LMFDB - Embedded newform 23.13.b.b.22.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 13);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$21.0218577974$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{69}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 17 $ x^2 - x - 17
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			22.2
Root			$4.65331$ 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)

$f(q)$	$=$	$q+126.720 q^{2} -1255.61 q^{3} +11961.8 q^{4} -159110. q^{6} +996756. q^{8} +1.04511e6 q^{9} +O(q^{10})$	$q+126.720 q^{2} -1255.61 q^{3} +11961.8 q^{4} -159110. q^{6} +996756. q^{8} +1.04511e6 q^{9} -1.50194e7 q^{12} +8.23202e6 q^{13} +7.73128e7 q^{16} +1.32436e8 q^{18} +1.48036e8 q^{23} -1.25153e9 q^{24} +2.44141e8 q^{25} +1.04316e9 q^{26} -6.44963e8 q^{27} -8.86136e8 q^{29} -1.34218e9 q^{31} +5.71433e9 q^{32} +1.25014e10 q^{36} -1.03362e10 q^{39} -1.14279e9 q^{41} +1.87590e10 q^{46} -1.57890e10 q^{47} -9.70745e10 q^{48} +1.38413e10 q^{49} +3.09374e10 q^{50} +9.84701e10 q^{52} -8.17294e10 q^{54} -1.12291e11 q^{58} -1.98745e10 q^{59} -1.70080e11 q^{62} +4.07444e11 q^{64} -1.85875e11 q^{69} +5.54476e10 q^{71} +1.04172e12 q^{72} +6.57018e10 q^{73} -3.06545e11 q^{75} -1.30980e12 q^{78} +2.54407e11 q^{81} -1.44814e11 q^{82} +1.11264e12 q^{87} +1.77078e12 q^{92} +1.68525e12 q^{93} -2.00077e12 q^{94} -7.17495e12 q^{96} +1.75396e12 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 79 q^{2} + 14 q^{3} + 10143 q^{4} - 219695 q^{6} + 1279010 q^{8} + 2125568 q^{9}+O(q^{10}) $ 2 * q + 79 * q^2 + 14 * q^3 + 10143 * q^4 - 219695 * q^6 + 1279010 * q^8 + 2125568 * q^9	$ 2 q + 79 q^{2} + 14 q^{3} + 10143 q^{4} - 219695 q^{6} + 1279010 q^{8} + 2125568 q^{9} - 17328591 q^{12} + 8482894 q^{13} + 71293759 q^{16} + 80876464 q^{18} + 296071778 q^{23} - 893182738 q^{24} + 488281250 q^{25} + 1031185505 q^{26} + 52075730 q^{27} + 244330126 q^{29} - 1677025154 q^{31} + 4845442959 q^{32} + 10536223824 q^{36} - 10017657950 q^{39} + 7596282526 q^{41} + 11694835231 q^{46} - 20606906306 q^{47} - 104716276511 q^{48} + 27682574402 q^{49} + 19287109375 q^{50} + 98013781089 q^{52} - 114991826033 q^{54} - 166236126415 q^{58} - 39749055836 q^{59} - 154101345055 q^{62} + 473560750658 q^{64} + 2072502446 q^{69} - 188893891874 q^{71} + 1346681462528 q^{72} - 223017449186 q^{73} + 3417968750 q^{75} - 1324994929537 q^{78} + 565171521854 q^{81} - 561839119135 q^{82} + 2547886433890 q^{87} + 1501528022127 q^{92} + 1260122292850 q^{93} - 1770863233615 q^{94} - 8278093192383 q^{96} + 1093461688879 q^{98}+O(q^{100}) $ 2 * q + 79 * q^2 + 14 * q^3 + 10143 * q^4 - 219695 * q^6 + 1279010 * q^8 + 2125568 * q^9 - 17328591 * q^12 + 8482894 * q^13 + 71293759 * q^16 + 80876464 * q^18 + 296071778 * q^23 - 893182738 * q^24 + 488281250 * q^25 + 1031185505 * q^26 + 52075730 * q^27 + 244330126 * q^29 - 1677025154 * q^31 + 4845442959 * q^32 + 10536223824 * q^36 - 10017657950 * q^39 + 7596282526 * q^41 + 11694835231 * q^46 - 20606906306 * q^47 - 104716276511 * q^48 + 27682574402 * q^49 + 19287109375 * q^50 + 98013781089 * q^52 - 114991826033 * q^54 - 166236126415 * q^58 - 39749055836 * q^59 - 154101345055 * q^62 + 473560750658 * q^64 + 2072502446 * q^69 - 188893891874 * q^71 + 1346681462528 * q^72 - 223017449186 * q^73 + 3417968750 * q^75 - 1324994929537 * q^78 + 565171521854 * q^81 - 561839119135 * q^82 + 2547886433890 * q^87 + 1501528022127 * q^92 + 1260122292850 * q^93 - 1770863233615 * q^94 - 8278093192383 * q^96 + 1093461688879 * q^98

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 $

$2$	126.720	1.97999	0.989996	−	0.141092i	$-0.0450613\pi$
$2$	126.720	1.97999	0.989996	+	0.141092i	$0.0450613\pi$
$3$	−1255.61	−1.72237	−0.861184	−	0.508293i	$-0.830277\pi$
$3$	−1255.61	−1.72237	−0.861184	+	0.508293i	$0.830277\pi$
$4$	11961.8	2.92037
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−159110.	−3.41028
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	996756.	3.80232
$9$	1.04511e6	1.96655
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.50194e7	−5.02996
$13$	8.23202e6	1.70548	0.852739	−	0.522337i	$-0.174940\pi$
$13$	8.23202e6	1.70548	0.852739	+	0.522337i	$0.174940\pi$
$14$	0	0
$15$	0	0
$16$	7.73128e7	4.60820
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.32436e8	3.89376
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.48036e8	1.00000
$23$	1.48036e8	1.00000
$24$	−1.25153e9	−6.54900
$25$	2.44141e8	1.00000
$26$	1.04316e9	3.37683
$27$	−6.44963e8	−1.66476
$28$	0	0
$29$	−8.86136e8	−1.48975	−0.744873	−	0.667206i	$-0.767490\pi$
$29$	−8.86136e8	−1.48975	−0.744873	+	0.667206i	$0.767490\pi$
$30$	0	0
$31$	−1.34218e9	−1.51231	−0.756153	−	0.654395i	$-0.772924\pi$
$31$	−1.34218e9	−1.51231	−0.756153	+	0.654395i	$0.772924\pi$
$32$	5.71433e9	5.32188
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1.25014e10	5.74307
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−1.03362e10	−2.93746
$40$	0	0
$41$	−1.14279e9	−0.240583	−0.120291	−	0.992739i	$-0.538383\pi$
$41$	−1.14279e9	−0.240583	−0.120291	+	0.992739i	$0.538383\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	1.87590e10	1.97999
$47$	−1.57890e10	−1.46476	−0.732381	−	0.680895i	$-0.761591\pi$
$47$	−1.57890e10	−1.46476	−0.732381	+	0.680895i	$0.761591\pi$
$48$	−9.70745e10	−7.93702
$49$	1.38413e10	1.00000
$50$	3.09374e10	1.97999
$51$	0	0
$52$	9.84701e10	4.98063
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−8.17294e10	−3.29622
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−1.12291e11	−2.94969
$59$	−1.98745e10	−0.471178	−0.235589	−	0.971853i	$-0.575702\pi$
$59$	−1.98745e10	−0.471178	−0.235589	+	0.971853i	$0.575702\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−1.70080e11	−2.99436
$63$	0	0
$64$	4.07444e11	5.92909
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−1.85875e11	−1.72237
$70$	0	0
$71$	5.54476e10	0.432845	0.216423	−	0.976300i	$-0.430561\pi$
$71$	5.54476e10	0.432845	0.216423	+	0.976300i	$0.430561\pi$
$72$	1.04172e12	7.47747
$73$	6.57018e10	0.434150	0.217075	−	0.976155i	$-0.430348\pi$
$73$	6.57018e10	0.434150	0.217075	+	0.976155i	$0.430348\pi$
$74$	0	0
$75$	−3.06545e11	−1.72237
$76$	0	0
$77$	0	0
$78$	−1.30980e12	−5.81615
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	2.54407e11	0.900781
$82$	−1.44814e11	−0.476352
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.11264e12	2.56589
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	1.77078e12	2.92037
$93$	1.68525e12	2.60475
$94$	−2.00077e12	−2.90022
$95$	0	0
$96$	−7.17495e12	−9.16625
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.75396e12	1.97999
$99$	0	0
$100$	2.92037e12	2.92037
$101$	−1.86720e12	−1.75899	−0.879496	−	0.475907i	$-0.842120\pi$
$101$	−1.86720e12	−1.75899	−0.879496	+	0.475907i	$0.842120\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	8.20531e12	6.48478
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−7.71495e12	−4.86173
$109$	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
$116$	−1.05998e13	−4.35062
$117$	8.60334e12	3.35391
$118$	−2.51849e12	−0.932929
$119$	0	0
$120$	0	0
$121$	3.13843e12	1.00000
$122$	0	0
$123$	1.43490e12	0.414372
$124$	−1.60549e13	−4.41650
$125$	0	0
$126$	0	0
$127$	−6.66992e12	−1.58964	−0.794819	−	0.606846i	$-0.792434\pi$
$127$	−6.66992e12	−1.58964	−0.794819	+	0.606846i	$0.792434\pi$
$128$	2.82252e13	6.41767
$129$	0	0
$130$	0	0
$131$	8.25493e12	1.63337	0.816687	−	0.577082i	$-0.195809\pi$
$131$	8.25493e12	1.63337	0.816687	+	0.577082i	$0.195809\pi$
$132$	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			$\pi$
$138$	−2.35540e13	−3.41028
$139$	−2.80606e11	−0.0389052	−0.0194526	−	0.999811i	$-0.506192\pi$
$139$	−2.80606e11	−0.0389052	−0.0194526	+	0.999811i	$0.506192\pi$
$140$	0	0
$141$	1.98248e13	2.52286
$142$	7.02629e12	0.857030
$143$	0	0
$144$	8.08002e13	9.06228
$145$	0	0
$146$	8.32570e12	0.859614
$147$	−1.73792e13	−1.72237
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−3.88452e13	−3.41028
$151$	−2.34759e13	−1.98044	−0.990218	−	0.139532i	$-0.955440\pi$
$151$	−2.34759e13	−1.98044	−0.990218	+	0.139532i	$0.955440\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−1.23640e14	−8.57848
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	3.22384e13	1.78354
$163$	−3.16070e13	−1.68522	−0.842611	−	0.538522i	$-0.818983\pi$
$163$	−3.16070e13	−1.68522	−0.842611	+	0.538522i	$0.818983\pi$
$164$	−1.36699e13	−0.702592
$165$	0	0
$166$	0	0
$167$	−6.47920e12	−0.298691	−0.149346	−	0.988785i	$-0.547717\pi$
$167$	−6.47920e12	−0.298691	−0.149346	+	0.988785i	$0.547717\pi$
$168$	0	0
$169$	4.44680e13	1.90865
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.97576e13	−1.85602	−0.928010	−	0.372555i	$-0.878482\pi$
$173$	−4.97576e13	−1.85602	−0.928010	+	0.372555i	$0.878482\pi$
$174$	1.40993e14	5.08045
$175$	0	0
$176$	0	0
$177$	2.49546e13	0.811542
$178$	0	0
$179$	−1.13756e13	−0.345824	−0.172912	−	0.984937i	$-0.555318\pi$
$179$	−1.13756e13	−0.345824	−0.172912	+	0.984937i	$0.555318\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	1.47556e14	3.80232
$185$	0	0
$186$	2.13554e14	5.15739
$187$	0	0
$188$	−1.88865e14	−4.27765
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−5.11589e14	−10.2121
$193$	−4.03406e12	−0.0780546	−0.0390273	−	0.999238i	$-0.512426\pi$
$193$	−4.03406e12	−0.0780546	−0.0390273	+	0.999238i	$0.512426\pi$
$194$	0	0
$195$	0	0
$196$	1.65567e14	2.92037
$197$	−1.40553e13	−0.240461	−0.120230	−	0.992746i	$-0.538363\pi$
$197$	−1.40553e13	−0.240461	−0.120230	+	0.992746i	$0.538363\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	2.43349e14	3.80232
$201$	0	0
$202$	−2.36611e14	−3.48279
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.54713e14	1.96655
$208$	6.36440e14	7.85919
$209$	0	0
$210$	0	0
$211$	−1.52822e13	−0.173177	−0.0865886	−	0.996244i	$-0.527597\pi$
$211$	−1.52822e13	−0.173177	−0.0865886	+	0.996244i	$0.527597\pi$
$212$	0	0
$213$	−6.96204e13	−0.745519
$214$	0	0
$215$	0	0
$216$	−6.42871e14	−6.32997
$217$	0	0
$218$	0	0
$219$	−8.24956e13	−0.747767
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.44403e14	−1.98737	−0.993683	−	0.112222i	$-0.964203\pi$
$223$	−2.44403e14	−1.98737	−0.993683	+	0.112222i	$0.964203\pi$
$224$	0	0
$225$	2.55153e14	1.96655
$226$	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
$232$	−8.83262e14	−5.66450
$233$	2.92260e14	1.82656	0.913279	−	0.407335i	$-0.133542\pi$
$233$	2.92260e14	1.82656	0.913279	+	0.407335i	$0.133542\pi$
$234$	1.09021e15	6.64073
$235$	0	0
$236$	−2.37736e14	−1.37601
$237$	0	0
$238$	0	0
$239$	−8.02446e13	−0.430555	−0.215277	−	0.976553i	$-0.569066\pi$
$239$	−8.02446e13	−0.430555	−0.215277	+	0.976553i	$0.569066\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	3.97700e14	1.97999
$243$	2.33245e13	0.113285
$244$	0	0
$245$	0	0
$246$	1.81830e14	0.820455
$247$	0	0
$248$	−1.33782e15	−5.75028
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	−8.45209e14	−3.14747
$255$	0	0
$256$	1.90780e15	6.77786
$257$	−4.78385e14	−1.66027	−0.830134	−	0.557563i	$-0.811736\pi$
$257$	−4.78385e14	−1.66027	−0.830134	+	0.557563i	$0.811736\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−9.26108e14	−2.92967
$262$	1.04606e15	3.23407
$263$	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
$269$	−5.17870e13	−0.136681	−0.0683404	−	0.997662i	$-0.521770\pi$
$269$	−5.17870e13	−0.136681	−0.0683404	+	0.997662i	$0.521770\pi$
$270$	0	0
$271$	7.77423e14	1.96265	0.981323	−	0.192370i	$-0.0616172\pi$
$271$	7.77423e14	1.96265	0.981323	+	0.192370i	$0.0616172\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	−2.22341e15	−5.02996
$277$	−8.76598e14	−1.94054	−0.970269	−	0.242030i	$-0.922187\pi$
$277$	−8.76598e14	−1.94054	−0.970269	+	0.242030i	$0.922187\pi$
$278$	−3.55582e13	−0.0770320
$279$	−1.40272e15	−2.97403
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	2.51218e15	4.99524
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	6.63255e14	1.26407
$285$	0	0
$286$	0	0
$287$	0	0
$288$	5.97209e15	10.4658
$289$	5.82622e14	1.00000
$290$	0	0
$291$	0	0
$292$	7.85914e14	1.26788
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−2.20229e15	−3.41028
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.21863e15	1.70548
$300$	−3.66684e15	−5.02996
$301$	0	0
$302$	−2.97486e15	−3.92125
$303$	2.34447e15	3.02963
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.96848e14	1.07124	0.535622	−	0.844458i	$-0.320077\pi$
$307$	8.96848e14	1.07124	0.535622	+	0.844458i	$0.320077\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.78703e15	1.97501	0.987503	−	0.157599i	$-0.0503754\pi$
$311$	1.78703e15	1.97501	0.987503	+	0.157599i	$0.0503754\pi$
$312$	−1.03026e16	−11.1692
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.73669e15	1.71146	0.855728	−	0.517426i	$-0.173110\pi$
$317$	1.73669e15	1.71146	0.855728	+	0.517426i	$0.173110\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	3.04318e15	2.63062
$325$	2.00977e15	1.70548
$326$	−4.00522e15	−3.33673
$327$	0	0
$328$	−1.13909e15	−0.914774
$329$	0	0
$330$	0	0
$331$	−1.33684e15	−1.01651	−0.508253	−	0.861208i	$-0.669709\pi$
$331$	−1.33684e15	−1.01651	−0.508253	+	0.861208i	$0.669709\pi$
$332$	0	0
$333$	0	0
$334$	−8.21041e14	−0.591406
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	5.63496e15	3.77912
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−6.30526e15	−3.67491
$347$	−3.49096e15	−1.99971	−0.999856	−	0.0169578i	$-0.994602\pi$
$347$	−3.49096e15	−1.99971	−0.999856	+	0.0169578i	$0.994602\pi$
$348$	1.33092e16	7.49336
$349$	2.45694e15	1.35970	0.679848	−	0.733353i	$-0.262046\pi$
$349$	2.45694e15	1.35970	0.679848	+	0.733353i	$0.262046\pi$
$350$	0	0
$351$	−5.30935e15	−2.83922
$352$	0	0
$353$	1.92741e15	0.996151	0.498075	−	0.867134i	$-0.334040\pi$
$353$	1.92741e15	0.996151	0.498075	+	0.867134i	$0.334040\pi$
$354$	3.16223e15	1.60685
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−1.44151e15	−0.684729
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	2.21331e15	1.00000
$362$	0	0
$363$	−3.94063e15	−1.72237
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	1.14451e16	4.60820
$369$	−1.19434e15	−0.473119
$370$	0	0
$371$	0	0
$372$	2.01587e16	7.60684
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	−1.57378e16	−5.56950
$377$	−7.29469e15	−2.54073
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	8.37480e15	2.73794
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−3.54398e16	−11.0536
$385$	0	0
$386$	−5.11195e14	−0.154548
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	1.37964e16	3.80232
$393$	−1.03649e16	−2.81327
$394$	−1.78109e15	−0.476111
$395$	0	0
$396$	0	0
$397$	5.06392e15	1.29343	0.646716	−	0.762731i	$-0.276142\pi$
$397$	5.06392e15	1.29343	0.646716	+	0.762731i	$0.276142\pi$
$398$	0	0
$399$	0	0
$400$	1.88752e16	4.60820
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−1.10488e16	−2.57920
$404$	−2.23352e16	−5.13691
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	6.06751e15	1.29620	0.648098	−	0.761557i	$-0.275565\pi$
$409$	6.06751e15	1.29620	0.648098	+	0.761557i	$0.275565\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	1.96052e16	3.89376
$415$	0	0
$416$	4.70404e16	9.07635
$417$	3.52330e14	0.0670091
$418$	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			$\pi$
$422$	−1.93655e15	−0.342890
$423$	−1.65012e16	−2.88053
$424$	0	0
$425$	0	0
$426$	−8.82226e15	−1.47612
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−4.98639e16	−7.67156
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−1.04538e16	−1.48057
$439$	1.27956e16	1.78762	0.893809	−	0.448448i	$-0.148023\pi$
$439$	1.27956e16	1.78762	0.893809	+	0.448448i	$0.148023\pi$
$440$	0	0
$441$	1.44656e16	1.96655
$442$	0	0
$443$	−1.49070e16	−1.97228	−0.986138	−	0.165927i	$-0.946938\pi$
$443$	−1.49070e16	−1.97228	−0.986138	+	0.165927i	$0.946938\pi$
$444$	0	0
$445$	0	0
$446$	−3.09707e16	−3.93497
$447$	0	0
$448$	0	0
$449$	8.58859e15	1.04820	0.524100	−	0.851657i	$-0.324402\pi$
$449$	8.58859e15	1.04820	0.524100	+	0.851657i	$0.324402\pi$
$450$	3.23329e16	3.89376
$451$	0	0
$452$	0	0
$453$	2.94765e16	3.41104
$454$	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
$461$	−1.75837e16	−1.83192	−0.915959	−	0.401272i	$-0.868568\pi$
$461$	−1.75837e16	−1.83192	−0.915959	+	0.401272i	$0.868568\pi$
$462$	0	0
$463$	−1.58479e16	−1.60874	−0.804371	−	0.594128i	$-0.797497\pi$
$463$	−1.58479e16	−1.60874	−0.804371	+	0.594128i	$0.797497\pi$
$464$	−6.85097e16	−6.86505
$465$	0	0
$466$	3.70350e16	3.61657
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	1.02912e17	9.79468
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.98101e16	−1.79157
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−1.01686e16	−0.852496
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	3.75414e16	2.92037
$485$	0	0
$486$	2.95567e15	0.224304
$487$	2.41916e16	1.81339	0.906695	−	0.421787i	$-0.138597\pi$
$487$	2.41916e16	1.81339	0.906695	+	0.421787i	$0.138597\pi$
$488$	0	0
$489$	3.96859e16	2.90258
$490$	0	0
$491$	−2.73448e16	−1.95158	−0.975788	−	0.218718i	$-0.929813\pi$
$491$	−2.73448e16	−1.95158	−0.975788	+	0.218718i	$0.929813\pi$
$492$	1.71640e16	1.21012
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.03767e17	−6.96901
$497$	0	0
$498$	0	0
$499$	1.52750e16	0.989416	0.494708	−	0.869059i	$-0.335275\pi$
$499$	1.52750e16	0.989416	0.494708	+	0.869059i	$0.335275\pi$
$500$	0	0
$501$	8.13532e15	0.514456
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5.58343e16	−3.28741
$508$	−7.97846e16	−4.64234
$509$	−3.24410e16	−1.86547	−0.932734	−	0.360566i	$-0.882584\pi$
$509$	−3.24410e16	−1.86547	−0.932734	+	0.360566i	$0.882584\pi$
$510$	0	0
$511$	0	0
$512$	1.26145e17	7.00244
$513$	0	0
$514$	−6.06207e16	−3.28732
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.24760e16	3.19675
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−1.17356e17	−5.80072
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	9.87441e16	4.77006
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	2.19146e16	1.00000
$530$	0	0
$531$	−2.07710e16	−0.926597
$532$	0	0
$533$	−9.40750e15	−0.410309
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.42832e16	0.595636
$538$	−6.56243e15	−0.270627
$539$	0	0
$540$	0	0
$541$	−3.12201e16	−1.24523	−0.622617	−	0.782526i	$-0.713931\pi$
$541$	−3.12201e16	−1.24523	−0.622617	+	0.782526i	$0.713931\pi$
$542$	9.85147e16	3.88602
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.19671e16	1.94001	0.970006	−	0.243083i	$-0.0781586\pi$
$547$	5.19671e16	1.94001	0.970006	+	0.243083i	$0.0781586\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−1.85272e17	−6.54900
$553$	0	0
$554$	−1.11082e17	−3.84225
$555$	0	0
$556$	−3.35656e15	−0.113618
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	−1.77752e17	−5.88856
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	2.37141e17	7.36769
$565$	0	0
$566$	0	0
$567$	0	0
$568$	5.52677e16	1.64582
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.61416e16	1.00000
$576$	4.25823e17	11.6599
$577$	3.70815e15	0.100485	0.0502426	−	0.998737i	$-0.484001\pi$
$577$	3.70815e15	0.100485	0.0502426	+	0.998737i	$0.484001\pi$
$578$	7.38296e16	1.97999
$579$	5.06520e15	0.134439
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	6.54886e16	1.65078
$585$	0	0
$586$	0	0
$587$	−6.01536e16	−1.47039	−0.735196	−	0.677854i	$-0.762910\pi$
$587$	−6.01536e16	−1.47039	−0.735196	+	0.677854i	$0.762910\pi$
$588$	−2.07887e17	−5.02996
$589$	0	0
$590$	0	0
$591$	1.76480e16	0.414162
$592$	0	0
$593$	−9.50505e15	−0.218588	−0.109294	−	0.994009i	$-0.534859\pi$
$593$	−9.50505e15	−0.218588	−0.109294	+	0.994009i	$0.534859\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	1.54425e17	3.37683
$599$	−6.60099e16	−1.42905	−0.714526	−	0.699609i	$-0.753358\pi$
$599$	−6.60099e16	−1.42905	−0.714526	+	0.699609i	$0.753358\pi$
$600$	−3.05550e17	−6.54900
$601$	−4.34675e16	−0.922397	−0.461199	−	0.887297i	$-0.652580\pi$
$601$	−4.34675e16	−0.922397	−0.461199	+	0.887297i	$0.652580\pi$
$602$	0	0
$603$	0	0
$604$	−2.80815e17	−5.78361
$605$	0	0
$606$	2.97091e17	5.99865
$607$	3.62355e16	0.724439	0.362220	−	0.932093i	$-0.382019\pi$
$607$	3.62355e16	0.724439	0.362220	+	0.932093i	$0.382019\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.29975e17	−2.49812
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	1.13648e17	2.12106
$615$	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
$621$	−9.54777e16	−1.66476
$622$	2.26451e17	3.91050
$623$	0	0
$624$	−7.99119e17	−13.5364
$625$	5.96046e16	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	1.91884e16	0.298275
$634$	2.20072e17	3.38867
$635$	0	0
$636$	0	0
$637$	1.13942e17	1.70548
$638$	0	0
$639$	5.79487e16	0.851213
$640$	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
$647$	1.46662e17	1.99936	0.999682	−	0.0252337i	$-0.00803300\pi$
$647$	1.46662e17	1.99936	0.999682	+	0.0252337i	$0.00803300\pi$
$648$	2.53582e17	3.42506
$649$	0	0
$650$	2.54677e17	3.37683
$651$	0	0
$652$	−3.78078e17	−4.92148
$653$	4.06448e16	0.524235	0.262118	−	0.965036i	$-0.415579\pi$
$653$	4.06448e16	0.524235	0.262118	+	0.965036i	$0.415579\pi$
$654$	0	0
$655$	0	0
$656$	−8.83526e16	−1.10865
$657$	6.86654e16	0.853780
$658$	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			$\pi$
$662$	−1.69403e17	−2.01268
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.31180e17	−1.48975
$668$	−7.75032e16	−0.872289
$669$	3.06874e17	3.42298
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−1.85483e17	−1.99624	−0.998122	−	0.0612591i	$-0.980488\pi$
$673$	−1.85483e17	−1.99624	−0.998122	+	0.0612591i	$0.980488\pi$
$674$	0	0
$675$	−1.57462e17	−1.66476
$676$	5.31919e17	5.57398
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.78324e17	−1.75665	−0.878325	−	0.478063i	$-0.841339\pi$
$683$	−1.78324e17	−1.75665	−0.878325	+	0.478063i	$0.841339\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.71085e17	1.57160	0.785801	−	0.618479i	$-0.212251\pi$
$691$	1.71085e17	1.57160	0.785801	+	0.618479i	$0.212251\pi$
$692$	−5.95193e17	−5.42027
$693$	0	0
$694$	−4.42372e17	−3.95942
$695$	0	0
$696$	1.10903e18	9.75636
$697$	0	0
$698$	3.11342e17	2.69219
$699$	−3.66963e17	−3.14601
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−6.72798e17	−5.62163
$703$	0	0
$704$	0	0
$705$	0	0
$706$	2.44240e17	1.97237
$707$	0	0
$708$	2.98503e17	2.37000
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.98690e17	−1.51231
$714$	0	0
$715$	0	0
$716$	−1.36073e17	−1.00993
$717$	1.00756e17	0.741574
$718$	0	0
$719$	2.08599e17	1.50987	0.754934	−	0.655801i	$-0.227669\pi$
$719$	2.08599e17	1.50987	0.754934	+	0.655801i	$0.227669\pi$
$720$	0	0
$721$	0	0
$722$	2.80470e17	1.97999
$723$	0	0
$724$	0	0
$725$	−2.16342e17	−1.48975
$726$	−4.99355e17	−3.41028
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−1.64489e17	−1.09590
$730$	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
$736$	8.45926e17	5.32188
$737$	0	0
$738$	−1.51347e17	−0.936773
$739$	6.79332e16	0.417076	0.208538	−	0.978014i	$-0.433130\pi$
$739$	6.79332e16	0.417076	0.208538	+	0.978014i	$0.433130\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	1.67978e18	9.90410
$745$	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			$\pi$
$752$	−1.22069e18	−6.74992
$753$	0	0
$754$	−9.24379e17	−5.03063
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.18524e17	1.63996	0.819981	−	0.572391i	$-0.193984\pi$
$761$	3.18524e17	1.63996	0.819981	+	0.572391i	$0.193984\pi$
$762$	1.06125e18	5.42111
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.63607e17	−0.803583
$768$	−2.39544e18	−11.6740
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	6.00663e17	2.85959
$772$	−4.82548e16	−0.227949
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−3.27680e17	−1.51231
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5.71525e17	2.48008
$784$	1.07011e18	4.60820
$785$	0	0
$786$	−1.31344e18	−5.57026
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1.68128e17	−0.702235
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	6.41697e17	2.56099
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	1.39510e18	5.32188
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−1.40010e18	−5.10681
$807$	6.50242e16	0.235415
$808$	−1.86115e18	−6.68825
$809$	−4.80287e17	−1.71321	−0.856603	−	0.515976i	$-0.827429\pi$
$809$	−4.80287e17	−1.71321	−0.856603	+	0.515976i	$0.827429\pi$
$810$	0	0
$811$	−1.30857e17	−0.459909	−0.229955	−	0.973201i	$-0.573858\pi$
$811$	−1.30857e17	−0.459909	−0.229955	+	0.973201i	$0.573858\pi$
$812$	0	0
$813$	−9.76138e17	−3.38040
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	7.68872e17	2.56646
$819$	0	0
$820$	0	0
$821$	−3.54020e17	−1.15603	−0.578014	−	0.816027i	$-0.696172\pi$
$821$	−3.54020e17	−1.15603	−0.578014	+	0.816027i	$0.696172\pi$
$822$	0	0
$823$	5.23863e17	1.68585	0.842924	−	0.538032i	$-0.180832\pi$
$823$	5.23863e17	1.68585	0.842924	+	0.538032i	$0.180832\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	1.85066e18	5.74307
$829$	5.93576e17	1.82873	0.914364	−	0.404892i	$-0.132691\pi$
$829$	5.93576e17	1.82873	0.914364	+	0.404892i	$0.132691\pi$
$830$	0	0
$831$	1.10066e18	3.34232
$832$	3.35408e18	10.1119
$833$	0	0
$834$	4.46471e16	0.132678
$835$	0	0
$836$	0	0
$837$	8.65655e17	2.51763
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	4.31423e17	1.21935
$842$	0	0
$843$	0	0
$844$	−1.82803e17	−0.505742
$845$	0	0
$846$	−2.09102e18	−5.70343
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−8.32788e17	−2.17719
$853$	−3.65113e17	−0.947835	−0.473917	−	0.880569i	$-0.657160\pi$
$853$	−3.65113e17	−0.947835	−0.473917	+	0.880569i	$0.657160\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.75190e17	1.95669	0.978347	−	0.206969i	$-0.0663600\pi$
$857$	7.75190e17	1.95669	0.978347	+	0.206969i	$0.0663600\pi$
$858$	0	0
$859$	−3.64629e17	−0.907594	−0.453797	−	0.891105i	$-0.649931\pi$
$859$	−3.64629e17	−0.907594	−0.453797	+	0.891105i	$0.649931\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.56727e17	1.83178	0.915892	−	0.401424i	$-0.131485\pi$
$863$	7.56727e17	1.83178	0.915892	+	0.401424i	$0.131485\pi$
$864$	−3.68553e18	−8.85967
$865$	0	0
$866$	0	0
$867$	−7.31544e17	−1.72237
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−9.86799e17	−2.18376
$877$	−8.75030e17	−1.92320	−0.961602	−	0.274446i	$-0.911505\pi$
$877$	−8.75030e17	−1.92320	−0.961602	+	0.274446i	$0.911505\pi$
$878$	1.62146e18	3.53947
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.83308e18	3.89376
$883$	9.25511e17	1.95262	0.976309	−	0.216382i	$-0.0694257\pi$
$883$	9.25511e17	1.95262	0.976309	+	0.216382i	$0.0694257\pi$
$884$	0	0
$885$	0	0
$886$	−1.88901e18	−3.90509
$887$	−3.31514e17	−0.680706	−0.340353	−	0.940298i	$-0.610547\pi$
$887$	−3.31514e17	−0.680706	−0.340353	+	0.940298i	$0.610547\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−2.92351e18	−5.80385
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.53012e18	−2.93746
$898$	1.08834e18	2.07543
$899$	1.18935e18	2.25295
$900$	3.05210e18	5.74307
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	3.73525e18	6.75383
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	−1.95143e18	−3.45915
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−1.12609e18	−1.84508
$922$	−2.22820e18	−3.62718
$923$	4.56445e17	0.738208
$924$	0	0
$925$	0	0
$926$	−2.00824e18	−3.18530
$927$	0	0
$928$	−5.06367e18	−7.92826
$929$	−7.07656e17	−1.10085	−0.550424	−	0.834885i	$-0.685534\pi$
$929$	−7.07656e17	−1.10085	−0.550424	+	0.834885i	$0.685534\pi$
$930$	0	0
$931$	0	0
$932$	3.49597e18	5.33423
$933$	−2.24380e18	−3.40169
$934$	0	0
$935$	0	0
$936$	8.57543e18	12.7527
$937$	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
$943$	−1.69175e17	−0.240583
$944$	−1.53656e18	−2.17128
$945$	0	0
$946$	0	0
$947$	1.05473e18	1.46232	0.731158	−	0.682208i	$-0.238980\pi$
$947$	1.05473e18	1.46232	0.731158	+	0.682208i	$0.238980\pi$
$948$	0	0
$949$	5.40858e17	0.740433
$950$	0	0
$951$	−2.18059e18	−2.94776
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	−9.59874e17	−1.25738
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.01378e18	1.28707
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.26573e16	0.0888629	0.0444314	−	0.999012i	$-0.485852\pi$
$967$	7.26573e16	0.0888629	0.0444314	+	0.999012i	$0.485852\pi$
$968$	3.12825e18	3.80232
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	2.79004e17	0.330836
$973$	0	0
$974$	3.06555e18	3.59050
$975$	−2.52348e18	−2.93746
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	5.02898e18	5.74708
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−3.46512e18	−3.86411
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	1.43025e18	1.57558
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1.57698e18	1.66488	0.832442	−	0.554112i	$-0.186942\pi$
$991$	1.57698e18	1.66488	0.832442	+	0.554112i	$0.186942\pi$
$992$	−7.66964e18	−8.04832
$993$	1.67854e18	1.75080
$994$	0	0
$995$	0	0
$996$	0	0
$997$	7.27897e17	0.741138	0.370569	−	0.928805i	$-0.379163\pi$
$997$	7.27897e17	0.741138	0.370569	+	0.928805i	$0.379163\pi$
$998$	1.93565e18	1.95904
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.13.b.b.22.2	✓	2
23.22	odd	2	CM	23.13.b.b.22.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.13.b.b.22.2	✓	2	1.1	even	1	trivial
23.13.b.b.22.2	✓	2	23.22	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+126.720 q^{2} -1255.61 q^{3} +11961.8 q^{4} -159110. q^{6} +996756. q^{8} +1.04511e6 q^{9} +O(q^{10})\)	\(q+126.720 q^{2} -1255.61 q^{3} +11961.8 q^{4} -159110. q^{6} +996756. q^{8} +1.04511e6 q^{9} -1.50194e7 q^{12} +8.23202e6 q^{13} +7.73128e7 q^{16} +1.32436e8 q^{18} +1.48036e8 q^{23} -1.25153e9 q^{24} +2.44141e8 q^{25} +1.04316e9 q^{26} -6.44963e8 q^{27} -8.86136e8 q^{29} -1.34218e9 q^{31} +5.71433e9 q^{32} +1.25014e10 q^{36} -1.03362e10 q^{39} -1.14279e9 q^{41} +1.87590e10 q^{46} -1.57890e10 q^{47} -9.70745e10 q^{48} +1.38413e10 q^{49} +3.09374e10 q^{50} +9.84701e10 q^{52} -8.17294e10 q^{54} -1.12291e11 q^{58} -1.98745e10 q^{59} -1.70080e11 q^{62} +4.07444e11 q^{64} -1.85875e11 q^{69} +5.54476e10 q^{71} +1.04172e12 q^{72} +6.57018e10 q^{73} -3.06545e11 q^{75} -1.30980e12 q^{78} +2.54407e11 q^{81} -1.44814e11 q^{82} +1.11264e12 q^{87} +1.77078e12 q^{92} +1.68525e12 q^{93} -2.00077e12 q^{94} -7.17495e12 q^{96} +1.75396e12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 79 q^{2} + 14 q^{3} + 10143 q^{4} - 219695 q^{6} + 1279010 q^{8} + 2125568 q^{9}+O(q^{10}) \) 2 * q + 79 * q^2 + 14 * q^3 + 10143 * q^4 - 219695 * q^6 + 1279010 * q^8 + 2125568 * q^9	\( 2 q + 79 q^{2} + 14 q^{3} + 10143 q^{4} - 219695 q^{6} + 1279010 q^{8} + 2125568 q^{9} - 17328591 q^{12} + 8482894 q^{13} + 71293759 q^{16} + 80876464 q^{18} + 296071778 q^{23} - 893182738 q^{24} + 488281250 q^{25} + 1031185505 q^{26} + 52075730 q^{27} + 244330126 q^{29} - 1677025154 q^{31} + 4845442959 q^{32} + 10536223824 q^{36} - 10017657950 q^{39} + 7596282526 q^{41} + 11694835231 q^{46} - 20606906306 q^{47} - 104716276511 q^{48} + 27682574402 q^{49} + 19287109375 q^{50} + 98013781089 q^{52} - 114991826033 q^{54} - 166236126415 q^{58} - 39749055836 q^{59} - 154101345055 q^{62} + 473560750658 q^{64} + 2072502446 q^{69} - 188893891874 q^{71} + 1346681462528 q^{72} - 223017449186 q^{73} + 3417968750 q^{75} - 1324994929537 q^{78} + 565171521854 q^{81} - 561839119135 q^{82} + 2547886433890 q^{87} + 1501528022127 q^{92} + 1260122292850 q^{93} - 1770863233615 q^{94} - 8278093192383 q^{96} + 1093461688879 q^{98}+O(q^{100}) \) 2 * q + 79 * q^2 + 14 * q^3 + 10143 * q^4 - 219695 * q^6 + 1279010 * q^8 + 2125568 * q^9 - 17328591 * q^12 + 8482894 * q^13 + 71293759 * q^16 + 80876464 * q^18 + 296071778 * q^23 - 893182738 * q^24 + 488281250 * q^25 + 1031185505 * q^26 + 52075730 * q^27 + 244330126 * q^29 - 1677025154 * q^31 + 4845442959 * q^32 + 10536223824 * q^36 - 10017657950 * q^39 + 7596282526 * q^41 + 11694835231 * q^46 - 20606906306 * q^47 - 104716276511 * q^48 + 27682574402 * q^49 + 19287109375 * q^50 + 98013781089 * q^52 - 114991826033 * q^54 - 166236126415 * q^58 - 39749055836 * q^59 - 154101345055 * q^62 + 473560750658 * q^64 + 2072502446 * q^69 - 188893891874 * q^71 + 1346681462528 * q^72 - 223017449186 * q^73 + 3417968750 * q^75 - 1324994929537 * q^78 + 565171521854 * q^81 - 561839119135 * q^82 + 2547886433890 * q^87 + 1501528022127 * q^92 + 1260122292850 * q^93 - 1770863233615 * q^94 - 8278093192383 * q^96 + 1093461688879 * q^98

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