LMFDB - Embedded newform 1512.2.bs.a.1097.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(521,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1512.521"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.bs (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$12.0733807856$
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			1097.15
Character	$\chi$	$=$	1512.1097
Dual form			1512.2.bs.a.521.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.527910 + 0.914367i) q^{5} +(0.781227 + 2.52778i) q^{7} +(-5.40824 - 3.12245i) q^{11} +(0.872074 + 0.503492i) q^{13} +(3.26821 + 5.66070i) q^{17} +(1.73329 + 1.00071i) q^{19} +(-3.81168 + 2.20067i) q^{23} +(1.94262 - 3.36472i) q^{25} +(-6.12821 + 3.53813i) q^{29} +2.39266i q^{31} +(-1.89890 + 2.04877i) q^{35} +(-3.64197 + 6.30808i) q^{37} +(1.80119 - 3.11974i) q^{41} +(1.60595 + 2.78159i) q^{43} +3.74047 q^{47} +(-5.77937 + 3.94954i) q^{49} +(-6.02455 + 3.47827i) q^{53} -6.59348i q^{55} -13.3417 q^{59} +8.20683i q^{61} +1.06319i q^{65} +0.122792 q^{67} -5.37678i q^{71} +(14.4429 - 8.33860i) q^{73} +(3.66781 - 16.1102i) q^{77} -8.86272 q^{79} +(-1.07668 - 1.86486i) q^{83} +(-3.45064 + 5.97668i) q^{85} +(-2.23201 + 3.86595i) q^{89} +(-0.591431 + 2.59775i) q^{91} +2.11315i q^{95} +(-0.960756 + 0.554693i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 12 q^{23} - 24 q^{25} - 18 q^{29} - 6 q^{41} - 6 q^{43} - 36 q^{47} + 6 q^{49} - 12 q^{53} + 36 q^{77} - 12 q^{79} - 18 q^{89} + 6 q^{91}+O(q^{100}) $ 48 * q + 12 * q^23 - 24 * q^25 - 18 * q^29 - 6 * q^41 - 6 * q^43 - 36 * q^47 + 6 * q^49 - 12 * q^53 + 36 * q^77 - 12 * q^79 - 18 * q^89 + 6 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{5}{6}\right)$	$1$

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
$5$	0.527910	+	0.914367i	0.236089	+	0.408917i	0.959588	−	0.281407i	$-0.0908011\pi$
$5$	0.527910	+	0.914367i	0.236089	+	0.408917i	−0.723500	+	0.690324i	$0.757468\pi$
$6$		0			0
$7$	0.781227	+	2.52778i	0.295276	+	0.955412i
$7$	0.781227	+	2.52778i	0.295276	+	0.955412i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.40824	−	3.12245i	−1.63065	−	0.941453i	−0.983894	−	0.178752i	$-0.942794\pi$
$11$	−5.40824	−	3.12245i	−1.63065	−	0.941453i	−0.646751	−	0.762701i	$-0.723873\pi$
$12$		0			0
$13$	0.872074	+	0.503492i	0.241870	+	0.139644i	0.616036	−	0.787718i	$-0.288738\pi$
$13$	0.872074	+	0.503492i	0.241870	+	0.139644i	−0.374166	+	0.927362i	$0.622071\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.26821	+	5.66070i	0.792656	+	1.37292i	0.924317	+	0.381626i	$0.124636\pi$
$17$	3.26821	+	5.66070i	0.792656	+	1.37292i	−0.131660	+	0.991295i	$0.542031\pi$
$18$		0			0
$19$	1.73329	+	1.00071i	0.397643	+	0.229579i	0.685466	−	0.728104i	$-0.259598\pi$
$19$	1.73329	+	1.00071i	0.397643	+	0.229579i	−0.287823	+	0.957683i	$0.592932\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.81168	+	2.20067i	−0.794790	+	0.458872i	−0.841646	−	0.540029i	$-0.818413\pi$
$23$	−3.81168	+	2.20067i	−0.794790	+	0.458872i	0.0468562	+	0.998902i	$0.485080\pi$
$24$		0			0
$25$	1.94262	−	3.36472i	0.388524	−	0.672944i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.12821	+	3.53813i	−1.13798	+	0.657014i	−0.945930	−	0.324371i	$-0.894847\pi$
$29$	−6.12821	+	3.53813i	−1.13798	+	0.657014i	−0.192051	+	0.981385i	$0.561514\pi$
$30$		0			0
$31$			2.39266i			0.429734i	0.976643	+	0.214867i	$0.0689317\pi$
$31$			2.39266i			0.429734i	−0.976643	+	0.214867i	$0.931068\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.89890	+	2.04877i	−0.320973	+	0.346305i
$36$		0			0
$37$	−3.64197	+	6.30808i	−0.598737	+	1.03704i	0.394271	+	0.918994i	$0.370997\pi$
$37$	−3.64197	+	6.30808i	−0.598737	+	1.03704i	−0.993008	+	0.118048i	$0.962336\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.80119	−	3.11974i	0.281298	−	0.487222i	−0.690407	−	0.723421i	$-0.742568\pi$
$41$	1.80119	−	3.11974i	0.281298	−	0.487222i	0.971705	+	0.236199i	$0.0759018\pi$
$42$		0			0
$43$	1.60595	+	2.78159i	0.244906	+	0.424189i	0.962105	−	0.272679i	$-0.0879097\pi$
$43$	1.60595	+	2.78159i	0.244906	+	0.424189i	−0.717199	+	0.696868i	$0.754576\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.74047			0.545604			0.272802	−	0.962070i	$-0.412050\pi$
$47$	3.74047			0.545604			0.272802	+	0.962070i	$0.412050\pi$
$48$		0			0
$49$	−5.77937	+	3.94954i	−0.825624	+	0.564220i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.02455	+	3.47827i	−0.827535	+	0.477778i	−0.853008	−	0.521898i	$-0.825224\pi$
$53$	−6.02455	+	3.47827i	−0.827535	+	0.477778i	0.0254729	+	0.999676i	$0.491891\pi$
$54$		0			0
$55$		−	6.59348i		−	0.889065i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−13.3417			−1.73694			−0.868469	−	0.495744i	$-0.834896\pi$
$59$	−13.3417			−1.73694			−0.868469	+	0.495744i	$0.834896\pi$
$60$		0			0
$61$			8.20683i			1.05078i	0.850862	+	0.525389i	$0.176080\pi$
$61$			8.20683i			1.05078i	−0.850862	+	0.525389i	$0.823920\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			1.06319i			0.131873i
$66$		0			0
$67$	0.122792			0.0150015			0.00750074	−	0.999972i	$-0.497612\pi$
$67$	0.122792			0.0150015			0.00750074	+	0.999972i	$0.497612\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.37678i		−	0.638107i	−0.947737	−	0.319053i	$-0.896635\pi$
$71$		−	5.37678i		−	0.638107i	0.947737	−	0.319053i	$-0.103365\pi$
$72$		0			0
$73$	14.4429	−	8.33860i	1.69041	−	0.975959i	0.736228	−	0.676734i	$-0.236605\pi$
$73$	14.4429	−	8.33860i	1.69041	−	0.975959i	0.954182	−	0.299225i	$-0.0967282\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.66781	−	16.1102i	0.417986	−	1.83593i
$78$		0			0
$79$	−8.86272			−0.997134			−0.498567	−	0.866851i	$-0.666140\pi$
$79$	−8.86272			−0.997134			−0.498567	+	0.866851i	$0.666140\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.07668	−	1.86486i	−0.118181	−	0.204695i	0.800866	−	0.598844i	$-0.204373\pi$
$83$	−1.07668	−	1.86486i	−0.118181	−	0.204695i	−0.919047	+	0.394149i	$0.871040\pi$
$84$		0			0
$85$	−3.45064	+	5.97668i	−0.374274	+	0.648262i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.23201	+	3.86595i	−0.236592	+	0.409790i	−0.959734	−	0.280910i	$-0.909364\pi$
$89$	−2.23201	+	3.86595i	−0.236592	+	0.409790i	0.723142	+	0.690699i	$0.242697\pi$
$90$		0			0
$91$	−0.591431	+	2.59775i	−0.0619988	+	0.272319i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			2.11315i			0.216804i
$96$		0			0
$97$	−0.960756	+	0.554693i	−0.0975500	+	0.0563205i	−0.547981	−	0.836491i	$-0.684604\pi$
$97$	−0.960756	+	0.554693i	−0.0975500	+	0.0563205i	0.450431	+	0.892811i	$0.351270\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.89446	+	5.01335i	−0.288009	+	0.498847i	−0.973335	−	0.229390i	$-0.926327\pi$
$101$	−2.89446	+	5.01335i	−0.288009	+	0.498847i	0.685325	+	0.728237i	$0.259660\pi$
$102$		0			0
$103$	−17.1208	+	9.88467i	−1.68696	+	0.973966i	−0.730131	+	0.683307i	$0.760541\pi$
$103$	−17.1208	+	9.88467i	−1.68696	+	0.973966i	−0.956827	+	0.290658i	$0.906126\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.50275	+	3.17701i	0.531971	+	0.307133i	0.741819	−	0.670601i	$-0.233964\pi$
$107$	5.50275	+	3.17701i	0.531971	+	0.307133i	−0.209848	+	0.977734i	$0.567297\pi$
$108$		0			0
$109$	−1.03951	−	1.80048i	−0.0995669	−	0.172455i	0.811939	−	0.583743i	$-0.198412\pi$
$109$	−1.03951	−	1.80048i	−0.0995669	−	0.172455i	−0.911505	+	0.411288i	$0.865079\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.9040	+	7.45015i	1.21391	+	0.700851i	0.963609	−	0.267317i	$-0.0861371\pi$
$113$	12.9040	+	7.45015i	1.21391	+	0.700851i	0.250301	+	0.968168i	$0.419470\pi$
$114$		0			0
$115$	−4.02445	−	2.32352i	−0.375282	−	0.216669i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−11.7558	+	12.6836i	−1.07765	+	1.16270i
$120$		0			0
$121$	13.9994	+	24.2476i	1.27267	+	2.20433i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.38122			0.839082
$126$		0			0
$127$	18.6841			1.65794			0.828971	−	0.559292i	$-0.188927\pi$
$127$	18.6841			1.65794			0.828971	+	0.559292i	$0.188927\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.27063	+	7.39694i	0.373126	+	0.646274i	0.990045	−	0.140753i	$-0.0449524\pi$
$131$	4.27063	+	7.39694i	0.373126	+	0.646274i	−0.616918	+	0.787027i	$0.711619\pi$
$132$		0			0
$133$	−1.17550	+	5.16315i	−0.101928	+	0.447702i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.7370	+	6.19899i	0.917321	+	0.529616i	0.882779	−	0.469788i	$-0.155670\pi$
$137$	10.7370	+	6.19899i	0.917321	+	0.529616i	0.0345416	+	0.999403i	$0.489003\pi$
$138$		0			0
$139$	−15.4789	−	8.93676i	−1.31291	−	0.758006i	−0.330329	−	0.943866i	$-0.607160\pi$
$139$	−15.4789	−	8.93676i	−1.31291	−	0.758006i	−0.982576	+	0.185860i	$0.940493\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.14425	−	5.44601i	−0.262936	−	0.455418i
$144$		0			0
$145$	−6.47029	−	3.73562i	−0.537328	−	0.310227i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.39571	−	3.69257i	0.523957	−	0.302507i	−0.214595	−	0.976703i	$-0.568843\pi$
$149$	6.39571	−	3.69257i	0.523957	−	0.302507i	0.738552	+	0.674196i	$0.235510\pi$
$150$		0			0
$151$	−5.13984	+	8.90246i	−0.418274	+	0.724472i	−0.995766	−	0.0919243i	$-0.970698\pi$
$151$	−5.13984	+	8.90246i	−0.418274	+	0.724472i	0.577492	+	0.816397i	$0.304032\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.18776	+	1.26311i	−0.175725	+	0.101455i
$156$		0			0
$157$			15.5379i			1.24006i	0.784579	+	0.620029i	$0.212879\pi$
$157$			15.5379i			1.24006i	−0.784579	+	0.620029i	$0.787121\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.54061	−	7.91587i	−0.673094	−	0.623858i
$162$		0			0
$163$	1.98761	−	3.44264i	0.155682	−	0.269649i	−0.777625	−	0.628728i	$-0.783576\pi$
$163$	1.98761	−	3.44264i	0.155682	−	0.269649i	0.933307	+	0.359079i	$0.116909\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.65088	+	6.32351i	−0.282513	+	0.489328i	−0.972003	−	0.234968i	$-0.924502\pi$
$167$	−3.65088	+	6.32351i	−0.282513	+	0.489328i	0.689490	+	0.724295i	$0.257835\pi$
$168$		0			0
$169$	−5.99299	−	10.3802i	−0.460999	−	0.798474i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.45426			0.642765			0.321383	−	0.946949i	$-0.395852\pi$
$173$	8.45426			0.642765			0.321383	+	0.946949i	$0.395852\pi$
$174$		0			0
$175$	10.0229	+	2.28192i	0.757661	+	0.172497i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.0439783	+	0.0253909i	−0.00328709	+	0.00189780i	−0.501643	−	0.865075i	$-0.667271\pi$
$179$	−0.0439783	+	0.0253909i	−0.00328709	+	0.00189780i	0.498356	+	0.866973i	$0.333937\pi$
$180$		0			0
$181$		−	2.64284i		−	0.196440i	−0.995165	−	0.0982202i	$-0.968685\pi$
$181$		−	2.64284i		−	0.196440i	0.995165	−	0.0982202i	$-0.0313149\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.69054			−0.565420
$186$		0			0
$187$		−	40.8192i		−	2.98500i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	24.2855i		−	1.75724i	−0.477523	−	0.878619i	$-0.658465\pi$
$191$		−	24.2855i		−	1.75724i	0.477523	−	0.878619i	$-0.341535\pi$
$192$		0			0
$193$	−5.97870			−0.430356			−0.215178	−	0.976575i	$-0.569033\pi$
$193$	−5.97870			−0.430356			−0.215178	+	0.976575i	$0.569033\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	0.110985i		−	0.00790737i	−0.999992	−	0.00395369i	$-0.998741\pi$
$197$		−	0.110985i		−	0.00790737i	0.999992	−	0.00395369i	$-0.00125850\pi$
$198$		0			0
$199$	7.95580	−	4.59329i	0.563972	−	0.325609i	−0.190766	−	0.981636i	$-0.561097\pi$
$199$	7.95580	−	4.59329i	0.563972	−	0.325609i	0.754738	+	0.656026i	$0.227764\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−13.7311	−	12.7267i	−0.963737	−	0.893240i
$204$		0			0
$205$	3.80345			0.265645
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.24935	−	10.8242i	−0.432276	−	0.748725i
$210$		0			0
$211$	7.95208	−	13.7734i	0.547443	−	0.948200i	−0.451005	−	0.892521i	$-0.648934\pi$
$211$	7.95208	−	13.7734i	0.547443	−	0.948200i	0.998449	−	0.0556786i	$-0.0177322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.69560	+	2.93686i	−0.115639	+	0.200292i
$216$		0			0
$217$	−6.04811	+	1.86921i	−0.410573	+	0.126890i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			6.58206i			0.442757i
$222$		0			0
$223$	−2.79336	+	1.61275i	−0.187057	+	0.107998i	−0.590604	−	0.806961i	$-0.701111\pi$
$223$	−2.79336	+	1.61275i	−0.187057	+	0.107998i	0.403547	+	0.914959i	$0.367777\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.09633	−	5.36299i	0.205510	−	0.355954i	−0.744785	−	0.667305i	$-0.767448\pi$
$227$	3.09633	−	5.36299i	0.205510	−	0.355954i	0.950295	+	0.311350i	$0.100781\pi$
$228$		0			0
$229$	16.6167	−	9.59364i	1.09806	−	0.633965i	0.162349	−	0.986733i	$-0.448093\pi$
$229$	16.6167	−	9.59364i	1.09806	−	0.633965i	0.935711	+	0.352768i	$0.114760\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.45069	+	1.99225i	0.226062	+	0.130517i	0.608754	−	0.793359i	$-0.291670\pi$
$233$	3.45069	+	1.99225i	0.226062	+	0.130517i	−0.382692	+	0.923876i	$0.625003\pi$
$234$		0			0
$235$	1.97463	+	3.42016i	0.128811	+	0.223107i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.2139	+	7.62905i	0.854736	+	0.493482i	0.862246	−	0.506490i	$-0.169057\pi$
$239$	13.2139	+	7.62905i	0.854736	+	0.493482i	−0.00750987	+	0.999972i	$0.502390\pi$
$240$		0			0
$241$	−7.61826	−	4.39841i	−0.490735	−	0.283326i	0.234144	−	0.972202i	$-0.424771\pi$
$241$	−7.61826	−	4.39841i	−0.490735	−	0.283326i	−0.724880	+	0.688876i	$0.758105\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.66232	−	3.19946i	−0.425640	−	0.204406i
$246$		0			0
$247$	1.00770	+	1.74539i	0.0641185	+	0.111057i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.38549			0.592407			0.296204	−	0.955125i	$-0.404279\pi$
$251$	9.38549			0.592407			0.296204	+	0.955125i	$0.404279\pi$
$252$		0			0
$253$	27.4860			1.72803
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.32591	−	4.02860i	−0.145087	−	0.251297i	0.784319	−	0.620358i	$-0.213013\pi$
$257$	−2.32591	−	4.02860i	−0.145087	−	0.251297i	−0.929405	+	0.369061i	$0.879679\pi$
$258$		0			0
$259$	−18.7907	−	4.27807i	−1.16760	−	0.265827i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	18.2883	+	10.5587i	1.12770	+	0.651079i	0.943356	−	0.331782i	$-0.107650\pi$
$263$	18.2883	+	10.5587i	1.12770	+	0.651079i	0.184346	+	0.982861i	$0.440983\pi$
$264$		0			0
$265$	−6.36084	−	3.67243i	−0.390743	−	0.225596i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.87450	−	10.1749i	−0.358175	−	0.620377i	0.629481	−	0.777016i	$-0.283267\pi$
$269$	−5.87450	−	10.1749i	−0.358175	−	0.620377i	−0.987656	+	0.156639i	$0.949934\pi$
$270$		0			0
$271$	6.24601	+	3.60613i	0.379418	+	0.219057i	0.677565	−	0.735463i	$-0.263035\pi$
$271$	6.24601	+	3.60613i	0.379418	+	0.219057i	−0.298147	+	0.954520i	$0.596369\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−21.0123	+	12.1315i	−1.26709	+	0.731555i
$276$		0			0
$277$	−0.137930	+	0.238902i	−0.00828743	+	0.0143542i	−0.870139	−	0.492806i	$-0.835971\pi$
$277$	−0.137930	+	0.238902i	−0.00828743	+	0.0143542i	0.861852	+	0.507160i	$0.169305\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.7254	−	8.50170i	0.878442	−	0.507169i	0.00829763	−	0.999966i	$-0.497359\pi$
$281$	14.7254	−	8.50170i	0.878442	−	0.507169i	0.870144	+	0.492797i	$0.164025\pi$
$282$		0			0
$283$			4.78319i			0.284331i	0.989843	+	0.142165i	$0.0454065\pi$
$283$			4.78319i			0.284331i	−0.989843	+	0.142165i	$0.954594\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.29317	+	2.11578i	0.548558	+	0.124890i
$288$		0			0
$289$	−12.8623	+	22.2782i	−0.756608	+	1.31048i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.8580	−	20.5386i	0.692750	−	1.19988i	−0.278184	−	0.960528i	$-0.589732\pi$
$293$	11.8580	−	20.5386i	0.692750	−	1.19988i	0.970933	−	0.239350i	$-0.0769343\pi$
$294$		0			0
$295$	−7.04320	−	12.1992i	−0.410071	−	0.710264i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.43209			−0.256314
$300$		0			0
$301$	−5.77665	+	6.23256i	−0.332961	+	0.359239i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.50406	+	4.33247i	−0.429681	+	0.248076i
$306$		0			0
$307$		−	8.38070i		−	0.478312i	−0.970981	−	0.239156i	$-0.923129\pi$
$307$		−	8.38070i		−	0.478312i	0.970981	−	0.239156i	$-0.0768707\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	16.7917			0.952170			0.476085	−	0.879399i	$-0.342055\pi$
$311$	16.7917			0.952170			0.476085	+	0.879399i	$0.342055\pi$
$312$		0			0
$313$		−	12.5799i		−	0.711057i	−0.934665	−	0.355528i	$-0.884301\pi$
$313$		−	12.5799i		−	0.711057i	0.934665	−	0.355528i	$-0.115699\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			11.1060i			0.623775i	0.950119	+	0.311887i	$0.100961\pi$
$317$			11.1060i			0.623775i	−0.950119	+	0.311887i	$0.899039\pi$
$318$		0			0
$319$	44.1905			2.47419
$320$		0			0
$321$		0			0
$322$		0			0
$323$			13.0821i			0.727910i
$324$		0			0
$325$	3.38822	−	1.95619i	0.187945	−	0.108510i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.92216	+	9.45510i	0.161104	+	0.521277i
$330$		0			0
$331$	−4.14254			−0.227695			−0.113847	−	0.993498i	$-0.536317\pi$
$331$	−4.14254			−0.227695			−0.113847	+	0.993498i	$0.536317\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.0648233	+	0.112277i	0.00354168	+	0.00613437i
$336$		0			0
$337$	14.7897	−	25.6166i	0.805649	−	1.39542i	−0.110204	−	0.993909i	$-0.535150\pi$
$337$	14.7897	−	25.6166i	0.805649	−	1.39542i	0.915852	−	0.401515i	$-0.131516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.47094	−	12.9400i	0.404574	−	0.700743i
$342$		0			0
$343$	−14.4986	−	11.5235i	−0.782850	−	0.622211i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			1.64188i			0.0881407i	0.999028	+	0.0440704i	$0.0140326\pi$
$347$			1.64188i			0.0881407i	−0.999028	+	0.0440704i	$0.985967\pi$
$348$		0			0
$349$	17.5869	−	10.1538i	0.941405	−	0.543520i	0.0510046	−	0.998698i	$-0.483758\pi$
$349$	17.5869	−	10.1538i	0.941405	−	0.543520i	0.890400	+	0.455178i	$0.150424\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8.86687	+	15.3579i	−0.471936	+	0.817417i	−0.999484	−	0.0321077i	$-0.989778\pi$
$353$	−8.86687	+	15.3579i	−0.471936	+	0.817417i	0.527548	+	0.849525i	$0.323111\pi$
$354$		0			0
$355$	4.91635	−	2.83846i	0.260933	−	0.150650i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−23.6573	−	13.6585i	−1.24858	−	0.720871i	−0.277757	−	0.960651i	$-0.589591\pi$
$359$	−23.6573	−	13.6585i	−1.24858	−	0.720871i	−0.970827	+	0.239781i	$0.922924\pi$
$360$		0			0
$361$	−7.49715	−	12.9854i	−0.394587	−	0.683444i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.2491	+	8.80406i	0.798173	+	0.460825i
$366$		0			0
$367$	−1.63678	−	0.944994i	−0.0854391	−	0.0493283i	0.456672	−	0.889635i	$-0.349041\pi$
$367$	−1.63678	−	0.944994i	−0.0854391	−	0.0493283i	−0.542111	+	0.840307i	$0.682375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−13.4989	−	12.5114i	−0.700826	−	0.649561i
$372$		0			0
$373$	−9.05866	−	15.6901i	−0.469040	−	0.812401i	0.530334	−	0.847789i	$-0.322067\pi$
$373$	−9.05866	−	15.6901i	−0.469040	−	0.812401i	−0.999374	+	0.0353881i	$0.988733\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.12567			−0.366991
$378$		0			0
$379$	−3.05133			−0.156736			−0.0783682	−	0.996924i	$-0.524971\pi$
$379$	−3.05133			−0.156736			−0.0783682	+	0.996924i	$0.524971\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.86967	+	15.3627i	0.453219	+	0.784999i	0.998584	−	0.0532002i	$-0.0169422\pi$
$383$	8.86967	+	15.3627i	0.453219	+	0.784999i	−0.545365	+	0.838199i	$0.683609\pi$
$384$		0			0
$385$	16.6669	−	5.15101i	0.849424	−	0.262520i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.2651	+	10.5453i	0.926076	+	0.534670i	0.885568	−	0.464509i	$-0.153769\pi$
$389$	18.2651	+	10.5453i	0.926076	+	0.534670i	0.0405073	+	0.999179i	$0.487103\pi$
$390$		0			0
$391$	−24.9147	−	14.3845i	−1.25999	−	0.727456i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.67872	−	8.10378i	−0.235412	−	0.407745i
$396$		0			0
$397$	19.7684	+	11.4133i	0.992147	+	0.572816i	0.905915	−	0.423459i	$-0.139184\pi$
$397$	19.7684	+	11.4133i	0.992147	+	0.572816i	0.0862315	+	0.996275i	$0.472518\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.39595	+	0.805949i	−0.0697102	+	0.0402472i	−0.534450	−	0.845200i	$-0.679481\pi$
$401$	−1.39595	+	0.805949i	−0.0697102	+	0.0402472i	0.464740	+	0.885447i	$0.346148\pi$
$402$		0			0
$403$	−1.20468	+	2.08657i	−0.0600095	+	0.103940i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	39.3933	−	22.7437i	1.95265	−	1.12737i
$408$		0			0
$409$			22.9811i			1.13634i	0.822910	+	0.568172i	$0.192349\pi$
$409$			22.9811i			1.13634i	−0.822910	+	0.568172i	$0.807651\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−10.4229	−	33.7248i	−0.512876	−	1.65949i
$414$		0			0
$415$	1.13678	−	1.96896i	0.0558022	−	0.0966522i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.46660	−	16.3966i	0.462474	−	0.801028i	−0.536610	−	0.843830i	$-0.680295\pi$
$419$	9.46660	−	16.3966i	0.462474	−	0.801028i	0.999084	+	0.0428027i	$0.0136287\pi$
$420$		0			0
$421$	−9.69784	−	16.7971i	−0.472644	−	0.818643i	0.526866	−	0.849948i	$-0.323367\pi$
$421$	−9.69784	−	16.7971i	−0.472644	−	0.818643i	−0.999510	+	0.0313053i	$0.990034\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	25.3956			1.23187
$426$		0			0
$427$	−20.7451	+	6.41140i	−1.00393	+	0.310269i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	21.5507	−	12.4423i	1.03806	−	0.599323i	0.118776	−	0.992921i	$-0.462103\pi$
$431$	21.5507	−	12.4423i	1.03806	−	0.599323i	0.919283	+	0.393598i	$0.128770\pi$
$432$		0			0
$433$		−	26.4665i		−	1.27190i	−0.771731	−	0.635949i	$-0.780609\pi$
$433$		−	26.4665i		−	1.27190i	0.771731	−	0.635949i	$-0.219391\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.80897			−0.421390
$438$		0			0
$439$			27.3799i			1.30677i	0.757025	+	0.653386i	$0.226652\pi$
$439$			27.3799i			1.30677i	−0.757025	+	0.653386i	$0.773348\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			29.6427i			1.40837i	0.710018	+	0.704183i	$0.248687\pi$
$443$			29.6427i			1.40837i	−0.710018	+	0.704183i	$0.751313\pi$
$444$		0			0
$445$	−4.71320			−0.223427
$446$		0			0
$447$		0			0
$448$		0			0
$449$			18.2726i			0.862339i	0.902271	+	0.431170i	$0.141899\pi$
$449$			18.2726i			0.862339i	−0.902271	+	0.431170i	$0.858101\pi$
$450$		0			0
$451$	−19.4825	+	11.2482i	−0.917394	+	0.529658i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.68752	+	0.830595i	−0.125993	+	0.0389389i
$456$		0			0
$457$	−17.9646			−0.840351			−0.420175	−	0.907443i	$-0.638031\pi$
$457$	−17.9646			−0.840351			−0.420175	+	0.907443i	$0.638031\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.51625	−	11.2865i	−0.303492	−	0.525664i	0.673432	−	0.739249i	$-0.264819\pi$
$461$	−6.51625	−	11.2865i	−0.303492	−	0.525664i	−0.976924	+	0.213585i	$0.931486\pi$
$462$		0			0
$463$	−4.24422	+	7.35121i	−0.197246	+	0.341640i	−0.947634	−	0.319357i	$-0.896533\pi$
$463$	−4.24422	+	7.35121i	−0.197246	+	0.341640i	0.750389	+	0.660997i	$0.229866\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.895025	−	1.55023i	0.0414168	−	0.0717360i	−0.844574	−	0.535439i	$-0.820146\pi$
$467$	0.895025	−	1.55023i	0.0414168	−	0.0717360i	0.885991	+	0.463703i	$0.153480\pi$
$468$		0			0
$469$	0.0959287	+	0.310393i	0.00442958	+	0.0143326i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	20.0580i		−	0.922269i
$474$		0			0
$475$	6.73424	−	3.88801i	0.308988	−	0.178394i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.3189	−	21.3370i	0.562867	−	0.974914i	−0.434378	−	0.900731i	$-0.643032\pi$
$479$	12.3189	−	21.3370i	0.562867	−	0.974914i	0.997245	−	0.0741834i	$-0.0236350\pi$
$480$		0			0
$481$	−6.35214	+	3.66741i	−0.289633	+	0.167219i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.01439	−	0.585656i	−0.0460609	−	0.0265933i
$486$		0			0
$487$	13.1846	+	22.8364i	0.597450	+	1.03481i	0.993196	+	0.116454i	$0.0371528\pi$
$487$	13.1846	+	22.8364i	0.597450	+	1.03481i	−0.395746	+	0.918360i	$0.629514\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−21.0365	−	12.1454i	−0.949365	−	0.548116i	−0.0564812	−	0.998404i	$-0.517988\pi$
$491$	−21.0365	−	12.1454i	−0.949365	−	0.548116i	−0.892884	+	0.450288i	$0.851321\pi$
$492$		0			0
$493$	−40.0565	−	23.1267i	−1.80406	−	1.04157i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.5913	−	4.20049i	0.609655	−	0.188418i
$498$		0			0
$499$	16.5220	+	28.6169i	0.739625	+	1.28107i	0.952664	+	0.304024i	$0.0983303\pi$
$499$	16.5220	+	28.6169i	0.739625	+	1.28107i	−0.213040	+	0.977044i	$0.568336\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	31.5400			1.40630			0.703149	−	0.711043i	$-0.251777\pi$
$503$	31.5400			1.40630			0.703149	+	0.711043i	$0.251777\pi$
$504$		0			0
$505$	−6.11205			−0.271983
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.36236	+	9.28789i	0.237683	+	0.411678i	0.960049	−	0.279832i	$-0.0902788\pi$
$509$	5.36236	+	9.28789i	0.237683	+	0.411678i	−0.722366	+	0.691511i	$0.756946\pi$
$510$		0			0
$511$	32.3613	+	29.9941i	1.43158	+	1.32686i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−18.0764	−	10.4364i	−0.796543	−	0.459884i
$516$		0			0
$517$	−20.2294	−	11.6794i	−0.889686	−	0.513661i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	7.64350	+	13.2389i	0.334868	+	0.580008i	0.983459	−	0.181128i	$-0.0579749\pi$
$521$	7.64350	+	13.2389i	0.334868	+	0.580008i	−0.648591	+	0.761137i	$0.724642\pi$
$522$		0			0
$523$	−28.6654	−	16.5500i	−1.25345	−	0.723681i	−0.281659	−	0.959515i	$-0.590885\pi$
$523$	−28.6654	−	16.5500i	−1.25345	−	0.723681i	−0.971793	+	0.235834i	$0.924218\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.5441	+	7.81969i	−0.589990	+	0.340631i
$528$		0			0
$529$	−1.81407	+	3.14206i	−0.0788726	+	0.136611i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.14153	−	1.81376i	0.136075	−	0.0785629i
$534$		0			0
$535$			6.70871i			0.290043i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	43.5884	−	3.31429i	1.87749	−	0.142756i
$540$		0			0
$541$	−16.2012	+	28.0613i	−0.696543	+	1.20645i	0.273115	+	0.961981i	$0.411946\pi$
$541$	−16.2012	+	28.0613i	−0.696543	+	1.20645i	−0.969658	+	0.244466i	$0.921387\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.09753	−	1.90099i	0.0470132	−	0.0814293i
$546$		0			0
$547$	−4.77988	−	8.27900i	−0.204373	−	0.353984i	0.745560	−	0.666439i	$-0.232182\pi$
$547$	−4.77988	−	8.27900i	−0.204373	−	0.353984i	−0.949933	+	0.312454i	$0.898849\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−14.1626			−0.603347
$552$		0			0
$553$	−6.92380	−	22.4030i	−0.294430	−	0.952674i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16.6963	+	9.63963i	−0.707446	+	0.408444i	−0.810115	−	0.586271i	$-0.800595\pi$
$557$	−16.6963	+	9.63963i	−0.707446	+	0.408444i	0.102669	+	0.994716i	$0.467262\pi$
$558$		0			0
$559$			3.23434i			0.136798i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.43920			−0.144945			−0.0724725	−	0.997370i	$-0.523089\pi$
$563$	−3.43920			−0.144945			−0.0724725	+	0.997370i	$0.523089\pi$
$564$		0			0
$565$			15.7320i			0.661852i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			34.5725i			1.44935i	0.689089	+	0.724677i	$0.258011\pi$
$569$			34.5725i			1.44935i	−0.689089	+	0.724677i	$0.741989\pi$
$570$		0			0
$571$	47.4948			1.98759			0.993797	−	0.111205i	$-0.0354710\pi$
$571$	47.4948			1.98759			0.993797	+	0.111205i	$0.0354710\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			17.1003i			0.713132i
$576$		0			0
$577$	16.6404	−	9.60732i	0.692748	−	0.399958i	−0.111893	−	0.993720i	$-0.535691\pi$
$577$	16.6404	−	9.60732i	0.692748	−	0.399958i	0.804641	+	0.593762i	$0.202358\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.87283	−	4.17848i	0.160672	−	0.173353i
$582$		0			0
$583$	43.4429			1.79922
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.1279	−	21.0061i	−0.500570	−	0.867013i	−1.00000		0.000658772i	$-0.999790\pi$
$587$	−12.1279	−	21.0061i	−0.500570	−	0.867013i	0.499429	−	0.866355i	$-0.333543\pi$
$588$		0			0
$589$	−2.39436	+	4.14716i	−0.0986579	+	0.170881i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.31832	+	2.28340i	−0.0541371	+	0.0937681i	−0.891824	−	0.452383i	$-0.850574\pi$
$593$	−1.31832	+	2.28340i	−0.0541371	+	0.0937681i	0.837687	+	0.546151i	$0.183907\pi$
$594$		0			0
$595$	−17.8035	−	4.05332i	−0.729871	−	0.166170i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			37.9371i			1.55007i	0.631919	+	0.775034i	$0.282267\pi$
$599$			37.9371i			1.55007i	−0.631919	+	0.775034i	$0.717733\pi$
$600$		0			0
$601$	−26.3440	+	15.2097i	−1.07459	+	0.620417i	−0.929433	−	0.368991i	$-0.879703\pi$
$601$	−26.3440	+	15.2097i	−1.07459	+	0.620417i	−0.145161	+	0.989408i	$0.546370\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.7808	+	25.6011i	−0.600925	+	1.04083i
$606$		0			0
$607$	−38.5356	+	22.2485i	−1.56411	+	0.903040i	−0.567277	+	0.823527i	$0.692003\pi$
$607$	−38.5356	+	22.2485i	−1.56411	+	0.903040i	−0.996834	+	0.0795126i	$0.974664\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.26197	+	1.88330i	0.131965	+	0.0761901i
$612$		0			0
$613$	−23.2533	−	40.2759i	−0.939192	−	1.62673i	−0.766982	−	0.641668i	$-0.778243\pi$
$613$	−23.2533	−	40.2759i	−0.939192	−	1.62673i	−0.172210	−	0.985060i	$-0.555091\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.9058	−	10.9153i	−0.761118	−	0.439432i	0.0685792	−	0.997646i	$-0.478153\pi$
$617$	−18.9058	−	10.9153i	−0.761118	−	0.439432i	−0.829697	+	0.558214i	$0.811487\pi$
$618$		0			0
$619$	6.29883	+	3.63663i	0.253171	+	0.146169i	0.621216	−	0.783640i	$-0.286639\pi$
$619$	6.29883	+	3.63663i	0.253171	+	0.146169i	−0.368044	+	0.929808i	$0.619972\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.5160	−	2.62185i	−0.461378	−	0.105042i
$624$		0			0
$625$	−4.76067	−	8.24573i	−0.190427	−	0.329829i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−47.6109			−1.89837
$630$		0			0
$631$	39.2496			1.56250			0.781252	−	0.624216i	$-0.214582\pi$
$631$	39.2496			1.56250			0.781252	+	0.624216i	$0.214582\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.86350	+	17.0841i	0.391421	+	0.677961i
$636$		0			0
$637$	−7.02860	+	0.534426i	−0.278483	+	0.0211747i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−29.8786	−	17.2504i	−1.18013	−	0.681351i	−0.224088	−	0.974569i	$-0.571940\pi$
$641$	−29.8786	−	17.2504i	−1.18013	−	0.681351i	−0.956046	+	0.293218i	$0.905274\pi$
$642$		0			0
$643$	5.08992	+	2.93867i	0.200727	+	0.115890i	0.596994	−	0.802245i	$-0.296361\pi$
$643$	5.08992	+	2.93867i	0.200727	+	0.115890i	−0.396268	+	0.918135i	$0.629695\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.6539	+	25.3814i	0.576106	+	0.997844i	0.995921	+	0.0902346i	$0.0287617\pi$
$647$	14.6539	+	25.3814i	0.576106	+	0.997844i	−0.419815	+	0.907610i	$0.637905\pi$
$648$		0			0
$649$	72.1549	+	41.6587i	2.83233	+	1.63525i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−36.8999	+	21.3042i	−1.44400	+	0.833696i	−0.998114	−	0.0613917i	$-0.980446\pi$
$653$	−36.8999	+	21.3042i	−1.44400	+	0.833696i	−0.445890	+	0.895088i	$0.647113\pi$
$654$		0			0
$655$	−4.50901	+	7.80984i	−0.176182	+	0.305156i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.0891	−	11.0211i	0.743608	−	0.429322i	−0.0797719	−	0.996813i	$-0.525419\pi$
$659$	19.0891	−	11.0211i	0.743608	−	0.429322i	0.823380	+	0.567491i	$0.192086\pi$
$660$		0			0
$661$		−	14.6856i		−	0.571204i	−0.958348	−	0.285602i	$-0.907806\pi$
$661$		−	14.6856i		−	0.571204i	0.958348	−	0.285602i	$-0.0921935\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.34157	+	1.65085i	−0.207137	+	0.0640170i
$666$		0			0
$667$	15.5725	−	26.9724i	0.602971	−	1.04438i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	25.6254	−	44.3845i	0.989258	−	1.71344i
$672$		0			0
$673$	18.8645	+	32.6743i	0.727173	+	1.25950i	0.958073	+	0.286524i	$0.0924997\pi$
$673$	18.8645	+	32.6743i	0.727173	+	1.25950i	−0.230900	+	0.972978i	$0.574167\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−21.5971			−0.830042			−0.415021	−	0.909812i	$-0.636226\pi$
$677$	−21.5971			−0.830042			−0.415021	+	0.909812i	$0.636226\pi$
$678$		0			0
$679$	−2.15271	−	1.99524i	−0.0826135	−	0.0765704i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−15.8213	+	9.13444i	−0.605386	+	0.349520i	−0.771157	−	0.636645i	$-0.780322\pi$
$683$	−15.8213	+	9.13444i	−0.605386	+	0.349520i	0.165772	+	0.986164i	$0.446989\pi$
$684$		0			0
$685$			13.0900i			0.500145i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.00513			−0.266874
$690$		0			0
$691$		−	9.91882i		−	0.377330i	−0.982042	−	0.188665i	$-0.939584\pi$
$691$		−	9.91882i		−	0.377330i	0.982042	−	0.188665i	$-0.0604160\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	18.8712i		−	0.715826i
$696$		0			0
$697$	23.5466			0.891890
$698$		0			0
$699$		0			0
$700$		0			0
$701$			4.41270i			0.166665i	0.996522	+	0.0833327i	$0.0265564\pi$
$701$			4.41270i			0.166665i	−0.996522	+	0.0833327i	$0.973444\pi$
$702$		0			0
$703$	−12.6252	+	7.28914i	−0.476167	+	0.274915i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14.9339	−	3.40000i	−0.561647	−	0.127870i
$708$		0			0
$709$	33.5026			1.25822			0.629108	−	0.777318i	$-0.283420\pi$
$709$	33.5026			1.25822			0.629108	+	0.777318i	$0.283420\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.26545	−	9.12003i	−0.197193	−	0.341548i
$714$		0			0
$715$	3.31977	−	5.75000i	0.124152	−	0.215038i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.57296	−	6.18855i	0.133249	−	0.230794i	−0.791678	−	0.610938i	$-0.790792\pi$
$719$	3.57296	−	6.18855i	0.133249	−	0.230794i	0.924927	+	0.380144i	$0.124126\pi$
$720$		0			0
$721$	−38.3615	−	35.5554i	−1.42866	−	1.32415i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			27.4930i			1.02106i
$726$		0			0
$727$	8.71161	−	5.02965i	0.323096	−	0.186539i	−0.329676	−	0.944094i	$-0.606939\pi$
$727$	8.71161	−	5.02965i	0.323096	−	0.186539i	0.652772	+	0.757555i	$0.273606\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10.4972	+	18.1816i	−0.388252	+	0.672473i
$732$		0			0
$733$	21.8681	−	12.6256i	0.807717	−	0.466335i	−0.0384457	−	0.999261i	$-0.512241\pi$
$733$	21.8681	−	12.6256i	0.807717	−	0.466335i	0.846162	+	0.532925i	$0.178907\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.664091	−	0.383413i	−0.0244621	−	0.0141232i
$738$		0			0
$739$	7.29244	+	12.6309i	0.268257	+	0.464634i	0.968412	−	0.249357i	$-0.0802191\pi$
$739$	7.29244	+	12.6309i	0.268257	+	0.464634i	−0.700155	+	0.713991i	$0.746886\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	26.0383	+	15.0332i	0.955251	+	0.551515i	0.894708	−	0.446651i	$-0.147383\pi$
$743$	26.0383	+	15.0332i	0.955251	+	0.551515i	0.0605431	+	0.998166i	$0.480717\pi$
$744$		0			0
$745$	6.75272	+	3.89869i	0.247401	+	0.142837i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.73190	+	16.3917i	−0.136361	+	0.598940i
$750$		0			0
$751$	−2.76386	−	4.78715i	−0.100855	−	0.174686i	0.811182	−	0.584793i	$-0.198824\pi$
$751$	−2.76386	−	4.78715i	−0.100855	−	0.174686i	−0.912037	+	0.410108i	$0.865491\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.8535			−0.394999
$756$		0			0
$757$	−22.2176			−0.807512			−0.403756	−	0.914867i	$-0.632296\pi$
$757$	−22.2176			−0.807512			−0.403756	+	0.914867i	$0.632296\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19.7735	−	34.2487i	−0.716789	−	1.24151i	−0.962265	−	0.272112i	$-0.912278\pi$
$761$	−19.7735	−	34.2487i	−0.716789	−	1.24151i	0.245476	−	0.969403i	$-0.421056\pi$
$762$		0			0
$763$	3.73914	−	4.03424i	0.135366	−	0.146049i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−11.6349	−	6.71742i	−0.420113	−	0.242552i
$768$		0			0
$769$	−27.1448	−	15.6720i	−0.978865	−	0.565148i	−0.0769379	−	0.997036i	$-0.524514\pi$
$769$	−27.1448	−	15.6720i	−0.978865	−	0.565148i	−0.901927	+	0.431888i	$0.857848\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.11009	−	8.85093i	−0.183797	−	0.318346i	0.759373	−	0.650655i	$-0.225506\pi$
$773$	−5.11009	−	8.85093i	−0.183797	−	0.318346i	−0.943171	+	0.332309i	$0.892172\pi$
$774$		0			0
$775$	8.05062	+	4.64803i	0.289187	+	0.166962i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.24394	−	3.60494i	0.223712	−	0.129160i
$780$		0			0
$781$	−16.7887	+	29.0789i	−0.600748	+	1.04053i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−14.2073	+	8.20260i	−0.507081	+	0.292763i
$786$		0			0
$787$			47.8397i			1.70530i	0.522482	+	0.852650i	$0.325006\pi$
$787$			47.8397i			1.70530i	−0.522482	+	0.852650i	$0.674994\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8.75138	+	38.4389i	−0.311163	+	1.36673i
$792$		0			0
$793$	−4.13207	+	7.15696i	−0.146734	+	0.254151i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.7811	−	32.5297i	0.665259	−	1.15226i	−0.313956	−	0.949438i	$-0.601654\pi$
$797$	18.7811	−	32.5297i	0.665259	−	1.15226i	0.979215	−	0.202825i	$-0.0650123\pi$
$798$		0			0
$799$	12.2246	+	21.1737i	0.432476	+	0.749071i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−104.147			−3.67528
$804$		0			0
$805$	2.72934	−	11.9881i	0.0961965	−	0.422526i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	4.88303	−	2.81922i	0.171678	−	0.0991184i	−0.411699	−	0.911320i	$-0.635064\pi$
$809$	4.88303	−	2.81922i	0.171678	−	0.0991184i	0.583377	+	0.812202i	$0.301731\pi$
$810$		0			0
$811$			29.2693i			1.02778i	0.857855	+	0.513892i	$0.171797\pi$
$811$			29.2693i			1.02778i	−0.857855	+	0.513892i	$0.828203\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.19712			0.147019
$816$		0			0
$817$			6.42840i			0.224901i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			47.6951i			1.66457i	0.554346	+	0.832286i	$0.312968\pi$
$821$			47.6951i			1.66457i	−0.554346	+	0.832286i	$0.687032\pi$
$822$		0			0
$823$	−39.5423			−1.37836			−0.689180	−	0.724591i	$-0.742029\pi$
$823$	−39.5423			−1.37836			−0.689180	+	0.724591i	$0.742029\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			27.9149i			0.970696i	0.874321	+	0.485348i	$0.161307\pi$
$827$			27.9149i			0.970696i	−0.874321	+	0.485348i	$0.838693\pi$
$828$		0			0
$829$	−19.5489	+	11.2866i	−0.678962	+	0.391999i	−0.799464	−	0.600714i	$-0.794883\pi$
$829$	−19.5489	+	11.2866i	−0.678962	+	0.391999i	0.120502	+	0.992713i	$0.461550\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−41.2453	−	19.8074i	−1.42907	−	0.686284i
$834$		0			0
$835$	−7.70934			−0.266793
$836$		0			0
$837$		0			0
$838$		0			0
$839$	25.2760	+	43.7794i	0.872625	+	1.51143i	0.859271	+	0.511520i	$0.170918\pi$
$839$	25.2760	+	43.7794i	0.872625	+	1.51143i	0.0133541	+	0.999911i	$0.495749\pi$
$840$		0			0
$841$	10.5367	−	18.2501i	0.363334	−	0.629312i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.32752	−	10.9596i	0.217673	−	0.377021i
$846$		0			0
$847$	−50.3560	+	54.3302i	−1.73025	+	1.86681i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	32.0592i		−	1.09897i
$852$		0			0
$853$	31.0102	−	17.9037i	1.06177	−	0.613013i	0.135848	−	0.990730i	$-0.456624\pi$
$853$	31.0102	−	17.9037i	1.06177	−	0.613013i	0.925921	+	0.377717i	$0.123291\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	13.8199	−	23.9368i	0.472079	−	0.817664i	−0.527411	−	0.849610i	$-0.676837\pi$
$857$	13.8199	−	23.9368i	0.472079	−	0.817664i	0.999490	+	0.0319461i	$0.0101705\pi$
$858$		0			0
$859$	28.0584	−	16.1995i	0.957340	−	0.552721i	0.0619868	−	0.998077i	$-0.480256\pi$
$859$	28.0584	−	16.1995i	0.957340	−	0.552721i	0.895353	+	0.445356i	$0.146923\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14.5080	−	8.37622i	−0.493859	−	0.285130i	0.232315	−	0.972641i	$-0.425370\pi$
$863$	−14.5080	−	8.37622i	−0.493859	−	0.285130i	−0.726174	+	0.687511i	$0.758703\pi$
$864$		0			0
$865$	4.46309	+	7.73029i	0.151749	+	0.262838i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	47.9317	+	27.6734i	1.62597	+	0.938755i
$870$		0			0
$871$	0.107084	+	0.0618250i	0.00362840	+	0.00209486i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.32886	+	23.7137i	0.247761	+	0.801669i
$876$		0			0
$877$	13.0090	+	22.5322i	0.439282	+	0.760859i	0.997634	−	0.0687448i	$-0.0218994\pi$
$877$	13.0090	+	22.5322i	0.439282	+	0.760859i	−0.558352	+	0.829604i	$0.688566\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−8.83562			−0.297680			−0.148840	−	0.988861i	$-0.547554\pi$
$881$	−8.83562			−0.297680			−0.148840	+	0.988861i	$0.547554\pi$
$882$		0			0
$883$	−12.1308			−0.408234			−0.204117	−	0.978946i	$-0.565432\pi$
$883$	−12.1308			−0.408234			−0.204117	+	0.978946i	$0.565432\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.2553	−	17.7626i	−0.344338	−	0.596411i	0.640895	−	0.767628i	$-0.278563\pi$
$887$	−10.2553	−	17.7626i	−0.344338	−	0.596411i	−0.985233	+	0.171218i	$0.945230\pi$
$888$		0			0
$889$	14.5965	+	47.2292i	0.489550	+	1.58402i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.48331	+	3.74314i	0.216956	+	0.125259i
$894$		0			0
$895$	−0.0464332	−	0.0268082i	−0.00155209	−	0.000896100i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8.46552	−	14.6627i	−0.282341	−	0.489029i
$900$		0			0
$901$	−39.3789	−	22.7354i	−1.31190	−	0.757427i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.41652	−	1.39518i	0.0803279	−	0.0463773i
$906$		0			0
$907$	−21.2066	+	36.7308i	−0.704152	+	1.21963i	0.262845	+	0.964838i	$0.415339\pi$
$907$	−21.2066	+	36.7308i	−0.704152	+	1.21963i	−0.966997	+	0.254789i	$0.917994\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.19281	−	3.57542i	0.205177	−	0.118459i	−0.393891	−	0.919157i	$-0.628871\pi$
$911$	6.19281	−	3.57542i	0.205177	−	0.118459i	0.599068	+	0.800698i	$0.295538\pi$
$912$		0			0
$913$			13.4475i			0.445046i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−15.3615	+	16.5739i	−0.507283	+	0.547319i
$918$		0			0
$919$	3.31427	−	5.74049i	0.109328	−	0.189361i	−0.806170	−	0.591683i	$-0.798464\pi$
$919$	3.31427	−	5.74049i	0.109328	−	0.189361i	0.915498	+	0.402322i	$0.131797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.70717	−	4.68895i	0.0891075	−	0.154339i
$924$		0			0
$925$	14.1500	+	24.5084i	0.465248	+	0.805833i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12.9656			0.425387			0.212693	−	0.977119i	$-0.431776\pi$
$929$	12.9656			0.425387			0.212693	+	0.977119i	$0.431776\pi$
$930$		0			0
$931$	−13.9697	+	1.06220i	−0.457837	+	0.0348121i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	37.3237	−	21.5489i	1.22062	−	0.704723i
$936$		0			0
$937$			29.9409i			0.978127i	0.872248	+	0.489063i	$0.162661\pi$
$937$			29.9409i			0.978127i	−0.872248	+	0.489063i	$0.837339\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	54.8921			1.78943			0.894715	−	0.446638i	$-0.147379\pi$
$941$	54.8921			1.78943			0.894715	+	0.446638i	$0.147379\pi$
$942$		0			0
$943$			15.8553i			0.516319i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	2.96187i		−	0.0962478i	−0.998841	−	0.0481239i	$-0.984676\pi$
$947$		−	2.96187i		−	0.0962478i	0.998841	−	0.0481239i	$-0.0153242\pi$
$948$		0			0
$949$	16.7937			0.545145
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	25.3530i		−	0.821264i	−0.911801	−	0.410632i	$-0.865308\pi$
$953$		−	25.3530i		−	0.821264i	0.911801	−	0.410632i	$-0.134692\pi$
$954$		0			0
$955$	22.2059	−	12.8206i	0.718565	−	0.414864i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.28170	+	31.9835i	−0.235138	+	1.03280i
$960$		0			0
$961$	25.2752			0.815329
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.15621	−	5.46672i	−0.101602	−	0.175980i
$966$		0			0
$967$	10.3196	−	17.8741i	0.331857	−	0.574794i	−0.651019	−	0.759062i	$-0.725658\pi$
$967$	10.3196	−	17.8741i	0.331857	−	0.574794i	0.982876	+	0.184268i	$0.0589914\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.1851	+	26.3013i	−0.487312	+	0.844050i	−0.999894	−	0.0145888i	$-0.995356\pi$
$971$	−15.1851	+	26.3013i	−0.487312	+	0.844050i	0.512581	+	0.858639i	$0.328689\pi$
$972$		0			0
$973$	10.4976	−	46.1090i	0.336539	−	1.47819i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			5.11686i			0.163703i	0.996645	+	0.0818514i	$0.0260833\pi$
$977$			5.11686i			0.163703i	−0.996645	+	0.0818514i	$0.973917\pi$
$978$		0			0
$979$	24.1425	−	13.9387i	0.771596	−	0.445481i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.94884	+	3.37549i	−0.0621584	+	0.107661i	−0.895430	−	0.445202i	$-0.853132\pi$
$983$	−1.94884	+	3.37549i	−0.0621584	+	0.107661i	0.833272	+	0.552864i	$0.186465\pi$
$984$		0			0
$985$	0.101481	−	0.0585902i	0.00323346	−	0.00186684i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.2428	−	7.06836i	−0.389297	−	0.224761i
$990$		0			0
$991$	−7.39964	−	12.8166i	−0.235057	−	0.407131i	0.724232	−	0.689556i	$-0.242194\pi$
$991$	−7.39964	−	12.8166i	−0.235057	−	0.407131i	−0.959289	+	0.282425i	$0.908861\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.39990	+	4.84968i	0.266295	+	0.153745i
$996$		0			0
$997$	27.2673	+	15.7428i	0.863563	+	0.498578i	0.865204	−	0.501420i	$-0.167189\pi$
$997$	27.2673	+	15.7428i	0.863563	+	0.498578i	−0.00164102	+	0.999999i	$0.500522\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.bs.a.1097.15		48
3.2	odd	2		504.2.bs.a.257.7	✓	48
4.3	odd	2		3024.2.ca.e.2609.15		48
7.3	odd	6		1512.2.cx.a.17.15		48
9.2	odd	6		1512.2.cx.a.89.15		48
9.7	even	3		504.2.cx.a.425.15	yes	48
12.11	even	2		1008.2.ca.e.257.18		48
21.17	even	6		504.2.cx.a.185.15	yes	48
28.3	even	6		3024.2.df.e.17.15		48
36.7	odd	6		1008.2.df.e.929.10		48
36.11	even	6		3024.2.df.e.1601.15		48
63.38	even	6	inner	1512.2.bs.a.521.15		48
63.52	odd	6		504.2.bs.a.353.7	yes	48
84.59	odd	6		1008.2.df.e.689.10		48
252.115	even	6		1008.2.ca.e.353.18		48
252.227	odd	6		3024.2.ca.e.2033.15		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bs.a.257.7	✓	48	3.2	odd	2
504.2.bs.a.353.7	yes	48	63.52	odd	6
504.2.cx.a.185.15	yes	48	21.17	even	6
504.2.cx.a.425.15	yes	48	9.7	even	3
1008.2.ca.e.257.18		48	12.11	even	2
1008.2.ca.e.353.18		48	252.115	even	6
1008.2.df.e.689.10		48	84.59	odd	6
1008.2.df.e.929.10		48	36.7	odd	6
1512.2.bs.a.521.15		48	63.38	even	6	inner
1512.2.bs.a.1097.15		48	1.1	even	1	trivial
1512.2.cx.a.17.15		48	7.3	odd	6
1512.2.cx.a.89.15		48	9.2	odd	6
3024.2.ca.e.2033.15		48	252.227	odd	6
3024.2.ca.e.2609.15		48	4.3	odd	2
3024.2.df.e.17.15		48	28.3	even	6
3024.2.df.e.1601.15		48	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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.bs (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.527910 + 0.914367i) q^{5} +(0.781227 + 2.52778i) q^{7} +(-5.40824 - 3.12245i) q^{11} +(0.872074 + 0.503492i) q^{13} +(3.26821 + 5.66070i) q^{17} +(1.73329 + 1.00071i) q^{19} +(-3.81168 + 2.20067i) q^{23} +(1.94262 - 3.36472i) q^{25} +(-6.12821 + 3.53813i) q^{29} +2.39266i q^{31} +(-1.89890 + 2.04877i) q^{35} +(-3.64197 + 6.30808i) q^{37} +(1.80119 - 3.11974i) q^{41} +(1.60595 + 2.78159i) q^{43} +3.74047 q^{47} +(-5.77937 + 3.94954i) q^{49} +(-6.02455 + 3.47827i) q^{53} -6.59348i q^{55} -13.3417 q^{59} +8.20683i q^{61} +1.06319i q^{65} +0.122792 q^{67} -5.37678i q^{71} +(14.4429 - 8.33860i) q^{73} +(3.66781 - 16.1102i) q^{77} -8.86272 q^{79} +(-1.07668 - 1.86486i) q^{83} +(-3.45064 + 5.97668i) q^{85} +(-2.23201 + 3.86595i) q^{89} +(-0.591431 + 2.59775i) q^{91} +2.11315i q^{95} +(-0.960756 + 0.554693i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 12 q^{23} - 24 q^{25} - 18 q^{29} - 6 q^{41} - 6 q^{43} - 36 q^{47} + 6 q^{49} - 12 q^{53} + 36 q^{77} - 12 q^{79} - 18 q^{89} + 6 q^{91}+O(q^{100}) \) 48 * q + 12 * q^23 - 24 * q^25 - 18 * q^29 - 6 * q^41 - 6 * q^43 - 36 * q^47 + 6 * q^49 - 12 * q^53 + 36 * q^77 - 12 * q^79 - 18 * q^89 + 6 * q^91

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