LMFDB - Embedded newform 144.12.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,12,Mod(1,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("144.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	144.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:	$110.641418001$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.72015$ of defining polynomial
Character	$\chi$	$=$	144.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11952.2 q^{5} -17464.5 q^{7} +O(q^{10})$	$q-11952.2 q^{5} -17464.5 q^{7} +542588. q^{11} -108196. q^{13} -7.94782e6 q^{17} +7.86836e6 q^{19} +3.44328e7 q^{23} +9.40259e7 q^{25} +1.54208e8 q^{29} -5.14924e7 q^{31} +2.08738e8 q^{35} +9.22788e7 q^{37} -1.68236e8 q^{41} +2.39284e8 q^{43} -6.90244e7 q^{47} -1.67232e9 q^{49} +4.25291e9 q^{53} -6.48510e9 q^{55} -5.88462e9 q^{59} +1.82883e9 q^{61} +1.29318e9 q^{65} -2.15955e10 q^{67} +1.55898e10 q^{71} +7.70566e9 q^{73} -9.47601e9 q^{77} +1.66608e9 q^{79} -4.50587e10 q^{83} +9.49936e10 q^{85} +2.50490e10 q^{89} +1.88959e9 q^{91} -9.40439e10 q^{95} +4.20468e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 7868 q^{5} - 91056 q^{7}+O(q^{10}) $ 2 * q - 7868 * q^5 - 91056 * q^7	$ 2 q - 7868 q^{5} - 91056 q^{7} + 159080 q^{11} + 1050476 q^{13} - 1430884 q^{17} + 21866600 q^{19} + 35806736 q^{23} + 61878094 q^{25} + 228827700 q^{29} - 64722112 q^{31} - 91821408 q^{35} + 75558780 q^{37} - 1201214196 q^{41} + 45519832 q^{43} - 1229079264 q^{47} + 1766069074 q^{49} + 3808549924 q^{53} - 8051409968 q^{55} - 6012926584 q^{59} + 9789792908 q^{61} + 6025376344 q^{65} - 14703095224 q^{67} + 4319991088 q^{71} + 11055639476 q^{73} + 18746968128 q^{77} - 51957623264 q^{79} - 108227975912 q^{83} + 121609729720 q^{85} - 71188291860 q^{89} - 83378892576 q^{91} - 36872875568 q^{95} - 1699807676 q^{97}+O(q^{100}) $ 2 * q - 7868 * q^5 - 91056 * q^7 + 159080 * q^11 + 1050476 * q^13 - 1430884 * q^17 + 21866600 * q^19 + 35806736 * q^23 + 61878094 * q^25 + 228827700 * q^29 - 64722112 * q^31 - 91821408 * q^35 + 75558780 * q^37 - 1201214196 * q^41 + 45519832 * q^43 - 1229079264 * q^47 + 1766069074 * q^49 + 3808549924 * q^53 - 8051409968 * q^55 - 6012926584 * q^59 + 9789792908 * q^61 + 6025376344 * q^65 - 14703095224 * q^67 + 4319991088 * q^71 + 11055639476 * q^73 + 18746968128 * q^77 - 51957623264 * q^79 - 108227975912 * q^83 + 121609729720 * q^85 - 71188291860 * q^89 - 83378892576 * q^91 - 36872875568 * q^95 - 1699807676 * q^97

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
$5$	−11952.2	−1.71045	−0.855227	−	0.518254i	$-0.826582\pi$
$5$	−11952.2	−1.71045	−0.855227	+	0.518254i	$0.826582\pi$
$6$	0	0
$7$	−17464.5	−0.392750	−0.196375	−	0.980529i	$-0.562917\pi$
$7$	−17464.5	−0.392750	−0.196375	+	0.980529i	$0.562917\pi$
$8$	0	0
$9$	0	0
$10$	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
$15$	0	0
$16$	0	0
$17$	−7.94782e6	−1.35762	−0.678811	−	0.734313i	$-0.737504\pi$
$17$	−7.94782e6	−1.35762	−0.678811	+	0.734313i	$0.737504\pi$
$18$	0	0
$19$	7.86836e6	0.729020	0.364510	−	0.931199i	$-0.381237\pi$
$19$	7.86836e6	0.729020	0.364510	+	0.931199i	$0.381237\pi$
$20$	0	0
$21$	0	0
$22$	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
$27$	0	0
$28$	0	0
$29$	1.54208e8	1.39610	0.698052	−	0.716047i	$-0.254050\pi$
$29$	1.54208e8	1.39610	0.698052	+	0.716047i	$0.254050\pi$
$30$	0	0
$31$	−5.14924e7	−0.323038	−0.161519	−	0.986870i	$-0.551639\pi$
$31$	−5.14924e7	−0.323038	−0.161519	+	0.986870i	$0.551639\pi$
$32$	0	0
$33$	0	0
$34$	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
$39$	0	0
$40$	0	0
$41$	−1.68236e8	−0.226782	−0.113391	−	0.993550i	$-0.536171\pi$
$41$	−1.68236e8	−0.226782	−0.113391	+	0.993550i	$0.536171\pi$
$42$	0	0
$43$	2.39284e8	0.248220	0.124110	−	0.992268i	$-0.460392\pi$
$43$	2.39284e8	0.248220	0.124110	+	0.992268i	$0.460392\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	0	0
$53$	4.25291e9	1.39691	0.698456	−	0.715653i	$-0.253871\pi$
$53$	4.25291e9	1.39691	0.698456	+	0.715653i	$0.253871\pi$
$54$	0	0
$55$	−6.48510e9	−1.73749
$56$	0	0
$57$	0	0
$58$	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
$63$	0	0
$64$	0	0
$65$	1.29318e9	0.138240
$66$	0	0
$67$	−2.15955e10	−1.95412	−0.977062	−	0.212953i	$-0.931692\pi$
$67$	−2.15955e10	−1.95412	−0.977062	+	0.212953i	$0.931692\pi$
$68$	0	0
$69$	0	0
$70$	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
$75$	0	0
$76$	0	0
$77$	−9.47601e9	−0.398958
$78$	0	0
$79$	1.66608e9	0.0609182	0.0304591	−	0.999536i	$-0.490303\pi$
$79$	1.66608e9	0.0609182	0.0304591	+	0.999536i	$0.490303\pi$
$80$	0	0
$81$	0	0
$82$	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
$87$	0	0
$88$	0	0
$89$	2.50490e10	0.475495	0.237748	−	0.971327i	$-0.423591\pi$
$89$	2.50490e10	0.475495	0.237748	+	0.971327i	$0.423591\pi$
$90$	0	0
$91$	1.88959e9	0.0317424
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	4.42685e10	0.419109	0.209554	−	0.977797i	$-0.432799\pi$
$101$	4.42685e10	0.419109	0.209554	+	0.977797i	$0.432799\pi$
$102$	0	0
$103$	6.88268e9	0.0584996	0.0292498	−	0.999572i	$-0.490688\pi$
$103$	6.88268e9	0.0584996	0.0292498	+	0.999572i	$0.490688\pi$
$104$	0	0
$105$	0	0
$106$	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
$111$	0	0
$112$	0	0
$113$	−3.05173e11	−1.55817	−0.779084	−	0.626920i	$-0.784315\pi$
$113$	−3.05173e11	−1.55817	−0.779084	+	0.626920i	$0.784315\pi$
$114$	0	0
$115$	−4.11546e11	−1.90801
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.38804e11	0.533206
$120$	0	0
$121$	9.09058e9	0.0318619
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.40211e11	−1.58328
$126$	0	0
$127$	6.59571e11	1.77150	0.885750	−	0.464163i	$-0.153645\pi$
$127$	6.59571e11	1.77150	0.885750	+	0.464163i	$0.153645\pi$
$128$	0	0
$129$	0	0
$130$	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
$135$	0	0
$136$	0	0
$137$	3.28605e11	0.581717	0.290858	−	0.956766i	$-0.406059\pi$
$137$	3.28605e11	0.581717	0.290858	+	0.956766i	$0.406059\pi$
$138$	0	0
$139$	−1.06744e12	−1.74487	−0.872435	−	0.488730i	$-0.837460\pi$
$139$	−1.06744e12	−1.74487	−0.872435	+	0.488730i	$0.837460\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.87061e10	−0.0820984
$144$	0	0
$145$	−1.84312e12	−2.38797
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.92629e11	−0.661087	−0.330543	−	0.943791i	$-0.607232\pi$
$149$	−5.92629e11	−0.661087	−0.330543	+	0.943791i	$0.607232\pi$
$150$	0	0
$151$	1.16984e12	1.21270	0.606352	−	0.795196i	$-0.292632\pi$
$151$	1.16984e12	1.21270	0.606352	+	0.795196i	$0.292632\pi$
$152$	0	0
$153$	0	0
$154$	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
$159$	0	0
$160$	0	0
$161$	−6.01350e11	−0.438112
$162$	0	0
$163$	1.91683e12	1.30483	0.652413	−	0.757864i	$-0.273757\pi$
$163$	1.91683e12	1.30483	0.652413	+	0.757864i	$0.273757\pi$
$164$	0	0
$165$	0	0
$166$	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
$171$	0	0
$172$	0	0
$173$	−2.16256e12	−1.06100	−0.530499	−	0.847686i	$-0.677995\pi$
$173$	−2.16256e12	−1.06100	−0.530499	+	0.847686i	$0.677995\pi$
$174$	0	0
$175$	−1.64211e12	−0.756299
$176$	0	0
$177$	0	0
$178$	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
$183$	0	0
$184$	0	0
$185$	−1.10293e12	−0.374200
$186$	0	0
$187$	−4.31239e12	−1.37908
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	0	0
$197$	−6.13716e12	−1.47368	−0.736839	−	0.676068i	$-0.763683\pi$
$197$	−6.13716e12	−1.47368	−0.736839	+	0.676068i	$0.763683\pi$
$198$	0	0
$199$	−7.12259e12	−1.61788	−0.808940	−	0.587892i	$-0.799958\pi$
$199$	−7.12259e12	−1.61788	−0.808940	+	0.587892i	$0.799958\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.69316e12	−0.548319
$204$	0	0
$205$	2.01078e12	0.387899
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.26928e12	0.740543
$210$	0	0
$211$	7.64441e12	1.25832	0.629159	−	0.777277i	$-0.283400\pi$
$211$	7.64441e12	1.25832	0.629159	+	0.777277i	$0.283400\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.85996e12	−0.424569
$216$	0	0
$217$	8.99286e11	0.126873
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.59924e11	0.109724
$222$	0	0
$223$	−2.56098e12	−0.310978	−0.155489	−	0.987838i	$-0.549695\pi$
$223$	−2.56098e12	−0.310978	−0.155489	+	0.987838i	$0.549695\pi$
$224$	0	0
$225$	0	0
$226$	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
$231$	0	0
$232$	0	0
$233$	9.99552e12	0.953560	0.476780	−	0.879023i	$-0.341804\pi$
$233$	9.99552e12	0.953560	0.476780	+	0.879023i	$0.341804\pi$
$234$	0	0
$235$	8.24991e11	0.0750889
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	1.99878e13	1.44661
$246$	0	0
$247$	−8.51327e11	−0.0589201
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	3.00608e12	0.167251	0.0836254	−	0.996497i	$-0.473350\pi$
$257$	3.00608e12	0.167251	0.0836254	+	0.996497i	$0.473350\pi$
$258$	0	0
$259$	−1.61160e12	−0.0859228
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	0	0
$269$	−4.05862e13	−1.75688	−0.878438	−	0.477857i	$-0.841414\pi$
$269$	−4.05862e13	−1.75688	−0.878438	+	0.477857i	$0.841414\pi$
$270$	0	0
$271$	−2.32350e13	−0.965633	−0.482817	−	0.875721i	$-0.660386\pi$
$271$	−2.32350e13	−0.965633	−0.482817	+	0.875721i	$0.660386\pi$
$272$	0	0
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	1.97996e13	0.674174	0.337087	−	0.941473i	$-0.390558\pi$
$281$	1.97996e13	0.674174	0.337087	+	0.941473i	$0.390558\pi$
$282$	0	0
$283$	2.71536e13	0.889207	0.444603	−	0.895728i	$-0.353345\pi$
$283$	2.71536e13	0.889207	0.444603	+	0.895728i	$0.353345\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.93815e12	0.0890684
$288$	0	0
$289$	2.88959e13	0.843137
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.98350e13	0.807150	0.403575	−	0.914947i	$-0.367768\pi$
$293$	2.98350e13	0.807150	0.403575	+	0.914947i	$0.367768\pi$
$294$	0	0
$295$	7.03339e13	1.83292
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.72550e12	−0.0901557
$300$	0	0
$301$	−4.17897e12	−0.0974885
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.18584e13	−0.474209
$306$	0	0
$307$	2.24780e13	0.470432	0.235216	−	0.971943i	$-0.424420\pi$
$307$	2.24780e13	0.470432	0.235216	+	0.971943i	$0.424420\pi$
$308$	0	0
$309$	0	0
$310$	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
$315$	0	0
$316$	0	0
$317$	6.88050e13	1.20724	0.603620	−	0.797272i	$-0.293724\pi$
$317$	6.88050e13	1.20724	0.603620	+	0.797272i	$0.293724\pi$
$318$	0	0
$319$	8.36715e13	1.41817
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.25363e13	−0.989734
$324$	0	0
$325$	−1.01733e13	−0.155633
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.20547e12	0.0172417
$330$	0	0
$331$	−1.84782e13	−0.255626	−0.127813	−	0.991798i	$-0.540796\pi$
$331$	−1.84782e13	−0.255626	−0.127813	+	0.991798i	$0.540796\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	−2.79392e13	−0.328144
$342$	0	0
$343$	6.37391e13	0.724917
$344$	0	0
$345$	0	0
$346$	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
$351$	0	0
$352$	0	0
$353$	4.56680e13	0.443456	0.221728	−	0.975109i	$-0.428830\pi$
$353$	4.56680e13	0.443456	0.221728	+	0.975109i	$0.428830\pi$
$354$	0	0
$355$	−1.86332e14	−1.75401
$356$	0	0
$357$	0	0
$358$	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
$363$	0	0
$364$	0	0
$365$	−9.20993e13	−0.744124
$366$	0	0
$367$	−1.36650e14	−1.07139	−0.535693	−	0.844413i	$-0.679950\pi$
$367$	−1.36650e14	−1.07139	−0.535693	+	0.844413i	$0.679950\pi$
$368$	0	0
$369$	0	0
$370$	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
$375$	0	0
$376$	0	0
$377$	−1.66847e13	−0.112834
$378$	0	0
$379$	−1.58566e14	−1.04158	−0.520791	−	0.853684i	$-0.674363\pi$
$379$	−1.58566e14	−1.04158	−0.520791	+	0.853684i	$0.674363\pi$
$380$	0	0
$381$	0	0
$382$	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
$387$	0	0
$388$	0	0
$389$	1.92046e14	1.09315	0.546577	−	0.837409i	$-0.315931\pi$
$389$	1.92046e14	1.09315	0.546577	+	0.837409i	$0.315931\pi$
$390$	0	0
$391$	−2.73666e14	−1.51443
$392$	0	0
$393$	0	0
$394$	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
$399$	0	0
$400$	0	0
$401$	1.72882e13	0.0832637	0.0416318	−	0.999133i	$-0.486744\pi$
$401$	1.72882e13	0.0832637	0.0416318	+	0.999133i	$0.486744\pi$
$402$	0	0
$403$	5.57128e12	0.0261082
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	1.02772e14	0.420870
$414$	0	0
$415$	5.38548e14	2.14763
$416$	0	0
$417$	0	0
$418$	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
$423$	0	0
$424$	0	0
$425$	−7.47301e14	−2.61431
$426$	0	0
$427$	−3.19395e13	−0.108887
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	0	0
$437$	2.70930e14	0.813221
$438$	0	0
$439$	−1.16021e14	−0.339610	−0.169805	−	0.985478i	$-0.554314\pi$
$439$	−1.16021e14	−0.339610	−0.169805	+	0.985478i	$0.554314\pi$
$440$	0	0
$441$	0	0
$442$	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
$447$	0	0
$448$	0	0
$449$	−1.46433e14	−0.378690	−0.189345	−	0.981911i	$-0.560636\pi$
$449$	−1.46433e14	−0.378690	−0.189345	+	0.981911i	$0.560636\pi$
$450$	0	0
$451$	−9.12830e13	−0.230366
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	−8.46403e14	−1.89331	−0.946655	−	0.322248i	$-0.895562\pi$
$461$	−8.46403e14	−1.89331	−0.946655	+	0.322248i	$0.895562\pi$
$462$	0	0
$463$	1.75496e14	0.383330	0.191665	−	0.981460i	$-0.438611\pi$
$463$	1.75496e14	0.383330	0.191665	+	0.981460i	$0.438611\pi$
$464$	0	0
$465$	0	0
$466$	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
$471$	0	0
$472$	0	0
$473$	1.29833e14	0.252144
$474$	0	0
$475$	7.39830e14	1.40384
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	−5.02550e14	−0.850354
$486$	0	0
$487$	−8.99264e14	−1.48757	−0.743786	−	0.668418i	$-0.766972\pi$
$487$	−8.99264e14	−1.48757	−0.743786	+	0.668418i	$0.766972\pi$
$488$	0	0
$489$	0	0
$490$	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
$495$	0	0
$496$	0	0
$497$	−2.72268e14	−0.402751
$498$	0	0
$499$	7.02134e14	1.01594	0.507969	−	0.861375i	$-0.330396\pi$
$499$	7.02134e14	1.01594	0.507969	+	0.861375i	$0.330396\pi$
$500$	0	0
$501$	0	0
$502$	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
$507$	0	0
$508$	0	0
$509$	2.67254e14	0.346718	0.173359	−	0.984859i	$-0.444538\pi$
$509$	2.67254e14	0.346718	0.173359	+	0.984859i	$0.444538\pi$
$510$	0	0
$511$	−1.34575e14	−0.170864
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.22628e13	−0.100061
$516$	0	0
$517$	−3.74519e13	−0.0445939
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.63788e14	1.09995	0.549976	−	0.835180i	$-0.314637\pi$
$521$	9.63788e14	1.09995	0.549976	+	0.835180i	$0.314637\pi$
$522$	0	0
$523$	−4.44881e14	−0.497146	−0.248573	−	0.968613i	$-0.579962\pi$
$523$	−4.44881e14	−0.497146	−0.248573	+	0.968613i	$0.579962\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.09252e14	0.438563
$528$	0	0
$529$	2.32808e14	0.244338
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.82025e13	0.0183287
$534$	0	0
$535$	1.34540e15	1.32710
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	0	0
$545$	2.95443e15	2.63204
$546$	0	0
$547$	−1.74722e15	−1.52552	−0.762760	−	0.646682i	$-0.776156\pi$
$547$	−1.74722e15	−1.52552	−0.762760	+	0.646682i	$0.776156\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.21336e15	1.01779
$552$	0	0
$553$	−2.90972e13	−0.0239256
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.28921e14	0.576072	0.288036	−	0.957620i	$-0.406998\pi$
$557$	7.28921e14	0.576072	0.288036	+	0.957620i	$0.406998\pi$
$558$	0	0
$559$	−2.58897e13	−0.0200614
$560$	0	0
$561$	0	0
$562$	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
$567$	0	0
$568$	0	0
$569$	−2.73759e15	−1.92420	−0.962101	−	0.272694i	$-0.912085\pi$
$569$	−2.73759e15	−1.92420	−0.962101	+	0.272694i	$0.912085\pi$
$570$	0	0
$571$	−1.04213e15	−0.718491	−0.359245	−	0.933243i	$-0.616966\pi$
$571$	−1.04213e15	−0.718491	−0.359245	+	0.933243i	$0.616966\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	7.86925e14	0.493134
$582$	0	0
$583$	2.30758e15	1.41899
$584$	0	0
$585$	0	0
$586$	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
$591$	0	0
$592$	0	0
$593$	2.51449e15	1.40815	0.704075	−	0.710126i	$-0.251362\pi$
$593$	2.51449e15	1.40815	0.704075	+	0.710126i	$0.251362\pi$
$594$	0	0
$595$	−1.65901e15	−0.912023
$596$	0	0
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	−1.08652e14	−0.0544983
$606$	0	0
$607$	8.40427e14	0.413964	0.206982	−	0.978345i	$-0.433636\pi$
$607$	8.40427e14	0.413964	0.206982	+	0.978345i	$0.433636\pi$
$608$	0	0
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	8.78027e14	0.395312	0.197656	−	0.980271i	$-0.436667\pi$
$617$	8.78027e14	0.395312	0.197656	+	0.980271i	$0.436667\pi$
$618$	0	0
$619$	−2.73116e15	−1.20795	−0.603975	−	0.797003i	$-0.706417\pi$
$619$	−2.73116e15	−1.20795	−0.603975	+	0.797003i	$0.706417\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.37468e14	−0.186751
$624$	0	0
$625$	1.86557e15	0.782479
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.33415e14	−0.297010
$630$	0	0
$631$	2.11332e15	0.841016	0.420508	−	0.907289i	$-0.361852\pi$
$631$	2.11332e15	0.841016	0.420508	+	0.907289i	$0.361852\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7.88329e15	−3.03007
$636$	0	0
$637$	1.80939e14	0.0683541
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.52374e15	−0.556149	−0.278074	−	0.960560i	$-0.589696\pi$
$641$	−1.52374e15	−0.556149	−0.278074	+	0.960560i	$0.589696\pi$
$642$	0	0
$643$	−1.62810e15	−0.584145	−0.292073	−	0.956396i	$-0.594345\pi$
$643$	−1.62810e15	−0.584145	−0.292073	+	0.956396i	$0.594345\pi$
$644$	0	0
$645$	0	0
$646$	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
$651$	0	0
$652$	0	0
$653$	5.05508e14	0.166612	0.0833059	−	0.996524i	$-0.473452\pi$
$653$	5.05508e14	0.166612	0.0833059	+	0.996524i	$0.473452\pi$
$654$	0	0
$655$	4.88186e15	1.58219
$656$	0	0
$657$	0	0
$658$	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
$663$	0	0
$664$	0	0
$665$	1.64242e15	0.489741
$666$	0	0
$667$	5.30981e15	1.55735
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	−2.21202e15	−0.597793	−0.298897	−	0.954285i	$-0.596619\pi$
$677$	−2.21202e15	−0.597793	−0.298897	+	0.954285i	$0.596619\pi$
$678$	0	0
$679$	−7.34325e14	−0.195256
$680$	0	0
$681$	0	0
$682$	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
$687$	0	0
$688$	0	0
$689$	−4.60149e14	−0.112900
$690$	0	0
$691$	−4.62101e14	−0.111586	−0.0557928	−	0.998442i	$-0.517769\pi$
$691$	−4.62101e14	−0.111586	−0.0557928	+	0.998442i	$0.517769\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.27582e16	2.98452
$696$	0	0
$697$	1.33711e15	0.307884
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.26266e15	−1.84362	−0.921808	−	0.387646i	$-0.873288\pi$
$701$	−8.26266e15	−1.84362	−0.921808	+	0.387646i	$0.873288\pi$
$702$	0	0
$703$	7.26083e14	0.159489
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	−1.77303e15	−0.360349
$714$	0	0
$715$	7.01664e14	0.140425
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	1.44995e16	2.68841
$726$	0	0
$727$	−7.54547e15	−1.37799	−0.688996	−	0.724765i	$-0.741948\pi$
$727$	−7.54547e15	−1.37799	−0.688996	+	0.724765i	$0.741948\pi$
$728$	0	0
$729$	0	0
$730$	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
$735$	0	0
$736$	0	0
$737$	−1.17175e16	−1.98501
$738$	0	0
$739$	−4.56861e15	−0.762500	−0.381250	−	0.924472i	$-0.624506\pi$
$739$	−4.56861e15	−0.762500	−0.381250	+	0.924472i	$0.624506\pi$
$740$	0	0
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	1.96589e15	0.304726
$750$	0	0
$751$	−2.26878e15	−0.346555	−0.173277	−	0.984873i	$-0.555436\pi$
$751$	−2.26878e15	−0.346555	−0.173277	+	0.984873i	$0.555436\pi$
$752$	0	0
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	1.57459e15	0.223642	0.111821	−	0.993728i	$-0.464332\pi$
$761$	1.57459e15	0.223642	0.111821	+	0.993728i	$0.464332\pi$
$762$	0	0
$763$	4.31701e15	0.604363
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	−6.47184e15	−0.843413	−0.421706	−	0.906732i	$-0.638569\pi$
$773$	−6.47184e15	−0.843413	−0.421706	+	0.906732i	$0.638569\pi$
$774$	0	0
$775$	−4.84162e15	−0.622058
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.32374e15	−0.165328
$780$	0	0
$781$	8.45886e15	1.04167
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.40727e15	−0.527699
$786$	0	0
$787$	−1.32055e15	−0.155918	−0.0779589	−	0.996957i	$-0.524840\pi$
$787$	−1.32055e15	−0.155918	−0.0779589	+	0.996957i	$0.524840\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.32967e15	0.611970
$792$	0	0
$793$	−1.97872e14	−0.0224069
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.95580e15	−0.325578	−0.162789	−	0.986661i	$-0.552049\pi$
$797$	−2.95580e15	−0.325578	−0.162789	+	0.986661i	$0.552049\pi$
$798$	0	0
$799$	5.48594e14	0.0595996
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.18100e15	0.441921
$804$	0	0
$805$	7.18743e15	0.749370
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.44052e15	−0.247608	−0.123804	−	0.992307i	$-0.539509\pi$
$809$	−2.44052e15	−0.247608	−0.123804	+	0.992307i	$0.539509\pi$
$810$	0	0
$811$	−1.17315e16	−1.17419	−0.587094	−	0.809518i	$-0.699728\pi$
$811$	−1.17315e16	−1.17419	−0.587094	+	0.809518i	$0.699728\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.29103e16	−2.23184
$816$	0	0
$817$	1.88277e15	0.180958
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.13455e16	−1.06154	−0.530771	−	0.847515i	$-0.678098\pi$
$821$	−1.13455e16	−1.06154	−0.530771	+	0.847515i	$0.678098\pi$
$822$	0	0
$823$	−4.78756e14	−0.0441993	−0.0220997	−	0.999756i	$-0.507035\pi$
$823$	−4.78756e14	−0.0441993	−0.0220997	+	0.999756i	$0.507035\pi$
$824$	0	0
$825$	0	0
$826$	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
$831$	0	0
$832$	0	0
$833$	1.32913e16	1.14821
$834$	0	0
$835$	3.40002e15	0.289872
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	0	0
$845$	2.12803e16	1.69928
$846$	0	0
$847$	−1.58762e14	−0.0125138
$848$	0	0
$849$	0	0
$850$	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
$855$	0	0
$856$	0	0
$857$	3.86464e15	0.285571	0.142786	−	0.989754i	$-0.454394\pi$
$857$	3.86464e15	0.285571	0.142786	+	0.989754i	$0.454394\pi$
$858$	0	0
$859$	8.04819e15	0.587132	0.293566	−	0.955939i	$-0.405158\pi$
$859$	8.04819e15	0.587132	0.293566	+	0.955939i	$0.405158\pi$
$860$	0	0
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	9.03996e14	0.0618811
$870$	0	0
$871$	2.33656e15	0.157934
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	−1.88870e16	−1.19894	−0.599468	−	0.800399i	$-0.704621\pi$
$881$	−1.88870e16	−1.19894	−0.599468	+	0.800399i	$0.704621\pi$
$882$	0	0
$883$	−2.73933e16	−1.71736	−0.858679	−	0.512513i	$-0.828715\pi$
$883$	−2.73933e16	−1.71736	−0.858679	+	0.512513i	$0.828715\pi$
$884$	0	0
$885$	0	0
$886$	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
$891$	0	0
$892$	0	0
$893$	−5.43109e14	−0.0320040
$894$	0	0
$895$	5.51761e16	3.21162
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.94053e15	−0.450995
$900$	0	0
$901$	−3.38014e16	−1.89648
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.49267e14	0.0355514
$906$	0	0
$907$	1.51062e15	0.0817175	0.0408588	−	0.999165i	$-0.486991\pi$
$907$	1.51062e15	0.0817175	0.0408588	+	0.999165i	$0.486991\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	7.13336e15	0.363298
$918$	0	0
$919$	−8.52224e15	−0.428863	−0.214431	−	0.976739i	$-0.568790\pi$
$919$	−8.52224e15	−0.428863	−0.214431	+	0.976739i	$0.568790\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.68676e15	−0.0828790
$924$	0	0
$925$	8.67660e15	0.421279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.39800e16	0.662856	0.331428	−	0.943480i	$-0.392470\pi$
$929$	1.39800e16	0.662856	0.331428	+	0.943480i	$0.392470\pi$
$930$	0	0
$931$	−1.31584e16	−0.616567
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	1.84912e16	0.817002	0.408501	−	0.912758i	$-0.366052\pi$
$941$	1.84912e16	0.817002	0.408501	+	0.912758i	$0.366052\pi$
$942$	0	0
$943$	−5.79284e15	−0.252975
$944$	0	0
$945$	0	0
$946$	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
$951$	0	0
$952$	0	0
$953$	1.74368e16	0.718550	0.359275	−	0.933232i	$-0.383024\pi$
$953$	1.74368e16	0.718550	0.359275	+	0.933232i	$0.383024\pi$
$954$	0	0
$955$	2.78355e16	1.13392
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.73892e15	−0.228469
$960$	0	0
$961$	−2.27570e16	−0.895646
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.98176e16	−1.14703
$966$	0	0
$967$	−2.45449e16	−0.933503	−0.466752	−	0.884388i	$-0.654576\pi$
$967$	−2.45449e16	−0.933503	−0.466752	+	0.884388i	$0.654576\pi$
$968$	0	0
$969$	0	0
$970$	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
$975$	0	0
$976$	0	0
$977$	2.20624e16	0.792925	0.396462	−	0.918051i	$-0.370238\pi$
$977$	2.20624e16	0.792925	0.396462	+	0.918051i	$0.370238\pi$
$978$	0	0
$979$	1.35913e16	0.483011
$980$	0	0
$981$	0	0
$982$	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
$987$	0	0
$988$	0	0
$989$	8.23922e15	0.276890
$990$	0	0
$991$	2.97458e16	0.988600	0.494300	−	0.869292i	$-0.335425\pi$
$991$	2.97458e16	0.988600	0.494300	+	0.869292i	$0.335425\pi$
$992$	0	0
$993$	0	0
$994$	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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.12.a.p.1.1		2
3.2	odd	2		16.12.a.d.1.2		2
4.3	odd	2		72.12.a.e.1.1		2
12.11	even	2		8.12.a.b.1.1	✓	2
24.5	odd	2		64.12.a.k.1.1		2
24.11	even	2		64.12.a.h.1.2		2
48.5	odd	4		256.12.b.k.129.2		4
48.11	even	4		256.12.b.h.129.3		4
48.29	odd	4		256.12.b.k.129.3		4
48.35	even	4		256.12.b.h.129.2		4
60.23	odd	4		200.12.c.c.49.2		4
60.47	odd	4		200.12.c.c.49.3		4
60.59	even	2		200.12.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.12.a.b.1.1	✓	2	12.11	even	2
16.12.a.d.1.2		2	3.2	odd	2
64.12.a.h.1.2		2	24.11	even	2
64.12.a.k.1.1		2	24.5	odd	2
72.12.a.e.1.1		2	4.3	odd	2
144.12.a.p.1.1		2	1.1	even	1	trivial
200.12.a.d.1.2		2	60.59	even	2
200.12.c.c.49.2		4	60.23	odd	4
200.12.c.c.49.3		4	60.47	odd	4
256.12.b.h.129.2		4	48.35	even	4
256.12.b.h.129.3		4	48.11	even	4
256.12.b.k.129.2		4	48.5	odd	4
256.12.b.k.129.3		4	48.29	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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

\(f(q)\)	\(=\)	\(q-11952.2 q^{5} -17464.5 q^{7} +O(q^{10})\)	\(q-11952.2 q^{5} -17464.5 q^{7} +542588. q^{11} -108196. q^{13} -7.94782e6 q^{17} +7.86836e6 q^{19} +3.44328e7 q^{23} +9.40259e7 q^{25} +1.54208e8 q^{29} -5.14924e7 q^{31} +2.08738e8 q^{35} +9.22788e7 q^{37} -1.68236e8 q^{41} +2.39284e8 q^{43} -6.90244e7 q^{47} -1.67232e9 q^{49} +4.25291e9 q^{53} -6.48510e9 q^{55} -5.88462e9 q^{59} +1.82883e9 q^{61} +1.29318e9 q^{65} -2.15955e10 q^{67} +1.55898e10 q^{71} +7.70566e9 q^{73} -9.47601e9 q^{77} +1.66608e9 q^{79} -4.50587e10 q^{83} +9.49936e10 q^{85} +2.50490e10 q^{89} +1.88959e9 q^{91} -9.40439e10 q^{95} +4.20468e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 7868 q^{5} - 91056 q^{7}+O(q^{10}) \) 2 * q - 7868 * q^5 - 91056 * q^7	\( 2 q - 7868 q^{5} - 91056 q^{7} + 159080 q^{11} + 1050476 q^{13} - 1430884 q^{17} + 21866600 q^{19} + 35806736 q^{23} + 61878094 q^{25} + 228827700 q^{29} - 64722112 q^{31} - 91821408 q^{35} + 75558780 q^{37} - 1201214196 q^{41} + 45519832 q^{43} - 1229079264 q^{47} + 1766069074 q^{49} + 3808549924 q^{53} - 8051409968 q^{55} - 6012926584 q^{59} + 9789792908 q^{61} + 6025376344 q^{65} - 14703095224 q^{67} + 4319991088 q^{71} + 11055639476 q^{73} + 18746968128 q^{77} - 51957623264 q^{79} - 108227975912 q^{83} + 121609729720 q^{85} - 71188291860 q^{89} - 83378892576 q^{91} - 36872875568 q^{95} - 1699807676 q^{97}+O(q^{100}) \) 2 * q - 7868 * q^5 - 91056 * q^7 + 159080 * q^11 + 1050476 * q^13 - 1430884 * q^17 + 21866600 * q^19 + 35806736 * q^23 + 61878094 * q^25 + 228827700 * q^29 - 64722112 * q^31 - 91821408 * q^35 + 75558780 * q^37 - 1201214196 * q^41 + 45519832 * q^43 - 1229079264 * q^47 + 1766069074 * q^49 + 3808549924 * q^53 - 8051409968 * q^55 - 6012926584 * q^59 + 9789792908 * q^61 + 6025376344 * q^65 - 14703095224 * q^67 + 4319991088 * q^71 + 11055639476 * q^73 + 18746968128 * q^77 - 51957623264 * q^79 - 108227975912 * q^83 + 121609729720 * q^85 - 71188291860 * q^89 - 83378892576 * q^91 - 36872875568 * q^95 - 1699807676 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more