LMFDB - Embedded newform 23.9.b.a.22.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,9,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, 9, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 9 $
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:	$9.36970803141$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			22.2
Root			$-0.523976$ of defining polynomial
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)

$f(q)$	$=$	$q+2.56227 q^{2} -161.997 q^{3} -249.435 q^{4} -415.079 q^{6} -1295.06 q^{8} +19681.9 q^{9} +O(q^{10})$	$q+2.56227 q^{2} -161.997 q^{3} -249.435 q^{4} -415.079 q^{6} -1295.06 q^{8} +19681.9 q^{9} +40407.6 q^{12} -43115.2 q^{13} +60537.0 q^{16} +50430.4 q^{18} +279841. q^{23} +209795. q^{24} +390625. q^{25} -110473. q^{26} -2.12555e6 q^{27} +532166. q^{29} -1.25379e6 q^{31} +486647. q^{32} -4.90936e6 q^{36} +6.98453e6 q^{39} +5.13832e6 q^{41} +717028. q^{46} -3.11431e6 q^{47} -9.80680e6 q^{48} +5.76480e6 q^{49} +1.00089e6 q^{50} +1.07544e7 q^{52} -5.44622e6 q^{54} +1.36355e6 q^{58} +1.52791e7 q^{59} -3.21255e6 q^{62} -1.42506e7 q^{64} -4.53333e7 q^{69} -4.32563e7 q^{71} -2.54893e7 q^{72} +4.99806e7 q^{73} -6.32800e7 q^{75} +1.78962e7 q^{78} +2.15198e8 q^{81} +1.31658e7 q^{82} -8.62091e7 q^{87} -6.98021e7 q^{92} +2.03110e8 q^{93} -7.97969e6 q^{94} -7.88352e7 q^{96} +1.47710e7 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 768 q^{4} - 573 q^{6} - 5853 q^{8} + 19683 q^{9}+O(q^{10}) $ 3 * q + 768 * q^4 - 573 * q^6 - 5853 * q^8 + 19683 * q^9	$ 3 q + 768 q^{4} - 573 q^{6} - 5853 q^{8} + 19683 q^{9} + 122691 q^{12} + 196608 q^{16} + 58467 q^{18} + 839523 q^{23} - 146688 q^{24} + 1171875 q^{25} - 1958493 q^{26} - 3188058 q^{27} - 1498368 q^{32} - 4929405 q^{36} + 10418502 q^{39} + 1117923 q^{48} + 17294403 q^{49} + 37281507 q^{52} - 3759453 q^{54} - 60663549 q^{58} + 45837222 q^{59} - 69712509 q^{62} - 38912445 q^{64} - 23433981 q^{72} - 130646013 q^{78} + 129140163 q^{81} + 132337347 q^{82} - 131352378 q^{87} + 214917888 q^{92} + 302555622 q^{93} + 430562787 q^{94} - 239370141 q^{96}+O(q^{100}) $ 3 * q + 768 * q^4 - 573 * q^6 - 5853 * q^8 + 19683 * q^9 + 122691 * q^12 + 196608 * q^16 + 58467 * q^18 + 839523 * q^23 - 146688 * q^24 + 1171875 * q^25 - 1958493 * q^26 - 3188058 * q^27 - 1498368 * q^32 - 4929405 * q^36 + 10418502 * q^39 + 1117923 * q^48 + 17294403 * q^49 + 37281507 * q^52 - 3759453 * q^54 - 60663549 * q^58 + 45837222 * q^59 - 69712509 * q^62 - 38912445 * q^64 - 23433981 * q^72 - 130646013 * q^78 + 129140163 * q^81 + 132337347 * q^82 - 131352378 * q^87 + 214917888 * q^92 + 302555622 * q^93 + 430562787 * q^94 - 239370141 * q^96

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 $

$2$	2.56227	0.160142	0.0800709	−	0.996789i	$-0.474485\pi$
$2$	2.56227	0.160142	0.0800709	+	0.996789i	$0.474485\pi$
$3$	−161.997	−1.99996	−0.999980	−	0.00640152i	$-0.997962\pi$
$3$	−161.997	−1.99996	−0.999980	+	0.00640152i	$0.997962\pi$
$4$	−249.435	−0.974355
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−415.079	−0.320277
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−1295.06	−0.316177
$9$	19681.9	2.99984
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	40407.6	1.94867
$13$	−43115.2	−1.50958	−0.754792	−	0.655964i	$-0.772262\pi$
$13$	−43115.2	−1.50958	−0.754792	+	0.655964i	$0.772262\pi$
$14$	0	0
$15$	0	0
$16$	60537.0	0.923722
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	50430.4	0.480399
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	279841.	1.00000
$23$	279841.	1.00000
$24$	209795.	0.632340
$25$	390625.	1.00000
$26$	−110473.	−0.241748
$27$	−2.12555e6	−3.99959
$28$	0	0
$29$	532166.	0.752410	0.376205	−	0.926536i	$-0.377229\pi$
$29$	532166.	0.752410	0.376205	+	0.926536i	$0.377229\pi$
$30$	0	0
$31$	−1.25379e6	−1.35762	−0.678811	−	0.734313i	$-0.737505\pi$
$31$	−1.25379e6	−1.35762	−0.678811	+	0.734313i	$0.737505\pi$
$32$	486647.	0.464103
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−4.90936e6	−2.92290
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	6.98453e6	3.01911
$40$	0	0
$41$	5.13832e6	1.81839	0.909193	−	0.416375i	$-0.136700\pi$
$41$	5.13832e6	1.81839	0.909193	+	0.416375i	$0.136700\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	717028.	0.160142
$47$	−3.11431e6	−0.638220	−0.319110	−	0.947718i	$-0.603384\pi$
$47$	−3.11431e6	−0.638220	−0.319110	+	0.947718i	$0.603384\pi$
$48$	−9.80680e6	−1.84741
$49$	5.76480e6	1.00000
$50$	1.00089e6	0.160142
$51$	0	0
$52$	1.07544e7	1.47087
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−5.44622e6	−0.640501
$55$	0	0
$56$	0	0
$57$	0	0
$58$	1.36355e6	0.120492
$59$	1.52791e7	1.26092	0.630462	−	0.776220i	$-0.282865\pi$
$59$	1.52791e7	1.26092	0.630462	+	0.776220i	$0.282865\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−3.21255e6	−0.217412
$63$	0	0
$64$	−1.42506e7	−0.849399
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−4.53333e7	−1.99996
$70$	0	0
$71$	−4.32563e7	−1.70222	−0.851111	−	0.524986i	$-0.824070\pi$
$71$	−4.32563e7	−1.70222	−0.851111	+	0.524986i	$0.824070\pi$
$72$	−2.54893e7	−0.948478
$73$	4.99806e7	1.75999	0.879995	−	0.474982i	$-0.157546\pi$
$73$	4.99806e7	1.75999	0.879995	+	0.474982i	$0.157546\pi$
$74$	0	0
$75$	−6.32800e7	−1.99996
$76$	0	0
$77$	0	0
$78$	1.78962e7	0.483485
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	2.15198e8	4.99918
$82$	1.31658e7	0.291199
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.62091e7	−1.50479
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	−6.98021e7	−0.974355
$93$	2.03110e8	2.71519
$94$	−7.97969e6	−0.102206
$95$	0	0
$96$	−7.88352e7	−0.928187
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.47710e7	0.160142
$99$	0	0
$100$	−9.74355e7	−0.974355
$101$	−1.56941e8	−1.50817	−0.754086	−	0.656775i	$-0.771920\pi$
$101$	−1.56941e8	−1.50817	−0.754086	+	0.656775i	$0.771920\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	5.58368e7	0.477295
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	5.30185e8	3.89702
$109$	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
$116$	−1.32741e8	−0.733115
$117$	−8.48591e8	−4.52851
$118$	3.91491e7	0.201927
$119$	0	0
$120$	0	0
$121$	2.14359e8	1.00000
$122$	0	0
$123$	−8.32391e8	−3.63670
$124$	3.12740e8	1.32281
$125$	0	0
$126$	0	0
$127$	2.97740e8	1.14452	0.572259	−	0.820073i	$-0.306067\pi$
$127$	2.97740e8	1.14452	0.572259	+	0.820073i	$0.306067\pi$
$128$	−1.61095e8	−0.600127
$129$	0	0
$130$	0	0
$131$	2.62838e8	0.892489	0.446244	−	0.894911i	$-0.352761\pi$
$131$	2.62838e8	0.892489	0.446244	+	0.894911i	$0.352761\pi$
$132$	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			$\pi$
$138$	−1.16156e8	−0.320277
$139$	7.04829e8	1.88810	0.944049	−	0.329805i	$-0.106983\pi$
$139$	7.04829e8	1.88810	0.944049	+	0.329805i	$0.106983\pi$
$140$	0	0
$141$	5.04508e8	1.27641
$142$	−1.10834e8	−0.272597
$143$	0	0
$144$	1.19148e9	2.77101
$145$	0	0
$146$	1.28064e8	0.281848
$147$	−9.33879e8	−1.99996
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−1.62140e8	−0.320277
$151$	8.55329e8	1.64522	0.822612	−	0.568603i	$-0.192516\pi$
$151$	8.55329e8	1.64522	0.822612	+	0.568603i	$0.192516\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−1.74218e9	−2.94168
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	5.51396e8	0.800578
$163$	7.42336e8	1.05160	0.525799	−	0.850609i	$-0.323766\pi$
$163$	7.42336e8	1.05160	0.525799	+	0.850609i	$0.323766\pi$
$164$	−1.28168e9	−1.77175
$165$	0	0
$166$	0	0
$167$	−1.54783e9	−1.99002	−0.995010	−	0.0997713i	$-0.968189\pi$
$167$	−1.54783e9	−1.99002	−0.995010	+	0.0997713i	$0.968189\pi$
$168$	0	0
$169$	1.04319e9	1.27885
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.73378e9	1.93558	0.967788	−	0.251768i	$-0.0810121\pi$
$173$	1.73378e9	1.93558	0.967788	+	0.251768i	$0.0810121\pi$
$174$	−2.20891e8	−0.240980
$175$	0	0
$176$	0	0
$177$	−2.47516e9	−2.52180
$178$	0	0
$179$	−1.20407e8	−0.117285	−0.0586423	−	0.998279i	$-0.518677\pi$
$179$	−1.20407e8	−0.117285	−0.0586423	+	0.998279i	$0.518677\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−3.62411e8	−0.316177
$185$	0	0
$186$	5.20423e8	0.434815
$187$	0	0
$188$	7.76817e8	0.621852
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	2.30854e9	1.69876
$193$	−2.17145e9	−1.56502	−0.782512	−	0.622636i	$-0.786062\pi$
$193$	−2.17145e9	−1.56502	−0.782512	+	0.622636i	$0.786062\pi$
$194$	0	0
$195$	0	0
$196$	−1.43794e9	−0.974355
$197$	−7.59486e8	−0.504260	−0.252130	−	0.967693i	$-0.581131\pi$
$197$	−7.59486e8	−0.504260	−0.252130	+	0.967693i	$0.581131\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−5.05883e8	−0.316177
$201$	0	0
$202$	−4.02125e8	−0.241521
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.50781e9	2.99984
$208$	−2.61007e9	−1.39444
$209$	0	0
$210$	0	0
$211$	1.78049e9	0.898277	0.449138	−	0.893462i	$-0.351731\pi$
$211$	1.78049e9	0.898277	0.449138	+	0.893462i	$0.351731\pi$
$212$	0	0
$213$	7.00738e9	3.40437
$214$	0	0
$215$	0	0
$216$	2.75271e9	1.26458
$217$	0	0
$218$	0	0
$219$	−8.09670e9	−3.51991
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.78686e9	−1.12693	−0.563463	−	0.826141i	$-0.690531\pi$
$223$	−2.78686e9	−1.12693	−0.563463	+	0.826141i	$0.690531\pi$
$224$	0	0
$225$	7.68825e9	2.99984
$226$	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
$232$	−6.89186e8	−0.237895
$233$	−6.18712e8	−0.209925	−0.104963	−	0.994476i	$-0.533472\pi$
$233$	−6.18712e8	−0.209925	−0.104963	+	0.994476i	$0.533472\pi$
$234$	−2.17432e9	−0.725203
$235$	0	0
$236$	−3.81113e9	−1.22859
$237$	0	0
$238$	0	0
$239$	6.38975e9	1.95836	0.979180	−	0.202995i	$-0.0650674\pi$
$239$	6.38975e9	1.95836	0.979180	+	0.202995i	$0.0650674\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	5.49245e8	0.160142
$243$	−2.09157e10	−5.99857
$244$	0	0
$245$	0	0
$246$	−2.13281e9	−0.582387
$247$	0	0
$248$	1.62374e9	0.429249
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	7.62890e8	0.183285
$255$	0	0
$256$	3.23537e9	0.753294
$257$	4.01635e9	0.920659	0.460329	−	0.887748i	$-0.347731\pi$
$257$	4.01635e9	0.920659	0.460329	+	0.887748i	$0.347731\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.04740e10	2.25711
$262$	6.73461e8	0.142925
$263$	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
$269$	−2.28667e9	−0.436710	−0.218355	−	0.975869i	$-0.570069\pi$
$269$	−2.28667e9	−0.436710	−0.218355	+	0.975869i	$0.570069\pi$
$270$	0	0
$271$	−6.55097e9	−1.21459	−0.607294	−	0.794478i	$-0.707745\pi$
$271$	−6.55097e9	−1.21459	−0.607294	+	0.794478i	$0.707745\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	1.13077e10	1.94867
$277$	1.36127e8	0.0231220	0.0115610	−	0.999933i	$-0.496320\pi$
$277$	1.36127e8	0.0231220	0.0115610	+	0.999933i	$0.496320\pi$
$278$	1.80596e9	0.302363
$279$	−2.46771e10	−4.07265
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	1.29268e9	0.204407
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	1.07896e10	1.65857
$285$	0	0
$286$	0	0
$287$	0	0
$288$	9.57815e9	1.39223
$289$	6.97576e9	1.00000
$290$	0	0
$291$	0	0
$292$	−1.24669e10	−1.71486
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−2.39285e9	−0.320277
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.20654e10	−1.50958
$300$	1.57842e10	1.94867
$301$	0	0
$302$	2.19158e9	0.263469
$303$	2.54239e10	3.01628
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.59811e9	0.292486	0.146243	−	0.989249i	$-0.453282\pi$
$307$	2.59811e9	0.292486	0.146243	+	0.989249i	$0.453282\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.55194e10	1.65895	0.829477	−	0.558541i	$-0.188639\pi$
$311$	1.55194e10	1.65895	0.829477	+	0.558541i	$0.188639\pi$
$312$	−9.04538e9	−0.954571
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.23831e9	−0.320687	−0.160343	−	0.987061i	$-0.551260\pi$
$317$	−3.23831e9	−0.320687	−0.160343	+	0.987061i	$0.551260\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−5.36779e10	−4.87097
$325$	−1.68419e10	−1.50958
$326$	1.90206e9	0.168405
$327$	0	0
$328$	−6.65443e9	−0.574931
$329$	0	0
$330$	0	0
$331$	−1.18709e10	−0.988945	−0.494472	−	0.869193i	$-0.664639\pi$
$331$	−1.18709e10	−0.988945	−0.494472	+	0.869193i	$0.664639\pi$
$332$	0	0
$333$	0	0
$334$	−3.96596e9	−0.318685
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	2.67294e9	0.204797
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	4.44241e9	0.309966
$347$	−1.42135e10	−0.980354	−0.490177	−	0.871623i	$-0.663068\pi$
$347$	−1.42135e10	−0.980354	−0.490177	+	0.871623i	$0.663068\pi$
$348$	2.15035e10	1.46620
$349$	−1.08482e10	−0.731232	−0.365616	−	0.930766i	$-0.619142\pi$
$349$	−1.08482e10	−0.731232	−0.365616	+	0.930766i	$0.619142\pi$
$350$	0	0
$351$	9.16434e10	6.03772
$352$	0	0
$353$	−2.91662e10	−1.87837	−0.939185	−	0.343411i	$-0.888418\pi$
$353$	−2.91662e10	−1.87837	−0.939185	+	0.343411i	$0.888418\pi$
$354$	−6.34202e9	−0.403845
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−3.08516e8	−0.0187822
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.69836e10	1.00000
$362$	0	0
$363$	−3.47254e10	−1.99996
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	1.69407e10	0.923722
$369$	1.01132e11	5.45486
$370$	0	0
$371$	0	0
$372$	−5.06628e10	−2.64556
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	4.03321e9	0.201790
$377$	−2.29444e10	−1.13583
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−4.82329e10	−2.28899
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	2.60969e10	1.20023
$385$	0	0
$386$	−5.56384e9	−0.250626
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−7.46576e9	−0.316177
$393$	−4.25789e10	−1.78494
$394$	−1.94601e9	−0.0807531
$395$	0	0
$396$	0	0
$397$	1.44117e10	0.580167	0.290084	−	0.957001i	$-0.406317\pi$
$397$	1.44117e10	0.580167	0.290084	+	0.957001i	$0.406317\pi$
$398$	0	0
$399$	0	0
$400$	2.36473e10	0.923722
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	5.40576e10	2.04945
$404$	3.91466e10	1.46950
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	3.82170e10	1.36573	0.682863	−	0.730547i	$-0.260735\pi$
$409$	3.82170e10	1.36573	0.682863	+	0.730547i	$0.260735\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	1.41125e10	0.480399
$415$	0	0
$416$	−2.09819e10	−0.700603
$417$	−1.14180e11	−3.77612
$418$	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			$\pi$
$422$	4.56210e9	0.143852
$423$	−6.12956e10	−1.91455
$424$	0	0
$425$	0	0
$426$	1.79548e10	0.545182
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.28674e11	−3.69451
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−2.07459e10	−0.563684
$439$	3.96244e10	1.06685	0.533426	−	0.845847i	$-0.320904\pi$
$439$	3.96244e10	1.06685	0.533426	+	0.845847i	$0.320904\pi$
$440$	0	0
$441$	1.13462e11	2.99984
$442$	0	0
$443$	2.17059e10	0.563591	0.281795	−	0.959475i	$-0.409070\pi$
$443$	2.17059e10	0.563591	0.281795	+	0.959475i	$0.409070\pi$
$444$	0	0
$445$	0	0
$446$	−7.14068e9	−0.180468
$447$	0	0
$448$	0	0
$449$	−7.58504e10	−1.86626	−0.933131	−	0.359537i	$-0.882935\pi$
$449$	−7.58504e10	−1.86626	−0.933131	+	0.359537i	$0.882935\pi$
$450$	1.96994e10	0.480399
$451$	0	0
$452$	0	0
$453$	−1.38560e11	−3.29038
$454$	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
$461$	8.23739e10	1.82384	0.911918	−	0.410371i	$-0.134601\pi$
$461$	8.23739e10	1.82384	0.911918	+	0.410371i	$0.134601\pi$
$462$	0	0
$463$	8.37652e10	1.82280	0.911401	−	0.411519i	$-0.135002\pi$
$463$	8.37652e10	1.82280	0.911401	+	0.411519i	$0.135002\pi$
$464$	3.22157e10	0.695018
$465$	0	0
$466$	−1.58531e9	−0.0336178
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	2.11668e11	4.41237
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.97873e10	−0.398675
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	1.63723e10	0.313615
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−5.34686e10	−0.974355
$485$	0	0
$486$	−5.35916e10	−0.960621
$487$	5.04277e10	0.896505	0.448253	−	0.893907i	$-0.352046\pi$
$487$	5.04277e10	0.896505	0.448253	+	0.893907i	$0.352046\pi$
$488$	0	0
$489$	−1.20256e11	−2.10315
$490$	0	0
$491$	−1.13875e11	−1.95931	−0.979654	−	0.200693i	$-0.935681\pi$
$491$	−1.13875e11	−1.95931	−0.979654	+	0.200693i	$0.935681\pi$
$492$	2.07627e11	3.54343
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−7.59009e10	−1.25407
$497$	0	0
$498$	0	0
$499$	−6.24377e10	−1.00704	−0.503518	−	0.863985i	$-0.667961\pi$
$499$	−6.24377e10	−1.00704	−0.503518	+	0.863985i	$0.667961\pi$
$500$	0	0
$501$	2.50743e11	3.97996
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.68994e11	−2.55764
$508$	−7.42667e10	−1.11517
$509$	5.47952e10	0.816341	0.408171	−	0.912906i	$-0.366167\pi$
$509$	5.47952e10	0.816341	0.408171	+	0.912906i	$0.366167\pi$
$510$	0	0
$511$	0	0
$512$	4.95303e10	0.720761
$513$	0	0
$514$	1.02910e10	0.147436
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.80867e11	−3.87107
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	2.68373e10	0.361457
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−6.55609e10	−0.869600
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.83110e10	1.00000
$530$	0	0
$531$	3.00722e11	3.78257
$532$	0	0
$533$	−2.21540e11	−2.74501
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.95056e10	0.234564
$538$	−5.85905e9	−0.0699356
$539$	0	0
$540$	0	0
$541$	−9.99060e10	−1.16628	−0.583140	−	0.812372i	$-0.698176\pi$
$541$	−9.99060e10	−1.16628	−0.583140	+	0.812372i	$0.698176\pi$
$542$	−1.67854e10	−0.194506
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.54084e11	1.72111	0.860556	−	0.509356i	$-0.170117\pi$
$547$	1.54084e11	1.72111	0.860556	+	0.509356i	$0.170117\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	5.87093e10	0.632340
$553$	0	0
$554$	3.48794e8	0.00370280
$555$	0	0
$556$	−1.75809e11	−1.83968
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−6.32292e10	−0.652201
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−1.25842e11	−1.24368
$565$	0	0
$566$	0	0
$567$	0	0
$568$	5.60195e10	0.538203
$569$	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
$575$	1.09313e11	1.00000
$576$	−2.80478e11	−2.54806
$577$	−4.57875e10	−0.413089	−0.206545	−	0.978437i	$-0.566222\pi$
$577$	−4.57875e10	−0.413089	−0.206545	+	0.978437i	$0.566222\pi$
$578$	1.78738e10	0.160142
$579$	3.51768e11	3.12998
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−6.47279e10	−0.556468
$585$	0	0
$586$	0	0
$587$	2.32648e11	1.95951	0.979753	−	0.200209i	$-0.0641623\pi$
$587$	2.32648e11	1.95951	0.979753	+	0.200209i	$0.0641623\pi$
$588$	2.32942e11	1.94867
$589$	0	0
$590$	0	0
$591$	1.23034e11	1.00850
$592$	0	0
$593$	1.38951e11	1.12368	0.561839	−	0.827247i	$-0.310094\pi$
$593$	1.38951e11	1.12368	0.561839	+	0.827247i	$0.310094\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−3.09148e10	−0.241748
$599$	−1.81057e9	−0.0140640	−0.00703200	−	0.999975i	$-0.502238\pi$
$599$	−1.81057e9	−0.0140640	−0.00703200	+	0.999975i	$0.502238\pi$
$600$	8.19513e10	0.632340
$601$	−1.37075e11	−1.05066	−0.525328	−	0.850900i	$-0.676057\pi$
$601$	−1.37075e11	−1.05066	−0.525328	+	0.850900i	$0.676057\pi$
$602$	0	0
$603$	0	0
$604$	−2.13349e11	−1.60303
$605$	0	0
$606$	6.51429e10	0.483033
$607$	7.41194e10	0.545981	0.272990	−	0.962017i	$-0.411987\pi$
$607$	7.41194e10	0.545981	0.272990	+	0.962017i	$0.411987\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.34274e11	0.963446
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	6.65706e9	0.0468392
$615$	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
$621$	−5.94815e11	−3.99959
$622$	3.97649e10	0.265668
$623$	0	0
$624$	4.22822e11	2.78881
$625$	1.52588e11	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
$633$	−2.88434e11	−1.79652
$634$	−8.29742e9	−0.0513554
$635$	0	0
$636$	0	0
$637$	−2.48551e11	−1.50958
$638$	0	0
$639$	−8.51367e11	−5.10638
$640$	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
$647$	6.66558e10	0.380383	0.190192	−	0.981747i	$-0.439089\pi$
$647$	6.66558e10	0.380383	0.190192	+	0.981747i	$0.439089\pi$
$648$	−2.78695e11	−1.58062
$649$	0	0
$650$	−4.31534e10	−0.241748
$651$	0	0
$652$	−1.85164e11	−1.02463
$653$	−2.33114e11	−1.28208	−0.641041	−	0.767507i	$-0.721497\pi$
$653$	−2.33114e11	−1.28208	−0.641041	+	0.767507i	$0.721497\pi$
$654$	0	0
$655$	0	0
$656$	3.11059e11	1.67968
$657$	9.83715e11	5.27968
$658$	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			$\pi$
$662$	−3.04165e10	−0.158371
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.48922e11	0.752410
$668$	3.86083e11	1.93899
$669$	4.51462e11	2.25381
$670$	0	0
$671$	0	0
$672$	0	0
$673$	8.76937e10	0.427472	0.213736	−	0.976891i	$-0.431437\pi$
$673$	8.76937e10	0.427472	0.213736	+	0.976891i	$0.431437\pi$
$674$	0	0
$675$	−8.30291e11	−3.99959
$676$	−2.60209e11	−1.24605
$677$	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
$683$	4.06424e11	1.86765	0.933826	−	0.357727i	$-0.116448\pi$
$683$	4.06424e11	1.86765	0.933826	+	0.357727i	$0.116448\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−3.60126e10	−0.157958	−0.0789791	−	0.996876i	$-0.525166\pi$
$691$	−3.60126e10	−0.157958	−0.0789791	+	0.996876i	$0.525166\pi$
$692$	−4.32465e11	−1.88594
$693$	0	0
$694$	−3.64188e10	−0.156996
$695$	0	0
$696$	1.11646e11	0.475779
$697$	0	0
$698$	−2.77959e10	−0.117101
$699$	1.00229e11	0.419842
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	2.34815e11	0.966891
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−7.47317e10	−0.300806
$707$	0	0
$708$	6.17391e11	2.45712
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.50863e11	−1.35762
$714$	0	0
$715$	0	0
$716$	3.00338e10	0.114277
$717$	−1.03512e12	−3.91664
$718$	0	0
$719$	−4.49887e11	−1.68340	−0.841702	−	0.539943i	$-0.818446\pi$
$719$	−4.49887e11	−1.68340	−0.841702	+	0.539943i	$0.818446\pi$
$720$	0	0
$721$	0	0
$722$	4.35164e10	0.160142
$723$	0	0
$724$	0	0
$725$	2.07877e11	0.752410
$726$	−8.89759e10	−0.320277
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	1.97636e12	6.99771
$730$	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
$736$	1.36184e11	0.464103
$737$	0	0
$738$	2.59128e11	0.873551
$739$	5.83514e11	1.95647	0.978236	−	0.207494i	$-0.0665308\pi$
$739$	5.83514e11	1.95647	0.978236	+	0.207494i	$0.0665308\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−2.63040e11	−0.858479
$745$	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			$\pi$
$752$	−1.88531e11	−0.589537
$753$	0	0
$754$	−5.87898e10	−0.181893
$755$	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
$761$	−6.69665e11	−1.99673	−0.998364	−	0.0571821i	$-0.981788\pi$
$761$	−6.69665e11	−1.99673	−0.998364	+	0.0571821i	$0.981788\pi$
$762$	−1.23586e11	−0.366562
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.58761e11	−1.90347
$768$	−5.24120e11	−1.50656
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−6.50635e11	−1.84128
$772$	5.41635e11	1.52489
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−4.89763e11	−1.35762
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.13114e12	−3.00933
$784$	3.48984e11	0.923722
$785$	0	0
$786$	−1.09098e11	−0.285844
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	1.89442e11	0.491328
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	3.69266e10	0.0929090
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	1.90097e11	0.464103
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	1.38510e11	0.328202
$807$	3.70432e11	0.873403
$808$	2.03248e11	0.476849
$809$	8.01341e11	1.87078	0.935392	−	0.353613i	$-0.115047\pi$
$809$	8.01341e11	1.87078	0.935392	+	0.353613i	$0.115047\pi$
$810$	0	0
$811$	−5.69179e11	−1.31573	−0.657863	−	0.753138i	$-0.728539\pi$
$811$	−5.69179e11	−1.31573	−0.657863	+	0.753138i	$0.728539\pi$
$812$	0	0
$813$	1.06124e12	2.42912
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	9.79222e10	0.218710
$819$	0	0
$820$	0	0
$821$	−8.33041e11	−1.83355	−0.916777	−	0.399400i	$-0.869219\pi$
$821$	−8.33041e11	−1.83355	−0.916777	+	0.399400i	$0.869219\pi$
$822$	0	0
$823$	4.82144e11	1.05094	0.525469	−	0.850812i	$-0.323890\pi$
$823$	4.82144e11	1.05094	0.525469	+	0.850812i	$0.323890\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−1.37384e12	−2.92290
$829$	9.08358e11	1.92326	0.961632	−	0.274344i	$-0.0884607\pi$
$829$	9.08358e11	1.92326	0.961632	+	0.274344i	$0.0884607\pi$
$830$	0	0
$831$	−2.20521e10	−0.0462431
$832$	6.14416e11	1.28224
$833$	0	0
$834$	−2.92560e11	−0.604714
$835$	0	0
$836$	0	0
$837$	2.66499e12	5.42993
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−2.17046e11	−0.433879
$842$	0	0
$843$	0	0
$844$	−4.44117e11	−0.875240
$845$	0	0
$846$	−1.57056e11	−0.306600
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−1.74788e12	−3.31707
$853$	2.04587e11	0.386439	0.193220	−	0.981156i	$-0.438107\pi$
$853$	2.04587e11	0.386439	0.193220	+	0.981156i	$0.438107\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.52876e11	−1.76650	−0.883249	−	0.468904i	$-0.844649\pi$
$857$	−9.52876e11	−1.76650	−0.883249	+	0.468904i	$0.844649\pi$
$858$	0	0
$859$	1.08827e12	1.99877	0.999386	−	0.0350442i	$-0.0111572\pi$
$859$	1.08827e12	1.99877	0.999386	+	0.0350442i	$0.0111572\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.01170e12	1.82393	0.911964	−	0.410271i	$-0.134566\pi$
$863$	1.01170e12	1.82393	0.911964	+	0.410271i	$0.134566\pi$
$864$	−1.03439e12	−1.85622
$865$	0	0
$866$	0	0
$867$	−1.13005e12	−1.99996
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	2.01960e12	3.42964
$877$	−3.92609e11	−0.663685	−0.331843	−	0.943335i	$-0.607670\pi$
$877$	−3.92609e11	−0.663685	−0.331843	+	0.943335i	$0.607670\pi$
$878$	1.01528e11	0.170848
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	2.90721e11	0.480399
$883$	−7.54063e11	−1.24041	−0.620205	−	0.784440i	$-0.712950\pi$
$883$	−7.54063e11	−1.24041	−0.620205	+	0.784440i	$0.712950\pi$
$884$	0	0
$885$	0	0
$886$	5.56164e10	0.0902544
$887$	1.22947e12	1.98620	0.993100	−	0.117267i	$-0.0374134\pi$
$887$	1.22947e12	1.98620	0.993100	+	0.117267i	$0.0374134\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	6.95139e11	1.09803
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.95456e12	3.01911
$898$	−1.94349e11	−0.298866
$899$	−6.67225e11	−1.02149
$900$	−1.91772e12	−2.92290
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−3.55029e11	−0.526928
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−3.08890e12	−4.52427
$910$	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
$921$	−4.20886e11	−0.584959
$922$	2.11064e11	0.292072
$923$	1.86501e12	2.56965
$924$	0	0
$925$	0	0
$926$	2.14629e11	0.291907
$927$	0	0
$928$	2.58977e11	0.349196
$929$	−1.41888e12	−1.90494	−0.952471	−	0.304630i	$-0.901467\pi$
$929$	−1.41888e12	−1.90494	−0.952471	+	0.304630i	$0.901467\pi$
$930$	0	0
$931$	0	0
$932$	1.54328e11	0.204542
$933$	−2.51410e12	−3.31784
$934$	0	0
$935$	0	0
$936$	1.09898e12	1.43181
$937$	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
$943$	1.43791e12	1.81839
$944$	9.24950e11	1.16474
$945$	0	0
$946$	0	0
$947$	−1.59012e12	−1.97711	−0.988553	−	0.150874i	$-0.951791\pi$
$947$	−1.59012e12	−1.97711	−0.988553	+	0.150874i	$0.951791\pi$
$948$	0	0
$949$	−2.15493e12	−2.65685
$950$	0	0
$951$	5.24595e11	0.641361
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	−1.59383e12	−1.90814
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.19106e11	0.843139
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.52517e11	−0.288791	−0.144396	−	0.989520i	$-0.546124\pi$
$967$	−2.52517e11	−0.288791	−0.144396	+	0.989520i	$0.546124\pi$
$968$	−2.77607e11	−0.316177
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	5.21710e12	5.84473
$973$	0	0
$974$	1.29209e11	0.143568
$975$	2.72833e12	3.01911
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	−3.08128e11	−0.336803
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−2.91778e11	−0.313767
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	1.07800e12	1.14984
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.52898e12	−1.58528	−0.792641	−	0.609688i	$-0.791295\pi$
$991$	−1.52898e12	−1.58528	−0.792641	+	0.609688i	$0.791295\pi$
$992$	−6.10155e11	−0.630077
$993$	1.92305e12	1.97785
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.91316e12	−1.93629	−0.968146	−	0.250388i	$-0.919442\pi$
$997$	−1.91316e12	−1.93629	−0.968146	+	0.250388i	$0.919442\pi$
$998$	−1.59982e11	−0.161268
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.9.b.a.22.2	✓	3
3.2	odd	2		207.9.d.a.91.2		3
4.3	odd	2		368.9.f.a.321.3		3
23.22	odd	2	CM	23.9.b.a.22.2	✓	3
69.68	even	2		207.9.d.a.91.2		3
92.91	even	2		368.9.f.a.321.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.9.b.a.22.2	✓	3	1.1	even	1	trivial
23.9.b.a.22.2	✓	3	23.22	odd	2	CM
207.9.d.a.91.2		3	3.2	odd	2
207.9.d.a.91.2		3	69.68	even	2
368.9.f.a.321.3		3	4.3	odd	2
368.9.f.a.321.3		3	92.91	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.56227 q^{2} -161.997 q^{3} -249.435 q^{4} -415.079 q^{6} -1295.06 q^{8} +19681.9 q^{9} +O(q^{10})\)	\(q+2.56227 q^{2} -161.997 q^{3} -249.435 q^{4} -415.079 q^{6} -1295.06 q^{8} +19681.9 q^{9} +40407.6 q^{12} -43115.2 q^{13} +60537.0 q^{16} +50430.4 q^{18} +279841. q^{23} +209795. q^{24} +390625. q^{25} -110473. q^{26} -2.12555e6 q^{27} +532166. q^{29} -1.25379e6 q^{31} +486647. q^{32} -4.90936e6 q^{36} +6.98453e6 q^{39} +5.13832e6 q^{41} +717028. q^{46} -3.11431e6 q^{47} -9.80680e6 q^{48} +5.76480e6 q^{49} +1.00089e6 q^{50} +1.07544e7 q^{52} -5.44622e6 q^{54} +1.36355e6 q^{58} +1.52791e7 q^{59} -3.21255e6 q^{62} -1.42506e7 q^{64} -4.53333e7 q^{69} -4.32563e7 q^{71} -2.54893e7 q^{72} +4.99806e7 q^{73} -6.32800e7 q^{75} +1.78962e7 q^{78} +2.15198e8 q^{81} +1.31658e7 q^{82} -8.62091e7 q^{87} -6.98021e7 q^{92} +2.03110e8 q^{93} -7.97969e6 q^{94} -7.88352e7 q^{96} +1.47710e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 768 q^{4} - 573 q^{6} - 5853 q^{8} + 19683 q^{9}+O(q^{10}) \) 3 * q + 768 * q^4 - 573 * q^6 - 5853 * q^8 + 19683 * q^9	\( 3 q + 768 q^{4} - 573 q^{6} - 5853 q^{8} + 19683 q^{9} + 122691 q^{12} + 196608 q^{16} + 58467 q^{18} + 839523 q^{23} - 146688 q^{24} + 1171875 q^{25} - 1958493 q^{26} - 3188058 q^{27} - 1498368 q^{32} - 4929405 q^{36} + 10418502 q^{39} + 1117923 q^{48} + 17294403 q^{49} + 37281507 q^{52} - 3759453 q^{54} - 60663549 q^{58} + 45837222 q^{59} - 69712509 q^{62} - 38912445 q^{64} - 23433981 q^{72} - 130646013 q^{78} + 129140163 q^{81} + 132337347 q^{82} - 131352378 q^{87} + 214917888 q^{92} + 302555622 q^{93} + 430562787 q^{94} - 239370141 q^{96}+O(q^{100}) \) 3 * q + 768 * q^4 - 573 * q^6 - 5853 * q^8 + 19683 * q^9 + 122691 * q^12 + 196608 * q^16 + 58467 * q^18 + 839523 * q^23 - 146688 * q^24 + 1171875 * q^25 - 1958493 * q^26 - 3188058 * q^27 - 1498368 * q^32 - 4929405 * q^36 + 10418502 * q^39 + 1117923 * q^48 + 17294403 * q^49 + 37281507 * q^52 - 3759453 * q^54 - 60663549 * q^58 + 45837222 * q^59 - 69712509 * q^62 - 38912445 * q^64 - 23433981 * q^72 - 130646013 * q^78 + 129140163 * q^81 + 132337347 * q^82 - 131352378 * q^87 + 214917888 * q^92 + 302555622 * q^93 + 430562787 * q^94 - 239370141 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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