LMFDB - Embedded newform 55.9.d.a.54.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,9,Mod(54,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("55.54");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 55 = 5 \cdot 11 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	55.d (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:	yes
Analytic conductor:	$22.4058235534$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			54.1
Character	$\chi$	$=$	55.54

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-23.0000 q^{2} +273.000 q^{4} +625.000 q^{5} +1282.00 q^{7} -391.000 q^{8} +6561.00 q^{9} +O(q^{10})$

$q-23.0000 q^{2} +273.000 q^{4} +625.000 q^{5} +1282.00 q^{7} -391.000 q^{8} +6561.00 q^{9} -14375.0 q^{10} +14641.0 q^{11} -30878.0 q^{13} -29486.0 q^{14} -60895.0 q^{16} -5438.00 q^{17} -150903. q^{18} +170625. q^{20} -336743. q^{22} +390625. q^{25} +710194. q^{26} +349986. q^{28} +706562. q^{31} +1.50068e6 q^{32} +125074. q^{34} +801250. q^{35} +1.79115e6 q^{36} -244375. q^{40} +5.56688e6 q^{43} +3.99699e6 q^{44} +4.10062e6 q^{45} -4.12128e6 q^{49} -8.98438e6 q^{50} -8.42969e6 q^{52} +9.15062e6 q^{55} -501262. q^{56} -1.23874e7 q^{59} -1.62509e7 q^{62} +8.41120e6 q^{63} -1.89265e7 q^{64} -1.92988e7 q^{65} -1.48457e6 q^{68} -1.84288e7 q^{70} -3.48394e7 q^{71} -2.56535e6 q^{72} +5.63706e7 q^{73} +1.87698e7 q^{77} -3.80594e7 q^{80} +4.30467e7 q^{81} +9.00978e7 q^{83} -3.39875e6 q^{85} -1.28038e8 q^{86} -5.72463e6 q^{88} +1.25358e8 q^{89} -9.43144e7 q^{90} -3.95856e7 q^{91} +9.47894e7 q^{98} +9.60596e7 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$.

$n$	$12$	$46$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−23.0000	−1.43750	−0.718750	−	0.695269i	$-0.755285\pi$
$2$	−23.0000	−1.43750	−0.718750	+	0.695269i	$0.755285\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	273.000	1.06641
$5$	625.000	1.00000
$5$	625.000	1.00000
$6$	0	0
$7$	1282.00	0.533944	0.266972	−	0.963704i	$-0.413977\pi$
$7$	1282.00	0.533944	0.266972	+	0.963704i	$0.413977\pi$
$8$	−391.000	−0.0954590
$9$	6561.00	1.00000
$10$	−14375.0	−1.43750
$11$	14641.0	1.00000
$11$	14641.0	1.00000
$12$	0	0
$13$	−30878.0	−1.08112	−0.540562	−	0.841304i	$-0.681788\pi$
$13$	−30878.0	−1.08112	−0.540562	+	0.841304i	$0.681788\pi$
$14$	−29486.0	−0.767545
$15$	0	0
$16$	−60895.0	−0.929184
$17$	−5438.00	−0.0651094	−0.0325547	−	0.999470i	$-0.510364\pi$
$17$	−5438.00	−0.0651094	−0.0325547	+	0.999470i	$0.510364\pi$
$18$	−150903.	−1.43750
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	170625.	1.06641
$21$	0	0
$22$	−336743.	−1.43750
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	390625.	1.00000
$26$	710194.	1.55412
$27$	0	0
$28$	349986.	0.569401
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	706562.	0.765074	0.382537	−	0.923940i	$-0.375050\pi$
$31$	706562.	0.765074	0.382537	+	0.923940i	$0.375050\pi$
$32$	1.50068e6	1.43116
$33$	0	0
$34$	125074.	0.0935947
$35$	801250.	0.533944
$36$	1.79115e6	1.06641
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	−244375.	−0.0954590
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	5.56688e6	1.62831	0.814157	−	0.580645i	$-0.197199\pi$
$43$	5.56688e6	1.62831	0.814157	+	0.580645i	$0.197199\pi$
$44$	3.99699e6	1.06641
$45$	4.10062e6	1.00000
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−4.12128e6	−0.714904
$50$	−8.98438e6	−1.43750
$51$	0	0
$52$	−8.42969e6	−1.15292
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	9.15062e6	1.00000
$56$	−501262.	−0.0509698
$57$	0	0
$58$	0	0
$59$	−1.23874e7	−1.02228	−0.511141	−	0.859497i	$-0.670777\pi$
$59$	−1.23874e7	−1.02228	−0.511141	+	0.859497i	$0.670777\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−1.62509e7	−1.09979
$63$	8.41120e6	0.533944
$64$	−1.89265e7	−1.12811
$65$	−1.92988e7	−1.08112
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.48457e6	−0.0694330
$69$	0	0
$70$	−1.84288e7	−0.767545
$71$	−3.48394e7	−1.37100	−0.685499	−	0.728074i	$-0.740416\pi$
$71$	−3.48394e7	−1.37100	−0.685499	+	0.728074i	$0.740416\pi$
$72$	−2.56535e6	−0.0954590
$73$	5.63706e7	1.98500	0.992501	−	0.122237i	$-0.0390068\pi$
$73$	5.63706e7	1.98500	0.992501	+	0.122237i	$0.0390068\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.87698e7	0.533944
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−3.80594e7	−0.929184
$81$	4.30467e7	1.00000
$82$	0	0
$83$	9.00978e7	1.89846	0.949230	−	0.314582i	$-0.101864\pi$
$83$	9.00978e7	1.89846	0.949230	+	0.314582i	$0.101864\pi$
$84$	0	0
$85$	−3.39875e6	−0.0651094
$86$	−1.28038e8	−2.34070
$87$	0	0
$88$	−5.72463e6	−0.0954590
$89$	1.25358e8	1.99798	0.998990	−	0.0449296i	$-0.0143064\pi$
$89$	1.25358e8	1.99798	0.998990	+	0.0449296i	$0.0143064\pi$
$90$	−9.43144e7	−1.43750
$91$	−3.95856e7	−0.577260
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	9.47894e7	1.02767
$99$	9.60596e7	1.00000
$100$	1.06641e8	1.06641
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	1.20733e7	0.103203
$105$	0	0
$106$	0	0
$107$	−2.59593e8	−1.98042	−0.990211	−	0.139582i	$-0.955424\pi$
$107$	−2.59593e8	−1.98042	−0.990211	+	0.139582i	$0.955424\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−2.10464e8	−1.43750
$111$	0	0
$112$	−7.80674e7	−0.496132
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.02591e8	−1.08112
$118$	2.84909e8	1.46953
$119$	−6.97152e6	−0.0347648
$120$	0	0
$121$	2.14359e8	1.00000
$122$	0	0
$123$	0	0
$124$	1.92891e8	0.815880
$125$	2.44141e8	1.00000
$126$	−1.93458e8	−0.767545
$127$	4.58024e8	1.76065	0.880326	−	0.474370i	$-0.157324\pi$
$127$	4.58024e8	1.76065	0.880326	+	0.474370i	$0.157324\pi$
$128$	5.11362e7	0.190497
$129$	0	0
$130$	4.43871e8	1.55412
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	2.12626e6	0.00621527
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	2.18741e8	0.569401
$141$	0	0
$142$	8.01305e8	1.97081
$143$	−4.52085e8	−1.08112
$144$	−3.99532e8	−0.929184
$145$	0	0
$146$	−1.29652e9	−2.85344
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−3.56787e7	−0.0651094
$154$	−4.31705e8	−0.767545
$155$	4.41601e8	0.765074
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	9.37926e8	1.43116
$161$	0	0
$162$	−9.90075e8	−1.43750
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	−2.07225e9	−2.72904
$167$	−1.49541e9	−1.92262	−0.961310	−	0.275470i	$-0.911166\pi$
$167$	−1.49541e9	−1.92262	−0.961310	+	0.275470i	$0.911166\pi$
$168$	0	0
$169$	1.37720e8	0.168830
$170$	7.81712e7	0.0935947
$171$	0	0
$172$	1.51976e9	1.73644
$173$	−1.77796e9	−1.98490	−0.992449	−	0.122659i	$-0.960858\pi$
$173$	−1.77796e9	−1.98490	−0.992449	+	0.122659i	$0.960858\pi$
$174$	0	0
$175$	5.00781e8	0.533944
$176$	−8.91564e8	−0.929184
$177$	0	0
$178$	−2.88323e9	−2.87210
$179$	1.87027e9	1.82176	0.910881	−	0.412669i	$-0.135403\pi$
$179$	1.87027e9	1.82176	0.910881	+	0.412669i	$0.135403\pi$
$180$	1.11947e9	1.06641
$181$	4.99713e8	0.465593	0.232797	−	0.972525i	$-0.425212\pi$
$181$	4.99713e8	0.465593	0.232797	+	0.972525i	$0.425212\pi$
$182$	9.10469e8	0.829812
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.96178e7	−0.0651094
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.57523e8	−0.719475	−0.359738	−	0.933054i	$-0.617134\pi$
$191$	−9.57523e8	−0.719475	−0.359738	+	0.933054i	$0.617134\pi$
$192$	0	0
$193$	2.62992e9	1.89545	0.947727	−	0.319082i	$-0.103374\pi$
$193$	2.62992e9	1.89545	0.947727	+	0.319082i	$0.103374\pi$
$194$	0	0
$195$	0	0
$196$	−1.12511e9	−0.762378
$197$	1.94360e9	1.29045	0.645227	−	0.763991i	$-0.276763\pi$
$197$	1.94360e9	1.29045	0.645227	+	0.763991i	$0.276763\pi$
$198$	−2.20937e9	−1.43750
$199$	−8.78518e8	−0.560194	−0.280097	−	0.959972i	$-0.590367\pi$
$199$	−8.78518e8	−0.560194	−0.280097	+	0.959972i	$0.590367\pi$
$200$	−1.52734e8	−0.0954590
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	1.88032e9	1.00456
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	5.97063e9	2.84686
$215$	3.47930e9	1.62831
$216$	0	0
$217$	9.05812e8	0.408507
$218$	0	0
$219$	0	0
$220$	2.49812e9	1.06641
$221$	1.67915e8	0.0703913
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	1.92387e9	0.764160
$225$	2.56289e9	1.00000
$226$	0	0
$227$	−9.44174e8	−0.355589	−0.177795	−	0.984068i	$-0.556896\pi$
$227$	−9.44174e8	−0.355589	−0.177795	+	0.984068i	$0.556896\pi$
$228$	0	0
$229$	−1.41584e8	−0.0514841	−0.0257420	−	0.999669i	$-0.508195\pi$
$229$	−1.41584e8	−0.0514841	−0.0257420	+	0.999669i	$0.508195\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.22732e9	−0.755716	−0.377858	−	0.925863i	$-0.623339\pi$
$233$	−2.22732e9	−0.755716	−0.377858	+	0.925863i	$0.623339\pi$
$234$	4.65958e9	1.55412
$235$	0	0
$236$	−3.38175e9	−1.09017
$237$	0	0
$238$	1.60345e8	0.0499744
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−4.93025e9	−1.43750
$243$	0	0
$244$	0	0
$245$	−2.57580e9	−0.714904
$246$	0	0
$247$	0	0
$248$	−2.76266e8	−0.0730332
$249$	0	0
$250$	−5.61523e9	−1.43750
$251$	−5.77805e9	−1.45575	−0.727874	−	0.685711i	$-0.759491\pi$
$251$	−5.77805e9	−1.45575	−0.727874	+	0.685711i	$0.759491\pi$
$252$	2.29626e9	0.569401
$253$	0	0
$254$	−1.05346e10	−2.53094
$255$	0	0
$256$	3.66906e9	0.854270
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	−5.26856e9	−1.15292
$261$	0	0
$262$	0	0
$263$	4.13013e9	0.863258	0.431629	−	0.902051i	$-0.357939\pi$
$263$	4.13013e9	0.863258	0.431629	+	0.902051i	$0.357939\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.70172e9	−1.85285	−0.926424	−	0.376482i	$-0.877134\pi$
$269$	−9.70172e9	−1.85285	−0.926424	+	0.376482i	$0.877134\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	3.31147e8	0.0604986
$273$	0	0
$274$	0	0
$275$	5.71914e9	1.00000
$276$	0	0
$277$	−1.15289e10	−1.95825	−0.979125	−	0.203260i	$-0.934846\pi$
$277$	−1.15289e10	−1.95825	−0.979125	+	0.203260i	$0.934846\pi$
$278$	0	0
$279$	4.63575e9	0.765074
$280$	−3.13289e8	−0.0509698
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	5.58002e9	0.869942	0.434971	−	0.900444i	$-0.356759\pi$
$283$	5.58002e9	0.869942	0.434971	+	0.900444i	$0.356759\pi$
$284$	−9.51114e9	−1.46204
$285$	0	0
$286$	1.03980e10	1.55412
$287$	0	0
$288$	9.84597e9	1.43116
$289$	−6.94619e9	−0.995761
$290$	0	0
$291$	0	0
$292$	1.53892e10	2.11682
$293$	1.46797e10	1.99180	0.995902	−	0.0904403i	$-0.0288274\pi$
$293$	1.46797e10	1.99180	0.995902	+	0.0904403i	$0.0288274\pi$
$294$	0	0
$295$	−7.74210e9	−1.02228
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	7.13674e9	0.869429
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	8.20611e8	0.0935947
$307$	−1.61270e10	−1.81551	−0.907756	−	0.419500i	$-0.862206\pi$
$307$	−1.61270e10	−1.81551	−0.907756	+	0.419500i	$0.862206\pi$
$308$	5.12415e9	0.569401
$309$	0	0
$310$	−1.01568e10	−1.09979
$311$	−1.76962e10	−1.89164	−0.945822	−	0.324685i	$-0.894742\pi$
$311$	−1.76962e10	−1.89164	−0.945822	+	0.324685i	$0.894742\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	5.25700e9	0.533944
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−1.18291e10	−1.12811
$321$	0	0
$322$	0	0
$323$	0	0
$324$	1.17518e10	1.06641
$325$	−1.20617e10	−1.08112
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.49727e9	0.208043	0.104021	−	0.994575i	$-0.466829\pi$
$331$	2.49727e9	0.208043	0.104021	+	0.994575i	$0.466829\pi$
$332$	2.45967e10	2.02453
$333$	0	0
$334$	3.43943e10	2.76376
$335$	0	0
$336$	0	0
$337$	−2.10737e10	−1.63388	−0.816941	−	0.576721i	$-0.804332\pi$
$337$	−2.10737e10	−1.63388	−0.816941	+	0.576721i	$0.804332\pi$
$338$	−3.16756e9	−0.242694
$339$	0	0
$340$	−9.27859e8	−0.0694330
$341$	1.03448e10	0.765074
$342$	0	0
$343$	−1.26740e10	−0.915663
$344$	−2.17665e9	−0.155437
$345$	0	0
$346$	4.08931e10	2.85329
$347$	−2.38583e10	−1.64559	−0.822796	−	0.568336i	$-0.807587\pi$
$347$	−2.38583e10	−1.64559	−0.822796	+	0.568336i	$0.807587\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−1.15180e10	−0.767545
$351$	0	0
$352$	2.19715e10	1.43116
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−2.17746e10	−1.37100
$356$	3.42227e10	2.13066
$357$	0	0
$358$	−4.30162e10	−2.61878
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−1.60334e9	−0.0954590
$361$	1.69836e10	1.00000
$362$	−1.14934e10	−0.669290
$363$	0	0
$364$	−1.08069e10	−0.615594
$365$	3.52316e10	1.98500
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7.39178e9	0.381868	0.190934	−	0.981603i	$-0.438848\pi$
$373$	7.39178e9	0.381868	0.190934	+	0.981603i	$0.438848\pi$
$374$	1.83121e9	0.0935947
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−4.12608e10	−1.99977	−0.999887	−	0.0150510i	$-0.995209\pi$
$379$	−4.12608e10	−1.99977	−0.999887	+	0.0150510i	$0.995209\pi$
$380$	0	0
$381$	0	0
$382$	2.20230e10	1.03425
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	1.17311e10	0.533944
$386$	−6.04882e10	−2.72472
$387$	3.65243e10	1.62831
$388$	0	0
$389$	3.96070e10	1.72971	0.864855	−	0.502022i	$-0.167411\pi$
$389$	3.96070e10	1.72971	0.864855	+	0.502022i	$0.167411\pi$
$390$	0	0
$391$	0	0
$392$	1.61142e9	0.0682440
$393$	0	0
$394$	−4.47028e10	−1.85503
$395$	0	0
$396$	2.62243e10	1.06641
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	2.02059e10	0.805279
$399$	0	0
$400$	−2.37871e10	−0.929184
$401$	3.23408e10	1.25076	0.625380	−	0.780321i	$-0.284944\pi$
$401$	3.23408e10	1.25076	0.625380	+	0.780321i	$0.284944\pi$
$402$	0	0
$403$	−2.18172e10	−0.827140
$404$	0	0
$405$	2.69042e10	1.00000
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.58806e10	−0.545841
$414$	0	0
$415$	5.63111e10	1.89846
$416$	−4.63380e10	−1.54726
$417$	0	0
$418$	0	0
$419$	−3.73828e10	−1.21287	−0.606437	−	0.795132i	$-0.707402\pi$
$419$	−3.73828e10	−1.21287	−0.606437	+	0.795132i	$0.707402\pi$
$420$	0	0
$421$	4.96899e10	1.58176	0.790879	−	0.611973i	$-0.209624\pi$
$421$	4.96899e10	1.58176	0.790879	+	0.611973i	$0.209624\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.12422e9	−0.0651094
$426$	0	0
$427$	0	0
$428$	−7.08688e10	−2.11193
$429$	0	0
$430$	−8.00239e10	−2.34070
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−2.08337e10	−0.587229
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	−3.57789e9	−0.0954590
$441$	−2.70397e10	−0.714904
$442$	−3.86203e9	−0.101188
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	7.83486e10	1.99798
$446$	0	0
$447$	0	0
$448$	−2.42638e10	−0.602348
$449$	−6.73426e10	−1.65693	−0.828465	−	0.560040i	$-0.810786\pi$
$449$	−6.73426e10	−1.65693	−0.828465	+	0.560040i	$0.810786\pi$
$450$	−5.89465e10	−1.43750
$451$	0	0
$452$	0	0
$453$	0	0
$454$	2.17160e10	0.511159
$455$	−2.47410e10	−0.577260
$456$	0	0
$457$	−3.09729e10	−0.710096	−0.355048	−	0.934848i	$-0.615536\pi$
$457$	−3.09729e10	−0.710096	−0.355048	+	0.934848i	$0.615536\pi$
$458$	3.25644e9	0.0740083
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	5.12284e10	1.08634
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−5.53072e10	−1.15292
$469$	0	0
$470$	0	0
$471$	0	0
$472$	4.84346e9	0.0975860
$473$	8.15047e10	1.62831
$474$	0	0
$475$	0	0
$476$	−1.90322e9	−0.0370734
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	5.85200e10	1.06641
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	5.92434e10	1.02767
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	6.00373e10	1.00000
$496$	−4.30261e10	−0.710895
$497$	−4.46641e10	−0.732036
$498$	0	0
$499$	9.34532e10	1.50727	0.753637	−	0.657291i	$-0.228298\pi$
$499$	9.34532e10	1.50727	0.753637	+	0.657291i	$0.228298\pi$
$500$	6.66504e10	1.06641
$501$	0	0
$502$	1.32895e11	2.09264
$503$	1.72848e10	0.270018	0.135009	−	0.990844i	$-0.456894\pi$
$503$	1.72848e10	0.270018	0.135009	+	0.990844i	$0.456894\pi$
$504$	−3.28878e9	−0.0509698
$505$	0	0
$506$	0	0
$507$	0	0
$508$	1.25041e11	1.87757
$509$	1.14873e11	1.71138	0.855690	−	0.517489i	$-0.173133\pi$
$509$	1.14873e11	1.71138	0.855690	+	0.517489i	$0.173133\pi$
$510$	0	0
$511$	7.22671e10	1.05988
$512$	−9.74793e10	−1.41851
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	7.54581e9	0.103203
$521$	−8.53007e10	−1.15771	−0.578857	−	0.815429i	$-0.696501\pi$
$521$	−8.53007e10	−1.15771	−0.578857	+	0.815429i	$0.696501\pi$
$522$	0	0
$523$	8.41573e10	1.12483	0.562413	−	0.826857i	$-0.309873\pi$
$523$	8.41573e10	1.12483	0.562413	+	0.826857i	$0.309873\pi$
$524$	0	0
$525$	0	0
$526$	−9.49930e10	−1.24093
$527$	−3.84228e9	−0.0498135
$528$	0	0
$529$	7.83110e10	1.00000
$530$	0	0
$531$	−8.12735e10	−1.02228
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1.62245e11	−1.98042
$536$	0	0
$537$	0	0
$538$	2.23140e11	2.66347
$539$	−6.03396e10	−0.714904
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−8.16070e9	−0.0931820
$545$	0	0
$546$	0	0
$547$	2.38099e10	0.265955	0.132978	−	0.991119i	$-0.457546\pi$
$547$	2.38099e10	0.265955	0.132978	+	0.991119i	$0.457546\pi$
$548$	0	0
$549$	0	0
$550$	−1.31540e11	−1.43750
$551$	0	0
$552$	0	0
$553$	0	0
$554$	2.65164e11	2.81498
$555$	0	0
$556$	0	0
$557$	−1.21370e11	−1.26093	−0.630466	−	0.776217i	$-0.717136\pi$
$557$	−1.21370e11	−1.26093	−0.630466	+	0.776217i	$0.717136\pi$
$558$	−1.06622e11	−1.09979
$559$	−1.71894e11	−1.76041
$560$	−4.87921e10	−0.496132
$561$	0	0
$562$	0	0
$563$	−2.00486e11	−1.99549	−0.997746	−	0.0671044i	$-0.978624\pi$
$563$	−2.00486e11	−1.99549	−0.997746	+	0.0671044i	$0.978624\pi$
$564$	0	0
$565$	0	0
$566$	−1.28341e11	−1.25054
$567$	5.51859e10	0.533944
$568$	1.36222e10	0.130874
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	−1.23419e11	−1.15292
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−1.24177e11	−1.12811
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1.59762e11	1.43141
$579$	0	0
$580$	0	0
$581$	1.15505e11	1.01367
$582$	0	0
$583$	0	0
$584$	−2.20409e10	−0.189486
$585$	−1.26619e11	−1.08112
$586$	−3.37633e11	−2.86322
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	1.78068e11	1.46953
$591$	0	0
$592$	0	0
$593$	2.29888e11	1.85908	0.929538	−	0.368726i	$-0.120206\pi$
$593$	2.29888e11	1.85908	0.929538	+	0.368726i	$0.120206\pi$
$594$	0	0
$595$	−4.35720e9	−0.0347648
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.51941e11	−1.95701	−0.978503	−	0.206234i	$-0.933879\pi$
$599$	−2.51941e11	−1.95701	−0.978503	+	0.206234i	$0.933879\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	−1.64145e11	−1.24980
$603$	0	0
$604$	0	0
$605$	1.33974e11	1.00000
$606$	0	0
$607$	−1.40814e11	−1.03727	−0.518634	−	0.854996i	$-0.673559\pi$
$607$	−1.40814e11	−1.03727	−0.518634	+	0.854996i	$0.673559\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−9.74029e9	−0.0694330
$613$	−2.63116e11	−1.86340	−0.931700	−	0.363228i	$-0.881674\pi$
$613$	−2.63116e11	−1.86340	−0.931700	+	0.363228i	$0.881674\pi$
$614$	3.70920e11	2.60980
$615$	0	0
$616$	−7.33898e9	−0.0509698
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−2.87787e11	−1.96024	−0.980120	−	0.198406i	$-0.936424\pi$
$619$	−2.87787e11	−1.96024	−0.980120	+	0.198406i	$0.936424\pi$
$620$	1.20557e11	0.815880
$621$	0	0
$622$	4.07014e11	2.71924
$623$	1.60709e11	1.06681
$624$	0	0
$625$	1.52588e11	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−1.20911e11	−0.767545
$631$	−2.33288e11	−1.47155	−0.735774	−	0.677227i	$-0.763181\pi$
$631$	−2.33288e11	−1.47155	−0.735774	+	0.677227i	$0.763181\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.86265e11	1.76065
$636$	0	0
$637$	1.27257e11	0.772900
$638$	0	0
$639$	−2.28581e11	−1.37100
$640$	3.19601e10	0.190497
$641$	1.96069e11	1.16138	0.580692	−	0.814123i	$-0.302782\pi$
$641$	1.96069e11	1.16138	0.580692	+	0.814123i	$0.302782\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.68313e10	−0.0954590
$649$	−1.81363e11	−1.02228
$650$	2.77420e11	1.55412
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.69847e11	1.98500
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	7.85325e10	0.411380	0.205690	−	0.978617i	$-0.434056\pi$
$661$	7.85325e10	0.411380	0.205690	+	0.978617i	$0.434056\pi$
$662$	−5.74371e10	−0.299062
$663$	0	0
$664$	−3.52282e10	−0.181225
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−4.08246e11	−2.05029
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.09942e11	−1.99831	−0.999153	−	0.0411501i	$-0.986898\pi$
$673$	−4.09942e11	−1.99831	−0.999153	+	0.0411501i	$0.986898\pi$
$674$	4.84694e11	2.34870
$675$	0	0
$676$	3.75976e10	0.180042
$677$	3.81542e11	1.81630	0.908151	−	0.418644i	$-0.137494\pi$
$677$	3.81542e11	1.81630	0.908151	+	0.418644i	$0.137494\pi$
$678$	0	0
$679$	0	0
$680$	1.32891e9	0.00621527
$681$	0	0
$682$	−2.37930e11	−1.09979
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	2.91501e11	1.31627
$687$	0	0
$688$	−3.38995e11	−1.51300
$689$	0	0
$690$	0	0
$691$	−9.91396e10	−0.434846	−0.217423	−	0.976078i	$-0.569765\pi$
$691$	−9.91396e10	−0.434846	−0.217423	+	0.976078i	$0.569765\pi$
$692$	−4.85384e11	−2.11671
$693$	1.23148e11	0.533944
$694$	5.48742e11	2.36554
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	1.36713e11	0.569401
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	−2.77104e11	−1.12811
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.95304e11	1.56440	0.782198	−	0.623030i	$-0.214099\pi$
$709$	3.95304e11	1.56440	0.782198	+	0.623030i	$0.214099\pi$
$710$	5.00816e11	1.97081
$711$	0	0
$712$	−4.90149e10	−0.190725
$713$	0	0
$714$	0	0
$715$	−2.82553e11	−1.08112
$716$	5.10583e11	1.94274
$717$	0	0
$718$	0	0
$719$	−2.65761e11	−0.994433	−0.497216	−	0.867627i	$-0.665645\pi$
$719$	−2.65761e11	−0.994433	−0.497216	+	0.867627i	$0.665645\pi$
$720$	−2.49708e11	−0.929184
$721$	0	0
$722$	−3.90622e11	−1.43750
$723$	0	0
$724$	1.36422e11	0.496511
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	1.54780e10	0.0551047
$729$	2.82430e11	1.00000
$730$	−8.10327e11	−2.85344
$731$	−3.02727e10	−0.106019
$732$	0	0
$733$	−5.78092e10	−0.200254	−0.100127	−	0.994975i	$-0.531925\pi$
$733$	−5.78092e10	−0.200254	−0.100127	+	0.994975i	$0.531925\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.62478e11	−1.84566	−0.922828	−	0.385213i	$-0.874128\pi$
$743$	−5.62478e11	−1.84566	−0.922828	+	0.385213i	$0.874128\pi$
$744$	0	0
$745$	0	0
$746$	−1.70011e11	−0.548936
$747$	5.91131e11	1.89846
$748$	−2.17356e10	−0.0694330
$749$	−3.32798e11	−1.05743
$750$	0	0
$751$	1.72938e11	0.543665	0.271833	−	0.962345i	$-0.412370\pi$
$751$	1.72938e11	0.543665	0.271833	+	0.962345i	$0.412370\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	9.48998e11	2.87467
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	−2.61404e11	−0.767253
$765$	−2.22992e10	−0.0651094
$766$	0	0
$767$	3.82497e11	1.10521
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	−2.69815e11	−0.767545
$771$	0	0
$772$	7.17968e11	2.02132
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−8.40059e11	−2.34070
$775$	2.76001e11	0.765074
$776$	0	0
$777$	0	0
$778$	−9.10960e11	−2.48646
$779$	0	0
$780$	0	0
$781$	−5.10083e11	−1.37100
$782$	0	0
$783$	0	0
$784$	2.50965e11	0.664277
$785$	0	0
$786$	0	0
$787$	−4.07899e10	−0.106330	−0.0531648	−	0.998586i	$-0.516931\pi$
$787$	−4.07899e10	−0.106330	−0.0531648	+	0.998586i	$0.516931\pi$
$788$	5.30603e11	1.37615
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−3.75593e10	−0.0954590
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−2.39835e11	−0.597394
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	5.86204e11	1.43116
$801$	8.22472e11	1.99798
$802$	−7.43839e11	−1.79797
$803$	8.25321e11	1.98500
$804$	0	0
$805$	0	0
$806$	5.01796e11	1.18901
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−6.18797e11	−1.43750
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.59721e11	−0.577260
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	3.65254e11	0.784647
$827$	−9.34829e11	−1.99853	−0.999264	−	0.0383566i	$-0.987788\pi$
$827$	−9.34829e11	−1.99853	−0.999264	+	0.0383566i	$0.987788\pi$
$828$	0	0
$829$	−9.43477e11	−1.99762	−0.998811	−	0.0487573i	$-0.984474\pi$
$829$	−9.43477e11	−1.99762	−0.998811	+	0.0487573i	$0.984474\pi$
$830$	−1.29516e12	−2.72904
$831$	0	0
$832$	5.84414e11	1.21963
$833$	2.24115e10	0.0465469
$834$	0	0
$835$	−9.34629e11	−1.92262
$836$	0	0
$837$	0	0
$838$	8.59804e11	1.74351
$839$	−4.75598e11	−0.959824	−0.479912	−	0.877317i	$-0.659331\pi$
$839$	−4.75598e11	−0.959824	−0.479912	+	0.877317i	$0.659331\pi$
$840$	0	0
$841$	5.00246e11	1.00000
$842$	−1.14287e12	−2.27378
$843$	0	0
$844$	0	0
$845$	8.60751e10	0.168830
$846$	0	0
$847$	2.74808e11	0.533944
$848$	0	0
$849$	0	0
$850$	4.88570e10	0.0935947
$851$	0	0
$852$	0	0
$853$	−9.31522e11	−1.75953	−0.879766	−	0.475408i	$-0.842301\pi$
$853$	−9.31522e11	−1.75953	−0.879766	+	0.475408i	$0.842301\pi$
$854$	0	0
$855$	0	0
$856$	1.01501e11	0.189049
$857$	−9.72243e11	−1.80240	−0.901201	−	0.433402i	$-0.857313\pi$
$857$	−9.72243e11	−1.80240	−0.901201	+	0.433402i	$0.857313\pi$
$858$	0	0
$859$	5.59509e11	1.02762	0.513812	−	0.857903i	$-0.328233\pi$
$859$	5.59509e11	1.02762	0.513812	+	0.857903i	$0.328233\pi$
$860$	9.49849e11	1.73644
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−1.11123e12	−1.98490
$866$	0	0
$867$	0	0
$868$	2.47287e11	0.435634
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.12988e11	0.533944
$876$	0	0
$877$	1.01344e12	1.71316	0.856580	−	0.516014i	$-0.172585\pi$
$877$	1.01344e12	1.71316	0.856580	+	0.516014i	$0.172585\pi$
$878$	0	0
$879$	0	0
$880$	−5.57227e11	−0.929184
$881$	1.16134e12	1.92777	0.963883	−	0.266325i	$-0.0858093\pi$
$881$	1.16134e12	1.92777	0.963883	+	0.266325i	$0.0858093\pi$
$882$	6.21913e11	1.02767
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	4.58407e10	0.0750658
$885$	0	0
$886$	0	0
$887$	−3.94108e11	−0.636679	−0.318340	−	0.947977i	$-0.603125\pi$
$887$	−3.94108e11	−0.636679	−0.318340	+	0.947977i	$0.603125\pi$
$888$	0	0
$889$	5.87187e11	0.940090
$890$	−1.80202e12	−2.87210
$891$	6.30247e11	1.00000
$892$	0	0
$893$	0	0
$894$	0	0
$895$	1.16892e12	1.82176
$896$	6.55565e10	0.101715
$897$	0	0
$898$	1.54888e12	2.38184
$899$	0	0
$900$	6.99669e11	1.06641
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.12321e11	0.465593
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−2.57759e11	−0.379202
$909$	0	0
$910$	5.69043e11	0.829812
$911$	5.12323e11	0.743824	0.371912	−	0.928268i	$-0.378702\pi$
$911$	5.12323e11	0.743824	0.371912	+	0.928268i	$0.378702\pi$
$912$	0	0
$913$	1.31912e12	1.89846
$914$	7.12377e11	1.02076
$915$	0	0
$916$	−3.86525e10	−0.0549029
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.07577e12	1.48222
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.16048e11	−0.558574	−0.279287	−	0.960208i	$-0.590098\pi$
$929$	−4.16048e11	−0.558574	−0.279287	+	0.960208i	$0.590098\pi$
$930$	0	0
$931$	0	0
$932$	−6.08058e11	−0.805901
$933$	0	0
$934$	0	0
$935$	−4.97611e10	−0.0651094
$936$	7.92129e10	0.103203
$937$	5.90180e11	0.765643	0.382822	−	0.923822i	$-0.374952\pi$
$937$	5.90180e11	0.765643	0.382822	+	0.923822i	$0.374952\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	7.54328e11	0.949888
$945$	0	0
$946$	−1.87461e12	−2.34070
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	−1.74061e12	−2.14603
$950$	0	0
$951$	0	0
$952$	2.72586e9	0.00331861
$953$	−2.00269e11	−0.242796	−0.121398	−	0.992604i	$-0.538738\pi$
$953$	−2.00269e11	−0.242796	−0.121398	+	0.992604i	$0.538738\pi$
$954$	0	0
$955$	−5.98452e11	−0.719475
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−3.53661e11	−0.414662
$962$	0	0
$963$	−1.70319e12	−1.98042
$964$	0	0
$965$	1.64370e12	1.89545
$966$	0	0
$967$	7.29707e11	0.834531	0.417266	−	0.908785i	$-0.362988\pi$
$967$	7.29707e11	0.834531	0.417266	+	0.908785i	$0.362988\pi$
$968$	−8.38143e10	−0.0954590
$969$	0	0
$970$	0	0
$971$	−1.22257e12	−1.37530	−0.687650	−	0.726042i	$-0.741358\pi$
$971$	−1.22257e12	−1.37530	−0.687650	+	0.726042i	$0.741358\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	1.83536e12	1.99798
$980$	−7.03193e11	−0.762378
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	1.21475e12	1.29045
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−1.38086e12	−1.43750
$991$	1.28116e12	1.32834	0.664168	−	0.747583i	$-0.268786\pi$
$991$	1.28116e12	1.32834	0.664168	+	0.747583i	$0.268786\pi$
$992$	1.06032e12	1.09494
$993$	0	0
$994$	1.02727e12	1.05230
$995$	−5.49074e11	−0.560194
$996$	0	0
$997$	−1.96549e12	−1.98925	−0.994627	−	0.103520i	$-0.966990\pi$
$997$	−1.96549e12	−1.98925	−0.994627	+	0.103520i	$0.966990\pi$
$998$	−2.14942e12	−2.16671
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.9.d.a.54.1	✓	1
5.4	even	2		55.9.d.b.54.1	yes	1
11.10	odd	2		55.9.d.b.54.1	yes	1
55.54	odd	2	CM	55.9.d.a.54.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.9.d.a.54.1	✓	1	1.1	even	1	trivial
55.9.d.a.54.1	✓	1	55.54	odd	2	CM
55.9.d.b.54.1	yes	1	5.4	even	2
55.9.d.b.54.1	yes	1	11.10	odd	2

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:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	55.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(12\)	\(46\)
\(\chi(n)\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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