LMFDB - Embedded newform 504.4.bl.a.17.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,4,Mod(17,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("504.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	504.bl (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$29.7369626429$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.3
Character	$\chi$	$=$	504.17
Dual form			504.4.bl.a.89.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.48442 - 14.6954i) q^{5} +(3.52711 - 18.1813i) q^{7} +O(q^{10})$	$q+(-8.48442 - 14.6954i) q^{5} +(3.52711 - 18.1813i) q^{7} +(60.3894 + 34.8658i) q^{11} +38.2009i q^{13} +(52.9872 - 91.7766i) q^{17} +(50.3954 - 29.0958i) q^{19} +(107.941 - 62.3199i) q^{23} +(-81.4706 + 141.111i) q^{25} -66.5957i q^{29} +(-136.580 - 78.8547i) q^{31} +(-297.108 + 102.425i) q^{35} +(-107.191 - 185.660i) q^{37} -448.509 q^{41} -320.784 q^{43} +(-87.8142 - 152.099i) q^{47} +(-318.119 - 128.255i) q^{49} +(585.770 + 338.194i) q^{53} -1183.26i q^{55} +(343.373 - 594.740i) q^{59} +(-91.1567 + 52.6294i) q^{61} +(561.379 - 324.112i) q^{65} +(-426.641 + 738.964i) q^{67} +21.6022i q^{71} +(296.300 + 171.069i) q^{73} +(846.906 - 974.982i) q^{77} +(-156.927 - 271.805i) q^{79} -627.751 q^{83} -1798.26 q^{85} +(207.480 + 359.367i) q^{89} +(694.541 + 134.739i) q^{91} +(-855.150 - 493.721i) q^{95} -223.956i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 24 q^{7}+O(q^{10}) $ 48 * q - 24 * q^7	$ 48 q - 24 q^{7} + 540 q^{19} - 924 q^{25} + 648 q^{31} - 132 q^{37} - 792 q^{43} + 672 q^{49} - 12 q^{67} + 2412 q^{73} + 1680 q^{79} + 480 q^{85} + 1404 q^{91}+O(q^{100}) $ 48 * q - 24 * q^7 + 540 * q^19 - 924 * q^25 + 648 * q^31 - 132 * q^37 - 792 * q^43 + 672 * q^49 - 12 * q^67 + 2412 * q^73 + 1680 * q^79 + 480 * q^85 + 1404 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$	$-1$

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
$5$	−8.48442	−	14.6954i	−0.758869	−	1.31440i	−0.943428	−	0.331578i	$-0.892419\pi$
$5$	−8.48442	−	14.6954i	−0.758869	−	1.31440i	0.184558	−	0.982822i	$-0.440914\pi$
$6$		0			0
$7$	3.52711	−	18.1813i	0.190446	−	0.981698i
$7$	3.52711	−	18.1813i	0.190446	−	0.981698i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	60.3894	+	34.8658i	1.65528	+	0.955677i	0.974849	+	0.222865i	$0.0715409\pi$
$11$	60.3894	+	34.8658i	1.65528	+	0.955677i	0.680431	+	0.732812i	$0.261792\pi$
$12$		0			0
$13$			38.2009i			0.815001i	0.913205	+	0.407501i	$0.133600\pi$
$13$			38.2009i			0.815001i	−0.913205	+	0.407501i	$0.866400\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	52.9872	−	91.7766i	0.755958	−	1.30936i	−0.188938	−	0.981989i	$-0.560505\pi$
$17$	52.9872	−	91.7766i	0.755958	−	1.30936i	0.944897	−	0.327369i	$-0.106162\pi$
$18$		0			0
$19$	50.3954	−	29.0958i	0.608499	−	0.351317i	−0.163879	−	0.986481i	$-0.552401\pi$
$19$	50.3954	−	29.0958i	0.608499	−	0.351317i	0.772378	+	0.635163i	$0.219067\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	107.941	−	62.3199i	0.978578	−	0.564982i	0.0767375	−	0.997051i	$-0.475550\pi$
$23$	107.941	−	62.3199i	0.978578	−	0.564982i	0.901841	+	0.432069i	$0.142216\pi$
$24$		0			0
$25$	−81.4706	+	141.111i	−0.651765	+	1.12889i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	66.5957i		−	0.426431i	−0.977005	−	0.213216i	$-0.931606\pi$
$29$		−	66.5957i		−	0.426431i	0.977005	−	0.213216i	$-0.0683937\pi$
$30$		0			0
$31$	−136.580	−	78.8547i	−0.791309	−	0.456862i	0.0491145	−	0.998793i	$-0.484360\pi$
$31$	−136.580	−	78.8547i	−0.791309	−	0.456862i	−0.840423	+	0.541931i	$0.817693\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−297.108	+	102.425i	−1.43487	+	0.494658i
$36$		0			0
$37$	−107.191	−	185.660i	−0.476271	−	0.824926i	0.523359	−	0.852112i	$-0.324679\pi$
$37$	−107.191	−	185.660i	−0.476271	−	0.824926i	−0.999630	+	0.0271864i	$0.991345\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−448.509			−1.70842			−0.854212	−	0.519925i	$-0.825960\pi$
$41$	−448.509			−1.70842			−0.854212	+	0.519925i	$0.825960\pi$
$42$		0			0
$43$	−320.784			−1.13765			−0.568827	−	0.822457i	$-0.692603\pi$
$43$	−320.784			−1.13765			−0.568827	+	0.822457i	$0.692603\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−87.8142	−	152.099i	−0.272532	−	0.472040i	0.696977	−	0.717093i	$-0.254528\pi$
$47$	−87.8142	−	152.099i	−0.272532	−	0.472040i	−0.969510	+	0.245054i	$0.921194\pi$
$48$		0			0
$49$	−318.119	−	128.255i	−0.927461	−	0.373921i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	585.770	+	338.194i	1.51815	+	0.876502i	0.999772	+	0.0213470i	$0.00679548\pi$
$53$	585.770	+	338.194i	1.51815	+	0.876502i	0.518373	+	0.855155i	$0.326538\pi$
$54$		0			0
$55$		−	1183.26i		−	2.90093i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	343.373	−	594.740i	0.757685	−	1.31235i	−0.186344	−	0.982485i	$-0.559664\pi$
$59$	343.373	−	594.740i	0.757685	−	1.31235i	0.944028	−	0.329864i	$-0.107003\pi$
$60$		0			0
$61$	−91.1567	+	52.6294i	−0.191335	+	0.110467i	−0.592607	−	0.805492i	$-0.701901\pi$
$61$	−91.1567	+	52.6294i	−0.191335	+	0.110467i	0.401272	+	0.915959i	$0.368568\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	561.379	−	324.112i	1.07124	−	0.618479i
$66$		0			0
$67$	−426.641	+	738.964i	−0.777948	+	1.34745i	0.155174	+	0.987887i	$0.450406\pi$
$67$	−426.641	+	738.964i	−0.777948	+	1.34745i	−0.933122	+	0.359559i	$0.882927\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			21.6022i			0.0361086i	0.999837	+	0.0180543i	$0.00574718\pi$
$71$			21.6022i			0.0361086i	−0.999837	+	0.0180543i	$0.994253\pi$
$72$		0			0
$73$	296.300	+	171.069i	0.475059	+	0.274276i	0.718355	−	0.695676i	$-0.244895\pi$
$73$	296.300	+	171.069i	0.475059	+	0.274276i	−0.243296	+	0.969952i	$0.578229\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	846.906	−	974.982i	1.25343	−	1.44298i
$78$		0			0
$79$	−156.927	−	271.805i	−0.223489	−	0.387095i	0.732376	−	0.680901i	$-0.238411\pi$
$79$	−156.927	−	271.805i	−0.223489	−	0.387095i	−0.955865	+	0.293806i	$0.905078\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−627.751			−0.830176			−0.415088	−	0.909781i	$-0.636249\pi$
$83$	−627.751			−0.830176			−0.415088	+	0.909781i	$0.636249\pi$
$84$		0			0
$85$	−1798.26			−2.29469
$86$		0			0
$87$		0			0
$88$		0			0
$89$	207.480	+	359.367i	0.247111	+	0.428009i	0.962723	−	0.270489i	$-0.0871854\pi$
$89$	207.480	+	359.367i	0.247111	+	0.428009i	−0.715612	+	0.698498i	$0.753852\pi$
$90$		0			0
$91$	694.541	+	134.739i	0.800085	+	0.155214i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−855.150	−	493.721i	−0.923543	−	0.533208i
$96$		0			0
$97$		−	223.956i		−	0.234426i	−0.993107	−	0.117213i	$-0.962604\pi$
$97$		−	223.956i		−	0.234426i	0.993107	−	0.117213i	$-0.0373960\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	974.570	−	1688.01i	0.960133	−	1.66300i	0.237973	−	0.971272i	$-0.423517\pi$
$101$	974.570	−	1688.01i	0.960133	−	1.66300i	0.722160	−	0.691727i	$-0.243150\pi$
$102$		0			0
$103$	−755.905	+	436.422i	−0.723122	+	0.417495i	−0.815901	−	0.578192i	$-0.803758\pi$
$103$	−755.905	+	436.422i	−0.723122	+	0.417495i	0.0927787	+	0.995687i	$0.470425\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	490.143	−	282.984i	0.442840	−	0.255674i	−0.261961	−	0.965078i	$-0.584369\pi$
$107$	490.143	−	282.984i	0.442840	−	0.255674i	0.704802	+	0.709404i	$0.251036\pi$
$108$		0			0
$109$	781.523	−	1353.64i	0.686755	−	1.18949i	−0.286127	−	0.958192i	$-0.592368\pi$
$109$	781.523	−	1353.64i	0.686755	−	1.18949i	0.972882	−	0.231303i	$-0.0742988\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1609.24i			1.33969i	0.742502	+	0.669844i	$0.233639\pi$
$113$			1609.24i			1.33969i	−0.742502	+	0.669844i	$0.766361\pi$
$114$		0			0
$115$	−1831.64	−	1057.50i	−1.48523	−	0.857495i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1481.72	−	1287.08i	−1.14142	−	0.991484i
$120$		0			0
$121$	1765.75	+	3058.37i	1.32664	+	2.29780i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	643.818			0.460679
$126$		0			0
$127$	−741.194			−0.517877			−0.258938	−	0.965894i	$-0.583373\pi$
$127$	−741.194			−0.517877			−0.258938	+	0.965894i	$0.583373\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	132.345	+	229.229i	0.0882676	+	0.152884i	0.906779	−	0.421606i	$-0.138534\pi$
$131$	132.345	+	229.229i	0.0882676	+	0.152884i	−0.818511	+	0.574490i	$0.805200\pi$
$132$		0			0
$133$	−351.249	−	1018.88i	−0.229001	−	0.664269i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1028.68	+	593.907i	0.641502	+	0.370372i	0.785193	−	0.619251i	$-0.212564\pi$
$137$	1028.68	+	593.907i	0.641502	+	0.370372i	−0.143691	+	0.989623i	$0.545897\pi$
$138$		0			0
$139$		−	1080.39i		−	0.659262i	−0.944110	−	0.329631i	$-0.893076\pi$
$139$		−	1080.39i		−	0.659262i	0.944110	−	0.329631i	$-0.106924\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1331.91	+	2306.93i	−0.778878	+	1.34906i
$144$		0			0
$145$	−978.653	+	565.026i	−0.560501	+	0.323606i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.97793	−	4.60606i	0.00438643	−	0.00253251i	−0.497805	−	0.867289i	$-0.665861\pi$
$149$	7.97793	−	4.60606i	0.00438643	−	0.00253251i	0.502192	+	0.864756i	$0.332527\pi$
$150$		0			0
$151$	978.571	−	1694.94i	0.527384	−	0.913456i	−0.472106	−	0.881542i	$-0.656506\pi$
$151$	978.571	−	1694.94i	0.527384	−	0.913456i	0.999491	−	0.0319147i	$-0.0101605\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			2676.15i			1.38679i
$156$		0			0
$157$	−188.970	−	109.102i	−0.0960600	−	0.0554603i	0.451200	−	0.892423i	$-0.350996\pi$
$157$	−188.970	−	109.102i	−0.0960600	−	0.0554603i	−0.547260	+	0.836962i	$0.684329\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−752.336	−	2182.32i	−0.368276	−	1.06827i
$162$		0			0
$163$	−1095.52	−	1897.50i	−0.526430	−	0.911803i	−0.999526	−	0.0307921i	$-0.990197\pi$
$163$	−1095.52	−	1897.50i	−0.526430	−	0.911803i	0.473096	−	0.881011i	$-0.343136\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1567.81			−0.726471			−0.363235	−	0.931697i	$-0.618328\pi$
$167$	−1567.81			−0.726471			−0.363235	+	0.931697i	$0.618328\pi$
$168$		0			0
$169$	737.693			0.335773
$170$		0			0
$171$		0			0
$172$		0			0
$173$	388.338	+	672.621i	0.170664	+	0.295598i	0.938652	−	0.344866i	$-0.112076\pi$
$173$	388.338	+	672.621i	0.170664	+	0.295598i	−0.767989	+	0.640464i	$0.778742\pi$
$174$		0			0
$175$	2278.23	+	1978.96i	0.984103	+	0.854829i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1513.06	−	873.568i	−0.631797	−	0.364768i	0.149650	−	0.988739i	$-0.452185\pi$
$179$	−1513.06	−	873.568i	−0.631797	−	0.364768i	−0.781448	+	0.623971i	$0.785518\pi$
$180$		0			0
$181$			1787.57i			0.734085i	0.930204	+	0.367042i	$0.119630\pi$
$181$			1787.57i			0.734085i	−0.930204	+	0.367042i	$0.880370\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1818.90	+	3150.43i	−0.722855	+	1.25202i
$186$		0			0
$187$	6399.73	−	3694.89i	2.50265	−	1.44490i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	588.184	−	339.588i	0.222825	−	0.128648i	−0.384433	−	0.923153i	$-0.625603\pi$
$191$	588.184	−	339.588i	0.222825	−	0.128648i	0.607258	+	0.794505i	$0.292270\pi$
$192$		0			0
$193$	−1698.73	+	2942.29i	−0.633562	+	1.09736i	0.353255	+	0.935527i	$0.385075\pi$
$193$	−1698.73	+	2942.29i	−0.633562	+	1.09736i	−0.986818	+	0.161835i	$0.948259\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3407.09i		−	1.23221i	−0.787665	−	0.616104i	$-0.788710\pi$
$197$		−	3407.09i		−	1.23221i	0.787665	−	0.616104i	$-0.211290\pi$
$198$		0			0
$199$	181.152	+	104.588i	0.0645302	+	0.0372565i	0.531918	−	0.846796i	$-0.321471\pi$
$199$	181.152	+	104.588i	0.0645302	+	0.0372565i	−0.467388	+	0.884052i	$0.654805\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1210.80	−	234.890i	−0.418627	−	0.0812122i
$204$		0			0
$205$	3805.34	+	6591.04i	1.29647	+	2.24555i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4057.79			1.34298
$210$		0			0
$211$	2275.52			0.742434			0.371217	−	0.928546i	$-0.378941\pi$
$211$	2275.52			0.742434			0.371217	+	0.928546i	$0.378941\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2721.67	+	4714.06i	0.863331	+	1.49533i
$216$		0			0
$217$	−1915.42	+	2205.08i	−0.599202	+	0.689818i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3505.94	+	2024.16i	1.06713	+	0.616107i
$222$		0			0
$223$			3401.49i			1.02144i	0.859747	+	0.510719i	$0.170621\pi$
$223$			3401.49i			1.02144i	−0.859747	+	0.510719i	$0.829379\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1502.78	−	2602.90i	0.439397	−	0.761058i	−0.558246	−	0.829676i	$-0.688525\pi$
$227$	1502.78	−	2602.90i	0.439397	−	0.761058i	0.997643	+	0.0686172i	$0.0218587\pi$
$228$		0			0
$229$	−397.955	+	229.759i	−0.114837	+	0.0663009i	−0.556318	−	0.830969i	$-0.687786\pi$
$229$	−397.955	+	229.759i	−0.114837	+	0.0663009i	0.441482	+	0.897270i	$0.354453\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5044.86	+	2912.65i	−1.41845	+	0.818944i	−0.996163	−	0.0875166i	$-0.972107\pi$
$233$	−5044.86	+	2912.65i	−1.41845	+	0.818944i	−0.422290	+	0.906461i	$0.638774\pi$
$234$		0			0
$235$	−1490.10	+	2580.94i	−0.413633	+	0.716433i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4219.42i		−	1.14197i	−0.820959	−	0.570987i	$-0.806561\pi$
$239$		−	4219.42i		−	1.14197i	0.820959	−	0.570987i	$-0.193439\pi$
$240$		0			0
$241$	2837.81	+	1638.41i	0.758503	+	0.437922i	0.828758	−	0.559607i	$-0.189048\pi$
$241$	2837.81	+	1638.41i	0.758503	+	0.437922i	−0.0702548	+	0.997529i	$0.522381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	814.292	+	5763.07i	0.212340	+	1.50281i
$246$		0			0
$247$	1111.48	+	1925.15i	0.286324	+	0.495928i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3221.43			−0.810100			−0.405050	−	0.914294i	$-0.632746\pi$
$251$	−3221.43			−0.810100			−0.405050	+	0.914294i	$0.632746\pi$
$252$		0			0
$253$	8691.34			2.15976
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−411.892	−	713.418i	−0.0999732	−	0.173159i	0.811700	−	0.584074i	$-0.198542\pi$
$257$	−411.892	−	713.418i	−0.0999732	−	0.173159i	−0.911673	+	0.410916i	$0.865209\pi$
$258$		0			0
$259$	−3753.61	+	1294.02i	−0.900532	+	0.310450i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2955.61	−	1706.42i	−0.692969	−	0.400086i	0.111754	−	0.993736i	$-0.464353\pi$
$263$	−2955.61	−	1706.42i	−0.692969	−	0.400086i	−0.804723	+	0.593650i	$0.797686\pi$
$264$		0			0
$265$		−	11477.5i		−	2.66060i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3209.91	−	5559.73i	0.727553	−	1.26016i	−0.230361	−	0.973105i	$-0.573991\pi$
$269$	3209.91	−	5559.73i	0.727553	−	1.26016i	0.957914	−	0.287054i	$-0.0926758\pi$
$270$		0			0
$271$	7375.18	−	4258.06i	1.65317	−	0.954461i	0.677419	−	0.735597i	$-0.263098\pi$
$271$	7375.18	−	4258.06i	1.65317	−	0.954461i	0.975756	−	0.218863i	$-0.0702349\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9839.92	+	5681.08i	−2.15771	+	1.24575i
$276$		0			0
$277$	3028.50	−	5245.52i	0.656914	−	1.13781i	−0.324497	−	0.945887i	$-0.605195\pi$
$277$	3028.50	−	5245.52i	0.656914	−	1.13781i	0.981410	−	0.191921i	$-0.0614717\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			4217.81i			0.895421i	0.894179	+	0.447710i	$0.147760\pi$
$281$			4217.81i			0.895421i	−0.894179	+	0.447710i	$0.852240\pi$
$282$		0			0
$283$	−708.451	−	409.024i	−0.148809	−	0.0859151i	0.423747	−	0.905781i	$-0.360715\pi$
$283$	−708.451	−	409.024i	−0.148809	−	0.0859151i	−0.572556	+	0.819866i	$0.694048\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1581.94	+	8154.48i	−0.325363	+	1.67716i
$288$		0			0
$289$	−3158.79	−	5471.19i	−0.642945	−	1.11361i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−699.920			−0.139556			−0.0697778	−	0.997563i	$-0.522229\pi$
$293$	−699.920			−0.139556			−0.0697778	+	0.997563i	$0.522229\pi$
$294$		0			0
$295$	−11653.3			−2.29993
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2380.67	+	4123.45i	0.460461	+	0.797542i
$300$		0			0
$301$	−1131.44	+	5832.27i	−0.216662	+	1.11683i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1546.82	+	893.059i	0.290396	+	0.167660i
$306$		0			0
$307$			8523.88i			1.58464i	0.610108	+	0.792319i	$0.291126\pi$
$307$			8523.88i			1.58464i	−0.610108	+	0.792319i	$0.708874\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1170.89	−	2028.04i	0.213489	−	0.369774i	−0.739315	−	0.673360i	$-0.764851\pi$
$311$	1170.89	−	2028.04i	0.213489	−	0.369774i	0.952804	+	0.303586i	$0.0981839\pi$
$312$		0			0
$313$	3462.94	−	1999.33i	0.625357	−	0.361050i	−0.153595	−	0.988134i	$-0.549085\pi$
$313$	3462.94	−	1999.33i	0.625357	−	0.361050i	0.778952	+	0.627084i	$0.215752\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6056.74	+	3496.86i	−1.07312	+	0.619569i	−0.929034	−	0.369996i	$-0.879359\pi$
$317$	−6056.74	+	3496.86i	−1.07312	+	0.619569i	−0.144091	+	0.989564i	$0.546026\pi$
$318$		0			0
$319$	2321.91	−	4021.67i	0.407531	−	0.705864i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	6166.82i		−	1.06232i
$324$		0			0
$325$	−5390.57	−	3112.25i	−0.920047	−	0.531189i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3075.08	+	1060.11i	−0.515303	+	0.177646i
$330$		0			0
$331$	2095.81	+	3630.05i	0.348025	+	0.602797i	0.985899	−	0.167344i	$-0.0535192\pi$
$331$	2095.81	+	3630.05i	0.348025	+	0.602797i	−0.637874	+	0.770141i	$0.720186\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	14479.2			2.36144
$336$		0			0
$337$	2226.74			0.359936			0.179968	−	0.983672i	$-0.442401\pi$
$337$	2226.74			0.359936			0.179968	+	0.983672i	$0.442401\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5498.67	−	9523.98i	−0.873225	−	1.51247i
$342$		0			0
$343$	−3453.88	+	5331.45i	−0.543708	+	0.839274i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1073.31	−	619.673i	−0.166046	−	0.0958668i	0.414674	−	0.909970i	$-0.363896\pi$
$347$	−1073.31	−	619.673i	−0.166046	−	0.0958668i	−0.580720	+	0.814103i	$0.697229\pi$
$348$		0			0
$349$			8806.73i			1.35075i	0.737473	+	0.675377i	$0.236019\pi$
$349$			8806.73i			1.35075i	−0.737473	+	0.675377i	$0.763981\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2424.56	−	4199.45i	0.365570	−	0.633185i	−0.623298	−	0.781985i	$-0.714208\pi$
$353$	2424.56	−	4199.45i	0.365570	−	0.633185i	0.988867	+	0.148799i	$0.0475409\pi$
$354$		0			0
$355$	317.454	−	183.282i	0.0474612	−	0.0274017i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1691.26	−	976.450i	0.248639	−	0.143552i	−0.370502	−	0.928832i	$-0.620814\pi$
$359$	1691.26	−	976.450i	0.248639	−	0.143552i	0.619141	+	0.785280i	$0.287481\pi$
$360$		0			0
$361$	−1736.37	+	3007.49i	−0.253152	+	0.438473i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	5805.69i		−	0.832557i
$366$		0			0
$367$	−1159.84	−	669.633i	−0.164968	−	0.0952440i	0.415243	−	0.909710i	$-0.363696\pi$
$367$	−1159.84	−	669.633i	−0.164968	−	0.0952440i	−0.580211	+	0.814466i	$0.697030\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8214.89	−	9457.21i	1.14958	−	1.32343i
$372$		0			0
$373$	3493.37	+	6050.69i	0.484933	+	0.839928i	0.999850	−	0.0173119i	$-0.00551082\pi$
$373$	3493.37	+	6050.69i	0.484933	+	0.839928i	−0.514918	+	0.857240i	$0.672177\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2544.01			0.347542
$378$		0			0
$379$	5328.03			0.722117			0.361059	−	0.932543i	$-0.382415\pi$
$379$	5328.03			0.722117			0.361059	+	0.932543i	$0.382415\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2600.62	+	4504.40i	0.346959	+	0.600951i	0.985708	−	0.168465i	$-0.0538809\pi$
$383$	2600.62	+	4504.40i	0.346959	+	0.600951i	−0.638749	+	0.769416i	$0.720548\pi$
$384$		0			0
$385$	−21513.3	−	4173.51i	−2.84784	−	0.552472i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2739.35	+	1581.56i	0.357045	+	0.206140i	0.667784	−	0.744355i	$-0.267243\pi$
$389$	2739.35	+	1581.56i	0.357045	+	0.206140i	−0.310739	+	0.950495i	$0.600576\pi$
$390$		0			0
$391$		−	13208.6i		−	1.70841i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2662.87	+	4612.22i	−0.339198	+	0.587509i
$396$		0			0
$397$	6689.56	−	3862.22i	0.845692	−	0.488260i	−0.0135032	−	0.999909i	$-0.504298\pi$
$397$	6689.56	−	3862.22i	0.845692	−	0.488260i	0.859195	+	0.511649i	$0.170965\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8214.37	+	4742.57i	−1.02296	+	0.590605i	−0.914959	−	0.403546i	$-0.867778\pi$
$401$	−8214.37	+	4742.57i	−1.02296	+	0.590605i	−0.107999	+	0.994151i	$0.534444\pi$
$402$		0			0
$403$	3012.32	−	5217.49i	0.372343	−	0.644917i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	14949.2i		−	1.82064i
$408$		0			0
$409$	−13564.1	−	7831.22i	−1.63985	−	0.946770i	−0.980882	−	0.194602i	$-0.937659\pi$
$409$	−13564.1	−	7831.22i	−1.63985	−	0.946770i	−0.658971	−	0.752168i	$-0.729008\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9602.03	−	8340.69i	−1.14403	−	0.993749i
$414$		0			0
$415$	5326.10	+	9225.07i	0.629995	+	1.09118i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13214.3			1.54071			0.770357	−	0.637613i	$-0.220078\pi$
$419$	13214.3			1.54071			0.770357	+	0.637613i	$0.220078\pi$
$420$		0			0
$421$	−3985.81			−0.461417			−0.230709	−	0.973023i	$-0.574104\pi$
$421$	−3985.81			−0.461417			−0.230709	+	0.973023i	$0.574104\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	8633.80	+	14954.2i	0.985414	+	1.70679i
$426$		0			0
$427$	635.350	+	1842.98i	0.0720064	+	0.208871i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	875.377	+	505.399i	0.0978316	+	0.0564831i	0.548118	−	0.836401i	$-0.315345\pi$
$431$	875.377	+	505.399i	0.0978316	+	0.0564831i	−0.450286	+	0.892884i	$0.648678\pi$
$432$		0			0
$433$		−	10704.5i		−	1.18805i	−0.804446	−	0.594026i	$-0.797538\pi$
$433$		−	10704.5i		−	1.18805i	0.804446	−	0.594026i	$-0.202462\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3626.49	−	6281.26i	0.396976	−	0.687583i
$438$		0			0
$439$	−5935.42	+	3426.82i	−0.645289	+	0.372558i	−0.786649	−	0.617400i	$-0.788186\pi$
$439$	−5935.42	+	3426.82i	−0.645289	+	0.372558i	0.141360	+	0.989958i	$0.454853\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13821.3	−	7979.76i	1.48233	−	0.855823i	0.482531	−	0.875879i	$-0.339718\pi$
$443$	13821.3	−	7979.76i	1.48233	−	0.855823i	0.999799	+	0.0200558i	$0.00638437\pi$
$444$		0			0
$445$	3520.70	−	6098.03i	0.375050	−	0.649605i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	2486.47i		−	0.261345i	−0.991426	−	0.130673i	$-0.958286\pi$
$449$		−	2486.47i		−	0.261345i	0.991426	−	0.130673i	$-0.0417137\pi$
$450$		0			0
$451$	−27085.2	−	15637.6i	−2.82792	−	1.63270i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3912.73	−	11349.8i	−0.403147	−	1.16942i
$456$		0			0
$457$	−2649.67	−	4589.36i	−0.271217	−	0.469762i	0.697957	−	0.716140i	$-0.254093\pi$
$457$	−2649.67	−	4589.36i	−0.271217	−	0.469762i	−0.969174	+	0.246378i	$0.920759\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−633.711			−0.0640236			−0.0320118	−	0.999487i	$-0.510191\pi$
$461$	−633.711			−0.0640236			−0.0320118	+	0.999487i	$0.510191\pi$
$462$		0			0
$463$	1208.19			0.121273			0.0606366	−	0.998160i	$-0.480687\pi$
$463$	1208.19			0.121273			0.0606366	+	0.998160i	$0.480687\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2211.57	+	3830.55i	0.219142	+	0.379564i	0.954546	−	0.298064i	$-0.0963410\pi$
$467$	2211.57	+	3830.55i	0.219142	+	0.379564i	−0.735404	+	0.677629i	$0.763008\pi$
$468$		0			0
$469$	11930.5	+	10363.3i	1.17463	+	1.02033i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−19372.0	−	11184.4i	−1.88314	−	1.08723i
$474$		0			0
$475$			9481.80i			0.915905i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1210.61	+	2096.84i	−0.115479	+	0.200015i	−0.917971	−	0.396648i	$-0.870173\pi$
$479$	−1210.61	+	2096.84i	−0.115479	+	0.200015i	0.802492	+	0.596662i	$0.203507\pi$
$480$		0			0
$481$	7092.36	−	4094.78i	0.672315	−	0.388162i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3291.14	+	1900.14i	−0.308130	+	0.177899i
$486$		0			0
$487$	−5304.79	+	9188.16i	−0.493599	+	0.854939i	−0.999973	−	0.00737514i	$-0.997652\pi$
$487$	−5304.79	+	9188.16i	−0.493599	+	0.854939i	0.506373	+	0.862314i	$0.330986\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			16314.2i			1.49949i	0.661728	+	0.749744i	$0.269824\pi$
$491$			16314.2i			1.49949i	−0.661728	+	0.749744i	$0.730176\pi$
$492$		0			0
$493$	−6111.92	−	3528.72i	−0.558351	−	0.322364i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	392.756	+	76.1934i	0.0354478	+	0.00687675i
$498$		0			0
$499$	1876.73	+	3250.60i	0.168365	+	0.291617i	0.937845	−	0.347054i	$-0.112818\pi$
$499$	1876.73	+	3250.60i	0.168365	+	0.291617i	−0.769480	+	0.638671i	$0.779485\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9135.50			0.809805			0.404903	−	0.914360i	$-0.367305\pi$
$503$	9135.50			0.809805			0.404903	+	0.914360i	$0.367305\pi$
$504$		0			0
$505$	−33074.6			−2.91446
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.73217	−	3.00021i	−0.000150839	−	0.000261261i	0.865950	−	0.500131i	$-0.166715\pi$
$509$	−1.73217	−	3.00021i	−0.000150839	−	0.000261261i	−0.866101	+	0.499869i	$0.833381\pi$
$510$		0			0
$511$	4155.34	−	4783.75i	0.359729	−	0.414130i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12826.8	+	7405.57i	1.09751	+	0.633648i
$516$		0			0
$517$		−	12246.9i		−	1.04181i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5027.11	+	8707.20i	−0.422728	+	0.732187i	−0.996205	−	0.0870347i	$-0.972261\pi$
$521$	−5027.11	+	8707.20i	−0.422728	+	0.732187i	0.573477	+	0.819222i	$0.305594\pi$
$522$		0			0
$523$	−1142.12	+	659.405i	−0.0954905	+	0.0551314i	−0.546985	−	0.837143i	$-0.684224\pi$
$523$	−1142.12	+	659.405i	−0.0954905	+	0.0551314i	0.451494	+	0.892274i	$0.350891\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14474.0	+	8356.59i	−1.19639	+	0.690737i
$528$		0			0
$529$	1684.04	−	2916.83i	0.138410	−	0.239733i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	17133.4i		−	1.39237i
$534$		0			0
$535$	−8317.16	−	4801.91i	−0.672116	−	0.388046i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−14739.3	−	18836.7i	−1.17786	−	1.50530i
$540$		0			0
$541$	2144.41	+	3714.23i	0.170417	+	0.295170i	0.938566	−	0.345101i	$-0.112155\pi$
$541$	2144.41	+	3714.23i	0.170417	+	0.295170i	−0.768149	+	0.640271i	$0.778822\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−26523.1			−2.08463
$546$		0			0
$547$	−8456.92			−0.661045			−0.330523	−	0.943798i	$-0.607225\pi$
$547$	−8456.92			−0.661045			−0.330523	+	0.943798i	$0.607225\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1937.65	−	3356.11i	−0.149813	−	0.259483i
$552$		0			0
$553$	−5495.27	+	1894.45i	−0.422573	+	0.145678i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12043.6	+	6953.36i	0.916163	+	0.528947i	0.882409	−	0.470483i	$-0.155920\pi$
$557$	12043.6	+	6953.36i	0.916163	+	0.528947i	0.0337539	+	0.999430i	$0.489254\pi$
$558$		0			0
$559$		−	12254.2i		−	0.927190i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9052.19	−	15678.8i	0.677627	−	1.17369i	−0.298066	−	0.954545i	$-0.596342\pi$
$563$	9052.19	−	15678.8i	0.677627	−	1.17369i	0.975693	−	0.219140i	$-0.0703251\pi$
$564$		0			0
$565$	23648.5	−	13653.5i	1.76089	−	1.01665i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	664.706	−	383.768i	0.0489735	−	0.0282749i	−0.475313	−	0.879817i	$-0.657665\pi$
$569$	664.706	−	383.768i	0.0489735	−	0.0282749i	0.524287	+	0.851542i	$0.324332\pi$
$570$		0			0
$571$	10674.7	−	18489.2i	0.782354	−	1.35508i	−0.148213	−	0.988955i	$-0.547352\pi$
$571$	10674.7	−	18489.2i	0.782354	−	1.35508i	0.930567	−	0.366121i	$-0.119314\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			20309.0i			1.47294i
$576$		0			0
$577$	14315.8	+	8265.21i	1.03288	+	0.596335i	0.917809	−	0.397022i	$-0.129956\pi$
$577$	14315.8	+	8265.21i	1.03288	+	0.596335i	0.115073	+	0.993357i	$0.463290\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2214.15	+	11413.3i	−0.158104	+	0.814982i
$582$		0			0
$583$	23582.9	+	40846.7i	1.67530	+	2.90171i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10940.7			0.769286			0.384643	−	0.923065i	$-0.374325\pi$
$587$	10940.7			0.769286			0.384643	+	0.923065i	$0.374325\pi$
$588$		0			0
$589$	−9177.36			−0.642014
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11096.9	+	19220.4i	0.768459	+	1.33101i	0.938398	+	0.345555i	$0.112309\pi$
$593$	11096.9	+	19220.4i	0.768459	+	1.33101i	−0.169940	+	0.985454i	$0.554357\pi$
$594$		0			0
$595$	−6342.67	+	32694.7i	−0.437015	+	2.25270i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11509.9	−	6645.23i	−0.785110	−	0.453283i	0.0531284	−	0.998588i	$-0.483081\pi$
$599$	−11509.9	−	6645.23i	−0.785110	−	0.453283i	−0.838238	+	0.545304i	$0.816414\pi$
$600$		0			0
$601$		−	9590.78i		−	0.650942i	−0.945552	−	0.325471i	$-0.894477\pi$
$601$		−	9590.78i		−	0.650942i	0.945552	−	0.325471i	$-0.105523\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	29962.8	−	51897.0i	2.01349	−	3.48746i
$606$		0			0
$607$	−3936.82	+	2272.93i	−0.263247	+	0.151985i	−0.625815	−	0.779972i	$-0.715233\pi$
$607$	−3936.82	+	2272.93i	−0.263247	+	0.151985i	0.362568	+	0.931957i	$0.381900\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5810.30	−	3354.58i	0.384713	−	0.222114i
$612$		0			0
$613$	−10676.4	+	18492.1i	−0.703451	+	1.21841i	0.263796	+	0.964578i	$0.415025\pi$
$613$	−10676.4	+	18492.1i	−0.703451	+	1.21841i	−0.967248	+	0.253835i	$0.918308\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			9496.27i			0.619620i	0.950798	+	0.309810i	$0.100265\pi$
$617$			9496.27i			0.619620i	−0.950798	+	0.309810i	$0.899735\pi$
$618$		0			0
$619$	15773.7	+	9106.96i	1.02423	+	0.591340i	0.915327	−	0.402712i	$-0.131932\pi$
$619$	15773.7	+	9106.96i	1.02423	+	0.591340i	0.108905	+	0.994052i	$0.465266\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7265.56	−	2504.74i	0.467237	−	0.161076i
$624$		0			0
$625$	4721.41	+	8177.71i	0.302170	+	0.523374i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−22718.9			−1.44016
$630$		0			0
$631$	−6711.25			−0.423409			−0.211704	−	0.977334i	$-0.567901\pi$
$631$	−6711.25			−0.423409			−0.211704	+	0.977334i	$0.567901\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6288.60	+	10892.2i	0.393001	+	0.680697i
$636$		0			0
$637$	4899.45	−	12152.4i	0.304746	−	0.755882i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25619.6	+	14791.5i	1.57865	+	0.911434i	0.995048	+	0.0993938i	$0.0316904\pi$
$641$	25619.6	+	14791.5i	1.57865	+	0.911434i	0.583602	+	0.812040i	$0.301643\pi$
$642$		0			0
$643$			3793.02i			0.232631i	0.993212	+	0.116316i	$0.0371084\pi$
$643$			3793.02i			0.232631i	−0.993212	+	0.116316i	$0.962892\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5797.58	−	10041.7i	0.352282	−	0.610170i	−0.634367	−	0.773032i	$-0.718739\pi$
$647$	5797.58	−	10041.7i	0.352282	−	0.610170i	0.986649	+	0.162862i	$0.0520725\pi$
$648$		0			0
$649$	41472.2	−	23944.0i	2.50836	−	1.44820i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18076.2	−	10436.3i	1.08327	−	0.625427i	0.151494	−	0.988458i	$-0.451591\pi$
$653$	18076.2	−	10436.3i	1.08327	−	0.625427i	0.931777	+	0.363031i	$0.118258\pi$
$654$		0			0
$655$	2245.74	−	3889.74i	0.133967	−	0.232038i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			26007.1i			1.53732i	0.639659	+	0.768658i	$0.279075\pi$
$659$			26007.1i			1.53732i	−0.639659	+	0.768658i	$0.720925\pi$
$660$		0			0
$661$	14619.8	+	8440.73i	0.860277	+	0.496681i	0.864105	−	0.503312i	$-0.167885\pi$
$661$	14619.8	+	8440.73i	0.860277	+	0.496681i	−0.00382819	+	0.999993i	$0.501219\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−11992.7	+	13806.3i	−0.699334	+	0.805092i
$666$		0			0
$667$	−4150.24	−	7188.42i	−0.240926	−	0.417296i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7339.87			−0.422284
$672$		0			0
$673$	2688.00			0.153960			0.0769799	−	0.997033i	$-0.475472\pi$
$673$	2688.00			0.153960			0.0769799	+	0.997033i	$0.475472\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10001.4	+	17323.0i	0.567779	+	0.983422i	0.996785	+	0.0801203i	$0.0255305\pi$
$677$	10001.4	+	17323.0i	0.567779	+	0.983422i	−0.429006	+	0.903301i	$0.641136\pi$
$678$		0			0
$679$	−4071.82	−	789.919i	−0.230135	−	0.0446455i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9088.08	−	5247.00i	−0.509144	−	0.293955i	0.223337	−	0.974741i	$-0.428305\pi$
$683$	−9088.08	−	5247.00i	−0.509144	−	0.293955i	−0.732482	+	0.680787i	$0.761638\pi$
$684$		0			0
$685$		−	20155.8i		−	1.12425i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12919.3	+	22376.9i	−0.714350	+	1.23729i
$690$		0			0
$691$	2514.74	−	1451.88i	0.138444	−	0.0799310i	−0.429178	−	0.903220i	$-0.641197\pi$
$691$	2514.74	−	1451.88i	0.138444	−	0.0799310i	0.567622	+	0.823289i	$0.307863\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−15876.8	+	9166.47i	−0.866534	+	0.500294i
$696$		0			0
$697$	−23765.3	+	41162.6i	−1.29150	+	2.23694i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			5829.71i			0.314102i	0.987591	+	0.157051i	$0.0501986\pi$
$701$			5829.71i			0.314102i	−0.987591	+	0.157051i	$0.949801\pi$
$702$		0			0
$703$	−10803.8	−	6237.59i	−0.579621	−	0.334644i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−27252.7	−	23672.7i	−1.44971	−	1.25927i
$708$		0			0
$709$	7940.28	+	13753.0i	0.420597	+	0.728495i	0.995998	−	0.0893760i	$-0.0284873\pi$
$709$	7940.28	+	13753.0i	0.420597	+	0.728495i	−0.575401	+	0.817872i	$0.695154\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−19656.9			−1.03248
$714$		0			0
$715$	45201.7			2.36427
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2601.36	−	4505.68i	−0.134929	−	0.233705i	0.790641	−	0.612280i	$-0.209747\pi$
$719$	−2601.36	−	4505.68i	−0.134929	−	0.233705i	−0.925570	+	0.378575i	$0.876414\pi$
$720$		0			0
$721$	5268.56	+	15282.6i	0.272138	+	0.789397i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9397.40	+	5425.59i	0.481394	+	0.277933i
$726$		0			0
$727$		−	37948.0i		−	1.93592i	−0.251107	−	0.967959i	$-0.580794\pi$
$727$		−	37948.0i		−	1.93592i	0.251107	−	0.967959i	$-0.419206\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−16997.5	+	29440.5i	−0.860019	+	1.48960i
$732$		0			0
$733$	−15869.6	+	9162.32i	−0.799669	+	0.461689i	−0.843355	−	0.537356i	$-0.819423\pi$
$733$	−15869.6	+	9162.32i	−0.799669	+	0.461689i	0.0436864	+	0.999045i	$0.486090\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−51529.2	+	29750.4i	−2.57545	+	1.48693i
$738$		0			0
$739$	9963.10	−	17256.6i	0.495939	−	0.858991i	−0.504050	−	0.863674i	$-0.668157\pi$
$739$	9963.10	−	17256.6i	0.495939	−	0.858991i	0.999989	+	0.00468324i	$0.00149073\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	7621.89i		−	0.376339i	−0.982137	−	0.188170i	$-0.939745\pi$
$743$		−	7621.89i		−	0.376339i	0.982137	−	0.188170i	$-0.0602555\pi$
$744$		0			0
$745$	−135.376	−	78.1595i	−0.00665745	−	0.00384368i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3416.23	−	9909.55i	−0.166657	−	0.483428i
$750$		0			0
$751$	17180.8	+	29758.1i	0.834803	+	1.44592i	0.894191	+	0.447687i	$0.147752\pi$
$751$	17180.8	+	29758.1i	0.834803	+	1.44592i	−0.0593874	+	0.998235i	$0.518915\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−33210.4			−1.60086
$756$		0			0
$757$	13327.2			0.639874			0.319937	−	0.947439i	$-0.396338\pi$
$757$	13327.2			0.639874			0.319937	+	0.947439i	$0.396338\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11743.9	−	20341.0i	−0.559414	−	0.968934i	−0.997545	−	0.0700230i	$-0.977693\pi$
$761$	−11743.9	−	20341.0i	−0.559414	−	0.968934i	0.438131	−	0.898911i	$-0.355641\pi$
$762$		0			0
$763$	−21854.4	−	18983.5i	−1.03693	−	0.900720i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22719.6	+	13117.2i	1.06957	+	0.617514i
$768$		0			0
$769$			27145.7i			1.27295i	0.771297	+	0.636476i	$0.219609\pi$
$769$			27145.7i			1.27295i	−0.771297	+	0.636476i	$0.780391\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10441.9	−	18085.9i	0.485859	−	0.841532i	−0.514009	−	0.857785i	$-0.671840\pi$
$773$	10441.9	−	18085.9i	0.485859	−	0.841532i	0.999868	+	0.0162524i	$0.00517352\pi$
$774$		0			0
$775$	22254.6	−	12848.7i	1.03149	−	0.595533i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−22602.8	+	13049.7i	−1.03957	+	0.600199i
$780$		0			0
$781$	−753.180	+	1304.55i	−0.0345082	+	0.0597699i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			3702.66i			0.168348i
$786$		0			0
$787$	−690.527	−	398.676i	−0.0312766	−	0.0180575i	0.484280	−	0.874913i	$-0.339082\pi$
$787$	−690.527	−	398.676i	−0.0312766	−	0.0180575i	−0.515557	+	0.856855i	$0.672415\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	29258.1	+	5675.97i	1.31517	+	0.255138i
$792$		0			0
$793$	−2010.49	−	3482.27i	−0.0900309	−	0.155938i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14974.0			0.665504			0.332752	−	0.943014i	$-0.392023\pi$
$797$	14974.0			0.665504			0.332752	+	0.943014i	$0.392023\pi$
$798$		0			0
$799$	−18612.1			−0.824092
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11928.9	+	20661.5i	0.524238	+	0.908006i
$804$		0			0
$805$	−25687.0	+	29571.6i	−1.12466	+	1.29474i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	35121.5	+	20277.4i	1.52634	+	0.881231i	0.999511	+	0.0312567i	$0.00995093\pi$
$809$	35121.5	+	20277.4i	1.52634	+	0.881231i	0.526825	+	0.849974i	$0.323382\pi$
$810$		0			0
$811$			17327.0i			0.750227i	0.926979	+	0.375113i	$0.122396\pi$
$811$			17327.0i			0.750227i	−0.926979	+	0.375113i	$0.877604\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18589.8	+	32198.4i	−0.798982	+	1.38388i
$816$		0			0
$817$	−16166.0	+	9333.46i	−0.692262	+	0.399678i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	34173.6	−	19730.1i	1.45270	−	0.838717i	0.454066	−	0.890968i	$-0.349973\pi$
$821$	34173.6	−	19730.1i	1.45270	−	0.838717i	0.998634	+	0.0522514i	$0.0166397\pi$
$822$		0			0
$823$	14242.4	−	24668.6i	0.603232	−	1.04483i	−0.389096	−	0.921197i	$-0.627213\pi$
$823$	14242.4	−	24668.6i	0.603232	−	1.04483i	0.992328	−	0.123631i	$-0.0394540\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	23460.2i		−	0.986446i	−0.869903	−	0.493223i	$-0.835819\pi$
$827$		−	23460.2i		−	0.986446i	0.869903	−	0.493223i	$-0.164181\pi$
$828$		0			0
$829$	7503.24	+	4332.00i	0.314353	+	0.181492i	0.648873	−	0.760897i	$-0.275241\pi$
$829$	7503.24	+	4332.00i	0.314353	+	0.181492i	−0.334520	+	0.942389i	$0.608574\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28627.0	+	22400.0i	−1.19072	+	0.931709i
$834$		0			0
$835$	13301.9	+	23039.6i	0.551296	+	0.954873i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	45107.5			1.85612			0.928060	−	0.372430i	$-0.121475\pi$
$839$	45107.5			1.85612			0.928060	+	0.372430i	$0.121475\pi$
$840$		0			0
$841$	19954.0			0.818156
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6258.90	−	10840.7i	−0.254808	−	0.441340i
$846$		0			0
$847$	61833.2	−	21316.4i	2.50840	−	0.864748i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−23140.6	−	13360.2i	−0.932137	−	0.538170i
$852$		0			0
$853$			8466.89i			0.339860i	0.985456	+	0.169930i	$0.0543542\pi$
$853$			8466.89i			0.339860i	−0.985456	+	0.169930i	$0.945646\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18419.2	−	31902.9i	0.734173	−	1.27163i	−0.220912	−	0.975294i	$-0.570903\pi$
$857$	18419.2	−	31902.9i	0.734173	−	1.27163i	0.955085	−	0.296332i	$-0.0957635\pi$
$858$		0			0
$859$	−2523.02	+	1456.67i	−0.100215	+	0.0578589i	−0.549270	−	0.835645i	$-0.685094\pi$
$859$	−2523.02	+	1456.67i	−0.100215	+	0.0578589i	0.449055	+	0.893504i	$0.351761\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6013.10	+	3471.67i	−0.237182	+	0.136937i	−0.613881	−	0.789399i	$-0.710393\pi$
$863$	−6013.10	+	3471.67i	−0.237182	+	0.136937i	0.376699	+	0.926336i	$0.377059\pi$
$864$		0			0
$865$	6589.64	−	11413.6i	0.259023	−	0.448640i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	21885.5i		−	0.854334i
$870$		0			0
$871$	−28229.1	−	16298.1i	−1.09817	−	0.634029i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2270.82	−	11705.4i	0.0877344	−	0.452247i
$876$		0			0
$877$	6098.71	+	10563.3i	0.234822	+	0.406723i	0.959221	−	0.282658i	$-0.0912160\pi$
$877$	6098.71	+	10563.3i	0.234822	+	0.406723i	−0.724399	+	0.689381i	$0.757883\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	49702.0			1.90068			0.950341	−	0.311210i	$-0.100734\pi$
$881$	49702.0			1.90068			0.950341	+	0.311210i	$0.100734\pi$
$882$		0			0
$883$	−45661.4			−1.74024			−0.870118	−	0.492843i	$-0.835958\pi$
$883$	−45661.4			−1.74024			−0.870118	+	0.492843i	$0.835958\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13631.5	+	23610.4i	0.516009	+	0.893754i	0.999827	+	0.0185857i	$0.00591634\pi$
$887$	13631.5	+	23610.4i	0.516009	+	0.893754i	−0.483818	+	0.875169i	$0.660750\pi$
$888$		0			0
$889$	−2614.27	+	13475.9i	−0.0986276	+	0.508398i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8850.85	−	5110.04i	−0.331671	−	0.191490i
$894$		0			0
$895$			29646.9i			1.10725i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5251.39	+	9095.67i	−0.194820	+	0.337439i
$900$		0			0
$901$	62076.6	−	35840.0i	2.29531	−	1.32520i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	26269.2	−	15166.5i	0.964881	−	0.557074i
$906$		0			0
$907$	24726.1	−	42826.9i	0.905200	−	1.56785i	0.0845520	−	0.996419i	$-0.473054\pi$
$907$	24726.1	−	42826.9i	0.905200	−	1.56785i	0.820648	−	0.571434i	$-0.193613\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	18037.2i		−	0.655982i	−0.944681	−	0.327991i	$-0.893628\pi$
$911$		−	18037.2i		−	0.655982i	0.944681	−	0.327991i	$-0.106372\pi$
$912$		0			0
$913$	−37909.5	−	21887.0i	−1.37417	−	0.793380i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4634.47	−	1597.69i	0.166896	−	0.0575359i
$918$		0			0
$919$	−10193.8	−	17656.2i	−0.365900	−	0.633757i	0.623020	−	0.782206i	$-0.285905\pi$
$919$	−10193.8	−	17656.2i	−0.365900	−	0.633757i	−0.988920	+	0.148448i	$0.952572\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−825.224			−0.0294286
$924$		0			0
$925$	34931.5			1.24167
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2997.17	+	5191.26i	0.105849	+	0.183337i	0.914085	−	0.405523i	$-0.132911\pi$
$929$	2997.17	+	5191.26i	0.105849	+	0.183337i	−0.808236	+	0.588859i	$0.799577\pi$
$930$		0			0
$931$	−19763.4	+	2792.47i	−0.695724	+	0.0983023i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−108596.	−	62697.9i	−3.79836	−	2.19299i
$936$		0			0
$937$		−	6529.41i		−	0.227648i	−0.993501	−	0.113824i	$-0.963690\pi$
$937$		−	6529.41i		−	0.227648i	0.993501	−	0.113824i	$-0.0363101\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14309.7	+	24785.2i	−0.495732	+	0.858634i	−0.999988	−	0.00492088i	$-0.998434\pi$
$941$	−14309.7	+	24785.2i	−0.495732	+	0.858634i	0.504256	+	0.863554i	$0.331767\pi$
$942$		0			0
$943$	−48412.6	+	27951.0i	−1.67183	+	0.965229i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5709.78	−	3296.54i	0.195927	−	0.113119i	−0.398827	−	0.917026i	$-0.630583\pi$
$947$	5709.78	−	3296.54i	0.195927	−	0.113119i	0.594754	+	0.803908i	$0.297249\pi$
$948$		0			0
$949$	−6534.99	+	11318.9i	−0.223535	+	0.387174i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			168.973i			0.00574351i	0.999996	+	0.00287175i	$0.000914109\pi$
$953$			168.973i			0.00574351i	−0.999996	+	0.00287175i	$0.999086\pi$
$954$		0			0
$955$	−9980.80	−	5762.42i	−0.338190	−	0.195254i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14426.3	−	16607.9i	0.485765	−	0.559226i
$960$		0			0
$961$	−2459.36	−	4259.74i	−0.0825539	−	0.142988i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	57651.1			1.92316
$966$		0			0
$967$	−37762.6			−1.25580			−0.627902	−	0.778292i	$-0.716086\pi$
$967$	−37762.6			−1.25580			−0.627902	+	0.778292i	$0.716086\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12773.5	−	22124.4i	−0.422165	−	0.731212i	0.573986	−	0.818865i	$-0.305397\pi$
$971$	−12773.5	−	22124.4i	−0.422165	−	0.731212i	−0.996151	+	0.0876535i	$0.972063\pi$
$972$		0			0
$973$	−19642.9	−	3810.65i	−0.647196	−	0.125554i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5397.11	+	3116.02i	0.176734	+	0.102037i	0.585757	−	0.810487i	$-0.300797\pi$
$977$	5397.11	+	3116.02i	0.176734	+	0.102037i	−0.409023	+	0.912524i	$0.634131\pi$
$978$		0			0
$979$			28935.9i			0.944633i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−22199.8	+	38451.1i	−0.720308	+	1.24761i	0.240568	+	0.970632i	$0.422666\pi$
$983$	−22199.8	+	38451.1i	−0.720308	+	1.24761i	−0.960876	+	0.276978i	$0.910667\pi$
$984$		0			0
$985$	−50068.6	+	28907.1i	−1.61961	+	0.935084i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−34625.8	+	19991.2i	−1.11328	+	0.642755i
$990$		0			0
$991$	−23356.8	+	40455.2i	−0.748693	+	1.29677i	0.199756	+	0.979846i	$0.435985\pi$
$991$	−23356.8	+	40455.2i	−0.748693	+	1.29677i	−0.948449	+	0.316929i	$0.897348\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	3549.47i		−	0.113091i
$996$		0			0
$997$	26568.0	+	15339.0i	0.843949	+	0.487254i	0.858604	−	0.512639i	$-0.171332\pi$
$997$	26568.0	+	15339.0i	0.843949	+	0.487254i	−0.0146559	+	0.999893i	$0.504665\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.4.bl.a.17.3	✓	48
3.2	odd	2	inner	504.4.bl.a.17.22	yes	48
4.3	odd	2		1008.4.bt.d.17.3		48
7.5	odd	6	inner	504.4.bl.a.89.22	yes	48
12.11	even	2		1008.4.bt.d.17.22		48
21.5	even	6	inner	504.4.bl.a.89.3	yes	48
28.19	even	6		1008.4.bt.d.593.22		48
84.47	odd	6		1008.4.bt.d.593.3		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.3	✓	48	1.1	even	1	trivial
504.4.bl.a.17.22	yes	48	3.2	odd	2	inner
504.4.bl.a.89.3	yes	48	21.5	even	6	inner
504.4.bl.a.89.22	yes	48	7.5	odd	6	inner
1008.4.bt.d.17.3		48	4.3	odd	2
1008.4.bt.d.17.22		48	12.11	even	2
1008.4.bt.d.593.3		48	84.47	odd	6
1008.4.bt.d.593.22		48	28.19	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	504.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.48442 - 14.6954i) q^{5} +(3.52711 - 18.1813i) q^{7} +O(q^{10})\)	\(q+(-8.48442 - 14.6954i) q^{5} +(3.52711 - 18.1813i) q^{7} +(60.3894 + 34.8658i) q^{11} +38.2009i q^{13} +(52.9872 - 91.7766i) q^{17} +(50.3954 - 29.0958i) q^{19} +(107.941 - 62.3199i) q^{23} +(-81.4706 + 141.111i) q^{25} -66.5957i q^{29} +(-136.580 - 78.8547i) q^{31} +(-297.108 + 102.425i) q^{35} +(-107.191 - 185.660i) q^{37} -448.509 q^{41} -320.784 q^{43} +(-87.8142 - 152.099i) q^{47} +(-318.119 - 128.255i) q^{49} +(585.770 + 338.194i) q^{53} -1183.26i q^{55} +(343.373 - 594.740i) q^{59} +(-91.1567 + 52.6294i) q^{61} +(561.379 - 324.112i) q^{65} +(-426.641 + 738.964i) q^{67} +21.6022i q^{71} +(296.300 + 171.069i) q^{73} +(846.906 - 974.982i) q^{77} +(-156.927 - 271.805i) q^{79} -627.751 q^{83} -1798.26 q^{85} +(207.480 + 359.367i) q^{89} +(694.541 + 134.739i) q^{91} +(-855.150 - 493.721i) q^{95} -223.956i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 24 q^{7}+O(q^{10}) \) 48 * q - 24 * q^7	\( 48 q - 24 q^{7} + 540 q^{19} - 924 q^{25} + 648 q^{31} - 132 q^{37} - 792 q^{43} + 672 q^{49} - 12 q^{67} + 2412 q^{73} + 1680 q^{79} + 480 q^{85} + 1404 q^{91}+O(q^{100}) \) 48 * q - 24 * q^7 + 540 * q^19 - 924 * q^25 + 648 * q^31 - 132 * q^37 - 792 * q^43 + 672 * q^49 - 12 * q^67 + 2412 * q^73 + 1680 * q^79 + 480 * q^85 + 1404 * q^91

Introduction

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