LMFDB - Embedded newform 12.12.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [12,12,Mod(1,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

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

Newform invariants

comment:select newform

sage:traces = [1,0,-243] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$9.22011816672$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	12.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-243.000 q^{3} +9990.00 q^{5} -86128.0 q^{7} +59049.0 q^{9} -806004. q^{11} -960250. q^{13} -2.42757e6 q^{15} -4.30688e6 q^{17} +401300. q^{19} +2.09291e7 q^{21} +1.77515e7 q^{23} +5.09720e7 q^{25} -1.43489e7 q^{27} -8.47050e7 q^{29} +1.40931e8 q^{31} +1.95859e8 q^{33} -8.60419e8 q^{35} -4.13507e8 q^{37} +2.33341e8 q^{39} +1.50095e8 q^{41} +7.06702e8 q^{43} +5.89900e8 q^{45} -2.47573e9 q^{47} +5.44071e9 q^{49} +1.04657e9 q^{51} +1.60012e9 q^{53} -8.05198e9 q^{55} -9.75159e7 q^{57} +3.94549e9 q^{59} -8.85973e8 q^{61} -5.08577e9 q^{63} -9.59290e9 q^{65} -4.88160e9 q^{67} -4.31362e9 q^{69} +1.26315e10 q^{71} +1.42334e9 q^{73} -1.23862e10 q^{75} +6.94195e10 q^{77} +6.67408e8 q^{79} +3.48678e9 q^{81} +5.71607e9 q^{83} -4.30257e10 q^{85} +2.05833e10 q^{87} -8.57387e10 q^{89} +8.27044e10 q^{91} -3.42461e10 q^{93} +4.00899e9 q^{95} -5.23026e10 q^{97} -4.75937e10 q^{99} +O(q^{100})$

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
$5$	9990.00	1.42965	0.714826	−	0.699302i	$-0.246506\pi$
$5$	9990.00	1.42965	0.714826	+	0.699302i	$0.246506\pi$
$6$	0	0
$7$	−86128.0	−1.93689	−0.968445	−	0.249226i	$-0.919824\pi$
$7$	−86128.0	−1.93689	−0.968445	+	0.249226i	$0.919824\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	−806004.	−1.50896	−0.754479	−	0.656324i	$-0.772111\pi$
$11$	−806004.	−1.50896	−0.754479	+	0.656324i	$0.772111\pi$
$12$	0	0
$13$	−960250.	−0.717292	−0.358646	−	0.933474i	$-0.616761\pi$
$13$	−960250.	−0.717292	−0.358646	+	0.933474i	$0.616761\pi$
$14$	0	0
$15$	−2.42757e6	−0.825410
$16$	0	0
$17$	−4.30688e6	−0.735688	−0.367844	−	0.929888i	$-0.619904\pi$
$17$	−4.30688e6	−0.735688	−0.367844	+	0.929888i	$0.619904\pi$
$18$	0	0
$19$	401300.	0.0371813	0.0185906	−	0.999827i	$-0.494082\pi$
$19$	401300.	0.0371813	0.0185906	+	0.999827i	$0.494082\pi$
$20$	0	0
$21$	2.09291e7	1.11826
$22$	0	0
$23$	1.77515e7	0.575086	0.287543	−	0.957768i	$-0.407162\pi$
$23$	1.77515e7	0.575086	0.287543	+	0.957768i	$0.407162\pi$
$24$	0	0
$25$	5.09720e7	1.04391
$26$	0	0
$27$	−1.43489e7	−0.192450
$28$	0	0
$29$	−8.47050e7	−0.766867	−0.383433	−	0.923568i	$-0.625258\pi$
$29$	−8.47050e7	−0.766867	−0.383433	+	0.923568i	$0.625258\pi$
$30$	0	0
$31$	1.40931e8	0.884129	0.442065	−	0.896983i	$-0.354246\pi$
$31$	1.40931e8	0.884129	0.442065	+	0.896983i	$0.354246\pi$
$32$	0	0
$33$	1.95859e8	0.871198
$34$	0	0
$35$	−8.60419e8	−2.76908
$36$	0	0
$37$	−4.13507e8	−0.980331	−0.490166	−	0.871629i	$-0.663064\pi$
$37$	−4.13507e8	−0.980331	−0.490166	+	0.871629i	$0.663064\pi$
$38$	0	0
$39$	2.33341e8	0.414129
$40$	0	0
$41$	1.50095e8	0.202327	0.101164	−	0.994870i	$-0.467743\pi$
$41$	1.50095e8	0.202327	0.101164	+	0.994870i	$0.467743\pi$
$42$	0	0
$43$	7.06702e8	0.733094	0.366547	−	0.930399i	$-0.380540\pi$
$43$	7.06702e8	0.733094	0.366547	+	0.930399i	$0.380540\pi$
$44$	0	0
$45$	5.89900e8	0.476551
$46$	0	0
$47$	−2.47573e9	−1.57458	−0.787289	−	0.616584i	$-0.788516\pi$
$47$	−2.47573e9	−1.57458	−0.787289	+	0.616584i	$0.788516\pi$
$48$	0	0
$49$	5.44071e9	2.75155
$50$	0	0
$51$	1.04657e9	0.424749
$52$	0	0
$53$	1.60012e9	0.525577	0.262789	−	0.964853i	$-0.415358\pi$
$53$	1.60012e9	0.525577	0.262789	+	0.964853i	$0.415358\pi$
$54$	0	0
$55$	−8.05198e9	−2.15729
$56$	0	0
$57$	−9.75159e7	−0.0214666
$58$	0	0
$59$	3.94549e9	0.718481	0.359240	−	0.933245i	$-0.383036\pi$
$59$	3.94549e9	0.718481	0.359240	+	0.933245i	$0.383036\pi$
$60$	0	0
$61$	−8.85973e8	−0.134309	−0.0671547	−	0.997743i	$-0.521392\pi$
$61$	−8.85973e8	−0.134309	−0.0671547	+	0.997743i	$0.521392\pi$
$62$	0	0
$63$	−5.08577e9	−0.645630
$64$	0	0
$65$	−9.59290e9	−1.02548
$66$	0	0
$67$	−4.88160e9	−0.441724	−0.220862	−	0.975305i	$-0.570887\pi$
$67$	−4.88160e9	−0.441724	−0.220862	+	0.975305i	$0.570887\pi$
$68$	0	0
$69$	−4.31362e9	−0.332026
$70$	0	0
$71$	1.26315e10	0.830870	0.415435	−	0.909623i	$-0.363629\pi$
$71$	1.26315e10	0.830870	0.415435	+	0.909623i	$0.363629\pi$
$72$	0	0
$73$	1.42334e9	0.0803584	0.0401792	−	0.999192i	$-0.487207\pi$
$73$	1.42334e9	0.0803584	0.0401792	+	0.999192i	$0.487207\pi$
$74$	0	0
$75$	−1.23862e10	−0.602699
$76$	0	0
$77$	6.94195e10	2.92269
$78$	0	0
$79$	6.67408e8	0.0244029	0.0122015	−	0.999926i	$-0.496116\pi$
$79$	6.67408e8	0.0244029	0.0122015	+	0.999926i	$0.496116\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	5.71607e9	0.159283	0.0796413	−	0.996824i	$-0.474623\pi$
$83$	5.71607e9	0.159283	0.0796413	+	0.996824i	$0.474623\pi$
$84$	0	0
$85$	−4.30257e10	−1.05178
$86$	0	0
$87$	2.05833e10	0.442751
$88$	0	0
$89$	−8.57387e10	−1.62754	−0.813771	−	0.581186i	$-0.802589\pi$
$89$	−8.57387e10	−1.62754	−0.813771	+	0.581186i	$0.802589\pi$
$90$	0	0
$91$	8.27044e10	1.38932
$92$	0	0
$93$	−3.42461e10	−0.510452
$94$	0	0
$95$	4.00899e9	0.0531563
$96$	0	0
$97$	−5.23026e10	−0.618414	−0.309207	−	0.950995i	$-0.600064\pi$
$97$	−5.23026e10	−0.618414	−0.309207	+	0.950995i	$0.600064\pi$
$98$	0	0
$99$	−4.75937e10	−0.502986
$100$	0	0
$101$	−1.61154e11	−1.52572	−0.762859	−	0.646564i	$-0.776205\pi$
$101$	−1.61154e11	−1.52572	−0.762859	+	0.646564i	$0.776205\pi$
$102$	0	0
$103$	−6.77986e10	−0.576257	−0.288128	−	0.957592i	$-0.593033\pi$
$103$	−6.77986e10	−0.576257	−0.288128	+	0.957592i	$0.593033\pi$
$104$	0	0
$105$	2.09082e11	1.59873
$106$	0	0
$107$	8.55967e10	0.589992	0.294996	−	0.955498i	$-0.404682\pi$
$107$	8.55967e10	0.589992	0.294996	+	0.955498i	$0.404682\pi$
$108$	0	0
$109$	1.82523e11	1.13624	0.568122	−	0.822945i	$-0.307670\pi$
$109$	1.82523e11	1.13624	0.568122	+	0.822945i	$0.307670\pi$
$110$	0	0
$111$	1.00482e11	0.565995
$112$	0	0
$113$	−3.44055e11	−1.75669	−0.878347	−	0.478024i	$-0.841353\pi$
$113$	−3.44055e11	−1.75669	−0.878347	+	0.478024i	$0.841353\pi$
$114$	0	0
$115$	1.77338e11	0.822173
$116$	0	0
$117$	−5.67018e10	−0.239097
$118$	0	0
$119$	3.70943e11	1.42495
$120$	0	0
$121$	3.64331e11	1.27696
$122$	0	0
$123$	−3.64731e10	−0.116814
$124$	0	0
$125$	2.14171e10	0.0627704
$126$	0	0
$127$	−6.89240e11	−1.85119	−0.925593	−	0.378521i	$-0.876433\pi$
$127$	−6.89240e11	−1.85119	−0.925593	+	0.378521i	$0.876433\pi$
$128$	0	0
$129$	−1.71729e11	−0.423252
$130$	0	0
$131$	−6.30533e11	−1.42796	−0.713979	−	0.700167i	$-0.753109\pi$
$131$	−6.30533e11	−1.42796	−0.713979	+	0.700167i	$0.753109\pi$
$132$	0	0
$133$	−3.45632e10	−0.0720161
$134$	0	0
$135$	−1.43346e11	−0.275137
$136$	0	0
$137$	3.98781e11	0.705946	0.352973	−	0.935634i	$-0.385171\pi$
$137$	3.98781e11	0.705946	0.352973	+	0.935634i	$0.385171\pi$
$138$	0	0
$139$	−4.20521e11	−0.687395	−0.343697	−	0.939080i	$-0.611679\pi$
$139$	−4.20521e11	−0.687395	−0.343697	+	0.939080i	$0.611679\pi$
$140$	0	0
$141$	6.01601e11	0.909083
$142$	0	0
$143$	7.73965e11	1.08236
$144$	0	0
$145$	−8.46203e11	−1.09635
$146$	0	0
$147$	−1.32209e12	−1.58861
$148$	0	0
$149$	9.37126e11	1.04538	0.522689	−	0.852523i	$-0.324929\pi$
$149$	9.37126e11	1.04538	0.522689	+	0.852523i	$0.324929\pi$
$150$	0	0
$151$	3.26258e11	0.338212	0.169106	−	0.985598i	$-0.445912\pi$
$151$	3.26258e11	0.338212	0.169106	+	0.985598i	$0.445912\pi$
$152$	0	0
$153$	−2.54317e11	−0.245229
$154$	0	0
$155$	1.40790e12	1.26400
$156$	0	0
$157$	2.13610e12	1.78720	0.893601	−	0.448863i	$-0.148171\pi$
$157$	2.13610e12	1.78720	0.893601	+	0.448863i	$0.148171\pi$
$158$	0	0
$159$	−3.88830e11	−0.303442
$160$	0	0
$161$	−1.52890e12	−1.11388
$162$	0	0
$163$	−7.90837e10	−0.0538338	−0.0269169	−	0.999638i	$-0.508569\pi$
$163$	−7.90837e10	−0.0538338	−0.0269169	+	0.999638i	$0.508569\pi$
$164$	0	0
$165$	1.95663e12	1.24551
$166$	0	0
$167$	1.56078e12	0.929824	0.464912	−	0.885357i	$-0.346086\pi$
$167$	1.56078e12	0.929824	0.464912	+	0.885357i	$0.346086\pi$
$168$	0	0
$169$	−8.70080e11	−0.485492
$170$	0	0
$171$	2.36964e10	0.0123938
$172$	0	0
$173$	−3.15093e12	−1.54591	−0.772956	−	0.634460i	$-0.781222\pi$
$173$	−3.15093e12	−1.54591	−0.772956	+	0.634460i	$0.781222\pi$
$174$	0	0
$175$	−4.39011e12	−2.02193
$176$	0	0
$177$	−9.58755e11	−0.414815
$178$	0	0
$179$	4.59872e11	0.187045	0.0935223	−	0.995617i	$-0.470187\pi$
$179$	4.59872e11	0.187045	0.0935223	+	0.995617i	$0.470187\pi$
$180$	0	0
$181$	2.50019e12	0.956623	0.478311	−	0.878190i	$-0.341249\pi$
$181$	2.50019e12	0.956623	0.478311	+	0.878190i	$0.341249\pi$
$182$	0	0
$183$	2.15292e11	0.0775436
$184$	0	0
$185$	−4.13093e12	−1.40153
$186$	0	0
$187$	3.47136e12	1.11012
$188$	0	0
$189$	1.23584e12	0.372755
$190$	0	0
$191$	−4.25760e12	−1.21194	−0.605970	−	0.795487i	$-0.707215\pi$
$191$	−4.25760e12	−1.21194	−0.605970	+	0.795487i	$0.707215\pi$
$192$	0	0
$193$	−9.32441e10	−0.0250643	−0.0125322	−	0.999921i	$-0.503989\pi$
$193$	−9.32441e10	−0.0250643	−0.0125322	+	0.999921i	$0.503989\pi$
$194$	0	0
$195$	2.33107e12	0.592060
$196$	0	0
$197$	−2.24661e12	−0.539466	−0.269733	−	0.962935i	$-0.586935\pi$
$197$	−2.24661e12	−0.539466	−0.269733	+	0.962935i	$0.586935\pi$
$198$	0	0
$199$	6.18786e12	1.40556	0.702779	−	0.711408i	$-0.251942\pi$
$199$	6.18786e12	1.40556	0.702779	+	0.711408i	$0.251942\pi$
$200$	0	0
$201$	1.18623e12	0.255029
$202$	0	0
$203$	7.29547e12	1.48534
$204$	0	0
$205$	1.49945e12	0.289258
$206$	0	0
$207$	1.04821e12	0.191695
$208$	0	0
$209$	−3.23449e11	−0.0561050
$210$	0	0
$211$	−5.04749e12	−0.830849	−0.415425	−	0.909628i	$-0.636367\pi$
$211$	−5.04749e12	−0.830849	−0.415425	+	0.909628i	$0.636367\pi$
$212$	0	0
$213$	−3.06945e12	−0.479703
$214$	0	0
$215$	7.05995e12	1.04807
$216$	0	0
$217$	−1.21381e13	−1.71246
$218$	0	0
$219$	−3.45870e11	−0.0463950
$220$	0	0
$221$	4.13568e12	0.527703
$222$	0	0
$223$	7.12235e12	0.864862	0.432431	−	0.901667i	$-0.357656\pi$
$223$	7.12235e12	0.864862	0.432431	+	0.901667i	$0.357656\pi$
$224$	0	0
$225$	3.00984e12	0.347969
$226$	0	0
$227$	−8.82409e12	−0.971690	−0.485845	−	0.874045i	$-0.661488\pi$
$227$	−8.82409e12	−0.971690	−0.485845	+	0.874045i	$0.661488\pi$
$228$	0	0
$229$	1.22311e13	1.28343	0.641713	−	0.766945i	$-0.278224\pi$
$229$	1.22311e13	1.28343	0.641713	+	0.766945i	$0.278224\pi$
$230$	0	0
$231$	−1.68689e13	−1.68742
$232$	0	0
$233$	−1.26728e13	−1.20897	−0.604485	−	0.796616i	$-0.706621\pi$
$233$	−1.26728e13	−1.20897	−0.604485	+	0.796616i	$0.706621\pi$
$234$	0	0
$235$	−2.47325e13	−2.25110
$236$	0	0
$237$	−1.62180e11	−0.0140890
$238$	0	0
$239$	8.39071e11	0.0696002	0.0348001	−	0.999394i	$-0.488921\pi$
$239$	8.39071e11	0.0696002	0.0348001	+	0.999394i	$0.488921\pi$
$240$	0	0
$241$	−4.50963e12	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$241$	−4.50963e12	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$242$	0	0
$243$	−8.47289e11	−0.0641500
$244$	0	0
$245$	5.43526e13	3.93375
$246$	0	0
$247$	−3.85348e11	−0.0266698
$248$	0	0
$249$	−1.38901e12	−0.0919618
$250$	0	0
$251$	9.93358e12	0.629362	0.314681	−	0.949197i	$-0.398102\pi$
$251$	9.93358e12	0.629362	0.314681	+	0.949197i	$0.398102\pi$
$252$	0	0
$253$	−1.43078e13	−0.867781
$254$	0	0
$255$	1.04552e13	0.607244
$256$	0	0
$257$	−1.23707e13	−0.688277	−0.344139	−	0.938919i	$-0.611829\pi$
$257$	−1.23707e13	−0.688277	−0.344139	+	0.938919i	$0.611829\pi$
$258$	0	0
$259$	3.56145e13	1.89880
$260$	0	0
$261$	−5.00175e12	−0.255622
$262$	0	0
$263$	−3.12644e13	−1.53212	−0.766061	−	0.642768i	$-0.777786\pi$
$263$	−3.12644e13	−1.53212	−0.766061	+	0.642768i	$0.777786\pi$
$264$	0	0
$265$	1.59852e13	0.751393
$266$	0	0
$267$	2.08345e13	0.939661
$268$	0	0
$269$	1.65927e13	0.718256	0.359128	−	0.933288i	$-0.383074\pi$
$269$	1.65927e13	0.718256	0.359128	+	0.933288i	$0.383074\pi$
$270$	0	0
$271$	−9.74246e12	−0.404890	−0.202445	−	0.979294i	$-0.564889\pi$
$271$	−9.74246e12	−0.404890	−0.202445	+	0.979294i	$0.564889\pi$
$272$	0	0
$273$	−2.00972e13	−0.802122
$274$	0	0
$275$	−4.10836e13	−1.57521
$276$	0	0
$277$	2.96465e13	1.09228	0.546140	−	0.837694i	$-0.316097\pi$
$277$	2.96465e13	1.09228	0.546140	+	0.837694i	$0.316097\pi$
$278$	0	0
$279$	8.32181e12	0.294710
$280$	0	0
$281$	1.49493e12	0.0509020	0.0254510	−	0.999676i	$-0.491898\pi$
$281$	1.49493e12	0.0509020	0.0254510	+	0.999676i	$0.491898\pi$
$282$	0	0
$283$	−4.24492e13	−1.39009	−0.695047	−	0.718964i	$-0.744617\pi$
$283$	−4.24492e13	−1.39009	−0.695047	+	0.718964i	$0.744617\pi$
$284$	0	0
$285$	−9.74184e11	−0.0306898
$286$	0	0
$287$	−1.29274e13	−0.391886
$288$	0	0
$289$	−1.57227e13	−0.458764
$290$	0	0
$291$	1.27095e13	0.357041
$292$	0	0
$293$	−5.60424e13	−1.51616	−0.758080	−	0.652162i	$-0.773862\pi$
$293$	−5.60424e13	−1.51616	−0.758080	+	0.652162i	$0.773862\pi$
$294$	0	0
$295$	3.94155e13	1.02718
$296$	0	0
$297$	1.15653e13	0.290399
$298$	0	0
$299$	−1.70459e13	−0.412504
$300$	0	0
$301$	−6.08668e13	−1.41992
$302$	0	0
$303$	3.91605e13	0.880874
$304$	0	0
$305$	−8.85088e12	−0.192016
$306$	0	0
$307$	5.76774e13	1.20710	0.603552	−	0.797324i	$-0.293752\pi$
$307$	5.76774e13	1.20710	0.603552	+	0.797324i	$0.293752\pi$
$308$	0	0
$309$	1.64751e13	0.332702
$310$	0	0
$311$	5.64879e13	1.10096	0.550482	−	0.834847i	$-0.314444\pi$
$311$	5.64879e13	1.10096	0.550482	+	0.834847i	$0.314444\pi$
$312$	0	0
$313$	2.86939e13	0.539878	0.269939	−	0.962877i	$-0.412996\pi$
$313$	2.86939e13	0.539878	0.269939	+	0.962877i	$0.412996\pi$
$314$	0	0
$315$	−5.08069e13	−0.923027
$316$	0	0
$317$	1.24332e13	0.218151	0.109075	−	0.994033i	$-0.465211\pi$
$317$	1.24332e13	0.218151	0.109075	+	0.994033i	$0.465211\pi$
$318$	0	0
$319$	6.82726e13	1.15717
$320$	0	0
$321$	−2.08000e13	−0.340632
$322$	0	0
$323$	−1.72835e12	−0.0273538
$324$	0	0
$325$	−4.89458e13	−0.748785
$326$	0	0
$327$	−4.43530e13	−0.656010
$328$	0	0
$329$	2.13229e14	3.04979
$330$	0	0
$331$	9.38818e13	1.29876	0.649378	−	0.760466i	$-0.275029\pi$
$331$	9.38818e13	1.29876	0.649378	+	0.760466i	$0.275029\pi$
$332$	0	0
$333$	−2.44172e13	−0.326777
$334$	0	0
$335$	−4.87672e13	−0.631511
$336$	0	0
$337$	−1.47739e14	−1.85153	−0.925766	−	0.378096i	$-0.876579\pi$
$337$	−1.47739e14	−1.85153	−0.925766	+	0.378096i	$0.876579\pi$
$338$	0	0
$339$	8.36053e13	1.01423
$340$	0	0
$341$	−1.13591e14	−1.33411
$342$	0	0
$343$	−2.98294e14	−3.39255
$344$	0	0
$345$	−4.30931e13	−0.474682
$346$	0	0
$347$	2.44519e12	0.0260916	0.0130458	−	0.999915i	$-0.495847\pi$
$347$	2.44519e12	0.0260916	0.0130458	+	0.999915i	$0.495847\pi$
$348$	0	0
$349$	1.08907e14	1.12594	0.562971	−	0.826477i	$-0.309658\pi$
$349$	1.08907e14	1.12594	0.562971	+	0.826477i	$0.309658\pi$
$350$	0	0
$351$	1.37785e13	0.138043
$352$	0	0
$353$	4.05373e13	0.393635	0.196817	−	0.980440i	$-0.436939\pi$
$353$	4.05373e13	0.393635	0.196817	+	0.980440i	$0.436939\pi$
$354$	0	0
$355$	1.26188e14	1.18785
$356$	0	0
$357$	−9.01391e13	−0.822693
$358$	0	0
$359$	1.67076e14	1.47875	0.739374	−	0.673295i	$-0.235122\pi$
$359$	1.67076e14	1.47875	0.739374	+	0.673295i	$0.235122\pi$
$360$	0	0
$361$	−1.16329e14	−0.998618
$362$	0	0
$363$	−8.85324e13	−0.737252
$364$	0	0
$365$	1.42191e13	0.114885
$366$	0	0
$367$	−8.13245e12	−0.0637614	−0.0318807	−	0.999492i	$-0.510150\pi$
$367$	−8.13245e12	−0.0637614	−0.0318807	+	0.999492i	$0.510150\pi$
$368$	0	0
$369$	8.86295e12	0.0674425
$370$	0	0
$371$	−1.37815e14	−1.01799
$372$	0	0
$373$	7.11946e13	0.510562	0.255281	−	0.966867i	$-0.417832\pi$
$373$	7.11946e13	0.510562	0.255281	+	0.966867i	$0.417832\pi$
$374$	0	0
$375$	−5.20435e12	−0.0362405
$376$	0	0
$377$	8.13380e13	0.550067
$378$	0	0
$379$	−1.32772e14	−0.872148	−0.436074	−	0.899911i	$-0.643631\pi$
$379$	−1.32772e14	−0.872148	−0.436074	+	0.899911i	$0.643631\pi$
$380$	0	0
$381$	1.67485e14	1.06878
$382$	0	0
$383$	−8.42537e13	−0.522391	−0.261196	−	0.965286i	$-0.584117\pi$
$383$	−8.42537e13	−0.522391	−0.261196	+	0.965286i	$0.584117\pi$
$384$	0	0
$385$	6.93501e14	4.17843
$386$	0	0
$387$	4.17300e13	0.244365
$388$	0	0
$389$	−3.15986e13	−0.179864	−0.0899321	−	0.995948i	$-0.528665\pi$
$389$	−3.15986e13	−0.179864	−0.0899321	+	0.995948i	$0.528665\pi$
$390$	0	0
$391$	−7.64537e13	−0.423084
$392$	0	0
$393$	1.53220e14	0.824432
$394$	0	0
$395$	6.66740e12	0.0348877
$396$	0	0
$397$	−2.87957e14	−1.46548	−0.732739	−	0.680510i	$-0.761758\pi$
$397$	−2.87957e14	−1.46548	−0.732739	+	0.680510i	$0.761758\pi$
$398$	0	0
$399$	8.39885e12	0.0415785
$400$	0	0
$401$	−3.36601e14	−1.62114	−0.810571	−	0.585640i	$-0.800843\pi$
$401$	−3.36601e14	−1.62114	−0.810571	+	0.585640i	$0.800843\pi$
$402$	0	0
$403$	−1.35329e14	−0.634179
$404$	0	0
$405$	3.48330e13	0.158850
$406$	0	0
$407$	3.33288e14	1.47928
$408$	0	0
$409$	2.67590e13	0.115609	0.0578046	−	0.998328i	$-0.481590\pi$
$409$	2.67590e13	0.115609	0.0578046	+	0.998328i	$0.481590\pi$
$410$	0	0
$411$	−9.69038e13	−0.407578
$412$	0	0
$413$	−3.39817e14	−1.39162
$414$	0	0
$415$	5.71036e13	0.227719
$416$	0	0
$417$	1.02187e14	0.396867
$418$	0	0
$419$	−3.04568e14	−1.15214	−0.576072	−	0.817399i	$-0.695415\pi$
$419$	−3.04568e14	−1.15214	−0.576072	+	0.817399i	$0.695415\pi$
$420$	0	0
$421$	−2.19907e13	−0.0810377	−0.0405188	−	0.999179i	$-0.512901\pi$
$421$	−2.19907e13	−0.0810377	−0.0405188	+	0.999179i	$0.512901\pi$
$422$	0	0
$423$	−1.46189e14	−0.524859
$424$	0	0
$425$	−2.19530e14	−0.767989
$426$	0	0
$427$	7.63071e13	0.260143
$428$	0	0
$429$	−1.88074e14	−0.624903
$430$	0	0
$431$	−1.84103e14	−0.596261	−0.298130	−	0.954525i	$-0.596363\pi$
$431$	−1.84103e14	−0.596261	−0.298130	+	0.954525i	$0.596363\pi$
$432$	0	0
$433$	1.87650e14	0.592469	0.296234	−	0.955115i	$-0.404269\pi$
$433$	1.87650e14	0.592469	0.296234	+	0.955115i	$0.404269\pi$
$434$	0	0
$435$	2.05627e14	0.632980
$436$	0	0
$437$	7.12369e12	0.0213824
$438$	0	0
$439$	5.22200e14	1.52856	0.764279	−	0.644885i	$-0.223095\pi$
$439$	5.22200e14	1.52856	0.764279	+	0.644885i	$0.223095\pi$
$440$	0	0
$441$	3.21268e14	0.917182
$442$	0	0
$443$	4.02043e14	1.11957	0.559785	−	0.828638i	$-0.310884\pi$
$443$	4.02043e14	1.11957	0.559785	+	0.828638i	$0.310884\pi$
$444$	0	0
$445$	−8.56530e14	−2.32682
$446$	0	0
$447$	−2.27722e14	−0.603550
$448$	0	0
$449$	5.62244e14	1.45402	0.727009	−	0.686628i	$-0.240910\pi$
$449$	5.62244e14	1.45402	0.727009	+	0.686628i	$0.240910\pi$
$450$	0	0
$451$	−1.20977e14	−0.305304
$452$	0	0
$453$	−7.92808e13	−0.195267
$454$	0	0
$455$	8.26217e14	1.98624
$456$	0	0
$457$	−7.00143e13	−0.164304	−0.0821519	−	0.996620i	$-0.526179\pi$
$457$	−7.00143e13	−0.164304	−0.0821519	+	0.996620i	$0.526179\pi$
$458$	0	0
$459$	6.17990e13	0.141583
$460$	0	0
$461$	3.63273e14	0.812603	0.406301	−	0.913739i	$-0.366818\pi$
$461$	3.63273e14	0.812603	0.406301	+	0.913739i	$0.366818\pi$
$462$	0	0
$463$	−1.81837e14	−0.397179	−0.198589	−	0.980083i	$-0.563636\pi$
$463$	−1.81837e14	−0.397179	−0.198589	+	0.980083i	$0.563636\pi$
$464$	0	0
$465$	−3.42119e14	−0.729769
$466$	0	0
$467$	1.99364e14	0.415339	0.207670	−	0.978199i	$-0.433412\pi$
$467$	1.99364e14	0.415339	0.207670	+	0.978199i	$0.433412\pi$
$468$	0	0
$469$	4.20442e14	0.855570
$470$	0	0
$471$	−5.19072e14	−1.03184
$472$	0	0
$473$	−5.69605e14	−1.10621
$474$	0	0
$475$	2.04551e13	0.0388138
$476$	0	0
$477$	9.44857e13	0.175192
$478$	0	0
$479$	7.99891e13	0.144939	0.0724696	−	0.997371i	$-0.476912\pi$
$479$	7.99891e13	0.144939	0.0724696	+	0.997371i	$0.476912\pi$
$480$	0	0
$481$	3.97070e14	0.703184
$482$	0	0
$483$	3.71524e14	0.643098
$484$	0	0
$485$	−5.22503e14	−0.884117
$486$	0	0
$487$	−8.53441e14	−1.41177	−0.705885	−	0.708326i	$-0.749451\pi$
$487$	−8.53441e14	−1.41177	−0.705885	+	0.708326i	$0.749451\pi$
$488$	0	0
$489$	1.92173e13	0.0310810
$490$	0	0
$491$	5.39015e14	0.852418	0.426209	−	0.904625i	$-0.359849\pi$
$491$	5.39015e14	0.852418	0.426209	+	0.904625i	$0.359849\pi$
$492$	0	0
$493$	3.64814e14	0.564175
$494$	0	0
$495$	−4.75461e14	−0.719096
$496$	0	0
$497$	−1.08792e15	−1.60930
$498$	0	0
$499$	−5.46806e14	−0.791189	−0.395594	−	0.918425i	$-0.629461\pi$
$499$	−5.46806e14	−0.791189	−0.395594	+	0.918425i	$0.629461\pi$
$500$	0	0
$501$	−3.79269e14	−0.536834
$502$	0	0
$503$	5.85282e14	0.810477	0.405239	−	0.914211i	$-0.367188\pi$
$503$	5.85282e14	0.810477	0.405239	+	0.914211i	$0.367188\pi$
$504$	0	0
$505$	−1.60993e15	−2.18125
$506$	0	0
$507$	2.11430e14	0.280299
$508$	0	0
$509$	2.30032e14	0.298429	0.149214	−	0.988805i	$-0.452326\pi$
$509$	2.30032e14	0.298429	0.149214	+	0.988805i	$0.452326\pi$
$510$	0	0
$511$	−1.22589e14	−0.155646
$512$	0	0
$513$	−5.75822e12	−0.00715554
$514$	0	0
$515$	−6.77308e14	−0.823847
$516$	0	0
$517$	1.99544e15	2.37597
$518$	0	0
$519$	7.65675e14	0.892532
$520$	0	0
$521$	1.44985e15	1.65469	0.827345	−	0.561694i	$-0.189850\pi$
$521$	1.44985e15	1.65469	0.827345	+	0.561694i	$0.189850\pi$
$522$	0	0
$523$	8.87051e14	0.991264	0.495632	−	0.868533i	$-0.334936\pi$
$523$	8.87051e14	0.991264	0.495632	+	0.868533i	$0.334936\pi$
$524$	0	0
$525$	1.06680e15	1.16736
$526$	0	0
$527$	−6.06970e14	−0.650443
$528$	0	0
$529$	−6.37693e14	−0.669276
$530$	0	0
$531$	2.32977e14	0.239494
$532$	0	0
$533$	−1.44129e14	−0.145128
$534$	0	0
$535$	8.55112e14	0.843484
$536$	0	0
$537$	−1.11749e14	−0.107990
$538$	0	0
$539$	−4.38523e15	−4.15197
$540$	0	0
$541$	1.21596e15	1.12807	0.564034	−	0.825752i	$-0.309249\pi$
$541$	1.21596e15	1.12807	0.564034	+	0.825752i	$0.309249\pi$
$542$	0	0
$543$	−6.07546e14	−0.552306
$544$	0	0
$545$	1.82340e15	1.62443
$546$	0	0
$547$	1.77148e15	1.54670	0.773348	−	0.633982i	$-0.218581\pi$
$547$	1.77148e15	1.54670	0.773348	+	0.633982i	$0.218581\pi$
$548$	0	0
$549$	−5.23158e13	−0.0447698
$550$	0	0
$551$	−3.39921e13	−0.0285131
$552$	0	0
$553$	−5.74825e13	−0.0472658
$554$	0	0
$555$	1.00382e15	0.809176
$556$	0	0
$557$	1.12842e15	0.891801	0.445900	−	0.895083i	$-0.352884\pi$
$557$	1.12842e15	0.891801	0.445900	+	0.895083i	$0.352884\pi$
$558$	0	0
$559$	−6.78611e14	−0.525843
$560$	0	0
$561$	−8.43541e14	−0.640930
$562$	0	0
$563$	−6.21047e14	−0.462731	−0.231365	−	0.972867i	$-0.574319\pi$
$563$	−6.21047e14	−0.462731	−0.231365	+	0.972867i	$0.574319\pi$
$564$	0	0
$565$	−3.43711e15	−2.51146
$566$	0	0
$567$	−3.00310e14	−0.215210
$568$	0	0
$569$	−1.21509e15	−0.854068	−0.427034	−	0.904236i	$-0.640441\pi$
$569$	−1.21509e15	−0.854068	−0.427034	+	0.904236i	$0.640441\pi$
$570$	0	0
$571$	−1.36046e15	−0.937964	−0.468982	−	0.883208i	$-0.655379\pi$
$571$	−1.36046e15	−0.937964	−0.468982	+	0.883208i	$0.655379\pi$
$572$	0	0
$573$	1.03460e15	0.699714
$574$	0	0
$575$	9.04830e14	0.600336
$576$	0	0
$577$	8.26592e14	0.538052	0.269026	−	0.963133i	$-0.413298\pi$
$577$	8.26592e14	0.538052	0.269026	+	0.963133i	$0.413298\pi$
$578$	0	0
$579$	2.26583e13	0.0144709
$580$	0	0
$581$	−4.92314e14	−0.308513
$582$	0	0
$583$	−1.28971e15	−0.793074
$584$	0	0
$585$	−5.66451e14	−0.341826
$586$	0	0
$587$	4.88505e14	0.289307	0.144654	−	0.989482i	$-0.453793\pi$
$587$	4.88505e14	0.289307	0.144654	+	0.989482i	$0.453793\pi$
$588$	0	0
$589$	5.65554e13	0.0328731
$590$	0	0
$591$	5.45927e14	0.311461
$592$	0	0
$593$	−1.06917e15	−0.598751	−0.299376	−	0.954135i	$-0.596778\pi$
$593$	−1.06917e15	−0.598751	−0.299376	+	0.954135i	$0.596778\pi$
$594$	0	0
$595$	3.70572e15	2.03718
$596$	0	0
$597$	−1.50365e15	−0.811499
$598$	0	0
$599$	−6.36019e14	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$599$	−6.36019e14	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$600$	0	0
$601$	−7.52046e14	−0.391232	−0.195616	−	0.980681i	$-0.562671\pi$
$601$	−7.52046e14	−0.391232	−0.195616	+	0.980681i	$0.562671\pi$
$602$	0	0
$603$	−2.88253e14	−0.147241
$604$	0	0
$605$	3.63966e15	1.82560
$606$	0	0
$607$	−3.11325e15	−1.53347	−0.766737	−	0.641962i	$-0.778121\pi$
$607$	−3.11325e15	−1.53347	−0.766737	+	0.641962i	$0.778121\pi$
$608$	0	0
$609$	−1.77280e15	−0.857560
$610$	0	0
$611$	2.37732e15	1.12943
$612$	0	0
$613$	3.57054e15	1.66610	0.833051	−	0.553197i	$-0.186592\pi$
$613$	3.57054e15	1.66610	0.833051	+	0.553197i	$0.186592\pi$
$614$	0	0
$615$	−3.64366e14	−0.167003
$616$	0	0
$617$	1.11912e15	0.503858	0.251929	−	0.967746i	$-0.418935\pi$
$617$	1.11912e15	0.503858	0.251929	+	0.967746i	$0.418935\pi$
$618$	0	0
$619$	−4.16601e14	−0.184256	−0.0921280	−	0.995747i	$-0.529367\pi$
$619$	−4.16601e14	−0.184256	−0.0921280	+	0.995747i	$0.529367\pi$
$620$	0	0
$621$	−2.54715e14	−0.110675
$622$	0	0
$623$	7.38451e15	3.15237
$624$	0	0
$625$	−2.27491e15	−0.954166
$626$	0	0
$627$	7.85982e13	0.0323923
$628$	0	0
$629$	1.78092e15	0.721218
$630$	0	0
$631$	3.55936e15	1.41648	0.708240	−	0.705972i	$-0.249490\pi$
$631$	3.55936e15	1.41648	0.708240	+	0.705972i	$0.249490\pi$
$632$	0	0
$633$	1.22654e15	0.479691
$634$	0	0
$635$	−6.88551e15	−2.64655
$636$	0	0
$637$	−5.22444e15	−1.97366
$638$	0	0
$639$	7.45876e14	0.276957
$640$	0	0
$641$	−5.00307e15	−1.82607	−0.913036	−	0.407880i	$-0.866268\pi$
$641$	−5.00307e15	−1.82607	−0.913036	+	0.407880i	$0.866268\pi$
$642$	0	0
$643$	4.51393e15	1.61955	0.809775	−	0.586740i	$-0.199589\pi$
$643$	4.51393e15	1.61955	0.809775	+	0.586740i	$0.199589\pi$
$644$	0	0
$645$	−1.71557e15	−0.605104
$646$	0	0
$647$	2.87451e15	0.996758	0.498379	−	0.866959i	$-0.333929\pi$
$647$	2.87451e15	0.996758	0.498379	+	0.866959i	$0.333929\pi$
$648$	0	0
$649$	−3.18008e15	−1.08416
$650$	0	0
$651$	2.94955e15	0.988690
$652$	0	0
$653$	9.33676e14	0.307733	0.153866	−	0.988092i	$-0.450827\pi$
$653$	9.33676e14	0.307733	0.153866	+	0.988092i	$0.450827\pi$
$654$	0	0
$655$	−6.29902e15	−2.04148
$656$	0	0
$657$	8.40465e13	0.0267861
$658$	0	0
$659$	−5.86905e15	−1.83949	−0.919746	−	0.392515i	$-0.871605\pi$
$659$	−5.86905e15	−1.83949	−0.919746	+	0.392515i	$0.871605\pi$
$660$	0	0
$661$	2.97430e15	0.916804	0.458402	−	0.888745i	$-0.348422\pi$
$661$	2.97430e15	0.916804	0.458402	+	0.888745i	$0.348422\pi$
$662$	0	0
$663$	−1.00497e15	−0.304669
$664$	0	0
$665$	−3.45286e14	−0.102958
$666$	0	0
$667$	−1.50364e15	−0.441014
$668$	0	0
$669$	−1.73073e15	−0.499328
$670$	0	0
$671$	7.14098e14	0.202667
$672$	0	0
$673$	3.98702e15	1.11318	0.556590	−	0.830787i	$-0.312109\pi$
$673$	3.98702e15	1.11318	0.556590	+	0.830787i	$0.312109\pi$
$674$	0	0
$675$	−7.31392e14	−0.200900
$676$	0	0
$677$	2.91335e15	0.787326	0.393663	−	0.919255i	$-0.371208\pi$
$677$	2.91335e15	0.787326	0.393663	+	0.919255i	$0.371208\pi$
$678$	0	0
$679$	4.50472e15	1.19780
$680$	0	0
$681$	2.14425e15	0.561005
$682$	0	0
$683$	−7.17859e15	−1.84810	−0.924049	−	0.382273i	$-0.875141\pi$
$683$	−7.17859e15	−1.84810	−0.924049	+	0.382273i	$0.875141\pi$
$684$	0	0
$685$	3.98382e15	1.00926
$686$	0	0
$687$	−2.97216e15	−0.740987
$688$	0	0
$689$	−1.53652e15	−0.376992
$690$	0	0
$691$	−1.13258e15	−0.273488	−0.136744	−	0.990606i	$-0.543664\pi$
$691$	−1.13258e15	−0.273488	−0.136744	+	0.990606i	$0.543664\pi$
$692$	0	0
$693$	4.09915e15	0.974230
$694$	0	0
$695$	−4.20100e15	−0.982735
$696$	0	0
$697$	−6.46440e14	−0.148850
$698$	0	0
$699$	3.07949e15	0.697999
$700$	0	0
$701$	−6.88948e15	−1.53722	−0.768612	−	0.639715i	$-0.779052\pi$
$701$	−6.88948e15	−1.53722	−0.768612	+	0.639715i	$0.779052\pi$
$702$	0	0
$703$	−1.65940e14	−0.0364500
$704$	0	0
$705$	6.01000e15	1.29967
$706$	0	0
$707$	1.38799e16	2.95515
$708$	0	0
$709$	−3.12864e15	−0.655846	−0.327923	−	0.944704i	$-0.606349\pi$
$709$	−3.12864e15	−0.655846	−0.327923	+	0.944704i	$0.606349\pi$
$710$	0	0
$711$	3.94097e13	0.00813431
$712$	0	0
$713$	2.50173e15	0.508450
$714$	0	0
$715$	7.73191e15	1.54740
$716$	0	0
$717$	−2.03894e14	−0.0401837
$718$	0	0
$719$	1.24806e15	0.242228	0.121114	−	0.992639i	$-0.461353\pi$
$719$	1.24806e15	0.242228	0.121114	+	0.992639i	$0.461353\pi$
$720$	0	0
$721$	5.83936e15	1.11615
$722$	0	0
$723$	1.09584e15	0.206294
$724$	0	0
$725$	−4.31758e15	−0.800537
$726$	0	0
$727$	−6.33188e15	−1.15636	−0.578181	−	0.815909i	$-0.696237\pi$
$727$	−6.33188e15	−1.15636	−0.578181	+	0.815909i	$0.696237\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−3.04368e15	−0.539328
$732$	0	0
$733$	−4.73279e15	−0.826124	−0.413062	−	0.910703i	$-0.635541\pi$
$733$	−4.73279e15	−0.826124	−0.413062	+	0.910703i	$0.635541\pi$
$734$	0	0
$735$	−1.32077e16	−2.27115
$736$	0	0
$737$	3.93459e15	0.666543
$738$	0	0
$739$	−6.08206e15	−1.01509	−0.507547	−	0.861624i	$-0.669448\pi$
$739$	−6.08206e15	−1.01509	−0.507547	+	0.861624i	$0.669448\pi$
$740$	0	0
$741$	9.36396e13	0.0153978
$742$	0	0
$743$	4.07307e15	0.659909	0.329954	−	0.943997i	$-0.392967\pi$
$743$	4.07307e15	0.659909	0.329954	+	0.943997i	$0.392967\pi$
$744$	0	0
$745$	9.36189e15	1.49453
$746$	0	0
$747$	3.37528e14	0.0530942
$748$	0	0
$749$	−7.37228e15	−1.14275
$750$	0	0
$751$	−8.73635e15	−1.33447	−0.667237	−	0.744845i	$-0.732523\pi$
$751$	−8.73635e15	−1.33447	−0.667237	+	0.744845i	$0.732523\pi$
$752$	0	0
$753$	−2.41386e15	−0.363362
$754$	0	0
$755$	3.25932e15	0.483525
$756$	0	0
$757$	7.84148e15	1.14649	0.573245	−	0.819384i	$-0.305684\pi$
$757$	7.84148e15	1.14649	0.573245	+	0.819384i	$0.305684\pi$
$758$	0	0
$759$	3.47680e15	0.501014
$760$	0	0
$761$	1.57189e15	0.223258	0.111629	−	0.993750i	$-0.464393\pi$
$761$	1.57189e15	0.223258	0.111629	+	0.993750i	$0.464393\pi$
$762$	0	0
$763$	−1.57203e16	−2.20078
$764$	0	0
$765$	−2.54063e15	−0.350593
$766$	0	0
$767$	−3.78866e15	−0.515360
$768$	0	0
$769$	−8.76621e14	−0.117548	−0.0587742	−	0.998271i	$-0.518719\pi$
$769$	−8.76621e14	−0.117548	−0.0587742	+	0.998271i	$0.518719\pi$
$770$	0	0
$771$	3.00609e15	0.397377
$772$	0	0
$773$	−5.53836e15	−0.721762	−0.360881	−	0.932612i	$-0.617524\pi$
$773$	−5.53836e15	−0.721762	−0.360881	+	0.932612i	$0.617524\pi$
$774$	0	0
$775$	7.18351e15	0.922948
$776$	0	0
$777$	−8.65432e15	−1.09627
$778$	0	0
$779$	6.02331e13	0.00752279
$780$	0	0
$781$	−1.01810e16	−1.25375
$782$	0	0
$783$	1.21542e15	0.147584
$784$	0	0
$785$	2.13396e16	2.55508
$786$	0	0
$787$	1.11368e15	0.131493	0.0657463	−	0.997836i	$-0.479057\pi$
$787$	1.11368e15	0.131493	0.0657463	+	0.997836i	$0.479057\pi$
$788$	0	0
$789$	7.59724e15	0.884571
$790$	0	0
$791$	2.96327e16	3.40252
$792$	0	0
$793$	8.50756e14	0.0963391
$794$	0	0
$795$	−3.88441e15	−0.433817
$796$	0	0
$797$	−1.19780e15	−0.131936	−0.0659681	−	0.997822i	$-0.521014\pi$
$797$	−1.19780e15	−0.131936	−0.0659681	+	0.997822i	$0.521014\pi$
$798$	0	0
$799$	1.06626e16	1.15840
$800$	0	0
$801$	−5.06279e15	−0.542514
$802$	0	0
$803$	−1.14721e15	−0.121258
$804$	0	0
$805$	−1.52737e16	−1.59246
$806$	0	0
$807$	−4.03202e15	−0.414685
$808$	0	0
$809$	−9.08896e15	−0.922141	−0.461070	−	0.887364i	$-0.652534\pi$
$809$	−9.08896e15	−0.922141	−0.461070	+	0.887364i	$0.652534\pi$
$810$	0	0
$811$	4.75285e15	0.475707	0.237854	−	0.971301i	$-0.423556\pi$
$811$	4.75285e15	0.475707	0.237854	+	0.971301i	$0.423556\pi$
$812$	0	0
$813$	2.36742e15	0.233764
$814$	0	0
$815$	−7.90046e14	−0.0769637
$816$	0	0
$817$	2.83600e14	0.0272574
$818$	0	0
$819$	4.88361e15	0.463105
$820$	0	0
$821$	−7.46358e15	−0.698328	−0.349164	−	0.937062i	$-0.613534\pi$
$821$	−7.46358e15	−0.698328	−0.349164	+	0.937062i	$0.613534\pi$
$822$	0	0
$823$	1.33712e16	1.23444	0.617222	−	0.786789i	$-0.288258\pi$
$823$	1.33712e16	1.23444	0.617222	+	0.786789i	$0.288258\pi$
$824$	0	0
$825$	9.98332e15	0.909449
$826$	0	0
$827$	−8.28236e15	−0.744515	−0.372258	−	0.928129i	$-0.621416\pi$
$827$	−8.28236e15	−0.744515	−0.372258	+	0.928129i	$0.621416\pi$
$828$	0	0
$829$	1.81760e16	1.61231	0.806153	−	0.591707i	$-0.201546\pi$
$829$	1.81760e16	1.61231	0.806153	+	0.591707i	$0.201546\pi$
$830$	0	0
$831$	−7.20409e15	−0.630628
$832$	0	0
$833$	−2.34325e16	−2.02428
$834$	0	0
$835$	1.55922e16	1.32933
$836$	0	0
$837$	−2.02220e15	−0.170151
$838$	0	0
$839$	7.04343e15	0.584916	0.292458	−	0.956278i	$-0.405527\pi$
$839$	7.04343e15	0.584916	0.292458	+	0.956278i	$0.405527\pi$
$840$	0	0
$841$	−5.02557e15	−0.411915
$842$	0	0
$843$	−3.63267e14	−0.0293883
$844$	0	0
$845$	−8.69210e15	−0.694085
$846$	0	0
$847$	−3.13791e16	−2.47333
$848$	0	0
$849$	1.03152e16	0.802571
$850$	0	0
$851$	−7.34037e15	−0.563775
$852$	0	0
$853$	−1.02559e16	−0.777597	−0.388798	−	0.921323i	$-0.627110\pi$
$853$	−1.02559e16	−0.777597	−0.388798	+	0.921323i	$0.627110\pi$
$854$	0	0
$855$	2.36727e14	0.0177188
$856$	0	0
$857$	1.30722e16	0.965952	0.482976	−	0.875634i	$-0.339556\pi$
$857$	1.30722e16	0.965952	0.482976	+	0.875634i	$0.339556\pi$
$858$	0	0
$859$	3.07866e15	0.224594	0.112297	−	0.993675i	$-0.464179\pi$
$859$	3.07866e15	0.224594	0.112297	+	0.993675i	$0.464179\pi$
$860$	0	0
$861$	3.14135e15	0.226256
$862$	0	0
$863$	−1.90413e16	−1.35406	−0.677029	−	0.735956i	$-0.736733\pi$
$863$	−1.90413e16	−1.35406	−0.677029	+	0.735956i	$0.736733\pi$
$864$	0	0
$865$	−3.14777e16	−2.21012
$866$	0	0
$867$	3.82062e15	0.264867
$868$	0	0
$869$	−5.37933e14	−0.0368230
$870$	0	0
$871$	4.68755e15	0.316845
$872$	0	0
$873$	−3.08842e15	−0.206138
$874$	0	0
$875$	−1.84461e15	−0.121579
$876$	0	0
$877$	1.88173e16	1.22479	0.612394	−	0.790553i	$-0.290207\pi$
$877$	1.88173e16	1.22479	0.612394	+	0.790553i	$0.290207\pi$
$878$	0	0
$879$	1.36183e16	0.875355
$880$	0	0
$881$	−2.40222e15	−0.152491	−0.0762456	−	0.997089i	$-0.524293\pi$
$881$	−2.40222e15	−0.152491	−0.0762456	+	0.997089i	$0.524293\pi$
$882$	0	0
$883$	1.31415e15	0.0823877	0.0411938	−	0.999151i	$-0.486884\pi$
$883$	1.31415e15	0.0823877	0.0411938	+	0.999151i	$0.486884\pi$
$884$	0	0
$885$	−9.57796e15	−0.593041
$886$	0	0
$887$	−4.02583e15	−0.246193	−0.123096	−	0.992395i	$-0.539282\pi$
$887$	−4.02583e15	−0.246193	−0.123096	+	0.992395i	$0.539282\pi$
$888$	0	0
$889$	5.93629e16	3.58554
$890$	0	0
$891$	−2.81036e15	−0.167662
$892$	0	0
$893$	−9.93509e14	−0.0585448
$894$	0	0
$895$	4.59412e15	0.267409
$896$	0	0
$897$	4.14215e15	0.238159
$898$	0	0
$899$	−1.19375e16	−0.678009
$900$	0	0
$901$	−6.89154e15	−0.386661
$902$	0	0
$903$	1.47906e16	0.819793
$904$	0	0
$905$	2.49769e16	1.36764
$906$	0	0
$907$	1.01871e16	0.551076	0.275538	−	0.961290i	$-0.411144\pi$
$907$	1.01871e16	0.551076	0.275538	+	0.961290i	$0.411144\pi$
$908$	0	0
$909$	−9.51600e15	−0.508573
$910$	0	0
$911$	−2.10881e16	−1.11349	−0.556745	−	0.830683i	$-0.687950\pi$
$911$	−2.10881e16	−1.11349	−0.556745	+	0.830683i	$0.687950\pi$
$912$	0	0
$913$	−4.60718e15	−0.240351
$914$	0	0
$915$	2.15076e15	0.110860
$916$	0	0
$917$	5.43065e16	2.76580
$918$	0	0
$919$	−1.64106e16	−0.825829	−0.412914	−	0.910770i	$-0.635489\pi$
$919$	−1.64106e16	−0.825829	−0.412914	+	0.910770i	$0.635489\pi$
$920$	0	0
$921$	−1.40156e16	−0.696922
$922$	0	0
$923$	−1.21294e16	−0.595976
$924$	0	0
$925$	−2.10772e16	−1.02337
$926$	0	0
$927$	−4.00344e15	−0.192086
$928$	0	0
$929$	−6.18346e15	−0.293188	−0.146594	−	0.989197i	$-0.546831\pi$
$929$	−6.18346e15	−0.293188	−0.146594	+	0.989197i	$0.546831\pi$
$930$	0	0
$931$	2.18336e15	0.102306
$932$	0	0
$933$	−1.37266e16	−0.635642
$934$	0	0
$935$	3.46789e16	1.58709
$936$	0	0
$937$	−2.53085e16	−1.14472	−0.572358	−	0.820004i	$-0.693971\pi$
$937$	−2.53085e16	−1.14472	−0.572358	+	0.820004i	$0.693971\pi$
$938$	0	0
$939$	−6.97262e15	−0.311699
$940$	0	0
$941$	2.95742e16	1.30668	0.653341	−	0.757064i	$-0.273367\pi$
$941$	2.95742e16	1.30668	0.653341	+	0.757064i	$0.273367\pi$
$942$	0	0
$943$	2.66441e15	0.116356
$944$	0	0
$945$	1.23461e16	0.532910
$946$	0	0
$947$	1.62003e16	0.691192	0.345596	−	0.938383i	$-0.387677\pi$
$947$	1.62003e16	0.691192	0.345596	+	0.938383i	$0.387677\pi$
$948$	0	0
$949$	−1.36676e15	−0.0576404
$950$	0	0
$951$	−3.02126e15	−0.125949
$952$	0	0
$953$	−2.76004e16	−1.13738	−0.568688	−	0.822554i	$-0.692549\pi$
$953$	−2.76004e16	−1.13738	−0.568688	+	0.822554i	$0.692549\pi$
$954$	0	0
$955$	−4.25334e16	−1.73265
$956$	0	0
$957$	−1.65902e16	−0.668093
$958$	0	0
$959$	−3.43462e16	−1.36734
$960$	0	0
$961$	−5.54707e15	−0.218316
$962$	0	0
$963$	5.05440e15	0.196664
$964$	0	0
$965$	−9.31508e14	−0.0358333
$966$	0	0
$967$	−4.38626e16	−1.66820	−0.834102	−	0.551610i	$-0.814014\pi$
$967$	−4.38626e16	−1.66820	−0.834102	+	0.551610i	$0.814014\pi$
$968$	0	0
$969$	4.19989e14	0.0157927
$970$	0	0
$971$	−3.19932e16	−1.18946	−0.594732	−	0.803924i	$-0.702742\pi$
$971$	−3.19932e16	−1.18946	−0.594732	+	0.803924i	$0.702742\pi$
$972$	0	0
$973$	3.62186e16	1.33141
$974$	0	0
$975$	1.18938e16	0.432311
$976$	0	0
$977$	3.96188e16	1.42391	0.711954	−	0.702226i	$-0.247811\pi$
$977$	3.96188e16	1.42391	0.711954	+	0.702226i	$0.247811\pi$
$978$	0	0
$979$	6.91058e16	2.45589
$980$	0	0
$981$	1.07778e16	0.378748
$982$	0	0
$983$	4.01905e15	0.139662	0.0698312	−	0.997559i	$-0.477754\pi$
$983$	4.01905e15	0.139662	0.0698312	+	0.997559i	$0.477754\pi$
$984$	0	0
$985$	−2.24437e16	−0.771249
$986$	0	0
$987$	−5.18147e16	−1.76079
$988$	0	0
$989$	1.25450e16	0.421592
$990$	0	0
$991$	2.49658e16	0.829737	0.414869	−	0.909881i	$-0.363828\pi$
$991$	2.49658e16	0.829737	0.414869	+	0.909881i	$0.363828\pi$
$992$	0	0
$993$	−2.28133e16	−0.749837
$994$	0	0
$995$	6.18167e16	2.00946
$996$	0	0
$997$	−1.79381e16	−0.576704	−0.288352	−	0.957525i	$-0.593107\pi$
$997$	−1.79381e16	−0.576704	−0.288352	+	0.957525i	$0.593107\pi$
$998$	0	0
$999$	5.93337e15	0.188665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.12.a.a.1.1	✓	1
3.2	odd	2		36.12.a.a.1.1		1
4.3	odd	2		48.12.a.i.1.1		1
8.3	odd	2		192.12.a.a.1.1		1
8.5	even	2		192.12.a.k.1.1		1
12.11	even	2		144.12.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.12.a.a.1.1	✓	1	1.1	even	1	trivial
36.12.a.a.1.1		1	3.2	odd	2
48.12.a.i.1.1		1	4.3	odd	2
144.12.a.a.1.1		1	12.11	even	2
192.12.a.a.1.1		1	8.3	odd	2
192.12.a.k.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 $\F_{q}$
Belyi maps

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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