LMFDB - Embedded newform 261.12.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,12,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, 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("261.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	261.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:	$200.537570126$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} + \cdots - 75\!\cdots\!58 $ x^11 - x^10 - 15790x^9 + 14666x^8 + 87206462x^7 - 14008334x^6 - 203974096304x^5 - 388180519304x^4 + 193065378004825x^3 + 1279291654973975x^2 - 65244901875230266*x - 758324542468966858
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$81.6399$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-78.6399 q^{2} +4136.23 q^{4} +12464.4 q^{5} -230.276 q^{7} -164218. q^{8} +O(q^{10})$	\(q-78.6399 q^{2} +4136.23 q^{4} +12464.4 q^{5} -230.276 q^{7} -164218. q^{8} -980202. q^{10} +26731.2 q^{11} -1.35410e6 q^{13} +18108.8 q^{14} +4.44308e6 q^{16} -3.19725e6 q^{17} -9.37456e6 q^{19} +5.15558e7 q^{20} -2.10214e6 q^{22} +1.17129e7 q^{23} +1.06534e8 q^{25} +1.06486e8 q^{26} -952472. q^{28} -2.05111e7 q^{29} +1.69267e8 q^{31} -1.30850e7 q^{32} +2.51431e8 q^{34} -2.87026e6 q^{35} -6.21479e8 q^{37} +7.37214e8 q^{38} -2.04688e9 q^{40} +9.80521e8 q^{41} +1.55730e9 q^{43} +1.10566e8 q^{44} -9.21103e8 q^{46} -7.66462e8 q^{47} -1.97727e9 q^{49} -8.37783e9 q^{50} -5.60086e9 q^{52} +1.24890e7 q^{53} +3.33190e8 q^{55} +3.78154e7 q^{56} +1.61299e9 q^{58} -3.94786e9 q^{59} -6.43946e9 q^{61} -1.33112e10 q^{62} -8.07043e9 q^{64} -1.68781e10 q^{65} +4.26072e9 q^{67} -1.32246e10 q^{68} +2.25716e8 q^{70} -2.70019e10 q^{71} -8.00729e9 q^{73} +4.88730e10 q^{74} -3.87753e10 q^{76} -6.15555e6 q^{77} +4.92961e10 q^{79} +5.53805e10 q^{80} -7.71080e10 q^{82} +7.07907e10 q^{83} -3.98519e10 q^{85} -1.22466e11 q^{86} -4.38975e9 q^{88} +8.54214e10 q^{89} +3.11816e8 q^{91} +4.84473e10 q^{92} +6.02744e10 q^{94} -1.16849e11 q^{95} -4.48857e10 q^{97} +1.55493e11 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 32 q^{2} + 9146 q^{4} + 2740 q^{5} - 49432 q^{7} + 150054 q^{8}+O(q^{10}) $ 11 * q + 32 * q^2 + 9146 * q^4 + 2740 * q^5 - 49432 * q^7 + 150054 * q^8	\( 11 q + 32 q^{2} + 9146 q^{4} + 2740 q^{5} - 49432 q^{7} + 150054 q^{8} - 685834 q^{10} + 612246 q^{11} + 1510364 q^{13} - 3955400 q^{14} + 3024818 q^{16} + 3291098 q^{17} - 44121388 q^{19} + 49472662 q^{20} - 43435618 q^{22} + 88684076 q^{23} - 44195521 q^{25} + 324999762 q^{26} - 391274848 q^{28} - 225622639 q^{29} - 292235934 q^{31} + 632542514 q^{32} - 1113307936 q^{34} + 1312820120 q^{35} - 1380429338 q^{37} + 1222857284 q^{38} - 2713154106 q^{40} + 1062067494 q^{41} + 74588594 q^{43} - 52891466 q^{44} - 87670324 q^{46} + 1821239394 q^{47} + 4692522003 q^{49} - 9494259926 q^{50} + 3266669866 q^{52} - 7818635688 q^{53} - 191002682 q^{55} - 11263587512 q^{56} - 656356768 q^{58} - 1230002712 q^{59} - 18602654230 q^{61} - 22075953162 q^{62} + 11813658086 q^{64} - 32245789334 q^{65} + 27481284652 q^{67} - 29588811820 q^{68} + 42862666712 q^{70} + 20347168516 q^{71} - 57740010478 q^{73} + 2640709564 q^{74} - 33350650772 q^{76} - 871959792 q^{77} - 120245016462 q^{79} + 84319695274 q^{80} - 111495532412 q^{82} + 142463983824 q^{83} - 181628566552 q^{85} - 47870165542 q^{86} - 180608014462 q^{88} + 96700717270 q^{89} - 355162031176 q^{91} + 22429477796 q^{92} + 172608565078 q^{94} + 195922150708 q^{95} - 303190852014 q^{97} + 123776497136 q^{98}+O(q^{100}) \) 11 * q + 32 * q^2 + 9146 * q^4 + 2740 * q^5 - 49432 * q^7 + 150054 * q^8 - 685834 * q^10 + 612246 * q^11 + 1510364 * q^13 - 3955400 * q^14 + 3024818 * q^16 + 3291098 * q^17 - 44121388 * q^19 + 49472662 * q^20 - 43435618 * q^22 + 88684076 * q^23 - 44195521 * q^25 + 324999762 * q^26 - 391274848 * q^28 - 225622639 * q^29 - 292235934 * q^31 + 632542514 * q^32 - 1113307936 * q^34 + 1312820120 * q^35 - 1380429338 * q^37 + 1222857284 * q^38 - 2713154106 * q^40 + 1062067494 * q^41 + 74588594 * q^43 - 52891466 * q^44 - 87670324 * q^46 + 1821239394 * q^47 + 4692522003 * q^49 - 9494259926 * q^50 + 3266669866 * q^52 - 7818635688 * q^53 - 191002682 * q^55 - 11263587512 * q^56 - 656356768 * q^58 - 1230002712 * q^59 - 18602654230 * q^61 - 22075953162 * q^62 + 11813658086 * q^64 - 32245789334 * q^65 + 27481284652 * q^67 - 29588811820 * q^68 + 42862666712 * q^70 + 20347168516 * q^71 - 57740010478 * q^73 + 2640709564 * q^74 - 33350650772 * q^76 - 871959792 * q^77 - 120245016462 * q^79 + 84319695274 * q^80 - 111495532412 * q^82 + 142463983824 * q^83 - 181628566552 * q^85 - 47870165542 * q^86 - 180608014462 * q^88 + 96700717270 * q^89 - 355162031176 * q^91 + 22429477796 * q^92 + 172608565078 * q^94 + 195922150708 * q^95 - 303190852014 * q^97 + 123776497136 * q^98

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 $

$2$	−78.6399	−1.73771	−0.868856	−	0.495065i	$-0.835144\pi$
$2$	−78.6399	−1.73771	−0.868856	+	0.495065i	$0.835144\pi$
$3$	0	0
$3$	0	0
$4$	4136.23	2.01964
$5$	12464.4	1.78377	0.891883	−	0.452267i	$-0.149385\pi$
$5$	12464.4	1.78377	0.891883	+	0.452267i	$0.149385\pi$
$6$	0	0
$7$	−230.276	−0.00517856	−0.00258928	−	0.999997i	$-0.500824\pi$
$7$	−230.276	−0.00517856	−0.00258928	+	0.999997i	$0.500824\pi$
$8$	−164218.	−1.77184
$9$	0	0
$10$	−980202.	−3.09967
$11$	26731.2	0.0500448	0.0250224	−	0.999687i	$-0.492034\pi$
$11$	26731.2	0.0500448	0.0250224	+	0.999687i	$0.492034\pi$
$12$	0	0
$13$	−1.35410e6	−1.01149	−0.505745	−	0.862683i	$-0.668782\pi$
$13$	−1.35410e6	−1.01149	−0.505745	+	0.862683i	$0.668782\pi$
$14$	18108.8	0.00899884
$15$	0	0
$16$	4.44308e6	1.05931
$17$	−3.19725e6	−0.546144	−0.273072	−	0.961994i	$-0.588040\pi$
$17$	−3.19725e6	−0.546144	−0.273072	+	0.961994i	$0.588040\pi$
$18$	0	0
$19$	−9.37456e6	−0.868573	−0.434286	−	0.900775i	$-0.642999\pi$
$19$	−9.37456e6	−0.868573	−0.434286	+	0.900775i	$0.642999\pi$
$20$	5.15558e7	3.60257
$21$	0	0
$22$	−2.10214e6	−0.0869635
$23$	1.17129e7	0.379457	0.189728	−	0.981837i	$-0.439239\pi$
$23$	1.17129e7	0.379457	0.189728	+	0.981837i	$0.439239\pi$
$24$	0	0
$25$	1.06534e8	2.18182
$26$	1.06486e8	1.75768
$27$	0	0
$28$	−952472.	−0.0104588
$29$	−2.05111e7	−0.185695
$29$	−2.05111e7	−0.185695
$30$	0	0
$31$	1.69267e8	1.06190	0.530951	−	0.847403i	$-0.321835\pi$
$31$	1.69267e8	1.06190	0.530951	+	0.847403i	$0.321835\pi$
$32$	−1.30850e7	−0.0689364
$33$	0	0
$34$	2.51431e8	0.949041
$35$	−2.87026e6	−0.00923733
$36$	0	0
$37$	−6.21479e8	−1.47339	−0.736694	−	0.676227i	$-0.763614\pi$
$37$	−6.21479e8	−1.47339	−0.736694	+	0.676227i	$0.763614\pi$
$38$	7.37214e8	1.50933
$39$	0	0
$40$	−2.04688e9	−3.16055
$41$	9.80521e8	1.32174	0.660869	−	0.750501i	$-0.270188\pi$
$41$	9.80521e8	1.32174	0.660869	+	0.750501i	$0.270188\pi$
$42$	0	0
$43$	1.55730e9	1.61546	0.807729	−	0.589554i	$-0.200697\pi$
$43$	1.55730e9	1.61546	0.807729	+	0.589554i	$0.200697\pi$
$44$	1.10566e8	0.101073
$45$	0	0
$46$	−9.21103e8	−0.659386
$47$	−7.66462e8	−0.487475	−0.243737	−	0.969841i	$-0.578374\pi$
$47$	−7.66462e8	−0.487475	−0.243737	+	0.969841i	$0.578374\pi$
$48$	0	0
$49$	−1.97727e9	−0.999973
$50$	−8.37783e9	−3.79137
$51$	0	0
$52$	−5.60086e9	−2.04285
$53$	1.24890e7	0.00410215	0.00205108	−	0.999998i	$-0.499347\pi$
$53$	1.24890e7	0.00410215	0.00205108	+	0.999998i	$0.499347\pi$
$54$	0	0
$55$	3.33190e8	0.0892682
$56$	3.78154e7	0.00917560
$57$	0	0
$58$	1.61299e9	0.322685
$59$	−3.94786e9	−0.718911	−0.359456	−	0.933162i	$-0.617038\pi$
$59$	−3.94786e9	−0.718911	−0.359456	+	0.933162i	$0.617038\pi$
$60$	0	0
$61$	−6.43946e9	−0.976193	−0.488096	−	0.872790i	$-0.662309\pi$
$61$	−6.43946e9	−0.976193	−0.488096	+	0.872790i	$0.662309\pi$
$62$	−1.33112e10	−1.84528
$63$	0	0
$64$	−8.07043e9	−0.939522
$65$	−1.68781e10	−1.80426
$66$	0	0
$67$	4.26072e9	0.385542	0.192771	−	0.981244i	$-0.438253\pi$
$67$	4.26072e9	0.385542	0.192771	+	0.981244i	$0.438253\pi$
$68$	−1.32246e10	−1.10302
$69$	0	0
$70$	2.25716e8	0.0160518
$71$	−2.70019e10	−1.77612	−0.888062	−	0.459724i	$-0.847948\pi$
$71$	−2.70019e10	−1.77612	−0.888062	+	0.459724i	$0.847948\pi$
$72$	0	0
$73$	−8.00729e9	−0.452075	−0.226037	−	0.974119i	$-0.572577\pi$
$73$	−8.00729e9	−0.452075	−0.226037	+	0.974119i	$0.572577\pi$
$74$	4.88730e10	2.56032
$75$	0	0
$76$	−3.87753e10	−1.75421
$77$	−6.15555e6	−0.000259160 0
$78$	0	0
$79$	4.92961e10	1.80245	0.901225	−	0.433351i	$-0.142669\pi$
$79$	4.92961e10	1.80245	0.901225	+	0.433351i	$0.142669\pi$
$80$	5.53805e10	1.88957
$81$	0	0
$82$	−7.71080e10	−2.29680
$83$	7.07907e10	1.97264	0.986318	−	0.164856i	$-0.0527160\pi$
$83$	7.07907e10	1.97264	0.986318	+	0.164856i	$0.0527160\pi$
$84$	0	0
$85$	−3.98519e10	−0.974193
$86$	−1.22466e11	−2.80720
$87$	0	0
$88$	−4.38975e9	−0.0886717
$89$	8.54214e10	1.62152	0.810758	−	0.585381i	$-0.199055\pi$
$89$	8.54214e10	1.62152	0.810758	+	0.585381i	$0.199055\pi$
$90$	0	0
$91$	3.11816e8	0.00523806
$92$	4.84473e10	0.766367
$93$	0	0
$94$	6.02744e10	0.847091
$95$	−1.16849e11	−1.54933
$96$	0	0
$97$	−4.48857e10	−0.530718	−0.265359	−	0.964150i	$-0.585490\pi$
$97$	−4.48857e10	−0.530718	−0.265359	+	0.964150i	$0.585490\pi$
$98$	1.55493e11	1.73767
$99$	0	0
$100$	4.40649e11	4.40649
$101$	6.57247e9	0.0622245	0.0311122	−	0.999516i	$-0.490095\pi$
$101$	6.57247e9	0.0622245	0.0311122	+	0.999516i	$0.490095\pi$
$102$	0	0
$103$	−1.46925e11	−1.24880	−0.624398	−	0.781106i	$-0.714656\pi$
$103$	−1.46925e11	−1.24880	−0.624398	+	0.781106i	$0.714656\pi$
$104$	2.22367e11	1.79220
$105$	0	0
$106$	−9.82136e8	−0.00712836
$107$	1.16620e11	0.803825	0.401912	−	0.915678i	$-0.368346\pi$
$107$	1.16620e11	0.803825	0.401912	+	0.915678i	$0.368346\pi$
$108$	0	0
$109$	−2.91271e11	−1.81322	−0.906612	−	0.421966i	$-0.861340\pi$
$109$	−2.91271e11	−1.81322	−0.906612	+	0.421966i	$0.861340\pi$
$110$	−2.62020e10	−0.155122
$111$	0	0
$112$	−1.02313e9	−0.00548571
$113$	3.60464e10	0.184048	0.0920239	−	0.995757i	$-0.470666\pi$
$113$	3.60464e10	0.184048	0.0920239	+	0.995757i	$0.470666\pi$
$114$	0	0
$115$	1.45995e11	0.676862
$116$	−8.48388e10	−0.375038
$117$	0	0
$118$	3.10459e11	1.24926
$119$	7.36248e8	0.00282824
$120$	0	0
$121$	−2.84597e11	−0.997496
$122$	5.06398e11	1.69634
$123$	0	0
$124$	7.00129e11	2.14466
$125$	7.19272e11	2.10809
$126$	0	0
$127$	−3.25875e11	−0.875247	−0.437624	−	0.899158i	$-0.644180\pi$
$127$	−3.25875e11	−0.875247	−0.437624	+	0.899158i	$0.644180\pi$
$128$	6.61456e11	1.70155
$129$	0	0
$130$	1.32729e12	3.13529
$131$	−1.38770e11	−0.314271	−0.157136	−	0.987577i	$-0.550226\pi$
$131$	−1.38770e11	−0.314271	−0.157136	+	0.987577i	$0.550226\pi$
$132$	0	0
$133$	2.15873e9	0.00449795
$134$	−3.35063e11	−0.669961
$135$	0	0
$136$	5.25046e11	0.967683
$137$	−8.88362e11	−1.57263	−0.786315	−	0.617825i	$-0.788014\pi$
$137$	−8.88362e11	−1.57263	−0.786315	+	0.617825i	$0.788014\pi$
$138$	0	0
$139$	−5.26116e11	−0.860003	−0.430001	−	0.902828i	$-0.641487\pi$
$139$	−5.26116e11	−0.860003	−0.430001	+	0.902828i	$0.641487\pi$
$140$	−1.18720e10	−0.0186561
$141$	0	0
$142$	2.12342e12	3.08639
$143$	−3.61967e10	−0.0506199
$144$	0	0
$145$	−2.55660e11	−0.331237
$146$	6.29692e11	0.785575
$147$	0	0
$148$	−2.57058e12	−2.97572
$149$	2.24927e11	0.250910	0.125455	−	0.992099i	$-0.459961\pi$
$149$	2.24927e11	0.250910	0.125455	+	0.992099i	$0.459961\pi$
$150$	0	0
$151$	7.38831e11	0.765900	0.382950	−	0.923769i	$-0.374908\pi$
$151$	7.38831e11	0.765900	0.382950	+	0.923769i	$0.374908\pi$
$152$	1.53947e12	1.53898
$153$	0	0
$154$	4.84072e8	0.000450345 0
$155$	2.10982e12	1.89418
$156$	0	0
$157$	1.40898e12	1.17884	0.589422	−	0.807825i	$-0.299355\pi$
$157$	1.40898e12	1.17884	0.589422	+	0.807825i	$0.299355\pi$
$158$	−3.87664e12	−3.13214
$159$	0	0
$160$	−1.63097e11	−0.122966
$161$	−2.69720e9	−0.00196504
$162$	0	0
$163$	−9.72492e11	−0.661994	−0.330997	−	0.943632i	$-0.607385\pi$
$163$	−9.72492e11	−0.661994	−0.330997	+	0.943632i	$0.607385\pi$
$164$	4.05566e12	2.66944
$165$	0	0
$166$	−5.56697e12	−3.42787
$167$	3.91628e10	0.0233310	0.0116655	−	0.999932i	$-0.496287\pi$
$167$	3.91628e10	0.0233310	0.0116655	+	0.999932i	$0.496287\pi$
$168$	0	0
$169$	4.14236e10	0.0231138
$170$	3.13395e12	1.69287
$171$	0	0
$172$	6.44134e12	3.26265
$173$	2.21018e12	1.08436	0.542181	−	0.840262i	$-0.317599\pi$
$173$	2.21018e12	1.08436	0.542181	+	0.840262i	$0.317599\pi$
$174$	0	0
$175$	−2.45322e10	−0.0112987
$176$	1.18769e11	0.0530132
$177$	0	0
$178$	−6.71752e12	−2.81773
$179$	−2.23234e12	−0.907965	−0.453983	−	0.891010i	$-0.649997\pi$
$179$	−2.23234e12	−0.907965	−0.453983	+	0.891010i	$0.649997\pi$
$180$	0	0
$181$	−7.82120e10	−0.0299255	−0.0149627	−	0.999888i	$-0.504763\pi$
$181$	−7.82120e10	−0.0299255	−0.0149627	+	0.999888i	$0.504763\pi$
$182$	−2.45212e10	−0.00910224
$183$	0	0
$184$	−1.92347e12	−0.672338
$185$	−7.74638e12	−2.62818
$186$	0	0
$187$	−8.54664e10	−0.0273317
$188$	−3.17026e12	−0.984525
$189$	0	0
$190$	9.18896e12	2.69229
$191$	−2.06678e12	−0.588317	−0.294158	−	0.955757i	$-0.595039\pi$
$191$	−2.06678e12	−0.588317	−0.294158	+	0.955757i	$0.595039\pi$
$192$	0	0
$193$	2.83320e12	0.761574	0.380787	−	0.924663i	$-0.375653\pi$
$193$	2.83320e12	0.761574	0.380787	+	0.924663i	$0.375653\pi$
$194$	3.52981e12	0.922235
$195$	0	0
$196$	−8.17845e12	−2.01959
$197$	−4.94999e12	−1.18861	−0.594305	−	0.804239i	$-0.702573\pi$
$197$	−4.94999e12	−1.18861	−0.594305	+	0.804239i	$0.702573\pi$
$198$	0	0
$199$	−3.92808e12	−0.892253	−0.446127	−	0.894970i	$-0.647197\pi$
$199$	−3.92808e12	−0.892253	−0.446127	+	0.894970i	$0.647197\pi$
$200$	−1.74948e13	−3.86584
$201$	0	0
$202$	−5.16858e11	−0.108128
$203$	4.72322e9	0.000961634 0
$204$	0	0
$205$	1.22216e13	2.35767
$206$	1.15542e13	2.17005
$207$	0	0
$208$	−6.01637e12	−1.07149
$209$	−2.50594e11	−0.0434676
$210$	0	0
$211$	−6.85663e12	−1.12865	−0.564323	−	0.825554i	$-0.690863\pi$
$211$	−6.85663e12	−1.12865	−0.564323	+	0.825554i	$0.690863\pi$
$212$	5.16575e10	0.00828488
$213$	0	0
$214$	−9.17096e12	−1.39682
$215$	1.94109e13	2.88160
$216$	0	0
$217$	−3.89782e10	−0.00549911
$218$	2.29055e13	3.15086
$219$	0	0
$220$	1.37815e12	0.180290
$221$	4.32939e12	0.552420
$222$	0	0
$223$	−4.05692e12	−0.492629	−0.246314	−	0.969190i	$-0.579220\pi$
$223$	−4.05692e12	−0.492629	−0.246314	+	0.969190i	$0.579220\pi$
$224$	3.01315e9	0.000356991 0
$225$	0	0
$226$	−2.83469e12	−0.319822
$227$	−1.11423e13	−1.22696	−0.613481	−	0.789709i	$-0.710231\pi$
$227$	−1.11423e13	−1.22696	−0.613481	+	0.789709i	$0.710231\pi$
$228$	0	0
$229$	5.75727e12	0.604118	0.302059	−	0.953289i	$-0.402326\pi$
$229$	5.75727e12	0.604118	0.302059	+	0.953289i	$0.402326\pi$
$230$	−1.14810e13	−1.17619
$231$	0	0
$232$	3.36830e12	0.329023
$233$	−3.46144e12	−0.330217	−0.165108	−	0.986275i	$-0.552797\pi$
$233$	−3.46144e12	−0.330217	−0.165108	+	0.986275i	$0.552797\pi$
$234$	0	0
$235$	−9.55351e12	−0.869541
$236$	−1.63292e13	−1.45194
$237$	0	0
$238$	−5.78985e10	−0.00491466
$239$	−1.86431e13	−1.54643	−0.773215	−	0.634144i	$-0.781353\pi$
$239$	−1.86431e13	−1.54643	−0.773215	+	0.634144i	$0.781353\pi$
$240$	0	0
$241$	−3.22030e12	−0.255154	−0.127577	−	0.991829i	$-0.540720\pi$
$241$	−3.22030e12	−0.255154	−0.127577	+	0.991829i	$0.540720\pi$
$242$	2.23807e13	1.73336
$243$	0	0
$244$	−2.66351e13	−1.97156
$245$	−2.46456e13	−1.78372
$246$	0	0
$247$	1.26941e13	0.878553
$248$	−2.77967e13	−1.88152
$249$	0	0
$250$	−5.65635e13	−3.66325
$251$	−1.10256e13	−0.698547	−0.349273	−	0.937021i	$-0.613572\pi$
$251$	−1.10256e13	−0.698547	−0.349273	+	0.937021i	$0.613572\pi$
$252$	0	0
$253$	3.13101e11	0.0189899
$254$	2.56268e13	1.52093
$255$	0	0
$256$	−3.54885e13	−2.01729
$257$	1.84306e13	1.02543	0.512716	−	0.858558i	$-0.328639\pi$
$257$	1.84306e13	1.02543	0.512716	+	0.858558i	$0.328639\pi$
$258$	0	0
$259$	1.43111e11	0.00763002
$260$	−6.98116e13	−3.64396
$261$	0	0
$262$	1.09129e13	0.546113
$263$	−1.30599e13	−0.640005	−0.320003	−	0.947417i	$-0.603684\pi$
$263$	−1.30599e13	−0.640005	−0.320003	+	0.947417i	$0.603684\pi$
$264$	0	0
$265$	1.55669e11	0.00731728
$266$	−1.69762e11	−0.00781614
$267$	0	0
$268$	1.76233e13	0.778657
$269$	3.11337e13	1.34770	0.673850	−	0.738868i	$-0.264639\pi$
$269$	3.11337e13	1.34770	0.673850	+	0.738868i	$0.264639\pi$
$270$	0	0
$271$	2.84357e13	1.18177	0.590885	−	0.806756i	$-0.298779\pi$
$271$	2.84357e13	1.18177	0.590885	+	0.806756i	$0.298779\pi$
$272$	−1.42056e13	−0.578538
$273$	0	0
$274$	6.98607e13	2.73278
$275$	2.84779e12	0.109189
$276$	0	0
$277$	−2.26144e13	−0.833193	−0.416597	−	0.909091i	$-0.636777\pi$
$277$	−2.26144e13	−0.833193	−0.416597	+	0.909091i	$0.636777\pi$
$278$	4.13737e13	1.49444
$279$	0	0
$280$	4.71347e11	0.0163671
$281$	−3.67511e13	−1.25137	−0.625684	−	0.780076i	$-0.715180\pi$
$281$	−3.67511e13	−1.25137	−0.625684	+	0.780076i	$0.715180\pi$
$282$	0	0
$283$	1.41159e13	0.462255	0.231128	−	0.972923i	$-0.425759\pi$
$283$	1.41159e13	0.462255	0.231128	+	0.972923i	$0.425759\pi$
$284$	−1.11686e14	−3.58713
$285$	0	0
$286$	2.84651e12	0.0879628
$287$	−2.25790e11	−0.00684470
$288$	0	0
$289$	−2.40495e13	−0.701726
$290$	2.01051e13	0.575594
$291$	0	0
$292$	−3.31200e13	−0.913029
$293$	3.80422e12	0.102918	0.0514592	−	0.998675i	$-0.483613\pi$
$293$	3.80422e12	0.102918	0.0514592	+	0.998675i	$0.483613\pi$
$294$	0	0
$295$	−4.92078e13	−1.28237
$296$	1.02058e14	2.61061
$297$	0	0
$298$	−1.76882e13	−0.436009
$299$	−1.58605e13	−0.383817
$300$	0	0
$301$	−3.58608e11	−0.00836574
$302$	−5.81016e13	−1.33091
$303$	0	0
$304$	−4.16519e13	−0.920090
$305$	−8.02643e13	−1.74130
$306$	0	0
$307$	7.16897e13	1.50036	0.750180	−	0.661233i	$-0.229967\pi$
$307$	7.16897e13	1.50036	0.750180	+	0.661233i	$0.229967\pi$
$308$	−2.54608e10	−0.000523411 0
$309$	0	0
$310$	−1.65916e14	−3.29154
$311$	−5.89462e13	−1.14888	−0.574439	−	0.818547i	$-0.694780\pi$
$311$	−5.89462e13	−1.14888	−0.574439	+	0.818547i	$0.694780\pi$
$312$	0	0
$313$	−3.15570e13	−0.593747	−0.296874	−	0.954917i	$-0.595944\pi$
$313$	−3.15570e13	−0.593747	−0.296874	+	0.954917i	$0.595944\pi$
$314$	−1.10802e14	−2.04849
$315$	0	0
$316$	2.03900e14	3.64030
$317$	−4.00150e13	−0.702096	−0.351048	−	0.936357i	$-0.614175\pi$
$317$	−4.00150e13	−0.702096	−0.351048	+	0.936357i	$0.614175\pi$
$318$	0	0
$319$	−5.48288e11	−0.00929309
$320$	−1.00593e14	−1.67589
$321$	0	0
$322$	2.12107e11	0.00341467
$323$	2.99728e13	0.474366
$324$	0	0
$325$	−1.44258e14	−2.20689
$326$	7.64766e13	1.15035
$327$	0	0
$328$	−1.61019e14	−2.34192
$329$	1.76497e11	0.00252442
$330$	0	0
$331$	−7.96381e13	−1.10171	−0.550855	−	0.834601i	$-0.685698\pi$
$331$	−7.96381e13	−1.10171	−0.550855	+	0.834601i	$0.685698\pi$
$332$	2.92807e14	3.98402
$333$	0	0
$334$	−3.07976e12	−0.0405425
$335$	5.31075e13	0.687717
$336$	0	0
$337$	1.59262e13	0.199594	0.0997971	−	0.995008i	$-0.468181\pi$
$337$	1.59262e13	0.199594	0.0997971	+	0.995008i	$0.468181\pi$
$338$	−3.25755e12	−0.0401651
$339$	0	0
$340$	−1.64837e14	−1.96752
$341$	4.52473e12	0.0531427
$342$	0	0
$343$	9.10648e11	0.0103570
$344$	−2.55736e14	−2.86234
$345$	0	0
$346$	−1.73808e14	−1.88431
$347$	1.48839e14	1.58820	0.794100	−	0.607787i	$-0.207943\pi$
$347$	1.48839e14	1.58820	0.794100	+	0.607787i	$0.207943\pi$
$348$	0	0
$349$	1.26082e14	1.30351	0.651755	−	0.758430i	$-0.274033\pi$
$349$	1.26082e14	1.30351	0.651755	+	0.758430i	$0.274033\pi$
$350$	1.92921e12	0.0196338
$351$	0	0
$352$	−3.49778e11	−0.00344991
$353$	7.45266e12	0.0723687	0.0361843	−	0.999345i	$-0.488480\pi$
$353$	7.45266e12	0.0723687	0.0361843	+	0.999345i	$0.488480\pi$
$354$	0	0
$355$	−3.36563e14	−3.16819
$356$	3.53322e14	3.27488
$357$	0	0
$358$	1.75551e14	1.57778
$359$	3.60657e13	0.319209	0.159604	−	0.987181i	$-0.448978\pi$
$359$	3.60657e13	0.319209	0.159604	+	0.987181i	$0.448978\pi$
$360$	0	0
$361$	−2.86079e13	−0.245582
$362$	6.15058e12	0.0520019
$363$	0	0
$364$	1.28974e12	0.0105790
$365$	−9.98064e13	−0.806395
$366$	0	0
$367$	4.67133e13	0.366250	0.183125	−	0.983090i	$-0.441379\pi$
$367$	4.67133e13	0.366250	0.183125	+	0.983090i	$0.441379\pi$
$368$	5.20415e13	0.401964
$369$	0	0
$370$	6.09175e14	4.56701
$371$	−2.87592e9	−2.12432e−5 0
$372$	0	0
$373$	−1.54301e14	−1.10655	−0.553274	−	0.832999i	$-0.686622\pi$
$373$	−1.54301e14	−1.10655	−0.553274	+	0.832999i	$0.686622\pi$
$374$	6.72107e12	0.0474946
$375$	0	0
$376$	1.25867e14	0.863730
$377$	2.77741e13	0.187829
$378$	0	0
$379$	8.46962e13	0.556350	0.278175	−	0.960530i	$-0.410270\pi$
$379$	8.46962e13	0.556350	0.278175	+	0.960530i	$0.410270\pi$
$380$	−4.83313e14	−3.12909
$381$	0	0
$382$	1.62531e14	1.02233
$383$	−3.82978e13	−0.237455	−0.118727	−	0.992927i	$-0.537881\pi$
$383$	−3.82978e13	−0.237455	−0.118727	+	0.992927i	$0.537881\pi$
$384$	0	0
$385$	−7.67255e10	−0.000462281 0
$386$	−2.22802e14	−1.32340
$387$	0	0
$388$	−1.85658e14	−1.07186
$389$	−2.06383e14	−1.17476	−0.587381	−	0.809310i	$-0.699841\pi$
$389$	−2.06383e14	−1.17476	−0.587381	+	0.809310i	$0.699841\pi$
$390$	0	0
$391$	−3.74491e13	−0.207238
$392$	3.24704e14	1.77180
$393$	0	0
$394$	3.89266e14	2.06546
$395$	6.14448e14	3.21515
$396$	0	0
$397$	−2.05709e12	−0.0104690	−0.00523452	−	0.999986i	$-0.501666\pi$
$397$	−2.05709e12	−0.0104690	−0.00523452	+	0.999986i	$0.501666\pi$
$398$	3.08903e14	1.55048
$399$	0	0
$400$	4.73340e14	2.31123
$401$	3.68472e14	1.77464	0.887320	−	0.461154i	$-0.152564\pi$
$401$	3.68472e14	1.77464	0.887320	+	0.461154i	$0.152564\pi$
$402$	0	0
$403$	−2.29205e14	−1.07410
$404$	2.71852e13	0.125671
$405$	0	0
$406$	−3.71433e11	−0.00167104
$407$	−1.66129e13	−0.0737354
$408$	0	0
$409$	1.82553e14	0.788700	0.394350	−	0.918960i	$-0.370970\pi$
$409$	1.82553e14	0.788700	0.394350	+	0.918960i	$0.370970\pi$
$410$	−9.61108e14	−4.09695
$411$	0	0
$412$	−6.07716e14	−2.52212
$413$	9.09095e11	0.00372292
$414$	0	0
$415$	8.82366e14	3.51872
$416$	1.77184e13	0.0697285
$417$	0	0
$418$	1.97066e13	0.0755341
$419$	3.84878e14	1.45595	0.727974	−	0.685604i	$-0.240462\pi$
$419$	3.84878e14	1.45595	0.727974	+	0.685604i	$0.240462\pi$
$420$	0	0
$421$	1.47090e14	0.542042	0.271021	−	0.962573i	$-0.412639\pi$
$421$	1.47090e14	0.542042	0.271021	+	0.962573i	$0.412639\pi$
$422$	5.39205e14	1.96126
$423$	0	0
$424$	−2.05092e12	−0.00726838
$425$	−3.40616e14	−1.19159
$426$	0	0
$427$	1.48285e12	0.00505527
$428$	4.82366e14	1.62344
$429$	0	0
$430$	−1.52647e15	−5.00739
$431$	−2.42526e14	−0.785477	−0.392738	−	0.919650i	$-0.628472\pi$
$431$	−2.42526e14	−0.785477	−0.392738	+	0.919650i	$0.628472\pi$
$432$	0	0
$433$	−7.22429e13	−0.228093	−0.114046	−	0.993475i	$-0.536381\pi$
$433$	−7.22429e13	−0.228093	−0.114046	+	0.993475i	$0.536381\pi$
$434$	3.06524e12	0.00955588
$435$	0	0
$436$	−1.20476e15	−3.66206
$437$	−1.09804e14	−0.329586
$438$	0	0
$439$	4.25961e14	1.24685	0.623426	−	0.781882i	$-0.285740\pi$
$439$	4.25961e14	1.24685	0.623426	+	0.781882i	$0.285740\pi$
$440$	−5.47158e13	−0.158169
$441$	0	0
$442$	−3.40463e14	−0.959947
$443$	2.61970e14	0.729509	0.364754	−	0.931104i	$-0.381153\pi$
$443$	2.61970e14	0.729509	0.364754	+	0.931104i	$0.381153\pi$
$444$	0	0
$445$	1.06473e15	2.89240
$446$	3.19036e14	0.856047
$447$	0	0
$448$	1.85842e12	0.00486537
$449$	−5.02452e14	−1.29939	−0.649695	−	0.760195i	$-0.725103\pi$
$449$	−5.02452e14	−1.29939	−0.649695	+	0.760195i	$0.725103\pi$
$450$	0	0
$451$	2.62105e13	0.0661462
$452$	1.49096e14	0.371711
$453$	0	0
$454$	8.76226e14	2.13211
$455$	3.88661e12	0.00934347
$456$	0	0
$457$	−3.12924e14	−0.734344	−0.367172	−	0.930153i	$-0.619674\pi$
$457$	−3.12924e14	−0.734344	−0.367172	+	0.930153i	$0.619674\pi$
$458$	−4.52751e14	−1.04978
$459$	0	0
$460$	6.03869e14	1.36702
$461$	4.17648e14	0.934234	0.467117	−	0.884196i	$-0.345293\pi$
$461$	4.17648e14	0.934234	0.467117	+	0.884196i	$0.345293\pi$
$462$	0	0
$463$	−2.26782e14	−0.495351	−0.247676	−	0.968843i	$-0.579667\pi$
$463$	−2.26782e14	−0.495351	−0.247676	+	0.968843i	$0.579667\pi$
$464$	−9.11327e13	−0.196710
$465$	0	0
$466$	2.72207e14	0.573821
$467$	−9.21047e14	−1.91884	−0.959421	−	0.281979i	$-0.909009\pi$
$467$	−9.21047e14	−1.91884	−0.959421	+	0.281979i	$0.909009\pi$
$468$	0	0
$469$	−9.81140e11	−0.00199655
$470$	7.51287e14	1.51101
$471$	0	0
$472$	6.48309e14	1.27380
$473$	4.16285e13	0.0808453
$474$	0	0
$475$	−9.98710e14	−1.89507
$476$	3.04529e12	0.00571203
$477$	0	0
$478$	1.46609e15	2.68725
$479$	−2.79890e14	−0.507156	−0.253578	−	0.967315i	$-0.581607\pi$
$479$	−2.79890e14	−0.507156	−0.253578	+	0.967315i	$0.581607\pi$
$480$	0	0
$481$	8.41544e14	1.49032
$482$	2.53244e14	0.443384
$483$	0	0
$484$	−1.17716e15	−2.01458
$485$	−5.59476e14	−0.946676
$486$	0	0
$487$	−3.65476e14	−0.604574	−0.302287	−	0.953217i	$-0.597750\pi$
$487$	−3.65476e14	−0.604574	−0.302287	+	0.953217i	$0.597750\pi$
$488$	1.05748e15	1.72966
$489$	0	0
$490$	1.93813e15	3.09959
$491$	−8.51205e14	−1.34613	−0.673063	−	0.739585i	$-0.735022\pi$
$491$	−8.51205e14	−1.34613	−0.673063	+	0.739585i	$0.735022\pi$
$492$	0	0
$493$	6.55793e13	0.101416
$494$	−9.98261e14	−1.52667
$495$	0	0
$496$	7.52069e14	1.12489
$497$	6.21787e12	0.00919776
$498$	0	0
$499$	−1.13855e15	−1.64740	−0.823702	−	0.567023i	$-0.808095\pi$
$499$	−1.13855e15	−1.64740	−0.823702	+	0.567023i	$0.808095\pi$
$500$	2.97507e15	4.25758
$501$	0	0
$502$	8.67049e14	1.21387
$503$	1.06306e15	1.47208	0.736041	−	0.676936i	$-0.236693\pi$
$503$	1.06306e15	1.47208	0.736041	+	0.676936i	$0.236693\pi$
$504$	0	0
$505$	8.19222e13	0.110994
$506$	−2.46222e13	−0.0329989
$507$	0	0
$508$	−1.34789e15	−1.76769
$509$	5.69558e14	0.738908	0.369454	−	0.929249i	$-0.379545\pi$
$509$	5.69558e14	0.738908	0.369454	+	0.929249i	$0.379545\pi$
$510$	0	0
$511$	1.84388e12	0.00234109
$512$	1.43615e15	1.80391
$513$	0	0
$514$	−1.44938e15	−1.78190
$515$	−1.83134e15	−2.22756
$516$	0	0
$517$	−2.04885e13	−0.0243956
$518$	−1.12543e13	−0.0132588
$519$	0	0
$520$	2.77168e15	3.19687
$521$	5.33971e14	0.609411	0.304706	−	0.952447i	$-0.401442\pi$
$521$	5.33971e14	0.609411	0.304706	+	0.952447i	$0.401442\pi$
$522$	0	0
$523$	−1.01418e15	−1.13333	−0.566663	−	0.823949i	$-0.691766\pi$
$523$	−1.01418e15	−1.13333	−0.566663	+	0.823949i	$0.691766\pi$
$524$	−5.73986e14	−0.634716
$525$	0	0
$526$	1.02703e15	1.11214
$527$	−5.41190e14	−0.579951
$528$	0	0
$529$	−8.15617e14	−0.856013
$530$	−1.22418e13	−0.0127153
$531$	0	0
$532$	8.92901e12	0.00908425
$533$	−1.32772e15	−1.33693
$534$	0	0
$535$	1.45360e15	1.43383
$536$	−6.99687e14	−0.683121
$537$	0	0
$538$	−2.44835e15	−2.34191
$539$	−5.28550e13	−0.0500435
$540$	0	0
$541$	−2.04525e15	−1.89741	−0.948707	−	0.316158i	$-0.897607\pi$
$541$	−2.04525e15	−1.89741	−0.948707	+	0.316158i	$0.897607\pi$
$542$	−2.23618e15	−2.05358
$543$	0	0
$544$	4.18360e13	0.0376492
$545$	−3.63053e15	−3.23436
$546$	0	0
$547$	−2.04513e15	−1.78562	−0.892811	−	0.450431i	$-0.851270\pi$
$547$	−2.04513e15	−1.78562	−0.892811	+	0.450431i	$0.851270\pi$
$548$	−3.67447e15	−3.17615
$549$	0	0
$550$	−2.23950e14	−0.189739
$551$	1.92283e14	0.161290
$552$	0	0
$553$	−1.13517e13	−0.00933409
$554$	1.77839e15	1.44785
$555$	0	0
$556$	−2.17613e15	−1.73690
$557$	3.44808e14	0.272504	0.136252	−	0.990674i	$-0.456494\pi$
$557$	3.44808e14	0.272504	0.136252	+	0.990674i	$0.456494\pi$
$558$	0	0
$559$	−2.10874e15	−1.63402
$560$	−1.27528e13	−0.00978522
$561$	0	0
$562$	2.89010e15	2.17452
$563$	4.52289e14	0.336992	0.168496	−	0.985702i	$-0.446109\pi$
$563$	4.52289e14	0.336992	0.168496	+	0.985702i	$0.446109\pi$
$564$	0	0
$565$	4.49298e14	0.328298
$566$	−1.11007e15	−0.803266
$567$	0	0
$568$	4.43419e15	3.14702
$569$	−3.63369e14	−0.255405	−0.127703	−	0.991812i	$-0.540760\pi$
$569$	−3.63369e14	−0.255405	−0.127703	+	0.991812i	$0.540760\pi$
$570$	0	0
$571$	−2.17931e15	−1.50252	−0.751261	−	0.660005i	$-0.770554\pi$
$571$	−2.17931e15	−1.50252	−0.751261	+	0.660005i	$0.770554\pi$
$572$	−1.49718e14	−0.102234
$573$	0	0
$574$	1.77561e13	0.0118941
$575$	1.24783e15	0.827906
$576$	0	0
$577$	1.34691e15	0.876740	0.438370	−	0.898795i	$-0.355556\pi$
$577$	1.34691e15	0.876740	0.438370	+	0.898795i	$0.355556\pi$
$578$	1.89125e15	1.21940
$579$	0	0
$580$	−1.05747e15	−0.668980
$581$	−1.63014e13	−0.0102154
$582$	0	0
$583$	3.33847e11	0.000205292 0
$584$	1.31494e15	0.801006
$585$	0	0
$586$	−2.99163e14	−0.178843
$587$	−2.46936e15	−1.46243	−0.731214	−	0.682148i	$-0.761046\pi$
$587$	−2.46936e15	−1.46243	−0.731214	+	0.682148i	$0.761046\pi$
$588$	0	0
$589$	−1.58681e15	−0.922338
$590$	3.86970e15	2.22839
$591$	0	0
$592$	−2.76128e15	−1.56078
$593$	8.09156e14	0.453139	0.226569	−	0.973995i	$-0.427249\pi$
$593$	8.09156e14	0.453139	0.226569	+	0.973995i	$0.427249\pi$
$594$	0	0
$595$	9.17692e12	0.00504491
$596$	9.30350e14	0.506748
$597$	0	0
$598$	1.24726e15	0.666963
$599$	−6.45581e14	−0.342061	−0.171030	−	0.985266i	$-0.554710\pi$
$599$	−6.45581e14	−0.342061	−0.171030	+	0.985266i	$0.554710\pi$
$600$	0	0
$601$	1.25215e14	0.0651397	0.0325698	−	0.999469i	$-0.489631\pi$
$601$	1.25215e14	0.0651397	0.0325698	+	0.999469i	$0.489631\pi$
$602$	2.82009e13	0.0145372
$603$	0	0
$604$	3.05597e15	1.54684
$605$	−3.54734e15	−1.77930
$606$	0	0
$607$	6.50089e14	0.320210	0.160105	−	0.987100i	$-0.448817\pi$
$607$	6.50089e14	0.320210	0.160105	+	0.987100i	$0.448817\pi$
$608$	1.22666e14	0.0598762
$609$	0	0
$610$	6.31197e15	3.02587
$611$	1.03786e15	0.493076
$612$	0	0
$613$	2.17950e14	0.101701	0.0508504	−	0.998706i	$-0.483807\pi$
$613$	2.17950e14	0.101701	0.0508504	+	0.998706i	$0.483807\pi$
$614$	−5.63767e15	−2.60720
$615$	0	0
$616$	1.01085e12	0.000459191 0
$617$	1.34231e15	0.604343	0.302172	−	0.953254i	$-0.402288\pi$
$617$	1.34231e15	0.604343	0.302172	+	0.953254i	$0.402288\pi$
$618$	0	0
$619$	−1.52758e15	−0.675624	−0.337812	−	0.941214i	$-0.609687\pi$
$619$	−1.52758e15	−0.675624	−0.337812	+	0.941214i	$0.609687\pi$
$620$	8.72671e15	3.82557
$621$	0	0
$622$	4.63552e15	1.99642
$623$	−1.96705e13	−0.00839711
$624$	0	0
$625$	3.76347e15	1.57851
$626$	2.48164e15	1.03176
$627$	0	0
$628$	5.82786e15	2.38084
$629$	1.98702e15	0.804682
$630$	0	0
$631$	−3.33450e15	−1.32699	−0.663497	−	0.748179i	$-0.730928\pi$
$631$	−3.33450e15	−1.32699	−0.663497	+	0.748179i	$0.730928\pi$
$632$	−8.09530e15	−3.19366
$633$	0	0
$634$	3.14677e15	1.22004
$635$	−4.06185e15	−1.56124
$636$	0	0
$637$	2.67742e15	1.01146
$638$	4.31173e13	0.0161487
$639$	0	0
$640$	8.24467e15	3.03517
$641$	−5.75395e14	−0.210013	−0.105007	−	0.994472i	$-0.533486\pi$
$641$	−5.75395e14	−0.210013	−0.105007	+	0.994472i	$0.533486\pi$
$642$	0	0
$643$	−5.23490e14	−0.187823	−0.0939113	−	0.995581i	$-0.529937\pi$
$643$	−5.23490e14	−0.187823	−0.0939113	+	0.995581i	$0.529937\pi$
$644$	−1.11562e13	−0.00396867
$645$	0	0
$646$	−2.35706e15	−0.824311
$647$	4.35727e15	1.51092	0.755458	−	0.655197i	$-0.227414\pi$
$647$	4.35727e15	1.51092	0.755458	+	0.655197i	$0.227414\pi$
$648$	0	0
$649$	−1.05531e14	−0.0359778
$650$	1.13444e16	3.83494
$651$	0	0
$652$	−4.02245e15	−1.33699
$653$	3.06751e14	0.101103	0.0505514	−	0.998721i	$-0.483902\pi$
$653$	3.06751e14	0.101103	0.0505514	+	0.998721i	$0.483902\pi$
$654$	0	0
$655$	−1.72970e15	−0.560586
$656$	4.35653e15	1.40014
$657$	0	0
$658$	−1.38797e13	−0.00438671
$659$	−5.67369e14	−0.177826	−0.0889132	−	0.996039i	$-0.528339\pi$
$659$	−5.67369e14	−0.177826	−0.0889132	+	0.996039i	$0.528339\pi$
$660$	0	0
$661$	5.48457e15	1.69057	0.845287	−	0.534312i	$-0.179429\pi$
$661$	5.48457e15	1.69057	0.845287	+	0.534312i	$0.179429\pi$
$662$	6.26273e15	1.91445
$663$	0	0
$664$	−1.16251e16	−3.49520
$665$	2.69074e13	0.00802329
$666$	0	0
$667$	−2.40246e14	−0.0704633
$668$	1.61986e14	0.0471202
$669$	0	0
$670$	−4.17637e15	−1.19505
$671$	−1.72135e14	−0.0488534
$672$	0	0
$673$	6.70532e15	1.87213	0.936066	−	0.351825i	$-0.114439\pi$
$673$	6.70532e15	1.87213	0.936066	+	0.351825i	$0.114439\pi$
$674$	−1.25243e15	−0.346837
$675$	0	0
$676$	1.71338e14	0.0466816
$677$	−6.29656e14	−0.170163	−0.0850816	−	0.996374i	$-0.527115\pi$
$677$	−6.29656e14	−0.170163	−0.0850816	+	0.996374i	$0.527115\pi$
$678$	0	0
$679$	1.03361e13	0.00274835
$680$	6.54440e15	1.72612
$681$	0	0
$682$	−3.55824e14	−0.0923467
$683$	−3.10297e15	−0.798846	−0.399423	−	0.916767i	$-0.630790\pi$
$683$	−3.10297e15	−0.798846	−0.399423	+	0.916767i	$0.630790\pi$
$684$	0	0
$685$	−1.10729e16	−2.80520
$686$	−7.16132e13	−0.0179974
$687$	0	0
$688$	6.91921e15	1.71128
$689$	−1.69114e13	−0.00414929
$690$	0	0
$691$	4.40450e15	1.06357	0.531786	−	0.846879i	$-0.321521\pi$
$691$	4.40450e15	1.06357	0.531786	+	0.846879i	$0.321521\pi$
$692$	9.14182e15	2.19002
$693$	0	0
$694$	−1.17047e16	−2.75983
$695$	−6.55774e15	−1.53404
$696$	0	0
$697$	−3.13497e15	−0.721860
$698$	−9.91509e15	−2.26512
$699$	0	0
$700$	−1.01471e14	−0.0228193
$701$	−2.42636e15	−0.541385	−0.270693	−	0.962666i	$-0.587253\pi$
$701$	−2.42636e15	−0.541385	−0.270693	+	0.962666i	$0.587253\pi$
$702$	0	0
$703$	5.82609e15	1.27974
$704$	−2.15733e14	−0.0470182
$705$	0	0
$706$	−5.86076e14	−0.125756
$707$	−1.51348e12	−0.000322233 0
$708$	0	0
$709$	−6.40975e15	−1.34365	−0.671826	−	0.740709i	$-0.734490\pi$
$709$	−6.40975e15	−1.34365	−0.671826	+	0.740709i	$0.734490\pi$
$710$	2.64673e16	5.50540
$711$	0	0
$712$	−1.40277e16	−2.87308
$713$	1.98262e15	0.402946
$714$	0	0
$715$	−4.51172e14	−0.0902940
$716$	−9.23348e15	−1.83377
$717$	0	0
$718$	−2.83620e15	−0.554693
$719$	−8.38597e14	−0.162759	−0.0813794	−	0.996683i	$-0.525933\pi$
$719$	−8.38597e14	−0.162759	−0.0813794	+	0.996683i	$0.525933\pi$
$720$	0	0
$721$	3.38333e13	0.00646696
$722$	2.24972e15	0.426750
$723$	0	0
$724$	−3.23503e14	−0.0604388
$725$	−2.18514e15	−0.405153
$726$	0	0
$727$	−8.88989e15	−1.62352	−0.811759	−	0.583993i	$-0.801490\pi$
$727$	−8.88989e15	−1.62352	−0.811759	+	0.583993i	$0.801490\pi$
$728$	−5.12058e13	−0.00928103
$729$	0	0
$730$	7.84876e15	1.40128
$731$	−4.97907e15	−0.882273
$732$	0	0
$733$	3.14574e15	0.549099	0.274550	−	0.961573i	$-0.411471\pi$
$733$	3.14574e15	0.549099	0.274550	+	0.961573i	$0.411471\pi$
$734$	−3.67353e15	−0.636437
$735$	0	0
$736$	−1.53263e14	−0.0261584
$737$	1.13894e14	0.0192944
$738$	0	0
$739$	2.14266e15	0.357609	0.178804	−	0.983885i	$-0.442777\pi$
$739$	2.14266e15	0.357609	0.178804	+	0.983885i	$0.442777\pi$
$740$	−3.20408e16	−5.30798
$741$	0	0
$742$	2.26162e11	3.69146e−5 0
$743$	−3.63029e15	−0.588170	−0.294085	−	0.955779i	$-0.595015\pi$
$743$	−3.63029e15	−0.588170	−0.294085	+	0.955779i	$0.595015\pi$
$744$	0	0
$745$	2.80359e15	0.447564
$746$	1.21342e16	1.92286
$747$	0	0
$748$	−3.53509e14	−0.0552003
$749$	−2.68547e13	−0.00416265
$750$	0	0
$751$	−4.09016e15	−0.624770	−0.312385	−	0.949956i	$-0.601128\pi$
$751$	−4.09016e15	−0.624770	−0.312385	+	0.949956i	$0.601128\pi$
$752$	−3.40545e15	−0.516389
$753$	0	0
$754$	−2.18415e15	−0.326393
$755$	9.20911e15	1.36619
$756$	0	0
$757$	−8.56928e15	−1.25290	−0.626451	−	0.779461i	$-0.715493\pi$
$757$	−8.56928e15	−1.25290	−0.626451	+	0.779461i	$0.715493\pi$
$758$	−6.66050e15	−0.966777
$759$	0	0
$760$	1.91886e16	2.74517
$761$	−3.52288e15	−0.500360	−0.250180	−	0.968199i	$-0.580490\pi$
$761$	−3.52288e15	−0.500360	−0.250180	+	0.968199i	$0.580490\pi$
$762$	0	0
$763$	6.70726e13	0.00938988
$764$	−8.54868e15	−1.18819
$765$	0	0
$766$	3.01174e15	0.412628
$767$	5.34579e15	0.727172
$768$	0	0
$769$	−3.44010e15	−0.461292	−0.230646	−	0.973038i	$-0.574084\pi$
$769$	−3.44010e15	−0.461292	−0.230646	+	0.973038i	$0.574084\pi$
$770$	6.03368e12	0.000803310 0
$771$	0	0
$772$	1.17188e16	1.53811
$773$	1.09067e16	1.42137	0.710686	−	0.703510i	$-0.248385\pi$
$773$	1.09067e16	1.42137	0.710686	+	0.703510i	$0.248385\pi$
$774$	0	0
$775$	1.80328e16	2.31688
$776$	7.37104e15	0.940350
$777$	0	0
$778$	1.62299e16	2.04140
$779$	−9.19195e15	−1.14803
$780$	0	0
$781$	−7.21794e14	−0.0888858
$782$	2.94500e15	0.360120
$783$	0	0
$784$	−8.78519e15	−1.05928
$785$	1.75621e16	2.10278
$786$	0	0
$787$	−8.17365e15	−0.965061	−0.482531	−	0.875879i	$-0.660282\pi$
$787$	−8.17365e15	−0.965061	−0.482531	+	0.875879i	$0.660282\pi$
$788$	−2.04743e16	−2.40057
$789$	0	0
$790$	−4.83201e16	−5.58700
$791$	−8.30061e12	−0.000953102 0
$792$	0	0
$793$	8.71967e15	0.987410
$794$	1.61770e14	0.0181922
$795$	0	0
$796$	−1.62474e16	−1.80203
$797$	−1.09849e16	−1.20997	−0.604984	−	0.796238i	$-0.706820\pi$
$797$	−1.09849e16	−1.20997	−0.604984	+	0.796238i	$0.706820\pi$
$798$	0	0
$799$	2.45057e15	0.266232
$800$	−1.39400e15	−0.150407
$801$	0	0
$802$	−2.89766e16	−3.08381
$803$	−2.14045e14	−0.0226240
$804$	0	0
$805$	−3.36191e13	−0.00350517
$806$	1.80246e16	1.86648
$807$	0	0
$808$	−1.07932e15	−0.110252
$809$	−7.37110e15	−0.747851	−0.373926	−	0.927459i	$-0.621989\pi$
$809$	−7.37110e15	−0.747851	−0.373926	+	0.927459i	$0.621989\pi$
$810$	0	0
$811$	−1.58534e16	−1.58674	−0.793372	−	0.608737i	$-0.791676\pi$
$811$	−1.58534e16	−1.58674	−0.793372	+	0.608737i	$0.791676\pi$
$812$	1.95363e13	0.00194216
$813$	0	0
$814$	1.30644e15	0.128131
$815$	−1.21216e16	−1.18084
$816$	0	0
$817$	−1.45990e16	−1.40314
$818$	−1.43560e16	−1.37053
$819$	0	0
$820$	5.05515e16	4.76165
$821$	5.60490e15	0.524422	0.262211	−	0.965011i	$-0.415548\pi$
$821$	5.60490e15	0.524422	0.262211	+	0.965011i	$0.415548\pi$
$822$	0	0
$823$	−8.99004e15	−0.829971	−0.414985	−	0.909828i	$-0.636213\pi$
$823$	−8.99004e15	−0.829971	−0.414985	+	0.909828i	$0.636213\pi$
$824$	2.41278e16	2.21267
$825$	0	0
$826$	−7.14911e13	−0.00646937
$827$	−1.65577e16	−1.48840	−0.744199	−	0.667958i	$-0.767168\pi$
$827$	−1.65577e16	−1.48840	−0.744199	+	0.667958i	$0.767168\pi$
$828$	0	0
$829$	−5.71365e14	−0.0506832	−0.0253416	−	0.999679i	$-0.508067\pi$
$829$	−5.71365e14	−0.0506832	−0.0253416	+	0.999679i	$0.508067\pi$
$830$	−6.93892e16	−6.11452
$831$	0	0
$832$	1.09282e16	0.950318
$833$	6.32184e15	0.546130
$834$	0	0
$835$	4.88142e14	0.0416170
$836$	−1.03651e15	−0.0877890
$837$	0	0
$838$	−3.02668e16	−2.53002
$839$	−1.42940e16	−1.18703	−0.593517	−	0.804821i	$-0.702261\pi$
$839$	−1.42940e16	−1.18703	−0.593517	+	0.804821i	$0.702261\pi$
$840$	0	0
$841$	4.20707e14	0.0344828
$842$	−1.15672e16	−0.941912
$843$	0	0
$844$	−2.83606e16	−2.27946
$845$	5.16322e14	0.0412296
$846$	0	0
$847$	6.55358e13	0.00516559
$848$	5.54898e13	0.00434546
$849$	0	0
$850$	2.67860e16	2.07064
$851$	−7.27933e15	−0.559087
$852$	0	0
$853$	1.62880e16	1.23495	0.617473	−	0.786592i	$-0.288156\pi$
$853$	1.62880e16	1.23495	0.617473	+	0.786592i	$0.288156\pi$
$854$	−1.16611e14	−0.00878460
$855$	0	0
$856$	−1.91511e16	−1.42425
$857$	1.41268e16	1.04388	0.521940	−	0.852982i	$-0.325209\pi$
$857$	1.41268e16	1.04388	0.521940	+	0.852982i	$0.325209\pi$
$858$	0	0
$859$	−1.09143e16	−0.796224	−0.398112	−	0.917337i	$-0.630335\pi$
$859$	−1.09143e16	−0.796224	−0.398112	+	0.917337i	$0.630335\pi$
$860$	8.02877e16	5.81980
$861$	0	0
$862$	1.90722e16	1.36493
$863$	−6.50838e15	−0.462822	−0.231411	−	0.972856i	$-0.574334\pi$
$863$	−6.50838e15	−0.462822	−0.231411	+	0.972856i	$0.574334\pi$
$864$	0	0
$865$	2.75487e16	1.93425
$866$	5.68117e15	0.396359
$867$	0	0
$868$	−1.61223e14	−0.0111062
$869$	1.31774e15	0.0902033
$870$	0	0
$871$	−5.76944e15	−0.389972
$872$	4.78319e16	3.21275
$873$	0	0
$874$	8.63493e15	0.572725
$875$	−1.65631e14	−0.0109168
$876$	0	0
$877$	5.86025e15	0.381433	0.190716	−	0.981645i	$-0.438919\pi$
$877$	5.86025e15	0.381433	0.190716	+	0.981645i	$0.438919\pi$
$878$	−3.34975e16	−2.16667
$879$	0	0
$880$	1.48039e15	0.0945630
$881$	1.13320e16	0.719346	0.359673	−	0.933078i	$-0.382888\pi$
$881$	1.13320e16	0.719346	0.359673	+	0.933078i	$0.382888\pi$
$882$	0	0
$883$	4.59592e15	0.288130	0.144065	−	0.989568i	$-0.453983\pi$
$883$	4.59592e15	0.288130	0.144065	+	0.989568i	$0.453983\pi$
$884$	1.79074e16	1.11569
$885$	0	0
$886$	−2.06013e16	−1.26768
$887$	−1.85397e16	−1.13376	−0.566881	−	0.823800i	$-0.691850\pi$
$887$	−1.85397e16	−1.13376	−0.566881	+	0.823800i	$0.691850\pi$
$888$	0	0
$889$	7.50411e13	0.00453252
$890$	−8.37302e16	−5.02617
$891$	0	0
$892$	−1.67804e16	−0.994934
$893$	7.18524e15	0.423407
$894$	0	0
$895$	−2.78249e16	−1.61960
$896$	−1.52317e14	−0.00881159
$897$	0	0
$898$	3.95127e16	2.25796
$899$	−3.47187e15	−0.197190
$900$	0	0
$901$	−3.99306e13	−0.00224037
$902$	−2.06119e15	−0.114943
$903$	0	0
$904$	−5.91947e15	−0.326104
$905$	−9.74868e14	−0.0533800
$906$	0	0
$907$	1.52365e16	0.824226	0.412113	−	0.911133i	$-0.364791\pi$
$907$	1.52365e16	0.824226	0.412113	+	0.911133i	$0.364791\pi$
$908$	−4.60869e16	−2.47802
$909$	0	0
$910$	−3.05642e14	−0.0162363
$911$	2.85826e16	1.50921	0.754607	−	0.656177i	$-0.227828\pi$
$911$	2.85826e16	1.50921	0.754607	+	0.656177i	$0.227828\pi$
$912$	0	0
$913$	1.89232e15	0.0987202
$914$	2.46083e16	1.27608
$915$	0	0
$916$	2.38134e16	1.22010
$917$	3.19554e13	0.00162747
$918$	0	0
$919$	3.83197e15	0.192836	0.0964178	−	0.995341i	$-0.469262\pi$
$919$	3.83197e15	0.192836	0.0964178	+	0.995341i	$0.469262\pi$
$920$	−2.39750e16	−1.19929
$921$	0	0
$922$	−3.28438e16	−1.62343
$923$	3.65632e16	1.79653
$924$	0	0
$925$	−6.62087e16	−3.21466
$926$	1.78341e16	0.860778
$927$	0	0
$928$	2.68388e14	0.0128012
$929$	−1.12285e15	−0.0532396	−0.0266198	−	0.999646i	$-0.508474\pi$
$929$	−1.12285e15	−0.0532396	−0.0266198	+	0.999646i	$0.508474\pi$
$930$	0	0
$931$	1.85361e16	0.868549
$932$	−1.43173e16	−0.666920
$933$	0	0
$934$	7.24310e16	3.33439
$935$	−1.06529e15	−0.0487533
$936$	0	0
$937$	2.64634e16	1.19696	0.598478	−	0.801139i	$-0.295772\pi$
$937$	2.64634e16	1.19696	0.598478	+	0.801139i	$0.295772\pi$
$938$	7.71567e13	0.00346943
$939$	0	0
$940$	−3.95155e16	−1.75616
$941$	5.56710e15	0.245972	0.122986	−	0.992408i	$-0.460753\pi$
$941$	5.56710e15	0.245972	0.122986	+	0.992408i	$0.460753\pi$
$942$	0	0
$943$	1.14848e16	0.501543
$944$	−1.75407e16	−0.761552
$945$	0	0
$946$	−3.27366e15	−0.140486
$947$	4.76013e15	0.203093	0.101546	−	0.994831i	$-0.467621\pi$
$947$	4.76013e15	0.203093	0.101546	+	0.994831i	$0.467621\pi$
$948$	0	0
$949$	1.08427e16	0.457269
$950$	7.85384e16	3.29308
$951$	0	0
$952$	−1.20905e14	−0.00501120
$953$	2.30580e16	0.950189	0.475095	−	0.879935i	$-0.342414\pi$
$953$	2.30580e16	0.950189	0.475095	+	0.879935i	$0.342414\pi$
$954$	0	0
$955$	−2.57613e16	−1.04942
$956$	−7.71122e16	−3.12323
$957$	0	0
$958$	2.20105e16	0.881291
$959$	2.04568e14	0.00814396
$960$	0	0
$961$	3.24299e15	0.127634
$962$	−6.61789e16	−2.58974
$963$	0	0
$964$	−1.33199e16	−0.515320
$965$	3.53142e16	1.35847
$966$	0	0
$967$	4.58017e16	1.74195	0.870975	−	0.491327i	$-0.163488\pi$
$967$	4.58017e16	1.74195	0.870975	+	0.491327i	$0.163488\pi$
$968$	4.67360e16	1.76741
$969$	0	0
$970$	4.39971e16	1.64505
$971$	−3.23841e16	−1.20400	−0.601999	−	0.798497i	$-0.705629\pi$
$971$	−3.23841e16	−1.20400	−0.601999	+	0.798497i	$0.705629\pi$
$972$	0	0
$973$	1.21152e14	0.00445357
$974$	2.87410e16	1.05058
$975$	0	0
$976$	−2.86111e16	−1.03409
$977$	1.53051e16	0.550069	0.275035	−	0.961434i	$-0.411311\pi$
$977$	1.53051e16	0.550069	0.275035	+	0.961434i	$0.411311\pi$
$978$	0	0
$979$	2.28342e15	0.0811485
$980$	−1.01940e17	−3.60247
$981$	0	0
$982$	6.69386e16	2.33918
$983$	3.08928e16	1.07353	0.536764	−	0.843732i	$-0.319646\pi$
$983$	3.08928e16	1.07353	0.536764	+	0.843732i	$0.319646\pi$
$984$	0	0
$985$	−6.16988e16	−2.12020
$986$	−5.15714e15	−0.176233
$987$	0	0
$988$	5.25056e16	1.77436
$989$	1.82405e16	0.612996
$990$	0	0
$991$	−1.59520e16	−0.530163	−0.265082	−	0.964226i	$-0.585399\pi$
$991$	−1.59520e16	−0.530163	−0.265082	+	0.964226i	$0.585399\pi$
$992$	−2.21486e15	−0.0732036
$993$	0	0
$994$	−4.88973e14	−0.0159830
$995$	−4.89613e16	−1.59157
$996$	0	0
$997$	−1.48585e16	−0.477697	−0.238848	−	0.971057i	$-0.576770\pi$
$997$	−1.48585e16	−0.477697	−0.238848	+	0.971057i	$0.576770\pi$
$998$	8.95356e16	2.86271
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.12.a.a.1.1		11
3.2	odd	2		29.12.a.a.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.12.a.a.1.11	✓	11	3.2	odd	2
261.12.a.a.1.1		11	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q-78.6399 q^{2} +4136.23 q^{4} +12464.4 q^{5} -230.276 q^{7} -164218. q^{8} +O(q^{10})\)	\(q-78.6399 q^{2} +4136.23 q^{4} +12464.4 q^{5} -230.276 q^{7} -164218. q^{8} -980202. q^{10} +26731.2 q^{11} -1.35410e6 q^{13} +18108.8 q^{14} +4.44308e6 q^{16} -3.19725e6 q^{17} -9.37456e6 q^{19} +5.15558e7 q^{20} -2.10214e6 q^{22} +1.17129e7 q^{23} +1.06534e8 q^{25} +1.06486e8 q^{26} -952472. q^{28} -2.05111e7 q^{29} +1.69267e8 q^{31} -1.30850e7 q^{32} +2.51431e8 q^{34} -2.87026e6 q^{35} -6.21479e8 q^{37} +7.37214e8 q^{38} -2.04688e9 q^{40} +9.80521e8 q^{41} +1.55730e9 q^{43} +1.10566e8 q^{44} -9.21103e8 q^{46} -7.66462e8 q^{47} -1.97727e9 q^{49} -8.37783e9 q^{50} -5.60086e9 q^{52} +1.24890e7 q^{53} +3.33190e8 q^{55} +3.78154e7 q^{56} +1.61299e9 q^{58} -3.94786e9 q^{59} -6.43946e9 q^{61} -1.33112e10 q^{62} -8.07043e9 q^{64} -1.68781e10 q^{65} +4.26072e9 q^{67} -1.32246e10 q^{68} +2.25716e8 q^{70} -2.70019e10 q^{71} -8.00729e9 q^{73} +4.88730e10 q^{74} -3.87753e10 q^{76} -6.15555e6 q^{77} +4.92961e10 q^{79} +5.53805e10 q^{80} -7.71080e10 q^{82} +7.07907e10 q^{83} -3.98519e10 q^{85} -1.22466e11 q^{86} -4.38975e9 q^{88} +8.54214e10 q^{89} +3.11816e8 q^{91} +4.84473e10 q^{92} +6.02744e10 q^{94} -1.16849e11 q^{95} -4.48857e10 q^{97} +1.55493e11 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 32 q^{2} + 9146 q^{4} + 2740 q^{5} - 49432 q^{7} + 150054 q^{8}+O(q^{10}) \) 11 * q + 32 * q^2 + 9146 * q^4 + 2740 * q^5 - 49432 * q^7 + 150054 * q^8	\( 11 q + 32 q^{2} + 9146 q^{4} + 2740 q^{5} - 49432 q^{7} + 150054 q^{8} - 685834 q^{10} + 612246 q^{11} + 1510364 q^{13} - 3955400 q^{14} + 3024818 q^{16} + 3291098 q^{17} - 44121388 q^{19} + 49472662 q^{20} - 43435618 q^{22} + 88684076 q^{23} - 44195521 q^{25} + 324999762 q^{26} - 391274848 q^{28} - 225622639 q^{29} - 292235934 q^{31} + 632542514 q^{32} - 1113307936 q^{34} + 1312820120 q^{35} - 1380429338 q^{37} + 1222857284 q^{38} - 2713154106 q^{40} + 1062067494 q^{41} + 74588594 q^{43} - 52891466 q^{44} - 87670324 q^{46} + 1821239394 q^{47} + 4692522003 q^{49} - 9494259926 q^{50} + 3266669866 q^{52} - 7818635688 q^{53} - 191002682 q^{55} - 11263587512 q^{56} - 656356768 q^{58} - 1230002712 q^{59} - 18602654230 q^{61} - 22075953162 q^{62} + 11813658086 q^{64} - 32245789334 q^{65} + 27481284652 q^{67} - 29588811820 q^{68} + 42862666712 q^{70} + 20347168516 q^{71} - 57740010478 q^{73} + 2640709564 q^{74} - 33350650772 q^{76} - 871959792 q^{77} - 120245016462 q^{79} + 84319695274 q^{80} - 111495532412 q^{82} + 142463983824 q^{83} - 181628566552 q^{85} - 47870165542 q^{86} - 180608014462 q^{88} + 96700717270 q^{89} - 355162031176 q^{91} + 22429477796 q^{92} + 172608565078 q^{94} + 195922150708 q^{95} - 303190852014 q^{97} + 123776497136 q^{98}+O(q^{100}) \) 11 * q + 32 * q^2 + 9146 * q^4 + 2740 * q^5 - 49432 * q^7 + 150054 * q^8 - 685834 * q^10 + 612246 * q^11 + 1510364 * q^13 - 3955400 * q^14 + 3024818 * q^16 + 3291098 * q^17 - 44121388 * q^19 + 49472662 * q^20 - 43435618 * q^22 + 88684076 * q^23 - 44195521 * q^25 + 324999762 * q^26 - 391274848 * q^28 - 225622639 * q^29 - 292235934 * q^31 + 632542514 * q^32 - 1113307936 * q^34 + 1312820120 * q^35 - 1380429338 * q^37 + 1222857284 * q^38 - 2713154106 * q^40 + 1062067494 * q^41 + 74588594 * q^43 - 52891466 * q^44 - 87670324 * q^46 + 1821239394 * q^47 + 4692522003 * q^49 - 9494259926 * q^50 + 3266669866 * q^52 - 7818635688 * q^53 - 191002682 * q^55 - 11263587512 * q^56 - 656356768 * q^58 - 1230002712 * q^59 - 18602654230 * q^61 - 22075953162 * q^62 + 11813658086 * q^64 - 32245789334 * q^65 + 27481284652 * q^67 - 29588811820 * q^68 + 42862666712 * q^70 + 20347168516 * q^71 - 57740010478 * q^73 + 2640709564 * q^74 - 33350650772 * q^76 - 871959792 * q^77 - 120245016462 * q^79 + 84319695274 * q^80 - 111495532412 * q^82 + 142463983824 * q^83 - 181628566552 * q^85 - 47870165542 * q^86 - 180608014462 * q^88 + 96700717270 * q^89 - 355162031176 * q^91 + 22429477796 * q^92 + 172608565078 * q^94 + 195922150708 * q^95 - 303190852014 * q^97 + 123776497136 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more