LMFDB - Embedded newform 864.2.bh.a.527.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(47,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 7]))

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

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

chi := DirichletCharacter("864.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.bh (of order $18$, degree $6$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.89907473464$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			527.1
Root			$-0.483690 + 1.32893i$ of defining polynomial
Character	$\chi$	$=$	864.527
Dual form			864.2.bh.a.623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.142995 - 1.72614i) q^{3} +(-2.95911 + 0.493657i) q^{9} +O(q^{10})$	$q+(-0.142995 - 1.72614i) q^{3} +(-2.95911 + 0.493657i) q^{9} +(-0.389357 - 0.464017i) q^{11} +(-5.96791 - 3.44557i) q^{17} +(-1.96107 - 3.39668i) q^{19} +(-0.868241 - 4.92404i) q^{25} +(1.27526 + 5.03723i) q^{27} +(-0.745282 + 0.738435i) q^{33} +(-12.3333 - 2.17469i) q^{41} +(-5.07113 + 4.25518i) q^{43} +(5.36231 + 4.49951i) q^{49} +(-5.09416 + 10.7941i) q^{51} +(-5.58271 + 3.87079i) q^{57} +(4.48459 - 5.34452i) q^{59} +(2.78841 - 15.8139i) q^{67} +(8.18272 + 14.1729i) q^{73} +(-8.37542 + 2.20281i) q^{75} +(8.51261 - 2.92156i) q^{81} +(-13.9589 + 2.46133i) q^{83} +(-15.9495 + 9.20844i) q^{89} +(11.6448 - 9.77114i) q^{97} +(1.38121 + 1.18087i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 18 q^{11} + 30 q^{27} + 6 q^{33} - 18 q^{41} - 30 q^{43} + 12 q^{51} + 42 q^{57} - 36 q^{59} + 42 q^{67} - 162 q^{89} + 30 q^{97}+O(q^{100}) $ 12 * q - 18 * q^11 + 30 * q^27 + 6 * q^33 - 18 * q^41 - 30 * q^43 + 12 * q^51 + 42 * q^57 - 36 * q^59 + 42 * q^67 - 162 * q^89 + 30 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{17}{18}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.142995	−	1.72614i	−0.0825579	−	0.996586i
$3$	−0.142995	−	1.72614i	−0.0825579	−	0.996586i
$4$		0			0
$5$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$5$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$6$		0			0
$7$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$7$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$8$		0			0
$9$	−2.95911	+	0.493657i	−0.986368	+	0.164552i
$10$		0			0
$11$	−0.389357	−	0.464017i	−0.117395	−	0.139906i	0.704146	−	0.710055i	$-0.251330\pi$
$11$	−0.389357	−	0.464017i	−0.117395	−	0.139906i	−0.821541	+	0.570149i	$0.806886\pi$
$12$		0			0
$13$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$13$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.96791	−	3.44557i	−1.44743	−	0.835675i	−0.449103	−	0.893480i	$-0.648256\pi$
$17$	−5.96791	−	3.44557i	−1.44743	−	0.835675i	−0.998328	+	0.0578055i	$0.981590\pi$
$18$		0			0
$19$	−1.96107	−	3.39668i	−0.449901	−	0.779251i	0.548478	−	0.836165i	$-0.315207\pi$
$19$	−1.96107	−	3.39668i	−0.449901	−	0.779251i	−0.998379	+	0.0569137i	$0.981874\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$23$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$24$		0			0
$25$	−0.868241	−	4.92404i	−0.173648	−	0.984808i
$26$		0			0
$27$	1.27526	+	5.03723i	0.245423	+	0.969416i
$28$		0			0
$29$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$29$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$30$		0			0
$31$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$31$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$32$		0			0
$33$	−0.745282	+	0.738435i	−0.129737	+	0.128545i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−12.3333	−	2.17469i	−1.92613	−	0.339629i	−0.926780	−	0.375604i	$-0.877435\pi$
$41$	−12.3333	−	2.17469i	−1.92613	−	0.339629i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	−5.07113	+	4.25518i	−0.773340	+	0.648909i	−0.941562	−	0.336840i	$-0.890642\pi$
$43$	−5.07113	+	4.25518i	−0.773340	+	0.648909i	0.168222	+	0.985749i	$0.446197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$47$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$48$		0			0
$49$	5.36231	+	4.49951i	0.766044	+	0.642788i
$50$		0			0
$51$	−5.09416	+	10.7941i	−0.713325	+	1.51148i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.58271	+	3.87079i	−0.739448	+	0.512698i
$58$		0			0
$59$	4.48459	−	5.34452i	0.583843	−	0.695798i	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	4.48459	−	5.34452i	0.583843	−	0.695798i	0.974410	+	0.224777i	$0.0721654\pi$
$60$		0			0
$61$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$61$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.78841	−	15.8139i	0.340659	−	1.93197i	−0.0212861	−	0.999773i	$-0.506776\pi$
$67$	2.78841	−	15.8139i	0.340659	−	1.93197i	0.361945	−	0.932199i	$-0.382113\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$		0			0
$73$	8.18272	+	14.1729i	0.957715	+	1.65881i	0.728028	+	0.685547i	$0.240437\pi$
$73$	8.18272	+	14.1729i	0.957715	+	1.65881i	0.229687	+	0.973265i	$0.426230\pi$
$74$		0			0
$75$	−8.37542	+	2.20281i	−0.967110	+	0.254359i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$79$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$80$		0			0
$81$	8.51261	−	2.92156i	0.945845	−	0.324618i
$82$		0			0
$83$	−13.9589	+	2.46133i	−1.53219	+	0.270166i	−0.875210	−	0.483743i	$-0.839277\pi$
$83$	−13.9589	+	2.46133i	−1.53219	+	0.270166i	−0.656979	+	0.753909i	$0.728166\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−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
$96$		0			0
$97$	11.6448	−	9.77114i	1.18235	−	0.992109i	0.182389	−	0.983226i	$-0.441617\pi$
$97$	11.6448	−	9.77114i	1.18235	−	0.992109i	0.999961	−	0.00888289i	$-0.00282755\pi$
$98$		0			0
$99$	1.38121	+	1.18087i	0.138817	+	0.118682i
$100$		0			0
$101$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$101$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$102$		0			0
$103$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$103$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.47079i		−	0.238860i	−0.992843	−	0.119430i	$-0.961893\pi$
$107$		−	2.47079i		−	0.238860i	0.992843	−	0.119430i	$-0.0381067\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.38391	−	7.60805i	0.600548	−	0.715705i	−0.377048	−	0.926194i	$-0.623061\pi$
$113$	6.38391	−	7.60805i	0.600548	−	0.715705i	0.977596	+	0.210488i	$0.0675054\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.84642	−	10.4715i	0.167856	−	0.951959i
$122$		0			0
$123$	−1.99022	+	21.5999i	−0.179452	+	1.94760i
$124$		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	$0.333333\pi$
$128$		0			0
$129$	8.07018	+	8.14500i	0.710539	+	0.717127i
$130$		0			0
$131$	−7.75001	−	21.2930i	−0.677122	−	1.86038i	−0.471871	−	0.881668i	$-0.656421\pi$
$131$	−7.75001	−	21.2930i	−0.677122	−	1.86038i	−0.205251	−	0.978709i	$-0.565801\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	22.9607	−	4.04860i	1.96167	−	0.345895i	0.965250	−	0.261329i	$-0.0841608\pi$
$137$	22.9607	−	4.04860i	1.96167	−	0.345895i	0.996418	−	0.0845659i	$-0.0269503\pi$
$138$		0			0
$139$	16.6994	−	6.07807i	1.41642	−	0.515536i	0.483414	−	0.875392i	$-0.339396\pi$
$139$	16.6994	−	6.07807i	1.41642	−	0.515536i	0.933008	+	0.359856i	$0.117174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.00000	−	9.89949i	0.577350	−	0.816497i
$148$		0			0
$149$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$149$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$150$		0			0
$151$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$151$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$152$		0			0
$153$	19.3606	+	7.24972i	1.56521	+	0.586105i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$157$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	23.0454			1.80506			0.902528	−	0.430632i	$-0.141709\pi$
$163$	23.0454			1.80506			0.902528	+	0.430632i	$0.141709\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$167$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$168$		0			0
$169$	−12.2160	−	4.44626i	−0.939693	−	0.342020i
$170$		0			0
$171$	7.47981	+	9.08303i	0.571995	+	0.694597i
$172$		0			0
$173$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$173$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−9.86666	−	6.97678i	−0.741623	−	0.524407i
$178$		0			0
$179$	−4.92679	−	2.84448i	−0.368245	−	0.212607i	0.304446	−	0.952529i	$-0.401529\pi$
$179$	−4.92679	−	2.84448i	−0.368245	−	0.212607i	−0.672692	+	0.739923i	$0.734862\pi$
$180$		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
$187$	0.724839	+	4.11077i	0.0530055	+	0.300609i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$191$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$192$		0			0
$193$	5.58601	−	2.03314i	0.402090	−	0.146349i	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	5.58601	−	2.03314i	0.402090	−	0.146349i	0.535146	+	0.844760i	$0.320257\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$		0			0
$201$	−27.6957	−	2.55189i	−1.95350	−	0.179996i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.812560	+	2.23249i	−0.0562059	+	0.154425i
$210$		0			0
$211$	22.2501	+	18.6700i	1.53176	+	1.28530i	0.779233	+	0.626734i	$0.215609\pi$
$211$	22.2501	+	18.6700i	1.53176	+	1.28530i	0.752525	+	0.658564i	$0.228836\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	23.2943	−	16.1512i	1.57408	−	1.09139i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$223$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$224$		0			0
$225$	5.00000	+	14.1421i	0.333333	+	0.942809i
$226$		0			0
$227$	19.3064	+	23.0084i	1.28141	+	1.52712i	0.702327	+	0.711854i	$0.252144\pi$
$227$	19.3064	+	23.0084i	1.28141	+	1.52712i	0.579082	+	0.815270i	$0.303411\pi$
$228$		0			0
$229$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$229$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−20.4529	−	11.8085i	−1.33992	−	0.773602i	−0.353122	−	0.935577i	$-0.614880\pi$
$233$	−20.4529	−	11.8085i	−1.33992	−	0.773602i	−0.986795	+	0.161976i	$0.948213\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$239$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$240$		0			0
$241$	−3.23467	−	18.3447i	−0.208364	−	1.18169i	−0.892058	−	0.451920i	$-0.850739\pi$
$241$	−3.23467	−	18.3447i	−0.208364	−	1.18169i	0.683695	−	0.729768i	$-0.260372\pi$
$242$		0			0
$243$	−6.26028	−	14.2762i	−0.401597	−	0.915817i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	6.24464	+	23.7430i	0.395738	+	1.50465i
$250$		0			0
$251$	−9.79605	+	5.65575i	−0.618322	+	0.356988i	−0.776215	−	0.630468i	$-0.782863\pi$
$251$	−9.79605	+	5.65575i	−0.618322	+	0.356988i	0.157894	+	0.987456i	$0.449530\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	20.5746	+	3.62786i	1.28341	+	0.226300i	0.773427	−	0.633885i	$-0.218541\pi$
$257$	20.5746	+	3.62786i	1.28341	+	0.226300i	0.509982	+	0.860185i	$0.329652\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$263$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	18.1757	+	26.2143i	1.11234	+	1.60429i
$268$		0			0
$269$		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			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.94678	+	2.32009i	−0.117395	+	0.139906i
$276$		0			0
$277$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$277$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.929678	−	1.10795i	−0.0554599	−	0.0660946i	0.737601	−	0.675236i	$-0.235958\pi$
$281$	−0.929678	−	1.10795i	−0.0554599	−	0.0660946i	−0.793061	+	0.609142i	$0.791514\pi$
$282$		0			0
$283$	1.91801	−	10.8776i	0.114014	−	0.646606i	−0.873219	−	0.487327i	$-0.837972\pi$
$283$	1.91801	−	10.8776i	0.114014	−	0.646606i	0.987234	−	0.159279i	$-0.0509170\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	15.2440	+	26.4033i	0.896704	+	1.55314i
$290$		0			0
$291$	−18.5315	−	18.7033i	−1.08633	−	1.09641i
$292$		0			0
$293$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$293$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	1.84083	−	2.55302i	0.106816	−	0.148141i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−15.7499	+	27.2796i	−0.898893	+	1.55693i	−0.0699810	+	0.997548i	$0.522294\pi$
$307$	−15.7499	+	27.2796i	−0.898893	+	1.55693i	−0.828912	+	0.559379i	$0.811039\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$311$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$312$		0			0
$313$	−22.5810	+	18.9477i	−1.27635	+	1.07099i	−0.282617	+	0.959233i	$0.591202\pi$
$313$	−22.5810	+	18.9477i	−1.27635	+	1.07099i	−0.993736	+	0.111754i	$0.964353\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$317$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−4.26492	+	0.353309i	−0.238045	+	0.0197198i
$322$		0			0
$323$			27.0281i			1.50388i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−32.9319	−	11.9862i	−1.81010	−	0.658823i	−0.997061	−	0.0766165i	$-0.975588\pi$
$331$	−32.9319	−	11.9862i	−1.81010	−	0.658823i	−0.813041	−	0.582207i	$-0.802189\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	6.22643	−	35.3118i	0.339175	−	1.92356i	−0.0421393	−	0.999112i	$-0.513417\pi$
$337$	6.22643	−	35.3118i	0.339175	−	1.92356i	0.381314	−	0.924445i	$-0.375472\pi$
$338$		0			0
$339$	−14.0454	−	9.93160i	−0.762842	−	0.539411i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.9528	−	30.0926i	−0.587977	−	1.61545i	−0.774197	−	0.632945i	$-0.781846\pi$
$347$	−10.9528	−	30.0926i	−0.587977	−	1.61545i	0.186220	−	0.982508i	$-0.440376\pi$
$348$		0			0
$349$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$349$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	25.2259	−	4.44800i	1.34264	−	0.236743i	0.544268	−	0.838911i	$-0.316807\pi$
$353$	25.2259	−	4.44800i	1.34264	−	0.236743i	0.798369	+	0.602168i	$0.205696\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	1.80839	−	3.13222i	0.0951784	−	0.164854i
$362$		0			0
$363$	−18.3394	−	1.68980i	−0.962567	−	0.0886913i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$367$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$368$		0			0
$369$	37.5690	+	0.346732i	1.95576	+	0.0180501i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$373$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−23.6555			−1.21510			−0.607550	−	0.794282i	$-0.707847\pi$
$379$	−23.6555			−1.21510			−0.607550	+	0.794282i	$0.707847\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$383$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	12.9054	−	15.0949i	0.656019	−	0.767318i
$388$		0			0
$389$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$389$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−35.6464	+	16.4224i	−1.79812	+	0.828399i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		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
$400$		0			0
$401$	−9.05570	−	24.8803i	−0.452220	−	1.24246i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−9.05570	−	24.8803i	−0.452220	−	1.24246i	0.478938	−	0.877849i	$-0.341022\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−27.8198	+	10.1256i	−1.37560	+	0.500678i	−0.920842	−	0.389937i	$-0.872497\pi$
$409$	−27.8198	+	10.1256i	−1.37560	+	0.500678i	−0.454759	+	0.890614i	$0.650275\pi$
$410$		0			0
$411$	−10.2717	−	39.0545i	−0.506665	−	1.92641i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−12.8795	−	27.9563i	−0.630713	−	1.36903i
$418$		0			0
$419$	−2.75384	−	0.485576i	−0.134534	−	0.0237219i	0.105976	−	0.994369i	$-0.466203\pi$
$419$	−2.75384	−	0.485576i	−0.134534	−	0.0237219i	−0.240509	+	0.970647i	$0.577315\pi$
$420$		0			0
$421$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$421$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−11.7846	+	32.3778i	−0.571635	+	1.57055i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−10.1141			−0.486053			−0.243027	−	0.970020i	$-0.578140\pi$
$433$	−10.1141			−0.486053			−0.243027	+	0.970020i	$0.578140\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$439$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$440$		0			0
$441$	−18.0889	−	10.6674i	−0.861374	−	0.507971i
$442$		0			0
$443$	18.7461	+	22.3408i	0.890655	+	1.06144i	0.997740	+	0.0671913i	$0.0214038\pi$
$443$	18.7461	+	22.3408i	0.890655	+	1.06144i	−0.107085	+	0.994250i	$0.534152\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17.0439	+	9.84028i	0.804349	+	0.464391i	0.844990	−	0.534783i	$-0.179606\pi$
$449$	17.0439	+	9.84028i	0.804349	+	0.464391i	−0.0406404	+	0.999174i	$0.512940\pi$
$450$		0			0
$451$	3.79295	+	6.56958i	0.178603	+	0.309349i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.329893	−	1.87092i	−0.0154318	−	0.0875178i	0.976119	−	0.217236i	$-0.0697040\pi$
$457$	−0.329893	−	1.87092i	−0.0154318	−	0.0875178i	−0.991551	+	0.129718i	$0.958593\pi$
$458$		0			0
$459$	9.74556	−	34.4557i	0.454884	−	1.60826i
$460$		0			0
$461$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$461$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$462$		0			0
$463$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$463$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−37.3407	+	21.5587i	−1.72792	+	0.997616i	−0.829458	+	0.558569i	$0.811350\pi$
$467$	−37.3407	+	21.5587i	−1.72792	+	0.997616i	−0.898464	+	0.439047i	$0.855316\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.94895	+	0.696307i	0.181573	+	0.0320162i
$474$		0			0
$475$	−15.0227	+	12.6055i	−0.689288	+	0.578381i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$479$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−3.29537	−	39.7796i	−0.149022	−	1.79889i
$490$		0			0
$491$	28.1655	−	33.5663i	1.27109	−	1.51483i	0.520871	−	0.853635i	$-0.325607\pi$
$491$	28.1655	−	33.5663i	1.27109	−	1.51483i	0.750218	−	0.661190i	$-0.229948\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.83319	−	38.7530i	0.305896	−	1.73482i	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	6.83319	−	38.7530i	0.305896	−	1.73482i	0.619259	−	0.785187i	$-0.287433\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		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
$507$	−5.92804	+	21.7223i	−0.263274	+	0.964721i
$508$		0			0
$509$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$509$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	14.6090	−	14.2100i	0.645003	−	0.627387i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.68838	+	0.974786i	−0.0739693	+	0.0427062i	−0.536528	−	0.843882i	$-0.680265\pi$
$521$	−1.68838	+	0.974786i	−0.0739693	+	0.0427062i	0.462559	+	0.886588i	$0.346931\pi$
$522$		0			0
$523$	−20.5227	+	35.5464i	−0.897395	+	1.55433i	−0.0665832	+	0.997781i	$0.521210\pi$
$523$	−20.5227	+	35.5464i	−0.897395	+	1.55433i	−0.830812	+	0.556553i	$0.812124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−17.6190	+	14.7841i	−0.766044	+	0.642788i
$530$		0			0
$531$	−10.6320	+	18.0288i	−0.461390	+	0.782385i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−4.20546	+	8.91106i	−0.181479	+	0.384541i
$538$		0			0
$539$		−	4.24012i		−	0.182635i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.346328	+	0.126053i	0.0148079	+	0.00538964i	0.349413	−	0.936969i	$-0.386381\pi$
$547$	0.346328	+	0.126053i	0.0148079	+	0.00538964i	−0.334606	+	0.942358i	$0.608603\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		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
$561$	6.99211	−	1.83899i	0.295207	−	0.0776422i
$562$		0			0
$563$	15.3268	+	42.1101i	0.645949	+	1.77473i	0.632175	+	0.774826i	$0.282163\pi$
$563$	15.3268	+	42.1101i	0.645949	+	1.77473i	0.0137747	+	0.999905i	$0.495615\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9.51666	−	1.67804i	0.398959	−	0.0703473i	0.0294311	−	0.999567i	$-0.490630\pi$
$569$	9.51666	−	1.67804i	0.398959	−	0.0703473i	0.369528	+	0.929220i	$0.379519\pi$
$570$		0			0
$571$	−42.8520	+	15.5968i	−1.79330	+	0.652708i	−0.794322	+	0.607497i	$0.792174\pi$
$571$	−42.8520	+	15.5968i	−1.79330	+	0.652708i	−0.998978	+	0.0452101i	$0.985604\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−10.1705	+	17.6158i	−0.423403	+	0.733356i	−0.996270	−	0.0862925i	$-0.972498\pi$
$577$	−10.1705	+	17.6158i	−0.423403	+	0.733356i	0.572866	+	0.819649i	$0.305831\pi$
$578$		0			0
$579$	−4.30825	−	9.35150i	−0.179045	−	0.388635i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.1555	−	44.3870i	0.666810	−	1.83205i	0.123823	−	0.992304i	$-0.460484\pi$
$587$	16.1555	−	44.3870i	0.666810	−	1.83205i	0.542987	−	0.839741i	$-0.317293\pi$
$588$		0			0
$589$		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	$0.953834\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$599$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$600$		0			0
$601$	−46.0733	−	16.7693i	−1.87937	−	0.684034i	−0.941178	−	0.337912i	$-0.890279\pi$
$601$	−46.0733	−	16.7693i	−1.87937	−	0.684034i	−0.938190	−	0.346122i	$-0.887498\pi$
$602$		0			0
$603$	−0.444584	+	48.1714i	−0.0181049	+	1.96169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$607$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.21721	+	25.3241i	0.371071	+	1.01951i	0.974948	+	0.222432i	$0.0713995\pi$
$617$	9.21721	+	25.3241i	0.371071	+	1.01951i	−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	−8.19417	−	46.4714i	−0.329351	−	1.86784i	−0.477143	−	0.878826i	$-0.658328\pi$
$619$	−8.19417	−	46.4714i	−0.329351	−	1.86784i	0.147791	−	0.989019i	$-0.452784\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.4923	+	8.55050i	−0.939693	+	0.342020i
$626$		0			0
$627$	3.96978	+	1.08336i	0.158538	+	0.0432651i
$628$		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
$641$	−16.6441	+	45.7293i	−0.657402	+	1.80620i	−0.0690201	+	0.997615i	$0.521987\pi$
$641$	−16.6441	+	45.7293i	−0.657402	+	1.80620i	−0.588382	+	0.808583i	$0.700235\pi$
$642$		0			0
$643$	38.2155	+	32.0666i	1.50707	+	1.26458i	0.869222	+	0.494422i	$0.164620\pi$
$643$	38.2155	+	32.0666i	1.50707	+	1.26458i	0.637850	+	0.770161i	$0.279824\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−4.22605			−0.165887
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$653$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−31.2101	−	37.8996i	−1.21762	−	1.47861i
$658$		0			0
$659$	25.4737	+	30.3584i	0.992316	+	1.18260i	0.983180	+	0.182637i	$0.0584634\pi$
$659$	25.4737	+	30.3584i	0.992316	+	1.18260i	0.00913528	+	0.999958i	$0.497092\pi$
$660$		0			0
$661$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$661$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8.52453	+	48.3450i	0.328597	+	1.86356i	0.483092	+	0.875570i	$0.339514\pi$
$673$	8.52453	+	48.3450i	0.328597	+	1.86356i	−0.154495	+	0.987994i	$0.549375\pi$
$674$		0			0
$675$	23.6963	−	10.6529i	0.912071	−	0.410032i
$676$		0			0
$677$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$677$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	36.9550	−	36.6155i	1.41612	−	1.40311i
$682$		0			0
$683$	37.7598	−	21.8006i	1.44484	−	0.834177i	0.446670	−	0.894699i	$-0.352610\pi$
$683$	37.7598	−	21.8006i	1.44484	−	0.834177i	0.998167	+	0.0605215i	$0.0192764\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.731260	+	0.613600i	−0.0278184	+	0.0233424i	−0.656591	−	0.754247i	$-0.728002\pi$
$691$	−0.731260	+	0.613600i	−0.0278184	+	0.0233424i	0.628772	+	0.777589i	$0.283558\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	66.1108	+	55.4735i	2.50412	+	2.10121i
$698$		0			0
$699$	−17.4585	+	36.9932i	−0.660340	+	1.39921i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$709$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		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
$722$		0			0
$723$	−31.2030	+	8.20669i	−1.16045	+	0.305210i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$727$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$728$		0			0
$729$	−23.7474	+	12.8475i	−0.879535	+	0.475834i
$730$		0			0
$731$	44.9256	−	7.92159i	1.66163	−	0.292991i
$732$		0			0
$733$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$733$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.42360	+	4.86337i	−0.310287	+	0.179144i
$738$		0			0
$739$	23.2301	−	40.2357i	0.854534	−	1.48010i	−0.0225433	−	0.999746i	$-0.507176\pi$
$739$	23.2301	−	40.2357i	0.854534	−	1.48010i	0.877077	−	0.480350i	$-0.159490\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$743$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	40.0908	−	14.1742i	1.46685	−	0.518608i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$751$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$752$		0			0
$753$	11.1634	+	16.1006i	0.406817	+	0.586739i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.27231	−	8.66680i	0.263621	−	0.314171i	−0.617955	−	0.786214i	$-0.712039\pi$
$761$	7.27231	−	8.66680i	0.263621	−	0.314171i	0.881576	+	0.472042i	$0.156483\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	9.56642	−	54.2539i	0.344974	−	1.95644i	0.0587868	−	0.998271i	$-0.481277\pi$
$769$	9.56642	−	54.2539i	0.344974	−	1.95644i	0.286187	−	0.958174i	$-0.407612\pi$
$770$		0			0
$771$	3.32013	−	36.0334i	0.119572	−	1.29771i
$772$		0			0
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.7997	+	46.1569i	0.601912	+	1.65374i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−44.2082	+	16.0905i	−1.57585	+	0.573564i	−0.974297	−	0.225267i	$-0.927675\pi$
$787$	−44.2082	+	16.0905i	−1.57585	+	0.573564i	−0.601556	+	0.798831i	$0.705452\pi$
$788$		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		−0.984808	−	0.173648i	$-0.944444\pi$
$797$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	42.6504	−	35.1223i	1.50698	−	1.24099i
$802$		0			0
$803$	3.39047	−	9.31523i	0.119647	−	0.328727i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			15.7319i			0.553103i	0.960999	+	0.276552i	$0.0891917\pi$
$809$			15.7319i			0.553103i	−0.960999	+	0.276552i	$0.910808\pi$
$810$		0			0
$811$	21.1976			0.744350			0.372175	−	0.928163i	$-0.378612\pi$
$811$	21.1976			0.744350			0.372175	+	0.928163i	$0.378612\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	24.3983	+	8.88027i	0.853590	+	0.310681i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$821$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$822$		0			0
$823$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$823$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$824$		0			0
$825$	4.28317	+	3.02866i	0.149121	+	0.105444i
$826$		0			0
$827$	−17.1464	−	9.89949i	−0.596240	−	0.344239i	0.171321	−	0.985215i	$-0.445196\pi$
$827$	−17.1464	−	9.89949i	−0.596240	−	0.344239i	−0.767561	+	0.640976i	$0.778530\pi$
$828$		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
$833$	−16.4984	−	45.3289i	−0.571635	−	1.57055i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$839$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$840$		0			0
$841$	27.2511	−	9.91858i	0.939693	−	0.342020i
$842$		0			0
$843$	−1.77953	+	1.76318i	−0.0612903	+	0.0607272i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−19.0505	−	1.75532i	−0.653812	−	0.0602425i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$853$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.73903	−	21.2628i	0.264360	−	0.726324i	−0.734501	−	0.678608i	$-0.762584\pi$
$857$	7.73903	−	21.2628i	0.264360	−	0.726324i	0.998861	−	0.0477160i	$-0.0151942\pi$
$858$		0			0
$859$	−39.5279	−	33.1679i	−1.34868	−	1.13167i	−0.979305	−	0.202391i	$-0.935129\pi$
$859$	−39.5279	−	33.1679i	−1.34868	−	1.13167i	−0.369370	−	0.929282i	$-0.620427\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	43.3960	−	30.0887i	1.47380	−	1.02187i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−29.6346	+	34.6624i	−1.00298	+	1.17314i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$877$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$878$		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	$0.401939\pi$
$882$		0			0
$883$	29.4209	+	50.9586i	0.990094	+	1.71489i	0.616644	+	0.787242i	$0.288492\pi$
$883$	29.4209	+	50.9586i	0.990094	+	1.71489i	0.373450	+	0.927650i	$0.378175\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$887$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−4.67009	−	2.81246i	−0.156454	−	0.0942211i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	8.36370	−	7.01798i	0.277712	−	0.233028i	−0.493283	−	0.869869i	$-0.664203\pi$
$907$	8.36370	−	7.01798i	0.277712	−	0.233028i	0.770996	+	0.636841i	$0.219759\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$911$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$912$		0			0
$913$	6.57709	+	5.51883i	0.217670	+	0.182647i
$914$		0			0
$915$		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
$921$	49.3405	+	23.2856i	1.62582	+	0.767287i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18.1808	−	21.6670i	−0.596492	−	0.710872i	0.380348	−	0.924844i	$-0.375804\pi$
$929$	−18.1808	−	21.6670i	−0.596492	−	0.710872i	−0.976840	+	0.213972i	$0.931360\pi$
$930$		0			0
$931$	4.76751	−	27.0379i	0.156249	−	0.886132i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		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
$939$	35.9353	+	36.2685i	1.17270	+	1.18358i
$940$		0			0
$941$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$941$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−38.2855	+	6.75076i	−1.24411	+	0.219370i	−0.756677	−	0.653789i	$-0.773178\pi$
$947$	−38.2855	+	6.75076i	−1.24411	+	0.219370i	−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24.3880	−	14.0804i	0.790003	−	0.456109i	−0.0499603	−	0.998751i	$-0.515909\pi$
$953$	24.3880	−	14.0804i	0.790003	−	0.456109i	0.839964	+	0.542643i	$0.182576\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	23.7474	−	19.9264i	0.766044	−	0.642788i
$962$		0			0
$963$	1.21972	+	7.31132i	0.0393049	+	0.235604i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$967$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$968$		0			0
$969$	46.6542	−	3.86487i	1.49875	−	0.124157i
$970$		0			0
$971$		−	31.1127i		−	0.998454i	−0.866471	−	0.499227i	$-0.833617\pi$
$971$		−	31.1127i		−	0.998454i	0.866471	−	0.499227i	$-0.166383\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	33.1191	−	39.4698i	1.05957	−	1.26275i	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	33.1191	−	39.4698i	1.05957	−	1.26275i	0.963595	−	0.267367i	$-0.0861536\pi$
$978$		0			0
$979$	10.4829	+	3.81547i	0.335035	+	0.121943i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$983$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$	−15.9808	+	58.5590i	−0.507136	+	1.85831i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$997$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bh.a.527.1		12
4.3	odd	2		216.2.v.a.203.2	yes	12
8.3	odd	2	CM	864.2.bh.a.527.1		12
8.5	even	2		216.2.v.a.203.2	yes	12
12.11	even	2		648.2.v.a.395.1		12
24.5	odd	2		648.2.v.a.395.1		12
27.2	odd	18	inner	864.2.bh.a.623.1		12
108.79	odd	18		648.2.v.a.251.1		12
108.83	even	18		216.2.v.a.83.2	✓	12
216.29	odd	18		216.2.v.a.83.2	✓	12
216.83	even	18	inner	864.2.bh.a.623.1		12
216.133	even	18		648.2.v.a.251.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.a.83.2	✓	12	108.83	even	18
216.2.v.a.83.2	✓	12	216.29	odd	18
216.2.v.a.203.2	yes	12	4.3	odd	2
216.2.v.a.203.2	yes	12	8.5	even	2
648.2.v.a.251.1		12	108.79	odd	18
648.2.v.a.251.1		12	216.133	even	18
648.2.v.a.395.1		12	12.11	even	2
648.2.v.a.395.1		12	24.5	odd	2
864.2.bh.a.527.1		12	1.1	even	1	trivial
864.2.bh.a.527.1		12	8.3	odd	2	CM
864.2.bh.a.623.1		12	27.2	odd	18	inner
864.2.bh.a.623.1		12	216.83	even	18	inner

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.bh (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.142995 - 1.72614i) q^{3} +(-2.95911 + 0.493657i) q^{9} +O(q^{10})\)	\(q+(-0.142995 - 1.72614i) q^{3} +(-2.95911 + 0.493657i) q^{9} +(-0.389357 - 0.464017i) q^{11} +(-5.96791 - 3.44557i) q^{17} +(-1.96107 - 3.39668i) q^{19} +(-0.868241 - 4.92404i) q^{25} +(1.27526 + 5.03723i) q^{27} +(-0.745282 + 0.738435i) q^{33} +(-12.3333 - 2.17469i) q^{41} +(-5.07113 + 4.25518i) q^{43} +(5.36231 + 4.49951i) q^{49} +(-5.09416 + 10.7941i) q^{51} +(-5.58271 + 3.87079i) q^{57} +(4.48459 - 5.34452i) q^{59} +(2.78841 - 15.8139i) q^{67} +(8.18272 + 14.1729i) q^{73} +(-8.37542 + 2.20281i) q^{75} +(8.51261 - 2.92156i) q^{81} +(-13.9589 + 2.46133i) q^{83} +(-15.9495 + 9.20844i) q^{89} +(11.6448 - 9.77114i) q^{97} +(1.38121 + 1.18087i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 18 q^{11} + 30 q^{27} + 6 q^{33} - 18 q^{41} - 30 q^{43} + 12 q^{51} + 42 q^{57} - 36 q^{59} + 42 q^{67} - 162 q^{89} + 30 q^{97}+O(q^{100}) \) 12 * q - 18 * q^11 + 30 * q^27 + 6 * q^33 - 18 * q^41 - 30 * q^43 + 12 * q^51 + 42 * q^57 - 36 * q^59 + 42 * q^67 - 162 * q^89 + 30 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more