LMFDB - Embedded newform 361.3.f.a.262.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [361,3,Mod(116,361)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(361, base_ring=CyclotomicField(18))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("361.116");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 361 = 19^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	361.f (of order $18$, degree $6$, not 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:	$9.83653754341$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			262.1
Root			$-0.766044 - 0.642788i$ of defining polynomial
Character	$\chi$	$=$	361.262
Dual form			361.3.f.a.299.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.06418 + 2.57115i) q^{4} +(-6.89440 + 5.78509i) q^{5} +(2.50000 - 4.33013i) q^{7} +(-8.45723 + 3.07818i) q^{9} +(-1.50000 - 2.59808i) q^{11} +(2.77837 + 15.7569i) q^{16} +(-14.0954 - 5.13030i) q^{17} -36.0000 q^{20} +(-22.9813 - 19.2836i) q^{23} +(9.72430 - 55.1492i) q^{25} +(18.7939 - 6.84040i) q^{28} +(7.81417 + 44.3163i) q^{35} +(-33.8289 - 12.3127i) q^{36} +(-65.1138 + 54.6369i) q^{43} +(2.08378 - 11.8177i) q^{44} +(40.5000 - 70.1481i) q^{45} +(-70.4769 + 25.6515i) q^{47} +(12.0000 + 20.7846i) q^{49} +(25.3717 + 9.23454i) q^{55} +(78.9026 + 66.2071i) q^{61} +(-7.81417 + 44.3163i) q^{63} +(-32.0000 + 55.4256i) q^{64} +(-30.0000 - 51.9615i) q^{68} +(-4.34120 - 24.6202i) q^{73} -15.0000 q^{77} +(-110.310 - 92.5614i) q^{80} +(62.0496 - 52.0658i) q^{81} +(-45.0000 + 77.9423i) q^{83} +(126.859 - 46.1727i) q^{85} +(-20.8378 - 118.177i) q^{92} +(20.6832 + 17.3553i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 15 q^{7} - 9 q^{11} - 216 q^{20} + 243 q^{45} + 72 q^{49} - 192 q^{64} - 180 q^{68} - 90 q^{77} - 270 q^{83}+O(q^{100}) $ 6 * q + 15 * q^7 - 9 * q^11 - 216 * q^20 + 243 * q^45 + 72 * q^49 - 192 * q^64 - 180 * q^68 - 90 * q^77 - 270 * q^83

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{11}{18}\right)$

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		−0.939693	−	0.342020i	$-0.888889\pi$
$2$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$3$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$3$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$4$	3.06418	+	2.57115i	0.766044	+	0.642788i
$5$	−6.89440	+	5.78509i	−1.37888	+	1.15702i	−0.409255	+	0.912420i	$0.634211\pi$
$5$	−6.89440	+	5.78509i	−1.37888	+	1.15702i	−0.969625	+	0.244598i	$0.921344\pi$
$6$		0			0
$7$	2.50000	−	4.33013i	0.357143	−	0.618590i	−0.630339	−	0.776320i	$-0.717084\pi$
$7$	2.50000	−	4.33013i	0.357143	−	0.618590i	0.987482	+	0.157730i	$0.0504176\pi$
$8$		0			0
$9$	−8.45723	+	3.07818i	−0.939693	+	0.342020i
$10$		0			0
$11$	−1.50000	−	2.59808i	−0.136364	−	0.236189i	0.789754	−	0.613424i	$-0.210208\pi$
$11$	−1.50000	−	2.59808i	−0.136364	−	0.236189i	−0.926118	+	0.377235i	$0.876875\pi$
$12$		0			0
$13$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$13$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$14$		0			0
$15$		0			0
$16$	2.77837	+	15.7569i	0.173648	+	0.984808i
$17$	−14.0954	−	5.13030i	−0.829141	−	0.301782i	−0.107634	−	0.994191i	$-0.534328\pi$
$17$	−14.0954	−	5.13030i	−0.829141	−	0.301782i	−0.721506	+	0.692408i	$0.756550\pi$
$18$		0			0
$19$		0			0
$19$		0			0
$20$	−36.0000			−1.80000
$21$		0			0
$22$		0			0
$23$	−22.9813	−	19.2836i	−0.999188	−	0.838419i	−0.0123166	−	0.999924i	$-0.503921\pi$
$23$	−22.9813	−	19.2836i	−0.999188	−	0.838419i	−0.986872	+	0.161506i	$0.948365\pi$
$24$		0			0
$25$	9.72430	−	55.1492i	0.388972	−	2.20597i
$26$		0			0
$27$		0			0
$28$	18.7939	−	6.84040i	0.671209	−	0.244300i
$29$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$29$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.81417	+	44.3163i	0.223262	+	1.26618i
$36$	−33.8289	−	12.3127i	−0.939693	−	0.342020i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$41$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$42$		0			0
$43$	−65.1138	+	54.6369i	−1.51427	+	1.27063i	−0.659398	+	0.751794i	$0.729189\pi$
$43$	−65.1138	+	54.6369i	−1.51427	+	1.27063i	−0.854876	+	0.518833i	$0.826367\pi$
$44$	2.08378	−	11.8177i	0.0473586	−	0.268584i
$45$	40.5000	−	70.1481i	0.900000	−	1.55885i
$46$		0			0
$47$	−70.4769	+	25.6515i	−1.49951	+	0.545777i	−0.955934	−	0.293583i	$-0.905152\pi$
$47$	−70.4769	+	25.6515i	−1.49951	+	0.545777i	−0.543576	+	0.839360i	$0.682930\pi$
$48$		0			0
$49$	12.0000	+	20.7846i	0.244898	+	0.424176i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$53$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$54$		0			0
$55$	25.3717	+	9.23454i	0.461304	+	0.167901i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$59$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$60$		0			0
$61$	78.9026	+	66.2071i	1.29348	+	1.08536i	0.991232	+	0.132135i	$0.0421832\pi$
$61$	78.9026	+	66.2071i	1.29348	+	1.08536i	0.302253	+	0.953228i	$0.402261\pi$
$62$		0			0
$63$	−7.81417	+	44.3163i	−0.124034	+	0.703434i
$64$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$67$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$68$	−30.0000	−	51.9615i	−0.441176	−	0.764140i
$69$		0			0
$70$		0			0
$71$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$71$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$72$		0			0
$73$	−4.34120	−	24.6202i	−0.0594686	−	0.337263i	0.940528	−	0.339715i	$-0.110331\pi$
$73$	−4.34120	−	24.6202i	−0.0594686	−	0.337263i	−0.999997	+	0.00245203i	$0.999219\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−15.0000			−0.194805
$78$		0			0
$79$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$79$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$80$	−110.310	−	92.5614i	−1.37888	−	1.15702i
$81$	62.0496	−	52.0658i	0.766044	−	0.642788i
$82$		0			0
$83$	−45.0000	+	77.9423i	−0.542169	+	0.939064i	0.456611	+	0.889667i	$0.349063\pi$
$83$	−45.0000	+	77.9423i	−0.542169	+	0.939064i	−0.998779	+	0.0493970i	$0.984270\pi$
$84$		0			0
$85$	126.859	−	46.1727i	1.49245	−	0.543208i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$89$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$90$		0			0
$91$		0			0
$92$	−20.8378	−	118.177i	−0.226498	−	1.28453i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$97$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$98$		0			0
$99$	20.6832	+	17.3553i	0.208921	+	0.175306i
$100$	171.594	−	143.984i	1.71594	−	1.43984i
$101$	−17.7121	+	100.450i	−0.175367	+	0.994558i	0.762351	+	0.647163i	$0.224045\pi$
$101$	−17.7121	+	100.450i	−0.175367	+	0.994558i	−0.937719	+	0.347395i	$0.887066\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$109$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$110$		0			0
$111$		0			0
$112$	75.1754	+	27.3616i	0.671209	+	0.244300i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	270.000			2.34783
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−57.4533	+	48.2091i	−0.482801	+	0.405118i
$120$		0			0
$121$	56.0000	−	96.9948i	0.462810	−	0.801610i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	139.500	+	241.621i	1.11600	+	1.93297i
$126$		0			0
$127$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$127$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	200.155	+	72.8503i	1.52790	+	0.556109i	0.963105	−	0.269124i	$-0.0867343\pi$
$131$	200.155	+	72.8503i	1.52790	+	0.556109i	0.564792	+	0.825233i	$0.308956\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	195.341	+	163.911i	1.42585	+	1.19643i	0.948116	+	0.317924i	$0.102986\pi$
$137$	195.341	+	163.911i	1.42585	+	1.19643i	0.477733	+	0.878505i	$0.341459\pi$
$138$		0			0
$139$	−34.2087	+	194.007i	−0.246106	+	1.39573i	0.571805	+	0.820390i	$0.306243\pi$
$139$	−34.2087	+	194.007i	−0.246106	+	1.39573i	−0.817911	+	0.575345i	$0.804868\pi$
$140$	−90.0000	+	155.885i	−0.642857	+	1.11346i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−72.0000	−	124.708i	−0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−30.7357	−	174.311i	−0.206280	−	1.16987i	−0.895413	−	0.445237i	$-0.853119\pi$
$149$	−30.7357	−	174.311i	−0.206280	−	1.16987i	0.689133	−	0.724635i	$-0.257992\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	135.000			0.882353
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.66044	−	6.42788i	0.0487926	−	0.0409419i	−0.618065	−	0.786127i	$-0.712083\pi$
$157$	7.66044	−	6.42788i	0.0487926	−	0.0409419i	0.666858	+	0.745185i	$0.267639\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−140.954	+	51.3030i	−0.875490	+	0.318652i
$162$		0			0
$163$	−125.000	−	216.506i	−0.766871	−	1.32826i	−0.939252	−	0.343229i	$-0.888479\pi$
$163$	−125.000	−	216.506i	−0.766871	−	1.32826i	0.172380	−	0.985030i	$-0.444854\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$167$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$168$		0			0
$169$	−158.808	−	57.8014i	−0.939693	−	0.342020i
$170$		0			0
$171$		0			0
$172$	−340.000			−1.97674
$173$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$173$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$174$		0			0
$175$	−214.492	−	179.981i	−1.22567	−	1.02846i
$176$	36.7701	−	30.8538i	0.208921	−	0.175306i
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$	304.460	−	110.815i	1.69145	−	0.615636i
$181$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$181$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	7.81417	+	44.3163i	0.0417870	+	0.236986i
$188$	−281.908	−	102.606i	−1.49951	−	0.545777i
$189$		0			0
$190$		0			0
$191$	−93.0000			−0.486911			−0.243455	−	0.969912i	$-0.578281\pi$
$191$	−93.0000			−0.486911			−0.243455	+	0.969912i	$0.578281\pi$
$192$		0			0
$193$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$193$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$194$		0			0
$195$		0			0
$196$	−16.6702	+	94.5415i	−0.0850522	+	0.482355i
$197$	−45.0000	+	77.9423i	−0.228426	+	0.395646i	−0.957342	−	0.288958i	$-0.906691\pi$
$197$	−45.0000	+	77.9423i	−0.228426	+	0.395646i	0.728916	+	0.684604i	$0.240025\pi$
$198$		0			0
$199$	−213.310	+	77.6386i	−1.07191	+	0.390144i	−0.816891	−	0.576792i	$-0.804304\pi$
$199$	−213.310	+	77.6386i	−1.07191	+	0.390144i	−0.255020	+	0.966936i	$0.582082\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	253.717	+	92.3454i	1.22569	+	0.446113i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$211$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	132.841	−	753.378i	0.617864	−	3.50408i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	54.0000	+	93.5307i	0.245455	+	0.425140i
$221$		0			0
$222$		0			0
$223$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$223$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$224$		0			0
$225$	87.5187	+	496.343i	0.388972	+	2.20597i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	−17.0000			−0.0742358			−0.0371179	−	0.999311i	$-0.511818\pi$
$229$	−17.0000			−0.0742358			−0.0371179	+	0.999311i	$0.511818\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−356.211	+	298.896i	−1.52880	+	1.28282i	−0.722313	+	0.691567i	$0.756921\pi$
$233$	−356.211	+	298.896i	−1.52880	+	1.28282i	−0.806488	+	0.591250i	$0.798635\pi$
$234$		0			0
$235$	337.500	−	584.567i	1.43617	−	2.48752i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	226.500	+	392.310i	0.947699	+	1.64146i	0.750255	+	0.661148i	$0.229930\pi$
$239$	226.500	+	392.310i	0.947699	+	1.64146i	0.197443	+	0.980314i	$0.436736\pi$
$240$		0			0
$241$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$241$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$242$		0			0
$243$		0			0
$244$	71.5430	+	405.741i	0.293209	+	1.66287i
$245$	−202.974	−	73.8764i	−0.828464	−	0.301536i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.6832	+	17.3553i	0.0824032	+	0.0691445i	0.683059	−	0.730363i	$-0.260649\pi$
$251$	20.6832	+	17.3553i	0.0824032	+	0.0691445i	−0.600656	+	0.799508i	$0.705094\pi$
$252$	−137.888	+	115.702i	−0.547175	+	0.459134i
$253$	−15.6283	+	88.6327i	−0.0617721	+	0.350327i
$254$		0			0
$255$		0			0
$256$	−240.561	+	87.5572i	−0.939693	+	0.342020i
$257$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$257$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−70.3275	−	398.847i	−0.267405	−	1.51653i	−0.762098	−	0.647461i	$-0.775831\pi$
$263$	−70.3275	−	398.847i	−0.267405	−	1.51653i	0.494693	−	0.869068i	$-0.335280\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$269$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$270$		0			0
$271$	−108.778	+	91.2758i	−0.401396	+	0.336811i	−0.821033	−	0.570881i	$-0.806602\pi$
$271$	−108.778	+	91.2758i	−0.401396	+	0.336811i	0.419637	+	0.907692i	$0.362157\pi$
$272$	41.6756	−	236.354i	0.153219	−	0.868948i
$273$		0			0
$274$		0			0
$275$	−157.868	+	57.4594i	−0.574067	+	0.208943i
$276$		0			0
$277$	−267.500	−	463.324i	−0.965704	−	1.67265i	−0.707712	−	0.706501i	$-0.750272\pi$
$277$	−267.500	−	463.324i	−0.965704	−	1.67265i	−0.257992	−	0.966147i	$-0.583061\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$281$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$282$		0			0
$283$	−371.179	−	135.098i	−1.31159	−	0.477378i	−0.410833	−	0.911711i	$-0.634762\pi$
$283$	−371.179	−	135.098i	−1.31159	−	0.477378i	−0.900752	+	0.434333i	$0.856984\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−49.0268	−	41.1384i	−0.169643	−	0.142347i
$290$		0			0
$291$		0			0
$292$	50.0000	−	86.6025i	0.171233	−	0.296584i
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	73.8005	+	418.543i	0.245184	+	1.39051i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−927.000			−3.03934
$306$		0			0
$307$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$307$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$308$	−45.9627	−	38.5673i	−0.149229	−	0.125218i
$309$		0			0
$310$		0			0
$311$	−301.500	+	522.213i	−0.969453	+	1.67914i	−0.272312	+	0.962209i	$0.587788\pi$
$311$	−301.500	+	522.213i	−0.969453	+	1.67914i	−0.697142	+	0.716933i	$0.745545\pi$
$312$		0			0
$313$	554.419	−	201.792i	1.77131	−	0.644703i	0.771340	−	0.636423i	$-0.219587\pi$
$313$	554.419	−	201.792i	1.77131	−	0.644703i	0.999966	−	0.00827904i	$-0.00263533\pi$
$314$		0			0
$315$	−202.500	−	350.740i	−0.642857	−	1.11346i
$316$		0			0
$317$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$317$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$318$		0			0
$319$		0			0
$320$	−100.021	−	567.249i	−0.312567	−	1.77265i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	324.000			1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−65.1181	+	369.303i	−0.197927	+	1.12250i
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	−338.289	+	123.127i	−1.01894	+	0.370865i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$337$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$338$		0			0
$339$		0			0
$340$	507.434	+	184.691i	1.49245	+	0.543208i
$341$		0			0
$342$		0			0
$343$	365.000			1.06414
$344$		0			0
$345$		0			0
$346$		0			0
$347$	517.080	−	433.882i	1.49014	−	1.25038i	0.595694	−	0.803211i	$-0.296877\pi$
$347$	517.080	−	433.882i	1.49014	−	1.25038i	0.894450	−	0.447168i	$-0.147567\pi$
$348$		0			0
$349$	−263.500	+	456.395i	−0.755014	+	1.30772i	0.190353	+	0.981716i	$0.439037\pi$
$349$	−263.500	+	456.395i	−0.755014	+	1.30772i	−0.945367	+	0.326007i	$0.894297\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	255.000	+	441.673i	0.722380	+	1.25120i	0.960044	+	0.279851i	$0.0902850\pi$
$353$	255.000	+	441.673i	0.722380	+	1.25120i	−0.237664	+	0.971347i	$0.576382\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−228.345	−	83.1109i	−0.636059	−	0.231507i	0.00380715	−	0.999993i	$-0.498788\pi$
$359$	−228.345	−	83.1109i	−0.636059	−	0.231507i	−0.639867	+	0.768486i	$0.721010\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$	172.360	+	144.627i	0.472219	+	0.396239i
$366$		0			0
$367$	8.68241	−	49.2404i	0.0236578	−	0.134170i	−0.970691	−	0.240330i	$-0.922744\pi$
$367$	8.68241	−	49.2404i	0.0236578	−	0.134170i	0.994349	+	0.106160i	$0.0338555\pi$
$368$	240.000	−	415.692i	0.652174	−	1.12960i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		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
$383$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$383$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$384$		0			0
$385$	103.416	−	86.7763i	0.268613	−	0.225393i
$386$		0			0
$387$	382.500	−	662.509i	0.988372	−	1.71191i
$388$		0			0
$389$	143.773	−	52.3291i	0.369596	−	0.134522i	−0.150543	−	0.988603i	$-0.548102\pi$
$389$	143.773	−	52.3291i	0.369596	−	0.134522i	0.520139	+	0.854081i	$0.325880\pi$
$390$		0			0
$391$	225.000	+	389.711i	0.575448	+	0.996704i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	18.7540	+	106.359i	0.0473586	+	0.268584i
$397$	700.071	+	254.805i	1.76340	+	0.641826i	0.999992	−	0.00408637i	$-0.00130074\pi$
$397$	700.071	+	254.805i	1.76340	+	0.641826i	0.763411	+	0.645913i	$0.223523\pi$
$398$		0			0
$399$		0			0
$400$	896.000			2.24000
$401$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$401$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$402$		0			0
$403$		0			0
$404$	−312.546	+	262.257i	−0.773629	+	0.649152i
$405$	−126.590	+	717.925i	−0.312567	+	1.77265i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$409$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−140.655	−	797.694i	−0.338928	−	1.92215i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	762.000			1.81862			0.909308	−	0.416124i	$-0.136612\pi$
$419$	762.000			1.81862			0.909308	+	0.416124i	$0.136612\pi$
$420$		0			0
$421$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$421$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$422$		0			0
$423$	517.080	−	433.882i	1.22241	−	1.02572i
$424$		0			0
$425$	−420.000	+	727.461i	−0.988235	+	1.71167i
$426$		0			0
$427$	483.942	−	176.140i	1.13335	−	0.412507i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$431$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$432$		0			0
$433$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$433$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$439$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$440$		0			0
$441$	−165.466	−	138.842i	−0.375205	−	0.314835i
$442$		0			0
$443$	−7.81417	+	44.3163i	−0.0176392	+	0.100037i	−0.992356	−	0.123406i	$-0.960618\pi$
$443$	−7.81417	+	44.3163i	−0.0176392	+	0.100037i	0.974717	+	0.223443i	$0.0717295\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	160.000	+	277.128i	0.357143	+	0.618590i
$449$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$449$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−625.000			−1.36761			−0.683807	−	0.729663i	$-0.739677\pi$
$457$	−625.000			−1.36761			−0.683807	+	0.729663i	$0.739677\pi$
$458$		0			0
$459$		0			0
$460$	827.328	+	694.211i	1.79854	+	1.50915i
$461$	342.422	−	287.326i	0.742781	−	0.623267i	−0.190802	−	0.981628i	$-0.561109\pi$
$461$	342.422	−	287.326i	0.742781	−	0.623267i	0.933583	+	0.358362i	$0.116664\pi$
$462$		0			0
$463$	−377.500	+	653.849i	−0.815335	+	1.41220i	0.0937525	+	0.995596i	$0.470114\pi$
$463$	−377.500	+	653.849i	−0.815335	+	1.41220i	−0.909087	+	0.416606i	$0.863220\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−457.500	−	792.413i	−0.979657	−	1.69682i	−0.663620	−	0.748070i	$-0.730981\pi$
$467$	−457.500	−	792.413i	−0.979657	−	1.69682i	−0.316037	−	0.948747i	$-0.602352\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	239.622	+	87.2151i	0.506600	+	0.184387i
$474$		0			0
$475$		0			0
$476$	−300.000			−0.630252
$477$		0			0
$478$		0			0
$479$	−721.614	−	605.506i	−1.50650	−	1.26410i	−0.870238	−	0.492632i	$-0.836035\pi$
$479$	−721.614	−	605.506i	−1.50650	−	1.26410i	−0.636263	−	0.771472i	$-0.719521\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	420.982	−	153.225i	0.869798	−	0.316581i
$485$		0			0
$486$		0			0
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−159.409	−	904.054i	−0.324662	−	1.84125i	−0.512040	−	0.858962i	$-0.671110\pi$
$491$	−159.409	−	904.054i	−0.324662	−	1.84125i	0.187378	−	0.982288i	$-0.440001\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−243.000			−0.490909
$496$		0			0
$497$		0			0
$498$		0			0
$499$	400.641	−	336.178i	0.802888	−	0.673703i	−0.146011	−	0.989283i	$-0.546643\pi$
$499$	400.641	−	336.178i	0.802888	−	0.673703i	0.948899	+	0.315580i	$0.102199\pi$
$500$	−193.791	+	1099.05i	−0.387583	+	2.19809i
$501$		0			0
$502$		0			0
$503$	−873.914	+	318.079i	−1.73740	+	0.632363i	−0.999113	−	0.0421188i	$-0.986589\pi$
$503$	−873.914	+	318.079i	−1.73740	+	0.632363i	−0.738291	+	0.674482i	$0.764367\pi$
$504$		0			0
$505$	−459.000	−	795.011i	−0.908911	−	1.57428i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$509$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$510$		0			0
$511$	−117.462	−	42.7525i	−0.229866	−	0.0836644i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	172.360	+	144.627i	0.333385	+	0.279743i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$523$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$524$	426.000	+	737.854i	0.812977	+	1.40812i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	64.4235	+	365.364i	0.121784	+	0.690669i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	36.0000	−	62.3538i	0.0667904	−	0.115684i
$540$		0			0
$541$	429.440	−	156.303i	0.793788	−	0.288915i	0.0868785	−	0.996219i	$-0.472311\pi$
$541$	429.440	−	156.303i	0.793788	−	0.288915i	0.706910	+	0.707304i	$0.250089\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$547$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$548$	177.121	+	1004.50i	0.323214	+	1.83304i
$549$	−871.095	−	317.053i	−1.58669	−	0.577509i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−603.643	+	506.517i	−1.08569	+	0.911001i
$557$	190.145	−	1078.36i	0.341373	−	1.93602i	−0.0104229	−	0.999946i	$-0.503318\pi$
$557$	190.145	−	1078.36i	0.341373	−	1.93602i	0.351796	−	0.936077i	$-0.385571\pi$
$558$		0			0
$559$		0			0
$560$	−676.579	+	246.255i	−1.20818	+	0.439740i
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−70.3275	−	398.847i	−0.124034	−	0.703434i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	458.000			0.802102			0.401051	−	0.916056i	$-0.368645\pi$
$571$	458.000			0.802102			0.401051	+	0.916056i	$0.368645\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1286.95	+	1079.88i	−2.23818	+	1.87806i
$576$	100.021	−	567.249i	0.173648	−	0.984808i
$577$	572.500	−	991.599i	0.992201	−	1.71854i	0.388152	−	0.921595i	$-0.373113\pi$
$577$	572.500	−	991.599i	0.992201	−	1.71854i	0.604049	−	0.796947i	$-0.293553\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	225.000	+	389.711i	0.387263	+	0.670760i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1057.15	+	384.773i	1.80094	+	0.655490i	0.998252	+	0.0590960i	$0.0188218\pi$
$587$	1057.15	+	384.773i	1.80094	+	0.655490i	0.802692	+	0.596394i	$0.203400\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.9813	−	19.2836i	−0.0387544	−	0.0325188i	0.623205	−	0.782059i	$-0.285830\pi$
$593$	−22.9813	−	19.2836i	−0.0387544	−	0.0325188i	−0.661959	+	0.749540i	$0.730275\pi$
$594$		0			0
$595$	117.213	−	664.745i	0.196996	−	1.11722i
$596$	354.000	−	613.146i	0.593960	−	1.02877i
$597$		0			0
$598$		0			0
$599$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$599$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$600$		0			0
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	175.037	+	992.686i	0.289318	+	1.64080i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	413.664	+	347.105i	0.675922	+	0.567166i
$613$	225.983	−	189.622i	0.368651	−	0.309335i	−0.439577	−	0.898205i	$-0.644871\pi$
$613$	225.983	−	189.622i	0.368651	−	0.309335i	0.808228	+	0.588870i	$0.200427\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1000.77	−	364.251i	1.62200	−	0.590359i	0.638236	−	0.769840i	$-0.279664\pi$
$617$	1000.77	−	364.251i	1.62200	−	0.590359i	0.983761	+	0.179481i	$0.0574420\pi$
$618$		0			0
$619$	331.000	+	573.309i	0.534733	+	0.926185i	0.999176	+	0.0405823i	$0.0129213\pi$
$619$	331.000	+	573.309i	0.534733	+	0.926185i	−0.464443	+	0.885603i	$0.653745\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1044.00	−	379.984i	−1.67040	−	0.607975i
$626$		0			0
$627$		0			0
$628$	40.0000			0.0636943
$629$		0			0
$630$		0			0
$631$	−794.388	−	666.571i	−1.25894	−	1.05637i	−0.995795	−	0.0916145i	$-0.970797\pi$
$631$	−794.388	−	666.571i	−1.25894	−	1.05637i	−0.263141	−	0.964757i	$-0.584758\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$641$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$642$		0			0
$643$	193.618	+	1098.06i	0.301116	+	1.70771i	0.641245	+	0.767336i	$0.278418\pi$
$643$	193.618	+	1098.06i	0.301116	+	1.70771i	−0.340129	+	0.940379i	$0.610471\pi$
$644$	−563.816	−	205.212i	−0.875490	−	0.318652i
$645$		0			0
$646$		0			0
$647$	−1005.00			−1.55332			−0.776662	−	0.629918i	$-0.783088\pi$
$647$	−1005.00			−1.55332			−0.776662	+	0.629918i	$0.783088\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	173.648	−	984.808i	0.266332	−	1.51044i
$653$	−187.500	+	324.760i	−0.287136	+	0.497335i	−0.973125	−	0.230278i	$-0.926037\pi$
$653$	−187.500	+	324.760i	−0.287136	+	0.497335i	0.685989	+	0.727612i	$0.259370\pi$
$654$		0			0
$655$	−1801.39	+	655.653i	−2.75021	+	1.00100i
$656$		0			0
$657$	112.500	+	194.856i	0.171233	+	0.296584i
$658$		0			0
$659$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$659$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$660$		0			0
$661$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$661$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	53.6573	−	304.306i	0.0799662	−	0.453511i
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$		0			0
$676$	−338.000	−	585.433i	−0.500000	−	0.866025i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	−2295.00			−3.35036
$686$		0			0
$687$		0			0
$688$	−1041.82	−	874.191i	−1.51427	−	1.27063i
$689$		0			0
$690$		0			0
$691$	78.5000	−	135.966i	0.113603	−	0.196767i	−0.803617	−	0.595147i	$-0.797094\pi$
$691$	78.5000	−	135.966i	0.113603	−	0.196767i	0.917221	+	0.398380i	$0.130427\pi$
$692$		0			0
$693$	126.859	−	46.1727i	0.183057	−	0.0666273i
$694$		0			0
$695$	−886.500	−	1535.46i	−1.27554	−	2.20930i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−194.486	−	1102.98i	−0.277837	−	1.57569i
$701$	−1031.78	−	375.538i	−1.47187	−	0.535718i	−0.523264	−	0.852171i	$-0.675286\pi$
$701$	−1031.78	−	375.538i	−1.47187	−	0.535718i	−0.948608	+	0.316453i	$0.897508\pi$
$702$		0			0
$703$		0			0
$704$	192.000			0.272727
$705$		0			0
$706$		0			0
$707$	390.683	+	327.822i	0.552592	+	0.463680i
$708$		0			0
$709$	−228.868	+	1297.98i	−0.322804	+	1.83071i	0.201871	+	0.979412i	$0.435298\pi$
$709$	−228.868	+	1297.98i	−0.322804	+	1.83071i	−0.524675	+	0.851302i	$0.675813\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	167.223	+	948.370i	0.232577	+	1.31901i	0.847656	+	0.530546i	$0.178013\pi$
$719$	167.223	+	948.370i	0.232577	+	1.31901i	−0.615078	+	0.788466i	$0.710876\pi$
$720$	1217.84	+	443.258i	1.69145	+	0.615636i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−65.1138	+	54.6369i	−0.0895650	+	0.0751540i	−0.686471	−	0.727157i	$-0.740841\pi$
$727$	−65.1138	+	54.6369i	−0.0895650	+	0.0751540i	0.596906	+	0.802311i	$0.296397\pi$
$728$		0			0
$729$	−364.500	+	631.333i	−0.500000	+	0.866025i
$730$		0			0
$731$	1198.11	−	436.076i	1.63900	−	0.596547i
$732$		0			0
$733$	635.000	+	1099.85i	0.866303	+	1.50048i	0.865748	+	0.500481i	$0.166843\pi$
$733$	635.000	+	1099.85i	0.866303	+	1.50048i	0.000555189		1.00000i	$0.499823\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−514.012	−	187.085i	−0.695551	−	0.253160i	−0.0300406	−	0.999549i	$-0.509564\pi$
$739$	−514.012	−	187.085i	−0.695551	−	0.253160i	−0.665510	+	0.746389i	$0.731786\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$743$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$744$		0			0
$745$	1220.31	+	1023.96i	1.63800	+	1.37444i
$746$		0			0
$747$	140.655	−	797.694i	0.188293	−	1.06786i
$748$	−90.0000	+	155.885i	−0.120321	+	0.208402i
$749$		0			0
$750$		0			0
$751$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$751$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$752$	−600.000	−	1039.23i	−0.797872	−	1.38196i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−136.314	−	773.074i	−0.180071	−	1.02123i	−0.932126	−	0.362134i	$-0.882048\pi$
$757$	−136.314	−	773.074i	−0.180071	−	1.02123i	0.752055	−	0.659100i	$-0.229063\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1503.00			1.97503			0.987516	−	0.157516i	$-0.0503486\pi$
$761$	1503.00			1.97503			0.987516	+	0.157516i	$0.0503486\pi$
$762$		0			0
$763$		0			0
$764$	−284.969	−	239.117i	−0.372995	−	0.312980i
$765$	−930.744	+	780.987i	−1.21666	+	1.02090i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−998.893	+	363.567i	−1.29895	+	0.472779i	−0.896655	−	0.442730i	$-0.854010\pi$
$769$	−998.893	+	363.567i	−1.29895	+	0.472779i	−0.402296	+	0.915510i	$0.631788\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$773$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−294.161	+	246.830i	−0.375205	+	0.314835i
$785$	−15.6283	+	88.6327i	−0.0199087	+	0.112908i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−338.289	+	123.127i	−0.429301	+	0.156253i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−853.241	−	310.554i	−1.07191	−	0.390144i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	1125.00			1.40801
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−57.4533	+	48.2091i	−0.0715484	+	0.0600362i
$804$		0			0
$805$	675.000	−	1169.13i	0.838509	−	1.45234i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	796.500	+	1379.58i	0.984549	+	1.70529i	0.643924	+	0.765089i	$0.277305\pi$
$809$	796.500	+	1379.58i	0.984549	+	1.70529i	0.340624	+	0.940199i	$0.389362\pi$
$810$		0			0
$811$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$811$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2114.31	+	769.545i	2.59424	+	0.944227i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	893.974	+	750.133i	1.08888	+	0.913682i	0.996628	−	0.0820530i	$-0.0261477\pi$
$821$	893.974	+	750.133i	1.08888	+	0.913682i	0.0922562	+	0.995735i	$0.470592\pi$
$822$		0			0
$823$	−271.759	+	1541.22i	−0.330206	+	1.87269i	0.140027	+	0.990148i	$0.455281\pi$
$823$	−271.759	+	1541.22i	−0.330206	+	1.87269i	−0.470232	+	0.882543i	$0.655830\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$827$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$828$	540.000	+	935.307i	0.652174	+	1.12960i
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−62.5133	−	354.531i	−0.0750460	−	0.425607i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$839$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$840$		0			0
$841$	644.243	−	540.584i	0.766044	−	0.642788i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1429.27	−	520.213i	1.69145	−	0.615636i
$846$		0			0
$847$	−280.000	−	484.974i	−0.330579	−	0.572579i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	967.883	+	352.281i	1.13468	+	0.412990i	0.839990	−	0.542602i	$-0.182561\pi$
$853$	967.883	+	352.281i	1.13468	+	0.412990i	0.294692	+	0.955592i	$0.404783\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$857$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$858$		0			0
$859$	−1143.70	−	959.682i	−1.33144	−	1.11721i	−0.983739	−	0.179602i	$-0.942519\pi$
$859$	−1143.70	−	959.682i	−1.33144	−	1.11721i	−0.347698	−	0.937607i	$-0.613036\pi$
$860$	2344.10	−	1966.93i	2.72569	−	2.28713i
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1395.00			1.59429
$876$		0			0
$877$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$877$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$878$		0			0
$879$		0			0
$880$	−75.0160	+	425.437i	−0.0852455	+	0.483451i
$881$	268.500	−	465.056i	0.304767	−	0.527872i	−0.672442	−	0.740150i	$-0.734755\pi$
$881$	268.500	−	465.056i	0.304767	−	0.527872i	0.977209	+	0.212277i	$0.0680880\pi$
$882$		0			0
$883$	−784.643	+	285.587i	−0.888611	+	0.323428i	−0.745680	−	0.666305i	$-0.767875\pi$
$883$	−784.643	+	285.587i	−0.888611	+	0.323428i	−0.142931	+	0.989733i	$0.545653\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$887$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−228.345	−	83.1109i	−0.256280	−	0.0932782i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1008.00	+	1745.91i	−1.12000	+	1.93990i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$907$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$908$		0			0
$909$	−159.409	−	904.054i	−0.175367	−	0.994558i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	270.000			0.295728
$914$		0			0
$915$		0			0
$916$	−52.0910	−	43.7096i	−0.0568679	−	0.0477179i
$917$	815.837	−	684.569i	0.889681	−	0.746531i
$918$		0			0
$919$	−881.000	+	1525.94i	−0.958651	+	1.66043i	−0.232867	+	0.972509i	$0.574811\pi$
$919$	−881.000	+	1525.94i	−0.958651	+	1.66043i	−0.725783	+	0.687923i	$0.758523\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−603.283	−	219.577i	−0.649389	−	0.236358i	−0.00374067	−	0.999993i	$-0.501191\pi$
$929$	−603.283	−	219.577i	−0.649389	−	0.236358i	−0.645649	+	0.763635i	$0.723413\pi$
$930$		0			0
$931$		0			0
$932$	−1860.00			−1.99571
$933$		0			0
$934$		0			0
$935$	−310.248	−	260.329i	−0.331816	−	0.278427i
$936$		0			0
$937$	58.1721	−	329.911i	0.0620834	−	0.352092i	−0.937903	−	0.346897i	$-0.887235\pi$
$937$	58.1721	−	329.911i	0.0620834	−	0.352092i	0.999987	−	0.00519510i	$-0.00165366\pi$
$938$		0			0
$939$		0			0
$940$	2537.17	−	923.454i	2.69912	−	0.982398i
$941$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$941$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−317.776	−	1802.20i	−0.335561	−	1.90306i	−0.421624	−	0.906771i	$-0.638540\pi$
$947$	−317.776	−	1802.20i	−0.335561	−	1.90306i	0.0860632	−	0.996290i	$-0.472571\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$953$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$954$		0			0
$955$	641.179	−	538.013i	0.671392	−	0.563365i
$956$	−314.650	+	1784.47i	−0.329132	+	1.86660i
$957$		0			0
$958$		0			0
$959$	1198.11	−	436.076i	1.24933	−	0.454719i
$960$		0			0
$961$	−480.500	−	832.250i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1682.05	+	612.216i	1.73945	+	0.633109i	0.999229	−	0.0392535i	$-0.0124980\pi$
$967$	1682.05	+	612.216i	1.73945	+	0.633109i	0.740222	+	0.672362i	$0.234720\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$971$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$972$		0			0
$973$	754.554	+	633.146i	0.775492	+	0.650715i
$974$		0			0
$975$		0			0
$976$	−824.000	+	1427.21i	−0.844262	+	1.46231i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$	−432.000	−	748.246i	−0.440816	−	0.763516i
$981$		0			0
$982$		0			0
$983$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$983$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$984$		0			0
$985$	−140.655	−	797.694i	−0.142797	−	0.809842i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2550.00			2.57836
$990$		0			0
$991$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$991$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1021.50	−	1769.29i	1.02663	−	1.77818i
$996$		0			0
$997$	−1855.89	+	675.490i	−1.86148	+	0.677522i	−0.883636	+	0.468174i	$0.844912\pi$
$997$	−1855.89	+	675.490i	−1.86148	+	0.677522i	−0.977841	+	0.209348i	$0.932866\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	361.3.f.a.262.1		6
19.2	odd	18	inner	361.3.f.a.307.1		6
19.3	odd	18	inner	361.3.f.a.116.1		6
19.4	even	9		361.3.d.a.69.1		2
19.5	even	9	inner	361.3.f.a.299.1		6
19.6	even	9		361.3.d.a.293.1		2
19.7	even	3	inner	361.3.f.a.127.1		6
19.8	odd	6	inner	361.3.f.a.333.1		6
19.9	even	9		19.3.b.a.18.1	✓	1
19.10	odd	18		19.3.b.a.18.1	✓	1
19.11	even	3	inner	361.3.f.a.333.1		6
19.12	odd	6	inner	361.3.f.a.127.1		6
19.13	odd	18		361.3.d.a.293.1		2
19.14	odd	18	inner	361.3.f.a.299.1		6
19.15	odd	18		361.3.d.a.69.1		2
19.16	even	9	inner	361.3.f.a.116.1		6
19.17	even	9	inner	361.3.f.a.307.1		6
19.18	odd	2	CM	361.3.f.a.262.1		6
57.29	even	18		171.3.c.a.37.1		1
57.47	odd	18		171.3.c.a.37.1		1
76.47	odd	18		304.3.e.a.113.1		1
76.67	even	18		304.3.e.a.113.1		1
95.9	even	18		475.3.c.a.151.1		1
95.28	odd	36		475.3.d.a.474.2		2
95.29	odd	18		475.3.c.a.151.1		1
95.47	odd	36		475.3.d.a.474.1		2
95.48	even	36		475.3.d.a.474.2		2
95.67	even	36		475.3.d.a.474.1		2
152.29	odd	18		1216.3.e.a.1025.1		1
152.67	even	18		1216.3.e.b.1025.1		1
152.85	even	18		1216.3.e.a.1025.1		1
152.123	odd	18		1216.3.e.b.1025.1		1
228.47	even	18		2736.3.o.a.721.1		1
228.143	odd	18		2736.3.o.a.721.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.b.a.18.1	✓	1	19.9	even	9
19.3.b.a.18.1	✓	1	19.10	odd	18
171.3.c.a.37.1		1	57.29	even	18
171.3.c.a.37.1		1	57.47	odd	18
304.3.e.a.113.1		1	76.47	odd	18
304.3.e.a.113.1		1	76.67	even	18
361.3.d.a.69.1		2	19.4	even	9
361.3.d.a.69.1		2	19.15	odd	18
361.3.d.a.293.1		2	19.6	even	9
361.3.d.a.293.1		2	19.13	odd	18
361.3.f.a.116.1		6	19.3	odd	18	inner
361.3.f.a.116.1		6	19.16	even	9	inner
361.3.f.a.127.1		6	19.7	even	3	inner
361.3.f.a.127.1		6	19.12	odd	6	inner
361.3.f.a.262.1		6	1.1	even	1	trivial
361.3.f.a.262.1		6	19.18	odd	2	CM
361.3.f.a.299.1		6	19.5	even	9	inner
361.3.f.a.299.1		6	19.14	odd	18	inner
361.3.f.a.307.1		6	19.2	odd	18	inner
361.3.f.a.307.1		6	19.17	even	9	inner
361.3.f.a.333.1		6	19.8	odd	6	inner
361.3.f.a.333.1		6	19.11	even	3	inner
475.3.c.a.151.1		1	95.9	even	18
475.3.c.a.151.1		1	95.29	odd	18
475.3.d.a.474.1		2	95.47	odd	36
475.3.d.a.474.1		2	95.67	even	36
475.3.d.a.474.2		2	95.28	odd	36
475.3.d.a.474.2		2	95.48	even	36
1216.3.e.a.1025.1		1	152.29	odd	18
1216.3.e.a.1025.1		1	152.85	even	18
1216.3.e.b.1025.1		1	152.67	even	18
1216.3.e.b.1025.1		1	152.123	odd	18
2736.3.o.a.721.1		1	228.47	even	18
2736.3.o.a.721.1		1	228.143	odd	18

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:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	361.f (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.06418 + 2.57115i) q^{4} +(-6.89440 + 5.78509i) q^{5} +(2.50000 - 4.33013i) q^{7} +(-8.45723 + 3.07818i) q^{9} +(-1.50000 - 2.59808i) q^{11} +(2.77837 + 15.7569i) q^{16} +(-14.0954 - 5.13030i) q^{17} -36.0000 q^{20} +(-22.9813 - 19.2836i) q^{23} +(9.72430 - 55.1492i) q^{25} +(18.7939 - 6.84040i) q^{28} +(7.81417 + 44.3163i) q^{35} +(-33.8289 - 12.3127i) q^{36} +(-65.1138 + 54.6369i) q^{43} +(2.08378 - 11.8177i) q^{44} +(40.5000 - 70.1481i) q^{45} +(-70.4769 + 25.6515i) q^{47} +(12.0000 + 20.7846i) q^{49} +(25.3717 + 9.23454i) q^{55} +(78.9026 + 66.2071i) q^{61} +(-7.81417 + 44.3163i) q^{63} +(-32.0000 + 55.4256i) q^{64} +(-30.0000 - 51.9615i) q^{68} +(-4.34120 - 24.6202i) q^{73} -15.0000 q^{77} +(-110.310 - 92.5614i) q^{80} +(62.0496 - 52.0658i) q^{81} +(-45.0000 + 77.9423i) q^{83} +(126.859 - 46.1727i) q^{85} +(-20.8378 - 118.177i) q^{92} +(20.6832 + 17.3553i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 15 q^{7} - 9 q^{11} - 216 q^{20} + 243 q^{45} + 72 q^{49} - 192 q^{64} - 180 q^{68} - 90 q^{77} - 270 q^{83}+O(q^{100}) \) 6 * q + 15 * q^7 - 9 * q^11 - 216 * q^20 + 243 * q^45 + 72 * q^49 - 192 * q^64 - 180 * q^68 - 90 * q^77 - 270 * q^83

Introduction

L-functions

Modular forms

Varieties

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Database

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