LMFDB - Embedded newform 18.32.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,32,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 32, names="a")

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

chi := DirichletCharacter("18.1");

S:= CuspForms(chi, 32);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 32 $
Character orbit:	$[\chi]$	$=$	18.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:	$109.578839074$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-32768.0 q^{2} +1.07374e9 q^{4} -4.29517e10 q^{5} -1.68354e13 q^{7} -3.51844e13 q^{8} +O(q^{10})$

\(q-32768.0 q^{2} +1.07374e9 q^{4} -4.29517e10 q^{5} -1.68354e13 q^{7} -3.51844e13 q^{8} +1.40744e15 q^{10} +7.20783e15 q^{11} -2.70054e17 q^{13} +5.51661e17 q^{14} +1.15292e18 q^{16} +1.62755e19 q^{17} +1.09087e20 q^{19} -4.61190e19 q^{20} -2.36186e20 q^{22} +3.05935e20 q^{23} -2.81176e21 q^{25} +8.84912e21 q^{26} -1.80768e22 q^{28} -3.42166e22 q^{29} +1.62779e23 q^{31} -3.77789e22 q^{32} -5.33315e23 q^{34} +7.23107e23 q^{35} -2.50895e24 q^{37} -3.57457e24 q^{38} +1.51123e24 q^{40} +9.29498e24 q^{41} +5.86009e24 q^{43} +7.73935e24 q^{44} -1.00249e25 q^{46} +9.16365e25 q^{47} +1.25654e26 q^{49} +9.21359e25 q^{50} -2.89968e26 q^{52} -5.88384e26 q^{53} -3.09589e26 q^{55} +5.92342e26 q^{56} +1.12121e27 q^{58} +3.22410e27 q^{59} +4.04424e27 q^{61} -5.33394e27 q^{62} +1.23794e27 q^{64} +1.15993e28 q^{65} +1.64520e28 q^{67} +1.74757e28 q^{68} -2.36948e28 q^{70} +6.89178e27 q^{71} -2.34635e28 q^{73} +8.22132e28 q^{74} +1.17132e29 q^{76} -1.21346e29 q^{77} -6.35600e28 q^{79} -4.95199e28 q^{80} -3.04578e29 q^{82} +2.80657e29 q^{83} -6.99060e29 q^{85} -1.92023e29 q^{86} -2.53603e29 q^{88} -1.97625e30 q^{89} +4.54645e30 q^{91} +3.28496e29 q^{92} -3.00274e30 q^{94} -4.68549e30 q^{95} -7.78054e30 q^{97} -4.11743e30 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−32768.0	−0.707107
$2$	−32768.0	−0.707107
$3$	0	0
$3$	0	0
$4$	1.07374e9	0.500000
$5$	−4.29517e10	−0.629427	−0.314714	−	0.949187i	$-0.601908\pi$
$5$	−4.29517e10	−0.629427	−0.314714	+	0.949187i	$0.601908\pi$
$6$	0	0
$7$	−1.68354e13	−1.34030	−0.670151	−	0.742225i	$-0.733771\pi$
$7$	−1.68354e13	−1.34030	−0.670151	+	0.742225i	$0.733771\pi$
$8$	−3.51844e13	−0.353553
$9$	0	0
$10$	1.40744e15	0.445072
$11$	7.20783e15	0.520257	0.260129	−	0.965574i	$-0.416235\pi$
$11$	7.20783e15	0.520257	0.260129	+	0.965574i	$0.416235\pi$
$12$	0	0
$13$	−2.70054e17	−1.46328	−0.731641	−	0.681690i	$-0.761245\pi$
$13$	−2.70054e17	−1.46328	−0.731641	+	0.681690i	$0.761245\pi$
$14$	5.51661e17	0.947737
$15$	0	0
$16$	1.15292e18	0.250000
$17$	1.62755e19	1.37904	0.689518	−	0.724269i	$-0.257822\pi$
$17$	1.62755e19	1.37904	0.689518	+	0.724269i	$0.257822\pi$
$18$	0	0
$19$	1.09087e20	1.64852	0.824259	−	0.566214i	$-0.191592\pi$
$19$	1.09087e20	1.64852	0.824259	+	0.566214i	$0.191592\pi$
$20$	−4.61190e19	−0.314714
$21$	0	0
$22$	−2.36186e20	−0.367877
$23$	3.05935e20	0.239248	0.119624	−	0.992819i	$-0.461831\pi$
$23$	3.05935e20	0.239248	0.119624	+	0.992819i	$0.461831\pi$
$24$	0	0
$25$	−2.81176e21	−0.603822
$26$	8.84912e21	1.03470
$27$	0	0
$28$	−1.80768e22	−0.670151
$29$	−3.42166e22	−0.736322	−0.368161	−	0.929762i	$-0.620012\pi$
$29$	−3.42166e22	−0.736322	−0.368161	+	0.929762i	$0.620012\pi$
$30$	0	0
$31$	1.62779e23	1.24593	0.622963	−	0.782251i	$-0.285929\pi$
$31$	1.62779e23	1.24593	0.622963	+	0.782251i	$0.285929\pi$
$32$	−3.77789e22	−0.176777
$33$	0	0
$34$	−5.33315e23	−0.975126
$35$	7.23107e23	0.843622
$36$	0	0
$37$	−2.50895e24	−1.23699	−0.618493	−	0.785790i	$-0.712257\pi$
$37$	−2.50895e24	−1.23699	−0.618493	+	0.785790i	$0.712257\pi$
$38$	−3.57457e24	−1.16568
$39$	0	0
$40$	1.51123e24	0.222536
$41$	9.29498e24	0.933466	0.466733	−	0.884398i	$-0.345431\pi$
$41$	9.29498e24	0.933466	0.466733	+	0.884398i	$0.345431\pi$
$42$	0	0
$43$	5.86009e24	0.281283	0.140641	−	0.990061i	$-0.455084\pi$
$43$	5.86009e24	0.281283	0.140641	+	0.990061i	$0.455084\pi$
$44$	7.73935e24	0.260129
$45$	0	0
$46$	−1.00249e25	−0.169174
$47$	9.16365e25	1.10803	0.554015	−	0.832507i	$-0.313095\pi$
$47$	9.16365e25	1.10803	0.554015	+	0.832507i	$0.313095\pi$
$48$	0	0
$49$	1.25654e26	0.796410
$50$	9.21359e25	0.426966
$51$	0	0
$52$	−2.89968e26	−0.731641
$53$	−5.88384e26	−1.10506	−0.552528	−	0.833494i	$-0.686337\pi$
$53$	−5.88384e26	−1.10506	−0.552528	+	0.833494i	$0.686337\pi$
$54$	0	0
$55$	−3.09589e26	−0.327464
$56$	5.92342e26	0.473868
$57$	0	0
$58$	1.12121e27	0.520658
$59$	3.22410e27	1.14869	0.574344	−	0.818614i	$-0.305257\pi$
$59$	3.22410e27	1.14869	0.574344	+	0.818614i	$0.305257\pi$
$60$	0	0
$61$	4.04424e27	0.859458	0.429729	−	0.902958i	$-0.358609\pi$
$61$	4.04424e27	0.859458	0.429729	+	0.902958i	$0.358609\pi$
$62$	−5.33394e27	−0.881003
$63$	0	0
$64$	1.23794e27	0.125000
$65$	1.15993e28	0.921029
$66$	0	0
$67$	1.64520e28	0.816692	0.408346	−	0.912827i	$-0.366106\pi$
$67$	1.64520e28	0.816692	0.408346	+	0.912827i	$0.366106\pi$
$68$	1.74757e28	0.689518
$69$	0	0
$70$	−2.36948e28	−0.596531
$71$	6.89178e27	0.139260	0.0696301	−	0.997573i	$-0.477818\pi$
$71$	6.89178e27	0.139260	0.0696301	+	0.997573i	$0.477818\pi$
$72$	0	0
$73$	−2.34635e28	−0.308240	−0.154120	−	0.988052i	$-0.549254\pi$
$73$	−2.34635e28	−0.308240	−0.154120	+	0.988052i	$0.549254\pi$
$74$	8.22132e28	0.874681
$75$	0	0
$76$	1.17132e29	0.824259
$77$	−1.21346e29	−0.697302
$78$	0	0
$79$	−6.35600e28	−0.245451	−0.122725	−	0.992441i	$-0.539163\pi$
$79$	−6.35600e28	−0.245451	−0.122725	+	0.992441i	$0.539163\pi$
$80$	−4.95199e28	−0.157357
$81$	0	0
$82$	−3.04578e29	−0.660060
$83$	2.80657e29	0.504041	0.252020	−	0.967722i	$-0.418905\pi$
$83$	2.80657e29	0.504041	0.252020	+	0.967722i	$0.418905\pi$
$84$	0	0
$85$	−6.99060e29	−0.868003
$86$	−1.92023e29	−0.198897
$87$	0	0
$88$	−2.53603e29	−0.183939
$89$	−1.97625e30	−1.20308	−0.601542	−	0.798841i	$-0.705447\pi$
$89$	−1.97625e30	−1.20308	−0.601542	+	0.798841i	$0.705447\pi$
$90$	0	0
$91$	4.54645e30	1.96124
$92$	3.28496e29	0.119624
$93$	0	0
$94$	−3.00274e30	−0.783496
$95$	−4.68549e30	−1.03762
$96$	0	0
$97$	−7.78054e30	−1.24752	−0.623760	−	0.781616i	$-0.714396\pi$
$97$	−7.78054e30	−1.24752	−0.623760	+	0.781616i	$0.714396\pi$
$98$	−4.11743e30	−0.563147
$99$	0	0
$100$	−3.01911e30	−0.301911
$101$	7.73050e30	0.662562	0.331281	−	0.943532i	$-0.392519\pi$
$101$	7.73050e30	0.662562	0.331281	+	0.943532i	$0.392519\pi$
$102$	0	0
$103$	−1.41437e31	−0.894512	−0.447256	−	0.894406i	$-0.647599\pi$
$103$	−1.41437e31	−0.894512	−0.447256	+	0.894406i	$0.647599\pi$
$104$	9.50167e30	0.517348
$105$	0	0
$106$	1.92802e31	0.781393
$107$	−3.26395e31	−1.14365	−0.571827	−	0.820374i	$-0.693765\pi$
$107$	−3.26395e31	−1.14365	−0.571827	+	0.820374i	$0.693765\pi$
$108$	0	0
$109$	−3.26173e31	−0.857704	−0.428852	−	0.903375i	$-0.641082\pi$
$109$	−3.26173e31	−0.857704	−0.428852	+	0.903375i	$0.641082\pi$
$110$	1.01446e31	0.231552
$111$	0	0
$112$	−1.94098e31	−0.335076
$113$	1.08807e32	1.63659	0.818293	−	0.574801i	$-0.194920\pi$
$113$	1.08807e32	1.63659	0.818293	+	0.574801i	$0.194920\pi$
$114$	0	0
$115$	−1.31404e31	−0.150589
$116$	−3.67398e31	−0.368161
$117$	0	0
$118$	−1.05647e32	−0.812245
$119$	−2.74004e32	−1.84833
$120$	0	0
$121$	−1.39991e32	−0.729332
$122$	−1.32522e32	−0.607729
$123$	0	0
$124$	1.74783e32	0.622963
$125$	3.20780e32	1.00949
$126$	0	0
$127$	−2.53110e32	−0.622805	−0.311403	−	0.950278i	$-0.600799\pi$
$127$	−2.53110e32	−0.622805	−0.311403	+	0.950278i	$0.600799\pi$
$128$	−4.05648e31	−0.0883883
$129$	0	0
$130$	−3.80085e32	−0.651266
$131$	−4.62256e32	−0.703360	−0.351680	−	0.936120i	$-0.614389\pi$
$131$	−4.62256e32	−0.703360	−0.351680	+	0.936120i	$0.614389\pi$
$132$	0	0
$133$	−1.83652e33	−2.20951
$134$	−5.39098e32	−0.577488
$135$	0	0
$136$	−5.72643e32	−0.487563
$137$	−2.41393e33	−1.83466	−0.917332	−	0.398123i	$-0.869662\pi$
$137$	−2.41393e33	−1.83466	−0.917332	+	0.398123i	$0.869662\pi$
$138$	0	0
$139$	−1.53762e33	−0.933510	−0.466755	−	0.884387i	$-0.654577\pi$
$139$	−1.53762e33	−0.933510	−0.466755	+	0.884387i	$0.654577\pi$
$140$	7.76431e32	0.421811
$141$	0	0
$142$	−2.25830e32	−0.0984718
$143$	−1.94650e33	−0.761283
$144$	0	0
$145$	1.46966e33	0.463461
$146$	7.68852e32	0.217958
$147$	0	0
$148$	−2.69396e33	−0.618493
$149$	7.06717e33	1.46170	0.730850	−	0.682538i	$-0.239124\pi$
$149$	7.06717e33	1.46170	0.730850	+	0.682538i	$0.239124\pi$
$150$	0	0
$151$	−6.31239e33	−1.06182	−0.530911	−	0.847428i	$-0.678150\pi$
$151$	−6.31239e33	−1.06182	−0.530911	+	0.847428i	$0.678150\pi$
$152$	−3.83817e33	−0.582839
$153$	0	0
$154$	3.97628e33	0.493067
$155$	−6.99164e33	−0.784219
$156$	0	0
$157$	9.11534e33	0.838163	0.419082	−	0.907949i	$-0.362352\pi$
$157$	9.11534e33	0.838163	0.419082	+	0.907949i	$0.362352\pi$
$158$	2.08273e33	0.173560
$159$	0	0
$160$	1.62267e33	0.111268
$161$	−5.15053e33	−0.320665
$162$	0	0
$163$	−4.96877e33	−0.255471	−0.127735	−	0.991808i	$-0.540771\pi$
$163$	−4.96877e33	−0.255471	−0.127735	+	0.991808i	$0.540771\pi$
$164$	9.98041e33	0.466733
$165$	0	0
$166$	−9.19657e33	−0.356411
$167$	2.74735e34	0.970082	0.485041	−	0.874491i	$-0.338805\pi$
$167$	2.74735e34	0.970082	0.485041	+	0.874491i	$0.338805\pi$
$168$	0	0
$169$	3.88690e34	1.14119
$170$	2.29068e34	0.613770
$171$	0	0
$172$	6.29222e33	0.140641
$173$	−1.32315e34	−0.270331	−0.135165	−	0.990823i	$-0.543157\pi$
$173$	−1.32315e34	−0.270331	−0.135165	+	0.990823i	$0.543157\pi$
$174$	0	0
$175$	4.73370e34	0.809304
$176$	8.31007e33	0.130064
$177$	0	0
$178$	6.47576e34	0.850708
$179$	5.48512e34	0.660639	0.330319	−	0.943869i	$-0.392844\pi$
$179$	5.48512e34	0.660639	0.330319	+	0.943869i	$0.392844\pi$
$180$	0	0
$181$	−6.85442e34	−0.694949	−0.347475	−	0.937689i	$-0.612961\pi$
$181$	−6.85442e34	−0.694949	−0.347475	+	0.937689i	$0.612961\pi$
$182$	−1.48978e35	−1.38681
$183$	0	0
$184$	−1.07641e34	−0.0845869
$185$	1.07764e35	0.778593
$186$	0	0
$187$	1.17311e35	0.717453
$188$	9.83939e34	0.554015
$189$	0	0
$190$	1.53534e35	0.733709
$191$	−2.17438e35	−0.957895	−0.478947	−	0.877844i	$-0.658982\pi$
$191$	−2.17438e35	−0.957895	−0.478947	+	0.877844i	$0.658982\pi$
$192$	0	0
$193$	3.50352e34	0.131331	0.0656653	−	0.997842i	$-0.479083\pi$
$193$	3.50352e34	0.131331	0.0656653	+	0.997842i	$0.479083\pi$
$194$	2.54953e35	0.882130
$195$	0	0
$196$	1.34920e35	0.398205
$197$	−4.67654e35	−1.27555	−0.637776	−	0.770222i	$-0.720145\pi$
$197$	−4.67654e35	−1.27555	−0.637776	+	0.770222i	$0.720145\pi$
$198$	0	0
$199$	3.55886e35	0.830020	0.415010	−	0.909817i	$-0.363778\pi$
$199$	3.55886e35	0.830020	0.415010	+	0.909817i	$0.363778\pi$
$200$	9.89301e34	0.213483
$201$	0	0
$202$	−2.53313e35	−0.468502
$203$	5.76049e35	0.986894
$204$	0	0
$205$	−3.99235e35	−0.587549
$206$	4.63461e35	0.632515
$207$	0	0
$208$	−3.11351e35	−0.365821
$209$	7.86283e35	0.857653
$210$	0	0
$211$	9.00169e35	0.847122	0.423561	−	0.905868i	$-0.360780\pi$
$211$	9.00169e35	0.847122	0.423561	+	0.905868i	$0.360780\pi$
$212$	−6.31772e35	−0.552528
$213$	0	0
$214$	1.06953e36	0.808685
$215$	−2.51701e35	−0.177047
$216$	0	0
$217$	−2.74044e36	−1.66992
$218$	1.06880e36	0.606489
$219$	0	0
$220$	−3.32418e35	−0.163732
$221$	−4.39525e36	−2.01792
$222$	0	0
$223$	−1.53232e36	−0.611820	−0.305910	−	0.952060i	$-0.598961\pi$
$223$	−1.53232e36	−0.611820	−0.305910	+	0.952060i	$0.598961\pi$
$224$	6.36022e35	0.236934
$225$	0	0
$226$	−3.56537e36	−1.15724
$227$	−1.78639e36	−0.541470	−0.270735	−	0.962654i	$-0.587267\pi$
$227$	−1.78639e36	−0.541470	−0.270735	+	0.962654i	$0.587267\pi$
$228$	0	0
$229$	−4.03297e36	−1.06703	−0.533514	−	0.845791i	$-0.679129\pi$
$229$	−4.03297e36	−1.06703	−0.533514	+	0.845791i	$0.679129\pi$
$230$	4.30586e35	0.106483
$231$	0	0
$232$	1.20389e36	0.260329
$233$	−4.83929e36	−0.978959	−0.489480	−	0.872015i	$-0.662813\pi$
$233$	−4.83929e36	−0.978959	−0.489480	+	0.872015i	$0.662813\pi$
$234$	0	0
$235$	−3.93594e36	−0.697424
$236$	3.46185e36	0.574344
$237$	0	0
$238$	8.97855e36	1.30696
$239$	−7.98646e36	−1.08940	−0.544699	−	0.838632i	$-0.683356\pi$
$239$	−7.98646e36	−1.08940	−0.544699	+	0.838632i	$0.683356\pi$
$240$	0	0
$241$	5.53361e36	0.663353	0.331676	−	0.943393i	$-0.392386\pi$
$241$	5.53361e36	0.663353	0.331676	+	0.943393i	$0.392386\pi$
$242$	4.58721e36	0.515716
$243$	0	0
$244$	4.34247e36	0.429729
$245$	−5.39705e36	−0.501282
$246$	0	0
$247$	−2.94594e37	−2.41225
$248$	−5.72728e36	−0.440501
$249$	0	0
$250$	−1.05113e37	−0.713816
$251$	1.82546e37	1.16528	0.582640	−	0.812730i	$-0.302020\pi$
$251$	1.82546e37	1.16528	0.582640	+	0.812730i	$0.302020\pi$
$252$	0	0
$253$	2.20513e36	0.124470
$254$	8.29392e36	0.440390
$255$	0	0
$256$	1.32923e36	0.0625000
$257$	3.53190e36	0.156331	0.0781654	−	0.996940i	$-0.475094\pi$
$257$	3.53190e36	0.156331	0.0781654	+	0.996940i	$0.475094\pi$
$258$	0	0
$259$	4.22390e37	1.65794
$260$	1.24546e37	0.460515
$261$	0	0
$262$	1.51472e37	0.497350
$263$	−3.77552e37	−1.16859	−0.584296	−	0.811541i	$-0.698629\pi$
$263$	−3.77552e37	−1.16859	−0.584296	+	0.811541i	$0.698629\pi$
$264$	0	0
$265$	2.52721e37	0.695553
$266$	6.01792e37	1.56236
$267$	0	0
$268$	1.76652e37	0.408346
$269$	1.73593e37	0.378767	0.189384	−	0.981903i	$-0.439351\pi$
$269$	1.73593e37	0.378767	0.189384	+	0.981903i	$0.439351\pi$
$270$	0	0
$271$	6.46880e36	0.125835	0.0629173	−	0.998019i	$-0.479960\pi$
$271$	6.46880e36	0.125835	0.0629173	+	0.998019i	$0.479960\pi$
$272$	1.87644e37	0.344759
$273$	0	0
$274$	7.90998e37	1.29730
$275$	−2.02667e37	−0.314143
$276$	0	0
$277$	−1.98649e37	−0.275201	−0.137601	−	0.990488i	$-0.543939\pi$
$277$	−1.98649e37	−0.275201	−0.137601	+	0.990488i	$0.543939\pi$
$278$	5.03846e37	0.660091
$279$	0	0
$280$	−2.54421e37	−0.298266
$281$	−5.48253e37	−0.608181	−0.304091	−	0.952643i	$-0.598353\pi$
$281$	−5.48253e37	−0.608181	−0.304091	+	0.952643i	$0.598353\pi$
$282$	0	0
$283$	−2.62737e37	−0.261115	−0.130557	−	0.991441i	$-0.541677\pi$
$283$	−2.62737e37	−0.261115	−0.130557	+	0.991441i	$0.541677\pi$
$284$	7.39999e36	0.0696301
$285$	0	0
$286$	6.37830e37	0.538308
$287$	−1.56484e38	−1.25113
$288$	0	0
$289$	1.25602e38	0.901740
$290$	−4.81579e37	−0.327716
$291$	0	0
$292$	−2.51937e37	−0.154120
$293$	1.33198e38	0.772768	0.386384	−	0.922338i	$-0.373724\pi$
$293$	1.33198e38	0.772768	0.386384	+	0.922338i	$0.373724\pi$
$294$	0	0
$295$	−1.38481e38	−0.723016
$296$	8.82757e37	0.437341
$297$	0	0
$298$	−2.31577e38	−1.03358
$299$	−8.26190e37	−0.350087
$300$	0	0
$301$	−9.86567e37	−0.377004
$302$	2.06844e38	0.750821
$303$	0	0
$304$	1.25769e38	0.412129
$305$	−1.73707e38	−0.540966
$306$	0	0
$307$	−4.38543e37	−0.123415	−0.0617075	−	0.998094i	$-0.519655\pi$
$307$	−4.38543e37	−0.123415	−0.0617075	+	0.998094i	$0.519655\pi$
$308$	−1.30295e38	−0.348651
$309$	0	0
$310$	2.29102e38	0.554527
$311$	2.86015e38	0.658571	0.329285	−	0.944230i	$-0.393192\pi$
$311$	2.86015e38	0.658571	0.329285	+	0.944230i	$0.393192\pi$
$312$	0	0
$313$	−6.01224e38	−1.25343	−0.626714	−	0.779250i	$-0.715600\pi$
$313$	−6.01224e38	−1.25343	−0.626714	+	0.779250i	$0.715600\pi$
$314$	−2.98692e38	−0.592671
$315$	0	0
$316$	−6.82470e37	−0.122725
$317$	−9.02679e38	−1.54566	−0.772831	−	0.634612i	$-0.781160\pi$
$317$	−9.02679e38	−1.54566	−0.772831	+	0.634612i	$0.781160\pi$
$318$	0	0
$319$	−2.46627e38	−0.383077
$320$	−5.31716e37	−0.0786784
$321$	0	0
$322$	1.68773e38	0.226744
$323$	1.77545e39	2.27336
$324$	0	0
$325$	7.59327e38	0.883561
$326$	1.62817e38	0.180645
$327$	0	0
$328$	−3.27038e38	−0.330030
$329$	−1.54273e39	−1.48510
$330$	0	0
$331$	−1.37329e39	−1.20345	−0.601725	−	0.798704i	$-0.705520\pi$
$331$	−1.37329e39	−1.20345	−0.601725	+	0.798704i	$0.705520\pi$
$332$	3.01353e38	0.252020
$333$	0	0
$334$	−9.00252e38	−0.685952
$335$	−7.06640e38	−0.514048
$336$	0	0
$337$	1.29341e37	0.00857975	0.00428987	−	0.999991i	$-0.498634\pi$
$337$	1.29341e37	0.00857975	0.00428987	+	0.999991i	$0.498634\pi$
$338$	−1.27366e39	−0.806946
$339$	0	0
$340$	−7.50610e38	−0.434001
$341$	1.17328e39	0.648202
$342$	0	0
$343$	5.40776e38	0.272872
$344$	−2.06183e38	−0.0994485
$345$	0	0
$346$	4.33571e38	0.191153
$347$	1.27078e39	0.535753	0.267876	−	0.963453i	$-0.413678\pi$
$347$	1.27078e39	0.535753	0.267876	+	0.963453i	$0.413678\pi$
$348$	0	0
$349$	1.38237e39	0.533128	0.266564	−	0.963817i	$-0.414112\pi$
$349$	1.38237e39	0.533128	0.266564	+	0.963817i	$0.414112\pi$
$350$	−1.55114e39	−0.572264
$351$	0	0
$352$	−2.72304e38	−0.0919693
$353$	5.17037e39	1.67114	0.835571	−	0.549382i	$-0.185137\pi$
$353$	5.17037e39	1.67114	0.835571	+	0.549382i	$0.185137\pi$
$354$	0	0
$355$	−2.96014e38	−0.0876541
$356$	−2.12198e39	−0.601542
$357$	0	0
$358$	−1.79736e39	−0.467142
$359$	−6.53823e39	−1.62741	−0.813704	−	0.581279i	$-0.802552\pi$
$359$	−6.53823e39	−1.62741	−0.813704	+	0.581279i	$0.802552\pi$
$360$	0	0
$361$	7.52118e39	1.71761
$362$	2.24606e39	0.491403
$363$	0	0
$364$	4.88171e39	0.980620
$365$	1.00780e39	0.194014
$366$	0	0
$367$	7.94764e39	1.40577	0.702885	−	0.711304i	$-0.251895\pi$
$367$	7.94764e39	1.40577	0.702885	+	0.711304i	$0.251895\pi$
$368$	3.52719e38	0.0598120
$369$	0	0
$370$	−3.53120e39	−0.550548
$371$	9.90565e39	1.48111
$372$	0	0
$373$	−7.09424e39	−0.975931	−0.487966	−	0.872863i	$-0.662261\pi$
$373$	−7.09424e39	−0.975931	−0.487966	+	0.872863i	$0.662261\pi$
$374$	−3.84405e39	−0.507316
$375$	0	0
$376$	−3.22417e39	−0.391748
$377$	9.24032e39	1.07745
$378$	0	0
$379$	8.95619e39	0.962089	0.481045	−	0.876696i	$-0.340257\pi$
$379$	8.95619e39	0.962089	0.481045	+	0.876696i	$0.340257\pi$
$380$	−5.03100e39	−0.518811
$381$	0	0
$382$	7.12500e39	0.677334
$383$	1.43173e40	1.30702	0.653509	−	0.756919i	$-0.273296\pi$
$383$	1.43173e40	1.30702	0.653509	+	0.756919i	$0.273296\pi$
$384$	0	0
$385$	5.21204e39	0.438901
$386$	−1.14803e39	−0.0928647
$387$	0	0
$388$	−8.35429e39	−0.623760
$389$	−7.93003e39	−0.568926	−0.284463	−	0.958687i	$-0.591815\pi$
$389$	−7.93003e39	−0.568926	−0.284463	+	0.958687i	$0.591815\pi$
$390$	0	0
$391$	4.97925e39	0.329932
$392$	−4.42105e39	−0.281574
$393$	0	0
$394$	1.53241e40	0.901951
$395$	2.73001e39	0.154493
$396$	0	0
$397$	−2.07260e40	−1.08458	−0.542292	−	0.840190i	$-0.682444\pi$
$397$	−2.07260e40	−1.08458	−0.542292	+	0.840190i	$0.682444\pi$
$398$	−1.16617e40	−0.586913
$399$	0	0
$400$	−3.24174e39	−0.150955
$401$	−3.44745e38	−0.0154440	−0.00772201	−	0.999970i	$-0.502458\pi$
$401$	−3.44745e38	−0.0154440	−0.00772201	+	0.999970i	$0.502458\pi$
$402$	0	0
$403$	−4.39591e40	−1.82314
$404$	8.30056e39	0.331281
$405$	0	0
$406$	−1.88760e40	−0.697840
$407$	−1.80841e40	−0.643551
$408$	0	0
$409$	2.68695e40	0.886235	0.443118	−	0.896463i	$-0.353872\pi$
$409$	2.68695e40	0.886235	0.443118	+	0.896463i	$0.353872\pi$
$410$	1.30821e40	0.415460
$411$	0	0
$412$	−1.51867e40	−0.447256
$413$	−5.42789e40	−1.53959
$414$	0	0
$415$	−1.20547e40	−0.317257
$416$	1.02023e40	0.258674
$417$	0	0
$418$	−2.57649e40	−0.606452
$419$	1.18872e40	0.269627	0.134814	−	0.990871i	$-0.456956\pi$
$419$	1.18872e40	0.269627	0.134814	+	0.990871i	$0.456956\pi$
$420$	0	0
$421$	3.74735e40	0.789499	0.394749	−	0.918789i	$-0.370831\pi$
$421$	3.74735e40	0.789499	0.394749	+	0.918789i	$0.370831\pi$
$422$	−2.94967e40	−0.599005
$423$	0	0
$424$	2.07019e40	0.390697
$425$	−4.57628e40	−0.832692
$426$	0	0
$427$	−6.80863e40	−1.15193
$428$	−3.50464e40	−0.571827
$429$	0	0
$430$	8.24773e39	0.125191
$431$	−2.80334e40	−0.410466	−0.205233	−	0.978713i	$-0.565795\pi$
$431$	−2.80334e40	−0.410466	−0.205233	+	0.978713i	$0.565795\pi$
$432$	0	0
$433$	−1.18599e41	−1.61629	−0.808146	−	0.588982i	$-0.799529\pi$
$433$	−1.18599e41	−1.61629	−0.808146	+	0.588982i	$0.799529\pi$
$434$	8.97989e40	1.18081
$435$	0	0
$436$	−3.50226e40	−0.428852
$437$	3.33737e40	0.394404
$438$	0	0
$439$	2.22996e40	0.245526	0.122763	−	0.992436i	$-0.460825\pi$
$439$	2.22996e40	0.245526	0.122763	+	0.992436i	$0.460825\pi$
$440$	1.08927e40	0.115776
$441$	0	0
$442$	1.44024e41	1.42688
$443$	4.19902e40	0.401689	0.200845	−	0.979623i	$-0.435631\pi$
$443$	4.19902e40	0.401689	0.200845	+	0.979623i	$0.435631\pi$
$444$	0	0
$445$	8.48831e40	0.757253
$446$	5.02111e40	0.432622
$447$	0	0
$448$	−2.08412e40	−0.167538
$449$	9.25651e40	0.718835	0.359417	−	0.933177i	$-0.382975\pi$
$449$	9.25651e40	0.718835	0.359417	+	0.933177i	$0.382975\pi$
$450$	0	0
$451$	6.69967e40	0.485643
$452$	1.16830e41	0.818293
$453$	0	0
$454$	5.85363e40	0.382877
$455$	−1.95278e41	−1.23446
$456$	0	0
$457$	−4.96857e40	−0.293447	−0.146724	−	0.989178i	$-0.546873\pi$
$457$	−4.96857e40	−0.293447	−0.146724	+	0.989178i	$0.546873\pi$
$458$	1.32153e41	0.754502
$459$	0	0
$460$	−1.41094e40	−0.0752946
$461$	1.25763e40	0.0648916	0.0324458	−	0.999473i	$-0.489670\pi$
$461$	1.25763e40	0.0648916	0.0324458	+	0.999473i	$0.489670\pi$
$462$	0	0
$463$	2.97996e41	1.43782	0.718912	−	0.695102i	$-0.244641\pi$
$463$	2.97996e41	1.43782	0.718912	+	0.695102i	$0.244641\pi$
$464$	−3.94490e40	−0.184081
$465$	0	0
$466$	1.58574e41	0.692229
$467$	−1.08239e41	−0.457060	−0.228530	−	0.973537i	$-0.573392\pi$
$467$	−1.08239e41	−0.457060	−0.228530	+	0.973537i	$0.573392\pi$
$468$	0	0
$469$	−2.76975e41	−1.09461
$470$	1.28973e41	0.493153
$471$	0	0
$472$	−1.13438e41	−0.406123
$473$	4.22385e40	0.146339
$474$	0	0
$475$	−3.06728e41	−0.995410
$476$	−2.94209e41	−0.924163
$477$	0	0
$478$	2.61700e41	0.770321
$479$	−2.71815e41	−0.774591	−0.387296	−	0.921956i	$-0.626591\pi$
$479$	−2.71815e41	−0.774591	−0.387296	+	0.921956i	$0.626591\pi$
$480$	0	0
$481$	6.77550e41	1.81006
$482$	−1.81325e41	−0.469061
$483$	0	0
$484$	−1.50314e41	−0.364666
$485$	3.34187e41	0.785223
$486$	0	0
$487$	6.38745e41	1.40808	0.704042	−	0.710158i	$-0.251376\pi$
$487$	6.38745e41	1.40808	0.704042	+	0.710158i	$0.251376\pi$
$488$	−1.42294e41	−0.303864
$489$	0	0
$490$	1.76851e41	0.354460
$491$	3.73865e41	0.726025	0.363012	−	0.931784i	$-0.381748\pi$
$491$	3.73865e41	0.726025	0.363012	+	0.931784i	$0.381748\pi$
$492$	0	0
$493$	−5.56892e41	−1.01541
$494$	9.65326e41	1.70572
$495$	0	0
$496$	1.87671e41	0.311481
$497$	−1.16026e41	−0.186651
$498$	0	0
$499$	−9.33430e41	−1.41099	−0.705494	−	0.708716i	$-0.749275\pi$
$499$	−9.33430e41	−1.41099	−0.705494	+	0.708716i	$0.749275\pi$
$500$	3.44434e41	0.504744
$501$	0	0
$502$	−5.98168e41	−0.823978
$503$	−9.86263e41	−1.31731	−0.658657	−	0.752443i	$-0.728875\pi$
$503$	−9.86263e41	−1.31731	−0.658657	+	0.752443i	$0.728875\pi$
$504$	0	0
$505$	−3.32038e41	−0.417034
$506$	−7.22577e40	−0.0880139
$507$	0	0
$508$	−2.71775e41	−0.311403
$509$	−7.59362e41	−0.843962	−0.421981	−	0.906605i	$-0.638665\pi$
$509$	−7.59362e41	−0.843962	−0.421981	+	0.906605i	$0.638665\pi$
$510$	0	0
$511$	3.95017e41	0.413134
$512$	−4.35561e40	−0.0441942
$513$	0	0
$514$	−1.15733e41	−0.110543
$515$	6.07496e41	0.563030
$516$	0	0
$517$	6.60500e41	0.576461
$518$	−1.38409e42	−1.17234
$519$	0	0
$520$	−4.08113e41	−0.325633
$521$	1.05689e42	0.818553	0.409277	−	0.912410i	$-0.365781\pi$
$521$	1.05689e42	0.818553	0.409277	+	0.912410i	$0.365781\pi$
$522$	0	0
$523$	−1.00854e42	−0.736064	−0.368032	−	0.929813i	$-0.619968\pi$
$523$	−1.00854e42	−0.736064	−0.368032	+	0.929813i	$0.619968\pi$
$524$	−4.96344e41	−0.351680
$525$	0	0
$526$	1.23716e42	0.826319
$527$	2.64931e42	1.71818
$528$	0	0
$529$	−1.54157e42	−0.942760
$530$	−8.28116e41	−0.491830
$531$	0	0
$532$	−1.97195e42	−1.10476
$533$	−2.51014e42	−1.36592
$534$	0	0
$535$	1.40192e42	0.719847
$536$	−5.78852e41	−0.288744
$537$	0	0
$538$	−5.68830e41	−0.267829
$539$	9.05693e41	0.414338
$540$	0	0
$541$	−5.23504e41	−0.226132	−0.113066	−	0.993587i	$-0.536067\pi$
$541$	−5.23504e41	−0.226132	−0.113066	+	0.993587i	$0.536067\pi$
$542$	−2.11970e41	−0.0889785
$543$	0	0
$544$	−6.14870e41	−0.243781
$545$	1.40097e42	0.539862
$546$	0	0
$547$	2.05991e42	0.749970	0.374985	−	0.927031i	$-0.377648\pi$
$547$	2.05991e42	0.749970	0.374985	+	0.927031i	$0.377648\pi$
$548$	−2.59194e42	−0.917332
$549$	0	0
$550$	6.64100e41	0.222132
$551$	−3.73260e42	−1.21384
$552$	0	0
$553$	1.07006e42	0.328978
$554$	6.50934e41	0.194597
$555$	0	0
$556$	−1.65100e42	−0.466755
$557$	−1.26865e40	−0.00348807	−0.00174404	−	0.999998i	$-0.500555\pi$
$557$	−1.26865e40	−0.00348807	−0.00174404	+	0.999998i	$0.500555\pi$
$558$	0	0
$559$	−1.58254e42	−0.411596
$560$	8.33686e41	0.210906
$561$	0	0
$562$	1.79652e42	0.430049
$563$	4.54148e42	1.05759	0.528795	−	0.848749i	$-0.322644\pi$
$563$	4.54148e42	1.05759	0.528795	+	0.848749i	$0.322644\pi$
$564$	0	0
$565$	−4.67343e42	−1.03011
$566$	8.60937e41	0.184636
$567$	0	0
$568$	−2.42483e41	−0.0492359
$569$	7.78015e42	1.53726	0.768631	−	0.639693i	$-0.220938\pi$
$569$	7.78015e42	1.53726	0.768631	+	0.639693i	$0.220938\pi$
$570$	0	0
$571$	4.96641e42	0.929357	0.464679	−	0.885479i	$-0.346170\pi$
$571$	4.96641e42	0.929357	0.464679	+	0.885479i	$0.346170\pi$
$572$	−2.09004e42	−0.380642
$573$	0	0
$574$	5.12768e42	0.884680
$575$	−8.60218e41	−0.144463
$576$	0	0
$577$	8.08612e42	1.28681	0.643406	−	0.765525i	$-0.277521\pi$
$577$	8.08612e42	1.28681	0.643406	+	0.765525i	$0.277521\pi$
$578$	−4.11574e42	−0.637627
$579$	0	0
$580$	1.57804e42	0.231731
$581$	−4.72496e42	−0.675567
$582$	0	0
$583$	−4.24097e42	−0.574914
$584$	8.25549e41	0.108979
$585$	0	0
$586$	−4.36463e42	−0.546429
$587$	3.64148e42	0.444005	0.222003	−	0.975046i	$-0.428741\pi$
$587$	3.64148e42	0.444005	0.222003	+	0.975046i	$0.428741\pi$
$588$	0	0
$589$	1.77571e43	2.05393
$590$	4.53773e42	0.511249
$591$	0	0
$592$	−2.89262e42	−0.309247
$593$	1.59482e43	1.66098	0.830491	−	0.557032i	$-0.188060\pi$
$593$	1.59482e43	1.66098	0.830491	+	0.557032i	$0.188060\pi$
$594$	0	0
$595$	1.17689e43	1.16339
$596$	7.58832e42	0.730850
$597$	0	0
$598$	2.70726e42	0.247549
$599$	−1.05341e43	−0.938600	−0.469300	−	0.883039i	$-0.655494\pi$
$599$	−1.05341e43	−0.938600	−0.469300	+	0.883039i	$0.655494\pi$
$600$	0	0
$601$	4.64301e42	0.392867	0.196434	−	0.980517i	$-0.437064\pi$
$601$	4.64301e42	0.392867	0.196434	+	0.980517i	$0.437064\pi$
$602$	3.23278e42	0.266582
$603$	0	0
$604$	−6.77788e42	−0.530911
$605$	6.01283e42	0.459062
$606$	0	0
$607$	−2.17421e43	−1.57717	−0.788583	−	0.614929i	$-0.789185\pi$
$607$	−2.17421e43	−1.57717	−0.788583	+	0.614929i	$0.789185\pi$
$608$	−4.12120e42	−0.291419
$609$	0	0
$610$	5.69204e42	0.382521
$611$	−2.47468e43	−1.62136
$612$	0	0
$613$	−1.21288e43	−0.755409	−0.377704	−	0.925926i	$-0.623286\pi$
$613$	−1.21288e43	−0.755409	−0.377704	+	0.925926i	$0.623286\pi$
$614$	1.43702e42	0.0872675
$615$	0	0
$616$	4.26950e42	0.246533
$617$	−3.00516e43	−1.69219	−0.846093	−	0.533036i	$-0.821051\pi$
$617$	−3.00516e43	−1.69219	−0.846093	+	0.533036i	$0.821051\pi$
$618$	0	0
$619$	−2.57828e43	−1.38078	−0.690391	−	0.723436i	$-0.742562\pi$
$619$	−2.57828e43	−1.38078	−0.690391	+	0.723436i	$0.742562\pi$
$620$	−7.50721e42	−0.392110
$621$	0	0
$622$	−9.37213e42	−0.465680
$623$	3.32708e43	1.61250
$624$	0	0
$625$	−6.84734e41	−0.0315778
$626$	1.97009e43	0.886307
$627$	0	0
$628$	9.78753e42	0.419082
$629$	−4.08343e43	−1.70585
$630$	0	0
$631$	−4.05493e43	−1.61261	−0.806303	−	0.591503i	$-0.798535\pi$
$631$	−4.05493e43	−1.61261	−0.806303	+	0.591503i	$0.798535\pi$
$632$	2.23632e42	0.0867799
$633$	0	0
$634$	2.95790e43	1.09295
$635$	1.08715e43	0.392011
$636$	0	0
$637$	−3.39333e43	−1.16537
$638$	8.08149e42	0.270876
$639$	0	0
$640$	1.74233e42	0.0556340
$641$	−2.77200e43	−0.863960	−0.431980	−	0.901883i	$-0.642185\pi$
$641$	−2.77200e43	−0.863960	−0.431980	+	0.901883i	$0.642185\pi$
$642$	0	0
$643$	4.13799e43	1.22891	0.614454	−	0.788952i	$-0.289376\pi$
$643$	4.13799e43	1.22891	0.614454	+	0.788952i	$0.289376\pi$
$644$	−5.53034e42	−0.160332
$645$	0	0
$646$	−5.81779e43	−1.60751
$647$	1.37833e43	0.371823	0.185912	−	0.982566i	$-0.440476\pi$
$647$	1.37833e43	0.371823	0.185912	+	0.982566i	$0.440476\pi$
$648$	0	0
$649$	2.32388e43	0.597613
$650$	−2.48816e43	−0.624772
$651$	0	0
$652$	−5.33518e42	−0.127735
$653$	−1.72282e43	−0.402797	−0.201398	−	0.979509i	$-0.564549\pi$
$653$	−1.72282e43	−0.402797	−0.201398	+	0.979509i	$0.564549\pi$
$654$	0	0
$655$	1.98547e43	0.442714
$656$	1.07164e43	0.233367
$657$	0	0
$658$	5.05523e43	1.05012
$659$	−8.45482e43	−1.71546	−0.857730	−	0.514101i	$-0.828126\pi$
$659$	−8.45482e43	−1.71546	−0.857730	+	0.514101i	$0.828126\pi$
$660$	0	0
$661$	−6.65139e43	−1.28763	−0.643813	−	0.765183i	$-0.722648\pi$
$661$	−6.65139e43	−1.28763	−0.643813	+	0.765183i	$0.722648\pi$
$662$	4.49999e43	0.850967
$663$	0	0
$664$	−9.87474e42	−0.178205
$665$	7.88818e43	1.39073
$666$	0	0
$667$	−1.04681e43	−0.176164
$668$	2.94994e43	0.485041
$669$	0	0
$670$	2.31552e43	0.363487
$671$	2.91502e43	0.447139
$672$	0	0
$673$	−8.33123e43	−1.22033	−0.610163	−	0.792276i	$-0.708896\pi$
$673$	−8.33123e43	−1.22033	−0.610163	+	0.792276i	$0.708896\pi$
$674$	−4.23826e41	−0.00606680
$675$	0	0
$676$	4.17353e43	0.570597
$677$	1.76795e42	0.0236235	0.0118118	−	0.999930i	$-0.496240\pi$
$677$	1.76795e42	0.0236235	0.0118118	+	0.999930i	$0.496240\pi$
$678$	0	0
$679$	1.30988e44	1.67205
$680$	2.45960e43	0.306885
$681$	0	0
$682$	−3.84462e43	−0.458348
$683$	−3.16064e43	−0.368344	−0.184172	−	0.982894i	$-0.558960\pi$
$683$	−3.16064e43	−0.368344	−0.184172	+	0.982894i	$0.558960\pi$
$684$	0	0
$685$	1.03683e44	1.15479
$686$	−1.77202e43	−0.192950
$687$	0	0
$688$	6.75622e42	0.0703207
$689$	1.58895e44	1.61701
$690$	0	0
$691$	1.14213e44	1.11124	0.555618	−	0.831438i	$-0.312482\pi$
$691$	1.14213e44	1.11124	0.555618	+	0.831438i	$0.312482\pi$
$692$	−1.42073e43	−0.135165
$693$	0	0
$694$	−4.16410e43	−0.378834
$695$	6.60433e43	0.587576
$696$	0	0
$697$	1.51280e44	1.28728
$698$	−4.52977e43	−0.376979
$699$	0	0
$700$	5.08278e43	0.404652
$701$	−1.40003e44	−1.09020	−0.545102	−	0.838370i	$-0.683509\pi$
$701$	−1.40003e44	−1.09020	−0.545102	+	0.838370i	$0.683509\pi$
$702$	0	0
$703$	−2.73694e44	−2.03919
$704$	8.92286e42	0.0650321
$705$	0	0
$706$	−1.69423e44	−1.18168
$707$	−1.30146e44	−0.888033
$708$	0	0
$709$	−3.96655e43	−0.259057	−0.129529	−	0.991576i	$-0.541346\pi$
$709$	−3.96655e43	−0.259057	−0.129529	+	0.991576i	$0.541346\pi$
$710$	9.69977e42	0.0619808
$711$	0	0
$712$	6.95329e43	0.425354
$713$	4.97999e43	0.298085
$714$	0	0
$715$	8.36056e43	0.479172
$716$	5.88960e43	0.330319
$717$	0	0
$718$	2.14245e44	1.15075
$719$	1.62203e44	0.852629	0.426314	−	0.904575i	$-0.359812\pi$
$719$	1.62203e44	0.852629	0.426314	+	0.904575i	$0.359812\pi$
$720$	0	0
$721$	2.38114e44	1.19892
$722$	−2.46454e44	−1.21453
$723$	0	0
$724$	−7.35988e43	−0.347475
$725$	9.62090e43	0.444607
$726$	0	0
$727$	1.45218e44	0.643036	0.321518	−	0.946903i	$-0.395807\pi$
$727$	1.45218e44	0.643036	0.321518	+	0.946903i	$0.395807\pi$
$728$	−1.59964e44	−0.693403
$729$	0	0
$730$	−3.30235e43	−0.137189
$731$	9.53758e43	0.387899
$732$	0	0
$733$	−1.34214e44	−0.523222	−0.261611	−	0.965173i	$-0.584254\pi$
$733$	−1.34214e44	−0.523222	−0.261611	+	0.965173i	$0.584254\pi$
$734$	−2.60428e44	−0.994029
$735$	0	0
$736$	−1.15579e43	−0.0422935
$737$	1.18583e44	0.424890
$738$	0	0
$739$	−3.10772e44	−1.06771	−0.533855	−	0.845576i	$-0.679257\pi$
$739$	−3.10772e44	−1.06771	−0.533855	+	0.845576i	$0.679257\pi$
$740$	1.15710e44	0.389296
$741$	0	0
$742$	−3.24588e44	−1.04730
$743$	−3.12688e44	−0.988062	−0.494031	−	0.869444i	$-0.664477\pi$
$743$	−3.12688e44	−0.988062	−0.494031	+	0.869444i	$0.664477\pi$
$744$	0	0
$745$	−3.03547e44	−0.920034
$746$	2.32464e44	0.690088
$747$	0	0
$748$	1.25962e44	0.358727
$749$	5.49497e44	1.53284
$750$	0	0
$751$	−3.16352e44	−0.846742	−0.423371	−	0.905956i	$-0.639153\pi$
$751$	−3.16352e44	−0.846742	−0.423371	+	0.905956i	$0.639153\pi$
$752$	1.05650e44	0.277008
$753$	0	0
$754$	−3.02787e44	−0.761870
$755$	2.71128e44	0.668339
$756$	0	0
$757$	4.10509e44	0.971263	0.485632	−	0.874164i	$-0.338590\pi$
$757$	4.10509e44	0.971263	0.485632	+	0.874164i	$0.338590\pi$
$758$	−2.93476e44	−0.680300
$759$	0	0
$760$	1.64856e44	0.366854
$761$	−3.57723e44	−0.779982	−0.389991	−	0.920819i	$-0.627522\pi$
$761$	−3.57723e44	−0.779982	−0.389991	+	0.920819i	$0.627522\pi$
$762$	0	0
$763$	5.49124e44	1.14958
$764$	−2.33472e44	−0.478947
$765$	0	0
$766$	−4.69151e44	−0.924201
$767$	−8.70680e44	−1.68086
$768$	0	0
$769$	−1.12924e44	−0.209377	−0.104689	−	0.994505i	$-0.533385\pi$
$769$	−1.12924e44	−0.209377	−0.104689	+	0.994505i	$0.533385\pi$
$770$	−1.70788e44	−0.310350
$771$	0	0
$772$	3.76187e43	0.0656653
$773$	4.52518e44	0.774201	0.387100	−	0.922038i	$-0.373477\pi$
$773$	4.52518e44	0.774201	0.387100	+	0.922038i	$0.373477\pi$
$774$	0	0
$775$	−4.57696e44	−0.752317
$776$	2.73753e44	0.441065
$777$	0	0
$778$	2.59851e44	0.402291
$779$	1.01396e45	1.53884
$780$	0	0
$781$	4.96748e43	0.0724511
$782$	−1.63160e44	−0.233297
$783$	0	0
$784$	1.44869e44	0.199103
$785$	−3.91520e44	−0.527562
$786$	0	0
$787$	−8.95840e44	−1.16044	−0.580219	−	0.814460i	$-0.697033\pi$
$787$	−8.95840e44	−1.16044	−0.580219	+	0.814460i	$0.697033\pi$
$788$	−5.02139e44	−0.637776
$789$	0	0
$790$	−8.94570e43	−0.109243
$791$	−1.83180e45	−2.19352
$792$	0	0
$793$	−1.09216e45	−1.25763
$794$	6.79151e44	0.766917
$795$	0	0
$796$	3.82130e44	0.415010
$797$	4.97034e44	0.529398	0.264699	−	0.964331i	$-0.414727\pi$
$797$	4.97034e44	0.529398	0.264699	+	0.964331i	$0.414727\pi$
$798$	0	0
$799$	1.49143e45	1.52801
$800$	1.06225e44	0.106742
$801$	0	0
$802$	1.12966e43	0.0109206
$803$	−1.69121e44	−0.160364
$804$	0	0
$805$	2.21224e44	0.201835
$806$	1.44045e45	1.28916
$807$	0	0
$808$	−2.71993e44	−0.234251
$809$	−1.66443e45	−1.40625	−0.703127	−	0.711064i	$-0.748214\pi$
$809$	−1.66443e45	−1.40625	−0.703127	+	0.711064i	$0.748214\pi$
$810$	0	0
$811$	1.88658e45	1.53409	0.767046	−	0.641592i	$-0.221726\pi$
$811$	1.88658e45	1.53409	0.767046	+	0.641592i	$0.221726\pi$
$812$	6.18528e44	0.493447
$813$	0	0
$814$	5.92579e44	0.455059
$815$	2.13417e44	0.160800
$816$	0	0
$817$	6.39261e44	0.463699
$818$	−8.80461e44	−0.626663
$819$	0	0
$820$	−4.28676e44	−0.293774
$821$	8.30695e44	0.558628	0.279314	−	0.960200i	$-0.409893\pi$
$821$	8.30695e44	0.558628	0.279314	+	0.960200i	$0.409893\pi$
$822$	0	0
$823$	−2.11118e45	−1.36719	−0.683594	−	0.729862i	$-0.739584\pi$
$823$	−2.11118e45	−1.36719	−0.683594	+	0.729862i	$0.739584\pi$
$824$	4.97637e44	0.316258
$825$	0	0
$826$	1.77861e45	1.08865
$827$	2.31980e45	1.39353	0.696764	−	0.717301i	$-0.254623\pi$
$827$	2.31980e45	1.39353	0.696764	+	0.717301i	$0.254623\pi$
$828$	0	0
$829$	2.73984e45	1.58537	0.792684	−	0.609633i	$-0.208683\pi$
$829$	2.73984e45	1.58537	0.792684	+	0.609633i	$0.208683\pi$
$830$	3.95008e44	0.224334
$831$	0	0
$832$	−3.34310e44	−0.182910
$833$	2.04508e45	1.09828
$834$	0	0
$835$	−1.18003e45	−0.610596
$836$	8.44265e44	0.428826
$837$	0	0
$838$	−3.89521e44	−0.190655
$839$	2.08003e44	0.0999444	0.0499722	−	0.998751i	$-0.484087\pi$
$839$	2.08003e44	0.0999444	0.0499722	+	0.998751i	$0.484087\pi$
$840$	0	0
$841$	−9.88649e44	−0.457830
$842$	−1.22793e45	−0.558260
$843$	0	0
$844$	9.66549e44	0.423561
$845$	−1.66949e45	−0.718299
$846$	0	0
$847$	2.35679e45	0.977526
$848$	−6.78360e44	−0.276264
$849$	0	0
$850$	1.49956e45	0.588802
$851$	−7.67576e44	−0.295946
$852$	0	0
$853$	−8.05595e44	−0.299507	−0.149753	−	0.988723i	$-0.547848\pi$
$853$	−8.05595e44	−0.299507	−0.149753	+	0.988723i	$0.547848\pi$
$854$	2.23105e45	0.814540
$855$	0	0
$856$	1.14840e45	0.404343
$857$	−8.81281e44	−0.304728	−0.152364	−	0.988324i	$-0.548689\pi$
$857$	−8.81281e44	−0.304728	−0.152364	+	0.988324i	$0.548689\pi$
$858$	0	0
$859$	−3.21045e45	−1.07071	−0.535356	−	0.844627i	$-0.679822\pi$
$859$	−3.21045e45	−1.07071	−0.535356	+	0.844627i	$0.679822\pi$
$860$	−2.70262e44	−0.0885235
$861$	0	0
$862$	9.18597e44	0.290243
$863$	−3.88645e45	−1.20611	−0.603053	−	0.797701i	$-0.706049\pi$
$863$	−3.88645e45	−1.20611	−0.603053	+	0.797701i	$0.706049\pi$
$864$	0	0
$865$	5.68317e44	0.170154
$866$	3.88626e45	1.14289
$867$	0	0
$868$	−2.94253e45	−0.834959
$869$	−4.58130e44	−0.127697
$870$	0	0
$871$	−4.44291e45	−1.19505
$872$	1.14762e45	0.303244
$873$	0	0
$874$	−1.09359e45	−0.278886
$875$	−5.40044e45	−1.35302
$876$	0	0
$877$	2.77423e45	0.670886	0.335443	−	0.942061i	$-0.391114\pi$
$877$	2.77423e45	0.670886	0.335443	+	0.942061i	$0.391114\pi$
$878$	−7.30714e44	−0.173613
$879$	0	0
$880$	−3.56932e44	−0.0818660
$881$	−1.10273e45	−0.248508	−0.124254	−	0.992250i	$-0.539654\pi$
$881$	−1.10273e45	−0.248508	−0.124254	+	0.992250i	$0.539654\pi$
$882$	0	0
$883$	−4.69663e45	−1.02187	−0.510935	−	0.859619i	$-0.670701\pi$
$883$	−4.69663e45	−1.02187	−0.510935	+	0.859619i	$0.670701\pi$
$884$	−4.71937e45	−1.00896
$885$	0	0
$886$	−1.37594e45	−0.284037
$887$	−1.19670e45	−0.242755	−0.121378	−	0.992606i	$-0.538731\pi$
$887$	−1.19670e45	−0.242755	−0.121378	+	0.992606i	$0.538731\pi$
$888$	0	0
$889$	4.26120e45	0.834747
$890$	−2.78145e45	−0.535459
$891$	0	0
$892$	−1.64532e45	−0.305910
$893$	9.99638e45	1.82661
$894$	0	0
$895$	−2.35595e45	−0.415824
$896$	6.82923e44	0.118467
$897$	0	0
$898$	−3.03317e45	−0.508293
$899$	−5.56974e45	−0.917403
$900$	0	0
$901$	−9.57623e45	−1.52391
$902$	−2.19535e45	−0.343401
$903$	0	0
$904$	−3.82829e45	−0.578621
$905$	2.94409e45	0.437420
$906$	0	0
$907$	3.04922e45	0.437800	0.218900	−	0.975747i	$-0.429753\pi$
$907$	3.04922e45	0.437800	0.218900	+	0.975747i	$0.429753\pi$
$908$	−1.91812e45	−0.270735
$909$	0	0
$910$	6.39886e45	0.872893
$911$	3.67350e45	0.492657	0.246329	−	0.969186i	$-0.420776\pi$
$911$	3.67350e45	0.492657	0.246329	+	0.969186i	$0.420776\pi$
$912$	0	0
$913$	2.02293e45	0.262231
$914$	1.62810e45	0.207499
$915$	0	0
$916$	−4.33037e45	−0.533514
$917$	7.78224e45	0.942714
$918$	0	0
$919$	−1.23482e45	−0.144614	−0.0723072	−	0.997382i	$-0.523036\pi$
$919$	−1.23482e45	−0.144614	−0.0723072	+	0.997382i	$0.523036\pi$
$920$	4.62338e44	0.0532413
$921$	0	0
$922$	−4.12100e44	−0.0458853
$923$	−1.86115e45	−0.203777
$924$	0	0
$925$	7.05457e45	0.746919
$926$	−9.76474e45	−1.01669
$927$	0	0
$928$	1.29267e45	0.130165
$929$	1.71673e44	0.0170003	0.00850017	−	0.999964i	$-0.497294\pi$
$929$	1.71673e44	0.0170003	0.00850017	+	0.999964i	$0.497294\pi$
$930$	0	0
$931$	1.37072e46	1.31290
$932$	−5.19615e45	−0.489480
$933$	0	0
$934$	3.54679e45	0.323191
$935$	−5.03871e45	−0.451585
$936$	0	0
$937$	−2.90729e45	−0.252072	−0.126036	−	0.992026i	$-0.540226\pi$
$937$	−2.90729e45	−0.252072	−0.126036	+	0.992026i	$0.540226\pi$
$938$	9.07591e45	0.774009
$939$	0	0
$940$	−4.22619e45	−0.348712
$941$	2.30709e45	0.187251	0.0936257	−	0.995607i	$-0.470154\pi$
$941$	2.30709e45	0.187251	0.0936257	+	0.995607i	$0.470154\pi$
$942$	0	0
$943$	2.84366e45	0.223330
$944$	3.71713e45	0.287172
$945$	0	0
$946$	−1.38407e45	−0.103478
$947$	1.36872e45	0.100668	0.0503340	−	0.998732i	$-0.483971\pi$
$947$	1.36872e45	0.100668	0.0503340	+	0.998732i	$0.483971\pi$
$948$	0	0
$949$	6.33641e45	0.451042
$950$	1.00509e46	0.703861
$951$	0	0
$952$	9.64064e45	0.653482
$953$	2.04354e46	1.36284	0.681418	−	0.731894i	$-0.261363\pi$
$953$	2.04354e46	1.36284	0.681418	+	0.731894i	$0.261363\pi$
$954$	0	0
$955$	9.33932e45	0.602925
$956$	−8.57539e45	−0.544699
$957$	0	0
$958$	8.90683e45	0.547719
$959$	4.06394e46	2.45900
$960$	0	0
$961$	9.42784e45	0.552332
$962$	−2.22020e46	−1.27991
$963$	0	0
$964$	5.94167e45	0.331676
$965$	−1.50482e45	−0.0826630
$966$	0	0
$967$	−2.05882e46	−1.09524	−0.547619	−	0.836728i	$-0.684466\pi$
$967$	−2.05882e46	−1.09524	−0.547619	+	0.836728i	$0.684466\pi$
$968$	4.92548e45	0.257858
$969$	0	0
$970$	−1.09507e46	−0.555237
$971$	−3.11280e46	−1.55329	−0.776647	−	0.629937i	$-0.783081\pi$
$971$	−3.11280e46	−1.55329	−0.776647	+	0.629937i	$0.783081\pi$
$972$	0	0
$973$	2.58863e46	1.25119
$974$	−2.09304e46	−0.995666
$975$	0	0
$976$	4.66269e45	0.214865
$977$	2.53889e46	1.15154	0.575769	−	0.817613i	$-0.304703\pi$
$977$	2.53889e46	1.15154	0.575769	+	0.817613i	$0.304703\pi$
$978$	0	0
$979$	−1.42444e46	−0.625913
$980$	−5.79504e45	−0.250641
$981$	0	0
$982$	−1.22508e46	−0.513377
$983$	−4.44662e46	−1.83422	−0.917109	−	0.398637i	$-0.869483\pi$
$983$	−4.44662e46	−1.83422	−0.917109	+	0.398637i	$0.869483\pi$
$984$	0	0
$985$	2.00865e46	0.802866
$986$	1.82482e46	0.718007
$987$	0	0
$988$	−3.16318e46	−1.20612
$989$	1.79281e45	0.0672963
$990$	0	0
$991$	−3.72910e46	−1.35663	−0.678317	−	0.734769i	$-0.737291\pi$
$991$	−3.72910e46	−1.35663	−0.678317	+	0.734769i	$0.737291\pi$
$992$	−6.14962e45	−0.220251
$993$	0	0
$994$	3.80192e45	0.131982
$995$	−1.52859e46	−0.522437
$996$	0	0
$997$	−9.69190e45	−0.321095	−0.160548	−	0.987028i	$-0.551326\pi$
$997$	−9.69190e45	−0.321095	−0.160548	+	0.987028i	$0.551326\pi$
$998$	3.05866e46	0.997719
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.32.a.a.1.1		1
3.2	odd	2		2.32.a.a.1.1	✓	1
12.11	even	2		16.32.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.32.a.a.1.1	✓	1	3.2	odd	2
16.32.a.a.1.1		1	12.11	even	2
18.32.a.a.1.1		1	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:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 32 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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