LMFDB - Embedded newform 72.16.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,16,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("72.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	72.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:	$102.739323672$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	72.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+251890. q^{5} +1.37407e6 q^{7} +O(q^{10})$

$q+251890. q^{5} +1.37407e6 q^{7} +4.32867e7 q^{11} -3.23161e8 q^{13} +1.91654e8 q^{17} -6.51546e9 q^{19} -2.38808e10 q^{23} +3.29310e10 q^{25} -1.76821e11 q^{29} -1.52007e11 q^{31} +3.46115e11 q^{35} +2.15812e10 q^{37} +2.45334e11 q^{41} +2.76996e12 q^{43} -2.81177e12 q^{47} -2.85949e12 q^{49} +3.49141e12 q^{53} +1.09035e13 q^{55} +1.58278e13 q^{59} -2.46090e13 q^{61} -8.14011e13 q^{65} -2.07062e13 q^{67} +7.19983e11 q^{71} +2.98830e13 q^{73} +5.94791e13 q^{77} -1.48101e14 q^{79} +3.02807e14 q^{83} +4.82756e13 q^{85} +4.96151e14 q^{89} -4.44047e14 q^{91} -1.64118e15 q^{95} +3.09183e14 q^{97} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	251890.	1.44190	0.720951	−	0.692986i	$-0.243705\pi$
$5$	251890.	1.44190	0.720951	+	0.692986i	$0.243705\pi$
$6$	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
$9$	0	0
$10$	0	0
$11$	4.32867e7	0.669745	0.334873	−	0.942263i	$-0.391307\pi$
$11$	4.32867e7	0.669745	0.334873	+	0.942263i	$0.391307\pi$
$12$	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
$15$	0	0
$16$	0	0
$17$	1.91654e8	0.113279	0.0566396	−	0.998395i	$-0.481961\pi$
$17$	1.91654e8	0.113279	0.0566396	+	0.998395i	$0.481961\pi$
$18$	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
$21$	0	0
$22$	0	0
$23$	−2.38808e10	−1.46248	−0.731240	−	0.682120i	$-0.761058\pi$
$23$	−2.38808e10	−1.46248	−0.731240	+	0.682120i	$0.761058\pi$
$24$	0	0
$25$	3.29310e10	1.07908
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.76821e11	−1.90348	−0.951739	−	0.306909i	$-0.900705\pi$
$29$	−1.76821e11	−1.90348	−0.951739	+	0.306909i	$0.900705\pi$
$30$	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
$33$	0	0
$34$	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
$39$	0	0
$40$	0	0
$41$	2.45334e11	0.196734	0.0983671	−	0.995150i	$-0.468638\pi$
$41$	2.45334e11	0.196734	0.0983671	+	0.995150i	$0.468638\pi$
$42$	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
$45$	0	0
$46$	0	0
$47$	−2.81177e12	−0.809555	−0.404777	−	0.914415i	$-0.632651\pi$
$47$	−2.81177e12	−0.809555	−0.404777	+	0.914415i	$0.632651\pi$
$48$	0	0
$49$	−2.85949e12	−0.602307
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.49141e12	0.408256	0.204128	−	0.978944i	$-0.434564\pi$
$53$	3.49141e12	0.408256	0.204128	+	0.978944i	$0.434564\pi$
$54$	0	0
$55$	1.09035e13	0.965707
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.58278e13	0.828000	0.414000	−	0.910277i	$-0.364131\pi$
$59$	1.58278e13	0.828000	0.414000	+	0.910277i	$0.364131\pi$
$60$	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
$63$	0	0
$64$	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
$69$	0	0
$70$	0	0
$71$	7.19983e11	0.00939473	0.00469737	−	0.999989i	$-0.498505\pi$
$71$	7.19983e11	0.00939473	0.00469737	+	0.999989i	$0.498505\pi$
$72$	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
$75$	0	0
$76$	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
$81$	0	0
$82$	0	0
$83$	3.02807e14	1.22484	0.612421	−	0.790532i	$-0.290196\pi$
$83$	3.02807e14	1.22484	0.612421	+	0.790532i	$0.290196\pi$
$84$	0	0
$85$	4.82756e13	0.163338
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.96151e14	1.18902	0.594509	−	0.804089i	$-0.297346\pi$
$89$	4.96151e14	1.18902	0.594509	+	0.804089i	$0.297346\pi$
$90$	0	0
$91$	−4.44047e14	−0.900780
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	−1.48859e15	−1.38155	−0.690773	−	0.723072i	$-0.742729\pi$
$101$	−1.48859e15	−1.38155	−0.690773	+	0.723072i	$0.742729\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	1.17587e15	0.707916	0.353958	−	0.935261i	$-0.384836\pi$
$107$	1.17587e15	0.707916	0.353958	+	0.935261i	$0.384836\pi$
$108$	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
$111$	0	0
$112$	0	0
$113$	−2.87232e15	−1.14854	−0.574269	−	0.818667i	$-0.694714\pi$
$113$	−2.87232e15	−1.14854	−0.574269	+	0.818667i	$0.694714\pi$
$114$	0	0
$115$	−6.01534e15	−2.10875
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.63346e14	0.0714372
$120$	0	0
$121$	−2.30351e15	−0.551442
$122$	0	0
$123$	0	0
$124$	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
$129$	0	0
$130$	0	0
$131$	2.79931e15	0.369416	0.184708	−	0.982793i	$-0.440866\pi$
$131$	2.79931e15	0.369416	0.184708	+	0.982793i	$0.440866\pi$
$132$	0	0
$133$	−8.95271e15	−1.05455
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.99920e15	−0.660153	−0.330076	−	0.943954i	$-0.607074\pi$
$137$	−6.99920e15	−0.660153	−0.330076	+	0.943954i	$0.607074\pi$
$138$	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
$141$	0	0
$142$	0	0
$143$	−1.39886e16	−0.956652
$144$	0	0
$145$	−4.45393e16	−2.74463
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.09882e16	0.552114	0.276057	−	0.961141i	$-0.410972\pi$
$149$	1.09882e16	0.552114	0.276057	+	0.961141i	$0.410972\pi$
$150$	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
$153$	0	0
$154$	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
$159$	0	0
$160$	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
$165$	0	0
$166$	0	0
$167$	3.23639e16	0.691333	0.345667	−	0.938357i	$-0.387653\pi$
$167$	3.23639e16	0.691333	0.345667	+	0.938357i	$0.387653\pi$
$168$	0	0
$169$	5.32474e16	1.04028
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.66152e16	−0.436299	−0.218149	−	0.975915i	$-0.570002\pi$
$173$	−2.66152e16	−0.436299	−0.218149	+	0.975915i	$0.570002\pi$
$174$	0	0
$175$	4.52496e16	0.680501
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.67028e16	0.719791	0.359896	−	0.932993i	$-0.382812\pi$
$179$	5.67028e16	0.719791	0.359896	+	0.932993i	$0.382812\pi$
$180$	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
$183$	0	0
$184$	0	0
$185$	5.43610e15	0.0538889
$186$	0	0
$187$	8.29606e15	0.0758682
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.81483e16	0.609774	0.304887	−	0.952388i	$-0.401381\pi$
$191$	7.81483e16	0.609774	0.304887	+	0.952388i	$0.401381\pi$
$192$	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
$195$	0	0
$196$	0	0
$197$	1.00451e16	0.0621521	0.0310761	−	0.999517i	$-0.490107\pi$
$197$	1.00451e16	0.0621521	0.0310761	+	0.999517i	$0.490107\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	−2.42964e17	−1.20039
$204$	0	0
$205$	6.17973e16	0.283671
$206$	0	0
$207$	0	0
$208$	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
$213$	0	0
$214$	0	0
$215$	6.97726e17	2.24076
$216$	0	0
$217$	−2.08869e17	−0.625786
$218$	0	0
$219$	0	0
$220$	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
$225$	0	0
$226$	0	0
$227$	7.13606e17	1.52498	0.762491	−	0.646999i	$-0.223976\pi$
$227$	7.13606e17	1.52498	0.762491	+	0.646999i	$0.223976\pi$
$228$	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
$231$	0	0
$232$	0	0
$233$	1.97002e17	0.346179	0.173089	−	0.984906i	$-0.444625\pi$
$233$	1.97002e17	0.346179	0.173089	+	0.984906i	$0.444625\pi$
$234$	0	0
$235$	−7.08257e17	−1.16730
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.90558e17	1.14802	0.574010	−	0.818848i	$-0.305387\pi$
$239$	7.90558e17	1.14802	0.574010	+	0.818848i	$0.305387\pi$
$240$	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
$243$	0	0
$244$	0	0
$245$	−7.20276e17	−0.868467
$246$	0	0
$247$	2.10554e18	2.38857
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.87746e17	−0.993327	−0.496663	−	0.867943i	$-0.665442\pi$
$251$	−9.87746e17	−0.993327	−0.496663	+	0.867943i	$0.665442\pi$
$252$	0	0
$253$	−1.03372e18	−0.979489
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.41281e16	−0.0119011	−0.00595054	−	0.999982i	$-0.501894\pi$
$257$	−1.41281e16	−0.0119011	−0.00595054	+	0.999982i	$0.501894\pi$
$258$	0	0
$259$	2.96542e16	0.0235688
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.20775e17	0.227265	0.113632	−	0.993523i	$-0.463751\pi$
$263$	3.20775e17	0.227265	0.113632	+	0.993523i	$0.463751\pi$
$264$	0	0
$265$	8.79452e17	0.588665
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.83450e17	−0.408851	−0.204425	−	0.978882i	$-0.565533\pi$
$269$	−6.83450e17	−0.408851	−0.204425	+	0.978882i	$0.565533\pi$
$270$	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
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	−1.22333e18	−0.527528	−0.263764	−	0.964587i	$-0.584964\pi$
$281$	−1.22333e18	−0.527528	−0.263764	+	0.964587i	$0.584964\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	3.37107e17	0.124066
$288$	0	0
$289$	−2.82569e18	−0.987168
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.24549e18	−0.707625	−0.353813	−	0.935316i	$-0.615115\pi$
$293$	−2.24549e18	−0.707625	−0.353813	+	0.935316i	$0.615115\pi$
$294$	0	0
$295$	3.98686e18	1.19390
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.71735e18	2.08898
$300$	0	0
$301$	3.80613e18	0.980019
$302$	0	0
$303$	0	0
$304$	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
$309$	0	0
$310$	0	0
$311$	−2.12389e18	−0.427986	−0.213993	−	0.976835i	$-0.568647\pi$
$311$	−2.12389e18	−0.427986	−0.213993	+	0.976835i	$0.568647\pi$
$312$	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
$315$	0	0
$316$	0	0
$317$	1.93512e18	0.337882	0.168941	−	0.985626i	$-0.445965\pi$
$317$	1.93512e18	0.337882	0.168941	+	0.985626i	$0.445965\pi$
$318$	0	0
$319$	−7.65398e18	−1.27484
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.24871e18	−0.189427
$324$	0	0
$325$	−1.06420e19	−1.54134
$326$	0	0
$327$	0	0
$328$	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
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	−6.57989e18	−0.664601
$342$	0	0
$343$	−1.04526e19	−1.01046
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.24789e18	−0.110587	−0.0552937	−	0.998470i	$-0.517609\pi$
$347$	−1.24789e18	−0.110587	−0.0552937	+	0.998470i	$0.517609\pi$
$348$	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
$351$	0	0
$352$	0	0
$353$	−2.38127e19	−1.85566	−0.927831	−	0.373001i	$-0.878329\pi$
$353$	−2.38127e19	−1.85566	−0.927831	+	0.373001i	$0.878329\pi$
$354$	0	0
$355$	1.81356e17	0.0135463
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.02973e19	0.707158	0.353579	−	0.935405i	$-0.384964\pi$
$359$	1.02973e19	0.707158	0.353579	+	0.935405i	$0.384964\pi$
$360$	0	0
$361$	2.72700e19	1.79631
$362$	0	0
$363$	0	0
$364$	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
$369$	0	0
$370$	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
$375$	0	0
$376$	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
$381$	0	0
$382$	0	0
$383$	1.76036e19	0.744064	0.372032	−	0.928220i	$-0.378661\pi$
$383$	1.76036e19	0.744064	0.372032	+	0.928220i	$0.378661\pi$
$384$	0	0
$385$	1.49822e19	0.609003
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.16034e19	0.436479	0.218239	−	0.975895i	$-0.429969\pi$
$389$	1.16034e19	0.436479	0.218239	+	0.975895i	$0.429969\pi$
$390$	0	0
$391$	−4.57684e18	−0.165669
$392$	0	0
$393$	0	0
$394$	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
$399$	0	0
$400$	0	0
$401$	−5.21606e19	−1.56228	−0.781142	−	0.624353i	$-0.785363\pi$
$401$	−5.21606e19	−1.56228	−0.781142	+	0.624353i	$0.785363\pi$
$402$	0	0
$403$	4.91229e19	1.41741
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	2.17485e19	0.522161
$414$	0	0
$415$	7.62740e19	1.76610
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.17874e18	0.0469462	0.0234731	−	0.999724i	$-0.492528\pi$
$419$	2.17874e18	0.0469462	0.0234731	+	0.999724i	$0.492528\pi$
$420$	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
$423$	0	0
$424$	0	0
$425$	6.31135e18	0.122238
$426$	0	0
$427$	−3.38146e19	−0.632259
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.41503e19	−1.11846	−0.559228	−	0.829014i	$-0.688902\pi$
$431$	−6.41503e19	−1.11846	−0.559228	+	0.829014i	$0.688902\pi$
$432$	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
$435$	0	0
$436$	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
$441$	0	0
$442$	0	0
$443$	1.08737e20	1.54295	0.771474	−	0.636261i	$-0.219520\pi$
$443$	1.08737e20	1.54295	0.771474	+	0.636261i	$0.219520\pi$
$444$	0	0
$445$	1.24975e20	1.71445
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.76125e19	0.867320	0.433660	−	0.901077i	$-0.357222\pi$
$449$	6.76125e19	0.867320	0.433660	+	0.901077i	$0.357222\pi$
$450$	0	0
$451$	1.06197e19	0.131762
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	−1.50713e20	−1.58633	−0.793163	−	0.609009i	$-0.791567\pi$
$461$	−1.50713e20	−1.58633	−0.793163	+	0.609009i	$0.791567\pi$
$462$	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
$465$	0	0
$466$	0	0
$467$	−8.76843e19	−0.837616	−0.418808	−	0.908075i	$-0.637552\pi$
$467$	−8.76843e19	−0.837616	−0.418808	+	0.908075i	$0.637552\pi$
$468$	0	0
$469$	−2.84519e19	−0.263217
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.19903e20	1.04081
$474$	0	0
$475$	−2.14560e20	−1.80446
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.54724e19	−0.359114	−0.179557	−	0.983748i	$-0.557466\pi$
$479$	−4.54724e19	−0.359114	−0.179557	+	0.983748i	$0.557466\pi$
$480$	0	0
$481$	−6.97422e18	−0.0533836
$482$	0	0
$483$	0	0
$484$	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
$489$	0	0
$490$	0	0
$491$	−8.83155e19	−0.579330	−0.289665	−	0.957128i	$-0.593544\pi$
$491$	−8.83155e19	−0.579330	−0.289665	+	0.957128i	$0.593544\pi$
$492$	0	0
$493$	−3.38883e19	−0.215624
$494$	0	0
$495$	0	0
$496$	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
$501$	0	0
$502$	0	0
$503$	−7.83316e19	−0.428723	−0.214362	−	0.976754i	$-0.568767\pi$
$503$	−7.83316e19	−0.428723	−0.214362	+	0.976754i	$0.568767\pi$
$504$	0	0
$505$	−3.74961e20	−1.99205
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.00108e20	−0.501288	−0.250644	−	0.968079i	$-0.580642\pi$
$509$	−1.00108e20	−0.501288	−0.250644	+	0.968079i	$0.580642\pi$
$510$	0	0
$511$	4.10614e19	0.199654
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7.53995e19	−0.345791
$516$	0	0
$517$	−1.21712e20	−0.542195
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.49686e20	0.629361	0.314680	−	0.949198i	$-0.398103\pi$
$521$	1.49686e20	0.629361	0.314680	+	0.949198i	$0.398103\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	−2.91327e19	−0.112409
$528$	0	0
$529$	3.03657e20	1.13885
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.92827e19	−0.281012
$534$	0	0
$535$	2.96191e20	1.02075
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	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
$549$	0	0
$550$	0	0
$551$	1.15207e21	3.18303
$552$	0	0
$553$	−2.03501e20	−0.547178
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.47728e19	−0.0631046	−0.0315523	−	0.999502i	$-0.510045\pi$
$557$	−2.47728e19	−0.0631046	−0.0315523	+	0.999502i	$0.510045\pi$
$558$	0	0
$559$	−8.95145e20	−2.21975
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.09568e20	−0.962750	−0.481375	−	0.876515i	$-0.659862\pi$
$563$	−4.09568e20	−0.962750	−0.481375	+	0.876515i	$0.659862\pi$
$564$	0	0
$565$	−7.23510e20	−1.65608
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.29562e20	−0.715476	−0.357738	−	0.933822i	$-0.616452\pi$
$569$	−3.29562e20	−0.715476	−0.357738	+	0.933822i	$0.616452\pi$
$570$	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
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	4.16078e20	0.772421
$582$	0	0
$583$	1.51132e20	0.273427
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.89721e20	−0.497960	−0.248980	−	0.968509i	$-0.580095\pi$
$587$	−2.89721e20	−0.497960	−0.248980	+	0.968509i	$0.580095\pi$
$588$	0	0
$589$	9.90396e20	1.65937
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.21501e20	0.671256	0.335628	−	0.941995i	$-0.391052\pi$
$593$	4.21501e20	0.671256	0.335628	+	0.941995i	$0.391052\pi$
$594$	0	0
$595$	6.63342e19	0.103005
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.12785e20	−0.166553	−0.0832764	−	0.996526i	$-0.526538\pi$
$599$	−1.12785e20	−0.166553	−0.0832764	+	0.996526i	$0.526538\pi$
$600$	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
$603$	0	0
$604$	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
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	1.11031e21	1.31312	0.656559	−	0.754275i	$-0.272011\pi$
$617$	1.11031e21	1.31312	0.656559	+	0.754275i	$0.272011\pi$
$618$	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
$621$	0	0
$622$	0	0
$623$	6.81747e20	0.749830
$624$	0	0
$625$	−8.51846e20	−0.914663
$626$	0	0
$627$	0	0
$628$	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
$633$	0	0
$634$	0	0
$635$	2.65540e21	2.53120
$636$	0	0
$637$	9.24076e20	0.860324
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.12045e20	−0.543686	−0.271843	−	0.962342i	$-0.587633\pi$
$641$	−6.12045e20	−0.543686	−0.271843	+	0.962342i	$0.587633\pi$
$642$	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
$645$	0	0
$646$	0	0
$647$	−2.83600e20	−0.234922	−0.117461	−	0.993078i	$-0.537475\pi$
$647$	−2.83600e20	−0.234922	−0.117461	+	0.993078i	$0.537475\pi$
$648$	0	0
$649$	6.85134e20	0.554549
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.90607e21	−1.47330	−0.736648	−	0.676276i	$-0.763593\pi$
$653$	−1.90607e21	−1.47330	−0.736648	+	0.676276i	$0.763593\pi$
$654$	0	0
$655$	7.05118e20	0.532662
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.01338e21	1.45306	0.726532	−	0.687133i	$-0.241131\pi$
$659$	2.01338e21	1.45306	0.726532	+	0.687133i	$0.241131\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	−2.25510e21	−1.52056
$666$	0	0
$667$	4.22262e21	2.78380
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	−2.62800e19	−0.0154957	−0.00774785	−	0.999970i	$-0.502466\pi$
$677$	−2.62800e19	−0.0154957	−0.00774785	+	0.999970i	$0.502466\pi$
$678$	0	0
$679$	4.24840e20	0.245020
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.75876e21	0.970623	0.485311	−	0.874341i	$-0.338706\pi$
$683$	1.75876e21	0.970623	0.485311	+	0.874341i	$0.338706\pi$
$684$	0	0
$685$	−1.76303e21	−0.951876
$686$	0	0
$687$	0	0
$688$	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
$693$	0	0
$694$	0	0
$695$	−4.25068e21	−2.05860
$696$	0	0
$697$	4.70193e19	0.0222859
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.49002e21	1.58468	0.792342	−	0.610078i	$-0.208862\pi$
$701$	3.49002e21	1.58468	0.792342	+	0.610078i	$0.208862\pi$
$702$	0	0
$703$	−1.40612e20	−0.0624966
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	3.63005e21	1.45125
$714$	0	0
$715$	−3.52359e21	−1.37940
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.39878e21	1.65145	0.825726	−	0.564071i	$-0.190766\pi$
$719$	4.39878e21	1.65145	0.825726	+	0.564071i	$0.190766\pi$
$720$	0	0
$721$	−4.11308e20	−0.151235
$722$	0	0
$723$	0	0
$724$	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
$729$	0	0
$730$	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
$735$	0	0
$736$	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
$741$	0	0
$742$	0	0
$743$	1.92644e21	0.565378	0.282689	−	0.959212i	$-0.408774\pi$
$743$	1.92644e21	0.565378	0.282689	+	0.959212i	$0.408774\pi$
$744$	0	0
$745$	2.76782e21	0.796095
$746$	0	0
$747$	0	0
$748$	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
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	7.86875e20	0.192984	0.0964918	−	0.995334i	$-0.469238\pi$
$761$	7.86875e20	0.192984	0.0964918	+	0.995334i	$0.469238\pi$
$762$	0	0
$763$	−2.57544e21	−0.619323
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	−4.72043e21	−1.02952	−0.514761	−	0.857334i	$-0.672119\pi$
$773$	−4.72043e21	−1.02952	−0.514761	+	0.857334i	$0.672119\pi$
$774$	0	0
$775$	−5.00575e21	−1.07079
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.59847e21	−0.328982
$780$	0	0
$781$	3.11657e19	0.00629207
$782$	0	0
$783$	0	0
$784$	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
$789$	0	0
$790$	0	0
$791$	−3.94678e21	−0.724302
$792$	0	0
$793$	7.95270e21	1.43207
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.02993e22	1.78595	0.892975	−	0.450106i	$-0.148614\pi$
$797$	1.02993e22	1.78595	0.892975	+	0.450106i	$0.148614\pi$
$798$	0	0
$799$	−5.38886e20	−0.0917057
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.29354e21	0.212038
$804$	0	0
$805$	−8.26550e21	−1.32984
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.32458e21	−1.29047	−0.645236	−	0.763984i	$-0.723241\pi$
$809$	−8.32458e21	−1.29047	−0.645236	+	0.763984i	$0.723241\pi$
$810$	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
$813$	0	0
$814$	0	0
$815$	−1.17337e22	−1.72089
$816$	0	0
$817$	−1.80476e22	−2.59868
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.11082e20	0.0709442	0.0354721	−	0.999371i	$-0.488707\pi$
$821$	5.11082e20	0.0709442	0.0354721	+	0.999371i	$0.488707\pi$
$822$	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
$825$	0	0
$826$	0	0
$827$	−2.98458e20	−0.0392276	−0.0196138	−	0.999808i	$-0.506244\pi$
$827$	−2.98458e20	−0.0392276	−0.0196138	+	0.999808i	$0.506244\pi$
$828$	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
$831$	0	0
$832$	0	0
$833$	−5.48031e20	−0.0682288
$834$	0	0
$835$	8.15215e21	0.996835
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.01678e21	1.06374	0.531872	−	0.846825i	$-0.321489\pi$
$839$	9.01678e21	1.06374	0.531872	+	0.846825i	$0.321489\pi$
$840$	0	0
$841$	2.26363e22	2.62323
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.34125e22	1.49998
$846$	0	0
$847$	−3.16519e21	−0.347755
$848$	0	0
$849$	0	0
$850$	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
$855$	0	0
$856$	0	0
$857$	1.75411e22	1.76482	0.882412	−	0.470478i	$-0.155918\pi$
$857$	1.75411e22	1.76482	0.882412	+	0.470478i	$0.155918\pi$
$858$	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
$861$	0	0
$862$	0	0
$863$	3.02914e21	0.289226	0.144613	−	0.989488i	$-0.453806\pi$
$863$	3.02914e21	0.289226	0.144613	+	0.989488i	$0.453806\pi$
$864$	0	0
$865$	−6.70410e21	−0.629100
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6.41080e21	−0.581118
$870$	0	0
$871$	6.69146e21	0.596190
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	−1.07941e22	−0.882809	−0.441404	−	0.897308i	$-0.645520\pi$
$881$	−1.07941e22	−0.882809	−0.441404	+	0.897308i	$0.645520\pi$
$882$	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
$885$	0	0
$886$	0	0
$887$	−1.24547e22	−0.968069	−0.484034	−	0.875049i	$-0.660829\pi$
$887$	−1.24547e22	−0.968069	−0.484034	+	0.875049i	$0.660829\pi$
$888$	0	0
$889$	1.44853e22	1.10704
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.83200e22	1.35375
$894$	0	0
$895$	1.42829e22	1.03787
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.68780e22	1.88886
$900$	0	0
$901$	6.69142e20	0.0462469
$902$	0	0
$903$	0	0
$904$	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
$909$	0	0
$910$	0	0
$911$	−3.11450e22	−1.98153	−0.990765	−	0.135588i	$-0.956708\pi$
$911$	−3.11450e22	−1.98153	−0.990765	+	0.135588i	$0.956708\pi$
$912$	0	0
$913$	1.31075e22	0.820331
$914$	0	0
$915$	0	0
$916$	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
$921$	0	0
$922$	0	0
$923$	−2.32671e20	−0.0134193
$924$	0	0
$925$	7.10691e20	0.0403291
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.80710e22	0.992806	0.496403	−	0.868092i	$-0.334654\pi$
$929$	1.80710e22	0.992806	0.496403	+	0.868092i	$0.334654\pi$
$930$	0	0
$931$	1.86309e22	1.00719
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	4.01869e21	0.200522	0.100261	−	0.994961i	$-0.468032\pi$
$941$	4.01869e21	0.200522	0.100261	+	0.994961i	$0.468032\pi$
$942$	0	0
$943$	−5.85878e21	−0.287720
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.63291e22	1.25260	0.626298	−	0.779583i	$-0.284569\pi$
$947$	2.63291e22	1.25260	0.626298	+	0.779583i	$0.284569\pi$
$948$	0	0
$949$	−9.65705e21	−0.452218
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.44141e22	1.56149	0.780747	−	0.624847i	$-0.214839\pi$
$953$	3.44141e22	1.56149	0.780747	+	0.624847i	$0.214839\pi$
$954$	0	0
$955$	1.96848e22	0.879235
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.61741e21	−0.416312
$960$	0	0
$961$	−3.59075e20	−0.0153024
$962$	0	0
$963$	0	0
$964$	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
$969$	0	0
$970$	0	0
$971$	1.49909e22	0.591130	0.295565	−	0.955323i	$-0.404492\pi$
$971$	1.49909e22	0.591130	0.295565	+	0.955323i	$0.404492\pi$
$972$	0	0
$973$	−2.31877e22	−0.900349
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.01531e22	−1.13533	−0.567665	−	0.823259i	$-0.692153\pi$
$977$	−3.01531e22	−1.13533	−0.567665	+	0.823259i	$0.692153\pi$
$978$	0	0
$979$	2.14767e22	0.796340
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9.62494e21	−0.346136	−0.173068	−	0.984910i	$-0.555368\pi$
$983$	−9.62494e21	−0.346136	−0.173068	+	0.984910i	$0.555368\pi$
$984$	0	0
$985$	2.53025e21	0.0896173
$986$	0	0
$987$	0	0
$988$	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
$993$	0	0
$994$	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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.16.a.b.1.1	✓	1	3.2	odd	2
16.16.a.b.1.1		1	12.11	even	2
64.16.a.d.1.1		1	24.5	odd	2
64.16.a.h.1.1		1	24.11	even	2
72.16.a.c.1.1		1	1.1	even	1	trivial
144.16.a.m.1.1		1	4.3	odd	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

Introduction

L-functions

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Fields

Representations

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Database

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