LMFDB - Embedded newform 504.3.cu.a.233.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,3,Mod(233,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("504.233"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	504.cu (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$13.7330053238$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 90x^{14} + 2793x^{12} + 37090x^{10} + 214104x^{8} + 463326x^{6} + 257641x^{4} + 28374x^{2} + 9 $ x^16 + 90x^14 + 2793x^12 + 37090x^10 + 214104x^8 + 463326x^6 + 257641x^4 + 28374*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			233.1
Root			$-3.52391i$ of defining polynomial
Character	$\chi$	$=$	504.233
Dual form			504.3.cu.a.305.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.75278 - 4.47607i) q^{5} +(-5.17399 - 4.71485i) q^{7} +(-1.91794 + 1.10732i) q^{11} +15.7846 q^{13} +(-15.1138 + 8.72595i) q^{17} +(-1.79193 + 3.10371i) q^{19} +(6.65847 + 3.84427i) q^{23} +(27.5704 + 47.7533i) q^{25} +24.6165i q^{29} +(22.9586 + 39.7654i) q^{31} +(19.0088 + 59.7123i) q^{35} +(-2.46331 + 4.26658i) q^{37} -28.9060i q^{41} -35.9761 q^{43} +(-41.2203 - 23.7985i) q^{47} +(4.54035 + 48.7892i) q^{49} +(50.1884 - 28.9763i) q^{53} +19.8258 q^{55} +(-100.481 + 58.0128i) q^{59} +(53.9476 - 93.4400i) q^{61} +(-122.374 - 70.6529i) q^{65} +(65.9631 + 114.252i) q^{67} +115.889i q^{71} +(-23.1873 - 40.1615i) q^{73} +(15.1443 + 3.31353i) q^{77} +(-10.0824 + 17.4631i) q^{79} -5.93622i q^{83} +156.232 q^{85} +(-67.5241 - 38.9850i) q^{89} +(-81.6693 - 74.4220i) q^{91} +(27.7848 - 16.0416i) q^{95} -87.3734 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{7} - 24 q^{13} - 12 q^{19} + 92 q^{25} + 32 q^{31} - 132 q^{37} + 24 q^{43} + 64 q^{49} - 440 q^{55} + 184 q^{61} + 332 q^{67} + 188 q^{73} - 112 q^{79} + 256 q^{85} - 252 q^{91} - 1032 q^{97}+O(q^{100}) $ 16 * q - 24 * q^7 - 24 * q^13 - 12 * q^19 + 92 * q^25 + 32 * q^31 - 132 * q^37 + 24 * q^43 + 64 * q^49 - 440 * q^55 + 184 * q^61 + 332 * q^67 + 188 * q^73 - 112 * q^79 + 256 * q^85 - 252 * q^91 - 1032 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$-1$

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
$5$	−7.75278	−	4.47607i	−1.55056	−	0.895214i	−0.998096	−	0.0616742i	$-0.980356\pi$
$5$	−7.75278	−	4.47607i	−1.55056	−	0.895214i	−0.552460	−	0.833540i	$-0.686311\pi$
$6$		0			0
$7$	−5.17399	−	4.71485i	−0.739141	−	0.673550i
$7$	−5.17399	−	4.71485i	−0.739141	−	0.673550i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.91794	+	1.10732i	−0.174358	+	0.100666i	−0.584639	−	0.811293i	$-0.698764\pi$
$11$	−1.91794	+	1.10732i	−0.174358	+	0.100666i	0.410281	+	0.911959i	$0.365431\pi$
$12$		0			0
$13$	15.7846			1.21420			0.607100	−	0.794626i	$-0.292333\pi$
$13$	15.7846			1.21420			0.607100	+	0.794626i	$0.292333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.1138	+	8.72595i	−0.889046	+	0.513291i	−0.873630	−	0.486590i	$-0.838241\pi$
$17$	−15.1138	+	8.72595i	−0.889046	+	0.513291i	−0.0154157	+	0.999881i	$0.504907\pi$
$18$		0			0
$19$	−1.79193	+	3.10371i	−0.0943120	+	0.163353i	−0.909321	−	0.416095i	$-0.863398\pi$
$19$	−1.79193	+	3.10371i	−0.0943120	+	0.163353i	0.815009	+	0.579448i	$0.196732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.65847	+	3.84427i	0.289498	+	0.167142i	0.637716	−	0.770272i	$-0.279879\pi$
$23$	6.65847	+	3.84427i	0.289498	+	0.167142i	−0.348217	+	0.937414i	$0.613213\pi$
$24$		0			0
$25$	27.5704	+	47.7533i	1.10282	+	1.91013i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			24.6165i			0.848845i	0.905464	+	0.424423i	$0.139523\pi$
$29$			24.6165i			0.848845i	−0.905464	+	0.424423i	$0.860477\pi$
$30$		0			0
$31$	22.9586	+	39.7654i	0.740600	+	1.28276i	0.952223	+	0.305405i	$0.0987917\pi$
$31$	22.9586	+	39.7654i	0.740600	+	1.28276i	−0.211623	+	0.977351i	$0.567875\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	19.0088	+	59.7123i	0.543109	+	1.70607i
$36$		0			0
$37$	−2.46331	+	4.26658i	−0.0665760	+	0.115313i	−0.897392	−	0.441234i	$-0.854541\pi$
$37$	−2.46331	+	4.26658i	−0.0665760	+	0.115313i	0.830816	+	0.556547i	$0.187874\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	28.9060i		−	0.705024i	−0.935807	−	0.352512i	$-0.885328\pi$
$41$		−	28.9060i		−	0.705024i	0.935807	−	0.352512i	$-0.114672\pi$
$42$		0			0
$43$	−35.9761			−0.836653			−0.418327	−	0.908297i	$-0.637383\pi$
$43$	−35.9761			−0.836653			−0.418327	+	0.908297i	$0.637383\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−41.2203	−	23.7985i	−0.877027	−	0.506352i	−0.00734985	−	0.999973i	$-0.502340\pi$
$47$	−41.2203	−	23.7985i	−0.877027	−	0.506352i	−0.869677	+	0.493621i	$0.835673\pi$
$48$		0			0
$49$	4.54035	+	48.7892i	0.0926602	+	0.995698i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	50.1884	−	28.9763i	0.946952	−	0.546723i	0.0548190	−	0.998496i	$-0.482542\pi$
$53$	50.1884	−	28.9763i	0.946952	−	0.546723i	0.892133	+	0.451773i	$0.149208\pi$
$54$		0			0
$55$	19.8258			0.360470
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−100.481	+	58.0128i	−1.70307	+	0.983268i	−0.760453	+	0.649393i	$0.775023\pi$
$59$	−100.481	+	58.0128i	−1.70307	+	0.983268i	−0.942617	+	0.333875i	$0.891644\pi$
$60$		0			0
$61$	53.9476	−	93.4400i	0.884387	−	1.53180i	0.0379718	−	0.999279i	$-0.487910\pi$
$61$	53.9476	−	93.4400i	0.884387	−	1.53180i	0.846415	−	0.532524i	$-0.178756\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−122.374	−	70.6529i	−1.88268	−	1.08697i
$66$		0			0
$67$	65.9631	+	114.252i	0.984524	+	1.70525i	0.644031	+	0.765000i	$0.277261\pi$
$67$	65.9631	+	114.252i	0.984524	+	1.70525i	0.340494	+	0.940247i	$0.389406\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			115.889i			1.63224i	0.577883	+	0.816119i	$0.303879\pi$
$71$			115.889i			1.63224i	−0.577883	+	0.816119i	$0.696121\pi$
$72$		0			0
$73$	−23.1873	−	40.1615i	−0.317634	−	0.550158i	0.662360	−	0.749186i	$-0.269555\pi$
$73$	−23.1873	−	40.1615i	−0.317634	−	0.550158i	−0.979994	+	0.199028i	$0.936222\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	15.1443	+	3.31353i	0.196679	+	0.0430328i
$78$		0			0
$79$	−10.0824	+	17.4631i	−0.127625	+	0.221053i	−0.922756	−	0.385385i	$-0.874069\pi$
$79$	−10.0824	+	17.4631i	−0.127625	+	0.221053i	0.795131	+	0.606438i	$0.207402\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	5.93622i		−	0.0715208i	−0.999360	−	0.0357604i	$-0.988615\pi$
$83$		−	5.93622i		−	0.0715208i	0.999360	−	0.0357604i	$-0.0113853\pi$
$84$		0			0
$85$	156.232			1.83802
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−67.5241	−	38.9850i	−0.758697	−	0.438034i	0.0701305	−	0.997538i	$-0.477658\pi$
$89$	−67.5241	−	38.9850i	−0.758697	−	0.438034i	−0.828828	+	0.559504i	$0.810992\pi$
$90$		0			0
$91$	−81.6693	−	74.4220i	−0.897465	−	0.817824i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	27.7848	−	16.0416i	0.292472	−	0.168859i
$96$		0			0
$97$	−87.3734			−0.900757			−0.450378	−	0.892838i	$-0.648711\pi$
$97$	−87.3734			−0.900757			−0.450378	+	0.892838i	$0.648711\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	138.974	−	80.2369i	1.37598	−	0.794425i	0.384311	−	0.923204i	$-0.374439\pi$
$101$	138.974	−	80.2369i	1.37598	−	0.794425i	0.991673	+	0.128778i	$0.0411056\pi$
$102$		0			0
$103$	−8.05119	+	13.9451i	−0.0781669	+	0.135389i	−0.902459	−	0.430776i	$-0.858240\pi$
$103$	−8.05119	+	13.9451i	−0.0781669	+	0.135389i	0.824292	+	0.566165i	$0.191573\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	99.1075	+	57.2198i	0.926239	+	0.534764i	0.885620	−	0.464411i	$-0.153734\pi$
$107$	99.1075	+	57.2198i	0.926239	+	0.534764i	0.0406186	+	0.999175i	$0.487067\pi$
$108$		0			0
$109$	25.8973	+	44.8555i	0.237590	+	0.411518i	0.960022	−	0.279924i	$-0.0903092\pi$
$109$	25.8973	+	44.8555i	0.237590	+	0.411518i	−0.722432	+	0.691442i	$0.756976\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			131.309i			1.16202i	0.813895	+	0.581012i	$0.197343\pi$
$113$			131.309i			1.16202i	−0.813895	+	0.581012i	$0.802657\pi$
$114$		0			0
$115$	−34.4144	−	59.6075i	−0.299256	−	0.518326i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	119.340	+	26.1113i	1.00286	+	0.219423i
$120$		0			0
$121$	−58.0477	+	100.542i	−0.479733	+	0.830922i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	269.825i		−	2.15860i
$126$		0			0
$127$	−219.227			−1.72620			−0.863099	−	0.505035i	$-0.831480\pi$
$127$	−219.227			−1.72620			−0.863099	+	0.505035i	$0.831480\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	174.192	+	100.570i	1.32971	+	0.767710i	0.985255	−	0.171092i	$-0.0547297\pi$
$131$	174.192	+	100.570i	1.32971	+	0.767710i	0.344457	+	0.938802i	$0.388063\pi$
$132$		0			0
$133$	23.9049	−	7.60989i	0.179736	−	0.0572172i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−62.4737	+	36.0692i	−0.456012	+	0.263279i	−0.710366	−	0.703832i	$-0.751471\pi$
$137$	−62.4737	+	36.0692i	−0.456012	+	0.263279i	0.254354	+	0.967111i	$0.418137\pi$
$138$		0			0
$139$	−199.392			−1.43447			−0.717236	−	0.696830i	$-0.754593\pi$
$139$	−199.392			−1.43447			−0.717236	+	0.696830i	$0.754593\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−30.2739	+	17.4787i	−0.211706	+	0.122228i
$144$		0			0
$145$	110.185	−	190.846i	0.759898	−	1.31618i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	48.6559	+	28.0915i	0.326549	+	0.188533i	0.654308	−	0.756228i	$-0.272960\pi$
$149$	48.6559	+	28.0915i	0.326549	+	0.188533i	−0.327759	+	0.944761i	$0.606293\pi$
$150$		0			0
$151$	−53.9534	−	93.4501i	−0.357307	−	0.618875i	0.630203	−	0.776431i	$-0.282972\pi$
$151$	−53.9534	−	93.4501i	−0.357307	−	0.618875i	−0.987510	+	0.157556i	$0.949638\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	411.057i		−	2.65198i
$156$		0			0
$157$	−3.73199	−	6.46400i	−0.0237707	−	0.0411720i	0.853895	−	0.520445i	$-0.174234\pi$
$157$	−3.73199	−	6.46400i	−0.0237707	−	0.0411720i	−0.877666	+	0.479273i	$0.840900\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.3257	−	51.2839i	−0.101402	−	0.318533i
$162$		0			0
$163$	85.9193	−	148.817i	0.527112	−	0.912986i	−0.472388	−	0.881391i	$-0.656608\pi$
$163$	85.9193	−	148.817i	0.527112	−	0.912986i	0.999501	−	0.0315950i	$-0.0100587\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			4.17115i			0.0249769i	0.999922	+	0.0124885i	$0.00397531\pi$
$167$			4.17115i			0.0249769i	−0.999922	+	0.0124885i	$0.996025\pi$
$168$		0			0
$169$	80.1534			0.474281
$170$		0			0
$171$		0			0
$172$		0			0
$173$	111.963	+	64.6418i	0.647185	+	0.373652i	0.787377	−	0.616472i	$-0.211439\pi$
$173$	111.963	+	64.6418i	0.647185	+	0.373652i	−0.140192	+	0.990124i	$0.544772\pi$
$174$		0			0
$175$	82.5009	−	377.066i	0.471434	−	2.15466i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−233.083	+	134.571i	−1.30214	+	0.751791i	−0.980771	−	0.195163i	$-0.937476\pi$
$179$	−233.083	+	134.571i	−1.30214	+	0.751791i	−0.321369	+	0.946954i	$0.604143\pi$
$180$		0			0
$181$	−218.196			−1.20550			−0.602752	−	0.797929i	$-0.705929\pi$
$181$	−218.196			−1.20550			−0.602752	+	0.797929i	$0.705929\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	38.1950	−	22.0519i	0.206460	−	0.119199i
$186$		0			0
$187$	19.3249	−	33.4717i	0.103342	−	0.178993i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−210.766	−	121.686i	−1.10349	−	0.637099i	−0.166353	−	0.986066i	$-0.553199\pi$
$191$	−210.766	−	121.686i	−1.10349	−	0.637099i	−0.937135	+	0.348967i	$0.886532\pi$
$192$		0			0
$193$	−149.368	−	258.713i	−0.773926	−	1.34048i	−0.935396	−	0.353602i	$-0.884957\pi$
$193$	−149.368	−	258.713i	−0.773926	−	1.34048i	0.161470	−	0.986878i	$-0.448377\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	47.0566i		−	0.238866i	−0.992842	−	0.119433i	$-0.961892\pi$
$197$		−	47.0566i		−	0.238866i	0.992842	−	0.119433i	$-0.0381077\pi$
$198$		0			0
$199$	137.324	+	237.852i	0.690069	+	1.19523i	0.971815	+	0.235745i	$0.0757532\pi$
$199$	137.324	+	237.852i	0.690069	+	1.19523i	−0.281746	+	0.959489i	$0.590914\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	116.063	−	127.366i	0.571740	−	0.627417i
$204$		0			0
$205$	−129.385	+	224.102i	−0.631147	+	1.09318i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	7.93698i		−	0.0379760i
$210$		0			0
$211$	166.122			0.787310			0.393655	−	0.919258i	$-0.371210\pi$
$211$	166.122			0.787310			0.393655	+	0.919258i	$0.371210\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	278.915	+	161.031i	1.29728	+	0.748983i
$216$		0			0
$217$	68.7006	−	313.992i	0.316593	−	1.44697i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−238.565	+	137.736i	−1.07948	+	0.623238i
$222$		0			0
$223$	−300.037			−1.34546			−0.672728	−	0.739890i	$-0.734877\pi$
$223$	−300.037			−1.34546			−0.672728	+	0.739890i	$0.734877\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.92278	−	5.72892i	0.0437127	−	0.0252375i	−0.477984	−	0.878368i	$-0.658632\pi$
$227$	9.92278	−	5.72892i	0.0437127	−	0.0252375i	0.521697	+	0.853131i	$0.325299\pi$
$228$		0			0
$229$	−39.4844	+	68.3890i	−0.172421	+	0.298642i	−0.939266	−	0.343191i	$-0.888492\pi$
$229$	−39.4844	+	68.3890i	−0.172421	+	0.298642i	0.766845	+	0.641833i	$0.221826\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	184.800	+	106.694i	0.793133	+	0.457915i	0.841064	−	0.540935i	$-0.181930\pi$
$233$	184.800	+	106.694i	0.793133	+	0.457915i	−0.0479315	+	0.998851i	$0.515263\pi$
$234$		0			0
$235$	213.048	+	369.009i	0.906586	+	1.57025i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.32278i		−	0.0348234i	−0.999848	−	0.0174117i	$-0.994457\pi$
$239$		−	8.32278i		−	0.0348234i	0.999848	−	0.0174117i	$-0.00554259\pi$
$240$		0			0
$241$	−58.6994	−	101.670i	−0.243566	−	0.421868i	0.718162	−	0.695876i	$-0.244984\pi$
$241$	−58.6994	−	101.670i	−0.243566	−	0.421868i	−0.961727	+	0.274008i	$0.911651\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	183.183	−	398.575i	0.747688	−	1.62684i
$246$		0			0
$247$	−28.2848	+	48.9908i	−0.114514	+	0.198343i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	33.1840i		−	0.132207i	−0.997813	−	0.0661036i	$-0.978943\pi$
$251$		−	33.1840i		−	0.132207i	0.997813	−	0.0661036i	$-0.0210568\pi$
$252$		0			0
$253$	−17.0274			−0.0673020
$254$		0			0
$255$		0			0
$256$		0			0
$257$	258.612	+	149.310i	1.00627	+	0.580971i	0.910098	−	0.414394i	$-0.136006\pi$
$257$	258.612	+	149.310i	1.00627	+	0.580971i	0.0961736	+	0.995365i	$0.469340\pi$
$258$		0			0
$259$	32.8614	−	10.4611i	0.126878	−	0.0403903i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	17.6251	−	10.1759i	0.0670157	−	0.0386915i	−0.466118	−	0.884723i	$-0.654348\pi$
$263$	17.6251	−	10.1759i	0.0670157	−	0.0386915i	0.533133	+	0.846031i	$0.321014\pi$
$264$		0			0
$265$	−518.800			−1.95774
$266$		0			0
$267$		0			0
$268$		0			0
$269$	197.643	−	114.109i	0.734732	−	0.424198i	−0.0854188	−	0.996345i	$-0.527223\pi$
$269$	197.643	−	114.109i	0.734732	−	0.424198i	0.820151	+	0.572147i	$0.193889\pi$
$270$		0			0
$271$	−228.622	+	395.985i	−0.843624	+	1.46120i	0.0431875	+	0.999067i	$0.486249\pi$
$271$	−228.622	+	395.985i	−0.843624	+	1.46120i	−0.886811	+	0.462132i	$0.847085\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−105.757	−	61.0587i	−0.384570	−	0.222032i
$276$		0			0
$277$	194.978	+	337.711i	0.703890	+	1.21917i	0.967090	+	0.254433i	$0.0818887\pi$
$277$	194.978	+	337.711i	0.703890	+	1.21917i	−0.263200	+	0.964741i	$0.584778\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			301.317i			1.07230i	0.844122	+	0.536151i	$0.180122\pi$
$281$			301.317i			1.07230i	−0.844122	+	0.536151i	$0.819878\pi$
$282$		0			0
$283$	−107.357	−	185.949i	−0.379355	−	0.657062i	0.611613	−	0.791157i	$-0.290521\pi$
$283$	−107.357	−	185.949i	−0.379355	−	0.657062i	−0.990969	+	0.134094i	$0.957187\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−136.287	+	149.559i	−0.474869	+	0.521112i
$288$		0			0
$289$	7.78430	−	13.4828i	0.0269353	−	0.0466533i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			257.705i			0.879539i	0.898111	+	0.439770i	$0.144940\pi$
$293$			257.705i			0.879539i	−0.898111	+	0.439770i	$0.855060\pi$
$294$		0			0
$295$	1038.68			3.52094
$296$		0			0
$297$		0			0
$298$		0			0
$299$	105.101	+	60.6802i	0.351509	+	0.202944i
$300$		0			0
$301$	186.140	+	169.622i	0.618405	+	0.563528i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−836.488	+	482.946i	−2.74258	+	1.58343i
$306$		0			0
$307$	163.197			0.531586			0.265793	−	0.964030i	$-0.414366\pi$
$307$	163.197			0.531586			0.265793	+	0.964030i	$0.414366\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−260.213	+	150.234i	−0.836698	+	0.483068i	−0.856140	−	0.516743i	$-0.827144\pi$
$311$	−260.213	+	150.234i	−0.836698	+	0.483068i	0.0194425	+	0.999811i	$0.493811\pi$
$312$		0			0
$313$	233.217	−	403.944i	0.745103	−	1.29056i	−0.205043	−	0.978753i	$-0.565734\pi$
$313$	233.217	−	403.944i	0.745103	−	1.29056i	0.950147	−	0.311804i	$-0.100933\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−220.520	−	127.318i	−0.695648	−	0.401633i	0.110076	−	0.993923i	$-0.464890\pi$
$317$	−220.520	−	127.318i	−0.695648	−	0.401633i	−0.805725	+	0.592290i	$0.798224\pi$
$318$		0			0
$319$	−27.2585	−	47.2130i	−0.0854497	−	0.148003i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	62.5451i		−	0.193638i
$324$		0			0
$325$	435.188	+	753.767i	1.33904	+	2.31928i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	101.067	+	317.481i	0.307194	+	0.964987i
$330$		0			0
$331$	−288.476	+	499.655i	−0.871529	+	1.50953i	−0.0111141	+	0.999938i	$0.503538\pi$
$331$	−288.476	+	499.655i	−0.871529	+	1.50953i	−0.860415	+	0.509594i	$0.829796\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1181.02i		−	3.52544i
$336$		0			0
$337$	−248.660			−0.737865			−0.368932	−	0.929456i	$-0.620277\pi$
$337$	−248.660			−0.737865			−0.368932	+	0.929456i	$0.620277\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−88.0665	−	50.8452i	−0.258259	−	0.149106i
$342$		0			0
$343$	206.542	−	273.842i	0.602163	−	0.798373i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−204.684	+	118.174i	−0.589868	+	0.340560i	−0.765045	−	0.643977i	$-0.777283\pi$
$347$	−204.684	+	118.174i	−0.589868	+	0.340560i	0.175178	+	0.984537i	$0.443950\pi$
$348$		0			0
$349$	−148.888			−0.426612			−0.213306	−	0.976985i	$-0.568423\pi$
$349$	−148.888			−0.426612			−0.213306	+	0.976985i	$0.568423\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−413.411	+	238.683i	−1.17113	+	0.676155i	−0.953947	−	0.299975i	$-0.903022\pi$
$353$	−413.411	+	238.683i	−1.17113	+	0.676155i	−0.217188	+	0.976130i	$0.569688\pi$
$354$		0			0
$355$	518.727	−	898.461i	1.46120	−	2.53088i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−317.904	−	183.542i	−0.885527	−	0.511259i	−0.0130503	−	0.999915i	$-0.504154\pi$
$359$	−317.904	−	183.542i	−0.885527	−	0.511259i	−0.872477	+	0.488655i	$0.837488\pi$
$360$		0			0
$361$	174.078	+	301.512i	0.482211	+	0.835213i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			415.151i			1.13740i
$366$		0			0
$367$	−52.0227	−	90.1060i	−0.141751	−	0.245521i	0.786405	−	0.617711i	$-0.211940\pi$
$367$	−52.0227	−	90.1060i	−0.141751	−	0.245521i	−0.928156	+	0.372191i	$0.878607\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−396.294	−	86.7079i	−1.06818	−	0.233714i
$372$		0			0
$373$	−184.661	+	319.842i	−0.495070	+	0.857486i	−0.999984	−	0.00568377i	$-0.998191\pi$
$373$	−184.661	+	319.842i	−0.495070	+	0.857486i	0.504914	+	0.863170i	$0.331524\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			388.562i			1.03067i
$378$		0			0
$379$	−91.9709			−0.242667			−0.121334	−	0.992612i	$-0.538717\pi$
$379$	−91.9709			−0.242667			−0.121334	+	0.992612i	$0.538717\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	235.682	+	136.071i	0.615358	+	0.355277i	0.775059	−	0.631888i	$-0.217720\pi$
$383$	235.682	+	136.071i	0.615358	+	0.355277i	−0.159702	+	0.987165i	$0.551053\pi$
$384$		0			0
$385$	−102.579	−	93.4759i	−0.266438	−	0.242795i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	386.028	−	222.874i	0.992361	−	0.572940i	0.0863819	−	0.996262i	$-0.472469\pi$
$389$	386.028	−	222.874i	0.992361	−	0.572940i	0.905979	+	0.423322i	$0.139136\pi$
$390$		0			0
$391$	−134.179			−0.343170
$392$		0			0
$393$		0			0
$394$		0			0
$395$	156.333	−	90.2586i	0.395779	−	0.228503i
$396$		0			0
$397$	23.1619	−	40.1176i	0.0583423	−	0.101052i	−0.835379	−	0.549674i	$-0.814752\pi$
$397$	23.1619	−	40.1176i	0.0583423	−	0.101052i	0.893721	+	0.448622i	$0.148085\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.21519	+	1.85629i	0.00801792	+	0.00462915i	0.504004	−	0.863702i	$-0.331860\pi$
$401$	3.21519	+	1.85629i	0.00801792	+	0.00462915i	−0.495986	+	0.868331i	$0.665193\pi$
$402$		0			0
$403$	362.392	+	627.681i	0.899236	+	1.55752i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	10.9107i		−	0.0268077i
$408$		0			0
$409$	142.249	+	246.383i	0.347798	+	0.602403i	0.985858	−	0.167583i	$-0.0535963\pi$
$409$	142.249	+	246.383i	0.347798	+	0.602403i	−0.638060	+	0.769986i	$0.720263\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	793.410	+	173.596i	1.92109	+	0.420329i
$414$		0			0
$415$	−26.5709	+	46.0222i	−0.0640264	+	0.110897i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			129.962i			0.310172i	0.987901	+	0.155086i	$0.0495654\pi$
$419$			129.962i			0.310172i	−0.987901	+	0.155086i	$0.950435\pi$
$420$		0			0
$421$	−307.217			−0.729732			−0.364866	−	0.931060i	$-0.618885\pi$
$421$	−307.217			−0.729732			−0.364866	+	0.931060i	$0.618885\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−833.386	−	481.156i	−1.96091	−	1.13213i
$426$		0			0
$427$	−719.680	+	229.103i	−1.68543	+	0.536540i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	404.222	−	233.377i	0.937869	−	0.541479i	0.0485774	−	0.998819i	$-0.484531\pi$
$431$	404.222	−	233.377i	0.937869	−	0.541479i	0.889292	+	0.457340i	$0.151198\pi$
$432$		0			0
$433$	1.18036			0.00272601			0.00136300	−	0.999999i	$-0.499566\pi$
$433$	1.18036			0.00272601			0.00136300	+	0.999999i	$0.499566\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−23.8630	+	13.7773i	−0.0546063	+	0.0315270i
$438$		0			0
$439$	−184.775	+	320.040i	−0.420900	+	0.729020i	−0.996028	−	0.0890436i	$-0.971619\pi$
$439$	−184.775	+	320.040i	−0.420900	+	0.729020i	0.575128	+	0.818064i	$0.304952\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	22.3910	+	12.9274i	0.0505440	+	0.0291816i	0.525059	−	0.851066i	$-0.324043\pi$
$443$	22.3910	+	12.9274i	0.0505440	+	0.0291816i	−0.474515	+	0.880247i	$0.657377\pi$
$444$		0			0
$445$	348.999	+	604.485i	0.784268	+	1.35839i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			690.627i			1.53814i	0.639162	+	0.769072i	$0.279281\pi$
$449$			690.627i			1.53814i	−0.639162	+	0.769072i	$0.720719\pi$
$450$		0			0
$451$	32.0083	+	55.4400i	0.0709718	+	0.122927i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	300.046	+	942.535i	0.659442	+	2.07151i
$456$		0			0
$457$	119.412	−	206.828i	0.261295	−	0.452577i	−0.705291	−	0.708918i	$-0.749184\pi$
$457$	119.412	−	206.828i	0.261295	−	0.452577i	0.966586	+	0.256341i	$0.0825170\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	665.309i		−	1.44319i	−0.692318	−	0.721593i	$-0.743410\pi$
$461$		−	665.309i		−	1.44319i	0.692318	−	0.721593i	$-0.256590\pi$
$462$		0			0
$463$	852.096			1.84038			0.920190	−	0.391473i	$-0.128034\pi$
$463$	852.096			1.84038			0.920190	+	0.391473i	$0.128034\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	241.957	+	139.694i	0.518110	+	0.299131i	0.736161	−	0.676806i	$-0.236637\pi$
$467$	241.957	+	139.694i	0.518110	+	0.299131i	−0.218051	+	0.975937i	$0.569970\pi$
$468$		0			0
$469$	197.386	−	902.143i	0.420866	−	1.92355i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	69.0000	−	39.8372i	0.145877	−	0.0842224i
$474$		0			0
$475$	−197.617			−0.416035
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−225.090	+	129.956i	−0.469917	+	0.271307i	−0.716205	−	0.697890i	$-0.754122\pi$
$479$	−225.090	+	129.956i	−0.469917	+	0.271307i	0.246288	+	0.969197i	$0.420789\pi$
$480$		0			0
$481$	−38.8824	+	67.3462i	−0.0808365	+	0.140013i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	677.387	+	391.089i	1.39667	+	0.806370i
$486$		0			0
$487$	153.354	+	265.618i	0.314896	+	0.545416i	0.979415	−	0.201855i	$-0.0646970\pi$
$487$	153.354	+	265.618i	0.314896	+	0.545416i	−0.664519	+	0.747271i	$0.731364\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			107.088i			0.218102i	0.994036	+	0.109051i	$0.0347811\pi$
$491$			107.088i			0.218102i	−0.994036	+	0.109051i	$0.965219\pi$
$492$		0			0
$493$	−214.802	−	372.049i	−0.435705	−	0.754662i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	546.399	−	599.608i	1.09939	−	1.20646i
$498$		0			0
$499$	203.015	−	351.632i	0.406843	−	0.704673i	−0.587691	−	0.809086i	$-0.699963\pi$
$499$	203.015	−	351.632i	0.406843	−	0.704673i	0.994534	+	0.104412i	$0.0332962\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	300.937i		−	0.598283i	−0.954209	−	0.299142i	$-0.903300\pi$
$503$		−	300.937i		−	0.598283i	0.954209	−	0.299142i	$-0.0967003\pi$
$504$		0			0
$505$	−1436.58			−2.84472
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−392.436	−	226.573i	−0.770993	−	0.445133i	0.0622355	−	0.998061i	$-0.480177\pi$
$509$	−392.436	−	226.573i	−0.770993	−	0.445133i	−0.833229	+	0.552928i	$0.813510\pi$
$510$		0			0
$511$	−69.3850	+	317.120i	−0.135783	+	0.620587i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	124.838	−	72.0753i	0.242404	−	0.139952i
$516$		0			0
$517$	105.411			0.203889
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−338.021	+	195.156i	−0.648792	+	0.374580i	−0.787993	−	0.615684i	$-0.788880\pi$
$521$	−338.021	+	195.156i	−0.648792	+	0.374580i	0.139201	+	0.990264i	$0.455546\pi$
$522$		0			0
$523$	170.504	−	295.322i	0.326012	−	0.564670i	−0.655705	−	0.755018i	$-0.727628\pi$
$523$	170.504	−	295.322i	0.326012	−	0.564670i	0.981717	+	0.190348i	$0.0609616\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−693.982	−	400.671i	−1.31685	−	0.760286i
$528$		0			0
$529$	−234.943	−	406.934i	−0.444127	−	0.769251i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	456.269i		−	0.856040i
$534$		0			0
$535$	−512.239	−	887.224i	−0.957456	−	1.65836i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−62.7336	−	88.5472i	−0.116389	−	0.164280i
$540$		0			0
$541$	455.204	−	788.436i	0.841412	−	1.45737i	−0.0472887	−	0.998881i	$-0.515058\pi$
$541$	455.204	−	788.436i	0.841412	−	1.45737i	0.888701	−	0.458487i	$-0.151609\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	463.673i		−	0.850776i
$546$		0			0
$547$	−93.6660			−0.171236			−0.0856179	−	0.996328i	$-0.527286\pi$
$547$	−93.6660			−0.171236			−0.0856179	+	0.996328i	$0.527286\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−76.4025	−	44.1110i	−0.138662	−	0.0800563i
$552$		0			0
$553$	134.502	−	42.8174i	0.243223	−	0.0774274i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	336.858	−	194.485i	0.604772	−	0.349165i	−0.166145	−	0.986101i	$-0.553132\pi$
$557$	336.858	−	194.485i	0.604772	−	0.349165i	0.770916	+	0.636936i	$0.219799\pi$
$558$		0			0
$559$	−567.868			−1.01586
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−201.709	+	116.457i	−0.358275	+	0.206850i	−0.668324	−	0.743870i	$-0.732988\pi$
$563$	−201.709	+	116.457i	−0.358275	+	0.206850i	0.310049	+	0.950721i	$0.399655\pi$
$564$		0			0
$565$	587.747	−	1018.01i	1.04026	−	1.80178i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−422.193	−	243.753i	−0.741990	−	0.428388i	0.0808021	−	0.996730i	$-0.474252\pi$
$569$	−422.193	−	243.753i	−0.741990	−	0.428388i	−0.822793	+	0.568342i	$0.807585\pi$
$570$		0			0
$571$	380.009	+	658.195i	0.665515	+	1.15271i	0.979145	+	0.203161i	$0.0651216\pi$
$571$	380.009	+	658.195i	0.665515	+	1.15271i	−0.313630	+	0.949545i	$0.601545\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			423.952i			0.737307i
$576$		0			0
$577$	224.653	+	389.110i	0.389346	+	0.674368i	0.992362	−	0.123362i	$-0.0393676\pi$
$577$	224.653	+	389.110i	0.389346	+	0.674368i	−0.603015	+	0.797729i	$0.706034\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−27.9884	+	30.7140i	−0.0481728	+	0.0528640i
$582$		0			0
$583$	−64.1723	+	111.150i	−0.110073	+	0.190651i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	39.3268i		−	0.0669962i	−0.999439	−	0.0334981i	$-0.989335\pi$
$587$		−	39.3268i		−	0.0669962i	0.999439	−	0.0334981i	$-0.0106648\pi$
$588$		0			0
$589$	−164.560			−0.279390
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−364.581	−	210.491i	−0.614807	−	0.354959i	0.160037	−	0.987111i	$-0.448839\pi$
$593$	−364.581	−	210.491i	−0.614807	−	0.354959i	−0.774844	+	0.632152i	$0.782172\pi$
$594$		0			0
$595$	−808.342	−	736.610i	−1.35856	−	1.23800i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	833.627	−	481.295i	1.39170	−	0.803497i	0.398195	−	0.917301i	$-0.369637\pi$
$599$	833.627	−	481.295i	1.39170	−	0.803497i	0.993503	+	0.113804i	$0.0363036\pi$
$600$		0			0
$601$	−313.194			−0.521121			−0.260560	−	0.965458i	$-0.583907\pi$
$601$	−313.194			−0.521121			−0.260560	+	0.965458i	$0.583907\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	900.062	−	519.651i	1.48771	−	0.858927i
$606$		0			0
$607$	−9.21009	+	15.9523i	−0.0151731	+	0.0262806i	−0.873512	−	0.486802i	$-0.838163\pi$
$607$	−9.21009	+	15.9523i	−0.0151731	+	0.0262806i	0.858339	+	0.513083i	$0.171497\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−650.645	−	375.650i	−1.06489	−	0.614812i
$612$		0			0
$613$	−497.804	−	862.221i	−0.812078	−	1.40656i	−0.911407	−	0.411505i	$-0.865003\pi$
$613$	−497.804	−	862.221i	−0.812078	−	1.40656i	0.0993295	−	0.995055i	$-0.468330\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	425.645i		−	0.689863i	−0.938628	−	0.344931i	$-0.887902\pi$
$617$		−	425.645i		−	0.689863i	0.938628	−	0.344931i	$-0.112098\pi$
$618$		0			0
$619$	−433.536	−	750.907i	−0.700382	−	1.21310i	−0.968332	−	0.249664i	$-0.919680\pi$
$619$	−433.536	−	750.907i	−0.700382	−	1.21310i	0.267950	−	0.963433i	$-0.413654\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	165.560	+	520.074i	0.265747	+	0.834790i
$624$		0			0
$625$	−518.494	+	898.057i	−0.829590	+	1.43689i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	85.9789i		−	0.136691i
$630$		0			0
$631$	−657.164			−1.04146			−0.520732	−	0.853720i	$-0.674341\pi$
$631$	−657.164			−1.04146			−0.520732	+	0.853720i	$0.674341\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1699.62	+	981.276i	2.67657	+	1.54532i
$636$		0			0
$637$	71.6676	+	770.118i	0.112508	+	1.20898i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−127.940	+	73.8664i	−0.199595	+	0.115236i	−0.596467	−	0.802638i	$-0.703429\pi$
$641$	−127.940	+	73.8664i	−0.199595	+	0.115236i	0.396872	+	0.917874i	$0.370096\pi$
$642$		0			0
$643$	85.1602			0.132442			0.0662210	−	0.997805i	$-0.478906\pi$
$643$	85.1602			0.132442			0.0662210	+	0.997805i	$0.478906\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−304.830	+	175.993i	−0.471143	+	0.272015i	−0.716718	−	0.697363i	$-0.754357\pi$
$647$	−304.830	+	175.993i	−0.471143	+	0.272015i	0.245575	+	0.969378i	$0.421023\pi$
$648$		0			0
$649$	128.478	−	222.530i	0.197963	−	0.342882i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	770.964	+	445.116i	1.18065	+	0.681648i	0.956165	−	0.292829i	$-0.0945968\pi$
$653$	770.964	+	445.116i	1.18065	+	0.681648i	0.224485	+	0.974478i	$0.427930\pi$
$654$		0			0
$655$	−900.316	−	1559.39i	−1.37453	−	2.38075i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	598.195i		−	0.907732i	−0.891070	−	0.453866i	$-0.850044\pi$
$659$		−	598.195i		−	0.907732i	0.891070	−	0.453866i	$-0.149956\pi$
$660$		0			0
$661$	373.678	+	647.229i	0.565322	+	0.979167i	0.997020	+	0.0771485i	$0.0245816\pi$
$661$	373.678	+	647.229i	0.565322	+	0.979167i	−0.431697	+	0.902019i	$0.642085\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−219.392	−	48.0024i	−0.329913	−	0.0721840i
$666$		0			0
$667$	−94.6324	+	163.908i	−0.141878	+	0.245739i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			238.950i			0.356110i
$672$		0			0
$673$	1200.21			1.78337			0.891687	−	0.452652i	$-0.149522\pi$
$673$	1200.21			1.78337			0.891687	+	0.452652i	$0.149522\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−157.531	−	90.9508i	−0.232690	−	0.134344i	0.379122	−	0.925347i	$-0.376226\pi$
$677$	−157.531	−	90.9508i	−0.232690	−	0.134344i	−0.611813	+	0.791003i	$0.709559\pi$
$678$		0			0
$679$	452.069	+	411.953i	0.665786	+	0.606705i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	520.787	−	300.676i	0.762499	−	0.440229i	−0.0676932	−	0.997706i	$-0.521564\pi$
$683$	520.787	−	300.676i	0.762499	−	0.440229i	0.830192	+	0.557477i	$0.188231\pi$
$684$		0			0
$685$	645.793			0.942764
$686$		0			0
$687$		0			0
$688$		0			0
$689$	792.204	−	457.379i	1.14979	−	0.663831i
$690$		0			0
$691$	−313.485	+	542.972i	−0.453669	+	0.785777i	−0.998611	−	0.0526968i	$-0.983218\pi$
$691$	−313.485	+	542.972i	−0.453669	+	0.785777i	0.544942	+	0.838474i	$0.316552\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1545.84	+	892.491i	2.22423	+	1.28416i
$696$		0			0
$697$	252.232	+	436.879i	0.361882	+	0.626799i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			277.434i			0.395769i	0.980225	+	0.197885i	$0.0634071\pi$
$701$			277.434i			0.395769i	−0.980225	+	0.197885i	$0.936593\pi$
$702$		0			0
$703$	−8.82815	−	15.2908i	−0.0125578	−	0.0217508i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1097.36	−	240.099i	−1.55213	−	0.339602i
$708$		0			0
$709$	−320.286	+	554.751i	−0.451743	+	0.782441i	−0.998494	−	0.0548534i	$-0.982531\pi$
$709$	−320.286	+	554.751i	−0.451743	+	0.782441i	0.546752	+	0.837295i	$0.315864\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			353.036i			0.495141i
$714$		0			0
$715$	312.943			0.437682
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−102.376	−	59.1067i	−0.142386	−	0.0822068i	0.427114	−	0.904198i	$-0.359530\pi$
$719$	−102.376	−	59.1067i	−0.142386	−	0.0822068i	−0.569501	+	0.821991i	$0.692863\pi$
$720$		0			0
$721$	107.406	−	34.1915i	0.148968	−	0.0474223i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1175.52	+	678.687i	−1.62141	+	0.936120i
$726$		0			0
$727$	−623.825			−0.858081			−0.429041	−	0.903285i	$-0.641148\pi$
$727$	−623.825			−0.858081			−0.429041	+	0.903285i	$0.641148\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	543.735	−	313.925i	0.743823	−	0.429447i
$732$		0			0
$733$	76.1467	−	131.890i	0.103884	−	0.179932i	−0.809398	−	0.587261i	$-0.800206\pi$
$733$	76.1467	−	131.890i	0.103884	−	0.179932i	0.913282	+	0.407329i	$0.133540\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−253.027	−	146.085i	−0.343320	−	0.198216i
$738$		0			0
$739$	220.788	+	382.415i	0.298765	+	0.517477i	0.975854	−	0.218425i	$-0.0700918\pi$
$739$	220.788	+	382.415i	0.298765	+	0.517477i	−0.677088	+	0.735902i	$0.736759\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	82.3447i		−	0.110827i	−0.998463	−	0.0554137i	$-0.982352\pi$
$743$		−	82.3447i		−	0.110827i	0.998463	−	0.0554137i	$-0.0176478\pi$
$744$		0			0
$745$	−251.479	−	435.574i	−0.337555	−	0.584663i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−242.999	−	763.332i	−0.324431	−	1.01913i
$750$		0			0
$751$	205.466	−	355.877i	0.273590	−	0.473871i	−0.696189	−	0.717859i	$-0.745122\pi$
$751$	205.466	−	355.877i	0.273590	−	0.473871i	0.969778	+	0.243988i	$0.0784557\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			965.997i			1.27947i
$756$		0			0
$757$	219.158			0.289508			0.144754	−	0.989468i	$-0.453761\pi$
$757$	219.158			0.289508			0.144754	+	0.989468i	$0.453761\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−411.177	−	237.393i	−0.540311	−	0.311949i	0.204894	−	0.978784i	$-0.434315\pi$
$761$	−411.177	−	237.393i	−0.540311	−	0.311949i	−0.745205	+	0.666835i	$0.767648\pi$
$762$		0			0
$763$	77.4944	−	354.184i	0.101565	−	0.464199i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1586.05	+	915.709i	−2.06787	+	1.19388i
$768$		0			0
$769$	109.532			0.142434			0.0712172	−	0.997461i	$-0.477312\pi$
$769$	109.532			0.142434			0.0712172	+	0.997461i	$0.477312\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	614.807	−	354.959i	0.795351	−	0.459196i	−0.0464917	−	0.998919i	$-0.514804\pi$
$773$	614.807	−	354.959i	0.795351	−	0.459196i	0.841843	+	0.539722i	$0.181471\pi$
$774$		0			0
$775$	−1265.95	+	2192.70i	−1.63349	+	2.82929i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	89.7158	+	51.7974i	0.115168	+	0.0664922i
$780$		0			0
$781$	−128.327	−	222.268i	−0.164311	−	0.284594i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			66.8186i			0.0851193i
$786$		0			0
$787$	−167.606	−	290.303i	−0.212969	−	0.368873i	0.739674	−	0.672966i	$-0.234980\pi$
$787$	−167.606	−	290.303i	−0.212969	−	0.368873i	−0.952642	+	0.304093i	$0.901647\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	619.101	−	679.390i	0.782682	−	0.858900i
$792$		0			0
$793$	851.541	−	1474.91i	1.07382	−	1.85991i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	342.591i		−	0.429851i	−0.976630	−	0.214925i	$-0.931049\pi$
$797$		−	342.591i		−	0.429851i	0.976630	−	0.214925i	$-0.0689509\pi$
$798$		0			0
$799$	830.659			1.03962
$800$		0			0
$801$		0			0
$802$		0			0
$803$	88.9437	+	51.3517i	0.110764	+	0.0639498i
$804$		0			0
$805$	−102.981	+	470.667i	−0.127926	+	0.584680i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−306.230	+	176.802i	−0.378530	+	0.218544i	−0.677178	−	0.735819i	$-0.736797\pi$
$809$	−306.230	+	176.802i	−0.378530	+	0.218544i	0.298649	+	0.954363i	$0.403464\pi$
$810$		0			0
$811$	588.302			0.725404			0.362702	−	0.931905i	$-0.381854\pi$
$811$	588.302			0.725404			0.362702	+	0.931905i	$0.381854\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1332.23	+	769.162i	−1.63463	+	0.943757i
$816$		0			0
$817$	64.4665	−	111.659i	0.0789064	−	0.136670i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1244.63	−	718.587i	−1.51599	−	0.875259i	−0.999824	−	0.0187718i	$-0.994024\pi$
$821$	−1244.63	−	718.587i	−1.51599	−	0.875259i	−0.516169	−	0.856487i	$-0.672642\pi$
$822$		0			0
$823$	−690.118	−	1195.32i	−0.838540	−	1.45239i	−0.891115	−	0.453777i	$-0.850076\pi$
$823$	−690.118	−	1195.32i	−0.838540	−	1.45239i	0.0525755	−	0.998617i	$-0.483257\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	713.927i		−	0.863273i	−0.902048	−	0.431637i	$-0.857936\pi$
$827$		−	713.927i		−	0.863273i	0.902048	−	0.431637i	$-0.142064\pi$
$828$		0			0
$829$	133.228	+	230.758i	0.160710	+	0.278358i	0.935123	−	0.354322i	$-0.115288\pi$
$829$	133.228	+	230.758i	0.160710	+	0.278358i	−0.774414	+	0.632680i	$0.781955\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−494.354	−	697.770i	−0.593462	−	0.837660i
$834$		0			0
$835$	18.6703	−	32.3380i	0.0223597	−	0.0387281i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			258.193i			0.307739i	0.988091	+	0.153869i	$0.0491735\pi$
$839$			258.193i			0.307739i	−0.988091	+	0.153869i	$0.950827\pi$
$840$		0			0
$841$	235.028			0.279462
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−621.412	−	358.772i	−0.735399	−	0.424583i
$846$		0			0
$847$	774.376	−	246.515i	0.914258	−	0.291044i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−32.8037	+	18.9392i	−0.0385473	+	0.0222553i
$852$		0			0
$853$	276.361			0.323987			0.161993	−	0.986792i	$-0.448208\pi$
$853$	276.361			0.323987			0.161993	+	0.986792i	$0.448208\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	509.114	−	293.937i	0.594065	−	0.342984i	−0.172638	−	0.984985i	$-0.555229\pi$
$857$	509.114	−	293.937i	0.594065	−	0.342984i	0.766703	+	0.642002i	$0.221896\pi$
$858$		0			0
$859$	−407.998	+	706.673i	−0.474968	+	0.822669i	−0.999589	−	0.0286670i	$-0.990874\pi$
$859$	−407.998	+	706.673i	−0.474968	+	0.822669i	0.524621	+	0.851336i	$0.324207\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−370.696	−	214.022i	−0.429544	−	0.247997i	0.269608	−	0.962970i	$-0.413106\pi$
$863$	−370.696	−	214.022i	−0.429544	−	0.247997i	−0.699152	+	0.714973i	$0.746439\pi$
$864$		0			0
$865$	−578.683	−	1002.31i	−0.668997	−	1.15874i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	44.6577i		−	0.0513898i
$870$		0			0
$871$	1041.20	+	1803.41i	1.19541	+	2.07051i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1272.18	+	1396.07i	−1.45392	+	1.59551i
$876$		0			0
$877$	−381.148	+	660.167i	−0.434604	+	0.752756i	−0.997263	−	0.0739326i	$-0.976445\pi$
$877$	−381.148	+	660.167i	−0.434604	+	0.752756i	0.562659	+	0.826689i	$0.309778\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			995.350i			1.12980i	0.825161	+	0.564898i	$0.191085\pi$
$881$			995.350i			1.12980i	−0.825161	+	0.564898i	$0.808915\pi$
$882$		0			0
$883$	−874.928			−0.990859			−0.495429	−	0.868648i	$-0.664989\pi$
$883$	−874.928			−0.990859			−0.495429	+	0.868648i	$0.664989\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	725.318	+	418.762i	0.817720	+	0.472111i	0.849630	−	0.527380i	$-0.176825\pi$
$887$	725.318	+	418.762i	0.817720	+	0.472111i	−0.0319094	+	0.999491i	$0.510159\pi$
$888$		0			0
$889$	1134.28	+	1033.62i	1.27590	+	1.16268i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	147.727	−	85.2905i	0.165428	−	0.0955101i
$894$		0			0
$895$	2409.39			2.69205
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−978.886	+	565.160i	−1.08886	+	0.628654i
$900$		0			0
$901$	−505.691	+	875.883i	−0.561256	+	0.972124i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1691.63	+	976.661i	1.86920	+	1.07918i
$906$		0			0
$907$	633.977	+	1098.08i	0.698982	+	1.21067i	0.968820	+	0.247767i	$0.0796966\pi$
$907$	633.977	+	1098.08i	0.698982	+	1.21067i	−0.269838	+	0.962906i	$0.586970\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1262.85i		−	1.38623i	−0.720829	−	0.693113i	$-0.756239\pi$
$911$		−	1262.85i		−	1.38623i	0.720829	−	0.693113i	$-0.243761\pi$
$912$		0			0
$913$	6.57332	+	11.3853i	0.00719970	+	0.0124702i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−427.097	−	1341.64i	−0.465754	−	1.46307i
$918$		0			0
$919$	−32.8984	+	56.9817i	−0.0357980	+	0.0620040i	−0.883369	−	0.468678i	$-0.844731\pi$
$919$	−32.8984	+	56.9817i	−0.0357980	+	0.0620040i	0.847571	+	0.530682i	$0.178064\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1829.26i			1.98186i
$924$		0			0
$925$	−271.658			−0.293684
$926$		0			0
$927$		0			0
$928$		0			0
$929$	962.951	+	555.960i	1.03655	+	0.598450i	0.918853	−	0.394600i	$-0.129117\pi$
$929$	962.951	+	555.960i	1.03655	+	0.598450i	0.117693	+	0.993050i	$0.462450\pi$
$930$		0			0
$931$	−159.563	−	73.3348i	−0.171389	−	0.0787699i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−299.643	+	172.999i	−0.320474	+	0.185026i
$936$		0			0
$937$	636.217			0.678993			0.339497	−	0.940607i	$-0.389743\pi$
$937$	636.217			0.678993			0.339497	+	0.940607i	$0.389743\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1429.27	+	825.189i	−1.51888	+	0.876927i	−0.519130	+	0.854695i	$0.673744\pi$
$941$	−1429.27	+	825.189i	−1.51888	+	0.876927i	−0.999753	+	0.0222319i	$0.992923\pi$
$942$		0			0
$943$	111.122	−	192.469i	0.117839	−	0.204103i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	615.578	+	355.404i	0.650029	+	0.375294i	0.788467	−	0.615077i	$-0.210875\pi$
$947$	615.578	+	355.404i	0.650029	+	0.375294i	−0.138438	+	0.990371i	$0.544208\pi$
$948$		0			0
$949$	−366.002	−	633.934i	−0.385671	−	0.668002i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			839.240i			0.880629i	0.897844	+	0.440315i	$0.145133\pi$
$953$			839.240i			0.880629i	−0.897844	+	0.440315i	$0.854867\pi$
$954$		0			0
$955$	1089.35	+	1886.81i	1.14068	+	1.97572i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	493.299	+	107.932i	0.514389	+	0.112547i
$960$		0			0
$961$	−573.693	+	993.666i	−0.596975	+	1.03399i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2674.32i			2.77132i
$966$		0			0
$967$	−848.150			−0.877094			−0.438547	−	0.898708i	$-0.644507\pi$
$967$	−848.150			−0.877094			−0.438547	+	0.898708i	$0.644507\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−483.894	−	279.376i	−0.498346	−	0.287720i	0.229685	−	0.973265i	$-0.426231\pi$
$971$	−483.894	−	279.376i	−0.498346	−	0.287720i	−0.728030	+	0.685545i	$0.759564\pi$
$972$		0			0
$973$	1031.65	+	940.102i	1.06028	+	0.966189i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1052.41	−	607.609i	1.07719	−	0.621913i	0.147049	−	0.989129i	$-0.453022\pi$
$977$	1052.41	−	607.609i	1.07719	−	0.621913i	0.930136	+	0.367216i	$0.119689\pi$
$978$		0			0
$979$	172.676			0.176380
$980$		0			0
$981$		0			0
$982$		0			0
$983$	937.994	−	541.551i	0.954216	−	0.550917i	0.0598280	−	0.998209i	$-0.480945\pi$
$983$	937.994	−	541.551i	0.954216	−	0.550917i	0.894388	+	0.447292i	$0.147611\pi$
$984$		0			0
$985$	−210.629	+	364.820i	−0.213836	+	0.370375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−239.545	−	138.302i	−0.242210	−	0.139840i
$990$		0			0
$991$	350.453	+	607.003i	0.353636	+	0.612516i	0.986884	−	0.161434i	$-0.0516118\pi$
$991$	350.453	+	607.003i	0.353636	+	0.612516i	−0.633248	+	0.773949i	$0.718278\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	2458.68i		−	2.47104i
$996$		0			0
$997$	−265.177	−	459.300i	−0.265975	−	0.460682i	0.701844	−	0.712331i	$-0.252360\pi$
$997$	−265.177	−	459.300i	−0.265975	−	0.460682i	−0.967819	+	0.251649i	$0.919027\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.3.cu.a.233.1	✓	16
3.2	odd	2	inner	504.3.cu.a.233.8	yes	16
4.3	odd	2		1008.3.dc.g.737.1		16
7.2	even	3		3528.3.d.g.1961.8		8
7.4	even	3	inner	504.3.cu.a.305.8	yes	16
7.5	odd	6		3528.3.d.j.1961.1		8
12.11	even	2		1008.3.dc.g.737.8		16
21.2	odd	6		3528.3.d.g.1961.1		8
21.5	even	6		3528.3.d.j.1961.8		8
21.11	odd	6	inner	504.3.cu.a.305.1	yes	16
28.11	odd	6		1008.3.dc.g.305.8		16
84.11	even	6		1008.3.dc.g.305.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.cu.a.233.1	✓	16	1.1	even	1	trivial
504.3.cu.a.233.8	yes	16	3.2	odd	2	inner
504.3.cu.a.305.1	yes	16	21.11	odd	6	inner
504.3.cu.a.305.8	yes	16	7.4	even	3	inner
1008.3.dc.g.305.1		16	84.11	even	6
1008.3.dc.g.305.8		16	28.11	odd	6
1008.3.dc.g.737.1		16	4.3	odd	2
1008.3.dc.g.737.8		16	12.11	even	2
3528.3.d.g.1961.1		8	21.2	odd	6
3528.3.d.g.1961.8		8	7.2	even	3
3528.3.d.j.1961.1		8	7.5	odd	6
3528.3.d.j.1961.8		8	21.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	504.cu (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.75278 - 4.47607i) q^{5} +(-5.17399 - 4.71485i) q^{7} +(-1.91794 + 1.10732i) q^{11} +15.7846 q^{13} +(-15.1138 + 8.72595i) q^{17} +(-1.79193 + 3.10371i) q^{19} +(6.65847 + 3.84427i) q^{23} +(27.5704 + 47.7533i) q^{25} +24.6165i q^{29} +(22.9586 + 39.7654i) q^{31} +(19.0088 + 59.7123i) q^{35} +(-2.46331 + 4.26658i) q^{37} -28.9060i q^{41} -35.9761 q^{43} +(-41.2203 - 23.7985i) q^{47} +(4.54035 + 48.7892i) q^{49} +(50.1884 - 28.9763i) q^{53} +19.8258 q^{55} +(-100.481 + 58.0128i) q^{59} +(53.9476 - 93.4400i) q^{61} +(-122.374 - 70.6529i) q^{65} +(65.9631 + 114.252i) q^{67} +115.889i q^{71} +(-23.1873 - 40.1615i) q^{73} +(15.1443 + 3.31353i) q^{77} +(-10.0824 + 17.4631i) q^{79} -5.93622i q^{83} +156.232 q^{85} +(-67.5241 - 38.9850i) q^{89} +(-81.6693 - 74.4220i) q^{91} +(27.7848 - 16.0416i) q^{95} -87.3734 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{7} - 24 q^{13} - 12 q^{19} + 92 q^{25} + 32 q^{31} - 132 q^{37} + 24 q^{43} + 64 q^{49} - 440 q^{55} + 184 q^{61} + 332 q^{67} + 188 q^{73} - 112 q^{79} + 256 q^{85} - 252 q^{91} - 1032 q^{97}+O(q^{100}) \) 16 * q - 24 * q^7 - 24 * q^13 - 12 * q^19 + 92 * q^25 + 32 * q^31 - 132 * q^37 + 24 * q^43 + 64 * q^49 - 440 * q^55 + 184 * q^61 + 332 * q^67 + 188 * q^73 - 112 * q^79 + 256 * q^85 - 252 * q^91 - 1032 * q^97

Introduction

L-functions

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