117	+	0.586268i	$0.800596\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.8829	1.29156	0.645781	−	0.763523i	$-0.276532\pi$
$71$	10.8829	1.29156	0.645781	+	0.763523i	$0.276532\pi$
$72$	0	0
$73$	1.87621	0.219594	0.109797	−	0.993954i	$-0.464980\pi$
$73$	1.87621	0.219594	0.109797	+	0.993954i	$0.464980\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	23.7783	2.70979
$78$	0	0
$79$	−7.46410	−0.839777	−0.419889	−	0.907576i	$-0.637931\pi$
$79$	−7.46410	−0.839777	−0.419889	+	0.907576i	$0.637931\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.80019	0.746418	0.373209	−	0.927747i	$-0.378257\pi$
$83$	6.80019	0.746418	0.373209	+	0.927747i	$0.378257\pi$
$84$	0	0
$85$	−2.46199	−0.267041
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.1493	−1.28782	−0.643912	−	0.765100i	$-0.722690\pi$
$89$	−12.1493	−1.28782	−0.643912	+	0.765100i	$0.722690\pi$
$90$	0	0
$91$	7.89961	0.828104
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.33820	0.342492
$96$	0	0
$97$	−0.663907	−0.0674095	−0.0337048	−	0.999432i	$-0.510731\pi$
$97$	−0.663907	−0.0674095	−0.0337048	+	0.999432i	$0.510731\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	19.3616	1.92655	0.963276	−	0.268515i	$-0.0865327\pi$
$101$	19.3616	1.92655	0.963276	+	0.268515i	$0.0865327\pi$
$102$	0	0
$103$	13.2913	1.30964	0.654818	−	0.755787i	$-0.272745\pi$
$103$	13.2913	1.30964	0.654818	+	0.755787i	$0.272745\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.39230	0.617967	0.308984	−	0.951067i	$-0.400011\pi$
$107$	6.39230	0.617967	0.308984	+	0.951067i	$0.400011\pi$
$108$	0	0
$109$	7.29500	0.698734	0.349367	−	0.936986i	$-0.386397\pi$
$109$	7.29500	0.698734	0.349367	+	0.936986i	$0.386397\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.24969	0.305705	0.152853	−	0.988249i	$-0.451154\pi$
$113$	3.24969	0.305705	0.152853	+	0.988249i	$0.451154\pi$
$114$	0	0
$115$	−0.830899	−0.0774817
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.60461	−0.880453
$120$	0	0
$121$	26.1514	2.37740
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.8294	−1.31589	−0.657946	−	0.753065i	$-0.728575\pi$
$127$	−14.8294	−1.31589	−0.657946	+	0.753065i	$0.728575\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.0483	−1.14004	−0.570019	−	0.821631i	$-0.693064\pi$
$131$	−13.0483	−1.14004	−0.570019	+	0.821631i	$0.693064\pi$
$132$	0	0
$133$	13.0228	1.12922
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.8023	1.00834	0.504169	−	0.863605i	$-0.331799\pi$
$137$	11.8023	1.00834	0.504169	+	0.863605i	$0.331799\pi$
$138$	0	0
$139$	16.7044	1.41685	0.708423	−	0.705788i	$-0.249407\pi$
$139$	16.7044	1.41685	0.708423	+	0.705788i	$0.249407\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.3424	1.03213
$144$	0	0
$145$	2.63109	0.218500
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.2664	1.74221	0.871106	−	0.491095i	$-0.163403\pi$
$149$	21.2664	1.74221	0.871106	+	0.491095i	$0.163403\pi$
$150$	0	0
$151$	11.6831	0.950756	0.475378	−	0.879782i	$-0.342311\pi$
$151$	11.6831	0.950756	0.475378	+	0.879782i	$0.342311\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.83090	0.147061
$156$	0	0
$157$	−20.0915	−1.60348	−0.801740	−	0.597673i	$-0.796092\pi$
$157$	−20.0915	−1.60348	−0.801740	+	0.597673i	$0.796092\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.24146	−0.255463
$162$	0	0
$163$	−2.99789	−0.234813	−0.117406	−	0.993084i	$-0.537458\pi$
$163$	−2.99789	−0.234813	−0.117406	+	0.993084i	$0.537458\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.5119	0.813434	0.406717	−	0.913554i	$-0.366674\pi$
$167$	10.5119	0.813434	0.406717	+	0.913554i	$0.366674\pi$
$168$	0	0
$169$	−8.89961	−0.684585
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.7591	0.894028	0.447014	−	0.894527i	$-0.352487\pi$
$173$	11.7591	0.894028	0.447014	+	0.894527i	$0.352487\pi$
$174$	0	0
$175$	−3.90115	−0.294899
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.65604	−0.497496	−0.248748	−	0.968568i	$-0.580019\pi$
$179$	−6.65604	−0.497496	−0.248748	+	0.968568i	$0.580019\pi$
$180$	0	0
$181$	8.50730	0.632343	0.316171	−	0.948702i	$-0.397603\pi$
$181$	8.50730	0.632343	0.316171	+	0.948702i	$0.397603\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.58789	−0.410830
$186$	0	0
$187$	−15.0063	−1.09737
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.5052	1.26663	0.633316	−	0.773894i	$-0.281693\pi$
$191$	17.5052	1.26663	0.633316	+	0.773894i	$0.281693\pi$
$192$	0	0
$193$	26.1165	1.87991	0.939953	−	0.341304i	$-0.110869\pi$
$193$	26.1165	1.87991	0.939953	+	0.341304i	$0.110869\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.9215	0.920620	0.460310	−	0.887758i	$-0.347738\pi$
$197$	12.9215	0.920620	0.460310	+	0.887758i	$0.347738\pi$
$198$	0	0
$199$	−16.9781	−1.20354	−0.601772	−	0.798668i	$-0.705539\pi$
$199$	−16.9781	−1.20354	−0.601772	+	0.798668i	$0.705539\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.2643	0.720412
$204$	0	0
$205$	6.65604	0.464878
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.3470	1.40743
$210$	0	0
$211$	−11.4422	−0.787713	−0.393856	−	0.919172i	$-0.628859\pi$
$211$	−11.4422	−0.787713	−0.393856	+	0.919172i	$0.628859\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.0520	0.753740
$216$	0	0
$217$	7.14261	0.484872
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.98539	−0.335354
$222$	0	0
$223$	−15.6545	−1.04830	−0.524151	−	0.851625i	$-0.675617\pi$
$223$	−15.6545	−1.04830	−0.524151	+	0.851625i	$0.675617\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.0520	−0.999036	−0.499518	−	0.866304i	$-0.666490\pi$
$227$	−15.0520	−0.999036	−0.499518	+	0.866304i	$0.666490\pi$
$228$	0	0
$229$	16.6764	1.10201	0.551004	−	0.834503i	$-0.314245\pi$
$229$	16.6764	1.10201	0.551004	+	0.834503i	$0.314245\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.99789	0.196398	0.0981992	−	0.995167i	$-0.468692\pi$
$233$	2.99789	0.196398	0.0981992	+	0.995167i	$0.468692\pi$
$234$	0	0
$235$	−0.166992	−0.0108934
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.4641	1.00029	0.500145	−	0.865942i	$-0.333280\pi$
$239$	15.4641	1.00029	0.500145	+	0.865942i	$0.333280\pi$
$240$	0	0
$241$	16.3054	1.05032	0.525161	−	0.851003i	$-0.324005\pi$
$241$	16.3054	1.05032	0.525161	+	0.851003i	$0.324005\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−8.21899	−0.525092
$246$	0	0
$247$	6.75967	0.430107
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.6295	1.74396	0.871981	−	0.489540i	$-0.162835\pi$
$251$	27.6295	1.74396	0.871981	+	0.489540i	$0.162835\pi$
$252$	0	0
$253$	−5.06449	−0.318402
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.14781	0.133977	0.0669884	−	0.997754i	$-0.478661\pi$
$257$	2.14781	0.133977	0.0669884	+	0.997754i	$0.478661\pi$
$258$	0	0
$259$	−21.7992	−1.35454
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.60707	−0.345747	−0.172873	−	0.984944i	$-0.555305\pi$
$263$	−5.60707	−0.345747	−0.172873	+	0.984944i	$0.555305\pi$
$264$	0	0
$265$	12.0234	0.738592
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.36469	0.327091	0.163545	−	0.986536i	$-0.447707\pi$
$269$	5.36469	0.327091	0.163545	+	0.986536i	$0.447707\pi$
$270$	0	0
$271$	8.59000	0.521805	0.260903	−	0.965365i	$-0.415980\pi$
$271$	8.59000	0.521805	0.260903	+	0.965365i	$0.415980\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.09520	−0.367554
$276$	0	0
$277$	−24.7679	−1.48816	−0.744081	−	0.668090i	$-0.767112\pi$
$277$	−24.7679	−1.48816	−0.744081	+	0.668090i	$0.767112\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−25.4276	−1.51689	−0.758443	−	0.651740i	$-0.774039\pi$
$281$	−25.4276	−1.51689	−0.758443	+	0.651740i	$0.774039\pi$
$282$	0	0
$283$	−2.87410	−0.170848	−0.0854238	−	0.996345i	$-0.527224\pi$
$283$	−2.87410	−0.170848	−0.0854238	+	0.996345i	$0.527224\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	25.9662	1.53274
$288$	0	0
$289$	−10.9386	−0.643447
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.53317	−0.556934	−0.278467	−	0.960446i	$-0.589826\pi$
$293$	−9.53317	−0.556934	−0.278467	+	0.960446i	$0.589826\pi$
$294$	0	0
$295$	1.73205	0.100844
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.68252	−0.0973028
$300$	0	0
$301$	43.1155	2.48514
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.3163	0.762489
$306$	0	0
$307$	−4.02402	−0.229663	−0.114831	−	0.993385i	$-0.536633\pi$
$307$	−4.02402	−0.229663	−0.114831	+	0.993385i	$0.536633\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.65722	0.547611	0.273805	−	0.961785i	$-0.411718\pi$
$311$	9.65722	0.547611	0.273805	+	0.961785i	$0.411718\pi$
$312$	0	0
$313$	−34.0925	−1.92702	−0.963510	−	0.267672i	$-0.913746\pi$
$313$	−34.0925	−1.92702	−0.963510	+	0.267672i	$0.913746\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.5052	1.03936	0.519678	−	0.854362i	$-0.326052\pi$
$317$	18.5052	1.03936	0.519678	+	0.854362i	$0.326052\pi$
$318$	0	0
$319$	16.0370	0.897901
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.21863	−0.457297
$324$	0	0
$325$	−2.02494	−0.112324
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.651462	−0.0359163
$330$	0	0
$331$	−18.3230	−1.00712	−0.503561	−	0.863960i	$-0.667977\pi$
$331$	−18.3230	−1.00712	−0.503561	+	0.863960i	$0.667977\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.2622	0.724591
$336$	0	0
$337$	−28.0468	−1.52781	−0.763903	−	0.645331i	$-0.776720\pi$
$337$	−28.0468	−1.52781	−0.763903	+	0.645331i	$0.776720\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11.1597	0.604331
$342$	0	0
$343$	−4.75545	−0.256770
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.07812	−0.111560	−0.0557798	−	0.998443i	$-0.517764\pi$
$347$	−2.07812	−0.111560	−0.0557798	+	0.998443i	$0.517764\pi$
$348$	0	0
$349$	15.3814	0.823348	0.411674	−	0.911331i	$-0.364944\pi$
$349$	15.3814	0.823348	0.411674	+	0.911331i	$0.364944\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.55261	0.135862	0.0679310	−	0.997690i	$-0.478360\pi$
$353$	2.55261	0.135862	0.0679310	+	0.997690i	$0.478360\pi$
$354$	0	0
$355$	−10.8829	−0.577604
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.57699	0.241564	0.120782	−	0.992679i	$-0.461460\pi$
$359$	4.57699	0.241564	0.120782	+	0.992679i	$0.461460\pi$
$360$	0	0
$361$	−7.85641	−0.413495
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.87621	−0.0982053
$366$	0	0
$367$	−22.4921	−1.17408	−0.587038	−	0.809559i	$-0.699706\pi$
$367$	−22.4921	−1.17408	−0.587038	+	0.809559i	$0.699706\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	46.9051	2.43519
$372$	0	0
$373$	−31.1311	−1.61191	−0.805953	−	0.591979i	$-0.798347\pi$
$373$	−31.1311	−1.61191	−0.805953	+	0.591979i	$0.798347\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.32781	0.274396
$378$	0	0
$379$	35.4136	1.81907	0.909537	−	0.415623i	$-0.136436\pi$
$379$	35.4136	1.81907	0.909537	+	0.415623i	$0.136436\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.8735	−0.862194	−0.431097	−	0.902305i	$-0.641873\pi$
$383$	−16.8735	−0.862194	−0.431097	+	0.902305i	$0.641873\pi$
$384$	0	0
$385$	−23.7783	−1.21185
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.4787	0.531292	0.265646	−	0.964071i	$-0.414415\pi$
$389$	10.4787	0.531292	0.265646	+	0.964071i	$0.414415\pi$
$390$	0	0
$391$	2.04567	0.103454
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.46410	0.375560
$396$	0	0
$397$	−1.55781	−0.0781843	−0.0390921	−	0.999236i	$-0.512447\pi$
$397$	−1.55781	−0.0781843	−0.0390921	+	0.999236i	$0.512447\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.65568	−0.332369	−0.166184	−	0.986095i	$-0.553145\pi$
$401$	−6.65568	−0.332369	−0.166184	+	0.986095i	$0.553145\pi$
$402$	0	0
$403$	3.70746	0.184682
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−34.0593	−1.68826
$408$	0	0
$409$	−35.6904	−1.76478	−0.882388	−	0.470522i	$-0.844066\pi$
$409$	−35.6904	−1.76478	−0.882388	+	0.470522i	$0.844066\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.75699	0.332490
$414$	0	0
$415$	−6.80019	−0.333808
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.8741	−0.628941	−0.314470	−	0.949267i	$-0.601827\pi$
$419$	−12.8741	−0.628941	−0.314470	+	0.949267i	$0.601827\pi$
$420$	0	0
$421$	−14.1733	−0.690765	−0.345383	−	0.938462i	$-0.612251\pi$
$421$	−14.1733	−0.690765	−0.345383	+	0.938462i	$0.612251\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.46199	0.119424
$426$	0	0
$427$	51.9489	2.51398
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.786211	0.0378704	0.0189352	−	0.999821i	$-0.493972\pi$
$431$	0.786211	0.0378704	0.0189352	+	0.999821i	$0.493972\pi$
$432$	0	0
$433$	31.2184	1.50026	0.750129	−	0.661291i	$-0.229991\pi$
$433$	31.2184	1.50026	0.750129	+	0.661291i	$0.229991\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.77371	−0.132684
$438$	0	0
$439$	29.0359	1.38581	0.692904	−	0.721030i	$-0.256331\pi$
$439$	29.0359	1.38581	0.692904	+	0.721030i	$0.256331\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.0885	−0.811900	−0.405950	−	0.913895i	$-0.633059\pi$
$443$	−17.0885	−0.811900	−0.405950	+	0.913895i	$0.633059\pi$
$444$	0	0
$445$	12.1493	0.575932
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.5082	−0.873457	−0.436729	−	0.899593i	$-0.643863\pi$
$449$	−18.5082	−0.873457	−0.436729	+	0.899593i	$0.643863\pi$
$450$	0	0
$451$	40.5698	1.91036
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.89961	−0.370339
$456$	0	0
$457$	0.123791	0.00579068	0.00289534	−	0.999996i	$-0.499078\pi$
$457$	0.123791	0.00579068	0.00289534	+	0.999996i	$0.499078\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16.8737	−0.785889	−0.392944	−	0.919562i	$-0.628543\pi$
$461$	−16.8737	−0.785889	−0.392944	+	0.919562i	$0.628543\pi$
$462$	0	0
$463$	4.56506	0.212156	0.106078	−	0.994358i	$-0.466171\pi$
$463$	4.56506	0.212156	0.106078	+	0.994358i	$0.466171\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−34.8793	−1.61402	−0.807011	−	0.590537i	$-0.798916\pi$
$467$	−34.8793	−1.61402	−0.807011	+	0.590537i	$0.798916\pi$
$468$	0	0
$469$	51.7378	2.38903
$470$	0	0
$471$	0	0
$472$	0	0
$473$	67.3641	3.09740
$474$	0	0
$475$	−3.33820	−0.153167
$476$	0	0
$477$	0	0
$478$	0	0
$479$	42.4303	1.93869	0.969345	−	0.245703i	$-0.0790188\pi$
$479$	42.4303	1.93869	0.969345	+	0.245703i	$0.0790188\pi$
$480$	0	0
$481$	−11.3152	−0.515927
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.663907	0.0301465
$486$	0	0
$487$	−18.1331	−0.821691	−0.410846	−	0.911705i	$-0.634767\pi$
$487$	−18.1331	−0.821691	−0.410846	+	0.911705i	$0.634767\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.5082	0.835264	0.417632	−	0.908616i	$-0.362860\pi$
$491$	18.5082	0.835264	0.417632	+	0.908616i	$0.362860\pi$
$492$	0	0
$493$	−6.47773	−0.291742
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−42.4558	−1.90440
$498$	0	0
$499$	2.16664	0.0969919	0.0484960	−	0.998823i	$-0.484557\pi$
$499$	2.16664	0.0969919	0.0484960	+	0.998823i	$0.484557\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.6088	−0.963490	−0.481745	−	0.876311i	$-0.659997\pi$
$503$	−21.6088	−0.963490	−0.481745	+	0.876311i	$0.659997\pi$
$504$	0	0
$505$	−19.3616	−0.861580
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.35281	−0.148611	−0.0743053	−	0.997236i	$-0.523674\pi$
$509$	−3.35281	−0.148611	−0.0743053	+	0.997236i	$0.523674\pi$
$510$	0	0
$511$	−7.31938	−0.323790
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.2913	−0.585687
$516$	0	0
$517$	−1.01785	−0.0447650
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−33.4775	−1.46668	−0.733338	−	0.679864i	$-0.762039\pi$
$521$	−33.4775	−1.46668	−0.733338	+	0.679864i	$0.762039\pi$
$522$	0	0
$523$	32.8230	1.43525	0.717624	−	0.696431i	$-0.245230\pi$
$523$	32.8230	1.43525	0.717624	+	0.696431i	$0.245230\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.50766	−0.196357
$528$	0	0
$529$	−22.3096	−0.969983
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13.4781	0.583801
$534$	0	0
$535$	−6.39230	−0.276363
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−50.0963	−2.15780
$540$	0	0
$541$	−24.9038	−1.07070	−0.535350	−	0.844631i	$-0.679820\pi$
$541$	−24.9038	−1.07070	−0.535350	+	0.844631i	$0.679820\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.29500	−0.312483
$546$	0	0
$547$	−40.5255	−1.73275	−0.866373	−	0.499398i	$-0.833555\pi$
$547$	−40.5255	−1.73275	−0.866373	+	0.499398i	$0.833555\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.78312	0.374174
$552$	0	0
$553$	29.1186	1.23825
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.8673	1.09603	0.548016	−	0.836468i	$-0.315383\pi$
$557$	25.8673	1.09603	0.548016	+	0.836468i	$0.315383\pi$
$558$	0	0
$559$	22.3797	0.946558
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.03008	0.338428	0.169214	−	0.985579i	$-0.445877\pi$
$563$	8.03008	0.338428	0.169214	+	0.985579i	$0.445877\pi$
$564$	0	0
$565$	−3.24969	−0.136715
$566$	0	0
$567$	0	0
$568$	0	0
$569$	33.3865	1.39964	0.699818	−	0.714321i	$-0.253264\pi$
$569$	33.3865	1.39964	0.699818	+	0.714321i	$0.253264\pi$
$570$	0	0
$571$	−27.9318	−1.16891	−0.584455	−	0.811426i	$-0.698692\pi$
$571$	−27.9318	−1.16891	−0.584455	+	0.811426i	$0.698692\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.830899	0.0346509
$576$	0	0
$577$	−2.74187	−0.114146	−0.0570729	−	0.998370i	$-0.518177\pi$
$577$	−2.74187	−0.114146	−0.0570729	+	0.998370i	$0.518177\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−26.5286	−1.10059
$582$	0	0
$583$	73.2850	3.03515
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.523401	−0.0216031	−0.0108015	−	0.999942i	$-0.503438\pi$
$587$	−0.523401	−0.0216031	−0.0108015	+	0.999942i	$0.503438\pi$
$588$	0	0
$589$	6.11191	0.251837
$590$	0	0
$591$	0	0
$592$	0	0
$593$	39.5973	1.62607	0.813033	−	0.582217i	$-0.197815\pi$
$593$	39.5973	1.62607	0.813033	+	0.582217i	$0.197815\pi$
$594$	0	0
$595$	9.60461	0.393751
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22.5873	0.922891	0.461445	−	0.887169i	$-0.347331\pi$
$599$	22.5873	0.922891	0.461445	+	0.887169i	$0.347331\pi$
$600$	0	0
$601$	13.7720	0.561770	0.280885	−	0.959741i	$-0.409372\pi$
$601$	13.7720	0.561770	0.280885	+	0.959741i	$0.409372\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−26.1514	−1.06321
$606$	0	0
$607$	26.1748	1.06240	0.531201	−	0.847246i	$-0.321741\pi$
$607$	26.1748	1.06240	0.531201	+	0.847246i	$0.321741\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.338150	−0.0136801
$612$	0	0
$613$	1.67943	0.0678317	0.0339159	−	0.999425i	$-0.489202\pi$
$613$	1.67943	0.0678317	0.0339159	+	0.999425i	$0.489202\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.90629	−0.0767444	−0.0383722	−	0.999264i	$-0.512217\pi$
$617$	−1.90629	−0.0767444	−0.0383722	+	0.999264i	$0.512217\pi$
$618$	0	0
$619$	−10.9848	−0.441515	−0.220758	−	0.975329i	$-0.570853\pi$
$619$	−10.9848	−0.441515	−0.220758	+	0.975329i	$0.570853\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	47.3963	1.89889
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.7573	0.548541
$630$	0	0
$631$	0.318756	0.0126895	0.00634473	−	0.999980i	$-0.497980\pi$
$631$	0.318756	0.0126895	0.00634473	+	0.999980i	$0.497980\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.8294	0.588485
$636$	0	0
$637$	−16.6430	−0.659419
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.13469	−0.281803	−0.140902	−	0.990024i	$-0.545000\pi$
$641$	−7.13469	−0.281803	−0.140902	+	0.990024i	$0.545000\pi$
$642$	0	0
$643$	−11.5806	−0.456694	−0.228347	−	0.973580i	$-0.573332\pi$
$643$	−11.5806	−0.456694	−0.228347	+	0.973580i	$0.573332\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.68555	−0.341464	−0.170732	−	0.985317i	$-0.554613\pi$
$647$	−8.68555	−0.341464	−0.170732	+	0.985317i	$0.554613\pi$
$648$	0	0
$649$	10.5572	0.414406
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−45.2184	−1.76953	−0.884766	−	0.466036i	$-0.845682\pi$
$653$	−45.2184	−1.76953	−0.884766	+	0.466036i	$0.845682\pi$
$654$	0	0
$655$	13.0483	0.509841
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.9957	0.428333	0.214166	−	0.976797i	$-0.431297\pi$
$659$	10.9957	0.428333	0.214166	+	0.976797i	$0.431297\pi$
$660$	0	0
$661$	23.1107	0.898901	0.449450	−	0.893305i	$-0.351620\pi$
$661$	23.1107	0.898901	0.449450	+	0.893305i	$0.351620\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−13.0228	−0.505004
$666$	0	0
$667$	−2.18617	−0.0846490
$668$	0	0
$669$	0	0
$670$	0	0
$671$	81.1654	3.13336
$672$	0	0
$673$	−5.98848	−0.230839	−0.115419	−	0.993317i	$-0.536821\pi$
$673$	−5.98848	−0.230839	−0.115419	+	0.993317i	$0.536821\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.5541	−0.674659	−0.337329	−	0.941387i	$-0.609524\pi$
$677$	−17.5541	−0.674659	−0.337329	+	0.941387i	$0.609524\pi$
$678$	0	0
$679$	2.59000	0.0993951
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.9458	−0.954524	−0.477262	−	0.878761i	$-0.658371\pi$
$683$	−24.9458	−0.954524	−0.477262	+	0.878761i	$0.658371\pi$
$684$	0	0
$685$	−11.8023	−0.450943
$686$	0	0
$687$	0	0
$688$	0	0
$689$	24.3467	0.927535
$690$	0	0
$691$	35.0712	1.33417	0.667085	−	0.744981i	$-0.267542\pi$
$691$	35.0712	1.33417	0.667085	+	0.744981i	$0.267542\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.7044	−0.633633
$696$	0	0
$697$	−16.3871	−0.620706
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.6530	−0.440128	−0.220064	−	0.975485i	$-0.570627\pi$
$701$	−11.6530	−0.440128	−0.220064	+	0.975485i	$0.570627\pi$
$702$	0	0
$703$	−18.6535	−0.703531
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−75.5325	−2.84069
$708$	0	0
$709$	21.5589	0.809663	0.404832	−	0.914391i	$-0.367330\pi$
$709$	21.5589	0.809663	0.404832	+	0.914391i	$0.367330\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.52129	−0.0569728
$714$	0	0
$715$	−12.3424	−0.461580
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.4199	−1.02259	−0.511295	−	0.859405i	$-0.670834\pi$
$719$	−27.4199	−1.02259	−0.511295	+	0.859405i	$0.670834\pi$
$720$	0	0
$721$	−51.8516	−1.93105
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.63109	−0.0977164
$726$	0	0
$727$	−12.8449	−0.476392	−0.238196	−	0.971217i	$-0.576556\pi$
$727$	−12.8449	−0.476392	−0.238196	+	0.971217i	$0.576556\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−27.2099	−1.00640
$732$	0	0
$733$	6.59000	0.243407	0.121704	−	0.992566i	$-0.461164\pi$
$733$	6.59000	0.243407	0.121704	+	0.992566i	$0.461164\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	80.8356	2.97762
$738$	0	0
$739$	−33.5238	−1.23319	−0.616596	−	0.787280i	$-0.711489\pi$
$739$	−33.5238	−1.23319	−0.616596	+	0.787280i	$0.711489\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.6409	1.34422	0.672111	−	0.740450i	$-0.265388\pi$
$743$	36.6409	1.34422	0.672111	+	0.740450i	$0.265388\pi$
$744$	0	0
$745$	−21.2664	−0.779141
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.9374	−0.911191
$750$	0	0
$751$	1.65336	0.0603320	0.0301660	−	0.999545i	$-0.490396\pi$
$751$	1.65336	0.0603320	0.0301660	+	0.999545i	$0.490396\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.6831	−0.425191
$756$	0	0
$757$	−24.9447	−0.906631	−0.453315	−	0.891350i	$-0.649759\pi$
$757$	−24.9447	−0.906631	−0.453315	+	0.891350i	$0.649759\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.7278	−0.715133	−0.357567	−	0.933888i	$-0.616394\pi$
$761$	−19.7278	−0.715133	−0.357567	+	0.933888i	$0.616394\pi$
$762$	0	0
$763$	−28.4589	−1.03028
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.50730	0.126641
$768$	0	0
$769$	−46.3698	−1.67214	−0.836068	−	0.548625i	$-0.815151\pi$
$769$	−46.3698	−1.67214	−0.836068	+	0.548625i	$0.815151\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−24.2163	−0.870998	−0.435499	−	0.900189i	$-0.643428\pi$
$773$	−24.2163	−0.870998	−0.435499	+	0.900189i	$0.643428\pi$
$774$	0	0
$775$	−1.83090	−0.0657678
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22.2192	0.796085
$780$	0	0
$781$	−66.3334	−2.37359
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20.0915	0.717098
$786$	0	0
$787$	3.44008	0.122626	0.0613128	−	0.998119i	$-0.480471\pi$
$787$	3.44008	0.122626	0.0613128	+	0.998119i	$0.480471\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.6775	−0.450761
$792$	0	0
$793$	26.9647	0.957545
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−46.9750	−1.66394	−0.831970	−	0.554821i	$-0.812787\pi$
$797$	−46.9750	−1.66394	−0.831970	+	0.554821i	$0.812787\pi$
$798$	0	0
$799$	0.411134	0.0145449
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11.4359	−0.403563
$804$	0	0
$805$	3.24146	0.114247
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.8829	0.874838	0.437419	−	0.899258i	$-0.355893\pi$
$809$	24.8829	0.874838	0.437419	+	0.899258i	$0.355893\pi$
$810$	0	0
$811$	40.1794	1.41089	0.705444	−	0.708765i	$-0.250747\pi$
$811$	40.1794	1.41089	0.705444	+	0.708765i	$0.250747\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.99789	0.105012
$816$	0	0
$817$	36.8938	1.29075
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.56660	−0.0546748	−0.0273374	−	0.999626i	$-0.508703\pi$
$821$	−1.56660	−0.0546748	−0.0273374	+	0.999626i	$0.508703\pi$
$822$	0	0
$823$	31.4975	1.09793	0.548967	−	0.835844i	$-0.315021\pi$
$823$	31.4975	1.09793	0.548967	+	0.835844i	$0.315021\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.7013	0.685081	0.342540	−	0.939503i	$-0.388713\pi$
$827$	19.7013	0.685081	0.342540	+	0.939503i	$0.388713\pi$
$828$	0	0
$829$	−15.2950	−0.531217	−0.265609	−	0.964081i	$-0.585573\pi$
$829$	−15.2950	−0.531217	−0.265609	+	0.964081i	$0.585573\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	20.2351	0.701104
$834$	0	0
$835$	−10.5119	−0.363779
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.1201	1.21248	0.606240	−	0.795282i	$-0.292677\pi$
$839$	35.1201	1.21248	0.606240	+	0.795282i	$0.292677\pi$
$840$	0	0
$841$	−22.0773	−0.761288
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.89961	0.306156
$846$	0	0
$847$	−102.021	−3.50547
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.64297	0.159159
$852$	0	0
$853$	25.6305	0.877571	0.438785	−	0.898592i	$-0.355409\pi$
$853$	25.6305	0.877571	0.438785	+	0.898592i	$0.355409\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.2944	−0.556605	−0.278303	−	0.960493i	$-0.589772\pi$
$857$	−16.2944	−0.556605	−0.278303	+	0.960493i	$0.589772\pi$
$858$	0	0
$859$	23.1046	0.788319	0.394160	−	0.919042i	$-0.371036\pi$
$859$	23.1046	0.788319	0.394160	+	0.919042i	$0.371036\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.3148	0.623443	0.311722	−	0.950173i	$-0.399094\pi$
$863$	18.3148	0.623443	0.311722	+	0.950173i	$0.399094\pi$
$864$	0	0
$865$	−11.7591	−0.399821
$866$	0	0
$867$	0	0
$868$	0	0
$869$	45.4952	1.54332
$870$	0	0
$871$	26.8552	0.909953
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.90115	0.131883
$876$	0	0
$877$	−7.83146	−0.264450	−0.132225	−	0.991220i	$-0.542212\pi$
$877$	−7.83146	−0.264450	−0.132225	+	0.991220i	$0.542212\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.17430	−0.106945	−0.0534724	−	0.998569i	$-0.517029\pi$
$881$	−3.17430	−0.106945	−0.0534724	+	0.998569i	$0.517029\pi$
$882$	0	0
$883$	34.4132	1.15810	0.579049	−	0.815293i	$-0.303424\pi$
$883$	34.4132	1.15810	0.579049	+	0.815293i	$0.303424\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−55.6706	−1.86924	−0.934618	−	0.355654i	$-0.884258\pi$
$887$	−55.6706	−1.86924	−0.934618	+	0.355654i	$0.884258\pi$
$888$	0	0
$889$	57.8516	1.94028
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.557454	−0.0186545
$894$	0	0
$895$	6.65604	0.222487
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.81727	0.160665
$900$	0	0
$901$	−29.6015	−0.986170
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.50730	−0.282792
$906$	0	0
$907$	28.0292	0.930695	0.465347	−	0.885128i	$-0.345929\pi$
$907$	28.0292	0.930695	0.465347	+	0.885128i	$0.345929\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.9236	0.825757	0.412878	−	0.910786i	$-0.364523\pi$
$911$	24.9236	0.825757	0.412878	+	0.910786i	$0.364523\pi$
$912$	0	0
$913$	−41.4485	−1.37175
$914$	0	0
$915$	0	0
$916$	0	0
$917$	50.9036	1.68098
$918$	0	0
$919$	−13.2512	−0.437116	−0.218558	−	0.975824i	$-0.570135\pi$
$919$	−13.2512	−0.437116	−0.218558	+	0.975824i	$0.570135\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−22.0372	−0.725364
$924$	0	0
$925$	5.58789	0.183729
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.6913	1.40065	0.700327	−	0.713822i	$-0.253037\pi$
$929$	42.6913	1.40065	0.700327	+	0.713822i	$0.253037\pi$
$930$	0	0
$931$	−27.4366	−0.899199
$932$	0	0
$933$	0	0
$934$	0	0
$935$	15.0063	0.490759
$936$	0	0
$937$	−11.5620	−0.377715	−0.188857	−	0.982005i	$-0.560478\pi$
$937$	−11.5620	−0.377715	−0.188857	+	0.982005i	$0.560478\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	38.7731	1.26397	0.631983	−	0.774982i	$-0.282241\pi$
$941$	38.7731	1.26397	0.631983	+	0.774982i	$0.282241\pi$
$942$	0	0
$943$	−5.53049	−0.180098
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−50.3393	−1.63581	−0.817904	−	0.575354i	$-0.804864\pi$
$947$	−50.3393	−1.63581	−0.817904	+	0.575354i	$0.804864\pi$
$948$	0	0
$949$	−3.79922	−0.123328
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.4118	0.369666	0.184833	−	0.982770i	$-0.440826\pi$
$953$	11.4118	0.369666	0.184833	+	0.982770i	$0.440826\pi$
$954$	0	0
$955$	−17.5052	−0.566455
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−46.0426	−1.48679
$960$	0	0
$961$	−27.6478	−0.891865
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26.1165	−0.840719
$966$	0	0
$967$	7.63285	0.245456	0.122728	−	0.992440i	$-0.460836\pi$
$967$	7.63285	0.245456	0.122728	+	0.992440i	$0.460836\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.5124	0.658276	0.329138	−	0.944282i	$-0.393242\pi$
$971$	20.5124	0.658276	0.329138	+	0.944282i	$0.393242\pi$
$972$	0	0
$973$	−65.1663	−2.08914
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.93859	0.253978	0.126989	−	0.991904i	$-0.459469\pi$
$977$	7.93859	0.253978	0.126989	+	0.991904i	$0.459469\pi$
$978$	0	0
$979$	74.0523	2.36672
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.7253	0.756720	0.378360	−	0.925659i	$-0.376488\pi$
$983$	23.7253	0.756720	0.378360	+	0.925659i	$0.376488\pi$
$984$	0	0
$985$	−12.9215	−0.411714
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.18309	−0.292005
$990$	0	0
$991$	−4.04618	−0.128531	−0.0642656	−	0.997933i	$-0.520470\pi$
$991$	−4.04618	−0.128531	−0.0642656	+	0.997933i	$0.520470\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.9781	0.538242
$996$	0	0
$997$	9.08115	0.287603	0.143802	−	0.989607i	$-0.454067\pi$
$997$	9.08115	0.287603	0.143802	+	0.989607i	$0.454067\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.by.1.1		4
3.2	odd	2		6480.2.a.ca.1.1		4
4.3	odd	2		3240.2.a.t.1.4	✓	4
12.11	even	2		3240.2.a.v.1.4	yes	4
36.7	odd	6		3240.2.q.bh.1081.1		8
36.11	even	6		3240.2.q.bg.1081.1		8
36.23	even	6		3240.2.q.bg.2161.1		8
36.31	odd	6		3240.2.q.bh.2161.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.t.1.4	✓	4	4.3	odd	2
3240.2.a.v.1.4	yes	4	12.11	even	2
3240.2.q.bg.1081.1		8	36.11	even	6
3240.2.q.bg.2161.1		8	36.23	even	6
3240.2.q.bh.1081.1		8	36.7	odd	6
3240.2.q.bh.2161.1		8	36.31	odd	6
6480.2.a.by.1.1		4	1.1	even	1	trivial
6480.2.a.ca.1.1		4	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.90115 q^{7} -6.09520 q^{11} -2.02494 q^{13} +2.46199 q^{17} -3.33820 q^{19} +0.830899 q^{23} +1.00000 q^{25} -2.63109 q^{29} -1.83090 q^{31} +3.90115 q^{35} +5.58789 q^{37} -6.65604 q^{41} -11.0520 q^{43} +0.166992 q^{47} +8.21899 q^{49} -12.0234 q^{53} +6.09520 q^{55} -1.73205 q^{59} -13.3163 q^{61} +2.02494 q^{65} -13.2622 q^{67} +10.8829 q^{71} +1.87621 q^{73} +23.7783 q^{77} -7.46410 q^{79} +6.80019 q^{83} -2.46199 q^{85} -12.1493 q^{89} +7.89961 q^{91} +3.33820 q^{95} -0.663907 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 2 q^{7} + 4 q^{11} + 2 q^{17} + 10 q^{23} + 4 q^{25} + 4 q^{29} - 14 q^{31} + 2 q^{35} + 14 q^{37} - 4 q^{41} - 22 q^{43} + 10 q^{49} + 8 q^{53} - 4 q^{55} + 4 q^{61} - 24 q^{67} + 28 q^{71}+ \cdots - 10 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 2 * q^7 + 4 * q^11 + 2 * q^17 + 10 * q^23 + 4 * q^25 + 4 * q^29 - 14 * q^31 + 2 * q^35 + 14 * q^37 - 4 * q^41 - 22 * q^43 + 10 * q^49 + 8 * q^53 - 4 * q^55 + 4 * q^61 - 24 * q^67 + 28 * q^71 + 2 * q^73 + 30 * q^77 - 16 * q^79 + 6 * q^83 - 2 * q^85 + 8 * q^89 - 30 * q^91 - 10 * q^97

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