LMFDB - Embedded newform 24.12.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,12,Mod(1,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("24.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	24.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:	$18.4402363334$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	24.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-243.000 q^{3} +1190.00 q^{5} +18480.0 q^{7} +59049.0 q^{9} +O(q^{10})$

$q-243.000 q^{3} +1190.00 q^{5} +18480.0 q^{7} +59049.0 q^{9} +135884. q^{11} -848186. q^{13} -289170. q^{15} -7.12461e6 q^{17} -5.04632e6 q^{19} -4.49064e6 q^{21} -1.48912e7 q^{23} -4.74120e7 q^{25} -1.43489e7 q^{27} -1.15001e8 q^{29} -1.63991e8 q^{31} -3.30198e7 q^{33} +2.19912e7 q^{35} -2.23622e8 q^{37} +2.06109e8 q^{39} +1.05358e8 q^{41} +1.41948e9 q^{43} +7.02683e7 q^{45} +2.46928e9 q^{47} -1.63582e9 q^{49} +1.73128e9 q^{51} -4.83705e8 q^{53} +1.61702e8 q^{55} +1.22625e9 q^{57} +6.15184e9 q^{59} -7.53273e9 q^{61} +1.09123e9 q^{63} -1.00934e9 q^{65} -8.76495e9 q^{67} +3.61857e9 q^{69} -1.04016e10 q^{71} -3.17384e10 q^{73} +1.15211e10 q^{75} +2.51114e9 q^{77} -3.98800e10 q^{79} +3.48678e9 q^{81} +1.35133e10 q^{83} -8.47828e9 q^{85} +2.79453e10 q^{87} +8.15145e10 q^{89} -1.56745e10 q^{91} +3.98497e10 q^{93} -6.00512e9 q^{95} +3.07830e10 q^{97} +8.02381e9 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	0	0
$5$	1190.00	0.170299	0.0851495	−	0.996368i	$-0.472863\pi$
$5$	1190.00	0.170299	0.0851495	+	0.996368i	$0.472863\pi$
$6$	0	0
$7$	18480.0	0.415588	0.207794	−	0.978173i	$-0.433372\pi$
$7$	18480.0	0.415588	0.207794	+	0.978173i	$0.433372\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	135884.	0.254395	0.127197	−	0.991877i	$-0.459402\pi$
$11$	135884.	0.254395	0.127197	+	0.991877i	$0.459402\pi$
$12$	0	0
$13$	−848186.	−0.633582	−0.316791	−	0.948495i	$-0.602605\pi$
$13$	−848186.	−0.633582	−0.316791	+	0.948495i	$0.602605\pi$
$14$	0	0
$15$	−289170.	−0.0983221
$16$	0	0
$17$	−7.12461e6	−1.21700	−0.608502	−	0.793553i	$-0.708229\pi$
$17$	−7.12461e6	−1.21700	−0.608502	+	0.793553i	$0.708229\pi$
$18$	0	0
$19$	−5.04632e6	−0.467552	−0.233776	−	0.972291i	$-0.575108\pi$
$19$	−5.04632e6	−0.467552	−0.233776	+	0.972291i	$0.575108\pi$
$20$	0	0
$21$	−4.49064e6	−0.239940
$22$	0	0
$23$	−1.48912e7	−0.482422	−0.241211	−	0.970473i	$-0.577545\pi$
$23$	−1.48912e7	−0.482422	−0.241211	+	0.970473i	$0.577545\pi$
$24$	0	0
$25$	−4.74120e7	−0.970998
$26$	0	0
$27$	−1.43489e7	−0.192450
$28$	0	0
$29$	−1.15001e8	−1.04115	−0.520576	−	0.853815i	$-0.674283\pi$
$29$	−1.15001e8	−1.04115	−0.520576	+	0.853815i	$0.674283\pi$
$30$	0	0
$31$	−1.63991e8	−1.02880	−0.514398	−	0.857551i	$-0.671985\pi$
$31$	−1.63991e8	−1.02880	−0.514398	+	0.857551i	$0.671985\pi$
$32$	0	0
$33$	−3.30198e7	−0.146875
$34$	0	0
$35$	2.19912e7	0.0707742
$36$	0	0
$37$	−2.23622e8	−0.530158	−0.265079	−	0.964227i	$-0.585398\pi$
$37$	−2.23622e8	−0.530158	−0.265079	+	0.964227i	$0.585398\pi$
$38$	0	0
$39$	2.06109e8	0.365799
$40$	0	0
$41$	1.05358e8	0.142023	0.0710113	−	0.997476i	$-0.477377\pi$
$41$	1.05358e8	0.142023	0.0710113	+	0.997476i	$0.477377\pi$
$42$	0	0
$43$	1.41948e9	1.47249	0.736244	−	0.676717i	$-0.236598\pi$
$43$	1.41948e9	1.47249	0.736244	+	0.676717i	$0.236598\pi$
$44$	0	0
$45$	7.02683e7	0.0567663
$46$	0	0
$47$	2.46928e9	1.57048	0.785239	−	0.619193i	$-0.212540\pi$
$47$	2.46928e9	1.57048	0.785239	+	0.619193i	$0.212540\pi$
$48$	0	0
$49$	−1.63582e9	−0.827287
$50$	0	0
$51$	1.73128e9	0.702637
$52$	0	0
$53$	−4.83705e8	−0.158878	−0.0794389	−	0.996840i	$-0.525313\pi$
$53$	−4.83705e8	−0.158878	−0.0794389	+	0.996840i	$0.525313\pi$
$54$	0	0
$55$	1.61702e8	0.0433232
$56$	0	0
$57$	1.22625e9	0.269941
$58$	0	0
$59$	6.15184e9	1.12026	0.560130	−	0.828404i	$-0.310751\pi$
$59$	6.15184e9	1.12026	0.560130	+	0.828404i	$0.310751\pi$
$60$	0	0
$61$	−7.53273e9	−1.14193	−0.570964	−	0.820975i	$-0.693430\pi$
$61$	−7.53273e9	−1.14193	−0.570964	+	0.820975i	$0.693430\pi$
$62$	0	0
$63$	1.09123e9	0.138529
$64$	0	0
$65$	−1.00934e9	−0.107898
$66$	0	0
$67$	−8.76495e9	−0.793118	−0.396559	−	0.918009i	$-0.629796\pi$
$67$	−8.76495e9	−0.793118	−0.396559	+	0.918009i	$0.629796\pi$
$68$	0	0
$69$	3.61857e9	0.278527
$70$	0	0
$71$	−1.04016e10	−0.684196	−0.342098	−	0.939664i	$-0.611137\pi$
$71$	−1.04016e10	−0.684196	−0.342098	+	0.939664i	$0.611137\pi$
$72$	0	0
$73$	−3.17384e10	−1.79188	−0.895941	−	0.444174i	$-0.853497\pi$
$73$	−3.17384e10	−1.79188	−0.895941	+	0.444174i	$0.853497\pi$
$74$	0	0
$75$	1.15211e10	0.560606
$76$	0	0
$77$	2.51114e9	0.105723
$78$	0	0
$79$	−3.98800e10	−1.45816	−0.729082	−	0.684426i	$-0.760053\pi$
$79$	−3.98800e10	−1.45816	−0.729082	+	0.684426i	$0.760053\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	1.35133e10	0.376559	0.188279	−	0.982116i	$-0.439709\pi$
$83$	1.35133e10	0.376559	0.188279	+	0.982116i	$0.439709\pi$
$84$	0	0
$85$	−8.47828e9	−0.207254
$86$	0	0
$87$	2.79453e10	0.601109
$88$	0	0
$89$	8.15145e10	1.54735	0.773677	−	0.633580i	$-0.218415\pi$
$89$	8.15145e10	1.54735	0.773677	+	0.633580i	$0.218415\pi$
$90$	0	0
$91$	−1.56745e10	−0.263309
$92$	0	0
$93$	3.98497e10	0.593976
$94$	0	0
$95$	−6.00512e9	−0.0796236
$96$	0	0
$97$	3.07830e10	0.363971	0.181986	−	0.983301i	$-0.441748\pi$
$97$	3.07830e10	0.363971	0.181986	+	0.983301i	$0.441748\pi$
$98$	0	0
$99$	8.02381e9	0.0847983
$100$	0	0
$101$	1.61449e11	1.52850	0.764252	−	0.644918i	$-0.223108\pi$
$101$	1.61449e11	1.52850	0.764252	+	0.644918i	$0.223108\pi$
$102$	0	0
$103$	−5.18595e10	−0.440782	−0.220391	−	0.975412i	$-0.570733\pi$
$103$	−5.18595e10	−0.440782	−0.220391	+	0.975412i	$0.570733\pi$
$104$	0	0
$105$	−5.34386e9	−0.0408615
$106$	0	0
$107$	−1.20810e11	−0.832709	−0.416355	−	0.909202i	$-0.636693\pi$
$107$	−1.20810e11	−0.832709	−0.416355	+	0.909202i	$0.636693\pi$
$108$	0	0
$109$	4.25401e10	0.264821	0.132410	−	0.991195i	$-0.457728\pi$
$109$	4.25401e10	0.264821	0.132410	+	0.991195i	$0.457728\pi$
$110$	0	0
$111$	5.43402e10	0.306087
$112$	0	0
$113$	3.25294e10	0.166090	0.0830452	−	0.996546i	$-0.473535\pi$
$113$	3.25294e10	0.166090	0.0830452	+	0.996546i	$0.473535\pi$
$114$	0	0
$115$	−1.77206e10	−0.0821560
$116$	0	0
$117$	−5.00845e10	−0.211194
$118$	0	0
$119$	−1.31663e11	−0.505772
$120$	0	0
$121$	−2.66847e11	−0.935283
$122$	0	0
$123$	−2.56021e10	−0.0819968
$124$	0	0
$125$	−1.14526e11	−0.335659
$126$	0	0
$127$	3.71230e11	0.997063	0.498532	−	0.866872i	$-0.333873\pi$
$127$	3.71230e11	0.997063	0.498532	+	0.866872i	$0.333873\pi$
$128$	0	0
$129$	−3.44933e11	−0.850141
$130$	0	0
$131$	−5.37142e11	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$131$	−5.37142e11	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$132$	0	0
$133$	−9.32559e10	−0.194309
$134$	0	0
$135$	−1.70752e10	−0.0327740
$136$	0	0
$137$	−3.62059e11	−0.640938	−0.320469	−	0.947259i	$-0.603841\pi$
$137$	−3.62059e11	−0.640938	−0.320469	+	0.947259i	$0.603841\pi$
$138$	0	0
$139$	7.19954e11	1.17686	0.588428	−	0.808550i	$-0.299747\pi$
$139$	7.19954e11	1.17686	0.588428	+	0.808550i	$0.299747\pi$
$140$	0	0
$141$	−6.00034e11	−0.906715
$142$	0	0
$143$	−1.15255e11	−0.161180
$144$	0	0
$145$	−1.36852e11	−0.177307
$146$	0	0
$147$	3.97503e11	0.477634
$148$	0	0
$149$	1.19007e12	1.32754	0.663769	−	0.747938i	$-0.268956\pi$
$149$	1.19007e12	1.32754	0.663769	+	0.747938i	$0.268956\pi$
$150$	0	0
$151$	1.50709e12	1.56231	0.781153	−	0.624339i	$-0.214632\pi$
$151$	1.50709e12	1.56231	0.781153	+	0.624339i	$0.214632\pi$
$152$	0	0
$153$	−4.20701e11	−0.405668
$154$	0	0
$155$	−1.95149e11	−0.175203
$156$	0	0
$157$	1.52866e12	1.27898	0.639490	−	0.768800i	$-0.279146\pi$
$157$	1.52866e12	1.27898	0.639490	+	0.768800i	$0.279146\pi$
$158$	0	0
$159$	1.17540e11	0.0917282
$160$	0	0
$161$	−2.75190e11	−0.200489
$162$	0	0
$163$	1.53695e12	1.04623	0.523116	−	0.852261i	$-0.324769\pi$
$163$	1.53695e12	1.04623	0.523116	+	0.852261i	$0.324769\pi$
$164$	0	0
$165$	−3.92936e10	−0.0250127
$166$	0	0
$167$	−9.64628e11	−0.574671	−0.287336	−	0.957830i	$-0.592769\pi$
$167$	−9.64628e11	−0.574671	−0.287336	+	0.957830i	$0.592769\pi$
$168$	0	0
$169$	−1.07274e12	−0.598574
$170$	0	0
$171$	−2.97980e11	−0.155851
$172$	0	0
$173$	1.93014e12	0.946966	0.473483	−	0.880803i	$-0.342997\pi$
$173$	1.93014e12	0.946966	0.473483	+	0.880803i	$0.342997\pi$
$174$	0	0
$175$	−8.76174e11	−0.403535
$176$	0	0
$177$	−1.49490e12	−0.646783
$178$	0	0
$179$	−1.70894e12	−0.695080	−0.347540	−	0.937665i	$-0.612983\pi$
$179$	−1.70894e12	−0.695080	−0.347540	+	0.937665i	$0.612983\pi$
$180$	0	0
$181$	1.30945e11	0.0501021	0.0250511	−	0.999686i	$-0.492025\pi$
$181$	1.30945e11	0.0501021	0.0250511	+	0.999686i	$0.492025\pi$
$182$	0	0
$183$	1.83045e12	0.659292
$184$	0	0
$185$	−2.66110e11	−0.0902854
$186$	0	0
$187$	−9.68120e11	−0.309600
$188$	0	0
$189$	−2.65168e11	−0.0799799
$190$	0	0
$191$	−5.05880e12	−1.44000	−0.720002	−	0.693972i	$-0.755859\pi$
$191$	−5.05880e12	−1.44000	−0.720002	+	0.693972i	$0.755859\pi$
$192$	0	0
$193$	−4.84580e12	−1.30257	−0.651283	−	0.758835i	$-0.725769\pi$
$193$	−4.84580e12	−1.30257	−0.651283	+	0.758835i	$0.725769\pi$
$194$	0	0
$195$	2.45270e11	0.0622951
$196$	0	0
$197$	7.83197e12	1.88064	0.940322	−	0.340286i	$-0.110524\pi$
$197$	7.83197e12	1.88064	0.940322	+	0.340286i	$0.110524\pi$
$198$	0	0
$199$	−5.02421e12	−1.14124	−0.570618	−	0.821216i	$-0.693296\pi$
$199$	−5.02421e12	−1.14124	−0.570618	+	0.821216i	$0.693296\pi$
$200$	0	0
$201$	2.12988e12	0.457907
$202$	0	0
$203$	−2.12522e12	−0.432690
$204$	0	0
$205$	1.25376e11	0.0241863
$206$	0	0
$207$	−8.79312e11	−0.160807
$208$	0	0
$209$	−6.85714e11	−0.118943
$210$	0	0
$211$	−5.40292e12	−0.889355	−0.444677	−	0.895691i	$-0.646682\pi$
$211$	−5.40292e12	−0.889355	−0.444677	+	0.895691i	$0.646682\pi$
$212$	0	0
$213$	2.52760e12	0.395021
$214$	0	0
$215$	1.68918e12	0.250763
$216$	0	0
$217$	−3.03055e12	−0.427555
$218$	0	0
$219$	7.71243e12	1.03454
$220$	0	0
$221$	6.04299e12	0.771071
$222$	0	0
$223$	2.65520e12	0.322419	0.161210	−	0.986920i	$-0.448460\pi$
$223$	2.65520e12	0.322419	0.161210	+	0.986920i	$0.448460\pi$
$224$	0	0
$225$	−2.79963e12	−0.323666
$226$	0	0
$227$	1.00927e13	1.11138	0.555691	−	0.831389i	$-0.312454\pi$
$227$	1.00927e13	1.11138	0.555691	+	0.831389i	$0.312454\pi$
$228$	0	0
$229$	8.75264e12	0.918425	0.459213	−	0.888326i	$-0.348132\pi$
$229$	8.75264e12	0.918425	0.459213	+	0.888326i	$0.348132\pi$
$230$	0	0
$231$	−6.10206e11	−0.0610395
$232$	0	0
$233$	−3.29016e12	−0.313877	−0.156938	−	0.987608i	$-0.550162\pi$
$233$	−3.29016e12	−0.313877	−0.156938	+	0.987608i	$0.550162\pi$
$234$	0	0
$235$	2.93844e12	0.267451
$236$	0	0
$237$	9.69084e12	0.841871
$238$	0	0
$239$	1.73649e13	1.44040	0.720200	−	0.693767i	$-0.244050\pi$
$239$	1.73649e13	1.44040	0.720200	+	0.693767i	$0.244050\pi$
$240$	0	0
$241$	−1.88851e13	−1.49632	−0.748162	−	0.663516i	$-0.769063\pi$
$241$	−1.88851e13	−1.49632	−0.748162	+	0.663516i	$0.769063\pi$
$242$	0	0
$243$	−8.47289e11	−0.0641500
$244$	0	0
$245$	−1.94662e12	−0.140886
$246$	0	0
$247$	4.28021e12	0.296232
$248$	0	0
$249$	−3.28374e12	−0.217406
$250$	0	0
$251$	−3.73484e12	−0.236628	−0.118314	−	0.992976i	$-0.537749\pi$
$251$	−3.73484e12	−0.236628	−0.118314	+	0.992976i	$0.537749\pi$
$252$	0	0
$253$	−2.02348e12	−0.122726
$254$	0	0
$255$	2.06022e12	0.119658
$256$	0	0
$257$	−2.55617e13	−1.42219	−0.711096	−	0.703095i	$-0.751801\pi$
$257$	−2.55617e13	−1.42219	−0.711096	+	0.703095i	$0.751801\pi$
$258$	0	0
$259$	−4.13254e12	−0.220327
$260$	0	0
$261$	−6.79071e12	−0.347051
$262$	0	0
$263$	3.71976e12	0.182288	0.0911441	−	0.995838i	$-0.470948\pi$
$263$	3.71976e12	0.182288	0.0911441	+	0.995838i	$0.470948\pi$
$264$	0	0
$265$	−5.75609e11	−0.0270567
$266$	0	0
$267$	−1.98080e13	−0.893366
$268$	0	0
$269$	−2.32714e13	−1.00736	−0.503680	−	0.863890i	$-0.668021\pi$
$269$	−2.32714e13	−1.00736	−0.503680	+	0.863890i	$0.668021\pi$
$270$	0	0
$271$	−1.69620e12	−0.0704931	−0.0352465	−	0.999379i	$-0.511222\pi$
$271$	−1.69620e12	−0.0704931	−0.0352465	+	0.999379i	$0.511222\pi$
$272$	0	0
$273$	3.80890e12	0.152021
$274$	0	0
$275$	−6.44254e12	−0.247017
$276$	0	0
$277$	3.22077e13	1.18664	0.593322	−	0.804965i	$-0.297816\pi$
$277$	3.22077e13	1.18664	0.593322	+	0.804965i	$0.297816\pi$
$278$	0	0
$279$	−9.68348e12	−0.342932
$280$	0	0
$281$	−2.19732e13	−0.748185	−0.374092	−	0.927391i	$-0.622046\pi$
$281$	−2.19732e13	−0.748185	−0.374092	+	0.927391i	$0.622046\pi$
$282$	0	0
$283$	6.26529e12	0.205171	0.102585	−	0.994724i	$-0.467289\pi$
$283$	6.26529e12	0.205171	0.102585	+	0.994724i	$0.467289\pi$
$284$	0	0
$285$	1.45924e12	0.0459707
$286$	0	0
$287$	1.94702e12	0.0590229
$288$	0	0
$289$	1.64881e13	0.481097
$290$	0	0
$291$	−7.48028e12	−0.210139
$292$	0	0
$293$	4.30466e12	0.116457	0.0582286	−	0.998303i	$-0.481455\pi$
$293$	4.30466e12	0.116457	0.0582286	+	0.998303i	$0.481455\pi$
$294$	0	0
$295$	7.32069e12	0.190779
$296$	0	0
$297$	−1.94979e12	−0.0489583
$298$	0	0
$299$	1.26305e13	0.305654
$300$	0	0
$301$	2.62319e13	0.611948
$302$	0	0
$303$	−3.92320e13	−0.882482
$304$	0	0
$305$	−8.96395e12	−0.194469
$306$	0	0
$307$	1.49844e13	0.313601	0.156800	−	0.987630i	$-0.449882\pi$
$307$	1.49844e13	0.313601	0.156800	+	0.987630i	$0.449882\pi$
$308$	0	0
$309$	1.26019e13	0.254486
$310$	0	0
$311$	1.92805e13	0.375783	0.187891	−	0.982190i	$-0.439835\pi$
$311$	1.92805e13	0.375783	0.187891	+	0.982190i	$0.439835\pi$
$312$	0	0
$313$	9.38955e13	1.76665	0.883325	−	0.468760i	$-0.155299\pi$
$313$	9.38955e13	1.76665	0.883325	+	0.468760i	$0.155299\pi$
$314$	0	0
$315$	1.29856e12	0.0235914
$316$	0	0
$317$	−1.75293e13	−0.307566	−0.153783	−	0.988105i	$-0.549146\pi$
$317$	−1.75293e13	−0.307566	−0.153783	+	0.988105i	$0.549146\pi$
$318$	0	0
$319$	−1.56268e13	−0.264864
$320$	0	0
$321$	2.93569e13	0.480765
$322$	0	0
$323$	3.59530e13	0.569012
$324$	0	0
$325$	4.02142e13	0.615207
$326$	0	0
$327$	−1.03372e13	−0.152894
$328$	0	0
$329$	4.56322e13	0.652671
$330$	0	0
$331$	−2.01778e13	−0.279139	−0.139569	−	0.990212i	$-0.544572\pi$
$331$	−2.01778e13	−0.279139	−0.139569	+	0.990212i	$0.544572\pi$
$332$	0	0
$333$	−1.32047e13	−0.176719
$334$	0	0
$335$	−1.04303e13	−0.135067
$336$	0	0
$337$	2.21846e13	0.278027	0.139014	−	0.990290i	$-0.455607\pi$
$337$	2.21846e13	0.278027	0.139014	+	0.990290i	$0.455607\pi$
$338$	0	0
$339$	−7.90464e12	−0.0958923
$340$	0	0
$341$	−2.22837e13	−0.261721
$342$	0	0
$343$	−6.67709e13	−0.759398
$344$	0	0
$345$	4.30610e12	0.0474328
$346$	0	0
$347$	−7.47701e13	−0.797840	−0.398920	−	0.916986i	$-0.630615\pi$
$347$	−7.47701e13	−0.797840	−0.398920	+	0.916986i	$0.630615\pi$
$348$	0	0
$349$	−1.57514e14	−1.62847	−0.814237	−	0.580533i	$-0.802844\pi$
$349$	−1.57514e14	−1.62847	−0.814237	+	0.580533i	$0.802844\pi$
$350$	0	0
$351$	1.21705e13	0.121933
$352$	0	0
$353$	−2.54892e13	−0.247512	−0.123756	−	0.992313i	$-0.539494\pi$
$353$	−2.54892e13	−0.247512	−0.123756	+	0.992313i	$0.539494\pi$
$354$	0	0
$355$	−1.23779e13	−0.116518
$356$	0	0
$357$	3.19940e13	0.292007
$358$	0	0
$359$	9.41138e12	0.0832979	0.0416489	−	0.999132i	$-0.486739\pi$
$359$	9.41138e12	0.0832979	0.0416489	+	0.999132i	$0.486739\pi$
$360$	0	0
$361$	−9.10250e13	−0.781395
$362$	0	0
$363$	6.48439e13	0.539986
$364$	0	0
$365$	−3.77687e13	−0.305155
$366$	0	0
$367$	−1.24601e14	−0.976922	−0.488461	−	0.872586i	$-0.662441\pi$
$367$	−1.24601e14	−0.976922	−0.488461	+	0.872586i	$0.662441\pi$
$368$	0	0
$369$	6.22130e12	0.0473409
$370$	0	0
$371$	−8.93887e12	−0.0660277
$372$	0	0
$373$	−2.05385e14	−1.47289	−0.736446	−	0.676496i	$-0.763498\pi$
$373$	−2.05385e14	−1.47289	−0.736446	+	0.676496i	$0.763498\pi$
$374$	0	0
$375$	2.78298e13	0.193793
$376$	0	0
$377$	9.75425e13	0.659655
$378$	0	0
$379$	7.18102e13	0.471705	0.235852	−	0.971789i	$-0.424212\pi$
$379$	7.18102e13	0.471705	0.235852	+	0.971789i	$0.424212\pi$
$380$	0	0
$381$	−9.02089e13	−0.575655
$382$	0	0
$383$	2.94282e14	1.82461	0.912307	−	0.409508i	$-0.134300\pi$
$383$	2.94282e14	1.82461	0.912307	+	0.409508i	$0.134300\pi$
$384$	0	0
$385$	2.98825e12	0.0180046
$386$	0	0
$387$	8.38186e13	0.490829
$388$	0	0
$389$	−1.34095e14	−0.763291	−0.381646	−	0.924309i	$-0.624643\pi$
$389$	−1.34095e14	−0.763291	−0.381646	+	0.924309i	$0.624643\pi$
$390$	0	0
$391$	1.06094e14	0.587109
$392$	0	0
$393$	1.30525e14	0.702322
$394$	0	0
$395$	−4.74572e13	−0.248324
$396$	0	0
$397$	−1.17524e14	−0.598109	−0.299054	−	0.954236i	$-0.596671\pi$
$397$	−1.17524e14	−0.598109	−0.299054	+	0.954236i	$0.596671\pi$
$398$	0	0
$399$	2.26612e13	0.112184
$400$	0	0
$401$	2.56980e14	1.23767	0.618835	−	0.785521i	$-0.287605\pi$
$401$	2.56980e14	1.23767	0.618835	+	0.785521i	$0.287605\pi$
$402$	0	0
$403$	1.39094e14	0.651827
$404$	0	0
$405$	4.14927e12	0.0189221
$406$	0	0
$407$	−3.03867e13	−0.134870
$408$	0	0
$409$	−1.40372e13	−0.0606460	−0.0303230	−	0.999540i	$-0.509654\pi$
$409$	−1.40372e13	−0.0606460	−0.0303230	+	0.999540i	$0.509654\pi$
$410$	0	0
$411$	8.79804e13	0.370046
$412$	0	0
$413$	1.13686e14	0.465567
$414$	0	0
$415$	1.60809e13	0.0641275
$416$	0	0
$417$	−1.74949e14	−0.679458
$418$	0	0
$419$	−9.05803e13	−0.342655	−0.171327	−	0.985214i	$-0.554806\pi$
$419$	−9.05803e13	−0.342655	−0.171327	+	0.985214i	$0.554806\pi$
$420$	0	0
$421$	−4.25794e13	−0.156909	−0.0784546	−	0.996918i	$-0.524999\pi$
$421$	−4.25794e13	−0.156909	−0.0784546	+	0.996918i	$0.524999\pi$
$422$	0	0
$423$	1.45808e14	0.523492
$424$	0	0
$425$	3.37792e14	1.18171
$426$	0	0
$427$	−1.39205e14	−0.474571
$428$	0	0
$429$	2.80069e13	0.0930573
$430$	0	0
$431$	−2.45393e14	−0.794763	−0.397381	−	0.917654i	$-0.630081\pi$
$431$	−2.45393e14	−0.794763	−0.397381	+	0.917654i	$0.630081\pi$
$432$	0	0
$433$	3.45088e14	1.08955	0.544775	−	0.838583i	$-0.316615\pi$
$433$	3.45088e14	1.08955	0.544775	+	0.838583i	$0.316615\pi$
$434$	0	0
$435$	3.32549e13	0.102368
$436$	0	0
$437$	7.51458e13	0.225557
$438$	0	0
$439$	1.66129e14	0.486283	0.243142	−	0.969991i	$-0.421822\pi$
$439$	1.66129e14	0.486283	0.243142	+	0.969991i	$0.421822\pi$
$440$	0	0
$441$	−9.65933e13	−0.275762
$442$	0	0
$443$	−5.76583e14	−1.60562	−0.802808	−	0.596238i	$-0.796662\pi$
$443$	−5.76583e14	−1.60562	−0.802808	+	0.596238i	$0.796662\pi$
$444$	0	0
$445$	9.70023e13	0.263513
$446$	0	0
$447$	−2.89186e14	−0.766454
$448$	0	0
$449$	3.12410e13	0.0807923	0.0403962	−	0.999184i	$-0.487138\pi$
$449$	3.12410e13	0.0807923	0.0403962	+	0.999184i	$0.487138\pi$
$450$	0	0
$451$	1.43165e13	0.0361298
$452$	0	0
$453$	−3.66223e14	−0.901998
$454$	0	0
$455$	−1.86526e13	−0.0448412
$456$	0	0
$457$	−8.36591e13	−0.196324	−0.0981622	−	0.995170i	$-0.531296\pi$
$457$	−8.36591e13	−0.196324	−0.0981622	+	0.995170i	$0.531296\pi$
$458$	0	0
$459$	1.02230e14	0.234212
$460$	0	0
$461$	−3.12974e14	−0.700090	−0.350045	−	0.936733i	$-0.613834\pi$
$461$	−3.12974e14	−0.700090	−0.350045	+	0.936733i	$0.613834\pi$
$462$	0	0
$463$	1.73544e13	0.0379065	0.0189532	−	0.999820i	$-0.493967\pi$
$463$	1.73544e13	0.0379065	0.0189532	+	0.999820i	$0.493967\pi$
$464$	0	0
$465$	4.74211e13	0.101153
$466$	0	0
$467$	−6.41703e14	−1.33688	−0.668439	−	0.743767i	$-0.733037\pi$
$467$	−6.41703e14	−1.33688	−0.668439	+	0.743767i	$0.733037\pi$
$468$	0	0
$469$	−1.61976e14	−0.329610
$470$	0	0
$471$	−3.71465e14	−0.738419
$472$	0	0
$473$	1.92884e14	0.374593
$474$	0	0
$475$	2.39256e14	0.453992
$476$	0	0
$477$	−2.85623e13	−0.0529593
$478$	0	0
$479$	−3.54411e14	−0.642187	−0.321094	−	0.947047i	$-0.604050\pi$
$479$	−3.54411e14	−0.642187	−0.321094	+	0.947047i	$0.604050\pi$
$480$	0	0
$481$	1.89673e14	0.335898
$482$	0	0
$483$	6.68711e13	0.115752
$484$	0	0
$485$	3.66318e13	0.0619839
$486$	0	0
$487$	−1.20917e14	−0.200022	−0.100011	−	0.994986i	$-0.531888\pi$
$487$	−1.20917e14	−0.200022	−0.100011	+	0.994986i	$0.531888\pi$
$488$	0	0
$489$	−3.73479e14	−0.604043
$490$	0	0
$491$	−2.44195e14	−0.386179	−0.193090	−	0.981181i	$-0.561851\pi$
$491$	−2.44195e14	−0.386179	−0.193090	+	0.981181i	$0.561851\pi$
$492$	0	0
$493$	8.19339e14	1.26708
$494$	0	0
$495$	9.54834e12	0.0144411
$496$	0	0
$497$	−1.92222e14	−0.284343
$498$	0	0
$499$	8.60223e14	1.24468	0.622340	−	0.782747i	$-0.286182\pi$
$499$	8.60223e14	1.24468	0.622340	+	0.782747i	$0.286182\pi$
$500$	0	0
$501$	2.34405e14	0.331787
$502$	0	0
$503$	1.21648e15	1.68453	0.842266	−	0.539062i	$-0.181221\pi$
$503$	1.21648e15	1.68453	0.842266	+	0.539062i	$0.181221\pi$
$504$	0	0
$505$	1.92124e14	0.260303
$506$	0	0
$507$	2.60676e14	0.345587
$508$	0	0
$509$	9.57278e14	1.24191	0.620955	−	0.783846i	$-0.286745\pi$
$509$	9.57278e14	1.24191	0.620955	+	0.783846i	$0.286745\pi$
$510$	0	0
$511$	−5.86525e14	−0.744684
$512$	0	0
$513$	7.24091e13	0.0899804
$514$	0	0
$515$	−6.17129e13	−0.0750647
$516$	0	0
$517$	3.35535e14	0.399521
$518$	0	0
$519$	−4.69023e14	−0.546731
$520$	0	0
$521$	−1.39668e15	−1.59400	−0.797000	−	0.603979i	$-0.793581\pi$
$521$	−1.39668e15	−1.59400	−0.797000	+	0.603979i	$0.793581\pi$
$522$	0	0
$523$	1.14451e15	1.27897	0.639483	−	0.768805i	$-0.279148\pi$
$523$	1.14451e15	1.27897	0.639483	+	0.768805i	$0.279148\pi$
$524$	0	0
$525$	2.12910e14	0.232981
$526$	0	0
$527$	1.16837e15	1.25205
$528$	0	0
$529$	−7.31061e14	−0.767269
$530$	0	0
$531$	3.63260e14	0.373420
$532$	0	0
$533$	−8.93634e13	−0.0899830
$534$	0	0
$535$	−1.43764e14	−0.141809
$536$	0	0
$537$	4.15272e14	0.401305
$538$	0	0
$539$	−2.22281e14	−0.210458
$540$	0	0
$541$	−9.32189e14	−0.864807	−0.432403	−	0.901680i	$-0.642334\pi$
$541$	−9.32189e14	−0.864807	−0.432403	+	0.901680i	$0.642334\pi$
$542$	0	0
$543$	−3.18196e13	−0.0289265
$544$	0	0
$545$	5.06227e13	0.0450987
$546$	0	0
$547$	−6.90223e13	−0.0602642	−0.0301321	−	0.999546i	$-0.509593\pi$
$547$	−6.90223e13	−0.0602642	−0.0301321	+	0.999546i	$0.509593\pi$
$548$	0	0
$549$	−4.44800e14	−0.380642
$550$	0	0
$551$	5.80333e14	0.486792
$552$	0	0
$553$	−7.36983e14	−0.605995
$554$	0	0
$555$	6.46648e13	0.0521263
$556$	0	0
$557$	−2.24654e14	−0.177546	−0.0887728	−	0.996052i	$-0.528294\pi$
$557$	−2.24654e14	−0.177546	−0.0887728	+	0.996052i	$0.528294\pi$
$558$	0	0
$559$	−1.20398e15	−0.932941
$560$	0	0
$561$	2.35253e14	0.178747
$562$	0	0
$563$	2.12470e15	1.58307	0.791537	−	0.611122i	$-0.209281\pi$
$563$	2.12470e15	1.58307	0.791537	+	0.611122i	$0.209281\pi$
$564$	0	0
$565$	3.87100e13	0.0282850
$566$	0	0
$567$	6.44358e13	0.0461764
$568$	0	0
$569$	−2.52164e15	−1.77241	−0.886206	−	0.463291i	$-0.846669\pi$
$569$	−2.52164e15	−1.77241	−0.886206	+	0.463291i	$0.846669\pi$
$570$	0	0
$571$	−2.83532e15	−1.95481	−0.977403	−	0.211386i	$-0.932202\pi$
$571$	−2.83532e15	−1.95481	−0.977403	+	0.211386i	$0.932202\pi$
$572$	0	0
$573$	1.22929e15	0.831387
$574$	0	0
$575$	7.06023e14	0.468431
$576$	0	0
$577$	−1.62921e15	−1.06050	−0.530250	−	0.847842i	$-0.677902\pi$
$577$	−1.62921e15	−1.06050	−0.530250	+	0.847842i	$0.677902\pi$
$578$	0	0
$579$	1.17753e15	0.752037
$580$	0	0
$581$	2.49726e14	0.156493
$582$	0	0
$583$	−6.57278e13	−0.0404177
$584$	0	0
$585$	−5.96006e13	−0.0359661
$586$	0	0
$587$	−2.99007e15	−1.77081	−0.885404	−	0.464822i	$-0.846118\pi$
$587$	−2.99007e15	−1.77081	−0.885404	+	0.464822i	$0.846118\pi$
$588$	0	0
$589$	8.27548e14	0.481016
$590$	0	0
$591$	−1.90317e15	−1.08579
$592$	0	0
$593$	2.26562e14	0.126878	0.0634389	−	0.997986i	$-0.479793\pi$
$593$	2.26562e14	0.126878	0.0634389	+	0.997986i	$0.479793\pi$
$594$	0	0
$595$	−1.56679e14	−0.0861324
$596$	0	0
$597$	1.22088e15	0.658893
$598$	0	0
$599$	3.50028e15	1.85462	0.927311	−	0.374293i	$-0.122114\pi$
$599$	3.50028e15	1.85462	0.927311	+	0.374293i	$0.122114\pi$
$600$	0	0
$601$	−1.20578e15	−0.627275	−0.313638	−	0.949543i	$-0.601548\pi$
$601$	−1.20578e15	−0.627275	−0.313638	+	0.949543i	$0.601548\pi$
$602$	0	0
$603$	−5.17561e14	−0.264373
$604$	0	0
$605$	−3.17548e14	−0.159278
$606$	0	0
$607$	−2.22942e15	−1.09813	−0.549067	−	0.835779i	$-0.685017\pi$
$607$	−2.22942e15	−1.09813	−0.549067	+	0.835779i	$0.685017\pi$
$608$	0	0
$609$	5.16430e14	0.249814
$610$	0	0
$611$	−2.09441e15	−0.995026
$612$	0	0
$613$	3.15369e15	1.47159	0.735795	−	0.677204i	$-0.236809\pi$
$613$	3.15369e15	1.47159	0.735795	+	0.677204i	$0.236809\pi$
$614$	0	0
$615$	−3.04665e13	−0.0139640
$616$	0	0
$617$	1.64703e15	0.741539	0.370770	−	0.928725i	$-0.379094\pi$
$617$	1.64703e15	0.741539	0.370770	+	0.928725i	$0.379094\pi$
$618$	0	0
$619$	−1.94862e14	−0.0861845	−0.0430922	−	0.999071i	$-0.513721\pi$
$619$	−1.94862e14	−0.0861845	−0.0430922	+	0.999071i	$0.513721\pi$
$620$	0	0
$621$	2.13673e14	0.0928422
$622$	0	0
$623$	1.50639e15	0.643062
$624$	0	0
$625$	2.17875e15	0.913836
$626$	0	0
$627$	1.66628e14	0.0686717
$628$	0	0
$629$	1.59322e15	0.645204
$630$	0	0
$631$	−1.38596e15	−0.551555	−0.275777	−	0.961222i	$-0.588935\pi$
$631$	−1.38596e15	−0.551555	−0.275777	+	0.961222i	$0.588935\pi$
$632$	0	0
$633$	1.31291e15	0.513469
$634$	0	0
$635$	4.41764e14	0.169799
$636$	0	0
$637$	1.38748e15	0.524154
$638$	0	0
$639$	−6.14206e14	−0.228065
$640$	0	0
$641$	−2.58508e15	−0.943529	−0.471764	−	0.881725i	$-0.656383\pi$
$641$	−2.58508e15	−0.943529	−0.471764	+	0.881725i	$0.656383\pi$
$642$	0	0
$643$	4.96300e15	1.78067	0.890337	−	0.455303i	$-0.150469\pi$
$643$	4.96300e15	1.78067	0.890337	+	0.455303i	$0.150469\pi$
$644$	0	0
$645$	−4.10470e14	−0.144778
$646$	0	0
$647$	3.12484e15	1.08356	0.541782	−	0.840519i	$-0.317750\pi$
$647$	3.12484e15	1.08356	0.541782	+	0.840519i	$0.317750\pi$
$648$	0	0
$649$	8.35937e14	0.284989
$650$	0	0
$651$	7.36423e14	0.246849
$652$	0	0
$653$	4.29342e15	1.41508	0.707540	−	0.706674i	$-0.249805\pi$
$653$	4.29342e15	1.41508	0.707540	+	0.706674i	$0.249805\pi$
$654$	0	0
$655$	−6.39199e14	−0.207161
$656$	0	0
$657$	−1.87412e15	−0.597294
$658$	0	0
$659$	−4.93625e15	−1.54713	−0.773566	−	0.633716i	$-0.781529\pi$
$659$	−4.93625e15	−1.54713	−0.773566	+	0.633716i	$0.781529\pi$
$660$	0	0
$661$	−9.32360e14	−0.287393	−0.143696	−	0.989622i	$-0.545899\pi$
$661$	−9.32360e14	−0.287393	−0.143696	+	0.989622i	$0.545899\pi$
$662$	0	0
$663$	−1.46845e15	−0.445178
$664$	0	0
$665$	−1.10975e14	−0.0330906
$666$	0	0
$667$	1.71251e15	0.502275
$668$	0	0
$669$	−6.45214e14	−0.186149
$670$	0	0
$671$	−1.02358e15	−0.290501
$672$	0	0
$673$	1.51685e15	0.423507	0.211753	−	0.977323i	$-0.432083\pi$
$673$	1.51685e15	0.423507	0.211753	+	0.977323i	$0.432083\pi$
$674$	0	0
$675$	6.80311e14	0.186869
$676$	0	0
$677$	−6.42102e15	−1.73527	−0.867634	−	0.497203i	$-0.834360\pi$
$677$	−6.42102e15	−1.73527	−0.867634	+	0.497203i	$0.834360\pi$
$678$	0	0
$679$	5.68870e14	0.151262
$680$	0	0
$681$	−2.45252e15	−0.641657
$682$	0	0
$683$	−3.67035e15	−0.944916	−0.472458	−	0.881353i	$-0.656633\pi$
$683$	−3.67035e15	−0.944916	−0.472458	+	0.881353i	$0.656633\pi$
$684$	0	0
$685$	−4.30850e14	−0.109151
$686$	0	0
$687$	−2.12689e15	−0.530253
$688$	0	0
$689$	4.10272e14	0.100662
$690$	0	0
$691$	2.15581e15	0.520573	0.260286	−	0.965531i	$-0.416183\pi$
$691$	2.15581e15	0.520573	0.260286	+	0.965531i	$0.416183\pi$
$692$	0	0
$693$	1.48280e14	0.0352411
$694$	0	0
$695$	8.56745e14	0.200417
$696$	0	0
$697$	−7.50636e14	−0.172842
$698$	0	0
$699$	7.99508e14	0.181217
$700$	0	0
$701$	−6.84907e15	−1.52821	−0.764104	−	0.645094i	$-0.776818\pi$
$701$	−6.84907e15	−1.52821	−0.764104	+	0.645094i	$0.776818\pi$
$702$	0	0
$703$	1.12847e15	0.247876
$704$	0	0
$705$	−7.14041e14	−0.154413
$706$	0	0
$707$	2.98357e15	0.635227
$708$	0	0
$709$	−9.12308e14	−0.191244	−0.0956218	−	0.995418i	$-0.530484\pi$
$709$	−9.12308e14	−0.191244	−0.0956218	+	0.995418i	$0.530484\pi$
$710$	0	0
$711$	−2.35488e15	−0.486055
$712$	0	0
$713$	2.44202e15	0.496314
$714$	0	0
$715$	−1.37153e14	−0.0274488
$716$	0	0
$717$	−4.21966e15	−0.831615
$718$	0	0
$719$	−3.99411e15	−0.775195	−0.387597	−	0.921829i	$-0.626695\pi$
$719$	−3.99411e15	−0.775195	−0.387597	+	0.921829i	$0.626695\pi$
$720$	0	0
$721$	−9.58364e14	−0.183184
$722$	0	0
$723$	4.58908e15	0.863903
$724$	0	0
$725$	5.45245e15	1.01096
$726$	0	0
$727$	3.31749e15	0.605858	0.302929	−	0.953013i	$-0.402036\pi$
$727$	3.31749e15	0.605858	0.302929	+	0.953013i	$0.402036\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−1.01132e16	−1.79202
$732$	0	0
$733$	−2.60662e15	−0.454994	−0.227497	−	0.973779i	$-0.573054\pi$
$733$	−2.60662e15	−0.454994	−0.227497	+	0.973779i	$0.573054\pi$
$734$	0	0
$735$	4.73029e14	0.0813406
$736$	0	0
$737$	−1.19102e15	−0.201765
$738$	0	0
$739$	7.46259e15	1.24550	0.622752	−	0.782419i	$-0.286015\pi$
$739$	7.46259e15	1.24550	0.622752	+	0.782419i	$0.286015\pi$
$740$	0	0
$741$	−1.04009e15	−0.171030
$742$	0	0
$743$	1.12334e16	1.82001	0.910003	−	0.414602i	$-0.136079\pi$
$743$	1.12334e16	1.82001	0.910003	+	0.414602i	$0.136079\pi$
$744$	0	0
$745$	1.41618e15	0.226078
$746$	0	0
$747$	7.97948e14	0.125520
$748$	0	0
$749$	−2.23258e15	−0.346064
$750$	0	0
$751$	−4.15071e15	−0.634020	−0.317010	−	0.948422i	$-0.602679\pi$
$751$	−4.15071e15	−0.634020	−0.317010	+	0.948422i	$0.602679\pi$
$752$	0	0
$753$	9.07565e14	0.136617
$754$	0	0
$755$	1.79344e15	0.266059
$756$	0	0
$757$	−9.93212e15	−1.45216	−0.726080	−	0.687610i	$-0.758660\pi$
$757$	−9.93212e15	−1.45216	−0.726080	+	0.687610i	$0.758660\pi$
$758$	0	0
$759$	4.91705e14	0.0708558
$760$	0	0
$761$	1.37276e16	1.94974	0.974872	−	0.222765i	$-0.0715082\pi$
$761$	1.37276e16	1.94974	0.974872	+	0.222765i	$0.0715082\pi$
$762$	0	0
$763$	7.86140e14	0.110056
$764$	0	0
$765$	−5.00634e14	−0.0690848
$766$	0	0
$767$	−5.21791e15	−0.709777
$768$	0	0
$769$	2.85267e15	0.382522	0.191261	−	0.981539i	$-0.438742\pi$
$769$	2.85267e15	0.382522	0.191261	+	0.981539i	$0.438742\pi$
$770$	0	0
$771$	6.21150e15	0.821103
$772$	0	0
$773$	5.44730e15	0.709895	0.354948	−	0.934886i	$-0.384499\pi$
$773$	5.44730e15	0.709895	0.354948	+	0.934886i	$0.384499\pi$
$774$	0	0
$775$	7.77512e15	0.998960
$776$	0	0
$777$	1.00421e15	0.127206
$778$	0	0
$779$	−5.31671e14	−0.0664029
$780$	0	0
$781$	−1.41341e15	−0.174056
$782$	0	0
$783$	1.65014e15	0.200370
$784$	0	0
$785$	1.81911e15	0.217809
$786$	0	0
$787$	3.33904e15	0.394240	0.197120	−	0.980379i	$-0.436841\pi$
$787$	3.33904e15	0.394240	0.197120	+	0.980379i	$0.436841\pi$
$788$	0	0
$789$	−9.03902e14	−0.105244
$790$	0	0
$791$	6.01143e14	0.0690251
$792$	0	0
$793$	6.38916e15	0.723504
$794$	0	0
$795$	1.39873e14	0.0156212
$796$	0	0
$797$	3.31632e15	0.365289	0.182644	−	0.983179i	$-0.441534\pi$
$797$	3.31632e15	0.365289	0.182644	+	0.983179i	$0.441534\pi$
$798$	0	0
$799$	−1.75926e16	−1.91128
$800$	0	0
$801$	4.81335e15	0.515785
$802$	0	0
$803$	−4.31274e15	−0.455846
$804$	0	0
$805$	−3.27476e14	−0.0341430
$806$	0	0
$807$	5.65495e15	0.581599
$808$	0	0
$809$	8.29315e15	0.841400	0.420700	−	0.907200i	$-0.361785\pi$
$809$	8.29315e15	0.841400	0.420700	+	0.907200i	$0.361785\pi$
$810$	0	0
$811$	−1.67813e16	−1.67962	−0.839811	−	0.542879i	$-0.817334\pi$
$811$	−1.67813e16	−1.67962	−0.839811	+	0.542879i	$0.817334\pi$
$812$	0	0
$813$	4.12177e14	0.0406992
$814$	0	0
$815$	1.82897e15	0.178172
$816$	0	0
$817$	−7.16312e15	−0.688464
$818$	0	0
$819$	−9.25562e14	−0.0877696
$820$	0	0
$821$	−2.03242e15	−0.190163	−0.0950813	−	0.995470i	$-0.530311\pi$
$821$	−2.03242e15	−0.190163	−0.0950813	+	0.995470i	$0.530311\pi$
$822$	0	0
$823$	−5.59910e15	−0.516915	−0.258457	−	0.966023i	$-0.583214\pi$
$823$	−5.59910e15	−0.516915	−0.258457	+	0.966023i	$0.583214\pi$
$824$	0	0
$825$	1.56554e15	0.142615
$826$	0	0
$827$	−2.09682e16	−1.88486	−0.942432	−	0.334399i	$-0.891467\pi$
$827$	−2.09682e16	−1.88486	−0.942432	+	0.334399i	$0.891467\pi$
$828$	0	0
$829$	−1.75589e16	−1.55757	−0.778783	−	0.627293i	$-0.784163\pi$
$829$	−1.75589e16	−1.55757	−0.778783	+	0.627293i	$0.784163\pi$
$830$	0	0
$831$	−7.82646e15	−0.685109
$832$	0	0
$833$	1.16545e16	1.00681
$834$	0	0
$835$	−1.14791e15	−0.0978659
$836$	0	0
$837$	2.35309e15	0.197992
$838$	0	0
$839$	6.17140e14	0.0512499	0.0256249	−	0.999672i	$-0.491842\pi$
$839$	6.17140e14	0.0512499	0.0256249	+	0.999672i	$0.491842\pi$
$840$	0	0
$841$	1.02480e15	0.0839965
$842$	0	0
$843$	5.33949e15	0.431965
$844$	0	0
$845$	−1.27656e15	−0.101937
$846$	0	0
$847$	−4.93134e15	−0.388692
$848$	0	0
$849$	−1.52246e15	−0.118455
$850$	0	0
$851$	3.33001e15	0.255760
$852$	0	0
$853$	7.46292e15	0.565834	0.282917	−	0.959144i	$-0.408698\pi$
$853$	7.46292e15	0.565834	0.282917	+	0.959144i	$0.408698\pi$
$854$	0	0
$855$	−3.54596e14	−0.0265412
$856$	0	0
$857$	−2.16990e16	−1.60341	−0.801707	−	0.597717i	$-0.796075\pi$
$857$	−2.16990e16	−1.60341	−0.801707	+	0.597717i	$0.796075\pi$
$858$	0	0
$859$	−5.63489e15	−0.411077	−0.205538	−	0.978649i	$-0.565895\pi$
$859$	−5.63489e15	−0.411077	−0.205538	+	0.978649i	$0.565895\pi$
$860$	0	0
$861$	−4.73126e14	−0.0340769
$862$	0	0
$863$	8.74062e15	0.621560	0.310780	−	0.950482i	$-0.399410\pi$
$863$	8.74062e15	0.621560	0.310780	+	0.950482i	$0.399410\pi$
$864$	0	0
$865$	2.29686e15	0.161267
$866$	0	0
$867$	−4.00661e15	−0.277762
$868$	0	0
$869$	−5.41906e15	−0.370950
$870$	0	0
$871$	7.43431e15	0.502505
$872$	0	0
$873$	1.81771e15	0.121324
$874$	0	0
$875$	−2.11644e15	−0.139496
$876$	0	0
$877$	5.97195e15	0.388704	0.194352	−	0.980932i	$-0.437740\pi$
$877$	5.97195e15	0.388704	0.194352	+	0.980932i	$0.437740\pi$
$878$	0	0
$879$	−1.04603e15	−0.0672366
$880$	0	0
$881$	3.75775e15	0.238540	0.119270	−	0.992862i	$-0.461945\pi$
$881$	3.75775e15	0.238540	0.119270	+	0.992862i	$0.461945\pi$
$882$	0	0
$883$	1.34954e16	0.846061	0.423030	−	0.906115i	$-0.360966\pi$
$883$	1.34954e16	0.846061	0.423030	+	0.906115i	$0.360966\pi$
$884$	0	0
$885$	−1.77893e15	−0.110146
$886$	0	0
$887$	3.17242e16	1.94004	0.970020	−	0.243025i	$-0.0781398\pi$
$887$	3.17242e16	1.94004	0.970020	+	0.243025i	$0.0781398\pi$
$888$	0	0
$889$	6.86033e15	0.414367
$890$	0	0
$891$	4.73798e14	0.0282661
$892$	0	0
$893$	−1.24608e16	−0.734279
$894$	0	0
$895$	−2.03364e15	−0.118371
$896$	0	0
$897$	−3.06922e15	−0.176469
$898$	0	0
$899$	1.88591e16	1.07113
$900$	0	0
$901$	3.44621e15	0.193355
$902$	0	0
$903$	−6.37436e15	−0.353308
$904$	0	0
$905$	1.55824e14	0.00853234
$906$	0	0
$907$	1.10011e16	0.595110	0.297555	−	0.954705i	$-0.403829\pi$
$907$	1.10011e16	0.595110	0.297555	+	0.954705i	$0.403829\pi$
$908$	0	0
$909$	9.53337e15	0.509501
$910$	0	0
$911$	−2.68524e16	−1.41786	−0.708928	−	0.705281i	$-0.750821\pi$
$911$	−2.68524e16	−1.41786	−0.708928	+	0.705281i	$0.750821\pi$
$912$	0	0
$913$	1.83624e15	0.0957946
$914$	0	0
$915$	2.17824e15	0.112277
$916$	0	0
$917$	−9.92638e15	−0.505545
$918$	0	0
$919$	3.69307e16	1.85846	0.929228	−	0.369506i	$-0.120473\pi$
$919$	3.69307e16	1.85846	0.929228	+	0.369506i	$0.120473\pi$
$920$	0	0
$921$	−3.64120e15	−0.181058
$922$	0	0
$923$	8.82252e15	0.433494
$924$	0	0
$925$	1.06024e16	0.514783
$926$	0	0
$927$	−3.06225e15	−0.146927
$928$	0	0
$929$	−1.35447e16	−0.642221	−0.321110	−	0.947042i	$-0.604056\pi$
$929$	−1.35447e16	−0.642221	−0.321110	+	0.947042i	$0.604056\pi$
$930$	0	0
$931$	8.25485e15	0.386799
$932$	0	0
$933$	−4.68517e15	−0.216958
$934$	0	0
$935$	−1.15206e15	−0.0527245
$936$	0	0
$937$	−6.01961e15	−0.272270	−0.136135	−	0.990690i	$-0.543468\pi$
$937$	−6.01961e15	−0.272270	−0.136135	+	0.990690i	$0.543468\pi$
$938$	0	0
$939$	−2.28166e16	−1.01998
$940$	0	0
$941$	8.63146e15	0.381365	0.190683	−	0.981652i	$-0.438930\pi$
$941$	8.63146e15	0.381365	0.190683	+	0.981652i	$0.438930\pi$
$942$	0	0
$943$	−1.56891e15	−0.0685149
$944$	0	0
$945$	−3.15550e14	−0.0136205
$946$	0	0
$947$	3.74754e16	1.59890	0.799450	−	0.600733i	$-0.205124\pi$
$947$	3.74754e16	1.59890	0.799450	+	0.600733i	$0.205124\pi$
$948$	0	0
$949$	2.69201e16	1.13530
$950$	0	0
$951$	4.25961e15	0.177573
$952$	0	0
$953$	−3.58712e16	−1.47821	−0.739104	−	0.673592i	$-0.764751\pi$
$953$	−3.58712e16	−1.47821	−0.739104	+	0.673592i	$0.764751\pi$
$954$	0	0
$955$	−6.01997e15	−0.245231
$956$	0	0
$957$	3.79732e15	0.152919
$958$	0	0
$959$	−6.69085e15	−0.266366
$960$	0	0
$961$	1.48442e15	0.0584224
$962$	0	0
$963$	−7.13373e15	−0.277570
$964$	0	0
$965$	−5.76650e15	−0.221826
$966$	0	0
$967$	−1.66314e16	−0.632534	−0.316267	−	0.948670i	$-0.602429\pi$
$967$	−1.66314e16	−0.632534	−0.316267	+	0.948670i	$0.602429\pi$
$968$	0	0
$969$	−8.73658e15	−0.328519
$970$	0	0
$971$	−3.14015e15	−0.116747	−0.0583734	−	0.998295i	$-0.518591\pi$
$971$	−3.14015e15	−0.116747	−0.0583734	+	0.998295i	$0.518591\pi$
$972$	0	0
$973$	1.33047e16	0.489087
$974$	0	0
$975$	−9.77205e15	−0.355190
$976$	0	0
$977$	−2.43733e16	−0.875980	−0.437990	−	0.898980i	$-0.644310\pi$
$977$	−2.43733e16	−0.875980	−0.437990	+	0.898980i	$0.644310\pi$
$978$	0	0
$979$	1.10765e16	0.393639
$980$	0	0
$981$	2.51195e15	0.0882736
$982$	0	0
$983$	−1.37201e16	−0.476775	−0.238387	−	0.971170i	$-0.576619\pi$
$983$	−1.37201e16	−0.476775	−0.238387	+	0.971170i	$0.576619\pi$
$984$	0	0
$985$	9.32004e15	0.320272
$986$	0	0
$987$	−1.10886e16	−0.376820
$988$	0	0
$989$	−2.11377e16	−0.710360
$990$	0	0
$991$	−2.95853e16	−0.983267	−0.491633	−	0.870802i	$-0.663600\pi$
$991$	−2.95853e16	−0.983267	−0.491633	+	0.870802i	$0.663600\pi$
$992$	0	0
$993$	4.90321e15	0.161161
$994$	0	0
$995$	−5.97880e15	−0.194351
$996$	0	0
$997$	−1.90504e16	−0.612465	−0.306233	−	0.951957i	$-0.599069\pi$
$997$	−1.90504e16	−0.612465	−0.306233	+	0.951957i	$0.599069\pi$
$998$	0	0
$999$	3.20873e15	0.102029

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.12.a.b.1.1	✓	1
3.2	odd	2		72.12.a.b.1.1		1
4.3	odd	2		48.12.a.g.1.1		1
8.3	odd	2		192.12.a.f.1.1		1
8.5	even	2		192.12.a.p.1.1		1
12.11	even	2		144.12.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.12.a.b.1.1	✓	1	1.1	even	1	trivial
48.12.a.g.1.1		1	4.3	odd	2
72.12.a.b.1.1		1	3.2	odd	2
144.12.a.h.1.1		1	12.11	even	2
192.12.a.f.1.1		1	8.3	odd	2
192.12.a.p.1.1		1	8.5	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:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	24.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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