LMFDB - Embedded newform 1323.2.c.d.1322.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1322,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1323.1322");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.c (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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 $ x^12 - 9x^10 + 59x^8 - 180x^6 + 403x^4 - 198*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 189)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.12
Root			$-0.617942 + 0.356769i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.d.1322.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49086i q^{2} -4.20440 q^{4} +1.23588 q^{5} -5.49086i q^{8} +O(q^{10})$	$q+2.49086i q^{2} -4.20440 q^{4} +1.23588 q^{5} -5.49086i q^{8} +3.07842i q^{10} +5.49086i q^{11} -2.96793i q^{13} +5.26819 q^{16} +4.31430 q^{17} +5.55019i q^{19} -5.19615 q^{20} -13.6770 q^{22} +1.63148i q^{23} -3.47259 q^{25} +7.39272 q^{26} -0.509136i q^{29} +8.13244i q^{31} +2.14061i q^{32} +10.7463i q^{34} -3.06379 q^{37} -13.8248 q^{38} -6.78607i q^{40} +0.354034 q^{41} -0.0637877 q^{43} -23.0858i q^{44} -4.06379 q^{46} -9.86449 q^{47} -8.64975i q^{50} +12.4784i q^{52} -4.12234i q^{53} +6.78607i q^{55} +1.26819 q^{58} -3.07842 q^{59} +6.89655i q^{61} -20.2568 q^{62} +5.20440 q^{64} -3.66802i q^{65} -12.3320 q^{67} -18.1391 q^{68} +4.63148i q^{71} -7.03869i q^{73} -7.63148i q^{74} -23.3352i q^{76} -0.331977 q^{79} +6.51087 q^{80} +0.881850i q^{82} +7.39272 q^{83} +5.33198 q^{85} -0.158887i q^{86} +30.1496 q^{88} +3.43245 q^{89} -6.85939i q^{92} -24.5711i q^{94} +6.85939i q^{95} +4.45644i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{4}+O(q^{10}) $ 12 * q - 16 * q^4	$ 12 q - 16 q^{4} + 8 q^{16} - 40 q^{22} + 48 q^{25} - 16 q^{37} + 20 q^{43} - 28 q^{46} - 40 q^{58} + 28 q^{64} - 72 q^{67} + 72 q^{79} - 12 q^{85} + 148 q^{88}+O(q^{100}) $ 12 * q - 16 * q^4 + 8 * q^16 - 40 * q^22 + 48 * q^25 - 16 * q^37 + 20 * q^43 - 28 * q^46 - 40 * q^58 + 28 * q^64 - 72 * q^67 + 72 * q^79 - 12 * q^85 + 148 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

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 $

$2$	2.49086i	1.76131i	0.473761	+	0.880653i	$0.342896\pi$
$2$	2.49086i	1.76131i	−0.473761	+	0.880653i	$0.657104\pi$
$3$	0	0
$3$	0	0
$4$	−4.20440	−2.10220
$5$	1.23588	0.552704	0.276352	−	0.961056i	$-0.410874\pi$
$5$	1.23588	0.552704	0.276352	+	0.961056i	$0.410874\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 5.49086i	− 1.94131i
$9$	0	0
$10$	3.07842i	0.973481i
$11$	5.49086i	1.65556i	0.561055	+	0.827779i	$0.310396\pi$
$11$	5.49086i	1.65556i	−0.561055	+	0.827779i	$0.689604\pi$
$12$	0	0
$13$	− 2.96793i	− 0.823157i	−0.911374	−	0.411578i	$-0.864978\pi$
$13$	− 2.96793i	− 0.823157i	0.911374	−	0.411578i	$-0.135022\pi$
$14$	0	0
$15$	0	0
$16$	5.26819	1.31705
$17$	4.31430	1.04637	0.523186	−	0.852219i	$-0.324743\pi$
$17$	4.31430	1.04637	0.523186	+	0.852219i	$0.324743\pi$
$18$	0	0
$19$	5.55019i	1.27330i	0.771153	+	0.636650i	$0.219680\pi$
$19$	5.55019i	1.27330i	−0.771153	+	0.636650i	$0.780320\pi$
$20$	−5.19615	−1.16190
$21$	0	0
$22$	−13.6770	−2.91594
$23$	1.63148i	0.340187i	0.985428	+	0.170093i	$0.0544069\pi$
$23$	1.63148i	0.340187i	−0.985428	+	0.170093i	$0.945593\pi$
$24$	0	0
$25$	−3.47259	−0.694518
$26$	7.39272	1.44983
$27$	0	0
$28$	0	0
$29$	− 0.509136i	− 0.0945443i	−0.998882	−	0.0472721i	$-0.984947\pi$
$29$	− 0.509136i	− 0.0945443i	0.998882	−	0.0472721i	$-0.0150528\pi$
$30$	0	0
$31$	8.13244i	1.46063i	0.683111	+	0.730314i	$0.260626\pi$
$31$	8.13244i	1.46063i	−0.683111	+	0.730314i	$0.739374\pi$
$32$	2.14061i	0.378411i
$33$	0	0
$34$	10.7463i	1.84298i
$35$	0	0
$36$	0	0
$37$	−3.06379	−0.503684	−0.251842	−	0.967768i	$-0.581036\pi$
$37$	−3.06379	−0.503684	−0.251842	+	0.967768i	$0.581036\pi$
$38$	−13.8248	−2.24267
$39$	0	0
$40$	− 6.78607i	− 1.07297i
$41$	0.354034	0.0552908	0.0276454	−	0.999618i	$-0.491199\pi$
$41$	0.354034	0.0552908	0.0276454	+	0.999618i	$0.491199\pi$
$42$	0	0
$43$	−0.0637877	−0.00972754	−0.00486377	−	0.999988i	$-0.501548\pi$
$43$	−0.0637877	−0.00972754	−0.00486377	+	0.999988i	$0.501548\pi$
$44$	− 23.0858i	− 3.48031i
$45$	0	0
$46$	−4.06379	−0.599173
$47$	−9.86449	−1.43888	−0.719442	−	0.694553i	$-0.755602\pi$
$47$	−9.86449	−1.43888	−0.719442	+	0.694553i	$0.755602\pi$
$48$	0	0
$49$	0	0
$50$	− 8.64975i	− 1.22326i
$51$	0	0
$52$	12.4784i	1.73044i
$53$	− 4.12234i	− 0.566247i	−0.959083	−	0.283124i	$-0.908629\pi$
$53$	− 4.12234i	− 0.566247i	0.959083	−	0.283124i	$-0.0913706\pi$
$54$	0	0
$55$	6.78607i	0.915034i
$56$	0	0
$57$	0	0
$58$	1.26819	0.166521
$59$	−3.07842	−0.400776	−0.200388	−	0.979717i	$-0.564220\pi$
$59$	−3.07842	−0.400776	−0.200388	+	0.979717i	$0.564220\pi$
$60$	0	0
$61$	6.89655i	0.883013i	0.897258	+	0.441507i	$0.145556\pi$
$61$	6.89655i	0.883013i	−0.897258	+	0.441507i	$0.854444\pi$
$62$	−20.2568	−2.57262
$63$	0	0
$64$	5.20440	0.650550
$65$	− 3.66802i	− 0.454962i
$66$	0	0
$67$	−12.3320	−1.50659	−0.753295	−	0.657682i	$-0.771537\pi$
$67$	−12.3320	−1.50659	−0.753295	+	0.657682i	$0.771537\pi$
$68$	−18.1391	−2.19968
$69$	0	0
$70$	0	0
$71$	4.63148i	0.549655i	0.961494	+	0.274828i	$0.0886208\pi$
$71$	4.63148i	0.549655i	−0.961494	+	0.274828i	$0.911379\pi$
$72$	0	0
$73$	− 7.03869i	− 0.823816i	−0.911225	−	0.411908i	$-0.864863\pi$
$73$	− 7.03869i	− 0.823816i	0.911225	−	0.411908i	$-0.135137\pi$
$74$	− 7.63148i	− 0.887141i
$75$	0	0
$76$	− 23.3352i	− 2.67673i
$77$	0	0
$78$	0	0
$79$	−0.331977	−0.0373503	−0.0186752	−	0.999826i	$-0.505945\pi$
$79$	−0.331977	−0.0373503	−0.0186752	+	0.999826i	$0.505945\pi$
$80$	6.51087	0.727937
$81$	0	0
$82$	0.881850i	0.0973841i
$83$	7.39272	0.811457	0.405728	−	0.913994i	$-0.367018\pi$
$83$	7.39272	0.811457	0.405728	+	0.913994i	$0.367018\pi$
$84$	0	0
$85$	5.33198	0.578334
$86$	− 0.158887i	− 0.0171332i
$87$	0	0
$88$	30.1496	3.21396
$89$	3.43245	0.363839	0.181920	−	0.983313i	$-0.441769\pi$
$89$	3.43245	0.363839	0.181920	+	0.983313i	$0.441769\pi$
$90$	0	0
$91$	0	0
$92$	− 6.85939i	− 0.715140i
$93$	0	0
$94$	− 24.5711i	− 2.53432i
$95$	6.85939i	0.703758i
$96$	0	0
$97$	4.45644i	0.452482i	0.974071	+	0.226241i	$0.0726438\pi$
$97$	4.45644i	0.452482i	−0.974071	+	0.226241i	$0.927356\pi$
$98$	0	0
$99$	0	0
$100$	14.6002	1.46002
$101$	9.15642	0.911098	0.455549	−	0.890211i	$-0.349443\pi$
$101$	9.15642	0.911098	0.455549	+	0.890211i	$0.349443\pi$
$102$	0	0
$103$	4.20382i	0.414215i	0.978318	+	0.207107i	$0.0664049\pi$
$103$	4.20382i	0.414215i	−0.978318	+	0.207107i	$0.933595\pi$
$104$	−16.2965	−1.59801
$105$	0	0
$106$	10.2682	0.997335
$107$	− 10.3137i	− 0.997063i	−0.866871	−	0.498532i	$-0.833873\pi$
$107$	− 10.3137i	− 0.997063i	0.866871	−	0.498532i	$-0.166127\pi$
$108$	0	0
$109$	0.127575	0.0122195	0.00610976	−	0.999981i	$-0.498055\pi$
$109$	0.127575	0.0122195	0.00610976	+	0.999981i	$0.498055\pi$
$110$	−16.9032	−1.61165
$111$	0	0
$112$	0	0
$113$	2.29950i	0.216319i	0.994134	+	0.108159i	$0.0344957\pi$
$113$	2.29950i	0.216319i	−0.994134	+	0.108159i	$0.965504\pi$
$114$	0	0
$115$	2.01632i	0.188023i
$116$	2.14061i	0.198751i
$117$	0	0
$118$	− 7.66792i	− 0.705889i
$119$	0	0
$120$	0	0
$121$	−19.1496	−1.74087
$122$	−17.1784	−1.55526
$123$	0	0
$124$	− 34.1920i	− 3.07054i
$125$	−10.4711	−0.936567
$126$	0	0
$127$	−0.386795	−0.0343225	−0.0171613	−	0.999853i	$-0.505463\pi$
$127$	−0.386795	−0.0343225	−0.0171613	+	0.999853i	$0.505463\pi$
$128$	17.2447i	1.52423i
$129$	0	0
$130$	9.13654	0.801328
$131$	20.8634	1.82285	0.911424	−	0.411469i	$-0.134984\pi$
$131$	20.8634	1.82285	0.911424	+	0.411469i	$0.134984\pi$
$132$	0	0
$133$	0	0
$134$	− 30.7173i	− 2.65357i
$135$	0	0
$136$	− 23.6892i	− 2.03134i
$137$	− 0.700500i	− 0.0598477i	−0.999552	−	0.0299239i	$-0.990474\pi$
$137$	− 0.700500i	− 0.0598477i	0.999552	−	0.0299239i	$-0.00952648\pi$
$138$	0	0
$139$	11.5965i	0.983606i	0.870707	+	0.491803i	$0.163662\pi$
$139$	11.5965i	0.983606i	−0.870707	+	0.491803i	$0.836338\pi$
$140$	0	0
$141$	0	0
$142$	−11.5364	−0.968111
$143$	16.2965	1.36278
$144$	0	0
$145$	− 0.629233i	− 0.0522550i
$146$	17.5324	1.45099
$147$	0	0
$148$	12.8814	1.05884
$149$	0.350250i	0.0286936i	0.999897	+	0.0143468i	$0.00456688\pi$
$149$	0.350250i	0.0286936i	−0.999897	+	0.0143468i	$0.995433\pi$
$150$	0	0
$151$	4.65498	0.378817	0.189409	−	0.981898i	$-0.439343\pi$
$151$	4.65498	0.378817	0.189409	+	0.981898i	$0.439343\pi$
$152$	30.4753	2.47187
$153$	0	0
$154$	0	0
$155$	10.0507i	0.807296i
$156$	0	0
$157$	− 1.23588i	− 0.0986343i	−0.998783	−	0.0493171i	$-0.984295\pi$
$157$	− 1.23588i	− 0.0986343i	0.998783	−	0.0493171i	$-0.0157045\pi$
$158$	− 0.826910i	− 0.0657854i
$159$	0	0
$160$	2.64555i	0.209149i
$161$	0	0
$162$	0	0
$163$	−19.6132	−1.53622	−0.768112	−	0.640315i	$-0.778804\pi$
$163$	−19.6132	−1.53622	−0.768112	+	0.640315i	$0.778804\pi$
$164$	−1.48850	−0.116232
$165$	0	0
$166$	18.4143i	1.42922i
$167$	23.6892	1.83313	0.916564	−	0.399887i	$-0.130951\pi$
$167$	23.6892	1.83313	0.916564	+	0.399887i	$0.130951\pi$
$168$	0	0
$169$	4.19136	0.322413
$170$	13.2812i	1.01862i
$171$	0	0
$172$	0.268189	0.0204492
$173$	11.9822	0.910992	0.455496	−	0.890238i	$-0.349462\pi$
$173$	11.9822	0.910992	0.455496	+	0.890238i	$0.349462\pi$
$174$	0	0
$175$	0	0
$176$	28.9269i	2.18045i
$177$	0	0
$178$	8.54977i	0.640832i
$179$	22.3853i	1.67316i	0.547848	+	0.836578i	$0.315447\pi$
$179$	22.3853i	1.67316i	−0.547848	+	0.836578i	$0.684553\pi$
$180$	0	0
$181$	− 8.90380i	− 0.661815i	−0.943663	−	0.330907i	$-0.892645\pi$
$181$	− 8.90380i	− 0.661815i	0.943663	−	0.330907i	$-0.107355\pi$
$182$	0	0
$183$	0	0
$184$	8.95822	0.660409
$185$	−3.78649	−0.278388
$186$	0	0
$187$	23.6892i	1.73233i
$188$	41.4743	3.02482
$189$	0	0
$190$	−17.0858	−1.23953
$191$	11.4909i	0.831450i	0.909490	+	0.415725i	$0.136472\pi$
$191$	11.4909i	0.831450i	−0.909490	+	0.415725i	$0.863528\pi$
$192$	0	0
$193$	11.0220	0.793381	0.396691	−	0.917952i	$-0.370159\pi$
$193$	11.0220	0.793381	0.396691	+	0.917952i	$0.370159\pi$
$194$	−11.1004	−0.796960
$195$	0	0
$196$	0	0
$197$	19.8228i	1.41232i	0.708053	+	0.706159i	$0.249574\pi$
$197$	19.8228i	1.41232i	−0.708053	+	0.706159i	$0.750426\pi$
$198$	0	0
$199$	− 8.66025i	− 0.613909i	−0.951724	−	0.306955i	$-0.900690\pi$
$199$	− 8.66025i	− 0.613909i	0.951724	−	0.306955i	$-0.0993100\pi$
$200$	19.0675i	1.34828i
$201$	0	0
$202$	22.8074i	1.60472i
$203$	0	0
$204$	0	0
$205$	0.437545	0.0305595
$206$	−10.4711	−0.729559
$207$	0	0
$208$	− 15.6356i	− 1.08414i
$209$	−30.4753	−2.10802
$210$	0	0
$211$	22.1626	1.52574	0.762869	−	0.646553i	$-0.223790\pi$
$211$	22.1626	1.52574	0.762869	+	0.646553i	$0.223790\pi$
$212$	17.3320i	1.19037i
$213$	0	0
$214$	25.6900	1.75613
$215$	−0.0788343	−0.00537645
$216$	0	0
$217$	0	0
$218$	0.317773i	0.0215223i
$219$	0	0
$220$	− 28.5314i	− 1.92358i
$221$	− 12.8046i	− 0.861328i
$222$	0	0
$223$	− 18.7457i	− 1.25531i	−0.778493	−	0.627653i	$-0.784016\pi$
$223$	− 18.7457i	− 1.25531i	0.778493	−	0.627653i	$-0.215984\pi$
$224$	0	0
$225$	0	0
$226$	−5.72774	−0.381004
$227$	−14.1788	−0.941079	−0.470540	−	0.882379i	$-0.655941\pi$
$227$	−14.1788	−0.941079	−0.470540	+	0.882379i	$0.655941\pi$
$228$	0	0
$229$	4.07075i	0.269003i	0.990913	+	0.134501i	$0.0429433\pi$
$229$	4.07075i	0.269003i	−0.990913	+	0.134501i	$0.957057\pi$
$230$	−5.02237	−0.331165
$231$	0	0
$232$	−2.79560	−0.183540
$233$	6.96345i	0.456191i	0.973639	+	0.228096i	$0.0732499\pi$
$233$	6.96345i	0.456191i	−0.973639	+	0.228096i	$0.926750\pi$
$234$	0	0
$235$	−12.1914	−0.795277
$236$	12.9429	0.842511
$237$	0	0
$238$	0	0
$239$	− 26.5401i	− 1.71674i	−0.513033	−	0.858369i	$-0.671478\pi$
$239$	− 26.5401i	− 1.71674i	0.513033	−	0.858369i	$-0.328522\pi$
$240$	0	0
$241$	13.6038i	0.876297i	0.898903	+	0.438149i	$0.144366\pi$
$241$	13.6038i	0.876297i	−0.898903	+	0.438149i	$0.855634\pi$
$242$	− 47.6990i	− 3.06621i
$243$	0	0
$244$	− 28.9959i	− 1.85627i
$245$	0	0
$246$	0	0
$247$	16.4726	1.04813
$248$	44.6541	2.83554
$249$	0	0
$250$	− 26.0822i	− 1.64958i
$251$	−2.55060	−0.160993	−0.0804963	−	0.996755i	$-0.525651\pi$
$251$	−2.55060	−0.160993	−0.0804963	+	0.996755i	$0.525651\pi$
$252$	0	0
$253$	−8.95822	−0.563198
$254$	− 0.963454i	− 0.0604525i
$255$	0	0
$256$	−32.5453	−2.03408
$257$	−18.0602	−1.12657	−0.563283	−	0.826264i	$-0.690462\pi$
$257$	−18.0602	−1.12657	−0.563283	+	0.826264i	$0.690462\pi$
$258$	0	0
$259$	0	0
$260$	15.4218i	0.956422i
$261$	0	0
$262$	51.9680i	3.21059i
$263$	8.06902i	0.497557i	0.968560	+	0.248779i	$0.0800292\pi$
$263$	8.06902i	0.497557i	−0.968560	+	0.248779i	$0.919971\pi$
$264$	0	0
$265$	− 5.09474i	− 0.312967i
$266$	0	0
$267$	0	0
$268$	51.8486	3.16716
$269$	2.09515	0.127744	0.0638718	−	0.997958i	$-0.479655\pi$
$269$	2.09515	0.127744	0.0638718	+	0.997958i	$0.479655\pi$
$270$	0	0
$271$	− 24.6027i	− 1.49451i	−0.664537	−	0.747255i	$-0.731371\pi$
$271$	− 24.6027i	− 1.49451i	0.664537	−	0.747255i	$-0.268629\pi$
$272$	22.7286	1.37812
$273$	0	0
$274$	1.74485	0.105410
$275$	− 19.0675i	− 1.14981i
$276$	0	0
$277$	−21.6990	−1.30377	−0.651883	−	0.758319i	$-0.726021\pi$
$277$	−21.6990	−1.30377	−0.651883	+	0.758319i	$0.726021\pi$
$278$	−28.8854	−1.73243
$279$	0	0
$280$	0	0
$281$	22.6315i	1.35008i	0.737781	+	0.675040i	$0.235874\pi$
$281$	22.6315i	1.35008i	−0.737781	+	0.675040i	$0.764126\pi$
$282$	0	0
$283$	− 9.54210i	− 0.567219i	−0.958940	−	0.283610i	$-0.908468\pi$
$283$	− 9.54210i	− 0.567219i	0.958940	−	0.283610i	$-0.0915320\pi$
$284$	− 19.4726i	− 1.15549i
$285$	0	0
$286$	40.5924i	2.40028i
$287$	0	0
$288$	0	0
$289$	1.61320	0.0948944
$290$	1.56733	0.0920371
$291$	0	0
$292$	29.5935i	1.73183i
$293$	26.3348	1.53850	0.769248	−	0.638951i	$-0.220631\pi$
$293$	26.3348	1.53850	0.769248	+	0.638951i	$0.220631\pi$
$294$	0	0
$295$	−3.80457	−0.221511
$296$	16.8228i	0.977808i
$297$	0	0
$298$	−0.872425	−0.0505382
$299$	4.84212	0.280027
$300$	0	0
$301$	0	0
$302$	11.5949i	0.667213i
$303$	0	0
$304$	29.2394i	1.67700i
$305$	8.52334i	0.488045i
$306$	0	0
$307$	6.81772i	0.389108i	0.980892	+	0.194554i	$0.0623259\pi$
$307$	6.81772i	0.389108i	−0.980892	+	0.194554i	$0.937674\pi$
$308$	0	0
$309$	0	0
$310$	−25.0350	−1.42190
$311$	26.9414	1.52771	0.763855	−	0.645388i	$-0.223304\pi$
$311$	26.9414	1.52771	0.763855	+	0.645388i	$0.223304\pi$
$312$	0	0
$313$	− 23.1614i	− 1.30916i	−0.755992	−	0.654581i	$-0.772845\pi$
$313$	− 23.1614i	− 1.30916i	0.755992	−	0.654581i	$-0.227155\pi$
$314$	3.07842	0.173725
$315$	0	0
$316$	1.39576	0.0785179
$317$	− 25.0675i	− 1.40793i	−0.710234	−	0.703966i	$-0.751411\pi$
$317$	− 25.0675i	− 1.40793i	0.710234	−	0.703966i	$-0.248589\pi$
$318$	0	0
$319$	2.79560	0.156523
$320$	6.43204	0.359562
$321$	0	0
$322$	0	0
$323$	23.9452i	1.33235i
$324$	0	0
$325$	10.3064i	0.571697i
$326$	− 48.8538i	− 2.70576i
$327$	0	0
$328$	− 1.94395i	− 0.107337i
$329$	0	0
$330$	0	0
$331$	−8.39576	−0.461473	−0.230736	−	0.973016i	$-0.574114\pi$
$331$	−8.39576	−0.461473	−0.230736	+	0.973016i	$0.574114\pi$
$332$	−31.0820	−1.70584
$333$	0	0
$334$	59.0067i	3.22870i
$335$	−15.2409	−0.832699
$336$	0	0
$337$	−28.2902	−1.54107	−0.770533	−	0.637401i	$-0.780010\pi$
$337$	−28.2902	−1.54107	−0.770533	+	0.637401i	$0.780010\pi$
$338$	10.4401i	0.567867i
$339$	0	0
$340$	−22.4178	−1.21577
$341$	−44.6541	−2.41816
$342$	0	0
$343$	0	0
$344$	0.350250i	0.0188842i
$345$	0	0
$346$	29.8461i	1.60454i
$347$	9.96345i	0.534866i	0.963576	+	0.267433i	$0.0861754\pi$
$347$	9.96345i	0.534866i	−0.963576	+	0.267433i	$0.913825\pi$
$348$	0	0
$349$	22.4624i	1.20239i	0.799104	+	0.601193i	$0.205308\pi$
$349$	22.4624i	1.20239i	−0.799104	+	0.601193i	$0.794692\pi$
$350$	0	0
$351$	0	0
$352$	−11.7538	−0.626481
$353$	32.0652	1.70666	0.853330	−	0.521371i	$-0.174579\pi$
$353$	32.0652	1.70666	0.853330	+	0.521371i	$0.174579\pi$
$354$	0	0
$355$	5.72397i	0.303797i
$356$	−14.4314	−0.764863
$357$	0	0
$358$	−55.7587	−2.94694
$359$	11.8736i	0.626664i	0.949644	+	0.313332i	$0.101445\pi$
$359$	11.8736i	0.626664i	−0.949644	+	0.313332i	$0.898555\pi$
$360$	0	0
$361$	−11.8046	−0.621293
$362$	22.1782	1.16566
$363$	0	0
$364$	0	0
$365$	− 8.69900i	− 0.455326i
$366$	0	0
$367$	− 13.5721i	− 0.708460i	−0.935158	−	0.354230i	$-0.884743\pi$
$367$	− 13.5721i	− 0.708460i	0.935158	−	0.354230i	$-0.115257\pi$
$368$	8.59493i	0.448042i
$369$	0	0
$370$	− 9.43162i	− 0.490327i
$371$	0	0
$372$	0	0
$373$	30.1716	1.56223	0.781113	−	0.624390i	$-0.214652\pi$
$373$	30.1716	1.56223	0.781113	+	0.624390i	$0.214652\pi$
$374$	−59.0067	−3.05116
$375$	0	0
$376$	54.1646i	2.79332i
$377$	−1.51108	−0.0778248
$378$	0	0
$379$	−12.6770	−0.651173	−0.325587	−	0.945512i	$-0.605562\pi$
$379$	−12.6770	−0.651173	−0.325587	+	0.945512i	$0.605562\pi$
$380$	− 28.8396i	− 1.47944i
$381$	0	0
$382$	−28.6222	−1.46444
$383$	31.9864	1.63443	0.817214	−	0.576334i	$-0.195517\pi$
$383$	31.9864	1.63443	0.817214	+	0.576334i	$0.195517\pi$
$384$	0	0
$385$	0	0
$386$	27.4543i	1.39739i
$387$	0	0
$388$	− 18.7366i	− 0.951209i
$389$	7.57666i	0.384152i	0.981380	+	0.192076i	$0.0615220\pi$
$389$	7.57666i	0.384152i	−0.981380	+	0.192076i	$0.938478\pi$
$390$	0	0
$391$	7.03869i	0.355962i
$392$	0	0
$393$	0	0
$394$	−49.3760	−2.48753
$395$	−0.410285	−0.0206437
$396$	0	0
$397$	0.951618i	0.0477603i	0.999715	+	0.0238802i	$0.00760202\pi$
$397$	0.951618i	0.0477603i	−0.999715	+	0.0238802i	$0.992398\pi$
$398$	21.5715	1.08128
$399$	0	0
$400$	−18.2943	−0.914713
$401$	− 21.0325i	− 1.05031i	−0.851006	−	0.525156i	$-0.824007\pi$
$401$	− 21.0325i	− 1.05031i	0.851006	−	0.525156i	$-0.175993\pi$
$402$	0	0
$403$	24.1365	1.20233
$404$	−38.4973	−1.91531
$405$	0	0
$406$	0	0
$407$	− 16.8228i	− 0.833877i
$408$	0	0
$409$	20.1463i	0.996171i	0.867128	+	0.498085i	$0.165963\pi$
$409$	20.1463i	0.996171i	−0.867128	+	0.498085i	$0.834037\pi$
$410$	1.08986i	0.0538246i
$411$	0	0
$412$	− 17.6745i	− 0.870762i
$413$	0	0
$414$	0	0
$415$	9.13654	0.448495
$416$	6.35320	0.311491
$417$	0	0
$418$	− 75.9099i	− 3.71287i
$419$	9.58929	0.468467	0.234234	−	0.972180i	$-0.424742\pi$
$419$	9.58929	0.468467	0.234234	+	0.972180i	$0.424742\pi$
$420$	0	0
$421$	8.61320	0.419782	0.209891	−	0.977725i	$-0.432689\pi$
$421$	8.61320	0.419782	0.209891	+	0.977725i	$0.432689\pi$
$422$	55.2041i	2.68729i
$423$	0	0
$424$	−22.6352	−1.09926
$425$	−14.9818	−0.726724
$426$	0	0
$427$	0	0
$428$	43.3630i	2.09603i
$429$	0	0
$430$	− 0.196365i	− 0.00946958i
$431$	11.5457i	0.556136i	0.960561	+	0.278068i	$0.0896940\pi$
$431$	11.5457i	0.556136i	−0.960561	+	0.278068i	$0.910306\pi$
$432$	0	0
$433$	22.8707i	1.09910i	0.835462	+	0.549548i	$0.185200\pi$
$433$	22.8707i	1.09910i	−0.835462	+	0.549548i	$0.814800\pi$
$434$	0	0
$435$	0	0
$436$	−0.536379	−0.0256879
$437$	−9.05500	−0.433160
$438$	0	0
$439$	− 0.252617i	− 0.0120567i	−0.999982	−	0.00602837i	$-0.998081\pi$
$439$	− 0.252617i	− 0.0120567i	0.999982	−	0.00602837i	$-0.00191890\pi$
$440$	37.2614	1.77637
$441$	0	0
$442$	31.8944	1.51706
$443$	21.7721i	1.03442i	0.855858	+	0.517212i	$0.173030\pi$
$443$	21.7721i	1.03442i	−0.855858	+	0.517212i	$0.826970\pi$
$444$	0	0
$445$	4.24211	0.201095
$446$	46.6930	2.21098
$447$	0	0
$448$	0	0
$449$	− 10.7904i	− 0.509229i	−0.967043	−	0.254614i	$-0.918051\pi$
$449$	− 10.7904i	− 0.509229i	0.967043	−	0.254614i	$-0.0819485\pi$
$450$	0	0
$451$	1.94395i	0.0915371i
$452$	− 9.66802i	− 0.454746i
$453$	0	0
$454$	− 35.3174i	− 1.65753i
$455$	0	0
$456$	0	0
$457$	35.0220	1.63826	0.819130	−	0.573608i	$-0.194457\pi$
$457$	35.0220	1.63826	0.819130	+	0.573608i	$0.194457\pi$
$458$	−10.1397	−0.473797
$459$	0	0
$460$	− 8.47741i	− 0.395261i
$461$	32.9471	1.53450	0.767249	−	0.641349i	$-0.221625\pi$
$461$	32.9471	1.53450	0.767249	+	0.641349i	$0.221625\pi$
$462$	0	0
$463$	−5.22641	−0.242892	−0.121446	−	0.992598i	$-0.538753\pi$
$463$	−5.22641	−0.242892	−0.121446	+	0.992598i	$0.538753\pi$
$464$	− 2.68223i	− 0.124519i
$465$	0	0
$466$	−17.3450	−0.803492
$467$	21.6665	1.00260	0.501302	−	0.865272i	$-0.332855\pi$
$467$	21.6665	1.00260	0.501302	+	0.865272i	$0.332855\pi$
$468$	0	0
$469$	0	0
$470$	− 30.3670i	− 1.40073i
$471$	0	0
$472$	16.9032i	0.778032i
$473$	− 0.350250i	− 0.0161045i
$474$	0	0
$475$	− 19.2735i	− 0.884330i
$476$	0	0
$477$	0	0
$478$	66.1078	3.02370
$479$	22.4308	1.02489	0.512444	−	0.858721i	$-0.328740\pi$
$479$	22.4308	1.02489	0.512444	+	0.858721i	$0.328740\pi$
$480$	0	0
$481$	9.09312i	0.414611i
$482$	−33.8852	−1.54343
$483$	0	0
$484$	80.5125	3.65966
$485$	5.50764i	0.250089i
$486$	0	0
$487$	22.1626	1.00428	0.502142	−	0.864785i	$-0.332545\pi$
$487$	22.1626	1.00428	0.502142	+	0.864785i	$0.332545\pi$
$488$	37.8680	1.71421
$489$	0	0
$490$	0	0
$491$	− 15.4543i	− 0.697444i	−0.937226	−	0.348722i	$-0.886616\pi$
$491$	− 15.4543i	− 0.697444i	0.937226	−	0.348722i	$-0.113384\pi$
$492$	0	0
$493$	− 2.19657i	− 0.0989285i
$494$	41.0310i	1.84607i
$495$	0	0
$496$	42.8432i	1.92372i
$497$	0	0
$498$	0	0
$499$	−1.34502	−0.0602112	−0.0301056	−	0.999547i	$-0.509584\pi$
$499$	−1.34502	−0.0602112	−0.0301056	+	0.999547i	$0.509584\pi$
$500$	44.0249	1.96885
$501$	0	0
$502$	− 6.35320i	− 0.283557i
$503$	−18.1391	−0.808781	−0.404390	−	0.914586i	$-0.632516\pi$
$503$	−18.1391	−0.808781	−0.404390	+	0.914586i	$0.632516\pi$
$504$	0	0
$505$	11.3163	0.503568
$506$	− 22.3137i	− 0.991965i
$507$	0	0
$508$	1.62624	0.0721529
$509$	28.1838	1.24922	0.624612	−	0.780935i	$-0.285257\pi$
$509$	28.1838	1.24922	0.624612	+	0.780935i	$0.285257\pi$
$510$	0	0
$511$	0	0
$512$	− 46.5767i	− 2.05842i
$513$	0	0
$514$	− 44.9856i	− 1.98423i
$515$	5.19543i	0.228938i
$516$	0	0
$517$	− 54.1646i	− 2.38215i
$518$	0	0
$519$	0	0
$520$	−20.1406	−0.883224
$521$	−34.1830	−1.49758	−0.748792	−	0.662806i	$-0.769366\pi$
$521$	−34.1830	−1.49758	−0.748792	+	0.662806i	$0.769366\pi$
$522$	0	0
$523$	− 36.8531i	− 1.61147i	−0.592273	−	0.805737i	$-0.701769\pi$
$523$	− 36.8531i	− 1.61147i	0.592273	−	0.805737i	$-0.298231\pi$
$524$	−87.7183	−3.83199
$525$	0	0
$526$	−20.0988	−0.876351
$527$	35.0858i	1.52836i
$528$	0	0
$529$	20.3383	0.884273
$530$	12.6903	0.551231
$531$	0	0
$532$	0	0
$533$	− 1.05075i	− 0.0455130i
$534$	0	0
$535$	− 12.7465i	− 0.551081i
$536$	67.7132i	2.92476i
$537$	0	0
$538$	5.21874i	0.224996i
$539$	0	0
$540$	0	0
$541$	−38.4178	−1.65171	−0.825855	−	0.563883i	$-0.809307\pi$
$541$	−38.4178	−1.65171	−0.825855	+	0.563883i	$0.809307\pi$
$542$	61.2821	2.63229
$543$	0	0
$544$	9.23526i	0.395958i
$545$	0.157669	0.00675378
$546$	0	0
$547$	11.9959	0.512909	0.256454	−	0.966556i	$-0.417446\pi$
$547$	11.9959	0.512909	0.256454	+	0.966556i	$0.417446\pi$
$548$	2.94518i	0.125812i
$549$	0	0
$550$	47.4946	2.02518
$551$	2.82580	0.120383
$552$	0	0
$553$	0	0
$554$	− 54.0492i	− 2.29633i
$555$	0	0
$556$	− 48.7565i	− 2.06774i
$557$	− 8.70457i	− 0.368824i	−0.982849	−	0.184412i	$-0.940962\pi$
$557$	− 8.70457i	− 0.368824i	0.982849	−	0.184412i	$-0.0590381\pi$
$558$	0	0
$559$	0.189318i	0.00800729i
$560$	0	0
$561$	0	0
$562$	−56.3719	−2.37791
$563$	−23.4366	−0.987736	−0.493868	−	0.869537i	$-0.664417\pi$
$563$	−23.4366	−0.987736	−0.493868	+	0.869537i	$0.664417\pi$
$564$	0	0
$565$	2.84192i	0.119560i
$566$	23.7681	0.999047
$567$	0	0
$568$	25.4308	1.06705
$569$	− 34.3137i	− 1.43851i	−0.694749	−	0.719253i	$-0.744484\pi$
$569$	− 34.3137i	− 1.43851i	0.694749	−	0.719253i	$-0.255516\pi$
$570$	0	0
$571$	21.4726	0.898600	0.449300	−	0.893381i	$-0.351673\pi$
$571$	21.4726	0.898600	0.449300	+	0.893381i	$0.351673\pi$
$572$	−68.5171	−2.86485
$573$	0	0
$574$	0	0
$575$	− 5.66545i	− 0.236266i
$576$	0	0
$577$	− 28.6193i	− 1.19144i	−0.803194	−	0.595718i	$-0.796868\pi$
$577$	− 28.6193i	− 1.19144i	0.803194	−	0.595718i	$-0.203132\pi$
$578$	4.01827i	0.167138i
$579$	0	0
$580$	2.64555i	0.109850i
$581$	0	0
$582$	0	0
$583$	22.6352	0.937455
$584$	−38.6485	−1.59929
$585$	0	0
$586$	65.5964i	2.70976i
$587$	18.5157	0.764224	0.382112	−	0.924116i	$-0.375197\pi$
$587$	18.5157	0.764224	0.382112	+	0.924116i	$0.375197\pi$
$588$	0	0
$589$	−45.1365	−1.85982
$590$	− 9.47666i	− 0.390148i
$591$	0	0
$592$	−16.1406	−0.663375
$593$	−18.7457	−0.769794	−0.384897	−	0.922960i	$-0.625763\pi$
$593$	−18.7457	−0.769794	−0.384897	+	0.922960i	$0.625763\pi$
$594$	0	0
$595$	0	0
$596$	− 1.47259i	− 0.0603197i
$597$	0	0
$598$	12.0611i	0.493213i
$599$	− 25.8721i	− 1.05710i	−0.848901	−	0.528552i	$-0.822735\pi$
$599$	− 25.8721i	− 1.05710i	0.848901	−	0.528552i	$-0.177265\pi$
$600$	0	0
$601$	− 14.5735i	− 0.594467i	−0.954805	−	0.297234i	$-0.903936\pi$
$601$	− 14.5735i	− 0.594467i	0.954805	−	0.297234i	$-0.0960640\pi$
$602$	0	0
$603$	0	0
$604$	−19.5714	−0.796350
$605$	−23.6667	−0.962187
$606$	0	0
$607$	− 22.2796i	− 0.904300i	−0.891942	−	0.452150i	$-0.850657\pi$
$607$	− 22.2796i	− 0.904300i	0.891942	−	0.452150i	$-0.149343\pi$
$608$	−11.8808	−0.481830
$609$	0	0
$610$	−21.2305	−0.859597
$611$	29.2772i	1.18443i
$612$	0	0
$613$	25.2223	1.01872	0.509360	−	0.860553i	$-0.329882\pi$
$613$	25.2223	1.01872	0.509360	+	0.860553i	$0.329882\pi$
$614$	−16.9820	−0.685338
$615$	0	0
$616$	0	0
$617$	− 2.03655i	− 0.0819882i	−0.999159	−	0.0409941i	$-0.986948\pi$
$617$	− 2.03655i	− 0.0819882i	0.999159	−	0.0409941i	$-0.0130525\pi$
$618$	0	0
$619$	38.0011i	1.52739i	0.645575	+	0.763697i	$0.276618\pi$
$619$	38.0011i	1.52739i	−0.645575	+	0.763697i	$0.723382\pi$
$620$	− 42.2574i	− 1.69710i
$621$	0	0
$622$	67.1075i	2.69076i
$623$	0	0
$624$	0	0
$625$	4.42184	0.176874
$626$	57.6920	2.30583
$627$	0	0
$628$	5.19615i	0.207349i
$629$	−13.2181	−0.527040
$630$	0	0
$631$	2.80457	0.111648	0.0558240	−	0.998441i	$-0.482221\pi$
$631$	2.80457	0.111648	0.0558240	+	0.998441i	$0.482221\pi$
$632$	1.82284i	0.0725087i
$633$	0	0
$634$	62.4398	2.47980
$635$	−0.478034	−0.0189702
$636$	0	0
$637$	0	0
$638$	6.96345i	0.275686i
$639$	0	0
$640$	21.3124i	0.842448i
$641$	1.87766i	0.0741631i	0.999312	+	0.0370815i	$0.0118061\pi$
$641$	1.87766i	0.0741631i	−0.999312	+	0.0370815i	$0.988194\pi$
$642$	0	0
$643$	14.5735i	0.574724i	0.957822	+	0.287362i	$0.0927783\pi$
$643$	14.5735i	0.574724i	−0.957822	+	0.287362i	$0.907222\pi$
$644$	0	0
$645$	0	0
$646$	−59.6442	−2.34667
$647$	36.4584	1.43333	0.716663	−	0.697419i	$-0.245668\pi$
$647$	36.4584	1.43333	0.716663	+	0.697419i	$0.245668\pi$
$648$	0	0
$649$	− 16.9032i	− 0.663508i
$650$	−25.6719	−1.00693
$651$	0	0
$652$	82.4618	3.22945
$653$	− 15.0716i	− 0.589797i	−0.955529	−	0.294898i	$-0.904714\pi$
$653$	− 15.0716i	− 0.589797i	0.955529	−	0.294898i	$-0.0952858\pi$
$654$	0	0
$655$	25.7848	1.00750
$656$	1.86512	0.0728206
$657$	0	0
$658$	0	0
$659$	31.4670i	1.22578i	0.790168	+	0.612891i	$0.209993\pi$
$659$	31.4670i	1.22578i	−0.790168	+	0.612891i	$0.790007\pi$
$660$	0	0
$661$	− 13.7524i	− 0.534906i	−0.963571	−	0.267453i	$-0.913818\pi$
$661$	− 13.7524i	− 0.534906i	0.963571	−	0.267453i	$-0.0861820\pi$
$662$	− 20.9127i	− 0.812795i
$663$	0	0
$664$	− 40.5924i	− 1.57529i
$665$	0	0
$666$	0	0
$667$	0.830645	0.0321627
$668$	−99.5991	−3.85360
$669$	0	0
$670$	− 37.9630i	− 1.46664i
$671$	−37.8680	−1.46188
$672$	0	0
$673$	11.7448	0.452731	0.226365	−	0.974042i	$-0.427316\pi$
$673$	11.7448	0.452731	0.226365	+	0.974042i	$0.427316\pi$
$674$	− 70.4670i	− 2.71429i
$675$	0	0
$676$	−17.6222	−0.677776
$677$	−21.7453	−0.835740	−0.417870	−	0.908507i	$-0.637223\pi$
$677$	−21.7453	−0.835740	−0.417870	+	0.908507i	$0.637223\pi$
$678$	0	0
$679$	0	0
$680$	− 29.2772i	− 1.12273i
$681$	0	0
$682$	− 111.227i	− 4.25911i
$683$	− 12.8919i	− 0.493293i	−0.969105	−	0.246647i	$-0.920671\pi$
$683$	− 12.8919i	− 0.493293i	0.969105	−	0.246647i	$-0.0793287\pi$
$684$	0	0
$685$	− 0.865736i	− 0.0330781i
$686$	0	0
$687$	0	0
$688$	−0.336046	−0.0128116
$689$	−12.2348	−0.466110
$690$	0	0
$691$	10.3381i	0.393279i	0.980476	+	0.196639i	$0.0630028\pi$
$691$	10.3381i	0.393279i	−0.980476	+	0.196639i	$0.936997\pi$
$692$	−50.3781	−1.91509
$693$	0	0
$694$	−24.8176	−0.942063
$695$	14.3320i	0.543643i
$696$	0	0
$697$	1.52741	0.0578547
$698$	−55.9508	−2.11777
$699$	0	0
$700$	0	0
$701$	41.2056i	1.55631i	0.628071	+	0.778156i	$0.283845\pi$
$701$	41.2056i	1.55631i	−0.628071	+	0.778156i	$0.716155\pi$
$702$	0	0
$703$	− 17.0046i	− 0.641340i
$704$	28.5767i	1.07702i
$705$	0	0
$706$	79.8701i	3.00595i
$707$	0	0
$708$	0	0
$709$	28.6002	1.07410	0.537051	−	0.843550i	$-0.319538\pi$
$709$	28.6002	1.07410	0.537051	+	0.843550i	$0.319538\pi$
$710$	−14.2576	−0.535079
$711$	0	0
$712$	− 18.8471i	− 0.706326i
$713$	−13.2679	−0.496886
$714$	0	0
$715$	20.1406	0.753216
$716$	− 94.1168i	− 3.51731i
$717$	0	0
$718$	−29.5755	−1.10375
$719$	−1.76370	−0.0657749	−0.0328875	−	0.999459i	$-0.510470\pi$
$719$	−1.76370	−0.0657749	−0.0328875	+	0.999459i	$0.510470\pi$
$720$	0	0
$721$	0	0
$722$	− 29.4036i	− 1.09429i
$723$	0	0
$724$	37.4352i	1.39127i
$725$	1.76802i	0.0656627i
$726$	0	0
$727$	− 20.0358i	− 0.743088i	−0.928415	−	0.371544i	$-0.878829\pi$
$727$	− 20.0358i	− 0.743088i	0.928415	−	0.371544i	$-0.121171\pi$
$728$	0	0
$729$	0	0
$730$	21.6680	0.801970
$731$	−0.275200	−0.0101786
$732$	0	0
$733$	− 15.4237i	− 0.569689i	−0.958574	−	0.284844i	$-0.908058\pi$
$733$	− 15.4237i	− 0.569689i	0.958574	−	0.284844i	$-0.0919419\pi$
$734$	33.8064	1.24782
$735$	0	0
$736$	−3.49236	−0.128730
$737$	− 67.7132i	− 2.49425i
$738$	0	0
$739$	−32.4569	−1.19395	−0.596973	−	0.802261i	$-0.703630\pi$
$739$	−32.4569	−1.19395	−0.596973	+	0.802261i	$0.703630\pi$
$740$	15.9199	0.585227
$741$	0	0
$742$	0	0
$743$	7.52184i	0.275950i	0.990436	+	0.137975i	$0.0440593\pi$
$743$	7.52184i	0.275950i	−0.990436	+	0.137975i	$0.955941\pi$
$744$	0	0
$745$	0.432868i	0.0158591i
$746$	75.1533i	2.75156i
$747$	0	0
$748$	− 99.5991i	− 3.64170i
$749$	0	0
$750$	0	0
$751$	25.8463	0.943147	0.471573	−	0.881827i	$-0.343686\pi$
$751$	25.8463	0.943147	0.471573	+	0.881827i	$0.343686\pi$
$752$	−51.9680	−1.89508
$753$	0	0
$754$	− 3.76390i	− 0.137073i
$755$	5.75302	0.209374
$756$	0	0
$757$	33.5103	1.21795	0.608976	−	0.793188i	$-0.291580\pi$
$757$	33.5103	1.21795	0.608976	+	0.793188i	$0.291580\pi$
$758$	− 31.5767i	− 1.14692i
$759$	0	0
$760$	37.6640	1.36622
$761$	−8.73002	−0.316463	−0.158232	−	0.987402i	$-0.550579\pi$
$761$	−8.73002	−0.316463	−0.158232	+	0.987402i	$0.550579\pi$
$762$	0	0
$763$	0	0
$764$	− 48.3122i	− 1.74787i
$765$	0	0
$766$	79.6738i	2.87873i
$767$	9.13654i	0.329902i
$768$	0	0
$769$	5.04495i	0.181926i	0.995854	+	0.0909628i	$0.0289944\pi$
$769$	5.04495i	0.181926i	−0.995854	+	0.0909628i	$0.971006\pi$
$770$	0	0
$771$	0	0
$772$	−46.3409	−1.66785
$773$	5.80280	0.208712	0.104356	−	0.994540i	$-0.466722\pi$
$773$	5.80280	0.208712	0.104356	+	0.994540i	$0.466722\pi$
$774$	0	0
$775$	− 28.2406i	− 1.01443i
$776$	24.4697	0.878410
$777$	0	0
$778$	−18.8724	−0.676609
$779$	1.96495i	0.0704018i
$780$	0	0
$781$	−25.4308	−0.909986
$782$	−17.5324	−0.626958
$783$	0	0
$784$	0	0
$785$	− 1.52741i	− 0.0545156i
$786$	0	0
$787$	− 24.7042i	− 0.880608i	−0.897849	−	0.440304i	$-0.854871\pi$
$787$	− 24.7042i	− 0.880608i	0.897849	−	0.440304i	$-0.145129\pi$
$788$	− 83.3432i	− 2.96898i
$789$	0	0
$790$	− 1.02196i	− 0.0363599i
$791$	0	0
$792$	0	0
$793$	20.4685	0.726859
$794$	−2.37035	−0.0841206
$795$	0	0
$796$	36.4112i	1.29056i
$797$	4.39314	0.155613	0.0778064	−	0.996968i	$-0.475208\pi$
$797$	4.39314	0.155613	0.0778064	+	0.996968i	$0.475208\pi$
$798$	0	0
$799$	−42.5584	−1.50561
$800$	− 7.43348i	− 0.262813i
$801$	0	0
$802$	52.3890	1.84992
$803$	38.6485	1.36387
$804$	0	0
$805$	0	0
$806$	60.1208i	2.11767i
$807$	0	0
$808$	− 50.2767i	− 1.76873i
$809$	− 50.7315i	− 1.78362i	−0.452406	−	0.891812i	$-0.649434\pi$
$809$	− 50.7315i	− 1.78362i	0.452406	−	0.891812i	$-0.350566\pi$
$810$	0	0
$811$	− 31.8126i	− 1.11709i	−0.829474	−	0.558546i	$-0.811359\pi$
$811$	− 31.8126i	− 1.11709i	0.829474	−	0.558546i	$-0.188641\pi$
$812$	0	0
$813$	0	0
$814$	41.9034	1.46871
$815$	−24.2396	−0.849078
$816$	0	0
$817$	− 0.354034i	− 0.0123861i
$818$	−50.1817	−1.75456
$819$	0	0
$820$	−1.83961	−0.0642421
$821$	25.9437i	0.905441i	0.891653	+	0.452720i	$0.149546\pi$
$821$	25.9437i	0.905441i	−0.891653	+	0.452720i	$0.850454\pi$
$822$	0	0
$823$	−0.140614	−0.00490149	−0.00245074	−	0.999997i	$-0.500780\pi$
$823$	−0.140614	−0.00490149	−0.00245074	+	0.999997i	$0.500780\pi$
$824$	23.0826	0.804120
$825$	0	0
$826$	0	0
$827$	− 16.1772i	− 0.562535i	−0.959629	−	0.281267i	$-0.909245\pi$
$827$	− 16.1772i	− 0.562535i	0.959629	−	0.281267i	$-0.0907548\pi$
$828$	0	0
$829$	− 33.3237i	− 1.15738i	−0.815548	−	0.578690i	$-0.803564\pi$
$829$	− 33.3237i	− 1.15738i	0.815548	−	0.578690i	$-0.196436\pi$
$830$	22.7579i	0.789938i
$831$	0	0
$832$	− 15.4463i	− 0.535505i
$833$	0	0
$834$	0	0
$835$	29.2772	1.01318
$836$	128.130	4.43148
$837$	0	0
$838$	23.8856i	0.825115i
$839$	−21.4701	−0.741230	−0.370615	−	0.928787i	$-0.620853\pi$
$839$	−21.4701	−0.741230	−0.370615	+	0.928787i	$0.620853\pi$
$840$	0	0
$841$	28.7408	0.991061
$842$	21.4543i	0.739365i
$843$	0	0
$844$	−93.1806	−3.20741
$845$	5.18004	0.178199
$846$	0	0
$847$	0	0
$848$	− 21.7173i	− 0.745774i
$849$	0	0
$850$	− 37.3176i	− 1.27998i
$851$	− 4.99850i	− 0.171346i
$852$	0	0
$853$	25.1842i	0.862291i	0.902282	+	0.431146i	$0.141890\pi$
$853$	25.1842i	0.862291i	−0.902282	+	0.431146i	$0.858110\pi$
$854$	0	0
$855$	0	0
$856$	−56.6311	−1.93561
$857$	−29.0366	−0.991871	−0.495936	−	0.868359i	$-0.665175\pi$
$857$	−29.0366	−0.991871	−0.495936	+	0.868359i	$0.665175\pi$
$858$	0	0
$859$	23.8314i	0.813116i	0.913625	+	0.406558i	$0.133271\pi$
$859$	23.8314i	0.813116i	−0.913625	+	0.406558i	$0.866729\pi$
$860$	0.331451	0.0113024
$861$	0	0
$862$	−28.7587	−0.979526
$863$	31.8777i	1.08513i	0.840014	+	0.542564i	$0.182546\pi$
$863$	31.8777i	1.08513i	−0.840014	+	0.542564i	$0.817454\pi$
$864$	0	0
$865$	14.8086	0.503509
$866$	−56.9678	−1.93584
$867$	0	0
$868$	0	0
$869$	− 1.82284i	− 0.0618356i
$870$	0	0
$871$	36.6005i	1.24016i
$872$	− 0.700500i	− 0.0237219i
$873$	0	0
$874$	− 22.5548i	− 0.762927i
$875$	0	0
$876$	0	0
$877$	−2.28123	−0.0770316	−0.0385158	−	0.999258i	$-0.512263\pi$
$877$	−2.28123	−0.0770316	−0.0385158	+	0.999258i	$0.512263\pi$
$878$	0.629233	0.0212356
$879$	0	0
$880$	35.7503i	1.20514i
$881$	11.8808	0.400275	0.200137	−	0.979768i	$-0.435861\pi$
$881$	11.8808	0.400275	0.200137	+	0.979768i	$0.435861\pi$
$882$	0	0
$883$	−49.7120	−1.67294	−0.836472	−	0.548010i	$-0.815385\pi$
$883$	−49.7120	−1.67294	−0.836472	+	0.548010i	$0.815385\pi$
$884$	53.8355i	1.81069i
$885$	0	0
$886$	−54.2313	−1.82194
$887$	12.3588	0.414969	0.207485	−	0.978238i	$-0.433472\pi$
$887$	12.3588	0.414969	0.207485	+	0.978238i	$0.433472\pi$
$888$	0	0
$889$	0	0
$890$	10.5665i	0.354191i
$891$	0	0
$892$	78.8145i	2.63890i
$893$	− 54.7497i	− 1.83213i
$894$	0	0
$895$	27.6656i	0.924760i
$896$	0	0
$897$	0	0
$898$	26.8773	0.896908
$899$	4.14052	0.138094
$900$	0	0
$901$	− 17.7850i	− 0.592505i
$902$	−4.84212	−0.161225
$903$	0	0
$904$	12.6262	0.419943
$905$	− 11.0041i	− 0.365788i
$906$	0	0
$907$	−35.7628	−1.18748	−0.593742	−	0.804656i	$-0.702350\pi$
$907$	−35.7628	−1.18748	−0.593742	+	0.804656i	$0.702350\pi$
$908$	59.6133	1.97834
$909$	0	0
$910$	0	0
$911$	− 47.9228i	− 1.58775i	−0.608078	−	0.793877i	$-0.708059\pi$
$911$	− 47.9228i	− 1.58775i	0.608078	−	0.793877i	$-0.291941\pi$
$912$	0	0
$913$	40.5924i	1.34341i
$914$	87.2350i	2.88548i
$915$	0	0
$916$	− 17.1151i	− 0.565498i
$917$	0	0
$918$	0	0
$919$	−34.4816	−1.13744	−0.568721	−	0.822531i	$-0.692562\pi$
$919$	−34.4816	−1.13744	−0.568721	+	0.822531i	$0.692562\pi$
$920$	11.0713	0.365011
$921$	0	0
$922$	82.0667i	2.70272i
$923$	13.7459	0.452453
$924$	0	0
$925$	10.6393	0.349817
$926$	− 13.0183i	− 0.427807i
$927$	0	0
$928$	1.08986	0.0357766
$929$	−58.3549	−1.91456	−0.957280	−	0.289161i	$-0.906624\pi$
$929$	−58.3549	−1.91456	−0.957280	+	0.289161i	$0.906624\pi$
$930$	0	0
$931$	0	0
$932$	− 29.2772i	− 0.959005i
$933$	0	0
$934$	53.9682i	1.76589i
$935$	29.2772i	0.957465i
$936$	0	0
$937$	29.8596i	0.975471i	0.872992	+	0.487735i	$0.162177\pi$
$937$	29.8596i	0.975471i	−0.872992	+	0.487735i	$0.837823\pi$
$938$	0	0
$939$	0	0
$940$	51.2574	1.67183
$941$	16.2177	0.528682	0.264341	−	0.964429i	$-0.414846\pi$
$941$	16.2177	0.528682	0.264341	+	0.964429i	$0.414846\pi$
$942$	0	0
$943$	0.577598i	0.0188092i
$944$	−16.2177	−0.527841
$945$	0	0
$946$	0.872425	0.0283650
$947$	28.1939i	0.916180i	0.888906	+	0.458090i	$0.151466\pi$
$947$	28.1939i	0.916180i	−0.888906	+	0.458090i	$0.848534\pi$
$948$	0	0
$949$	−20.8904	−0.678130
$950$	48.0077	1.55758
$951$	0	0
$952$	0	0
$953$	− 56.3488i	− 1.82532i	−0.408725	−	0.912658i	$-0.634027\pi$
$953$	− 56.3488i	− 1.82532i	0.408725	−	0.912658i	$-0.365973\pi$
$954$	0	0
$955$	14.2014i	0.459546i
$956$	111.585i	3.60893i
$957$	0	0
$958$	55.8720i	1.80514i
$959$	0	0
$960$	0	0
$961$	−35.1365	−1.13344
$962$	−22.6497	−0.730257
$963$	0	0
$964$	− 57.1958i	− 1.84215i
$965$	13.6219	0.438505
$966$	0	0
$967$	34.0310	1.09436	0.547181	−	0.837014i	$-0.315701\pi$
$967$	34.0310	1.09436	0.547181	+	0.837014i	$0.315701\pi$
$968$	105.148i	3.37958i
$969$	0	0
$970$	−13.7188	−0.440483
$971$	32.6945	1.04922	0.524608	−	0.851344i	$-0.324212\pi$
$971$	32.6945	1.04922	0.524608	+	0.851344i	$0.324212\pi$
$972$	0	0
$973$	0	0
$974$	55.2041i	1.76885i
$975$	0	0
$976$	36.3324i	1.16297i
$977$	24.6625i	0.789022i	0.918891	+	0.394511i	$0.129086\pi$
$977$	24.6625i	0.789022i	−0.918891	+	0.394511i	$0.870914\pi$
$978$	0	0
$979$	18.8471i	0.602357i
$980$	0	0
$981$	0	0
$982$	38.4946	1.22841
$983$	−17.7915	−0.567461	−0.283730	−	0.958904i	$-0.591572\pi$
$983$	−17.7915	−0.567461	−0.283730	+	0.958904i	$0.591572\pi$
$984$	0	0
$985$	24.4987i	0.780594i
$986$	5.47135	0.174243
$987$	0	0
$988$	−69.2574	−2.20337
$989$	− 0.104068i	− 0.00330918i
$990$	0	0
$991$	−31.9034	−1.01344	−0.506722	−	0.862109i	$-0.669143\pi$
$991$	−31.9034	−1.01344	−0.506722	+	0.862109i	$0.669143\pi$
$992$	−17.4084	−0.552718
$993$	0	0
$994$	0	0
$995$	− 10.7031i	− 0.339310i
$996$	0	0
$997$	− 38.9553i	− 1.23373i	−0.787070	−	0.616864i	$-0.788403\pi$
$997$	− 38.9553i	− 1.23373i	0.787070	−	0.616864i	$-0.211597\pi$
$998$	− 3.35025i	− 0.106050i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.d.1322.12		12
3.2	odd	2	inner	1323.2.c.d.1322.1		12
7.4	even	3		189.2.p.d.26.1	✓	12
7.5	odd	6		189.2.p.d.80.6	yes	12
7.6	odd	2	inner	1323.2.c.d.1322.11		12
21.5	even	6		189.2.p.d.80.1	yes	12
21.11	odd	6		189.2.p.d.26.6	yes	12
21.20	even	2	inner	1323.2.c.d.1322.2		12
63.4	even	3		567.2.s.f.26.6		12
63.5	even	6		567.2.i.f.269.1		12
63.11	odd	6		567.2.i.f.215.1		12
63.25	even	3		567.2.i.f.215.6		12
63.32	odd	6		567.2.s.f.26.1		12
63.40	odd	6		567.2.i.f.269.6		12
63.47	even	6		567.2.s.f.458.6		12
63.61	odd	6		567.2.s.f.458.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.p.d.26.1	✓	12	7.4	even	3
189.2.p.d.26.6	yes	12	21.11	odd	6
189.2.p.d.80.1	yes	12	21.5	even	6
189.2.p.d.80.6	yes	12	7.5	odd	6
567.2.i.f.215.1		12	63.11	odd	6
567.2.i.f.215.6		12	63.25	even	3
567.2.i.f.269.1		12	63.5	even	6
567.2.i.f.269.6		12	63.40	odd	6
567.2.s.f.26.1		12	63.32	odd	6
567.2.s.f.26.6		12	63.4	even	3
567.2.s.f.458.1		12	63.61	odd	6
567.2.s.f.458.6		12	63.47	even	6
1323.2.c.d.1322.1		12	3.2	odd	2	inner
1323.2.c.d.1322.2		12	21.20	even	2	inner
1323.2.c.d.1322.11		12	7.6	odd	2	inner
1323.2.c.d.1322.12		12	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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.49086i q^{2} -4.20440 q^{4} +1.23588 q^{5} -5.49086i q^{8} +O(q^{10})\)	\(q+2.49086i q^{2} -4.20440 q^{4} +1.23588 q^{5} -5.49086i q^{8} +3.07842i q^{10} +5.49086i q^{11} -2.96793i q^{13} +5.26819 q^{16} +4.31430 q^{17} +5.55019i q^{19} -5.19615 q^{20} -13.6770 q^{22} +1.63148i q^{23} -3.47259 q^{25} +7.39272 q^{26} -0.509136i q^{29} +8.13244i q^{31} +2.14061i q^{32} +10.7463i q^{34} -3.06379 q^{37} -13.8248 q^{38} -6.78607i q^{40} +0.354034 q^{41} -0.0637877 q^{43} -23.0858i q^{44} -4.06379 q^{46} -9.86449 q^{47} -8.64975i q^{50} +12.4784i q^{52} -4.12234i q^{53} +6.78607i q^{55} +1.26819 q^{58} -3.07842 q^{59} +6.89655i q^{61} -20.2568 q^{62} +5.20440 q^{64} -3.66802i q^{65} -12.3320 q^{67} -18.1391 q^{68} +4.63148i q^{71} -7.03869i q^{73} -7.63148i q^{74} -23.3352i q^{76} -0.331977 q^{79} +6.51087 q^{80} +0.881850i q^{82} +7.39272 q^{83} +5.33198 q^{85} -0.158887i q^{86} +30.1496 q^{88} +3.43245 q^{89} -6.85939i q^{92} -24.5711i q^{94} +6.85939i q^{95} +4.45644i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{4}+O(q^{10}) \) 12 * q - 16 * q^4	\( 12 q - 16 q^{4} + 8 q^{16} - 40 q^{22} + 48 q^{25} - 16 q^{37} + 20 q^{43} - 28 q^{46} - 40 q^{58} + 28 q^{64} - 72 q^{67} + 72 q^{79} - 12 q^{85} + 148 q^{88}+O(q^{100}) \) 12 * q - 16 * q^4 + 8 * q^16 - 40 * q^22 + 48 * q^25 - 16 * q^37 + 20 * q^43 - 28 * q^46 - 40 * q^58 + 28 * q^64 - 72 * q^67 + 72 * q^79 - 12 * q^85 + 148 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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