LMFDB - Embedded newform 252.12.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,12,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("252.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	252.a (trivial)

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:	yes
Analytic conductor:	$193.622481501$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 5681627x^{4} + 7706355585775x^{2} - 2456393975347843125 $ x^6 - 5681627x^4 + 7706355585775x^2 - 2456393975347843125
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{10}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-682.125$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4092.75 q^{5} +16807.0 q^{7} +O(q^{10})$	$q-4092.75 q^{5} +16807.0 q^{7} +830540. q^{11} -1.80593e6 q^{13} +530738. q^{17} -2.80774e6 q^{19} +3.08982e7 q^{23} -3.20775e7 q^{25} +9.85991e7 q^{29} +2.90909e7 q^{31} -6.87868e7 q^{35} +5.27655e8 q^{37} -7.13596e8 q^{41} +4.10771e8 q^{43} -2.25052e9 q^{47} +2.82475e8 q^{49} -7.14707e8 q^{53} -3.39919e9 q^{55} -2.36553e9 q^{59} +4.98924e8 q^{61} +7.39122e9 q^{65} -8.06486e9 q^{67} -1.35074e9 q^{71} +9.62552e9 q^{73} +1.39589e10 q^{77} +1.41245e10 q^{79} -3.52430e10 q^{83} -2.17218e9 q^{85} +1.78519e10 q^{89} -3.03523e10 q^{91} +1.14914e10 q^{95} +7.17271e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 100842 q^{7}+O(q^{10}) $ 6 * q + 100842 * q^7	$ 6 q + 100842 q^{7} + 1227468 q^{13} - 6532728 q^{19} + 116108394 q^{25} + 293608488 q^{31} - 74878260 q^{37} + 1432813080 q^{43} + 1694851494 q^{49} + 6150010104 q^{55} - 11105239740 q^{61} - 3754055112 q^{67} - 31466324076 q^{73} - 58630542864 q^{79} + 94916392200 q^{85} + 20630054676 q^{91} - 6046820556 q^{97}+O(q^{100}) $ 6 * q + 100842 * q^7 + 1227468 * q^13 - 6532728 * q^19 + 116108394 * q^25 + 293608488 * q^31 - 74878260 * q^37 + 1432813080 * q^43 + 1694851494 * q^49 + 6150010104 * q^55 - 11105239740 * q^61 - 3754055112 * q^67 - 31466324076 * q^73 - 58630542864 * q^79 + 94916392200 * q^85 + 20630054676 * q^91 - 6046820556 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−4092.75	−0.585707	−0.292853	−	0.956157i	$-0.594605\pi$
$5$	−4092.75	−0.585707	−0.292853	+	0.956157i	$0.594605\pi$
$6$	0	0
$7$	16807.0	0.377964
$7$	16807.0	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	830540.	1.55489	0.777447	−	0.628949i	$-0.216514\pi$
$11$	830540.	1.55489	0.777447	+	0.628949i	$0.216514\pi$
$12$	0	0
$13$	−1.80593e6	−1.34900	−0.674501	−	0.738274i	$-0.735641\pi$
$13$	−1.80593e6	−1.34900	−0.674501	+	0.738274i	$0.735641\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	530738.	0.0906591	0.0453295	−	0.998972i	$-0.485566\pi$
$17$	530738.	0.0906591	0.0453295	+	0.998972i	$0.485566\pi$
$18$	0	0
$19$	−2.80774e6	−0.260143	−0.130072	−	0.991505i	$-0.541521\pi$
$19$	−2.80774e6	−0.260143	−0.130072	+	0.991505i	$0.541521\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.08982e7	1.00099	0.500496	−	0.865739i	$-0.333151\pi$
$23$	3.08982e7	1.00099	0.500496	+	0.865739i	$0.333151\pi$
$24$	0	0
$25$	−3.20775e7	−0.656948
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.85991e7	0.892655	0.446328	−	0.894870i	$-0.352732\pi$
$29$	9.85991e7	0.892655	0.446328	+	0.894870i	$0.352732\pi$
$30$	0	0
$31$	2.90909e7	0.182502	0.0912509	−	0.995828i	$-0.470913\pi$
$31$	2.90909e7	0.182502	0.0912509	+	0.995828i	$0.470913\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.87868e7	−0.221376
$36$	0	0
$37$	5.27655e8	1.25095	0.625475	−	0.780244i	$-0.284905\pi$
$37$	5.27655e8	1.25095	0.625475	+	0.780244i	$0.284905\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.13596e8	−0.961926	−0.480963	−	0.876741i	$-0.659713\pi$
$41$	−7.13596e8	−0.961926	−0.480963	+	0.876741i	$0.659713\pi$
$42$	0	0
$43$	4.10771e8	0.426111	0.213056	−	0.977040i	$-0.431658\pi$
$43$	4.10771e8	0.426111	0.213056	+	0.977040i	$0.431658\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.25052e9	−1.43135	−0.715673	−	0.698435i	$-0.753880\pi$
$47$	−2.25052e9	−1.43135	−0.715673	+	0.698435i	$0.753880\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.14707e8	−0.234753	−0.117376	−	0.993087i	$-0.537448\pi$
$53$	−7.14707e8	−0.234753	−0.117376	+	0.993087i	$0.537448\pi$
$54$	0	0
$55$	−3.39919e9	−0.910712
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.36553e9	−0.430768	−0.215384	−	0.976529i	$-0.569100\pi$
$59$	−2.36553e9	−0.430768	−0.215384	+	0.976529i	$0.569100\pi$
$60$	0	0
$61$	4.98924e8	0.0756346	0.0378173	−	0.999285i	$-0.487960\pi$
$61$	4.98924e8	0.0756346	0.0378173	+	0.999285i	$0.487960\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.39122e9	0.790120
$66$	0	0
$67$	−8.06486e9	−0.729769	−0.364885	−	0.931053i	$-0.618892\pi$
$67$	−8.06486e9	−0.729769	−0.364885	+	0.931053i	$0.618892\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.35074e9	−0.0888489	−0.0444245	−	0.999013i	$-0.514145\pi$
$71$	−1.35074e9	−0.0888489	−0.0444245	+	0.999013i	$0.514145\pi$
$72$	0	0
$73$	9.62552e9	0.543436	0.271718	−	0.962377i	$-0.412408\pi$
$73$	9.62552e9	0.543436	0.271718	+	0.962377i	$0.412408\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.39589e10	0.587695
$78$	0	0
$79$	1.41245e10	0.516446	0.258223	−	0.966085i	$-0.416863\pi$
$79$	1.41245e10	0.516446	0.258223	+	0.966085i	$0.416863\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.52430e10	−0.982071	−0.491035	−	0.871140i	$-0.663381\pi$
$83$	−3.52430e10	−0.982071	−0.491035	+	0.871140i	$0.663381\pi$
$84$	0	0
$85$	−2.17218e9	−0.0530996
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.78519e10	0.338875	0.169437	−	0.985541i	$-0.445805\pi$
$89$	1.78519e10	0.338875	0.169437	+	0.985541i	$0.445805\pi$
$90$	0	0
$91$	−3.03523e10	−0.509875
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.14914e10	0.152368
$96$	0	0
$97$	7.17271e10	0.848083	0.424042	−	0.905643i	$-0.360611\pi$
$97$	7.17271e10	0.848083	0.424042	+	0.905643i	$0.360611\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.25159e10	−0.497191	−0.248595	−	0.968607i	$-0.579969\pi$
$101$	−5.25159e10	−0.497191	−0.248595	+	0.968607i	$0.579969\pi$
$102$	0	0
$103$	−8.89878e10	−0.756355	−0.378178	−	0.925733i	$-0.623449\pi$
$103$	−8.89878e10	−0.756355	−0.378178	+	0.925733i	$0.623449\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.69394e10	−0.254612	−0.127306	−	0.991863i	$-0.540633\pi$
$107$	−3.69394e10	−0.254612	−0.127306	+	0.991863i	$0.540633\pi$
$108$	0	0
$109$	1.54862e10	0.0964046	0.0482023	−	0.998838i	$-0.484651\pi$
$109$	1.54862e10	0.0964046	0.0482023	+	0.998838i	$0.484651\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.21124e11	1.63961	0.819807	−	0.572640i	$-0.194081\pi$
$113$	3.21124e11	1.63961	0.819807	+	0.572640i	$0.194081\pi$
$114$	0	0
$115$	−1.26459e11	−0.586288
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.92012e9	0.0342659
$120$	0	0
$121$	4.04485e11	1.41769
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.31127e11	0.970485
$126$	0	0
$127$	3.50031e11	0.940127	0.470064	−	0.882633i	$-0.344231\pi$
$127$	3.50031e11	0.940127	0.470064	+	0.882633i	$0.344231\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.26851e11	−0.287277	−0.143638	−	0.989630i	$-0.545880\pi$
$131$	−1.26851e11	−0.287277	−0.143638	+	0.989630i	$0.545880\pi$
$132$	0	0
$133$	−4.71898e10	−0.0983250
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.45733e11	0.435011	0.217505	−	0.976059i	$-0.430208\pi$
$137$	2.45733e11	0.435011	0.217505	+	0.976059i	$0.430208\pi$
$138$	0	0
$139$	2.05088e11	0.335242	0.167621	−	0.985851i	$-0.446391\pi$
$139$	2.05088e11	0.335242	0.167621	+	0.985851i	$0.446391\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.49990e12	−2.09756
$144$	0	0
$145$	−4.03541e11	−0.522834
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.20954e12	−1.34926	−0.674628	−	0.738158i	$-0.735696\pi$
$149$	−1.20954e12	−1.34926	−0.674628	+	0.738158i	$0.735696\pi$
$150$	0	0
$151$	−4.63728e11	−0.480717	−0.240359	−	0.970684i	$-0.577265\pi$
$151$	−4.63728e11	−0.480717	−0.240359	+	0.970684i	$0.577265\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.19062e11	−0.106893
$156$	0	0
$157$	−1.57086e12	−1.31428	−0.657141	−	0.753768i	$-0.728234\pi$
$157$	−1.57086e12	−1.31428	−0.657141	+	0.753768i	$0.728234\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.19307e11	0.378339
$162$	0	0
$163$	2.72802e12	1.85701	0.928507	−	0.371315i	$-0.121093\pi$
$163$	2.72802e12	1.85701	0.928507	+	0.371315i	$0.121093\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.75396e12	1.04491	0.522454	−	0.852667i	$-0.325017\pi$
$167$	1.75396e12	1.04491	0.522454	+	0.852667i	$0.325017\pi$
$168$	0	0
$169$	1.46923e12	0.819807
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.12834e12	−1.04421	−0.522105	−	0.852881i	$-0.674853\pi$
$173$	−2.12834e12	−1.04421	−0.522105	+	0.852881i	$0.674853\pi$
$174$	0	0
$175$	−5.39127e11	−0.248303
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.60161e12	1.87162	0.935811	−	0.352501i	$-0.114669\pi$
$179$	4.60161e12	1.87162	0.935811	+	0.352501i	$0.114669\pi$
$180$	0	0
$181$	4.58552e12	1.75451	0.877257	−	0.480022i	$-0.159371\pi$
$181$	4.58552e12	1.75451	0.877257	+	0.480022i	$0.159371\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.15956e12	−0.732690
$186$	0	0
$187$	4.40799e11	0.140965
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.66222e12	−0.757811	−0.378905	−	0.925435i	$-0.623699\pi$
$191$	−2.66222e12	−0.757811	−0.378905	+	0.925435i	$0.623699\pi$
$192$	0	0
$193$	1.74291e12	0.468501	0.234251	−	0.972176i	$-0.424736\pi$
$193$	1.74291e12	0.468501	0.234251	+	0.972176i	$0.424736\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.98298e12	1.91690	0.958452	−	0.285252i	$-0.0920775\pi$
$197$	7.98298e12	1.91690	0.958452	+	0.285252i	$0.0920775\pi$
$198$	0	0
$199$	5.60430e12	1.27300	0.636501	−	0.771276i	$-0.280381\pi$
$199$	5.60430e12	1.27300	0.636501	+	0.771276i	$0.280381\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.65715e12	0.337392
$204$	0	0
$205$	2.92057e12	0.563406
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.33194e12	−0.404495
$210$	0	0
$211$	−2.01162e12	−0.331125	−0.165562	−	0.986199i	$-0.552944\pi$
$211$	−2.01162e12	−0.331125	−0.165562	+	0.986199i	$0.552944\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.68118e12	−0.249576
$216$	0	0
$217$	4.88930e11	0.0689792
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.58476e11	−0.122299
$222$	0	0
$223$	−8.52777e11	−0.103552	−0.0517760	−	0.998659i	$-0.516488\pi$
$223$	−8.52777e11	−0.103552	−0.0517760	+	0.998659i	$0.516488\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.42846e13	−1.57298	−0.786492	−	0.617600i	$-0.788105\pi$
$227$	−1.42846e13	−1.57298	−0.786492	+	0.617600i	$0.788105\pi$
$228$	0	0
$229$	1.10263e13	1.15701	0.578504	−	0.815680i	$-0.303637\pi$
$229$	1.10263e13	1.15701	0.578504	+	0.815680i	$0.303637\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00065e12	0.572454	0.286227	−	0.958162i	$-0.407599\pi$
$233$	6.00065e12	0.572454	0.286227	+	0.958162i	$0.407599\pi$
$234$	0	0
$235$	9.21082e12	0.838349
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.36144e12	0.444726	0.222363	−	0.974964i	$-0.428623\pi$
$239$	5.36144e12	0.444726	0.222363	+	0.974964i	$0.428623\pi$
$240$	0	0
$241$	1.01653e13	0.805424	0.402712	−	0.915327i	$-0.368068\pi$
$241$	1.01653e13	0.805424	0.402712	+	0.915327i	$0.368068\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.15610e12	−0.0836724
$246$	0	0
$247$	5.07059e12	0.350934
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.38095e11	−0.0594349	−0.0297174	−	0.999558i	$-0.509461\pi$
$251$	−9.38095e11	−0.0594349	−0.0297174	+	0.999558i	$0.509461\pi$
$252$	0	0
$253$	2.56622e13	1.55644
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.31851e13	0.733585	0.366792	−	0.930303i	$-0.380456\pi$
$257$	1.31851e13	0.733585	0.366792	+	0.930303i	$0.380456\pi$
$258$	0	0
$259$	8.86829e12	0.472815
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.64215e12	−0.374506	−0.187253	−	0.982312i	$-0.559958\pi$
$263$	−7.64215e12	−0.374506	−0.187253	+	0.982312i	$0.559958\pi$
$264$	0	0
$265$	2.92512e12	0.137496
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.94095e13	1.27306	0.636531	−	0.771251i	$-0.280369\pi$
$269$	2.94095e13	1.27306	0.636531	+	0.771251i	$0.280369\pi$
$270$	0	0
$271$	3.29724e13	1.37031	0.685155	−	0.728397i	$-0.259734\pi$
$271$	3.29724e13	1.37031	0.685155	+	0.728397i	$0.259734\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.66417e13	−1.02148
$276$	0	0
$277$	3.17820e13	1.17096	0.585481	−	0.810686i	$-0.300906\pi$
$277$	3.17820e13	1.17096	0.585481	+	0.810686i	$0.300906\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.57714e12	−0.258000	−0.129000	−	0.991645i	$-0.541177\pi$
$281$	−7.57714e12	−0.258000	−0.129000	+	0.991645i	$0.541177\pi$
$282$	0	0
$283$	4.26503e13	1.39668	0.698339	−	0.715767i	$-0.253923\pi$
$283$	4.26503e13	1.39668	0.698339	+	0.715767i	$0.253923\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.19934e13	−0.363574
$288$	0	0
$289$	−3.39902e13	−0.991781
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.78448e13	−1.02384	−0.511922	−	0.859032i	$-0.671066\pi$
$293$	−3.78448e13	−1.02384	−0.511922	+	0.859032i	$0.671066\pi$
$294$	0	0
$295$	9.68154e12	0.252303
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.58001e13	−1.35034
$300$	0	0
$301$	6.90382e12	0.161055
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.04197e12	−0.0442997
$306$	0	0
$307$	5.00088e13	1.04661	0.523306	−	0.852145i	$-0.324699\pi$
$307$	5.00088e13	1.04661	0.523306	+	0.852145i	$0.324699\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.01628e14	1.98076	0.990378	−	0.138387i	$-0.0441918\pi$
$311$	1.01628e14	1.98076	0.990378	+	0.138387i	$0.0441918\pi$
$312$	0	0
$313$	3.15307e13	0.593252	0.296626	−	0.954994i	$-0.404138\pi$
$313$	3.15307e13	0.593252	0.296626	+	0.954994i	$0.404138\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.14493e13	0.902721	0.451361	−	0.892342i	$-0.350939\pi$
$317$	5.14493e13	0.902721	0.451361	+	0.892342i	$0.350939\pi$
$318$	0	0
$319$	8.18904e13	1.38798
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.49018e12	−0.0235844
$324$	0	0
$325$	5.79298e13	0.886224
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.78245e13	−0.540998
$330$	0	0
$331$	4.46744e13	0.618024	0.309012	−	0.951058i	$-0.400002\pi$
$331$	4.46744e13	0.618024	0.309012	+	0.951058i	$0.400002\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.30075e13	0.427431
$336$	0	0
$337$	3.54263e13	0.443978	0.221989	−	0.975049i	$-0.428745\pi$
$337$	3.54263e13	0.443978	0.221989	+	0.975049i	$0.428745\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.41611e13	0.283771
$342$	0	0
$343$	4.74756e12	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.08777e13	0.116071	0.0580355	−	0.998315i	$-0.481516\pi$
$347$	1.08777e13	0.116071	0.0580355	+	0.998315i	$0.481516\pi$
$348$	0	0
$349$	1.82282e13	0.188453	0.0942267	−	0.995551i	$-0.469962\pi$
$349$	1.82282e13	0.188453	0.0942267	+	0.995551i	$0.469962\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.13544e14	1.10256	0.551282	−	0.834319i	$-0.314139\pi$
$353$	1.13544e14	1.10256	0.551282	+	0.834319i	$0.314139\pi$
$354$	0	0
$355$	5.52826e12	0.0520394
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.92040e13	0.524001	0.262000	−	0.965068i	$-0.415618\pi$
$359$	5.92040e13	0.524001	0.262000	+	0.965068i	$0.415618\pi$
$360$	0	0
$361$	−1.08607e14	−0.932325
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.93948e13	−0.318294
$366$	0	0
$367$	−7.73944e13	−0.606801	−0.303401	−	0.952863i	$-0.598122\pi$
$367$	−7.73944e13	−0.606801	−0.303401	+	0.952863i	$0.598122\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.20121e13	−0.0887282
$372$	0	0
$373$	2.29038e14	1.64251	0.821257	−	0.570559i	$-0.193273\pi$
$373$	2.29038e14	1.64251	0.821257	+	0.570559i	$0.193273\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.78063e14	−1.20419
$378$	0	0
$379$	−1.19364e12	−0.00784077	−0.00392038	−	0.999992i	$-0.501248\pi$
$379$	−1.19364e12	−0.00784077	−0.00392038	+	0.999992i	$0.501248\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.71717e13	−0.168471	−0.0842353	−	0.996446i	$-0.526845\pi$
$383$	−2.71717e13	−0.168471	−0.0842353	+	0.996446i	$0.526845\pi$
$384$	0	0
$385$	−5.71302e13	−0.344217
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.31638e14	−1.31852	−0.659261	−	0.751914i	$-0.729131\pi$
$389$	−2.31638e14	−1.31852	−0.659261	+	0.751914i	$0.729131\pi$
$390$	0	0
$391$	1.63989e13	0.0907490
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.78082e13	−0.302486
$396$	0	0
$397$	7.75801e12	0.0394823	0.0197411	−	0.999805i	$-0.493716\pi$
$397$	7.75801e12	0.0394823	0.0197411	+	0.999805i	$0.493716\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.36205e14	−1.13761	−0.568807	−	0.822471i	$-0.692595\pi$
$401$	−2.36205e14	−1.13761	−0.568807	+	0.822471i	$0.692595\pi$
$402$	0	0
$403$	−5.25361e13	−0.246195
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.38238e14	1.94510
$408$	0	0
$409$	−2.19317e13	−0.0947534	−0.0473767	−	0.998877i	$-0.515086\pi$
$409$	−2.19317e13	−0.0947534	−0.0473767	+	0.998877i	$0.515086\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.97575e13	−0.162815
$414$	0	0
$415$	1.44241e14	0.575205
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.29986e14	−1.24830	−0.624149	−	0.781305i	$-0.714554\pi$
$419$	−3.29986e14	−1.24830	−0.624149	+	0.781305i	$0.714554\pi$
$420$	0	0
$421$	9.24207e13	0.340579	0.170289	−	0.985394i	$-0.445530\pi$
$421$	9.24207e13	0.340579	0.170289	+	0.985394i	$0.445530\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.70248e13	−0.0595583
$426$	0	0
$427$	8.38542e12	0.0285872
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.81299e14	1.55880	0.779400	−	0.626527i	$-0.215524\pi$
$431$	4.81299e14	1.55880	0.779400	+	0.626527i	$0.215524\pi$
$432$	0	0
$433$	6.90008e13	0.217857	0.108928	−	0.994050i	$-0.465258\pi$
$433$	6.90008e13	0.217857	0.108928	+	0.994050i	$0.465258\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.67544e13	−0.260401
$438$	0	0
$439$	−3.40041e14	−0.995352	−0.497676	−	0.867363i	$-0.665813\pi$
$439$	−3.40041e14	−0.995352	−0.497676	+	0.867363i	$0.665813\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.28996e13	−0.175157	−0.0875785	−	0.996158i	$-0.527913\pi$
$443$	−6.28996e13	−0.175157	−0.0875785	+	0.996158i	$0.527913\pi$
$444$	0	0
$445$	−7.30633e13	−0.198481
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.46662e14	1.15511	0.577556	−	0.816351i	$-0.304007\pi$
$449$	4.46662e14	1.15511	0.577556	+	0.816351i	$0.304007\pi$
$450$	0	0
$451$	−5.92670e14	−1.49569
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.24224e14	0.298637
$456$	0	0
$457$	−5.15190e14	−1.20901	−0.604503	−	0.796603i	$-0.706628\pi$
$457$	−5.15190e14	−1.20901	−0.604503	+	0.796603i	$0.706628\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.14819e14	1.37528	0.687642	−	0.726050i	$-0.258646\pi$
$461$	6.14819e14	1.37528	0.687642	+	0.726050i	$0.258646\pi$
$462$	0	0
$463$	−4.90882e14	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$463$	−4.90882e14	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.09562e14	−1.26992	−0.634958	−	0.772547i	$-0.718982\pi$
$467$	−6.09562e14	−1.26992	−0.634958	+	0.772547i	$0.718982\pi$
$468$	0	0
$469$	−1.35546e14	−0.275827
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.41161e14	0.662557
$474$	0	0
$475$	9.00655e13	0.170901
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.23465e14	−1.12971	−0.564855	−	0.825190i	$-0.691068\pi$
$479$	−6.23465e14	−1.12971	−0.564855	+	0.825190i	$0.691068\pi$
$480$	0	0
$481$	−9.52908e14	−1.68754
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.93561e14	−0.496728
$486$	0	0
$487$	−3.79159e14	−0.627209	−0.313604	−	0.949554i	$-0.601537\pi$
$487$	−3.79159e14	−0.627209	−0.313604	+	0.949554i	$0.601537\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.78496e14	−0.440424	−0.220212	−	0.975452i	$-0.570675\pi$
$491$	−2.78496e14	−0.440424	−0.220212	+	0.975452i	$0.570675\pi$
$492$	0	0
$493$	5.23303e13	0.0809273
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.27020e13	−0.0335817
$498$	0	0
$499$	−2.40491e14	−0.347974	−0.173987	−	0.984748i	$-0.555665\pi$
$499$	−2.40491e14	−0.347974	−0.173987	+	0.984748i	$0.555665\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.69290e14	0.511379	0.255690	−	0.966759i	$-0.417697\pi$
$503$	3.69290e14	0.511379	0.255690	+	0.966759i	$0.417697\pi$
$504$	0	0
$505$	2.14934e14	0.291208
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.51457e14	−0.326225	−0.163112	−	0.986608i	$-0.552153\pi$
$509$	−2.51457e14	−0.326225	−0.163112	+	0.986608i	$0.552153\pi$
$510$	0	0
$511$	1.61776e14	0.205399
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.64205e14	0.443002
$516$	0	0
$517$	−1.86915e15	−2.22559
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.42500e14	−0.505017	−0.252509	−	0.967595i	$-0.581256\pi$
$521$	−4.42500e14	−0.505017	−0.252509	+	0.967595i	$0.581256\pi$
$522$	0	0
$523$	1.56754e15	1.75170	0.875851	−	0.482581i	$-0.160301\pi$
$523$	1.56754e15	1.75170	0.875851	+	0.482581i	$0.160301\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.54396e13	0.0165454
$528$	0	0
$529$	1.89113e12	0.00198479
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.28871e15	1.29764
$534$	0	0
$535$	1.51184e14	0.149128
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.34607e14	0.222128
$540$	0	0
$541$	−1.86585e15	−1.73098	−0.865491	−	0.500924i	$-0.832994\pi$
$541$	−1.86585e15	−1.73098	−0.865491	+	0.500924i	$0.832994\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.33809e13	−0.0564648
$546$	0	0
$547$	−1.63916e15	−1.43117	−0.715584	−	0.698527i	$-0.753839\pi$
$547$	−1.63916e15	−1.43117	−0.715584	+	0.698527i	$0.753839\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.76841e14	−0.232218
$552$	0	0
$553$	2.37391e14	0.195198
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.36064e14	0.186563	0.0932815	−	0.995640i	$-0.470264\pi$
$557$	2.36064e14	0.186563	0.0932815	+	0.995640i	$0.470264\pi$
$558$	0	0
$559$	−7.41823e14	−0.574825
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.32765e15	0.989204	0.494602	−	0.869120i	$-0.335314\pi$
$563$	1.32765e15	0.989204	0.494602	+	0.869120i	$0.335314\pi$
$564$	0	0
$565$	−1.31428e15	−0.960333
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.15483e15	−1.51459	−0.757296	−	0.653072i	$-0.773480\pi$
$569$	−2.15483e15	−1.51459	−0.757296	+	0.653072i	$0.773480\pi$
$570$	0	0
$571$	−9.19522e14	−0.633963	−0.316981	−	0.948432i	$-0.602669\pi$
$571$	−9.19522e14	−0.633963	−0.316981	+	0.948432i	$0.602669\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.91139e14	−0.657599
$576$	0	0
$577$	2.73277e14	0.177884	0.0889418	−	0.996037i	$-0.471651\pi$
$577$	2.73277e14	0.177884	0.0889418	+	0.996037i	$0.471651\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.92328e14	−0.371188
$582$	0	0
$583$	−5.93593e14	−0.365016
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.60533e15	−0.950726	−0.475363	−	0.879790i	$-0.657683\pi$
$587$	−1.60533e15	−0.950726	−0.475363	+	0.879790i	$0.657683\pi$
$588$	0	0
$589$	−8.16797e13	−0.0474767
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.73515e15	1.53172	0.765862	−	0.643005i	$-0.222313\pi$
$593$	2.73515e15	1.53172	0.765862	+	0.643005i	$0.222313\pi$
$594$	0	0
$595$	−3.65078e13	−0.0200698
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.52174e15	−1.33614	−0.668070	−	0.744098i	$-0.732879\pi$
$599$	−2.52174e15	−1.33614	−0.668070	+	0.744098i	$0.732879\pi$
$600$	0	0
$601$	1.15017e14	0.0598348	0.0299174	−	0.999552i	$-0.490476\pi$
$601$	1.15017e14	0.0598348	0.0299174	+	0.999552i	$0.490476\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.65546e15	−0.830353
$606$	0	0
$607$	6.86714e14	0.338250	0.169125	−	0.985595i	$-0.445906\pi$
$607$	6.86714e14	0.338250	0.169125	+	0.985595i	$0.445906\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.06429e15	1.93089
$612$	0	0
$613$	1.14096e15	0.532400	0.266200	−	0.963918i	$-0.414232\pi$
$613$	1.14096e15	0.532400	0.266200	+	0.963918i	$0.414232\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.37621e15	−0.619606	−0.309803	−	0.950801i	$-0.600263\pi$
$617$	−1.37621e15	−0.619606	−0.309803	+	0.950801i	$0.600263\pi$
$618$	0	0
$619$	5.62215e14	0.248659	0.124329	−	0.992241i	$-0.460322\pi$
$619$	5.62215e14	0.248659	0.124329	+	0.992241i	$0.460322\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.00037e14	0.128083
$624$	0	0
$625$	2.11067e14	0.0885281
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.80046e14	0.113410
$630$	0	0
$631$	−2.01912e15	−0.803527	−0.401763	−	0.915744i	$-0.631603\pi$
$631$	−2.01912e15	−0.803527	−0.401763	+	0.915744i	$0.631603\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.43259e15	−0.550639
$636$	0	0
$637$	−5.10131e14	−0.192715
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.77556e15	−1.01305	−0.506526	−	0.862225i	$-0.669070\pi$
$641$	−2.77556e15	−1.01305	−0.506526	+	0.862225i	$0.669070\pi$
$642$	0	0
$643$	3.20918e15	1.15142	0.575711	−	0.817653i	$-0.304725\pi$
$643$	3.20918e15	1.15142	0.575711	+	0.817653i	$0.304725\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.15084e15	1.78609	0.893047	−	0.449963i	$-0.148563\pi$
$647$	5.15084e15	1.78609	0.893047	+	0.449963i	$0.148563\pi$
$648$	0	0
$649$	−1.96467e15	−0.669798
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.52018e15	1.48982	0.744909	−	0.667166i	$-0.232493\pi$
$653$	4.52018e15	1.48982	0.744909	+	0.667166i	$0.232493\pi$
$654$	0	0
$655$	5.19168e14	0.168260
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4.41803e15	−1.38471	−0.692355	−	0.721557i	$-0.743427\pi$
$659$	−4.41803e15	−1.38471	−0.692355	+	0.721557i	$0.743427\pi$
$660$	0	0
$661$	−1.72014e15	−0.530219	−0.265109	−	0.964218i	$-0.585408\pi$
$661$	−1.72014e15	−0.530219	−0.265109	+	0.964218i	$0.585408\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.93136e14	0.0575896
$666$	0	0
$667$	3.04654e15	0.893541
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.14376e14	0.117604
$672$	0	0
$673$	4.98000e15	1.39042	0.695210	−	0.718806i	$-0.255311\pi$
$673$	4.98000e15	1.39042	0.695210	+	0.718806i	$0.255311\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.68874e15	1.80762	0.903809	−	0.427937i	$-0.140760\pi$
$677$	6.68874e15	1.80762	0.903809	+	0.427937i	$0.140760\pi$
$678$	0	0
$679$	1.20552e15	0.320545
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.56856e15	0.918712	0.459356	−	0.888252i	$-0.348080\pi$
$683$	3.56856e15	0.918712	0.459356	+	0.888252i	$0.348080\pi$
$684$	0	0
$685$	−1.00572e15	−0.254789
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.29071e15	0.316682
$690$	0	0
$691$	2.14336e15	0.517565	0.258783	−	0.965936i	$-0.416679\pi$
$691$	2.14336e15	0.517565	0.258783	+	0.965936i	$0.416679\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.39374e14	−0.196354
$696$	0	0
$697$	−3.78733e14	−0.0872073
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.24854e15	−0.501709	−0.250854	−	0.968025i	$-0.580712\pi$
$701$	−2.24854e15	−0.501709	−0.250854	+	0.968025i	$0.580712\pi$
$702$	0	0
$703$	−1.48152e15	−0.325427
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.82635e14	−0.187920
$708$	0	0
$709$	5.22905e15	1.09615	0.548073	−	0.836430i	$-0.315362\pi$
$709$	5.22905e15	1.09615	0.548073	+	0.836430i	$0.315362\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.98856e14	0.182683
$714$	0	0
$715$	6.13871e15	1.22855
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.50990e15	−1.84572	−0.922862	−	0.385130i	$-0.874157\pi$
$719$	−9.50990e15	−1.84572	−0.922862	+	0.385130i	$0.874157\pi$
$720$	0	0
$721$	−1.49562e15	−0.285875
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.16281e15	−0.586428
$726$	0	0
$727$	6.89574e15	1.25934	0.629668	−	0.776864i	$-0.283191\pi$
$727$	6.89574e15	1.25934	0.629668	+	0.776864i	$0.283191\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.18012e14	0.0386308
$732$	0	0
$733$	2.94966e14	0.0514873	0.0257437	−	0.999669i	$-0.491805\pi$
$733$	2.94966e14	0.0514873	0.0257437	+	0.999669i	$0.491805\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.69819e15	−1.13471
$738$	0	0
$739$	4.86549e15	0.812049	0.406024	−	0.913862i	$-0.366915\pi$
$739$	4.86549e15	0.812049	0.406024	+	0.913862i	$0.366915\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.08184e15	−1.14738	−0.573691	−	0.819072i	$-0.694489\pi$
$743$	−7.08184e15	−1.14738	−0.573691	+	0.819072i	$0.694489\pi$
$744$	0	0
$745$	4.95033e15	0.790268
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.20841e14	−0.0962344
$750$	0	0
$751$	1.22105e16	1.86516	0.932578	−	0.360970i	$-0.117554\pi$
$751$	1.22105e16	1.86516	0.932578	+	0.360970i	$0.117554\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.89792e15	0.281559
$756$	0	0
$757$	−4.78568e14	−0.0699708	−0.0349854	−	0.999388i	$-0.511138\pi$
$757$	−4.78568e14	−0.0699708	−0.0349854	+	0.999388i	$0.511138\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.63155e15	1.08392	0.541960	−	0.840404i	$-0.317682\pi$
$761$	7.63155e15	1.08392	0.541960	+	0.840404i	$0.317682\pi$
$762$	0	0
$763$	2.60276e14	0.0364375
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.27199e15	0.581107
$768$	0	0
$769$	−2.41515e15	−0.323854	−0.161927	−	0.986803i	$-0.551771\pi$
$769$	−2.41515e15	−0.323854	−0.161927	+	0.986803i	$0.551771\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.18425e15	−1.19690	−0.598448	−	0.801162i	$-0.704216\pi$
$773$	−9.18425e15	−1.19690	−0.598448	+	0.801162i	$0.704216\pi$
$774$	0	0
$775$	−9.33163e14	−0.119894
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.00360e15	0.250239
$780$	0	0
$781$	−1.12185e15	−0.138151
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.42913e15	0.769784
$786$	0	0
$787$	3.66038e15	0.432181	0.216090	−	0.976373i	$-0.430669\pi$
$787$	3.66038e15	0.432181	0.216090	+	0.976373i	$0.430669\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.39714e15	0.619716
$792$	0	0
$793$	−9.01022e14	−0.102031
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.00946e16	1.11191	0.555955	−	0.831213i	$-0.312353\pi$
$797$	1.00946e16	1.11191	0.555955	+	0.831213i	$0.312353\pi$
$798$	0	0
$799$	−1.19444e15	−0.129765
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.99438e15	0.844985
$804$	0	0
$805$	−2.12539e15	−0.221596
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9.61973e15	−0.975991	−0.487996	−	0.872846i	$-0.662272\pi$
$809$	−9.61973e15	−0.975991	−0.487996	+	0.872846i	$0.662272\pi$
$810$	0	0
$811$	8.27986e15	0.828721	0.414360	−	0.910113i	$-0.364005\pi$
$811$	8.27986e15	0.828721	0.414360	+	0.910113i	$0.364005\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.11651e16	−1.08767
$816$	0	0
$817$	−1.15334e15	−0.110850
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.70462e14	0.0720882	0.0360441	−	0.999350i	$-0.488524\pi$
$821$	7.70462e14	0.0720882	0.0360441	+	0.999350i	$0.488524\pi$
$822$	0	0
$823$	9.99603e15	0.922845	0.461422	−	0.887181i	$-0.347339\pi$
$823$	9.99603e15	0.922845	0.461422	+	0.887181i	$0.347339\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.14907e15	0.283075	0.141538	−	0.989933i	$-0.454795\pi$
$827$	3.14907e15	0.283075	0.141538	+	0.989933i	$0.454795\pi$
$828$	0	0
$829$	1.01966e16	0.904492	0.452246	−	0.891893i	$-0.350623\pi$
$829$	1.01966e16	0.904492	0.452246	+	0.891893i	$0.350623\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.49920e14	0.0129513
$834$	0	0
$835$	−7.17851e15	−0.612010
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.25600e15	0.270392	0.135196	−	0.990819i	$-0.456834\pi$
$839$	3.25600e15	0.270392	0.135196	+	0.990819i	$0.456834\pi$
$840$	0	0
$841$	−2.47874e15	−0.203167
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6.01317e15	−0.480166
$846$	0	0
$847$	6.79818e15	0.535838
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.63036e16	1.25219
$852$	0	0
$853$	2.28981e15	0.173612	0.0868059	−	0.996225i	$-0.472334\pi$
$853$	2.28981e15	0.173612	0.0868059	+	0.996225i	$0.472334\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.66403e16	−1.22961	−0.614804	−	0.788680i	$-0.710765\pi$
$857$	−1.66403e16	−1.22961	−0.614804	+	0.788680i	$0.710765\pi$
$858$	0	0
$859$	1.53295e16	1.11832	0.559161	−	0.829059i	$-0.311124\pi$
$859$	1.53295e16	1.11832	0.559161	+	0.829059i	$0.311124\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.75654e15	0.338245	0.169123	−	0.985595i	$-0.445907\pi$
$863$	4.75654e15	0.338245	0.169123	+	0.985595i	$0.445907\pi$
$864$	0	0
$865$	8.71077e15	0.611601
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.17310e16	0.803019
$870$	0	0
$871$	1.45646e16	0.984460
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.56524e15	0.366809
$876$	0	0
$877$	−1.63252e16	−1.06258	−0.531290	−	0.847190i	$-0.678292\pi$
$877$	−1.63252e16	−1.06258	−0.531290	+	0.847190i	$0.678292\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.54149e16	−0.978527	−0.489263	−	0.872136i	$-0.662734\pi$
$881$	−1.54149e16	−0.978527	−0.489263	+	0.872136i	$0.662734\pi$
$882$	0	0
$883$	2.11017e15	0.132292	0.0661459	−	0.997810i	$-0.478930\pi$
$883$	2.11017e15	0.132292	0.0661459	+	0.997810i	$0.478930\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.88101e16	−1.15030	−0.575150	−	0.818048i	$-0.695057\pi$
$887$	−1.88101e16	−1.15030	−0.575150	+	0.818048i	$0.695057\pi$
$888$	0	0
$889$	5.88298e15	0.355335
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.31889e15	0.372355
$894$	0	0
$895$	−1.88332e16	−1.09622
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.86833e15	0.162911
$900$	0	0
$901$	−3.79322e14	−0.0212825
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.87674e16	−1.02763
$906$	0	0
$907$	−4.93810e15	−0.267128	−0.133564	−	0.991040i	$-0.542642\pi$
$907$	−4.93810e15	−0.267128	−0.133564	+	0.991040i	$0.542642\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.93620e16	1.02235	0.511173	−	0.859478i	$-0.329211\pi$
$911$	1.93620e16	1.02235	0.511173	+	0.859478i	$0.329211\pi$
$912$	0	0
$913$	−2.92707e16	−1.52702
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.13198e15	−0.108580
$918$	0	0
$919$	9.96111e15	0.501271	0.250635	−	0.968082i	$-0.419360\pi$
$919$	9.96111e15	0.501271	0.250635	+	0.968082i	$0.419360\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.43935e15	0.119857
$924$	0	0
$925$	−1.69259e16	−0.821809
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.34406e16	0.637283	0.318641	−	0.947875i	$-0.396773\pi$
$929$	1.34406e16	0.637283	0.318641	+	0.947875i	$0.396773\pi$
$930$	0	0
$931$	−7.93118e14	−0.0371633
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.80408e15	−0.0825643
$936$	0	0
$937$	−3.04621e16	−1.37782	−0.688909	−	0.724848i	$-0.741910\pi$
$937$	−3.04621e16	−1.37782	−0.688909	+	0.724848i	$0.741910\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.66381e16	1.17696	0.588479	−	0.808513i	$-0.299727\pi$
$941$	2.66381e16	1.17696	0.588479	+	0.808513i	$0.299727\pi$
$942$	0	0
$943$	−2.20489e16	−0.962880
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.37707e15	0.101418	0.0507092	−	0.998713i	$-0.483852\pi$
$947$	2.37707e15	0.101418	0.0507092	+	0.998713i	$0.483852\pi$
$948$	0	0
$949$	−1.73830e16	−0.733096
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.12063e16	−0.873885	−0.436943	−	0.899489i	$-0.643939\pi$
$953$	−2.12063e16	−0.873885	−0.436943	+	0.899489i	$0.643939\pi$
$954$	0	0
$955$	1.08958e16	0.443855
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.13003e15	0.164419
$960$	0	0
$961$	−2.45622e16	−0.966693
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.13331e15	−0.274404
$966$	0	0
$967$	3.68791e15	0.140260	0.0701301	−	0.997538i	$-0.477659\pi$
$967$	3.68791e15	0.140260	0.0701301	+	0.997538i	$0.477659\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.58950e16	−0.590954	−0.295477	−	0.955350i	$-0.595479\pi$
$971$	−1.58950e16	−0.590954	−0.295477	+	0.955350i	$0.595479\pi$
$972$	0	0
$973$	3.44691e15	0.126710
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.31393e16	0.472229	0.236115	−	0.971725i	$-0.424126\pi$
$977$	1.31393e16	0.472229	0.236115	+	0.971725i	$0.424126\pi$
$978$	0	0
$979$	1.48267e16	0.526914
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.40343e15	−0.0487692	−0.0243846	−	0.999703i	$-0.507763\pi$
$983$	−1.40343e15	−0.0487692	−0.0243846	+	0.999703i	$0.507763\pi$
$984$	0	0
$985$	−3.26723e16	−1.12274
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.26921e16	0.426534
$990$	0	0
$991$	4.27483e16	1.42074	0.710368	−	0.703830i	$-0.248528\pi$
$991$	4.27483e16	1.42074	0.710368	+	0.703830i	$0.248528\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.29370e16	−0.745606
$996$	0	0
$997$	−2.35197e16	−0.756151	−0.378076	−	0.925775i	$-0.623414\pi$
$997$	−2.35197e16	−0.756151	−0.378076	+	0.925775i	$0.623414\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.a.i.1.3	✓	6
3.2	odd	2	inner	252.12.a.i.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.12.a.i.1.3	✓	6	1.1	even	1	trivial
252.12.a.i.1.4	yes	6	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q-4092.75 q^{5} +16807.0 q^{7} +O(q^{10})\)	\(q-4092.75 q^{5} +16807.0 q^{7} +830540. q^{11} -1.80593e6 q^{13} +530738. q^{17} -2.80774e6 q^{19} +3.08982e7 q^{23} -3.20775e7 q^{25} +9.85991e7 q^{29} +2.90909e7 q^{31} -6.87868e7 q^{35} +5.27655e8 q^{37} -7.13596e8 q^{41} +4.10771e8 q^{43} -2.25052e9 q^{47} +2.82475e8 q^{49} -7.14707e8 q^{53} -3.39919e9 q^{55} -2.36553e9 q^{59} +4.98924e8 q^{61} +7.39122e9 q^{65} -8.06486e9 q^{67} -1.35074e9 q^{71} +9.62552e9 q^{73} +1.39589e10 q^{77} +1.41245e10 q^{79} -3.52430e10 q^{83} -2.17218e9 q^{85} +1.78519e10 q^{89} -3.03523e10 q^{91} +1.14914e10 q^{95} +7.17271e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 100842 q^{7}+O(q^{10}) \) 6 * q + 100842 * q^7	\( 6 q + 100842 q^{7} + 1227468 q^{13} - 6532728 q^{19} + 116108394 q^{25} + 293608488 q^{31} - 74878260 q^{37} + 1432813080 q^{43} + 1694851494 q^{49} + 6150010104 q^{55} - 11105239740 q^{61} - 3754055112 q^{67} - 31466324076 q^{73} - 58630542864 q^{79} + 94916392200 q^{85} + 20630054676 q^{91} - 6046820556 q^{97}+O(q^{100}) \) 6 * q + 100842 * q^7 + 1227468 * q^13 - 6532728 * q^19 + 116108394 * q^25 + 293608488 * q^31 - 74878260 * q^37 + 1432813080 * q^43 + 1694851494 * q^49 + 6150010104 * q^55 - 11105239740 * q^61 - 3754055112 * q^67 - 31466324076 * q^73 - 58630542864 * q^79 + 94916392200 * q^85 + 20630054676 * q^91 - 6046820556 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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