LMFDB - Embedded newform 23.5.b.a.22.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,5,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, 5, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 5 $
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:	$2.37750915093$
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:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			22.1
Root			$-2.14510$ 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)

$f(q)$	$=$	$q-7.63824 q^{2} -15.6172 q^{3} +42.3427 q^{4} +119.288 q^{6} -201.212 q^{8} +162.896 q^{9} +O(q^{10})$	$q-7.63824 q^{2} -15.6172 q^{3} +42.3427 q^{4} +119.288 q^{6} -201.212 q^{8} +162.896 q^{9} -661.274 q^{12} +215.011 q^{13} +859.421 q^{16} -1244.24 q^{18} +529.000 q^{23} +3142.36 q^{24} +625.000 q^{25} -1642.30 q^{26} -1278.99 q^{27} +115.883 q^{29} -1139.89 q^{31} -3345.07 q^{32} +6897.48 q^{36} -3357.86 q^{39} +2262.80 q^{41} -4040.63 q^{46} +4396.19 q^{47} -13421.7 q^{48} +2401.00 q^{49} -4773.90 q^{50} +9104.14 q^{52} +9769.25 q^{54} -885.145 q^{58} -6286.00 q^{59} +8706.73 q^{62} +11799.7 q^{64} -8261.49 q^{69} +9775.40 q^{71} -32776.7 q^{72} -7427.60 q^{73} -9760.74 q^{75} +25648.2 q^{78} +6779.64 q^{81} -17283.8 q^{82} -1809.77 q^{87} +22399.3 q^{92} +17801.8 q^{93} -33579.2 q^{94} +52240.6 q^{96} -18339.4 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9	$ 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9} - 861 q^{12} + 768 q^{16} - 1437 q^{18} + 1587 q^{23} + 2352 q^{24} + 1875 q^{25} - 1533 q^{26} - 42 q^{27} - 3792 q^{32} + 3315 q^{36} - 8538 q^{39} - 11613 q^{48} + 7203 q^{49} + 15603 q^{52} + 11907 q^{54} + 4659 q^{58} - 18858 q^{59} + 7539 q^{62} + 6435 q^{64} - 42189 q^{72} + 16563 q^{78} + 19683 q^{81} - 34797 q^{82} - 25242 q^{87} + 25392 q^{92} + 47478 q^{93} - 51933 q^{94} + 68019 q^{96}+O(q^{100}) $ 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9 - 861 * q^12 + 768 * q^16 - 1437 * q^18 + 1587 * q^23 + 2352 * q^24 + 1875 * q^25 - 1533 * q^26 - 42 * q^27 - 3792 * q^32 + 3315 * q^36 - 8538 * q^39 - 11613 * q^48 + 7203 * q^49 + 15603 * q^52 + 11907 * q^54 + 4659 * q^58 - 18858 * q^59 + 7539 * q^62 + 6435 * q^64 - 42189 * q^72 + 16563 * q^78 + 19683 * q^81 - 34797 * q^82 - 25242 * q^87 + 25392 * q^92 + 47478 * q^93 - 51933 * q^94 + 68019 * 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 $

$2$	−7.63824	−1.90956	−0.954780	−	0.297314i	$-0.903909\pi$
$2$	−7.63824	−1.90956	−0.954780	+	0.297314i	$0.903909\pi$
$3$	−15.6172	−1.73524	−0.867621	−	0.497225i	$-0.834352\pi$
$3$	−15.6172	−1.73524	−0.867621	+	0.497225i	$0.834352\pi$
$4$	42.3427	2.64642
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	119.288	3.31355
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−201.212	−3.14393
$9$	162.896	2.01107
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−661.274	−4.59218
$13$	215.011	1.27225	0.636127	−	0.771585i	$-0.280536\pi$
$13$	215.011	1.27225	0.636127	+	0.771585i	$0.280536\pi$
$14$	0	0
$15$	0	0
$16$	859.421	3.35711
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−1244.24	−3.84025
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	529.000	1.00000
$23$	529.000	1.00000
$24$	3142.36	5.45549
$25$	625.000	1.00000
$26$	−1642.30	−2.42944
$27$	−1278.99	−1.75445
$28$	0	0
$29$	115.883	0.137792	0.0688962	−	0.997624i	$-0.478052\pi$
$29$	115.883	0.137792	0.0688962	+	0.997624i	$0.478052\pi$
$30$	0	0
$31$	−1139.89	−1.18615	−0.593073	−	0.805148i	$-0.702086\pi$
$31$	−1139.89	−1.18615	−0.593073	+	0.805148i	$0.702086\pi$
$32$	−3345.07	−3.26667
$33$	0	0
$34$	0	0
$35$	0	0
$36$	6897.48	5.32213
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−3357.86	−2.20767
$40$	0	0
$41$	2262.80	1.34610	0.673051	−	0.739596i	$-0.264983\pi$
$41$	2262.80	1.34610	0.673051	+	0.739596i	$0.264983\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−4040.63	−1.90956
$47$	4396.19	1.99013	0.995063	−	0.0992408i	$-0.0316414\pi$
$47$	4396.19	1.99013	0.995063	+	0.0992408i	$0.0316414\pi$
$48$	−13421.7	−5.82541
$49$	2401.00	1.00000
$50$	−4773.90	−1.90956
$51$	0	0
$52$	9104.14	3.36692
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	9769.25	3.35022
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−885.145	−0.263123
$59$	−6286.00	−1.80580	−0.902901	−	0.429848i	$-0.858567\pi$
$59$	−6286.00	−1.80580	−0.902901	+	0.429848i	$0.858567\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	8706.73	2.26502
$63$	0	0
$64$	11799.7	2.88079
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−8261.49	−1.73524
$70$	0	0
$71$	9775.40	1.93918	0.969589	−	0.244738i	$-0.0787019\pi$
$71$	9775.40	1.93918	0.969589	+	0.244738i	$0.0787019\pi$
$72$	−32776.7	−6.32266
$73$	−7427.60	−1.39381	−0.696904	−	0.717165i	$-0.745439\pi$
$73$	−7427.60	−1.39381	−0.696904	+	0.717165i	$0.745439\pi$
$74$	0	0
$75$	−9760.74	−1.73524
$76$	0	0
$77$	0	0
$78$	25648.2	4.21568
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	6779.64	1.03332
$82$	−17283.8	−2.57046
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1809.77	−0.239103
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	22399.3	2.64642
$93$	17801.8	2.05825
$94$	−33579.2	−3.80027
$95$	0	0
$96$	52240.6	5.66847
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−18339.4	−1.90956
$99$	0	0
$100$	26464.2	2.64642
$101$	7154.00	0.701304	0.350652	−	0.936506i	$-0.385960\pi$
$101$	7154.00	0.701304	0.350652	+	0.936506i	$0.385960\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−43262.7	−3.99988
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−54156.0	−4.64300
$109$	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
$116$	4906.81	0.364656
$117$	35024.5	2.55859
$118$	48014.0	3.44829
$119$	0	0
$120$	0	0
$121$	14641.0	1.00000
$122$	0	0
$123$	−35338.5	−2.33581
$124$	−48265.9	−3.13904
$125$	0	0
$126$	0	0
$127$	27220.0	1.68765	0.843823	−	0.536621i	$-0.180300\pi$
$127$	27220.0	1.68765	0.843823	+	0.536621i	$0.180300\pi$
$128$	−36608.0	−2.23438
$129$	0	0
$130$	0	0
$131$	−1047.25	−0.0610251	−0.0305125	−	0.999534i	$-0.509714\pi$
$131$	−1047.25	−0.0610251	−0.0305125	+	0.999534i	$0.509714\pi$
$132$	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			$\pi$
$138$	63103.2	3.31355
$139$	−24646.1	−1.27561	−0.637807	−	0.770197i	$-0.720158\pi$
$139$	−24646.1	−1.27561	−0.637807	+	0.770197i	$0.720158\pi$
$140$	0	0
$141$	−68656.1	−3.45335
$142$	−74666.8	−3.70298
$143$	0	0
$144$	139997.	6.75138
$145$	0	0
$146$	56733.8	2.66156
$147$	−37496.9	−1.73524
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	74554.9	3.31355
$151$	−10004.9	−0.438793	−0.219396	−	0.975636i	$-0.570409\pi$
$151$	−10004.9	−0.438793	−0.219396	+	0.975636i	$0.570409\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−142181.	−5.84242
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−51784.5	−1.97319
$163$	45614.4	1.71683	0.858414	−	0.512958i	$-0.171450\pi$
$163$	45614.4	1.71683	0.858414	+	0.512958i	$0.171450\pi$
$164$	95812.9	3.56235
$165$	0	0
$166$	0	0
$167$	2786.00	0.0998960	0.0499480	−	0.998752i	$-0.484094\pi$
$167$	2786.00	0.0998960	0.0499480	+	0.998752i	$0.484094\pi$
$168$	0	0
$169$	17668.7	0.618629
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−59374.0	−1.98383	−0.991914	−	0.126910i	$-0.959494\pi$
$173$	−59374.0	−1.98383	−0.991914	+	0.126910i	$0.959494\pi$
$174$	13823.5	0.456582
$175$	0	0
$176$	0	0
$177$	98169.6	3.13351
$178$	0	0
$179$	18394.2	0.574084	0.287042	−	0.957918i	$-0.407328\pi$
$179$	18394.2	0.574084	0.287042	+	0.957918i	$0.407328\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−106441.	−3.14393
$185$	0	0
$186$	−135975.	−3.93036
$187$	0	0
$188$	186147.	5.26671
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−184279.	−4.99888
$193$	−73191.7	−1.96493	−0.982465	−	0.186448i	$-0.940302\pi$
$193$	−73191.7	−1.96493	−0.982465	+	0.186448i	$0.940302\pi$
$194$	0	0
$195$	0	0
$196$	101665.	2.64642
$197$	−29454.9	−0.758970	−0.379485	−	0.925198i	$-0.623899\pi$
$197$	−29454.9	−0.758970	−0.379485	+	0.925198i	$0.623899\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−125757.	−3.14393
$201$	0	0
$202$	−54644.0	−1.33918
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	86172.2	2.01107
$208$	184785.	4.27110
$209$	0	0
$210$	0	0
$211$	75794.0	1.70243	0.851216	−	0.524815i	$-0.175866\pi$
$211$	75794.0	1.70243	0.851216	+	0.524815i	$0.175866\pi$
$212$	0	0
$213$	−152664.	−3.36495
$214$	0	0
$215$	0	0
$216$	257348.	5.51587
$217$	0	0
$218$	0	0
$219$	115998.	2.41859
$220$	0	0
$221$	0	0
$222$	0	0
$223$	46466.0	0.934384	0.467192	−	0.884156i	$-0.345266\pi$
$223$	46466.0	0.934384	0.467192	+	0.884156i	$0.345266\pi$
$224$	0	0
$225$	101810.	2.01107
$226$	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
$232$	−23317.1	−0.433210
$233$	−106210.	−1.95639	−0.978193	−	0.207698i	$-0.933403\pi$
$233$	−106210.	−1.95639	−0.978193	+	0.207698i	$0.933403\pi$
$234$	−267526.	−4.88578
$235$	0	0
$236$	−266166.	−4.77891
$237$	0	0
$238$	0	0
$239$	−66917.4	−1.17150	−0.585751	−	0.810491i	$-0.699200\pi$
$239$	−66917.4	−1.17150	−0.585751	+	0.810491i	$0.699200\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−111831.	−1.90956
$243$	−2280.55	−0.0386213
$244$	0	0
$245$	0	0
$246$	269924.	4.46038
$247$	0	0
$248$	229359.	3.72917
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	−207913.	−3.22266
$255$	0	0
$256$	90825.1	1.38588
$257$	2987.32	0.0452288	0.0226144	−	0.999744i	$-0.492801\pi$
$257$	2987.32	0.0452288	0.0226144	+	0.999744i	$0.492801\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	18877.0	0.277110
$262$	7999.15	0.116531
$263$	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
$269$	−52585.1	−0.726705	−0.363352	−	0.931652i	$-0.618368\pi$
$269$	−52585.1	−0.726705	−0.363352	+	0.931652i	$0.618368\pi$
$270$	0	0
$271$	−65086.0	−0.886235	−0.443118	−	0.896463i	$-0.646128\pi$
$271$	−65086.0	−0.886235	−0.443118	+	0.896463i	$0.646128\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−349814.	−4.59218
$277$	147997.	1.92883	0.964414	−	0.264398i	$-0.0851734\pi$
$277$	147997.	1.92883	0.964414	+	0.264398i	$0.0851734\pi$
$278$	188253.	2.43586
$279$	−185684.	−2.38542
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	524412.	6.59438
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	413917.	5.13188
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−544901.	−6.56950
$289$	83521.0	1.00000
$290$	0	0
$291$	0	0
$292$	−314505.	−3.68860
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	286410.	3.31355
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	113741.	1.27225
$300$	−413296.	−4.59218
$301$	0	0
$302$	76419.9	0.837901
$303$	−111725.	−1.21693
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−142702.	−1.51410	−0.757048	−	0.653359i	$-0.773359\pi$
$307$	−142702.	−1.51410	−0.757048	+	0.653359i	$0.773359\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	43589.1	0.450668	0.225334	−	0.974282i	$-0.427653\pi$
$311$	43589.1	0.450668	0.225334	+	0.974282i	$0.427653\pi$
$312$	675642.	6.94077
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−130222.	−1.29588	−0.647942	−	0.761690i	$-0.724370\pi$
$317$	−130222.	−1.29588	−0.647942	+	0.761690i	$0.724370\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	287068.	2.73461
$325$	134382.	1.27225
$326$	−348414.	−3.27839
$327$	0	0
$328$	−455302.	−4.23206
$329$	0	0
$330$	0	0
$331$	219121.	1.99999	0.999995	−	0.00318553i	$-0.00101399\pi$
$331$	219121.	1.99999	0.999995	+	0.00318553i	$0.00101399\pi$
$332$	0	0
$333$	0	0
$334$	−21280.1	−0.190757
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−134958.	−1.18131
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	453513.	3.78824
$347$	121586.	1.00978	0.504888	−	0.863185i	$-0.331534\pi$
$347$	121586.	1.00978	0.504888	+	0.863185i	$0.331534\pi$
$348$	−76630.6	−0.632767
$349$	−242924.	−1.99443	−0.997215	−	0.0745829i	$-0.976237\pi$
$349$	−242924.	−1.99443	−0.997215	+	0.0745829i	$0.976237\pi$
$350$	0	0
$351$	−274997.	−2.23210
$352$	0	0
$353$	234251.	1.87989	0.939946	−	0.341324i	$-0.110875\pi$
$353$	234251.	1.87989	0.939946	+	0.341324i	$0.110875\pi$
$354$	−749843.	−5.98362
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−140499.	−1.09625
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	130321.	1.00000
$362$	0	0
$363$	−228651.	−1.73524
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	454634.	3.35711
$369$	368602.	2.70710
$370$	0	0
$371$	0	0
$372$	753777.	5.44700
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−884565.	−6.25683
$377$	24916.2	0.175307
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−425100.	−2.92848
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	571714.	3.87719
$385$	0	0
$386$	559055.	3.75215
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−483110.	−3.14393
$393$	16355.1	0.105893
$394$	224983.	1.44930
$395$	0	0
$396$	0	0
$397$	−36058.2	−0.228783	−0.114391	−	0.993436i	$-0.536492\pi$
$397$	−36058.2	−0.228783	−0.114391	+	0.993436i	$0.536492\pi$
$398$	0	0
$399$	0	0
$400$	537138.	3.35711
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−245088.	−1.50908
$404$	302920.	1.85594
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−268822.	−1.60701	−0.803505	−	0.595298i	$-0.797034\pi$
$409$	−268822.	−1.60701	−0.803505	+	0.595298i	$0.797034\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−658204.	−3.84025
$415$	0	0
$416$	−719227.	−4.15604
$417$	384903.	2.21350
$418$	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			$\pi$
$422$	−578933.	−3.25090
$423$	716124.	4.00228
$424$	0	0
$425$	0	0
$426$	1.16609e6	6.42556
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.09919e6	−5.88988
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−886022.	−4.61845
$439$	7524.73	0.0390447	0.0195223	−	0.999809i	$-0.493785\pi$
$439$	7524.73	0.0390447	0.0195223	+	0.999809i	$0.493785\pi$
$440$	0	0
$441$	391114.	2.01107
$442$	0	0
$443$	360803.	1.83849	0.919247	−	0.393680i	$-0.128798\pi$
$443$	360803.	1.83849	0.919247	+	0.393680i	$0.128798\pi$
$444$	0	0
$445$	0	0
$446$	−354918.	−1.78426
$447$	0	0
$448$	0	0
$449$	−73726.0	−0.365703	−0.182851	−	0.983141i	$-0.558533\pi$
$449$	−73726.0	−0.365703	−0.182851	+	0.983141i	$0.558533\pi$
$450$	−777651.	−3.84025
$451$	0	0
$452$	0	0
$453$	156249.	0.761412
$454$	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
$461$	−130540.	−0.614245	−0.307123	−	0.951670i	$-0.599366\pi$
$461$	−130540.	−0.614245	−0.307123	+	0.951670i	$0.599366\pi$
$462$	0	0
$463$	−419134.	−1.95520	−0.977599	−	0.210474i	$-0.932499\pi$
$463$	−419134.	−1.95520	−0.977599	+	0.210474i	$0.932499\pi$
$464$	99592.6	0.462584
$465$	0	0
$466$	811259.	3.73584
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	1.48303e6	6.77109
$469$	0	0
$470$	0	0
$471$	0	0
$472$	1.26482e6	5.67733
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	511131.	2.23705
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	619939.	2.64642
$485$	0	0
$486$	17419.4	0.0737497
$487$	−417582.	−1.76069	−0.880346	−	0.474331i	$-0.842690\pi$
$487$	−417582.	−1.76069	−0.880346	+	0.474331i	$0.842690\pi$
$488$	0	0
$489$	−712369.	−2.97911
$490$	0	0
$491$	−391120.	−1.62236	−0.811179	−	0.584798i	$-0.801174\pi$
$491$	−391120.	−1.62236	−0.811179	+	0.584798i	$0.801174\pi$
$492$	−1.49633e6	−6.18154
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−979643.	−3.98203
$497$	0	0
$498$	0	0
$499$	−498001.	−2.00000	−0.999998	−	0.00203337i	$-0.999353\pi$
$499$	−498001.	−2.00000	−0.999998	+	0.00203337i	$0.999353\pi$
$500$	0	0
$501$	−43509.5	−0.173344
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−275935.	−1.07347
$508$	1.15257e6	4.46622
$509$	461501.	1.78130	0.890650	−	0.454690i	$-0.150250\pi$
$509$	461501.	1.78130	0.890650	+	0.454690i	$0.150250\pi$
$510$	0	0
$511$	0	0
$512$	−108015.	−0.412046
$513$	0	0
$514$	−22817.8	−0.0863671
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	927255.	3.44242
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−144187.	−0.529158
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−44343.4	−0.161498
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	279841.	1.00000
$530$	0	0
$531$	−1.02397e6	−3.63159
$532$	0	0
$533$	486526.	1.71258
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−287266.	−0.996175
$538$	401657.	1.38769
$539$	0	0
$540$	0	0
$541$	−317403.	−1.08447	−0.542234	−	0.840228i	$-0.682421\pi$
$541$	−317403.	−1.08447	−0.542234	+	0.840228i	$0.682421\pi$
$542$	497142.	1.69232
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	151747.	0.507162	0.253581	−	0.967314i	$-0.418392\pi$
$547$	151747.	0.507162	0.253581	+	0.967314i	$0.418392\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	1.66231e6	5.45549
$553$	0	0
$554$	−1.13044e6	−3.68321
$555$	0	0
$556$	−1.04358e6	−3.37581
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	1.41830e6	4.55510
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−2.90709e6	−9.13902
$565$	0	0
$566$	0	0
$567$	0	0
$568$	−1.96693e6	−6.09665
$569$	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
$575$	330625.	1.00000
$576$	1.92213e6	5.79347
$577$	238188.	0.715431	0.357716	−	0.933831i	$-0.383556\pi$
$577$	238188.	0.715431	0.357716	+	0.933831i	$0.383556\pi$
$578$	−637953.	−1.90956
$579$	1.14305e6	3.40963
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	1.49452e6	4.38204
$585$	0	0
$586$	0	0
$587$	−282772.	−0.820655	−0.410327	−	0.911938i	$-0.634586\pi$
$587$	−282772.	−0.820655	−0.410327	+	0.911938i	$0.634586\pi$
$588$	−1.58772e6	−4.59218
$589$	0	0
$590$	0	0
$591$	460002.	1.31700
$592$	0	0
$593$	−621502.	−1.76739	−0.883697	−	0.468060i	$-0.844953\pi$
$593$	−621502.	−1.76739	−0.883697	+	0.468060i	$0.844953\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−868779.	−2.42944
$599$	505634.	1.40923	0.704616	−	0.709589i	$-0.251119\pi$
$599$	505634.	1.40923	0.704616	+	0.709589i	$0.251119\pi$
$600$	1.96398e6	5.45549
$601$	722323.	1.99978	0.999891	−	0.0147505i	$-0.00469541\pi$
$601$	722323.	1.99978	0.999891	+	0.0147505i	$0.00469541\pi$
$602$	0	0
$603$	0	0
$604$	−423635.	−1.16123
$605$	0	0
$606$	853385.	2.32380
$607$	−587902.	−1.59561	−0.797806	−	0.602914i	$-0.794006\pi$
$607$	−587902.	−1.59561	−0.797806	+	0.602914i	$0.794006\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	945229.	2.53195
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.08999e6	2.89126
$615$	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
$621$	−676587.	−1.75445
$622$	−332944.	−0.860578
$623$	0	0
$624$	−2.88582e6	−7.41139
$625$	390625.	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
$633$	−1.18369e6	−2.95413
$634$	994667.	2.47457
$635$	0	0
$636$	0	0
$637$	516241.	1.27225
$638$	0	0
$639$	1.59238e6	3.89982
$640$	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
$647$	−784291.	−1.87356	−0.936782	−	0.349913i	$-0.886211\pi$
$647$	−784291.	−1.87356	−0.936782	+	0.349913i	$0.886211\pi$
$648$	−1.36414e6	−3.24870
$649$	0	0
$650$	−1.02644e6	−2.42944
$651$	0	0
$652$	1.93144e6	4.54345
$653$	−849657.	−1.99259	−0.996293	−	0.0860212i	$-0.972585\pi$
$653$	−849657.	−1.99259	−0.996293	+	0.0860212i	$0.972585\pi$
$654$	0	0
$655$	0	0
$656$	1.94470e6	4.51902
$657$	−1.20993e6	−2.80304
$658$	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			$\pi$
$662$	−1.67370e6	−3.81910
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	61302.3	0.137792
$668$	117967.	0.264367
$669$	−725668.	−1.62138
$670$	0	0
$671$	0	0
$672$	0	0
$673$	844720.	1.86502	0.932509	−	0.361148i	$-0.117615\pi$
$673$	844720.	1.86502	0.932509	+	0.361148i	$0.117615\pi$
$674$	0	0
$675$	−799370.	−1.75445
$676$	748139.	1.63715
$677$	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
$683$	−311736.	−0.668261	−0.334130	−	0.942527i	$-0.608443\pi$
$683$	−311736.	−0.668261	−0.334130	+	0.942527i	$0.608443\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−648046.	−1.35722	−0.678609	−	0.734500i	$-0.737417\pi$
$691$	−648046.	−1.35722	−0.678609	+	0.734500i	$0.737417\pi$
$692$	−2.51406e6	−5.25004
$693$	0	0
$694$	−928703.	−1.92823
$695$	0	0
$696$	364148.	0.751725
$697$	0	0
$698$	1.85551e6	3.80848
$699$	1.65870e6	3.39480
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	2.10049e6	4.26233
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−1.78927e6	−3.58976
$707$	0	0
$708$	4.15677e6	8.29257
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−603000.	−1.18615
$714$	0	0
$715$	0	0
$716$	778861.	1.51927
$717$	1.04506e6	2.03284
$718$	0	0
$719$	−290878.	−0.562669	−0.281335	−	0.959610i	$-0.590777\pi$
$719$	−290878.	−0.562669	−0.281335	+	0.959610i	$0.590777\pi$
$720$	0	0
$721$	0	0
$722$	−995423.	−1.90956
$723$	0	0
$724$	0	0
$725$	72427.1	0.137792
$726$	1.74649e6	3.31355
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−513535.	−0.966307
$730$	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
$736$	−1.76954e6	−3.26667
$737$	0	0
$738$	−2.81547e6	−5.16937
$739$	641815.	1.17523	0.587613	−	0.809142i	$-0.300068\pi$
$739$	641815.	1.17523	0.587613	+	0.809142i	$0.300068\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−3.58194e6	−6.47101
$745$	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			$\pi$
$752$	3.77818e6	6.68108
$753$	0	0
$754$	−190316.	−0.334759
$755$	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
$761$	1.01922e6	1.75994	0.879972	−	0.475025i	$-0.157561\pi$
$761$	1.01922e6	1.75994	0.879972	+	0.475025i	$0.157561\pi$
$762$	3.24702e6	5.59210
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.35156e6	−2.29744
$768$	−1.41843e6	−2.40484
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−46653.5	−0.0784830
$772$	−3.09913e6	−5.20003
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−712429.	−1.18615
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−148214.	−0.241749
$784$	2.06347e6	3.35711
$785$	0	0
$786$	−124924.	−0.202210
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1.24720e6	−2.00855
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	275421.	0.436874
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−2.09067e6	−3.26667
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	1.87204e6	2.88168
$807$	821231.	1.26101
$808$	−1.43947e6	−2.20485
$809$	−1.28765e6	−1.96743	−0.983715	−	0.179733i	$-0.942476\pi$
$809$	−1.28765e6	−1.96743	−0.983715	+	0.179733i	$0.942476\pi$
$810$	0	0
$811$	1.30923e6	1.99056	0.995281	−	0.0970363i	$-0.0309363\pi$
$811$	1.30923e6	1.99056	0.995281	+	0.0970363i	$0.0309363\pi$
$812$	0	0
$813$	1.01646e6	1.53783
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	2.05333e6	3.06868
$819$	0	0
$820$	0	0
$821$	274994.	0.407978	0.203989	−	0.978973i	$-0.434609\pi$
$821$	274994.	0.407978	0.203989	+	0.978973i	$0.434609\pi$
$822$	0	0
$823$	−1.16299e6	−1.71703	−0.858513	−	0.512791i	$-0.828612\pi$
$823$	−1.16299e6	−1.71703	−0.858513	+	0.512791i	$0.828612\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	3.64876e6	5.32213
$829$	1.36123e6	1.98072	0.990361	−	0.138507i	$-0.0442303\pi$
$829$	1.36123e6	1.98072	0.990361	+	0.138507i	$0.0442303\pi$
$830$	0	0
$831$	−2.31130e6	−3.34698
$832$	2.53707e6	3.66510
$833$	0	0
$834$	−2.93998e6	−4.22681
$835$	0	0
$836$	0	0
$837$	1.45791e6	2.08103
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−693852.	−0.981013
$842$	0	0
$843$	0	0
$844$	3.20932e6	4.50535
$845$	0	0
$846$	−5.46992e6	−7.64259
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−6.46421e6	−8.90505
$853$	1.12402e6	1.54481	0.772405	−	0.635130i	$-0.219053\pi$
$853$	1.12402e6	1.54481	0.772405	+	0.635130i	$0.219053\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.05696e6	1.43913	0.719563	−	0.694427i	$-0.244342\pi$
$857$	1.05696e6	1.43913	0.719563	+	0.694427i	$0.244342\pi$
$858$	0	0
$859$	−760165.	−1.03020	−0.515100	−	0.857130i	$-0.672245\pi$
$859$	−760165.	−1.03020	−0.515100	+	0.857130i	$0.672245\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	457549.	0.614351	0.307175	−	0.951653i	$-0.400616\pi$
$863$	457549.	0.614351	0.307175	+	0.951653i	$0.400616\pi$
$864$	4.27832e6	5.73120
$865$	0	0
$866$	0	0
$867$	−1.30436e6	−1.73524
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	4.91168e6	6.40061
$877$	889106.	1.15599	0.577995	−	0.816040i	$-0.303835\pi$
$877$	889106.	1.15599	0.577995	+	0.816040i	$0.303835\pi$
$878$	−57475.7	−0.0745582
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−2.98743e6	−3.84025
$883$	−679534.	−0.871545	−0.435772	−	0.900057i	$-0.643525\pi$
$883$	−679534.	−0.871545	−0.435772	+	0.900057i	$0.643525\pi$
$884$	0	0
$885$	0	0
$886$	−2.75590e6	−3.51072
$887$	−865450.	−1.10001	−0.550003	−	0.835163i	$-0.685373\pi$
$887$	−865450.	−1.10001	−0.550003	+	0.835163i	$0.685373\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	1.96750e6	2.47277
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.77631e6	−2.20767
$898$	563137.	0.698331
$899$	−132094.	−0.163442
$900$	4.31092e6	5.32213
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−1.19346e6	−1.45396
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	1.16536e6	1.41037
$910$	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
$921$	2.22860e6	2.62732
$922$	997096.	1.17294
$923$	2.10182e6	2.46713
$924$	0	0
$925$	0	0
$926$	3.20145e6	3.73357
$927$	0	0
$928$	−387638.	−0.450123
$929$	−1.61001e6	−1.86550	−0.932752	−	0.360519i	$-0.882600\pi$
$929$	−1.61001e6	−1.86550	−0.932752	+	0.360519i	$0.882600\pi$
$930$	0	0
$931$	0	0
$932$	−4.49723e6	−5.17742
$933$	−680738.	−0.782018
$934$	0	0
$935$	0	0
$936$	−7.04734e6	−8.04403
$937$	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
$943$	1.19702e6	1.34610
$944$	−5.40232e6	−6.06228
$945$	0	0
$946$	0	0
$947$	1.61671e6	1.80274	0.901370	−	0.433049i	$-0.142562\pi$
$947$	1.61671e6	1.80274	0.901370	+	0.433049i	$0.142562\pi$
$948$	0	0
$949$	−1.59701e6	−1.77328
$950$	0	0
$951$	2.03370e6	2.24867
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	−2.83346e6	−3.10028
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	375822.	0.406944
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−613533.	−0.656122	−0.328061	−	0.944656i	$-0.606395\pi$
$967$	−613533.	−0.656122	−0.328061	+	0.944656i	$0.606395\pi$
$968$	−2.94594e6	−3.14393
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−96564.7	−0.102208
$973$	0	0
$974$	3.18959e6	3.36215
$975$	−2.09867e6	−2.20767
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	5.44124e6	5.68879
$979$	0	0
$980$	0	0
$981$	0	0
$982$	2.98746e6	3.09799
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	7.11053e6	7.34364
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−632446.	−0.643986	−0.321993	−	0.946742i	$-0.604353\pi$
$991$	−632446.	−0.643986	−0.321993	+	0.946742i	$0.604353\pi$
$992$	3.81301e6	3.87475
$993$	−3.42205e6	−3.47047
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−250894.	−0.252406	−0.126203	−	0.992004i	$-0.540279\pi$
$997$	−250894.	−0.252406	−0.126203	+	0.992004i	$0.540279\pi$
$998$	3.80385e6	3.81911
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.5.b.a.22.1	✓	3
3.2	odd	2		207.5.d.a.91.3		3
4.3	odd	2		368.5.f.a.321.3		3
23.22	odd	2	CM	23.5.b.a.22.1	✓	3
69.68	even	2		207.5.d.a.91.3		3
92.91	even	2		368.5.f.a.321.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.5.b.a.22.1	✓	3	1.1	even	1	trivial
23.5.b.a.22.1	✓	3	23.22	odd	2	CM
207.5.d.a.91.3		3	3.2	odd	2
207.5.d.a.91.3		3	69.68	even	2
368.5.f.a.321.3		3	4.3	odd	2
368.5.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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.63824 q^{2} -15.6172 q^{3} +42.3427 q^{4} +119.288 q^{6} -201.212 q^{8} +162.896 q^{9} +O(q^{10})\)	\(q-7.63824 q^{2} -15.6172 q^{3} +42.3427 q^{4} +119.288 q^{6} -201.212 q^{8} +162.896 q^{9} -661.274 q^{12} +215.011 q^{13} +859.421 q^{16} -1244.24 q^{18} +529.000 q^{23} +3142.36 q^{24} +625.000 q^{25} -1642.30 q^{26} -1278.99 q^{27} +115.883 q^{29} -1139.89 q^{31} -3345.07 q^{32} +6897.48 q^{36} -3357.86 q^{39} +2262.80 q^{41} -4040.63 q^{46} +4396.19 q^{47} -13421.7 q^{48} +2401.00 q^{49} -4773.90 q^{50} +9104.14 q^{52} +9769.25 q^{54} -885.145 q^{58} -6286.00 q^{59} +8706.73 q^{62} +11799.7 q^{64} -8261.49 q^{69} +9775.40 q^{71} -32776.7 q^{72} -7427.60 q^{73} -9760.74 q^{75} +25648.2 q^{78} +6779.64 q^{81} -17283.8 q^{82} -1809.77 q^{87} +22399.3 q^{92} +17801.8 q^{93} -33579.2 q^{94} +52240.6 q^{96} -18339.4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9	\( 3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9} - 861 q^{12} + 768 q^{16} - 1437 q^{18} + 1587 q^{23} + 2352 q^{24} + 1875 q^{25} - 1533 q^{26} - 42 q^{27} - 3792 q^{32} + 3315 q^{36} - 8538 q^{39} - 11613 q^{48} + 7203 q^{49} + 15603 q^{52} + 11907 q^{54} + 4659 q^{58} - 18858 q^{59} + 7539 q^{62} + 6435 q^{64} - 42189 q^{72} + 16563 q^{78} + 19683 q^{81} - 34797 q^{82} - 25242 q^{87} + 25392 q^{92} + 47478 q^{93} - 51933 q^{94} + 68019 q^{96}+O(q^{100}) \) 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9 - 861 * q^12 + 768 * q^16 - 1437 * q^18 + 1587 * q^23 + 2352 * q^24 + 1875 * q^25 - 1533 * q^26 - 42 * q^27 - 3792 * q^32 + 3315 * q^36 - 8538 * q^39 - 11613 * q^48 + 7203 * q^49 + 15603 * q^52 + 11907 * q^54 + 4659 * q^58 - 18858 * q^59 + 7539 * q^62 + 6435 * q^64 - 42189 * q^72 + 16563 * q^78 + 19683 * q^81 - 34797 * q^82 - 25242 * q^87 + 25392 * q^92 + 47478 * q^93 - 51933 * q^94 + 68019 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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