LMFDB - Embedded newform 864.2.bh.a.623.2

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			623.2
Root			$0.483690 + 1.32893i$ of defining polynomial
Character	$\chi$	$=$	864.623
Dual form			864.2.bh.a.527.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.67508 - 0.440563i) q^{3} +(2.61181 - 1.47596i) q^{9} +O(q^{10})$	$q+(1.67508 - 0.440563i) q^{3} +(2.61181 - 1.47596i) q^{9} +(3.02751 - 3.60805i) q^{11} +(-4.26651 + 2.46327i) q^{17} +(3.49316 - 6.05033i) q^{19} +(-0.868241 + 4.92404i) q^{25} +(3.72474 - 3.62302i) q^{27} +(3.48176 - 7.37759i) q^{33} +(4.73700 - 0.835261i) q^{41} +(7.73157 + 6.48756i) q^{43} +(5.36231 - 4.49951i) q^{49} +(-6.06154 + 6.00585i) q^{51} +(3.18578 - 11.6738i) q^{57} +(-9.44270 - 11.2534i) q^{59} +(1.78051 + 10.0978i) q^{67} +(-8.53002 + 14.7744i) q^{73} +(0.714973 + 8.63069i) q^{75} +(4.64309 - 7.70985i) q^{81} +(-16.7444 - 2.95248i) q^{83} +(-11.0505 - 6.38002i) q^{89} +(2.75213 + 2.30931i) q^{97} +(2.58195 - 13.8920i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

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

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
$3$	1.67508	−	0.440563i	0.967110	−	0.254359i
$3$	1.67508	−	0.440563i	0.967110	−	0.254359i
$4$		0			0
$5$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$5$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$6$		0			0
$7$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$7$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$8$		0			0
$9$	2.61181	−	1.47596i	0.870603	−	0.491986i
$10$		0			0
$11$	3.02751	−	3.60805i	0.912829	−	1.08787i	−0.0829925	−	0.996550i	$-0.526448\pi$
$11$	3.02751	−	3.60805i	0.912829	−	1.08787i	0.995822	−	0.0913174i	$-0.0291078\pi$
$12$		0			0
$13$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$13$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.26651	+	2.46327i	−1.03478	+	0.597431i	−0.918351	−	0.395768i	$-0.870479\pi$
$17$	−4.26651	+	2.46327i	−1.03478	+	0.597431i	−0.116431	+	0.993199i	$0.537145\pi$
$18$		0			0
$19$	3.49316	−	6.05033i	0.801386	−	1.38804i	−0.117318	−	0.993094i	$-0.537430\pi$
$19$	3.49316	−	6.05033i	0.801386	−	1.38804i	0.918704	−	0.394947i	$-0.129237\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$23$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$24$		0			0
$25$	−0.868241	+	4.92404i	−0.173648	+	0.984808i
$26$		0			0
$27$	3.72474	−	3.62302i	0.716827	−	0.697251i
$28$		0			0
$29$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$29$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$30$		0			0
$31$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$31$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$32$		0			0
$33$	3.48176	−	7.37759i	0.606097	−	1.28427i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.73700	−	0.835261i	0.739795	−	0.130446i	0.208964	−	0.977923i	$-0.432991\pi$
$41$	4.73700	−	0.835261i	0.739795	−	0.130446i	0.530831	+	0.847477i	$0.321880\pi$
$42$		0			0
$43$	7.73157	+	6.48756i	1.17905	+	0.989344i	0.999985	+	0.00550722i	$0.00175301\pi$
$43$	7.73157	+	6.48756i	1.17905	+	0.989344i	0.179069	+	0.983836i	$0.442691\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$47$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$48$		0			0
$49$	5.36231	−	4.49951i	0.766044	−	0.642788i
$50$		0			0
$51$	−6.06154	+	6.00585i	−0.848785	+	0.840988i
$52$		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
$57$	3.18578	−	11.6738i	0.421967	−	1.54623i
$58$		0			0
$59$	−9.44270	−	11.2534i	−1.22933	−	1.46506i	−0.838768	−	0.544489i	$-0.816724\pi$
$59$	−9.44270	−	11.2534i	−1.22933	−	1.46506i	−0.390567	−	0.920575i	$-0.627721\pi$
$60$		0			0
$61$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$61$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.78051	+	10.0978i	0.217524	+	1.23364i	0.876472	+	0.481452i	$0.159891\pi$
$67$	1.78051	+	10.0978i	0.217524	+	1.23364i	−0.658948	+	0.752188i	$0.728998\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$		0			0
$73$	−8.53002	+	14.7744i	−0.998363	+	1.72922i	−0.449652	+	0.893204i	$0.648452\pi$
$73$	−8.53002	+	14.7744i	−0.998363	+	1.72922i	−0.548711	+	0.836012i	$0.684881\pi$
$74$		0			0
$75$	0.714973	+	8.63069i	0.0825579	+	0.996586i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$79$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$80$		0			0
$81$	4.64309	−	7.70985i	0.515899	−	0.856649i
$82$		0			0
$83$	−16.7444	−	2.95248i	−1.83793	−	0.324077i	−0.856539	−	0.516083i	$-0.827390\pi$
$83$	−16.7444	−	2.95248i	−1.83793	−	0.324077i	−0.981394	+	0.192006i	$0.938501\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.0505	−	6.38002i	−1.17135	−	0.676280i	−0.217354	−	0.976093i	$-0.569742\pi$
$89$	−11.0505	−	6.38002i	−1.17135	−	0.676280i	−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
$97$	2.75213	+	2.30931i	0.279437	+	0.234475i	0.771724	−	0.635958i	$-0.219395\pi$
$97$	2.75213	+	2.30931i	0.279437	+	0.234475i	−0.492287	+	0.870433i	$0.663839\pi$
$98$		0			0
$99$	2.58195	−	13.8920i	0.259496	−	1.39620i
$100$		0			0
$101$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$101$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$102$		0			0
$103$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$103$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			9.34691i			0.903599i	0.892119	+	0.451800i	$0.149218\pi$
$107$			9.34691i			0.903599i	−0.892119	+	0.451800i	$0.850782\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.6562	+	16.2749i	1.28467	+	1.53101i	0.671087	+	0.741379i	$0.265828\pi$
$113$	13.6562	+	16.2749i	1.28467	+	1.53101i	0.613583	+	0.789630i	$0.289728\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.94205	−	11.0139i	−0.176550	−	1.00127i
$122$		0			0
$123$	7.56689	−	3.48608i	0.682283	−	0.314329i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$		0			0
$129$	15.8092	+	7.46096i	1.39192	+	0.656901i
$130$		0			0
$131$	−2.91312	+	8.00373i	−0.254520	+	0.699289i	0.744962	+	0.667107i	$0.232468\pi$
$131$	−2.91312	+	8.00373i	−0.254520	+	0.699289i	−0.999482	+	0.0321817i	$0.989754\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−18.9188	−	3.33590i	−1.61635	−	0.285005i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	−18.9188	−	3.33590i	−1.61635	−	0.285005i	−0.907403	+	0.420261i	$0.861939\pi$
$138$		0			0
$139$	22.1536	+	8.06325i	1.87904	+	0.683916i	0.946036	+	0.324060i	$0.105048\pi$
$139$	22.1536	+	8.06325i	1.87904	+	0.683916i	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	$-0.0555556\pi$
$149$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$150$		0			0
$151$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$151$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$152$		0			0
$153$	−7.50763	+	12.7308i	−0.606956	+	1.02922i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$157$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−21.0454			−1.64840			−0.824202	−	0.566296i	$-0.808376\pi$
$163$	−21.0454			−1.64840			−0.824202	+	0.566296i	$0.808376\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$167$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$168$		0			0
$169$	−12.2160	+	4.44626i	−0.939693	+	0.342020i
$170$		0			0
$171$	0.193426	−	20.9581i	0.0147917	−	1.60270i
$172$		0			0
$173$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$173$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−20.7751	−	14.6902i	−1.56155	−	1.10419i
$178$		0			0
$179$	−22.0732	+	12.7440i	−1.64983	+	0.952529i	−0.672692	+	0.739923i	$0.734862\pi$
$179$	−22.0732	+	12.7440i	−1.64983	+	0.952529i	−0.977138	+	0.212607i	$0.931805\pi$
$180$		0			0
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−4.02931	+	22.8514i	−0.294653	+	1.67106i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$191$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$192$		0			0
$193$	26.0872	+	9.49497i	1.87780	+	0.683463i	0.952947	+	0.303139i	$0.0980345\pi$
$193$	26.0872	+	9.49497i	1.87780	+	0.683463i	0.924853	+	0.380325i	$0.124188\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$		0			0
$201$	7.43121	+	16.1302i	0.524157	+	1.13774i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−11.2543	−	30.9210i	−0.778477	−	2.13885i
$210$		0			0
$211$	−11.5255	+	9.67100i	−0.793445	+	0.665779i	−0.946595	−	0.322424i	$-0.895502\pi$
$211$	−11.5255	+	9.67100i	−0.793445	+	0.665779i	0.153151	+	0.988203i	$0.451058\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−7.77943	+	28.5064i	−0.525685	+	1.92628i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$223$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$224$		0			0
$225$	5.00000	+	14.1421i	0.333333	+	0.942809i
$226$		0			0
$227$	18.6750	−	22.2560i	1.23950	−	1.47718i	0.416503	−	0.909134i	$-0.363255\pi$
$227$	18.6750	−	22.2560i	1.23950	−	1.47718i	0.822997	−	0.568045i	$-0.192300\pi$
$228$		0			0
$229$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$229$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.9473	+	7.47511i	−0.848204	+	0.489711i	−0.860045	−	0.510219i	$-0.829564\pi$
$233$	−12.9473	+	7.47511i	−0.848204	+	0.489711i	0.0118403	+	0.999930i	$0.496231\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$239$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$240$		0			0
$241$	−5.25048	+	29.7769i	−0.338213	+	1.91810i	0.0546547	+	0.998505i	$0.482594\pi$
$241$	−5.25048	+	29.7769i	−0.338213	+	1.91810i	−0.392868	+	0.919595i	$0.628517\pi$
$242$		0			0
$243$	4.38089	−	14.9602i	0.281034	−	0.959698i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−29.3490	+	2.43129i	−1.85991	+	0.154077i
$250$		0			0
$251$	−0.438367	−	0.253092i	−0.0276695	−	0.0159750i	0.486101	−	0.873902i	$-0.338419\pi$
$251$	−0.438367	−	0.253092i	−0.0276695	−	0.0159750i	−0.513771	+	0.857927i	$0.671752\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.365171	+	0.0643895i	−0.0227787	+	0.00401650i	−0.185026	−	0.982734i	$-0.559237\pi$
$257$	−0.365171	+	0.0643895i	−0.0227787	+	0.00401650i	0.162247	+	0.986750i	$0.448126\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$263$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−21.3213	−	5.81861i	−1.30484	−	0.356093i
$268$		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
$275$	15.1376	+	18.0402i	0.912829	+	1.08787i
$276$		0			0
$277$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$277$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−19.1105	+	22.7750i	−1.14003	+	1.35864i	−0.215971	+	0.976400i	$0.569292\pi$
$281$	−19.1105	+	22.7750i	−1.14003	+	1.35864i	−0.924063	+	0.382240i	$0.875153\pi$
$282$		0			0
$283$	−5.73827	−	32.5434i	−0.341105	−	1.93450i	−0.355677	−	0.934609i	$-0.615750\pi$
$283$	−5.73827	−	32.5434i	−0.341105	−	1.93450i	0.0145720	−	0.999894i	$-0.495361\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	3.63542	−	6.29673i	0.213848	−	0.370396i
$290$		0			0
$291$	5.62745	+	2.65580i	0.329887	+	0.155686i
$292$		0			0
$293$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$293$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.79532	−	24.4078i	−0.104175	−	1.41628i
$298$		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
$307$	−10.2956	−	17.8326i	−0.587603	−	1.01776i	−0.994545	−	0.104305i	$-0.966738\pi$
$307$	−10.2956	−	17.8326i	−0.587603	−	1.01776i	0.406942	−	0.913454i	$-0.366595\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$311$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$312$		0			0
$313$	10.8445	+	9.09962i	0.612967	+	0.514341i	0.895584	−	0.444892i	$-0.146758\pi$
$313$	10.8445	+	9.09962i	0.612967	+	0.514341i	−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$317$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	4.11790	+	15.6568i	0.229839	+	0.873880i
$322$		0			0
$323$			34.4184i			1.91509i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8.49990	−	3.09371i	0.467197	−	0.170046i	−0.0976852	−	0.995217i	$-0.531144\pi$
$331$	8.49990	−	3.09371i	0.467197	−	0.170046i	0.564882	+	0.825172i	$0.308922\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5.38212	−	30.5235i	−0.293183	−	1.66272i	−0.674497	−	0.738277i	$-0.735640\pi$
$337$	−5.38212	−	30.5235i	−0.293183	−	1.66272i	0.381314	−	0.924445i	$-0.375472\pi$
$338$		0			0
$339$	30.0454	+	21.2453i	1.63184	+	1.15389i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.31464	−	22.8443i	0.446353	−	1.22635i	−0.488892	−	0.872345i	$-0.662599\pi$
$347$	8.31464	−	22.8443i	0.446353	−	1.22635i	0.935245	−	0.354001i	$-0.115179\pi$
$348$		0			0
$349$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$349$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	32.9649	+	5.81260i	1.75454	+	0.309374i	0.956175	−	0.292794i	$-0.0945851\pi$
$353$	32.9649	+	5.81260i	1.75454	+	0.309374i	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	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	−14.9044	−	25.8151i	−0.784440	−	1.35869i
$362$		0			0
$363$	−8.10543	−	17.5937i	−0.425425	−	0.923428i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$367$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$368$		0			0
$369$	11.1393	−	9.17316i	0.579890	−	0.477536i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$373$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−34.5639			−1.77543			−0.887715	−	0.460394i	$-0.847708\pi$
$379$	−34.5639			−1.77543			−0.887715	+	0.460394i	$0.847708\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$383$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	29.7688	+	5.53278i	1.51323	+	0.281247i
$388$		0			0
$389$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$389$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−1.35357	+	14.6903i	−0.0682787	+	0.741029i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	11.6939	−	32.1286i	0.583963	−	1.60443i	−0.197382	−	0.980327i	$-0.563244\pi$
$401$	11.6939	−	32.1286i	0.583963	−	1.60443i	0.781345	−	0.624099i	$-0.214534\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	34.9996	+	12.7388i	1.73062	+	0.629893i	0.998674	−	0.0514740i	$-0.0163919\pi$
$409$	34.9996	+	12.7388i	1.73062	+	0.629893i	0.731942	+	0.681367i	$0.238614\pi$
$410$		0			0
$411$	−33.1603	+	2.74702i	−1.63568	+	0.135501i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	40.6615	+	3.74657i	1.99120	+	0.183470i
$418$		0			0
$419$	33.4571	−	5.89939i	1.63449	−	0.288204i	0.720350	−	0.693611i	$-0.243981\pi$
$419$	33.4571	−	5.89939i	1.63449	−	0.288204i	0.914136	+	0.405407i	$0.132870\pi$
$420$		0			0
$421$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$421$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.42489	−	23.1472i	−0.408667	−	1.12280i
$426$		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
$433$	23.3114			1.12027			0.560137	−	0.828400i	$-0.310749\pi$
$433$	23.3114			1.12027			0.560137	+	0.828400i	$0.310749\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$439$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$440$		0			0
$441$	7.36423	−	19.6664i	0.350678	−	0.936496i
$442$		0			0
$443$	15.9607	−	19.0212i	0.758314	−	0.903723i	−0.239426	−	0.970915i	$-0.576959\pi$
$443$	15.9607	−	19.0212i	0.758314	−	0.903723i	0.997740	+	0.0671913i	$0.0214038\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.83679	−	4.52457i	0.369841	−	0.213528i	−0.303548	−	0.952816i	$-0.598171\pi$
$449$	7.83679	−	4.52457i	0.369841	−	0.213528i	0.673389	+	0.739288i	$0.264838\pi$
$450$		0			0
$451$	11.3277	−	19.6201i	0.533399	−	0.923874i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.24705	−	41.1001i	0.339003	−	1.92258i	−0.0444340	−	0.999012i	$-0.514148\pi$
$457$	7.24705	−	41.1001i	0.339003	−	1.92258i	0.383437	−	0.923567i	$-0.374740\pi$
$458$		0			0
$459$	−6.96719	+	24.6327i	−0.325200	+	1.14976i
$460$		0			0
$461$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$461$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$462$		0			0
$463$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$463$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.94049	+	2.27504i	0.182344	+	0.105276i	0.588393	−	0.808575i	$-0.299761\pi$
$467$	3.94049	+	2.27504i	0.182344	+	0.105276i	−0.406050	+	0.913851i	$0.633094\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	46.8149	−	8.25472i	2.15255	−	0.379553i
$474$		0			0
$475$	26.7592	+	22.4536i	1.22779	+	1.03024i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$479$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$480$		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
$489$	−35.2528	+	9.27182i	−1.59419	+	0.419286i
$490$		0			0
$491$	25.0084	+	29.8039i	1.12861	+	1.34503i	0.931115	+	0.364726i	$0.118837\pi$
$491$	25.0084	+	29.8039i	1.12861	+	1.34503i	0.197499	+	0.980303i	$0.436718\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−7.67749	−	43.5412i	−0.343692	−	1.94917i	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−7.67749	−	43.5412i	−0.343692	−	1.94917i	−0.0303288	−	0.999540i	$-0.509655\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−18.5040	+	12.8298i	−0.821790	+	0.569790i
$508$		0			0
$509$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$509$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−8.90934	−	35.1917i	−0.393357	−	1.55375i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11.9228	+	6.88363i	0.522347	+	0.301577i	0.737895	−	0.674916i	$-0.235820\pi$
$521$	11.9228	+	6.88363i	0.522347	+	0.301577i	−0.215547	+	0.976493i	$0.569153\pi$
$522$		0			0
$523$	1.52270	+	2.63740i	0.0665832	+	0.115325i	0.897395	−	0.441228i	$-0.145457\pi$
$523$	1.52270	+	2.63740i	0.0665832	+	0.115325i	−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
$531$	−41.2720	−	15.4546i	−1.79105	−	0.670673i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−31.3600	+	31.0719i	−1.35328	+	1.34085i
$538$		0			0
$539$		−	32.9698i		−	1.42011i
$540$		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
$547$	−15.3585	+	5.59004i	−0.656683	+	0.239013i	−0.648803	−	0.760956i	$-0.724730\pi$
$547$	−15.3585	+	5.59004i	−0.656683	+	0.239013i	−0.00787913	+	0.999969i	$0.502508\pi$
$548$		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	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	3.31803	+	40.0531i	0.140087	+	1.69104i
$562$		0			0
$563$	−8.30817	+	22.8265i	−0.350148	+	0.962023i	0.632175	+	0.774826i	$0.282163\pi$
$563$	−8.30817	+	22.8265i	−0.350148	+	0.962023i	−0.982322	+	0.187197i	$0.940060\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	43.6572	+	7.69794i	1.83021	+	0.322715i	0.979270	−	0.202557i	$-0.0649251\pi$
$569$	43.6572	+	7.69794i	1.83021	+	0.322715i	0.850935	+	0.525271i	$0.176036\pi$
$570$		0			0
$571$	35.6722	+	12.9836i	1.49284	+	0.543348i	0.954195	−	0.299187i	$-0.0967154\pi$
$571$	35.6722	+	12.9836i	1.49284	+	0.543348i	0.538642	+	0.842535i	$0.318938\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−21.7790	−	37.7224i	−0.906674	−	1.57040i	−0.818655	−	0.574286i	$-0.805280\pi$
$577$	−21.7790	−	37.7224i	−0.906674	−	1.57040i	−0.0880190	−	0.996119i	$-0.528054\pi$
$578$		0			0
$579$	47.8814	+	4.41181i	1.98988	+	0.183349i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.7518	−	40.5302i	−0.608872	−	1.67286i	−0.732695	−	0.680557i	$-0.761738\pi$
$587$	−14.7518	−	40.5302i	−0.608872	−	1.67286i	0.123823	−	0.992304i	$-0.460484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			38.2159i			1.56934i	0.619915	+	0.784669i	$0.287167\pi$
$593$			38.2159i			1.56934i	−0.619915	+	0.784669i	$0.712833\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$599$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$600$		0			0
$601$	−35.1648	+	12.7989i	−1.43440	+	0.522080i	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−35.1648	+	12.7989i	−1.43440	+	0.522080i	−0.496212	+	0.868201i	$0.665276\pi$
$602$		0			0
$603$	19.5543	+	23.7455i	0.796312	+	0.966992i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$607$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.2359	+	44.6077i	−0.653632	+	1.79584i	−0.0497546	+	0.998761i	$0.515844\pi$
$617$	−16.2359	+	44.6077i	−0.653632	+	1.79584i	−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	1.27701	−	7.24229i	0.0513274	−	0.291092i	−0.948329	−	0.317287i	$-0.897228\pi$
$619$	1.27701	−	7.24229i	0.0513274	−	0.291092i	0.999657	+	0.0261952i	$0.00833914\pi$
$620$		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
$627$	−32.4745	−	46.8369i	−1.29691	−	1.87049i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$	−15.0454	+	21.2774i	−0.598001	+	0.845702i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.82299	−	5.00863i	−0.0720038	−	0.197829i	0.898470	−	0.439034i	$-0.144679\pi$
$641$	−1.82299	−	5.00863i	−0.0720038	−	0.197829i	−0.970474	+	0.241206i	$0.922457\pi$
$642$		0			0
$643$	33.7691	−	28.3357i	1.33173	−	1.11745i	0.348054	−	0.937474i	$-0.386843\pi$
$643$	33.7691	−	28.3357i	1.33173	−	1.11745i	0.983671	−	0.179976i	$-0.0576019\pi$
$644$		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
$649$	−69.1906			−2.71597
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$653$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.472332	+	51.1779i	−0.0184274	+	1.99664i
$658$		0			0
$659$	−5.43360	+	6.47551i	−0.211663	+	0.252250i	−0.861422	−	0.507891i	$-0.830425\pi$
$659$	−5.43360	+	6.47551i	−0.211663	+	0.252250i	0.649759	+	0.760140i	$0.274870\pi$
$660$		0			0
$661$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$661$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$662$		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
$673$	−6.78805	+	38.4969i	−0.261660	+	1.48395i	0.516720	+	0.856154i	$0.327153\pi$
$673$	−6.78805	+	38.4969i	−0.261660	+	1.48395i	−0.778380	+	0.627793i	$0.783958\pi$
$674$		0			0
$675$	14.6059	+	21.4864i	0.562182	+	0.827014i
$676$		0			0
$677$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$677$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	21.4770	−	45.5081i	0.822999	−	1.74387i
$682$		0			0
$683$	−12.8791	−	7.43576i	−0.492806	−	0.284522i	0.232932	−	0.972493i	$-0.425168\pi$
$683$	−12.8791	−	7.43576i	−0.492806	−	0.284522i	−0.725738	+	0.687971i	$0.758501\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−34.5068	−	28.9546i	−1.31270	−	1.10149i	−0.987798	−	0.155738i	$-0.950224\pi$
$691$	−34.5068	−	28.9546i	−1.31270	−	1.10149i	−0.324902	−	0.945748i	$-0.605331\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−18.1530	+	15.2322i	−0.687594	+	0.576960i
$698$		0			0
$699$	−18.3945	+	18.2255i	−0.695744	+	0.689353i
$700$		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	$-0.111111\pi$
$709$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$710$		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	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	4.32362	+	52.1920i	0.160797	+	1.94104i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$727$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$728$		0			0
$729$	0.747449	−	26.9897i	0.0276833	−	0.999617i
$730$		0			0
$731$	−48.9675	−	8.63429i	−1.81113	−	0.319351i
$732$		0			0
$733$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$733$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	41.8238	+	24.1470i	1.54060	+	0.889466i
$738$		0			0
$739$	8.71943	+	15.1025i	0.320749	+	0.555554i	0.980643	−	0.195805i	$-0.0627319\pi$
$739$	8.71943	+	15.1025i	0.320749	+	0.555554i	−0.659893	+	0.751359i	$0.729399\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$743$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−48.0908	+	17.0027i	−1.75955	+	0.622095i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$751$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$752$		0			0
$753$	−0.845805	−	0.230821i	−0.0308228	−	0.00841159i
$754$		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	$-0.287961\pi$
$761$	−7.27231	−	8.66680i	−0.263621	−	0.314171i	−0.881576	+	0.472042i	$0.843517\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−5.74616	−	32.5881i	−0.207212	−	1.17516i	−0.893921	−	0.448224i	$-0.852057\pi$
$769$	−5.74616	−	32.5881i	−0.207212	−	1.17516i	0.686709	−	0.726932i	$-0.259055\pi$
$770$		0			0
$771$	−0.583324	+	0.268738i	−0.0210079	+	0.00967838i
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11.4935	−	31.5781i	0.411797	−	1.13140i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2.77641	−	1.01053i	−0.0989683	−	0.0360215i	0.292061	−	0.956400i	$-0.405659\pi$
$787$	−2.77641	−	1.01053i	−0.0989683	−	0.0360215i	−0.391030	+	0.920378i	$0.627881\pi$
$788$		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	$-0.0555556\pi$
$797$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−38.2785	−	0.353280i	−1.35250	−	0.0124825i
$802$		0			0
$803$	27.4821	+	75.5065i	0.969823	+	2.66457i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			3.91418i			0.137615i	0.997630	+	0.0688076i	$0.0219195\pi$
$809$			3.91418i			0.137615i	−0.997630	+	0.0688076i	$0.978081\pi$
$810$		0			0
$811$	50.2190			1.76343			0.881714	−	0.471784i	$-0.156390\pi$
$811$	50.2190			1.76343			0.881714	+	0.471784i	$0.156390\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	66.2595	−	24.1165i	2.31813	−	0.843729i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$821$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$822$		0			0
$823$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$823$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$824$		0			0
$825$	33.3045	+	23.5499i	1.15952	+	0.819901i
$826$		0			0
$827$	17.1464	−	9.89949i	0.596240	−	0.344239i	−0.171321	−	0.985215i	$-0.554804\pi$
$827$	17.1464	−	9.89949i	0.596240	−	0.344239i	0.767561	+	0.640976i	$0.221470\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−11.7948	+	32.4061i	−0.408667	+	1.12280i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$839$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$840$		0			0
$841$	27.2511	+	9.91858i	0.939693	+	0.342020i
$842$		0			0
$843$	−21.9778	+	46.5693i	−0.756956	+	1.60393i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−23.9495	−	51.9848i	−0.821944	−	1.78411i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$853$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.73903	−	21.2628i	−0.264360	−	0.726324i	−0.998861	−	0.0477160i	$-0.984806\pi$
$857$	−7.73903	−	21.2628i	−0.264360	−	0.726324i	0.734501	−	0.678608i	$-0.237416\pi$
$858$		0			0
$859$	−43.9743	+	36.8988i	−1.50038	+	1.25897i	−0.620097	+	0.784525i	$0.712907\pi$
$859$	−43.9743	+	36.8988i	−1.50038	+	1.25897i	−0.880285	+	0.474445i	$0.842649\pi$
$860$		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
$867$	3.31552	−	12.1492i	0.112601	−	0.412608i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	10.5965	+	1.96945i	0.358637	+	0.0666558i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$877$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.9949	+	6.34791i	−0.370428	+	0.213866i	−0.673645	−	0.739055i	$-0.735272\pi$
$881$	−10.9949	+	6.34791i	−0.370428	+	0.213866i	0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	−29.0736	+	50.3570i	−0.978406	+	1.69465i	−0.310203	+	0.950670i	$0.600397\pi$
$883$	−29.0736	+	50.3570i	−0.978406	+	1.69465i	−0.668203	+	0.743979i	$0.732936\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$887$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−13.7605	−	40.0941i	−0.460994	−	1.34320i
$892$		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
$907$	−22.7606	−	19.0984i	−0.755754	−	0.634153i	0.181264	−	0.983435i	$-0.441981\pi$
$907$	−22.7606	−	19.0984i	−0.755754	−	0.634153i	−0.937018	+	0.349281i	$0.886426\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$911$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$912$		0			0
$913$	−61.3465	+	51.4758i	−2.03027	+	1.70360i
$914$		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
$921$	−25.1024	−	25.3352i	−0.827153	−	0.834822i
$922$		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	$-0.624196\pi$
$929$	18.1808	−	21.6670i	0.596492	−	0.710872i	0.976840	+	0.213972i	$0.0686401\pi$
$930$		0			0
$931$	−8.49214	−	48.1613i	−0.278318	−	1.57842i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	13.5454	−	23.4613i	0.442509	−	0.766448i	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	13.5454	−	23.4613i	0.442509	−	0.766448i	0.997875	+	0.0651578i	$0.0207551\pi$
$938$		0			0
$939$	22.1744	+	10.4649i	0.723634	+	0.341510i
$940$		0			0
$941$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$941$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−19.9053	−	3.50984i	−0.646835	−	0.114054i	−0.159400	−	0.987214i	$-0.550956\pi$
$947$	−19.9053	−	3.50984i	−0.646835	−	0.114054i	−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−49.2686	−	28.4452i	−1.59597	−	0.921432i	−0.992253	−	0.124230i	$-0.960354\pi$
$953$	−49.2686	−	28.4452i	−1.59597	−	0.921432i	−0.603713	−	0.797202i	$-0.706313\pi$
$954$		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
$963$	13.7956	+	24.4123i	0.444559	+	0.786676i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$967$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$968$		0			0
$969$	15.1635	+	57.6537i	0.487121	+	1.85210i
$970$		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
$977$	−28.1610	−	33.5609i	−0.900949	−	1.07371i	−0.996928	−	0.0783261i	$-0.975042\pi$
$977$	−28.1610	−	33.5609i	−0.900949	−	1.07371i	0.0959785	−	0.995383i	$-0.469402\pi$
$978$		0			0
$979$	−56.4750	+	20.5552i	−1.80495	+	0.656947i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$983$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$984$		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	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	12.8751	−	8.92697i	0.408578	−	0.283289i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$997$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$998$		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.623.2		12
4.3	odd	2		216.2.v.a.83.1	✓	12
8.3	odd	2	CM	864.2.bh.a.623.2		12
8.5	even	2		216.2.v.a.83.1	✓	12
12.11	even	2		648.2.v.a.251.2		12
24.5	odd	2		648.2.v.a.251.2		12
27.14	odd	18	inner	864.2.bh.a.527.2		12
108.67	odd	18		648.2.v.a.395.2		12
108.95	even	18		216.2.v.a.203.1	yes	12
216.13	even	18		648.2.v.a.395.2		12
216.149	odd	18		216.2.v.a.203.1	yes	12
216.203	even	18	inner	864.2.bh.a.527.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.v.a.83.1	✓	12	4.3	odd	2
216.2.v.a.83.1	✓	12	8.5	even	2
216.2.v.a.203.1	yes	12	108.95	even	18
216.2.v.a.203.1	yes	12	216.149	odd	18
648.2.v.a.251.2		12	12.11	even	2
648.2.v.a.251.2		12	24.5	odd	2
648.2.v.a.395.2		12	108.67	odd	18
648.2.v.a.395.2		12	216.13	even	18
864.2.bh.a.527.2		12	27.14	odd	18	inner
864.2.bh.a.527.2		12	216.203	even	18	inner
864.2.bh.a.623.2		12	1.1	even	1	trivial
864.2.bh.a.623.2		12	8.3	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(1.67508 - 0.440563i) q^{3} +(2.61181 - 1.47596i) q^{9} +O(q^{10})\)	\(q+(1.67508 - 0.440563i) q^{3} +(2.61181 - 1.47596i) q^{9} +(3.02751 - 3.60805i) q^{11} +(-4.26651 + 2.46327i) q^{17} +(3.49316 - 6.05033i) q^{19} +(-0.868241 + 4.92404i) q^{25} +(3.72474 - 3.62302i) q^{27} +(3.48176 - 7.37759i) q^{33} +(4.73700 - 0.835261i) q^{41} +(7.73157 + 6.48756i) q^{43} +(5.36231 - 4.49951i) q^{49} +(-6.06154 + 6.00585i) q^{51} +(3.18578 - 11.6738i) q^{57} +(-9.44270 - 11.2534i) q^{59} +(1.78051 + 10.0978i) q^{67} +(-8.53002 + 14.7744i) q^{73} +(0.714973 + 8.63069i) q^{75} +(4.64309 - 7.70985i) q^{81} +(-16.7444 - 2.95248i) q^{83} +(-11.0505 - 6.38002i) q^{89} +(2.75213 + 2.30931i) q^{97} +(2.58195 - 13.8920i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 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

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