LMFDB - Embedded newform 16.12.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,12,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	16.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:	$12.2934908890$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.72015$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-696.180 q^{3} -4084.16 q^{5} -73591.5 q^{7} +307519. q^{9} +O(q^{10})$	$q-696.180 q^{3} -4084.16 q^{5} -73591.5 q^{7} +307519. q^{9} +383508. q^{11} +1.15867e6 q^{13} +2.84331e6 q^{15} -6.51693e6 q^{17} +1.39982e7 q^{19} +5.12329e7 q^{21} -1.37394e6 q^{23} -3.21478e7 q^{25} -9.07624e7 q^{27} -7.46197e7 q^{29} -1.32297e7 q^{31} -2.66991e8 q^{33} +3.00559e8 q^{35} -1.67200e7 q^{37} -8.06644e8 q^{39} +1.03298e9 q^{41} -1.93764e8 q^{43} -1.25596e9 q^{45} +1.16005e9 q^{47} +3.43839e9 q^{49} +4.53696e9 q^{51} +4.44363e8 q^{53} -1.56631e9 q^{55} -9.74529e9 q^{57} +1.28304e8 q^{59} +7.96097e9 q^{61} -2.26308e10 q^{63} -4.73220e9 q^{65} +6.89243e9 q^{67} +9.56509e8 q^{69} +1.12698e10 q^{71} +3.34998e9 q^{73} +2.23806e10 q^{75} -2.82230e10 q^{77} -5.36237e10 q^{79} +8.71083e9 q^{81} +6.31693e10 q^{83} +2.66162e10 q^{85} +5.19487e10 q^{87} +9.62373e10 q^{89} -8.52685e10 q^{91} +9.21027e9 q^{93} -5.71710e10 q^{95} -4.37466e10 q^{97} +1.17936e11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 56 q^{3} + 7868 q^{5} - 91056 q^{7} + 540202 q^{9}+O(q^{10}) $ 2 * q - 56 * q^3 + 7868 * q^5 - 91056 * q^7 + 540202 * q^9	$ 2 q - 56 q^{3} + 7868 q^{5} - 91056 q^{7} + 540202 q^{9} - 159080 q^{11} + 1050476 q^{13} + 10494832 q^{15} + 1430884 q^{17} + 21866600 q^{19} + 40052544 q^{21} - 35806736 q^{23} + 61878094 q^{25} - 55209392 q^{27} - 228827700 q^{29} - 64722112 q^{31} - 614344864 q^{33} + 91821408 q^{35} + 75558780 q^{37} - 875909072 q^{39} + 1201214196 q^{41} + 45519832 q^{43} + 1525107052 q^{45} + 1229079264 q^{47} + 1766069074 q^{49} + 9624987920 q^{51} - 3808549924 q^{53} - 8051409968 q^{55} - 4708125536 q^{57} + 6012926584 q^{59} + 9789792908 q^{61} - 26694483312 q^{63} - 6025376344 q^{65} - 14703095224 q^{67} - 21086664256 q^{69} - 4319991088 q^{71} + 11055639476 q^{73} + 82574103416 q^{75} - 18746968128 q^{77} - 51957623264 q^{79} - 9747960302 q^{81} + 108227975912 q^{83} + 121609729720 q^{85} - 46772044368 q^{87} + 71188291860 q^{89} - 83378892576 q^{91} - 23754095360 q^{93} + 36872875568 q^{95} - 1699807676 q^{97} - 8314917256 q^{99}+O(q^{100}) $ 2 * q - 56 * q^3 + 7868 * q^5 - 91056 * q^7 + 540202 * q^9 - 159080 * q^11 + 1050476 * q^13 + 10494832 * q^15 + 1430884 * q^17 + 21866600 * q^19 + 40052544 * q^21 - 35806736 * q^23 + 61878094 * q^25 - 55209392 * q^27 - 228827700 * q^29 - 64722112 * q^31 - 614344864 * q^33 + 91821408 * q^35 + 75558780 * q^37 - 875909072 * q^39 + 1201214196 * q^41 + 45519832 * q^43 + 1525107052 * q^45 + 1229079264 * q^47 + 1766069074 * q^49 + 9624987920 * q^51 - 3808549924 * q^53 - 8051409968 * q^55 - 4708125536 * q^57 + 6012926584 * q^59 + 9789792908 * q^61 - 26694483312 * q^63 - 6025376344 * q^65 - 14703095224 * q^67 - 21086664256 * q^69 - 4319991088 * q^71 + 11055639476 * q^73 + 82574103416 * q^75 - 18746968128 * q^77 - 51957623264 * q^79 - 9747960302 * q^81 + 108227975912 * q^83 + 121609729720 * q^85 - 46772044368 * q^87 + 71188291860 * q^89 - 83378892576 * q^91 - 23754095360 * q^93 + 36872875568 * q^95 - 1699807676 * q^97 - 8314917256 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−696.180	−1.65407	−0.827036	−	0.562149i	$-0.809975\pi$
$3$	−696.180	−1.65407	−0.827036	+	0.562149i	$0.809975\pi$
$4$	0	0
$5$	−4084.16	−0.584477	−0.292238	−	0.956346i	$-0.594400\pi$
$5$	−4084.16	−0.584477	−0.292238	+	0.956346i	$0.594400\pi$
$6$	0	0
$7$	−73591.5	−1.65496	−0.827482	−	0.561492i	$-0.810228\pi$
$7$	−73591.5	−1.65496	−0.827482	+	0.561492i	$0.810228\pi$
$8$	0	0
$9$	307519.	1.73595
$10$	0	0
$11$	383508.	0.717985	0.358992	−	0.933340i	$-0.383120\pi$
$11$	383508.	0.717985	0.358992	+	0.933340i	$0.383120\pi$
$12$	0	0
$13$	1.15867e6	0.865510	0.432755	−	0.901512i	$-0.357541\pi$
$13$	1.15867e6	0.865510	0.432755	+	0.901512i	$0.357541\pi$
$14$	0	0
$15$	2.84331e6	0.966767
$16$	0	0
$17$	−6.51693e6	−1.11320	−0.556601	−	0.830780i	$-0.687895\pi$
$17$	−6.51693e6	−1.11320	−0.556601	+	0.830780i	$0.687895\pi$
$18$	0	0
$19$	1.39982e7	1.29697	0.648483	−	0.761229i	$-0.275404\pi$
$19$	1.39982e7	1.29697	0.648483	+	0.761229i	$0.275404\pi$
$20$	0	0
$21$	5.12329e7	2.73743
$22$	0	0
$23$	−1.37394e6	−0.0445107	−0.0222554	−	0.999752i	$-0.507085\pi$
$23$	−1.37394e6	−0.0445107	−0.0222554	+	0.999752i	$0.507085\pi$
$24$	0	0
$25$	−3.21478e7	−0.658387
$26$	0	0
$27$	−9.07624e7	−1.21732
$28$	0	0
$29$	−7.46197e7	−0.675561	−0.337781	−	0.941225i	$-0.609676\pi$
$29$	−7.46197e7	−0.675561	−0.337781	+	0.941225i	$0.609676\pi$
$30$	0	0
$31$	−1.32297e7	−0.0829969	−0.0414985	−	0.999139i	$-0.513213\pi$
$31$	−1.32297e7	−0.0829969	−0.0414985	+	0.999139i	$0.513213\pi$
$32$	0	0
$33$	−2.66991e8	−1.18760
$34$	0	0
$35$	3.00559e8	0.967288
$36$	0	0
$37$	−1.67200e7	−0.0396394	−0.0198197	−	0.999804i	$-0.506309\pi$
$37$	−1.67200e7	−0.0396394	−0.0198197	+	0.999804i	$0.506309\pi$
$38$	0	0
$39$	−8.06644e8	−1.43162
$40$	0	0
$41$	1.03298e9	1.39245	0.696225	−	0.717823i	$-0.254861\pi$
$41$	1.03298e9	1.39245	0.696225	+	0.717823i	$0.254861\pi$
$42$	0	0
$43$	−1.93764e8	−0.201001	−0.100500	−	0.994937i	$-0.532044\pi$
$43$	−1.93764e8	−0.201001	−0.100500	+	0.994937i	$0.532044\pi$
$44$	0	0
$45$	−1.25596e9	−1.01462
$46$	0	0
$47$	1.16005e9	0.737803	0.368901	−	0.929469i	$-0.379734\pi$
$47$	1.16005e9	0.737803	0.368901	+	0.929469i	$0.379734\pi$
$48$	0	0
$49$	3.43839e9	1.73891
$50$	0	0
$51$	4.53696e9	1.84132
$52$	0	0
$53$	4.44363e8	0.145956	0.0729778	−	0.997334i	$-0.476750\pi$
$53$	4.44363e8	0.145956	0.0729778	+	0.997334i	$0.476750\pi$
$54$	0	0
$55$	−1.56631e9	−0.419645
$56$	0	0
$57$	−9.74529e9	−2.14528
$58$	0	0
$59$	1.28304e8	0.0233644	0.0116822	−	0.999932i	$-0.496281\pi$
$59$	1.28304e8	0.0233644	0.0116822	+	0.999932i	$0.496281\pi$
$60$	0	0
$61$	7.96097e9	1.20685	0.603423	−	0.797421i	$-0.293803\pi$
$61$	7.96097e9	1.20685	0.603423	+	0.797421i	$0.293803\pi$
$62$	0	0
$63$	−2.26308e10	−2.87294
$64$	0	0
$65$	−4.73220e9	−0.505871
$66$	0	0
$67$	6.89243e9	0.623678	0.311839	−	0.950135i	$-0.399055\pi$
$67$	6.89243e9	0.623678	0.311839	+	0.950135i	$0.399055\pi$
$68$	0	0
$69$	9.56509e8	0.0736240
$70$	0	0
$71$	1.12698e10	0.741305	0.370652	−	0.928772i	$-0.379134\pi$
$71$	1.12698e10	0.741305	0.370652	+	0.928772i	$0.379134\pi$
$72$	0	0
$73$	3.34998e9	0.189132	0.0945662	−	0.995519i	$-0.469854\pi$
$73$	3.34998e9	0.189132	0.0945662	+	0.995519i	$0.469854\pi$
$74$	0	0
$75$	2.23806e10	1.08902
$76$	0	0
$77$	−2.82230e10	−1.18824
$78$	0	0
$79$	−5.36237e10	−1.96068	−0.980342	−	0.197304i	$-0.936782\pi$
$79$	−5.36237e10	−1.96068	−0.980342	+	0.197304i	$0.936782\pi$
$80$	0	0
$81$	8.71083e9	0.277583
$82$	0	0
$83$	6.31693e10	1.76026	0.880130	−	0.474733i	$-0.157455\pi$
$83$	6.31693e10	1.76026	0.880130	+	0.474733i	$0.157455\pi$
$84$	0	0
$85$	2.66162e10	0.650641
$86$	0	0
$87$	5.19487e10	1.11743
$88$	0	0
$89$	9.62373e10	1.82683	0.913416	−	0.407028i	$-0.133435\pi$
$89$	9.62373e10	1.82683	0.913416	+	0.407028i	$0.133435\pi$
$90$	0	0
$91$	−8.52685e10	−1.43239
$92$	0	0
$93$	9.21027e9	0.137283
$94$	0	0
$95$	−5.71710e10	−0.758047
$96$	0	0
$97$	−4.37466e10	−0.517250	−0.258625	−	0.965978i	$-0.583269\pi$
$97$	−4.37466e10	−0.517250	−0.258625	+	0.965978i	$0.583269\pi$
$98$	0	0
$99$	1.17936e11	1.24639
$100$	0	0
$101$	−1.68061e11	−1.59111	−0.795554	−	0.605882i	$-0.792820\pi$
$101$	−1.68061e11	−1.59111	−0.795554	+	0.605882i	$0.792820\pi$
$102$	0	0
$103$	−6.02325e10	−0.511948	−0.255974	−	0.966684i	$-0.582396\pi$
$103$	−6.02325e10	−0.511948	−0.255974	+	0.966684i	$0.582396\pi$
$104$	0	0
$105$	−2.09243e11	−1.59996
$106$	0	0
$107$	−1.39380e11	−0.960701	−0.480351	−	0.877077i	$-0.659491\pi$
$107$	−1.39380e11	−0.960701	−0.480351	+	0.877077i	$0.659491\pi$
$108$	0	0
$109$	1.19601e11	0.744540	0.372270	−	0.928124i	$-0.378579\pi$
$109$	1.19601e11	0.744540	0.372270	+	0.928124i	$0.378579\pi$
$110$	0	0
$111$	1.16401e10	0.0655664
$112$	0	0
$113$	2.69277e10	0.137489	0.0687445	−	0.997634i	$-0.478101\pi$
$113$	2.69277e10	0.137489	0.0687445	+	0.997634i	$0.478101\pi$
$114$	0	0
$115$	5.61139e9	0.0260155
$116$	0	0
$117$	3.56314e11	1.50249
$118$	0	0
$119$	4.79591e11	1.84231
$120$	0	0
$121$	−1.38233e11	−0.484498
$122$	0	0
$123$	−7.19138e11	−2.30321
$124$	0	0
$125$	3.30718e11	0.969289
$126$	0	0
$127$	1.63631e11	0.439486	0.219743	−	0.975558i	$-0.429478\pi$
$127$	1.63631e11	0.439486	0.219743	+	0.975558i	$0.429478\pi$
$128$	0	0
$129$	1.34895e11	0.332469
$130$	0	0
$131$	2.54332e11	0.575981	0.287991	−	0.957633i	$-0.407013\pi$
$131$	2.54332e11	0.575981	0.287991	+	0.957633i	$0.407013\pi$
$132$	0	0
$133$	−1.03015e12	−2.14643
$134$	0	0
$135$	3.70688e11	0.711496
$136$	0	0
$137$	1.05270e12	1.86355	0.931776	−	0.363032i	$-0.118259\pi$
$137$	1.05270e12	1.86355	0.931776	+	0.363032i	$0.118259\pi$
$138$	0	0
$139$	9.02335e11	1.47498	0.737491	−	0.675357i	$-0.236011\pi$
$139$	9.02335e11	1.47498	0.737491	+	0.675357i	$0.236011\pi$
$140$	0	0
$141$	−8.07607e11	−1.22038
$142$	0	0
$143$	4.44361e11	0.621423
$144$	0	0
$145$	3.04759e11	0.394850
$146$	0	0
$147$	−2.39374e12	−2.87628
$148$	0	0
$149$	1.08288e12	1.20797	0.603986	−	0.796995i	$-0.293578\pi$
$149$	1.08288e12	1.20797	0.603986	+	0.796995i	$0.293578\pi$
$150$	0	0
$151$	−8.99725e11	−0.932688	−0.466344	−	0.884603i	$-0.654429\pi$
$151$	−8.99725e11	−0.932688	−0.466344	+	0.884603i	$0.654429\pi$
$152$	0	0
$153$	−2.00408e12	−1.93247
$154$	0	0
$155$	5.40323e10	0.0485098
$156$	0	0
$157$	−6.05620e11	−0.506701	−0.253351	−	0.967375i	$-0.581533\pi$
$157$	−6.05620e11	−0.506701	−0.253351	+	0.967375i	$0.581533\pi$
$158$	0	0
$159$	−3.09356e11	−0.241421
$160$	0	0
$161$	1.01110e11	0.0736637
$162$	0	0
$163$	−5.92020e11	−0.402999	−0.201500	−	0.979489i	$-0.564582\pi$
$163$	−5.92020e11	−0.402999	−0.201500	+	0.979489i	$0.564582\pi$
$164$	0	0
$165$	1.09043e12	0.694124
$166$	0	0
$167$	5.67821e10	0.0338276	0.0169138	−	0.999857i	$-0.494616\pi$
$167$	5.67821e10	0.0338276	0.0169138	+	0.999857i	$0.494616\pi$
$168$	0	0
$169$	−4.49639e11	−0.250892
$170$	0	0
$171$	4.30473e12	2.25147
$172$	0	0
$173$	2.27930e12	1.11828	0.559138	−	0.829075i	$-0.311132\pi$
$173$	2.27930e12	1.11828	0.559138	+	0.829075i	$0.311132\pi$
$174$	0	0
$175$	2.36581e12	1.08961
$176$	0	0
$177$	−8.93228e10	−0.0386464
$178$	0	0
$179$	1.91742e12	0.779876	0.389938	−	0.920841i	$-0.372497\pi$
$179$	1.91742e12	0.779876	0.389938	+	0.920841i	$0.372497\pi$
$180$	0	0
$181$	−3.61293e12	−1.38238	−0.691190	−	0.722673i	$-0.742913\pi$
$181$	−3.61293e12	−1.38238	−0.691190	+	0.722673i	$0.742913\pi$
$182$	0	0
$183$	−5.54226e12	−1.99621
$184$	0	0
$185$	6.82871e10	0.0231683
$186$	0	0
$187$	−2.49930e12	−0.799263
$188$	0	0
$189$	6.67934e12	2.01462
$190$	0	0
$191$	6.07217e11	0.172846	0.0864232	−	0.996259i	$-0.472456\pi$
$191$	6.07217e11	0.172846	0.0864232	+	0.996259i	$0.472456\pi$
$192$	0	0
$193$	4.64252e12	1.24793	0.623963	−	0.781454i	$-0.285522\pi$
$193$	4.64252e12	1.24793	0.623963	+	0.781454i	$0.285522\pi$
$194$	0	0
$195$	3.29446e12	0.836746
$196$	0	0
$197$	−4.86139e12	−1.16734	−0.583668	−	0.811992i	$-0.698383\pi$
$197$	−4.86139e12	−1.16734	−0.583668	+	0.811992i	$0.698383\pi$
$198$	0	0
$199$	−6.52349e12	−1.48180	−0.740898	−	0.671618i	$-0.765600\pi$
$199$	−6.52349e12	−1.48180	−0.740898	+	0.671618i	$0.765600\pi$
$200$	0	0
$201$	−4.79837e12	−1.03161
$202$	0	0
$203$	5.49138e12	1.11803
$204$	0	0
$205$	−4.21884e12	−0.813855
$206$	0	0
$207$	−4.22513e11	−0.0772686
$208$	0	0
$209$	5.36844e12	0.931202
$210$	0	0
$211$	−1.89227e12	−0.311479	−0.155740	−	0.987798i	$-0.549776\pi$
$211$	−1.89227e12	−0.311479	−0.155740	+	0.987798i	$0.549776\pi$
$212$	0	0
$213$	−7.84583e12	−1.22617
$214$	0	0
$215$	7.91364e11	0.117480
$216$	0	0
$217$	9.73597e11	0.137357
$218$	0	0
$219$	−2.33219e12	−0.312839
$220$	0	0
$221$	−7.55099e12	−0.963488
$222$	0	0
$223$	−6.33591e12	−0.769365	−0.384682	−	0.923049i	$-0.625689\pi$
$223$	−6.33591e12	−0.769365	−0.384682	+	0.923049i	$0.625689\pi$
$224$	0	0
$225$	−9.88606e12	−1.14293
$226$	0	0
$227$	−4.95535e12	−0.545673	−0.272837	−	0.962060i	$-0.587962\pi$
$227$	−4.95535e12	−0.545673	−0.272837	+	0.962060i	$0.587962\pi$
$228$	0	0
$229$	1.55285e13	1.62943	0.814714	−	0.579863i	$-0.196894\pi$
$229$	1.55285e13	1.62943	0.814714	+	0.579863i	$0.196894\pi$
$230$	0	0
$231$	1.96483e13	1.96543
$232$	0	0
$233$	7.75664e12	0.739973	0.369986	−	0.929037i	$-0.379362\pi$
$233$	7.75664e12	0.739973	0.369986	+	0.929037i	$0.379362\pi$
$234$	0	0
$235$	−4.73784e12	−0.431229
$236$	0	0
$237$	3.73317e13	3.24311
$238$	0	0
$239$	−4.18729e12	−0.347332	−0.173666	−	0.984805i	$-0.555561\pi$
$239$	−4.18729e12	−0.347332	−0.173666	+	0.984805i	$0.555561\pi$
$240$	0	0
$241$	8.28156e12	0.656173	0.328087	−	0.944648i	$-0.393596\pi$
$241$	8.28156e12	0.656173	0.328087	+	0.944648i	$0.393596\pi$
$242$	0	0
$243$	1.00140e13	0.758180
$244$	0	0
$245$	−1.40429e13	−1.01635
$246$	0	0
$247$	1.62194e13	1.12254
$248$	0	0
$249$	−4.39772e13	−2.91160
$250$	0	0
$251$	6.31791e11	0.0400284	0.0200142	−	0.999800i	$-0.493629\pi$
$251$	6.31791e11	0.0400284	0.0200142	+	0.999800i	$0.493629\pi$
$252$	0	0
$253$	−5.26918e11	−0.0319580
$254$	0	0
$255$	−1.85296e13	−1.07621
$256$	0	0
$257$	1.76085e13	0.979691	0.489845	−	0.871809i	$-0.337053\pi$
$257$	1.76085e13	0.979691	0.489845	+	0.871809i	$0.337053\pi$
$258$	0	0
$259$	1.23045e12	0.0656018
$260$	0	0
$261$	−2.29470e13	−1.17274
$262$	0	0
$263$	2.18141e13	1.06901	0.534505	−	0.845165i	$-0.320498\pi$
$263$	2.18141e13	1.06901	0.534505	+	0.845165i	$0.320498\pi$
$264$	0	0
$265$	−1.81485e12	−0.0853076
$266$	0	0
$267$	−6.69985e13	−3.02171
$268$	0	0
$269$	−1.09453e13	−0.473795	−0.236897	−	0.971535i	$-0.576131\pi$
$269$	−1.09453e13	−0.473795	−0.236897	+	0.971535i	$0.576131\pi$
$270$	0	0
$271$	4.38029e13	1.82042	0.910210	−	0.414146i	$-0.135920\pi$
$271$	4.38029e13	1.82042	0.910210	+	0.414146i	$0.135920\pi$
$272$	0	0
$273$	5.93622e13	2.36927
$274$	0	0
$275$	−1.23290e13	−0.472712
$276$	0	0
$277$	−3.83236e13	−1.41198	−0.705988	−	0.708224i	$-0.749497\pi$
$277$	−3.83236e13	−1.41198	−0.705988	+	0.708224i	$0.749497\pi$
$278$	0	0
$279$	−4.06840e12	−0.144079
$280$	0	0
$281$	−1.88363e13	−0.641372	−0.320686	−	0.947186i	$-0.603913\pi$
$281$	−1.88363e13	−0.641372	−0.320686	+	0.947186i	$0.603913\pi$
$282$	0	0
$283$	3.59883e12	0.117852	0.0589258	−	0.998262i	$-0.481232\pi$
$283$	3.59883e12	0.117852	0.0589258	+	0.998262i	$0.481232\pi$
$284$	0	0
$285$	3.98013e13	1.25386
$286$	0	0
$287$	−7.60185e13	−2.30446
$288$	0	0
$289$	8.19853e12	0.239220
$290$	0	0
$291$	3.04555e13	0.855568
$292$	0	0
$293$	−1.89960e13	−0.513913	−0.256957	−	0.966423i	$-0.582720\pi$
$293$	−1.89960e13	−0.513913	−0.256957	+	0.966423i	$0.582720\pi$
$294$	0	0
$295$	−5.24015e11	−0.0136560
$296$	0	0
$297$	−3.48081e13	−0.874018
$298$	0	0
$299$	−1.59195e12	−0.0385245
$300$	0	0
$301$	1.42594e13	0.332649
$302$	0	0
$303$	1.17001e14	2.63181
$304$	0	0
$305$	−3.25138e13	−0.705373
$306$	0	0
$307$	5.07622e13	1.06238	0.531189	−	0.847253i	$-0.321745\pi$
$307$	5.07622e13	1.06238	0.531189	+	0.847253i	$0.321745\pi$
$308$	0	0
$309$	4.19326e13	0.846799
$310$	0	0
$311$	−4.89908e13	−0.954844	−0.477422	−	0.878674i	$-0.658429\pi$
$311$	−4.89908e13	−0.954844	−0.477422	+	0.878674i	$0.658429\pi$
$312$	0	0
$313$	2.02897e13	0.381752	0.190876	−	0.981614i	$-0.438867\pi$
$313$	2.02897e13	0.381752	0.190876	+	0.981614i	$0.438867\pi$
$314$	0	0
$315$	9.24277e13	1.67917
$316$	0	0
$317$	5.80299e13	1.01818	0.509091	−	0.860712i	$-0.329982\pi$
$317$	5.80299e13	1.01818	0.509091	+	0.860712i	$0.329982\pi$
$318$	0	0
$319$	−2.86173e13	−0.485043
$320$	0	0
$321$	9.70332e13	1.58907
$322$	0	0
$323$	−9.12256e13	−1.44379
$324$	0	0
$325$	−3.72488e13	−0.569841
$326$	0	0
$327$	−8.32636e13	−1.23152
$328$	0	0
$329$	−8.53702e13	−1.22104
$330$	0	0
$331$	1.94810e13	0.269500	0.134750	−	0.990880i	$-0.456977\pi$
$331$	1.94810e13	0.269500	0.134750	+	0.990880i	$0.456977\pi$
$332$	0	0
$333$	−5.14172e12	−0.0688122
$334$	0	0
$335$	−2.81497e13	−0.364526
$336$	0	0
$337$	−1.11994e14	−1.40356	−0.701782	−	0.712392i	$-0.747612\pi$
$337$	−1.11994e14	−1.40356	−0.701782	+	0.712392i	$0.747612\pi$
$338$	0	0
$339$	−1.87465e13	−0.227417
$340$	0	0
$341$	−5.07372e12	−0.0595905
$342$	0	0
$343$	−1.07522e14	−1.22287
$344$	0	0
$345$	−3.90653e12	−0.0430315
$346$	0	0
$347$	1.43917e14	1.53568	0.767841	−	0.640640i	$-0.221331\pi$
$347$	1.43917e14	1.53568	0.767841	+	0.640640i	$0.221331\pi$
$348$	0	0
$349$	1.17413e14	1.21389	0.606943	−	0.794745i	$-0.292396\pi$
$349$	1.17413e14	1.21389	0.606943	+	0.794745i	$0.292396\pi$
$350$	0	0
$351$	−1.05164e14	−1.05360
$352$	0	0
$353$	−3.14959e12	−0.0305839	−0.0152919	−	0.999883i	$-0.504868\pi$
$353$	−3.14959e12	−0.0305839	−0.0152919	+	0.999883i	$0.504868\pi$
$354$	0	0
$355$	−4.60278e13	−0.433275
$356$	0	0
$357$	−3.33882e14	−3.04732
$358$	0	0
$359$	2.16645e14	1.91748	0.958738	−	0.284291i	$-0.0917583\pi$
$359$	2.16645e14	1.91748	0.958738	+	0.284291i	$0.0917583\pi$
$360$	0	0
$361$	7.94605e13	0.682121
$362$	0	0
$363$	9.62349e13	0.801394
$364$	0	0
$365$	−1.36818e13	−0.110544
$366$	0	0
$367$	1.32924e14	1.04217	0.521085	−	0.853505i	$-0.325527\pi$
$367$	1.32924e14	1.04217	0.521085	+	0.853505i	$0.325527\pi$
$368$	0	0
$369$	3.17660e14	2.41723
$370$	0	0
$371$	−3.27014e13	−0.241551
$372$	0	0
$373$	4.97449e13	0.356738	0.178369	−	0.983964i	$-0.442918\pi$
$373$	4.97449e13	0.356738	0.178369	+	0.983964i	$0.442918\pi$
$374$	0	0
$375$	−2.30239e14	−1.60327
$376$	0	0
$377$	−8.64598e13	−0.584705
$378$	0	0
$379$	−6.41630e13	−0.421472	−0.210736	−	0.977543i	$-0.567586\pi$
$379$	−6.41630e13	−0.421472	−0.210736	+	0.977543i	$0.567586\pi$
$380$	0	0
$381$	−1.13917e14	−0.726942
$382$	0	0
$383$	−1.53711e14	−0.953040	−0.476520	−	0.879164i	$-0.658102\pi$
$383$	−1.53711e14	−0.953040	−0.476520	+	0.879164i	$0.658102\pi$
$384$	0	0
$385$	1.15267e14	0.694498
$386$	0	0
$387$	−5.95862e13	−0.348928
$388$	0	0
$389$	3.08115e14	1.75384	0.876921	−	0.480635i	$-0.159594\pi$
$389$	3.08115e14	1.75384	0.876921	+	0.480635i	$0.159594\pi$
$390$	0	0
$391$	8.95388e12	0.0495495
$392$	0	0
$393$	−1.77061e14	−0.952714
$394$	0	0
$395$	2.19008e14	1.14597
$396$	0	0
$397$	2.15252e14	1.09547	0.547734	−	0.836653i	$-0.315491\pi$
$397$	2.15252e14	1.09547	0.547734	+	0.836653i	$0.315491\pi$
$398$	0	0
$399$	7.17171e14	3.55035
$400$	0	0
$401$	−4.29414e13	−0.206815	−0.103407	−	0.994639i	$-0.532975\pi$
$401$	−4.29414e13	−0.206815	−0.103407	+	0.994639i	$0.532975\pi$
$402$	0	0
$403$	−1.53289e13	−0.0718347
$404$	0	0
$405$	−3.55764e13	−0.162241
$406$	0	0
$407$	−6.41227e12	−0.0284605
$408$	0	0
$409$	−1.78142e14	−0.769639	−0.384819	−	0.922992i	$-0.625736\pi$
$409$	−1.78142e14	−0.769639	−0.384819	+	0.922992i	$0.625736\pi$
$410$	0	0
$411$	−7.32869e14	−3.08245
$412$	0	0
$413$	−9.44211e12	−0.0386673
$414$	0	0
$415$	−2.57993e14	−1.02883
$416$	0	0
$417$	−6.28187e14	−2.43972
$418$	0	0
$419$	3.36601e14	1.27332	0.636662	−	0.771143i	$-0.280315\pi$
$419$	3.36601e14	1.27332	0.636662	+	0.771143i	$0.280315\pi$
$420$	0	0
$421$	6.36951e13	0.234722	0.117361	−	0.993089i	$-0.462557\pi$
$421$	6.36951e13	0.234722	0.117361	+	0.993089i	$0.462557\pi$
$422$	0	0
$423$	3.56739e14	1.28079
$424$	0	0
$425$	2.09505e14	0.732918
$426$	0	0
$427$	−5.85860e14	−1.99729
$428$	0	0
$429$	−3.09355e14	−1.02788
$430$	0	0
$431$	−1.38853e14	−0.449707	−0.224853	−	0.974393i	$-0.572190\pi$
$431$	−1.38853e14	−0.449707	−0.224853	+	0.974393i	$0.572190\pi$
$432$	0	0
$433$	−3.07253e14	−0.970093	−0.485046	−	0.874488i	$-0.661197\pi$
$433$	−3.07253e14	−0.970093	−0.485046	+	0.874488i	$0.661197\pi$
$434$	0	0
$435$	−2.12167e14	−0.653110
$436$	0	0
$437$	−1.92328e13	−0.0577289
$438$	0	0
$439$	2.37747e14	0.695921	0.347961	−	0.937509i	$-0.386874\pi$
$439$	2.37747e14	0.695921	0.347961	+	0.937509i	$0.386874\pi$
$440$	0	0
$441$	1.05737e15	3.01866
$442$	0	0
$443$	6.82803e13	0.190141	0.0950703	−	0.995471i	$-0.469692\pi$
$443$	6.82803e13	0.190141	0.0950703	+	0.995471i	$0.469692\pi$
$444$	0	0
$445$	−3.93048e14	−1.06774
$446$	0	0
$447$	−7.53881e14	−1.99807
$448$	0	0
$449$	7.19557e14	1.86084	0.930422	−	0.366490i	$-0.119440\pi$
$449$	7.19557e14	1.86084	0.930422	+	0.366490i	$0.119440\pi$
$450$	0	0
$451$	3.96156e14	0.999758
$452$	0	0
$453$	6.26370e14	1.54273
$454$	0	0
$455$	3.48250e14	0.837198
$456$	0	0
$457$	−5.95803e14	−1.39818	−0.699091	−	0.715033i	$-0.746412\pi$
$457$	−5.95803e14	−1.39818	−0.699091	+	0.715033i	$0.746412\pi$
$458$	0	0
$459$	5.91492e14	1.35513
$460$	0	0
$461$	−2.48547e14	−0.555973	−0.277987	−	0.960585i	$-0.589667\pi$
$461$	−2.48547e14	−0.555973	−0.277987	+	0.960585i	$0.589667\pi$
$462$	0	0
$463$	5.55882e14	1.21419	0.607096	−	0.794629i	$-0.292334\pi$
$463$	5.55882e14	1.21419	0.607096	+	0.794629i	$0.292334\pi$
$464$	0	0
$465$	−3.76162e13	−0.0802386
$466$	0	0
$467$	−5.11081e13	−0.106475	−0.0532374	−	0.998582i	$-0.516954\pi$
$467$	−5.11081e13	−0.106475	−0.0532374	+	0.998582i	$0.516954\pi$
$468$	0	0
$469$	−5.07224e14	−1.03217
$470$	0	0
$471$	4.21620e14	0.838121
$472$	0	0
$473$	−7.43103e13	−0.144315
$474$	0	0
$475$	−4.50013e14	−0.853906
$476$	0	0
$477$	1.36650e14	0.253372
$478$	0	0
$479$	−6.79608e14	−1.23144	−0.615720	−	0.787965i	$-0.711135\pi$
$479$	−6.79608e14	−1.23144	−0.615720	+	0.787965i	$0.711135\pi$
$480$	0	0
$481$	−1.93730e13	−0.0343083
$482$	0	0
$483$	−7.03910e13	−0.121845
$484$	0	0
$485$	1.78668e14	0.302320
$486$	0	0
$487$	3.50601e14	0.579967	0.289984	−	0.957032i	$-0.406350\pi$
$487$	3.50601e14	0.579967	0.289984	+	0.957032i	$0.406350\pi$
$488$	0	0
$489$	4.12152e14	0.666590
$490$	0	0
$491$	5.60616e14	0.886579	0.443289	−	0.896379i	$-0.353811\pi$
$491$	5.60616e14	0.886579	0.443289	+	0.896379i	$0.353811\pi$
$492$	0	0
$493$	4.86292e14	0.752037
$494$	0	0
$495$	−4.81670e14	−0.728485
$496$	0	0
$497$	−8.29365e14	−1.22683
$498$	0	0
$499$	2.60119e14	0.376374	0.188187	−	0.982133i	$-0.439739\pi$
$499$	2.60119e14	0.376374	0.188187	+	0.982133i	$0.439739\pi$
$500$	0	0
$501$	−3.95306e13	−0.0559533
$502$	0	0
$503$	−1.34386e15	−1.86093	−0.930466	−	0.366377i	$-0.880598\pi$
$503$	−1.34386e15	−1.86093	−0.930466	+	0.366377i	$0.880598\pi$
$504$	0	0
$505$	6.86388e14	0.929966
$506$	0	0
$507$	3.13029e14	0.414994
$508$	0	0
$509$	3.28205e14	0.425792	0.212896	−	0.977075i	$-0.431710\pi$
$509$	3.28205e14	0.425792	0.212896	+	0.977075i	$0.431710\pi$
$510$	0	0
$511$	−2.46530e14	−0.313007
$512$	0	0
$513$	−1.27051e15	−1.57882
$514$	0	0
$515$	2.45999e14	0.299222
$516$	0	0
$517$	4.44891e14	0.529731
$518$	0	0
$519$	−1.58681e15	−1.84971
$520$	0	0
$521$	3.73999e14	0.426838	0.213419	−	0.976961i	$-0.431540\pi$
$521$	3.73999e14	0.426838	0.213419	+	0.976961i	$0.431540\pi$
$522$	0	0
$523$	−7.77988e13	−0.0869388	−0.0434694	−	0.999055i	$-0.513841\pi$
$523$	−7.77988e13	−0.0869388	−0.0434694	+	0.999055i	$0.513841\pi$
$524$	0	0
$525$	−1.64703e15	−1.80229
$526$	0	0
$527$	8.62173e13	0.0923924
$528$	0	0
$529$	−9.50922e14	−0.998019
$530$	0	0
$531$	3.94560e13	0.0405596
$532$	0	0
$533$	1.19688e15	1.20518
$534$	0	0
$535$	5.69248e14	0.561507
$536$	0	0
$537$	−1.33487e15	−1.28997
$538$	0	0
$539$	1.31865e15	1.24851
$540$	0	0
$541$	−1.17909e15	−1.09386	−0.546929	−	0.837179i	$-0.684203\pi$
$541$	−1.17909e15	−1.09386	−0.546929	+	0.837179i	$0.684203\pi$
$542$	0	0
$543$	2.51525e15	2.28656
$544$	0	0
$545$	−4.88468e14	−0.435167
$546$	0	0
$547$	−7.58961e14	−0.662658	−0.331329	−	0.943515i	$-0.607497\pi$
$547$	−7.58961e14	−0.662658	−0.331329	+	0.943515i	$0.607497\pi$
$548$	0	0
$549$	2.44815e15	2.09503
$550$	0	0
$551$	−1.04455e15	−0.876180
$552$	0	0
$553$	3.94625e15	3.24486
$554$	0	0
$555$	−4.75401e13	−0.0383221
$556$	0	0
$557$	−9.88953e14	−0.781578	−0.390789	−	0.920480i	$-0.627798\pi$
$557$	−9.88953e14	−0.781578	−0.390789	+	0.920480i	$0.627798\pi$
$558$	0	0
$559$	−2.24509e14	−0.173968
$560$	0	0
$561$	1.73996e15	1.32204
$562$	0	0
$563$	1.70471e15	1.27015	0.635075	−	0.772450i	$-0.280969\pi$
$563$	1.70471e15	1.27015	0.635075	+	0.772450i	$0.280969\pi$
$564$	0	0
$565$	−1.09977e14	−0.0803591
$566$	0	0
$567$	−6.41044e14	−0.459389
$568$	0	0
$569$	5.26889e14	0.370341	0.185170	−	0.982706i	$-0.440716\pi$
$569$	5.26889e14	0.370341	0.185170	+	0.982706i	$0.440716\pi$
$570$	0	0
$571$	6.85464e14	0.472592	0.236296	−	0.971681i	$-0.424067\pi$
$571$	6.85464e14	0.472592	0.236296	+	0.971681i	$0.424067\pi$
$572$	0	0
$573$	−4.22732e14	−0.285900
$574$	0	0
$575$	4.41692e13	0.0293053
$576$	0	0
$577$	1.25421e15	0.816402	0.408201	−	0.912892i	$-0.366156\pi$
$577$	1.25421e15	0.816402	0.408201	+	0.912892i	$0.366156\pi$
$578$	0	0
$579$	−3.23203e15	−2.06416
$580$	0	0
$581$	−4.64873e15	−2.91317
$582$	0	0
$583$	1.70417e14	0.104794
$584$	0	0
$585$	−1.45524e15	−0.878168
$586$	0	0
$587$	−2.56777e15	−1.52071	−0.760356	−	0.649507i	$-0.774975\pi$
$587$	−2.56777e15	−1.52071	−0.760356	+	0.649507i	$0.774975\pi$
$588$	0	0
$589$	−1.85193e14	−0.107644
$590$	0	0
$591$	3.38440e15	1.93086
$592$	0	0
$593$	−1.05019e15	−0.588123	−0.294062	−	0.955786i	$-0.595007\pi$
$593$	−1.05019e15	−0.588123	−0.294062	+	0.955786i	$0.595007\pi$
$594$	0	0
$595$	−1.95873e15	−1.07679
$596$	0	0
$597$	4.54152e15	2.45100
$598$	0	0
$599$	−1.16345e14	−0.0616455	−0.0308227	−	0.999525i	$-0.509813\pi$
$599$	−1.16345e14	−0.0616455	−0.0308227	+	0.999525i	$0.509813\pi$
$600$	0	0
$601$	−9.27714e14	−0.482619	−0.241310	−	0.970448i	$-0.577577\pi$
$601$	−9.27714e14	−0.482619	−0.241310	+	0.970448i	$0.577577\pi$
$602$	0	0
$603$	2.11955e15	1.08268
$604$	0	0
$605$	5.64565e14	0.283178
$606$	0	0
$607$	9.37322e14	0.461691	0.230845	−	0.972990i	$-0.425851\pi$
$607$	9.37322e14	0.461691	0.230845	+	0.972990i	$0.425851\pi$
$608$	0	0
$609$	−3.82299e15	−1.84930
$610$	0	0
$611$	1.34412e15	0.638576
$612$	0	0
$613$	9.73439e13	0.0454230	0.0227115	−	0.999742i	$-0.492770\pi$
$613$	9.73439e13	0.0454230	0.0227115	+	0.999742i	$0.492770\pi$
$614$	0	0
$615$	2.93707e15	1.34618
$616$	0	0
$617$	−2.70501e15	−1.21787	−0.608935	−	0.793220i	$-0.708403\pi$
$617$	−2.70501e15	−1.21787	−0.608935	+	0.793220i	$0.708403\pi$
$618$	0	0
$619$	−5.22797e14	−0.231225	−0.115612	−	0.993294i	$-0.536883\pi$
$619$	−5.22797e14	−0.231225	−0.115612	+	0.993294i	$0.536883\pi$
$620$	0	0
$621$	1.24702e14	0.0541839
$622$	0	0
$623$	−7.08225e15	−3.02334
$624$	0	0
$625$	2.19012e14	0.0918603
$626$	0	0
$627$	−3.73740e15	−1.54027
$628$	0	0
$629$	1.08963e14	0.0441267
$630$	0	0
$631$	1.58461e15	0.630611	0.315305	−	0.948990i	$-0.397893\pi$
$631$	1.58461e15	0.630611	0.315305	+	0.948990i	$0.397893\pi$
$632$	0	0
$633$	1.31736e15	0.515209
$634$	0	0
$635$	−6.68295e14	−0.256869
$636$	0	0
$637$	3.98397e15	1.50504
$638$	0	0
$639$	3.46569e15	1.28687
$640$	0	0
$641$	3.67621e15	1.34178	0.670890	−	0.741557i	$-0.265912\pi$
$641$	3.67621e15	1.34178	0.670890	+	0.741557i	$0.265912\pi$
$642$	0	0
$643$	−1.40268e15	−0.503268	−0.251634	−	0.967822i	$-0.580968\pi$
$643$	−1.40268e15	−0.503268	−0.251634	+	0.967822i	$0.580968\pi$
$644$	0	0
$645$	−5.50931e14	−0.194321
$646$	0	0
$647$	−7.65400e14	−0.265409	−0.132704	−	0.991156i	$-0.542366\pi$
$647$	−7.65400e14	−0.265409	−0.132704	+	0.991156i	$0.542366\pi$
$648$	0	0
$649$	4.92058e13	0.0167753
$650$	0	0
$651$	−6.77798e14	−0.227198
$652$	0	0
$653$	−3.06409e15	−1.00990	−0.504951	−	0.863148i	$-0.668490\pi$
$653$	−3.06409e15	−1.00990	−0.504951	+	0.863148i	$0.668490\pi$
$654$	0	0
$655$	−1.03873e15	−0.336648
$656$	0	0
$657$	1.03018e15	0.328325
$658$	0	0
$659$	5.49550e15	1.72241	0.861207	−	0.508254i	$-0.169709\pi$
$659$	5.49550e15	1.72241	0.861207	+	0.508254i	$0.169709\pi$
$660$	0	0
$661$	−4.42413e15	−1.36370	−0.681851	−	0.731491i	$-0.738825\pi$
$661$	−4.42413e15	−1.36370	−0.681851	+	0.731491i	$0.738825\pi$
$662$	0	0
$663$	5.25685e15	1.59368
$664$	0	0
$665$	4.20730e15	1.25454
$666$	0	0
$667$	1.02523e14	0.0300697
$668$	0	0
$669$	4.41093e15	1.27258
$670$	0	0
$671$	3.05310e15	0.866497
$672$	0	0
$673$	−3.72652e15	−1.04045	−0.520225	−	0.854029i	$-0.674152\pi$
$673$	−3.72652e15	−1.04045	−0.520225	+	0.854029i	$0.674152\pi$
$674$	0	0
$675$	2.91781e15	0.801468
$676$	0	0
$677$	3.80105e15	1.02723	0.513613	−	0.858022i	$-0.328307\pi$
$677$	3.80105e15	1.02723	0.513613	+	0.858022i	$0.328307\pi$
$678$	0	0
$679$	3.21938e15	0.856030
$680$	0	0
$681$	3.44982e15	0.902582
$682$	0	0
$683$	−4.06733e15	−1.04712	−0.523558	−	0.851990i	$-0.675396\pi$
$683$	−4.06733e15	−1.04712	−0.523558	+	0.851990i	$0.675396\pi$
$684$	0	0
$685$	−4.29939e15	−1.08920
$686$	0	0
$687$	−1.08106e16	−2.69519
$688$	0	0
$689$	5.14871e14	0.126326
$690$	0	0
$691$	5.14416e15	1.24218	0.621091	−	0.783739i	$-0.286690\pi$
$691$	5.14416e15	1.24218	0.621091	+	0.783739i	$0.286690\pi$
$692$	0	0
$693$	−8.67910e15	−2.06273
$694$	0	0
$695$	−3.68528e15	−0.862092
$696$	0	0
$697$	−6.73185e15	−1.55008
$698$	0	0
$699$	−5.40001e15	−1.22397
$700$	0	0
$701$	−2.86572e15	−0.639417	−0.319708	−	0.947516i	$-0.603585\pi$
$701$	−2.86572e15	−0.639417	−0.319708	+	0.947516i	$0.603585\pi$
$702$	0	0
$703$	−2.34051e14	−0.0514110
$704$	0	0
$705$	3.29839e15	0.713283
$706$	0	0
$707$	1.23679e16	2.63323
$708$	0	0
$709$	5.54221e15	1.16179	0.580896	−	0.813978i	$-0.302702\pi$
$709$	5.54221e15	1.16179	0.580896	+	0.813978i	$0.302702\pi$
$710$	0	0
$711$	−1.64903e16	−3.40366
$712$	0	0
$713$	1.81769e13	0.00369425
$714$	0	0
$715$	−1.81484e15	−0.363207
$716$	0	0
$717$	2.91511e15	0.574512
$718$	0	0
$719$	−4.73660e15	−0.919301	−0.459651	−	0.888100i	$-0.652025\pi$
$719$	−4.73660e15	−0.919301	−0.459651	+	0.888100i	$0.652025\pi$
$720$	0	0
$721$	4.43260e15	0.847256
$722$	0	0
$723$	−5.76545e15	−1.08536
$724$	0	0
$725$	2.39886e15	0.444781
$726$	0	0
$727$	1.01777e16	1.85870	0.929348	−	0.369205i	$-0.120370\pi$
$727$	1.01777e16	1.85870	0.929348	+	0.369205i	$0.120370\pi$
$728$	0	0
$729$	−8.51462e15	−1.53167
$730$	0	0
$731$	1.26275e15	0.223754
$732$	0	0
$733$	5.29486e15	0.924235	0.462117	−	0.886819i	$-0.347090\pi$
$733$	5.29486e15	0.924235	0.462117	+	0.886819i	$0.347090\pi$
$734$	0	0
$735$	9.77639e15	1.68112
$736$	0	0
$737$	2.64330e15	0.447792
$738$	0	0
$739$	−4.55855e15	−0.760821	−0.380411	−	0.924818i	$-0.624217\pi$
$739$	−4.55855e15	−0.760821	−0.380411	+	0.924818i	$0.624217\pi$
$740$	0	0
$741$	−1.12916e16	−1.85676
$742$	0	0
$743$	−9.91668e14	−0.160667	−0.0803337	−	0.996768i	$-0.525599\pi$
$743$	−9.91668e14	−0.160667	−0.0803337	+	0.996768i	$0.525599\pi$
$744$	0	0
$745$	−4.42266e15	−0.706032
$746$	0	0
$747$	1.94258e16	3.05573
$748$	0	0
$749$	1.02572e16	1.58993
$750$	0	0
$751$	−5.23831e15	−0.800151	−0.400075	−	0.916482i	$-0.631016\pi$
$751$	−5.23831e15	−0.800151	−0.400075	+	0.916482i	$0.631016\pi$
$752$	0	0
$753$	−4.39840e14	−0.0662098
$754$	0	0
$755$	3.67461e15	0.545134
$756$	0	0
$757$	−7.31380e15	−1.06934	−0.534670	−	0.845061i	$-0.679564\pi$
$757$	−7.31380e15	−1.06934	−0.534670	+	0.845061i	$0.679564\pi$
$758$	0	0
$759$	3.66829e14	0.0528609
$760$	0	0
$761$	−7.81344e15	−1.10975	−0.554877	−	0.831932i	$-0.687235\pi$
$761$	−7.81344e15	−1.10975	−0.554877	+	0.831932i	$0.687235\pi$
$762$	0	0
$763$	−8.80161e15	−1.23219
$764$	0	0
$765$	8.18498e15	1.12948
$766$	0	0
$767$	1.48663e14	0.0202222
$768$	0	0
$769$	2.19554e15	0.294405	0.147203	−	0.989106i	$-0.452973\pi$
$769$	2.19554e15	0.294405	0.147203	+	0.989106i	$0.452973\pi$
$770$	0	0
$771$	−1.22586e16	−1.62048
$772$	0	0
$773$	1.32704e16	1.72940	0.864702	−	0.502286i	$-0.167507\pi$
$773$	1.32704e16	1.72940	0.864702	+	0.502286i	$0.167507\pi$
$774$	0	0
$775$	4.25307e14	0.0546441
$776$	0	0
$777$	−8.56615e14	−0.108510
$778$	0	0
$779$	1.44599e16	1.80596
$780$	0	0
$781$	4.32208e15	0.532246
$782$	0	0
$783$	6.77266e15	0.822375
$784$	0	0
$785$	2.47345e15	0.296155
$786$	0	0
$787$	−1.40226e15	−0.165565	−0.0827826	−	0.996568i	$-0.526381\pi$
$787$	−1.40226e15	−0.165565	−0.0827826	+	0.996568i	$0.526381\pi$
$788$	0	0
$789$	−1.51866e16	−1.76822
$790$	0	0
$791$	−1.98165e15	−0.227539
$792$	0	0
$793$	9.22415e15	1.04454
$794$	0	0
$795$	1.26346e15	0.141105
$796$	0	0
$797$	3.09627e15	0.341049	0.170525	−	0.985353i	$-0.445454\pi$
$797$	3.09627e15	0.341049	0.170525	+	0.985353i	$0.445454\pi$
$798$	0	0
$799$	−7.56000e15	−0.821324
$800$	0	0
$801$	2.95948e16	3.17130
$802$	0	0
$803$	1.28474e15	0.135794
$804$	0	0
$805$	−4.12951e14	−0.0430547
$806$	0	0
$807$	7.61990e15	0.783691
$808$	0	0
$809$	−1.62765e16	−1.65137	−0.825683	−	0.564135i	$-0.809210\pi$
$809$	−1.62765e16	−1.65137	−0.825683	+	0.564135i	$0.809210\pi$
$810$	0	0
$811$	−1.55106e16	−1.55244	−0.776219	−	0.630463i	$-0.782865\pi$
$811$	−1.55106e16	−1.55244	−0.776219	+	0.630463i	$0.782865\pi$
$812$	0	0
$813$	−3.04947e16	−3.01111
$814$	0	0
$815$	2.41790e15	0.235544
$816$	0	0
$817$	−2.71236e15	−0.260691
$818$	0	0
$819$	−2.62217e16	−2.48656
$820$	0	0
$821$	−5.57057e15	−0.521209	−0.260605	−	0.965446i	$-0.583922\pi$
$821$	−5.57057e15	−0.521209	−0.260605	+	0.965446i	$0.583922\pi$
$822$	0	0
$823$	−4.05461e15	−0.374326	−0.187163	−	0.982329i	$-0.559929\pi$
$823$	−4.05461e15	−0.374326	−0.187163	+	0.982329i	$0.559929\pi$
$824$	0	0
$825$	8.58317e15	0.781899
$826$	0	0
$827$	−9.59579e14	−0.0862582	−0.0431291	−	0.999070i	$-0.513733\pi$
$827$	−9.59579e14	−0.0862582	−0.0431291	+	0.999070i	$0.513733\pi$
$828$	0	0
$829$	−1.05269e16	−0.933791	−0.466895	−	0.884313i	$-0.654627\pi$
$829$	−1.05269e16	−0.933791	−0.466895	+	0.884313i	$0.654627\pi$
$830$	0	0
$831$	2.66801e16	2.33551
$832$	0	0
$833$	−2.24078e16	−1.93576
$834$	0	0
$835$	−2.31907e14	−0.0197714
$836$	0	0
$837$	1.20076e15	0.101034
$838$	0	0
$839$	−6.15316e15	−0.510984	−0.255492	−	0.966811i	$-0.582238\pi$
$839$	−6.15316e15	−0.510984	−0.255492	+	0.966811i	$0.582238\pi$
$840$	0	0
$841$	−6.63240e15	−0.543617
$842$	0	0
$843$	1.31134e16	1.06087
$844$	0	0
$845$	1.83640e15	0.146641
$846$	0	0
$847$	1.01728e16	0.801827
$848$	0	0
$849$	−2.50543e15	−0.194935
$850$	0	0
$851$	2.29723e13	0.00176438
$852$	0	0
$853$	1.77560e16	1.34625	0.673126	−	0.739528i	$-0.264951\pi$
$853$	1.77560e16	1.34625	0.673126	+	0.739528i	$0.264951\pi$
$854$	0	0
$855$	−1.75812e16	−1.31593
$856$	0	0
$857$	2.26793e16	1.67585	0.837925	−	0.545785i	$-0.183769\pi$
$857$	2.26793e16	1.67585	0.837925	+	0.545785i	$0.183769\pi$
$858$	0	0
$859$	1.65429e15	0.120684	0.0603421	−	0.998178i	$-0.480781\pi$
$859$	1.65429e15	0.120684	0.0603421	+	0.998178i	$0.480781\pi$
$860$	0	0
$861$	5.29225e16	3.81174
$862$	0	0
$863$	2.31182e15	0.164397	0.0821986	−	0.996616i	$-0.473806\pi$
$863$	2.31182e15	0.164397	0.0821986	+	0.996616i	$0.473806\pi$
$864$	0	0
$865$	−9.30903e15	−0.653606
$866$	0	0
$867$	−5.70765e15	−0.395688
$868$	0	0
$869$	−2.05651e16	−1.40774
$870$	0	0
$871$	7.98606e15	0.539800
$872$	0	0
$873$	−1.34529e16	−0.897921
$874$	0	0
$875$	−2.43381e16	−1.60414
$876$	0	0
$877$	1.39330e16	0.906874	0.453437	−	0.891288i	$-0.350198\pi$
$877$	1.39330e16	0.906874	0.453437	+	0.891288i	$0.350198\pi$
$878$	0	0
$879$	1.32246e16	0.850050
$880$	0	0
$881$	4.04171e15	0.256565	0.128283	−	0.991738i	$-0.459054\pi$
$881$	4.04171e15	0.256565	0.128283	+	0.991738i	$0.459054\pi$
$882$	0	0
$883$	1.26327e16	0.791975	0.395988	−	0.918256i	$-0.370402\pi$
$883$	1.26327e16	0.791975	0.395988	+	0.918256i	$0.370402\pi$
$884$	0	0
$885$	3.64808e14	0.0225880
$886$	0	0
$887$	1.96489e16	1.20160	0.600798	−	0.799401i	$-0.294849\pi$
$887$	1.96489e16	1.20160	0.600798	+	0.799401i	$0.294849\pi$
$888$	0	0
$889$	−1.20419e16	−0.727334
$890$	0	0
$891$	3.34068e15	0.199300
$892$	0	0
$893$	1.62387e16	0.956905
$894$	0	0
$895$	−7.83104e15	−0.455819
$896$	0	0
$897$	1.10828e15	0.0637223
$898$	0	0
$899$	9.87200e14	0.0560695
$900$	0	0
$901$	−2.89588e15	−0.162478
$902$	0	0
$903$	−9.92712e15	−0.550225
$904$	0	0
$905$	1.47558e16	0.807969
$906$	0	0
$907$	−1.75609e15	−0.0949961	−0.0474981	−	0.998871i	$-0.515125\pi$
$907$	−1.75609e15	−0.0949961	−0.0474981	+	0.998871i	$0.515125\pi$
$908$	0	0
$909$	−5.16820e16	−2.76209
$910$	0	0
$911$	−2.30636e16	−1.21780	−0.608899	−	0.793247i	$-0.708389\pi$
$911$	−2.30636e16	−1.21780	−0.608899	+	0.793247i	$0.708389\pi$
$912$	0	0
$913$	2.42260e16	1.26384
$914$	0	0
$915$	2.26355e16	1.16674
$916$	0	0
$917$	−1.87167e16	−0.953229
$918$	0	0
$919$	−9.41404e14	−0.0473741	−0.0236870	−	0.999719i	$-0.507541\pi$
$919$	−9.41404e14	−0.0473741	−0.0236870	+	0.999719i	$0.507541\pi$
$920$	0	0
$921$	−3.53396e16	−1.75725
$922$	0	0
$923$	1.30581e16	0.641607
$924$	0	0
$925$	5.37512e14	0.0260981
$926$	0	0
$927$	−1.85226e16	−0.888718
$928$	0	0
$929$	1.77556e16	0.841876	0.420938	−	0.907089i	$-0.361701\pi$
$929$	1.77556e16	0.841876	0.420938	+	0.907089i	$0.361701\pi$
$930$	0	0
$931$	4.81314e16	2.25530
$932$	0	0
$933$	3.41064e16	1.57938
$934$	0	0
$935$	1.02075e16	0.467150
$936$	0	0
$937$	1.10107e16	0.498020	0.249010	−	0.968501i	$-0.419895\pi$
$937$	1.10107e16	0.498020	0.249010	+	0.968501i	$0.419895\pi$
$938$	0	0
$939$	−1.41253e16	−0.631446
$940$	0	0
$941$	−2.50908e16	−1.10859	−0.554295	−	0.832320i	$-0.687012\pi$
$941$	−2.50908e16	−1.10859	−0.554295	+	0.832320i	$0.687012\pi$
$942$	0	0
$943$	−1.41925e15	−0.0619790
$944$	0	0
$945$	−2.72795e16	−1.17750
$946$	0	0
$947$	−1.97365e16	−0.842064	−0.421032	−	0.907046i	$-0.638332\pi$
$947$	−1.97365e16	−0.842064	−0.421032	+	0.907046i	$0.638332\pi$
$948$	0	0
$949$	3.88152e15	0.163696
$950$	0	0
$951$	−4.03992e16	−1.68415
$952$	0	0
$953$	1.28343e16	0.528884	0.264442	−	0.964402i	$-0.414812\pi$
$953$	1.28343e16	0.528884	0.264442	+	0.964402i	$0.414812\pi$
$954$	0	0
$955$	−2.47997e15	−0.101025
$956$	0	0
$957$	1.99228e16	0.802296
$958$	0	0
$959$	−7.74699e16	−3.08411
$960$	0	0
$961$	−2.52335e16	−0.993112
$962$	0	0
$963$	−4.28619e16	−1.66773
$964$	0	0
$965$	−1.89608e16	−0.729383
$966$	0	0
$967$	4.73110e16	1.79935	0.899676	−	0.436558i	$-0.143803\pi$
$967$	4.73110e16	1.79935	0.899676	+	0.436558i	$0.143803\pi$
$968$	0	0
$969$	6.35094e16	2.38813
$970$	0	0
$971$	−2.69052e16	−1.00030	−0.500149	−	0.865939i	$-0.666722\pi$
$971$	−2.69052e16	−1.00030	−0.500149	+	0.865939i	$0.666722\pi$
$972$	0	0
$973$	−6.64042e16	−2.44104
$974$	0	0
$975$	2.59318e16	0.942557
$976$	0	0
$977$	−1.19615e16	−0.429899	−0.214950	−	0.976625i	$-0.568959\pi$
$977$	−1.19615e16	−0.429899	−0.214950	+	0.976625i	$0.568959\pi$
$978$	0	0
$979$	3.69078e16	1.31164
$980$	0	0
$981$	3.67795e16	1.29249
$982$	0	0
$983$	4.93179e16	1.71380	0.856900	−	0.515483i	$-0.172387\pi$
$983$	4.93179e16	1.71380	0.856900	+	0.515483i	$0.172387\pi$
$984$	0	0
$985$	1.98547e16	0.682281
$986$	0	0
$987$	5.94330e16	2.01968
$988$	0	0
$989$	2.66221e14	0.00894669
$990$	0	0
$991$	−4.42856e16	−1.47183	−0.735915	−	0.677074i	$-0.763247\pi$
$991$	−4.42856e16	−1.47183	−0.735915	+	0.677074i	$0.763247\pi$
$992$	0	0
$993$	−1.35623e16	−0.445772
$994$	0	0
$995$	2.66430e16	0.866075
$996$	0	0
$997$	2.10581e16	0.677011	0.338506	−	0.940964i	$-0.390079\pi$
$997$	2.10581e16	0.677011	0.338506	+	0.940964i	$0.390079\pi$
$998$	0	0
$999$	1.51755e15	0.0482539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.12.a.d.1.1		2
3.2	odd	2		144.12.a.p.1.2		2
4.3	odd	2		8.12.a.b.1.2	✓	2
8.3	odd	2		64.12.a.h.1.1		2
8.5	even	2		64.12.a.k.1.2		2
12.11	even	2		72.12.a.e.1.2		2
16.3	odd	4		256.12.b.h.129.4		4
16.5	even	4		256.12.b.k.129.4		4
16.11	odd	4		256.12.b.h.129.1		4
16.13	even	4		256.12.b.k.129.1		4
20.3	even	4		200.12.c.c.49.4		4
20.7	even	4		200.12.c.c.49.1		4
20.19	odd	2		200.12.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.12.a.b.1.2	✓	2	4.3	odd	2
16.12.a.d.1.1		2	1.1	even	1	trivial
64.12.a.h.1.1		2	8.3	odd	2
64.12.a.k.1.2		2	8.5	even	2
72.12.a.e.1.2		2	12.11	even	2
144.12.a.p.1.2		2	3.2	odd	2
200.12.a.d.1.1		2	20.19	odd	2
200.12.c.c.49.1		4	20.7	even	4
200.12.c.c.49.4		4	20.3	even	4
256.12.b.h.129.1		4	16.11	odd	4
256.12.b.h.129.4		4	16.3	odd	4
256.12.b.k.129.1		4	16.13	even	4
256.12.b.k.129.4		4	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-696.180 q^{3} -4084.16 q^{5} -73591.5 q^{7} +307519. q^{9} +O(q^{10})\)	\(q-696.180 q^{3} -4084.16 q^{5} -73591.5 q^{7} +307519. q^{9} +383508. q^{11} +1.15867e6 q^{13} +2.84331e6 q^{15} -6.51693e6 q^{17} +1.39982e7 q^{19} +5.12329e7 q^{21} -1.37394e6 q^{23} -3.21478e7 q^{25} -9.07624e7 q^{27} -7.46197e7 q^{29} -1.32297e7 q^{31} -2.66991e8 q^{33} +3.00559e8 q^{35} -1.67200e7 q^{37} -8.06644e8 q^{39} +1.03298e9 q^{41} -1.93764e8 q^{43} -1.25596e9 q^{45} +1.16005e9 q^{47} +3.43839e9 q^{49} +4.53696e9 q^{51} +4.44363e8 q^{53} -1.56631e9 q^{55} -9.74529e9 q^{57} +1.28304e8 q^{59} +7.96097e9 q^{61} -2.26308e10 q^{63} -4.73220e9 q^{65} +6.89243e9 q^{67} +9.56509e8 q^{69} +1.12698e10 q^{71} +3.34998e9 q^{73} +2.23806e10 q^{75} -2.82230e10 q^{77} -5.36237e10 q^{79} +8.71083e9 q^{81} +6.31693e10 q^{83} +2.66162e10 q^{85} +5.19487e10 q^{87} +9.62373e10 q^{89} -8.52685e10 q^{91} +9.21027e9 q^{93} -5.71710e10 q^{95} -4.37466e10 q^{97} +1.17936e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 56 q^{3} + 7868 q^{5} - 91056 q^{7} + 540202 q^{9}+O(q^{10}) \) 2 * q - 56 * q^3 + 7868 * q^5 - 91056 * q^7 + 540202 * q^9	\( 2 q - 56 q^{3} + 7868 q^{5} - 91056 q^{7} + 540202 q^{9} - 159080 q^{11} + 1050476 q^{13} + 10494832 q^{15} + 1430884 q^{17} + 21866600 q^{19} + 40052544 q^{21} - 35806736 q^{23} + 61878094 q^{25} - 55209392 q^{27} - 228827700 q^{29} - 64722112 q^{31} - 614344864 q^{33} + 91821408 q^{35} + 75558780 q^{37} - 875909072 q^{39} + 1201214196 q^{41} + 45519832 q^{43} + 1525107052 q^{45} + 1229079264 q^{47} + 1766069074 q^{49} + 9624987920 q^{51} - 3808549924 q^{53} - 8051409968 q^{55} - 4708125536 q^{57} + 6012926584 q^{59} + 9789792908 q^{61} - 26694483312 q^{63} - 6025376344 q^{65} - 14703095224 q^{67} - 21086664256 q^{69} - 4319991088 q^{71} + 11055639476 q^{73} + 82574103416 q^{75} - 18746968128 q^{77} - 51957623264 q^{79} - 9747960302 q^{81} + 108227975912 q^{83} + 121609729720 q^{85} - 46772044368 q^{87} + 71188291860 q^{89} - 83378892576 q^{91} - 23754095360 q^{93} + 36872875568 q^{95} - 1699807676 q^{97} - 8314917256 q^{99}+O(q^{100}) \) 2 * q - 56 * q^3 + 7868 * q^5 - 91056 * q^7 + 540202 * q^9 - 159080 * q^11 + 1050476 * q^13 + 10494832 * q^15 + 1430884 * q^17 + 21866600 * q^19 + 40052544 * q^21 - 35806736 * q^23 + 61878094 * q^25 - 55209392 * q^27 - 228827700 * q^29 - 64722112 * q^31 - 614344864 * q^33 + 91821408 * q^35 + 75558780 * q^37 - 875909072 * q^39 + 1201214196 * q^41 + 45519832 * q^43 + 1525107052 * q^45 + 1229079264 * q^47 + 1766069074 * q^49 + 9624987920 * q^51 - 3808549924 * q^53 - 8051409968 * q^55 - 4708125536 * q^57 + 6012926584 * q^59 + 9789792908 * q^61 - 26694483312 * q^63 - 6025376344 * q^65 - 14703095224 * q^67 - 21086664256 * q^69 - 4319991088 * q^71 + 11055639476 * q^73 + 82574103416 * q^75 - 18746968128 * q^77 - 51957623264 * q^79 - 9747960302 * q^81 + 108227975912 * q^83 + 121609729720 * q^85 - 46772044368 * q^87 + 71188291860 * q^89 - 83378892576 * q^91 - 23754095360 * q^93 + 36872875568 * q^95 - 1699807676 * q^97 - 8314917256 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more