LMFDB - Embedded newform 756.3.m.a.557.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,3,Mod(557,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 4]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("756.557");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.m (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$20.5995079856$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			557.2
Character	$\chi$	$=$	756.557
Dual form			756.3.m.a.737.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.05008i q^{5} +(-6.58302 - 2.37987i) q^{7} +O(q^{10})$	$q-8.05008i q^{5} +(-6.58302 - 2.37987i) q^{7} -19.2134i q^{11} +(0.782606 - 1.35551i) q^{13} +(8.39721 + 4.84813i) q^{17} +(5.84871 + 10.1303i) q^{19} +2.07535i q^{23} -39.8037 q^{25} +(-40.2492 + 23.2379i) q^{29} +(11.1747 + 19.3552i) q^{31} +(-19.1582 + 52.9938i) q^{35} +(-30.6213 - 53.0376i) q^{37} +(-48.9545 - 28.2639i) q^{41} +(21.6300 + 37.4643i) q^{43} +(5.50180 + 3.17647i) q^{47} +(37.6724 + 31.3335i) q^{49} +(55.3510 + 31.9569i) q^{53} -154.669 q^{55} +(-27.7798 + 16.0387i) q^{59} +(37.2708 - 64.5549i) q^{61} +(-10.9120 - 6.30004i) q^{65} +(-2.38096 - 4.12394i) q^{67} +33.1772i q^{71} +(6.60726 - 11.4441i) q^{73} +(-45.7254 + 126.482i) q^{77} +(-23.1230 + 40.0502i) q^{79} +(-81.0445 + 46.7910i) q^{83} +(39.0278 - 67.5982i) q^{85} +(-55.9009 + 32.2744i) q^{89} +(-8.37786 + 7.06087i) q^{91} +(81.5494 - 47.0826i) q^{95} +(-96.6875 - 167.468i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - q^{7}+O(q^{10}) $ 32 * q - q^7	$ 32 q - q^{7} - 5 q^{13} + 27 q^{17} - 14 q^{19} - 160 q^{25} - 36 q^{29} - 8 q^{31} + 45 q^{35} - 11 q^{37} - 72 q^{41} + 16 q^{43} + 108 q^{47} + 35 q^{49} - 180 q^{53} - 24 q^{55} - 45 q^{59} - 41 q^{61} + 81 q^{65} - 35 q^{67} - 98 q^{73} - 225 q^{77} - 71 q^{79} - 30 q^{85} + 189 q^{89} + 109 q^{91} + 288 q^{95} + 19 q^{97}+O(q^{100}) $ 32 * q - q^7 - 5 * q^13 + 27 * q^17 - 14 * q^19 - 160 * q^25 - 36 * q^29 - 8 * q^31 + 45 * q^35 - 11 * q^37 - 72 * q^41 + 16 * q^43 + 108 * q^47 + 35 * q^49 - 180 * q^53 - 24 * q^55 - 45 * q^59 - 41 * q^61 + 81 * q^65 - 35 * q^67 - 98 * q^73 - 225 * q^77 - 71 * q^79 - 30 * q^85 + 189 * q^89 + 109 * q^91 + 288 * q^95 + 19 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$	$1$

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
$5$		−	8.05008i		−	1.61002i	−0.593264	−	0.805008i	$-0.702161\pi$
$5$		−	8.05008i		−	1.61002i	0.593264	−	0.805008i	$-0.297839\pi$
$6$		0			0
$7$	−6.58302	−	2.37987i	−0.940432	−	0.339982i
$7$	−6.58302	−	2.37987i	−0.940432	−	0.339982i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	19.2134i		−	1.74667i	−0.487120	−	0.873335i	$-0.661952\pi$
$11$		−	19.2134i		−	1.74667i	0.487120	−	0.873335i	$-0.338048\pi$
$12$		0			0
$13$	0.782606	−	1.35551i	0.0602005	−	0.104270i	−0.834354	−	0.551228i	$-0.814159\pi$
$13$	0.782606	−	1.35551i	0.0602005	−	0.104270i	0.894555	+	0.446958i	$0.147493\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	8.39721	+	4.84813i	0.493953	+	0.285184i	0.726213	−	0.687470i	$-0.241279\pi$
$17$	8.39721	+	4.84813i	0.493953	+	0.285184i	−0.232260	+	0.972654i	$0.574612\pi$
$18$		0			0
$19$	5.84871	+	10.1303i	0.307827	+	0.533172i	0.977887	−	0.209135i	$-0.0670649\pi$
$19$	5.84871	+	10.1303i	0.307827	+	0.533172i	−0.670060	+	0.742307i	$0.733732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.07535i			0.0902328i	0.998982	+	0.0451164i	$0.0143659\pi$
$23$			2.07535i			0.0902328i	−0.998982	+	0.0451164i	$0.985634\pi$
$24$		0			0
$25$	−39.8037			−1.59215
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−40.2492	+	23.2379i	−1.38790	+	0.801306i	−0.993079	−	0.117451i	$-0.962528\pi$
$29$	−40.2492	+	23.2379i	−1.38790	+	0.801306i	−0.394823	+	0.918757i	$0.629194\pi$
$30$		0			0
$31$	11.1747	+	19.3552i	0.360474	+	0.624360i	0.988039	−	0.154205i	$-0.0492815\pi$
$31$	11.1747	+	19.3552i	0.360474	+	0.624360i	−0.627564	+	0.778565i	$0.715948\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−19.1582	+	52.9938i	−0.547376	+	1.51411i
$36$		0			0
$37$	−30.6213	−	53.0376i	−0.827603	−	1.43345i	−0.899914	−	0.436068i	$-0.856371\pi$
$37$	−30.6213	−	53.0376i	−0.827603	−	1.43345i	0.0723114	−	0.997382i	$-0.476962\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−48.9545	−	28.2639i	−1.19401	−	0.689363i	−0.234797	−	0.972044i	$-0.575443\pi$
$41$	−48.9545	−	28.2639i	−1.19401	−	0.689363i	−0.959214	+	0.282682i	$0.908776\pi$
$42$		0			0
$43$	21.6300	+	37.4643i	0.503023	+	0.871262i	0.999994	+	0.00349442i	$0.00111231\pi$
$43$	21.6300	+	37.4643i	0.503023	+	0.871262i	−0.496971	+	0.867767i	$0.665554\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.50180	+	3.17647i	0.117060	+	0.0675844i	0.557387	−	0.830253i	$-0.311804\pi$
$47$	5.50180	+	3.17647i	0.117060	+	0.0675844i	−0.440327	+	0.897837i	$0.645137\pi$
$48$		0			0
$49$	37.6724	+	31.3335i	0.768824	+	0.639460i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	55.3510	+	31.9569i	1.04436	+	0.602960i	0.921065	−	0.389410i	$-0.127321\pi$
$53$	55.3510	+	31.9569i	1.04436	+	0.602960i	0.123293	+	0.992370i	$0.460654\pi$
$54$		0			0
$55$	−154.669			−2.81217
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−27.7798	+	16.0387i	−0.470844	+	0.271842i	−0.716593	−	0.697492i	$-0.754299\pi$
$59$	−27.7798	+	16.0387i	−0.470844	+	0.271842i	0.245749	+	0.969333i	$0.420966\pi$
$60$		0			0
$61$	37.2708	−	64.5549i	0.610997	−	1.05828i	−0.380076	−	0.924955i	$-0.624102\pi$
$61$	37.2708	−	64.5549i	0.610997	−	1.05828i	0.991073	−	0.133322i	$-0.0425645\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.9120	−	6.30004i	−0.167877	−	0.0969237i
$66$		0			0
$67$	−2.38096	−	4.12394i	−0.0355367	−	0.0615514i	0.847710	−	0.530460i	$-0.177981\pi$
$67$	−2.38096	−	4.12394i	−0.0355367	−	0.0615514i	−0.883247	+	0.468908i	$0.844647\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			33.1772i			0.467284i	0.972323	+	0.233642i	$0.0750644\pi$
$71$			33.1772i			0.467284i	−0.972323	+	0.233642i	$0.924936\pi$
$72$		0			0
$73$	6.60726	−	11.4441i	0.0905104	−	0.156769i	−0.817216	−	0.576332i	$-0.804483\pi$
$73$	6.60726	−	11.4441i	0.0905104	−	0.156769i	0.907726	+	0.419563i	$0.137817\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−45.7254	+	126.482i	−0.593837	+	1.64262i
$78$		0			0
$79$	−23.1230	+	40.0502i	−0.292696	+	0.506965i	−0.974446	−	0.224620i	$-0.927886\pi$
$79$	−23.1230	+	40.0502i	−0.292696	+	0.506965i	0.681750	+	0.731585i	$0.261219\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−81.0445	+	46.7910i	−0.976439	+	0.563748i	−0.901193	−	0.433417i	$-0.857308\pi$
$83$	−81.0445	+	46.7910i	−0.976439	+	0.563748i	−0.0752462	+	0.997165i	$0.523974\pi$
$84$		0			0
$85$	39.0278	−	67.5982i	0.459151	−	0.795273i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−55.9009	+	32.2744i	−0.628099	+	0.362633i	−0.780016	−	0.625760i	$-0.784789\pi$
$89$	−55.9009	+	32.2744i	−0.628099	+	0.362633i	0.151916	+	0.988393i	$0.451456\pi$
$90$		0			0
$91$	−8.37786	+	7.06087i	−0.0920644	+	0.0775920i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	81.5494	−	47.0826i	0.858415	−	0.495606i
$96$		0			0
$97$	−96.6875	−	167.468i	−0.996779	−	1.72647i	−0.567846	−	0.823135i	$-0.692223\pi$
$97$	−96.6875	−	167.468i	−0.996779	−	1.72647i	−0.428933	−	0.903337i	$-0.641110\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	67.4181i		−	0.667506i	−0.942661	−	0.333753i	$-0.891685\pi$
$101$		−	67.4181i		−	0.667506i	0.942661	−	0.333753i	$-0.108315\pi$
$102$		0			0
$103$	−80.2840			−0.779456			−0.389728	−	0.920930i	$-0.627431\pi$
$103$	−80.2840			−0.779456			−0.389728	+	0.920930i	$0.627431\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	108.086	−	62.4035i	1.01015	−	0.583210i	0.0989143	−	0.995096i	$-0.468463\pi$
$107$	108.086	−	62.4035i	1.01015	−	0.583210i	0.911236	+	0.411886i	$0.135130\pi$
$108$		0			0
$109$	80.8079	−	139.963i	0.741357	−	1.28407i	−0.210521	−	0.977589i	$-0.567516\pi$
$109$	80.8079	−	139.963i	0.741357	−	1.28407i	0.951878	−	0.306478i	$-0.0991506\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−28.7804	−	16.6164i	−0.254694	−	0.147048i	0.367218	−	0.930135i	$-0.380310\pi$
$113$	−28.7804	−	16.6164i	−0.254694	−	0.147048i	−0.621912	+	0.783087i	$0.713644\pi$
$114$		0			0
$115$	16.7068			0.145276
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−43.7411	−	51.8997i	−0.367572	−	0.436132i
$120$		0			0
$121$	−248.154			−2.05086
$122$		0			0
$123$		0			0
$124$		0			0
$125$			119.171i			0.953370i
$126$		0			0
$127$	−194.763			−1.53357			−0.766785	−	0.641904i	$-0.778145\pi$
$127$	−194.763			−1.53357			−0.766785	+	0.641904i	$0.778145\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			117.044i			0.893463i	0.894668	+	0.446731i	$0.147412\pi$
$131$			117.044i			0.893463i	−0.894668	+	0.446731i	$0.852588\pi$
$132$		0			0
$133$	−14.3934	−	80.6069i	−0.108221	−	0.606067i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	79.8132i		−	0.582578i	−0.956635	−	0.291289i	$-0.905916\pi$
$137$		−	79.8132i		−	0.582578i	0.956635	−	0.291289i	$-0.0940841\pi$
$138$		0			0
$139$	−37.9101	+	65.6622i	−0.272734	+	0.472390i	−0.969561	−	0.244850i	$-0.921261\pi$
$139$	−37.9101	+	65.6622i	−0.272734	+	0.472390i	0.696827	+	0.717240i	$0.254595\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−26.0440	−	15.0365i	−0.182126	−	0.105150i
$144$		0			0
$145$	187.067	+	324.009i	1.29011	+	2.23454i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	108.954i		−	0.731236i	−0.930765	−	0.365618i	$-0.880858\pi$
$149$		−	108.954i		−	0.731236i	0.930765	−	0.365618i	$-0.119142\pi$
$150$		0			0
$151$	163.434			1.08234			0.541172	−	0.840912i	$-0.317981\pi$
$151$	163.434			1.08234			0.541172	+	0.840912i	$0.317981\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	155.811	−	89.9573i	1.00523	−	0.580369i
$156$		0			0
$157$	15.7565	+	27.2910i	0.100360	+	0.173828i	0.911833	−	0.410561i	$-0.134667\pi$
$157$	15.7565	+	27.2910i	0.100360	+	0.173828i	−0.811473	+	0.584390i	$0.801334\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.93908	−	13.6621i	0.0306775	−	0.0848578i
$162$		0			0
$163$	149.467	+	258.884i	0.916974	+	1.58824i	0.803986	+	0.594648i	$0.202709\pi$
$163$	149.467	+	258.884i	0.916974	+	1.58824i	0.112988	+	0.993596i	$0.463958\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−226.397	−	130.710i	−1.35567	−	0.782695i	−0.366632	−	0.930366i	$-0.619489\pi$
$167$	−226.397	−	130.710i	−1.35567	−	0.782695i	−0.989037	+	0.147671i	$0.952822\pi$
$168$		0			0
$169$	83.2751	+	144.237i	0.492752	+	0.853471i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−166.695	−	96.2412i	−0.963553	−	0.556307i	−0.0662881	−	0.997801i	$-0.521116\pi$
$173$	−166.695	−	96.2412i	−0.963553	−	0.556307i	−0.897265	+	0.441493i	$0.854449\pi$
$174$		0			0
$175$	262.029	+	94.7279i	1.49731	+	0.541302i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	303.708	+	175.346i	1.69669	+	0.979585i	0.948860	+	0.315696i	$0.102238\pi$
$179$	303.708	+	175.346i	1.69669	+	0.979585i	0.747831	+	0.663889i	$0.231095\pi$
$180$		0			0
$181$	227.913			1.25919			0.629594	−	0.776925i	$-0.283221\pi$
$181$	227.913			1.25919			0.629594	+	0.776925i	$0.283221\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−426.957	+	246.504i	−2.30788	+	1.33245i
$186$		0			0
$187$	93.1490	−	161.339i	0.498123	−	0.862774i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	162.483	+	93.8095i	0.850695	+	0.491149i	0.860885	−	0.508799i	$-0.169910\pi$
$191$	162.483	+	93.8095i	0.850695	+	0.491149i	−0.0101901	+	0.999948i	$0.503244\pi$
$192$		0			0
$193$	17.9056	+	31.0135i	0.0927753	+	0.160691i	0.908678	−	0.417498i	$-0.137093\pi$
$193$	17.9056	+	31.0135i	0.0927753	+	0.160691i	−0.815903	+	0.578189i	$0.803760\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	110.071i		−	0.558736i	−0.960184	−	0.279368i	$-0.909875\pi$
$197$		−	110.071i		−	0.558736i	0.960184	−	0.279368i	$-0.0901250\pi$
$198$		0			0
$199$	34.5156	−	59.7829i	0.173445	−	0.300416i	−0.766177	−	0.642630i	$-0.777843\pi$
$199$	34.5156	−	59.7829i	0.173445	−	0.300416i	0.939622	+	0.342214i	$0.111177\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	320.264	−	57.1875i	1.57766	−	0.281712i
$204$		0			0
$205$	−227.526	+	394.087i	−1.10988	+	1.92238i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	194.637	−	112.373i	0.931275	−	0.537672i
$210$		0			0
$211$	122.482	−	212.145i	0.580484	−	1.00543i	−0.414938	−	0.909849i	$-0.636197\pi$
$211$	122.482	−	212.145i	0.580484	−	1.00543i	0.995422	−	0.0955775i	$-0.0304698\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	301.590	−	174.123i	1.40274	−	0.809875i
$216$		0			0
$217$	−27.5005	−	154.010i	−0.126730	−	0.709723i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.1434	−	7.58835i	0.0594724	−	0.0343364i
$222$		0			0
$223$	−50.2344	−	87.0085i	−0.225266	−	0.390173i	0.731133	−	0.682235i	$-0.238992\pi$
$223$	−50.2344	−	87.0085i	−0.225266	−	0.390173i	−0.956399	+	0.292062i	$0.905659\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	166.877i		−	0.735142i	−0.929996	−	0.367571i	$-0.880190\pi$
$227$		−	166.877i		−	0.735142i	0.929996	−	0.367571i	$-0.119810\pi$
$228$		0			0
$229$	−277.529			−1.21192			−0.605959	−	0.795496i	$-0.707210\pi$
$229$	−277.529			−1.21192			−0.605959	+	0.795496i	$0.707210\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−41.2378	+	23.8087i	−0.176986	+	0.102183i	−0.585876	−	0.810401i	$-0.699249\pi$
$233$	−41.2378	+	23.8087i	−0.176986	+	0.102183i	0.408890	+	0.912584i	$0.365916\pi$
$234$		0			0
$235$	25.5708	−	44.2899i	0.108812	−	0.188468i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−199.849	−	115.383i	−0.836187	−	0.482773i	0.0197796	−	0.999804i	$-0.493704\pi$
$239$	−199.849	−	115.383i	−0.836187	−	0.482773i	−0.855966	+	0.517032i	$0.827037\pi$
$240$		0			0
$241$	297.477			1.23434			0.617172	−	0.786828i	$-0.288278\pi$
$241$	297.477			1.23434			0.617172	+	0.786828i	$0.288278\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	252.237	−	303.266i	1.02954	−	1.23782i
$246$		0			0
$247$	18.3089			0.0741252
$248$		0			0
$249$		0			0
$250$		0			0
$251$			85.2548i			0.339661i	0.985473	+	0.169830i	$0.0543220\pi$
$251$			85.2548i			0.339661i	−0.985473	+	0.169830i	$0.945678\pi$
$252$		0			0
$253$	39.8746			0.157607
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	71.7566i		−	0.279208i	−0.990207	−	0.139604i	$-0.955417\pi$
$257$		−	71.7566i		−	0.279208i	0.990207	−	0.139604i	$-0.0445830\pi$
$258$		0			0
$259$	75.3578	+	422.023i	0.290957	+	1.62943i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	341.324i		−	1.29781i	−0.760869	−	0.648906i	$-0.775227\pi$
$263$		−	341.324i		−	1.29781i	0.760869	−	0.648906i	$-0.224773\pi$
$264$		0			0
$265$	257.255	−	445.580i	0.970775	−	1.68143i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−182.681	−	105.471i	−0.679111	−	0.392085i	0.120409	−	0.992724i	$-0.461579\pi$
$269$	−182.681	−	105.471i	−0.679111	−	0.392085i	−0.799520	+	0.600639i	$0.794913\pi$
$270$		0			0
$271$	−129.425	−	224.171i	−0.477584	−	0.827199i	0.522086	−	0.852893i	$-0.325154\pi$
$271$	−129.425	−	224.171i	−0.477584	−	0.827199i	−0.999670	+	0.0256934i	$0.991821\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			764.764i			2.78096i
$276$		0			0
$277$	−409.257			−1.47746			−0.738731	−	0.674000i	$-0.764575\pi$
$277$	−409.257			−1.47746			−0.738731	+	0.674000i	$0.764575\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	277.104	−	159.986i	0.986134	−	0.569344i	0.0820173	−	0.996631i	$-0.473864\pi$
$281$	277.104	−	159.986i	0.986134	−	0.569344i	0.904116	+	0.427286i	$0.140530\pi$
$282$		0			0
$283$	−61.8838	−	107.186i	−0.218671	−	0.378749i	0.735731	−	0.677274i	$-0.236839\pi$
$283$	−61.8838	−	107.186i	−0.218671	−	0.378749i	−0.954402	+	0.298525i	$0.903505\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	255.004	+	302.567i	0.888515	+	1.05424i
$288$		0			0
$289$	−97.4913	−	168.860i	−0.337340	−	0.584290i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.0474	−	9.26498i	−0.0547693	−	0.0316211i	0.472365	−	0.881403i	$-0.343400\pi$
$293$	−16.0474	−	9.26498i	−0.0547693	−	0.0316211i	−0.527135	+	0.849782i	$0.676734\pi$
$294$		0			0
$295$	129.112	+	223.629i	0.437669	+	0.758066i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.81317	+	1.62418i	0.00940860	+	0.00543206i
$300$		0			0
$301$	−53.2306	−	298.105i	−0.176846	−	0.990381i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−519.672	−	300.033i	−1.70384	−	0.983714i
$306$		0			0
$307$	363.739			1.18482			0.592408	−	0.805638i	$-0.298177\pi$
$307$	363.739			1.18482			0.592408	+	0.805638i	$0.298177\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	130.067	−	75.0941i	0.418221	−	0.241460i	−0.276095	−	0.961130i	$-0.589040\pi$
$311$	130.067	−	75.0941i	0.418221	−	0.241460i	0.694316	+	0.719670i	$0.255707\pi$
$312$		0			0
$313$	117.107	−	202.836i	0.374145	−	0.648038i	−0.616054	−	0.787704i	$-0.711270\pi$
$313$	117.107	−	202.836i	0.374145	−	0.648038i	0.990199	+	0.139666i	$0.0446029\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	340.075	+	196.342i	1.07279	+	0.619377i	0.928943	−	0.370223i	$-0.120719\pi$
$317$	340.075	+	196.342i	1.07279	+	0.619377i	0.143849	+	0.989600i	$0.454052\pi$
$318$		0			0
$319$	446.478	+	773.322i	1.39962	+	2.42421i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			113.421i			0.351149i
$324$		0			0
$325$	−31.1506	+	53.9545i	−0.0958481	+	0.166014i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−28.6589	−	34.0044i	−0.0871091	−	0.103357i
$330$		0			0
$331$	−76.0134	+	131.659i	−0.229648	+	0.397761i	−0.957704	−	0.287756i	$-0.907091\pi$
$331$	−76.0134	+	131.659i	−0.229648	+	0.397761i	0.728056	+	0.685518i	$0.240424\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−33.1981	+	19.1669i	−0.0990987	+	0.0572147i
$336$		0			0
$337$	−142.286	+	246.447i	−0.422214	+	0.731295i	−0.996156	−	0.0876006i	$-0.972080\pi$
$337$	−142.286	+	246.447i	−0.422214	+	0.731295i	0.573942	+	0.818896i	$0.305413\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	371.878	−	214.704i	1.09055	−	0.629630i
$342$		0			0
$343$	−173.428	−	295.925i	−0.505622	−	0.862755i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−115.858	+	66.8907i	−0.333885	+	0.192768i	−0.657565	−	0.753398i	$-0.728413\pi$
$347$	−115.858	+	66.8907i	−0.333885	+	0.192768i	0.323680	+	0.946167i	$0.395080\pi$
$348$		0			0
$349$	−231.018	−	400.135i	−0.661943	−	1.14652i	−0.980105	−	0.198482i	$-0.936399\pi$
$349$	−231.018	−	400.135i	−0.661943	−	1.14652i	0.318162	−	0.948036i	$-0.396934\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			644.199i			1.82492i	0.409160	+	0.912462i	$0.365822\pi$
$353$			644.199i			1.82492i	−0.409160	+	0.912462i	$0.634178\pi$
$354$		0			0
$355$	267.079			0.752335
$356$		0			0
$357$		0			0
$358$		0			0
$359$	157.335	−	90.8373i	0.438259	−	0.253029i	−0.264600	−	0.964358i	$-0.585240\pi$
$359$	157.335	−	90.8373i	0.438259	−	0.253029i	0.702859	+	0.711329i	$0.251907\pi$
$360$		0			0
$361$	112.085	−	194.137i	0.310485	−	0.537776i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−92.1260	−	53.1889i	−0.252400	−	0.145723i
$366$		0			0
$367$	−563.242			−1.53472			−0.767360	−	0.641217i	$-0.778430\pi$
$367$	−563.242			−1.53472			−0.767360	+	0.641217i	$0.778430\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−288.323	−	342.101i	−0.777152	−	0.922106i
$372$		0			0
$373$	−44.1540			−0.118375			−0.0591877	−	0.998247i	$-0.518851\pi$
$373$	−44.1540			−0.118375			−0.0591877	+	0.998247i	$0.518851\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			72.7444i			0.192956i
$378$		0			0
$379$	106.066			0.279857			0.139928	−	0.990162i	$-0.455313\pi$
$379$	106.066			0.279857			0.139928	+	0.990162i	$0.455313\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	313.826i		−	0.819390i	−0.912223	−	0.409695i	$-0.865635\pi$
$383$		−	313.826i		−	0.819390i	0.912223	−	0.409695i	$-0.134365\pi$
$384$		0			0
$385$	1018.19	+	368.093i	2.64465	+	0.956086i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			61.5101i			0.158124i	0.996870	+	0.0790618i	$0.0251924\pi$
$389$			61.5101i			0.158124i	−0.996870	+	0.0790618i	$0.974808\pi$
$390$		0			0
$391$	−10.0616	+	17.4272i	−0.0257330	+	0.0445708i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	322.407	+	186.142i	0.816221	+	0.471245i
$396$		0			0
$397$	−65.6615	−	113.729i	−0.165394	−	0.286471i	0.771401	−	0.636349i	$-0.219556\pi$
$397$	−65.6615	−	113.729i	−0.165394	−	0.286471i	−0.936795	+	0.349878i	$0.886223\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			209.667i			0.522861i	0.965222	+	0.261431i	$0.0841943\pi$
$401$			209.667i			0.522861i	−0.965222	+	0.261431i	$0.915806\pi$
$402$		0			0
$403$	34.9816			0.0868029
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1019.03	+	588.338i	−2.50376	+	1.44555i
$408$		0			0
$409$	−136.488	−	236.404i	−0.333711	−	0.578005i	0.649525	−	0.760340i	$-0.274968\pi$
$409$	−136.488	−	236.404i	−0.333711	−	0.578005i	−0.983236	+	0.182335i	$0.941634\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	221.045	−	39.4705i	0.535218	−	0.0955702i
$414$		0			0
$415$	376.672	+	652.414i	0.907642	+	1.57208i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−93.8050	−	54.1584i	−0.223878	−	0.129256i	0.383866	−	0.923389i	$-0.374592\pi$
$419$	−93.8050	−	54.1584i	−0.223878	−	0.129256i	−0.607745	+	0.794132i	$0.707926\pi$
$420$		0			0
$421$	−268.137	−	464.427i	−0.636905	−	1.10315i	−0.986108	−	0.166104i	$-0.946881\pi$
$421$	−268.137	−	464.427i	−0.636905	−	1.10315i	0.349203	−	0.937047i	$-0.386452\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−334.240	−	192.974i	−0.786448	−	0.454056i
$426$		0			0
$427$	−398.987	+	336.267i	−0.934396	+	0.787510i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−235.401	−	135.909i	−0.546173	−	0.315333i	0.201404	−	0.979508i	$-0.435450\pi$
$431$	−235.401	−	135.909i	−0.546173	−	0.315333i	−0.747577	+	0.664175i	$0.768783\pi$
$432$		0			0
$433$	−428.323			−0.989199			−0.494600	−	0.869121i	$-0.664685\pi$
$433$	−428.323			−0.989199			−0.494600	+	0.869121i	$0.664685\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.0239	+	12.1381i	−0.0481096	+	0.0277761i
$438$		0			0
$439$	153.290	−	265.505i	0.349179	−	0.604795i	−0.636925	−	0.770926i	$-0.719794\pi$
$439$	153.290	−	265.505i	0.349179	−	0.604795i	0.986104	+	0.166130i	$0.0531272\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	251.820	+	145.388i	0.568443	+	0.328191i	0.756527	−	0.653962i	$-0.226895\pi$
$443$	251.820	+	145.388i	0.568443	+	0.328191i	−0.188084	+	0.982153i	$0.560228\pi$
$444$		0			0
$445$	259.811	+	450.006i	0.583845	+	1.01125i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	278.892i		−	0.621140i	−0.950550	−	0.310570i	$-0.899480\pi$
$449$		−	278.892i		−	0.621140i	0.950550	−	0.310570i	$-0.100520\pi$
$450$		0			0
$451$	−543.044	+	940.580i	−1.20409	+	2.08554i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	56.8406	+	67.4425i	0.124924	+	0.148225i
$456$		0			0
$457$	−221.608	+	383.836i	−0.484919	+	0.839904i	−0.999850	−	0.0173275i	$-0.994484\pi$
$457$	−221.608	+	383.836i	−0.484919	+	0.839904i	0.514931	+	0.857232i	$0.327818\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−134.497	+	77.6519i	−0.291751	+	0.168442i	−0.638731	−	0.769430i	$-0.720540\pi$
$461$	−134.497	+	77.6519i	−0.291751	+	0.168442i	0.346980	+	0.937872i	$0.387207\pi$
$462$		0			0
$463$	201.958	−	349.802i	0.436195	−	0.755511i	−0.561198	−	0.827682i	$-0.689659\pi$
$463$	201.958	−	349.802i	0.436195	−	0.755511i	0.997392	+	0.0721704i	$0.0229925\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−358.548	+	207.008i	−0.767769	+	0.443272i	−0.832078	−	0.554658i	$-0.812849\pi$
$467$	−358.548	+	207.008i	−0.767769	+	0.443272i	0.0643092	+	0.997930i	$0.479516\pi$
$468$		0			0
$469$	5.85945	+	32.8144i	0.0124935	+	0.0699668i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	719.815	−	415.585i	1.52181	−	0.878616i
$474$		0			0
$475$	−232.800	−	403.222i	−0.490106	−	0.848889i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	832.818i		−	1.73866i	−0.494232	−	0.869330i	$-0.664551\pi$
$479$		−	832.818i		−	1.73866i	0.494232	−	0.869330i	$-0.335449\pi$
$480$		0			0
$481$	−95.8576			−0.199288
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1348.13	+	778.342i	−2.77965	+	1.60483i
$486$		0			0
$487$	90.7601	−	157.201i	0.186366	−	0.322795i	−0.757670	−	0.652638i	$-0.773662\pi$
$487$	90.7601	−	157.201i	0.186366	−	0.322795i	0.944036	+	0.329843i	$0.106996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	213.443	+	123.232i	0.434712	+	0.250981i	0.701352	−	0.712815i	$-0.252580\pi$
$491$	213.443	+	123.232i	0.434712	+	0.250981i	−0.266640	+	0.963796i	$0.585914\pi$
$492$		0			0
$493$	−450.641			−0.914079
$494$		0			0
$495$		0			0
$496$		0			0
$497$	78.9575	−	218.406i	0.158868	−	0.439449i
$498$		0			0
$499$	−927.638			−1.85899			−0.929497	−	0.368829i	$-0.879759\pi$
$499$	−927.638			−1.85899			−0.929497	+	0.368829i	$0.879759\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			569.020i			1.13125i	0.824662	+	0.565626i	$0.191366\pi$
$503$			569.020i			1.13125i	−0.824662	+	0.565626i	$0.808634\pi$
$504$		0			0
$505$	−542.721			−1.07470
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	631.624i		−	1.24091i	−0.784242	−	0.620456i	$-0.786948\pi$
$509$		−	631.624i		−	1.24091i	0.784242	−	0.620456i	$-0.213052\pi$
$510$		0			0
$511$	−70.7313	+	59.6124i	−0.138417	+	0.116658i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			646.292i			1.25494i
$516$		0			0
$517$	61.0307	−	105.708i	0.118048	−	0.204465i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	278.992	+	161.076i	0.535494	+	0.309168i	0.743251	−	0.669013i	$-0.233283\pi$
$521$	278.992	+	161.076i	0.535494	+	0.309168i	−0.207757	+	0.978181i	$0.566616\pi$
$522$		0			0
$523$	−204.092	−	353.498i	−0.390233	−	0.675904i	0.602247	−	0.798310i	$-0.294272\pi$
$523$	−204.092	−	353.498i	−0.390233	−	0.675904i	−0.992480	+	0.122406i	$0.960939\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			216.706i			0.411206i
$528$		0			0
$529$	524.693			0.991858
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−76.6241	+	44.2389i	−0.143760	+	0.0829999i
$534$		0			0
$535$	−502.353	−	870.101i	−0.938977	−	1.62636i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	602.023	−	723.814i	1.11693	−	1.34288i
$540$		0			0
$541$	−182.747	−	316.528i	−0.337795	−	0.585079i	0.646222	−	0.763149i	$-0.276348\pi$
$541$	−182.747	−	316.528i	−0.337795	−	0.585079i	−0.984018	+	0.178070i	$0.943015\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1126.72	−	650.510i	−2.06737	−	1.19360i
$546$		0			0
$547$	−294.880	−	510.747i	−0.539086	−	0.933724i	−0.998954	−	0.0457367i	$-0.985436\pi$
$547$	−294.880	−	510.747i	−0.539086	−	0.933724i	0.459868	−	0.887987i	$-0.347897\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−470.811	−	271.823i	−0.854467	−	0.493327i
$552$		0			0
$553$	247.534	−	208.622i	0.447620	−	0.377254i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	426.103	+	246.011i	0.764996	+	0.441671i	0.831087	−	0.556143i	$-0.187719\pi$
$557$	426.103	+	246.011i	0.764996	+	0.441671i	−0.0660907	+	0.997814i	$0.521053\pi$
$558$		0			0
$559$	67.7111			0.121129
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.3180	−	10.5759i	0.0325365	−	0.0187850i	−0.483644	−	0.875265i	$-0.660687\pi$
$563$	18.3180	−	10.5759i	0.0325365	−	0.0187850i	0.516180	+	0.856480i	$0.327354\pi$
$564$		0			0
$565$	−133.763	+	231.685i	−0.236749	+	0.410061i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	133.437	+	77.0401i	0.234512	+	0.135396i	0.612652	−	0.790353i	$-0.290103\pi$
$569$	133.437	+	77.0401i	0.234512	+	0.135396i	−0.378140	+	0.925749i	$0.623436\pi$
$570$		0			0
$571$	−22.5846	−	39.1178i	−0.0395528	−	0.0685075i	0.845571	−	0.533862i	$-0.179260\pi$
$571$	−22.5846	−	39.1178i	−0.0395528	−	0.0685075i	−0.885124	+	0.465355i	$0.845927\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	82.6069i		−	0.143664i
$576$		0			0
$577$	13.1264	−	22.7356i	0.0227494	−	0.0394032i	−0.854427	−	0.519572i	$-0.826091\pi$
$577$	13.1264	−	22.7356i	0.0227494	−	0.0394032i	0.877176	+	0.480169i	$0.159425\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	644.875	−	115.151i	1.10994	−	0.198194i
$582$		0			0
$583$	614.000	−	1063.48i	1.05317	−	1.82415i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−606.599	+	350.220i	−1.03339	+	0.596627i	−0.917953	−	0.396688i	$-0.870159\pi$
$587$	−606.599	+	350.220i	−1.03339	+	0.596627i	−0.115434	+	0.993315i	$0.536826\pi$
$588$		0			0
$589$	−130.715	+	226.405i	−0.221927	+	0.384389i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−679.184	+	392.127i	−1.14534	+	0.661260i	−0.947746	−	0.319025i	$-0.896645\pi$
$593$	−679.184	+	392.127i	−1.14534	+	0.661260i	−0.197589	+	0.980285i	$0.563311\pi$
$594$		0			0
$595$	−417.796	+	352.119i	−0.702179	+	0.591797i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	449.279	−	259.391i	0.750048	−	0.433041i	−0.0756631	−	0.997133i	$-0.524107\pi$
$599$	449.279	−	259.391i	0.750048	−	0.433041i	0.825711	+	0.564093i	$0.190774\pi$
$600$		0			0
$601$	1.71229	+	2.96577i	0.00284907	+	0.00493473i	0.867446	−	0.497531i	$-0.165760\pi$
$601$	1.71229	+	2.96577i	0.00284907	+	0.00493473i	−0.864597	+	0.502465i	$0.832426\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			1997.66i			3.30191i
$606$		0			0
$607$	439.277			0.723686			0.361843	−	0.932239i	$-0.382148\pi$
$607$	439.277			0.723686			0.361843	+	0.932239i	$0.382148\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.61149	−	4.97184i	0.0140941	−	0.00813722i
$612$		0			0
$613$	56.2801	−	97.4800i	0.0918109	−	0.159021i	−0.816462	−	0.577399i	$-0.804068\pi$
$613$	56.2801	−	97.4800i	0.0918109	−	0.159021i	0.908273	+	0.418378i	$0.137401\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	259.198	+	149.648i	0.420094	+	0.242541i	0.695117	−	0.718896i	$-0.255352\pi$
$617$	259.198	+	149.648i	0.420094	+	0.242541i	−0.275024	+	0.961437i	$0.588686\pi$
$618$		0			0
$619$	286.982			0.463621			0.231811	−	0.972761i	$-0.425535\pi$
$619$	286.982			0.463621			0.231811	+	0.972761i	$0.425535\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	444.806	−	79.4259i	0.713974	−	0.127489i
$624$		0			0
$625$	−35.7559			−0.0572095
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	593.824i		−	0.944077i
$630$		0			0
$631$	−263.760			−0.418003			−0.209001	−	0.977915i	$-0.567021\pi$
$631$	−263.760			−0.418003			−0.209001	+	0.977915i	$0.567021\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1567.86i			2.46907i
$636$		0			0
$637$	71.9557	−	26.5436i	0.112960	−	0.0416697i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	120.631i		−	0.188192i	−0.995563	−	0.0940960i	$-0.970004\pi$
$641$		−	120.631i		−	0.188192i	0.995563	−	0.0940960i	$-0.0299961\pi$
$642$		0			0
$643$	−477.894	+	827.737i	−0.743226	+	1.28730i	0.207794	+	0.978173i	$0.433372\pi$
$643$	−477.894	+	827.737i	−0.743226	+	1.28730i	−0.951019	+	0.309132i	$0.899962\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−340.484	−	196.578i	−0.526250	−	0.303831i	0.213238	−	0.977000i	$-0.431599\pi$
$647$	−340.484	−	196.578i	−0.526250	−	0.303831i	−0.739488	+	0.673170i	$0.764932\pi$
$648$		0			0
$649$	308.157	+	533.743i	0.474818	+	0.822409i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			693.093i			1.06140i	0.847560	+	0.530699i	$0.178071\pi$
$653$			693.093i			1.06140i	−0.847560	+	0.530699i	$0.821929\pi$
$654$		0			0
$655$	942.210			1.43849
$656$		0			0
$657$		0			0
$658$		0			0
$659$	120.779	−	69.7321i	0.183277	−	0.105815i	−0.405554	−	0.914071i	$-0.632922\pi$
$659$	120.779	−	69.7321i	0.183277	−	0.105815i	0.588831	+	0.808256i	$0.299588\pi$
$660$		0			0
$661$	101.266	+	175.398i	0.153201	+	0.265352i	0.932403	−	0.361421i	$-0.117708\pi$
$661$	101.266	+	175.398i	0.153201	+	0.265352i	−0.779201	+	0.626774i	$0.784375\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−648.892	+	115.868i	−0.975778	+	0.174238i
$666$		0			0
$667$	−48.2268	−	83.5313i	−0.0723040	−	0.125234i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1240.32	−	716.098i	−1.84846	−	1.06721i
$672$		0			0
$673$	−319.572	−	553.515i	−0.474847	−	0.822459i	0.524738	−	0.851264i	$-0.324163\pi$
$673$	−319.572	−	553.515i	−0.474847	−	0.822459i	−0.999585	+	0.0288047i	$0.990830\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	298.599	+	172.396i	0.441062	+	0.254647i	0.704048	−	0.710152i	$-0.251374\pi$
$677$	298.599	+	172.396i	0.441062	+	0.254647i	−0.262986	+	0.964800i	$0.584707\pi$
$678$		0			0
$679$	237.944	+	1332.55i	0.350433	+	1.96252i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−392.616	−	226.677i	−0.574840	−	0.331884i	0.184240	−	0.982881i	$-0.441018\pi$
$683$	−392.616	−	226.677i	−0.574840	−	0.331884i	−0.759080	+	0.650997i	$0.774351\pi$
$684$		0			0
$685$	−642.502			−0.937959
$686$		0			0
$687$		0			0
$688$		0			0
$689$	86.6360	−	50.0193i	0.125742	−	0.0725970i
$690$		0			0
$691$	10.8436	−	18.7816i	0.0156926	−	0.0271804i	−0.858072	−	0.513529i	$-0.828338\pi$
$691$	10.8436	−	18.7816i	0.0156926	−	0.0271804i	0.873765	+	0.486348i	$0.161671\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	528.586	+	305.179i	0.760555	+	0.439107i
$696$		0			0
$697$	−274.054	−	474.675i	−0.393191	−	0.681026i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			988.932i			1.41074i	0.708837	+	0.705372i	$0.249220\pi$
$701$			988.932i			1.41074i	−0.708837	+	0.705372i	$0.750780\pi$
$702$		0			0
$703$	358.190	−	620.403i	0.509516	−	0.882508i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−160.447	+	443.815i	−0.226940	+	0.627744i
$708$		0			0
$709$	−328.607	+	569.164i	−0.463480	+	0.802771i	−0.999131	−	0.0416683i	$-0.986733\pi$
$709$	−328.607	+	569.164i	−0.463480	+	0.802771i	0.535652	+	0.844439i	$0.320066\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−40.1688	+	23.1915i	−0.0563378	+	0.0325266i
$714$		0			0
$715$	−121.045	+	209.656i	−0.169294	+	0.293225i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	490.350	−	283.104i	0.681988	−	0.393746i	−0.118615	−	0.992940i	$-0.537846\pi$
$719$	490.350	−	283.104i	0.681988	−	0.393746i	0.800604	+	0.599194i	$0.204512\pi$
$720$		0			0
$721$	528.511	+	191.066i	0.733025	+	0.265001i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1602.07	−	924.954i	2.20975	−	1.27580i
$726$		0			0
$727$	61.6116	+	106.714i	0.0847478	+	0.146787i	0.905284	−	0.424807i	$-0.139658\pi$
$727$	61.6116	+	106.714i	0.0847478	+	0.146787i	−0.820536	+	0.571595i	$0.806325\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			419.460i			0.573817i
$732$		0			0
$733$	1037.13			1.41492			0.707458	−	0.706756i	$-0.249842\pi$
$733$	1037.13			1.41492			0.707458	+	0.706756i	$0.249842\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−79.2349	+	45.7463i	−0.107510	+	0.0620709i
$738$		0			0
$739$	536.011	−	928.397i	0.725319	−	1.25629i	−0.233524	−	0.972351i	$-0.575026\pi$
$739$	536.011	−	928.397i	0.725319	−	1.25629i	0.958843	−	0.283938i	$-0.0916409\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	645.464	+	372.659i	0.868727	+	0.501560i	0.866925	−	0.498439i	$-0.166093\pi$
$743$	645.464	+	372.659i	0.868727	+	0.501560i	0.00180187	+	0.999998i	$0.499426\pi$
$744$		0			0
$745$	−877.090			−1.17730
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−860.045	+	153.573i	−1.14826	+	0.205037i
$750$		0			0
$751$	−6.37835			−0.00849314			−0.00424657	−	0.999991i	$-0.501352\pi$
$751$	−6.37835			−0.00849314			−0.00424657	+	0.999991i	$0.501352\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1315.66i		−	1.74259i
$756$		0			0
$757$	1187.30			1.56843			0.784216	−	0.620488i	$-0.213066\pi$
$757$	1187.30			1.56843			0.784216	+	0.620488i	$0.213066\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	271.035i		−	0.356157i	−0.984016	−	0.178078i	$-0.943012\pi$
$761$		−	271.035i		−	0.356157i	0.984016	−	0.178078i	$-0.0569880\pi$
$762$		0			0
$763$	−865.055	+	729.069i	−1.13376	+	0.955530i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			50.2078i			0.0654600i
$768$		0			0
$769$	−12.4068	+	21.4892i	−0.0161337	+	0.0279443i	−0.873980	−	0.485963i	$-0.838469\pi$
$769$	−12.4068	+	21.4892i	−0.0161337	+	0.0279443i	0.857846	+	0.513907i	$0.171802\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−780.685	−	450.729i	−1.00994	−	0.583090i	−0.0987672	−	0.995111i	$-0.531490\pi$
$773$	−780.685	−	450.729i	−1.00994	−	0.583090i	−0.911175	+	0.412020i	$0.864823\pi$
$774$		0			0
$775$	−444.795	−	770.408i	−0.573929	−	0.994075i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	661.229i		−	0.848817i
$780$		0			0
$781$	637.446			0.816192
$782$		0			0
$783$		0			0
$784$		0			0
$785$	219.695	−	126.841i	0.279866	−	0.161581i
$786$		0			0
$787$	−389.427	−	674.508i	−0.494825	−	0.857063i	0.505157	−	0.863028i	$-0.331435\pi$
$787$	−389.427	−	674.508i	−0.494825	−	0.857063i	−0.999982	+	0.00596498i	$0.998101\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	149.917	+	177.880i	0.189529	+	0.224880i
$792$		0			0
$793$	−58.3367	−	101.042i	−0.0735646	−	0.127418i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−492.791	−	284.513i	−0.618307	−	0.356980i	0.157903	−	0.987455i	$-0.449527\pi$
$797$	−492.791	−	284.513i	−0.618307	−	0.356980i	−0.776210	+	0.630475i	$0.782860\pi$
$798$		0			0
$799$	30.7999	+	53.3469i	0.0385480	+	0.0667671i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−219.880	−	126.948i	−0.273823	−	0.158092i
$804$		0			0
$805$	−109.981	−	39.7600i	−0.136622	−	0.0493913i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−888.209	−	512.808i	−1.09791	−	0.633878i	−0.162238	−	0.986752i	$-0.551871\pi$
$809$	−888.209	−	512.808i	−1.09791	−	0.633878i	−0.935671	+	0.352873i	$0.885205\pi$
$810$		0			0
$811$	−276.040			−0.340370			−0.170185	−	0.985412i	$-0.554437\pi$
$811$	−276.040			−0.340370			−0.170185	+	0.985412i	$0.554437\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2084.04	−	1203.22i	2.55710	−	1.47634i
$816$		0			0
$817$	−253.015	+	438.235i	−0.309688	+	0.536395i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	176.195	+	101.726i	0.214610	+	0.123905i	0.603452	−	0.797399i	$-0.293792\pi$
$821$	176.195	+	101.726i	0.214610	+	0.123905i	−0.388842	+	0.921304i	$0.627125\pi$
$822$		0			0
$823$	−410.181	−	710.454i	−0.498397	−	0.863249i	0.501602	−	0.865099i	$-0.332744\pi$
$823$	−410.181	−	710.454i	−0.498397	−	0.863249i	−0.999998	+	0.00185026i	$0.999411\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1580.90i		−	1.91161i	−0.293998	−	0.955806i	$-0.594986\pi$
$827$		−	1580.90i		−	1.91161i	0.293998	−	0.955806i	$-0.405014\pi$
$828$		0			0
$829$	51.7377	−	89.6124i	0.0624098	−	0.108097i	−0.833132	−	0.553074i	$-0.813455\pi$
$829$	51.7377	−	89.6124i	0.0624098	−	0.108097i	0.895542	+	0.444977i	$0.146788\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	164.434	+	445.755i	0.197400	+	0.535120i
$834$		0			0
$835$	−1052.23	+	1822.51i	−1.26015	+	2.18265i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1258.61	−	726.657i	1.50013	−	0.866099i	0.500127	−	0.865952i	$-0.333287\pi$
$839$	1258.61	−	726.657i	1.50013	−	0.866099i	1.00000		0.000147057i	$-4.68097e-5\pi$
$840$		0			0
$841$	659.496	−	1142.28i	0.784181	−	1.35824i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1161.12	−	670.371i	1.37410	−	0.793338i
$846$		0			0
$847$	1633.60	+	590.575i	1.92869	+	0.697255i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	110.072	−	63.5500i	0.129344	−	0.0746769i
$852$		0			0
$853$	−52.3396	−	90.6549i	−0.0613595	−	0.106278i	0.833714	−	0.552197i	$-0.186210\pi$
$853$	−52.3396	−	90.6549i	−0.0613595	−	0.106278i	−0.895073	+	0.445919i	$0.852877\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	225.718i		−	0.263382i	−0.991291	−	0.131691i	$-0.957959\pi$
$857$		−	225.718i		−	0.263382i	0.991291	−	0.131691i	$-0.0420407\pi$
$858$		0			0
$859$	583.238			0.678973			0.339486	−	0.940611i	$-0.389747\pi$
$859$	583.238			0.678973			0.339486	+	0.940611i	$0.389747\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	306.120	−	176.739i	0.354716	−	0.204796i	−0.312044	−	0.950068i	$-0.601014\pi$
$863$	306.120	−	176.739i	0.354716	−	0.204796i	0.666761	+	0.745272i	$0.267680\pi$
$864$		0			0
$865$	−774.749	+	1341.90i	−0.895664	+	1.55133i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	769.500	+	444.271i	0.885500	+	0.511244i
$870$		0			0
$871$	−7.45341			−0.00855731
$872$		0			0
$873$		0			0
$874$		0			0
$875$	283.613	−	784.507i	0.324129	−	0.896579i
$876$		0			0
$877$	916.182			1.04468			0.522339	−	0.852738i	$-0.325060\pi$
$877$	916.182			1.04468			0.522339	+	0.852738i	$0.325060\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1041.09i			1.18172i	0.806775	+	0.590859i	$0.201211\pi$
$881$			1041.09i			1.18172i	−0.806775	+	0.590859i	$0.798789\pi$
$882$		0			0
$883$	247.115			0.279858			0.139929	−	0.990162i	$-0.455313\pi$
$883$	247.115			0.279858			0.139929	+	0.990162i	$0.455313\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1077.91i			1.21523i	0.794231	+	0.607616i	$0.207874\pi$
$887$			1077.91i			1.21523i	−0.794231	+	0.607616i	$0.792126\pi$
$888$		0			0
$889$	1282.13	+	463.512i	1.44222	+	0.521386i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			74.3129i			0.0832172i
$894$		0			0
$895$	1411.55	−	2444.87i	1.57715	−	2.73170i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−899.545	−	519.353i	−1.00061	−	0.577700i
$900$		0			0
$901$	309.862	+	536.697i	0.343909	+	0.595669i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1834.72i		−	2.02731i
$906$		0			0
$907$	−867.280			−0.956208			−0.478104	−	0.878303i	$-0.658676\pi$
$907$	−867.280			−0.956208			−0.478104	+	0.878303i	$0.658676\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−987.899	+	570.364i	−1.08441	+	0.626085i	−0.932083	−	0.362245i	$-0.882010\pi$
$911$	−987.899	+	570.364i	−1.08441	+	0.626085i	−0.152328	+	0.988330i	$0.548677\pi$
$912$		0			0
$913$	899.014	+	1557.14i	0.984681	+	1.70552i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	278.549	−	770.501i	0.303761	−	0.840241i
$918$		0			0
$919$	308.533	+	534.394i	0.335726	+	0.581495i	0.983624	−	0.180232i	$-0.0576849\pi$
$919$	308.533	+	534.394i	0.335726	+	0.581495i	−0.647898	+	0.761727i	$0.724352\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	44.9721	+	25.9647i	0.0487239	+	0.0281307i
$924$		0			0
$925$	1218.84	+	2111.10i	1.31767	+	2.28227i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	460.934	+	266.120i	0.496161	+	0.286459i	0.727127	−	0.686503i	$-0.240855\pi$
$929$	460.934	+	266.120i	0.496161	+	0.286459i	−0.230966	+	0.972962i	$0.574189\pi$
$930$		0			0
$931$	−97.0820	+	564.892i	−0.104277	+	0.606758i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1298.79	−	749.856i	−1.38908	−	0.801985i
$936$		0			0
$937$	−1506.24			−1.60751			−0.803755	−	0.594960i	$-0.797168\pi$
$937$	−1506.24			−1.60751			−0.803755	+	0.594960i	$0.797168\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−176.771	+	102.059i	−0.187855	+	0.108458i	−0.590978	−	0.806688i	$-0.701258\pi$
$941$	−176.771	+	102.059i	−0.187855	+	0.108458i	0.403123	+	0.915146i	$0.367925\pi$
$942$		0			0
$943$	58.6575	−	101.598i	0.0622031	−	0.107739i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	430.653	+	248.638i	0.454755	+	0.262553i	0.709836	−	0.704367i	$-0.248769\pi$
$947$	430.653	+	248.638i	0.454755	+	0.262553i	−0.255081	+	0.966920i	$0.582102\pi$
$948$		0			0
$949$	−10.3418	−	17.9125i	−0.0108975	−	0.0188751i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	582.144i		−	0.610854i	−0.952215	−	0.305427i	$-0.901201\pi$
$953$		−	582.144i		−	0.610854i	0.952215	−	0.305427i	$-0.0987992\pi$
$954$		0			0
$955$	755.174	−	1308.00i	0.790758	−	1.36963i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−189.945	+	525.412i	−0.198066	+	0.547875i
$960$		0			0
$961$	230.752	−	399.674i	0.240116	−	0.415894i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	249.661	−	144.142i	0.258716	−	0.149370i
$966$		0			0
$967$	−239.897	+	415.514i	−0.248084	+	0.429694i	−0.962994	−	0.269522i	$-0.913134\pi$
$967$	−239.897	+	415.514i	−0.248084	+	0.429694i	0.714910	+	0.699216i	$0.246468\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	311.099	−	179.613i	0.320391	−	0.184978i	−0.331176	−	0.943569i	$-0.607445\pi$
$971$	311.099	−	179.613i	0.320391	−	0.184978i	0.651567	+	0.758591i	$0.274112\pi$
$972$		0			0
$973$	405.831	−	342.034i	0.417092	−	0.351526i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.54385	+	4.93279i	−0.00874498	+	0.00504892i	−0.504366	−	0.863490i	$-0.668274\pi$
$977$	−8.54385	+	4.93279i	−0.00874498	+	0.00504892i	0.495621	+	0.868539i	$0.334940\pi$
$978$		0			0
$979$	620.100	+	1074.04i	0.633401	+	1.09708i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			205.281i			0.208831i	0.994534	+	0.104415i	$0.0332972\pi$
$983$			205.281i			0.208831i	−0.994534	+	0.104415i	$0.966703\pi$
$984$		0			0
$985$	−886.080			−0.899574
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−77.7516	+	44.8899i	−0.0786164	+	0.0453892i
$990$		0			0
$991$	−204.103	+	353.517i	−0.205957	+	0.356728i	−0.950437	−	0.310917i	$-0.899364\pi$
$991$	−204.103	+	353.517i	−0.205957	+	0.356728i	0.744480	+	0.667645i	$0.232697\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−481.257	−	277.854i	−0.483675	−	0.279250i
$996$		0			0
$997$	−650.652			−0.652610			−0.326305	−	0.945264i	$-0.605804\pi$
$997$	−650.652			−0.652610			−0.326305	+	0.945264i	$0.605804\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.m.a.557.2		32
3.2	odd	2		252.3.m.a.221.16	yes	32
7.2	even	3		756.3.bh.a.233.2		32
9.2	odd	6		756.3.bh.a.305.2		32
9.7	even	3		252.3.bh.a.137.6	yes	32
21.2	odd	6		252.3.bh.a.149.6	yes	32
63.2	odd	6	inner	756.3.m.a.737.15		32
63.16	even	3		252.3.m.a.65.16	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.m.a.65.16	✓	32	63.16	even	3
252.3.m.a.221.16	yes	32	3.2	odd	2
252.3.bh.a.137.6	yes	32	9.7	even	3
252.3.bh.a.149.6	yes	32	21.2	odd	6
756.3.m.a.557.2		32	1.1	even	1	trivial
756.3.m.a.737.15		32	63.2	odd	6	inner
756.3.bh.a.233.2		32	7.2	even	3
756.3.bh.a.305.2		32	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	756.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-8.05008i q^{5} +(-6.58302 - 2.37987i) q^{7} +O(q^{10})\)	\(q-8.05008i q^{5} +(-6.58302 - 2.37987i) q^{7} -19.2134i q^{11} +(0.782606 - 1.35551i) q^{13} +(8.39721 + 4.84813i) q^{17} +(5.84871 + 10.1303i) q^{19} +2.07535i q^{23} -39.8037 q^{25} +(-40.2492 + 23.2379i) q^{29} +(11.1747 + 19.3552i) q^{31} +(-19.1582 + 52.9938i) q^{35} +(-30.6213 - 53.0376i) q^{37} +(-48.9545 - 28.2639i) q^{41} +(21.6300 + 37.4643i) q^{43} +(5.50180 + 3.17647i) q^{47} +(37.6724 + 31.3335i) q^{49} +(55.3510 + 31.9569i) q^{53} -154.669 q^{55} +(-27.7798 + 16.0387i) q^{59} +(37.2708 - 64.5549i) q^{61} +(-10.9120 - 6.30004i) q^{65} +(-2.38096 - 4.12394i) q^{67} +33.1772i q^{71} +(6.60726 - 11.4441i) q^{73} +(-45.7254 + 126.482i) q^{77} +(-23.1230 + 40.0502i) q^{79} +(-81.0445 + 46.7910i) q^{83} +(39.0278 - 67.5982i) q^{85} +(-55.9009 + 32.2744i) q^{89} +(-8.37786 + 7.06087i) q^{91} +(81.5494 - 47.0826i) q^{95} +(-96.6875 - 167.468i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - q^{7}+O(q^{10}) \) 32 * q - q^7	\( 32 q - q^{7} - 5 q^{13} + 27 q^{17} - 14 q^{19} - 160 q^{25} - 36 q^{29} - 8 q^{31} + 45 q^{35} - 11 q^{37} - 72 q^{41} + 16 q^{43} + 108 q^{47} + 35 q^{49} - 180 q^{53} - 24 q^{55} - 45 q^{59} - 41 q^{61} + 81 q^{65} - 35 q^{67} - 98 q^{73} - 225 q^{77} - 71 q^{79} - 30 q^{85} + 189 q^{89} + 109 q^{91} + 288 q^{95} + 19 q^{97}+O(q^{100}) \) 32 * q - q^7 - 5 * q^13 + 27 * q^17 - 14 * q^19 - 160 * q^25 - 36 * q^29 - 8 * q^31 + 45 * q^35 - 11 * q^37 - 72 * q^41 + 16 * q^43 + 108 * q^47 + 35 * q^49 - 180 * q^53 - 24 * q^55 - 45 * q^59 - 41 * q^61 + 81 * q^65 - 35 * q^67 - 98 * q^73 - 225 * q^77 - 71 * q^79 - 30 * q^85 + 189 * q^89 + 109 * q^91 + 288 * q^95 + 19 * q^97

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