LMFDB - Embedded newform 23.13.b.a.22.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,13,Mod(22,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 13, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 13);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	23.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:	yes
Analytic conductor:	$21.0218577974$
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			22.1
Character	$\chi$	$=$	23.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-79.0000 q^{2} -14.0000 q^{3} +2145.00 q^{4} +1106.00 q^{6} +154129. q^{8} -531245. q^{9} +O(q^{10})$

$q-79.0000 q^{2} -14.0000 q^{3} +2145.00 q^{4} +1106.00 q^{6} +154129. q^{8} -531245. q^{9} -30030.0 q^{12} -8.48289e6 q^{13} -2.09621e7 q^{16} +4.19684e7 q^{18} +1.48036e8 q^{23} -2.15781e6 q^{24} +2.44141e8 q^{25} +6.70149e8 q^{26} +1.48776e7 q^{27} -2.44330e8 q^{29} +1.67703e9 q^{31} +1.02469e9 q^{32} -1.13952e9 q^{36} +1.18761e8 q^{39} -7.59628e9 q^{41} -1.16948e10 q^{46} +2.06069e10 q^{47} +2.93470e8 q^{48} +1.38413e10 q^{49} -1.92871e10 q^{50} -1.81958e10 q^{52} -1.17533e9 q^{54} +1.93021e10 q^{58} -1.98745e10 q^{59} -1.32485e11 q^{62} +4.90995e9 q^{64} -2.07250e9 q^{69} +1.88894e11 q^{71} -8.18803e10 q^{72} +2.23017e11 q^{73} -3.41797e9 q^{75} -9.38208e9 q^{78} +2.82117e11 q^{81} +6.00106e11 q^{82} +3.42062e9 q^{87} +3.17537e11 q^{92} -2.34784e10 q^{93} -1.62795e12 q^{94} -1.43457e10 q^{96} -1.09346e12 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

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 $

$2$	−79.0000	−1.23438	−0.617188	−	0.786816i	$-0.711728\pi$
$2$	−79.0000	−1.23438	−0.617188	+	0.786816i	$0.711728\pi$
$3$	−14.0000	−0.0192044	−0.00960219	−	0.999954i	$-0.503057\pi$
$3$	−14.0000	−0.0192044	−0.00960219	+	0.999954i	$0.503057\pi$
$4$	2145.00	0.523682
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	1106.00	0.0237054
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	154129.	0.587955
$9$	−531245.	−0.999631
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−30030.0	−0.0100570
$13$	−8.48289e6	−1.75745	−0.878727	−	0.477325i	$-0.841607\pi$
$13$	−8.48289e6	−1.75745	−0.878727	+	0.477325i	$0.841607\pi$
$14$	0	0
$15$	0	0
$16$	−2.09621e7	−1.24944
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	4.19684e7	1.23392
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.48036e8	1.00000
$23$	1.48036e8	1.00000
$24$	−2.15781e6	−0.0112913
$25$	2.44141e8	1.00000
$26$	6.70149e8	2.16936
$27$	1.48776e7	0.0384017
$28$	0	0
$29$	−2.44330e8	−0.410761	−0.205380	−	0.978682i	$-0.565843\pi$
$29$	−2.44330e8	−0.410761	−0.205380	+	0.978682i	$0.565843\pi$
$30$	0	0
$31$	1.67703e9	1.88960	0.944799	−	0.327651i	$-0.106257\pi$
$31$	1.67703e9	1.88960	0.944799	+	0.327651i	$0.106257\pi$
$32$	1.02469e9	0.954321
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−1.13952e9	−0.523489
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	1.18761e8	0.0337508
$40$	0	0
$41$	−7.59628e9	−1.59918	−0.799591	−	0.600545i	$-0.794950\pi$
$41$	−7.59628e9	−1.59918	−0.799591	+	0.600545i	$0.794950\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−1.16948e10	−1.23438
$47$	2.06069e10	1.91173	0.955863	−	0.293813i	$-0.0949242\pi$
$47$	2.06069e10	1.91173	0.955863	+	0.293813i	$0.0949242\pi$
$48$	2.93470e8	0.0239947
$49$	1.38413e10	1.00000
$50$	−1.92871e10	−1.23438
$51$	0	0
$52$	−1.81958e10	−0.920346
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−1.17533e9	−0.0474021
$55$	0	0
$56$	0	0
$57$	0	0
$58$	1.93021e10	0.507033
$59$	−1.98745e10	−0.471178	−0.235589	−	0.971853i	$-0.575702\pi$
$59$	−1.98745e10	−0.471178	−0.235589	+	0.971853i	$0.575702\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−1.32485e11	−2.33247
$63$	0	0
$64$	4.90995e9	0.0714492
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−2.07250e9	−0.0192044
$70$	0	0
$71$	1.88894e11	1.47458	0.737289	−	0.675577i	$-0.236106\pi$
$71$	1.88894e11	1.47458	0.737289	+	0.675577i	$0.236106\pi$
$72$	−8.18803e10	−0.587739
$73$	2.23017e11	1.47367	0.736837	−	0.676070i	$-0.236318\pi$
$73$	2.23017e11	1.47367	0.736837	+	0.676070i	$0.236318\pi$
$74$	0	0
$75$	−3.41797e9	−0.0192044
$76$	0	0
$77$	0	0
$78$	−9.38208e9	−0.0416612
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	2.82117e11	0.998894
$82$	6.00106e11	1.97399
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.42062e9	0.00788841
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	3.17537e11	0.523682
$93$	−2.34784e10	−0.0362886
$94$	−1.62795e12	−2.35979
$95$	0	0
$96$	−1.43457e10	−0.0183272
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.09346e12	−1.23438
$99$	0	0
$100$	5.23682e11	0.523682
$101$	−1.86720e12	−1.75899	−0.879496	−	0.475907i	$-0.842120\pi$
$101$	−1.86720e12	−1.75899	−0.879496	+	0.475907i	$0.842120\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−1.30746e12	−1.03330
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	3.19125e10	0.0201103
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−5.24088e11	−0.215108
$117$	4.50650e12	1.75681
$118$	1.57009e12	0.581610
$119$	0	0
$120$	0	0
$121$	3.13843e12	1.00000
$122$	0	0
$123$	1.06348e11	0.0307113
$124$	3.59722e12	0.989548
$125$	0	0
$126$	0	0
$127$	−1.07527e12	−0.256269	−0.128135	−	0.991757i	$-0.540899\pi$
$127$	−1.07527e12	−0.256269	−0.128135	+	0.991757i	$0.540899\pi$
$128$	−4.58503e12	−1.04252
$129$	0	0
$130$	0	0
$131$	9.24098e11	0.182848	0.0914240	−	0.995812i	$-0.470858\pi$
$131$	9.24098e11	0.182848	0.0914240	+	0.995812i	$0.470858\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	1.63728e11	0.0237054
$139$	1.26304e13	1.75118	0.875588	−	0.483059i	$-0.160474\pi$
$139$	1.26304e13	1.75118	0.875588	+	0.483059i	$0.160474\pi$
$140$	0	0
$141$	−2.88497e11	−0.0367135
$142$	−1.49226e13	−1.82018
$143$	0	0
$144$	1.11360e13	1.24898
$145$	0	0
$146$	−1.76184e13	−1.81907
$147$	−1.93778e11	−0.0192044
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	2.70020e11	0.0237054
$151$	1.46028e13	1.23189	0.615947	−	0.787788i	$-0.288774\pi$
$151$	1.46028e13	1.23189	0.615947	+	0.787788i	$0.288774\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	2.54741e11	0.0176747
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−2.22872e13	−1.23301
$163$	−1.69054e12	−0.0901362	−0.0450681	−	0.998984i	$-0.514350\pi$
$163$	−1.69054e12	−0.0901362	−0.0450681	+	0.998984i	$0.514350\pi$
$164$	−1.62940e13	−0.837462
$165$	0	0
$166$	0	0
$167$	−6.47920e12	−0.298691	−0.149346	−	0.988785i	$-0.547717\pi$
$167$	−6.47920e12	−0.298691	−0.149346	+	0.988785i	$0.547717\pi$
$168$	0	0
$169$	4.86614e13	2.08864
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.97576e13	−1.85602	−0.928010	−	0.372555i	$-0.878482\pi$
$173$	−4.97576e13	−1.85602	−0.928010	+	0.372555i	$0.878482\pi$
$174$	−2.70229e11	−0.00973726
$175$	0	0
$176$	0	0
$177$	2.78243e11	0.00904868
$178$	0	0
$179$	−5.04283e13	−1.53305	−0.766525	−	0.642215i	$-0.778016\pi$
$179$	−5.04283e13	−1.53305	−0.766525	+	0.642215i	$0.778016\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	2.28166e13	0.587955
$185$	0	0
$186$	1.85479e12	0.0447937
$187$	0	0
$188$	4.42018e13	1.00114
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−6.87393e10	−0.00137214
$193$	−8.74316e13	−1.69170	−0.845852	−	0.533418i	$-0.820907\pi$
$193$	−8.74316e13	−1.69170	−0.845852	+	0.533418i	$0.820907\pi$
$194$	0	0
$195$	0	0
$196$	2.96896e13	0.523682
$197$	1.07535e14	1.83972	0.919858	−	0.392250i	$-0.128303\pi$
$197$	1.07535e14	1.83972	0.919858	+	0.392250i	$0.128303\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	3.76292e13	0.587955
$201$	0	0
$202$	1.47509e14	2.17125
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.86433e13	−0.999631
$208$	1.77819e14	2.19583
$209$	0	0
$210$	0	0
$211$	−1.52822e13	−0.173177	−0.0865886	−	0.996244i	$-0.527597\pi$
$211$	−1.52822e13	−0.173177	−0.0865886	+	0.996244i	$0.527597\pi$
$212$	0	0
$213$	−2.64451e12	−0.0283184
$214$	0	0
$215$	0	0
$216$	2.29307e12	0.0225785
$217$	0	0
$218$	0	0
$219$	−3.12224e12	−0.0283010
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.44403e14	−1.98737	−0.993683	−	0.112222i	$-0.964203\pi$
$223$	−2.44403e14	−1.98737	−0.993683	+	0.112222i	$0.964203\pi$
$224$	0	0
$225$	−1.29698e14	−0.999631
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	−3.76584e13	−0.241509
$233$	−2.59018e14	−1.61880	−0.809402	−	0.587255i	$-0.800208\pi$
$233$	−2.59018e14	−1.61880	−0.809402	+	0.587255i	$0.800208\pi$
$234$	−3.56013e14	−2.16856
$235$	0	0
$236$	−4.26309e13	−0.246747
$237$	0	0
$238$	0	0
$239$	3.55364e14	1.90672	0.953358	−	0.301841i	$-0.0976011\pi$
$239$	3.55364e14	1.90672	0.953358	+	0.301841i	$0.0976011\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−2.47936e14	−1.23438
$243$	−1.18562e13	−0.0575848
$244$	0	0
$245$	0	0
$246$	−8.40149e12	−0.0379093
$247$	0	0
$248$	2.58478e14	1.11100
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	8.49465e13	0.316332
$255$	0	0
$256$	3.42107e14	1.21541
$257$	−3.90696e13	−0.135594	−0.0677970	−	0.997699i	$-0.521597\pi$
$257$	−3.90696e13	−0.135594	−0.0677970	+	0.997699i	$0.521597\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.29799e14	0.410609
$262$	−7.30037e13	−0.225703
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.80617e14	1.79634	0.898171	−	0.439647i	$-0.144896\pi$
$269$	6.80617e14	1.79634	0.898171	+	0.439647i	$0.144896\pi$
$270$	0	0
$271$	7.77423e14	1.96265	0.981323	−	0.192370i	$-0.0616172\pi$
$271$	7.77423e14	1.96265	0.981323	+	0.192370i	$0.0616172\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−4.44552e12	−0.0100570
$277$	6.27668e14	1.38948	0.694738	−	0.719262i	$-0.255520\pi$
$277$	6.27668e14	1.38948	0.694738	+	0.719262i	$0.255520\pi$
$278$	−9.97805e14	−2.16161
$279$	−8.90911e14	−1.88890
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	2.27912e13	0.0453183
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	4.05177e14	0.772210
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−5.44364e14	−0.953969
$289$	5.82622e14	1.00000
$290$	0	0
$291$	0	0
$292$	4.78372e14	0.771736
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	1.53085e13	0.0237054
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.25577e15	−1.75745
$300$	−7.33154e12	−0.0100570
$301$	0	0
$302$	−1.15362e15	−1.52062
$303$	2.61409e13	0.0337804
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.96848e14	1.07124	0.535622	−	0.844458i	$-0.320077\pi$
$307$	8.96848e14	1.07124	0.535622	+	0.844458i	$0.320077\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.14050e15	−1.26047	−0.630236	−	0.776403i	$-0.717042\pi$
$311$	−1.14050e15	−1.26047	−0.630236	+	0.776403i	$0.717042\pi$
$312$	1.83044e13	0.0198440
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.73669e15	1.71146	0.855728	−	0.517426i	$-0.173110\pi$
$317$	1.73669e15	1.71146	0.855728	+	0.517426i	$0.173110\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	6.05141e14	0.523102
$325$	−2.07102e15	−1.75745
$326$	1.33553e14	0.111262
$327$	0	0
$328$	−1.17081e15	−0.940248
$329$	0	0
$330$	0	0
$331$	2.63014e15	1.99991	0.999954	−	0.00955646i	$-0.00304196\pi$
$331$	2.63014e15	1.99991	0.999954	+	0.00955646i	$0.00304196\pi$
$332$	0	0
$333$	0	0
$334$	5.11857e14	0.368697
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−3.84425e15	−2.57817
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	3.93085e15	2.29103
$347$	−3.49096e15	−1.99971	−0.999856	−	0.0169578i	$-0.994602\pi$
$347$	−3.49096e15	−1.99971	−0.999856	+	0.0169578i	$0.994602\pi$
$348$	7.33723e12	0.00413102
$349$	−3.52370e15	−1.95005	−0.975026	−	0.222089i	$-0.928712\pi$
$349$	−3.52370e15	−1.95005	−0.975026	+	0.222089i	$0.928712\pi$
$350$	0	0
$351$	−1.26205e14	−0.0674892
$352$	0	0
$353$	1.94229e15	1.00384	0.501922	−	0.864913i	$-0.332626\pi$
$353$	1.94229e15	1.00384	0.501922	+	0.864913i	$0.332626\pi$
$354$	−2.19812e13	−0.0111695
$355$	0	0
$356$	0	0
$357$	0	0
$358$	3.98384e15	1.89236
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	2.21331e15	1.00000
$362$	0	0
$363$	−4.39380e13	−0.0192044
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−3.10314e15	−1.24944
$369$	4.03549e15	1.59859
$370$	0	0
$371$	0	0
$372$	−5.03611e13	−0.0190037
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	3.17612e15	1.12401
$377$	2.07263e15	0.721893
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	1.50538e13	0.00492149
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	6.41905e13	0.0200209
$385$	0	0
$386$	6.90709e15	2.08820
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	2.13334e15	0.587955
$393$	−1.29374e13	−0.00351148
$394$	−8.49524e15	−2.27090
$395$	0	0
$396$	0	0
$397$	2.64024e15	0.674373	0.337186	−	0.941438i	$-0.390525\pi$
$397$	2.64024e15	0.674373	0.337186	+	0.941438i	$0.390525\pi$
$398$	0	0
$399$	0	0
$400$	−5.11770e15	−1.24944
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−1.42260e16	−3.32088
$404$	−4.00515e15	−0.921151
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	3.14076e15	0.670957	0.335479	−	0.942048i	$-0.391102\pi$
$409$	3.14076e15	0.670957	0.335479	+	0.942048i	$0.391102\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	6.21282e15	1.23392
$415$	0	0
$416$	−8.69237e15	−1.67718
$417$	−1.76826e14	−0.0336303
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	1.20729e15	0.213766
$423$	−1.09473e16	−1.91102
$424$	0	0
$425$	0	0
$426$	2.08917e14	0.0349555
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−3.11866e14	−0.0479806
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	2.46657e14	0.0349341
$439$	−8.38011e14	−0.117075	−0.0585373	−	0.998285i	$-0.518644\pi$
$439$	−8.38011e14	−0.117075	−0.0585373	+	0.998285i	$0.518644\pi$
$440$	0	0
$441$	−7.35311e15	−0.999631
$442$	0	0
$443$	5.28130e15	0.698745	0.349372	−	0.936984i	$-0.386395\pi$
$443$	5.28130e15	0.698745	0.349372	+	0.936984i	$0.386395\pi$
$444$	0	0
$445$	0	0
$446$	1.93079e16	2.45316
$447$	0	0
$448$	0	0
$449$	8.58859e15	1.04820	0.524100	−	0.851657i	$-0.324402\pi$
$449$	8.58859e15	1.04820	0.524100	+	0.851657i	$0.324402\pi$
$450$	1.02462e16	1.23392
$451$	0	0
$452$	0	0
$453$	−2.04439e14	−0.0236578
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.54631e16	1.61098	0.805491	−	0.592607i	$-0.201901\pi$
$461$	1.54631e16	1.61098	0.805491	+	0.592607i	$0.201901\pi$
$462$	0	0
$463$	−1.58479e16	−1.60874	−0.804371	−	0.594128i	$-0.797497\pi$
$463$	−1.58479e16	−1.60874	−0.804371	+	0.594128i	$0.797497\pi$
$464$	5.12168e15	0.513221
$465$	0	0
$466$	2.04624e16	1.99821
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	9.66643e15	0.920007
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−3.06324e15	−0.277032
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−2.80738e16	−2.35360
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	6.73193e15	0.523682
$485$	0	0
$486$	9.36640e14	0.0710813
$487$	−2.34979e15	−0.176139	−0.0880694	−	0.996114i	$-0.528070\pi$
$487$	−2.34979e15	−0.176139	−0.0880694	+	0.996114i	$0.528070\pi$
$488$	0	0
$489$	2.36675e13	0.00173101
$490$	0	0
$491$	8.36436e15	0.596958	0.298479	−	0.954416i	$-0.403521\pi$
$491$	8.36436e15	0.596958	0.298479	+	0.954416i	$0.403521\pi$
$492$	2.28116e14	0.0160830
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−3.51540e16	−2.36094
$497$	0	0
$498$	0	0
$499$	−3.08763e16	−1.99996	−0.999981	−	0.00610007i	$-0.998058\pi$
$499$	−3.08763e16	−1.99996	−0.999981	+	0.00610007i	$0.998058\pi$
$500$	0	0
$501$	9.07088e13	0.00573618
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.81260e14	−0.0401111
$508$	−2.30646e15	−0.134203
$509$	5.35996e15	0.308216	0.154108	−	0.988054i	$-0.450750\pi$
$509$	5.35996e15	0.308216	0.154108	+	0.988054i	$0.450750\pi$
$510$	0	0
$511$	0	0
$512$	−8.24612e15	−0.457751
$513$	0	0
$514$	3.08650e15	0.167374
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.96606e14	0.0356437
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1.02541e16	−0.506846
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	1.98219e15	0.0957541
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	2.19146e16	1.00000
$530$	0	0
$531$	1.05582e16	0.471004
$532$	0	0
$533$	6.44385e16	2.81049
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.05996e14	0.0294413
$538$	−5.37687e16	−2.21736
$539$	0	0
$540$	0	0
$541$	4.95916e16	1.97799	0.988996	−	0.147939i	$-0.0472640\pi$
$541$	4.95916e16	1.97799	0.988996	+	0.147939i	$0.0472640\pi$
$542$	−6.14164e16	−2.42264
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.72617e16	−1.39104	−0.695518	−	0.718508i	$-0.744825\pi$
$547$	−3.72617e16	−1.39104	−0.695518	+	0.718508i	$0.744825\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−3.19433e14	−0.0112913
$553$	0	0
$554$	−4.95858e16	−1.71514
$555$	0	0
$556$	2.70923e16	0.917059
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	7.03820e16	2.33161
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−6.18825e14	−0.0192262
$565$	0	0
$566$	0	0
$567$	0	0
$568$	2.91140e16	0.866986
$569$	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			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.61416e16	1.00000
$576$	−2.60839e15	−0.0714228
$577$	−6.56902e16	−1.78011	−0.890053	−	0.455857i	$-0.849333\pi$
$577$	−6.56902e16	−1.78011	−0.890053	+	0.455857i	$0.849333\pi$
$578$	−4.60272e16	−1.23438
$579$	1.22404e15	0.0324881
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	3.43735e16	0.866455
$585$	0	0
$586$	0	0
$587$	7.81083e16	1.90927	0.954637	−	0.297772i	$-0.0962434\pi$
$587$	7.81083e16	1.90927	0.954637	+	0.297772i	$0.0962434\pi$
$588$	−4.15654e14	−0.0100570
$589$	0	0
$590$	0	0
$591$	−1.50548e15	−0.0353306
$592$	0	0
$593$	−9.50505e15	−0.218588	−0.109294	−	0.994009i	$-0.534859\pi$
$593$	−9.50505e15	−0.218588	−0.109294	+	0.994009i	$0.534859\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	9.92060e16	2.16936
$599$	−6.60099e16	−1.42905	−0.714526	−	0.699609i	$-0.753358\pi$
$599$	−6.60099e16	−1.42905	−0.714526	+	0.699609i	$0.753358\pi$
$600$	−5.26808e14	−0.0112913
$601$	9.41567e16	1.99804	0.999021	−	0.0442388i	$-0.0140862\pi$
$601$	9.41567e16	1.99804	0.999021	+	0.0442388i	$0.0140862\pi$
$602$	0	0
$603$	0	0
$604$	3.13229e16	0.645120
$605$	0	0
$606$	−2.06513e15	−0.0416976
$607$	3.62355e16	0.724439	0.362220	−	0.932093i	$-0.382019\pi$
$607$	3.62355e16	0.724439	0.362220	+	0.932093i	$0.382019\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.74806e17	−3.35977
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−7.08510e16	−1.32232
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	2.20242e15	0.0384017
$622$	9.00996e16	1.55590
$623$	0	0
$624$	−2.48947e15	−0.0421696
$625$	5.96046e16	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	2.13951e14	0.00332576
$634$	−1.37198e17	−2.11258
$635$	0	0
$636$	0	0
$637$	−1.17414e17	−1.75745
$638$	0	0
$639$	−1.00349e17	−1.47403
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.01249e16	−0.955975	−0.477988	−	0.878367i	$-0.658634\pi$
$647$	−7.01249e16	−0.955975	−0.477988	+	0.878367i	$0.658634\pi$
$648$	4.34824e16	0.587305
$649$	0	0
$650$	1.63611e17	2.16936
$651$	0	0
$652$	−3.62620e15	−0.0472027
$653$	−1.49916e17	−1.93361	−0.966804	−	0.255517i	$-0.917754\pi$
$653$	−1.49916e17	−1.93361	−0.966804	+	0.255517i	$0.917754\pi$
$654$	0	0
$655$	0	0
$656$	1.59234e17	1.99808
$657$	−1.18477e17	−1.47313
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−2.07781e17	−2.46864
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.61696e16	−0.410761
$668$	−1.38979e16	−0.156419
$669$	3.42165e15	0.0381662
$670$	0	0
$671$	0	0
$672$	0	0
$673$	8.28827e16	0.892018	0.446009	−	0.895028i	$-0.352845\pi$
$673$	8.28827e16	0.892018	0.446009	+	0.895028i	$0.352845\pi$
$674$	0	0
$675$	3.63223e15	0.0384017
$676$	1.04379e17	1.09378
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.73218e17	1.70636	0.853178	−	0.521620i	$-0.174672\pi$
$683$	1.73218e17	1.70636	0.853178	+	0.521620i	$0.174672\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.71085e17	1.57160	0.785801	−	0.618479i	$-0.212251\pi$
$691$	1.71085e17	1.57160	0.785801	+	0.618479i	$0.212251\pi$
$692$	−1.06730e17	−0.971964
$693$	0	0
$694$	2.75786e17	2.46840
$695$	0	0
$696$	5.27217e14	0.00463803
$697$	0	0
$698$	2.78372e17	2.40710
$699$	3.62625e15	0.0310881
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	9.97021e15	0.0833070
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−1.53441e17	−1.23912
$707$	0	0
$708$	5.96832e14	0.00473863
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.48260e17	1.88960
$714$	0	0
$715$	0	0
$716$	−1.08169e17	−0.802830
$717$	−4.97510e15	−0.0366173
$718$	0	0
$719$	2.08599e17	1.50987	0.754934	−	0.655801i	$-0.227669\pi$
$719$	2.08599e17	1.50987	0.754934	+	0.655801i	$0.227669\pi$
$720$	0	0
$721$	0	0
$722$	−1.74852e17	−1.23438
$723$	0	0
$724$	0	0
$725$	−5.96509e16	−0.410761
$726$	3.47110e15	0.0237054
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.49763e17	−0.997788
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	1.51692e17	0.954321
$737$	0	0
$738$	−3.18803e17	−1.97326
$739$	−3.09880e17	−1.90251	−0.951254	−	0.308408i	$-0.900204\pi$
$739$	−3.09880e17	−1.90251	−0.951254	+	0.308408i	$0.900204\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−3.61869e15	−0.0213361
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−4.31964e17	−2.38859
$753$	0	0
$754$	−1.63737e17	−0.891087
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.32960e16	0.171429	0.0857146	−	0.996320i	$-0.472683\pi$
$761$	3.32960e16	0.171429	0.0857146	+	0.996320i	$0.472683\pi$
$762$	−1.18925e15	−0.00607497
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.68594e17	0.828073
$768$	−4.78949e15	−0.0233411
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	5.46974e14	0.00260400
$772$	−1.87541e17	−0.885914
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	4.09430e17	1.88960
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−3.63505e15	−0.0157739
$784$	−2.90143e17	−1.24944
$785$	0	0
$786$	1.02205e15	0.00433449
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	2.30662e17	0.963426
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−2.08579e17	−0.832429
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	2.50170e17	0.954321
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	1.12386e18	4.09921
$807$	−9.52863e15	−0.0344976
$808$	−2.87790e17	−1.03421
$809$	−4.80287e17	−1.71321	−0.856603	−	0.515976i	$-0.827429\pi$
$809$	−4.80287e17	−1.71321	−0.856603	+	0.515976i	$0.827429\pi$
$810$	0	0
$811$	5.45039e17	1.91559	0.957794	−	0.287454i	$-0.0928089\pi$
$811$	5.45039e17	1.91559	0.957794	+	0.287454i	$0.0928089\pi$
$812$	0	0
$813$	−1.08839e16	−0.0376914
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−2.48120e17	−0.828213
$819$	0	0
$820$	0	0
$821$	−3.54020e17	−1.15603	−0.578014	−	0.816027i	$-0.696172\pi$
$821$	−3.54020e17	−1.15603	−0.578014	+	0.816027i	$0.696172\pi$
$822$	0	0
$823$	2.76481e16	0.0889745	0.0444872	−	0.999010i	$-0.485835\pi$
$823$	2.76481e16	0.0889745	0.0444872	+	0.999010i	$0.485835\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−1.68690e17	−0.523489
$829$	5.93576e17	1.82873	0.914364	−	0.404892i	$-0.132691\pi$
$829$	5.93576e17	1.82873	0.914364	+	0.404892i	$0.132691\pi$
$830$	0	0
$831$	−8.78735e15	−0.0266841
$832$	−4.16506e16	−0.125569
$833$	0	0
$834$	1.39693e16	0.0415124
$835$	0	0
$836$	0	0
$837$	2.49501e16	0.0725638
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−2.94118e17	−0.831276
$842$	0	0
$843$	0	0
$844$	−3.27803e16	−0.0906897
$845$	0	0
$846$	8.64838e17	2.35892
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−5.67248e15	−0.0148298
$853$	−3.65113e17	−0.947835	−0.473917	−	0.880569i	$-0.657160\pi$
$853$	−3.65113e17	−0.947835	−0.473917	+	0.880569i	$0.657160\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.29616e17	−1.33683	−0.668414	−	0.743789i	$-0.733027\pi$
$857$	−5.29616e17	−1.33683	−0.668414	+	0.743789i	$0.733027\pi$
$858$	0	0
$859$	8.02395e17	1.99724	0.998618	−	0.0525528i	$-0.0167358\pi$
$859$	8.02395e17	1.99724	0.998618	+	0.0525528i	$0.0167358\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.65593e17	−1.61118	−0.805590	−	0.592474i	$-0.798151\pi$
$863$	−6.65593e17	−1.61118	−0.805590	+	0.592474i	$0.798151\pi$
$864$	1.52450e16	0.0366475
$865$	0	0
$866$	0	0
$867$	−8.15671e15	−0.0192044
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−6.69721e15	−0.0148207
$877$	−8.75030e17	−1.92320	−0.961602	−	0.274446i	$-0.911505\pi$
$877$	−8.75030e17	−1.92320	−0.961602	+	0.274446i	$0.911505\pi$
$878$	6.62028e16	0.144514
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	5.80896e17	1.23392
$883$	9.25511e17	1.95262	0.976309	−	0.216382i	$-0.0694257\pi$
$883$	9.25511e17	1.95262	0.976309	+	0.216382i	$0.0694257\pi$
$884$	0	0
$885$	0	0
$886$	−4.17223e17	−0.862513
$887$	9.58929e17	1.96900	0.984498	−	0.175394i	$-0.0561201\pi$
$887$	9.58929e17	1.96900	0.984498	+	0.175394i	$0.0561201\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−5.24245e17	−1.04075
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.75808e16	0.0337508
$898$	−6.78499e17	−1.29387
$899$	−4.09748e17	−0.776173
$900$	−2.78203e17	−0.523489
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	1.61506e16	0.0292025
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	9.91943e17	1.75834
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1.25559e16	−0.0205726
$922$	−1.22158e18	−1.98856
$923$	−1.60237e18	−2.59150
$924$	0	0
$925$	0	0
$926$	1.25199e18	1.98579
$927$	0	0
$928$	−2.50364e17	−0.391998
$929$	−5.75741e17	−0.895639	−0.447819	−	0.894124i	$-0.647799\pi$
$929$	−5.75741e17	−0.895639	−0.447819	+	0.894124i	$0.647799\pi$
$930$	0	0
$931$	0	0
$932$	−5.55593e17	−0.847738
$933$	1.59670e16	0.0242066
$934$	0	0
$935$	0	0
$936$	6.94582e17	1.03292
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−1.12452e18	−1.59918
$944$	4.16612e17	0.588708
$945$	0	0
$946$	0	0
$947$	3.24905e17	0.450460	0.225230	−	0.974306i	$-0.427687\pi$
$947$	3.24905e17	0.450460	0.225230	+	0.974306i	$0.427687\pi$
$948$	0	0
$949$	−1.89183e18	−2.58992
$950$	0	0
$951$	−2.43136e16	−0.0328675
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	7.62256e17	0.998513
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.02475e18	2.57058
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.37846e18	1.68591	0.842954	−	0.537985i	$-0.180814\pi$
$967$	1.37846e18	1.68591	0.842954	+	0.537985i	$0.180814\pi$
$968$	4.83723e17	0.587955
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−2.54316e16	−0.0301561
$973$	0	0
$974$	1.85633e17	0.217421
$975$	2.89943e16	0.0337508
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−1.86974e15	−0.00213672
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−6.60784e17	−0.736870
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	1.63913e16	0.0180569
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1.57698e18	1.66488	0.832442	−	0.554112i	$-0.186942\pi$
$991$	1.57698e18	1.66488	0.832442	+	0.554112i	$0.186942\pi$
$992$	1.71844e18	1.80328
$993$	−3.68219e16	−0.0384070
$994$	0	0
$995$	0	0
$996$	0	0
$997$	7.27897e17	0.741138	0.370569	−	0.928805i	$-0.379163\pi$
$997$	7.27897e17	0.741138	0.370569	+	0.928805i	$0.379163\pi$
$998$	2.43923e18	2.46870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.a.22.1	✓	1
23.22	odd	2	CM	23.13.b.a.22.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.a.22.1	✓	1	1.1	even	1	trivial
23.13.b.a.22.1	✓	1	23.22	odd	2	CM

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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(n\)	\(5\)
\(\chi(n)\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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