LMFDB - Embedded newform 5184.2.f.d.2591.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(2591,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("5184.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.f (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$41.3944484078$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.33418400425706520576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 $ x^16 - 8x^14 + 49x^12 - 104x^10 + 160x^8 - 104x^6 + 49x^4 - 8*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.15
Root			$0.367543 - 0.212201i$ of defining polynomial
Character	$\chi$	$=$	5184.2591
Dual form			5184.2.f.d.2591.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.78066 q^{5} +0.732051i q^{7} +O(q^{10})$	$q+2.78066 q^{5} +0.732051i q^{7} +2.03558i q^{11} -2.49307i q^{13} -1.55291i q^{17} -6.81119 q^{19} -4.24264 q^{23} +2.73205 q^{25} +4.81624 q^{29} +1.46410i q^{31} +2.03558i q^{35} +9.30426i q^{37} +7.34847i q^{41} +6.81119 q^{43} +8.48528 q^{47} +6.46410 q^{49} +7.59689 q^{53} +5.66025i q^{55} +9.63248i q^{59} -4.31812i q^{61} -6.93237i q^{65} +11.7973 q^{67} +7.34847 q^{71} +6.66025 q^{73} -1.49015 q^{77} +6.19615i q^{79} +1.49015i q^{83} -4.31812i q^{85} -8.90138i q^{89} +1.82505 q^{91} -18.9396 q^{95} +8.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{25} + 48 q^{49} - 32 q^{73} + 32 q^{97}+O(q^{100}) $ 16 * q + 16 * q^25 + 48 * q^49 - 32 * q^73 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	2.78066	1.24355	0.621774	−	0.783197i	$-0.286412\pi$
$5$	2.78066	1.24355	0.621774	+	0.783197i	$0.286412\pi$
$6$	0	0
$7$	0.732051i	0.276689i	0.990384	+	0.138345i	$0.0441781\pi$
$7$	0.732051i	0.276689i	−0.990384	+	0.138345i	$0.955822\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.03558i	0.613751i	0.951750	+	0.306876i	$0.0992835\pi$
$11$	2.03558i	0.613751i	−0.951750	+	0.306876i	$0.900716\pi$
$12$	0	0
$13$	− 2.49307i	− 0.691453i	−0.938335	−	0.345726i	$-0.887633\pi$
$13$	− 2.49307i	− 0.691453i	0.938335	−	0.345726i	$-0.112367\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.55291i	− 0.376637i	−0.982108	−	0.188319i	$-0.939696\pi$
$17$	− 1.55291i	− 0.376637i	0.982108	−	0.188319i	$-0.0603037\pi$
$18$	0	0
$19$	−6.81119	−1.56259	−0.781297	−	0.624159i	$-0.785442\pi$
$19$	−6.81119	−1.56259	−0.781297	+	0.624159i	$0.785442\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.24264	−0.884652	−0.442326	−	0.896854i	$-0.645847\pi$
$23$	−4.24264	−0.884652	−0.442326	+	0.896854i	$0.645847\pi$
$24$	0	0
$25$	2.73205	0.546410
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.81624	0.894353	0.447177	−	0.894446i	$-0.352430\pi$
$29$	4.81624	0.894353	0.447177	+	0.894446i	$0.352430\pi$
$30$	0	0
$31$	1.46410i	0.262960i	0.991319	+	0.131480i	$0.0419730\pi$
$31$	1.46410i	0.262960i	−0.991319	+	0.131480i	$0.958027\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.03558i	0.344076i
$36$	0	0
$37$	9.30426i	1.52961i	0.644261	+	0.764805i	$0.277165\pi$
$37$	9.30426i	1.52961i	−0.644261	+	0.764805i	$0.722835\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.34847i	1.14764i	0.818982	+	0.573819i	$0.194539\pi$
$41$	7.34847i	1.14764i	−0.818982	+	0.573819i	$0.805461\pi$
$42$	0	0
$43$	6.81119	1.03870	0.519348	−	0.854563i	$-0.326175\pi$
$43$	6.81119	1.03870	0.519348	+	0.854563i	$0.326175\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.48528	1.23771	0.618853	−	0.785507i	$-0.287598\pi$
$47$	8.48528	1.23771	0.618853	+	0.785507i	$0.287598\pi$
$48$	0	0
$49$	6.46410	0.923443
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.59689	1.04351	0.521757	−	0.853094i	$-0.325277\pi$
$53$	7.59689	1.04351	0.521757	+	0.853094i	$0.325277\pi$
$54$	0	0
$55$	5.66025i	0.763228i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.63248i	1.25404i	0.779002	+	0.627021i	$0.215726\pi$
$59$	9.63248i	1.25404i	−0.779002	+	0.627021i	$0.784274\pi$
$60$	0	0
$61$	− 4.31812i	− 0.552879i	−0.961031	−	0.276439i	$-0.910845\pi$
$61$	− 4.31812i	− 0.552879i	0.961031	−	0.276439i	$-0.0891545\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6.93237i	− 0.859854i
$66$	0	0
$67$	11.7973	1.44127	0.720636	−	0.693313i	$-0.243850\pi$
$67$	11.7973	1.44127	0.720636	+	0.693313i	$0.243850\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.34847	0.872103	0.436051	−	0.899922i	$-0.356377\pi$
$71$	7.34847	0.872103	0.436051	+	0.899922i	$0.356377\pi$
$72$	0	0
$73$	6.66025	0.779524	0.389762	−	0.920916i	$-0.372557\pi$
$73$	6.66025	0.779524	0.389762	+	0.920916i	$0.372557\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.49015	−0.169818
$78$	0	0
$79$	6.19615i	0.697122i	0.937286	+	0.348561i	$0.113330\pi$
$79$	6.19615i	0.697122i	−0.937286	+	0.348561i	$0.886670\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.49015i	0.163565i	0.996650	+	0.0817826i	$0.0260613\pi$
$83$	1.49015i	0.163565i	−0.996650	+	0.0817826i	$0.973939\pi$
$84$	0	0
$85$	− 4.31812i	− 0.468366i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 8.90138i	− 0.943545i	−0.881720	−	0.471772i	$-0.843614\pi$
$89$	− 8.90138i	− 0.943545i	0.881720	−	0.471772i	$-0.156386\pi$
$90$	0	0
$91$	1.82505	0.191318
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−18.9396	−1.94316
$96$	0	0
$97$	8.92820	0.906522	0.453261	−	0.891378i	$-0.350261\pi$
$97$	8.92820	0.906522	0.453261	+	0.891378i	$0.350261\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.03558	−0.202548	−0.101274	−	0.994859i	$-0.532292\pi$
$101$	−2.03558	−0.202548	−0.101274	+	0.994859i	$0.532292\pi$
$102$	0	0
$103$	14.3923i	1.41812i	0.705150	+	0.709058i	$0.250880\pi$
$103$	14.3923i	1.41812i	−0.705150	+	0.709058i	$0.749120\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.1226i	1.07526i	0.843179	+	0.537632i	$0.180681\pi$
$107$	11.1226i	1.07526i	−0.843179	+	0.537632i	$0.819319\pi$
$108$	0	0
$109$	− 2.49307i	− 0.238793i	−0.992847	−	0.119396i	$-0.961904\pi$
$109$	− 2.49307i	− 0.238793i	0.992847	−	0.119396i	$-0.0380959\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.5235i	1.74254i	0.490802	+	0.871271i	$0.336704\pi$
$113$	18.5235i	1.74254i	−0.490802	+	0.871271i	$0.663296\pi$
$114$	0	0
$115$	−11.7973	−1.10011
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.13681	0.104211
$120$	0	0
$121$	6.85641	0.623310
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.30639	−0.564060
$126$	0	0
$127$	6.73205i	0.597373i	0.954351	+	0.298686i	$0.0965485\pi$
$127$	6.73205i	0.597373i	−0.954351	+	0.298686i	$0.903452\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 17.2294i	− 1.50534i	−0.658400	−	0.752669i	$-0.728766\pi$
$131$	− 17.2294i	− 1.50534i	0.658400	−	0.752669i	$-0.271234\pi$
$132$	0	0
$133$	− 4.98614i	− 0.432353i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.2808i	− 1.22009i	−0.792365	−	0.610047i	$-0.791151\pi$
$137$	− 14.2808i	− 1.22009i	0.792365	−	0.610047i	$-0.208849\pi$
$138$	0	0
$139$	−18.6085	−1.57835	−0.789177	−	0.614166i	$-0.789493\pi$
$139$	−18.6085	−1.57835	−0.789177	+	0.614166i	$0.789493\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.07484	0.424380
$144$	0	0
$145$	13.3923	1.11217
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.745075	0.0610389	0.0305194	−	0.999534i	$-0.490284\pi$
$149$	0.745075	0.0610389	0.0305194	+	0.999534i	$0.490284\pi$
$150$	0	0
$151$	19.3205i	1.57228i	0.618048	+	0.786140i	$0.287924\pi$
$151$	19.3205i	1.57228i	−0.618048	+	0.786140i	$0.712076\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.07116i	0.327004i
$156$	0	0
$157$	9.30426i	0.742561i	0.928521	+	0.371280i	$0.121081\pi$
$157$	9.30426i	0.742561i	−0.928521	+	0.371280i	$0.878919\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3.10583i	− 0.244774i
$162$	0	0
$163$	4.98614	0.390544	0.195272	−	0.980749i	$-0.437441\pi$
$163$	4.98614	0.390544	0.195272	+	0.980749i	$0.437441\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.8338	−1.22525	−0.612626	−	0.790373i	$-0.709887\pi$
$167$	−15.8338	−1.22525	−0.612626	+	0.790373i	$0.709887\pi$
$168$	0	0
$169$	6.78461	0.521893
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.9389	−1.21181	−0.605905	−	0.795537i	$-0.707189\pi$
$173$	−15.9389	−1.21181	−0.605905	+	0.795537i	$0.707189\pi$
$174$	0	0
$175$	2.00000i	0.151186i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 24.8263i	− 1.85560i	−0.373075	−	0.927801i	$-0.621697\pi$
$179$	− 24.8263i	− 1.85560i	0.373075	−	0.927801i	$-0.378303\pi$
$180$	0	0
$181$	− 18.6085i	− 1.38316i	−0.722300	−	0.691580i	$-0.756915\pi$
$181$	− 18.6085i	− 1.38316i	0.722300	−	0.691580i	$-0.243085\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	25.8719i	1.90214i
$186$	0	0
$187$	3.16108	0.231161
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.3190	1.75966	0.879832	−	0.475285i	$-0.157655\pi$
$191$	24.3190	1.75966	0.879832	+	0.475285i	$0.157655\pi$
$192$	0	0
$193$	5.92820	0.426721	0.213361	−	0.976973i	$-0.431559\pi$
$193$	5.92820	0.426721	0.213361	+	0.976973i	$0.431559\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.5199	−1.31949	−0.659743	−	0.751491i	$-0.729335\pi$
$197$	−18.5199	−1.31949	−0.659743	+	0.751491i	$0.729335\pi$
$198$	0	0
$199$	23.8564i	1.69114i	0.533868	+	0.845568i	$0.320738\pi$
$199$	23.8564i	1.69114i	−0.533868	+	0.845568i	$0.679262\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.52573i	0.247458i
$204$	0	0
$205$	20.4336i	1.42714i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 13.8647i	− 0.959044i
$210$	0	0
$211$	1.82505	0.125642	0.0628209	−	0.998025i	$-0.479990\pi$
$211$	1.82505	0.125642	0.0628209	+	0.998025i	$0.479990\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.9396	1.29167
$216$	0	0
$217$	−1.07180	−0.0727583
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.87152	−0.260427
$222$	0	0
$223$	− 6.19615i	− 0.414925i	−0.978243	−	0.207463i	$-0.933479\pi$
$223$	− 6.19615i	− 0.414925i	0.978243	−	0.207463i	$-0.0665205\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 29.8421i	− 1.98069i	−0.138614	−	0.990346i	$-0.544265\pi$
$227$	− 29.8421i	− 1.98069i	0.138614	−	0.990346i	$-0.455735\pi$
$228$	0	0
$229$	11.1293i	0.735446i	0.929935	+	0.367723i	$0.119862\pi$
$229$	11.1293i	0.735446i	−0.929935	+	0.367723i	$0.880138\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 8.90138i	− 0.583149i	−0.956548	−	0.291575i	$-0.905821\pi$
$233$	− 8.90138i	− 0.583149i	0.956548	−	0.291575i	$-0.0941791\pi$
$234$	0	0
$235$	23.5947	1.53915
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.2880	−1.70043	−0.850216	−	0.526434i	$-0.823529\pi$
$239$	−26.2880	−1.70043	−0.850216	+	0.526434i	$0.823529\pi$
$240$	0	0
$241$	15.1962	0.978870	0.489435	−	0.872040i	$-0.337203\pi$
$241$	15.1962	0.978870	0.489435	+	0.872040i	$0.337203\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	17.9744	1.14835
$246$	0	0
$247$	16.9808i	1.08046i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 10.5772i	− 0.667627i	−0.942639	−	0.333813i	$-0.891665\pi$
$251$	− 10.5772i	− 0.667627i	0.942639	−	0.333813i	$-0.108335\pi$
$252$	0	0
$253$	− 8.63624i	− 0.542956i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.3867i	1.08455i	0.840201	+	0.542275i	$0.182437\pi$
$257$	17.3867i	1.08455i	−0.840201	+	0.542275i	$0.817563\pi$
$258$	0	0
$259$	−6.81119	−0.423227
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.21166	−0.383027	−0.191514	−	0.981490i	$-0.561340\pi$
$263$	−6.21166	−0.383027	−0.191514	+	0.981490i	$0.561340\pi$
$264$	0	0
$265$	21.1244	1.29766
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.3934	0.938554	0.469277	−	0.883051i	$-0.344515\pi$
$269$	15.3934	0.938554	0.469277	+	0.883051i	$0.344515\pi$
$270$	0	0
$271$	− 6.87564i	− 0.417666i	−0.977951	−	0.208833i	$-0.933034\pi$
$271$	− 6.87564i	− 0.417666i	0.977951	−	0.208833i	$-0.0669665\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.56131i	0.335360i
$276$	0	0
$277$	18.6085i	1.11808i	0.829142	+	0.559039i	$0.188830\pi$
$277$	18.6085i	1.11808i	−0.829142	+	0.559039i	$0.811170\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 6.93237i	− 0.413550i	−0.978388	−	0.206775i	$-0.933703\pi$
$281$	− 6.93237i	− 0.413550i	0.978388	−	0.206775i	$-0.0662969\pi$
$282$	0	0
$283$	−18.6085	−1.10616	−0.553081	−	0.833128i	$-0.686548\pi$
$283$	−18.6085	−1.10616	−0.553081	+	0.833128i	$0.686548\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.37945	−0.317539
$288$	0	0
$289$	14.5885	0.858145
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−25.0259	−1.46203	−0.731015	−	0.682362i	$-0.760953\pi$
$293$	−25.0259	−1.46203	−0.731015	+	0.682362i	$0.760953\pi$
$294$	0	0
$295$	26.7846i	1.55946i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.5772i	0.611695i
$300$	0	0
$301$	4.98614i	0.287396i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 12.0072i	− 0.687531i
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.37945	−0.305041	−0.152520	−	0.988300i	$-0.548739\pi$
$311$	−5.37945	−0.305041	−0.152520	+	0.988300i	$0.548739\pi$
$312$	0	0
$313$	6.07180	0.343198	0.171599	−	0.985167i	$-0.445107\pi$
$313$	6.07180	0.343198	0.171599	+	0.985167i	$0.445107\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.36167	−0.301141	−0.150571	−	0.988599i	$-0.548111\pi$
$317$	−5.36167	−0.301141	−0.150571	+	0.988599i	$0.548111\pi$
$318$	0	0
$319$	9.80385i	0.548910i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.5772i	0.588531i
$324$	0	0
$325$	− 6.81119i	− 0.377817i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.21166i	0.342460i
$330$	0	0
$331$	6.81119	0.374377	0.187188	−	0.982324i	$-0.440063\pi$
$331$	6.81119	0.374377	0.187188	+	0.982324i	$0.440063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	32.8043	1.79229
$336$	0	0
$337$	6.39230	0.348211	0.174106	−	0.984727i	$-0.444297\pi$
$337$	6.39230	0.348211	0.174106	+	0.984727i	$0.444297\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.98030	−0.161392
$342$	0	0
$343$	9.85641i	0.532196i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 25.7710i	− 1.38346i	−0.722157	−	0.691729i	$-0.756849\pi$
$347$	− 25.7710i	− 1.38346i	0.722157	−	0.691729i	$-0.243151\pi$
$348$	0	0
$349$	− 18.6085i	− 0.996091i	−0.867151	−	0.498046i	$-0.834051\pi$
$349$	− 18.6085i	− 0.996091i	0.867151	−	0.498046i	$-0.165949\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 4.24264i	− 0.225813i	−0.993606	−	0.112906i	$-0.963984\pi$
$353$	− 4.24264i	− 0.225813i	0.993606	−	0.112906i	$-0.0360161\pi$
$354$	0	0
$355$	20.4336	1.08450
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.1074	0.955671	0.477835	−	0.878449i	$-0.341422\pi$
$359$	18.1074	0.955671	0.477835	+	0.878449i	$0.341422\pi$
$360$	0	0
$361$	27.3923	1.44170
$362$	0	0
$363$	0	0
$364$	0	0
$365$	18.5199	0.969375
$366$	0	0
$367$	− 4.53590i	− 0.236772i	−0.992968	−	0.118386i	$-0.962228\pi$
$367$	− 4.53590i	− 0.236772i	0.992968	−	0.118386i	$-0.0377720\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.56131i	0.288729i
$372$	0	0
$373$	− 13.6224i	− 0.705340i	−0.935748	−	0.352670i	$-0.885274\pi$
$373$	− 13.6224i	− 0.705340i	0.935748	−	0.352670i	$-0.114726\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0072i	− 0.618403i
$378$	0	0
$379$	−9.97227	−0.512241	−0.256121	−	0.966645i	$-0.582444\pi$
$379$	−9.97227	−0.512241	−0.256121	+	0.966645i	$0.582444\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−27.4249	−1.40134	−0.700672	−	0.713483i	$-0.747117\pi$
$383$	−27.4249	−1.40134	−0.700672	+	0.713483i	$0.747117\pi$
$384$	0	0
$385$	−4.14359	−0.211177
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−33.9133	−1.71947	−0.859737	−	0.510738i	$-0.829372\pi$
$389$	−33.9133	−1.71947	−0.859737	+	0.510738i	$0.829372\pi$
$390$	0	0
$391$	6.58846i	0.333193i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	17.2294i	0.866904i
$396$	0	0
$397$	17.9405i	0.900408i	0.892926	+	0.450204i	$0.148649\pi$
$397$	17.9405i	0.900408i	−0.892926	+	0.450204i	$0.851351\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.90138i	0.444514i	0.974988	+	0.222257i	$0.0713424\pi$
$401$	8.90138i	0.444514i	−0.974988	+	0.222257i	$0.928658\pi$
$402$	0	0
$403$	3.65011	0.181825
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.9396	−0.938800
$408$	0	0
$409$	−14.1244	−0.698404	−0.349202	−	0.937047i	$-0.613547\pi$
$409$	−14.1244	−0.698404	−0.349202	+	0.937047i	$0.613547\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.05146	−0.346980
$414$	0	0
$415$	4.14359i	0.203401i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.63248i	0.470577i	0.971926	+	0.235289i	$0.0756035\pi$
$419$	9.63248i	0.470577i	−0.971926	+	0.235289i	$0.924396\pi$
$420$	0	0
$421$	− 38.3741i	− 1.87024i	−0.354334	−	0.935119i	$-0.615292\pi$
$421$	− 38.3741i	− 1.87024i	0.354334	−	0.935119i	$-0.384708\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.24264i	− 0.205798i
$426$	0	0
$427$	3.16108	0.152976
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.3500	−1.07656	−0.538281	−	0.842765i	$-0.680926\pi$
$431$	−22.3500	−1.07656	−0.538281	+	0.842765i	$0.680926\pi$
$432$	0	0
$433$	−22.3205	−1.07266	−0.536328	−	0.844010i	$-0.680189\pi$
$433$	−22.3205	−1.07266	−0.536328	+	0.844010i	$0.680189\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	28.8974	1.38235
$438$	0	0
$439$	− 6.14359i	− 0.293218i	−0.989195	−	0.146609i	$-0.953164\pi$
$439$	− 6.14359i	− 0.293218i	0.989195	−	0.146609i	$-0.0468359\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	− 24.7517i	− 1.17334i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.13681i	0.0536495i	0.999640	+	0.0268247i	$0.00853960\pi$
$449$	1.13681i	0.0536495i	−0.999640	+	0.0268247i	$0.991460\pi$
$450$	0	0
$451$	−14.9584	−0.704364
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.07484	0.237912
$456$	0	0
$457$	24.6603	1.15356	0.576779	−	0.816900i	$-0.304309\pi$
$457$	24.6603	1.15356	0.576779	+	0.816900i	$0.304309\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	33.9133	1.57950	0.789750	−	0.613429i	$-0.210210\pi$
$461$	33.9133	1.57950	0.789750	+	0.613429i	$0.210210\pi$
$462$	0	0
$463$	− 31.4641i	− 1.46226i	−0.682238	−	0.731130i	$-0.738993\pi$
$463$	− 31.4641i	− 1.46226i	0.682238	−	0.731130i	$-0.261007\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.1226i	0.514694i	0.966319	+	0.257347i	$0.0828483\pi$
$467$	11.1226i	0.514694i	−0.966319	+	0.257347i	$0.917152\pi$
$468$	0	0
$469$	8.63624i	0.398785i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.8647i	0.637501i
$474$	0	0
$475$	−18.6085	−0.853817
$476$	0	0
$477$	0	0
$478$	0	0
$479$	33.6365	1.53689	0.768446	−	0.639915i	$-0.221030\pi$
$479$	33.6365	1.53689	0.768446	+	0.639915i	$0.221030\pi$
$480$	0	0
$481$	23.1962	1.05765
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.8263	1.12730
$486$	0	0
$487$	4.78461i	0.216811i	0.994107	+	0.108406i	$0.0345745\pi$
$487$	4.78461i	0.216811i	−0.994107	+	0.108406i	$0.965425\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.7710i	1.16303i	0.813536	+	0.581514i	$0.197539\pi$
$491$	25.7710i	1.16303i	−0.813536	+	0.581514i	$0.802461\pi$
$492$	0	0
$493$	− 7.47921i	− 0.336846i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.37945i	0.241301i
$498$	0	0
$499$	3.16108	0.141510	0.0707548	−	0.997494i	$-0.477459\pi$
$499$	3.16108	0.141510	0.0707548	+	0.997494i	$0.477459\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.0764	0.895162	0.447581	−	0.894243i	$-0.352286\pi$
$503$	20.0764	0.895162	0.447581	+	0.894243i	$0.352286\pi$
$504$	0	0
$505$	−5.66025	−0.251878
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.7195	−0.829728	−0.414864	−	0.909883i	$-0.636171\pi$
$509$	−18.7195	−0.829728	−0.414864	+	0.909883i	$0.636171\pi$
$510$	0	0
$511$	4.87564i	0.215686i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	40.0201i	1.76349i
$516$	0	0
$517$	17.2725i	0.759643i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.5307i	1.33757i	0.743454	+	0.668787i	$0.233186\pi$
$521$	30.5307i	1.33757i	−0.743454	+	0.668787i	$0.766814\pi$
$522$	0	0
$523$	−29.0698	−1.27113	−0.635567	−	0.772046i	$-0.719234\pi$
$523$	−29.0698	−1.27113	−0.635567	+	0.772046i	$0.719234\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.27362	0.0990406
$528$	0	0
$529$	−5.00000	−0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	18.3202	0.793538
$534$	0	0
$535$	30.9282i	1.33714i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13.1582i	0.566764i
$540$	0	0
$541$	34.7240i	1.49290i	0.665442	+	0.746450i	$0.268243\pi$
$541$	34.7240i	1.49290i	−0.665442	+	0.746450i	$0.731757\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 6.93237i	− 0.296950i
$546$	0	0
$547$	23.5947	1.00883	0.504417	−	0.863460i	$-0.331707\pi$
$547$	23.5947	1.00883	0.504417	+	0.863460i	$0.331707\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−32.8043	−1.39751
$552$	0	0
$553$	−4.53590	−0.192886
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.32609	−0.140931	−0.0704655	−	0.997514i	$-0.522448\pi$
$557$	−3.32609	−0.140931	−0.0704655	+	0.997514i	$0.522448\pi$
$558$	0	0
$559$	− 16.9808i	− 0.718210i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.1938i	0.640342i	0.947360	+	0.320171i	$0.103740\pi$
$563$	15.1938i	0.640342i	−0.947360	+	0.320171i	$0.896260\pi$
$564$	0	0
$565$	51.5074i	2.16693i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.8704i	0.455711i	0.973695	+	0.227855i	$0.0731714\pi$
$569$	10.8704i	0.455711i	−0.973695	+	0.227855i	$0.926829\pi$
$570$	0	0
$571$	42.2032	1.76615	0.883074	−	0.469234i	$-0.155470\pi$
$571$	42.2032	1.76615	0.883074	+	0.469234i	$0.155470\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.5911	−0.483383
$576$	0	0
$577$	17.3923	0.724051	0.362026	−	0.932168i	$-0.382085\pi$
$577$	17.3923	0.724051	0.362026	+	0.932168i	$0.382085\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.09087	−0.0452567
$582$	0	0
$583$	15.4641i	0.640458i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.08704i	− 0.375062i	−0.982259	−	0.187531i	$-0.939951\pi$
$587$	− 9.08704i	− 0.375062i	0.982259	−	0.187531i	$-0.0600486\pi$
$588$	0	0
$589$	− 9.97227i	− 0.410900i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 19.6603i	− 0.807351i	−0.914902	−	0.403676i	$-0.867732\pi$
$593$	− 19.6603i	− 0.807351i	0.914902	−	0.403676i	$-0.132268\pi$
$594$	0	0
$595$	3.16108	0.129592
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.1822	0.947200	0.473600	−	0.880740i	$-0.342954\pi$
$599$	23.1822	0.947200	0.473600	+	0.880740i	$0.342954\pi$
$600$	0	0
$601$	−32.8564	−1.34024	−0.670120	−	0.742252i	$-0.733758\pi$
$601$	−32.8564	−1.34024	−0.670120	+	0.742252i	$0.733758\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19.0653	0.775115
$606$	0	0
$607$	− 20.9808i	− 0.851583i	−0.904821	−	0.425791i	$-0.859996\pi$
$607$	− 20.9808i	− 0.851583i	0.904821	−	0.425791i	$-0.140004\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 21.1544i	− 0.855815i
$612$	0	0
$613$	47.1893i	1.90596i	0.303036	+	0.952979i	$0.402000\pi$
$613$	47.1893i	1.90596i	−0.303036	+	0.952979i	$0.598000\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 28.9778i	− 1.16660i	−0.812256	−	0.583301i	$-0.801761\pi$
$617$	− 28.9778i	− 1.16660i	0.812256	−	0.583301i	$-0.198239\pi$
$618$	0	0
$619$	8.63624	0.347120	0.173560	−	0.984823i	$-0.444473\pi$
$619$	8.63624	0.347120	0.173560	+	0.984823i	$0.444473\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.51626	0.261069
$624$	0	0
$625$	−31.1962	−1.24785
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.4487	0.576108
$630$	0	0
$631$	31.3205i	1.24685i	0.781883	+	0.623425i	$0.214259\pi$
$631$	31.3205i	1.24685i	−0.781883	+	0.623425i	$0.785741\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	18.7195i	0.742862i
$636$	0	0
$637$	− 16.1154i	− 0.638517i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 6.62776i	− 0.261781i	−0.991397	−	0.130890i	$-0.958216\pi$
$641$	− 6.62776i	− 0.261781i	0.991397	−	0.130890i	$-0.0417836\pi$
$642$	0	0
$643$	−32.2309	−1.27106	−0.635531	−	0.772075i	$-0.719219\pi$
$643$	−32.2309	−1.27106	−0.635531	+	0.772075i	$0.719219\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.62209	0.378284	0.189142	−	0.981950i	$-0.439429\pi$
$647$	9.62209	0.378284	0.189142	+	0.981950i	$0.439429\pi$
$648$	0	0
$649$	−19.6077	−0.769669
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.7195	0.732551	0.366276	−	0.930506i	$-0.380633\pi$
$653$	18.7195	0.732551	0.366276	+	0.930506i	$0.380633\pi$
$654$	0	0
$655$	− 47.9090i	− 1.87196i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 16.1385i	− 0.628667i	−0.949313	−	0.314333i	$-0.898219\pi$
$659$	− 16.1385i	− 0.628667i	0.949313	−	0.314333i	$-0.101781\pi$
$660$	0	0
$661$	− 4.31812i	− 0.167955i	−0.996468	−	0.0839777i	$-0.973238\pi$
$661$	− 4.31812i	− 0.167955i	0.996468	−	0.0839777i	$-0.0267625\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 13.8647i	− 0.537651i
$666$	0	0
$667$	−20.4336	−0.791191
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.78989	0.339330
$672$	0	0
$673$	1.67949	0.0647397	0.0323698	−	0.999476i	$-0.489695\pi$
$673$	1.67949	0.0647397	0.0323698	+	0.999476i	$0.489695\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.68776	−0.333898	−0.166949	−	0.985966i	$-0.553391\pi$
$677$	−8.68776	−0.333898	−0.166949	+	0.985966i	$0.553391\pi$
$678$	0	0
$679$	6.53590i	0.250825i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 38.5299i	− 1.47431i	−0.675726	−	0.737153i	$-0.736170\pi$
$683$	− 38.5299i	− 1.47431i	0.675726	−	0.737153i	$-0.263830\pi$
$684$	0	0
$685$	− 39.7101i	− 1.51724i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 18.9396i	− 0.721541i
$690$	0	0
$691$	21.7696	0.828155	0.414077	−	0.910242i	$-0.364104\pi$
$691$	21.7696	0.828155	0.414077	+	0.910242i	$0.364104\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−51.7439	−1.96276
$696$	0	0
$697$	11.4115	0.432243
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0.199642	0.00754038	0.00377019	−	0.999993i	$-0.498800\pi$
$701$	0.199642	0.00754038	0.00377019	+	0.999993i	$0.498800\pi$
$702$	0	0
$703$	− 63.3731i	− 2.39016i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.49015i	− 0.0560428i
$708$	0	0
$709$	− 26.0877i	− 0.979745i	−0.871794	−	0.489872i	$-0.837043\pi$
$709$	− 26.0877i	− 0.979745i	0.871794	−	0.489872i	$-0.162957\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 6.21166i	− 0.232628i
$714$	0	0
$715$	14.1114	0.527736
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.3190	−0.906947	−0.453473	−	0.891270i	$-0.649815\pi$
$719$	−24.3190	−0.906947	−0.453473	+	0.891270i	$0.649815\pi$
$720$	0	0
$721$	−10.5359	−0.392377
$722$	0	0
$723$	0	0
$724$	0	0
$725$	13.1582	0.488684
$726$	0	0
$727$	24.7321i	0.917261i	0.888627	+	0.458630i	$0.151660\pi$
$727$	24.7321i	0.917261i	−0.888627	+	0.458630i	$0.848340\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 10.5772i	− 0.391212i
$732$	0	0
$733$	− 1.33603i	− 0.0493474i	−0.999696	−	0.0246737i	$-0.992145\pi$
$733$	− 1.33603i	− 0.0493474i	0.999696	−	0.0246737i	$-0.00785469\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.0144i	0.884583i
$738$	0	0
$739$	4.98614	0.183418	0.0917090	−	0.995786i	$-0.470767\pi$
$739$	4.98614	0.183418	0.0917090	+	0.995786i	$0.470767\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.4997	1.19230	0.596149	−	0.802874i	$-0.296697\pi$
$743$	32.4997	1.19230	0.596149	+	0.802874i	$0.296697\pi$
$744$	0	0
$745$	2.07180	0.0759048
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.14233	−0.297514
$750$	0	0
$751$	6.87564i	0.250896i	0.992100	+	0.125448i	$0.0400368\pi$
$751$	6.87564i	0.250896i	−0.992100	+	0.125448i	$0.959963\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	53.7237i	1.95521i
$756$	0	0
$757$	− 32.2309i	− 1.17145i	−0.810509	−	0.585726i	$-0.800810\pi$
$757$	− 32.2309i	− 1.17145i	0.810509	−	0.585726i	$-0.199190\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 15.1130i	− 0.547847i	−0.961751	−	0.273924i	$-0.911678\pi$
$761$	− 15.1130i	− 0.547847i	0.961751	−	0.273924i	$-0.0883216\pi$
$762$	0	0
$763$	1.82505	0.0660713
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.0144	0.867111
$768$	0	0
$769$	−28.4641	−1.02644	−0.513221	−	0.858257i	$-0.671548\pi$
$769$	−28.4641	−1.02644	−0.513221	+	0.858257i	$0.671548\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33.7137	−1.21260	−0.606298	−	0.795237i	$-0.707346\pi$
$773$	−33.7137	−1.21260	−0.606298	+	0.795237i	$0.707346\pi$
$774$	0	0
$775$	4.00000i	0.143684i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 50.0518i	− 1.79329i
$780$	0	0
$781$	14.9584i	0.535254i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	25.8719i	0.923409i
$786$	0	0
$787$	−44.0282	−1.56944	−0.784718	−	0.619853i	$-0.787192\pi$
$787$	−44.0282	−1.56944	−0.784718	+	0.619853i	$0.787192\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.5601	−0.482143
$792$	0	0
$793$	−10.7654	−0.382290
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.30639	−0.223384	−0.111692	−	0.993743i	$-0.535627\pi$
$797$	−6.30639	−0.223384	−0.111692	+	0.993743i	$0.535627\pi$
$798$	0	0
$799$	− 13.1769i	− 0.466166i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.5575i	0.478434i
$804$	0	0
$805$	− 8.63624i	− 0.304388i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	49.0542i	1.72465i	0.506353	+	0.862326i	$0.330993\pi$
$809$	49.0542i	1.72465i	−0.506353	+	0.862326i	$0.669007\pi$
$810$	0	0
$811$	3.65011	0.128173	0.0640863	−	0.997944i	$-0.479587\pi$
$811$	3.65011	0.128173	0.0640863	+	0.997944i	$0.479587\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.8647	0.485660
$816$	0	0
$817$	−46.3923	−1.62306
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.87152	−0.135117	−0.0675585	−	0.997715i	$-0.521521\pi$
$821$	−3.87152	−0.135117	−0.0675585	+	0.997715i	$0.521521\pi$
$822$	0	0
$823$	− 31.4641i	− 1.09677i	−0.836226	−	0.548385i	$-0.815243\pi$
$823$	− 31.4641i	− 1.09677i	0.836226	−	0.548385i	$-0.184757\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.68776i	− 0.302103i	−0.988526	−	0.151052i	$-0.951734\pi$
$827$	− 8.68776i	− 0.302103i	0.988526	−	0.151052i	$-0.0482659\pi$
$828$	0	0
$829$	− 22.2586i	− 0.773074i	−0.922274	−	0.386537i	$-0.873671\pi$
$829$	− 22.2586i	− 0.773074i	0.922274	−	0.386537i	$-0.126329\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 10.0382i	− 0.347803i
$834$	0	0
$835$	−44.0282	−1.52366
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−35.0779	−1.21102	−0.605512	−	0.795836i	$-0.707032\pi$
$839$	−35.0779	−1.21102	−0.605512	+	0.795836i	$0.707032\pi$
$840$	0	0
$841$	−5.80385	−0.200133
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.8657	0.648999
$846$	0	0
$847$	5.01924i	0.172463i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 39.4746i	− 1.35317i
$852$	0	0
$853$	17.2725i	0.591399i	0.955281	+	0.295699i	$0.0955526\pi$
$853$	17.2725i	0.591399i	−0.955281	+	0.295699i	$0.904447\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.8719i	0.883769i	0.897072	+	0.441884i	$0.145690\pi$
$857$	25.8719i	0.883769i	−0.897072	+	0.441884i	$0.854310\pi$
$858$	0	0
$859$	27.2448	0.929579	0.464790	−	0.885421i	$-0.346130\pi$
$859$	27.2448	0.929579	0.464790	+	0.885421i	$0.346130\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.24264	−0.144421	−0.0722106	−	0.997389i	$-0.523005\pi$
$863$	−4.24264	−0.144421	−0.0722106	+	0.997389i	$0.523005\pi$
$864$	0	0
$865$	−44.3205	−1.50694
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.6128	−0.427859
$870$	0	0
$871$	− 29.4115i	− 0.996572i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4.61660i	− 0.156069i
$876$	0	0
$877$	24.2627i	0.819292i	0.912245	+	0.409646i	$0.134348\pi$
$877$	24.2627i	0.819292i	−0.912245	+	0.409646i	$0.865652\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.0454i	0.742729i	0.928487	+	0.371364i	$0.121110\pi$
$881$	22.0454i	0.742729i	−0.928487	+	0.371364i	$0.878890\pi$
$882$	0	0
$883$	35.8810	1.20749	0.603746	−	0.797177i	$-0.293674\pi$
$883$	35.8810	1.20749	0.603746	+	0.797177i	$0.293674\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.3335	1.62288	0.811439	−	0.584437i	$-0.198684\pi$
$887$	48.3335	1.62288	0.811439	+	0.584437i	$0.198684\pi$
$888$	0	0
$889$	−4.92820	−0.165287
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−57.7949	−1.93403
$894$	0	0
$895$	− 69.0333i	− 2.30753i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.05146i	0.235179i
$900$	0	0
$901$	− 11.7973i	− 0.393026i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 51.7439i	− 1.72003i
$906$	0	0
$907$	−28.5808	−0.949010	−0.474505	−	0.880253i	$-0.657373\pi$
$907$	−28.5808	−0.949010	−0.474505	+	0.880253i	$0.657373\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	13.5601	0.449267	0.224634	−	0.974443i	$-0.427882\pi$
$911$	13.5601	0.449267	0.224634	+	0.974443i	$0.427882\pi$
$912$	0	0
$913$	−3.03332	−0.100388
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.6128	0.416511
$918$	0	0
$919$	− 15.6603i	− 0.516584i	−0.966067	−	0.258292i	$-0.916840\pi$
$919$	− 15.6603i	− 0.516584i	0.966067	−	0.258292i	$-0.0831597\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 18.3202i	− 0.603018i
$924$	0	0
$925$	25.4197i	0.835795i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23.9029i	0.784230i	0.919916	+	0.392115i	$0.128256\pi$
$929$	23.9029i	0.784230i	−0.919916	+	0.392115i	$0.871744\pi$
$930$	0	0
$931$	−44.0282	−1.44297
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.78989	0.287460
$936$	0	0
$937$	1.67949	0.0548666	0.0274333	−	0.999624i	$-0.491267\pi$
$937$	1.67949	0.0548666	0.0274333	+	0.999624i	$0.491267\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29.6425	−0.966318	−0.483159	−	0.875533i	$-0.660511\pi$
$941$	−29.6425	−0.966318	−0.483159	+	0.875533i	$0.660511\pi$
$942$	0	0
$943$	− 31.1769i	− 1.01526i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.01588i	0.162994i	0.996674	+	0.0814971i	$0.0259701\pi$
$947$	5.01588i	0.162994i	−0.996674	+	0.0814971i	$0.974030\pi$
$948$	0	0
$949$	− 16.6045i	− 0.539004i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 51.3278i	− 1.66267i	−0.555771	−	0.831335i	$-0.687577\pi$
$953$	− 51.3278i	− 1.66267i	0.555771	−	0.831335i	$-0.312423\pi$
$954$	0	0
$955$	67.6229	2.18822
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.4543	0.337587
$960$	0	0
$961$	28.8564	0.930852
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.4843	0.530648
$966$	0	0
$967$	33.6603i	1.08244i	0.840881	+	0.541220i	$0.182038\pi$
$967$	33.6603i	1.08244i	−0.840881	+	0.541220i	$0.817962\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 36.4943i	− 1.17116i	−0.810615	−	0.585579i	$-0.800867\pi$
$971$	− 36.4943i	− 1.17116i	0.810615	−	0.585579i	$-0.199133\pi$
$972$	0	0
$973$	− 13.6224i	− 0.436713i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.3190i	− 0.778035i	−0.921230	−	0.389017i	$-0.872815\pi$
$977$	− 24.3190i	− 0.778035i	0.921230	−	0.389017i	$-0.127185\pi$
$978$	0	0
$979$	18.1195	0.579102
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57.9555	−1.84849	−0.924247	−	0.381794i	$-0.875306\pi$
$983$	−57.9555	−1.84849	−0.924247	+	0.381794i	$0.875306\pi$
$984$	0	0
$985$	−51.4974	−1.64084
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−28.8974	−0.918885
$990$	0	0
$991$	− 11.1244i	− 0.353377i	−0.984267	−	0.176688i	$-0.943462\pi$
$991$	− 11.1244i	− 0.353377i	0.984267	−	0.176688i	$-0.0565385\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	66.3365i	2.10301i
$996$	0	0
$997$	− 51.5074i	− 1.63126i	−0.578576	−	0.815628i	$-0.696391\pi$
$997$	− 51.5074i	− 1.63126i	0.578576	−	0.815628i	$-0.303609\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.f.d.2591.15	yes	16
3.2	odd	2	inner	5184.2.f.d.2591.3	yes	16
4.3	odd	2	inner	5184.2.f.d.2591.13	yes	16
8.3	odd	2	inner	5184.2.f.d.2591.2	yes	16
8.5	even	2	inner	5184.2.f.d.2591.4	yes	16
12.11	even	2	inner	5184.2.f.d.2591.1	✓	16
24.5	odd	2	inner	5184.2.f.d.2591.16	yes	16
24.11	even	2	inner	5184.2.f.d.2591.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5184.2.f.d.2591.1	✓	16	12.11	even	2	inner
5184.2.f.d.2591.2	yes	16	8.3	odd	2	inner
5184.2.f.d.2591.3	yes	16	3.2	odd	2	inner
5184.2.f.d.2591.4	yes	16	8.5	even	2	inner
5184.2.f.d.2591.13	yes	16	4.3	odd	2	inner
5184.2.f.d.2591.14	yes	16	24.11	even	2	inner
5184.2.f.d.2591.15	yes	16	1.1	even	1	trivial
5184.2.f.d.2591.16	yes	16	24.5	odd	2	inner

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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.78066 q^{5} +0.732051i q^{7} +O(q^{10})\)	\(q+2.78066 q^{5} +0.732051i q^{7} +2.03558i q^{11} -2.49307i q^{13} -1.55291i q^{17} -6.81119 q^{19} -4.24264 q^{23} +2.73205 q^{25} +4.81624 q^{29} +1.46410i q^{31} +2.03558i q^{35} +9.30426i q^{37} +7.34847i q^{41} +6.81119 q^{43} +8.48528 q^{47} +6.46410 q^{49} +7.59689 q^{53} +5.66025i q^{55} +9.63248i q^{59} -4.31812i q^{61} -6.93237i q^{65} +11.7973 q^{67} +7.34847 q^{71} +6.66025 q^{73} -1.49015 q^{77} +6.19615i q^{79} +1.49015i q^{83} -4.31812i q^{85} -8.90138i q^{89} +1.82505 q^{91} -18.9396 q^{95} +8.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{25} + 48 q^{49} - 32 q^{73} + 32 q^{97}+O(q^{100}) \) 16 * q + 16 * q^25 + 48 * q^49 - 32 * q^73 + 32 * q^97

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