LMFDB - Embedded newform 288.3.t.a.79.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [288,3,Mod(79,288)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("288.79"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	288.t (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$7.84743161358$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			79.1
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	288.79
Dual form			288.3.t.a.175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.94949 - 0.548188i) q^{3} +(8.39898 + 3.23375i) q^{9} +(3.84847 - 6.66574i) q^{11} -30.3939 q^{17} -31.6969 q^{19} +(-12.5000 + 21.6506i) q^{25} +(-23.0000 - 14.1421i) q^{27} +(-15.0051 + 17.5509i) q^{33} +(-40.8939 - 70.8303i) q^{41} +(-40.2423 + 69.7018i) q^{43} +(-24.5000 - 42.4352i) q^{49} +(89.6464 + 16.6616i) q^{51} +(93.4898 + 17.3759i) q^{57} +(-16.2423 - 28.1326i) q^{59} +(35.9393 + 62.2487i) q^{67} +41.6061 q^{73} +(48.7372 - 57.0060i) q^{75} +(60.0857 + 54.3204i) q^{81} +(79.0000 - 136.832i) q^{83} +146.000 q^{89} +(-96.9847 + 167.982i) q^{97} +(53.8786 - 43.5405i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 14 q^{9} - 14 q^{11} - 4 q^{17} - 68 q^{19} - 50 q^{25} - 92 q^{27} - 158 q^{33} - 46 q^{41} - 14 q^{43} - 98 q^{49} + 290 q^{51} + 178 q^{57} + 82 q^{59} - 62 q^{67} + 284 q^{73} - 50 q^{75}+ \cdots - 196 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 14 * q^9 - 14 * q^11 - 4 * q^17 - 68 * q^19 - 50 * q^25 - 92 * q^27 - 158 * q^33 - 46 * q^41 - 14 * q^43 - 98 * q^49 + 290 * q^51 + 178 * q^57 + 82 * q^59 - 62 * q^67 + 284 * q^73 - 50 * q^75 - 34 * q^81 + 316 * q^83 + 584 * q^89 - 94 * q^97 - 196 * q^99

Character values

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

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

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
$3$	−2.94949	−	0.548188i	−0.983163	−	0.182729i
$3$	−2.94949	−	0.548188i	−0.983163	−	0.182729i
$4$		0			0
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		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
$9$	8.39898	+	3.23375i	0.933220	+	0.359306i
$10$		0			0
$11$	3.84847	−	6.66574i	0.349861	−	0.605977i	−0.636364	−	0.771389i	$-0.719562\pi$
$11$	3.84847	−	6.66574i	0.349861	−	0.605977i	0.986224	+	0.165412i	$0.0528955\pi$
$12$		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
$17$	−30.3939			−1.78788			−0.893938	−	0.448192i	$-0.852068\pi$
$17$	−30.3939			−1.78788			−0.893938	+	0.448192i	$0.852068\pi$
$18$		0			0
$19$	−31.6969			−1.66826			−0.834130	−	0.551568i	$-0.814030\pi$
$19$	−31.6969			−1.66826			−0.834130	+	0.551568i	$0.814030\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$26$		0			0
$27$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		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
$33$	−15.0051	+	17.5509i	−0.454700	+	0.531844i
$34$		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
$41$	−40.8939	−	70.8303i	−0.997412	−	1.72757i	−0.560976	−	0.827832i	$-0.689574\pi$
$41$	−40.8939	−	70.8303i	−0.997412	−	1.72757i	−0.436436	−	0.899735i	$-0.643759\pi$
$42$		0			0
$43$	−40.2423	+	69.7018i	−0.935869	+	1.62097i	−0.162791	+	0.986661i	$0.552050\pi$
$43$	−40.2423	+	69.7018i	−0.935869	+	1.62097i	−0.773078	+	0.634311i	$0.781284\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	−24.5000	−	42.4352i	−0.500000	−	0.866025i
$50$		0			0
$51$	89.6464	+	16.6616i	1.75777	+	0.326697i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	93.4898	+	17.3759i	1.64017	+	0.304840i
$58$		0			0
$59$	−16.2423	−	28.1326i	−0.275294	−	0.476823i	0.694915	−	0.719092i	$-0.255442\pi$
$59$	−16.2423	−	28.1326i	−0.275294	−	0.476823i	−0.970209	+	0.242268i	$0.922109\pi$
$60$		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
$67$	35.9393	+	62.2487i	0.536407	+	0.929085i	0.999094	+	0.0425626i	$0.0135522\pi$
$67$	35.9393	+	62.2487i	0.536407	+	0.929085i	−0.462687	+	0.886522i	$0.653114\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	41.6061			0.569947			0.284973	−	0.958535i	$-0.408015\pi$
$73$	41.6061			0.569947			0.284973	+	0.958535i	$0.408015\pi$
$74$		0			0
$75$	48.7372	−	57.0060i	0.649830	−	0.760080i
$76$		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
$81$	60.0857	+	54.3204i	0.741799	+	0.670622i
$82$		0			0
$83$	79.0000	−	136.832i	0.951807	−	1.64858i	0.210296	−	0.977638i	$-0.432557\pi$
$83$	79.0000	−	136.832i	0.951807	−	1.64858i	0.741511	−	0.670941i	$-0.234110\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	146.000			1.64045			0.820225	−	0.572041i	$-0.193848\pi$
$89$	146.000			1.64045			0.820225	+	0.572041i	$0.193848\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−96.9847	+	167.982i	−0.999842	+	1.73178i	−0.484536	+	0.874771i	$0.661012\pi$
$97$	−96.9847	+	167.982i	−0.999842	+	1.73178i	−0.515306	+	0.857006i	$0.672322\pi$
$98$		0			0
$99$	53.8786	−	43.5405i	0.544228	−	0.439803i
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		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
$107$	−191.879			−1.79326			−0.896629	−	0.442783i	$-0.853991\pi$
$107$	−191.879			−1.79326			−0.896629	+	0.442783i	$0.853991\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−49.0000	−	84.8705i	−0.433628	−	0.751066i	0.563554	−	0.826079i	$-0.309434\pi$
$113$	−49.0000	−	84.8705i	−0.433628	−	0.751066i	−0.997183	+	0.0750128i	$0.976100\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	30.8786	+	53.4833i	0.255195	+	0.442010i
$122$		0			0
$123$	81.7878	+	231.331i	0.664941	+	1.88074i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	156.904	−	183.524i	1.21631	−	1.42267i
$130$		0			0
$131$	31.0000	+	53.6936i	0.236641	+	0.409875i	0.959748	−	0.280861i	$-0.0906201\pi$
$131$	31.0000	+	53.6936i	0.236641	+	0.409875i	−0.723107	+	0.690736i	$0.757287\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.712246	+	1.23365i	−0.00519888	+	0.00900472i	−0.868613	−	0.495491i	$-0.834988\pi$
$137$	−0.712246	+	1.23365i	−0.00519888	+	0.00900472i	0.863414	+	0.504496i	$0.168322\pi$
$138$		0			0
$139$	−132.333	−	229.208i	−0.952037	−	1.64898i	−0.741007	−	0.671497i	$-0.765652\pi$
$139$	−132.333	−	229.208i	−0.952037	−	1.64898i	−0.211030	−	0.977480i	$-0.567682\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	49.0000	+	138.593i	0.333333	+	0.942809i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		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
$153$	−255.278	−	98.2862i	−1.66848	−	0.642394i
$154$		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
$163$	322.000			1.97546			0.987730	−	0.156171i	$-0.0499150\pi$
$163$	322.000			1.97546			0.987730	+	0.156171i	$0.0499150\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−84.5000	+	146.358i	−0.500000	+	0.866025i
$170$		0			0
$171$	−266.222	−	102.500i	−1.55685	−	0.599415i
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	32.4847	+	91.8806i	0.183529	+	0.519099i
$178$		0			0
$179$	34.0000			0.189944			0.0949721	−	0.995480i	$-0.469724\pi$
$179$	34.0000			0.189944			0.0949721	+	0.995480i	$0.469724\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−116.970	+	202.598i	−0.625507	+	1.08341i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−137.166	−	237.579i	−0.710706	−	1.23098i	−0.964592	−	0.263745i	$-0.915042\pi$
$193$	−137.166	−	237.579i	−0.710706	−	1.23098i	0.253886	−	0.967234i	$-0.418291\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	−71.8786	−	203.303i	−0.357605	−	1.01146i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−121.985	+	211.284i	−0.583659	+	1.01093i
$210$		0			0
$211$	−113.000	−	195.722i	−0.535545	−	0.927591i	−0.999137	−	0.0415423i	$-0.986773\pi$
$211$	−113.000	−	195.722i	−0.535545	−	0.927591i	0.463592	−	0.886049i	$-0.346560\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−122.717	−	22.8080i	−0.560351	−	0.104146i
$220$		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
$225$	−175.000	+	141.421i	−0.777778	+	0.628539i
$226$		0			0
$227$	−148.242	+	256.763i	−0.653050	+	1.13112i	0.329329	+	0.944215i	$0.393178\pi$
$227$	−148.242	+	256.763i	−0.653050	+	1.13112i	−0.982379	+	0.186900i	$0.940156\pi$
$228$		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
$233$	−70.0306			−0.300561			−0.150280	−	0.988643i	$-0.548018\pi$
$233$	−70.0306			−0.300561			−0.150280	+	0.988643i	$0.548018\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	239.560	−	414.930i	0.994026	−	1.72170i	0.402490	−	0.915425i	$-0.368145\pi$
$241$	239.560	−	414.930i	0.994026	−	1.72170i	0.591536	−	0.806279i	$-0.298522\pi$
$242$		0			0
$243$	−147.444	−	193.156i	−0.606767	−	0.794880i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−308.019	+	360.278i	−1.23703	+	1.44690i
$250$		0			0
$251$	−71.3337			−0.284198			−0.142099	−	0.989852i	$-0.545385\pi$
$251$	−71.3337			−0.284198			−0.142099	+	0.989852i	$0.545385\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	243.469	+	421.701i	0.947352	+	1.64086i	0.750973	+	0.660333i	$0.229585\pi$
$257$	243.469	+	421.701i	0.947352	+	1.64086i	0.196379	+	0.980528i	$0.437082\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−430.626	−	80.0355i	−1.61283	−	0.299758i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	96.2117	+	166.644i	0.349861	+	0.605977i
$276$		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
$281$	119.000	−	206.114i	0.423488	−	0.733502i	−0.572790	−	0.819702i	$-0.694139\pi$
$281$	119.000	−	206.114i	0.423488	−	0.733502i	0.996278	+	0.0862000i	$0.0274724\pi$
$282$		0			0
$283$	−41.0000	−	71.0141i	−0.144876	−	0.250933i	0.784450	−	0.620191i	$-0.212945\pi$
$283$	−41.0000	−	71.0141i	−0.144876	−	0.250933i	−0.929327	+	0.369258i	$0.879612\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	634.788			2.19650
$290$		0			0
$291$	378.141	−	442.297i	1.29945	−	1.51992i
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−182.783	+	98.8865i	−0.615430	+	0.332951i
$298$		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
$307$	520.848			1.69657			0.848287	−	0.529537i	$-0.177634\pi$
$307$	520.848			1.69657			0.848287	+	0.529537i	$0.177634\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	15.4694	−	26.7938i	0.0494230	−	0.0856031i	−0.840256	−	0.542191i	$-0.817595\pi$
$313$	15.4694	−	26.7938i	0.0494230	−	0.0856031i	0.889679	+	0.456587i	$0.150928\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	565.944	+	105.186i	1.76307	+	0.327681i
$322$		0			0
$323$	963.393			2.98264
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7.00000	−	12.1244i	0.0211480	−	0.0366295i	−0.855258	−	0.518203i	$-0.826601\pi$
$331$	7.00000	−	12.1244i	0.0211480	−	0.0366295i	0.876406	+	0.481573i	$0.159935\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−325.257	−	563.362i	−0.965155	−	1.67170i	−0.709199	−	0.705009i	$-0.750943\pi$
$337$	−325.257	−	563.362i	−0.965155	−	1.67170i	−0.255956	−	0.966688i	$-0.582390\pi$
$338$		0			0
$339$	98.0000	+	277.186i	0.289086	+	0.817657i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	260.030	+	450.385i	0.749366	+	1.29794i	0.948127	+	0.317892i	$0.102975\pi$
$347$	260.030	+	450.385i	0.749366	+	1.29794i	−0.198761	+	0.980048i	$0.563692\pi$
$348$		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
$353$	−245.439	+	425.112i	−0.695294	+	1.20428i	0.274788	+	0.961505i	$0.411392\pi$
$353$	−245.439	+	425.112i	−0.695294	+	1.20428i	−0.970081	+	0.242779i	$0.921941\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	643.696			1.78309
$362$		0			0
$363$	−61.7571	−	174.676i	−0.170130	−	0.481200i
$364$		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
$369$	−114.419	−	727.143i	−0.310080	−	1.97058i
$370$		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
$379$	63.7582			0.168227			0.0841137	−	0.996456i	$-0.473194\pi$
$379$	63.7582			0.168227			0.0841137	+	0.996456i	$0.473194\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−563.393	+	455.290i	−1.45580	+	1.17646i
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−62.0000	−	175.362i	−0.157761	−	0.446215i
$394$		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
$401$	−88.6214	−	153.497i	−0.221001	−	0.382785i	0.734111	−	0.679029i	$-0.237599\pi$
$401$	−88.6214	−	153.497i	−0.221001	−	0.382785i	−0.955112	+	0.296244i	$0.904266\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	239.833	+	415.402i	0.586388	+	1.01565i	0.994701	+	0.102812i	$0.0327839\pi$
$409$	239.833	+	415.402i	0.586388	+	1.01565i	−0.408313	+	0.912842i	$0.633883\pi$
$410$		0			0
$411$	2.77703	−	3.24818i	0.00675677	−	0.00790312i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	264.666	+	748.589i	0.634691	+	1.79518i
$418$		0			0
$419$	−257.000	−	445.137i	−0.613365	−	1.06238i	−0.990669	−	0.136290i	$-0.956482\pi$
$419$	−257.000	−	445.137i	−0.613365	−	1.06238i	0.377304	−	0.926090i	$-0.376851\pi$
$420$		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
$425$	379.923	−	658.047i	0.893938	−	1.54835i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−847.484			−1.95724			−0.978619	−	0.205684i	$-0.934058\pi$
$433$	−847.484			−1.95724			−0.978619	+	0.205684i	$0.934058\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−68.5500	−	435.640i	−0.155442	−	0.987845i
$442$		0			0
$443$	−168.061	+	291.090i	−0.379370	+	0.657087i	−0.990971	−	0.134079i	$-0.957192\pi$
$443$	−168.061	+	291.090i	−0.379370	+	0.657087i	0.611601	+	0.791166i	$0.290526\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−638.757			−1.42262			−0.711311	−	0.702878i	$-0.751898\pi$
$449$	−638.757			−1.42262			−0.711311	+	0.702878i	$0.751898\pi$
$450$		0			0
$451$	−629.515			−1.39582
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−441.620	+	764.909i	−0.966347	+	1.67376i	−0.260394	+	0.965502i	$0.583852\pi$
$457$	−441.620	+	764.909i	−0.966347	+	1.67376i	−0.705953	+	0.708259i	$0.749481\pi$
$458$		0			0
$459$	699.059	+	429.834i	1.52300	+	0.936458i
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		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
$467$	−825.332			−1.76731			−0.883653	−	0.468143i	$-0.844923\pi$
$467$	−825.332			−1.76731			−0.883653	+	0.468143i	$0.844923\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	309.743	+	536.490i	0.654847	+	1.13423i
$474$		0			0
$475$	396.212	−	686.259i	0.834130	−	1.44476i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		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
$489$	−949.736	−	176.517i	−1.94220	−	0.360975i
$490$		0			0
$491$	−452.696	−	784.093i	−0.921989	−	1.59693i	−0.796334	−	0.604857i	$-0.793230\pi$
$491$	−452.696	−	784.093i	−0.921989	−	1.59693i	−0.125655	−	0.992074i	$-0.540103\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−56.6964	−	98.2011i	−0.113620	−	0.196796i	0.803607	−	0.595160i	$-0.202911\pi$
$499$	−56.6964	−	98.2011i	−0.113620	−	0.196796i	−0.917227	+	0.398364i	$0.869578\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	329.464	−	385.360i	0.649830	−	0.760080i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	729.030	+	448.262i	1.42111	+	0.873806i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	738.151			1.41680			0.708398	−	0.705813i	$-0.249418\pi$
$521$	738.151			1.41680			0.708398	+	0.705813i	$0.249418\pi$
$522$		0			0
$523$	−398.000			−0.760994			−0.380497	−	0.924782i	$-0.624247\pi$
$523$	−398.000			−0.760994			−0.380497	+	0.924782i	$0.624247\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	+	458.127i	−0.500000	+	0.866025i
$530$		0			0
$531$	−45.4454	−	288.809i	−0.0855846	−	0.543896i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−100.283	−	18.6384i	−0.186746	−	0.0347084i
$538$		0			0
$539$	−377.150			−0.699722
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−424.515	+	735.281i	−0.776078	+	1.34421i	0.158108	+	0.987422i	$0.449461\pi$
$547$	−424.515	+	735.281i	−0.776078	+	1.34421i	−0.934186	+	0.356785i	$0.883873\pi$
$548$		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		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	456.063	−	533.439i	0.812947	−	0.950871i
$562$		0			0
$563$	−421.150	−	729.454i	−0.748047	−	1.29566i	−0.948758	−	0.316005i	$-0.897658\pi$
$563$	−421.150	−	729.454i	−0.748047	−	1.29566i	0.200710	−	0.979651i	$-0.435675\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	568.014	−	983.830i	0.998268	−	1.72905i	0.448180	−	0.893943i	$-0.352072\pi$
$569$	568.014	−	983.830i	0.998268	−	1.72905i	0.550088	−	0.835107i	$-0.314594\pi$
$570$		0			0
$571$	568.666	+	984.958i	0.995912	+	1.72497i	0.576182	+	0.817321i	$0.304542\pi$
$571$	568.666	+	984.958i	0.995912	+	1.72497i	0.419730	+	0.907649i	$0.362125\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1000.39			−1.73378			−0.866891	−	0.498498i	$-0.833885\pi$
$577$	−1000.39			−1.73378			−0.866891	+	0.498498i	$0.833885\pi$
$578$		0			0
$579$	274.333	+	775.930i	0.473804	+	1.34012i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	159.576	−	276.394i	0.271850	−	0.470858i	−0.697485	−	0.716599i	$-0.745698\pi$
$587$	159.576	−	276.394i	0.271850	−	0.470858i	0.969336	+	0.245741i	$0.0790312\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−862.000			−1.45363			−0.726813	−	0.686836i	$-0.758999\pi$
$593$	−862.000			−1.45363			−0.726813	+	0.686836i	$0.758999\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	−109.530	+	189.711i	−0.182246	+	0.315659i	−0.942645	−	0.333797i	$-0.891670\pi$
$601$	−109.530	+	189.711i	−0.182246	+	0.315659i	0.760399	+	0.649456i	$0.225003\pi$
$602$		0			0
$603$	100.557	+	639.044i	0.166761	+	1.05977i
$604$		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
$617$	−597.893	−	1035.58i	−0.969032	−	1.67841i	−0.698368	−	0.715739i	$-0.746090\pi$
$617$	−597.893	−	1035.58i	−0.969032	−	1.67841i	−0.270665	−	0.962674i	$-0.587243\pi$
$618$		0			0
$619$	−337.150	+	583.962i	−0.544670	+	0.943395i	0.453958	+	0.891023i	$0.350012\pi$
$619$	−337.150	+	583.962i	−0.544670	+	0.943395i	−0.998628	+	0.0523724i	$0.983322\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	−	541.266i	−0.500000	−	0.866025i
$626$		0			0
$627$	475.616	−	556.308i	0.758558	−	0.887254i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	226.000	+	639.225i	0.357030	+	1.00983i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−393.893	+	682.242i	−0.614497	+	1.06434i	0.375975	+	0.926630i	$0.377308\pi$
$641$	−393.893	+	682.242i	−0.614497	+	1.06434i	−0.990472	+	0.137711i	$0.956025\pi$
$642$		0			0
$643$	−119.788	−	207.479i	−0.186296	−	0.322674i	0.757717	−	0.652584i	$-0.226315\pi$
$643$	−119.788	−	207.479i	−0.186296	−	0.322674i	−0.944012	+	0.329910i	$0.892982\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−250.033			−0.385258
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	349.449	+	134.544i	0.531886	+	0.204785i
$658$		0			0
$659$	−497.000	+	860.829i	−0.754173	+	1.30627i	0.191611	+	0.981471i	$0.438629\pi$
$659$	−497.000	+	860.829i	−0.754173	+	1.30627i	−0.945784	+	0.324795i	$0.894705\pi$
$660$		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
$673$	623.000	−	1079.07i	0.925706	−	1.60337i	0.135284	−	0.990807i	$-0.456805\pi$
$673$	623.000	−	1079.07i	0.925706	−	1.60337i	0.790422	−	0.612563i	$-0.209861\pi$
$674$		0			0
$675$	593.686	−	321.188i	0.879535	−	0.475834i
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	577.994	−	676.056i	0.848743	−	0.992740i
$682$		0			0
$683$	1330.66			1.94826			0.974132	−	0.225980i	$-0.0725584\pi$
$683$	1330.66			1.94826			0.974132	+	0.225980i	$0.0725584\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	367.000	−	635.663i	0.531114	−	0.919917i	−0.468226	−	0.883609i	$-0.655107\pi$
$691$	367.000	−	635.663i	0.531114	−	0.919917i	0.999341	−	0.0363084i	$-0.0115599\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1242.92	+	2152.81i	1.78325	+	3.08868i
$698$		0			0
$699$	206.555	+	38.3900i	0.295500	+	0.0549213i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		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		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−934.040	+	1092.51i	−1.29190	+	1.51108i
$724$		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
$729$	329.000	+	650.538i	0.451303	+	0.892371i
$730$		0			0
$731$	1223.12	−	2118.51i	1.67322	−	2.89810i
$732$		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
$737$	553.245			0.750672
$738$		0			0
$739$	−1410.24			−1.90831			−0.954154	−	0.299316i	$-0.903242\pi$
$739$	−1410.24			−1.90831			−0.954154	+	0.299316i	$0.903242\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1106.00	−	893.783i	1.48059	−	1.19650i
$748$		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
$753$	210.398	+	39.1043i	0.279413	+	0.0519313i
$754$		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
$761$	−697.000	−	1207.24i	−0.915900	−	1.58639i	−0.805578	−	0.592490i	$-0.798145\pi$
$761$	−697.000	−	1207.24i	−0.915900	−	1.58639i	−0.110322	−	0.993896i	$-0.535188\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	527.000	+	912.791i	0.685306	+	1.18698i	0.973341	+	0.229364i	$0.0736647\pi$
$769$	527.000	+	912.791i	0.685306	+	1.18698i	−0.288035	+	0.957620i	$0.593002\pi$
$770$		0			0
$771$	−486.939	−	1377.27i	−0.631568	−	1.78634i
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1296.21	+	2245.10i	1.66394	+	2.88203i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	463.000	+	801.940i	0.588310	+	1.01898i	0.994454	+	0.105174i	$0.0335399\pi$
$787$	463.000	+	801.940i	0.588310	+	1.01898i	−0.406144	+	0.913809i	$0.633127\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	1226.25	+	472.128i	1.53090	+	0.589423i
$802$		0			0
$803$	160.120	−	277.336i	0.199402	−	0.345375i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	497.061			0.614414			0.307207	−	0.951643i	$-0.400606\pi$
$809$	497.061			0.614414			0.307207	+	0.951643i	$0.400606\pi$
$810$		0			0
$811$	1307.21			1.61185			0.805924	−	0.592019i	$-0.201669\pi$
$811$	1307.21			1.61185			0.805924	+	0.592019i	$0.201669\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1275.56	−	2209.33i	1.56127	−	2.70420i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		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
$825$	−192.423	−	544.256i	−0.233241	−	0.659704i
$826$		0			0
$827$	−1262.00			−1.52600			−0.762999	−	0.646400i	$-0.776274\pi$
$827$	−1262.00			−1.52600			−0.762999	+	0.646400i	$0.776274\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	744.650	+	1289.77i	0.893938	+	1.54835i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$		0			0
$843$	−463.979	+	542.697i	−0.550390	+	0.643769i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	82.0000	+	231.931i	0.0965842	+	0.273181i
$850$		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
$857$	−601.000	+	1040.96i	−0.701284	+	1.21466i	0.266733	+	0.963771i	$0.414056\pi$
$857$	−601.000	+	1040.96i	−0.701284	+	1.21466i	−0.968016	+	0.250888i	$0.919277\pi$
$858$		0			0
$859$	−624.606	−	1081.85i	−0.727131	−	1.25943i	−0.958091	−	0.286465i	$-0.907520\pi$
$859$	−624.606	−	1081.85i	−0.727131	−	1.25943i	0.230960	−	0.972963i	$-0.425813\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1872.30	−	347.983i	−2.15952	−	0.401365i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1357.79	+	1097.26i	−1.55531	+	1.25688i
$874$		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
$881$	−1438.00			−1.63224			−0.816118	−	0.577885i	$-0.803878\pi$
$881$	−1438.00			−1.63224			−0.816118	+	0.577885i	$0.803878\pi$
$882$		0			0
$883$	−983.879			−1.11425			−0.557123	−	0.830430i	$-0.688095\pi$
$883$	−983.879			−1.11425			−0.557123	+	0.830430i	$0.688095\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	593.324	−	191.466i	0.665908	−	0.214888i
$892$		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
$907$	171.304	−	296.706i	0.188868	−	0.327130i	−0.756005	−	0.654566i	$-0.772851\pi$
$907$	171.304	−	296.706i	0.188868	−	0.327130i	0.944873	+	0.327436i	$0.106185\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$	−608.058	−	1053.19i	−0.666000	−	1.15355i
$914$		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
$921$	−1536.24	−	285.523i	−1.66801	−	0.310014i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−529.000	+	916.255i	−0.569429	+	0.986281i	0.427193	+	0.904160i	$0.359503\pi$
$929$	−529.000	+	916.255i	−0.569429	+	0.986281i	−0.996622	+	0.0821203i	$0.973831\pi$
$930$		0			0
$931$	776.575	+	1345.07i	0.834130	+	1.44476i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−718.000			−0.766275			−0.383138	−	0.923691i	$-0.625157\pi$
$937$	−718.000			−0.766275			−0.383138	+	0.923691i	$0.625157\pi$
$938$		0			0
$939$	−60.3148	+	70.5478i	−0.0642330	+	0.0751308i
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−449.605	+	778.738i	−0.474767	+	0.822321i	−0.999582	−	0.0288952i	$-0.990801\pi$
$947$	−449.605	+	778.738i	−0.474767	+	0.822321i	0.524815	+	0.851216i	$0.324134\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1717.06			1.80174			0.900869	−	0.434090i	$-0.142930\pi$
$953$	1717.06			1.80174			0.900869	+	0.434090i	$0.142930\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−480.500	+	832.250i	−0.500000	+	0.866025i
$962$		0			0
$963$	−1611.58	−	620.487i	−1.67350	−	0.644328i
$964$		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
$969$	−2841.52	−	528.121i	−2.93242	−	0.545016i
$970$		0			0
$971$	−974.000			−1.00309			−0.501545	−	0.865132i	$-0.667235\pi$
$971$	−974.000			−1.00309			−0.501545	+	0.865132i	$0.667235\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−641.166	−	1110.53i	−0.656260	−	1.13668i	−0.981576	−	0.191071i	$-0.938804\pi$
$977$	−641.166	−	1110.53i	−0.656260	−	1.13668i	0.325316	−	0.945605i	$-0.394529\pi$
$978$		0			0
$979$	561.877	−	973.199i	0.573929	−	0.994074i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		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
$993$	−27.2929	+	31.9233i	−0.0274853	+	0.0321484i
$994$		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	288.3.t.a.79.1		4
3.2	odd	2		864.3.t.a.559.1		4
4.3	odd	2		72.3.p.a.43.2	✓	4
8.3	odd	2	CM	288.3.t.a.79.1		4
8.5	even	2		72.3.p.a.43.2	✓	4
9.2	odd	6		2592.3.b.a.1135.2		2
9.4	even	3	inner	288.3.t.a.175.1		4
9.5	odd	6		864.3.t.a.847.1		4
9.7	even	3		2592.3.b.b.1135.1		2
12.11	even	2		216.3.p.a.19.2		4
24.5	odd	2		216.3.p.a.19.2		4
24.11	even	2		864.3.t.a.559.1		4
36.7	odd	6		648.3.b.a.163.2		2
36.11	even	6		648.3.b.b.163.1		2
36.23	even	6		216.3.p.a.91.2		4
36.31	odd	6		72.3.p.a.67.2	yes	4
72.5	odd	6		216.3.p.a.91.2		4
72.11	even	6		2592.3.b.a.1135.2		2
72.13	even	6		72.3.p.a.67.2	yes	4
72.29	odd	6		648.3.b.b.163.1		2
72.43	odd	6		2592.3.b.b.1135.1		2
72.59	even	6		864.3.t.a.847.1		4
72.61	even	6		648.3.b.a.163.2		2
72.67	odd	6	inner	288.3.t.a.175.1		4

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

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.94949 - 0.548188i) q^{3} +(8.39898 + 3.23375i) q^{9} +(3.84847 - 6.66574i) q^{11} -30.3939 q^{17} -31.6969 q^{19} +(-12.5000 + 21.6506i) q^{25} +(-23.0000 - 14.1421i) q^{27} +(-15.0051 + 17.5509i) q^{33} +(-40.8939 - 70.8303i) q^{41} +(-40.2423 + 69.7018i) q^{43} +(-24.5000 - 42.4352i) q^{49} +(89.6464 + 16.6616i) q^{51} +(93.4898 + 17.3759i) q^{57} +(-16.2423 - 28.1326i) q^{59} +(35.9393 + 62.2487i) q^{67} +41.6061 q^{73} +(48.7372 - 57.0060i) q^{75} +(60.0857 + 54.3204i) q^{81} +(79.0000 - 136.832i) q^{83} +146.000 q^{89} +(-96.9847 + 167.982i) q^{97} +(53.8786 - 43.5405i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 14 q^{9} - 14 q^{11} - 4 q^{17} - 68 q^{19} - 50 q^{25} - 92 q^{27} - 158 q^{33} - 46 q^{41} - 14 q^{43} - 98 q^{49} + 290 q^{51} + 178 q^{57} + 82 q^{59} - 62 q^{67} + 284 q^{73} - 50 q^{75}+ \cdots - 196 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 14 * q^9 - 14 * q^11 - 4 * q^17 - 68 * q^19 - 50 * q^25 - 92 * q^27 - 158 * q^33 - 46 * q^41 - 14 * q^43 - 98 * q^49 + 290 * q^51 + 178 * q^57 + 82 * q^59 - 62 * q^67 + 284 * q^73 - 50 * q^75 - 34 * q^81 + 316 * q^83 + 584 * q^89 - 94 * q^97 - 196 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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