LMFDB - Embedded newform 9.12.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,12,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	9.a (trivial)

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:	$6.91508862504$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{70}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 70 $ x^2 - 70
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.36660$ of defining polynomial
Character	$\chi$	$=$	9.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-50.1996 q^{2} +472.000 q^{4} -11244.7 q^{5} +58100.0 q^{7} +79114.6 q^{8} +O(q^{10})$	\(q-50.1996 q^{2} +472.000 q^{4} -11244.7 q^{5} +58100.0 q^{7} +79114.6 q^{8} +564480. q^{10} +160639. q^{11} +762650. q^{13} -2.91660e6 q^{14} -4.93818e6 q^{16} +8.97328e6 q^{17} -1.03017e7 q^{19} -5.30750e6 q^{20} -8.06400e6 q^{22} -1.03965e7 q^{23} +7.76154e7 q^{25} -3.82847e7 q^{26} +2.74232e7 q^{28} +1.41001e8 q^{29} +1.06160e8 q^{31} +8.58678e7 q^{32} -4.50455e8 q^{34} -6.53318e8 q^{35} -9.57445e6 q^{37} +5.17141e8 q^{38} -8.89620e8 q^{40} +1.08511e8 q^{41} +1.59070e9 q^{43} +7.58215e7 q^{44} +5.21902e8 q^{46} +1.44049e9 q^{47} +1.39828e9 q^{49} -3.89626e9 q^{50} +3.59971e8 q^{52} -1.05136e9 q^{53} -1.80634e9 q^{55} +4.59656e9 q^{56} -7.07818e9 q^{58} +5.77753e9 q^{59} -3.09262e9 q^{61} -5.32917e9 q^{62} +5.80285e9 q^{64} -8.57578e9 q^{65} -9.11382e9 q^{67} +4.23539e9 q^{68} +3.27963e10 q^{70} -3.35028e9 q^{71} +6.20143e8 q^{73} +4.80634e8 q^{74} -4.86240e9 q^{76} +9.33311e9 q^{77} +1.06185e10 q^{79} +5.55284e10 q^{80} -5.44723e9 q^{82} -6.00030e10 q^{83} -1.00902e11 q^{85} -7.98524e10 q^{86} +1.27089e10 q^{88} +6.14130e10 q^{89} +4.43100e10 q^{91} -4.90717e9 q^{92} -7.23121e10 q^{94} +1.15840e11 q^{95} +1.31873e11 q^{97} -7.01933e10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 944 q^{4} + 116200 q^{7}+O(q^{10}) $ 2 * q + 944 * q^4 + 116200 * q^7	$ 2 q + 944 q^{4} + 116200 q^{7} + 1128960 q^{10} + 1525300 q^{13} - 9876352 q^{16} - 20603408 q^{19} - 16128000 q^{22} + 155230790 q^{25} + 54846400 q^{28} + 212319016 q^{31} - 900910080 q^{34} - 19148900 q^{37} - 1779240960 q^{40} + 3181394800 q^{43} + 1043804160 q^{46} + 2796566514 q^{49} + 719941600 q^{52} - 3612672000 q^{55} - 14156352000 q^{58} - 6185242196 q^{61} + 11605707776 q^{64} - 18227640800 q^{67} + 65592576000 q^{70} + 1240285900 q^{73} - 9724808576 q^{76} + 21236972968 q^{79} - 10894464000 q^{82} - 201803857920 q^{85} + 25417728000 q^{88} + 88619930000 q^{91} - 144624291840 q^{94} + 263745804700 q^{97}+O(q^{100}) $ 2 * q + 944 * q^4 + 116200 * q^7 + 1128960 * q^10 + 1525300 * q^13 - 9876352 * q^16 - 20603408 * q^19 - 16128000 * q^22 + 155230790 * q^25 + 54846400 * q^28 + 212319016 * q^31 - 900910080 * q^34 - 19148900 * q^37 - 1779240960 * q^40 + 3181394800 * q^43 + 1043804160 * q^46 + 2796566514 * q^49 + 719941600 * q^52 - 3612672000 * q^55 - 14156352000 * q^58 - 6185242196 * q^61 + 11605707776 * q^64 - 18227640800 * q^67 + 65592576000 * q^70 + 1240285900 * q^73 - 9724808576 * q^76 + 21236972968 * q^79 - 10894464000 * q^82 - 201803857920 * q^85 + 25417728000 * q^88 + 88619930000 * q^91 - 144624291840 * q^94 + 263745804700 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−50.1996	−1.10926	−0.554632	−	0.832095i	$-0.687141\pi$
$2$	−50.1996	−1.10926	−0.554632	+	0.832095i	$0.687141\pi$
$3$	0	0
$3$	0	0
$4$	472.000	0.230469
$5$	−11244.7	−1.60921	−0.804606	−	0.593809i	$-0.797623\pi$
$5$	−11244.7	−1.60921	−0.804606	+	0.593809i	$0.797623\pi$
$6$	0	0
$7$	58100.0	1.30658	0.653291	−	0.757107i	$-0.273388\pi$
$7$	58100.0	1.30658	0.653291	+	0.757107i	$0.273388\pi$
$8$	79114.6	0.853614
$9$	0	0
$10$	564480.	1.78504
$11$	160639.	0.300740	0.150370	−	0.988630i	$-0.451954\pi$
$11$	160639.	0.300740	0.150370	+	0.988630i	$0.451954\pi$
$12$	0	0
$13$	762650.	0.569688	0.284844	−	0.958574i	$-0.408058\pi$
$13$	762650.	0.569688	0.284844	+	0.958574i	$0.408058\pi$
$14$	−2.91660e6	−1.44935
$15$	0	0
$16$	−4.93818e6	−1.17735
$17$	8.97328e6	1.53279	0.766394	−	0.642371i	$-0.222049\pi$
$17$	8.97328e6	1.53279	0.766394	+	0.642371i	$0.222049\pi$
$18$	0	0
$19$	−1.03017e7	−0.954474	−0.477237	−	0.878775i	$-0.658362\pi$
$19$	−1.03017e7	−0.954474	−0.477237	+	0.878775i	$0.658362\pi$
$20$	−5.30750e6	−0.370873
$21$	0	0
$22$	−8.06400e6	−0.333600
$23$	−1.03965e7	−0.336811	−0.168405	−	0.985718i	$-0.553862\pi$
$23$	−1.03965e7	−0.336811	−0.168405	+	0.985718i	$0.553862\pi$
$24$	0	0
$25$	7.76154e7	1.58956
$26$	−3.82847e7	−0.631935
$27$	0	0
$28$	2.74232e7	0.301126
$29$	1.41001e8	1.27653	0.638267	−	0.769815i	$-0.279652\pi$
$29$	1.41001e8	1.27653	0.638267	+	0.769815i	$0.279652\pi$
$30$	0	0
$31$	1.06160e8	0.665993	0.332996	−	0.942928i	$-0.391940\pi$
$31$	1.06160e8	0.665993	0.332996	+	0.942928i	$0.391940\pi$
$32$	8.58678e7	0.452382
$33$	0	0
$34$	−4.50455e8	−1.70027
$35$	−6.53318e8	−2.10257
$36$	0	0
$37$	−9.57445e6	−0.0226989	−0.0113494	−	0.999936i	$-0.503613\pi$
$37$	−9.57445e6	−0.0226989	−0.0113494	+	0.999936i	$0.503613\pi$
$38$	5.17141e8	1.05876
$39$	0	0
$40$	−8.89620e8	−1.37365
$41$	1.08511e8	0.146273	0.0731365	−	0.997322i	$-0.476699\pi$
$41$	1.08511e8	0.146273	0.0731365	+	0.997322i	$0.476699\pi$
$42$	0	0
$43$	1.59070e9	1.65010	0.825052	−	0.565058i	$-0.191146\pi$
$43$	1.59070e9	1.65010	0.825052	+	0.565058i	$0.191146\pi$
$44$	7.58215e7	0.0693111
$45$	0	0
$46$	5.21902e8	0.373612
$47$	1.44049e9	0.916163	0.458082	−	0.888910i	$-0.348537\pi$
$47$	1.44049e9	0.916163	0.458082	+	0.888910i	$0.348537\pi$
$48$	0	0
$49$	1.39828e9	0.707158
$50$	−3.89626e9	−1.76325
$51$	0	0
$52$	3.59971e8	0.131295
$53$	−1.05136e9	−0.345331	−0.172665	−	0.984981i	$-0.555238\pi$
$53$	−1.05136e9	−0.345331	−0.172665	+	0.984981i	$0.555238\pi$
$54$	0	0
$55$	−1.80634e9	−0.483954
$56$	4.59656e9	1.11532
$57$	0	0
$58$	−7.07818e9	−1.41601
$59$	5.77753e9	1.05210	0.526049	−	0.850454i	$-0.323673\pi$
$59$	5.77753e9	1.05210	0.526049	+	0.850454i	$0.323673\pi$
$60$	0	0
$61$	−3.09262e9	−0.468827	−0.234414	−	0.972137i	$-0.575317\pi$
$61$	−3.09262e9	−0.468827	−0.234414	+	0.972137i	$0.575317\pi$
$62$	−5.32917e9	−0.738763
$63$	0	0
$64$	5.80285e9	0.675541
$65$	−8.57578e9	−0.916748
$66$	0	0
$67$	−9.11382e9	−0.824687	−0.412343	−	0.911028i	$-0.635290\pi$
$67$	−9.11382e9	−0.824687	−0.412343	+	0.911028i	$0.635290\pi$
$68$	4.23539e9	0.353260
$69$	0	0
$70$	3.27963e10	2.33231
$71$	−3.35028e9	−0.220374	−0.110187	−	0.993911i	$-0.535145\pi$
$71$	−3.35028e9	−0.220374	−0.110187	+	0.993911i	$0.535145\pi$
$72$	0	0
$73$	6.20143e8	0.0350119	0.0175060	−	0.999847i	$-0.494427\pi$
$73$	6.20143e8	0.0350119	0.0175060	+	0.999847i	$0.494427\pi$
$74$	4.80634e8	0.0251791
$75$	0	0
$76$	−4.86240e9	−0.219977
$77$	9.33311e9	0.392941
$78$	0	0
$79$	1.06185e10	0.388252	0.194126	−	0.980977i	$-0.437813\pi$
$79$	1.06185e10	0.388252	0.194126	+	0.980977i	$0.437813\pi$
$80$	5.55284e10	1.89461
$81$	0	0
$82$	−5.44723e9	−0.162256
$83$	−6.00030e10	−1.67203	−0.836014	−	0.548709i	$-0.815120\pi$
$83$	−6.00030e10	−1.67203	−0.836014	+	0.548709i	$0.815120\pi$
$84$	0	0
$85$	−1.00902e11	−2.46658
$86$	−7.98524e10	−1.83040
$87$	0	0
$88$	1.27089e10	0.256715
$89$	6.14130e10	1.16578	0.582888	−	0.812552i	$-0.301923\pi$
$89$	6.14130e10	1.16578	0.582888	+	0.812552i	$0.301923\pi$
$90$	0	0
$91$	4.43100e10	0.744344
$92$	−4.90717e9	−0.0776243
$93$	0	0
$94$	−7.23121e10	−1.01627
$95$	1.15840e11	1.53595
$96$	0	0
$97$	1.31873e11	1.55923	0.779617	−	0.626257i	$-0.215414\pi$
$97$	1.31873e11	1.55923	0.779617	+	0.626257i	$0.215414\pi$
$98$	−7.01933e10	−0.784426
$99$	0	0
$100$	3.66345e10	0.366345
$101$	−2.72206e9	−0.0257710	−0.0128855	−	0.999917i	$-0.504102\pi$
$101$	−2.72206e9	−0.0257710	−0.0128855	+	0.999917i	$0.504102\pi$
$102$	0	0
$103$	7.95338e10	0.676000	0.338000	−	0.941146i	$-0.390250\pi$
$103$	7.95338e10	0.676000	0.338000	+	0.941146i	$0.390250\pi$
$104$	6.03367e10	0.486293
$105$	0	0
$106$	5.27780e10	0.383063
$107$	−1.98741e11	−1.36986	−0.684931	−	0.728608i	$-0.740168\pi$
$107$	−1.98741e11	−1.36986	−0.684931	+	0.728608i	$0.740168\pi$
$108$	0	0
$109$	−2.89337e10	−0.180119	−0.0900594	−	0.995936i	$-0.528706\pi$
$109$	−2.89337e10	−0.180119	−0.0900594	+	0.995936i	$0.528706\pi$
$110$	9.06773e10	0.536833
$111$	0	0
$112$	−2.86908e11	−1.53831
$113$	−2.05831e11	−1.05094	−0.525471	−	0.850811i	$-0.676111\pi$
$113$	−2.05831e11	−1.05094	−0.525471	+	0.850811i	$0.676111\pi$
$114$	0	0
$115$	1.16906e11	0.542000
$116$	6.65523e10	0.294201
$117$	0	0
$118$	−2.90030e11	−1.16706
$119$	5.21348e11	2.00271
$120$	0	0
$121$	−2.59507e11	−0.909556
$122$	1.55248e11	0.520053
$123$	0	0
$124$	5.01073e10	0.153491
$125$	−3.23705e11	−0.948732
$126$	0	0
$127$	4.49318e11	1.20680	0.603398	−	0.797441i	$-0.293813\pi$
$127$	4.49318e11	1.20680	0.603398	+	0.797441i	$0.293813\pi$
$128$	−4.67158e11	−1.20174
$129$	0	0
$130$	4.30501e11	1.01692
$131$	−4.01940e11	−0.910267	−0.455134	−	0.890423i	$-0.650409\pi$
$131$	−4.01940e11	−0.910267	−0.455134	+	0.890423i	$0.650409\pi$
$132$	0	0
$133$	−5.98529e11	−1.24710
$134$	4.57510e11	0.914796
$135$	0	0
$136$	7.09917e11	1.30841
$137$	4.60579e11	0.815345	0.407673	−	0.913128i	$-0.366341\pi$
$137$	4.60579e11	0.815345	0.407673	+	0.913128i	$0.366341\pi$
$138$	0	0
$139$	7.81318e11	1.27716	0.638581	−	0.769554i	$-0.279522\pi$
$139$	7.81318e11	1.27716	0.638581	+	0.769554i	$0.279522\pi$
$140$	−3.08366e11	−0.484576
$141$	0	0
$142$	1.68183e11	0.244453
$143$	1.22511e11	0.171328
$144$	0	0
$145$	−1.58551e12	−2.05421
$146$	−3.11309e10	−0.0388375
$147$	0	0
$148$	−4.51914e9	−0.00523138
$149$	1.45811e12	1.62654	0.813272	−	0.581883i	$-0.197684\pi$
$149$	1.45811e12	1.62654	0.813272	+	0.581883i	$0.197684\pi$
$150$	0	0
$151$	−4.60586e11	−0.477461	−0.238730	−	0.971086i	$-0.576731\pi$
$151$	−4.60586e11	−0.477461	−0.238730	+	0.971086i	$0.576731\pi$
$152$	−8.15015e11	−0.814753
$153$	0	0
$154$	−4.68518e11	−0.435876
$155$	−1.19373e12	−1.07172
$156$	0	0
$157$	−4.17640e11	−0.349425	−0.174713	−	0.984619i	$-0.555900\pi$
$157$	−4.17640e11	−0.349425	−0.174713	+	0.984619i	$0.555900\pi$
$158$	−5.33044e11	−0.430674
$159$	0	0
$160$	−9.65559e11	−0.727979
$161$	−6.04039e11	−0.440071
$162$	0	0
$163$	5.66142e11	0.385384	0.192692	−	0.981259i	$-0.438278\pi$
$163$	5.66142e11	0.385384	0.192692	+	0.981259i	$0.438278\pi$
$164$	5.12174e10	0.0337114
$165$	0	0
$166$	3.01213e12	1.85472
$167$	8.35744e11	0.497889	0.248945	−	0.968518i	$-0.419916\pi$
$167$	8.35744e11	0.497889	0.248945	+	0.968518i	$0.419916\pi$
$168$	0	0
$169$	−1.21053e12	−0.675456
$170$	5.06524e12	2.73609
$171$	0	0
$172$	7.50809e11	0.380297
$173$	−2.25677e12	−1.10722	−0.553609	−	0.832777i	$-0.686750\pi$
$173$	−2.25677e12	−1.10722	−0.553609	+	0.832777i	$0.686750\pi$
$174$	0	0
$175$	4.50945e12	2.07690
$176$	−7.93262e11	−0.354077
$177$	0	0
$178$	−3.08291e12	−1.29315
$179$	−1.28902e12	−0.524284	−0.262142	−	0.965029i	$-0.584429\pi$
$179$	−1.28902e12	−0.524284	−0.262142	+	0.965029i	$0.584429\pi$
$180$	0	0
$181$	1.14166e12	0.436823	0.218411	−	0.975857i	$-0.429913\pi$
$181$	1.14166e12	0.436823	0.218411	+	0.975857i	$0.429913\pi$
$182$	−2.22434e12	−0.825675
$183$	0	0
$184$	−8.22518e11	−0.287506
$185$	1.07662e11	0.0365273
$186$	0	0
$187$	1.44146e12	0.460970
$188$	6.79912e11	0.211147
$189$	0	0
$190$	−5.81511e12	−1.70378
$191$	3.96741e12	1.12934	0.564668	−	0.825318i	$-0.309004\pi$
$191$	3.96741e12	1.12934	0.564668	+	0.825318i	$0.309004\pi$
$192$	0	0
$193$	−6.13149e12	−1.64817	−0.824083	−	0.566469i	$-0.808309\pi$
$193$	−6.13149e12	−1.64817	−0.824083	+	0.566469i	$0.808309\pi$
$194$	−6.61997e12	−1.72960
$195$	0	0
$196$	6.59990e11	0.162978
$197$	−3.47714e11	−0.0834945	−0.0417472	−	0.999128i	$-0.513292\pi$
$197$	−3.47714e11	−0.0834945	−0.0417472	+	0.999128i	$0.513292\pi$
$198$	0	0
$199$	4.26854e12	0.969590	0.484795	−	0.874628i	$-0.338894\pi$
$199$	4.26854e12	0.969590	0.484795	+	0.874628i	$0.338894\pi$
$200$	6.14051e12	1.35687
$201$	0	0
$202$	1.36646e11	0.0285868
$203$	8.19214e12	1.66790
$204$	0	0
$205$	−1.22018e12	−0.235384
$206$	−3.99257e12	−0.749864
$207$	0	0
$208$	−3.76610e12	−0.670723
$209$	−1.65485e12	−0.287048
$210$	0	0
$211$	6.86132e12	1.12942	0.564708	−	0.825290i	$-0.308989\pi$
$211$	6.86132e12	1.12942	0.564708	+	0.825290i	$0.308989\pi$
$212$	−4.96243e11	−0.0795880
$213$	0	0
$214$	9.97672e12	1.51954
$215$	−1.78869e13	−2.65537
$216$	0	0
$217$	6.16787e12	0.870175
$218$	1.45246e12	0.199799
$219$	0	0
$220$	−8.52591e11	−0.111536
$221$	6.84347e12	0.873211
$222$	0	0
$223$	−1.10053e13	−1.33637	−0.668183	−	0.743997i	$-0.732928\pi$
$223$	−1.10053e13	−1.33637	−0.668183	+	0.743997i	$0.732928\pi$
$224$	4.98892e12	0.591075
$225$	0	0
$226$	1.03326e13	1.16577
$227$	8.87118e12	0.976876	0.488438	−	0.872599i	$-0.337567\pi$
$227$	8.87118e12	0.976876	0.488438	+	0.872599i	$0.337567\pi$
$228$	0	0
$229$	−1.13895e13	−1.19511	−0.597556	−	0.801827i	$-0.703862\pi$
$229$	−1.13895e13	−1.19511	−0.597556	+	0.801827i	$0.703862\pi$
$230$	−5.86864e12	−0.601221
$231$	0	0
$232$	1.11552e13	1.08967
$233$	9.26940e12	0.884288	0.442144	−	0.896944i	$-0.354218\pi$
$233$	9.26940e12	0.884288	0.442144	+	0.896944i	$0.354218\pi$
$234$	0	0
$235$	−1.61979e13	−1.47430
$236$	2.72700e12	0.242476
$237$	0	0
$238$	−2.61714e13	−2.22154
$239$	−2.05536e13	−1.70490	−0.852452	−	0.522806i	$-0.824885\pi$
$239$	−2.05536e13	−1.70490	−0.852452	+	0.522806i	$0.824885\pi$
$240$	0	0
$241$	1.28698e13	1.01971	0.509857	−	0.860259i	$-0.329698\pi$
$241$	1.28698e13	1.01971	0.509857	+	0.860259i	$0.329698\pi$
$242$	1.30271e13	1.00894
$243$	0	0
$244$	−1.45972e12	−0.108050
$245$	−1.57233e13	−1.13797
$246$	0	0
$247$	−7.85659e12	−0.543752
$248$	8.39876e12	0.568501
$249$	0	0
$250$	1.62498e13	1.05240
$251$	3.57196e12	0.226309	0.113154	−	0.993577i	$-0.463905\pi$
$251$	3.57196e12	0.226309	0.113154	+	0.993577i	$0.463905\pi$
$252$	0	0
$253$	−1.67009e12	−0.101292
$254$	−2.25556e13	−1.33866
$255$	0	0
$256$	1.15669e13	0.657503
$257$	−2.79858e12	−0.155706	−0.0778530	−	0.996965i	$-0.524806\pi$
$257$	−2.79858e12	−0.155706	−0.0778530	+	0.996965i	$0.524806\pi$
$258$	0	0
$259$	−5.56276e11	−0.0296580
$260$	−4.04777e12	−0.211282
$261$	0	0
$262$	2.01772e13	1.00973
$263$	2.99339e13	1.46692	0.733460	−	0.679733i	$-0.237904\pi$
$263$	2.99339e13	1.46692	0.733460	+	0.679733i	$0.237904\pi$
$264$	0	0
$265$	1.18223e13	0.555710
$266$	3.00459e13	1.38336
$267$	0	0
$268$	−4.30172e12	−0.190065
$269$	2.64097e13	1.14321	0.571606	−	0.820529i	$-0.306321\pi$
$269$	2.64097e13	1.14321	0.571606	+	0.820529i	$0.306321\pi$
$270$	0	0
$271$	2.96897e13	1.23388	0.616942	−	0.787009i	$-0.288371\pi$
$271$	2.96897e13	1.23388	0.616942	+	0.787009i	$0.288371\pi$
$272$	−4.43116e13	−1.80463
$273$	0	0
$274$	−2.31209e13	−0.904434
$275$	1.24680e13	0.478044
$276$	0	0
$277$	9.53508e12	0.351306	0.175653	−	0.984452i	$-0.443796\pi$
$277$	9.53508e12	0.351306	0.175653	+	0.984452i	$0.443796\pi$
$278$	−3.92218e13	−1.41671
$279$	0	0
$280$	−5.16869e13	−1.79478
$281$	−3.89335e13	−1.32568	−0.662839	−	0.748762i	$-0.730649\pi$
$281$	−3.89335e13	−1.32568	−0.662839	+	0.748762i	$0.730649\pi$
$282$	0	0
$283$	−3.71087e13	−1.21521	−0.607603	−	0.794241i	$-0.707869\pi$
$283$	−3.71087e13	−1.21521	−0.607603	+	0.794241i	$0.707869\pi$
$284$	−1.58133e12	−0.0507893
$285$	0	0
$286$	−6.15001e12	−0.190048
$287$	6.30452e12	0.191118
$288$	0	0
$289$	4.62478e13	1.34944
$290$	7.95920e13	2.27867
$291$	0	0
$292$	2.92707e11	0.00806916
$293$	−3.39516e13	−0.918518	−0.459259	−	0.888302i	$-0.651885\pi$
$293$	−3.39516e13	−0.918518	−0.459259	+	0.888302i	$0.651885\pi$
$294$	0	0
$295$	−6.49667e13	−1.69305
$296$	−7.57479e11	−0.0193761
$297$	0	0
$298$	−7.31966e13	−1.80427
$299$	−7.92892e12	−0.191877
$300$	0	0
$301$	9.24195e13	2.15600
$302$	2.31212e13	0.529630
$303$	0	0
$304$	5.08716e13	1.12375
$305$	3.47756e13	0.754442
$306$	0	0
$307$	3.85370e12	0.0806524	0.0403262	−	0.999187i	$-0.487160\pi$
$307$	3.85370e12	0.0806524	0.0403262	+	0.999187i	$0.487160\pi$
$308$	4.40523e12	0.0905606
$309$	0	0
$310$	5.99249e13	1.18883
$311$	−5.07944e13	−0.989996	−0.494998	−	0.868894i	$-0.664831\pi$
$311$	−5.07944e13	−0.989996	−0.494998	+	0.868894i	$0.664831\pi$
$312$	0	0
$313$	−2.78127e13	−0.523299	−0.261649	−	0.965163i	$-0.584266\pi$
$313$	−2.78127e13	−0.523299	−0.261649	+	0.965163i	$0.584266\pi$
$314$	2.09654e13	0.387605
$315$	0	0
$316$	5.01193e12	0.0894799
$317$	2.62079e13	0.459840	0.229920	−	0.973210i	$-0.426154\pi$
$317$	2.62079e13	0.459840	0.229920	+	0.973210i	$0.426154\pi$
$318$	0	0
$319$	2.26502e13	0.383904
$320$	−6.52514e13	−1.08709
$321$	0	0
$322$	3.03225e13	0.488155
$323$	−9.24401e13	−1.46301
$324$	0	0
$325$	5.91934e13	0.905555
$326$	−2.84201e13	−0.427493
$327$	0	0
$328$	8.58484e12	0.124861
$329$	8.36926e13	1.19704
$330$	0	0
$331$	−1.36923e14	−1.89418	−0.947091	−	0.320964i	$-0.895993\pi$
$331$	−1.36923e14	−1.89418	−0.947091	+	0.320964i	$0.895993\pi$
$332$	−2.83214e13	−0.385350
$333$	0	0
$334$	−4.19540e13	−0.552291
$335$	1.02482e14	1.32710
$336$	0	0
$337$	−5.25144e13	−0.658134	−0.329067	−	0.944307i	$-0.606734\pi$
$337$	−5.25144e13	−0.658134	−0.329067	+	0.944307i	$0.606734\pi$
$338$	6.07679e13	0.749260
$339$	0	0
$340$	−4.76257e13	−0.568470
$341$	1.70533e13	0.200290
$342$	0	0
$343$	−3.36424e13	−0.382622
$344$	1.25847e14	1.40855
$345$	0	0
$346$	1.13289e14	1.22820
$347$	−3.79900e13	−0.405375	−0.202687	−	0.979243i	$-0.564968\pi$
$347$	−3.79900e13	−0.405375	−0.202687	+	0.979243i	$0.564968\pi$
$348$	0	0
$349$	−1.48524e14	−1.53552	−0.767760	−	0.640737i	$-0.778629\pi$
$349$	−1.48524e14	−1.53552	−0.767760	+	0.640737i	$0.778629\pi$
$350$	−2.26373e14	−2.30383
$351$	0	0
$352$	1.37937e13	0.136049
$353$	−1.45298e14	−1.41091	−0.705455	−	0.708755i	$-0.749257\pi$
$353$	−1.45298e14	−1.41091	−0.705455	+	0.708755i	$0.749257\pi$
$354$	0	0
$355$	3.76729e13	0.354628
$356$	2.89869e13	0.268675
$357$	0	0
$358$	6.47081e13	0.581570
$359$	1.66188e14	1.47089	0.735443	−	0.677587i	$-0.236974\pi$
$359$	1.66188e14	1.47089	0.735443	+	0.677587i	$0.236974\pi$
$360$	0	0
$361$	−1.03652e13	−0.0889787
$362$	−5.73110e13	−0.484552
$363$	0	0
$364$	2.09143e13	0.171548
$365$	−6.97333e12	−0.0563416
$366$	0	0
$367$	1.64228e14	1.28761	0.643804	−	0.765191i	$-0.277355\pi$
$367$	1.64228e14	1.28761	0.643804	+	0.765191i	$0.277355\pi$
$368$	5.13399e13	0.396545
$369$	0	0
$370$	−5.40459e12	−0.0405185
$371$	−6.10842e13	−0.451203
$372$	0	0
$373$	1.49640e14	1.07312	0.536561	−	0.843862i	$-0.319723\pi$
$373$	1.49640e14	1.07312	0.536561	+	0.843862i	$0.319723\pi$
$374$	−7.23605e13	−0.511338
$375$	0	0
$376$	1.13964e14	0.782050
$377$	1.07534e14	0.727225
$378$	0	0
$379$	7.84646e12	0.0515416	0.0257708	−	0.999668i	$-0.491796\pi$
$379$	7.84646e12	0.0515416	0.0257708	+	0.999668i	$0.491796\pi$
$380$	5.46763e13	0.353989
$381$	0	0
$382$	−1.99162e14	−1.25273
$383$	5.62251e13	0.348608	0.174304	−	0.984692i	$-0.444233\pi$
$383$	5.62251e13	0.348608	0.174304	+	0.984692i	$0.444233\pi$
$384$	0	0
$385$	−1.04948e14	−0.632325
$386$	3.07799e14	1.82825
$387$	0	0
$388$	6.22440e13	0.359354
$389$	−2.38179e14	−1.35575	−0.677876	−	0.735177i	$-0.737099\pi$
$389$	−2.38179e14	−1.35575	−0.677876	+	0.735177i	$0.737099\pi$
$390$	0	0
$391$	−9.32910e13	−0.516259
$392$	1.10625e14	0.603640
$393$	0	0
$394$	1.74551e13	0.0926175
$395$	−1.19402e14	−0.624780
$396$	0	0
$397$	−1.01185e13	−0.0514956	−0.0257478	−	0.999668i	$-0.508197\pi$
$397$	−1.01185e13	−0.0514956	−0.0257478	+	0.999668i	$0.508197\pi$
$398$	−2.14279e14	−1.07553
$399$	0	0
$400$	−3.83278e14	−1.87148
$401$	−3.13474e14	−1.50976	−0.754878	−	0.655865i	$-0.772304\pi$
$401$	−3.13474e14	−1.50976	−0.754878	+	0.655865i	$0.772304\pi$
$402$	0	0
$403$	8.09625e13	0.379408
$404$	−1.28481e12	−0.00593940
$405$	0	0
$406$	−4.11242e14	−1.85014
$407$	−1.53803e12	−0.00682645
$408$	0	0
$409$	9.25258e13	0.399747	0.199873	−	0.979822i	$-0.435947\pi$
$409$	9.25258e13	0.399747	0.199873	+	0.979822i	$0.435947\pi$
$410$	6.12525e13	0.261104
$411$	0	0
$412$	3.75400e13	0.155797
$413$	3.35675e14	1.37465
$414$	0	0
$415$	6.74716e14	2.69065
$416$	6.54871e13	0.257717
$417$	0	0
$418$	8.30729e13	0.318412
$419$	2.36795e14	0.895769	0.447884	−	0.894091i	$-0.352178\pi$
$419$	2.36795e14	0.895769	0.447884	+	0.894091i	$0.352178\pi$
$420$	0	0
$421$	−4.05554e14	−1.49450	−0.747252	−	0.664541i	$-0.768627\pi$
$421$	−4.05554e14	−1.49450	−0.747252	+	0.664541i	$0.768627\pi$
$422$	−3.44436e14	−1.25282
$423$	0	0
$424$	−8.31781e13	−0.294779
$425$	6.96465e14	2.43646
$426$	0	0
$427$	−1.79681e14	−0.612561
$428$	−9.38058e13	−0.315710
$429$	0	0
$430$	8.97917e14	2.94550
$431$	−4.86438e14	−1.57544	−0.787720	−	0.616033i	$-0.788739\pi$
$431$	−4.86438e14	−1.57544	−0.787720	+	0.616033i	$0.788739\pi$
$432$	0	0
$433$	1.85422e14	0.585434	0.292717	−	0.956199i	$-0.405441\pi$
$433$	1.85422e14	0.585434	0.292717	+	0.956199i	$0.405441\pi$
$434$	−3.09624e14	−0.965254
$435$	0	0
$436$	−1.36567e13	−0.0415117
$437$	1.07102e14	0.321477
$438$	0	0
$439$	5.01757e14	1.46872	0.734359	−	0.678762i	$-0.237483\pi$
$439$	5.01757e14	1.46872	0.734359	+	0.678762i	$0.237483\pi$
$440$	−1.42907e14	−0.413110
$441$	0	0
$442$	−3.43540e14	−0.968622
$443$	1.34268e14	0.373896	0.186948	−	0.982370i	$-0.440140\pi$
$443$	1.34268e14	0.373896	0.186948	+	0.982370i	$0.440140\pi$
$444$	0	0
$445$	−6.90571e14	−1.87598
$446$	5.52462e14	1.48238
$447$	0	0
$448$	3.37146e14	0.882650
$449$	−6.49358e14	−1.67931	−0.839653	−	0.543124i	$-0.817241\pi$
$449$	−6.49358e14	−1.67931	−0.839653	+	0.543124i	$0.817241\pi$
$450$	0	0
$451$	1.74311e13	0.0439901
$452$	−9.71522e13	−0.242209
$453$	0	0
$454$	−4.45330e14	−1.08361
$455$	−4.98253e14	−1.19781
$456$	0	0
$457$	−4.64508e14	−1.09007	−0.545035	−	0.838413i	$-0.683484\pi$
$457$	−4.64508e14	−1.09007	−0.545035	+	0.838413i	$0.683484\pi$
$458$	5.71748e14	1.32570
$459$	0	0
$460$	5.51797e13	0.124914
$461$	2.00000e14	0.447379	0.223690	−	0.974660i	$-0.428190\pi$
$461$	2.00000e14	0.447379	0.223690	+	0.974660i	$0.428190\pi$
$462$	0	0
$463$	2.71686e14	0.593433	0.296717	−	0.954966i	$-0.404108\pi$
$463$	2.71686e14	0.593433	0.296717	+	0.954966i	$0.404108\pi$
$464$	−6.96286e14	−1.50293
$465$	0	0
$466$	−4.65320e14	−0.980910
$467$	8.06950e14	1.68114	0.840570	−	0.541703i	$-0.182220\pi$
$467$	8.06950e14	1.68114	0.840570	+	0.541703i	$0.182220\pi$
$468$	0	0
$469$	−5.29513e14	−1.07752
$470$	8.13129e14	1.63539
$471$	0	0
$472$	4.57087e14	0.898086
$473$	2.55528e14	0.496251
$474$	0	0
$475$	−7.99571e14	−1.51720
$476$	2.46076e14	0.461563
$477$	0	0
$478$	1.03178e15	1.89119
$479$	−2.97087e14	−0.538317	−0.269158	−	0.963096i	$-0.586745\pi$
$479$	−2.97087e14	−0.538317	−0.269158	+	0.963096i	$0.586745\pi$
$480$	0	0
$481$	−7.30195e12	−0.0129313
$482$	−6.46060e14	−1.13113
$483$	0	0
$484$	−1.22487e14	−0.209624
$485$	−1.48287e15	−2.50914
$486$	0	0
$487$	−9.67456e13	−0.160038	−0.0800188	−	0.996793i	$-0.525498\pi$
$487$	−9.67456e13	−0.160038	−0.0800188	+	0.996793i	$0.525498\pi$
$488$	−2.44671e14	−0.400197
$489$	0	0
$490$	7.89303e14	1.26231
$491$	4.52873e14	0.716189	0.358095	−	0.933685i	$-0.383426\pi$
$491$	4.52873e14	0.716189	0.358095	+	0.933685i	$0.383426\pi$
$492$	0	0
$493$	1.26524e15	1.95665
$494$	3.94398e14	0.603165
$495$	0	0
$496$	−5.24234e14	−0.784109
$497$	−1.94651e14	−0.287937
$498$	0	0
$499$	2.84707e14	0.411950	0.205975	−	0.978557i	$-0.433963\pi$
$499$	2.84707e14	0.411950	0.205975	+	0.978557i	$0.433963\pi$
$500$	−1.52789e14	−0.218653
$501$	0	0
$502$	−1.79311e14	−0.251036
$503$	4.60548e14	0.637750	0.318875	−	0.947797i	$-0.396695\pi$
$503$	4.60548e14	0.637750	0.318875	+	0.947797i	$0.396695\pi$
$504$	0	0
$505$	3.06088e13	0.0414709
$506$	8.38377e13	0.112360
$507$	0	0
$508$	2.12078e14	0.278129
$509$	1.07972e15	1.40076	0.700378	−	0.713772i	$-0.253015\pi$
$509$	1.07972e15	1.40076	0.700378	+	0.713772i	$0.253015\pi$
$510$	0	0
$511$	3.60303e13	0.0457460
$512$	3.76086e14	0.472391
$513$	0	0
$514$	1.40487e14	0.172719
$515$	−8.94335e14	−1.08783
$516$	0	0
$517$	2.31399e14	0.275526
$518$	2.79248e13	0.0328985
$519$	0	0
$520$	−6.78469e14	−0.782549
$521$	−8.11121e12	−0.00925717	−0.00462859	−	0.999989i	$-0.501473\pi$
$521$	−8.11121e12	−0.00925717	−0.00462859	+	0.999989i	$0.501473\pi$
$522$	0	0
$523$	−1.00543e15	−1.12355	−0.561774	−	0.827291i	$-0.689881\pi$
$523$	−1.00543e15	−1.12355	−0.561774	+	0.827291i	$0.689881\pi$
$524$	−1.89716e14	−0.209788
$525$	0	0
$526$	−1.50267e15	−1.62720
$527$	9.52599e14	1.02083
$528$	0	0
$529$	−8.44722e14	−0.886559
$530$	−5.93473e14	−0.616430
$531$	0	0
$532$	−2.82506e14	−0.287418
$533$	8.27563e13	0.0833300
$534$	0	0
$535$	2.23479e15	2.20440
$536$	−7.21036e14	−0.703964
$537$	0	0
$538$	−1.32576e15	−1.26812
$539$	2.24618e14	0.212670
$540$	0	0
$541$	8.81901e14	0.818153	0.409077	−	0.912500i	$-0.365851\pi$
$541$	8.81901e14	0.818153	0.409077	+	0.912500i	$0.365851\pi$
$542$	−1.49041e15	−1.36870
$543$	0	0
$544$	7.70516e14	0.693406
$545$	3.25352e14	0.289849
$546$	0	0
$547$	−1.79872e15	−1.57048	−0.785241	−	0.619190i	$-0.787461\pi$
$547$	−1.79872e15	−1.57048	−0.785241	+	0.619190i	$0.787461\pi$
$548$	2.17393e14	0.187912
$549$	0	0
$550$	−6.25891e14	−0.530278
$551$	−1.45255e15	−1.21842
$552$	0	0
$553$	6.16934e14	0.507283
$554$	−4.78657e14	−0.389691
$555$	0	0
$556$	3.68782e14	0.294346
$557$	1.41741e15	1.12019	0.560095	−	0.828428i	$-0.310764\pi$
$557$	1.41741e15	1.12019	0.560095	+	0.828428i	$0.310764\pi$
$558$	0	0
$559$	1.21315e15	0.940043
$560$	3.22620e15	2.47547
$561$	0	0
$562$	1.95444e15	1.47053
$563$	−1.42508e15	−1.06180	−0.530900	−	0.847435i	$-0.678146\pi$
$563$	−1.42508e15	−1.06180	−0.530900	+	0.847435i	$0.678146\pi$
$564$	0	0
$565$	2.31451e15	1.69119
$566$	1.86284e15	1.34799
$567$	0	0
$568$	−2.65056e14	−0.188114
$569$	1.45634e15	1.02364	0.511819	−	0.859093i	$-0.328972\pi$
$569$	1.45634e15	1.02364	0.511819	+	0.859093i	$0.328972\pi$
$570$	0	0
$571$	−1.56479e15	−1.07884	−0.539421	−	0.842036i	$-0.681357\pi$
$571$	−1.56479e15	−1.07884	−0.539421	+	0.842036i	$0.681357\pi$
$572$	5.78253e13	0.0394857
$573$	0	0
$574$	−3.16484e14	−0.212000
$575$	−8.06931e14	−0.535382
$576$	0	0
$577$	9.10934e14	0.592952	0.296476	−	0.955040i	$-0.404188\pi$
$577$	9.10934e14	0.592952	0.296476	+	0.955040i	$0.404188\pi$
$578$	−2.32162e15	−1.49689
$579$	0	0
$580$	−7.48361e14	−0.473432
$581$	−3.48617e15	−2.18464
$582$	0	0
$583$	−1.68890e14	−0.103855
$584$	4.90623e13	0.0298867
$585$	0	0
$586$	1.70435e15	1.01888
$587$	−2.65329e14	−0.157136	−0.0785680	−	0.996909i	$-0.525035\pi$
$587$	−2.65329e14	−0.157136	−0.0785680	+	0.996909i	$0.525035\pi$
$588$	0	0
$589$	−1.09362e15	−0.635673
$590$	3.26130e15	1.87804
$591$	0	0
$592$	4.72803e13	0.0267246
$593$	−2.50632e15	−1.40358	−0.701788	−	0.712386i	$-0.747615\pi$
$593$	−2.50632e15	−1.40358	−0.701788	+	0.712386i	$0.747615\pi$
$594$	0	0
$595$	−5.86240e15	−3.22279
$596$	6.88228e14	0.374868
$597$	0	0
$598$	3.98029e14	0.212842
$599$	3.47914e15	1.84342	0.921710	−	0.387881i	$-0.126793\pi$
$599$	3.47914e15	1.84342	0.921710	+	0.387881i	$0.126793\pi$
$600$	0	0
$601$	1.81057e15	0.941901	0.470950	−	0.882160i	$-0.343911\pi$
$601$	1.81057e15	0.941901	0.470950	+	0.882160i	$0.343911\pi$
$602$	−4.63942e15	−2.39157
$603$	0	0
$604$	−2.17397e14	−0.110040
$605$	2.91808e15	1.46367
$606$	0	0
$607$	−1.64117e15	−0.808380	−0.404190	−	0.914675i	$-0.632446\pi$
$607$	−1.64117e15	−0.808380	−0.404190	+	0.914675i	$0.632446\pi$
$608$	−8.84585e14	−0.431787
$609$	0	0
$610$	−1.74572e15	−0.836876
$611$	1.09859e15	0.521927
$612$	0	0
$613$	−6.47250e14	−0.302022	−0.151011	−	0.988532i	$-0.548253\pi$
$613$	−6.47250e14	−0.302022	−0.151011	+	0.988532i	$0.548253\pi$
$614$	−1.93454e14	−0.0894649
$615$	0	0
$616$	7.38385e14	0.335420
$617$	−1.60917e14	−0.0724493	−0.0362247	−	0.999344i	$-0.511533\pi$
$617$	−1.60917e14	−0.0724493	−0.0362247	+	0.999344i	$0.511533\pi$
$618$	0	0
$619$	1.16338e15	0.514545	0.257272	−	0.966339i	$-0.417176\pi$
$619$	1.16338e15	0.514545	0.257272	+	0.966339i	$0.417176\pi$
$620$	−5.63442e14	−0.246999
$621$	0	0
$622$	2.54986e15	1.09817
$623$	3.56809e15	1.52318
$624$	0	0
$625$	−1.49850e14	−0.0628518
$626$	1.39619e15	0.580477
$627$	0	0
$628$	−1.97126e14	−0.0805315
$629$	−8.59142e13	−0.0347926
$630$	0	0
$631$	2.12985e15	0.847594	0.423797	−	0.905757i	$-0.360697\pi$
$631$	2.12985e15	0.847594	0.423797	+	0.905757i	$0.360697\pi$
$632$	8.40077e14	0.331417
$633$	0	0
$634$	−1.31563e15	−0.510084
$635$	−5.05245e15	−1.94199
$636$	0	0
$637$	1.06640e15	0.402859
$638$	−1.13703e15	−0.425851
$639$	0	0
$640$	5.25306e15	1.93385
$641$	−2.33718e15	−0.853046	−0.426523	−	0.904477i	$-0.640262\pi$
$641$	−2.33718e15	−0.853046	−0.426523	+	0.904477i	$0.640262\pi$
$642$	0	0
$643$	−4.93057e15	−1.76904	−0.884518	−	0.466505i	$-0.845513\pi$
$643$	−4.93057e15	−1.76904	−0.884518	+	0.466505i	$0.845513\pi$
$644$	−2.85106e14	−0.101423
$645$	0	0
$646$	4.64045e15	1.62286
$647$	2.00854e15	0.696478	0.348239	−	0.937406i	$-0.386780\pi$
$647$	2.00854e15	0.696478	0.348239	+	0.937406i	$0.386780\pi$
$648$	0	0
$649$	9.28095e14	0.316408
$650$	−2.97148e15	−1.00450
$651$	0	0
$652$	2.67219e14	0.0888189
$653$	−9.77111e14	−0.322048	−0.161024	−	0.986950i	$-0.551480\pi$
$653$	−9.77111e14	−0.322048	−0.161024	+	0.986950i	$0.551480\pi$
$654$	0	0
$655$	4.51970e15	1.46481
$656$	−5.35849e14	−0.172215
$657$	0	0
$658$	−4.20134e15	−1.32784
$659$	−1.02367e15	−0.320842	−0.160421	−	0.987049i	$-0.551285\pi$
$659$	−1.02367e15	−0.320842	−0.160421	+	0.987049i	$0.551285\pi$
$660$	0	0
$661$	−1.85889e15	−0.572989	−0.286495	−	0.958082i	$-0.592490\pi$
$661$	−1.85889e15	−0.572989	−0.286495	+	0.958082i	$0.592490\pi$
$662$	6.87347e15	2.10115
$663$	0	0
$664$	−4.74711e15	−1.42727
$665$	6.73029e15	2.00685
$666$	0	0
$667$	−1.46592e15	−0.429950
$668$	3.94471e14	0.114748
$669$	0	0
$670$	−5.14457e15	−1.47210
$671$	−4.96795e14	−0.140995
$672$	0	0
$673$	1.22633e15	0.342392	0.171196	−	0.985237i	$-0.445237\pi$
$673$	1.22633e15	0.342392	0.171196	+	0.985237i	$0.445237\pi$
$674$	2.63620e15	0.730045
$675$	0	0
$676$	−5.71368e14	−0.155671
$677$	−2.91034e15	−0.786513	−0.393257	−	0.919429i	$-0.628652\pi$
$677$	−2.91034e15	−0.786513	−0.393257	+	0.919429i	$0.628652\pi$
$678$	0	0
$679$	7.66182e15	2.03727
$680$	−7.98281e15	−2.10551
$681$	0	0
$682$	−8.56070e14	−0.222175
$683$	1.19523e15	0.307707	0.153854	−	0.988094i	$-0.450832\pi$
$683$	1.19523e15	0.307707	0.153854	+	0.988094i	$0.450832\pi$
$684$	0	0
$685$	−5.17908e15	−1.31206
$686$	1.68884e15	0.424429
$687$	0	0
$688$	−7.85514e15	−1.94275
$689$	−8.01822e14	−0.196731
$690$	0	0
$691$	−2.23559e14	−0.0539836	−0.0269918	−	0.999636i	$-0.508593\pi$
$691$	−2.23559e14	−0.0539836	−0.0269918	+	0.999636i	$0.508593\pi$
$692$	−1.06519e15	−0.255179
$693$	0	0
$694$	1.90708e15	0.449668
$695$	−8.78569e15	−2.05523
$696$	0	0
$697$	9.73704e14	0.224206
$698$	7.45582e15	1.70330
$699$	0	0
$700$	2.12846e15	0.478660
$701$	1.27005e15	0.283382	0.141691	−	0.989911i	$-0.454746\pi$
$701$	1.27005e15	0.283382	0.141691	+	0.989911i	$0.454746\pi$
$702$	0	0
$703$	9.86331e13	0.0216655
$704$	9.32163e14	0.203162
$705$	0	0
$706$	7.29391e15	1.56507
$707$	−1.58152e14	−0.0336719
$708$	0	0
$709$	3.53998e15	0.742073	0.371036	−	0.928618i	$-0.379003\pi$
$709$	3.53998e15	0.742073	0.371036	+	0.928618i	$0.379003\pi$
$710$	−1.89117e15	−0.393377
$711$	0	0
$712$	4.85866e15	0.995123
$713$	−1.10369e15	−0.224313
$714$	0	0
$715$	−1.37760e15	−0.275702
$716$	−6.08416e14	−0.120831
$717$	0	0
$718$	−8.34255e15	−1.63160
$719$	−6.36514e15	−1.23538	−0.617688	−	0.786423i	$-0.711931\pi$
$719$	−6.36514e15	−1.23538	−0.617688	+	0.786423i	$0.711931\pi$
$720$	0	0
$721$	4.62091e15	0.883251
$722$	5.20327e14	0.0987010
$723$	0	0
$724$	5.38864e14	0.100674
$725$	1.09438e16	2.02913
$726$	0	0
$727$	8.81856e15	1.61049	0.805246	−	0.592941i	$-0.202033\pi$
$727$	8.81856e15	1.61049	0.805246	+	0.592941i	$0.202033\pi$
$728$	3.50556e15	0.635383
$729$	0	0
$730$	3.50058e14	0.0624978
$731$	1.42738e16	2.52926
$732$	0	0
$733$	1.75733e15	0.306747	0.153374	−	0.988168i	$-0.450986\pi$
$733$	1.75733e15	0.306747	0.153374	+	0.988168i	$0.450986\pi$
$734$	−8.24417e15	−1.42830
$735$	0	0
$736$	−8.92728e14	−0.152367
$737$	−1.46403e15	−0.248016
$738$	0	0
$739$	−3.76801e15	−0.628879	−0.314440	−	0.949277i	$-0.601817\pi$
$739$	−3.76801e15	−0.628879	−0.314440	+	0.949277i	$0.601817\pi$
$740$	5.08164e13	0.00841840
$741$	0	0
$742$	3.06640e15	0.500504
$743$	2.68502e15	0.435019	0.217510	−	0.976058i	$-0.430207\pi$
$743$	2.68502e15	0.435019	0.217510	+	0.976058i	$0.430207\pi$
$744$	0	0
$745$	−1.63960e16	−2.61746
$746$	−7.51186e15	−1.19038
$747$	0	0
$748$	6.80367e14	0.106239
$749$	−1.15469e16	−1.78984
$750$	0	0
$751$	3.28092e15	0.501159	0.250580	−	0.968096i	$-0.419379\pi$
$751$	3.28092e15	0.501159	0.250580	+	0.968096i	$0.419379\pi$
$752$	−7.11341e15	−1.07865
$753$	0	0
$754$	−5.39817e15	−0.806685
$755$	5.17916e15	0.768335
$756$	0	0
$757$	−1.15379e16	−1.68695	−0.843473	−	0.537172i	$-0.819492\pi$
$757$	−1.15379e16	−1.68695	−0.843473	+	0.537172i	$0.819492\pi$
$758$	−3.93889e14	−0.0571733
$759$	0	0
$760$	9.16461e15	1.31111
$761$	−5.19209e15	−0.737440	−0.368720	−	0.929541i	$-0.620204\pi$
$761$	−5.19209e15	−0.737440	−0.368720	+	0.929541i	$0.620204\pi$
$762$	0	0
$763$	−1.68105e15	−0.235340
$764$	1.87262e15	0.260277
$765$	0	0
$766$	−2.82248e15	−0.386698
$767$	4.40624e15	0.599367
$768$	0	0
$769$	−8.28529e15	−1.11100	−0.555498	−	0.831518i	$-0.687472\pi$
$769$	−8.28529e15	−1.11100	−0.555498	+	0.831518i	$0.687472\pi$
$770$	5.26835e15	0.701416
$771$	0	0
$772$	−2.89406e15	−0.379851
$773$	−1.13272e15	−0.147617	−0.0738083	−	0.997272i	$-0.523515\pi$
$773$	−1.13272e15	−0.147617	−0.0738083	+	0.997272i	$0.523515\pi$
$774$	0	0
$775$	8.23961e15	1.05864
$776$	1.04331e16	1.33098
$777$	0	0
$778$	1.19565e16	1.50389
$779$	−1.11785e15	−0.139614
$780$	0	0
$781$	−5.38185e14	−0.0662752
$782$	4.68317e15	0.572668
$783$	0	0
$784$	−6.90497e15	−0.832575
$785$	4.69624e15	0.562299
$786$	0	0
$787$	−2.83408e15	−0.334619	−0.167309	−	0.985904i	$-0.553508\pi$
$787$	−2.83408e15	−0.334619	−0.167309	+	0.985904i	$0.553508\pi$
$788$	−1.64121e14	−0.0192429
$789$	0	0
$790$	5.99392e15	0.693046
$791$	−1.19588e16	−1.37314
$792$	0	0
$793$	−2.35859e15	−0.267085
$794$	5.07947e14	0.0571223
$795$	0	0
$796$	2.01475e15	0.223460
$797$	−4.32095e15	−0.475947	−0.237973	−	0.971272i	$-0.576483\pi$
$797$	−4.32095e15	−0.475947	−0.237973	+	0.971272i	$0.576483\pi$
$798$	0	0
$799$	1.29259e16	1.40428
$800$	6.66467e15	0.719090
$801$	0	0
$802$	1.57362e16	1.67472
$803$	9.96190e13	0.0105295
$804$	0	0
$805$	6.79224e15	0.708167
$806$	−4.06429e15	−0.420864
$807$	0	0
$808$	−2.15355e14	−0.0219985
$809$	−4.11714e15	−0.417714	−0.208857	−	0.977946i	$-0.566974\pi$
$809$	−4.11714e15	−0.417714	−0.208857	+	0.977946i	$0.566974\pi$
$810$	0	0
$811$	−1.04760e15	−0.104853	−0.0524266	−	0.998625i	$-0.516696\pi$
$811$	−1.04760e15	−0.104853	−0.0524266	+	0.998625i	$0.516696\pi$
$812$	3.86669e15	0.384398
$813$	0	0
$814$	7.72084e13	0.00757234
$815$	−6.36610e15	−0.620164
$816$	0	0
$817$	−1.63869e16	−1.57498
$818$	−4.64476e15	−0.443425
$819$	0	0
$820$	−5.75925e14	−0.0542487
$821$	7.49519e15	0.701286	0.350643	−	0.936509i	$-0.385963\pi$
$821$	7.49519e15	0.701286	0.350643	+	0.936509i	$0.385963\pi$
$822$	0	0
$823$	6.22230e15	0.574449	0.287225	−	0.957863i	$-0.407267\pi$
$823$	6.22230e15	0.574449	0.287225	+	0.957863i	$0.407267\pi$
$824$	6.29228e15	0.577043
$825$	0	0
$826$	−1.68507e16	−1.52485
$827$	−8.83500e15	−0.794193	−0.397097	−	0.917777i	$-0.629982\pi$
$827$	−8.83500e15	−0.794193	−0.397097	+	0.917777i	$0.629982\pi$
$828$	0	0
$829$	8.47566e15	0.751837	0.375918	−	0.926653i	$-0.377327\pi$
$829$	8.47566e15	0.751837	0.375918	+	0.926653i	$0.377327\pi$
$830$	−3.38705e16	−2.98464
$831$	0	0
$832$	4.42555e15	0.384847
$833$	1.25472e16	1.08392
$834$	0	0
$835$	−9.39770e15	−0.801209
$836$	−7.81090e14	−0.0661556
$837$	0	0
$838$	−1.18870e16	−0.993645
$839$	1.76797e16	1.46819	0.734097	−	0.679044i	$-0.237606\pi$
$839$	1.76797e16	1.46819	0.734097	+	0.679044i	$0.237606\pi$
$840$	0	0
$841$	7.68067e15	0.629537
$842$	2.03586e16	1.65780
$843$	0	0
$844$	3.23854e15	0.260295
$845$	1.36120e16	1.08695
$846$	0	0
$847$	−1.50773e16	−1.18841
$848$	5.19181e15	0.406576
$849$	0	0
$850$	−3.49622e16	−2.70268
$851$	9.95411e13	0.00764522
$852$	0	0
$853$	1.41010e16	1.06913	0.534565	−	0.845127i	$-0.320475\pi$
$853$	1.41010e16	1.06913	0.534565	+	0.845127i	$0.320475\pi$
$854$	9.01993e15	0.679493
$855$	0	0
$856$	−1.57233e16	−1.16933
$857$	−1.44396e16	−1.06699	−0.533494	−	0.845804i	$-0.679121\pi$
$857$	−1.44396e16	−1.06699	−0.533494	+	0.845804i	$0.679121\pi$
$858$	0	0
$859$	1.93032e15	0.140821	0.0704104	−	0.997518i	$-0.477569\pi$
$859$	1.93032e15	0.140821	0.0704104	+	0.997518i	$0.477569\pi$
$860$	−8.44263e15	−0.611979
$861$	0	0
$862$	2.44190e16	1.74758
$863$	−1.33130e16	−0.946709	−0.473354	−	0.880872i	$-0.656957\pi$
$863$	−1.33130e16	−0.946709	−0.473354	+	0.880872i	$0.656957\pi$
$864$	0	0
$865$	2.53767e16	1.78175
$866$	−9.30811e15	−0.649401
$867$	0	0
$868$	2.91123e15	0.200548
$869$	1.70574e15	0.116763
$870$	0	0
$871$	−6.95066e15	−0.469814
$872$	−2.28908e15	−0.153752
$873$	0	0
$874$	−5.37648e15	−0.356603
$875$	−1.88072e16	−1.23960
$876$	0	0
$877$	−2.19539e16	−1.42894	−0.714470	−	0.699666i	$-0.753332\pi$
$877$	−2.19539e16	−1.42894	−0.714470	+	0.699666i	$0.753332\pi$
$878$	−2.51880e16	−1.62920
$879$	0	0
$880$	8.92001e15	0.569784
$881$	2.10443e16	1.33588	0.667939	−	0.744216i	$-0.267177\pi$
$881$	2.10443e16	1.33588	0.667939	+	0.744216i	$0.267177\pi$
$882$	0	0
$883$	2.63651e16	1.65290	0.826449	−	0.563012i	$-0.190357\pi$
$883$	2.63651e16	1.65290	0.826449	+	0.563012i	$0.190357\pi$
$884$	3.23012e15	0.201248
$885$	0	0
$886$	−6.74019e15	−0.414750
$887$	2.03473e16	1.24431	0.622154	−	0.782895i	$-0.286258\pi$
$887$	2.03473e16	1.24431	0.622154	+	0.782895i	$0.286258\pi$
$888$	0	0
$889$	2.61054e16	1.57678
$890$	3.46664e16	2.08096
$891$	0	0
$892$	−5.19450e15	−0.307991
$893$	−1.48395e16	−0.874454
$894$	0	0
$895$	1.44946e16	0.843685
$896$	−2.71419e16	−1.57017
$897$	0	0
$898$	3.25975e16	1.86279
$899$	1.49686e16	0.850162
$900$	0	0
$901$	−9.43417e15	−0.529319
$902$	−8.75036e14	−0.0487967
$903$	0	0
$904$	−1.62842e16	−0.897099
$905$	−1.28377e16	−0.702941
$906$	0	0
$907$	−2.57952e16	−1.39540	−0.697699	−	0.716391i	$-0.745793\pi$
$907$	−2.57952e16	−1.39540	−0.697699	+	0.716391i	$0.745793\pi$
$908$	4.18720e15	0.225139
$909$	0	0
$910$	2.50121e16	1.32869
$911$	2.51182e16	1.32629	0.663143	−	0.748493i	$-0.269222\pi$
$911$	2.51182e16	1.32629	0.663143	+	0.748493i	$0.269222\pi$
$912$	0	0
$913$	−9.63880e15	−0.502845
$914$	2.33181e16	1.20918
$915$	0	0
$916$	−5.37584e15	−0.275436
$917$	−2.33527e16	−1.18934
$918$	0	0
$919$	−2.49514e16	−1.25562	−0.627812	−	0.778365i	$-0.716049\pi$
$919$	−2.49514e16	−1.25562	−0.627812	+	0.778365i	$0.716049\pi$
$920$	9.24897e15	0.462658
$921$	0	0
$922$	−1.00399e16	−0.496262
$923$	−2.55509e15	−0.125544
$924$	0	0
$925$	−7.43125e14	−0.0360813
$926$	−1.36385e16	−0.658275
$927$	0	0
$928$	1.21074e16	0.577481
$929$	−1.47918e15	−0.0701350	−0.0350675	−	0.999385i	$-0.511165\pi$
$929$	−1.47918e15	−0.0701350	−0.0350675	+	0.999385i	$0.511165\pi$
$930$	0	0
$931$	−1.44047e16	−0.674965
$932$	4.37516e15	0.203801
$933$	0	0
$934$	−4.05086e16	−1.86483
$935$	−1.62088e16	−0.741798
$936$	0	0
$937$	2.25881e16	1.02167	0.510836	−	0.859678i	$-0.329336\pi$
$937$	2.25881e16	1.02167	0.510836	+	0.859678i	$0.329336\pi$
$938$	2.65813e16	1.19526
$939$	0	0
$940$	−7.64542e15	−0.339780
$941$	−1.81217e16	−0.800676	−0.400338	−	0.916368i	$-0.631107\pi$
$941$	−1.81217e16	−0.800676	−0.400338	+	0.916368i	$0.631107\pi$
$942$	0	0
$943$	−1.12814e15	−0.0492663
$944$	−2.85305e16	−1.23869
$945$	0	0
$946$	−1.28274e16	−0.550474
$947$	−4.12129e16	−1.75836	−0.879182	−	0.476487i	$-0.841910\pi$
$947$	−4.12129e16	−1.75836	−0.879182	+	0.476487i	$0.841910\pi$
$948$	0	0
$949$	4.72952e14	0.0199459
$950$	4.01381e16	1.68297
$951$	0	0
$952$	4.12462e16	1.70955
$953$	−3.38222e16	−1.39377	−0.696884	−	0.717184i	$-0.745431\pi$
$953$	−3.38222e16	−1.39377	−0.696884	+	0.717184i	$0.745431\pi$
$954$	0	0
$955$	−4.46123e16	−1.81734
$956$	−9.70131e15	−0.392927
$957$	0	0
$958$	1.49136e16	0.597136
$959$	2.67597e16	1.06532
$960$	0	0
$961$	−1.41386e16	−0.556453
$962$	3.66555e14	0.0143442
$963$	0	0
$964$	6.07455e15	0.235012
$965$	6.89469e16	2.65225
$966$	0	0
$967$	−2.90802e16	−1.10599	−0.552996	−	0.833184i	$-0.686515\pi$
$967$	−2.90802e16	−1.10599	−0.552996	+	0.833184i	$0.686515\pi$
$968$	−2.05308e16	−0.776410
$969$	0	0
$970$	7.44396e16	2.78330
$971$	2.93231e16	1.09020	0.545098	−	0.838373i	$-0.316493\pi$
$971$	2.93231e16	1.09020	0.545098	+	0.838373i	$0.316493\pi$
$972$	0	0
$973$	4.53946e16	1.66872
$974$	4.85659e15	0.177524
$975$	0	0
$976$	1.52719e16	0.551975
$977$	4.17408e16	1.50017	0.750086	−	0.661341i	$-0.230012\pi$
$977$	4.17408e16	1.50017	0.750086	+	0.661341i	$0.230012\pi$
$978$	0	0
$979$	9.86530e15	0.350595
$980$	−7.42139e15	−0.262266
$981$	0	0
$982$	−2.27340e16	−0.794444
$983$	9.55672e15	0.332097	0.166048	−	0.986118i	$-0.446899\pi$
$983$	9.55672e15	0.332097	0.166048	+	0.986118i	$0.446899\pi$
$984$	0	0
$985$	3.90994e15	0.134360
$986$	−6.35144e16	−2.17045
$987$	0	0
$988$	−3.70831e15	−0.125318
$989$	−1.65377e16	−0.555772
$990$	0	0
$991$	−2.90356e15	−0.0964997	−0.0482499	−	0.998835i	$-0.515364\pi$
$991$	−2.90356e15	−0.0964997	−0.0482499	+	0.998835i	$0.515364\pi$
$992$	9.11569e15	0.301283
$993$	0	0
$994$	9.77142e15	0.319398
$995$	−4.79986e16	−1.56028
$996$	0	0
$997$	−3.74887e15	−0.120525	−0.0602625	−	0.998183i	$-0.519194\pi$
$997$	−3.74887e15	−0.120525	−0.0602625	+	0.998183i	$0.519194\pi$
$998$	−1.42922e16	−0.456962
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.12.a.c.1.1	✓	2
3.2	odd	2	inner	9.12.a.c.1.2	yes	2
4.3	odd	2		144.12.a.r.1.1		2
5.2	odd	4		225.12.b.g.199.2		4
5.3	odd	4		225.12.b.g.199.3		4
5.4	even	2		225.12.a.j.1.2		2
9.2	odd	6		81.12.c.g.28.1		4
9.4	even	3		81.12.c.g.55.2		4
9.5	odd	6		81.12.c.g.55.1		4
9.7	even	3		81.12.c.g.28.2		4
12.11	even	2		144.12.a.r.1.2		2
15.2	even	4		225.12.b.g.199.4		4
15.8	even	4		225.12.b.g.199.1		4
15.14	odd	2		225.12.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.12.a.c.1.1	✓	2	1.1	even	1	trivial
9.12.a.c.1.2	yes	2	3.2	odd	2	inner
81.12.c.g.28.1		4	9.2	odd	6
81.12.c.g.28.2		4	9.7	even	3
81.12.c.g.55.1		4	9.5	odd	6
81.12.c.g.55.2		4	9.4	even	3
144.12.a.r.1.1		2	4.3	odd	2
144.12.a.r.1.2		2	12.11	even	2
225.12.a.j.1.1		2	15.14	odd	2
225.12.a.j.1.2		2	5.4	even	2
225.12.b.g.199.1		4	15.8	even	4
225.12.b.g.199.2		4	5.2	odd	4
225.12.b.g.199.3		4	5.3	odd	4
225.12.b.g.199.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

\(f(q)\)	\(=\)	\(q-50.1996 q^{2} +472.000 q^{4} -11244.7 q^{5} +58100.0 q^{7} +79114.6 q^{8} +O(q^{10})\)	\(q-50.1996 q^{2} +472.000 q^{4} -11244.7 q^{5} +58100.0 q^{7} +79114.6 q^{8} +564480. q^{10} +160639. q^{11} +762650. q^{13} -2.91660e6 q^{14} -4.93818e6 q^{16} +8.97328e6 q^{17} -1.03017e7 q^{19} -5.30750e6 q^{20} -8.06400e6 q^{22} -1.03965e7 q^{23} +7.76154e7 q^{25} -3.82847e7 q^{26} +2.74232e7 q^{28} +1.41001e8 q^{29} +1.06160e8 q^{31} +8.58678e7 q^{32} -4.50455e8 q^{34} -6.53318e8 q^{35} -9.57445e6 q^{37} +5.17141e8 q^{38} -8.89620e8 q^{40} +1.08511e8 q^{41} +1.59070e9 q^{43} +7.58215e7 q^{44} +5.21902e8 q^{46} +1.44049e9 q^{47} +1.39828e9 q^{49} -3.89626e9 q^{50} +3.59971e8 q^{52} -1.05136e9 q^{53} -1.80634e9 q^{55} +4.59656e9 q^{56} -7.07818e9 q^{58} +5.77753e9 q^{59} -3.09262e9 q^{61} -5.32917e9 q^{62} +5.80285e9 q^{64} -8.57578e9 q^{65} -9.11382e9 q^{67} +4.23539e9 q^{68} +3.27963e10 q^{70} -3.35028e9 q^{71} +6.20143e8 q^{73} +4.80634e8 q^{74} -4.86240e9 q^{76} +9.33311e9 q^{77} +1.06185e10 q^{79} +5.55284e10 q^{80} -5.44723e9 q^{82} -6.00030e10 q^{83} -1.00902e11 q^{85} -7.98524e10 q^{86} +1.27089e10 q^{88} +6.14130e10 q^{89} +4.43100e10 q^{91} -4.90717e9 q^{92} -7.23121e10 q^{94} +1.15840e11 q^{95} +1.31873e11 q^{97} -7.01933e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 944 q^{4} + 116200 q^{7}+O(q^{10}) \) 2 * q + 944 * q^4 + 116200 * q^7	\( 2 q + 944 q^{4} + 116200 q^{7} + 1128960 q^{10} + 1525300 q^{13} - 9876352 q^{16} - 20603408 q^{19} - 16128000 q^{22} + 155230790 q^{25} + 54846400 q^{28} + 212319016 q^{31} - 900910080 q^{34} - 19148900 q^{37} - 1779240960 q^{40} + 3181394800 q^{43} + 1043804160 q^{46} + 2796566514 q^{49} + 719941600 q^{52} - 3612672000 q^{55} - 14156352000 q^{58} - 6185242196 q^{61} + 11605707776 q^{64} - 18227640800 q^{67} + 65592576000 q^{70} + 1240285900 q^{73} - 9724808576 q^{76} + 21236972968 q^{79} - 10894464000 q^{82} - 201803857920 q^{85} + 25417728000 q^{88} + 88619930000 q^{91} - 144624291840 q^{94} + 263745804700 q^{97}+O(q^{100}) \) 2 * q + 944 * q^4 + 116200 * q^7 + 1128960 * q^10 + 1525300 * q^13 - 9876352 * q^16 - 20603408 * q^19 - 16128000 * q^22 + 155230790 * q^25 + 54846400 * q^28 + 212319016 * q^31 - 900910080 * q^34 - 19148900 * q^37 - 1779240960 * q^40 + 3181394800 * q^43 + 1043804160 * q^46 + 2796566514 * q^49 + 719941600 * q^52 - 3612672000 * q^55 - 14156352000 * q^58 - 6185242196 * q^61 + 11605707776 * q^64 - 18227640800 * q^67 + 65592576000 * q^70 + 1240285900 * q^73 - 9724808576 * q^76 + 21236972968 * q^79 - 10894464000 * q^82 - 201803857920 * q^85 + 25417728000 * q^88 + 88619930000 * q^91 - 144624291840 * q^94 + 263745804700 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more