LMFDB - Embedded newform 2160.2.h.g.431.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,2,Mod(431,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.431"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1731891456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.4
Root			$-2.21837 + 1.28078i$ of defining polynomial
Character	$\chi$	$=$	2160.431
Dual form			2160.2.h.g.431.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} +4.43674i q^{7} -6.16879 q^{11} +1.00000 q^{13} +4.68466i q^{17} -4.43674i q^{19} -6.16879 q^{23} -1.00000 q^{25} -10.6847i q^{29} -6.16879i q^{31} +4.43674 q^{35} +9.68466 q^{37} -6.00000i q^{41} -6.16879i q^{43} -4.22351 q^{47} -12.6847 q^{49} -6.00000i q^{53} +6.16879i q^{55} +12.3376 q^{59} -5.68466 q^{61} -1.00000i q^{65} +9.84612i q^{67} +1.94528 q^{71} -3.68466 q^{73} -27.3693i q^{77} -0.213225i q^{79} -10.3923 q^{83} +4.68466 q^{85} -12.0000i q^{89} +4.43674i q^{91} -4.43674 q^{95} +5.68466 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{13} - 8 q^{25} + 28 q^{37} - 52 q^{49} + 4 q^{61} + 20 q^{73} - 12 q^{85} - 4 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 - 8 * q^25 + 28 * q^37 - 52 * q^49 + 4 * q^61 + 20 * q^73 - 12 * q^85 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	4.43674i	1.67693i	0.544955	+	0.838465i	$0.316547\pi$
$7$	4.43674i	1.67693i	−0.544955	+	0.838465i	$0.683453\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.16879	−1.85996	−0.929980	−	0.367610i	$-0.880176\pi$
$11$	−6.16879	−1.85996	−0.929980	+	0.367610i	$0.880176\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.68466i	1.13620i	0.822961	+	0.568098i	$0.192321\pi$
$17$	4.68466i	1.13620i	−0.822961	+	0.568098i	$0.807679\pi$
$18$	0	0
$19$	− 4.43674i	− 1.01786i	−0.860809	−	0.508929i	$-0.830042\pi$
$19$	− 4.43674i	− 1.01786i	0.860809	−	0.508929i	$-0.169958\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.16879	−1.28628	−0.643141	−	0.765748i	$-0.722369\pi$
$23$	−6.16879	−1.28628	−0.643141	+	0.765748i	$0.722369\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 10.6847i	− 1.98409i	−0.125879	−	0.992046i	$-0.540175\pi$
$29$	− 10.6847i	− 1.98409i	0.125879	−	0.992046i	$-0.459825\pi$
$30$	0	0
$31$	− 6.16879i	− 1.10795i	−0.832534	−	0.553974i	$-0.813111\pi$
$31$	− 6.16879i	− 1.10795i	0.832534	−	0.553974i	$-0.186889\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.43674	0.749946
$36$	0	0
$37$	9.68466	1.59215	0.796074	−	0.605199i	$-0.206907\pi$
$37$	9.68466	1.59215	0.796074	+	0.605199i	$0.206907\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	0	0
$43$	− 6.16879i	− 0.940732i	−0.882472	−	0.470366i	$-0.844122\pi$
$43$	− 6.16879i	− 0.940732i	0.882472	−	0.470366i	$-0.155878\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.22351	−0.616063	−0.308031	−	0.951376i	$-0.599670\pi$
$47$	−4.22351	−0.616063	−0.308031	+	0.951376i	$0.599670\pi$
$48$	0	0
$49$	−12.6847	−1.81209
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	6.16879i	0.831800i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.3376	1.60622	0.803108	−	0.595833i	$-0.203178\pi$
$59$	12.3376	1.60622	0.803108	+	0.595833i	$0.203178\pi$
$60$	0	0
$61$	−5.68466	−0.727846	−0.363923	−	0.931429i	$-0.618563\pi$
$61$	−5.68466	−0.727846	−0.363923	+	0.931429i	$0.618563\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.00000i	− 0.124035i
$66$	0	0
$67$	9.84612i	1.20289i	0.798912	+	0.601447i	$0.205409\pi$
$67$	9.84612i	1.20289i	−0.798912	+	0.601447i	$0.794591\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.94528	0.230862	0.115431	−	0.993316i	$-0.463175\pi$
$71$	1.94528	0.230862	0.115431	+	0.993316i	$0.463175\pi$
$72$	0	0
$73$	−3.68466	−0.431257	−0.215628	−	0.976476i	$-0.569180\pi$
$73$	−3.68466	−0.431257	−0.215628	+	0.976476i	$0.569180\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 27.3693i	− 3.11902i
$78$	0	0
$79$	− 0.213225i	− 0.0239897i	−0.999928	−	0.0119949i	$-0.996182\pi$
$79$	− 0.213225i	− 0.0239897i	0.999928	−	0.0119949i	$-0.00381817\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.3923	−1.14070	−0.570352	−	0.821401i	$-0.693193\pi$
$83$	−10.3923	−1.14070	−0.570352	+	0.821401i	$0.693193\pi$
$84$	0	0
$85$	4.68466	0.508123
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.0000i	− 1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$	− 12.0000i	− 1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$	0	0
$91$	4.43674i	0.465097i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.43674	−0.455200
$96$	0	0
$97$	5.68466	0.577190	0.288595	−	0.957451i	$-0.406812\pi$
$97$	5.68466	0.577190	0.288595	+	0.957451i	$0.406812\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.68466i	0.466141i	0.972460	+	0.233070i	$0.0748773\pi$
$101$	4.68466i	0.466141i	−0.972460	+	0.233070i	$0.925123\pi$
$102$	0	0
$103$	− 7.90084i	− 0.778493i	−0.921134	−	0.389247i	$-0.872735\pi$
$103$	− 7.90084i	− 0.778493i	0.921134	−	0.389247i	$-0.127265\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.94528	−0.188057	−0.0940285	−	0.995570i	$-0.529974\pi$
$107$	−1.94528	−0.188057	−0.0940285	+	0.995570i	$0.529974\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.31534i	− 0.123737i	−0.998084	−	0.0618685i	$-0.980294\pi$
$113$	− 1.31534i	− 0.123737i	0.998084	−	0.0618685i	$-0.0197059\pi$
$114$	0	0
$115$	6.16879i	0.575243i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−20.7846	−1.90532
$120$	0	0
$121$	27.0540	2.45945
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	10.3923i	0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$	10.3923i	0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.22351	0.369010	0.184505	−	0.982832i	$-0.440932\pi$
$131$	4.22351	0.369010	0.184505	+	0.982832i	$0.440932\pi$
$132$	0	0
$133$	19.6847	1.70688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	2.49146i	0.211323i	0.994402	+	0.105662i	$0.0336960\pi$
$139$	2.49146i	0.211323i	−0.994402	+	0.105662i	$0.966304\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.16879	−0.515860
$144$	0	0
$145$	−10.6847	−0.887313
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.0540i	1.15135i	0.817680	+	0.575673i	$0.195260\pi$
$149$	14.0540i	1.15135i	−0.817680	+	0.575673i	$0.804740\pi$
$150$	0	0
$151$	3.67733i	0.299257i	0.988742	+	0.149628i	$0.0478077\pi$
$151$	3.67733i	0.299257i	−0.988742	+	0.149628i	$0.952192\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.16879	−0.495489
$156$	0	0
$157$	−24.0540	−1.91972	−0.959858	−	0.280486i	$-0.909504\pi$
$157$	−24.0540	−1.91972	−0.959858	+	0.280486i	$0.909504\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 27.3693i	− 2.15700i
$162$	0	0
$163$	− 12.1244i	− 0.949653i	−0.880079	−	0.474826i	$-0.842511\pi$
$163$	− 12.1244i	− 0.949653i	0.880079	−	0.474826i	$-0.157489\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.3376	−0.954711	−0.477355	−	0.878710i	$-0.658405\pi$
$167$	−12.3376	−0.954711	−0.477355	+	0.878710i	$0.658405\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	− 4.43674i	− 0.335386i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.2829	−1.06755	−0.533775	−	0.845626i	$-0.679227\pi$
$179$	−14.2829	−1.06755	−0.533775	+	0.845626i	$0.679227\pi$
$180$	0	0
$181$	−19.0540	−1.41627	−0.708135	−	0.706077i	$-0.750463\pi$
$181$	−19.0540	−1.41627	−0.708135	+	0.706077i	$0.750463\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 9.68466i	− 0.712030i
$186$	0	0
$187$	− 28.8987i	− 2.11328i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.7846	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$191$	−20.7846	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$192$	0	0
$193$	12.3153	0.886478	0.443239	−	0.896404i	$-0.353829\pi$
$193$	12.3153	0.886478	0.443239	+	0.896404i	$0.353829\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	− 1.73205i	− 0.122782i	−0.998114	−	0.0613909i	$-0.980446\pi$
$199$	− 1.73205i	− 0.122782i	0.998114	−	0.0613909i	$-0.0195536\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	47.4050	3.32718
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	27.3693i	1.89318i
$210$	0	0
$211$	− 9.84612i	− 0.677835i	−0.940816	−	0.338917i	$-0.889939\pi$
$211$	− 9.84612i	− 0.677835i	0.940816	−	0.338917i	$-0.110061\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.16879	−0.420708
$216$	0	0
$217$	27.3693	1.85795
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.68466i	0.315124i
$222$	0	0
$223$	1.94528i	0.130265i	0.997877	+	0.0651327i	$0.0207471\pi$
$223$	1.94528i	0.130265i	−0.997877	+	0.0651327i	$0.979253\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.94528	0.129113	0.0645563	−	0.997914i	$-0.479437\pi$
$227$	1.94528	0.129113	0.0645563	+	0.997914i	$0.479437\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 21.3693i	− 1.39995i	−0.714167	−	0.699975i	$-0.753194\pi$
$233$	− 21.3693i	− 1.39995i	0.714167	−	0.699975i	$-0.246806\pi$
$234$	0	0
$235$	4.22351i	0.275512i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.94528	−0.125829	−0.0629147	−	0.998019i	$-0.520040\pi$
$239$	−1.94528	−0.125829	−0.0629147	+	0.998019i	$0.520040\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.6847i	0.810393i
$246$	0	0
$247$	− 4.43674i	− 0.282303i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.6158	0.922542	0.461271	−	0.887259i	$-0.347394\pi$
$251$	14.6158	0.922542	0.461271	+	0.887259i	$0.347394\pi$
$252$	0	0
$253$	38.0540	2.39243
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.68466i	0.292221i	0.989268	+	0.146110i	$0.0466755\pi$
$257$	4.68466i	0.292221i	−0.989268	+	0.146110i	$0.953324\pi$
$258$	0	0
$259$	42.9683i	2.66992i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.44703	−0.520866	−0.260433	−	0.965492i	$-0.583865\pi$
$263$	−8.44703	−0.520866	−0.260433	+	0.965492i	$0.583865\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 14.0540i	− 0.856886i	−0.903569	−	0.428443i	$-0.859062\pi$
$269$	− 14.0540i	− 0.856886i	0.903569	−	0.428443i	$-0.140938\pi$
$270$	0	0
$271$	− 12.8838i	− 0.782633i	−0.920256	−	0.391317i	$-0.872020\pi$
$271$	− 12.8838i	− 0.782633i	0.920256	−	0.391317i	$-0.127980\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.16879	0.371992
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.63068i	0.156933i	0.996917	+	0.0784667i	$0.0250024\pi$
$281$	2.63068i	0.156933i	−0.996917	+	0.0784667i	$0.974998\pi$
$282$	0	0
$283$	14.2829i	0.849028i	0.905421	+	0.424514i	$0.139555\pi$
$283$	14.2829i	0.849028i	−0.905421	+	0.424514i	$0.860445\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26.6204	1.57135
$288$	0	0
$289$	−4.94602	−0.290943
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 15.3693i	− 0.897885i	−0.893561	−	0.448943i	$-0.851801\pi$
$293$	− 15.3693i	− 0.897885i	0.893561	−	0.448943i	$-0.148199\pi$
$294$	0	0
$295$	− 12.3376i	− 0.718322i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.16879	−0.356750
$300$	0	0
$301$	27.3693	1.57754
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.68466i	0.325503i
$306$	0	0
$307$	− 10.0593i	− 0.574117i	−0.957913	−	0.287058i	$-0.907323\pi$
$307$	− 10.0593i	− 0.574117i	0.957913	−	0.287058i	$-0.0926774\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.3376	0.699600	0.349800	−	0.936824i	$-0.386250\pi$
$311$	12.3376	0.699600	0.349800	+	0.936824i	$0.386250\pi$
$312$	0	0
$313$	−9.05398	−0.511761	−0.255880	−	0.966708i	$-0.582365\pi$
$313$	−9.05398	−0.511761	−0.255880	+	0.966708i	$0.582365\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.3693i	1.20022i	0.799917	+	0.600110i	$0.204877\pi$
$317$	21.3693i	1.20022i	−0.799917	+	0.600110i	$0.795123\pi$
$318$	0	0
$319$	65.9114i	3.69033i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	20.7846	1.15649
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 18.7386i	− 1.03309i
$330$	0	0
$331$	2.49146i	0.136943i	0.997653	+	0.0684716i	$0.0218123\pi$
$331$	2.49146i	0.136943i	−0.997653	+	0.0684716i	$0.978188\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.84612	0.537951
$336$	0	0
$337$	−25.0540	−1.36478	−0.682388	−	0.730990i	$-0.739059\pi$
$337$	−25.0540	−1.36478	−0.682388	+	0.730990i	$0.739059\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	38.0540i	2.06074i
$342$	0	0
$343$	− 25.2213i	− 1.36182i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.1769	1.67366	0.836832	−	0.547459i	$-0.184405\pi$
$347$	31.1769	1.67366	0.836832	+	0.547459i	$0.184405\pi$
$348$	0	0
$349$	−8.31534	−0.445110	−0.222555	−	0.974920i	$-0.571440\pi$
$349$	−8.31534	−0.445110	−0.222555	+	0.974920i	$0.571440\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.3153i	1.02805i	0.857775	+	0.514026i	$0.171847\pi$
$353$	19.3153i	1.02805i	−0.857775	+	0.514026i	$0.828153\pi$
$354$	0	0
$355$	− 1.94528i	− 0.103245i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.2829	0.753820	0.376910	−	0.926250i	$-0.376987\pi$
$359$	14.2829	0.753820	0.376910	+	0.926250i	$0.376987\pi$
$360$	0	0
$361$	−0.684658	−0.0360347
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.68466i	0.192864i
$366$	0	0
$367$	− 20.2384i	− 1.05644i	−0.849108	−	0.528219i	$-0.822860\pi$
$367$	− 20.2384i	− 1.05644i	0.849108	−	0.528219i	$-0.177140\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	26.6204	1.38206
$372$	0	0
$373$	26.3693	1.36535	0.682676	−	0.730721i	$-0.260816\pi$
$373$	26.3693	1.36535	0.682676	+	0.730721i	$0.260816\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10.6847i	− 0.550288i
$378$	0	0
$379$	− 0.546188i	− 0.0280558i	−0.999902	−	0.0140279i	$-0.995535\pi$
$379$	− 0.546188i	− 0.0280558i	0.999902	−	0.0140279i	$-0.00446536\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.4516	−1.04503	−0.522515	−	0.852630i	$-0.675006\pi$
$383$	−20.4516	−1.04503	−0.522515	+	0.852630i	$0.675006\pi$
$384$	0	0
$385$	−27.3693	−1.39487
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.31534i	0.370903i	0.982653	+	0.185451i	$0.0593747\pi$
$389$	7.31534i	0.370903i	−0.982653	+	0.185451i	$0.940625\pi$
$390$	0	0
$391$	− 28.8987i	− 1.46147i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.213225	−0.0107285
$396$	0	0
$397$	−12.0540	−0.604972	−0.302486	−	0.953154i	$-0.597816\pi$
$397$	−12.0540	−0.604972	−0.302486	+	0.953154i	$0.597816\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.00000i	0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$	6.00000i	0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$	0	0
$403$	− 6.16879i	− 0.307289i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−59.7426	−2.96133
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	54.7386i	2.69351i
$414$	0	0
$415$	10.3923i	0.510138i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.6705	0.618997	0.309498	−	0.950900i	$-0.399839\pi$
$419$	12.6705	0.618997	0.309498	+	0.950900i	$0.399839\pi$
$420$	0	0
$421$	2.31534	0.112843	0.0564214	−	0.998407i	$-0.482031\pi$
$421$	2.31534	0.112843	0.0564214	+	0.998407i	$0.482031\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.68466i	− 0.227239i
$426$	0	0
$427$	− 25.2213i	− 1.22055i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.1734	−0.875382	−0.437691	−	0.899125i	$-0.644204\pi$
$431$	−18.1734	−0.875382	−0.437691	+	0.899125i	$0.644204\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.3693i	1.30925i
$438$	0	0
$439$	6.50175i	0.310312i	0.987890	+	0.155156i	$0.0495880\pi$
$439$	6.50175i	0.310312i	−0.987890	+	0.155156i	$0.950412\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−40.9033	−1.94337	−0.971687	−	0.236271i	$-0.924075\pi$
$443$	−40.9033	−1.94337	−0.971687	+	0.236271i	$0.924075\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.7386i	1.45065i	0.688409	+	0.725323i	$0.258310\pi$
$449$	30.7386i	1.45065i	−0.688409	+	0.725323i	$0.741690\pi$
$450$	0	0
$451$	37.0127i	1.74286i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.43674	0.207998
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.63068i	0.122523i	0.998122	+	0.0612616i	$0.0195124\pi$
$461$	2.63068i	0.122523i	−0.998122	+	0.0612616i	$0.980488\pi$
$462$	0	0
$463$	− 7.90084i	− 0.367183i	−0.983003	−	0.183592i	$-0.941228\pi$
$463$	− 7.90084i	− 0.367183i	0.983003	−	0.183592i	$-0.0587724\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.3376	0.570915	0.285458	−	0.958391i	$-0.407854\pi$
$467$	12.3376	0.570915	0.285458	+	0.958391i	$0.407854\pi$
$468$	0	0
$469$	−43.6847	−2.01717
$470$	0	0
$471$	0	0
$472$	0	0
$473$	38.0540i	1.74972i
$474$	0	0
$475$	4.43674i	0.203572i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.3923	0.474837	0.237418	−	0.971408i	$-0.423699\pi$
$479$	10.3923	0.474837	0.237418	+	0.971408i	$0.423699\pi$
$480$	0	0
$481$	9.68466	0.441582
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 5.68466i	− 0.258127i
$486$	0	0
$487$	21.3308i	0.966591i	0.875457	+	0.483295i	$0.160560\pi$
$487$	21.3308i	0.966591i	−0.875457	+	0.483295i	$0.839440\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.3923	0.468998	0.234499	−	0.972116i	$-0.424655\pi$
$491$	10.3923	0.468998	0.234499	+	0.972116i	$0.424655\pi$
$492$	0	0
$493$	50.0540	2.25432
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.63068i	0.387139i
$498$	0	0
$499$	− 18.1734i	− 0.813554i	−0.913528	−	0.406777i	$-0.866653\pi$
$499$	− 18.1734i	− 0.813554i	0.913528	−	0.406777i	$-0.133347\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.27824	−0.101582	−0.0507908	−	0.998709i	$-0.516174\pi$
$503$	−2.27824	−0.101582	−0.0507908	+	0.998709i	$0.516174\pi$
$504$	0	0
$505$	4.68466	0.208465
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 22.6847i	− 1.00548i	−0.864438	−	0.502740i	$-0.832325\pi$
$509$	− 22.6847i	− 1.00548i	0.864438	−	0.502740i	$-0.167675\pi$
$510$	0	0
$511$	− 16.3479i	− 0.723187i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7.90084	−0.348153
$516$	0	0
$517$	26.0540	1.14585
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.7386i	1.08382i	0.840437	+	0.541910i	$0.182298\pi$
$521$	24.7386i	1.08382i	−0.840437	+	0.541910i	$0.817702\pi$
$522$	0	0
$523$	− 6.71498i	− 0.293625i	−0.989164	−	0.146813i	$-0.953099\pi$
$523$	− 6.71498i	− 0.293625i	0.989164	−	0.146813i	$-0.0469015\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.8987	1.25885
$528$	0	0
$529$	15.0540	0.654521
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	1.94528i	0.0841016i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	78.2490	3.37042
$540$	0	0
$541$	11.6847	0.502363	0.251181	−	0.967940i	$-0.419181\pi$
$541$	11.6847	0.502363	0.251181	+	0.967940i	$0.419181\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14.0000i	0.599694i
$546$	0	0
$547$	− 20.5714i	− 0.879569i	−0.898103	−	0.439784i	$-0.855055\pi$
$547$	− 20.5714i	− 0.879569i	0.898103	−	0.439784i	$-0.144945\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−47.4050	−2.01952
$552$	0	0
$553$	0.946025	0.0402291
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.6307i	0.619922i	0.950749	+	0.309961i	$0.100316\pi$
$557$	14.6307i	0.619922i	−0.950749	+	0.309961i	$0.899684\pi$
$558$	0	0
$559$	− 6.16879i	− 0.260912i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.5657	−1.20390	−0.601951	−	0.798533i	$-0.705610\pi$
$563$	−28.5657	−1.20390	−0.601951	+	0.798533i	$0.705610\pi$
$564$	0	0
$565$	−1.31534	−0.0553368
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.0000i	0.503066i	0.967849	+	0.251533i	$0.0809347\pi$
$569$	12.0000i	0.503066i	−0.967849	+	0.251533i	$0.919065\pi$
$570$	0	0
$571$	32.5760i	1.36326i	0.731695	+	0.681632i	$0.238729\pi$
$571$	32.5760i	1.36326i	−0.731695	+	0.681632i	$0.761271\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.16879	0.257256
$576$	0	0
$577$	24.3153	1.01226	0.506130	−	0.862457i	$-0.331075\pi$
$577$	24.3153	1.01226	0.506130	+	0.862457i	$0.331075\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 46.1080i	− 1.91288i
$582$	0	0
$583$	37.0127i	1.53291i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.1769	1.28681	0.643404	−	0.765526i	$-0.277521\pi$
$587$	31.1769	1.28681	0.643404	+	0.765526i	$0.277521\pi$
$588$	0	0
$589$	−27.3693	−1.12773
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1.31534i	− 0.0540146i	−0.999635	−	0.0270073i	$-0.991402\pi$
$593$	− 1.31534i	− 0.0540146i	0.999635	−	0.0270073i	$-0.00859774\pi$
$594$	0	0
$595$	20.7846i	0.852086i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.83583	0.238445	0.119223	−	0.992868i	$-0.461960\pi$
$599$	5.83583	0.238445	0.119223	+	0.992868i	$0.461960\pi$
$600$	0	0
$601$	40.0540	1.63384	0.816918	−	0.576754i	$-0.195681\pi$
$601$	40.0540	1.63384	0.816918	+	0.576754i	$0.195681\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 27.0540i	− 1.09990i
$606$	0	0
$607$	− 7.90084i	− 0.320685i	−0.987061	−	0.160343i	$-0.948740\pi$
$607$	− 7.90084i	− 0.320685i	0.987061	−	0.160343i	$-0.0512599\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.22351	−0.170865
$612$	0	0
$613$	−4.36932	−0.176475	−0.0882375	−	0.996099i	$-0.528123\pi$
$613$	−4.36932	−0.176475	−0.0882375	+	0.996099i	$0.528123\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.6847i	0.913250i	0.889659	+	0.456625i	$0.150942\pi$
$617$	22.6847i	0.913250i	−0.889659	+	0.456625i	$0.849058\pi$
$618$	0	0
$619$	32.5760i	1.30934i	0.755915	+	0.654670i	$0.227192\pi$
$619$	32.5760i	1.30934i	−0.755915	+	0.654670i	$0.772808\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	53.2409	2.13305
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	45.3693i	1.80899i
$630$	0	0
$631$	− 37.5589i	− 1.49520i	−0.664151	−	0.747599i	$-0.731207\pi$
$631$	− 37.5589i	− 1.49520i	0.664151	−	0.747599i	$-0.268793\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.3923	0.412406
$636$	0	0
$637$	−12.6847	−0.502584
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.7386i	1.68807i	0.536285	+	0.844037i	$0.319827\pi$
$641$	42.7386i	1.68807i	−0.536285	+	0.844037i	$0.680173\pi$
$642$	0	0
$643$	− 39.2910i	− 1.54949i	−0.632277	−	0.774743i	$-0.717879\pi$
$643$	− 39.2910i	− 1.54949i	0.632277	−	0.774743i	$-0.282121\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.9580	−1.53160	−0.765799	−	0.643080i	$-0.777656\pi$
$647$	−38.9580	−1.53160	−0.765799	+	0.643080i	$0.777656\pi$
$648$	0	0
$649$	−76.1080	−2.98750
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 21.3693i	− 0.836246i	−0.908390	−	0.418123i	$-0.862688\pi$
$653$	− 21.3693i	− 0.836246i	0.908390	−	0.418123i	$-0.137312\pi$
$654$	0	0
$655$	− 4.22351i	− 0.165026i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.2281	0.632158	0.316079	−	0.948733i	$-0.397633\pi$
$659$	16.2281	0.632158	0.316079	+	0.948733i	$0.397633\pi$
$660$	0	0
$661$	−24.4233	−0.949956	−0.474978	−	0.879998i	$-0.657544\pi$
$661$	−24.4233	−0.949956	−0.474978	+	0.879998i	$0.657544\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 19.6847i	− 0.763338i
$666$	0	0
$667$	65.9114i	2.55210i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	35.0675	1.35376
$672$	0	0
$673$	33.6847	1.29845	0.649224	−	0.760597i	$-0.275094\pi$
$673$	33.6847	1.29845	0.649224	+	0.760597i	$0.275094\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 42.7386i	− 1.64258i	−0.570512	−	0.821290i	$-0.693255\pi$
$677$	− 42.7386i	− 1.64258i	0.570512	−	0.821290i	$-0.306745\pi$
$678$	0	0
$679$	25.2213i	0.967907i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	35.0675	1.34182	0.670910	−	0.741539i	$-0.265904\pi$
$683$	35.0675	1.34182	0.670910	+	0.741539i	$0.265904\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 6.00000i	− 0.228582i
$690$	0	0
$691$	− 5.83583i	− 0.222005i	−0.993820	−	0.111003i	$-0.964594\pi$
$691$	− 5.83583i	− 0.222005i	0.993820	−	0.111003i	$-0.0354062\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.49146	0.0945066
$696$	0	0
$697$	28.1080	1.06466
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 20.0540i	− 0.757428i	−0.925514	−	0.378714i	$-0.876366\pi$
$701$	− 20.0540i	− 0.757428i	0.925514	−	0.378714i	$-0.123634\pi$
$702$	0	0
$703$	− 42.9683i	− 1.62058i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.7846	−0.781686
$708$	0	0
$709$	−40.4233	−1.51813	−0.759064	−	0.651016i	$-0.774343\pi$
$709$	−40.4233	−1.51813	−0.759064	+	0.651016i	$0.774343\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	38.0540i	1.42513i
$714$	0	0
$715$	6.16879i	0.230700i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−38.9580	−1.45289	−0.726444	−	0.687225i	$-0.758829\pi$
$719$	−38.9580	−1.45289	−0.726444	+	0.687225i	$0.758829\pi$
$720$	0	0
$721$	35.0540	1.30548
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.6847i	0.396818i
$726$	0	0
$727$	18.1734i	0.674014i	0.941502	+	0.337007i	$0.109415\pi$
$727$	18.1734i	0.674014i	−0.941502	+	0.337007i	$0.890585\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	28.8987	1.06886
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 60.7386i	− 2.23734i
$738$	0	0
$739$	− 47.4050i	− 1.74382i	−0.489664	−	0.871911i	$-0.662880\pi$
$739$	− 47.4050i	− 1.74382i	0.489664	−	0.871911i	$-0.337120\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−0.332962	−0.0122152	−0.00610760	−	0.999981i	$-0.501944\pi$
$743$	−0.332962	−0.0122152	−0.00610760	+	0.999981i	$0.501944\pi$
$744$	0	0
$745$	14.0540	0.514898
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 8.63068i	− 0.315358i
$750$	0	0
$751$	− 4.76970i	− 0.174049i	−0.996206	−	0.0870244i	$-0.972264\pi$
$751$	− 4.76970i	− 0.174049i	0.996206	−	0.0870244i	$-0.0277358\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.67733	0.133832
$756$	0	0
$757$	43.7386	1.58971	0.794854	−	0.606801i	$-0.207547\pi$
$757$	43.7386	1.58971	0.794854	+	0.606801i	$0.207547\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	48.7386i	1.76677i	0.468644	+	0.883387i	$0.344743\pi$
$761$	48.7386i	1.76677i	−0.468644	+	0.883387i	$0.655257\pi$
$762$	0	0
$763$	− 62.1144i	− 2.24869i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.3376	0.445484
$768$	0	0
$769$	39.1080	1.41027	0.705134	−	0.709074i	$-0.250887\pi$
$769$	39.1080	1.41027	0.705134	+	0.709074i	$0.250887\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 34.1080i	− 1.22678i	−0.789781	−	0.613389i	$-0.789806\pi$
$773$	− 34.1080i	− 1.22678i	0.789781	−	0.613389i	$-0.210194\pi$
$774$	0	0
$775$	6.16879i	0.221589i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26.6204	−0.953776
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.0540i	0.858523i
$786$	0	0
$787$	− 27.4996i	− 0.980254i	−0.871651	−	0.490127i	$-0.836950\pi$
$787$	− 27.4996i	− 0.980254i	0.871651	−	0.490127i	$-0.163050\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.83583	0.207498
$792$	0	0
$793$	−5.68466	−0.201868
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.7386i	1.08882i	0.838820	+	0.544409i	$0.183246\pi$
$797$	30.7386i	1.08882i	−0.838820	+	0.544409i	$0.816754\pi$
$798$	0	0
$799$	− 19.7857i	− 0.699969i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	22.7299	0.802120
$804$	0	0
$805$	−27.3693	−0.964642
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 45.3693i	− 1.59510i	−0.603253	−	0.797550i	$-0.706129\pi$
$809$	− 45.3693i	− 1.59510i	0.603253	−	0.797550i	$-0.293871\pi$
$810$	0	0
$811$	− 55.1862i	− 1.93785i	−0.247359	−	0.968924i	$-0.579563\pi$
$811$	− 55.1862i	− 1.93785i	0.247359	−	0.968924i	$-0.420437\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.1244	−0.424698
$816$	0	0
$817$	−27.3693	−0.957531
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 26.6307i	− 0.929417i	−0.885464	−	0.464709i	$-0.846159\pi$
$821$	− 26.6307i	− 0.929417i	0.885464	−	0.464709i	$-0.153841\pi$
$822$	0	0
$823$	− 6.38202i	− 0.222463i	−0.993795	−	0.111232i	$-0.964520\pi$
$823$	− 6.38202i	− 0.222463i	0.993795	−	0.111232i	$-0.0354795\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−24.6752	−0.858039	−0.429020	−	0.903295i	$-0.641141\pi$
$827$	−24.6752	−0.858039	−0.429020	+	0.903295i	$0.641141\pi$
$828$	0	0
$829$	18.4233	0.639867	0.319934	−	0.947440i	$-0.396339\pi$
$829$	18.4233	0.639867	0.319934	+	0.947440i	$0.396339\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 59.4233i	− 2.05890i
$834$	0	0
$835$	12.3376i	0.426960i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−49.3503	−1.70376	−0.851881	−	0.523735i	$-0.824538\pi$
$839$	−49.3503	−1.70376	−0.851881	+	0.523735i	$0.824538\pi$
$840$	0	0
$841$	−85.1619	−2.93662
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.0000i	0.412813i
$846$	0	0
$847$	120.031i	4.12433i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−59.7426	−2.04795
$852$	0	0
$853$	−35.1080	−1.20207	−0.601037	−	0.799221i	$-0.705245\pi$
$853$	−35.1080	−1.20207	−0.601037	+	0.799221i	$0.705245\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.7386i	0.435143i	0.976044	+	0.217572i	$0.0698136\pi$
$857$	12.7386i	0.435143i	−0.976044	+	0.217572i	$0.930186\pi$
$858$	0	0
$859$	− 34.5213i	− 1.17785i	−0.808187	−	0.588925i	$-0.799551\pi$
$859$	− 34.5213i	− 1.17785i	0.808187	−	0.588925i	$-0.200449\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.3422	−0.828618	−0.414309	−	0.910136i	$-0.635977\pi$
$863$	−24.3422	−0.828618	−0.414309	+	0.910136i	$0.635977\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.31534i	0.0446199i
$870$	0	0
$871$	9.84612i	0.333623i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.43674	−0.149989
$876$	0	0
$877$	14.3693	0.485217	0.242609	−	0.970124i	$-0.421997\pi$
$877$	14.3693	0.485217	0.242609	+	0.970124i	$0.421997\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 24.7386i	− 0.833466i	−0.909029	−	0.416733i	$-0.863175\pi$
$881$	− 24.7386i	− 0.833466i	0.909029	−	0.416733i	$-0.136825\pi$
$882$	0	0
$883$	− 0.119737i	− 0.00402948i	−0.999998	−	0.00201474i	$-0.999359\pi$
$883$	− 0.119737i	− 0.00402948i	0.999998	−	0.00201474i	$-0.000641312\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.6798	1.23159	0.615793	−	0.787908i	$-0.288836\pi$
$887$	36.6798	1.23159	0.615793	+	0.787908i	$0.288836\pi$
$888$	0	0
$889$	−46.1080	−1.54641
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.7386i	0.627064i
$894$	0	0
$895$	14.2829i	0.477423i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−65.9114	−2.19827
$900$	0	0
$901$	28.1080	0.936412
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.0540i	0.633376i
$906$	0	0
$907$	− 15.1620i	− 0.503446i	−0.967799	−	0.251723i	$-0.919003\pi$
$907$	− 15.1620i	− 0.503446i	0.967799	−	0.251723i	$-0.0809972\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	29.2316	0.968487	0.484244	−	0.874933i	$-0.339095\pi$
$911$	29.2316	0.968487	0.484244	+	0.874933i	$0.339095\pi$
$912$	0	0
$913$	64.1080	2.12166
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.7386i	0.618804i
$918$	0	0
$919$	1.61231i	0.0531853i	0.999646	+	0.0265927i	$0.00846570\pi$
$919$	1.61231i	0.0531853i	−0.999646	+	0.0265927i	$0.991534\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.94528	0.0640295
$924$	0	0
$925$	−9.68466	−0.318430
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 24.7386i	− 0.811648i	−0.913951	−	0.405824i	$-0.866985\pi$
$929$	− 24.7386i	− 0.811648i	0.913951	−	0.405824i	$-0.133015\pi$
$930$	0	0
$931$	56.2785i	1.84445i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−28.8987	−0.945088
$936$	0	0
$937$	24.3153	0.794348	0.397174	−	0.917743i	$-0.369991\pi$
$937$	24.3153	0.794348	0.397174	+	0.917743i	$0.369991\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.42329i	0.176794i	0.996085	+	0.0883971i	$0.0281745\pi$
$941$	5.42329i	0.176794i	−0.996085	+	0.0883971i	$0.971826\pi$
$942$	0	0
$943$	37.0127i	1.20530i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.55648	−0.148066	−0.0740328	−	0.997256i	$-0.523587\pi$
$947$	−4.55648	−0.148066	−0.0740328	+	0.997256i	$0.523587\pi$
$948$	0	0
$949$	−3.68466	−0.119609
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 31.3153i	− 1.01440i	−0.861827	−	0.507202i	$-0.830680\pi$
$953$	− 31.3153i	− 1.01440i	0.861827	−	0.507202i	$-0.169320\pi$
$954$	0	0
$955$	20.7846i	0.672574i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−26.6204	−0.859619
$960$	0	0
$961$	−7.05398	−0.227548
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 12.3153i	− 0.396445i
$966$	0	0
$967$	18.2931i	0.588268i	0.955764	+	0.294134i	$0.0950312\pi$
$967$	18.2931i	0.588268i	−0.955764	+	0.294134i	$0.904969\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.4516	0.656325	0.328162	−	0.944621i	$-0.393571\pi$
$971$	20.4516	0.656325	0.328162	+	0.944621i	$0.393571\pi$
$972$	0	0
$973$	−11.0540	−0.354374
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10.6847i	− 0.341833i	−0.985286	−	0.170916i	$-0.945327\pi$
$977$	− 10.6847i	− 0.341833i	0.985286	−	0.170916i	$-0.0546728\pi$
$978$	0	0
$979$	74.0255i	2.36586i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10.0593	−0.320843	−0.160422	−	0.987049i	$-0.551285\pi$
$983$	−10.0593	−0.320843	−0.160422	+	0.987049i	$0.551285\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	38.0540i	1.21005i
$990$	0	0
$991$	− 51.0824i	− 1.62269i	−0.584571	−	0.811343i	$-0.698737\pi$
$991$	− 51.0824i	− 1.62269i	0.584571	−	0.811343i	$-0.301263\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.73205	−0.0549097
$996$	0	0
$997$	−7.94602	−0.251653	−0.125827	−	0.992052i	$-0.540158\pi$
$997$	−7.94602	−0.251653	−0.125827	+	0.992052i	$0.540158\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.h.g.431.4	yes	8
3.2	odd	2	inner	2160.2.h.g.431.8	yes	8
4.3	odd	2	inner	2160.2.h.g.431.1	✓	8
12.11	even	2	inner	2160.2.h.g.431.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.2.h.g.431.1	✓	8	4.3	odd	2	inner
2160.2.h.g.431.4	yes	8	1.1	even	1	trivial
2160.2.h.g.431.5	yes	8	12.11	even	2	inner
2160.2.h.g.431.8	yes	8	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} +4.43674i q^{7} -6.16879 q^{11} +1.00000 q^{13} +4.68466i q^{17} -4.43674i q^{19} -6.16879 q^{23} -1.00000 q^{25} -10.6847i q^{29} -6.16879i q^{31} +4.43674 q^{35} +9.68466 q^{37} -6.00000i q^{41} -6.16879i q^{43} -4.22351 q^{47} -12.6847 q^{49} -6.00000i q^{53} +6.16879i q^{55} +12.3376 q^{59} -5.68466 q^{61} -1.00000i q^{65} +9.84612i q^{67} +1.94528 q^{71} -3.68466 q^{73} -27.3693i q^{77} -0.213225i q^{79} -10.3923 q^{83} +4.68466 q^{85} -12.0000i q^{89} +4.43674i q^{91} -4.43674 q^{95} +5.68466 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{13} - 8 q^{25} + 28 q^{37} - 52 q^{49} + 4 q^{61} + 20 q^{73} - 12 q^{85} - 4 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 - 8 * q^25 + 28 * q^37 - 52 * q^49 + 4 * q^61 + 20 * q^73 - 12 * q^85 - 4 * q^97

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