LMFDB - Embedded newform 1008.4.bt.d.593.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(17,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.bt (of order $6$, degree $2$, not 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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.17
Character	$\chi$	$=$	1008.593
Dual form			1008.4.bt.d.17.17

$q$-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 $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	3.34783	−	5.79860i	0.299439	−	0.518643i	−0.676569	−	0.736379i	$-0.736534\pi$
$5$	3.34783	−	5.79860i	0.299439	−	0.518643i	0.976008	+	0.217736i	$0.0698673\pi$
$6$		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	0.954361	−	0.298656i	$-0.0965383\pi$
$17$	15.3179	+	26.5313i	0.218537	+	0.378517i	−0.735824	+	0.677173i	$0.763205\pi$
$18$		0			0
$19$	21.8820	+	12.6336i	0.264215	+	0.152544i	0.626256	−	0.779618i	$-0.284587\pi$
$19$	21.8820	+	12.6336i	0.264215	+	0.152544i	−0.362041	+	0.932162i	$0.617920\pi$
$20$		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	$0.114198\pi$
$29$			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	$-0.960709\pi$
$31$	−238.527	+	137.714i	−1.38196	+	0.797875i	−0.389568	+	0.920998i	$0.627376\pi$
$32$		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	$-0.385485\pi$
$41$	184.846			0.704100			0.352050	+	0.935981i	$0.385485\pi$
$42$		0			0
$43$	−364.766			−1.29363			−0.646817	−	0.762645i	$-0.723900\pi$
$43$	−364.766			−1.29363			−0.646817	+	0.762645i	$0.723900\pi$
$44$		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	−0.992222	−	0.124483i	$-0.960273\pi$
$53$	−532.671	+	307.538i	−1.38053	+	0.797048i	−0.388306	+	0.921531i	$0.626940\pi$
$54$		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	0.706777	−	0.707436i	$-0.250148\pi$
$67$	−142.188	−	246.277i	−0.259269	−	0.449068i	−0.966046	+	0.258369i	$0.916815\pi$
$68$		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	−0.978822	−	0.204713i	$-0.934374\pi$
$79$	−219.163	+	379.602i	−0.312124	+	0.540615i	0.666698	+	0.745328i	$0.267707\pi$
$80$		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	$-0.972941\pi$
$89$	−356.559	+	617.579i	−0.424665	+	0.735542i	0.571724	+	0.820446i	$0.306275\pi$
$90$		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	$0.222069\pi$
$101$	953.656	+	1651.78i	0.939527	+	1.62731i	0.173173	+	0.984891i	$0.444598\pi$
$102$		0			0
$103$	−1177.37	−	679.756i	−1.12631	−	0.650276i	−0.183306	−	0.983056i	$-0.558680\pi$
$103$	−1177.37	−	679.756i	−1.12631	−	0.650276i	−0.943004	+	0.332780i	$0.892013\pi$
$104$		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	$0.0155666\pi$
$113$			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	$-0.442878\pi$
$127$	510.925			0.356986			0.178493	+	0.983941i	$0.442878\pi$
$128$		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	−0.208033	−	0.978122i	$-0.566706\pi$
$137$	857.943	−	495.334i	0.535029	−	0.308899i	0.743062	+	0.669223i	$0.233373\pi$
$138$		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	$-0.224661\pi$
$149$	1054.73	+	608.947i	0.579910	+	0.334811i	−0.181188	+	0.983449i	$0.557994\pi$
$150$		0			0
$151$	764.206	+	1323.64i	0.411856	+	0.713355i	0.995093	−	0.0989470i	$-0.0315474\pi$
$151$	764.206	+	1323.64i	0.411856	+	0.713355i	−0.583237	+	0.812302i	$0.698214\pi$
$152$		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	−0.993113	−	0.117157i	$-0.962622\pi$
$163$	−822.212	+	1424.11i	−0.395096	+	0.684326i	0.598018	+	0.801483i	$0.295955\pi$
$164$		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	$-0.824364\pi$
$173$	64.1081	−	111.038i	0.0281737	−	0.0487983i	0.879768	+	0.475402i	$0.157698\pi$
$174$		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	$0.178317\pi$
$197$			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	$-0.467299\pi$
$199$	2850.16	−	1645.54i	1.01529	−	0.586178i	0.912736	+	0.408550i	$0.133965\pi$
$200$		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	$-0.440815\pi$
$211$	1133.21			0.369733			0.184867	+	0.982764i	$0.440815\pi$
$212$		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	0.767760	−	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	0.928629	−	0.371011i	$-0.120989\pi$
$233$	3811.38	+	2200.50i	1.07164	+	0.618710i	0.143009	+	0.989721i	$0.454322\pi$
$234$		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	$-0.656691\pi$
$257$	2170.80	−	3759.94i	0.526891	−	0.912602i	0.999509	−	0.0313343i	$-0.00997565\pi$
$258$		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	$-0.761117\pi$
$269$	−4219.12	−	7307.74i	−0.956299	−	1.65636i	−0.224933	−	0.974374i	$-0.572216\pi$
$270$		0			0
$271$	−4544.68	−	2623.87i	−1.01871	−	0.588150i	−0.104977	−	0.994475i	$-0.533477\pi$
$271$	−4544.68	−	2623.87i	−1.01871	−	0.588150i	−0.913729	+	0.406324i	$0.866810\pi$
$272$		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	$-0.812076\pi$
$281$		−	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	$-0.500430\pi$
$283$	4113.33	−	2374.83i	0.864000	−	0.498830i	0.865350	+	0.501169i	$0.167096\pi$
$284$		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	$-0.616753\pi$
$293$	−3597.22			−0.717242			−0.358621	+	0.933483i	$0.616753\pi$
$294$		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	0.803745	−	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	0.992376	−	0.123245i	$-0.0393301\pi$
$317$	7799.09	+	4502.81i	1.38183	+	0.797801i	0.389455	+	0.921046i	$0.372663\pi$
$318$		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	$-0.896876\pi$
$331$	−1194.18	+	2068.38i	−0.198302	+	0.343470i	0.749676	+	0.661805i	$0.230209\pi$
$332$		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	0.835396	−	0.549648i	$-0.185238\pi$
$353$	−386.737	−	669.848i	−0.0583114	−	0.100998i	−0.893708	+	0.448650i	$0.851905\pi$
$354$		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	−0.115038	−	0.993361i	$-0.536699\pi$
$367$	4835.15	−	2791.58i	0.687719	−	0.397055i	0.802757	+	0.596306i	$0.203366\pi$
$368$		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	$-0.475847\pi$
$379$	1118.64			0.151611			0.0758057	+	0.997123i	$0.475847\pi$
$380$		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	−0.930867	−	0.365359i	$-0.880946\pi$
$389$	−8285.22	+	4783.47i	−1.07989	+	0.623475i	−0.149023	+	0.988834i	$0.547613\pi$
$390$		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	−0.320735	−	0.947169i	$-0.603930\pi$
$401$	−10450.1	−	6033.35i	−1.30137	−	0.751349i	−0.980640	+	0.195820i	$0.937263\pi$
$402$		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	−0.237080	−	0.971490i	$-0.576190\pi$
$439$	−11009.7	−	6356.44i	−1.19695	−	0.691062i	−0.959875	+	0.280428i	$0.909524\pi$
$440$		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	$-0.806170\pi$
$449$		−	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	0.186295	−	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	$-0.424410\pi$
$461$	4657.01			0.470496			0.235248	+	0.971935i	$0.424410\pi$
$462$		0			0
$463$	5088.08			0.510720			0.255360	−	0.966846i	$-0.417806\pi$
$463$	5088.08			0.510720			0.255360	+	0.966846i	$0.417806\pi$
$464$		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	$-0.973692\pi$
$487$	−6123.60	−	10606.4i	−0.569788	−	0.986903i	0.426798	−	0.904347i	$-0.359642\pi$
$488$		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	$0.492471\pi$
$499$	−9518.91	+	16487.2i	−0.853958	+	1.47910i	−0.877609	+	0.479378i	$0.840862\pi$
$500$		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	$-0.955342\pi$
$509$	−4294.66	+	7438.57i	−0.373983	+	0.647758i	0.616191	+	0.787597i	$0.288675\pi$
$510$		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	$-0.766810\pi$
$521$	−11308.3	−	19586.6i	−0.950917	−	1.64704i	−0.207471	−	0.978241i	$-0.566523\pi$
$522$		0			0
$523$	−122.551	−	70.7547i	−0.0102462	−	0.00591566i	0.494868	−	0.868968i	$-0.335216\pi$
$523$	−122.551	−	70.7547i	−0.0102462	−	0.00591566i	−0.505114	+	0.863052i	$0.668550\pi$
$524$		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	$-0.653819\pi$
$547$	−11888.7			−0.929294			−0.464647	+	0.885496i	$0.653819\pi$
$548$		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	$-0.966403\pi$
$557$	−18409.4	+	10628.7i	−1.40042	+	0.808530i	−0.405980	+	0.913882i	$0.633070\pi$
$558$		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	0.932394	−	0.361445i	$-0.117716\pi$
$569$	14734.2	+	8506.79i	1.08557	+	0.626754i	0.153176	+	0.988199i	$0.451050\pi$
$570$		0			0
$571$	1134.69	+	1965.34i	0.0831617	+	0.144040i	0.904606	−	0.426248i	$-0.140165\pi$
$571$	1134.69	+	1965.34i	0.0831617	+	0.144040i	−0.821445	+	0.570288i	$0.806832\pi$
$572$		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	−0.759823	−	0.650131i	$-0.774714\pi$
$593$	2644.32	−	4580.09i	0.183118	−	0.317170i	0.942941	+	0.332960i	$0.108048\pi$
$594$		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	0.996320	−	0.0857094i	$-0.0273157\pi$
$607$	21239.7	+	12262.8i	1.42025	+	0.819984i	0.423934	+	0.905693i	$0.360649\pi$
$608$		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	$-0.716149\pi$
$617$		−	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	$-0.482060\pi$
$619$	14617.4	−	8439.34i	0.949147	−	0.547990i	0.892816	+	0.450422i	$0.148726\pi$
$620$		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	$-0.287665\pi$
$631$	19613.0			1.23737			0.618686	+	0.785638i	$0.287665\pi$
$632$		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	$-0.581427\pi$
$641$	7437.62	−	4294.11i	0.458297	−	0.264598i	0.711328	+	0.702860i	$0.248094\pi$
$642$		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	0.955607	−	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	0.590681	−	0.806905i	$-0.298859\pi$
$653$	3124.09	+	1803.69i	0.187221	+	0.108092i	−0.403460	+	0.914997i	$0.632193\pi$
$654$		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	−0.997047	−	0.0767925i	$-0.975532\pi$
$677$	−7610.03	+	13181.0i	−0.432019	+	0.748279i	0.565028	+	0.825072i	$0.308865\pi$
$678$		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	0.386385	−	0.922337i	$-0.373724\pi$
$691$	−3981.41	−	2298.67i	−0.219190	−	0.126549i	−0.605575	+	0.795788i	$0.707057\pi$
$692$		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	−0.745849	−	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	$0.209150\pi$
$727$			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	0.880052	−	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	0.991165	−	0.132633i	$-0.0423433\pi$
$739$	7648.41	+	13247.4i	0.380719	+	0.659424i	−0.610446	+	0.792058i	$0.709010\pi$
$740$		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	$0.582449\pi$
$751$	−19864.5	+	34406.4i	−0.965203	+	1.67178i	−0.709069	+	0.705139i	$0.750884\pi$
$752$		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	$-0.680677\pi$
$761$	9686.45	−	16777.4i	0.461411	−	0.799187i	0.999032	+	0.0440000i	$0.0140102\pi$
$762$		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	0.615376	−	0.788234i	$-0.289004\pi$
$773$	−8058.12	−	13957.1i	−0.374942	−	0.649419i	−0.990318	+	0.138815i	$0.955671\pi$
$774$		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	$-0.500770\pi$
$787$	19040.0	−	10992.8i	0.862394	−	0.497903i	0.864813	+	0.502094i	$0.167437\pi$
$788$		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	$-0.483785\pi$
$797$	2291.42			0.101840			0.0509198	+	0.998703i	$0.483785\pi$
$798$		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	$-0.562114\pi$
$809$	12856.7	−	7422.83i	0.558737	−	0.322587i	0.752638	+	0.658434i	$0.228781\pi$
$810$		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	$-0.0621417\pi$
$821$	30664.0	+	17703.9i	1.30351	+	0.752582i	0.322505	+	0.946568i	$0.395475\pi$
$822$		0			0
$823$	2744.14	+	4752.98i	0.116227	+	0.201311i	0.918269	−	0.395956i	$-0.129587\pi$
$823$	2744.14	+	4752.98i	0.116227	+	0.201311i	−0.802043	+	0.597267i	$0.796253\pi$
$824$		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	0.618925	−	0.785450i	$-0.287569\pi$
$857$	−9301.67	−	16111.0i	−0.370757	−	0.642171i	−0.989682	+	0.143280i	$0.954235\pi$
$858$		0			0
$859$	7453.04	+	4303.02i	0.296036	+	0.170916i	0.640661	−	0.767824i	$-0.278661\pi$
$859$	7453.04	+	4303.02i	0.296036	+	0.170916i	−0.344625	+	0.938740i	$0.611994\pi$
$860$		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	$-0.292821\pi$
$881$	31686.9			1.21176			0.605879	+	0.795556i	$0.292821\pi$
$882$		0			0
$883$	−21170.2			−0.806832			−0.403416	−	0.915017i	$-0.632177\pi$
$883$	−21170.2			−0.806832			−0.403416	+	0.915017i	$0.632177\pi$
$884$		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	$-0.996043\pi$
$907$	−13950.8	−	24163.5i	−0.510726	−	0.884604i	0.489196	−	0.872174i	$-0.337290\pi$
$908$		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	$0.367777\pi$
$919$	−16454.0	+	28499.1i	−0.590605	+	1.02296i	−0.994151	+	0.107998i	$0.965556\pi$
$920$		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	$-0.833669\pi$
$929$	−29.8811	+	51.7556i	−0.00105529	+	0.00182782i	0.865497	+	0.500914i	$0.167003\pi$
$930$		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	$0.112800\pi$
$941$	22210.7	+	38470.0i	0.769444	+	1.33272i	−0.168421	+	0.985715i	$0.553867\pi$
$942$		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	$-0.886696\pi$
$953$		−	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	$-0.486558\pi$
$967$	2538.92			0.0844325			0.0422162	+	0.999108i	$0.486558\pi$
$968$		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	−0.859008	−	0.511962i	$-0.828919\pi$
$977$	−25809.0	+	14900.8i	−0.845140	+	0.487942i	0.0138678	+	0.999904i	$0.495586\pi$
$978$		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	$-0.898988\pi$
$991$	−23250.0	−	40270.2i	−0.745268	−	1.29084i	0.204801	−	0.978804i	$-0.434345\pi$
$992$		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	1008.4.bt.d.593.17		48
3.2	odd	2	inner	1008.4.bt.d.593.8		48
4.3	odd	2		504.4.bl.a.89.17	yes	48
7.3	odd	6	inner	1008.4.bt.d.17.8		48
12.11	even	2		504.4.bl.a.89.8	yes	48
21.17	even	6	inner	1008.4.bt.d.17.17		48
28.3	even	6		504.4.bl.a.17.8	✓	48
84.59	odd	6		504.4.bl.a.17.17	yes	48

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

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