LMFDB - Embedded newform 48.12.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,12,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, 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("48.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	48.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:	$36.8804726669$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+243.000 q^{3} -7130.00 q^{5} +19536.0 q^{7} +59049.0 q^{9} +O(q^{10})$

$q+243.000 q^{3} -7130.00 q^{5} +19536.0 q^{7} +59049.0 q^{9} +196148. q^{11} +361414. q^{13} -1.73259e6 q^{15} -130942. q^{17} -1.85167e7 q^{19} +4.74725e6 q^{21} -2.15609e7 q^{23} +2.00878e6 q^{25} +1.43489e7 q^{27} +1.91664e8 q^{29} -2.07934e8 q^{31} +4.76640e7 q^{33} -1.39292e8 q^{35} -2.00785e8 q^{37} +8.78236e7 q^{39} -1.43526e9 q^{41} -7.12703e8 q^{43} -4.21019e8 q^{45} +4.96082e8 q^{47} -1.59567e9 q^{49} -3.18189e7 q^{51} -3.35011e9 q^{53} -1.39854e9 q^{55} -4.49956e9 q^{57} -4.58322e9 q^{59} +3.42750e9 q^{61} +1.15358e9 q^{63} -2.57688e9 q^{65} -1.70794e10 q^{67} -5.23929e9 q^{69} +7.91508e9 q^{71} +3.15597e10 q^{73} +4.88132e8 q^{75} +3.83195e9 q^{77} -4.10236e10 q^{79} +3.48678e9 q^{81} +1.99747e10 q^{83} +9.33616e8 q^{85} +4.65743e10 q^{87} -1.06402e10 q^{89} +7.06058e9 q^{91} -5.05279e10 q^{93} +1.32024e11 q^{95} +6.44111e9 q^{97} +1.15823e10 q^{99} +O(q^{100})$

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
$5$	−7130.00	−1.02036	−0.510181	−	0.860067i	$-0.670422\pi$
$5$	−7130.00	−1.02036	−0.510181	+	0.860067i	$0.670422\pi$
$6$	0	0
$7$	19536.0	0.439336	0.219668	−	0.975575i	$-0.429503\pi$
$7$	19536.0	0.439336	0.219668	+	0.975575i	$0.429503\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	196148.	0.367218	0.183609	−	0.982999i	$-0.441222\pi$
$11$	196148.	0.367218	0.183609	+	0.982999i	$0.441222\pi$
$12$	0	0
$13$	361414.	0.269971	0.134985	−	0.990848i	$-0.456901\pi$
$13$	361414.	0.269971	0.134985	+	0.990848i	$0.456901\pi$
$14$	0	0
$15$	−1.73259e6	−0.589107
$16$	0	0
$17$	−130942.	−0.0223671	−0.0111836	−	0.999937i	$-0.503560\pi$
$17$	−130942.	−0.0223671	−0.0111836	+	0.999937i	$0.503560\pi$
$18$	0	0
$19$	−1.85167e7	−1.71561	−0.857805	−	0.513975i	$-0.828172\pi$
$19$	−1.85167e7	−1.71561	−0.857805	+	0.513975i	$0.828172\pi$
$20$	0	0
$21$	4.74725e6	0.253651
$22$	0	0
$23$	−2.15609e7	−0.698495	−0.349247	−	0.937031i	$-0.613563\pi$
$23$	−2.15609e7	−0.698495	−0.349247	+	0.937031i	$0.613563\pi$
$24$	0	0
$25$	2.00878e6	0.0411397
$26$	0	0
$27$	1.43489e7	0.192450
$28$	0	0
$29$	1.91664e8	1.73521	0.867603	−	0.497258i	$-0.165660\pi$
$29$	1.91664e8	1.73521	0.867603	+	0.497258i	$0.165660\pi$
$30$	0	0
$31$	−2.07934e8	−1.30448	−0.652238	−	0.758015i	$-0.726170\pi$
$31$	−2.07934e8	−1.30448	−0.652238	+	0.758015i	$0.726170\pi$
$32$	0	0
$33$	4.76640e7	0.212013
$34$	0	0
$35$	−1.39292e8	−0.448282
$36$	0	0
$37$	−2.00785e8	−0.476016	−0.238008	−	0.971263i	$-0.576494\pi$
$37$	−2.00785e8	−0.476016	−0.238008	+	0.971263i	$0.576494\pi$
$38$	0	0
$39$	8.78236e7	0.155868
$40$	0	0
$41$	−1.43526e9	−1.93472	−0.967360	−	0.253404i	$-0.918450\pi$
$41$	−1.43526e9	−1.93472	−0.967360	+	0.253404i	$0.918450\pi$
$42$	0	0
$43$	−7.12703e8	−0.739319	−0.369660	−	0.929167i	$-0.620526\pi$
$43$	−7.12703e8	−0.739319	−0.369660	+	0.929167i	$0.620526\pi$
$44$	0	0
$45$	−4.21019e8	−0.340121
$46$	0	0
$47$	4.96082e8	0.315512	0.157756	−	0.987478i	$-0.449574\pi$
$47$	4.96082e8	0.315512	0.157756	+	0.987478i	$0.449574\pi$
$48$	0	0
$49$	−1.59567e9	−0.806984
$50$	0	0
$51$	−3.18189e7	−0.0129137
$52$	0	0
$53$	−3.35011e9	−1.10038	−0.550190	−	0.835040i	$-0.685445\pi$
$53$	−3.35011e9	−1.10038	−0.550190	+	0.835040i	$0.685445\pi$
$54$	0	0
$55$	−1.39854e9	−0.374696
$56$	0	0
$57$	−4.49956e9	−0.990508
$58$	0	0
$59$	−4.58322e9	−0.834613	−0.417306	−	0.908766i	$-0.637026\pi$
$59$	−4.58322e9	−0.834613	−0.417306	+	0.908766i	$0.637026\pi$
$60$	0	0
$61$	3.42750e9	0.519593	0.259797	−	0.965663i	$-0.416344\pi$
$61$	3.42750e9	0.519593	0.259797	+	0.965663i	$0.416344\pi$
$62$	0	0
$63$	1.15358e9	0.146445
$64$	0	0
$65$	−2.57688e9	−0.275468
$66$	0	0
$67$	−1.70794e10	−1.54547	−0.772735	−	0.634729i	$-0.781112\pi$
$67$	−1.70794e10	−1.54547	−0.772735	+	0.634729i	$0.781112\pi$
$68$	0	0
$69$	−5.23929e9	−0.403276
$70$	0	0
$71$	7.91508e9	0.520636	0.260318	−	0.965523i	$-0.416173\pi$
$71$	7.91508e9	0.520636	0.260318	+	0.965523i	$0.416173\pi$
$72$	0	0
$73$	3.15597e10	1.78179	0.890895	−	0.454209i	$-0.150078\pi$
$73$	3.15597e10	1.78179	0.890895	+	0.454209i	$0.150078\pi$
$74$	0	0
$75$	4.88132e8	0.0237520
$76$	0	0
$77$	3.83195e9	0.161332
$78$	0	0
$79$	−4.10236e10	−1.49998	−0.749988	−	0.661451i	$-0.769941\pi$
$79$	−4.10236e10	−1.49998	−0.749988	+	0.661451i	$0.769941\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	1.99747e10	0.556609	0.278304	−	0.960493i	$-0.410228\pi$
$83$	1.99747e10	0.556609	0.278304	+	0.960493i	$0.410228\pi$
$84$	0	0
$85$	9.33616e8	0.0228226
$86$	0	0
$87$	4.65743e10	1.00182
$88$	0	0
$89$	−1.06402e10	−0.201978	−0.100989	−	0.994888i	$-0.532201\pi$
$89$	−1.06402e10	−0.201978	−0.100989	+	0.994888i	$0.532201\pi$
$90$	0	0
$91$	7.06058e9	0.118608
$92$	0	0
$93$	−5.05279e10	−0.753139
$94$	0	0
$95$	1.32024e11	1.75054
$96$	0	0
$97$	6.44111e9	0.0761581	0.0380790	−	0.999275i	$-0.487876\pi$
$97$	6.44111e9	0.0761581	0.0380790	+	0.999275i	$0.487876\pi$
$98$	0	0
$99$	1.15823e10	0.122406
$100$	0	0
$101$	1.06774e11	1.01088	0.505439	−	0.862863i	$-0.331331\pi$
$101$	1.06774e11	1.01088	0.505439	+	0.862863i	$0.331331\pi$
$102$	0	0
$103$	2.79038e10	0.237169	0.118585	−	0.992944i	$-0.462164\pi$
$103$	2.79038e10	0.237169	0.118585	+	0.992944i	$0.462164\pi$
$104$	0	0
$105$	−3.38479e10	−0.258816
$106$	0	0
$107$	1.49670e11	1.03163	0.515817	−	0.856699i	$-0.327489\pi$
$107$	1.49670e11	1.03163	0.515817	+	0.856699i	$0.327489\pi$
$108$	0	0
$109$	2.05730e11	1.28072	0.640358	−	0.768077i	$-0.278786\pi$
$109$	2.05730e11	1.28072	0.640358	+	0.768077i	$0.278786\pi$
$110$	0	0
$111$	−4.87907e10	−0.274828
$112$	0	0
$113$	6.03547e8	0.00308163	0.00154081	−	0.999999i	$-0.499510\pi$
$113$	6.03547e8	0.00308163	0.00154081	+	0.999999i	$0.499510\pi$
$114$	0	0
$115$	1.53729e11	0.712718
$116$	0	0
$117$	2.13411e10	0.0899902
$118$	0	0
$119$	−2.55808e9	−0.00982667
$120$	0	0
$121$	−2.46838e11	−0.865151
$122$	0	0
$123$	−3.48767e11	−1.11701
$124$	0	0
$125$	3.33822e11	0.978385
$126$	0	0
$127$	−6.94981e11	−1.86661	−0.933303	−	0.359090i	$-0.883087\pi$
$127$	−6.94981e11	−1.86661	−0.933303	+	0.359090i	$0.883087\pi$
$128$	0	0
$129$	−1.73187e11	−0.426846
$130$	0	0
$131$	−5.01509e11	−1.13576	−0.567880	−	0.823111i	$-0.692236\pi$
$131$	−5.01509e11	−1.13576	−0.567880	+	0.823111i	$0.692236\pi$
$132$	0	0
$133$	−3.61742e11	−0.753729
$134$	0	0
$135$	−1.02308e11	−0.196369
$136$	0	0
$137$	3.96827e11	0.702486	0.351243	−	0.936284i	$-0.385759\pi$
$137$	3.96827e11	0.702486	0.351243	+	0.936284i	$0.385759\pi$
$138$	0	0
$139$	5.87790e11	0.960817	0.480409	−	0.877045i	$-0.340488\pi$
$139$	5.87790e11	0.960817	0.480409	+	0.877045i	$0.340488\pi$
$140$	0	0
$141$	1.20548e11	0.182161
$142$	0	0
$143$	7.08906e10	0.0991381
$144$	0	0
$145$	−1.36656e12	−1.77054
$146$	0	0
$147$	−3.87748e11	−0.465913
$148$	0	0
$149$	5.49939e11	0.613466	0.306733	−	0.951796i	$-0.400764\pi$
$149$	5.49939e11	0.613466	0.306733	+	0.951796i	$0.400764\pi$
$150$	0	0
$151$	−1.09467e10	−0.0113478	−0.00567390	−	0.999984i	$-0.501806\pi$
$151$	−1.09467e10	−0.0113478	−0.00567390	+	0.999984i	$0.501806\pi$
$152$	0	0
$153$	−7.73199e9	−0.00745570
$154$	0	0
$155$	1.48257e12	1.33104
$156$	0	0
$157$	−1.98667e12	−1.66218	−0.831088	−	0.556141i	$-0.812282\pi$
$157$	−1.98667e12	−1.66218	−0.831088	+	0.556141i	$0.812282\pi$
$158$	0	0
$159$	−8.14078e11	−0.635304
$160$	0	0
$161$	−4.21213e11	−0.306874
$162$	0	0
$163$	−5.08774e11	−0.346332	−0.173166	−	0.984893i	$-0.555400\pi$
$163$	−5.08774e11	−0.346332	−0.173166	+	0.984893i	$0.555400\pi$
$164$	0	0
$165$	−3.39844e11	−0.216331
$166$	0	0
$167$	1.66583e12	0.992405	0.496203	−	0.868207i	$-0.334727\pi$
$167$	1.66583e12	0.992405	0.496203	+	0.868207i	$0.334727\pi$
$168$	0	0
$169$	−1.66154e12	−0.927116
$170$	0	0
$171$	−1.09339e12	−0.571870
$172$	0	0
$173$	2.93849e12	1.44169	0.720843	−	0.693098i	$-0.243755\pi$
$173$	2.93849e12	1.44169	0.720843	+	0.693098i	$0.243755\pi$
$174$	0	0
$175$	3.92434e10	0.0180741
$176$	0	0
$177$	−1.11372e12	−0.481864
$178$	0	0
$179$	−8.48882e11	−0.345267	−0.172634	−	0.984986i	$-0.555228\pi$
$179$	−8.48882e11	−0.345267	−0.172634	+	0.984986i	$0.555228\pi$
$180$	0	0
$181$	5.03014e12	1.92463	0.962317	−	0.271932i	$-0.0876625\pi$
$181$	5.03014e12	1.92463	0.962317	+	0.271932i	$0.0876625\pi$
$182$	0	0
$183$	8.32883e11	0.299987
$184$	0	0
$185$	1.43160e12	0.485709
$186$	0	0
$187$	−2.56840e10	−0.00821361
$188$	0	0
$189$	2.80320e11	0.0845502
$190$	0	0
$191$	1.40796e12	0.400781	0.200390	−	0.979716i	$-0.435779\pi$
$191$	1.40796e12	0.400781	0.200390	+	0.979716i	$0.435779\pi$
$192$	0	0
$193$	−1.94985e11	−0.0524127	−0.0262063	−	0.999657i	$-0.508343\pi$
$193$	−1.94985e11	−0.0524127	−0.0262063	+	0.999657i	$0.508343\pi$
$194$	0	0
$195$	−6.26182e11	−0.159041
$196$	0	0
$197$	5.16140e12	1.23938	0.619688	−	0.784848i	$-0.287259\pi$
$197$	5.16140e12	1.23938	0.619688	+	0.784848i	$0.287259\pi$
$198$	0	0
$199$	1.02078e12	0.231867	0.115934	−	0.993257i	$-0.463014\pi$
$199$	1.02078e12	0.231867	0.115934	+	0.993257i	$0.463014\pi$
$200$	0	0
$201$	−4.15029e12	−0.892278
$202$	0	0
$203$	3.74434e12	0.762338
$204$	0	0
$205$	1.02334e13	1.97412
$206$	0	0
$207$	−1.27315e12	−0.232832
$208$	0	0
$209$	−3.63201e12	−0.630003
$210$	0	0
$211$	−9.12166e12	−1.50148	−0.750742	−	0.660596i	$-0.770304\pi$
$211$	−9.12166e12	−1.50148	−0.750742	+	0.660596i	$0.770304\pi$
$212$	0	0
$213$	1.92336e12	0.300589
$214$	0	0
$215$	5.08157e12	0.754374
$216$	0	0
$217$	−4.06219e12	−0.573102
$218$	0	0
$219$	7.66900e12	1.02872
$220$	0	0
$221$	−4.73243e10	−0.00603846
$222$	0	0
$223$	4.68108e11	0.0568419	0.0284210	−	0.999596i	$-0.490952\pi$
$223$	4.68108e11	0.0568419	0.0284210	+	0.999596i	$0.490952\pi$
$224$	0	0
$225$	1.18616e11	0.0137132
$226$	0	0
$227$	1.32196e13	1.45571	0.727857	−	0.685729i	$-0.240517\pi$
$227$	1.32196e13	1.45571	0.727857	+	0.685729i	$0.240517\pi$
$228$	0	0
$229$	−8.47551e11	−0.0889346	−0.0444673	−	0.999011i	$-0.514159\pi$
$229$	−8.47551e11	−0.0889346	−0.0444673	+	0.999011i	$0.514159\pi$
$230$	0	0
$231$	9.31163e11	0.0931451
$232$	0	0
$233$	−1.77637e13	−1.69464	−0.847318	−	0.531086i	$-0.821784\pi$
$233$	−1.77637e13	−1.69464	−0.847318	+	0.531086i	$0.821784\pi$
$234$	0	0
$235$	−3.53707e12	−0.321936
$236$	0	0
$237$	−9.96873e12	−0.866012
$238$	0	0
$239$	−1.10633e13	−0.917690	−0.458845	−	0.888516i	$-0.651737\pi$
$239$	−1.10633e13	−0.917690	−0.458845	+	0.888516i	$0.651737\pi$
$240$	0	0
$241$	−1.31452e13	−1.04153	−0.520765	−	0.853700i	$-0.674353\pi$
$241$	−1.31452e13	−1.04153	−0.520765	+	0.853700i	$0.674353\pi$
$242$	0	0
$243$	8.47289e11	0.0641500
$244$	0	0
$245$	1.13771e13	0.823416
$246$	0	0
$247$	−6.69219e12	−0.463164
$248$	0	0
$249$	4.85385e12	0.321358
$250$	0	0
$251$	−2.49029e13	−1.57777	−0.788887	−	0.614539i	$-0.789342\pi$
$251$	−2.49029e13	−1.57777	−0.788887	+	0.614539i	$0.789342\pi$
$252$	0	0
$253$	−4.22912e12	−0.256500
$254$	0	0
$255$	2.26869e11	0.0131766
$256$	0	0
$257$	−1.90572e13	−1.06030	−0.530148	−	0.847905i	$-0.677864\pi$
$257$	−1.90572e13	−1.06030	−0.530148	+	0.847905i	$0.677864\pi$
$258$	0	0
$259$	−3.92253e12	−0.209131
$260$	0	0
$261$	1.13176e13	0.578402
$262$	0	0
$263$	−2.58113e13	−1.26489	−0.632447	−	0.774604i	$-0.717949\pi$
$263$	−2.58113e13	−1.26489	−0.632447	+	0.774604i	$0.717949\pi$
$264$	0	0
$265$	2.38863e13	1.12279
$266$	0	0
$267$	−2.58556e12	−0.116612
$268$	0	0
$269$	−1.60114e13	−0.693093	−0.346547	−	0.938033i	$-0.612646\pi$
$269$	−1.60114e13	−0.693093	−0.346547	+	0.938033i	$0.612646\pi$
$270$	0	0
$271$	2.14306e13	0.890644	0.445322	−	0.895371i	$-0.353089\pi$
$271$	2.14306e13	0.890644	0.445322	+	0.895371i	$0.353089\pi$
$272$	0	0
$273$	1.71572e12	0.0684782
$274$	0	0
$275$	3.94017e11	0.0151072
$276$	0	0
$277$	2.25644e13	0.831352	0.415676	−	0.909513i	$-0.363545\pi$
$277$	2.25644e13	0.831352	0.415676	+	0.909513i	$0.363545\pi$
$278$	0	0
$279$	−1.22783e13	−0.434825
$280$	0	0
$281$	1.45643e13	0.495912	0.247956	−	0.968771i	$-0.420241\pi$
$281$	1.45643e13	0.495912	0.247956	+	0.968771i	$0.420241\pi$
$282$	0	0
$283$	2.43495e13	0.797380	0.398690	−	0.917086i	$-0.369465\pi$
$283$	2.43495e13	0.797380	0.398690	+	0.917086i	$0.369465\pi$
$284$	0	0
$285$	3.20818e13	1.01068
$286$	0	0
$287$	−2.80392e13	−0.849992
$288$	0	0
$289$	−3.42548e13	−0.999500
$290$	0	0
$291$	1.56519e12	0.0439699
$292$	0	0
$293$	−4.32740e11	−0.0117073	−0.00585363	−	0.999983i	$-0.501863\pi$
$293$	−4.32740e11	−0.0117073	−0.00585363	+	0.999983i	$0.501863\pi$
$294$	0	0
$295$	3.26784e13	0.851607
$296$	0	0
$297$	2.81451e12	0.0706712
$298$	0	0
$299$	−7.79240e12	−0.188573
$300$	0	0
$301$	−1.39234e13	−0.324809
$302$	0	0
$303$	2.59461e13	0.583630
$304$	0	0
$305$	−2.44381e13	−0.530174
$306$	0	0
$307$	5.89356e13	1.23344	0.616718	−	0.787185i	$-0.288462\pi$
$307$	5.89356e13	1.23344	0.616718	+	0.787185i	$0.288462\pi$
$308$	0	0
$309$	6.78062e12	0.136930
$310$	0	0
$311$	−1.29359e13	−0.252125	−0.126062	−	0.992022i	$-0.540234\pi$
$311$	−1.29359e13	−0.252125	−0.126062	+	0.992022i	$0.540234\pi$
$312$	0	0
$313$	−2.56949e13	−0.483452	−0.241726	−	0.970345i	$-0.577714\pi$
$313$	−2.56949e13	−0.483452	−0.241726	+	0.970345i	$0.577714\pi$
$314$	0	0
$315$	−8.22503e12	−0.149427
$316$	0	0
$317$	1.33665e13	0.234527	0.117263	−	0.993101i	$-0.462588\pi$
$317$	1.33665e13	0.234527	0.117263	+	0.993101i	$0.462588\pi$
$318$	0	0
$319$	3.75945e13	0.637199
$320$	0	0
$321$	3.63699e13	0.595614
$322$	0	0
$323$	2.42461e12	0.0383732
$324$	0	0
$325$	7.25999e11	0.0111065
$326$	0	0
$327$	4.99925e13	0.739422
$328$	0	0
$329$	9.69147e12	0.138616
$330$	0	0
$331$	6.25245e12	0.0864960	0.0432480	−	0.999064i	$-0.486229\pi$
$331$	6.25245e12	0.0864960	0.0432480	+	0.999064i	$0.486229\pi$
$332$	0	0
$333$	−1.18561e13	−0.158672
$334$	0	0
$335$	1.21776e14	1.57694
$336$	0	0
$337$	6.09170e13	0.763439	0.381720	−	0.924278i	$-0.375332\pi$
$337$	6.09170e13	0.763439	0.381720	+	0.924278i	$0.375332\pi$
$338$	0	0
$339$	1.46662e11	0.00177918
$340$	0	0
$341$	−4.07858e13	−0.479027
$342$	0	0
$343$	−6.98021e13	−0.793873
$344$	0	0
$345$	3.73562e13	0.411488
$346$	0	0
$347$	−3.85685e13	−0.411548	−0.205774	−	0.978600i	$-0.565971\pi$
$347$	−3.85685e13	−0.411548	−0.205774	+	0.978600i	$0.565971\pi$
$348$	0	0
$349$	3.48853e13	0.360664	0.180332	−	0.983606i	$-0.442283\pi$
$349$	3.48853e13	0.360664	0.180332	+	0.983606i	$0.442283\pi$
$350$	0	0
$351$	5.18590e12	0.0519559
$352$	0	0
$353$	9.67569e13	0.939553	0.469777	−	0.882785i	$-0.344335\pi$
$353$	9.67569e13	0.939553	0.469777	+	0.882785i	$0.344335\pi$
$354$	0	0
$355$	−5.64345e13	−0.531238
$356$	0	0
$357$	−6.21614e11	−0.00567343
$358$	0	0
$359$	−1.64650e14	−1.45728	−0.728638	−	0.684899i	$-0.759846\pi$
$359$	−1.64650e14	−1.45728	−0.728638	+	0.684899i	$0.759846\pi$
$360$	0	0
$361$	2.26378e14	1.94332
$362$	0	0
$363$	−5.99815e13	−0.499495
$364$	0	0
$365$	−2.25020e14	−1.81807
$366$	0	0
$367$	−1.19183e14	−0.934435	−0.467218	−	0.884142i	$-0.654744\pi$
$367$	−1.19183e14	−0.934435	−0.467218	+	0.884142i	$0.654744\pi$
$368$	0	0
$369$	−8.47505e13	−0.644907
$370$	0	0
$371$	−6.54478e13	−0.483436
$372$	0	0
$373$	−1.61183e14	−1.15590	−0.577949	−	0.816073i	$-0.696147\pi$
$373$	−1.61183e14	−1.15590	−0.577949	+	0.816073i	$0.696147\pi$
$374$	0	0
$375$	8.11187e13	0.564871
$376$	0	0
$377$	6.92700e13	0.468455
$378$	0	0
$379$	−1.33388e14	−0.876195	−0.438098	−	0.898927i	$-0.644348\pi$
$379$	−1.33388e14	−0.876195	−0.438098	+	0.898927i	$0.644348\pi$
$380$	0	0
$381$	−1.68880e14	−1.07769
$382$	0	0
$383$	6.38158e13	0.395671	0.197836	−	0.980235i	$-0.436609\pi$
$383$	6.38158e13	0.395671	0.197836	+	0.980235i	$0.436609\pi$
$384$	0	0
$385$	−2.73218e13	−0.164617
$386$	0	0
$387$	−4.20844e13	−0.246440
$388$	0	0
$389$	−3.85909e13	−0.219665	−0.109833	−	0.993950i	$-0.535032\pi$
$389$	−3.85909e13	−0.219665	−0.109833	+	0.993950i	$0.535032\pi$
$390$	0	0
$391$	2.82322e12	0.0156233
$392$	0	0
$393$	−1.21867e14	−0.655731
$394$	0	0
$395$	2.92498e14	1.53052
$396$	0	0
$397$	−2.64317e13	−0.134517	−0.0672585	−	0.997736i	$-0.521425\pi$
$397$	−2.64317e13	−0.134517	−0.0672585	+	0.997736i	$0.521425\pi$
$398$	0	0
$399$	−8.79033e13	−0.435165
$400$	0	0
$401$	4.59354e13	0.221235	0.110617	−	0.993863i	$-0.464717\pi$
$401$	4.59354e13	0.221235	0.110617	+	0.993863i	$0.464717\pi$
$402$	0	0
$403$	−7.51502e13	−0.352170
$404$	0	0
$405$	−2.48608e13	−0.113374
$406$	0	0
$407$	−3.93836e13	−0.174802
$408$	0	0
$409$	4.23890e13	0.183136	0.0915682	−	0.995799i	$-0.470812\pi$
$409$	4.23890e13	0.183136	0.0915682	+	0.995799i	$0.470812\pi$
$410$	0	0
$411$	9.64289e13	0.405581
$412$	0	0
$413$	−8.95378e13	−0.366675
$414$	0	0
$415$	−1.42419e14	−0.567943
$416$	0	0
$417$	1.42833e14	0.554728
$418$	0	0
$419$	4.41929e14	1.67177	0.835883	−	0.548907i	$-0.184956\pi$
$419$	4.41929e14	1.67177	0.835883	+	0.548907i	$0.184956\pi$
$420$	0	0
$421$	−3.10369e14	−1.14374	−0.571870	−	0.820344i	$-0.693782\pi$
$421$	−3.10369e14	−1.14374	−0.571870	+	0.820344i	$0.693782\pi$
$422$	0	0
$423$	2.92932e13	0.105171
$424$	0	0
$425$	−2.63033e11	−0.000920177 0
$426$	0	0
$427$	6.69597e13	0.228276
$428$	0	0
$429$	1.72264e13	0.0572374
$430$	0	0
$431$	−3.06558e13	−0.0992860	−0.0496430	−	0.998767i	$-0.515808\pi$
$431$	−3.06558e13	−0.0992860	−0.0496430	+	0.998767i	$0.515808\pi$
$432$	0	0
$433$	−2.06583e14	−0.652245	−0.326122	−	0.945328i	$-0.605742\pi$
$433$	−2.06583e14	−0.652245	−0.326122	+	0.945328i	$0.605742\pi$
$434$	0	0
$435$	−3.32075e14	−1.02222
$436$	0	0
$437$	3.99236e14	1.19834
$438$	0	0
$439$	1.97201e14	0.577238	0.288619	−	0.957444i	$-0.406804\pi$
$439$	1.97201e14	0.577238	0.288619	+	0.957444i	$0.406804\pi$
$440$	0	0
$441$	−9.42228e13	−0.268995
$442$	0	0
$443$	1.98874e14	0.553805	0.276903	−	0.960898i	$-0.410692\pi$
$443$	1.98874e14	0.553805	0.276903	+	0.960898i	$0.410692\pi$
$444$	0	0
$445$	7.58644e13	0.206090
$446$	0	0
$447$	1.33635e14	0.354185
$448$	0	0
$449$	6.87389e14	1.77766	0.888828	−	0.458240i	$-0.151520\pi$
$449$	6.87389e14	1.77766	0.888828	+	0.458240i	$0.151520\pi$
$450$	0	0
$451$	−2.81523e14	−0.710465
$452$	0	0
$453$	−2.66006e12	−0.00655166
$454$	0	0
$455$	−5.03420e13	−0.121023
$456$	0	0
$457$	3.89942e14	0.915085	0.457543	−	0.889188i	$-0.348730\pi$
$457$	3.89942e14	0.915085	0.457543	+	0.889188i	$0.348730\pi$
$458$	0	0
$459$	−1.87887e12	−0.00430455
$460$	0	0
$461$	7.08640e14	1.58515	0.792576	−	0.609774i	$-0.208740\pi$
$461$	7.08640e14	1.58515	0.792576	+	0.609774i	$0.208740\pi$
$462$	0	0
$463$	6.00543e14	1.31174	0.655871	−	0.754873i	$-0.272301\pi$
$463$	6.00543e14	1.31174	0.655871	+	0.754873i	$0.272301\pi$
$464$	0	0
$465$	3.60264e14	0.768475
$466$	0	0
$467$	2.95277e13	0.0615159	0.0307579	−	0.999527i	$-0.490208\pi$
$467$	2.95277e13	0.0615159	0.0307579	+	0.999527i	$0.490208\pi$
$468$	0	0
$469$	−3.33663e14	−0.678980
$470$	0	0
$471$	−4.82760e14	−0.959658
$472$	0	0
$473$	−1.39795e14	−0.271492
$474$	0	0
$475$	−3.71959e13	−0.0705797
$476$	0	0
$477$	−1.97821e14	−0.366793
$478$	0	0
$479$	−1.00961e15	−1.82940	−0.914702	−	0.404128i	$-0.867575\pi$
$479$	−1.00961e15	−1.82940	−0.914702	+	0.404128i	$0.867575\pi$
$480$	0	0
$481$	−7.25665e13	−0.128510
$482$	0	0
$483$	−1.02355e14	−0.177174
$484$	0	0
$485$	−4.59251e13	−0.0777088
$486$	0	0
$487$	8.21837e14	1.35949	0.679745	−	0.733448i	$-0.262090\pi$
$487$	8.21837e14	1.35949	0.679745	+	0.733448i	$0.262090\pi$
$488$	0	0
$489$	−1.23632e14	−0.199955
$490$	0	0
$491$	8.64714e14	1.36749	0.683745	−	0.729721i	$-0.260350\pi$
$491$	8.64714e14	1.36749	0.683745	+	0.729721i	$0.260350\pi$
$492$	0	0
$493$	−2.50968e13	−0.0388115
$494$	0	0
$495$	−8.25821e13	−0.124899
$496$	0	0
$497$	1.54629e14	0.228734
$498$	0	0
$499$	−6.07789e14	−0.879427	−0.439714	−	0.898138i	$-0.644920\pi$
$499$	−6.07789e14	−0.879427	−0.439714	+	0.898138i	$0.644920\pi$
$500$	0	0
$501$	4.04796e14	0.572965
$502$	0	0
$503$	4.90522e14	0.679257	0.339629	−	0.940560i	$-0.389699\pi$
$503$	4.90522e14	0.679257	0.339629	+	0.940560i	$0.389699\pi$
$504$	0	0
$505$	−7.61299e14	−1.03146
$506$	0	0
$507$	−4.03754e14	−0.535271
$508$	0	0
$509$	−1.07479e15	−1.39436	−0.697179	−	0.716897i	$-0.745562\pi$
$509$	−1.07479e15	−1.39436	−0.697179	+	0.716897i	$0.745562\pi$
$510$	0	0
$511$	6.16549e14	0.782804
$512$	0	0
$513$	−2.65694e14	−0.330169
$514$	0	0
$515$	−1.98954e14	−0.241999
$516$	0	0
$517$	9.73056e13	0.115862
$518$	0	0
$519$	7.14053e14	0.832358
$520$	0	0
$521$	−8.25510e13	−0.0942139	−0.0471069	−	0.998890i	$-0.515000\pi$
$521$	−8.25510e13	−0.0942139	−0.0471069	+	0.998890i	$0.515000\pi$
$522$	0	0
$523$	6.87128e14	0.767854	0.383927	−	0.923363i	$-0.374571\pi$
$523$	6.87128e14	0.767854	0.383927	+	0.923363i	$0.374571\pi$
$524$	0	0
$525$	9.53615e12	0.0104351
$526$	0	0
$527$	2.72273e13	0.0291773
$528$	0	0
$529$	−4.87939e14	−0.512105
$530$	0	0
$531$	−2.70635e14	−0.278204
$532$	0	0
$533$	−5.18722e14	−0.522318
$534$	0	0
$535$	−1.06715e15	−1.05264
$536$	0	0
$537$	−2.06278e14	−0.199340
$538$	0	0
$539$	−3.12988e14	−0.296339
$540$	0	0
$541$	1.39934e15	1.29819	0.649093	−	0.760709i	$-0.275149\pi$
$541$	1.39934e15	1.29819	0.649093	+	0.760709i	$0.275149\pi$
$542$	0	0
$543$	1.22232e15	1.11119
$544$	0	0
$545$	−1.46686e15	−1.30679
$546$	0	0
$547$	−1.41257e15	−1.23333	−0.616667	−	0.787224i	$-0.711517\pi$
$547$	−1.41257e15	−1.23333	−0.616667	+	0.787224i	$0.711517\pi$
$548$	0	0
$549$	2.02391e14	0.173198
$550$	0	0
$551$	−3.54898e15	−2.97694
$552$	0	0
$553$	−8.01437e14	−0.658993
$554$	0	0
$555$	3.47878e14	0.280424
$556$	0	0
$557$	1.02686e15	0.811539	0.405769	−	0.913975i	$-0.367004\pi$
$557$	1.02686e15	0.811539	0.405769	+	0.913975i	$0.367004\pi$
$558$	0	0
$559$	−2.57581e14	−0.199595
$560$	0	0
$561$	−6.24121e12	−0.00474213
$562$	0	0
$563$	1.08846e15	0.810992	0.405496	−	0.914097i	$-0.367099\pi$
$563$	1.08846e15	0.810992	0.405496	+	0.914097i	$0.367099\pi$
$564$	0	0
$565$	−4.30329e12	−0.00314438
$566$	0	0
$567$	6.81178e13	0.0488151
$568$	0	0
$569$	−1.95900e15	−1.37695	−0.688474	−	0.725261i	$-0.741719\pi$
$569$	−1.95900e15	−1.37695	−0.688474	+	0.725261i	$0.741719\pi$
$570$	0	0
$571$	−1.58327e15	−1.09158	−0.545791	−	0.837921i	$-0.683771\pi$
$571$	−1.58327e15	−1.09158	−0.545791	+	0.837921i	$0.683771\pi$
$572$	0	0
$573$	3.42134e14	0.231391
$574$	0	0
$575$	−4.33109e13	−0.0287359
$576$	0	0
$577$	2.69378e14	0.175346	0.0876728	−	0.996149i	$-0.472057\pi$
$577$	2.69378e14	0.175346	0.0876728	+	0.996149i	$0.472057\pi$
$578$	0	0
$579$	−4.73814e13	−0.0302605
$580$	0	0
$581$	3.90225e14	0.244538
$582$	0	0
$583$	−6.57118e14	−0.404079
$584$	0	0
$585$	−1.52162e14	−0.0918226
$586$	0	0
$587$	7.20364e14	0.426621	0.213311	−	0.976984i	$-0.431575\pi$
$587$	7.20364e14	0.426621	0.213311	+	0.976984i	$0.431575\pi$
$588$	0	0
$589$	3.85025e15	2.23797
$590$	0	0
$591$	1.25422e15	0.715554
$592$	0	0
$593$	−1.27192e15	−0.712294	−0.356147	−	0.934430i	$-0.615910\pi$
$593$	−1.27192e15	−0.712294	−0.356147	+	0.934430i	$0.615910\pi$
$594$	0	0
$595$	1.82391e13	0.0100268
$596$	0	0
$597$	2.48049e14	0.133869
$598$	0	0
$599$	−4.83469e14	−0.256166	−0.128083	−	0.991763i	$-0.540882\pi$
$599$	−4.83469e14	−0.256166	−0.128083	+	0.991763i	$0.540882\pi$
$600$	0	0
$601$	3.68175e15	1.91533	0.957667	−	0.287880i	$-0.0929503\pi$
$601$	3.68175e15	1.91533	0.957667	+	0.287880i	$0.0929503\pi$
$602$	0	0
$603$	−1.00852e15	−0.515157
$604$	0	0
$605$	1.75995e15	0.882768
$606$	0	0
$607$	2.30279e15	1.13427	0.567135	−	0.823625i	$-0.308052\pi$
$607$	2.30279e15	1.13427	0.567135	+	0.823625i	$0.308052\pi$
$608$	0	0
$609$	9.09875e14	0.440136
$610$	0	0
$611$	1.79291e14	0.0851789
$612$	0	0
$613$	1.23428e14	0.0575947	0.0287973	−	0.999585i	$-0.490832\pi$
$613$	1.23428e14	0.0575947	0.0287973	+	0.999585i	$0.490832\pi$
$614$	0	0
$615$	2.48671e15	1.13976
$616$	0	0
$617$	−2.94167e15	−1.32442	−0.662210	−	0.749318i	$-0.730381\pi$
$617$	−2.94167e15	−1.32442	−0.662210	+	0.749318i	$0.730381\pi$
$618$	0	0
$619$	3.62405e15	1.60286	0.801430	−	0.598088i	$-0.204073\pi$
$619$	3.62405e15	1.60286	0.801430	+	0.598088i	$0.204073\pi$
$620$	0	0
$621$	−3.09375e14	−0.134425
$622$	0	0
$623$	−2.07866e14	−0.0887360
$624$	0	0
$625$	−2.47824e15	−1.03945
$626$	0	0
$627$	−8.82579e14	−0.363732
$628$	0	0
$629$	2.62912e13	0.0106471
$630$	0	0
$631$	−1.12294e15	−0.446884	−0.223442	−	0.974717i	$-0.571729\pi$
$631$	−1.12294e15	−0.446884	−0.223442	+	0.974717i	$0.571729\pi$
$632$	0	0
$633$	−2.21656e15	−0.866882
$634$	0	0
$635$	4.95522e15	1.90461
$636$	0	0
$637$	−5.76698e14	−0.217862
$638$	0	0
$639$	4.67377e14	0.173545
$640$	0	0
$641$	3.60734e15	1.31664	0.658321	−	0.752738i	$-0.271267\pi$
$641$	3.60734e15	1.31664	0.658321	+	0.752738i	$0.271267\pi$
$642$	0	0
$643$	4.13436e15	1.48337	0.741683	−	0.670751i	$-0.234028\pi$
$643$	4.13436e15	1.48337	0.741683	+	0.670751i	$0.234028\pi$
$644$	0	0
$645$	1.23482e15	0.435538
$646$	0	0
$647$	−5.00765e15	−1.73644	−0.868221	−	0.496177i	$-0.834737\pi$
$647$	−5.00765e15	−1.73644	−0.868221	+	0.496177i	$0.834737\pi$
$648$	0	0
$649$	−8.98990e14	−0.306485
$650$	0	0
$651$	−9.87113e14	−0.330881
$652$	0	0
$653$	−5.80675e15	−1.91386	−0.956931	−	0.290314i	$-0.906240\pi$
$653$	−5.80675e15	−1.91386	−0.956931	+	0.290314i	$0.906240\pi$
$654$	0	0
$655$	3.57576e15	1.15889
$656$	0	0
$657$	1.86357e15	0.593930
$658$	0	0
$659$	−3.09336e15	−0.969529	−0.484765	−	0.874645i	$-0.661095\pi$
$659$	−3.09336e15	−0.969529	−0.484765	+	0.874645i	$0.661095\pi$
$660$	0	0
$661$	−3.99504e14	−0.123144	−0.0615719	−	0.998103i	$-0.519611\pi$
$661$	−3.99504e14	−0.123144	−0.0615719	+	0.998103i	$0.519611\pi$
$662$	0	0
$663$	−1.14998e13	−0.00348631
$664$	0	0
$665$	2.57922e15	0.769076
$666$	0	0
$667$	−4.13244e15	−1.21203
$668$	0	0
$669$	1.13750e14	0.0328177
$670$	0	0
$671$	6.72298e14	0.190804
$672$	0	0
$673$	−2.16111e15	−0.603384	−0.301692	−	0.953405i	$-0.597551\pi$
$673$	−2.16111e15	−0.603384	−0.301692	+	0.953405i	$0.597551\pi$
$674$	0	0
$675$	2.88237e13	0.00791734
$676$	0	0
$677$	4.72622e15	1.27725	0.638626	−	0.769518i	$-0.279503\pi$
$677$	4.72622e15	1.27725	0.638626	+	0.769518i	$0.279503\pi$
$678$	0	0
$679$	1.25833e14	0.0334590
$680$	0	0
$681$	3.21236e15	0.840456
$682$	0	0
$683$	4.37911e15	1.12738	0.563692	−	0.825985i	$-0.309380\pi$
$683$	4.37911e15	1.12738	0.563692	+	0.825985i	$0.309380\pi$
$684$	0	0
$685$	−2.82938e15	−0.716791
$686$	0	0
$687$	−2.05955e14	−0.0513464
$688$	0	0
$689$	−1.21078e15	−0.297070
$690$	0	0
$691$	−7.37871e15	−1.78177	−0.890884	−	0.454230i	$-0.849914\pi$
$691$	−7.37871e15	−1.78177	−0.890884	+	0.454230i	$0.849914\pi$
$692$	0	0
$693$	2.26273e14	0.0537773
$694$	0	0
$695$	−4.19094e15	−0.980382
$696$	0	0
$697$	1.87935e14	0.0432741
$698$	0	0
$699$	−4.31659e15	−0.978399
$700$	0	0
$701$	2.97202e15	0.663136	0.331568	−	0.943431i	$-0.392422\pi$
$701$	2.97202e15	0.663136	0.331568	+	0.943431i	$0.392422\pi$
$702$	0	0
$703$	3.71787e15	0.816658
$704$	0	0
$705$	−8.59507e14	−0.185870
$706$	0	0
$707$	2.08594e15	0.444114
$708$	0	0
$709$	−3.84420e15	−0.805845	−0.402923	−	0.915234i	$-0.632006\pi$
$709$	−3.84420e15	−0.805845	−0.402923	+	0.915234i	$0.632006\pi$
$710$	0	0
$711$	−2.42240e15	−0.499992
$712$	0	0
$713$	4.48323e15	0.911169
$714$	0	0
$715$	−5.05450e14	−0.101157
$716$	0	0
$717$	−2.68838e15	−0.529829
$718$	0	0
$719$	−3.49014e15	−0.677383	−0.338691	−	0.940898i	$-0.609984\pi$
$719$	−3.49014e15	−0.677383	−0.338691	+	0.940898i	$0.609984\pi$
$720$	0	0
$721$	5.45128e14	0.104197
$722$	0	0
$723$	−3.19427e15	−0.601328
$724$	0	0
$725$	3.85009e14	0.0713859
$726$	0	0
$727$	−1.26477e15	−0.230978	−0.115489	−	0.993309i	$-0.536844\pi$
$727$	−1.26477e15	−0.230978	−0.115489	+	0.993309i	$0.536844\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	9.33228e13	0.0165364
$732$	0	0
$733$	−7.50679e15	−1.31033	−0.655167	−	0.755484i	$-0.727402\pi$
$733$	−7.50679e15	−1.31033	−0.655167	+	0.755484i	$0.727402\pi$
$734$	0	0
$735$	2.76464e15	0.475400
$736$	0	0
$737$	−3.35009e15	−0.567525
$738$	0	0
$739$	−7.20650e15	−1.20276	−0.601381	−	0.798962i	$-0.705383\pi$
$739$	−7.20650e15	−1.20276	−0.601381	+	0.798962i	$0.705383\pi$
$740$	0	0
$741$	−1.62620e15	−0.267408
$742$	0	0
$743$	1.39426e15	0.225894	0.112947	−	0.993601i	$-0.463971\pi$
$743$	1.39426e15	0.225894	0.112947	+	0.993601i	$0.463971\pi$
$744$	0	0
$745$	−3.92107e15	−0.625958
$746$	0	0
$747$	1.17948e15	0.185536
$748$	0	0
$749$	2.92396e15	0.453233
$750$	0	0
$751$	7.66385e15	1.17065	0.585326	−	0.810798i	$-0.300967\pi$
$751$	7.66385e15	1.17065	0.585326	+	0.810798i	$0.300967\pi$
$752$	0	0
$753$	−6.05141e15	−0.910928
$754$	0	0
$755$	7.80503e13	0.0115789
$756$	0	0
$757$	2.12920e15	0.311307	0.155654	−	0.987812i	$-0.450252\pi$
$757$	2.12920e15	0.311307	0.155654	+	0.987812i	$0.450252\pi$
$758$	0	0
$759$	−1.02768e15	−0.148090
$760$	0	0
$761$	2.44003e14	0.0346561	0.0173280	−	0.999850i	$-0.494484\pi$
$761$	2.44003e14	0.0346561	0.0173280	+	0.999850i	$0.494484\pi$
$762$	0	0
$763$	4.01915e15	0.562664
$764$	0	0
$765$	5.51291e13	0.00760752
$766$	0	0
$767$	−1.65644e15	−0.225321
$768$	0	0
$769$	7.87277e15	1.05568	0.527840	−	0.849344i	$-0.323002\pi$
$769$	7.87277e15	1.05568	0.527840	+	0.849344i	$0.323002\pi$
$770$	0	0
$771$	−4.63090e15	−0.612162
$772$	0	0
$773$	−1.12137e16	−1.46138	−0.730691	−	0.682709i	$-0.760802\pi$
$773$	−1.12137e16	−1.46138	−0.730691	+	0.682709i	$0.760802\pi$
$774$	0	0
$775$	−4.17692e14	−0.0536657
$776$	0	0
$777$	−9.53176e14	−0.120742
$778$	0	0
$779$	2.65762e16	3.31923
$780$	0	0
$781$	1.55253e15	0.191187
$782$	0	0
$783$	2.75017e15	0.333940
$784$	0	0
$785$	1.41649e16	1.69602
$786$	0	0
$787$	−1.34408e16	−1.58695	−0.793475	−	0.608603i	$-0.791730\pi$
$787$	−1.34408e16	−1.58695	−0.793475	+	0.608603i	$0.791730\pi$
$788$	0	0
$789$	−6.27215e15	−0.730286
$790$	0	0
$791$	1.17909e13	0.00135387
$792$	0	0
$793$	1.23875e15	0.140275
$794$	0	0
$795$	5.80437e15	0.648241
$796$	0	0
$797$	−1.64398e16	−1.81082	−0.905408	−	0.424542i	$-0.860435\pi$
$797$	−1.64398e16	−1.81082	−0.905408	+	0.424542i	$0.860435\pi$
$798$	0	0
$799$	−6.49580e13	−0.00705709
$800$	0	0
$801$	−6.28291e14	−0.0673259
$802$	0	0
$803$	6.19036e15	0.654306
$804$	0	0
$805$	3.00325e15	0.313122
$806$	0	0
$807$	−3.89077e15	−0.400158
$808$	0	0
$809$	1.37074e16	1.39072	0.695358	−	0.718664i	$-0.255246\pi$
$809$	1.37074e16	1.39072	0.695358	+	0.718664i	$0.255246\pi$
$810$	0	0
$811$	4.58825e15	0.459232	0.229616	−	0.973281i	$-0.426253\pi$
$811$	4.58825e15	0.459232	0.229616	+	0.973281i	$0.426253\pi$
$812$	0	0
$813$	5.20764e15	0.514213
$814$	0	0
$815$	3.62756e15	0.353385
$816$	0	0
$817$	1.31969e16	1.26838
$818$	0	0
$819$	4.16920e14	0.0395359
$820$	0	0
$821$	1.03921e16	0.972331	0.486165	−	0.873867i	$-0.338395\pi$
$821$	1.03921e16	0.972331	0.486165	+	0.873867i	$0.338395\pi$
$822$	0	0
$823$	−5.90788e15	−0.545422	−0.272711	−	0.962096i	$-0.587920\pi$
$823$	−5.90788e15	−0.545422	−0.272711	+	0.962096i	$0.587920\pi$
$824$	0	0
$825$	9.57462e13	0.00872217
$826$	0	0
$827$	−6.40538e15	−0.575791	−0.287895	−	0.957662i	$-0.592956\pi$
$827$	−6.40538e15	−0.575791	−0.287895	+	0.957662i	$0.592956\pi$
$828$	0	0
$829$	−6.31696e15	−0.560348	−0.280174	−	0.959949i	$-0.590392\pi$
$829$	−6.31696e15	−0.560348	−0.280174	+	0.959949i	$0.590392\pi$
$830$	0	0
$831$	5.48315e15	0.479981
$832$	0	0
$833$	2.08940e14	0.0180499
$834$	0	0
$835$	−1.18773e16	−1.01261
$836$	0	0
$837$	−2.98362e15	−0.251046
$838$	0	0
$839$	−1.09652e16	−0.910598	−0.455299	−	0.890339i	$-0.650468\pi$
$839$	−1.09652e16	−0.910598	−0.455299	+	0.890339i	$0.650468\pi$
$840$	0	0
$841$	2.45345e16	2.01094
$842$	0	0
$843$	3.53912e15	0.286315
$844$	0	0
$845$	1.18468e16	0.945994
$846$	0	0
$847$	−4.82222e15	−0.380092
$848$	0	0
$849$	5.91694e15	0.460368
$850$	0	0
$851$	4.32910e15	0.332495
$852$	0	0
$853$	3.39896e15	0.257707	0.128854	−	0.991664i	$-0.458870\pi$
$853$	3.39896e15	0.257707	0.128854	+	0.991664i	$0.458870\pi$
$854$	0	0
$855$	7.79589e15	0.583515
$856$	0	0
$857$	1.05370e16	0.778614	0.389307	−	0.921108i	$-0.372715\pi$
$857$	1.05370e16	0.778614	0.389307	+	0.921108i	$0.372715\pi$
$858$	0	0
$859$	1.29250e16	0.942903	0.471451	−	0.881892i	$-0.343730\pi$
$859$	1.29250e16	0.942903	0.471451	+	0.881892i	$0.343730\pi$
$860$	0	0
$861$	−6.81352e15	−0.490743
$862$	0	0
$863$	1.45598e16	1.03537	0.517685	−	0.855571i	$-0.326794\pi$
$863$	1.45598e16	1.03537	0.517685	+	0.855571i	$0.326794\pi$
$864$	0	0
$865$	−2.09514e16	−1.47104
$866$	0	0
$867$	−8.32390e15	−0.577061
$868$	0	0
$869$	−8.04669e15	−0.550819
$870$	0	0
$871$	−6.17273e15	−0.417232
$872$	0	0
$873$	3.80341e14	0.0253860
$874$	0	0
$875$	6.52155e15	0.429839
$876$	0	0
$877$	−6.57548e15	−0.427986	−0.213993	−	0.976835i	$-0.568647\pi$
$877$	−6.57548e15	−0.427986	−0.213993	+	0.976835i	$0.568647\pi$
$878$	0	0
$879$	−1.05156e14	−0.00675919
$880$	0	0
$881$	−2.17457e16	−1.38040	−0.690202	−	0.723617i	$-0.742478\pi$
$881$	−2.17457e16	−1.38040	−0.690202	+	0.723617i	$0.742478\pi$
$882$	0	0
$883$	−1.89298e15	−0.118676	−0.0593378	−	0.998238i	$-0.518899\pi$
$883$	−1.89298e15	−0.118676	−0.0593378	+	0.998238i	$0.518899\pi$
$884$	0	0
$885$	7.94085e15	0.491676
$886$	0	0
$887$	−2.03579e16	−1.24495	−0.622475	−	0.782639i	$-0.713873\pi$
$887$	−2.03579e16	−1.24495	−0.622475	+	0.782639i	$0.713873\pi$
$888$	0	0
$889$	−1.35772e16	−0.820066
$890$	0	0
$891$	6.83926e14	0.0408020
$892$	0	0
$893$	−9.18581e15	−0.541295
$894$	0	0
$895$	6.05253e15	0.352298
$896$	0	0
$897$	−1.89355e15	−0.108873
$898$	0	0
$899$	−3.98534e16	−2.26353
$900$	0	0
$901$	4.38671e14	0.0246123
$902$	0	0
$903$	−3.38338e15	−0.187529
$904$	0	0
$905$	−3.58649e16	−1.96382
$906$	0	0
$907$	2.08169e16	1.12610	0.563049	−	0.826424i	$-0.309628\pi$
$907$	2.08169e16	1.12610	0.563049	+	0.826424i	$0.309628\pi$
$908$	0	0
$909$	6.30490e15	0.336959
$910$	0	0
$911$	−1.56740e15	−0.0827616	−0.0413808	−	0.999143i	$-0.513176\pi$
$911$	−1.56740e15	−0.0827616	−0.0413808	+	0.999143i	$0.513176\pi$
$912$	0	0
$913$	3.91799e15	0.204397
$914$	0	0
$915$	−5.93846e15	−0.306096
$916$	0	0
$917$	−9.79748e15	−0.498980
$918$	0	0
$919$	1.45715e15	0.0733278	0.0366639	−	0.999328i	$-0.488327\pi$
$919$	1.45715e15	0.0733278	0.0366639	+	0.999328i	$0.488327\pi$
$920$	0	0
$921$	1.43213e16	0.712124
$922$	0	0
$923$	2.86062e15	0.140556
$924$	0	0
$925$	−4.03332e14	−0.0195832
$926$	0	0
$927$	1.64769e15	0.0790564
$928$	0	0
$929$	−1.70074e16	−0.806400	−0.403200	−	0.915112i	$-0.632102\pi$
$929$	−1.70074e16	−0.806400	−0.403200	+	0.915112i	$0.632102\pi$
$930$	0	0
$931$	2.95466e16	1.38447
$932$	0	0
$933$	−3.14343e15	−0.145564
$934$	0	0
$935$	1.83127e14	0.00838086
$936$	0	0
$937$	−3.71528e16	−1.68044	−0.840222	−	0.542242i	$-0.817576\pi$
$937$	−3.71528e16	−1.68044	−0.840222	+	0.542242i	$0.817576\pi$
$938$	0	0
$939$	−6.24387e15	−0.279121
$940$	0	0
$941$	2.09136e16	0.924031	0.462016	−	0.886872i	$-0.347126\pi$
$941$	2.09136e16	0.924031	0.462016	+	0.886872i	$0.347126\pi$
$942$	0	0
$943$	3.09454e16	1.35139
$944$	0	0
$945$	−1.99868e15	−0.0862718
$946$	0	0
$947$	−3.53699e16	−1.50907	−0.754533	−	0.656262i	$-0.772137\pi$
$947$	−3.53699e16	−1.50907	−0.754533	+	0.656262i	$0.772137\pi$
$948$	0	0
$949$	1.14061e16	0.481031
$950$	0	0
$951$	3.24807e15	0.135404
$952$	0	0
$953$	1.45761e16	0.600661	0.300331	−	0.953835i	$-0.402903\pi$
$953$	1.45761e16	0.600661	0.300331	+	0.953835i	$0.402903\pi$
$954$	0	0
$955$	−1.00388e16	−0.408941
$956$	0	0
$957$	9.13545e15	0.367887
$958$	0	0
$959$	7.75241e15	0.308627
$960$	0	0
$961$	1.78280e16	0.701655
$962$	0	0
$963$	8.83789e15	0.343878
$964$	0	0
$965$	1.39024e15	0.0534800
$966$	0	0
$967$	−1.57202e16	−0.597878	−0.298939	−	0.954272i	$-0.596633\pi$
$967$	−1.57202e16	−0.597878	−0.298939	+	0.954272i	$0.596633\pi$
$968$	0	0
$969$	5.89181e14	0.0221548
$970$	0	0
$971$	−1.34980e16	−0.501837	−0.250918	−	0.968008i	$-0.580733\pi$
$971$	−1.34980e16	−0.501837	−0.250918	+	0.968008i	$0.580733\pi$
$972$	0	0
$973$	1.14831e16	0.422121
$974$	0	0
$975$	1.76418e14	0.00641235
$976$	0	0
$977$	−3.80905e15	−0.136898	−0.0684489	−	0.997655i	$-0.521805\pi$
$977$	−3.80905e15	−0.136898	−0.0684489	+	0.997655i	$0.521805\pi$
$978$	0	0
$979$	−2.08705e15	−0.0741698
$980$	0	0
$981$	1.21482e16	0.426905
$982$	0	0
$983$	−5.42344e16	−1.88465	−0.942325	−	0.334698i	$-0.891366\pi$
$983$	−5.42344e16	−1.88465	−0.942325	+	0.334698i	$0.891366\pi$
$984$	0	0
$985$	−3.68008e16	−1.26461
$986$	0	0
$987$	2.35503e15	0.0800297
$988$	0	0
$989$	1.53665e16	0.516411
$990$	0	0
$991$	−1.20030e16	−0.398919	−0.199459	−	0.979906i	$-0.563919\pi$
$991$	−1.20030e16	−0.398919	−0.199459	+	0.979906i	$0.563919\pi$
$992$	0	0
$993$	1.51934e15	0.0499385
$994$	0	0
$995$	−7.27815e15	−0.236589
$996$	0	0
$997$	2.90012e15	0.0932379	0.0466189	−	0.998913i	$-0.485155\pi$
$997$	2.90012e15	0.0932379	0.0466189	+	0.998913i	$0.485155\pi$
$998$	0	0
$999$	−2.88104e15	−0.0916093

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.12.a.e.1.1		1
3.2	odd	2		144.12.a.m.1.1		1
4.3	odd	2		24.12.a.a.1.1	✓	1
8.3	odd	2		192.12.a.s.1.1		1
8.5	even	2		192.12.a.i.1.1		1
12.11	even	2		72.12.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.12.a.a.1.1	✓	1	4.3	odd	2
48.12.a.e.1.1		1	1.1	even	1	trivial
72.12.a.d.1.1		1	12.11	even	2
144.12.a.m.1.1		1	3.2	odd	2
192.12.a.i.1.1		1	8.5	even	2
192.12.a.s.1.1		1	8.3	odd	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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