LMFDB - Embedded newform 8.12.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,12,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, 12, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 12 $
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:	$6.14674544448$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.72015$ of defining polynomial
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)

$f(q)$	$=$	$q-640.180 q^{3} +11952.2 q^{5} +17464.5 q^{7} +232683. q^{9} +O(q^{10})$	$q-640.180 q^{3} +11952.2 q^{5} +17464.5 q^{7} +232683. q^{9} +542588. q^{11} -108196. q^{13} -7.65153e6 q^{15} +7.94782e6 q^{17} -7.86836e6 q^{19} -1.11804e7 q^{21} +3.44328e7 q^{23} +9.40259e7 q^{25} -3.55530e7 q^{27} -1.54208e8 q^{29} +5.14924e7 q^{31} -3.47354e8 q^{33} +2.08738e8 q^{35} +9.22788e7 q^{37} +6.92651e7 q^{39} +1.68236e8 q^{41} -2.39284e8 q^{43} +2.78106e9 q^{45} -6.90244e7 q^{47} -1.67232e9 q^{49} -5.08803e9 q^{51} -4.25291e9 q^{53} +6.48510e9 q^{55} +5.03716e9 q^{57} -5.88462e9 q^{59} +1.82883e9 q^{61} +4.06368e9 q^{63} -1.29318e9 q^{65} +2.15955e10 q^{67} -2.20432e10 q^{69} +1.55898e10 q^{71} +7.70566e9 q^{73} -6.01935e10 q^{75} +9.47601e9 q^{77} -1.66608e9 q^{79} -1.84588e10 q^{81} -4.50587e10 q^{83} +9.49936e10 q^{85} +9.87208e10 q^{87} -2.50490e10 q^{89} -1.88959e9 q^{91} -3.29644e10 q^{93} -9.40439e10 q^{95} +4.20468e10 q^{97} +1.26251e11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 56 q^{3} + 7868 q^{5} + 91056 q^{7} + 540202 q^{9}+O(q^{10}) $ 2 * q + 56 * q^3 + 7868 * q^5 + 91056 * q^7 + 540202 * q^9	$ 2 q + 56 q^{3} + 7868 q^{5} + 91056 q^{7} + 540202 q^{9} + 159080 q^{11} + 1050476 q^{13} - 10494832 q^{15} + 1430884 q^{17} - 21866600 q^{19} + 40052544 q^{21} + 35806736 q^{23} + 61878094 q^{25} + 55209392 q^{27} - 228827700 q^{29} + 64722112 q^{31} - 614344864 q^{33} - 91821408 q^{35} + 75558780 q^{37} + 875909072 q^{39} + 1201214196 q^{41} - 45519832 q^{43} + 1525107052 q^{45} - 1229079264 q^{47} + 1766069074 q^{49} - 9624987920 q^{51} - 3808549924 q^{53} + 8051409968 q^{55} - 4708125536 q^{57} - 6012926584 q^{59} + 9789792908 q^{61} + 26694483312 q^{63} - 6025376344 q^{65} + 14703095224 q^{67} - 21086664256 q^{69} + 4319991088 q^{71} + 11055639476 q^{73} - 82574103416 q^{75} - 18746968128 q^{77} + 51957623264 q^{79} - 9747960302 q^{81} - 108227975912 q^{83} + 121609729720 q^{85} + 46772044368 q^{87} + 71188291860 q^{89} + 83378892576 q^{91} - 23754095360 q^{93} - 36872875568 q^{95} - 1699807676 q^{97} + 8314917256 q^{99}+O(q^{100}) $ 2 * q + 56 * q^3 + 7868 * q^5 + 91056 * q^7 + 540202 * q^9 + 159080 * q^11 + 1050476 * q^13 - 10494832 * q^15 + 1430884 * q^17 - 21866600 * q^19 + 40052544 * q^21 + 35806736 * q^23 + 61878094 * q^25 + 55209392 * q^27 - 228827700 * q^29 + 64722112 * q^31 - 614344864 * q^33 - 91821408 * q^35 + 75558780 * q^37 + 875909072 * q^39 + 1201214196 * q^41 - 45519832 * q^43 + 1525107052 * q^45 - 1229079264 * q^47 + 1766069074 * q^49 - 9624987920 * q^51 - 3808549924 * q^53 + 8051409968 * q^55 - 4708125536 * q^57 - 6012926584 * q^59 + 9789792908 * q^61 + 26694483312 * q^63 - 6025376344 * q^65 + 14703095224 * q^67 - 21086664256 * q^69 + 4319991088 * q^71 + 11055639476 * q^73 - 82574103416 * q^75 - 18746968128 * q^77 + 51957623264 * q^79 - 9747960302 * q^81 - 108227975912 * q^83 + 121609729720 * q^85 + 46772044368 * q^87 + 71188291860 * q^89 + 83378892576 * q^91 - 23754095360 * q^93 - 36872875568 * q^95 - 1699807676 * q^97 + 8314917256 * q^99

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
$3$	−640.180	−1.52102	−0.760510	−	0.649326i	$-0.775051\pi$
$3$	−640.180	−1.52102	−0.760510	+	0.649326i	$0.775051\pi$
$4$	0	0
$5$	11952.2	1.71045	0.855227	−	0.518254i	$-0.173418\pi$
$5$	11952.2	1.71045	0.855227	+	0.518254i	$0.173418\pi$
$6$	0	0
$7$	17464.5	0.392750	0.196375	−	0.980529i	$-0.437083\pi$
$7$	17464.5	0.392750	0.196375	+	0.980529i	$0.437083\pi$
$8$	0	0
$9$	232683.	1.31350
$10$	0	0
$11$	542588.	1.01581	0.507903	−	0.861414i	$-0.330421\pi$
$11$	542588.	1.01581	0.507903	+	0.861414i	$0.330421\pi$
$12$	0	0
$13$	−108196.	−0.0808209	−0.0404105	−	0.999183i	$-0.512867\pi$
$13$	−108196.	−0.0808209	−0.0404105	+	0.999183i	$0.512867\pi$
$14$	0	0
$15$	−7.65153e6	−2.60163
$16$	0	0
$17$	7.94782e6	1.35762	0.678811	−	0.734313i	$-0.262496\pi$
$17$	7.94782e6	1.35762	0.678811	+	0.734313i	$0.262496\pi$
$18$	0	0
$19$	−7.86836e6	−0.729020	−0.364510	−	0.931199i	$-0.618763\pi$
$19$	−7.86836e6	−0.729020	−0.364510	+	0.931199i	$0.618763\pi$
$20$	0	0
$21$	−1.11804e7	−0.597380
$22$	0	0
$23$	3.44328e7	1.11550	0.557749	−	0.830009i	$-0.311665\pi$
$23$	3.44328e7	1.11550	0.557749	+	0.830009i	$0.311665\pi$
$24$	0	0
$25$	9.40259e7	1.92565
$26$	0	0
$27$	−3.55530e7	−0.476843
$28$	0	0
$29$	−1.54208e8	−1.39610	−0.698052	−	0.716047i	$-0.745950\pi$
$29$	−1.54208e8	−1.39610	−0.698052	+	0.716047i	$0.745950\pi$
$30$	0	0
$31$	5.14924e7	0.323038	0.161519	−	0.986870i	$-0.448361\pi$
$31$	5.14924e7	0.323038	0.161519	+	0.986870i	$0.448361\pi$
$32$	0	0
$33$	−3.47354e8	−1.54506
$34$	0	0
$35$	2.08738e8	0.671780
$36$	0	0
$37$	9.22788e7	0.218772	0.109386	−	0.993999i	$-0.465111\pi$
$37$	9.22788e7	0.218772	0.109386	+	0.993999i	$0.465111\pi$
$38$	0	0
$39$	6.92651e7	0.122930
$40$	0	0
$41$	1.68236e8	0.226782	0.113391	−	0.993550i	$-0.463829\pi$
$41$	1.68236e8	0.226782	0.113391	+	0.993550i	$0.463829\pi$
$42$	0	0
$43$	−2.39284e8	−0.248220	−0.124110	−	0.992268i	$-0.539608\pi$
$43$	−2.39284e8	−0.248220	−0.124110	+	0.992268i	$0.539608\pi$
$44$	0	0
$45$	2.78106e9	2.24668
$46$	0	0
$47$	−6.90244e7	−0.0439000	−0.0219500	−	0.999759i	$-0.506987\pi$
$47$	−6.90244e7	−0.0439000	−0.0219500	+	0.999759i	$0.506987\pi$
$48$	0	0
$49$	−1.67232e9	−0.845748
$50$	0	0
$51$	−5.08803e9	−2.06497
$52$	0	0
$53$	−4.25291e9	−1.39691	−0.698456	−	0.715653i	$-0.746129\pi$
$53$	−4.25291e9	−1.39691	−0.698456	+	0.715653i	$0.746129\pi$
$54$	0	0
$55$	6.48510e9	1.73749
$56$	0	0
$57$	5.03716e9	1.10885
$58$	0	0
$59$	−5.88462e9	−1.07160	−0.535800	−	0.844345i	$-0.679990\pi$
$59$	−5.88462e9	−1.07160	−0.535800	+	0.844345i	$0.679990\pi$
$60$	0	0
$61$	1.82883e9	0.277242	0.138621	−	0.990346i	$-0.455733\pi$
$61$	1.82883e9	0.277242	0.138621	+	0.990346i	$0.455733\pi$
$62$	0	0
$63$	4.06368e9	0.515878
$64$	0	0
$65$	−1.29318e9	−0.138240
$66$	0	0
$67$	2.15955e10	1.95412	0.977062	−	0.212953i	$-0.0683082\pi$
$67$	2.15955e10	1.95412	0.977062	+	0.212953i	$0.0683082\pi$
$68$	0	0
$69$	−2.20432e10	−1.69670
$70$	0	0
$71$	1.55898e10	1.02546	0.512732	−	0.858549i	$-0.328633\pi$
$71$	1.55898e10	1.02546	0.512732	+	0.858549i	$0.328633\pi$
$72$	0	0
$73$	7.70566e9	0.435045	0.217523	−	0.976055i	$-0.430202\pi$
$73$	7.70566e9	0.435045	0.217523	+	0.976055i	$0.430202\pi$
$74$	0	0
$75$	−6.01935e10	−2.92895
$76$	0	0
$77$	9.47601e9	0.398958
$78$	0	0
$79$	−1.66608e9	−0.0609182	−0.0304591	−	0.999536i	$-0.509697\pi$
$79$	−1.66608e9	−0.0609182	−0.0304591	+	0.999536i	$0.509697\pi$
$80$	0	0
$81$	−1.84588e10	−0.588214
$82$	0	0
$83$	−4.50587e10	−1.25559	−0.627796	−	0.778378i	$-0.716043\pi$
$83$	−4.50587e10	−1.25559	−0.627796	+	0.778378i	$0.716043\pi$
$84$	0	0
$85$	9.49936e10	2.32215
$86$	0	0
$87$	9.87208e10	2.12350
$88$	0	0
$89$	−2.50490e10	−0.475495	−0.237748	−	0.971327i	$-0.576409\pi$
$89$	−2.50490e10	−0.475495	−0.237748	+	0.971327i	$0.576409\pi$
$90$	0	0
$91$	−1.88959e9	−0.0317424
$92$	0	0
$93$	−3.29644e10	−0.491347
$94$	0	0
$95$	−9.40439e10	−1.24695
$96$	0	0
$97$	4.20468e10	0.497151	0.248576	−	0.968612i	$-0.420038\pi$
$97$	4.20468e10	0.497151	0.248576	+	0.968612i	$0.420038\pi$
$98$	0	0
$99$	1.26251e11	1.33426
$100$	0	0
$101$	−4.42685e10	−0.419109	−0.209554	−	0.977797i	$-0.567201\pi$
$101$	−4.42685e10	−0.419109	−0.209554	+	0.977797i	$0.567201\pi$
$102$	0	0
$103$	−6.88268e9	−0.0584996	−0.0292498	−	0.999572i	$-0.509312\pi$
$103$	−6.88268e9	−0.0584996	−0.0292498	+	0.999572i	$0.509312\pi$
$104$	0	0
$105$	−1.33630e11	−1.02179
$106$	0	0
$107$	−1.12565e11	−0.775878	−0.387939	−	0.921685i	$-0.626813\pi$
$107$	−1.12565e11	−0.775878	−0.387939	+	0.921685i	$0.626813\pi$
$108$	0	0
$109$	−2.47188e11	−1.53880	−0.769399	−	0.638768i	$-0.779444\pi$
$109$	−2.47188e11	−1.53880	−0.769399	+	0.638768i	$0.779444\pi$
$110$	0	0
$111$	−5.90750e10	−0.332757
$112$	0	0
$113$	3.05173e11	1.55817	0.779084	−	0.626920i	$-0.215685\pi$
$113$	3.05173e11	1.55817	0.779084	+	0.626920i	$0.215685\pi$
$114$	0	0
$115$	4.11546e11	1.90801
$116$	0	0
$117$	−2.51754e10	−0.106158
$118$	0	0
$119$	1.38804e11	0.533206
$120$	0	0
$121$	9.09058e9	0.0318619
$122$	0	0
$123$	−1.07701e11	−0.344939
$124$	0	0
$125$	5.40211e11	1.58328
$126$	0	0
$127$	−6.59571e11	−1.77150	−0.885750	−	0.464163i	$-0.846355\pi$
$127$	−6.59571e11	−1.77150	−0.885750	+	0.464163i	$0.846355\pi$
$128$	0	0
$129$	1.53185e11	0.377548
$130$	0	0
$131$	−4.08450e11	−0.925011	−0.462505	−	0.886616i	$-0.653049\pi$
$131$	−4.08450e11	−0.925011	−0.462505	+	0.886616i	$0.653049\pi$
$132$	0	0
$133$	−1.37417e11	−0.286322
$134$	0	0
$135$	−4.24935e11	−0.815617
$136$	0	0
$137$	−3.28605e11	−0.581717	−0.290858	−	0.956766i	$-0.593941\pi$
$137$	−3.28605e11	−0.581717	−0.290858	+	0.956766i	$0.593941\pi$
$138$	0	0
$139$	1.06744e12	1.74487	0.872435	−	0.488730i	$-0.162540\pi$
$139$	1.06744e12	1.74487	0.872435	+	0.488730i	$0.162540\pi$
$140$	0	0
$141$	4.41880e10	0.0667728
$142$	0	0
$143$	−5.87061e10	−0.0820984
$144$	0	0
$145$	−1.84312e12	−2.38797
$146$	0	0
$147$	1.07058e12	1.28640
$148$	0	0
$149$	5.92629e11	0.661087	0.330543	−	0.943791i	$-0.392768\pi$
$149$	5.92629e11	0.661087	0.330543	+	0.943791i	$0.392768\pi$
$150$	0	0
$151$	−1.16984e12	−1.21270	−0.606352	−	0.795196i	$-0.707368\pi$
$151$	−1.16984e12	−1.21270	−0.606352	+	0.795196i	$0.707368\pi$
$152$	0	0
$153$	1.84932e12	1.78324
$154$	0	0
$155$	6.15445e11	0.552541
$156$	0	0
$157$	3.68743e11	0.308514	0.154257	−	0.988031i	$-0.450702\pi$
$157$	3.68743e11	0.308514	0.154257	+	0.988031i	$0.450702\pi$
$158$	0	0
$159$	2.72263e12	2.12473
$160$	0	0
$161$	6.01350e11	0.438112
$162$	0	0
$163$	−1.91683e12	−1.30483	−0.652413	−	0.757864i	$-0.726243\pi$
$163$	−1.91683e12	−1.30483	−0.652413	+	0.757864i	$0.726243\pi$
$164$	0	0
$165$	−4.15163e12	−2.64276
$166$	0	0
$167$	−2.84469e11	−0.169471	−0.0847354	−	0.996403i	$-0.527005\pi$
$167$	−2.84469e11	−0.169471	−0.0847354	+	0.996403i	$0.527005\pi$
$168$	0	0
$169$	−1.78045e12	−0.993468
$170$	0	0
$171$	−1.83083e12	−0.957569
$172$	0	0
$173$	2.16256e12	1.06100	0.530499	−	0.847686i	$-0.322005\pi$
$173$	2.16256e12	1.06100	0.530499	+	0.847686i	$0.322005\pi$
$174$	0	0
$175$	1.64211e12	0.756299
$176$	0	0
$177$	3.76722e12	1.62992
$178$	0	0
$179$	−4.61641e12	−1.87764	−0.938822	−	0.344404i	$-0.888081\pi$
$179$	−4.61641e12	−1.87764	−0.938822	+	0.344404i	$0.888081\pi$
$180$	0	0
$181$	−5.43222e10	−0.0207848	−0.0103924	−	0.999946i	$-0.503308\pi$
$181$	−5.43222e10	−0.0207848	−0.0103924	+	0.999946i	$0.503308\pi$
$182$	0	0
$183$	−1.17078e12	−0.421690
$184$	0	0
$185$	1.10293e12	0.374200
$186$	0	0
$187$	4.31239e12	1.37908
$188$	0	0
$189$	−6.20913e11	−0.187280
$190$	0	0
$191$	−2.32891e12	−0.662933	−0.331466	−	0.943467i	$-0.607543\pi$
$191$	−2.32891e12	−0.662933	−0.331466	+	0.943467i	$0.607543\pi$
$192$	0	0
$193$	2.49475e12	0.670597	0.335299	−	0.942112i	$-0.391163\pi$
$193$	2.49475e12	0.670597	0.335299	+	0.942112i	$0.391163\pi$
$194$	0	0
$195$	8.27867e11	0.210266
$196$	0	0
$197$	6.13716e12	1.47368	0.736839	−	0.676068i	$-0.236317\pi$
$197$	6.13716e12	1.47368	0.736839	+	0.676068i	$0.236317\pi$
$198$	0	0
$199$	7.12259e12	1.61788	0.808940	−	0.587892i	$-0.200042\pi$
$199$	7.12259e12	1.61788	0.808940	+	0.587892i	$0.200042\pi$
$200$	0	0
$201$	−1.38250e13	−2.97226
$202$	0	0
$203$	−2.69316e12	−0.548319
$204$	0	0
$205$	2.01078e12	0.387899
$206$	0	0
$207$	8.01192e12	1.46521
$208$	0	0
$209$	−4.26928e12	−0.740543
$210$	0	0
$211$	−7.64441e12	−1.25832	−0.629159	−	0.777277i	$-0.716600\pi$
$211$	−7.64441e12	−1.25832	−0.629159	+	0.777277i	$0.716600\pi$
$212$	0	0
$213$	−9.98029e12	−1.55975
$214$	0	0
$215$	−2.85996e12	−0.424569
$216$	0	0
$217$	8.99286e11	0.126873
$218$	0	0
$219$	−4.93301e12	−0.661712
$220$	0	0
$221$	−8.59924e11	−0.109724
$222$	0	0
$223$	2.56098e12	0.310978	0.155489	−	0.987838i	$-0.450305\pi$
$223$	2.56098e12	0.310978	0.155489	+	0.987838i	$0.450305\pi$
$224$	0	0
$225$	2.18782e13	2.52935
$226$	0	0
$227$	1.15359e13	1.27031	0.635156	−	0.772384i	$-0.280936\pi$
$227$	1.15359e13	1.27031	0.635156	+	0.772384i	$0.280936\pi$
$228$	0	0
$229$	8.77117e12	0.920370	0.460185	−	0.887823i	$-0.347783\pi$
$229$	8.77117e12	0.920370	0.460185	+	0.887823i	$0.347783\pi$
$230$	0	0
$231$	−6.06635e12	−0.606822
$232$	0	0
$233$	−9.99552e12	−0.953560	−0.476780	−	0.879023i	$-0.658196\pi$
$233$	−9.99552e12	−0.953560	−0.476780	+	0.879023i	$0.658196\pi$
$234$	0	0
$235$	−8.24991e11	−0.0750889
$236$	0	0
$237$	1.06659e12	0.0926578
$238$	0	0
$239$	−9.88633e12	−0.820062	−0.410031	−	0.912072i	$-0.634482\pi$
$239$	−9.88633e12	−0.820062	−0.410031	+	0.912072i	$0.634482\pi$
$240$	0	0
$241$	1.46080e12	0.115744	0.0578718	−	0.998324i	$-0.481569\pi$
$241$	1.46080e12	0.115744	0.0578718	+	0.998324i	$0.481569\pi$
$242$	0	0
$243$	1.81150e13	1.37153
$244$	0	0
$245$	−1.99878e13	−1.44661
$246$	0	0
$247$	8.51327e11	0.0589201
$248$	0	0
$249$	2.88456e13	1.90978
$250$	0	0
$251$	−5.71128e12	−0.361850	−0.180925	−	0.983497i	$-0.557909\pi$
$251$	−5.71128e12	−0.361850	−0.180925	+	0.983497i	$0.557909\pi$
$252$	0	0
$253$	1.86828e13	1.13313
$254$	0	0
$255$	−6.08129e13	−3.53204
$256$	0	0
$257$	−3.00608e12	−0.167251	−0.0836254	−	0.996497i	$-0.526650\pi$
$257$	−3.00608e12	−0.167251	−0.0836254	+	0.996497i	$0.526650\pi$
$258$	0	0
$259$	1.61160e12	0.0859228
$260$	0	0
$261$	−3.58816e13	−1.83379
$262$	0	0
$263$	2.94339e13	1.44242	0.721210	−	0.692717i	$-0.243586\pi$
$263$	2.94339e13	1.44242	0.721210	+	0.692717i	$0.243586\pi$
$264$	0	0
$265$	−5.08315e13	−2.38935
$266$	0	0
$267$	1.60359e13	0.723238
$268$	0	0
$269$	4.05862e13	1.75688	0.878438	−	0.477857i	$-0.158586\pi$
$269$	4.05862e13	1.75688	0.878438	+	0.477857i	$0.158586\pi$
$270$	0	0
$271$	2.32350e13	0.965633	0.482817	−	0.875721i	$-0.339614\pi$
$271$	2.32350e13	0.965633	0.482817	+	0.875721i	$0.339614\pi$
$272$	0	0
$273$	1.20968e12	0.0482808
$274$	0	0
$275$	5.10174e13	1.95609
$276$	0	0
$277$	−1.78412e13	−0.657331	−0.328666	−	0.944446i	$-0.606599\pi$
$277$	−1.78412e13	−0.657331	−0.328666	+	0.944446i	$0.606599\pi$
$278$	0	0
$279$	1.19814e13	0.424311
$280$	0	0
$281$	−1.97996e13	−0.674174	−0.337087	−	0.941473i	$-0.609442\pi$
$281$	−1.97996e13	−0.674174	−0.337087	+	0.941473i	$0.609442\pi$
$282$	0	0
$283$	−2.71536e13	−0.889207	−0.444603	−	0.895728i	$-0.646655\pi$
$283$	−2.71536e13	−0.889207	−0.444603	+	0.895728i	$0.646655\pi$
$284$	0	0
$285$	6.02050e13	1.89664
$286$	0	0
$287$	2.93815e12	0.0890684
$288$	0	0
$289$	2.88959e13	0.843137
$290$	0	0
$291$	−2.69175e13	−0.756177
$292$	0	0
$293$	−2.98350e13	−0.807150	−0.403575	−	0.914947i	$-0.632232\pi$
$293$	−2.98350e13	−0.807150	−0.403575	+	0.914947i	$0.632232\pi$
$294$	0	0
$295$	−7.03339e13	−1.83292
$296$	0	0
$297$	−1.92906e13	−0.484380
$298$	0	0
$299$	−3.72550e12	−0.0901557
$300$	0	0
$301$	−4.17897e12	−0.0974885
$302$	0	0
$303$	2.83398e13	0.637473
$304$	0	0
$305$	2.18584e13	0.474209
$306$	0	0
$307$	−2.24780e13	−0.470432	−0.235216	−	0.971943i	$-0.575580\pi$
$307$	−2.24780e13	−0.470432	−0.235216	+	0.971943i	$0.575580\pi$
$308$	0	0
$309$	4.40615e12	0.0889790
$310$	0	0
$311$	−2.13685e13	−0.416477	−0.208239	−	0.978078i	$-0.566773\pi$
$311$	−2.13685e13	−0.416477	−0.208239	+	0.978078i	$0.566773\pi$
$312$	0	0
$313$	2.43422e13	0.458000	0.229000	−	0.973426i	$-0.426454\pi$
$313$	2.43422e13	0.458000	0.229000	+	0.973426i	$0.426454\pi$
$314$	0	0
$315$	4.85697e13	0.882384
$316$	0	0
$317$	−6.88050e13	−1.20724	−0.603620	−	0.797272i	$-0.706276\pi$
$317$	−6.88050e13	−1.20724	−0.603620	+	0.797272i	$0.706276\pi$
$318$	0	0
$319$	−8.36715e13	−1.41817
$320$	0	0
$321$	7.20620e13	1.18013
$322$	0	0
$323$	−6.25363e13	−0.989734
$324$	0	0
$325$	−1.01733e13	−0.155633
$326$	0	0
$327$	1.58245e14	2.34054
$328$	0	0
$329$	−1.20547e12	−0.0172417
$330$	0	0
$331$	1.84782e13	0.255626	0.127813	−	0.991798i	$-0.459204\pi$
$331$	1.84782e13	0.255626	0.127813	+	0.991798i	$0.459204\pi$
$332$	0	0
$333$	2.14717e13	0.287358
$334$	0	0
$335$	2.58113e14	3.34244
$336$	0	0
$337$	1.00646e14	1.26134	0.630669	−	0.776052i	$-0.282781\pi$
$337$	1.00646e14	1.26134	0.630669	+	0.776052i	$0.282781\pi$
$338$	0	0
$339$	−1.95365e14	−2.37000
$340$	0	0
$341$	2.79392e13	0.328144
$342$	0	0
$343$	−6.37391e13	−0.724917
$344$	0	0
$345$	−2.63463e14	−2.90212
$346$	0	0
$347$	−8.02297e13	−0.856097	−0.428049	−	0.903756i	$-0.640799\pi$
$347$	−8.02297e13	−0.856097	−0.428049	+	0.903756i	$0.640799\pi$
$348$	0	0
$349$	−1.83187e13	−0.189389	−0.0946947	−	0.995506i	$-0.530187\pi$
$349$	−1.83187e13	−0.189389	−0.0946947	+	0.995506i	$0.530187\pi$
$350$	0	0
$351$	3.84670e12	0.0385389
$352$	0	0
$353$	−4.56680e13	−0.443456	−0.221728	−	0.975109i	$-0.571170\pi$
$353$	−4.56680e13	−0.443456	−0.221728	+	0.975109i	$0.571170\pi$
$354$	0	0
$355$	1.86332e14	1.75401
$356$	0	0
$357$	−8.88597e13	−0.811016
$358$	0	0
$359$	−4.60991e13	−0.408012	−0.204006	−	0.978970i	$-0.565396\pi$
$359$	−4.60991e13	−0.408012	−0.204006	+	0.978970i	$0.565396\pi$
$360$	0	0
$361$	−5.45792e13	−0.468530
$362$	0	0
$363$	−5.81961e12	−0.0484626
$364$	0	0
$365$	9.20993e13	0.744124
$366$	0	0
$367$	1.36650e14	1.07139	0.535693	−	0.844413i	$-0.320050\pi$
$367$	1.36650e14	1.07139	0.535693	+	0.844413i	$0.320050\pi$
$368$	0	0
$369$	3.91457e13	0.297878
$370$	0	0
$371$	−7.42748e13	−0.548637
$372$	0	0
$373$	9.26839e13	0.664669	0.332335	−	0.943162i	$-0.392164\pi$
$373$	9.26839e13	0.664669	0.332335	+	0.943162i	$0.392164\pi$
$374$	0	0
$375$	−3.45832e14	−2.40820
$376$	0	0
$377$	1.66847e13	0.112834
$378$	0	0
$379$	1.58566e14	1.04158	0.520791	−	0.853684i	$-0.325637\pi$
$379$	1.58566e14	1.04158	0.520791	+	0.853684i	$0.325637\pi$
$380$	0	0
$381$	4.22244e14	2.69449
$382$	0	0
$383$	−2.03453e14	−1.26145	−0.630727	−	0.776005i	$-0.717243\pi$
$383$	−2.03453e14	−1.26145	−0.630727	+	0.776005i	$0.717243\pi$
$384$	0	0
$385$	1.13259e14	0.682398
$386$	0	0
$387$	−5.56773e13	−0.326038
$388$	0	0
$389$	−1.92046e14	−1.09315	−0.546577	−	0.837409i	$-0.684069\pi$
$389$	−1.92046e14	−1.09315	−0.546577	+	0.837409i	$0.684069\pi$
$390$	0	0
$391$	2.73666e14	1.51443
$392$	0	0
$393$	2.61481e14	1.40696
$394$	0	0
$395$	−1.99133e13	−0.104198
$396$	0	0
$397$	−3.51854e14	−1.79067	−0.895334	−	0.445396i	$-0.853063\pi$
$397$	−3.51854e14	−1.79067	−0.895334	+	0.445396i	$0.853063\pi$
$398$	0	0
$399$	8.79713e13	0.435502
$400$	0	0
$401$	−1.72882e13	−0.0832637	−0.0416318	−	0.999133i	$-0.513256\pi$
$401$	−1.72882e13	−0.0832637	−0.0416318	+	0.999133i	$0.513256\pi$
$402$	0	0
$403$	−5.57128e12	−0.0261082
$404$	0	0
$405$	−2.20622e14	−1.00611
$406$	0	0
$407$	5.00694e13	0.222230
$408$	0	0
$409$	−4.41360e13	−0.190684	−0.0953422	−	0.995445i	$-0.530395\pi$
$409$	−4.41360e13	−0.190684	−0.0953422	+	0.995445i	$0.530395\pi$
$410$	0	0
$411$	2.10367e14	0.884803
$412$	0	0
$413$	−1.02772e14	−0.420870
$414$	0	0
$415$	−5.38548e14	−2.14763
$416$	0	0
$417$	−6.83355e14	−2.65398
$418$	0	0
$419$	−1.09214e14	−0.413144	−0.206572	−	0.978431i	$-0.566231\pi$
$419$	−1.09214e14	−0.413144	−0.206572	+	0.978431i	$0.566231\pi$
$420$	0	0
$421$	2.33050e14	0.858811	0.429406	−	0.903112i	$-0.358723\pi$
$421$	2.33050e14	0.858811	0.429406	+	0.903112i	$0.358723\pi$
$422$	0	0
$423$	−1.60608e13	−0.0576627
$424$	0	0
$425$	7.47301e14	2.61431
$426$	0	0
$427$	3.19395e13	0.108887
$428$	0	0
$429$	3.75824e13	0.124873
$430$	0	0
$431$	9.80730e13	0.317632	0.158816	−	0.987308i	$-0.449232\pi$
$431$	9.80730e13	0.317632	0.158816	+	0.987308i	$0.449232\pi$
$432$	0	0
$433$	1.44428e14	0.456003	0.228001	−	0.973661i	$-0.426781\pi$
$433$	1.44428e14	0.456003	0.228001	+	0.973661i	$0.426781\pi$
$434$	0	0
$435$	1.17993e15	3.63215
$436$	0	0
$437$	−2.70930e14	−0.813221
$438$	0	0
$439$	1.16021e14	0.339610	0.169805	−	0.985478i	$-0.445686\pi$
$439$	1.16021e14	0.339610	0.169805	+	0.985478i	$0.445686\pi$
$440$	0	0
$441$	−3.89120e14	−1.11089
$442$	0	0
$443$	−3.51848e14	−0.979795	−0.489897	−	0.871780i	$-0.662966\pi$
$443$	−3.51848e14	−0.979795	−0.489897	+	0.871780i	$0.662966\pi$
$444$	0	0
$445$	−2.99390e14	−0.813312
$446$	0	0
$447$	−3.79389e14	−1.00553
$448$	0	0
$449$	1.46433e14	0.378690	0.189345	−	0.981911i	$-0.439364\pi$
$449$	1.46433e14	0.378690	0.189345	+	0.981911i	$0.439364\pi$
$450$	0	0
$451$	9.12830e13	0.230366
$452$	0	0
$453$	7.48910e14	1.84455
$454$	0	0
$455$	−2.25847e13	−0.0542939
$456$	0	0
$457$	−3.55422e14	−0.834076	−0.417038	−	0.908889i	$-0.636932\pi$
$457$	−3.55422e14	−0.834076	−0.417038	+	0.908889i	$0.636932\pi$
$458$	0	0
$459$	−2.82569e14	−0.647372
$460$	0	0
$461$	8.46403e14	1.89331	0.946655	−	0.322248i	$-0.104438\pi$
$461$	8.46403e14	1.89331	0.946655	+	0.322248i	$0.104438\pi$
$462$	0	0
$463$	−1.75496e14	−0.383330	−0.191665	−	0.981460i	$-0.561389\pi$
$463$	−1.75496e14	−0.383330	−0.191665	+	0.981460i	$0.561389\pi$
$464$	0	0
$465$	−3.93995e14	−0.840427
$466$	0	0
$467$	−7.53411e13	−0.156960	−0.0784800	−	0.996916i	$-0.525007\pi$
$467$	−7.53411e13	−0.156960	−0.0784800	+	0.996916i	$0.525007\pi$
$468$	0	0
$469$	3.77154e14	0.767482
$470$	0	0
$471$	−2.36062e14	−0.469257
$472$	0	0
$473$	−1.29833e14	−0.252144
$474$	0	0
$475$	−7.39830e14	−1.40384
$476$	0	0
$477$	−9.89580e14	−1.83485
$478$	0	0
$479$	−4.09390e14	−0.741809	−0.370905	−	0.928671i	$-0.620952\pi$
$479$	−4.09390e14	−0.741809	−0.370905	+	0.928671i	$0.620952\pi$
$480$	0	0
$481$	−9.98422e12	−0.0176814
$482$	0	0
$483$	−3.84972e14	−0.666377
$484$	0	0
$485$	5.02550e14	0.850354
$486$	0	0
$487$	8.99264e14	1.48757	0.743786	−	0.668418i	$-0.233028\pi$
$487$	8.99264e14	1.48757	0.743786	+	0.668418i	$0.233028\pi$
$488$	0	0
$489$	1.22712e15	1.98467
$490$	0	0
$491$	−1.50544e14	−0.238075	−0.119038	−	0.992890i	$-0.537981\pi$
$491$	−1.50544e14	−0.238075	−0.119038	+	0.992890i	$0.537981\pi$
$492$	0	0
$493$	−1.22562e15	−1.89538
$494$	0	0
$495$	1.50897e15	2.28219
$496$	0	0
$497$	2.72268e14	0.402751
$498$	0	0
$499$	−7.02134e14	−1.01594	−0.507969	−	0.861375i	$-0.669604\pi$
$499$	−7.02134e14	−1.01594	−0.507969	+	0.861375i	$0.669604\pi$
$500$	0	0
$501$	1.82112e14	0.257769
$502$	0	0
$503$	1.24917e15	1.72981	0.864903	−	0.501939i	$-0.167380\pi$
$503$	1.24917e15	1.72981	0.864903	+	0.501939i	$0.167380\pi$
$504$	0	0
$505$	−5.29104e14	−0.716866
$506$	0	0
$507$	1.13981e15	1.51108
$508$	0	0
$509$	−2.67254e14	−0.346718	−0.173359	−	0.984859i	$-0.555462\pi$
$509$	−2.67254e14	−0.346718	−0.173359	+	0.984859i	$0.555462\pi$
$510$	0	0
$511$	1.34575e14	0.170864
$512$	0	0
$513$	2.79744e14	0.347628
$514$	0	0
$515$	−8.22628e13	−0.100061
$516$	0	0
$517$	−3.74519e13	−0.0445939
$518$	0	0
$519$	−1.38443e15	−1.61380
$520$	0	0
$521$	−9.63788e14	−1.09995	−0.549976	−	0.835180i	$-0.685363\pi$
$521$	−9.63788e14	−1.09995	−0.549976	+	0.835180i	$0.685363\pi$
$522$	0	0
$523$	4.44881e14	0.497146	0.248573	−	0.968613i	$-0.420038\pi$
$523$	4.44881e14	0.497146	0.248573	+	0.968613i	$0.420038\pi$
$524$	0	0
$525$	−1.05125e15	−1.15035
$526$	0	0
$527$	4.09252e14	0.438563
$528$	0	0
$529$	2.32808e14	0.244338
$530$	0	0
$531$	−1.36925e15	−1.40755
$532$	0	0
$533$	−1.82025e13	−0.0183287
$534$	0	0
$535$	−1.34540e15	−1.32710
$536$	0	0
$537$	2.95533e15	2.85593
$538$	0	0
$539$	−9.07381e14	−0.859116
$540$	0	0
$541$	9.18626e14	0.852224	0.426112	−	0.904671i	$-0.359883\pi$
$541$	9.18626e14	0.852224	0.426112	+	0.904671i	$0.359883\pi$
$542$	0	0
$543$	3.47760e13	0.0316140
$544$	0	0
$545$	−2.95443e15	−2.63204
$546$	0	0
$547$	1.74722e15	1.52552	0.762760	−	0.646682i	$-0.223844\pi$
$547$	1.74722e15	1.52552	0.762760	+	0.646682i	$0.223844\pi$
$548$	0	0
$549$	4.25537e14	0.364157
$550$	0	0
$551$	1.21336e15	1.01779
$552$	0	0
$553$	−2.90972e13	−0.0239256
$554$	0	0
$555$	−7.06074e14	−0.569166
$556$	0	0
$557$	−7.28921e14	−0.576072	−0.288036	−	0.957620i	$-0.593002\pi$
$557$	−7.28921e14	−0.576072	−0.288036	+	0.957620i	$0.593002\pi$
$558$	0	0
$559$	2.58897e13	0.0200614
$560$	0	0
$561$	−2.76071e15	−2.09761
$562$	0	0
$563$	−1.77150e15	−1.31991	−0.659956	−	0.751304i	$-0.729425\pi$
$563$	−1.77150e15	−1.31991	−0.659956	+	0.751304i	$0.729425\pi$
$564$	0	0
$565$	3.64747e15	2.66517
$566$	0	0
$567$	−3.22373e14	−0.231021
$568$	0	0
$569$	2.73759e15	1.92420	0.962101	−	0.272694i	$-0.0879146\pi$
$569$	2.73759e15	1.92420	0.962101	+	0.272694i	$0.0879146\pi$
$570$	0	0
$571$	1.04213e15	0.718491	0.359245	−	0.933243i	$-0.383034\pi$
$571$	1.04213e15	0.718491	0.359245	+	0.933243i	$0.383034\pi$
$572$	0	0
$573$	1.49092e15	1.00833
$574$	0	0
$575$	3.23757e15	2.14806
$576$	0	0
$577$	1.89448e15	1.23317	0.616585	−	0.787288i	$-0.288516\pi$
$577$	1.89448e15	1.23317	0.616585	+	0.787288i	$0.288516\pi$
$578$	0	0
$579$	−1.59709e15	−1.01999
$580$	0	0
$581$	−7.86925e14	−0.493134
$582$	0	0
$583$	−2.30758e15	−1.41899
$584$	0	0
$585$	−3.00901e14	−0.181579
$586$	0	0
$587$	1.08643e15	0.643419	0.321709	−	0.946838i	$-0.395743\pi$
$587$	1.08643e15	0.643419	0.321709	+	0.946838i	$0.395743\pi$
$588$	0	0
$589$	−4.05161e14	−0.235501
$590$	0	0
$591$	−3.92888e15	−2.24150
$592$	0	0
$593$	−2.51449e15	−1.40815	−0.704075	−	0.710126i	$-0.748638\pi$
$593$	−2.51449e15	−1.40815	−0.704075	+	0.710126i	$0.748638\pi$
$594$	0	0
$595$	1.65901e15	0.912023
$596$	0	0
$597$	−4.55974e15	−2.46083
$598$	0	0
$599$	3.28321e14	0.173961	0.0869803	−	0.996210i	$-0.472278\pi$
$599$	3.28321e14	0.173961	0.0869803	+	0.996210i	$0.472278\pi$
$600$	0	0
$601$	−2.14854e15	−1.11772	−0.558861	−	0.829261i	$-0.688762\pi$
$601$	−2.14854e15	−1.11772	−0.558861	+	0.829261i	$0.688762\pi$
$602$	0	0
$603$	5.02491e15	2.56675
$604$	0	0
$605$	1.08652e14	0.0544983
$606$	0	0
$607$	−8.40427e14	−0.413964	−0.206982	−	0.978345i	$-0.566364\pi$
$607$	−8.40427e14	−0.413964	−0.206982	+	0.978345i	$0.566364\pi$
$608$	0	0
$609$	1.72410e15	0.834005
$610$	0	0
$611$	7.46819e12	0.00354804
$612$	0	0
$613$	−9.20771e14	−0.429654	−0.214827	−	0.976652i	$-0.568919\pi$
$613$	−9.20771e14	−0.429654	−0.214827	+	0.976652i	$0.568919\pi$
$614$	0	0
$615$	−1.28726e15	−0.590003
$616$	0	0
$617$	−8.78027e14	−0.395312	−0.197656	−	0.980271i	$-0.563333\pi$
$617$	−8.78027e14	−0.395312	−0.197656	+	0.980271i	$0.563333\pi$
$618$	0	0
$619$	2.73116e15	1.20795	0.603975	−	0.797003i	$-0.293583\pi$
$619$	2.73116e15	1.20795	0.603975	+	0.797003i	$0.293583\pi$
$620$	0	0
$621$	−1.22419e15	−0.531918
$622$	0	0
$623$	−4.37468e14	−0.186751
$624$	0	0
$625$	1.86557e15	0.782479
$626$	0	0
$627$	2.73311e15	1.12638
$628$	0	0
$629$	7.33415e14	0.297010
$630$	0	0
$631$	−2.11332e15	−0.841016	−0.420508	−	0.907289i	$-0.638148\pi$
$631$	−2.11332e15	−0.841016	−0.420508	+	0.907289i	$0.638148\pi$
$632$	0	0
$633$	4.89379e15	1.91393
$634$	0	0
$635$	−7.88329e15	−3.03007
$636$	0	0
$637$	1.80939e14	0.0683541
$638$	0	0
$639$	3.62749e15	1.34695
$640$	0	0
$641$	1.52374e15	0.556149	0.278074	−	0.960560i	$-0.410304\pi$
$641$	1.52374e15	0.556149	0.278074	+	0.960560i	$0.410304\pi$
$642$	0	0
$643$	1.62810e15	0.584145	0.292073	−	0.956396i	$-0.405655\pi$
$643$	1.62810e15	0.584145	0.292073	+	0.956396i	$0.405655\pi$
$644$	0	0
$645$	1.83089e15	0.645779
$646$	0	0
$647$	2.46240e15	0.853856	0.426928	−	0.904286i	$-0.359596\pi$
$647$	2.46240e15	0.853856	0.426928	+	0.904286i	$0.359596\pi$
$648$	0	0
$649$	−3.19293e15	−1.08854
$650$	0	0
$651$	−5.75705e14	−0.192976
$652$	0	0
$653$	−5.05508e14	−0.166612	−0.0833059	−	0.996524i	$-0.526548\pi$
$653$	−5.05508e14	−0.166612	−0.0833059	+	0.996524i	$0.526548\pi$
$654$	0	0
$655$	−4.88186e15	−1.58219
$656$	0	0
$657$	1.79298e15	0.571433
$658$	0	0
$659$	3.21458e15	1.00752	0.503761	−	0.863843i	$-0.331949\pi$
$659$	3.21458e15	1.00752	0.503761	+	0.863843i	$0.331949\pi$
$660$	0	0
$661$	4.00382e15	1.23414	0.617072	−	0.786906i	$-0.288319\pi$
$661$	4.00382e15	1.23414	0.617072	+	0.786906i	$0.288319\pi$
$662$	0	0
$663$	5.50506e14	0.166893
$664$	0	0
$665$	−1.64242e15	−0.489741
$666$	0	0
$667$	−5.30981e15	−1.55735
$668$	0	0
$669$	−1.63949e15	−0.473004
$670$	0	0
$671$	9.92300e14	0.281624
$672$	0	0
$673$	−1.69811e15	−0.474115	−0.237058	−	0.971496i	$-0.576183\pi$
$673$	−1.69811e15	−0.474115	−0.237058	+	0.971496i	$0.576183\pi$
$674$	0	0
$675$	−3.34290e15	−0.918233
$676$	0	0
$677$	2.21202e15	0.597793	0.298897	−	0.954285i	$-0.403381\pi$
$677$	2.21202e15	0.597793	0.298897	+	0.954285i	$0.403381\pi$
$678$	0	0
$679$	7.34325e14	0.195256
$680$	0	0
$681$	−7.38506e15	−1.93217
$682$	0	0
$683$	−2.27584e15	−0.585906	−0.292953	−	0.956127i	$-0.594638\pi$
$683$	−2.27584e15	−0.585906	−0.292953	+	0.956127i	$0.594638\pi$
$684$	0	0
$685$	−3.92754e15	−0.994999
$686$	0	0
$687$	−5.61512e15	−1.39990
$688$	0	0
$689$	4.60149e14	0.112900
$690$	0	0
$691$	4.62101e14	0.111586	0.0557928	−	0.998442i	$-0.482231\pi$
$691$	4.62101e14	0.111586	0.0557928	+	0.998442i	$0.482231\pi$
$692$	0	0
$693$	2.20491e15	0.524031
$694$	0	0
$695$	1.27582e16	2.98452
$696$	0	0
$697$	1.33711e15	0.307884
$698$	0	0
$699$	6.39893e15	1.45038
$700$	0	0
$701$	8.26266e15	1.84362	0.921808	−	0.387646i	$-0.126712\pi$
$701$	8.26266e15	1.84362	0.921808	+	0.387646i	$0.126712\pi$
$702$	0	0
$703$	−7.26083e14	−0.159489
$704$	0	0
$705$	5.28142e14	0.114212
$706$	0	0
$707$	−7.73125e14	−0.164605
$708$	0	0
$709$	−4.65932e15	−0.976716	−0.488358	−	0.872643i	$-0.662404\pi$
$709$	−4.65932e15	−0.976716	−0.488358	+	0.872643i	$0.662404\pi$
$710$	0	0
$711$	−3.87668e14	−0.0800162
$712$	0	0
$713$	1.77303e15	0.360349
$714$	0	0
$715$	−7.01664e14	−0.140425
$716$	0	0
$717$	6.32902e15	1.24733
$718$	0	0
$719$	−2.02905e15	−0.393808	−0.196904	−	0.980423i	$-0.563089\pi$
$719$	−2.02905e15	−0.393808	−0.196904	+	0.980423i	$0.563089\pi$
$720$	0	0
$721$	−1.20202e14	−0.0229757
$722$	0	0
$723$	−9.35175e14	−0.176048
$724$	0	0
$725$	−1.44995e16	−2.68841
$726$	0	0
$727$	7.54547e15	1.37799	0.688996	−	0.724765i	$-0.258052\pi$
$727$	7.54547e15	1.37799	0.688996	+	0.724765i	$0.258052\pi$
$728$	0	0
$729$	−8.32696e15	−1.49791
$730$	0	0
$731$	−1.90179e15	−0.336989
$732$	0	0
$733$	−5.66204e15	−0.988328	−0.494164	−	0.869369i	$-0.664526\pi$
$733$	−5.66204e15	−0.988328	−0.494164	+	0.869369i	$0.664526\pi$
$734$	0	0
$735$	1.27958e16	2.20033
$736$	0	0
$737$	1.17175e16	1.98501
$738$	0	0
$739$	4.56861e15	0.762500	0.381250	−	0.924472i	$-0.375494\pi$
$739$	4.56861e15	0.762500	0.381250	+	0.924472i	$0.375494\pi$
$740$	0	0
$741$	−5.45002e14	−0.0896186
$742$	0	0
$743$	−9.19545e15	−1.48982	−0.744911	−	0.667164i	$-0.767508\pi$
$743$	−9.19545e15	−1.48982	−0.744911	+	0.667164i	$0.767508\pi$
$744$	0	0
$745$	7.08319e15	1.13076
$746$	0	0
$747$	−1.04844e16	−1.64922
$748$	0	0
$749$	−1.96589e15	−0.304726
$750$	0	0
$751$	2.26878e15	0.346555	0.173277	−	0.984873i	$-0.444564\pi$
$751$	2.26878e15	0.346555	0.173277	+	0.984873i	$0.444564\pi$
$752$	0	0
$753$	3.65625e15	0.550381
$754$	0	0
$755$	−1.39822e16	−2.07427
$756$	0	0
$757$	−9.88961e15	−1.44595	−0.722973	−	0.690877i	$-0.757225\pi$
$757$	−9.88961e15	−1.44595	−0.722973	+	0.690877i	$0.757225\pi$
$758$	0	0
$759$	−1.19604e16	−1.72351
$760$	0	0
$761$	−1.57459e15	−0.223642	−0.111821	−	0.993728i	$-0.535668\pi$
$761$	−1.57459e15	−0.223642	−0.111821	+	0.993728i	$0.535668\pi$
$762$	0	0
$763$	−4.31701e15	−0.604363
$764$	0	0
$765$	2.21034e16	3.05015
$766$	0	0
$767$	6.36694e14	0.0866077
$768$	0	0
$769$	−6.31836e15	−0.847246	−0.423623	−	0.905839i	$-0.639242\pi$
$769$	−6.31836e15	−0.847246	−0.423623	+	0.905839i	$0.639242\pi$
$770$	0	0
$771$	1.92443e15	0.254392
$772$	0	0
$773$	6.47184e15	0.843413	0.421706	−	0.906732i	$-0.361431\pi$
$773$	6.47184e15	0.843413	0.421706	+	0.906732i	$0.361431\pi$
$774$	0	0
$775$	4.84162e15	0.622058
$776$	0	0
$777$	−1.03171e15	−0.130690
$778$	0	0
$779$	−1.32374e15	−0.165328
$780$	0	0
$781$	8.45886e15	1.04167
$782$	0	0
$783$	5.48255e15	0.665722
$784$	0	0
$785$	4.40727e15	0.527699
$786$	0	0
$787$	1.32055e15	0.155918	0.0779589	−	0.996957i	$-0.475160\pi$
$787$	1.32055e15	0.155918	0.0779589	+	0.996957i	$0.475160\pi$
$788$	0	0
$789$	−1.88430e16	−2.19395
$790$	0	0
$791$	5.32967e15	0.611970
$792$	0	0
$793$	−1.97872e14	−0.0224069
$794$	0	0
$795$	3.25413e16	3.63425
$796$	0	0
$797$	2.95580e15	0.325578	0.162789	−	0.986661i	$-0.447951\pi$
$797$	2.95580e15	0.325578	0.162789	+	0.986661i	$0.447951\pi$
$798$	0	0
$799$	−5.48594e14	−0.0595996
$800$	0	0
$801$	−5.82848e15	−0.624564
$802$	0	0
$803$	4.18100e15	0.441921
$804$	0	0
$805$	7.18743e15	0.749370
$806$	0	0
$807$	−2.59825e16	−2.67224
$808$	0	0
$809$	2.44052e15	0.247608	0.123804	−	0.992307i	$-0.460491\pi$
$809$	2.44052e15	0.247608	0.123804	+	0.992307i	$0.460491\pi$
$810$	0	0
$811$	1.17315e16	1.17419	0.587094	−	0.809518i	$-0.300272\pi$
$811$	1.17315e16	1.17419	0.587094	+	0.809518i	$0.300272\pi$
$812$	0	0
$813$	−1.48746e16	−1.46875
$814$	0	0
$815$	−2.29103e16	−2.23184
$816$	0	0
$817$	1.88277e15	0.180958
$818$	0	0
$819$	−4.39675e14	−0.0416937
$820$	0	0
$821$	1.13455e16	1.06154	0.530771	−	0.847515i	$-0.321902\pi$
$821$	1.13455e16	1.06154	0.530771	+	0.847515i	$0.321902\pi$
$822$	0	0
$823$	4.78756e14	0.0441993	0.0220997	−	0.999756i	$-0.492965\pi$
$823$	4.78756e14	0.0441993	0.0220997	+	0.999756i	$0.492965\pi$
$824$	0	0
$825$	−3.26603e16	−2.97525
$826$	0	0
$827$	−1.04383e16	−0.938315	−0.469157	−	0.883115i	$-0.655442\pi$
$827$	−1.04383e16	−0.938315	−0.469157	+	0.883115i	$0.655442\pi$
$828$	0	0
$829$	1.91489e16	1.69861	0.849306	−	0.527902i	$-0.177021\pi$
$829$	1.91489e16	1.69861	0.849306	+	0.527902i	$0.177021\pi$
$830$	0	0
$831$	1.14215e16	0.999814
$832$	0	0
$833$	−1.32913e16	−1.14821
$834$	0	0
$835$	−3.40002e15	−0.289872
$836$	0	0
$837$	−1.83071e15	−0.154038
$838$	0	0
$839$	1.67404e16	1.39019	0.695097	−	0.718916i	$-0.255361\pi$
$839$	1.67404e16	1.39019	0.695097	+	0.718916i	$0.255361\pi$
$840$	0	0
$841$	1.15796e16	0.949107
$842$	0	0
$843$	1.26753e16	1.02543
$844$	0	0
$845$	−2.12803e16	−1.69928
$846$	0	0
$847$	1.58762e14	0.0125138
$848$	0	0
$849$	1.73832e16	1.35250
$850$	0	0
$851$	3.17742e15	0.244040
$852$	0	0
$853$	−1.25299e16	−0.950009	−0.475005	−	0.879983i	$-0.657554\pi$
$853$	−1.25299e16	−0.950009	−0.475005	+	0.879983i	$0.657554\pi$
$854$	0	0
$855$	−2.18824e16	−1.63788
$856$	0	0
$857$	−3.86464e15	−0.285571	−0.142786	−	0.989754i	$-0.545606\pi$
$857$	−3.86464e15	−0.285571	−0.142786	+	0.989754i	$0.545606\pi$
$858$	0	0
$859$	−8.04819e15	−0.587132	−0.293566	−	0.955939i	$-0.594842\pi$
$859$	−8.04819e15	−0.587132	−0.293566	+	0.955939i	$0.594842\pi$
$860$	0	0
$861$	−1.88094e15	−0.135475
$862$	0	0
$863$	1.48813e16	1.05823	0.529116	−	0.848550i	$-0.322524\pi$
$863$	1.48813e16	1.05823	0.529116	+	0.848550i	$0.322524\pi$
$864$	0	0
$865$	2.58472e16	1.81479
$866$	0	0
$867$	−1.84986e16	−1.28243
$868$	0	0
$869$	−9.03996e14	−0.0618811
$870$	0	0
$871$	−2.33656e15	−0.157934
$872$	0	0
$873$	9.78358e15	0.653009
$874$	0	0
$875$	9.43449e15	0.621833
$876$	0	0
$877$	−4.76485e15	−0.310136	−0.155068	−	0.987904i	$-0.549560\pi$
$877$	−4.76485e15	−0.310136	−0.155068	+	0.987904i	$0.549560\pi$
$878$	0	0
$879$	1.90998e16	1.22769
$880$	0	0
$881$	1.88870e16	1.19894	0.599468	−	0.800399i	$-0.295379\pi$
$881$	1.88870e16	1.19894	0.599468	+	0.800399i	$0.295379\pi$
$882$	0	0
$883$	2.73933e16	1.71736	0.858679	−	0.512513i	$-0.171285\pi$
$883$	2.73933e16	1.71736	0.858679	+	0.512513i	$0.171285\pi$
$884$	0	0
$885$	4.50263e16	2.78791
$886$	0	0
$887$	−1.46672e16	−0.896950	−0.448475	−	0.893795i	$-0.648033\pi$
$887$	−1.46672e16	−0.896950	−0.448475	+	0.893795i	$0.648033\pi$
$888$	0	0
$889$	−1.15190e16	−0.695756
$890$	0	0
$891$	−1.00155e16	−0.597512
$892$	0	0
$893$	5.43109e14	0.0320040
$894$	0	0
$895$	−5.51761e16	−3.21162
$896$	0	0
$897$	2.38499e15	0.137129
$898$	0	0
$899$	−7.94053e15	−0.450995
$900$	0	0
$901$	−3.38014e16	−1.89648
$902$	0	0
$903$	2.67529e15	0.148282
$904$	0	0
$905$	−6.49267e14	−0.0355514
$906$	0	0
$907$	−1.51062e15	−0.0817175	−0.0408588	−	0.999165i	$-0.513009\pi$
$907$	−1.51062e15	−0.0817175	−0.0408588	+	0.999165i	$0.513009\pi$
$908$	0	0
$909$	−1.03005e16	−0.550500
$910$	0	0
$911$	1.53728e16	0.811714	0.405857	−	0.913937i	$-0.366973\pi$
$911$	1.53728e16	0.811714	0.405857	+	0.913937i	$0.366973\pi$
$912$	0	0
$913$	−2.44483e16	−1.27544
$914$	0	0
$915$	−1.39933e16	−0.721281
$916$	0	0
$917$	−7.13336e15	−0.363298
$918$	0	0
$919$	8.52224e15	0.428863	0.214431	−	0.976739i	$-0.431210\pi$
$919$	8.52224e15	0.428863	0.214431	+	0.976739i	$0.431210\pi$
$920$	0	0
$921$	1.43900e16	0.715536
$922$	0	0
$923$	−1.68676e15	−0.0828790
$924$	0	0
$925$	8.67660e15	0.421279
$926$	0	0
$927$	−1.60148e15	−0.0768393
$928$	0	0
$929$	−1.39800e16	−0.662856	−0.331428	−	0.943480i	$-0.607530\pi$
$929$	−1.39800e16	−0.662856	−0.331428	+	0.943480i	$0.607530\pi$
$930$	0	0
$931$	1.31584e16	0.616567
$932$	0	0
$933$	1.36797e16	0.633470
$934$	0	0
$935$	5.15424e16	2.35885
$936$	0	0
$937$	−2.51788e16	−1.13885	−0.569426	−	0.822043i	$-0.692834\pi$
$937$	−2.51788e16	−1.13885	−0.569426	+	0.822043i	$0.692834\pi$
$938$	0	0
$939$	−1.55834e16	−0.696628
$940$	0	0
$941$	−1.84912e16	−0.817002	−0.408501	−	0.912758i	$-0.633948\pi$
$941$	−1.84912e16	−0.817002	−0.408501	+	0.912758i	$0.633948\pi$
$942$	0	0
$943$	5.79284e15	0.252975
$944$	0	0
$945$	−7.42125e15	−0.320333
$946$	0	0
$947$	2.54291e16	1.08494	0.542470	−	0.840075i	$-0.317489\pi$
$947$	2.54291e16	1.08494	0.542470	+	0.840075i	$0.317489\pi$
$948$	0	0
$949$	−8.33724e14	−0.0351608
$950$	0	0
$951$	4.40475e16	1.83624
$952$	0	0
$953$	−1.74368e16	−0.718550	−0.359275	−	0.933232i	$-0.616976\pi$
$953$	−1.74368e16	−0.718550	−0.359275	+	0.933232i	$0.616976\pi$
$954$	0	0
$955$	−2.78355e16	−1.13392
$956$	0	0
$957$	5.35648e16	2.15707
$958$	0	0
$959$	−5.73892e15	−0.228469
$960$	0	0
$961$	−2.27570e16	−0.895646
$962$	0	0
$963$	−2.61920e16	−1.01912
$964$	0	0
$965$	2.98176e16	1.14703
$966$	0	0
$967$	2.45449e16	0.933503	0.466752	−	0.884388i	$-0.345424\pi$
$967$	2.45449e16	0.933503	0.466752	+	0.884388i	$0.345424\pi$
$968$	0	0
$969$	4.00345e16	1.50540
$970$	0	0
$971$	−3.90567e16	−1.45208	−0.726038	−	0.687655i	$-0.758640\pi$
$971$	−3.90567e16	−1.45208	−0.726038	+	0.687655i	$0.758640\pi$
$972$	0	0
$973$	1.86423e16	0.685297
$974$	0	0
$975$	6.51271e15	0.236721
$976$	0	0
$977$	−2.20624e16	−0.792925	−0.396462	−	0.918051i	$-0.629762\pi$
$977$	−2.20624e16	−0.792925	−0.396462	+	0.918051i	$0.629762\pi$
$978$	0	0
$979$	−1.35913e16	−0.483011
$980$	0	0
$981$	−5.75164e16	−2.02121
$982$	0	0
$983$	−2.11862e16	−0.736220	−0.368110	−	0.929782i	$-0.619995\pi$
$983$	−2.11862e16	−0.736220	−0.368110	+	0.929782i	$0.619995\pi$
$984$	0	0
$985$	7.33522e16	2.52066
$986$	0	0
$987$	7.71720e14	0.0262250
$988$	0	0
$989$	−8.23922e15	−0.276890
$990$	0	0
$991$	−2.97458e16	−0.988600	−0.494300	−	0.869292i	$-0.664575\pi$
$991$	−2.97458e16	−0.988600	−0.494300	+	0.869292i	$0.664575\pi$
$992$	0	0
$993$	−1.18294e16	−0.388813
$994$	0	0
$995$	8.51303e16	2.76731
$996$	0	0
$997$	5.21562e16	1.67680	0.838402	−	0.545052i	$-0.183490\pi$
$997$	5.21562e16	1.67680	0.838402	+	0.545052i	$0.183490\pi$
$998$	0	0
$999$	−3.28079e15	−0.104320

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.12.a.b.1.1	✓	2
3.2	odd	2		72.12.a.e.1.1		2
4.3	odd	2		16.12.a.d.1.2		2
5.2	odd	4		200.12.c.c.49.3		4
5.3	odd	4		200.12.c.c.49.2		4
5.4	even	2		200.12.a.d.1.2		2
8.3	odd	2		64.12.a.k.1.1		2
8.5	even	2		64.12.a.h.1.2		2
12.11	even	2		144.12.a.p.1.1		2
16.3	odd	4		256.12.b.k.129.3		4
16.5	even	4		256.12.b.h.129.3		4
16.11	odd	4		256.12.b.k.129.2		4
16.13	even	4		256.12.b.h.129.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.12.a.b.1.1	✓	2	1.1	even	1	trivial
16.12.a.d.1.2		2	4.3	odd	2
64.12.a.h.1.2		2	8.5	even	2
64.12.a.k.1.1		2	8.3	odd	2
72.12.a.e.1.1		2	3.2	odd	2
144.12.a.p.1.1		2	12.11	even	2
200.12.a.d.1.2		2	5.4	even	2
200.12.c.c.49.2		4	5.3	odd	4
200.12.c.c.49.3		4	5.2	odd	4
256.12.b.h.129.2		4	16.13	even	4
256.12.b.h.129.3		4	16.5	even	4
256.12.b.k.129.2		4	16.11	odd	4
256.12.b.k.129.3		4	16.3	odd	4

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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q-640.180 q^{3} +11952.2 q^{5} +17464.5 q^{7} +232683. q^{9} +O(q^{10})\)	\(q-640.180 q^{3} +11952.2 q^{5} +17464.5 q^{7} +232683. q^{9} +542588. q^{11} -108196. q^{13} -7.65153e6 q^{15} +7.94782e6 q^{17} -7.86836e6 q^{19} -1.11804e7 q^{21} +3.44328e7 q^{23} +9.40259e7 q^{25} -3.55530e7 q^{27} -1.54208e8 q^{29} +5.14924e7 q^{31} -3.47354e8 q^{33} +2.08738e8 q^{35} +9.22788e7 q^{37} +6.92651e7 q^{39} +1.68236e8 q^{41} -2.39284e8 q^{43} +2.78106e9 q^{45} -6.90244e7 q^{47} -1.67232e9 q^{49} -5.08803e9 q^{51} -4.25291e9 q^{53} +6.48510e9 q^{55} +5.03716e9 q^{57} -5.88462e9 q^{59} +1.82883e9 q^{61} +4.06368e9 q^{63} -1.29318e9 q^{65} +2.15955e10 q^{67} -2.20432e10 q^{69} +1.55898e10 q^{71} +7.70566e9 q^{73} -6.01935e10 q^{75} +9.47601e9 q^{77} -1.66608e9 q^{79} -1.84588e10 q^{81} -4.50587e10 q^{83} +9.49936e10 q^{85} +9.87208e10 q^{87} -2.50490e10 q^{89} -1.88959e9 q^{91} -3.29644e10 q^{93} -9.40439e10 q^{95} +4.20468e10 q^{97} +1.26251e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 56 q^{3} + 7868 q^{5} + 91056 q^{7} + 540202 q^{9}+O(q^{10}) \) 2 * q + 56 * q^3 + 7868 * q^5 + 91056 * q^7 + 540202 * q^9	\( 2 q + 56 q^{3} + 7868 q^{5} + 91056 q^{7} + 540202 q^{9} + 159080 q^{11} + 1050476 q^{13} - 10494832 q^{15} + 1430884 q^{17} - 21866600 q^{19} + 40052544 q^{21} + 35806736 q^{23} + 61878094 q^{25} + 55209392 q^{27} - 228827700 q^{29} + 64722112 q^{31} - 614344864 q^{33} - 91821408 q^{35} + 75558780 q^{37} + 875909072 q^{39} + 1201214196 q^{41} - 45519832 q^{43} + 1525107052 q^{45} - 1229079264 q^{47} + 1766069074 q^{49} - 9624987920 q^{51} - 3808549924 q^{53} + 8051409968 q^{55} - 4708125536 q^{57} - 6012926584 q^{59} + 9789792908 q^{61} + 26694483312 q^{63} - 6025376344 q^{65} + 14703095224 q^{67} - 21086664256 q^{69} + 4319991088 q^{71} + 11055639476 q^{73} - 82574103416 q^{75} - 18746968128 q^{77} + 51957623264 q^{79} - 9747960302 q^{81} - 108227975912 q^{83} + 121609729720 q^{85} + 46772044368 q^{87} + 71188291860 q^{89} + 83378892576 q^{91} - 23754095360 q^{93} - 36872875568 q^{95} - 1699807676 q^{97} + 8314917256 q^{99}+O(q^{100}) \) 2 * q + 56 * q^3 + 7868 * q^5 + 91056 * q^7 + 540202 * q^9 + 159080 * q^11 + 1050476 * q^13 - 10494832 * q^15 + 1430884 * q^17 - 21866600 * q^19 + 40052544 * q^21 + 35806736 * q^23 + 61878094 * q^25 + 55209392 * q^27 - 228827700 * q^29 + 64722112 * q^31 - 614344864 * q^33 - 91821408 * q^35 + 75558780 * q^37 + 875909072 * q^39 + 1201214196 * q^41 - 45519832 * q^43 + 1525107052 * q^45 - 1229079264 * q^47 + 1766069074 * q^49 - 9624987920 * q^51 - 3808549924 * q^53 + 8051409968 * q^55 - 4708125536 * q^57 - 6012926584 * q^59 + 9789792908 * q^61 + 26694483312 * q^63 - 6025376344 * q^65 + 14703095224 * q^67 - 21086664256 * q^69 + 4319991088 * q^71 + 11055639476 * q^73 - 82574103416 * q^75 - 18746968128 * q^77 + 51957623264 * q^79 - 9747960302 * q^81 - 108227975912 * q^83 + 121609729720 * q^85 + 46772044368 * q^87 + 71188291860 * q^89 + 83378892576 * q^91 - 23754095360 * q^93 - 36872875568 * q^95 - 1699807676 * q^97 + 8314917256 * q^99

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