LMFDB - Embedded newform 1008.3.dc.f.737.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(305,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.305"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.dc (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
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} + 38 x^{14} - 120 x^{13} + 1059 x^{12} - 3540 x^{11} + 20690 x^{10} - 73200 x^{9} + 269971 x^{8} + \cdots + 352836 $ x^16 + 38x^14 - 120x^13 + 1059x^12 - 3540x^11 + 20690x^10 - 73200x^9 + 269971x^8 - 612180x^7 + 1101306x^6 - 1098360x^5 + 603810x^4 + 832680x^3 - 439020x^2 - 320760x + 352836
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			737.1
Root			$0.326392 + 1.97954i$ of defining polynomial
Character	$\chi$	$=$	1008.737
Dual form			1008.3.dc.f.305.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.42523 - 4.28696i) q^{5} +(-1.80762 - 6.76258i) q^{7} +(12.5934 - 7.27083i) q^{11} +10.9055 q^{13} +(23.2217 - 13.4070i) q^{17} +(-3.98799 + 6.90741i) q^{19} +(-23.2217 - 13.4070i) q^{23} +(24.2560 + 42.0126i) q^{25} -35.7359i q^{29} +(11.9223 + 20.6501i) q^{31} +(-15.5689 + 57.9629i) q^{35} +(19.1827 - 33.2254i) q^{37} -27.8548i q^{41} -68.4986 q^{43} +(-61.5951 - 35.5620i) q^{47} +(-42.4651 + 24.4483i) q^{49} +(-54.9706 + 31.7373i) q^{53} -124.679 q^{55} +(-7.72657 + 4.46094i) q^{59} +(2.88709 - 5.00059i) q^{61} +(-80.9760 - 46.7515i) q^{65} +(-17.3360 - 30.0269i) q^{67} +68.6199i q^{71} +(65.6321 + 113.678i) q^{73} +(-71.9336 - 72.0213i) q^{77} +(15.0321 - 26.0363i) q^{79} -86.5943i q^{83} -229.901 q^{85} +(111.334 + 64.2790i) q^{89} +(-19.7130 - 73.7495i) q^{91} +(59.2235 - 34.1927i) q^{95} +80.0096 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 40 q^{13} + 20 q^{19} - 4 q^{25} + 56 q^{31} + 76 q^{37} - 72 q^{43} - 48 q^{49} - 648 q^{55} - 72 q^{61} + 156 q^{67} + 124 q^{73} - 184 q^{79} - 864 q^{85} + 116 q^{91} + 1112 q^{97}+O(q^{100}) $ 16 * q + 40 * q^13 + 20 * q^19 - 4 * q^25 + 56 * q^31 + 76 * q^37 - 72 * q^43 - 48 * q^49 - 648 * q^55 - 72 * q^61 + 156 * q^67 + 124 * q^73 - 184 * q^79 - 864 * q^85 + 116 * q^91 + 1112 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$-1$

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
$5$	−7.42523	−	4.28696i	−1.48505	−	0.857392i	−0.485191	−	0.874408i	$-0.661250\pi$
$5$	−7.42523	−	4.28696i	−1.48505	−	0.857392i	−0.999855	+	0.0170169i	$0.994583\pi$
$6$		0			0
$7$	−1.80762	−	6.76258i	−0.258231	−	0.966083i
$7$	−1.80762	−	6.76258i	−0.258231	−	0.966083i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	12.5934	−	7.27083i	1.14486	−	0.660984i	0.197229	−	0.980357i	$-0.436806\pi$
$11$	12.5934	−	7.27083i	1.14486	−	0.660984i	0.947629	+	0.319373i	$0.103472\pi$
$12$		0			0
$13$	10.9055			0.838886			0.419443	−	0.907782i	$-0.362225\pi$
$13$	10.9055			0.838886			0.419443	+	0.907782i	$0.362225\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	23.2217	−	13.4070i	1.36598	−	0.788649i	0.375568	−	0.926795i	$-0.377448\pi$
$17$	23.2217	−	13.4070i	1.36598	−	0.788649i	0.990412	+	0.138146i	$0.0441143\pi$
$18$		0			0
$19$	−3.98799	+	6.90741i	−0.209894	+	0.363548i	−0.951681	−	0.307088i	$-0.900645\pi$
$19$	−3.98799	+	6.90741i	−0.209894	+	0.363548i	0.741787	+	0.670636i	$0.233979\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−23.2217	−	13.4070i	−1.00964	−	0.582914i	−0.0985518	−	0.995132i	$-0.531421\pi$
$23$	−23.2217	−	13.4070i	−1.00964	−	0.582914i	−0.911085	+	0.412218i	$0.864754\pi$
$24$		0			0
$25$	24.2560	+	42.0126i	0.970241	+	1.68051i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	35.7359i		−	1.23227i	−0.787639	−	0.616137i	$-0.788697\pi$
$29$		−	35.7359i		−	1.23227i	0.787639	−	0.616137i	$-0.211303\pi$
$30$		0			0
$31$	11.9223	+	20.6501i	0.384591	+	0.666131i	0.991712	−	0.128478i	$-0.0410091\pi$
$31$	11.9223	+	20.6501i	0.384591	+	0.666131i	−0.607121	+	0.794609i	$0.707676\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−15.5689	+	57.9629i	−0.444827	+	1.65608i
$36$		0			0
$37$	19.1827	−	33.2254i	0.518451	−	0.897984i	−0.481319	−	0.876546i	$-0.659842\pi$
$37$	19.1827	−	33.2254i	0.518451	−	0.897984i	0.999770	−	0.0214386i	$-0.00682464\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	27.8548i		−	0.679386i	−0.940536	−	0.339693i	$-0.889677\pi$
$41$		−	27.8548i		−	0.679386i	0.940536	−	0.339693i	$-0.110323\pi$
$42$		0			0
$43$	−68.4986			−1.59299			−0.796496	−	0.604644i	$-0.793315\pi$
$43$	−68.4986			−1.59299			−0.796496	+	0.604644i	$0.793315\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−61.5951	−	35.5620i	−1.31053	−	0.756637i	−0.328350	−	0.944556i	$-0.606492\pi$
$47$	−61.5951	−	35.5620i	−1.31053	−	0.756637i	−0.982185	+	0.187919i	$0.939826\pi$
$48$		0			0
$49$	−42.4651	+	24.4483i	−0.866634	+	0.498945i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−54.9706	+	31.7373i	−1.03718	+	0.598817i	−0.919034	−	0.394179i	$-0.871029\pi$
$53$	−54.9706	+	31.7373i	−1.03718	+	0.598817i	−0.118148	+	0.992996i	$0.537696\pi$
$54$		0			0
$55$	−124.679			−2.26689
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.72657	+	4.46094i	−0.130959	+	0.0756091i	−0.564048	−	0.825742i	$-0.690757\pi$
$59$	−7.72657	+	4.46094i	−0.130959	+	0.0756091i	0.433089	+	0.901351i	$0.357423\pi$
$60$		0			0
$61$	2.88709	−	5.00059i	0.0473294	−	0.0819770i	−0.841390	−	0.540428i	$-0.818262\pi$
$61$	2.88709	−	5.00059i	0.0473294	−	0.0819770i	0.888720	+	0.458451i	$0.151596\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−80.9760	−	46.7515i	−1.24578	−	0.719254i
$66$		0			0
$67$	−17.3360	−	30.0269i	−0.258746	−	0.448162i	0.707160	−	0.707054i	$-0.249976\pi$
$67$	−17.3360	−	30.0269i	−0.258746	−	0.448162i	−0.965906	+	0.258892i	$0.916643\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			68.6199i			0.966478i	0.875488	+	0.483239i	$0.160540\pi$
$71$			68.6199i			0.966478i	−0.875488	+	0.483239i	$0.839460\pi$
$72$		0			0
$73$	65.6321	+	113.678i	0.899069	+	1.55723i	0.828686	+	0.559714i	$0.189089\pi$
$73$	65.6321	+	113.678i	0.899069	+	1.55723i	0.0703831	+	0.997520i	$0.477578\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−71.9336	−	72.0213i	−0.934203	−	0.935342i
$78$		0			0
$79$	15.0321	−	26.0363i	0.190280	−	0.329574i	−0.755063	−	0.655652i	$-0.772394\pi$
$79$	15.0321	−	26.0363i	0.190280	−	0.329574i	0.945343	+	0.326078i	$0.105727\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	86.5943i		−	1.04331i	−0.853158	−	0.521653i	$-0.825316\pi$
$83$		−	86.5943i		−	1.04331i	0.853158	−	0.521653i	$-0.174684\pi$
$84$		0			0
$85$	−229.901			−2.70472
$86$		0			0
$87$		0			0
$88$		0			0
$89$	111.334	+	64.2790i	1.25095	+	0.722236i	0.971298	−	0.237866i	$-0.0764480\pi$
$89$	111.334	+	64.2790i	1.25095	+	0.722236i	0.279651	+	0.960102i	$0.409781\pi$
$90$		0			0
$91$	−19.7130	−	73.7495i	−0.216626	−	0.810434i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	59.2235	−	34.1927i	0.623406	−	0.359923i
$96$		0			0
$97$	80.0096			0.824841			0.412421	−	0.910994i	$-0.364683\pi$
$97$	80.0096			0.824841			0.412421	+	0.910994i	$0.364683\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	52.2199	−	30.1492i	0.517029	−	0.298507i	−0.218689	−	0.975795i	$-0.570178\pi$
$101$	52.2199	−	30.1492i	0.517029	−	0.298507i	0.735718	+	0.677288i	$0.236845\pi$
$102$		0			0
$103$	−68.3079	+	118.313i	−0.663184	+	1.14867i	0.316591	+	0.948562i	$0.397462\pi$
$103$	−68.3079	+	118.313i	−0.663184	+	1.14867i	−0.979774	+	0.200106i	$0.935871\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−76.6949	−	44.2798i	−0.716775	−	0.413830i	0.0967895	−	0.995305i	$-0.469143\pi$
$107$	−76.6949	−	44.2798i	−0.716775	−	0.413830i	−0.813565	+	0.581475i	$0.802476\pi$
$108$		0			0
$109$	−60.2202	−	104.304i	−0.552479	−	0.956921i	−0.998095	−	0.0616973i	$-0.980349\pi$
$109$	−60.2202	−	104.304i	−0.552479	−	0.956921i	0.445616	−	0.895224i	$-0.352985\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			6.02151i			0.0532877i	0.999645	+	0.0266438i	$0.00848200\pi$
$113$			6.02151i			0.0532877i	−0.999645	+	0.0266438i	$0.991518\pi$
$114$		0			0
$115$	114.951	+	199.101i	0.999572	+	1.73131i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−132.642	−	132.804i	−1.11464	−	1.11600i
$120$		0			0
$121$	45.2298	−	78.3403i	0.373800	−	0.647440i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	201.590i		−	1.61272i
$126$		0			0
$127$	63.3782			0.499041			0.249521	−	0.968369i	$-0.419727\pi$
$127$	63.3782			0.499041			0.249521	+	0.968369i	$0.419727\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−32.7469	−	18.9064i	−0.249976	−	0.144324i	0.369777	−	0.929121i	$-0.379434\pi$
$131$	−32.7469	−	18.9064i	−0.249976	−	0.144324i	−0.619753	+	0.784797i	$0.712767\pi$
$132$		0			0
$133$	53.9207	+	14.4832i	0.405419	+	0.108896i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−56.2178	+	32.4573i	−0.410349	+	0.236915i	−0.690940	−	0.722913i	$-0.742803\pi$
$137$	−56.2178	+	32.4573i	−0.410349	+	0.236915i	0.280591	+	0.959827i	$0.409470\pi$
$138$		0			0
$139$	124.568			0.896171			0.448086	−	0.893991i	$-0.352106\pi$
$139$	124.568			0.896171			0.448086	+	0.893991i	$0.352106\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	137.338	−	79.2921i	0.960406	−	0.554490i
$144$		0			0
$145$	−153.198	+	265.347i	−1.05654	+	1.82998i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−135.169	−	78.0397i	−0.907173	−	0.523757i	−0.0276527	−	0.999618i	$-0.508803\pi$
$149$	−135.169	−	78.0397i	−0.907173	−	0.523757i	−0.879521	+	0.475861i	$0.842137\pi$
$150$		0			0
$151$	88.2247	+	152.810i	0.584270	+	1.01198i	0.994966	+	0.100213i	$0.0319523\pi$
$151$	88.2247	+	152.810i	0.584270	+	1.01198i	−0.410696	+	0.911772i	$0.634714\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	204.442i		−	1.31898i
$156$		0			0
$157$	−21.7282	−	37.6344i	−0.138396	−	0.239710i	0.788493	−	0.615043i	$-0.210861\pi$
$157$	−21.7282	−	37.6344i	−0.138396	−	0.239710i	−0.926890	+	0.375334i	$0.877528\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−48.6903	+	181.273i	−0.302424	+	1.12592i
$162$		0			0
$163$	−78.1675	+	135.390i	−0.479555	+	0.830614i	−0.999725	−	0.0234489i	$-0.992535\pi$
$163$	−78.1675	+	135.390i	−0.479555	+	0.830614i	0.520170	+	0.854063i	$0.325869\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			137.999i			0.826343i	0.910653	+	0.413172i	$0.135579\pi$
$167$			137.999i			0.826343i	−0.910653	+	0.413172i	$0.864421\pi$
$168$		0			0
$169$	−50.0696			−0.296270
$170$		0			0
$171$		0			0
$172$		0			0
$173$	53.5540	+	30.9194i	0.309560	+	0.178725i	0.646730	−	0.762719i	$-0.276136\pi$
$173$	53.5540	+	30.9194i	0.309560	+	0.178725i	−0.337169	+	0.941444i	$0.609469\pi$
$174$		0			0
$175$	240.268	−	239.976i	1.37296	−	1.37129i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	168.258	−	97.1438i	0.939988	−	0.542703i	0.0500318	−	0.998748i	$-0.484068\pi$
$179$	168.258	−	97.1438i	0.939988	−	0.542703i	0.889957	+	0.456045i	$0.150734\pi$
$180$		0			0
$181$	23.8714			0.131886			0.0659430	−	0.997823i	$-0.478994\pi$
$181$	23.8714			0.131886			0.0659430	+	0.997823i	$0.478994\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−284.872	+	164.471i	−1.53985	+	0.889032i
$186$		0			0
$187$	194.960	−	337.681i	1.04257	−	1.80578i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−142.176	−	82.0853i	−0.744376	−	0.429766i	0.0792821	−	0.996852i	$-0.474737\pi$
$191$	−142.176	−	82.0853i	−0.744376	−	0.429766i	−0.823658	+	0.567086i	$0.808071\pi$
$192$		0			0
$193$	−146.469	−	253.693i	−0.758909	−	1.31447i	−0.943407	−	0.331636i	$-0.892399\pi$
$193$	−146.469	−	253.693i	−0.758909	−	1.31447i	0.184498	−	0.982833i	$-0.440934\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			232.573i			1.18058i	0.807193	+	0.590288i	$0.200986\pi$
$197$			232.573i			1.18058i	−0.807193	+	0.590288i	$0.799014\pi$
$198$		0			0
$199$	12.0562	+	20.8820i	0.0605840	+	0.104935i	0.894727	−	0.446614i	$-0.147370\pi$
$199$	12.0562	+	20.8820i	0.0605840	+	0.104935i	−0.834143	+	0.551549i	$0.814037\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−241.667	+	64.5968i	−1.19048	+	0.318211i
$204$		0			0
$205$	−119.412	+	206.828i	−0.582500	+	1.00892i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			115.984i			0.554947i
$210$		0			0
$211$	−393.303			−1.86400			−0.931998	−	0.362464i	$-0.881936\pi$
$211$	−393.303			−1.86400			−0.931998	+	0.362464i	$0.881936\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	508.618	+	293.651i	2.36567	+	1.36582i
$216$		0			0
$217$	118.097	−	117.953i	0.544225	−	0.543563i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	253.244	−	146.211i	1.14590	−	0.661587i
$222$		0			0
$223$	−198.609			−0.890623			−0.445311	−	0.895376i	$-0.646907\pi$
$223$	−198.609			−0.890623			−0.445311	+	0.895376i	$0.646907\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−135.809	+	78.4096i	−0.598280	+	0.345417i	−0.768364	−	0.640013i	$-0.778929\pi$
$227$	−135.809	+	78.4096i	−0.598280	+	0.345417i	0.170085	+	0.985429i	$0.445596\pi$
$228$		0			0
$229$	10.2857	−	17.8154i	0.0449158	−	0.0777964i	−0.842694	−	0.538394i	$-0.819031\pi$
$229$	10.2857	−	17.8154i	0.0449158	−	0.0777964i	0.887609	+	0.460597i	$0.152365\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−64.0512	−	36.9800i	−0.274898	−	0.158712i	0.356214	−	0.934405i	$-0.384068\pi$
$233$	−64.0512	−	36.9800i	−0.274898	−	0.158712i	−0.631111	+	0.775692i	$0.717401\pi$
$234$		0			0
$235$	304.905	+	528.111i	1.29747	+	2.24728i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	238.052i		−	0.996035i	−0.867167	−	0.498017i	$-0.834062\pi$
$239$		−	238.052i		−	0.996035i	0.867167	−	0.498017i	$-0.165938\pi$
$240$		0			0
$241$	−100.701	−	174.420i	−0.417847	−	0.723733i	0.577875	−	0.816125i	$-0.303882\pi$
$241$	−100.701	−	174.420i	−0.417847	−	0.723733i	−0.995723	+	0.0923922i	$0.970549\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	420.122	+	0.511700i	1.71478	+	0.00208857i
$246$		0			0
$247$	−43.4912	+	75.3289i	−0.176078	+	0.304975i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			260.896i			1.03942i	0.854341	+	0.519712i	$0.173961\pi$
$251$			260.896i			1.03942i	−0.854341	+	0.519712i	$0.826039\pi$
$252$		0			0
$253$	−389.921			−1.54119
$254$		0			0
$255$		0			0
$256$		0			0
$257$	125.244	+	72.3099i	0.487332	+	0.281361i	0.723467	−	0.690359i	$-0.242547\pi$
$257$	125.244	+	72.3099i	0.487332	+	0.281361i	−0.236135	+	0.971720i	$0.575881\pi$
$258$		0			0
$259$	−259.365	−	69.6658i	−1.00141	−	0.268980i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−138.511	+	79.9692i	−0.526657	+	0.304065i	−0.739654	−	0.672987i	$-0.765011\pi$
$263$	−138.511	+	79.9692i	−0.526657	+	0.304065i	0.212997	+	0.977053i	$0.431677\pi$
$264$		0			0
$265$	544.226			2.05368
$266$		0			0
$267$		0			0
$268$		0			0
$269$	103.688	−	59.8641i	0.385456	−	0.222543i	−0.294733	−	0.955580i	$-0.595231\pi$
$269$	103.688	−	59.8641i	0.385456	−	0.222543i	0.680189	+	0.733036i	$0.261897\pi$
$270$		0			0
$271$	252.711	−	437.707i	0.932511	−	1.61516i	0.153498	−	0.988149i	$-0.450946\pi$
$271$	252.711	−	437.707i	0.932511	−	1.61516i	0.779013	−	0.627008i	$-0.215721\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	610.933	+	352.722i	2.22158	+	1.28263i
$276$		0			0
$277$	−145.701	−	252.361i	−0.525995	−	0.911050i	−0.999541	−	0.0302812i	$-0.990360\pi$
$277$	−145.701	−	252.361i	−0.525995	−	0.911050i	0.473546	−	0.880769i	$-0.342974\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			371.722i			1.32285i	0.750010	+	0.661427i	$0.230049\pi$
$281$			371.722i			1.32285i	−0.750010	+	0.661427i	$0.769951\pi$
$282$		0			0
$283$	−132.228	−	229.025i	−0.467236	−	0.809277i	0.532063	−	0.846705i	$-0.321417\pi$
$283$	−132.228	−	229.025i	−0.467236	−	0.809277i	−0.999299	+	0.0374281i	$0.988083\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−188.371	+	50.3508i	−0.656343	+	0.175438i
$288$		0			0
$289$	214.997	−	372.386i	0.743934	−	1.28853i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	559.211i		−	1.90857i	−0.298897	−	0.954285i	$-0.596619\pi$
$293$		−	559.211i		−	1.90857i	0.298897	−	0.954285i	$-0.403381\pi$
$294$		0			0
$295$	76.4954			0.259306
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−253.244	−	146.211i	−0.846971	−	0.488999i
$300$		0			0
$301$	123.819	+	463.228i	0.411359	+	1.53896i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−42.8747	+	24.7537i	−0.140573	+	0.0811597i
$306$		0			0
$307$	−230.693			−0.751442			−0.375721	−	0.926733i	$-0.622605\pi$
$307$	−230.693			−0.751442			−0.375721	+	0.926733i	$0.622605\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−147.705	+	85.2775i	−0.474936	+	0.274204i	−0.718304	−	0.695730i	$-0.755081\pi$
$311$	−147.705	+	85.2775i	−0.474936	+	0.274204i	0.243368	+	0.969934i	$0.421748\pi$
$312$		0			0
$313$	131.354	−	227.512i	0.419662	−	0.726875i	−0.576244	−	0.817278i	$-0.695482\pi$
$313$	131.354	−	227.512i	0.419662	−	0.726875i	0.995905	+	0.0904028i	$0.0288154\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	176.831	+	102.094i	0.557827	+	0.322062i	0.752273	−	0.658852i	$-0.228957\pi$
$317$	176.831	+	102.094i	0.557827	+	0.322062i	−0.194446	+	0.980913i	$0.562291\pi$
$318$		0			0
$319$	−259.830	−	450.038i	−0.814513	−	1.41078i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			213.869i			0.662132i
$324$		0			0
$325$	264.524	+	458.170i	0.813921	+	1.40975i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−129.150	+	480.824i	−0.392554	+	1.46147i
$330$		0			0
$331$	81.7030	−	141.514i	0.246837	−	0.427534i	−0.715810	−	0.698296i	$-0.753942\pi$
$331$	81.7030	−	141.514i	0.246837	−	0.427534i	0.962646	+	0.270761i	$0.0872755\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			297.275i			0.887388i
$336$		0			0
$337$	588.142			1.74523			0.872614	−	0.488411i	$-0.162423\pi$
$337$	588.142			1.74523			0.872614	+	0.488411i	$0.162423\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	300.286	+	173.370i	0.880605	+	0.508417i
$342$		0			0
$343$	242.094	+	242.980i	0.705814	+	0.708397i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	480.652	−	277.505i	1.38516	−	0.799725i	0.392399	−	0.919795i	$-0.371645\pi$
$347$	480.652	−	277.505i	1.38516	−	0.799725i	0.992765	+	0.120070i	$0.0383118\pi$
$348$		0			0
$349$	248.425			0.711819			0.355910	−	0.934520i	$-0.384171\pi$
$349$	248.425			0.711819			0.355910	+	0.934520i	$0.384171\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−502.632	+	290.195i	−1.42389	+	0.822081i	−0.996628	−	0.0820486i	$-0.973854\pi$
$353$	−502.632	+	290.195i	−1.42389	+	0.822081i	−0.427258	+	0.904130i	$0.640520\pi$
$354$		0			0
$355$	294.171	−	509.519i	0.828650	−	1.43526i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	167.450	+	96.6772i	0.466434	+	0.269296i	0.714746	−	0.699384i	$-0.246542\pi$
$359$	167.450	+	96.6772i	0.466434	+	0.269296i	−0.248312	+	0.968680i	$0.579876\pi$
$360$		0			0
$361$	148.692	+	257.542i	0.411889	+	0.713412i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	1125.45i		−	3.08342i
$366$		0			0
$367$	166.464	+	288.325i	0.453582	+	0.785626i	0.998605	−	0.0527943i	$-0.0168128\pi$
$367$	166.464	+	288.325i	0.453582	+	0.785626i	−0.545024	+	0.838421i	$0.683479\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	313.992	+	314.374i	0.846339	+	0.847371i
$372$		0			0
$373$	226.498	−	392.306i	0.607233	−	1.05176i	−0.384461	−	0.923141i	$-0.625613\pi$
$373$	226.498	−	392.306i	0.607233	−	1.05176i	0.991694	−	0.128617i	$-0.0410539\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	389.719i		−	1.03374i
$378$		0			0
$379$	598.216			1.57841			0.789203	−	0.614133i	$-0.210494\pi$
$379$	598.216			1.57841			0.789203	+	0.614133i	$0.210494\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	251.702	+	145.320i	0.657185	+	0.379426i	0.791204	−	0.611553i	$-0.209455\pi$
$383$	251.702	+	145.320i	0.657185	+	0.379426i	−0.134019	+	0.990979i	$0.542788\pi$
$384$		0			0
$385$	225.371	+	843.151i	0.585380	+	2.19000i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	80.6352	−	46.5548i	0.207289	−	0.119678i	−0.392762	−	0.919640i	$-0.628480\pi$
$389$	80.6352	−	46.5548i	0.207289	−	0.119678i	0.600051	+	0.799962i	$0.295147\pi$
$390$		0			0
$391$	−718.994			−1.83886
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−223.233	+	128.884i	−0.565148	+	0.326288i
$396$		0			0
$397$	150.183	−	260.124i	0.378294	−	0.655225i	−0.612520	−	0.790455i	$-0.709844\pi$
$397$	150.183	−	260.124i	0.378294	−	0.655225i	0.990814	+	0.135230i	$0.0431774\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	191.329	+	110.464i	0.477130	+	0.275471i	0.719220	−	0.694783i	$-0.244499\pi$
$401$	191.329	+	110.464i	0.477130	+	0.275471i	−0.242089	+	0.970254i	$0.577833\pi$
$402$		0			0
$403$	130.019	+	225.200i	0.322628	+	0.558809i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	557.896i		−	1.37075i
$408$		0			0
$409$	−70.8825	−	122.772i	−0.173307	−	0.300176i	0.766267	−	0.642522i	$-0.222112\pi$
$409$	−70.8825	−	122.772i	−0.173307	−	0.300176i	−0.939574	+	0.342346i	$0.888779\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	44.1341	+	44.1879i	0.106862	+	0.106993i
$414$		0			0
$415$	−371.226	+	642.983i	−0.894521	+	1.54936i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			455.434i			1.08695i	0.839424	+	0.543477i	$0.182893\pi$
$419$			455.434i			1.08695i	−0.839424	+	0.543477i	$0.817107\pi$
$420$		0			0
$421$	123.182			0.292593			0.146296	−	0.989241i	$-0.453265\pi$
$421$	123.182			0.292593			0.146296	+	0.989241i	$0.453265\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1126.53	+	650.402i	2.65066	+	1.53036i
$426$		0			0
$427$	−39.0357	−	10.4851i	−0.0914185	−	0.0245552i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−90.5387	+	52.2726i	−0.210067	+	0.121282i	−0.601342	−	0.798991i	$-0.705367\pi$
$431$	−90.5387	+	52.2726i	−0.210067	+	0.121282i	0.391276	+	0.920274i	$0.372034\pi$
$432$		0			0
$433$	249.952			0.577256			0.288628	−	0.957441i	$-0.406801\pi$
$433$	249.952			0.577256			0.288628	+	0.957441i	$0.406801\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	185.216	−	106.934i	0.423834	−	0.244701i
$438$		0			0
$439$	−181.672	+	314.666i	−0.413832	+	0.716779i	−0.995305	−	0.0967872i	$-0.969143\pi$
$439$	−181.672	+	314.666i	−0.413832	+	0.716779i	0.581473	+	0.813566i	$0.302477\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−627.067	−	362.037i	−1.41550	−	0.817240i	−0.419602	−	0.907708i	$-0.637830\pi$
$443$	−627.067	−	362.037i	−1.41550	−	0.817240i	−0.995899	+	0.0904680i	$0.971164\pi$
$444$		0			0
$445$	−551.122	−	954.572i	−1.23848	−	2.14511i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	498.027i		−	1.10919i	−0.832120	−	0.554596i	$-0.812873\pi$
$449$		−	498.027i		−	1.10919i	0.832120	−	0.554596i	$-0.187127\pi$
$450$		0			0
$451$	−202.528	−	350.788i	−0.449063	−	0.777800i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−169.788	+	632.116i	−0.373159	+	1.38927i
$456$		0			0
$457$	−318.373	+	551.437i	−0.696658	+	1.20665i	0.272961	+	0.962025i	$0.411997\pi$
$457$	−318.373	+	551.437i	−0.696658	+	1.20665i	−0.969619	+	0.244621i	$0.921336\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			597.813i			1.29678i	0.761310	+	0.648388i	$0.224556\pi$
$461$			597.813i			1.29678i	−0.761310	+	0.648388i	$0.775444\pi$
$462$		0			0
$463$	710.671			1.53493			0.767463	−	0.641093i	$-0.221519\pi$
$463$	710.671			1.53493			0.767463	+	0.641093i	$0.221519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−180.557	−	104.245i	−0.386632	−	0.223222i	0.294068	−	0.955784i	$-0.404991\pi$
$467$	−180.557	−	104.245i	−0.386632	−	0.223222i	−0.680700	+	0.732563i	$0.738324\pi$
$468$		0			0
$469$	−171.722	+	171.513i	−0.366145	+	0.365700i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−862.634	+	498.042i	−1.82375	+	1.05294i
$474$		0			0
$475$	−386.931			−0.814592
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−270.160	+	155.977i	−0.564008	+	0.325630i	−0.754753	−	0.656009i	$-0.772243\pi$
$479$	−270.160	+	155.977i	−0.564008	+	0.325630i	0.190745	+	0.981640i	$0.438910\pi$
$480$		0			0
$481$	209.197	−	362.341i	0.434922	−	0.753307i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−594.090	−	342.998i	−1.22493	−	0.707212i
$486$		0			0
$487$	−477.588	−	827.206i	−0.980673	−	1.69858i	−0.659778	−	0.751461i	$-0.729350\pi$
$487$	−477.588	−	827.206i	−0.980673	−	1.69858i	−0.320895	−	0.947115i	$-0.603984\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	131.161i		−	0.267131i	−0.991040	−	0.133565i	$-0.957357\pi$
$491$		−	131.161i		−	0.267131i	0.991040	−	0.133565i	$-0.0426426\pi$
$492$		0			0
$493$	−479.113	−	829.848i	−0.971831	−	1.68326i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	464.048	−	124.038i	0.933698	−	0.249574i
$498$		0			0
$499$	174.152	−	301.641i	0.349003	−	0.604491i	−0.637070	−	0.770806i	$-0.719854\pi$
$499$	174.152	−	301.641i	0.349003	−	0.604491i	0.986073	+	0.166315i	$0.0531870\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	602.739i		−	1.19829i	−0.800641	−	0.599144i	$-0.795508\pi$
$503$		−	602.739i		−	1.19829i	0.800641	−	0.599144i	$-0.204492\pi$
$504$		0			0
$505$	−516.993			−1.02375
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.2012	+	7.04438i	0.0239710	+	0.0138397i	0.511938	−	0.859023i	$-0.328928\pi$
$509$	12.2012	+	7.04438i	0.0239710	+	0.0138397i	−0.487967	+	0.872862i	$0.662261\pi$
$510$		0			0
$511$	650.120	−	649.328i	1.27225	−	1.27070i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1014.40	−	585.666i	1.96972	−	1.13722i
$516$		0			0
$517$	−1034.26			−2.00050
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−41.4161	+	23.9116i	−0.0794934	+	0.0458956i	−0.539220	−	0.842165i	$-0.681281\pi$
$521$	−41.4161	+	23.9116i	−0.0794934	+	0.0458956i	0.459726	+	0.888061i	$0.347947\pi$
$522$		0			0
$523$	192.487	−	333.397i	0.368044	−	0.637471i	−0.621216	−	0.783640i	$-0.713361\pi$
$523$	192.487	−	333.397i	0.368044	−	0.637471i	0.989260	+	0.146169i	$0.0466943\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	553.712	+	319.686i	1.05069	+	0.606615i
$528$		0			0
$529$	94.9969	+	164.539i	0.179578	+	0.311039i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	303.771i		−	0.569927i
$534$		0			0
$535$	379.652	+	657.576i	0.709629	+	1.22911i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−357.022	+	616.644i	−0.662378	+	1.14405i
$540$		0			0
$541$	−200.591	+	347.435i	−0.370779	+	0.642208i	−0.989686	−	0.143256i	$-0.954243\pi$
$541$	−200.591	+	347.435i	−0.370779	+	0.642208i	0.618907	+	0.785465i	$0.287576\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1032.65i			1.89476i
$546$		0			0
$547$	−826.934			−1.51176			−0.755881	−	0.654709i	$-0.772791\pi$
$547$	−826.934			−1.51176			−0.755881	+	0.654709i	$0.772791\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	246.843	+	142.515i	0.447990	+	0.258647i
$552$		0			0
$553$	−203.245	−	54.5921i	−0.367532	−	0.0987198i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−620.464	+	358.225i	−1.11394	+	0.643133i	−0.939847	−	0.341596i	$-0.889033\pi$
$557$	−620.464	+	358.225i	−1.11394	+	0.643133i	−0.174093	+	0.984729i	$0.555699\pi$
$558$		0			0
$559$	−747.014			−1.33634
$560$		0			0
$561$		0			0
$562$		0			0
$563$	215.447	−	124.388i	0.382676	−	0.220938i	−0.296306	−	0.955093i	$-0.595755\pi$
$563$	215.447	−	124.388i	0.382676	−	0.220938i	0.678982	+	0.734155i	$0.262421\pi$
$564$		0			0
$565$	25.8140	−	44.7111i	0.0456884	−	0.0791346i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−47.1007	−	27.1936i	−0.0827780	−	0.0477919i	0.458040	−	0.888932i	$-0.348552\pi$
$569$	−47.1007	−	27.1936i	−0.0827780	−	0.0477919i	−0.540818	+	0.841140i	$0.681885\pi$
$570$		0			0
$571$	210.364	+	364.362i	0.368414	+	0.638111i	0.989318	−	0.145775i	$-0.0465676\pi$
$571$	210.364	+	364.362i	0.368414	+	0.638111i	−0.620904	+	0.783887i	$0.713234\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1300.80i		−	2.26227i
$576$		0			0
$577$	356.399	+	617.301i	0.617676	+	1.06985i	0.989909	+	0.141707i	$0.0452591\pi$
$577$	356.399	+	617.301i	0.617676	+	1.06985i	−0.372232	+	0.928140i	$0.621408\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−585.601	+	156.529i	−1.00792	+	0.269413i
$582$		0			0
$583$	−461.513	+	799.363i	−0.791617	+	1.37112i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			35.9882i			0.0613088i	0.999530	+	0.0306544i	$0.00975912\pi$
$587$			35.9882i			0.0613088i	−0.999530	+	0.0306544i	$0.990241\pi$
$588$		0			0
$589$	−190.185			−0.322894
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−355.876	−	205.465i	−0.600128	−	0.346484i	0.168964	−	0.985622i	$-0.445958\pi$
$593$	−355.876	−	205.465i	−0.600128	−	0.346484i	−0.769092	+	0.639138i	$0.779291\pi$
$594$		0			0
$595$	415.573	+	1554.73i	0.698443	+	2.61299i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−432.401	+	249.647i	−0.721871	+	0.416773i	−0.815441	−	0.578840i	$-0.803506\pi$
$599$	−432.401	+	249.647i	−0.721871	+	0.416773i	0.0935698	+	0.995613i	$0.470172\pi$
$600$		0			0
$601$	70.6569			0.117566			0.0587828	−	0.998271i	$-0.481278\pi$
$601$	70.6569			0.117566			0.0587828	+	0.998271i	$0.481278\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−671.683	+	387.796i	−1.11022	+	0.640986i
$606$		0			0
$607$	415.264	−	719.258i	0.684125	−	1.18494i	−0.289586	−	0.957152i	$-0.593517\pi$
$607$	415.264	−	719.258i	0.684125	−	1.18494i	0.973711	−	0.227788i	$-0.0731492\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−671.727	−	387.822i	−1.09939	−	0.634733i
$612$		0			0
$613$	113.459	+	196.516i	0.185088	+	0.320582i	0.943606	−	0.331070i	$-0.107410\pi$
$613$	113.459	+	196.516i	0.185088	+	0.320582i	−0.758518	+	0.651652i	$0.774076\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	418.262i		−	0.677895i	−0.940805	−	0.338948i	$-0.889929\pi$
$617$		−	418.262i		−	0.677895i	0.940805	−	0.338948i	$-0.110071\pi$
$618$		0			0
$619$	256.600	+	444.445i	0.414540	+	0.718005i	0.995380	−	0.0960133i	$-0.0306091\pi$
$619$	256.600	+	444.445i	0.414540	+	0.718005i	−0.580840	+	0.814018i	$0.697276\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	233.442	−	869.100i	0.374706	−	1.39502i
$624$		0			0
$625$	−257.808	+	446.537i	−0.412493	+	0.714458i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1028.73i		−	1.63550i
$630$		0			0
$631$	976.542			1.54761			0.773805	−	0.633424i	$-0.218351\pi$
$631$	976.542			1.54761			0.773805	+	0.633424i	$0.218351\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−470.598	−	271.700i	−0.741099	−	0.427874i
$636$		0			0
$637$	−463.104	+	266.621i	−0.727007	+	0.418558i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	824.693	−	476.137i	1.28657	−	0.742803i	0.308532	−	0.951214i	$-0.400162\pi$
$641$	824.693	−	476.137i	1.28657	−	0.742803i	0.978041	+	0.208411i	$0.0668291\pi$
$642$		0			0
$643$	910.267			1.41566			0.707828	−	0.706385i	$-0.249675\pi$
$643$	910.267			1.41566			0.707828	+	0.706385i	$0.249675\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−671.283	+	387.565i	−1.03753	+	0.599019i	−0.919133	−	0.393947i	$-0.871109\pi$
$647$	−671.283	+	387.565i	−1.03753	+	0.599019i	−0.118398	+	0.992966i	$0.537776\pi$
$648$		0			0
$649$	−64.8694	+	112.357i	−0.0999528	+	0.173123i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	200.155	+	115.559i	0.306516	+	0.176967i	0.645366	−	0.763873i	$-0.276705\pi$
$653$	200.155	+	115.559i	0.306516	+	0.176967i	−0.338851	+	0.940840i	$0.610038\pi$
$654$		0			0
$655$	162.102	+	280.769i	0.247484	+	0.428655i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	731.379i		−	1.10983i	−0.831907	−	0.554915i	$-0.812751\pi$
$659$		−	731.379i		−	1.10983i	0.831907	−	0.554915i	$-0.187249\pi$
$660$		0			0
$661$	−313.850	−	543.605i	−0.474811	−	0.822397i	0.524773	−	0.851243i	$-0.324150\pi$
$661$	−313.850	−	543.605i	−0.474811	−	0.822397i	−0.999584	+	0.0288451i	$0.990817\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−338.284	−	338.697i	−0.508698	−	0.509318i
$666$		0			0
$667$	−479.113	+	829.848i	−0.718310	+	1.24415i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	83.9662i		−	0.125136i
$672$		0			0
$673$	−536.968			−0.797872			−0.398936	−	0.916979i	$-0.630620\pi$
$673$	−536.968			−0.797872			−0.398936	+	0.916979i	$0.630620\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	360.666	+	208.231i	0.532742	+	0.307579i	0.742132	−	0.670254i	$-0.233815\pi$
$677$	360.666	+	208.231i	0.532742	+	0.307579i	−0.209391	+	0.977832i	$0.567148\pi$
$678$		0			0
$679$	−144.627	−	541.072i	−0.212999	−	0.796865i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	725.980	−	419.145i	1.06293	−	0.613682i	0.136688	−	0.990614i	$-0.456354\pi$
$683$	725.980	−	419.145i	1.06293	−	0.613682i	0.926241	+	0.376932i	$0.123021\pi$
$684$		0			0
$685$	556.573			0.812515
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−599.483	+	346.112i	−0.870077	+	0.502339i
$690$		0			0
$691$	206.175	−	357.106i	0.298372	−	0.516796i	−0.677391	−	0.735623i	$-0.736890\pi$
$691$	206.175	−	357.106i	0.298372	−	0.516796i	0.975764	+	0.218827i	$0.0702230\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−924.945	−	534.017i	−1.33086	−	0.768370i
$696$		0			0
$697$	−373.450	−	646.835i	−0.535797	−	0.928027i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	19.5457i		−	0.0278825i	−0.999903	−	0.0139413i	$-0.995562\pi$
$701$		−	19.5457i		−	0.0278825i	0.999903	−	0.0139413i	$-0.00443779\pi$
$702$		0			0
$703$	153.001	+	265.006i	0.217640	+	0.376964i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−298.280	−	298.643i	−0.421895	−	0.422409i
$708$		0			0
$709$	60.3562	−	104.540i	0.0851287	−	0.147447i	−0.820317	−	0.571909i	$-0.806203\pi$
$709$	60.3562	−	104.540i	0.0851287	−	0.147447i	0.905446	+	0.424461i	$0.139537\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	639.372i		−	0.896735i
$714$		0			0
$715$	−1359.69			−1.90166
$716$		0			0
$717$		0			0
$718$		0			0
$719$	765.605	+	442.022i	1.06482	+	0.614773i	0.926761	−	0.375651i	$-0.122581\pi$
$719$	765.605	+	442.022i	1.06482	+	0.614773i	0.138057	+	0.990424i	$0.455914\pi$
$720$		0			0
$721$	923.574	+	248.074i	1.28096	+	0.344069i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1501.36	−	866.811i	2.07084	−	1.19560i
$726$		0			0
$727$	1322.62			1.81929			0.909643	−	0.415391i	$-0.136355\pi$
$727$	1322.62			1.81929			0.909643	+	0.415391i	$0.136355\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1590.65	+	918.363i	−2.17599	+	1.25631i
$732$		0			0
$733$	162.392	−	281.271i	0.221544	−	0.383726i	−0.733733	−	0.679438i	$-0.762224\pi$
$733$	162.392	−	281.271i	0.221544	−	0.383726i	0.955277	+	0.295712i	$0.0955570\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−436.640	−	252.094i	−0.592456	−	0.342055i
$738$		0			0
$739$	−82.2495	−	142.460i	−0.111298	−	0.192774i	0.804996	−	0.593281i	$-0.202168\pi$
$739$	−82.2495	−	142.460i	−0.111298	−	0.192774i	−0.916294	+	0.400506i	$0.868834\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			247.995i			0.333775i	0.985976	+	0.166888i	$0.0533717\pi$
$743$			247.995i			0.333775i	−0.985976	+	0.166888i	$0.946628\pi$
$744$		0			0
$745$	669.106	+	1158.93i	0.898129	+	1.55561i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−160.811	+	598.697i	−0.214701	+	0.799328i
$750$		0			0
$751$	312.140	−	540.642i	0.415632	−	0.719896i	−0.579862	−	0.814714i	$-0.696894\pi$
$751$	312.140	−	540.642i	0.415632	−	0.719896i	0.995495	+	0.0948184i	$0.0302270\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1512.86i		−	2.00379i
$756$		0			0
$757$	−1091.35			−1.44167			−0.720836	−	0.693105i	$-0.756242\pi$
$757$	−1091.35			−1.44167			−0.720836	+	0.693105i	$0.756242\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−418.488	−	241.614i	−0.549919	−	0.317496i	0.199171	−	0.979965i	$-0.436175\pi$
$761$	−418.488	−	241.614i	−0.549919	−	0.317496i	−0.749089	+	0.662469i	$0.769509\pi$
$762$		0			0
$763$	−596.512	+	595.786i	−0.781799	+	0.780847i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−84.2623	+	48.6489i	−0.109860	+	0.0634274i
$768$		0			0
$769$	−1025.78			−1.33391			−0.666957	−	0.745096i	$-0.732404\pi$
$769$	−1025.78			−1.33391			−0.666957	+	0.745096i	$0.732404\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1032.07	+	595.867i	−1.33515	+	0.770851i	−0.986084	−	0.166247i	$-0.946835\pi$
$773$	−1032.07	+	595.867i	−1.33515	+	0.770851i	−0.349068	+	0.937097i	$0.613502\pi$
$774$		0			0
$775$	−578.376	+	1001.78i	−0.746292	+	1.29262i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	192.405	+	111.085i	0.246989	+	0.142599i
$780$		0			0
$781$	498.924	+	864.161i	0.638827	+	1.10648i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			372.592i			0.474640i
$786$		0			0
$787$	−705.626	−	1222.18i	−0.896602	−	1.55296i	−0.831809	−	0.555062i	$-0.812695\pi$
$787$	−705.626	−	1222.18i	−0.896602	−	1.55296i	−0.0647927	−	0.997899i	$-0.520639\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	40.7209	−	10.8846i	0.0514803	−	0.0137605i
$792$		0			0
$793$	31.4853	−	54.5341i	0.0397040	−	0.0687693i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1495.22i		−	1.87607i	−0.346547	−	0.938033i	$-0.612646\pi$
$797$		−	1495.22i		−	1.87607i	0.346547	−	0.938033i	$-0.387354\pi$
$798$		0			0
$799$	−1907.12			−2.38688
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1653.07	+	954.398i	2.05861	+	1.18854i
$804$		0			0
$805$	1138.65	−	1137.26i	1.41447	−	1.41275i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	539.657	−	311.571i	0.667067	−	0.385131i	−0.127897	−	0.991787i	$-0.540823\pi$
$809$	539.657	−	311.571i	0.667067	−	0.385131i	0.794964	+	0.606656i	$0.207489\pi$
$810$		0			0
$811$	215.591			0.265833			0.132917	−	0.991127i	$-0.457566\pi$
$811$	215.591			0.265833			0.132917	+	0.991127i	$0.457566\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1160.82	−	670.201i	1.42432	−	0.822333i
$816$		0			0
$817$	273.172	−	473.148i	0.334360	−	0.579129i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1155.79	−	667.294i	−1.40778	−	0.812781i	−0.412605	−	0.910910i	$-0.635381\pi$
$821$	−1155.79	−	667.294i	−1.40778	−	0.812781i	−0.995174	+	0.0981286i	$0.968714\pi$
$822$		0			0
$823$	84.4535	+	146.278i	0.102617	+	0.177737i	0.912762	−	0.408492i	$-0.133945\pi$
$823$	84.4535	+	146.278i	0.102617	+	0.177737i	−0.810145	+	0.586229i	$0.800612\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			336.132i			0.406447i	0.979132	+	0.203224i	$0.0651419\pi$
$827$			336.132i			0.406447i	−0.979132	+	0.203224i	$0.934858\pi$
$828$		0			0
$829$	625.046	+	1082.61i	0.753976	+	1.30593i	0.945882	+	0.324512i	$0.105200\pi$
$829$	625.046	+	1082.61i	0.753976	+	1.30593i	−0.191905	+	0.981413i	$0.561467\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−658.330	+	1137.06i	−0.790312	+	1.36502i
$834$		0			0
$835$	591.597	−	1024.68i	0.708500	−	1.22716i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	173.924i		−	0.207299i	−0.994614	−	0.103649i	$-0.966948\pi$
$839$		−	173.924i		−	0.207299i	0.994614	−	0.103649i	$-0.0330520\pi$
$840$		0			0
$841$	−436.057			−0.518498
$842$		0			0
$843$		0			0
$844$		0			0
$845$	371.778	+	214.646i	0.439974	+	0.254019i
$846$		0			0
$847$	−611.541	−	164.261i	−0.722008	−	0.193933i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−890.908	+	514.366i	−1.04690	+	0.604426i
$852$		0			0
$853$	−74.2360			−0.0870293			−0.0435147	−	0.999053i	$-0.513856\pi$
$853$	−74.2360			−0.0870293			−0.0435147	+	0.999053i	$0.513856\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	769.828	−	444.460i	0.898282	−	0.518624i	0.0216398	−	0.999766i	$-0.493111\pi$
$857$	769.828	−	444.460i	0.898282	−	0.518624i	0.876642	+	0.481142i	$0.159778\pi$
$858$		0			0
$859$	−31.0197	+	53.7276i	−0.0361114	+	0.0625467i	−0.883516	−	0.468401i	$-0.844830\pi$
$859$	−31.0197	+	53.7276i	−0.0361114	+	0.0625467i	0.847405	+	0.530947i	$0.178164\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	567.763	+	327.798i	0.657894	+	0.379835i	0.791474	−	0.611203i	$-0.209314\pi$
$863$	567.763	+	327.798i	0.657894	+	0.379835i	−0.133580	+	0.991038i	$0.542647\pi$
$864$		0			0
$865$	−265.100	−	459.167i	−0.306474	−	0.530829i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	437.183i		−	0.503087i
$870$		0			0
$871$	−189.058	−	327.458i	−0.217059	−	0.375957i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1363.27	+	364.397i	−1.55802	+	0.416454i
$876$		0			0
$877$	−633.622	+	1097.46i	−0.722488	+	1.25139i	0.237512	+	0.971385i	$0.423668\pi$
$877$	−633.622	+	1097.46i	−0.722488	+	1.25139i	−0.960000	+	0.280001i	$0.909665\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	6.20047i		−	0.00703800i	−0.999994	−	0.00351900i	$-0.998880\pi$
$881$		−	6.20047i		−	0.00703800i	0.999994	−	0.00351900i	$-0.00112013\pi$
$882$		0			0
$883$	−1122.22			−1.27092			−0.635459	−	0.772135i	$-0.719189\pi$
$883$	−1122.22			−1.27092			−0.635459	+	0.772135i	$0.719189\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−416.132	−	240.254i	−0.469146	−	0.270861i	0.246736	−	0.969083i	$-0.420642\pi$
$887$	−416.132	−	240.254i	−0.469146	−	0.270861i	−0.715882	+	0.698221i	$0.753975\pi$
$888$		0			0
$889$	−114.563	−	428.601i	−0.128868	−	0.482115i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	491.282	−	283.642i	0.550148	−	0.317628i
$894$		0			0
$895$	−1665.80			−1.86123
$896$		0			0
$897$		0			0
$898$		0			0
$899$	737.950	−	426.056i	0.820856	−	0.473922i
$900$		0			0
$901$	−851.006	+	1473.99i	−0.944513	+	1.63594i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−177.250	−	102.336i	−0.195857	−	0.113078i
$906$		0			0
$907$	−85.2244	−	147.613i	−0.0939630	−	0.162749i	0.815212	−	0.579162i	$-0.196620\pi$
$907$	−85.2244	−	147.613i	−0.0939630	−	0.162749i	−0.909175	+	0.416414i	$0.863287\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1180.21i			1.29552i	0.761847	+	0.647758i	$0.224293\pi$
$911$			1180.21i			1.29552i	−0.761847	+	0.647758i	$0.775707\pi$
$912$		0			0
$913$	−629.612	−	1090.52i	−0.689608	−	1.19444i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−68.6625	+	255.629i	−0.0748773	+	0.278767i
$918$		0			0
$919$	267.840	−	463.913i	0.291448	−	0.504802i	−0.682705	−	0.730695i	$-0.739196\pi$
$919$	267.840	−	463.913i	0.291448	−	0.504802i	0.974152	+	0.225892i	$0.0725297\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			748.336i			0.810765i
$924$		0			0
$925$	1861.18			2.01209
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−427.479	−	246.805i	−0.460149	−	0.265667i	0.251958	−	0.967738i	$-0.418926\pi$
$929$	−427.479	−	246.805i	−0.460149	−	0.265667i	−0.712107	+	0.702071i	$0.752259\pi$
$930$		0			0
$931$	0.476015	−	390.823i	0.000511294	−	0.419788i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2895.25	+	1671.57i	−3.09652	+	1.78778i
$936$		0			0
$937$	−196.477			−0.209687			−0.104844	−	0.994489i	$-0.533434\pi$
$937$	−196.477			−0.209687			−0.104844	+	0.994489i	$0.533434\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	589.750	−	340.492i	0.626727	−	0.361841i	−0.152756	−	0.988264i	$-0.548815\pi$
$941$	589.750	−	340.492i	0.626727	−	0.361841i	0.779483	+	0.626423i	$0.215482\pi$
$942$		0			0
$943$	−373.450	+	646.835i	−0.396024	+	0.685933i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−565.269	−	326.358i	−0.596905	−	0.344623i	0.170918	−	0.985285i	$-0.445327\pi$
$947$	−565.269	−	326.358i	−0.596905	−	0.344623i	−0.767823	+	0.640662i	$0.778660\pi$
$948$		0			0
$949$	715.752	+	1239.72i	0.754217	+	1.30634i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	153.185i		−	0.160740i	−0.996765	−	0.0803701i	$-0.974390\pi$
$953$		−	153.185i		−	0.160740i	0.996765	−	0.0803701i	$-0.0256102\pi$
$954$		0			0
$955$	703.792	+	1219.00i	0.736955	+	1.27644i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	321.116	+	321.507i	0.334844	+	0.335252i
$960$		0			0
$961$	196.216	−	339.856i	0.204179	−	0.353649i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2511.63i			2.60273i
$966$		0			0
$967$	−1697.56			−1.75550			−0.877748	−	0.479122i	$-0.840955\pi$
$967$	−1697.56			−1.75550			−0.877748	+	0.479122i	$0.840955\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	439.999	+	254.033i	0.453140	+	0.261620i	0.709155	−	0.705052i	$-0.249076\pi$
$971$	439.999	+	254.033i	0.453140	+	0.261620i	−0.256016	+	0.966673i	$0.582410\pi$
$972$		0			0
$973$	−225.171	−	842.400i	−0.231419	−	0.865776i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	545.858	−	315.151i	0.558709	−	0.322571i	−0.193918	−	0.981018i	$-0.562120\pi$
$977$	545.858	−	315.151i	0.558709	−	0.322571i	0.752627	+	0.658447i	$0.228786\pi$
$978$		0			0
$979$	1869.44			1.90954
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−243.752	+	140.730i	−0.247968	+	0.143164i	−0.618833	−	0.785522i	$-0.712394\pi$
$983$	−243.752	+	140.730i	−0.247968	+	0.143164i	0.370866	+	0.928687i	$0.379061\pi$
$984$		0			0
$985$	997.032	−	1726.91i	1.01222	−	1.75321i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1590.65	+	918.363i	1.60834	+	0.928578i
$990$		0			0
$991$	−120.510	−	208.730i	−0.121605	−	0.210626i	0.798796	−	0.601602i	$-0.205471\pi$
$991$	−120.510	−	208.730i	−0.121605	−	0.210626i	−0.920401	+	0.390976i	$0.872137\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	206.738i		−	0.207777i
$996$		0			0
$997$	712.685	+	1234.41i	0.714830	+	1.23812i	0.963025	+	0.269411i	$0.0868291\pi$
$997$	712.685	+	1234.41i	0.714830	+	1.23812i	−0.248196	+	0.968710i	$0.579838\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.dc.f.737.1		16
3.2	odd	2	inner	1008.3.dc.f.737.8		16
4.3	odd	2		504.3.cu.b.233.1	✓	16
7.4	even	3	inner	1008.3.dc.f.305.8		16
12.11	even	2		504.3.cu.b.233.8	yes	16
21.11	odd	6	inner	1008.3.dc.f.305.1		16
28.11	odd	6		504.3.cu.b.305.8	yes	16
28.19	even	6		3528.3.d.f.1961.1		8
28.23	odd	6		3528.3.d.k.1961.8		8
84.11	even	6		504.3.cu.b.305.1	yes	16
84.23	even	6		3528.3.d.k.1961.1		8
84.47	odd	6		3528.3.d.f.1961.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.cu.b.233.1	✓	16	4.3	odd	2
504.3.cu.b.233.8	yes	16	12.11	even	2
504.3.cu.b.305.1	yes	16	84.11	even	6
504.3.cu.b.305.8	yes	16	28.11	odd	6
1008.3.dc.f.305.1		16	21.11	odd	6	inner
1008.3.dc.f.305.8		16	7.4	even	3	inner
1008.3.dc.f.737.1		16	1.1	even	1	trivial
1008.3.dc.f.737.8		16	3.2	odd	2	inner
3528.3.d.f.1961.1		8	28.19	even	6
3528.3.d.f.1961.8		8	84.47	odd	6
3528.3.d.k.1961.1		8	84.23	even	6
3528.3.d.k.1961.8		8	28.23	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-7.42523 - 4.28696i) q^{5} +(-1.80762 - 6.76258i) q^{7} +(12.5934 - 7.27083i) q^{11} +10.9055 q^{13} +(23.2217 - 13.4070i) q^{17} +(-3.98799 + 6.90741i) q^{19} +(-23.2217 - 13.4070i) q^{23} +(24.2560 + 42.0126i) q^{25} -35.7359i q^{29} +(11.9223 + 20.6501i) q^{31} +(-15.5689 + 57.9629i) q^{35} +(19.1827 - 33.2254i) q^{37} -27.8548i q^{41} -68.4986 q^{43} +(-61.5951 - 35.5620i) q^{47} +(-42.4651 + 24.4483i) q^{49} +(-54.9706 + 31.7373i) q^{53} -124.679 q^{55} +(-7.72657 + 4.46094i) q^{59} +(2.88709 - 5.00059i) q^{61} +(-80.9760 - 46.7515i) q^{65} +(-17.3360 - 30.0269i) q^{67} +68.6199i q^{71} +(65.6321 + 113.678i) q^{73} +(-71.9336 - 72.0213i) q^{77} +(15.0321 - 26.0363i) q^{79} -86.5943i q^{83} -229.901 q^{85} +(111.334 + 64.2790i) q^{89} +(-19.7130 - 73.7495i) q^{91} +(59.2235 - 34.1927i) q^{95} +80.0096 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 40 q^{13} + 20 q^{19} - 4 q^{25} + 56 q^{31} + 76 q^{37} - 72 q^{43} - 48 q^{49} - 648 q^{55} - 72 q^{61} + 156 q^{67} + 124 q^{73} - 184 q^{79} - 864 q^{85} + 116 q^{91} + 1112 q^{97}+O(q^{100}) \) 16 * q + 40 * q^13 + 20 * q^19 - 4 * q^25 + 56 * q^31 + 76 * q^37 - 72 * q^43 - 48 * q^49 - 648 * q^55 - 72 * q^61 + 156 * q^67 + 124 * q^73 - 184 * q^79 - 864 * q^85 + 116 * q^91 + 1112 * q^97

Introduction

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