LMFDB - Embedded newform 23.19.b.a.22.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,19,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, 19, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 19);

N := Newforms(S);

Level:	$N$	$=$	$23$
Weight:	$k$	$=$	$19$
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:	$47.2388116732$
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+1001.00 q^{2} +28234.0 q^{3} +739857. q^{4} +2.82622e7 q^{6} +4.78191e8 q^{8} +4.09738e8 q^{9} +2.08891e10 q^{12} -1.44011e10 q^{13} +2.84720e11 q^{16} +4.10148e11 q^{18} -1.80115e12 q^{23} +1.35012e13 q^{24} +3.81470e12 q^{25} -1.44155e13 q^{26} +6.30120e11 q^{27} -2.58006e13 q^{29} +4.63875e13 q^{31} +1.59650e14 q^{32} +3.03148e14 q^{36} -4.06601e14 q^{39} -5.38722e14 q^{41} -1.80295e15 q^{46} -2.01804e15 q^{47} +8.03878e15 q^{48} +1.62841e15 q^{49} +3.81851e15 q^{50} -1.06548e16 q^{52} +6.30750e14 q^{54} -2.58264e16 q^{58} +1.57584e16 q^{59} +4.64339e16 q^{62} +8.51718e16 q^{64} -5.08537e16 q^{69} +4.05587e16 q^{71} +1.95933e17 q^{72} +5.19733e16 q^{73} +1.07704e17 q^{75} -4.07007e17 q^{78} -1.40950e17 q^{81} -5.39261e17 q^{82} -7.28454e17 q^{87} -1.33260e18 q^{92} +1.30971e18 q^{93} -2.02006e18 q^{94} +4.50755e18 q^{96} +1.63004e18 q^{98} +O(q^{100})

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1001.00	1.95508	0.977539	−	0.210754i	$-0.0675920\pi$
$2$	1001.00	1.95508	0.977539	+	0.210754i	$0.0675920\pi$
$3$	28234.0	1.43444	0.717218	−	0.696849i	$-0.245415\pi$
$3$	28234.0	1.43444	0.717218	+	0.696849i	$0.245415\pi$
$4$	739857.	2.82233
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	2.82622e7	2.80443
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	4.78191e8	3.56280
$9$	4.09738e8	1.05761
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	2.08891e10	4.04845
$13$	−1.44011e10	−1.35802	−0.679009	−	0.734130i	$-0.737590\pi$
$13$	−1.44011e10	−1.35802	−0.679009	+	0.734130i	$0.737590\pi$
$14$	0	0
$15$	0	0
$16$	2.84720e11	4.14322
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	4.10148e11	2.06770
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.80115e12	−1.00000
$23$	−1.80115e12	−1.00000
$24$	1.35012e13	5.11061
$25$	3.81470e12	1.00000
$26$	−1.44155e13	−2.65503
$27$	6.30120e11	0.0826322
$28$	0	0
$29$	−2.58006e13	−1.77848	−0.889238	−	0.457445i	$-0.848765\pi$
$29$	−2.58006e13	−1.77848	−0.889238	+	0.457445i	$0.848765\pi$
$30$	0	0
$31$	4.63875e13	1.75447	0.877235	−	0.480060i	$-0.159385\pi$
$31$	4.63875e13	1.75447	0.877235	+	0.480060i	$0.159385\pi$
$32$	1.59650e14	4.53752
$33$	0	0
$34$	0	0
$35$	0	0
$36$	3.03148e14	2.98491
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−4.06601e14	−1.94799
$40$	0	0
$41$	−5.38722e14	−1.64555	−0.822773	−	0.568370i	$-0.807574\pi$
$41$	−5.38722e14	−1.64555	−0.822773	+	0.568370i	$0.807574\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−1.80295e15	−1.95508
$47$	−2.01804e15	−1.80322	−0.901610	−	0.432551i	$-0.857614\pi$
$47$	−2.01804e15	−1.80322	−0.901610	+	0.432551i	$0.857614\pi$
$48$	8.03878e15	5.94318
$49$	1.62841e15	1.00000
$50$	3.81851e15	1.95508
$51$	0	0
$52$	−1.06548e16	−3.83277
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	6.30750e14	0.161552
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−2.58264e16	−3.47706
$59$	1.57584e16	1.81905	0.909523	−	0.415654i	$-0.136447\pi$
$59$	1.57584e16	1.81905	0.909523	+	0.415654i	$0.136447\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	4.64339e16	3.43013
$63$	0	0
$64$	8.51718e16	4.72798
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−5.08537e16	−1.43444
$70$	0	0
$71$	4.05587e16	0.884624	0.442312	−	0.896861i	$-0.354158\pi$
$71$	4.05587e16	0.884624	0.442312	+	0.896861i	$0.354158\pi$
$72$	1.95933e17	3.76804
$73$	5.19733e16	0.882826	0.441413	−	0.897304i	$-0.354477\pi$
$73$	5.19733e16	0.882826	0.441413	+	0.897304i	$0.354477\pi$
$74$	0	0
$75$	1.07704e17	1.43444
$76$	0	0
$77$	0	0
$78$	−4.07007e17	−3.80847
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−1.40950e17	−0.939075
$82$	−5.39261e17	−3.21717
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.28454e17	−2.55111
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−1.33260e18	−2.82233
$93$	1.30971e18	2.51668
$94$	−2.02006e18	−3.52543
$95$	0	0
$96$	4.50755e18	6.50878
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.63004e18	1.95508
$99$	0	0
$100$	2.82233e18	2.82233
$101$	−1.48135e18	−1.35446	−0.677231	−	0.735771i	$-0.736820\pi$
$101$	−1.48135e18	−1.35446	−0.677231	+	0.735771i	$0.736820\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−6.88647e18	−4.83834
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	4.66199e17	0.233215
$109$	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
$116$	−1.90888e19	−5.01945
$117$	−5.90068e18	−1.43625
$118$	1.57741e19	3.55638
$119$	0	0
$120$	0	0
$121$	5.55992e18	1.00000
$122$	0	0
$123$	−1.52103e19	−2.36043
$124$	3.43201e19	4.95170
$125$	0	0
$126$	0	0
$127$	−1.42579e19	−1.65891	−0.829454	−	0.558575i	$-0.811348\pi$
$127$	−1.42579e19	−1.65891	−0.829454	+	0.558575i	$0.811348\pi$
$128$	4.34057e19	4.70606
$129$	0	0
$130$	0	0
$131$	−1.37169e19	−1.20730	−0.603649	−	0.797250i	$-0.706287\pi$
$131$	−1.37169e19	−1.20730	−0.603649	+	0.797250i	$0.706287\pi$
$132$	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			$\pi$
$138$	−5.09046e19	−2.80443
$139$	−2.81811e19	−1.45487	−0.727437	−	0.686175i	$-0.759289\pi$
$139$	−2.81811e19	−1.45487	−0.727437	+	0.686175i	$0.759289\pi$
$140$	0	0
$141$	−5.69773e19	−2.58660
$142$	4.05993e19	1.72951
$143$	0	0
$144$	1.16661e20	4.38189
$145$	0	0
$146$	5.20253e19	1.72599
$147$	4.59766e19	1.43444
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	1.07812e20	2.80443
$151$	1.70141e19	0.416886	0.208443	−	0.978035i	$-0.433160\pi$
$151$	1.70141e19	0.416886	0.208443	+	0.978035i	$0.433160\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−3.00826e20	−5.49787
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−1.41091e20	−1.83597
$163$	1.22369e20	1.50654	0.753272	−	0.657709i	$-0.228474\pi$
$163$	1.22369e20	1.50654	0.753272	+	0.657709i	$0.228474\pi$
$164$	−3.98577e20	−4.64428
$165$	0	0
$166$	0	0
$167$	1.71138e20	1.69394	0.846969	−	0.531642i	$-0.178425\pi$
$167$	1.71138e20	1.69394	0.846969	+	0.531642i	$0.178425\pi$
$168$	0	0
$169$	9.49362e19	0.844212
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.50427e20	−1.08371	−0.541853	−	0.840473i	$-0.682277\pi$
$173$	−1.50427e20	−1.08371	−0.541853	+	0.840473i	$0.682277\pi$
$174$	−7.29183e20	−4.98762
$175$	0	0
$176$	0	0
$177$	4.44922e20	2.60930
$178$	0	0
$179$	3.26555e20	1.73093	0.865464	−	0.500971i	$-0.167024\pi$
$179$	3.26555e20	1.73093	0.865464	+	0.500971i	$0.167024\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−8.61294e20	−3.56280
$185$	0	0
$186$	1.31102e21	4.92030
$187$	0	0
$188$	−1.49306e21	−5.08928
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	2.40474e21	6.78199
$193$	−5.55299e20	−1.49455	−0.747276	−	0.664513i	$-0.768639\pi$
$193$	−5.55299e20	−1.49455	−0.747276	+	0.664513i	$0.768639\pi$
$194$	0	0
$195$	0	0
$196$	1.20479e21	2.82233
$197$	7.35324e20	1.64544	0.822721	−	0.568446i	$-0.192455\pi$
$197$	7.35324e20	1.64544	0.822721	+	0.568446i	$0.192455\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	1.82415e21	3.56280
$201$	0	0
$202$	−1.48284e21	−2.64808
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.38001e20	−1.05761
$208$	−4.10028e21	−5.62656
$209$	0	0
$210$	0	0
$211$	1.31448e21	1.58567	0.792833	−	0.609439i	$-0.208605\pi$
$211$	1.31448e21	1.58567	0.792833	+	0.609439i	$0.208605\pi$
$212$	0	0
$213$	1.14513e21	1.26894
$214$	0	0
$215$	0	0
$216$	3.01318e20	0.294402
$217$	0	0
$218$	0	0
$219$	1.46742e21	1.26636
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−4.57929e20	−0.335780	−0.167890	−	0.985806i	$-0.553695\pi$
$223$	−4.57929e20	−0.335780	−0.167890	+	0.985806i	$0.553695\pi$
$224$	0	0
$225$	1.56303e21	1.05761
$226$	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
$232$	−1.23376e22	−6.33635
$233$	−3.27251e21	−1.61688	−0.808439	−	0.588580i	$-0.799687\pi$
$233$	−3.27251e21	−1.61688	−0.808439	+	0.588580i	$0.799687\pi$
$234$	−5.90658e21	−2.80798
$235$	0	0
$236$	1.16590e22	5.13395
$237$	0	0
$238$	0	0
$239$	4.55994e21	1.79216	0.896082	−	0.443889i	$-0.146402\pi$
$239$	4.55994e21	1.79216	0.896082	+	0.443889i	$0.146402\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	5.56548e21	1.95508
$243$	−4.22371e21	−1.42968
$244$	0	0
$245$	0	0
$246$	−1.52255e22	−4.61483
$247$	0	0
$248$	2.21821e22	6.25083
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	−1.42722e22	−3.24330
$255$	0	0
$256$	2.11219e22	4.47273
$257$	7.58388e21	1.55058	0.775288	−	0.631607i	$-0.217604\pi$
$257$	7.58388e21	1.55058	0.775288	+	0.631607i	$0.217604\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.05715e22	−1.88093
$262$	−1.37306e22	−2.36036
$263$	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
$269$	−1.14433e22	−1.55161	−0.775804	−	0.630974i	$-0.782656\pi$
$269$	−1.14433e22	−1.55161	−0.775804	+	0.630974i	$0.782656\pi$
$270$	0	0
$271$	1.51071e22	1.91628	0.958140	−	0.286301i	$-0.0924258\pi$
$271$	1.51071e22	1.91628	0.958140	+	0.286301i	$0.0924258\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−3.76245e22	−4.04845
$277$	−6.88440e21	−0.717048	−0.358524	−	0.933521i	$-0.616720\pi$
$277$	−6.88440e21	−0.717048	−0.358524	+	0.933521i	$0.616720\pi$
$278$	−2.82093e22	−2.84439
$279$	1.90068e22	1.85554
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	−5.70343e22	−5.05701
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	3.00076e22	2.49670
$285$	0	0
$286$	0	0
$287$	0	0
$288$	6.54146e22	4.79891
$289$	1.40631e22	1.00000
$290$	0	0
$291$	0	0
$292$	3.84528e22	2.49163
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	4.60226e22	2.80443
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.59386e22	1.35802
$300$	7.96857e22	4.04845
$301$	0	0
$302$	1.70311e22	0.815044
$303$	−4.18246e22	−1.94289
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.02450e21	0.124856	0.0624278	−	0.998049i	$-0.480116\pi$
$307$	3.02450e21	0.124856	0.0624278	+	0.998049i	$0.480116\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.29078e22	1.94391	0.971955	−	0.235165i	$-0.0755630\pi$
$311$	5.29078e22	1.94391	0.971955	+	0.235165i	$0.0755630\pi$
$312$	−1.94433e23	−6.94029
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.43051e22	−1.37063	−0.685315	−	0.728247i	$-0.740335\pi$
$317$	−4.43051e22	−1.37063	−0.685315	+	0.728247i	$0.740335\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−1.04283e23	−2.65038
$325$	−5.49358e22	−1.35802
$326$	1.22491e23	2.94541
$327$	0	0
$328$	−2.57612e23	−5.86275
$329$	0	0
$330$	0	0
$331$	9.53756e22	1.99979	0.999897	−	0.0143344i	$-0.00456294\pi$
$331$	9.53756e22	1.99979	0.999897	+	0.0143344i	$0.00456294\pi$
$332$	0	0
$333$	0	0
$334$	1.71309e23	3.31178
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	9.50312e22	1.65050
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−1.50578e23	−2.11873
$347$	−3.71047e21	−0.0508703	−0.0254351	−	0.999676i	$-0.508097\pi$
$347$	−3.71047e21	−0.0508703	−0.0254351	+	0.999676i	$0.508097\pi$
$348$	−5.38952e23	−7.20007
$349$	−5.06424e22	−0.659304	−0.329652	−	0.944103i	$-0.606931\pi$
$349$	−5.06424e22	−0.659304	−0.329652	+	0.944103i	$0.606931\pi$
$350$	0	0
$351$	−9.07442e21	−0.112216
$352$	0	0
$353$	−5.67058e20	−0.00666277	−0.00333139	−	0.999994i	$-0.501060\pi$
$353$	−5.67058e20	−0.00666277	−0.00333139	+	0.999994i	$0.501060\pi$
$354$	4.45367e23	5.10139
$355$	0	0
$356$	0	0
$357$	0	0
$358$	3.26882e23	3.38410
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.04127e23	1.00000
$362$	0	0
$363$	1.56979e23	1.43444
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	−5.12824e23	−4.14322
$369$	−2.20735e23	−1.74034
$370$	0	0
$371$	0	0
$372$	9.68995e23	7.10289
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−9.65007e23	−6.42451
$377$	3.71557e23	2.41520
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−4.02558e23	−2.37960
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.22552e24	6.75054
$385$	0	0
$386$	−5.55855e23	−2.92197
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	7.78692e23	3.56280
$393$	−3.87283e23	−1.73179
$394$	7.36059e23	3.21697
$395$	0	0
$396$	0	0
$397$	−1.30451e23	−0.532514	−0.266257	−	0.963902i	$-0.585787\pi$
$397$	−1.30451e23	−0.532514	−0.266257	+	0.963902i	$0.585787\pi$
$398$	0	0
$399$	0	0
$400$	1.08612e24	4.14322
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−6.68032e23	−2.38260
$404$	−1.09599e24	−3.82274
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	1.72225e23	0.537757	0.268878	−	0.963174i	$-0.413347\pi$
$409$	1.72225e23	0.537757	0.268878	+	0.963174i	$0.413347\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−7.38739e23	−2.06770
$415$	0	0
$416$	−2.29913e24	−6.16203
$417$	−7.95666e23	−2.08692
$418$	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			$\pi$
$422$	1.31579e24	3.10010
$423$	−8.26867e23	−1.90710
$424$	0	0
$425$	0	0
$426$	1.14628e24	2.48087
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.79408e23	0.342363
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.46888e24	2.47583
$439$	9.28281e23	1.53285	0.766423	−	0.642336i	$-0.222035\pi$
$439$	9.28281e23	1.53285	0.766423	+	0.642336i	$0.222035\pi$
$440$	0	0
$441$	6.67223e23	1.05761
$442$	0	0
$443$	−3.25199e23	−0.494898	−0.247449	−	0.968901i	$-0.579592\pi$
$443$	−3.25199e23	−0.494898	−0.247449	+	0.968901i	$0.579592\pi$
$444$	0	0
$445$	0	0
$446$	−4.58387e23	−0.656475
$447$	0	0
$448$	0	0
$449$	6.24135e22	0.0841514	0.0420757	−	0.999114i	$-0.486603\pi$
$449$	6.24135e22	0.0841514	0.0420757	+	0.999114i	$0.486603\pi$
$450$	1.56459e24	2.06770
$451$	0	0
$452$	0	0
$453$	4.80377e23	0.597996
$454$	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
$461$	−1.09182e24	−1.16103	−0.580513	−	0.814251i	$-0.697148\pi$
$461$	−1.09182e24	−1.16103	−0.580513	+	0.814251i	$0.697148\pi$
$462$	0	0
$463$	1.59548e24	1.63178	0.815892	−	0.578204i	$-0.196246\pi$
$463$	1.59548e24	1.63178	0.815892	+	0.578204i	$0.196246\pi$
$464$	−7.34595e24	−7.36862
$465$	0	0
$466$	−3.27578e24	−3.16112
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−4.36566e24	−4.05357
$469$	0	0
$470$	0	0
$471$	0	0
$472$	7.53551e24	6.48089
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	4.56450e24	3.50382
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	4.11354e24	2.82233
$485$	0	0
$486$	−4.22793e24	−2.79513
$487$	−2.44746e24	−1.58838	−0.794190	−	0.607669i	$-0.792105\pi$
$487$	−2.44746e24	−1.58838	−0.794190	+	0.607669i	$0.792105\pi$
$488$	0	0
$489$	3.45496e24	2.16104
$490$	0	0
$491$	−1.07725e24	−0.649506	−0.324753	−	0.945799i	$-0.605281\pi$
$491$	−1.07725e24	−0.649506	−0.324753	+	0.945799i	$0.605281\pi$
$492$	−1.12534e25	−6.66192
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.32075e25	7.26916
$497$	0	0
$498$	0	0
$499$	−3.51041e22	−0.0183001	−0.00915003	−	0.999958i	$-0.502913\pi$
$499$	−3.51041e22	−0.0183001	−0.00915003	+	0.999958i	$0.502913\pi$
$500$	0	0
$501$	4.83190e24	2.42985
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.68043e24	1.21097
$508$	−1.05488e25	−4.68199
$509$	2.41029e24	1.05101	0.525507	−	0.850789i	$-0.323875\pi$
$509$	2.41029e24	1.05101	0.525507	+	0.850789i	$0.323875\pi$
$510$	0	0
$511$	0	0
$512$	9.76444e24	4.03848
$513$	0	0
$514$	7.59146e24	3.03150
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−4.24716e24	−1.55451
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1.05821e25	−3.67736
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.01486e25	−3.40740
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	3.24415e24	1.00000
$530$	0	0
$531$	6.45681e24	1.92383
$532$	0	0
$533$	7.75819e24	2.23468
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.21996e24	2.48291
$538$	−1.14548e25	−3.03352
$539$	0	0
$540$	0	0
$541$	−7.74360e24	−1.95060	−0.975299	−	0.220890i	$-0.929104\pi$
$541$	−7.74360e24	−1.95060	−0.975299	+	0.220890i	$0.929104\pi$
$542$	1.51223e25	3.74648
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.18027e24	−1.86587	−0.932936	−	0.360043i	$-0.882762\pi$
$547$	−8.18027e24	−1.86587	−0.932936	+	0.360043i	$0.882762\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−2.43178e25	−5.11061
$553$	0	0
$554$	−6.89128e24	−1.40188
$555$	0	0
$556$	−2.08500e25	−4.10613
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1.90258e25	3.62772
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−4.21550e25	−7.30025
$565$	0	0
$566$	0	0
$567$	0	0
$568$	1.93948e25	3.15174
$569$	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
$575$	−6.87085e24	−1.00000
$576$	3.48981e25	5.00034
$577$	−9.24167e24	−1.30367	−0.651836	−	0.758360i	$-0.726001\pi$
$577$	−9.24167e24	−1.30367	−0.651836	+	0.758360i	$0.726001\pi$
$578$	1.40771e25	1.95508
$579$	−1.56783e25	−2.14384
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	2.48532e25	3.14533
$585$	0	0
$586$	0	0
$587$	−1.48760e25	−1.79781	−0.898903	−	0.438147i	$-0.855635\pi$
$587$	−1.48760e25	−1.79781	−0.898903	+	0.438147i	$0.855635\pi$
$588$	3.40161e25	4.04845
$589$	0	0
$590$	0	0
$591$	2.07611e25	2.36028
$592$	0	0
$593$	−1.47479e25	−1.62644	−0.813222	−	0.581954i	$-0.802288\pi$
$593$	−1.47479e25	−1.62644	−0.813222	+	0.581954i	$0.802288\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	2.59645e25	2.65503
$599$	−1.82212e25	−1.83542	−0.917709	−	0.397253i	$-0.869964\pi$
$599$	−1.82212e25	−1.83542	−0.917709	+	0.397253i	$0.869964\pi$
$600$	5.15031e25	5.11061
$601$	−2.04147e25	−1.99560	−0.997798	−	0.0663311i	$-0.978871\pi$
$601$	−2.04147e25	−1.99560	−0.997798	+	0.0663311i	$0.978871\pi$
$602$	0	0
$603$	0	0
$604$	1.25880e25	1.17659
$605$	0	0
$606$	−4.18664e25	−3.79850
$607$	5.08809e24	0.454837	0.227419	−	0.973797i	$-0.426971\pi$
$607$	5.08809e24	0.454837	0.227419	+	0.973797i	$0.426971\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.90620e25	2.44880
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	3.02752e24	0.244103
$615$	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
$621$	−1.13494e24	−0.0826322
$622$	5.29607e25	3.80050
$623$	0	0
$624$	−1.15767e26	−8.07095
$625$	1.45519e25	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
$633$	3.71130e25	2.27454
$634$	−4.43494e25	−2.67969
$635$	0	0
$636$	0	0
$637$	−2.34509e25	−1.35802
$638$	0	0
$639$	1.66185e25	0.935584
$640$	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
$647$	−3.97061e25	−1.99857	−0.999284	−	0.0378456i	$-0.987950\pi$
$647$	−3.97061e25	−1.99857	−0.999284	+	0.0378456i	$0.987950\pi$
$648$	−6.74011e25	−3.34574
$649$	0	0
$650$	−5.49908e25	−2.65503
$651$	0	0
$652$	9.05354e25	4.25197
$653$	1.63183e25	0.755888	0.377944	−	0.925829i	$-0.376631\pi$
$653$	1.63183e25	0.755888	0.377944	+	0.925829i	$0.376631\pi$
$654$	0	0
$655$	0	0
$656$	−1.53385e26	−6.81786
$657$	2.12955e25	0.933682
$658$	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			$\pi$
$662$	9.54710e25	3.90975
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.64708e25	1.77848
$668$	1.26617e26	4.78085
$669$	−1.29292e25	−0.481654
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−5.20100e24	−0.183633	−0.0918167	−	0.995776i	$-0.529267\pi$
$673$	−5.20100e24	−0.183633	−0.0918167	+	0.995776i	$0.529267\pi$
$674$	0	0
$675$	2.40372e24	0.0826322
$676$	7.02392e25	2.38265
$677$	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
$683$	4.39828e25	1.35987	0.679934	−	0.733273i	$-0.262008\pi$
$683$	4.39828e25	1.35987	0.679934	+	0.733273i	$0.262008\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−3.87997e25	−1.08025	−0.540126	−	0.841584i	$-0.681624\pi$
$691$	−3.87997e25	−1.08025	−0.540126	+	0.841584i	$0.681624\pi$
$692$	−1.11295e26	−3.05858
$693$	0	0
$694$	−3.71418e24	−0.0994554
$695$	0	0
$696$	−3.48340e26	−9.08909
$697$	0	0
$698$	−5.06931e25	−1.28899
$699$	−9.23960e25	−2.31931
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−9.08350e24	−0.219391
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−5.67625e23	−0.0130262
$707$	0	0
$708$	3.29179e26	7.36432
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.35510e25	−1.75447
$714$	0	0
$715$	0	0
$716$	2.41604e26	4.88525
$717$	1.28745e26	2.57074
$718$	0	0
$719$	4.90528e25	0.955220	0.477610	−	0.878572i	$-0.341503\pi$
$719$	4.90528e25	0.955220	0.477610	+	0.878572i	$0.341503\pi$
$720$	0	0
$721$	0	0
$722$	1.04231e26	1.95508
$723$	0	0
$724$	0	0
$725$	−9.84215e25	−1.77848
$726$	1.57136e26	2.80443
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−6.46452e25	−1.11170
$730$	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
$736$	−2.87554e26	−4.53752
$737$	0	0
$738$	−2.20956e26	−3.40250
$739$	−5.95740e25	−0.906269	−0.453134	−	0.891442i	$-0.649694\pi$
$739$	−5.95740e25	−0.906269	−0.453134	+	0.891442i	$0.649694\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	6.26289e26	8.96641
$745$	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			$\pi$
$752$	−5.74575e26	−7.47113
$753$	0	0
$754$	3.71929e26	4.72191
$755$	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
$761$	−1.04512e26	−1.22096	−0.610481	−	0.792031i	$-0.709024\pi$
$761$	−1.04512e26	−1.22096	−0.610481	+	0.792031i	$0.709024\pi$
$762$	−4.02960e26	−4.65230
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.26938e26	−2.47030
$768$	5.96355e26	6.41584
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	2.14123e26	2.22420
$772$	−4.10842e26	−4.21812
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	1.76954e26	1.75447
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.62575e25	−0.146959
$784$	4.63642e26	4.14322
$785$	0	0
$786$	−3.87671e26	−3.38579
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	5.44034e26	4.64398
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−1.30581e26	−1.04111
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	6.09015e26	4.53752
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−6.68700e26	−4.65817
$807$	−3.23091e26	−2.22568
$808$	−7.08370e26	−4.82567
$809$	2.15677e26	1.45301	0.726504	−	0.687163i	$-0.241144\pi$
$809$	2.15677e26	1.45301	0.726504	+	0.687163i	$0.241144\pi$
$810$	0	0
$811$	−2.74971e26	−1.81175	−0.905877	−	0.423542i	$-0.860787\pi$
$811$	−2.74971e26	−1.81175	−0.905877	+	0.423542i	$0.860787\pi$
$812$	0	0
$813$	4.26535e26	2.74878
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	1.72397e26	1.05136
$819$	0	0
$820$	0	0
$821$	3.35666e26	1.98070	0.990349	−	0.138594i	$-0.0442582\pi$
$821$	3.35666e26	1.98070	0.990349	+	0.138594i	$0.0442582\pi$
$822$	0	0
$823$	2.28084e26	1.31673	0.658366	−	0.752698i	$-0.271248\pi$
$823$	2.28084e26	1.31673	0.658366	+	0.752698i	$0.271248\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−5.46015e26	−2.98491
$829$	−2.99868e26	−1.62158	−0.810792	−	0.585334i	$-0.800964\pi$
$829$	−2.99868e26	−1.62158	−0.810792	+	0.585334i	$0.800964\pi$
$830$	0	0
$831$	−1.94374e26	−1.02856
$832$	−1.22657e27	−6.42069
$833$	0	0
$834$	−7.96462e26	−4.08010
$835$	0	0
$836$	0	0
$837$	2.92297e25	0.144976
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	4.55214e26	2.16298
$842$	0	0
$843$	0	0
$844$	9.72527e26	4.47527
$845$	0	0
$846$	−8.27694e26	−3.72852
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	8.47236e26	3.58136
$853$	4.77678e26	1.99799	0.998997	−	0.0447818i	$-0.0142593\pi$
$853$	4.77678e26	1.99799	0.998997	+	0.0447818i	$0.0142593\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.74534e26	1.90300	0.951502	−	0.307642i	$-0.0995401\pi$
$857$	4.74534e26	1.90300	0.951502	+	0.307642i	$0.0995401\pi$
$858$	0	0
$859$	5.07711e26	1.99378	0.996892	−	0.0787838i	$-0.0251037\pi$
$859$	5.07711e26	1.99378	0.996892	+	0.0787838i	$0.0251037\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.32323e26	−1.62821	−0.814107	−	0.580715i	$-0.802773\pi$
$863$	−4.32323e26	−1.62821	−0.814107	+	0.580715i	$0.802773\pi$
$864$	1.00599e26	0.374945
$865$	0	0
$866$	0	0
$867$	3.97057e26	1.43444
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	1.08568e27	3.57408
$877$	2.48612e26	0.810076	0.405038	−	0.914300i	$-0.367258\pi$
$877$	2.48612e26	0.810076	0.405038	+	0.914300i	$0.367258\pi$
$878$	9.29209e26	2.99683
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	6.67891e26	2.06770
$883$	−6.18027e26	−1.89392	−0.946959	−	0.321356i	$-0.895861\pi$
$883$	−6.18027e26	−1.89392	−0.946959	+	0.321356i	$0.895861\pi$
$884$	0	0
$885$	0	0
$886$	−3.25524e26	−0.967564
$887$	−6.56108e26	−1.93047	−0.965234	−	0.261388i	$-0.915820\pi$
$887$	−6.56108e26	−1.93047	−0.965234	+	0.261388i	$0.915820\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−3.38802e26	−0.947681
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.32350e26	1.94799
$898$	6.24759e25	0.164523
$899$	−1.19683e27	−3.12028
$900$	1.15642e27	2.98491
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	4.80857e26	1.16913
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−6.06968e26	−1.43249
$910$	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
$921$	8.53937e25	0.179097
$922$	−1.09291e27	−2.26990
$923$	−5.84090e26	−1.20134
$924$	0	0
$925$	0	0
$926$	1.59708e27	3.19027
$927$	0	0
$928$	−4.11906e27	−8.06987
$929$	−1.02672e27	−1.99210	−0.996050	−	0.0887934i	$-0.971699\pi$
$929$	−1.02672e27	−1.99210	−0.996050	+	0.0887934i	$0.971699\pi$
$930$	0	0
$931$	0	0
$932$	−2.42119e27	−4.56337
$933$	1.49380e27	2.78842
$934$	0	0
$935$	0	0
$936$	−2.82165e27	−5.11706
$937$	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
$943$	9.70321e26	1.64555
$944$	4.48673e27	7.53670
$945$	0	0
$946$	0	0
$947$	5.26954e26	0.860247	0.430123	−	0.902770i	$-0.358470\pi$
$947$	5.26954e26	0.860247	0.430123	+	0.902770i	$0.358470\pi$
$948$	0	0
$949$	−7.48473e26	−1.19889
$950$	0	0
$951$	−1.25091e27	−1.96608
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	3.37370e27	5.05808
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.45275e27	2.07817
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−9.73592e26	−1.31686	−0.658429	−	0.752643i	$-0.728779\pi$
$967$	−9.73592e26	−1.31686	−0.658429	+	0.752643i	$0.728779\pi$
$968$	2.65870e27	3.56280
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−3.12494e27	−4.03502
$973$	0	0
$974$	−2.44990e27	−3.10541
$975$	−1.55106e27	−1.94799
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	3.45841e27	4.22500
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−1.07833e27	−1.26983
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	−7.27342e27	−8.40974
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.17338e27	−1.27285	−0.636424	−	0.771339i	$-0.719587\pi$
$991$	−1.17338e27	−1.27285	−0.636424	+	0.771339i	$0.719587\pi$
$992$	7.40576e27	7.96094
$993$	2.69283e27	2.86858
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.17148e26	−0.428582	−0.214291	−	0.976770i	$-0.568744\pi$
$997$	−4.17148e26	−0.428582	−0.214291	+	0.976770i	$0.568744\pi$
$998$	−3.51392e25	−0.0357780
$999$	0	0

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.19.b.a.22.1	✓	1
23.22	odd	2	CM	23.19.b.a.22.1	✓	1

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

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Classical	Maass
Hilbert	Bianchi

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

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Groups

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	23.19.b.a.22.1

Level	$23$
Weight	$19$
Character	23.22
Self dual	yes
Analytic conductor	$47.239$
Analytic rank	$0$
Dimension	$1$
CM discriminant	-23
Inner twists	$2$