LMFDB - Embedded newform 216.2.v.a.131.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,2,Mod(11,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 13]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("216.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	216.v (of order $18$, degree $6$, 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:	$1.72476868366$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			131.1
Root			$-1.39273 - 0.245576i$ of defining polynomial
Character	$\chi$	$=$	216.131
Dual form			216.2.v.a.155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.483690 + 1.32893i) q^{2} +(-1.56638 - 0.739232i) q^{3} +(-1.53209 - 1.28558i) q^{4} +(1.74002 - 1.72404i) q^{6} +(2.44949 - 1.41421i) q^{8} +(1.90707 + 2.31583i) q^{9} +O(q^{10})$	\(q+(-0.483690 + 1.32893i) q^{2} +(-1.56638 - 0.739232i) q^{3} +(-1.53209 - 1.28558i) q^{4} +(1.74002 - 1.72404i) q^{6} +(2.44949 - 1.41421i) q^{8} +(1.90707 + 2.31583i) q^{9} +(5.93192 - 1.04596i) q^{11} +(1.44949 + 3.14626i) q^{12} +(0.694593 + 3.93923i) q^{16} +(6.38072 + 3.68391i) q^{17} +(-4.00000 + 1.41421i) q^{18} +(-4.35183 - 7.53760i) q^{19} +(-1.47921 + 8.38900i) q^{22} +(-4.88226 + 0.404450i) q^{24} +(4.69846 + 1.71010i) q^{25} +(-1.27526 - 5.03723i) q^{27} +(-5.57091 - 0.982302i) q^{32} +(-10.0648 - 2.74670i) q^{33} +(-7.98194 + 6.69764i) q^{34} +(0.0553729 - 5.99974i) q^{36} +(12.1218 - 2.13741i) q^{38} +(-1.34901 - 3.70637i) q^{41} +(1.12781 + 6.39612i) q^{43} +(-10.4329 - 6.02343i) q^{44} +(1.82401 - 6.68378i) q^{48} +(1.21554 - 6.89365i) q^{49} +(-4.54519 + 5.41675i) q^{50} +(-7.27135 - 10.4872i) q^{51} +(7.31094 + 0.741737i) q^{54} +(1.24458 + 15.0237i) q^{57} +(8.23753 + 1.45250i) q^{59} +(4.00000 - 6.92820i) q^{64} +(8.51841 - 12.0469i) q^{66} +(-4.95255 + 1.80258i) q^{67} +(-5.03989 - 13.8470i) q^{68} +(7.94643 + 2.97560i) q^{72} +(-1.96244 - 3.39905i) q^{73} +(-6.09540 - 6.15192i) q^{75} +(-3.02275 + 17.1429i) q^{76} +(-1.72616 + 8.83292i) q^{81} +5.57800 q^{82} +(-4.84788 + 13.3194i) q^{83} +(-9.04548 - 1.59496i) q^{86} +(13.0510 - 10.9511i) q^{88} +(-15.9495 + 9.20844i) q^{89} +(8.00000 + 5.65685i) q^{96} +(0.564013 + 3.19868i) q^{97} +(8.57321 + 4.94975i) q^{98} +(13.7349 + 11.7426i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 18 q^{11} - 12 q^{12} - 48 q^{18} + 12 q^{22} - 30 q^{27} + 6 q^{33} - 24 q^{34} + 72 q^{38} - 18 q^{41} + 30 q^{43} - 12 q^{51} + 42 q^{57} + 36 q^{59} + 48 q^{64} - 42 q^{67} - 72 q^{68} - 24 q^{76} + 36 q^{86} + 48 q^{88} - 162 q^{89} + 96 q^{96} + 30 q^{97}+O(q^{100}) $ 12 * q + 18 * q^11 - 12 * q^12 - 48 * q^18 + 12 * q^22 - 30 * q^27 + 6 * q^33 - 24 * q^34 + 72 * q^38 - 18 * q^41 + 30 * q^43 - 12 * q^51 + 42 * q^57 + 36 * q^59 + 48 * q^64 - 42 * q^67 - 72 * q^68 - 24 * q^76 + 36 * q^86 + 48 * q^88 - 162 * q^89 + 96 * q^96 + 30 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{11}{18}\right)$

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 $

$2$	−0.483690	+	1.32893i	−0.342020	+	0.939693i
$2$	−0.483690	+	1.32893i	−0.342020	+	0.939693i
$3$	−1.56638	−	0.739232i	−0.904348	−	0.426796i
$3$	−1.56638	−	0.739232i	−0.904348	−	0.426796i
$4$	−1.53209	−	1.28558i	−0.766044	−	0.642788i
$5$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$5$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$6$	1.74002	−	1.72404i	0.710362	−	0.703836i
$7$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$7$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$8$	2.44949	−	1.41421i	0.866025	−	0.500000i
$9$	1.90707	+	2.31583i	0.635691	+	0.771944i
$10$		0			0
$11$	5.93192	−	1.04596i	1.78854	−	0.315368i	0.821541	−	0.570149i	$-0.193114\pi$
$11$	5.93192	−	1.04596i	1.78854	−	0.315368i	0.966999	+	0.254781i	$0.0820033\pi$
$12$	1.44949	+	3.14626i	0.418432	+	0.908248i
$13$		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
$16$	0.694593	+	3.93923i	0.173648	+	0.984808i
$17$	6.38072	+	3.68391i	1.54755	+	0.893480i	0.998328	+	0.0578055i	$0.0184103\pi$
$17$	6.38072	+	3.68391i	1.54755	+	0.893480i	0.549225	+	0.835675i	$0.314923\pi$
$18$	−4.00000	+	1.41421i	−0.942809	+	0.333333i
$19$	−4.35183	−	7.53760i	−0.998379	−	1.72924i	−0.548478	−	0.836165i	$-0.684793\pi$
$19$	−4.35183	−	7.53760i	−0.998379	−	1.72924i	−0.449901	−	0.893079i	$-0.648541\pi$
$20$		0			0
$21$		0			0
$22$	−1.47921	+	8.38900i	−0.315368	+	1.78854i
$23$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$23$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$24$	−4.88226	+	0.404450i	−0.996586	+	0.0825579i
$25$	4.69846	+	1.71010i	0.939693	+	0.342020i
$26$		0			0
$27$	−1.27526	−	5.03723i	−0.245423	−	0.969416i
$28$		0			0
$29$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$29$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$30$		0			0
$31$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$31$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$32$	−5.57091	−	0.982302i	−0.984808	−	0.173648i
$33$	−10.0648	−	2.74670i	−1.75206	−	0.478139i
$34$	−7.98194	+	6.69764i	−1.36889	+	1.14864i
$35$		0			0
$36$	0.0553729	−	5.99974i	0.00922881	−	0.999957i
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	12.1218	−	2.13741i	1.96642	−	0.346733i
$39$		0			0
$40$		0			0
$41$	−1.34901	−	3.70637i	−0.210680	−	0.578838i	0.788673	−	0.614813i	$-0.210769\pi$
$41$	−1.34901	−	3.70637i	−0.210680	−	0.578838i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	1.12781	+	6.39612i	0.171989	+	0.975399i	0.941562	+	0.336840i	$0.109358\pi$
$43$	1.12781	+	6.39612i	0.171989	+	0.975399i	−0.769573	+	0.638559i	$0.779531\pi$
$44$	−10.4329	−	6.02343i	−1.57282	−	0.908066i
$45$		0			0
$46$		0			0
$47$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$47$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$48$	1.82401	−	6.68378i	0.263274	−	0.964721i
$49$	1.21554	−	6.89365i	0.173648	−	0.984808i
$50$	−4.54519	+	5.41675i	−0.642788	+	0.766044i
$51$	−7.27135	−	10.4872i	−1.01819	−	1.46851i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	7.31094	+	0.741737i	0.994893	+	0.100938i
$55$		0			0
$56$		0			0
$57$	1.24458	+	15.0237i	0.164848	+	1.98994i
$58$		0			0
$59$	8.23753	+	1.45250i	1.07243	+	0.189099i	0.681868	−	0.731475i	$-0.261168\pi$
$59$	8.23753	+	1.45250i	1.07243	+	0.189099i	0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$61$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$62$		0			0
$63$		0			0
$64$	4.00000	−	6.92820i	0.500000	−	0.866025i
$65$		0			0
$66$	8.51841	−	12.0469i	1.04854	−	1.48287i
$67$	−4.95255	+	1.80258i	−0.605050	+	0.220220i	−0.626336	−	0.779553i	$-0.715446\pi$
$67$	−4.95255	+	1.80258i	−0.605050	+	0.220220i	0.0212861	+	0.999773i	$0.493224\pi$
$68$	−5.03989	−	13.8470i	−0.611176	−	1.67919i
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$	7.94643	+	2.97560i	0.936496	+	0.350678i
$73$	−1.96244	−	3.39905i	−0.229687	−	0.397829i	0.728028	−	0.685547i	$-0.240437\pi$
$73$	−1.96244	−	3.39905i	−0.229687	−	0.397829i	−0.957715	+	0.287718i	$0.907104\pi$
$74$		0			0
$75$	−6.09540	−	6.15192i	−0.703836	−	0.710362i
$76$	−3.02275	+	17.1429i	−0.346733	+	1.96642i
$77$		0			0
$78$		0			0
$79$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$79$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$80$		0			0
$81$	−1.72616	+	8.83292i	−0.191795	+	0.981435i
$82$	5.57800			0.615987
$83$	−4.84788	+	13.3194i	−0.532124	+	1.46200i	0.324415	+	0.945915i	$0.394833\pi$
$83$	−4.84788	+	13.3194i	−0.532124	+	1.46200i	−0.856539	+	0.516083i	$0.827390\pi$
$84$		0			0
$85$		0			0
$86$	−9.04548	−	1.59496i	−0.975399	−	0.171989i
$87$		0			0
$88$	13.0510	−	10.9511i	1.39124	−	1.16739i
$89$	−15.9495	+	9.20844i	−1.69064	+	0.976093i	−0.736644	+	0.676280i	$0.763591\pi$
$89$	−15.9495	+	9.20844i	−1.69064	+	0.976093i	−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	8.00000	+	5.65685i	0.816497	+	0.577350i
$97$	0.564013	+	3.19868i	0.0572669	+	0.324777i	0.999961	−	0.00888289i	$-0.00282755\pi$
$97$	0.564013	+	3.19868i	0.0572669	+	0.324777i	−0.942694	+	0.333659i	$0.891716\pi$
$98$	8.57321	+	4.94975i	0.866025	+	0.500000i
$99$	13.7349	+	11.7426i	1.38040	+	1.18018i
$100$	−5.00000	−	8.66025i	−0.500000	−	0.866025i
$101$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$101$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$102$	17.4538	−	4.59052i	1.72819	−	0.454529i
$103$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$103$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			16.5528i			1.60022i	0.599851	+	0.800112i	$0.295227\pi$
$107$			16.5528i			1.60022i	−0.599851	+	0.800112i	$0.704773\pi$
$108$	−4.52194	+	9.35693i	−0.435124	+	0.900371i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.78072	−	1.72460i	−0.920093	−	0.162237i	−0.306510	−	0.951867i	$-0.599161\pi$
$113$	−9.78072	−	1.72460i	−0.920093	−	0.162237i	−0.613583	+	0.789630i	$0.710272\pi$
$114$	−20.5674	−	5.61287i	−1.92632	−	0.525694i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−5.91467	+	10.2445i	−0.544489	+	0.943083i
$119$		0			0
$120$		0			0
$121$	23.7570	−	8.64684i	2.15973	−	0.786076i
$122$		0			0
$123$	−0.626813	+	6.80281i	−0.0565179	+	0.613389i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$	7.27231	+	8.66680i	0.642788	+	0.766044i
$129$	2.96164	−	10.8524i	0.260758	−	0.955504i
$130$		0			0
$131$	14.5653	−	17.3582i	1.27257	−	1.51659i	0.527611	−	0.849486i	$-0.323088\pi$
$131$	14.5653	−	17.3582i	1.27257	−	1.51659i	0.744962	−	0.667107i	$-0.232468\pi$
$132$	11.8891	+	17.1473i	1.03481	+	1.49248i
$133$		0			0
$134$		−	7.45345i		−	0.643880i
$135$		0			0
$136$	20.8394			1.78696
$137$	4.60936	−	12.6641i	0.393805	−	1.08197i	−0.571445	−	0.820640i	$-0.693617\pi$
$137$	4.60936	−	12.6641i	0.393805	−	1.08197i	0.965250	−	0.261329i	$-0.0841608\pi$
$138$		0			0
$139$	−17.0883	−	14.3388i	−1.44941	−	1.21620i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−17.0883	−	14.3388i	−1.44941	−	1.21620i	−0.516404	−	0.856345i	$-0.672730\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−7.79796	+	9.12096i	−0.649830	+	0.760080i
$145$		0			0
$146$	5.46630	−	0.963857i	0.452395	−	0.0797694i
$147$	−7.00000	+	9.89949i	−0.577350	+	0.816497i
$148$		0			0
$149$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$149$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$150$	11.1237	−	5.12472i	0.908248	−	0.418432i
$151$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$151$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$152$	−21.3195	−	12.3088i	−1.72924	−	0.998379i
$153$	3.63718	+	21.8022i	0.294048	+	1.76260i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$157$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−10.9034	−	6.56632i	−0.856649	−	0.515899i
$163$	−23.0454			−1.80506			−0.902528	−	0.430632i	$-0.858291\pi$
$163$	−23.0454			−1.80506			−0.902528	+	0.430632i	$0.858291\pi$
$164$	−2.69802	+	7.41275i	−0.210680	+	0.578838i
$165$		0			0
$166$	−15.3557	−	12.8849i	−1.19183	−	1.00007i
$167$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$167$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$168$		0			0
$169$	9.95858	−	8.35624i	0.766044	−	0.642788i
$170$		0			0
$171$	9.15655	−	24.4529i	0.700219	−	1.86996i
$172$	6.49479	−	11.2493i	0.495223	−	0.857751i
$173$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$173$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$174$		0			0
$175$		0			0
$176$	8.24053	+	22.6407i	0.621154	+	1.70661i
$177$	−11.8293	−	8.36460i	−0.889147	−	0.628722i
$178$	−4.52274	−	25.6497i	−0.338994	−	1.92253i
$179$	4.92679	+	2.84448i	0.368245	+	0.212607i	0.672692	−	0.739923i	$-0.265138\pi$
$179$	4.92679	+	2.84448i	0.368245	+	0.212607i	−0.304446	+	0.952529i	$0.598471\pi$
$180$		0			0
$181$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$181$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	41.7031	+	15.1787i	3.04964	+	1.10998i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$191$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$192$	−11.3871	+	7.89525i	−0.821790	+	0.569790i
$193$	−15.7292	−	13.1984i	−1.13221	−	0.950039i	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	−15.7292	−	13.1984i	−1.13221	−	0.950039i	−0.999156	+	0.0410699i	$0.986923\pi$
$194$	−4.52361	−	0.797635i	−0.324777	−	0.0572669i
$195$		0			0
$196$	−10.7246	+	8.99903i	−0.766044	+	0.642788i
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$	−22.2485	+	12.5728i	−1.58113	+	0.893512i
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	13.9273	−	2.45576i	0.984808	−	0.173648i
$201$	9.09008	+	0.837563i	0.641164	+	0.0590771i
$202$		0			0
$203$		0			0
$204$	−2.34177	+	25.4152i	−0.163957	+	1.77942i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−33.6987	−	40.1606i	−2.33099	−	2.77796i
$210$		0			0
$211$	−5.04368	+	28.6041i	−0.347221	+	1.96919i	−0.153151	+	0.988203i	$0.548942\pi$
$211$	−5.04368	+	28.6041i	−0.347221	+	1.96919i	−0.194071	+	0.980988i	$0.562169\pi$
$212$		0			0
$213$		0			0
$214$	−21.9975	−	8.00644i	−1.50372	−	0.547309i
$215$		0			0
$216$	−10.2474	−	10.5352i	−0.697251	−	0.716827i
$217$		0			0
$218$		0			0
$219$	0.561238	+	6.77490i	0.0379249	+	0.457805i
$220$		0			0
$221$		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
$225$	5.00000	+	14.1421i	0.333333	+	0.942809i
$226$	7.02270	−	12.1637i	0.467143	−	0.809116i
$227$	−12.7222	+	2.24327i	−0.844402	+	0.148891i	−0.579082	−	0.815270i	$-0.696589\pi$
$227$	−12.7222	+	2.24327i	−0.844402	+	0.148891i	−0.265320	+	0.964160i	$0.585478\pi$
$228$	17.4073	−	24.6177i	1.15283	−	1.63035i
$229$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$229$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	24.7354	+	14.2810i	1.62047	+	0.935577i	0.986795	+	0.161976i	$0.0517866\pi$
$233$	24.7354	+	14.2810i	1.62047	+	0.935577i	0.633672	+	0.773602i	$0.281547\pi$
$234$		0			0
$235$		0			0
$236$	−10.7533	−	12.8153i	−0.699982	−	0.834206i
$237$		0			0
$238$		0			0
$239$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$239$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$240$		0			0
$241$	−28.9666	−	10.5430i	−1.86590	−	0.679133i	−0.973845	−	0.227213i	$-0.927039\pi$
$241$	−28.9666	−	10.5430i	−1.86590	−	0.679133i	−0.892058	−	0.451920i	$-0.850739\pi$
$242$			35.7537i			2.29833i
$243$	9.23338	−	12.5596i	0.592322	−	0.805701i
$244$		0			0
$245$		0			0
$246$	−8.73725	−	4.12344i	−0.557067	−	0.262901i
$247$		0			0
$248$		0			0
$249$	17.4398	−	17.2795i	1.10520	−	1.09505i
$250$		0			0
$251$	−27.0966	+	15.6442i	−1.71032	+	0.987456i	−0.776215	+	0.630468i	$0.782863\pi$
$251$	−27.0966	+	15.6442i	−1.71032	+	0.987456i	−0.934109	+	0.356988i	$0.883804\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−15.0351	+	5.47232i	−0.939693	+	0.342020i
$257$	−3.63117	−	9.97655i	−0.226506	−	0.622320i	0.773427	−	0.633885i	$-0.218541\pi$
$257$	−3.63117	−	9.97655i	−0.226506	−	0.622320i	−0.999933	+	0.0115651i	$0.996319\pi$
$258$	12.9896	+	9.18502i	0.808696	+	0.571834i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	16.0227	+	27.7521i	0.989886	+	1.71453i
$263$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$263$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$264$	−28.5381	+	7.50579i	−1.75640	+	0.461950i
$265$		0			0
$266$		0			0
$267$	31.7901	−	2.63351i	1.94552	−	0.161168i
$268$	9.90509	+	3.60516i	0.605050	+	0.220220i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−10.0798	+	27.6940i	−0.611176	+	1.67919i
$273$		0			0
$274$	14.6002	+	12.2510i	0.882029	+	0.740111i
$275$	29.6596	+	5.22979i	1.78854	+	0.315368i
$276$		0			0
$277$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$277$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$278$	27.3206	−	15.7736i	1.63858	−	0.946036i
$279$		0			0
$280$		0			0
$281$	1.42435	−	0.251151i	0.0849695	−	0.0149824i	−0.131002	−	0.991382i	$-0.541819\pi$
$281$	1.42435	−	0.251151i	0.0849695	−	0.0149824i	0.215971	+	0.976400i	$0.430708\pi$
$282$		0			0
$283$	10.3793	−	3.77775i	0.616985	−	0.224564i	−0.0145720	−	0.999894i	$-0.504639\pi$
$283$	10.3793	−	3.77775i	0.616985	−	0.224564i	0.631557	+	0.775330i	$0.282416\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−8.34928	−	14.7746i	−0.491986	−	0.870603i
$289$	18.6424	+	32.2896i	1.09661	+	1.89939i
$290$		0			0
$291$	1.48111	−	5.42727i	0.0868241	−	0.318152i
$292$	−1.36310	+	7.73052i	−0.0797694	+	0.452395i
$293$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$293$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$294$	−9.76987	−	14.0908i	−0.569790	−	0.821790i
$295$		0			0
$296$		0			0
$297$	−12.8334	−	28.5466i	−0.744672	−	1.65644i
$298$		0			0
$299$		0			0
$300$	1.42995	+	17.2614i	0.0825579	+	0.996586i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	26.6696	−	22.3784i	1.52961	−	1.28349i
$305$		0			0
$306$	−30.7327	−	5.71194i	−1.75687	−	0.326530i
$307$	−1.22617	+	2.12378i	−0.0699810	+	0.121211i	−0.898893	−	0.438169i	$-0.855627\pi$
$307$	−1.22617	+	2.12378i	−0.0699810	+	0.121211i	0.828912	+	0.559379i	$0.188961\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$311$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$312$		0			0
$313$	5.50274	+	31.2076i	0.311033	+	1.76396i	0.593650	+	0.804723i	$0.297686\pi$
$313$	5.50274	+	31.2076i	0.311033	+	1.76396i	−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$317$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	12.2364	−	25.9280i	0.682969	−	1.44716i
$322$		0			0
$323$		−	64.1271i		−	3.56813i
$324$	14.0000	−	11.3137i	0.777778	−	0.628539i
$325$		0			0
$326$	11.1468	−	30.6256i	0.617365	−	1.69620i
$327$		0			0
$328$	−8.54599	−	7.17094i	−0.471873	−	0.395949i
$329$		0			0
$330$		0			0
$331$	−26.8463	+	22.5268i	−1.47561	+	1.23818i	−0.564882	+	0.825172i	$0.691078\pi$
$331$	−26.8463	+	22.5268i	−1.47561	+	1.23818i	−0.910726	+	0.413011i	$0.864477\pi$
$332$	24.5505	−	14.1742i	1.34738	−	0.777913i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	23.2708	−	8.46987i	1.26764	−	0.461383i	0.381314	−	0.924445i	$-0.375472\pi$
$337$	23.2708	−	8.46987i	1.26764	−	0.461383i	0.886326	+	0.463062i	$0.153249\pi$
$338$	6.28796	+	17.2760i	0.342020	+	0.939693i
$339$	14.0454	+	9.93160i	0.762842	+	0.539411i
$340$		0			0
$341$		0			0
$342$	28.0671	+	23.9960i	1.51770	+	1.29755i
$343$		0			0
$344$	11.8080	+	14.0723i	0.636646	+	0.758726i
$345$		0			0
$346$		0			0
$347$	0.306044	−	0.364729i	0.0164293	−	0.0195797i	−0.757767	−	0.652525i	$-0.773710\pi$
$347$	0.306044	−	0.364729i	0.0164293	−	0.0195797i	0.774197	+	0.632945i	$0.218154\pi$
$348$		0			0
$349$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$349$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$350$		0			0
$351$		0			0
$352$	−34.0736			−1.81613
$353$	−3.76298	+	10.3387i	−0.200283	+	0.550273i	−0.998653	−	0.0518946i	$-0.983474\pi$
$353$	−3.76298	+	10.3387i	−0.200283	+	0.550273i	0.798369	+	0.602168i	$0.205696\pi$
$354$	16.8377	−	11.6744i	0.894912	−	0.620489i
$355$		0			0
$356$	36.2742	+	6.39612i	1.92253	+	0.338994i
$357$		0			0
$358$	−6.16314	+	5.17149i	−0.325732	+	0.273322i
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−28.3769	+	49.1503i	−1.49352	+	2.58686i
$362$		0			0
$363$	−43.6044	−	4.01773i	−2.28864	−	0.210876i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$367$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$368$		0			0
$369$	6.01068	−	10.1924i	0.312904	−	0.530595i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$373$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$374$	−40.3427	+	48.0786i	−2.08607	+	2.48609i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	14.9550			0.768187			0.384093	−	0.923294i	$-0.374514\pi$
$379$	14.9550			0.768187			0.384093	+	0.923294i	$0.374514\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$383$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$384$	−4.98440	−	18.9514i	−0.254359	−	0.967110i
$385$		0			0
$386$	25.1477	−	14.5190i	1.27998	−	0.738999i
$387$	−12.6615	+	14.8097i	−0.643621	+	0.752818i
$388$	3.24802	−	5.62574i	0.164893	−	0.285604i
$389$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$389$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$390$		0			0
$391$		0			0
$392$	−6.77165	−	18.6050i	−0.342020	−	0.939693i
$393$	−35.6464	+	16.4224i	−1.79812	+	0.828399i
$394$		0			0
$395$		0			0
$396$	−5.94701	−	35.6479i	−0.298848	−	1.79137i
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$	−3.47296	+	19.6962i	−0.173648	+	0.984808i
$401$	−8.21802	+	9.79385i	−0.410388	+	0.489082i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−8.21802	+	9.79385i	−0.410388	+	0.489082i	0.520770	+	0.853697i	$0.325645\pi$
$402$	−5.50983	+	11.6749i	−0.274806	+	0.582292i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−32.6423	−	15.4051i	−1.61603	−	0.762667i
$409$	6.94395	+	5.82667i	0.343356	+	0.288110i	0.798116	−	0.602504i	$-0.205830\pi$
$409$	6.94395	+	5.82667i	0.343356	+	0.288110i	−0.454759	+	0.890614i	$0.650275\pi$
$410$		0			0
$411$	−16.5817	+	16.4294i	−0.817916	+	0.810403i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	16.1670	+	35.0922i	0.791703	+	1.71847i
$418$	69.6701	−	25.3579i	3.40768	−	1.24029i
$419$	−0.956397	−	2.62768i	−0.0467231	−	0.128371i	0.914136	−	0.405407i	$-0.132870\pi$
$419$	−0.956397	−	2.62768i	−0.0467231	−	0.128371i	−0.960860	+	0.277036i	$0.910648\pi$
$420$		0			0
$421$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$421$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$422$	−35.5732	−	20.5382i	−1.73168	−	0.999784i
$423$		0			0
$424$		0			0
$425$	23.6797	+	28.2204i	1.14864	+	1.36889i
$426$		0			0
$427$		0			0
$428$	21.2799	−	25.3604i	1.02860	−	1.22584i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	18.9570	−	8.52235i	0.912071	−	0.410032i
$433$	−29.9040			−1.43710			−0.718548	−	0.695477i	$-0.755193\pi$
$433$	−29.9040			−1.43710			−0.718548	+	0.695477i	$0.755193\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−9.27481	−	2.53111i	−0.443167	−	0.120941i
$439$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$439$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$440$		0			0
$441$	18.2827	−	10.3317i	0.870603	−	0.491986i
$442$		0			0
$443$	−40.2499	+	7.09714i	−1.91233	+	0.337195i	−0.997740	−	0.0671913i	$-0.978596\pi$
$443$	−40.2499	+	7.09714i	−1.91233	+	0.337195i	−0.914588	+	0.404386i	$0.867485\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	19.6273	+	11.3318i	0.926271	+	0.534783i	0.885630	−	0.464391i	$-0.153727\pi$
$449$	19.6273	+	11.3318i	0.926271	+	0.534783i	0.0406404	+	0.999174i	$0.487060\pi$
$450$	−21.2123	−	0.195773i	−0.999957	−	0.00922881i
$451$	−11.8789	−	20.5749i	−0.559357	−	0.968834i
$452$	12.7678	+	15.2161i	0.600548	+	0.715705i
$453$		0			0
$454$	3.17246	−	17.9919i	0.148891	−	0.844402i
$455$		0			0
$456$	24.2953	+	35.0404i	1.13773	+	1.64092i
$457$	−35.6523	−	12.9764i	−1.66774	−	0.607008i	−0.676191	−	0.736726i	$-0.736371\pi$
$457$	−35.6523	−	12.9764i	−1.66774	−	0.607008i	−0.991551	+	0.129718i	$0.958593\pi$
$458$		0			0
$459$	10.4197	−	36.8391i	0.486349	−	1.71950i
$460$		0			0
$461$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$461$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$462$		0			0
$463$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$463$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$464$		0			0
$465$		0			0
$466$	−30.9426	+	25.9639i	−1.43339	+	1.20275i
$467$	−20.9072	+	12.0708i	−0.967470	+	0.558569i	−0.898464	−	0.439047i	$-0.855316\pi$
$467$	−20.9072	+	12.0708i	−0.967470	+	0.558569i	−0.0690063	+	0.997616i	$0.521983\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	22.2319	−	8.09174i	1.02331	−	0.372453i
$473$	13.3801	+	36.7616i	0.615219	+	1.69030i
$474$		0			0
$475$	−7.55688	−	42.8572i	−0.346733	−	1.96642i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$479$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$480$		0			0
$481$		0			0
$482$	28.0217	−	33.3950i	1.27635	−	1.52110i
$483$		0			0
$484$	−47.5140	−	17.2937i	−2.15973	−	0.786076i
$485$		0			0
$486$	12.2247	+	18.3455i	0.554526	+	0.832167i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	36.0978	+	17.0359i	1.63240	+	0.770390i
$490$		0			0
$491$	−27.2340	−	4.80209i	−1.22905	−	0.216715i	−0.478834	−	0.877905i	$-0.658940\pi$
$491$	−27.2340	−	4.80209i	−1.22905	−	0.216715i	−0.750218	+	0.661190i	$0.770052\pi$
$492$	9.70586	−	9.61669i	0.437574	−	0.433554i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	14.5278	+	31.5341i	0.651007	+	1.41308i
$499$	−1.27327	+	0.463434i	−0.0569995	+	0.0207461i	−0.370363	−	0.928887i	$-0.620767\pi$
$499$	−1.27327	+	0.463434i	−0.0569995	+	0.0207461i	0.313363	+	0.949633i	$0.398544\pi$
$500$		0			0
$501$		0			0
$502$	−7.68369	−	43.5764i	−0.342940	−	1.94491i
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−21.7761	+	5.72732i	−0.967110	+	0.254359i
$508$		0			0
$509$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$509$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$	−32.4189	+	31.5336i	−1.43133	+	1.39224i
$514$	15.0145			0.662259
$515$		0			0
$516$	−18.4891	+	12.8195i	−0.813939	+	0.564347i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	35.0511	−	20.2368i	1.53562	−	0.886588i	0.536528	−	0.843882i	$-0.319735\pi$
$521$	35.0511	−	20.2368i	1.53562	−	0.886588i	0.999088	−	0.0427062i	$-0.0135979\pi$
$522$		0			0
$523$	20.5227	−	35.5464i	0.897395	−	1.55433i	0.0665832	−	0.997781i	$-0.478790\pi$
$523$	20.5227	−	35.5464i	0.897395	−	1.55433i	0.830812	−	0.556553i	$-0.187876\pi$
$524$	−44.6306	+	7.86957i	−1.94969	+	0.343784i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	3.82894	−	41.5555i	0.166633	−	1.80847i
$529$	−3.99391	−	22.6506i	−0.173648	−	0.984808i
$530$		0			0
$531$	12.3458	+	21.8467i	0.535763	+	0.948068i
$532$		0			0
$533$		0			0
$534$	−11.8768	+	43.5205i	−0.513959	+	1.88332i
$535$		0			0
$536$	−9.58198	+	11.4194i	−0.413878	+	0.493241i
$537$	−5.61447	−	8.09757i	−0.242282	−	0.349436i
$538$		0			0
$539$		−	42.1640i		−	1.81613i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	−31.9277	−	26.7906i	−1.36889	−	1.14864i
$545$		0			0
$546$		0			0
$547$	30.8898	−	25.9196i	1.32075	−	1.10824i	0.334606	−	0.942358i	$-0.391397\pi$
$547$	30.8898	−	25.9196i	1.32075	−	1.10824i	0.986145	−	0.165883i	$-0.0530475\pi$
$548$	−23.3426	+	13.4769i	−0.997148	+	0.575704i
$549$		0			0
$550$	−21.2960	+	36.8858i	−0.908066	+	1.57282i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	7.74721	+	43.9366i	0.328555	+	1.86333i
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−54.1022	−	54.6039i	−2.28420	−	2.30538i
$562$	−0.355181	+	2.01433i	−0.0149824	+	0.0849695i
$563$	5.71019	−	6.80514i	0.240656	−	0.286803i	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	5.71019	−	6.80514i	0.240656	−	0.286803i	0.872831	+	0.488023i	$0.162282\pi$
$564$		0			0
$565$		0			0
$566$			15.6206i			0.656581i
$567$		0			0
$568$		0			0
$569$	15.4905	−	42.5597i	0.649395	−	1.78420i	0.0294311	−	0.999567i	$-0.490630\pi$
$569$	15.4905	−	42.5597i	0.649395	−	1.78420i	0.619964	−	0.784631i	$-0.287147\pi$
$570$		0			0
$571$	26.9524	+	22.6158i	1.12792	+	0.946441i	0.998978	−	0.0452101i	$-0.0143957\pi$
$571$	26.9524	+	22.6158i	1.12792	+	0.946441i	0.128947	+	0.991651i	$0.458840\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	23.6728	−	3.94925i	0.986368	−	0.164552i
$577$	23.9312	−	41.4501i	0.996270	−	1.72559i	0.423403	−	0.905941i	$-0.360835\pi$
$577$	23.9312	−	41.4501i	0.996270	−	1.72559i	0.572866	−	0.819649i	$-0.305831\pi$
$578$	−51.9277	+	9.15625i	−2.15991	+	0.380850i
$579$	14.8812	+	32.3011i	0.618441	+	1.34239i
$580$		0			0
$581$		0			0
$582$	6.49605	+	4.59340i	0.269270	+	0.190402i
$583$		0			0
$584$	−9.61398	−	5.55063i	−0.397829	−	0.229687i
$585$		0			0
$586$		0			0
$587$	21.1973	+	25.2620i	0.874907	+	1.04267i	0.998731	+	0.0503697i	$0.0160400\pi$
$587$	21.1973	+	25.2620i	0.874907	+	1.04267i	−0.123823	+	0.992304i	$0.539516\pi$
$588$	23.4512	−	6.16788i	0.967110	−	0.254359i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			7.03896i			0.289055i	0.989501	+	0.144528i	$0.0461663\pi$
$593$			7.03896i			0.289055i	−0.989501	+	0.144528i	$0.953834\pi$
$594$	44.1437	−	3.24700i	1.81124	−	0.133226i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$599$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$600$	−23.6307	−	6.44886i	−0.964721	−	0.263274i
$601$	−18.6375	+	15.6388i	−0.760241	+	0.637918i	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−18.6375	+	15.6388i	−0.760241	+	0.637918i	0.177949	+	0.984040i	$0.443054\pi$
$602$		0			0
$603$	−13.6193	−	8.03161i	−0.554622	−	0.327073i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$607$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$608$	16.8395	+	46.2661i	0.682932	+	1.87634i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	22.4559	−	38.0787i	0.907724	−	1.53924i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	−2.22927	−	2.65674i	−0.0899659	−	0.107217i
$615$		0			0
$616$		0			0
$617$	−31.8935	+	38.0092i	−1.28398	+	1.53019i	−0.603877	+	0.797077i	$0.706378\pi$
$617$	−31.8935	+	38.0092i	−1.28398	+	1.53019i	−0.680106	+	0.733114i	$0.738066\pi$
$618$		0			0
$619$	35.0195	+	12.7461i	1.40755	+	0.512308i	0.930411	−	0.366518i	$-0.119450\pi$
$619$	35.0195	+	12.7461i	1.40755	+	0.512308i	0.477143	+	0.878826i	$0.341672\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	19.1511	+	16.0697i	0.766044	+	0.642788i
$626$	−44.1342	−	7.78205i	−1.76396	−	0.311033i
$627$	23.0969	+	87.8178i	0.922401	+	3.50710i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$631$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$	29.0454	−	41.0764i	1.15445	−	1.63264i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.4299	+	27.9226i	0.925424	+	1.10288i	0.994444	+	0.105263i	$0.0335683\pi$
$641$	23.4299	+	27.9226i	0.925424	+	1.10288i	−0.0690201	+	0.997615i	$0.521987\pi$
$642$	28.5378	+	28.8024i	1.12630	+	1.13674i
$643$	5.70399	−	32.3490i	0.224944	−	1.27572i	−0.637850	−	0.770161i	$-0.720176\pi$
$643$	5.70399	−	32.3490i	0.224944	−	1.27572i	0.862793	−	0.505557i	$-0.168713\pi$
$644$		0			0
$645$		0			0
$646$	85.2202	+	31.0176i	3.35294	+	1.22037i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	8.26343	+	24.0773i	0.324618	+	0.945845i
$649$	50.3836			1.97773
$650$		0			0
$651$		0			0
$652$	35.3076	+	29.6266i	1.38275	+	1.16027i
$653$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$653$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$654$		0			0
$655$		0			0
$656$	13.6633	−	7.88848i	0.533460	−	0.307994i
$657$	4.12911	−	11.0269i	0.161092	−	0.430202i
$658$		0			0
$659$	39.0280	−	6.88169i	1.52032	−	0.268073i	0.649759	−	0.760140i	$-0.274870\pi$
$659$	39.0280	−	6.88169i	1.52032	−	0.268073i	0.870557	+	0.492068i	$0.163759\pi$
$660$		0			0
$661$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$661$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$662$	−16.9511	−	46.5728i	−0.658823	−	1.81010i
$663$		0			0
$664$	6.96170	+	39.4817i	0.270166	+	1.53219i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−46.1303	−	16.7900i	−1.77819	−	0.647209i	−0.999812	−	0.0194154i	$-0.993820\pi$
$673$	−46.1303	−	16.7900i	−1.77819	−	0.647209i	−0.778380	−	0.627793i	$-0.783958\pi$
$674$			35.0219i			1.34899i
$675$	2.62244	−	25.8481i	0.100938	−	0.994893i
$676$	−26.0000			−1.00000
$677$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$677$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$678$	−19.9920	+	13.8615i	−0.767787	+	0.532348i
$679$		0			0
$680$		0			0
$681$	21.5860	+	5.89086i	0.827179	+	0.225738i
$682$		0			0
$683$	−2.73956	+	1.58168i	−0.104826	+	0.0605215i	−0.551497	−	0.834177i	$-0.685943\pi$
$683$	−2.73956	+	1.58168i	−0.104826	+	0.0605215i	0.446670	+	0.894699i	$0.352610\pi$
$684$	−45.4646	+	25.6925i	−1.73838	+	0.982378i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−24.4124	+	8.88539i	−0.930715	+	0.338752i
$689$		0			0
$690$		0			0
$691$	0.165763	+	0.940090i	0.00630593	+	0.0357627i	0.987798	−	0.155738i	$-0.0497756\pi$
$691$	0.165763	+	0.940090i	0.00630593	+	0.0357627i	−0.981492	+	0.191501i	$0.938665\pi$
$692$		0			0
$693$		0			0
$694$	0.336667	+	0.583125i	0.0127797	+	0.0221351i
$695$		0			0
$696$		0			0
$697$	5.04630	−	28.6190i	0.191142	−	1.08402i
$698$		0			0
$699$	−28.1879	−	40.6545i	−1.06617	−	1.53770i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	16.4811	−	45.2814i	0.621154	−	1.70661i
$705$		0			0
$706$	−11.9193	−	10.0014i	−0.448587	−	0.376409i
$707$		0			0
$708$	7.37027	+	28.0228i	0.276992	+	1.05316i
$709$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$709$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$710$		0			0
$711$		0			0
$712$	−26.0454	+	45.1120i	−0.976093	+	1.69064i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−3.89148	−	10.6918i	−0.145431	−	0.399570i
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$	−51.5914	−	61.4843i	−1.92003	−	2.28821i
$723$	37.5789	+	37.9273i	1.39757	+	1.41053i
$724$		0			0
$725$		0			0
$726$	26.4303	−	56.0037i	0.980919	−	2.07849i
$727$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$727$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$728$		0			0
$729$	−23.7474	+	12.8475i	−0.879535	+	0.475834i
$730$		0			0
$731$	−16.3665	+	44.9666i	−0.605337	+	1.66315i
$732$		0			0
$733$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$733$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−27.4927	+	15.8729i	−1.01271	+	0.584686i
$738$	10.6376	+	12.9177i	0.391577	+	0.475508i
$739$	−0.612829	+	1.06145i	−0.0225433	+	0.0390461i	−0.877077	−	0.480350i	$-0.840510\pi$
$739$	−0.612829	+	1.06145i	−0.0225433	+	0.0390461i	0.854534	+	0.519396i	$0.173843\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$743$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−40.0908	+	14.1742i	−1.46685	+	0.518608i
$748$	−44.3796	−	76.8676i	−1.62268	−	2.81056i
$749$		0			0
$750$		0			0
$751$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$751$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$752$		0			0
$753$	54.0083	−	4.47408i	1.96817	−	0.163045i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−7.23358	+	19.8741i	−0.262735	+	0.721859i
$759$		0			0
$760$		0			0
$761$	−11.1418	−	1.96460i	−0.403891	−	0.0712169i	−0.0319875	−	0.999488i	$-0.510184\pi$
$761$	−11.1418	−	1.96460i	−0.403891	−	0.0712169i	−0.371903	+	0.928271i	$0.621295\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.5959	+	2.54270i	0.995782	+	0.0917517i
$769$	−51.7684	+	18.8422i	−1.86682	+	0.679466i	−0.893921	+	0.448224i	$0.852057\pi$
$769$	−51.7684	+	18.8422i	−1.86682	+	0.679466i	−0.972896	+	0.231242i	$0.925721\pi$
$770$		0			0
$771$	−1.68721	+	18.3113i	−0.0607634	+	0.659466i
$772$	7.13104	+	40.4421i	0.256652	+	1.45554i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$	−13.5567	−	23.9895i	−0.487286	−	0.862285i
$775$		0			0
$776$	5.90516	+	7.03749i	0.211983	+	0.252631i
$777$		0			0
$778$		0			0
$779$	−22.0665	+	26.2978i	−0.790614	+	0.942217i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$		0			0
$786$	−4.58232	−	55.3148i	−0.163446	−	1.97301i
$787$	−36.0389	−	30.2402i	−1.28465	−	1.07795i	−0.992586	−	0.121547i	$-0.961214\pi$
$787$	−36.0389	−	30.2402i	−1.28465	−	1.07795i	−0.292061	−	0.956400i	$-0.594341\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	50.2499	+	9.33938i	1.78555	+	0.331861i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$797$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$798$		0			0
$799$		0			0
$800$	−24.4949	−	14.1421i	−0.866025	−	0.500000i
$801$	−51.7420	−	19.3752i	−1.82821	−	0.684588i
$802$	−9.04033	−	15.6583i	−0.319225	−	0.552914i
$803$	−15.1963	−	18.1103i	−0.536267	−	0.639098i
$804$	−12.8501	−	12.9692i	−0.453186	−	0.457388i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	55.2092i		−	1.94105i	−0.240999	−	0.970525i	$-0.577475\pi$
$809$		−	55.2092i		−	1.94105i	0.240999	−	0.970525i	$-0.422525\pi$
$810$		0			0
$811$	56.3809			1.97980			0.989900	−	0.141768i	$-0.0452789\pi$
$811$	56.3809			1.97980			0.989900	+	0.141768i	$0.0452789\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	36.2610	−	35.9279i	1.26939	−	1.25773i
$817$	43.3033	−	36.3358i	1.51499	−	1.27123i
$818$	−11.1019	+	6.40970i	−0.388170	+	0.224110i
$819$		0			0
$820$		0			0
$821$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$821$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$822$	−13.8130	−	29.9826i	−0.481785	−	1.04576i
$823$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$823$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$824$		0			0
$825$	−42.5921	−	30.1171i	−1.48287	−	1.04854i
$826$		0			0
$827$	17.1464	+	9.89949i	0.596240	+	0.344239i	0.767561	−	0.640976i	$-0.221470\pi$
$827$	17.1464	+	9.89949i	0.596240	+	0.344239i	−0.171321	+	0.985215i	$0.554804\pi$
$828$		0			0
$829$		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
$833$	33.1516	−	39.5086i	1.14864	−	1.36889i
$834$	−54.4548	+	4.51107i	−1.88561	+	0.156206i
$835$		0			0
$836$			104.852i			3.62638i
$837$		0			0
$838$	3.95459			0.136609
$839$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$839$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$840$		0			0
$841$	−22.2153	−	18.6408i	−0.766044	−	0.642788i
$842$		0			0
$843$	−2.41673	−	0.659527i	−0.0832365	−	0.0227153i
$844$	44.5001	−	37.3401i	1.53176	−	1.28530i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−19.0505	−	1.75532i	−0.653812	−	0.0602425i
$850$	−48.9565	+	17.8187i	−1.67919	+	0.611176i
$851$		0			0
$852$		0			0
$853$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$853$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$854$		0			0
$855$		0			0
$856$	23.4093	+	40.5460i	0.800112	+	1.38583i
$857$	14.5446	+	17.3336i	0.496835	+	0.592105i	0.954942	−	0.296792i	$-0.0959169\pi$
$857$	14.5446	+	17.3336i	0.496835	+	0.592105i	−0.458107	+	0.888897i	$0.651472\pi$
$858$		0			0
$859$	−8.66241	+	49.1269i	−0.295557	+	1.67619i	0.369370	+	0.929282i	$0.379573\pi$
$859$	−8.66241	+	49.1269i	−0.295557	+	1.67619i	−0.664928	+	0.746908i	$0.731538\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	2.15625	+	29.3147i	0.0733571	+	0.997306i
$865$		0			0
$866$	14.4643	−	39.7403i	0.491516	−	1.35043i
$867$	−5.33153	−	64.3588i	−0.181068	−	2.18574i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−6.33199	+	7.40627i	−0.214305	+	0.250664i
$874$		0			0
$875$		0			0
$876$	7.84978	−	11.1013i	0.265219	−	0.375077i
$877$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$877$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	37.9949	+	21.9364i	1.28008	+	0.739055i	0.976863	−	0.213866i	$-0.0686057\pi$
$881$	37.9949	+	21.9364i	1.28008	+	0.739055i	0.303218	+	0.952921i	$0.401939\pi$
$882$	4.88695	+	29.2936i	0.164552	+	0.986368i
$883$	11.0972	+	19.2209i	0.373450	+	0.646834i	0.990094	−	0.140408i	$-0.0448416\pi$
$883$	11.0972	+	19.2209i	0.373450	+	0.646834i	−0.616644	+	0.787242i	$0.711508\pi$
$884$		0			0
$885$		0			0
$886$	10.0369	−	56.9219i	0.337195	−	1.91233i
$887$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$887$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.00056	+	54.2016i	−0.0335200	+	1.81582i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−24.5527	+	20.6022i	−0.819334	+	0.687503i
$899$		0			0
$900$	10.5203	−	28.0949i	0.350678	−	0.936496i
$901$		0			0
$902$	33.0882	−	5.83435i	1.10172	−	0.194263i
$903$		0			0
$904$	−26.3967	+	9.60762i	−0.877942	+	0.319545i
$905$		0			0
$906$		0			0
$907$	−7.96005	−	45.1437i	−0.264309	−	1.49897i	−0.770996	−	0.636841i	$-0.780241\pi$
$907$	−7.96005	−	45.1437i	−0.264309	−	1.49897i	0.506687	−	0.862130i	$-0.330870\pi$
$908$	22.3754	+	12.9185i	0.742554	+	0.428714i
$909$		0			0
$910$		0			0
$911$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$911$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$912$	−58.3175	+	15.3380i	−1.93108	+	0.507894i
$913$	−14.8257	+	84.0804i	−0.490657	+	2.78266i
$914$	34.4892	−	41.1027i	1.14080	−	1.35956i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	43.9166	+	31.6657i	1.44946	+	1.04512i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	3.49061	−	2.42022i	0.115019	−	0.0797490i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27.8546	−	4.91151i	0.913879	−	0.161141i	0.303115	−	0.952954i	$-0.401973\pi$
$929$	27.8546	−	4.91151i	0.913879	−	0.161141i	0.610764	+	0.791813i	$0.290862\pi$
$930$		0			0
$931$	−57.2514	+	20.8378i	−1.87634	+	0.682932i
$932$	−19.5375	−	53.6789i	−0.639972	−	1.75831i
$933$		0			0
$934$	−5.92858	−	33.6227i	−0.193989	−	1.10017i
$935$		0			0
$936$		0			0
$937$	−30.5454	−	52.9062i	−0.997875	−	1.72837i	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−30.5454	−	52.9062i	−0.997875	−	1.72837i	−0.442509	−	0.896764i	$-0.645912\pi$
$938$		0			0
$939$	14.4503	−	52.9506i	0.471567	−	1.72798i
$940$		0			0
$941$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$941$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$942$		0			0
$943$		0			0
$944$			33.4584i			1.08898i
$945$		0			0
$946$	−55.3253			−1.79878
$947$	20.7811	−	57.0955i	0.675294	−	1.85536i	0.187860	−	0.982196i	$-0.439845\pi$
$947$	20.7811	−	57.0955i	0.675294	−	1.85536i	0.487435	−	0.873160i	$-0.337933\pi$
$948$		0			0
$949$		0			0
$950$	60.6092	+	10.6870i	1.96642	+	0.346733i
$951$		0			0
$952$		0			0
$953$	−53.4028	+	30.8321i	−1.72989	+	0.998751i	−0.839964	+	0.542643i	$0.817424\pi$
$953$	−53.4028	+	30.8321i	−1.72989	+	0.998751i	−0.889924	+	0.456109i	$0.849243\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	5.38309	+	30.5290i	0.173648	+	0.984808i
$962$		0			0
$963$	−38.3336	+	31.5675i	−1.23528	+	1.01725i
$964$	30.8256	+	53.3915i	0.992826	+	1.71963i
$965$		0			0
$966$		0			0
$967$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$967$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$968$	45.9640	−	54.7778i	1.47734	−	1.76063i
$969$	−47.4048	+	100.447i	−1.52286	+	3.22683i
$970$		0			0
$971$			31.1127i			0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$			31.1127i			0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$	−30.2927	+	7.37228i	−0.971640	+	0.236466i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.82209	−	0.850265i	−0.154272	−	0.0272024i	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	−4.82209	−	0.850265i	−0.154272	−	0.0272024i	−0.250251	+	0.968181i	$0.580513\pi$
$978$	−40.0996	+	39.7312i	−1.28224	+	1.27046i
$979$	−84.9794	+	71.3062i	−2.71595	+	2.27896i
$980$		0			0
$981$		0			0
$982$	19.5544	−	33.8692i	0.624006	−	1.08081i
$983$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$983$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$984$	8.08525	+	17.5499i	0.257748	+	0.559469i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$	58.7040	−	15.4397i	1.86291	−	0.489964i
$994$		0			0
$995$		0			0
$996$	−48.9334	+	4.05368i	−1.55051	+	0.128446i
$997$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$997$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$998$		−	1.91624i		−	0.0606576i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.v.a.131.1	✓	12
3.2	odd	2		648.2.v.a.179.2		12
4.3	odd	2		864.2.bh.a.239.2		12
8.3	odd	2	CM	216.2.v.a.131.1	✓	12
8.5	even	2		864.2.bh.a.239.2		12
24.11	even	2		648.2.v.a.179.2		12
27.7	even	9		648.2.v.a.467.2		12
27.20	odd	18	inner	216.2.v.a.155.1	yes	12
108.47	even	18		864.2.bh.a.47.2		12
216.101	odd	18		864.2.bh.a.47.2		12
216.115	odd	18		648.2.v.a.467.2		12
216.155	even	18	inner	216.2.v.a.155.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.a.131.1	✓	12	1.1	even	1	trivial
216.2.v.a.131.1	✓	12	8.3	odd	2	CM
216.2.v.a.155.1	yes	12	27.20	odd	18	inner
216.2.v.a.155.1	yes	12	216.155	even	18	inner
648.2.v.a.179.2		12	3.2	odd	2
648.2.v.a.179.2		12	24.11	even	2
648.2.v.a.467.2		12	27.7	even	9
648.2.v.a.467.2		12	216.115	odd	18
864.2.bh.a.47.2		12	108.47	even	18
864.2.bh.a.47.2		12	216.101	odd	18
864.2.bh.a.239.2		12	4.3	odd	2
864.2.bh.a.239.2		12	8.5	even	2

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	216.v (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.483690 + 1.32893i) q^{2} +(-1.56638 - 0.739232i) q^{3} +(-1.53209 - 1.28558i) q^{4} +(1.74002 - 1.72404i) q^{6} +(2.44949 - 1.41421i) q^{8} +(1.90707 + 2.31583i) q^{9} +O(q^{10})\)	\(q+(-0.483690 + 1.32893i) q^{2} +(-1.56638 - 0.739232i) q^{3} +(-1.53209 - 1.28558i) q^{4} +(1.74002 - 1.72404i) q^{6} +(2.44949 - 1.41421i) q^{8} +(1.90707 + 2.31583i) q^{9} +(5.93192 - 1.04596i) q^{11} +(1.44949 + 3.14626i) q^{12} +(0.694593 + 3.93923i) q^{16} +(6.38072 + 3.68391i) q^{17} +(-4.00000 + 1.41421i) q^{18} +(-4.35183 - 7.53760i) q^{19} +(-1.47921 + 8.38900i) q^{22} +(-4.88226 + 0.404450i) q^{24} +(4.69846 + 1.71010i) q^{25} +(-1.27526 - 5.03723i) q^{27} +(-5.57091 - 0.982302i) q^{32} +(-10.0648 - 2.74670i) q^{33} +(-7.98194 + 6.69764i) q^{34} +(0.0553729 - 5.99974i) q^{36} +(12.1218 - 2.13741i) q^{38} +(-1.34901 - 3.70637i) q^{41} +(1.12781 + 6.39612i) q^{43} +(-10.4329 - 6.02343i) q^{44} +(1.82401 - 6.68378i) q^{48} +(1.21554 - 6.89365i) q^{49} +(-4.54519 + 5.41675i) q^{50} +(-7.27135 - 10.4872i) q^{51} +(7.31094 + 0.741737i) q^{54} +(1.24458 + 15.0237i) q^{57} +(8.23753 + 1.45250i) q^{59} +(4.00000 - 6.92820i) q^{64} +(8.51841 - 12.0469i) q^{66} +(-4.95255 + 1.80258i) q^{67} +(-5.03989 - 13.8470i) q^{68} +(7.94643 + 2.97560i) q^{72} +(-1.96244 - 3.39905i) q^{73} +(-6.09540 - 6.15192i) q^{75} +(-3.02275 + 17.1429i) q^{76} +(-1.72616 + 8.83292i) q^{81} +5.57800 q^{82} +(-4.84788 + 13.3194i) q^{83} +(-9.04548 - 1.59496i) q^{86} +(13.0510 - 10.9511i) q^{88} +(-15.9495 + 9.20844i) q^{89} +(8.00000 + 5.65685i) q^{96} +(0.564013 + 3.19868i) q^{97} +(8.57321 + 4.94975i) q^{98} +(13.7349 + 11.7426i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 18 q^{11} - 12 q^{12} - 48 q^{18} + 12 q^{22} - 30 q^{27} + 6 q^{33} - 24 q^{34} + 72 q^{38} - 18 q^{41} + 30 q^{43} - 12 q^{51} + 42 q^{57} + 36 q^{59} + 48 q^{64} - 42 q^{67} - 72 q^{68} - 24 q^{76} + 36 q^{86} + 48 q^{88} - 162 q^{89} + 96 q^{96} + 30 q^{97}+O(q^{100}) \) 12 * q + 18 * q^11 - 12 * q^12 - 48 * q^18 + 12 * q^22 - 30 * q^27 + 6 * q^33 - 24 * q^34 + 72 * q^38 - 18 * q^41 + 30 * q^43 - 12 * q^51 + 42 * q^57 + 36 * q^59 + 48 * q^64 - 42 * q^67 - 72 * q^68 - 24 * q^76 + 36 * q^86 + 48 * q^88 - 162 * q^89 + 96 * q^96 + 30 * q^97

Introduction

L-functions

Modular forms

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Database

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