LMFDB - Embedded newform 3060.2.z.g.829.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(829,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 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("3060.829"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.z (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 1020)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			829.13
Character	$\chi$	$=$	3060.829
Dual form			3060.2.z.g.2809.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.880193 + 2.05554i) q^{5} +(-1.19618 - 1.19618i) q^{7} +(-0.103233 + 0.103233i) q^{11} +0.176789i q^{13} +(-4.11792 + 0.206669i) q^{17} -1.74915i q^{19} +(1.53429 + 1.53429i) q^{23} +(-3.45052 + 3.61855i) q^{25} +(6.66831 + 6.66831i) q^{29} +(-5.47293 - 5.47293i) q^{31} +(1.40593 - 3.51166i) q^{35} +(-5.73720 + 5.73720i) q^{37} +(-0.296143 + 0.296143i) q^{41} -4.71656 q^{43} -6.04051i q^{47} -4.13832i q^{49} -7.52346 q^{53} +(-0.303064 - 0.121335i) q^{55} +12.8952i q^{59} +(-6.00478 + 6.00478i) q^{61} +(-0.363397 + 0.155608i) q^{65} +11.1502i q^{67} +(2.57123 + 2.57123i) q^{71} +(-0.427746 + 0.427746i) q^{73} +0.246969 q^{77} +(2.74478 - 2.74478i) q^{79} -10.2946 q^{83} +(-4.04938 - 8.28266i) q^{85} -11.1039 q^{89} +(0.211470 - 0.211470i) q^{91} +(3.59546 - 1.53959i) q^{95} +(9.87044 - 9.87044i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 4 q^{5} + 8 q^{11} - 24 q^{29} - 16 q^{31} - 8 q^{35} + 8 q^{41} + 28 q^{55} + 16 q^{61} - 56 q^{71} + 16 q^{79} - 40 q^{85} - 32 q^{89} + 64 q^{91} - 64 q^{95}+O(q^{100}) $ 40 * q - 4 * q^5 + 8 * q^11 - 24 * q^29 - 16 * q^31 - 8 * q^35 + 8 * q^41 + 28 * q^55 + 16 * q^61 - 56 * q^71 + 16 * q^79 - 40 * q^85 - 32 * q^89 + 64 * q^91 - 64 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.880193	+	2.05554i	0.393634	+	0.919267i
$5$	0.880193	+	2.05554i	0.393634	+	0.919267i
$6$		0			0
$7$	−1.19618	−	1.19618i	−0.452112	−	0.452112i	0.443943	−	0.896055i	$-0.353579\pi$
$7$	−1.19618	−	1.19618i	−0.452112	−	0.452112i	−0.896055	+	0.443943i	$0.853579\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.103233	+	0.103233i	−0.0311259	+	0.0311259i	−0.722498	−	0.691373i	$-0.757006\pi$
$11$	−0.103233	+	0.103233i	−0.0311259	+	0.0311259i	0.691373	+	0.722498i	$0.257006\pi$
$12$		0			0
$13$			0.176789i			0.0490323i	0.999699	+	0.0245162i	$0.00780452\pi$
$13$			0.176789i			0.0490323i	−0.999699	+	0.0245162i	$0.992195\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.11792	+	0.206669i	−0.998743	+	0.0501245i
$17$	−4.11792	+	0.206669i	−0.998743	+	0.0501245i
$18$		0			0
$19$		−	1.74915i		−	0.401283i	−0.979665	−	0.200641i	$-0.935697\pi$
$19$		−	1.74915i		−	0.401283i	0.979665	−	0.200641i	$-0.0643026\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.53429	+	1.53429i	0.319921	+	0.319921i	0.848737	−	0.528816i	$-0.177364\pi$
$23$	1.53429	+	1.53429i	0.319921	+	0.319921i	−0.528816	+	0.848737i	$0.677364\pi$
$24$		0			0
$25$	−3.45052	+	3.61855i	−0.690104	+	0.723710i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.66831	+	6.66831i	1.23827	+	1.23827i	0.960706	+	0.277568i	$0.0895283\pi$
$29$	6.66831	+	6.66831i	1.23827	+	1.23827i	0.277568	+	0.960706i	$0.410472\pi$
$30$		0			0
$31$	−5.47293	−	5.47293i	−0.982967	−	0.982967i	0.0168908	−	0.999857i	$-0.494623\pi$
$31$	−5.47293	−	5.47293i	−0.982967	−	0.982967i	−0.999857	+	0.0168908i	$0.994623\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.40593	−	3.51166i	0.237645	−	0.593579i
$36$		0			0
$37$	−5.73720	+	5.73720i	−0.943189	+	0.943189i	−0.998471	−	0.0552817i	$-0.982394\pi$
$37$	−5.73720	+	5.73720i	−0.943189	+	0.943189i	0.0552817	+	0.998471i	$0.482394\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.296143	+	0.296143i	−0.0462498	+	0.0462498i	−0.729853	−	0.683604i	$-0.760412\pi$
$41$	−0.296143	+	0.296143i	−0.0462498	+	0.0462498i	0.683604	+	0.729853i	$0.260412\pi$
$42$		0			0
$43$	−4.71656			−0.719269			−0.359634	−	0.933093i	$-0.617099\pi$
$43$	−4.71656			−0.719269			−0.359634	+	0.933093i	$0.617099\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.04051i		−	0.881098i	−0.897728	−	0.440549i	$-0.854784\pi$
$47$		−	6.04051i		−	0.881098i	0.897728	−	0.440549i	$-0.145216\pi$
$48$		0			0
$49$		−	4.13832i		−	0.591189i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.52346			−1.03343			−0.516713	−	0.856158i	$-0.672845\pi$
$53$	−7.52346			−1.03343			−0.516713	+	0.856158i	$0.672845\pi$
$54$		0			0
$55$	−0.303064	−	0.121335i	−0.0408652	−	0.0163608i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			12.8952i			1.67881i	0.543504	+	0.839407i	$0.317097\pi$
$59$			12.8952i			1.67881i	−0.543504	+	0.839407i	$0.682903\pi$
$60$		0			0
$61$	−6.00478	+	6.00478i	−0.768834	+	0.768834i	−0.977901	−	0.209067i	$-0.932957\pi$
$61$	−6.00478	+	6.00478i	−0.768834	+	0.768834i	0.209067	+	0.977901i	$0.432957\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.363397	+	0.155608i	−0.0450738	+	0.0193008i
$66$		0			0
$67$			11.1502i			1.36221i	0.732185	+	0.681106i	$0.238501\pi$
$67$			11.1502i			1.36221i	−0.732185	+	0.681106i	$0.761499\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.57123	+	2.57123i	0.305149	+	0.305149i	0.843024	−	0.537875i	$-0.180773\pi$
$71$	2.57123	+	2.57123i	0.305149	+	0.305149i	−0.537875	+	0.843024i	$0.680773\pi$
$72$		0			0
$73$	−0.427746	+	0.427746i	−0.0500638	+	0.0500638i	−0.731695	−	0.681632i	$-0.761271\pi$
$73$	−0.427746	+	0.427746i	−0.0500638	+	0.0500638i	0.681632	+	0.731695i	$0.261271\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.246969			0.0281448
$78$		0			0
$79$	2.74478	−	2.74478i	0.308812	−	0.308812i	−0.535637	−	0.844449i	$-0.679928\pi$
$79$	2.74478	−	2.74478i	0.308812	−	0.308812i	0.844449	+	0.535637i	$0.179928\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.2946			−1.12997			−0.564987	−	0.825099i	$-0.691119\pi$
$83$	−10.2946			−1.12997			−0.564987	+	0.825099i	$0.691119\pi$
$84$		0			0
$85$	−4.04938	−	8.28266i	−0.439217	−	0.898381i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.1039			−1.17701			−0.588505	−	0.808494i	$-0.700283\pi$
$89$	−11.1039			−1.17701			−0.588505	+	0.808494i	$0.700283\pi$
$90$		0			0
$91$	0.211470	−	0.211470i	0.0221681	−	0.0221681i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.59546	−	1.53959i	0.368886	−	0.157959i
$96$		0			0
$97$	9.87044	−	9.87044i	1.00219	−	1.00219i	0.00219390	−	0.999998i	$-0.499302\pi$
$97$	9.87044	−	9.87044i	1.00219	−	1.00219i	0.999998	−	0.00219390i	$-0.000698340\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.41755			−0.837578			−0.418789	−	0.908084i	$-0.637545\pi$
$101$	−8.41755			−0.837578			−0.418789	+	0.908084i	$0.637545\pi$
$102$		0			0
$103$		−	12.2496i		−	1.20699i	−0.797367	−	0.603495i	$-0.793774\pi$
$103$		−	12.2496i		−	1.20699i	0.797367	−	0.603495i	$-0.206226\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.29983	−	2.29983i	0.222333	−	0.222333i	−0.587147	−	0.809480i	$-0.699749\pi$
$107$	2.29983	−	2.29983i	0.222333	−	0.222333i	0.809480	+	0.587147i	$0.199749\pi$
$108$		0			0
$109$	3.45370	−	3.45370i	0.330804	−	0.330804i	−0.522088	−	0.852892i	$-0.674847\pi$
$109$	3.45370	−	3.45370i	0.330804	−	0.330804i	0.852892	+	0.522088i	$0.174847\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.92805	−	8.92805i	−0.839880	−	0.839880i	0.148963	−	0.988843i	$-0.452407\pi$
$113$	−8.92805	−	8.92805i	−0.839880	−	0.839880i	−0.988843	+	0.148963i	$0.952407\pi$
$114$		0			0
$115$	−1.80333	+	4.50426i	−0.168161	+	0.420024i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.17298	+	4.67855i	0.474206	+	0.428882i
$120$		0			0
$121$			10.9787i			0.998062i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.4752	−	3.90768i	−0.936931	−	0.349514i
$126$		0			0
$127$	0.423483			0.0375780			0.0187890	−	0.999823i	$-0.494019\pi$
$127$	0.423483			0.0375780			0.0187890	+	0.999823i	$0.494019\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.199067	−	0.199067i	−0.0173926	−	0.0173926i	0.698357	−	0.715750i	$-0.253915\pi$
$131$	−0.199067	−	0.199067i	−0.0173926	−	0.0173926i	−0.715750	+	0.698357i	$0.753915\pi$
$132$		0			0
$133$	−2.09229	+	2.09229i	−0.181425	+	0.181425i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			18.7514i			1.60204i	0.598636	+	0.801021i	$0.295710\pi$
$137$			18.7514i			1.60204i	−0.598636	+	0.801021i	$0.704290\pi$
$138$		0			0
$139$	−11.2085	−	11.2085i	−0.950696	−	0.950696i	0.0481445	−	0.998840i	$-0.484669\pi$
$139$	−11.2085	−	11.2085i	−0.950696	−	0.950696i	−0.998840	+	0.0481445i	$0.984669\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.0182504	−	0.0182504i	−0.00152617	−	0.00152617i
$144$		0			0
$145$	−7.83761	+	19.5764i	−0.650878	+	1.62573i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.1247			−0.911374			−0.455687	−	0.890140i	$-0.650606\pi$
$149$	−11.1247			−0.911374			−0.455687	+	0.890140i	$0.650606\pi$
$150$		0			0
$151$		−	8.37121i		−	0.681240i	−0.940201	−	0.340620i	$-0.889363\pi$
$151$		−	8.37121i		−	0.681240i	0.940201	−	0.340620i	$-0.110637\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.43261	−	16.0671i	0.516680	−	1.29054i
$156$		0			0
$157$		−	2.35837i		−	0.188218i	−0.995562	−	0.0941092i	$-0.970000\pi$
$157$		−	2.35837i		−	0.188218i	0.995562	−	0.0941092i	$-0.0300003\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	3.67055i		−	0.289280i
$162$		0			0
$163$	−2.01291	−	2.01291i	−0.157663	−	0.157663i	0.623867	−	0.781530i	$-0.285561\pi$
$163$	−2.01291	−	2.01291i	−0.157663	−	0.157663i	−0.781530	+	0.623867i	$0.785561\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.69058	+	5.69058i	−0.440351	+	0.440351i	−0.892130	−	0.451779i	$-0.850789\pi$
$167$	−5.69058	+	5.69058i	−0.440351	+	0.440351i	0.451779	+	0.892130i	$0.350789\pi$
$168$		0			0
$169$	12.9687			0.997596
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.67771	+	8.67771i	−0.659754	+	0.659754i	−0.955322	−	0.295568i	$-0.904491\pi$
$173$	−8.67771	+	8.67771i	−0.659754	+	0.659754i	0.295568	+	0.955322i	$0.404491\pi$
$174$		0			0
$175$	8.45586	−	0.200990i	0.639203	−	0.0151934i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			7.01151i			0.524065i	0.965059	+	0.262032i	$0.0843927\pi$
$179$			7.01151i			0.524065i	−0.965059	+	0.262032i	$0.915607\pi$
$180$		0			0
$181$	−7.00501	+	7.00501i	−0.520678	+	0.520678i	−0.917776	−	0.397098i	$-0.870017\pi$
$181$	−7.00501	+	7.00501i	−0.520678	+	0.520678i	0.397098	+	0.917776i	$0.370017\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.8429	−	6.74322i	−1.23831	−	0.495771i
$186$		0			0
$187$	0.403770	−	0.446440i	0.0295266	−	0.0326469i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.6361			0.841957			0.420979	−	0.907071i	$-0.361687\pi$
$191$	11.6361			0.841957			0.420979	+	0.907071i	$0.361687\pi$
$192$		0			0
$193$	−4.30087	−	4.30087i	−0.309584	−	0.309584i	0.535164	−	0.844748i	$-0.320250\pi$
$193$	−4.30087	−	4.30087i	−0.309584	−	0.309584i	−0.844748	+	0.535164i	$0.820250\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.87864	−	9.87864i	−0.703824	−	0.703824i	0.261405	−	0.965229i	$-0.415814\pi$
$197$	−9.87864	−	9.87864i	−0.703824	−	0.703824i	−0.965229	+	0.261405i	$0.915814\pi$
$198$		0			0
$199$	−6.81270	−	6.81270i	−0.482940	−	0.482940i	0.423130	−	0.906069i	$-0.360932\pi$
$199$	−6.81270	−	6.81270i	−0.482940	−	0.482940i	−0.906069	+	0.423130i	$0.860932\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	15.9530i		−	1.11968i
$204$		0			0
$205$	−0.869398	−	0.348072i	−0.0607214	−	0.0243104i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.180570	+	0.180570i	0.0124903	+	0.0124903i
$210$		0			0
$211$	−6.43332	+	6.43332i	−0.442888	+	0.442888i	−0.892981	−	0.450093i	$-0.851391\pi$
$211$	−6.43332	+	6.43332i	−0.442888	+	0.442888i	0.450093	+	0.892981i	$0.351391\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.15148	−	9.69510i	−0.283129	−	0.661200i
$216$		0			0
$217$			13.0932i			0.888823i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.0365367	−	0.728002i	−0.00245772	−	0.0489707i
$222$		0			0
$223$	11.9249			0.798552			0.399276	−	0.916831i	$-0.369261\pi$
$223$	11.9249			0.798552			0.399276	+	0.916831i	$0.369261\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.09203	+	5.09203i	0.337970	+	0.337970i	0.855603	−	0.517633i	$-0.173187\pi$
$227$	5.09203	+	5.09203i	0.337970	+	0.337970i	−0.517633	+	0.855603i	$0.673187\pi$
$228$		0			0
$229$			4.72050i			0.311939i	0.987762	+	0.155970i	$0.0498502\pi$
$229$			4.72050i			0.311939i	−0.987762	+	0.155970i	$0.950150\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.8073	+	11.8073i	−0.773522	+	0.773522i	−0.978720	−	0.205199i	$-0.934216\pi$
$233$	−11.8073	+	11.8073i	−0.773522	+	0.773522i	0.205199	+	0.978720i	$0.434216\pi$
$234$		0			0
$235$	12.4165	−	5.31681i	0.809965	−	0.346830i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.2088			−0.789719			−0.394860	−	0.918741i	$-0.629207\pi$
$239$	−12.2088			−0.789719			−0.394860	+	0.918741i	$0.629207\pi$
$240$		0			0
$241$	19.8743	+	19.8743i	1.28022	+	1.28022i	0.940544	+	0.339672i	$0.110316\pi$
$241$	19.8743	+	19.8743i	1.28022	+	1.28022i	0.339672	+	0.940544i	$0.389684\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.50650	−	3.64252i	0.543461	−	0.232712i
$246$		0			0
$247$	0.309230			0.0196758
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−14.5298			−0.917112			−0.458556	−	0.888666i	$-0.651633\pi$
$251$	−14.5298			−0.917112			−0.458556	+	0.888666i	$0.651633\pi$
$252$		0			0
$253$	−0.316777			−0.0199156
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.4730			−1.27707			−0.638537	−	0.769591i	$-0.720460\pi$
$257$	−20.4730			−1.27707			−0.638537	+	0.769591i	$0.720460\pi$
$258$		0			0
$259$	13.7254			0.852855
$260$		0			0
$261$		0			0
$262$		0			0
$263$	20.3812			1.25676			0.628380	−	0.777907i	$-0.283718\pi$
$263$	20.3812			1.25676			0.628380	+	0.777907i	$0.283718\pi$
$264$		0			0
$265$	−6.62209	−	15.4648i	−0.406792	−	0.949995i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.8943	+	10.8943i	0.664237	+	0.664237i	0.956376	−	0.292139i	$-0.0943670\pi$
$269$	10.8943	+	10.8943i	0.664237	+	0.664237i	−0.292139	+	0.956376i	$0.594367\pi$
$270$		0			0
$271$	13.2692			0.806046			0.403023	−	0.915190i	$-0.367959\pi$
$271$	13.2692			0.806046			0.403023	+	0.915190i	$0.367959\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.0173459	−	0.729760i	−0.00104599	−	0.0440062i
$276$		0			0
$277$	−5.10335	+	5.10335i	−0.306630	+	0.306630i	−0.843601	−	0.536971i	$-0.819569\pi$
$277$	−5.10335	+	5.10335i	−0.306630	+	0.306630i	0.536971	+	0.843601i	$0.319569\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	28.6602i		−	1.70972i	−0.518856	−	0.854862i	$-0.673642\pi$
$281$		−	28.6602i		−	1.70972i	0.518856	−	0.854862i	$-0.326358\pi$
$282$		0			0
$283$	−10.9346	−	10.9346i	−0.649994	−	0.649994i	0.302997	−	0.952991i	$-0.402013\pi$
$283$	−10.9346	−	10.9346i	−0.649994	−	0.649994i	−0.952991	+	0.302997i	$0.902013\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.708479			0.0418202
$288$		0			0
$289$	16.9146	−	1.70209i	0.994975	−	0.100123i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	10.5786i		−	0.618007i	−0.951061	−	0.309003i	$-0.900005\pi$
$293$		−	10.5786i		−	0.618007i	0.951061	−	0.309003i	$-0.0999955\pi$
$294$		0			0
$295$	−26.5067	+	11.3503i	−1.54328	+	0.660838i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.271244	+	0.271244i	−0.0156865	+	0.0156865i
$300$		0			0
$301$	5.64184	+	5.64184i	0.325190	+	0.325190i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−17.6285	−	7.05773i	−1.00940	−	0.404125i
$306$		0			0
$307$			33.2947i			1.90023i	0.311906	+	0.950113i	$0.399033\pi$
$307$			33.2947i			1.90023i	−0.311906	+	0.950113i	$0.600967\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4.15895	+	4.15895i	0.235832	+	0.235832i	0.815122	−	0.579290i	$-0.196670\pi$
$311$	4.15895	+	4.15895i	0.235832	+	0.235832i	−0.579290	+	0.815122i	$0.696670\pi$
$312$		0			0
$313$	−6.04575	−	6.04575i	−0.341726	−	0.341726i	0.515290	−	0.857016i	$-0.327684\pi$
$313$	−6.04575	−	6.04575i	−0.341726	−	0.341726i	−0.857016	+	0.515290i	$0.827684\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.01519	−	3.01519i	−0.169350	−	0.169350i	0.617344	−	0.786694i	$-0.288209\pi$
$317$	−3.01519	−	3.01519i	−0.169350	−	0.169350i	−0.786694	+	0.617344i	$0.788209\pi$
$318$		0			0
$319$	−1.37678			−0.0770847
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.361495	+	7.20287i	0.0201141	+	0.400778i
$324$		0			0
$325$	−0.639718	−	0.610013i	−0.0354852	−	0.0338374i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.22551	+	7.22551i	−0.398356	+	0.398356i
$330$		0			0
$331$		−	1.89956i		−	0.104409i	−0.998636	−	0.0522046i	$-0.983375\pi$
$331$		−	1.89956i		−	0.104409i	0.998636	−	0.0522046i	$-0.0166248\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−22.9197	+	9.81430i	−1.25224	+	0.536213i
$336$		0			0
$337$	10.3241	−	10.3241i	0.562391	−	0.562391i	−0.367595	−	0.929986i	$-0.619819\pi$
$337$	10.3241	−	10.3241i	0.562391	−	0.562391i	0.929986	+	0.367595i	$0.119819\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.12997			0.0611914
$342$		0			0
$343$	−13.3234	+	13.3234i	−0.719396	+	0.719396i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	22.1896	+	22.1896i	1.19120	+	1.19120i	0.976730	+	0.214470i	$0.0688025\pi$
$347$	22.1896	+	22.1896i	1.19120	+	1.19120i	0.214470	+	0.976730i	$0.431197\pi$
$348$		0			0
$349$		−	6.27989i		−	0.336155i	−0.985774	−	0.168077i	$-0.946244\pi$
$349$		−	6.27989i		−	0.336155i	0.985774	−	0.168077i	$-0.0537558\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	5.89369i		−	0.313689i	−0.987623	−	0.156845i	$-0.949868\pi$
$353$		−	5.89369i		−	0.313689i	0.987623	−	0.156845i	$-0.0501322\pi$
$354$		0			0
$355$	−3.02210	+	7.54846i	−0.160397	+	0.400631i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			12.5830i			0.664106i	0.943261	+	0.332053i	$0.107741\pi$
$359$			12.5830i			0.664106i	−0.943261	+	0.332053i	$0.892259\pi$
$360$		0			0
$361$	15.9405			0.838972
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.25575	−	0.502752i	−0.0657289	−	0.0263152i
$366$		0			0
$367$	8.19212	+	8.19212i	0.427625	+	0.427625i	0.887819	−	0.460193i	$-0.152220\pi$
$367$	8.19212	+	8.19212i	0.427625	+	0.427625i	−0.460193	+	0.887819i	$0.652220\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.99939	+	8.99939i	0.467225	+	0.467225i
$372$		0			0
$373$			24.2522i			1.25573i	0.778322	+	0.627865i	$0.216071\pi$
$373$			24.2522i			1.25573i	−0.778322	+	0.627865i	$0.783929\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.17888	+	1.17888i	−0.0607155	+	0.0607155i
$378$		0			0
$379$	3.63829	+	3.63829i	0.186887	+	0.186887i	0.794349	−	0.607462i	$-0.207812\pi$
$379$	3.63829	+	3.63829i	0.186887	+	0.186887i	−0.607462	+	0.794349i	$0.707812\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.1330			0.671064			0.335532	−	0.942029i	$-0.391084\pi$
$383$	13.1330			0.671064			0.335532	+	0.942029i	$0.391084\pi$
$384$		0			0
$385$	0.217381	+	0.507656i	0.0110787	+	0.0258726i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			23.3568i			1.18424i	0.805850	+	0.592120i	$0.201709\pi$
$389$			23.3568i			1.18424i	−0.805850	+	0.592120i	$0.798291\pi$
$390$		0			0
$391$	−6.63516	−	6.00098i	−0.335554	−	0.303483i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.05795	+	3.22608i	0.405439	+	0.162322i
$396$		0			0
$397$	16.9194	+	16.9194i	0.849159	+	0.849159i	0.990028	−	0.140869i	$-0.0449897\pi$
$397$	16.9194	+	16.9194i	0.849159	+	0.849159i	−0.140869	+	0.990028i	$0.544990\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.79279	−	5.79279i	0.289278	−	0.289278i	−0.547517	−	0.836795i	$-0.684427\pi$
$401$	5.79279	−	5.79279i	0.289278	−	0.289278i	0.836795	+	0.547517i	$0.184427\pi$
$402$		0			0
$403$	0.967551	−	0.967551i	0.0481971	−	0.0481971i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.18453i		−	0.0587151i
$408$		0			0
$409$	−30.5267			−1.50945			−0.754725	−	0.656041i	$-0.772230\pi$
$409$	−30.5267			−1.50945			−0.754725	+	0.656041i	$0.772230\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	15.4250	−	15.4250i	0.759012	−	0.759012i
$414$		0			0
$415$	−9.06120	−	21.1609i	−0.444797	−	1.03875i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.8969	−	21.8969i	1.06973	−	1.06973i	0.0723530	−	0.997379i	$-0.476949\pi$
$419$	21.8969	−	21.8969i	1.06973	−	1.06973i	0.997379	−	0.0723530i	$-0.0230508\pi$
$420$		0			0
$421$	−34.2282			−1.66818			−0.834089	−	0.551630i	$-0.814006\pi$
$421$	−34.2282			−1.66818			−0.834089	+	0.551630i	$0.814006\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13.4611	−	15.6140i	0.652961	−	0.757391i
$426$		0			0
$427$	14.3656			0.695199
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.8061	+	20.8061i	−1.00219	+	1.00219i	−0.00219615	+	0.999998i	$0.500699\pi$
$431$	−20.8061	+	20.8061i	−1.00219	+	1.00219i	−0.999998	+	0.00219615i	$0.999301\pi$
$432$		0			0
$433$	−3.84853			−0.184949			−0.0924744	−	0.995715i	$-0.529478\pi$
$433$	−3.84853			−0.184949			−0.0924744	+	0.995715i	$0.529478\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.68370	−	2.68370i	0.128379	−	0.128379i
$438$		0			0
$439$	−22.1944	−	22.1944i	−1.05928	−	1.05928i	−0.998128	−	0.0611539i	$-0.980522\pi$
$439$	−22.1944	−	22.1944i	−1.05928	−	1.05928i	−0.0611539	−	0.998128i	$-0.519478\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12.8469i			0.610375i	0.952292	+	0.305187i	$0.0987191\pi$
$443$			12.8469i			0.610375i	−0.952292	+	0.305187i	$0.901281\pi$
$444$		0			0
$445$	−9.77356	−	22.8245i	−0.463311	−	1.08199i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.27581	+	5.27581i	−0.248981	+	0.248981i	−0.820552	−	0.571571i	$-0.806334\pi$
$449$	−5.27581	+	5.27581i	−0.248981	+	0.248981i	0.571571	+	0.820552i	$0.306334\pi$
$450$		0			0
$451$		−	0.0611434i		−	0.00287913i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.620821	+	0.248552i	0.0291046	+	0.0116523i
$456$		0			0
$457$	13.4961			0.631321			0.315660	−	0.948872i	$-0.397774\pi$
$457$	13.4961			0.631321			0.315660	+	0.948872i	$0.397774\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			8.36859i			0.389764i	0.980827	+	0.194882i	$0.0624324\pi$
$461$			8.36859i			0.389764i	−0.980827	+	0.194882i	$0.937568\pi$
$462$		0			0
$463$		−	25.2792i		−	1.17483i	−0.809287	−	0.587413i	$-0.800146\pi$
$463$		−	25.2792i		−	1.17483i	0.809287	−	0.587413i	$-0.199854\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	36.3353			1.68140			0.840698	−	0.541505i	$-0.182145\pi$
$467$	36.3353			1.68140			0.840698	+	0.541505i	$0.182145\pi$
$468$		0			0
$469$	13.3376	−	13.3376i	0.615873	−	0.615873i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.486904	−	0.486904i	0.0223879	−	0.0223879i
$474$		0			0
$475$	6.32939	+	6.03548i	0.290412	+	0.276927i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.44239	−	9.44239i	−0.431434	−	0.431434i	0.457682	−	0.889116i	$-0.348680\pi$
$479$	−9.44239	−	9.44239i	−0.431434	−	0.431434i	−0.889116	+	0.457682i	$0.848680\pi$
$480$		0			0
$481$	−1.01427	−	1.01427i	−0.0462468	−	0.0462468i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	28.9770	+	11.6012i	1.31578	+	0.526785i
$486$		0			0
$487$	25.9251	+	25.9251i	1.17478	+	1.17478i	0.981056	+	0.193722i	$0.0620560\pi$
$487$	25.9251	+	25.9251i	1.17478	+	1.17478i	0.193722	+	0.981056i	$0.437944\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	11.0838i		−	0.500206i	−0.968219	−	0.250103i	$-0.919536\pi$
$491$		−	11.0838i		−	0.500206i	0.968219	−	0.250103i	$-0.0804644\pi$
$492$		0			0
$493$	−28.8377	−	26.0814i	−1.29879	−	1.17465i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	6.15130i		−	0.275923i
$498$		0			0
$499$	6.42411	−	6.42411i	0.287583	−	0.287583i	−0.548541	−	0.836124i	$-0.684817\pi$
$499$	6.42411	−	6.42411i	0.287583	−	0.287583i	0.836124	+	0.548541i	$0.184817\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−20.6348	−	20.6348i	−0.920060	−	0.920060i	0.0769732	−	0.997033i	$-0.475474\pi$
$503$	−20.6348	−	20.6348i	−0.920060	−	0.920060i	−0.997033	+	0.0769732i	$0.975474\pi$
$504$		0			0
$505$	−7.40907	−	17.3027i	−0.329699	−	0.769958i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.15274			0.139743			0.0698714	−	0.997556i	$-0.477741\pi$
$509$	3.15274			0.139743			0.0698714	+	0.997556i	$0.477741\pi$
$510$		0			0
$511$	1.02332			0.0452690
$512$		0			0
$513$		0			0
$514$		0			0
$515$	25.1796	−	10.7820i	1.10955	−	0.475112i
$516$		0			0
$517$	0.623578	+	0.623578i	0.0274249	+	0.0274249i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	17.5124	−	17.5124i	0.767231	−	0.767231i	−0.210387	−	0.977618i	$-0.567472\pi$
$521$	17.5124	−	17.5124i	0.767231	−	0.767231i	0.977618	+	0.210387i	$0.0674724\pi$
$522$		0			0
$523$			26.1056i			1.14152i	0.821117	+	0.570760i	$0.193351\pi$
$523$			26.1056i			1.14152i	−0.821117	+	0.570760i	$0.806649\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	23.6682	+	21.4060i	1.03100	+	0.932460i
$528$		0			0
$529$		−	18.2919i		−	0.795301i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.0523547	−	0.0523547i	−0.00226773	−	0.00226773i
$534$		0			0
$535$	6.75169	+	2.70311i	0.291901	+	0.116866i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.427211	+	0.427211i	0.0184013	+	0.0184013i
$540$		0			0
$541$	−13.6036	−	13.6036i	−0.584862	−	0.584862i	0.351373	−	0.936236i	$-0.385715\pi$
$541$	−13.6036	−	13.6036i	−0.584862	−	0.584862i	−0.936236	+	0.351373i	$0.885715\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.1391	+	4.05931i	0.434313	+	0.173882i
$546$		0			0
$547$	9.31869	−	9.31869i	0.398439	−	0.398439i	−0.479243	−	0.877682i	$-0.659089\pi$
$547$	9.31869	−	9.31869i	0.398439	−	0.398439i	0.877682	+	0.479243i	$0.159089\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	11.6639	−	11.6639i	0.496898	−	0.496898i
$552$		0			0
$553$	−6.56648			−0.279235
$554$		0			0
$555$		0			0
$556$		0			0
$557$			45.1155i			1.91161i	0.294009	+	0.955803i	$0.405010\pi$
$557$			45.1155i			1.91161i	−0.294009	+	0.955803i	$0.594990\pi$
$558$		0			0
$559$		−	0.833834i		−	0.0352674i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.89079			0.206122			0.103061	−	0.994675i	$-0.467136\pi$
$563$	4.89079			0.206122			0.103061	+	0.994675i	$0.467136\pi$
$564$		0			0
$565$	10.4936	−	26.2104i	0.441469	−	1.10268i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			15.0025i			0.628939i	0.949268	+	0.314469i	$0.101827\pi$
$569$			15.0025i			0.628939i	−0.949268	+	0.314469i	$0.898173\pi$
$570$		0			0
$571$	26.1952	−	26.1952i	1.09623	−	1.09623i	0.101387	−	0.994847i	$-0.467672\pi$
$571$	26.1952	−	26.1952i	1.09623	−	1.09623i	0.994847	−	0.101387i	$-0.0323279\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.8460	+	0.257801i	−0.452308	+	0.0107510i
$576$		0			0
$577$			29.9615i			1.24731i	0.781698	+	0.623657i	$0.214354\pi$
$577$			29.9615i			1.24731i	−0.781698	+	0.623657i	$0.785646\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.3141	+	12.3141i	0.510876	+	0.510876i
$582$		0			0
$583$	0.776668	−	0.776668i	0.0321663	−	0.0321663i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	39.2420			1.61969			0.809846	−	0.586642i	$-0.199551\pi$
$587$	39.2420			1.61969			0.809846	+	0.586642i	$0.199551\pi$
$588$		0			0
$589$	−9.57297	+	9.57297i	−0.394448	+	0.394448i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7.78605			0.319735			0.159867	−	0.987138i	$-0.448893\pi$
$593$	7.78605			0.319735			0.159867	+	0.987138i	$0.448893\pi$
$594$		0			0
$595$	−5.06375	+	14.7513i	−0.207594	+	0.604745i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11.5486			0.471862			0.235931	−	0.971770i	$-0.424186\pi$
$599$	11.5486			0.471862			0.235931	+	0.971770i	$0.424186\pi$
$600$		0			0
$601$	−2.93848	+	2.93848i	−0.119863	+	0.119863i	−0.764494	−	0.644631i	$-0.777011\pi$
$601$	−2.93848	+	2.93848i	−0.119863	+	0.119863i	0.644631	+	0.764494i	$0.277011\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.5672	+	9.66336i	−0.917486	+	0.392871i
$606$		0			0
$607$	17.4679	−	17.4679i	0.709000	−	0.709000i	−0.257325	−	0.966325i	$-0.582841\pi$
$607$	17.4679	−	17.4679i	0.709000	−	0.709000i	0.966325	+	0.257325i	$0.0828410\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.06789			0.0432023
$612$		0			0
$613$		−	13.8618i		−	0.559875i	−0.960018	−	0.279937i	$-0.909686\pi$
$613$		−	13.8618i		−	0.559875i	0.960018	−	0.279937i	$-0.0903137\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.2659	−	17.2659i	0.695098	−	0.695098i	−0.268251	−	0.963349i	$-0.586446\pi$
$617$	17.2659	−	17.2659i	0.695098	−	0.695098i	0.963349	+	0.268251i	$0.0864457\pi$
$618$		0			0
$619$	−0.532233	+	0.532233i	−0.0213923	+	0.0213923i	−0.717722	−	0.696330i	$-0.754815\pi$
$619$	−0.532233	+	0.532233i	−0.0213923	+	0.0213923i	0.696330	+	0.717722i	$0.254815\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.2822	+	13.2822i	0.532141	+	0.532141i
$624$		0			0
$625$	−1.18779	−	24.9718i	−0.0475117	−	0.998871i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	22.4396	−	24.8110i	0.894727	−	0.989280i
$630$		0			0
$631$		−	14.1421i		−	0.562987i	−0.959563	−	0.281493i	$-0.909170\pi$
$631$		−	14.1421i		−	0.562987i	0.959563	−	0.281493i	$-0.0908297\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.372747	+	0.870488i	0.0147920	+	0.0345443i
$636$		0			0
$637$	0.731608			0.0289874
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.7475	−	10.7475i	−0.424501	−	0.424501i	0.462249	−	0.886750i	$-0.347043\pi$
$641$	−10.7475	−	10.7475i	−0.424501	−	0.424501i	−0.886750	+	0.462249i	$0.847043\pi$
$642$		0			0
$643$	34.2913	−	34.2913i	1.35232	−	1.35232i	0.469256	−	0.883062i	$-0.344522\pi$
$643$	34.2913	−	34.2913i	1.35232	−	1.35232i	0.883062	−	0.469256i	$-0.155478\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	24.6744i		−	0.970052i	−0.874500	−	0.485026i	$-0.838810\pi$
$647$		−	24.6744i		−	0.970052i	0.874500	−	0.485026i	$-0.161190\pi$
$648$		0			0
$649$	−1.33121	−	1.33121i	−0.0522545	−	0.0522545i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20.5203	+	20.5203i	0.803023	+	0.803023i	0.983567	−	0.180544i	$-0.0577857\pi$
$653$	20.5203	+	20.5203i	0.803023	+	0.803023i	−0.180544	+	0.983567i	$0.557786\pi$
$654$		0			0
$655$	0.233974	−	0.584409i	0.00914212	−	0.0228347i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.79365			−0.381507			−0.190753	−	0.981638i	$-0.561093\pi$
$659$	−9.79365			−0.381507			−0.190753	+	0.981638i	$0.561093\pi$
$660$		0			0
$661$		−	37.9631i		−	1.47659i	−0.674477	−	0.738296i	$-0.735631\pi$
$661$		−	37.9631i		−	1.47659i	0.674477	−	0.738296i	$-0.264369\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.14242	−	2.45918i	−0.238193	−	0.0953629i
$666$		0			0
$667$			20.4622i			0.792299i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1.23978i		−	0.0478612i
$672$		0			0
$673$	24.0348	+	24.0348i	0.926473	+	0.926473i	0.997476	−	0.0710032i	$-0.0226201\pi$
$673$	24.0348	+	24.0348i	0.926473	+	0.926473i	−0.0710032	+	0.997476i	$0.522620\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.38222	+	5.38222i	−0.206856	+	0.206856i	−0.802930	−	0.596074i	$-0.796727\pi$
$677$	−5.38222	+	5.38222i	−0.206856	+	0.206856i	0.596074	+	0.802930i	$0.296727\pi$
$678$		0			0
$679$	−23.6136			−0.906206
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.56311	−	4.56311i	0.174602	−	0.174602i	−0.614396	−	0.788998i	$-0.710600\pi$
$683$	4.56311	−	4.56311i	0.174602	−	0.174602i	0.788998	+	0.614396i	$0.210600\pi$
$684$		0			0
$685$	−38.5444	+	16.5049i	−1.47270	+	0.630618i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1.33006i		−	0.0506713i
$690$		0			0
$691$	15.4739	−	15.4739i	0.588654	−	0.588654i	−0.348613	−	0.937267i	$-0.613347\pi$
$691$	15.4739	−	15.4739i	0.588654	−	0.588654i	0.937267	+	0.348613i	$0.113347\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13.1740	−	32.9053i	0.499717	−	1.24817i
$696$		0			0
$697$	1.15829	−	1.28070i	0.0438734	−	0.0485099i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	23.2466			0.878013			0.439007	−	0.898484i	$-0.355330\pi$
$701$	23.2466			0.878013			0.439007	+	0.898484i	$0.355330\pi$
$702$		0			0
$703$	10.0352	+	10.0352i	0.378486	+	0.378486i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	10.0689	+	10.0689i	0.378679	+	0.378679i
$708$		0			0
$709$	−4.35978	−	4.35978i	−0.163735	−	0.163735i	0.620484	−	0.784219i	$-0.286936\pi$
$709$	−4.35978	−	4.35978i	−0.163735	−	0.163735i	−0.784219	+	0.620484i	$0.786936\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	16.7941i		−	0.628943i
$714$		0			0
$715$	0.0214506	−	0.0535783i	0.000802207	−	0.00200372i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−34.6966	−	34.6966i	−1.29396	−	1.29396i	−0.932314	−	0.361649i	$-0.882214\pi$
$719$	−34.6966	−	34.6966i	−1.29396	−	1.29396i	−0.361649	−	0.932314i	$-0.617786\pi$
$720$		0			0
$721$	−14.6527	+	14.6527i	−0.545695	+	0.545695i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−47.1387	+	1.12045i	−1.75069	+	0.0416126i
$726$		0			0
$727$			44.3742i			1.64575i	0.568224	+	0.822874i	$0.307631\pi$
$727$			44.3742i			1.64575i	−0.568224	+	0.822874i	$0.692369\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	19.4224	−	0.974766i	0.718365	−	0.0360530i
$732$		0			0
$733$	41.2535			1.52373			0.761866	−	0.647735i	$-0.224284\pi$
$733$	41.2535			1.52373			0.761866	+	0.647735i	$0.224284\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.15106	−	1.15106i	−0.0424000	−	0.0424000i
$738$		0			0
$739$			47.9117i			1.76246i	0.472688	+	0.881230i	$0.343284\pi$
$739$			47.9117i			1.76246i	−0.472688	+	0.881230i	$0.656716\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	23.1660	−	23.1660i	0.849880	−	0.849880i	−0.140238	−	0.990118i	$-0.544787\pi$
$743$	23.1660	−	23.1660i	0.849880	−	0.849880i	0.990118	+	0.140238i	$0.0447869\pi$
$744$		0			0
$745$	−9.79191	−	22.8674i	−0.358748	−	0.837796i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.50201			−0.201039
$750$		0			0
$751$	1.19624	+	1.19624i	0.0436513	+	0.0436513i	0.728596	−	0.684944i	$-0.240173\pi$
$751$	1.19624	+	1.19624i	0.0436513	+	0.0436513i	−0.684944	+	0.728596i	$0.740173\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17.2074	−	7.36828i	0.626241	−	0.268159i
$756$		0			0
$757$	−35.9669			−1.30724			−0.653620	−	0.756823i	$-0.726750\pi$
$757$	−35.9669			−1.30724			−0.653620	+	0.756823i	$0.726750\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−31.3043			−1.13478			−0.567390	−	0.823449i	$-0.692047\pi$
$761$	−31.3043			−1.13478			−0.567390	+	0.823449i	$0.692047\pi$
$762$		0			0
$763$	−8.26247			−0.299121
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.27973			−0.0823161
$768$		0			0
$769$	33.8570			1.22091			0.610457	−	0.792049i	$-0.290986\pi$
$769$	33.8570			1.22091			0.610457	+	0.792049i	$0.290986\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10.8896			0.391672			0.195836	−	0.980637i	$-0.437258\pi$
$773$	10.8896			0.391672			0.195836	+	0.980637i	$0.437258\pi$
$774$		0			0
$775$	38.6885	−	0.919597i	1.38973	−	0.0330329i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.517999	+	0.517999i	0.0185592	+	0.0185592i
$780$		0			0
$781$	−0.530871			−0.0189961
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.84773	−	2.07582i	0.173023	−	0.0740891i
$786$		0			0
$787$	3.04641	−	3.04641i	0.108593	−	0.108593i	−0.650723	−	0.759316i	$-0.725534\pi$
$787$	3.04641	−	3.04641i	0.108593	−	0.108593i	0.759316	+	0.650723i	$0.225534\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			21.3590i			0.759440i
$792$		0			0
$793$	−1.06158	−	1.06158i	−0.0376977	−	0.0376977i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7.09190			−0.251208			−0.125604	−	0.992080i	$-0.540087\pi$
$797$	−7.09190			−0.251208			−0.125604	+	0.992080i	$0.540087\pi$
$798$		0			0
$799$	1.24838	+	24.8743i	0.0441646	+	0.879991i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	0.0883148i		−	0.00311656i
$804$		0			0
$805$	7.54499	−	3.23079i	0.265926	−	0.113871i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	19.6646	−	19.6646i	0.691370	−	0.691370i	−0.271163	−	0.962533i	$-0.587408\pi$
$809$	19.6646	−	19.6646i	0.691370	−	0.691370i	0.962533	+	0.271163i	$0.0874083\pi$
$810$		0			0
$811$	22.4158	+	22.4158i	0.787124	+	0.787124i	0.981022	−	0.193898i	$-0.0621129\pi$
$811$	22.4158	+	22.4158i	0.787124	+	0.787124i	−0.193898	+	0.981022i	$0.562113\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.36587	−	5.90936i	0.0828729	−	0.206996i
$816$		0			0
$817$			8.24998i			0.288630i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.25498	+	3.25498i	0.113600	+	0.113600i	0.761622	−	0.648022i	$-0.224404\pi$
$821$	3.25498	+	3.25498i	0.113600	+	0.113600i	−0.648022	+	0.761622i	$0.724404\pi$
$822$		0			0
$823$	−31.5835	−	31.5835i	−1.10093	−	1.10093i	−0.994299	−	0.106631i	$-0.965994\pi$
$823$	−31.5835	−	31.5835i	−1.10093	−	1.10093i	−0.106631	−	0.994299i	$-0.534006\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9.25340	+	9.25340i	0.321772	+	0.321772i	0.849447	−	0.527675i	$-0.176936\pi$
$827$	9.25340	+	9.25340i	0.321772	+	0.321772i	−0.527675	+	0.849447i	$0.676936\pi$
$828$		0			0
$829$	−7.40453			−0.257170			−0.128585	−	0.991698i	$-0.541044\pi$
$829$	−7.40453			−0.257170			−0.128585	+	0.991698i	$0.541044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.855262	+	17.0413i	0.0296331	+	0.590446i
$834$		0			0
$835$	−16.7061	−	6.68844i	−0.578137	−	0.231463i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22.7230	−	22.7230i	0.784486	−	0.784486i	−0.196098	−	0.980584i	$-0.562827\pi$
$839$	22.7230	−	22.7230i	0.784486	−	0.784486i	0.980584	+	0.196098i	$0.0628271\pi$
$840$		0			0
$841$			59.9327i			2.06664i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.4150	+	26.6578i	0.392688	+	0.917057i
$846$		0			0
$847$	13.1325	−	13.1325i	0.451236	−	0.451236i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−17.6050			−0.603491
$852$		0			0
$853$	14.6526	−	14.6526i	0.501695	−	0.501695i	−0.410269	−	0.911964i	$-0.634565\pi$
$853$	14.6526	−	14.6526i	0.501695	−	0.501695i	0.911964	+	0.410269i	$0.134565\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	27.8195	+	27.8195i	0.950297	+	0.950297i	0.998822	−	0.0485252i	$-0.0154521\pi$
$857$	27.8195	+	27.8195i	0.950297	+	0.950297i	−0.0485252	+	0.998822i	$0.515452\pi$
$858$		0			0
$859$			26.7984i			0.914349i	0.889377	+	0.457175i	$0.151139\pi$
$859$			26.7984i			0.914349i	−0.889377	+	0.457175i	$0.848861\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	14.8401i		−	0.505164i	−0.967575	−	0.252582i	$-0.918720\pi$
$863$		−	14.8401i		−	0.505164i	0.967575	−	0.252582i	$-0.0812798\pi$
$864$		0			0
$865$	−25.4755	−	10.1994i	−0.866192	−	0.346789i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			0.566703i			0.0192241i
$870$		0			0
$871$	−1.97122			−0.0667924
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.85593	+	17.2045i	0.265579	+	0.581618i
$876$		0			0
$877$	4.22641	+	4.22641i	0.142716	+	0.142716i	0.774855	−	0.632139i	$-0.217823\pi$
$877$	4.22641	+	4.22641i	0.142716	+	0.142716i	−0.632139	+	0.774855i	$0.717823\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.00976	−	7.00976i	−0.236165	−	0.236165i	0.579095	−	0.815260i	$-0.303406\pi$
$881$	−7.00976	−	7.00976i	−0.236165	−	0.236165i	−0.815260	+	0.579095i	$0.803406\pi$
$882$		0			0
$883$			26.1862i			0.881237i	0.897695	+	0.440618i	$0.145241\pi$
$883$			26.1862i			0.881237i	−0.897695	+	0.440618i	$0.854759\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−22.6947	+	22.6947i	−0.762014	+	0.762014i	−0.976686	−	0.214673i	$-0.931132\pi$
$887$	−22.6947	+	22.6947i	−0.762014	+	0.762014i	0.214673	+	0.976686i	$0.431132\pi$
$888$		0			0
$889$	−0.506560	−	0.506560i	−0.0169895	−	0.0169895i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.5658			−0.353570
$894$		0			0
$895$	−14.4125	+	6.17148i	−0.481756	+	0.206290i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	72.9903i		−	2.43436i
$900$		0			0
$901$	30.9810	−	1.55486i	1.03213	−	0.0518000i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.5649	−	8.23335i	−0.683599	−	0.273686i
$906$		0			0
$907$	22.9163	+	22.9163i	0.760923	+	0.760923i	0.976489	−	0.215566i	$-0.0691597\pi$
$907$	22.9163	+	22.9163i	0.760923	+	0.760923i	−0.215566	+	0.976489i	$0.569160\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18.3002	+	18.3002i	−0.606314	+	0.606314i	−0.941981	−	0.335667i	$-0.891038\pi$
$911$	−18.3002	+	18.3002i	−0.606314	+	0.606314i	0.335667	+	0.941981i	$0.391038\pi$
$912$		0			0
$913$	1.06274	−	1.06274i	0.0351714	−	0.0351714i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			0.476239i			0.0157268i
$918$		0			0
$919$	42.7392			1.40984			0.704919	−	0.709288i	$-0.250983\pi$
$919$	42.7392			1.40984			0.704919	+	0.709288i	$0.250983\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.454565	+	0.454565i	−0.0149622	+	0.0149622i
$924$		0			0
$925$	−0.964001	−	40.5566i	−0.0316962	−	1.33349i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.0696	+	17.0696i	−0.560034	+	0.560034i	−0.929317	−	0.369283i	$-0.879603\pi$
$929$	−17.0696	+	17.0696i	−0.560034	+	0.560034i	0.369283	+	0.929317i	$0.379603\pi$
$930$		0			0
$931$	−7.23855			−0.237234
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.27307	+	0.437014i	0.0416339	+	0.0142919i
$936$		0			0
$937$	41.9491			1.37042			0.685209	−	0.728347i	$-0.259711\pi$
$937$	41.9491			1.37042			0.685209	+	0.728347i	$0.259711\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.99112	+	7.99112i	−0.260503	+	0.260503i	−0.825258	−	0.564755i	$-0.808971\pi$
$941$	−7.99112	+	7.99112i	−0.260503	+	0.260503i	0.564755	+	0.825258i	$0.308971\pi$
$942$		0			0
$943$	−0.908736			−0.0295925
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.9449	+	11.9449i	−0.388158	+	0.388158i	−0.874030	−	0.485872i	$-0.838502\pi$
$947$	−11.9449	+	11.9449i	−0.388158	+	0.388158i	0.485872	+	0.874030i	$0.338502\pi$
$948$		0			0
$949$	−0.0756206	−	0.0756206i	−0.00245475	−	0.00245475i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			47.4871i			1.53826i	0.639093	+	0.769129i	$0.279310\pi$
$953$			47.4871i			1.53826i	−0.639093	+	0.769129i	$0.720690\pi$
$954$		0			0
$955$	10.2420	+	23.9185i	0.331423	+	0.773984i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	22.4300	−	22.4300i	0.724303	−	0.724303i
$960$		0			0
$961$			28.9058i			0.932446i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.05504	−	12.6262i	0.162727	−	0.406453i
$966$		0			0
$967$	−0.157252			−0.00505688			−0.00252844	−	0.999997i	$-0.500805\pi$
$967$	−0.157252			−0.00505688			−0.00252844	+	0.999997i	$0.500805\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	5.70175i		−	0.182978i	−0.995806	−	0.0914889i	$-0.970837\pi$
$971$		−	5.70175i		−	0.182978i	0.995806	−	0.0914889i	$-0.0291626\pi$
$972$		0			0
$973$			26.8148i			0.859643i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	4.92404			0.157534			0.0787671	−	0.996893i	$-0.474902\pi$
$977$	4.92404			0.157534			0.0787671	+	0.996893i	$0.474902\pi$
$978$		0			0
$979$	1.14629	−	1.14629i	0.0366355	−	0.0366355i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−32.3248	+	32.3248i	−1.03100	+	1.03100i	−0.0314964	+	0.999504i	$0.510027\pi$
$983$	−32.3248	+	32.3248i	−1.03100	+	1.03100i	−0.999504	+	0.0314964i	$0.989973\pi$
$984$		0			0
$985$	11.6109	−	29.0011i	0.369953	−	0.924051i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.23655	−	7.23655i	−0.230109	−	0.230109i
$990$		0			0
$991$	9.09429	+	9.09429i	0.288890	+	0.288890i	0.836641	−	0.547752i	$-0.184516\pi$
$991$	9.09429	+	9.09429i	0.288890	+	0.288890i	−0.547752	+	0.836641i	$0.684516\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.00732	−	20.0003i	0.253849	−	0.634052i
$996$		0			0
$997$	11.9136	+	11.9136i	0.377307	+	0.377307i	0.870130	−	0.492823i	$-0.164035\pi$
$997$	11.9136	+	11.9136i	0.377307	+	0.377307i	−0.492823	+	0.870130i	$0.664035\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.z.g.829.13		40
3.2	odd	2		1020.2.y.a.829.13	yes	40
5.4	even	2	inner	3060.2.z.g.829.18		40
15.14	odd	2		1020.2.y.a.829.8	yes	40
17.4	even	4	inner	3060.2.z.g.2809.18		40
51.38	odd	4		1020.2.y.a.769.8	✓	40
85.4	even	4	inner	3060.2.z.g.2809.13		40
255.89	odd	4		1020.2.y.a.769.13	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.y.a.769.8	✓	40	51.38	odd	4
1020.2.y.a.769.13	yes	40	255.89	odd	4
1020.2.y.a.829.8	yes	40	15.14	odd	2
1020.2.y.a.829.13	yes	40	3.2	odd	2
3060.2.z.g.829.13		40	1.1	even	1	trivial
3060.2.z.g.829.18		40	5.4	even	2	inner
3060.2.z.g.2809.13		40	85.4	even	4	inner
3060.2.z.g.2809.18		40	17.4	even	4	inner

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:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.z (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.880193 + 2.05554i) q^{5} +(-1.19618 - 1.19618i) q^{7} +(-0.103233 + 0.103233i) q^{11} +0.176789i q^{13} +(-4.11792 + 0.206669i) q^{17} -1.74915i q^{19} +(1.53429 + 1.53429i) q^{23} +(-3.45052 + 3.61855i) q^{25} +(6.66831 + 6.66831i) q^{29} +(-5.47293 - 5.47293i) q^{31} +(1.40593 - 3.51166i) q^{35} +(-5.73720 + 5.73720i) q^{37} +(-0.296143 + 0.296143i) q^{41} -4.71656 q^{43} -6.04051i q^{47} -4.13832i q^{49} -7.52346 q^{53} +(-0.303064 - 0.121335i) q^{55} +12.8952i q^{59} +(-6.00478 + 6.00478i) q^{61} +(-0.363397 + 0.155608i) q^{65} +11.1502i q^{67} +(2.57123 + 2.57123i) q^{71} +(-0.427746 + 0.427746i) q^{73} +0.246969 q^{77} +(2.74478 - 2.74478i) q^{79} -10.2946 q^{83} +(-4.04938 - 8.28266i) q^{85} -11.1039 q^{89} +(0.211470 - 0.211470i) q^{91} +(3.59546 - 1.53959i) q^{95} +(9.87044 - 9.87044i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 4 q^{5} + 8 q^{11} - 24 q^{29} - 16 q^{31} - 8 q^{35} + 8 q^{41} + 28 q^{55} + 16 q^{61} - 56 q^{71} + 16 q^{79} - 40 q^{85} - 32 q^{89} + 64 q^{91} - 64 q^{95}+O(q^{100}) \) 40 * q - 4 * q^5 + 8 * q^11 - 24 * q^29 - 16 * q^31 - 8 * q^35 + 8 * q^41 + 28 * q^55 + 16 * q^61 - 56 * q^71 + 16 * q^79 - 40 * q^85 - 32 * q^89 + 64 * q^91 - 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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