LMFDB - Embedded newform 504.4.bl.a.89.8

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			89.8
Character	$\chi$	$=$	504.89
Dual form			504.4.bl.a.17.8

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

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$-1$

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	−0.976008	−	0.217736i	$-0.930133\pi$
$5$	−3.34783	+	5.79860i	−0.299439	+	0.518643i	0.676569	+	0.736379i	$0.263466\pi$
$6$		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	−0.0592734	−	0.998242i	$-0.518878\pi$
$11$	28.2958	−	16.3366i	0.775593	−	0.447789i	0.834866	+	0.550453i	$0.185545\pi$
$12$		0			0
$13$			67.9019i			1.44866i	0.689452	+	0.724331i	$0.257851\pi$
$13$			67.9019i			1.44866i	−0.689452	+	0.724331i	$0.742149\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.3179	−	26.5313i	−0.218537	−	0.378517i	0.735824	−	0.677173i	$-0.236795\pi$
$17$	−15.3179	−	26.5313i	−0.218537	−	0.378517i	−0.954361	+	0.298656i	$0.903462\pi$
$18$		0			0
$19$	−21.8820	−	12.6336i	−0.264215	−	0.152544i	0.362041	−	0.932162i	$-0.382080\pi$
$19$	−21.8820	−	12.6336i	−0.264215	−	0.152544i	−0.626256	+	0.779618i	$0.715413\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−68.6216	−	39.6187i	−0.622113	−	0.359177i	0.155578	−	0.987824i	$-0.450276\pi$
$23$	−68.6216	−	39.6187i	−0.622113	−	0.359177i	−0.777691	+	0.628647i	$0.783609\pi$
$24$		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	$-0.885802\pi$
$29$		−	109.668i		−	0.702236i	0.936331	−	0.351118i	$-0.114198\pi$
$30$		0			0
$31$	238.527	−	137.714i	1.38196	−	0.797875i	0.389568	−	0.920998i	$-0.372624\pi$
$31$	238.527	−	137.714i	1.38196	−	0.797875i	0.992391	+	0.123123i	$0.0392910\pi$
$32$		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	−0.252246	−	0.967663i	$-0.581169\pi$
$37$	160.221	−	277.511i	0.711898	−	1.23304i	0.964144	−	0.265380i	$-0.0854975\pi$
$38$		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	−0.823257	−	0.567669i	$-0.807846\pi$
$47$	25.7730	−	44.6402i	0.0799868	−	0.138541i	0.903244	+	0.429127i	$0.141179\pi$
$48$		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	0.388306	−	0.921531i	$-0.373060\pi$
$53$	532.671	−	307.538i	1.38053	−	0.797048i	0.992222	+	0.124483i	$0.0397271\pi$
$54$		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	0.998889	−	0.0471340i	$-0.0150088\pi$
$59$	207.843	+	359.995i	0.458625	+	0.794362i	−0.540263	+	0.841496i	$0.681675\pi$
$60$		0			0
$61$	411.761	+	237.730i	0.864271	+	0.498987i	0.865440	−	0.501012i	$-0.167039\pi$
$61$	411.761	+	237.730i	0.864271	+	0.498987i	−0.00116918	+	0.999999i	$0.500372\pi$
$62$		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	0.966046	−	0.258369i	$-0.0831849\pi$
$67$	142.188	+	246.277i	0.259269	+	0.449068i	−0.706777	+	0.707436i	$0.749852\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	965.404i		−	1.61370i	−0.590759	−	0.806848i	$-0.701172\pi$
$71$		−	965.404i		−	1.61370i	0.590759	−	0.806848i	$-0.298828\pi$
$72$		0			0
$73$	225.387	−	130.127i	0.361364	−	0.208633i	−0.308315	−	0.951284i	$-0.599765\pi$
$73$	225.387	−	130.127i	0.361364	−	0.208633i	0.669679	+	0.742651i	$0.266432\pi$
$74$		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	−0.666698	−	0.745328i	$-0.732293\pi$
$79$	219.163	−	379.602i	0.312124	−	0.540615i	0.978822	+	0.204713i	$0.0656262\pi$
$80$		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	−0.571724	−	0.820446i	$-0.693725\pi$
$89$	356.559	−	617.579i	0.424665	−	0.735542i	0.996389	+	0.0849045i	$0.0270585\pi$
$90$		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	$0.0689430\pi$
$97$			410.607i			0.429803i	−0.976636	+	0.214901i	$0.931057\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−953.656	−	1651.78i	−0.939527	−	1.62731i	−0.766355	−	0.642418i	$-0.777931\pi$
$101$	−953.656	−	1651.78i	−0.939527	−	1.62731i	−0.173173	−	0.984891i	$-0.555402\pi$
$102$		0			0
$103$	1177.37	+	679.756i	1.12631	+	0.650276i	0.943004	−	0.332780i	$-0.107987\pi$
$103$	1177.37	+	679.756i	1.12631	+	0.650276i	0.183306	+	0.983056i	$0.441320\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1555.77	+	898.226i	1.40563	+	0.811540i	0.994963	−	0.100246i	$-0.0319629\pi$
$107$	1555.77	+	898.226i	1.40563	+	0.811540i	0.410666	+	0.911786i	$0.365296\pi$
$108$		0			0
$109$	42.0727	+	72.8720i	0.0369709	+	0.0640355i	0.883919	−	0.467640i	$-0.154896\pi$
$109$	42.0727	+	72.8720i	0.0369709	+	0.0640355i	−0.846948	+	0.531676i	$0.821562\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	117.441i		−	0.0977688i	−0.998804	−	0.0488844i	$-0.984433\pi$
$113$		−	117.441i		−	0.0977688i	0.998804	−	0.0488844i	$-0.0155666\pi$
$114$		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	−0.319189	−	0.947691i	$-0.603411\pi$
$131$	991.274	−	1716.94i	0.661130	−	1.14511i	0.980319	−	0.197419i	$-0.0632560\pi$
$132$		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	−0.743062	−	0.669223i	$-0.766627\pi$
$137$	−857.943	+	495.334i	−0.535029	+	0.308899i	0.208033	+	0.978122i	$0.433294\pi$
$138$		0			0
$139$		−	369.921i		−	0.225729i	−0.993610	−	0.112864i	$-0.963997\pi$
$139$		−	369.921i		−	0.225729i	0.993610	−	0.112864i	$-0.0360026\pi$
$140$		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	0.181188	−	0.983449i	$-0.442006\pi$
$149$	−1054.73	−	608.947i	−0.579910	−	0.334811i	−0.761098	+	0.648637i	$0.775339\pi$
$150$		0			0
$151$	−764.206	−	1323.64i	−0.411856	−	0.713355i	0.583237	−	0.812302i	$-0.301786\pi$
$151$	−764.206	−	1323.64i	−0.411856	−	0.713355i	−0.995093	+	0.0989470i	$0.968453\pi$
$152$		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	−0.984855	−	0.173379i	$-0.944531\pi$
$157$	−2610.74	+	1507.31i	−1.32713	+	0.766220i	−0.342277	+	0.939599i	$0.611198\pi$
$158$		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	−0.598018	−	0.801483i	$-0.704045\pi$
$163$	822.212	−	1424.11i	0.395096	−	0.684326i	0.993113	+	0.117157i	$0.0373780\pi$
$164$		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	−0.879768	−	0.475402i	$-0.842302\pi$
$173$	−64.1081	+	111.038i	−0.0281737	+	0.0487983i	0.851595	+	0.524201i	$0.175636\pi$
$174$		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	−0.807408	−	0.589994i	$-0.799130\pi$
$179$	−1676.79	+	968.093i	−0.700161	+	0.404238i	0.107246	+	0.994233i	$0.465797\pi$
$180$		0			0
$181$			1596.97i			0.655810i	0.944711	+	0.327905i	$0.106343\pi$
$181$			1596.97i			0.655810i	−0.944711	+	0.327905i	$0.893657\pi$
$182$		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	0.358356	−	0.933585i	$-0.383338\pi$
$191$	−715.285	−	412.970i	−0.270975	−	0.156448i	−0.629331	+	0.777138i	$0.716671\pi$
$192$		0			0
$193$	927.072	+	1605.74i	0.345762	+	0.598878i	0.985492	−	0.169722i	$-0.0542870\pi$
$193$	927.072	+	1605.74i	0.345762	+	0.598878i	−0.639730	+	0.768600i	$0.720954\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2938.42i		−	1.06271i	−0.847150	−	0.531355i	$-0.821683\pi$
$197$		−	2938.42i		−	1.06271i	0.847150	−	0.531355i	$-0.178317\pi$
$198$		0			0
$199$	−2850.16	+	1645.54i	−1.01529	+	0.586178i	−0.912736	−	0.408550i	$-0.866035\pi$
$199$	−2850.16	+	1645.54i	−1.01529	+	0.586178i	−0.102554	+	0.994727i	$0.532701\pi$
$200$		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	$-0.921238\pi$
$223$		−	1631.23i		−	0.489845i	0.969543	−	0.244922i	$-0.0787624\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−584.886	−	1013.05i	−0.171014	−	0.296205i	0.767760	−	0.640737i	$-0.221371\pi$
$227$	−584.886	−	1013.05i	−0.171014	−	0.296205i	−0.938775	+	0.344532i	$0.888038\pi$
$228$		0			0
$229$	−4457.87	−	2573.75i	−1.28640	−	0.742701i	−0.308386	−	0.951261i	$-0.599789\pi$
$229$	−4457.87	−	2573.75i	−1.28640	−	0.742701i	−0.978009	+	0.208561i	$0.933122\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3811.38	−	2200.50i	−1.07164	−	0.618710i	−0.143009	−	0.989721i	$-0.545678\pi$
$233$	−3811.38	−	2200.50i	−1.07164	−	0.618710i	−0.928629	+	0.371011i	$0.879011\pi$
$234$		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	$0.301487\pi$
$239$			5998.62i			1.62351i	−0.583998	+	0.811755i	$0.698513\pi$
$240$		0			0
$241$	796.548	−	459.887i	0.212905	−	0.122921i	−0.389756	−	0.920918i	$-0.627441\pi$
$241$	796.548	−	459.887i	0.212905	−	0.122921i	0.602661	+	0.797997i	$0.294107\pi$
$242$		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	0.472618	+	0.881267i	$0.343309\pi$
$257$	−2170.80	+	3759.94i	−0.526891	+	0.912602i	−0.999509	+	0.0313343i	$0.990024\pi$
$258$		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	−0.0135429	−	0.999908i	$-0.504311\pi$
$263$	3606.74	−	2082.35i	0.845632	−	0.488226i	0.859175	+	0.511683i	$0.170978\pi$
$264$		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	0.731366	+	0.681985i	$0.238883\pi$
$269$	4219.12	+	7307.74i	0.956299	+	1.65636i	0.224933	+	0.974374i	$0.427784\pi$
$270$		0			0
$271$	4544.68	+	2623.87i	1.01871	+	0.588150i	0.913729	−	0.406324i	$-0.133190\pi$
$271$	4544.68	+	2623.87i	1.01871	+	0.588150i	0.104977	+	0.994475i	$0.466523\pi$
$272$		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	0.886718	−	0.462311i	$-0.152980\pi$
$277$	198.173	+	343.245i	0.0429857	+	0.0744534i	−0.843732	+	0.536765i	$0.819646\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5244.36i			1.11335i	0.830729	+	0.556677i	$0.187924\pi$
$281$			5244.36i			1.11335i	−0.830729	+	0.556677i	$0.812076\pi$
$282$		0			0
$283$	−4113.33	+	2374.83i	−0.864000	+	0.498830i	−0.865350	−	0.501169i	$-0.832904\pi$
$283$	−4113.33	+	2374.83i	−0.864000	+	0.498830i	0.00134996	+	0.999999i	$0.499570\pi$
$284$		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	$-0.999184\pi$
$307$		−	27.5911i		−	0.00512933i	0.999997	−	0.00256467i	$-0.000816359\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−621.896	−	1077.15i	−0.113391	−	0.196398i	0.803745	−	0.594974i	$-0.202838\pi$
$311$	−621.896	−	1077.15i	−0.113391	−	0.196398i	−0.917135	+	0.398576i	$0.869504\pi$
$312$		0			0
$313$	7541.55	+	4354.12i	1.36190	+	0.786292i	0.989876	−	0.141933i	$-0.0453317\pi$
$313$	7541.55	+	4354.12i	1.36190	+	0.786292i	0.372021	+	0.928224i	$0.378665\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7799.09	−	4502.81i	−1.38183	−	0.797801i	−0.389455	−	0.921046i	$-0.627337\pi$
$317$	−7799.09	−	4502.81i	−1.38183	−	0.797801i	−0.992376	+	0.123245i	$0.960670\pi$
$318$		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	−0.749676	−	0.661805i	$-0.769791\pi$
$331$	1194.18	−	2068.38i	0.198302	−	0.343470i	0.947978	+	0.318336i	$0.103124\pi$
$332$		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	−0.683414	−	0.730031i	$-0.739506\pi$
$347$	−2539.63	+	1466.26i	−0.392895	+	0.226838i	0.290519	+	0.956869i	$0.406172\pi$
$348$		0			0
$349$		−	63.0106i		−	0.00966442i	−0.999988	−	0.00483221i	$-0.998462\pi$
$349$		−	63.0106i		−	0.00966442i	0.999988	−	0.00483221i	$-0.00153815\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	386.737	+	669.848i	0.0583114	+	0.100998i	0.893708	−	0.448650i	$-0.148095\pi$
$353$	386.737	+	669.848i	0.0583114	+	0.100998i	−0.835396	+	0.549648i	$0.814762\pi$
$354$		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	−0.997850	−	0.0655458i	$-0.979121\pi$
$359$	−10567.3	−	6101.03i	−1.55354	−	0.896936i	−0.555689	−	0.831390i	$-0.687546\pi$
$360$		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	−0.802757	−	0.596306i	$-0.796634\pi$
$367$	−4835.15	+	2791.58i	−0.687719	+	0.397055i	0.115038	+	0.993361i	$0.463301\pi$
$368$		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	−0.727872	−	0.685713i	$-0.759491\pi$
$373$	1656.22	−	2868.66i	0.229909	−	0.398213i	0.957781	+	0.287499i	$0.0928240\pi$
$374$		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	−0.409337	−	0.912383i	$-0.634240\pi$
$383$	4388.43	−	7600.98i	0.585479	−	1.01408i	0.994816	−	0.101695i	$-0.0324267\pi$
$384$		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	0.149023	−	0.988834i	$-0.452387\pi$
$389$	8285.22	−	4783.47i	1.07989	−	0.623475i	0.930867	+	0.365359i	$0.119054\pi$
$390$		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	0.928399	−	0.371584i	$-0.121185\pi$
$397$	8470.20	+	4890.27i	1.07080	+	0.618226i	0.142399	+	0.989809i	$0.454518\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10450.1	+	6033.35i	1.30137	+	0.751349i	0.980640	−	0.195820i	$-0.0627369\pi$
$401$	10450.1	+	6033.35i	1.30137	+	0.751349i	0.320735	+	0.947169i	$0.396070\pi$
$402$		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	−0.0763859	−	0.997078i	$-0.524338\pi$
$409$	6194.67	−	3576.49i	0.748916	−	0.432387i	0.825302	+	0.564691i	$0.191005\pi$
$410$		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	−0.989015	−	0.147814i	$-0.952776\pi$
$431$	−12128.8	+	7002.59i	−1.35551	+	0.782605i	−0.366497	+	0.930419i	$0.619443\pi$
$432$		0			0
$433$			3410.83i			0.378554i	0.981924	+	0.189277i	$0.0606145\pi$
$433$			3410.83i			0.378554i	−0.981924	+	0.189277i	$0.939386\pi$
$434$		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	0.959875	−	0.280428i	$-0.0904764\pi$
$439$	11009.7	+	6356.44i	1.19695	+	0.691062i	0.237080	+	0.971490i	$0.423810\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10853.8	−	6266.44i	−1.16406	−	0.672071i	−0.211787	−	0.977316i	$-0.567928\pi$
$443$	−10853.8	−	6266.44i	−1.16406	−	0.672071i	−0.952274	+	0.305245i	$0.901262\pi$
$444$		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	$0.193830\pi$
$449$			10884.1i			1.14399i	−0.820258	+	0.571993i	$0.806170\pi$
$450$		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	0.186295	−	0.982494i	$-0.440352\pi$
$457$	9222.57	−	15974.0i	0.944012	−	1.63508i	0.757717	−	0.652583i	$-0.226315\pi$
$458$		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	−0.139577	−	0.990211i	$-0.544574\pi$
$467$	7950.04	−	13769.9i	0.787760	−	1.36444i	0.927336	−	0.374229i	$-0.122092\pi$
$468$		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	−0.790818	−	0.612051i	$-0.790345\pi$
$479$	−9702.00	−	16804.4i	−0.925461	−	1.60295i	−0.134643	−	0.990894i	$-0.542989\pi$
$480$		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	0.996586	+	0.0825556i	$0.0263082\pi$
$487$	6123.60	+	10606.4i	0.569788	+	0.986903i	−0.426798	+	0.904347i	$0.640358\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7773.35i		−	0.714473i	−0.934014	−	0.357237i	$-0.883719\pi$
$491$		−	7773.35i		−	0.714473i	0.934014	−	0.357237i	$-0.116281\pi$
$492$		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	−0.0236507	−	0.999720i	$-0.507529\pi$
$499$	9518.91	−	16487.2i	0.853958	−	1.47910i	0.877609	−	0.479378i	$-0.159138\pi$
$500$		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	−0.616191	−	0.787597i	$-0.711325\pi$
$509$	4294.66	−	7438.57i	0.373983	−	0.647758i	0.990174	+	0.139839i	$0.0446585\pi$
$510$		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	0.743447	+	0.668795i	$0.233190\pi$
$521$	11308.3	+	19586.6i	0.950917	+	1.64704i	0.207471	+	0.978241i	$0.433477\pi$
$522$		0			0
$523$	122.551	+	70.7547i	0.0102462	+	0.00591566i	0.505114	−	0.863052i	$-0.331450\pi$
$523$	122.551	+	70.7547i	0.0102462	+	0.00591566i	−0.494868	+	0.868968i	$0.664784\pi$
$524$		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	−0.619561	−	0.784949i	$-0.712689\pi$
$541$	4655.90	−	8064.25i	0.370005	−	0.640867i	0.989566	+	0.144081i	$0.0460227\pi$
$542$		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	0.405980	−	0.913882i	$-0.366930\pi$
$557$	18409.4	−	10628.7i	1.40042	−	0.808530i	0.994435	+	0.105352i	$0.0335969\pi$
$558$		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	0.986061	−	0.166385i	$-0.0532093\pi$
$563$	4661.33	+	8073.66i	0.348937	+	0.604377i	−0.637124	+	0.770762i	$0.719876\pi$
$564$		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	−0.153176	−	0.988199i	$-0.548950\pi$
$569$	−14734.2	−	8506.79i	−1.08557	−	0.626754i	−0.932394	+	0.361445i	$0.882284\pi$
$570$		0			0
$571$	−1134.69	−	1965.34i	−0.0831617	−	0.144040i	0.821445	−	0.570288i	$-0.193168\pi$
$571$	−1134.69	−	1965.34i	−0.0831617	−	0.144040i	−0.904606	+	0.426248i	$0.859835\pi$
$572$		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	−0.399889	−	0.916564i	$-0.630951\pi$
$577$	2687.93	−	1551.88i	0.193934	−	0.111968i	0.593823	+	0.804596i	$0.297618\pi$
$578$		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	−0.942941	−	0.332960i	$-0.891952\pi$
$593$	−2644.32	+	4580.09i	−0.183118	+	0.317170i	0.759823	+	0.650131i	$0.225286\pi$
$594$		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	−0.317472	−	0.948268i	$-0.602834\pi$
$599$	5058.01	−	2920.24i	0.345016	−	0.199195i	0.662488	+	0.749073i	$0.269501\pi$
$600$		0			0
$601$			10801.2i			0.733098i	0.930399	+	0.366549i	$0.119461\pi$
$601$			10801.2i			0.733098i	−0.930399	+	0.366549i	$0.880539\pi$
$602$		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	−0.423934	−	0.905693i	$-0.639351\pi$
$607$	−21239.7	−	12262.8i	−1.42025	−	0.819984i	−0.996320	+	0.0857094i	$0.972684\pi$
$608$		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	−0.998799	−	0.0489865i	$-0.984401\pi$
$613$	−8223.34	−	14243.2i	−0.541823	−	0.938465i	0.456976	−	0.889479i	$-0.348932\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			23852.4i			1.55634i	0.628055	+	0.778169i	$0.283851\pi$
$617$			23852.4i			1.55634i	−0.628055	+	0.778169i	$0.716149\pi$
$618$		0			0
$619$	−14617.4	+	8439.34i	−0.949147	+	0.547990i	−0.892816	−	0.450422i	$-0.851274\pi$
$619$	−14617.4	+	8439.34i	−0.949147	+	0.547990i	−0.0563309	+	0.998412i	$0.517940\pi$
$620$		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	−0.711328	−	0.702860i	$-0.751906\pi$
$641$	−7437.62	+	4294.11i	−0.458297	+	0.264598i	0.253031	+	0.967458i	$0.418573\pi$
$642$		0			0
$643$			11966.4i			0.733920i	0.930237	+	0.366960i	$0.119601\pi$
$643$			11966.4i			0.733920i	−0.930237	+	0.366960i	$0.880399\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3663.93	+	6346.11i	0.222634	+	0.385613i	0.955607	−	0.294645i	$-0.0952013\pi$
$647$	3663.93	+	6346.11i	0.222634	+	0.385613i	−0.732973	+	0.680257i	$0.761868\pi$
$648$		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	0.403460	−	0.914997i	$-0.367807\pi$
$653$	−3124.09	−	1803.69i	−0.187221	−	0.108092i	−0.590681	+	0.806905i	$0.701141\pi$
$654$		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	$0.0305716\pi$
$659$			3244.57i			0.191792i	−0.995391	+	0.0958958i	$0.969428\pi$
$660$		0			0
$661$	−12548.2	+	7244.69i	−0.738377	+	0.426302i	−0.821479	−	0.570239i	$-0.806851\pi$
$661$	−12548.2	+	7244.69i	−0.738377	+	0.426302i	0.0831019	+	0.996541i	$0.473517\pi$
$662$		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	−0.565028	−	0.825072i	$-0.691135\pi$
$677$	7610.03	−	13181.0i	0.432019	−	0.748279i	0.997047	+	0.0767925i	$0.0244679\pi$
$678$		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	−0.0357376	−	0.999361i	$-0.511378\pi$
$683$	14491.6	−	8366.71i	0.811866	−	0.468731i	0.847603	+	0.530630i	$0.178045\pi$
$684$		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	0.605575	−	0.795788i	$-0.292943\pi$
$691$	3981.41	+	2298.67i	0.219190	+	0.126549i	−0.386385	+	0.922337i	$0.626276\pi$
$692$		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	$0.0658217\pi$
$701$			7621.26i			0.410629i	−0.978696	+	0.205315i	$0.934178\pi$
$702$		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	−0.745849	−	0.666115i	$-0.767956\pi$
$709$	3850.25	−	6668.83i	0.203948	−	0.353249i	0.949797	+	0.312867i	$0.101289\pi$
$710$		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	−0.966755	−	0.255703i	$-0.917693\pi$
$719$	−5049.89	+	8746.67i	−0.261932	+	0.453680i	0.704823	+	0.709383i	$0.251026\pi$
$720$		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	$-0.790850\pi$
$727$		−	23945.7i		−	1.22159i	0.791789	−	0.610795i	$-0.209150\pi$
$728$		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	0.880052	−	0.474878i	$-0.157508\pi$
$733$	18035.8	+	10413.0i	0.908822	+	0.524708i	0.0287698	+	0.999586i	$0.490841\pi$
$734$		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	0.610446	−	0.792058i	$-0.290990\pi$
$739$	−7648.41	−	13247.4i	−0.380719	−	0.659424i	−0.991165	+	0.132633i	$0.957657\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			27249.8i			1.34549i	0.739875	+	0.672744i	$0.234885\pi$
$743$			27249.8i			1.34549i	−0.739875	+	0.672744i	$0.765115\pi$
$744$		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	0.256134	−	0.966641i	$-0.417551\pi$
$751$	19864.5	−	34406.4i	0.965203	−	1.67178i	0.709069	−	0.705139i	$-0.249116\pi$
$752$		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	−0.999032	−	0.0440000i	$-0.985990\pi$
$761$	−9686.45	+	16777.4i	−0.461411	+	0.799187i	0.537621	+	0.843187i	$0.319323\pi$
$762$		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	$0.385866\pi$
$769$			39937.6i			1.87280i	−0.350930	+	0.936402i	$0.614134\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8058.12	+	13957.1i	0.374942	+	0.649419i	0.990318	−	0.138815i	$-0.0443292\pi$
$773$	8058.12	+	13957.1i	0.374942	+	0.649419i	−0.615376	+	0.788234i	$0.710996\pi$
$774$		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	−0.864813	−	0.502094i	$-0.832563\pi$
$787$	−19040.0	+	10992.8i	−0.862394	+	0.497903i	0.00241936	+	0.999997i	$0.499230\pi$
$788$		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	−0.752638	−	0.658434i	$-0.771219\pi$
$809$	−12856.7	+	7422.83i	−0.558737	+	0.322587i	0.193901	+	0.981021i	$0.437886\pi$
$810$		0			0
$811$		−	41094.7i		−	1.77932i	−0.456621	−	0.889661i	$-0.650941\pi$
$811$		−	41094.7i		−	1.77932i	0.456621	−	0.889661i	$-0.349059\pi$
$812$		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	−0.322505	−	0.946568i	$-0.604525\pi$
$821$	−30664.0	−	17703.9i	−1.30351	−	0.752582i	−0.981004	+	0.193986i	$0.937858\pi$
$822$		0			0
$823$	−2744.14	−	4752.98i	−0.116227	−	0.201311i	0.802043	−	0.597267i	$-0.203747\pi$
$823$	−2744.14	−	4752.98i	−0.116227	−	0.201311i	−0.918269	+	0.395956i	$0.870413\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	96.9958i		−	0.00407844i	−0.999998	−	0.00203922i	$-0.999351\pi$
$827$		−	96.9958i		−	0.00407844i	0.999998	−	0.00203922i	$-0.000649105\pi$
$828$		0			0
$829$	−9914.41	+	5724.09i	−0.415370	+	0.239814i	−0.693094	−	0.720847i	$-0.743753\pi$
$829$	−9914.41	+	5724.09i	−0.415370	+	0.239814i	0.277724	+	0.960661i	$0.410420\pi$
$830$		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	$0.418363\pi$
$853$			48196.0i			1.93458i	−0.253668	+	0.967291i	$0.581637\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9301.67	+	16111.0i	0.370757	+	0.642171i	0.989682	−	0.143280i	$-0.0457648\pi$
$857$	9301.67	+	16111.0i	0.370757	+	0.642171i	−0.618925	+	0.785450i	$0.712431\pi$
$858$		0			0
$859$	−7453.04	−	4303.02i	−0.296036	−	0.170916i	0.344625	−	0.938740i	$-0.388006\pi$
$859$	−7453.04	−	4303.02i	−0.296036	−	0.170916i	−0.640661	+	0.767824i	$0.721339\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	37188.6	+	21470.8i	1.46687	+	0.846901i	0.999313	−	0.0370598i	$-0.0117992\pi$
$863$	37188.6	+	21470.8i	1.46687	+	0.846901i	0.467562	+	0.883960i	$0.345133\pi$
$864$		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	0.183659	+	0.982990i	$0.441206\pi$
$877$	−19724.6	+	34163.9i	−0.759465	+	1.31543i	−0.943124	+	0.332442i	$0.892127\pi$
$878$		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	0.0413598	−	0.999144i	$-0.486831\pi$
$887$	23404.6	−	40538.0i	0.885964	−	1.53454i	0.844604	−	0.535391i	$-0.179836\pi$
$888$		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	0.999923	+	0.0124304i	$0.00395683\pi$
$907$	13950.8	+	24163.5i	0.510726	+	0.884604i	−0.489196	+	0.872174i	$0.662710\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			22732.2i			0.826732i	0.910565	+	0.413366i	$0.135647\pi$
$911$			22732.2i			0.826732i	−0.910565	+	0.413366i	$0.864353\pi$
$912$		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	−0.403546	−	0.914959i	$-0.632223\pi$
$919$	16454.0	−	28499.1i	0.590605	−	1.02296i	0.994151	−	0.107998i	$-0.0344442\pi$
$920$		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	−0.865497	−	0.500914i	$-0.832997\pi$
$929$	29.8811	−	51.7556i	0.00105529	−	0.00182782i	0.866553	+	0.499086i	$0.166331\pi$
$930$		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	$-0.727024\pi$
$937$		−	43382.1i		−	1.51252i	0.654270	−	0.756261i	$-0.272976\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22210.7	−	38470.0i	−0.769444	−	1.33272i	−0.937865	−	0.347001i	$-0.887200\pi$
$941$	−22210.7	−	38470.0i	−0.769444	−	1.33272i	0.168421	−	0.985715i	$-0.446133\pi$
$942$		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	−0.0439922	−	0.999032i	$-0.514008\pi$
$947$	−27136.7	−	15667.4i	−0.931175	−	0.537614i	−0.887183	+	0.461418i	$0.847341\pi$
$948$		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	$0.113304\pi$
$953$			20504.7i			0.696971i	−0.937314	+	0.348486i	$0.886696\pi$
$954$		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	−0.970442	−	0.241333i	$-0.922415\pi$
$971$	−8357.65	+	14475.9i	−0.276220	+	0.478427i	0.694222	+	0.719761i	$0.255749\pi$
$972$		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	−0.0138678	−	0.999904i	$-0.504414\pi$
$977$	25809.0	−	14900.8i	0.845140	−	0.487942i	0.859008	+	0.511962i	$0.171081\pi$
$978$		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	−0.983792	−	0.179314i	$-0.942612\pi$
$983$	−19946.2	−	34547.8i	−0.647186	−	1.12096i	0.336605	−	0.941646i	$-0.390721\pi$
$984$		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	0.950069	+	0.312039i	$0.101012\pi$
$991$	23250.0	+	40270.2i	0.745268	+	1.29084i	−0.204801	+	0.978804i	$0.565655\pi$
$992$		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	0.0900114	−	0.995941i	$-0.471310\pi$
$997$	31402.7	−	18130.4i	0.997527	−	0.575923i	0.907516	+	0.420018i	$0.137976\pi$
$998$		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.89.8	yes	48
3.2	odd	2	inner	504.4.bl.a.89.17	yes	48
4.3	odd	2		1008.4.bt.d.593.8		48
7.3	odd	6	inner	504.4.bl.a.17.17	yes	48
12.11	even	2		1008.4.bt.d.593.17		48
21.17	even	6	inner	504.4.bl.a.17.8	✓	48
28.3	even	6		1008.4.bt.d.17.17		48
84.59	odd	6		1008.4.bt.d.17.8		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.8	✓	48	21.17	even	6	inner
504.4.bl.a.17.17	yes	48	7.3	odd	6	inner
504.4.bl.a.89.8	yes	48	1.1	even	1	trivial
504.4.bl.a.89.17	yes	48	3.2	odd	2	inner
1008.4.bt.d.17.8		48	84.59	odd	6
1008.4.bt.d.17.17		48	28.3	even	6
1008.4.bt.d.593.8		48	4.3	odd	2
1008.4.bt.d.593.17		48	12.11	even	2

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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more