LMFDB - Embedded newform 3744.2.g.e.1873.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1873,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("3744.1873");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.g (of order $2$, degree $1$, not 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:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 2x^14 - 4x^13 + 9x^12 - 10x^11 + 2x^10 - 8x^9 + 28x^8 - 16x^7 + 8x^6 - 80x^5 + 144x^4 - 128x^3 + 128x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1873.9
Root			$1.40722 - 0.140463i$ of defining polynomial
Character	$\chi$	$=$	3744.1873
Dual form			3744.2.g.e.1873.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.218531i q^{5} +4.47783 q^{7} +O(q^{10})$	$q+0.218531i q^{5} +4.47783 q^{7} +3.58286i q^{11} -1.00000i q^{13} +7.60367 q^{17} -4.65804i q^{19} -5.21219 q^{23} +4.95224 q^{25} -3.39240i q^{29} +1.71291 q^{31} +0.978543i q^{35} -3.06703i q^{37} -1.17387 q^{41} +6.53664i q^{43} +6.69636 q^{47} +13.0509 q^{49} -1.18362i q^{53} -0.782966 q^{55} +9.33654i q^{59} -14.6809i q^{61} +0.218531 q^{65} -2.10688i q^{67} -11.9731 q^{71} -13.8518 q^{73} +16.0434i q^{77} -2.12278 q^{79} +11.9086i q^{83} +1.66164i q^{85} +10.2015 q^{89} -4.47783i q^{91} +1.01792 q^{95} +16.0227 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7}+O(q^{10}) $ 16 * q - 4 * q^7	$ 16 q - 4 q^{7} - 16 q^{17} - 8 q^{23} - 32 q^{25} + 4 q^{31} + 36 q^{41} + 24 q^{47} + 48 q^{49} - 24 q^{55} - 4 q^{65} - 32 q^{73} + 60 q^{89} - 24 q^{95} + 40 q^{97}+O(q^{100}) $ 16 * q - 4 * q^7 - 16 * q^17 - 8 * q^23 - 32 * q^25 + 4 * q^31 + 36 * q^41 + 24 * q^47 + 48 * q^49 - 24 * q^55 - 4 * q^65 - 32 * q^73 + 60 * q^89 - 24 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	0.218531i	0.0977299i	0.998805	+	0.0488650i	$0.0155604\pi$
$5$	0.218531i	0.0977299i	−0.998805	+	0.0488650i	$0.984440\pi$
$6$	0	0
$7$	4.47783	1.69246	0.846230	−	0.532818i	$-0.178867\pi$
$7$	4.47783	1.69246	0.846230	+	0.532818i	$0.178867\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.58286i	1.08027i	0.841577	+	0.540137i	$0.181628\pi$
$11$	3.58286i	1.08027i	−0.841577	+	0.540137i	$0.818372\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.60367	1.84416	0.922080	−	0.386998i	$-0.126488\pi$
$17$	7.60367	1.84416	0.922080	+	0.386998i	$0.126488\pi$
$18$	0	0
$19$	− 4.65804i	− 1.06863i	−0.845286	−	0.534313i	$-0.820570\pi$
$19$	− 4.65804i	− 1.06863i	0.845286	−	0.534313i	$-0.179430\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.21219	−1.08682	−0.543409	−	0.839468i	$-0.682867\pi$
$23$	−5.21219	−1.08682	−0.543409	+	0.839468i	$0.682867\pi$
$24$	0	0
$25$	4.95224	0.990449
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.39240i	− 0.629953i	−0.949099	−	0.314977i	$-0.898003\pi$
$29$	− 3.39240i	− 0.629953i	0.949099	−	0.314977i	$-0.101997\pi$
$30$	0	0
$31$	1.71291	0.307647	0.153824	−	0.988098i	$-0.450841\pi$
$31$	1.71291	0.307647	0.153824	+	0.988098i	$0.450841\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.978543i	0.165404i
$36$	0	0
$37$	− 3.06703i	− 0.504217i	−0.967699	−	0.252108i	$-0.918876\pi$
$37$	− 3.06703i	− 0.504217i	0.967699	−	0.252108i	$-0.0811239\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.17387	−0.183328	−0.0916638	−	0.995790i	$-0.529219\pi$
$41$	−1.17387	−0.183328	−0.0916638	+	0.995790i	$0.529219\pi$
$42$	0	0
$43$	6.53664i	0.996828i	0.866939	+	0.498414i	$0.166084\pi$
$43$	6.53664i	0.996828i	−0.866939	+	0.498414i	$0.833916\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.69636	0.976764	0.488382	−	0.872630i	$-0.337587\pi$
$47$	6.69636	0.976764	0.488382	+	0.872630i	$0.337587\pi$
$48$	0	0
$49$	13.0509	1.86442
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.18362i	− 0.162583i	−0.996690	−	0.0812914i	$-0.974096\pi$
$53$	− 1.18362i	− 0.162583i	0.996690	−	0.0812914i	$-0.0259044\pi$
$54$	0	0
$55$	−0.782966	−0.105575
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.33654i	1.21551i	0.794123	+	0.607757i	$0.207930\pi$
$59$	9.33654i	1.21551i	−0.794123	+	0.607757i	$0.792070\pi$
$60$	0	0
$61$	− 14.6809i	− 1.87970i	−0.341590	−	0.939849i	$-0.610965\pi$
$61$	− 14.6809i	− 1.87970i	0.341590	−	0.939849i	$-0.389035\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.218531	0.0271054
$66$	0	0
$67$	− 2.10688i	− 0.257397i	−0.991684	−	0.128698i	$-0.958920\pi$
$67$	− 2.10688i	− 0.257397i	0.991684	−	0.128698i	$-0.0410799\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.9731	−1.42094	−0.710470	−	0.703728i	$-0.751518\pi$
$71$	−11.9731	−1.42094	−0.710470	+	0.703728i	$0.751518\pi$
$72$	0	0
$73$	−13.8518	−1.62123	−0.810617	−	0.585576i	$-0.800868\pi$
$73$	−13.8518	−1.62123	−0.810617	+	0.585576i	$0.800868\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.0434i	1.82832i
$78$	0	0
$79$	−2.12278	−0.238832	−0.119416	−	0.992844i	$-0.538102\pi$
$79$	−2.12278	−0.238832	−0.119416	+	0.992844i	$0.538102\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.9086i	1.30713i	0.756869	+	0.653567i	$0.226728\pi$
$83$	11.9086i	1.30713i	−0.756869	+	0.653567i	$0.773272\pi$
$84$	0	0
$85$	1.66164i	0.180230i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.2015	1.08136	0.540679	−	0.841229i	$-0.318167\pi$
$89$	10.2015	1.08136	0.540679	+	0.841229i	$0.318167\pi$
$90$	0	0
$91$	− 4.47783i	− 0.469404i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.01792	0.104437
$96$	0	0
$97$	16.0227	1.62686	0.813429	−	0.581665i	$-0.197598\pi$
$97$	16.0227	1.62686	0.813429	+	0.581665i	$0.197598\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 3.90916i	− 0.388976i	−0.980905	−	0.194488i	$-0.937695\pi$
$101$	− 3.90916i	− 0.388976i	0.980905	−	0.194488i	$-0.0623045\pi$
$102$	0	0
$103$	3.38273	0.333310	0.166655	−	0.986015i	$-0.446703\pi$
$103$	3.38273	0.333310	0.166655	+	0.986015i	$0.446703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.01021i	− 0.194334i	−0.995268	−	0.0971672i	$-0.969022\pi$
$107$	− 2.01021i	− 0.194334i	0.995268	−	0.0971672i	$-0.0309781\pi$
$108$	0	0
$109$	16.0032i	1.53283i	0.642347	+	0.766414i	$0.277961\pi$
$109$	16.0032i	1.53283i	−0.642347	+	0.766414i	$0.722039\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.90667	−0.555653	−0.277826	−	0.960631i	$-0.589614\pi$
$113$	−5.90667	−0.555653	−0.277826	+	0.960631i	$0.589614\pi$
$114$	0	0
$115$	− 1.13902i	− 0.106215i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	34.0479	3.12117
$120$	0	0
$121$	−1.83691	−0.166992
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.17487i	0.194526i
$126$	0	0
$127$	−6.89436	−0.611776	−0.305888	−	0.952068i	$-0.598953\pi$
$127$	−6.89436	−0.611776	−0.305888	+	0.952068i	$0.598953\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.98979i	− 0.173849i	−0.996215	−	0.0869244i	$-0.972296\pi$
$131$	− 1.98979i	− 0.173849i	0.996215	−	0.0869244i	$-0.0277039\pi$
$132$	0	0
$133$	− 20.8579i	− 1.80861i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.961081	0.0821107	0.0410554	−	0.999157i	$-0.486928\pi$
$137$	0.961081	0.0821107	0.0410554	+	0.999157i	$0.486928\pi$
$138$	0	0
$139$	5.54489i	0.470312i	0.971958	+	0.235156i	$0.0755601\pi$
$139$	5.54489i	0.470312i	−0.971958	+	0.235156i	$0.924440\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.58286	0.299614
$144$	0	0
$145$	0.741344	0.0615653
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 20.9305i	− 1.71469i	−0.514739	−	0.857347i	$-0.672111\pi$
$149$	− 20.9305i	− 1.71469i	0.514739	−	0.857347i	$-0.327889\pi$
$150$	0	0
$151$	0.401184	0.0326479	0.0163239	−	0.999867i	$-0.494804\pi$
$151$	0.401184	0.0326479	0.0163239	+	0.999867i	$0.494804\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.374323i	0.0300663i
$156$	0	0
$157$	18.9079i	1.50901i	0.656292	+	0.754507i	$0.272124\pi$
$157$	18.9079i	1.50901i	−0.656292	+	0.754507i	$0.727876\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−23.3393	−1.83939
$162$	0	0
$163$	− 18.5887i	− 1.45598i	−0.685589	−	0.727989i	$-0.740455\pi$
$163$	− 18.5887i	− 1.45598i	0.685589	−	0.727989i	$-0.259545\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.9204	−1.23196	−0.615979	−	0.787763i	$-0.711239\pi$
$167$	−15.9204	−1.23196	−0.615979	+	0.787763i	$0.711239\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.39369i	− 0.258018i	−0.991643	−	0.129009i	$-0.958820\pi$
$173$	− 3.39369i	− 0.258018i	0.991643	−	0.129009i	$-0.0411796\pi$
$174$	0	0
$175$	22.1753	1.67629
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 0.978543i	− 0.0731397i	−0.999331	−	0.0365699i	$-0.988357\pi$
$179$	− 0.978543i	− 0.0731397i	0.999331	−	0.0365699i	$-0.0116431\pi$
$180$	0	0
$181$	12.2039i	0.907111i	0.891228	+	0.453555i	$0.149845\pi$
$181$	12.2039i	0.907111i	−0.891228	+	0.453555i	$0.850155\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.670241	0.0492771
$186$	0	0
$187$	27.2429i	1.99220i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.11614	0.659621	0.329810	−	0.944047i	$-0.393015\pi$
$191$	9.11614	0.659621	0.329810	+	0.944047i	$0.393015\pi$
$192$	0	0
$193$	7.98107	0.574490	0.287245	−	0.957857i	$-0.407261\pi$
$193$	7.98107	0.574490	0.287245	+	0.957857i	$0.407261\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.6000i	− 0.897713i	−0.893604	−	0.448856i	$-0.851831\pi$
$197$	− 12.6000i	− 0.897713i	0.893604	−	0.448856i	$-0.148169\pi$
$198$	0	0
$199$	19.1593	1.35816	0.679082	−	0.734062i	$-0.262378\pi$
$199$	19.1593	1.35816	0.679082	+	0.734062i	$0.262378\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 15.1906i	− 1.06617i
$204$	0	0
$205$	− 0.256527i	− 0.0179166i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.6891	1.15441
$210$	0	0
$211$	17.4547i	1.20163i	0.799386	+	0.600817i	$0.205158\pi$
$211$	17.4547i	1.20163i	−0.799386	+	0.600817i	$0.794842\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.42846	−0.0974200
$216$	0	0
$217$	7.67010	0.520681
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 7.60367i	− 0.511478i
$222$	0	0
$223$	8.42115	0.563922	0.281961	−	0.959426i	$-0.409015\pi$
$223$	8.42115	0.563922	0.281961	+	0.959426i	$0.409015\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.0875i	− 1.39963i	−0.714326	−	0.699813i	$-0.753267\pi$
$227$	− 21.0875i	− 1.39963i	0.714326	−	0.699813i	$-0.246733\pi$
$228$	0	0
$229$	− 12.0289i	− 0.794894i	−0.917625	−	0.397447i	$-0.869896\pi$
$229$	− 12.0289i	− 0.794894i	0.917625	−	0.397447i	$-0.130104\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.3048	0.937140	0.468570	−	0.883426i	$-0.344769\pi$
$233$	14.3048	0.937140	0.468570	+	0.883426i	$0.344769\pi$
$234$	0	0
$235$	1.46336i	0.0954591i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.22763	−0.338147	−0.169074	−	0.985603i	$-0.554078\pi$
$239$	−5.22763	−0.338147	−0.169074	+	0.985603i	$0.554078\pi$
$240$	0	0
$241$	−2.04161	−0.131512	−0.0657559	−	0.997836i	$-0.520946\pi$
$241$	−2.04161	−0.131512	−0.0657559	+	0.997836i	$0.520946\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.85203i	0.182210i
$246$	0	0
$247$	−4.65804	−0.296384
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.1961i	1.40100i	0.713650	+	0.700502i	$0.247041\pi$
$251$	22.1961i	1.40100i	−0.713650	+	0.700502i	$0.752959\pi$
$252$	0	0
$253$	− 18.6746i	− 1.17406i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.71742	0.231886	0.115943	−	0.993256i	$-0.463011\pi$
$257$	3.71742	0.231886	0.115943	+	0.993256i	$0.463011\pi$
$258$	0	0
$259$	− 13.7336i	− 0.853367i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.09245	−0.129026	−0.0645130	−	0.997917i	$-0.520549\pi$
$263$	−2.09245	−0.129026	−0.0645130	+	0.997917i	$0.520549\pi$
$264$	0	0
$265$	0.258657	0.0158892
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.16604i	0.314979i	0.987521	+	0.157490i	$0.0503401\pi$
$269$	5.16604i	0.314979i	−0.987521	+	0.157490i	$0.949660\pi$
$270$	0	0
$271$	12.0412	0.731453	0.365727	−	0.930722i	$-0.380821\pi$
$271$	12.0412	0.731453	0.365727	+	0.930722i	$0.380821\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	17.7432i	1.06996i
$276$	0	0
$277$	16.5584i	0.994901i	0.867492	+	0.497450i	$0.165730\pi$
$277$	16.5584i	0.994901i	−0.867492	+	0.497450i	$0.834270\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.99219	−0.297809	−0.148905	−	0.988852i	$-0.547575\pi$
$281$	−4.99219	−0.297809	−0.148905	+	0.988852i	$0.547575\pi$
$282$	0	0
$283$	4.99175i	0.296729i	0.988933	+	0.148364i	$0.0474008\pi$
$283$	4.99175i	0.296729i	−0.988933	+	0.148364i	$0.952599\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.25639	−0.310275
$288$	0	0
$289$	40.8158	2.40093
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3.02283i	− 0.176596i	−0.996094	−	0.0882979i	$-0.971857\pi$
$293$	− 3.02283i	− 0.176596i	0.996094	−	0.0882979i	$-0.0281427\pi$
$294$	0	0
$295$	−2.04032	−0.118792
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.21219i	0.301429i
$300$	0	0
$301$	29.2699i	1.68709i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.20823	0.183703
$306$	0	0
$307$	22.8741i	1.30549i	0.757576	+	0.652747i	$0.226383\pi$
$307$	22.8741i	1.30549i	−0.757576	+	0.652747i	$0.773617\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.1117	1.25384	0.626920	−	0.779084i	$-0.284315\pi$
$311$	22.1117	1.25384	0.626920	+	0.779084i	$0.284315\pi$
$312$	0	0
$313$	−15.6954	−0.887155	−0.443577	−	0.896236i	$-0.646291\pi$
$313$	−15.6954	−0.887155	−0.443577	+	0.896236i	$0.646291\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.6083i	1.38214i	0.722788	+	0.691070i	$0.242860\pi$
$317$	24.6083i	1.38214i	−0.722788	+	0.691070i	$0.757140\pi$
$318$	0	0
$319$	12.1545	0.680522
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 35.4182i	− 1.97072i
$324$	0	0
$325$	− 4.95224i	− 0.274701i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	29.9851	1.65313
$330$	0	0
$331$	6.29568i	0.346042i	0.984918	+	0.173021i	$0.0553529\pi$
$331$	6.29568i	0.346042i	−0.984918	+	0.173021i	$0.944647\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.460419	0.0251554
$336$	0	0
$337$	−16.8841	−0.919737	−0.459869	−	0.887987i	$-0.652104\pi$
$337$	−16.8841	−0.919737	−0.459869	+	0.887987i	$0.652104\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.13711i	0.332343i
$342$	0	0
$343$	27.0951	1.46300
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 29.1187i	− 1.56317i	−0.623798	−	0.781586i	$-0.714411\pi$
$347$	− 29.1187i	− 1.56317i	0.623798	−	0.781586i	$-0.285589\pi$
$348$	0	0
$349$	7.81462i	0.418307i	0.977883	+	0.209154i	$0.0670709\pi$
$349$	7.81462i	0.418307i	−0.977883	+	0.209154i	$0.932929\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11.3088	−0.601908	−0.300954	−	0.953639i	$-0.597305\pi$
$353$	−11.3088	−0.601908	−0.300954	+	0.953639i	$0.597305\pi$
$354$	0	0
$355$	− 2.61648i	− 0.138868i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.0361	0.951910	0.475955	−	0.879470i	$-0.342102\pi$
$359$	18.0361	0.951910	0.475955	+	0.879470i	$0.342102\pi$
$360$	0	0
$361$	−2.69730	−0.141963
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.02705i	− 0.158443i
$366$	0	0
$367$	18.7918	0.980924	0.490462	−	0.871463i	$-0.336828\pi$
$367$	18.7918	0.980924	0.490462	+	0.871463i	$0.336828\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 5.30005i	− 0.275165i
$372$	0	0
$373$	− 4.04291i	− 0.209334i	−0.994507	−	0.104667i	$-0.966622\pi$
$373$	− 4.04291i	− 0.209334i	0.994507	−	0.104667i	$-0.0333777\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.39240	−0.174718
$378$	0	0
$379$	18.3308i	0.941590i	0.882243	+	0.470795i	$0.156033\pi$
$379$	18.3308i	0.941590i	−0.882243	+	0.470795i	$0.843967\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.24438	0.0635851	0.0317925	−	0.999494i	$-0.489878\pi$
$383$	1.24438	0.0635851	0.0317925	+	0.999494i	$0.489878\pi$
$384$	0	0
$385$	−3.50599	−0.178682
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 18.8394i	− 0.955196i	−0.878579	−	0.477598i	$-0.841508\pi$
$389$	− 18.8394i	− 0.955196i	0.878579	−	0.477598i	$-0.158492\pi$
$390$	0	0
$391$	−39.6318	−2.00427
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 0.463893i	− 0.0233410i
$396$	0	0
$397$	22.7274i	1.14065i	0.821418	+	0.570327i	$0.193183\pi$
$397$	22.7274i	1.14065i	−0.821418	+	0.570327i	$0.806817\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.720524	0.0359813	0.0179906	−	0.999838i	$-0.494273\pi$
$401$	0.720524	0.0359813	0.0179906	+	0.999838i	$0.494273\pi$
$402$	0	0
$403$	− 1.71291i	− 0.0853260i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.9888	0.544692
$408$	0	0
$409$	−13.4116	−0.663163	−0.331582	−	0.943427i	$-0.607582\pi$
$409$	−13.4116	−0.663163	−0.331582	+	0.943427i	$0.607582\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	41.8074i	2.05721i
$414$	0	0
$415$	−2.60238	−0.127746
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 21.7292i	− 1.06154i	−0.847516	−	0.530770i	$-0.821903\pi$
$419$	− 21.7292i	− 1.06154i	0.847516	−	0.530770i	$-0.178097\pi$
$420$	0	0
$421$	17.1215i	0.834452i	0.908803	+	0.417226i	$0.136998\pi$
$421$	17.1215i	0.834452i	−0.908803	+	0.417226i	$0.863002\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	37.6552	1.82655
$426$	0	0
$427$	− 65.7386i	− 3.18131i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−35.9214	−1.73027	−0.865137	−	0.501536i	$-0.832768\pi$
$431$	−35.9214	−1.73027	−0.865137	+	0.501536i	$0.832768\pi$
$432$	0	0
$433$	−20.7974	−0.999460	−0.499730	−	0.866181i	$-0.666567\pi$
$433$	−20.7974	−0.999460	−0.499730	+	0.866181i	$0.666567\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.2786i	1.16140i
$438$	0	0
$439$	−8.05946	−0.384657	−0.192328	−	0.981331i	$-0.561604\pi$
$439$	−8.05946	−0.384657	−0.192328	+	0.981331i	$0.561604\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 7.84426i	− 0.372692i	−0.982484	−	0.186346i	$-0.940335\pi$
$443$	− 7.84426i	− 0.372692i	0.982484	−	0.186346i	$-0.0596646\pi$
$444$	0	0
$445$	2.22935i	0.105681i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−28.1265	−1.32737	−0.663685	−	0.748012i	$-0.731008\pi$
$449$	−28.1265	−1.32737	−0.663685	+	0.748012i	$0.731008\pi$
$450$	0	0
$451$	− 4.20582i	− 0.198044i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.978543	0.0458748
$456$	0	0
$457$	7.83861	0.366675	0.183337	−	0.983050i	$-0.441310\pi$
$457$	7.83861	0.366675	0.183337	+	0.983050i	$0.441310\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 11.2498i	− 0.523955i	−0.965074	−	0.261977i	$-0.915625\pi$
$461$	− 11.2498i	− 0.523955i	0.965074	−	0.261977i	$-0.0843746\pi$
$462$	0	0
$463$	−16.7795	−0.779811	−0.389905	−	0.920855i	$-0.627492\pi$
$463$	−16.7795	−0.779811	−0.389905	+	0.920855i	$0.627492\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 39.3481i	− 1.82081i	−0.413713	−	0.910407i	$-0.635768\pi$
$467$	− 39.3481i	− 1.82081i	0.413713	−	0.910407i	$-0.364232\pi$
$468$	0	0
$469$	− 9.43426i	− 0.435634i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−23.4199	−1.07685
$474$	0	0
$475$	− 23.0677i	− 1.05842i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.7196	−0.581173	−0.290586	−	0.956849i	$-0.593850\pi$
$479$	−12.7196	−0.581173	−0.290586	+	0.956849i	$0.593850\pi$
$480$	0	0
$481$	−3.06703	−0.139845
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.50145i	0.158993i
$486$	0	0
$487$	−36.3811	−1.64859	−0.824293	−	0.566164i	$-0.808427\pi$
$487$	−36.3811	−1.64859	−0.824293	+	0.566164i	$0.808427\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.3927i	0.965440i	0.875775	+	0.482720i	$0.160351\pi$
$491$	21.3927i	0.965440i	−0.875775	+	0.482720i	$0.839649\pi$
$492$	0	0
$493$	− 25.7947i	− 1.16173i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−53.6133	−2.40488
$498$	0	0
$499$	− 8.39795i	− 0.375944i	−0.982174	−	0.187972i	$-0.939809\pi$
$499$	− 8.39795i	− 0.375944i	0.982174	−	0.187972i	$-0.0601914\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.1422	−1.65609	−0.828044	−	0.560663i	$-0.810547\pi$
$503$	−37.1422	−1.65609	−0.828044	+	0.560663i	$0.810547\pi$
$504$	0	0
$505$	0.854272	0.0380146
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 6.89677i	− 0.305694i	−0.988250	−	0.152847i	$-0.951156\pi$
$509$	− 6.89677i	− 0.305694i	0.988250	−	0.152847i	$-0.0488442\pi$
$510$	0	0
$511$	−62.0261	−2.74387
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.739231i	0.0325744i
$516$	0	0
$517$	23.9921i	1.05517i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.62254	0.333949	0.166975	−	0.985961i	$-0.446600\pi$
$521$	7.62254	0.333949	0.166975	+	0.985961i	$0.446600\pi$
$522$	0	0
$523$	− 14.7801i	− 0.646287i	−0.946350	−	0.323144i	$-0.895260\pi$
$523$	− 14.7801i	− 0.646287i	0.946350	−	0.323144i	$-0.104740\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.0244	0.567351
$528$	0	0
$529$	4.16695	0.181172
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.17387i	0.0508460i
$534$	0	0
$535$	0.439293	0.0189923
$536$	0	0
$537$	0	0
$538$	0	0
$539$	46.7597i	2.01408i
$540$	0	0
$541$	− 37.4916i	− 1.61189i	−0.591990	−	0.805945i	$-0.701658\pi$
$541$	− 37.4916i	− 1.61189i	0.591990	−	0.805945i	$-0.298342\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.49719	−0.149803
$546$	0	0
$547$	15.7598i	0.673840i	0.941533	+	0.336920i	$0.109385\pi$
$547$	15.7598i	0.673840i	−0.941533	+	0.336920i	$0.890615\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−15.8019	−0.673185
$552$	0	0
$553$	−9.50545	−0.404213
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.0153i	− 1.01756i	−0.860896	−	0.508781i	$-0.830096\pi$
$557$	− 24.0153i	− 1.01756i	0.860896	−	0.508781i	$-0.169904\pi$
$558$	0	0
$559$	6.53664	0.276470
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.9917i	1.13757i	0.822488	+	0.568783i	$0.192586\pi$
$563$	26.9917i	1.13757i	−0.822488	+	0.568783i	$0.807414\pi$
$564$	0	0
$565$	− 1.29079i	− 0.0543039i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	39.0770	1.63819	0.819096	−	0.573657i	$-0.194476\pi$
$569$	39.0770	1.63819	0.819096	+	0.573657i	$0.194476\pi$
$570$	0	0
$571$	− 38.5630i	− 1.61381i	−0.590679	−	0.806907i	$-0.701140\pi$
$571$	− 38.5630i	− 1.61381i	0.590679	−	0.806907i	$-0.298860\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−25.8120	−1.07644
$576$	0	0
$577$	29.4706	1.22688	0.613438	−	0.789743i	$-0.289786\pi$
$577$	29.4706	1.22688	0.613438	+	0.789743i	$0.289786\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	53.3244i	2.21227i
$582$	0	0
$583$	4.24075	0.175634
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.19553i	0.296991i	0.988913	+	0.148496i	$0.0474431\pi$
$587$	7.19553i	0.296991i	−0.988913	+	0.148496i	$0.952557\pi$
$588$	0	0
$589$	− 7.97879i	− 0.328760i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.68075	−0.315411	−0.157705	−	0.987486i	$-0.550410\pi$
$593$	−7.68075	−0.315411	−0.157705	+	0.987486i	$0.550410\pi$
$594$	0	0
$595$	7.44052i	0.305032i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.3642	0.995493	0.497747	−	0.867322i	$-0.334161\pi$
$599$	24.3642	0.995493	0.497747	+	0.867322i	$0.334161\pi$
$600$	0	0
$601$	8.34674	0.340471	0.170235	−	0.985403i	$-0.445547\pi$
$601$	8.34674	0.340471	0.170235	+	0.985403i	$0.445547\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 0.401422i	− 0.0163201i
$606$	0	0
$607$	−25.7099	−1.04353	−0.521767	−	0.853088i	$-0.674727\pi$
$607$	−25.7099	−1.04353	−0.521767	+	0.853088i	$0.674727\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 6.69636i	− 0.270906i
$612$	0	0
$613$	12.4181i	0.501564i	0.968044	+	0.250782i	$0.0806877\pi$
$613$	12.4181i	0.501564i	−0.968044	+	0.250782i	$0.919312\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.8886	−1.04224	−0.521118	−	0.853485i	$-0.674485\pi$
$617$	−25.8886	−1.04224	−0.521118	+	0.853485i	$0.674485\pi$
$618$	0	0
$619$	35.4019i	1.42292i	0.702726	+	0.711461i	$0.251966\pi$
$619$	35.4019i	1.42292i	−0.702726	+	0.711461i	$0.748034\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	45.6806	1.83016
$624$	0	0
$625$	24.2859	0.971438
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 23.3207i	− 0.929857i
$630$	0	0
$631$	−15.1930	−0.604824	−0.302412	−	0.953177i	$-0.597792\pi$
$631$	−15.1930	−0.604824	−0.302412	+	0.953177i	$0.597792\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.50663i	− 0.0597888i
$636$	0	0
$637$	− 13.0509i	− 0.517097i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−33.9555	−1.34116	−0.670582	−	0.741836i	$-0.733955\pi$
$641$	−33.9555	−1.34116	−0.670582	+	0.741836i	$0.733955\pi$
$642$	0	0
$643$	− 11.8070i	− 0.465623i	−0.972522	−	0.232812i	$-0.925207\pi$
$643$	− 11.8070i	− 0.465623i	0.972522	−	0.232812i	$-0.0747925\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.7307	−0.775695	−0.387848	−	0.921724i	$-0.626781\pi$
$647$	−19.7307	−0.775695	−0.387848	+	0.921724i	$0.626781\pi$
$648$	0	0
$649$	−33.4515	−1.31309
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 14.9225i	− 0.583963i	−0.956424	−	0.291982i	$-0.905685\pi$
$653$	− 14.9225i	− 0.583963i	0.956424	−	0.291982i	$-0.0943146\pi$
$654$	0	0
$655$	0.434830	0.0169902
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.7477i	1.35358i	0.736177	+	0.676789i	$0.236629\pi$
$659$	34.7477i	1.35358i	−0.736177	+	0.676789i	$0.763371\pi$
$660$	0	0
$661$	− 42.5576i	− 1.65530i	−0.561245	−	0.827650i	$-0.689677\pi$
$661$	− 42.5576i	− 1.65530i	0.561245	−	0.827650i	$-0.310323\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.55809	0.176755
$666$	0	0
$667$	17.6818i	0.684644i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	52.5997	2.03059
$672$	0	0
$673$	−30.8129	−1.18775	−0.593875	−	0.804557i	$-0.702403\pi$
$673$	−30.8129	−1.18775	−0.593875	+	0.804557i	$0.702403\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.1235i	1.08087i	0.841385	+	0.540436i	$0.181741\pi$
$677$	28.1235i	1.08087i	−0.841385	+	0.540436i	$0.818259\pi$
$678$	0	0
$679$	71.7468	2.75339
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 10.1192i	− 0.387199i	−0.981081	−	0.193600i	$-0.937984\pi$
$683$	− 10.1192i	− 0.387199i	0.981081	−	0.193600i	$-0.0620163\pi$
$684$	0	0
$685$	0.210026i	0.00802468i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.18362	−0.0450923
$690$	0	0
$691$	− 25.1776i	− 0.957803i	−0.877869	−	0.478901i	$-0.841035\pi$
$691$	− 25.1776i	− 0.957803i	0.877869	−	0.478901i	$-0.158965\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.21173	−0.0459635
$696$	0	0
$697$	−8.92572	−0.338086
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.53851i	0.284725i	0.989815	+	0.142363i	$0.0454699\pi$
$701$	7.53851i	0.284725i	−0.989815	+	0.142363i	$0.954530\pi$
$702$	0	0
$703$	−14.2863	−0.538820
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17.5045i	− 0.658326i
$708$	0	0
$709$	13.5851i	0.510201i	0.966915	+	0.255100i	$0.0821085\pi$
$709$	13.5851i	0.510201i	−0.966915	+	0.255100i	$0.917892\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.92800	−0.334356
$714$	0	0
$715$	0.782966i	0.0292813i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.6991	−0.771947	−0.385973	−	0.922510i	$-0.626134\pi$
$719$	−20.6991	−0.771947	−0.385973	+	0.922510i	$0.626134\pi$
$720$	0	0
$721$	15.1473	0.564114
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 16.8000i	− 0.623936i
$726$	0	0
$727$	−15.2057	−0.563947	−0.281974	−	0.959422i	$-0.590989\pi$
$727$	−15.2057	−0.563947	−0.281974	+	0.959422i	$0.590989\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	49.7025i	1.83831i
$732$	0	0
$733$	43.4506i	1.60488i	0.596730	+	0.802442i	$0.296466\pi$
$733$	43.4506i	1.60488i	−0.596730	+	0.802442i	$0.703534\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.54868	0.278059
$738$	0	0
$739$	− 4.52267i	− 0.166369i	−0.996534	−	0.0831845i	$-0.973491\pi$
$739$	− 4.52267i	− 0.166369i	0.996534	−	0.0831845i	$-0.0265091\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.1002	−1.32439	−0.662194	−	0.749332i	$-0.730375\pi$
$743$	−36.1002	−1.32439	−0.662194	+	0.749332i	$0.730375\pi$
$744$	0	0
$745$	4.57396	0.167577
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 9.00137i	− 0.328903i
$750$	0	0
$751$	−17.6770	−0.645043	−0.322522	−	0.946562i	$-0.604531\pi$
$751$	−17.6770	−0.645043	−0.322522	+	0.946562i	$0.604531\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.0876710i	0.00319067i
$756$	0	0
$757$	− 30.7650i	− 1.11817i	−0.829110	−	0.559086i	$-0.811152\pi$
$757$	− 30.7650i	− 1.11817i	0.829110	−	0.559086i	$-0.188848\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36.6808	1.32968	0.664838	−	0.746987i	$-0.268500\pi$
$761$	36.6808	1.32968	0.664838	+	0.746987i	$0.268500\pi$
$762$	0	0
$763$	71.6595i	2.59425i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.33654	0.337123
$768$	0	0
$769$	−43.9664	−1.58547	−0.792735	−	0.609566i	$-0.791344\pi$
$769$	−43.9664	−1.58547	−0.792735	+	0.609566i	$0.791344\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.1079i	0.651296i	0.945491	+	0.325648i	$0.105582\pi$
$773$	18.1079i	0.651296i	−0.945491	+	0.325648i	$0.894418\pi$
$774$	0	0
$775$	8.48274	0.304709
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.46793i	0.195909i
$780$	0	0
$781$	− 42.8978i	− 1.53500i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.13196	−0.147476
$786$	0	0
$787$	− 34.2703i	− 1.22160i	−0.791784	−	0.610801i	$-0.790847\pi$
$787$	− 34.2703i	− 1.22160i	0.791784	−	0.610801i	$-0.209153\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−26.4491	−0.940420
$792$	0	0
$793$	−14.6809	−0.521334
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 23.3407i	− 0.826770i	−0.910556	−	0.413385i	$-0.864346\pi$
$797$	− 23.3407i	− 0.826770i	0.910556	−	0.413385i	$-0.135654\pi$
$798$	0	0
$799$	50.9169	1.80131
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 49.6292i	− 1.75138i
$804$	0	0
$805$	− 5.10035i	− 0.179764i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−48.3125	−1.69858	−0.849289	−	0.527928i	$-0.822969\pi$
$809$	−48.3125	−1.69858	−0.849289	+	0.527928i	$0.822969\pi$
$810$	0	0
$811$	25.5793i	0.898210i	0.893479	+	0.449105i	$0.148257\pi$
$811$	25.5793i	0.898210i	−0.893479	+	0.449105i	$0.851743\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.06220	0.142293
$816$	0	0
$817$	30.4479	1.06524
$818$	0	0
$819$	0	0
$820$	0	0
$821$	49.7894i	1.73766i	0.495108	+	0.868832i	$0.335129\pi$
$821$	49.7894i	1.73766i	−0.495108	+	0.868832i	$0.664871\pi$
$822$	0	0
$823$	−20.4403	−0.712504	−0.356252	−	0.934390i	$-0.615946\pi$
$823$	−20.4403	−0.712504	−0.356252	+	0.934390i	$0.615946\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.4382i	1.26708i	0.773710	+	0.633540i	$0.218399\pi$
$827$	36.4382i	1.26708i	−0.773710	+	0.633540i	$0.781601\pi$
$828$	0	0
$829$	− 45.7206i	− 1.58794i	−0.607956	−	0.793971i	$-0.708010\pi$
$829$	− 45.7206i	− 1.58794i	0.607956	−	0.793971i	$-0.291990\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	99.2351	3.43829
$834$	0	0
$835$	− 3.47910i	− 0.120399i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−17.2215	−0.594554	−0.297277	−	0.954791i	$-0.596078\pi$
$839$	−17.2215	−0.594554	−0.297277	+	0.954791i	$0.596078\pi$
$840$	0	0
$841$	17.4916	0.603159
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 0.218531i	− 0.00751769i
$846$	0	0
$847$	−8.22539	−0.282628
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15.9860i	0.547991i
$852$	0	0
$853$	39.1565i	1.34069i	0.742048	+	0.670347i	$0.233855\pi$
$853$	39.1565i	1.34069i	−0.742048	+	0.670347i	$0.766145\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.8926	−1.43102	−0.715511	−	0.698601i	$-0.753806\pi$
$857$	−41.8926	−1.43102	−0.715511	+	0.698601i	$0.753806\pi$
$858$	0	0
$859$	− 10.2063i	− 0.348236i	−0.984725	−	0.174118i	$-0.944293\pi$
$859$	− 10.2063i	− 0.348236i	0.984725	−	0.174118i	$-0.0557074\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.6974	−0.636466	−0.318233	−	0.948013i	$-0.603089\pi$
$863$	−18.6974	−0.636466	−0.318233	+	0.948013i	$0.603089\pi$
$864$	0	0
$865$	0.741626	0.0252160
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 7.60564i	− 0.258004i
$870$	0	0
$871$	−2.10688	−0.0713890
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.73870i	0.329228i
$876$	0	0
$877$	− 0.583295i	− 0.0196965i	−0.999952	−	0.00984824i	$-0.996865\pi$
$877$	− 0.583295i	− 0.0196965i	0.999952	−	0.00984824i	$-0.00313484\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.7506	−1.03601	−0.518007	−	0.855377i	$-0.673326\pi$
$881$	−30.7506	−1.03601	−0.518007	+	0.855377i	$0.673326\pi$
$882$	0	0
$883$	− 2.72627i	− 0.0917464i	−0.998947	−	0.0458732i	$-0.985393\pi$
$883$	− 2.72627i	− 0.0917464i	0.998947	−	0.0458732i	$-0.0146070\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.52841	0.185626	0.0928130	−	0.995684i	$-0.470414\pi$
$887$	5.52841	0.185626	0.0928130	+	0.995684i	$0.470414\pi$
$888$	0	0
$889$	−30.8718	−1.03541
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 31.1919i	− 1.04380i
$894$	0	0
$895$	0.213842	0.00714794
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 5.81087i	− 0.193803i
$900$	0	0
$901$	− 8.99986i	− 0.299829i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.66693	−0.0886519
$906$	0	0
$907$	− 41.3739i	− 1.37380i	−0.726752	−	0.686900i	$-0.758971\pi$
$907$	− 41.3739i	− 1.37380i	0.726752	−	0.686900i	$-0.241029\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−29.6950	−0.983839	−0.491920	−	0.870641i	$-0.663705\pi$
$911$	−29.6950	−0.983839	−0.491920	+	0.870641i	$0.663705\pi$
$912$	0	0
$913$	−42.6667	−1.41206
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.90994i	− 0.294232i
$918$	0	0
$919$	−52.9480	−1.74659	−0.873297	−	0.487188i	$-0.838023\pi$
$919$	−52.9480	−1.74659	−0.873297	+	0.487188i	$0.838023\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.9731i	0.394098i
$924$	0	0
$925$	− 15.1887i	− 0.499401i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35.6986	−1.17123	−0.585617	−	0.810588i	$-0.699148\pi$
$929$	−35.6986	−1.17123	−0.585617	+	0.810588i	$0.699148\pi$
$930$	0	0
$931$	− 60.7918i	− 1.99237i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.95342	−0.194697
$936$	0	0
$937$	−59.7049	−1.95047	−0.975237	−	0.221163i	$-0.929015\pi$
$937$	−59.7049	−1.95047	−0.975237	+	0.221163i	$0.929015\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.1999i	0.723697i	0.932237	+	0.361849i	$0.117854\pi$
$941$	22.1999i	0.723697i	−0.932237	+	0.361849i	$0.882146\pi$
$942$	0	0
$943$	6.11844	0.199244
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.89010i	− 0.288889i	−0.989513	−	0.144445i	$-0.953860\pi$
$947$	− 8.89010i	− 0.288889i	0.989513	−	0.144445i	$-0.0461396\pi$
$948$	0	0
$949$	13.8518i	0.449649i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.55786	0.277216	0.138608	−	0.990347i	$-0.455737\pi$
$953$	8.55786	0.277216	0.138608	+	0.990347i	$0.455737\pi$
$954$	0	0
$955$	1.99216i	0.0644647i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.30356	0.138969
$960$	0	0
$961$	−28.0659	−0.905353
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.74411i	0.0561449i
$966$	0	0
$967$	47.2162	1.51837	0.759186	−	0.650874i	$-0.225597\pi$
$967$	47.2162	1.51837	0.759186	+	0.650874i	$0.225597\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.5324i	0.915648i	0.889043	+	0.457824i	$0.151371\pi$
$971$	28.5324i	0.915648i	−0.889043	+	0.457824i	$0.848629\pi$
$972$	0	0
$973$	24.8291i	0.795984i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.2487	1.35165	0.675827	−	0.737060i	$-0.263787\pi$
$977$	42.2487	1.35165	0.675827	+	0.737060i	$0.263787\pi$
$978$	0	0
$979$	36.5507i	1.16816i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.6115	0.912565	0.456283	−	0.889835i	$-0.349181\pi$
$983$	28.6115	0.912565	0.456283	+	0.889835i	$0.349181\pi$
$984$	0	0
$985$	2.75349	0.0877334
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 34.0702i	− 1.08337i
$990$	0	0
$991$	−51.5651	−1.63802	−0.819009	−	0.573780i	$-0.805476\pi$
$991$	−51.5651	−1.63802	−0.819009	+	0.573780i	$0.805476\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.18689i	0.132733i
$996$	0	0
$997$	37.0106i	1.17214i	0.810261	+	0.586069i	$0.199325\pi$
$997$	37.0106i	1.17214i	−0.810261	+	0.586069i	$0.800675\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.g.e.1873.9		16
3.2	odd	2		1248.2.g.b.625.4		16
4.3	odd	2		936.2.g.e.469.10		16
8.3	odd	2		936.2.g.e.469.9		16
8.5	even	2	inner	3744.2.g.e.1873.8		16
12.11	even	2		312.2.g.b.157.7	✓	16
24.5	odd	2		1248.2.g.b.625.13		16
24.11	even	2		312.2.g.b.157.8	yes	16
48.5	odd	4		9984.2.a.bs.1.5		8
48.11	even	4		9984.2.a.bu.1.5		8
48.29	odd	4		9984.2.a.bv.1.4		8
48.35	even	4		9984.2.a.bt.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.b.157.7	✓	16	12.11	even	2
312.2.g.b.157.8	yes	16	24.11	even	2
936.2.g.e.469.9		16	8.3	odd	2
936.2.g.e.469.10		16	4.3	odd	2
1248.2.g.b.625.4		16	3.2	odd	2
1248.2.g.b.625.13		16	24.5	odd	2
3744.2.g.e.1873.8		16	8.5	even	2	inner
3744.2.g.e.1873.9		16	1.1	even	1	trivial
9984.2.a.bs.1.5		8	48.5	odd	4
9984.2.a.bt.1.4		8	48.35	even	4
9984.2.a.bu.1.5		8	48.11	even	4
9984.2.a.bv.1.4		8	48.29	odd	4

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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.218531i q^{5} +4.47783 q^{7} +O(q^{10})\)	\(q+0.218531i q^{5} +4.47783 q^{7} +3.58286i q^{11} -1.00000i q^{13} +7.60367 q^{17} -4.65804i q^{19} -5.21219 q^{23} +4.95224 q^{25} -3.39240i q^{29} +1.71291 q^{31} +0.978543i q^{35} -3.06703i q^{37} -1.17387 q^{41} +6.53664i q^{43} +6.69636 q^{47} +13.0509 q^{49} -1.18362i q^{53} -0.782966 q^{55} +9.33654i q^{59} -14.6809i q^{61} +0.218531 q^{65} -2.10688i q^{67} -11.9731 q^{71} -13.8518 q^{73} +16.0434i q^{77} -2.12278 q^{79} +11.9086i q^{83} +1.66164i q^{85} +10.2015 q^{89} -4.47783i q^{91} +1.01792 q^{95} +16.0227 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7}+O(q^{10}) \) 16 * q - 4 * q^7	\( 16 q - 4 q^{7} - 16 q^{17} - 8 q^{23} - 32 q^{25} + 4 q^{31} + 36 q^{41} + 24 q^{47} + 48 q^{49} - 24 q^{55} - 4 q^{65} - 32 q^{73} + 60 q^{89} - 24 q^{95} + 40 q^{97}+O(q^{100}) \) 16 * q - 4 * q^7 - 16 * q^17 - 8 * q^23 - 32 * q^25 + 4 * q^31 + 36 * q^41 + 24 * q^47 + 48 * q^49 - 24 * q^55 - 4 * q^65 - 32 * q^73 + 60 * q^89 - 24 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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