LMFDB - Embedded newform 48.22.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,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, 22, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
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:	$134.149125258$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
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)

$f(q)$

$=$

$q+59049.0 q^{3} -4.15128e7 q^{5} -5.38430e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$

$q+59049.0 q^{3} -4.15128e7 q^{5} -5.38430e8 q^{7} +3.48678e9 q^{9} +6.41130e10 q^{11} -1.30980e11 q^{13} -2.45129e12 q^{15} +8.24203e12 q^{17} -1.34921e13 q^{19} -3.17937e13 q^{21} +2.33185e14 q^{23} +1.24647e15 q^{25} +2.05891e14 q^{27} -2.02456e15 q^{29} +6.86919e15 q^{31} +3.78581e15 q^{33} +2.23517e16 q^{35} +3.44400e15 q^{37} -7.73424e15 q^{39} -2.18424e16 q^{41} +7.17928e16 q^{43} -1.44746e17 q^{45} -2.83545e17 q^{47} -2.68639e17 q^{49} +4.86684e17 q^{51} -2.17229e18 q^{53} -2.66151e18 q^{55} -7.96695e17 q^{57} -1.53483e18 q^{59} +4.31159e18 q^{61} -1.87739e18 q^{63} +5.43735e18 q^{65} -9.24391e18 q^{67} +1.37693e19 q^{69} +2.03874e19 q^{71} +1.66178e19 q^{73} +7.36030e19 q^{75} -3.45204e19 q^{77} -6.79403e19 q^{79} +1.21577e19 q^{81} -3.95037e19 q^{83} -3.42149e20 q^{85} -1.19548e20 q^{87} +4.16117e19 q^{89} +7.05236e19 q^{91} +4.05619e20 q^{93} +5.60095e20 q^{95} +5.71815e19 q^{97} +2.23548e20 q^{99} +O(q^{100})$

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
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	0	0
$5$	−4.15128e7	−1.90106	−0.950532	−	0.310627i	$-0.899461\pi$
$5$	−4.15128e7	−1.90106	−0.950532	+	0.310627i	$0.899461\pi$
$6$	0	0
$7$	−5.38430e8	−0.720443	−0.360222	−	0.932867i	$-0.617299\pi$
$7$	−5.38430e8	−0.720443	−0.360222	+	0.932867i	$0.617299\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	6.41130e10	0.745286	0.372643	−	0.927975i	$-0.378452\pi$
$11$	6.41130e10	0.745286	0.372643	+	0.927975i	$0.378452\pi$
$12$	0	0
$13$	−1.30980e11	−0.263512	−0.131756	−	0.991282i	$-0.542062\pi$
$13$	−1.30980e11	−0.263512	−0.131756	+	0.991282i	$0.542062\pi$
$14$	0	0
$15$	−2.45129e12	−1.09758
$16$	0	0
$17$	8.24203e12	0.991563	0.495782	−	0.868447i	$-0.334882\pi$
$17$	8.24203e12	0.991563	0.495782	+	0.868447i	$0.334882\pi$
$18$	0	0
$19$	−1.34921e13	−0.504855	−0.252428	−	0.967616i	$-0.581229\pi$
$19$	−1.34921e13	−0.504855	−0.252428	+	0.967616i	$0.581229\pi$
$20$	0	0
$21$	−3.17937e13	−0.415948
$22$	0	0
$23$	2.33185e14	1.17370	0.586851	−	0.809695i	$-0.300367\pi$
$23$	2.33185e14	1.17370	0.586851	+	0.809695i	$0.300367\pi$
$24$	0	0
$25$	1.24647e15	2.61404
$26$	0	0
$27$	2.05891e14	0.192450
$28$	0	0
$29$	−2.02456e15	−0.893618	−0.446809	−	0.894629i	$-0.647440\pi$
$29$	−2.02456e15	−0.893618	−0.446809	+	0.894629i	$0.647440\pi$
$30$	0	0
$31$	6.86919e15	1.50525	0.752624	−	0.658451i	$-0.228788\pi$
$31$	6.86919e15	1.50525	0.752624	+	0.658451i	$0.228788\pi$
$32$	0	0
$33$	3.78581e15	0.430291
$34$	0	0
$35$	2.23517e16	1.36961
$36$	0	0
$37$	3.44400e15	0.117746	0.0588728	−	0.998265i	$-0.481249\pi$
$37$	3.44400e15	0.117746	0.0588728	+	0.998265i	$0.481249\pi$
$38$	0	0
$39$	−7.73424e15	−0.152139
$40$	0	0
$41$	−2.18424e16	−0.254138	−0.127069	−	0.991894i	$-0.540557\pi$
$41$	−2.18424e16	−0.254138	−0.127069	+	0.991894i	$0.540557\pi$
$42$	0	0
$43$	7.17928e16	0.506597	0.253298	−	0.967388i	$-0.418485\pi$
$43$	7.17928e16	0.506597	0.253298	+	0.967388i	$0.418485\pi$
$44$	0	0
$45$	−1.44746e17	−0.633688
$46$	0	0
$47$	−2.83545e17	−0.786310	−0.393155	−	0.919472i	$-0.628617\pi$
$47$	−2.83545e17	−0.786310	−0.393155	+	0.919472i	$0.628617\pi$
$48$	0	0
$49$	−2.68639e17	−0.480962
$50$	0	0
$51$	4.86684e17	0.572479
$52$	0	0
$53$	−2.17229e18	−1.70616	−0.853081	−	0.521779i	$-0.825269\pi$
$53$	−2.17229e18	−1.70616	−0.853081	+	0.521779i	$0.825269\pi$
$54$	0	0
$55$	−2.66151e18	−1.41684
$56$	0	0
$57$	−7.96695e17	−0.291478
$58$	0	0
$59$	−1.53483e18	−0.390944	−0.195472	−	0.980709i	$-0.562624\pi$
$59$	−1.53483e18	−0.390944	−0.195472	+	0.980709i	$0.562624\pi$
$60$	0	0
$61$	4.31159e18	0.773881	0.386940	−	0.922105i	$-0.373532\pi$
$61$	4.31159e18	0.773881	0.386940	+	0.922105i	$0.373532\pi$
$62$	0	0
$63$	−1.87739e18	−0.240148
$64$	0	0
$65$	5.43735e18	0.500953
$66$	0	0
$67$	−9.24391e18	−0.619541	−0.309771	−	0.950811i	$-0.600252\pi$
$67$	−9.24391e18	−0.619541	−0.309771	+	0.950811i	$0.600252\pi$
$68$	0	0
$69$	1.37693e19	0.677637
$70$	0	0
$71$	2.03874e19	0.743273	0.371636	−	0.928378i	$-0.378797\pi$
$71$	2.03874e19	0.743273	0.371636	+	0.928378i	$0.378797\pi$
$72$	0	0
$73$	1.66178e19	0.452566	0.226283	−	0.974062i	$-0.427343\pi$
$73$	1.66178e19	0.452566	0.226283	+	0.974062i	$0.427343\pi$
$74$	0	0
$75$	7.36030e19	1.50922
$76$	0	0
$77$	−3.45204e19	−0.536936
$78$	0	0
$79$	−6.79403e19	−0.807315	−0.403658	−	0.914910i	$-0.632261\pi$
$79$	−6.79403e19	−0.807315	−0.403658	+	0.914910i	$0.632261\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	−3.95037e19	−0.279459	−0.139730	−	0.990190i	$-0.544623\pi$
$83$	−3.95037e19	−0.279459	−0.139730	+	0.990190i	$0.544623\pi$
$84$	0	0
$85$	−3.42149e20	−1.88503
$86$	0	0
$87$	−1.19548e20	−0.515931
$88$	0	0
$89$	4.16117e19	0.141456	0.0707278	−	0.997496i	$-0.477468\pi$
$89$	4.16117e19	0.141456	0.0707278	+	0.997496i	$0.477468\pi$
$90$	0	0
$91$	7.05236e19	0.189845
$92$	0	0
$93$	4.05619e20	0.869055
$94$	0	0
$95$	5.60095e20	0.959762
$96$	0	0
$97$	5.71815e19	0.0787322	0.0393661	−	0.999225i	$-0.487466\pi$
$97$	5.71815e19	0.0787322	0.0393661	+	0.999225i	$0.487466\pi$
$98$	0	0
$99$	2.23548e20	0.248429
$100$	0	0
$101$	4.32417e20	0.389518	0.194759	−	0.980851i	$-0.437607\pi$
$101$	4.32417e20	0.389518	0.194759	+	0.980851i	$0.437607\pi$
$102$	0	0
$103$	−1.84123e21	−1.34995	−0.674974	−	0.737841i	$-0.735845\pi$
$103$	−1.84123e21	−1.34995	−0.674974	+	0.737841i	$0.735845\pi$
$104$	0	0
$105$	1.31985e21	0.790744
$106$	0	0
$107$	2.43805e21	1.19815	0.599077	−	0.800691i	$-0.295534\pi$
$107$	2.43805e21	1.19815	0.599077	+	0.800691i	$0.295534\pi$
$108$	0	0
$109$	−4.13676e21	−1.67372	−0.836859	−	0.547418i	$-0.815611\pi$
$109$	−4.13676e21	−1.67372	−0.836859	+	0.547418i	$0.815611\pi$
$110$	0	0
$111$	2.03365e20	0.0679805
$112$	0	0
$113$	3.47910e21	0.964146	0.482073	−	0.876131i	$-0.339884\pi$
$113$	3.47910e21	0.964146	0.482073	+	0.876131i	$0.339884\pi$
$114$	0	0
$115$	−9.68015e21	−2.23128
$116$	0	0
$117$	−4.56699e20	−0.0878373
$118$	0	0
$119$	−4.43775e21	−0.714365
$120$	0	0
$121$	−3.28977e21	−0.444548
$122$	0	0
$123$	−1.28977e21	−0.146727
$124$	0	0
$125$	−3.19497e22	−3.06840
$126$	0	0
$127$	−1.37141e21	−0.111488	−0.0557438	−	0.998445i	$-0.517753\pi$
$127$	−1.37141e21	−0.111488	−0.0557438	+	0.998445i	$0.517753\pi$
$128$	0	0
$129$	4.23929e21	0.292484
$130$	0	0
$131$	2.45276e22	1.43981	0.719907	−	0.694071i	$-0.244185\pi$
$131$	2.45276e22	1.43981	0.719907	+	0.694071i	$0.244185\pi$
$132$	0	0
$133$	7.26455e21	0.363719
$134$	0	0
$135$	−8.54711e21	−0.365860
$136$	0	0
$137$	1.02835e22	0.377204	0.188602	−	0.982054i	$-0.439604\pi$
$137$	1.02835e22	0.377204	0.188602	+	0.982054i	$0.439604\pi$
$138$	0	0
$139$	−8.70692e21	−0.274289	−0.137145	−	0.990551i	$-0.543793\pi$
$139$	−8.70692e21	−0.274289	−0.137145	+	0.990551i	$0.543793\pi$
$140$	0	0
$141$	−1.67430e22	−0.453977
$142$	0	0
$143$	−8.39753e21	−0.196392
$144$	0	0
$145$	8.40452e22	1.69883
$146$	0	0
$147$	−1.58629e22	−0.277683
$148$	0	0
$149$	−9.03997e22	−1.37313	−0.686564	−	0.727069i	$-0.740882\pi$
$149$	−9.03997e22	−1.37313	−0.686564	+	0.727069i	$0.740882\pi$
$150$	0	0
$151$	4.75206e22	0.627514	0.313757	−	0.949503i	$-0.398412\pi$
$151$	4.75206e22	0.627514	0.313757	+	0.949503i	$0.398412\pi$
$152$	0	0
$153$	2.87382e22	0.330521
$154$	0	0
$155$	−2.85159e23	−2.86157
$156$	0	0
$157$	−1.50901e23	−1.32356	−0.661781	−	0.749697i	$-0.730199\pi$
$157$	−1.50901e23	−1.32356	−0.661781	+	0.749697i	$0.730199\pi$
$158$	0	0
$159$	−1.28271e23	−0.985053
$160$	0	0
$161$	−1.25554e23	−0.845586
$162$	0	0
$163$	4.83503e22	0.286042	0.143021	−	0.989720i	$-0.454318\pi$
$163$	4.83503e22	0.286042	0.143021	+	0.989720i	$0.454318\pi$
$164$	0	0
$165$	−1.57159e23	−0.818011
$166$	0	0
$167$	−4.78731e20	−0.00219568	−0.00109784	−	0.999999i	$-0.500349\pi$
$167$	−4.78731e20	−0.00219568	−0.00109784	+	0.999999i	$0.500349\pi$
$168$	0	0
$169$	−2.29909e23	−0.930562
$170$	0	0
$171$	−4.70440e22	−0.168285
$172$	0	0
$173$	−1.61804e23	−0.512277	−0.256139	−	0.966640i	$-0.582450\pi$
$173$	−1.61804e23	−0.512277	−0.256139	+	0.966640i	$0.582450\pi$
$174$	0	0
$175$	−6.71138e23	−1.88327
$176$	0	0
$177$	−9.06303e22	−0.225712
$178$	0	0
$179$	8.76377e22	0.193970	0.0969849	−	0.995286i	$-0.469080\pi$
$179$	8.76377e22	0.193970	0.0969849	+	0.995286i	$0.469080\pi$
$180$	0	0
$181$	9.36624e22	0.184476	0.0922381	−	0.995737i	$-0.470598\pi$
$181$	9.36624e22	0.184476	0.0922381	+	0.995737i	$0.470598\pi$
$182$	0	0
$183$	2.54595e23	0.446800
$184$	0	0
$185$	−1.42970e23	−0.223842
$186$	0	0
$187$	5.28422e23	0.738999
$188$	0	0
$189$	−1.10858e23	−0.138649
$190$	0	0
$191$	−1.20858e24	−1.35340	−0.676699	−	0.736260i	$-0.736590\pi$
$191$	−1.20858e24	−1.35340	−0.676699	+	0.736260i	$0.736590\pi$
$192$	0	0
$193$	−1.78822e24	−1.79502	−0.897509	−	0.440997i	$-0.854625\pi$
$193$	−1.78822e24	−1.79502	−0.897509	+	0.440997i	$0.854625\pi$
$194$	0	0
$195$	3.21070e23	0.289225
$196$	0	0
$197$	1.90963e24	1.54545	0.772723	−	0.634743i	$-0.218894\pi$
$197$	1.90963e24	1.54545	0.772723	+	0.634743i	$0.218894\pi$
$198$	0	0
$199$	−1.44254e24	−1.04995	−0.524977	−	0.851116i	$-0.675926\pi$
$199$	−1.44254e24	−1.04995	−0.524977	+	0.851116i	$0.675926\pi$
$200$	0	0
$201$	−5.45844e23	−0.357692
$202$	0	0
$203$	1.09008e24	0.643801
$204$	0	0
$205$	9.06739e23	0.483133
$206$	0	0
$207$	8.13065e23	0.391234
$208$	0	0
$209$	−8.65020e23	−0.376262
$210$	0	0
$211$	−3.98848e24	−1.56979	−0.784895	−	0.619629i	$-0.787283\pi$
$211$	−3.98848e24	−1.56979	−0.784895	+	0.619629i	$0.787283\pi$
$212$	0	0
$213$	1.20385e24	0.429129
$214$	0	0
$215$	−2.98032e24	−0.963072
$216$	0	0
$217$	−3.69858e24	−1.08445
$218$	0	0
$219$	9.81262e23	0.261289
$220$	0	0
$221$	−1.07954e24	−0.261289
$222$	0	0
$223$	4.62963e24	1.01940	0.509700	−	0.860352i	$-0.329756\pi$
$223$	4.62963e24	1.01940	0.509700	+	0.860352i	$0.329756\pi$
$224$	0	0
$225$	4.34618e24	0.871348
$226$	0	0
$227$	3.43010e24	0.626664	0.313332	−	0.949644i	$-0.398555\pi$
$227$	3.43010e24	0.626664	0.313332	+	0.949644i	$0.398555\pi$
$228$	0	0
$229$	8.11792e23	0.135261	0.0676304	−	0.997710i	$-0.478456\pi$
$229$	8.11792e23	0.135261	0.0676304	+	0.997710i	$0.478456\pi$
$230$	0	0
$231$	−2.03839e24	−0.310000
$232$	0	0
$233$	8.22188e23	0.114218	0.0571089	−	0.998368i	$-0.481812\pi$
$233$	8.22188e23	0.114218	0.0571089	+	0.998368i	$0.481812\pi$
$234$	0	0
$235$	1.17707e25	1.49483
$236$	0	0
$237$	−4.01181e24	−0.466104
$238$	0	0
$239$	8.85525e24	0.941940	0.470970	−	0.882149i	$-0.343904\pi$
$239$	8.85525e24	0.941940	0.470970	+	0.882149i	$0.343904\pi$
$240$	0	0
$241$	7.46934e24	0.727953	0.363977	−	0.931408i	$-0.381419\pi$
$241$	7.46934e24	0.727953	0.363977	+	0.931408i	$0.381419\pi$
$242$	0	0
$243$	7.17898e23	0.0641500
$244$	0	0
$245$	1.11520e25	0.914339
$246$	0	0
$247$	1.76720e24	0.133035
$248$	0	0
$249$	−2.33266e24	−0.161346
$250$	0	0
$251$	−9.46474e23	−0.0601914	−0.0300957	−	0.999547i	$-0.509581\pi$
$251$	−9.46474e23	−0.0601914	−0.0300957	+	0.999547i	$0.509581\pi$
$252$	0	0
$253$	1.49502e25	0.874744
$254$	0	0
$255$	−2.02036e25	−1.08832
$256$	0	0
$257$	−1.91825e25	−0.951936	−0.475968	−	0.879463i	$-0.657902\pi$
$257$	−1.91825e25	−0.951936	−0.475968	+	0.879463i	$0.657902\pi$
$258$	0	0
$259$	−1.85435e24	−0.0848290
$260$	0	0
$261$	−7.05921e24	−0.297873
$262$	0	0
$263$	−8.88429e23	−0.0346009	−0.0173004	−	0.999850i	$-0.505507\pi$
$263$	−8.88429e23	−0.0346009	−0.0173004	+	0.999850i	$0.505507\pi$
$264$	0	0
$265$	9.01776e25	3.24352
$266$	0	0
$267$	2.45713e24	0.0816694
$268$	0	0
$269$	−2.13847e25	−0.657211	−0.328605	−	0.944467i	$-0.606579\pi$
$269$	−2.13847e25	−0.657211	−0.328605	+	0.944467i	$0.606579\pi$
$270$	0	0
$271$	1.56435e25	0.444791	0.222395	−	0.974957i	$-0.428612\pi$
$271$	1.56435e25	0.444791	0.222395	+	0.974957i	$0.428612\pi$
$272$	0	0
$273$	4.16435e24	0.109607
$274$	0	0
$275$	7.99152e25	1.94821
$276$	0	0
$277$	−8.04973e25	−1.81863	−0.909313	−	0.416112i	$-0.863392\pi$
$277$	−8.04973e25	−1.81863	−0.909313	+	0.416112i	$0.863392\pi$
$278$	0	0
$279$	2.39514e25	0.501749
$280$	0	0
$281$	8.33171e25	1.61926	0.809632	−	0.586938i	$-0.199667\pi$
$281$	8.33171e25	1.61926	0.809632	+	0.586938i	$0.199667\pi$
$282$	0	0
$283$	4.46130e24	0.0804829	0.0402415	−	0.999190i	$-0.487187\pi$
$283$	4.46130e24	0.0804829	0.0402415	+	0.999190i	$0.487187\pi$
$284$	0	0
$285$	3.30730e25	0.554119
$286$	0	0
$287$	1.17606e25	0.183092
$288$	0	0
$289$	−1.16088e24	−0.0168020
$290$	0	0
$291$	3.37651e24	0.0454560
$292$	0	0
$293$	9.67128e25	1.21164	0.605820	−	0.795602i	$-0.292845\pi$
$293$	9.67128e25	1.21164	0.605820	+	0.795602i	$0.292845\pi$
$294$	0	0
$295$	6.37151e25	0.743209
$296$	0	0
$297$	1.32003e25	0.143430
$298$	0	0
$299$	−3.05426e25	−0.309284
$300$	0	0
$301$	−3.86554e25	−0.364974
$302$	0	0
$303$	2.55338e25	0.224889
$304$	0	0
$305$	−1.78986e26	−1.47120
$306$	0	0
$307$	−1.68163e26	−1.29056	−0.645278	−	0.763948i	$-0.723258\pi$
$307$	−1.68163e26	−1.29056	−0.645278	+	0.763948i	$0.723258\pi$
$308$	0	0
$309$	−1.08723e26	−0.779393
$310$	0	0
$311$	−2.30370e26	−1.54327	−0.771636	−	0.636065i	$-0.780561\pi$
$311$	−2.30370e26	−1.54327	−0.771636	+	0.636065i	$0.780561\pi$
$312$	0	0
$313$	−2.79658e26	−1.75151	−0.875753	−	0.482759i	$-0.839635\pi$
$313$	−2.79658e26	−1.75151	−0.875753	+	0.482759i	$0.839635\pi$
$314$	0	0
$315$	7.79356e25	0.456536
$316$	0	0
$317$	2.98501e25	0.163615	0.0818075	−	0.996648i	$-0.473931\pi$
$317$	2.98501e25	0.163615	0.0818075	+	0.996648i	$0.473931\pi$
$318$	0	0
$319$	−1.29801e26	−0.666002
$320$	0	0
$321$	1.43964e26	0.691755
$322$	0	0
$323$	−1.11202e26	−0.500596
$324$	0	0
$325$	−1.63263e26	−0.688831
$326$	0	0
$327$	−2.44272e26	−0.966322
$328$	0	0
$329$	1.52669e26	0.566492
$330$	0	0
$331$	−2.55594e26	−0.889933	−0.444967	−	0.895547i	$-0.646784\pi$
$331$	−2.55594e26	−0.889933	−0.444967	+	0.895547i	$0.646784\pi$
$332$	0	0
$333$	1.20085e25	0.0392485
$334$	0	0
$335$	3.83740e26	1.17779
$336$	0	0
$337$	−4.91931e25	−0.141837	−0.0709187	−	0.997482i	$-0.522593\pi$
$337$	−4.91931e25	−0.141837	−0.0709187	+	0.997482i	$0.522593\pi$
$338$	0	0
$339$	2.05437e26	0.556650
$340$	0	0
$341$	4.40405e26	1.12184
$342$	0	0
$343$	4.45381e26	1.06695
$344$	0	0
$345$	−5.71603e26	−1.28823
$346$	0	0
$347$	−2.98136e26	−0.632345	−0.316173	−	0.948702i	$-0.602398\pi$
$347$	−2.98136e26	−0.632345	−0.316173	+	0.948702i	$0.602398\pi$
$348$	0	0
$349$	7.72834e26	1.54319	0.771595	−	0.636115i	$-0.219459\pi$
$349$	7.72834e26	1.54319	0.771595	+	0.636115i	$0.219459\pi$
$350$	0	0
$351$	−2.69676e25	−0.0507129
$352$	0	0
$353$	−7.30755e26	−1.29461	−0.647303	−	0.762233i	$-0.724103\pi$
$353$	−7.30755e26	−1.29461	−0.647303	+	0.762233i	$0.724103\pi$
$354$	0	0
$355$	−8.46336e26	−1.41301
$356$	0	0
$357$	−2.62045e26	−0.412439
$358$	0	0
$359$	1.58936e25	0.0235901	0.0117951	−	0.999930i	$-0.496245\pi$
$359$	1.58936e25	0.0235901	0.0117951	+	0.999930i	$0.496245\pi$
$360$	0	0
$361$	−5.32173e26	−0.745121
$362$	0	0
$363$	−1.94258e26	−0.256660
$364$	0	0
$365$	−6.89849e26	−0.860358
$366$	0	0
$367$	1.40734e27	1.65732	0.828660	−	0.559752i	$-0.189104\pi$
$367$	1.40734e27	1.65732	0.828660	+	0.559752i	$0.189104\pi$
$368$	0	0
$369$	−7.61598e25	−0.0847127
$370$	0	0
$371$	1.16962e27	1.22919
$372$	0	0
$373$	−9.30077e26	−0.923797	−0.461898	−	0.886933i	$-0.652831\pi$
$373$	−9.30077e26	−0.923797	−0.461898	+	0.886933i	$0.652831\pi$
$374$	0	0
$375$	−1.88660e27	−1.77154
$376$	0	0
$377$	2.65177e26	0.235479
$378$	0	0
$379$	−2.18541e27	−1.83578	−0.917892	−	0.396830i	$-0.870110\pi$
$379$	−2.18541e27	−1.83578	−0.917892	+	0.396830i	$0.870110\pi$
$380$	0	0
$381$	−8.09801e25	−0.0643674
$382$	0	0
$383$	−2.10347e27	−1.58252	−0.791258	−	0.611482i	$-0.790574\pi$
$383$	−2.10347e27	−1.58252	−0.791258	+	0.611482i	$0.790574\pi$
$384$	0	0
$385$	1.43304e27	1.02075
$386$	0	0
$387$	2.50326e26	0.168866
$388$	0	0
$389$	−2.97815e26	−0.190316	−0.0951582	−	0.995462i	$-0.530336\pi$
$389$	−2.97815e26	−0.190316	−0.0951582	+	0.995462i	$0.530336\pi$
$390$	0	0
$391$	1.92192e27	1.16380
$392$	0	0
$393$	1.44833e27	0.831277
$394$	0	0
$395$	2.82039e27	1.53476
$396$	0	0
$397$	6.36504e26	0.328474	0.164237	−	0.986421i	$-0.447484\pi$
$397$	6.36504e26	0.328474	0.164237	+	0.986421i	$0.447484\pi$
$398$	0	0
$399$	4.28964e26	0.209993
$400$	0	0
$401$	2.43888e27	1.13286	0.566428	−	0.824111i	$-0.308325\pi$
$401$	2.43888e27	1.13286	0.566428	+	0.824111i	$0.308325\pi$
$402$	0	0
$403$	−8.99728e26	−0.396651
$404$	0	0
$405$	−5.04698e26	−0.211229
$406$	0	0
$407$	2.20805e26	0.0877542
$408$	0	0
$409$	5.48032e26	0.206876	0.103438	−	0.994636i	$-0.467016\pi$
$409$	5.48032e26	0.206876	0.103438	+	0.994636i	$0.467016\pi$
$410$	0	0
$411$	6.07232e26	0.217779
$412$	0	0
$413$	8.26399e26	0.281653
$414$	0	0
$415$	1.63991e27	0.531269
$416$	0	0
$417$	−5.14135e26	−0.158361
$418$	0	0
$419$	6.08246e27	1.78169	0.890844	−	0.454309i	$-0.150114\pi$
$419$	6.08246e27	1.78169	0.890844	+	0.454309i	$0.150114\pi$
$420$	0	0
$421$	−4.05990e27	−1.13124	−0.565618	−	0.824667i	$-0.691362\pi$
$421$	−4.05990e27	−1.13124	−0.565618	+	0.824667i	$0.691362\pi$
$422$	0	0
$423$	−9.88659e26	−0.262103
$424$	0	0
$425$	1.02735e28	2.59199
$426$	0	0
$427$	−2.32149e27	−0.557537
$428$	0	0
$429$	−4.95866e26	−0.113387
$430$	0	0
$431$	7.87214e27	1.71428	0.857140	−	0.515084i	$-0.172239\pi$
$431$	7.87214e27	1.71428	0.857140	+	0.515084i	$0.172239\pi$
$432$	0	0
$433$	−1.73785e27	−0.360486	−0.180243	−	0.983622i	$-0.557688\pi$
$433$	−1.73785e27	−0.360486	−0.180243	+	0.983622i	$0.557688\pi$
$434$	0	0
$435$	4.96278e27	0.980817
$436$	0	0
$437$	−3.14615e27	−0.592550
$438$	0	0
$439$	−8.37416e27	−1.50336	−0.751681	−	0.659526i	$-0.770757\pi$
$439$	−8.37416e27	−1.50336	−0.751681	+	0.659526i	$0.770757\pi$
$440$	0	0
$441$	−9.36687e26	−0.160321
$442$	0	0
$443$	3.30286e25	0.00539077	0.00269539	−	0.999996i	$-0.499142\pi$
$443$	3.30286e25	0.00539077	0.00269539	+	0.999996i	$0.499142\pi$
$444$	0	0
$445$	−1.72742e27	−0.268916
$446$	0	0
$447$	−5.33801e27	−0.792776
$448$	0	0
$449$	5.21713e27	0.739341	0.369670	−	0.929163i	$-0.379471\pi$
$449$	5.21713e27	0.739341	0.369670	+	0.929163i	$0.379471\pi$
$450$	0	0
$451$	−1.40038e27	−0.189406
$452$	0	0
$453$	2.80604e27	0.362296
$454$	0	0
$455$	−2.92763e27	−0.360908
$456$	0	0
$457$	2.15211e26	0.0253363	0.0126682	−	0.999920i	$-0.495967\pi$
$457$	2.15211e26	0.0253363	0.0126682	+	0.999920i	$0.495967\pi$
$458$	0	0
$459$	1.69696e27	0.190826
$460$	0	0
$461$	1.68699e28	1.81239	0.906197	−	0.422855i	$-0.138972\pi$
$461$	1.68699e28	1.81239	0.906197	+	0.422855i	$0.138972\pi$
$462$	0	0
$463$	1.90352e28	1.95415	0.977074	−	0.212898i	$-0.0682901\pi$
$463$	1.90352e28	1.95415	0.977074	+	0.212898i	$0.0682901\pi$
$464$	0	0
$465$	−1.68384e28	−1.65213
$466$	0	0
$467$	1.21027e28	1.13515	0.567576	−	0.823321i	$-0.307881\pi$
$467$	1.21027e28	1.13515	0.567576	+	0.823321i	$0.307881\pi$
$468$	0	0
$469$	4.97720e27	0.446344
$470$	0	0
$471$	−8.91053e27	−0.764159
$472$	0	0
$473$	4.60286e27	0.377559
$474$	0	0
$475$	−1.68175e28	−1.31971
$476$	0	0
$477$	−7.57429e27	−0.568721
$478$	0	0
$479$	−6.95253e27	−0.499597	−0.249798	−	0.968298i	$-0.580364\pi$
$479$	−6.95253e27	−0.499597	−0.249798	+	0.968298i	$0.580364\pi$
$480$	0	0
$481$	−4.51095e26	−0.0310274
$482$	0	0
$483$	−7.41382e27	−0.488199
$484$	0	0
$485$	−2.37376e27	−0.149675
$486$	0	0
$487$	1.06412e28	0.642596	0.321298	−	0.946978i	$-0.395881\pi$
$487$	1.06412e28	0.642596	0.321298	+	0.946978i	$0.395881\pi$
$488$	0	0
$489$	2.85504e27	0.165146
$490$	0	0
$491$	−1.68064e28	−0.931361	−0.465681	−	0.884953i	$-0.654190\pi$
$491$	−1.68064e28	−0.931361	−0.465681	+	0.884953i	$0.654190\pi$
$492$	0	0
$493$	−1.66865e28	−0.886079
$494$	0	0
$495$	−9.28011e27	−0.472279
$496$	0	0
$497$	−1.09772e28	−0.535486
$498$	0	0
$499$	5.12285e27	0.239583	0.119792	−	0.992799i	$-0.461777\pi$
$499$	5.12285e27	0.239583	0.119792	+	0.992799i	$0.461777\pi$
$500$	0	0
$501$	−2.82686e25	−0.00126768
$502$	0	0
$503$	−1.99606e28	−0.858442	−0.429221	−	0.903200i	$-0.641212\pi$
$503$	−1.99606e28	−0.858442	−0.429221	+	0.903200i	$0.641212\pi$
$504$	0	0
$505$	−1.79508e28	−0.740499
$506$	0	0
$507$	−1.35759e28	−0.537260
$508$	0	0
$509$	−2.57966e27	−0.0979550	−0.0489775	−	0.998800i	$-0.515596\pi$
$509$	−2.57966e27	−0.0979550	−0.0489775	+	0.998800i	$0.515596\pi$
$510$	0	0
$511$	−8.94749e27	−0.326048
$512$	0	0
$513$	−2.77790e27	−0.0971594
$514$	0	0
$515$	7.64346e28	2.56634
$516$	0	0
$517$	−1.81789e28	−0.586026
$518$	0	0
$519$	−9.55436e27	−0.295763
$520$	0	0
$521$	−2.61230e28	−0.776652	−0.388326	−	0.921522i	$-0.626947\pi$
$521$	−2.61230e28	−0.776652	−0.388326	+	0.921522i	$0.626947\pi$
$522$	0	0
$523$	−6.70750e28	−1.91555	−0.957774	−	0.287523i	$-0.907168\pi$
$523$	−6.70750e28	−1.91555	−0.957774	+	0.287523i	$0.907168\pi$
$524$	0	0
$525$	−3.96300e28	−1.08731
$526$	0	0
$527$	5.66161e28	1.49255
$528$	0	0
$529$	1.49036e28	0.377577
$530$	0	0
$531$	−5.35163e27	−0.130315
$532$	0	0
$533$	2.86092e27	0.0669684
$534$	0	0
$535$	−1.01210e29	−2.27777
$536$	0	0
$537$	5.17492e27	0.111988
$538$	0	0
$539$	−1.72233e28	−0.358454
$540$	0	0
$541$	−2.15196e28	−0.430787	−0.215394	−	0.976527i	$-0.569103\pi$
$541$	−2.15196e28	−0.430787	−0.215394	+	0.976527i	$0.569103\pi$
$542$	0	0
$543$	5.53067e27	0.106507
$544$	0	0
$545$	1.71728e29	3.18185
$546$	0	0
$547$	7.46789e28	1.33147	0.665734	−	0.746189i	$-0.268118\pi$
$547$	7.46789e28	1.33147	0.665734	+	0.746189i	$0.268118\pi$
$548$	0	0
$549$	1.50336e28	0.257960
$550$	0	0
$551$	2.73156e28	0.451148
$552$	0	0
$553$	3.65811e28	0.581625
$554$	0	0
$555$	−8.44223e27	−0.129235
$556$	0	0
$557$	−7.95166e28	−1.17214	−0.586068	−	0.810262i	$-0.699325\pi$
$557$	−7.95166e28	−1.17214	−0.586068	+	0.810262i	$0.699325\pi$
$558$	0	0
$559$	−9.40343e27	−0.133494
$560$	0	0
$561$	3.12028e28	0.426661
$562$	0	0
$563$	−5.46305e28	−0.719609	−0.359805	−	0.933028i	$-0.617157\pi$
$563$	−5.46305e28	−0.719609	−0.359805	+	0.933028i	$0.617157\pi$
$564$	0	0
$565$	−1.44427e29	−1.83290
$566$	0	0
$567$	−6.54605e27	−0.0800492
$568$	0	0
$569$	9.43478e28	1.11187	0.555933	−	0.831227i	$-0.312361\pi$
$569$	9.43478e28	1.11187	0.555933	+	0.831227i	$0.312361\pi$
$570$	0	0
$571$	−8.05027e28	−0.914390	−0.457195	−	0.889367i	$-0.651146\pi$
$571$	−8.05027e28	−0.914390	−0.457195	+	0.889367i	$0.651146\pi$
$572$	0	0
$573$	−7.13655e28	−0.781384
$574$	0	0
$575$	2.90659e29	3.06811
$576$	0	0
$577$	−1.67132e28	−0.170104	−0.0850519	−	0.996377i	$-0.527106\pi$
$577$	−1.67132e28	−0.170104	−0.0850519	+	0.996377i	$0.527106\pi$
$578$	0	0
$579$	−1.05592e29	−1.03635
$580$	0	0
$581$	2.12700e28	0.201334
$582$	0	0
$583$	−1.39272e29	−1.27158
$584$	0	0
$585$	1.89589e28	0.166984
$586$	0	0
$587$	−3.15730e28	−0.268297	−0.134149	−	0.990961i	$-0.542830\pi$
$587$	−3.15730e28	−0.268297	−0.134149	+	0.990961i	$0.542830\pi$
$588$	0	0
$589$	−9.26799e28	−0.759932
$590$	0	0
$591$	1.12762e29	0.892264
$592$	0	0
$593$	−5.48493e27	−0.0418887	−0.0209443	−	0.999781i	$-0.506667\pi$
$593$	−5.48493e27	−0.0418887	−0.0209443	+	0.999781i	$0.506667\pi$
$594$	0	0
$595$	1.84223e29	1.35805
$596$	0	0
$597$	−8.51805e28	−0.606191
$598$	0	0
$599$	−1.25621e29	−0.863136	−0.431568	−	0.902080i	$-0.642040\pi$
$599$	−1.25621e29	−0.863136	−0.431568	+	0.902080i	$0.642040\pi$
$600$	0	0
$601$	3.99325e28	0.264938	0.132469	−	0.991187i	$-0.457709\pi$
$601$	3.99325e28	0.264938	0.132469	+	0.991187i	$0.457709\pi$
$602$	0	0
$603$	−3.22315e28	−0.206514
$604$	0	0
$605$	1.36567e29	0.845115
$606$	0	0
$607$	2.46990e29	1.47638	0.738189	−	0.674594i	$-0.235681\pi$
$607$	2.46990e29	1.47638	0.738189	+	0.674594i	$0.235681\pi$
$608$	0	0
$609$	6.43684e28	0.371699
$610$	0	0
$611$	3.71387e28	0.207202
$612$	0	0
$613$	2.63911e28	0.142273	0.0711364	−	0.997467i	$-0.477337\pi$
$613$	2.63911e28	0.142273	0.0711364	+	0.997467i	$0.477337\pi$
$614$	0	0
$615$	5.35420e28	0.278937
$616$	0	0
$617$	3.09820e29	1.55997	0.779984	−	0.625800i	$-0.215227\pi$
$617$	3.09820e29	1.55997	0.779984	+	0.625800i	$0.215227\pi$
$618$	0	0
$619$	2.50758e29	1.22040	0.610202	−	0.792246i	$-0.291088\pi$
$619$	2.50758e29	1.22040	0.610202	+	0.792246i	$0.291088\pi$
$620$	0	0
$621$	4.80107e28	0.225879
$622$	0	0
$623$	−2.24050e28	−0.101911
$624$	0	0
$625$	7.31956e29	3.21918
$626$	0	0
$627$	−5.10785e28	−0.217235
$628$	0	0
$629$	2.83855e28	0.116752
$630$	0	0
$631$	4.32770e28	0.172167	0.0860833	−	0.996288i	$-0.472565\pi$
$631$	4.32770e28	0.172167	0.0860833	+	0.996288i	$0.472565\pi$
$632$	0	0
$633$	−2.35516e29	−0.906318
$634$	0	0
$635$	5.69309e28	0.211945
$636$	0	0
$637$	3.51864e28	0.126739
$638$	0	0
$639$	7.10863e28	0.247758
$640$	0	0
$641$	−8.73381e28	−0.294574	−0.147287	−	0.989094i	$-0.547054\pi$
$641$	−8.73381e28	−0.294574	−0.147287	+	0.989094i	$0.547054\pi$
$642$	0	0
$643$	4.72013e29	1.54077	0.770386	−	0.637578i	$-0.220063\pi$
$643$	4.72013e29	1.54077	0.770386	+	0.637578i	$0.220063\pi$
$644$	0	0
$645$	−1.75985e29	−0.556030
$646$	0	0
$647$	−1.26799e28	−0.0387812	−0.0193906	−	0.999812i	$-0.506173\pi$
$647$	−1.26799e28	−0.0387812	−0.0193906	+	0.999812i	$0.506173\pi$
$648$	0	0
$649$	−9.84027e28	−0.291365
$650$	0	0
$651$	−2.18397e29	−0.626105
$652$	0	0
$653$	−2.76226e29	−0.766790	−0.383395	−	0.923584i	$-0.625245\pi$
$653$	−2.76226e29	−0.766790	−0.383395	+	0.923584i	$0.625245\pi$
$654$	0	0
$655$	−1.01821e30	−2.73718
$656$	0	0
$657$	5.79425e28	0.150855
$658$	0	0
$659$	−6.46511e28	−0.163034	−0.0815172	−	0.996672i	$-0.525977\pi$
$659$	−6.46511e28	−0.163034	−0.0815172	+	0.996672i	$0.525977\pi$
$660$	0	0
$661$	−2.27730e29	−0.556295	−0.278147	−	0.960538i	$-0.589720\pi$
$661$	−2.27730e29	−0.556295	−0.278147	+	0.960538i	$0.589720\pi$
$662$	0	0
$663$	−6.37459e28	−0.150855
$664$	0	0
$665$	−3.01572e29	−0.691454
$666$	0	0
$667$	−4.72097e29	−1.04884
$668$	0	0
$669$	2.73375e29	0.588551
$670$	0	0
$671$	2.76429e29	0.576763
$672$	0	0
$673$	−3.79243e29	−0.766936	−0.383468	−	0.923554i	$-0.625270\pi$
$673$	−3.79243e29	−0.766936	−0.383468	+	0.923554i	$0.625270\pi$
$674$	0	0
$675$	2.56638e29	0.503073
$676$	0	0
$677$	−3.39717e29	−0.645559	−0.322780	−	0.946474i	$-0.604617\pi$
$677$	−3.39717e29	−0.645559	−0.322780	+	0.946474i	$0.604617\pi$
$678$	0	0
$679$	−3.07882e28	−0.0567220
$680$	0	0
$681$	2.02544e29	0.361805
$682$	0	0
$683$	−4.54742e29	−0.787677	−0.393838	−	0.919180i	$-0.628853\pi$
$683$	−4.54742e29	−0.787677	−0.393838	+	0.919180i	$0.628853\pi$
$684$	0	0
$685$	−4.26897e29	−0.717088
$686$	0	0
$687$	4.79355e28	0.0780929
$688$	0	0
$689$	2.84526e29	0.449594
$690$	0	0
$691$	3.71838e29	0.569947	0.284974	−	0.958535i	$-0.408015\pi$
$691$	3.71838e29	0.569947	0.284974	+	0.958535i	$0.408015\pi$
$692$	0	0
$693$	−1.20365e29	−0.178979
$694$	0	0
$695$	3.61448e29	0.521442
$696$	0	0
$697$	−1.80026e29	−0.251994
$698$	0	0
$699$	4.85494e28	0.0659437
$700$	0	0
$701$	−9.37969e29	−1.23637	−0.618186	−	0.786032i	$-0.712132\pi$
$701$	−9.37969e29	−1.23637	−0.618186	+	0.786032i	$0.712132\pi$
$702$	0	0
$703$	−4.64668e28	−0.0594445
$704$	0	0
$705$	6.95050e29	0.863038
$706$	0	0
$707$	−2.32826e29	−0.280626
$708$	0	0
$709$	−7.54578e29	−0.882914	−0.441457	−	0.897282i	$-0.645538\pi$
$709$	−7.54578e29	−0.882914	−0.441457	+	0.897282i	$0.645538\pi$
$710$	0	0
$711$	−2.36893e29	−0.269105
$712$	0	0
$713$	1.60179e30	1.76671
$714$	0	0
$715$	3.48605e29	0.373353
$716$	0	0
$717$	5.22894e29	0.543829
$718$	0	0
$719$	−1.22754e30	−1.23989	−0.619944	−	0.784646i	$-0.712845\pi$
$719$	−1.22754e30	−1.23989	−0.619944	+	0.784646i	$0.712845\pi$
$720$	0	0
$721$	9.91374e29	0.972561
$722$	0	0
$723$	4.41057e29	0.420284
$724$	0	0
$725$	−2.52356e30	−2.33596
$726$	0	0
$727$	−9.04407e29	−0.813303	−0.406652	−	0.913583i	$-0.633304\pi$
$727$	−9.04407e29	−0.813303	−0.406652	+	0.913583i	$0.633304\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	5.91719e29	0.502323
$732$	0	0
$733$	−1.38874e30	−1.14559	−0.572795	−	0.819699i	$-0.694141\pi$
$733$	−1.38874e30	−1.14559	−0.572795	+	0.819699i	$0.694141\pi$
$734$	0	0
$735$	6.58512e29	0.527894
$736$	0	0
$737$	−5.92655e29	−0.461736
$738$	0	0
$739$	−2.11506e30	−1.60161	−0.800804	−	0.598927i	$-0.795594\pi$
$739$	−2.11506e30	−1.60161	−0.800804	+	0.598927i	$0.795594\pi$
$740$	0	0
$741$	1.04351e29	0.0768080
$742$	0	0
$743$	−2.26805e30	−1.62282	−0.811411	−	0.584476i	$-0.801300\pi$
$743$	−2.26805e30	−1.62282	−0.811411	+	0.584476i	$0.801300\pi$
$744$	0	0
$745$	3.75274e30	2.61040
$746$	0	0
$747$	−1.37741e29	−0.0931530
$748$	0	0
$749$	−1.31272e30	−0.863202
$750$	0	0
$751$	6.98349e29	0.446533	0.223266	−	0.974757i	$-0.428328\pi$
$751$	6.98349e29	0.446533	0.223266	+	0.974757i	$0.428328\pi$
$752$	0	0
$753$	−5.58883e28	−0.0347515
$754$	0	0
$755$	−1.97271e30	−1.19294
$756$	0	0
$757$	−3.79778e29	−0.223369	−0.111685	−	0.993744i	$-0.535625\pi$
$757$	−3.79778e29	−0.223369	−0.111685	+	0.993744i	$0.535625\pi$
$758$	0	0
$759$	8.82794e29	0.505034
$760$	0	0
$761$	−7.50371e29	−0.417577	−0.208789	−	0.977961i	$-0.566952\pi$
$761$	−7.50371e29	−0.417577	−0.208789	+	0.977961i	$0.566952\pi$
$762$	0	0
$763$	2.22736e30	1.20582
$764$	0	0
$765$	−1.19300e30	−0.628342
$766$	0	0
$767$	2.01032e29	0.103018
$768$	0	0
$769$	2.16884e29	0.108144	0.0540719	−	0.998537i	$-0.482780\pi$
$769$	2.16884e29	0.108144	0.0540719	+	0.998537i	$0.482780\pi$
$770$	0	0
$771$	−1.13271e30	−0.549600
$772$	0	0
$773$	−3.08128e30	−1.45495	−0.727473	−	0.686136i	$-0.759305\pi$
$773$	−3.08128e30	−1.45495	−0.727473	+	0.686136i	$0.759305\pi$
$774$	0	0
$775$	8.56227e30	3.93478
$776$	0	0
$777$	−1.09498e29	−0.0489761
$778$	0	0
$779$	2.94700e29	0.128303
$780$	0	0
$781$	1.30710e30	0.553951
$782$	0	0
$783$	−4.16839e29	−0.171977
$784$	0	0
$785$	6.26430e30	2.51618
$786$	0	0
$787$	−1.46460e30	−0.572775	−0.286387	−	0.958114i	$-0.592454\pi$
$787$	−1.46460e30	−0.572775	−0.286387	+	0.958114i	$0.592454\pi$
$788$	0	0
$789$	−5.24608e28	−0.0199768
$790$	0	0
$791$	−1.87325e30	−0.694612
$792$	0	0
$793$	−5.64732e29	−0.203927
$794$	0	0
$795$	5.32490e30	1.87265
$796$	0	0
$797$	1.27731e30	0.437505	0.218752	−	0.975780i	$-0.429801\pi$
$797$	1.27731e30	0.437505	0.218752	+	0.975780i	$0.429801\pi$
$798$	0	0
$799$	−2.33698e30	−0.779677
$800$	0	0
$801$	1.45091e29	0.0471519
$802$	0	0
$803$	1.06541e30	0.337292
$804$	0	0
$805$	5.21208e30	1.60751
$806$	0	0
$807$	−1.26275e30	−0.379441
$808$	0	0
$809$	−4.24975e30	−1.24424	−0.622120	−	0.782922i	$-0.713728\pi$
$809$	−4.24975e30	−1.24424	−0.622120	+	0.782922i	$0.713728\pi$
$810$	0	0
$811$	−2.05863e30	−0.587298	−0.293649	−	0.955913i	$-0.594870\pi$
$811$	−2.05863e30	−0.587298	−0.293649	+	0.955913i	$0.594870\pi$
$812$	0	0
$813$	9.23732e29	0.256800
$814$	0	0
$815$	−2.00716e30	−0.543784
$816$	0	0
$817$	−9.68636e29	−0.255758
$818$	0	0
$819$	2.45901e29	0.0632818
$820$	0	0
$821$	−1.80717e29	−0.0453309	−0.0226655	−	0.999743i	$-0.507215\pi$
$821$	−1.80717e29	−0.0453309	−0.0226655	+	0.999743i	$0.507215\pi$
$822$	0	0
$823$	6.23532e30	1.52462	0.762308	−	0.647214i	$-0.224066\pi$
$823$	6.23532e30	1.52462	0.762308	+	0.647214i	$0.224066\pi$
$824$	0	0
$825$	4.71891e30	1.12480
$826$	0	0
$827$	−3.37179e30	−0.783524	−0.391762	−	0.920067i	$-0.628134\pi$
$827$	−3.37179e30	−0.783524	−0.391762	+	0.920067i	$0.628134\pi$
$828$	0	0
$829$	−2.70711e29	−0.0613315	−0.0306658	−	0.999530i	$-0.509763\pi$
$829$	−2.70711e29	−0.0613315	−0.0306658	+	0.999530i	$0.509763\pi$
$830$	0	0
$831$	−4.75328e30	−1.04998
$832$	0	0
$833$	−2.21413e30	−0.476904
$834$	0	0
$835$	1.98735e28	0.00417413
$836$	0	0
$837$	1.41431e30	0.289685
$838$	0	0
$839$	7.45457e30	1.48909	0.744547	−	0.667570i	$-0.232666\pi$
$839$	7.45457e30	1.48909	0.744547	+	0.667570i	$0.232666\pi$
$840$	0	0
$841$	−1.03399e30	−0.201446
$842$	0	0
$843$	4.91979e30	0.934882
$844$	0	0
$845$	9.54415e30	1.76906
$846$	0	0
$847$	1.77131e30	0.320272
$848$	0	0
$849$	2.63435e29	0.0464668
$850$	0	0
$851$	8.03088e29	0.138198
$852$	0	0
$853$	−6.10653e30	−1.02525	−0.512625	−	0.858613i	$-0.671327\pi$
$853$	−6.10653e30	−1.02525	−0.512625	+	0.858613i	$0.671327\pi$
$854$	0	0
$855$	1.95293e30	0.319921
$856$	0	0
$857$	−4.08307e30	−0.652662	−0.326331	−	0.945256i	$-0.605812\pi$
$857$	−4.08307e30	−0.652662	−0.326331	+	0.945256i	$0.605812\pi$
$858$	0	0
$859$	−5.27189e30	−0.822316	−0.411158	−	0.911564i	$-0.634875\pi$
$859$	−5.27189e30	−0.822316	−0.411158	+	0.911564i	$0.634875\pi$
$860$	0	0
$861$	6.94452e29	0.105708
$862$	0	0
$863$	−1.05537e31	−1.56780	−0.783902	−	0.620884i	$-0.786774\pi$
$863$	−1.05537e31	−1.56780	−0.783902	+	0.620884i	$0.786774\pi$
$864$	0	0
$865$	6.71693e30	0.973872
$866$	0	0
$867$	−6.85488e28	−0.00970062
$868$	0	0
$869$	−4.35586e30	−0.601681
$870$	0	0
$871$	1.21077e30	0.163256
$872$	0	0
$873$	1.99379e29	0.0262441
$874$	0	0
$875$	1.72027e31	2.21061
$876$	0	0
$877$	7.16069e30	0.898378	0.449189	−	0.893437i	$-0.351713\pi$
$877$	7.16069e30	0.898378	0.449189	+	0.893437i	$0.351713\pi$
$878$	0	0
$879$	5.71079e30	0.699541
$880$	0	0
$881$	−1.05943e31	−1.26714	−0.633568	−	0.773687i	$-0.718410\pi$
$881$	−1.05943e31	−1.26714	−0.633568	+	0.773687i	$0.718410\pi$
$882$	0	0
$883$	−6.60744e30	−0.771695	−0.385848	−	0.922562i	$-0.626091\pi$
$883$	−6.60744e30	−0.771695	−0.385848	+	0.922562i	$0.626091\pi$
$884$	0	0
$885$	3.76231e30	0.429092
$886$	0	0
$887$	2.74467e30	0.305698	0.152849	−	0.988250i	$-0.451155\pi$
$887$	2.74467e30	0.305698	0.152849	+	0.988250i	$0.451155\pi$
$888$	0	0
$889$	7.38406e29	0.0803205
$890$	0	0
$891$	7.79465e29	0.0828096
$892$	0	0
$893$	3.82561e30	0.396973
$894$	0	0
$895$	−3.63808e30	−0.368749
$896$	0	0
$897$	−1.80351e30	−0.178565
$898$	0	0
$899$	−1.39071e31	−1.34512
$900$	0	0
$901$	−1.79040e31	−1.69177
$902$	0	0
$903$	−2.28256e30	−0.210718
$904$	0	0
$905$	−3.88819e30	−0.350701
$906$	0	0
$907$	−5.45568e30	−0.480809	−0.240404	−	0.970673i	$-0.577280\pi$
$907$	−5.45568e30	−0.480809	−0.240404	+	0.970673i	$0.577280\pi$
$908$	0	0
$909$	1.50774e30	0.129839
$910$	0	0
$911$	8.75739e30	0.736940	0.368470	−	0.929640i	$-0.379882\pi$
$911$	8.75739e30	0.736940	0.368470	+	0.929640i	$0.379882\pi$
$912$	0	0
$913$	−2.53270e30	−0.208277
$914$	0	0
$915$	−1.05689e31	−0.849396
$916$	0	0
$917$	−1.32064e31	−1.03730
$918$	0	0
$919$	−1.24295e31	−0.954206	−0.477103	−	0.878847i	$-0.658313\pi$
$919$	−1.24295e31	−0.954206	−0.477103	+	0.878847i	$0.658313\pi$
$920$	0	0
$921$	−9.92984e30	−0.745102
$922$	0	0
$923$	−2.67034e30	−0.195861
$924$	0	0
$925$	4.29285e30	0.307792
$926$	0	0
$927$	−6.41997e30	−0.449983
$928$	0	0
$929$	2.24310e31	1.53703	0.768517	−	0.639829i	$-0.220995\pi$
$929$	2.24310e31	1.53703	0.768517	+	0.639829i	$0.220995\pi$
$930$	0	0
$931$	3.62451e30	0.242816
$932$	0	0
$933$	−1.36031e31	−0.891008
$934$	0	0
$935$	−2.19362e31	−1.40488
$936$	0	0
$937$	−1.10052e31	−0.689176	−0.344588	−	0.938754i	$-0.611981\pi$
$937$	−1.10052e31	−0.689176	−0.344588	+	0.938754i	$0.611981\pi$
$938$	0	0
$939$	−1.65135e31	−1.01123
$940$	0	0
$941$	−2.38036e31	−1.42545	−0.712725	−	0.701444i	$-0.752539\pi$
$941$	−2.38036e31	−1.42545	−0.712725	+	0.701444i	$0.752539\pi$
$942$	0	0
$943$	−5.09332e30	−0.298283
$944$	0	0
$945$	4.60202e30	0.263581
$946$	0	0
$947$	−8.09762e30	−0.453610	−0.226805	−	0.973940i	$-0.572828\pi$
$947$	−8.09762e30	−0.453610	−0.226805	+	0.973940i	$0.572828\pi$
$948$	0	0
$949$	−2.17660e30	−0.119257
$950$	0	0
$951$	1.76262e30	0.0944631
$952$	0	0
$953$	−3.42232e30	−0.179410	−0.0897048	−	0.995968i	$-0.528592\pi$
$953$	−3.42232e30	−0.179410	−0.0897048	+	0.995968i	$0.528592\pi$
$954$	0	0
$955$	5.01715e31	2.57290
$956$	0	0
$957$	−7.66461e30	−0.384516
$958$	0	0
$959$	−5.53695e30	−0.271754
$960$	0	0
$961$	2.63603e31	1.26577
$962$	0	0
$963$	8.50095e30	0.399385
$964$	0	0
$965$	7.42339e31	3.41244
$966$	0	0
$967$	−1.33121e31	−0.598780	−0.299390	−	0.954131i	$-0.596783\pi$
$967$	−1.33121e31	−0.598780	−0.299390	+	0.954131i	$0.596783\pi$
$968$	0	0
$969$	−6.56638e30	−0.289019
$970$	0	0
$971$	−9.71774e30	−0.418565	−0.209283	−	0.977855i	$-0.567113\pi$
$971$	−9.71774e30	−0.418565	−0.209283	+	0.977855i	$0.567113\pi$
$972$	0	0
$973$	4.68806e30	0.197610
$974$	0	0
$975$	−9.64053e30	−0.397697
$976$	0	0
$977$	−1.40637e31	−0.567816	−0.283908	−	0.958851i	$-0.591631\pi$
$977$	−1.40637e31	−0.567816	−0.283908	+	0.958851i	$0.591631\pi$
$978$	0	0
$979$	2.66785e30	0.105425
$980$	0	0
$981$	−1.44240e31	−0.557906
$982$	0	0
$983$	4.87291e31	1.84492	0.922458	−	0.386098i	$-0.126177\pi$
$983$	4.87291e31	1.84492	0.922458	+	0.386098i	$0.126177\pi$
$984$	0	0
$985$	−7.92741e31	−2.93799
$986$	0	0
$987$	9.01495e30	0.327064
$988$	0	0
$989$	1.67410e31	0.594594
$990$	0	0
$991$	−1.17847e30	−0.0409773	−0.0204887	−	0.999790i	$-0.506522\pi$
$991$	−1.17847e30	−0.0409773	−0.0204887	+	0.999790i	$0.506522\pi$
$992$	0	0
$993$	−1.50926e31	−0.513803
$994$	0	0
$995$	5.98838e31	1.99603
$996$	0	0
$997$	5.05438e31	1.64956	0.824781	−	0.565453i	$-0.191299\pi$
$997$	5.05438e31	1.64956	0.824781	+	0.565453i	$0.191299\pi$
$998$	0	0
$999$	7.09089e29	0.0226602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.a.d.1.1		1
4.3	odd	2		3.22.a.b.1.1	✓	1
12.11	even	2		9.22.a.a.1.1		1
20.3	even	4		75.22.b.b.49.1		2
20.7	even	4		75.22.b.b.49.2		2
20.19	odd	2		75.22.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.22.a.b.1.1	✓	1	4.3	odd	2
9.22.a.a.1.1		1	12.11	even	2
48.22.a.d.1.1		1	1.1	even	1	trivial
75.22.a.a.1.1		1	20.19	odd	2
75.22.b.b.49.1		2	20.3	even	4
75.22.b.b.49.2		2	20.7	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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