LMFDB - Embedded newform 1170.2.b.d.181.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(181,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1170.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.b (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:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.4
Root			$2.30278i$ of defining polynomial
Character	$\chi$	$=$	1170.181
Dual form			1170.2.b.d.181.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +4.60555i q^{7} -1.00000i q^{8} -1.00000 q^{10} +3.60555 q^{13} -4.60555 q^{14} +1.00000 q^{16} -4.60555 q^{17} +4.60555i q^{19} -1.00000i q^{20} +1.39445 q^{23} -1.00000 q^{25} +3.60555i q^{26} -4.60555i q^{28} -4.60555 q^{29} -6.00000i q^{31} +1.00000i q^{32} -4.60555i q^{34} -4.60555 q^{35} +9.21110i q^{37} -4.60555 q^{38} +1.00000 q^{40} +3.21110i q^{41} +8.00000 q^{43} +1.39445i q^{46} -9.21110i q^{47} -14.2111 q^{49} -1.00000i q^{50} -3.60555 q^{52} -6.00000 q^{53} +4.60555 q^{56} -4.60555i q^{58} -9.21110i q^{59} -11.2111 q^{61} +6.00000 q^{62} -1.00000 q^{64} +3.60555i q^{65} +3.21110i q^{67} +4.60555 q^{68} -4.60555i q^{70} +9.21110i q^{71} +1.39445i q^{73} -9.21110 q^{74} -4.60555i q^{76} -14.4222 q^{79} +1.00000i q^{80} -3.21110 q^{82} +2.78890i q^{83} -4.60555i q^{85} +8.00000i q^{86} +15.2111i q^{89} +16.6056i q^{91} -1.39445 q^{92} +9.21110 q^{94} -4.60555 q^{95} -1.39445i q^{97} -14.2111i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{10} - 4 q^{14} + 4 q^{16} - 4 q^{17} + 20 q^{23} - 4 q^{25} - 4 q^{29} - 4 q^{35} - 4 q^{38} + 4 q^{40} + 32 q^{43} - 28 q^{49} - 24 q^{53} + 4 q^{56} - 16 q^{61} + 24 q^{62} - 4 q^{64}+ \cdots - 4 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^10 - 4 * q^14 + 4 * q^16 - 4 * q^17 + 20 * q^23 - 4 * q^25 - 4 * q^29 - 4 * q^35 - 4 * q^38 + 4 * q^40 + 32 * q^43 - 28 * q^49 - 24 * q^53 + 4 * q^56 - 16 * q^61 + 24 * q^62 - 4 * q^64 + 4 * q^68 - 8 * q^74 + 16 * q^82 - 20 * q^92 + 8 * q^94 - 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	4.60555i	1.74073i	0.492403	+	0.870367i	$0.336119\pi$
$7$	4.60555i	1.74073i	−0.492403	+	0.870367i	$0.663881\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	3.60555	1.00000
$13$	3.60555	1.00000
$14$	−4.60555	−1.23089
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.60555	−1.11701	−0.558505	−	0.829501i	$-0.688625\pi$
$17$	−4.60555	−1.11701	−0.558505	+	0.829501i	$0.688625\pi$
$18$	0	0
$19$	4.60555i	1.05659i	0.849062	+	0.528293i	$0.177168\pi$
$19$	4.60555i	1.05659i	−0.849062	+	0.528293i	$0.822832\pi$
$20$	− 1.00000i	− 0.223607i
$21$	0	0
$22$	0	0
$23$	1.39445	0.290763	0.145381	−	0.989376i	$-0.453559\pi$
$23$	1.39445	0.290763	0.145381	+	0.989376i	$0.453559\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	3.60555i	0.707107i
$27$	0	0
$28$	− 4.60555i	− 0.870367i
$29$	−4.60555	−0.855229	−0.427615	−	0.903961i	$-0.640646\pi$
$29$	−4.60555	−0.855229	−0.427615	+	0.903961i	$0.640646\pi$
$30$	0	0
$31$	− 6.00000i	− 1.07763i	−0.842424	−	0.538816i	$-0.818872\pi$
$31$	− 6.00000i	− 1.07763i	0.842424	−	0.538816i	$-0.181128\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 4.60555i	− 0.789846i
$35$	−4.60555	−0.778480
$36$	0	0
$37$	9.21110i	1.51430i	0.653243	+	0.757148i	$0.273408\pi$
$37$	9.21110i	1.51430i	−0.653243	+	0.757148i	$0.726592\pi$
$38$	−4.60555	−0.747119
$39$	0	0
$40$	1.00000	0.158114
$41$	3.21110i	0.501490i	0.968053	+	0.250745i	$0.0806756\pi$
$41$	3.21110i	0.501490i	−0.968053	+	0.250745i	$0.919324\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	1.39445i	0.205600i
$47$	− 9.21110i	− 1.34358i	−0.740743	−	0.671789i	$-0.765526\pi$
$47$	− 9.21110i	− 1.34358i	0.740743	−	0.671789i	$-0.234474\pi$
$48$	0	0
$49$	−14.2111	−2.03016
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	−3.60555	−0.500000
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	4.60555	0.615443
$57$	0	0
$58$	− 4.60555i	− 0.604739i
$59$	− 9.21110i	− 1.19918i	−0.800306	−	0.599592i	$-0.795330\pi$
$59$	− 9.21110i	− 1.19918i	0.800306	−	0.599592i	$-0.204670\pi$
$60$	0	0
$61$	−11.2111	−1.43543	−0.717717	−	0.696335i	$-0.754813\pi$
$61$	−11.2111	−1.43543	−0.717717	+	0.696335i	$0.754813\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	−1.00000	−0.125000
$65$	3.60555i	0.447214i
$66$	0	0
$67$	3.21110i	0.392299i	0.980574	+	0.196149i	$0.0628437\pi$
$67$	3.21110i	0.392299i	−0.980574	+	0.196149i	$0.937156\pi$
$68$	4.60555	0.558505
$69$	0	0
$70$	− 4.60555i	− 0.550469i
$71$	9.21110i	1.09316i	0.837408	+	0.546578i	$0.184070\pi$
$71$	9.21110i	1.09316i	−0.837408	+	0.546578i	$0.815930\pi$
$72$	0	0
$73$	1.39445i	0.163208i	0.996665	+	0.0816039i	$0.0260043\pi$
$73$	1.39445i	0.163208i	−0.996665	+	0.0816039i	$0.973996\pi$
$74$	−9.21110	−1.07077
$75$	0	0
$76$	− 4.60555i	− 0.528293i
$77$	0	0
$78$	0	0
$79$	−14.4222	−1.62262	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	−14.4222	−1.62262	−0.811312	+	0.584613i	$0.801246\pi$
$80$	1.00000i	0.111803i
$81$	0	0
$82$	−3.21110	−0.354607
$83$	2.78890i	0.306121i	0.988217	+	0.153061i	$0.0489130\pi$
$83$	2.78890i	0.306121i	−0.988217	+	0.153061i	$0.951087\pi$
$84$	0	0
$85$	− 4.60555i	− 0.499542i
$86$	8.00000i	0.862662i
$87$	0	0
$88$	0	0
$89$	15.2111i	1.61237i	0.591661	+	0.806187i	$0.298472\pi$
$89$	15.2111i	1.61237i	−0.591661	+	0.806187i	$0.701528\pi$
$90$	0	0
$91$	16.6056i	1.74073i
$92$	−1.39445	−0.145381
$93$	0	0
$94$	9.21110	0.950053
$95$	−4.60555	−0.472520
$96$	0	0
$97$	− 1.39445i	− 0.141585i	−0.997491	−	0.0707924i	$-0.977447\pi$
$97$	− 1.39445i	− 0.141585i	0.997491	−	0.0707924i	$-0.0225528\pi$
$98$	− 14.2111i	− 1.43554i
$99$	0	0
$100$	1.00000	0.100000
$101$	7.39445	0.735775	0.367888	−	0.929870i	$-0.380081\pi$
$101$	7.39445	0.735775	0.367888	+	0.929870i	$0.380081\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	− 3.60555i	− 0.353553i
$105$	0	0
$106$	− 6.00000i	− 0.582772i
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	− 1.39445i	− 0.133564i	−0.997768	−	0.0667820i	$-0.978727\pi$
$109$	− 1.39445i	− 0.133564i	0.997768	−	0.0667820i	$-0.0212732\pi$
$110$	0	0
$111$	0	0
$112$	4.60555i	0.435184i
$113$	13.8167	1.29976	0.649881	−	0.760036i	$-0.274819\pi$
$113$	13.8167	1.29976	0.649881	+	0.760036i	$0.274819\pi$
$114$	0	0
$115$	1.39445i	0.130033i
$116$	4.60555	0.427615
$117$	0	0
$118$	9.21110	0.847951
$119$	− 21.2111i	− 1.94442i
$120$	0	0
$121$	11.0000	1.00000
$122$	− 11.2111i	− 1.01501i
$123$	0	0
$124$	6.00000i	0.538816i
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	1.21110	0.107468	0.0537340	−	0.998555i	$-0.482888\pi$
$127$	1.21110	0.107468	0.0537340	+	0.998555i	$0.482888\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	−3.60555	−0.316228
$131$	−22.6056	−1.97506	−0.987528	−	0.157443i	$-0.949675\pi$
$131$	−22.6056	−1.97506	−0.987528	+	0.157443i	$0.949675\pi$
$132$	0	0
$133$	−21.2111	−1.83924
$134$	−3.21110	−0.277397
$135$	0	0
$136$	4.60555i	0.394923i
$137$	3.21110i	0.274343i	0.990547	+	0.137172i	$0.0438011\pi$
$137$	3.21110i	0.274343i	−0.990547	+	0.137172i	$0.956199\pi$
$138$	0	0
$139$	17.2111	1.45983	0.729913	−	0.683540i	$-0.239560\pi$
$139$	17.2111	1.45983	0.729913	+	0.683540i	$0.239560\pi$
$140$	4.60555	0.389240
$141$	0	0
$142$	−9.21110	−0.772979
$143$	0	0
$144$	0	0
$145$	− 4.60555i	− 0.382470i
$146$	−1.39445	−0.115405
$147$	0	0
$148$	− 9.21110i	− 0.757148i
$149$	15.2111i	1.24614i	0.782165	+	0.623071i	$0.214115\pi$
$149$	15.2111i	1.24614i	−0.782165	+	0.623071i	$0.785885\pi$
$150$	0	0
$151$	− 6.00000i	− 0.488273i	−0.969741	−	0.244137i	$-0.921495\pi$
$151$	− 6.00000i	− 0.488273i	0.969741	−	0.244137i	$-0.0785045\pi$
$152$	4.60555	0.373560
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	20.4222	1.62987	0.814935	−	0.579553i	$-0.196773\pi$
$157$	20.4222	1.62987	0.814935	+	0.579553i	$0.196773\pi$
$158$	− 14.4222i	− 1.14737i
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	6.42221i	0.506141i
$162$	0	0
$163$	− 24.4222i	− 1.91289i	−0.291905	−	0.956447i	$-0.594289\pi$
$163$	− 24.4222i	− 1.91289i	0.291905	−	0.956447i	$-0.405711\pi$
$164$	− 3.21110i	− 0.250745i
$165$	0	0
$166$	−2.78890	−0.216460
$167$	9.21110i	0.712777i	0.934338	+	0.356388i	$0.115992\pi$
$167$	9.21110i	0.712777i	−0.934338	+	0.356388i	$0.884008\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	4.60555	0.353230
$171$	0	0
$172$	−8.00000	−0.609994
$173$	12.4222	0.944443	0.472221	−	0.881480i	$-0.343452\pi$
$173$	12.4222	0.944443	0.472221	+	0.881480i	$0.343452\pi$
$174$	0	0
$175$	− 4.60555i	− 0.348147i
$176$	0	0
$177$	0	0
$178$	−15.2111	−1.14012
$179$	19.8167	1.48117	0.740583	−	0.671965i	$-0.234549\pi$
$179$	19.8167	1.48117	0.740583	+	0.671965i	$0.234549\pi$
$180$	0	0
$181$	8.42221	0.626018	0.313009	−	0.949750i	$-0.398663\pi$
$181$	8.42221	0.626018	0.313009	+	0.949750i	$0.398663\pi$
$182$	−16.6056	−1.23089
$183$	0	0
$184$	− 1.39445i	− 0.102800i
$185$	−9.21110	−0.677214
$186$	0	0
$187$	0	0
$188$	9.21110i	0.671789i
$189$	0	0
$190$	− 4.60555i	− 0.334122i
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	− 7.81665i	− 0.562655i	−0.959612	−	0.281328i	$-0.909225\pi$
$193$	− 7.81665i	− 0.562655i	0.959612	−	0.281328i	$-0.0907747\pi$
$194$	1.39445	0.100116
$195$	0	0
$196$	14.2111	1.01508
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	22.4222	1.58947	0.794734	−	0.606958i	$-0.207610\pi$
$199$	22.4222	1.58947	0.794734	+	0.606958i	$0.207610\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	7.39445i	0.520272i
$203$	− 21.2111i	− 1.48873i
$204$	0	0
$205$	−3.21110	−0.224273
$206$	− 4.00000i	− 0.278693i
$207$	0	0
$208$	3.60555	0.250000
$209$	0	0
$210$	0	0
$211$	−17.2111	−1.18486	−0.592431	−	0.805622i	$-0.701832\pi$
$211$	−17.2111	−1.18486	−0.592431	+	0.805622i	$0.701832\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	0	0
$215$	8.00000i	0.545595i
$216$	0	0
$217$	27.6333	1.87587
$218$	1.39445	0.0944440
$219$	0	0
$220$	0	0
$221$	−16.6056	−1.11701
$222$	0	0
$223$	− 1.81665i	− 0.121652i	−0.998148	−	0.0608261i	$-0.980627\pi$
$223$	− 1.81665i	− 0.121652i	0.998148	−	0.0608261i	$-0.0193735\pi$
$224$	−4.60555	−0.307721
$225$	0	0
$226$	13.8167i	0.919070i
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	19.8167i	1.30952i	0.755836	+	0.654761i	$0.227231\pi$
$229$	19.8167i	1.30952i	−0.755836	+	0.654761i	$0.772769\pi$
$230$	−1.39445	−0.0919472
$231$	0	0
$232$	4.60555i	0.302369i
$233$	1.81665	0.119013	0.0595065	−	0.998228i	$-0.481047\pi$
$233$	1.81665	0.119013	0.0595065	+	0.998228i	$0.481047\pi$
$234$	0	0
$235$	9.21110	0.600866
$236$	9.21110i	0.599592i
$237$	0	0
$238$	21.2111	1.37491
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	6.42221i	0.413691i	0.978374	+	0.206845i	$0.0663197\pi$
$241$	6.42221i	0.413691i	−0.978374	+	0.206845i	$0.933680\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	11.2111	0.717717
$245$	− 14.2111i	− 0.907914i
$246$	0	0
$247$	16.6056i	1.05659i
$248$	−6.00000	−0.381000
$249$	0	0
$250$	1.00000	0.0632456
$251$	13.3944	0.845450	0.422725	−	0.906258i	$-0.361074\pi$
$251$	13.3944	0.845450	0.422725	+	0.906258i	$0.361074\pi$
$252$	0	0
$253$	0	0
$254$	1.21110i	0.0759913i
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.6056	1.78437	0.892183	−	0.451675i	$-0.149173\pi$
$257$	28.6056	1.78437	0.892183	+	0.451675i	$0.149173\pi$
$258$	0	0
$259$	−42.4222	−2.63599
$260$	− 3.60555i	− 0.223607i
$261$	0	0
$262$	− 22.6056i	− 1.39658i
$263$	7.81665	0.481996	0.240998	−	0.970526i	$-0.422525\pi$
$263$	7.81665	0.481996	0.240998	+	0.970526i	$0.422525\pi$
$264$	0	0
$265$	− 6.00000i	− 0.368577i
$266$	− 21.2111i	− 1.30054i
$267$	0	0
$268$	− 3.21110i	− 0.196149i
$269$	−25.8167	−1.57407	−0.787035	−	0.616909i	$-0.788385\pi$
$269$	−25.8167	−1.57407	−0.787035	+	0.616909i	$0.788385\pi$
$270$	0	0
$271$	0.422205i	0.0256471i	0.999918	+	0.0128236i	$0.00408198\pi$
$271$	0.422205i	0.0256471i	−0.999918	+	0.0128236i	$0.995918\pi$
$272$	−4.60555	−0.279253
$273$	0	0
$274$	−3.21110	−0.193990
$275$	0	0
$276$	0	0
$277$	−16.4222	−0.986715	−0.493357	−	0.869827i	$-0.664230\pi$
$277$	−16.4222	−0.986715	−0.493357	+	0.869827i	$0.664230\pi$
$278$	17.2111i	1.03225i
$279$	0	0
$280$	4.60555i	0.275234i
$281$	27.2111i	1.62328i	0.584159	+	0.811639i	$0.301424\pi$
$281$	27.2111i	1.62328i	−0.584159	+	0.811639i	$0.698576\pi$
$282$	0	0
$283$	10.4222	0.619536	0.309768	−	0.950812i	$-0.399749\pi$
$283$	10.4222	0.619536	0.309768	+	0.950812i	$0.399749\pi$
$284$	− 9.21110i	− 0.546578i
$285$	0	0
$286$	0	0
$287$	−14.7889	−0.872961
$288$	0	0
$289$	4.21110	0.247712
$290$	4.60555	0.270447
$291$	0	0
$292$	− 1.39445i	− 0.0816039i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	9.21110	0.536291
$296$	9.21110	0.535384
$297$	0	0
$298$	−15.2111	−0.881156
$299$	5.02776	0.290763
$300$	0	0
$301$	36.8444i	2.12368i
$302$	6.00000	0.345261
$303$	0	0
$304$	4.60555i	0.264146i
$305$	− 11.2111i	− 0.641946i
$306$	0	0
$307$	8.78890i	0.501609i	0.968038	+	0.250804i	$0.0806951\pi$
$307$	8.78890i	0.501609i	−0.968038	+	0.250804i	$0.919305\pi$
$308$	0	0
$309$	0	0
$310$	6.00000i	0.340777i
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	3.57779	0.202229	0.101114	−	0.994875i	$-0.467759\pi$
$313$	3.57779	0.202229	0.101114	+	0.994875i	$0.467759\pi$
$314$	20.4222i	1.15249i
$315$	0	0
$316$	14.4222	0.811312
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	0	0
$320$	− 1.00000i	− 0.0559017i
$321$	0	0
$322$	−6.42221	−0.357895
$323$	− 21.2111i	− 1.18022i
$324$	0	0
$325$	−3.60555	−0.200000
$326$	24.4222	1.35262
$327$	0	0
$328$	3.21110	0.177303
$329$	42.4222	2.33881
$330$	0	0
$331$	16.6056i	0.912724i	0.889794	+	0.456362i	$0.150848\pi$
$331$	16.6056i	0.912724i	−0.889794	+	0.456362i	$0.849152\pi$
$332$	− 2.78890i	− 0.153061i
$333$	0	0
$334$	−9.21110	−0.504009
$335$	−3.21110	−0.175441
$336$	0	0
$337$	13.6333	0.742654	0.371327	−	0.928502i	$-0.378903\pi$
$337$	13.6333	0.742654	0.371327	+	0.928502i	$0.378903\pi$
$338$	13.0000i	0.707107i
$339$	0	0
$340$	4.60555i	0.249771i
$341$	0	0
$342$	0	0
$343$	− 33.2111i	− 1.79323i
$344$	− 8.00000i	− 0.431331i
$345$	0	0
$346$	12.4222i	0.667822i
$347$	27.6333	1.48343	0.741717	−	0.670713i	$-0.234012\pi$
$347$	27.6333	1.48343	0.741717	+	0.670713i	$0.234012\pi$
$348$	0	0
$349$	7.81665i	0.418416i	0.977871	+	0.209208i	$0.0670886\pi$
$349$	7.81665i	0.418416i	−0.977871	+	0.209208i	$0.932911\pi$
$350$	4.60555	0.246177
$351$	0	0
$352$	0	0
$353$	− 8.78890i	− 0.467786i	−0.972262	−	0.233893i	$-0.924853\pi$
$353$	− 8.78890i	− 0.467786i	0.972262	−	0.233893i	$-0.0751465\pi$
$354$	0	0
$355$	−9.21110	−0.488875
$356$	− 15.2111i	− 0.806187i
$357$	0	0
$358$	19.8167i	1.04734i
$359$	15.6333i	0.825094i	0.910936	+	0.412547i	$0.135361\pi$
$359$	15.6333i	0.825094i	−0.910936	+	0.412547i	$0.864639\pi$
$360$	0	0
$361$	−2.21110	−0.116374
$362$	8.42221i	0.442661i
$363$	0	0
$364$	− 16.6056i	− 0.870367i
$365$	−1.39445	−0.0729888
$366$	0	0
$367$	−19.6333	−1.02485	−0.512425	−	0.858732i	$-0.671253\pi$
$367$	−19.6333	−1.02485	−0.512425	+	0.858732i	$0.671253\pi$
$368$	1.39445	0.0726907
$369$	0	0
$370$	− 9.21110i	− 0.478862i
$371$	− 27.6333i	− 1.43465i
$372$	0	0
$373$	−20.4222	−1.05742	−0.528711	−	0.848802i	$-0.677324\pi$
$373$	−20.4222	−1.05742	−0.528711	+	0.848802i	$0.677324\pi$
$374$	0	0
$375$	0	0
$376$	−9.21110	−0.475026
$377$	−16.6056	−0.855229
$378$	0	0
$379$	− 35.0278i	− 1.79925i	−0.436658	−	0.899627i	$-0.643838\pi$
$379$	− 35.0278i	− 1.79925i	0.436658	−	0.899627i	$-0.356162\pi$
$380$	4.60555	0.236260
$381$	0	0
$382$	12.0000i	0.613973i
$383$	− 27.6333i	− 1.41200i	−0.708214	−	0.705998i	$-0.750499\pi$
$383$	− 27.6333i	− 1.41200i	0.708214	−	0.705998i	$-0.249501\pi$
$384$	0	0
$385$	0	0
$386$	7.81665	0.397857
$387$	0	0
$388$	1.39445i	0.0707924i
$389$	−4.60555	−0.233511	−0.116755	−	0.993161i	$-0.537249\pi$
$389$	−4.60555	−0.233511	−0.116755	+	0.993161i	$0.537249\pi$
$390$	0	0
$391$	−6.42221	−0.324785
$392$	14.2111i	0.717769i
$393$	0	0
$394$	6.00000	0.302276
$395$	− 14.4222i	− 0.725660i
$396$	0	0
$397$	3.63331i	0.182350i	0.995835	+	0.0911752i	$0.0290623\pi$
$397$	3.63331i	0.182350i	−0.995835	+	0.0911752i	$0.970938\pi$
$398$	22.4222i	1.12392i
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	8.78890i	0.438897i	0.975624	+	0.219448i	$0.0704257\pi$
$401$	8.78890i	0.438897i	−0.975624	+	0.219448i	$0.929574\pi$
$402$	0	0
$403$	− 21.6333i	− 1.07763i
$404$	−7.39445	−0.367888
$405$	0	0
$406$	21.2111	1.05269
$407$	0	0
$408$	0	0
$409$	14.7889i	0.731264i	0.930760	+	0.365632i	$0.119147\pi$
$409$	14.7889i	0.731264i	−0.930760	+	0.365632i	$0.880853\pi$
$410$	− 3.21110i	− 0.158585i
$411$	0	0
$412$	4.00000	0.197066
$413$	42.4222	2.08746
$414$	0	0
$415$	−2.78890	−0.136902
$416$	3.60555i	0.176777i
$417$	0	0
$418$	0	0
$419$	−4.18335	−0.204370	−0.102185	−	0.994765i	$-0.532583\pi$
$419$	−4.18335	−0.204370	−0.102185	+	0.994765i	$0.532583\pi$
$420$	0	0
$421$	19.8167i	0.965805i	0.875674	+	0.482902i	$0.160417\pi$
$421$	19.8167i	0.965805i	−0.875674	+	0.482902i	$0.839583\pi$
$422$	− 17.2111i	− 0.837823i
$423$	0	0
$424$	6.00000i	0.291386i
$425$	4.60555	0.223402
$426$	0	0
$427$	− 51.6333i	− 2.49871i
$428$	0	0
$429$	0	0
$430$	−8.00000	−0.385794
$431$	12.0000i	0.578020i	0.957326	+	0.289010i	$0.0933260\pi$
$431$	12.0000i	0.578020i	−0.957326	+	0.289010i	$0.906674\pi$
$432$	0	0
$433$	19.2111	0.923227	0.461613	−	0.887081i	$-0.347271\pi$
$433$	19.2111	0.923227	0.461613	+	0.887081i	$0.347271\pi$
$434$	27.6333i	1.32644i
$435$	0	0
$436$	1.39445i	0.0667820i
$437$	6.42221i	0.307216i
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	− 16.6056i	− 0.789846i
$443$	−15.6333	−0.742761	−0.371380	−	0.928481i	$-0.621115\pi$
$443$	−15.6333	−0.742761	−0.371380	+	0.928481i	$0.621115\pi$
$444$	0	0
$445$	−15.2111	−0.721075
$446$	1.81665	0.0860211
$447$	0	0
$448$	− 4.60555i	− 0.217592i
$449$	33.6333i	1.58725i	0.608405	+	0.793627i	$0.291810\pi$
$449$	33.6333i	1.58725i	−0.608405	+	0.793627i	$0.708190\pi$
$450$	0	0
$451$	0	0
$452$	−13.8167	−0.649881
$453$	0	0
$454$	−24.0000	−1.12638
$455$	−16.6056	−0.778480
$456$	0	0
$457$	− 38.2389i	− 1.78874i	−0.447330	−	0.894369i	$-0.647625\pi$
$457$	− 38.2389i	− 1.78874i	0.447330	−	0.894369i	$-0.352375\pi$
$458$	−19.8167	−0.925971
$459$	0	0
$460$	− 1.39445i	− 0.0650165i
$461$	− 33.6333i	− 1.56646i	−0.621733	−	0.783230i	$-0.713571\pi$
$461$	− 33.6333i	− 1.56646i	0.621733	−	0.783230i	$-0.286429\pi$
$462$	0	0
$463$	31.3944i	1.45902i	0.683968	+	0.729512i	$0.260253\pi$
$463$	31.3944i	1.45902i	−0.683968	+	0.729512i	$0.739747\pi$
$464$	−4.60555	−0.213807
$465$	0	0
$466$	1.81665i	0.0841549i
$467$	30.4222	1.40777	0.703886	−	0.710313i	$-0.251447\pi$
$467$	30.4222	1.40777	0.703886	+	0.710313i	$0.251447\pi$
$468$	0	0
$469$	−14.7889	−0.682888
$470$	9.21110i	0.424876i
$471$	0	0
$472$	−9.21110	−0.423975
$473$	0	0
$474$	0	0
$475$	− 4.60555i	− 0.211317i
$476$	21.2111i	0.972209i
$477$	0	0
$478$	0	0
$479$	5.57779i	0.254856i	0.991848	+	0.127428i	$0.0406722\pi$
$479$	5.57779i	0.254856i	−0.991848	+	0.127428i	$0.959328\pi$
$480$	0	0
$481$	33.2111i	1.51430i
$482$	−6.42221	−0.292523
$483$	0	0
$484$	−11.0000	−0.500000
$485$	1.39445	0.0633187
$486$	0	0
$487$	0.972244i	0.0440566i	0.999757	+	0.0220283i	$0.00701239\pi$
$487$	0.972244i	0.0440566i	−0.999757	+	0.0220283i	$0.992988\pi$
$488$	11.2111i	0.507503i
$489$	0	0
$490$	14.2111	0.641992
$491$	−7.81665	−0.352761	−0.176380	−	0.984322i	$-0.556439\pi$
$491$	−7.81665	−0.352761	−0.176380	+	0.984322i	$0.556439\pi$
$492$	0	0
$493$	21.2111	0.955300
$494$	−16.6056	−0.747119
$495$	0	0
$496$	− 6.00000i	− 0.269408i
$497$	−42.4222	−1.90290
$498$	0	0
$499$	− 23.0278i	− 1.03086i	−0.856930	−	0.515432i	$-0.827631\pi$
$499$	− 23.0278i	− 1.03086i	0.856930	−	0.515432i	$-0.172369\pi$
$500$	1.00000i	0.0447214i
$501$	0	0
$502$	13.3944i	0.597824i
$503$	−23.4500	−1.04558	−0.522791	−	0.852461i	$-0.675109\pi$
$503$	−23.4500	−1.04558	−0.522791	+	0.852461i	$0.675109\pi$
$504$	0	0
$505$	7.39445i	0.329049i
$506$	0	0
$507$	0	0
$508$	−1.21110	−0.0537340
$509$	− 33.6333i	− 1.49077i	−0.666634	−	0.745385i	$-0.732266\pi$
$509$	− 33.6333i	− 1.49077i	0.666634	−	0.745385i	$-0.267734\pi$
$510$	0	0
$511$	−6.42221	−0.284102
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	28.6056i	1.26174i
$515$	− 4.00000i	− 0.176261i
$516$	0	0
$517$	0	0
$518$	− 42.4222i	− 1.86392i
$519$	0	0
$520$	3.60555	0.158114
$521$	−21.6333	−0.947772	−0.473886	−	0.880586i	$-0.657149\pi$
$521$	−21.6333	−0.947772	−0.473886	+	0.880586i	$0.657149\pi$
$522$	0	0
$523$	32.8444	1.43619	0.718093	−	0.695947i	$-0.245015\pi$
$523$	32.8444	1.43619	0.718093	+	0.695947i	$0.245015\pi$
$524$	22.6056	0.987528
$525$	0	0
$526$	7.81665i	0.340822i
$527$	27.6333i	1.20373i
$528$	0	0
$529$	−21.0555	−0.915457
$530$	6.00000	0.260623
$531$	0	0
$532$	21.2111	0.919618
$533$	11.5778i	0.501490i
$534$	0	0
$535$	0	0
$536$	3.21110	0.138699
$537$	0	0
$538$	− 25.8167i	− 1.11303i
$539$	0	0
$540$	0	0
$541$	− 6.97224i	− 0.299760i	−0.988704	−	0.149880i	$-0.952111\pi$
$541$	− 6.97224i	− 0.299760i	0.988704	−	0.149880i	$-0.0478888\pi$
$542$	−0.422205	−0.0181353
$543$	0	0
$544$	− 4.60555i	− 0.197461i
$545$	1.39445	0.0597316
$546$	0	0
$547$	14.4222	0.616649	0.308324	−	0.951281i	$-0.400232\pi$
$547$	14.4222	0.616649	0.308324	+	0.951281i	$0.400232\pi$
$548$	− 3.21110i	− 0.137172i
$549$	0	0
$550$	0	0
$551$	− 21.2111i	− 0.903623i
$552$	0	0
$553$	− 66.4222i	− 2.82456i
$554$	− 16.4222i	− 0.697713i
$555$	0	0
$556$	−17.2111	−0.729913
$557$	11.5778i	0.490567i	0.969451	+	0.245283i	$0.0788810\pi$
$557$	11.5778i	0.490567i	−0.969451	+	0.245283i	$0.921119\pi$
$558$	0	0
$559$	28.8444	1.21999
$560$	−4.60555	−0.194620
$561$	0	0
$562$	−27.2111	−1.14783
$563$	−34.0555	−1.43527	−0.717634	−	0.696420i	$-0.754775\pi$
$563$	−34.0555	−1.43527	−0.717634	+	0.696420i	$0.754775\pi$
$564$	0	0
$565$	13.8167i	0.581271i
$566$	10.4222i	0.438078i
$567$	0	0
$568$	9.21110	0.386489
$569$	−33.6333	−1.40998	−0.704991	−	0.709216i	$-0.749049\pi$
$569$	−33.6333	−1.40998	−0.704991	+	0.709216i	$0.749049\pi$
$570$	0	0
$571$	−30.0555	−1.25778	−0.628892	−	0.777493i	$-0.716491\pi$
$571$	−30.0555	−1.25778	−0.628892	+	0.777493i	$0.716491\pi$
$572$	0	0
$573$	0	0
$574$	− 14.7889i	− 0.617277i
$575$	−1.39445	−0.0581525
$576$	0	0
$577$	37.3944i	1.55675i	0.627799	+	0.778376i	$0.283956\pi$
$577$	37.3944i	1.55675i	−0.627799	+	0.778376i	$0.716044\pi$
$578$	4.21110i	0.175159i
$579$	0	0
$580$	4.60555i	0.191235i
$581$	−12.8444	−0.532876
$582$	0	0
$583$	0	0
$584$	1.39445	0.0577027
$585$	0	0
$586$	−18.0000	−0.743573
$587$	− 6.42221i	− 0.265073i	−0.991178	−	0.132536i	$-0.957688\pi$
$587$	− 6.42221i	− 0.265073i	0.991178	−	0.132536i	$-0.0423121\pi$
$588$	0	0
$589$	27.6333	1.13861
$590$	9.21110i	0.379215i
$591$	0	0
$592$	9.21110i	0.378574i
$593$	− 24.4222i	− 1.00290i	−0.865187	−	0.501450i	$-0.832800\pi$
$593$	− 24.4222i	− 1.00290i	0.865187	−	0.501450i	$-0.167200\pi$
$594$	0	0
$595$	21.2111	0.869570
$596$	− 15.2111i	− 0.623071i
$597$	0	0
$598$	5.02776i	0.205600i
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	1.63331	0.0666240	0.0333120	−	0.999445i	$-0.489394\pi$
$601$	1.63331	0.0666240	0.0333120	+	0.999445i	$0.489394\pi$
$602$	−36.8444	−1.50167
$603$	0	0
$604$	6.00000i	0.244137i
$605$	11.0000i	0.447214i
$606$	0	0
$607$	17.2111	0.698577	0.349289	−	0.937015i	$-0.386423\pi$
$607$	17.2111	0.698577	0.349289	+	0.937015i	$0.386423\pi$
$608$	−4.60555	−0.186780
$609$	0	0
$610$	11.2111	0.453924
$611$	− 33.2111i	− 1.34358i
$612$	0	0
$613$	33.2111i	1.34138i	0.741736	+	0.670692i	$0.234003\pi$
$613$	33.2111i	1.34138i	−0.741736	+	0.670692i	$0.765997\pi$
$614$	−8.78890	−0.354691
$615$	0	0
$616$	0	0
$617$	12.4222i	0.500099i	0.968233	+	0.250050i	$0.0804469\pi$
$617$	12.4222i	0.500099i	−0.968233	+	0.250050i	$0.919553\pi$
$618$	0	0
$619$	− 25.8167i	− 1.03766i	−0.854878	−	0.518829i	$-0.826368\pi$
$619$	− 25.8167i	− 1.03766i	0.854878	−	0.518829i	$-0.173632\pi$
$620$	−6.00000	−0.240966
$621$	0	0
$622$	− 12.0000i	− 0.481156i
$623$	−70.0555	−2.80671
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.57779i	0.142997i
$627$	0	0
$628$	−20.4222	−0.814935
$629$	− 42.4222i	− 1.69148i
$630$	0	0
$631$	− 3.21110i	− 0.127832i	−0.997955	−	0.0639160i	$-0.979641\pi$
$631$	− 3.21110i	− 0.127832i	0.997955	−	0.0639160i	$-0.0203590\pi$
$632$	14.4222i	0.573685i
$633$	0	0
$634$	18.0000	0.714871
$635$	1.21110i	0.0480611i
$636$	0	0
$637$	−51.2389	−2.03016
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	−0.422205	−0.0166761	−0.00833805	−	0.999965i	$-0.502654\pi$
$641$	−0.422205	−0.0166761	−0.00833805	+	0.999965i	$0.502654\pi$
$642$	0	0
$643$	− 9.63331i	− 0.379901i	−0.981794	−	0.189950i	$-0.939167\pi$
$643$	− 9.63331i	− 0.379901i	0.981794	−	0.189950i	$-0.0608327\pi$
$644$	− 6.42221i	− 0.253070i
$645$	0	0
$646$	21.2111	0.834540
$647$	34.6056	1.36048	0.680242	−	0.732987i	$-0.261875\pi$
$647$	34.6056	1.36048	0.680242	+	0.732987i	$0.261875\pi$
$648$	0	0
$649$	0	0
$650$	− 3.60555i	− 0.141421i
$651$	0	0
$652$	24.4222i	0.956447i
$653$	−39.2111	−1.53445	−0.767225	−	0.641379i	$-0.778363\pi$
$653$	−39.2111	−1.53445	−0.767225	+	0.641379i	$0.778363\pi$
$654$	0	0
$655$	− 22.6056i	− 0.883272i
$656$	3.21110i	0.125372i
$657$	0	0
$658$	42.4222i	1.65379i
$659$	26.2389	1.02212	0.511060	−	0.859545i	$-0.329253\pi$
$659$	26.2389	1.02212	0.511060	+	0.859545i	$0.329253\pi$
$660$	0	0
$661$	− 50.2389i	− 1.95407i	−0.213090	−	0.977033i	$-0.568353\pi$
$661$	− 50.2389i	− 1.95407i	0.213090	−	0.977033i	$-0.431647\pi$
$662$	−16.6056	−0.645393
$663$	0	0
$664$	2.78890	0.108230
$665$	− 21.2111i	− 0.822531i
$666$	0	0
$667$	−6.42221	−0.248669
$668$	− 9.21110i	− 0.356388i
$669$	0	0
$670$	− 3.21110i	− 0.124056i
$671$	0	0
$672$	0	0
$673$	−37.6333	−1.45066	−0.725329	−	0.688403i	$-0.758312\pi$
$673$	−37.6333	−1.45066	−0.725329	+	0.688403i	$0.758312\pi$
$674$	13.6333i	0.525135i
$675$	0	0
$676$	−13.0000	−0.500000
$677$	28.0555	1.07826	0.539130	−	0.842222i	$-0.318753\pi$
$677$	28.0555	1.07826	0.539130	+	0.842222i	$0.318753\pi$
$678$	0	0
$679$	6.42221	0.246462
$680$	−4.60555	−0.176615
$681$	0	0
$682$	0	0
$683$	9.21110i	0.352453i	0.984350	+	0.176227i	$0.0563891\pi$
$683$	9.21110i	0.352453i	−0.984350	+	0.176227i	$0.943611\pi$
$684$	0	0
$685$	−3.21110	−0.122690
$686$	33.2111	1.26801
$687$	0	0
$688$	8.00000	0.304997
$689$	−21.6333	−0.824163
$690$	0	0
$691$	− 20.2389i	− 0.769922i	−0.922933	−	0.384961i	$-0.874215\pi$
$691$	− 20.2389i	− 0.769922i	0.922933	−	0.384961i	$-0.125785\pi$
$692$	−12.4222	−0.472221
$693$	0	0
$694$	27.6333i	1.04895i
$695$	17.2111i	0.652854i
$696$	0	0
$697$	− 14.7889i	− 0.560169i
$698$	−7.81665	−0.295865
$699$	0	0
$700$	4.60555i	0.174073i
$701$	−47.0278	−1.77621	−0.888107	−	0.459637i	$-0.847980\pi$
$701$	−47.0278	−1.77621	−0.888107	+	0.459637i	$0.847980\pi$
$702$	0	0
$703$	−42.4222	−1.59998
$704$	0	0
$705$	0	0
$706$	8.78890	0.330775
$707$	34.0555i	1.28079i
$708$	0	0
$709$	1.39445i	0.0523696i	0.999657	+	0.0261848i	$0.00833584\pi$
$709$	1.39445i	0.0523696i	−0.999657	+	0.0261848i	$0.991664\pi$
$710$	− 9.21110i	− 0.345687i
$711$	0	0
$712$	15.2111	0.570060
$713$	− 8.36669i	− 0.313335i
$714$	0	0
$715$	0	0
$716$	−19.8167	−0.740583
$717$	0	0
$718$	−15.6333	−0.583430
$719$	51.6333	1.92560	0.962799	−	0.270220i	$-0.0870963\pi$
$719$	51.6333	1.92560	0.962799	+	0.270220i	$0.0870963\pi$
$720$	0	0
$721$	− 18.4222i	− 0.686079i
$722$	− 2.21110i	− 0.0822887i
$723$	0	0
$724$	−8.42221	−0.313009
$725$	4.60555	0.171046
$726$	0	0
$727$	−14.4222	−0.534890	−0.267445	−	0.963573i	$-0.586179\pi$
$727$	−14.4222	−0.534890	−0.267445	+	0.963573i	$0.586179\pi$
$728$	16.6056	0.615443
$729$	0	0
$730$	− 1.39445i	− 0.0516109i
$731$	−36.8444	−1.36274
$732$	0	0
$733$	34.0555i	1.25787i	0.777458	+	0.628935i	$0.216509\pi$
$733$	34.0555i	1.25787i	−0.777458	+	0.628935i	$0.783491\pi$
$734$	− 19.6333i	− 0.724679i
$735$	0	0
$736$	1.39445i	0.0514001i
$737$	0	0
$738$	0	0
$739$	− 20.2389i	− 0.744498i	−0.928133	−	0.372249i	$-0.878587\pi$
$739$	− 20.2389i	− 0.744498i	0.928133	−	0.372249i	$-0.121413\pi$
$740$	9.21110	0.338607
$741$	0	0
$742$	27.6333	1.01445
$743$	36.8444i	1.35169i	0.737044	+	0.675845i	$0.236221\pi$
$743$	36.8444i	1.35169i	−0.737044	+	0.675845i	$0.763779\pi$
$744$	0	0
$745$	−15.2111	−0.557292
$746$	− 20.4222i	− 0.747710i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.4222	−0.380312	−0.190156	−	0.981754i	$-0.560899\pi$
$751$	−10.4222	−0.380312	−0.190156	+	0.981754i	$0.560899\pi$
$752$	− 9.21110i	− 0.335894i
$753$	0	0
$754$	− 16.6056i	− 0.604739i
$755$	6.00000	0.218362
$756$	0	0
$757$	12.7889	0.464820	0.232410	−	0.972618i	$-0.425339\pi$
$757$	12.7889	0.464820	0.232410	+	0.972618i	$0.425339\pi$
$758$	35.0278	1.27227
$759$	0	0
$760$	4.60555i	0.167061i
$761$	33.6333i	1.21921i	0.792707	+	0.609603i	$0.208671\pi$
$761$	33.6333i	1.21921i	−0.792707	+	0.609603i	$0.791329\pi$
$762$	0	0
$763$	6.42221	0.232499
$764$	−12.0000	−0.434145
$765$	0	0
$766$	27.6333	0.998432
$767$	− 33.2111i	− 1.19918i
$768$	0	0
$769$	12.8444i	0.463181i	0.972813	+	0.231591i	$0.0743930\pi$
$769$	12.8444i	0.463181i	−0.972813	+	0.231591i	$0.925607\pi$
$770$	0	0
$771$	0	0
$772$	7.81665i	0.281328i
$773$	30.0000i	1.07903i	0.841978	+	0.539513i	$0.181391\pi$
$773$	30.0000i	1.07903i	−0.841978	+	0.539513i	$0.818609\pi$
$774$	0	0
$775$	6.00000i	0.215526i
$776$	−1.39445	−0.0500578
$777$	0	0
$778$	− 4.60555i	− 0.165117i
$779$	−14.7889	−0.529867
$780$	0	0
$781$	0	0
$782$	− 6.42221i	− 0.229658i
$783$	0	0
$784$	−14.2111	−0.507539
$785$	20.4222i	0.728900i
$786$	0	0
$787$	− 49.2666i	− 1.75617i	−0.478509	−	0.878083i	$-0.658823\pi$
$787$	− 49.2666i	− 1.75617i	0.478509	−	0.878083i	$-0.341177\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	14.4222	0.513119
$791$	63.6333i	2.26254i
$792$	0	0
$793$	−40.4222	−1.43543
$794$	−3.63331	−0.128941
$795$	0	0
$796$	−22.4222	−0.794734
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	42.4222i	1.50079i
$800$	− 1.00000i	− 0.0353553i
$801$	0	0
$802$	−8.78890	−0.310347
$803$	0	0
$804$	0	0
$805$	−6.42221	−0.226353
$806$	21.6333	0.762001
$807$	0	0
$808$	− 7.39445i	− 0.260136i
$809$	6.84441	0.240637	0.120318	−	0.992735i	$-0.461608\pi$
$809$	6.84441	0.240637	0.120318	+	0.992735i	$0.461608\pi$
$810$	0	0
$811$	− 32.2389i	− 1.13206i	−0.824385	−	0.566030i	$-0.808479\pi$
$811$	− 32.2389i	− 1.13206i	0.824385	−	0.566030i	$-0.191521\pi$
$812$	21.2111i	0.744364i
$813$	0	0
$814$	0	0
$815$	24.4222	0.855473
$816$	0	0
$817$	36.8444i	1.28902i
$818$	−14.7889	−0.517082
$819$	0	0
$820$	3.21110	0.112137
$821$	− 3.21110i	− 0.112068i	−0.998429	−	0.0560341i	$-0.982154\pi$
$821$	− 3.21110i	− 0.112068i	0.998429	−	0.0560341i	$-0.0178456\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	4.00000i	0.139347i
$825$	0	0
$826$	42.4222i	1.47606i
$827$	− 27.6333i	− 0.960904i	−0.877021	−	0.480452i	$-0.840473\pi$
$827$	− 27.6333i	− 0.960904i	0.877021	−	0.480452i	$-0.159527\pi$
$828$	0	0
$829$	46.8444	1.62697	0.813487	−	0.581583i	$-0.197567\pi$
$829$	46.8444	1.62697	0.813487	+	0.581583i	$0.197567\pi$
$830$	− 2.78890i	− 0.0968040i
$831$	0	0
$832$	−3.60555	−0.125000
$833$	65.4500	2.26771
$834$	0	0
$835$	−9.21110	−0.318763
$836$	0	0
$837$	0	0
$838$	− 4.18335i	− 0.144511i
$839$	− 18.4222i	− 0.636005i	−0.948090	−	0.318003i	$-0.896988\pi$
$839$	− 18.4222i	− 0.636005i	0.948090	−	0.318003i	$-0.103012\pi$
$840$	0	0
$841$	−7.78890	−0.268583
$842$	−19.8167	−0.682927
$843$	0	0
$844$	17.2111	0.592431
$845$	13.0000i	0.447214i
$846$	0	0
$847$	50.6611i	1.74073i
$848$	−6.00000	−0.206041
$849$	0	0
$850$	4.60555i	0.157969i
$851$	12.8444i	0.440301i
$852$	0	0
$853$	14.7889i	0.506362i	0.967419	+	0.253181i	$0.0814769\pi$
$853$	14.7889i	0.506362i	−0.967419	+	0.253181i	$0.918523\pi$
$854$	51.6333	1.76686
$855$	0	0
$856$	0	0
$857$	23.0278	0.786613	0.393307	−	0.919407i	$-0.371331\pi$
$857$	23.0278	0.786613	0.393307	+	0.919407i	$0.371331\pi$
$858$	0	0
$859$	25.2111	0.860192	0.430096	−	0.902783i	$-0.358480\pi$
$859$	25.2111	0.860192	0.430096	+	0.902783i	$0.358480\pi$
$860$	− 8.00000i	− 0.272798i
$861$	0	0
$862$	−12.0000	−0.408722
$863$	51.6333i	1.75762i	0.477173	+	0.878809i	$0.341661\pi$
$863$	51.6333i	1.75762i	−0.477173	+	0.878809i	$0.658339\pi$
$864$	0	0
$865$	12.4222i	0.422368i
$866$	19.2111i	0.652820i
$867$	0	0
$868$	−27.6333	−0.937936
$869$	0	0
$870$	0	0
$871$	11.5778i	0.392299i
$872$	−1.39445	−0.0472220
$873$	0	0
$874$	−6.42221	−0.217234
$875$	4.60555	0.155696
$876$	0	0
$877$	24.8444i	0.838936i	0.907770	+	0.419468i	$0.137783\pi$
$877$	24.8444i	0.838936i	−0.907770	+	0.419468i	$0.862217\pi$
$878$	8.00000i	0.269987i
$879$	0	0
$880$	0	0
$881$	−39.2111	−1.32106	−0.660528	−	0.750802i	$-0.729667\pi$
$881$	−39.2111	−1.32106	−0.660528	+	0.750802i	$0.729667\pi$
$882$	0	0
$883$	9.57779	0.322318	0.161159	−	0.986928i	$-0.448477\pi$
$883$	9.57779	0.322318	0.161159	+	0.986928i	$0.448477\pi$
$884$	16.6056	0.558505
$885$	0	0
$886$	− 15.6333i	− 0.525211i
$887$	−6.97224	−0.234105	−0.117053	−	0.993126i	$-0.537345\pi$
$887$	−6.97224	−0.234105	−0.117053	+	0.993126i	$0.537345\pi$
$888$	0	0
$889$	5.57779i	0.187073i
$890$	− 15.2111i	− 0.509877i
$891$	0	0
$892$	1.81665i	0.0608261i
$893$	42.4222	1.41960
$894$	0	0
$895$	19.8167i	0.662398i
$896$	4.60555	0.153861
$897$	0	0
$898$	−33.6333	−1.12236
$899$	27.6333i	0.921622i
$900$	0	0
$901$	27.6333	0.920599
$902$	0	0
$903$	0	0
$904$	− 13.8167i	− 0.459535i
$905$	8.42221i	0.279964i
$906$	0	0
$907$	21.5778	0.716479	0.358239	−	0.933630i	$-0.383377\pi$
$907$	21.5778	0.716479	0.358239	+	0.933630i	$0.383377\pi$
$908$	− 24.0000i	− 0.796468i
$909$	0	0
$910$	− 16.6056i	− 0.550469i
$911$	27.6333	0.915532	0.457766	−	0.889073i	$-0.348650\pi$
$911$	27.6333	0.915532	0.457766	+	0.889073i	$0.348650\pi$
$912$	0	0
$913$	0	0
$914$	38.2389	1.26483
$915$	0	0
$916$	− 19.8167i	− 0.654761i
$917$	− 104.111i	− 3.43805i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	1.39445	0.0459736
$921$	0	0
$922$	33.6333	1.10765
$923$	33.2111i	1.09316i
$924$	0	0
$925$	− 9.21110i	− 0.302859i
$926$	−31.3944	−1.03169
$927$	0	0
$928$	− 4.60555i	− 0.151185i
$929$	39.2111i	1.28647i	0.765667	+	0.643237i	$0.222409\pi$
$929$	39.2111i	1.28647i	−0.765667	+	0.643237i	$0.777591\pi$
$930$	0	0
$931$	− 65.4500i	− 2.14504i
$932$	−1.81665	−0.0595065
$933$	0	0
$934$	30.4222i	0.995445i
$935$	0	0
$936$	0	0
$937$	−10.3667	−0.338665	−0.169333	−	0.985559i	$-0.554161\pi$
$937$	−10.3667	−0.338665	−0.169333	+	0.985559i	$0.554161\pi$
$938$	− 14.7889i	− 0.482875i
$939$	0	0
$940$	−9.21110	−0.300433
$941$	− 54.0000i	− 1.76035i	−0.474650	−	0.880175i	$-0.657425\pi$
$941$	− 54.0000i	− 1.76035i	0.474650	−	0.880175i	$-0.342575\pi$
$942$	0	0
$943$	4.47772i	0.145815i
$944$	− 9.21110i	− 0.299796i
$945$	0	0
$946$	0	0
$947$	− 15.6333i	− 0.508014i	−0.967202	−	0.254007i	$-0.918251\pi$
$947$	− 15.6333i	− 0.508014i	0.967202	−	0.254007i	$-0.0817487\pi$
$948$	0	0
$949$	5.02776i	0.163208i
$950$	4.60555	0.149424
$951$	0	0
$952$	−21.2111	−0.687456
$953$	20.2389	0.655601	0.327800	−	0.944747i	$-0.393693\pi$
$953$	20.2389	0.655601	0.327800	+	0.944747i	$0.393693\pi$
$954$	0	0
$955$	12.0000i	0.388311i
$956$	0	0
$957$	0	0
$958$	−5.57779	−0.180210
$959$	−14.7889	−0.477558
$960$	0	0
$961$	−5.00000	−0.161290
$962$	−33.2111	−1.07077
$963$	0	0
$964$	− 6.42221i	− 0.206845i
$965$	7.81665	0.251627
$966$	0	0
$967$	8.23886i	0.264944i	0.991187	+	0.132472i	$0.0422914\pi$
$967$	8.23886i	0.264944i	−0.991187	+	0.132472i	$0.957709\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	1.39445i	0.0447731i
$971$	53.0278	1.70174	0.850871	−	0.525375i	$-0.176075\pi$
$971$	53.0278	1.70174	0.850871	+	0.525375i	$0.176075\pi$
$972$	0	0
$973$	79.2666i	2.54117i
$974$	−0.972244	−0.0311527
$975$	0	0
$976$	−11.2111	−0.358859
$977$	18.8444i	0.602886i	0.953484	+	0.301443i	$0.0974683\pi$
$977$	18.8444i	0.602886i	−0.953484	+	0.301443i	$0.902532\pi$
$978$	0	0
$979$	0	0
$980$	14.2111i	0.453957i
$981$	0	0
$982$	− 7.81665i	− 0.249439i
$983$	− 42.4222i	− 1.35306i	−0.736416	−	0.676529i	$-0.763483\pi$
$983$	− 42.4222i	− 1.35306i	0.736416	−	0.676529i	$-0.236517\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	21.2111i	0.675499i
$987$	0	0
$988$	− 16.6056i	− 0.528293i
$989$	11.1556	0.354727
$990$	0	0
$991$	−22.4222	−0.712265	−0.356132	−	0.934436i	$-0.615905\pi$
$991$	−22.4222	−0.712265	−0.356132	+	0.934436i	$0.615905\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	− 42.4222i	− 1.34555i
$995$	22.4222i	0.710832i
$996$	0	0
$997$	−16.4222	−0.520096	−0.260048	−	0.965596i	$-0.583738\pi$
$997$	−16.4222	−0.520096	−0.260048	+	0.965596i	$0.583738\pi$
$998$	23.0278	0.728931
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.b.d.181.4		4
3.2	odd	2		390.2.b.c.181.2	✓	4
12.11	even	2		3120.2.g.q.961.1		4
13.12	even	2	inner	1170.2.b.d.181.1		4
15.2	even	4		1950.2.f.n.649.3		4
15.8	even	4		1950.2.f.m.649.2		4
15.14	odd	2		1950.2.b.k.1351.3		4
39.5	even	4		5070.2.a.z.1.1		2
39.8	even	4		5070.2.a.bf.1.2		2
39.38	odd	2		390.2.b.c.181.3	yes	4
156.155	even	2		3120.2.g.q.961.4		4
195.38	even	4		1950.2.f.n.649.1		4
195.77	even	4		1950.2.f.m.649.4		4
195.194	odd	2		1950.2.b.k.1351.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.c.181.2	✓	4	3.2	odd	2
390.2.b.c.181.3	yes	4	39.38	odd	2
1170.2.b.d.181.1		4	13.12	even	2	inner
1170.2.b.d.181.4		4	1.1	even	1	trivial
1950.2.b.k.1351.2		4	195.194	odd	2
1950.2.b.k.1351.3		4	15.14	odd	2
1950.2.f.m.649.2		4	15.8	even	4
1950.2.f.m.649.4		4	195.77	even	4
1950.2.f.n.649.1		4	195.38	even	4
1950.2.f.n.649.3		4	15.2	even	4
3120.2.g.q.961.1		4	12.11	even	2
3120.2.g.q.961.4		4	156.155	even	2
5070.2.a.z.1.1		2	39.5	even	4
5070.2.a.bf.1.2		2	39.8	even	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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +4.60555i q^{7} -1.00000i q^{8} -1.00000 q^{10} +3.60555 q^{13} -4.60555 q^{14} +1.00000 q^{16} -4.60555 q^{17} +4.60555i q^{19} -1.00000i q^{20} +1.39445 q^{23} -1.00000 q^{25} +3.60555i q^{26} -4.60555i q^{28} -4.60555 q^{29} -6.00000i q^{31} +1.00000i q^{32} -4.60555i q^{34} -4.60555 q^{35} +9.21110i q^{37} -4.60555 q^{38} +1.00000 q^{40} +3.21110i q^{41} +8.00000 q^{43} +1.39445i q^{46} -9.21110i q^{47} -14.2111 q^{49} -1.00000i q^{50} -3.60555 q^{52} -6.00000 q^{53} +4.60555 q^{56} -4.60555i q^{58} -9.21110i q^{59} -11.2111 q^{61} +6.00000 q^{62} -1.00000 q^{64} +3.60555i q^{65} +3.21110i q^{67} +4.60555 q^{68} -4.60555i q^{70} +9.21110i q^{71} +1.39445i q^{73} -9.21110 q^{74} -4.60555i q^{76} -14.4222 q^{79} +1.00000i q^{80} -3.21110 q^{82} +2.78890i q^{83} -4.60555i q^{85} +8.00000i q^{86} +15.2111i q^{89} +16.6056i q^{91} -1.39445 q^{92} +9.21110 q^{94} -4.60555 q^{95} -1.39445i q^{97} -14.2111i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{10} - 4 q^{14} + 4 q^{16} - 4 q^{17} + 20 q^{23} - 4 q^{25} - 4 q^{29} - 4 q^{35} - 4 q^{38} + 4 q^{40} + 32 q^{43} - 28 q^{49} - 24 q^{53} + 4 q^{56} - 16 q^{61} + 24 q^{62} - 4 q^{64}+ \cdots - 4 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^10 - 4 * q^14 + 4 * q^16 - 4 * q^17 + 20 * q^23 - 4 * q^25 - 4 * q^29 - 4 * q^35 - 4 * q^38 + 4 * q^40 + 32 * q^43 - 28 * q^49 - 24 * q^53 + 4 * q^56 - 16 * q^61 + 24 * q^62 - 4 * q^64 + 4 * q^68 - 8 * q^74 + 16 * q^82 - 20 * q^92 + 8 * q^94 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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