LMFDB - Embedded newform 588.8.a.m.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,8,Mod(1,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("588.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	588.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:	$183.682394985$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} - 20109 x^{6} - 684862 x^{5} + 88807787 x^{4} + 5739442252 x^{3} + 94265489852 x^{2} + \cdots - 3335964735712 $ x^8 - 2x^7 - 20109x^6 - 684862x^5 + 88807787x^4 + 5739442252x^3 + 94265489852x^2 + 181078465152*x - 3335964735712
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{17}\cdot 3^{4}\cdot 7^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$91.7291$ of defining polynomial
Character	$\chi$	$=$	588.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} +451.992 q^{5} +729.000 q^{9} -8081.02 q^{11} +3487.92 q^{13} -12203.8 q^{15} +15434.2 q^{17} +36856.6 q^{19} -61237.4 q^{23} +126172. q^{25} -19683.0 q^{27} -250085. q^{29} +122535. q^{31} +218188. q^{33} +120983. q^{37} -94173.9 q^{39} -395217. q^{41} +378681. q^{43} +329502. q^{45} -669990. q^{47} -416723. q^{51} -2.11871e6 q^{53} -3.65256e6 q^{55} -995128. q^{57} +987048. q^{59} +2.20973e6 q^{61} +1.57651e6 q^{65} -3.69012e6 q^{67} +1.65341e6 q^{69} -3.16876e6 q^{71} +1.68346e6 q^{73} -3.40664e6 q^{75} +2.33267e6 q^{79} +531441. q^{81} -4.57725e6 q^{83} +6.97612e6 q^{85} +6.75230e6 q^{87} +1.56713e6 q^{89} -3.30845e6 q^{93} +1.66589e7 q^{95} -3.36078e6 q^{97} -5.89106e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 216 q^{3} + 5832 q^{9} - 41184 q^{17} + 17280 q^{19} - 24000 q^{23} + 161064 q^{25} - 157464 q^{27} - 137568 q^{29} - 194400 q^{31} + 120320 q^{37} - 635040 q^{41} + 666400 q^{43} - 1004832 q^{47}+ \cdots - 20236608 q^{97}+O(q^{100}) $ 8 * q - 216 * q^3 + 5832 * q^9 - 41184 * q^17 + 17280 * q^19 - 24000 * q^23 + 161064 * q^25 - 157464 * q^27 - 137568 * q^29 - 194400 * q^31 + 120320 * q^37 - 635040 * q^41 + 666400 * q^43 - 1004832 * q^47 + 1111968 * q^51 + 611280 * q^53 - 550368 * q^55 - 466560 * q^57 + 987840 * q^59 + 4259520 * q^61 + 8519760 * q^65 + 4710240 * q^67 + 648000 * q^69 + 3941952 * q^71 + 3556224 * q^73 - 4348728 * q^75 + 14245312 * q^79 + 4251528 * q^81 - 2573280 * q^83 + 1923280 * q^85 + 3714336 * q^87 - 1378080 * q^89 + 5248800 * q^93 + 29140416 * q^95 - 20236608 * q^97

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	451.992	1.61710	0.808548	−	0.588430i	$-0.200254\pi$
$5$	451.992	1.61710	0.808548	+	0.588430i	$0.200254\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−8081.02	−1.83059	−0.915296	−	0.402781i	$-0.868044\pi$
$11$	−8081.02	−1.83059	−0.915296	+	0.402781i	$0.868044\pi$
$12$	0	0
$13$	3487.92	0.440316	0.220158	−	0.975464i	$-0.429343\pi$
$13$	3487.92	0.440316	0.220158	+	0.975464i	$0.429343\pi$
$14$	0	0
$15$	−12203.8	−0.933631
$16$	0	0
$17$	15434.2	0.761925	0.380962	−	0.924591i	$-0.375593\pi$
$17$	15434.2	0.761925	0.380962	+	0.924591i	$0.375593\pi$
$18$	0	0
$19$	36856.6	1.23276	0.616379	−	0.787450i	$-0.288599\pi$
$19$	36856.6	1.23276	0.616379	+	0.787450i	$0.288599\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−61237.4	−1.04947	−0.524734	−	0.851266i	$-0.675835\pi$
$23$	−61237.4	−1.04947	−0.524734	+	0.851266i	$0.675835\pi$
$24$	0	0
$25$	126172.	1.61500
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−250085.	−1.90412	−0.952062	−	0.305907i	$-0.901040\pi$
$29$	−250085.	−1.90412	−0.952062	+	0.305907i	$0.901040\pi$
$30$	0	0
$31$	122535.	0.738746	0.369373	−	0.929281i	$-0.379573\pi$
$31$	122535.	0.738746	0.369373	+	0.929281i	$0.379573\pi$
$32$	0	0
$33$	218188.	1.05689
$34$	0	0
$35$	0	0
$36$	0	0
$37$	120983.	0.392661	0.196330	−	0.980538i	$-0.437097\pi$
$37$	120983.	0.392661	0.196330	+	0.980538i	$0.437097\pi$
$38$	0	0
$39$	−94173.9	−0.254217
$40$	0	0
$41$	−395217.	−0.895554	−0.447777	−	0.894145i	$-0.647784\pi$
$41$	−395217.	−0.895554	−0.447777	+	0.894145i	$0.647784\pi$
$42$	0	0
$43$	378681.	0.726330	0.363165	−	0.931725i	$-0.381696\pi$
$43$	378681.	0.726330	0.363165	+	0.931725i	$0.381696\pi$
$44$	0	0
$45$	329502.	0.539032
$46$	0	0
$47$	−669990.	−0.941295	−0.470647	−	0.882321i	$-0.655980\pi$
$47$	−669990.	−0.941295	−0.470647	+	0.882321i	$0.655980\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−416723.	−0.439897
$52$	0	0
$53$	−2.11871e6	−1.95482	−0.977409	−	0.211358i	$-0.932212\pi$
$53$	−2.11871e6	−1.95482	−0.977409	+	0.211358i	$0.932212\pi$
$54$	0	0
$55$	−3.65256e6	−2.96024
$56$	0	0
$57$	−995128.	−0.711733
$58$	0	0
$59$	987048.	0.625686	0.312843	−	0.949805i	$-0.398719\pi$
$59$	987048.	0.625686	0.312843	+	0.949805i	$0.398719\pi$
$60$	0	0
$61$	2.20973e6	1.24648	0.623239	−	0.782031i	$-0.285816\pi$
$61$	2.20973e6	1.24648	0.623239	+	0.782031i	$0.285816\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.57651e6	0.712034
$66$	0	0
$67$	−3.69012e6	−1.49892	−0.749461	−	0.662049i	$-0.769687\pi$
$67$	−3.69012e6	−1.49892	−0.749461	+	0.662049i	$0.769687\pi$
$68$	0	0
$69$	1.65341e6	0.605911
$70$	0	0
$71$	−3.16876e6	−1.05072	−0.525358	−	0.850881i	$-0.676068\pi$
$71$	−3.16876e6	−1.05072	−0.525358	+	0.850881i	$0.676068\pi$
$72$	0	0
$73$	1.68346e6	0.506492	0.253246	−	0.967402i	$-0.418502\pi$
$73$	1.68346e6	0.506492	0.253246	+	0.967402i	$0.418502\pi$
$74$	0	0
$75$	−3.40664e6	−0.932421
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.33267e6	0.532303	0.266152	−	0.963931i	$-0.414248\pi$
$79$	2.33267e6	0.532303	0.266152	+	0.963931i	$0.414248\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−4.57725e6	−0.878682	−0.439341	−	0.898320i	$-0.644788\pi$
$83$	−4.57725e6	−0.878682	−0.439341	+	0.898320i	$0.644788\pi$
$84$	0	0
$85$	6.97612e6	1.23211
$86$	0	0
$87$	6.75230e6	1.09935
$88$	0	0
$89$	1.56713e6	0.235635	0.117817	−	0.993035i	$-0.462410\pi$
$89$	1.56713e6	0.235635	0.117817	+	0.993035i	$0.462410\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.30845e6	−0.426515
$94$	0	0
$95$	1.66589e7	1.99349
$96$	0	0
$97$	−3.36078e6	−0.373886	−0.186943	−	0.982371i	$-0.559858\pi$
$97$	−3.36078e6	−0.373886	−0.186943	+	0.982371i	$0.559858\pi$
$98$	0	0
$99$	−5.89106e6	−0.610198
$100$	0	0
$101$	1.58607e7	1.53178	0.765892	−	0.642969i	$-0.222298\pi$
$101$	1.58607e7	1.53178	0.765892	+	0.642969i	$0.222298\pi$
$102$	0	0
$103$	−1.49495e6	−0.134802	−0.0674008	−	0.997726i	$-0.521471\pi$
$103$	−1.49495e6	−0.134802	−0.0674008	+	0.997726i	$0.521471\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.63742e7	−1.29216	−0.646081	−	0.763269i	$-0.723593\pi$
$107$	−1.63742e7	−1.29216	−0.646081	+	0.763269i	$0.723593\pi$
$108$	0	0
$109$	8.19567e6	0.606166	0.303083	−	0.952964i	$-0.401984\pi$
$109$	8.19567e6	0.606166	0.303083	+	0.952964i	$0.401984\pi$
$110$	0	0
$111$	−3.26654e6	−0.226703
$112$	0	0
$113$	−3.84294e6	−0.250547	−0.125273	−	0.992122i	$-0.539981\pi$
$113$	−3.84294e6	−0.250547	−0.125273	+	0.992122i	$0.539981\pi$
$114$	0	0
$115$	−2.76788e7	−1.69709
$116$	0	0
$117$	2.54269e6	0.146772
$118$	0	0
$119$	0	0
$120$	0	0
$121$	4.58157e7	2.35107
$122$	0	0
$123$	1.06708e7	0.517048
$124$	0	0
$125$	2.17168e7	0.994515
$126$	0	0
$127$	3.97392e7	1.72150	0.860748	−	0.509032i	$-0.169996\pi$
$127$	3.97392e7	1.72150	0.860748	+	0.509032i	$0.169996\pi$
$128$	0	0
$129$	−1.02244e7	−0.419347
$130$	0	0
$131$	−2.75018e7	−1.06884	−0.534418	−	0.845220i	$-0.679469\pi$
$131$	−2.75018e7	−1.06884	−0.534418	+	0.845220i	$0.679469\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.89656e6	−0.311210
$136$	0	0
$137$	2.65279e7	0.881415	0.440708	−	0.897651i	$-0.354728\pi$
$137$	2.65279e7	0.881415	0.440708	+	0.897651i	$0.354728\pi$
$138$	0	0
$139$	1.43539e7	0.453335	0.226668	−	0.973972i	$-0.427217\pi$
$139$	1.43539e7	0.453335	0.226668	+	0.973972i	$0.427217\pi$
$140$	0	0
$141$	1.80897e7	0.543457
$142$	0	0
$143$	−2.81860e7	−0.806040
$144$	0	0
$145$	−1.13037e8	−3.07915
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.49065e7	0.369168	0.184584	−	0.982817i	$-0.440906\pi$
$149$	1.49065e7	0.369168	0.184584	+	0.982817i	$0.440906\pi$
$150$	0	0
$151$	−5.46913e7	−1.29270	−0.646351	−	0.763040i	$-0.723706\pi$
$151$	−5.46913e7	−1.29270	−0.646351	+	0.763040i	$0.723706\pi$
$152$	0	0
$153$	1.12515e7	0.253975
$154$	0	0
$155$	5.53850e7	1.19462
$156$	0	0
$157$	2.09060e7	0.431144	0.215572	−	0.976488i	$-0.430838\pi$
$157$	2.09060e7	0.431144	0.215572	+	0.976488i	$0.430838\pi$
$158$	0	0
$159$	5.72052e7	1.12861
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−7.99281e7	−1.44558	−0.722791	−	0.691067i	$-0.757141\pi$
$163$	−7.99281e7	−1.44558	−0.722791	+	0.691067i	$0.757141\pi$
$164$	0	0
$165$	9.86191e7	1.70910
$166$	0	0
$167$	−2.74846e7	−0.456648	−0.228324	−	0.973585i	$-0.573324\pi$
$167$	−2.74846e7	−0.456648	−0.228324	+	0.973585i	$0.573324\pi$
$168$	0	0
$169$	−5.05829e7	−0.806121
$170$	0	0
$171$	2.68685e7	0.410919
$172$	0	0
$173$	1.04806e8	1.53896	0.769479	−	0.638672i	$-0.220516\pi$
$173$	1.04806e8	1.53896	0.769479	+	0.638672i	$0.220516\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.66503e7	−0.361240
$178$	0	0
$179$	3.73656e7	0.486952	0.243476	−	0.969907i	$-0.421712\pi$
$179$	3.73656e7	0.486952	0.243476	+	0.969907i	$0.421712\pi$
$180$	0	0
$181$	−1.21511e8	−1.52314	−0.761572	−	0.648080i	$-0.775572\pi$
$181$	−1.21511e8	−1.52314	−0.761572	+	0.648080i	$0.775572\pi$
$182$	0	0
$183$	−5.96627e7	−0.719655
$184$	0	0
$185$	5.46833e7	0.634970
$186$	0	0
$187$	−1.24724e8	−1.39477
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.79444e8	−1.86343	−0.931715	−	0.363189i	$-0.881688\pi$
$191$	−1.79444e8	−1.86343	−0.931715	+	0.363189i	$0.881688\pi$
$192$	0	0
$193$	−1.13939e8	−1.14083	−0.570417	−	0.821356i	$-0.693218\pi$
$193$	−1.13939e8	−1.14083	−0.570417	+	0.821356i	$0.693218\pi$
$194$	0	0
$195$	−4.25658e7	−0.411093
$196$	0	0
$197$	4.15855e7	0.387534	0.193767	−	0.981048i	$-0.437929\pi$
$197$	4.15855e7	0.387534	0.193767	+	0.981048i	$0.437929\pi$
$198$	0	0
$199$	−1.61371e8	−1.45158	−0.725789	−	0.687917i	$-0.758525\pi$
$199$	−1.61371e8	−1.45158	−0.725789	+	0.687917i	$0.758525\pi$
$200$	0	0
$201$	9.96333e7	0.865403
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.78635e8	−1.44820
$206$	0	0
$207$	−4.46421e7	−0.349823
$208$	0	0
$209$	−2.97839e8	−2.25668
$210$	0	0
$211$	5.72672e7	0.419679	0.209840	−	0.977736i	$-0.432706\pi$
$211$	5.72672e7	0.419679	0.209840	+	0.977736i	$0.432706\pi$
$212$	0	0
$213$	8.55565e7	0.606631
$214$	0	0
$215$	1.71161e8	1.17455
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.54534e7	−0.292423
$220$	0	0
$221$	5.38332e7	0.335488
$222$	0	0
$223$	4.48478e7	0.270816	0.135408	−	0.990790i	$-0.456765\pi$
$223$	4.48478e7	0.270816	0.135408	+	0.990790i	$0.456765\pi$
$224$	0	0
$225$	9.19793e7	0.538333
$226$	0	0
$227$	−2.32612e7	−0.131990	−0.0659951	−	0.997820i	$-0.521022\pi$
$227$	−2.32612e7	−0.131990	−0.0659951	+	0.997820i	$0.521022\pi$
$228$	0	0
$229$	−2.26500e8	−1.24636	−0.623181	−	0.782077i	$-0.714160\pi$
$229$	−2.26500e8	−1.24636	−0.623181	+	0.782077i	$0.714160\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.29764e8	−0.672059	−0.336030	−	0.941851i	$-0.609084\pi$
$233$	−1.29764e8	−0.672059	−0.336030	+	0.941851i	$0.609084\pi$
$234$	0	0
$235$	−3.02830e8	−1.52216
$236$	0	0
$237$	−6.29822e7	−0.307325
$238$	0	0
$239$	−3.78090e8	−1.79144	−0.895721	−	0.444616i	$-0.853340\pi$
$239$	−3.78090e8	−1.79144	−0.895721	+	0.444616i	$0.853340\pi$
$240$	0	0
$241$	−1.23504e8	−0.568359	−0.284180	−	0.958771i	$-0.591721\pi$
$241$	−1.23504e8	−0.568359	−0.284180	+	0.958771i	$0.591721\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.28553e8	0.542803
$248$	0	0
$249$	1.23586e8	0.507307
$250$	0	0
$251$	−8.74207e7	−0.348945	−0.174472	−	0.984662i	$-0.555822\pi$
$251$	−8.74207e7	−0.348945	−0.174472	+	0.984662i	$0.555822\pi$
$252$	0	0
$253$	4.94861e8	1.92115
$254$	0	0
$255$	−1.88355e8	−0.711356
$256$	0	0
$257$	−5.22709e8	−1.92085	−0.960426	−	0.278536i	$-0.910151\pi$
$257$	−5.22709e8	−1.92085	−0.960426	+	0.278536i	$0.910151\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.82312e8	−0.634708
$262$	0	0
$263$	5.32122e7	0.180371	0.0901854	−	0.995925i	$-0.471254\pi$
$263$	5.32122e7	0.180371	0.0901854	+	0.995925i	$0.471254\pi$
$264$	0	0
$265$	−9.57641e8	−3.16113
$266$	0	0
$267$	−4.23124e7	−0.136044
$268$	0	0
$269$	−1.51009e7	−0.0473010	−0.0236505	−	0.999720i	$-0.507529\pi$
$269$	−1.51009e7	−0.0473010	−0.0236505	+	0.999720i	$0.507529\pi$
$270$	0	0
$271$	−1.46208e7	−0.0446250	−0.0223125	−	0.999751i	$-0.507103\pi$
$271$	−1.46208e7	−0.0446250	−0.0223125	+	0.999751i	$0.507103\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.01960e9	−2.95641
$276$	0	0
$277$	2.84940e8	0.805515	0.402758	−	0.915307i	$-0.368052\pi$
$277$	2.84940e8	0.805515	0.402758	+	0.915307i	$0.368052\pi$
$278$	0	0
$279$	8.93282e7	0.246249
$280$	0	0
$281$	5.90301e8	1.58709	0.793544	−	0.608512i	$-0.208233\pi$
$281$	5.90301e8	1.58709	0.793544	+	0.608512i	$0.208233\pi$
$282$	0	0
$283$	−2.42565e8	−0.636174	−0.318087	−	0.948062i	$-0.603040\pi$
$283$	−2.42565e8	−0.636174	−0.318087	+	0.948062i	$0.603040\pi$
$284$	0	0
$285$	−4.49790e8	−1.15094
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.72125e8	−0.419471
$290$	0	0
$291$	9.07411e7	0.215863
$292$	0	0
$293$	3.11797e8	0.724162	0.362081	−	0.932147i	$-0.382066\pi$
$293$	3.11797e8	0.724162	0.362081	+	0.932147i	$0.382066\pi$
$294$	0	0
$295$	4.46138e8	1.01179
$296$	0	0
$297$	1.59059e8	0.352298
$298$	0	0
$299$	−2.13591e8	−0.462098
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.28239e8	−0.884376
$304$	0	0
$305$	9.98781e8	2.01568
$306$	0	0
$307$	8.03650e8	1.58519	0.792597	−	0.609746i	$-0.208728\pi$
$307$	8.03650e8	1.58519	0.792597	+	0.609746i	$0.208728\pi$
$308$	0	0
$309$	4.03635e7	0.0778278
$310$	0	0
$311$	−3.73497e8	−0.704087	−0.352044	−	0.935984i	$-0.614513\pi$
$311$	−3.73497e8	−0.704087	−0.352044	+	0.935984i	$0.614513\pi$
$312$	0	0
$313$	7.41945e8	1.36762	0.683812	−	0.729658i	$-0.260321\pi$
$313$	7.41945e8	1.36762	0.683812	+	0.729658i	$0.260321\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.58238e8	0.984264	0.492132	−	0.870521i	$-0.336218\pi$
$317$	5.58238e8	0.984264	0.492132	+	0.870521i	$0.336218\pi$
$318$	0	0
$319$	2.02094e9	3.48567
$320$	0	0
$321$	4.42104e8	0.746030
$322$	0	0
$323$	5.68851e8	0.939268
$324$	0	0
$325$	4.40078e8	0.711111
$326$	0	0
$327$	−2.21283e8	−0.349970
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.12116e9	−1.69930	−0.849652	−	0.527344i	$-0.823188\pi$
$331$	−1.12116e9	−1.69930	−0.849652	+	0.527344i	$0.823188\pi$
$332$	0	0
$333$	8.81964e7	0.130887
$334$	0	0
$335$	−1.66791e9	−2.42390
$336$	0	0
$337$	1.50796e8	0.214627	0.107313	−	0.994225i	$-0.465775\pi$
$337$	1.50796e8	0.214627	0.107313	+	0.994225i	$0.465775\pi$
$338$	0	0
$339$	1.03759e8	0.144653
$340$	0	0
$341$	−9.90209e8	−1.35234
$342$	0	0
$343$	0	0
$344$	0	0
$345$	7.47329e8	0.979817
$346$	0	0
$347$	−1.30570e9	−1.67761	−0.838806	−	0.544430i	$-0.816746\pi$
$347$	−1.30570e9	−1.67761	−0.838806	+	0.544430i	$0.816746\pi$
$348$	0	0
$349$	−7.75432e8	−0.976460	−0.488230	−	0.872715i	$-0.662357\pi$
$349$	−7.75432e8	−0.976460	−0.488230	+	0.872715i	$0.662357\pi$
$350$	0	0
$351$	−6.86527e7	−0.0847389
$352$	0	0
$353$	3.31269e8	0.400839	0.200419	−	0.979710i	$-0.435770\pi$
$353$	3.31269e8	0.400839	0.200419	+	0.979710i	$0.435770\pi$
$354$	0	0
$355$	−1.43225e9	−1.69911
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.26793e7	−0.0600910	−0.0300455	−	0.999549i	$-0.509565\pi$
$359$	−5.26793e7	−0.0600910	−0.0300455	+	0.999549i	$0.509565\pi$
$360$	0	0
$361$	4.64537e8	0.519691
$362$	0	0
$363$	−1.23702e9	−1.35739
$364$	0	0
$365$	7.60911e8	0.819047
$366$	0	0
$367$	−5.82310e8	−0.614926	−0.307463	−	0.951560i	$-0.599480\pi$
$367$	−5.82310e8	−0.614926	−0.307463	+	0.951560i	$0.599480\pi$
$368$	0	0
$369$	−2.88113e8	−0.298518
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−9.18792e8	−0.916719	−0.458359	−	0.888767i	$-0.651563\pi$
$373$	−9.18792e8	−0.916719	−0.458359	+	0.888767i	$0.651563\pi$
$374$	0	0
$375$	−5.86354e8	−0.574183
$376$	0	0
$377$	−8.72277e8	−0.838417
$378$	0	0
$379$	−9.14421e7	−0.0862798	−0.0431399	−	0.999069i	$-0.513736\pi$
$379$	−9.14421e7	−0.0862798	−0.0431399	+	0.999069i	$0.513736\pi$
$380$	0	0
$381$	−1.07296e9	−0.993906
$382$	0	0
$383$	−9.01192e8	−0.819637	−0.409819	−	0.912167i	$-0.634408\pi$
$383$	−9.01192e8	−0.819637	−0.409819	+	0.912167i	$0.634408\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.76059e8	0.242110
$388$	0	0
$389$	−1.32341e9	−1.13991	−0.569955	−	0.821676i	$-0.693039\pi$
$389$	−1.32341e9	−1.13991	−0.569955	+	0.821676i	$0.693039\pi$
$390$	0	0
$391$	−9.45149e8	−0.799616
$392$	0	0
$393$	7.42548e8	0.617093
$394$	0	0
$395$	1.05435e9	0.860785
$396$	0	0
$397$	4.77243e8	0.382801	0.191400	−	0.981512i	$-0.438697\pi$
$397$	4.77243e8	0.382801	0.191400	+	0.981512i	$0.438697\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.20207e9	−0.930943	−0.465471	−	0.885063i	$-0.654115\pi$
$401$	−1.20207e9	−0.930943	−0.465471	+	0.885063i	$0.654115\pi$
$402$	0	0
$403$	4.27393e8	0.325282
$404$	0	0
$405$	2.40207e8	0.179677
$406$	0	0
$407$	−9.77664e8	−0.718802
$408$	0	0
$409$	2.37094e9	1.71352	0.856758	−	0.515718i	$-0.172475\pi$
$409$	2.37094e9	1.71352	0.856758	+	0.515718i	$0.172475\pi$
$410$	0	0
$411$	−7.16253e8	−0.508885
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.06888e9	−1.42091
$416$	0	0
$417$	−3.87557e8	−0.261733
$418$	0	0
$419$	−1.31313e9	−0.872087	−0.436044	−	0.899926i	$-0.643621\pi$
$419$	−1.31313e9	−0.872087	−0.436044	+	0.899926i	$0.643621\pi$
$420$	0	0
$421$	−1.94154e9	−1.26811	−0.634057	−	0.773286i	$-0.718612\pi$
$421$	−1.94154e9	−1.26811	−0.634057	+	0.773286i	$0.718612\pi$
$422$	0	0
$423$	−4.88422e8	−0.313765
$424$	0	0
$425$	1.94736e9	1.23051
$426$	0	0
$427$	0	0
$428$	0	0
$429$	7.61021e8	0.465367
$430$	0	0
$431$	1.16634e9	0.701705	0.350852	−	0.936431i	$-0.385892\pi$
$431$	1.16634e9	0.701705	0.350852	+	0.936431i	$0.385892\pi$
$432$	0	0
$433$	−1.43804e9	−0.851260	−0.425630	−	0.904897i	$-0.639947\pi$
$433$	−1.43804e9	−0.851260	−0.425630	+	0.904897i	$0.639947\pi$
$434$	0	0
$435$	3.05199e9	1.77775
$436$	0	0
$437$	−2.25700e9	−1.29374
$438$	0	0
$439$	8.07279e8	0.455405	0.227702	−	0.973731i	$-0.426879\pi$
$439$	8.07279e8	0.455405	0.227702	+	0.973731i	$0.426879\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.35867e8	0.238199	0.119100	−	0.992882i	$-0.461999\pi$
$443$	4.35867e8	0.238199	0.119100	+	0.992882i	$0.461999\pi$
$444$	0	0
$445$	7.08329e8	0.381044
$446$	0	0
$447$	−4.02475e8	−0.213139
$448$	0	0
$449$	−3.29463e8	−0.171769	−0.0858845	−	0.996305i	$-0.527372\pi$
$449$	−3.29463e8	−0.171769	−0.0858845	+	0.996305i	$0.527372\pi$
$450$	0	0
$451$	3.19375e9	1.63939
$452$	0	0
$453$	1.47666e9	0.746342
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.41800e9	−0.694976	−0.347488	−	0.937684i	$-0.612965\pi$
$457$	−1.41800e9	−0.694976	−0.347488	+	0.937684i	$0.612965\pi$
$458$	0	0
$459$	−3.03791e8	−0.146632
$460$	0	0
$461$	−2.40881e9	−1.14512	−0.572558	−	0.819864i	$-0.694049\pi$
$461$	−2.40881e9	−1.14512	−0.572558	+	0.819864i	$0.694049\pi$
$462$	0	0
$463$	−3.46303e9	−1.62152	−0.810762	−	0.585376i	$-0.800947\pi$
$463$	−3.46303e9	−1.62152	−0.810762	+	0.585376i	$0.800947\pi$
$464$	0	0
$465$	−1.49539e9	−0.689716
$466$	0	0
$467$	6.52909e8	0.296649	0.148325	−	0.988939i	$-0.452612\pi$
$467$	6.52909e8	0.296649	0.148325	+	0.988939i	$0.452612\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.64462e8	−0.248921
$472$	0	0
$473$	−3.06013e9	−1.32962
$474$	0	0
$475$	4.65027e9	1.99090
$476$	0	0
$477$	−1.54454e9	−0.651606
$478$	0	0
$479$	2.95873e9	1.23007	0.615036	−	0.788499i	$-0.289141\pi$
$479$	2.95873e9	1.23007	0.615036	+	0.788499i	$0.289141\pi$
$480$	0	0
$481$	4.21978e8	0.172895
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.51905e9	−0.604609
$486$	0	0
$487$	1.11825e9	0.438720	0.219360	−	0.975644i	$-0.429603\pi$
$487$	1.11825e9	0.438720	0.219360	+	0.975644i	$0.429603\pi$
$488$	0	0
$489$	2.15806e9	0.834607
$490$	0	0
$491$	8.64451e8	0.329576	0.164788	−	0.986329i	$-0.447306\pi$
$491$	8.64451e8	0.329576	0.164788	+	0.986329i	$0.447306\pi$
$492$	0	0
$493$	−3.85986e9	−1.45080
$494$	0	0
$495$	−2.66271e9	−0.986748
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.12167e9	1.84527	0.922634	−	0.385677i	$-0.126032\pi$
$499$	5.12167e9	1.84527	0.922634	+	0.385677i	$0.126032\pi$
$500$	0	0
$501$	7.42083e8	0.263646
$502$	0	0
$503$	−4.49361e8	−0.157437	−0.0787186	−	0.996897i	$-0.525083\pi$
$503$	−4.49361e8	−0.157437	−0.0787186	+	0.996897i	$0.525083\pi$
$504$	0	0
$505$	7.16891e9	2.47704
$506$	0	0
$507$	1.36574e9	0.465414
$508$	0	0
$509$	3.08554e9	1.03710	0.518549	−	0.855048i	$-0.326472\pi$
$509$	3.08554e9	1.03710	0.518549	+	0.855048i	$0.326472\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.25448e8	−0.237244
$514$	0	0
$515$	−6.75704e8	−0.217987
$516$	0	0
$517$	5.41420e9	1.72313
$518$	0	0
$519$	−2.82977e9	−0.888518
$520$	0	0
$521$	−2.85515e9	−0.884498	−0.442249	−	0.896892i	$-0.645819\pi$
$521$	−2.85515e9	−0.884498	−0.442249	+	0.896892i	$0.645819\pi$
$522$	0	0
$523$	1.82294e8	0.0557205	0.0278603	−	0.999612i	$-0.491131\pi$
$523$	1.82294e8	0.0557205	0.0278603	+	0.999612i	$0.491131\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.89123e9	0.562869
$528$	0	0
$529$	3.45197e8	0.101385
$530$	0	0
$531$	7.19558e8	0.208562
$532$	0	0
$533$	−1.37848e9	−0.394327
$534$	0	0
$535$	−7.40101e9	−2.08955
$536$	0	0
$537$	−1.00887e9	−0.281142
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.95494e9	−1.34539	−0.672694	−	0.739920i	$-0.734863\pi$
$541$	−4.95494e9	−1.34539	−0.672694	+	0.739920i	$0.734863\pi$
$542$	0	0
$543$	3.28080e9	0.879388
$544$	0	0
$545$	3.70438e9	0.980229
$546$	0	0
$547$	9.68687e8	0.253063	0.126531	−	0.991963i	$-0.459616\pi$
$547$	9.68687e8	0.253063	0.126531	+	0.991963i	$0.459616\pi$
$548$	0	0
$549$	1.61089e9	0.415493
$550$	0	0
$551$	−9.21729e9	−2.34732
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.47645e9	−0.366600
$556$	0	0
$557$	−3.02958e9	−0.742829	−0.371415	−	0.928467i	$-0.621127\pi$
$557$	−3.02958e9	−0.742829	−0.371415	+	0.928467i	$0.621127\pi$
$558$	0	0
$559$	1.32081e9	0.319815
$560$	0	0
$561$	3.36754e9	0.805273
$562$	0	0
$563$	−5.40383e9	−1.27621	−0.638105	−	0.769949i	$-0.720281\pi$
$563$	−5.40383e9	−1.27621	−0.638105	+	0.769949i	$0.720281\pi$
$564$	0	0
$565$	−1.73698e9	−0.405158
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.42982e9	1.00807	0.504037	−	0.863682i	$-0.331847\pi$
$569$	4.42982e9	1.00807	0.504037	+	0.863682i	$0.331847\pi$
$570$	0	0
$571$	−1.50129e9	−0.337473	−0.168737	−	0.985661i	$-0.553969\pi$
$571$	−1.50129e9	−0.337473	−0.168737	+	0.985661i	$0.553969\pi$
$572$	0	0
$573$	4.84500e9	1.07585
$574$	0	0
$575$	−7.72644e9	−1.69489
$576$	0	0
$577$	−5.66760e9	−1.22824	−0.614121	−	0.789212i	$-0.710489\pi$
$577$	−5.66760e9	−1.22824	−0.614121	+	0.789212i	$0.710489\pi$
$578$	0	0
$579$	3.07636e9	0.658660
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.71213e10	3.57848
$584$	0	0
$585$	1.14928e9	0.237345
$586$	0	0
$587$	5.00583e9	1.02151	0.510755	−	0.859726i	$-0.329366\pi$
$587$	5.00583e9	1.02151	0.510755	+	0.859726i	$0.329366\pi$
$588$	0	0
$589$	4.51623e9	0.910694
$590$	0	0
$591$	−1.12281e9	−0.223743
$592$	0	0
$593$	9.33774e9	1.83887	0.919433	−	0.393246i	$-0.128648\pi$
$593$	9.33774e9	1.83887	0.919433	+	0.393246i	$0.128648\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.35703e9	0.838069
$598$	0	0
$599$	−3.39260e9	−0.644969	−0.322485	−	0.946575i	$-0.604518\pi$
$599$	−3.39260e9	−0.644969	−0.322485	+	0.946575i	$0.604518\pi$
$600$	0	0
$601$	−3.29915e9	−0.619928	−0.309964	−	0.950748i	$-0.600317\pi$
$601$	−3.29915e9	−0.619928	−0.309964	+	0.950748i	$0.600317\pi$
$602$	0	0
$603$	−2.69010e9	−0.499641
$604$	0	0
$605$	2.07083e10	3.80191
$606$	0	0
$607$	7.78056e9	1.41205	0.706026	−	0.708186i	$-0.250486\pi$
$607$	7.78056e9	1.41205	0.706026	+	0.708186i	$0.250486\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.33687e9	−0.414468
$612$	0	0
$613$	−2.54469e9	−0.446193	−0.223096	−	0.974796i	$-0.571616\pi$
$613$	−2.54469e9	−0.446193	−0.223096	+	0.974796i	$0.571616\pi$
$614$	0	0
$615$	4.82314e9	0.836117
$616$	0	0
$617$	−9.21447e9	−1.57933	−0.789664	−	0.613540i	$-0.789745\pi$
$617$	−9.21447e9	−1.57933	−0.789664	+	0.613540i	$0.789745\pi$
$618$	0	0
$619$	8.12717e9	1.37728	0.688639	−	0.725104i	$-0.258208\pi$
$619$	8.12717e9	1.37728	0.688639	+	0.725104i	$0.258208\pi$
$620$	0	0
$621$	1.20534e9	0.201970
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.13481e7	−0.00677447
$626$	0	0
$627$	8.04165e9	1.30289
$628$	0	0
$629$	1.86727e9	0.299178
$630$	0	0
$631$	−4.05269e9	−0.642156	−0.321078	−	0.947053i	$-0.604045\pi$
$631$	−4.05269e9	−0.642156	−0.321078	+	0.947053i	$0.604045\pi$
$632$	0	0
$633$	−1.54621e9	−0.242302
$634$	0	0
$635$	1.79618e10	2.78382
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.31003e9	−0.350238
$640$	0	0
$641$	−3.49281e8	−0.0523807	−0.0261903	−	0.999657i	$-0.508338\pi$
$641$	−3.49281e8	−0.0523807	−0.0261903	+	0.999657i	$0.508338\pi$
$642$	0	0
$643$	−7.03731e8	−0.104392	−0.0521961	−	0.998637i	$-0.516622\pi$
$643$	−7.03731e8	−0.104392	−0.0521961	+	0.998637i	$0.516622\pi$
$644$	0	0
$645$	−4.62135e9	−0.678125
$646$	0	0
$647$	1.23333e10	1.79025	0.895125	−	0.445816i	$-0.147086\pi$
$647$	1.23333e10	1.79025	0.895125	+	0.445816i	$0.147086\pi$
$648$	0	0
$649$	−7.97635e9	−1.14538
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.73708e9	−0.665756	−0.332878	−	0.942970i	$-0.608020\pi$
$653$	−4.73708e9	−0.665756	−0.332878	+	0.942970i	$0.608020\pi$
$654$	0	0
$655$	−1.24306e10	−1.72841
$656$	0	0
$657$	1.22724e9	0.168831
$658$	0	0
$659$	−2.34642e9	−0.319380	−0.159690	−	0.987167i	$-0.551049\pi$
$659$	−2.34642e9	−0.319380	−0.159690	+	0.987167i	$0.551049\pi$
$660$	0	0
$661$	−5.45268e9	−0.734353	−0.367176	−	0.930151i	$-0.619675\pi$
$661$	−5.45268e9	−0.734353	−0.367176	+	0.930151i	$0.619675\pi$
$662$	0	0
$663$	−1.45350e9	−0.193694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.53146e10	1.99832
$668$	0	0
$669$	−1.21089e9	−0.156356
$670$	0	0
$671$	−1.78569e10	−2.28179
$672$	0	0
$673$	4.55675e9	0.576239	0.288120	−	0.957594i	$-0.406970\pi$
$673$	4.55675e9	0.576239	0.288120	+	0.957594i	$0.406970\pi$
$674$	0	0
$675$	−2.48344e9	−0.310807
$676$	0	0
$677$	4.62323e9	0.572645	0.286322	−	0.958133i	$-0.407567\pi$
$677$	4.62323e9	0.572645	0.286322	+	0.958133i	$0.407567\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6.28052e8	0.0762045
$682$	0	0
$683$	−1.35281e9	−0.162467	−0.0812335	−	0.996695i	$-0.525886\pi$
$683$	−1.35281e9	−0.162467	−0.0812335	+	0.996695i	$0.525886\pi$
$684$	0	0
$685$	1.19904e10	1.42533
$686$	0	0
$687$	6.11551e9	0.719588
$688$	0	0
$689$	−7.38990e9	−0.860738
$690$	0	0
$691$	−1.32531e10	−1.52807	−0.764037	−	0.645172i	$-0.776786\pi$
$691$	−1.32531e10	−1.52807	−0.764037	+	0.645172i	$0.776786\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.48787e9	0.733087
$696$	0	0
$697$	−6.09984e9	−0.682344
$698$	0	0
$699$	3.50362e9	0.388014
$700$	0	0
$701$	−1.15263e10	−1.26380	−0.631900	−	0.775050i	$-0.717725\pi$
$701$	−1.15263e10	−1.26380	−0.631900	+	0.775050i	$0.717725\pi$
$702$	0	0
$703$	4.45901e9	0.484055
$704$	0	0
$705$	8.17641e9	0.878822
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.35442e9	0.774973	0.387487	−	0.921875i	$-0.373343\pi$
$709$	7.35442e9	0.774973	0.387487	+	0.921875i	$0.373343\pi$
$710$	0	0
$711$	1.70052e9	0.177434
$712$	0	0
$713$	−7.50374e9	−0.775291
$714$	0	0
$715$	−1.27398e10	−1.30344
$716$	0	0
$717$	1.02084e10	1.03429
$718$	0	0
$719$	6.01484e9	0.603494	0.301747	−	0.953388i	$-0.402430\pi$
$719$	6.01484e9	0.603494	0.301747	+	0.953388i	$0.402430\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.33462e9	0.328142
$724$	0	0
$725$	−3.15537e10	−3.07516
$726$	0	0
$727$	1.92599e10	1.85902	0.929509	−	0.368800i	$-0.120231\pi$
$727$	1.92599e10	1.85902	0.929509	+	0.368800i	$0.120231\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	5.84463e9	0.553409
$732$	0	0
$733$	−1.88260e10	−1.76561	−0.882804	−	0.469741i	$-0.844347\pi$
$733$	−1.88260e10	−1.76561	−0.882804	+	0.469741i	$0.844347\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.98199e10	2.74392
$738$	0	0
$739$	−4.33280e9	−0.394924	−0.197462	−	0.980311i	$-0.563270\pi$
$739$	−4.33280e9	−0.394924	−0.197462	+	0.980311i	$0.563270\pi$
$740$	0	0
$741$	−3.47093e9	−0.313388
$742$	0	0
$743$	−3.47938e8	−0.0311201	−0.0155600	−	0.999879i	$-0.504953\pi$
$743$	−3.47938e8	−0.0311201	−0.0155600	+	0.999879i	$0.504953\pi$
$744$	0	0
$745$	6.73762e9	0.596980
$746$	0	0
$747$	−3.33682e9	−0.292894
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−6.35798e9	−0.547746	−0.273873	−	0.961766i	$-0.588305\pi$
$751$	−6.35798e9	−0.547746	−0.273873	+	0.961766i	$0.588305\pi$
$752$	0	0
$753$	2.36036e9	0.201463
$754$	0	0
$755$	−2.47200e10	−2.09042
$756$	0	0
$757$	−6.43605e9	−0.539242	−0.269621	−	0.962966i	$-0.586898\pi$
$757$	−6.43605e9	−0.539242	−0.269621	+	0.962966i	$0.586898\pi$
$758$	0	0
$759$	−1.33612e10	−1.10918
$760$	0	0
$761$	1.94038e10	1.59602	0.798012	−	0.602642i	$-0.205885\pi$
$761$	1.94038e10	1.59602	0.798012	+	0.602642i	$0.205885\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.08559e9	0.410702
$766$	0	0
$767$	3.44274e9	0.275500
$768$	0	0
$769$	1.02717e10	0.814520	0.407260	−	0.913312i	$-0.366484\pi$
$769$	1.02717e10	0.814520	0.407260	+	0.913312i	$0.366484\pi$
$770$	0	0
$771$	1.41131e10	1.10900
$772$	0	0
$773$	2.40473e10	1.87257	0.936287	−	0.351237i	$-0.114239\pi$
$773$	2.40473e10	1.87257	0.936287	+	0.351237i	$0.114239\pi$
$774$	0	0
$775$	1.54605e10	1.19307
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.45663e10	−1.10400
$780$	0	0
$781$	2.56068e10	1.92343
$782$	0	0
$783$	4.92243e9	0.366449
$784$	0	0
$785$	9.44934e9	0.697201
$786$	0	0
$787$	−1.64642e10	−1.20401	−0.602003	−	0.798494i	$-0.705631\pi$
$787$	−1.64642e10	−1.20401	−0.602003	+	0.798494i	$0.705631\pi$
$788$	0	0
$789$	−1.43673e9	−0.104137
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.70736e9	0.548845
$794$	0	0
$795$	2.58563e10	1.82508
$796$	0	0
$797$	2.76494e10	1.93456	0.967279	−	0.253716i	$-0.0816530\pi$
$797$	2.76494e10	1.93456	0.967279	+	0.253716i	$0.0816530\pi$
$798$	0	0
$799$	−1.03407e10	−0.717196
$800$	0	0
$801$	1.14244e9	0.0785449
$802$	0	0
$803$	−1.36041e10	−0.927181
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.07725e8	0.0273093
$808$	0	0
$809$	−4.64507e9	−0.308441	−0.154221	−	0.988036i	$-0.549287\pi$
$809$	−4.64507e9	−0.308441	−0.154221	+	0.988036i	$0.549287\pi$
$810$	0	0
$811$	1.23997e10	0.816280	0.408140	−	0.912919i	$-0.366178\pi$
$811$	1.23997e10	0.816280	0.408140	+	0.912919i	$0.366178\pi$
$812$	0	0
$813$	3.94762e8	0.0257643
$814$	0	0
$815$	−3.61269e10	−2.33764
$816$	0	0
$817$	1.39569e10	0.895389
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.27726e9	0.332819	0.166409	−	0.986057i	$-0.446783\pi$
$821$	5.27726e9	0.332819	0.166409	+	0.986057i	$0.446783\pi$
$822$	0	0
$823$	7.08717e9	0.443173	0.221587	−	0.975141i	$-0.428876\pi$
$823$	7.08717e9	0.443173	0.221587	+	0.975141i	$0.428876\pi$
$824$	0	0
$825$	2.75291e10	1.70688
$826$	0	0
$827$	−2.82391e10	−1.73613	−0.868064	−	0.496453i	$-0.834636\pi$
$827$	−2.82391e10	−1.73613	−0.868064	+	0.496453i	$0.834636\pi$
$828$	0	0
$829$	2.22198e9	0.135456	0.0677282	−	0.997704i	$-0.478425\pi$
$829$	2.22198e9	0.135456	0.0677282	+	0.997704i	$0.478425\pi$
$830$	0	0
$831$	−7.69337e9	−0.465065
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.24228e10	−0.738443
$836$	0	0
$837$	−2.41186e9	−0.142172
$838$	0	0
$839$	−6.11614e9	−0.357529	−0.178764	−	0.983892i	$-0.557210\pi$
$839$	−6.11614e9	−0.357529	−0.178764	+	0.983892i	$0.557210\pi$
$840$	0	0
$841$	4.52927e10	2.62568
$842$	0	0
$843$	−1.59381e10	−0.916306
$844$	0	0
$845$	−2.28631e10	−1.30358
$846$	0	0
$847$	0	0
$848$	0	0
$849$	6.54926e9	0.367295
$850$	0	0
$851$	−7.40867e9	−0.412085
$852$	0	0
$853$	5.40282e9	0.298057	0.149028	−	0.988833i	$-0.452385\pi$
$853$	5.40282e9	0.298057	0.149028	+	0.988833i	$0.452385\pi$
$854$	0	0
$855$	1.21443e10	0.664496
$856$	0	0
$857$	2.98161e10	1.61815	0.809073	−	0.587708i	$-0.199970\pi$
$857$	2.98161e10	1.61815	0.809073	+	0.587708i	$0.199970\pi$
$858$	0	0
$859$	2.62604e10	1.41360	0.706799	−	0.707415i	$-0.250139\pi$
$859$	2.62604e10	1.41360	0.706799	+	0.707415i	$0.250139\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−7.98245e9	−0.422764	−0.211382	−	0.977404i	$-0.567796\pi$
$863$	−7.98245e9	−0.422764	−0.211382	+	0.977404i	$0.567796\pi$
$864$	0	0
$865$	4.73717e10	2.48864
$866$	0	0
$867$	4.64738e9	0.242182
$868$	0	0
$869$	−1.88504e10	−0.974430
$870$	0	0
$871$	−1.28709e10	−0.660000
$872$	0	0
$873$	−2.45001e9	−0.124629
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.31271e10	−1.65838	−0.829191	−	0.558966i	$-0.811198\pi$
$877$	−3.31271e10	−1.65838	−0.829191	+	0.558966i	$0.811198\pi$
$878$	0	0
$879$	−8.41853e9	−0.418095
$880$	0	0
$881$	−3.69116e10	−1.81864	−0.909321	−	0.416095i	$-0.863398\pi$
$881$	−3.69116e10	−1.81864	−0.909321	+	0.416095i	$0.863398\pi$
$882$	0	0
$883$	−1.50171e9	−0.0734045	−0.0367022	−	0.999326i	$-0.511685\pi$
$883$	−1.50171e9	−0.0734045	−0.0367022	+	0.999326i	$0.511685\pi$
$884$	0	0
$885$	−1.20457e10	−0.584159
$886$	0	0
$887$	1.04158e10	0.501141	0.250570	−	0.968098i	$-0.419382\pi$
$887$	1.04158e10	0.501141	0.250570	+	0.968098i	$0.419382\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.29459e9	−0.203399
$892$	0	0
$893$	−2.46935e10	−1.16039
$894$	0	0
$895$	1.68890e10	0.787448
$896$	0	0
$897$	5.76697e9	0.266793
$898$	0	0
$899$	−3.06442e10	−1.40666
$900$	0	0
$901$	−3.27005e10	−1.48942
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.49221e10	−2.46307
$906$	0	0
$907$	2.13957e10	0.952141	0.476071	−	0.879407i	$-0.342061\pi$
$907$	2.13957e10	0.952141	0.476071	+	0.879407i	$0.342061\pi$
$908$	0	0
$909$	1.15624e10	0.510595
$910$	0	0
$911$	2.57900e10	1.13015	0.565077	−	0.825038i	$-0.308846\pi$
$911$	2.57900e10	1.13015	0.565077	+	0.825038i	$0.308846\pi$
$912$	0	0
$913$	3.69889e10	1.60851
$914$	0	0
$915$	−2.69671e10	−1.16375
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.71215e10	0.727674	0.363837	−	0.931463i	$-0.381467\pi$
$919$	1.71215e10	0.727674	0.363837	+	0.931463i	$0.381467\pi$
$920$	0	0
$921$	−2.16985e10	−0.915212
$922$	0	0
$923$	−1.10524e10	−0.462647
$924$	0	0
$925$	1.52646e10	0.634147
$926$	0	0
$927$	−1.08982e9	−0.0449339
$928$	0	0
$929$	1.88157e10	0.769955	0.384977	−	0.922926i	$-0.374209\pi$
$929$	1.88157e10	0.769955	0.384977	+	0.922926i	$0.374209\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.00844e10	0.406505
$934$	0	0
$935$	−5.63742e10	−2.25548
$936$	0	0
$937$	−3.85068e10	−1.52915	−0.764574	−	0.644537i	$-0.777050\pi$
$937$	−3.85068e10	−1.52915	−0.764574	+	0.644537i	$0.777050\pi$
$938$	0	0
$939$	−2.00325e10	−0.789598
$940$	0	0
$941$	3.68708e10	1.44251	0.721254	−	0.692670i	$-0.243566\pi$
$941$	3.68708e10	1.44251	0.721254	+	0.692670i	$0.243566\pi$
$942$	0	0
$943$	2.42020e10	0.939856
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.57121e10	0.983813	0.491907	−	0.870648i	$-0.336300\pi$
$947$	2.57121e10	0.983813	0.491907	+	0.870648i	$0.336300\pi$
$948$	0	0
$949$	5.87178e9	0.223017
$950$	0	0
$951$	−1.50724e10	−0.568265
$952$	0	0
$953$	1.04009e10	0.389265	0.194632	−	0.980876i	$-0.437649\pi$
$953$	1.04009e10	0.389265	0.194632	+	0.980876i	$0.437649\pi$
$954$	0	0
$955$	−8.11075e10	−3.01335
$956$	0	0
$957$	−5.45655e10	−2.01245
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.24977e10	−0.454255
$962$	0	0
$963$	−1.19368e10	−0.430721
$964$	0	0
$965$	−5.14996e10	−1.84484
$966$	0	0
$967$	−4.78225e9	−0.170075	−0.0850374	−	0.996378i	$-0.527101\pi$
$967$	−4.78225e9	−0.170075	−0.0850374	+	0.996378i	$0.527101\pi$
$968$	0	0
$969$	−1.53590e10	−0.542287
$970$	0	0
$971$	1.26459e10	0.443286	0.221643	−	0.975128i	$-0.428858\pi$
$971$	1.26459e10	0.443286	0.221643	+	0.975128i	$0.428858\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.18821e10	−0.410560
$976$	0	0
$977$	−4.90124e10	−1.68142	−0.840708	−	0.541488i	$-0.817861\pi$
$977$	−4.90124e10	−1.68142	−0.840708	+	0.541488i	$0.817861\pi$
$978$	0	0
$979$	−1.26640e10	−0.431351
$980$	0	0
$981$	5.97464e9	0.202055
$982$	0	0
$983$	5.23089e10	1.75646	0.878229	−	0.478240i	$-0.158725\pi$
$983$	5.23089e10	1.75646	0.878229	+	0.478240i	$0.158725\pi$
$984$	0	0
$985$	1.87963e10	0.626680
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.31895e10	−0.762261
$990$	0	0
$991$	3.15694e10	1.03041	0.515203	−	0.857068i	$-0.327716\pi$
$991$	3.15694e10	1.03041	0.515203	+	0.857068i	$0.327716\pi$
$992$	0	0
$993$	3.02714e10	0.981094
$994$	0	0
$995$	−7.29386e10	−2.34734
$996$	0	0
$997$	2.52871e9	0.0808102	0.0404051	−	0.999183i	$-0.487135\pi$
$997$	2.52871e9	0.0808102	0.0404051	+	0.999183i	$0.487135\pi$
$998$	0	0
$999$	−2.38130e9	−0.0755676

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.8.a.m.1.8	✓	8
7.2	even	3		588.8.i.r.361.1		16
7.3	odd	6		588.8.i.q.373.8		16
7.4	even	3		588.8.i.r.373.1		16
7.5	odd	6		588.8.i.q.361.8		16
7.6	odd	2		588.8.a.n.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.8.a.m.1.8	✓	8	1.1	even	1	trivial
588.8.a.n.1.1	yes	8	7.6	odd	2
588.8.i.q.361.8		16	7.5	odd	6
588.8.i.q.373.8		16	7.3	odd	6
588.8.i.r.361.1		16	7.2	even	3
588.8.i.r.373.1		16	7.4	even	3

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	588.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} +451.992 q^{5} +729.000 q^{9} -8081.02 q^{11} +3487.92 q^{13} -12203.8 q^{15} +15434.2 q^{17} +36856.6 q^{19} -61237.4 q^{23} +126172. q^{25} -19683.0 q^{27} -250085. q^{29} +122535. q^{31} +218188. q^{33} +120983. q^{37} -94173.9 q^{39} -395217. q^{41} +378681. q^{43} +329502. q^{45} -669990. q^{47} -416723. q^{51} -2.11871e6 q^{53} -3.65256e6 q^{55} -995128. q^{57} +987048. q^{59} +2.20973e6 q^{61} +1.57651e6 q^{65} -3.69012e6 q^{67} +1.65341e6 q^{69} -3.16876e6 q^{71} +1.68346e6 q^{73} -3.40664e6 q^{75} +2.33267e6 q^{79} +531441. q^{81} -4.57725e6 q^{83} +6.97612e6 q^{85} +6.75230e6 q^{87} +1.56713e6 q^{89} -3.30845e6 q^{93} +1.66589e7 q^{95} -3.36078e6 q^{97} -5.89106e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 216 q^{3} + 5832 q^{9} - 41184 q^{17} + 17280 q^{19} - 24000 q^{23} + 161064 q^{25} - 157464 q^{27} - 137568 q^{29} - 194400 q^{31} + 120320 q^{37} - 635040 q^{41} + 666400 q^{43} - 1004832 q^{47}+ \cdots - 20236608 q^{97}+O(q^{100}) \) 8 * q - 216 * q^3 + 5832 * q^9 - 41184 * q^17 + 17280 * q^19 - 24000 * q^23 + 161064 * q^25 - 157464 * q^27 - 137568 * q^29 - 194400 * q^31 + 120320 * q^37 - 635040 * q^41 + 666400 * q^43 - 1004832 * q^47 + 1111968 * q^51 + 611280 * q^53 - 550368 * q^55 - 466560 * q^57 + 987840 * q^59 + 4259520 * q^61 + 8519760 * q^65 + 4710240 * q^67 + 648000 * q^69 + 3941952 * q^71 + 3556224 * q^73 - 4348728 * q^75 + 14245312 * q^79 + 4251528 * q^81 - 2573280 * q^83 + 1923280 * q^85 + 3714336 * q^87 - 1378080 * q^89 + 5248800 * q^93 + 29140416 * q^95 - 20236608 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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