LMFDB - Embedded newform 117.4.b.c.64.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.8112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 5x^{2} + 3 $ x^4 + 5*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			64.3
Root			$2.07431i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.4.b.c.64.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.52058i q^{2} -4.39445 q^{4} +9.17304i q^{5} +12.6936i q^{8} -32.2944 q^{10} +20.4780i q^{11} -46.8722 q^{13} -79.8444 q^{16} -40.3104i q^{20} -72.0942 q^{22} +40.8554 q^{25} -165.017i q^{26} -179.549i q^{32} -116.439 q^{40} +196.898i q^{41} +452.000 q^{43} -89.9894i q^{44} +640.490i q^{47} +343.000 q^{49} +143.834i q^{50} +205.977 q^{52} -187.845 q^{55} +579.506i q^{59} +944.654 q^{61} -6.63840 q^{64} -429.960i q^{65} -1190.09i q^{71} -418.244 q^{79} -732.416i q^{80} -693.193 q^{82} -94.6440i q^{83} +1591.30i q^{86} -259.939 q^{88} -1672.06i q^{89} -2254.89 q^{94} +1207.56i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} + 116 q^{10} - 204 q^{16} + 476 q^{22} - 500 q^{25} - 884 q^{40} + 1808 q^{43} + 1372 q^{49} - 676 q^{52} - 3232 q^{55} + 1632 q^{64} + 2924 q^{82} - 2756 q^{88} - 5140 q^{94}+O(q^{100}) $ 4 * q - 32 * q^4 + 116 * q^10 - 204 * q^16 + 476 * q^22 - 500 * q^25 - 884 * q^40 + 1808 * q^43 + 1372 * q^49 - 676 * q^52 - 3232 * q^55 + 1632 * q^64 + 2924 * q^82 - 2756 * q^88 - 5140 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.52058i	1.24471i	0.782735	+	0.622356i	$0.213824\pi$
$2$	3.52058i	1.24471i	−0.782735	+	0.622356i	$0.786176\pi$
$3$	0	0
$3$	0	0
$4$	−4.39445	−0.549306
$5$	9.17304i	0.820462i	0.911982	+	0.410231i	$0.134552\pi$
$5$	9.17304i	0.820462i	−0.911982	+	0.410231i	$0.865448\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	12.6936i	0.560984i
$9$	0	0
$10$	−32.2944	−1.02124
$11$	20.4780i	0.561304i	0.959810	+	0.280652i	$0.0905506\pi$
$11$	20.4780i	0.561304i	−0.959810	+	0.280652i	$0.909449\pi$
$12$	0	0
$13$	−46.8722	−1.00000
$13$	−46.8722	−1.00000
$14$	0	0
$15$	0	0
$16$	−79.8444	−1.24757
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	− 40.3104i	− 0.450685i
$21$	0	0
$22$	−72.0942	−0.698661
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	40.8554	0.326843
$26$	− 165.017i	− 1.24471i
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	− 179.549i	− 0.991879i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	−116.439	−0.460266
$41$	196.898i	0.750006i	0.927024	+	0.375003i	$0.122358\pi$
$41$	196.898i	0.750006i	−0.927024	+	0.375003i	$0.877642\pi$
$42$	0	0
$43$	452.000	1.60301	0.801504	−	0.597989i	$-0.204033\pi$
$43$	452.000	1.60301	0.801504	+	0.597989i	$0.204033\pi$
$44$	− 89.9894i	− 0.308327i
$45$	0	0
$46$	0	0
$47$	640.490i	1.98777i	0.110431	+	0.993884i	$0.464777\pi$
$47$	640.490i	1.98777i	−0.110431	+	0.993884i	$0.535223\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	143.834i	0.406825i
$51$	0	0
$52$	205.977	0.549306
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−187.845	−0.460528
$56$	0	0
$57$	0	0
$58$	0	0
$59$	579.506i	1.27873i	0.768902	+	0.639367i	$0.220803\pi$
$59$	579.506i	1.27873i	−0.768902	+	0.639367i	$0.779197\pi$
$60$	0	0
$61$	944.654	1.98280	0.991398	−	0.130879i	$-0.0417798\pi$
$61$	944.654	1.98280	0.991398	+	0.130879i	$0.0417798\pi$
$62$	0	0
$63$	0	0
$64$	−6.63840	−0.0129656
$65$	− 429.960i	− 0.820462i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 1190.09i	− 1.98926i	−0.103474	−	0.994632i	$-0.532996\pi$
$71$	− 1190.09i	− 1.98926i	0.103474	−	0.994632i	$-0.467004\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−418.244	−0.595647	−0.297824	−	0.954621i	$-0.596261\pi$
$79$	−418.244	−0.595647	−0.297824	+	0.954621i	$0.596261\pi$
$80$	− 732.416i	− 1.02358i
$81$	0	0
$82$	−693.193	−0.933541
$83$	− 94.6440i	− 0.125163i	−0.998040	−	0.0625815i	$-0.980067\pi$
$83$	− 94.6440i	− 0.125163i	0.998040	−	0.0625815i	$-0.0199333\pi$
$84$	0	0
$85$	0	0
$86$	1591.30i	1.99528i
$87$	0	0
$88$	−259.939	−0.314882
$89$	− 1672.06i	− 1.99143i	−0.0924493	−	0.995717i	$-0.529470\pi$
$89$	− 1672.06i	− 1.99143i	0.0924493	−	0.995717i	$-0.470530\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−2254.89	−2.47420
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1207.56i	1.24471i
$99$	0	0
$100$	−179.537	−0.179537
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	848.000	0.811223	0.405611	−	0.914046i	$-0.367059\pi$
$103$	848.000	0.811223	0.405611	+	0.914046i	$0.367059\pi$
$104$	− 594.977i	− 0.560984i
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	− 661.323i	− 0.573224i
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−2040.20	−1.59165
$119$	0	0
$120$	0	0
$121$	911.653	0.684938
$122$	3325.73i	2.46801i
$123$	0	0
$124$	0	0
$125$	1521.40i	1.08862i
$126$	0	0
$127$	−2495.04	−1.74330	−0.871650	−	0.490129i	$-0.836950\pi$
$127$	−2495.04	−1.74330	−0.871650	+	0.490129i	$0.836950\pi$
$128$	− 1459.77i	− 1.00802i
$129$	0	0
$130$	1513.71	1.02124
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3018.79i	1.88258i	0.337604	+	0.941288i	$0.390384\pi$
$137$	3018.79i	1.88258i	−0.337604	+	0.941288i	$0.609616\pi$
$138$	0	0
$139$	−340.000	−0.207471	−0.103735	−	0.994605i	$-0.533079\pi$
$139$	−340.000	−0.207471	−0.103735	+	0.994605i	$0.533079\pi$
$140$	0	0
$141$	0	0
$142$	4189.80	2.47606
$143$	− 959.847i	− 0.561304i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2477.09i	− 1.36196i	−0.732304	−	0.680978i	$-0.761555\pi$
$149$	− 2477.09i	− 1.36196i	0.732304	−	0.680978i	$-0.238445\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−934.000	−0.474785	−0.237393	−	0.971414i	$-0.576293\pi$
$157$	−934.000	−0.474785	−0.237393	+	0.971414i	$0.576293\pi$
$158$	− 1472.46i	− 0.741409i
$159$	0	0
$160$	1647.01	0.813799
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 865.256i	− 0.411983i
$165$	0	0
$166$	333.201	0.155792
$167$	− 2446.51i	− 1.13363i	−0.823845	−	0.566815i	$-0.808175\pi$
$167$	− 2446.51i	− 1.13363i	0.823845	−	0.566815i	$-0.191825\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	−1986.29	−0.880542
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	− 1635.05i	− 0.700265i
$177$	0	0
$178$	5886.60	2.47876
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	4430.00	1.81922	0.909611	−	0.415460i	$-0.136379\pi$
$181$	4430.00	1.81922	0.909611	+	0.415460i	$0.136379\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 2814.60i	− 1.09189i
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	−1507.30	−0.549306
$197$	3977.33i	1.43844i	0.694782	+	0.719220i	$0.255501\pi$
$197$	3977.33i	1.43844i	−0.694782	+	0.719220i	$0.744499\pi$
$198$	0	0
$199$	−5610.24	−1.99849	−0.999244	−	0.0388706i	$-0.987624\pi$
$199$	−5610.24	−1.99849	−0.999244	+	0.0388706i	$0.987624\pi$
$200$	518.602i	0.183354i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1806.15	−0.615351
$206$	2985.45i	1.00974i
$207$	0	0
$208$	3742.48	1.24757
$209$	0	0
$210$	0	0
$211$	−5740.04	−1.87280	−0.936399	−	0.350937i	$-0.885863\pi$
$211$	−5740.04	−1.87280	−0.936399	+	0.350937i	$0.885863\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4146.21i	1.31521i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	825.476	0.252971
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6668.65i	1.94984i	0.222554	+	0.974920i	$0.428561\pi$
$227$	6668.65i	1.94984i	−0.222554	+	0.974920i	$0.571439\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−5875.24	−1.63089
$236$	− 2546.61i	− 0.702416i
$237$	0	0
$238$	0	0
$239$	− 6085.88i	− 1.64712i	−0.567226	−	0.823562i	$-0.691984\pi$
$239$	− 6085.88i	− 1.64712i	0.567226	−	0.823562i	$-0.308016\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	3209.54i	0.852550i
$243$	0	0
$244$	−4151.24	−1.08916
$245$	3146.35i	0.820462i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−5356.19	−1.35502
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	− 8783.98i	− 2.16991i
$255$	0	0
$256$	5086.11	1.24173
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	1889.44i	0.450685i
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	−10627.9	−2.34326
$275$	836.635i	0.183458i
$276$	0	0
$277$	7634.00	1.65589	0.827947	−	0.560806i	$-0.189509\pi$
$277$	7634.00	1.65589	0.827947	+	0.560806i	$0.189509\pi$
$278$	− 1197.00i	− 0.258241i
$279$	0	0
$280$	0	0
$281$	− 8255.10i	− 1.75252i	−0.481839	−	0.876260i	$-0.660031\pi$
$281$	− 8255.10i	− 1.75252i	0.481839	−	0.876260i	$-0.339969\pi$
$282$	0	0
$283$	490.355	0.102999	0.0514993	−	0.998673i	$-0.483600\pi$
$283$	490.355	0.102999	0.0514993	+	0.998673i	$0.483600\pi$
$284$	5229.79i	1.09272i
$285$	0	0
$286$	3379.21	0.698661
$287$	0	0
$288$	0	0
$289$	−4913.00	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 9933.00i	− 1.98052i	−0.139232	−	0.990260i	$-0.544463\pi$
$293$	− 9933.00i	− 1.98052i	0.139232	−	0.990260i	$-0.455537\pi$
$294$	0	0
$295$	−5315.83	−1.04915
$296$	0	0
$297$	0	0
$298$	8720.79	1.69524
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8665.35i	1.62681i
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	6396.25	1.15507	0.577536	−	0.816365i	$-0.304014\pi$
$313$	6396.25	1.15507	0.577536	+	0.816365i	$0.304014\pi$
$314$	− 3288.22i	− 0.590971i
$315$	0	0
$316$	1837.95	0.327193
$317$	− 7133.53i	− 1.26391i	−0.775006	−	0.631954i	$-0.782253\pi$
$317$	− 7133.53i	− 1.26391i	0.775006	−	0.631954i	$-0.217747\pi$
$318$	0	0
$319$	0	0
$320$	− 60.8943i	− 0.0106378i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1914.98	−0.326843
$326$	0	0
$327$	0	0
$328$	−2499.34	−0.420741
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	415.908i	0.0687528i
$333$	0	0
$334$	8613.11	1.41104
$335$	0	0
$336$	0	0
$337$	10420.0	1.68432	0.842160	−	0.539228i	$-0.181284\pi$
$337$	10420.0	1.68432	0.842160	+	0.539228i	$0.181284\pi$
$338$	7734.70i	1.24471i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	5737.51i	0.899262i
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	3676.81	0.556745
$353$	7331.69i	1.10546i	0.833361	+	0.552728i	$0.186413\pi$
$353$	7331.69i	1.10546i	−0.833361	+	0.552728i	$0.813587\pi$
$354$	0	0
$355$	10916.7	1.63211
$356$	7347.77i	1.09391i
$357$	0	0
$358$	0	0
$359$	− 11077.3i	− 1.62852i	−0.580502	−	0.814259i	$-0.697143\pi$
$359$	− 11077.3i	− 1.62852i	0.580502	−	0.814259i	$-0.302857\pi$
$360$	0	0
$361$	6859.00	1.00000
$362$	15596.1i	2.26441i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8296.00	−1.17997	−0.589983	−	0.807416i	$-0.700866\pi$
$367$	−8296.00	−1.17997	−0.589983	+	0.807416i	$0.700866\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6526.05	0.905914	0.452957	−	0.891532i	$-0.350369\pi$
$373$	6526.05	0.905914	0.452957	+	0.891532i	$0.350369\pi$
$374$	0	0
$375$	0	0
$376$	−8130.13	−1.11511
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12702.2i	1.69465i	0.531071	+	0.847327i	$0.321790\pi$
$383$	12702.2i	1.69465i	−0.531071	+	0.847327i	$0.678210\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	4353.91i	0.560984i
$393$	0	0
$394$	−14002.5	−1.79044
$395$	− 3836.57i	− 0.488706i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	− 19751.3i	− 2.48754i
$399$	0	0
$400$	−3262.07	−0.407759
$401$	− 15342.6i	− 1.91066i	−0.295549	−	0.955328i	$-0.595503\pi$
$401$	− 15342.6i	− 1.91066i	0.295549	−	0.955328i	$-0.404497\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	− 6358.68i	− 0.765934i
$411$	0	0
$412$	−3726.49	−0.445609
$413$	0	0
$414$	0	0
$415$	868.173	0.102691
$416$	8415.87i	0.991879i
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	− 20208.2i	− 2.33109i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−14597.1	−1.63705
$431$	− 320.305i	− 0.0357971i	−0.999840	−	0.0178985i	$-0.994302\pi$
$431$	− 320.305i	− 0.0357971i	0.999840	−	0.0178985i	$-0.00569759\pi$
$432$	0	0
$433$	36.0555	0.00400166	0.00200083	−	0.999998i	$-0.499363\pi$
$433$	36.0555	0.00400166	0.00200083	+	0.999998i	$0.499363\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−11320.0	−1.23069	−0.615346	−	0.788257i	$-0.710984\pi$
$439$	−11320.0	−1.23069	−0.615346	+	0.788257i	$0.710984\pi$
$440$	− 2384.43i	− 0.258349i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	15337.8	1.63390
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 333.683i	− 0.0350723i	−0.999846	−	0.0175361i	$-0.994418\pi$
$449$	− 333.683i	− 0.0350723i	0.999846	−	0.0175361i	$-0.00558221\pi$
$450$	0	0
$451$	−4032.06	−0.420981
$452$	0	0
$453$	0	0
$454$	−23477.5	−2.42699
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17433.8i	1.76133i	0.473738	+	0.880666i	$0.342905\pi$
$461$	17433.8i	1.76133i	−0.473738	+	0.880666i	$0.657095\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	− 20684.2i	− 2.02998i
$471$	0	0
$472$	−7356.03	−0.717349
$473$	9256.04i	0.899774i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	21425.8	2.05019
$479$	5200.84i	0.496101i	0.968747	+	0.248050i	$0.0797898\pi$
$479$	5200.84i	0.496101i	−0.968747	+	0.248050i	$0.920210\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−4006.21	−0.376241
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	11991.1i	1.11232i
$489$	0	0
$490$	−11077.0	−1.02124
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 6685.70i	− 0.597988i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	10964.3	0.957605
$509$	22966.8i	1.99997i	0.00508931	+	0.999987i	$0.498380\pi$
$509$	22966.8i	1.99997i	−0.00508931	+	0.999987i	$0.501620\pi$
$510$	0	0
$511$	0	0
$512$	6227.90i	0.537572i
$513$	0	0
$514$	0	0
$515$	7778.74i	0.665577i
$516$	0	0
$517$	−13115.9	−1.11574
$518$	0	0
$519$	0	0
$520$	5457.75	0.460266
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	16148.0	1.35010	0.675050	−	0.737772i	$-0.264122\pi$
$523$	16148.0	1.35010	0.675050	+	0.737772i	$0.264122\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 9229.02i	− 0.750006i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	7023.94i	0.561304i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23996.0	1.87568	0.937838	−	0.347073i	$-0.112824\pi$
$547$	23996.0	1.87568	0.937838	+	0.347073i	$0.112824\pi$
$548$	− 13265.9i	− 1.03411i
$549$	0	0
$550$	−2945.44	−0.228352
$551$	0	0
$552$	0	0
$553$	0	0
$554$	26876.1i	2.06111i
$555$	0	0
$556$	1494.11	0.113965
$557$	− 22258.1i	− 1.69319i	−0.532237	−	0.846595i	$-0.678648\pi$
$557$	− 22258.1i	− 1.69319i	0.532237	−	0.846595i	$-0.321352\pi$
$558$	0	0
$559$	−21186.2	−1.60301
$560$	0	0
$561$	0	0
$562$	29062.7	2.18138
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	1726.33i	0.128203i
$567$	0	0
$568$	15106.6	1.11595
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	25411.9	1.86244	0.931222	−	0.364451i	$-0.118743\pi$
$571$	25411.9	1.86244	0.931222	+	0.364451i	$0.118743\pi$
$572$	4218.00i	0.308327i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 17296.6i	− 1.24471i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	34969.9	2.46517
$587$	− 19338.0i	− 1.35974i	−0.733335	−	0.679868i	$-0.762037\pi$
$587$	− 19338.0i	− 1.35974i	0.733335	−	0.679868i	$-0.237963\pi$
$588$	0	0
$589$	0	0
$590$	− 18714.8i	− 1.30589i
$591$	0	0
$592$	0	0
$593$	25958.2i	1.79760i	0.438363	+	0.898798i	$0.355558\pi$
$593$	25958.2i	1.79760i	−0.438363	+	0.898798i	$0.644442\pi$
$594$	0	0
$595$	0	0
$596$	10885.5i	0.748131i
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	12626.6	0.856991	0.428495	−	0.903544i	$-0.359044\pi$
$601$	12626.6	0.856991	0.428495	+	0.903544i	$0.359044\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8362.63i	0.561966i
$606$	0	0
$607$	−25504.0	−1.70540	−0.852698	−	0.522404i	$-0.825035\pi$
$607$	−25504.0	−1.70540	−0.852698	+	0.522404i	$0.825035\pi$
$608$	0	0
$609$	0	0
$610$	−30507.0	−2.02491
$611$	− 30021.2i	− 1.98777i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 29859.9i	− 1.94832i	−0.225860	−	0.974160i	$-0.572519\pi$
$617$	− 29859.9i	− 1.94832i	0.225860	−	0.974160i	$-0.427481\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8848.92	−0.566331
$626$	22518.5i	1.43773i
$627$	0	0
$628$	4104.42	0.260803
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	− 5309.03i	− 0.334148i
$633$	0	0
$634$	25114.1	1.57320
$635$	− 22887.1i	− 1.43031i
$636$	0	0
$637$	−16077.2	−1.00000
$638$	0	0
$639$	0	0
$640$	13390.5	0.827040
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−11867.1	−0.717758
$650$	− 6741.83i	− 0.406825i
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	− 15721.2i	− 0.935684i
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	1201.37	0.0702144
$665$	0	0
$666$	0	0
$667$	0	0
$668$	10751.0i	0.622710i
$669$	0	0
$670$	0	0
$671$	19344.6i	1.11295i
$672$	0	0
$673$	−25522.0	−1.46181	−0.730907	−	0.682477i	$-0.760903\pi$
$673$	−25522.0	−1.46181	−0.730907	+	0.682477i	$0.760903\pi$
$674$	36684.5i	2.09649i
$675$	0	0
$676$	−9654.60	−0.549306
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32750.6i	1.83480i	0.397968	+	0.917399i	$0.369715\pi$
$683$	32750.6i	1.83480i	−0.397968	+	0.917399i	$0.630285\pi$
$684$	0	0
$685$	−27691.5	−1.54458
$686$	0	0
$687$	0	0
$688$	−36089.7	−1.99986
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 3118.83i	− 0.170222i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	− 135.941i	− 0.00727765i
$705$	0	0
$706$	−25811.8	−1.37597
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	38433.2i	2.03151i
$711$	0	0
$712$	21224.4	1.11716
$713$	0	0
$714$	0	0
$715$	8804.71	0.460528
$716$	0	0
$717$	0	0
$718$	38998.5	2.02704
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	24147.6i	1.24471i
$723$	0	0
$724$	−19467.4	−0.999310
$725$	0	0
$726$	0	0
$727$	−28455.0	−1.45163	−0.725817	−	0.687888i	$-0.758538\pi$
$727$	−28455.0	−1.45163	−0.725817	+	0.687888i	$0.758538\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	− 29206.7i	− 1.46872i
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	19936.4i	0.984382i	0.870487	+	0.492191i	$0.163804\pi$
$743$	19936.4i	0.984382i	−0.870487	+	0.492191i	$0.836196\pi$
$744$	0	0
$745$	22722.5	1.11743
$746$	22975.4i	1.12760i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26120.0	1.26915	0.634575	−	0.772861i	$-0.281175\pi$
$751$	26120.0	1.26915	0.634575	+	0.772861i	$0.281175\pi$
$752$	− 51139.6i	− 2.47988i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−35009.9	−1.68092	−0.840460	−	0.541873i	$-0.817715\pi$
$757$	−35009.9	−1.68092	−0.840460	+	0.541873i	$0.817715\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 28936.4i	− 1.37838i	−0.724582	−	0.689189i	$-0.757967\pi$
$761$	− 28936.4i	− 1.37838i	0.724582	−	0.689189i	$-0.242033\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−44719.1	−2.10936
$767$	− 27162.7i	− 1.27873i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3484.00i	0.162110i	0.996710	+	0.0810548i	$0.0258289\pi$
$773$	3484.00i	0.162110i	−0.996710	+	0.0810548i	$0.974171\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	24370.6	1.11658
$782$	0	0
$783$	0	0
$784$	−27386.6	−1.24757
$785$	− 8567.62i	− 0.389543i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 17478.2i	− 0.790144i
$789$	0	0
$790$	13506.9	0.608297
$791$	0	0
$792$	0	0
$793$	−44278.0	−1.98280
$794$	0	0
$795$	0	0
$796$	24653.9	1.09778
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	− 7335.55i	− 0.324189i
$801$	0	0
$802$	54014.8	2.37821
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	7937.03	0.338016
$821$	− 33048.4i	− 1.40487i	−0.711749	−	0.702433i	$-0.752097\pi$
$821$	− 33048.4i	− 1.40487i	0.711749	−	0.702433i	$-0.247903\pi$
$822$	0	0
$823$	37352.0	1.58203	0.791014	−	0.611798i	$-0.209554\pi$
$823$	37352.0	1.58203	0.791014	+	0.611798i	$0.209554\pi$
$824$	10764.2i	0.455083i
$825$	0	0
$826$	0	0
$827$	− 31167.2i	− 1.31051i	−0.755409	−	0.655253i	$-0.772562\pi$
$827$	− 31167.2i	− 1.31051i	0.755409	−	0.655253i	$-0.227438\pi$
$828$	0	0
$829$	−3079.14	−0.129002	−0.0645012	−	0.997918i	$-0.520546\pi$
$829$	−3079.14	−0.129002	−0.0645012	+	0.997918i	$0.520546\pi$
$830$	3056.47i	0.127821i
$831$	0	0
$832$	311.156	0.0129656
$833$	0	0
$834$	0	0
$835$	22441.9	0.930101
$836$	0	0
$837$	0	0
$838$	0	0
$839$	46872.5i	1.92875i	0.264543	+	0.964374i	$0.414779\pi$
$839$	46872.5i	1.92875i	−0.264543	+	0.964374i	$0.585221\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	25224.3	1.02874
$845$	20153.2i	0.820462i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	43324.3	1.72085	0.860423	−	0.509581i	$-0.170200\pi$
$859$	43324.3	1.72085	0.860423	+	0.509581i	$0.170200\pi$
$860$	− 18220.3i	− 0.722451i
$861$	0	0
$862$	1127.66	0.0445571
$863$	23594.8i	0.930678i	0.885133	+	0.465339i	$0.154068\pi$
$863$	23594.8i	0.930678i	−0.885133	+	0.465339i	$0.845932\pi$
$864$	0	0
$865$	0	0
$866$	126.936i	0.00498091i
$867$	0	0
$868$	0	0
$869$	− 8564.79i	− 0.334339i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	− 39852.9i	− 1.53186i
$879$	0	0
$880$	14998.4	0.574540
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	52410.3	1.99745	0.998724	−	0.0504988i	$-0.0160811\pi$
$883$	52410.3	1.99745	0.998724	+	0.0504988i	$0.0160811\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	53998.0i	2.03373i
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	1174.75	0.0436549
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 14195.2i	− 0.524000i
$903$	0	0
$904$	0	0
$905$	40636.6i	1.49260i
$906$	0	0
$907$	35017.1	1.28195	0.640973	−	0.767564i	$-0.278531\pi$
$907$	35017.1	1.28195	0.640973	+	0.767564i	$0.278531\pi$
$908$	− 29305.0i	− 1.07106i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	1938.12	0.0702545
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−36762.2	−1.31956	−0.659779	−	0.751460i	$-0.729350\pi$
$919$	−36762.2	−1.31956	−0.659779	+	0.751460i	$0.729350\pi$
$920$	0	0
$921$	0	0
$922$	−61377.1	−2.19235
$923$	55782.1i	1.98926i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7352.53i	0.259665i	0.991536	+	0.129832i	$0.0414439\pi$
$929$	7352.53i	0.259665i	−0.991536	+	0.129832i	$0.958556\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13275.6	0.462856	0.231428	−	0.972852i	$-0.425660\pi$
$937$	13275.6	0.462856	0.231428	+	0.972852i	$0.425660\pi$
$938$	0	0
$939$	0	0
$940$	25818.4	0.895856
$941$	15462.1i	0.535652i	0.963467	+	0.267826i	$0.0863052\pi$
$941$	15462.1i	0.535652i	−0.963467	+	0.267826i	$0.913695\pi$
$942$	0	0
$943$	0	0
$944$	− 46270.3i	− 1.59531i
$945$	0	0
$946$	−32586.6	−1.11996
$947$	56498.4i	1.93870i	0.245678	+	0.969352i	$0.420989\pi$
$947$	56498.4i	1.93870i	−0.245678	+	0.969352i	$0.579011\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	26744.1i	0.904775i
$957$	0	0
$958$	−18309.9	−0.617502
$959$	0	0
$960$	0	0
$961$	29791.0	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	11572.2i	0.384239i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−75425.4	−2.47368
$977$	52089.2i	1.70571i	0.522146	+	0.852856i	$0.325132\pi$
$977$	52089.2i	1.70571i	−0.522146	+	0.852856i	$0.674868\pi$
$978$	0	0
$979$	34240.3	1.11780
$980$	− 13826.5i	− 0.450685i
$981$	0	0
$982$	0	0
$983$	− 60639.0i	− 1.96753i	−0.179454	−	0.983766i	$-0.557433\pi$
$983$	− 60639.0i	− 1.96753i	0.179454	−	0.983766i	$-0.442567\pi$
$984$	0	0
$985$	−36484.2	−1.18019
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−23272.0	−0.745973	−0.372987	−	0.927837i	$-0.621666\pi$
$991$	−23272.0	−0.745973	−0.372987	+	0.927837i	$0.621666\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 51462.9i	− 1.63968i
$996$	0	0
$997$	−21634.0	−0.687217	−0.343609	−	0.939113i	$-0.611649\pi$
$997$	−21634.0	−0.687217	−0.343609	+	0.939113i	$0.611649\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.b.c.64.3	yes	4
3.2	odd	2	inner	117.4.b.c.64.2	✓	4
13.5	odd	4		1521.4.a.y.1.3		4
13.8	odd	4		1521.4.a.y.1.2		4
13.12	even	2	inner	117.4.b.c.64.2	✓	4
39.5	even	4		1521.4.a.y.1.2		4
39.8	even	4		1521.4.a.y.1.3		4
39.38	odd	2	CM	117.4.b.c.64.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.b.c.64.2	✓	4	3.2	odd	2	inner
117.4.b.c.64.2	✓	4	13.12	even	2	inner
117.4.b.c.64.3	yes	4	1.1	even	1	trivial
117.4.b.c.64.3	yes	4	39.38	odd	2	CM
1521.4.a.y.1.2		4	13.8	odd	4
1521.4.a.y.1.2		4	39.5	even	4
1521.4.a.y.1.3		4	13.5	odd	4
1521.4.a.y.1.3		4	39.8	even	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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.52058i q^{2} -4.39445 q^{4} +9.17304i q^{5} +12.6936i q^{8} -32.2944 q^{10} +20.4780i q^{11} -46.8722 q^{13} -79.8444 q^{16} -40.3104i q^{20} -72.0942 q^{22} +40.8554 q^{25} -165.017i q^{26} -179.549i q^{32} -116.439 q^{40} +196.898i q^{41} +452.000 q^{43} -89.9894i q^{44} +640.490i q^{47} +343.000 q^{49} +143.834i q^{50} +205.977 q^{52} -187.845 q^{55} +579.506i q^{59} +944.654 q^{61} -6.63840 q^{64} -429.960i q^{65} -1190.09i q^{71} -418.244 q^{79} -732.416i q^{80} -693.193 q^{82} -94.6440i q^{83} +1591.30i q^{86} -259.939 q^{88} -1672.06i q^{89} -2254.89 q^{94} +1207.56i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} + 116 q^{10} - 204 q^{16} + 476 q^{22} - 500 q^{25} - 884 q^{40} + 1808 q^{43} + 1372 q^{49} - 676 q^{52} - 3232 q^{55} + 1632 q^{64} + 2924 q^{82} - 2756 q^{88} - 5140 q^{94}+O(q^{100}) \) 4 * q - 32 * q^4 + 116 * q^10 - 204 * q^16 + 476 * q^22 - 500 * q^25 - 884 * q^40 + 1808 * q^43 + 1372 * q^49 - 676 * q^52 - 3232 * q^55 + 1632 * q^64 + 2924 * q^82 - 2756 * q^88 - 5140 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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