LMFDB - Embedded newform 288.2.p.a.47.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,2,Mod(47,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, 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("288.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	288.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:	$2.29969157821$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			47.2
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	288.47
Dual form			288.2.p.a.239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.724745 + 1.57313i) q^{3} +(-1.94949 + 2.28024i) q^{9} +O(q^{10})$	$q+(0.724745 + 1.57313i) q^{3} +(-1.94949 + 2.28024i) q^{9} +(5.72474 + 3.30518i) q^{11} +2.36773i q^{17} -6.34847 q^{19} +(2.50000 - 4.33013i) q^{25} +(-5.00000 - 1.41421i) q^{27} +(-1.05051 + 11.4012i) q^{33} +(9.39898 - 5.42650i) q^{41} +(6.17423 - 10.6941i) q^{43} +(-3.50000 - 6.06218i) q^{49} +(-3.72474 + 1.71600i) q^{51} +(-4.60102 - 9.98698i) q^{57} +(1.62372 - 0.937458i) q^{59} +(0.174235 + 0.301783i) q^{67} -15.6969 q^{73} +(8.62372 + 0.794593i) q^{75} +(-1.39898 - 8.89060i) q^{81} +(-2.44949 - 1.41421i) q^{83} +5.65685i q^{89} +(4.84847 - 8.39780i) q^{97} +(-18.6969 + 6.61037i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 2 * q^9	$ 4 q - 2 q^{3} + 2 q^{9} + 18 q^{11} + 4 q^{19} + 10 q^{25} - 20 q^{27} - 14 q^{33} + 18 q^{41} + 10 q^{43} - 14 q^{49} - 10 q^{51} - 38 q^{57} - 18 q^{59} - 14 q^{67} - 4 q^{73} + 10 q^{75} + 14 q^{81} - 10 q^{97} - 16 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 2 * q^9 + 18 * q^11 + 4 * q^19 + 10 * q^25 - 20 * q^27 - 14 * q^33 + 18 * q^41 + 10 * q^43 - 14 * q^49 - 10 * q^51 - 38 * q^57 - 18 * q^59 - 14 * q^67 - 4 * q^73 + 10 * q^75 + 14 * q^81 - 10 * q^97 - 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$-1$

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
$3$	0.724745	+	1.57313i	0.418432	+	0.908248i
$3$	0.724745	+	1.57313i	0.418432	+	0.908248i
$4$		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
$9$	−1.94949	+	2.28024i	−0.649830	+	0.760080i
$10$		0			0
$11$	5.72474	+	3.30518i	1.72608	+	0.996550i	0.904534	+	0.426401i	$0.140219\pi$
$11$	5.72474	+	3.30518i	1.72608	+	0.996550i	0.821541	+	0.570149i	$0.193114\pi$
$12$		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	$0.0927008\pi$
$17$			2.36773i			0.574258i	−0.957892	+	0.287129i	$0.907299\pi$
$18$		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
$27$	−5.00000	−	1.41421i	−0.962250	−	0.272166i
$28$		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
$33$	−1.05051	+	11.4012i	−0.182870	+	1.98469i
$34$		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	0.468521	−	0.883452i	$-0.344787\pi$
$41$	9.39898	−	5.42650i	1.46787	−	0.847477i	0.999353	+	0.0359748i	$0.0114536\pi$
$42$		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
$51$	−3.72474	+	1.71600i	−0.521569	+	0.240288i
$52$		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
$57$	−4.60102	−	9.98698i	−0.609420	−	1.32281i
$58$		0			0
$59$	1.62372	−	0.937458i	0.211391	−	0.122047i	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	1.62372	−	0.937458i	0.211391	−	0.122047i	0.601958	+	0.798528i	$0.294388\pi$
$60$		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
$75$	8.62372	+	0.794593i	0.995782	+	0.0917517i
$76$		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
$81$	−1.39898	−	8.89060i	−0.155442	−	0.987845i
$82$		0			0
$83$	−2.44949	−	1.41421i	−0.268866	−	0.155230i	0.359506	−	0.933143i	$-0.382945\pi$
$83$	−2.44949	−	1.41421i	−0.268866	−	0.155230i	−0.628372	+	0.777913i	$0.716279\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			5.65685i			0.599625i	0.953998	+	0.299813i	$0.0969242\pi$
$89$			5.65685i			0.599625i	−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
$97$	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
$99$	−18.6969	+	6.61037i	−1.87911	+	0.664367i
$100$		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	$-0.739671\pi$
$107$		−	15.0956i		−	1.45935i	0.683793	−	0.729676i	$-0.260329\pi$
$108$		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	−0.884182	−	0.467143i	$-0.845283\pi$
$113$	−9.79796	+	5.65685i	−0.921714	+	0.532152i	−0.0375328	+	0.999295i	$0.511950\pi$
$114$		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
$123$	15.3485	+	10.8530i	1.38392	+	0.978583i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	21.2980	+	1.96240i	1.87518	+	0.172780i
$130$		0			0
$131$	−12.2474	+	7.07107i	−1.07006	+	0.617802i	−0.928199	−	0.372084i	$-0.878643\pi$
$131$	−12.2474	+	7.07107i	−1.07006	+	0.617802i	−0.141865	+	0.989886i	$0.545310\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.29796	+	3.05878i	0.452635	+	0.261329i	0.708942	−	0.705266i	$-0.249173\pi$
$137$	5.29796	+	3.05878i	0.452635	+	0.261329i	−0.256307	+	0.966595i	$0.582506\pi$
$138$		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
$147$	7.00000	−	9.89949i	0.577350	−	0.816497i
$148$		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
$153$	−5.39898	−	4.61586i	−0.436482	−	0.373170i
$154$		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
$171$	12.3763	−	14.4760i	0.946437	−	1.10701i
$172$		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
$177$	2.65153	+	1.87492i	0.199301	+	0.140927i
$178$		0			0
$179$			19.7990i			1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$			19.7990i			1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$		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
$201$	−0.348469	+	0.492810i	−0.0245791	+	0.0347601i
$202$		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
$219$	−11.3763	−	24.6934i	−0.768737	−	1.66862i
$220$		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
$225$	5.00000	+	14.1421i	0.333333	+	0.942809i
$226$		0			0
$227$	−21.2753	−	12.2833i	−1.41209	−	0.815270i	−0.416503	−	0.909134i	$-0.636745\pi$
$227$	−21.2753	−	12.2833i	−1.41209	−	0.815270i	−0.995585	+	0.0938647i	$0.970078\pi$
$228$		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	$-0.607342\pi$
$233$		−	28.8092i		−	1.88735i	0.330870	−	0.943676i	$-0.392658\pi$
$234$		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
$243$	12.9722	−	8.64420i	0.832167	−	0.554526i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.449490	−	4.87832i	0.0284853	−	0.309151i
$250$		0			0
$251$			10.3602i			0.653930i	0.945036	+	0.326965i	$0.106026\pi$
$251$			10.3602i			0.653930i	−0.945036	+	0.326965i	$0.893974\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−17.6010	+	10.1620i	−1.09792	+	0.633885i	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−17.6010	+	10.1620i	−1.09792	+	0.633885i	−0.162247	+	0.986750i	$0.551874\pi$
$258$		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
$267$	−8.89898	+	4.09978i	−0.544609	+	0.250902i
$268$		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	0.999069	−	0.0431402i	$-0.0137362\pi$
$281$	24.4949	+	14.1421i	1.46124	+	0.843649i	0.462174	+	0.886789i	$0.347070\pi$
$282$		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
$291$	16.7247	+	1.54102i	0.980422	+	0.0903364i
$292$		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
$297$	−23.9495	−	24.6219i	−1.38969	−	1.42871i
$298$		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
$321$	23.7474	−	10.9405i	1.32545	−	0.610639i
$322$		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
$339$	−16.0000	−	11.3137i	−0.869001	−	0.614476i
$340$		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	−0.935245	−	0.354001i	$-0.884821\pi$
$347$	−20.4217	+	11.7905i	−1.09629	+	0.632945i	−0.161048	+	0.986947i	$0.551488\pi$
$348$		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	$0.127637\pi$
$353$	32.2980	+	18.6472i	1.71905	+	0.992492i	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			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	21.3031			1.12121
$362$		0			0
$363$	−32.6969	+	46.2405i	−1.71614	+	2.42699i
$364$		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
$369$	−5.94949	+	32.0108i	−0.309718	+	1.66642i
$370$		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
$387$	12.3485	+	34.9267i	0.627708	+	1.77543i
$388$		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
$393$	−20.0000	−	14.1421i	−1.00887	−	0.713376i
$394$		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	−0.781345	−	0.624099i	$-0.785466\pi$
$401$	−12.6464	+	7.30142i	−0.631532	+	0.364615i	0.149813	+	0.988714i	$0.452133\pi$
$402$		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
$411$	−0.972194	+	10.5512i	−0.0479548	+	0.520453i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−18.3485	+	25.9487i	−0.898528	+	1.27071i
$418$		0			0
$419$	31.8434	−	18.3848i	1.55565	−	0.898155i	0.557986	−	0.829851i	$-0.311574\pi$
$419$	31.8434	−	18.3848i	1.55565	−	0.898155i	0.997665	−	0.0683046i	$-0.0217590\pi$
$420$		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
$441$	20.6464	+	3.83732i	0.983163	+	0.182729i
$442$		0			0
$443$	32.7247	+	18.8936i	1.55480	+	0.897664i	0.997740	+	0.0671913i	$0.0214038\pi$
$443$	32.7247	+	18.8936i	1.55480	+	0.897664i	0.557059	+	0.830473i	$0.311930\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			33.5446i			1.58307i	0.611124	+	0.791535i	$0.290718\pi$
$449$			33.5446i			1.58307i	−0.611124	+	0.791535i	$0.709282\pi$
$450$		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
$459$	3.34847	−	11.8386i	0.156293	−	0.552580i
$460$		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	$0.410872\pi$
$467$			41.5371i			1.92211i	−0.276360	+	0.961054i	$0.589128\pi$
$468$		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
$489$	−1.44949	−	3.14626i	−0.0655482	−	0.142279i
$490$		0			0
$491$	−25.3763	+	14.6510i	−1.14522	+	0.661190i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−25.3763	+	14.6510i	−1.14522	+	0.661190i	−0.197499	+	0.980303i	$0.563282\pi$
$492$		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
$507$	−22.4217	−	2.06594i	−0.995782	−	0.0917517i
$508$		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
$513$	31.7423	+	8.97809i	1.40146	+	0.396392i
$514$		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	$0.208624\pi$
$521$			27.8236i			1.21897i	−0.792797	+	0.609486i	$0.791376\pi$
$522$		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
$531$	−1.02781	+	5.53004i	−0.0446030	+	0.239983i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−31.1464	+	14.3492i	−1.34407	+	0.619214i
$538$		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
$561$	−26.9949	−	2.48732i	−1.13973	−	0.105015i
$562$		0			0
$563$	6.57832	−	3.79799i	0.277243	−	0.160066i	−0.354932	−	0.934892i	$-0.615496\pi$
$563$	6.57832	−	3.79799i	0.277243	−	0.160066i	0.632175	+	0.774826i	$0.282163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.7020	−	12.5297i	−0.909797	−	0.525271i	−0.0294311	−	0.999567i	$-0.509370\pi$
$569$	−21.7020	−	12.5297i	−0.909797	−	0.525271i	−0.880366	+	0.474295i	$0.842703\pi$
$570$		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
$579$	−3.69694	+	5.22826i	−0.153640	+	0.217279i
$580$		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	0.123823	−	0.992304i	$-0.460484\pi$
$587$	−16.3207	−	9.42274i	−0.673626	−	0.388918i	−0.797449	+	0.603386i	$0.793818\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	45.2548i		−	1.85839i	−0.369586	−	0.929197i	$-0.620500\pi$
$593$		−	45.2548i		−	1.85839i	0.369586	−	0.929197i	$-0.379500\pi$
$594$		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
$603$	−1.02781	−	0.191027i	−0.0418555	−	0.00777921i
$604$		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	$0.706378\pi$
$617$	−39.6464	+	22.8899i	−1.59611	+	0.921512i	−0.992228	+	0.124434i	$0.960288\pi$
$618$		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
$627$	6.66913	−	72.3801i	0.266339	−	2.89058i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−14.0000	+	19.7990i	−0.556450	+	0.786939i
$634$		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	$-0.855321\pi$
$641$	−43.7474	−	25.2576i	−1.72792	−	0.997615i	−0.829450	−	0.558581i	$-0.811346\pi$
$642$		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
$657$	30.6010	−	35.7928i	1.19386	−	1.39641i
$658$		0			0
$659$	41.6413	+	24.0416i	1.62212	+	0.936529i	0.986353	+	0.164644i	$0.0526477\pi$
$659$	41.6413	+	24.0416i	1.62212	+	0.936529i	0.635763	+	0.771885i	$0.280686\pi$
$660$		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
$675$	−18.6237	+	18.1151i	−0.716827	+	0.697251i
$676$		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
$681$	3.90408	−	42.3710i	0.149605	−	1.62366i
$682$		0			0
$683$		−	20.8167i		−	0.796530i	−0.917270	−	0.398265i	$-0.869613\pi$
$683$		−	20.8167i		−	0.796530i	0.917270	−	0.398265i	$-0.130387\pi$
$684$		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
$699$	45.3207	−	20.8793i	1.71418	−	0.789728i
$700$		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
$723$	47.7702	+	4.40156i	1.77659	+	0.163696i
$724$		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
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		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
$747$	8.00000	−	2.82843i	0.292705	−	0.103487i
$748$		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
$753$	−16.2980	+	7.50850i	−0.593931	+	0.273625i
$754$		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	−0.666962	−	0.745091i	$-0.732406\pi$
$761$	−9.79796	+	5.65685i	−0.355176	+	0.205061i	0.311787	+	0.950152i	$0.399073\pi$
$762$		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
$771$	−28.7423	−	20.3239i	−1.03513	−	0.731948i
$772$		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
$801$	−12.8990	−	11.0280i	−0.455763	−	0.389654i
$802$		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	$-0.866969\pi$
$809$		−	23.0881i		−	0.811735i	0.913932	−	0.405868i	$-0.133031\pi$
$810$		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
$825$	46.7423	+	33.0518i	1.62736	+	1.15072i
$826$		0			0
$827$			19.7990i			0.688478i	0.938882	+	0.344239i	$0.111863\pi$
$827$			19.7990i			0.688478i	−0.938882	+	0.344239i	$0.888137\pi$
$828$		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
$843$	−4.49490	+	48.7832i	−0.154812	+	1.68018i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	22.0000	−	31.1127i	0.755038	−	1.06779i
$850$		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	0.126459	−	0.991972i	$-0.459639\pi$
$857$	−19.5959	−	11.3137i	−0.669384	−	0.386469i	−0.795843	+	0.605503i	$0.792972\pi$
$858$		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
$867$	8.25765	+	17.9241i	0.280445	+	0.608733i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	9.69694	+	27.4271i	0.328192	+	0.928266i
$874$		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	$0.401939\pi$
$881$			56.5685i			1.90584i	−0.303218	+	0.952921i	$0.598061\pi$
$882$		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
$891$	21.3763	−	55.5203i	0.716132	−	1.86000i
$892$		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
$921$	−17.6464	−	38.3034i	−0.581470	−	1.26214i
$922$		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	0.844746	−	0.535167i	$-0.179751\pi$
$929$	24.4949	+	14.1421i	0.803652	+	0.463988i	−0.0410949	+	0.999155i	$0.513085\pi$
$930$		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
$939$	−59.3207	−	5.46583i	−1.93586	−	0.178371i
$940$		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	0.512461	−	0.858710i	$-0.328734\pi$
$947$	0.770153	+	0.444648i	0.0250266	+	0.0144491i	−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			59.0005i			1.91121i	0.294646	+	0.955607i	$0.404798\pi$
$953$			59.0005i			1.91121i	−0.294646	+	0.955607i	$0.595202\pi$
$954$		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
$963$	34.4217	+	29.4288i	1.10922	+	0.948330i
$964$		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
$969$	23.6464	−	10.8940i	0.759633	−	0.349964i
$970$		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
$977$	31.4444	−	18.1544i	1.00600	−	0.580812i	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	31.4444	−	18.1544i	1.00600	−	0.580812i	0.910017	+	0.414572i	$0.136069\pi$
$978$		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
$993$	44.8434	+	4.13188i	1.42306	+	0.131121i
$994$		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	288.2.p.a.47.2		4
3.2	odd	2		864.2.p.a.143.1		4
4.3	odd	2		72.2.l.a.11.2	✓	4
8.3	odd	2	CM	288.2.p.a.47.2		4
8.5	even	2		72.2.l.a.11.2	✓	4
9.2	odd	6		2592.2.f.a.1295.4		4
9.4	even	3		864.2.p.a.719.1		4
9.5	odd	6	inner	288.2.p.a.239.2		4
9.7	even	3		2592.2.f.a.1295.1		4
12.11	even	2		216.2.l.a.35.1		4
24.5	odd	2		216.2.l.a.35.1		4
24.11	even	2		864.2.p.a.143.1		4
36.7	odd	6		648.2.f.a.323.2		4
36.11	even	6		648.2.f.a.323.3		4
36.23	even	6		72.2.l.a.59.2	yes	4
36.31	odd	6		216.2.l.a.179.1		4
72.5	odd	6		72.2.l.a.59.2	yes	4
72.11	even	6		2592.2.f.a.1295.4		4
72.13	even	6		216.2.l.a.179.1		4
72.29	odd	6		648.2.f.a.323.3		4
72.43	odd	6		2592.2.f.a.1295.1		4
72.59	even	6	inner	288.2.p.a.239.2		4
72.61	even	6		648.2.f.a.323.2		4
72.67	odd	6		864.2.p.a.719.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.l.a.11.2	✓	4	4.3	odd	2
72.2.l.a.11.2	✓	4	8.5	even	2
72.2.l.a.59.2	yes	4	36.23	even	6
72.2.l.a.59.2	yes	4	72.5	odd	6
216.2.l.a.35.1		4	12.11	even	2
216.2.l.a.35.1		4	24.5	odd	2
216.2.l.a.179.1		4	36.31	odd	6
216.2.l.a.179.1		4	72.13	even	6
288.2.p.a.47.2		4	1.1	even	1	trivial
288.2.p.a.47.2		4	8.3	odd	2	CM
288.2.p.a.239.2		4	9.5	odd	6	inner
288.2.p.a.239.2		4	72.59	even	6	inner
648.2.f.a.323.2		4	36.7	odd	6
648.2.f.a.323.2		4	72.61	even	6
648.2.f.a.323.3		4	36.11	even	6
648.2.f.a.323.3		4	72.29	odd	6
864.2.p.a.143.1		4	3.2	odd	2
864.2.p.a.143.1		4	24.11	even	2
864.2.p.a.719.1		4	9.4	even	3
864.2.p.a.719.1		4	72.67	odd	6
2592.2.f.a.1295.1		4	9.7	even	3
2592.2.f.a.1295.1		4	72.43	odd	6
2592.2.f.a.1295.4		4	9.2	odd	6
2592.2.f.a.1295.4		4	72.11	even	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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	288.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.724745 + 1.57313i) q^{3} +(-1.94949 + 2.28024i) q^{9} +O(q^{10})\)	\(q+(0.724745 + 1.57313i) q^{3} +(-1.94949 + 2.28024i) q^{9} +(5.72474 + 3.30518i) q^{11} +2.36773i q^{17} -6.34847 q^{19} +(2.50000 - 4.33013i) q^{25} +(-5.00000 - 1.41421i) q^{27} +(-1.05051 + 11.4012i) q^{33} +(9.39898 - 5.42650i) q^{41} +(6.17423 - 10.6941i) q^{43} +(-3.50000 - 6.06218i) q^{49} +(-3.72474 + 1.71600i) q^{51} +(-4.60102 - 9.98698i) q^{57} +(1.62372 - 0.937458i) q^{59} +(0.174235 + 0.301783i) q^{67} -15.6969 q^{73} +(8.62372 + 0.794593i) q^{75} +(-1.39898 - 8.89060i) q^{81} +(-2.44949 - 1.41421i) q^{83} +5.65685i q^{89} +(4.84847 - 8.39780i) q^{97} +(-18.6969 + 6.61037i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 2 * q^9	\( 4 q - 2 q^{3} + 2 q^{9} + 18 q^{11} + 4 q^{19} + 10 q^{25} - 20 q^{27} - 14 q^{33} + 18 q^{41} + 10 q^{43} - 14 q^{49} - 10 q^{51} - 38 q^{57} - 18 q^{59} - 14 q^{67} - 4 q^{73} + 10 q^{75} + 14 q^{81} - 10 q^{97} - 16 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 2 * q^9 + 18 * q^11 + 4 * q^19 + 10 * q^25 - 20 * q^27 - 14 * q^33 + 18 * q^41 + 10 * q^43 - 14 * q^49 - 10 * q^51 - 38 * q^57 - 18 * q^59 - 14 * q^67 - 4 * q^73 + 10 * q^75 + 14 * q^81 - 10 * q^97 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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