LMFDB - Embedded newform 16.32.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,32,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 32);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 32 $
Character orbit:	$[\chi]$	$=$	16.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$97.4034125104$
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	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.99842e7 q^{3} +4.29517e10 q^{5} +1.68354e13 q^{7} -2.18305e14 q^{9} +O(q^{10})$

$q+1.99842e7 q^{3} +4.29517e10 q^{5} +1.68354e13 q^{7} -2.18305e14 q^{9} +7.20783e15 q^{11} -2.70054e17 q^{13} +8.58356e17 q^{15} -1.62755e19 q^{17} -1.09087e20 q^{19} +3.36441e20 q^{21} +3.05935e20 q^{23} -2.81176e21 q^{25} -1.67064e22 q^{27} +3.42166e22 q^{29} -1.62779e23 q^{31} +1.44043e23 q^{33} +7.23107e23 q^{35} -2.50895e24 q^{37} -5.39681e24 q^{39} -9.29498e24 q^{41} -5.86009e24 q^{43} -9.37656e24 q^{45} +9.16365e25 q^{47} +1.25654e26 q^{49} -3.25253e26 q^{51} +5.88384e26 q^{53} +3.09589e26 q^{55} -2.18002e27 q^{57} +3.22410e27 q^{59} +4.04424e27 q^{61} -3.67524e27 q^{63} -1.15993e28 q^{65} -1.64520e28 q^{67} +6.11388e27 q^{69} +6.89178e27 q^{71} -2.34635e28 q^{73} -5.61909e28 q^{75} +1.21346e29 q^{77} +6.35600e28 q^{79} -1.99023e29 q^{81} +2.80657e29 q^{83} -6.99060e29 q^{85} +6.83792e29 q^{87} +1.97625e30 q^{89} -4.54645e30 q^{91} -3.25301e30 q^{93} -4.68549e30 q^{95} -7.78054e30 q^{97} -1.57350e30 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	1.99842e7	0.804095	0.402048	−	0.915619i	$-0.368299\pi$
$3$	1.99842e7	0.804095	0.402048	+	0.915619i	$0.368299\pi$
$4$	0	0
$5$	4.29517e10	0.629427	0.314714	−	0.949187i	$-0.398092\pi$
$5$	4.29517e10	0.629427	0.314714	+	0.949187i	$0.398092\pi$
$6$	0	0
$7$	1.68354e13	1.34030	0.670151	−	0.742225i	$-0.266229\pi$
$7$	1.68354e13	1.34030	0.670151	+	0.742225i	$0.266229\pi$
$8$	0	0
$9$	−2.18305e14	−0.353431
$10$	0	0
$11$	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	$0.761245\pi$
$14$	0	0
$15$	8.58356e17	0.506119
$16$	0	0
$17$	−1.62755e19	−1.37904	−0.689518	−	0.724269i	$-0.742178\pi$
$17$	−1.62755e19	−1.37904	−0.689518	+	0.724269i	$0.742178\pi$
$18$	0	0
$19$	−1.09087e20	−1.64852	−0.824259	−	0.566214i	$-0.808408\pi$
$19$	−1.09087e20	−1.64852	−0.824259	+	0.566214i	$0.808408\pi$
$20$	0	0
$21$	3.36441e20	1.07773
$22$	0	0
$23$	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
$26$	0	0
$27$	−1.67064e22	−1.08829
$28$	0	0
$29$	3.42166e22	0.736322	0.368161	−	0.929762i	$-0.379988\pi$
$29$	3.42166e22	0.736322	0.368161	+	0.929762i	$0.379988\pi$
$30$	0	0
$31$	−1.62779e23	−1.24593	−0.622963	−	0.782251i	$-0.714071\pi$
$31$	−1.62779e23	−1.24593	−0.622963	+	0.782251i	$0.714071\pi$
$32$	0	0
$33$	1.44043e23	0.418336
$34$	0	0
$35$	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	$0.712257\pi$
$38$	0	0
$39$	−5.39681e24	−1.17662
$40$	0	0
$41$	−9.29498e24	−0.933466	−0.466733	−	0.884398i	$-0.654569\pi$
$41$	−9.29498e24	−0.933466	−0.466733	+	0.884398i	$0.654569\pi$
$42$	0	0
$43$	−5.86009e24	−0.281283	−0.140641	−	0.990061i	$-0.544916\pi$
$43$	−5.86009e24	−0.281283	−0.140641	+	0.990061i	$0.544916\pi$
$44$	0	0
$45$	−9.37656e24	−0.222459
$46$	0	0
$47$	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
$50$	0	0
$51$	−3.25253e26	−1.10888
$52$	0	0
$53$	5.88384e26	1.10506	0.552528	−	0.833494i	$-0.313663\pi$
$53$	5.88384e26	1.10506	0.552528	+	0.833494i	$0.313663\pi$
$54$	0	0
$55$	3.09589e26	0.327464
$56$	0	0
$57$	−2.18002e27	−1.32556
$58$	0	0
$59$	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	$0.358609\pi$
$62$	0	0
$63$	−3.67524e27	−0.473704
$64$	0	0
$65$	−1.15993e28	−0.921029
$66$	0	0
$67$	−1.64520e28	−0.816692	−0.408346	−	0.912827i	$-0.633894\pi$
$67$	−1.64520e28	−0.816692	−0.408346	+	0.912827i	$0.633894\pi$
$68$	0	0
$69$	6.11388e27	0.192378
$70$	0	0
$71$	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	$0.549254\pi$
$74$	0	0
$75$	−5.61909e28	−0.485530
$76$	0	0
$77$	1.21346e29	0.697302
$78$	0	0
$79$	6.35600e28	0.245451	0.122725	−	0.992441i	$-0.460837\pi$
$79$	6.35600e28	0.245451	0.122725	+	0.992441i	$0.460837\pi$
$80$	0	0
$81$	−1.99023e29	−0.521656
$82$	0	0
$83$	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
$86$	0	0
$87$	6.83792e29	0.592073
$88$	0	0
$89$	1.97625e30	1.20308	0.601542	−	0.798841i	$-0.294553\pi$
$89$	1.97625e30	1.20308	0.601542	+	0.798841i	$0.294553\pi$
$90$	0	0
$91$	−4.54645e30	−1.96124
$92$	0	0
$93$	−3.25301e30	−1.00184
$94$	0	0
$95$	−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	$0.714396\pi$
$98$	0	0
$99$	−1.57350e30	−0.183875
$100$	0	0
$101$	−7.73050e30	−0.662562	−0.331281	−	0.943532i	$-0.607481\pi$
$101$	−7.73050e30	−0.662562	−0.331281	+	0.943532i	$0.607481\pi$
$102$	0	0
$103$	1.41437e31	0.894512	0.447256	−	0.894406i	$-0.352401\pi$
$103$	1.41437e31	0.894512	0.447256	+	0.894406i	$0.352401\pi$
$104$	0	0
$105$	1.44507e31	0.678353
$106$	0	0
$107$	−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	$0.641082\pi$
$110$	0	0
$111$	−5.01393e31	−0.994655
$112$	0	0
$113$	−1.08807e32	−1.63659	−0.818293	−	0.574801i	$-0.805080\pi$
$113$	−1.08807e32	−1.63659	−0.818293	+	0.574801i	$0.805080\pi$
$114$	0	0
$115$	1.31404e31	0.150589
$116$	0	0
$117$	5.89540e31	0.517169
$118$	0	0
$119$	−2.74004e32	−1.84833
$120$	0	0
$121$	−1.39991e32	−0.729332
$122$	0	0
$123$	−1.85753e32	−0.750596
$124$	0	0
$125$	−3.20780e32	−1.00949
$126$	0	0
$127$	2.53110e32	0.622805	0.311403	−	0.950278i	$-0.399201\pi$
$127$	2.53110e32	0.622805	0.311403	+	0.950278i	$0.399201\pi$
$128$	0	0
$129$	−1.17109e32	−0.226178
$130$	0	0
$131$	−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
$134$	0	0
$135$	−7.17567e32	−0.684997
$136$	0	0
$137$	2.41393e33	1.83466	0.917332	−	0.398123i	$-0.130338\pi$
$137$	2.41393e33	1.83466	0.917332	+	0.398123i	$0.130338\pi$
$138$	0	0
$139$	1.53762e33	0.933510	0.466755	−	0.884387i	$-0.345423\pi$
$139$	1.53762e33	0.933510	0.466755	+	0.884387i	$0.345423\pi$
$140$	0	0
$141$	1.83128e33	0.890962
$142$	0	0
$143$	−1.94650e33	−0.761283
$144$	0	0
$145$	1.46966e33	0.463461
$146$	0	0
$147$	2.51109e33	0.640390
$148$	0	0
$149$	−7.06717e33	−1.46170	−0.730850	−	0.682538i	$-0.760876\pi$
$149$	−7.06717e33	−1.46170	−0.730850	+	0.682538i	$0.760876\pi$
$150$	0	0
$151$	6.31239e33	1.06182	0.530911	−	0.847428i	$-0.321850\pi$
$151$	6.31239e33	1.06182	0.530911	+	0.847428i	$0.321850\pi$
$152$	0	0
$153$	3.55301e33	0.487393
$154$	0	0
$155$	−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	$0.362352\pi$
$158$	0	0
$159$	1.17584e34	0.888571
$160$	0	0
$161$	5.15053e33	0.320665
$162$	0	0
$163$	4.96877e33	0.255471	0.127735	−	0.991808i	$-0.459229\pi$
$163$	4.96877e33	0.255471	0.127735	+	0.991808i	$0.459229\pi$
$164$	0	0
$165$	6.18689e33	0.263312
$166$	0	0
$167$	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
$170$	0	0
$171$	2.38143e34	0.582636
$172$	0	0
$173$	1.32315e34	0.270331	0.135165	−	0.990823i	$-0.456843\pi$
$173$	1.32315e34	0.270331	0.135165	+	0.990823i	$0.456843\pi$
$174$	0	0
$175$	−4.73370e34	−0.809304
$176$	0	0
$177$	6.44311e34	0.923655
$178$	0	0
$179$	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	$0.612961\pi$
$182$	0	0
$183$	8.08210e34	0.691086
$184$	0	0
$185$	−1.07764e35	−0.778593
$186$	0	0
$187$	−1.17311e35	−0.717453
$188$	0	0
$189$	−2.81258e35	−1.45863
$190$	0	0
$191$	−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	$0.479083\pi$
$194$	0	0
$195$	−2.31802e35	−0.740595
$196$	0	0
$197$	4.67654e35	1.27555	0.637776	−	0.770222i	$-0.279855\pi$
$197$	4.67654e35	1.27555	0.637776	+	0.770222i	$0.279855\pi$
$198$	0	0
$199$	−3.55886e35	−0.830020	−0.415010	−	0.909817i	$-0.636222\pi$
$199$	−3.55886e35	−0.830020	−0.415010	+	0.909817i	$0.636222\pi$
$200$	0	0
$201$	−3.28780e35	−0.656698
$202$	0	0
$203$	5.76049e35	0.986894
$204$	0	0
$205$	−3.99235e35	−0.587549
$206$	0	0
$207$	−6.67871e34	−0.0845575
$208$	0	0
$209$	−7.86283e35	−0.857653
$210$	0	0
$211$	−9.00169e35	−0.847122	−0.423561	−	0.905868i	$-0.639220\pi$
$211$	−9.00169e35	−0.847122	−0.423561	+	0.905868i	$0.639220\pi$
$212$	0	0
$213$	1.37727e35	0.111978
$214$	0	0
$215$	−2.51701e35	−0.177047
$216$	0	0
$217$	−2.74044e36	−1.66992
$218$	0	0
$219$	−4.68900e35	−0.247854
$220$	0	0
$221$	4.39525e36	2.01792
$222$	0	0
$223$	1.53232e36	0.611820	0.305910	−	0.952060i	$-0.401039\pi$
$223$	1.53232e36	0.611820	0.305910	+	0.952060i	$0.401039\pi$
$224$	0	0
$225$	6.13821e35	0.213409
$226$	0	0
$227$	−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	$0.679129\pi$
$230$	0	0
$231$	2.42501e36	0.560697
$232$	0	0
$233$	4.83929e36	0.978959	0.489480	−	0.872015i	$-0.337187\pi$
$233$	4.83929e36	0.978959	0.489480	+	0.872015i	$0.337187\pi$
$234$	0	0
$235$	3.93594e36	0.697424
$236$	0	0
$237$	1.27020e36	0.197366
$238$	0	0
$239$	−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	$0.392386\pi$
$242$	0	0
$243$	6.34177e36	0.668826
$244$	0	0
$245$	5.39705e36	0.501282
$246$	0	0
$247$	2.94594e37	2.41225
$248$	0	0
$249$	5.60871e36	0.405297
$250$	0	0
$251$	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
$254$	0	0
$255$	−1.39702e37	−0.697957
$256$	0	0
$257$	−3.53190e36	−0.156331	−0.0781654	−	0.996940i	$-0.524906\pi$
$257$	−3.53190e36	−0.156331	−0.0781654	+	0.996940i	$0.524906\pi$
$258$	0	0
$259$	−4.22390e37	−1.65794
$260$	0	0
$261$	−7.46964e36	−0.260239
$262$	0	0
$263$	−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
$266$	0	0
$267$	3.94937e37	0.967394
$268$	0	0
$269$	−1.73593e37	−0.378767	−0.189384	−	0.981903i	$-0.560649\pi$
$269$	−1.73593e37	−0.378767	−0.189384	+	0.981903i	$0.560649\pi$
$270$	0	0
$271$	−6.46880e36	−0.125835	−0.0629173	−	0.998019i	$-0.520040\pi$
$271$	−6.46880e36	−0.125835	−0.0629173	+	0.998019i	$0.520040\pi$
$272$	0	0
$273$	−9.08572e37	−1.57702
$274$	0	0
$275$	−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	$0.543939\pi$
$278$	0	0
$279$	3.55354e37	0.440348
$280$	0	0
$281$	5.48253e37	0.608181	0.304091	−	0.952643i	$-0.401647\pi$
$281$	5.48253e37	0.608181	0.304091	+	0.952643i	$0.401647\pi$
$282$	0	0
$283$	2.62737e37	0.261115	0.130557	−	0.991441i	$-0.458323\pi$
$283$	2.62737e37	0.261115	0.130557	+	0.991441i	$0.458323\pi$
$284$	0	0
$285$	−9.36358e37	−0.834346
$286$	0	0
$287$	−1.56484e38	−1.25113
$288$	0	0
$289$	1.25602e38	0.901740
$290$	0	0
$291$	−1.55488e38	−1.00313
$292$	0	0
$293$	−1.33198e38	−0.772768	−0.386384	−	0.922338i	$-0.626276\pi$
$293$	−1.33198e38	−0.772768	−0.386384	+	0.922338i	$0.626276\pi$
$294$	0	0
$295$	1.38481e38	0.723016
$296$	0	0
$297$	−1.20417e38	−0.566189
$298$	0	0
$299$	−8.26190e37	−0.350087
$300$	0	0
$301$	−9.86567e37	−0.377004
$302$	0	0
$303$	−1.54488e38	−0.532763
$304$	0	0
$305$	1.73707e38	0.540966
$306$	0	0
$307$	4.38543e37	0.123415	0.0617075	−	0.998094i	$-0.480345\pi$
$307$	4.38543e37	0.123415	0.0617075	+	0.998094i	$0.480345\pi$
$308$	0	0
$309$	2.82651e38	0.719273
$310$	0	0
$311$	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	$0.715600\pi$
$314$	0	0
$315$	−1.57858e38	−0.298162
$316$	0	0
$317$	9.02679e38	1.54566	0.772831	−	0.634612i	$-0.218840\pi$
$317$	9.02679e38	1.54566	0.772831	+	0.634612i	$0.218840\pi$
$318$	0	0
$319$	2.46627e38	0.383077
$320$	0	0
$321$	−6.52274e38	−0.919607
$322$	0	0
$323$	1.77545e39	2.27336
$324$	0	0
$325$	7.59327e38	0.883561
$326$	0	0
$327$	−6.51832e38	−0.689676
$328$	0	0
$329$	1.54273e39	1.48510
$330$	0	0
$331$	1.37329e39	1.20345	0.601725	−	0.798704i	$-0.294480\pi$
$331$	1.37329e39	1.20345	0.601725	+	0.798704i	$0.294480\pi$
$332$	0	0
$333$	5.47715e38	0.437189
$334$	0	0
$335$	−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	$0.498634\pi$
$338$	0	0
$339$	−2.17441e39	−1.31597
$340$	0	0
$341$	−1.17328e39	−0.648202
$342$	0	0
$343$	−5.40776e38	−0.272872
$344$	0	0
$345$	2.62601e38	0.121088
$346$	0	0
$347$	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	$0.414112\pi$
$350$	0	0
$351$	4.51161e39	1.59247
$352$	0	0
$353$	−5.17037e39	−1.67114	−0.835571	−	0.549382i	$-0.814863\pi$
$353$	−5.17037e39	−1.67114	−0.835571	+	0.549382i	$0.814863\pi$
$354$	0	0
$355$	2.96014e38	0.0876541
$356$	0	0
$357$	−5.47575e39	−1.48623
$358$	0	0
$359$	−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
$362$	0	0
$363$	−2.79760e39	−0.586453
$364$	0	0
$365$	−1.00780e39	−0.194014
$366$	0	0
$367$	−7.94764e39	−1.40577	−0.702885	−	0.711304i	$-0.748105\pi$
$367$	−7.94764e39	−1.40577	−0.702885	+	0.711304i	$0.748105\pi$
$368$	0	0
$369$	2.02914e39	0.329916
$370$	0	0
$371$	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	$0.662261\pi$
$374$	0	0
$375$	−6.41053e39	−0.811725
$376$	0	0
$377$	−9.24032e39	−1.07745
$378$	0	0
$379$	−8.95619e39	−0.962089	−0.481045	−	0.876696i	$-0.659743\pi$
$379$	−8.95619e39	−0.962089	−0.481045	+	0.876696i	$0.659743\pi$
$380$	0	0
$381$	5.05821e39	0.500795
$382$	0	0
$383$	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
$386$	0	0
$387$	1.27928e39	0.0994139
$388$	0	0
$389$	7.93003e39	0.568926	0.284463	−	0.958687i	$-0.408185\pi$
$389$	7.93003e39	0.568926	0.284463	+	0.958687i	$0.408185\pi$
$390$	0	0
$391$	−4.97925e39	−0.329932
$392$	0	0
$393$	−9.23782e39	−0.565568
$394$	0	0
$395$	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	$0.682444\pi$
$398$	0	0
$399$	−3.67015e40	−1.77666
$400$	0	0
$401$	3.44745e38	0.0154440	0.00772201	−	0.999970i	$-0.497542\pi$
$401$	3.44745e38	0.0154440	0.00772201	+	0.999970i	$0.497542\pi$
$402$	0	0
$403$	4.39591e40	1.82314
$404$	0	0
$405$	−8.54836e39	−0.328345
$406$	0	0
$407$	−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	$0.353872\pi$
$410$	0	0
$411$	4.82406e40	1.47525
$412$	0	0
$413$	5.42789e40	1.53959
$414$	0	0
$415$	1.20547e40	0.317257
$416$	0	0
$417$	3.07281e40	0.750631
$418$	0	0
$419$	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	$0.370831\pi$
$422$	0	0
$423$	−2.00047e40	−0.391612
$424$	0	0
$425$	4.57628e40	0.832692
$426$	0	0
$427$	6.80863e40	1.15193
$428$	0	0
$429$	−3.88993e40	−0.612144
$430$	0	0
$431$	−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	$0.799529\pi$
$434$	0	0
$435$	2.93700e40	0.372667
$436$	0	0
$437$	−3.33737e40	−0.394404
$438$	0	0
$439$	−2.22996e40	−0.245526	−0.122763	−	0.992436i	$-0.539175\pi$
$439$	−2.22996e40	−0.245526	−0.122763	+	0.992436i	$0.539175\pi$
$440$	0	0
$441$	−2.74308e40	−0.281476
$442$	0	0
$443$	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
$446$	0	0
$447$	−1.41232e41	−1.17535
$448$	0	0
$449$	−9.25651e40	−0.718835	−0.359417	−	0.933177i	$-0.617025\pi$
$449$	−9.25651e40	−0.718835	−0.359417	+	0.933177i	$0.617025\pi$
$450$	0	0
$451$	−6.69967e40	−0.485643
$452$	0	0
$453$	1.26148e41	0.853805
$454$	0	0
$455$	−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	$0.546873\pi$
$458$	0	0
$459$	2.71904e41	1.50079
$460$	0	0
$461$	−1.25763e40	−0.0648916	−0.0324458	−	0.999473i	$-0.510330\pi$
$461$	−1.25763e40	−0.0648916	−0.0324458	+	0.999473i	$0.510330\pi$
$462$	0	0
$463$	−2.97996e41	−1.43782	−0.718912	−	0.695102i	$-0.755359\pi$
$463$	−2.97996e41	−1.43782	−0.718912	+	0.695102i	$0.755359\pi$
$464$	0	0
$465$	−1.39722e41	−0.630587
$466$	0	0
$467$	−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
$470$	0	0
$471$	1.82163e41	0.673963
$472$	0	0
$473$	−4.22385e40	−0.146339
$474$	0	0
$475$	3.06728e41	0.995410
$476$	0	0
$477$	−1.28447e41	−0.390561
$478$	0	0
$479$	−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
$482$	0	0
$483$	1.02929e41	0.257845
$484$	0	0
$485$	−3.34187e41	−0.785223
$486$	0	0
$487$	−6.38745e41	−1.40808	−0.704042	−	0.710158i	$-0.748624\pi$
$487$	−6.38745e41	−1.40808	−0.704042	+	0.710158i	$0.748624\pi$
$488$	0	0
$489$	9.92970e40	0.205423
$490$	0	0
$491$	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
$494$	0	0
$495$	−6.75847e40	−0.115736
$496$	0	0
$497$	1.16026e41	0.186651
$498$	0	0
$499$	9.33430e41	1.41099	0.705494	−	0.708716i	$-0.250725\pi$
$499$	9.33430e41	1.41099	0.705494	+	0.708716i	$0.250725\pi$
$500$	0	0
$501$	5.49036e41	0.780039
$502$	0	0
$503$	−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
$506$	0	0
$507$	7.76767e41	0.917629
$508$	0	0
$509$	7.59362e41	0.843962	0.421981	−	0.906605i	$-0.361335\pi$
$509$	7.59362e41	0.843962	0.421981	+	0.906605i	$0.361335\pi$
$510$	0	0
$511$	−3.95017e41	−0.413134
$512$	0	0
$513$	1.82245e42	1.79406
$514$	0	0
$515$	6.07496e41	0.563030
$516$	0	0
$517$	6.60500e41	0.576461
$518$	0	0
$519$	2.64422e41	0.217372
$520$	0	0
$521$	−1.05689e42	−0.818553	−0.409277	−	0.912410i	$-0.634219\pi$
$521$	−1.05689e42	−0.818553	−0.409277	+	0.912410i	$0.634219\pi$
$522$	0	0
$523$	1.00854e42	0.736064	0.368032	−	0.929813i	$-0.380032\pi$
$523$	1.00854e42	0.736064	0.368032	+	0.929813i	$0.380032\pi$
$524$	0	0
$525$	−9.45994e41	−0.650757
$526$	0	0
$527$	2.64931e42	1.71818
$528$	0	0
$529$	−1.54157e42	−0.942760
$530$	0	0
$531$	−7.03836e41	−0.405982
$532$	0	0
$533$	2.51014e42	1.36592
$534$	0	0
$535$	−1.40192e42	−0.719847
$536$	0	0
$537$	1.09616e42	0.531216
$538$	0	0
$539$	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	$0.536067\pi$
$542$	0	0
$543$	−1.36980e42	−0.558805
$544$	0	0
$545$	−1.40097e42	−0.539862
$546$	0	0
$547$	−2.05991e42	−0.749970	−0.374985	−	0.927031i	$-0.622352\pi$
$547$	−2.05991e42	−0.749970	−0.374985	+	0.927031i	$0.622352\pi$
$548$	0	0
$549$	−8.82877e41	−0.303759
$550$	0	0
$551$	−3.73260e42	−1.21384
$552$	0	0
$553$	1.07006e42	0.328978
$554$	0	0
$555$	−2.15357e42	−0.626063
$556$	0	0
$557$	1.26865e40	0.00348807	0.00174404	−	0.999998i	$-0.499445\pi$
$557$	1.26865e40	0.00348807	0.00174404	+	0.999998i	$0.499445\pi$
$558$	0	0
$559$	1.58254e42	0.411596
$560$	0	0
$561$	−2.34437e42	−0.576901
$562$	0	0
$563$	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
$566$	0	0
$567$	−3.35062e42	−0.699177
$568$	0	0
$569$	−7.78015e42	−1.53726	−0.768631	−	0.639693i	$-0.779062\pi$
$569$	−7.78015e42	−1.53726	−0.768631	+	0.639693i	$0.779062\pi$
$570$	0	0
$571$	−4.96641e42	−0.929357	−0.464679	−	0.885479i	$-0.653830\pi$
$571$	−4.96641e42	−0.929357	−0.464679	+	0.885479i	$0.653830\pi$
$572$	0	0
$573$	−4.34532e42	−0.770239
$574$	0	0
$575$	−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	$0.277521\pi$
$578$	0	0
$579$	7.00150e41	0.105602
$580$	0	0
$581$	4.72496e42	0.675567
$582$	0	0
$583$	4.24097e42	0.574914
$584$	0	0
$585$	2.53217e42	0.325520
$586$	0	0
$587$	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
$590$	0	0
$591$	9.34569e42	1.02566
$592$	0	0
$593$	−1.59482e43	−1.66098	−0.830491	−	0.557032i	$-0.811940\pi$
$593$	−1.59482e43	−1.66098	−0.830491	+	0.557032i	$0.811940\pi$
$594$	0	0
$595$	−1.17689e43	−1.16339
$596$	0	0
$597$	−7.11211e42	−0.667415
$598$	0	0
$599$	−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	$0.437064\pi$
$602$	0	0
$603$	3.59154e42	0.288644
$604$	0	0
$605$	−6.01283e42	−0.459062
$606$	0	0
$607$	2.17421e43	1.57717	0.788583	−	0.614929i	$-0.210815\pi$
$607$	2.17421e43	1.57717	0.788583	+	0.614929i	$0.210815\pi$
$608$	0	0
$609$	1.15119e43	0.793557
$610$	0	0
$611$	−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	$0.623286\pi$
$614$	0	0
$615$	−7.97840e42	−0.472445
$616$	0	0
$617$	3.00516e43	1.69219	0.846093	−	0.533036i	$-0.178949\pi$
$617$	3.00516e43	1.69219	0.846093	+	0.533036i	$0.178949\pi$
$618$	0	0
$619$	2.57828e43	1.38078	0.690391	−	0.723436i	$-0.257438\pi$
$619$	2.57828e43	1.38078	0.690391	+	0.723436i	$0.257438\pi$
$620$	0	0
$621$	−5.11107e42	−0.260371
$622$	0	0
$623$	3.32708e43	1.61250
$624$	0	0
$625$	−6.84734e41	−0.0315778
$626$	0	0
$627$	−1.57132e43	−0.689635
$628$	0	0
$629$	4.08343e43	1.70585
$630$	0	0
$631$	4.05493e43	1.61261	0.806303	−	0.591503i	$-0.201465\pi$
$631$	4.05493e43	1.61261	0.806303	+	0.591503i	$0.201465\pi$
$632$	0	0
$633$	−1.79892e43	−0.681167
$634$	0	0
$635$	1.08715e43	0.392011
$636$	0	0
$637$	−3.39333e43	−1.16537
$638$	0	0
$639$	−1.50451e42	−0.0492188
$640$	0	0
$641$	2.77200e43	0.863960	0.431980	−	0.901883i	$-0.357815\pi$
$641$	2.77200e43	0.863960	0.431980	+	0.901883i	$0.357815\pi$
$642$	0	0
$643$	−4.13799e43	−1.22891	−0.614454	−	0.788952i	$-0.710624\pi$
$643$	−4.13799e43	−1.22891	−0.614454	+	0.788952i	$0.710624\pi$
$644$	0	0
$645$	−5.03004e42	−0.142363
$646$	0	0
$647$	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
$650$	0	0
$651$	−5.47656e43	−1.34277
$652$	0	0
$653$	1.72282e43	0.402797	0.201398	−	0.979509i	$-0.435451\pi$
$653$	1.72282e43	0.402797	0.201398	+	0.979509i	$0.435451\pi$
$654$	0	0
$655$	−1.98547e43	−0.442714
$656$	0	0
$657$	5.12219e42	0.108941
$658$	0	0
$659$	−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	$0.722648\pi$
$662$	0	0
$663$	8.78357e43	1.62260
$664$	0	0
$665$	−7.88818e43	−1.39073
$666$	0	0
$667$	1.04681e43	0.176164
$668$	0	0
$669$	3.06222e43	0.491962
$670$	0	0
$671$	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	$0.708896\pi$
$674$	0	0
$675$	4.69743e43	0.657131
$676$	0	0
$677$	−1.76795e42	−0.0236235	−0.0118118	−	0.999930i	$-0.503760\pi$
$677$	−1.76795e42	−0.0236235	−0.0118118	+	0.999930i	$0.503760\pi$
$678$	0	0
$679$	−1.30988e44	−1.67205
$680$	0	0
$681$	−3.56995e43	−0.435394
$682$	0	0
$683$	−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
$686$	0	0
$687$	−8.05958e43	−0.857992
$688$	0	0
$689$	−1.58895e44	−1.61701
$690$	0	0
$691$	−1.14213e44	−1.11124	−0.555618	−	0.831438i	$-0.687518\pi$
$691$	−1.14213e44	−1.11124	−0.555618	+	0.831438i	$0.687518\pi$
$692$	0	0
$693$	−2.64905e43	−0.246448
$694$	0	0
$695$	6.60433e43	0.587576
$696$	0	0
$697$	1.51280e44	1.28728
$698$	0	0
$699$	9.67093e43	0.787177
$700$	0	0
$701$	1.40003e44	1.09020	0.545102	−	0.838370i	$-0.316491\pi$
$701$	1.40003e44	1.09020	0.545102	+	0.838370i	$0.316491\pi$
$702$	0	0
$703$	2.73694e44	2.03919
$704$	0	0
$705$	7.86567e43	0.560796
$706$	0	0
$707$	−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	$0.541346\pi$
$710$	0	0
$711$	−1.38754e43	−0.0867497
$712$	0	0
$713$	−4.97999e43	−0.298085
$714$	0	0
$715$	−8.36056e43	−0.479172
$716$	0	0
$717$	−1.59603e44	−0.875980
$718$	0	0
$719$	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
$722$	0	0
$723$	1.10585e44	0.533399
$724$	0	0
$725$	−9.62090e43	−0.444607
$726$	0	0
$727$	−1.45218e44	−0.643036	−0.321518	−	0.946903i	$-0.604193\pi$
$727$	−1.45218e44	−0.643036	−0.321518	+	0.946903i	$0.604193\pi$
$728$	0	0
$729$	2.49666e44	1.05946
$730$	0	0
$731$	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	$0.584254\pi$
$734$	0	0
$735$	1.07856e44	0.403079
$736$	0	0
$737$	−1.18583e44	−0.424890
$738$	0	0
$739$	3.10772e44	1.06771	0.533855	−	0.845576i	$-0.320743\pi$
$739$	3.10772e44	1.06771	0.533855	+	0.845576i	$0.320743\pi$
$740$	0	0
$741$	5.88723e44	1.93968
$742$	0	0
$743$	−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
$746$	0	0
$747$	−6.12687e43	−0.178143
$748$	0	0
$749$	−5.49497e44	−1.53284
$750$	0	0
$751$	3.16352e44	0.846742	0.423371	−	0.905956i	$-0.360847\pi$
$751$	3.16352e44	0.846742	0.423371	+	0.905956i	$0.360847\pi$
$752$	0	0
$753$	3.64804e44	0.936997
$754$	0	0
$755$	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	$0.338590\pi$
$758$	0	0
$759$	4.40678e43	0.100086
$760$	0	0
$761$	3.57723e44	0.779982	0.389991	−	0.920819i	$-0.372478\pi$
$761$	3.57723e44	0.779982	0.389991	+	0.920819i	$0.372478\pi$
$762$	0	0
$763$	−5.49124e44	−1.14958
$764$	0	0
$765$	1.52608e44	0.306779
$766$	0	0
$767$	−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	$0.533385\pi$
$770$	0	0
$771$	−7.05822e43	−0.125705
$772$	0	0
$773$	−4.52518e44	−0.774201	−0.387100	−	0.922038i	$-0.626523\pi$
$773$	−4.52518e44	−0.774201	−0.387100	+	0.922038i	$0.626523\pi$
$774$	0	0
$775$	4.57696e44	0.752317
$776$	0	0
$777$	−8.44114e44	−1.33314
$778$	0	0
$779$	1.01396e45	1.53884
$780$	0	0
$781$	4.96748e43	0.0724511
$782$	0	0
$783$	−5.71635e44	−0.801330
$784$	0	0
$785$	3.91520e44	0.527562
$786$	0	0
$787$	8.95840e44	1.16044	0.580219	−	0.814460i	$-0.302967\pi$
$787$	8.95840e44	1.16044	0.580219	+	0.814460i	$0.302967\pi$
$788$	0	0
$789$	−7.54508e44	−0.939659
$790$	0	0
$791$	−1.83180e45	−2.19352
$792$	0	0
$793$	−1.09216e45	−1.25763
$794$	0	0
$795$	5.05043e44	0.559291
$796$	0	0
$797$	−4.97034e44	−0.529398	−0.264699	−	0.964331i	$-0.585273\pi$
$797$	−4.97034e44	−0.529398	−0.264699	+	0.964331i	$0.585273\pi$
$798$	0	0
$799$	−1.49143e45	−1.52801
$800$	0	0
$801$	−4.31424e44	−0.425206
$802$	0	0
$803$	−1.69121e44	−0.160364
$804$	0	0
$805$	2.21224e44	0.201835
$806$	0	0
$807$	−3.46912e44	−0.304565
$808$	0	0
$809$	1.66443e45	1.40625	0.703127	−	0.711064i	$-0.251786\pi$
$809$	1.66443e45	1.40625	0.703127	+	0.711064i	$0.251786\pi$
$810$	0	0
$811$	−1.88658e45	−1.53409	−0.767046	−	0.641592i	$-0.778274\pi$
$811$	−1.88658e45	−1.53409	−0.767046	+	0.641592i	$0.778274\pi$
$812$	0	0
$813$	−1.29274e44	−0.101183
$814$	0	0
$815$	2.13417e44	0.160800
$816$	0	0
$817$	6.39261e44	0.463699
$818$	0	0
$819$	9.92511e44	0.693162
$820$	0	0
$821$	−8.30695e44	−0.558628	−0.279314	−	0.960200i	$-0.590107\pi$
$821$	−8.30695e44	−0.558628	−0.279314	+	0.960200i	$0.590107\pi$
$822$	0	0
$823$	2.11118e45	1.36719	0.683594	−	0.729862i	$-0.260416\pi$
$823$	2.11118e45	1.36719	0.683594	+	0.729862i	$0.260416\pi$
$824$	0	0
$825$	−4.05014e44	−0.252601
$826$	0	0
$827$	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	$0.208683\pi$
$830$	0	0
$831$	−3.96985e44	−0.221288
$832$	0	0
$833$	−2.04508e45	−1.09828
$834$	0	0
$835$	1.18003e45	0.610596
$836$	0	0
$837$	2.71945e45	1.35593
$838$	0	0
$839$	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
$842$	0	0
$843$	1.09564e45	0.489036
$844$	0	0
$845$	1.66949e45	0.718299
$846$	0	0
$847$	−2.35679e45	−0.977526
$848$	0	0
$849$	5.25059e44	0.209961
$850$	0	0
$851$	−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	$0.547848\pi$
$854$	0	0
$855$	1.02286e45	0.366727
$856$	0	0
$857$	8.81281e44	0.304728	0.152364	−	0.988324i	$-0.451311\pi$
$857$	8.81281e44	0.304728	0.152364	+	0.988324i	$0.451311\pi$
$858$	0	0
$859$	3.21045e45	1.07071	0.535356	−	0.844627i	$-0.320178\pi$
$859$	3.21045e45	1.07071	0.535356	+	0.844627i	$0.320178\pi$
$860$	0	0
$861$	−3.12722e45	−1.00603
$862$	0	0
$863$	−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
$866$	0	0
$867$	2.51007e45	0.725085
$868$	0	0
$869$	4.58130e44	0.127697
$870$	0	0
$871$	4.44291e45	1.19505
$872$	0	0
$873$	1.69853e45	0.440912
$874$	0	0
$875$	−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	$0.391114\pi$
$878$	0	0
$879$	−2.66185e45	−0.621379
$880$	0	0
$881$	1.10273e45	0.248508	0.124254	−	0.992250i	$-0.460346\pi$
$881$	1.10273e45	0.248508	0.124254	+	0.992250i	$0.460346\pi$
$882$	0	0
$883$	4.69663e45	1.02187	0.510935	−	0.859619i	$-0.329299\pi$
$883$	4.69663e45	1.02187	0.510935	+	0.859619i	$0.329299\pi$
$884$	0	0
$885$	2.76743e45	0.581373
$886$	0	0
$887$	−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
$890$	0	0
$891$	−1.43452e45	−0.271395
$892$	0	0
$893$	−9.99638e45	−1.82661
$894$	0	0
$895$	2.35595e45	0.415824
$896$	0	0
$897$	−1.65107e45	−0.281504
$898$	0	0
$899$	−5.56974e45	−0.917403
$900$	0	0
$901$	−9.57623e45	−1.52391
$902$	0	0
$903$	−1.97158e45	−0.303147
$904$	0	0
$905$	−2.94409e45	−0.437420
$906$	0	0
$907$	−3.04922e45	−0.437800	−0.218900	−	0.975747i	$-0.570247\pi$
$907$	−3.04922e45	−0.437800	−0.218900	+	0.975747i	$0.570247\pi$
$908$	0	0
$909$	1.68760e45	0.234170
$910$	0	0
$911$	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
$914$	0	0
$915$	3.47140e45	0.434988
$916$	0	0
$917$	−7.78224e45	−0.942714
$918$	0	0
$919$	1.23482e45	0.144614	0.0723072	−	0.997382i	$-0.476964\pi$
$919$	1.23482e45	0.144614	0.0723072	+	0.997382i	$0.476964\pi$
$920$	0	0
$921$	8.76394e44	0.0992374
$922$	0	0
$923$	−1.86115e45	−0.203777
$924$	0	0
$925$	7.05457e45	0.746919
$926$	0	0
$927$	−3.08764e45	−0.316148
$928$	0	0
$929$	−1.71673e44	−0.0170003	−0.00850017	−	0.999964i	$-0.502706\pi$
$929$	−1.71673e44	−0.0170003	−0.00850017	+	0.999964i	$0.502706\pi$
$930$	0	0
$931$	−1.37072e46	−1.31290
$932$	0	0
$933$	5.71578e45	0.529554
$934$	0	0
$935$	−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	$0.540226\pi$
$938$	0	0
$939$	−1.20150e46	−1.00787
$940$	0	0
$941$	−2.30709e45	−0.187251	−0.0936257	−	0.995607i	$-0.529846\pi$
$941$	−2.30709e45	−0.187251	−0.0936257	+	0.995607i	$0.529846\pi$
$942$	0	0
$943$	−2.84366e45	−0.223330
$944$	0	0
$945$	−1.20805e46	−0.918104
$946$	0	0
$947$	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
$950$	0	0
$951$	1.80393e46	1.24286
$952$	0	0
$953$	−2.04354e46	−1.36284	−0.681418	−	0.731894i	$-0.738637\pi$
$953$	−2.04354e46	−1.36284	−0.681418	+	0.731894i	$0.738637\pi$
$954$	0	0
$955$	−9.33932e45	−0.602925
$956$	0	0
$957$	4.92866e45	0.308030
$958$	0	0
$959$	4.06394e46	2.45900
$960$	0	0
$961$	9.42784e45	0.552332
$962$	0	0
$963$	7.12535e45	0.404202
$964$	0	0
$965$	1.50482e45	0.0826630
$966$	0	0
$967$	2.05882e46	1.09524	0.547619	−	0.836728i	$-0.315534\pi$
$967$	2.05882e46	1.09524	0.547619	+	0.836728i	$0.315534\pi$
$968$	0	0
$969$	3.54809e46	1.82800
$970$	0	0
$971$	−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
$974$	0	0
$975$	1.51746e46	0.710468
$976$	0	0
$977$	−2.53889e46	−1.15154	−0.575769	−	0.817613i	$-0.695297\pi$
$977$	−2.53889e46	−1.15154	−0.575769	+	0.817613i	$0.695297\pi$
$978$	0	0
$979$	1.42444e46	0.625913
$980$	0	0
$981$	7.12051e45	0.303139
$982$	0	0
$983$	−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
$986$	0	0
$987$	3.08303e46	1.19416
$988$	0	0
$989$	−1.79281e45	−0.0672963
$990$	0	0
$991$	3.72910e46	1.35663	0.678317	−	0.734769i	$-0.262709\pi$
$991$	3.72910e46	1.35663	0.678317	+	0.734769i	$0.262709\pi$
$992$	0	0
$993$	2.74441e46	0.967688
$994$	0	0
$995$	−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	$0.551326\pi$
$998$	0	0
$999$	4.19154e46	1.34620

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.32.a.a.1.1	✓	1	4.3	odd	2
16.32.a.a.1.1		1	1.1	even	1	trivial
18.32.a.a.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 32 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more