LMFDB - Embedded newform 23.25.b.a.22.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,25,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, 25, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 25);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 25 $
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:	$83.9424450193$
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)

$f(q)$

$=$

$q-1951.00 q^{2} -1.06269e6 q^{3} -1.29708e7 q^{4} +2.07330e9 q^{6} +5.80384e10 q^{8} +8.46872e11 q^{9} +O(q^{10})$

$q-1951.00 q^{2} -1.06269e6 q^{3} -1.29708e7 q^{4} +2.07330e9 q^{6} +5.80384e10 q^{8} +8.46872e11 q^{9} +1.37839e13 q^{12} +2.53633e13 q^{13} +1.04381e14 q^{16} -1.65225e15 q^{18} +2.19146e16 q^{23} -6.16766e16 q^{24} +5.96046e16 q^{25} -4.94838e16 q^{26} -5.99825e17 q^{27} -6.47932e17 q^{29} +1.23709e18 q^{31} -1.17737e18 q^{32} -1.09846e19 q^{36} -2.69532e19 q^{39} +1.25765e19 q^{41} -4.27554e19 q^{46} +1.92262e20 q^{47} -1.10924e20 q^{48} +1.91581e20 q^{49} -1.16289e20 q^{50} -3.28983e20 q^{52} +1.17026e21 q^{54} +1.26412e21 q^{58} -3.16340e21 q^{59} -2.41356e21 q^{62} +5.45824e20 q^{64} -2.32884e22 q^{69} +2.86154e21 q^{71} +4.91511e22 q^{72} +3.93269e21 q^{73} -6.33410e22 q^{75} +5.25858e22 q^{78} +3.98244e23 q^{81} -2.45368e22 q^{82} +6.88549e23 q^{87} -2.84251e23 q^{92} -1.31464e24 q^{93} -3.75102e23 q^{94} +1.25118e24 q^{96} -3.73775e23 q^{98} +O(q^{100})$

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 $

$2$	−1951.00	−0.476318	−0.238159	−	0.971226i	$-0.576544\pi$
$2$	−1951.00	−0.476318	−0.238159	+	0.971226i	$0.576544\pi$
$3$	−1.06269e6	−1.99963	−0.999816	−	0.0192035i	$-0.993887\pi$
$3$	−1.06269e6	−1.99963	−0.999816	+	0.0192035i	$0.993887\pi$
$4$	−1.29708e7	−0.773121
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	2.07330e9	0.952461
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	5.80384e10	0.844570
$9$	8.46872e11	2.99852
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1.37839e13	1.54596
$13$	2.53633e13	1.08864	0.544322	−	0.838876i	$-0.316787\pi$
$13$	2.53633e13	1.08864	0.544322	+	0.838876i	$0.316787\pi$
$14$	0	0
$15$	0	0
$16$	1.04381e14	0.370837
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−1.65225e15	−1.42825
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.19146e16	1.00000
$23$	2.19146e16	1.00000
$24$	−6.16766e16	−1.68883
$25$	5.96046e16	1.00000
$26$	−4.94838e16	−0.518541
$27$	−5.99825e17	−3.99631
$28$	0	0
$29$	−6.47932e17	−1.83128	−0.915638	−	0.402004i	$-0.868314\pi$
$29$	−6.47932e17	−1.83128	−0.915638	+	0.402004i	$0.868314\pi$
$30$	0	0
$31$	1.23709e18	1.57058	0.785290	−	0.619128i	$-0.212514\pi$
$31$	1.23709e18	1.57058	0.785290	+	0.619128i	$0.212514\pi$
$32$	−1.17737e18	−1.02121
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−1.09846e19	−2.31822
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−2.69532e19	−2.17689
$40$	0	0
$41$	1.25765e19	0.557384	0.278692	−	0.960381i	$-0.410099\pi$
$41$	1.25765e19	0.557384	0.278692	+	0.960381i	$0.410099\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−4.27554e19	−0.476318
$47$	1.92262e20	1.65470	0.827348	−	0.561689i	$-0.189848\pi$
$47$	1.92262e20	1.65470	0.827348	+	0.561689i	$0.189848\pi$
$48$	−1.10924e20	−0.741536
$49$	1.91581e20	1.00000
$50$	−1.16289e20	−0.476318
$51$	0	0
$52$	−3.28983e20	−0.841653
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	1.17026e21	1.90352
$55$	0	0
$56$	0	0
$57$	0	0
$58$	1.26412e21	0.872270
$59$	−3.16340e21	−1.77799	−0.888996	−	0.457915i	$-0.848596\pi$
$59$	−3.16340e21	−1.77799	−0.888996	+	0.457915i	$0.848596\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−2.41356e21	−0.748096
$63$	0	0
$64$	5.45824e20	0.115583
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−2.32884e22	−1.99963
$70$	0	0
$71$	2.86154e21	0.174381	0.0871905	−	0.996192i	$-0.472211\pi$
$71$	2.86154e21	0.174381	0.0871905	+	0.996192i	$0.472211\pi$
$72$	4.91511e22	2.53246
$73$	3.93269e21	0.171718	0.0858588	−	0.996307i	$-0.472637\pi$
$73$	3.93269e21	0.171718	0.0858588	+	0.996307i	$0.472637\pi$
$74$	0	0
$75$	−6.33410e22	−1.99963
$76$	0	0
$77$	0	0
$78$	5.25858e22	1.03689
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	3.98244e23	4.99263
$82$	−2.45368e22	−0.265492
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.88549e23	3.66188
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−2.84251e23	−0.773121
$93$	−1.31464e24	−3.14058
$94$	−3.75102e23	−0.788162
$95$	0	0
$96$	1.25118e24	2.04204
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−3.73775e23	−0.476318
$99$	0	0
$100$	−7.73121e23	−0.773121
$101$	1.23280e24	1.09405	0.547025	−	0.837116i	$-0.315760\pi$
$101$	1.23280e24	1.09405	0.547025	+	0.837116i	$0.315760\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	1.47205e24	0.919436
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	7.78022e24	3.08963
$109$	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
$116$	8.40421e24	1.41580
$117$	2.14795e25	3.26433
$118$	6.17179e24	0.846890
$119$	0	0
$120$	0	0
$121$	9.84973e24	1.00000
$122$	0	0
$123$	−1.33649e25	−1.11456
$124$	−1.60460e25	−1.21425
$125$	0	0
$126$	0	0
$127$	−3.40545e25	−1.93433	−0.967163	−	0.254157i	$-0.918202\pi$
$127$	−3.40545e25	−1.93433	−0.967163	+	0.254157i	$0.918202\pi$
$128$	1.86881e25	0.966152
$129$	0	0
$130$	0	0
$131$	−5.02301e25	−1.96657	−0.983283	−	0.182082i	$-0.941716\pi$
$131$	−5.02301e25	−1.96657	−0.983283	+	0.182082i	$0.941716\pi$
$132$	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			$\pi$
$138$	4.54356e25	0.952461
$139$	5.54863e25	1.06662	0.533308	−	0.845921i	$-0.320949\pi$
$139$	5.54863e25	1.06662	0.533308	+	0.845921i	$0.320949\pi$
$140$	0	0
$141$	−2.04314e26	−3.30878
$142$	−5.58286e24	−0.0830609
$143$	0	0
$144$	8.83975e25	1.11196
$145$	0	0
$146$	−7.67267e24	−0.0817923
$147$	−2.03591e26	−1.99963
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.23578e26	0.952461
$151$	−6.77900e25	−0.482439	−0.241219	−	0.970471i	$-0.577547\pi$
$151$	−6.77900e25	−0.482439	−0.241219	+	0.970471i	$0.577547\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	3.49606e26	1.68300
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−7.76974e26	−2.37808
$163$	−7.00670e26	−1.99188	−0.995938	−	0.0900446i	$-0.971299\pi$
$163$	−7.00670e26	−1.99188	−0.995938	+	0.0900446i	$0.971299\pi$
$164$	−1.63128e26	−0.430925
$165$	0	0
$166$	0	0
$167$	−8.99102e26	−1.91078	−0.955392	−	0.295341i	$-0.904567\pi$
$167$	−8.99102e26	−1.91078	−0.955392	+	0.295341i	$0.904567\pi$
$168$	0	0
$169$	1.00497e26	0.185146
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.03840e27	1.44481	0.722406	−	0.691469i	$-0.243036\pi$
$173$	1.03840e27	1.44481	0.722406	+	0.691469i	$0.243036\pi$
$174$	−1.34336e27	−1.74422
$175$	0	0
$176$	0	0
$177$	3.36170e27	3.55533
$178$	0	0
$179$	3.78968e26	0.350241	0.175120	−	0.984547i	$-0.443969\pi$
$179$	3.78968e26	0.350241	0.175120	+	0.984547i	$0.443969\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	1.27189e27	0.844570
$185$	0	0
$186$	2.56485e27	1.49592
$187$	0	0
$188$	−2.49379e27	−1.27928
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−5.80039e26	−0.231123
$193$	2.30211e27	0.861862	0.430931	−	0.902385i	$-0.358185\pi$
$193$	2.30211e27	0.861862	0.430931	+	0.902385i	$0.358185\pi$
$194$	0	0
$195$	0	0
$196$	−2.48496e27	−0.773121
$197$	4.73049e27	1.38456	0.692279	−	0.721630i	$-0.256607\pi$
$197$	4.73049e27	1.38456	0.692279	+	0.721630i	$0.256607\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	3.45936e27	0.844570
$201$	0	0
$202$	−2.40520e27	−0.521116
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.85589e28	2.99852
$208$	2.64745e27	0.403709
$209$	0	0
$210$	0	0
$211$	−1.53411e28	−1.97001	−0.985005	−	0.172527i	$-0.944807\pi$
$211$	−1.53411e28	−1.97001	−0.985005	+	0.172527i	$0.944807\pi$
$212$	0	0
$213$	−3.04092e27	−0.348698
$214$	0	0
$215$	0	0
$216$	−3.48129e28	−3.37517
$217$	0	0
$218$	0	0
$219$	−4.17921e27	−0.343372
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.94856e28	1.94962	0.974812	−	0.223027i	$-0.0715937\pi$
$223$	2.94856e28	1.94962	0.974812	+	0.223027i	$0.0715937\pi$
$224$	0	0
$225$	5.04775e28	2.99852
$226$	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
$232$	−3.76050e28	−1.54664
$233$	1.58866e28	0.620525	0.310262	−	0.950651i	$-0.399583\pi$
$233$	1.58866e28	0.620525	0.310262	+	0.950651i	$0.399583\pi$
$234$	−4.19065e28	−1.55486
$235$	0	0
$236$	4.10318e28	1.37460
$237$	0	0
$238$	0	0
$239$	5.68125e28	1.63557	0.817784	−	0.575525i	$-0.195202\pi$
$239$	5.68125e28	1.63557	0.817784	+	0.575525i	$0.195202\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−1.92168e28	−0.476318
$243$	−2.53800e29	−5.98710
$244$	0	0
$245$	0	0
$246$	2.60749e28	0.530887
$247$	0	0
$248$	7.17986e28	1.32647
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	6.64403e28	0.921355
$255$	0	0
$256$	−4.56179e28	−0.575779
$257$	−1.64519e29	−1.98161	−0.990807	−	0.135282i	$-0.956806\pi$
$257$	−1.64519e29	−1.98161	−0.990807	+	0.135282i	$0.956806\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5.48716e29	−5.49113
$262$	9.79990e28	0.936712
$263$	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
$269$	1.76123e29	1.22684	0.613422	−	0.789755i	$-0.289793\pi$
$269$	1.76123e29	1.22684	0.613422	+	0.789755i	$0.289793\pi$
$270$	0	0
$271$	2.90581e29	1.85198	0.925988	−	0.377553i	$-0.123234\pi$
$271$	2.90581e29	1.85198	0.925988	+	0.377553i	$0.123234\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	3.02069e29	1.54596
$277$	−1.41523e28	−0.0693537	−0.0346768	−	0.999399i	$-0.511040\pi$
$277$	−1.41523e28	−0.0693537	−0.0346768	+	0.999399i	$0.511040\pi$
$278$	−1.08254e29	−0.508049
$279$	1.04766e30	4.70942
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	3.98616e29	1.57603
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−3.71165e28	−0.134818
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−9.97082e29	−3.06211
$289$	3.39449e29	1.00000
$290$	0	0
$291$	0	0
$292$	−5.10101e28	−0.132759
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	3.97205e29	0.952461
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.55828e29	1.08864
$300$	8.21585e29	1.54596
$301$	0	0
$302$	1.32258e29	0.229794
$303$	−1.31008e30	−2.18770
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5.97478e29	−0.852435	−0.426218	−	0.904621i	$-0.640154\pi$
$307$	−5.97478e29	−0.852435	−0.426218	+	0.904621i	$0.640154\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.36656e29	−0.411208	−0.205604	−	0.978635i	$-0.565916\pi$
$311$	−3.36656e29	−0.411208	−0.205604	+	0.978635i	$0.565916\pi$
$312$	−1.56432e30	−1.83853
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.56676e29	0.929081	0.464541	−	0.885552i	$-0.346219\pi$
$317$	9.56676e29	0.929081	0.464541	+	0.885552i	$0.346219\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−5.16555e30	−3.85990
$325$	1.51177e30	1.08864
$326$	1.36701e30	0.948767
$327$	0	0
$328$	7.29922e29	0.470750
$329$	0	0
$330$	0	0
$331$	3.45849e30	1.99963	0.999817	−	0.0191120i	$-0.00608392\pi$
$331$	3.45849e30	1.99963	0.999817	+	0.0191120i	$0.00608392\pi$
$332$	0	0
$333$	0	0
$334$	1.75415e30	0.910141
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−1.96070e29	−0.0881883
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−2.02592e30	−0.688190
$347$	6.09163e30	1.99885	0.999425	−	0.0339107i	$-0.0107962\pi$
$347$	6.09163e30	1.99885	0.999425	+	0.0339107i	$0.0107962\pi$
$348$	−8.93104e30	−2.83107
$349$	5.88613e30	1.80271	0.901353	−	0.433085i	$-0.142575\pi$
$349$	5.88613e30	1.80271	0.901353	+	0.433085i	$0.142575\pi$
$350$	0	0
$351$	−1.52136e31	−4.35056
$352$	0	0
$353$	−3.71482e30	−0.992297	−0.496148	−	0.868238i	$-0.665253\pi$
$353$	−3.71482e30	−0.992297	−0.496148	+	0.868238i	$0.665253\pi$
$354$	−6.55867e30	−1.69347
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−7.39367e29	−0.166826
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	4.89876e30	1.00000
$362$	0	0
$363$	−1.04672e31	−1.99963
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	2.28748e30	0.370837
$369$	1.06507e31	1.67133
$370$	0	0
$371$	0	0
$372$	1.70519e31	2.42805
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	1.11586e31	1.39751
$377$	−1.64337e31	−1.99361
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	3.61892e31	3.86794
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.98596e31	−1.93195
$385$	0	0
$386$	−4.49141e30	−0.410521
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	1.11191e31	0.844570
$393$	5.33788e31	3.93241
$394$	−9.22918e30	−0.659491
$395$	0	0
$396$	0	0
$397$	−2.36852e31	−1.54522	−0.772611	−	0.634880i	$-0.781049\pi$
$397$	−2.36852e31	−1.54522	−0.772611	+	0.634880i	$0.781049\pi$
$398$	0	0
$399$	0	0
$400$	6.22161e30	0.370837
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	3.13767e31	1.70980
$404$	−1.59905e31	−0.845833
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−3.39594e31	−1.54982	−0.774908	−	0.632074i	$-0.782204\pi$
$409$	−3.39594e31	−1.54982	−0.774908	+	0.632074i	$0.782204\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−3.62084e31	−1.42825
$415$	0	0
$416$	−2.98620e31	−1.11173
$417$	−5.89645e31	−2.13284
$418$	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			$\pi$
$422$	2.99306e31	0.938352
$423$	1.62821e32	4.96165
$424$	0	0
$425$	0	0
$426$	5.93283e30	0.166091
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−6.26105e31	−1.48198
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	8.15364e30	0.163554
$439$	−1.01770e32	−1.98629	−0.993147	−	0.116874i	$-0.962713\pi$
$439$	−1.01770e32	−1.98629	−0.993147	+	0.116874i	$0.962713\pi$
$440$	0	0
$441$	1.62245e32	2.99852
$442$	0	0
$443$	−8.63627e31	−1.51176	−0.755878	−	0.654713i	$-0.772790\pi$
$443$	−8.63627e31	−1.51176	−0.755878	+	0.654713i	$0.772790\pi$
$444$	0	0
$445$	0	0
$446$	−5.75263e31	−0.928642
$447$	0	0
$448$	0	0
$449$	−6.05083e31	−0.901278	−0.450639	−	0.892706i	$-0.648804\pi$
$449$	−6.05083e31	−0.901278	−0.450639	+	0.892706i	$0.648804\pi$
$450$	−9.84816e31	−1.42825
$451$	0	0
$452$	0	0
$453$	7.20395e31	0.964699
$454$	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
$461$	5.48431e31	0.595266	0.297633	−	0.954680i	$-0.403803\pi$
$461$	5.48431e31	0.595266	0.297633	+	0.954680i	$0.403803\pi$
$462$	0	0
$463$	5.70671e31	0.588050	0.294025	−	0.955798i	$-0.405005\pi$
$463$	5.70671e31	0.588050	0.294025	+	0.955798i	$0.405005\pi$
$464$	−6.76320e31	−0.679104
$465$	0	0
$466$	−3.09947e31	−0.295567
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−2.78606e32	−2.52372
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.83599e32	−1.50164
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−1.10841e32	−0.779051
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−1.27759e32	−0.773121
$485$	0	0
$486$	4.95164e32	2.85177
$487$	−3.50419e32	−1.96898	−0.984488	−	0.175454i	$-0.943861\pi$
$487$	−3.50419e32	−1.96898	−0.984488	+	0.175454i	$0.943861\pi$
$488$	0	0
$489$	7.44592e32	3.98302
$490$	0	0
$491$	−3.22690e32	−1.64364	−0.821821	−	0.569746i	$-0.807042\pi$
$491$	−3.22690e32	−1.64364	−0.821821	+	0.569746i	$0.807042\pi$
$492$	1.73354e32	0.861691
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.29129e32	0.582429
$497$	0	0
$498$	0	0
$499$	4.76655e32	1.99985	0.999926	−	0.0121999i	$-0.00388345\pi$
$499$	4.76655e32	1.99985	0.999926	+	0.0121999i	$0.00388345\pi$
$500$	0	0
$501$	9.55464e32	3.82086
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.06797e32	−0.370223
$508$	4.41714e32	1.49547
$509$	−5.76115e32	−1.90500	−0.952502	−	0.304534i	$-0.901499\pi$
$509$	−5.76115e32	−1.90500	−0.952502	+	0.304534i	$0.901499\pi$
$510$	0	0
$511$	0	0
$512$	−2.24534e32	−0.691898
$513$	0	0
$514$	3.20977e32	0.943879
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−1.10349e33	−2.88909
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	1.07054e33	2.61552
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	6.51526e32	1.52039
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	4.80251e32	1.00000
$530$	0	0
$531$	−2.67899e33	−5.33135
$532$	0	0
$533$	3.18982e32	0.606793
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.02724e32	−0.700352
$538$	−3.43616e32	−0.584368
$539$	0	0
$540$	0	0
$541$	1.20215e33	1.91246	0.956228	−	0.292623i	$-0.0945281\pi$
$541$	1.20215e33	1.91246	0.956228	+	0.292623i	$0.0945281\pi$
$542$	−5.66923e32	−0.882130
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−4.66521e31	−0.0650164	−0.0325082	−	0.999471i	$-0.510350\pi$
$547$	−4.66521e31	−0.0650164	−0.0325082	+	0.999471i	$0.510350\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−1.35162e33	−1.68883
$553$	0	0
$554$	2.76111e31	0.0330344
$555$	0	0
$556$	−7.19703e32	−0.824623
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−2.04397e33	−2.24319
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	2.65012e33	2.55809
$565$	0	0
$566$	0	0
$567$	0	0
$568$	1.66079e32	0.147277
$569$	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
$575$	1.30621e33	1.00000
$576$	4.62243e32	0.346578
$577$	1.59163e33	1.16878	0.584388	−	0.811474i	$-0.301334\pi$
$577$	1.59163e33	1.16878	0.584388	+	0.811474i	$0.301334\pi$
$578$	−6.62264e32	−0.476318
$579$	−2.44642e33	−1.72341
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	2.28247e32	0.145028
$585$	0	0
$586$	0	0
$587$	2.75366e33	1.64533	0.822664	−	0.568528i	$-0.192487\pi$
$587$	2.75366e33	1.64533	0.822664	+	0.568528i	$0.192487\pi$
$588$	2.64074e33	1.54596
$589$	0	0
$590$	0	0
$591$	−5.02702e33	−2.76861
$592$	0	0
$593$	−3.69135e33	−1.95222	−0.976110	−	0.217279i	$-0.930282\pi$
$593$	−3.69135e33	−1.95222	−0.976110	+	0.217279i	$0.930282\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−1.08442e33	−0.518541
$599$	9.00167e31	0.0421892	0.0210946	−	0.999777i	$-0.493285\pi$
$599$	9.00167e31	0.0421892	0.0210946	+	0.999777i	$0.493285\pi$
$600$	−3.67621e33	−1.68883
$601$	4.42405e33	1.99217	0.996086	−	0.0883909i	$-0.0281725\pi$
$601$	4.42405e33	1.99217	0.996086	+	0.0883909i	$0.0281725\pi$
$602$	0	0
$603$	0	0
$604$	8.79291e32	0.372983
$605$	0	0
$606$	2.55597e33	1.04204
$607$	−3.69072e33	−1.47519	−0.737594	−	0.675244i	$-0.764038\pi$
$607$	−3.69072e33	−1.47519	−0.737594	+	0.675244i	$0.764038\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.87639e33	1.80138
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.16568e33	0.406031
$615$	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
$621$	−1.31449e34	−3.99631
$622$	6.56816e32	0.195866
$623$	0	0
$624$	−2.81341e33	−0.807269
$625$	3.55271e33	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
$633$	1.63028e34	3.93929
$634$	−1.86647e33	−0.442538
$635$	0	0
$636$	0	0
$637$	4.85914e33	1.08864
$638$	0	0
$639$	2.42336e33	0.522886
$640$	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
$647$	−5.84421e33	−1.08611	−0.543055	−	0.839697i	$-0.682733\pi$
$647$	−5.84421e33	−1.08611	−0.543055	+	0.839697i	$0.682733\pi$
$648$	2.31135e34	4.21662
$649$	0	0
$650$	−2.94947e33	−0.518541
$651$	0	0
$652$	9.08826e33	1.53996
$653$	1.04525e34	1.73884	0.869422	−	0.494071i	$-0.164492\pi$
$653$	1.04525e34	1.73884	0.869422	+	0.494071i	$0.164492\pi$
$654$	0	0
$655$	0	0
$656$	1.31275e33	0.206698
$657$	3.33048e33	0.514900
$658$	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			$\pi$
$662$	−6.74751e33	−0.952463
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.41992e34	−1.83128
$668$	1.16621e34	1.47727
$669$	−3.13339e34	−3.89853
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.03972e34	−1.20430	−0.602152	−	0.798382i	$-0.705690\pi$
$673$	−1.03972e34	−1.20430	−0.602152	+	0.798382i	$0.705690\pi$
$674$	0	0
$675$	−3.57524e34	−3.99631
$676$	−1.30353e33	−0.143140
$677$	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
$683$	9.39455e33	0.911649	0.455824	−	0.890070i	$-0.349344\pi$
$683$	9.39455e33	0.911649	0.455824	+	0.890070i	$0.349344\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	5.56894e33	0.469933	0.234967	−	0.972003i	$-0.424502\pi$
$691$	5.56894e33	0.469933	0.234967	+	0.972003i	$0.424502\pi$
$692$	−1.34689e34	−1.11701
$693$	0	0
$694$	−1.18848e34	−0.952089
$695$	0	0
$696$	3.99623e34	3.09271
$697$	0	0
$698$	−1.14838e34	−0.858662
$699$	−1.68824e34	−1.24082
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	2.96816e34	2.07225
$703$	0	0
$704$	0	0
$705$	0	0
$706$	7.24762e33	0.472649
$707$	0	0
$708$	−4.36040e34	−2.74870
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.71103e34	1.57058
$714$	0	0
$715$	0	0
$716$	−4.91553e33	−0.270778
$717$	−6.03738e34	−3.27053
$718$	0	0
$719$	5.33879e33	0.279702	0.139851	−	0.990173i	$-0.455338\pi$
$719$	5.33879e33	0.279702	0.139851	+	0.990173i	$0.455338\pi$
$720$	0	0
$721$	0	0
$722$	−9.55749e33	−0.476318
$723$	0	0
$724$	0	0
$725$	−3.86198e34	−1.83128
$726$	2.04215e34	0.952461
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.57234e35	6.97936
$730$	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
$736$	−2.58016e34	−1.02121
$737$	0	0
$738$	−2.07795e34	−0.796085
$739$	4.29660e34	1.61954	0.809769	−	0.586749i	$-0.199592\pi$
$739$	4.29660e34	1.61954	0.809769	+	0.586749i	$0.199592\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−7.62994e34	−2.65244
$745$	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			$\pi$
$752$	2.00685e34	0.613622
$753$	0	0
$754$	3.20622e34	0.949592
$755$	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
$761$	−7.43391e34	−1.97061	−0.985306	−	0.170798i	$-0.945365\pi$
$761$	−7.43391e34	−1.97061	−0.985306	+	0.170798i	$0.945365\pi$
$762$	−7.06052e34	−1.84237
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.02343e34	−1.93560
$768$	4.84775e34	1.15135
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	1.74832e35	3.96250
$772$	−2.98602e34	−0.666324
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	7.37362e34	1.57058
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.88646e35	7.31835
$784$	1.99975e34	0.370837
$785$	0	0
$786$	−1.04142e35	−1.87308
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−6.13583e34	−1.07043
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	4.62098e34	0.736017
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−7.01768e34	−1.02121
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−6.12159e34	−0.814411
$807$	−1.87164e35	−2.45323
$808$	7.15499e34	0.924002
$809$	7.34901e34	0.935075	0.467538	−	0.883973i	$-0.345141\pi$
$809$	7.34901e34	0.935075	0.467538	+	0.883973i	$0.345141\pi$
$810$	0	0
$811$	1.35155e35	1.66948	0.834740	−	0.550644i	$-0.185618\pi$
$811$	1.35155e35	1.66948	0.834740	+	0.550644i	$0.185618\pi$
$812$	0	0
$813$	−3.08796e35	−3.70327
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	6.62548e34	0.738206
$819$	0	0
$820$	0	0
$821$	−6.22334e34	−0.663599	−0.331800	−	0.943350i	$-0.607656\pi$
$821$	−6.22334e34	−0.663599	−0.331800	+	0.943350i	$0.607656\pi$
$822$	0	0
$823$	−1.92356e35	−1.99208	−0.996042	−	0.0888864i	$-0.971669\pi$
$823$	−1.92356e35	−1.99208	−0.996042	+	0.0888864i	$0.971669\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−2.40724e35	−2.31822
$829$	1.41623e35	1.34425	0.672124	−	0.740438i	$-0.265382\pi$
$829$	1.41623e35	1.34425	0.672124	+	0.740438i	$0.265382\pi$
$830$	0	0
$831$	1.50394e34	0.138682
$832$	1.38439e34	0.125828
$833$	0	0
$834$	1.15040e35	1.01591
$835$	0	0
$836$	0	0
$837$	−7.42036e35	−6.27653
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.94631e35	2.35357
$842$	0	0
$843$	0	0
$844$	1.98987e35	1.52306
$845$	0	0
$846$	−3.17664e35	−2.36332
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	3.94431e34	0.269585
$853$	−1.63462e35	−1.10161	−0.550805	−	0.834634i	$-0.685679\pi$
$853$	−1.63462e35	−1.10161	−0.550805	+	0.834634i	$0.685679\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.34135e34	−0.212889	−0.106444	−	0.994319i	$-0.533947\pi$
$857$	−3.34135e34	−0.212889	−0.106444	+	0.994319i	$0.533947\pi$
$858$	0	0
$859$	3.21028e35	1.98895	0.994476	−	0.104960i	$-0.0334716\pi$
$859$	3.21028e35	1.98895	0.994476	+	0.104960i	$0.0334716\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.01696e35	0.595900	0.297950	−	0.954582i	$-0.403697\pi$
$863$	1.01696e35	0.595900	0.297950	+	0.954582i	$0.403697\pi$
$864$	7.06216e35	4.08106
$865$	0	0
$866$	0	0
$867$	−3.60727e35	−1.99963
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	5.42078e34	0.265468
$877$	3.51655e35	1.69872	0.849359	−	0.527816i	$-0.176989\pi$
$877$	3.51655e35	1.69872	0.849359	+	0.527816i	$0.176989\pi$
$878$	1.98552e35	0.946108
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−3.16540e35	−1.42825
$883$	4.07247e35	1.81272	0.906358	−	0.422512i	$-0.138851\pi$
$883$	4.07247e35	1.81272	0.906358	+	0.422512i	$0.138851\pi$
$884$	0	0
$885$	0	0
$886$	1.68494e35	0.720077
$887$	4.45180e35	1.87695	0.938474	−	0.345351i	$-0.112240\pi$
$887$	4.45180e35	1.87695	0.938474	+	0.345351i	$0.112240\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−3.82452e35	−1.50730
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5.90670e35	−2.17689
$898$	1.18052e35	0.429295
$899$	−8.01549e35	−2.87617
$900$	−6.54734e35	−2.31822
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−1.40549e35	−0.459504
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1.04403e36	3.28054
$910$	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
$921$	6.34932e35	1.70456
$922$	−1.06999e35	−0.283536
$923$	7.25781e34	0.189839
$924$	0	0
$925$	0	0
$926$	−1.11338e35	−0.280099
$927$	0	0
$928$	7.62857e35	1.87011
$929$	−4.94976e35	−1.19783	−0.598915	−	0.800812i	$-0.704401\pi$
$929$	−4.94976e35	−1.19783	−0.598915	+	0.800812i	$0.704401\pi$
$930$	0	0
$931$	0	0
$932$	−2.06062e35	−0.479740
$933$	3.57760e35	0.822265
$934$	0	0
$935$	0	0
$936$	1.24664e36	2.75695
$937$	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
$943$	2.75610e35	0.557384
$944$	−3.30199e35	−0.659344
$945$	0	0
$946$	0	0
$947$	−9.34907e35	−1.79709	−0.898543	−	0.438886i	$-0.855373\pi$
$947$	−9.34907e35	−1.79709	−0.898543	+	0.438886i	$0.855373\pi$
$948$	0	0
$949$	9.97460e34	0.186939
$950$	0	0
$951$	−1.01665e36	−1.85782
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	−7.36904e35	−1.26449
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	9.09974e35	1.46672
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5.63091e35	0.842289	0.421144	−	0.906994i	$-0.361629\pi$
$967$	5.63091e35	0.842289	0.421144	+	0.906994i	$0.361629\pi$
$968$	5.71663e35	0.844570
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	3.29199e36	4.62875
$973$	0	0
$974$	6.83668e35	0.937859
$975$	−1.60654e36	−2.17689
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−1.45270e36	−1.89718
$979$	0	0
$980$	0	0
$981$	0	0
$982$	6.29567e35	0.782897
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−7.75678e35	−0.941326
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	6.92487e35	0.771841	0.385921	−	0.922532i	$-0.373884\pi$
$991$	6.92487e35	0.771841	0.385921	+	0.922532i	$0.373884\pi$
$992$	−1.45651e36	−1.60389
$993$	−3.67529e36	−3.99853
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.39934e36	−1.45071	−0.725357	−	0.688373i	$-0.758325\pi$
$997$	−1.39934e36	−1.45071	−0.725357	+	0.688373i	$0.758325\pi$
$998$	−9.29954e35	−0.952566
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.25.b.a.22.1	✓	1
23.22	odd	2	CM	23.25.b.a.22.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.25.b.a.22.1	✓	1	1.1	even	1	trivial
23.25.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:	\( k \)	\(=\)	\( 25 \)
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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