LMFDB - Embedded newform 216.3.p.a.91.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,3,Mod(19,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4]))

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

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

chi := DirichletCharacter("216.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	216.p (of order $6$, degree $2$, 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:	$5.88557371018$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			91.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	216.91
Dual form			216.3.p.a.19.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +8.00000 q^{8} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +8.00000 q^{8} +(3.84847 + 6.66574i) q^{11} +(-8.00000 - 13.8564i) q^{16} +30.3939 q^{17} +31.6969 q^{19} +(7.69694 - 13.3315i) q^{22} +(-12.5000 - 21.6506i) q^{25} +(-16.0000 + 27.7128i) q^{32} +(-30.3939 - 52.6437i) q^{34} +(-31.6969 - 54.9007i) q^{38} +(40.8939 - 70.8303i) q^{41} +(40.2423 + 69.7018i) q^{43} -30.7878 q^{44} +(-24.5000 + 42.4352i) q^{49} +(-25.0000 + 43.3013i) q^{50} +(-16.2423 + 28.1326i) q^{59} +64.0000 q^{64} +(-35.9393 + 62.2487i) q^{67} +(-60.7878 + 105.287i) q^{68} +41.6061 q^{73} +(-63.3939 + 109.801i) q^{76} -163.576 q^{82} +(79.0000 + 136.832i) q^{83} +(80.4847 - 139.404i) q^{86} +(30.7878 + 53.3260i) q^{88} -146.000 q^{89} +(-96.9847 - 167.982i) q^{97} +98.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8	$ 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 14 q^{11} - 32 q^{16} + 4 q^{17} + 68 q^{19} - 28 q^{22} - 50 q^{25} - 64 q^{32} - 4 q^{34} - 68 q^{38} + 46 q^{41} + 14 q^{43} + 112 q^{44} - 98 q^{49} - 100 q^{50} + 82 q^{59} + 256 q^{64} + 62 q^{67} - 8 q^{68} + 284 q^{73} - 136 q^{76} - 184 q^{82} + 316 q^{83} + 28 q^{86} - 112 q^{88} - 584 q^{89} - 94 q^{97} + 392 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 - 14 * q^11 - 32 * q^16 + 4 * q^17 + 68 * q^19 - 28 * q^22 - 50 * q^25 - 64 * q^32 - 4 * q^34 - 68 * q^38 + 46 * q^41 + 14 * q^43 + 112 * q^44 - 98 * q^49 - 100 * q^50 + 82 * q^59 + 256 * q^64 + 62 * q^67 - 8 * q^68 + 284 * q^73 - 136 * q^76 - 184 * q^82 + 316 * q^83 + 28 * q^86 - 112 * q^88 - 584 * q^89 - 94 * q^97 + 392 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$55$	$109$	$137$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$2$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	8.00000			1.00000
$9$		0			0
$10$		0			0
$11$	3.84847	+	6.66574i	0.349861	+	0.605977i	0.986224	−	0.165412i	$-0.0528955\pi$
$11$	3.84847	+	6.66574i	0.349861	+	0.605977i	−0.636364	+	0.771389i	$0.719562\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$17$	30.3939			1.78788			0.893938	−	0.448192i	$-0.147932\pi$
$17$	30.3939			1.78788			0.893938	+	0.448192i	$0.147932\pi$
$18$		0			0
$19$	31.6969			1.66826			0.834130	−	0.551568i	$-0.185970\pi$
$19$	31.6969			1.66826			0.834130	+	0.551568i	$0.185970\pi$
$20$		0			0
$21$		0			0
$22$	7.69694	−	13.3315i	0.349861	−	0.605977i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−16.0000	+	27.7128i	−0.500000	+	0.866025i
$33$		0			0
$34$	−30.3939	−	52.6437i	−0.893938	−	1.54835i
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−31.6969	−	54.9007i	−0.834130	−	1.44476i
$39$		0			0
$40$		0			0
$41$	40.8939	−	70.8303i	0.997412	−	1.72757i	0.436436	−	0.899735i	$-0.356241\pi$
$41$	40.8939	−	70.8303i	0.997412	−	1.72757i	0.560976	−	0.827832i	$-0.310426\pi$
$42$		0			0
$43$	40.2423	+	69.7018i	0.935869	+	1.62097i	0.773078	+	0.634311i	$0.218716\pi$
$43$	40.2423	+	69.7018i	0.935869	+	1.62097i	0.162791	+	0.986661i	$0.447950\pi$
$44$	−30.7878			−0.699722
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	−24.5000	+	42.4352i	−0.500000	+	0.866025i
$50$	−25.0000	+	43.3013i	−0.500000	+	0.866025i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−16.2423	+	28.1326i	−0.275294	+	0.476823i	−0.970209	−	0.242268i	$-0.922109\pi$
$59$	−16.2423	+	28.1326i	−0.275294	+	0.476823i	0.694915	+	0.719092i	$0.255442\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$		0			0
$66$		0			0
$67$	−35.9393	+	62.2487i	−0.536407	+	0.929085i	0.462687	+	0.886522i	$0.346886\pi$
$67$	−35.9393	+	62.2487i	−0.536407	+	0.929085i	−0.999094	+	0.0425626i	$0.986448\pi$
$68$	−60.7878	+	105.287i	−0.893938	+	1.54835i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	41.6061			0.569947			0.284973	−	0.958535i	$-0.408015\pi$
$73$	41.6061			0.569947			0.284973	+	0.958535i	$0.408015\pi$
$74$		0			0
$75$		0			0
$76$	−63.3939	+	109.801i	−0.834130	+	1.44476i
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$		0			0
$82$	−163.576			−1.99482
$83$	79.0000	+	136.832i	0.951807	+	1.64858i	0.741511	+	0.670941i	$0.234110\pi$
$83$	79.0000	+	136.832i	0.951807	+	1.64858i	0.210296	+	0.977638i	$0.432557\pi$
$84$		0			0
$85$		0			0
$86$	80.4847	−	139.404i	0.935869	−	1.62097i
$87$		0			0
$88$	30.7878	+	53.3260i	0.349861	+	0.605977i
$89$	−146.000			−1.64045			−0.820225	−	0.572041i	$-0.806152\pi$
$89$	−146.000			−1.64045			−0.820225	+	0.572041i	$0.806152\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−96.9847	−	167.982i	−0.999842	−	1.73178i	−0.515306	−	0.857006i	$-0.672322\pi$
$97$	−96.9847	−	167.982i	−0.999842	−	1.73178i	−0.484536	−	0.874771i	$-0.661012\pi$
$98$	98.0000			1.00000
$99$		0			0
$100$	100.000			1.00000
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−191.879			−1.79326			−0.896629	−	0.442783i	$-0.853991\pi$
$107$	−191.879			−1.79326			−0.896629	+	0.442783i	$0.853991\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	49.0000	−	84.8705i	0.433628	−	0.751066i	−0.563554	−	0.826079i	$-0.690566\pi$
$113$	49.0000	−	84.8705i	0.433628	−	0.751066i	0.997183	+	0.0750128i	$0.0238998\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	64.9694			0.550588
$119$		0			0
$120$		0			0
$121$	30.8786	−	53.4833i	0.255195	−	0.442010i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−64.0000	−	110.851i	−0.500000	−	0.866025i
$129$		0			0
$130$		0			0
$131$	31.0000	−	53.6936i	0.236641	−	0.409875i	−0.723107	−	0.690736i	$-0.757287\pi$
$131$	31.0000	−	53.6936i	0.236641	−	0.409875i	0.959748	+	0.280861i	$0.0906201\pi$
$132$		0			0
$133$		0			0
$134$	143.757			1.07281
$135$		0			0
$136$	243.151			1.78788
$137$	0.712246	+	1.23365i	0.00519888	+	0.00900472i	0.868613	−	0.495491i	$-0.165012\pi$
$137$	0.712246	+	1.23365i	0.00519888	+	0.00900472i	−0.863414	+	0.504496i	$0.831678\pi$
$138$		0			0
$139$	132.333	−	229.208i	0.952037	−	1.64898i	0.211030	−	0.977480i	$-0.432318\pi$
$139$	132.333	−	229.208i	0.952037	−	1.64898i	0.741007	−	0.671497i	$-0.234348\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−41.6061	−	72.0639i	−0.284973	−	0.493588i
$147$		0			0
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	253.576			1.66826
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−322.000			−1.97546			−0.987730	−	0.156171i	$-0.950085\pi$
$163$	−322.000			−1.97546			−0.987730	+	0.156171i	$0.950085\pi$
$164$	163.576	+	283.321i	0.997412	+	1.72757i
$165$		0			0
$166$	158.000	−	273.664i	0.951807	−	1.64858i
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−84.5000	−	146.358i	−0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$	−321.939			−1.87174
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$	61.5755	−	106.652i	0.349861	−	0.605977i
$177$		0			0
$178$	146.000	+	252.879i	0.820225	+	1.42067i
$179$	34.0000			0.189944			0.0949721	−	0.995480i	$-0.469724\pi$
$179$	34.0000			0.189944			0.0949721	+	0.995480i	$0.469724\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	116.970	+	202.598i	0.625507	+	1.08341i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−137.166	+	237.579i	−0.710706	+	1.23098i	0.253886	+	0.967234i	$0.418291\pi$
$193$	−137.166	+	237.579i	−0.710706	+	1.23098i	−0.964592	+	0.263745i	$0.915042\pi$
$194$	−193.969	+	335.965i	−0.999842	+	1.73178i
$195$		0			0
$196$	−98.0000	−	169.741i	−0.500000	−	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−100.000	−	173.205i	−0.500000	−	0.866025i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	121.985	+	211.284i	0.583659	+	1.01093i
$210$		0			0
$211$	113.000	−	195.722i	0.535545	−	0.927591i	−0.463592	−	0.886049i	$-0.653440\pi$
$211$	113.000	−	195.722i	0.535545	−	0.927591i	0.999137	−	0.0415423i	$-0.0132271\pi$
$212$		0			0
$213$		0			0
$214$	191.879	+	332.343i	0.896629	+	1.55301i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		0			0
$225$		0			0
$226$	−196.000			−0.867257
$227$	−148.242	−	256.763i	−0.653050	−	1.13112i	−0.982379	−	0.186900i	$-0.940156\pi$
$227$	−148.242	−	256.763i	−0.653050	−	1.13112i	0.329329	−	0.944215i	$-0.393178\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	70.0306			0.300561			0.150280	−	0.988643i	$-0.451982\pi$
$233$	70.0306			0.300561			0.150280	+	0.988643i	$0.451982\pi$
$234$		0			0
$235$		0			0
$236$	−64.9694	−	112.530i	−0.275294	−	0.476823i
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	239.560	+	414.930i	0.994026	+	1.72170i	0.591536	+	0.806279i	$0.298522\pi$
$241$	239.560	+	414.930i	0.994026	+	1.72170i	0.402490	+	0.915425i	$0.368145\pi$
$242$	−123.514			−0.510390
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−71.3337			−0.284198			−0.142099	−	0.989852i	$-0.545385\pi$
$251$	−71.3337			−0.284198			−0.142099	+	0.989852i	$0.545385\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$	−243.469	+	421.701i	−0.947352	+	1.64086i	−0.196379	+	0.980528i	$0.562918\pi$
$257$	−243.469	+	421.701i	−0.947352	+	1.64086i	−0.750973	+	0.660333i	$0.770415\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−124.000			−0.473282
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−143.757	−	248.995i	−0.536407	−	0.929085i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−243.151	−	421.150i	−0.893938	−	1.54835i
$273$		0			0
$274$	1.42449	−	2.46729i	0.00519888	−	0.00900472i
$275$	96.2117	−	166.644i	0.349861	−	0.605977i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$	−529.333			−1.90407
$279$		0			0
$280$		0			0
$281$	−119.000	−	206.114i	−0.423488	−	0.733502i	0.572790	−	0.819702i	$-0.305861\pi$
$281$	−119.000	−	206.114i	−0.423488	−	0.733502i	−0.996278	+	0.0862000i	$0.972528\pi$
$282$		0			0
$283$	41.0000	−	71.0141i	0.144876	−	0.250933i	−0.784450	−	0.620191i	$-0.787055\pi$
$283$	41.0000	−	71.0141i	0.144876	−	0.250933i	0.929327	+	0.369258i	$0.120388\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	634.788			2.19650
$290$		0			0
$291$		0			0
$292$	−83.2122	+	144.128i	−0.284973	+	0.493588i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−253.576	−	439.206i	−0.834130	−	1.44476i
$305$		0			0
$306$		0			0
$307$	−520.848			−1.69657			−0.848287	−	0.529537i	$-0.822366\pi$
$307$	−520.848			−1.69657			−0.848287	+	0.529537i	$0.822366\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	15.4694	+	26.7938i	0.0494230	+	0.0856031i	0.889679	−	0.456587i	$-0.150928\pi$
$313$	15.4694	+	26.7938i	0.0494230	+	0.0856031i	−0.840256	+	0.542191i	$0.817595\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	963.393			2.98264
$324$		0			0
$325$		0			0
$326$	322.000	+	557.720i	0.987730	+	1.71080i
$327$		0			0
$328$	327.151	−	566.642i	0.997412	−	1.72757i
$329$		0			0
$330$		0			0
$331$	−7.00000	−	12.1244i	−0.0211480	−	0.0366295i	0.855258	−	0.518203i	$-0.173399\pi$
$331$	−7.00000	−	12.1244i	−0.0211480	−	0.0366295i	−0.876406	+	0.481573i	$0.840065\pi$
$332$	−632.000			−1.90361
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−325.257	+	563.362i	−0.965155	+	1.67170i	−0.255956	+	0.966688i	$0.582390\pi$
$337$	−325.257	+	563.362i	−0.965155	+	1.67170i	−0.709199	+	0.705009i	$0.750943\pi$
$338$	−169.000	+	292.717i	−0.500000	+	0.866025i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	321.939	+	557.614i	0.935869	+	1.62097i
$345$		0			0
$346$		0			0
$347$	260.030	−	450.385i	0.749366	−	1.29794i	−0.198761	−	0.980048i	$-0.563692\pi$
$347$	260.030	−	450.385i	0.749366	−	1.29794i	0.948127	−	0.317892i	$-0.102975\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$	−246.302			−0.699722
$353$	245.439	+	425.112i	0.695294	+	1.20428i	0.970081	+	0.242779i	$0.0780591\pi$
$353$	245.439	+	425.112i	0.695294	+	1.20428i	−0.274788	+	0.961505i	$0.588608\pi$
$354$		0			0
$355$		0			0
$356$	292.000	−	505.759i	0.820225	−	1.42067i
$357$		0			0
$358$	−34.0000	−	58.8897i	−0.0949721	−	0.164496i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	643.696			1.78309
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$		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	$0.666667\pi$
$374$	233.940	−	405.196i	0.625507	−	1.08341i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−63.7582			−0.168227			−0.0841137	−	0.996456i	$-0.526806\pi$
$379$	−63.7582			−0.168227			−0.0841137	+	0.996456i	$0.526806\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$	548.665			1.42141
$387$		0			0
$388$	775.878			1.99968
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	−196.000	+	339.482i	−0.500000	+	0.866025i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−200.000	+	346.410i	−0.500000	+	0.866025i
$401$	88.6214	−	153.497i	0.221001	−	0.382785i	−0.734111	−	0.679029i	$-0.762401\pi$
$401$	88.6214	−	153.497i	0.221001	−	0.382785i	0.955112	+	0.296244i	$0.0957342\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	239.833	−	415.402i	0.586388	−	1.01565i	−0.408313	−	0.912842i	$-0.633883\pi$
$409$	239.833	−	415.402i	0.586388	−	1.01565i	0.994701	−	0.102812i	$-0.0327839\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	243.969	−	422.567i	0.583659	−	1.01093i
$419$	−257.000	+	445.137i	−0.613365	+	1.06238i	0.377304	+	0.926090i	$0.376851\pi$
$419$	−257.000	+	445.137i	−0.613365	+	1.06238i	−0.990669	+	0.136290i	$0.956482\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$	−452.000			−1.07109
$423$		0			0
$424$		0			0
$425$	−379.923	−	658.047i	−0.893938	−	1.54835i
$426$		0			0
$427$		0			0
$428$	383.757	−	664.687i	0.896629	−	1.55301i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−847.484			−1.95724			−0.978619	−	0.205684i	$-0.934058\pi$
$433$	−847.484			−1.95724			−0.978619	+	0.205684i	$0.934058\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−168.061	−	291.090i	−0.379370	−	0.657087i	0.611601	−	0.791166i	$-0.290526\pi$
$443$	−168.061	−	291.090i	−0.379370	−	0.657087i	−0.990971	+	0.134079i	$0.957192\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	638.757			1.42262			0.711311	−	0.702878i	$-0.248102\pi$
$449$	638.757			1.42262			0.711311	+	0.702878i	$0.248102\pi$
$450$		0			0
$451$	629.515			1.39582
$452$	196.000	+	339.482i	0.433628	+	0.751066i
$453$		0			0
$454$	−296.485	+	513.527i	−0.653050	+	1.13112i
$455$		0			0
$456$		0			0
$457$	−441.620	−	764.909i	−0.966347	−	1.67376i	−0.705953	−	0.708259i	$-0.749481\pi$
$457$	−441.620	−	764.909i	−0.966347	−	1.67376i	−0.260394	−	0.965502i	$-0.583852\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$		0			0
$465$		0			0
$466$	−70.0306	−	121.297i	−0.150280	−	0.260293i
$467$	−825.332			−1.76731			−0.883653	−	0.468143i	$-0.844923\pi$
$467$	−825.332			−1.76731			−0.883653	+	0.468143i	$0.844923\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−129.939	+	225.061i	−0.275294	+	0.476823i
$473$	−309.743	+	536.490i	−0.654847	+	1.13423i
$474$		0			0
$475$	−396.212	−	686.259i	−0.834130	−	1.44476i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$	479.120	−	829.861i	0.994026	−	1.72170i
$483$		0			0
$484$	123.514	+	213.933i	0.255195	+	0.442010i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−452.696	+	784.093i	−0.921989	+	1.59693i	−0.125655	+	0.992074i	$0.540103\pi$
$491$	−452.696	+	784.093i	−0.921989	+	1.59693i	−0.796334	+	0.604857i	$0.793230\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	56.6964	−	98.2011i	0.113620	−	0.196796i	−0.803607	−	0.595160i	$-0.797089\pi$
$499$	56.6964	−	98.2011i	0.113620	−	0.196796i	0.917227	+	0.398364i	$0.130422\pi$
$500$		0			0
$501$		0			0
$502$	71.3337	+	123.554i	0.142099	+	0.246123i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	512.000			1.00000
$513$		0			0
$514$	973.878			1.89470
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−738.151			−1.41680			−0.708398	−	0.705813i	$-0.750582\pi$
$521$	−738.151			−1.41680			−0.708398	+	0.705813i	$0.750582\pi$
$522$		0			0
$523$	398.000			0.760994			0.380497	−	0.924782i	$-0.375753\pi$
$523$	398.000			0.760994			0.380497	+	0.924782i	$0.375753\pi$
$524$	124.000	+	214.774i	0.236641	+	0.409875i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	−	458.127i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−287.514	+	497.989i	−0.536407	+	0.929085i
$537$		0			0
$538$		0			0
$539$	−377.150			−0.699722
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	−486.302	+	842.300i	−0.893938	+	1.54835i
$545$		0			0
$546$		0			0
$547$	424.515	+	735.281i	0.776078	+	1.34421i	0.934186	+	0.356785i	$0.116127\pi$
$547$	424.515	+	735.281i	0.776078	+	1.34421i	−0.158108	+	0.987422i	$0.550539\pi$
$548$	−5.69797			−0.0103978
$549$		0			0
$550$	−384.847			−0.699722
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	529.333	+	916.831i	0.952037	+	1.64898i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−238.000	+	412.228i	−0.423488	+	0.733502i
$563$	−421.150	+	729.454i	−0.748047	+	1.29566i	0.200710	+	0.979651i	$0.435675\pi$
$563$	−421.150	+	729.454i	−0.748047	+	1.29566i	−0.948758	+	0.316005i	$0.897658\pi$
$564$		0			0
$565$		0			0
$566$	−164.000			−0.289753
$567$		0			0
$568$		0			0
$569$	−568.014	−	983.830i	−0.998268	−	1.72905i	−0.550088	−	0.835107i	$-0.685406\pi$
$569$	−568.014	−	983.830i	−0.998268	−	1.72905i	−0.448180	−	0.893943i	$-0.647928\pi$
$570$		0			0
$571$	−568.666	+	984.958i	−0.995912	+	1.72497i	−0.419730	+	0.907649i	$0.637875\pi$
$571$	−568.666	+	984.958i	−0.995912	+	1.72497i	−0.576182	+	0.817321i	$0.695458\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1000.39			−1.73378			−0.866891	−	0.498498i	$-0.833885\pi$
$577$	−1000.39			−1.73378			−0.866891	+	0.498498i	$0.833885\pi$
$578$	−634.788	−	1099.48i	−1.09825	−	1.90222i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	332.849			0.569947
$585$		0			0
$586$		0			0
$587$	159.576	+	276.394i	0.271850	+	0.470858i	0.969336	−	0.245741i	$-0.0790312\pi$
$587$	159.576	+	276.394i	0.271850	+	0.470858i	−0.697485	+	0.716599i	$0.745698\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	862.000			1.45363			0.726813	−	0.686836i	$-0.241001\pi$
$593$	862.000			1.45363			0.726813	+	0.686836i	$0.241001\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	−109.530	−	189.711i	−0.182246	−	0.315659i	0.760399	−	0.649456i	$-0.225003\pi$
$601$	−109.530	−	189.711i	−0.182246	−	0.315659i	−0.942645	+	0.333797i	$0.891670\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−507.151	+	878.411i	−0.834130	+	1.44476i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	520.848	+	902.135i	0.848287	+	1.46928i
$615$		0			0
$616$		0			0
$617$	597.893	−	1035.58i	0.969032	−	1.67841i	0.270665	−	0.962674i	$-0.412757\pi$
$617$	597.893	−	1035.58i	0.969032	−	1.67841i	0.698368	−	0.715739i	$-0.253910\pi$
$618$		0			0
$619$	337.150	+	583.962i	0.544670	+	0.943395i	0.998628	+	0.0523724i	$0.0166783\pi$
$619$	337.150	+	583.962i	0.544670	+	0.943395i	−0.453958	+	0.891023i	$0.649988\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	+	541.266i	−0.500000	+	0.866025i
$626$	30.9388	−	53.5875i	0.0494230	−	0.0856031i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		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
$641$	393.893	+	682.242i	0.614497	+	1.06434i	0.990472	+	0.137711i	$0.0439745\pi$
$641$	393.893	+	682.242i	0.614497	+	1.06434i	−0.375975	+	0.926630i	$0.622692\pi$
$642$		0			0
$643$	119.788	−	207.479i	0.186296	−	0.322674i	−0.757717	−	0.652584i	$-0.773685\pi$
$643$	119.788	−	207.479i	0.186296	−	0.322674i	0.944012	+	0.329910i	$0.107018\pi$
$644$		0			0
$645$		0			0
$646$	−963.393	−	1668.65i	−1.49132	−	2.58304i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−250.033			−0.385258
$650$		0			0
$651$		0			0
$652$	644.000	−	1115.44i	0.987730	−	1.71080i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$	−1308.60			−1.99482
$657$		0			0
$658$		0			0
$659$	−497.000	−	860.829i	−0.754173	−	1.30627i	−0.945784	−	0.324795i	$-0.894705\pi$
$659$	−497.000	−	860.829i	−0.754173	−	1.30627i	0.191611	−	0.981471i	$-0.438629\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	−14.0000	+	24.2487i	−0.0211480	+	0.0366295i
$663$		0			0
$664$	632.000	+	1094.66i	0.951807	+	1.64858i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	623.000	+	1079.07i	0.925706	+	1.60337i	0.790422	+	0.612563i	$0.209861\pi$
$673$	623.000	+	1079.07i	0.925706	+	1.60337i	0.135284	+	0.990807i	$0.456805\pi$
$674$	1301.03			1.93031
$675$		0			0
$676$	676.000			1.00000
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1330.66			1.94826			0.974132	−	0.225980i	$-0.0725584\pi$
$683$	1330.66			1.94826			0.974132	+	0.225980i	$0.0725584\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	643.878	−	1115.23i	0.935869	−	1.62097i
$689$		0			0
$690$		0			0
$691$	−367.000	−	635.663i	−0.531114	−	0.919917i	−0.999341	−	0.0363084i	$-0.988440\pi$
$691$	−367.000	−	635.663i	−0.531114	−	0.919917i	0.468226	−	0.883609i	$-0.344893\pi$
$692$		0			0
$693$		0			0
$694$	−1040.12			−1.49873
$695$		0			0
$696$		0			0
$697$	1242.92	−	2152.81i	1.78325	−	3.08868i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	246.302	+	426.608i	0.349861	+	0.605977i
$705$		0			0
$706$	490.878	−	850.225i	0.695294	−	1.20428i
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$	−1168.00			−1.64045
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−68.0000	+	117.779i	−0.0949721	+	0.164496i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−643.696	−	1114.91i	−0.891546	−	1.54420i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1223.12	+	2118.51i	1.67322	+	2.89810i
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−553.245			−0.750672
$738$		0			0
$739$	1410.24			1.90831			0.954154	−	0.299316i	$-0.0967584\pi$
$739$	1410.24			1.90831			0.954154	+	0.299316i	$0.0967584\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	−935.759			−1.25101
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	63.7582	+	110.432i	0.0841137	+	0.145689i
$759$		0			0
$760$		0			0
$761$	697.000	−	1207.24i	0.915900	−	1.58639i	0.110322	−	0.993896i	$-0.464812\pi$
$761$	697.000	−	1207.24i	0.915900	−	1.58639i	0.805578	−	0.592490i	$-0.201855\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	527.000	−	912.791i	0.685306	−	1.18698i	−0.288035	−	0.957620i	$-0.593002\pi$
$769$	527.000	−	912.791i	0.685306	−	1.18698i	0.973341	−	0.229364i	$-0.0736647\pi$
$770$		0			0
$771$		0			0
$772$	−548.665	−	950.316i	−0.710706	−	1.23098i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$	−775.878	−	1343.86i	−0.999842	−	1.73178i
$777$		0			0
$778$		0			0
$779$	1296.21	−	2245.10i	1.66394	−	2.88203i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	784.000			1.00000
$785$		0			0
$786$		0			0
$787$	−463.000	+	801.940i	−0.588310	+	1.01898i	0.406144	+	0.913809i	$0.366873\pi$
$787$	−463.000	+	801.940i	−0.588310	+	1.01898i	−0.994454	+	0.105174i	$0.966460\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$	800.000			1.00000
$801$		0			0
$802$	−354.486			−0.442002
$803$	160.120	+	277.336i	0.199402	+	0.345375i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−497.061			−0.614414			−0.307207	−	0.951643i	$-0.599394\pi$
$809$	−497.061			−0.614414			−0.307207	+	0.951643i	$0.599394\pi$
$810$		0			0
$811$	−1307.21			−1.61185			−0.805924	−	0.592019i	$-0.798331\pi$
$811$	−1307.21			−1.61185			−0.805924	+	0.592019i	$0.798331\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1275.56	+	2209.33i	1.56127	+	2.70420i
$818$	−959.331			−1.17278
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1262.00			−1.52600			−0.762999	−	0.646400i	$-0.776274\pi$
$827$	−1262.00			−1.52600			−0.762999	+	0.646400i	$0.776274\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−744.650	+	1289.77i	−0.893938	+	1.54835i
$834$		0			0
$835$		0			0
$836$	−975.878			−1.16732
$837$		0			0
$838$	1028.00			1.22673
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−420.500	+	728.327i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	452.000	+	782.887i	0.535545	+	0.927591i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$	−759.847	+	1316.09i	−0.893938	+	1.54835i
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	−1535.03			−1.79326
$857$	601.000	+	1040.96i	0.701284	+	1.21466i	0.968016	+	0.250888i	$0.0807226\pi$
$857$	601.000	+	1040.96i	0.701284	+	1.21466i	−0.266733	+	0.963771i	$0.585944\pi$
$858$		0			0
$859$	624.606	−	1081.85i	0.727131	−	1.25943i	−0.230960	−	0.972963i	$-0.574187\pi$
$859$	624.606	−	1081.85i	0.727131	−	1.25943i	0.958091	−	0.286465i	$-0.0924801\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$	847.484	+	1467.88i	0.978619	+	1.69502i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1438.00			1.63224			0.816118	−	0.577885i	$-0.196122\pi$
$881$	1438.00			1.63224			0.816118	+	0.577885i	$0.196122\pi$
$882$		0			0
$883$	983.879			1.11425			0.557123	−	0.830430i	$-0.311905\pi$
$883$	983.879			1.11425			0.557123	+	0.830430i	$0.311905\pi$
$884$		0			0
$885$		0			0
$886$	−336.121	+	582.179i	−0.379370	+	0.657087i
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−638.757	−	1106.36i	−0.711311	−	1.23203i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−629.515	−	1090.35i	−0.697911	−	1.20882i
$903$		0			0
$904$	392.000	−	678.964i	0.433628	−	0.751066i
$905$		0			0
$906$		0			0
$907$	−171.304	−	296.706i	−0.188868	−	0.327130i	0.756005	−	0.654566i	$-0.227149\pi$
$907$	−171.304	−	296.706i	−0.188868	−	0.327130i	−0.944873	+	0.327436i	$0.893815\pi$
$908$	1185.94			1.30610
$909$		0			0
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$	−608.058	+	1053.19i	−0.666000	+	1.15355i
$914$	−883.241	+	1529.82i	−0.966347	+	1.67376i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		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
$929$	529.000	+	916.255i	0.569429	+	0.986281i	0.996622	+	0.0821203i	$0.0261692\pi$
$929$	529.000	+	916.255i	0.569429	+	0.986281i	−0.427193	+	0.904160i	$0.640497\pi$
$930$		0			0
$931$	−776.575	+	1345.07i	−0.834130	+	1.44476i
$932$	−140.061	+	242.593i	−0.150280	+	0.260293i
$933$		0			0
$934$	825.332	+	1429.52i	0.883653	+	1.53053i
$935$		0			0
$936$		0			0
$937$	−718.000			−0.766275			−0.383138	−	0.923691i	$-0.625157\pi$
$937$	−718.000			−0.766275			−0.383138	+	0.923691i	$0.625157\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	519.755			0.550588
$945$		0			0
$946$	1238.97			1.30969
$947$	−449.605	−	778.738i	−0.474767	−	0.822321i	0.524815	−	0.851216i	$-0.324134\pi$
$947$	−449.605	−	778.738i	−0.474767	−	0.822321i	−0.999582	+	0.0288952i	$0.990801\pi$
$948$		0			0
$949$		0			0
$950$	−792.423	+	1372.52i	−0.834130	+	1.44476i
$951$		0			0
$952$		0			0
$953$	−1717.06			−1.80174			−0.900869	−	0.434090i	$-0.857070\pi$
$953$	−1717.06			−1.80174			−0.900869	+	0.434090i	$0.857070\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−480.500	−	832.250i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$	−1916.48			−1.98805
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$	247.029	−	427.866i	0.255195	−	0.442010i
$969$		0			0
$970$		0			0
$971$	−974.000			−1.00309			−0.501545	−	0.865132i	$-0.667235\pi$
$971$	−974.000			−1.00309			−0.501545	+	0.865132i	$0.667235\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	641.166	−	1110.53i	0.656260	−	1.13668i	−0.325316	−	0.945605i	$-0.605471\pi$
$977$	641.166	−	1110.53i	0.656260	−	1.13668i	0.981576	−	0.191071i	$-0.0611960\pi$
$978$		0			0
$979$	−561.877	−	973.199i	−0.573929	−	0.994074i
$980$		0			0
$981$		0			0
$982$	1810.79			1.84398
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	−226.786			−0.227240
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.p.a.91.2		4
3.2	odd	2		72.3.p.a.67.2	yes	4
4.3	odd	2		864.3.t.a.847.1		4
8.3	odd	2	CM	216.3.p.a.91.2		4
8.5	even	2		864.3.t.a.847.1		4
9.2	odd	6		72.3.p.a.43.2	✓	4
9.4	even	3		648.3.b.b.163.1		2
9.5	odd	6		648.3.b.a.163.2		2
9.7	even	3	inner	216.3.p.a.19.2		4
12.11	even	2		288.3.t.a.175.1		4
24.5	odd	2		288.3.t.a.175.1		4
24.11	even	2		72.3.p.a.67.2	yes	4
36.7	odd	6		864.3.t.a.559.1		4
36.11	even	6		288.3.t.a.79.1		4
36.23	even	6		2592.3.b.b.1135.1		2
36.31	odd	6		2592.3.b.a.1135.2		2
72.5	odd	6		2592.3.b.b.1135.1		2
72.11	even	6		72.3.p.a.43.2	✓	4
72.13	even	6		2592.3.b.a.1135.2		2
72.29	odd	6		288.3.t.a.79.1		4
72.43	odd	6	inner	216.3.p.a.19.2		4
72.59	even	6		648.3.b.a.163.2		2
72.61	even	6		864.3.t.a.559.1		4
72.67	odd	6		648.3.b.b.163.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.p.a.43.2	✓	4	9.2	odd	6
72.3.p.a.43.2	✓	4	72.11	even	6
72.3.p.a.67.2	yes	4	3.2	odd	2
72.3.p.a.67.2	yes	4	24.11	even	2
216.3.p.a.19.2		4	9.7	even	3	inner
216.3.p.a.19.2		4	72.43	odd	6	inner
216.3.p.a.91.2		4	1.1	even	1	trivial
216.3.p.a.91.2		4	8.3	odd	2	CM
288.3.t.a.79.1		4	36.11	even	6
288.3.t.a.79.1		4	72.29	odd	6
288.3.t.a.175.1		4	12.11	even	2
288.3.t.a.175.1		4	24.5	odd	2
648.3.b.a.163.2		2	9.5	odd	6
648.3.b.a.163.2		2	72.59	even	6
648.3.b.b.163.1		2	9.4	even	3
648.3.b.b.163.1		2	72.67	odd	6
864.3.t.a.559.1		4	36.7	odd	6
864.3.t.a.559.1		4	72.61	even	6
864.3.t.a.847.1		4	4.3	odd	2
864.3.t.a.847.1		4	8.5	even	2
2592.3.b.a.1135.2		2	36.31	odd	6
2592.3.b.a.1135.2		2	72.13	even	6
2592.3.b.b.1135.1		2	36.23	even	6
2592.3.b.b.1135.1		2	72.5	odd	6

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +8.00000 q^{8} +(3.84847 + 6.66574i) q^{11} +(-8.00000 - 13.8564i) q^{16} +30.3939 q^{17} +31.6969 q^{19} +(7.69694 - 13.3315i) q^{22} +(-12.5000 - 21.6506i) q^{25} +(-16.0000 + 27.7128i) q^{32} +(-30.3939 - 52.6437i) q^{34} +(-31.6969 - 54.9007i) q^{38} +(40.8939 - 70.8303i) q^{41} +(40.2423 + 69.7018i) q^{43} -30.7878 q^{44} +(-24.5000 + 42.4352i) q^{49} +(-25.0000 + 43.3013i) q^{50} +(-16.2423 + 28.1326i) q^{59} +64.0000 q^{64} +(-35.9393 + 62.2487i) q^{67} +(-60.7878 + 105.287i) q^{68} +41.6061 q^{73} +(-63.3939 + 109.801i) q^{76} -163.576 q^{82} +(79.0000 + 136.832i) q^{83} +(80.4847 - 139.404i) q^{86} +(30.7878 + 53.3260i) q^{88} -146.000 q^{89} +(-96.9847 - 167.982i) q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8	\( 4 q - 4 q^{2} - 8 q^{4} + 32 q^{8} - 14 q^{11} - 32 q^{16} + 4 q^{17} + 68 q^{19} - 28 q^{22} - 50 q^{25} - 64 q^{32} - 4 q^{34} - 68 q^{38} + 46 q^{41} + 14 q^{43} + 112 q^{44} - 98 q^{49} - 100 q^{50} + 82 q^{59} + 256 q^{64} + 62 q^{67} - 8 q^{68} + 284 q^{73} - 136 q^{76} - 184 q^{82} + 316 q^{83} + 28 q^{86} - 112 q^{88} - 584 q^{89} - 94 q^{97} + 392 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 8 * q^4 + 32 * q^8 - 14 * q^11 - 32 * q^16 + 4 * q^17 + 68 * q^19 - 28 * q^22 - 50 * q^25 - 64 * q^32 - 4 * q^34 - 68 * q^38 + 46 * q^41 + 14 * q^43 + 112 * q^44 - 98 * q^49 - 100 * q^50 + 82 * q^59 + 256 * q^64 + 62 * q^67 - 8 * q^68 + 284 * q^73 - 136 * q^76 - 184 * q^82 + 316 * q^83 + 28 * q^86 - 112 * q^88 - 584 * q^89 - 94 * q^97 + 392 * q^98

Introduction

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