LMFDB - Embedded newform 126.10.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,10,Mod(1,126)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("126.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.8945153566$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	126.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+16.0000 q^{2} +256.000 q^{4} -1590.00 q^{5} +2401.00 q^{7} +4096.00 q^{8} +O(q^{10})$

\(q+16.0000 q^{2} +256.000 q^{4} -1590.00 q^{5} +2401.00 q^{7} +4096.00 q^{8} -25440.0 q^{10} +22668.0 q^{11} -64186.0 q^{13} +38416.0 q^{14} +65536.0 q^{16} -29946.0 q^{17} +301484. q^{19} -407040. q^{20} +362688. q^{22} -1.23749e6 q^{23} +574975. q^{25} -1.02698e6 q^{26} +614656. q^{28} -391806. q^{29} +6.80269e6 q^{31} +1.04858e6 q^{32} -479136. q^{34} -3.81759e6 q^{35} +1.92798e7 q^{37} +4.82374e6 q^{38} -6.51264e6 q^{40} +1.04874e7 q^{41} +3.84209e7 q^{43} +5.80301e6 q^{44} -1.97998e7 q^{46} +2928.00 q^{47} +5.76480e6 q^{49} +9.19960e6 q^{50} -1.64316e7 q^{52} +6.81907e7 q^{53} -3.60421e7 q^{55} +9.83450e6 q^{56} -6.26890e6 q^{58} -7.96367e7 q^{59} -3.72034e7 q^{61} +1.08843e8 q^{62} +1.67772e7 q^{64} +1.02056e8 q^{65} +5.82341e7 q^{67} -7.66618e6 q^{68} -6.10814e7 q^{70} +4.93494e7 q^{71} +3.45748e8 q^{73} +3.08476e8 q^{74} +7.71799e7 q^{76} +5.44259e7 q^{77} +4.55982e8 q^{79} -1.04202e8 q^{80} +1.67798e8 q^{82} -4.46212e8 q^{83} +4.76141e7 q^{85} +6.14734e8 q^{86} +9.28481e7 q^{88} +5.71902e8 q^{89} -1.54111e8 q^{91} -3.16797e8 q^{92} +46848.0 q^{94} -4.79360e8 q^{95} +2.44250e8 q^{97} +9.22368e7 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	16.0000	0.707107
$2$	16.0000	0.707107
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	−1590.00	−1.13771	−0.568856	−	0.822437i	$-0.692614\pi$
$5$	−1590.00	−1.13771	−0.568856	+	0.822437i	$0.692614\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	4096.00	0.353553
$9$	0	0
$10$	−25440.0	−0.804483
$11$	22668.0	0.466816	0.233408	−	0.972379i	$-0.425012\pi$
$11$	22668.0	0.466816	0.233408	+	0.972379i	$0.425012\pi$
$12$	0	0
$13$	−64186.0	−0.623297	−0.311649	−	0.950197i	$-0.600881\pi$
$13$	−64186.0	−0.623297	−0.311649	+	0.950197i	$0.600881\pi$
$14$	38416.0	0.267261
$15$	0	0
$16$	65536.0	0.250000
$17$	−29946.0	−0.0869598	−0.0434799	−	0.999054i	$-0.513844\pi$
$17$	−29946.0	−0.0869598	−0.0434799	+	0.999054i	$0.513844\pi$
$18$	0	0
$19$	301484.	0.530729	0.265365	−	0.964148i	$-0.414508\pi$
$19$	301484.	0.530729	0.265365	+	0.964148i	$0.414508\pi$
$20$	−407040.	−0.568856
$21$	0	0
$22$	362688.	0.330089
$23$	−1.23749e6	−0.922074	−0.461037	−	0.887381i	$-0.652522\pi$
$23$	−1.23749e6	−0.922074	−0.461037	+	0.887381i	$0.652522\pi$
$24$	0	0
$25$	574975.	0.294387
$26$	−1.02698e6	−0.440738
$27$	0	0
$28$	614656.	0.188982
$29$	−391806.	−0.102868	−0.0514340	−	0.998676i	$-0.516379\pi$
$29$	−391806.	−0.102868	−0.0514340	+	0.998676i	$0.516379\pi$
$30$	0	0
$31$	6.80269e6	1.32298	0.661489	−	0.749954i	$-0.269925\pi$
$31$	6.80269e6	1.32298	0.661489	+	0.749954i	$0.269925\pi$
$32$	1.04858e6	0.176777
$33$	0	0
$34$	−479136.	−0.0614899
$35$	−3.81759e6	−0.430014
$36$	0	0
$37$	1.92798e7	1.69120	0.845598	−	0.533820i	$-0.179244\pi$
$37$	1.92798e7	1.69120	0.845598	+	0.533820i	$0.179244\pi$
$38$	4.82374e6	0.375282
$39$	0	0
$40$	−6.51264e6	−0.402242
$41$	1.04874e7	0.579614	0.289807	−	0.957085i	$-0.406409\pi$
$41$	1.04874e7	0.579614	0.289807	+	0.957085i	$0.406409\pi$
$42$	0	0
$43$	3.84209e7	1.71380	0.856898	−	0.515486i	$-0.172388\pi$
$43$	3.84209e7	1.71380	0.856898	+	0.515486i	$0.172388\pi$
$44$	5.80301e6	0.233408
$45$	0	0
$46$	−1.97998e7	−0.652005
$47$	2928.00	8.75247e−5 0	4.37624e−5	−	1.00000i	$-0.499986\pi$
$47$	2928.00	8.75247e−5 0	4.37624e−5		1.00000i	$0.499986\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	9.19960e6	0.208163
$51$	0	0
$52$	−1.64316e7	−0.311649
$53$	6.81907e7	1.18709	0.593545	−	0.804801i	$-0.297728\pi$
$53$	6.81907e7	1.18709	0.593545	+	0.804801i	$0.297728\pi$
$54$	0	0
$55$	−3.60421e7	−0.531102
$56$	9.83450e6	0.133631
$57$	0	0
$58$	−6.26890e6	−0.0727386
$59$	−7.96367e7	−0.855617	−0.427808	−	0.903870i	$-0.640714\pi$
$59$	−7.96367e7	−0.855617	−0.427808	+	0.903870i	$0.640714\pi$
$60$	0	0
$61$	−3.72034e7	−0.344031	−0.172016	−	0.985094i	$-0.555028\pi$
$61$	−3.72034e7	−0.344031	−0.172016	+	0.985094i	$0.555028\pi$
$62$	1.08843e8	0.935487
$63$	0	0
$64$	1.67772e7	0.125000
$65$	1.02056e8	0.709132
$66$	0	0
$67$	5.82341e7	0.353053	0.176527	−	0.984296i	$-0.443514\pi$
$67$	5.82341e7	0.353053	0.176527	+	0.984296i	$0.443514\pi$
$68$	−7.66618e6	−0.0434799
$69$	0	0
$70$	−6.10814e7	−0.304066
$71$	4.93494e7	0.230472	0.115236	−	0.993338i	$-0.463237\pi$
$71$	4.93494e7	0.230472	0.115236	+	0.993338i	$0.463237\pi$
$72$	0	0
$73$	3.45748e8	1.42497	0.712486	−	0.701686i	$-0.247569\pi$
$73$	3.45748e8	1.42497	0.712486	+	0.701686i	$0.247569\pi$
$74$	3.08476e8	1.19586
$75$	0	0
$76$	7.71799e7	0.265365
$77$	5.44259e7	0.176440
$78$	0	0
$79$	4.55982e8	1.31712	0.658561	−	0.752528i	$-0.271166\pi$
$79$	4.55982e8	1.31712	0.658561	+	0.752528i	$0.271166\pi$
$80$	−1.04202e8	−0.284428
$81$	0	0
$82$	1.67798e8	0.409849
$83$	−4.46212e8	−1.03202	−0.516012	−	0.856581i	$-0.672584\pi$
$83$	−4.46212e8	−1.03202	−0.516012	+	0.856581i	$0.672584\pi$
$84$	0	0
$85$	4.76141e7	0.0989352
$86$	6.14734e8	1.21184
$87$	0	0
$88$	9.28481e7	0.165045
$89$	5.71902e8	0.966199	0.483099	−	0.875566i	$-0.339511\pi$
$89$	5.71902e8	0.966199	0.483099	+	0.875566i	$0.339511\pi$
$90$	0	0
$91$	−1.54111e8	−0.235584
$92$	−3.16797e8	−0.461037
$93$	0	0
$94$	46848.0	6.18893e−5 0
$95$	−4.79360e8	−0.603817
$96$	0	0
$97$	2.44250e8	0.280132	0.140066	−	0.990142i	$-0.455269\pi$
$97$	2.44250e8	0.280132	0.140066	+	0.990142i	$0.455269\pi$
$98$	9.22368e7	0.101015
$99$	0	0
$100$	1.47194e8	0.147194
$101$	−1.31391e9	−1.25637	−0.628185	−	0.778064i	$-0.716202\pi$
$101$	−1.31391e9	−1.25637	−0.628185	+	0.778064i	$0.716202\pi$
$102$	0	0
$103$	−1.53762e8	−0.134611	−0.0673056	−	0.997732i	$-0.521440\pi$
$103$	−1.53762e8	−0.134611	−0.0673056	+	0.997732i	$0.521440\pi$
$104$	−2.62906e8	−0.220369
$105$	0	0
$106$	1.09105e9	0.839399
$107$	2.11407e8	0.155917	0.0779584	−	0.996957i	$-0.475160\pi$
$107$	2.11407e8	0.155917	0.0779584	+	0.996957i	$0.475160\pi$
$108$	0	0
$109$	2.68503e9	1.82192	0.910960	−	0.412495i	$-0.135343\pi$
$109$	2.68503e9	1.82192	0.910960	+	0.412495i	$0.135343\pi$
$110$	−5.76674e8	−0.375546
$111$	0	0
$112$	1.57352e8	0.0944911
$113$	−9.52202e8	−0.549384	−0.274692	−	0.961532i	$-0.588576\pi$
$113$	−9.52202e8	−0.549384	−0.274692	+	0.961532i	$0.588576\pi$
$114$	0	0
$115$	1.96761e9	1.04905
$116$	−1.00302e8	−0.0514340
$117$	0	0
$118$	−1.27419e9	−0.605012
$119$	−7.19003e7	−0.0328677
$120$	0	0
$121$	−1.84411e9	−0.782082
$122$	−5.95254e8	−0.243267
$123$	0	0
$124$	1.74149e9	0.661489
$125$	2.19126e9	0.802784
$126$	0	0
$127$	−9.28823e8	−0.316823	−0.158411	−	0.987373i	$-0.550637\pi$
$127$	−9.28823e8	−0.316823	−0.158411	+	0.987373i	$0.550637\pi$
$128$	2.68435e8	0.0883883
$129$	0	0
$130$	1.63289e9	0.501432
$131$	4.75201e9	1.40980	0.704899	−	0.709308i	$-0.250992\pi$
$131$	4.75201e9	1.40980	0.704899	+	0.709308i	$0.250992\pi$
$132$	0	0
$133$	7.23863e8	0.200597
$134$	9.31745e8	0.249646
$135$	0	0
$136$	−1.22659e8	−0.0307449
$137$	6.96787e9	1.68989	0.844943	−	0.534856i	$-0.179634\pi$
$137$	6.96787e9	1.68989	0.844943	+	0.534856i	$0.179634\pi$
$138$	0	0
$139$	−6.39056e9	−1.45202	−0.726009	−	0.687685i	$-0.758627\pi$
$139$	−6.39056e9	−1.45202	−0.726009	+	0.687685i	$0.758627\pi$
$140$	−9.77303e8	−0.215007
$141$	0	0
$142$	7.89590e8	0.162969
$143$	−1.45497e9	−0.290965
$144$	0	0
$145$	6.22972e8	0.117034
$146$	5.53196e9	1.00761
$147$	0	0
$148$	4.93562e9	0.845598
$149$	1.00114e10	1.66402	0.832008	−	0.554763i	$-0.187191\pi$
$149$	1.00114e10	1.66402	0.832008	+	0.554763i	$0.187191\pi$
$150$	0	0
$151$	4.95258e9	0.775238	0.387619	−	0.921820i	$-0.373298\pi$
$151$	4.95258e9	0.775238	0.387619	+	0.921820i	$0.373298\pi$
$152$	1.23488e9	0.187641
$153$	0	0
$154$	8.70814e8	0.124762
$155$	−1.08163e10	−1.50517
$156$	0	0
$157$	−4.30789e9	−0.565869	−0.282934	−	0.959139i	$-0.591308\pi$
$157$	−4.30789e9	−0.565869	−0.282934	+	0.959139i	$0.591308\pi$
$158$	7.29571e9	0.931345
$159$	0	0
$160$	−1.66724e9	−0.201121
$161$	−2.97121e9	−0.348511
$162$	0	0
$163$	−1.24554e10	−1.38202	−0.691011	−	0.722845i	$-0.742834\pi$
$163$	−1.24554e10	−1.38202	−0.691011	+	0.722845i	$0.742834\pi$
$164$	2.68476e9	0.289807
$165$	0	0
$166$	−7.13940e9	−0.729752
$167$	−1.67083e10	−1.66230	−0.831149	−	0.556050i	$-0.812316\pi$
$167$	−1.67083e10	−1.66230	−0.831149	+	0.556050i	$0.812316\pi$
$168$	0	0
$169$	−6.48466e9	−0.611501
$170$	7.61826e8	0.0699577
$171$	0	0
$172$	9.83574e9	0.856898
$173$	−2.07893e10	−1.76455	−0.882274	−	0.470736i	$-0.843988\pi$
$173$	−2.07893e10	−1.76455	−0.882274	+	0.470736i	$0.843988\pi$
$174$	0	0
$175$	1.38051e9	0.111268
$176$	1.48557e9	0.116704
$177$	0	0
$178$	9.15043e9	0.683206
$179$	−1.30135e10	−0.947447	−0.473723	−	0.880674i	$-0.657090\pi$
$179$	−1.30135e10	−0.947447	−0.473723	+	0.880674i	$0.657090\pi$
$180$	0	0
$181$	2.55798e9	0.177151	0.0885755	−	0.996069i	$-0.471769\pi$
$181$	2.55798e9	0.177151	0.0885755	+	0.996069i	$0.471769\pi$
$182$	−2.46577e9	−0.166583
$183$	0	0
$184$	−5.06875e9	−0.326002
$185$	−3.06548e10	−1.92409
$186$	0	0
$187$	−6.78816e8	−0.0405943
$188$	749568.	4.37624e−5 0
$189$	0	0
$190$	−7.66975e9	−0.426963
$191$	9.94011e9	0.540432	0.270216	−	0.962800i	$-0.412905\pi$
$191$	9.94011e9	0.540432	0.270216	+	0.962800i	$0.412905\pi$
$192$	0	0
$193$	−1.07119e10	−0.555724	−0.277862	−	0.960621i	$-0.589626\pi$
$193$	−1.07119e10	−0.555724	−0.277862	+	0.960621i	$0.589626\pi$
$194$	3.90800e9	0.198083
$195$	0	0
$196$	1.47579e9	0.0714286
$197$	1.14767e10	0.542899	0.271449	−	0.962453i	$-0.412497\pi$
$197$	1.14767e10	0.542899	0.271449	+	0.962453i	$0.412497\pi$
$198$	0	0
$199$	1.20785e10	0.545975	0.272987	−	0.962018i	$-0.411988\pi$
$199$	1.20785e10	0.545975	0.272987	+	0.962018i	$0.411988\pi$
$200$	2.35510e9	0.104082
$201$	0	0
$202$	−2.10225e10	−0.888388
$203$	−9.40726e8	−0.0388804
$204$	0	0
$205$	−1.66749e10	−0.659433
$206$	−2.46019e9	−0.0951845
$207$	0	0
$208$	−4.20649e9	−0.155824
$209$	6.83404e9	0.247753
$210$	0	0
$211$	−5.39229e10	−1.87285	−0.936423	−	0.350873i	$-0.885885\pi$
$211$	−5.39229e10	−1.87285	−0.936423	+	0.350873i	$0.885885\pi$
$212$	1.74568e10	0.593545
$213$	0	0
$214$	3.38251e9	0.110250
$215$	−6.10892e10	−1.94981
$216$	0	0
$217$	1.63333e10	0.500039
$218$	4.29604e10	1.28829
$219$	0	0
$220$	−9.22678e9	−0.265551
$221$	1.92211e9	0.0542018
$222$	0	0
$223$	1.54752e10	0.419049	0.209525	−	0.977803i	$-0.432808\pi$
$223$	1.54752e10	0.419049	0.209525	+	0.977803i	$0.432808\pi$
$224$	2.51763e9	0.0668153
$225$	0	0
$226$	−1.52352e10	−0.388473
$227$	−6.84122e10	−1.71008	−0.855041	−	0.518560i	$-0.826468\pi$
$227$	−6.84122e10	−1.71008	−0.855041	+	0.518560i	$0.826468\pi$
$228$	0	0
$229$	6.07017e9	0.145862	0.0729309	−	0.997337i	$-0.476765\pi$
$229$	6.07017e9	0.145862	0.0729309	+	0.997337i	$0.476765\pi$
$230$	3.14817e10	0.741793
$231$	0	0
$232$	−1.60484e9	−0.0363693
$233$	2.11862e10	0.470924	0.235462	−	0.971884i	$-0.424340\pi$
$233$	2.11862e10	0.470924	0.235462	+	0.971884i	$0.424340\pi$
$234$	0	0
$235$	−4.65552e6	−9.95779e−5 0
$236$	−2.03870e10	−0.427808
$237$	0	0
$238$	−1.15041e9	−0.0232410
$239$	3.06668e10	0.607965	0.303982	−	0.952678i	$-0.401684\pi$
$239$	3.06668e10	0.607965	0.303982	+	0.952678i	$0.401684\pi$
$240$	0	0
$241$	−5.87077e10	−1.12103	−0.560516	−	0.828143i	$-0.689397\pi$
$241$	−5.87077e10	−1.12103	−0.560516	+	0.828143i	$0.689397\pi$
$242$	−2.95058e10	−0.553016
$243$	0	0
$244$	−9.52406e9	−0.172016
$245$	−9.16603e9	−0.162530
$246$	0	0
$247$	−1.93511e10	−0.330802
$248$	2.78638e10	0.467744
$249$	0	0
$250$	3.50601e10	0.567654
$251$	8.13561e10	1.29377	0.646887	−	0.762586i	$-0.276071\pi$
$251$	8.13561e10	1.29377	0.646887	+	0.762586i	$0.276071\pi$
$252$	0	0
$253$	−2.80514e10	−0.430439
$254$	−1.48612e10	−0.224027
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	3.38969e9	0.0484686	0.0242343	−	0.999706i	$-0.492285\pi$
$257$	3.38969e9	0.0484686	0.0242343	+	0.999706i	$0.492285\pi$
$258$	0	0
$259$	4.62907e10	0.639212
$260$	2.61263e10	0.354566
$261$	0	0
$262$	7.60322e10	0.996878
$263$	−7.18542e10	−0.926086	−0.463043	−	0.886336i	$-0.653242\pi$
$263$	−7.18542e10	−0.926086	−0.463043	+	0.886336i	$0.653242\pi$
$264$	0	0
$265$	−1.08423e11	−1.35057
$266$	1.15818e10	0.141843
$267$	0	0
$268$	1.49079e10	0.176527
$269$	−1.39264e10	−0.162164	−0.0810820	−	0.996707i	$-0.525838\pi$
$269$	−1.39264e10	−0.162164	−0.0810820	+	0.996707i	$0.525838\pi$
$270$	0	0
$271$	1.31656e11	1.48279	0.741394	−	0.671070i	$-0.234165\pi$
$271$	1.31656e11	1.48279	0.741394	+	0.671070i	$0.234165\pi$
$272$	−1.96254e9	−0.0217400
$273$	0	0
$274$	1.11486e11	1.19493
$275$	1.30335e10	0.137425
$276$	0	0
$277$	−3.63674e10	−0.371154	−0.185577	−	0.982630i	$-0.559415\pi$
$277$	−3.63674e10	−0.371154	−0.185577	+	0.982630i	$0.559415\pi$
$278$	−1.02249e11	−1.02673
$279$	0	0
$280$	−1.56368e10	−0.152033
$281$	1.16969e11	1.11916	0.559580	−	0.828776i	$-0.310962\pi$
$281$	1.16969e11	1.11916	0.559580	+	0.828776i	$0.310962\pi$
$282$	0	0
$283$	1.57635e11	1.46087	0.730437	−	0.682980i	$-0.239316\pi$
$283$	1.57635e11	1.46087	0.730437	+	0.682980i	$0.239316\pi$
$284$	1.26334e10	0.115236
$285$	0	0
$286$	−2.32795e10	−0.205744
$287$	2.51801e10	0.219073
$288$	0	0
$289$	−1.17691e11	−0.992438
$290$	9.96754e9	0.0827556
$291$	0	0
$292$	8.85114e10	0.712486
$293$	1.34540e11	1.06647	0.533235	−	0.845967i	$-0.320976\pi$
$293$	1.34540e11	1.06647	0.533235	+	0.845967i	$0.320976\pi$
$294$	0	0
$295$	1.26622e11	0.973445
$296$	7.89699e10	0.597928
$297$	0	0
$298$	1.60183e11	1.17664
$299$	7.94294e10	0.574726
$300$	0	0
$301$	9.22485e10	0.647754
$302$	7.92412e10	0.548176
$303$	0	0
$304$	1.97581e10	0.132682
$305$	5.91534e10	0.391408
$306$	0	0
$307$	2.81460e11	1.80840	0.904199	−	0.427111i	$-0.140469\pi$
$307$	2.81460e11	1.80840	0.904199	+	0.427111i	$0.140469\pi$
$308$	1.39330e10	0.0882200
$309$	0	0
$310$	−1.73060e11	−1.06431
$311$	−1.84533e11	−1.11854	−0.559270	−	0.828986i	$-0.688918\pi$
$311$	−1.84533e11	−1.11854	−0.559270	+	0.828986i	$0.688918\pi$
$312$	0	0
$313$	−1.61632e10	−0.0951872	−0.0475936	−	0.998867i	$-0.515155\pi$
$313$	−1.61632e10	−0.0951872	−0.0475936	+	0.998867i	$0.515155\pi$
$314$	−6.89262e10	−0.400130
$315$	0	0
$316$	1.16731e11	0.658561
$317$	8.95885e9	0.0498294	0.0249147	−	0.999690i	$-0.492069\pi$
$317$	8.95885e9	0.0498294	0.0249147	+	0.999690i	$0.492069\pi$
$318$	0	0
$319$	−8.88146e9	−0.0480204
$320$	−2.66758e10	−0.142214
$321$	0	0
$322$	−4.75393e10	−0.246435
$323$	−9.02824e9	−0.0461521
$324$	0	0
$325$	−3.69053e10	−0.183491
$326$	−1.99287e11	−0.977237
$327$	0	0
$328$	4.29562e10	0.204924
$329$	7.03013e6	3.30812e−5 0
$330$	0	0
$331$	−3.77109e10	−0.172680	−0.0863398	−	0.996266i	$-0.527517\pi$
$331$	−3.77109e10	−0.172680	−0.0863398	+	0.996266i	$0.527517\pi$
$332$	−1.14230e11	−0.516012
$333$	0	0
$334$	−2.67333e11	−1.17542
$335$	−9.25921e10	−0.401673
$336$	0	0
$337$	2.02981e11	0.857278	0.428639	−	0.903476i	$-0.358993\pi$
$337$	2.02981e11	0.857278	0.428639	+	0.903476i	$0.358993\pi$
$338$	−1.03755e11	−0.432396
$339$	0	0
$340$	1.21892e10	0.0494676
$341$	1.54203e11	0.617588
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	1.57372e11	0.605919
$345$	0	0
$346$	−3.32630e11	−1.24772
$347$	1.88559e11	0.698177	0.349089	−	0.937090i	$-0.386491\pi$
$347$	1.88559e11	0.698177	0.349089	+	0.937090i	$0.386491\pi$
$348$	0	0
$349$	−1.40280e11	−0.506154	−0.253077	−	0.967446i	$-0.581443\pi$
$349$	−1.40280e11	−0.506154	−0.253077	+	0.967446i	$0.581443\pi$
$350$	2.20882e10	0.0786783
$351$	0	0
$352$	2.37691e10	0.0825223
$353$	−7.44306e10	−0.255132	−0.127566	−	0.991830i	$-0.540717\pi$
$353$	−7.44306e10	−0.255132	−0.127566	+	0.991830i	$0.540717\pi$
$354$	0	0
$355$	−7.84655e10	−0.262211
$356$	1.46407e11	0.483099
$357$	0	0
$358$	−2.08216e11	−0.669946
$359$	2.31256e11	0.734798	0.367399	−	0.930063i	$-0.380248\pi$
$359$	2.31256e11	0.734798	0.367399	+	0.930063i	$0.380248\pi$
$360$	0	0
$361$	−2.31795e11	−0.718326
$362$	4.09277e10	0.125265
$363$	0	0
$364$	−3.94523e10	−0.117792
$365$	−5.49739e11	−1.62121
$366$	0	0
$367$	−2.94257e11	−0.846701	−0.423351	−	0.905966i	$-0.639146\pi$
$367$	−2.94257e11	−0.846701	−0.423351	+	0.905966i	$0.639146\pi$
$368$	−8.11000e10	−0.230518
$369$	0	0
$370$	−4.90477e11	−1.36054
$371$	1.63726e11	0.448678
$372$	0	0
$373$	2.70035e11	0.722322	0.361161	−	0.932503i	$-0.382380\pi$
$373$	2.70035e11	0.722322	0.361161	+	0.932503i	$0.382380\pi$
$374$	−1.08611e10	−0.0287045
$375$	0	0
$376$	1.19931e7	3.09447e−5 0
$377$	2.51485e10	0.0641173
$378$	0	0
$379$	1.88183e11	0.468494	0.234247	−	0.972177i	$-0.424737\pi$
$379$	1.88183e11	0.468494	0.234247	+	0.972177i	$0.424737\pi$
$380$	−1.22716e11	−0.301908
$381$	0	0
$382$	1.59042e11	0.382143
$383$	2.82829e11	0.671628	0.335814	−	0.941928i	$-0.390989\pi$
$383$	2.82829e11	0.671628	0.335814	+	0.941928i	$0.390989\pi$
$384$	0	0
$385$	−8.65371e10	−0.200738
$386$	−1.71391e11	−0.392956
$387$	0	0
$388$	6.25281e10	0.140066
$389$	7.58058e11	1.67853	0.839265	−	0.543722i	$-0.182986\pi$
$389$	7.58058e11	1.67853	0.839265	+	0.543722i	$0.182986\pi$
$390$	0	0
$391$	3.70578e10	0.0801834
$392$	2.36126e10	0.0505076
$393$	0	0
$394$	1.83627e11	0.383887
$395$	−7.25011e11	−1.49850
$396$	0	0
$397$	5.58142e11	1.12768	0.563842	−	0.825882i	$-0.309323\pi$
$397$	5.58142e11	1.12768	0.563842	+	0.825882i	$0.309323\pi$
$398$	1.93255e11	0.386063
$399$	0	0
$400$	3.76816e10	0.0735968
$401$	−7.13587e11	−1.37815	−0.689076	−	0.724689i	$-0.741983\pi$
$401$	−7.13587e11	−1.37815	−0.689076	+	0.724689i	$0.741983\pi$
$402$	0	0
$403$	−4.36637e11	−0.824609
$404$	−3.36360e11	−0.628185
$405$	0	0
$406$	−1.50516e10	−0.0274926
$407$	4.37034e11	0.789478
$408$	0	0
$409$	−6.71417e11	−1.18642	−0.593208	−	0.805049i	$-0.702139\pi$
$409$	−6.71417e11	−1.18642	−0.593208	+	0.805049i	$0.702139\pi$
$410$	−2.66798e11	−0.466290
$411$	0	0
$412$	−3.93630e10	−0.0673056
$413$	−1.91208e11	−0.323393
$414$	0	0
$415$	7.09477e11	1.17415
$416$	−6.73039e10	−0.110184
$417$	0	0
$418$	1.09345e11	0.175188
$419$	−7.85557e11	−1.24513	−0.622565	−	0.782568i	$-0.713909\pi$
$419$	−7.85557e11	−1.24513	−0.622565	+	0.782568i	$0.713909\pi$
$420$	0	0
$421$	8.41170e11	1.30501	0.652506	−	0.757784i	$-0.273718\pi$
$421$	8.41170e11	1.30501	0.652506	+	0.757784i	$0.273718\pi$
$422$	−8.62766e11	−1.32430
$423$	0	0
$424$	2.79309e11	0.419700
$425$	−1.72182e10	−0.0255999
$426$	0	0
$427$	−8.93253e10	−0.130032
$428$	5.41202e10	0.0779584
$429$	0	0
$430$	−9.77427e11	−1.37872
$431$	9.91535e11	1.38408	0.692039	−	0.721861i	$-0.256713\pi$
$431$	9.91535e11	1.38408	0.692039	+	0.721861i	$0.256713\pi$
$432$	0	0
$433$	−2.09922e11	−0.286987	−0.143493	−	0.989651i	$-0.545834\pi$
$433$	−2.09922e11	−0.286987	−0.143493	+	0.989651i	$0.545834\pi$
$434$	2.61332e11	0.353581
$435$	0	0
$436$	6.87367e11	0.910960
$437$	−3.73083e11	−0.489372
$438$	0	0
$439$	−1.05061e12	−1.35005	−0.675026	−	0.737794i	$-0.735868\pi$
$439$	−1.05061e12	−1.35005	−0.675026	+	0.737794i	$0.735868\pi$
$440$	−1.47629e11	−0.187773
$441$	0	0
$442$	3.07538e10	0.0383265
$443$	1.71164e11	0.211153	0.105576	−	0.994411i	$-0.466331\pi$
$443$	1.71164e11	0.211153	0.105576	+	0.994411i	$0.466331\pi$
$444$	0	0
$445$	−9.09324e11	−1.09926
$446$	2.47604e11	0.296313
$447$	0	0
$448$	4.02821e10	0.0472456
$449$	8.92199e11	1.03598	0.517992	−	0.855385i	$-0.326680\pi$
$449$	8.92199e11	1.03598	0.517992	+	0.855385i	$0.326680\pi$
$450$	0	0
$451$	2.37727e11	0.270573
$452$	−2.43764e11	−0.274692
$453$	0	0
$454$	−1.09459e12	−1.20921
$455$	2.45036e11	0.268027
$456$	0	0
$457$	1.93571e11	0.207595	0.103798	−	0.994598i	$-0.466901\pi$
$457$	1.93571e11	0.207595	0.103798	+	0.994598i	$0.466901\pi$
$458$	9.71228e10	0.103140
$459$	0	0
$460$	5.03707e11	0.524527
$461$	−1.20264e12	−1.24017	−0.620087	−	0.784533i	$-0.712903\pi$
$461$	−1.20264e12	−1.24017	−0.620087	+	0.784533i	$0.712903\pi$
$462$	0	0
$463$	−1.28520e12	−1.29974	−0.649870	−	0.760045i	$-0.725177\pi$
$463$	−1.28520e12	−1.29974	−0.649870	+	0.760045i	$0.725177\pi$
$464$	−2.56774e10	−0.0257170
$465$	0	0
$466$	3.38979e11	0.332994
$467$	1.37619e12	1.33891	0.669457	−	0.742851i	$-0.266527\pi$
$467$	1.37619e12	1.33891	0.669457	+	0.742851i	$0.266527\pi$
$468$	0	0
$469$	1.39820e11	0.133442
$470$	−7.44883e7	−7.04122e−5 0
$471$	0	0
$472$	−3.26192e11	−0.302506
$473$	8.70924e11	0.800028
$474$	0	0
$475$	1.73346e11	0.156240
$476$	−1.84065e10	−0.0164339
$477$	0	0
$478$	4.90669e11	0.429896
$479$	1.15599e12	1.00333	0.501665	−	0.865062i	$-0.332721\pi$
$479$	1.15599e12	1.00333	0.501665	+	0.865062i	$0.332721\pi$
$480$	0	0
$481$	−1.23749e12	−1.05412
$482$	−9.39323e11	−0.792690
$483$	0	0
$484$	−4.72092e11	−0.391041
$485$	−3.88358e11	−0.318709
$486$	0	0
$487$	−6.72883e11	−0.542075	−0.271038	−	0.962569i	$-0.587367\pi$
$487$	−6.72883e11	−0.542075	−0.271038	+	0.962569i	$0.587367\pi$
$488$	−1.52385e11	−0.121633
$489$	0	0
$490$	−1.46657e11	−0.114926
$491$	7.90636e11	0.613917	0.306958	−	0.951723i	$-0.400689\pi$
$491$	7.90636e11	0.613917	0.306958	+	0.951723i	$0.400689\pi$
$492$	0	0
$493$	1.17330e10	0.00894538
$494$	−3.09617e11	−0.233912
$495$	0	0
$496$	4.45821e11	0.330745
$497$	1.18488e11	0.0871104
$498$	0	0
$499$	2.73134e11	0.197207	0.0986037	−	0.995127i	$-0.468562\pi$
$499$	2.73134e11	0.197207	0.0986037	+	0.995127i	$0.468562\pi$
$500$	5.60962e11	0.401392
$501$	0	0
$502$	1.30170e12	0.914836
$503$	−1.57083e12	−1.09414	−0.547072	−	0.837086i	$-0.684258\pi$
$503$	−1.57083e12	−1.09414	−0.547072	+	0.837086i	$0.684258\pi$
$504$	0	0
$505$	2.08911e12	1.42939
$506$	−4.48822e11	−0.304367
$507$	0	0
$508$	−2.37779e11	−0.158411
$509$	1.24142e12	0.819767	0.409883	−	0.912138i	$-0.365569\pi$
$509$	1.24142e12	0.819767	0.409883	+	0.912138i	$0.365569\pi$
$510$	0	0
$511$	8.30140e11	0.538589
$512$	6.87195e10	0.0441942
$513$	0	0
$514$	5.42350e10	0.0342725
$515$	2.44481e11	0.153149
$516$	0	0
$517$	6.63719e7	4.08580e−5 0
$518$	7.40651e11	0.451991
$519$	0	0
$520$	4.18020e11	0.250716
$521$	−6.88837e11	−0.409587	−0.204794	−	0.978805i	$-0.565652\pi$
$521$	−6.88837e11	−0.409587	−0.204794	+	0.978805i	$0.565652\pi$
$522$	0	0
$523$	−4.93203e11	−0.288249	−0.144125	−	0.989560i	$-0.546037\pi$
$523$	−4.93203e11	−0.288249	−0.144125	+	0.989560i	$0.546037\pi$
$524$	1.21652e12	0.704899
$525$	0	0
$526$	−1.14967e12	−0.654841
$527$	−2.03713e11	−0.115046
$528$	0	0
$529$	−2.69776e11	−0.149780
$530$	−1.73477e12	−0.954994
$531$	0	0
$532$	1.85309e11	0.100298
$533$	−6.73142e11	−0.361272
$534$	0	0
$535$	−3.36137e11	−0.177388
$536$	2.38527e11	0.124823
$537$	0	0
$538$	−2.22823e11	−0.114667
$539$	1.30677e11	0.0666881
$540$	0	0
$541$	3.70983e12	1.86194	0.930972	−	0.365091i	$-0.118962\pi$
$541$	3.70983e12	1.86194	0.930972	+	0.365091i	$0.118962\pi$
$542$	2.10650e12	1.04849
$543$	0	0
$544$	−3.14007e10	−0.0153725
$545$	−4.26919e12	−2.07282
$546$	0	0
$547$	3.35488e12	1.60226	0.801132	−	0.598487i	$-0.204231\pi$
$547$	3.35488e12	1.60226	0.801132	+	0.598487i	$0.204231\pi$
$548$	1.78378e12	0.844943
$549$	0	0
$550$	2.08537e11	0.0971740
$551$	−1.18123e11	−0.0545950
$552$	0	0
$553$	1.09481e12	0.497825
$554$	−5.81879e11	−0.262445
$555$	0	0
$556$	−1.63598e12	−0.726009
$557$	−4.39669e12	−1.93543	−0.967714	−	0.252050i	$-0.918895\pi$
$557$	−4.39669e12	−1.93543	−0.967714	+	0.252050i	$0.918895\pi$
$558$	0	0
$559$	−2.46608e12	−1.06820
$560$	−2.50190e11	−0.107504
$561$	0	0
$562$	1.87150e12	0.791366
$563$	−2.18836e9	−0.000917974 0	−0.000458987	−	1.00000i	$-0.500146\pi$
$563$	−2.18836e9	−0.000917974 0	−0.000458987		1.00000i	$0.500146\pi$
$564$	0	0
$565$	1.51400e12	0.625041
$566$	2.52215e12	1.03299
$567$	0	0
$568$	2.02135e11	0.0814843
$569$	6.93422e11	0.277327	0.138664	−	0.990340i	$-0.455719\pi$
$569$	6.93422e11	0.277327	0.138664	+	0.990340i	$0.455719\pi$
$570$	0	0
$571$	−1.71495e12	−0.675131	−0.337565	−	0.941302i	$-0.609603\pi$
$571$	−1.71495e12	−0.675131	−0.337565	+	0.941302i	$0.609603\pi$
$572$	−3.72472e11	−0.145483
$573$	0	0
$574$	4.02882e11	0.154908
$575$	−7.11525e11	−0.271447
$576$	0	0
$577$	6.18396e11	0.232261	0.116130	−	0.993234i	$-0.462951\pi$
$577$	6.18396e11	0.232261	0.116130	+	0.993234i	$0.462951\pi$
$578$	−1.88306e12	−0.701760
$579$	0	0
$580$	1.59481e11	0.0585170
$581$	−1.07136e12	−0.390069
$582$	0	0
$583$	1.54575e12	0.554153
$584$	1.41618e12	0.503804
$585$	0	0
$586$	2.15265e12	0.754108
$587$	4.54366e12	1.57955	0.789776	−	0.613395i	$-0.210197\pi$
$587$	4.54366e12	1.57955	0.789776	+	0.613395i	$0.210197\pi$
$588$	0	0
$589$	2.05090e12	0.702144
$590$	2.02596e12	0.688329
$591$	0	0
$592$	1.26352e12	0.422799
$593$	−4.23622e12	−1.40680	−0.703400	−	0.710795i	$-0.748336\pi$
$593$	−4.23622e12	−1.40680	−0.703400	+	0.710795i	$0.748336\pi$
$594$	0	0
$595$	1.14322e11	0.0373940
$596$	2.56292e12	0.832008
$597$	0	0
$598$	1.27087e12	0.406393
$599$	3.37403e12	1.07085	0.535424	−	0.844583i	$-0.320152\pi$
$599$	3.37403e12	1.07085	0.535424	+	0.844583i	$0.320152\pi$
$600$	0	0
$601$	−3.09334e12	−0.967146	−0.483573	−	0.875304i	$-0.660661\pi$
$601$	−3.09334e12	−0.967146	−0.483573	+	0.875304i	$0.660661\pi$
$602$	1.47598e12	0.458031
$603$	0	0
$604$	1.26786e12	0.387619
$605$	2.93213e12	0.889784
$606$	0	0
$607$	4.37823e11	0.130903	0.0654514	−	0.997856i	$-0.479151\pi$
$607$	4.37823e11	0.130903	0.0654514	+	0.997856i	$0.479151\pi$
$608$	3.16129e11	0.0938206
$609$	0	0
$610$	9.46454e11	0.276768
$611$	−1.87937e8	−5.45539e−5 0
$612$	0	0
$613$	−2.16858e12	−0.620303	−0.310152	−	0.950687i	$-0.600380\pi$
$613$	−2.16858e12	−0.620303	−0.310152	+	0.950687i	$0.600380\pi$
$614$	4.50336e12	1.27873
$615$	0	0
$616$	2.22928e11	0.0623810
$617$	−1.32098e12	−0.366956	−0.183478	−	0.983024i	$-0.558736\pi$
$617$	−1.32098e12	−0.366956	−0.183478	+	0.983024i	$0.558736\pi$
$618$	0	0
$619$	3.97904e12	1.08936	0.544678	−	0.838645i	$-0.316652\pi$
$619$	3.97904e12	1.08936	0.544678	+	0.838645i	$0.316652\pi$
$620$	−2.76897e12	−0.752584
$621$	0	0
$622$	−2.95252e12	−0.790927
$623$	1.37314e12	0.365189
$624$	0	0
$625$	−4.60710e12	−1.20772
$626$	−2.58612e11	−0.0673075
$627$	0	0
$628$	−1.10282e12	−0.282934
$629$	−5.77352e11	−0.147066
$630$	0	0
$631$	3.79318e12	0.952515	0.476257	−	0.879306i	$-0.341993\pi$
$631$	3.79318e12	0.952515	0.476257	+	0.879306i	$0.341993\pi$
$632$	1.86770e12	0.465673
$633$	0	0
$634$	1.43342e11	0.0352347
$635$	1.47683e12	0.360453
$636$	0	0
$637$	−3.70020e11	−0.0890425
$638$	−1.42103e11	−0.0339556
$639$	0	0
$640$	−4.26812e11	−0.100560
$641$	−7.02589e12	−1.64377	−0.821884	−	0.569655i	$-0.807077\pi$
$641$	−7.02589e12	−1.64377	−0.821884	+	0.569655i	$0.807077\pi$
$642$	0	0
$643$	−6.14886e12	−1.41855	−0.709276	−	0.704931i	$-0.750978\pi$
$643$	−6.14886e12	−1.41855	−0.709276	+	0.704931i	$0.750978\pi$
$644$	−7.60629e11	−0.174256
$645$	0	0
$646$	−1.44452e11	−0.0326345
$647$	−2.17149e12	−0.487179	−0.243589	−	0.969878i	$-0.578325\pi$
$647$	−2.17149e12	−0.487179	−0.243589	+	0.969878i	$0.578325\pi$
$648$	0	0
$649$	−1.80520e12	−0.399416
$650$	−5.90486e11	−0.129748
$651$	0	0
$652$	−3.18859e12	−0.691011
$653$	5.47584e12	1.17853	0.589266	−	0.807939i	$-0.299417\pi$
$653$	5.47584e12	1.17853	0.589266	+	0.807939i	$0.299417\pi$
$654$	0	0
$655$	−7.55570e12	−1.60394
$656$	6.87299e11	0.144903
$657$	0	0
$658$	1.12482e8	2.33920e−5 0
$659$	4.67490e12	0.965580	0.482790	−	0.875736i	$-0.339624\pi$
$659$	4.67490e12	0.965580	0.482790	+	0.875736i	$0.339624\pi$
$660$	0	0
$661$	3.34777e11	0.0682103	0.0341051	−	0.999418i	$-0.489142\pi$
$661$	3.34777e11	0.0682103	0.0341051	+	0.999418i	$0.489142\pi$
$662$	−6.03375e11	−0.122103
$663$	0	0
$664$	−1.82769e12	−0.364876
$665$	−1.15094e12	−0.228221
$666$	0	0
$667$	4.84855e11	0.0948519
$668$	−4.27733e12	−0.831149
$669$	0	0
$670$	−1.48147e12	−0.284026
$671$	−8.43326e11	−0.160599
$672$	0	0
$673$	−2.47029e12	−0.464173	−0.232087	−	0.972695i	$-0.574555\pi$
$673$	−2.47029e12	−0.464173	−0.232087	+	0.972695i	$0.574555\pi$
$674$	3.24770e12	0.606187
$675$	0	0
$676$	−1.66007e12	−0.305750
$677$	4.48933e12	0.821359	0.410679	−	0.911780i	$-0.365292\pi$
$677$	4.48933e12	0.821359	0.410679	+	0.911780i	$0.365292\pi$
$678$	0	0
$679$	5.86445e11	0.105880
$680$	1.95028e11	0.0349789
$681$	0	0
$682$	2.46725e12	0.436701
$683$	−8.32962e12	−1.46464	−0.732322	−	0.680958i	$-0.761563\pi$
$683$	−8.32962e12	−1.46464	−0.732322	+	0.680958i	$0.761563\pi$
$684$	0	0
$685$	−1.10789e13	−1.92260
$686$	2.21461e11	0.0381802
$687$	0	0
$688$	2.51795e12	0.428449
$689$	−4.37689e12	−0.739910
$690$	0	0
$691$	1.06352e13	1.77458	0.887292	−	0.461209i	$-0.152584\pi$
$691$	1.06352e13	1.77458	0.887292	+	0.461209i	$0.152584\pi$
$692$	−5.32207e12	−0.882274
$693$	0	0
$694$	3.01695e12	0.493686
$695$	1.01610e13	1.65198
$696$	0	0
$697$	−3.14054e11	−0.0504031
$698$	−2.24449e12	−0.357905
$699$	0	0
$700$	3.53412e11	0.0556340
$701$	−5.02284e12	−0.785630	−0.392815	−	0.919618i	$-0.628499\pi$
$701$	−5.02284e12	−0.785630	−0.392815	+	0.919618i	$0.628499\pi$
$702$	0	0
$703$	5.81254e12	0.897567
$704$	3.80306e11	0.0583521
$705$	0	0
$706$	−1.19089e12	−0.180406
$707$	−3.15469e12	−0.474864
$708$	0	0
$709$	−1.03005e13	−1.53091	−0.765456	−	0.643488i	$-0.777487\pi$
$709$	−1.03005e13	−1.53091	−0.765456	+	0.643488i	$0.777487\pi$
$710$	−1.25545e12	−0.185411
$711$	0	0
$712$	2.34251e12	0.341603
$713$	−8.41824e12	−1.21988
$714$	0	0
$715$	2.31340e12	0.331035
$716$	−3.33145e12	−0.473723
$717$	0	0
$718$	3.70010e12	0.519581
$719$	5.03660e11	0.0702841	0.0351421	−	0.999382i	$-0.488812\pi$
$719$	5.03660e11	0.0702841	0.0351421	+	0.999382i	$0.488812\pi$
$720$	0	0
$721$	−3.69182e11	−0.0508782
$722$	−3.70872e12	−0.507933
$723$	0	0
$724$	6.54843e11	0.0885755
$725$	−2.25279e11	−0.0302830
$726$	0	0
$727$	−1.04863e13	−1.39225	−0.696126	−	0.717920i	$-0.745094\pi$
$727$	−1.04863e13	−1.39225	−0.696126	+	0.717920i	$0.745094\pi$
$728$	−6.31237e11	−0.0832916
$729$	0	0
$730$	−8.79582e12	−1.14637
$731$	−1.15055e12	−0.149031
$732$	0	0
$733$	1.29295e13	1.65430	0.827150	−	0.561981i	$-0.189961\pi$
$733$	1.29295e13	1.65430	0.827150	+	0.561981i	$0.189961\pi$
$734$	−4.70812e12	−0.598708
$735$	0	0
$736$	−1.29760e12	−0.163001
$737$	1.32005e12	0.164811
$738$	0	0
$739$	−5.31822e12	−0.655944	−0.327972	−	0.944687i	$-0.606365\pi$
$739$	−5.31822e12	−0.655944	−0.327972	+	0.944687i	$0.606365\pi$
$740$	−7.84764e12	−0.962047
$741$	0	0
$742$	2.61961e12	0.317263
$743$	2.50527e11	0.0301582	0.0150791	−	0.999886i	$-0.495200\pi$
$743$	2.50527e11	0.0301582	0.0150791	+	0.999886i	$0.495200\pi$
$744$	0	0
$745$	−1.59182e13	−1.89317
$746$	4.32057e12	0.510759
$747$	0	0
$748$	−1.73777e11	−0.0202971
$749$	5.07589e11	0.0589310
$750$	0	0
$751$	1.43882e13	1.65054	0.825269	−	0.564739i	$-0.191023\pi$
$751$	1.43882e13	1.65054	0.825269	+	0.564739i	$0.191023\pi$
$752$	1.91889e8	2.18812e−5 0
$753$	0	0
$754$	4.02375e11	0.0453378
$755$	−7.87460e12	−0.881997
$756$	0	0
$757$	−4.39087e12	−0.485981	−0.242991	−	0.970029i	$-0.578128\pi$
$757$	−4.39087e12	−0.485981	−0.242991	+	0.970029i	$0.578128\pi$
$758$	3.01093e12	0.331276
$759$	0	0
$760$	−1.96346e12	−0.213481
$761$	4.31046e12	0.465900	0.232950	−	0.972489i	$-0.425162\pi$
$761$	4.31046e12	0.465900	0.232950	+	0.972489i	$0.425162\pi$
$762$	0	0
$763$	6.44675e12	0.688621
$764$	2.54467e12	0.270216
$765$	0	0
$766$	4.52526e12	0.474913
$767$	5.11156e12	0.533303
$768$	0	0
$769$	−1.27531e13	−1.31506	−0.657530	−	0.753428i	$-0.728399\pi$
$769$	−1.27531e13	−1.31506	−0.657530	+	0.753428i	$0.728399\pi$
$770$	−1.38459e12	−0.141943
$771$	0	0
$772$	−2.74225e12	−0.277862
$773$	−1.26121e13	−1.27051	−0.635256	−	0.772301i	$-0.719106\pi$
$773$	−1.26121e13	−1.27051	−0.635256	+	0.772301i	$0.719106\pi$
$774$	0	0
$775$	3.91138e12	0.389468
$776$	1.00045e12	0.0990415
$777$	0	0
$778$	1.21289e13	1.18690
$779$	3.16177e12	0.307618
$780$	0	0
$781$	1.11865e12	0.107588
$782$	5.92925e11	0.0566982
$783$	0	0
$784$	3.77802e11	0.0357143
$785$	6.84954e12	0.643795
$786$	0	0
$787$	2.63165e12	0.244535	0.122267	−	0.992497i	$-0.460983\pi$
$787$	2.63165e12	0.244535	0.122267	+	0.992497i	$0.460983\pi$
$788$	2.93803e12	0.271449
$789$	0	0
$790$	−1.16002e13	−1.05960
$791$	−2.28624e12	−0.207648
$792$	0	0
$793$	2.38794e12	0.214434
$794$	8.93028e12	0.797394
$795$	0	0
$796$	3.09209e12	0.272987
$797$	−2.11384e10	−0.00185571	−0.000927854	−	1.00000i	$-0.500295\pi$
$797$	−2.11384e10	−0.00185571	−0.000927854		1.00000i	$0.500295\pi$
$798$	0	0
$799$	−8.76819e7	−7.61113e−6 0
$800$	6.02905e11	0.0520408
$801$	0	0
$802$	−1.14174e13	−0.974500
$803$	7.83741e12	0.665200
$804$	0	0
$805$	4.72422e12	0.396505
$806$	−6.98620e12	−0.583087
$807$	0	0
$808$	−5.38176e12	−0.444194
$809$	−6.73901e12	−0.553131	−0.276565	−	0.960995i	$-0.589196\pi$
$809$	−6.73901e12	−0.553131	−0.276565	+	0.960995i	$0.589196\pi$
$810$	0	0
$811$	−1.43465e12	−0.116454	−0.0582269	−	0.998303i	$-0.518545\pi$
$811$	−1.43465e12	−0.116454	−0.0582269	+	0.998303i	$0.518545\pi$
$812$	−2.40826e11	−0.0194402
$813$	0	0
$814$	6.99254e12	0.558245
$815$	1.98041e13	1.57234
$816$	0	0
$817$	1.15833e13	0.909562
$818$	−1.07427e13	−0.838923
$819$	0	0
$820$	−4.26877e12	−0.329717
$821$	1.38553e13	1.06432	0.532160	−	0.846644i	$-0.321381\pi$
$821$	1.38553e13	1.06432	0.532160	+	0.846644i	$0.321381\pi$
$822$	0	0
$823$	3.12155e12	0.237176	0.118588	−	0.992944i	$-0.462163\pi$
$823$	3.12155e12	0.237176	0.118588	+	0.992944i	$0.462163\pi$
$824$	−6.29809e11	−0.0475922
$825$	0	0
$826$	−3.05932e12	−0.228673
$827$	9.34575e12	0.694767	0.347384	−	0.937723i	$-0.387070\pi$
$827$	9.34575e12	0.694767	0.347384	+	0.937723i	$0.387070\pi$
$828$	0	0
$829$	−1.11481e13	−0.819799	−0.409899	−	0.912131i	$-0.634436\pi$
$829$	−1.11481e13	−0.819799	−0.409899	+	0.912131i	$0.634436\pi$
$830$	1.13516e13	0.830247
$831$	0	0
$832$	−1.07686e12	−0.0779122
$833$	−1.72633e11	−0.0124228
$834$	0	0
$835$	2.65662e13	1.89122
$836$	1.74951e12	0.123877
$837$	0	0
$838$	−1.25689e13	−0.880440
$839$	2.44315e13	1.70224	0.851121	−	0.524970i	$-0.175923\pi$
$839$	2.44315e13	1.70224	0.851121	+	0.524970i	$0.175923\pi$
$840$	0	0
$841$	−1.43536e13	−0.989418
$842$	1.34587e13	0.922783
$843$	0	0
$844$	−1.38043e13	−0.936423
$845$	1.03106e13	0.695711
$846$	0	0
$847$	−4.42771e12	−0.295599
$848$	4.46894e12	0.296772
$849$	0	0
$850$	−2.75491e11	−0.0181018
$851$	−2.38585e13	−1.55941
$852$	0	0
$853$	1.67379e13	1.08251	0.541253	−	0.840859i	$-0.317950\pi$
$853$	1.67379e13	1.08251	0.541253	+	0.840859i	$0.317950\pi$
$854$	−1.42920e12	−0.0919462
$855$	0	0
$856$	8.65924e11	0.0551249
$857$	1.79355e13	1.13579	0.567896	−	0.823100i	$-0.307758\pi$
$857$	1.79355e13	1.13579	0.567896	+	0.823100i	$0.307758\pi$
$858$	0	0
$859$	1.78417e13	1.11807	0.559033	−	0.829145i	$-0.311172\pi$
$859$	1.78417e13	1.11807	0.559033	+	0.829145i	$0.311172\pi$
$860$	−1.56388e13	−0.974903
$861$	0	0
$862$	1.58646e13	0.978690
$863$	−1.32676e13	−0.814225	−0.407112	−	0.913378i	$-0.633464\pi$
$863$	−1.32676e13	−0.814225	−0.407112	+	0.913378i	$0.633464\pi$
$864$	0	0
$865$	3.30551e13	2.00755
$866$	−3.35875e12	−0.202930
$867$	0	0
$868$	4.18131e12	0.250020
$869$	1.03362e13	0.614854
$870$	0	0
$871$	−3.73781e12	−0.220057
$872$	1.09979e13	0.644146
$873$	0	0
$874$	−5.96933e12	−0.346038
$875$	5.26121e12	0.303424
$876$	0	0
$877$	2.51717e13	1.43686	0.718430	−	0.695599i	$-0.244861\pi$
$877$	2.51717e13	1.43686	0.718430	+	0.695599i	$0.244861\pi$
$878$	−1.68097e13	−0.954632
$879$	0	0
$880$	−2.36206e12	−0.132776
$881$	−2.12069e13	−1.18601	−0.593003	−	0.805201i	$-0.702058\pi$
$881$	−2.12069e13	−1.18601	−0.593003	+	0.805201i	$0.702058\pi$
$882$	0	0
$883$	2.79753e13	1.54864	0.774322	−	0.632792i	$-0.218091\pi$
$883$	2.79753e13	1.54864	0.774322	+	0.632792i	$0.218091\pi$
$884$	4.92061e11	0.0271009
$885$	0	0
$886$	2.73863e12	0.149307
$887$	7.69155e12	0.417213	0.208606	−	0.978000i	$-0.433107\pi$
$887$	7.69155e12	0.417213	0.208606	+	0.978000i	$0.433107\pi$
$888$	0	0
$889$	−2.23010e12	−0.119748
$890$	−1.45492e13	−0.777291
$891$	0	0
$892$	3.96166e12	0.209525
$893$	8.82745e8	4.64519e−5 0
$894$	0	0
$895$	2.06914e13	1.07792
$896$	6.44514e11	0.0334077
$897$	0	0
$898$	1.42752e13	0.732552
$899$	−2.66533e12	−0.136092
$900$	0	0
$901$	−2.04204e12	−0.103229
$902$	3.80364e12	0.191324
$903$	0	0
$904$	−3.90022e12	−0.194237
$905$	−4.06719e12	−0.201547
$906$	0	0
$907$	−8.03188e12	−0.394080	−0.197040	−	0.980395i	$-0.563133\pi$
$907$	−8.03188e12	−0.394080	−0.197040	+	0.980395i	$0.563133\pi$
$908$	−1.75135e13	−0.855041
$909$	0	0
$910$	3.92057e12	0.189524
$911$	3.70764e13	1.78347	0.891733	−	0.452561i	$-0.149490\pi$
$911$	3.70764e13	1.78347	0.891733	+	0.452561i	$0.149490\pi$
$912$	0	0
$913$	−1.01147e13	−0.481766
$914$	3.09713e12	0.146792
$915$	0	0
$916$	1.55396e12	0.0729309
$917$	1.14096e13	0.532853
$918$	0	0
$919$	−2.32387e13	−1.07471	−0.537357	−	0.843355i	$-0.680577\pi$
$919$	−2.32387e13	−1.07471	−0.537357	+	0.843355i	$0.680577\pi$
$920$	8.05931e12	0.370897
$921$	0	0
$922$	−1.92423e13	−0.876936
$923$	−3.16754e12	−0.143653
$924$	0	0
$925$	1.10854e13	0.497867
$926$	−2.05632e13	−0.919055
$927$	0	0
$928$	−4.10838e11	−0.0181847
$929$	1.47386e13	0.649211	0.324605	−	0.945850i	$-0.394769\pi$
$929$	1.47386e13	0.649211	0.324605	+	0.945850i	$0.394769\pi$
$930$	0	0
$931$	1.73800e12	0.0758185
$932$	5.42366e12	0.235462
$933$	0	0
$934$	2.20191e13	0.946756
$935$	1.07932e12	0.0461846
$936$	0	0
$937$	−3.86396e13	−1.63759	−0.818795	−	0.574087i	$-0.805357\pi$
$937$	−3.86396e13	−1.63759	−0.818795	+	0.574087i	$0.805357\pi$
$938$	2.23712e12	0.0943575
$939$	0	0
$940$	−1.19181e9	−4.97889e−5 0
$941$	−4.12643e13	−1.71562	−0.857810	−	0.513967i	$-0.828175\pi$
$941$	−4.12643e13	−1.71562	−0.857810	+	0.513967i	$0.828175\pi$
$942$	0	0
$943$	−1.29780e13	−0.534447
$944$	−5.21907e12	−0.213904
$945$	0	0
$946$	1.39348e13	0.565705
$947$	−3.25255e13	−1.31416	−0.657081	−	0.753820i	$-0.728209\pi$
$947$	−3.25255e13	−1.31416	−0.657081	+	0.753820i	$0.728209\pi$
$948$	0	0
$949$	−2.21922e13	−0.888181
$950$	2.77353e12	0.110478
$951$	0	0
$952$	−2.94504e11	−0.0116205
$953$	−2.30788e13	−0.906349	−0.453174	−	0.891422i	$-0.649709\pi$
$953$	−2.30788e13	−0.906349	−0.453174	+	0.891422i	$0.649709\pi$
$954$	0	0
$955$	−1.58048e13	−0.614855
$956$	7.85071e12	0.303982
$957$	0	0
$958$	1.84958e13	0.709461
$959$	1.67299e13	0.638717
$960$	0	0
$961$	1.98369e13	0.750273
$962$	−1.97999e13	−0.745374
$963$	0	0
$964$	−1.50292e13	−0.560516
$965$	1.70319e13	0.632254
$966$	0	0
$967$	−1.53358e13	−0.564011	−0.282005	−	0.959413i	$-0.591000\pi$
$967$	−1.53358e13	−0.564011	−0.282005	+	0.959413i	$0.591000\pi$
$968$	−7.55347e12	−0.276508
$969$	0	0
$970$	−6.21373e12	−0.225361
$971$	−1.98971e13	−0.718297	−0.359149	−	0.933280i	$-0.616933\pi$
$971$	−1.98971e13	−0.718297	−0.359149	+	0.933280i	$0.616933\pi$
$972$	0	0
$973$	−1.53437e13	−0.548811
$974$	−1.07661e13	−0.383305
$975$	0	0
$976$	−2.43816e12	−0.0860078
$977$	−1.16887e13	−0.410432	−0.205216	−	0.978717i	$-0.565790\pi$
$977$	−1.16887e13	−0.410432	−0.205216	+	0.978717i	$0.565790\pi$
$978$	0	0
$979$	1.29639e13	0.451037
$980$	−2.34650e12	−0.0812651
$981$	0	0
$982$	1.26502e13	0.434105
$983$	3.68758e13	1.25965	0.629826	−	0.776736i	$-0.283126\pi$
$983$	3.68758e13	1.25965	0.629826	+	0.776736i	$0.283126\pi$
$984$	0	0
$985$	−1.82479e13	−0.617662
$986$	1.87728e11	0.00632534
$987$	0	0
$988$	−4.95387e12	−0.165401
$989$	−4.75454e13	−1.58025
$990$	0	0
$991$	−4.10347e13	−1.35151	−0.675756	−	0.737126i	$-0.736183\pi$
$991$	−4.10347e13	−1.35151	−0.675756	+	0.737126i	$0.736183\pi$
$992$	7.13314e12	0.233872
$993$	0	0
$994$	1.89581e12	0.0615964
$995$	−1.92047e13	−0.621162
$996$	0	0
$997$	−4.99823e11	−0.0160209	−0.00801047	−	0.999968i	$-0.502550\pi$
$997$	−4.99823e11	−0.0160209	−0.00801047	+	0.999968i	$0.502550\pi$
$998$	4.37014e12	0.139447
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.10.a.d.1.1		1
3.2	odd	2		42.10.a.b.1.1	✓	1
12.11	even	2		336.10.a.g.1.1		1
21.20	even	2		294.10.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.10.a.b.1.1	✓	1	3.2	odd	2
126.10.a.d.1.1		1	1.1	even	1	trivial
294.10.a.e.1.1		1	21.20	even	2
336.10.a.g.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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