LMFDB - Embedded newform 196.10.a.g.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [196,10,Mod(1,196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("196.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 196 = 2^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0] 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:	$100.947023888$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 23330x^{8} + 114080917x^{6} - 121201507892x^{4} + 31086921022884x^{2} - 2278042380749088 $ x^10 - 23330x^8 + 114080917x^6 - 121201507892x^4 + 31086921022884x^2 - 2278042380749088
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{4}\cdot 7^{12} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-130.574$ of defining polynomial
Character	$\chi$	$=$	196.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-70.3781 q^{3} +162.776 q^{5} -14729.9 q^{9} +90868.8 q^{11} -167897. q^{13} -11455.9 q^{15} +9826.94 q^{17} +475528. q^{19} -221044. q^{23} -1.92663e6 q^{25} +2.42191e6 q^{27} +820028. q^{29} -2.83174e6 q^{31} -6.39517e6 q^{33} -2.71305e6 q^{37} +1.18163e7 q^{39} -1.90289e7 q^{41} +2.58212e7 q^{43} -2.39768e6 q^{45} +3.51051e7 q^{47} -691601. q^{51} -8.20924e7 q^{53} +1.47913e7 q^{55} -3.34668e7 q^{57} -1.33986e8 q^{59} -1.01652e8 q^{61} -2.73296e7 q^{65} +1.48020e8 q^{67} +1.55567e7 q^{69} +1.94170e8 q^{71} +6.20986e7 q^{73} +1.35592e8 q^{75} +4.01444e8 q^{79} +1.19480e8 q^{81} -1.49637e8 q^{83} +1.59959e6 q^{85} -5.77120e7 q^{87} +6.54169e8 q^{89} +1.99292e8 q^{93} +7.74046e7 q^{95} +1.51683e9 q^{97} -1.33849e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 93550 q^{9} + 27740 q^{11} + 912088 q^{15} - 2921096 q^{23} + 3484998 q^{25} - 835748 q^{29} - 31928756 q^{37} - 4930944 q^{39} + 13414340 q^{43} + 206013028 q^{51} + 137301788 q^{53} + 72246332 q^{57}+ \cdots + 5316067012 q^{99}+O(q^{100}) $ 10 * q + 93550 * q^9 + 27740 * q^11 + 912088 * q^15 - 2921096 * q^23 + 3484998 * q^25 - 835748 * q^29 - 31928756 * q^37 - 4930944 * q^39 + 13414340 * q^43 + 206013028 * q^51 + 137301788 * q^53 + 72246332 * q^57 + 519063672 * q^65 - 342262536 * q^67 - 531036408 * q^71 + 462186584 * q^79 + 2495014982 * q^81 + 1236603584 * q^85 + 5733362712 * q^93 + 2984918936 * q^95 + 5316067012 * q^99

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
$3$	−70.3781	−0.501639	−0.250820	−	0.968034i	$-0.580700\pi$
$3$	−70.3781	−0.501639	−0.250820	+	0.968034i	$0.580700\pi$
$4$	0	0
$5$	162.776	0.116473	0.0582365	−	0.998303i	$-0.481452\pi$
$5$	162.776	0.116473	0.0582365	+	0.998303i	$0.481452\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−14729.9	−0.748358
$10$	0	0
$11$	90868.8	1.87132	0.935659	−	0.352905i	$-0.114806\pi$
$11$	90868.8	1.87132	0.935659	+	0.352905i	$0.114806\pi$
$12$	0	0
$13$	−167897.	−1.63041	−0.815206	−	0.579171i	$-0.803377\pi$
$13$	−167897.	−1.63041	−0.815206	+	0.579171i	$0.803377\pi$
$14$	0	0
$15$	−11455.9	−0.0584275
$16$	0	0
$17$	9826.94	0.0285363	0.0142682	−	0.999898i	$-0.495458\pi$
$17$	9826.94	0.0285363	0.0142682	+	0.999898i	$0.495458\pi$
$18$	0	0
$19$	475528.	0.837115	0.418558	−	0.908190i	$-0.362536\pi$
$19$	475528.	0.837115	0.418558	+	0.908190i	$0.362536\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−221044.	−0.164704	−0.0823519	−	0.996603i	$-0.526243\pi$
$23$	−221044.	−0.164704	−0.0823519	+	0.996603i	$0.526243\pi$
$24$	0	0
$25$	−1.92663e6	−0.986434
$26$	0	0
$27$	2.42191e6	0.877045
$28$	0	0
$29$	820028.	0.215297	0.107648	−	0.994189i	$-0.465668\pi$
$29$	820028.	0.215297	0.107648	+	0.994189i	$0.465668\pi$
$30$	0	0
$31$	−2.83174e6	−0.550714	−0.275357	−	0.961342i	$-0.588796\pi$
$31$	−2.83174e6	−0.550714	−0.275357	+	0.961342i	$0.588796\pi$
$32$	0	0
$33$	−6.39517e6	−0.938727
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.71305e6	−0.237986	−0.118993	−	0.992895i	$-0.537967\pi$
$37$	−2.71305e6	−0.237986	−0.118993	+	0.992895i	$0.537967\pi$
$38$	0	0
$39$	1.18163e7	0.817879
$40$	0	0
$41$	−1.90289e7	−1.05169	−0.525845	−	0.850581i	$-0.676251\pi$
$41$	−1.90289e7	−1.05169	−0.525845	+	0.850581i	$0.676251\pi$
$42$	0	0
$43$	2.58212e7	1.15178	0.575889	−	0.817528i	$-0.304656\pi$
$43$	2.58212e7	1.15178	0.575889	+	0.817528i	$0.304656\pi$
$44$	0	0
$45$	−2.39768e6	−0.0871635
$46$	0	0
$47$	3.51051e7	1.04937	0.524687	−	0.851295i	$-0.324182\pi$
$47$	3.51051e7	1.04937	0.524687	+	0.851295i	$0.324182\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−691601.	−0.0143150
$52$	0	0
$53$	−8.20924e7	−1.42910	−0.714549	−	0.699586i	$-0.753368\pi$
$53$	−8.20924e7	−1.42910	−0.714549	+	0.699586i	$0.753368\pi$
$54$	0	0
$55$	1.47913e7	0.217958
$56$	0	0
$57$	−3.34668e7	−0.419930
$58$	0	0
$59$	−1.33986e8	−1.43954	−0.719770	−	0.694212i	$-0.755753\pi$
$59$	−1.33986e8	−1.43954	−0.719770	+	0.694212i	$0.755753\pi$
$60$	0	0
$61$	−1.01652e8	−0.940011	−0.470006	−	0.882663i	$-0.655748\pi$
$61$	−1.01652e8	−0.940011	−0.470006	+	0.882663i	$0.655748\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.73296e7	−0.189899
$66$	0	0
$67$	1.48020e8	0.897398	0.448699	−	0.893683i	$-0.351887\pi$
$67$	1.48020e8	0.897398	0.448699	+	0.893683i	$0.351887\pi$
$68$	0	0
$69$	1.55567e7	0.0826219
$70$	0	0
$71$	1.94170e8	0.906816	0.453408	−	0.891303i	$-0.350208\pi$
$71$	1.94170e8	0.906816	0.453408	+	0.891303i	$0.350208\pi$
$72$	0	0
$73$	6.20986e7	0.255935	0.127967	−	0.991778i	$-0.459155\pi$
$73$	6.20986e7	0.255935	0.127967	+	0.991778i	$0.459155\pi$
$74$	0	0
$75$	1.35592e8	0.494834
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.01444e8	1.15959	0.579793	−	0.814764i	$-0.303134\pi$
$79$	4.01444e8	1.15959	0.579793	+	0.814764i	$0.303134\pi$
$80$	0	0
$81$	1.19480e8	0.308398
$82$	0	0
$83$	−1.49637e8	−0.346088	−0.173044	−	0.984914i	$-0.555360\pi$
$83$	−1.49637e8	−0.346088	−0.173044	+	0.984914i	$0.555360\pi$
$84$	0	0
$85$	1.59959e6	0.00332372
$86$	0	0
$87$	−5.77120e7	−0.108001
$88$	0	0
$89$	6.54169e8	1.10519	0.552593	−	0.833451i	$-0.313639\pi$
$89$	6.54169e8	1.10519	0.552593	+	0.833451i	$0.313639\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.99292e8	0.276260
$94$	0	0
$95$	7.74046e7	0.0975013
$96$	0	0
$97$	1.51683e9	1.73966	0.869830	−	0.493352i	$-0.164229\pi$
$97$	1.51683e9	1.73966	0.869830	+	0.493352i	$0.164229\pi$
$98$	0	0
$99$	−1.33849e9	−1.40042
$100$	0	0
$101$	−1.65819e9	−1.58558	−0.792792	−	0.609493i	$-0.791373\pi$
$101$	−1.65819e9	−1.58558	−0.792792	+	0.609493i	$0.791373\pi$
$102$	0	0
$103$	−5.25830e8	−0.460339	−0.230169	−	0.973151i	$-0.573928\pi$
$103$	−5.25830e8	−0.460339	−0.230169	+	0.973151i	$0.573928\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.26015e8	0.387946	0.193973	−	0.981007i	$-0.437863\pi$
$107$	5.26015e8	0.387946	0.193973	+	0.981007i	$0.437863\pi$
$108$	0	0
$109$	1.98795e9	1.34892	0.674459	−	0.738312i	$-0.264377\pi$
$109$	1.98795e9	1.34892	0.674459	+	0.738312i	$0.264377\pi$
$110$	0	0
$111$	1.90940e8	0.119383
$112$	0	0
$113$	1.09466e9	0.631577	0.315789	−	0.948830i	$-0.397731\pi$
$113$	1.09466e9	0.631577	0.315789	+	0.948830i	$0.397731\pi$
$114$	0	0
$115$	−3.59807e7	−0.0191836
$116$	0	0
$117$	2.47311e9	1.22013
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.89919e9	2.50183
$122$	0	0
$123$	1.33922e9	0.527569
$124$	0	0
$125$	−6.31531e8	−0.231366
$126$	0	0
$127$	3.18245e9	1.08554	0.542769	−	0.839882i	$-0.317376\pi$
$127$	3.18245e9	1.08554	0.542769	+	0.839882i	$0.317376\pi$
$128$	0	0
$129$	−1.81725e9	−0.577777
$130$	0	0
$131$	2.47096e9	0.733068	0.366534	−	0.930405i	$-0.380544\pi$
$131$	2.47096e9	0.733068	0.366534	+	0.930405i	$0.380544\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.94230e8	0.102152
$136$	0	0
$137$	1.02753e9	0.249202	0.124601	−	0.992207i	$-0.460235\pi$
$137$	1.02753e9	0.249202	0.124601	+	0.992207i	$0.460235\pi$
$138$	0	0
$139$	6.01240e9	1.36610	0.683048	−	0.730373i	$-0.260654\pi$
$139$	6.01240e9	1.36610	0.683048	+	0.730373i	$0.260654\pi$
$140$	0	0
$141$	−2.47063e9	−0.526407
$142$	0	0
$143$	−1.52566e10	−3.05102
$144$	0	0
$145$	1.33481e8	0.0250763
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.01062e9	0.500400	0.250200	−	0.968194i	$-0.419504\pi$
$149$	3.01062e9	0.500400	0.250200	+	0.968194i	$0.419504\pi$
$150$	0	0
$151$	4.59068e9	0.718590	0.359295	−	0.933224i	$-0.383017\pi$
$151$	4.59068e9	0.718590	0.359295	+	0.933224i	$0.383017\pi$
$152$	0	0
$153$	−1.44750e8	−0.0213554
$154$	0	0
$155$	−4.60939e8	−0.0641433
$156$	0	0
$157$	−8.29932e9	−1.09017	−0.545085	−	0.838381i	$-0.683503\pi$
$157$	−8.29932e9	−1.09017	−0.545085	+	0.838381i	$0.683503\pi$
$158$	0	0
$159$	5.77751e9	0.716891
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.23181e10	1.36679	0.683393	−	0.730051i	$-0.260503\pi$
$163$	1.23181e10	1.36679	0.683393	+	0.730051i	$0.260503\pi$
$164$	0	0
$165$	−1.04098e9	−0.109336
$166$	0	0
$167$	1.95830e9	0.194830	0.0974150	−	0.995244i	$-0.468943\pi$
$167$	1.95830e9	0.194830	0.0974150	+	0.995244i	$0.468943\pi$
$168$	0	0
$169$	1.75849e10	1.65825
$170$	0	0
$171$	−7.00450e9	−0.626462
$172$	0	0
$173$	1.50206e10	1.27491	0.637457	−	0.770486i	$-0.279986\pi$
$173$	1.50206e10	1.27491	0.637457	+	0.770486i	$0.279986\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.42964e9	0.722130
$178$	0	0
$179$	2.13087e10	1.55138	0.775692	−	0.631112i	$-0.217401\pi$
$179$	2.13087e10	1.55138	0.775692	+	0.631112i	$0.217401\pi$
$180$	0	0
$181$	6.65134e9	0.460634	0.230317	−	0.973116i	$-0.426024\pi$
$181$	6.65134e9	0.460634	0.230317	+	0.973116i	$0.426024\pi$
$182$	0	0
$183$	7.15409e9	0.471547
$184$	0	0
$185$	−4.41620e8	−0.0277189
$186$	0	0
$187$	8.92962e8	0.0534006
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.78305e10	0.969422	0.484711	−	0.874674i	$-0.338925\pi$
$191$	1.78305e10	0.969422	0.484711	+	0.874674i	$0.338925\pi$
$192$	0	0
$193$	4.40234e8	0.0228389	0.0114195	−	0.999935i	$-0.496365\pi$
$193$	4.40234e8	0.0228389	0.0114195	+	0.999935i	$0.496365\pi$
$194$	0	0
$195$	1.92340e9	0.0952609
$196$	0	0
$197$	−2.01458e10	−0.952988	−0.476494	−	0.879178i	$-0.658093\pi$
$197$	−2.01458e10	−0.952988	−0.476494	+	0.879178i	$0.658093\pi$
$198$	0	0
$199$	−3.23466e10	−1.46214	−0.731071	−	0.682301i	$-0.760979\pi$
$199$	−3.23466e10	−1.46214	−0.731071	+	0.682301i	$0.760979\pi$
$200$	0	0
$201$	−1.04174e10	−0.450170
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.09746e9	−0.122493
$206$	0	0
$207$	3.25596e9	0.123257
$208$	0	0
$209$	4.32107e10	1.56651
$210$	0	0
$211$	−3.85763e10	−1.33983	−0.669915	−	0.742438i	$-0.733670\pi$
$211$	−3.85763e10	−1.33983	−0.669915	+	0.742438i	$0.733670\pi$
$212$	0	0
$213$	−1.36653e10	−0.454894
$214$	0	0
$215$	4.20308e9	0.134151
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.37038e9	−0.128387
$220$	0	0
$221$	−1.64991e9	−0.0465260
$222$	0	0
$223$	−3.40626e10	−0.922371	−0.461186	−	0.887304i	$-0.652576\pi$
$223$	−3.40626e10	−0.922371	−0.461186	+	0.887304i	$0.652576\pi$
$224$	0	0
$225$	2.83791e10	0.738206
$226$	0	0
$227$	6.93803e10	1.73428	0.867141	−	0.498063i	$-0.165955\pi$
$227$	6.93803e10	1.73428	0.867141	+	0.498063i	$0.165955\pi$
$228$	0	0
$229$	7.06757e10	1.69828	0.849142	−	0.528164i	$-0.177119\pi$
$229$	7.06757e10	1.69828	0.849142	+	0.528164i	$0.177119\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.33248e10	1.18530	0.592649	−	0.805461i	$-0.298082\pi$
$233$	5.33248e10	1.18530	0.592649	+	0.805461i	$0.298082\pi$
$234$	0	0
$235$	5.71427e9	0.122224
$236$	0	0
$237$	−2.82528e10	−0.581694
$238$	0	0
$239$	2.61227e10	0.517878	0.258939	−	0.965894i	$-0.416627\pi$
$239$	2.61227e10	0.517878	0.258939	+	0.965894i	$0.416627\pi$
$240$	0	0
$241$	2.43771e10	0.465485	0.232743	−	0.972538i	$-0.425230\pi$
$241$	2.43771e10	0.465485	0.232743	+	0.972538i	$0.425230\pi$
$242$	0	0
$243$	−5.60793e10	−1.03175
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.98397e10	−1.36484
$248$	0	0
$249$	1.05311e10	0.173612
$250$	0	0
$251$	6.65520e9	0.105835	0.0529175	−	0.998599i	$-0.483148\pi$
$251$	6.65520e9	0.105835	0.0529175	+	0.998599i	$0.483148\pi$
$252$	0	0
$253$	−2.00860e10	−0.308213
$254$	0	0
$255$	−1.12576e8	−0.00166731
$256$	0	0
$257$	−1.22130e11	−1.74632	−0.873160	−	0.487433i	$-0.837933\pi$
$257$	−1.22130e11	−1.74632	−0.873160	+	0.487433i	$0.837933\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.20790e10	−0.161119
$262$	0	0
$263$	−3.13988e10	−0.404680	−0.202340	−	0.979315i	$-0.564855\pi$
$263$	−3.13988e10	−0.404680	−0.202340	+	0.979315i	$0.564855\pi$
$264$	0	0
$265$	−1.33627e10	−0.166451
$266$	0	0
$267$	−4.60392e10	−0.554405
$268$	0	0
$269$	1.26966e10	0.147844	0.0739220	−	0.997264i	$-0.476448\pi$
$269$	1.26966e10	0.147844	0.0739220	+	0.997264i	$0.476448\pi$
$270$	0	0
$271$	6.42236e9	0.0723324	0.0361662	−	0.999346i	$-0.488485\pi$
$271$	6.42236e9	0.0723324	0.0361662	+	0.999346i	$0.488485\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.75070e11	−1.84593
$276$	0	0
$277$	−1.54290e11	−1.57463	−0.787317	−	0.616549i	$-0.788530\pi$
$277$	−1.54290e11	−1.57463	−0.787317	+	0.616549i	$0.788530\pi$
$278$	0	0
$279$	4.17113e10	0.412131
$280$	0	0
$281$	6.22022e10	0.595151	0.297576	−	0.954698i	$-0.403822\pi$
$281$	6.22022e10	0.595151	0.297576	+	0.954698i	$0.403822\pi$
$282$	0	0
$283$	5.21640e10	0.483428	0.241714	−	0.970348i	$-0.422290\pi$
$283$	5.21640e10	0.483428	0.241714	+	0.970348i	$0.422290\pi$
$284$	0	0
$285$	−5.44759e9	−0.0489105
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.18491e11	−0.999186
$290$	0	0
$291$	−1.06752e11	−0.872682
$292$	0	0
$293$	−1.98645e11	−1.57461	−0.787307	−	0.616561i	$-0.788525\pi$
$293$	−1.98645e11	−1.57461	−0.787307	+	0.616561i	$0.788525\pi$
$294$	0	0
$295$	−2.18096e10	−0.167668
$296$	0	0
$297$	2.20076e11	1.64123
$298$	0	0
$299$	3.71126e10	0.268535
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.16700e11	0.795391
$304$	0	0
$305$	−1.65466e10	−0.109486
$306$	0	0
$307$	−1.72725e11	−1.10977	−0.554885	−	0.831927i	$-0.687238\pi$
$307$	−1.72725e11	−1.10977	−0.554885	+	0.831927i	$0.687238\pi$
$308$	0	0
$309$	3.70069e10	0.230924
$310$	0	0
$311$	1.30572e10	0.0791462	0.0395731	−	0.999217i	$-0.487400\pi$
$311$	1.30572e10	0.0791462	0.0395731	+	0.999217i	$0.487400\pi$
$312$	0	0
$313$	−1.68471e11	−0.992145	−0.496072	−	0.868281i	$-0.665225\pi$
$313$	−1.68471e11	−0.992145	−0.496072	+	0.868281i	$0.665225\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.24039e11	−1.80231	−0.901157	−	0.433492i	$-0.857281\pi$
$317$	−3.24039e11	−1.80231	−0.901157	+	0.433492i	$0.857281\pi$
$318$	0	0
$319$	7.45150e10	0.402889
$320$	0	0
$321$	−3.70199e10	−0.194609
$322$	0	0
$323$	4.67299e9	0.0238882
$324$	0	0
$325$	3.23475e11	1.60829
$326$	0	0
$327$	−1.39908e11	−0.676670
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8.25555e10	−0.378025	−0.189012	−	0.981975i	$-0.560529\pi$
$331$	−8.25555e10	−0.378025	−0.189012	+	0.981975i	$0.560529\pi$
$332$	0	0
$333$	3.99631e10	0.178098
$334$	0	0
$335$	2.40942e10	0.104523
$336$	0	0
$337$	−5.14965e10	−0.217492	−0.108746	−	0.994070i	$-0.534683\pi$
$337$	−5.14965e10	−0.217492	−0.108746	+	0.994070i	$0.534683\pi$
$338$	0	0
$339$	−7.70401e10	−0.316824
$340$	0	0
$341$	−2.57317e11	−1.03056
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.53225e9	0.00962323
$346$	0	0
$347$	−3.99878e11	−1.48062	−0.740312	−	0.672263i	$-0.765322\pi$
$347$	−3.99878e11	−1.48062	−0.740312	+	0.672263i	$0.765322\pi$
$348$	0	0
$349$	5.32897e11	1.92278	0.961388	−	0.275195i	$-0.0887426\pi$
$349$	5.32897e11	1.92278	0.961388	+	0.275195i	$0.0887426\pi$
$350$	0	0
$351$	−4.06632e11	−1.42995
$352$	0	0
$353$	2.83386e10	0.0971387	0.0485694	−	0.998820i	$-0.484534\pi$
$353$	2.83386e10	0.0971387	0.0485694	+	0.998820i	$0.484534\pi$
$354$	0	0
$355$	3.16062e10	0.105620
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.32309e10	−0.169137	−0.0845686	−	0.996418i	$-0.526951\pi$
$359$	−5.32309e10	−0.169137	−0.0845686	+	0.996418i	$0.526951\pi$
$360$	0	0
$361$	−9.65605e10	−0.299238
$362$	0	0
$363$	−4.15173e11	−1.25502
$364$	0	0
$365$	1.01082e10	0.0298095
$366$	0	0
$367$	5.16128e11	1.48511	0.742557	−	0.669783i	$-0.233613\pi$
$367$	5.16128e11	1.48511	0.742557	+	0.669783i	$0.233613\pi$
$368$	0	0
$369$	2.80295e11	0.787040
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.51851e11	0.941173	0.470586	−	0.882354i	$-0.344042\pi$
$373$	3.51851e11	0.941173	0.470586	+	0.882354i	$0.344042\pi$
$374$	0	0
$375$	4.44459e10	0.116062
$376$	0	0
$377$	−1.37680e11	−0.351023
$378$	0	0
$379$	3.25484e11	0.810313	0.405157	−	0.914247i	$-0.367217\pi$
$379$	3.25484e11	0.810313	0.405157	+	0.914247i	$0.367217\pi$
$380$	0	0
$381$	−2.23975e11	−0.544548
$382$	0	0
$383$	7.65132e11	1.81695	0.908473	−	0.417943i	$-0.137249\pi$
$383$	7.65132e11	1.81695	0.908473	+	0.417943i	$0.137249\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.80345e11	−0.861942
$388$	0	0
$389$	7.85896e11	1.74017	0.870085	−	0.492902i	$-0.164064\pi$
$389$	7.85896e11	1.74017	0.870085	+	0.492902i	$0.164064\pi$
$390$	0	0
$391$	−2.17219e9	−0.00470005
$392$	0	0
$393$	−1.73901e11	−0.367736
$394$	0	0
$395$	6.53454e10	0.135060
$396$	0	0
$397$	−7.39214e11	−1.49353	−0.746763	−	0.665090i	$-0.768393\pi$
$397$	−7.39214e11	−1.49353	−0.746763	+	0.665090i	$0.768393\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.29587e11	0.636531	0.318266	−	0.948002i	$-0.396900\pi$
$401$	3.29587e11	0.636531	0.318266	+	0.948002i	$0.396900\pi$
$402$	0	0
$403$	4.75440e11	0.897890
$404$	0	0
$405$	1.94484e10	0.0359200
$406$	0	0
$407$	−2.46532e11	−0.445347
$408$	0	0
$409$	−9.14043e11	−1.61515	−0.807573	−	0.589768i	$-0.799219\pi$
$409$	−9.14043e11	−1.61515	−0.807573	+	0.589768i	$0.799219\pi$
$410$	0	0
$411$	−7.23156e10	−0.125010
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.43573e10	−0.0403100
$416$	0	0
$417$	−4.23141e11	−0.685288
$418$	0	0
$419$	6.65088e11	1.05418	0.527092	−	0.849808i	$-0.323282\pi$
$419$	6.65088e11	1.05418	0.527092	+	0.849808i	$0.323282\pi$
$420$	0	0
$421$	8.13818e11	1.26258	0.631288	−	0.775548i	$-0.282527\pi$
$421$	8.13818e11	1.26258	0.631288	+	0.775548i	$0.282527\pi$
$422$	0	0
$423$	−5.17096e11	−0.785307
$424$	0	0
$425$	−1.89329e10	−0.0281492
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.07373e12	1.53051
$430$	0	0
$431$	−5.61579e11	−0.783905	−0.391953	−	0.919985i	$-0.628200\pi$
$431$	−5.61579e11	−0.783905	−0.391953	+	0.919985i	$0.628200\pi$
$432$	0	0
$433$	5.13223e11	0.701633	0.350817	−	0.936444i	$-0.385904\pi$
$433$	5.13223e11	0.701633	0.350817	+	0.936444i	$0.385904\pi$
$434$	0	0
$435$	−9.39413e9	−0.0125793
$436$	0	0
$437$	−1.05113e11	−0.137876
$438$	0	0
$439$	1.03560e12	1.33076	0.665382	−	0.746503i	$-0.268268\pi$
$439$	1.03560e12	1.33076	0.665382	+	0.746503i	$0.268268\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.57316e11	0.810881	0.405441	−	0.914121i	$-0.367118\pi$
$443$	6.57316e11	0.810881	0.405441	+	0.914121i	$0.367118\pi$
$444$	0	0
$445$	1.06483e11	0.128724
$446$	0	0
$447$	−2.11881e11	−0.251020
$448$	0	0
$449$	2.82701e11	0.328260	0.164130	−	0.986439i	$-0.447518\pi$
$449$	2.82701e11	0.328260	0.164130	+	0.986439i	$0.447518\pi$
$450$	0	0
$451$	−1.72914e12	−1.96805
$452$	0	0
$453$	−3.23083e11	−0.360473
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.99719e10	−0.0643168	−0.0321584	−	0.999483i	$-0.510238\pi$
$457$	−5.99719e10	−0.0643168	−0.0321584	+	0.999483i	$0.510238\pi$
$458$	0	0
$459$	2.38000e10	0.0250277
$460$	0	0
$461$	5.38434e11	0.555237	0.277618	−	0.960691i	$-0.410455\pi$
$461$	5.38434e11	0.555237	0.277618	+	0.960691i	$0.410455\pi$
$462$	0	0
$463$	4.90683e10	0.0496234	0.0248117	−	0.999692i	$-0.492101\pi$
$463$	4.90683e10	0.0496234	0.0248117	+	0.999692i	$0.492101\pi$
$464$	0	0
$465$	3.24400e10	0.0321768
$466$	0	0
$467$	1.51457e12	1.47355	0.736774	−	0.676139i	$-0.236348\pi$
$467$	1.51457e12	1.47355	0.736774	+	0.676139i	$0.236348\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.84090e11	0.546872
$472$	0	0
$473$	2.34634e12	2.15534
$474$	0	0
$475$	−9.16167e11	−0.825759
$476$	0	0
$477$	1.20922e12	1.06948
$478$	0	0
$479$	−1.44132e12	−1.25098	−0.625490	−	0.780232i	$-0.715101\pi$
$479$	−1.44132e12	−1.25098	−0.625490	+	0.780232i	$0.715101\pi$
$480$	0	0
$481$	4.55513e11	0.388015
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.46904e11	0.202623
$486$	0	0
$487$	7.52521e11	0.606232	0.303116	−	0.952954i	$-0.401973\pi$
$487$	7.52521e11	0.606232	0.303116	+	0.952954i	$0.401973\pi$
$488$	0	0
$489$	−8.66926e11	−0.685634
$490$	0	0
$491$	−1.05719e12	−0.820896	−0.410448	−	0.911884i	$-0.634628\pi$
$491$	−1.05719e12	−0.820896	−0.410448	+	0.911884i	$0.634628\pi$
$492$	0	0
$493$	8.05837e9	0.00614379
$494$	0	0
$495$	−2.17874e11	−0.163111
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.10866e11	0.657661	0.328830	−	0.944389i	$-0.393346\pi$
$499$	9.10866e11	0.657661	0.328830	+	0.944389i	$0.393346\pi$
$500$	0	0
$501$	−1.37822e11	−0.0977343
$502$	0	0
$503$	7.10909e11	0.495174	0.247587	−	0.968866i	$-0.420362\pi$
$503$	7.10909e11	0.495174	0.247587	+	0.968866i	$0.420362\pi$
$504$	0	0
$505$	−2.69914e11	−0.184678
$506$	0	0
$507$	−1.23759e12	−0.831841
$508$	0	0
$509$	−1.22197e12	−0.806919	−0.403459	−	0.914998i	$-0.632192\pi$
$509$	−1.22197e12	−0.806919	−0.403459	+	0.914998i	$0.632192\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.15169e12	0.734188
$514$	0	0
$515$	−8.55925e10	−0.0536171
$516$	0	0
$517$	3.18996e12	1.96371
$518$	0	0
$519$	−1.05712e12	−0.639547
$520$	0	0
$521$	6.45178e11	0.383627	0.191814	−	0.981431i	$-0.438563\pi$
$521$	6.45178e11	0.383627	0.191814	+	0.981431i	$0.438563\pi$
$522$	0	0
$523$	2.56126e12	1.49691	0.748456	−	0.663184i	$-0.230795\pi$
$523$	2.56126e12	1.49691	0.748456	+	0.663184i	$0.230795\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.78274e10	−0.0157154
$528$	0	0
$529$	−1.75229e12	−0.972873
$530$	0	0
$531$	1.97360e12	1.07729
$532$	0	0
$533$	3.19490e12	1.71469
$534$	0	0
$535$	8.56226e10	0.0451852
$536$	0	0
$537$	−1.49967e12	−0.778235
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.62574e10	−0.0382731	−0.0191366	−	0.999817i	$-0.506092\pi$
$541$	−7.62574e10	−0.0382731	−0.0191366	+	0.999817i	$0.506092\pi$
$542$	0	0
$543$	−4.68109e11	−0.231072
$544$	0	0
$545$	3.23590e11	0.157113
$546$	0	0
$547$	−3.32708e12	−1.58899	−0.794494	−	0.607272i	$-0.792264\pi$
$547$	−3.32708e12	−1.58899	−0.794494	+	0.607272i	$0.792264\pi$
$548$	0	0
$549$	1.49733e12	0.703465
$550$	0	0
$551$	3.89947e11	0.180228
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.10804e10	0.0139049
$556$	0	0
$557$	1.65349e12	0.727867	0.363933	−	0.931425i	$-0.381434\pi$
$557$	1.65349e12	0.727867	0.363933	+	0.931425i	$0.381434\pi$
$558$	0	0
$559$	−4.33530e12	−1.87787
$560$	0	0
$561$	−6.28450e10	−0.0267878
$562$	0	0
$563$	1.44327e12	0.605424	0.302712	−	0.953082i	$-0.402108\pi$
$563$	1.44327e12	0.605424	0.302712	+	0.953082i	$0.402108\pi$
$564$	0	0
$565$	1.78185e11	0.0735618
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.96648e12	0.786474	0.393237	−	0.919437i	$-0.371355\pi$
$569$	1.96648e12	0.786474	0.393237	+	0.919437i	$0.371355\pi$
$570$	0	0
$571$	3.54964e12	1.39740	0.698701	−	0.715414i	$-0.253762\pi$
$571$	3.54964e12	1.39740	0.698701	+	0.715414i	$0.253762\pi$
$572$	0	0
$573$	−1.25488e12	−0.486300
$574$	0	0
$575$	4.25870e11	0.162469
$576$	0	0
$577$	−3.31017e11	−0.124325	−0.0621625	−	0.998066i	$-0.519800\pi$
$577$	−3.31017e11	−0.124325	−0.0621625	+	0.998066i	$0.519800\pi$
$578$	0	0
$579$	−3.09828e10	−0.0114569
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.45964e12	−2.67430
$584$	0	0
$585$	4.02563e11	0.142113
$586$	0	0
$587$	−3.23387e12	−1.12422	−0.562109	−	0.827063i	$-0.690010\pi$
$587$	−3.23387e12	−1.12422	−0.562109	+	0.827063i	$0.690010\pi$
$588$	0	0
$589$	−1.34657e12	−0.461011
$590$	0	0
$591$	1.41782e12	0.478056
$592$	0	0
$593$	−3.28172e12	−1.08982	−0.544911	−	0.838494i	$-0.683437\pi$
$593$	−3.28172e12	−1.08982	−0.544911	+	0.838494i	$0.683437\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.27649e12	0.733468
$598$	0	0
$599$	−4.82955e12	−1.53280	−0.766401	−	0.642362i	$-0.777955\pi$
$599$	−4.82955e12	−1.53280	−0.766401	+	0.642362i	$0.777955\pi$
$600$	0	0
$601$	2.74359e12	0.857797	0.428898	−	0.903353i	$-0.358902\pi$
$601$	2.74359e12	0.857797	0.428898	+	0.903353i	$0.358902\pi$
$602$	0	0
$603$	−2.18033e12	−0.671575
$604$	0	0
$605$	9.60246e11	0.291396
$606$	0	0
$607$	5.89792e11	0.176340	0.0881698	−	0.996105i	$-0.471898\pi$
$607$	5.89792e11	0.176340	0.0881698	+	0.996105i	$0.471898\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.89404e12	−1.71091
$612$	0	0
$613$	−4.93280e12	−1.41098	−0.705491	−	0.708719i	$-0.749274\pi$
$613$	−4.93280e12	−1.41098	−0.705491	+	0.708719i	$0.749274\pi$
$614$	0	0
$615$	2.17993e11	0.0614475
$616$	0	0
$617$	4.74933e12	1.31932	0.659658	−	0.751566i	$-0.270701\pi$
$617$	4.74933e12	1.31932	0.659658	+	0.751566i	$0.270701\pi$
$618$	0	0
$619$	3.54867e12	0.971532	0.485766	−	0.874089i	$-0.338541\pi$
$619$	3.54867e12	0.971532	0.485766	+	0.874089i	$0.338541\pi$
$620$	0	0
$621$	−5.35350e11	−0.144453
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.66015e12	0.959486
$626$	0	0
$627$	−3.04108e12	−0.785822
$628$	0	0
$629$	−2.66610e10	−0.00679124
$630$	0	0
$631$	1.34343e11	0.0337351	0.0168676	−	0.999858i	$-0.494631\pi$
$631$	1.34343e11	0.0337351	0.0168676	+	0.999858i	$0.494631\pi$
$632$	0	0
$633$	2.71493e12	0.672112
$634$	0	0
$635$	5.18027e11	0.126436
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.86011e12	−0.678623
$640$	0	0
$641$	−3.82754e12	−0.895486	−0.447743	−	0.894162i	$-0.647772\pi$
$641$	−3.82754e12	−0.895486	−0.447743	+	0.894162i	$0.647772\pi$
$642$	0	0
$643$	−1.30004e12	−0.299921	−0.149960	−	0.988692i	$-0.547915\pi$
$643$	−1.30004e12	−0.299921	−0.149960	+	0.988692i	$0.547915\pi$
$644$	0	0
$645$	−2.95804e11	−0.0672955
$646$	0	0
$647$	−2.26204e12	−0.507494	−0.253747	−	0.967271i	$-0.581663\pi$
$647$	−2.26204e12	−0.507494	−0.253747	+	0.967271i	$0.581663\pi$
$648$	0	0
$649$	−1.21751e13	−2.69384
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.05082e12	−1.73273	−0.866365	−	0.499411i	$-0.833550\pi$
$653$	−8.05082e12	−1.73273	−0.866365	+	0.499411i	$0.833550\pi$
$654$	0	0
$655$	4.02213e11	0.0853827
$656$	0	0
$657$	−9.14709e11	−0.191531
$658$	0	0
$659$	−3.50688e12	−0.724330	−0.362165	−	0.932114i	$-0.617962\pi$
$659$	−3.50688e12	−0.724330	−0.362165	+	0.932114i	$0.617962\pi$
$660$	0	0
$661$	6.86224e12	1.39817	0.699084	−	0.715040i	$-0.253591\pi$
$661$	6.86224e12	1.39817	0.699084	+	0.715040i	$0.253591\pi$
$662$	0	0
$663$	1.16118e11	0.0233393
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.81262e11	−0.0354602
$668$	0	0
$669$	2.39726e12	0.462698
$670$	0	0
$671$	−9.23702e12	−1.75906
$672$	0	0
$673$	8.29182e12	1.55805	0.779026	−	0.626991i	$-0.215714\pi$
$673$	8.29182e12	1.55805	0.779026	+	0.626991i	$0.215714\pi$
$674$	0	0
$675$	−4.66613e12	−0.865147
$676$	0	0
$677$	4.00553e12	0.732843	0.366422	−	0.930449i	$-0.380583\pi$
$677$	4.00553e12	0.732843	0.366422	+	0.930449i	$0.380583\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4.88285e12	−0.869984
$682$	0	0
$683$	4.36723e12	0.767914	0.383957	−	0.923351i	$-0.374561\pi$
$683$	4.36723e12	0.767914	0.383957	+	0.923351i	$0.374561\pi$
$684$	0	0
$685$	1.67257e11	0.0290254
$686$	0	0
$687$	−4.97402e12	−0.851927
$688$	0	0
$689$	1.37831e13	2.33002
$690$	0	0
$691$	−4.60531e11	−0.0768436	−0.0384218	−	0.999262i	$-0.512233\pi$
$691$	−4.60531e11	−0.0768436	−0.0384218	+	0.999262i	$0.512233\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.78675e11	0.159113
$696$	0	0
$697$	−1.86996e11	−0.0300114
$698$	0	0
$699$	−3.75289e12	−0.594592
$700$	0	0
$701$	−8.47672e11	−0.132586	−0.0662928	−	0.997800i	$-0.521117\pi$
$701$	−8.47672e11	−0.132586	−0.0662928	+	0.997800i	$0.521117\pi$
$702$	0	0
$703$	−1.29013e12	−0.199221
$704$	0	0
$705$	−4.02159e11	−0.0613122
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5.76926e12	−0.857457	−0.428728	−	0.903433i	$-0.641038\pi$
$709$	−5.76926e12	−0.857457	−0.428728	+	0.903433i	$0.641038\pi$
$710$	0	0
$711$	−5.91324e12	−0.867785
$712$	0	0
$713$	6.25940e11	0.0907046
$714$	0	0
$715$	−2.48341e12	−0.355362
$716$	0	0
$717$	−1.83846e12	−0.259788
$718$	0	0
$719$	7.53837e12	1.05196	0.525978	−	0.850498i	$-0.323700\pi$
$719$	7.53837e12	1.05196	0.525978	+	0.850498i	$0.323700\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.71562e12	−0.233506
$724$	0	0
$725$	−1.57989e12	−0.212376
$726$	0	0
$727$	9.73268e12	1.29219	0.646097	−	0.763255i	$-0.276400\pi$
$727$	9.73268e12	1.29219	0.646097	+	0.763255i	$0.276400\pi$
$728$	0	0
$729$	1.59504e12	0.209169
$730$	0	0
$731$	2.53744e11	0.0328675
$732$	0	0
$733$	−8.35470e12	−1.06896	−0.534482	−	0.845180i	$-0.679493\pi$
$733$	−8.35470e12	−1.06896	−0.534482	+	0.845180i	$0.679493\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.34504e13	1.67932
$738$	0	0
$739$	−1.42736e13	−1.76050	−0.880248	−	0.474515i	$-0.842624\pi$
$739$	−1.42736e13	−1.76050	−0.880248	+	0.474515i	$0.842624\pi$
$740$	0	0
$741$	5.61896e12	0.684659
$742$	0	0
$743$	−1.54557e13	−1.86054	−0.930271	−	0.366873i	$-0.880428\pi$
$743$	−1.54557e13	−1.86054	−0.930271	+	0.366873i	$0.880428\pi$
$744$	0	0
$745$	4.90056e11	0.0582831
$746$	0	0
$747$	2.20414e12	0.258998
$748$	0	0
$749$	0	0
$750$	0	0
$751$	3.84446e12	0.441017	0.220509	−	0.975385i	$-0.429228\pi$
$751$	3.84446e12	0.441017	0.220509	+	0.975385i	$0.429228\pi$
$752$	0	0
$753$	−4.68380e11	−0.0530910
$754$	0	0
$755$	7.47253e11	0.0836964
$756$	0	0
$757$	5.43161e12	0.601169	0.300585	−	0.953755i	$-0.402818\pi$
$757$	5.43161e12	0.601169	0.300585	+	0.953755i	$0.402818\pi$
$758$	0	0
$759$	1.41361e12	0.154612
$760$	0	0
$761$	−8.40759e12	−0.908742	−0.454371	−	0.890813i	$-0.650136\pi$
$761$	−8.40759e12	−0.908742	−0.454371	+	0.890813i	$0.650136\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.35619e10	−0.00248733
$766$	0	0
$767$	2.24958e13	2.34704
$768$	0	0
$769$	−1.01254e12	−0.104411	−0.0522053	−	0.998636i	$-0.516625\pi$
$769$	−1.01254e12	−0.104411	−0.0522053	+	0.998636i	$0.516625\pi$
$770$	0	0
$771$	8.59529e12	0.876023
$772$	0	0
$773$	−1.41775e12	−0.142821	−0.0714106	−	0.997447i	$-0.522750\pi$
$773$	−1.41775e12	−0.142821	−0.0714106	+	0.997447i	$0.522750\pi$
$774$	0	0
$775$	5.45571e12	0.543243
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.04880e12	−0.880385
$780$	0	0
$781$	1.76440e13	1.69694
$782$	0	0
$783$	1.98604e12	0.188825
$784$	0	0
$785$	−1.35093e12	−0.126975
$786$	0	0
$787$	−5.91661e12	−0.549777	−0.274889	−	0.961476i	$-0.588641\pi$
$787$	−5.91661e12	−0.549777	−0.274889	+	0.961476i	$0.588641\pi$
$788$	0	0
$789$	2.20979e12	0.203004
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.70671e13	1.53261
$794$	0	0
$795$	9.40440e11	0.0834985
$796$	0	0
$797$	−3.00078e12	−0.263434	−0.131717	−	0.991287i	$-0.542049\pi$
$797$	−3.00078e12	−0.263434	−0.131717	+	0.991287i	$0.542049\pi$
$798$	0	0
$799$	3.44976e11	0.0299453
$800$	0	0
$801$	−9.63587e12	−0.827074
$802$	0	0
$803$	5.64283e12	0.478935
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.93565e11	−0.0741643
$808$	0	0
$809$	5.32817e12	0.437331	0.218665	−	0.975800i	$-0.429830\pi$
$809$	5.32817e12	0.437331	0.218665	+	0.975800i	$0.429830\pi$
$810$	0	0
$811$	−1.28093e13	−1.03976	−0.519878	−	0.854241i	$-0.674022\pi$
$811$	−1.28093e13	−1.03976	−0.519878	+	0.854241i	$0.674022\pi$
$812$	0	0
$813$	−4.51993e11	−0.0362848
$814$	0	0
$815$	2.00510e12	0.159194
$816$	0	0
$817$	1.22787e13	0.964171
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.24492e13	1.72447	0.862237	−	0.506504i	$-0.169063\pi$
$821$	2.24492e13	1.72447	0.862237	+	0.506504i	$0.169063\pi$
$822$	0	0
$823$	−1.34606e13	−1.02274	−0.511371	−	0.859360i	$-0.670862\pi$
$823$	−1.34606e13	−1.02274	−0.511371	+	0.859360i	$0.670862\pi$
$824$	0	0
$825$	1.23211e13	0.925992
$826$	0	0
$827$	1.35017e13	1.00372	0.501861	−	0.864948i	$-0.332649\pi$
$827$	1.35017e13	1.00372	0.501861	+	0.864948i	$0.332649\pi$
$828$	0	0
$829$	−1.81692e13	−1.33610	−0.668052	−	0.744115i	$-0.732872\pi$
$829$	−1.81692e13	−1.33610	−0.668052	+	0.744115i	$0.732872\pi$
$830$	0	0
$831$	1.08586e13	0.789898
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.18765e11	0.0226924
$836$	0	0
$837$	−6.85823e12	−0.483001
$838$	0	0
$839$	6.58386e12	0.458724	0.229362	−	0.973341i	$-0.426336\pi$
$839$	6.58386e12	0.458724	0.229362	+	0.973341i	$0.426336\pi$
$840$	0	0
$841$	−1.38347e13	−0.953647
$842$	0	0
$843$	−4.37767e12	−0.298551
$844$	0	0
$845$	2.86239e12	0.193141
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.67120e12	−0.242506
$850$	0	0
$851$	5.99705e11	0.0391972
$852$	0	0
$853$	2.09998e13	1.35814	0.679070	−	0.734073i	$-0.262383\pi$
$853$	2.09998e13	1.35814	0.679070	+	0.734073i	$0.262383\pi$
$854$	0	0
$855$	−1.14016e12	−0.0729659
$856$	0	0
$857$	−2.50798e13	−1.58822	−0.794108	−	0.607776i	$-0.792062\pi$
$857$	−2.50798e13	−1.58822	−0.794108	+	0.607776i	$0.792062\pi$
$858$	0	0
$859$	1.78667e13	1.11963	0.559816	−	0.828617i	$-0.310872\pi$
$859$	1.78667e13	1.11963	0.559816	+	0.828617i	$0.310872\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.57104e13	0.964139	0.482070	−	0.876133i	$-0.339885\pi$
$863$	1.57104e13	0.964139	0.482070	+	0.876133i	$0.339885\pi$
$864$	0	0
$865$	2.44500e12	0.148493
$866$	0	0
$867$	8.33919e12	0.501231
$868$	0	0
$869$	3.64787e13	2.16995
$870$	0	0
$871$	−2.48522e13	−1.46313
$872$	0	0
$873$	−2.23428e13	−1.30189
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.27499e13	−0.727793	−0.363896	−	0.931439i	$-0.618554\pi$
$877$	−1.27499e13	−0.727793	−0.363896	+	0.931439i	$0.618554\pi$
$878$	0	0
$879$	1.39803e13	0.789889
$880$	0	0
$881$	−1.77457e13	−0.992432	−0.496216	−	0.868199i	$-0.665278\pi$
$881$	−1.77457e13	−0.992432	−0.496216	+	0.868199i	$0.665278\pi$
$882$	0	0
$883$	−2.35402e13	−1.30313	−0.651565	−	0.758593i	$-0.725887\pi$
$883$	−2.35402e13	−1.30313	−0.651565	+	0.758593i	$0.725887\pi$
$884$	0	0
$885$	1.53492e12	0.0841087
$886$	0	0
$887$	−7.05989e12	−0.382950	−0.191475	−	0.981498i	$-0.561327\pi$
$887$	−7.05989e12	−0.382950	−0.191475	+	0.981498i	$0.561327\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.08570e13	0.577110
$892$	0	0
$893$	1.66935e13	0.878446
$894$	0	0
$895$	3.46855e12	0.180694
$896$	0	0
$897$	−2.61191e12	−0.134708
$898$	0	0
$899$	−2.32211e12	−0.118567
$900$	0	0
$901$	−8.06718e11	−0.0407812
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.08268e12	0.0536514
$906$	0	0
$907$	4.68299e12	0.229768	0.114884	−	0.993379i	$-0.463350\pi$
$907$	4.68299e12	0.229768	0.114884	+	0.993379i	$0.463350\pi$
$908$	0	0
$909$	2.44251e13	1.18658
$910$	0	0
$911$	2.77029e13	1.33258	0.666289	−	0.745694i	$-0.267882\pi$
$911$	2.77029e13	1.33258	0.666289	+	0.745694i	$0.267882\pi$
$912$	0	0
$913$	−1.35973e13	−0.647641
$914$	0	0
$915$	1.16451e12	0.0549225
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.34343e13	1.54622	0.773112	−	0.634269i	$-0.218699\pi$
$919$	3.34343e13	1.54622	0.773112	+	0.634269i	$0.218699\pi$
$920$	0	0
$921$	1.21561e13	0.556705
$922$	0	0
$923$	−3.26005e13	−1.47848
$924$	0	0
$925$	5.22705e12	0.234757
$926$	0	0
$927$	7.74544e12	0.344498
$928$	0	0
$929$	3.43828e13	1.51450	0.757252	−	0.653123i	$-0.226542\pi$
$929$	3.43828e13	1.51450	0.757252	+	0.653123i	$0.226542\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−9.18944e11	−0.0397028
$934$	0	0
$935$	1.45353e11	0.00621973
$936$	0	0
$937$	−2.90342e13	−1.23050	−0.615250	−	0.788332i	$-0.710945\pi$
$937$	−2.90342e13	−1.23050	−0.615250	+	0.788332i	$0.710945\pi$
$938$	0	0
$939$	1.18566e13	0.497699
$940$	0	0
$941$	2.31361e13	0.961916	0.480958	−	0.876744i	$-0.340289\pi$
$941$	2.31361e13	0.961916	0.480958	+	0.876744i	$0.340289\pi$
$942$	0	0
$943$	4.20624e12	0.173217
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.38426e13	0.559299	0.279649	−	0.960102i	$-0.409782\pi$
$947$	1.38426e13	0.559299	0.279649	+	0.960102i	$0.409782\pi$
$948$	0	0
$949$	−1.04262e13	−0.417279
$950$	0	0
$951$	2.28052e13	0.904112
$952$	0	0
$953$	−1.60424e13	−0.630016	−0.315008	−	0.949089i	$-0.602007\pi$
$953$	−1.60424e13	−0.630016	−0.315008	+	0.949089i	$0.602007\pi$
$954$	0	0
$955$	2.90238e12	0.112912
$956$	0	0
$957$	−5.24422e12	−0.202105
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.84209e13	−0.696715
$962$	0	0
$963$	−7.74816e12	−0.290322
$964$	0	0
$965$	7.16595e10	0.00266012
$966$	0	0
$967$	−2.69595e13	−0.991502	−0.495751	−	0.868465i	$-0.665107\pi$
$967$	−2.69595e13	−0.991502	−0.495751	+	0.868465i	$0.665107\pi$
$968$	0	0
$969$	−3.28876e11	−0.0119833
$970$	0	0
$971$	1.26793e12	0.0457728	0.0228864	−	0.999738i	$-0.492714\pi$
$971$	1.26793e12	0.0457728	0.0228864	+	0.999738i	$0.492714\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.27655e13	−0.806784
$976$	0	0
$977$	−2.08710e13	−0.732855	−0.366428	−	0.930447i	$-0.619419\pi$
$977$	−2.08710e13	−0.732855	−0.366428	+	0.930447i	$0.619419\pi$
$978$	0	0
$979$	5.94436e13	2.06815
$980$	0	0
$981$	−2.92823e13	−1.00947
$982$	0	0
$983$	−2.63946e13	−0.901623	−0.450811	−	0.892619i	$-0.648865\pi$
$983$	−2.63946e13	−0.901623	−0.450811	+	0.892619i	$0.648865\pi$
$984$	0	0
$985$	−3.27926e12	−0.110997
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.70763e12	−0.189702
$990$	0	0
$991$	−2.33283e13	−0.768338	−0.384169	−	0.923263i	$-0.625512\pi$
$991$	−2.33283e13	−0.768338	−0.384169	+	0.923263i	$0.625512\pi$
$992$	0	0
$993$	5.81010e12	0.189632
$994$	0	0
$995$	−5.26525e12	−0.170300
$996$	0	0
$997$	3.62240e13	1.16110	0.580548	−	0.814226i	$-0.302838\pi$
$997$	3.62240e13	1.16110	0.580548	+	0.814226i	$0.302838\pi$
$998$	0	0
$999$	−6.57079e12	−0.208724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.10.a.g.1.4	✓	10
7.2	even	3		196.10.e.i.165.7		20
7.3	odd	6		196.10.e.i.177.4		20
7.4	even	3		196.10.e.i.177.7		20
7.5	odd	6		196.10.e.i.165.4		20
7.6	odd	2	inner	196.10.a.g.1.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.10.a.g.1.4	✓	10	1.1	even	1	trivial
196.10.a.g.1.7	yes	10	7.6	odd	2	inner
196.10.e.i.165.4		20	7.5	odd	6
196.10.e.i.165.7		20	7.2	even	3
196.10.e.i.177.4		20	7.3	odd	6
196.10.e.i.177.7		20	7.4	even	3

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:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	196.a (trivial)

\(f(q)\)	\(=\)	\(q-70.3781 q^{3} +162.776 q^{5} -14729.9 q^{9} +90868.8 q^{11} -167897. q^{13} -11455.9 q^{15} +9826.94 q^{17} +475528. q^{19} -221044. q^{23} -1.92663e6 q^{25} +2.42191e6 q^{27} +820028. q^{29} -2.83174e6 q^{31} -6.39517e6 q^{33} -2.71305e6 q^{37} +1.18163e7 q^{39} -1.90289e7 q^{41} +2.58212e7 q^{43} -2.39768e6 q^{45} +3.51051e7 q^{47} -691601. q^{51} -8.20924e7 q^{53} +1.47913e7 q^{55} -3.34668e7 q^{57} -1.33986e8 q^{59} -1.01652e8 q^{61} -2.73296e7 q^{65} +1.48020e8 q^{67} +1.55567e7 q^{69} +1.94170e8 q^{71} +6.20986e7 q^{73} +1.35592e8 q^{75} +4.01444e8 q^{79} +1.19480e8 q^{81} -1.49637e8 q^{83} +1.59959e6 q^{85} -5.77120e7 q^{87} +6.54169e8 q^{89} +1.99292e8 q^{93} +7.74046e7 q^{95} +1.51683e9 q^{97} -1.33849e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 93550 q^{9} + 27740 q^{11} + 912088 q^{15} - 2921096 q^{23} + 3484998 q^{25} - 835748 q^{29} - 31928756 q^{37} - 4930944 q^{39} + 13414340 q^{43} + 206013028 q^{51} + 137301788 q^{53} + 72246332 q^{57}+ \cdots + 5316067012 q^{99}+O(q^{100}) \) 10 * q + 93550 * q^9 + 27740 * q^11 + 912088 * q^15 - 2921096 * q^23 + 3484998 * q^25 - 835748 * q^29 - 31928756 * q^37 - 4930944 * q^39 + 13414340 * q^43 + 206013028 * q^51 + 137301788 * q^53 + 72246332 * q^57 + 519063672 * q^65 - 342262536 * q^67 - 531036408 * q^71 + 462186584 * q^79 + 2495014982 * q^81 + 1236603584 * q^85 + 5733362712 * q^93 + 2984918936 * q^95 + 5316067012 * q^99

Introduction

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Modular forms

Varieties

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