LMFDB - Embedded newform 294.8.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,8,Mod(1,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("294.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	294.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:	$91.8411974923$
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	$\chi$	$=$	294.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -470.000 q^{5} -216.000 q^{6} +512.000 q^{8} +729.000 q^{9} -3760.00 q^{10} -7268.00 q^{11} -1728.00 q^{12} -11382.0 q^{13} +12690.0 q^{15} +4096.00 q^{16} -20842.0 q^{17} +5832.00 q^{18} +13684.0 q^{19} -30080.0 q^{20} -58144.0 q^{22} -46736.0 q^{23} -13824.0 q^{24} +142775. q^{25} -91056.0 q^{26} -19683.0 q^{27} +2774.00 q^{29} +101520. q^{30} +11232.0 q^{31} +32768.0 q^{32} +196236. q^{33} -166736. q^{34} +46656.0 q^{36} -239698. q^{37} +109472. q^{38} +307314. q^{39} -240640. q^{40} -52962.0 q^{41} -874732. q^{43} -465152. q^{44} -342630. q^{45} -373888. q^{46} +451296. q^{47} -110592. q^{48} +1.14220e6 q^{50} +562734. q^{51} -728448. q^{52} -1.94354e6 q^{53} -157464. q^{54} +3.41596e6 q^{55} -369468. q^{57} +22192.0 q^{58} -1.56391e6 q^{59} +812160. q^{60} +3.18369e6 q^{61} +89856.0 q^{62} +262144. q^{64} +5.34954e6 q^{65} +1.56989e6 q^{66} +2.12868e6 q^{67} -1.33389e6 q^{68} +1.26187e6 q^{69} +5.69454e6 q^{71} +373248. q^{72} +2.25388e6 q^{73} -1.91758e6 q^{74} -3.85492e6 q^{75} +875776. q^{76} +2.45851e6 q^{78} +132912. q^{79} -1.92512e6 q^{80} +531441. q^{81} -423696. q^{82} -6.95052e6 q^{83} +9.79574e6 q^{85} -6.99786e6 q^{86} -74898.0 q^{87} -3.72122e6 q^{88} +1.06265e7 q^{89} -2.74104e6 q^{90} -2.99110e6 q^{92} -303264. q^{93} +3.61037e6 q^{94} -6.43148e6 q^{95} -884736. q^{96} -2.40709e6 q^{97} -5.29837e6 q^{99} +O(q^{100})\)

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	−470.000	−1.68152	−0.840762	−	0.541406i	$-0.817892\pi$
$5$	−470.000	−1.68152	−0.840762	+	0.541406i	$0.817892\pi$
$6$	−216.000	−0.408248
$7$	0	0
$7$	0	0
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−3760.00	−1.18902
$11$	−7268.00	−1.64642	−0.823210	−	0.567737i	$-0.807819\pi$
$11$	−7268.00	−1.64642	−0.823210	+	0.567737i	$0.807819\pi$
$12$	−1728.00	−0.288675
$13$	−11382.0	−1.43687	−0.718434	−	0.695595i	$-0.755141\pi$
$13$	−11382.0	−1.43687	−0.718434	+	0.695595i	$0.755141\pi$
$14$	0	0
$15$	12690.0	0.970828
$16$	4096.00	0.250000
$17$	−20842.0	−1.02889	−0.514444	−	0.857524i	$-0.672002\pi$
$17$	−20842.0	−1.02889	−0.514444	+	0.857524i	$0.672002\pi$
$18$	5832.00	0.235702
$19$	13684.0	0.457694	0.228847	−	0.973462i	$-0.426504\pi$
$19$	13684.0	0.457694	0.228847	+	0.973462i	$0.426504\pi$
$20$	−30080.0	−0.840762
$21$	0	0
$22$	−58144.0	−1.16419
$23$	−46736.0	−0.800948	−0.400474	−	0.916308i	$-0.631154\pi$
$23$	−46736.0	−0.800948	−0.400474	+	0.916308i	$0.631154\pi$
$24$	−13824.0	−0.204124
$25$	142775.	1.82752
$26$	−91056.0	−1.01602
$27$	−19683.0	−0.192450
$28$	0	0
$29$	2774.00	0.0211210	0.0105605	−	0.999944i	$-0.496638\pi$
$29$	2774.00	0.0211210	0.0105605	+	0.999944i	$0.496638\pi$
$30$	101520.	0.686479
$31$	11232.0	0.0677160	0.0338580	−	0.999427i	$-0.489221\pi$
$31$	11232.0	0.0677160	0.0338580	+	0.999427i	$0.489221\pi$
$32$	32768.0	0.176777
$33$	196236.	0.950561
$34$	−166736.	−0.727534
$35$	0	0
$36$	46656.0	0.166667
$37$	−239698.	−0.777962	−0.388981	−	0.921246i	$-0.627173\pi$
$37$	−239698.	−0.777962	−0.388981	+	0.921246i	$0.627173\pi$
$38$	109472.	0.323639
$39$	307314.	0.829576
$40$	−240640.	−0.594508
$41$	−52962.0	−0.120011	−0.0600055	−	0.998198i	$-0.519112\pi$
$41$	−52962.0	−0.120011	−0.0600055	+	0.998198i	$0.519112\pi$
$42$	0	0
$43$	−874732.	−1.67778	−0.838891	−	0.544300i	$-0.816795\pi$
$43$	−874732.	−1.67778	−0.838891	+	0.544300i	$0.816795\pi$
$44$	−465152.	−0.823210
$45$	−342630.	−0.560508
$46$	−373888.	−0.566355
$47$	451296.	0.634043	0.317022	−	0.948418i	$-0.397317\pi$
$47$	451296.	0.634043	0.317022	+	0.948418i	$0.397317\pi$
$48$	−110592.	−0.144338
$49$	0	0
$50$	1.14220e6	1.29225
$51$	562734.	0.594029
$52$	−728448.	−0.718434
$53$	−1.94354e6	−1.79319	−0.896597	−	0.442846i	$-0.853969\pi$
$53$	−1.94354e6	−1.79319	−0.896597	+	0.442846i	$0.853969\pi$
$54$	−157464.	−0.136083
$55$	3.41596e6	2.76849
$56$	0	0
$57$	−369468.	−0.264250
$58$	22192.0	0.0149348
$59$	−1.56391e6	−0.991355	−0.495677	−	0.868507i	$-0.665080\pi$
$59$	−1.56391e6	−0.991355	−0.495677	+	0.868507i	$0.665080\pi$
$60$	812160.	0.485414
$61$	3.18369e6	1.79588	0.897938	−	0.440122i	$-0.145065\pi$
$61$	3.18369e6	1.79588	0.897938	+	0.440122i	$0.145065\pi$
$62$	89856.0	0.0478824
$63$	0	0
$64$	262144.	0.125000
$65$	5.34954e6	2.41613
$66$	1.56989e6	0.672148
$67$	2.12868e6	0.864668	0.432334	−	0.901714i	$-0.357690\pi$
$67$	2.12868e6	0.864668	0.432334	+	0.901714i	$0.357690\pi$
$68$	−1.33389e6	−0.514444
$69$	1.26187e6	0.462427
$70$	0	0
$71$	5.69454e6	1.88823	0.944115	−	0.329617i	$-0.106920\pi$
$71$	5.69454e6	1.88823	0.944115	+	0.329617i	$0.106920\pi$
$72$	373248.	0.117851
$73$	2.25388e6	0.678110	0.339055	−	0.940766i	$-0.389893\pi$
$73$	2.25388e6	0.678110	0.339055	+	0.940766i	$0.389893\pi$
$74$	−1.91758e6	−0.550102
$75$	−3.85492e6	−1.05512
$76$	875776.	0.228847
$77$	0	0
$78$	2.45851e6	0.586599
$79$	132912.	0.0303298	0.0151649	−	0.999885i	$-0.495173\pi$
$79$	132912.	0.0303298	0.0151649	+	0.999885i	$0.495173\pi$
$80$	−1.92512e6	−0.420381
$81$	531441.	0.111111
$82$	−423696.	−0.0848606
$83$	−6.95052e6	−1.33427	−0.667136	−	0.744936i	$-0.732480\pi$
$83$	−6.95052e6	−1.33427	−0.667136	+	0.744936i	$0.732480\pi$
$84$	0	0
$85$	9.79574e6	1.73010
$86$	−6.99786e6	−1.18637
$87$	−74898.0	−0.0121942
$88$	−3.72122e6	−0.582097
$89$	1.06265e7	1.59781	0.798906	−	0.601456i	$-0.205412\pi$
$89$	1.06265e7	1.59781	0.798906	+	0.601456i	$0.205412\pi$
$90$	−2.74104e6	−0.396339
$91$	0	0
$92$	−2.99110e6	−0.400474
$93$	−303264.	−0.0390958
$94$	3.61037e6	0.448336
$95$	−6.43148e6	−0.769623
$96$	−884736.	−0.102062
$97$	−2.40709e6	−0.267788	−0.133894	−	0.990996i	$-0.542748\pi$
$97$	−2.40709e6	−0.267788	−0.133894	+	0.990996i	$0.542748\pi$
$98$	0	0
$99$	−5.29837e6	−0.548807
$100$	9.13760e6	0.913760
$101$	−7.77575e6	−0.750961	−0.375481	−	0.926830i	$-0.622522\pi$
$101$	−7.77575e6	−0.750961	−0.375481	+	0.926830i	$0.622522\pi$
$102$	4.50187e6	0.420042
$103$	1.32340e7	1.19333	0.596666	−	0.802489i	$-0.296492\pi$
$103$	1.32340e7	1.19333	0.596666	+	0.802489i	$0.296492\pi$
$104$	−5.82758e6	−0.508010
$105$	0	0
$106$	−1.55483e7	−1.26798
$107$	−7.20944e6	−0.568930	−0.284465	−	0.958686i	$-0.591816\pi$
$107$	−7.20944e6	−0.568930	−0.284465	+	0.958686i	$0.591816\pi$
$108$	−1.25971e6	−0.0962250
$109$	3.89037e6	0.287739	0.143869	−	0.989597i	$-0.454045\pi$
$109$	3.89037e6	0.287739	0.143869	+	0.989597i	$0.454045\pi$
$110$	2.73277e7	1.95762
$111$	6.47185e6	0.449157
$112$	0	0
$113$	1.46647e7	0.956091	0.478046	−	0.878335i	$-0.341345\pi$
$113$	1.46647e7	0.956091	0.478046	+	0.878335i	$0.341345\pi$
$114$	−2.95574e6	−0.186853
$115$	2.19659e7	1.34681
$116$	177536.	0.0105605
$117$	−8.29748e6	−0.478956
$118$	−1.25113e7	−0.700994
$119$	0	0
$120$	6.49728e6	0.343239
$121$	3.33367e7	1.71070
$122$	2.54695e7	1.26988
$123$	1.42997e6	0.0692884
$124$	718848.	0.0338580
$125$	−3.03855e7	−1.39149
$126$	0	0
$127$	3.87331e6	0.167791	0.0838957	−	0.996475i	$-0.473264\pi$
$127$	3.87331e6	0.167791	0.0838957	+	0.996475i	$0.473264\pi$
$128$	2.09715e6	0.0883883
$129$	2.36178e7	0.968668
$130$	4.27963e7	1.70846
$131$	−2.27476e7	−0.884070	−0.442035	−	0.896998i	$-0.645743\pi$
$131$	−2.27476e7	−0.884070	−0.442035	+	0.896998i	$0.645743\pi$
$132$	1.25591e7	0.475280
$133$	0	0
$134$	1.70295e7	0.611413
$135$	9.25101e6	0.323609
$136$	−1.06711e7	−0.363767
$137$	−4.22382e7	−1.40341	−0.701703	−	0.712469i	$-0.747577\pi$
$137$	−4.22382e7	−1.40341	−0.701703	+	0.712469i	$0.747577\pi$
$138$	1.00950e7	0.326985
$139$	2.02320e6	0.0638978	0.0319489	−	0.999490i	$-0.489829\pi$
$139$	2.02320e6	0.0638978	0.0319489	+	0.999490i	$0.489829\pi$
$140$	0	0
$141$	−1.21850e7	−0.366065
$142$	4.55564e7	1.33518
$143$	8.27244e7	2.36569
$144$	2.98598e6	0.0833333
$145$	−1.30378e6	−0.0355154
$146$	1.80310e7	0.479496
$147$	0	0
$148$	−1.53407e7	−0.388981
$149$	6.33017e7	1.56770	0.783851	−	0.620949i	$-0.213253\pi$
$149$	6.33017e7	1.56770	0.783851	+	0.620949i	$0.213253\pi$
$150$	−3.08394e7	−0.746082
$151$	−5.39626e7	−1.27548	−0.637739	−	0.770252i	$-0.720130\pi$
$151$	−5.39626e7	−1.27548	−0.637739	+	0.770252i	$0.720130\pi$
$152$	7.00621e6	0.161819
$153$	−1.51938e7	−0.342963
$154$	0	0
$155$	−5.27904e6	−0.113866
$156$	1.96681e7	0.414788
$157$	−5.83367e7	−1.20308	−0.601538	−	0.798844i	$-0.705445\pi$
$157$	−5.83367e7	−1.20308	−0.601538	+	0.798844i	$0.705445\pi$
$158$	1.06330e6	0.0214464
$159$	5.24755e7	1.03530
$160$	−1.54010e7	−0.297254
$161$	0	0
$162$	4.25153e6	0.0785674
$163$	6.05845e7	1.09573	0.547867	−	0.836566i	$-0.315440\pi$
$163$	6.05845e7	1.09573	0.547867	+	0.836566i	$0.315440\pi$
$164$	−3.38957e6	−0.0600055
$165$	−9.22309e7	−1.59839
$166$	−5.56042e7	−0.943472
$167$	−5.94193e7	−0.987234	−0.493617	−	0.869680i	$-0.664325\pi$
$167$	−5.94193e7	−0.987234	−0.493617	+	0.869680i	$0.664325\pi$
$168$	0	0
$169$	6.68014e7	1.06459
$170$	7.83659e7	1.22336
$171$	9.97564e6	0.152565
$172$	−5.59828e7	−0.838891
$173$	−5.58566e7	−0.820188	−0.410094	−	0.912043i	$-0.634504\pi$
$173$	−5.58566e7	−0.820188	−0.410094	+	0.912043i	$0.634504\pi$
$174$	−599184.	−0.00862259
$175$	0	0
$176$	−2.97697e7	−0.411605
$177$	4.22255e7	0.572359
$178$	8.50121e7	1.12982
$179$	8.02252e7	1.04550	0.522752	−	0.852485i	$-0.324906\pi$
$179$	8.02252e7	1.04550	0.522752	+	0.852485i	$0.324906\pi$
$180$	−2.19283e7	−0.280254
$181$	7.18632e7	0.900807	0.450403	−	0.892825i	$-0.351280\pi$
$181$	7.18632e7	0.900807	0.450403	+	0.892825i	$0.351280\pi$
$182$	0	0
$183$	−8.59596e7	−1.03685
$184$	−2.39288e7	−0.283178
$185$	1.12658e8	1.30816
$186$	−2.42611e6	−0.0276449
$187$	1.51480e8	1.69398
$188$	2.88829e7	0.317022
$189$	0	0
$190$	−5.14518e7	−0.544206
$191$	−1.67435e7	−0.173872	−0.0869361	−	0.996214i	$-0.527708\pi$
$191$	−1.67435e7	−0.173872	−0.0869361	+	0.996214i	$0.527708\pi$
$192$	−7.07789e6	−0.0721688
$193$	−1.34702e8	−1.34873	−0.674364	−	0.738399i	$-0.735582\pi$
$193$	−1.34702e8	−1.34873	−0.674364	+	0.738399i	$0.735582\pi$
$194$	−1.92567e7	−0.189355
$195$	−1.44438e8	−1.39495
$196$	0	0
$197$	−5.89413e7	−0.549273	−0.274636	−	0.961548i	$-0.588557\pi$
$197$	−5.89413e7	−0.549273	−0.274636	+	0.961548i	$0.588557\pi$
$198$	−4.23870e7	−0.388065
$199$	−1.31373e8	−1.18174	−0.590869	−	0.806768i	$-0.701215\pi$
$199$	−1.31373e8	−1.18174	−0.590869	+	0.806768i	$0.701215\pi$
$200$	7.31008e7	0.646126
$201$	−5.74745e7	−0.499216
$202$	−6.22060e7	−0.531010
$203$	0	0
$204$	3.60150e7	0.297014
$205$	2.48921e7	0.201801
$206$	1.05872e8	0.843814
$207$	−3.40705e7	−0.266983
$208$	−4.66207e7	−0.359217
$209$	−9.94553e7	−0.753557
$210$	0	0
$211$	−2.96600e7	−0.217362	−0.108681	−	0.994077i	$-0.534663\pi$
$211$	−2.96600e7	−0.217362	−0.108681	+	0.994077i	$0.534663\pi$
$212$	−1.24386e8	−0.896597
$213$	−1.53753e8	−1.09017
$214$	−5.76756e7	−0.402294
$215$	4.11124e8	2.82123
$216$	−1.00777e7	−0.0680414
$217$	0	0
$218$	3.11230e7	0.203462
$219$	−6.08547e7	−0.391507
$220$	2.18621e8	1.38425
$221$	2.37224e8	1.47838
$222$	5.17748e7	0.317602
$223$	−3.16037e8	−1.90841	−0.954203	−	0.299161i	$-0.903293\pi$
$223$	−3.16037e8	−1.90841	−0.954203	+	0.299161i	$0.903293\pi$
$224$	0	0
$225$	1.04083e8	0.609173
$226$	1.17318e8	0.676059
$227$	−4.76056e7	−0.270127	−0.135063	−	0.990837i	$-0.543124\pi$
$227$	−4.76056e7	−0.270127	−0.135063	+	0.990837i	$0.543124\pi$
$228$	−2.36460e7	−0.132125
$229$	1.04289e8	0.573870	0.286935	−	0.957950i	$-0.407364\pi$
$229$	1.04289e8	0.573870	0.286935	+	0.957950i	$0.407364\pi$
$230$	1.75727e8	0.952340
$231$	0	0
$232$	1.42029e6	0.00746738
$233$	2.35059e8	1.21739	0.608697	−	0.793403i	$-0.291693\pi$
$233$	2.35059e8	1.21739	0.608697	+	0.793403i	$0.291693\pi$
$234$	−6.63798e7	−0.338673
$235$	−2.12109e8	−1.06616
$236$	−1.00090e8	−0.495677
$237$	−3.58862e6	−0.0175109
$238$	0	0
$239$	3.74362e7	0.177378	0.0886888	−	0.996059i	$-0.471732\pi$
$239$	3.74362e7	0.177378	0.0886888	+	0.996059i	$0.471732\pi$
$240$	5.19782e7	0.242707
$241$	−9.87667e6	−0.0454518	−0.0227259	−	0.999742i	$-0.507235\pi$
$241$	−9.87667e6	−0.0454518	−0.0227259	+	0.999742i	$0.507235\pi$
$242$	2.66693e8	1.20965
$243$	−1.43489e7	−0.0641500
$244$	2.03756e8	0.897938
$245$	0	0
$246$	1.14398e7	0.0489943
$247$	−1.55751e8	−0.657646
$248$	5.75078e6	0.0239412
$249$	1.87664e8	0.770342
$250$	−2.43084e8	−0.983935
$251$	−1.32438e8	−0.528634	−0.264317	−	0.964436i	$-0.585147\pi$
$251$	−1.32438e8	−0.528634	−0.264317	+	0.964436i	$0.585147\pi$
$252$	0	0
$253$	3.39677e8	1.31870
$254$	3.09865e7	0.118646
$255$	−2.64485e8	−0.998873
$256$	1.67772e7	0.0625000
$257$	−3.32667e8	−1.22249	−0.611243	−	0.791443i	$-0.709330\pi$
$257$	−3.32667e8	−1.22249	−0.611243	+	0.791443i	$0.709330\pi$
$258$	1.88942e8	0.684951
$259$	0	0
$260$	3.42371e8	1.20806
$261$	2.02225e6	0.00704032
$262$	−1.81981e8	−0.625132
$263$	−9.51228e7	−0.322433	−0.161217	−	0.986919i	$-0.551542\pi$
$263$	−9.51228e7	−0.322433	−0.161217	+	0.986919i	$0.551542\pi$
$264$	1.00473e8	0.336074
$265$	9.13463e8	3.01530
$266$	0	0
$267$	−2.86916e8	−0.922497
$268$	1.36236e8	0.432334
$269$	2.68855e8	0.842141	0.421071	−	0.907028i	$-0.361654\pi$
$269$	2.68855e8	0.842141	0.421071	+	0.907028i	$0.361654\pi$
$270$	7.40081e7	0.228826
$271$	−4.52884e8	−1.38227	−0.691137	−	0.722724i	$-0.742890\pi$
$271$	−4.52884e8	−1.38227	−0.691137	+	0.722724i	$0.742890\pi$
$272$	−8.53688e7	−0.257222
$273$	0	0
$274$	−3.37906e8	−0.992358
$275$	−1.03769e9	−3.00886
$276$	8.07598e7	0.231214
$277$	−1.64709e7	−0.0465628	−0.0232814	−	0.999729i	$-0.507411\pi$
$277$	−1.64709e7	−0.0465628	−0.0232814	+	0.999729i	$0.507411\pi$
$278$	1.61856e7	0.0451826
$279$	8.18813e6	0.0225720
$280$	0	0
$281$	5.08113e8	1.36612	0.683059	−	0.730363i	$-0.260649\pi$
$281$	5.08113e8	1.36612	0.683059	+	0.730363i	$0.260649\pi$
$282$	−9.74799e7	−0.258847
$283$	2.57496e8	0.675333	0.337666	−	0.941266i	$-0.390362\pi$
$283$	2.57496e8	0.675333	0.337666	+	0.941266i	$0.390362\pi$
$284$	3.64451e8	0.944115
$285$	1.73650e8	0.444342
$286$	6.61795e8	1.67279
$287$	0	0
$288$	2.38879e7	0.0589256
$289$	2.40503e7	0.0586108
$290$	−1.04302e7	−0.0251132
$291$	6.49914e7	0.154608
$292$	1.44248e8	0.339055
$293$	−7.33847e7	−0.170439	−0.0852195	−	0.996362i	$-0.527159\pi$
$293$	−7.33847e7	−0.170439	−0.0852195	+	0.996362i	$0.527159\pi$
$294$	0	0
$295$	7.35037e8	1.66699
$296$	−1.22725e8	−0.275051
$297$	1.43056e8	0.316854
$298$	5.06414e8	1.10853
$299$	5.31949e8	1.15086
$300$	−2.46715e8	−0.527560
$301$	0	0
$302$	−4.31701e8	−0.901900
$303$	2.09945e8	0.433568
$304$	5.60497e7	0.114424
$305$	−1.49633e9	−3.01981
$306$	−1.21551e8	−0.242511
$307$	9.31624e7	0.183762	0.0918811	−	0.995770i	$-0.470712\pi$
$307$	9.31624e7	0.183762	0.0918811	+	0.995770i	$0.470712\pi$
$308$	0	0
$309$	−3.57319e8	−0.688971
$310$	−4.22323e7	−0.0805154
$311$	−6.10934e8	−1.15168	−0.575842	−	0.817561i	$-0.695326\pi$
$311$	−6.10934e8	−1.15168	−0.575842	+	0.817561i	$0.695326\pi$
$312$	1.57345e8	0.293299
$313$	2.12231e8	0.391204	0.195602	−	0.980683i	$-0.437334\pi$
$313$	2.12231e8	0.391204	0.195602	+	0.980683i	$0.437334\pi$
$314$	−4.66694e8	−0.850703
$315$	0	0
$316$	8.50637e6	0.0151649
$317$	−1.10394e9	−1.94644	−0.973218	−	0.229885i	$-0.926165\pi$
$317$	−1.10394e9	−1.94644	−0.973218	+	0.229885i	$0.926165\pi$
$318$	4.19804e8	0.732069
$319$	−2.01614e7	−0.0347739
$320$	−1.23208e8	−0.210190
$321$	1.94655e8	0.328472
$322$	0	0
$323$	−2.85202e8	−0.470916
$324$	3.40122e7	0.0555556
$325$	−1.62507e9	−2.62590
$326$	4.84676e8	0.774800
$327$	−1.05040e8	−0.166126
$328$	−2.71165e7	−0.0424303
$329$	0	0
$330$	−7.37847e8	−1.13023
$331$	−6.70588e8	−1.01638	−0.508192	−	0.861244i	$-0.669686\pi$
$331$	−6.70588e8	−1.01638	−0.508192	+	0.861244i	$0.669686\pi$
$332$	−4.44834e8	−0.667136
$333$	−1.74740e8	−0.259321
$334$	−4.75354e8	−0.698080
$335$	−1.00048e9	−1.45396
$336$	0	0
$337$	−6.84122e8	−0.973709	−0.486854	−	0.873483i	$-0.661856\pi$
$337$	−6.84122e8	−0.973709	−0.486854	+	0.873483i	$0.661856\pi$
$338$	5.34411e8	0.752778
$339$	−3.95947e8	−0.551999
$340$	6.26927e8	0.865050
$341$	−8.16342e7	−0.111489
$342$	7.98051e7	0.107880
$343$	0	0
$344$	−4.47863e8	−0.593185
$345$	−5.93080e8	−0.777582
$346$	−4.46853e8	−0.579960
$347$	4.37553e8	0.562183	0.281091	−	0.959681i	$-0.409304\pi$
$347$	4.37553e8	0.562183	0.281091	+	0.959681i	$0.409304\pi$
$348$	−4.79347e6	−0.00609709
$349$	−9.39833e6	−0.0118348	−0.00591741	−	0.999982i	$-0.501884\pi$
$349$	−9.39833e6	−0.0118348	−0.00591741	+	0.999982i	$0.501884\pi$
$350$	0	0
$351$	2.24032e8	0.276525
$352$	−2.38158e8	−0.291049
$353$	−1.41472e9	−1.71182	−0.855911	−	0.517123i	$-0.827003\pi$
$353$	−1.41472e9	−1.71182	−0.855911	+	0.517123i	$0.827003\pi$
$354$	3.37804e8	0.404719
$355$	−2.67644e9	−3.17510
$356$	6.80097e8	0.798906
$357$	0	0
$358$	6.41802e8	0.739282
$359$	−3.07583e8	−0.350858	−0.175429	−	0.984492i	$-0.556131\pi$
$359$	−3.07583e8	−0.350858	−0.175429	+	0.984492i	$0.556131\pi$
$360$	−1.75427e8	−0.198169
$361$	−7.06620e8	−0.790516
$362$	5.74906e8	0.636967
$363$	−9.00090e8	−0.987672
$364$	0	0
$365$	−1.05932e9	−1.14026
$366$	−6.87677e8	−0.733163
$367$	1.23127e9	1.30024	0.650119	−	0.759832i	$-0.274719\pi$
$367$	1.23127e9	1.30024	0.650119	+	0.759832i	$0.274719\pi$
$368$	−1.91431e8	−0.200237
$369$	−3.86093e7	−0.0400037
$370$	9.01264e8	0.925010
$371$	0	0
$372$	−1.94089e7	−0.0195479
$373$	1.83592e9	1.83177	0.915887	−	0.401435i	$-0.131488\pi$
$373$	1.83592e9	1.83177	0.915887	+	0.401435i	$0.131488\pi$
$374$	1.21184e9	1.19783
$375$	8.20408e8	0.803379
$376$	2.31064e8	0.224168
$377$	−3.15737e7	−0.0303480
$378$	0	0
$379$	−6.48934e8	−0.612298	−0.306149	−	0.951984i	$-0.599041\pi$
$379$	−6.48934e8	−0.612298	−0.306149	+	0.951984i	$0.599041\pi$
$380$	−4.11615e8	−0.384812
$381$	−1.04579e8	−0.0968744
$382$	−1.33948e8	−0.122946
$383$	6.80321e8	0.618754	0.309377	−	0.950939i	$-0.399880\pi$
$383$	6.80321e8	0.618754	0.309377	+	0.950939i	$0.399880\pi$
$384$	−5.66231e7	−0.0510310
$385$	0	0
$386$	−1.07762e9	−0.953694
$387$	−6.37680e8	−0.559261
$388$	−1.54054e8	−0.133894
$389$	2.73705e8	0.235754	0.117877	−	0.993028i	$-0.462391\pi$
$389$	2.73705e8	0.235754	0.117877	+	0.993028i	$0.462391\pi$
$390$	−1.15550e9	−0.986380
$391$	9.74072e8	0.824086
$392$	0	0
$393$	6.14186e8	0.510418
$394$	−4.71531e8	−0.388394
$395$	−6.24686e7	−0.0510002
$396$	−3.39096e8	−0.274403
$397$	1.21034e9	0.970825	0.485413	−	0.874285i	$-0.338669\pi$
$397$	1.21034e9	0.970825	0.485413	+	0.874285i	$0.338669\pi$
$398$	−1.05099e9	−0.835615
$399$	0	0
$400$	5.84806e8	0.456880
$401$	−5.97102e8	−0.462427	−0.231214	−	0.972903i	$-0.574270\pi$
$401$	−5.97102e8	−0.462427	−0.231214	+	0.972903i	$0.574270\pi$
$402$	−4.59796e8	−0.352999
$403$	−1.27843e8	−0.0972989
$404$	−4.97648e8	−0.375481
$405$	−2.49777e8	−0.186836
$406$	0	0
$407$	1.74213e9	1.28085
$408$	2.88120e8	0.210021
$409$	1.19767e9	0.865578	0.432789	−	0.901495i	$-0.357530\pi$
$409$	1.19767e9	0.865578	0.432789	+	0.901495i	$0.357530\pi$
$410$	1.99137e8	0.142695
$411$	1.14043e9	0.810257
$412$	8.46978e8	0.596666
$413$	0	0
$414$	−2.72564e8	−0.188785
$415$	3.26675e9	2.24361
$416$	−3.72965e8	−0.254005
$417$	−5.46263e7	−0.0368914
$418$	−7.95642e8	−0.532845
$419$	1.68680e9	1.12025	0.560125	−	0.828408i	$-0.310753\pi$
$419$	1.68680e9	1.12025	0.560125	+	0.828408i	$0.310753\pi$
$420$	0	0
$421$	−9.60356e8	−0.627256	−0.313628	−	0.949546i	$-0.601544\pi$
$421$	−9.60356e8	−0.627256	−0.313628	+	0.949546i	$0.601544\pi$
$422$	−2.37280e8	−0.153698
$423$	3.28995e8	0.211348
$424$	−9.95091e8	−0.633990
$425$	−2.97572e9	−1.88031
$426$	−1.23002e9	−0.770866
$427$	0	0
$428$	−4.61404e8	−0.284465
$429$	−2.23356e9	−1.36583
$430$	3.28899e9	1.99491
$431$	−1.03166e8	−0.0620677	−0.0310338	−	0.999518i	$-0.509880\pi$
$431$	−1.03166e8	−0.0620677	−0.0310338	+	0.999518i	$0.509880\pi$
$432$	−8.06216e7	−0.0481125
$433$	2.33278e9	1.38091	0.690455	−	0.723375i	$-0.257410\pi$
$433$	2.33278e9	1.38091	0.690455	+	0.723375i	$0.257410\pi$
$434$	0	0
$435$	3.52021e7	0.0205048
$436$	2.48984e8	0.143869
$437$	−6.39535e8	−0.366589
$438$	−4.86838e8	−0.276837
$439$	1.56159e9	0.880928	0.440464	−	0.897770i	$-0.354814\pi$
$439$	1.56159e9	0.880928	0.440464	+	0.897770i	$0.354814\pi$
$440$	1.74897e9	0.978810
$441$	0	0
$442$	1.89779e9	1.04537
$443$	1.62481e9	0.887951	0.443975	−	0.896039i	$-0.353568\pi$
$443$	1.62481e9	0.887951	0.443975	+	0.896039i	$0.353568\pi$
$444$	4.14198e8	0.224578
$445$	−4.99446e9	−2.68676
$446$	−2.52829e9	−1.34945
$447$	−1.70915e9	−0.905113
$448$	0	0
$449$	−8.18478e8	−0.426722	−0.213361	−	0.976973i	$-0.568441\pi$
$449$	−8.18478e8	−0.426722	−0.213361	+	0.976973i	$0.568441\pi$
$450$	8.32664e8	0.430751
$451$	3.84928e8	0.197588
$452$	9.38542e8	0.478046
$453$	1.45699e9	0.736398
$454$	−3.80845e8	−0.191008
$455$	0	0
$456$	−1.89168e8	−0.0934264
$457$	−1.53326e9	−0.751463	−0.375732	−	0.926728i	$-0.622609\pi$
$457$	−1.53326e9	−0.751463	−0.375732	+	0.926728i	$0.622609\pi$
$458$	8.34311e8	0.405787
$459$	4.10233e8	0.198010
$460$	1.40582e9	0.673406
$461$	−3.43507e9	−1.63299	−0.816494	−	0.577354i	$-0.804085\pi$
$461$	−3.43507e9	−1.63299	−0.816494	+	0.577354i	$0.804085\pi$
$462$	0	0
$463$	1.73200e9	0.810990	0.405495	−	0.914097i	$-0.367099\pi$
$463$	1.73200e9	0.810990	0.405495	+	0.914097i	$0.367099\pi$
$464$	1.13623e7	0.00528024
$465$	1.42534e8	0.0657406
$466$	1.88047e9	0.860827
$467$	4.81488e8	0.218764	0.109382	−	0.994000i	$-0.465113\pi$
$467$	4.81488e8	0.218764	0.109382	+	0.994000i	$0.465113\pi$
$468$	−5.31039e8	−0.239478
$469$	0	0
$470$	−1.69687e9	−0.753888
$471$	1.57509e9	0.694596
$472$	−8.00721e8	−0.350497
$473$	6.35755e9	2.76233
$474$	−2.87090e7	−0.0123821
$475$	1.95373e9	0.836445
$476$	0	0
$477$	−1.41684e9	−0.597732
$478$	2.99489e8	0.125425
$479$	1.65939e9	0.689882	0.344941	−	0.938624i	$-0.387899\pi$
$479$	1.65939e9	0.689882	0.344941	+	0.938624i	$0.387899\pi$
$480$	4.15826e8	0.171620
$481$	2.72824e9	1.11783
$482$	−7.90134e7	−0.0321393
$483$	0	0
$484$	2.13355e9	0.855349
$485$	1.13133e9	0.450292
$486$	−1.14791e8	−0.0453609
$487$	1.72017e9	0.674868	0.337434	−	0.941349i	$-0.390441\pi$
$487$	1.72017e9	0.674868	0.337434	+	0.941349i	$0.390441\pi$
$488$	1.63005e9	0.634938
$489$	−1.63578e9	−0.632622
$490$	0	0
$491$	−6.67098e8	−0.254334	−0.127167	−	0.991881i	$-0.540588\pi$
$491$	−6.67098e8	−0.254334	−0.127167	+	0.991881i	$0.540588\pi$
$492$	9.15183e7	0.0346442
$493$	−5.78157e7	−0.0217311
$494$	−1.24601e9	−0.465026
$495$	2.49023e9	0.922831
$496$	4.60063e7	0.0169290
$497$	0	0
$498$	1.50131e9	0.544714
$499$	−5.45977e8	−0.196708	−0.0983541	−	0.995151i	$-0.531358\pi$
$499$	−5.45977e8	−0.196708	−0.0983541	+	0.995151i	$0.531358\pi$
$500$	−1.94467e9	−0.695747
$501$	1.60432e9	0.569980
$502$	−1.05951e9	−0.373801
$503$	−4.13284e9	−1.44797	−0.723987	−	0.689813i	$-0.757693\pi$
$503$	−4.13284e9	−1.44797	−0.723987	+	0.689813i	$0.757693\pi$
$504$	0	0
$505$	3.65460e9	1.26276
$506$	2.71742e9	0.932459
$507$	−1.80364e9	−0.614641
$508$	2.47892e8	0.0838957
$509$	1.91213e9	0.642695	0.321348	−	0.946961i	$-0.395864\pi$
$509$	1.91213e9	0.642695	0.321348	+	0.946961i	$0.395864\pi$
$510$	−2.11588e9	−0.706310
$511$	0	0
$512$	1.34218e8	0.0441942
$513$	−2.69342e8	−0.0880833
$514$	−2.66134e9	−0.864428
$515$	−6.21999e9	−2.00662
$516$	1.51154e9	0.484334
$517$	−3.28002e9	−1.04390
$518$	0	0
$519$	1.50813e9	0.473536
$520$	2.73896e9	0.854230
$521$	−6.21174e7	−0.0192434	−0.00962169	−	0.999954i	$-0.503063\pi$
$521$	−6.21174e7	−0.0192434	−0.00962169	+	0.999954i	$0.503063\pi$
$522$	1.61780e7	0.00497826
$523$	−6.82075e8	−0.208486	−0.104243	−	0.994552i	$-0.533242\pi$
$523$	−6.82075e8	−0.208486	−0.104243	+	0.994552i	$0.533242\pi$
$524$	−1.45585e9	−0.442035
$525$	0	0
$526$	−7.60983e8	−0.227995
$527$	−2.34097e8	−0.0696722
$528$	8.03783e8	0.237640
$529$	−1.22057e9	−0.358483
$530$	7.30770e9	2.13214
$531$	−1.14009e9	−0.330452
$532$	0	0
$533$	6.02813e8	0.172440
$534$	−2.29533e9	−0.652304
$535$	3.38844e9	0.956668
$536$	1.08989e9	0.305706
$537$	−2.16608e9	−0.603621
$538$	2.15084e9	0.595484
$539$	0	0
$540$	5.92065e8	0.161805
$541$	2.20541e8	0.0598824	0.0299412	−	0.999552i	$-0.490468\pi$
$541$	2.20541e8	0.0598824	0.0299412	+	0.999552i	$0.490468\pi$
$542$	−3.62307e9	−0.977415
$543$	−1.94031e9	−0.520081
$544$	−6.82951e8	−0.181883
$545$	−1.82848e9	−0.483840
$546$	0	0
$547$	5.39461e9	1.40930	0.704652	−	0.709553i	$-0.251103\pi$
$547$	5.39461e9	1.40930	0.704652	+	0.709553i	$0.251103\pi$
$548$	−2.70324e9	−0.701703
$549$	2.32091e9	0.598625
$550$	−8.30151e9	−2.12759
$551$	3.79594e7	0.00966694
$552$	6.46078e8	0.163493
$553$	0	0
$554$	−1.31768e8	−0.0329249
$555$	−3.04177e9	−0.755267
$556$	1.29485e8	0.0319489
$557$	−1.22387e9	−0.300083	−0.150041	−	0.988680i	$-0.547941\pi$
$557$	−1.22387e9	−0.300083	−0.150041	+	0.988680i	$0.547941\pi$
$558$	6.55050e7	0.0159608
$559$	9.95620e9	2.41075
$560$	0	0
$561$	−4.08995e9	−0.978021
$562$	4.06491e9	0.965992
$563$	3.67094e9	0.866957	0.433479	−	0.901164i	$-0.357286\pi$
$563$	3.67094e9	0.866957	0.433479	+	0.901164i	$0.357286\pi$
$564$	−7.79839e8	−0.183033
$565$	−6.89242e9	−1.60769
$566$	2.05997e9	0.477532
$567$	0	0
$568$	2.91561e9	0.667590
$569$	5.00370e9	1.13867	0.569335	−	0.822105i	$-0.307201\pi$
$569$	5.00370e9	1.13867	0.569335	+	0.822105i	$0.307201\pi$
$570$	1.38920e9	0.314197
$571$	1.51166e9	0.339803	0.169902	−	0.985461i	$-0.445655\pi$
$571$	1.51166e9	0.339803	0.169902	+	0.985461i	$0.445655\pi$
$572$	5.29436e9	1.18284
$573$	4.52075e8	0.100385
$574$	0	0
$575$	−6.67273e9	−1.46375
$576$	1.91103e8	0.0416667
$577$	4.39120e9	0.951629	0.475814	−	0.879546i	$-0.342153\pi$
$577$	4.39120e9	0.951629	0.475814	+	0.879546i	$0.342153\pi$
$578$	1.92402e8	0.0414441
$579$	3.63696e9	0.778688
$580$	−8.34419e7	−0.0177577
$581$	0	0
$582$	5.19931e8	0.109324
$583$	1.41256e10	2.95235
$584$	1.15399e9	0.239748
$585$	3.89981e9	0.805376
$586$	−5.87078e8	−0.120519
$587$	6.47628e9	1.32158	0.660789	−	0.750572i	$-0.270222\pi$
$587$	6.47628e9	1.32158	0.660789	+	0.750572i	$0.270222\pi$
$588$	0	0
$589$	1.53699e8	0.0309932
$590$	5.88029e9	1.17874
$591$	1.59142e9	0.317123
$592$	−9.81803e8	−0.194490
$593$	−5.28184e8	−0.104014	−0.0520072	−	0.998647i	$-0.516562\pi$
$593$	−5.28184e8	−0.104014	−0.0520072	+	0.998647i	$0.516562\pi$
$594$	1.14445e9	0.224049
$595$	0	0
$596$	4.05131e9	0.783851
$597$	3.54708e9	0.682277
$598$	4.25559e9	0.813778
$599$	−9.03423e9	−1.71750	−0.858751	−	0.512394i	$-0.828759\pi$
$599$	−9.03423e9	−1.71750	−0.858751	+	0.512394i	$0.828759\pi$
$600$	−1.97372e9	−0.373041
$601$	−5.03992e9	−0.947028	−0.473514	−	0.880786i	$-0.657015\pi$
$601$	−5.03992e9	−0.947028	−0.473514	+	0.880786i	$0.657015\pi$
$602$	0	0
$603$	1.55181e9	0.288223
$604$	−3.45360e9	−0.637739
$605$	−1.56682e10	−2.87658
$606$	1.67956e9	0.306579
$607$	−5.96562e9	−1.08267	−0.541334	−	0.840808i	$-0.682080\pi$
$607$	−5.96562e9	−1.08267	−0.541334	+	0.840808i	$0.682080\pi$
$608$	4.48397e8	0.0809097
$609$	0	0
$610$	−1.19707e10	−2.13533
$611$	−5.13665e9	−0.911037
$612$	−9.72404e8	−0.171481
$613$	5.42350e9	0.950973	0.475487	−	0.879723i	$-0.342272\pi$
$613$	5.42350e9	0.950973	0.475487	+	0.879723i	$0.342272\pi$
$614$	7.45299e8	0.129939
$615$	−6.72088e8	−0.116510
$616$	0	0
$617$	2.18073e9	0.373770	0.186885	−	0.982382i	$-0.440161\pi$
$617$	2.18073e9	0.373770	0.186885	+	0.982382i	$0.440161\pi$
$618$	−2.85855e9	−0.487176
$619$	5.35735e9	0.907889	0.453945	−	0.891030i	$-0.350016\pi$
$619$	5.35735e9	0.907889	0.453945	+	0.891030i	$0.350016\pi$
$620$	−3.37859e8	−0.0569330
$621$	9.19905e8	0.154142
$622$	−4.88747e9	−0.814363
$623$	0	0
$624$	1.25876e9	0.207394
$625$	3.12689e9	0.512309
$626$	1.69785e9	0.276623
$627$	2.68529e9	0.435066
$628$	−3.73355e9	−0.601538
$629$	4.99579e9	0.800436
$630$	0	0
$631$	1.14559e10	1.81521	0.907607	−	0.419822i	$-0.137907\pi$
$631$	1.14559e10	1.81521	0.907607	+	0.419822i	$0.137907\pi$
$632$	6.80509e7	0.0107232
$633$	8.00821e8	0.125494
$634$	−8.83156e9	−1.37634
$635$	−1.82046e9	−0.282145
$636$	3.35843e9	0.517651
$637$	0	0
$638$	−1.61291e8	−0.0245889
$639$	4.15132e9	0.629410
$640$	−9.85661e8	−0.148627
$641$	1.01287e10	1.51898	0.759491	−	0.650518i	$-0.225448\pi$
$641$	1.01287e10	1.51898	0.759491	+	0.650518i	$0.225448\pi$
$642$	1.55724e9	0.232265
$643$	4.99152e9	0.740448	0.370224	−	0.928943i	$-0.379281\pi$
$643$	4.99152e9	0.740448	0.370224	+	0.928943i	$0.379281\pi$
$644$	0	0
$645$	−1.11003e10	−1.62884
$646$	−2.28162e9	−0.332988
$647$	5.54237e9	0.804508	0.402254	−	0.915528i	$-0.368227\pi$
$647$	5.54237e9	0.804508	0.402254	+	0.915528i	$0.368227\pi$
$648$	2.72098e8	0.0392837
$649$	1.13665e10	1.63219
$650$	−1.30005e10	−1.85680
$651$	0	0
$652$	3.87741e9	0.547867
$653$	−7.42589e9	−1.04364	−0.521822	−	0.853054i	$-0.674748\pi$
$653$	−7.42589e9	−1.04364	−0.521822	+	0.853054i	$0.674748\pi$
$654$	−8.40321e8	−0.117469
$655$	1.06914e10	1.48658
$656$	−2.16932e8	−0.0300027
$657$	1.64308e9	0.226037
$658$	0	0
$659$	8.68127e8	0.118164	0.0590819	−	0.998253i	$-0.481183\pi$
$659$	8.68127e8	0.118164	0.0590819	+	0.998253i	$0.481183\pi$
$660$	−5.90278e9	−0.799195
$661$	−3.36920e9	−0.453755	−0.226878	−	0.973923i	$-0.572852\pi$
$661$	−3.36920e9	−0.453755	−0.226878	+	0.973923i	$0.572852\pi$
$662$	−5.36470e9	−0.718692
$663$	−6.40504e9	−0.853541
$664$	−3.55867e9	−0.471736
$665$	0	0
$666$	−1.39792e9	−0.183367
$667$	−1.29646e8	−0.0169168
$668$	−3.80284e9	−0.493617
$669$	8.53299e9	1.10182
$670$	−8.00385e9	−1.02810
$671$	−2.31391e10	−2.95677
$672$	0	0
$673$	−3.43406e9	−0.434265	−0.217133	−	0.976142i	$-0.569670\pi$
$673$	−3.43406e9	−0.434265	−0.217133	+	0.976142i	$0.569670\pi$
$674$	−5.47298e9	−0.688516
$675$	−2.81024e9	−0.351706
$676$	4.27529e9	0.532295
$677$	−1.28899e10	−1.59658	−0.798289	−	0.602275i	$-0.794261\pi$
$677$	−1.28899e10	−1.59658	−0.798289	+	0.602275i	$0.794261\pi$
$678$	−3.16758e9	−0.390323
$679$	0	0
$680$	5.01542e9	0.611682
$681$	1.28535e9	0.155958
$682$	−6.53073e8	−0.0788346
$683$	−6.52012e9	−0.783039	−0.391519	−	0.920170i	$-0.628050\pi$
$683$	−6.52012e9	−0.783039	−0.391519	+	0.920170i	$0.628050\pi$
$684$	6.38441e8	0.0762824
$685$	1.98520e10	2.35986
$686$	0	0
$687$	−2.81580e9	−0.331324
$688$	−3.58290e9	−0.419445
$689$	2.21213e10	2.57658
$690$	−4.74464e9	−0.549834
$691$	1.51264e10	1.74407	0.872035	−	0.489444i	$-0.162800\pi$
$691$	1.51264e10	1.74407	0.872035	+	0.489444i	$0.162800\pi$
$692$	−3.57482e9	−0.410094
$693$	0	0
$694$	3.50042e9	0.397523
$695$	−9.50902e8	−0.107446
$696$	−3.83478e7	−0.00431130
$697$	1.10383e9	0.123478
$698$	−7.51866e7	−0.00836848
$699$	−6.34659e9	−0.702862
$700$	0	0
$701$	−8.52704e9	−0.934943	−0.467472	−	0.884008i	$-0.654835\pi$
$701$	−8.52704e9	−0.934943	−0.467472	+	0.884008i	$0.654835\pi$
$702$	1.79226e9	0.195533
$703$	−3.28003e9	−0.356069
$704$	−1.90526e9	−0.205802
$705$	5.72695e9	0.615547
$706$	−1.13177e10	−1.21044
$707$	0	0
$708$	2.70243e9	0.286180
$709$	−9.84579e9	−1.03750	−0.518751	−	0.854925i	$-0.673603\pi$
$709$	−9.84579e9	−1.03750	−0.518751	+	0.854925i	$0.673603\pi$
$710$	−2.14115e10	−2.24514
$711$	9.68928e7	0.0101099
$712$	5.44077e9	0.564912
$713$	−5.24939e8	−0.0542370
$714$	0	0
$715$	−3.88805e10	−3.97796
$716$	5.13441e9	0.522752
$717$	−1.01078e9	−0.102409
$718$	−2.46066e9	−0.248094
$719$	−4.02796e9	−0.404142	−0.202071	−	0.979371i	$-0.564767\pi$
$719$	−4.02796e9	−0.404142	−0.202071	+	0.979371i	$0.564767\pi$
$720$	−1.40341e9	−0.140127
$721$	0	0
$722$	−5.65296e9	−0.558979
$723$	2.66670e8	0.0262416
$724$	4.59925e9	0.450403
$725$	3.96058e8	0.0385990
$726$	−7.20072e9	−0.698389
$727$	5.95463e9	0.574757	0.287379	−	0.957817i	$-0.407216\pi$
$727$	5.95463e9	0.574757	0.287379	+	0.957817i	$0.407216\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	−8.47458e9	−0.806284
$731$	1.82312e10	1.72625
$732$	−5.50142e9	−0.518425
$733$	−1.65545e10	−1.55257	−0.776286	−	0.630381i	$-0.782899\pi$
$733$	−1.65545e10	−1.55257	−0.776286	+	0.630381i	$0.782899\pi$
$734$	9.85018e9	0.919407
$735$	0	0
$736$	−1.53145e9	−0.141589
$737$	−1.54713e10	−1.42361
$738$	−3.08874e8	−0.0282869
$739$	1.14446e10	1.04315	0.521574	−	0.853206i	$-0.325345\pi$
$739$	1.14446e10	1.04315	0.521574	+	0.853206i	$0.325345\pi$
$740$	7.21012e9	0.654081
$741$	4.20528e9	0.379692
$742$	0	0
$743$	−4.25895e8	−0.0380927	−0.0190464	−	0.999819i	$-0.506063\pi$
$743$	−4.25895e8	−0.0380927	−0.0190464	+	0.999819i	$0.506063\pi$
$744$	−1.55271e8	−0.0138225
$745$	−2.97518e10	−2.63613
$746$	1.46873e10	1.29526
$747$	−5.06693e9	−0.444757
$748$	9.69470e9	0.846991
$749$	0	0
$750$	6.56327e9	0.568075
$751$	−2.95035e8	−0.0254176	−0.0127088	−	0.999919i	$-0.504045\pi$
$751$	−2.95035e8	−0.0254176	−0.0127088	+	0.999919i	$0.504045\pi$
$752$	1.84851e9	0.158511
$753$	3.57583e9	0.305207
$754$	−2.52589e8	−0.0214593
$755$	2.53624e10	2.14475
$756$	0	0
$757$	1.87377e10	1.56994	0.784968	−	0.619537i	$-0.212680\pi$
$757$	1.87377e10	1.56994	0.784968	+	0.619537i	$0.212680\pi$
$758$	−5.19147e9	−0.432960
$759$	−9.17129e9	−0.761349
$760$	−3.29292e9	−0.272103
$761$	−4.07496e9	−0.335179	−0.167590	−	0.985857i	$-0.553598\pi$
$761$	−4.07496e9	−0.335179	−0.167590	+	0.985857i	$0.553598\pi$
$762$	−8.36635e8	−0.0685005
$763$	0	0
$764$	−1.07159e9	−0.0869361
$765$	7.14109e9	0.576700
$766$	5.44257e9	0.437525
$767$	1.78004e10	1.42445
$768$	−4.52985e8	−0.0360844
$769$	−1.19604e10	−0.948425	−0.474213	−	0.880410i	$-0.657267\pi$
$769$	−1.19604e10	−0.948425	−0.474213	+	0.880410i	$0.657267\pi$
$770$	0	0
$771$	8.98201e9	0.705803
$772$	−8.62094e9	−0.674364
$773$	−4.67303e8	−0.0363890	−0.0181945	−	0.999834i	$-0.505792\pi$
$773$	−4.67303e8	−0.0363890	−0.0181945	+	0.999834i	$0.505792\pi$
$774$	−5.10144e9	−0.395457
$775$	1.60365e9	0.123752
$776$	−1.23243e9	−0.0946774
$777$	0	0
$778$	2.18964e9	0.166703
$779$	−7.24732e8	−0.0549283
$780$	−9.24401e9	−0.697476
$781$	−4.13879e10	−3.10882
$782$	7.79257e9	0.582716
$783$	−5.46006e7	−0.00406473
$784$	0	0
$785$	2.74182e10	2.02300
$786$	4.91349e9	0.360920
$787$	8.07987e9	0.590871	0.295436	−	0.955363i	$-0.404535\pi$
$787$	8.07987e9	0.590871	0.295436	+	0.955363i	$0.404535\pi$
$788$	−3.77224e9	−0.274636
$789$	2.56832e9	0.186157
$790$	−4.99749e8	−0.0360626
$791$	0	0
$792$	−2.71277e9	−0.194032
$793$	−3.62368e10	−2.58044
$794$	9.68273e9	0.686477
$795$	−2.46635e10	−1.74088
$796$	−8.40789e9	−0.590869
$797$	1.48799e10	1.04111	0.520555	−	0.853828i	$-0.325725\pi$
$797$	1.48799e10	1.04111	0.520555	+	0.853828i	$0.325725\pi$
$798$	0	0
$799$	−9.40591e9	−0.652360
$800$	4.67845e9	0.323063
$801$	7.74673e9	0.532604
$802$	−4.77682e9	−0.326985
$803$	−1.63812e10	−1.11645
$804$	−3.67837e9	−0.249608
$805$	0	0
$806$	−1.02274e9	−0.0688007
$807$	−7.25908e9	−0.486211
$808$	−3.98118e9	−0.265505
$809$	7.15132e9	0.474861	0.237431	−	0.971405i	$-0.423695\pi$
$809$	7.15132e9	0.474861	0.237431	+	0.971405i	$0.423695\pi$
$810$	−1.99822e9	−0.132113
$811$	−2.13058e9	−0.140257	−0.0701286	−	0.997538i	$-0.522341\pi$
$811$	−2.13058e9	−0.140257	−0.0701286	+	0.997538i	$0.522341\pi$
$812$	0	0
$813$	1.22279e10	0.798056
$814$	1.39370e10	0.905699
$815$	−2.84747e10	−1.84250
$816$	2.30496e9	0.148507
$817$	−1.19698e10	−0.767911
$818$	9.58137e9	0.612056
$819$	0	0
$820$	1.59310e9	0.100901
$821$	2.04052e9	0.128688	0.0643442	−	0.997928i	$-0.479504\pi$
$821$	2.04052e9	0.128688	0.0643442	+	0.997928i	$0.479504\pi$
$822$	9.12345e9	0.572938
$823$	2.99139e10	1.87057	0.935283	−	0.353900i	$-0.115145\pi$
$823$	2.99139e10	1.87057	0.935283	+	0.353900i	$0.115145\pi$
$824$	6.77582e9	0.421907
$825$	2.80176e10	1.73717
$826$	0	0
$827$	1.05103e10	0.646170	0.323085	−	0.946370i	$-0.395280\pi$
$827$	1.05103e10	0.646170	0.323085	+	0.946370i	$0.395280\pi$
$828$	−2.18051e9	−0.133491
$829$	−1.91146e10	−1.16526	−0.582632	−	0.812736i	$-0.697977\pi$
$829$	−1.91146e10	−1.16526	−0.582632	+	0.812736i	$0.697977\pi$
$830$	2.61340e10	1.58647
$831$	4.44716e8	0.0268831
$832$	−2.98372e9	−0.179608
$833$	0	0
$834$	−4.37010e8	−0.0260862
$835$	2.79271e10	1.66006
$836$	−6.36514e9	−0.376778
$837$	−2.21079e8	−0.0130319
$838$	1.34944e10	0.792136
$839$	−2.31635e9	−0.135406	−0.0677028	−	0.997706i	$-0.521567\pi$
$839$	−2.31635e9	−0.135406	−0.0677028	+	0.997706i	$0.521567\pi$
$840$	0	0
$841$	−1.72422e10	−0.999554
$842$	−7.68284e9	−0.443537
$843$	−1.37191e10	−0.788729
$844$	−1.89824e9	−0.108681
$845$	−3.13967e10	−1.79013
$846$	2.63196e9	0.149445
$847$	0	0
$848$	−7.96073e9	−0.448299
$849$	−6.95239e9	−0.389904
$850$	−2.38057e10	−1.32958
$851$	1.12025e10	0.623107
$852$	−9.84017e9	−0.545085
$853$	8.57597e8	0.0473110	0.0236555	−	0.999720i	$-0.492470\pi$
$853$	8.57597e8	0.0473110	0.0236555	+	0.999720i	$0.492470\pi$
$854$	0	0
$855$	−4.68855e9	−0.256541
$856$	−3.69124e9	−0.201147
$857$	2.73865e10	1.48629	0.743146	−	0.669130i	$-0.233333\pi$
$857$	2.73865e10	1.48629	0.743146	+	0.669130i	$0.233333\pi$
$858$	−1.78685e10	−0.965788
$859$	1.01916e10	0.548611	0.274306	−	0.961643i	$-0.411552\pi$
$859$	1.01916e10	0.548611	0.274306	+	0.961643i	$0.411552\pi$
$860$	2.63119e10	1.41061
$861$	0	0
$862$	−8.25327e8	−0.0438885
$863$	3.70675e9	0.196316	0.0981578	−	0.995171i	$-0.468705\pi$
$863$	3.70675e9	0.196316	0.0981578	+	0.995171i	$0.468705\pi$
$864$	−6.44973e8	−0.0340207
$865$	2.62526e10	1.37916
$866$	1.86622e10	0.976451
$867$	−6.49358e8	−0.0338390
$868$	0	0
$869$	−9.66004e8	−0.0499356
$870$	2.81616e8	0.0144991
$871$	−2.42287e10	−1.24241
$872$	1.99187e9	0.101731
$873$	−1.75477e9	−0.0892627
$874$	−5.11628e9	−0.259218
$875$	0	0
$876$	−3.89470e9	−0.195754
$877$	2.18681e9	0.109474	0.0547372	−	0.998501i	$-0.482568\pi$
$877$	2.18681e9	0.109474	0.0547372	+	0.998501i	$0.482568\pi$
$878$	1.24927e10	0.622910
$879$	1.98139e9	0.0984030
$880$	1.39918e10	0.692123
$881$	9.10532e9	0.448621	0.224310	−	0.974518i	$-0.427987\pi$
$881$	9.10532e9	0.448621	0.224310	+	0.974518i	$0.427987\pi$
$882$	0	0
$883$	1.99391e10	0.974639	0.487319	−	0.873224i	$-0.337975\pi$
$883$	1.99391e10	0.974639	0.487319	+	0.873224i	$0.337975\pi$
$884$	1.51823e10	0.739188
$885$	−1.98460e10	−0.962435
$886$	1.29985e10	0.627876
$887$	7.24425e9	0.348547	0.174273	−	0.984697i	$-0.444242\pi$
$887$	7.24425e9	0.348547	0.174273	+	0.984697i	$0.444242\pi$
$888$	3.31359e9	0.158801
$889$	0	0
$890$	−3.99557e10	−1.89982
$891$	−3.86251e9	−0.182936
$892$	−2.02264e10	−0.954203
$893$	6.17553e9	0.290198
$894$	−1.36732e10	−0.640012
$895$	−3.77059e10	−1.75804
$896$	0	0
$897$	−1.43626e10	−0.664447
$898$	−6.54782e9	−0.301738
$899$	3.11576e7	0.00143023
$900$	6.66131e9	0.304587
$901$	4.05072e10	1.84500
$902$	3.07942e9	0.139716
$903$	0	0
$904$	7.50834e9	0.338029
$905$	−3.37757e10	−1.51473
$906$	1.16559e10	0.520712
$907$	8.75334e9	0.389537	0.194768	−	0.980849i	$-0.437605\pi$
$907$	8.75334e9	0.389537	0.194768	+	0.980849i	$0.437605\pi$
$908$	−3.04676e9	−0.135063
$909$	−5.66852e9	−0.250320
$910$	0	0
$911$	1.99869e10	0.875853	0.437926	−	0.899011i	$-0.355713\pi$
$911$	1.99869e10	0.875853	0.437926	+	0.899011i	$0.355713\pi$
$912$	−1.51334e9	−0.0660625
$913$	5.05164e10	2.19677
$914$	−1.22660e10	−0.531365
$915$	4.04010e10	1.74349
$916$	6.67449e9	0.286935
$917$	0	0
$918$	3.28186e9	0.140014
$919$	−9.93075e9	−0.422063	−0.211032	−	0.977479i	$-0.567682\pi$
$919$	−9.93075e9	−0.422063	−0.211032	+	0.977479i	$0.567682\pi$
$920$	1.12466e10	0.476170
$921$	−2.51538e9	−0.106095
$922$	−2.74806e10	−1.15470
$923$	−6.48153e10	−2.71314
$924$	0	0
$925$	−3.42229e10	−1.42174
$926$	1.38560e10	0.573457
$927$	9.64760e9	0.397778
$928$	9.08984e7	0.00373369
$929$	−1.95908e10	−0.801671	−0.400836	−	0.916150i	$-0.631280\pi$
$929$	−1.95908e10	−0.801671	−0.400836	+	0.916150i	$0.631280\pi$
$930$	1.14027e9	0.0464856
$931$	0	0
$932$	1.50438e10	0.608697
$933$	1.64952e10	0.664925
$934$	3.85190e9	0.154690
$935$	−7.11954e10	−2.84847
$936$	−4.24831e9	−0.169337
$937$	−4.20286e10	−1.66900	−0.834500	−	0.551009i	$-0.814243\pi$
$937$	−4.20286e10	−1.66900	−0.834500	+	0.551009i	$0.814243\pi$
$938$	0	0
$939$	−5.73024e9	−0.225862
$940$	−1.35750e10	−0.533079
$941$	9.95430e9	0.389446	0.194723	−	0.980858i	$-0.437619\pi$
$941$	9.95430e9	0.389446	0.194723	+	0.980858i	$0.437619\pi$
$942$	1.26007e10	0.491154
$943$	2.47523e9	0.0961225
$944$	−6.40577e9	−0.247839
$945$	0	0
$946$	5.08604e10	1.95326
$947$	−2.43450e10	−0.931503	−0.465751	−	0.884916i	$-0.654216\pi$
$947$	−2.43450e10	−0.931503	−0.465751	+	0.884916i	$0.654216\pi$
$948$	−2.29672e8	−0.00875546
$949$	−2.56536e10	−0.974355
$950$	1.56299e10	0.591456
$951$	2.98065e10	1.12378
$952$	0	0
$953$	1.71198e10	0.640727	0.320364	−	0.947295i	$-0.396195\pi$
$953$	1.71198e10	0.640727	0.320364	+	0.947295i	$0.396195\pi$
$954$	−1.13347e10	−0.422660
$955$	7.86946e9	0.292370
$956$	2.39592e9	0.0886888
$957$	5.44359e8	0.0200767
$958$	1.32751e10	0.487821
$959$	0	0
$960$	3.32661e9	0.121353
$961$	−2.73865e10	−0.995415
$962$	2.18259e10	0.790424
$963$	−5.25568e9	−0.189643
$964$	−6.32107e8	−0.0227259
$965$	6.33100e10	2.26792
$966$	0	0
$967$	−5.29443e10	−1.88290	−0.941448	−	0.337157i	$-0.890535\pi$
$967$	−5.29443e10	−1.88290	−0.941448	+	0.337157i	$0.890535\pi$
$968$	1.70684e10	0.604823
$969$	7.70045e9	0.271884
$970$	9.05066e9	0.318404
$971$	2.44723e10	0.857841	0.428920	−	0.903342i	$-0.358894\pi$
$971$	2.44723e10	0.857841	0.428920	+	0.903342i	$0.358894\pi$
$972$	−9.18330e8	−0.0320750
$973$	0	0
$974$	1.37613e10	0.477204
$975$	4.38768e10	1.51607
$976$	1.30404e10	0.448969
$977$	−3.21678e10	−1.10355	−0.551774	−	0.833994i	$-0.686049\pi$
$977$	−3.21678e10	−1.10355	−0.551774	+	0.833994i	$0.686049\pi$
$978$	−1.30863e10	−0.447331
$979$	−7.72335e10	−2.63067
$980$	0	0
$981$	2.83608e9	0.0959130
$982$	−5.33678e9	−0.179841
$983$	−1.85564e10	−0.623097	−0.311549	−	0.950230i	$-0.600848\pi$
$983$	−1.85564e10	−0.623097	−0.311549	+	0.950230i	$0.600848\pi$
$984$	7.32147e8	0.0244971
$985$	2.77024e10	0.923615
$986$	−4.62526e8	−0.0153662
$987$	0	0
$988$	−9.96808e9	−0.328823
$989$	4.08815e10	1.34382
$990$	1.99219e10	0.652540
$991$	1.60149e10	0.522716	0.261358	−	0.965242i	$-0.415830\pi$
$991$	1.60149e10	0.522716	0.261358	+	0.965242i	$0.415830\pi$
$992$	3.68050e8	0.0119706
$993$	1.81059e10	0.586809
$994$	0	0
$995$	6.17454e10	1.98712
$996$	1.20105e10	0.385171
$997$	−3.20965e10	−1.02571	−0.512855	−	0.858475i	$-0.671412\pi$
$997$	−3.20965e10	−1.02571	−0.512855	+	0.858475i	$0.671412\pi$
$998$	−4.36782e9	−0.139094
$999$	4.71798e9	0.149719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.8.a.k.1.1		1
7.2	even	3		294.8.e.f.67.1		2
7.3	odd	6		294.8.e.a.79.1		2
7.4	even	3		294.8.e.f.79.1		2
7.5	odd	6		294.8.e.a.67.1		2
7.6	odd	2		42.8.a.f.1.1	✓	1
21.20	even	2		126.8.a.a.1.1		1
28.27	even	2		336.8.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.8.a.f.1.1	✓	1	7.6	odd	2
126.8.a.a.1.1		1	21.20	even	2
294.8.a.k.1.1		1	1.1	even	1	trivial
294.8.e.a.67.1		2	7.5	odd	6
294.8.e.a.79.1		2	7.3	odd	6
294.8.e.f.67.1		2	7.2	even	3
294.8.e.f.79.1		2	7.4	even	3
336.8.a.f.1.1		1	28.27	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:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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