LMFDB - Embedded newform 2160.2.h.f.431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(431,2160)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	no
Analytic conductor:	$17.2476868366$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2160.431
Dual form			2160.2.h.f.431.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} -1.26795i q^{7} +O(q^{10})$	$q-1.00000i q^{5} -1.26795i q^{7} +1.26795 q^{11} -4.19615 q^{13} +5.19615i q^{17} +6.46410i q^{19} -7.73205 q^{23} -1.00000 q^{25} -2.19615i q^{29} +6.46410i q^{31} -1.26795 q^{35} +2.00000 q^{37} -8.19615i q^{41} +11.6603i q^{43} +2.53590 q^{47} +5.39230 q^{49} +0.803848i q^{53} -1.26795i q^{55} +11.6603 q^{59} -1.00000 q^{61} +4.19615i q^{65} +10.3923i q^{67} +1.26795 q^{71} +6.19615 q^{73} -1.60770i q^{77} +3.92820i q^{79} -5.19615 q^{83} +5.19615 q^{85} +3.80385i q^{89} +5.32051i q^{91} +6.46410 q^{95} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{11} + 4 q^{13} - 24 q^{23} - 4 q^{25} - 12 q^{35} + 8 q^{37} + 24 q^{47} - 20 q^{49} + 12 q^{59} - 4 q^{61} + 12 q^{71} + 4 q^{73} + 12 q^{95} - 32 q^{97}+O(q^{100}) $ 4 * q + 12 * q^11 + 4 * q^13 - 24 * q^23 - 4 * q^25 - 12 * q^35 + 8 * q^37 + 24 * q^47 - 20 * q^49 + 12 * q^59 - 4 * q^61 + 12 * q^71 + 4 * q^73 + 12 * q^95 - 32 * q^97

Display 100 coefficients 10 coefficients

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
$7$	− 1.26795i	− 0.479240i	−0.970867	−	0.239620i	$-0.922977\pi$
$7$	− 1.26795i	− 0.479240i	0.970867	−	0.239620i	$-0.0770228\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	−4.19615	−1.16380	−0.581902	−	0.813259i	$-0.697691\pi$
$13$	−4.19615	−1.16380	−0.581902	+	0.813259i	$0.697691\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.19615i	1.26025i	0.776493	+	0.630126i	$0.216997\pi$
$17$	5.19615i	1.26025i	−0.776493	+	0.630126i	$0.783003\pi$
$18$	0	0
$19$	6.46410i	1.48297i	0.670971	+	0.741483i	$0.265877\pi$
$19$	6.46410i	1.48297i	−0.670971	+	0.741483i	$0.734123\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.73205	−1.61224	−0.806122	−	0.591749i	$-0.798438\pi$
$23$	−7.73205	−1.61224	−0.806122	+	0.591749i	$0.798438\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.19615i	− 0.407815i	−0.978990	−	0.203908i	$-0.934636\pi$
$29$	− 2.19615i	− 0.407815i	0.978990	−	0.203908i	$-0.0653642\pi$
$30$	0	0
$31$	6.46410i	1.16099i	0.814265	+	0.580493i	$0.197140\pi$
$31$	6.46410i	1.16099i	−0.814265	+	0.580493i	$0.802860\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.26795	−0.214323
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.19615i	− 1.28002i	−0.768365	−	0.640012i	$-0.778929\pi$
$41$	− 8.19615i	− 1.28002i	0.768365	−	0.640012i	$-0.221071\pi$
$42$	0	0
$43$	11.6603i	1.77817i	0.457740	+	0.889086i	$0.348659\pi$
$43$	11.6603i	1.77817i	−0.457740	+	0.889086i	$0.651341\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.53590	0.369899	0.184949	−	0.982748i	$-0.440788\pi$
$47$	2.53590	0.369899	0.184949	+	0.982748i	$0.440788\pi$
$48$	0	0
$49$	5.39230	0.770329
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.803848i	0.110417i	0.998475	+	0.0552085i	$0.0175823\pi$
$53$	0.803848i	0.110417i	−0.998475	+	0.0552085i	$0.982418\pi$
$54$	0	0
$55$	− 1.26795i	− 0.170970i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.6603	1.51804	0.759018	−	0.651070i	$-0.225679\pi$
$59$	11.6603	1.51804	0.759018	+	0.651070i	$0.225679\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.19615i	0.520469i
$66$	0	0
$67$	10.3923i	1.26962i	0.772667	+	0.634811i	$0.218922\pi$
$67$	10.3923i	1.26962i	−0.772667	+	0.634811i	$0.781078\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.26795	0.150478	0.0752389	−	0.997166i	$-0.476028\pi$
$71$	1.26795	0.150478	0.0752389	+	0.997166i	$0.476028\pi$
$72$	0	0
$73$	6.19615	0.725205	0.362602	−	0.931944i	$-0.381888\pi$
$73$	6.19615	0.725205	0.362602	+	0.931944i	$0.381888\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.60770i	− 0.183214i
$78$	0	0
$79$	3.92820i	0.441957i	0.975279	+	0.220979i	$0.0709251\pi$
$79$	3.92820i	0.441957i	−0.975279	+	0.220979i	$0.929075\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.19615	−0.570352	−0.285176	−	0.958475i	$-0.592052\pi$
$83$	−5.19615	−0.570352	−0.285176	+	0.958475i	$0.592052\pi$
$84$	0	0
$85$	5.19615	0.563602
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.80385i	0.403207i	0.979467	+	0.201604i	$0.0646152\pi$
$89$	3.80385i	0.403207i	−0.979467	+	0.201604i	$0.935385\pi$
$90$	0	0
$91$	5.32051i	0.557741i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.46410	0.663203
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.3923i	1.63110i	0.578690	+	0.815548i	$0.303564\pi$
$101$	16.3923i	1.63110i	−0.578690	+	0.815548i	$0.696436\pi$
$102$	0	0
$103$	− 2.53590i	− 0.249869i	−0.992165	−	0.124935i	$-0.960128\pi$
$103$	− 2.53590i	− 0.249869i	0.992165	−	0.124935i	$-0.0398722\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.07180	−0.490309	−0.245155	−	0.969484i	$-0.578839\pi$
$107$	−5.07180	−0.490309	−0.245155	+	0.969484i	$0.578839\pi$
$108$	0	0
$109$	−9.39230	−0.899620	−0.449810	−	0.893124i	$-0.648508\pi$
$109$	−9.39230	−0.899620	−0.449810	+	0.893124i	$0.648508\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	7.73205i	0.721017i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.58846	0.603963
$120$	0	0
$121$	−9.39230	−0.853846
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	18.0000i	1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$	18.0000i	1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.9282	−1.12954	−0.564771	−	0.825248i	$-0.691036\pi$
$131$	−12.9282	−1.12954	−0.564771	+	0.825248i	$0.691036\pi$
$132$	0	0
$133$	8.19615	0.710697
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.19615i	− 0.443937i	−0.975054	−	0.221969i	$-0.928752\pi$
$137$	− 5.19615i	− 0.443937i	0.975054	−	0.221969i	$-0.0712483\pi$
$138$	0	0
$139$	7.85641i	0.666372i	0.942861	+	0.333186i	$0.108124\pi$
$139$	7.85641i	0.666372i	−0.942861	+	0.333186i	$0.891876\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.32051	−0.444923
$144$	0	0
$145$	−2.19615	−0.182381
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 20.1962i	− 1.65453i	−0.561809	−	0.827267i	$-0.689895\pi$
$149$	− 20.1962i	− 1.65453i	0.561809	−	0.827267i	$-0.310105\pi$
$150$	0	0
$151$	5.07180i	0.412737i	0.978474	+	0.206368i	$0.0661646\pi$
$151$	5.07180i	0.412737i	−0.978474	+	0.206368i	$0.933835\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.46410	0.519209
$156$	0	0
$157$	5.80385	0.463197	0.231599	−	0.972811i	$-0.425604\pi$
$157$	5.80385	0.463197	0.231599	+	0.972811i	$0.425604\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.80385i	0.772651i
$162$	0	0
$163$	10.3923i	0.813988i	0.913431	+	0.406994i	$0.133423\pi$
$163$	10.3923i	0.813988i	−0.913431	+	0.406994i	$0.866577\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.66025	−0.205857	−0.102928	−	0.994689i	$-0.532821\pi$
$167$	−2.66025	−0.205857	−0.102928	+	0.994689i	$0.532821\pi$
$168$	0	0
$169$	4.60770	0.354438
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.80385i	0.517287i	0.965973	+	0.258643i	$0.0832755\pi$
$173$	6.80385i	0.517287i	−0.965973	+	0.258643i	$0.916725\pi$
$174$	0	0
$175$	1.26795i	0.0958479i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	25.8564	1.93260	0.966299	−	0.257421i	$-0.0828728\pi$
$179$	25.8564	1.93260	0.966299	+	0.257421i	$0.0828728\pi$
$180$	0	0
$181$	5.39230	0.400807	0.200403	−	0.979713i	$-0.435775\pi$
$181$	5.39230	0.400807	0.200403	+	0.979713i	$0.435775\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.00000i	− 0.147043i
$186$	0	0
$187$	6.58846i	0.481796i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	−12.1962	−0.877898	−0.438949	−	0.898512i	$-0.644649\pi$
$193$	−12.1962	−0.877898	−0.438949	+	0.898512i	$0.644649\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.80385i	0.484754i	0.970182	+	0.242377i	$0.0779271\pi$
$197$	6.80385i	0.484754i	−0.970182	+	0.242377i	$0.922073\pi$
$198$	0	0
$199$	− 10.3923i	− 0.736691i	−0.929689	−	0.368345i	$-0.879924\pi$
$199$	− 10.3923i	− 0.736691i	0.929689	−	0.368345i	$-0.120076\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.78461	−0.195441
$204$	0	0
$205$	−8.19615	−0.572444
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.19615i	0.566940i
$210$	0	0
$211$	9.00000i	0.619586i	0.950804	+	0.309793i	$0.100260\pi$
$211$	9.00000i	0.619586i	−0.950804	+	0.309793i	$0.899740\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.6603	0.795223
$216$	0	0
$217$	8.19615	0.556391
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 21.8038i	− 1.46669i
$222$	0	0
$223$	− 12.9282i	− 0.865737i	−0.901457	−	0.432868i	$-0.857502\pi$
$223$	− 12.9282i	− 0.865737i	0.901457	−	0.432868i	$-0.142498\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.73205	−0.513194	−0.256597	−	0.966518i	$-0.582601\pi$
$227$	−7.73205	−0.513194	−0.256597	+	0.966518i	$0.582601\pi$
$228$	0	0
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.7846i	1.75472i	0.479834	+	0.877359i	$0.340697\pi$
$233$	26.7846i	1.75472i	−0.479834	+	0.877359i	$0.659303\pi$
$234$	0	0
$235$	− 2.53590i	− 0.165424i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.32051	0.344155	0.172078	−	0.985083i	$-0.444952\pi$
$239$	5.32051	0.344155	0.172078	+	0.985083i	$0.444952\pi$
$240$	0	0
$241$	17.3923	1.12034	0.560168	−	0.828379i	$-0.310736\pi$
$241$	17.3923	1.12034	0.560168	+	0.828379i	$0.310736\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5.39230i	− 0.344502i
$246$	0	0
$247$	− 27.1244i	− 1.72588i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.3205	−1.47198	−0.735989	−	0.676994i	$-0.763282\pi$
$251$	−23.3205	−1.47198	−0.735989	+	0.676994i	$0.763282\pi$
$252$	0	0
$253$	−9.80385	−0.616363
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.5885i	− 1.34665i	−0.739346	−	0.673325i	$-0.764865\pi$
$257$	− 21.5885i	− 1.34665i	0.739346	−	0.673325i	$-0.235135\pi$
$258$	0	0
$259$	− 2.53590i	− 0.157573i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−23.3205	−1.43800	−0.719002	−	0.695008i	$-0.755401\pi$
$263$	−23.3205	−1.43800	−0.719002	+	0.695008i	$0.755401\pi$
$264$	0	0
$265$	0.803848	0.0493800
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.7846i	1.63309i	0.577284	+	0.816543i	$0.304112\pi$
$269$	26.7846i	1.63309i	−0.577284	+	0.816543i	$0.695888\pi$
$270$	0	0
$271$	− 24.4641i	− 1.48609i	−0.669242	−	0.743044i	$-0.733381\pi$
$271$	− 24.4641i	− 1.48609i	0.669242	−	0.743044i	$-0.266619\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.26795	−0.0764602
$276$	0	0
$277$	−17.8038	−1.06973	−0.534865	−	0.844938i	$-0.679637\pi$
$277$	−17.8038	−1.06973	−0.534865	+	0.844938i	$0.679637\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 18.5885i	− 1.10889i	−0.832219	−	0.554447i	$-0.812930\pi$
$281$	− 18.5885i	− 1.10889i	0.832219	−	0.554447i	$-0.187070\pi$
$282$	0	0
$283$	16.7321i	0.994617i	0.867574	+	0.497309i	$0.165678\pi$
$283$	16.7321i	0.994617i	−0.867574	+	0.497309i	$0.834322\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.3923	−0.613438
$288$	0	0
$289$	−10.0000	−0.588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 23.1962i	− 1.35513i	−0.735461	−	0.677567i	$-0.763034\pi$
$293$	− 23.1962i	− 1.35513i	0.735461	−	0.677567i	$-0.236966\pi$
$294$	0	0
$295$	− 11.6603i	− 0.678886i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	32.4449	1.87633
$300$	0	0
$301$	14.7846	0.852171
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.00000i	0.0572598i
$306$	0	0
$307$	− 7.85641i	− 0.448389i	−0.974544	−	0.224194i	$-0.928025\pi$
$307$	− 7.85641i	− 0.448389i	0.974544	−	0.224194i	$-0.0719751\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.2679	1.09259	0.546293	−	0.837594i	$-0.316039\pi$
$311$	19.2679	1.09259	0.546293	+	0.837594i	$0.316039\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.1962i	1.63982i	0.572493	+	0.819910i	$0.305976\pi$
$317$	29.1962i	1.63982i	−0.572493	+	0.819910i	$0.694024\pi$
$318$	0	0
$319$	− 2.78461i	− 0.155908i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−33.5885	−1.86891
$324$	0	0
$325$	4.19615	0.232761
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.21539i	− 0.177270i
$330$	0	0
$331$	15.4641i	0.849984i	0.905197	+	0.424992i	$0.139723\pi$
$331$	15.4641i	0.849984i	−0.905197	+	0.424992i	$0.860277\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.3923	0.567792
$336$	0	0
$337$	28.5885	1.55731	0.778656	−	0.627451i	$-0.215902\pi$
$337$	28.5885	1.55731	0.778656	+	0.627451i	$0.215902\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.19615i	0.443847i
$342$	0	0
$343$	− 15.7128i	− 0.848412i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.3923	0.557888	0.278944	−	0.960307i	$-0.410016\pi$
$347$	10.3923	0.557888	0.278944	+	0.960307i	$0.410016\pi$
$348$	0	0
$349$	33.3923	1.78745	0.893725	−	0.448616i	$-0.148083\pi$
$349$	33.3923	1.78745	0.893725	+	0.448616i	$0.148083\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.60770i	− 0.0855690i	−0.999084	−	0.0427845i	$-0.986377\pi$
$353$	− 1.60770i	− 0.0855690i	0.999084	−	0.0427845i	$-0.0136229\pi$
$354$	0	0
$355$	− 1.26795i	− 0.0672958i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.7128	1.77929	0.889647	−	0.456649i	$-0.150950\pi$
$359$	33.7128	1.77929	0.889647	+	0.456649i	$0.150950\pi$
$360$	0	0
$361$	−22.7846	−1.19919
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 6.19615i	− 0.324321i
$366$	0	0
$367$	− 24.5885i	− 1.28351i	−0.766911	−	0.641754i	$-0.778207\pi$
$367$	− 24.5885i	− 1.28351i	0.766911	−	0.641754i	$-0.221793\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.01924	0.0529162
$372$	0	0
$373$	12.3923	0.641649	0.320825	−	0.947139i	$-0.396040\pi$
$373$	12.3923	0.641649	0.320825	+	0.947139i	$0.396040\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.21539i	0.474617i
$378$	0	0
$379$	27.0000i	1.38690i	0.720506	+	0.693448i	$0.243909\pi$
$379$	27.0000i	1.38690i	−0.720506	+	0.693448i	$0.756091\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.6603	−1.05569	−0.527845	−	0.849341i	$-0.677000\pi$
$383$	−20.6603	−1.05569	−0.527845	+	0.849341i	$0.677000\pi$
$384$	0	0
$385$	−1.60770	−0.0819357
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.5885i	1.24668i	0.781949	+	0.623342i	$0.214226\pi$
$389$	24.5885i	1.24668i	−0.781949	+	0.623342i	$0.785774\pi$
$390$	0	0
$391$	− 40.1769i	− 2.03183i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.92820	0.197649
$396$	0	0
$397$	−4.58846	−0.230288	−0.115144	−	0.993349i	$-0.536733\pi$
$397$	−4.58846	−0.230288	−0.115144	+	0.993349i	$0.536733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.3923i	0.518967i	0.965748	+	0.259483i	$0.0835523\pi$
$401$	10.3923i	0.518967i	−0.965748	+	0.259483i	$0.916448\pi$
$402$	0	0
$403$	− 27.1244i	− 1.35116i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.53590	0.125700
$408$	0	0
$409$	−37.7846	−1.86833	−0.934164	−	0.356843i	$-0.883853\pi$
$409$	−37.7846	−1.86833	−0.934164	+	0.356843i	$0.883853\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 14.7846i	− 0.727503i
$414$	0	0
$415$	5.19615i	0.255069i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.9808	0.829565	0.414782	−	0.909921i	$-0.363858\pi$
$419$	16.9808	0.829565	0.414782	+	0.909921i	$0.363858\pi$
$420$	0	0
$421$	−30.1769	−1.47073	−0.735366	−	0.677670i	$-0.762990\pi$
$421$	−30.1769	−1.47073	−0.735366	+	0.677670i	$0.762990\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 5.19615i	− 0.252050i
$426$	0	0
$427$	1.26795i	0.0613604i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.12436	0.439505	0.219752	−	0.975556i	$-0.429475\pi$
$431$	9.12436	0.439505	0.219752	+	0.975556i	$0.429475\pi$
$432$	0	0
$433$	−4.58846	−0.220507	−0.110254	−	0.993903i	$-0.535166\pi$
$433$	−4.58846	−0.220507	−0.110254	+	0.993903i	$0.535166\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 49.9808i	− 2.39090i
$438$	0	0
$439$	− 24.7128i	− 1.17948i	−0.807594	−	0.589739i	$-0.799231\pi$
$439$	− 24.7128i	− 1.17948i	0.807594	−	0.589739i	$-0.200769\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−28.5167	−1.35487	−0.677434	−	0.735584i	$-0.736908\pi$
$443$	−28.5167	−1.35487	−0.677434	+	0.735584i	$0.736908\pi$
$444$	0	0
$445$	3.80385	0.180320
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 22.9808i	− 1.08453i	−0.840208	−	0.542265i	$-0.817567\pi$
$449$	− 22.9808i	− 1.08453i	0.840208	−	0.542265i	$-0.182433\pi$
$450$	0	0
$451$	− 10.3923i	− 0.489355i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.32051	0.249429
$456$	0	0
$457$	−40.7846	−1.90782	−0.953912	−	0.300087i	$-0.902984\pi$
$457$	−40.7846	−1.90782	−0.953912	+	0.300087i	$0.902984\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.60770i	0.0748778i	0.999299	+	0.0374389i	$0.0119200\pi$
$461$	1.60770i	0.0748778i	−0.999299	+	0.0374389i	$0.988080\pi$
$462$	0	0
$463$	− 12.9282i	− 0.600825i	−0.953809	−	0.300412i	$-0.902876\pi$
$463$	− 12.9282i	− 0.600825i	0.953809	−	0.300412i	$-0.0971243\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.2679	1.30808	0.654042	−	0.756458i	$-0.273072\pi$
$467$	28.2679	1.30808	0.654042	+	0.756458i	$0.273072\pi$
$468$	0	0
$469$	13.1769	0.608453
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.7846i	0.679797i
$474$	0	0
$475$	− 6.46410i	− 0.296593i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.1962	1.47108	0.735540	−	0.677481i	$-0.236929\pi$
$479$	32.1962	1.47108	0.735540	+	0.677481i	$0.236929\pi$
$480$	0	0
$481$	−8.39230	−0.382656
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.00000i	0.363261i
$486$	0	0
$487$	− 34.9808i	− 1.58513i	−0.609788	−	0.792565i	$-0.708745\pi$
$487$	− 34.9808i	− 1.58513i	0.609788	−	0.792565i	$-0.291255\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.7846	−1.75032	−0.875162	−	0.483829i	$-0.839246\pi$
$491$	−38.7846	−1.75032	−0.875162	+	0.483829i	$0.839246\pi$
$492$	0	0
$493$	11.4115	0.513950
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.60770i	− 0.0721150i
$498$	0	0
$499$	− 42.4641i	− 1.90095i	−0.310795	−	0.950477i	$-0.600595\pi$
$499$	− 42.4641i	− 1.90095i	0.310795	−	0.950477i	$-0.399405\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−33.5885	−1.49764	−0.748818	−	0.662776i	$-0.769378\pi$
$503$	−33.5885	−1.49764	−0.748818	+	0.662776i	$0.769378\pi$
$504$	0	0
$505$	16.3923	0.729448
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 16.3923i	− 0.726576i	−0.931677	−	0.363288i	$-0.881654\pi$
$509$	− 16.3923i	− 0.726576i	0.931677	−	0.363288i	$-0.118346\pi$
$510$	0	0
$511$	− 7.85641i	− 0.347547i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.53590	−0.111745
$516$	0	0
$517$	3.21539	0.141413
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.5885i	1.07724i	0.842549	+	0.538620i	$0.181054\pi$
$521$	24.5885i	1.07724i	−0.842549	+	0.538620i	$0.818946\pi$
$522$	0	0
$523$	− 6.33975i	− 0.277218i	−0.990347	−	0.138609i	$-0.955737\pi$
$523$	− 6.33975i	− 0.277218i	0.990347	−	0.138609i	$-0.0442631\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.5885	−1.46314
$528$	0	0
$529$	36.7846	1.59933
$530$	0	0
$531$	0	0
$532$	0	0
$533$	34.3923i	1.48970i
$534$	0	0
$535$	5.07180i	0.219273i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.83717	0.294498
$540$	0	0
$541$	16.0000	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9.39230i	0.402322i
$546$	0	0
$547$	19.2679i	0.823838i	0.911220	+	0.411919i	$0.135141\pi$
$547$	19.2679i	0.823838i	−0.911220	+	0.411919i	$0.864859\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14.1962	0.604776
$552$	0	0
$553$	4.98076	0.211804
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 13.6077i	− 0.576577i	−0.957544	−	0.288288i	$-0.906914\pi$
$557$	− 13.6077i	− 0.576577i	0.957544	−	0.288288i	$-0.0930861\pi$
$558$	0	0
$559$	− 48.9282i	− 2.06944i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.8564	−1.08972	−0.544859	−	0.838528i	$-0.683417\pi$
$563$	−25.8564	−1.08972	−0.544859	+	0.838528i	$0.683417\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 21.8038i	− 0.914065i	−0.889450	−	0.457032i	$-0.848912\pi$
$569$	− 21.8038i	− 0.914065i	0.889450	−	0.457032i	$-0.151088\pi$
$570$	0	0
$571$	6.46410i	0.270514i	0.990811	+	0.135257i	$0.0431860\pi$
$571$	6.46410i	0.270514i	−0.990811	+	0.135257i	$0.956814\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.73205	0.322449
$576$	0	0
$577$	26.5885	1.10689	0.553446	−	0.832885i	$-0.313313\pi$
$577$	26.5885	1.10689	0.553446	+	0.832885i	$0.313313\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.58846i	0.273335i
$582$	0	0
$583$	1.01924i	0.0422125i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.19615	0.214468	0.107234	−	0.994234i	$-0.465801\pi$
$587$	5.19615	0.214468	0.107234	+	0.994234i	$0.465801\pi$
$588$	0	0
$589$	−41.7846	−1.72170
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.8038i	0.525791i	0.964824	+	0.262896i	$0.0846774\pi$
$593$	12.8038i	0.525791i	−0.964824	+	0.262896i	$0.915323\pi$
$594$	0	0
$595$	− 6.58846i	− 0.270100i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	42.5885	1.74012	0.870059	−	0.492948i	$-0.164081\pi$
$599$	42.5885	1.74012	0.870059	+	0.492948i	$0.164081\pi$
$600$	0	0
$601$	30.1769	1.23094	0.615471	−	0.788160i	$-0.288966\pi$
$601$	30.1769	1.23094	0.615471	+	0.788160i	$0.288966\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.39230i	0.381851i
$606$	0	0
$607$	19.2679i	0.782062i	0.920378	+	0.391031i	$0.127881\pi$
$607$	19.2679i	0.782062i	−0.920378	+	0.391031i	$0.872119\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.6410	−0.430489
$612$	0	0
$613$	−29.1769	−1.17844	−0.589222	−	0.807971i	$-0.700566\pi$
$613$	−29.1769	−1.17844	−0.589222	+	0.807971i	$0.700566\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.5885i	1.11067i	0.831627	+	0.555335i	$0.187410\pi$
$617$	27.5885i	1.11067i	−0.831627	+	0.555335i	$0.812590\pi$
$618$	0	0
$619$	15.4641i	0.621555i	0.950483	+	0.310777i	$0.100589\pi$
$619$	15.4641i	0.621555i	−0.950483	+	0.310777i	$0.899411\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.82309	0.193233
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.3923i	0.414368i
$630$	0	0
$631$	6.21539i	0.247431i	0.992318	+	0.123715i	$0.0394810\pi$
$631$	6.21539i	0.247431i	−0.992318	+	0.123715i	$0.960519\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	18.0000	0.714308
$636$	0	0
$637$	−22.6269	−0.896512
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 4.39230i	− 0.173486i	−0.996231	−	0.0867428i	$-0.972354\pi$
$641$	− 4.39230i	− 0.173486i	0.996231	−	0.0867428i	$-0.0276458\pi$
$642$	0	0
$643$	− 24.5885i	− 0.969674i	−0.874605	−	0.484837i	$-0.838879\pi$
$643$	− 24.5885i	− 0.969674i	0.874605	−	0.484837i	$-0.161121\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−31.0526	−1.22080	−0.610401	−	0.792093i	$-0.708992\pi$
$647$	−31.0526	−1.22080	−0.610401	+	0.792093i	$0.708992\pi$
$648$	0	0
$649$	14.7846	0.580347
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 33.5885i	− 1.31442i	−0.753708	−	0.657209i	$-0.771737\pi$
$653$	− 33.5885i	− 1.31442i	0.753708	−	0.657209i	$-0.228263\pi$
$654$	0	0
$655$	12.9282i	0.505147i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.58846	0.256650	0.128325	−	0.991732i	$-0.459040\pi$
$659$	6.58846	0.256650	0.128325	+	0.991732i	$0.459040\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 8.19615i	− 0.317833i
$666$	0	0
$667$	16.9808i	0.657498i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.26795	−0.0489486
$672$	0	0
$673$	−16.5885	−0.639438	−0.319719	−	0.947512i	$-0.603589\pi$
$673$	−16.5885	−0.639438	−0.319719	+	0.947512i	$0.603589\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	10.1436i	0.389275i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.3397	0.586959	0.293480	−	0.955965i	$-0.405187\pi$
$683$	15.3397	0.586959	0.293480	+	0.955965i	$0.405187\pi$
$684$	0	0
$685$	−5.19615	−0.198535
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 3.37307i	− 0.128504i
$690$	0	0
$691$	− 27.0000i	− 1.02713i	−0.858051	−	0.513564i	$-0.828325\pi$
$691$	− 27.0000i	− 1.02713i	0.858051	−	0.513564i	$-0.171675\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.85641	0.298010
$696$	0	0
$697$	42.5885	1.61315
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 6.00000i	− 0.226617i	−0.993560	−	0.113308i	$-0.963855\pi$
$701$	− 6.00000i	− 0.226617i	0.993560	−	0.113308i	$-0.0361448\pi$
$702$	0	0
$703$	12.9282i	0.487596i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.7846	0.781686
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 49.9808i	− 1.87179i
$714$	0	0
$715$	5.32051i	0.198976i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.0718	−0.860433	−0.430216	−	0.902726i	$-0.641563\pi$
$719$	−23.0718	−0.860433	−0.430216	+	0.902726i	$0.641563\pi$
$720$	0	0
$721$	−3.21539	−0.119747
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.19615i	0.0815631i
$726$	0	0
$727$	− 15.7128i	− 0.582756i	−0.956608	−	0.291378i	$-0.905886\pi$
$727$	− 15.7128i	− 0.582756i	0.956608	−	0.291378i	$-0.0941137\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−60.5885	−2.24095
$732$	0	0
$733$	−15.6077	−0.576483	−0.288242	−	0.957558i	$-0.593071\pi$
$733$	−15.6077	−0.576483	−0.288242	+	0.957558i	$0.593071\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.1769i	0.485378i
$738$	0	0
$739$	− 1.39230i	− 0.0512168i	−0.999672	−	0.0256084i	$-0.991848\pi$
$739$	− 1.39230i	− 0.0512168i	0.999672	−	0.0256084i	$-0.00815229\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.1051	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$743$	−26.1051	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$744$	0	0
$745$	−20.1962	−0.739930
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.43078i	0.234976i
$750$	0	0
$751$	6.71281i	0.244954i	0.992471	+	0.122477i	$0.0390838\pi$
$751$	6.71281i	0.244954i	−0.992471	+	0.122477i	$0.960916\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.07180	0.184582
$756$	0	0
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 27.3731i	− 0.992273i	−0.868245	−	0.496136i	$-0.834752\pi$
$761$	− 27.3731i	− 0.992273i	0.868245	−	0.496136i	$-0.165248\pi$
$762$	0	0
$763$	11.9090i	0.431133i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−48.9282	−1.76670
$768$	0	0
$769$	−13.0000	−0.468792	−0.234396	−	0.972141i	$-0.575311\pi$
$769$	−13.0000	−0.468792	−0.234396	+	0.972141i	$0.575311\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 12.8038i	− 0.460522i	−0.973129	−	0.230261i	$-0.926042\pi$
$773$	− 12.8038i	− 0.460522i	0.973129	−	0.230261i	$-0.0739581\pi$
$774$	0	0
$775$	− 6.46410i	− 0.232197i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	52.9808	1.89823
$780$	0	0
$781$	1.60770	0.0575279
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 5.80385i	− 0.207148i
$786$	0	0
$787$	− 9.12436i	− 0.325248i	−0.986688	−	0.162624i	$-0.948004\pi$
$787$	− 9.12436i	− 0.325248i	0.986688	−	0.162624i	$-0.0519958\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.2154	−0.540997
$792$	0	0
$793$	4.19615	0.149010
$794$	0	0
$795$	0	0
$796$	0	0
$797$	52.7654i	1.86905i	0.355904	+	0.934523i	$0.384173\pi$
$797$	52.7654i	1.86905i	−0.355904	+	0.934523i	$0.615827\pi$
$798$	0	0
$799$	13.1769i	0.466166i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.85641	0.277247
$804$	0	0
$805$	9.80385	0.345540
$806$	0	0
$807$	0	0
$808$	0	0
$809$	36.5885i	1.28638i	0.765706	+	0.643191i	$0.222390\pi$
$809$	36.5885i	1.28638i	−0.765706	+	0.643191i	$0.777610\pi$
$810$	0	0
$811$	30.9282i	1.08604i	0.839721	+	0.543018i	$0.182718\pi$
$811$	30.9282i	1.08604i	−0.839721	+	0.543018i	$0.817282\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	10.3923	0.364027
$816$	0	0
$817$	−75.3731	−2.63697
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.3923i	0.362694i	0.983419	+	0.181347i	$0.0580457\pi$
$821$	10.3923i	0.362694i	−0.983419	+	0.181347i	$0.941954\pi$
$822$	0	0
$823$	− 14.1962i	− 0.494847i	−0.968907	−	0.247423i	$-0.920416\pi$
$823$	− 14.1962i	− 0.494847i	0.968907	−	0.247423i	$-0.0795839\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.2679	0.982973	0.491486	−	0.870885i	$-0.336454\pi$
$827$	28.2679	0.982973	0.491486	+	0.870885i	$0.336454\pi$
$828$	0	0
$829$	−33.1769	−1.15228	−0.576141	−	0.817350i	$-0.695442\pi$
$829$	−33.1769	−1.15228	−0.576141	+	0.817350i	$0.695442\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	28.0192i	0.970809i
$834$	0	0
$835$	2.66025i	0.0920619i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.07180	−0.175098	−0.0875489	−	0.996160i	$-0.527903\pi$
$839$	−5.07180	−0.175098	−0.0875489	+	0.996160i	$0.527903\pi$
$840$	0	0
$841$	24.1769	0.833687
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 4.60770i	− 0.158510i
$846$	0	0
$847$	11.9090i	0.409197i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−15.4641	−0.530103
$852$	0	0
$853$	40.7846	1.39644	0.698219	−	0.715884i	$-0.253976\pi$
$853$	40.7846	1.39644	0.698219	+	0.715884i	$0.253976\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 23.1962i	− 0.792365i	−0.918172	−	0.396183i	$-0.870335\pi$
$857$	− 23.1962i	− 0.792365i	0.918172	−	0.396183i	$-0.129665\pi$
$858$	0	0
$859$	45.2487i	1.54387i	0.635704	+	0.771933i	$0.280710\pi$
$859$	45.2487i	1.54387i	−0.635704	+	0.771933i	$0.719290\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30.8038	1.04858	0.524288	−	0.851541i	$-0.324332\pi$
$863$	30.8038	1.04858	0.524288	+	0.851541i	$0.324332\pi$
$864$	0	0
$865$	6.80385	0.231338
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.98076i	0.168961i
$870$	0	0
$871$	− 43.6077i	− 1.47759i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.26795	0.0428645
$876$	0	0
$877$	38.5885	1.30304	0.651520	−	0.758632i	$-0.274132\pi$
$877$	38.5885	1.30304	0.651520	+	0.758632i	$0.274132\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 42.5885i	− 1.43484i	−0.696640	−	0.717421i	$-0.745323\pi$
$881$	− 42.5885i	− 1.43484i	0.696640	−	0.717421i	$-0.254677\pi$
$882$	0	0
$883$	− 1.26795i	− 0.0426699i	−0.999772	−	0.0213349i	$-0.993208\pi$
$883$	− 1.26795i	− 0.0426699i	0.999772	−	0.0213349i	$-0.00679164\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−10.5167	−0.353115	−0.176557	−	0.984290i	$-0.556496\pi$
$887$	−10.5167	−0.353115	−0.176557	+	0.984290i	$0.556496\pi$
$888$	0	0
$889$	22.8231	0.765462
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.3923i	0.548548i
$894$	0	0
$895$	− 25.8564i	− 0.864284i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.1962	0.473468
$900$	0	0
$901$	−4.17691	−0.139153
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 5.39230i	− 0.179246i
$906$	0	0
$907$	− 5.07180i	− 0.168406i	−0.996449	−	0.0842031i	$-0.973166\pi$
$907$	− 5.07180i	− 0.168406i	0.996449	−	0.0842031i	$-0.0268345\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	37.5167	1.24298	0.621491	−	0.783421i	$-0.286527\pi$
$911$	37.5167	1.24298	0.621491	+	0.783421i	$0.286527\pi$
$912$	0	0
$913$	−6.58846	−0.218046
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.3923i	0.541322i
$918$	0	0
$919$	33.7128i	1.11208i	0.831155	+	0.556042i	$0.187680\pi$
$919$	33.7128i	1.11208i	−0.831155	+	0.556042i	$0.812320\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.32051	−0.175127
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 20.1962i	− 0.662614i	−0.943523	−	0.331307i	$-0.892510\pi$
$929$	− 20.1962i	− 0.662614i	0.943523	−	0.331307i	$-0.107490\pi$
$930$	0	0
$931$	34.8564i	1.14237i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.58846	0.215466
$936$	0	0
$937$	−29.1769	−0.953168	−0.476584	−	0.879129i	$-0.658125\pi$
$937$	−29.1769	−0.953168	−0.476584	+	0.879129i	$0.658125\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 7.17691i	− 0.233961i	−0.993134	−	0.116980i	$-0.962679\pi$
$941$	− 7.17691i	− 0.233961i	0.993134	−	0.116980i	$-0.0373215\pi$
$942$	0	0
$943$	63.3731i	2.06371i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.7654	1.51967	0.759835	−	0.650116i	$-0.225280\pi$
$947$	46.7654	1.51967	0.759835	+	0.650116i	$0.225280\pi$
$948$	0	0
$949$	−26.0000	−0.843996
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	18.0000i	0.582466i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.58846	−0.212752
$960$	0	0
$961$	−10.7846	−0.347891
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.1962i	0.392608i
$966$	0	0
$967$	5.32051i	0.171096i	0.996334	+	0.0855480i	$0.0272641\pi$
$967$	5.32051i	0.171096i	−0.996334	+	0.0855480i	$0.972736\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.32051	−0.170743	−0.0853716	−	0.996349i	$-0.527208\pi$
$971$	−5.32051	−0.170743	−0.0853716	+	0.996349i	$0.527208\pi$
$972$	0	0
$973$	9.96152	0.319352
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.1769i	0.997438i	0.866764	+	0.498719i	$0.166196\pi$
$977$	31.1769i	0.997438i	−0.866764	+	0.498719i	$0.833804\pi$
$978$	0	0
$979$	4.82309i	0.154146i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10.2679	−0.327497	−0.163748	−	0.986502i	$-0.552359\pi$
$983$	−10.2679	−0.327497	−0.163748	+	0.986502i	$0.552359\pi$
$984$	0	0
$985$	6.80385	0.216789
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 90.1577i	− 2.86685i
$990$	0	0
$991$	57.9282i	1.84015i	0.391742	+	0.920075i	$0.371873\pi$
$991$	57.9282i	1.84015i	−0.391742	+	0.920075i	$0.628127\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.3923	−0.329458
$996$	0	0
$997$	−37.8038	−1.19726	−0.598630	−	0.801026i	$-0.704288\pi$
$997$	−37.8038	−1.19726	−0.598630	+	0.801026i	$0.704288\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.h.f.431.2	yes	4
3.2	odd	2		2160.2.h.a.431.4	yes	4
4.3	odd	2		2160.2.h.a.431.1	✓	4
12.11	even	2	inner	2160.2.h.f.431.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.2.h.a.431.1	✓	4	4.3	odd	2
2160.2.h.a.431.4	yes	4	3.2	odd	2
2160.2.h.f.431.2	yes	4	1.1	even	1	trivial
2160.2.h.f.431.3	yes	4	12.11	even	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} -1.26795i q^{7} +O(q^{10})\)	\(q-1.00000i q^{5} -1.26795i q^{7} +1.26795 q^{11} -4.19615 q^{13} +5.19615i q^{17} +6.46410i q^{19} -7.73205 q^{23} -1.00000 q^{25} -2.19615i q^{29} +6.46410i q^{31} -1.26795 q^{35} +2.00000 q^{37} -8.19615i q^{41} +11.6603i q^{43} +2.53590 q^{47} +5.39230 q^{49} +0.803848i q^{53} -1.26795i q^{55} +11.6603 q^{59} -1.00000 q^{61} +4.19615i q^{65} +10.3923i q^{67} +1.26795 q^{71} +6.19615 q^{73} -1.60770i q^{77} +3.92820i q^{79} -5.19615 q^{83} +5.19615 q^{85} +3.80385i q^{89} +5.32051i q^{91} +6.46410 q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{11} + 4 q^{13} - 24 q^{23} - 4 q^{25} - 12 q^{35} + 8 q^{37} + 24 q^{47} - 20 q^{49} + 12 q^{59} - 4 q^{61} + 12 q^{71} + 4 q^{73} + 12 q^{95} - 32 q^{97}+O(q^{100}) \) 4 * q + 12 * q^11 + 4 * q^13 - 24 * q^23 - 4 * q^25 - 12 * q^35 + 8 * q^37 + 24 * q^47 - 20 * q^49 + 12 * q^59 - 4 * q^61 + 12 * q^71 + 4 * q^73 + 12 * q^95 - 32 * q^97

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