LMFDB - Embedded newform 648.5.m.g.593.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,5,Mod(377,648)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("648.377"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	648.m (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,-96] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$66.9837360783$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.4
Character	$\chi$	$=$	648.593
Dual form			648.5.m.g.377.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-18.7121 + 10.8034i) q^{5} +(-34.8578 + 60.3755i) q^{7} +(88.1176 + 50.8747i) q^{11} +(66.1822 + 114.631i) q^{13} -18.2471i q^{17} +715.640 q^{19} +(8.63567 - 4.98581i) q^{23} +(-79.0718 + 136.956i) q^{25} +(1273.16 + 735.059i) q^{29} +(-5.31449 - 9.20497i) q^{31} -1506.33i q^{35} -1588.06 q^{37} +(1127.97 - 651.236i) q^{41} +(-708.273 + 1226.77i) q^{43} +(-1065.38 - 615.098i) q^{47} +(-1229.63 - 2129.78i) q^{49} +4136.29i q^{53} -2198.49 q^{55} +(-1912.75 + 1104.33i) q^{59} +(-372.348 + 644.925i) q^{61} +(-2476.82 - 1429.99i) q^{65} +(274.153 + 474.847i) q^{67} -4094.86i q^{71} -7716.83 q^{73} +(-6143.17 + 3546.76i) q^{77} +(-1933.80 + 3349.43i) q^{79} +(5937.62 + 3428.09i) q^{83} +(197.131 + 341.440i) q^{85} -14349.6i q^{89} -9227.87 q^{91} +(-13391.1 + 7731.37i) q^{95} +(-4966.42 + 8602.09i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 96 q^{7} + 72 q^{13} + 672 q^{19} - 84 q^{25} + 1536 q^{31} - 984 q^{37} - 10128 q^{43} + 6828 q^{49} - 27936 q^{55} - 26268 q^{61} + 20784 q^{67} - 19968 q^{73} - 44592 q^{79} + 45348 q^{85} - 23616 q^{91}+ \cdots - 76608 q^{97}+O(q^{100}) $ 24 * q - 96 * q^7 + 72 * q^13 + 672 * q^19 - 84 * q^25 + 1536 * q^31 - 984 * q^37 - 10128 * q^43 + 6828 * q^49 - 27936 * q^55 - 26268 * q^61 + 20784 * q^67 - 19968 * q^73 - 44592 * q^79 + 45348 * q^85 - 23616 * q^91 - 76608 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	−18.7121	+	10.8034i	−0.748484	+	0.432137i	−0.825146	−	0.564920i	$-0.808907\pi$
$5$	−18.7121	+	10.8034i	−0.748484	+	0.432137i	0.0766622	+	0.997057i	$0.475574\pi$
$6$		0			0
$7$	−34.8578	+	60.3755i	−0.711384	+	1.23215i	0.252954	+	0.967478i	$0.418598\pi$
$7$	−34.8578	+	60.3755i	−0.711384	+	1.23215i	−0.964338	+	0.264674i	$0.914736\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	88.1176	+	50.8747i	0.728245	+	0.420452i	0.817780	−	0.575531i	$-0.195205\pi$
$11$	88.1176	+	50.8747i	0.728245	+	0.420452i	−0.0895348	+	0.995984i	$0.528538\pi$
$12$		0			0
$13$	66.1822	+	114.631i	0.391611	+	0.678290i	0.992662	−	0.120921i	$-0.0385846\pi$
$13$	66.1822	+	114.631i	0.391611	+	0.678290i	−0.601051	+	0.799210i	$0.705251\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	18.2471i		−	0.0631386i	−0.999502	−	0.0315693i	$-0.989950\pi$
$17$		−	18.2471i		−	0.0631386i	0.999502	−	0.0315693i	$-0.0100505\pi$
$18$		0			0
$19$	715.640			1.98238			0.991191	−	0.132437i	$-0.0422802\pi$
$19$	715.640			1.98238			0.991191	+	0.132437i	$0.0422802\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8.63567	−	4.98581i	0.0163245	−	0.00942496i	−0.491816	−	0.870699i	$-0.663667\pi$
$23$	8.63567	−	4.98581i	0.0163245	−	0.00942496i	0.508140	+	0.861274i	$0.330333\pi$
$24$		0			0
$25$	−79.0718	+	136.956i	−0.126515	+	0.219130i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1273.16	+	735.059i	1.51386	+	0.874030i	0.999868	+	0.0162377i	$0.00516883\pi$
$29$	1273.16	+	735.059i	1.51386	+	0.874030i	0.513996	+	0.857792i	$0.328165\pi$
$30$		0			0
$31$	−5.31449	−	9.20497i	−0.00553017	−	0.00957853i	0.863247	−	0.504782i	$-0.168427\pi$
$31$	−5.31449	−	9.20497i	−0.00553017	−	0.00957853i	−0.868777	+	0.495203i	$0.835094\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1506.33i		−	1.22966i
$36$		0			0
$37$	−1588.06			−1.16002			−0.580009	−	0.814610i	$-0.696951\pi$
$37$	−1588.06			−1.16002			−0.580009	+	0.814610i	$0.696951\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1127.97	−	651.236i	0.671013	−	0.387410i	−0.125447	−	0.992100i	$-0.540037\pi$
$41$	1127.97	−	651.236i	0.671013	−	0.387410i	0.796460	+	0.604691i	$0.206703\pi$
$42$		0			0
$43$	−708.273	+	1226.77i	−0.383058	+	0.663475i	−0.991498	−	0.130125i	$-0.958462\pi$
$43$	−708.273	+	1226.77i	−0.383058	+	0.663475i	0.608440	+	0.793600i	$0.291796\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1065.38	−	615.098i	−0.482291	−	0.278451i	0.239080	−	0.971000i	$-0.423154\pi$
$47$	−1065.38	−	615.098i	−0.482291	−	0.278451i	−0.721371	+	0.692549i	$0.756488\pi$
$48$		0			0
$49$	−1229.63	−	2129.78i	−0.512133	−	0.887040i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4136.29i			1.47251i	0.676703	+	0.736256i	$0.263408\pi$
$53$			4136.29i			1.47251i	−0.676703	+	0.736256i	$0.736592\pi$
$54$		0			0
$55$	−2198.49			−0.726772
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1912.75	+	1104.33i	−0.549484	+	0.317245i	−0.748914	−	0.662667i	$-0.769424\pi$
$59$	−1912.75	+	1104.33i	−0.549484	+	0.317245i	0.199430	+	0.979912i	$0.436091\pi$
$60$		0			0
$61$	−372.348	+	644.925i	−0.100067	+	0.173320i	−0.911712	−	0.410830i	$-0.865239\pi$
$61$	−372.348	+	644.925i	−0.100067	+	0.173320i	0.811645	+	0.584151i	$0.198572\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2476.82	−	1429.99i	−0.586229	−	0.338459i
$66$		0			0
$67$	274.153	+	474.847i	0.0610722	+	0.105780i	0.894945	−	0.446177i	$-0.147215\pi$
$67$	274.153	+	474.847i	0.0610722	+	0.105780i	−0.833873	+	0.551957i	$0.813881\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4094.86i		−	0.812311i	−0.913804	−	0.406155i	$-0.866869\pi$
$71$		−	4094.86i		−	0.812311i	0.913804	−	0.406155i	$-0.133131\pi$
$72$		0			0
$73$	−7716.83			−1.44808			−0.724042	−	0.689756i	$-0.757718\pi$
$73$	−7716.83			−1.44808			−0.724042	+	0.689756i	$0.757718\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6143.17	+	3546.76i	−1.03612	+	0.598206i
$78$		0			0
$79$	−1933.80	+	3349.43i	−0.309853	+	0.536682i	−0.978330	−	0.207051i	$-0.933613\pi$
$79$	−1933.80	+	3349.43i	−0.309853	+	0.536682i	0.668477	+	0.743733i	$0.266947\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5937.62	+	3428.09i	0.861899	+	0.497618i	0.864648	−	0.502378i	$-0.167542\pi$
$83$	5937.62	+	3428.09i	0.861899	+	0.497618i	−0.00274849	+	0.999996i	$0.500875\pi$
$84$		0			0
$85$	197.131	+	341.440i	0.0272845	+	0.0472582i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14349.6i		−	1.81159i	−0.423712	−	0.905797i	$-0.639273\pi$
$89$		−	14349.6i		−	1.81159i	0.423712	−	0.905797i	$-0.360727\pi$
$90$		0			0
$91$	−9227.87			−1.11434
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13391.1	+	7731.37i	−1.48378	+	0.856661i
$96$		0			0
$97$	−4966.42	+	8602.09i	−0.527837	+	0.914241i	0.471636	+	0.881793i	$0.343663\pi$
$97$	−4966.42	+	8602.09i	−0.527837	+	0.914241i	−0.999473	+	0.0324474i	$0.989670\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8669.57	+	5005.38i	0.849875	+	0.490675i	0.860609	−	0.509267i	$-0.170083\pi$
$101$	8669.57	+	5005.38i	0.849875	+	0.490675i	−0.0107339	+	0.999942i	$0.503417\pi$
$102$		0			0
$103$	7951.76	+	13772.8i	0.749529	+	1.29822i	0.948048	+	0.318126i	$0.103053\pi$
$103$	7951.76	+	13772.8i	0.749529	+	1.29822i	−0.198519	+	0.980097i	$0.563613\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	480.920i		−	0.0420054i	−0.999779	−	0.0210027i	$-0.993314\pi$
$107$		−	480.920i		−	0.0420054i	0.999779	−	0.0210027i	$-0.00668586\pi$
$108$		0			0
$109$	8398.92			0.706920			0.353460	−	0.935450i	$-0.385005\pi$
$109$	8398.92			0.706920			0.353460	+	0.935450i	$0.385005\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	11630.6	−	6714.94i	0.910848	−	0.525878i	0.0301440	−	0.999546i	$-0.490403\pi$
$113$	11630.6	−	6714.94i	0.910848	−	0.525878i	0.880704	+	0.473667i	$0.157070\pi$
$114$		0			0
$115$	−107.728	+	186.590i	−0.00814575	+	0.0141089i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1101.67	+	636.052i	0.0777964	+	0.0449158i
$120$		0			0
$121$	−2144.02	−	3713.55i	−0.146440	−	0.253641i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	16921.3i		−	1.08296i
$126$		0			0
$127$	−25615.2			−1.58815			−0.794073	−	0.607823i	$-0.792043\pi$
$127$	−25615.2			−1.58815			−0.794073	+	0.607823i	$0.792043\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7221.60	+	4169.39i	−0.420815	+	0.242958i	−0.695426	−	0.718598i	$-0.744784\pi$
$131$	−7221.60	+	4169.39i	−0.420815	+	0.242958i	0.274611	+	0.961555i	$0.411451\pi$
$132$		0			0
$133$	−24945.6	+	43207.1i	−1.41023	+	2.44260i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	29544.0	+	17057.2i	1.57408	+	0.908797i	0.995660	+	0.0930602i	$0.0296649\pi$
$137$	29544.0	+	17057.2i	1.57408	+	0.908797i	0.578423	+	0.815737i	$0.303668\pi$
$138$		0			0
$139$	−13497.3	−	23378.1i	−0.698584	−	1.20998i	−0.968957	−	0.247227i	$-0.920480\pi$
$139$	−13497.3	−	23378.1i	−0.698584	−	1.20998i	0.270373	−	0.962756i	$-0.412853\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			13468.0i			0.658615i
$144$		0			0
$145$	−31764.6			−1.51080
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−24067.4	+	13895.3i	−1.08407	+	0.625888i	−0.931991	−	0.362481i	$-0.881930\pi$
$149$	−24067.4	+	13895.3i	−1.08407	+	0.625888i	−0.152078	+	0.988368i	$0.548597\pi$
$150$		0			0
$151$	−12029.2	+	20835.2i	−0.527574	+	0.913785i	0.471909	+	0.881647i	$0.343565\pi$
$151$	−12029.2	+	20835.2i	−0.527574	+	0.913785i	−0.999483	+	0.0321380i	$0.989768\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	198.890	+	114.829i	0.00827848	+	0.00477958i
$156$		0			0
$157$	−15813.0	−	27389.0i	−0.641528	−	1.11116i	−0.985092	−	0.172030i	$-0.944967\pi$
$157$	−15813.0	−	27389.0i	−0.641528	−	1.11116i	0.343563	−	0.939130i	$-0.388366\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			695.177i			0.0268191i
$162$		0			0
$163$	−39116.7			−1.47227			−0.736133	−	0.676836i	$-0.763350\pi$
$163$	−39116.7			−1.47227			−0.736133	+	0.676836i	$0.763350\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13376.4	−	7722.87i	0.479630	−	0.276915i	−0.240632	−	0.970616i	$-0.577355\pi$
$167$	13376.4	−	7722.87i	0.479630	−	0.276915i	0.720262	+	0.693702i	$0.244021\pi$
$168$		0			0
$169$	5520.32	−	9561.48i	0.193282	−	0.334774i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	16608.2	+	9588.73i	0.554919	+	0.320382i	0.751103	−	0.660184i	$-0.229522\pi$
$173$	16608.2	+	9588.73i	0.554919	+	0.320382i	−0.196185	+	0.980567i	$0.562855\pi$
$174$		0			0
$175$	−5512.54	−	9548.00i	−0.180001	−	0.311771i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	61163.7i		−	1.90892i	−0.298341	−	0.954459i	$-0.596433\pi$
$179$		−	61163.7i		−	1.90892i	0.298341	−	0.954459i	$-0.403567\pi$
$180$		0			0
$181$	12496.9			0.381455			0.190728	−	0.981643i	$-0.438915\pi$
$181$	12496.9			0.381455			0.190728	+	0.981643i	$0.438915\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	29716.0	−	17156.5i	0.868254	−	0.501286i
$186$		0			0
$187$	928.314	−	1607.89i	0.0265468	−	0.0459804i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−37742.0	−	21790.4i	−1.03457	−	0.597308i	−0.116277	−	0.993217i	$-0.537096\pi$
$191$	−37742.0	−	21790.4i	−1.03457	−	0.597308i	−0.918290	+	0.395909i	$0.870430\pi$
$192$		0			0
$193$	25164.9	+	43586.9i	0.675585	+	1.17015i	0.976297	+	0.216433i	$0.0694424\pi$
$193$	25164.9	+	43586.9i	0.675585	+	1.17015i	−0.300712	+	0.953715i	$0.597224\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			33260.4i			0.857028i	0.903535	+	0.428514i	$0.140963\pi$
$197$			33260.4i			0.857028i	−0.903535	+	0.428514i	$0.859037\pi$
$198$		0			0
$199$	−52566.2			−1.32740			−0.663698	−	0.748000i	$-0.731014\pi$
$199$	−52566.2			−1.32740			−0.663698	+	0.748000i	$0.731014\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−88759.1	+	51245.1i	−2.15388	+	1.24354i
$204$		0			0
$205$	−14071.2	+	24372.0i	−0.334828	+	0.579940i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	63060.5	+	36408.0i	1.44366	+	0.833498i
$210$		0			0
$211$	25866.2	+	44801.6i	0.580989	+	1.00630i	0.995362	+	0.0961959i	$0.0306675\pi$
$211$	25866.2	+	44801.6i	0.580989	+	1.00630i	−0.414373	+	0.910107i	$0.635999\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	30607.1i		−	0.662134i
$216$		0			0
$217$	741.006			0.0157363
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2091.68	−	1207.63i	0.0428263	−	0.0247258i
$222$		0			0
$223$	−14463.1	+	25050.7i	−0.290837	+	0.503745i	−0.974008	−	0.226514i	$-0.927267\pi$
$223$	−14463.1	+	25050.7i	−0.290837	+	0.503745i	0.683171	+	0.730259i	$0.260601\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−58698.8	−	33889.7i	−1.13914	−	0.657683i	−0.192923	−	0.981214i	$-0.561797\pi$
$227$	−58698.8	−	33889.7i	−1.13914	−	0.657683i	−0.946218	+	0.323531i	$0.895130\pi$
$228$		0			0
$229$	−11215.9	−	19426.5i	−0.213877	−	0.370445i	0.739048	−	0.673653i	$-0.235276\pi$
$229$	−11215.9	−	19426.5i	−0.213877	−	0.370445i	−0.952924	+	0.303208i	$0.901942\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	58686.1i		−	1.08099i	−0.841346	−	0.540497i	$-0.818236\pi$
$233$		−	58686.1i		−	1.08099i	0.841346	−	0.540497i	$-0.181764\pi$
$234$		0			0
$235$	26580.7			0.481316
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−65780.8	+	37978.5i	−1.15160	+	0.664879i	−0.949278	−	0.314439i	$-0.898183\pi$
$239$	−65780.8	+	37978.5i	−1.15160	+	0.664879i	−0.202326	+	0.979318i	$0.564850\pi$
$240$		0			0
$241$	−18383.2	+	31840.7i	−0.316510	+	0.548212i	−0.979757	−	0.200189i	$-0.935845\pi$
$241$	−18383.2	+	31840.7i	−0.316510	+	0.548212i	0.663247	+	0.748401i	$0.269178\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	46017.9	+	26568.5i	0.766646	+	0.442623i
$246$		0			0
$247$	47362.7	+	82034.5i	0.776323	+	1.34463i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			12072.2i			0.191620i	0.995400	+	0.0958098i	$0.0305441\pi$
$251$			12072.2i			0.191620i	−0.995400	+	0.0958098i	$0.969456\pi$
$252$		0			0
$253$	1014.61			0.0158510
$254$		0			0
$255$		0			0
$256$		0			0
$257$	37575.7	−	21694.4i	0.568907	−	0.328459i	−0.187806	−	0.982206i	$-0.560138\pi$
$257$	37575.7	−	21694.4i	0.568907	−	0.328459i	0.756713	+	0.653748i	$0.226804\pi$
$258$		0			0
$259$	55356.4	−	95880.1i	0.825217	−	1.42932i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−19004.1	−	10972.0i	−0.274749	−	0.158626i	0.356295	−	0.934374i	$-0.384040\pi$
$263$	−19004.1	−	10972.0i	−0.274749	−	0.158626i	−0.631044	+	0.775747i	$0.717373\pi$
$264$		0			0
$265$	−44686.1	−	77398.6i	−0.636327	−	1.10215i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			82701.3i			1.14290i	0.820637	+	0.571450i	$0.193619\pi$
$269$			82701.3i			1.14290i	−0.820637	+	0.571450i	$0.806381\pi$
$270$		0			0
$271$	60271.4			0.820678			0.410339	−	0.911933i	$-0.365410\pi$
$271$	60271.4			0.820678			0.410339	+	0.911933i	$0.365410\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−13935.2	+	8045.52i	−0.184268	+	0.106387i
$276$		0			0
$277$	19226.9	−	33301.9i	0.250581	−	0.434020i	−0.713105	−	0.701058i	$-0.752712\pi$
$277$	19226.9	−	33301.9i	0.250581	−	0.434020i	0.963686	+	0.267038i	$0.0860449\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−53143.3	−	30682.3i	−0.673032	−	0.388575i	0.124193	−	0.992258i	$-0.460366\pi$
$281$	−53143.3	−	30682.3i	−0.673032	−	0.388575i	−0.797224	+	0.603683i	$0.793699\pi$
$282$		0			0
$283$	−43166.3	−	74766.3i	−0.538980	−	0.933540i	−0.998959	−	0.0456107i	$-0.985477\pi$
$283$	−43166.3	−	74766.3i	−0.538980	−	0.933540i	0.459980	−	0.887929i	$-0.347857\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			90802.6i			1.10239i
$288$		0			0
$289$	83188.0			0.996014
$290$		0			0
$291$		0			0
$292$		0			0
$293$	141974.	−	81968.6i	1.65376	−	0.954800i	0.678257	−	0.734825i	$-0.262736\pi$
$293$	141974.	−	81968.6i	1.65376	−	0.954800i	0.975505	−	0.219975i	$-0.0705977\pi$
$294$		0			0
$295$	23861.1	−	41328.6i	0.274187	−	0.474905i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1143.06	+	659.943i	0.0127857	+	0.00738184i
$300$		0			0
$301$	−49377.7	−	85524.7i	−0.545002	−	0.943971i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	16090.5i		−	0.172970i
$306$		0			0
$307$	−76361.8			−0.810214			−0.405107	−	0.914269i	$-0.632766\pi$
$307$	−76361.8			−0.810214			−0.405107	+	0.914269i	$0.632766\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	102170.	−	58988.0i	1.05634	−	0.609878i	0.131922	−	0.991260i	$-0.457885\pi$
$311$	102170.	−	58988.0i	1.05634	−	0.609878i	0.924417	+	0.381382i	$0.124552\pi$
$312$		0			0
$313$	−45897.5	+	79496.8i	−0.468490	+	0.811448i	−0.999351	−	0.0360104i	$-0.988535\pi$
$313$	−45897.5	+	79496.8i	−0.468490	+	0.811448i	0.530862	+	0.847458i	$0.321868\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1813.63	−	1047.10i	−0.0180481	−	0.0104201i	0.490949	−	0.871188i	$-0.336650\pi$
$317$	−1813.63	−	1047.10i	−0.0180481	−	0.0104201i	−0.508997	+	0.860768i	$0.669984\pi$
$318$		0			0
$319$	74791.9	+	129543.i	0.734976	+	1.27302i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	13058.3i		−	0.125165i
$324$		0			0
$325$	−20932.6			−0.198178
$326$		0			0
$327$		0			0
$328$		0			0
$329$	74273.6	−	42881.9i	0.686188	−	0.396171i
$330$		0			0
$331$	50913.3	−	88184.3i	0.464702	−	0.804888i	−0.534486	−	0.845178i	$-0.679495\pi$
$331$	50913.3	−	88184.3i	0.464702	−	0.804888i	0.999188	+	0.0402895i	$0.0128280\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10260.0	−	5923.59i	−0.0914231	−	0.0527831i
$336$		0			0
$337$	−55872.6	−	96774.1i	−0.491970	−	0.852117i	0.507987	−	0.861365i	$-0.330390\pi$
$337$	−55872.6	−	96774.1i	−0.491970	−	0.852117i	−0.999957	+	0.00924745i	$0.997056\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	1081.49i		−	0.00930069i
$342$		0			0
$343$	4061.82			0.0345249
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2342.92	−	1352.68i	0.0194580	−	0.0112341i	−0.490239	−	0.871588i	$-0.663091\pi$
$347$	2342.92	−	1352.68i	0.0194580	−	0.0112341i	0.509697	+	0.860354i	$0.329757\pi$
$348$		0			0
$349$	−522.710	+	905.360i	−0.00429151	+	0.00743311i	−0.868163	−	0.496279i	$-0.834699\pi$
$349$	−522.710	+	905.360i	−0.00429151	+	0.00743311i	0.863872	+	0.503712i	$0.168033\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	36663.1	+	21167.5i	0.294225	+	0.169871i	0.639846	−	0.768503i	$-0.278998\pi$
$353$	36663.1	+	21167.5i	0.294225	+	0.169871i	−0.345620	+	0.938374i	$0.612332\pi$
$354$		0			0
$355$	44238.5	+	76623.4i	0.351030	+	0.608001i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			156787.i			1.21652i	0.793737	+	0.608261i	$0.208133\pi$
$359$			156787.i			1.21652i	−0.793737	+	0.608261i	$0.791867\pi$
$360$		0			0
$361$	381820.			2.92984
$362$		0			0
$363$		0			0
$364$		0			0
$365$	144398.	−	83368.3i	1.08387	−	0.625771i
$366$		0			0
$367$	113067.	−	195839.i	0.839470	−	1.45401i	−0.0508679	−	0.998705i	$-0.516199\pi$
$367$	113067.	−	195839.i	0.839470	−	1.45401i	0.890338	−	0.455300i	$-0.150468\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−249730.	−	144182.i	−1.81436	−	1.04752i
$372$		0			0
$373$	24680.6	+	42748.1i	0.177394	+	0.307255i	0.940987	−	0.338442i	$-0.109900\pi$
$373$	24680.6	+	42748.1i	0.177394	+	0.307255i	−0.763593	+	0.645698i	$0.776567\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			194591.i			1.36912i
$378$		0			0
$379$	241069.			1.67828			0.839138	−	0.543918i	$-0.183060\pi$
$379$	241069.			1.67828			0.839138	+	0.543918i	$0.183060\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−231357.	+	133574.i	−1.57720	+	0.910595i	−0.581948	+	0.813226i	$0.697709\pi$
$383$	−231357.	+	133574.i	−1.57720	+	0.910595i	−0.995248	+	0.0973687i	$0.968957\pi$
$384$		0			0
$385$	76634.4	−	132735.i	0.517014	−	0.895494i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	106331.	+	61390.5i	0.702688	+	0.405697i	0.808348	−	0.588705i	$-0.200362\pi$
$389$	106331.	+	61390.5i	0.702688	+	0.405697i	−0.105660	+	0.994402i	$0.533695\pi$
$390$		0			0
$391$	−90.9763	−	157.575i	−0.000595079	−	0.00103071i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	83566.5i		−	0.535597i
$396$		0			0
$397$	261605.			1.65984			0.829919	−	0.557885i	$-0.188387\pi$
$397$	261605.			1.65984			0.829919	+	0.557885i	$0.188387\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−46531.4	+	26864.9i	−0.289373	+	0.167069i	−0.637659	−	0.770319i	$-0.720097\pi$
$401$	−46531.4	+	26864.9i	−0.289373	+	0.167069i	0.348286	+	0.937388i	$0.386764\pi$
$402$		0			0
$403$	703.450	−	1218.41i	0.00433135	−	0.00750211i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−139936.	−	80792.3i	−0.844777	−	0.487732i
$408$		0			0
$409$	94754.4	+	164119.i	0.566438	+	0.981100i	0.996914	+	0.0784979i	$0.0250124\pi$
$409$	94754.4	+	164119.i	0.566438	+	0.981100i	−0.430476	+	0.902602i	$0.641654\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	153978.i		−	0.902731i
$414$		0			0
$415$	−148140.			−0.860157
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2625.15	+	1515.63i	−0.0149529	+	0.00863308i	−0.507458	−	0.861677i	$-0.669415\pi$
$419$	−2625.15	+	1515.63i	−0.0149529	+	0.00863308i	0.492505	+	0.870310i	$0.336081\pi$
$420$		0			0
$421$	115223.	−	199573.i	0.650094	−	1.12600i	−0.333005	−	0.942925i	$-0.608063\pi$
$421$	115223.	−	199573.i	0.650094	−	1.12600i	0.983100	−	0.183071i	$-0.0586039\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2499.05	+	1442.83i	0.0138356	+	0.00798797i
$426$		0			0
$427$	−25958.4	−	44961.3i	−0.142371	−	0.246594i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	72632.4i		−	0.390999i	−0.980704	−	0.195499i	$-0.937367\pi$
$431$		−	72632.4i		−	0.390999i	0.980704	−	0.195499i	$-0.0626328\pi$
$432$		0			0
$433$	−135194.			−0.721076			−0.360538	−	0.932745i	$-0.617407\pi$
$433$	−135194.			−0.721076			−0.360538	+	0.932745i	$0.617407\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6180.03	−	3568.04i	0.0323614	−	0.0186839i
$438$		0			0
$439$	−353.757	+	612.724i	−0.00183559	+	0.00317933i	−0.866942	−	0.498409i	$-0.833918\pi$
$439$	−353.757	+	612.724i	−0.00183559	+	0.00317933i	0.865106	+	0.501589i	$0.167251\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	52890.0	+	30536.1i	0.269505	+	0.155599i	0.628663	−	0.777678i	$-0.283603\pi$
$443$	52890.0	+	30536.1i	0.269505	+	0.155599i	−0.359158	+	0.933277i	$0.616936\pi$
$444$		0			0
$445$	155025.	+	268512.i	0.782857	+	1.35595i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			312118.i			1.54820i	0.633064	+	0.774099i	$0.281797\pi$
$449$			312118.i			1.54820i	−0.633064	+	0.774099i	$0.718203\pi$
$450$		0			0
$451$	132526.			0.651549
$452$		0			0
$453$		0			0
$454$		0			0
$455$	172673.	−	99692.6i	0.834067	−	0.481549i
$456$		0			0
$457$	34795.4	−	60267.3i	0.166605	−	0.288569i	−0.770619	−	0.637296i	$-0.780053\pi$
$457$	34795.4	−	60267.3i	0.166605	−	0.288569i	0.937224	+	0.348727i	$0.113386\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−223746.	−	129180.i	−1.05282	−	0.607845i	−0.129381	−	0.991595i	$-0.541299\pi$
$461$	−223746.	−	129180.i	−1.05282	−	0.607845i	−0.923437	+	0.383750i	$0.874632\pi$
$462$		0			0
$463$	−90239.5	−	156299.i	−0.420954	−	0.729114i	0.575079	−	0.818098i	$-0.304971\pi$
$463$	−90239.5	−	156299.i	−0.420954	−	0.729114i	−0.996033	+	0.0889841i	$0.971638\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			109307.i			0.501203i	0.968090	+	0.250602i	$0.0806284\pi$
$467$			109307.i			0.501203i	−0.968090	+	0.250602i	$0.919372\pi$
$468$		0			0
$469$	−38225.5			−0.173783
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−124823.	+	72066.5i	−0.557920	+	0.322115i
$474$		0			0
$475$	−56587.0	+	98011.5i	−0.250801	+	0.434400i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−319965.	−	184732.i	−1.39454	−	0.805139i	−0.400728	−	0.916197i	$-0.631243\pi$
$479$	−319965.	−	184732.i	−1.39454	−	0.805139i	−0.993814	+	0.111058i	$0.964576\pi$
$480$		0			0
$481$	−105102.	−	182041.i	−0.454275	−	0.786828i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	214617.i		−	0.912392i
$486$		0			0
$487$	88956.5			0.375076			0.187538	−	0.982257i	$-0.439949\pi$
$487$	88956.5			0.375076			0.187538	+	0.982257i	$0.439949\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	296276.	−	171055.i	1.22895	−	0.709534i	0.262139	−	0.965030i	$-0.415572\pi$
$491$	296276.	−	171055.i	1.22895	−	0.709534i	0.966810	+	0.255496i	$0.0822387\pi$
$492$		0			0
$493$	13412.7	−	23231.4i	0.0551850	−	0.0955833i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	247229.	+	142738.i	1.00089	+	0.577864i
$498$		0			0
$499$	−22936.0	−	39726.3i	−0.0921120	−	0.159543i	0.816288	−	0.577646i	$-0.196028\pi$
$499$	−22936.0	−	39726.3i	−0.0921120	−	0.159543i	−0.908400	+	0.418103i	$0.862695\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			224886.i			0.888845i	0.895817	+	0.444422i	$0.146591\pi$
$503$			224886.i			0.888845i	−0.895817	+	0.444422i	$0.853409\pi$
$504$		0			0
$505$	−216301.			−0.848156
$506$		0			0
$507$		0			0
$508$		0			0
$509$	335248.	−	193555.i	1.29399	−	0.747085i	0.314630	−	0.949214i	$-0.398120\pi$
$509$	335248.	−	193555.i	1.29399	−	0.747085i	0.979359	+	0.202130i	$0.0647863\pi$
$510$		0			0
$511$	268992.	−	465908.i	1.03014	−	1.78426i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−297588.	−	171812.i	−1.12202	−	0.647799i
$516$		0			0
$517$	−62585.9	−	108402.i	−0.234151	−	0.405561i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			303979.i			1.11987i	0.828536	+	0.559936i	$0.189174\pi$
$521$			303979.i			1.11987i	−0.828536	+	0.559936i	$0.810826\pi$
$522$		0			0
$523$	−52206.3			−0.190862			−0.0954310	−	0.995436i	$-0.530423\pi$
$523$	−52206.3			−0.190862			−0.0954310	+	0.995436i	$0.530423\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−167.964	+	96.9738i	−0.000604775	+	0.000349167i
$528$		0			0
$529$	−139871.	+	242263.i	−0.499822	+	0.865718i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	149304.	+	86200.5i	0.525552	+	0.303428i
$534$		0			0
$535$	5195.59	+	8999.02i	0.0181521	+	0.0314404i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	250229.i		−	0.861310i
$540$		0			0
$541$	166809.			0.569935			0.284968	−	0.958537i	$-0.408017\pi$
$541$	166809.			0.569935			0.284968	+	0.958537i	$0.408017\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−157161.	+	90737.1i	−0.529118	+	0.305486i
$546$		0			0
$547$	−163116.	+	282524.i	−0.545156	+	0.944238i	0.453441	+	0.891286i	$0.350196\pi$
$547$	−163116.	+	282524.i	−0.545156	+	0.944238i	−0.998597	+	0.0529517i	$0.983137\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	911124.	+	526038.i	3.00106	+	1.73266i
$552$		0			0
$553$	−134816.	−	233508.i	−0.440849	−	0.763573i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	129945.i		−	0.418840i	−0.977826	−	0.209420i	$-0.932842\pi$
$557$		−	129945.i		−	0.418840i	0.977826	−	0.209420i	$-0.0671576\pi$
$558$		0			0
$559$	−187500.			−0.600038
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−224680.	+	129719.i	−0.708838	+	0.409248i	−0.810631	−	0.585558i	$-0.800876\pi$
$563$	−224680.	+	129719.i	−0.708838	+	0.409248i	0.101792	+	0.994806i	$0.467542\pi$
$564$		0			0
$565$	−145089.	+	251301.i	−0.454503	+	0.787222i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−315152.	−	181953.i	−0.973409	−	0.561998i	−0.0731354	−	0.997322i	$-0.523301\pi$
$569$	−315152.	−	181953.i	−0.973409	−	0.561998i	−0.900274	+	0.435324i	$0.856634\pi$
$570$		0			0
$571$	−98647.2	−	170862.i	−0.302561	−	0.524051i	0.674154	−	0.738590i	$-0.264508\pi$
$571$	−98647.2	−	170862.i	−0.302561	−	0.524051i	−0.976715	+	0.214540i	$0.931175\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1576.95i			0.00476959i
$576$		0			0
$577$	186408.			0.559904			0.279952	−	0.960014i	$-0.409682\pi$
$577$	186408.			0.559904			0.279952	+	0.960014i	$0.409682\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−413945.	+	238991.i	−1.22628	+	0.707994i
$582$		0			0
$583$	−210433.	+	364480.i	−0.619121	+	1.07235i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	88130.9	+	50882.4i	0.255771	+	0.147670i	0.622404	−	0.782696i	$-0.286156\pi$
$587$	88130.9	+	50882.4i	0.255771	+	0.147670i	−0.366633	+	0.930366i	$0.619489\pi$
$588$		0			0
$589$	−3803.26	−	6587.44i	−0.0109629	−	0.0189883i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	416289.i		−	1.18382i	−0.806004	−	0.591910i	$-0.798374\pi$
$593$		−	416289.i		−	1.18382i	0.806004	−	0.591910i	$-0.201626\pi$
$594$		0			0
$595$	−27486.2			−0.0776391
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−251341.	+	145112.i	−0.700504	+	0.404436i	−0.807535	−	0.589820i	$-0.799199\pi$
$599$	−251341.	+	145112.i	−0.700504	+	0.404436i	0.107031	+	0.994256i	$0.465865\pi$
$600$		0			0
$601$	17706.9	−	30669.2i	0.0490222	−	0.0849089i	−0.840473	−	0.541853i	$-0.817723\pi$
$601$	17706.9	−	30669.2i	0.0490222	−	0.0849089i	0.889495	+	0.456944i	$0.151056\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	80238.2	+	46325.6i	0.219215	+	0.126564i
$606$		0			0
$607$	290729.	+	503558.i	0.789063	+	1.36670i	0.926542	+	0.376192i	$0.122767\pi$
$607$	290729.	+	503558.i	0.789063	+	1.36670i	−0.137479	+	0.990505i	$0.543900\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	162834.i		−	0.436177i
$612$		0			0
$613$	−560291.			−1.49105			−0.745526	−	0.666476i	$-0.767802\pi$
$613$	−560291.			−1.49105			−0.745526	+	0.666476i	$0.767802\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	284756.	−	164404.i	0.748001	−	0.431859i	−0.0769699	−	0.997033i	$-0.524525\pi$
$617$	284756.	−	164404.i	0.748001	−	0.431859i	0.824971	+	0.565175i	$0.191191\pi$
$618$		0			0
$619$	−233653.	+	404699.i	−0.609804	+	1.05621i	0.381469	+	0.924382i	$0.375418\pi$
$619$	−233653.	+	404699.i	−0.609804	+	1.05621i	−0.991272	+	0.131829i	$0.957915\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	866366.	+	500197.i	2.23216	+	1.28874i
$624$		0			0
$625$	133388.	+	231035.i	0.341473	+	0.591449i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			28977.5i			0.0732418i
$630$		0			0
$631$	355301.			0.892356			0.446178	−	0.894944i	$-0.352785\pi$
$631$	355301.			0.892356			0.446178	+	0.894944i	$0.352785\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	479314.	−	276732.i	1.18870	−	0.686297i
$636$		0			0
$637$	162759.	−	281908.i	0.401114	−	0.694749i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−593060.	−	342403.i	−1.44339	−	0.833339i	−0.445311	−	0.895376i	$-0.646907\pi$
$641$	−593060.	−	342403.i	−1.44339	−	0.833339i	−0.998074	+	0.0620370i	$0.980240\pi$
$642$		0			0
$643$	178312.	+	308846.i	0.431280	+	0.746999i	0.996984	−	0.0776095i	$-0.0247287\pi$
$643$	178312.	+	308846.i	0.431280	+	0.746999i	−0.565704	+	0.824609i	$0.691395\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			192531.i			0.459930i	0.973199	+	0.229965i	$0.0738612\pi$
$647$			192531.i			0.459930i	−0.973199	+	0.229965i	$0.926139\pi$
$648$		0			0
$649$	−224730.			−0.533545
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−526760.	+	304125.i	−1.23534	+	0.713224i	−0.968138	−	0.250417i	$-0.919432\pi$
$653$	−526760.	+	304125.i	−1.23534	+	0.713224i	−0.267202	+	0.963641i	$0.586099\pi$
$654$		0			0
$655$	90087.5	−	156036.i	0.209982	−	0.363699i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−90891.9	−	52476.5i	−0.209293	−	0.120835i	0.391690	−	0.920097i	$-0.371891\pi$
$659$	−90891.9	−	52476.5i	−0.209293	−	0.120835i	−0.600983	+	0.799262i	$0.705224\pi$
$660$		0			0
$661$	203481.	+	352440.i	0.465716	+	0.806644i	0.999234	−	0.0391453i	$-0.0124635\pi$
$661$	203481.	+	352440.i	0.465716	+	0.806644i	−0.533518	+	0.845789i	$0.679130\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.07799e6i		−	2.43766i
$666$		0			0
$667$	14659.4			0.0329508
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−65620.8	+	37886.2i	−0.145746	+	0.0841464i
$672$		0			0
$673$	−81994.9	+	142019.i	−0.181033	+	0.313558i	−0.942232	−	0.334960i	$-0.891277\pi$
$673$	−81994.9	+	142019.i	−0.181033	+	0.313558i	0.761200	+	0.648517i	$0.224611\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−307402.	−	177479.i	−0.670703	−	0.387230i	0.125640	−	0.992076i	$-0.459901\pi$
$677$	−307402.	−	177479.i	−0.670703	−	0.387230i	−0.796343	+	0.604846i	$0.793235\pi$
$678$		0			0
$679$	−346237.	−	599700.i	−0.750989	−	1.30075i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	278372.i		−	0.596738i	−0.954451	−	0.298369i	$-0.903557\pi$
$683$		−	278372.i		−	0.596738i	0.954451	−	0.298369i	$-0.0964427\pi$
$684$		0			0
$685$	−737106.			−1.57090
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−474147.	+	273749.i	−0.998790	+	0.576652i
$690$		0			0
$691$	−360154.	+	623805.i	−0.754279	+	1.30645i	0.191453	+	0.981502i	$0.438680\pi$
$691$	−360154.	+	623805.i	−0.754279	+	1.30645i	−0.945732	+	0.324948i	$0.894653\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	505127.	+	291635.i	1.04576	+	0.603768i
$696$		0			0
$697$	−11883.1	−	20582.2i	−0.0244605	−	0.0423668i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			742057.i			1.51008i	0.655677	+	0.755042i	$0.272383\pi$
$701$			742057.i			1.51008i	−0.655677	+	0.755042i	$0.727617\pi$
$702$		0			0
$703$	−1.13648e6			−2.29960
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−604404.	+	348953.i	−1.20917	+	0.698117i
$708$		0			0
$709$	−317170.	+	549354.i	−0.630957	+	1.09285i	0.356400	+	0.934334i	$0.384004\pi$
$709$	−317170.	+	549354.i	−0.630957	+	1.09285i	−0.987356	+	0.158516i	$0.949329\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−91.7883	−	52.9940i	−0.000180555	−	0.000104243i
$714$		0			0
$715$	−145501.	−	252015.i	−0.284612	−	0.492962i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	256604.i		−	0.496369i	−0.968713	−	0.248185i	$-0.920166\pi$
$719$		−	256604.i		−	0.496369i	0.968713	−	0.248185i	$-0.0798340\pi$
$720$		0			0
$721$	−1.10872e6			−2.13281
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−201342.	+	116245.i	−0.383053	+	0.221156i
$726$		0			0
$727$	−75892.5	+	131450.i	−0.143592	+	0.248709i	−0.928847	−	0.370464i	$-0.879199\pi$
$727$	−75892.5	+	131450.i	−0.143592	+	0.248709i	0.785255	+	0.619173i	$0.212532\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	22384.9	+	12923.9i	0.0418909	+	0.0241857i
$732$		0			0
$733$	−307563.	−	532715.i	−0.572435	−	0.991486i	−0.996315	−	0.0857679i	$-0.972666\pi$
$733$	−307563.	−	532715.i	−0.572435	−	0.991486i	0.423880	−	0.905718i	$-0.360668\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			55789.9i			0.102712i
$738$		0			0
$739$	58065.2			0.106323			0.0531614	−	0.998586i	$-0.483070\pi$
$739$	58065.2			0.106323			0.0531614	+	0.998586i	$0.483070\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	61582.8	−	35554.8i	0.111553	−	0.0644052i	−0.443185	−	0.896430i	$-0.646152\pi$
$743$	61582.8	−	35554.8i	0.111553	−	0.0644052i	0.554738	+	0.832025i	$0.312818\pi$
$744$		0			0
$745$	300235.	−	520022.i	0.540939	−	0.936933i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	29035.8	+	16763.8i	0.0517571	+	0.0298820i
$750$		0			0
$751$	−259245.	−	449025.i	−0.459653	−	0.796142i	0.539289	−	0.842120i	$-0.318693\pi$
$751$	−259245.	−	449025.i	−0.459653	−	0.796142i	−0.998942	+	0.0459781i	$0.985360\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	519827.i		−	0.911937i
$756$		0			0
$757$	−185939.			−0.324473			−0.162237	−	0.986752i	$-0.551871\pi$
$757$	−185939.			−0.324473			−0.162237	+	0.986752i	$0.551871\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−165707.	+	95670.9i	−0.286135	+	0.165200i	−0.636198	−	0.771526i	$-0.719494\pi$
$761$	−165707.	+	95670.9i	−0.286135	+	0.165200i	0.350062	+	0.936726i	$0.386160\pi$
$762$		0			0
$763$	−292768.	+	507089.i	−0.502891	+	0.871033i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−253181.	−	146174.i	−0.430368	−	0.248473i
$768$		0			0
$769$	237735.	+	411769.i	0.402013	+	0.696308i	0.993969	−	0.109663i	$-0.0349772\pi$
$769$	237735.	+	411769.i	0.402013	+	0.696308i	−0.591955	+	0.805971i	$0.701644\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	13977.8i		−	0.0233926i	−0.999932	−	0.0116963i	$-0.996277\pi$
$773$		−	13977.8i		−	0.0233926i	0.999932	−	0.0116963i	$-0.00372313\pi$
$774$		0			0
$775$	1680.91			0.00279859
$776$		0			0
$777$		0			0
$778$		0			0
$779$	807223.	−	466050.i	1.33021	−	0.767994i
$780$		0			0
$781$	208325.	−	360829.i	0.341538	−	0.591561i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	591790.	+	341670.i	0.960347	+	0.554457i
$786$		0			0
$787$	−120389.	−	208520.i	−0.194374	−	0.336665i	0.752321	−	0.658796i	$-0.228934\pi$
$787$	−120389.	−	208520.i	−0.194374	−	0.336665i	−0.946695	+	0.322131i	$0.895601\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			936272.i			1.49640i
$792$		0			0
$793$	−98571.2			−0.156749
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−862831.	+	498155.i	−1.35834	+	0.784239i	−0.989400	−	0.145214i	$-0.953613\pi$
$797$	−862831.	+	498155.i	−1.35834	+	0.784239i	−0.368941	+	0.929453i	$0.620280\pi$
$798$		0			0
$799$	−11223.7	+	19440.1i	−0.0175810	+	0.0304512i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−679989.	−	392592.i	−1.05456	−	0.608850i
$804$		0			0
$805$	−7510.29	−	13008.2i	−0.0115895	−	0.0200736i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			204246.i			0.312073i	0.987751	+	0.156036i	$0.0498717\pi$
$809$			204246.i			0.312073i	−0.987751	+	0.156036i	$0.950128\pi$
$810$		0			0
$811$	489147.			0.743700			0.371850	−	0.928293i	$-0.378724\pi$
$811$	489147.			0.743700			0.371850	+	0.928293i	$0.378724\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	731954.	−	422594.i	1.10197	−	0.636221i
$816$		0			0
$817$	−506869.	+	877923.i	−0.759367	+	1.31526i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−92152.8	−	53204.4i	−0.136717	−	0.0789335i	0.430081	−	0.902790i	$-0.358485\pi$
$821$	−92152.8	−	53204.4i	−0.136717	−	0.0789335i	−0.566798	+	0.823857i	$0.691818\pi$
$822$		0			0
$823$	−75362.7	−	130532.i	−0.111265	−	0.192716i	0.805016	−	0.593253i	$-0.202157\pi$
$823$	−75362.7	−	130532.i	−0.111265	−	0.192716i	−0.916280	+	0.400538i	$0.868823\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.33944e6i		−	1.95846i	−0.202763	−	0.979228i	$-0.564992\pi$
$827$		−	1.33944e6i		−	1.95846i	0.202763	−	0.979228i	$-0.435008\pi$
$828$		0			0
$829$	345624.			0.502915			0.251457	−	0.967868i	$-0.419090\pi$
$829$	345624.			0.502915			0.251457	+	0.967868i	$0.419090\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−38862.3	+	22437.1i	−0.0560065	+	0.0323354i
$834$		0			0
$835$	−166867.	+	289022.i	−0.239330	+	0.414532i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.18780e6	+	685774.i	1.68740	+	0.974220i	0.956499	+	0.291735i	$0.0942327\pi$
$839$	1.18780e6	+	685774.i	1.68740	+	0.974220i	0.730900	+	0.682485i	$0.239101\pi$
$840$		0			0
$841$	726984.	+	1.25917e6i	1.02786	+	1.78030i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			238554.i			0.334097i
$846$		0			0
$847$	298943.			0.416699
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13714.0	+	7917.77i	−0.0189367	+	0.0109331i
$852$		0			0
$853$	−313290.	+	542634.i	−0.430575	+	0.745778i	−0.996923	−	0.0783886i	$-0.975022\pi$
$853$	−313290.	+	542634.i	−0.430575	+	0.745778i	0.566348	+	0.824166i	$0.308356\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	340966.	+	196857.i	0.464247	+	0.268033i	0.713828	−	0.700321i	$-0.246960\pi$
$857$	340966.	+	196857.i	0.464247	+	0.268033i	−0.249581	+	0.968354i	$0.580293\pi$
$858$		0			0
$859$	−480468.	−	832196.i	−0.651146	−	1.12782i	−0.982845	−	0.184433i	$-0.940955\pi$
$859$	−480468.	−	832196.i	−0.651146	−	1.12782i	0.331699	−	0.943385i	$-0.392378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			560804.i			0.752991i	0.926418	+	0.376495i	$0.122871\pi$
$863$			560804.i			0.752991i	−0.926418	+	0.376495i	$0.877129\pi$
$864$		0			0
$865$	−414365.			−0.553797
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−340803.	+	196763.i	−0.451298	+	0.260557i
$870$		0			0
$871$	−36288.1	+	62852.9i	−0.0478331	+	0.0828493i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.02163e6	+	589838.i	1.33437	+	0.770401i
$876$		0			0
$877$	195858.	+	339236.i	0.254649	+	0.441065i	0.964800	−	0.262984i	$-0.0847067\pi$
$877$	195858.	+	339236.i	0.254649	+	0.441065i	−0.710151	+	0.704049i	$0.751373\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1.48558e6i		−	1.91401i	−0.290079	−	0.957003i	$-0.593682\pi$
$881$		−	1.48558e6i		−	1.91401i	0.290079	−	0.957003i	$-0.406318\pi$
$882$		0			0
$883$	682700.			0.875605			0.437803	−	0.899071i	$-0.355757\pi$
$883$	682700.			0.875605			0.437803	+	0.899071i	$0.355757\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	850046.	−	490774.i	1.08043	−	0.623784i	0.149415	−	0.988775i	$-0.452261\pi$
$887$	850046.	−	490774.i	1.08043	−	0.623784i	0.931011	+	0.364990i	$0.118928\pi$
$888$		0			0
$889$	892889.	−	1.54653e6i	1.12978	−	1.95684i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−762429.	−	440189.i	−0.956085	−	0.551996i
$894$		0			0
$895$	660777.	+	1.14450e6i	0.824915	+	1.42879i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	15625.9i		−	0.0193341i
$900$		0			0
$901$	75475.1			0.0929724
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−233842.	+	135009.i	−0.285513	+	0.164841i
$906$		0			0
$907$	−470283.	+	814555.i	−0.571670	+	0.990161i	0.424725	+	0.905322i	$0.360371\pi$
$907$	−470283.	+	814555.i	−0.571670	+	0.990161i	−0.996395	+	0.0848385i	$0.972963\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	853855.	+	492973.i	1.02884	+	0.594000i	0.916652	−	0.399687i	$-0.130881\pi$
$911$	853855.	+	492973.i	1.02884	+	0.594000i	0.112187	+	0.993687i	$0.464215\pi$
$912$		0			0
$913$	348806.	+	604150.i	0.418449	+	0.724775i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	581343.i		−	0.691344i
$918$		0			0
$919$	−768111.			−0.909480			−0.454740	−	0.890624i	$-0.650268\pi$
$919$	−768111.			−0.909480			−0.454740	+	0.890624i	$0.650268\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	469398.	−	271007.i	0.550982	−	0.318110i
$924$		0			0
$925$	125571.	−	217495.i	0.146759	−	0.254195i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−113342.	−	65437.8i	−0.131328	−	0.0758223i	0.432897	−	0.901444i	$-0.357492\pi$
$929$	−113342.	−	65437.8i	−0.131328	−	0.0758223i	−0.564225	+	0.825621i	$0.690825\pi$
$930$		0			0
$931$	−879974.	−	1.52416e6i	−1.01524	−	1.75845i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			40115.9i			0.0458874i
$936$		0			0
$937$	1.16253e6			1.32411			0.662055	−	0.749455i	$-0.269684\pi$
$937$	1.16253e6			1.32411			0.662055	+	0.749455i	$0.269684\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	313077.	−	180755.i	0.353567	−	0.204132i	−0.312688	−	0.949856i	$-0.601230\pi$
$941$	313077.	−	180755.i	0.353567	−	0.204132i	0.666255	+	0.745724i	$0.267896\pi$
$942$		0			0
$943$	6493.87	−	11247.7i	0.00730264	−	0.0126486i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.11990e6	−	646573.i	−1.24876	−	0.720971i	−0.277897	−	0.960611i	$-0.589637\pi$
$947$	−1.11990e6	−	646573.i	−1.24876	−	0.720971i	−0.970862	+	0.239640i	$0.922971\pi$
$948$		0			0
$949$	−510717.	−	884588.i	−0.567085	−	0.982220i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.60264e6i			1.76462i	0.470672	+	0.882308i	$0.344011\pi$
$953$			1.60264e6i			1.76462i	−0.470672	+	0.882308i	$0.655989\pi$
$954$		0			0
$955$	941643.			1.03248
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.05967e6	+	1.18915e6i	−2.23955	+	1.29301i
$960$		0			0
$961$	461704.	−	799695.i	0.499939	−	0.865919i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−941775.	−	543734.i	−1.01133	−	0.583891i
$966$		0			0
$967$	425112.	+	736316.i	0.454622	+	0.787428i	0.998666	−	0.0516281i	$-0.0164410\pi$
$967$	425112.	+	736316.i	0.454622	+	0.787428i	−0.544044	+	0.839056i	$0.683108\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			819691.i			0.869384i	0.900579	+	0.434692i	$0.143143\pi$
$971$			819691.i			0.869384i	−0.900579	+	0.434692i	$0.856857\pi$
$972$		0			0
$973$	1.88195e6			1.98784
$974$		0			0
$975$		0			0
$976$		0			0
$977$	707362.	−	408395.i	0.741058	−	0.427850i	−0.0813957	−	0.996682i	$-0.525938\pi$
$977$	707362.	−	408395.i	0.741058	−	0.427850i	0.822454	+	0.568832i	$0.192604\pi$
$978$		0			0
$979$	730034.	−	1.26446e6i	0.761689	−	1.31928i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	278041.	+	160527.i	0.287742	+	0.166128i	0.636923	−	0.770928i	$-0.280207\pi$
$983$	278041.	+	160527.i	0.287742	+	0.166128i	−0.349181	+	0.937055i	$0.613540\pi$
$984$		0			0
$985$	−359326.	−	622371.i	−0.370353	−	0.641471i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			14125.3i			0.0144412i
$990$		0			0
$991$	−251405.			−0.255992			−0.127996	−	0.991775i	$-0.540854\pi$
$991$	−251405.			−0.255992			−0.127996	+	0.991775i	$0.540854\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	983624.	−	567896.i	0.993534	−	0.573617i
$996$		0			0
$997$	−53574.5	+	92793.8i	−0.0538974	+	0.0933531i	−0.891715	−	0.452597i	$-0.850498\pi$
$997$	−53574.5	+	92793.8i	−0.0538974	+	0.0933531i	0.837818	+	0.545950i	$0.183831\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.5.m.g.593.4		24
3.2	odd	2	inner	648.5.m.g.593.9		24
9.2	odd	6		648.5.e.b.161.9	yes	12
9.4	even	3	inner	648.5.m.g.377.9		24
9.5	odd	6	inner	648.5.m.g.377.4		24
9.7	even	3		648.5.e.b.161.4	✓	12
36.7	odd	6		1296.5.e.h.161.4		12
36.11	even	6		1296.5.e.h.161.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.5.e.b.161.4	✓	12	9.7	even	3
648.5.e.b.161.9	yes	12	9.2	odd	6
648.5.m.g.377.4		24	9.5	odd	6	inner
648.5.m.g.377.9		24	9.4	even	3	inner
648.5.m.g.593.4		24	1.1	even	1	trivial
648.5.m.g.593.9		24	3.2	odd	2	inner
1296.5.e.h.161.4		12	36.7	odd	6
1296.5.e.h.161.9		12	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	648.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-18.7121 + 10.8034i) q^{5} +(-34.8578 + 60.3755i) q^{7} +(88.1176 + 50.8747i) q^{11} +(66.1822 + 114.631i) q^{13} -18.2471i q^{17} +715.640 q^{19} +(8.63567 - 4.98581i) q^{23} +(-79.0718 + 136.956i) q^{25} +(1273.16 + 735.059i) q^{29} +(-5.31449 - 9.20497i) q^{31} -1506.33i q^{35} -1588.06 q^{37} +(1127.97 - 651.236i) q^{41} +(-708.273 + 1226.77i) q^{43} +(-1065.38 - 615.098i) q^{47} +(-1229.63 - 2129.78i) q^{49} +4136.29i q^{53} -2198.49 q^{55} +(-1912.75 + 1104.33i) q^{59} +(-372.348 + 644.925i) q^{61} +(-2476.82 - 1429.99i) q^{65} +(274.153 + 474.847i) q^{67} -4094.86i q^{71} -7716.83 q^{73} +(-6143.17 + 3546.76i) q^{77} +(-1933.80 + 3349.43i) q^{79} +(5937.62 + 3428.09i) q^{83} +(197.131 + 341.440i) q^{85} -14349.6i q^{89} -9227.87 q^{91} +(-13391.1 + 7731.37i) q^{95} +(-4966.42 + 8602.09i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 96 q^{7} + 72 q^{13} + 672 q^{19} - 84 q^{25} + 1536 q^{31} - 984 q^{37} - 10128 q^{43} + 6828 q^{49} - 27936 q^{55} - 26268 q^{61} + 20784 q^{67} - 19968 q^{73} - 44592 q^{79} + 45348 q^{85} - 23616 q^{91}+ \cdots - 76608 q^{97}+O(q^{100}) \) 24 * q - 96 * q^7 + 72 * q^13 + 672 * q^19 - 84 * q^25 + 1536 * q^31 - 984 * q^37 - 10128 * q^43 + 6828 * q^49 - 27936 * q^55 - 26268 * q^61 + 20784 * q^67 - 19968 * q^73 - 44592 * q^79 + 45348 * q^85 - 23616 * q^91 - 76608 * q^97

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