LMFDB - Embedded newform 48.12.a.a.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 6)
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)

$f(q)$

$=$

$q-243.000 q^{3} -11730.0 q^{5} +50008.0 q^{7} +59049.0 q^{9} +O(q^{10})$

$q-243.000 q^{3} -11730.0 q^{5} +50008.0 q^{7} +59049.0 q^{9} +531420. q^{11} +1.33257e6 q^{13} +2.85039e6 q^{15} -5.10968e6 q^{17} -2.90140e6 q^{19} -1.21519e7 q^{21} -3.05970e7 q^{23} +8.87648e7 q^{25} -1.43489e7 q^{27} -7.70066e7 q^{29} +2.39418e8 q^{31} -1.29135e8 q^{33} -5.86594e8 q^{35} -7.85042e8 q^{37} -3.23814e8 q^{39} +4.11253e8 q^{41} -3.51233e8 q^{43} -6.92645e8 q^{45} -9.58217e7 q^{47} +5.23473e8 q^{49} +1.24165e9 q^{51} -1.46586e9 q^{53} -6.23356e9 q^{55} +7.05041e8 q^{57} -5.62115e9 q^{59} -1.04736e10 q^{61} +2.95292e9 q^{63} -1.56310e10 q^{65} -4.51531e9 q^{67} +7.43507e9 q^{69} +8.50958e9 q^{71} +2.01250e9 q^{73} -2.15698e10 q^{75} +2.65753e10 q^{77} +2.22384e10 q^{79} +3.48678e9 q^{81} -6.32865e9 q^{83} +5.99365e10 q^{85} +1.87126e10 q^{87} -5.01237e10 q^{89} +6.66390e10 q^{91} -5.81787e10 q^{93} +3.40335e10 q^{95} +9.48060e10 q^{97} +3.13798e10 q^{99} +O(q^{100})$

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
$5$	−11730.0	−1.67866	−0.839330	−	0.543622i	$-0.817053\pi$
$5$	−11730.0	−1.67866	−0.839330	+	0.543622i	$0.817053\pi$
$6$	0	0
$7$	50008.0	1.12461	0.562303	−	0.826931i	$-0.309916\pi$
$7$	50008.0	1.12461	0.562303	+	0.826931i	$0.309916\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	531420.	0.994897	0.497449	−	0.867494i	$-0.334270\pi$
$11$	531420.	0.994897	0.497449	+	0.867494i	$0.334270\pi$
$12$	0	0
$13$	1.33257e6	0.995406	0.497703	−	0.867347i	$-0.334177\pi$
$13$	1.33257e6	0.995406	0.497703	+	0.867347i	$0.334177\pi$
$14$	0	0
$15$	2.85039e6	0.969175
$16$	0	0
$17$	−5.10968e6	−0.872820	−0.436410	−	0.899748i	$-0.643750\pi$
$17$	−5.10968e6	−0.872820	−0.436410	+	0.899748i	$0.643750\pi$
$18$	0	0
$19$	−2.90140e6	−0.268821	−0.134411	−	0.990926i	$-0.542914\pi$
$19$	−2.90140e6	−0.268821	−0.134411	+	0.990926i	$0.542914\pi$
$20$	0	0
$21$	−1.21519e7	−0.649291
$22$	0	0
$23$	−3.05970e7	−0.991233	−0.495616	−	0.868541i	$-0.665058\pi$
$23$	−3.05970e7	−0.991233	−0.495616	+	0.868541i	$0.665058\pi$
$24$	0	0
$25$	8.87648e7	1.81790
$26$	0	0
$27$	−1.43489e7	−0.192450
$28$	0	0
$29$	−7.70066e7	−0.697171	−0.348585	−	0.937277i	$-0.613338\pi$
$29$	−7.70066e7	−0.697171	−0.348585	+	0.937277i	$0.613338\pi$
$30$	0	0
$31$	2.39418e8	1.50199	0.750997	−	0.660306i	$-0.229573\pi$
$31$	2.39418e8	1.50199	0.750997	+	0.660306i	$0.229573\pi$
$32$	0	0
$33$	−1.29135e8	−0.574404
$34$	0	0
$35$	−5.86594e8	−1.88783
$36$	0	0
$37$	−7.85042e8	−1.86116	−0.930579	−	0.366091i	$-0.880696\pi$
$37$	−7.85042e8	−1.86116	−0.930579	+	0.366091i	$0.880696\pi$
$38$	0	0
$39$	−3.23814e8	−0.574698
$40$	0	0
$41$	4.11253e8	0.554368	0.277184	−	0.960817i	$-0.410599\pi$
$41$	4.11253e8	0.554368	0.277184	+	0.960817i	$0.410599\pi$
$42$	0	0
$43$	−3.51233e8	−0.364350	−0.182175	−	0.983266i	$-0.558314\pi$
$43$	−3.51233e8	−0.364350	−0.182175	+	0.983266i	$0.558314\pi$
$44$	0	0
$45$	−6.92645e8	−0.559554
$46$	0	0
$47$	−9.58217e7	−0.0609432	−0.0304716	−	0.999536i	$-0.509701\pi$
$47$	−9.58217e7	−0.0609432	−0.0304716	+	0.999536i	$0.509701\pi$
$48$	0	0
$49$	5.23473e8	0.264738
$50$	0	0
$51$	1.24165e9	0.503923
$52$	0	0
$53$	−1.46586e9	−0.481476	−0.240738	−	0.970590i	$-0.577389\pi$
$53$	−1.46586e9	−0.481476	−0.240738	+	0.970590i	$0.577389\pi$
$54$	0	0
$55$	−6.23356e9	−1.67009
$56$	0	0
$57$	7.05041e8	0.155204
$58$	0	0
$59$	−5.62115e9	−1.02362	−0.511811	−	0.859098i	$-0.671025\pi$
$59$	−5.62115e9	−1.02362	−0.511811	+	0.859098i	$0.671025\pi$
$60$	0	0
$61$	−1.04736e10	−1.58775	−0.793874	−	0.608083i	$-0.791939\pi$
$61$	−1.04736e10	−1.58775	−0.793874	+	0.608083i	$0.791939\pi$
$62$	0	0
$63$	2.95292e9	0.374869
$64$	0	0
$65$	−1.56310e10	−1.67095
$66$	0	0
$67$	−4.51531e9	−0.408579	−0.204289	−	0.978911i	$-0.565488\pi$
$67$	−4.51531e9	−0.408579	−0.204289	+	0.978911i	$0.565488\pi$
$68$	0	0
$69$	7.43507e9	0.572289
$70$	0	0
$71$	8.50958e9	0.559741	0.279871	−	0.960038i	$-0.409708\pi$
$71$	8.50958e9	0.559741	0.279871	+	0.960038i	$0.409708\pi$
$72$	0	0
$73$	2.01250e9	0.113621	0.0568106	−	0.998385i	$-0.481907\pi$
$73$	2.01250e9	0.113621	0.0568106	+	0.998385i	$0.481907\pi$
$74$	0	0
$75$	−2.15698e10	−1.04957
$76$	0	0
$77$	2.65753e10	1.11887
$78$	0	0
$79$	2.22384e10	0.813120	0.406560	−	0.913624i	$-0.366728\pi$
$79$	2.22384e10	0.813120	0.406560	+	0.913624i	$0.366728\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	−6.32865e9	−0.176352	−0.0881762	−	0.996105i	$-0.528104\pi$
$83$	−6.32865e9	−0.176352	−0.0881762	+	0.996105i	$0.528104\pi$
$84$	0	0
$85$	5.99365e10	1.46517
$86$	0	0
$87$	1.87126e10	0.402512
$88$	0	0
$89$	−5.01237e10	−0.951477	−0.475738	−	0.879587i	$-0.657819\pi$
$89$	−5.01237e10	−0.951477	−0.475738	+	0.879587i	$0.657819\pi$
$90$	0	0
$91$	6.66390e10	1.11944
$92$	0	0
$93$	−5.81787e10	−0.867176
$94$	0	0
$95$	3.40335e10	0.451260
$96$	0	0
$97$	9.48060e10	1.12096	0.560481	−	0.828167i	$-0.310616\pi$
$97$	9.48060e10	1.12096	0.560481	+	0.828167i	$0.310616\pi$
$98$	0	0
$99$	3.13798e10	0.331632
$100$	0	0
$101$	1.37952e10	0.130605	0.0653026	−	0.997866i	$-0.479199\pi$
$101$	1.37952e10	0.130605	0.0653026	+	0.997866i	$0.479199\pi$
$102$	0	0
$103$	−6.51836e10	−0.554031	−0.277015	−	0.960865i	$-0.589345\pi$
$103$	−6.51836e10	−0.554031	−0.277015	+	0.960865i	$0.589345\pi$
$104$	0	0
$105$	1.42542e11	1.08994
$106$	0	0
$107$	−9.33399e10	−0.643363	−0.321682	−	0.946848i	$-0.604248\pi$
$107$	−9.33399e10	−0.643363	−0.321682	+	0.946848i	$0.604248\pi$
$108$	0	0
$109$	−1.51369e11	−0.942307	−0.471154	−	0.882051i	$-0.656162\pi$
$109$	−1.51369e11	−0.942307	−0.471154	+	0.882051i	$0.656162\pi$
$110$	0	0
$111$	1.90765e11	1.07454
$112$	0	0
$113$	2.37349e11	1.21187	0.605935	−	0.795514i	$-0.292799\pi$
$113$	2.37349e11	1.21187	0.605935	+	0.795514i	$0.292799\pi$
$114$	0	0
$115$	3.58903e11	1.66394
$116$	0	0
$117$	7.86867e10	0.331802
$118$	0	0
$119$	−2.55525e11	−0.981578
$120$	0	0
$121$	−2.90445e9	−0.0101799
$122$	0	0
$123$	−9.99345e10	−0.320064
$124$	0	0
$125$	−4.68457e11	−1.37298
$126$	0	0
$127$	−5.14414e11	−1.38163	−0.690816	−	0.723031i	$-0.742749\pi$
$127$	−5.14414e11	−1.38163	−0.690816	+	0.723031i	$0.742749\pi$
$128$	0	0
$129$	8.53497e10	0.210358
$130$	0	0
$131$	−2.98572e11	−0.676171	−0.338086	−	0.941115i	$-0.609779\pi$
$131$	−2.98572e11	−0.676171	−0.338086	+	0.941115i	$0.609779\pi$
$132$	0	0
$133$	−1.45093e11	−0.302318
$134$	0	0
$135$	1.68313e11	0.323058
$136$	0	0
$137$	−6.16116e11	−1.09069	−0.545343	−	0.838213i	$-0.683600\pi$
$137$	−6.16116e11	−1.09069	−0.545343	+	0.838213i	$0.683600\pi$
$138$	0	0
$139$	5.22814e11	0.854606	0.427303	−	0.904109i	$-0.359464\pi$
$139$	5.22814e11	0.854606	0.427303	+	0.904109i	$0.359464\pi$
$140$	0	0
$141$	2.32847e10	0.0351856
$142$	0	0
$143$	7.08152e11	0.990327
$144$	0	0
$145$	9.03288e11	1.17031
$146$	0	0
$147$	−1.27204e11	−0.152846
$148$	0	0
$149$	−1.27015e12	−1.41687	−0.708434	−	0.705777i	$-0.750598\pi$
$149$	−1.27015e12	−1.41687	−0.708434	+	0.705777i	$0.750598\pi$
$150$	0	0
$151$	−1.25371e12	−1.29964	−0.649822	−	0.760086i	$-0.725157\pi$
$151$	−1.25371e12	−1.29964	−0.649822	+	0.760086i	$0.725157\pi$
$152$	0	0
$153$	−3.01721e11	−0.290940
$154$	0	0
$155$	−2.80838e12	−2.52134
$156$	0	0
$157$	6.06708e11	0.507611	0.253806	−	0.967255i	$-0.418318\pi$
$157$	6.06708e11	0.507611	0.253806	+	0.967255i	$0.418318\pi$
$158$	0	0
$159$	3.56203e11	0.277980
$160$	0	0
$161$	−1.53009e12	−1.11475
$162$	0	0
$163$	1.58401e12	1.07827	0.539133	−	0.842221i	$-0.318752\pi$
$163$	1.58401e12	1.07827	0.539133	+	0.842221i	$0.318752\pi$
$164$	0	0
$165$	1.51475e12	0.964230
$166$	0	0
$167$	−2.95272e12	−1.75907	−0.879533	−	0.475837i	$-0.842145\pi$
$167$	−2.95272e12	−1.75907	−0.879533	+	0.475837i	$0.842145\pi$
$168$	0	0
$169$	−1.64282e10	−0.00916673
$170$	0	0
$171$	−1.71325e11	−0.0896070
$172$	0	0
$173$	−2.65410e12	−1.30216	−0.651079	−	0.759010i	$-0.725684\pi$
$173$	−2.65410e12	−1.30216	−0.651079	+	0.759010i	$0.725684\pi$
$174$	0	0
$175$	4.43895e12	2.04442
$176$	0	0
$177$	1.36594e12	0.590988
$178$	0	0
$179$	4.06450e12	1.65316	0.826580	−	0.562819i	$-0.190283\pi$
$179$	4.06450e12	1.65316	0.826580	+	0.562819i	$0.190283\pi$
$180$	0	0
$181$	−1.68073e12	−0.643082	−0.321541	−	0.946896i	$-0.604201\pi$
$181$	−1.68073e12	−0.643082	−0.321541	+	0.946896i	$0.604201\pi$
$182$	0	0
$183$	2.54508e12	0.916686
$184$	0	0
$185$	9.20854e12	3.12425
$186$	0	0
$187$	−2.71539e12	−0.868366
$188$	0	0
$189$	−7.17560e11	−0.216430
$190$	0	0
$191$	−4.09135e11	−0.116462	−0.0582309	−	0.998303i	$-0.518546\pi$
$191$	−4.09135e11	−0.116462	−0.0582309	+	0.998303i	$0.518546\pi$
$192$	0	0
$193$	4.71906e12	1.26850	0.634249	−	0.773129i	$-0.281309\pi$
$193$	4.71906e12	1.26850	0.634249	+	0.773129i	$0.281309\pi$
$194$	0	0
$195$	3.79833e12	0.964723
$196$	0	0
$197$	−3.60899e12	−0.866605	−0.433303	−	0.901249i	$-0.642652\pi$
$197$	−3.60899e12	−0.866605	−0.433303	+	0.901249i	$0.642652\pi$
$198$	0	0
$199$	−1.72140e12	−0.391012	−0.195506	−	0.980703i	$-0.562635\pi$
$199$	−1.72140e12	−0.391012	−0.195506	+	0.980703i	$0.562635\pi$
$200$	0	0
$201$	1.09722e12	0.235893
$202$	0	0
$203$	−3.85095e12	−0.784042
$204$	0	0
$205$	−4.82400e12	−0.930595
$206$	0	0
$207$	−1.80672e12	−0.330411
$208$	0	0
$209$	−1.54186e12	−0.267449
$210$	0	0
$211$	−1.70345e12	−0.280398	−0.140199	−	0.990123i	$-0.544774\pi$
$211$	−1.70345e12	−0.280398	−0.140199	+	0.990123i	$0.544774\pi$
$212$	0	0
$213$	−2.06783e12	−0.323167
$214$	0	0
$215$	4.11997e12	0.611621
$216$	0	0
$217$	1.19728e13	1.68915
$218$	0	0
$219$	−4.89037e11	−0.0655993
$220$	0	0
$221$	−6.80898e12	−0.868810
$222$	0	0
$223$	1.46253e13	1.77594	0.887969	−	0.459904i	$-0.152116\pi$
$223$	1.46253e13	1.77594	0.887969	+	0.459904i	$0.152116\pi$
$224$	0	0
$225$	5.24147e12	0.605968
$226$	0	0
$227$	−3.36457e12	−0.370499	−0.185249	−	0.982692i	$-0.559309\pi$
$227$	−3.36457e12	−0.370499	−0.185249	+	0.982692i	$0.559309\pi$
$228$	0	0
$229$	1.89920e12	0.199286	0.0996429	−	0.995023i	$-0.468230\pi$
$229$	1.89920e12	0.199286	0.0996429	+	0.995023i	$0.468230\pi$
$230$	0	0
$231$	−6.45779e12	−0.645978
$232$	0	0
$233$	1.51901e12	0.144912	0.0724560	−	0.997372i	$-0.476916\pi$
$233$	1.51901e12	0.144912	0.0724560	+	0.997372i	$0.476916\pi$
$234$	0	0
$235$	1.12399e12	0.102303
$236$	0	0
$237$	−5.40393e12	−0.469455
$238$	0	0
$239$	−8.31497e12	−0.689719	−0.344859	−	0.938654i	$-0.612073\pi$
$239$	−8.31497e12	−0.689719	−0.344859	+	0.938654i	$0.612073\pi$
$240$	0	0
$241$	−1.30126e13	−1.03103	−0.515515	−	0.856881i	$-0.672399\pi$
$241$	−1.30126e13	−1.03103	−0.515515	+	0.856881i	$0.672399\pi$
$242$	0	0
$243$	−8.47289e11	−0.0641500
$244$	0	0
$245$	−6.14034e12	−0.444405
$246$	0	0
$247$	−3.86631e12	−0.267586
$248$	0	0
$249$	1.53786e12	0.101817
$250$	0	0
$251$	1.87138e13	1.18565	0.592825	−	0.805331i	$-0.298013\pi$
$251$	1.87138e13	1.18565	0.592825	+	0.805331i	$0.298013\pi$
$252$	0	0
$253$	−1.62599e13	−0.986175
$254$	0	0
$255$	−1.45646e13	−0.845915
$256$	0	0
$257$	−5.48427e12	−0.305131	−0.152566	−	0.988293i	$-0.548754\pi$
$257$	−5.48427e12	−0.305131	−0.152566	+	0.988293i	$0.548754\pi$
$258$	0	0
$259$	−3.92584e13	−2.09307
$260$	0	0
$261$	−4.54716e12	−0.232390
$262$	0	0
$263$	−1.95941e13	−0.960218	−0.480109	−	0.877209i	$-0.659403\pi$
$263$	−1.95941e13	−0.960218	−0.480109	+	0.877209i	$0.659403\pi$
$264$	0	0
$265$	1.71945e13	0.808235
$266$	0	0
$267$	1.21801e13	0.549335
$268$	0	0
$269$	3.57029e13	1.54549	0.772745	−	0.634717i	$-0.218883\pi$
$269$	3.57029e13	1.54549	0.772745	+	0.634717i	$0.218883\pi$
$270$	0	0
$271$	−3.60387e13	−1.49774	−0.748872	−	0.662714i	$-0.769404\pi$
$271$	−3.60387e13	−1.49774	−0.748872	+	0.662714i	$0.769404\pi$
$272$	0	0
$273$	−1.61933e13	−0.646309
$274$	0	0
$275$	4.71714e13	1.80863
$276$	0	0
$277$	−4.07304e13	−1.50065	−0.750326	−	0.661068i	$-0.770103\pi$
$277$	−4.07304e13	−1.50065	−0.750326	+	0.661068i	$0.770103\pi$
$278$	0	0
$279$	1.41374e13	0.500665
$280$	0	0
$281$	4.08826e12	0.139205	0.0696023	−	0.997575i	$-0.477827\pi$
$281$	4.08826e12	0.139205	0.0696023	+	0.997575i	$0.477827\pi$
$282$	0	0
$283$	1.33555e13	0.437356	0.218678	−	0.975797i	$-0.429826\pi$
$283$	1.33555e13	0.437356	0.218678	+	0.975797i	$0.429826\pi$
$284$	0	0
$285$	−8.27013e12	−0.260535
$286$	0	0
$287$	2.05659e13	0.623445
$288$	0	0
$289$	−8.16309e12	−0.238186
$290$	0	0
$291$	−2.30378e13	−0.647188
$292$	0	0
$293$	2.19433e13	0.593651	0.296825	−	0.954932i	$-0.404072\pi$
$293$	2.19433e13	0.593651	0.296825	+	0.954932i	$0.404072\pi$
$294$	0	0
$295$	6.59361e13	1.71831
$296$	0	0
$297$	−7.62530e12	−0.191468
$298$	0	0
$299$	−4.07725e13	−0.986679
$300$	0	0
$301$	−1.75645e13	−0.409751
$302$	0	0
$303$	−3.35223e12	−0.0754050
$304$	0	0
$305$	1.22855e14	2.66529
$306$	0	0
$307$	−2.52177e11	−0.00527770	−0.00263885	−	0.999997i	$-0.500840\pi$
$307$	−2.52177e11	−0.00527770	−0.00263885	+	0.999997i	$0.500840\pi$
$308$	0	0
$309$	1.58396e13	0.319870
$310$	0	0
$311$	4.95752e13	0.966234	0.483117	−	0.875556i	$-0.339505\pi$
$311$	4.95752e13	0.966234	0.483117	+	0.875556i	$0.339505\pi$
$312$	0	0
$313$	3.24318e13	0.610207	0.305104	−	0.952319i	$-0.401309\pi$
$313$	3.24318e13	0.610207	0.305104	+	0.952319i	$0.401309\pi$
$314$	0	0
$315$	−3.46378e13	−0.629277
$316$	0	0
$317$	−9.56422e13	−1.67812	−0.839061	−	0.544037i	$-0.816895\pi$
$317$	−9.56422e13	−1.67812	−0.839061	+	0.544037i	$0.816895\pi$
$318$	0	0
$319$	−4.09229e13	−0.693613
$320$	0	0
$321$	2.26816e13	0.371446
$322$	0	0
$323$	1.48252e13	0.234632
$324$	0	0
$325$	1.18285e14	1.80955
$326$	0	0
$327$	3.67828e13	0.544041
$328$	0	0
$329$	−4.79185e12	−0.0685371
$330$	0	0
$331$	1.21579e14	1.68192	0.840959	−	0.541099i	$-0.181992\pi$
$331$	1.21579e14	1.68192	0.840959	+	0.541099i	$0.181992\pi$
$332$	0	0
$333$	−4.63559e13	−0.620386
$334$	0	0
$335$	5.29646e13	0.685865
$336$	0	0
$337$	−8.38096e13	−1.05034	−0.525169	−	0.850998i	$-0.675998\pi$
$337$	−8.38096e13	−1.05034	−0.525169	+	0.850998i	$0.675998\pi$
$338$	0	0
$339$	−5.76758e13	−0.699673
$340$	0	0
$341$	1.27232e14	1.49433
$342$	0	0
$343$	−7.27043e13	−0.826880
$344$	0	0
$345$	−8.72134e13	−0.960679
$346$	0	0
$347$	7.79457e13	0.831725	0.415863	−	0.909427i	$-0.363480\pi$
$347$	7.79457e13	0.831725	0.415863	+	0.909427i	$0.363480\pi$
$348$	0	0
$349$	1.12179e14	1.15977	0.579884	−	0.814699i	$-0.303098\pi$
$349$	1.12179e14	1.15977	0.579884	+	0.814699i	$0.303098\pi$
$350$	0	0
$351$	−1.91209e13	−0.191566
$352$	0	0
$353$	−1.59721e14	−1.55096	−0.775480	−	0.631372i	$-0.782492\pi$
$353$	−1.59721e14	−1.55096	−0.775480	+	0.631372i	$0.782492\pi$
$354$	0	0
$355$	−9.98174e13	−0.939616
$356$	0	0
$357$	6.20925e13	0.566714
$358$	0	0
$359$	2.46378e12	0.0218063	0.0109032	−	0.999941i	$-0.496529\pi$
$359$	2.46378e12	0.0218063	0.0109032	+	0.999941i	$0.496529\pi$
$360$	0	0
$361$	−1.08072e14	−0.927735
$362$	0	0
$363$	7.05782e11	0.00587739
$364$	0	0
$365$	−2.36066e13	−0.190732
$366$	0	0
$367$	−1.34080e14	−1.05124	−0.525620	−	0.850720i	$-0.676167\pi$
$367$	−1.34080e14	−1.05124	−0.525620	+	0.850720i	$0.676167\pi$
$368$	0	0
$369$	2.42841e13	0.184789
$370$	0	0
$371$	−7.33046e13	−0.541470
$372$	0	0
$373$	−1.25684e14	−0.901328	−0.450664	−	0.892694i	$-0.648813\pi$
$373$	−1.25684e14	−0.901328	−0.450664	+	0.892694i	$0.648813\pi$
$374$	0	0
$375$	1.13835e14	0.792691
$376$	0	0
$377$	−1.02616e14	−0.693968
$378$	0	0
$379$	−5.62301e13	−0.369363	−0.184681	−	0.982798i	$-0.559125\pi$
$379$	−5.62301e13	−0.369363	−0.184681	+	0.982798i	$0.559125\pi$
$380$	0	0
$381$	1.25003e14	0.797686
$382$	0	0
$383$	−1.07826e14	−0.668543	−0.334272	−	0.942477i	$-0.608490\pi$
$383$	−1.07826e14	−0.668543	−0.334272	+	0.942477i	$0.608490\pi$
$384$	0	0
$385$	−3.11728e14	−1.87820
$386$	0	0
$387$	−2.07400e13	−0.121450
$388$	0	0
$389$	2.37629e14	1.35262	0.676311	−	0.736616i	$-0.263577\pi$
$389$	2.37629e14	1.35262	0.676311	+	0.736616i	$0.263577\pi$
$390$	0	0
$391$	1.56341e14	0.865168
$392$	0	0
$393$	7.25529e13	0.390388
$394$	0	0
$395$	−2.60857e14	−1.36495
$396$	0	0
$397$	2.85970e14	1.45537	0.727683	−	0.685914i	$-0.240597\pi$
$397$	2.85970e14	1.45537	0.727683	+	0.685914i	$0.240597\pi$
$398$	0	0
$399$	3.52577e13	0.174543
$400$	0	0
$401$	−9.31707e13	−0.448730	−0.224365	−	0.974505i	$-0.572031\pi$
$401$	−9.31707e13	−0.448730	−0.224365	+	0.974505i	$0.572031\pi$
$402$	0	0
$403$	3.19041e14	1.49509
$404$	0	0
$405$	−4.09000e13	−0.186518
$406$	0	0
$407$	−4.17187e14	−1.85166
$408$	0	0
$409$	2.56950e14	1.11012	0.555061	−	0.831810i	$-0.312695\pi$
$409$	2.56950e14	1.11012	0.555061	+	0.831810i	$0.312695\pi$
$410$	0	0
$411$	1.49716e14	0.629707
$412$	0	0
$413$	−2.81103e14	−1.15117
$414$	0	0
$415$	7.42350e13	0.296036
$416$	0	0
$417$	−1.27044e14	−0.493407
$418$	0	0
$419$	2.96091e14	1.12008	0.560039	−	0.828466i	$-0.310786\pi$
$419$	2.96091e14	1.12008	0.560039	+	0.828466i	$0.310786\pi$
$420$	0	0
$421$	−3.40485e14	−1.25472	−0.627360	−	0.778730i	$-0.715864\pi$
$421$	−3.40485e14	−1.25472	−0.627360	+	0.778730i	$0.715864\pi$
$422$	0	0
$423$	−5.65817e12	−0.0203144
$424$	0	0
$425$	−4.53559e14	−1.58670
$426$	0	0
$427$	−5.23763e14	−1.78559
$428$	0	0
$429$	−1.72081e14	−0.571765
$430$	0	0
$431$	−1.37789e14	−0.446261	−0.223130	−	0.974789i	$-0.571628\pi$
$431$	−1.37789e14	−0.446261	−0.223130	+	0.974789i	$0.571628\pi$
$432$	0	0
$433$	3.52377e14	1.11256	0.556280	−	0.830995i	$-0.312228\pi$
$433$	3.52377e14	1.11256	0.556280	+	0.830995i	$0.312228\pi$
$434$	0	0
$435$	−2.19499e14	−0.675681
$436$	0	0
$437$	8.87743e13	0.266464
$438$	0	0
$439$	4.29185e14	1.25629	0.628145	−	0.778097i	$-0.283815\pi$
$439$	4.29185e14	1.25629	0.628145	+	0.778097i	$0.283815\pi$
$440$	0	0
$441$	3.09106e13	0.0882460
$442$	0	0
$443$	3.61081e14	1.00551	0.502753	−	0.864430i	$-0.332321\pi$
$443$	3.61081e14	1.00551	0.502753	+	0.864430i	$0.332321\pi$
$444$	0	0
$445$	5.87951e14	1.59721
$446$	0	0
$447$	3.08646e14	0.818029
$448$	0	0
$449$	−6.27688e12	−0.0162326	−0.00811632	−	0.999967i	$-0.502584\pi$
$449$	−6.27688e12	−0.0162326	−0.00811632	+	0.999967i	$0.502584\pi$
$450$	0	0
$451$	2.18548e14	0.551539
$452$	0	0
$453$	3.04652e14	0.750350
$454$	0	0
$455$	−7.81675e14	−1.87916
$456$	0	0
$457$	4.60907e14	1.08162	0.540809	−	0.841146i	$-0.318118\pi$
$457$	4.60907e14	1.08162	0.540809	+	0.841146i	$0.318118\pi$
$458$	0	0
$459$	7.33183e13	0.167974
$460$	0	0
$461$	7.97399e14	1.78369	0.891847	−	0.452337i	$-0.149410\pi$
$461$	7.97399e14	1.78369	0.891847	+	0.452337i	$0.149410\pi$
$462$	0	0
$463$	5.80897e14	1.26883	0.634416	−	0.772992i	$-0.281241\pi$
$463$	5.80897e14	1.26883	0.634416	+	0.772992i	$0.281241\pi$
$464$	0	0
$465$	6.82436e14	1.45570
$466$	0	0
$467$	8.22532e14	1.71360	0.856801	−	0.515647i	$-0.172448\pi$
$467$	8.22532e14	1.71360	0.856801	+	0.515647i	$0.172448\pi$
$468$	0	0
$469$	−2.25801e14	−0.459490
$470$	0	0
$471$	−1.47430e14	−0.293070
$472$	0	0
$473$	−1.86652e14	−0.362491
$474$	0	0
$475$	−2.57542e14	−0.488691
$476$	0	0
$477$	−8.65574e13	−0.160492
$478$	0	0
$479$	4.88235e14	0.884675	0.442337	−	0.896849i	$-0.354149\pi$
$479$	4.88235e14	0.884675	0.442337	+	0.896849i	$0.354149\pi$
$480$	0	0
$481$	−1.04612e15	−1.85261
$482$	0	0
$483$	3.71813e14	0.643599
$484$	0	0
$485$	−1.11207e15	−1.88172
$486$	0	0
$487$	2.97281e13	0.0491766	0.0245883	−	0.999698i	$-0.492173\pi$
$487$	2.97281e13	0.0491766	0.0245883	+	0.999698i	$0.492173\pi$
$488$	0	0
$489$	−3.84914e14	−0.622537
$490$	0	0
$491$	−8.16135e14	−1.29067	−0.645333	−	0.763901i	$-0.723281\pi$
$491$	−8.16135e14	−1.29067	−0.645333	+	0.763901i	$0.723281\pi$
$492$	0	0
$493$	3.93479e14	0.608504
$494$	0	0
$495$	−3.68085e14	−0.556698
$496$	0	0
$497$	4.25547e14	0.629488
$498$	0	0
$499$	−6.09917e14	−0.882506	−0.441253	−	0.897383i	$-0.645466\pi$
$499$	−6.09917e14	−0.882506	−0.441253	+	0.897383i	$0.645466\pi$
$500$	0	0
$501$	7.17512e14	1.01560
$502$	0	0
$503$	−2.40472e14	−0.332998	−0.166499	−	0.986042i	$-0.553246\pi$
$503$	−2.40472e14	−0.332998	−0.166499	+	0.986042i	$0.553246\pi$
$504$	0	0
$505$	−1.61818e14	−0.219242
$506$	0	0
$507$	3.99206e12	0.00529241
$508$	0	0
$509$	−1.56720e14	−0.203318	−0.101659	−	0.994819i	$-0.532415\pi$
$509$	−1.56720e14	−0.203318	−0.101659	+	0.994819i	$0.532415\pi$
$510$	0	0
$511$	1.00641e14	0.127779
$512$	0	0
$513$	4.16320e13	0.0517347
$514$	0	0
$515$	7.64604e14	0.930030
$516$	0	0
$517$	−5.09216e13	−0.0606323
$518$	0	0
$519$	6.44947e14	0.751802
$520$	0	0
$521$	−1.00579e14	−0.114788	−0.0573942	−	0.998352i	$-0.518279\pi$
$521$	−1.00579e14	−0.114788	−0.0573942	+	0.998352i	$0.518279\pi$
$522$	0	0
$523$	−1.22443e15	−1.36827	−0.684137	−	0.729354i	$-0.739821\pi$
$523$	−1.22443e15	−1.36827	−0.684137	+	0.729354i	$0.739821\pi$
$524$	0	0
$525$	−1.07866e15	−1.18035
$526$	0	0
$527$	−1.22335e15	−1.31097
$528$	0	0
$529$	−1.66333e13	−0.0174572
$530$	0	0
$531$	−3.31923e14	−0.341207
$532$	0	0
$533$	5.48022e14	0.551821
$534$	0	0
$535$	1.09488e15	1.07999
$536$	0	0
$537$	−9.87672e14	−0.954452
$538$	0	0
$539$	2.78184e14	0.263387
$540$	0	0
$541$	−3.62345e14	−0.336154	−0.168077	−	0.985774i	$-0.553756\pi$
$541$	−3.62345e14	−0.336154	−0.168077	+	0.985774i	$0.553756\pi$
$542$	0	0
$543$	4.08418e14	0.371284
$544$	0	0
$545$	1.77556e15	1.58181
$546$	0	0
$547$	−1.32630e15	−1.15801	−0.579006	−	0.815324i	$-0.696559\pi$
$547$	−1.32630e15	−1.15801	−0.579006	+	0.815324i	$0.696559\pi$
$548$	0	0
$549$	−6.18455e14	−0.529249
$550$	0	0
$551$	2.23427e14	0.187414
$552$	0	0
$553$	1.11210e15	0.914440
$554$	0	0
$555$	−2.23767e15	−1.80379
$556$	0	0
$557$	−2.24474e15	−1.77404	−0.887018	−	0.461735i	$-0.847227\pi$
$557$	−2.24474e15	−1.77404	−0.887018	+	0.461735i	$0.847227\pi$
$558$	0	0
$559$	−4.68042e14	−0.362677
$560$	0	0
$561$	6.59839e14	0.501351
$562$	0	0
$563$	1.10178e15	0.820915	0.410457	−	0.911880i	$-0.365369\pi$
$563$	1.10178e15	0.820915	0.410457	+	0.911880i	$0.365369\pi$
$564$	0	0
$565$	−2.78410e15	−2.03432
$566$	0	0
$567$	1.74367e14	0.124956
$568$	0	0
$569$	1.69107e15	1.18863	0.594313	−	0.804234i	$-0.297424\pi$
$569$	1.69107e15	1.18863	0.594313	+	0.804234i	$0.297424\pi$
$570$	0	0
$571$	2.36575e15	1.63106	0.815529	−	0.578716i	$-0.196446\pi$
$571$	2.36575e15	1.63106	0.815529	+	0.578716i	$0.196446\pi$
$572$	0	0
$573$	9.94199e13	0.0672392
$574$	0	0
$575$	−2.71594e15	−1.80197
$576$	0	0
$577$	−3.25902e14	−0.212139	−0.106069	−	0.994359i	$-0.533827\pi$
$577$	−3.25902e14	−0.212139	−0.106069	+	0.994359i	$0.533827\pi$
$578$	0	0
$579$	−1.14673e15	−0.732368
$580$	0	0
$581$	−3.16483e14	−0.198327
$582$	0	0
$583$	−7.78986e14	−0.479019
$584$	0	0
$585$	−9.22995e14	−0.556983
$586$	0	0
$587$	2.76684e15	1.63861	0.819303	−	0.573361i	$-0.194361\pi$
$587$	2.76684e15	1.63861	0.819303	+	0.573361i	$0.194361\pi$
$588$	0	0
$589$	−6.94649e14	−0.403768
$590$	0	0
$591$	8.76984e14	0.500335
$592$	0	0
$593$	1.95966e15	1.09744	0.548719	−	0.836007i	$-0.315116\pi$
$593$	1.95966e15	1.09744	0.548719	+	0.836007i	$0.315116\pi$
$594$	0	0
$595$	2.99731e15	1.64774
$596$	0	0
$597$	4.18300e14	0.225751
$598$	0	0
$599$	3.37393e15	1.78767	0.893837	−	0.448393i	$-0.148003\pi$
$599$	3.37393e15	1.78767	0.893837	+	0.448393i	$0.148003\pi$
$600$	0	0
$601$	3.13585e15	1.63135	0.815673	−	0.578513i	$-0.196367\pi$
$601$	3.13585e15	1.63135	0.815673	+	0.578513i	$0.196367\pi$
$602$	0	0
$603$	−2.66624e14	−0.136193
$604$	0	0
$605$	3.40692e13	0.0170887
$606$	0	0
$607$	−3.67568e15	−1.81051	−0.905253	−	0.424873i	$-0.860319\pi$
$607$	−3.67568e15	−1.81051	−0.905253	+	0.424873i	$0.860319\pi$
$608$	0	0
$609$	9.35780e14	0.452667
$610$	0	0
$611$	−1.27689e14	−0.0606633
$612$	0	0
$613$	−9.78886e14	−0.456772	−0.228386	−	0.973571i	$-0.573345\pi$
$613$	−9.78886e14	−0.456772	−0.228386	+	0.973571i	$0.573345\pi$
$614$	0	0
$615$	1.17223e15	0.537279
$616$	0	0
$617$	−1.46255e15	−0.658481	−0.329241	−	0.944246i	$-0.606793\pi$
$617$	−1.46255e15	−0.658481	−0.329241	+	0.944246i	$0.606793\pi$
$618$	0	0
$619$	−4.30101e15	−1.90227	−0.951135	−	0.308776i	$-0.900081\pi$
$619$	−4.30101e15	−1.90227	−0.951135	+	0.308776i	$0.900081\pi$
$620$	0	0
$621$	4.39034e14	0.190763
$622$	0	0
$623$	−2.50659e15	−1.07004
$624$	0	0
$625$	1.16078e15	0.486867
$626$	0	0
$627$	3.74673e14	0.154412
$628$	0	0
$629$	4.01131e15	1.62446
$630$	0	0
$631$	5.95346e14	0.236924	0.118462	−	0.992959i	$-0.462204\pi$
$631$	5.95346e14	0.236924	0.118462	+	0.992959i	$0.462204\pi$
$632$	0	0
$633$	4.13937e14	0.161888
$634$	0	0
$635$	6.03408e15	2.31929
$636$	0	0
$637$	6.97563e14	0.263522
$638$	0	0
$639$	5.02482e14	0.186580
$640$	0	0
$641$	−7.51262e14	−0.274203	−0.137101	−	0.990557i	$-0.543779\pi$
$641$	−7.51262e14	−0.274203	−0.137101	+	0.990557i	$0.543779\pi$
$642$	0	0
$643$	−1.46144e15	−0.524351	−0.262175	−	0.965020i	$-0.584440\pi$
$643$	−1.46144e15	−0.524351	−0.262175	+	0.965020i	$0.584440\pi$
$644$	0	0
$645$	−1.00115e15	−0.353119
$646$	0	0
$647$	3.57862e15	1.24091	0.620457	−	0.784241i	$-0.286947\pi$
$647$	3.57862e15	1.24091	0.620457	+	0.784241i	$0.286947\pi$
$648$	0	0
$649$	−2.98719e15	−1.01840
$650$	0	0
$651$	−2.90940e15	−0.975232
$652$	0	0
$653$	7.31395e14	0.241063	0.120531	−	0.992710i	$-0.461540\pi$
$653$	7.31395e14	0.241063	0.120531	+	0.992710i	$0.461540\pi$
$654$	0	0
$655$	3.50225e15	1.13506
$656$	0	0
$657$	1.18836e14	0.0378737
$658$	0	0
$659$	1.69987e15	0.532778	0.266389	−	0.963866i	$-0.414169\pi$
$659$	1.69987e15	0.532778	0.266389	+	0.963866i	$0.414169\pi$
$660$	0	0
$661$	−1.78604e14	−0.0550534	−0.0275267	−	0.999621i	$-0.508763\pi$
$661$	−1.78604e14	−0.0550534	−0.0275267	+	0.999621i	$0.508763\pi$
$662$	0	0
$663$	1.65458e15	0.501608
$664$	0	0
$665$	1.70195e15	0.507489
$666$	0	0
$667$	2.35617e15	0.691059
$668$	0	0
$669$	−3.55394e15	−1.02534
$670$	0	0
$671$	−5.56587e15	−1.57965
$672$	0	0
$673$	−9.26676e14	−0.258729	−0.129364	−	0.991597i	$-0.541294\pi$
$673$	−9.26676e14	−0.258729	−0.129364	+	0.991597i	$0.541294\pi$
$674$	0	0
$675$	−1.27368e15	−0.349856
$676$	0	0
$677$	−1.49664e14	−0.0404465	−0.0202232	−	0.999795i	$-0.506438\pi$
$677$	−1.49664e14	−0.0404465	−0.0202232	+	0.999795i	$0.506438\pi$
$678$	0	0
$679$	4.74106e15	1.26064
$680$	0	0
$681$	8.17590e14	0.213908
$682$	0	0
$683$	−6.74856e15	−1.73739	−0.868694	−	0.495348i	$-0.835040\pi$
$683$	−6.74856e15	−1.73739	−0.868694	+	0.495348i	$0.835040\pi$
$684$	0	0
$685$	7.22704e15	1.83089
$686$	0	0
$687$	−4.61506e14	−0.115058
$688$	0	0
$689$	−1.95335e15	−0.479264
$690$	0	0
$691$	1.61272e15	0.389430	0.194715	−	0.980860i	$-0.437622\pi$
$691$	1.61272e15	0.389430	0.194715	+	0.980860i	$0.437622\pi$
$692$	0	0
$693$	1.56924e15	0.372956
$694$	0	0
$695$	−6.13261e15	−1.43459
$696$	0	0
$697$	−2.10137e15	−0.483863
$698$	0	0
$699$	−3.69121e14	−0.0836650
$700$	0	0
$701$	4.16261e14	0.0928789	0.0464394	−	0.998921i	$-0.485213\pi$
$701$	4.16261e14	0.0928789	0.0464394	+	0.998921i	$0.485213\pi$
$702$	0	0
$703$	2.27772e15	0.500319
$704$	0	0
$705$	−2.73129e14	−0.0590647
$706$	0	0
$707$	6.89870e14	0.146879
$708$	0	0
$709$	−2.82972e15	−0.593184	−0.296592	−	0.955004i	$-0.595850\pi$
$709$	−2.82972e15	−0.593184	−0.296592	+	0.955004i	$0.595850\pi$
$710$	0	0
$711$	1.31316e15	0.271040
$712$	0	0
$713$	−7.32548e15	−1.48883
$714$	0	0
$715$	−8.30663e15	−1.66242
$716$	0	0
$717$	2.02054e15	0.398209
$718$	0	0
$719$	−6.81122e15	−1.32195	−0.660977	−	0.750406i	$-0.729858\pi$
$719$	−6.81122e15	−1.32195	−0.660977	+	0.750406i	$0.729858\pi$
$720$	0	0
$721$	−3.25970e15	−0.623066
$722$	0	0
$723$	3.16207e15	0.595265
$724$	0	0
$725$	−6.83548e15	−1.26739
$726$	0	0
$727$	−2.73191e15	−0.498916	−0.249458	−	0.968386i	$-0.580252\pi$
$727$	−2.73191e15	−0.498916	−0.249458	+	0.968386i	$0.580252\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	1.79469e15	0.318012
$732$	0	0
$733$	−6.58368e15	−1.14920	−0.574602	−	0.818433i	$-0.694843\pi$
$733$	−6.58368e15	−1.14920	−0.574602	+	0.818433i	$0.694843\pi$
$734$	0	0
$735$	1.49210e15	0.256577
$736$	0	0
$737$	−2.39952e15	−0.406494
$738$	0	0
$739$	−4.37564e15	−0.730294	−0.365147	−	0.930950i	$-0.618981\pi$
$739$	−4.37564e15	−0.730294	−0.365147	+	0.930950i	$0.618981\pi$
$740$	0	0
$741$	9.39514e14	0.154491
$742$	0	0
$743$	−4.90059e15	−0.793981	−0.396990	−	0.917823i	$-0.629945\pi$
$743$	−4.90059e15	−0.793981	−0.396990	+	0.917823i	$0.629945\pi$
$744$	0	0
$745$	1.48988e16	2.37844
$746$	0	0
$747$	−3.73700e14	−0.0587841
$748$	0	0
$749$	−4.66774e15	−0.723530
$750$	0	0
$751$	7.04936e14	0.107679	0.0538394	−	0.998550i	$-0.482854\pi$
$751$	7.04936e14	0.107679	0.0538394	+	0.998550i	$0.482854\pi$
$752$	0	0
$753$	−4.54745e15	−0.684535
$754$	0	0
$755$	1.47060e16	2.18166
$756$	0	0
$757$	5.25516e15	0.768348	0.384174	−	0.923261i	$-0.374486\pi$
$757$	5.25516e15	0.768348	0.384174	+	0.923261i	$0.374486\pi$
$758$	0	0
$759$	3.95115e15	0.569368
$760$	0	0
$761$	3.81336e15	0.541617	0.270808	−	0.962633i	$-0.412709\pi$
$761$	3.81336e15	0.541617	0.270808	+	0.962633i	$0.412709\pi$
$762$	0	0
$763$	−7.56968e15	−1.05972
$764$	0	0
$765$	3.53919e15	0.488389
$766$	0	0
$767$	−7.49056e15	−1.01892
$768$	0	0
$769$	−9.81241e15	−1.31577	−0.657886	−	0.753117i	$-0.728549\pi$
$769$	−9.81241e15	−1.31577	−0.657886	+	0.753117i	$0.728549\pi$
$770$	0	0
$771$	1.33268e15	0.176168
$772$	0	0
$773$	7.09136e15	0.924149	0.462075	−	0.886841i	$-0.347105\pi$
$773$	7.09136e15	0.924149	0.462075	+	0.886841i	$0.347105\pi$
$774$	0	0
$775$	2.12519e16	2.73048
$776$	0	0
$777$	9.53978e15	1.20843
$778$	0	0
$779$	−1.19321e15	−0.149026
$780$	0	0
$781$	4.52216e15	0.556885
$782$	0	0
$783$	1.10496e15	0.134171
$784$	0	0
$785$	−7.11668e15	−0.852107
$786$	0	0
$787$	2.46400e15	0.290924	0.145462	−	0.989364i	$-0.453533\pi$
$787$	2.46400e15	0.290924	0.145462	+	0.989364i	$0.453533\pi$
$788$	0	0
$789$	4.76138e15	0.554382
$790$	0	0
$791$	1.18693e16	1.36288
$792$	0	0
$793$	−1.39567e16	−1.58045
$794$	0	0
$795$	−4.17827e15	−0.466634
$796$	0	0
$797$	3.04936e15	0.335883	0.167941	−	0.985797i	$-0.446288\pi$
$797$	3.04936e15	0.335883	0.167941	+	0.985797i	$0.446288\pi$
$798$	0	0
$799$	4.89618e14	0.0531925
$800$	0	0
$801$	−2.95975e15	−0.317159
$802$	0	0
$803$	1.06948e15	0.113041
$804$	0	0
$805$	1.79480e16	1.87128
$806$	0	0
$807$	−8.67580e15	−0.892289
$808$	0	0
$809$	−3.92265e15	−0.397982	−0.198991	−	0.980001i	$-0.563766\pi$
$809$	−3.92265e15	−0.397982	−0.198991	+	0.980001i	$0.563766\pi$
$810$	0	0
$811$	−1.75012e16	−1.75168	−0.875838	−	0.482606i	$-0.839690\pi$
$811$	−1.75012e16	−1.75168	−0.875838	+	0.482606i	$0.839690\pi$
$812$	0	0
$813$	8.75740e15	0.864723
$814$	0	0
$815$	−1.85804e16	−1.81004
$816$	0	0
$817$	1.01907e15	0.0979451
$818$	0	0
$819$	3.93496e15	0.373146
$820$	0	0
$821$	4.48148e15	0.419309	0.209655	−	0.977775i	$-0.432766\pi$
$821$	4.48148e15	0.419309	0.209655	+	0.977775i	$0.432766\pi$
$822$	0	0
$823$	2.38819e15	0.220480	0.110240	−	0.993905i	$-0.464838\pi$
$823$	2.38819e15	0.220480	0.110240	+	0.993905i	$0.464838\pi$
$824$	0	0
$825$	−1.14626e16	−1.04421
$826$	0	0
$827$	−3.42245e15	−0.307650	−0.153825	−	0.988098i	$-0.549159\pi$
$827$	−3.42245e15	−0.307650	−0.153825	+	0.988098i	$0.549159\pi$
$828$	0	0
$829$	7.18881e15	0.637686	0.318843	−	0.947808i	$-0.396706\pi$
$829$	7.18881e15	0.637686	0.318843	+	0.947808i	$0.396706\pi$
$830$	0	0
$831$	9.89748e15	0.866402
$832$	0	0
$833$	−2.67478e15	−0.231068
$834$	0	0
$835$	3.46355e16	2.95288
$836$	0	0
$837$	−3.43539e15	−0.289059
$838$	0	0
$839$	−1.25101e16	−1.03889	−0.519444	−	0.854504i	$-0.673861\pi$
$839$	−1.25101e16	−1.03889	−0.519444	+	0.854504i	$0.673861\pi$
$840$	0	0
$841$	−6.27049e15	−0.513953
$842$	0	0
$843$	−9.93447e14	−0.0803699
$844$	0	0
$845$	1.92703e14	0.0153878
$846$	0	0
$847$	−1.45246e14	−0.0114484
$848$	0	0
$849$	−3.24539e15	−0.252507
$850$	0	0
$851$	2.40199e16	1.84484
$852$	0	0
$853$	5.88516e15	0.446210	0.223105	−	0.974794i	$-0.428381\pi$
$853$	5.88516e15	0.446210	0.223105	+	0.974794i	$0.428381\pi$
$854$	0	0
$855$	2.00964e15	0.150420
$856$	0	0
$857$	2.57995e15	0.190641	0.0953207	−	0.995447i	$-0.469612\pi$
$857$	2.57995e15	0.190641	0.0953207	+	0.995447i	$0.469612\pi$
$858$	0	0
$859$	3.14874e15	0.229707	0.114854	−	0.993382i	$-0.463360\pi$
$859$	3.14874e15	0.229707	0.114854	+	0.993382i	$0.463360\pi$
$860$	0	0
$861$	−4.99752e15	−0.359946
$862$	0	0
$863$	2.28539e16	1.62518	0.812589	−	0.582837i	$-0.198058\pi$
$863$	2.28539e16	1.62518	0.812589	+	0.582837i	$0.198058\pi$
$864$	0	0
$865$	3.11326e16	2.18588
$866$	0	0
$867$	1.98363e15	0.137517
$868$	0	0
$869$	1.18179e16	0.808971
$870$	0	0
$871$	−6.01695e15	−0.406702
$872$	0	0
$873$	5.59820e15	0.373654
$874$	0	0
$875$	−2.34266e16	−1.54406
$876$	0	0
$877$	−1.83695e16	−1.19564	−0.597820	−	0.801630i	$-0.703966\pi$
$877$	−1.83695e16	−1.19564	−0.597820	+	0.801630i	$0.703966\pi$
$878$	0	0
$879$	−5.33223e15	−0.342744
$880$	0	0
$881$	−1.17507e16	−0.745925	−0.372963	−	0.927846i	$-0.621658\pi$
$881$	−1.17507e16	−0.745925	−0.372963	+	0.927846i	$0.621658\pi$
$882$	0	0
$883$	7.76791e15	0.486990	0.243495	−	0.969902i	$-0.421706\pi$
$883$	7.76791e15	0.486990	0.243495	+	0.969902i	$0.421706\pi$
$884$	0	0
$885$	−1.60225e16	−0.992068
$886$	0	0
$887$	−4.62750e15	−0.282987	−0.141493	−	0.989939i	$-0.545190\pi$
$887$	−4.62750e15	−0.282987	−0.141493	+	0.989939i	$0.545190\pi$
$888$	0	0
$889$	−2.57248e16	−1.55379
$890$	0	0
$891$	1.85295e15	0.110544
$892$	0	0
$893$	2.78017e14	0.0163828
$894$	0	0
$895$	−4.76765e16	−2.77510
$896$	0	0
$897$	9.90772e15	0.569660
$898$	0	0
$899$	−1.84368e16	−1.04715
$900$	0	0
$901$	7.49006e15	0.420241
$902$	0	0
$903$	4.26817e15	0.236570
$904$	0	0
$905$	1.97150e16	1.07952
$906$	0	0
$907$	−1.92511e16	−1.04140	−0.520698	−	0.853741i	$-0.674328\pi$
$907$	−1.92511e16	−1.04140	−0.520698	+	0.853741i	$0.674328\pi$
$908$	0	0
$909$	8.14593e14	0.0435351
$910$	0	0
$911$	1.80098e16	0.950952	0.475476	−	0.879729i	$-0.342276\pi$
$911$	1.80098e16	0.950952	0.475476	+	0.879729i	$0.342276\pi$
$912$	0	0
$913$	−3.36317e15	−0.175452
$914$	0	0
$915$	−2.98538e16	−1.53881
$916$	0	0
$917$	−1.49310e16	−0.760426
$918$	0	0
$919$	5.42882e15	0.273193	0.136597	−	0.990627i	$-0.456384\pi$
$919$	5.42882e15	0.273193	0.136597	+	0.990627i	$0.456384\pi$
$920$	0	0
$921$	6.12790e13	0.00304708
$922$	0	0
$923$	1.13396e16	0.557170
$924$	0	0
$925$	−6.96840e16	−3.38340
$926$	0	0
$927$	−3.84903e15	−0.184677
$928$	0	0
$929$	2.76310e16	1.31012	0.655059	−	0.755578i	$-0.272644\pi$
$929$	2.76310e16	1.31012	0.655059	+	0.755578i	$0.272644\pi$
$930$	0	0
$931$	−1.51881e15	−0.0711671
$932$	0	0
$933$	−1.20468e16	−0.557856
$934$	0	0
$935$	3.18515e16	1.45769
$936$	0	0
$937$	−2.46060e16	−1.11294	−0.556472	−	0.830867i	$-0.687845\pi$
$937$	−2.46060e16	−1.11294	−0.556472	+	0.830867i	$0.687845\pi$
$938$	0	0
$939$	−7.88093e15	−0.352303
$940$	0	0
$941$	3.11892e16	1.37804	0.689019	−	0.724743i	$-0.258042\pi$
$941$	3.11892e16	1.37804	0.689019	+	0.724743i	$0.258042\pi$
$942$	0	0
$943$	−1.25831e16	−0.549507
$944$	0	0
$945$	8.41698e15	0.363313
$946$	0	0
$947$	3.17189e16	1.35330	0.676649	−	0.736305i	$-0.263431\pi$
$947$	3.17189e16	1.35330	0.676649	+	0.736305i	$0.263431\pi$
$948$	0	0
$949$	2.68179e15	0.113099
$950$	0	0
$951$	2.32410e16	0.968864
$952$	0	0
$953$	9.40992e15	0.387771	0.193885	−	0.981024i	$-0.437891\pi$
$953$	9.40992e15	0.387771	0.193885	+	0.981024i	$0.437891\pi$
$954$	0	0
$955$	4.79916e15	0.195500
$956$	0	0
$957$	9.94426e15	0.400458
$958$	0	0
$959$	−3.08107e16	−1.22659
$960$	0	0
$961$	3.19127e16	1.25599
$962$	0	0
$963$	−5.51163e15	−0.214454
$964$	0	0
$965$	−5.53545e16	−2.12938
$966$	0	0
$967$	1.27731e16	0.485792	0.242896	−	0.970052i	$-0.421903\pi$
$967$	1.27731e16	0.485792	0.242896	+	0.970052i	$0.421903\pi$
$968$	0	0
$969$	−3.60253e15	−0.135465
$970$	0	0
$971$	−3.51105e16	−1.30536	−0.652681	−	0.757632i	$-0.726356\pi$
$971$	−3.51105e16	−1.30536	−0.652681	+	0.757632i	$0.726356\pi$
$972$	0	0
$973$	2.61449e16	0.961094
$974$	0	0
$975$	−2.87432e16	−1.04474
$976$	0	0
$977$	−5.08668e16	−1.82816	−0.914081	−	0.405532i	$-0.867086\pi$
$977$	−5.08668e16	−1.82816	−0.914081	+	0.405532i	$0.867086\pi$
$978$	0	0
$979$	−2.66367e16	−0.946621
$980$	0	0
$981$	−8.93821e15	−0.314102
$982$	0	0
$983$	4.25676e16	1.47923	0.739614	−	0.673032i	$-0.235008\pi$
$983$	4.25676e16	1.47923	0.739614	+	0.673032i	$0.235008\pi$
$984$	0	0
$985$	4.23334e16	1.45474
$986$	0	0
$987$	1.16442e15	0.0395699
$988$	0	0
$989$	1.07467e16	0.361156
$990$	0	0
$991$	−2.24124e16	−0.744873	−0.372437	−	0.928058i	$-0.621478\pi$
$991$	−2.24124e16	−0.744873	−0.372437	+	0.928058i	$0.621478\pi$
$992$	0	0
$993$	−2.95437e16	−0.971056
$994$	0	0
$995$	2.01920e16	0.656376
$996$	0	0
$997$	−2.03268e16	−0.653499	−0.326750	−	0.945111i	$-0.605953\pi$
$997$	−2.03268e16	−0.653499	−0.326750	+	0.945111i	$0.605953\pi$
$998$	0	0
$999$	1.12645e16	0.358180

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.12.a.a.1.1		1
3.2	odd	2		144.12.a.o.1.1		1
4.3	odd	2		6.12.a.b.1.1	✓	1
8.3	odd	2		192.12.a.j.1.1		1
8.5	even	2		192.12.a.t.1.1		1
12.11	even	2		18.12.a.e.1.1		1
20.3	even	4		150.12.c.b.49.2		2
20.7	even	4		150.12.c.b.49.1		2
20.19	odd	2		150.12.a.f.1.1		1
36.7	odd	6		162.12.c.j.109.1		2
36.11	even	6		162.12.c.a.109.1		2
36.23	even	6		162.12.c.a.55.1		2
36.31	odd	6		162.12.c.j.55.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.12.a.b.1.1	✓	1	4.3	odd	2
18.12.a.e.1.1		1	12.11	even	2
48.12.a.a.1.1		1	1.1	even	1	trivial
144.12.a.o.1.1		1	3.2	odd	2
150.12.a.f.1.1		1	20.19	odd	2
150.12.c.b.49.1		2	20.7	even	4
150.12.c.b.49.2		2	20.3	even	4
162.12.c.a.55.1		2	36.23	even	6
162.12.c.a.109.1		2	36.11	even	6
162.12.c.j.55.1		2	36.31	odd	6
162.12.c.j.109.1		2	36.7	odd	6
192.12.a.j.1.1		1	8.3	odd	2
192.12.a.t.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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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