LMFDB - Embedded newform 112.10.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(1,112)] 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("112.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,6,0,560] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.6840136504$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+6.00000 q^{3} +560.000 q^{5} +2401.00 q^{7} -19647.0 q^{9} +54152.0 q^{11} -113172. q^{13} +3360.00 q^{15} +6262.00 q^{17} -257078. q^{19} +14406.0 q^{21} +266000. q^{23} -1.63952e6 q^{25} -235980. q^{27} +1.57471e6 q^{29} +4.63748e6 q^{31} +324912. q^{33} +1.34456e6 q^{35} -1.19462e7 q^{37} -679032. q^{39} +2.19091e7 q^{41} -2.75206e7 q^{43} -1.10023e7 q^{45} -5.29278e7 q^{47} +5.76480e6 q^{49} +37572.0 q^{51} +1.62212e7 q^{53} +3.03251e7 q^{55} -1.54247e6 q^{57} +1.40510e8 q^{59} -2.02964e8 q^{61} -4.71724e7 q^{63} -6.33763e7 q^{65} -1.53735e8 q^{67} +1.59600e6 q^{69} -2.79656e8 q^{71} -4.04023e8 q^{73} -9.83715e6 q^{75} +1.30019e8 q^{77} +1.30690e8 q^{79} +3.85296e8 q^{81} -4.20134e8 q^{83} +3.50672e6 q^{85} +9.44828e6 q^{87} -4.69542e8 q^{89} -2.71726e8 q^{91} +2.78249e7 q^{93} -1.43964e8 q^{95} -8.72502e8 q^{97} -1.06392e9 q^{99} +O(q^{100})$

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
$3$	6.00000	0.0427667	0.0213833	−	0.999771i	$-0.493193\pi$
$3$	6.00000	0.0427667	0.0213833	+	0.999771i	$0.493193\pi$
$4$	0	0
$5$	560.000	0.400703	0.200352	−	0.979724i	$-0.435792\pi$
$5$	560.000	0.400703	0.200352	+	0.979724i	$0.435792\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	0	0
$9$	−19647.0	−0.998171
$10$	0	0
$11$	54152.0	1.11519	0.557593	−	0.830114i	$-0.311725\pi$
$11$	54152.0	1.11519	0.557593	+	0.830114i	$0.311725\pi$
$12$	0	0
$13$	−113172.	−1.09899	−0.549495	−	0.835497i	$-0.685180\pi$
$13$	−113172.	−1.09899	−0.549495	+	0.835497i	$0.685180\pi$
$14$	0	0
$15$	3360.00	0.0171368
$16$	0	0
$17$	6262.00	0.0181841	0.00909207	−	0.999959i	$-0.497106\pi$
$17$	6262.00	0.0181841	0.00909207	+	0.999959i	$0.497106\pi$
$18$	0	0
$19$	−257078.	−0.452557	−0.226279	−	0.974063i	$-0.572656\pi$
$19$	−257078.	−0.452557	−0.226279	+	0.974063i	$0.572656\pi$
$20$	0	0
$21$	14406.0	0.0161643
$22$	0	0
$23$	266000.	0.198201	0.0991006	−	0.995077i	$-0.468403\pi$
$23$	266000.	0.198201	0.0991006	+	0.995077i	$0.468403\pi$
$24$	0	0
$25$	−1.63952e6	−0.839437
$26$	0	0
$27$	−235980.	−0.0854552
$28$	0	0
$29$	1.57471e6	0.413438	0.206719	−	0.978400i	$-0.433721\pi$
$29$	1.57471e6	0.413438	0.206719	+	0.978400i	$0.433721\pi$
$30$	0	0
$31$	4.63748e6	0.901893	0.450946	−	0.892551i	$-0.351087\pi$
$31$	4.63748e6	0.901893	0.450946	+	0.892551i	$0.351087\pi$
$32$	0	0
$33$	324912.	0.0476928
$34$	0	0
$35$	1.34456e6	0.151452
$36$	0	0
$37$	−1.19462e7	−1.04791	−0.523954	−	0.851746i	$-0.675544\pi$
$37$	−1.19462e7	−1.04791	−0.523954	+	0.851746i	$0.675544\pi$
$38$	0	0
$39$	−679032.	−0.0470002
$40$	0	0
$41$	2.19091e7	1.21087	0.605435	−	0.795895i	$-0.292999\pi$
$41$	2.19091e7	1.21087	0.605435	+	0.795895i	$0.292999\pi$
$42$	0	0
$43$	−2.75206e7	−1.22758	−0.613790	−	0.789469i	$-0.710356\pi$
$43$	−2.75206e7	−1.22758	−0.613790	+	0.789469i	$0.710356\pi$
$44$	0	0
$45$	−1.10023e7	−0.399970
$46$	0	0
$47$	−5.29278e7	−1.58214	−0.791068	−	0.611728i	$-0.790475\pi$
$47$	−5.29278e7	−1.58214	−0.791068	+	0.611728i	$0.790475\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	37572.0	0.000777676 0
$52$	0	0
$53$	1.62212e7	0.282385	0.141193	−	0.989982i	$-0.454906\pi$
$53$	1.62212e7	0.282385	0.141193	+	0.989982i	$0.454906\pi$
$54$	0	0
$55$	3.03251e7	0.446859
$56$	0	0
$57$	−1.54247e6	−0.0193544
$58$	0	0
$59$	1.40510e8	1.50964	0.754818	−	0.655935i	$-0.227725\pi$
$59$	1.40510e8	1.50964	0.754818	+	0.655935i	$0.227725\pi$
$60$	0	0
$61$	−2.02964e8	−1.87687	−0.938434	−	0.345458i	$-0.887724\pi$
$61$	−2.02964e8	−1.87687	−0.938434	+	0.345458i	$0.887724\pi$
$62$	0	0
$63$	−4.71724e7	−0.377273
$64$	0	0
$65$	−6.33763e7	−0.440369
$66$	0	0
$67$	−1.53735e8	−0.932041	−0.466020	−	0.884774i	$-0.654313\pi$
$67$	−1.53735e8	−0.932041	−0.466020	+	0.884774i	$0.654313\pi$
$68$	0	0
$69$	1.59600e6	0.00847641
$70$	0	0
$71$	−2.79656e8	−1.30606	−0.653028	−	0.757334i	$-0.726501\pi$
$71$	−2.79656e8	−1.30606	−0.653028	+	0.757334i	$0.726501\pi$
$72$	0	0
$73$	−4.04023e8	−1.66515	−0.832574	−	0.553913i	$-0.813134\pi$
$73$	−4.04023e8	−1.66515	−0.832574	+	0.553913i	$0.813134\pi$
$74$	0	0
$75$	−9.83715e6	−0.0358999
$76$	0	0
$77$	1.30019e8	0.421501
$78$	0	0
$79$	1.30690e8	0.377503	0.188751	−	0.982025i	$-0.439556\pi$
$79$	1.30690e8	0.377503	0.188751	+	0.982025i	$0.439556\pi$
$80$	0	0
$81$	3.85296e8	0.994516
$82$	0	0
$83$	−4.20134e8	−0.971709	−0.485855	−	0.874040i	$-0.661492\pi$
$83$	−4.20134e8	−0.971709	−0.485855	+	0.874040i	$0.661492\pi$
$84$	0	0
$85$	3.50672e6	0.00728645
$86$	0	0
$87$	9.44828e6	0.0176814
$88$	0	0
$89$	−4.69542e8	−0.793268	−0.396634	−	0.917977i	$-0.629822\pi$
$89$	−4.69542e8	−0.793268	−0.396634	+	0.917977i	$0.629822\pi$
$90$	0	0
$91$	−2.71726e8	−0.415379
$92$	0	0
$93$	2.78249e7	0.0385710
$94$	0	0
$95$	−1.43964e8	−0.181341
$96$	0	0
$97$	−8.72502e8	−1.00068	−0.500338	−	0.865830i	$-0.666791\pi$
$97$	−8.72502e8	−1.00068	−0.500338	+	0.865830i	$0.666791\pi$
$98$	0	0
$99$	−1.06392e9	−1.11315
$100$	0	0
$101$	−1.20901e9	−1.15607	−0.578037	−	0.816011i	$-0.696181\pi$
$101$	−1.20901e9	−1.15607	−0.578037	+	0.816011i	$0.696181\pi$
$102$	0	0
$103$	−6.90563e8	−0.604555	−0.302277	−	0.953220i	$-0.597747\pi$
$103$	−6.90563e8	−0.604555	−0.302277	+	0.953220i	$0.597747\pi$
$104$	0	0
$105$	8.06736e6	0.00647708
$106$	0	0
$107$	−1.79499e8	−0.132384	−0.0661921	−	0.997807i	$-0.521085\pi$
$107$	−1.79499e8	−0.132384	−0.0661921	+	0.997807i	$0.521085\pi$
$108$	0	0
$109$	−1.60361e9	−1.08813	−0.544063	−	0.839044i	$-0.683115\pi$
$109$	−1.60361e9	−1.08813	−0.544063	+	0.839044i	$0.683115\pi$
$110$	0	0
$111$	−7.16774e7	−0.0448156
$112$	0	0
$113$	1.42785e9	0.823815	0.411908	−	0.911226i	$-0.364863\pi$
$113$	1.42785e9	0.823815	0.411908	+	0.911226i	$0.364863\pi$
$114$	0	0
$115$	1.48960e8	0.0794199
$116$	0	0
$117$	2.22349e9	1.09698
$118$	0	0
$119$	1.50351e7	0.00687296
$120$	0	0
$121$	5.74491e8	0.243640
$122$	0	0
$123$	1.31455e8	0.0517849
$124$	0	0
$125$	−2.01188e9	−0.737069
$126$	0	0
$127$	2.35873e9	0.804565	0.402282	−	0.915516i	$-0.368217\pi$
$127$	2.35873e9	0.804565	0.402282	+	0.915516i	$0.368217\pi$
$128$	0	0
$129$	−1.65124e8	−0.0524995
$130$	0	0
$131$	−6.01665e8	−0.178498	−0.0892492	−	0.996009i	$-0.528447\pi$
$131$	−6.01665e8	−0.178498	−0.0892492	+	0.996009i	$0.528447\pi$
$132$	0	0
$133$	−6.17244e8	−0.171051
$134$	0	0
$135$	−1.32149e8	−0.0342422
$136$	0	0
$137$	−5.16009e9	−1.25145	−0.625726	−	0.780043i	$-0.715197\pi$
$137$	−5.16009e9	−1.25145	−0.625726	+	0.780043i	$0.715197\pi$
$138$	0	0
$139$	7.14356e9	1.62311	0.811556	−	0.584275i	$-0.198621\pi$
$139$	7.14356e9	1.62311	0.811556	+	0.584275i	$0.198621\pi$
$140$	0	0
$141$	−3.17567e8	−0.0676627
$142$	0	0
$143$	−6.12849e9	−1.22558
$144$	0	0
$145$	8.81840e8	0.165666
$146$	0	0
$147$	3.45888e7	0.00610953
$148$	0	0
$149$	9.10424e9	1.51323	0.756616	−	0.653859i	$-0.226851\pi$
$149$	9.10424e9	1.51323	0.756616	+	0.653859i	$0.226851\pi$
$150$	0	0
$151$	2.89432e8	0.0453054	0.0226527	−	0.999743i	$-0.492789\pi$
$151$	2.89432e8	0.0453054	0.0226527	+	0.999743i	$0.492789\pi$
$152$	0	0
$153$	−1.23030e8	−0.0181509
$154$	0	0
$155$	2.59699e9	0.361391
$156$	0	0
$157$	1.39068e10	1.82675	0.913373	−	0.407124i	$-0.133468\pi$
$157$	1.39068e10	1.82675	0.913373	+	0.407124i	$0.133468\pi$
$158$	0	0
$159$	9.73273e7	0.0120767
$160$	0	0
$161$	6.38666e8	0.0749130
$162$	0	0
$163$	−1.66232e10	−1.84447	−0.922235	−	0.386629i	$-0.873639\pi$
$163$	−1.66232e10	−1.84447	−0.922235	+	0.386629i	$0.873639\pi$
$164$	0	0
$165$	1.81951e8	0.0191107
$166$	0	0
$167$	1.58019e10	1.57212	0.786061	−	0.618149i	$-0.212117\pi$
$167$	1.58019e10	1.57212	0.786061	+	0.618149i	$0.212117\pi$
$168$	0	0
$169$	2.20340e9	0.207780
$170$	0	0
$171$	5.05081e9	0.451730
$172$	0	0
$173$	3.23125e9	0.274260	0.137130	−	0.990553i	$-0.456212\pi$
$173$	3.23125e9	0.274260	0.137130	+	0.990553i	$0.456212\pi$
$174$	0	0
$175$	−3.93650e9	−0.317277
$176$	0	0
$177$	8.43058e8	0.0645621
$178$	0	0
$179$	2.41408e10	1.75757	0.878785	−	0.477218i	$-0.158355\pi$
$179$	2.41408e10	1.75757	0.878785	+	0.477218i	$0.158355\pi$
$180$	0	0
$181$	−3.89332e9	−0.269629	−0.134814	−	0.990871i	$-0.543044\pi$
$181$	−3.89332e9	−0.269629	−0.134814	+	0.990871i	$0.543044\pi$
$182$	0	0
$183$	−1.21778e9	−0.0802674
$184$	0	0
$185$	−6.68989e9	−0.419901
$186$	0	0
$187$	3.39100e8	0.0202787
$188$	0	0
$189$	−5.66588e8	−0.0322990
$190$	0	0
$191$	2.58988e10	1.40809	0.704043	−	0.710157i	$-0.251376\pi$
$191$	2.58988e10	1.40809	0.704043	+	0.710157i	$0.251376\pi$
$192$	0	0
$193$	1.59367e10	0.826783	0.413391	−	0.910553i	$-0.364344\pi$
$193$	1.59367e10	0.826783	0.413391	+	0.910553i	$0.364344\pi$
$194$	0	0
$195$	−3.80258e8	−0.0188331
$196$	0	0
$197$	−3.34685e9	−0.158321	−0.0791604	−	0.996862i	$-0.525224\pi$
$197$	−3.34685e9	−0.158321	−0.0791604	+	0.996862i	$0.525224\pi$
$198$	0	0
$199$	1.34261e10	0.606891	0.303445	−	0.952849i	$-0.401863\pi$
$199$	1.34261e10	0.606891	0.303445	+	0.952849i	$0.401863\pi$
$200$	0	0
$201$	−9.22407e8	−0.0398603
$202$	0	0
$203$	3.78089e9	0.156265
$204$	0	0
$205$	1.22691e10	0.485200
$206$	0	0
$207$	−5.22610e9	−0.197839
$208$	0	0
$209$	−1.39213e10	−0.504686
$210$	0	0
$211$	−3.01702e10	−1.04787	−0.523935	−	0.851759i	$-0.675536\pi$
$211$	−3.01702e10	−1.04787	−0.523935	+	0.851759i	$0.675536\pi$
$212$	0	0
$213$	−1.67794e9	−0.0558556
$214$	0	0
$215$	−1.54115e10	−0.491895
$216$	0	0
$217$	1.11346e10	0.340883
$218$	0	0
$219$	−2.42414e9	−0.0712129
$220$	0	0
$221$	−7.08683e8	−0.0199842
$222$	0	0
$223$	−5.35030e10	−1.44879	−0.724396	−	0.689384i	$-0.757881\pi$
$223$	−5.35030e10	−1.44879	−0.724396	+	0.689384i	$0.757881\pi$
$224$	0	0
$225$	3.22117e10	0.837901
$226$	0	0
$227$	4.02704e10	1.00663	0.503315	−	0.864103i	$-0.332114\pi$
$227$	4.02704e10	1.00663	0.503315	+	0.864103i	$0.332114\pi$
$228$	0	0
$229$	1.90247e10	0.457150	0.228575	−	0.973526i	$-0.426593\pi$
$229$	1.90247e10	0.457150	0.228575	+	0.973526i	$0.426593\pi$
$230$	0	0
$231$	7.80114e8	0.0180262
$232$	0	0
$233$	−3.67748e10	−0.817426	−0.408713	−	0.912663i	$-0.634022\pi$
$233$	−3.67748e10	−0.817426	−0.408713	+	0.912663i	$0.634022\pi$
$234$	0	0
$235$	−2.96396e10	−0.633967
$236$	0	0
$237$	7.84139e8	0.0161445
$238$	0	0
$239$	−6.56110e9	−0.130073	−0.0650363	−	0.997883i	$-0.520716\pi$
$239$	−6.56110e9	−0.130073	−0.0650363	+	0.997883i	$0.520716\pi$
$240$	0	0
$241$	−8.96818e10	−1.71249	−0.856244	−	0.516572i	$-0.827208\pi$
$241$	−8.96818e10	−1.71249	−0.856244	+	0.516572i	$0.827208\pi$
$242$	0	0
$243$	6.95657e9	0.127987
$244$	0	0
$245$	3.22829e9	0.0572433
$246$	0	0
$247$	2.90940e10	0.497356
$248$	0	0
$249$	−2.52080e9	−0.0415568
$250$	0	0
$251$	−5.33703e9	−0.0848727	−0.0424363	−	0.999099i	$-0.513512\pi$
$251$	−5.33703e9	−0.0848727	−0.0424363	+	0.999099i	$0.513512\pi$
$252$	0	0
$253$	1.44044e10	0.221031
$254$	0	0
$255$	2.10403e7	0.000311617 0
$256$	0	0
$257$	−8.35575e10	−1.19478	−0.597388	−	0.801952i	$-0.703795\pi$
$257$	−8.35575e10	−1.19478	−0.597388	+	0.801952i	$0.703795\pi$
$258$	0	0
$259$	−2.86829e10	−0.396072
$260$	0	0
$261$	−3.09384e10	−0.412682
$262$	0	0
$263$	−1.08635e11	−1.40013	−0.700065	−	0.714079i	$-0.746846\pi$
$263$	−1.08635e11	−1.40013	−0.700065	+	0.714079i	$0.746846\pi$
$264$	0	0
$265$	9.08388e9	0.113153
$266$	0	0
$267$	−2.81725e9	−0.0339254
$268$	0	0
$269$	1.41401e11	1.64652	0.823258	−	0.567668i	$-0.192154\pi$
$269$	1.41401e11	1.64652	0.823258	+	0.567668i	$0.192154\pi$
$270$	0	0
$271$	9.08353e10	1.02304	0.511520	−	0.859271i	$-0.329083\pi$
$271$	9.08353e10	1.02304	0.511520	+	0.859271i	$0.329083\pi$
$272$	0	0
$273$	−1.63036e9	−0.0177644
$274$	0	0
$275$	−8.87836e10	−0.936128
$276$	0	0
$277$	−2.65075e10	−0.270527	−0.135263	−	0.990810i	$-0.543188\pi$
$277$	−2.65075e10	−0.270527	−0.135263	+	0.990810i	$0.543188\pi$
$278$	0	0
$279$	−9.11126e10	−0.900243
$280$	0	0
$281$	−1.86968e11	−1.78891	−0.894455	−	0.447158i	$-0.852436\pi$
$281$	−1.86968e11	−1.78891	−0.894455	+	0.447158i	$0.852436\pi$
$282$	0	0
$283$	5.33413e9	0.0494338	0.0247169	−	0.999694i	$-0.492132\pi$
$283$	5.33413e9	0.0494338	0.0247169	+	0.999694i	$0.492132\pi$
$284$	0	0
$285$	−8.63782e8	−0.00775537
$286$	0	0
$287$	5.26038e10	0.457666
$288$	0	0
$289$	−1.18549e11	−0.999669
$290$	0	0
$291$	−5.23501e9	−0.0427956
$292$	0	0
$293$	7.65433e10	0.606741	0.303370	−	0.952873i	$-0.401888\pi$
$293$	7.65433e10	0.606741	0.303370	+	0.952873i	$0.401888\pi$
$294$	0	0
$295$	7.86854e10	0.604916
$296$	0	0
$297$	−1.27788e10	−0.0952984
$298$	0	0
$299$	−3.01038e10	−0.217821
$300$	0	0
$301$	−6.60769e10	−0.463982
$302$	0	0
$303$	−7.25409e9	−0.0494414
$304$	0	0
$305$	−1.13660e11	−0.752068
$306$	0	0
$307$	7.51944e10	0.483128	0.241564	−	0.970385i	$-0.422340\pi$
$307$	7.51944e10	0.483128	0.241564	+	0.970385i	$0.422340\pi$
$308$	0	0
$309$	−4.14338e9	−0.0258548
$310$	0	0
$311$	−2.15134e11	−1.30403	−0.652014	−	0.758207i	$-0.726076\pi$
$311$	−2.15134e11	−1.30403	−0.652014	+	0.758207i	$0.726076\pi$
$312$	0	0
$313$	9.59075e10	0.564811	0.282405	−	0.959295i	$-0.408868\pi$
$313$	9.59075e10	0.564811	0.282405	+	0.959295i	$0.408868\pi$
$314$	0	0
$315$	−2.64166e10	−0.151175
$316$	0	0
$317$	1.70586e11	0.948807	0.474403	−	0.880308i	$-0.342664\pi$
$317$	1.70586e11	0.948807	0.474403	+	0.880308i	$0.342664\pi$
$318$	0	0
$319$	8.52739e10	0.461061
$320$	0	0
$321$	−1.07700e9	−0.00566163
$322$	0	0
$323$	−1.60982e9	−0.00822937
$324$	0	0
$325$	1.85548e11	0.922533
$326$	0	0
$327$	−9.62165e9	−0.0465355
$328$	0	0
$329$	−1.27080e11	−0.597991
$330$	0	0
$331$	1.23992e11	0.567762	0.283881	−	0.958859i	$-0.408378\pi$
$331$	1.23992e11	0.567762	0.283881	+	0.958859i	$0.408378\pi$
$332$	0	0
$333$	2.34708e11	1.04599
$334$	0	0
$335$	−8.60914e10	−0.373472
$336$	0	0
$337$	−7.29335e10	−0.308030	−0.154015	−	0.988069i	$-0.549220\pi$
$337$	−7.29335e10	−0.308030	−0.154015	+	0.988069i	$0.549220\pi$
$338$	0	0
$339$	8.56710e9	0.0352318
$340$	0	0
$341$	2.51129e11	1.00578
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	0	0
$345$	8.93760e8	0.00339653
$346$	0	0
$347$	1.55720e11	0.576584	0.288292	−	0.957542i	$-0.406913\pi$
$347$	1.55720e11	0.576584	0.288292	+	0.957542i	$0.406913\pi$
$348$	0	0
$349$	1.08728e11	0.392310	0.196155	−	0.980573i	$-0.437154\pi$
$349$	1.08728e11	0.392310	0.196155	+	0.980573i	$0.437154\pi$
$350$	0	0
$351$	2.67063e10	0.0939144
$352$	0	0
$353$	3.25585e11	1.11604	0.558018	−	0.829829i	$-0.311562\pi$
$353$	3.25585e11	1.11604	0.558018	+	0.829829i	$0.311562\pi$
$354$	0	0
$355$	−1.56607e11	−0.523341
$356$	0	0
$357$	9.02104e7	0.000293934 0
$358$	0	0
$359$	2.27550e11	0.723022	0.361511	−	0.932368i	$-0.382261\pi$
$359$	2.27550e11	0.723022	0.361511	+	0.932368i	$0.382261\pi$
$360$	0	0
$361$	−2.56599e11	−0.795192
$362$	0	0
$363$	3.44695e9	0.0104197
$364$	0	0
$365$	−2.26253e11	−0.667231
$366$	0	0
$367$	4.21993e11	1.21425	0.607125	−	0.794607i	$-0.292323\pi$
$367$	4.21993e11	1.21425	0.607125	+	0.794607i	$0.292323\pi$
$368$	0	0
$369$	−4.30449e11	−1.20866
$370$	0	0
$371$	3.89472e10	0.106732
$372$	0	0
$373$	3.83283e11	1.02525	0.512625	−	0.858613i	$-0.328673\pi$
$373$	3.83283e11	1.02525	0.512625	+	0.858613i	$0.328673\pi$
$374$	0	0
$375$	−1.20713e10	−0.0315220
$376$	0	0
$377$	−1.78214e11	−0.454365
$378$	0	0
$379$	1.21462e11	0.302386	0.151193	−	0.988504i	$-0.451688\pi$
$379$	1.21462e11	0.302386	0.151193	+	0.988504i	$0.451688\pi$
$380$	0	0
$381$	1.41524e10	0.0344086
$382$	0	0
$383$	3.97721e11	0.944461	0.472230	−	0.881475i	$-0.343449\pi$
$383$	3.97721e11	0.944461	0.472230	+	0.881475i	$0.343449\pi$
$384$	0	0
$385$	7.28106e10	0.168897
$386$	0	0
$387$	5.40697e11	1.22533
$388$	0	0
$389$	6.75462e10	0.149564	0.0747821	−	0.997200i	$-0.476174\pi$
$389$	6.75462e10	0.149564	0.0747821	+	0.997200i	$0.476174\pi$
$390$	0	0
$391$	1.66569e9	0.00360412
$392$	0	0
$393$	−3.60999e9	−0.00763378
$394$	0	0
$395$	7.31863e10	0.151267
$396$	0	0
$397$	1.24656e11	0.251857	0.125929	−	0.992039i	$-0.459809\pi$
$397$	1.24656e11	0.251857	0.125929	+	0.992039i	$0.459809\pi$
$398$	0	0
$399$	−3.70347e9	−0.00731527
$400$	0	0
$401$	3.51196e11	0.678265	0.339133	−	0.940739i	$-0.389866\pi$
$401$	3.51196e11	0.678265	0.339133	+	0.940739i	$0.389866\pi$
$402$	0	0
$403$	−5.24833e11	−0.991171
$404$	0	0
$405$	2.15766e11	0.398506
$406$	0	0
$407$	−6.46913e11	−1.16861
$408$	0	0
$409$	−3.81956e10	−0.0674930	−0.0337465	−	0.999430i	$-0.510744\pi$
$409$	−3.81956e10	−0.0674930	−0.0337465	+	0.999430i	$0.510744\pi$
$410$	0	0
$411$	−3.09605e10	−0.0535205
$412$	0	0
$413$	3.37364e11	0.570588
$414$	0	0
$415$	−2.35275e11	−0.389367
$416$	0	0
$417$	4.28614e10	0.0694151
$418$	0	0
$419$	−2.15268e11	−0.341205	−0.170603	−	0.985340i	$-0.554571\pi$
$419$	−2.15268e11	−0.341205	−0.170603	+	0.985340i	$0.554571\pi$
$420$	0	0
$421$	1.19933e12	1.86066	0.930332	−	0.366718i	$-0.119519\pi$
$421$	1.19933e12	1.86066	0.930332	+	0.366718i	$0.119519\pi$
$422$	0	0
$423$	1.03987e12	1.57924
$424$	0	0
$425$	−1.02667e10	−0.0152644
$426$	0	0
$427$	−4.87316e11	−0.709390
$428$	0	0
$429$	−3.67709e10	−0.0524140
$430$	0	0
$431$	−7.91117e11	−1.10432	−0.552158	−	0.833740i	$-0.686195\pi$
$431$	−7.91117e11	−1.10432	−0.552158	+	0.833740i	$0.686195\pi$
$432$	0	0
$433$	−1.15451e12	−1.57834	−0.789170	−	0.614174i	$-0.789489\pi$
$433$	−1.15451e12	−1.57834	−0.789170	+	0.614174i	$0.789489\pi$
$434$	0	0
$435$	5.29104e9	0.00708499
$436$	0	0
$437$	−6.83827e10	−0.0896974
$438$	0	0
$439$	2.12728e11	0.273360	0.136680	−	0.990615i	$-0.456357\pi$
$439$	2.12728e11	0.273360	0.136680	+	0.990615i	$0.456357\pi$
$440$	0	0
$441$	−1.13261e11	−0.142596
$442$	0	0
$443$	6.48300e10	0.0799759	0.0399880	−	0.999200i	$-0.487268\pi$
$443$	6.48300e10	0.0799759	0.0399880	+	0.999200i	$0.487268\pi$
$444$	0	0
$445$	−2.62944e11	−0.317865
$446$	0	0
$447$	5.46255e10	0.0647159
$448$	0	0
$449$	−1.08031e12	−1.25441	−0.627207	−	0.778853i	$-0.715802\pi$
$449$	−1.08031e12	−1.25441	−0.627207	+	0.778853i	$0.715802\pi$
$450$	0	0
$451$	1.18642e12	1.35035
$452$	0	0
$453$	1.73659e9	0.00193756
$454$	0	0
$455$	−1.52167e11	−0.166444
$456$	0	0
$457$	−6.46725e10	−0.0693581	−0.0346790	−	0.999399i	$-0.511041\pi$
$457$	−6.46725e10	−0.0693581	−0.0346790	+	0.999399i	$0.511041\pi$
$458$	0	0
$459$	−1.47771e9	−0.00155393
$460$	0	0
$461$	4.29254e11	0.442649	0.221325	−	0.975200i	$-0.428962\pi$
$461$	4.29254e11	0.442649	0.221325	+	0.975200i	$0.428962\pi$
$462$	0	0
$463$	−1.61883e12	−1.63715	−0.818574	−	0.574401i	$-0.805235\pi$
$463$	−1.61883e12	−1.63715	−0.818574	+	0.574401i	$0.805235\pi$
$464$	0	0
$465$	1.55819e10	0.0154555
$466$	0	0
$467$	−3.27321e11	−0.318455	−0.159228	−	0.987242i	$-0.550900\pi$
$467$	−3.27321e11	−0.318455	−0.159228	+	0.987242i	$0.550900\pi$
$468$	0	0
$469$	−3.69117e11	−0.352278
$470$	0	0
$471$	8.34407e10	0.0781239
$472$	0	0
$473$	−1.49030e12	−1.36898
$474$	0	0
$475$	4.21486e11	0.379893
$476$	0	0
$477$	−3.18698e11	−0.281869
$478$	0	0
$479$	−2.84811e11	−0.247199	−0.123600	−	0.992332i	$-0.539444\pi$
$479$	−2.84811e11	−0.247199	−0.123600	+	0.992332i	$0.539444\pi$
$480$	0	0
$481$	1.35198e12	1.15164
$482$	0	0
$483$	3.83200e9	0.00320378
$484$	0	0
$485$	−4.88601e11	−0.400974
$486$	0	0
$487$	7.14776e11	0.575824	0.287912	−	0.957657i	$-0.407039\pi$
$487$	7.14776e11	0.575824	0.287912	+	0.957657i	$0.407039\pi$
$488$	0	0
$489$	−9.97395e10	−0.0788819
$490$	0	0
$491$	−1.01506e12	−0.788176	−0.394088	−	0.919073i	$-0.628939\pi$
$491$	−1.01506e12	−0.788176	−0.394088	+	0.919073i	$0.628939\pi$
$492$	0	0
$493$	9.86086e9	0.00751802
$494$	0	0
$495$	−5.95798e11	−0.446042
$496$	0	0
$497$	−6.71454e11	−0.493642
$498$	0	0
$499$	−1.33412e12	−0.963260	−0.481630	−	0.876375i	$-0.659955\pi$
$499$	−1.33412e12	−0.963260	−0.481630	+	0.876375i	$0.659955\pi$
$500$	0	0
$501$	9.48116e10	0.0672345
$502$	0	0
$503$	5.68445e11	0.395943	0.197971	−	0.980208i	$-0.436565\pi$
$503$	5.68445e11	0.395943	0.197971	+	0.980208i	$0.436565\pi$
$504$	0	0
$505$	−6.77048e11	−0.463243
$506$	0	0
$507$	1.32204e10	0.00888606
$508$	0	0
$509$	3.57173e11	0.235857	0.117928	−	0.993022i	$-0.462375\pi$
$509$	3.57173e11	0.235857	0.117928	+	0.993022i	$0.462375\pi$
$510$	0	0
$511$	−9.70059e11	−0.629367
$512$	0	0
$513$	6.06653e10	0.0386734
$514$	0	0
$515$	−3.86715e11	−0.242247
$516$	0	0
$517$	−2.86615e12	−1.76438
$518$	0	0
$519$	1.93875e10	0.0117292
$520$	0	0
$521$	2.17972e12	1.29608	0.648040	−	0.761606i	$-0.275589\pi$
$521$	2.17972e12	1.29608	0.648040	+	0.761606i	$0.275589\pi$
$522$	0	0
$523$	1.57081e12	0.918048	0.459024	−	0.888424i	$-0.348199\pi$
$523$	1.57081e12	0.918048	0.459024	+	0.888424i	$0.348199\pi$
$524$	0	0
$525$	−2.36190e10	−0.0135689
$526$	0	0
$527$	2.90399e10	0.0164001
$528$	0	0
$529$	−1.73040e12	−0.960716
$530$	0	0
$531$	−2.76059e12	−1.50687
$532$	0	0
$533$	−2.47950e12	−1.33074
$534$	0	0
$535$	−1.00520e11	−0.0530468
$536$	0	0
$537$	1.44845e11	0.0751655
$538$	0	0
$539$	3.12176e11	0.159312
$540$	0	0
$541$	2.24544e12	1.12697	0.563486	−	0.826126i	$-0.309460\pi$
$541$	2.24544e12	1.12697	0.563486	+	0.826126i	$0.309460\pi$
$542$	0	0
$543$	−2.33599e10	−0.0115311
$544$	0	0
$545$	−8.98021e11	−0.436016
$546$	0	0
$547$	3.86062e11	0.184380	0.0921899	−	0.995741i	$-0.470613\pi$
$547$	3.86062e11	0.184380	0.0921899	+	0.995741i	$0.470613\pi$
$548$	0	0
$549$	3.98763e12	1.87344
$550$	0	0
$551$	−4.04824e11	−0.187105
$552$	0	0
$553$	3.13786e11	0.142683
$554$	0	0
$555$	−4.01394e10	−0.0179578
$556$	0	0
$557$	−7.95102e11	−0.350005	−0.175003	−	0.984568i	$-0.555993\pi$
$557$	−7.95102e11	−0.350005	−0.175003	+	0.984568i	$0.555993\pi$
$558$	0	0
$559$	3.11456e12	1.34910
$560$	0	0
$561$	2.03460e9	0.000867253 0
$562$	0	0
$563$	2.13667e12	0.896292	0.448146	−	0.893960i	$-0.352084\pi$
$563$	2.13667e12	0.896292	0.448146	+	0.893960i	$0.352084\pi$
$564$	0	0
$565$	7.99596e11	0.330105
$566$	0	0
$567$	9.25096e11	0.375892
$568$	0	0
$569$	2.17461e12	0.869714	0.434857	−	0.900500i	$-0.356799\pi$
$569$	2.17461e12	0.869714	0.434857	+	0.900500i	$0.356799\pi$
$570$	0	0
$571$	−9.95075e11	−0.391736	−0.195868	−	0.980630i	$-0.562752\pi$
$571$	−9.95075e11	−0.391736	−0.195868	+	0.980630i	$0.562752\pi$
$572$	0	0
$573$	1.55393e11	0.0602192
$574$	0	0
$575$	−4.36114e11	−0.166377
$576$	0	0
$577$	4.30588e12	1.61723	0.808614	−	0.588340i	$-0.200218\pi$
$577$	4.30588e12	1.61723	0.808614	+	0.588340i	$0.200218\pi$
$578$	0	0
$579$	9.56204e10	0.0353588
$580$	0	0
$581$	−1.00874e12	−0.367272
$582$	0	0
$583$	8.78412e11	0.314912
$584$	0	0
$585$	1.24515e12	0.439564
$586$	0	0
$587$	−5.05762e12	−1.75822	−0.879112	−	0.476615i	$-0.841864\pi$
$587$	−5.05762e12	−1.75822	−0.879112	+	0.476615i	$0.841864\pi$
$588$	0	0
$589$	−1.19220e12	−0.408158
$590$	0	0
$591$	−2.00811e10	−0.00677085
$592$	0	0
$593$	−2.80300e12	−0.930844	−0.465422	−	0.885089i	$-0.654097\pi$
$593$	−2.80300e12	−0.930844	−0.465422	+	0.885089i	$0.654097\pi$
$594$	0	0
$595$	8.41963e9	0.00275402
$596$	0	0
$597$	8.05565e10	0.0259547
$598$	0	0
$599$	−3.20907e12	−1.01849	−0.509247	−	0.860620i	$-0.670076\pi$
$599$	−3.20907e12	−1.01849	−0.509247	+	0.860620i	$0.670076\pi$
$600$	0	0
$601$	7.49502e11	0.234335	0.117168	−	0.993112i	$-0.462619\pi$
$601$	7.49502e11	0.234335	0.117168	+	0.993112i	$0.462619\pi$
$602$	0	0
$603$	3.02042e12	0.930336
$604$	0	0
$605$	3.21715e11	0.0976275
$606$	0	0
$607$	1.74097e12	0.520526	0.260263	−	0.965538i	$-0.416191\pi$
$607$	1.74097e12	0.520526	0.260263	+	0.965538i	$0.416191\pi$
$608$	0	0
$609$	2.26853e10	0.00668294
$610$	0	0
$611$	5.98995e12	1.73875
$612$	0	0
$613$	−4.03977e12	−1.15554	−0.577770	−	0.816200i	$-0.696077\pi$
$613$	−4.03977e12	−1.15554	−0.577770	+	0.816200i	$0.696077\pi$
$614$	0	0
$615$	7.36147e10	0.0207504
$616$	0	0
$617$	−2.93367e12	−0.814945	−0.407472	−	0.913218i	$-0.633590\pi$
$617$	−2.93367e12	−0.814945	−0.407472	+	0.913218i	$0.633590\pi$
$618$	0	0
$619$	5.77691e12	1.58157	0.790784	−	0.612095i	$-0.209673\pi$
$619$	5.77691e12	1.58157	0.790784	+	0.612095i	$0.209673\pi$
$620$	0	0
$621$	−6.27707e10	−0.0169373
$622$	0	0
$623$	−1.12737e12	−0.299827
$624$	0	0
$625$	2.07554e12	0.544091
$626$	0	0
$627$	−8.35277e10	−0.0215837
$628$	0	0
$629$	−7.48073e10	−0.0190553
$630$	0	0
$631$	−3.99985e12	−1.00441	−0.502206	−	0.864748i	$-0.667478\pi$
$631$	−3.99985e12	−1.00441	−0.502206	+	0.864748i	$0.667478\pi$
$632$	0	0
$633$	−1.81021e11	−0.0448139
$634$	0	0
$635$	1.32089e12	0.322392
$636$	0	0
$637$	−6.52414e11	−0.156999
$638$	0	0
$639$	5.49440e12	1.30367
$640$	0	0
$641$	4.68328e12	1.09569	0.547846	−	0.836579i	$-0.315448\pi$
$641$	4.68328e12	1.09569	0.547846	+	0.836579i	$0.315448\pi$
$642$	0	0
$643$	1.54877e12	0.357304	0.178652	−	0.983912i	$-0.442826\pi$
$643$	1.54877e12	0.357304	0.178652	+	0.983912i	$0.442826\pi$
$644$	0	0
$645$	−9.24692e10	−0.0210367
$646$	0	0
$647$	8.14493e12	1.82733	0.913667	−	0.406463i	$-0.133238\pi$
$647$	8.14493e12	1.82733	0.913667	+	0.406463i	$0.133238\pi$
$648$	0	0
$649$	7.60888e12	1.68352
$650$	0	0
$651$	6.68076e10	0.0145785
$652$	0	0
$653$	−2.88925e12	−0.621836	−0.310918	−	0.950437i	$-0.600636\pi$
$653$	−2.88925e12	−0.621836	−0.310918	+	0.950437i	$0.600636\pi$
$654$	0	0
$655$	−3.36933e11	−0.0715249
$656$	0	0
$657$	7.93784e12	1.66210
$658$	0	0
$659$	−5.20255e12	−1.07456	−0.537281	−	0.843403i	$-0.680549\pi$
$659$	−5.20255e12	−1.07456	−0.537281	+	0.843403i	$0.680549\pi$
$660$	0	0
$661$	2.88973e12	0.588777	0.294388	−	0.955686i	$-0.404884\pi$
$661$	2.88973e12	0.588777	0.294388	+	0.955686i	$0.404884\pi$
$662$	0	0
$663$	−4.25210e9	−0.000854658 0
$664$	0	0
$665$	−3.45657e11	−0.0685406
$666$	0	0
$667$	4.18874e11	0.0819440
$668$	0	0
$669$	−3.21018e11	−0.0619600
$670$	0	0
$671$	−1.09909e13	−2.09306
$672$	0	0
$673$	9.46362e12	1.77824	0.889119	−	0.457677i	$-0.151318\pi$
$673$	9.46362e12	1.77824	0.889119	+	0.457677i	$0.151318\pi$
$674$	0	0
$675$	3.86895e11	0.0717342
$676$	0	0
$677$	−6.52268e12	−1.19338	−0.596688	−	0.802474i	$-0.703517\pi$
$677$	−6.52268e12	−1.19338	−0.596688	+	0.802474i	$0.703517\pi$
$678$	0	0
$679$	−2.09488e12	−0.378220
$680$	0	0
$681$	2.41622e11	0.0430502
$682$	0	0
$683$	−5.37240e12	−0.944660	−0.472330	−	0.881422i	$-0.656587\pi$
$683$	−5.37240e12	−0.944660	−0.472330	+	0.881422i	$0.656587\pi$
$684$	0	0
$685$	−2.88965e12	−0.501461
$686$	0	0
$687$	1.14148e11	0.0195508
$688$	0	0
$689$	−1.83579e12	−0.310339
$690$	0	0
$691$	−2.01563e12	−0.336325	−0.168163	−	0.985759i	$-0.553783\pi$
$691$	−2.01563e12	−0.336325	−0.168163	+	0.985759i	$0.553783\pi$
$692$	0	0
$693$	−2.55448e12	−0.420730
$694$	0	0
$695$	4.00040e12	0.650386
$696$	0	0
$697$	1.37195e11	0.0220186
$698$	0	0
$699$	−2.20649e11	−0.0349586
$700$	0	0
$701$	−1.06523e13	−1.66614	−0.833068	−	0.553171i	$-0.813418\pi$
$701$	−1.06523e13	−1.66614	−0.833068	+	0.553171i	$0.813418\pi$
$702$	0	0
$703$	3.07111e12	0.474239
$704$	0	0
$705$	−1.77838e11	−0.0271127
$706$	0	0
$707$	−2.90284e12	−0.436955
$708$	0	0
$709$	3.46187e12	0.514520	0.257260	−	0.966342i	$-0.417180\pi$
$709$	3.46187e12	0.514520	0.257260	+	0.966342i	$0.417180\pi$
$710$	0	0
$711$	−2.56766e12	−0.376812
$712$	0	0
$713$	1.23357e12	0.178756
$714$	0	0
$715$	−3.43195e12	−0.491094
$716$	0	0
$717$	−3.93666e10	−0.00556277
$718$	0	0
$719$	9.62025e12	1.34248	0.671238	−	0.741242i	$-0.265763\pi$
$719$	9.62025e12	1.34248	0.671238	+	0.741242i	$0.265763\pi$
$720$	0	0
$721$	−1.65804e12	−0.228500
$722$	0	0
$723$	−5.38091e11	−0.0732374
$724$	0	0
$725$	−2.58178e12	−0.347055
$726$	0	0
$727$	3.13479e12	0.416202	0.208101	−	0.978107i	$-0.433272\pi$
$727$	3.13479e12	0.416202	0.208101	+	0.978107i	$0.433272\pi$
$728$	0	0
$729$	−7.54204e12	−0.989043
$730$	0	0
$731$	−1.72334e11	−0.0223225
$732$	0	0
$733$	6.47775e12	0.828812	0.414406	−	0.910092i	$-0.363989\pi$
$733$	6.47775e12	0.828812	0.414406	+	0.910092i	$0.363989\pi$
$734$	0	0
$735$	1.93697e10	0.00244811
$736$	0	0
$737$	−8.32503e12	−1.03940
$738$	0	0
$739$	1.12510e12	0.138769	0.0693843	−	0.997590i	$-0.477897\pi$
$739$	1.12510e12	0.138769	0.0693843	+	0.997590i	$0.477897\pi$
$740$	0	0
$741$	1.74564e11	0.0212703
$742$	0	0
$743$	−1.65266e12	−0.198945	−0.0994725	−	0.995040i	$-0.531716\pi$
$743$	−1.65266e12	−0.198945	−0.0994725	+	0.995040i	$0.531716\pi$
$744$	0	0
$745$	5.09838e12	0.606357
$746$	0	0
$747$	8.25437e12	0.969932
$748$	0	0
$749$	−4.30978e11	−0.0500365
$750$	0	0
$751$	−6.03299e12	−0.692074	−0.346037	−	0.938221i	$-0.612473\pi$
$751$	−6.03299e12	−0.692074	−0.346037	+	0.938221i	$0.612473\pi$
$752$	0	0
$753$	−3.20222e10	−0.00362972
$754$	0	0
$755$	1.62082e11	0.0181540
$756$	0	0
$757$	−8.02798e12	−0.888535	−0.444268	−	0.895894i	$-0.646536\pi$
$757$	−8.02798e12	−0.888535	−0.444268	+	0.895894i	$0.646536\pi$
$758$	0	0
$759$	8.64266e10	0.00945278
$760$	0	0
$761$	−6.51923e12	−0.704637	−0.352318	−	0.935880i	$-0.614607\pi$
$761$	−6.51923e12	−0.704637	−0.352318	+	0.935880i	$0.614607\pi$
$762$	0	0
$763$	−3.85026e12	−0.411273
$764$	0	0
$765$	−6.88965e10	−0.00727312
$766$	0	0
$767$	−1.59018e13	−1.65907
$768$	0	0
$769$	−1.34250e13	−1.38435	−0.692175	−	0.721730i	$-0.743347\pi$
$769$	−1.34250e13	−1.38435	−0.692175	+	0.721730i	$0.743347\pi$
$770$	0	0
$771$	−5.01345e11	−0.0510966
$772$	0	0
$773$	7.85934e12	0.791733	0.395866	−	0.918308i	$-0.370444\pi$
$773$	7.85934e12	0.791733	0.395866	+	0.918308i	$0.370444\pi$
$774$	0	0
$775$	−7.60327e12	−0.757082
$776$	0	0
$777$	−1.72098e11	−0.0169387
$778$	0	0
$779$	−5.63235e12	−0.547988
$780$	0	0
$781$	−1.51439e13	−1.45649
$782$	0	0
$783$	−3.71601e11	−0.0353304
$784$	0	0
$785$	7.78780e12	0.731983
$786$	0	0
$787$	1.47720e12	0.137263	0.0686316	−	0.997642i	$-0.478137\pi$
$787$	1.47720e12	0.137263	0.0686316	+	0.997642i	$0.478137\pi$
$788$	0	0
$789$	−6.51809e11	−0.0598789
$790$	0	0
$791$	3.42827e12	0.311373
$792$	0	0
$793$	2.29698e13	2.06266
$794$	0	0
$795$	5.45033e10	0.00483917
$796$	0	0
$797$	−6.47327e12	−0.568278	−0.284139	−	0.958783i	$-0.591708\pi$
$797$	−6.47327e12	−0.568278	−0.284139	+	0.958783i	$0.591708\pi$
$798$	0	0
$799$	−3.31434e11	−0.0287698
$800$	0	0
$801$	9.22510e12	0.791817
$802$	0	0
$803$	−2.18786e13	−1.85695
$804$	0	0
$805$	3.57653e11	0.0300179
$806$	0	0
$807$	8.48403e11	0.0704160
$808$	0	0
$809$	−1.53631e13	−1.26099	−0.630493	−	0.776195i	$-0.717147\pi$
$809$	−1.53631e13	−1.26099	−0.630493	+	0.776195i	$0.717147\pi$
$810$	0	0
$811$	1.29056e13	1.04757	0.523787	−	0.851849i	$-0.324519\pi$
$811$	1.29056e13	1.04757	0.523787	+	0.851849i	$0.324519\pi$
$812$	0	0
$813$	5.45012e11	0.0437520
$814$	0	0
$815$	−9.30902e12	−0.739085
$816$	0	0
$817$	7.07494e12	0.555550
$818$	0	0
$819$	5.33860e12	0.414620
$820$	0	0
$821$	8.93771e11	0.0686566	0.0343283	−	0.999411i	$-0.489071\pi$
$821$	8.93771e11	0.0686566	0.0343283	+	0.999411i	$0.489071\pi$
$822$	0	0
$823$	2.29844e13	1.74636	0.873182	−	0.487394i	$-0.162053\pi$
$823$	2.29844e13	1.74636	0.873182	+	0.487394i	$0.162053\pi$
$824$	0	0
$825$	−5.32701e11	−0.0400351
$826$	0	0
$827$	2.75618e12	0.204896	0.102448	−	0.994738i	$-0.467333\pi$
$827$	2.75618e12	0.204896	0.102448	+	0.994738i	$0.467333\pi$
$828$	0	0
$829$	−3.15925e12	−0.232321	−0.116160	−	0.993230i	$-0.537059\pi$
$829$	−3.15925e12	−0.232321	−0.116160	+	0.993230i	$0.537059\pi$
$830$	0	0
$831$	−1.59045e11	−0.0115695
$832$	0	0
$833$	3.60992e10	0.00259774
$834$	0	0
$835$	8.84909e12	0.629955
$836$	0	0
$837$	−1.09435e12	−0.0770714
$838$	0	0
$839$	−1.92403e13	−1.34055	−0.670277	−	0.742111i	$-0.733825\pi$
$839$	−1.92403e13	−1.34055	−0.670277	+	0.742111i	$0.733825\pi$
$840$	0	0
$841$	−1.20274e13	−0.829069
$842$	0	0
$843$	−1.12181e12	−0.0765058
$844$	0	0
$845$	1.23391e12	0.0832581
$846$	0	0
$847$	1.37935e12	0.0920874
$848$	0	0
$849$	3.20048e10	0.00211412
$850$	0	0
$851$	−3.17770e12	−0.207697
$852$	0	0
$853$	−2.60804e13	−1.68672	−0.843362	−	0.537345i	$-0.819427\pi$
$853$	−2.60804e13	−1.68672	−0.843362	+	0.537345i	$0.819427\pi$
$854$	0	0
$855$	2.82845e12	0.181010
$856$	0	0
$857$	2.19177e13	1.38797	0.693986	−	0.719988i	$-0.255853\pi$
$857$	2.19177e13	1.38797	0.693986	+	0.719988i	$0.255853\pi$
$858$	0	0
$859$	3.55588e12	0.222832	0.111416	−	0.993774i	$-0.464461\pi$
$859$	3.55588e12	0.222832	0.111416	+	0.993774i	$0.464461\pi$
$860$	0	0
$861$	3.15623e11	0.0195729
$862$	0	0
$863$	−2.22084e13	−1.36292	−0.681458	−	0.731858i	$-0.738654\pi$
$863$	−2.22084e13	−1.36292	−0.681458	+	0.731858i	$0.738654\pi$
$864$	0	0
$865$	1.80950e12	0.109897
$866$	0	0
$867$	−7.11292e11	−0.0427525
$868$	0	0
$869$	7.07711e12	0.420986
$870$	0	0
$871$	1.73984e13	1.02430
$872$	0	0
$873$	1.71420e13	0.998846
$874$	0	0
$875$	−4.83053e12	−0.278586
$876$	0	0
$877$	3.38004e12	0.192941	0.0964703	−	0.995336i	$-0.469245\pi$
$877$	3.38004e12	0.192941	0.0964703	+	0.995336i	$0.469245\pi$
$878$	0	0
$879$	4.59260e11	0.0259483
$880$	0	0
$881$	5.25103e11	0.0293665	0.0146833	−	0.999892i	$-0.495326\pi$
$881$	5.25103e11	0.0293665	0.0146833	+	0.999892i	$0.495326\pi$
$882$	0	0
$883$	−3.33972e13	−1.84879	−0.924393	−	0.381441i	$-0.875428\pi$
$883$	−3.33972e13	−1.84879	−0.924393	+	0.381441i	$0.875428\pi$
$884$	0	0
$885$	4.72112e11	0.0258702
$886$	0	0
$887$	9.61964e12	0.521798	0.260899	−	0.965366i	$-0.415981\pi$
$887$	9.61964e12	0.521798	0.260899	+	0.965366i	$0.415981\pi$
$888$	0	0
$889$	5.66331e12	0.304097
$890$	0	0
$891$	2.08646e13	1.10907
$892$	0	0
$893$	1.36066e13	0.716007
$894$	0	0
$895$	1.35188e13	0.704264
$896$	0	0
$897$	−1.80623e11	−0.00931549
$898$	0	0
$899$	7.30271e12	0.372877
$900$	0	0
$901$	1.01577e11	0.00513494
$902$	0	0
$903$	−3.96462e11	−0.0198430
$904$	0	0
$905$	−2.18026e12	−0.108041
$906$	0	0
$907$	2.08341e13	1.02221	0.511107	−	0.859517i	$-0.329236\pi$
$907$	2.08341e13	1.02221	0.511107	+	0.859517i	$0.329236\pi$
$908$	0	0
$909$	2.37535e13	1.15396
$910$	0	0
$911$	1.33789e13	0.643560	0.321780	−	0.946814i	$-0.395719\pi$
$911$	1.33789e13	0.643560	0.321780	+	0.946814i	$0.395719\pi$
$912$	0	0
$913$	−2.27511e13	−1.08364
$914$	0	0
$915$	−6.81958e11	−0.0321634
$916$	0	0
$917$	−1.44460e12	−0.0674660
$918$	0	0
$919$	1.38352e13	0.639830	0.319915	−	0.947446i	$-0.396346\pi$
$919$	1.38352e13	0.639830	0.319915	+	0.947446i	$0.396346\pi$
$920$	0	0
$921$	4.51166e11	0.0206618
$922$	0	0
$923$	3.16492e13	1.43534
$924$	0	0
$925$	1.95862e13	0.879653
$926$	0	0
$927$	1.35675e13	0.603449
$928$	0	0
$929$	−1.74073e13	−0.766761	−0.383380	−	0.923591i	$-0.625240\pi$
$929$	−1.74073e13	−0.766761	−0.383380	+	0.923591i	$0.625240\pi$
$930$	0	0
$931$	−1.48200e12	−0.0646511
$932$	0	0
$933$	−1.29080e12	−0.0557690
$934$	0	0
$935$	1.89896e11	0.00812575
$936$	0	0
$937$	1.98361e12	0.0840676	0.0420338	−	0.999116i	$-0.486616\pi$
$937$	1.98361e12	0.0840676	0.0420338	+	0.999116i	$0.486616\pi$
$938$	0	0
$939$	5.75445e11	0.0241551
$940$	0	0
$941$	−2.33533e13	−0.970946	−0.485473	−	0.874252i	$-0.661353\pi$
$941$	−2.33533e13	−0.970946	−0.485473	+	0.874252i	$0.661353\pi$
$942$	0	0
$943$	5.82783e12	0.239996
$944$	0	0
$945$	−3.17289e11	−0.0129423
$946$	0	0
$947$	−3.59882e13	−1.45407	−0.727035	−	0.686600i	$-0.759102\pi$
$947$	−3.59882e13	−1.45407	−0.727035	+	0.686600i	$0.759102\pi$
$948$	0	0
$949$	4.57241e13	1.82998
$950$	0	0
$951$	1.02352e12	0.0405773
$952$	0	0
$953$	−1.37662e13	−0.540624	−0.270312	−	0.962773i	$-0.587127\pi$
$953$	−1.37662e13	−0.540624	−0.270312	+	0.962773i	$0.587127\pi$
$954$	0	0
$955$	1.45033e13	0.564225
$956$	0	0
$957$	5.11643e11	0.0197180
$958$	0	0
$959$	−1.23894e13	−0.473005
$960$	0	0
$961$	−4.93336e12	−0.186590
$962$	0	0
$963$	3.52662e12	0.132142
$964$	0	0
$965$	8.92457e12	0.331295
$966$	0	0
$967$	3.02718e12	0.111332	0.0556659	−	0.998449i	$-0.482272\pi$
$967$	3.02718e12	0.111332	0.0556659	+	0.998449i	$0.482272\pi$
$968$	0	0
$969$	−9.65893e9	−0.000351943 0
$970$	0	0
$971$	2.53183e13	0.914003	0.457002	−	0.889466i	$-0.348923\pi$
$971$	2.53183e13	0.914003	0.457002	+	0.889466i	$0.348923\pi$
$972$	0	0
$973$	1.71517e13	0.613478
$974$	0	0
$975$	1.11329e12	0.0394537
$976$	0	0
$977$	9.90729e12	0.347880	0.173940	−	0.984756i	$-0.444350\pi$
$977$	9.90729e12	0.347880	0.173940	+	0.984756i	$0.444350\pi$
$978$	0	0
$979$	−2.54267e13	−0.884641
$980$	0	0
$981$	3.15061e13	1.08614
$982$	0	0
$983$	−2.40523e13	−0.821610	−0.410805	−	0.911723i	$-0.634752\pi$
$983$	−2.40523e13	−0.821610	−0.410805	+	0.911723i	$0.634752\pi$
$984$	0	0
$985$	−1.87423e12	−0.0634397
$986$	0	0
$987$	−7.62478e11	−0.0255741
$988$	0	0
$989$	−7.32048e12	−0.243308
$990$	0	0
$991$	−3.15782e13	−1.04005	−0.520027	−	0.854150i	$-0.674078\pi$
$991$	−3.15782e13	−1.04005	−0.520027	+	0.854150i	$0.674078\pi$
$992$	0	0
$993$	7.43950e11	0.0242813
$994$	0	0
$995$	7.51860e12	0.243183
$996$	0	0
$997$	9.32241e12	0.298813	0.149407	−	0.988776i	$-0.452264\pi$
$997$	9.32241e12	0.298813	0.149407	+	0.988776i	$0.452264\pi$
$998$	0	0
$999$	2.81907e12	0.0895492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.a.b.1.1		1
4.3	odd	2		14.10.a.a.1.1	✓	1
12.11	even	2		126.10.a.e.1.1		1
20.3	even	4		350.10.c.b.99.2		2
20.7	even	4		350.10.c.b.99.1		2
20.19	odd	2		350.10.a.c.1.1		1
28.3	even	6		98.10.c.e.79.1		2
28.11	odd	6		98.10.c.f.79.1		2
28.19	even	6		98.10.c.e.67.1		2
28.23	odd	6		98.10.c.f.67.1		2
28.27	even	2		98.10.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.10.a.a.1.1	✓	1	4.3	odd	2
98.10.a.a.1.1		1	28.27	even	2
98.10.c.e.67.1		2	28.19	even	6
98.10.c.e.79.1		2	28.3	even	6
98.10.c.f.67.1		2	28.23	odd	6
98.10.c.f.79.1		2	28.11	odd	6
112.10.a.b.1.1		1	1.1	even	1	trivial
126.10.a.e.1.1		1	12.11	even	2
350.10.a.c.1.1		1	20.19	odd	2
350.10.c.b.99.1		2	20.7	even	4
350.10.c.b.99.2		2	20.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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