LMFDB - Embedded newform 8.16.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,16,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	8.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:	$11.4154804080$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2700.00 q^{3} -251890. q^{5} +1.37407e6 q^{7} -7.05891e6 q^{9} +O(q^{10})$

$q+2700.00 q^{3} -251890. q^{5} +1.37407e6 q^{7} -7.05891e6 q^{9} -4.32867e7 q^{11} -3.23161e8 q^{13} -6.80103e8 q^{15} -1.91654e8 q^{17} -6.51546e9 q^{19} +3.70999e9 q^{21} +2.38808e10 q^{23} +3.29310e10 q^{25} -5.78011e10 q^{27} +1.76821e11 q^{29} -1.52007e11 q^{31} -1.16874e11 q^{33} -3.46115e11 q^{35} +2.15812e10 q^{37} -8.72536e11 q^{39} -2.45334e11 q^{41} +2.76996e12 q^{43} +1.77807e12 q^{45} +2.81177e12 q^{47} -2.85949e12 q^{49} -5.17465e11 q^{51} -3.49141e12 q^{53} +1.09035e13 q^{55} -1.75917e13 q^{57} -1.58278e13 q^{59} -2.46090e13 q^{61} -9.69945e12 q^{63} +8.14011e13 q^{65} -2.07062e13 q^{67} +6.44782e13 q^{69} -7.19983e11 q^{71} +2.98830e13 q^{73} +8.89137e13 q^{75} -5.94791e13 q^{77} -1.48101e14 q^{79} -5.47754e13 q^{81} -3.02807e14 q^{83} +4.82756e13 q^{85} +4.77416e14 q^{87} -4.96151e14 q^{89} -4.44047e14 q^{91} -4.10419e14 q^{93} +1.64118e15 q^{95} +3.09183e14 q^{97} +3.05557e14 q^{99} +O(q^{100})$

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
$3$	2700.00	0.712778	0.356389	−	0.934338i	$-0.384008\pi$
$3$	2700.00	0.712778	0.356389	+	0.934338i	$0.384008\pi$
$4$	0	0
$5$	−251890.	−1.44190	−0.720951	−	0.692986i	$-0.756295\pi$
$5$	−251890.	−1.44190	−0.720951	+	0.692986i	$0.756295\pi$
$6$	0	0
$7$	1.37407e6	0.630629	0.315315	−	0.948987i	$-0.397890\pi$
$7$	1.37407e6	0.630629	0.315315	+	0.948987i	$0.397890\pi$
$8$	0	0
$9$	−7.05891e6	−0.491947
$10$	0	0
$11$	−4.32867e7	−0.669745	−0.334873	−	0.942263i	$-0.608693\pi$
$11$	−4.32867e7	−0.669745	−0.334873	+	0.942263i	$0.608693\pi$
$12$	0	0
$13$	−3.23161e8	−1.42838	−0.714191	−	0.699951i	$-0.753205\pi$
$13$	−3.23161e8	−1.42838	−0.714191	+	0.699951i	$0.753205\pi$
$14$	0	0
$15$	−6.80103e8	−1.02776
$16$	0	0
$17$	−1.91654e8	−0.113279	−0.0566396	−	0.998395i	$-0.518039\pi$
$17$	−1.91654e8	−0.113279	−0.0566396	+	0.998395i	$0.518039\pi$
$18$	0	0
$19$	−6.51546e9	−1.67222	−0.836109	−	0.548563i	$-0.815175\pi$
$19$	−6.51546e9	−1.67222	−0.836109	+	0.548563i	$0.815175\pi$
$20$	0	0
$21$	3.70999e9	0.449499
$22$	0	0
$23$	2.38808e10	1.46248	0.731240	−	0.682120i	$-0.238942\pi$
$23$	2.38808e10	1.46248	0.731240	+	0.682120i	$0.238942\pi$
$24$	0	0
$25$	3.29310e10	1.07908
$26$	0	0
$27$	−5.78011e10	−1.06343
$28$	0	0
$29$	1.76821e11	1.90348	0.951739	−	0.306909i	$-0.0992946\pi$
$29$	1.76821e11	1.90348	0.951739	+	0.306909i	$0.0992946\pi$
$30$	0	0
$31$	−1.52007e11	−0.992319	−0.496160	−	0.868231i	$-0.665257\pi$
$31$	−1.52007e11	−0.992319	−0.496160	+	0.868231i	$0.665257\pi$
$32$	0	0
$33$	−1.16874e11	−0.477380
$34$	0	0
$35$	−3.46115e11	−0.909306
$36$	0	0
$37$	2.15812e10	0.0373735	0.0186867	−	0.999825i	$-0.494051\pi$
$37$	2.15812e10	0.0373735	0.0186867	+	0.999825i	$0.494051\pi$
$38$	0	0
$39$	−8.72536e11	−1.01812
$40$	0	0
$41$	−2.45334e11	−0.196734	−0.0983671	−	0.995150i	$-0.531362\pi$
$41$	−2.45334e11	−0.196734	−0.0983671	+	0.995150i	$0.531362\pi$
$42$	0	0
$43$	2.76996e12	1.55403	0.777017	−	0.629480i	$-0.216732\pi$
$43$	2.76996e12	1.55403	0.777017	+	0.629480i	$0.216732\pi$
$44$	0	0
$45$	1.77807e12	0.709340
$46$	0	0
$47$	2.81177e12	0.809555	0.404777	−	0.914415i	$-0.367349\pi$
$47$	2.81177e12	0.809555	0.404777	+	0.914415i	$0.367349\pi$
$48$	0	0
$49$	−2.85949e12	−0.602307
$50$	0	0
$51$	−5.17465e11	−0.0807429
$52$	0	0
$53$	−3.49141e12	−0.408256	−0.204128	−	0.978944i	$-0.565436\pi$
$53$	−3.49141e12	−0.408256	−0.204128	+	0.978944i	$0.565436\pi$
$54$	0	0
$55$	1.09035e13	0.965707
$56$	0	0
$57$	−1.75917e13	−1.19192
$58$	0	0
$59$	−1.58278e13	−0.828000	−0.414000	−	0.910277i	$-0.635869\pi$
$59$	−1.58278e13	−0.828000	−0.414000	+	0.910277i	$0.635869\pi$
$60$	0	0
$61$	−2.46090e13	−1.00258	−0.501292	−	0.865278i	$-0.667142\pi$
$61$	−2.46090e13	−1.00258	−0.501292	+	0.865278i	$0.667142\pi$
$62$	0	0
$63$	−9.69945e12	−0.310236
$64$	0	0
$65$	8.14011e13	2.05959
$66$	0	0
$67$	−2.07062e13	−0.417388	−0.208694	−	0.977981i	$-0.566921\pi$
$67$	−2.07062e13	−0.417388	−0.208694	+	0.977981i	$0.566921\pi$
$68$	0	0
$69$	6.44782e13	1.04242
$70$	0	0
$71$	−7.19983e11	−0.00939473	−0.00469737	−	0.999989i	$-0.501495\pi$
$71$	−7.19983e11	−0.00939473	−0.00469737	+	0.999989i	$0.501495\pi$
$72$	0	0
$73$	2.98830e13	0.316594	0.158297	−	0.987392i	$-0.449400\pi$
$73$	2.98830e13	0.316594	0.158297	+	0.987392i	$0.449400\pi$
$74$	0	0
$75$	8.89137e13	0.769147
$76$	0	0
$77$	−5.94791e13	−0.422361
$78$	0	0
$79$	−1.48101e14	−0.867670	−0.433835	−	0.900992i	$-0.642840\pi$
$79$	−1.48101e14	−0.867670	−0.433835	+	0.900992i	$0.642840\pi$
$80$	0	0
$81$	−5.47754e13	−0.266040
$82$	0	0
$83$	−3.02807e14	−1.22484	−0.612421	−	0.790532i	$-0.709804\pi$
$83$	−3.02807e14	−1.22484	−0.612421	+	0.790532i	$0.709804\pi$
$84$	0	0
$85$	4.82756e13	0.163338
$86$	0	0
$87$	4.77416e14	1.35676
$88$	0	0
$89$	−4.96151e14	−1.18902	−0.594509	−	0.804089i	$-0.702654\pi$
$89$	−4.96151e14	−1.18902	−0.594509	+	0.804089i	$0.702654\pi$
$90$	0	0
$91$	−4.44047e14	−0.900780
$92$	0	0
$93$	−4.10419e14	−0.707303
$94$	0	0
$95$	1.64118e15	2.41118
$96$	0	0
$97$	3.09183e14	0.388533	0.194266	−	0.980949i	$-0.437767\pi$
$97$	3.09183e14	0.388533	0.194266	+	0.980949i	$0.437767\pi$
$98$	0	0
$99$	3.05557e14	0.329479
$100$	0	0
$101$	1.48859e15	1.38155	0.690773	−	0.723072i	$-0.257271\pi$
$101$	1.48859e15	1.38155	0.690773	+	0.723072i	$0.257271\pi$
$102$	0	0
$103$	−2.99335e14	−0.239816	−0.119908	−	0.992785i	$-0.538260\pi$
$103$	−2.99335e14	−0.239816	−0.119908	+	0.992785i	$0.538260\pi$
$104$	0	0
$105$	−9.34510e14	−0.648133
$106$	0	0
$107$	−1.17587e15	−0.707916	−0.353958	−	0.935261i	$-0.615164\pi$
$107$	−1.17587e15	−0.707916	−0.353958	+	0.935261i	$0.615164\pi$
$108$	0	0
$109$	−1.87431e15	−0.982072	−0.491036	−	0.871139i	$-0.663382\pi$
$109$	−1.87431e15	−0.982072	−0.491036	+	0.871139i	$0.663382\pi$
$110$	0	0
$111$	5.82693e13	0.0266390
$112$	0	0
$113$	2.87232e15	1.14854	0.574269	−	0.818667i	$-0.305286\pi$
$113$	2.87232e15	1.14854	0.574269	+	0.818667i	$0.305286\pi$
$114$	0	0
$115$	−6.01534e15	−2.10875
$116$	0	0
$117$	2.28117e15	0.702689
$118$	0	0
$119$	−2.63346e14	−0.0714372
$120$	0	0
$121$	−2.30351e15	−0.551442
$122$	0	0
$123$	−6.62403e14	−0.140228
$124$	0	0
$125$	−6.07915e14	−0.114030
$126$	0	0
$127$	1.05419e16	1.75546	0.877730	−	0.479156i	$-0.159057\pi$
$127$	1.05419e16	1.75546	0.877730	+	0.479156i	$0.159057\pi$
$128$	0	0
$129$	7.47890e15	1.10768
$130$	0	0
$131$	−2.79931e15	−0.369416	−0.184708	−	0.982793i	$-0.559134\pi$
$131$	−2.79931e15	−0.369416	−0.184708	+	0.982793i	$0.559134\pi$
$132$	0	0
$133$	−8.95271e15	−1.05455
$134$	0	0
$135$	1.45595e16	1.53336
$136$	0	0
$137$	6.99920e15	0.660153	0.330076	−	0.943954i	$-0.392926\pi$
$137$	6.99920e15	0.660153	0.330076	+	0.943954i	$0.392926\pi$
$138$	0	0
$139$	−1.68752e16	−1.42770	−0.713850	−	0.700299i	$-0.753050\pi$
$139$	−1.68752e16	−1.42770	−0.713850	+	0.700299i	$0.753050\pi$
$140$	0	0
$141$	7.59178e15	0.577033
$142$	0	0
$143$	1.39886e16	0.956652
$144$	0	0
$145$	−4.45393e16	−2.74463
$146$	0	0
$147$	−7.72062e15	−0.429311
$148$	0	0
$149$	−1.09882e16	−0.552114	−0.276057	−	0.961141i	$-0.589028\pi$
$149$	−1.09882e16	−0.552114	−0.276057	+	0.961141i	$0.589028\pi$
$150$	0	0
$151$	−1.42994e16	−0.650116	−0.325058	−	0.945694i	$-0.605384\pi$
$151$	−1.42994e16	−0.650116	−0.325058	+	0.945694i	$0.605384\pi$
$152$	0	0
$153$	1.35287e15	0.0557274
$154$	0	0
$155$	3.82891e16	1.43083
$156$	0	0
$157$	2.08915e16	0.709125	0.354563	−	0.935032i	$-0.384630\pi$
$157$	2.08915e16	0.709125	0.354563	+	0.935032i	$0.384630\pi$
$158$	0	0
$159$	−9.42682e15	−0.290996
$160$	0	0
$161$	3.28139e16	0.922283
$162$	0	0
$163$	−4.65828e16	−1.19349	−0.596745	−	0.802431i	$-0.703539\pi$
$163$	−4.65828e16	−1.19349	−0.596745	+	0.802431i	$0.703539\pi$
$164$	0	0
$165$	2.94394e16	0.688335
$166$	0	0
$167$	−3.23639e16	−0.691333	−0.345667	−	0.938357i	$-0.612347\pi$
$167$	−3.23639e16	−0.691333	−0.345667	+	0.938357i	$0.612347\pi$
$168$	0	0
$169$	5.32474e16	1.04028
$170$	0	0
$171$	4.59920e16	0.822643
$172$	0	0
$173$	2.66152e16	0.436299	0.218149	−	0.975915i	$-0.429998\pi$
$173$	2.66152e16	0.436299	0.218149	+	0.975915i	$0.429998\pi$
$174$	0	0
$175$	4.52496e16	0.680501
$176$	0	0
$177$	−4.27351e16	−0.590180
$178$	0	0
$179$	−5.67028e16	−0.719791	−0.359896	−	0.932993i	$-0.617188\pi$
$179$	−5.67028e16	−0.719791	−0.359896	+	0.932993i	$0.617188\pi$
$180$	0	0
$181$	−9.23330e16	−1.07837	−0.539185	−	0.842188i	$-0.681267\pi$
$181$	−9.23330e16	−1.07837	−0.539185	+	0.842188i	$0.681267\pi$
$182$	0	0
$183$	−6.64444e16	−0.714620
$184$	0	0
$185$	−5.43610e15	−0.0538889
$186$	0	0
$187$	8.29606e15	0.0758682
$188$	0	0
$189$	−7.94229e16	−0.670629
$190$	0	0
$191$	−7.81483e16	−0.609774	−0.304887	−	0.952388i	$-0.598619\pi$
$191$	−7.81483e16	−0.609774	−0.304887	+	0.952388i	$0.598619\pi$
$192$	0	0
$193$	−3.47351e16	−0.250662	−0.125331	−	0.992115i	$-0.539999\pi$
$193$	−3.47351e16	−0.250662	−0.125331	+	0.992115i	$0.539999\pi$
$194$	0	0
$195$	2.19783e17	1.46803
$196$	0	0
$197$	−1.00451e16	−0.0621521	−0.0310761	−	0.999517i	$-0.509893\pi$
$197$	−1.00451e16	−0.0621521	−0.0310761	+	0.999517i	$0.509893\pi$
$198$	0	0
$199$	6.10791e16	0.350344	0.175172	−	0.984538i	$-0.443952\pi$
$199$	6.10791e16	0.350344	0.175172	+	0.984538i	$0.443952\pi$
$200$	0	0
$201$	−5.59068e16	−0.297505
$202$	0	0
$203$	2.42964e17	1.20039
$204$	0	0
$205$	6.17973e16	0.283671
$206$	0	0
$207$	−1.68572e17	−0.719464
$208$	0	0
$209$	2.82033e17	1.11996
$210$	0	0
$211$	4.32583e17	1.59938	0.799689	−	0.600415i	$-0.204998\pi$
$211$	4.32583e17	1.59938	0.799689	+	0.600415i	$0.204998\pi$
$212$	0	0
$213$	−1.94395e15	−0.00669636
$214$	0	0
$215$	−6.97726e17	−2.24076
$216$	0	0
$217$	−2.08869e17	−0.625786
$218$	0	0
$219$	8.06842e16	0.225662
$220$	0	0
$221$	6.19351e16	0.161806
$222$	0	0
$223$	2.88493e17	0.704447	0.352223	−	0.935916i	$-0.385426\pi$
$223$	2.88493e17	0.704447	0.352223	+	0.935916i	$0.385426\pi$
$224$	0	0
$225$	−2.32457e17	−0.530852
$226$	0	0
$227$	−7.13606e17	−1.52498	−0.762491	−	0.646999i	$-0.776024\pi$
$227$	−7.13606e17	−1.52498	−0.762491	+	0.646999i	$0.776024\pi$
$228$	0	0
$229$	5.90635e17	1.18182	0.590912	−	0.806736i	$-0.298768\pi$
$229$	5.90635e17	1.18182	0.590912	+	0.806736i	$0.298768\pi$
$230$	0	0
$231$	−1.60593e17	−0.301050
$232$	0	0
$233$	−1.97002e17	−0.346179	−0.173089	−	0.984906i	$-0.555375\pi$
$233$	−1.97002e17	−0.346179	−0.173089	+	0.984906i	$0.555375\pi$
$234$	0	0
$235$	−7.08257e17	−1.16730
$236$	0	0
$237$	−3.99872e17	−0.618456
$238$	0	0
$239$	−7.90558e17	−1.14802	−0.574010	−	0.818848i	$-0.694613\pi$
$239$	−7.90558e17	−1.14802	−0.574010	+	0.818848i	$0.694613\pi$
$240$	0	0
$241$	1.08912e18	1.48576	0.742878	−	0.669427i	$-0.233460\pi$
$241$	1.08912e18	1.48576	0.742878	+	0.669427i	$0.233460\pi$
$242$	0	0
$243$	6.81489e17	0.873800
$244$	0	0
$245$	7.20276e17	0.868467
$246$	0	0
$247$	2.10554e18	2.38857
$248$	0	0
$249$	−8.17578e17	−0.873040
$250$	0	0
$251$	9.87746e17	0.993327	0.496663	−	0.867943i	$-0.334558\pi$
$251$	9.87746e17	0.993327	0.496663	+	0.867943i	$0.334558\pi$
$252$	0	0
$253$	−1.03372e18	−0.979489
$254$	0	0
$255$	1.30344e17	0.116423
$256$	0	0
$257$	1.41281e16	0.0119011	0.00595054	−	0.999982i	$-0.498106\pi$
$257$	1.41281e16	0.0119011	0.00595054	+	0.999982i	$0.498106\pi$
$258$	0	0
$259$	2.96542e16	0.0235688
$260$	0	0
$261$	−1.24816e18	−0.936411
$262$	0	0
$263$	−3.20775e17	−0.227265	−0.113632	−	0.993523i	$-0.536249\pi$
$263$	−3.20775e17	−0.227265	−0.113632	+	0.993523i	$0.536249\pi$
$264$	0	0
$265$	8.79452e17	0.588665
$266$	0	0
$267$	−1.33961e18	−0.847507
$268$	0	0
$269$	6.83450e17	0.408851	0.204425	−	0.978882i	$-0.434467\pi$
$269$	6.83450e17	0.408851	0.204425	+	0.978882i	$0.434467\pi$
$270$	0	0
$271$	6.18255e17	0.349863	0.174931	−	0.984581i	$-0.444030\pi$
$271$	6.18255e17	0.349863	0.174931	+	0.984581i	$0.444030\pi$
$272$	0	0
$273$	−1.19893e18	−0.642056
$274$	0	0
$275$	−1.42547e18	−0.722710
$276$	0	0
$277$	−3.71739e18	−1.78500	−0.892502	−	0.451042i	$-0.851052\pi$
$277$	−3.71739e18	−1.78500	−0.892502	+	0.451042i	$0.851052\pi$
$278$	0	0
$279$	1.07300e18	0.488169
$280$	0	0
$281$	1.22333e18	0.527528	0.263764	−	0.964587i	$-0.415036\pi$
$281$	1.22333e18	0.527528	0.263764	+	0.964587i	$0.415036\pi$
$282$	0	0
$283$	−1.64355e18	−0.672023	−0.336012	−	0.941858i	$-0.609078\pi$
$283$	−1.64355e18	−0.672023	−0.336012	+	0.941858i	$0.609078\pi$
$284$	0	0
$285$	4.43118e18	1.71863
$286$	0	0
$287$	−3.37107e17	−0.124066
$288$	0	0
$289$	−2.82569e18	−0.987168
$290$	0	0
$291$	8.34794e17	0.276938
$292$	0	0
$293$	2.24549e18	0.707625	0.353813	−	0.935316i	$-0.384885\pi$
$293$	2.24549e18	0.707625	0.353813	+	0.935316i	$0.384885\pi$
$294$	0	0
$295$	3.98686e18	1.19390
$296$	0	0
$297$	2.50202e18	0.712225
$298$	0	0
$299$	−7.71735e18	−2.08898
$300$	0	0
$301$	3.80613e18	0.980019
$302$	0	0
$303$	4.01920e18	0.984735
$304$	0	0
$305$	6.19877e18	1.44563
$306$	0	0
$307$	−2.88869e18	−0.641450	−0.320725	−	0.947172i	$-0.603927\pi$
$307$	−2.88869e18	−0.641450	−0.320725	+	0.947172i	$0.603927\pi$
$308$	0	0
$309$	−8.08205e17	−0.170936
$310$	0	0
$311$	2.12389e18	0.427986	0.213993	−	0.976835i	$-0.431353\pi$
$311$	2.12389e18	0.427986	0.213993	+	0.976835i	$0.431353\pi$
$312$	0	0
$313$	−8.59160e18	−1.65003	−0.825014	−	0.565112i	$-0.808833\pi$
$313$	−8.59160e18	−1.65003	−0.825014	+	0.565112i	$0.808833\pi$
$314$	0	0
$315$	2.44319e18	0.447331
$316$	0	0
$317$	−1.93512e18	−0.337882	−0.168941	−	0.985626i	$-0.554035\pi$
$317$	−1.93512e18	−0.337882	−0.168941	+	0.985626i	$0.554035\pi$
$318$	0	0
$319$	−7.65398e18	−1.27484
$320$	0	0
$321$	−3.17486e18	−0.504587
$322$	0	0
$323$	1.24871e18	0.189427
$324$	0	0
$325$	−1.06420e19	−1.54134
$326$	0	0
$327$	−5.06064e18	−0.699999
$328$	0	0
$329$	3.86358e18	0.510529
$330$	0	0
$331$	1.22509e19	1.54688	0.773441	−	0.633868i	$-0.218534\pi$
$331$	1.22509e19	1.54688	0.773441	+	0.633868i	$0.218534\pi$
$332$	0	0
$333$	−1.52340e17	−0.0183858
$334$	0	0
$335$	5.21569e18	0.601833
$336$	0	0
$337$	−1.59441e18	−0.175944	−0.0879721	−	0.996123i	$-0.528039\pi$
$337$	−1.59441e18	−0.175944	−0.0879721	+	0.996123i	$0.528039\pi$
$338$	0	0
$339$	7.75528e18	0.818652
$340$	0	0
$341$	6.57989e18	0.664601
$342$	0	0
$343$	−1.04526e19	−1.01046
$344$	0	0
$345$	−1.62414e19	−1.50307
$346$	0	0
$347$	1.24789e18	0.110587	0.0552937	−	0.998470i	$-0.482391\pi$
$347$	1.24789e18	0.110587	0.0552937	+	0.998470i	$0.482391\pi$
$348$	0	0
$349$	4.56769e17	0.0387709	0.0193854	−	0.999812i	$-0.493829\pi$
$349$	4.56769e17	0.0387709	0.0193854	+	0.999812i	$0.493829\pi$
$350$	0	0
$351$	1.86791e19	1.51898
$352$	0	0
$353$	2.38127e19	1.85566	0.927831	−	0.373001i	$-0.121671\pi$
$353$	2.38127e19	1.85566	0.927831	+	0.373001i	$0.121671\pi$
$354$	0	0
$355$	1.81356e17	0.0135463
$356$	0	0
$357$	−7.11034e17	−0.0509189
$358$	0	0
$359$	−1.02973e19	−0.707158	−0.353579	−	0.935405i	$-0.615036\pi$
$359$	−1.02973e19	−0.707158	−0.353579	+	0.935405i	$0.615036\pi$
$360$	0	0
$361$	2.72700e19	1.79631
$362$	0	0
$363$	−6.21947e18	−0.393055
$364$	0	0
$365$	−7.52724e18	−0.456498
$366$	0	0
$367$	−3.11070e19	−1.81077	−0.905385	−	0.424591i	$-0.860418\pi$
$367$	−3.11070e19	−1.81077	−0.905385	+	0.424591i	$0.860418\pi$
$368$	0	0
$369$	1.73179e18	0.0967828
$370$	0	0
$371$	−4.79745e18	−0.257458
$372$	0	0
$373$	−4.27668e18	−0.220440	−0.110220	−	0.993907i	$-0.535156\pi$
$373$	−4.27668e18	−0.220440	−0.110220	+	0.993907i	$0.535156\pi$
$374$	0	0
$375$	−1.64137e18	−0.0812779
$376$	0	0
$377$	−5.71416e19	−2.71889
$378$	0	0
$379$	2.21403e19	1.01249	0.506243	−	0.862391i	$-0.331034\pi$
$379$	2.21403e19	1.01249	0.506243	+	0.862391i	$0.331034\pi$
$380$	0	0
$381$	2.84631e19	1.25125
$382$	0	0
$383$	−1.76036e19	−0.744064	−0.372032	−	0.928220i	$-0.621339\pi$
$383$	−1.76036e19	−0.744064	−0.372032	+	0.928220i	$0.621339\pi$
$384$	0	0
$385$	1.49822e19	0.609003
$386$	0	0
$387$	−1.95529e19	−0.764503
$388$	0	0
$389$	−1.16034e19	−0.436479	−0.218239	−	0.975895i	$-0.570031\pi$
$389$	−1.16034e19	−0.436479	−0.218239	+	0.975895i	$0.570031\pi$
$390$	0	0
$391$	−4.57684e18	−0.165669
$392$	0	0
$393$	−7.55813e18	−0.263312
$394$	0	0
$395$	3.73051e19	1.25110
$396$	0	0
$397$	1.93839e19	0.625911	0.312955	−	0.949768i	$-0.398681\pi$
$397$	1.93839e19	0.625911	0.312955	+	0.949768i	$0.398681\pi$
$398$	0	0
$399$	−2.41723e19	−0.751660
$400$	0	0
$401$	5.21606e19	1.56228	0.781142	−	0.624353i	$-0.214637\pi$
$401$	5.21606e19	1.56228	0.781142	+	0.624353i	$0.214637\pi$
$402$	0	0
$403$	4.91229e19	1.41741
$404$	0	0
$405$	1.37974e19	0.383604
$406$	0	0
$407$	−9.34181e17	−0.0250307
$408$	0	0
$409$	−5.48323e19	−1.41616	−0.708079	−	0.706133i	$-0.750438\pi$
$409$	−5.48323e19	−1.41616	−0.708079	+	0.706133i	$0.750438\pi$
$410$	0	0
$411$	1.88979e19	0.470542
$412$	0	0
$413$	−2.17485e19	−0.522161
$414$	0	0
$415$	7.62740e19	1.76610
$416$	0	0
$417$	−4.55629e19	−1.01763
$418$	0	0
$419$	−2.17874e18	−0.0469462	−0.0234731	−	0.999724i	$-0.507472\pi$
$419$	−2.17874e18	−0.0469462	−0.0234731	+	0.999724i	$0.507472\pi$
$420$	0	0
$421$	−3.88732e19	−0.808229	−0.404114	−	0.914708i	$-0.632420\pi$
$421$	−3.88732e19	−0.808229	−0.404114	+	0.914708i	$0.632420\pi$
$422$	0	0
$423$	−1.98480e19	−0.398258
$424$	0	0
$425$	−6.31135e18	−0.122238
$426$	0	0
$427$	−3.38146e19	−0.632259
$428$	0	0
$429$	3.77692e19	0.681881
$430$	0	0
$431$	6.41503e19	1.11846	0.559228	−	0.829014i	$-0.311098\pi$
$431$	6.41503e19	1.11846	0.559228	+	0.829014i	$0.311098\pi$
$432$	0	0
$433$	9.51678e19	1.60262	0.801310	−	0.598249i	$-0.204137\pi$
$433$	9.51678e19	1.60262	0.801310	+	0.598249i	$0.204137\pi$
$434$	0	0
$435$	−1.20256e20	−1.95631
$436$	0	0
$437$	−1.55594e20	−2.44559
$438$	0	0
$439$	−1.23040e19	−0.186879	−0.0934397	−	0.995625i	$-0.529786\pi$
$439$	−1.23040e19	−0.186879	−0.0934397	+	0.995625i	$0.529786\pi$
$440$	0	0
$441$	2.01849e19	0.296303
$442$	0	0
$443$	−1.08737e20	−1.54295	−0.771474	−	0.636261i	$-0.780480\pi$
$443$	−1.08737e20	−1.54295	−0.771474	+	0.636261i	$0.780480\pi$
$444$	0	0
$445$	1.24975e20	1.71445
$446$	0	0
$447$	−2.96681e19	−0.393535
$448$	0	0
$449$	−6.76125e19	−0.867320	−0.433660	−	0.901077i	$-0.642778\pi$
$449$	−6.76125e19	−0.867320	−0.433660	+	0.901077i	$0.642778\pi$
$450$	0	0
$451$	1.06197e19	0.131762
$452$	0	0
$453$	−3.86084e19	−0.463389
$454$	0	0
$455$	1.11851e20	1.29884
$456$	0	0
$457$	1.01895e20	1.14494	0.572468	−	0.819927i	$-0.305986\pi$
$457$	1.01895e20	1.14494	0.572468	+	0.819927i	$0.305986\pi$
$458$	0	0
$459$	1.10778e19	0.120464
$460$	0	0
$461$	1.50713e20	1.58633	0.793163	−	0.609009i	$-0.208433\pi$
$461$	1.50713e20	1.58633	0.793163	+	0.609009i	$0.208433\pi$
$462$	0	0
$463$	−3.77378e19	−0.384520	−0.192260	−	0.981344i	$-0.561582\pi$
$463$	−3.77378e19	−0.384520	−0.192260	+	0.981344i	$0.561582\pi$
$464$	0	0
$465$	1.03381e20	1.01986
$466$	0	0
$467$	8.76843e19	0.837616	0.418808	−	0.908075i	$-0.362448\pi$
$467$	8.76843e19	0.837616	0.418808	+	0.908075i	$0.362448\pi$
$468$	0	0
$469$	−2.84519e19	−0.263217
$470$	0	0
$471$	5.64071e19	0.505449
$472$	0	0
$473$	−1.19903e20	−1.04081
$474$	0	0
$475$	−2.14560e20	−1.80446
$476$	0	0
$477$	2.46456e19	0.200840
$478$	0	0
$479$	4.54724e19	0.359114	0.179557	−	0.983748i	$-0.442534\pi$
$479$	4.54724e19	0.359114	0.179557	+	0.983748i	$0.442534\pi$
$480$	0	0
$481$	−6.97422e18	−0.0533836
$482$	0	0
$483$	8.85976e19	0.657383
$484$	0	0
$485$	−7.78801e19	−0.560227
$486$	0	0
$487$	4.24806e19	0.296294	0.148147	−	0.988965i	$-0.452669\pi$
$487$	4.24806e19	0.296294	0.148147	+	0.988965i	$0.452669\pi$
$488$	0	0
$489$	−1.25774e20	−0.850693
$490$	0	0
$491$	8.83155e19	0.579330	0.289665	−	0.957128i	$-0.406456\pi$
$491$	8.83155e19	0.579330	0.289665	+	0.957128i	$0.406456\pi$
$492$	0	0
$493$	−3.38883e19	−0.215624
$494$	0	0
$495$	−7.69667e19	−0.475077
$496$	0	0
$497$	−9.89308e17	−0.00592459
$498$	0	0
$499$	−2.33859e20	−1.35894	−0.679469	−	0.733704i	$-0.737790\pi$
$499$	−2.33859e20	−1.35894	−0.679469	+	0.733704i	$0.737790\pi$
$500$	0	0
$501$	−8.73826e19	−0.492767
$502$	0	0
$503$	7.83316e19	0.428723	0.214362	−	0.976754i	$-0.431233\pi$
$503$	7.83316e19	0.428723	0.214362	+	0.976754i	$0.431233\pi$
$504$	0	0
$505$	−3.74961e20	−1.99205
$506$	0	0
$507$	1.43768e20	0.741486
$508$	0	0
$509$	1.00108e20	0.501288	0.250644	−	0.968079i	$-0.419358\pi$
$509$	1.00108e20	0.501288	0.250644	+	0.968079i	$0.419358\pi$
$510$	0	0
$511$	4.10614e19	0.199654
$512$	0	0
$513$	3.76601e20	1.77828
$514$	0	0
$515$	7.53995e19	0.345791
$516$	0	0
$517$	−1.21712e20	−0.542195
$518$	0	0
$519$	7.18610e19	0.310984
$520$	0	0
$521$	−1.49686e20	−0.629361	−0.314680	−	0.949198i	$-0.601897\pi$
$521$	−1.49686e20	−0.629361	−0.314680	+	0.949198i	$0.601897\pi$
$522$	0	0
$523$	−1.64487e20	−0.671998	−0.335999	−	0.941862i	$-0.609074\pi$
$523$	−1.64487e20	−0.671998	−0.335999	+	0.941862i	$0.609074\pi$
$524$	0	0
$525$	1.22174e20	0.485046
$526$	0	0
$527$	2.91327e19	0.112409
$528$	0	0
$529$	3.03657e20	1.13885
$530$	0	0
$531$	1.11727e20	0.407333
$532$	0	0
$533$	7.92827e19	0.281012
$534$	0	0
$535$	2.96191e20	1.02075
$536$	0	0
$537$	−1.53097e20	−0.513051
$538$	0	0
$539$	1.23778e20	0.403392
$540$	0	0
$541$	9.38037e19	0.297331	0.148666	−	0.988888i	$-0.452502\pi$
$541$	9.38037e19	0.297331	0.148666	+	0.988888i	$0.452502\pi$
$542$	0	0
$543$	−2.49299e20	−0.768638
$544$	0	0
$545$	4.72121e20	1.41605
$546$	0	0
$547$	1.13080e20	0.329974	0.164987	−	0.986296i	$-0.447242\pi$
$547$	1.13080e20	0.329974	0.164987	+	0.986296i	$0.447242\pi$
$548$	0	0
$549$	1.73713e20	0.493219
$550$	0	0
$551$	−1.15207e21	−3.18303
$552$	0	0
$553$	−2.03501e20	−0.547178
$554$	0	0
$555$	−1.46775e19	−0.0384108
$556$	0	0
$557$	2.47728e19	0.0631046	0.0315523	−	0.999502i	$-0.489955\pi$
$557$	2.47728e19	0.0631046	0.0315523	+	0.999502i	$0.489955\pi$
$558$	0	0
$559$	−8.95145e20	−2.21975
$560$	0	0
$561$	2.23994e19	0.0540772
$562$	0	0
$563$	4.09568e20	0.962750	0.481375	−	0.876515i	$-0.340138\pi$
$563$	4.09568e20	0.962750	0.481375	+	0.876515i	$0.340138\pi$
$564$	0	0
$565$	−7.23510e20	−1.65608
$566$	0	0
$567$	−7.52653e19	−0.167773
$568$	0	0
$569$	3.29562e20	0.715476	0.357738	−	0.933822i	$-0.383548\pi$
$569$	3.29562e20	0.715476	0.357738	+	0.933822i	$0.383548\pi$
$570$	0	0
$571$	5.37338e20	1.13626	0.568128	−	0.822940i	$-0.307668\pi$
$571$	5.37338e20	1.13626	0.568128	+	0.822940i	$0.307668\pi$
$572$	0	0
$573$	−2.11000e20	−0.434634
$574$	0	0
$575$	7.86419e20	1.57814
$576$	0	0
$577$	−6.04184e19	−0.118127	−0.0590637	−	0.998254i	$-0.518812\pi$
$577$	−6.04184e19	−0.118127	−0.0590637	+	0.998254i	$0.518812\pi$
$578$	0	0
$579$	−9.37848e19	−0.178667
$580$	0	0
$581$	−4.16078e20	−0.772421
$582$	0	0
$583$	1.51132e20	0.273427
$584$	0	0
$585$	−5.74603e20	−1.01321
$586$	0	0
$587$	2.89721e20	0.497960	0.248980	−	0.968509i	$-0.419905\pi$
$587$	2.89721e20	0.497960	0.248980	+	0.968509i	$0.419905\pi$
$588$	0	0
$589$	9.90396e20	1.65937
$590$	0	0
$591$	−2.71217e19	−0.0443007
$592$	0	0
$593$	−4.21501e20	−0.671256	−0.335628	−	0.941995i	$-0.608948\pi$
$593$	−4.21501e20	−0.671256	−0.335628	+	0.941995i	$0.608948\pi$
$594$	0	0
$595$	6.63342e19	0.103005
$596$	0	0
$597$	1.64914e20	0.249717
$598$	0	0
$599$	1.12785e20	0.166553	0.0832764	−	0.996526i	$-0.473462\pi$
$599$	1.12785e20	0.166553	0.0832764	+	0.996526i	$0.473462\pi$
$600$	0	0
$601$	4.06529e19	0.0585508	0.0292754	−	0.999571i	$-0.490680\pi$
$601$	4.06529e19	0.0585508	0.0292754	+	0.999571i	$0.490680\pi$
$602$	0	0
$603$	1.46163e20	0.205333
$604$	0	0
$605$	5.80231e20	0.795125
$606$	0	0
$607$	1.03515e21	1.38385	0.691923	−	0.721972i	$-0.256764\pi$
$607$	1.03515e21	1.38385	0.691923	+	0.721972i	$0.256764\pi$
$608$	0	0
$609$	6.56003e20	0.855611
$610$	0	0
$611$	−9.08656e20	−1.15635
$612$	0	0
$613$	2.85246e20	0.354214	0.177107	−	0.984192i	$-0.443326\pi$
$613$	2.85246e20	0.354214	0.177107	+	0.984192i	$0.443326\pi$
$614$	0	0
$615$	1.66853e20	0.202195
$616$	0	0
$617$	−1.11031e21	−1.31312	−0.656559	−	0.754275i	$-0.727989\pi$
$617$	−1.11031e21	−1.31312	−0.656559	+	0.754275i	$0.727989\pi$
$618$	0	0
$619$	6.08628e20	0.702541	0.351271	−	0.936274i	$-0.385750\pi$
$619$	6.08628e20	0.702541	0.351271	+	0.936274i	$0.385750\pi$
$620$	0	0
$621$	−1.38034e21	−1.55524
$622$	0	0
$623$	−6.81747e20	−0.749830
$624$	0	0
$625$	−8.51846e20	−0.914663
$626$	0	0
$627$	7.61488e20	0.798283
$628$	0	0
$629$	−4.13612e18	−0.00423364
$630$	0	0
$631$	−1.09902e21	−1.09846	−0.549230	−	0.835671i	$-0.685079\pi$
$631$	−1.09902e21	−1.09846	−0.549230	+	0.835671i	$0.685079\pi$
$632$	0	0
$633$	1.16797e21	1.14000
$634$	0	0
$635$	−2.65540e21	−2.53120
$636$	0	0
$637$	9.24076e20	0.860324
$638$	0	0
$639$	5.08229e18	0.00462171
$640$	0	0
$641$	6.12045e20	0.543686	0.271843	−	0.962342i	$-0.412367\pi$
$641$	6.12045e20	0.543686	0.271843	+	0.962342i	$0.412367\pi$
$642$	0	0
$643$	2.77921e20	0.241179	0.120589	−	0.992702i	$-0.461522\pi$
$643$	2.77921e20	0.241179	0.120589	+	0.992702i	$0.461522\pi$
$644$	0	0
$645$	−1.88386e21	−1.59717
$646$	0	0
$647$	2.83600e20	0.234922	0.117461	−	0.993078i	$-0.462525\pi$
$647$	2.83600e20	0.234922	0.117461	+	0.993078i	$0.462525\pi$
$648$	0	0
$649$	6.85134e20	0.554549
$650$	0	0
$651$	−5.63946e20	−0.446046
$652$	0	0
$653$	1.90607e21	1.47330	0.736648	−	0.676276i	$-0.236407\pi$
$653$	1.90607e21	1.47330	0.736648	+	0.676276i	$0.236407\pi$
$654$	0	0
$655$	7.05118e20	0.532662
$656$	0	0
$657$	−2.10942e20	−0.155748
$658$	0	0
$659$	−2.01338e21	−1.45306	−0.726532	−	0.687133i	$-0.758869\pi$
$659$	−2.01338e21	−1.45306	−0.726532	+	0.687133i	$0.758869\pi$
$660$	0	0
$661$	−1.25330e20	−0.0884185	−0.0442092	−	0.999022i	$-0.514077\pi$
$661$	−1.25330e20	−0.0884185	−0.0442092	+	0.999022i	$0.514077\pi$
$662$	0	0
$663$	1.67225e20	0.115332
$664$	0	0
$665$	2.25510e21	1.52056
$666$	0	0
$667$	4.22262e21	2.78380
$668$	0	0
$669$	7.78930e20	0.502114
$670$	0	0
$671$	1.06524e21	0.671476
$672$	0	0
$673$	−7.25299e20	−0.447100	−0.223550	−	0.974693i	$-0.571765\pi$
$673$	−7.25299e20	−0.447100	−0.223550	+	0.974693i	$0.571765\pi$
$674$	0	0
$675$	−1.90345e21	−1.14753
$676$	0	0
$677$	2.62800e19	0.0154957	0.00774785	−	0.999970i	$-0.497534\pi$
$677$	2.62800e19	0.0154957	0.00774785	+	0.999970i	$0.497534\pi$
$678$	0	0
$679$	4.24840e20	0.245020
$680$	0	0
$681$	−1.92674e21	−1.08697
$682$	0	0
$683$	−1.75876e21	−0.970623	−0.485311	−	0.874341i	$-0.661294\pi$
$683$	−1.75876e21	−0.970623	−0.485311	+	0.874341i	$0.661294\pi$
$684$	0	0
$685$	−1.76303e21	−0.951876
$686$	0	0
$687$	1.59471e21	0.842379
$688$	0	0
$689$	1.12829e21	0.583145
$690$	0	0
$691$	−2.12559e21	−1.07496	−0.537482	−	0.843275i	$-0.680624\pi$
$691$	−2.12559e21	−1.07496	−0.537482	+	0.843275i	$0.680624\pi$
$692$	0	0
$693$	4.19857e20	0.207779
$694$	0	0
$695$	4.25068e21	2.05860
$696$	0	0
$697$	4.70193e19	0.0222859
$698$	0	0
$699$	−5.31904e20	−0.246748
$700$	0	0
$701$	−3.49002e21	−1.58468	−0.792342	−	0.610078i	$-0.791138\pi$
$701$	−3.49002e21	−1.58468	−0.792342	+	0.610078i	$0.791138\pi$
$702$	0	0
$703$	−1.40612e20	−0.0624966
$704$	0	0
$705$	−1.91229e21	−0.832025
$706$	0	0
$707$	2.04543e21	0.871243
$708$	0	0
$709$	7.51367e20	0.313332	0.156666	−	0.987652i	$-0.449925\pi$
$709$	7.51367e20	0.313332	0.156666	+	0.987652i	$0.449925\pi$
$710$	0	0
$711$	1.04543e21	0.426848
$712$	0	0
$713$	−3.63005e21	−1.45125
$714$	0	0
$715$	−3.52359e21	−1.37940
$716$	0	0
$717$	−2.13451e21	−0.818284
$718$	0	0
$719$	−4.39878e21	−1.65145	−0.825726	−	0.564071i	$-0.809234\pi$
$719$	−4.39878e21	−1.65145	−0.825726	+	0.564071i	$0.809234\pi$
$720$	0	0
$721$	−4.11308e20	−0.151235
$722$	0	0
$723$	2.94062e21	1.05901
$724$	0	0
$725$	5.82288e21	2.05401
$726$	0	0
$727$	−1.31541e21	−0.454520	−0.227260	−	0.973834i	$-0.572977\pi$
$727$	−1.31541e21	−0.454520	−0.227260	+	0.973834i	$0.572977\pi$
$728$	0	0
$729$	2.62599e21	0.888866
$730$	0	0
$731$	−5.30873e20	−0.176040
$732$	0	0
$733$	−6.19995e20	−0.201423	−0.100711	−	0.994916i	$-0.532112\pi$
$733$	−6.19995e20	−0.201423	−0.100711	+	0.994916i	$0.532112\pi$
$734$	0	0
$735$	1.94475e21	0.619025
$736$	0	0
$737$	8.96305e20	0.279544
$738$	0	0
$739$	2.77691e21	0.848649	0.424324	−	0.905510i	$-0.360512\pi$
$739$	2.77691e21	0.848649	0.424324	+	0.905510i	$0.360512\pi$
$740$	0	0
$741$	5.68497e21	1.70252
$742$	0	0
$743$	−1.92644e21	−0.565378	−0.282689	−	0.959212i	$-0.591226\pi$
$743$	−1.92644e21	−0.565378	−0.282689	+	0.959212i	$0.591226\pi$
$744$	0	0
$745$	2.76782e21	0.796095
$746$	0	0
$747$	2.13748e21	0.602558
$748$	0	0
$749$	−1.61573e21	−0.446433
$750$	0	0
$751$	−4.65899e21	−1.26180	−0.630902	−	0.775862i	$-0.717315\pi$
$751$	−4.65899e21	−1.26180	−0.630902	+	0.775862i	$0.717315\pi$
$752$	0	0
$753$	2.66691e21	0.708022
$754$	0	0
$755$	3.60188e21	0.937404
$756$	0	0
$757$	−3.09746e21	−0.790290	−0.395145	−	0.918619i	$-0.629306\pi$
$757$	−3.09746e21	−0.790290	−0.395145	+	0.918619i	$0.629306\pi$
$758$	0	0
$759$	−2.79105e21	−0.698158
$760$	0	0
$761$	−7.86875e20	−0.192984	−0.0964918	−	0.995334i	$-0.530762\pi$
$761$	−7.86875e20	−0.192984	−0.0964918	+	0.995334i	$0.530762\pi$
$762$	0	0
$763$	−2.57544e21	−0.619323
$764$	0	0
$765$	−3.40773e20	−0.0803535
$766$	0	0
$767$	5.11494e21	1.18270
$768$	0	0
$769$	−4.55474e21	−1.03280	−0.516400	−	0.856348i	$-0.672728\pi$
$769$	−4.55474e21	−1.03280	−0.516400	+	0.856348i	$0.672728\pi$
$770$	0	0
$771$	3.81460e19	0.00848283
$772$	0	0
$773$	4.72043e21	1.02952	0.514761	−	0.857334i	$-0.327881\pi$
$773$	4.72043e21	1.02952	0.514761	+	0.857334i	$0.327881\pi$
$774$	0	0
$775$	−5.00575e21	−1.07079
$776$	0	0
$777$	8.00663e19	0.0167993
$778$	0	0
$779$	1.59847e21	0.328982
$780$	0	0
$781$	3.11657e19	0.00629207
$782$	0	0
$783$	−1.02204e22	−2.02421
$784$	0	0
$785$	−5.26237e21	−1.02249
$786$	0	0
$787$	−1.96603e21	−0.374782	−0.187391	−	0.982285i	$-0.560003\pi$
$787$	−1.96603e21	−0.374782	−0.187391	+	0.982285i	$0.560003\pi$
$788$	0	0
$789$	−8.66092e20	−0.161989
$790$	0	0
$791$	3.94678e21	0.724302
$792$	0	0
$793$	7.95270e21	1.43207
$794$	0	0
$795$	2.37452e21	0.419587
$796$	0	0
$797$	−1.02993e22	−1.78595	−0.892975	−	0.450106i	$-0.851386\pi$
$797$	−1.02993e22	−1.78595	−0.892975	+	0.450106i	$0.851386\pi$
$798$	0	0
$799$	−5.38886e20	−0.0917057
$800$	0	0
$801$	3.50228e21	0.584935
$802$	0	0
$803$	−1.29354e21	−0.212038
$804$	0	0
$805$	−8.26550e21	−1.32984
$806$	0	0
$807$	1.84532e21	0.291420
$808$	0	0
$809$	8.32458e21	1.29047	0.645236	−	0.763984i	$-0.276759\pi$
$809$	8.32458e21	1.29047	0.645236	+	0.763984i	$0.276759\pi$
$810$	0	0
$811$	8.59463e21	1.30789	0.653944	−	0.756543i	$-0.273113\pi$
$811$	8.59463e21	1.30789	0.653944	+	0.756543i	$0.273113\pi$
$812$	0	0
$813$	1.66929e21	0.249374
$814$	0	0
$815$	1.17337e22	1.72089
$816$	0	0
$817$	−1.80476e22	−2.59868
$818$	0	0
$819$	3.13449e21	0.443136
$820$	0	0
$821$	−5.11082e20	−0.0709442	−0.0354721	−	0.999371i	$-0.511293\pi$
$821$	−5.11082e20	−0.0709442	−0.0354721	+	0.999371i	$0.511293\pi$
$822$	0	0
$823$	6.28710e21	0.856943	0.428471	−	0.903555i	$-0.359052\pi$
$823$	6.28710e21	0.856943	0.428471	+	0.903555i	$0.359052\pi$
$824$	0	0
$825$	−3.84878e21	−0.515132
$826$	0	0
$827$	2.98458e20	0.0392276	0.0196138	−	0.999808i	$-0.493756\pi$
$827$	2.98458e20	0.0392276	0.0196138	+	0.999808i	$0.493756\pi$
$828$	0	0
$829$	1.25860e22	1.62453	0.812264	−	0.583290i	$-0.198235\pi$
$829$	1.25860e22	1.62453	0.812264	+	0.583290i	$0.198235\pi$
$830$	0	0
$831$	−1.00369e22	−1.27231
$832$	0	0
$833$	5.48031e20	0.0682288
$834$	0	0
$835$	8.15215e21	0.996835
$836$	0	0
$837$	8.78618e21	1.05526
$838$	0	0
$839$	−9.01678e21	−1.06374	−0.531872	−	0.846825i	$-0.678511\pi$
$839$	−9.01678e21	−1.06374	−0.531872	+	0.846825i	$0.678511\pi$
$840$	0	0
$841$	2.26363e22	2.62323
$842$	0	0
$843$	3.30299e21	0.376011
$844$	0	0
$845$	−1.34125e22	−1.49998
$846$	0	0
$847$	−3.16519e21	−0.347755
$848$	0	0
$849$	−4.43758e21	−0.479004
$850$	0	0
$851$	5.15377e20	0.0546580
$852$	0	0
$853$	5.62091e21	0.585719	0.292859	−	0.956156i	$-0.405393\pi$
$853$	5.62091e21	0.585719	0.292859	+	0.956156i	$0.405393\pi$
$854$	0	0
$855$	−1.15849e22	−1.18617
$856$	0	0
$857$	−1.75411e22	−1.76482	−0.882412	−	0.470478i	$-0.844082\pi$
$857$	−1.75411e22	−1.76482	−0.882412	+	0.470478i	$0.844082\pi$
$858$	0	0
$859$	−1.44480e22	−1.42843	−0.714213	−	0.699928i	$-0.753215\pi$
$859$	−1.44480e22	−1.42843	−0.714213	+	0.699928i	$0.753215\pi$
$860$	0	0
$861$	−9.10190e20	−0.0884318
$862$	0	0
$863$	−3.02914e21	−0.289226	−0.144613	−	0.989488i	$-0.546194\pi$
$863$	−3.02914e21	−0.289226	−0.144613	+	0.989488i	$0.546194\pi$
$864$	0	0
$865$	−6.70410e21	−0.629100
$866$	0	0
$867$	−7.62937e21	−0.703632
$868$	0	0
$869$	6.41080e21	0.581118
$870$	0	0
$871$	6.69146e21	0.596190
$872$	0	0
$873$	−2.18249e21	−0.191138
$874$	0	0
$875$	−8.35319e20	−0.0719105
$876$	0	0
$877$	7.82261e21	0.661995	0.330998	−	0.943632i	$-0.392615\pi$
$877$	7.82261e21	0.661995	0.330998	+	0.943632i	$0.392615\pi$
$878$	0	0
$879$	6.06281e21	0.504380
$880$	0	0
$881$	1.07941e22	0.882809	0.441404	−	0.897308i	$-0.354480\pi$
$881$	1.07941e22	0.882809	0.441404	+	0.897308i	$0.354480\pi$
$882$	0	0
$883$	−2.10919e22	−1.69594	−0.847971	−	0.530043i	$-0.822176\pi$
$883$	−2.10919e22	−1.69594	−0.847971	+	0.530043i	$0.822176\pi$
$884$	0	0
$885$	1.07645e22	0.850983
$886$	0	0
$887$	1.24547e22	0.968069	0.484034	−	0.875049i	$-0.339171\pi$
$887$	1.24547e22	0.968069	0.484034	+	0.875049i	$0.339171\pi$
$888$	0	0
$889$	1.44853e22	1.10704
$890$	0	0
$891$	2.37105e21	0.178179
$892$	0	0
$893$	−1.83200e22	−1.35375
$894$	0	0
$895$	1.42829e22	1.03787
$896$	0	0
$897$	−2.08369e22	−1.48898
$898$	0	0
$899$	−2.68780e22	−1.88886
$900$	0	0
$901$	6.69142e20	0.0462469
$902$	0	0
$903$	1.02765e22	0.698536
$904$	0	0
$905$	2.32578e22	1.55490
$906$	0	0
$907$	5.58767e21	0.367431	0.183716	−	0.982979i	$-0.441187\pi$
$907$	5.58767e21	0.367431	0.183716	+	0.982979i	$0.441187\pi$
$908$	0	0
$909$	−1.05078e22	−0.679648
$910$	0	0
$911$	3.11450e22	1.98153	0.990765	−	0.135588i	$-0.0432924\pi$
$911$	3.11450e22	1.98153	0.990765	+	0.135588i	$0.0432924\pi$
$912$	0	0
$913$	1.31075e22	0.820331
$914$	0	0
$915$	1.67367e22	1.03041
$916$	0	0
$917$	−3.84645e21	−0.232965
$918$	0	0
$919$	−9.73672e21	−0.580158	−0.290079	−	0.957003i	$-0.593682\pi$
$919$	−9.73672e21	−0.580158	−0.290079	+	0.957003i	$0.593682\pi$
$920$	0	0
$921$	−7.79946e21	−0.457212
$922$	0	0
$923$	2.32671e20	0.0134193
$924$	0	0
$925$	7.10691e20	0.0403291
$926$	0	0
$927$	2.11298e21	0.117977
$928$	0	0
$929$	−1.80710e22	−0.992806	−0.496403	−	0.868092i	$-0.665346\pi$
$929$	−1.80710e22	−0.992806	−0.496403	+	0.868092i	$0.665346\pi$
$930$	0	0
$931$	1.86309e22	1.00719
$932$	0	0
$933$	5.73452e21	0.305059
$934$	0	0
$935$	−2.08969e21	−0.109395
$936$	0	0
$937$	−1.05347e22	−0.542719	−0.271360	−	0.962478i	$-0.587473\pi$
$937$	−1.05347e22	−0.542719	−0.271360	+	0.962478i	$0.587473\pi$
$938$	0	0
$939$	−2.31973e22	−1.17610
$940$	0	0
$941$	−4.01869e21	−0.200522	−0.100261	−	0.994961i	$-0.531968\pi$
$941$	−4.01869e21	−0.200522	−0.100261	+	0.994961i	$0.531968\pi$
$942$	0	0
$943$	−5.85878e21	−0.287720
$944$	0	0
$945$	2.00058e22	0.966981
$946$	0	0
$947$	−2.63291e22	−1.25260	−0.626298	−	0.779583i	$-0.715431\pi$
$947$	−2.63291e22	−1.25260	−0.626298	+	0.779583i	$0.715431\pi$
$948$	0	0
$949$	−9.65705e21	−0.452218
$950$	0	0
$951$	−5.22484e21	−0.240835
$952$	0	0
$953$	−3.44141e22	−1.56149	−0.780747	−	0.624847i	$-0.785161\pi$
$953$	−3.44141e22	−1.56149	−0.780747	+	0.624847i	$0.785161\pi$
$954$	0	0
$955$	1.96848e22	0.879235
$956$	0	0
$957$	−2.06658e22	−0.908682
$958$	0	0
$959$	9.61741e21	0.416312
$960$	0	0
$961$	−3.59075e20	−0.0153024
$962$	0	0
$963$	8.30037e21	0.348258
$964$	0	0
$965$	8.74943e21	0.361431
$966$	0	0
$967$	1.26402e21	0.0514110	0.0257055	−	0.999670i	$-0.491817\pi$
$967$	1.26402e21	0.0514110	0.0257055	+	0.999670i	$0.491817\pi$
$968$	0	0
$969$	3.37152e21	0.135020
$970$	0	0
$971$	−1.49909e22	−0.591130	−0.295565	−	0.955323i	$-0.595508\pi$
$971$	−1.49909e22	−0.591130	−0.295565	+	0.955323i	$0.595508\pi$
$972$	0	0
$973$	−2.31877e22	−0.900349
$974$	0	0
$975$	−2.87335e22	−1.09864
$976$	0	0
$977$	3.01531e22	1.13533	0.567665	−	0.823259i	$-0.307847\pi$
$977$	3.01531e22	1.13533	0.567665	+	0.823259i	$0.307847\pi$
$978$	0	0
$979$	2.14767e22	0.796340
$980$	0	0
$981$	1.32306e22	0.483128
$982$	0	0
$983$	9.62494e21	0.346136	0.173068	−	0.984910i	$-0.444632\pi$
$983$	9.62494e21	0.346136	0.173068	+	0.984910i	$0.444632\pi$
$984$	0	0
$985$	2.53025e21	0.0896173
$986$	0	0
$987$	1.04317e22	0.363894
$988$	0	0
$989$	6.61489e22	2.27274
$990$	0	0
$991$	−2.39779e22	−0.811444	−0.405722	−	0.913997i	$-0.632980\pi$
$991$	−2.39779e22	−0.811444	−0.405722	+	0.913997i	$0.632980\pi$
$992$	0	0
$993$	3.30774e22	1.10258
$994$	0	0
$995$	−1.53852e22	−0.505162
$996$	0	0
$997$	1.91661e21	0.0619899	0.0309949	−	0.999520i	$-0.490132\pi$
$997$	1.91661e21	0.0619899	0.0309949	+	0.999520i	$0.490132\pi$
$998$	0	0
$999$	−1.24742e21	−0.0397440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.16.a.b.1.1	✓	1
3.2	odd	2		72.16.a.c.1.1		1
4.3	odd	2		16.16.a.b.1.1		1
8.3	odd	2		64.16.a.h.1.1		1
8.5	even	2		64.16.a.d.1.1		1
12.11	even	2		144.16.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.16.a.b.1.1	✓	1	1.1	even	1	trivial
16.16.a.b.1.1		1	4.3	odd	2
64.16.a.d.1.1		1	8.5	even	2
64.16.a.h.1.1		1	8.3	odd	2
72.16.a.c.1.1		1	3.2	odd	2
144.16.a.m.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:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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