LMFDB - Embedded newform 864.2.p.a.143.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(143,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("864.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.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:	$6.89907473464$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			143.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	864.143
Dual form			864.2.p.a.719.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q+(-5.72474 - 3.30518i) q^{11} -2.36773i q^{17} -6.34847 q^{19} +(2.50000 - 4.33013i) q^{25} +(-9.39898 + 5.42650i) q^{41} +(6.17423 - 10.6941i) q^{43} +(-3.50000 - 6.06218i) q^{49} +(-1.62372 + 0.937458i) q^{59} +(0.174235 + 0.301783i) q^{67} -15.6969 q^{73} +(2.44949 + 1.41421i) q^{83} -5.65685i q^{89} +(4.84847 - 8.39780i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 18 q^{11} + 4 q^{19} + 10 q^{25} - 18 q^{41} + 10 q^{43} - 14 q^{49} + 18 q^{59} - 14 q^{67} - 4 q^{73} - 10 q^{97}+O(q^{100}) $ 4 * q - 18 * q^11 + 4 * q^19 + 10 * q^25 - 18 * q^41 + 10 * q^43 - 14 * q^49 + 18 * q^59 - 14 * q^67 - 4 * q^73 - 10 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.72474	−	3.30518i	−1.72608	−	0.996550i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−5.72474	−	3.30518i	−1.72608	−	0.996550i	−0.821541	−	0.570149i	$-0.806886\pi$
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.36773i		−	0.574258i	−0.957892	−	0.287129i	$-0.907299\pi$
$17$		−	2.36773i		−	0.574258i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		0			0
$19$	−6.34847			−1.45644			−0.728219	−	0.685344i	$-0.759652\pi$
$19$	−6.34847			−1.45644			−0.728219	+	0.685344i	$0.759652\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.39898	+	5.42650i	−1.46787	+	0.847477i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−9.39898	+	5.42650i	−1.46787	+	0.847477i	−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	6.17423	−	10.6941i	0.941562	−	1.63083i	0.179069	−	0.983836i	$-0.442691\pi$
$43$	6.17423	−	10.6941i	0.941562	−	1.63083i	0.762493	−	0.646997i	$-0.223975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		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
$57$		0			0
$58$		0			0
$59$	−1.62372	+	0.937458i	−0.211391	+	0.122047i	−0.601958	−	0.798528i	$-0.705612\pi$
$59$	−1.62372	+	0.937458i	−0.211391	+	0.122047i	0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.174235	+	0.301783i	0.0212861	+	0.0368687i	0.876472	−	0.481452i	$-0.159891\pi$
$67$	0.174235	+	0.301783i	0.0212861	+	0.0368687i	−0.855186	+	0.518321i	$0.826557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−15.6969			−1.83719			−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−15.6969			−1.83719			−0.918594	+	0.395203i	$0.870674\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.44949	+	1.41421i	0.268866	+	0.155230i	0.628372	−	0.777913i	$-0.283721\pi$
$83$	2.44949	+	1.41421i	0.268866	+	0.155230i	−0.359506	+	0.933143i	$0.617055\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	5.65685i		−	0.599625i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$		−	5.65685i		−	0.599625i	0.953998	−	0.299813i	$-0.0969242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.84847	−	8.39780i	0.492287	−	0.852667i	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	4.84847	−	8.39780i	0.492287	−	0.852667i	0.999961	+	0.00888289i	$0.00282755\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.0956i			1.45935i	0.683793	+	0.729676i	$0.260329\pi$
$107$			15.0956i			1.45935i	−0.683793	+	0.729676i	$0.739671\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.79796	−	5.65685i	0.921714	−	0.532152i	0.0375328	−	0.999295i	$-0.488050\pi$
$113$	9.79796	−	5.65685i	0.921714	−	0.532152i	0.884182	+	0.467143i	$0.154717\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	16.3485	+	28.3164i	1.48622	+	2.57422i
$122$		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			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.2474	−	7.07107i	1.07006	−	0.617802i	0.141865	−	0.989886i	$-0.454690\pi$
$131$	12.2474	−	7.07107i	1.07006	−	0.617802i	0.928199	+	0.372084i	$0.121357\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.29796	−	3.05878i	−0.452635	−	0.261329i	0.256307	−	0.966595i	$-0.417494\pi$
$137$	−5.29796	−	3.05878i	−0.452635	−	0.261329i	−0.708942	+	0.705266i	$0.750827\pi$
$138$		0			0
$139$	9.17423	+	15.8902i	0.778148	+	1.34779i	0.933008	+	0.359856i	$0.117174\pi$
$139$	9.17423	+	15.8902i	0.778148	+	1.34779i	−0.154859	+	0.987937i	$0.549492\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−6.50000	+	11.2583i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	19.7990i		−	1.47985i	−0.672692	−	0.739923i	$-0.734862\pi$
$179$		−	19.7990i		−	1.47985i	0.672692	−	0.739923i	$-0.265138\pi$
$180$		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
$187$	−7.82577	+	13.5546i	−0.572277	+	0.991212i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	1.84847	+	3.20164i	0.133056	+	0.230459i	0.924853	−	0.380325i	$-0.124188\pi$
$193$	1.84847	+	3.20164i	0.133056	+	0.230459i	−0.791797	+	0.610784i	$0.790854\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	36.3434	+	20.9829i	2.51392	+	1.45141i
$210$		0			0
$211$	7.00000	+	12.1244i	0.481900	+	0.834675i	0.999784	−	0.0207756i	$-0.00661356\pi$
$211$	7.00000	+	12.1244i	0.481900	+	0.834675i	−0.517884	+	0.855451i	$0.673280\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		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	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	21.2753	+	12.2833i	1.41209	+	0.815270i	0.995585	−	0.0938647i	$-0.0299221\pi$
$227$	21.2753	+	12.2833i	1.41209	+	0.815270i	0.416503	+	0.909134i	$0.363255\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			28.8092i			1.88735i	0.330870	+	0.943676i	$0.392658\pi$
$233$			28.8092i			1.88735i	−0.330870	+	0.943676i	$0.607342\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	13.8485	−	23.9863i	0.892058	−	1.54509i	0.0546547	−	0.998505i	$-0.482594\pi$
$241$	13.8485	−	23.9863i	0.892058	−	1.54509i	0.837404	−	0.546585i	$-0.184072\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	10.3602i		−	0.653930i	−0.945036	−	0.326965i	$-0.893974\pi$
$251$		−	10.3602i		−	0.653930i	0.945036	−	0.326965i	$-0.106026\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	17.6010	−	10.1620i	1.09792	−	0.633885i	0.162247	−	0.986750i	$-0.448126\pi$
$257$	17.6010	−	10.1620i	1.09792	−	0.633885i	0.935674	+	0.352865i	$0.114792\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		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
$275$	−28.6237	+	16.5259i	−1.72608	+	0.996550i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−24.4949	−	14.1421i	−1.46124	−	0.843649i	−0.462174	−	0.886789i	$-0.652930\pi$
$281$	−24.4949	−	14.1421i	−1.46124	−	0.843649i	−0.999069	+	0.0431402i	$0.986264\pi$
$282$		0			0
$283$	−11.0000	−	19.0526i	−0.653882	−	1.13256i	−0.982173	−	0.187980i	$-0.939806\pi$
$283$	−11.0000	−	19.0526i	−0.653882	−	1.13256i	0.328291	−	0.944577i	$-0.393527\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	11.3939			0.670228
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		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
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−24.3485			−1.38964			−0.694820	−	0.719183i	$-0.744516\pi$
$307$	−24.3485			−1.38964			−0.694820	+	0.719183i	$0.744516\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−17.1969	+	29.7860i	−0.972028	+	1.68360i	−0.282617	+	0.959233i	$0.591202\pi$
$313$	−17.1969	+	29.7860i	−0.972028	+	1.68360i	−0.689412	+	0.724370i	$0.742131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$			15.0314i			0.836371i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	13.0000	−	22.5167i	0.714545	−	1.23763i	−0.248590	−	0.968609i	$-0.579967\pi$
$331$	13.0000	−	22.5167i	0.714545	−	1.23763i	0.963135	−	0.269019i	$-0.0866994\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−11.1969	−	19.3937i	−0.609936	−	1.05644i	−0.991250	−	0.131995i	$-0.957862\pi$
$337$	−11.1969	−	19.3937i	−0.609936	−	1.05644i	0.381314	−	0.924445i	$-0.375472\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.4217	−	11.7905i	1.09629	−	0.632945i	0.161048	−	0.986947i	$-0.448512\pi$
$347$	20.4217	−	11.7905i	1.09629	−	0.632945i	0.935245	+	0.354001i	$0.115179\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−32.2980	−	18.6472i	−1.71905	−	0.992492i	−0.920677	−	0.390324i	$-0.872363\pi$
$353$	−32.2980	−	18.6472i	−1.71905	−	0.992492i	−0.798369	−	0.602168i	$-0.794304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	21.3031			1.12121
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	11.6515			0.598499			0.299249	−	0.954175i	$-0.403264\pi$
$379$	11.6515			0.598499			0.299249	+	0.954175i	$0.403264\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		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
$400$		0			0
$401$	12.6464	−	7.30142i	0.631532	−	0.364615i	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	12.6464	−	7.30142i	0.631532	−	0.364615i	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
$409$	−20.1969	−	34.9821i	−0.998674	−	1.72975i	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−20.1969	−	34.9821i	−0.998674	−	1.72975i	−0.454759	−	0.890614i	$-0.650275\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−31.8434	+	18.3848i	−1.55565	+	0.898155i	−0.557986	+	0.829851i	$0.688426\pi$
$419$	−31.8434	+	18.3848i	−1.55565	+	0.898155i	−0.997665	+	0.0683046i	$0.978241\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.2526	−	5.91931i	−0.497322	−	0.287129i
$426$		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
$433$	−33.6969			−1.61937			−0.809686	−	0.586864i	$-0.800362\pi$
$433$	−33.6969			−1.61937			−0.809686	+	0.586864i	$0.800362\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−32.7247	−	18.8936i	−1.55480	−	0.897664i	−0.997740	−	0.0671913i	$-0.978596\pi$
$443$	−32.7247	−	18.8936i	−1.55480	−	0.897664i	−0.557059	−	0.830473i	$-0.688070\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	33.5446i		−	1.58307i	−0.611124	−	0.791535i	$-0.709282\pi$
$449$		−	33.5446i		−	1.58307i	0.611124	−	0.791535i	$-0.290718\pi$
$450$		0			0
$451$	71.7423			3.37822
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.19694	+	14.1975i	−0.383437	+	0.664132i	−0.991551	−	0.129718i	$-0.958593\pi$
$457$	−8.19694	+	14.1975i	−0.383437	+	0.664132i	0.608114	+	0.793849i	$0.291926\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	41.5371i		−	1.92211i	−0.276360	−	0.961054i	$-0.589128\pi$
$467$		−	41.5371i		−	1.92211i	0.276360	−	0.961054i	$-0.410872\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−70.6918	+	40.8140i	−3.25041	+	1.87663i
$474$		0			0
$475$	−15.8712	+	27.4897i	−0.728219	+	1.26131i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		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
$489$		0			0
$490$		0			0
$491$	25.3763	−	14.6510i	1.14522	−	0.661190i	0.197499	−	0.980303i	$-0.436718\pi$
$491$	25.3763	−	14.6510i	1.14522	−	0.661190i	0.947717	+	0.319113i	$0.103385\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−21.8712	−	37.8820i	−0.979088	−	1.69583i	−0.665725	−	0.746197i	$-0.731878\pi$
$499$	−21.8712	−	37.8820i	−0.979088	−	1.69583i	−0.313363	−	0.949633i	$-0.601456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	27.8236i		−	1.21897i	−0.792797	−	0.609486i	$-0.791376\pi$
$521$		−	27.8236i		−	1.21897i	0.792797	−	0.609486i	$-0.208624\pi$
$522$		0			0
$523$	−38.0000			−1.66162			−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000			−1.66162			−0.830812	+	0.556553i	$0.812124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			46.2726i			1.99310i
$540$		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
$547$	15.1742	−	26.2825i	0.648803	−	1.12376i	−0.334606	−	0.942358i	$-0.608603\pi$
$547$	15.1742	−	26.2825i	0.648803	−	1.12376i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$		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			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.57832	+	3.79799i	−0.277243	+	0.160066i	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−6.57832	+	3.79799i	−0.277243	+	0.160066i	0.354932	+	0.934892i	$0.384504\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.7020	+	12.5297i	0.909797	+	0.525271i	0.880366	−	0.474295i	$-0.157297\pi$
$569$	21.7020	+	12.5297i	0.909797	+	0.525271i	0.0294311	+	0.999567i	$0.490630\pi$
$570$		0			0
$571$	−12.8712	−	22.2935i	−0.538642	−	0.932955i	−0.998978	−	0.0452101i	$-0.985604\pi$
$571$	−12.8712	−	22.2935i	−0.538642	−	0.932955i	0.460336	−	0.887745i	$-0.347729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	46.3939			1.93140			0.965701	−	0.259656i	$-0.0836092\pi$
$577$	46.3939			1.93140			0.965701	+	0.259656i	$0.0836092\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.3207	+	9.42274i	0.673626	+	0.388918i	0.797449	−	0.603386i	$-0.206182\pi$
$587$	16.3207	+	9.42274i	0.673626	+	0.388918i	−0.123823	+	0.992304i	$0.539516\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.2548i			1.85839i	0.369586	+	0.929197i	$0.379500\pi$
$593$			45.2548i			1.85839i	−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−4.15153	+	7.19066i	−0.169344	+	0.293313i	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−4.15153	+	7.19066i	−0.169344	+	0.293313i	0.768845	+	0.639435i	$0.220832\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	39.6464	−	22.8899i	1.59611	−	0.921512i	0.603877	−	0.797077i	$-0.293622\pi$
$617$	39.6464	−	22.8899i	1.59611	−	0.921512i	0.992228	−	0.124434i	$-0.0397116\pi$
$618$		0			0
$619$	−24.8712	+	43.0781i	−0.999657	+	1.73146i	−0.477143	+	0.878826i	$0.658328\pi$
$619$	−24.8712	+	43.0781i	−0.999657	+	1.73146i	−0.522514	+	0.852631i	$0.675006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		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
$641$	43.7474	+	25.2576i	1.72792	+	0.997615i	0.898470	+	0.439034i	$0.144679\pi$
$641$	43.7474	+	25.2576i	1.72792	+	0.997615i	0.829450	+	0.558581i	$0.188654\pi$
$642$		0			0
$643$	−8.82577	−	15.2867i	−0.348054	−	0.602848i	0.637850	−	0.770161i	$-0.279824\pi$
$643$	−8.82577	−	15.2867i	−0.348054	−	0.602848i	−0.985904	+	0.167313i	$0.946491\pi$
$644$		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
$649$	12.3939			0.486502
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−41.6413	−	24.0416i	−1.62212	−	0.936529i	−0.986353	−	0.164644i	$-0.947352\pi$
$659$	−41.6413	−	24.0416i	−1.62212	−	0.936529i	−0.635763	−	0.771885i	$-0.719314\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		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
$673$	5.00000	−	8.66025i	0.192736	−	0.333828i	−0.753420	−	0.657539i	$-0.771597\pi$
$673$	5.00000	−	8.66025i	0.192736	−	0.333828i	0.946156	+	0.323711i	$0.104931\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			20.8167i			0.796530i	0.917270	+	0.398265i	$0.130387\pi$
$683$			20.8167i			0.796530i	−0.917270	+	0.398265i	$0.869613\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−23.0000	+	39.8372i	−0.874961	+	1.51548i	−0.0181572	+	0.999835i	$0.505780\pi$
$691$	−23.0000	+	39.8372i	−0.874961	+	1.51548i	−0.856804	+	0.515642i	$0.827553\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	12.8485	+	22.2542i	0.486670	+	0.842938i
$698$		0			0
$699$		0			0
$700$		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		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		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			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−25.3207	−	14.6189i	−0.936519	−	0.540699i
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	2.30351i		−	0.0848508i
$738$		0			0
$739$	19.7423			0.726234			0.363117	−	0.931744i	$-0.381713\pi$
$739$	19.7423			0.726234			0.363117	+	0.931744i	$0.381713\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		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			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9.79796	−	5.65685i	0.355176	−	0.205061i	−0.311787	−	0.950152i	$-0.600927\pi$
$761$	9.79796	−	5.65685i	0.355176	−	0.205061i	0.666962	+	0.745091i	$0.267594\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	0.993313	−	0.115454i	$-0.0368323\pi$
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	−0.596643	+	0.802507i	$0.703499\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	59.6691	−	34.4500i	2.13787	−	1.23430i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	25.0000	+	43.3013i	0.891154	+	1.54352i	0.838494	+	0.544911i	$0.183437\pi$
$787$	25.0000	+	43.3013i	0.891154	+	1.54352i	0.0526599	+	0.998613i	$0.483230\pi$
$788$		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		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	89.8610	+	51.8813i	3.17112	+	1.83085i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			23.0881i			0.811735i	0.913932	+	0.405868i	$0.133031\pi$
$809$			23.0881i			0.811735i	−0.913932	+	0.405868i	$0.866969\pi$
$810$		0			0
$811$	55.7423			1.95738			0.978689	−	0.205347i	$-0.0658323\pi$
$811$	55.7423			1.95738			0.978689	+	0.205347i	$0.0658323\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−39.1969	+	67.8911i	−1.37133	+	2.37521i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	19.7990i		−	0.688478i	−0.938882	−	0.344239i	$-0.888137\pi$
$827$		−	19.7990i		−	0.688478i	0.938882	−	0.344239i	$-0.111863\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−14.3536	+	8.28704i	−0.497322	+	0.287129i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	14.5000	+	25.1147i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.5959	+	11.3137i	0.669384	+	0.386469i	0.795843	−	0.605503i	$-0.207028\pi$
$857$	19.5959	+	11.3137i	0.669384	+	0.386469i	−0.126459	+	0.991972i	$0.540361\pi$
$858$		0			0
$859$	18.1742	+	31.4787i	0.620097	+	1.07404i	0.989467	+	0.144757i	$0.0462401\pi$
$859$	18.1742	+	31.4787i	0.620097	+	1.07404i	−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
$867$		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	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	56.5685i		−	1.90584i	−0.303218	−	0.952921i	$-0.598061\pi$
$881$		−	56.5685i		−	1.90584i	0.303218	−	0.952921i	$-0.401939\pi$
$882$		0			0
$883$	−50.4393			−1.69742			−0.848709	−	0.528861i	$-0.822619\pi$
$883$	−50.4393			−1.69742			−0.848709	+	0.528861i	$0.822619\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		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
$898$		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
$907$	28.2196	−	48.8779i	0.937018	−	1.62296i	0.166022	−	0.986122i	$-0.446908\pi$
$907$	28.2196	−	48.8779i	0.937018	−	1.62296i	0.770996	−	0.636841i	$-0.219759\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$		0			0
$913$	−9.34847	−	16.1920i	−0.309389	−	0.535878i
$914$		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
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−24.4949	−	14.1421i	−0.803652	−	0.463988i	0.0410949	−	0.999155i	$-0.486915\pi$
$929$	−24.4949	−	14.1421i	−0.803652	−	0.463988i	−0.844746	+	0.535167i	$0.820249\pi$
$930$		0			0
$931$	22.2196	+	38.4855i	0.728219	+	1.26131i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.770153	−	0.444648i	−0.0250266	−	0.0144491i	0.487435	−	0.873160i	$-0.337933\pi$
$947$	−0.770153	−	0.444648i	−0.0250266	−	0.0144491i	−0.512461	+	0.858710i	$0.671266\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	59.0005i		−	1.91121i	−0.294646	−	0.955607i	$-0.595202\pi$
$953$		−	59.0005i		−	1.91121i	0.294646	−	0.955607i	$-0.404798\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.5000	+	26.8468i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			31.1127i			0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$			31.1127i			0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−31.4444	+	18.1544i	−1.00600	+	0.580812i	−0.910017	−	0.414572i	$-0.863931\pi$
$977$	−31.4444	+	18.1544i	−1.00600	+	0.580812i	−0.0959785	+	0.995383i	$0.530598\pi$
$978$		0			0
$979$	−18.6969	+	32.3840i	−0.597557	+	1.03500i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		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	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.p.a.143.1		4
3.2	odd	2		288.2.p.a.47.2		4
4.3	odd	2		216.2.l.a.35.1		4
8.3	odd	2	CM	864.2.p.a.143.1		4
8.5	even	2		216.2.l.a.35.1		4
9.2	odd	6		2592.2.f.a.1295.1		4
9.4	even	3		288.2.p.a.239.2		4
9.5	odd	6	inner	864.2.p.a.719.1		4
9.7	even	3		2592.2.f.a.1295.4		4
12.11	even	2		72.2.l.a.11.2	✓	4
24.5	odd	2		72.2.l.a.11.2	✓	4
24.11	even	2		288.2.p.a.47.2		4
36.7	odd	6		648.2.f.a.323.3		4
36.11	even	6		648.2.f.a.323.2		4
36.23	even	6		216.2.l.a.179.1		4
36.31	odd	6		72.2.l.a.59.2	yes	4
72.5	odd	6		216.2.l.a.179.1		4
72.11	even	6		2592.2.f.a.1295.1		4
72.13	even	6		72.2.l.a.59.2	yes	4
72.29	odd	6		648.2.f.a.323.2		4
72.43	odd	6		2592.2.f.a.1295.4		4
72.59	even	6	inner	864.2.p.a.719.1		4
72.61	even	6		648.2.f.a.323.3		4
72.67	odd	6		288.2.p.a.239.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.l.a.11.2	✓	4	12.11	even	2
72.2.l.a.11.2	✓	4	24.5	odd	2
72.2.l.a.59.2	yes	4	36.31	odd	6
72.2.l.a.59.2	yes	4	72.13	even	6
216.2.l.a.35.1		4	4.3	odd	2
216.2.l.a.35.1		4	8.5	even	2
216.2.l.a.179.1		4	36.23	even	6
216.2.l.a.179.1		4	72.5	odd	6
288.2.p.a.47.2		4	3.2	odd	2
288.2.p.a.47.2		4	24.11	even	2
288.2.p.a.239.2		4	9.4	even	3
288.2.p.a.239.2		4	72.67	odd	6
648.2.f.a.323.2		4	36.11	even	6
648.2.f.a.323.2		4	72.29	odd	6
648.2.f.a.323.3		4	36.7	odd	6
648.2.f.a.323.3		4	72.61	even	6
864.2.p.a.143.1		4	1.1	even	1	trivial
864.2.p.a.143.1		4	8.3	odd	2	CM
864.2.p.a.719.1		4	9.5	odd	6	inner
864.2.p.a.719.1		4	72.59	even	6	inner
2592.2.f.a.1295.1		4	9.2	odd	6
2592.2.f.a.1295.1		4	72.11	even	6
2592.2.f.a.1295.4		4	9.7	even	3
2592.2.f.a.1295.4		4	72.43	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q+(-5.72474 - 3.30518i) q^{11} -2.36773i q^{17} -6.34847 q^{19} +(2.50000 - 4.33013i) q^{25} +(-9.39898 + 5.42650i) q^{41} +(6.17423 - 10.6941i) q^{43} +(-3.50000 - 6.06218i) q^{49} +(-1.62372 + 0.937458i) q^{59} +(0.174235 + 0.301783i) q^{67} -15.6969 q^{73} +(2.44949 + 1.41421i) q^{83} -5.65685i q^{89} +(4.84847 - 8.39780i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 18 q^{11} + 4 q^{19} + 10 q^{25} - 18 q^{41} + 10 q^{43} - 14 q^{49} + 18 q^{59} - 14 q^{67} - 4 q^{73} - 10 q^{97}+O(q^{100}) \) 4 * q - 18 * q^11 + 4 * q^19 + 10 * q^25 - 18 * q^41 + 10 * q^43 - 14 * q^49 + 18 * q^59 - 14 * q^67 - 4 * q^73 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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