LMFDB - Embedded newform 361.3.f.a.116.1

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			116.1
Root			$0.939693 + 0.342020i$ of defining polynomial
Character	$\chi$	$=$	361.116
Dual form			361.3.f.a.333.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.75877 - 1.36808i) q^{4} +(8.45723 - 3.07818i) q^{5} +(2.50000 + 4.33013i) q^{7} +(1.56283 + 8.86327i) q^{9} +O(q^{10})$	$q+(-3.75877 - 1.36808i) q^{4} +(8.45723 - 3.07818i) q^{5} +(2.50000 + 4.33013i) q^{7} +(1.56283 + 8.86327i) q^{9} +(-1.50000 + 2.59808i) q^{11} +(12.2567 + 10.2846i) q^{16} +(2.60472 - 14.7721i) q^{17} -36.0000 q^{20} +(28.1908 + 10.2606i) q^{23} +(42.8985 - 35.9961i) q^{25} +(-3.47296 - 19.6962i) q^{28} +(34.4720 + 28.9254i) q^{35} +(6.25133 - 35.4531i) q^{36} +(79.8739 - 29.0717i) q^{43} +(9.19253 - 7.71345i) q^{44} +(40.5000 + 70.1481i) q^{45} +(13.0236 + 73.8606i) q^{47} +(12.0000 - 20.7846i) q^{49} +(-4.68850 + 26.5898i) q^{55} +(-96.7883 - 35.2281i) q^{61} +(-34.4720 + 28.9254i) q^{63} +(-32.0000 - 55.4256i) q^{64} +(-30.0000 + 51.9615i) q^{68} +(-19.1511 - 16.0697i) q^{73} -15.0000 q^{77} +(135.316 + 49.2509i) q^{80} +(-76.1151 + 27.7036i) q^{81} +(-45.0000 - 77.9423i) q^{83} +(-23.4425 - 132.949i) q^{85} +(-91.9253 - 77.1345i) q^{92} +(-25.3717 - 9.23454i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 15 q^{7}+O(q^{10}) $ 6 * q + 15 * q^7	$ 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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{18}\right)$

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		0.173648	−	0.984808i	$-0.444444\pi$
$2$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$3$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$3$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$4$	−3.75877	−	1.36808i	−0.939693	−	0.342020i
$5$	8.45723	−	3.07818i	1.69145	−	0.615636i	0.696640	−	0.717421i	$-0.254677\pi$
$5$	8.45723	−	3.07818i	1.69145	−	0.615636i	0.994806	+	0.101784i	$0.0324552\pi$
$6$		0			0
$7$	2.50000	+	4.33013i	0.357143	+	0.618590i	0.987482	−	0.157730i	$-0.0504176\pi$
$7$	2.50000	+	4.33013i	0.357143	+	0.618590i	−0.630339	+	0.776320i	$0.717084\pi$
$8$		0			0
$9$	1.56283	+	8.86327i	0.173648	+	0.984808i
$10$		0			0
$11$	−1.50000	+	2.59808i	−0.136364	+	0.236189i	−0.926118	−	0.377235i	$-0.876875\pi$
$11$	−1.50000	+	2.59808i	−0.136364	+	0.236189i	0.789754	+	0.613424i	$0.210208\pi$
$12$		0			0
$13$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$13$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$14$		0			0
$15$		0			0
$16$	12.2567	+	10.2846i	0.766044	+	0.642788i
$17$	2.60472	−	14.7721i	0.153219	−	0.868948i	−0.807177	−	0.590309i	$-0.799006\pi$
$17$	2.60472	−	14.7721i	0.153219	−	0.868948i	0.960396	−	0.278639i	$-0.0898832\pi$
$18$		0			0
$19$		0			0
$19$		0			0
$20$	−36.0000			−1.80000
$21$		0			0
$22$		0			0
$23$	28.1908	+	10.2606i	1.22569	+	0.446113i	0.872118	−	0.489296i	$-0.162746\pi$
$23$	28.1908	+	10.2606i	1.22569	+	0.446113i	0.353568	+	0.935409i	$0.384968\pi$
$24$		0			0
$25$	42.8985	−	35.9961i	1.71594	−	1.43984i
$26$		0			0
$27$		0			0
$28$	−3.47296	−	19.6962i	−0.124034	−	0.703434i
$29$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$29$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	34.4720	+	28.9254i	0.984914	+	0.826441i
$36$	6.25133	−	35.4531i	0.173648	−	0.984808i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$41$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$42$		0			0
$43$	79.8739	−	29.0717i	1.85753	−	0.676086i	0.876760	−	0.480928i	$-0.159700\pi$
$43$	79.8739	−	29.0717i	1.85753	−	0.676086i	0.980772	−	0.195159i	$-0.0625221\pi$
$44$	9.19253	−	7.71345i	0.208921	−	0.175306i
$45$	40.5000	+	70.1481i	0.900000	+	1.55885i
$46$		0			0
$47$	13.0236	+	73.8606i	0.277098	+	1.57150i	0.732217	+	0.681071i	$0.238486\pi$
$47$	13.0236	+	73.8606i	0.277098	+	1.57150i	−0.455119	+	0.890431i	$0.650403\pi$
$48$		0			0
$49$	12.0000	−	20.7846i	0.244898	−	0.424176i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$53$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$54$		0			0
$55$	−4.68850	+	26.5898i	−0.0852455	+	0.483451i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$59$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$60$		0			0
$61$	−96.7883	−	35.2281i	−1.58669	−	0.577509i	−0.610048	−	0.792364i	$-0.708850\pi$
$61$	−96.7883	−	35.2281i	−1.58669	−	0.577509i	−0.976646	+	0.214855i	$0.931072\pi$
$62$		0			0
$63$	−34.4720	+	28.9254i	−0.547175	+	0.459134i
$64$	−32.0000	−	55.4256i	−0.500000	−	0.866025i
$65$		0			0
$66$		0			0
$67$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$67$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$68$	−30.0000	+	51.9615i	−0.441176	+	0.764140i
$69$		0			0
$70$		0			0
$71$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$71$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$72$		0			0
$73$	−19.1511	−	16.0697i	−0.262344	−	0.220133i	0.502122	−	0.864797i	$-0.332553\pi$
$73$	−19.1511	−	16.0697i	−0.262344	−	0.220133i	−0.764466	+	0.644664i	$0.776997\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−15.0000			−0.194805
$78$		0			0
$79$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$79$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$80$	135.316	+	49.2509i	1.69145	+	0.615636i
$81$	−76.1151	+	27.7036i	−0.939693	+	0.342020i
$82$		0			0
$83$	−45.0000	−	77.9423i	−0.542169	−	0.939064i	−0.998779	−	0.0493970i	$-0.984270\pi$
$83$	−45.0000	−	77.9423i	−0.542169	−	0.939064i	0.456611	−	0.889667i	$-0.349063\pi$
$84$		0			0
$85$	−23.4425	−	132.949i	−0.275794	−	1.56411i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$89$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$90$		0			0
$91$		0			0
$92$	−91.9253	−	77.1345i	−0.999188	−	0.838419i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$97$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$98$		0			0
$99$	−25.3717	−	9.23454i	−0.256280	−	0.0932782i
$100$	−210.491	+	76.6125i	−2.10491	+	0.766125i
$101$	−78.1365	+	65.5643i	−0.773629	+	0.649152i	−0.941636	−	0.336634i	$-0.890711\pi$
$101$	−78.1365	+	65.5643i	−0.773629	+	0.649152i	0.168007	+	0.985786i	$0.446267\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$109$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$110$		0			0
$111$		0			0
$112$	−13.8919	+	78.7846i	−0.124034	+	0.703434i
$113$		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
$119$	70.4769	−	25.6515i	0.592243	−	0.215559i
$120$		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		0.766044	−	0.642788i	$-0.222222\pi$
$127$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−36.9871	+	209.764i	−0.282344	+	1.60125i	0.432277	+	0.901741i	$0.357710\pi$
$131$	−36.9871	+	209.764i	−0.282344	+	1.60125i	−0.714621	+	0.699512i	$0.753401\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−239.622	−	87.2151i	−1.74906	−	0.636607i	−0.749389	−	0.662130i	$-0.769653\pi$
$137$	−239.622	−	87.2151i	−1.74906	−	0.636607i	−0.999674	+	0.0255235i	$0.991875\pi$
$138$		0			0
$139$	−150.911	+	126.629i	−1.08569	+	0.911001i	−0.996381	−	0.0850029i	$-0.972910\pi$
$139$	−150.911	+	126.629i	−1.08569	+	0.911001i	−0.0893082	+	0.996004i	$0.528466\pi$
$140$	−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
$149$	−135.590	−	113.773i	−0.909999	−	0.763580i	0.0621197	−	0.998069i	$-0.480214\pi$
$149$	−135.590	−	113.773i	−0.909999	−	0.763580i	−0.972119	+	0.234489i	$0.924658\pi$
$150$		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
$157$	−9.39693	+	3.42020i	−0.0598530	+	0.0217847i	−0.371773	−	0.928324i	$-0.621250\pi$
$157$	−9.39693	+	3.42020i	−0.0598530	+	0.0217847i	0.311920	+	0.950108i	$0.399028\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	26.0472	+	147.721i	0.161784	+	0.917523i
$162$		0			0
$163$	−125.000	+	216.506i	−0.766871	+	1.32826i	0.172380	+	0.985030i	$0.444854\pi$
$163$	−125.000	+	216.506i	−0.766871	+	1.32826i	−0.939252	+	0.343229i	$0.888479\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$167$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$168$		0			0
$169$	29.3465	−	166.433i	0.173648	−	0.984808i
$170$		0			0
$171$		0			0
$172$	−340.000			−1.97674
$173$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$173$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$174$		0			0
$175$	263.114	+	95.7656i	1.50351	+	0.547232i
$176$	−45.1052	+	16.4170i	−0.256280	+	0.0932782i
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	−56.2620	−	319.078i	−0.312567	−	1.77265i
$181$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$181$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	34.4720	+	28.9254i	0.184342	+	0.154682i
$188$	52.0945	−	295.442i	0.277098	−	1.57150i
$189$		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		−0.766044	−	0.642788i	$-0.777778\pi$
$193$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$194$		0			0
$195$		0			0
$196$	−73.5403	+	61.7076i	−0.375205	+	0.314835i
$197$	−45.0000	−	77.9423i	−0.228426	−	0.395646i	0.728916	−	0.684604i	$-0.240025\pi$
$197$	−45.0000	−	77.9423i	−0.228426	−	0.395646i	−0.957342	+	0.288958i	$0.906691\pi$
$198$		0			0
$199$	39.4181	+	223.551i	0.198081	+	1.12337i	0.907962	+	0.419052i	$0.137638\pi$
$199$	39.4181	+	223.551i	0.198081	+	1.12337i	−0.709881	+	0.704322i	$0.751251\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−46.8850	+	265.898i	−0.226498	+	1.28453i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$211$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	586.024	−	491.733i	2.72569	−	2.28713i
$216$		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		0.939693	−	0.342020i	$-0.111111\pi$
$223$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$224$		0			0
$225$	386.086	+	323.965i	1.71594	+	1.43984i
$226$		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
$233$	436.957	−	159.039i	1.87535	−	0.682572i	0.915282	−	0.402814i	$-0.131968\pi$
$233$	436.957	−	159.039i	1.87535	−	0.682572i	0.960071	−	0.279758i	$-0.0902542\pi$
$234$		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	0.197443	−	0.980314i	$-0.436736\pi$
$239$	226.500	−	392.310i	0.947699	−	1.64146i	0.750255	−	0.661148i	$-0.229930\pi$
$240$		0			0
$241$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$241$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$242$		0			0
$243$		0			0
$244$	315.610	+	264.828i	1.29348	+	1.08536i
$245$	37.5080	−	212.718i	0.153094	−	0.868239i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−25.3717	−	9.23454i	−0.101082	−	0.0367910i	0.290984	−	0.956728i	$-0.406017\pi$
$251$	−25.3717	−	9.23454i	−0.101082	−	0.0367910i	−0.392066	+	0.919937i	$0.628240\pi$
$252$	169.145	−	61.5636i	0.671209	−	0.244300i
$253$	−68.9440	+	57.8509i	−0.272506	+	0.228660i
$254$		0			0
$255$		0			0
$256$	44.4539	+	252.111i	0.173648	+	0.984808i
$257$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$257$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−310.248	−	260.329i	−1.17965	−	0.989844i	−0.999981	−	0.00611690i	$-0.998053\pi$
$263$	−310.248	−	260.329i	−1.17965	−	0.989844i	−0.179669	−	0.983727i	$-0.557503\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$269$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$270$		0			0
$271$	133.436	−	48.5669i	0.492385	−	0.179214i	−0.0838808	−	0.996476i	$-0.526731\pi$
$271$	133.436	−	48.5669i	0.492385	−	0.179214i	0.576266	+	0.817262i	$0.304509\pi$
$272$	183.851	−	154.269i	0.675922	−	0.567166i
$273$		0			0
$274$		0			0
$275$	29.1729	+	165.448i	0.106083	+	0.601628i
$276$		0			0
$277$	−267.500	+	463.324i	−0.965704	+	1.67265i	−0.257992	+	0.966147i	$0.583061\pi$
$277$	−267.500	+	463.324i	−0.965704	+	1.67265i	−0.707712	+	0.706501i	$0.750272\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$281$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$282$		0			0
$283$	68.5910	−	388.999i	0.242371	−	1.37455i	−0.584148	−	0.811647i	$-0.698571\pi$
$283$	68.5910	−	388.999i	0.242371	−	1.37455i	0.826519	−	0.562908i	$-0.190318\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	60.1403	+	21.8893i	0.208098	+	0.0757415i
$290$		0			0
$291$		0			0
$292$	50.0000	+	86.6025i	0.171233	+	0.296584i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	325.569	+	273.185i	1.08162	+	0.907590i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−927.000			−3.03934
$306$		0			0
$307$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$307$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$308$	56.3816	+	20.5212i	0.183057	+	0.0666273i
$309$		0			0
$310$		0			0
$311$	−301.500	−	522.213i	−0.969453	−	1.67914i	−0.697142	−	0.716933i	$-0.745545\pi$
$311$	−301.500	−	522.213i	−0.969453	−	1.67914i	−0.272312	−	0.962209i	$-0.587788\pi$
$312$		0			0
$313$	−102.452	−	581.037i	−0.327324	−	1.85635i	−0.492813	−	0.870135i	$-0.664031\pi$
$313$	−102.452	−	581.037i	−0.327324	−	1.85635i	0.165489	−	0.986212i	$-0.447080\pi$
$314$		0			0
$315$	−202.500	+	350.740i	−0.642857	+	1.11346i
$316$		0			0
$317$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$317$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$318$		0			0
$319$		0			0
$320$	−441.242	−	370.246i	−1.37888	−	1.15702i
$321$		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
$329$	−287.267	+	241.045i	−0.873151	+	0.732661i
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	62.5133	+	354.531i	0.188293	+	1.06786i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$337$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$338$		0			0
$339$		0			0
$340$	−93.7700	+	531.796i	−0.275794	+	1.56411i
$341$		0			0
$342$		0			0
$343$	365.000			1.06414
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−634.293	+	230.864i	−1.82793	+	0.665313i	−0.834484	+	0.551032i	$0.814234\pi$
$347$	−634.293	+	230.864i	−1.82793	+	0.665313i	−0.993449	+	0.114280i	$0.963544\pi$
$348$		0			0
$349$	−263.500	−	456.395i	−0.755014	−	1.30772i	−0.945367	−	0.326007i	$-0.894297\pi$
$349$	−263.500	−	456.395i	−0.755014	−	1.30772i	0.190353	−	0.981716i	$-0.439037\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	255.000	−	441.673i	0.722380	−	1.25120i	−0.237664	−	0.971347i	$-0.576382\pi$
$353$	255.000	−	441.673i	0.722380	−	1.25120i	0.960044	−	0.279851i	$-0.0902850\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	42.1965	−	239.308i	0.117539	−	0.666597i	−0.867923	−	0.496699i	$-0.834545\pi$
$359$	42.1965	−	239.308i	0.117539	−	0.666597i	0.985462	−	0.169898i	$-0.0543437\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−211.431	−	76.9545i	−0.579263	−	0.210834i
$366$		0			0
$367$	38.3022	−	32.1394i	0.104366	−	0.0875732i	−0.589112	−	0.808052i	$-0.700522\pi$
$367$	38.3022	−	32.1394i	0.104366	−	0.0875732i	0.693477	+	0.720478i	$0.256078\pi$
$368$	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	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		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		−0.766044	−	0.642788i	$-0.777778\pi$
$383$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$384$		0			0
$385$	−126.859	+	46.1727i	−0.329503	+	0.119929i
$386$		0			0
$387$	382.500	+	662.509i	0.988372	+	1.71191i
$388$		0			0
$389$	−26.5682	−	150.676i	−0.0682986	−	0.387341i	−0.999726	−	0.0234132i	$-0.992547\pi$
$389$	−26.5682	−	150.676i	−0.0682986	−	0.387341i	0.931427	−	0.363928i	$-0.118564\pi$
$390$		0			0
$391$	225.000	−	389.711i	0.575448	−	0.996704i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	82.7328	+	69.4211i	0.208921	+	0.175306i
$397$	−129.368	+	733.682i	−0.325864	+	1.84806i	0.177671	+	0.984090i	$0.443144\pi$
$397$	−129.368	+	733.682i	−0.325864	+	1.84806i	−0.503535	+	0.863975i	$0.667967\pi$
$398$		0			0
$399$		0			0
$400$	896.000			2.24000
$401$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$401$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$402$		0			0
$403$		0			0
$404$	383.395	−	139.544i	0.948997	−	0.345406i
$405$	−558.446	+	468.592i	−1.37888	+	1.15702i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$409$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−620.496	−	520.658i	−1.49517	−	1.25460i
$416$		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		−0.766044	−	0.642788i	$-0.777778\pi$
$421$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$422$		0			0
$423$	−634.293	+	230.864i	−1.49951	+	0.545777i
$424$		0			0
$425$	−420.000	−	727.461i	−0.988235	−	1.71167i
$426$		0			0
$427$	−89.4288	−	507.176i	−0.209435	−	1.18777i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$431$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$432$		0			0
$433$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$433$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$439$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$440$		0			0
$441$	202.974	+	73.8764i	0.460258	+	0.167520i
$442$		0			0
$443$	−34.4720	+	28.9254i	−0.0778149	+	0.0652945i	−0.680865	−	0.732409i	$-0.738396\pi$
$443$	−34.4720	+	28.9254i	−0.0778149	+	0.0652945i	0.603051	+	0.797703i	$0.293952\pi$
$444$		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	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		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
$460$	−1014.87	−	369.382i	−2.20623	−	0.803004i
$461$	−420.043	+	152.883i	−0.911155	+	0.331633i	−0.754714	−	0.656054i	$-0.772224\pi$
$461$	−420.043	+	152.883i	−0.911155	+	0.331633i	−0.156441	+	0.987687i	$0.550002\pi$
$462$		0			0
$463$	−377.500	−	653.849i	−0.815335	−	1.41220i	−0.909087	−	0.416606i	$-0.863220\pi$
$463$	−377.500	−	653.849i	−0.815335	−	1.41220i	0.0937525	−	0.995596i	$-0.470114\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−457.500	+	792.413i	−0.979657	+	1.69682i	−0.316037	+	0.948747i	$0.602352\pi$
$467$	−457.500	+	792.413i	−0.979657	+	1.69682i	−0.663620	+	0.748070i	$0.730981\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−44.2803	+	251.126i	−0.0936158	+	0.530922i
$474$		0			0
$475$		0			0
$476$	−300.000			−0.630252
$477$		0			0
$478$		0			0
$479$	885.190	+	322.183i	1.84800	+	0.672616i	0.986246	+	0.165284i	$0.0528540\pi$
$479$	885.190	+	322.183i	1.84800	+	0.672616i	0.861751	+	0.507332i	$0.169368\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−77.7944	−	441.194i	−0.160732	−	0.911558i
$485$		0			0
$486$		0			0
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−703.229	−	590.079i	−1.43224	−	1.20179i	−0.944375	−	0.328870i	$-0.893332\pi$
$491$	−703.229	−	590.079i	−1.43224	−	1.20179i	−0.487863	−	0.872920i	$-0.662223\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−243.000			−0.490909
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−491.459	+	178.877i	−0.984888	+	0.358470i	−0.783739	−	0.621091i	$-0.786690\pi$
$499$	−491.459	+	178.877i	−0.984888	+	0.358470i	−0.201149	+	0.979561i	$0.564468\pi$
$500$	−854.906	+	717.351i	−1.70981	+	1.43470i
$501$		0			0
$502$		0			0
$503$	161.493	+	915.871i	0.321059	+	1.82082i	0.536032	+	0.844198i	$0.319923\pi$
$503$	161.493	+	915.871i	0.321059	+	1.82082i	−0.214973	+	0.976620i	$0.568966\pi$
$504$		0			0
$505$	−459.000	+	795.011i	−0.908911	+	1.57428i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$509$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$510$		0			0
$511$	21.7060	−	123.101i	0.0424775	−	0.240902i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−211.431	−	76.9545i	−0.408957	−	0.148848i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$523$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$524$	426.000	−	737.854i	0.812977	−	1.40812i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	284.202	+	238.474i	0.537245	+	0.450802i
$530$		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
$541$	−79.3572	−	450.057i	−0.146686	−	0.831899i	−0.965998	−	0.258550i	$-0.916755\pi$
$541$	−79.3572	−	450.057i	−0.146686	−	0.831899i	0.819312	−	0.573348i	$-0.194356\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$547$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$548$	781.365	+	655.643i	1.42585	+	1.19643i
$549$	160.972	−	912.917i	0.293209	−	1.66287i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	740.478	−	269.512i	1.33179	−	0.484734i
$557$	838.819	−	703.852i	1.50596	−	1.26365i	0.634768	−	0.772703i	$-0.281096\pi$
$557$	838.819	−	703.852i	1.50596	−	1.26365i	0.871190	−	0.490946i	$-0.163349\pi$
$558$		0			0
$559$		0			0
$560$	125.027	+	709.062i	0.223262	+	1.26618i
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−310.248	−	260.329i	−0.547175	−	0.459134i
$568$		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
$575$	1578.68	−	574.594i	2.74554	−	0.999294i
$576$	441.242	−	370.246i	0.766044	−	0.642788i
$577$	572.500	+	991.599i	0.992201	+	1.71854i	0.604049	+	0.796947i	$0.293553\pi$
$577$	572.500	+	991.599i	0.992201	+	1.71854i	0.388152	+	0.921595i	$0.373113\pi$
$578$		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
$587$	−195.354	+	1107.91i	−0.332801	+	1.88741i	0.115146	+	0.993349i	$0.463266\pi$
$587$	−195.354	+	1107.91i	−0.332801	+	1.88741i	−0.447948	+	0.894060i	$0.647845\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	28.1908	+	10.2606i	0.0475393	+	0.0173029i	0.365680	−	0.930741i	$-0.380836\pi$
$593$	28.1908	+	10.2606i	0.0475393	+	0.0173029i	−0.318141	+	0.948043i	$0.603059\pi$
$594$		0			0
$595$	517.080	−	433.882i	0.869042	−	0.729213i
$596$	354.000	+	613.146i	0.593960	+	1.02877i
$597$		0			0
$598$		0			0
$599$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$599$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	772.173	+	647.930i	1.27632	+	1.07096i
$606$		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
$612$	−507.434	−	184.691i	−0.829141	−	0.301782i
$613$	−277.209	+	100.896i	−0.452217	+	0.164594i	−0.558080	−	0.829787i	$-0.688462\pi$
$613$	−277.209	+	100.896i	−0.452217	+	0.164594i	0.105863	+	0.994381i	$0.466240\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−184.935	−	1048.82i	−0.299733	−	1.69987i	−0.647316	−	0.762222i	$-0.724109\pi$
$617$	−184.935	−	1048.82i	−0.299733	−	1.69987i	0.347583	−	0.937649i	$-0.387002\pi$
$618$		0			0
$619$	331.000	−	573.309i	0.534733	−	0.926185i	−0.464443	−	0.885603i	$-0.653745\pi$
$619$	331.000	−	573.309i	0.534733	−	0.926185i	0.999176	−	0.0405823i	$-0.0129213\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	192.923	−	1094.12i	0.308677	−	1.75059i
$626$		0			0
$627$		0			0
$628$	40.0000			0.0636943
$629$		0			0
$630$		0			0
$631$	974.461	+	354.675i	1.54431	+	0.562084i	0.967075	−	0.254492i	$-0.0819083\pi$
$631$	974.461	+	354.675i	1.54431	+	0.562084i	0.577238	+	0.816576i	$0.304131\pi$
$632$		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		0.939693	−	0.342020i	$-0.111111\pi$
$641$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$642$		0			0
$643$	854.140	+	716.708i	1.32837	+	1.11463i	0.984456	+	0.175629i	$0.0561959\pi$
$643$	854.140	+	716.708i	1.32837	+	1.11463i	0.343910	+	0.939003i	$0.388249\pi$
$644$	104.189	−	590.885i	0.161784	−	0.917523i
$645$		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
$652$	766.044	−	642.788i	1.17491	−	0.985871i
$653$	−187.500	−	324.760i	−0.287136	−	0.497335i	0.685989	−	0.727612i	$-0.259370\pi$
$653$	−187.500	−	324.760i	−0.287136	−	0.497335i	−0.973125	+	0.230278i	$0.926037\pi$
$654$		0			0
$655$	332.884	+	1887.88i	0.508219	+	2.88225i
$656$		0			0
$657$	112.500	−	194.856i	0.171233	−	0.296584i
$658$		0			0
$659$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$659$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$660$		0			0
$661$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$661$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$662$		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
$671$	236.708	−	198.621i	0.352769	−	0.296008i
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$		0			0
$676$	−338.000	+	585.433i	−0.500000	+	0.866025i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		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
$688$	1277.98	+	465.147i	1.85753	+	0.676086i
$689$		0			0
$690$		0			0
$691$	78.5000	+	135.966i	0.113603	+	0.196767i	0.917221	−	0.398380i	$-0.130427\pi$
$691$	78.5000	+	135.966i	0.113603	+	0.196767i	−0.803617	+	0.595147i	$0.797094\pi$
$692$		0			0
$693$	−23.4425	−	132.949i	−0.0338276	−	0.191846i
$694$		0			0
$695$	−886.500	+	1535.46i	−1.27554	+	2.20930i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−857.970	−	719.922i	−1.22567	−	1.02846i
$701$	190.666	−	1081.32i	0.271991	−	1.54254i	−0.476369	−	0.879245i	$-0.658047\pi$
$701$	190.666	−	1081.32i	0.271991	−	1.54254i	0.748360	−	0.663292i	$-0.230841\pi$
$702$		0			0
$703$		0			0
$704$	192.000			0.272727
$705$		0			0
$706$		0			0
$707$	−479.243	−	174.430i	−0.677855	−	0.246719i
$708$		0			0
$709$	−1009.65	+	847.194i	−1.42404	+	1.19491i	−0.474912	+	0.880033i	$0.657520\pi$
$709$	−1009.65	+	847.194i	−1.42404	+	1.19491i	−0.949131	+	0.314881i	$0.898035\pi$
$710$		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
$719$	737.701	+	619.004i	1.02601	+	0.860924i	0.990371	−	0.138441i	$-0.0442090\pi$
$719$	737.701	+	619.004i	1.02601	+	0.860924i	0.0356387	+	0.999365i	$0.488653\pi$
$720$	−225.048	+	1276.31i	−0.312567	+	1.77265i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	79.8739	−	29.0717i	0.109868	−	0.0399886i	−0.286501	−	0.958080i	$-0.592492\pi$
$727$	79.8739	−	29.0717i	0.109868	−	0.0399886i	0.396369	+	0.918091i	$0.370270\pi$
$728$		0			0
$729$	−364.500	−	631.333i	−0.500000	−	0.866025i
$730$		0			0
$731$	−221.401	−	1255.63i	−0.302875	−	1.71769i
$732$		0			0
$733$	635.000	−	1099.85i	0.866303	−	1.50048i	0.000555189	−	1.00000i	$-0.499823\pi$
$733$	635.000	−	1099.85i	0.866303	−	1.50048i	0.865748	−	0.500481i	$-0.166843\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	94.9856	−	538.690i	0.128533	−	0.728944i	−0.850614	−	0.525790i	$-0.823770\pi$
$739$	94.9856	−	538.690i	0.128533	−	0.728944i	0.979147	−	0.203154i	$-0.0651192\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$743$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$744$		0			0
$745$	−1496.93	−	544.838i	−2.00930	−	0.731326i
$746$		0			0
$747$	620.496	−	520.658i	0.830651	−	0.696999i
$748$	−90.0000	−	155.885i	−0.120321	−	0.208402i
$749$		0			0
$750$		0			0
$751$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$751$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$752$	−600.000	+	1039.23i	−0.797872	+	1.38196i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−601.345	−	504.588i	−0.794379	−	0.666563i	0.152446	−	0.988312i	$-0.451285\pi$
$757$	−601.345	−	504.588i	−0.794379	−	0.666563i	−0.946825	+	0.321749i	$0.895729\pi$
$758$		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
$764$	349.566	+	127.231i	0.457547	+	0.166533i
$765$	1141.73	−	415.554i	1.49245	−	0.543208i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	184.588	+	1046.85i	0.240036	+	1.36131i	0.831743	+	0.555161i	$0.187343\pi$
$769$	184.588	+	1046.85i	0.240036	+	1.36131i	−0.591707	+	0.806153i	$0.701546\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$773$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$774$		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
$784$	360.842	−	131.336i	0.460258	−	0.167520i
$785$	−68.9440	+	57.8509i	−0.0878268	+	0.0736954i
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	62.5133	+	354.531i	0.0793317	+	0.449912i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	157.673	−	894.205i	0.198081	−	1.12337i
$797$		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
$803$	70.4769	−	25.6515i	0.0877671	−	0.0319446i
$804$		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	0.340624	−	0.940199i	$-0.389362\pi$
$809$	796.500	−	1379.58i	0.984549	−	1.70529i	0.643924	−	0.765089i	$-0.277305\pi$
$810$		0			0
$811$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$811$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−390.708	+	2215.82i	−0.479397	+	2.71879i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1096.62	−	399.138i	−1.33571	−	0.486160i	−0.427254	−	0.904132i	$-0.640519\pi$
$821$	−1096.62	−	399.138i	−1.33571	−	0.486160i	−0.908460	+	0.417971i	$0.862741\pi$
$822$		0			0
$823$	−1198.86	+	1005.96i	−1.45669	+	1.22231i	−0.529188	+	0.848505i	$0.677503\pi$
$823$	−1198.86	+	1005.96i	−1.45669	+	1.22231i	−0.927506	+	0.373807i	$0.878052\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$827$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$828$	540.000	−	935.307i	0.652174	−	1.12960i
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−275.776	−	231.404i	−0.331064	−	0.277795i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$839$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$840$		0			0
$841$	−790.281	+	287.639i	−0.939693	+	0.342020i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−264.119	−	1497.89i	−0.312567	−	1.77265i
$846$		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
$853$	−178.858	+	1014.35i	−0.209681	+	1.18916i	0.680221	+	0.733007i	$0.261884\pi$
$853$	−178.858	+	1014.35i	−0.209681	+	1.18916i	−0.889902	+	0.456152i	$0.849227\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$857$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$858$		0			0
$859$	1402.96	+	510.636i	1.63325	+	0.594454i	0.985840	−	0.167688i	$-0.0536303\pi$
$859$	1402.96	+	510.636i	1.63325	+	0.594454i	0.647409	+	0.762143i	$0.275853\pi$
$860$	−2875.46	+	1046.58i	−3.34356	+	1.21696i
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		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		−0.766044	−	0.642788i	$-0.777778\pi$
$877$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$878$		0			0
$879$		0			0
$880$	−330.931	+	277.684i	−0.376058	+	0.315550i
$881$	268.500	+	465.056i	0.304767	+	0.527872i	0.977209	−	0.212277i	$-0.0680880\pi$
$881$	268.500	+	465.056i	0.304767	+	0.527872i	−0.672442	+	0.740150i	$0.734755\pi$
$882$		0			0
$883$	144.996	+	822.314i	0.164209	+	0.931273i	0.949877	+	0.312625i	$0.101208\pi$
$883$	144.996	+	822.314i	0.164209	+	0.931273i	−0.785668	+	0.618648i	$0.787681\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$887$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	42.1965	−	239.308i	0.0473586	−	0.268584i
$892$		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		0.939693	−	0.342020i	$-0.111111\pi$
$907$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$908$		0			0
$909$	−703.229	−	590.079i	−0.773629	−	0.649152i
$910$		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
$916$	63.8991	+	23.2574i	0.0697588	+	0.0253901i
$917$	−1000.77	+	364.251i	−1.09136	+	0.397221i
$918$		0			0
$919$	−881.000	−	1525.94i	−0.958651	−	1.66043i	−0.725783	−	0.687923i	$-0.758523\pi$
$919$	−881.000	−	1525.94i	−0.958651	−	1.66043i	−0.232867	−	0.972509i	$-0.574811\pi$
$920$		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
$929$	111.482	−	632.247i	0.120002	−	0.680567i	−0.864149	−	0.503236i	$-0.832143\pi$
$929$	111.482	−	632.247i	0.120002	−	0.680567i	0.984151	−	0.177331i	$-0.0567463\pi$
$930$		0			0
$931$		0			0
$932$	−1860.00			−1.99571
$933$		0			0
$934$		0			0
$935$	380.576	+	138.518i	0.407033	+	0.148148i
$936$		0			0
$937$	256.625	−	215.334i	0.273879	−	0.229812i	−0.495494	−	0.868611i	$-0.665013\pi$
$937$	256.625	−	215.334i	0.273879	−	0.229812i	0.769373	+	0.638799i	$0.220569\pi$
$938$		0			0
$939$		0			0
$940$	−468.850	−	2658.98i	−0.498777	−	2.82870i
$941$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$941$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1401.86	−	1176.30i	−1.48032	−	1.24213i	−0.905844	−	0.423612i	$-0.860762\pi$
$947$	−1401.86	−	1176.30i	−1.48032	−	1.24213i	−0.574474	−	0.818523i	$-0.694794\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$953$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$954$		0			0
$955$	−786.523	+	286.271i	−0.823584	+	0.299760i
$956$	−1388.07	+	1164.73i	−1.45196	+	1.21834i
$957$		0			0
$958$		0			0
$959$	−221.401	−	1255.63i	−0.230867	−	1.30931i
$960$		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
$967$	−310.830	+	1762.81i	−0.321438	+	1.82296i	0.212171	+	0.977232i	$0.431946\pi$
$967$	−310.830	+	1762.81i	−0.321438	+	1.82296i	−0.533609	+	0.845731i	$0.679165\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$971$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$972$		0			0
$973$	−925.597	−	336.890i	−0.951282	−	0.346238i
$974$		0			0
$975$		0			0
$976$	−824.000	−	1427.21i	−0.844262	−	1.46231i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$	−432.000	+	748.246i	−0.440816	+	0.763516i
$981$		0			0
$982$		0			0
$983$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$983$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$984$		0			0
$985$	−620.496	−	520.658i	−0.629945	−	0.528587i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2550.00			2.57836
$990$		0			0
$991$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$991$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1021.50	+	1769.29i	1.02663	+	1.77818i
$996$		0			0
$997$	342.955	+	1945.00i	0.343987	+	1.95085i	0.307620	+	0.951509i	$0.400468\pi$
$997$	342.955	+	1945.00i	0.343987	+	1.95085i	0.0363673	+	0.999338i	$0.488421\pi$
$998$		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.116.1		6
19.2	odd	18		361.3.d.a.293.1		2
19.3	odd	18		19.3.b.a.18.1	✓	1
19.4	even	9	inner	361.3.f.a.127.1		6
19.5	even	9		361.3.d.a.69.1		2
19.6	even	9	inner	361.3.f.a.262.1		6
19.7	even	3	inner	361.3.f.a.307.1		6
19.8	odd	6	inner	361.3.f.a.299.1		6
19.9	even	9	inner	361.3.f.a.333.1		6
19.10	odd	18	inner	361.3.f.a.333.1		6
19.11	even	3	inner	361.3.f.a.299.1		6
19.12	odd	6	inner	361.3.f.a.307.1		6
19.13	odd	18	inner	361.3.f.a.262.1		6
19.14	odd	18		361.3.d.a.69.1		2
19.15	odd	18	inner	361.3.f.a.127.1		6
19.16	even	9		19.3.b.a.18.1	✓	1
19.17	even	9		361.3.d.a.293.1		2
19.18	odd	2	CM	361.3.f.a.116.1		6
57.35	odd	18		171.3.c.a.37.1		1
57.41	even	18		171.3.c.a.37.1		1
76.3	even	18		304.3.e.a.113.1		1
76.35	odd	18		304.3.e.a.113.1		1
95.3	even	36		475.3.d.a.474.2		2
95.22	even	36		475.3.d.a.474.1		2
95.54	even	18		475.3.c.a.151.1		1
95.73	odd	36		475.3.d.a.474.2		2
95.79	odd	18		475.3.c.a.151.1		1
95.92	odd	36		475.3.d.a.474.1		2
152.3	even	18		1216.3.e.b.1025.1		1
152.35	odd	18		1216.3.e.b.1025.1		1
152.117	odd	18		1216.3.e.a.1025.1		1
152.149	even	18		1216.3.e.a.1025.1		1
228.35	even	18		2736.3.o.a.721.1		1
228.155	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	19.3	odd	18
19.3.b.a.18.1	✓	1	19.16	even	9
171.3.c.a.37.1		1	57.35	odd	18
171.3.c.a.37.1		1	57.41	even	18
304.3.e.a.113.1		1	76.3	even	18
304.3.e.a.113.1		1	76.35	odd	18
361.3.d.a.69.1		2	19.5	even	9
361.3.d.a.69.1		2	19.14	odd	18
361.3.d.a.293.1		2	19.2	odd	18
361.3.d.a.293.1		2	19.17	even	9
361.3.f.a.116.1		6	1.1	even	1	trivial
361.3.f.a.116.1		6	19.18	odd	2	CM
361.3.f.a.127.1		6	19.4	even	9	inner
361.3.f.a.127.1		6	19.15	odd	18	inner
361.3.f.a.262.1		6	19.6	even	9	inner
361.3.f.a.262.1		6	19.13	odd	18	inner
361.3.f.a.299.1		6	19.8	odd	6	inner
361.3.f.a.299.1		6	19.11	even	3	inner
361.3.f.a.307.1		6	19.7	even	3	inner
361.3.f.a.307.1		6	19.12	odd	6	inner
361.3.f.a.333.1		6	19.9	even	9	inner
361.3.f.a.333.1		6	19.10	odd	18	inner
475.3.c.a.151.1		1	95.54	even	18
475.3.c.a.151.1		1	95.79	odd	18
475.3.d.a.474.1		2	95.22	even	36
475.3.d.a.474.1		2	95.92	odd	36
475.3.d.a.474.2		2	95.3	even	36
475.3.d.a.474.2		2	95.73	odd	36
1216.3.e.a.1025.1		1	152.117	odd	18
1216.3.e.a.1025.1		1	152.149	even	18
1216.3.e.b.1025.1		1	152.3	even	18
1216.3.e.b.1025.1		1	152.35	odd	18
2736.3.o.a.721.1		1	228.35	even	18
2736.3.o.a.721.1		1	228.155	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)

\(f(q)\)	\(=\)	\(q+(-3.75877 - 1.36808i) q^{4} +(8.45723 - 3.07818i) q^{5} +(2.50000 + 4.33013i) q^{7} +(1.56283 + 8.86327i) q^{9} +O(q^{10})\)	\(q+(-3.75877 - 1.36808i) q^{4} +(8.45723 - 3.07818i) q^{5} +(2.50000 + 4.33013i) q^{7} +(1.56283 + 8.86327i) q^{9} +(-1.50000 + 2.59808i) q^{11} +(12.2567 + 10.2846i) q^{16} +(2.60472 - 14.7721i) q^{17} -36.0000 q^{20} +(28.1908 + 10.2606i) q^{23} +(42.8985 - 35.9961i) q^{25} +(-3.47296 - 19.6962i) q^{28} +(34.4720 + 28.9254i) q^{35} +(6.25133 - 35.4531i) q^{36} +(79.8739 - 29.0717i) q^{43} +(9.19253 - 7.71345i) q^{44} +(40.5000 + 70.1481i) q^{45} +(13.0236 + 73.8606i) q^{47} +(12.0000 - 20.7846i) q^{49} +(-4.68850 + 26.5898i) q^{55} +(-96.7883 - 35.2281i) q^{61} +(-34.4720 + 28.9254i) q^{63} +(-32.0000 - 55.4256i) q^{64} +(-30.0000 + 51.9615i) q^{68} +(-19.1511 - 16.0697i) q^{73} -15.0000 q^{77} +(135.316 + 49.2509i) q^{80} +(-76.1151 + 27.7036i) q^{81} +(-45.0000 - 77.9423i) q^{83} +(-23.4425 - 132.949i) q^{85} +(-91.9253 - 77.1345i) q^{92} +(-25.3717 - 9.23454i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 15 q^{7}+O(q^{10}) \) 6 * q + 15 * q^7	\( 6 q + 15 q^{7} - 9 q^{11} - 216 q^{20} + 243 q^{45} + 72 q^{49} - 192 q^{64} - 180 q^{68} - 90 q^{77} - 270 q^{83}+O(q^{100}) \) 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

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