LMFDB - Embedded newform 3744.2.g.e.1873.4

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			1873.4
Root			$0.791485 - 1.17199i$ of defining polynomial
Character	$\chi$	$=$	3744.1873
Dual form			3744.2.g.e.1873.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.05343i q^{5} +0.397397 q^{7} +O(q^{10})$	$q-3.05343i q^{5} +0.397397 q^{7} +0.332371i q^{11} +1.00000i q^{13} -4.78985 q^{17} +8.01683i q^{19} +2.73543 q^{23} -4.32345 q^{25} +2.88400i q^{29} -1.91940 q^{31} -1.21343i q^{35} +5.85389i q^{37} +2.16943 q^{41} +8.64374i q^{43} +5.45083 q^{47} -6.84208 q^{49} +9.73767i q^{53} +1.01487 q^{55} +7.96124i q^{59} -8.09505i q^{61} +3.05343 q^{65} -1.70003i q^{67} +5.78093 q^{71} -15.6219 q^{73} +0.132083i q^{77} +12.3143 q^{79} -2.71540i q^{83} +14.6255i q^{85} -13.5810 q^{89} +0.397397i q^{91} +24.4788 q^{95} +10.6487 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 3.05343i	− 1.36554i	−0.730635	−	0.682768i	$-0.760776\pi$
$5$	− 3.05343i	− 1.36554i	0.730635	−	0.682768i	$-0.239224\pi$
$6$	0	0
$7$	0.397397	0.150202	0.0751010	−	0.997176i	$-0.476072\pi$
$7$	0.397397	0.150202	0.0751010	+	0.997176i	$0.476072\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.332371i	0.100214i	0.998744	+	0.0501068i	$0.0159562\pi$
$11$	0.332371i	0.100214i	−0.998744	+	0.0501068i	$0.984044\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.78985	−1.16171	−0.580855	−	0.814007i	$-0.697282\pi$
$17$	−4.78985	−1.16171	−0.580855	+	0.814007i	$0.697282\pi$
$18$	0	0
$19$	8.01683i	1.83919i	0.392872	+	0.919593i	$0.371482\pi$
$19$	8.01683i	1.83919i	−0.392872	+	0.919593i	$0.628518\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.73543	0.570376	0.285188	−	0.958472i	$-0.407944\pi$
$23$	2.73543	0.570376	0.285188	+	0.958472i	$0.407944\pi$
$24$	0	0
$25$	−4.32345	−0.864690
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.88400i	0.535546i	0.963482	+	0.267773i	$0.0862877\pi$
$29$	2.88400i	0.535546i	−0.963482	+	0.267773i	$0.913712\pi$
$30$	0	0
$31$	−1.91940	−0.344734	−0.172367	−	0.985033i	$-0.555142\pi$
$31$	−1.91940	−0.344734	−0.172367	+	0.985033i	$0.555142\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.21343i	− 0.205106i
$36$	0	0
$37$	5.85389i	0.962373i	0.876618	+	0.481186i	$0.159794\pi$
$37$	5.85389i	0.962373i	−0.876618	+	0.481186i	$0.840206\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.16943	0.338808	0.169404	−	0.985547i	$-0.445816\pi$
$41$	2.16943	0.338808	0.169404	+	0.985547i	$0.445816\pi$
$42$	0	0
$43$	8.64374i	1.31816i	0.752074	+	0.659079i	$0.229054\pi$
$43$	8.64374i	1.31816i	−0.752074	+	0.659079i	$0.770946\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.45083	0.795085	0.397543	−	0.917584i	$-0.369863\pi$
$47$	5.45083	0.795085	0.397543	+	0.917584i	$0.369863\pi$
$48$	0	0
$49$	−6.84208	−0.977439
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.73767i	1.33757i	0.743455	+	0.668786i	$0.233186\pi$
$53$	9.73767i	1.33757i	−0.743455	+	0.668786i	$0.766814\pi$
$54$	0	0
$55$	1.01487	0.136845
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.96124i	1.03646i	0.855240	+	0.518232i	$0.173410\pi$
$59$	7.96124i	1.03646i	−0.855240	+	0.518232i	$0.826590\pi$
$60$	0	0
$61$	− 8.09505i	− 1.03647i	−0.855240	−	0.518233i	$-0.826590\pi$
$61$	− 8.09505i	− 1.03647i	0.855240	−	0.518233i	$-0.173410\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.05343	0.378732
$66$	0	0
$67$	− 1.70003i	− 0.207692i	−0.994593	−	0.103846i	$-0.966885\pi$
$67$	− 1.70003i	− 0.207692i	0.994593	−	0.103846i	$-0.0331149\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.78093	0.686070	0.343035	−	0.939323i	$-0.388545\pi$
$71$	5.78093	0.686070	0.343035	+	0.939323i	$0.388545\pi$
$72$	0	0
$73$	−15.6219	−1.82840	−0.914202	−	0.405258i	$-0.867182\pi$
$73$	−15.6219	−1.82840	−0.914202	+	0.405258i	$0.867182\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.132083i	0.0150523i
$78$	0	0
$79$	12.3143	1.38547	0.692734	−	0.721193i	$-0.256406\pi$
$79$	12.3143	1.38547	0.692734	+	0.721193i	$0.256406\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.71540i	− 0.298054i	−0.988833	−	0.149027i	$-0.952386\pi$
$83$	− 2.71540i	− 0.298054i	0.988833	−	0.149027i	$-0.0476142\pi$
$84$	0	0
$85$	14.6255i	1.58636i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.5810	−1.43958	−0.719789	−	0.694193i	$-0.755762\pi$
$89$	−13.5810	−1.43958	−0.719789	+	0.694193i	$0.755762\pi$
$90$	0	0
$91$	0.397397i	0.0416585i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	24.4788	2.51148
$96$	0	0
$97$	10.6487	1.08121	0.540605	−	0.841277i	$-0.318195\pi$
$97$	10.6487	1.08121	0.540605	+	0.841277i	$0.318195\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 11.2109i	− 1.11553i	−0.829999	−	0.557765i	$-0.811659\pi$
$101$	− 11.2109i	− 1.11553i	0.829999	−	0.557765i	$-0.188341\pi$
$102$	0	0
$103$	−9.72630	−0.958360	−0.479180	−	0.877717i	$-0.659066\pi$
$103$	−9.72630	−0.958360	−0.479180	+	0.877717i	$0.659066\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.1591i	− 1.07879i	−0.842053	−	0.539395i	$-0.818653\pi$
$107$	− 11.1591i	− 1.07879i	0.842053	−	0.539395i	$-0.181347\pi$
$108$	0	0
$109$	13.1655i	1.26103i	0.776178	+	0.630514i	$0.217156\pi$
$109$	13.1655i	1.26103i	−0.776178	+	0.630514i	$0.782844\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.39076	0.601192	0.300596	−	0.953752i	$-0.402814\pi$
$113$	6.39076	0.601192	0.300596	+	0.953752i	$0.402814\pi$
$114$	0	0
$115$	− 8.35244i	− 0.778869i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.90347	−0.174491
$120$	0	0
$121$	10.8895	0.989957
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.06580i	− 0.184771i
$126$	0	0
$127$	21.5128	1.90895	0.954475	−	0.298291i	$-0.0964166\pi$
$127$	21.5128	1.90895	0.954475	+	0.298291i	$0.0964166\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.1591i	1.32446i	0.749303	+	0.662228i	$0.230389\pi$
$131$	15.1591i	1.32446i	−0.749303	+	0.662228i	$0.769611\pi$
$132$	0	0
$133$	3.18586i	0.276250i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.73920	0.575769	0.287884	−	0.957665i	$-0.407048\pi$
$137$	6.73920	0.575769	0.287884	+	0.957665i	$0.407048\pi$
$138$	0	0
$139$	− 1.22480i	− 0.103886i	−0.998650	−	0.0519432i	$-0.983459\pi$
$139$	− 1.22480i	− 0.103886i	0.998650	−	0.0519432i	$-0.0165415\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.332371	−0.0277943
$144$	0	0
$145$	8.80611	0.731308
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 18.3349i	− 1.50205i	−0.660271	−	0.751027i	$-0.729559\pi$
$149$	− 18.3349i	− 1.50205i	0.660271	−	0.751027i	$-0.270441\pi$
$150$	0	0
$151$	5.52939	0.449976	0.224988	−	0.974362i	$-0.427766\pi$
$151$	5.52939	0.449976	0.224988	+	0.974362i	$0.427766\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.86075i	0.470747i
$156$	0	0
$157$	− 1.47135i	− 0.117426i	−0.998275	−	0.0587131i	$-0.981300\pi$
$157$	− 1.47135i	− 0.117426i	0.998275	−	0.0587131i	$-0.0186997\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.08705	0.0856716
$162$	0	0
$163$	13.6689i	1.07063i	0.844653	+	0.535315i	$0.179807\pi$
$163$	13.6689i	1.07063i	−0.844653	+	0.535315i	$0.820193\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.4879	−1.43063	−0.715317	−	0.698800i	$-0.753718\pi$
$167$	−18.4879	−1.43063	−0.715317	+	0.698800i	$0.753718\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.2781i	0.933489i	0.884392	+	0.466745i	$0.154573\pi$
$173$	12.2781i	0.933489i	−0.884392	+	0.466745i	$0.845427\pi$
$174$	0	0
$175$	−1.71813	−0.129878
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.21343i	0.0906957i	0.998971	+	0.0453478i	$0.0144396\pi$
$179$	1.21343i	0.0906957i	−0.998971	+	0.0453478i	$0.985560\pi$
$180$	0	0
$181$	13.6979i	1.01816i	0.860719	+	0.509080i	$0.170014\pi$
$181$	13.6979i	1.01816i	−0.860719	+	0.509080i	$0.829986\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	17.8744	1.31416
$186$	0	0
$187$	− 1.59201i	− 0.116419i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.77107	−0.634652	−0.317326	−	0.948316i	$-0.602785\pi$
$191$	−8.77107	−0.634652	−0.317326	+	0.948316i	$0.602785\pi$
$192$	0	0
$193$	19.5636	1.40822	0.704111	−	0.710090i	$-0.251346\pi$
$193$	19.5636	1.40822	0.704111	+	0.710090i	$0.251346\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.00942942i	0 0.000671819i	1.00000		0.000335909i	$0.000106923\pi$
$197$	0.00942942i	0 0.000671819i	−1.00000		0.000335909i	$0.999893\pi$
$198$	0	0
$199$	−12.9208	−0.915932	−0.457966	−	0.888970i	$-0.651422\pi$
$199$	−12.9208	−0.915932	−0.457966	+	0.888970i	$0.651422\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.14610i	0.0804401i
$204$	0	0
$205$	− 6.62420i	− 0.462654i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.66456	−0.184312
$210$	0	0
$211$	28.3315i	1.95042i	0.221283	+	0.975210i	$0.428976\pi$
$211$	28.3315i	1.95042i	−0.221283	+	0.975210i	$0.571024\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	26.3931	1.79999
$216$	0	0
$217$	−0.762764	−0.0517798
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4.78985i	− 0.322200i
$222$	0	0
$223$	17.6272	1.18040	0.590201	−	0.807256i	$-0.299048\pi$
$223$	17.6272	1.18040	0.590201	+	0.807256i	$0.299048\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.0171i	1.12947i	0.825274	+	0.564733i	$0.191021\pi$
$227$	17.0171i	1.12947i	−0.825274	+	0.564733i	$0.808979\pi$
$228$	0	0
$229$	− 26.4927i	− 1.75069i	−0.483503	−	0.875343i	$-0.660636\pi$
$229$	− 26.4927i	− 1.75069i	0.483503	−	0.875343i	$-0.339364\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.08799	0.529862	0.264931	−	0.964267i	$-0.414651\pi$
$233$	8.08799	0.529862	0.264931	+	0.964267i	$0.414651\pi$
$234$	0	0
$235$	− 16.6437i	− 1.08572i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.7165	−0.757876	−0.378938	−	0.925422i	$-0.623711\pi$
$239$	−11.7165	−0.757876	−0.378938	+	0.925422i	$0.623711\pi$
$240$	0	0
$241$	14.9150	0.960757	0.480379	−	0.877061i	$-0.340499\pi$
$241$	14.9150	0.960757	0.480379	+	0.877061i	$0.340499\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.8918i	1.33473i
$246$	0	0
$247$	−8.01683	−0.510099
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.5254i	0.979953i	0.871736	+	0.489976i	$0.162995\pi$
$251$	15.5254i	0.979953i	−0.871736	+	0.489976i	$0.837005\pi$
$252$	0	0
$253$	0.909176i	0.0571594i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.7173	−1.41706	−0.708532	−	0.705679i	$-0.750642\pi$
$257$	−22.7173	−1.41706	−0.708532	+	0.705679i	$0.750642\pi$
$258$	0	0
$259$	2.32632i	0.144550i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.6227	−1.51830	−0.759151	−	0.650914i	$-0.774386\pi$
$263$	−24.6227	−1.51830	−0.759151	+	0.650914i	$0.774386\pi$
$264$	0	0
$265$	29.7333	1.82650
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.64709i	0.283338i	0.989914	+	0.141669i	$0.0452469\pi$
$269$	4.64709i	0.283338i	−0.989914	+	0.141669i	$0.954753\pi$
$270$	0	0
$271$	−17.4259	−1.05855	−0.529274	−	0.848451i	$-0.677536\pi$
$271$	−17.4259	−1.05855	−0.529274	+	0.848451i	$0.677536\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.43699i	− 0.0866537i
$276$	0	0
$277$	− 6.23692i	− 0.374740i	−0.982289	−	0.187370i	$-0.940004\pi$
$277$	− 6.23692i	− 0.374740i	0.982289	−	0.187370i	$-0.0599963\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	28.5913	1.70562	0.852808	−	0.522225i	$-0.174898\pi$
$281$	28.5913	1.70562	0.852808	+	0.522225i	$0.174898\pi$
$282$	0	0
$283$	5.86854i	0.348849i	0.984671	+	0.174424i	$0.0558064\pi$
$283$	5.86854i	0.348849i	−0.984671	+	0.174424i	$0.944194\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.862125	0.0508896
$288$	0	0
$289$	5.94269	0.349570
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.6356i	1.67291i	0.548034	+	0.836456i	$0.315376\pi$
$293$	28.6356i	1.67291i	−0.548034	+	0.836456i	$0.684624\pi$
$294$	0	0
$295$	24.3091	1.41533
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.73543i	0.158194i
$300$	0	0
$301$	3.43500i	0.197990i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.7177	−1.41533
$306$	0	0
$307$	7.41077i	0.422955i	0.977383	+	0.211477i	$0.0678275\pi$
$307$	7.41077i	0.422955i	−0.977383	+	0.211477i	$0.932173\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.48025	−0.367461	−0.183731	−	0.982977i	$-0.558817\pi$
$311$	−6.48025	−0.367461	−0.183731	+	0.982977i	$0.558817\pi$
$312$	0	0
$313$	9.33064	0.527399	0.263699	−	0.964605i	$-0.415057\pi$
$313$	9.33064	0.527399	0.263699	+	0.964605i	$0.415057\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.4257i	− 1.31572i	−0.753140	−	0.657860i	$-0.771462\pi$
$317$	− 23.4257i	− 1.31572i	0.753140	−	0.657860i	$-0.228538\pi$
$318$	0	0
$319$	−0.958559	−0.0536690
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 38.3994i	− 2.13660i
$324$	0	0
$325$	− 4.32345i	− 0.239822i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.16614	0.119423
$330$	0	0
$331$	17.7455i	0.975378i	0.873017	+	0.487689i	$0.162160\pi$
$331$	17.7455i	0.975378i	−0.873017	+	0.487689i	$0.837840\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.19093	−0.283611
$336$	0	0
$337$	−1.28469	−0.0699815	−0.0349908	−	0.999388i	$-0.511140\pi$
$337$	−1.28469	−0.0699815	−0.0349908	+	0.999388i	$0.511140\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 0.637952i	− 0.0345471i
$342$	0	0
$343$	−5.50080	−0.297015
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 11.9901i	− 0.643663i	−0.946797	−	0.321832i	$-0.895702\pi$
$347$	− 11.9901i	− 0.643663i	0.946797	−	0.321832i	$-0.104298\pi$
$348$	0	0
$349$	14.4814i	0.775170i	0.921834	+	0.387585i	$0.126691\pi$
$349$	14.4814i	0.775170i	−0.921834	+	0.387585i	$0.873309\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.4003	−0.553555	−0.276777	−	0.960934i	$-0.589266\pi$
$353$	−10.4003	−0.553555	−0.276777	+	0.960934i	$0.589266\pi$
$354$	0	0
$355$	− 17.6517i	− 0.936854i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.3527	−1.44362	−0.721809	−	0.692092i	$-0.756689\pi$
$359$	−27.3527	−1.44362	−0.721809	+	0.692092i	$0.756689\pi$
$360$	0	0
$361$	−45.2695	−2.38261
$362$	0	0
$363$	0	0
$364$	0	0
$365$	47.7004i	2.49675i
$366$	0	0
$367$	26.3188	1.37383	0.686916	−	0.726737i	$-0.258964\pi$
$367$	26.3188	1.37383	0.686916	+	0.726737i	$0.258964\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.86972i	0.200906i
$372$	0	0
$373$	3.57315i	0.185011i	0.995712	+	0.0925053i	$0.0294875\pi$
$373$	3.57315i	0.185011i	−0.995712	+	0.0925053i	$0.970512\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.88400	−0.148534
$378$	0	0
$379$	− 23.8422i	− 1.22469i	−0.790590	−	0.612346i	$-0.790226\pi$
$379$	− 23.8422i	− 1.22469i	0.790590	−	0.612346i	$-0.209774\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.00932	−0.153769	−0.0768846	−	0.997040i	$-0.524497\pi$
$383$	−3.00932	−0.153769	−0.0768846	+	0.997040i	$0.524497\pi$
$384$	0	0
$385$	0.403307	0.0205544
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.4484i	1.03678i	0.855146	+	0.518388i	$0.173468\pi$
$389$	20.4484i	1.03678i	−0.855146	+	0.518388i	$0.826532\pi$
$390$	0	0
$391$	−13.1023	−0.662611
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 37.6009i	− 1.89191i
$396$	0	0
$397$	7.25972i	0.364355i	0.983266	+	0.182177i	$0.0583145\pi$
$397$	7.25972i	0.364355i	−0.983266	+	0.182177i	$0.941685\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.2669	−0.862269	−0.431134	−	0.902288i	$-0.641887\pi$
$401$	−17.2669	−0.862269	−0.431134	+	0.902288i	$0.641887\pi$
$402$	0	0
$403$	− 1.91940i	− 0.0956121i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.94566	−0.0964429
$408$	0	0
$409$	0.661985	0.0327330	0.0163665	−	0.999866i	$-0.494790\pi$
$409$	0.661985	0.0327330	0.0163665	+	0.999866i	$0.494790\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.16377i	0.155679i
$414$	0	0
$415$	−8.29130	−0.407004
$416$	0	0
$417$	0	0
$418$	0	0
$419$	36.1682i	1.76693i	0.468495	+	0.883466i	$0.344796\pi$
$419$	36.1682i	1.76693i	−0.468495	+	0.883466i	$0.655204\pi$
$420$	0	0
$421$	− 7.46196i	− 0.363673i	−0.983329	−	0.181837i	$-0.941796\pi$
$421$	− 7.46196i	− 0.363673i	0.983329	−	0.181837i	$-0.0582042\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	20.7087	1.00452
$426$	0	0
$427$	− 3.21695i	− 0.155679i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.1602	−0.489399	−0.244699	−	0.969599i	$-0.578689\pi$
$431$	−10.1602	−0.489399	−0.244699	+	0.969599i	$0.578689\pi$
$432$	0	0
$433$	7.67807	0.368984	0.184492	−	0.982834i	$-0.440936\pi$
$433$	7.67807	0.368984	0.184492	+	0.982834i	$0.440936\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.9294i	1.04903i
$438$	0	0
$439$	−27.5247	−1.31368	−0.656841	−	0.754029i	$-0.728108\pi$
$439$	−27.5247	−1.31368	−0.656841	+	0.754029i	$0.728108\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.22671i	0.105794i	0.998600	+	0.0528971i	$0.0168455\pi$
$443$	2.22671i	0.105794i	−0.998600	+	0.0528971i	$0.983154\pi$
$444$	0	0
$445$	41.4685i	1.96580i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−20.6012	−0.972231	−0.486115	−	0.873895i	$-0.661587\pi$
$449$	−20.6012	−0.972231	−0.486115	+	0.873895i	$0.661587\pi$
$450$	0	0
$451$	0.721055i	0.0339532i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.21343	0.0568863
$456$	0	0
$457$	−3.34457	−0.156452	−0.0782262	−	0.996936i	$-0.524926\pi$
$457$	−3.34457	−0.156452	−0.0782262	+	0.996936i	$0.524926\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 18.9452i	− 0.882365i	−0.897417	−	0.441183i	$-0.854559\pi$
$461$	− 18.9452i	− 0.882365i	0.897417	−	0.441183i	$-0.145441\pi$
$462$	0	0
$463$	3.69162	0.171564	0.0857821	−	0.996314i	$-0.472661\pi$
$463$	3.69162	0.171564	0.0857821	+	0.996314i	$0.472661\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.7180i	0.819891i	0.912110	+	0.409945i	$0.134452\pi$
$467$	17.7180i	0.819891i	−0.912110	+	0.409945i	$0.865548\pi$
$468$	0	0
$469$	− 0.675588i	− 0.0311958i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.87293	−0.132097
$474$	0	0
$475$	− 34.6603i	− 1.59033i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19.6699	0.898740	0.449370	−	0.893346i	$-0.351649\pi$
$479$	19.6699	0.898740	0.449370	+	0.893346i	$0.351649\pi$
$480$	0	0
$481$	−5.85389	−0.266914
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 32.5150i	− 1.47643i
$486$	0	0
$487$	25.5764	1.15898	0.579490	−	0.814979i	$-0.303252\pi$
$487$	25.5764	1.15898	0.579490	+	0.814979i	$0.303252\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 18.9017i	− 0.853020i	−0.904483	−	0.426510i	$-0.859743\pi$
$491$	− 18.9017i	− 0.853020i	0.904483	−	0.426510i	$-0.140257\pi$
$492$	0	0
$493$	− 13.8140i	− 0.622149i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.29733	0.103049
$498$	0	0
$499$	11.1909i	0.500973i	0.968120	+	0.250486i	$0.0805905\pi$
$499$	11.1909i	0.500973i	−0.968120	+	0.250486i	$0.919409\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.1313	0.897609	0.448805	−	0.893630i	$-0.351850\pi$
$503$	20.1313	0.897609	0.448805	+	0.893630i	$0.351850\pi$
$504$	0	0
$505$	−34.2319	−1.52330
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 23.6298i	− 1.04737i	−0.851912	−	0.523685i	$-0.824557\pi$
$509$	− 23.6298i	− 1.04737i	0.851912	−	0.523685i	$-0.175443\pi$
$510$	0	0
$511$	−6.20810	−0.274630
$512$	0	0
$513$	0	0
$514$	0	0
$515$	29.6986i	1.30868i
$516$	0	0
$517$	1.81170i	0.0796784i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.9890	1.00717	0.503584	−	0.863946i	$-0.332014\pi$
$521$	22.9890	1.00717	0.503584	+	0.863946i	$0.332014\pi$
$522$	0	0
$523$	32.7127i	1.43043i	0.698906	+	0.715213i	$0.253670\pi$
$523$	32.7127i	1.43043i	−0.698906	+	0.715213i	$0.746330\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.19364	0.400481
$528$	0	0
$529$	−15.5174	−0.674671
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.16943i	0.0939684i
$534$	0	0
$535$	−34.0735	−1.47313
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.27411i	− 0.0979527i
$540$	0	0
$541$	− 6.85961i	− 0.294918i	−0.989068	−	0.147459i	$-0.952891\pi$
$541$	− 6.85961i	− 0.294918i	0.989068	−	0.147459i	$-0.0471094\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	40.2000	1.72198
$546$	0	0
$547$	− 7.53129i	− 0.322015i	−0.986953	−	0.161007i	$-0.948526\pi$
$547$	− 7.53129i	− 0.322015i	0.986953	−	0.161007i	$-0.0514743\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−23.1206	−0.984969
$552$	0	0
$553$	4.89367	0.208100
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 18.2449i	− 0.773062i	−0.922276	−	0.386531i	$-0.873673\pi$
$557$	− 18.2449i	− 0.773062i	0.922276	−	0.386531i	$-0.126327\pi$
$558$	0	0
$559$	−8.64374	−0.365591
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.8811i	1.21719i	0.793479	+	0.608597i	$0.208268\pi$
$563$	28.8811i	1.21719i	−0.793479	+	0.608597i	$0.791732\pi$
$564$	0	0
$565$	− 19.5137i	− 0.820950i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.9778	0.879435	0.439718	−	0.898136i	$-0.355079\pi$
$569$	20.9778	0.879435	0.439718	+	0.898136i	$0.355079\pi$
$570$	0	0
$571$	− 44.0239i	− 1.84234i	−0.389155	−	0.921172i	$-0.627233\pi$
$571$	− 44.0239i	− 1.84234i	0.389155	−	0.921172i	$-0.372767\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.8265	−0.493198
$576$	0	0
$577$	15.4233	0.642078	0.321039	−	0.947066i	$-0.395968\pi$
$577$	15.4233	0.642078	0.321039	+	0.947066i	$0.395968\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1.07909i	− 0.0447683i
$582$	0	0
$583$	−3.23652	−0.134043
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 27.4900i	− 1.13463i	−0.823500	−	0.567317i	$-0.807981\pi$
$587$	− 27.4900i	− 1.13463i	0.823500	−	0.567317i	$-0.192019\pi$
$588$	0	0
$589$	− 15.3875i	− 0.634030i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.43105	0.0587660	0.0293830	−	0.999568i	$-0.490646\pi$
$593$	1.43105	0.0587660	0.0293830	+	0.999568i	$0.490646\pi$
$594$	0	0
$595$	5.81213i	0.238274i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	45.4512	1.85709	0.928543	−	0.371226i	$-0.121062\pi$
$599$	45.4512	1.85709	0.928543	+	0.371226i	$0.121062\pi$
$600$	0	0
$601$	−27.2351	−1.11094	−0.555471	−	0.831536i	$-0.687462\pi$
$601$	−27.2351	−1.11094	−0.555471	+	0.831536i	$0.687462\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 33.2504i	− 1.35182i
$606$	0	0
$607$	7.43305	0.301698	0.150849	−	0.988557i	$-0.451799\pi$
$607$	7.43305	0.301698	0.150849	+	0.988557i	$0.451799\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.45083i	0.220517i
$612$	0	0
$613$	− 29.6705i	− 1.19838i	−0.800607	−	0.599190i	$-0.795489\pi$
$613$	− 29.6705i	− 1.19838i	0.800607	−	0.599190i	$-0.204511\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.1404	−1.29392	−0.646961	−	0.762523i	$-0.723960\pi$
$617$	−32.1404	−1.29392	−0.646961	+	0.762523i	$0.723960\pi$
$618$	0	0
$619$	0.444539i	0.0178675i	0.999960	+	0.00893377i	$0.00284375\pi$
$619$	0.444539i	0.0178675i	−0.999960	+	0.00893377i	$0.997156\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.39703	−0.216228
$624$	0	0
$625$	−27.9250	−1.11700
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 28.0392i	− 1.11800i
$630$	0	0
$631$	−8.90456	−0.354485	−0.177242	−	0.984167i	$-0.556718\pi$
$631$	−8.90456	−0.354485	−0.177242	+	0.984167i	$0.556718\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 65.6878i	− 2.60674i
$636$	0	0
$637$	− 6.84208i	− 0.271093i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.4375	0.807233	0.403617	−	0.914928i	$-0.367753\pi$
$641$	20.4375	0.807233	0.403617	+	0.914928i	$0.367753\pi$
$642$	0	0
$643$	18.0779i	0.712922i	0.934310	+	0.356461i	$0.116017\pi$
$643$	18.0779i	0.712922i	−0.934310	+	0.356461i	$0.883983\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.8958	−0.428356	−0.214178	−	0.976795i	$-0.568707\pi$
$647$	−10.8958	−0.428356	−0.214178	+	0.976795i	$0.568707\pi$
$648$	0	0
$649$	−2.64608	−0.103868
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.9934i	0.469340i	0.972075	+	0.234670i	$0.0754009\pi$
$653$	11.9934i	0.469340i	−0.972075	+	0.234670i	$0.924599\pi$
$654$	0	0
$655$	46.2872	1.80859
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 12.8561i	− 0.500804i	−0.968142	−	0.250402i	$-0.919437\pi$
$659$	− 12.8561i	− 0.500804i	0.968142	−	0.250402i	$-0.0805628\pi$
$660$	0	0
$661$	46.4801i	1.80787i	0.427674	+	0.903933i	$0.359333\pi$
$661$	46.4801i	1.80787i	−0.427674	+	0.903933i	$0.640667\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.72782	0.377229
$666$	0	0
$667$	7.88898i	0.305463i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.69056	0.103868
$672$	0	0
$673$	42.7820	1.64913	0.824563	−	0.565770i	$-0.191421\pi$
$673$	42.7820	1.64913	0.824563	+	0.565770i	$0.191421\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.22226i	0.123842i	0.998081	+	0.0619208i	$0.0197226\pi$
$677$	3.22226i	0.123842i	−0.998081	+	0.0619208i	$0.980277\pi$
$678$	0	0
$679$	4.23176	0.162400
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 7.35539i	− 0.281446i	−0.990049	−	0.140723i	$-0.955057\pi$
$683$	− 7.35539i	− 0.281446i	0.990049	−	0.140723i	$-0.0449427\pi$
$684$	0	0
$685$	− 20.5777i	− 0.786233i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.73767	−0.370976
$690$	0	0
$691$	− 13.6229i	− 0.518240i	−0.965845	−	0.259120i	$-0.916567\pi$
$691$	− 13.6229i	− 0.518240i	0.965845	−	0.259120i	$-0.0834326\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.73985	−0.141861
$696$	0	0
$697$	−10.3912	−0.393596
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 38.1342i	− 1.44031i	−0.693814	−	0.720154i	$-0.744071\pi$
$701$	− 38.1342i	− 1.44031i	0.693814	−	0.720154i	$-0.255929\pi$
$702$	0	0
$703$	−46.9296	−1.76998
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.45520i	− 0.167555i
$708$	0	0
$709$	28.1895i	1.05868i	0.848410	+	0.529339i	$0.177560\pi$
$709$	28.1895i	1.05868i	−0.848410	+	0.529339i	$0.822440\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.25037	−0.196628
$714$	0	0
$715$	1.01487i	0.0379541i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12.3250	−0.459646	−0.229823	−	0.973233i	$-0.573815\pi$
$719$	−12.3250	−0.459646	−0.229823	+	0.973233i	$0.573815\pi$
$720$	0	0
$721$	−3.86520	−0.143948
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 12.4688i	− 0.463081i
$726$	0	0
$727$	4.04824	0.150141	0.0750704	−	0.997178i	$-0.476082\pi$
$727$	4.04824	0.150141	0.0750704	+	0.997178i	$0.476082\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 41.4022i	− 1.53132i
$732$	0	0
$733$	− 17.4117i	− 0.643115i	−0.946890	−	0.321558i	$-0.895794\pi$
$733$	− 17.4117i	− 0.643115i	0.946890	−	0.321558i	$-0.104206\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.565041	0.0208136
$738$	0	0
$739$	− 31.3372i	− 1.15276i	−0.817183	−	0.576379i	$-0.804465\pi$
$739$	− 31.3372i	− 1.15276i	0.817183	−	0.576379i	$-0.195535\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11.0758	0.406332	0.203166	−	0.979144i	$-0.434877\pi$
$743$	11.0758	0.406332	0.203166	+	0.979144i	$0.434877\pi$
$744$	0	0
$745$	−55.9844	−2.05111
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 4.43459i	− 0.162036i
$750$	0	0
$751$	−43.1463	−1.57443	−0.787216	−	0.616677i	$-0.788478\pi$
$751$	−43.1463	−1.57443	−0.787216	+	0.616677i	$0.788478\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 16.8836i	− 0.614458i
$756$	0	0
$757$	13.3272i	0.484387i	0.970228	+	0.242193i	$0.0778668\pi$
$757$	13.3272i	0.484387i	−0.970228	+	0.242193i	$0.922133\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	54.1469	1.96282	0.981412	−	0.191912i	$-0.0614687\pi$
$761$	54.1469	1.96282	0.981412	+	0.191912i	$0.0614687\pi$
$762$	0	0
$763$	5.23194i	0.189409i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.96124	−0.287464
$768$	0	0
$769$	47.5985	1.71645	0.858223	−	0.513276i	$-0.171568\pi$
$769$	47.5985	1.71645	0.858223	+	0.513276i	$0.171568\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 38.0639i	− 1.36906i	−0.728983	−	0.684531i	$-0.760007\pi$
$773$	− 38.0639i	− 1.36906i	0.728983	−	0.684531i	$-0.239993\pi$
$774$	0	0
$775$	8.29842	0.298088
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17.3919i	0.623131i
$780$	0	0
$781$	1.92141i	0.0687536i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.49266	−0.160350
$786$	0	0
$787$	− 26.8479i	− 0.957026i	−0.878081	−	0.478513i	$-0.841176\pi$
$787$	− 26.8479i	− 0.957026i	0.878081	−	0.478513i	$-0.158824\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.53967	0.0903003
$792$	0	0
$793$	8.09505	0.287464
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.6315i	1.68720i	0.536976	+	0.843598i	$0.319567\pi$
$797$	47.6315i	1.68720i	−0.536976	+	0.843598i	$0.680433\pi$
$798$	0	0
$799$	−26.1087	−0.923658
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 5.19226i	− 0.183231i
$804$	0	0
$805$	− 3.31924i	− 0.116988i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−54.2983	−1.90903	−0.954514	−	0.298168i	$-0.903625\pi$
$809$	−54.2983	−1.90903	−0.954514	+	0.298168i	$0.903625\pi$
$810$	0	0
$811$	− 9.06397i	− 0.318279i	−0.987256	−	0.159139i	$-0.949128\pi$
$811$	− 9.06397i	− 0.318279i	0.987256	−	0.159139i	$-0.0508719\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	41.7370	1.46198
$816$	0	0
$817$	−69.2954	−2.42434
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.33431i	0.151269i	0.997136	+	0.0756343i	$0.0240982\pi$
$821$	4.33431i	0.151269i	−0.997136	+	0.0756343i	$0.975902\pi$
$822$	0	0
$823$	−28.4454	−0.991543	−0.495772	−	0.868453i	$-0.665115\pi$
$823$	−28.4454	−0.991543	−0.495772	+	0.868453i	$0.665115\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41.1301i	1.43024i	0.699004	+	0.715118i	$0.253627\pi$
$827$	41.1301i	1.43024i	−0.699004	+	0.715118i	$0.746373\pi$
$828$	0	0
$829$	− 0.807025i	− 0.0280291i	−0.999902	−	0.0140146i	$-0.995539\pi$
$829$	− 0.807025i	− 0.0280291i	0.999902	−	0.0140146i	$-0.00446112\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	32.7725	1.13550
$834$	0	0
$835$	56.4514i	1.95358i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.32354	0.183789	0.0918944	−	0.995769i	$-0.470708\pi$
$839$	5.32354	0.183789	0.0918944	+	0.995769i	$0.470708\pi$
$840$	0	0
$841$	20.6825	0.713190
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.05343i	0.105041i
$846$	0	0
$847$	4.32747	0.148694
$848$	0	0
$849$	0	0
$850$	0	0
$851$	16.0129i	0.548914i
$852$	0	0
$853$	− 28.3780i	− 0.971644i	−0.874058	−	0.485822i	$-0.838520\pi$
$853$	− 28.3780i	− 0.971644i	0.874058	−	0.485822i	$-0.161480\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.0031	−1.60559	−0.802797	−	0.596253i	$-0.796656\pi$
$857$	−47.0031	−1.60559	−0.802797	+	0.596253i	$0.796656\pi$
$858$	0	0
$859$	− 2.44806i	− 0.0835268i	−0.999128	−	0.0417634i	$-0.986702\pi$
$859$	− 2.44806i	− 0.0835268i	0.999128	−	0.0417634i	$-0.0132976\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.8769	0.370252	0.185126	−	0.982715i	$-0.440731\pi$
$863$	10.8769	0.370252	0.185126	+	0.982715i	$0.440731\pi$
$864$	0	0
$865$	37.4905	1.27471
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.09292i	0.138843i
$870$	0	0
$871$	1.70003	0.0576034
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 0.820945i	− 0.0277530i
$876$	0	0
$877$	− 8.03845i	− 0.271439i	−0.990747	−	0.135720i	$-0.956665\pi$
$877$	− 8.03845i	− 0.271439i	0.990747	−	0.135720i	$-0.0433346\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−51.0077	−1.71849	−0.859247	−	0.511561i	$-0.829067\pi$
$881$	−51.0077	−1.71849	−0.859247	+	0.511561i	$0.829067\pi$
$882$	0	0
$883$	13.6570i	0.459596i	0.973238	+	0.229798i	$0.0738065\pi$
$883$	13.6570i	0.459596i	−0.973238	+	0.229798i	$0.926193\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.6301	0.356925	0.178462	−	0.983947i	$-0.442888\pi$
$887$	10.6301	0.356925	0.178462	+	0.983947i	$0.442888\pi$
$888$	0	0
$889$	8.54912	0.286728
$890$	0	0
$891$	0	0
$892$	0	0
$893$	43.6984i	1.46231i
$894$	0	0
$895$	3.70511	0.123848
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 5.53555i	− 0.184621i
$900$	0	0
$901$	− 46.6420i	− 1.55387i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	41.8258	1.39034
$906$	0	0
$907$	− 28.9517i	− 0.961326i	−0.876905	−	0.480663i	$-0.840396\pi$
$907$	− 28.9517i	− 0.961326i	0.876905	−	0.480663i	$-0.159604\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.8523	0.823394	0.411697	−	0.911321i	$-0.364936\pi$
$911$	24.8523	0.823394	0.411697	+	0.911321i	$0.364936\pi$
$912$	0	0
$913$	0.902521	0.0298691
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.02418i	0.198936i
$918$	0	0
$919$	42.7226	1.40929	0.704644	−	0.709561i	$-0.251107\pi$
$919$	42.7226	1.40929	0.704644	+	0.709561i	$0.251107\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.78093i	0.190282i
$924$	0	0
$925$	− 25.3090i	− 0.832154i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.2217	0.532218	0.266109	−	0.963943i	$-0.414262\pi$
$929$	16.2217	0.532218	0.266109	+	0.963943i	$0.414262\pi$
$930$	0	0
$931$	− 54.8517i	− 1.79769i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.86109	−0.158975
$936$	0	0
$937$	−30.6306	−1.00066	−0.500329	−	0.865835i	$-0.666788\pi$
$937$	−30.6306	−1.00066	−0.500329	+	0.865835i	$0.666788\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 45.0605i	− 1.46893i	−0.678646	−	0.734465i	$-0.737433\pi$
$941$	− 45.0605i	− 1.46893i	0.678646	−	0.734465i	$-0.262567\pi$
$942$	0	0
$943$	5.93431	0.193248
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.89625i	0.289089i	0.989498	+	0.144545i	$0.0461717\pi$
$947$	8.89625i	0.289089i	−0.989498	+	0.144545i	$0.953828\pi$
$948$	0	0
$949$	− 15.6219i	− 0.507108i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.0809	−0.456125	−0.228062	−	0.973647i	$-0.573239\pi$
$953$	−14.0809	−0.456125	−0.228062	+	0.973647i	$0.573239\pi$
$954$	0	0
$955$	26.7819i	0.866641i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.67814	0.0864817
$960$	0	0
$961$	−27.3159	−0.881158
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 59.7363i	− 1.92298i
$966$	0	0
$967$	25.8044	0.829813	0.414907	−	0.909864i	$-0.363814\pi$
$967$	25.8044	0.829813	0.414907	+	0.909864i	$0.363814\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 39.9722i	− 1.28277i	−0.767220	−	0.641384i	$-0.778360\pi$
$971$	− 39.9722i	− 1.28277i	0.767220	−	0.641384i	$-0.221640\pi$
$972$	0	0
$973$	− 0.486733i	− 0.0156040i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.7476	−0.407830	−0.203915	−	0.978989i	$-0.565367\pi$
$977$	−12.7476	−0.407830	−0.203915	+	0.978989i	$0.565367\pi$
$978$	0	0
$979$	− 4.51392i	− 0.144265i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11.8603	−0.378284	−0.189142	−	0.981950i	$-0.560571\pi$
$983$	−11.8603	−0.378284	−0.189142	+	0.981950i	$0.560571\pi$
$984$	0	0
$985$	0.0287921	0.000917393 0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23.6443i	0.751845i
$990$	0	0
$991$	−30.7094	−0.975516	−0.487758	−	0.872979i	$-0.662185\pi$
$991$	−30.7094	−0.975516	−0.487758	+	0.872979i	$0.662185\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	39.4528i	1.25074i
$996$	0	0
$997$	− 43.6457i	− 1.38227i	−0.722724	−	0.691137i	$-0.757110\pi$
$997$	− 43.6457i	− 1.38227i	0.722724	−	0.691137i	$-0.242890\pi$
$998$	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.1873.4		16
3.2	odd	2		1248.2.g.b.625.15		16
4.3	odd	2		936.2.g.e.469.3		16
8.3	odd	2		936.2.g.e.469.4		16
8.5	even	2	inner	3744.2.g.e.1873.13		16
12.11	even	2		312.2.g.b.157.14	yes	16
24.5	odd	2		1248.2.g.b.625.2		16
24.11	even	2		312.2.g.b.157.13	✓	16
48.5	odd	4		9984.2.a.bv.1.2		8
48.11	even	4		9984.2.a.bt.1.2		8
48.29	odd	4		9984.2.a.bs.1.7		8
48.35	even	4		9984.2.a.bu.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.b.157.13	✓	16	24.11	even	2
312.2.g.b.157.14	yes	16	12.11	even	2
936.2.g.e.469.3		16	4.3	odd	2
936.2.g.e.469.4		16	8.3	odd	2
1248.2.g.b.625.2		16	24.5	odd	2
1248.2.g.b.625.15		16	3.2	odd	2
3744.2.g.e.1873.4		16	1.1	even	1	trivial
3744.2.g.e.1873.13		16	8.5	even	2	inner
9984.2.a.bs.1.7		8	48.29	odd	4
9984.2.a.bt.1.2		8	48.11	even	4
9984.2.a.bu.1.7		8	48.35	even	4
9984.2.a.bv.1.2		8	48.5	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)

\(f(q)\)	\(=\)	\(q-3.05343i q^{5} +0.397397 q^{7} +O(q^{10})\)	\(q-3.05343i q^{5} +0.397397 q^{7} +0.332371i q^{11} +1.00000i q^{13} -4.78985 q^{17} +8.01683i q^{19} +2.73543 q^{23} -4.32345 q^{25} +2.88400i q^{29} -1.91940 q^{31} -1.21343i q^{35} +5.85389i q^{37} +2.16943 q^{41} +8.64374i q^{43} +5.45083 q^{47} -6.84208 q^{49} +9.73767i q^{53} +1.01487 q^{55} +7.96124i q^{59} -8.09505i q^{61} +3.05343 q^{65} -1.70003i q^{67} +5.78093 q^{71} -15.6219 q^{73} +0.132083i q^{77} +12.3143 q^{79} -2.71540i q^{83} +14.6255i q^{85} -13.5810 q^{89} +0.397397i q^{91} +24.4788 q^{95} +10.6487 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7}+O(q^{10}) \) 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

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Representations

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Database

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