LMFDB - Embedded newform 117.10.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,10,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, 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("117.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$N$	$=$	$117 = 3^{2} \cdot 13$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	117.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:	$60.2591928312$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580$ x^5 - 1438x^3 - 4164x^2 + 396957*x - 59580
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}\cdot 3$
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$35.1685$ of defining polynomial
Character	$\chi$	$=$	117.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-38.1685 q^{2} +944.833 q^{4} +109.762 q^{5} +5947.44 q^{7} -16520.6 q^{8} -4189.45 q^{10} +25205.7 q^{11} +28561.0 q^{13} -227005. q^{14} +146811. q^{16} -109318. q^{17} -904609. q^{19} +103707. q^{20} -962062. q^{22} +435749. q^{23} -1.94108e6 q^{25} -1.09013e6 q^{26} +5.61934e6 q^{28} -6.44791e6 q^{29} +6.62308e6 q^{31} +2.85499e6 q^{32} +4.17250e6 q^{34} +652804. q^{35} +4.14357e6 q^{37} +3.45275e7 q^{38} -1.81333e6 q^{40} -1.49568e7 q^{41} +4.01789e7 q^{43} +2.38151e7 q^{44} -1.66319e7 q^{46} -6.30151e6 q^{47} -4.98153e6 q^{49} +7.40880e7 q^{50} +2.69854e7 q^{52} -1.53111e7 q^{53} +2.76663e6 q^{55} -9.82552e7 q^{56} +2.46107e8 q^{58} +1.52760e8 q^{59} +8.66321e7 q^{61} -2.52793e8 q^{62} -1.84138e8 q^{64} +3.13492e6 q^{65} -1.01034e8 q^{67} -1.03287e8 q^{68} -2.49165e7 q^{70} -4.13122e8 q^{71} -3.14453e8 q^{73} -1.58154e8 q^{74} -8.54704e8 q^{76} +1.49909e8 q^{77} -2.00580e8 q^{79} +1.61143e7 q^{80} +5.70879e8 q^{82} -6.34578e7 q^{83} -1.19990e7 q^{85} -1.53357e9 q^{86} -4.16412e8 q^{88} -3.47074e7 q^{89} +1.69865e8 q^{91} +4.11710e8 q^{92} +2.40519e8 q^{94} -9.92918e7 q^{95} -1.25403e9 q^{97} +1.90137e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q - 15 q^{2} + 361 q^{4} - 1803 q^{5} + 10099 q^{7} - 23151 q^{8} + 84505 q^{10} - 121746 q^{11} + 142805 q^{13} - 8475 q^{14} - 322463 q^{16} + 495669 q^{17} - 840738 q^{19} + 1595607 q^{20} - 2023594 q^{22}+ \cdots - 1570614816 q^{98}+O(q^{100})$ 5 * q - 15 * q^2 + 361 * q^4 - 1803 * q^5 + 10099 * q^7 - 23151 * q^8 + 84505 * q^10 - 121746 * q^11 + 142805 * q^13 - 8475 * q^14 - 322463 * q^16 + 495669 * q^17 - 840738 * q^19 + 1595607 * q^20 - 2023594 * q^22 + 592152 * q^23 + 1670362 * q^25 - 428415 * q^26 + 2587955 * q^28 - 10678182 * q^29 + 12885296 * q^31 - 3282927 * q^32 - 9934079 * q^34 - 8380731 * q^35 + 7171823 * q^37 + 25568814 * q^38 - 54359445 * q^40 - 9294012 * q^41 + 12831975 * q^43 + 41479074 * q^44 - 59319696 * q^46 - 43354215 * q^47 + 25249488 * q^49 + 16270770 * q^50 + 10310521 * q^52 - 93231780 * q^53 + 99448846 * q^55 - 199599225 * q^56 + 151020970 * q^58 - 246496182 * q^59 - 132232612 * q^61 - 158135724 * q^62 + 91019105 * q^64 - 51495483 * q^65 - 369388534 * q^67 - 238172073 * q^68 - 144857425 * q^70 - 212150457 * q^71 - 252729806 * q^73 - 192105957 * q^74 - 953775990 * q^76 - 449666118 * q^77 - 1247271728 * q^79 - 900649725 * q^80 + 169559388 * q^82 - 1696894296 * q^83 - 775363765 * q^85 - 3291621459 * q^86 - 220227222 * q^88 + 753854382 * q^89 + 288437539 * q^91 - 13876128 * q^92 + 272071215 * q^94 - 1442632962 * q^95 + 3824606 * q^97 - 1570614816 * q^98

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

)

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

$2$	−38.1685	−1.68682	−0.843412	−	0.537267i	$-0.819457\pi$
$2$	−38.1685	−1.68682	−0.843412	+	0.537267i	$0.819457\pi$
$3$	0	0
$3$	0	0
$4$	944.833	1.84538
$5$	109.762	0.0785394	0.0392697	−	0.999229i	$-0.487497\pi$
$5$	109.762	0.0785394	0.0392697	+	0.999229i	$0.487497\pi$
$6$	0	0
$7$	5947.44	0.936244	0.468122	−	0.883664i	$-0.344931\pi$
$7$	5947.44	0.936244	0.468122	+	0.883664i	$0.344931\pi$
$8$	−16520.6	−1.42600
$9$	0	0
$10$	−4189.45	−0.132482
$11$	25205.7	0.519076	0.259538	−	0.965733i	$-0.416430\pi$
$11$	25205.7	0.519076	0.259538	+	0.965733i	$0.416430\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	−227005.	−1.57928
$15$	0	0
$16$	146811.	0.560039
$17$	−109318.	−0.317447	−0.158724	−	0.987323i	$-0.550738\pi$
$17$	−109318.	−0.317447	−0.158724	+	0.987323i	$0.550738\pi$
$18$	0	0
$19$	−904609.	−1.59246	−0.796232	−	0.604992i	$-0.793176\pi$
$19$	−904609.	−1.59246	−0.796232	+	0.604992i	$0.793176\pi$
$20$	103707.	0.144935
$21$	0	0
$22$	−962062.	−0.875590
$23$	435749.	0.324684	0.162342	−	0.986735i	$-0.448095\pi$
$23$	435749.	0.324684	0.162342	+	0.986735i	$0.448095\pi$
$24$	0	0
$25$	−1.94108e6	−0.993832
$26$	−1.09013e6	−0.467841
$27$	0	0
$28$	5.61934e6	1.72772
$29$	−6.44791e6	−1.69289	−0.846443	−	0.532479i	$-0.821260\pi$
$29$	−6.44791e6	−1.69289	−0.846443	+	0.532479i	$0.821260\pi$
$30$	0	0
$31$	6.62308e6	1.28805	0.644024	−	0.765005i	$-0.277264\pi$
$31$	6.62308e6	1.28805	0.644024	+	0.765005i	$0.277264\pi$
$32$	2.85499e6	0.481315
$33$	0	0
$34$	4.17250e6	0.535477
$35$	652804.	0.0735320
$36$	0	0
$37$	4.14357e6	0.363469	0.181734	−	0.983348i	$-0.441829\pi$
$37$	4.14357e6	0.363469	0.181734	+	0.983348i	$0.441829\pi$
$38$	3.45275e7	2.68621
$39$	0	0
$40$	−1.81333e6	−0.111997
$41$	−1.49568e7	−0.826632	−0.413316	−	0.910588i	$-0.635630\pi$
$41$	−1.49568e7	−0.826632	−0.413316	+	0.910588i	$0.635630\pi$
$42$	0	0
$43$	4.01789e7	1.79221	0.896107	−	0.443838i	$-0.146384\pi$
$43$	4.01789e7	1.79221	0.896107	+	0.443838i	$0.146384\pi$
$44$	2.38151e7	0.957891
$45$	0	0
$46$	−1.66319e7	−0.547685
$47$	−6.30151e6	−0.188367	−0.0941834	−	0.995555i	$-0.530024\pi$
$47$	−6.30151e6	−0.188367	−0.0941834	+	0.995555i	$0.530024\pi$
$48$	0	0
$49$	−4.98153e6	−0.123447
$50$	7.40880e7	1.67642
$51$	0	0
$52$	2.69854e7	0.511815
$53$	−1.53111e7	−0.266542	−0.133271	−	0.991080i	$-0.542548\pi$
$53$	−1.53111e7	−0.266542	−0.133271	+	0.991080i	$0.542548\pi$
$54$	0	0
$55$	2.76663e6	0.0407679
$56$	−9.82552e7	−1.33509
$57$	0	0
$58$	2.46107e8	2.85560
$59$	1.52760e8	1.64126	0.820629	−	0.571462i	$-0.193624\pi$
$59$	1.52760e8	1.64126	0.820629	+	0.571462i	$0.193624\pi$
$60$	0	0
$61$	8.66321e7	0.801114	0.400557	−	0.916272i	$-0.368817\pi$
$61$	8.66321e7	0.801114	0.400557	+	0.916272i	$0.368817\pi$
$62$	−2.52793e8	−2.17271
$63$	0	0
$64$	−1.84138e8	−1.37193
$65$	3.13492e6	0.0217829
$66$	0	0
$67$	−1.01034e8	−0.612537	−0.306268	−	0.951945i	$-0.599080\pi$
$67$	−1.01034e8	−0.612537	−0.306268	+	0.951945i	$0.599080\pi$
$68$	−1.03287e8	−0.585809
$69$	0	0
$70$	−2.49165e7	−0.124036
$71$	−4.13122e8	−1.92937	−0.964685	−	0.263406i	$-0.915154\pi$
$71$	−4.13122e8	−1.92937	−0.964685	+	0.263406i	$0.915154\pi$
$72$	0	0
$73$	−3.14453e8	−1.29599	−0.647997	−	0.761643i	$-0.724393\pi$
$73$	−3.14453e8	−1.29599	−0.647997	+	0.761643i	$0.724393\pi$
$74$	−1.58154e8	−0.613108
$75$	0	0
$76$	−8.54704e8	−2.93870
$77$	1.49909e8	0.485982
$78$	0	0
$79$	−2.00580e8	−0.579383	−0.289692	−	0.957120i	$-0.593553\pi$
$79$	−2.00580e8	−0.579383	−0.289692	+	0.957120i	$0.593553\pi$
$80$	1.61143e7	0.0439851
$81$	0	0
$82$	5.70879e8	1.39438
$83$	−6.34578e7	−0.146769	−0.0733843	−	0.997304i	$-0.523380\pi$
$83$	−6.34578e7	−0.146769	−0.0733843	+	0.997304i	$0.523380\pi$
$84$	0	0
$85$	−1.19990e7	−0.0249321
$86$	−1.53357e9	−3.02315
$87$	0	0
$88$	−4.16412e8	−0.740204
$89$	−3.47074e7	−0.0586364	−0.0293182	−	0.999570i	$-0.509334\pi$
$89$	−3.47074e7	−0.0586364	−0.0293182	+	0.999570i	$0.509334\pi$
$90$	0	0
$91$	1.69865e8	0.259667
$92$	4.11710e8	0.599164
$93$	0	0
$94$	2.40519e8	0.317742
$95$	−9.92918e7	−0.125071
$96$	0	0
$97$	−1.25403e9	−1.43825	−0.719127	−	0.694879i	$-0.755458\pi$
$97$	−1.25403e9	−1.43825	−0.719127	+	0.694879i	$0.755458\pi$
$98$	1.90137e8	0.208233
$99$	0	0
$100$	−1.83399e9	−1.83399
$101$	9.06459e8	0.866766	0.433383	−	0.901210i	$-0.357320\pi$
$101$	9.06459e8	0.866766	0.433383	+	0.901210i	$0.357320\pi$
$102$	0	0
$103$	−4.17013e8	−0.365075	−0.182537	−	0.983199i	$-0.558431\pi$
$103$	−4.17013e8	−0.365075	−0.182537	+	0.983199i	$0.558431\pi$
$104$	−4.71844e8	−0.395502
$105$	0	0
$106$	5.84402e8	0.449610
$107$	−6.71636e8	−0.495344	−0.247672	−	0.968844i	$-0.579666\pi$
$107$	−6.71636e8	−0.495344	−0.247672	+	0.968844i	$0.579666\pi$
$108$	0	0
$109$	1.66748e9	1.13147	0.565734	−	0.824588i	$-0.308593\pi$
$109$	1.66748e9	1.13147	0.565734	+	0.824588i	$0.308593\pi$
$110$	−1.05598e8	−0.0687683
$111$	0	0
$112$	8.73149e8	0.524333
$113$	1.84580e9	1.06496	0.532478	−	0.846444i	$-0.321261\pi$
$113$	1.84580e9	1.06496	0.532478	+	0.846444i	$0.321261\pi$
$114$	0	0
$115$	4.78287e7	0.0255005
$116$	−6.09219e9	−3.12401
$117$	0	0
$118$	−5.83063e9	−2.76851
$119$	−6.50162e8	−0.297208
$120$	0	0
$121$	−1.72262e9	−0.730560
$122$	−3.30661e9	−1.35134
$123$	0	0
$124$	6.25770e9	2.37694
$125$	−4.27436e8	−0.156594
$126$	0	0
$127$	1.87922e9	0.641006	0.320503	−	0.947248i	$-0.396148\pi$
$127$	1.87922e9	0.641006	0.320503	+	0.947248i	$0.396148\pi$
$128$	5.56650e9	1.83290
$129$	0	0
$130$	−1.19655e8	−0.0367439
$131$	−3.46045e9	−1.02663	−0.513313	−	0.858201i	$-0.671582\pi$
$131$	−3.46045e9	−1.02663	−0.513313	+	0.858201i	$0.671582\pi$
$132$	0	0
$133$	−5.38011e9	−1.49093
$134$	3.85632e9	1.03324
$135$	0	0
$136$	1.80600e9	0.452680
$137$	−5.04786e9	−1.22424	−0.612118	−	0.790766i	$-0.709682\pi$
$137$	−5.04786e9	−1.22424	−0.612118	+	0.790766i	$0.709682\pi$
$138$	0	0
$139$	3.59716e9	0.817322	0.408661	−	0.912686i	$-0.365996\pi$
$139$	3.59716e9	0.817322	0.408661	+	0.912686i	$0.365996\pi$
$140$	6.16791e8	0.135694
$141$	0	0
$142$	1.57682e10	3.25451
$143$	7.19899e8	0.143966
$144$	0	0
$145$	−7.07736e8	−0.132958
$146$	1.20022e10	2.18611
$147$	0	0
$148$	3.91498e9	0.670736
$149$	−1.27561e9	−0.212022	−0.106011	−	0.994365i	$-0.533808\pi$
$149$	−1.27561e9	−0.212022	−0.106011	+	0.994365i	$0.533808\pi$
$150$	0	0
$151$	4.35273e9	0.681342	0.340671	−	0.940183i	$-0.389346\pi$
$151$	4.35273e9	0.681342	0.340671	+	0.940183i	$0.389346\pi$
$152$	1.49447e10	2.27086
$153$	0	0
$154$	−5.72181e9	−0.819766
$155$	7.26963e8	0.101163
$156$	0	0
$157$	−1.41002e10	−1.85215	−0.926075	−	0.377339i	$-0.876839\pi$
$157$	−1.41002e10	−1.85215	−0.926075	+	0.377339i	$0.876839\pi$
$158$	7.65584e9	0.977318
$159$	0	0
$160$	3.13370e8	0.0378022
$161$	2.59159e9	0.303983
$162$	0	0
$163$	7.24812e8	0.0804231	0.0402116	−	0.999191i	$-0.487197\pi$
$163$	7.24812e8	0.0804231	0.0402116	+	0.999191i	$0.487197\pi$
$164$	−1.41317e10	−1.52545
$165$	0	0
$166$	2.42209e9	0.247573
$167$	8.33031e8	0.0828776	0.0414388	−	0.999141i	$-0.486806\pi$
$167$	8.33031e8	0.0828776	0.0414388	+	0.999141i	$0.486806\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	4.57983e8	0.0420561
$171$	0	0
$172$	3.79623e10	3.30731
$173$	−5.77955e8	−0.0490553	−0.0245277	−	0.999699i	$-0.507808\pi$
$173$	−5.77955e8	−0.0490553	−0.0245277	+	0.999699i	$0.507808\pi$
$174$	0	0
$175$	−1.15444e10	−0.930469
$176$	3.70046e9	0.290703
$177$	0	0
$178$	1.32473e9	0.0989094
$179$	1.56569e10	1.13990	0.569952	−	0.821678i	$-0.306962\pi$
$179$	1.56569e10	1.13990	0.569952	+	0.821678i	$0.306962\pi$
$180$	0	0
$181$	−2.18552e10	−1.51356	−0.756781	−	0.653668i	$-0.773229\pi$
$181$	−2.18552e10	−1.51356	−0.756781	+	0.653668i	$0.773229\pi$
$182$	−6.48349e9	−0.438013
$183$	0	0
$184$	−7.19882e9	−0.463000
$185$	4.54807e8	0.0285466
$186$	0	0
$187$	−2.75543e9	−0.164779
$188$	−5.95388e9	−0.347608
$189$	0	0
$190$	3.78982e9	0.210973
$191$	−1.67784e10	−0.912222	−0.456111	−	0.889923i	$-0.650758\pi$
$191$	−1.67784e10	−0.912222	−0.456111	+	0.889923i	$0.650758\pi$
$192$	0	0
$193$	−2.70036e10	−1.40092	−0.700462	−	0.713690i	$-0.747023\pi$
$193$	−2.70036e10	−1.40092	−0.700462	+	0.713690i	$0.747023\pi$
$194$	4.78645e10	2.42608
$195$	0	0
$196$	−4.70671e9	−0.227806
$197$	−1.58277e10	−0.748720	−0.374360	−	0.927283i	$-0.622138\pi$
$197$	−1.58277e10	−0.748720	−0.374360	+	0.927283i	$0.622138\pi$
$198$	0	0
$199$	8.80397e9	0.397960	0.198980	−	0.980004i	$-0.436237\pi$
$199$	8.80397e9	0.397960	0.198980	+	0.980004i	$0.436237\pi$
$200$	3.20677e10	1.41721
$201$	0	0
$202$	−3.45982e10	−1.46208
$203$	−3.83485e10	−1.58495
$204$	0	0
$205$	−1.64169e9	−0.0649232
$206$	1.59167e10	0.615817
$207$	0	0
$208$	4.19306e9	0.155327
$209$	−2.28013e10	−0.826610
$210$	0	0
$211$	−1.82054e10	−0.632308	−0.316154	−	0.948708i	$-0.602392\pi$
$211$	−1.82054e10	−0.632308	−0.316154	+	0.948708i	$0.602392\pi$
$212$	−1.44665e10	−0.491870
$213$	0	0
$214$	2.56353e10	0.835558
$215$	4.41012e9	0.140759
$216$	0	0
$217$	3.93904e10	1.20593
$218$	−6.36453e10	−1.90859
$219$	0	0
$220$	2.61400e9	0.0752322
$221$	−3.12223e9	−0.0880440
$222$	0	0
$223$	1.53511e10	0.415688	0.207844	−	0.978162i	$-0.433355\pi$
$223$	1.53511e10	0.415688	0.207844	+	0.978162i	$0.433355\pi$
$224$	1.69799e10	0.450628
$225$	0	0
$226$	−7.04514e10	−1.79639
$227$	−4.20620e10	−1.05141	−0.525707	−	0.850666i	$-0.676199\pi$
$227$	−4.20620e10	−1.05141	−0.525707	+	0.850666i	$0.676199\pi$
$228$	0	0
$229$	−6.68760e10	−1.60698	−0.803490	−	0.595318i	$-0.797026\pi$
$229$	−6.68760e10	−1.60698	−0.803490	+	0.595318i	$0.797026\pi$
$230$	−1.82555e9	−0.0430148
$231$	0	0
$232$	1.06523e11	2.41406
$233$	5.19268e10	1.15422	0.577112	−	0.816665i	$-0.304180\pi$
$233$	5.19268e10	1.15422	0.577112	+	0.816665i	$0.304180\pi$
$234$	0	0
$235$	−6.91668e8	−0.0147942
$236$	1.44333e11	3.02874
$237$	0	0
$238$	2.48157e10	0.501338
$239$	−7.45881e10	−1.47870	−0.739348	−	0.673323i	$-0.764866\pi$
$239$	−7.45881e10	−1.47870	−0.739348	+	0.673323i	$0.764866\pi$
$240$	0	0
$241$	5.74852e10	1.09769	0.548845	−	0.835924i	$-0.315068\pi$
$241$	5.74852e10	1.09769	0.548845	+	0.835924i	$0.315068\pi$
$242$	6.57499e10	1.23233
$243$	0	0
$244$	8.18528e10	1.47836
$245$	−5.46784e8	−0.00969545
$246$	0	0
$247$	−2.58365e10	−0.441670
$248$	−1.09417e11	−1.83676
$249$	0	0
$250$	1.63146e10	0.264147
$251$	1.07873e11	1.71546	0.857732	−	0.514097i	$-0.171873\pi$
$251$	1.07873e11	1.71546	0.857732	+	0.514097i	$0.171873\pi$
$252$	0	0
$253$	1.09833e10	0.168536
$254$	−7.17272e10	−1.08126
$255$	0	0
$256$	−1.18186e11	−1.71984
$257$	6.64074e10	0.949550	0.474775	−	0.880107i	$-0.342530\pi$
$257$	6.64074e10	0.949550	0.474775	+	0.880107i	$0.342530\pi$
$258$	0	0
$259$	2.46436e10	0.340295
$260$	2.96197e9	0.0401977
$261$	0	0
$262$	1.32080e11	1.73174
$263$	8.15356e10	1.05086	0.525432	−	0.850836i	$-0.323904\pi$
$263$	8.15356e10	1.05086	0.525432	+	0.850836i	$0.323904\pi$
$264$	0	0
$265$	−1.68058e9	−0.0209340
$266$	2.05351e11	2.51494
$267$	0	0
$268$	−9.54605e10	−1.13036
$269$	−1.00568e11	−1.17105	−0.585523	−	0.810656i	$-0.699111\pi$
$269$	−1.00568e11	−1.17105	−0.585523	+	0.810656i	$0.699111\pi$
$270$	0	0
$271$	−2.98757e10	−0.336477	−0.168239	−	0.985746i	$-0.553808\pi$
$271$	−2.98757e10	−0.336477	−0.168239	+	0.985746i	$0.553808\pi$
$272$	−1.60491e10	−0.177783
$273$	0	0
$274$	1.92669e11	2.06507
$275$	−4.89261e10	−0.515874
$276$	0	0
$277$	−3.17195e10	−0.323718	−0.161859	−	0.986814i	$-0.551749\pi$
$277$	−3.17195e10	−0.323718	−0.161859	+	0.986814i	$0.551749\pi$
$278$	−1.37298e11	−1.37868
$279$	0	0
$280$	−1.07847e10	−0.104857
$281$	−9.86953e10	−0.944318	−0.472159	−	0.881513i	$-0.656525\pi$
$281$	−9.86953e10	−0.944318	−0.472159	+	0.881513i	$0.656525\pi$
$282$	0	0
$283$	−1.06164e11	−0.983871	−0.491935	−	0.870632i	$-0.663710\pi$
$283$	−1.06164e11	−0.983871	−0.491935	+	0.870632i	$0.663710\pi$
$284$	−3.90331e11	−3.56042
$285$	0	0
$286$	−2.74774e10	−0.242845
$287$	−8.89549e10	−0.773929
$288$	0	0
$289$	−1.06637e11	−0.899227
$290$	2.70132e10	0.224277
$291$	0	0
$292$	−2.97106e11	−2.39160
$293$	−2.26355e11	−1.79426	−0.897132	−	0.441763i	$-0.854353\pi$
$293$	−2.26355e11	−1.79426	−0.897132	+	0.441763i	$0.854353\pi$
$294$	0	0
$295$	1.67673e10	0.128903
$296$	−6.84541e10	−0.518307
$297$	0	0
$298$	4.86881e10	0.357643
$299$	1.24454e10	0.0900511
$300$	0	0
$301$	2.38962e11	1.67795
$302$	−1.66137e11	−1.14930
$303$	0	0
$304$	−1.32806e11	−0.891841
$305$	9.50892e9	0.0629190
$306$	0	0
$307$	−2.23009e10	−0.143285	−0.0716424	−	0.997430i	$-0.522824\pi$
$307$	−2.23009e10	−0.143285	−0.0716424	+	0.997430i	$0.522824\pi$
$308$	1.41639e11	0.896820
$309$	0	0
$310$	−2.77471e10	−0.170643
$311$	2.71805e11	1.64754	0.823770	−	0.566925i	$-0.191867\pi$
$311$	2.71805e11	1.64754	0.823770	+	0.566925i	$0.191867\pi$
$312$	0	0
$313$	−7.90775e10	−0.465697	−0.232849	−	0.972513i	$-0.574805\pi$
$313$	−7.90775e10	−0.465697	−0.232849	+	0.972513i	$0.574805\pi$
$314$	5.38183e11	3.12425
$315$	0	0
$316$	−1.89515e11	−1.06918
$317$	−2.03032e11	−1.12927	−0.564636	−	0.825340i	$-0.690983\pi$
$317$	−2.03032e11	−1.12927	−0.564636	+	0.825340i	$0.690983\pi$
$318$	0	0
$319$	−1.62524e11	−0.878736
$320$	−2.02114e10	−0.107751
$321$	0	0
$322$	−9.89171e10	−0.512767
$323$	9.88899e10	0.505523
$324$	0	0
$325$	−5.54391e10	−0.275639
$326$	−2.76650e10	−0.135660
$327$	0	0
$328$	2.47095e11	1.17878
$329$	−3.74779e10	−0.176357
$330$	0	0
$331$	3.24137e11	1.48424	0.742118	−	0.670270i	$-0.233822\pi$
$331$	3.24137e11	1.48424	0.742118	+	0.670270i	$0.233822\pi$
$332$	−5.99570e10	−0.270844
$333$	0	0
$334$	−3.17955e10	−0.139800
$335$	−1.10897e10	−0.0481083
$336$	0	0
$337$	2.68510e11	1.13403	0.567017	−	0.823706i	$-0.308097\pi$
$337$	2.68510e11	1.13403	0.567017	+	0.823706i	$0.308097\pi$
$338$	−3.11352e10	−0.129756
$339$	0	0
$340$	−1.13370e10	−0.0460091
$341$	1.66939e11	0.668595
$342$	0	0
$343$	−2.69628e11	−1.05182
$344$	−6.63778e11	−2.55570
$345$	0	0
$346$	2.20597e10	0.0827478
$347$	−2.74715e11	−1.01719	−0.508593	−	0.861007i	$-0.669834\pi$
$347$	−2.74715e11	−1.01719	−0.508593	+	0.861007i	$0.669834\pi$
$348$	0	0
$349$	−3.54006e11	−1.27731	−0.638655	−	0.769494i	$-0.720509\pi$
$349$	−3.54006e11	−1.27731	−0.638655	+	0.769494i	$0.720509\pi$
$350$	4.40634e11	1.56954
$351$	0	0
$352$	7.19619e10	0.249839
$353$	2.25650e11	0.773481	0.386741	−	0.922189i	$-0.373601\pi$
$353$	2.25650e11	0.773481	0.386741	+	0.922189i	$0.373601\pi$
$354$	0	0
$355$	−4.53451e10	−0.151532
$356$	−3.27927e10	−0.108206
$357$	0	0
$358$	−5.97602e11	−1.92282
$359$	−3.39022e10	−0.107722	−0.0538608	−	0.998548i	$-0.517153\pi$
$359$	−3.39022e10	−0.107722	−0.0538608	+	0.998548i	$0.517153\pi$
$360$	0	0
$361$	4.95629e11	1.53594
$362$	8.34178e11	2.55312
$363$	0	0
$364$	1.60494e11	0.479184
$365$	−3.45150e10	−0.101787
$366$	0	0
$367$	−7.74387e10	−0.222823	−0.111412	−	0.993774i	$-0.535537\pi$
$367$	−7.74387e10	−0.222823	−0.111412	+	0.993774i	$0.535537\pi$
$368$	6.39726e10	0.181836
$369$	0	0
$370$	−1.73593e10	−0.0481531
$371$	−9.10620e10	−0.249548
$372$	0	0
$373$	6.50821e11	1.74089	0.870446	−	0.492264i	$-0.163831\pi$
$373$	6.50821e11	1.74089	0.870446	+	0.492264i	$0.163831\pi$
$374$	1.05171e11	0.277954
$375$	0	0
$376$	1.04105e11	0.268612
$377$	−1.84159e11	−0.469522
$378$	0	0
$379$	1.93989e11	0.482948	0.241474	−	0.970407i	$-0.422369\pi$
$379$	1.93989e11	0.482948	0.241474	+	0.970407i	$0.422369\pi$
$380$	−9.38141e10	−0.230803
$381$	0	0
$382$	6.40406e11	1.53876
$383$	−4.44645e11	−1.05589	−0.527946	−	0.849278i	$-0.677037\pi$
$383$	−4.44645e11	−1.05589	−0.527946	+	0.849278i	$0.677037\pi$
$384$	0	0
$385$	1.64544e10	0.0381687
$386$	1.03069e12	2.36311
$387$	0	0
$388$	−1.18485e12	−2.65412
$389$	1.69623e11	0.375589	0.187794	−	0.982208i	$-0.439866\pi$
$389$	1.69623e11	0.375589	0.187794	+	0.982208i	$0.439866\pi$
$390$	0	0
$391$	−4.76352e10	−0.103070
$392$	8.22978e10	0.176036
$393$	0	0
$394$	6.04119e11	1.26296
$395$	−2.20161e10	−0.0455044
$396$	0	0
$397$	2.94468e11	0.594951	0.297476	−	0.954729i	$-0.403855\pi$
$397$	2.94468e11	0.594951	0.297476	+	0.954729i	$0.403855\pi$
$398$	−3.36034e11	−0.671289
$399$	0	0
$400$	−2.84971e11	−0.556584
$401$	−4.43362e11	−0.856265	−0.428133	−	0.903716i	$-0.640828\pi$
$401$	−4.43362e11	−0.856265	−0.428133	+	0.903716i	$0.640828\pi$
$402$	0	0
$403$	1.89162e11	0.357240
$404$	8.56452e11	1.59951
$405$	0	0
$406$	1.46371e12	2.67354
$407$	1.04441e11	0.188668
$408$	0	0
$409$	−1.01253e11	−0.178918	−0.0894592	−	0.995990i	$-0.528514\pi$
$409$	−1.01253e11	−0.178918	−0.0894592	+	0.995990i	$0.528514\pi$
$410$	6.26610e10	0.109514
$411$	0	0
$412$	−3.94007e11	−0.673700
$413$	9.08533e11	1.53662
$414$	0	0
$415$	−6.96526e9	−0.0115271
$416$	8.15413e10	0.133493
$417$	0	0
$418$	8.70289e11	1.39435
$419$	−7.93982e10	−0.125848	−0.0629242	−	0.998018i	$-0.520043\pi$
$419$	−7.93982e10	−0.125848	−0.0629242	+	0.998018i	$0.520043\pi$
$420$	0	0
$421$	−6.06765e11	−0.941350	−0.470675	−	0.882307i	$-0.655990\pi$
$421$	−6.06765e11	−0.941350	−0.470675	+	0.882307i	$0.655990\pi$
$422$	6.94871e11	1.06659
$423$	0	0
$424$	2.52949e11	0.380089
$425$	2.12195e11	0.315489
$426$	0	0
$427$	5.15239e11	0.750038
$428$	−6.34583e11	−0.914096
$429$	0	0
$430$	−1.68328e11	−0.237436
$431$	2.72544e11	0.380442	0.190221	−	0.981741i	$-0.439080\pi$
$431$	2.72544e11	0.380442	0.190221	+	0.981741i	$0.439080\pi$
$432$	0	0
$433$	−1.15522e12	−1.57931	−0.789655	−	0.613551i	$-0.789740\pi$
$433$	−1.15522e12	−1.57931	−0.789655	+	0.613551i	$0.789740\pi$
$434$	−1.50347e12	−2.03419
$435$	0	0
$436$	1.57549e12	2.08799
$437$	−3.94182e11	−0.517047
$438$	0	0
$439$	1.78053e11	0.228801	0.114401	−	0.993435i	$-0.463505\pi$
$439$	1.78053e11	0.228801	0.114401	+	0.993435i	$0.463505\pi$
$440$	−4.57063e10	−0.0581351
$441$	0	0
$442$	1.19171e11	0.148515
$443$	6.55260e10	0.0808345	0.0404173	−	0.999183i	$-0.487131\pi$
$443$	6.55260e10	0.0808345	0.0404173	+	0.999183i	$0.487131\pi$
$444$	0	0
$445$	−3.80956e9	−0.00460527
$446$	−5.85928e11	−0.701192
$447$	0	0
$448$	−1.09515e12	−1.28446
$449$	7.98150e11	0.926779	0.463390	−	0.886155i	$-0.346633\pi$
$449$	7.98150e11	0.926779	0.463390	+	0.886155i	$0.346633\pi$
$450$	0	0
$451$	−3.76997e11	−0.429085
$452$	1.74397e12	1.96525
$453$	0	0
$454$	1.60544e12	1.77355
$455$	1.86447e10	0.0203941
$456$	0	0
$457$	−8.92736e11	−0.957415	−0.478708	−	0.877974i	$-0.658895\pi$
$457$	−8.92736e11	−0.957415	−0.478708	+	0.877974i	$0.658895\pi$
$458$	2.55256e12	2.71069
$459$	0	0
$460$	4.51901e10	0.0470580
$461$	−2.32490e11	−0.239745	−0.119872	−	0.992789i	$-0.538249\pi$
$461$	−2.32490e11	−0.239745	−0.119872	+	0.992789i	$0.538249\pi$
$462$	0	0
$463$	−1.54421e12	−1.56168	−0.780841	−	0.624730i	$-0.785209\pi$
$463$	−1.54421e12	−1.56168	−0.780841	+	0.624730i	$0.785209\pi$
$464$	−9.46622e11	−0.948082
$465$	0	0
$466$	−1.98197e12	−1.94697
$467$	−1.63317e12	−1.58893	−0.794465	−	0.607310i	$-0.792249\pi$
$467$	−1.63317e12	−1.58893	−0.794465	+	0.607310i	$0.792249\pi$
$468$	0	0
$469$	−6.00895e11	−0.573484
$470$	2.63999e10	0.0249553
$471$	0	0
$472$	−2.52369e12	−2.34044
$473$	1.01274e12	0.930295
$474$	0	0
$475$	1.75592e12	1.58264
$476$	−6.14295e11	−0.548461
$477$	0	0
$478$	2.84692e12	2.49430
$479$	−1.54414e12	−1.34022	−0.670112	−	0.742260i	$-0.733754\pi$
$479$	−1.54414e12	−1.34022	−0.670112	+	0.742260i	$0.733754\pi$
$480$	0	0
$481$	1.18344e11	0.100808
$482$	−2.19412e12	−1.85161
$483$	0	0
$484$	−1.62759e12	−1.34816
$485$	−1.37645e11	−0.112960
$486$	0	0
$487$	2.55439e11	0.205782	0.102891	−	0.994693i	$-0.467191\pi$
$487$	2.55439e11	0.205782	0.102891	+	0.994693i	$0.467191\pi$
$488$	−1.43121e12	−1.14239
$489$	0	0
$490$	2.08699e10	0.0163545
$491$	−1.58407e12	−1.23000	−0.615002	−	0.788526i	$-0.710845\pi$
$491$	−1.58407e12	−1.23000	−0.615002	+	0.788526i	$0.710845\pi$
$492$	0	0
$493$	7.04872e11	0.537402
$494$	9.86141e11	0.745020
$495$	0	0
$496$	9.72340e11	0.721357
$497$	−2.45702e12	−1.80636
$498$	0	0
$499$	−6.29948e11	−0.454833	−0.227417	−	0.973798i	$-0.573028\pi$
$499$	−6.29948e11	−0.454833	−0.227417	+	0.973798i	$0.573028\pi$
$500$	−4.03856e11	−0.288976
$501$	0	0
$502$	−4.11736e12	−2.89369
$503$	−5.49263e11	−0.382582	−0.191291	−	0.981533i	$-0.561267\pi$
$503$	−5.49263e11	−0.382582	−0.191291	+	0.981533i	$0.561267\pi$
$504$	0	0
$505$	9.94949e10	0.0680753
$506$	−4.19217e11	−0.284290
$507$	0	0
$508$	1.77555e12	1.18290
$509$	−1.36923e11	−0.0904165	−0.0452083	−	0.998978i	$-0.514395\pi$
$509$	−1.36923e11	−0.0904165	−0.0452083	+	0.998978i	$0.514395\pi$
$510$	0	0
$511$	−1.87019e12	−1.21337
$512$	1.66095e12	1.06817
$513$	0	0
$514$	−2.53467e12	−1.60172
$515$	−4.57722e10	−0.0286727
$516$	0	0
$517$	−1.58834e11	−0.0977767
$518$	−9.40610e11	−0.574018
$519$	0	0
$520$	−5.17906e10	−0.0310625
$521$	1.66627e12	0.990779	0.495389	−	0.868671i	$-0.335025\pi$
$521$	1.66627e12	0.990779	0.495389	+	0.868671i	$0.335025\pi$
$522$	0	0
$523$	1.97187e12	1.15244	0.576222	−	0.817293i	$-0.304526\pi$
$523$	1.97187e12	1.15244	0.576222	+	0.817293i	$0.304526\pi$
$524$	−3.26955e12	−1.89451
$525$	0	0
$526$	−3.11209e12	−1.77262
$527$	−7.24021e11	−0.408887
$528$	0	0
$529$	−1.61128e12	−0.894580
$530$	6.41452e10	0.0353121
$531$	0	0
$532$	−5.08330e12	−2.75134
$533$	−4.27182e11	−0.229266
$534$	0	0
$535$	−7.37202e10	−0.0389040
$536$	1.66914e12	0.873479
$537$	0	0
$538$	3.83852e12	1.97535
$539$	−1.25563e11	−0.0640784
$540$	0	0
$541$	1.54916e12	0.777513	0.388756	−	0.921341i	$-0.372905\pi$
$541$	1.54916e12	0.777513	0.388756	+	0.921341i	$0.372905\pi$
$542$	1.14031e12	0.567578
$543$	0	0
$544$	−3.12101e11	−0.152792
$545$	1.83027e11	0.0888648
$546$	0	0
$547$	1.98052e12	0.945879	0.472939	−	0.881095i	$-0.343193\pi$
$547$	1.98052e12	0.945879	0.472939	+	0.881095i	$0.343193\pi$
$548$	−4.76939e12	−2.25918
$549$	0	0
$550$	1.86744e12	0.870189
$551$	5.83283e12	2.69586
$552$	0	0
$553$	−1.19294e12	−0.542444
$554$	1.21068e12	0.546056
$555$	0	0
$556$	3.39871e12	1.50827
$557$	−8.86577e11	−0.390273	−0.195136	−	0.980776i	$-0.562515\pi$
$557$	−8.86577e11	−0.390273	−0.195136	+	0.980776i	$0.562515\pi$
$558$	0	0
$559$	1.14755e12	0.497071
$560$	9.58387e10	0.0411808
$561$	0	0
$562$	3.76705e12	1.59290
$563$	−3.89786e12	−1.63508	−0.817539	−	0.575873i	$-0.804662\pi$
$563$	−3.89786e12	−1.63508	−0.817539	+	0.575873i	$0.804662\pi$
$564$	0	0
$565$	2.02599e11	0.0836410
$566$	4.05212e12	1.65962
$567$	0	0
$568$	6.82501e12	2.75129
$569$	2.26590e12	0.906223	0.453112	−	0.891454i	$-0.350314\pi$
$569$	2.26590e12	0.906223	0.453112	+	0.891454i	$0.350314\pi$
$570$	0	0
$571$	−3.09546e12	−1.21860	−0.609302	−	0.792938i	$-0.708550\pi$
$571$	−3.09546e12	−1.21860	−0.609302	+	0.792938i	$0.708550\pi$
$572$	6.80184e11	0.265671
$573$	0	0
$574$	3.39527e12	1.30548
$575$	−8.45822e11	−0.322681
$576$	0	0
$577$	−3.23415e12	−1.21470	−0.607350	−	0.794434i	$-0.707767\pi$
$577$	−3.23415e12	−1.21470	−0.607350	+	0.794434i	$0.707767\pi$
$578$	4.07019e12	1.51684
$579$	0	0
$580$	−6.68692e11	−0.245358
$581$	−3.77411e11	−0.137411
$582$	0	0
$583$	−3.85927e11	−0.138356
$584$	5.19495e12	1.84809
$585$	0	0
$586$	8.63964e12	3.02661
$587$	8.78491e11	0.305398	0.152699	−	0.988273i	$-0.451204\pi$
$587$	8.78491e11	0.305398	0.152699	+	0.988273i	$0.451204\pi$
$588$	0	0
$589$	−5.99129e12	−2.05117
$590$	−6.39983e11	−0.217437
$591$	0	0
$592$	6.08321e11	0.203556
$593$	1.91730e12	0.636712	0.318356	−	0.947971i	$-0.396869\pi$
$593$	1.91730e12	0.636712	0.318356	+	0.947971i	$0.396869\pi$
$594$	0	0
$595$	−7.13632e10	−0.0233425
$596$	−1.20524e12	−0.391260
$597$	0	0
$598$	−4.75023e11	−0.151900
$599$	5.10464e12	1.62011	0.810054	−	0.586355i	$-0.199438\pi$
$599$	5.10464e12	1.62011	0.810054	+	0.586355i	$0.199438\pi$
$600$	0	0
$601$	3.00419e12	0.939273	0.469636	−	0.882860i	$-0.344385\pi$
$601$	3.00419e12	0.939273	0.469636	+	0.882860i	$0.344385\pi$
$602$	−9.12080e12	−2.83041
$603$	0	0
$604$	4.11260e12	1.25733
$605$	−1.89079e11	−0.0573778
$606$	0	0
$607$	1.18434e12	0.354101	0.177051	−	0.984202i	$-0.443344\pi$
$607$	1.18434e12	0.354101	0.177051	+	0.984202i	$0.443344\pi$
$608$	−2.58265e12	−0.766477
$609$	0	0
$610$	−3.62941e11	−0.106133
$611$	−1.79978e11	−0.0522436
$612$	0	0
$613$	6.42597e11	0.183809	0.0919045	−	0.995768i	$-0.470705\pi$
$613$	6.42597e11	0.183809	0.0919045	+	0.995768i	$0.470705\pi$
$614$	8.51193e11	0.241696
$615$	0	0
$616$	−2.47659e12	−0.693011
$617$	−5.05474e12	−1.40416	−0.702078	−	0.712100i	$-0.747744\pi$
$617$	−5.05474e12	−1.40416	−0.702078	+	0.712100i	$0.747744\pi$
$618$	0	0
$619$	−3.09492e11	−0.0847308	−0.0423654	−	0.999102i	$-0.513489\pi$
$619$	−3.09492e11	−0.0847308	−0.0423654	+	0.999102i	$0.513489\pi$
$620$	6.86859e11	0.186683
$621$	0	0
$622$	−1.03744e13	−2.77911
$623$	−2.06421e11	−0.0548980
$624$	0	0
$625$	3.74425e12	0.981533
$626$	3.01827e12	0.785549
$627$	0	0
$628$	−1.33223e13	−3.41792
$629$	−4.52966e11	−0.115382
$630$	0	0
$631$	−7.94237e11	−0.199443	−0.0997214	−	0.995015i	$-0.531795\pi$
$631$	−7.94237e11	−0.199443	−0.0997214	+	0.995015i	$0.531795\pi$
$632$	3.31370e12	0.826202
$633$	0	0
$634$	7.74944e12	1.90488
$635$	2.06268e11	0.0503442
$636$	0	0
$637$	−1.42278e11	−0.0342380
$638$	6.20328e12	1.48227
$639$	0	0
$640$	6.10991e11	0.143954
$641$	−2.30298e12	−0.538802	−0.269401	−	0.963028i	$-0.586826\pi$
$641$	−2.30298e12	−0.538802	−0.269401	+	0.963028i	$0.586826\pi$
$642$	0	0
$643$	−2.54593e12	−0.587350	−0.293675	−	0.955905i	$-0.594878\pi$
$643$	−2.54593e12	−0.587350	−0.293675	+	0.955905i	$0.594878\pi$
$644$	2.44862e12	0.560964
$645$	0	0
$646$	−3.77448e12	−0.852728
$647$	2.74568e11	0.0616001	0.0308000	−	0.999526i	$-0.490194\pi$
$647$	2.74568e11	0.0616001	0.0308000	+	0.999526i	$0.490194\pi$
$648$	0	0
$649$	3.85043e12	0.851937
$650$	2.11603e12	0.464955
$651$	0	0
$652$	6.84826e11	0.148411
$653$	6.05719e12	1.30365	0.651826	−	0.758369i	$-0.274003\pi$
$653$	6.05719e12	1.30365	0.651826	+	0.758369i	$0.274003\pi$
$654$	0	0
$655$	−3.79827e11	−0.0806306
$656$	−2.19582e12	−0.462946
$657$	0	0
$658$	1.43047e12	0.297484
$659$	−4.97718e12	−1.02801	−0.514007	−	0.857786i	$-0.671840\pi$
$659$	−4.97718e12	−1.02801	−0.514007	+	0.857786i	$0.671840\pi$
$660$	0	0
$661$	1.79834e12	0.366409	0.183204	−	0.983075i	$-0.441353\pi$
$661$	1.79834e12	0.366409	0.183204	+	0.983075i	$0.441353\pi$
$662$	−1.23718e13	−2.50364
$663$	0	0
$664$	1.04836e12	0.209292
$665$	−5.90532e11	−0.117097
$666$	0	0
$667$	−2.80967e12	−0.549653
$668$	7.87076e11	0.152940
$669$	0	0
$670$	4.23278e11	0.0811502
$671$	2.18362e12	0.415839
$672$	0	0
$673$	1.37693e12	0.258728	0.129364	−	0.991597i	$-0.458706\pi$
$673$	1.37693e12	0.258728	0.129364	+	0.991597i	$0.458706\pi$
$674$	−1.02486e13	−1.91292
$675$	0	0
$676$	7.70729e11	0.141952
$677$	−6.49642e11	−0.118857	−0.0594285	−	0.998233i	$-0.518928\pi$
$677$	−6.49642e11	−0.118857	−0.0594285	+	0.998233i	$0.518928\pi$
$678$	0	0
$679$	−7.45828e12	−1.34656
$680$	1.98230e11	0.0355532
$681$	0	0
$682$	−6.37181e12	−1.12780
$683$	1.10011e12	0.193438	0.0967190	−	0.995312i	$-0.469165\pi$
$683$	1.10011e12	0.193438	0.0967190	+	0.995312i	$0.469165\pi$
$684$	0	0
$685$	−5.54064e11	−0.0961507
$686$	1.02913e13	1.77424
$687$	0	0
$688$	5.89869e12	1.00371
$689$	−4.37301e11	−0.0739254
$690$	0	0
$691$	7.72289e12	1.28863	0.644315	−	0.764760i	$-0.277142\pi$
$691$	7.72289e12	1.28863	0.644315	+	0.764760i	$0.277142\pi$
$692$	−5.46071e11	−0.0905256
$693$	0	0
$694$	1.04855e13	1.71581
$695$	3.94832e11	0.0641920
$696$	0	0
$697$	1.63505e12	0.262412
$698$	1.35119e13	2.15460
$699$	0	0
$700$	−1.09076e13	−1.71707
$701$	−1.40436e12	−0.219658	−0.109829	−	0.993950i	$-0.535030\pi$
$701$	−1.40436e12	−0.219658	−0.109829	+	0.993950i	$0.535030\pi$
$702$	0	0
$703$	−3.74831e12	−0.578810
$704$	−4.64131e12	−0.712137
$705$	0	0
$706$	−8.61273e12	−1.30473
$707$	5.39111e12	0.811505
$708$	0	0
$709$	5.19764e12	0.772499	0.386250	−	0.922394i	$-0.373770\pi$
$709$	5.19764e12	0.772499	0.386250	+	0.922394i	$0.373770\pi$
$710$	1.73075e12	0.255607
$711$	0	0
$712$	5.73387e11	0.0836157
$713$	2.88600e12	0.418209
$714$	0	0
$715$	7.90176e10	0.0113070
$716$	1.47932e13	2.10355
$717$	0	0
$718$	1.29400e12	0.181707
$719$	1.27042e13	1.77284	0.886418	−	0.462886i	$-0.153186\pi$
$719$	1.27042e13	1.77284	0.886418	+	0.462886i	$0.153186\pi$
$720$	0	0
$721$	−2.48016e12	−0.341799
$722$	−1.89174e13	−2.59086
$723$	0	0
$724$	−2.06495e13	−2.79309
$725$	1.25159e13	1.68244
$726$	0	0
$727$	9.91998e11	0.131706	0.0658531	−	0.997829i	$-0.479023\pi$
$727$	9.91998e11	0.131706	0.0658531	+	0.997829i	$0.479023\pi$
$728$	−2.80627e12	−0.370286
$729$	0	0
$730$	1.31739e12	0.171696
$731$	−4.39227e12	−0.568933
$732$	0	0
$733$	5.54849e12	0.709916	0.354958	−	0.934882i	$-0.384495\pi$
$733$	5.54849e12	0.709916	0.354958	+	0.934882i	$0.384495\pi$
$734$	2.95572e12	0.375864
$735$	0	0
$736$	1.24406e12	0.156275
$737$	−2.54663e12	−0.317953
$738$	0	0
$739$	8.78597e12	1.08365	0.541826	−	0.840491i	$-0.317733\pi$
$739$	8.78597e12	1.08365	0.541826	+	0.840491i	$0.317733\pi$
$740$	4.29717e11	0.0526792
$741$	0	0
$742$	3.47570e12	0.420944
$743$	−4.04326e12	−0.486724	−0.243362	−	0.969936i	$-0.578250\pi$
$743$	−4.04326e12	−0.486724	−0.243362	+	0.969936i	$0.578250\pi$
$744$	0	0
$745$	−1.40014e11	−0.0166521
$746$	−2.48408e13	−2.93658
$747$	0	0
$748$	−2.60342e12	−0.304080
$749$	−3.99451e12	−0.463763
$750$	0	0
$751$	−2.52936e11	−0.0290156	−0.0145078	−	0.999895i	$-0.504618\pi$
$751$	−2.52936e11	−0.0290156	−0.0145078	+	0.999895i	$0.504618\pi$
$752$	−9.25130e11	−0.105493
$753$	0	0
$754$	7.02906e12	0.792001
$755$	4.77765e11	0.0535122
$756$	0	0
$757$	9.99638e12	1.10640	0.553199	−	0.833049i	$-0.313407\pi$
$757$	9.99638e12	1.10640	0.553199	+	0.833049i	$0.313407\pi$
$758$	−7.40426e12	−0.814649
$759$	0	0
$760$	1.64036e12	0.178352
$761$	2.98848e12	0.323013	0.161507	−	0.986872i	$-0.448365\pi$
$761$	2.98848e12	0.323013	0.161507	+	0.986872i	$0.448365\pi$
$762$	0	0
$763$	9.91726e12	1.05933
$764$	−1.58528e13	−1.68339
$765$	0	0
$766$	1.69714e13	1.78110
$767$	4.36299e12	0.455203
$768$	0	0
$769$	1.54887e13	1.59715	0.798576	−	0.601894i	$-0.205587\pi$
$769$	1.54887e13	1.59715	0.798576	+	0.601894i	$0.205587\pi$
$770$	−6.28038e11	−0.0643839
$771$	0	0
$772$	−2.55139e13	−2.58523
$773$	−1.07589e13	−1.08383	−0.541915	−	0.840434i	$-0.682300\pi$
$773$	−1.07589e13	−1.08383	−0.541915	+	0.840434i	$0.682300\pi$
$774$	0	0
$775$	−1.28559e13	−1.28010
$776$	2.07173e13	2.05095
$777$	0	0
$778$	−6.47427e12	−0.633552
$779$	1.35301e13	1.31638
$780$	0	0
$781$	−1.04130e13	−1.00149
$782$	1.81816e12	0.173861
$783$	0	0
$784$	−7.31343e11	−0.0691351
$785$	−1.54767e12	−0.145467
$786$	0	0
$787$	1.29304e13	1.20151	0.600755	−	0.799433i	$-0.294867\pi$
$787$	1.29304e13	1.20151	0.600755	+	0.799433i	$0.294867\pi$
$788$	−1.49545e13	−1.38167
$789$	0	0
$790$	8.40321e11	0.0767579
$791$	1.09778e13	0.997059
$792$	0	0
$793$	2.47430e12	0.222189
$794$	−1.12394e13	−1.00358
$795$	0	0
$796$	8.31828e12	0.734387
$797$	−1.24713e13	−1.09484	−0.547418	−	0.836859i	$-0.684389\pi$
$797$	−1.24713e13	−1.09484	−0.547418	+	0.836859i	$0.684389\pi$
$798$	0	0
$799$	6.88868e11	0.0597965
$800$	−5.54175e12	−0.478346
$801$	0	0
$802$	1.69224e13	1.44437
$803$	−7.92600e12	−0.672719
$804$	0	0
$805$	2.84459e11	0.0238747
$806$	−7.22002e12	−0.602602
$807$	0	0
$808$	−1.49752e13	−1.23601
$809$	2.40763e13	1.97616	0.988079	−	0.153948i	$-0.0491989\pi$
$809$	2.40763e13	1.97616	0.988079	+	0.153948i	$0.0491989\pi$
$810$	0	0
$811$	1.14686e13	0.930928	0.465464	−	0.885067i	$-0.345888\pi$
$811$	1.14686e13	0.930928	0.465464	+	0.885067i	$0.345888\pi$
$812$	−3.62330e13	−2.92484
$813$	0	0
$814$	−3.98637e12	−0.318249
$815$	7.95569e10	0.00631638
$816$	0	0
$817$	−3.63462e13	−2.85403
$818$	3.86469e12	0.301804
$819$	0	0
$820$	−1.55113e12	−0.119808
$821$	2.50113e13	1.92129	0.960643	−	0.277786i	$-0.0896005\pi$
$821$	2.50113e13	1.92129	0.960643	+	0.277786i	$0.0896005\pi$
$822$	0	0
$823$	1.29351e13	0.982808	0.491404	−	0.870932i	$-0.336484\pi$
$823$	1.29351e13	0.982808	0.491404	+	0.870932i	$0.336484\pi$
$824$	6.88929e12	0.520597
$825$	0	0
$826$	−3.46773e13	−2.59200
$827$	−1.00464e13	−0.746851	−0.373425	−	0.927660i	$-0.621817\pi$
$827$	−1.00464e13	−0.746851	−0.373425	+	0.927660i	$0.621817\pi$
$828$	0	0
$829$	8.77304e12	0.645141	0.322570	−	0.946545i	$-0.395453\pi$
$829$	8.77304e12	0.645141	0.322570	+	0.946545i	$0.395453\pi$
$830$	2.65853e11	0.0194442
$831$	0	0
$832$	−5.25916e12	−0.380506
$833$	5.44571e11	0.0391879
$834$	0	0
$835$	9.14353e10	0.00650916
$836$	−2.15434e13	−1.52541
$837$	0	0
$838$	3.03051e12	0.212284
$839$	1.54259e13	1.07478	0.537392	−	0.843332i	$-0.319410\pi$
$839$	1.54259e13	1.07478	0.537392	+	0.843332i	$0.319410\pi$
$840$	0	0
$841$	2.70683e13	1.86586
$842$	2.31593e13	1.58789
$843$	0	0
$844$	−1.72010e13	−1.16685
$845$	8.95364e10	0.00604149
$846$	0	0
$847$	−1.02452e13	−0.683983
$848$	−2.24784e12	−0.149274
$849$	0	0
$850$	−8.09914e12	−0.532174
$851$	1.80556e12	0.118012
$852$	0	0
$853$	−1.84928e12	−0.119600	−0.0598002	−	0.998210i	$-0.519046\pi$
$853$	−1.84928e12	−0.119600	−0.0598002	+	0.998210i	$0.519046\pi$
$854$	−1.96659e13	−1.26518
$855$	0	0
$856$	1.10958e13	0.706361
$857$	2.34715e13	1.48637	0.743185	−	0.669086i	$-0.233314\pi$
$857$	2.34715e13	1.48637	0.743185	+	0.669086i	$0.233314\pi$
$858$	0	0
$859$	1.25121e12	0.0784079	0.0392039	−	0.999231i	$-0.487518\pi$
$859$	1.25121e12	0.0784079	0.0392039	+	0.999231i	$0.487518\pi$
$860$	4.16683e12	0.259754
$861$	0	0
$862$	−1.04026e13	−0.641739
$863$	−2.38217e13	−1.46192	−0.730961	−	0.682419i	$-0.760928\pi$
$863$	−2.38217e13	−1.46192	−0.730961	+	0.682419i	$0.760928\pi$
$864$	0	0
$865$	−6.34375e10	−0.00385278
$866$	4.40928e13	2.66402
$867$	0	0
$868$	3.72173e13	2.22539
$869$	−5.05575e12	−0.300744
$870$	0	0
$871$	−2.88564e12	−0.169887
$872$	−2.75478e13	−1.61348
$873$	0	0
$874$	1.50453e13	0.872168
$875$	−2.54215e12	−0.146611
$876$	0	0
$877$	−1.35717e13	−0.774707	−0.387353	−	0.921931i	$-0.626611\pi$
$877$	−1.35717e13	−0.774707	−0.387353	+	0.921931i	$0.626611\pi$
$878$	−6.79601e12	−0.385948
$879$	0	0
$880$	4.06171e11	0.0228316
$881$	−2.38436e12	−0.133346	−0.0666730	−	0.997775i	$-0.521238\pi$
$881$	−2.38436e12	−0.133346	−0.0666730	+	0.997775i	$0.521238\pi$
$882$	0	0
$883$	−5.99779e11	−0.0332023	−0.0166011	−	0.999862i	$-0.505285\pi$
$883$	−5.99779e11	−0.0332023	−0.0166011	+	0.999862i	$0.505285\pi$
$884$	−2.94999e12	−0.162474
$885$	0	0
$886$	−2.50103e12	−0.136354
$887$	−1.02483e13	−0.555897	−0.277949	−	0.960596i	$-0.589654\pi$
$887$	−1.02483e13	−0.555897	−0.277949	+	0.960596i	$0.589654\pi$
$888$	0	0
$889$	1.11766e13	0.600138
$890$	1.45405e11	0.00776828
$891$	0	0
$892$	1.45042e13	0.767101
$893$	5.70040e12	0.299967
$894$	0	0
$895$	1.71854e12	0.0895274
$896$	3.31065e13	1.71604
$897$	0	0
$898$	−3.04642e13	−1.56331
$899$	−4.27050e13	−2.18052
$900$	0	0
$901$	1.67378e12	0.0846130
$902$	1.43894e13	0.723791
$903$	0	0
$904$	−3.04937e13	−1.51863
$905$	−2.39887e12	−0.118874
$906$	0	0
$907$	−2.43855e13	−1.19646	−0.598231	−	0.801323i	$-0.704130\pi$
$907$	−2.43855e13	−1.19646	−0.598231	+	0.801323i	$0.704130\pi$
$908$	−3.97416e13	−1.94026
$909$	0	0
$910$	−7.11641e11	−0.0344013
$911$	−2.90386e13	−1.39683	−0.698415	−	0.715693i	$-0.746111\pi$
$911$	−2.90386e13	−1.39683	−0.698415	+	0.715693i	$0.746111\pi$
$912$	0	0
$913$	−1.59949e12	−0.0761841
$914$	3.40744e13	1.61499
$915$	0	0
$916$	−6.31867e13	−2.96549
$917$	−2.05809e13	−0.961173
$918$	0	0
$919$	−1.41793e13	−0.655746	−0.327873	−	0.944722i	$-0.606332\pi$
$919$	−1.41793e13	−0.655746	−0.327873	+	0.944722i	$0.606332\pi$
$920$	−7.90158e11	−0.0363638
$921$	0	0
$922$	8.87378e12	0.404408
$923$	−1.17992e13	−0.535111
$924$	0	0
$925$	−8.04299e12	−0.361226
$926$	5.89402e13	2.63428
$927$	0	0
$928$	−1.84087e13	−0.814812
$929$	3.56246e13	1.56920	0.784601	−	0.620001i	$-0.212868\pi$
$929$	3.56246e13	1.56920	0.784601	+	0.620001i	$0.212868\pi$
$930$	0	0
$931$	4.50634e12	0.196585
$932$	4.90621e13	2.12998
$933$	0	0
$934$	6.23355e13	2.68025
$935$	−3.02442e11	−0.0129417
$936$	0	0
$937$	9.35575e12	0.396506	0.198253	−	0.980151i	$-0.436473\pi$
$937$	9.35575e12	0.396506	0.198253	+	0.980151i	$0.436473\pi$
$938$	2.29353e13	0.967366
$939$	0	0
$940$	−6.53510e11	−0.0273009
$941$	−1.23413e13	−0.513106	−0.256553	−	0.966530i	$-0.582587\pi$
$941$	−1.23413e13	−0.513106	−0.256553	+	0.966530i	$0.582587\pi$
$942$	0	0
$943$	−6.51742e12	−0.268394
$944$	2.24269e13	0.919168
$945$	0	0
$946$	−3.86546e13	−1.56924
$947$	−2.93722e12	−0.118676	−0.0593379	−	0.998238i	$-0.518899\pi$
$947$	−2.93722e12	−0.118676	−0.0593379	+	0.998238i	$0.518899\pi$
$948$	0	0
$949$	−8.98110e12	−0.359444
$950$	−6.70206e13	−2.66964
$951$	0	0
$952$	1.07411e13	0.423819
$953$	3.90544e12	0.153374	0.0766870	−	0.997055i	$-0.475566\pi$
$953$	3.90544e12	0.153374	0.0766870	+	0.997055i	$0.475566\pi$
$954$	0	0
$955$	−1.84163e12	−0.0716454
$956$	−7.04733e13	−2.72875
$957$	0	0
$958$	5.89375e13	2.26072
$959$	−3.00219e13	−1.14618
$960$	0	0
$961$	1.74255e13	0.659069
$962$	−4.51703e12	−0.170045
$963$	0	0
$964$	5.43139e13	2.02565
$965$	−2.96398e12	−0.110028
$966$	0	0
$967$	−1.06697e13	−0.392405	−0.196203	−	0.980563i	$-0.562861\pi$
$967$	−1.06697e13	−0.392405	−0.196203	+	0.980563i	$0.562861\pi$
$968$	2.84587e13	1.04178
$969$	0	0
$970$	5.25371e12	0.190543
$971$	3.10264e12	0.112007	0.0560034	−	0.998431i	$-0.482164\pi$
$971$	3.10264e12	0.112007	0.0560034	+	0.998431i	$0.482164\pi$
$972$	0	0
$973$	2.13939e13	0.765213
$974$	−9.74972e12	−0.347118
$975$	0	0
$976$	1.27185e13	0.448655
$977$	4.66167e13	1.63688	0.818438	−	0.574594i	$-0.194840\pi$
$977$	4.66167e13	1.63688	0.818438	+	0.574594i	$0.194840\pi$
$978$	0	0
$979$	−8.74824e11	−0.0304368
$980$	−5.16619e11	−0.0178918
$981$	0	0
$982$	6.04614e13	2.07480
$983$	3.22316e13	1.10101	0.550505	−	0.834832i	$-0.314435\pi$
$983$	3.22316e13	1.10101	0.550505	+	0.834832i	$0.314435\pi$
$984$	0	0
$985$	−1.73728e12	−0.0588040
$986$	−2.69039e13	−0.906502
$987$	0	0
$988$	−2.44112e13	−0.815047
$989$	1.75079e13	0.581903
$990$	0	0
$991$	−2.25008e13	−0.741081	−0.370540	−	0.928816i	$-0.620828\pi$
$991$	−2.25008e13	−0.741081	−0.370540	+	0.928816i	$0.620828\pi$
$992$	1.89088e13	0.619957
$993$	0	0
$994$	9.37807e13	3.04701
$995$	9.66342e11	0.0312556
$996$	0	0
$997$	−3.13756e13	−1.00569	−0.502844	−	0.864377i	$-0.667713\pi$
$997$	−3.13756e13	−1.00569	−0.502844	+	0.864377i	$0.667713\pi$
$998$	2.40442e13	0.767224
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.10.a.e.1.1		5
3.2	odd	2		13.10.a.b.1.5	✓	5
12.11	even	2		208.10.a.h.1.3		5
15.14	odd	2		325.10.a.b.1.1		5
39.38	odd	2		169.10.a.b.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.a.b.1.5	✓	5	3.2	odd	2
117.10.a.e.1.1		5	1.1	even	1	trivial
169.10.a.b.1.1		5	39.38	odd	2
208.10.a.h.1.3		5	12.11	even	2
325.10.a.b.1.1		5	15.14	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	117.10.a.e.1.1

Level	$117$
Weight	$10$
Character	117.1
Self dual	yes
Analytic conductor	$60.259$
Analytic rank	$1$
Dimension	$5$
CM	no
Inner twists	$1$