LMFDB - Embedded newform 23.9.b.a.22.1

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			22.1
Root			$2.66908$ 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)

$f(q)$	$=$	$q-28.9050 q^{2} +80.1002 q^{3} +579.497 q^{4} -2315.29 q^{6} -9350.67 q^{8} -144.953 q^{9} +O(q^{10})$	$q-28.9050 q^{2} +80.1002 q^{3} +579.497 q^{4} -2315.29 q^{6} -9350.67 q^{8} -144.953 q^{9} +46417.8 q^{12} +54007.6 q^{13} +121930. q^{16} +4189.86 q^{18} +279841. q^{23} -748991. q^{24} +390625. q^{25} -1.56109e6 q^{26} -537148. q^{27} +868967. q^{29} +1.80149e6 q^{31} -1.13060e6 q^{32} -83999.7 q^{36} +4.32602e6 q^{39} -4.60705e6 q^{41} -8.08879e6 q^{46} -6.45282e6 q^{47} +9.76659e6 q^{48} +5.76480e6 q^{49} -1.12910e7 q^{50} +3.12972e7 q^{52} +1.55263e7 q^{54} -2.51175e7 q^{58} +1.52791e7 q^{59} -5.20721e7 q^{62} +1.46594e6 q^{64} +2.24153e7 q^{69} -1.47877e6 q^{71} +1.35541e6 q^{72} -4.83534e7 q^{73} +3.12892e7 q^{75} -1.25043e8 q^{78} -4.20747e7 q^{81} +1.33167e8 q^{82} +6.96045e7 q^{87} +1.62167e8 q^{92} +1.44300e8 q^{93} +1.86519e8 q^{94} -9.05612e7 q^{96} -1.66631e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−28.9050	−1.80656	−0.903280	−	0.429051i	$-0.858848\pi$
$2$	−28.9050	−1.80656	−0.903280	+	0.429051i	$0.858848\pi$
$3$	80.1002	0.988892	0.494446	−	0.869208i	$-0.335371\pi$
$3$	80.1002	0.988892	0.494446	+	0.869208i	$0.335371\pi$
$4$	579.497	2.26366
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−2315.29	−1.78649
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	−9350.67	−2.28288
$9$	−144.953	−0.0220931
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	46417.8	2.23852
$13$	54007.6	1.89096	0.945478	−	0.325687i	$-0.105596\pi$
$13$	54007.6	1.89096	0.945478	+	0.325687i	$0.105596\pi$
$14$	0	0
$15$	0	0
$16$	121930.	1.86050
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	4189.86	0.0399125
$19$	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
$24$	−748991.	−2.25752
$25$	390625.	1.00000
$26$	−1.56109e6	−3.41612
$27$	−537148.	−1.01074
$28$	0	0
$29$	868967.	1.22860	0.614301	−	0.789071i	$-0.289438\pi$
$29$	868967.	1.22860	0.614301	+	0.789071i	$0.289438\pi$
$30$	0	0
$31$	1.80149e6	1.95068	0.975339	−	0.220711i	$-0.0708379\pi$
$31$	1.80149e6	1.95068	0.975339	+	0.220711i	$0.0708379\pi$
$32$	−1.13060e6	−1.07822
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−83999.7	−0.0500113
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	4.32602e6	1.86995
$40$	0	0
$41$	−4.60705e6	−1.63038	−0.815188	−	0.579197i	$-0.803366\pi$
$41$	−4.60705e6	−1.63038	−0.815188	+	0.579197i	$0.803366\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−8.08879e6	−1.80656
$47$	−6.45282e6	−1.32239	−0.661193	−	0.750216i	$-0.729949\pi$
$47$	−6.45282e6	−1.32239	−0.661193	+	0.750216i	$0.729949\pi$
$48$	9.76659e6	1.83983
$49$	5.76480e6	1.00000
$50$	−1.12910e7	−1.80656
$51$	0	0
$52$	3.12972e7	4.28048
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	1.55263e7	1.82596
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−2.51175e7	−2.21955
$59$	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			$\pi$
$62$	−5.20721e7	−3.52402
$63$	0	0
$64$	1.46594e6	0.0873770
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	2.24153e7	0.988892
$70$	0	0
$71$	−1.47877e6	−0.0581927	−0.0290963	−	0.999577i	$-0.509263\pi$
$71$	−1.47877e6	−0.0581927	−0.0290963	+	0.999577i	$0.509263\pi$
$72$	1.35541e6	0.0504359
$73$	−4.83534e7	−1.70269	−0.851344	−	0.524607i	$-0.824212\pi$
$73$	−4.83534e7	−1.70269	−0.851344	+	0.524607i	$0.824212\pi$
$74$	0	0
$75$	3.12892e7	0.988892
$76$	0	0
$77$	0	0
$78$	−1.25043e8	−3.37818
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−4.20747e7	−0.977419
$82$	1.33167e8	2.94537
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.96045e7	1.21496
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	1.62167e8	2.26366
$93$	1.44300e8	1.92901
$94$	1.86519e8	2.38897
$95$	0	0
$96$	−9.05612e7	−1.06625
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.66631e8	−1.80656
$99$	0	0
$100$	2.26366e8	2.26366
$101$	−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			$\pi$
$104$	−5.05007e8	−4.31682
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−3.11276e8	−2.28797
$109$	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
$116$	5.03564e8	2.78114
$117$	−7.82855e6	−0.0417771
$118$	−4.41641e8	−2.27794
$119$	0	0
$120$	0	0
$121$	2.14359e8	1.00000
$122$	0	0
$123$	−3.69026e8	−1.61227
$124$	1.04396e9	4.41567
$125$	0	0
$126$	0	0
$127$	−5.18382e8	−1.99267	−0.996334	−	0.0855538i	$-0.972734\pi$
$127$	−5.18382e8	−1.99267	−0.996334	+	0.0855538i	$0.972734\pi$
$128$	2.47060e8	0.920371
$129$	0	0
$130$	0	0
$131$	3.25065e8	1.10379	0.551894	−	0.833915i	$-0.313905\pi$
$131$	3.25065e8	1.10379	0.551894	+	0.833915i	$0.313905\pi$
$132$	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			$\pi$
$138$	−6.47914e8	−1.78649
$139$	−5.65658e8	−1.51529	−0.757644	−	0.652668i	$-0.773650\pi$
$139$	−5.65658e8	−1.51529	−0.757644	+	0.652668i	$0.773650\pi$
$140$	0	0
$141$	−5.16872e8	−1.30770
$142$	4.27439e7	0.105129
$143$	0	0
$144$	−1.76740e7	−0.0411042
$145$	0	0
$146$	1.39765e9	3.07601
$147$	4.61762e8	0.988892
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−9.04412e8	−1.78649
$151$	8.43443e7	0.162236	0.0811181	−	0.996704i	$-0.474151\pi$
$151$	8.43443e7	0.162236	0.0811181	+	0.996704i	$0.474151\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	2.50692e9	4.23293
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	1.21617e9	1.76577
$163$	−1.41119e9	−1.99910	−0.999548	−	0.0300511i	$-0.990433\pi$
$163$	−1.41119e9	−1.99910	−0.999548	+	0.0300511i	$0.990433\pi$
$164$	−2.66977e9	−3.69062
$165$	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
$169$	2.10109e9	2.57571
$170$	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	$0.0810121\pi$
$174$	−2.01192e9	−2.19489
$175$	0	0
$176$	0	0
$177$	1.22386e9	1.24692
$178$	0	0
$179$	1.83531e9	1.78771	0.893856	−	0.448354i	$-0.147990\pi$
$179$	1.83531e9	1.78771	0.893856	+	0.448354i	$0.147990\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−2.61670e9	−2.28288
$185$	0	0
$186$	−4.17099e9	−3.48487
$187$	0	0
$188$	−3.73939e9	−2.99343
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.17422e8	0.0864064
$193$	−4.10593e8	−0.295926	−0.147963	−	0.988993i	$-0.547272\pi$
$193$	−4.10593e8	−0.295926	−0.147963	+	0.988993i	$0.547272\pi$
$194$	0	0
$195$	0	0
$196$	3.34069e9	2.26366
$197$	2.90417e9	1.92822	0.964112	−	0.265496i	$-0.0855356\pi$
$197$	2.90417e9	1.92822	0.964112	+	0.265496i	$0.0855356\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−3.65261e9	−2.28288
$201$	0	0
$202$	4.53638e9	2.72461
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.05637e7	−0.0220931
$208$	6.58512e9	3.51812
$209$	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
$213$	−1.18450e8	−0.0575463
$214$	0	0
$215$	0	0
$216$	5.02270e9	2.30740
$217$	0	0
$218$	0	0
$219$	−3.87312e9	−1.68377
$220$	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
$225$	−5.66222e7	−0.0220931
$226$	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
$232$	−8.12543e9	−2.80475
$233$	−4.76731e9	−1.61752	−0.808760	−	0.588138i	$-0.799861\pi$
$233$	−4.76731e9	−1.61752	−0.808760	+	0.588138i	$0.799861\pi$
$234$	2.26284e8	0.0754728
$235$	0	0
$236$	8.85418e9	2.85430
$237$	0	0
$238$	0	0
$239$	−4.34207e9	−1.33078	−0.665388	−	0.746497i	$-0.731734\pi$
$239$	−4.34207e9	−1.33078	−0.665388	+	0.746497i	$0.731734\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−6.19604e9	−1.80656
$243$	1.54039e8	0.0441780
$244$	0	0
$245$	0	0
$246$	1.06667e10	2.91265
$247$	0	0
$248$	−1.68452e10	−4.45316
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	1.49838e10	3.59987
$255$	0	0
$256$	−7.51655e9	−1.75008
$257$	4.69967e9	1.07730	0.538648	−	0.842531i	$-0.318935\pi$
$257$	4.69967e9	1.07730	0.538648	+	0.842531i	$0.318935\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.25959e8	−0.0271436
$262$	−9.39600e9	−1.99406
$263$	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
$269$	9.99370e9	1.90861	0.954305	−	0.298833i	$-0.0965975\pi$
$269$	9.99370e9	1.90861	0.954305	+	0.298833i	$0.0965975\pi$
$270$	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
$276$	1.29896e10	2.23852
$277$	−1.02646e10	−1.74350	−0.871748	−	0.489954i	$-0.837013\pi$
$277$	−1.02646e10	−1.74350	−0.871748	+	0.489954i	$0.837013\pi$
$278$	1.63503e10	2.73746
$279$	−2.61131e8	−0.0430965
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	1.49402e10	2.36243
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	−8.56945e8	−0.131728
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.63883e8	0.0238213
$289$	6.97576e9	1.00000
$290$	0	0
$291$	0	0
$292$	−2.80206e10	−3.85431
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−1.33472e10	−1.78649
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.51135e10	1.89096
$300$	1.81320e10	2.23852
$301$	0	0
$302$	−2.43797e9	−0.293090
$303$	−1.25710e10	−1.49142
$304$	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
$311$	1.29046e9	0.137944	0.0689722	−	0.997619i	$-0.478028\pi$
$311$	1.29046e9	0.137944	0.0689722	+	0.997619i	$0.478028\pi$
$312$	−4.04512e10	−4.26887
$313$	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
$324$	−2.43821e10	−2.21254
$325$	2.10967e10	1.89096
$326$	4.07903e10	3.61149
$327$	0	0
$328$	4.30790e10	3.72195
$329$	0	0
$330$	0	0
$331$	−1.21358e10	−1.01101	−0.505507	−	0.862822i	$-0.668695\pi$
$331$	−1.21358e10	−1.01101	−0.505507	+	0.862822i	$0.668695\pi$
$332$	0	0
$333$	0	0
$334$	4.47400e10	3.59509
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−6.07318e10	−4.65318
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−5.01149e10	−3.49673
$347$	−1.42135e10	−0.980354	−0.490177	−	0.871623i	$-0.663068\pi$
$347$	−1.42135e10	−0.980354	−0.490177	+	0.871623i	$0.663068\pi$
$348$	4.03356e10	2.75025
$349$	−1.84927e10	−1.24652	−0.623259	−	0.782016i	$-0.714192\pi$
$349$	−1.84927e10	−1.24652	−0.623259	+	0.782016i	$0.714192\pi$
$350$	0	0
$351$	−2.90101e10	−1.91126
$352$	0	0
$353$	5.34732e9	0.344380	0.172190	−	0.985064i	$-0.444916\pi$
$353$	5.34732e9	0.344380	0.172190	+	0.985064i	$0.444916\pi$
$354$	−3.53756e10	−2.25263
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−5.30496e10	−3.22961
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.69836e10	1.00000
$362$	0	0
$363$	1.71702e10	0.988892
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	3.41209e10	1.86050
$369$	6.67805e8	0.0360200
$370$	0	0
$371$	0	0
$372$	8.36214e10	4.36662
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	6.03382e10	3.01885
$377$	4.69308e10	2.32323
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−4.15225e10	−1.97053
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.97896e10	0.910147
$385$	0	0
$386$	1.18682e10	0.534608
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	−5.39048e10	−2.28288
$393$	2.60378e10	1.09153
$394$	−8.39450e10	−3.48345
$395$	0	0
$396$	0	0
$397$	3.39693e10	1.36749	0.683746	−	0.729720i	$-0.260350\pi$
$397$	3.39693e10	1.36749	0.683746	+	0.729720i	$0.260350\pi$
$398$	0	0
$399$	0	0
$400$	4.76287e10	1.86050
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	9.72942e10	3.68865
$404$	−9.09469e10	−3.41399
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−5.45165e10	−1.94821	−0.974103	−	0.226103i	$-0.927401\pi$
$409$	−5.45165e10	−1.94821	−0.974103	+	0.226103i	$0.927401\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	1.17249e9	0.0399125
$415$	0	0
$416$	−6.10609e10	−2.03887
$417$	−4.53094e10	−1.49846
$418$	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			$\pi$
$422$	−5.14650e10	−1.62279
$423$	9.35354e8	0.0292156
$424$	0	0
$425$	0	0
$426$	3.42380e9	0.103961
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−6.54943e10	−1.88048
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.11952e11	3.04184
$439$	3.46018e10	0.931623	0.465812	−	0.884884i	$-0.345762\pi$
$439$	3.46018e10	0.931623	0.465812	+	0.884884i	$0.345762\pi$
$440$	0	0
$441$	−8.35624e8	−0.0220931
$442$	0	0
$443$	−7.48572e10	−1.94365	−0.971827	−	0.235695i	$-0.924263\pi$
$443$	−7.48572e10	−1.94365	−0.971827	+	0.235695i	$0.924263\pi$
$444$	0	0
$445$	0	0
$446$	8.05540e10	2.03586
$447$	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	$0.882935\pi$
$450$	1.63666e9	0.0399125
$451$	0	0
$452$	0	0
$453$	6.75600e9	0.160434
$454$	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
$461$	−9.08427e9	−0.201134	−0.100567	−	0.994930i	$-0.532066\pi$
$461$	−9.08427e9	−0.201134	−0.100567	+	0.994930i	$0.532066\pi$
$462$	0	0
$463$	8.37652e10	1.82280	0.911401	−	0.411519i	$-0.135002\pi$
$463$	8.37652e10	1.82280	0.911401	+	0.411519i	$0.135002\pi$
$464$	1.05953e11	2.28581
$465$	0	0
$466$	1.37799e11	2.92215
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−4.53662e9	−0.0945691
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−1.42870e11	−2.87854
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	1.25507e11	2.40413
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1.24220e11	2.26366
$485$	0	0
$486$	−4.45250e9	−0.0798103
$487$	−1.12304e11	−1.99654	−0.998272	−	0.0587552i	$-0.981287\pi$
$487$	−1.12304e11	−1.99654	−0.998272	+	0.0587552i	$0.981287\pi$
$488$	0	0
$489$	−1.13036e11	−1.97689
$490$	0	0
$491$	7.71407e10	1.32727	0.663633	−	0.748059i	$-0.269014\pi$
$491$	7.71407e10	1.32727	0.663633	+	0.748059i	$0.269014\pi$
$492$	−2.13849e11	−3.64962
$493$	0	0
$494$	0	0
$495$	0	0
$496$	2.19655e11	3.62923
$497$	0	0
$498$	0	0
$499$	−6.15643e10	−0.992948	−0.496474	−	0.868052i	$-0.665372\pi$
$499$	−6.15643e10	−0.992948	−0.496474	+	0.868052i	$0.665372\pi$
$500$	0	0
$501$	−1.23982e11	−1.96792
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.68298e11	2.54710
$508$	−3.00401e11	−4.51072
$509$	−1.33532e11	−1.98937	−0.994685	−	0.102967i	$-0.967166\pi$
$509$	−1.33532e11	−1.98937	−0.994685	+	0.102967i	$0.967166\pi$
$510$	0	0
$511$	0	0
$512$	1.54018e11	2.24126
$513$	0	0
$514$	−1.35844e11	−1.94620
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1.38876e11	1.91407
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	3.64085e9	0.0490366
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	1.88374e11	2.49860
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.83110e10	1.00000
$530$	0	0
$531$	−2.21474e9	−0.0278577
$532$	0	0
$533$	−2.48816e11	−3.08297
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.47009e11	1.76785
$538$	−2.88868e11	−3.44802
$539$	0	0
$540$	0	0
$541$	1.70486e11	1.99021	0.995105	−	0.0988280i	$-0.0315094\pi$
$541$	1.70486e11	1.99021	0.995105	+	0.0988280i	$0.0315094\pi$
$542$	1.89356e11	2.19423
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.94052e9	0.0216755	0.0108378	−	0.999941i	$-0.496550\pi$
$547$	1.94052e9	0.0216755	0.0108378	+	0.999941i	$0.496550\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−2.09598e11	−2.25752
$553$	0	0
$554$	2.96697e11	3.14973
$555$	0	0
$556$	−3.27797e11	−3.43010
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	7.54799e9	0.0778565
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	−2.99526e11	−2.96018
$565$	0	0
$566$	0	0
$567$	0	0
$568$	1.38275e10	0.132847
$569$	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
$576$	−2.12493e8	−0.00193043
$577$	2.10738e11	1.90125	0.950624	−	0.310346i	$-0.100445\pi$
$577$	2.10738e11	1.90125	0.950624	+	0.310346i	$0.100445\pi$
$578$	−2.01634e11	−1.80656
$579$	−3.28886e10	−0.292638
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	4.52136e11	3.88703
$585$	0	0
$586$	0	0
$587$	−7.51523e10	−0.632980	−0.316490	−	0.948596i	$-0.602504\pi$
$587$	−7.51523e10	−0.632980	−0.316490	+	0.948596i	$0.602504\pi$
$588$	2.67590e11	2.23852
$589$	0	0
$590$	0	0
$591$	2.32625e11	1.90680
$592$	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
$598$	−4.36856e11	−3.41612
$599$	−1.81057e9	−0.0140640	−0.00703200	−	0.999975i	$-0.502238\pi$
$599$	−1.81057e9	−0.0140640	−0.00703200	+	0.999975i	$0.502238\pi$
$600$	−2.92575e11	−2.25752
$601$	−1.23744e11	−0.948473	−0.474237	−	0.880397i	$-0.657276\pi$
$601$	−1.23744e11	−0.948473	−0.474237	+	0.880397i	$0.657276\pi$
$602$	0	0
$603$	0	0
$604$	4.88773e10	0.367248
$605$	0	0
$606$	3.63365e11	2.69434
$607$	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
$611$	−3.48501e11	−2.50057
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−7.50984e10	−0.528393
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−1.50316e11	−1.01074
$622$	−3.73008e10	−0.249205
$623$	0	0
$624$	5.27470e11	3.47904
$625$	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
$633$	1.42618e11	0.888298
$634$	9.36032e10	0.579340
$635$	0	0
$636$	0	0
$637$	3.11343e11	1.89096
$638$	0	0
$639$	2.14352e8	0.00128566
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.31301e11	−1.89063	−0.945314	−	0.326163i	$-0.894244\pi$
$647$	−3.31301e11	−1.89063	−0.945314	+	0.326163i	$0.894244\pi$
$648$	3.93426e11	2.23133
$649$	0	0
$650$	−6.09800e11	−3.41612
$651$	0	0
$652$	−8.17778e11	−4.52528
$653$	−1.25153e11	−0.688319	−0.344160	−	0.938911i	$-0.611836\pi$
$653$	−1.25153e11	−0.688319	−0.344160	+	0.938911i	$0.611836\pi$
$654$	0	0
$655$	0	0
$656$	−5.61736e11	−3.03331
$657$	7.00896e9	0.0376177
$658$	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			$\pi$
$662$	3.50786e11	1.82646
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.43173e11	1.22860
$668$	−8.96963e11	−4.50473
$669$	−2.23228e11	−1.11441
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−3.90957e11	−1.90576	−0.952881	−	0.303345i	$-0.901897\pi$
$673$	−3.90957e11	−1.90576	−0.952881	+	0.303345i	$0.901897\pi$
$674$	0	0
$675$	−2.09824e11	−1.01074
$676$	1.21757e12	5.83054
$677$	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
$683$	−6.83791e10	−0.314225	−0.157112	−	0.987581i	$-0.550219\pi$
$683$	−6.83791e10	−0.314225	−0.157112	+	0.987581i	$0.550219\pi$
$684$	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	$0.525166\pi$
$692$	1.00472e12	4.38148
$693$	0	0
$694$	4.10841e11	1.77107
$695$	0	0
$696$	−6.50849e11	−2.77360
$697$	0	0
$698$	5.34531e11	2.25191
$699$	−3.81863e11	−1.59955
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	8.38535e11	3.45281
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−1.54564e11	−0.622143
$707$	0	0
$708$	7.09222e11	2.82260
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.04131e11	1.95068
$714$	0	0
$715$	0	0
$716$	1.06356e12	4.04677
$717$	−3.47801e11	−1.31599
$718$	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
$722$	−4.90909e11	−1.80656
$723$	0	0
$724$	0	0
$725$	3.39440e11	1.22860
$726$	−4.96304e11	−1.78649
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	2.88391e11	1.02111
$730$	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
$736$	−3.16388e11	−1.07822
$737$	0	0
$738$	−1.93029e10	−0.0650724
$739$	−3.98945e11	−1.33763	−0.668813	−	0.743430i	$-0.733197\pi$
$739$	−3.98945e11	−1.33763	−0.668813	+	0.743430i	$0.733197\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−1.34930e12	−4.40370
$745$	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			$\pi$
$752$	−7.86790e11	−2.46030
$753$	0	0
$754$	−1.35653e12	−4.19706
$755$	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
$761$	3.01615e11	0.899322	0.449661	−	0.893199i	$-0.351545\pi$
$761$	3.01615e11	0.899322	0.449661	+	0.893199i	$0.351545\pi$
$762$	1.20021e12	3.55989
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.25186e11	2.38435
$768$	−6.02077e11	−1.73064
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	3.76445e11	1.06533
$772$	−2.37938e11	−0.669875
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	7.03708e11	1.95068
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4.66764e11	−1.24180
$784$	7.02900e11	1.86050
$785$	0	0
$786$	−7.52622e11	−1.97191
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	1.68296e12	4.36484
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−9.81881e11	−2.47046
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−4.41640e11	−1.07822
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−2.81229e12	−6.66376
$807$	8.00498e11	1.88741
$808$	1.46750e12	3.44298
$809$	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
$811$	−2.79722e11	−0.646611	−0.323305	−	0.946295i	$-0.604794\pi$
$811$	−2.79722e11	−0.646611	−0.323305	+	0.946295i	$0.604794\pi$
$812$	0	0
$813$	−5.24735e11	−1.20110
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	1.57580e12	3.51955
$819$	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
$823$	−9.17145e11	−1.99912	−0.999560	−	0.0296636i	$-0.990556\pi$
$823$	−9.17145e11	−1.99912	−0.999560	+	0.0296636i	$0.990556\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−2.35066e10	−0.0500113
$829$	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
$831$	−8.22193e11	−1.72413
$832$	7.91721e10	0.165226
$833$	0	0
$834$	1.30967e12	2.70705
$835$	0	0
$836$	0	0
$837$	−9.67669e11	−1.97163
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.54858e11	0.509465
$842$	0	0
$843$	0	0
$844$	1.03179e12	2.03339
$845$	0	0
$846$	−2.70364e10	−0.0527797
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	−6.86415e10	−0.130265
$853$	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
$857$	9.14533e11	1.69542	0.847708	−	0.530464i	$-0.177982\pi$
$857$	9.14533e11	1.69542	0.847708	+	0.530464i	$0.177982\pi$
$858$	0	0
$859$	−5.77182e11	−1.06008	−0.530042	−	0.847971i	$-0.677824\pi$
$859$	−5.77182e11	−1.06008	−0.530042	+	0.847971i	$0.677824\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.11687e11	−0.201354	−0.100677	−	0.994919i	$-0.532101\pi$
$863$	−1.11687e11	−0.201354	−0.100677	+	0.994919i	$0.532101\pi$
$864$	6.07299e11	1.08980
$865$	0	0
$866$	0	0
$867$	5.58760e11	0.988892
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	−2.24446e12	−3.81149
$877$	−3.92609e11	−0.663685	−0.331843	−	0.943335i	$-0.607670\pi$
$877$	−3.92609e11	−0.663685	−0.331843	+	0.943335i	$0.607670\pi$
$878$	−1.00016e12	−1.68303
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	2.41537e10	0.0399125
$883$	−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
$886$	2.16375e12	3.51133
$887$	−7.40462e11	−1.19621	−0.598107	−	0.801417i	$-0.704080\pi$
$887$	−7.40462e11	−1.19621	−0.598107	+	0.801417i	$0.704080\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−1.61498e12	−2.55098
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.21060e12	1.86995
$898$	2.19245e12	3.37151
$899$	1.56544e12	2.39661
$900$	−3.28124e10	−0.0500113
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−1.95282e11	−0.289834
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	2.27490e10	0.0333202
$910$	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
$921$	2.08109e11	0.289237
$922$	2.62580e11	0.363361
$923$	−7.98650e10	−0.110040
$924$	0	0
$925$	0	0
$926$	−2.42123e12	−3.29300
$927$	0	0
$928$	−9.82454e11	−1.32471
$929$	3.16435e11	0.424836	0.212418	−	0.977179i	$-0.431866\pi$
$929$	3.16435e11	0.424836	0.212418	+	0.977179i	$0.431866\pi$
$930$	0	0
$931$	0	0
$932$	−2.76264e12	−3.66152
$933$	1.03366e11	0.136412
$934$	0	0
$935$	0	0
$936$	7.32022e10	0.0953719
$937$	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
$943$	−1.28924e12	−1.63038
$944$	1.86297e12	2.34595
$945$	0	0
$946$	0	0
$947$	5.84887e11	0.727231	0.363615	−	0.931549i	$-0.381542\pi$
$947$	5.84887e11	0.727231	0.363615	+	0.931549i	$0.381542\pi$
$948$	0	0
$949$	−2.61145e12	−3.21971
$950$	0	0
$951$	−2.59389e11	−0.317125
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	−2.51622e12	−3.01243
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.39248e12	2.80515
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.62488e12	1.85829	0.929147	−	0.369710i	$-0.120543\pi$
$967$	1.62488e12	1.85829	0.929147	+	0.369710i	$0.120543\pi$
$968$	−2.00440e12	−2.28288
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	8.92653e10	0.100004
$973$	0	0
$974$	3.24614e12	3.60688
$975$	1.68985e12	1.86995
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	3.26731e12	3.57137
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−2.22975e12	−2.39778
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	3.45064e12	3.68061
$985$	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	$0.791295\pi$
$992$	−2.03677e12	−2.10327
$993$	−9.72083e11	−0.999784
$994$	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	$0.919442\pi$
$998$	1.77951e12	1.79382
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.9.b.a.22.1	✓	3
3.2	odd	2		207.9.d.a.91.3		3
4.3	odd	2		368.9.f.a.321.2		3
23.22	odd	2	CM	23.9.b.a.22.1	✓	3
69.68	even	2		207.9.d.a.91.3		3
92.91	even	2		368.9.f.a.321.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.9.b.a.22.1	✓	3	1.1	even	1	trivial
23.9.b.a.22.1	✓	3	23.22	odd	2	CM
207.9.d.a.91.3		3	3.2	odd	2
207.9.d.a.91.3		3	69.68	even	2
368.9.f.a.321.2		3	4.3	odd	2
368.9.f.a.321.2		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)

\(f(q)\)	\(=\)	\(q-28.9050 q^{2} +80.1002 q^{3} +579.497 q^{4} -2315.29 q^{6} -9350.67 q^{8} -144.953 q^{9} +O(q^{10})\)	\(q-28.9050 q^{2} +80.1002 q^{3} +579.497 q^{4} -2315.29 q^{6} -9350.67 q^{8} -144.953 q^{9} +46417.8 q^{12} +54007.6 q^{13} +121930. q^{16} +4189.86 q^{18} +279841. q^{23} -748991. q^{24} +390625. q^{25} -1.56109e6 q^{26} -537148. q^{27} +868967. q^{29} +1.80149e6 q^{31} -1.13060e6 q^{32} -83999.7 q^{36} +4.32602e6 q^{39} -4.60705e6 q^{41} -8.08879e6 q^{46} -6.45282e6 q^{47} +9.76659e6 q^{48} +5.76480e6 q^{49} -1.12910e7 q^{50} +3.12972e7 q^{52} +1.55263e7 q^{54} -2.51175e7 q^{58} +1.52791e7 q^{59} -5.20721e7 q^{62} +1.46594e6 q^{64} +2.24153e7 q^{69} -1.47877e6 q^{71} +1.35541e6 q^{72} -4.83534e7 q^{73} +3.12892e7 q^{75} -1.25043e8 q^{78} -4.20747e7 q^{81} +1.33167e8 q^{82} +6.96045e7 q^{87} +1.62167e8 q^{92} +1.44300e8 q^{93} +1.86519e8 q^{94} -9.05612e7 q^{96} -1.66631e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 768 q^{4} - 573 q^{6} - 5853 q^{8} + 19683 q^{9}+O(q^{10}) \) 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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