LMFDB - Embedded newform 504.4.bl.a.17.8

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			17.8
Character	$\chi$	$=$	504.17
Dual form			504.4.bl.a.89.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.34783 - 5.79860i) q^{5} +(-12.7404 + 13.4418i) q^{7} +O(q^{10})$	$q+(-3.34783 - 5.79860i) q^{5} +(-12.7404 + 13.4418i) q^{7} +(28.2958 + 16.3366i) q^{11} -67.9019i q^{13} +(-15.3179 + 26.5313i) q^{17} +(-21.8820 + 12.6336i) q^{19} +(-68.6216 + 39.6187i) q^{23} +(40.0841 - 69.4277i) q^{25} +109.668i q^{29} +(238.527 + 137.714i) q^{31} +(120.596 + 28.8753i) q^{35} +(160.221 + 277.511i) q^{37} -184.846 q^{41} +364.766 q^{43} +(25.7730 + 44.6402i) q^{47} +(-18.3666 - 342.508i) q^{49} +(532.671 + 307.538i) q^{53} -218.769i q^{55} +(207.843 - 359.995i) q^{59} +(411.761 - 237.730i) q^{61} +(-393.736 + 227.324i) q^{65} +(142.188 - 246.277i) q^{67} +965.404i q^{71} +(225.387 + 130.127i) q^{73} +(-580.093 + 172.214i) q^{77} +(219.163 + 379.602i) q^{79} +76.4726 q^{83} +205.126 q^{85} +(356.559 + 617.579i) q^{89} +(912.728 + 865.095i) q^{91} +(146.514 + 84.5900i) q^{95} -410.607i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.34783	−	5.79860i	−0.299439	−	0.518643i	0.676569	−	0.736379i	$-0.263466\pi$
$5$	−3.34783	−	5.79860i	−0.299439	−	0.518643i	−0.976008	+	0.217736i	$0.930133\pi$
$6$		0			0
$7$	−12.7404	+	13.4418i	−0.687915	+	0.725792i
$7$	−12.7404	+	13.4418i	−0.687915	+	0.725792i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	28.2958	+	16.3366i	0.775593	+	0.447789i	0.834866	−	0.550453i	$-0.185545\pi$
$11$	28.2958	+	16.3366i	0.775593	+	0.447789i	−0.0592734	+	0.998242i	$0.518878\pi$
$12$		0			0
$13$		−	67.9019i		−	1.44866i	−0.689452	−	0.724331i	$-0.742149\pi$
$13$		−	67.9019i		−	1.44866i	0.689452	−	0.724331i	$-0.257851\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.3179	+	26.5313i	−0.218537	+	0.378517i	−0.954361	−	0.298656i	$-0.903462\pi$
$17$	−15.3179	+	26.5313i	−0.218537	+	0.378517i	0.735824	+	0.677173i	$0.236795\pi$
$18$		0			0
$19$	−21.8820	+	12.6336i	−0.264215	+	0.152544i	−0.626256	−	0.779618i	$-0.715413\pi$
$19$	−21.8820	+	12.6336i	−0.264215	+	0.152544i	0.362041	+	0.932162i	$0.382080\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−68.6216	+	39.6187i	−0.622113	+	0.359177i	−0.777691	−	0.628647i	$-0.783609\pi$
$23$	−68.6216	+	39.6187i	−0.622113	+	0.359177i	0.155578	+	0.987824i	$0.450276\pi$
$24$		0			0
$25$	40.0841	−	69.4277i	0.320673	−	0.555422i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			109.668i			0.702236i	0.936331	+	0.351118i	$0.114198\pi$
$29$			109.668i			0.702236i	−0.936331	+	0.351118i	$0.885802\pi$
$30$		0			0
$31$	238.527	+	137.714i	1.38196	+	0.797875i	0.992391	−	0.123123i	$-0.0392910\pi$
$31$	238.527	+	137.714i	1.38196	+	0.797875i	0.389568	+	0.920998i	$0.372624\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	120.596	+	28.8753i	0.582415	+	0.139452i
$36$		0			0
$37$	160.221	+	277.511i	0.711898	+	1.23304i	0.964144	+	0.265380i	$0.0854975\pi$
$37$	160.221	+	277.511i	0.711898	+	1.23304i	−0.252246	+	0.967663i	$0.581169\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−184.846			−0.704100			−0.352050	−	0.935981i	$-0.614515\pi$
$41$	−184.846			−0.704100			−0.352050	+	0.935981i	$0.614515\pi$
$42$		0			0
$43$	364.766			1.29363			0.646817	−	0.762645i	$-0.276100\pi$
$43$	364.766			1.29363			0.646817	+	0.762645i	$0.276100\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	25.7730	+	44.6402i	0.0799868	+	0.138541i	0.903244	−	0.429127i	$-0.141179\pi$
$47$	25.7730	+	44.6402i	0.0799868	+	0.138541i	−0.823257	+	0.567669i	$0.807846\pi$
$48$		0			0
$49$	−18.3666	−	342.508i	−0.0535471	−	0.998565i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	532.671	+	307.538i	1.38053	+	0.797048i	0.992222	−	0.124483i	$-0.0397271\pi$
$53$	532.671	+	307.538i	1.38053	+	0.797048i	0.388306	+	0.921531i	$0.373060\pi$
$54$		0			0
$55$		−	218.769i		−	0.536341i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	207.843	−	359.995i	0.458625	−	0.794362i	−0.540263	−	0.841496i	$-0.681675\pi$
$59$	207.843	−	359.995i	0.458625	−	0.794362i	0.998889	+	0.0471340i	$0.0150088\pi$
$60$		0			0
$61$	411.761	−	237.730i	0.864271	−	0.498987i	−0.00116918	−	0.999999i	$-0.500372\pi$
$61$	411.761	−	237.730i	0.864271	−	0.498987i	0.865440	+	0.501012i	$0.167039\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−393.736	+	227.324i	−0.751338	+	0.433785i
$66$		0			0
$67$	142.188	−	246.277i	0.259269	−	0.449068i	−0.706777	−	0.707436i	$-0.749852\pi$
$67$	142.188	−	246.277i	0.259269	−	0.449068i	0.966046	+	0.258369i	$0.0831849\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			965.404i			1.61370i	0.590759	+	0.806848i	$0.298828\pi$
$71$			965.404i			1.61370i	−0.590759	+	0.806848i	$0.701172\pi$
$72$		0			0
$73$	225.387	+	130.127i	0.361364	+	0.208633i	0.669679	−	0.742651i	$-0.266432\pi$
$73$	225.387	+	130.127i	0.361364	+	0.208633i	−0.308315	+	0.951284i	$0.599765\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−580.093	+	172.214i	−0.858543	+	0.254878i
$78$		0			0
$79$	219.163	+	379.602i	0.312124	+	0.540615i	0.978822	−	0.204713i	$-0.0656262\pi$
$79$	219.163	+	379.602i	0.312124	+	0.540615i	−0.666698	+	0.745328i	$0.732293\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	76.4726			0.101132			0.0505660	−	0.998721i	$-0.483897\pi$
$83$	76.4726			0.101132			0.0505660	+	0.998721i	$0.483897\pi$
$84$		0			0
$85$	205.126			0.261753
$86$		0			0
$87$		0			0
$88$		0			0
$89$	356.559	+	617.579i	0.424665	+	0.735542i	0.996389	−	0.0849045i	$-0.0270585\pi$
$89$	356.559	+	617.579i	0.424665	+	0.735542i	−0.571724	+	0.820446i	$0.693725\pi$
$90$		0			0
$91$	912.728	+	865.095i	1.05143	+	0.996556i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	146.514	+	84.5900i	0.158232	+	0.0913553i
$96$		0			0
$97$		−	410.607i		−	0.429803i	−0.976636	−	0.214901i	$-0.931057\pi$
$97$		−	410.607i		−	0.429803i	0.976636	−	0.214901i	$-0.0689430\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−953.656	+	1651.78i	−0.939527	+	1.62731i	−0.173173	+	0.984891i	$0.555402\pi$
$101$	−953.656	+	1651.78i	−0.939527	+	1.62731i	−0.766355	+	0.642418i	$0.777931\pi$
$102$		0			0
$103$	1177.37	−	679.756i	1.12631	−	0.650276i	0.183306	−	0.983056i	$-0.441320\pi$
$103$	1177.37	−	679.756i	1.12631	−	0.650276i	0.943004	+	0.332780i	$0.107987\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1555.77	−	898.226i	1.40563	−	0.811540i	0.410666	−	0.911786i	$-0.365296\pi$
$107$	1555.77	−	898.226i	1.40563	−	0.811540i	0.994963	+	0.100246i	$0.0319629\pi$
$108$		0			0
$109$	42.0727	−	72.8720i	0.0369709	−	0.0640355i	−0.846948	−	0.531676i	$-0.821562\pi$
$109$	42.0727	−	72.8720i	0.0369709	−	0.0640355i	0.883919	+	0.467640i	$0.154896\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			117.441i			0.0977688i	0.998804	+	0.0488844i	$0.0155666\pi$
$113$			117.441i			0.0977688i	−0.998804	+	0.0488844i	$0.984433\pi$
$114$		0			0
$115$	459.466	+	265.273i	0.372569	+	0.215103i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−161.475	−	543.918i	−0.124390	−	0.418999i
$120$		0			0
$121$	−131.730	−	228.163i	−0.0989707	−	0.171422i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1373.74			−0.982965
$126$		0			0
$127$	−510.925			−0.356986			−0.178493	−	0.983941i	$-0.557122\pi$
$127$	−510.925			−0.356986			−0.178493	+	0.983941i	$0.557122\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	991.274	+	1716.94i	0.661130	+	1.14511i	0.980319	+	0.197419i	$0.0632560\pi$
$131$	991.274	+	1716.94i	0.661130	+	1.14511i	−0.319189	+	0.947691i	$0.603411\pi$
$132$		0			0
$133$	108.966	−	455.091i	0.0710416	−	0.296702i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−857.943	−	495.334i	−0.535029	−	0.308899i	0.208033	−	0.978122i	$-0.433294\pi$
$137$	−857.943	−	495.334i	−0.535029	−	0.308899i	−0.743062	+	0.669223i	$0.766627\pi$
$138$		0			0
$139$			369.921i			0.225729i	0.993610	+	0.112864i	$0.0360026\pi$
$139$			369.921i			0.225729i	−0.993610	+	0.112864i	$0.963997\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1109.29	−	1921.34i	0.648694	−	1.12357i
$144$		0			0
$145$	635.922	−	367.149i	0.364210	−	0.210277i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1054.73	+	608.947i	−0.579910	+	0.334811i	−0.761098	−	0.648637i	$-0.775339\pi$
$149$	−1054.73	+	608.947i	−0.579910	+	0.334811i	0.181188	+	0.983449i	$0.442006\pi$
$150$		0			0
$151$	−764.206	+	1323.64i	−0.411856	+	0.713355i	−0.995093	−	0.0989470i	$-0.968453\pi$
$151$	−764.206	+	1323.64i	−0.411856	+	0.713355i	0.583237	+	0.812302i	$0.301786\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	1844.17i		−	0.955658i
$156$		0			0
$157$	−2610.74	−	1507.31i	−1.32713	−	0.766220i	−0.342277	−	0.939599i	$-0.611198\pi$
$157$	−2610.74	−	1507.31i	−1.32713	−	0.766220i	−0.984855	+	0.173379i	$0.944531\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	341.715	−	1427.16i	0.167273	−	0.698607i
$162$		0			0
$163$	822.212	+	1424.11i	0.395096	+	0.684326i	0.993113	−	0.117157i	$-0.0373780\pi$
$163$	822.212	+	1424.11i	0.395096	+	0.684326i	−0.598018	+	0.801483i	$0.704045\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2783.64			1.28985			0.644923	−	0.764248i	$-0.276889\pi$
$167$	2783.64			1.28985			0.644923	+	0.764248i	$0.276889\pi$
$168$		0			0
$169$	−2413.67			−1.09862
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−64.1081	−	111.038i	−0.0281737	−	0.0487983i	0.851595	−	0.524201i	$-0.175636\pi$
$173$	−64.1081	−	111.038i	−0.0281737	−	0.0487983i	−0.879768	+	0.475402i	$0.842302\pi$
$174$		0			0
$175$	422.551	+	1423.34i	0.182525	+	0.614825i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1676.79	−	968.093i	−0.700161	−	0.404238i	0.107246	−	0.994233i	$-0.465797\pi$
$179$	−1676.79	−	968.093i	−0.700161	−	0.404238i	−0.807408	+	0.589994i	$0.799130\pi$
$180$		0			0
$181$		−	1596.97i		−	0.655810i	−0.944711	−	0.327905i	$-0.893657\pi$
$181$		−	1596.97i		−	0.655810i	0.944711	−	0.327905i	$-0.106343\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1072.79	−	1858.12i	0.426340	−	0.738442i
$186$		0			0
$187$	−866.863	+	500.484i	−0.338991	+	0.195717i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−715.285	+	412.970i	−0.270975	+	0.156448i	−0.629331	−	0.777138i	$-0.716671\pi$
$191$	−715.285	+	412.970i	−0.270975	+	0.156448i	0.358356	+	0.933585i	$0.383338\pi$
$192$		0			0
$193$	927.072	−	1605.74i	0.345762	−	0.598878i	−0.639730	−	0.768600i	$-0.720954\pi$
$193$	927.072	−	1605.74i	0.345762	−	0.598878i	0.985492	+	0.169722i	$0.0542870\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2938.42i			1.06271i	0.847150	+	0.531355i	$0.178317\pi$
$197$			2938.42i			1.06271i	−0.847150	+	0.531355i	$0.821683\pi$
$198$		0			0
$199$	−2850.16	−	1645.54i	−1.01529	−	0.586178i	−0.102554	−	0.994727i	$-0.532701\pi$
$199$	−2850.16	−	1645.54i	−1.01529	−	0.586178i	−0.912736	+	0.408550i	$0.866035\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1474.14	−	1397.21i	−0.509677	−	0.483078i
$204$		0			0
$205$	618.833	+	1071.85i	0.210835	+	0.365177i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−825.560			−0.273230
$210$		0			0
$211$	−1133.21			−0.369733			−0.184867	−	0.982764i	$-0.559185\pi$
$211$	−1133.21			−0.369733			−0.184867	+	0.982764i	$0.559185\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1221.17	−	2115.13i	−0.387364	−	0.670934i
$216$		0			0
$217$	−4890.05	+	1451.72i	−1.52976	+	0.454145i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1801.53	+	1040.11i	0.548343	+	0.316586i
$222$		0			0
$223$			1631.23i			0.489845i	0.969543	+	0.244922i	$0.0787624\pi$
$223$			1631.23i			0.489845i	−0.969543	+	0.244922i	$0.921238\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−584.886	+	1013.05i	−0.171014	+	0.296205i	−0.938775	−	0.344532i	$-0.888038\pi$
$227$	−584.886	+	1013.05i	−0.171014	+	0.296205i	0.767760	+	0.640737i	$0.221371\pi$
$228$		0			0
$229$	−4457.87	+	2573.75i	−1.28640	+	0.742701i	−0.978009	−	0.208561i	$-0.933122\pi$
$229$	−4457.87	+	2573.75i	−1.28640	+	0.742701i	−0.308386	+	0.951261i	$0.599789\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3811.38	+	2200.50i	−1.07164	+	0.618710i	−0.928629	−	0.371011i	$-0.879011\pi$
$233$	−3811.38	+	2200.50i	−1.07164	+	0.618710i	−0.143009	+	0.989721i	$0.545678\pi$
$234$		0			0
$235$	172.567	−	298.895i	0.0479023	−	0.0829692i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	5998.62i		−	1.62351i	−0.583998	−	0.811755i	$-0.698513\pi$
$239$		−	5998.62i		−	1.62351i	0.583998	−	0.811755i	$-0.301487\pi$
$240$		0			0
$241$	796.548	+	459.887i	0.212905	+	0.122921i	0.602661	−	0.797997i	$-0.294107\pi$
$241$	796.548	+	459.887i	0.212905	+	0.122921i	−0.389756	+	0.920918i	$0.627441\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1924.58	+	1253.16i	−0.501865	+	0.326781i
$246$		0			0
$247$	857.845	+	1485.83i	0.220985	+	0.382758i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	701.991			0.176531			0.0882655	−	0.996097i	$-0.471868\pi$
$251$	701.991			0.176531			0.0882655	+	0.996097i	$0.471868\pi$
$252$		0			0
$253$	−2588.94			−0.643341
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2170.80	−	3759.94i	−0.526891	−	0.912602i	−0.999509	−	0.0313343i	$-0.990024\pi$
$257$	−2170.80	−	3759.94i	−0.526891	−	0.912602i	0.472618	−	0.881267i	$-0.343309\pi$
$258$		0			0
$259$	−5771.54	−	1381.92i	−1.38466	−	0.331539i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3606.74	+	2082.35i	0.845632	+	0.488226i	0.859175	−	0.511683i	$-0.170978\pi$
$263$	3606.74	+	2082.35i	0.845632	+	0.488226i	−0.0135429	+	0.999908i	$0.504311\pi$
$264$		0			0
$265$		−	4118.33i		−	0.954668i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4219.12	−	7307.74i	0.956299	−	1.65636i	0.224933	−	0.974374i	$-0.427784\pi$
$269$	4219.12	−	7307.74i	0.956299	−	1.65636i	0.731366	−	0.681985i	$-0.238883\pi$
$270$		0			0
$271$	4544.68	−	2623.87i	1.01871	−	0.588150i	0.104977	−	0.994475i	$-0.466523\pi$
$271$	4544.68	−	2623.87i	1.01871	−	0.588150i	0.913729	+	0.406324i	$0.133190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2268.43	−	1309.68i	0.497423	−	0.287187i
$276$		0			0
$277$	198.173	−	343.245i	0.0429857	−	0.0744534i	−0.843732	−	0.536765i	$-0.819646\pi$
$277$	198.173	−	343.245i	0.0429857	−	0.0744534i	0.886718	+	0.462311i	$0.152980\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	5244.36i		−	1.11335i	−0.830729	−	0.556677i	$-0.812076\pi$
$281$		−	5244.36i		−	1.11335i	0.830729	−	0.556677i	$-0.187924\pi$
$282$		0			0
$283$	−4113.33	−	2374.83i	−0.864000	−	0.498830i	0.00134996	−	0.999999i	$-0.499570\pi$
$283$	−4113.33	−	2374.83i	−0.864000	−	0.498830i	−0.865350	+	0.501169i	$0.832904\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2355.01	−	2484.67i	0.484361	−	0.511030i
$288$		0			0
$289$	1987.23	+	3441.98i	0.404483	+	0.700586i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3597.22			0.717242			0.358621	−	0.933483i	$-0.383247\pi$
$293$	3597.22			0.717242			0.358621	+	0.933483i	$0.383247\pi$
$294$		0			0
$295$	−2783.29			−0.549320
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2690.19	+	4659.54i	0.520326	+	0.901231i
$300$		0			0
$301$	−4647.25	+	4903.13i	−0.889910	+	0.938909i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2757.00	−	1591.76i	−0.517592	−	0.298832i
$306$		0			0
$307$			27.5911i			0.00512933i	0.999997	+	0.00256467i	$0.000816359\pi$
$307$			27.5911i			0.00512933i	−0.999997	+	0.00256467i	$0.999184\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−621.896	+	1077.15i	−0.113391	+	0.196398i	−0.917135	−	0.398576i	$-0.869504\pi$
$311$	−621.896	+	1077.15i	−0.113391	+	0.196398i	0.803745	+	0.594974i	$0.202838\pi$
$312$		0			0
$313$	7541.55	−	4354.12i	1.36190	−	0.786292i	0.372021	−	0.928224i	$-0.378665\pi$
$313$	7541.55	−	4354.12i	1.36190	−	0.786292i	0.989876	+	0.141933i	$0.0453317\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7799.09	+	4502.81i	−1.38183	+	0.797801i	−0.992376	−	0.123245i	$-0.960670\pi$
$317$	−7799.09	+	4502.81i	−1.38183	+	0.797801i	−0.389455	+	0.921046i	$0.627337\pi$
$318$		0			0
$319$	−1791.60	+	3103.15i	−0.314453	+	0.544649i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	774.077i		−	0.133346i
$324$		0			0
$325$	−4714.28	−	2721.79i	−0.804619	−	0.464547i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−928.404	−	222.295i	−0.155576	−	0.0372508i
$330$		0			0
$331$	1194.18	+	2068.38i	0.198302	+	0.343470i	0.947978	−	0.318336i	$-0.103124\pi$
$331$	1194.18	+	2068.38i	0.198302	+	0.343470i	−0.749676	+	0.661805i	$0.769791\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1904.08			−0.310541
$336$		0			0
$337$	11637.2			1.88106			0.940531	−	0.339707i	$-0.110328\pi$
$337$	11637.2			1.88106			0.940531	+	0.339707i	$0.110328\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4499.55	+	7793.45i	0.714558	+	1.23765i
$342$		0			0
$343$	4837.94	+	4116.79i	0.761586	+	0.648064i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2539.63	−	1466.26i	−0.392895	−	0.226838i	0.290519	−	0.956869i	$-0.406172\pi$
$347$	−2539.63	−	1466.26i	−0.392895	−	0.226838i	−0.683414	+	0.730031i	$0.739506\pi$
$348$		0			0
$349$			63.0106i			0.00966442i	0.999988	+	0.00483221i	$0.00153815\pi$
$349$			63.0106i			0.00966442i	−0.999988	+	0.00483221i	$0.998462\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	386.737	−	669.848i	0.0583114	−	0.100998i	−0.835396	−	0.549648i	$-0.814762\pi$
$353$	386.737	−	669.848i	0.0583114	−	0.100998i	0.893708	+	0.448650i	$0.148095\pi$
$354$		0			0
$355$	5598.00	−	3232.00i	0.836932	−	0.483203i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10567.3	+	6101.03i	−1.55354	+	0.896936i	−0.555689	+	0.831390i	$0.687546\pi$
$359$	−10567.3	+	6101.03i	−1.55354	+	0.896936i	−0.997850	+	0.0655458i	$0.979121\pi$
$360$		0			0
$361$	−3110.29	+	5387.17i	−0.453460	+	0.785417i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	1742.57i		−	0.249892i
$366$		0			0
$367$	−4835.15	−	2791.58i	−0.687719	−	0.397055i	0.115038	−	0.993361i	$-0.463301\pi$
$367$	−4835.15	−	2791.58i	−0.687719	−	0.397055i	−0.802757	+	0.596306i	$0.796634\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10920.3	+	3241.94i	−1.52818	+	0.453674i
$372$		0			0
$373$	1656.22	+	2868.66i	0.229909	+	0.398213i	0.957781	−	0.287499i	$-0.0928240\pi$
$373$	1656.22	+	2868.66i	0.229909	+	0.398213i	−0.727872	+	0.685713i	$0.759491\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7446.67			1.01730
$378$		0			0
$379$	−1118.64			−0.151611			−0.0758057	−	0.997123i	$-0.524153\pi$
$379$	−1118.64			−0.151611			−0.0758057	+	0.997123i	$0.524153\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4388.43	+	7600.98i	0.585479	+	1.01408i	0.994816	+	0.101695i	$0.0324267\pi$
$383$	4388.43	+	7600.98i	0.585479	+	1.01408i	−0.409337	+	0.912383i	$0.634240\pi$
$384$		0			0
$385$	2940.65	+	2787.19i	0.389272	+	0.368957i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8285.22	+	4783.47i	1.07989	+	0.623475i	0.930867	−	0.365359i	$-0.119054\pi$
$389$	8285.22	+	4783.47i	1.07989	+	0.623475i	0.149023	+	0.988834i	$0.452387\pi$
$390$		0			0
$391$		−	2427.49i		−	0.313973i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1467.44	−	2541.68i	0.186924	−	0.323762i
$396$		0			0
$397$	8470.20	−	4890.27i	1.07080	−	0.618226i	0.142399	−	0.989809i	$-0.454518\pi$
$397$	8470.20	−	4890.27i	1.07080	−	0.618226i	0.928399	+	0.371584i	$0.121185\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10450.1	−	6033.35i	1.30137	−	0.751349i	0.320735	−	0.947169i	$-0.396070\pi$
$401$	10450.1	−	6033.35i	1.30137	−	0.751349i	0.980640	+	0.195820i	$0.0627369\pi$
$402$		0			0
$403$	9351.03	−	16196.5i	1.15585	−	2.00199i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			10469.9i			1.27512i
$408$		0			0
$409$	6194.67	+	3576.49i	0.748916	+	0.432387i	0.825302	−	0.564691i	$-0.191005\pi$
$409$	6194.67	+	3576.49i	0.748916	+	0.432387i	−0.0763859	+	0.997078i	$0.524338\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2191.00	+	7380.26i	0.261046	+	0.879319i
$414$		0			0
$415$	−256.017	−	443.434i	−0.0302828	−	0.0524514i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5703.45			0.664992			0.332496	−	0.943105i	$-0.392109\pi$
$419$	5703.45			0.664992			0.332496	+	0.943105i	$0.392109\pi$
$420$		0			0
$421$	−10027.7			−1.16086			−0.580428	−	0.814312i	$-0.697115\pi$
$421$	−10027.7			−1.16086			−0.580428	+	0.814312i	$0.697115\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1228.01	+	2126.97i	0.140158	+	0.242760i
$426$		0			0
$427$	−2050.44	+	8563.59i	−0.232384	+	0.970541i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12128.8	−	7002.59i	−1.35551	−	0.782605i	−0.366497	−	0.930419i	$-0.619443\pi$
$431$	−12128.8	−	7002.59i	−1.35551	−	0.782605i	−0.989015	+	0.147814i	$0.952776\pi$
$432$		0			0
$433$		−	3410.83i		−	0.378554i	−0.981924	−	0.189277i	$-0.939386\pi$
$433$		−	3410.83i		−	0.378554i	0.981924	−	0.189277i	$-0.0606145\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1001.05	−	1733.87i	0.109581	−	0.189800i
$438$		0			0
$439$	11009.7	−	6356.44i	1.19695	−	0.691062i	0.237080	−	0.971490i	$-0.423810\pi$
$439$	11009.7	−	6356.44i	1.19695	−	0.691062i	0.959875	+	0.280428i	$0.0904764\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10853.8	+	6266.44i	−1.16406	+	0.672071i	−0.952274	−	0.305245i	$-0.901262\pi$
$443$	−10853.8	+	6266.44i	−1.16406	+	0.672071i	−0.211787	+	0.977316i	$0.567928\pi$
$444$		0			0
$445$	2387.40	−	4135.09i	0.254322	−	0.440499i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	10884.1i		−	1.14399i	−0.820258	−	0.571993i	$-0.806170\pi$
$449$		−	10884.1i		−	1.14399i	0.820258	−	0.571993i	$-0.193830\pi$
$450$		0			0
$451$	−5230.38	−	3019.76i	−0.546095	−	0.315288i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1960.69	−	8188.73i	0.202019	−	0.843722i
$456$		0			0
$457$	9222.57	+	15974.0i	0.944012	+	1.63508i	0.757717	+	0.652583i	$0.226315\pi$
$457$	9222.57	+	15974.0i	0.944012	+	1.63508i	0.186295	+	0.982494i	$0.440352\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4657.01			−0.470496			−0.235248	−	0.971935i	$-0.575590\pi$
$461$	−4657.01			−0.470496			−0.235248	+	0.971935i	$0.575590\pi$
$462$		0			0
$463$	−5088.08			−0.510720			−0.255360	−	0.966846i	$-0.582194\pi$
$463$	−5088.08			−0.510720			−0.255360	+	0.966846i	$0.582194\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7950.04	+	13769.9i	0.787760	+	1.36444i	0.927336	+	0.374229i	$0.122092\pi$
$467$	7950.04	+	13769.9i	0.787760	+	1.36444i	−0.139577	+	0.990211i	$0.544574\pi$
$468$		0			0
$469$	1498.89	+	5048.93i	0.147574	+	0.497096i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10321.4	+	5959.04i	1.00333	+	0.579275i
$474$		0			0
$475$			2025.62i			0.195667i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9702.00	+	16804.4i	−0.925461	+	1.60295i	−0.134643	+	0.990894i	$0.542989\pi$
$479$	−9702.00	+	16804.4i	−0.925461	+	1.60295i	−0.790818	+	0.612051i	$0.790345\pi$
$480$		0			0
$481$	18843.6	−	10879.3i	1.78626	−	1.03130i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2380.95	+	1374.64i	−0.222914	+	0.128700i
$486$		0			0
$487$	6123.60	−	10606.4i	0.569788	−	0.986903i	−0.426798	−	0.904347i	$-0.640358\pi$
$487$	6123.60	−	10606.4i	0.569788	−	0.986903i	0.996586	−	0.0825556i	$-0.0263082\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			7773.35i			0.714473i	0.934014	+	0.357237i	$0.116281\pi$
$491$			7773.35i			0.714473i	−0.934014	+	0.357237i	$0.883719\pi$
$492$		0			0
$493$	−2909.64	−	1679.88i	−0.265808	−	0.153464i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12976.8	−	12299.6i	−1.17121	−	1.11008i
$498$		0			0
$499$	9518.91	+	16487.2i	0.853958	+	1.47910i	0.877609	+	0.479378i	$0.159138\pi$
$499$	9518.91	+	16487.2i	0.853958	+	1.47910i	−0.0236507	+	0.999720i	$0.507529\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	14323.9			1.26973			0.634863	−	0.772624i	$-0.281056\pi$
$503$	14323.9			1.26973			0.634863	+	0.772624i	$0.281056\pi$
$504$		0			0
$505$	12770.7			1.12532
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4294.66	+	7438.57i	0.373983	+	0.647758i	0.990174	−	0.139839i	$-0.0446585\pi$
$509$	4294.66	+	7438.57i	0.373983	+	0.647758i	−0.616191	+	0.787597i	$0.711325\pi$
$510$		0			0
$511$	−4620.66	+	1371.75i	−0.400012	+	0.118753i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7883.28	−	4551.41i	−0.674522	−	0.389435i
$516$		0			0
$517$			1684.17i			0.143269i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11308.3	−	19586.6i	0.950917	−	1.64704i	0.207471	−	0.978241i	$-0.433477\pi$
$521$	11308.3	−	19586.6i	0.950917	−	1.64704i	0.743447	−	0.668795i	$-0.233190\pi$
$522$		0			0
$523$	122.551	−	70.7547i	0.0102462	−	0.00591566i	−0.494868	−	0.868968i	$-0.664784\pi$
$523$	122.551	−	70.7547i	0.0102462	−	0.00591566i	0.505114	+	0.863052i	$0.331450\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7307.45	+	4218.96i	−0.604018	+	0.348730i
$528$		0			0
$529$	−2944.22	+	5099.53i	−0.241984	+	0.419128i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			12551.4i			1.02000i
$534$		0			0
$535$	−10416.9	−	6014.21i	−0.841799	−	0.486013i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5075.72	−	9991.60i	0.405615	−	0.798458i
$540$		0			0
$541$	4655.90	+	8064.25i	0.370005	+	0.640867i	0.989566	−	0.144081i	$-0.0460227\pi$
$541$	4655.90	+	8064.25i	0.370005	+	0.640867i	−0.619561	+	0.784949i	$0.712689\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−563.408			−0.0442821
$546$		0			0
$547$	11888.7			0.929294			0.464647	−	0.885496i	$-0.346181\pi$
$547$	11888.7			0.929294			0.464647	+	0.885496i	$0.346181\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1385.50	−	2399.76i	−0.107122	−	0.185541i
$552$		0			0
$553$	−7894.77	−	1890.30i	−0.607088	−	0.145360i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18409.4	+	10628.7i	1.40042	+	0.808530i	0.994435	−	0.105352i	$-0.0335969\pi$
$557$	18409.4	+	10628.7i	1.40042	+	0.808530i	0.405980	+	0.913882i	$0.366930\pi$
$558$		0			0
$559$		−	24768.3i		−	1.87404i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4661.33	−	8073.66i	0.348937	−	0.604377i	−0.637124	−	0.770762i	$-0.719876\pi$
$563$	4661.33	−	8073.66i	0.348937	−	0.604377i	0.986061	+	0.166385i	$0.0532093\pi$
$564$		0			0
$565$	680.991	−	393.170i	0.0507071	−	0.0292758i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14734.2	+	8506.79i	−1.08557	+	0.626754i	−0.932394	−	0.361445i	$-0.882284\pi$
$569$	−14734.2	+	8506.79i	−1.08557	+	0.626754i	−0.153176	+	0.988199i	$0.548950\pi$
$570$		0			0
$571$	−1134.69	+	1965.34i	−0.0831617	+	0.144040i	−0.904606	−	0.426248i	$-0.859835\pi$
$571$	−1134.69	+	1965.34i	−0.0831617	+	0.144040i	0.821445	+	0.570288i	$0.193168\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			6352.32i			0.460713i
$576$		0			0
$577$	2687.93	+	1551.88i	0.193934	+	0.111968i	0.593823	−	0.804596i	$-0.297618\pi$
$577$	2687.93	+	1551.88i	0.193934	+	0.111968i	−0.399889	+	0.916564i	$0.630951\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−974.288	+	1027.93i	−0.0695702	+	0.0734008i
$582$		0			0
$583$	10048.2	+	17404.1i	0.713818	+	1.23637i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12544.8			−0.882075			−0.441038	−	0.897489i	$-0.645389\pi$
$587$	−12544.8			−0.882075			−0.441038	+	0.897489i	$0.645389\pi$
$588$		0			0
$589$	−6959.27			−0.486845
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2644.32	−	4580.09i	−0.183118	−	0.317170i	0.759823	−	0.650131i	$-0.225286\pi$
$593$	−2644.32	−	4580.09i	−0.183118	−	0.317170i	−0.942941	+	0.332960i	$0.891952\pi$
$594$		0			0
$595$	−2613.38	+	2757.27i	−0.180064	+	0.189978i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5058.01	+	2920.24i	0.345016	+	0.199195i	0.662488	−	0.749073i	$-0.269501\pi$
$599$	5058.01	+	2920.24i	0.345016	+	0.199195i	−0.317472	+	0.948268i	$0.602834\pi$
$600$		0			0
$601$		−	10801.2i		−	0.733098i	−0.930399	−	0.366549i	$-0.880539\pi$
$601$		−	10801.2i		−	0.733098i	0.930399	−	0.366549i	$-0.119461\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−882.018	+	1527.70i	−0.0592713	+	0.102661i
$606$		0			0
$607$	−21239.7	+	12262.8i	−1.42025	+	0.819984i	−0.996320	−	0.0857094i	$-0.972684\pi$
$607$	−21239.7	+	12262.8i	−1.42025	+	0.819984i	−0.423934	+	0.905693i	$0.639351\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3031.15	−	1750.04i	0.200699	−	0.115874i
$612$		0			0
$613$	−8223.34	+	14243.2i	−0.541823	+	0.938465i	0.456976	+	0.889479i	$0.348932\pi$
$613$	−8223.34	+	14243.2i	−0.541823	+	0.938465i	−0.998799	+	0.0489865i	$0.984401\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	23852.4i		−	1.55634i	−0.628055	−	0.778169i	$-0.716149\pi$
$617$		−	23852.4i		−	1.55634i	0.628055	−	0.778169i	$-0.283851\pi$
$618$		0			0
$619$	−14617.4	−	8439.34i	−0.949147	−	0.547990i	−0.0563309	−	0.998412i	$-0.517940\pi$
$619$	−14617.4	−	8439.34i	−0.949147	−	0.547990i	−0.892816	+	0.450422i	$0.851274\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12844.1	−	3075.36i	−0.825983	−	0.197771i
$624$		0			0
$625$	−411.490	−	712.722i	−0.0263354	−	0.0456142i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9816.99			−0.622304
$630$		0			0
$631$	−19613.0			−1.23737			−0.618686	−	0.785638i	$-0.712335\pi$
$631$	−19613.0			−1.23737			−0.618686	+	0.785638i	$0.712335\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1710.49	+	2962.65i	0.106895	+	0.185148i
$636$		0			0
$637$	−23257.0	+	1247.13i	−1.44658	+	0.0775716i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7437.62	−	4294.11i	−0.458297	−	0.264598i	0.253031	−	0.967458i	$-0.418573\pi$
$641$	−7437.62	−	4294.11i	−0.458297	−	0.264598i	−0.711328	+	0.702860i	$0.751906\pi$
$642$		0			0
$643$		−	11966.4i		−	0.733920i	−0.930237	−	0.366960i	$-0.880399\pi$
$643$		−	11966.4i		−	0.733920i	0.930237	−	0.366960i	$-0.119601\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3663.93	−	6346.11i	0.222634	−	0.385613i	−0.732973	−	0.680257i	$-0.761868\pi$
$647$	3663.93	−	6346.11i	0.222634	−	0.385613i	0.955607	+	0.294645i	$0.0952013\pi$
$648$		0			0
$649$	11762.2	−	6790.91i	0.711412	−	0.410734i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3124.09	+	1803.69i	−0.187221	+	0.108092i	−0.590681	−	0.806905i	$-0.701141\pi$
$653$	−3124.09	+	1803.69i	−0.187221	+	0.108092i	0.403460	+	0.914997i	$0.367807\pi$
$654$		0			0
$655$	6637.22	−	11496.0i	0.395936	−	0.685780i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	3244.57i		−	0.191792i	−0.995391	−	0.0958958i	$-0.969428\pi$
$659$		−	3244.57i		−	0.191792i	0.995391	−	0.0958958i	$-0.0305716\pi$
$660$		0			0
$661$	−12548.2	−	7244.69i	−0.738377	−	0.426302i	0.0831019	−	0.996541i	$-0.473517\pi$
$661$	−12548.2	−	7244.69i	−0.738377	−	0.426302i	−0.821479	+	0.570239i	$0.806851\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3003.69	+	891.715i	−0.175155	+	0.0519989i
$666$		0			0
$667$	−4344.90	−	7525.60i	−0.252227	−	0.436870i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15534.8			0.893763
$672$		0			0
$673$	−27609.5			−1.58138			−0.790688	−	0.612219i	$-0.790277\pi$
$673$	−27609.5			−1.58138			−0.790688	+	0.612219i	$0.790277\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7610.03	+	13181.0i	0.432019	+	0.748279i	0.997047	−	0.0767925i	$-0.0244679\pi$
$677$	7610.03	+	13181.0i	0.432019	+	0.748279i	−0.565028	+	0.825072i	$0.691135\pi$
$678$		0			0
$679$	5519.32	+	5231.28i	0.311947	+	0.295668i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	14491.6	+	8366.71i	0.811866	+	0.468731i	0.847603	−	0.530630i	$-0.178045\pi$
$683$	14491.6	+	8366.71i	0.811866	+	0.468731i	−0.0357376	+	0.999361i	$0.511378\pi$
$684$		0			0
$685$			6633.16i			0.369986i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20882.4	−	36169.4i	1.15465	−	1.99992i
$690$		0			0
$691$	3981.41	−	2298.67i	0.219190	−	0.126549i	−0.386385	−	0.922337i	$-0.626276\pi$
$691$	3981.41	−	2298.67i	0.219190	−	0.126549i	0.605575	+	0.795788i	$0.292943\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2145.03	−	1238.43i	0.117073	−	0.0675920i
$696$		0			0
$697$	2831.45	−	4904.21i	0.153872	−	0.266514i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	7621.26i		−	0.410629i	−0.978696	−	0.205315i	$-0.934178\pi$
$701$		−	7621.26i		−	0.410629i	0.978696	−	0.205315i	$-0.0658217\pi$
$702$		0			0
$703$	−7011.93	−	4048.34i	−0.376188	−	0.217192i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10053.1	−	33863.2i	−0.534773	−	1.80135i
$708$		0			0
$709$	3850.25	+	6668.83i	0.203948	+	0.353249i	0.949797	−	0.312867i	$-0.101289\pi$
$709$	3850.25	+	6668.83i	0.203948	+	0.353249i	−0.745849	+	0.666115i	$0.767956\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−21824.2			−1.14631
$714$		0			0
$715$	−14854.8			−0.776977
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5049.89	−	8746.67i	−0.261932	−	0.453680i	0.704823	−	0.709383i	$-0.251026\pi$
$719$	−5049.89	−	8746.67i	−0.261932	−	0.453680i	−0.966755	+	0.255703i	$0.917693\pi$
$720$		0			0
$721$	−5862.96	+	24486.4i	−0.302841	+	1.26480i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	7614.00	+	4395.95i	0.390037	+	0.225188i
$726$		0			0
$727$			23945.7i			1.22159i	0.791789	+	0.610795i	$0.209150\pi$
$727$			23945.7i			1.22159i	−0.791789	+	0.610795i	$0.790850\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5587.43	+	9677.71i	−0.282707	+	0.489662i
$732$		0			0
$733$	18035.8	−	10413.0i	0.908822	−	0.524708i	0.0287698	−	0.999586i	$-0.490841\pi$
$733$	18035.8	−	10413.0i	0.908822	−	0.524708i	0.880052	+	0.474878i	$0.157508\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8046.67	−	4645.75i	0.402175	−	0.232196i
$738$		0			0
$739$	−7648.41	+	13247.4i	−0.380719	+	0.659424i	−0.991165	−	0.132633i	$-0.957657\pi$
$739$	−7648.41	+	13247.4i	−0.380719	+	0.659424i	0.610446	+	0.792058i	$0.290990\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	27249.8i		−	1.34549i	−0.739875	−	0.672744i	$-0.765115\pi$
$743$		−	27249.8i		−	1.34549i	0.739875	−	0.672744i	$-0.234885\pi$
$744$		0			0
$745$	7062.08	+	4077.29i	0.347295	+	0.200511i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7747.28	+	32356.2i	−0.377943	+	1.57846i
$750$		0			0
$751$	19864.5	+	34406.4i	0.965203	+	1.67178i	0.709069	+	0.705139i	$0.249116\pi$
$751$	19864.5	+	34406.4i	0.965203	+	1.67178i	0.256134	+	0.966641i	$0.417551\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10233.7			0.493302
$756$		0			0
$757$	−24557.2			−1.17906			−0.589530	−	0.807747i	$-0.700687\pi$
$757$	−24557.2			−1.17906			−0.589530	+	0.807747i	$0.700687\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9686.45	−	16777.4i	−0.461411	−	0.799187i	0.537621	−	0.843187i	$-0.319323\pi$
$761$	−9686.45	−	16777.4i	−0.461411	−	0.799187i	−0.999032	+	0.0440000i	$0.985990\pi$
$762$		0			0
$763$	443.514	+	1493.95i	0.0210436	+	0.0708842i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−24444.4	−	14113.0i	−1.15076	−	0.664393i
$768$		0			0
$769$		−	39937.6i		−	1.87280i	−0.350930	−	0.936402i	$-0.614134\pi$
$769$		−	39937.6i		−	1.87280i	0.350930	−	0.936402i	$-0.385866\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8058.12	−	13957.1i	0.374942	−	0.649419i	−0.615376	−	0.788234i	$-0.710996\pi$
$773$	8058.12	−	13957.1i	0.374942	−	0.649419i	0.990318	+	0.138815i	$0.0443292\pi$
$774$		0			0
$775$	19122.3	−	11040.3i	0.886314	−	0.511714i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4044.80	−	2335.27i	0.186034	−	0.107407i
$780$		0			0
$781$	−15771.4	+	27316.9i	−0.722594	+	1.25157i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			20184.9i			0.917744i
$786$		0			0
$787$	−19040.0	−	10992.8i	−0.862394	−	0.497903i	0.00241936	−	0.999997i	$-0.499230\pi$
$787$	−19040.0	−	10992.8i	−0.862394	−	0.497903i	−0.864813	+	0.502094i	$0.832563\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1578.62	−	1496.23i	−0.0709598	−	0.0672566i
$792$		0			0
$793$	−16142.3	−	27959.3i	−0.722864	−	1.25204i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2291.42			−0.101840			−0.0509198	−	0.998703i	$-0.516215\pi$
$797$	−2291.42			−0.101840			−0.0509198	+	0.998703i	$0.516215\pi$
$798$		0			0
$799$	−1579.15			−0.0699202
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4251.67	+	7364.12i	0.186847	+	0.323629i
$804$		0			0
$805$	−9419.52	+	2796.40i	−0.412416	+	0.122435i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12856.7	−	7422.83i	−0.558737	−	0.322587i	0.193901	−	0.981021i	$-0.437886\pi$
$809$	−12856.7	−	7422.83i	−0.558737	−	0.322587i	−0.752638	+	0.658434i	$0.771219\pi$
$810$		0			0
$811$			41094.7i			1.77932i	0.456621	+	0.889661i	$0.349059\pi$
$811$			41094.7i			1.77932i	−0.456621	+	0.889661i	$0.650941\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	5505.25	−	9535.37i	0.236614	−	0.409827i
$816$		0			0
$817$	−7981.81	+	4608.30i	−0.341797	+	0.197337i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−30664.0	+	17703.9i	−1.30351	+	0.752582i	−0.981004	−	0.193986i	$-0.937858\pi$
$821$	−30664.0	+	17703.9i	−1.30351	+	0.752582i	−0.322505	+	0.946568i	$0.604525\pi$
$822$		0			0
$823$	−2744.14	+	4752.98i	−0.116227	+	0.201311i	−0.918269	−	0.395956i	$-0.870413\pi$
$823$	−2744.14	+	4752.98i	−0.116227	+	0.201311i	0.802043	+	0.597267i	$0.203747\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			96.9958i			0.00407844i	0.999998	+	0.00203922i	$0.000649105\pi$
$827$			96.9958i			0.00407844i	−0.999998	+	0.00203922i	$0.999351\pi$
$828$		0			0
$829$	−9914.41	−	5724.09i	−0.415370	−	0.239814i	0.277724	−	0.960661i	$-0.410420\pi$
$829$	−9914.41	−	5724.09i	−0.415370	−	0.239814i	−0.693094	+	0.720847i	$0.743753\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9368.52	+	4759.19i	0.389676	+	0.197955i
$834$		0			0
$835$	−9319.13	−	16141.2i	−0.386230	−	0.668969i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	25083.7			1.03216			0.516082	−	0.856539i	$-0.327390\pi$
$839$	25083.7			1.03216			0.516082	+	0.856539i	$0.327390\pi$
$840$		0			0
$841$	12361.9			0.506865
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8080.56	+	13995.9i	0.328970	+	0.569793i
$846$		0			0
$847$	4745.22	+	1136.18i	0.192500	+	0.0460918i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−21989.3	−	12695.5i	−0.885762	−	0.511395i
$852$		0			0
$853$		−	48196.0i		−	1.93458i	−0.253668	−	0.967291i	$-0.581637\pi$
$853$		−	48196.0i		−	1.93458i	0.253668	−	0.967291i	$-0.418363\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9301.67	−	16111.0i	0.370757	−	0.642171i	−0.618925	−	0.785450i	$-0.712431\pi$
$857$	9301.67	−	16111.0i	0.370757	−	0.642171i	0.989682	+	0.143280i	$0.0457648\pi$
$858$		0			0
$859$	−7453.04	+	4303.02i	−0.296036	+	0.170916i	−0.640661	−	0.767824i	$-0.721339\pi$
$859$	−7453.04	+	4303.02i	−0.296036	+	0.170916i	0.344625	+	0.938740i	$0.388006\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	37188.6	−	21470.8i	1.46687	−	0.846901i	0.467562	−	0.883960i	$-0.345133\pi$
$863$	37188.6	−	21470.8i	1.46687	−	0.846901i	0.999313	+	0.0370598i	$0.0117992\pi$
$864$		0			0
$865$	−429.245	+	743.475i	−0.0168726	+	0.0292242i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			14321.5i			0.559062i
$870$		0			0
$871$	−16722.7	−	9654.85i	−0.650547	−	0.375594i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	17501.9	−	18465.5i	0.676196	−	0.713428i
$876$		0			0
$877$	−19724.6	−	34163.9i	−0.759465	−	1.31543i	−0.943124	−	0.332442i	$-0.892127\pi$
$877$	−19724.6	−	34163.9i	−0.759465	−	1.31543i	0.183659	−	0.982990i	$-0.441206\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−31686.9			−1.21176			−0.605879	−	0.795556i	$-0.707179\pi$
$881$	−31686.9			−1.21176			−0.605879	+	0.795556i	$0.707179\pi$
$882$		0			0
$883$	21170.2			0.806832			0.403416	−	0.915017i	$-0.367823\pi$
$883$	21170.2			0.806832			0.403416	+	0.915017i	$0.367823\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23404.6	+	40538.0i	0.885964	+	1.53454i	0.844604	+	0.535391i	$0.179836\pi$
$887$	23404.6	+	40538.0i	0.885964	+	1.53454i	0.0413598	+	0.999144i	$0.486831\pi$
$888$		0			0
$889$	6509.36	−	6867.77i	0.245576	−	0.259097i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1127.93	−	651.211i	−0.0422673	−	0.0244031i
$894$		0			0
$895$			12964.0i			0.484178i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15102.8	+	26158.8i	−0.560296	+	0.970462i
$900$		0			0
$901$	−16318.7	+	9421.63i	−0.603392	+	0.348369i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9260.18	+	5346.37i	−0.340131	+	0.196375i
$906$		0			0
$907$	13950.8	−	24163.5i	0.510726	−	0.884604i	−0.489196	−	0.872174i	$-0.662710\pi$
$907$	13950.8	−	24163.5i	0.510726	−	0.884604i	0.999923	−	0.0124304i	$-0.00395683\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	22732.2i		−	0.826732i	−0.910565	−	0.413366i	$-0.864353\pi$
$911$		−	22732.2i		−	0.826732i	0.910565	−	0.413366i	$-0.135647\pi$
$912$		0			0
$913$	2163.86	+	1249.30i	0.0784373	+	0.0452858i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−35708.0	−	8549.83i	−1.28591	−	0.307896i
$918$		0			0
$919$	16454.0	+	28499.1i	0.590605	+	1.02296i	0.994151	+	0.107998i	$0.0344442\pi$
$919$	16454.0	+	28499.1i	0.590605	+	1.02296i	−0.403546	+	0.914959i	$0.632223\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	65552.8			2.33770
$924$		0			0
$925$	25689.3			0.913146
$926$		0			0
$927$		0			0
$928$		0			0
$929$	29.8811	+	51.7556i	0.00105529	+	0.00182782i	0.866553	−	0.499086i	$-0.166331\pi$
$929$	29.8811	+	51.7556i	0.00105529	+	0.00182782i	−0.865497	+	0.500914i	$0.832997\pi$
$930$		0			0
$931$	4729.00	+	7262.72i	0.166473	+	0.255667i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5804.21	+	3351.06i	0.203014	+	0.117210i
$936$		0			0
$937$			43382.1i			1.51252i	0.654270	+	0.756261i	$0.272976\pi$
$937$			43382.1i			1.51252i	−0.654270	+	0.756261i	$0.727024\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22210.7	+	38470.0i	−0.769444	+	1.33272i	0.168421	+	0.985715i	$0.446133\pi$
$941$	−22210.7	+	38470.0i	−0.769444	+	1.33272i	−0.937865	+	0.347001i	$0.887200\pi$
$942$		0			0
$943$	12684.4	−	7323.36i	0.438030	−	0.252897i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27136.7	+	15667.4i	−0.931175	+	0.537614i	−0.887183	−	0.461418i	$-0.847341\pi$
$947$	−27136.7	+	15667.4i	−0.931175	+	0.537614i	−0.0439922	+	0.999032i	$0.514008\pi$
$948$		0			0
$949$	8835.89	−	15304.2i	0.302239	−	0.523494i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	20504.7i		−	0.696971i	−0.937314	−	0.348486i	$-0.886696\pi$
$953$		−	20504.7i		−	0.696971i	0.937314	−	0.348486i	$-0.113304\pi$
$954$		0			0
$955$	4789.30	+	2765.10i	0.162281	+	0.0936929i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17588.7	−	5221.61i	0.592251	−	0.175823i
$960$		0			0
$961$	23034.6	+	39897.2i	0.773208	+	1.33924i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12414.7			−0.414138
$966$		0			0
$967$	−2538.92			−0.0844325			−0.0422162	−	0.999108i	$-0.513442\pi$
$967$	−2538.92			−0.0844325			−0.0422162	+	0.999108i	$0.513442\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8357.65	−	14475.9i	−0.276220	−	0.478427i	0.694222	−	0.719761i	$-0.255749\pi$
$971$	−8357.65	−	14475.9i	−0.276220	−	0.478427i	−0.970442	+	0.241333i	$0.922415\pi$
$972$		0			0
$973$	−4972.43	−	4712.93i	−0.163832	−	0.155282i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25809.0	+	14900.8i	0.845140	+	0.487942i	0.859008	−	0.511962i	$-0.171081\pi$
$977$	25809.0	+	14900.8i	0.845140	+	0.487942i	−0.0138678	+	0.999904i	$0.504414\pi$
$978$		0			0
$979$			23299.9i			0.760641i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−19946.2	+	34547.8i	−0.647186	+	1.12096i	0.336605	+	0.941646i	$0.390721\pi$
$983$	−19946.2	+	34547.8i	−0.647186	+	1.12096i	−0.983792	+	0.179314i	$0.942612\pi$
$984$		0			0
$985$	17038.7	−	9837.31i	0.551167	−	0.318216i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−25030.8	+	14451.5i	−0.804786	+	0.464644i
$990$		0			0
$991$	23250.0	−	40270.2i	0.745268	−	1.29084i	−0.204801	−	0.978804i	$-0.565655\pi$
$991$	23250.0	−	40270.2i	0.745268	−	1.29084i	0.950069	−	0.312039i	$-0.101012\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			22036.0i			0.702097i
$996$		0			0
$997$	31402.7	+	18130.4i	0.997527	+	0.575923i	0.907516	−	0.420018i	$-0.137976\pi$
$997$	31402.7	+	18130.4i	0.997527	+	0.575923i	0.0900114	+	0.995941i	$0.471310\pi$
$998$		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.17.8	✓	48
3.2	odd	2	inner	504.4.bl.a.17.17	yes	48
4.3	odd	2		1008.4.bt.d.17.8		48
7.5	odd	6	inner	504.4.bl.a.89.17	yes	48
12.11	even	2		1008.4.bt.d.17.17		48
21.5	even	6	inner	504.4.bl.a.89.8	yes	48
28.19	even	6		1008.4.bt.d.593.17		48
84.47	odd	6		1008.4.bt.d.593.8		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.8	✓	48	1.1	even	1	trivial
504.4.bl.a.17.17	yes	48	3.2	odd	2	inner
504.4.bl.a.89.8	yes	48	21.5	even	6	inner
504.4.bl.a.89.17	yes	48	7.5	odd	6	inner
1008.4.bt.d.17.8		48	4.3	odd	2
1008.4.bt.d.17.17		48	12.11	even	2
1008.4.bt.d.593.8		48	84.47	odd	6
1008.4.bt.d.593.17		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)

\(f(q)\)	\(=\)	\(q+(-3.34783 - 5.79860i) q^{5} +(-12.7404 + 13.4418i) q^{7} +O(q^{10})\)	\(q+(-3.34783 - 5.79860i) q^{5} +(-12.7404 + 13.4418i) q^{7} +(28.2958 + 16.3366i) q^{11} -67.9019i q^{13} +(-15.3179 + 26.5313i) q^{17} +(-21.8820 + 12.6336i) q^{19} +(-68.6216 + 39.6187i) q^{23} +(40.0841 - 69.4277i) q^{25} +109.668i q^{29} +(238.527 + 137.714i) q^{31} +(120.596 + 28.8753i) q^{35} +(160.221 + 277.511i) q^{37} -184.846 q^{41} +364.766 q^{43} +(25.7730 + 44.6402i) q^{47} +(-18.3666 - 342.508i) q^{49} +(532.671 + 307.538i) q^{53} -218.769i q^{55} +(207.843 - 359.995i) q^{59} +(411.761 - 237.730i) q^{61} +(-393.736 + 227.324i) q^{65} +(142.188 - 246.277i) q^{67} +965.404i q^{71} +(225.387 + 130.127i) q^{73} +(-580.093 + 172.214i) q^{77} +(219.163 + 379.602i) q^{79} +76.4726 q^{83} +205.126 q^{85} +(356.559 + 617.579i) q^{89} +(912.728 + 865.095i) q^{91} +(146.514 + 84.5900i) q^{95} -410.607i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 24 q^{7}+O(q^{10}) \) 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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