LMFDB - Embedded newform 2160.2.h.f.431.4

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			431.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2160.431
Dual form			2160.2.h.f.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +4.73205i q^{7} +O(q^{10})$	$q+1.00000i q^{5} +4.73205i q^{7} +4.73205 q^{11} +6.19615 q^{13} +5.19615i q^{17} +0.464102i q^{19} -4.26795 q^{23} -1.00000 q^{25} -8.19615i q^{29} +0.464102i q^{31} -4.73205 q^{35} +2.00000 q^{37} -2.19615i q^{41} +5.66025i q^{43} +9.46410 q^{47} -15.3923 q^{49} -11.1962i q^{53} +4.73205i q^{55} -5.66025 q^{59} -1.00000 q^{61} +6.19615i q^{65} +10.3923i q^{67} +4.73205 q^{71} -4.19615 q^{73} +22.3923i q^{77} +9.92820i q^{79} +5.19615 q^{83} -5.19615 q^{85} -14.1962i q^{89} +29.3205i q^{91} -0.464102 q^{95} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.73205i	1.78855i	0.447521	+	0.894274i	$0.352307\pi$
$7$	4.73205i	1.78855i	−0.447521	+	0.894274i	$0.647693\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	0	0
$13$	6.19615	1.71850	0.859252	−	0.511553i	$-0.170930\pi$
$13$	6.19615	1.71850	0.859252	+	0.511553i	$0.170930\pi$
$14$	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
$19$	0.464102i	0.106472i	0.998582	+	0.0532361i	$0.0169536\pi$
$19$	0.464102i	0.106472i	−0.998582	+	0.0532361i	$0.983046\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.26795	−0.889929	−0.444964	−	0.895548i	$-0.646784\pi$
$23$	−4.26795	−0.889929	−0.444964	+	0.895548i	$0.646784\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.19615i	− 1.52199i	−0.648759	−	0.760994i	$-0.724712\pi$
$29$	− 8.19615i	− 1.52199i	0.648759	−	0.760994i	$-0.275288\pi$
$30$	0	0
$31$	0.464102i	0.0833551i	0.999131	+	0.0416776i	$0.0132702\pi$
$31$	0.464102i	0.0833551i	−0.999131	+	0.0416776i	$0.986730\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.73205	−0.799863
$36$	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
$41$	− 2.19615i	− 0.342981i	−0.985186	−	0.171491i	$-0.945142\pi$
$41$	− 2.19615i	− 0.342981i	0.985186	−	0.171491i	$-0.0548583\pi$
$42$	0	0
$43$	5.66025i	0.863181i	0.902070	+	0.431590i	$0.142047\pi$
$43$	5.66025i	0.863181i	−0.902070	+	0.431590i	$0.857953\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.46410	1.38048	0.690241	−	0.723580i	$-0.257505\pi$
$47$	9.46410	1.38048	0.690241	+	0.723580i	$0.257505\pi$
$48$	0	0
$49$	−15.3923	−2.19890
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.1962i	− 1.53791i	−0.639303	−	0.768955i	$-0.720777\pi$
$53$	− 11.1962i	− 1.53791i	0.639303	−	0.768955i	$-0.279223\pi$
$54$	0	0
$55$	4.73205i	0.638070i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.66025	−0.736902	−0.368451	−	0.929647i	$-0.620112\pi$
$59$	−5.66025	−0.736902	−0.368451	+	0.929647i	$0.620112\pi$
$60$	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
$65$	6.19615i	0.768538i
$66$	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
$71$	4.73205	0.561591	0.280796	−	0.959768i	$-0.409402\pi$
$71$	4.73205	0.561591	0.280796	+	0.959768i	$0.409402\pi$
$72$	0	0
$73$	−4.19615	−0.491122	−0.245561	−	0.969381i	$-0.578972\pi$
$73$	−4.19615	−0.491122	−0.245561	+	0.969381i	$0.578972\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	22.3923i	2.55184i
$78$	0	0
$79$	9.92820i	1.11701i	0.829501	+	0.558505i	$0.188625\pi$
$79$	9.92820i	1.11701i	−0.829501	+	0.558505i	$0.811375\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.19615	0.570352	0.285176	−	0.958475i	$-0.407948\pi$
$83$	5.19615	0.570352	0.285176	+	0.958475i	$0.407948\pi$
$84$	0	0
$85$	−5.19615	−0.563602
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 14.1962i	− 1.50479i	−0.658713	−	0.752395i	$-0.728899\pi$
$89$	− 14.1962i	− 1.50479i	0.658713	−	0.752395i	$-0.271101\pi$
$90$	0	0
$91$	29.3205i	3.07362i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.464102	−0.0476158
$96$	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
$101$	4.39230i	0.437051i	0.975831	+	0.218525i	$0.0701246\pi$
$101$	4.39230i	0.437051i	−0.975831	+	0.218525i	$0.929875\pi$
$102$	0	0
$103$	9.46410i	0.932526i	0.884646	+	0.466263i	$0.154400\pi$
$103$	9.46410i	0.932526i	−0.884646	+	0.466263i	$0.845600\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.9282	−1.82986	−0.914929	−	0.403614i	$-0.867754\pi$
$107$	−18.9282	−1.82986	−0.914929	+	0.403614i	$0.867754\pi$
$108$	0	0
$109$	11.3923	1.09118	0.545592	−	0.838051i	$-0.316305\pi$
$109$	11.3923	1.09118	0.545592	+	0.838051i	$0.316305\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	0	0
$115$	− 4.26795i	− 0.397988i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−24.5885	−2.25402
$120$	0	0
$121$	11.3923	1.03566
$122$	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	$-0.705563\pi$
$127$	− 18.0000i	− 1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.928203	0.0810975	0.0405487	−	0.999178i	$-0.487089\pi$
$131$	0.928203	0.0810975	0.0405487	+	0.999178i	$0.487089\pi$
$132$	0	0
$133$	−2.19615	−0.190431
$134$	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
$139$	19.8564i	1.68420i	0.539323	+	0.842099i	$0.318680\pi$
$139$	19.8564i	1.68420i	−0.539323	+	0.842099i	$0.681320\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	29.3205	2.45190
$144$	0	0
$145$	8.19615	0.680653
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.80385i	0.803162i	0.915823	+	0.401581i	$0.131539\pi$
$149$	9.80385i	0.803162i	−0.915823	+	0.401581i	$0.868461\pi$
$150$	0	0
$151$	− 18.9282i	− 1.54036i	−0.637829	−	0.770178i	$-0.720168\pi$
$151$	− 18.9282i	− 1.54036i	0.637829	−	0.770178i	$-0.279832\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.464102	−0.0372775
$156$	0	0
$157$	16.1962	1.29259	0.646297	−	0.763086i	$-0.276317\pi$
$157$	16.1962	1.29259	0.646297	+	0.763086i	$0.276317\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 20.1962i	− 1.59168i
$162$	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
$167$	14.6603	1.13444	0.567222	−	0.823565i	$-0.308018\pi$
$167$	14.6603	1.13444	0.567222	+	0.823565i	$0.308018\pi$
$168$	0	0
$169$	25.3923	1.95325
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 17.1962i	− 1.30740i	−0.756754	−	0.653700i	$-0.773216\pi$
$173$	− 17.1962i	− 1.30740i	0.756754	−	0.653700i	$-0.226784\pi$
$174$	0	0
$175$	− 4.73205i	− 0.357709i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.85641	−0.138754	−0.0693772	−	0.997591i	$-0.522101\pi$
$179$	−1.85641	−0.138754	−0.0693772	+	0.997591i	$0.522101\pi$
$180$	0	0
$181$	−15.3923	−1.14410	−0.572051	−	0.820218i	$-0.693852\pi$
$181$	−15.3923	−1.14410	−0.572051	+	0.820218i	$0.693852\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.00000i	0.147043i
$186$	0	0
$187$	24.5885i	1.79809i
$188$	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
$193$	−1.80385	−0.129844	−0.0649219	−	0.997890i	$-0.520680\pi$
$193$	−1.80385	−0.129844	−0.0649219	+	0.997890i	$0.520680\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.1962i	− 1.22518i	−0.790403	−	0.612588i	$-0.790129\pi$
$197$	− 17.1962i	− 1.22518i	0.790403	−	0.612588i	$-0.209871\pi$
$198$	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
$203$	38.7846	2.72215
$204$	0	0
$205$	2.19615	0.153386
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.19615i	0.151911i
$210$	0	0
$211$	− 9.00000i	− 0.619586i	−0.950804	−	0.309793i	$-0.899740\pi$
$211$	− 9.00000i	− 0.619586i	0.950804	−	0.309793i	$-0.100260\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.66025	−0.386026
$216$	0	0
$217$	−2.19615	−0.149085
$218$	0	0
$219$	0	0
$220$	0	0
$221$	32.1962i	2.16575i
$222$	0	0
$223$	− 0.928203i	− 0.0621571i	−0.999517	−	0.0310785i	$-0.990106\pi$
$223$	− 0.928203i	− 0.0621571i	0.999517	−	0.0310785i	$-0.00989420\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.26795	−0.283274	−0.141637	−	0.989919i	$-0.545237\pi$
$227$	−4.26795	−0.283274	−0.141637	+	0.989919i	$0.545237\pi$
$228$	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
$233$	14.7846i	0.968572i	0.874910	+	0.484286i	$0.160921\pi$
$233$	14.7846i	0.968572i	−0.874910	+	0.484286i	$0.839079\pi$
$234$	0	0
$235$	9.46410i	0.617370i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−29.3205	−1.89659	−0.948293	−	0.317396i	$-0.897191\pi$
$239$	−29.3205	−1.89659	−0.948293	+	0.317396i	$0.897191\pi$
$240$	0	0
$241$	−3.39230	−0.218518	−0.109259	−	0.994013i	$-0.534848\pi$
$241$	−3.39230	−0.218518	−0.109259	+	0.994013i	$0.534848\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 15.3923i	− 0.983378i
$246$	0	0
$247$	2.87564i	0.182973i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.3205	0.714544	0.357272	−	0.934000i	$-0.383707\pi$
$251$	11.3205	0.714544	0.357272	+	0.934000i	$0.383707\pi$
$252$	0	0
$253$	−20.1962	−1.26972
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.58846i	− 0.598112i	−0.954236	−	0.299056i	$-0.903328\pi$
$257$	− 9.58846i	− 0.598112i	0.954236	−	0.299056i	$-0.0966717\pi$
$258$	0	0
$259$	9.46410i	0.588071i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.3205	0.698052	0.349026	−	0.937113i	$-0.386512\pi$
$263$	11.3205	0.698052	0.349026	+	0.937113i	$0.386512\pi$
$264$	0	0
$265$	11.1962	0.687774
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.7846i	0.901434i	0.892667	+	0.450717i	$0.148832\pi$
$269$	14.7846i	0.901434i	−0.892667	+	0.450717i	$0.851168\pi$
$270$	0	0
$271$	17.5359i	1.06523i	0.846358	+	0.532615i	$0.178791\pi$
$271$	17.5359i	1.06523i	−0.846358	+	0.532615i	$0.821209\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.73205	−0.285353
$276$	0	0
$277$	−28.1962	−1.69414	−0.847071	−	0.531479i	$-0.821636\pi$
$277$	−28.1962	−1.69414	−0.847071	+	0.531479i	$0.821636\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 12.5885i	− 0.750964i	−0.926830	−	0.375482i	$-0.877477\pi$
$281$	− 12.5885i	− 0.750964i	0.926830	−	0.375482i	$-0.122523\pi$
$282$	0	0
$283$	− 13.2679i	− 0.788698i	−0.918961	−	0.394349i	$-0.870970\pi$
$283$	− 13.2679i	− 0.788698i	0.918961	−	0.394349i	$-0.129030\pi$
$284$	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
$293$	12.8038i	0.748009i	0.927427	+	0.374004i	$0.122016\pi$
$293$	12.8038i	0.748009i	−0.927427	+	0.374004i	$0.877984\pi$
$294$	0	0
$295$	− 5.66025i	− 0.329553i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−26.4449	−1.52935
$300$	0	0
$301$	−26.7846	−1.54384
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.00000i	− 0.0572598i
$306$	0	0
$307$	− 19.8564i	− 1.13326i	−0.823971	−	0.566632i	$-0.808246\pi$
$307$	− 19.8564i	− 1.13326i	0.823971	−	0.566632i	$-0.191754\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.7321	1.28902	0.644508	−	0.764597i	$-0.277062\pi$
$311$	22.7321	1.28902	0.644508	+	0.764597i	$0.277062\pi$
$312$	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
$317$	− 18.8038i	− 1.05613i	−0.849204	−	0.528065i	$-0.822918\pi$
$317$	− 18.8038i	− 1.05613i	0.849204	−	0.528065i	$-0.177082\pi$
$318$	0	0
$319$	− 38.7846i	− 2.17152i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.41154	−0.134182
$324$	0	0
$325$	−6.19615	−0.343701
$326$	0	0
$327$	0	0
$328$	0	0
$329$	44.7846i	2.46906i
$330$	0	0
$331$	− 8.53590i	− 0.469175i	−0.972095	−	0.234588i	$-0.924626\pi$
$331$	− 8.53590i	− 0.469175i	0.972095	−	0.234588i	$-0.0753740\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.3923	−0.567792
$336$	0	0
$337$	−2.58846	−0.141002	−0.0705011	−	0.997512i	$-0.522460\pi$
$337$	−2.58846	−0.141002	−0.0705011	+	0.997512i	$0.522460\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.19615i	0.118928i
$342$	0	0
$343$	− 39.7128i	− 2.14429i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.3923	−0.557888	−0.278944	−	0.960307i	$-0.589984\pi$
$347$	−10.3923	−0.557888	−0.278944	+	0.960307i	$0.589984\pi$
$348$	0	0
$349$	12.6077	0.674874	0.337437	−	0.941348i	$-0.390440\pi$
$349$	12.6077	0.674874	0.337437	+	0.941348i	$0.390440\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.3923i	1.19182i	0.803050	+	0.595911i	$0.203209\pi$
$353$	22.3923i	1.19182i	−0.803050	+	0.595911i	$0.796791\pi$
$354$	0	0
$355$	4.73205i	0.251151i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−21.7128	−1.14596	−0.572979	−	0.819570i	$-0.694212\pi$
$359$	−21.7128	−1.14596	−0.572979	+	0.819570i	$0.694212\pi$
$360$	0	0
$361$	18.7846	0.988664
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4.19615i	− 0.219637i
$366$	0	0
$367$	− 6.58846i	− 0.343915i	−0.985104	−	0.171957i	$-0.944991\pi$
$367$	− 6.58846i	− 0.343915i	0.985104	−	0.171957i	$-0.0550091\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	52.9808	2.75062
$372$	0	0
$373$	−8.39230	−0.434537	−0.217269	−	0.976112i	$-0.569715\pi$
$373$	−8.39230	−0.434537	−0.217269	+	0.976112i	$0.569715\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 50.7846i	− 2.61554i
$378$	0	0
$379$	− 27.0000i	− 1.38690i	−0.720506	−	0.693448i	$-0.756091\pi$
$379$	− 27.0000i	− 1.38690i	0.720506	−	0.693448i	$-0.243909\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.33975	−0.170653	−0.0853265	−	0.996353i	$-0.527193\pi$
$383$	−3.33975	−0.170653	−0.0853265	+	0.996353i	$0.527193\pi$
$384$	0	0
$385$	−22.3923	−1.14122
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.58846i	0.334048i	0.985953	+	0.167024i	$0.0534157\pi$
$389$	6.58846i	0.334048i	−0.985953	+	0.167024i	$0.946584\pi$
$390$	0	0
$391$	− 22.1769i	− 1.12153i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9.92820	−0.499542
$396$	0	0
$397$	26.5885	1.33444	0.667218	−	0.744862i	$-0.267485\pi$
$397$	26.5885	1.33444	0.667218	+	0.744862i	$0.267485\pi$
$398$	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
$403$	2.87564i	0.143246i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.46410	0.469118
$408$	0	0
$409$	3.78461	0.187137	0.0935685	−	0.995613i	$-0.470173\pi$
$409$	3.78461	0.187137	0.0935685	+	0.995613i	$0.470173\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 26.7846i	− 1.31798i
$414$	0	0
$415$	5.19615i	0.255069i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−34.9808	−1.70892	−0.854461	−	0.519515i	$-0.826112\pi$
$419$	−34.9808	−1.70892	−0.854461	+	0.519515i	$0.826112\pi$
$420$	0	0
$421$	32.1769	1.56821	0.784103	−	0.620630i	$-0.213123\pi$
$421$	32.1769	1.56821	0.784103	+	0.620630i	$0.213123\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 5.19615i	− 0.252050i
$426$	0	0
$427$	− 4.73205i	− 0.229000i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.1244	−0.728515	−0.364257	−	0.931298i	$-0.618677\pi$
$431$	−15.1244	−0.728515	−0.364257	+	0.931298i	$0.618677\pi$
$432$	0	0
$433$	26.5885	1.27776	0.638880	−	0.769307i	$-0.279398\pi$
$433$	26.5885	1.27776	0.638880	+	0.769307i	$0.279398\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.98076i	− 0.0947527i
$438$	0	0
$439$	− 30.7128i	− 1.46584i	−0.680313	−	0.732921i	$-0.738156\pi$
$439$	− 30.7128i	− 1.46584i	0.680313	−	0.732921i	$-0.261844\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.5167	0.784730	0.392365	−	0.919810i	$-0.371657\pi$
$443$	16.5167	0.784730	0.392365	+	0.919810i	$0.371657\pi$
$444$	0	0
$445$	14.1962	0.672962
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 28.9808i	− 1.36769i	−0.729629	−	0.683843i	$-0.760307\pi$
$449$	− 28.9808i	− 1.36769i	0.729629	−	0.683843i	$-0.239693\pi$
$450$	0	0
$451$	− 10.3923i	− 0.489355i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−29.3205	−1.37457
$456$	0	0
$457$	0.784610	0.0367025	0.0183512	−	0.999832i	$-0.494158\pi$
$457$	0.784610	0.0367025	0.0183512	+	0.999832i	$0.494158\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 22.3923i	− 1.04291i	−0.853278	−	0.521457i	$-0.825389\pi$
$461$	− 22.3923i	− 1.04291i	0.853278	−	0.521457i	$-0.174611\pi$
$462$	0	0
$463$	− 0.928203i	− 0.0431373i	−0.999767	−	0.0215686i	$-0.993134\pi$
$463$	− 0.928203i	− 0.0431373i	0.999767	−	0.0215686i	$-0.00686604\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.7321	1.46838	0.734192	−	0.678942i	$-0.237561\pi$
$467$	31.7321	1.46838	0.734192	+	0.678942i	$0.237561\pi$
$468$	0	0
$469$	−49.1769	−2.27078
$470$	0	0
$471$	0	0
$472$	0	0
$473$	26.7846i	1.23156i
$474$	0	0
$475$	− 0.464102i	− 0.0212944i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.8038	0.996243	0.498122	−	0.867107i	$-0.334023\pi$
$479$	21.8038	0.996243	0.498122	+	0.867107i	$0.334023\pi$
$480$	0	0
$481$	12.3923	0.565040
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 8.00000i	− 0.363261i
$486$	0	0
$487$	− 16.9808i	− 0.769472i	−0.923027	−	0.384736i	$-0.874293\pi$
$487$	− 16.9808i	− 0.769472i	0.923027	−	0.384736i	$-0.125707\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.78461	0.125668	0.0628338	−	0.998024i	$-0.479986\pi$
$491$	2.78461	0.125668	0.0628338	+	0.998024i	$0.479986\pi$
$492$	0	0
$493$	42.5885	1.91809
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.3923i	1.00443i
$498$	0	0
$499$	35.5359i	1.59081i	0.606081	+	0.795403i	$0.292741\pi$
$499$	35.5359i	1.59081i	−0.606081	+	0.795403i	$0.707259\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.41154	−0.107525	−0.0537627	−	0.998554i	$-0.517121\pi$
$503$	−2.41154	−0.107525	−0.0537627	+	0.998554i	$0.517121\pi$
$504$	0	0
$505$	−4.39230	−0.195455
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 4.39230i	− 0.194685i	−0.995251	−	0.0973427i	$-0.968966\pi$
$509$	− 4.39230i	− 0.194685i	0.995251	−	0.0973427i	$-0.0310343\pi$
$510$	0	0
$511$	− 19.8564i	− 0.878396i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.46410	−0.417038
$516$	0	0
$517$	44.7846	1.96962
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6.58846i	0.288646i	0.989531	+	0.144323i	$0.0461004\pi$
$521$	6.58846i	0.288646i	−0.989531	+	0.144323i	$0.953900\pi$
$522$	0	0
$523$	23.6603i	1.03459i	0.855807	+	0.517295i	$0.173061\pi$
$523$	23.6603i	1.03459i	−0.855807	+	0.517295i	$0.826939\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.41154	−0.105048
$528$	0	0
$529$	−4.78461	−0.208027
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 13.6077i	− 0.589415i
$534$	0	0
$535$	− 18.9282i	− 0.818338i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−72.8372	−3.13732
$540$	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
$545$	11.3923i	0.487993i
$546$	0	0
$547$	− 22.7321i	− 0.971952i	−0.873972	−	0.485976i	$-0.838464\pi$
$547$	− 22.7321i	− 0.971952i	0.873972	−	0.485976i	$-0.161536\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.80385	0.162049
$552$	0	0
$553$	−46.9808	−1.99783
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.3923i	1.45725i	0.684914	+	0.728624i	$0.259840\pi$
$557$	34.3923i	1.45725i	−0.684914	+	0.728624i	$0.740160\pi$
$558$	0	0
$559$	35.0718i	1.48338i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.85641	0.0782382	0.0391191	−	0.999235i	$-0.487545\pi$
$563$	1.85641	0.0782382	0.0391191	+	0.999235i	$0.487545\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.1962i	1.34973i	0.737940	+	0.674866i	$0.235799\pi$
$569$	32.1962i	1.34973i	−0.737940	+	0.674866i	$0.764201\pi$
$570$	0	0
$571$	0.464102i	0.0194220i	0.999953	+	0.00971102i	$0.00309116\pi$
$571$	0.464102i	0.0194220i	−0.999953	+	0.00971102i	$0.996909\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.26795	0.177986
$576$	0	0
$577$	−4.58846	−0.191020	−0.0955100	−	0.995428i	$-0.530448\pi$
$577$	−4.58846	−0.191020	−0.0955100	+	0.995428i	$0.530448\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24.5885i	1.02010i
$582$	0	0
$583$	− 52.9808i	− 2.19424i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.19615	−0.214468	−0.107234	−	0.994234i	$-0.534199\pi$
$587$	−5.19615	−0.214468	−0.107234	+	0.994234i	$0.534199\pi$
$588$	0	0
$589$	−0.215390	−0.00887500
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 23.1962i	− 0.952552i	−0.879296	−	0.476276i	$-0.841986\pi$
$593$	− 23.1962i	− 0.952552i	0.879296	−	0.476276i	$-0.158014\pi$
$594$	0	0
$595$	− 24.5885i	− 1.00803i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11.4115	0.466263	0.233131	−	0.972445i	$-0.425103\pi$
$599$	11.4115	0.466263	0.233131	+	0.972445i	$0.425103\pi$
$600$	0	0
$601$	−32.1769	−1.31252	−0.656262	−	0.754533i	$-0.727863\pi$
$601$	−32.1769	−1.31252	−0.656262	+	0.754533i	$0.727863\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11.3923i	0.463163i
$606$	0	0
$607$	− 22.7321i	− 0.922665i	−0.887227	−	0.461333i	$-0.847371\pi$
$607$	− 22.7321i	− 0.922665i	0.887227	−	0.461333i	$-0.152629\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	58.6410	2.37236
$612$	0	0
$613$	33.1769	1.34000	0.670001	−	0.742360i	$-0.266293\pi$
$613$	33.1769	1.34000	0.670001	+	0.742360i	$0.266293\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.58846i	0.144466i	0.997388	+	0.0722329i	$0.0230125\pi$
$617$	3.58846i	0.144466i	−0.997388	+	0.0722329i	$0.976988\pi$
$618$	0	0
$619$	− 8.53590i	− 0.343087i	−0.985177	−	0.171543i	$-0.945125\pi$
$619$	− 8.53590i	− 0.343087i	0.985177	−	0.171543i	$-0.0548754\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	67.1769	2.69139
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.3923i	0.414368i
$630$	0	0
$631$	− 47.7846i	− 1.90228i	−0.308767	−	0.951138i	$-0.599916\pi$
$631$	− 47.7846i	− 1.90228i	0.308767	−	0.951138i	$-0.400084\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	18.0000	0.714308
$636$	0	0
$637$	−95.3731	−3.77882
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 16.3923i	− 0.647457i	−0.946150	−	0.323729i	$-0.895064\pi$
$641$	− 16.3923i	− 0.647457i	0.946150	−	0.323729i	$-0.104936\pi$
$642$	0	0
$643$	− 6.58846i	− 0.259823i	−0.991526	−	0.129912i	$-0.958531\pi$
$643$	− 6.58846i	− 0.259823i	0.991526	−	0.129912i	$-0.0414694\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.05256	0.277265	0.138632	−	0.990344i	$-0.455729\pi$
$647$	7.05256	0.277265	0.138632	+	0.990344i	$0.455729\pi$
$648$	0	0
$649$	−26.7846	−1.05139
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.41154i	0.0943710i	0.998886	+	0.0471855i	$0.0150252\pi$
$653$	2.41154i	0.0943710i	−0.998886	+	0.0471855i	$0.984975\pi$
$654$	0	0
$655$	0.928203i	0.0362679i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.5885	−0.957830	−0.478915	−	0.877861i	$-0.658970\pi$
$659$	−24.5885	−0.957830	−0.478915	+	0.877861i	$0.658970\pi$
$660$	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
$665$	− 2.19615i	− 0.0851631i
$666$	0	0
$667$	34.9808i	1.35446i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.73205	−0.182679
$672$	0	0
$673$	14.5885	0.562344	0.281172	−	0.959657i	$-0.409277\pi$
$673$	14.5885	0.562344	0.281172	+	0.959657i	$0.409277\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	− 37.8564i	− 1.45280i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.6603	1.24971	0.624855	−	0.780741i	$-0.285158\pi$
$683$	32.6603	1.24971	0.624855	+	0.780741i	$0.285158\pi$
$684$	0	0
$685$	5.19615	0.198535
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 69.3731i	− 2.64290i
$690$	0	0
$691$	27.0000i	1.02713i	0.858051	+	0.513564i	$0.171675\pi$
$691$	27.0000i	1.02713i	−0.858051	+	0.513564i	$0.828325\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.8564	−0.753196
$696$	0	0
$697$	11.4115	0.432243
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.00000i	0.226617i	0.993560	+	0.113308i	$0.0361448\pi$
$701$	6.00000i	0.226617i	−0.993560	+	0.113308i	$0.963855\pi$
$702$	0	0
$703$	0.928203i	0.0350078i
$704$	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
$713$	− 1.98076i	− 0.0741801i
$714$	0	0
$715$	29.3205i	1.09652i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.9282	−1.37719	−0.688595	−	0.725146i	$-0.741772\pi$
$719$	−36.9282	−1.37719	−0.688595	+	0.725146i	$0.741772\pi$
$720$	0	0
$721$	−44.7846	−1.66787
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.19615i	0.304397i
$726$	0	0
$727$	− 39.7128i	− 1.47287i	−0.676510	−	0.736433i	$-0.736508\pi$
$727$	− 39.7128i	− 1.47287i	0.676510	−	0.736433i	$-0.263492\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−29.4115	−1.08783
$732$	0	0
$733$	−36.3923	−1.34418	−0.672090	−	0.740469i	$-0.734603\pi$
$733$	−36.3923	−1.34418	−0.672090	+	0.740469i	$0.734603\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	49.1769i	1.81145i
$738$	0	0
$739$	− 19.3923i	− 0.713357i	−0.934227	−	0.356679i	$-0.883909\pi$
$739$	− 19.3923i	− 0.713357i	0.934227	−	0.356679i	$-0.116091\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	50.1051	1.83818	0.919089	−	0.394049i	$-0.128926\pi$
$743$	50.1051	1.83818	0.919089	+	0.394049i	$0.128926\pi$
$744$	0	0
$745$	−9.80385	−0.359185
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 89.5692i	− 3.27279i
$750$	0	0
$751$	48.7128i	1.77756i	0.458338	+	0.888778i	$0.348445\pi$
$751$	48.7128i	1.77756i	−0.458338	+	0.888778i	$0.651555\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.9282	0.688868
$756$	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
$761$	− 45.3731i	− 1.64477i	−0.568930	−	0.822386i	$-0.692642\pi$
$761$	− 45.3731i	− 1.64477i	0.568930	−	0.822386i	$-0.307358\pi$
$762$	0	0
$763$	53.9090i	1.95164i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−35.0718	−1.26637
$768$	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
$773$	23.1962i	0.834308i	0.908836	+	0.417154i	$0.136972\pi$
$773$	23.1962i	0.834308i	−0.908836	+	0.417154i	$0.863028\pi$
$774$	0	0
$775$	− 0.464102i	− 0.0166710i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.01924	0.0365180
$780$	0	0
$781$	22.3923	0.801260
$782$	0	0
$783$	0	0
$784$	0	0
$785$	16.1962i	0.578065i
$786$	0	0
$787$	− 15.1244i	− 0.539125i	−0.962983	−	0.269563i	$-0.913121\pi$
$787$	− 15.1244i	− 0.539125i	0.962983	−	0.269563i	$-0.0868791\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−56.7846	−2.01903
$792$	0	0
$793$	−6.19615	−0.220032
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.7654i	1.44398i	0.691902	+	0.721992i	$0.256773\pi$
$797$	40.7654i	1.44398i	−0.691902	+	0.721992i	$0.743227\pi$
$798$	0	0
$799$	49.1769i	1.73975i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−19.8564	−0.700717
$804$	0	0
$805$	20.1962	0.711821
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 5.41154i	− 0.190260i	−0.995465	−	0.0951299i	$-0.969673\pi$
$809$	− 5.41154i	− 0.190260i	0.995465	−	0.0951299i	$-0.0303266\pi$
$810$	0	0
$811$	− 17.0718i	− 0.599472i	−0.954022	−	0.299736i	$-0.903101\pi$
$811$	− 17.0718i	− 0.599472i	0.954022	−	0.299736i	$-0.0968986\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.3923	−0.364027
$816$	0	0
$817$	−2.62693	−0.0919048
$818$	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
$823$	3.80385i	0.132594i	0.997800	+	0.0662969i	$0.0211184\pi$
$823$	3.80385i	0.132594i	−0.997800	+	0.0662969i	$0.978882\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.7321	1.10343	0.551716	−	0.834032i	$-0.313973\pi$
$827$	31.7321	1.10343	0.551716	+	0.834032i	$0.313973\pi$
$828$	0	0
$829$	29.1769	1.01336	0.506678	−	0.862135i	$-0.330873\pi$
$829$	29.1769	1.01336	0.506678	+	0.862135i	$0.330873\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 79.9808i	− 2.77117i
$834$	0	0
$835$	14.6603i	0.507339i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18.9282	−0.653474	−0.326737	−	0.945115i	$-0.605949\pi$
$839$	−18.9282	−0.653474	−0.326737	+	0.945115i	$0.605949\pi$
$840$	0	0
$841$	−38.1769	−1.31645
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25.3923i	0.873522i
$846$	0	0
$847$	53.9090i	1.85233i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.53590	−0.292607
$852$	0	0
$853$	−0.784610	−0.0268645	−0.0134323	−	0.999910i	$-0.504276\pi$
$853$	−0.784610	−0.0268645	−0.0134323	+	0.999910i	$0.504276\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.8038i	0.437371i	0.975795	+	0.218686i	$0.0701769\pi$
$857$	12.8038i	0.437371i	−0.975795	+	0.218686i	$0.929823\pi$
$858$	0	0
$859$	3.24871i	0.110845i	0.998463	+	0.0554223i	$0.0176505\pi$
$859$	3.24871i	0.110845i	−0.998463	+	0.0554223i	$0.982349\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.1962	1.40233	0.701167	−	0.712997i	$-0.252663\pi$
$863$	41.1962	1.40233	0.701167	+	0.712997i	$0.252663\pi$
$864$	0	0
$865$	17.1962	0.584687
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46.9808i	1.59371i
$870$	0	0
$871$	64.3923i	2.18185i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.73205	0.159973
$876$	0	0
$877$	7.41154	0.250270	0.125135	−	0.992140i	$-0.460064\pi$
$877$	7.41154	0.250270	0.125135	+	0.992140i	$0.460064\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.4115i	0.384465i	0.981349	+	0.192232i	$0.0615727\pi$
$881$	11.4115i	0.384465i	−0.981349	+	0.192232i	$0.938427\pi$
$882$	0	0
$883$	4.73205i	0.159246i	0.996825	+	0.0796231i	$0.0253717\pi$
$883$	4.73205i	0.159246i	−0.996825	+	0.0796231i	$0.974628\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.5167	1.15896	0.579478	−	0.814988i	$-0.303256\pi$
$887$	34.5167	1.15896	0.579478	+	0.814988i	$0.303256\pi$
$888$	0	0
$889$	85.1769	2.85674
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.39230i	0.146983i
$894$	0	0
$895$	− 1.85641i	− 0.0620528i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.80385	0.126865
$900$	0	0
$901$	58.1769	1.93815
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 15.3923i	− 0.511658i
$906$	0	0
$907$	18.9282i	0.628501i	0.949340	+	0.314250i	$0.101753\pi$
$907$	18.9282i	0.628501i	−0.949340	+	0.314250i	$0.898247\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.51666	−0.249038	−0.124519	−	0.992217i	$-0.539739\pi$
$911$	−7.51666	−0.249038	−0.124519	+	0.992217i	$0.539739\pi$
$912$	0	0
$913$	24.5885	0.813759
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.39230i	0.145047i
$918$	0	0
$919$	21.7128i	0.716240i	0.933676	+	0.358120i	$0.116582\pi$
$919$	21.7128i	0.716240i	−0.933676	+	0.358120i	$0.883418\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	29.3205	0.965096
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.80385i	0.321654i	0.986983	+	0.160827i	$0.0514161\pi$
$929$	9.80385i	0.321654i	−0.986983	+	0.160827i	$0.948584\pi$
$930$	0	0
$931$	− 7.14359i	− 0.234122i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.5885	−0.804129
$936$	0	0
$937$	33.1769	1.08384	0.541921	−	0.840429i	$-0.317697\pi$
$937$	33.1769	1.08384	0.541921	+	0.840429i	$0.317697\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 55.1769i	− 1.79872i	−0.437213	−	0.899358i	$-0.644034\pi$
$941$	− 55.1769i	− 1.79872i	0.437213	−	0.899358i	$-0.355966\pi$
$942$	0	0
$943$	9.37307i	0.305229i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.7654	−1.51967	−0.759835	−	0.650116i	$-0.774720\pi$
$947$	−46.7654	−1.51967	−0.759835	+	0.650116i	$0.774720\pi$
$948$	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	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	0	0
$955$	− 18.0000i	− 0.582466i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.5885	0.794003
$960$	0	0
$961$	30.7846	0.993052
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.80385i	− 0.0580679i
$966$	0	0
$967$	29.3205i	0.942884i	0.881897	+	0.471442i	$0.156266\pi$
$967$	29.3205i	0.942884i	−0.881897	+	0.471442i	$0.843734\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.3205	0.940940	0.470470	−	0.882416i	$-0.344084\pi$
$971$	29.3205	0.940940	0.470470	+	0.882416i	$0.344084\pi$
$972$	0	0
$973$	−93.9615	−3.01227
$974$	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
$979$	− 67.1769i	− 2.14698i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13.7321	−0.437984	−0.218992	−	0.975727i	$-0.570277\pi$
$983$	−13.7321	−0.437984	−0.218992	+	0.975727i	$0.570277\pi$
$984$	0	0
$985$	17.1962	0.547915
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 24.1577i	− 0.768169i
$990$	0	0
$991$	− 44.0718i	− 1.39999i	−0.714149	−	0.699993i	$-0.753186\pi$
$991$	− 44.0718i	− 1.39999i	0.714149	−	0.699993i	$-0.246814\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10.3923	0.329458
$996$	0	0
$997$	−48.1962	−1.52639	−0.763194	−	0.646170i	$-0.776370\pi$
$997$	−48.1962	−1.52639	−0.763194	+	0.646170i	$0.776370\pi$
$998$	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.431.4	yes	4
3.2	odd	2		2160.2.h.a.431.2	✓	4
4.3	odd	2		2160.2.h.a.431.3	yes	4
12.11	even	2	inner	2160.2.h.f.431.1	yes	4

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +4.73205i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} +4.73205i q^{7} +4.73205 q^{11} +6.19615 q^{13} +5.19615i q^{17} +0.464102i q^{19} -4.26795 q^{23} -1.00000 q^{25} -8.19615i q^{29} +0.464102i q^{31} -4.73205 q^{35} +2.00000 q^{37} -2.19615i q^{41} +5.66025i q^{43} +9.46410 q^{47} -15.3923 q^{49} -11.1962i q^{53} +4.73205i q^{55} -5.66025 q^{59} -1.00000 q^{61} +6.19615i q^{65} +10.3923i q^{67} +4.73205 q^{71} -4.19615 q^{73} +22.3923i q^{77} +9.92820i q^{79} +5.19615 q^{83} -5.19615 q^{85} -14.1962i q^{89} +29.3205i q^{91} -0.464102 q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 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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