LMFDB - Embedded newform 504.3.cu.a.305.1

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			305.1
Root			$3.52391i$ of defining polynomial
Character	$\chi$	$=$	504.305
Dual form			504.3.cu.a.233.1

$q$-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 $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$-1$

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	−0.552460	+	0.833540i	$0.686311\pi$
$5$	−7.75278	+	4.47607i	−1.55056	+	0.895214i	−0.998096	+	0.0616742i	$0.980356\pi$
$6$		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	0.410281	−	0.911959i	$-0.365431\pi$
$11$	−1.91794	−	1.10732i	−0.174358	−	0.100666i	−0.584639	+	0.811293i	$0.698764\pi$
$12$		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	−0.0154157	−	0.999881i	$-0.504907\pi$
$17$	−15.1138	−	8.72595i	−0.889046	−	0.513291i	−0.873630	+	0.486590i	$0.838241\pi$
$18$		0			0
$19$	−1.79193	−	3.10371i	−0.0943120	−	0.163353i	0.815009	−	0.579448i	$-0.196732\pi$
$19$	−1.79193	−	3.10371i	−0.0943120	−	0.163353i	−0.909321	+	0.416095i	$0.863398\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.65847	−	3.84427i	0.289498	−	0.167142i	−0.348217	−	0.937414i	$-0.613213\pi$
$23$	6.65847	−	3.84427i	0.289498	−	0.167142i	0.637716	+	0.770272i	$0.279879\pi$
$24$		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	$-0.860477\pi$
$29$		−	24.6165i		−	0.848845i	0.905464	−	0.424423i	$-0.139523\pi$
$30$		0			0
$31$	22.9586	−	39.7654i	0.740600	−	1.28276i	−0.211623	−	0.977351i	$-0.567875\pi$
$31$	22.9586	−	39.7654i	0.740600	−	1.28276i	0.952223	−	0.305405i	$-0.0987917\pi$
$32$		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	0.830816	−	0.556547i	$-0.187874\pi$
$37$	−2.46331	−	4.26658i	−0.0665760	−	0.115313i	−0.897392	+	0.441234i	$0.854541\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			28.9060i			0.705024i	0.935807	+	0.352512i	$0.114672\pi$
$41$			28.9060i			0.705024i	−0.935807	+	0.352512i	$0.885328\pi$
$42$		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	−0.869677	−	0.493621i	$-0.835673\pi$
$47$	−41.2203	+	23.7985i	−0.877027	+	0.506352i	−0.00734985	+	0.999973i	$0.502340\pi$
$48$		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	0.892133	−	0.451773i	$-0.149208\pi$
$53$	50.1884	+	28.9763i	0.946952	+	0.546723i	0.0548190	+	0.998496i	$0.482542\pi$
$54$		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	−0.942617	−	0.333875i	$-0.891644\pi$
$59$	−100.481	−	58.0128i	−1.70307	−	0.983268i	−0.760453	−	0.649393i	$-0.775023\pi$
$60$		0			0
$61$	53.9476	+	93.4400i	0.884387	+	1.53180i	0.846415	+	0.532524i	$0.178756\pi$
$61$	53.9476	+	93.4400i	0.884387	+	1.53180i	0.0379718	+	0.999279i	$0.487910\pi$
$62$		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	0.340494	−	0.940247i	$-0.389406\pi$
$67$	65.9631	−	114.252i	0.984524	−	1.70525i	0.644031	−	0.765000i	$-0.277261\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	115.889i		−	1.63224i	−0.577883	−	0.816119i	$-0.696121\pi$
$71$		−	115.889i		−	1.63224i	0.577883	−	0.816119i	$-0.303879\pi$
$72$		0			0
$73$	−23.1873	+	40.1615i	−0.317634	+	0.550158i	−0.979994	−	0.199028i	$-0.936222\pi$
$73$	−23.1873	+	40.1615i	−0.317634	+	0.550158i	0.662360	+	0.749186i	$0.269555\pi$
$74$		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	0.795131	−	0.606438i	$-0.207402\pi$
$79$	−10.0824	−	17.4631i	−0.127625	−	0.221053i	−0.922756	+	0.385385i	$0.874069\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.93622i			0.0715208i	0.999360	+	0.0357604i	$0.0113853\pi$
$83$			5.93622i			0.0715208i	−0.999360	+	0.0357604i	$0.988615\pi$
$84$		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	−0.828828	−	0.559504i	$-0.810992\pi$
$89$	−67.5241	+	38.9850i	−0.758697	+	0.438034i	0.0701305	+	0.997538i	$0.477658\pi$
$90$		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	0.991673	−	0.128778i	$-0.0411056\pi$
$101$	138.974	+	80.2369i	1.37598	+	0.794425i	0.384311	+	0.923204i	$0.374439\pi$
$102$		0			0
$103$	−8.05119	−	13.9451i	−0.0781669	−	0.135389i	0.824292	−	0.566165i	$-0.191573\pi$
$103$	−8.05119	−	13.9451i	−0.0781669	−	0.135389i	−0.902459	+	0.430776i	$0.858240\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	99.1075	−	57.2198i	0.926239	−	0.534764i	0.0406186	−	0.999175i	$-0.487067\pi$
$107$	99.1075	−	57.2198i	0.926239	−	0.534764i	0.885620	+	0.464411i	$0.153734\pi$
$108$		0			0
$109$	25.8973	−	44.8555i	0.237590	−	0.411518i	−0.722432	−	0.691442i	$-0.756976\pi$
$109$	25.8973	−	44.8555i	0.237590	−	0.411518i	0.960022	+	0.279924i	$0.0903092\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	131.309i		−	1.16202i	−0.813895	−	0.581012i	$-0.802657\pi$
$113$		−	131.309i		−	1.16202i	0.813895	−	0.581012i	$-0.197343\pi$
$114$		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	0.344457	−	0.938802i	$-0.388063\pi$
$131$	174.192	−	100.570i	1.32971	−	0.767710i	0.985255	+	0.171092i	$0.0547297\pi$
$132$		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	0.254354	−	0.967111i	$-0.418137\pi$
$137$	−62.4737	−	36.0692i	−0.456012	−	0.263279i	−0.710366	+	0.703832i	$0.751471\pi$
$138$		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	−0.327759	−	0.944761i	$-0.606293\pi$
$149$	48.6559	−	28.0915i	0.326549	−	0.188533i	0.654308	+	0.756228i	$0.272960\pi$
$150$		0			0
$151$	−53.9534	+	93.4501i	−0.357307	+	0.618875i	−0.987510	−	0.157556i	$-0.949638\pi$
$151$	−53.9534	+	93.4501i	−0.357307	+	0.618875i	0.630203	+	0.776431i	$0.282972\pi$
$152$		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	−0.877666	−	0.479273i	$-0.840900\pi$
$157$	−3.73199	+	6.46400i	−0.0237707	+	0.0411720i	0.853895	+	0.520445i	$0.174234\pi$
$158$		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	0.999501	+	0.0315950i	$0.0100587\pi$
$163$	85.9193	+	148.817i	0.527112	+	0.912986i	−0.472388	+	0.881391i	$0.656608\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.17115i		−	0.0249769i	−0.999922	−	0.0124885i	$-0.996025\pi$
$167$		−	4.17115i		−	0.0249769i	0.999922	−	0.0124885i	$-0.00397531\pi$
$168$		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	−0.140192	−	0.990124i	$-0.544772\pi$
$173$	111.963	−	64.6418i	0.647185	−	0.373652i	0.787377	+	0.616472i	$0.211439\pi$
$174$		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	−0.321369	−	0.946954i	$-0.604143\pi$
$179$	−233.083	−	134.571i	−1.30214	−	0.751791i	−0.980771	+	0.195163i	$0.937476\pi$
$180$		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	−0.937135	−	0.348967i	$-0.886532\pi$
$191$	−210.766	+	121.686i	−1.10349	+	0.637099i	−0.166353	+	0.986066i	$0.553199\pi$
$192$		0			0
$193$	−149.368	+	258.713i	−0.773926	+	1.34048i	0.161470	+	0.986878i	$0.448377\pi$
$193$	−149.368	+	258.713i	−0.773926	+	1.34048i	−0.935396	+	0.353602i	$0.884957\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			47.0566i			0.238866i	0.992842	+	0.119433i	$0.0381077\pi$
$197$			47.0566i			0.238866i	−0.992842	+	0.119433i	$0.961892\pi$
$198$		0			0
$199$	137.324	−	237.852i	0.690069	−	1.19523i	−0.281746	−	0.959489i	$-0.590914\pi$
$199$	137.324	−	237.852i	0.690069	−	1.19523i	0.971815	−	0.235745i	$-0.0757532\pi$
$200$		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	0.521697	−	0.853131i	$-0.325299\pi$
$227$	9.92278	+	5.72892i	0.0437127	+	0.0252375i	−0.477984	+	0.878368i	$0.658632\pi$
$228$		0			0
$229$	−39.4844	−	68.3890i	−0.172421	−	0.298642i	0.766845	−	0.641833i	$-0.221826\pi$
$229$	−39.4844	−	68.3890i	−0.172421	−	0.298642i	−0.939266	+	0.343191i	$0.888492\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	184.800	−	106.694i	0.793133	−	0.457915i	−0.0479315	−	0.998851i	$-0.515263\pi$
$233$	184.800	−	106.694i	0.793133	−	0.457915i	0.841064	+	0.540935i	$0.181930\pi$
$234$		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	$0.00554259\pi$
$239$			8.32278i			0.0348234i	−0.999848	+	0.0174117i	$0.994457\pi$
$240$		0			0
$241$	−58.6994	+	101.670i	−0.243566	+	0.421868i	−0.961727	−	0.274008i	$-0.911651\pi$
$241$	−58.6994	+	101.670i	−0.243566	+	0.421868i	0.718162	+	0.695876i	$0.244984\pi$
$242$		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	$0.0210568\pi$
$251$			33.1840i			0.132207i	−0.997813	+	0.0661036i	$0.978943\pi$
$252$		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	0.0961736	−	0.995365i	$-0.469340\pi$
$257$	258.612	−	149.310i	1.00627	−	0.580971i	0.910098	+	0.414394i	$0.136006\pi$
$258$		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	0.533133	−	0.846031i	$-0.321014\pi$
$263$	17.6251	+	10.1759i	0.0670157	+	0.0386915i	−0.466118	+	0.884723i	$0.654348\pi$
$264$		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	0.820151	−	0.572147i	$-0.193889\pi$
$269$	197.643	+	114.109i	0.734732	+	0.424198i	−0.0854188	+	0.996345i	$0.527223\pi$
$270$		0			0
$271$	−228.622	−	395.985i	−0.843624	−	1.46120i	−0.886811	−	0.462132i	$-0.847085\pi$
$271$	−228.622	−	395.985i	−0.843624	−	1.46120i	0.0431875	−	0.999067i	$-0.486249\pi$
$272$		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	−0.263200	−	0.964741i	$-0.584778\pi$
$277$	194.978	−	337.711i	0.703890	−	1.21917i	0.967090	−	0.254433i	$-0.0818887\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	301.317i		−	1.07230i	−0.844122	−	0.536151i	$-0.819878\pi$
$281$		−	301.317i		−	1.07230i	0.844122	−	0.536151i	$-0.180122\pi$
$282$		0			0
$283$	−107.357	+	185.949i	−0.379355	+	0.657062i	−0.990969	−	0.134094i	$-0.957187\pi$
$283$	−107.357	+	185.949i	−0.379355	+	0.657062i	0.611613	+	0.791157i	$0.290521\pi$
$284$		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	$-0.855060\pi$
$293$		−	257.705i		−	0.879539i	0.898111	−	0.439770i	$-0.144940\pi$
$294$		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	0.0194425	−	0.999811i	$-0.493811\pi$
$311$	−260.213	−	150.234i	−0.836698	−	0.483068i	−0.856140	+	0.516743i	$0.827144\pi$
$312$		0			0
$313$	233.217	+	403.944i	0.745103	+	1.29056i	0.950147	+	0.311804i	$0.100933\pi$
$313$	233.217	+	403.944i	0.745103	+	1.29056i	−0.205043	+	0.978753i	$0.565734\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−220.520	+	127.318i	−0.695648	+	0.401633i	−0.805725	−	0.592290i	$-0.798224\pi$
$317$	−220.520	+	127.318i	−0.695648	+	0.401633i	0.110076	+	0.993923i	$0.464890\pi$
$318$		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	−0.860415	−	0.509594i	$-0.829796\pi$
$331$	−288.476	−	499.655i	−0.871529	−	1.50953i	−0.0111141	−	0.999938i	$-0.503538\pi$
$332$		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	0.175178	−	0.984537i	$-0.443950\pi$
$347$	−204.684	−	118.174i	−0.589868	−	0.340560i	−0.765045	+	0.643977i	$0.777283\pi$
$348$		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	−0.217188	−	0.976130i	$-0.569688\pi$
$353$	−413.411	−	238.683i	−1.17113	−	0.676155i	−0.953947	+	0.299975i	$0.903022\pi$
$354$		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	−0.872477	−	0.488655i	$-0.837488\pi$
$359$	−317.904	+	183.542i	−0.885527	+	0.511259i	−0.0130503	+	0.999915i	$0.504154\pi$
$360$		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	−0.928156	−	0.372191i	$-0.878607\pi$
$367$	−52.0227	+	90.1060i	−0.141751	+	0.245521i	0.786405	+	0.617711i	$0.211940\pi$
$368$		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	0.504914	−	0.863170i	$-0.331524\pi$
$373$	−184.661	−	319.842i	−0.495070	−	0.857486i	−0.999984	+	0.00568377i	$0.998191\pi$
$374$		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	−0.159702	−	0.987165i	$-0.551053\pi$
$383$	235.682	−	136.071i	0.615358	−	0.355277i	0.775059	+	0.631888i	$0.217720\pi$
$384$		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	0.905979	−	0.423322i	$-0.139136\pi$
$389$	386.028	+	222.874i	0.992361	+	0.572940i	0.0863819	+	0.996262i	$0.472469\pi$
$390$		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	0.893721	−	0.448622i	$-0.148085\pi$
$397$	23.1619	+	40.1176i	0.0583423	+	0.101052i	−0.835379	+	0.549674i	$0.814752\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.21519	−	1.85629i	0.00801792	−	0.00462915i	−0.495986	−	0.868331i	$-0.665193\pi$
$401$	3.21519	−	1.85629i	0.00801792	−	0.00462915i	0.504004	+	0.863702i	$0.331860\pi$
$402$		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	−0.638060	−	0.769986i	$-0.720263\pi$
$409$	142.249	−	246.383i	0.347798	−	0.602403i	0.985858	+	0.167583i	$0.0535963\pi$
$410$		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	$-0.950435\pi$
$419$		−	129.962i		−	0.310172i	0.987901	−	0.155086i	$-0.0495654\pi$
$420$		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	0.889292	−	0.457340i	$-0.151198\pi$
$431$	404.222	+	233.377i	0.937869	+	0.541479i	0.0485774	+	0.998819i	$0.484531\pi$
$432$		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	0.575128	−	0.818064i	$-0.304952\pi$
$439$	−184.775	−	320.040i	−0.420900	−	0.729020i	−0.996028	+	0.0890436i	$0.971619\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	22.3910	−	12.9274i	0.0505440	−	0.0291816i	−0.474515	−	0.880247i	$-0.657377\pi$
$443$	22.3910	−	12.9274i	0.0505440	−	0.0291816i	0.525059	+	0.851066i	$0.324043\pi$
$444$		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	$-0.720719\pi$
$449$		−	690.627i		−	1.53814i	0.639162	−	0.769072i	$-0.279281\pi$
$450$		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	0.966586	−	0.256341i	$-0.0825170\pi$
$457$	119.412	+	206.828i	0.261295	+	0.452577i	−0.705291	+	0.708918i	$0.749184\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			665.309i			1.44319i	0.692318	+	0.721593i	$0.256590\pi$
$461$			665.309i			1.44319i	−0.692318	+	0.721593i	$0.743410\pi$
$462$		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	−0.218051	−	0.975937i	$-0.569970\pi$
$467$	241.957	−	139.694i	0.518110	−	0.299131i	0.736161	+	0.676806i	$0.236637\pi$
$468$		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	0.246288	−	0.969197i	$-0.420789\pi$
$479$	−225.090	−	129.956i	−0.469917	−	0.271307i	−0.716205	+	0.697890i	$0.754122\pi$
$480$		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	−0.664519	−	0.747271i	$-0.731364\pi$
$487$	153.354	−	265.618i	0.314896	−	0.545416i	0.979415	+	0.201855i	$0.0646970\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	107.088i		−	0.218102i	−0.994036	−	0.109051i	$-0.965219\pi$
$491$		−	107.088i		−	0.218102i	0.994036	−	0.109051i	$-0.0347811\pi$
$492$		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	0.994534	−	0.104412i	$-0.0332962\pi$
$499$	203.015	+	351.632i	0.406843	+	0.704673i	−0.587691	+	0.809086i	$0.699963\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			300.937i			0.598283i	0.954209	+	0.299142i	$0.0967003\pi$
$503$			300.937i			0.598283i	−0.954209	+	0.299142i	$0.903300\pi$
$504$		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	−0.833229	−	0.552928i	$-0.813510\pi$
$509$	−392.436	+	226.573i	−0.770993	+	0.445133i	0.0622355	+	0.998061i	$0.480177\pi$
$510$		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	0.139201	−	0.990264i	$-0.455546\pi$
$521$	−338.021	−	195.156i	−0.648792	−	0.374580i	−0.787993	+	0.615684i	$0.788880\pi$
$522$		0			0
$523$	170.504	+	295.322i	0.326012	+	0.564670i	0.981717	−	0.190348i	$-0.0609616\pi$
$523$	170.504	+	295.322i	0.326012	+	0.564670i	−0.655705	+	0.755018i	$0.727628\pi$
$524$		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	0.888701	+	0.458487i	$0.151609\pi$
$541$	455.204	+	788.436i	0.841412	+	1.45737i	−0.0472887	+	0.998881i	$0.515058\pi$
$542$		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	0.770916	−	0.636936i	$-0.219799\pi$
$557$	336.858	+	194.485i	0.604772	+	0.349165i	−0.166145	+	0.986101i	$0.553132\pi$
$558$		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	0.310049	−	0.950721i	$-0.399655\pi$
$563$	−201.709	−	116.457i	−0.358275	−	0.206850i	−0.668324	+	0.743870i	$0.732988\pi$
$564$		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	−0.822793	−	0.568342i	$-0.807585\pi$
$569$	−422.193	+	243.753i	−0.741990	+	0.428388i	0.0808021	+	0.996730i	$0.474252\pi$
$570$		0			0
$571$	380.009	−	658.195i	0.665515	−	1.15271i	−0.313630	−	0.949545i	$-0.601545\pi$
$571$	380.009	−	658.195i	0.665515	−	1.15271i	0.979145	−	0.203161i	$-0.0651216\pi$
$572$		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	−0.603015	−	0.797729i	$-0.706034\pi$
$577$	224.653	−	389.110i	0.389346	−	0.674368i	0.992362	+	0.123362i	$0.0393676\pi$
$578$		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	$0.0106648\pi$
$587$			39.3268i			0.0669962i	−0.999439	+	0.0334981i	$0.989335\pi$
$588$		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	−0.774844	−	0.632152i	$-0.782172\pi$
$593$	−364.581	+	210.491i	−0.614807	+	0.354959i	0.160037	+	0.987111i	$0.448839\pi$
$594$		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	0.993503	−	0.113804i	$-0.0363036\pi$
$599$	833.627	+	481.295i	1.39170	+	0.803497i	0.398195	+	0.917301i	$0.369637\pi$
$600$		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	0.858339	−	0.513083i	$-0.171497\pi$
$607$	−9.21009	−	15.9523i	−0.0151731	−	0.0262806i	−0.873512	+	0.486802i	$0.838163\pi$
$608$		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	0.0993295	+	0.995055i	$0.468330\pi$
$613$	−497.804	+	862.221i	−0.812078	+	1.40656i	−0.911407	+	0.411505i	$0.865003\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			425.645i			0.689863i	0.938628	+	0.344931i	$0.112098\pi$
$617$			425.645i			0.689863i	−0.938628	+	0.344931i	$0.887902\pi$
$618$		0			0
$619$	−433.536	+	750.907i	−0.700382	+	1.21310i	0.267950	+	0.963433i	$0.413654\pi$
$619$	−433.536	+	750.907i	−0.700382	+	1.21310i	−0.968332	+	0.249664i	$0.919680\pi$
$620$		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	0.396872	−	0.917874i	$-0.370096\pi$
$641$	−127.940	−	73.8664i	−0.199595	−	0.115236i	−0.596467	+	0.802638i	$0.703429\pi$
$642$		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	0.245575	−	0.969378i	$-0.421023\pi$
$647$	−304.830	−	175.993i	−0.471143	−	0.272015i	−0.716718	+	0.697363i	$0.754357\pi$
$648$		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	0.224485	−	0.974478i	$-0.427930\pi$
$653$	770.964	−	445.116i	1.18065	−	0.681648i	0.956165	+	0.292829i	$0.0945968\pi$
$654$		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	$0.149956\pi$
$659$			598.195i			0.907732i	−0.891070	+	0.453866i	$0.850044\pi$
$660$		0			0
$661$	373.678	−	647.229i	0.565322	−	0.979167i	−0.431697	−	0.902019i	$-0.642085\pi$
$661$	373.678	−	647.229i	0.565322	−	0.979167i	0.997020	−	0.0771485i	$-0.0245816\pi$
$662$		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	−0.611813	−	0.791003i	$-0.709559\pi$
$677$	−157.531	+	90.9508i	−0.232690	+	0.134344i	0.379122	+	0.925347i	$0.376226\pi$
$678$		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	0.830192	−	0.557477i	$-0.188231\pi$
$683$	520.787	+	300.676i	0.762499	+	0.440229i	−0.0676932	+	0.997706i	$0.521564\pi$
$684$		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	0.544942	−	0.838474i	$-0.316552\pi$
$691$	−313.485	−	542.972i	−0.453669	−	0.785777i	−0.998611	+	0.0526968i	$0.983218\pi$
$692$		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	$-0.936593\pi$
$701$		−	277.434i		−	0.395769i	0.980225	−	0.197885i	$-0.0634071\pi$
$702$		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	0.546752	−	0.837295i	$-0.315864\pi$
$709$	−320.286	−	554.751i	−0.451743	−	0.782441i	−0.998494	+	0.0548534i	$0.982531\pi$
$710$		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	−0.569501	−	0.821991i	$-0.692863\pi$
$719$	−102.376	+	59.1067i	−0.142386	+	0.0822068i	0.427114	+	0.904198i	$0.359530\pi$
$720$		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	0.913282	−	0.407329i	$-0.133540\pi$
$733$	76.1467	+	131.890i	0.103884	+	0.179932i	−0.809398	+	0.587261i	$0.800206\pi$
$734$		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	−0.677088	−	0.735902i	$-0.736759\pi$
$739$	220.788	−	382.415i	0.298765	−	0.517477i	0.975854	+	0.218425i	$0.0700918\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			82.3447i			0.110827i	0.998463	+	0.0554137i	$0.0176478\pi$
$743$			82.3447i			0.110827i	−0.998463	+	0.0554137i	$0.982352\pi$
$744$		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	0.969778	−	0.243988i	$-0.0784557\pi$
$751$	205.466	+	355.877i	0.273590	+	0.473871i	−0.696189	+	0.717859i	$0.745122\pi$
$752$		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	−0.745205	−	0.666835i	$-0.767648\pi$
$761$	−411.177	+	237.393i	−0.540311	+	0.311949i	0.204894	+	0.978784i	$0.434315\pi$
$762$		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	0.841843	−	0.539722i	$-0.181471\pi$
$773$	614.807	+	354.959i	0.795351	+	0.459196i	−0.0464917	+	0.998919i	$0.514804\pi$
$774$		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	−0.952642	−	0.304093i	$-0.901647\pi$
$787$	−167.606	+	290.303i	−0.212969	+	0.368873i	0.739674	+	0.672966i	$0.234980\pi$
$788$		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	$0.0689509\pi$
$797$			342.591i			0.429851i	−0.976630	+	0.214925i	$0.931049\pi$
$798$		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	0.298649	−	0.954363i	$-0.403464\pi$
$809$	−306.230	−	176.802i	−0.378530	−	0.218544i	−0.677178	+	0.735819i	$0.736797\pi$
$810$		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	−0.516169	+	0.856487i	$0.672642\pi$
$821$	−1244.63	+	718.587i	−1.51599	+	0.875259i	−0.999824	+	0.0187718i	$0.994024\pi$
$822$		0			0
$823$	−690.118	+	1195.32i	−0.838540	+	1.45239i	0.0525755	+	0.998617i	$0.483257\pi$
$823$	−690.118	+	1195.32i	−0.838540	+	1.45239i	−0.891115	+	0.453777i	$0.850076\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			713.927i			0.863273i	0.902048	+	0.431637i	$0.142064\pi$
$827$			713.927i			0.863273i	−0.902048	+	0.431637i	$0.857936\pi$
$828$		0			0
$829$	133.228	−	230.758i	0.160710	−	0.278358i	−0.774414	−	0.632680i	$-0.781955\pi$
$829$	133.228	−	230.758i	0.160710	−	0.278358i	0.935123	+	0.354322i	$0.115288\pi$
$830$		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	$-0.950827\pi$
$839$		−	258.193i		−	0.307739i	0.988091	−	0.153869i	$-0.0491735\pi$
$840$		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	0.766703	−	0.642002i	$-0.221896\pi$
$857$	509.114	+	293.937i	0.594065	+	0.342984i	−0.172638	+	0.984985i	$0.555229\pi$
$858$		0			0
$859$	−407.998	−	706.673i	−0.474968	−	0.822669i	0.524621	−	0.851336i	$-0.324207\pi$
$859$	−407.998	−	706.673i	−0.474968	−	0.822669i	−0.999589	+	0.0286670i	$0.990874\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−370.696	+	214.022i	−0.429544	+	0.247997i	−0.699152	−	0.714973i	$-0.746439\pi$
$863$	−370.696	+	214.022i	−0.429544	+	0.247997i	0.269608	+	0.962970i	$0.413106\pi$
$864$		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	0.562659	−	0.826689i	$-0.309778\pi$
$877$	−381.148	−	660.167i	−0.434604	−	0.752756i	−0.997263	+	0.0739326i	$0.976445\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	995.350i		−	1.12980i	−0.825161	−	0.564898i	$-0.808915\pi$
$881$		−	995.350i		−	1.12980i	0.825161	−	0.564898i	$-0.191085\pi$
$882$		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	−0.0319094	−	0.999491i	$-0.510159\pi$
$887$	725.318	−	418.762i	0.817720	−	0.472111i	0.849630	+	0.527380i	$0.176825\pi$
$888$		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	−0.269838	−	0.962906i	$-0.586970\pi$
$907$	633.977	−	1098.08i	0.698982	−	1.21067i	0.968820	−	0.247767i	$-0.0796966\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1262.85i			1.38623i	0.720829	+	0.693113i	$0.243761\pi$
$911$			1262.85i			1.38623i	−0.720829	+	0.693113i	$0.756239\pi$
$912$		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	0.847571	−	0.530682i	$-0.178064\pi$
$919$	−32.8984	−	56.9817i	−0.0357980	−	0.0620040i	−0.883369	+	0.468678i	$0.844731\pi$
$920$		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	0.117693	−	0.993050i	$-0.462450\pi$
$929$	962.951	−	555.960i	1.03655	−	0.598450i	0.918853	+	0.394600i	$0.129117\pi$
$930$		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	−0.999753	−	0.0222319i	$-0.992923\pi$
$941$	−1429.27	−	825.189i	−1.51888	−	0.876927i	−0.519130	−	0.854695i	$-0.673744\pi$
$942$		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	−0.138438	−	0.990371i	$-0.544208\pi$
$947$	615.578	−	355.404i	0.650029	−	0.375294i	0.788467	+	0.615077i	$0.210875\pi$
$948$		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	$-0.854867\pi$
$953$		−	839.240i		−	0.880629i	0.897844	−	0.440315i	$-0.145133\pi$
$954$		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	−0.728030	−	0.685545i	$-0.759564\pi$
$971$	−483.894	+	279.376i	−0.498346	+	0.287720i	0.229685	+	0.973265i	$0.426231\pi$
$972$		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	0.930136	−	0.367216i	$-0.119689\pi$
$977$	1052.41	+	607.609i	1.07719	+	0.621913i	0.147049	+	0.989129i	$0.453022\pi$
$978$		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	0.894388	−	0.447292i	$-0.147611\pi$
$983$	937.994	+	541.551i	0.954216	+	0.550917i	0.0598280	+	0.998209i	$0.480945\pi$
$984$		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	−0.633248	−	0.773949i	$-0.718278\pi$
$991$	350.453	−	607.003i	0.353636	−	0.612516i	0.986884	+	0.161434i	$0.0516118\pi$
$992$		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	−0.967819	−	0.251649i	$-0.919027\pi$
$997$	−265.177	+	459.300i	−0.265975	+	0.460682i	0.701844	+	0.712331i	$0.252360\pi$
$998$		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.305.1	yes	16
3.2	odd	2	inner	504.3.cu.a.305.8	yes	16
4.3	odd	2		1008.3.dc.g.305.1		16
7.2	even	3	inner	504.3.cu.a.233.8	yes	16
7.3	odd	6		3528.3.d.j.1961.8		8
7.4	even	3		3528.3.d.g.1961.1		8
12.11	even	2		1008.3.dc.g.305.8		16
21.2	odd	6	inner	504.3.cu.a.233.1	✓	16
21.11	odd	6		3528.3.d.g.1961.8		8
21.17	even	6		3528.3.d.j.1961.1		8
28.23	odd	6		1008.3.dc.g.737.8		16
84.23	even	6		1008.3.dc.g.737.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.cu.a.233.1	✓	16	21.2	odd	6	inner
504.3.cu.a.233.8	yes	16	7.2	even	3	inner
504.3.cu.a.305.1	yes	16	1.1	even	1	trivial
504.3.cu.a.305.8	yes	16	3.2	odd	2	inner
1008.3.dc.g.305.1		16	4.3	odd	2
1008.3.dc.g.305.8		16	12.11	even	2
1008.3.dc.g.737.1		16	84.23	even	6
1008.3.dc.g.737.8		16	28.23	odd	6
3528.3.d.g.1961.1		8	7.4	even	3
3528.3.d.g.1961.8		8	21.11	odd	6
3528.3.d.j.1961.1		8	21.17	even	6
3528.3.d.j.1961.8		8	7.3	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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
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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