LMFDB - Embedded newform 361.3.f.a.127.1

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			127.1
Root			$-0.173648 + 0.984808i$ of defining polynomial
Character	$\chi$	$=$	361.127
Dual form			361.3.f.a.307.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.694593 - 3.93923i) q^{4} +(-1.56283 - 8.86327i) q^{5} +(2.50000 - 4.33013i) q^{7} +(6.89440 + 5.78509i) q^{9} +O(q^{10})$	$q+(0.694593 - 3.93923i) q^{4} +(-1.56283 - 8.86327i) q^{5} +(2.50000 - 4.33013i) q^{7} +(6.89440 + 5.78509i) q^{9} +(-1.50000 - 2.59808i) q^{11} +(-15.0351 - 5.47232i) q^{16} +(11.4907 - 9.64181i) q^{17} -36.0000 q^{20} +(-5.20945 + 29.5442i) q^{23} +(-52.6228 + 19.1531i) q^{25} +(-15.3209 - 12.8558i) q^{28} +(-42.2862 - 15.3909i) q^{35} +(27.5776 - 23.1404i) q^{36} +(-14.7601 - 83.7087i) q^{43} +(-11.2763 + 4.10424i) q^{44} +(40.5000 - 70.1481i) q^{45} +(57.4533 + 48.2091i) q^{47} +(12.0000 + 20.7846i) q^{49} +(-20.6832 + 17.3553i) q^{55} +(17.8858 - 101.435i) q^{61} +(42.2862 - 15.3909i) q^{63} +(-32.0000 + 55.4256i) q^{64} +(-30.0000 - 51.9615i) q^{68} +(23.4923 + 8.55050i) q^{73} -15.0000 q^{77} +(-25.0053 + 141.812i) q^{80} +(14.0655 + 79.7694i) q^{81} +(-45.0000 + 77.9423i) q^{83} +(-103.416 - 86.7763i) q^{85} +(112.763 + 41.0424i) q^{92} +(4.68850 - 26.5898i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{5}{18}\right)$

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		0.766044	−	0.642788i	$-0.222222\pi$
$2$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$3$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$3$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$4$	0.694593	−	3.93923i	0.173648	−	0.984808i
$5$	−1.56283	−	8.86327i	−0.312567	−	1.77265i	−0.585551	−	0.810635i	$-0.699122\pi$
$5$	−1.56283	−	8.86327i	−0.312567	−	1.77265i	0.272984	−	0.962018i	$-0.411989\pi$
$6$		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
$9$	6.89440	+	5.78509i	0.766044	+	0.642788i
$10$		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		0.939693	−	0.342020i	$-0.111111\pi$
$13$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$14$		0			0
$15$		0			0
$16$	−15.0351	−	5.47232i	−0.939693	−	0.342020i
$17$	11.4907	−	9.64181i	0.675922	−	0.567166i	−0.238890	−	0.971047i	$-0.576784\pi$
$17$	11.4907	−	9.64181i	0.675922	−	0.567166i	0.914811	+	0.403881i	$0.132339\pi$
$18$		0			0
$19$		0			0
$19$		0			0
$20$	−36.0000			−1.80000
$21$		0			0
$22$		0			0
$23$	−5.20945	+	29.5442i	−0.226498	+	1.28453i	0.633304	+	0.773903i	$0.281698\pi$
$23$	−5.20945	+	29.5442i	−0.226498	+	1.28453i	−0.859801	+	0.510629i	$0.829413\pi$
$24$		0			0
$25$	−52.6228	+	19.1531i	−2.10491	+	0.766125i
$26$		0			0
$27$		0			0
$28$	−15.3209	−	12.8558i	−0.547175	−	0.459134i
$29$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$29$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$30$		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
$35$	−42.2862	−	15.3909i	−1.20818	−	0.439740i
$36$	27.5776	−	23.1404i	0.766044	−	0.642788i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$41$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$42$		0			0
$43$	−14.7601	−	83.7087i	−0.343258	−	1.94671i	−0.321374	−	0.946952i	$-0.604145\pi$
$43$	−14.7601	−	83.7087i	−0.343258	−	1.94671i	−0.0218844	−	0.999761i	$-0.506967\pi$
$44$	−11.2763	+	4.10424i	−0.256280	+	0.0932782i
$45$	40.5000	−	70.1481i	0.900000	−	1.55885i
$46$		0			0
$47$	57.4533	+	48.2091i	1.22241	+	1.02572i	0.998695	+	0.0510706i	$0.0162634\pi$
$47$	57.4533	+	48.2091i	1.22241	+	1.02572i	0.223716	+	0.974654i	$0.428181\pi$
$48$		0			0
$49$	12.0000	+	20.7846i	0.244898	+	0.424176i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$53$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$54$		0			0
$55$	−20.6832	+	17.3553i	−0.376058	+	0.315550i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$59$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$60$		0			0
$61$	17.8858	−	101.435i	0.293209	−	1.66287i	−0.381184	−	0.924499i	$-0.624483\pi$
$61$	17.8858	−	101.435i	0.293209	−	1.66287i	0.674393	−	0.738373i	$-0.264405\pi$
$62$		0			0
$63$	42.2862	−	15.3909i	0.671209	−	0.244300i
$64$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$67$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$68$	−30.0000	−	51.9615i	−0.441176	−	0.764140i
$69$		0			0
$70$		0			0
$71$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$71$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$72$		0			0
$73$	23.4923	+	8.55050i	0.321813	+	0.117130i	0.497875	−	0.867249i	$-0.334114\pi$
$73$	23.4923	+	8.55050i	0.321813	+	0.117130i	−0.176062	+	0.984379i	$0.556336\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−15.0000			−0.194805
$78$		0			0
$79$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$79$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$80$	−25.0053	+	141.812i	−0.312567	+	1.77265i
$81$	14.0655	+	79.7694i	0.173648	+	0.984808i
$82$		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
$85$	−103.416	−	86.7763i	−1.21666	−	1.02090i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$89$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$90$		0			0
$91$		0			0
$92$	112.763	+	41.0424i	1.22569	+	0.446113i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$97$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$98$		0			0
$99$	4.68850	−	26.5898i	0.0473586	−	0.268584i
$100$	38.8972	+	220.597i	0.388972	+	2.20597i
$101$	95.8486	−	34.8861i	0.948997	−	0.345406i	0.179284	−	0.983797i	$-0.442622\pi$
$101$	95.8486	−	34.8861i	0.948997	−	0.345406i	0.769712	+	0.638391i	$0.220400\pi$
$102$		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		−0.173648	−	0.984808i	$-0.555556\pi$
$109$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$110$		0			0
$111$		0			0
$112$	−61.2836	+	51.4230i	−0.547175	+	0.459134i
$113$		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
$119$	−13.0236	−	73.8606i	−0.109442	−	0.620677i
$120$		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		0.939693	−	0.342020i	$-0.111111\pi$
$127$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−163.167	+	136.914i	−1.24555	+	1.04514i	−0.248484	+	0.968636i	$0.579932\pi$
$131$	−163.167	+	136.914i	−1.24555	+	1.04514i	−0.997069	+	0.0765073i	$0.975623\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	44.2803	−	251.126i	0.323214	−	1.83304i	−0.198727	−	0.980055i	$-0.563681\pi$
$137$	44.2803	−	251.126i	0.323214	−	1.83304i	0.521941	−	0.852982i	$-0.325208\pi$
$138$		0			0
$139$	185.119	−	67.3780i	1.33179	−	0.484734i	0.424576	−	0.905392i	$-0.360423\pi$
$139$	185.119	−	67.3780i	1.33179	−	0.484734i	0.907219	+	0.420659i	$0.138201\pi$
$140$	−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
$149$	166.326	+	60.5376i	1.11628	+	0.406292i	0.833293	−	0.552832i	$-0.186453\pi$
$149$	166.326	+	60.5376i	1.11628	+	0.406292i	0.282986	+	0.959124i	$0.408675\pi$
$150$		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
$157$	1.73648	+	9.84808i	0.0110604	+	0.0627266i	0.989838	−	0.142197i	$-0.0454166\pi$
$157$	1.73648	+	9.84808i	0.0110604	+	0.0627266i	−0.978778	+	0.204923i	$0.934305\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	114.907	+	96.4181i	0.713706	+	0.598870i
$162$		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		0.173648	−	0.984808i	$-0.444444\pi$
$167$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$168$		0			0
$169$	129.462	−	108.631i	0.766044	−	0.642788i
$170$		0			0
$171$		0			0
$172$	−340.000			−1.97674
$173$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$173$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$174$		0			0
$175$	−48.6215	+	275.746i	−0.277837	+	1.57569i
$176$	8.33511	+	47.2708i	0.0473586	+	0.268584i
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$	−248.198	−	208.263i	−1.37888	−	1.15702i
$181$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$181$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−42.2862	−	15.3909i	−0.226129	−	0.0823043i
$188$	229.813	−	192.836i	1.22241	−	1.02572i
$189$		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		−0.939693	−	0.342020i	$-0.888889\pi$
$193$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$194$		0			0
$195$		0			0
$196$	90.2105	−	32.8339i	0.460258	−	0.167520i
$197$	−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
$199$	173.892	+	145.913i	0.873830	+	0.733230i	0.964901	−	0.262614i	$-0.0845847\pi$
$199$	173.892	+	145.913i	0.873830	+	0.733230i	−0.0910713	+	0.995844i	$0.529029\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−206.832	+	173.553i	−0.999188	+	0.838419i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$211$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−718.865	+	261.645i	−3.34356	+	1.21696i
$216$		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		−0.173648	−	0.984808i	$-0.555556\pi$
$223$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$224$		0			0
$225$	−473.605	−	172.378i	−2.10491	−	0.766125i
$226$		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
$233$	−80.7464	−	457.936i	−0.346551	−	1.96539i	−0.237758	−	0.971324i	$-0.576412\pi$
$233$	−80.7464	−	457.936i	−0.346551	−	1.96539i	−0.108793	−	0.994064i	$-0.534699\pi$
$234$		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		0.939693	−	0.342020i	$-0.111111\pi$
$241$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$242$		0			0
$243$		0			0
$244$	−387.153	−	140.912i	−1.58669	−	0.577509i
$245$	165.466	−	138.842i	0.675370	−	0.566703i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.68850	−	26.5898i	0.0186793	−	0.105935i	−0.974043	−	0.226365i	$-0.927316\pi$
$251$	4.68850	−	26.5898i	0.0186793	−	0.105935i	0.992722	+	0.120429i	$0.0384270\pi$
$252$	−31.2567	−	177.265i	−0.124034	−	0.703434i
$253$	84.5723	−	30.7818i	0.334278	−	0.121667i
$254$		0			0
$255$		0			0
$256$	196.107	+	164.554i	0.766044	+	0.642788i
$257$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$257$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	380.576	+	138.518i	1.44706	+	0.526685i	0.941767	−	0.336266i	$-0.109164\pi$
$263$	380.576	+	138.518i	1.44706	+	0.526685i	0.505288	+	0.862951i	$0.331386\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$269$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$270$		0			0
$271$	−24.6580	−	139.843i	−0.0909891	−	0.516025i	−0.995903	−	0.0904297i	$-0.971176\pi$
$271$	−24.6580	−	139.843i	−0.0909891	−	0.516025i	0.904914	−	0.425595i	$-0.139935\pi$
$272$	−225.526	+	82.0848i	−0.829141	+	0.301782i
$273$		0			0
$274$		0			0
$275$	128.695	+	107.988i	0.467984	+	0.392685i
$276$		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		0.173648	−	0.984808i	$-0.444444\pi$
$281$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$282$		0			0
$283$	302.588	−	253.901i	1.06921	−	0.897177i	0.0742330	−	0.997241i	$-0.476349\pi$
$283$	302.588	−	253.901i	1.06921	−	0.897177i	0.994981	+	0.100064i	$0.0319047\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−11.1135	+	63.0277i	−0.0384550	+	0.218089i
$290$		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
$301$	−399.369	−	145.359i	−1.32681	−	0.482919i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−927.000			−3.03934
$306$		0			0
$307$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$307$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$308$	−10.4189	+	59.0885i	−0.0338276	+	0.191846i
$309$		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
$313$	−451.966	−	379.245i	−1.44398	−	1.21164i	−0.936829	−	0.349788i	$-0.886254\pi$
$313$	−451.966	−	379.245i	−1.44398	−	1.21164i	−0.507153	−	0.861856i	$-0.669302\pi$
$314$		0			0
$315$	−202.500	−	350.740i	−0.642857	−	1.11346i
$316$		0			0
$317$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$317$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$318$		0			0
$319$		0			0
$320$	541.263	+	197.004i	1.69145	+	0.615636i
$321$		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
$329$	352.385	−	128.258i	1.07108	−	0.389841i
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	275.776	+	231.404i	0.830651	+	0.696999i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$337$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$338$		0			0
$339$		0			0
$340$	−413.664	+	347.105i	−1.21666	+	1.02090i
$341$		0			0
$342$		0			0
$343$	365.000			1.06414
$344$		0			0
$345$		0			0
$346$		0			0
$347$	117.213	+	664.745i	0.337788	+	1.91569i	0.397754	+	0.917492i	$0.369790\pi$
$347$	117.213	+	664.745i	0.337788	+	1.91569i	−0.0599662	+	0.998200i	$0.519099\pi$
$348$		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
$359$	186.149	−	156.197i	0.518520	−	0.435090i	−0.345595	−	0.938384i	$-0.612323\pi$
$359$	186.149	−	156.197i	0.518520	−	0.435090i	0.864116	+	0.503293i	$0.167879\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$	39.0708	−	221.582i	0.107043	−	0.607073i
$366$		0			0
$367$	−46.9846	+	17.1010i	−0.128024	+	0.0465967i	−0.405237	−	0.914211i	$-0.632811\pi$
$367$	−46.9846	+	17.1010i	−0.128024	+	0.0465967i	0.277214	+	0.960808i	$0.410589\pi$
$368$	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		−0.939693	−	0.342020i	$-0.888889\pi$
$383$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$384$		0			0
$385$	23.4425	+	132.949i	0.0608896	+	0.345322i
$386$		0			0
$387$	382.500	−	662.509i	0.988372	−	1.71191i
$388$		0			0
$389$	−117.205	−	98.3465i	−0.301298	−	0.252819i	0.479586	−	0.877495i	$-0.340787\pi$
$389$	−117.205	−	98.3465i	−0.301298	−	0.252819i	−0.780884	+	0.624676i	$0.785231\pi$
$390$		0			0
$391$	225.000	+	389.711i	0.575448	+	0.996704i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−101.487	−	36.9382i	−0.256280	−	0.0932782i
$397$	−570.703	+	478.877i	−1.43754	+	1.20624i	−0.496457	+	0.868061i	$0.665366\pi$
$397$	−570.703	+	478.877i	−1.43754	+	1.20624i	−0.941082	+	0.338177i	$0.890190\pi$
$398$		0			0
$399$		0			0
$400$	896.000			2.24000
$401$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$401$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$402$		0			0
$403$		0			0
$404$	−70.8485	−	401.802i	−0.175367	−	0.994558i
$405$	685.036	−	249.333i	1.69145	−	0.615636i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$409$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	761.151	+	277.036i	1.83410	+	0.667557i
$416$		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		−0.939693	−	0.342020i	$-0.888889\pi$
$421$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$422$		0			0
$423$	117.213	+	664.745i	0.277098	+	1.57150i
$424$		0			0
$425$	−420.000	+	727.461i	−0.988235	+	1.71167i
$426$		0			0
$427$	−394.513	−	331.036i	−0.923918	−	0.775259i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$431$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$432$		0			0
$433$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$433$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$439$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$440$		0			0
$441$	−37.5080	+	212.718i	−0.0850522	+	0.482355i
$442$		0			0
$443$	42.2862	−	15.3909i	0.0954541	−	0.0347425i	−0.293852	−	0.955851i	$-0.594937\pi$
$443$	42.2862	−	15.3909i	0.0954541	−	0.0347425i	0.389306	+	0.921109i	$0.372715\pi$
$444$		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
$460$	187.540	−	1063.59i	0.407696	−	2.31216i
$461$	77.6207	+	440.209i	0.168375	+	0.954900i	0.945516	+	0.325575i	$0.105558\pi$
$461$	77.6207	+	440.209i	0.168375	+	0.954900i	−0.777142	+	0.629326i	$0.783331\pi$
$462$		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
$473$	−195.341	+	163.911i	−0.412984	+	0.346535i
$474$		0			0
$475$		0			0
$476$	−300.000			−0.630252
$477$		0			0
$478$		0			0
$479$	−163.577	+	927.689i	−0.341496	+	1.93672i	0.00848697	+	0.999964i	$0.497298\pi$
$479$	−163.577	+	927.689i	−0.341496	+	1.93672i	−0.349983	+	0.936756i	$0.613813\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−343.188	−	287.969i	−0.709066	−	0.594977i
$485$		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
$491$	862.638	+	313.974i	1.75690	+	0.639459i	0.999903	−	0.0139588i	$-0.00444336\pi$
$491$	862.638	+	313.974i	1.75690	+	0.639459i	0.756997	+	0.653418i	$0.226666\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−243.000			−0.490909
$496$		0			0
$497$		0			0
$498$		0			0
$499$	90.8180	+	515.054i	0.182000	+	1.03217i	0.929750	+	0.368192i	$0.120023\pi$
$499$	90.8180	+	515.054i	0.182000	+	1.03217i	−0.747750	+	0.663981i	$0.768866\pi$
$500$	1048.70	−	381.694i	2.09739	−	0.763389i
$501$		0			0
$502$		0			0
$503$	712.421	+	597.792i	1.41634	+	1.18845i	0.953264	+	0.302138i	$0.0977003\pi$
$503$	712.421	+	597.792i	1.41634	+	1.18845i	0.463080	+	0.886316i	$0.346744\pi$
$504$		0			0
$505$	−459.000	−	795.011i	−0.908911	−	1.57428i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$509$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$510$		0			0
$511$	95.7556	−	80.3485i	0.187389	−	0.157238i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	39.0708	−	221.582i	0.0755722	−	0.428591i
$518$		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		−0.766044	−	0.642788i	$-0.777778\pi$
$523$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$524$	426.000	+	737.854i	0.812977	+	1.40812i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−348.626	−	126.889i	−0.659028	−	0.239867i
$530$		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
$541$	−350.082	−	293.754i	−0.647102	−	0.542983i	0.259088	−	0.965854i	$-0.416578\pi$
$541$	−350.082	−	293.754i	−0.647102	−	0.542983i	−0.906190	+	0.422871i	$0.861023\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$547$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$548$	−958.486	−	348.861i	−1.74906	−	0.636607i
$549$	710.123	−	595.864i	1.29348	−	1.08536i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−136.835	−	776.029i	−0.246106	−	1.39573i
$557$	−1028.96	+	374.512i	−1.84733	+	0.672374i	−0.860767	+	0.508999i	$0.830016\pi$
$557$	−1028.96	+	374.512i	−1.84733	+	0.672374i	−0.986564	+	0.163374i	$0.947762\pi$
$558$		0			0
$559$		0			0
$560$	551.552	+	462.807i	0.984914	+	0.826441i
$561$		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
$567$	380.576	+	138.518i	0.671209	+	0.244300i
$568$		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
$575$	−291.729	−	1654.48i	−0.507355	−	2.87735i
$576$	−541.263	+	197.004i	−0.939693	+	0.342020i
$577$	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
$587$	−861.800	+	723.136i	−1.46814	+	1.23192i	−0.550305	+	0.834964i	$0.685488\pi$
$587$	−861.800	+	723.136i	−1.46814	+	1.23192i	−0.917838	+	0.396954i	$0.870067\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.20945	+	29.5442i	−0.00878490	+	0.0498216i	−0.988885	−	0.148682i	$-0.952497\pi$
$593$	−5.20945	+	29.5442i	−0.00878490	+	0.0498216i	0.980100	+	0.198503i	$0.0636081\pi$
$594$		0			0
$595$	−634.293	+	230.864i	−1.06604	+	0.388006i
$596$	354.000	−	613.146i	0.593960	−	1.02877i
$597$		0			0
$598$		0			0
$599$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$599$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$600$		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
$605$	−947.210	−	344.756i	−1.56564	−	0.569845i
$606$		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
$612$	93.7700	−	531.796i	0.153219	−	0.868948i
$613$	51.2262	+	290.518i	0.0835664	+	0.473929i	0.997657	+	0.0684181i	$0.0217952\pi$
$613$	51.2262	+	290.518i	0.0835664	+	0.473929i	−0.914090	+	0.405511i	$0.867094\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−815.837	−	684.569i	−1.32226	−	1.10951i	−0.985820	−	0.167809i	$-0.946331\pi$
$617$	−815.837	−	684.569i	−1.32226	−	1.10951i	−0.336445	−	0.941703i	$-0.609225\pi$
$618$		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
$625$	851.075	−	714.137i	1.36172	−	1.14262i
$626$		0			0
$627$		0			0
$628$	40.0000			0.0636943
$629$		0			0
$630$		0			0
$631$	−180.073	+	1021.25i	−0.285377	+	1.61846i	0.418557	+	0.908191i	$0.362536\pi$
$631$	−180.073	+	1021.25i	−0.285377	+	1.61846i	−0.703934	+	0.710265i	$0.748575\pi$
$632$		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		−0.173648	−	0.984808i	$-0.555556\pi$
$641$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$642$		0			0
$643$	−1047.76	−	381.352i	−1.62948	−	0.593083i	−0.644327	−	0.764750i	$-0.722863\pi$
$643$	−1047.76	−	381.352i	−1.62948	−	0.593083i	−0.985155	+	0.171667i	$0.945085\pi$
$644$	459.627	−	385.673i	0.713706	−	0.598870i
$645$		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
$652$	−939.693	+	342.020i	−1.44125	+	0.524571i
$653$	−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
$655$	1468.51	+	1232.22i	2.24200	+	1.88126i
$656$		0			0
$657$	112.500	+	194.856i	0.171233	+	0.296584i
$658$		0			0
$659$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$659$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$660$		0			0
$661$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$661$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$662$		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
$671$	−290.365	+	105.684i	−0.432735	+	0.157503i
$672$		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
$688$	−236.162	+	1339.34i	−0.343258	+	1.94671i
$689$		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
$693$	−103.416	−	86.7763i	−0.149229	−	0.125218i
$694$		0			0
$695$	−886.500	−	1535.46i	−1.27554	−	2.20930i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1052.46	+	383.063i	1.50351	+	0.547232i
$701$	841.117	−	705.781i	1.19988	−	1.00682i	0.200248	−	0.979745i	$-0.435825\pi$
$701$	841.117	−	705.781i	1.19988	−	1.00682i	0.999633	−	0.0270747i	$-0.00861920\pi$
$702$		0			0
$703$		0			0
$704$	192.000			0.272727
$705$		0			0
$706$		0			0
$707$	88.5606	−	502.252i	0.125262	−	0.710399i
$708$		0			0
$709$	1238.51	−	450.783i	1.74685	−	0.635800i	0.747260	−	0.664531i	$-0.231369\pi$
$709$	1238.51	−	450.783i	1.74685	−	0.635800i	0.999587	+	0.0287309i	$0.00914660\pi$
$710$		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
$719$	−904.924	−	329.365i	−1.25859	−	0.458088i	−0.375292	−	0.926907i	$-0.622458\pi$
$719$	−904.924	−	329.365i	−1.25859	−	0.458088i	−0.883295	+	0.468818i	$0.844680\pi$
$720$	−992.794	+	833.053i	−1.37888	+	1.15702i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−14.7601	−	83.7087i	−0.0203027	−	0.115143i	0.972972	−	0.230922i	$-0.0741744\pi$
$727$	−14.7601	−	83.7087i	−0.0203027	−	0.115143i	−0.993275	+	0.115780i	$0.963063\pi$
$728$		0			0
$729$	−364.500	+	631.333i	−0.500000	+	0.866025i
$730$		0			0
$731$	−976.707	−	819.554i	−1.33612	−	1.12114i
$732$		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
$739$	419.026	−	351.605i	0.567018	−	0.475785i	−0.313637	−	0.949543i	$-0.601547\pi$
$739$	419.026	−	351.605i	0.567018	−	0.475785i	0.880655	+	0.473758i	$0.157103\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$743$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$744$		0			0
$745$	276.622	−	1568.80i	0.371304	−	2.10577i
$746$		0			0
$747$	−761.151	+	277.036i	−1.01894	+	0.370865i
$748$	−90.0000	+	155.885i	−0.120321	+	0.208402i
$749$		0			0
$750$		0			0
$751$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$751$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$752$	−600.000	−	1039.23i	−0.797872	−	1.38196i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	737.659	+	268.486i	0.974450	+	0.354671i	0.779680	−	0.626178i	$-0.215382\pi$
$757$	737.659	+	268.486i	0.974450	+	0.354671i	0.194770	+	0.980849i	$0.437604\pi$
$758$		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
$764$	−64.5971	+	366.348i	−0.0845512	+	0.479514i
$765$	−210.983	−	1196.54i	−0.275794	−	1.56411i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	814.305	+	683.283i	1.05891	+	0.888535i	0.994003	−	0.109356i	$-0.0348789\pi$
$769$	814.305	+	683.283i	1.05891	+	0.888535i	0.0649119	+	0.997891i	$0.479323\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$773$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$774$		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
$784$	−66.6809	−	378.166i	−0.0850522	−	0.482355i
$785$	84.5723	−	30.7818i	0.107735	−	0.0392125i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	275.776	+	231.404i	0.349970	+	0.293659i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	695.568	−	583.651i	0.873830	−	0.733230i
$797$		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
$803$	−13.0236	−	73.8606i	−0.0162187	−	0.0919808i
$804$		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		0.939693	−	0.342020i	$-0.111111\pi$
$811$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1723.60	+	1446.27i	−2.11485	+	1.77457i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	202.647	−	1149.27i	0.246830	−	1.39984i	−0.569374	−	0.822079i	$-0.692814\pi$
$821$	202.647	−	1149.27i	0.246830	−	1.39984i	0.816204	−	0.577764i	$-0.196075\pi$
$822$		0			0
$823$	1470.62	−	535.262i	1.78690	−	0.650379i	0.787480	−	0.616341i	$-0.211385\pi$
$823$	1470.62	−	535.262i	1.78690	−	0.650379i	0.999421	−	0.0340380i	$-0.0108367\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$827$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$828$	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
$833$	338.289	+	123.127i	0.406110	+	0.147812i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$839$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$840$		0			0
$841$	146.038	+	828.223i	0.173648	+	0.984808i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1165.15	−	977.680i	−1.37888	−	1.15702i
$846$		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
$853$	−789.026	+	662.071i	−0.925001	+	0.776168i	−0.974913	−	0.222586i	$-0.928550\pi$
$853$	−789.026	+	662.071i	−0.925001	+	0.776168i	0.0499122	+	0.998754i	$0.484106\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$857$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$858$		0			0
$859$	−259.257	+	1470.32i	−0.301812	+	1.71166i	0.336330	+	0.941744i	$0.390814\pi$
$859$	−259.257	+	1470.32i	−0.301812	+	1.71166i	−0.638143	+	0.769918i	$0.720297\pi$
$860$	531.363	+	3013.51i	0.617864	+	3.50408i
$861$		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		−0.939693	−	0.342020i	$-0.888889\pi$
$877$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$878$		0			0
$879$		0			0
$880$	405.947	−	147.753i	0.461304	−	0.167901i
$881$	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
$883$	639.647	+	536.728i	0.724402	+	0.607846i	0.928599	−	0.371084i	$-0.121014\pi$
$883$	639.647	+	536.728i	0.724402	+	0.607846i	−0.204197	+	0.978930i	$0.565458\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$887$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	186.149	−	156.197i	0.208921	−	0.175306i
$892$		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		−0.173648	−	0.984808i	$-0.555556\pi$
$907$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$908$		0			0
$909$	862.638	+	313.974i	0.948997	+	0.345406i
$910$		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
$916$	−11.8081	+	66.9669i	−0.0128909	+	0.0731080i
$917$	184.935	+	1048.82i	0.201674	+	1.14375i
$918$		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
$929$	491.801	−	412.670i	0.529387	−	0.444208i	−0.338503	−	0.940965i	$-0.609920\pi$
$929$	491.801	−	412.670i	0.529387	−	0.444208i	0.867890	+	0.496757i	$0.165476\pi$
$930$		0			0
$931$		0			0
$932$	−1860.00			−1.99571
$933$		0			0
$934$		0			0
$935$	−70.3275	+	398.847i	−0.0752166	+	0.426574i
$936$		0			0
$937$	−314.797	+	114.577i	−0.335963	+	0.122280i	−0.504492	−	0.863416i	$-0.668320\pi$
$937$	−314.797	+	114.577i	−0.335963	+	0.122280i	0.168530	+	0.985697i	$0.446098\pi$
$938$		0			0
$939$		0			0
$940$	−2068.32	−	1735.53i	−2.20034	−	1.84630i
$941$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$941$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1719.64	+	625.897i	1.81588	+	0.660926i	0.996099	+	0.0882482i	$0.0281269\pi$
$947$	1719.64	+	625.897i	1.81588	+	0.660926i	0.819781	+	0.572678i	$0.194095\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$953$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$954$		0			0
$955$	145.344	+	824.284i	0.152192	+	0.863125i
$956$	1702.72	−	619.740i	1.78109	−	0.648264i
$957$		0			0
$958$		0			0
$959$	−976.707	−	819.554i	−1.01846	−	0.854592i
$960$		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
$967$	−1371.22	+	1150.59i	−1.41801	+	1.18986i	−0.465620	+	0.884985i	$0.654169\pi$
$967$	−1371.22	+	1150.59i	−1.41801	+	1.18986i	−0.952394	+	0.304870i	$0.901387\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$971$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$972$		0			0
$973$	171.043	−	970.036i	0.175790	−	0.996953i
$974$		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		−0.173648	−	0.984808i	$-0.555556\pi$
$983$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$984$		0			0
$985$	761.151	+	277.036i	0.772742	+	0.281255i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2550.00			2.57836
$990$		0			0
$991$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$991$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1021.50	−	1769.29i	1.02663	−	1.77818i
$996$		0			0
$997$	1512.94	+	1269.51i	1.51749	+	1.27333i	0.847269	+	0.531164i	$0.178245\pi$
$997$	1512.94	+	1269.51i	1.51749	+	1.27333i	0.670221	+	0.742161i	$0.266199\pi$
$998$		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.127.1		6
19.2	odd	18	inner	361.3.f.a.299.1		6
19.3	odd	18	inner	361.3.f.a.307.1		6
19.4	even	9		19.3.b.a.18.1	✓	1
19.5	even	9	inner	361.3.f.a.116.1		6
19.6	even	9		361.3.d.a.69.1		2
19.7	even	3	inner	361.3.f.a.333.1		6
19.8	odd	6	inner	361.3.f.a.262.1		6
19.9	even	9		361.3.d.a.293.1		2
19.10	odd	18		361.3.d.a.293.1		2
19.11	even	3	inner	361.3.f.a.262.1		6
19.12	odd	6	inner	361.3.f.a.333.1		6
19.13	odd	18		361.3.d.a.69.1		2
19.14	odd	18	inner	361.3.f.a.116.1		6
19.15	odd	18		19.3.b.a.18.1	✓	1
19.16	even	9	inner	361.3.f.a.307.1		6
19.17	even	9	inner	361.3.f.a.299.1		6
19.18	odd	2	CM	361.3.f.a.127.1		6
57.23	odd	18		171.3.c.a.37.1		1
57.53	even	18		171.3.c.a.37.1		1
76.15	even	18		304.3.e.a.113.1		1
76.23	odd	18		304.3.e.a.113.1		1
95.4	even	18		475.3.c.a.151.1		1
95.23	odd	36		475.3.d.a.474.2		2
95.34	odd	18		475.3.c.a.151.1		1
95.42	odd	36		475.3.d.a.474.1		2
95.53	even	36		475.3.d.a.474.2		2
95.72	even	36		475.3.d.a.474.1		2
152.53	odd	18		1216.3.e.a.1025.1		1
152.61	even	18		1216.3.e.a.1025.1		1
152.91	even	18		1216.3.e.b.1025.1		1
152.99	odd	18		1216.3.e.b.1025.1		1
228.23	even	18		2736.3.o.a.721.1		1
228.167	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	19.4	even	9
19.3.b.a.18.1	✓	1	19.15	odd	18
171.3.c.a.37.1		1	57.23	odd	18
171.3.c.a.37.1		1	57.53	even	18
304.3.e.a.113.1		1	76.15	even	18
304.3.e.a.113.1		1	76.23	odd	18
361.3.d.a.69.1		2	19.6	even	9
361.3.d.a.69.1		2	19.13	odd	18
361.3.d.a.293.1		2	19.9	even	9
361.3.d.a.293.1		2	19.10	odd	18
361.3.f.a.116.1		6	19.5	even	9	inner
361.3.f.a.116.1		6	19.14	odd	18	inner
361.3.f.a.127.1		6	1.1	even	1	trivial
361.3.f.a.127.1		6	19.18	odd	2	CM
361.3.f.a.262.1		6	19.8	odd	6	inner
361.3.f.a.262.1		6	19.11	even	3	inner
361.3.f.a.299.1		6	19.2	odd	18	inner
361.3.f.a.299.1		6	19.17	even	9	inner
361.3.f.a.307.1		6	19.3	odd	18	inner
361.3.f.a.307.1		6	19.16	even	9	inner
361.3.f.a.333.1		6	19.7	even	3	inner
361.3.f.a.333.1		6	19.12	odd	6	inner
475.3.c.a.151.1		1	95.4	even	18
475.3.c.a.151.1		1	95.34	odd	18
475.3.d.a.474.1		2	95.42	odd	36
475.3.d.a.474.1		2	95.72	even	36
475.3.d.a.474.2		2	95.23	odd	36
475.3.d.a.474.2		2	95.53	even	36
1216.3.e.a.1025.1		1	152.53	odd	18
1216.3.e.a.1025.1		1	152.61	even	18
1216.3.e.b.1025.1		1	152.91	even	18
1216.3.e.b.1025.1		1	152.99	odd	18
2736.3.o.a.721.1		1	228.23	even	18
2736.3.o.a.721.1		1	228.167	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)

\(f(q)\)	\(=\)	\(q+(0.694593 - 3.93923i) q^{4} +(-1.56283 - 8.86327i) q^{5} +(2.50000 - 4.33013i) q^{7} +(6.89440 + 5.78509i) q^{9} +O(q^{10})\)	\(q+(0.694593 - 3.93923i) q^{4} +(-1.56283 - 8.86327i) q^{5} +(2.50000 - 4.33013i) q^{7} +(6.89440 + 5.78509i) q^{9} +(-1.50000 - 2.59808i) q^{11} +(-15.0351 - 5.47232i) q^{16} +(11.4907 - 9.64181i) q^{17} -36.0000 q^{20} +(-5.20945 + 29.5442i) q^{23} +(-52.6228 + 19.1531i) q^{25} +(-15.3209 - 12.8558i) q^{28} +(-42.2862 - 15.3909i) q^{35} +(27.5776 - 23.1404i) q^{36} +(-14.7601 - 83.7087i) q^{43} +(-11.2763 + 4.10424i) q^{44} +(40.5000 - 70.1481i) q^{45} +(57.4533 + 48.2091i) q^{47} +(12.0000 + 20.7846i) q^{49} +(-20.6832 + 17.3553i) q^{55} +(17.8858 - 101.435i) q^{61} +(42.2862 - 15.3909i) q^{63} +(-32.0000 + 55.4256i) q^{64} +(-30.0000 - 51.9615i) q^{68} +(23.4923 + 8.55050i) q^{73} -15.0000 q^{77} +(-25.0053 + 141.812i) q^{80} +(14.0655 + 79.7694i) q^{81} +(-45.0000 + 77.9423i) q^{83} +(-103.416 - 86.7763i) q^{85} +(112.763 + 41.0424i) q^{92} +(4.68850 - 26.5898i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 15 q^{7}+O(q^{10}) \) 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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Database

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