LMFDB - Embedded newform 117.4.b.c.64.1

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			64.1
Root			$-0.835000i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.4.b.c.64.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.42782i q^{2} -11.6056 q^{4} +20.3925i q^{5} +15.9647i q^{8} +O(q^{10})$	$q-4.42782i q^{2} -11.6056 q^{4} +20.3925i q^{5} +15.9647i q^{8} +90.2944 q^{10} +70.0332i q^{11} +46.8722 q^{13} -22.1556 q^{16} -236.667i q^{20} +310.094 q^{22} -290.855 q^{25} -207.541i q^{26} +225.819i q^{32} -325.561 q^{40} +486.739i q^{41} +452.000 q^{43} -812.774i q^{44} -71.1653i q^{47} +343.000 q^{49} +1287.85i q^{50} -543.977 q^{52} -1428.15 q^{55} -696.914i q^{59} -944.654 q^{61} +822.638 q^{64} +955.842i q^{65} +123.807i q^{71} +418.244 q^{79} -451.809i q^{80} +2155.19 q^{82} -1509.37i q^{83} -2001.37i q^{86} -1118.06 q^{88} +155.245i q^{89} -315.107 q^{94} -1518.74i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4}+O(q^{10}) $ 4 * q - 32 * q^4	$ 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

Display 100 coefficients 10 coefficients

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 $

$2$	− 4.42782i	− 1.56547i	−0.622356	−	0.782735i	$-0.713824\pi$
$2$	− 4.42782i	− 1.56547i	0.622356	−	0.782735i	$-0.286176\pi$
$3$	0	0
$3$	0	0
$4$	−11.6056	−1.45069
$5$	20.3925i	1.82396i	0.410231	+	0.911982i	$0.365448\pi$
$5$	20.3925i	1.82396i	−0.410231	+	0.911982i	$0.634552\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	15.9647i	0.705547i
$9$	0	0
$10$	90.2944	2.85536
$11$	70.0332i	1.91962i	0.280652	+	0.959810i	$0.409449\pi$
$11$	70.0332i	1.91962i	−0.280652	+	0.959810i	$0.590551\pi$
$12$	0	0
$13$	46.8722	1.00000
$13$	46.8722	1.00000
$14$	0	0
$15$	0	0
$16$	−22.1556	−0.346181
$17$	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			$\pi$
$20$	− 236.667i	− 2.64601i
$21$	0	0
$22$	310.094	3.00510
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−290.855	−2.32684
$26$	− 207.541i	− 1.56547i
$27$	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			$\pi$
$32$	225.819i	1.24748i
$33$	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
$40$	−325.561	−1.28689
$41$	486.739i	1.85405i	0.375003	+	0.927024i	$0.377642\pi$
$41$	486.739i	1.85405i	−0.375003	+	0.927024i	$0.622358\pi$
$42$	0	0
$43$	452.000	1.60301	0.801504	−	0.597989i	$-0.204033\pi$
$43$	452.000	1.60301	0.801504	+	0.597989i	$0.204033\pi$
$44$	− 812.774i	− 2.78478i
$45$	0	0
$46$	0	0
$47$	− 71.1653i	− 0.220862i	−0.993884	−	0.110431i	$-0.964777\pi$
$47$	− 71.1653i	− 0.220862i	0.993884	−	0.110431i	$-0.0352231\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	1287.85i	3.64260i
$51$	0	0
$52$	−543.977	−1.45069
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−1428.15	−3.50132
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 696.914i	− 1.53780i	−0.639367	−	0.768902i	$-0.720803\pi$
$59$	− 696.914i	− 1.53780i	0.639367	−	0.768902i	$-0.279197\pi$
$60$	0	0
$61$	−944.654	−1.98280	−0.991398	−	0.130879i	$-0.958220\pi$
$61$	−944.654	−1.98280	−0.991398	+	0.130879i	$0.958220\pi$
$62$	0	0
$63$	0	0
$64$	822.638	1.60672
$65$	955.842i	1.82396i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	123.807i	0.206947i	0.994632	+	0.103474i	$0.0329957\pi$
$71$	123.807i	0.206947i	−0.994632	+	0.103474i	$0.967004\pi$
$72$	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	$-0.403739\pi$
$79$	418.244	0.595647	0.297824	+	0.954621i	$0.403739\pi$
$80$	− 451.809i	− 0.631422i
$81$	0	0
$82$	2155.19	2.90245
$83$	− 1509.37i	− 1.99608i	−0.0625815	−	0.998040i	$-0.519933\pi$
$83$	− 1509.37i	− 1.99608i	0.0625815	−	0.998040i	$-0.480067\pi$
$84$	0	0
$85$	0	0
$86$	− 2001.37i	− 2.50946i
$87$	0	0
$88$	−1118.06	−1.35438
$89$	155.245i	0.184899i	0.995717	+	0.0924493i	$0.0294696\pi$
$89$	155.245i	0.184899i	−0.995717	+	0.0924493i	$0.970530\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−315.107	−0.345753
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	− 1518.74i	− 1.56547i
$99$	0	0
$100$	3375.54	3.37554
$101$	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	$0.367059\pi$
$104$	748.301i	0.705547i
$105$	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			$\pi$
$110$	6323.61i	5.48120i
$111$	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
$118$	−3085.80	−2.40738
$119$	0	0
$120$	0	0
$121$	−3573.65	−2.68494
$122$	4182.76i	3.10401i
$123$	0	0
$124$	0	0
$125$	− 3382.21i	− 2.42011i
$126$	0	0
$127$	2495.04	1.74330	0.871650	−	0.490129i	$-0.163050\pi$
$127$	2495.04	1.74330	0.871650	+	0.490129i	$0.163050\pi$
$128$	− 1835.94i	− 1.26778i
$129$	0	0
$130$	4232.29	2.85536
$131$	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
$137$	− 1082.73i	− 0.675208i	−0.941288	−	0.337604i	$-0.890384\pi$
$137$	− 1082.73i	− 0.675208i	0.941288	−	0.337604i	$-0.109616\pi$
$138$	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
$142$	548.197	0.323969
$143$	3282.61i	1.91962i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2663.80i	− 1.46461i	−0.680978	−	0.732304i	$-0.738445\pi$
$149$	− 2663.80i	− 1.46461i	0.680978	−	0.732304i	$-0.261555\pi$
$150$	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	$0.576293\pi$
$158$	− 1851.91i	− 0.932467i
$159$	0	0
$160$	−4605.01	−2.27536
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 5648.88i	− 2.68966i
$165$	0	0
$166$	−6683.20	−3.12480
$167$	3555.90i	1.64769i	0.566815	+	0.823845i	$0.308175\pi$
$167$	3555.90i	1.64769i	−0.566815	+	0.823845i	$0.691825\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	−5245.71	−2.32547
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	− 1551.63i	− 0.664536i
$177$	0	0
$178$	687.398	0.289453
$179$	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
$188$	825.912i	0.320403i
$189$	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
$196$	−3980.70	−1.45069
$197$	3842.18i	1.38956i	0.719220	+	0.694782i	$0.244499\pi$
$197$	3842.18i	1.38956i	−0.719220	+	0.694782i	$0.755501\pi$
$198$	0	0
$199$	5610.24	1.99849	0.999244	−	0.0388706i	$-0.0123760\pi$
$199$	5610.24	1.99849	0.999244	+	0.0388706i	$0.0123760\pi$
$200$	− 4643.42i	− 1.64170i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9925.85	−3.38171
$206$	− 3754.79i	− 1.26994i
$207$	0	0
$208$	−1038.48	−0.346181
$209$	0	0
$210$	0	0
$211$	5740.04	1.87280	0.936399	−	0.350937i	$-0.114137\pi$
$211$	5740.04	1.87280	0.936399	+	0.350937i	$0.114137\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9217.42i	2.92383i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	16574.5	5.07934
$221$	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
$227$	− 1522.31i	− 0.445107i	−0.974920	−	0.222554i	$-0.928561\pi$
$227$	− 1522.31i	− 0.445107i	0.974920	−	0.222554i	$-0.0714393\pi$
$228$	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
$235$	1451.24	0.402845
$236$	8088.07i	2.23088i
$237$	0	0
$238$	0	0
$239$	− 4191.63i	− 1.13445i	−0.823562	−	0.567226i	$-0.808016\pi$
$239$	− 4191.63i	− 1.13445i	0.823562	−	0.567226i	$-0.191984\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	15823.5i	4.20319i
$243$	0	0
$244$	10963.2	2.87643
$245$	6994.64i	1.82396i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−14975.8	−3.78861
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	− 11047.6i	− 2.72908i
$255$	0	0
$256$	−1548.11	−0.377956
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	− 11093.1i	− 2.64601i
$261$	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
$274$	−4794.11	−1.05702
$275$	− 20369.5i	− 4.46665i
$276$	0	0
$277$	7634.00	1.65589	0.827947	−	0.560806i	$-0.189509\pi$
$277$	7634.00	1.65589	0.827947	+	0.560806i	$0.189509\pi$
$278$	1505.46i	0.324789i
$279$	0	0
$280$	0	0
$281$	4539.33i	0.963678i	0.876260	+	0.481839i	$0.160031\pi$
$281$	4539.33i	0.963678i	−0.876260	+	0.481839i	$0.839969\pi$
$282$	0	0
$283$	−490.355	−0.102999	−0.0514993	−	0.998673i	$-0.516400\pi$
$283$	−490.355	−0.102999	−0.0514993	+	0.998673i	$0.516400\pi$
$284$	− 1436.85i	− 0.300217i
$285$	0	0
$286$	14534.8	3.00510
$287$	0	0
$288$	0	0
$289$	−4913.00	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1396.60i	− 0.278465i	−0.990260	−	0.139232i	$-0.955537\pi$
$293$	− 1396.60i	− 0.278465i	0.990260	−	0.139232i	$-0.0444635\pi$
$294$	0	0
$295$	14211.8	2.80490
$296$	0	0
$297$	0	0
$298$	−11794.8	−2.29280
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 19263.9i	− 3.61655i
$306$	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	$-0.695986\pi$
$313$	−6396.25	−1.15507	−0.577536	+	0.816365i	$0.695986\pi$
$314$	4135.58i	0.743262i
$315$	0	0
$316$	−4853.95	−0.864102
$317$	8748.31i	1.55001i	0.631954	+	0.775006i	$0.282253\pi$
$317$	8748.31i	1.55001i	−0.631954	+	0.775006i	$0.717747\pi$
$318$	0	0
$319$	0	0
$320$	16775.7i	2.93059i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−13633.0	−2.32684
$326$	0	0
$327$	0	0
$328$	−7770.66	−1.30812
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	17517.0i	2.89570i
$333$	0	0
$334$	15744.9	2.57941
$335$	0	0
$336$	0	0
$337$	−10420.0	−1.68432	−0.842160	−	0.539228i	$-0.818716\pi$
$337$	−10420.0	−1.68432	−0.842160	+	0.539228i	$0.818716\pi$
$338$	− 9727.91i	− 1.56547i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	7216.05i	1.13100i
$345$	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
$352$	−15814.8	−2.39469
$353$	11054.2i	1.66672i	0.552728	+	0.833361i	$0.313587\pi$
$353$	11054.2i	1.66672i	−0.552728	+	0.833361i	$0.686413\pi$
$354$	0	0
$355$	−2524.75	−0.377464
$356$	− 1801.71i	− 0.268231i
$357$	0	0
$358$	0	0
$359$	7897.23i	1.16100i	0.814259	+	0.580502i	$0.197143\pi$
$359$	7897.23i	1.16100i	−0.814259	+	0.580502i	$0.802857\pi$
$360$	0	0
$361$	6859.00	1.00000
$362$	− 19615.2i	− 2.84794i
$363$	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	$-0.649631\pi$
$373$	−6526.05	−0.905914	−0.452957	+	0.891532i	$0.649631\pi$
$374$	0	0
$375$	0	0
$376$	1136.13	0.155829
$377$	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
$383$	− 7961.24i	− 1.06214i	−0.847327	−	0.531071i	$-0.821790\pi$
$383$	− 7961.24i	− 1.06214i	0.847327	−	0.531071i	$-0.178210\pi$
$384$	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
$392$	5475.90i	0.705547i
$393$	0	0
$394$	17012.5	2.17532
$395$	8529.05i	1.08644i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	− 24841.1i	− 3.12857i
$399$	0	0
$400$	6444.07	0.805509
$401$	− 4746.53i	− 0.591098i	−0.955328	−	0.295549i	$-0.904497\pi$
$401$	− 4746.53i	− 0.591098i	0.955328	−	0.295549i	$-0.0955026\pi$
$402$	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			$\pi$
$410$	43949.8i	5.29397i
$411$	0	0
$412$	−9841.51	−1.17684
$413$	0	0
$414$	0	0
$415$	30779.8	3.64078
$416$	10584.6i	1.24748i
$417$	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			$\pi$
$422$	− 25415.8i	− 2.93181i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	40813.1	4.57716
$431$	− 17892.7i	− 1.99968i	−0.0178985	−	0.999840i	$-0.505698\pi$
$431$	− 17892.7i	− 1.99968i	0.0178985	−	0.999840i	$-0.494302\pi$
$432$	0	0
$433$	−36.0555	−0.00400166	−0.00200083	−	0.999998i	$-0.500637\pi$
$433$	−36.0555	−0.00400166	−0.00200083	+	0.999998i	$0.500637\pi$
$434$	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	$0.710984\pi$
$440$	− 22800.1i	− 2.47034i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−3165.84	−0.337248
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 19025.4i	− 1.99969i	−0.0175361	−	0.999846i	$-0.505582\pi$
$449$	− 19025.4i	− 1.99969i	0.0175361	−	0.999846i	$-0.494418\pi$
$450$	0	0
$451$	−34087.9	−3.55906
$452$	0	0
$453$	0	0
$454$	−6740.52	−0.696802
$455$	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
$461$	− 9378.19i	− 0.947475i	−0.880666	−	0.473738i	$-0.842905\pi$
$461$	− 9378.19i	− 0.947475i	0.880666	−	0.473738i	$-0.157095\pi$
$462$	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
$470$	− 6425.82i	− 0.630641i
$471$	0	0
$472$	11126.0	1.08499
$473$	31655.0i	3.07717i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−18559.8	−1.77595
$479$	20311.6i	1.93749i	0.248050	+	0.968747i	$0.420210\pi$
$479$	20311.6i	1.93749i	−0.248050	+	0.968747i	$0.579790\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	41474.2	3.89502
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	− 15081.1i	− 1.39896i
$489$	0	0
$490$	30971.0	2.85536
$491$	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			$\pi$
$500$	39252.4i	3.51084i
$501$	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
$508$	−28956.3	−2.52900
$509$	− 116.887i	− 0.0101786i	−0.999987	−	0.00508931i	$-0.998380\pi$
$509$	− 116.887i	− 0.0101786i	0.999987	−	0.00508931i	$-0.00161998\pi$
$510$	0	0
$511$	0	0
$512$	− 7832.81i	− 0.676102i
$513$	0	0
$514$	0	0
$515$	17292.9i	1.47964i
$516$	0	0
$517$	4983.93	0.423971
$518$	0	0
$519$	0	0
$520$	−15259.7	−1.28689
$521$	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
$533$	22814.5i	1.85405i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24021.4i	1.91962i
$540$	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	$0.112824\pi$
$548$	12565.6i	0.979520i
$549$	0	0
$550$	−90192.6	−6.99241
$551$	0	0
$552$	0	0
$553$	0	0
$554$	− 33801.9i	− 2.59225i
$555$	0	0
$556$	3945.89	0.300976
$557$	− 13993.2i	− 1.06447i	−0.846595	−	0.532237i	$-0.821352\pi$
$557$	− 13993.2i	− 1.06447i	0.846595	−	0.532237i	$-0.178648\pi$
$558$	0	0
$559$	21186.2	1.60301
$560$	0	0
$561$	0	0
$562$	20099.3	1.50861
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	2171.20i	0.161241i
$567$	0	0
$568$	−1976.55	−0.146011
$569$	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	$-0.881257\pi$
$571$	−25411.9	−1.86244	−0.931222	+	0.364451i	$0.881257\pi$
$572$	− 38096.5i	− 2.78478i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	21753.9i	1.56547i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−6183.88	−0.435928
$587$	− 20858.8i	− 1.46667i	−0.679868	−	0.733335i	$-0.737963\pi$
$587$	− 20858.8i	− 1.46667i	0.679868	−	0.733335i	$-0.262037\pi$
$588$	0	0
$589$	0	0
$590$	− 62927.4i	− 4.39098i
$591$	0	0
$592$	0	0
$593$	12660.4i	0.876726i	0.898798	+	0.438363i	$0.144442\pi$
$593$	12660.4i	0.876726i	−0.898798	+	0.438363i	$0.855558\pi$
$594$	0	0
$595$	0	0
$596$	30914.8i	2.12470i
$597$	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	$-0.640956\pi$
$601$	−12626.6	−0.856991	−0.428495	+	0.903544i	$0.640956\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 72875.8i	− 4.89723i
$606$	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
$610$	−85297.0	−5.66160
$611$	− 3335.67i	− 0.220862i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6923.04i	− 0.451719i	−0.974160	−	0.225860i	$-0.927481\pi$
$617$	− 6923.04i	− 0.451719i	0.974160	−	0.225860i	$-0.0725191\pi$
$618$	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
$625$	32614.9	2.08735
$626$	28321.4i	1.80823i
$627$	0	0
$628$	10839.6	0.688768
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	6677.15i	0.420257i
$633$	0	0
$634$	38735.9	2.42650
$635$	50880.2i	3.17972i
$636$	0	0
$637$	16077.2	1.00000
$638$	0	0
$639$	0	0
$640$	37439.5	2.31239
$641$	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
$649$	48807.1	2.95200
$650$	60364.5i	3.64260i
$651$	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
$656$	− 10784.0i	− 0.641836i
$657$	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
$664$	24096.6	1.40833
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 41268.2i	− 2.39029i
$669$	0	0
$670$	0	0
$671$	− 66157.2i	− 3.80622i
$672$	0	0
$673$	−25522.0	−1.46181	−0.730907	−	0.682477i	$-0.760903\pi$
$673$	−25522.0	−1.46181	−0.730907	+	0.682477i	$0.760903\pi$
$674$	46138.0i	2.63675i
$675$	0	0
$676$	−25497.4	−1.45069
$677$	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
$683$	14207.2i	0.795936i	0.917399	+	0.397968i	$0.130285\pi$
$683$	14207.2i	0.795936i	−0.917399	+	0.397968i	$0.869715\pi$
$684$	0	0
$685$	22079.5	1.23155
$686$	0	0
$687$	0	0
$688$	−10014.3	−0.554931
$689$	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
$695$	− 6933.46i	− 0.378419i
$696$	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
$704$	57612.0i	3.08428i
$705$	0	0
$706$	48945.8	2.60920
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	11179.1i	0.590908i
$711$	0	0
$712$	−2478.45	−0.130455
$713$	0	0
$714$	0	0
$715$	−66940.7	−3.50132
$716$	0	0
$717$	0	0
$718$	34967.5	1.81751
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	− 30370.4i	− 1.56547i
$723$	0	0
$724$	−51412.6	−2.63914
$725$	0	0
$726$	0	0
$727$	28455.0	1.45163	0.725817	−	0.687888i	$-0.241462\pi$
$727$	28455.0	1.45163	0.725817	+	0.687888i	$0.241462\pi$
$728$	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			$\pi$
$734$	36733.2i	1.84720i
$735$	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
$743$	− 35259.5i	− 1.74097i	−0.492191	−	0.870487i	$-0.663804\pi$
$743$	− 35259.5i	− 1.74097i	0.492191	−	0.870487i	$-0.336196\pi$
$744$	0	0
$745$	54321.5	2.67139
$746$	28896.1i	1.41818i
$747$	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	$0.281175\pi$
$752$	1576.71i	0.0764583i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35009.9	1.68092	0.840460	−	0.541873i	$-0.182285\pi$
$757$	35009.9	1.68092	0.840460	+	0.541873i	$0.182285\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30422.5i	1.44916i	0.689189	+	0.724582i	$0.257967\pi$
$761$	30422.5i	1.44916i	−0.689189	+	0.724582i	$0.742033\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−35250.9	−1.66275
$767$	− 32665.9i	− 1.53780i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	42841.8i	1.99342i	0.0810548	+	0.996710i	$0.474171\pi$
$773$	42841.8i	1.99342i	−0.0810548	+	0.996710i	$0.525829\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−8670.64	−0.397260
$782$	0	0
$783$	0	0
$784$	−7599.37	−0.346181
$785$	− 19046.6i	− 0.865991i
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	− 44590.6i	− 2.01583i
$789$	0	0
$790$	37765.1	1.70079
$791$	0	0
$792$	0	0
$793$	−44278.0	−1.98280
$794$	0	0
$795$	0	0
$796$	−65109.9	−2.89920
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	− 65680.5i	− 2.90270i
$801$	0	0
$802$	−21016.8	−0.925346
$803$	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
$820$	115195.	4.90583
$821$	33486.7i	1.42350i	0.702433	+	0.711749i	$0.252097\pi$
$821$	33486.7i	1.42350i	−0.702433	+	0.711749i	$0.747903\pi$
$822$	0	0
$823$	37352.0	1.58203	0.791014	−	0.611798i	$-0.209554\pi$
$823$	37352.0	1.58203	0.791014	+	0.611798i	$0.209554\pi$
$824$	13538.1i	0.572356i
$825$	0	0
$826$	0	0
$827$	− 35931.1i	− 1.51082i	−0.655253	−	0.755409i	$-0.727438\pi$
$827$	− 35931.1i	− 1.51082i	0.655253	−	0.755409i	$-0.272562\pi$
$828$	0	0
$829$	3079.14	0.129002	0.0645012	−	0.997918i	$-0.479454\pi$
$829$	3079.14	0.129002	0.0645012	+	0.997918i	$0.479454\pi$
$830$	− 136287.i	− 5.69952i
$831$	0	0
$832$	38558.8	1.60672
$833$	0	0
$834$	0	0
$835$	−72513.9	−3.00533
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12857.9i	0.529086i	0.964374	+	0.264543i	$0.0852210\pi$
$839$	12857.9i	0.529086i	−0.964374	+	0.264543i	$0.914779\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	−66616.3	−2.71686
$845$	44802.4i	1.82396i
$846$	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	$-0.829800\pi$
$859$	−43324.3	−1.72085	−0.860423	+	0.509581i	$0.829800\pi$
$860$	− 106973.i	− 4.24158i
$861$	0	0
$862$	−79225.7	−3.13044
$863$	44880.2i	1.77027i	0.465339	+	0.885133i	$0.345932\pi$
$863$	44880.2i	1.77027i	−0.465339	+	0.885133i	$0.654068\pi$
$864$	0	0
$865$	0	0
$866$	159.647i	0.00626447i
$867$	0	0
$868$	0	0
$869$	29291.0i	1.14342i
$870$	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			$\pi$
$878$	50122.9i	1.92661i
$879$	0	0
$880$	31641.6	1.21209
$881$	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	$-0.983919\pi$
$883$	−52410.3	−1.99745	−0.998724	+	0.0504988i	$0.983919\pi$
$884$	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
$890$	14017.8i	0.527952i
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−84240.8	−3.13046
$899$	0	0
$900$	0	0
$901$	0	0
$902$	150935.i	5.57161i
$903$	0	0
$904$	0	0
$905$	90338.9i	3.31820i
$906$	0	0
$907$	−35017.1	−1.28195	−0.640973	−	0.767564i	$-0.721469\pi$
$907$	−35017.1	−1.28195	−0.640973	+	0.767564i	$0.721469\pi$
$908$	17667.3i	0.645715i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	105706.	3.83171
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	36762.2	1.31956	0.659779	−	0.751460i	$-0.270650\pi$
$919$	36762.2	1.31956	0.659779	+	0.751460i	$0.270650\pi$
$920$	0	0
$921$	0	0
$922$	−41524.9	−1.48324
$923$	5803.12i	0.206947i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 56151.6i	− 1.98307i	−0.129832	−	0.991536i	$-0.541444\pi$
$929$	− 56151.6i	− 1.98307i	0.129832	−	0.991536i	$-0.458556\pi$
$930$	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	$-0.574340\pi$
$937$	−13275.6	−0.462856	−0.231428	+	0.972852i	$0.574340\pi$
$938$	0	0
$939$	0	0
$940$	−16842.4	−0.584404
$941$	− 55622.6i	− 1.92693i	−0.267826	−	0.963467i	$-0.586305\pi$
$941$	− 55622.6i	− 1.92693i	0.267826	−	0.963467i	$-0.413695\pi$
$942$	0	0
$943$	0	0
$944$	15440.5i	0.532359i
$945$	0	0
$946$	140163.	4.81721
$947$	− 14319.3i	− 0.491356i	−0.969352	−	0.245678i	$-0.920989\pi$
$947$	− 14319.3i	− 0.491356i	0.969352	−	0.245678i	$-0.0790105\pi$
$948$	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
$956$	48646.2i	1.64574i
$957$	0	0
$958$	89935.9	3.03309
$959$	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			$\pi$
$968$	− 57052.4i	− 1.89435i
$969$	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
$976$	20929.4	0.686407
$977$	− 31890.7i	− 1.04429i	−0.852856	−	0.522146i	$-0.825132\pi$
$977$	− 31890.7i	− 1.04429i	0.852856	−	0.522146i	$-0.174868\pi$
$978$	0	0
$979$	−10872.3	−0.354935
$980$	− 81176.6i	− 2.64601i
$981$	0	0
$982$	0	0
$983$	− 11061.5i	− 0.358908i	−0.983766	−	0.179454i	$-0.942567\pi$
$983$	− 11061.5i	− 0.358908i	0.983766	−	0.179454i	$-0.0574331\pi$
$984$	0	0
$985$	−78351.8	−2.53451
$986$	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
$995$	114407.i	3.64517i
$996$	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.64.1	✓	4
3.2	odd	2	inner	117.4.b.c.64.4	yes	4
13.5	odd	4		1521.4.a.y.1.1		4
13.8	odd	4		1521.4.a.y.1.4		4
13.12	even	2	inner	117.4.b.c.64.4	yes	4
39.5	even	4		1521.4.a.y.1.4		4
39.8	even	4		1521.4.a.y.1.1		4
39.38	odd	2	CM	117.4.b.c.64.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.b.c.64.1	✓	4	1.1	even	1	trivial
117.4.b.c.64.1	✓	4	39.38	odd	2	CM
117.4.b.c.64.4	yes	4	3.2	odd	2	inner
117.4.b.c.64.4	yes	4	13.12	even	2	inner
1521.4.a.y.1.1		4	13.5	odd	4
1521.4.a.y.1.1		4	39.8	even	4
1521.4.a.y.1.4		4	13.8	odd	4
1521.4.a.y.1.4		4	39.5	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)

\(f(q)\)	\(=\)	\(q-4.42782i q^{2} -11.6056 q^{4} +20.3925i q^{5} +15.9647i q^{8} +O(q^{10})\)	\(q-4.42782i q^{2} -11.6056 q^{4} +20.3925i q^{5} +15.9647i q^{8} +90.2944 q^{10} +70.0332i q^{11} +46.8722 q^{13} -22.1556 q^{16} -236.667i q^{20} +310.094 q^{22} -290.855 q^{25} -207.541i q^{26} +225.819i q^{32} -325.561 q^{40} +486.739i q^{41} +452.000 q^{43} -812.774i q^{44} -71.1653i q^{47} +343.000 q^{49} +1287.85i q^{50} -543.977 q^{52} -1428.15 q^{55} -696.914i q^{59} -944.654 q^{61} +822.638 q^{64} +955.842i q^{65} +123.807i q^{71} +418.244 q^{79} -451.809i q^{80} +2155.19 q^{82} -1509.37i q^{83} -2001.37i q^{86} -1118.06 q^{88} +155.245i q^{89} -315.107 q^{94} -1518.74i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4}+O(q^{10}) \) 4 * q - 32 * q^4	\( 4 q - 32 q^{4} + 116 q^{10} - 204 q^{16} + 476 q^{22} - 500 q^{25} - 884 q^{40} + 1808 q^{43} + 1372 q^{49} - 676 q^{52} - 3232 q^{55} + 1632 q^{64} + 2924 q^{82} - 2756 q^{88} - 5140 q^{94}+O(q^{100}) \) 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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