LMFDB - Embedded newform 16.30.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,30,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, 30, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 30);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 30 $
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:	$85.2448678129$
Analytic rank:	$0$
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)

$f(q)$

$=$

$q+2.79256e6 q^{3} +6.65186e9 q^{5} -1.43252e12 q^{7} -6.08320e13 q^{9} +O(q^{10})$

$q+2.79256e6 q^{3} +6.65186e9 q^{5} -1.43252e12 q^{7} -6.08320e13 q^{9} +7.77023e14 q^{11} -2.70958e16 q^{13} +1.85757e16 q^{15} +6.23720e17 q^{17} -2.33976e16 q^{19} -4.00039e18 q^{21} +1.01962e20 q^{23} -1.42017e20 q^{25} -3.61531e20 q^{27} -1.93843e20 q^{29} +4.41169e21 q^{31} +2.16988e21 q^{33} -9.52891e21 q^{35} +6.84647e22 q^{37} -7.56664e22 q^{39} -1.30597e23 q^{41} +6.55176e23 q^{43} -4.04646e23 q^{45} -1.88349e24 q^{47} -1.16780e24 q^{49} +1.74177e24 q^{51} +6.52990e24 q^{53} +5.16864e24 q^{55} -6.53390e22 q^{57} -4.48941e25 q^{59} +3.53091e25 q^{61} +8.71430e25 q^{63} -1.80237e26 q^{65} +3.83658e26 q^{67} +2.84733e26 q^{69} -7.28985e26 q^{71} +7.24287e26 q^{73} -3.96591e26 q^{75} -1.11310e27 q^{77} +2.01341e27 q^{79} +3.16533e27 q^{81} +7.23674e27 q^{83} +4.14890e27 q^{85} -5.41317e26 q^{87} +1.35782e28 q^{89} +3.88152e28 q^{91} +1.23199e28 q^{93} -1.55637e26 q^{95} +2.25184e28 q^{97} -4.72678e28 q^{99} +O(q^{100})$

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
$3$	2.79256e6	0.337088	0.168544	−	0.985694i	$-0.446093\pi$
$3$	2.79256e6	0.337088	0.168544	+	0.985694i	$0.446093\pi$
$4$	0	0
$5$	6.65186e9	0.487391	0.243696	−	0.969852i	$-0.421640\pi$
$5$	6.65186e9	0.487391	0.243696	+	0.969852i	$0.421640\pi$
$6$	0	0
$7$	−1.43252e12	−0.798323	−0.399162	−	0.916881i	$-0.630699\pi$
$7$	−1.43252e12	−0.798323	−0.399162	+	0.916881i	$0.630699\pi$
$8$	0	0
$9$	−6.08320e13	−0.886371
$10$	0	0
$11$	7.77023e14	0.616935	0.308468	−	0.951235i	$-0.400184\pi$
$11$	7.77023e14	0.616935	0.308468	+	0.951235i	$0.400184\pi$
$12$	0	0
$13$	−2.70958e16	−1.90863	−0.954317	−	0.298797i	$-0.903415\pi$
$13$	−2.70958e16	−1.90863	−0.954317	+	0.298797i	$0.903415\pi$
$14$	0	0
$15$	1.85757e16	0.164294
$16$	0	0
$17$	6.23720e17	0.898422	0.449211	−	0.893426i	$-0.351705\pi$
$17$	6.23720e17	0.898422	0.449211	+	0.893426i	$0.351705\pi$
$18$	0	0
$19$	−2.33976e16	−0.00671805	−0.00335903	−	0.999994i	$-0.501069\pi$
$19$	−2.33976e16	−0.00671805	−0.00335903	+	0.999994i	$0.501069\pi$
$20$	0	0
$21$	−4.00039e18	−0.269105
$22$	0	0
$23$	1.01962e20	1.83393	0.916965	−	0.398968i	$-0.130632\pi$
$23$	1.01962e20	1.83393	0.916965	+	0.398968i	$0.130632\pi$
$24$	0	0
$25$	−1.42017e20	−0.762450
$26$	0	0
$27$	−3.61531e20	−0.635874
$28$	0	0
$29$	−1.93843e20	−0.120970	−0.0604851	−	0.998169i	$-0.519265\pi$
$29$	−1.93843e20	−0.120970	−0.0604851	+	0.998169i	$0.519265\pi$
$30$	0	0
$31$	4.41169e21	1.04679	0.523396	−	0.852089i	$-0.324665\pi$
$31$	4.41169e21	1.04679	0.523396	+	0.852089i	$0.324665\pi$
$32$	0	0
$33$	2.16988e21	0.207962
$34$	0	0
$35$	−9.52891e21	−0.389096
$36$	0	0
$37$	6.84647e22	1.24894	0.624470	−	0.781049i	$-0.285315\pi$
$37$	6.84647e22	1.24894	0.624470	+	0.781049i	$0.285315\pi$
$38$	0	0
$39$	−7.56664e22	−0.643378
$40$	0	0
$41$	−1.30597e23	−0.537735	−0.268868	−	0.963177i	$-0.586649\pi$
$41$	−1.30597e23	−0.537735	−0.268868	+	0.963177i	$0.586649\pi$
$42$	0	0
$43$	6.55176e23	1.35228	0.676138	−	0.736775i	$-0.263652\pi$
$43$	6.55176e23	1.35228	0.676138	+	0.736775i	$0.263652\pi$
$44$	0	0
$45$	−4.04646e23	−0.432010
$46$	0	0
$47$	−1.88349e24	−1.07039	−0.535197	−	0.844727i	$-0.679763\pi$
$47$	−1.88349e24	−1.07039	−0.535197	+	0.844727i	$0.679763\pi$
$48$	0	0
$49$	−1.16780e24	−0.362680
$50$	0	0
$51$	1.74177e24	0.302848
$52$	0	0
$53$	6.52990e24	0.649990	0.324995	−	0.945716i	$-0.394637\pi$
$53$	6.52990e24	0.649990	0.324995	+	0.945716i	$0.394637\pi$
$54$	0	0
$55$	5.16864e24	0.300689
$56$	0	0
$57$	−6.53390e22	−0.00226458
$58$	0	0
$59$	−4.48941e25	−0.943702	−0.471851	−	0.881678i	$-0.656414\pi$
$59$	−4.48941e25	−0.943702	−0.471851	+	0.881678i	$0.656414\pi$
$60$	0	0
$61$	3.53091e25	0.457725	0.228862	−	0.973459i	$-0.426499\pi$
$61$	3.53091e25	0.457725	0.228862	+	0.973459i	$0.426499\pi$
$62$	0	0
$63$	8.71430e25	0.707611
$64$	0	0
$65$	−1.80237e26	−0.930252
$66$	0	0
$67$	3.83658e26	1.27603	0.638014	−	0.770025i	$-0.279756\pi$
$67$	3.83658e26	1.27603	0.638014	+	0.770025i	$0.279756\pi$
$68$	0	0
$69$	2.84733e26	0.618196
$70$	0	0
$71$	−7.28985e26	−1.04586	−0.522929	−	0.852376i	$-0.675161\pi$
$71$	−7.28985e26	−1.04586	−0.522929	+	0.852376i	$0.675161\pi$
$72$	0	0
$73$	7.24287e26	0.694591	0.347295	−	0.937756i	$-0.387100\pi$
$73$	7.24287e26	0.694591	0.347295	+	0.937756i	$0.387100\pi$
$74$	0	0
$75$	−3.96591e26	−0.257013
$76$	0	0
$77$	−1.11310e27	−0.492514
$78$	0	0
$79$	2.01341e27	0.614242	0.307121	−	0.951670i	$-0.400634\pi$
$79$	2.01341e27	0.614242	0.307121	+	0.951670i	$0.400634\pi$
$80$	0	0
$81$	3.16533e27	0.672026
$82$	0	0
$83$	7.23674e27	1.07872	0.539362	−	0.842074i	$-0.318665\pi$
$83$	7.23674e27	1.07872	0.539362	+	0.842074i	$0.318665\pi$
$84$	0	0
$85$	4.14890e27	0.437883
$86$	0	0
$87$	−5.41317e26	−0.0407777
$88$	0	0
$89$	1.35782e28	0.735674	0.367837	−	0.929890i	$-0.380098\pi$
$89$	1.35782e28	0.735674	0.367837	+	0.929890i	$0.380098\pi$
$90$	0	0
$91$	3.88152e28	1.52371
$92$	0	0
$93$	1.23199e28	0.352862
$94$	0	0
$95$	−1.55637e26	−0.00327432
$96$	0	0
$97$	2.25184e28	0.350225	0.175112	−	0.984548i	$-0.443971\pi$
$97$	2.25184e28	0.350225	0.175112	+	0.984548i	$0.443971\pi$
$98$	0	0
$99$	−4.72678e28	−0.546834
$100$	0	0
$101$	2.22681e29	1.92762	0.963812	−	0.266583i	$-0.0858945\pi$
$101$	2.22681e29	1.92762	0.963812	+	0.266583i	$0.0858945\pi$
$102$	0	0
$103$	2.93841e29	1.91413	0.957067	−	0.289865i	$-0.0936105\pi$
$103$	2.93841e29	1.91413	0.957067	+	0.289865i	$0.0936105\pi$
$104$	0	0
$105$	−2.66100e28	−0.131160
$106$	0	0
$107$	1.80718e29	0.677544	0.338772	−	0.940868i	$-0.389988\pi$
$107$	1.80718e29	0.677544	0.338772	+	0.940868i	$0.389988\pi$
$108$	0	0
$109$	−1.59004e29	−0.455747	−0.227874	−	0.973691i	$-0.573177\pi$
$109$	−1.59004e29	−0.455747	−0.227874	+	0.973691i	$0.573177\pi$
$110$	0	0
$111$	1.91192e29	0.421003
$112$	0	0
$113$	6.49105e29	1.10326	0.551630	−	0.834089i	$-0.314006\pi$
$113$	6.49105e29	1.10326	0.551630	+	0.834089i	$0.314006\pi$
$114$	0	0
$115$	6.78233e29	0.893841
$116$	0	0
$117$	1.64829e30	1.69176
$118$	0	0
$119$	−8.93491e29	−0.717231
$120$	0	0
$121$	−9.82545e29	−0.619391
$122$	0	0
$123$	−3.64701e29	−0.181264
$124$	0	0
$125$	−2.18368e30	−0.859003
$126$	0	0
$127$	3.49190e30	1.09121	0.545604	−	0.838043i	$-0.316300\pi$
$127$	3.49190e30	1.09121	0.545604	+	0.838043i	$0.316300\pi$
$128$	0	0
$129$	1.82962e30	0.455837
$130$	0	0
$131$	−5.19973e30	−1.03645	−0.518223	−	0.855245i	$-0.673406\pi$
$131$	−5.19973e30	−1.03645	−0.518223	+	0.855245i	$0.673406\pi$
$132$	0	0
$133$	3.35174e28	0.00536318
$134$	0	0
$135$	−2.40485e30	−0.309919
$136$	0	0
$137$	−1.42627e31	−1.48510	−0.742549	−	0.669792i	$-0.766383\pi$
$137$	−1.42627e31	−1.48510	−0.742549	+	0.669792i	$0.766383\pi$
$138$	0	0
$139$	−8.85960e28	−0.00747652	−0.00373826	−	0.999993i	$-0.501190\pi$
$139$	−8.85960e28	−0.00747652	−0.00373826	+	0.999993i	$0.501190\pi$
$140$	0	0
$141$	−5.25974e30	−0.360817
$142$	0	0
$143$	−2.10540e31	−1.17750
$144$	0	0
$145$	−1.28941e30	−0.0589599
$146$	0	0
$147$	−3.26114e30	−0.122255
$148$	0	0
$149$	1.66266e31	0.512391	0.256195	−	0.966625i	$-0.417531\pi$
$149$	1.66266e31	0.512391	0.256195	+	0.966625i	$0.417531\pi$
$150$	0	0
$151$	−3.87352e31	−0.983874	−0.491937	−	0.870631i	$-0.663711\pi$
$151$	−3.87352e31	−0.983874	−0.491937	+	0.870631i	$0.663711\pi$
$152$	0	0
$153$	−3.79422e31	−0.796336
$154$	0	0
$155$	2.93459e31	0.510198
$156$	0	0
$157$	−3.42328e31	−0.494194	−0.247097	−	0.968991i	$-0.579477\pi$
$157$	−3.42328e31	−0.494194	−0.247097	+	0.968991i	$0.579477\pi$
$158$	0	0
$159$	1.82351e31	0.219104
$160$	0	0
$161$	−1.46062e32	−1.46407
$162$	0	0
$163$	−1.22833e32	−1.02942	−0.514712	−	0.857363i	$-0.672101\pi$
$163$	−1.22833e32	−1.02942	−0.514712	+	0.857363i	$0.672101\pi$
$164$	0	0
$165$	1.44337e31	0.101359
$166$	0	0
$167$	1.08585e32	0.640299	0.320149	−	0.947367i	$-0.396267\pi$
$167$	1.08585e32	0.640299	0.320149	+	0.947367i	$0.396267\pi$
$168$	0	0
$169$	5.32642e32	2.64288
$170$	0	0
$171$	1.42332e30	0.00595469
$172$	0	0
$173$	−8.13504e30	−0.0287535	−0.0143768	−	0.999897i	$-0.504576\pi$
$173$	−8.13504e30	−0.0287535	−0.0143768	+	0.999897i	$0.504576\pi$
$174$	0	0
$175$	2.03442e32	0.608681
$176$	0	0
$177$	−1.25369e32	−0.318111
$178$	0	0
$179$	3.91205e32	0.843403	0.421702	−	0.906735i	$-0.361433\pi$
$179$	3.91205e32	0.843403	0.421702	+	0.906735i	$0.361433\pi$
$180$	0	0
$181$	1.26477e32	0.232099	0.116049	−	0.993243i	$-0.462977\pi$
$181$	1.26477e32	0.232099	0.116049	+	0.993243i	$0.462977\pi$
$182$	0	0
$183$	9.86027e31	0.154294
$184$	0	0
$185$	4.55417e32	0.608723
$186$	0	0
$187$	4.84645e32	0.554269
$188$	0	0
$189$	5.17900e32	0.507633
$190$	0	0
$191$	2.30536e32	0.193979	0.0969896	−	0.995285i	$-0.469079\pi$
$191$	2.30536e32	0.193979	0.0969896	+	0.995285i	$0.469079\pi$
$192$	0	0
$193$	−4.30456e32	−0.311421	−0.155711	−	0.987803i	$-0.549767\pi$
$193$	−4.30456e32	−0.311421	−0.155711	+	0.987803i	$0.549767\pi$
$194$	0	0
$195$	−5.03322e32	−0.313577
$196$	0	0
$197$	2.20727e32	0.118603	0.0593013	−	0.998240i	$-0.481113\pi$
$197$	2.20727e32	0.118603	0.0593013	+	0.998240i	$0.481113\pi$
$198$	0	0
$199$	1.28486e33	0.596331	0.298165	−	0.954514i	$-0.403625\pi$
$199$	1.28486e33	0.596331	0.298165	+	0.954514i	$0.403625\pi$
$200$	0	0
$201$	1.07139e33	0.430134
$202$	0	0
$203$	2.77683e32	0.0965734
$204$	0	0
$205$	−8.68715e32	−0.262088
$206$	0	0
$207$	−6.20252e33	−1.62554
$208$	0	0
$209$	−1.81804e31	−0.00414460
$210$	0	0
$211$	−9.42940e32	−0.187236	−0.0936178	−	0.995608i	$-0.529843\pi$
$211$	−9.42940e32	−0.187236	−0.0936178	+	0.995608i	$0.529843\pi$
$212$	0	0
$213$	−2.03573e33	−0.352547
$214$	0	0
$215$	4.35814e33	0.659088
$216$	0	0
$217$	−6.31983e33	−0.835679
$218$	0	0
$219$	2.02261e33	0.234138
$220$	0	0
$221$	−1.69002e34	−1.71476
$222$	0	0
$223$	−7.89801e33	−0.703229	−0.351615	−	0.936145i	$-0.614367\pi$
$223$	−7.89801e33	−0.703229	−0.351615	+	0.936145i	$0.614367\pi$
$224$	0	0
$225$	8.63920e33	0.675814
$226$	0	0
$227$	−1.89655e34	−1.30493	−0.652467	−	0.757817i	$-0.726266\pi$
$227$	−1.89655e34	−1.30493	−0.652467	+	0.757817i	$0.726266\pi$
$228$	0	0
$229$	1.80669e34	1.09463	0.547316	−	0.836926i	$-0.315650\pi$
$229$	1.80669e34	1.09463	0.547316	+	0.836926i	$0.315650\pi$
$230$	0	0
$231$	−3.10839e33	−0.166021
$232$	0	0
$233$	−2.63861e34	−1.24370	−0.621849	−	0.783137i	$-0.713618\pi$
$233$	−2.63861e34	−1.24370	−0.621849	+	0.783137i	$0.713618\pi$
$234$	0	0
$235$	−1.25287e34	−0.521701
$236$	0	0
$237$	5.62256e33	0.207054
$238$	0	0
$239$	1.60955e34	0.524728	0.262364	−	0.964969i	$-0.415498\pi$
$239$	1.60955e34	0.524728	0.262364	+	0.964969i	$0.415498\pi$
$240$	0	0
$241$	−5.02213e31	−0.00145091	−0.000725456	−	1.00000i	$-0.500231\pi$
$241$	−5.02213e31	−0.00145091	−0.000725456		1.00000i	$0.500231\pi$
$242$	0	0
$243$	3.36514e34	0.862406
$244$	0	0
$245$	−7.76802e33	−0.176767
$246$	0	0
$247$	6.33974e32	0.0128223
$248$	0	0
$249$	2.02090e34	0.363625
$250$	0	0
$251$	5.87216e34	0.940869	0.470434	−	0.882435i	$-0.344097\pi$
$251$	5.87216e34	0.940869	0.470434	+	0.882435i	$0.344097\pi$
$252$	0	0
$253$	7.92264e34	1.13142
$254$	0	0
$255$	1.15860e34	0.147605
$256$	0	0
$257$	1.32349e34	0.150553	0.0752767	−	0.997163i	$-0.476016\pi$
$257$	1.32349e34	0.150553	0.0752767	+	0.997163i	$0.476016\pi$
$258$	0	0
$259$	−9.80770e34	−0.997058
$260$	0	0
$261$	1.17918e34	0.107225
$262$	0	0
$263$	2.27613e35	1.85284	0.926422	−	0.376486i	$-0.122868\pi$
$263$	2.27613e35	1.85284	0.926422	+	0.376486i	$0.122868\pi$
$264$	0	0
$265$	4.34360e34	0.316799
$266$	0	0
$267$	3.79178e34	0.247987
$268$	0	0
$269$	8.55442e34	0.502091	0.251046	−	0.967975i	$-0.419226\pi$
$269$	8.55442e34	0.502091	0.251046	+	0.967975i	$0.419226\pi$
$270$	0	0
$271$	−1.12405e35	−0.592556	−0.296278	−	0.955102i	$-0.595746\pi$
$271$	−1.12405e35	−0.592556	−0.296278	+	0.955102i	$0.595746\pi$
$272$	0	0
$273$	1.08394e35	0.513624
$274$	0	0
$275$	−1.10351e35	−0.470382
$276$	0	0
$277$	−2.97851e35	−1.14299	−0.571495	−	0.820605i	$-0.693636\pi$
$277$	−2.97851e35	−1.14299	−0.571495	+	0.820605i	$0.693636\pi$
$278$	0	0
$279$	−2.68372e35	−0.927847
$280$	0	0
$281$	−3.63080e34	−0.113178	−0.0565889	−	0.998398i	$-0.518022\pi$
$281$	−3.63080e34	−0.113178	−0.0565889	+	0.998398i	$0.518022\pi$
$282$	0	0
$283$	5.66565e35	1.59348	0.796739	−	0.604324i	$-0.206557\pi$
$283$	5.66565e35	1.59348	0.796739	+	0.604324i	$0.206557\pi$
$284$	0	0
$285$	−4.34626e32	−0.00110374
$286$	0	0
$287$	1.87083e35	0.429286
$288$	0	0
$289$	−9.29415e34	−0.192837
$290$	0	0
$291$	6.28838e34	0.118057
$292$	0	0
$293$	5.56965e34	0.0946777	0.0473389	−	0.998879i	$-0.484926\pi$
$293$	5.56965e34	0.0946777	0.0473389	+	0.998879i	$0.484926\pi$
$294$	0	0
$295$	−2.98629e35	−0.459952
$296$	0	0
$297$	−2.80918e35	−0.392293
$298$	0	0
$299$	−2.76272e36	−3.50030
$300$	0	0
$301$	−9.38552e35	−1.07955
$302$	0	0
$303$	6.21848e35	0.649780
$304$	0	0
$305$	2.34871e35	0.223091
$306$	0	0
$307$	−2.86613e35	−0.247622	−0.123811	−	0.992306i	$-0.539512\pi$
$307$	−2.86613e35	−0.247622	−0.123811	+	0.992306i	$0.539512\pi$
$308$	0	0
$309$	8.20567e35	0.645232
$310$	0	0
$311$	1.26179e36	0.903569	0.451785	−	0.892127i	$-0.350788\pi$
$311$	1.26179e36	0.903569	0.451785	+	0.892127i	$0.350788\pi$
$312$	0	0
$313$	−1.73714e36	−1.13355	−0.566777	−	0.823871i	$-0.691810\pi$
$313$	−1.73714e36	−1.13355	−0.566777	+	0.823871i	$0.691810\pi$
$314$	0	0
$315$	5.79663e35	0.344883
$316$	0	0
$317$	−1.70738e36	−0.926767	−0.463383	−	0.886158i	$-0.653365\pi$
$317$	−1.70738e36	−0.926767	−0.463383	+	0.886158i	$0.653365\pi$
$318$	0	0
$319$	−1.50620e35	−0.0746309
$320$	0	0
$321$	5.04666e35	0.228392
$322$	0	0
$323$	−1.45935e34	−0.00603565
$324$	0	0
$325$	3.84807e36	1.45524
$326$	0	0
$327$	−4.44028e35	−0.153627
$328$	0	0
$329$	2.69813e36	0.854520
$330$	0	0
$331$	−4.49483e34	−0.0130379	−0.00651894	−	0.999979i	$-0.502075\pi$
$331$	−4.49483e34	−0.0130379	−0.00651894	+	0.999979i	$0.502075\pi$
$332$	0	0
$333$	−4.16485e36	−1.10703
$334$	0	0
$335$	2.55204e36	0.621925
$336$	0	0
$337$	2.90611e36	0.649648	0.324824	−	0.945775i	$-0.394695\pi$
$337$	2.90611e36	0.649648	0.324824	+	0.945775i	$0.394695\pi$
$338$	0	0
$339$	1.81266e36	0.371896
$340$	0	0
$341$	3.42799e36	0.645804
$342$	0	0
$343$	6.28546e36	1.08786
$344$	0	0
$345$	1.89400e36	0.301303
$346$	0	0
$347$	−6.84729e36	−1.00171	−0.500854	−	0.865532i	$-0.666981\pi$
$347$	−6.84729e36	−1.00171	−0.500854	+	0.865532i	$0.666981\pi$
$348$	0	0
$349$	8.63782e36	1.16261	0.581307	−	0.813685i	$-0.302542\pi$
$349$	8.63782e36	1.16261	0.581307	+	0.813685i	$0.302542\pi$
$350$	0	0
$351$	9.79595e36	1.21365
$352$	0	0
$353$	−6.18156e36	−0.705285	−0.352642	−	0.935758i	$-0.614717\pi$
$353$	−6.18156e36	−0.705285	−0.352642	+	0.935758i	$0.614717\pi$
$354$	0	0
$355$	−4.84911e36	−0.509742
$356$	0	0
$357$	−2.49512e36	−0.241770
$358$	0	0
$359$	6.34728e35	0.0567176	0.0283588	−	0.999598i	$-0.490972\pi$
$359$	6.34728e35	0.0567176	0.0283588	+	0.999598i	$0.490972\pi$
$360$	0	0
$361$	−1.21293e37	−0.999955
$362$	0	0
$363$	−2.74381e36	−0.208789
$364$	0	0
$365$	4.81785e36	0.338538
$366$	0	0
$367$	1.26942e37	0.824041	0.412020	−	0.911175i	$-0.364823\pi$
$367$	1.26942e37	0.824041	0.412020	+	0.911175i	$0.364823\pi$
$368$	0	0
$369$	7.94450e36	0.476633
$370$	0	0
$371$	−9.35421e36	−0.518902
$372$	0	0
$373$	−1.44149e37	−0.739663	−0.369831	−	0.929099i	$-0.620585\pi$
$373$	−1.44149e37	−0.739663	−0.369831	+	0.929099i	$0.620585\pi$
$374$	0	0
$375$	−6.09806e36	−0.289560
$376$	0	0
$377$	5.25232e36	0.230888
$378$	0	0
$379$	−4.26310e37	−1.73563	−0.867815	−	0.496887i	$-0.834476\pi$
$379$	−4.26310e37	−1.73563	−0.867815	+	0.496887i	$0.834476\pi$
$380$	0	0
$381$	9.75133e36	0.367834
$382$	0	0
$383$	6.28792e36	0.219849	0.109925	−	0.993940i	$-0.464939\pi$
$383$	6.28792e36	0.219849	0.109925	+	0.993940i	$0.464939\pi$
$384$	0	0
$385$	−7.40418e36	−0.240047
$386$	0	0
$387$	−3.98557e37	−1.19862
$388$	0	0
$389$	−3.51582e37	−0.981200	−0.490600	−	0.871385i	$-0.663222\pi$
$389$	−3.51582e37	−0.981200	−0.490600	+	0.871385i	$0.663222\pi$
$390$	0	0
$391$	6.35955e37	1.64764
$392$	0	0
$393$	−1.45205e37	−0.349374
$394$	0	0
$395$	1.33929e37	0.299376
$396$	0	0
$397$	−6.72150e37	−1.39638	−0.698190	−	0.715913i	$-0.746011\pi$
$397$	−6.72150e37	−1.39638	−0.698190	+	0.715913i	$0.746011\pi$
$398$	0	0
$399$	9.35993e34	0.00180786
$400$	0	0
$401$	1.74160e37	0.312863	0.156432	−	0.987689i	$-0.450001\pi$
$401$	1.74160e37	0.312863	0.156432	+	0.987689i	$0.450001\pi$
$402$	0	0
$403$	−1.19538e38	−1.99794
$404$	0	0
$405$	2.10553e37	0.327540
$406$	0	0
$407$	5.31986e37	0.770516
$408$	0	0
$409$	2.48596e37	0.335355	0.167678	−	0.985842i	$-0.446373\pi$
$409$	2.48596e37	0.335355	0.167678	+	0.985842i	$0.446373\pi$
$410$	0	0
$411$	−3.98295e37	−0.500609
$412$	0	0
$413$	6.43116e37	0.753379
$414$	0	0
$415$	4.81377e37	0.525761
$416$	0	0
$417$	−2.47409e35	−0.00252025
$418$	0	0
$419$	5.65207e37	0.537161	0.268580	−	0.963257i	$-0.413445\pi$
$419$	5.65207e37	0.537161	0.268580	+	0.963257i	$0.413445\pi$
$420$	0	0
$421$	1.35712e38	1.20373	0.601865	−	0.798598i	$-0.294425\pi$
$421$	1.35712e38	1.20373	0.601865	+	0.798598i	$0.294425\pi$
$422$	0	0
$423$	1.14576e38	0.948767
$424$	0	0
$425$	−8.85791e37	−0.685002
$426$	0	0
$427$	−5.05810e37	−0.365412
$428$	0	0
$429$	−5.87945e37	−0.396923
$430$	0	0
$431$	1.25590e38	0.792562	0.396281	−	0.918129i	$-0.370301\pi$
$431$	1.25590e38	0.792562	0.396281	+	0.918129i	$0.370301\pi$
$432$	0	0
$433$	2.60317e38	1.53613	0.768065	−	0.640371i	$-0.221220\pi$
$433$	2.60317e38	1.53613	0.768065	+	0.640371i	$0.221220\pi$
$434$	0	0
$435$	−3.60076e36	−0.0198747
$436$	0	0
$437$	−2.38565e36	−0.0123204
$438$	0	0
$439$	2.54018e38	1.22780	0.613901	−	0.789383i	$-0.289599\pi$
$439$	2.54018e38	1.22780	0.613901	+	0.789383i	$0.289599\pi$
$440$	0	0
$441$	7.10394e37	0.321470
$442$	0	0
$443$	−2.42259e38	−1.02666	−0.513329	−	0.858192i	$-0.671588\pi$
$443$	−2.42259e38	−1.02666	−0.513329	+	0.858192i	$0.671588\pi$
$444$	0	0
$445$	9.03199e37	0.358561
$446$	0	0
$447$	4.64306e37	0.172721
$448$	0	0
$449$	−4.91992e38	−1.71548	−0.857740	−	0.514083i	$-0.828132\pi$
$449$	−4.91992e38	−1.71548	−0.857740	+	0.514083i	$0.828132\pi$
$450$	0	0
$451$	−1.01477e38	−0.331748
$452$	0	0
$453$	−1.08170e38	−0.331652
$454$	0	0
$455$	2.58193e38	0.742641
$456$	0	0
$457$	2.07288e38	0.559487	0.279743	−	0.960075i	$-0.409751\pi$
$457$	2.07288e38	0.559487	0.279743	+	0.960075i	$0.409751\pi$
$458$	0	0
$459$	−2.25494e38	−0.571283
$460$	0	0
$461$	3.62984e38	0.863425	0.431712	−	0.902011i	$-0.357910\pi$
$461$	3.62984e38	0.863425	0.431712	+	0.902011i	$0.357910\pi$
$462$	0	0
$463$	−1.35939e38	−0.303682	−0.151841	−	0.988405i	$-0.548520\pi$
$463$	−1.35939e38	−0.303682	−0.151841	+	0.988405i	$0.548520\pi$
$464$	0	0
$465$	8.19502e37	0.171982
$466$	0	0
$467$	−5.71729e38	−1.12744	−0.563722	−	0.825964i	$-0.690631\pi$
$467$	−5.71729e38	−1.12744	−0.563722	+	0.825964i	$0.690631\pi$
$468$	0	0
$469$	−5.49598e38	−1.01868
$470$	0	0
$471$	−9.55970e37	−0.166587
$472$	0	0
$473$	5.09087e38	0.834267
$474$	0	0
$475$	3.32286e36	0.00512218
$476$	0	0
$477$	−3.97227e38	−0.576132
$478$	0	0
$479$	6.60236e38	0.901227	0.450614	−	0.892719i	$-0.351205\pi$
$479$	6.60236e38	0.901227	0.450614	+	0.892719i	$0.351205\pi$
$480$	0	0
$481$	−1.85510e39	−2.38377
$482$	0	0
$483$	−4.07886e38	−0.493520
$484$	0	0
$485$	1.49789e38	0.170697
$486$	0	0
$487$	2.40331e38	0.258012	0.129006	−	0.991644i	$-0.458821\pi$
$487$	2.40331e38	0.258012	0.129006	+	0.991644i	$0.458821\pi$
$488$	0	0
$489$	−3.43018e38	−0.347007
$490$	0	0
$491$	−1.56227e39	−1.48962	−0.744809	−	0.667278i	$-0.767459\pi$
$491$	−1.56227e39	−1.48962	−0.744809	+	0.667278i	$0.767459\pi$
$492$	0	0
$493$	−1.20904e38	−0.108682
$494$	0	0
$495$	−3.14419e38	−0.266522
$496$	0	0
$497$	1.04429e39	0.834933
$498$	0	0
$499$	1.40587e39	1.06044	0.530222	−	0.847859i	$-0.322109\pi$
$499$	1.40587e39	1.06044	0.530222	+	0.847859i	$0.322109\pi$
$500$	0	0
$501$	3.03231e38	0.215837
$502$	0	0
$503$	−1.47502e39	−0.990976	−0.495488	−	0.868615i	$-0.665011\pi$
$503$	−1.47502e39	−0.990976	−0.495488	+	0.868615i	$0.665011\pi$
$504$	0	0
$505$	1.48124e39	0.939507
$506$	0	0
$507$	1.48743e39	0.890885
$508$	0	0
$509$	7.11334e38	0.402407	0.201204	−	0.979549i	$-0.435515\pi$
$509$	7.11334e38	0.402407	0.201204	+	0.979549i	$0.435515\pi$
$510$	0	0
$511$	−1.03755e39	−0.554508
$512$	0	0
$513$	8.45894e36	0.00427183
$514$	0	0
$515$	1.95459e39	0.932933
$516$	0	0
$517$	−1.46351e39	−0.660364
$518$	0	0
$519$	−2.27175e37	−0.00969248
$520$	0	0
$521$	−3.51484e39	−1.41827	−0.709134	−	0.705073i	$-0.750914\pi$
$521$	−3.51484e39	−1.41827	−0.709134	+	0.705073i	$0.750914\pi$
$522$	0	0
$523$	−1.29307e38	−0.0493568	−0.0246784	−	0.999695i	$-0.507856\pi$
$523$	−1.29307e38	−0.0493568	−0.0246784	+	0.999695i	$0.507856\pi$
$524$	0	0
$525$	5.68124e38	0.205179
$526$	0	0
$527$	2.75166e39	0.940462
$528$	0	0
$529$	7.30509e39	2.36330
$530$	0	0
$531$	2.73100e39	0.836471
$532$	0	0
$533$	3.53863e39	1.02634
$534$	0	0
$535$	1.20211e39	0.330229
$536$	0	0
$537$	1.09246e39	0.284301
$538$	0	0
$539$	−9.07404e38	−0.223750
$540$	0	0
$541$	−2.95510e39	−0.690577	−0.345288	−	0.938497i	$-0.612219\pi$
$541$	−2.95510e39	−0.690577	−0.345288	+	0.938497i	$0.612219\pi$
$542$	0	0
$543$	3.53194e38	0.0782377
$544$	0	0
$545$	−1.05767e39	−0.222127
$546$	0	0
$547$	−5.64187e38	−0.112359	−0.0561793	−	0.998421i	$-0.517892\pi$
$547$	−5.64187e38	−0.112359	−0.0561793	+	0.998421i	$0.517892\pi$
$548$	0	0
$549$	−2.14793e39	−0.405714
$550$	0	0
$551$	4.53545e36	0.000812685 0
$552$	0	0
$553$	−2.88425e39	−0.490364
$554$	0	0
$555$	1.27178e39	0.205193
$556$	0	0
$557$	2.40790e39	0.368755	0.184378	−	0.982855i	$-0.440973\pi$
$557$	2.40790e39	0.368755	0.184378	+	0.982855i	$0.440973\pi$
$558$	0	0
$559$	−1.77525e40	−2.58100
$560$	0	0
$561$	1.35340e39	0.186837
$562$	0	0
$563$	8.33804e39	1.09318	0.546591	−	0.837400i	$-0.315925\pi$
$563$	8.33804e39	1.09318	0.546591	+	0.837400i	$0.315925\pi$
$564$	0	0
$565$	4.31776e39	0.537719
$566$	0	0
$567$	−4.53439e39	−0.536494
$568$	0	0
$569$	−7.78796e39	−0.875580	−0.437790	−	0.899077i	$-0.644239\pi$
$569$	−7.78796e39	−0.875580	−0.437790	+	0.899077i	$0.644239\pi$
$570$	0	0
$571$	1.07384e40	1.14740	0.573700	−	0.819066i	$-0.305508\pi$
$571$	1.07384e40	1.14740	0.573700	+	0.819066i	$0.305508\pi$
$572$	0	0
$573$	6.43785e38	0.0653881
$574$	0	0
$575$	−1.44803e40	−1.39828
$576$	0	0
$577$	−1.98468e40	−1.82239	−0.911193	−	0.411979i	$-0.864838\pi$
$577$	−1.98468e40	−1.82239	−0.911193	+	0.411979i	$0.864838\pi$
$578$	0	0
$579$	−1.20207e39	−0.104976
$580$	0	0
$581$	−1.03668e40	−0.861170
$582$	0	0
$583$	5.07388e39	0.401002
$584$	0	0
$585$	1.09642e40	0.824548
$586$	0	0
$587$	5.81533e39	0.416219	0.208110	−	0.978106i	$-0.433269\pi$
$587$	5.81533e39	0.416219	0.208110	+	0.978106i	$0.433269\pi$
$588$	0	0
$589$	−1.03223e38	−0.00703241
$590$	0	0
$591$	6.16391e38	0.0399796
$592$	0	0
$593$	2.08055e39	0.128495	0.0642474	−	0.997934i	$-0.479535\pi$
$593$	2.08055e39	0.128495	0.0642474	+	0.997934i	$0.479535\pi$
$594$	0	0
$595$	−5.94337e39	−0.349572
$596$	0	0
$597$	3.58805e39	0.201016
$598$	0	0
$599$	−2.19282e40	−1.17035	−0.585173	−	0.810908i	$-0.698973\pi$
$599$	−2.19282e40	−1.17035	−0.585173	+	0.810908i	$0.698973\pi$
$600$	0	0
$601$	1.16438e40	0.592129	0.296065	−	0.955168i	$-0.404326\pi$
$601$	1.16438e40	0.592129	0.296065	+	0.955168i	$0.404326\pi$
$602$	0	0
$603$	−2.33387e40	−1.13103
$604$	0	0
$605$	−6.53575e39	−0.301886
$606$	0	0
$607$	−9.01948e39	−0.397142	−0.198571	−	0.980087i	$-0.563630\pi$
$607$	−9.01948e39	−0.397142	−0.198571	+	0.980087i	$0.563630\pi$
$608$	0	0
$609$	7.75446e38	0.0325538
$610$	0	0
$611$	5.10345e40	2.04299
$612$	0	0
$613$	7.07487e39	0.270110	0.135055	−	0.990838i	$-0.456879\pi$
$613$	7.07487e39	0.270110	0.135055	+	0.990838i	$0.456879\pi$
$614$	0	0
$615$	−2.42594e39	−0.0883466
$616$	0	0
$617$	2.47470e39	0.0859782	0.0429891	−	0.999076i	$-0.486312\pi$
$617$	2.47470e39	0.0859782	0.0429891	+	0.999076i	$0.486312\pi$
$618$	0	0
$619$	5.55735e40	1.84227	0.921135	−	0.389243i	$-0.127263\pi$
$619$	5.55735e40	1.84227	0.921135	+	0.389243i	$0.127263\pi$
$620$	0	0
$621$	−3.68622e40	−1.16615
$622$	0	0
$623$	−1.94510e40	−0.587306
$624$	0	0
$625$	1.19272e40	0.343779
$626$	0	0
$627$	−5.07699e37	−0.00139710
$628$	0	0
$629$	4.27028e40	1.12208
$630$	0	0
$631$	−6.80135e39	−0.170675	−0.0853374	−	0.996352i	$-0.527197\pi$
$631$	−6.80135e39	−0.170675	−0.0853374	+	0.996352i	$0.527197\pi$
$632$	0	0
$633$	−2.63321e39	−0.0631149
$634$	0	0
$635$	2.32276e40	0.531846
$636$	0	0
$637$	3.16423e40	0.692224
$638$	0	0
$639$	4.43456e40	0.927019
$640$	0	0
$641$	5.50508e40	1.09982	0.549911	−	0.835223i	$-0.314662\pi$
$641$	5.50508e40	1.09982	0.549911	+	0.835223i	$0.314662\pi$
$642$	0	0
$643$	−4.13272e40	−0.789181	−0.394591	−	0.918857i	$-0.629114\pi$
$643$	−4.13272e40	−0.789181	−0.394591	+	0.918857i	$0.629114\pi$
$644$	0	0
$645$	1.21703e40	0.222171
$646$	0	0
$647$	7.12949e40	1.24436	0.622180	−	0.782874i	$-0.286247\pi$
$647$	7.12949e40	1.24436	0.622180	+	0.782874i	$0.286247\pi$
$648$	0	0
$649$	−3.48837e40	−0.582203
$650$	0	0
$651$	−1.76485e40	−0.281698
$652$	0	0
$653$	2.55638e40	0.390287	0.195144	−	0.980775i	$-0.437483\pi$
$653$	2.55638e40	0.390287	0.195144	+	0.980775i	$0.437483\pi$
$654$	0	0
$655$	−3.45878e40	−0.505155
$656$	0	0
$657$	−4.40598e40	−0.615665
$658$	0	0
$659$	−2.31450e40	−0.309470	−0.154735	−	0.987956i	$-0.549452\pi$
$659$	−2.31450e40	−0.309470	−0.154735	+	0.987956i	$0.549452\pi$
$660$	0	0
$661$	−4.73448e40	−0.605830	−0.302915	−	0.953018i	$-0.597960\pi$
$661$	−4.73448e40	−0.605830	−0.302915	+	0.953018i	$0.597960\pi$
$662$	0	0
$663$	−4.71947e40	−0.578025
$664$	0	0
$665$	2.22953e38	0.00261397
$666$	0	0
$667$	−1.97645e40	−0.221851
$668$	0	0
$669$	−2.20556e40	−0.237050
$670$	0	0
$671$	2.74360e40	0.282387
$672$	0	0
$673$	1.15677e41	1.14033	0.570164	−	0.821531i	$-0.306880\pi$
$673$	1.15677e41	1.14033	0.570164	+	0.821531i	$0.306880\pi$
$674$	0	0
$675$	5.13437e40	0.484822
$676$	0	0
$677$	−1.15938e41	−1.04879	−0.524397	−	0.851474i	$-0.675709\pi$
$677$	−1.15938e41	−1.04879	−0.524397	+	0.851474i	$0.675709\pi$
$678$	0	0
$679$	−3.22580e40	−0.279593
$680$	0	0
$681$	−5.29621e40	−0.439878
$682$	0	0
$683$	−2.27986e41	−1.81471	−0.907355	−	0.420365i	$-0.861902\pi$
$683$	−2.27986e41	−1.81471	−0.907355	+	0.420365i	$0.861902\pi$
$684$	0	0
$685$	−9.48737e40	−0.723824
$686$	0	0
$687$	5.04527e40	0.368988
$688$	0	0
$689$	−1.76933e41	−1.24059
$690$	0	0
$691$	1.28263e41	0.862319	0.431160	−	0.902276i	$-0.358105\pi$
$691$	1.28263e41	0.862319	0.431160	+	0.902276i	$0.358105\pi$
$692$	0	0
$693$	6.77121e40	0.436550
$694$	0	0
$695$	−5.89328e38	−0.00364399
$696$	0	0
$697$	−8.14562e40	−0.483113
$698$	0	0
$699$	−7.36847e40	−0.419236
$700$	0	0
$701$	2.62615e41	1.43354	0.716769	−	0.697311i	$-0.245620\pi$
$701$	2.62615e41	1.43354	0.716769	+	0.697311i	$0.245620\pi$
$702$	0	0
$703$	−1.60191e39	−0.00839045
$704$	0	0
$705$	−3.49871e40	−0.175859
$706$	0	0
$707$	−3.18994e41	−1.53887
$708$	0	0
$709$	−1.98212e41	−0.917824	−0.458912	−	0.888482i	$-0.651761\pi$
$709$	−1.98212e41	−0.917824	−0.458912	+	0.888482i	$0.651761\pi$
$710$	0	0
$711$	−1.22480e41	−0.544447
$712$	0	0
$713$	4.49823e41	1.91974
$714$	0	0
$715$	−1.40048e41	−0.573905
$716$	0	0
$717$	4.49475e40	0.176880
$718$	0	0
$719$	4.90474e41	1.85373	0.926867	−	0.375390i	$-0.122491\pi$
$719$	4.90474e41	1.85373	0.926867	+	0.375390i	$0.122491\pi$
$720$	0	0
$721$	−4.20933e41	−1.52810
$722$	0	0
$723$	−1.40246e38	−0.000489086 0
$724$	0	0
$725$	2.75290e40	0.0922338
$726$	0	0
$727$	4.24029e41	1.36504	0.682522	−	0.730865i	$-0.260883\pi$
$727$	4.24029e41	1.36504	0.682522	+	0.730865i	$0.260883\pi$
$728$	0	0
$729$	−1.23264e41	−0.381319
$730$	0	0
$731$	4.08647e41	1.21492
$732$	0	0
$733$	2.92123e41	0.834754	0.417377	−	0.908733i	$-0.362949\pi$
$733$	2.92123e41	0.834754	0.417377	+	0.908733i	$0.362949\pi$
$734$	0	0
$735$	−2.16926e40	−0.0595862
$736$	0	0
$737$	2.98111e41	0.787227
$738$	0	0
$739$	−4.81596e41	−1.22275	−0.611376	−	0.791340i	$-0.709384\pi$
$739$	−4.81596e41	−1.22275	−0.611376	+	0.791340i	$0.709384\pi$
$740$	0	0
$741$	1.77041e39	0.00432225
$742$	0	0
$743$	5.86417e41	1.37679	0.688397	−	0.725334i	$-0.258315\pi$
$743$	5.86417e41	1.37679	0.688397	+	0.725334i	$0.258315\pi$
$744$	0	0
$745$	1.10597e41	0.249735
$746$	0	0
$747$	−4.40225e41	−0.956150
$748$	0	0
$749$	−2.58883e41	−0.540899
$750$	0	0
$751$	−1.83133e41	−0.368118	−0.184059	−	0.982915i	$-0.558924\pi$
$751$	−1.83133e41	−0.368118	−0.184059	+	0.982915i	$0.558924\pi$
$752$	0	0
$753$	1.63983e41	0.317156
$754$	0	0
$755$	−2.57661e41	−0.479532
$756$	0	0
$757$	6.08700e41	1.09022	0.545109	−	0.838365i	$-0.316488\pi$
$757$	6.08700e41	1.09022	0.545109	+	0.838365i	$0.316488\pi$
$758$	0	0
$759$	2.21244e41	0.381387
$760$	0	0
$761$	1.50858e41	0.250317	0.125159	−	0.992137i	$-0.460056\pi$
$761$	1.50858e41	0.250317	0.125159	+	0.992137i	$0.460056\pi$
$762$	0	0
$763$	2.27776e41	0.363833
$764$	0	0
$765$	−2.52386e41	−0.388127
$766$	0	0
$767$	1.21644e42	1.80118
$768$	0	0
$769$	7.27607e41	1.03745	0.518723	−	0.854943i	$-0.326408\pi$
$769$	7.27607e41	1.03745	0.518723	+	0.854943i	$0.326408\pi$
$770$	0	0
$771$	3.69592e40	0.0507498
$772$	0	0
$773$	−3.92968e41	−0.519702	−0.259851	−	0.965649i	$-0.583673\pi$
$773$	−3.92968e41	−0.519702	−0.259851	+	0.965649i	$0.583673\pi$
$774$	0	0
$775$	−6.26537e41	−0.798127
$776$	0	0
$777$	−2.73885e41	−0.336097
$778$	0	0
$779$	3.05566e39	0.00361253
$780$	0	0
$781$	−5.66438e41	−0.645227
$782$	0	0
$783$	7.00802e40	0.0769218
$784$	0	0
$785$	−2.27712e41	−0.240866
$786$	0	0
$787$	−8.52629e41	−0.869214	−0.434607	−	0.900620i	$-0.643113\pi$
$787$	−8.52629e41	−0.869214	−0.434607	+	0.900620i	$0.643113\pi$
$788$	0	0
$789$	6.35622e41	0.624572
$790$	0	0
$791$	−9.29855e41	−0.880758
$792$	0	0
$793$	−9.56728e41	−0.873629
$794$	0	0
$795$	1.21297e41	0.106789
$796$	0	0
$797$	−9.95077e41	−0.844716	−0.422358	−	0.906429i	$-0.638798\pi$
$797$	−9.95077e41	−0.844716	−0.422358	+	0.906429i	$0.638798\pi$
$798$	0	0
$799$	−1.17477e42	−0.961666
$800$	0	0
$801$	−8.25986e41	−0.652081
$802$	0	0
$803$	5.62787e41	0.428518
$804$	0	0
$805$	−9.71582e41	−0.713574
$806$	0	0
$807$	2.38887e41	0.169249
$808$	0	0
$809$	7.25598e41	0.495955	0.247978	−	0.968766i	$-0.420234\pi$
$809$	7.25598e41	0.495955	0.247978	+	0.968766i	$0.420234\pi$
$810$	0	0
$811$	5.73901e41	0.378473	0.189236	−	0.981932i	$-0.439399\pi$
$811$	5.73901e41	0.378473	0.189236	+	0.981932i	$0.439399\pi$
$812$	0	0
$813$	−3.13896e41	−0.199744
$814$	0	0
$815$	−8.17067e41	−0.501732
$816$	0	0
$817$	−1.53295e40	−0.00908466
$818$	0	0
$819$	−2.36120e42	−1.35057
$820$	0	0
$821$	2.73841e42	1.51190	0.755950	−	0.654629i	$-0.227175\pi$
$821$	2.73841e42	1.51190	0.755950	+	0.654629i	$0.227175\pi$
$822$	0	0
$823$	5.81100e41	0.309709	0.154854	−	0.987937i	$-0.450509\pi$
$823$	5.81100e41	0.309709	0.154854	+	0.987937i	$0.450509\pi$
$824$	0	0
$825$	−3.08160e41	−0.158560
$826$	0	0
$827$	3.47953e41	0.172858	0.0864292	−	0.996258i	$-0.472454\pi$
$827$	3.47953e41	0.172858	0.0864292	+	0.996258i	$0.472454\pi$
$828$	0	0
$829$	5.41233e41	0.259623	0.129812	−	0.991539i	$-0.458563\pi$
$829$	5.41233e41	0.259623	0.129812	+	0.991539i	$0.458563\pi$
$830$	0	0
$831$	−8.31766e41	−0.385289
$832$	0	0
$833$	−7.28379e41	−0.325840
$834$	0	0
$835$	7.22294e41	0.312076
$836$	0	0
$837$	−1.59496e42	−0.665628
$838$	0	0
$839$	6.31678e41	0.254652	0.127326	−	0.991861i	$-0.459361\pi$
$839$	6.31678e41	0.254652	0.127326	+	0.991861i	$0.459361\pi$
$840$	0	0
$841$	−2.53011e42	−0.985366
$842$	0	0
$843$	−1.01392e41	−0.0381509
$844$	0	0
$845$	3.54306e42	1.28812
$846$	0	0
$847$	1.40751e42	0.494474
$848$	0	0
$849$	1.58216e42	0.537143
$850$	0	0
$851$	6.98076e42	2.29047
$852$	0	0
$853$	−3.80532e42	−1.20679	−0.603393	−	0.797444i	$-0.706185\pi$
$853$	−3.80532e42	−1.20679	−0.603393	+	0.797444i	$0.706185\pi$
$854$	0	0
$855$	9.46772e39	0.00290226
$856$	0	0
$857$	5.43316e42	1.61002	0.805009	−	0.593262i	$-0.202160\pi$
$857$	5.43316e42	1.61002	0.805009	+	0.593262i	$0.202160\pi$
$858$	0	0
$859$	1.23039e42	0.352486	0.176243	−	0.984347i	$-0.443605\pi$
$859$	1.23039e42	0.352486	0.176243	+	0.984347i	$0.443605\pi$
$860$	0	0
$861$	5.22440e41	0.144707
$862$	0	0
$863$	−1.94970e42	−0.522168	−0.261084	−	0.965316i	$-0.584080\pi$
$863$	−1.94970e42	−0.522168	−0.261084	+	0.965316i	$0.584080\pi$
$864$	0	0
$865$	−5.41131e40	−0.0140142
$866$	0	0
$867$	−2.59544e41	−0.0650032
$868$	0	0
$869$	1.56447e42	0.378948
$870$	0	0
$871$	−1.03955e43	−2.43547
$872$	0	0
$873$	−1.36984e42	−0.310429
$874$	0	0
$875$	3.12817e42	0.685762
$876$	0	0
$877$	2.90946e42	0.617047	0.308524	−	0.951217i	$-0.400165\pi$
$877$	2.90946e42	0.617047	0.308524	+	0.951217i	$0.400165\pi$
$878$	0	0
$879$	1.55536e41	0.0319148
$880$	0	0
$881$	7.45482e42	1.48008	0.740042	−	0.672561i	$-0.234806\pi$
$881$	7.45482e42	1.48008	0.740042	+	0.672561i	$0.234806\pi$
$882$	0	0
$883$	6.98315e42	1.34159	0.670797	−	0.741641i	$-0.265952\pi$
$883$	6.98315e42	1.34159	0.670797	+	0.741641i	$0.265952\pi$
$884$	0	0
$885$	−8.33938e41	−0.155045
$886$	0	0
$887$	3.03500e42	0.546093	0.273046	−	0.962001i	$-0.411969\pi$
$887$	3.03500e42	0.546093	0.273046	+	0.962001i	$0.411969\pi$
$888$	0	0
$889$	−5.00221e42	−0.871137
$890$	0	0
$891$	2.45953e42	0.414597
$892$	0	0
$893$	4.40690e40	0.00719096
$894$	0	0
$895$	2.60224e42	0.411067
$896$	0	0
$897$	−7.71506e42	−1.17991
$898$	0	0
$899$	−8.55175e41	−0.126631
$900$	0	0
$901$	4.07283e42	0.583965
$902$	0	0
$903$	−2.62096e42	−0.363905
$904$	0	0
$905$	8.41307e41	0.113123
$906$	0	0
$907$	−1.17002e43	−1.52366	−0.761828	−	0.647780i	$-0.775698\pi$
$907$	−1.17002e43	−1.52366	−0.761828	+	0.647780i	$0.775698\pi$
$908$	0	0
$909$	−1.35461e43	−1.70859
$910$	0	0
$911$	−1.40568e42	−0.171739	−0.0858696	−	0.996306i	$-0.527367\pi$
$911$	−1.40568e42	−0.171739	−0.0858696	+	0.996306i	$0.527367\pi$
$912$	0	0
$913$	5.62311e42	0.665503
$914$	0	0
$915$	6.55891e41	0.0752014
$916$	0	0
$917$	7.44871e42	0.827419
$918$	0	0
$919$	−9.41015e42	−1.01280	−0.506398	−	0.862300i	$-0.669023\pi$
$919$	−9.41015e42	−1.01280	−0.506398	+	0.862300i	$0.669023\pi$
$920$	0	0
$921$	−8.00384e41	−0.0834706
$922$	0	0
$923$	1.97524e43	1.99616
$924$	0	0
$925$	−9.72317e42	−0.952254
$926$	0	0
$927$	−1.78749e43	−1.69663
$928$	0	0
$929$	−1.48051e43	−1.36202	−0.681012	−	0.732272i	$-0.738460\pi$
$929$	−1.48051e43	−1.36202	−0.681012	+	0.732272i	$0.738460\pi$
$930$	0	0
$931$	2.73236e40	0.00243651
$932$	0	0
$933$	3.52362e42	0.304583
$934$	0	0
$935$	3.22379e42	0.270146
$936$	0	0
$937$	1.18857e42	0.0965606	0.0482803	−	0.998834i	$-0.484626\pi$
$937$	1.18857e42	0.0965606	0.0482803	+	0.998834i	$0.484626\pi$
$938$	0	0
$939$	−4.85106e42	−0.382108
$940$	0	0
$941$	4.56958e42	0.349001	0.174500	−	0.984657i	$-0.444169\pi$
$941$	4.56958e42	0.349001	0.174500	+	0.984657i	$0.444169\pi$
$942$	0	0
$943$	−1.33159e43	−0.986169
$944$	0	0
$945$	3.44500e42	0.247416
$946$	0	0
$947$	−3.71116e42	−0.258485	−0.129242	−	0.991613i	$-0.541254\pi$
$947$	−3.71116e42	−0.258485	−0.129242	+	0.991613i	$0.541254\pi$
$948$	0	0
$949$	−1.96251e43	−1.32572
$950$	0	0
$951$	−4.76795e42	−0.312402
$952$	0	0
$953$	1.84294e43	1.17129	0.585646	−	0.810567i	$-0.300841\pi$
$953$	1.84294e43	1.17129	0.585646	+	0.810567i	$0.300841\pi$
$954$	0	0
$955$	1.53349e42	0.0945438
$956$	0	0
$957$	−4.20615e41	−0.0251572
$958$	0	0
$959$	2.04316e43	1.18559
$960$	0	0
$961$	1.70115e42	0.0957752
$962$	0	0
$963$	−1.09935e43	−0.600556
$964$	0	0
$965$	−2.86333e42	−0.151784
$966$	0	0
$967$	2.84485e43	1.46344	0.731722	−	0.681604i	$-0.238717\pi$
$967$	2.84485e43	1.46344	0.731722	+	0.681604i	$0.238717\pi$
$968$	0	0
$969$	−4.07533e40	−0.00203455
$970$	0	0
$971$	−2.17992e43	−1.05624	−0.528119	−	0.849171i	$-0.677102\pi$
$971$	−2.17992e43	−1.05624	−0.528119	+	0.849171i	$0.677102\pi$
$972$	0	0
$973$	1.26915e41	0.00596868
$974$	0	0
$975$	1.07459e43	0.490543
$976$	0	0
$977$	−2.24726e43	−0.995822	−0.497911	−	0.867228i	$-0.665899\pi$
$977$	−2.24726e43	−0.995822	−0.497911	+	0.867228i	$0.665899\pi$
$978$	0	0
$979$	1.05505e43	0.453864
$980$	0	0
$981$	9.67253e42	0.403961
$982$	0	0
$983$	3.71193e43	1.50513	0.752565	−	0.658518i	$-0.228816\pi$
$983$	3.71193e43	1.50513	0.752565	+	0.658518i	$0.228816\pi$
$984$	0	0
$985$	1.46824e42	0.0578059
$986$	0	0
$987$	7.53468e42	0.288049
$988$	0	0
$989$	6.68027e43	2.47998
$990$	0	0
$991$	−3.41473e42	−0.123108	−0.0615542	−	0.998104i	$-0.519606\pi$
$991$	−3.41473e42	−0.123108	−0.0615542	+	0.998104i	$0.519606\pi$
$992$	0	0
$993$	−1.25521e41	−0.00439492
$994$	0	0
$995$	8.54672e42	0.290646
$996$	0	0
$997$	1.03265e43	0.341092	0.170546	−	0.985350i	$-0.445447\pi$
$997$	1.03265e43	0.341092	0.170546	+	0.985350i	$0.445447\pi$
$998$	0	0
$999$	−2.47521e43	−0.794168

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.30.a.b.1.1		1
4.3	odd	2		2.30.a.a.1.1	✓	1
12.11	even	2		18.30.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.30.a.a.1.1	✓	1	4.3	odd	2
16.30.a.b.1.1		1	1.1	even	1	trivial
18.30.a.c.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:	\( k \)	\(=\)	\( 30 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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