LMFDB - Embedded newform 2016.2.bu.c.431.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(431,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2016.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.bu (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:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			431.1
Character	$\chi$	$=$	2016.431
Dual form			2016.2.bu.c.1871.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.88256 - 3.26069i) q^{5} +(2.23426 + 1.41706i) q^{7} +O(q^{10})$	$q+(-1.88256 - 3.26069i) q^{5} +(2.23426 + 1.41706i) q^{7} +(1.47084 + 0.849192i) q^{11} +5.64222i q^{13} +(2.26012 + 1.30488i) q^{17} +(1.18076 + 2.04513i) q^{19} +(-0.653873 - 1.13254i) q^{23} +(-4.58807 + 7.94677i) q^{25} -6.80944 q^{29} +(-4.75107 - 2.74303i) q^{31} +(0.414472 - 9.95295i) q^{35} +(-5.20109 + 3.00285i) q^{37} +6.10456i q^{41} +6.49804 q^{43} +(3.53360 + 6.12038i) q^{47} +(2.98386 + 6.33219i) q^{49} +(0.488913 - 0.846821i) q^{53} -6.39462i q^{55} +(6.67395 + 3.85320i) q^{59} +(7.64484 - 4.41375i) q^{61} +(18.3975 - 10.6218i) q^{65} +(-6.69828 + 11.6018i) q^{67} +9.40544 q^{71} +(-1.09003 + 1.88798i) q^{73} +(2.08289 + 3.98160i) q^{77} +(15.0908 - 8.71267i) q^{79} +11.8864i q^{83} -9.82605i q^{85} +(-9.58642 + 5.53472i) q^{89} +(-7.99538 + 12.6062i) q^{91} +(4.44570 - 7.70018i) q^{95} +15.0631 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 32 q^{19} + 160 q^{43} + 56 q^{49} + 16 q^{73} + 32 q^{91} + 240 q^{97}+O(q^{100}) $ 48 * q + 32 * q^19 + 160 * q^43 + 56 * q^49 + 16 * q^73 + 32 * q^91 + 240 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.88256	−	3.26069i	−0.841907	−	1.45823i	−0.888281	−	0.459301i	$-0.848100\pi$
$5$	−1.88256	−	3.26069i	−0.841907	−	1.45823i	0.0463738	−	0.998924i	$-0.485233\pi$
$6$		0			0
$7$	2.23426	+	1.41706i	0.844472	+	0.535600i
$7$	2.23426	+	1.41706i	0.844472	+	0.535600i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.47084	+	0.849192i	0.443476	+	0.256041i	0.705071	−	0.709137i	$-0.250915\pi$
$11$	1.47084	+	0.849192i	0.443476	+	0.256041i	−0.261595	+	0.965178i	$0.584248\pi$
$12$		0			0
$13$			5.64222i			1.56487i	0.622733	+	0.782435i	$0.286022\pi$
$13$			5.64222i			1.56487i	−0.622733	+	0.782435i	$0.713978\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.26012	+	1.30488i	0.548158	+	0.316479i	0.748379	−	0.663271i	$-0.230833\pi$
$17$	2.26012	+	1.30488i	0.548158	+	0.316479i	−0.200220	+	0.979751i	$0.564166\pi$
$18$		0			0
$19$	1.18076	+	2.04513i	0.270885	+	0.469186i	0.969089	−	0.246713i	$-0.0793505\pi$
$19$	1.18076	+	2.04513i	0.270885	+	0.469186i	−0.698204	+	0.715899i	$0.746017\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.653873	−	1.13254i	−0.136342	−	0.236151i	0.789767	−	0.613406i	$-0.210201\pi$
$23$	−0.653873	−	1.13254i	−0.136342	−	0.236151i	−0.926109	+	0.377255i	$0.876868\pi$
$24$		0			0
$25$	−4.58807	+	7.94677i	−0.917614	+	1.58935i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.80944			−1.26448			−0.632241	−	0.774772i	$-0.717865\pi$
$29$	−6.80944			−1.26448			−0.632241	+	0.774772i	$0.717865\pi$
$30$		0			0
$31$	−4.75107	−	2.74303i	−0.853317	−	0.492663i	0.00845183	−	0.999964i	$-0.497310\pi$
$31$	−4.75107	−	2.74303i	−0.853317	−	0.492663i	−0.861769	+	0.507302i	$0.830643\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.414472	−	9.95295i	0.0700586	−	1.68236i
$36$		0			0
$37$	−5.20109	+	3.00285i	−0.855054	+	0.493666i	−0.862353	−	0.506308i	$-0.831010\pi$
$37$	−5.20109	+	3.00285i	−0.855054	+	0.493666i	0.00729899	+	0.999973i	$0.497677\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.10456i			0.953373i	0.879073	+	0.476686i	$0.158162\pi$
$41$			6.10456i			0.953373i	−0.879073	+	0.476686i	$0.841838\pi$
$42$		0			0
$43$	6.49804			0.990943			0.495471	−	0.868624i	$-0.334995\pi$
$43$	6.49804			0.990943			0.495471	+	0.868624i	$0.334995\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.53360	+	6.12038i	0.515429	+	0.892749i	0.999840	+	0.0179081i	$0.00570062\pi$
$47$	3.53360	+	6.12038i	0.515429	+	0.892749i	−0.484411	+	0.874841i	$0.660966\pi$
$48$		0			0
$49$	2.98386	+	6.33219i	0.426266	+	0.904598i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.488913	−	0.846821i	0.0671573	−	0.116320i	−0.830492	−	0.557031i	$-0.811940\pi$
$53$	0.488913	−	0.846821i	0.0671573	−	0.116320i	0.897649	+	0.440711i	$0.145274\pi$
$54$		0			0
$55$		−	6.39462i		−	0.862251i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.67395	+	3.85320i	0.868874	+	0.501645i	0.866974	−	0.498354i	$-0.166062\pi$
$59$	6.67395	+	3.85320i	0.868874	+	0.501645i	0.00190001	+	0.999998i	$0.499395\pi$
$60$		0			0
$61$	7.64484	−	4.41375i	0.978822	−	0.565123i	0.0769075	−	0.997038i	$-0.475495\pi$
$61$	7.64484	−	4.41375i	0.978822	−	0.565123i	0.901914	+	0.431915i	$0.142162\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	18.3975	−	10.6218i	2.28193	−	1.31747i
$66$		0			0
$67$	−6.69828	+	11.6018i	−0.818325	+	1.41738i	0.0885910	+	0.996068i	$0.471764\pi$
$67$	−6.69828	+	11.6018i	−0.818325	+	1.41738i	−0.906916	+	0.421312i	$0.861570\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	9.40544			1.11622			0.558110	−	0.829767i	$-0.311527\pi$
$71$	9.40544			1.11622			0.558110	+	0.829767i	$0.311527\pi$
$72$		0			0
$73$	−1.09003	+	1.88798i	−0.127578	+	0.220971i	−0.922738	−	0.385429i	$-0.874054\pi$
$73$	−1.09003	+	1.88798i	−0.127578	+	0.220971i	0.795160	+	0.606400i	$0.207387\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.08289	+	3.98160i	0.237368	+	0.453745i
$78$		0			0
$79$	15.0908	−	8.71267i	1.69785	−	0.980252i	0.750045	−	0.661386i	$-0.230032\pi$
$79$	15.0908	−	8.71267i	1.69785	−	0.980252i	0.947800	−	0.318865i	$-0.103302\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			11.8864i			1.30471i	0.757915	+	0.652353i	$0.226218\pi$
$83$			11.8864i			1.30471i	−0.757915	+	0.652353i	$0.773782\pi$
$84$		0			0
$85$		−	9.82605i		−	1.06578i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.58642	+	5.53472i	−1.01616	+	0.586679i	−0.912989	−	0.407984i	$-0.866232\pi$
$89$	−9.58642	+	5.53472i	−1.01616	+	0.586679i	−0.103170	+	0.994664i	$0.532898\pi$
$90$		0			0
$91$	−7.99538	+	12.6062i	−0.838144	+	1.32149i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.44570	−	7.70018i	0.456119	−	0.790021i
$96$		0			0
$97$	15.0631			1.52943			0.764715	−	0.644369i	$-0.222880\pi$
$97$	15.0631			1.52943			0.764715	+	0.644369i	$0.222880\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.777899	+	1.34736i	−0.0774039	+	0.134067i	−0.902129	−	0.431466i	$-0.857996\pi$
$101$	−0.777899	+	1.34736i	−0.0774039	+	0.134067i	0.824725	+	0.565534i	$0.191330\pi$
$102$		0			0
$103$	−6.94066	+	4.00719i	−0.683883	+	0.394840i	−0.801317	−	0.598240i	$-0.795867\pi$
$103$	−6.94066	+	4.00719i	−0.683883	+	0.394840i	0.117433	+	0.993081i	$0.462533\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.6257	−	7.28943i	1.22057	−	0.704696i	0.255530	−	0.966801i	$-0.417750\pi$
$107$	12.6257	−	7.28943i	1.22057	−	0.704696i	0.965039	+	0.262105i	$0.0844167\pi$
$108$		0			0
$109$	−0.655188	−	0.378273i	−0.0627556	−	0.0362320i	0.468294	−	0.883573i	$-0.344869\pi$
$109$	−0.655188	−	0.378273i	−0.0627556	−	0.0362320i	−0.531050	+	0.847341i	$0.678202\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.87390i		−	0.928859i	−0.885610	−	0.464429i	$-0.846259\pi$
$113$		−	9.87390i		−	0.928859i	0.885610	−	0.464429i	$-0.153741\pi$
$114$		0			0
$115$	−2.46191	+	4.26416i	−0.229574	+	0.397635i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.20060	+	6.11817i	0.293398	+	0.560852i
$120$		0			0
$121$	−4.05774	−	7.02822i	−0.368886	−	0.638929i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	15.7237			1.40637
$126$		0			0
$127$			4.50847i			0.400062i	0.979790	+	0.200031i	$0.0641043\pi$
$127$			4.50847i			0.400062i	−0.979790	+	0.200031i	$0.935896\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	14.0673	−	8.12176i	1.22907	−	0.709601i	0.262231	−	0.965005i	$-0.415542\pi$
$131$	14.0673	−	8.12176i	1.22907	−	0.709601i	0.966835	+	0.255404i	$0.0822084\pi$
$132$		0			0
$133$	−0.259961	+	6.24258i	−0.0225414	+	0.541300i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.63470	−	2.09849i	−0.310533	−	0.179286i	0.336632	−	0.941636i	$-0.390712\pi$
$137$	−3.63470	−	2.09849i	−0.310533	−	0.179286i	−0.647165	+	0.762350i	$0.724046\pi$
$138$		0			0
$139$	5.70274			0.483700			0.241850	−	0.970314i	$-0.422246\pi$
$139$	5.70274			0.483700			0.241850	+	0.970314i	$0.422246\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.79133	+	8.29882i	−0.400671	+	0.693982i
$144$		0			0
$145$	12.8192	+	22.2035i	1.06458	+	1.84390i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.74195	−	11.6774i	−0.552322	−	0.956650i	−0.998106	−	0.0615095i	$-0.980409\pi$
$149$	−6.74195	−	11.6774i	−0.552322	−	0.956650i	0.445784	−	0.895140i	$-0.352925\pi$
$150$		0			0
$151$	18.4727	+	10.6652i	1.50329	+	0.867924i	0.999993	+	0.00380878i	$0.00121238\pi$
$151$	18.4727	+	10.6652i	1.50329	+	0.867924i	0.503295	+	0.864115i	$0.332121\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			20.6557i			1.65910i
$156$		0			0
$157$	−7.97251	−	4.60293i	−0.636276	−	0.367354i	0.146903	−	0.989151i	$-0.453070\pi$
$157$	−7.97251	−	4.60293i	−0.636276	−	0.367354i	−0.783179	+	0.621797i	$0.786403\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.143959	−	3.45697i	0.0113456	−	0.272448i
$162$		0			0
$163$	−0.0600325	−	0.103979i	−0.00470211	−	0.00814429i	0.863665	−	0.504067i	$-0.168163\pi$
$163$	−0.0600325	−	0.103979i	−0.00470211	−	0.00814429i	−0.868367	+	0.495922i	$0.834830\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.9810			−1.00450			−0.502251	−	0.864722i	$-0.667495\pi$
$167$	−12.9810			−1.00450			−0.502251	+	0.864722i	$0.667495\pi$
$168$		0			0
$169$	−18.8346			−1.44882
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.44458	+	5.96618i	0.261886	+	0.453600i	0.966743	−	0.255749i	$-0.0823221\pi$
$173$	3.44458	+	5.96618i	0.261886	+	0.453600i	−0.704857	+	0.709350i	$0.748989\pi$
$174$		0			0
$175$	−21.5120	+	11.2536i	−1.62616	+	0.850691i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.6043	−	7.85445i	−1.01683	−	0.587070i	−0.103649	−	0.994614i	$-0.533052\pi$
$179$	−13.6043	−	7.85445i	−1.01683	−	0.587070i	−0.913185	+	0.407544i	$0.866385\pi$
$180$		0			0
$181$			15.6714i			1.16485i	0.812886	+	0.582423i	$0.197895\pi$
$181$			15.6714i			1.16485i	−0.812886	+	0.582423i	$0.802105\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	19.5827	+	11.3061i	1.43975	+	0.831241i
$186$		0			0
$187$	2.21618	+	3.83854i	0.162063	+	0.280702i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.0633	−	20.8943i	−0.872873	−	1.51186i	−0.859011	−	0.511957i	$-0.828921\pi$
$191$	−12.0633	−	20.8943i	−0.872873	−	1.51186i	−0.0138617	−	0.999904i	$-0.504412\pi$
$192$		0			0
$193$	−4.80306	+	8.31914i	−0.345732	+	0.598825i	−0.985486	−	0.169754i	$-0.945703\pi$
$193$	−4.80306	+	8.31914i	−0.345732	+	0.598825i	0.639755	+	0.768579i	$0.279036\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.05305			−0.146274			−0.0731371	−	0.997322i	$-0.523301\pi$
$197$	−2.05305			−0.146274			−0.0731371	+	0.997322i	$0.523301\pi$
$198$		0			0
$199$	−8.62103	−	4.97735i	−0.611128	−	0.352835i	0.162279	−	0.986745i	$-0.448116\pi$
$199$	−8.62103	−	4.97735i	−0.611128	−	0.352835i	−0.773407	+	0.633910i	$0.781449\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−15.2141	−	9.64941i	−1.06782	−	0.677256i
$204$		0			0
$205$	19.9051	−	11.4922i	1.39023	−	0.802651i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			4.01076i			0.277430i
$210$		0			0
$211$	13.9757			0.962128			0.481064	−	0.876685i	$-0.340250\pi$
$211$	13.9757			0.962128			0.481064	+	0.876685i	$0.340250\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.2330	−	21.1881i	−0.834281	−	1.44502i
$216$		0			0
$217$	−6.72808	−	12.8612i	−0.456732	−	0.873076i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.36241	+	12.7521i	−0.495249	+	0.857796i
$222$		0			0
$223$			15.8817i			1.06352i	0.846896	+	0.531759i	$0.178469\pi$
$223$			15.8817i			1.06352i	−0.846896	+	0.531759i	$0.821531\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.620059	−	0.357991i	−0.0411548	−	0.0237607i	0.479281	−	0.877661i	$-0.340897\pi$
$227$	−0.620059	−	0.357991i	−0.0411548	−	0.0237607i	−0.520436	+	0.853901i	$0.674231\pi$
$228$		0			0
$229$	3.89913	−	2.25117i	0.257662	−	0.148761i	−0.365606	−	0.930770i	$-0.619138\pi$
$229$	3.89913	−	2.25117i	0.257662	−	0.148761i	0.623268	+	0.782009i	$0.285805\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.69604	−	3.86596i	0.438672	−	0.253267i	−0.264362	−	0.964423i	$-0.585161\pi$
$233$	6.69604	−	3.86596i	0.438672	−	0.253267i	0.703034	+	0.711156i	$0.251828\pi$
$234$		0			0
$235$	13.3044	−	23.0440i	0.867886	−	1.50322i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.3788			−1.38288			−0.691440	−	0.722433i	$-0.743024\pi$
$239$	−21.3788			−1.38288			−0.691440	+	0.722433i	$0.743024\pi$
$240$		0			0
$241$	4.25688	−	7.37313i	0.274210	−	0.474945i	−0.695726	−	0.718308i	$-0.744917\pi$
$241$	4.25688	−	7.37313i	0.274210	−	0.474945i	0.969935	+	0.243362i	$0.0782504\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.0300	−	21.6502i	0.960232	−	1.38318i
$246$		0			0
$247$	−11.5391	+	6.66209i	−0.734215	+	0.423899i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			0.778871i			0.0491619i	0.999698	+	0.0245810i	$0.00782515\pi$
$251$			0.778871i			0.0491619i	−0.999698	+	0.0245810i	$0.992175\pi$
$252$		0			0
$253$		−	2.22106i		−	0.139637i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.6901	+	6.17195i	−0.666832	+	0.384996i	−0.794875	−	0.606773i	$-0.792464\pi$
$257$	−10.6901	+	6.17195i	−0.666832	+	0.384996i	0.128043	+	0.991769i	$0.459130\pi$
$258$		0			0
$259$	−15.8758	−	0.661120i	−0.986476	−	0.0410800i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.01432	+	1.75686i	−0.0625458	+	0.108333i	−0.895603	−	0.444855i	$-0.853255\pi$
$263$	−1.01432	+	1.75686i	−0.0625458	+	0.108333i	0.833057	+	0.553187i	$0.186589\pi$
$264$		0			0
$265$	−3.68163			−0.226161
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.1346	+	22.7498i	−0.800833	+	1.38708i	0.118236	+	0.992986i	$0.462276\pi$
$269$	−13.1346	+	22.7498i	−0.800833	+	1.38708i	−0.919069	+	0.394097i	$0.871057\pi$
$270$		0			0
$271$	7.01591	−	4.05064i	0.426186	−	0.246059i	−0.271535	−	0.962429i	$-0.587531\pi$
$271$	7.01591	−	4.05064i	0.426186	−	0.246059i	0.697720	+	0.716370i	$0.254198\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−13.4967	+	7.79231i	−0.813880	+	0.469894i
$276$		0			0
$277$	−8.45693	−	4.88261i	−0.508127	−	0.293368i	0.223936	−	0.974604i	$-0.428109\pi$
$277$	−8.45693	−	4.88261i	−0.508127	−	0.293368i	−0.732064	+	0.681236i	$0.761443\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.88752i			0.112600i	0.998414	+	0.0562999i	$0.0179303\pi$
$281$			1.88752i			0.112600i	−0.998414	+	0.0562999i	$0.982070\pi$
$282$		0			0
$283$	−3.28892	+	5.69657i	−0.195506	+	0.338626i	−0.947066	−	0.321038i	$-0.895968\pi$
$283$	−3.28892	+	5.69657i	−0.195506	+	0.338626i	0.751560	+	0.659664i	$0.229302\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8.65056	+	13.6392i	−0.510626	+	0.805097i
$288$		0			0
$289$	−5.09459	−	8.82408i	−0.299682	−	0.519064i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.1311			0.825545			0.412772	−	0.910834i	$-0.364560\pi$
$293$	14.1311			0.825545			0.412772	+	0.910834i	$0.364560\pi$
$294$		0			0
$295$		−	29.0156i		−	1.68935i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.39004	−	3.68929i	0.369546	−	0.213357i
$300$		0			0
$301$	14.5183	+	9.20814i	0.836823	+	0.530749i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−28.7838	−	16.6183i	−1.64815	−	0.951562i
$306$		0			0
$307$	17.5973			1.00433			0.502167	−	0.864771i	$-0.332536\pi$
$307$	17.5973			1.00433			0.502167	+	0.864771i	$0.332536\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.44683	+	9.43418i	−0.308861	+	0.534963i	−0.978114	−	0.208072i	$-0.933281\pi$
$311$	−5.44683	+	9.43418i	−0.308861	+	0.534963i	0.669252	+	0.743035i	$0.266615\pi$
$312$		0			0
$313$	−1.76535	−	3.05768i	−0.0997835	−	0.172830i	0.811811	−	0.583920i	$-0.198482\pi$
$313$	−1.76535	−	3.05768i	−0.0997835	−	0.172830i	−0.911595	+	0.411090i	$0.865148\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.09426	+	14.0197i	0.454619	+	0.787423i	0.998666	−	0.0516316i	$-0.0164422\pi$
$317$	8.09426	+	14.0197i	0.454619	+	0.787423i	−0.544047	+	0.839055i	$0.683109\pi$
$318$		0			0
$319$	−10.0156	−	5.78252i	−0.560767	−	0.323759i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6.16298i			0.342918i
$324$		0			0
$325$	−44.8374	−	25.8869i	−2.48713	−	1.43595i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.777972	+	18.6819i	−0.0428910	+	1.02996i
$330$		0			0
$331$	−5.79835	−	10.0430i	−0.318706	−	0.552016i	0.661512	−	0.749935i	$-0.269915\pi$
$331$	−5.79835	−	10.0430i	−0.318706	−	0.552016i	−0.980218	+	0.197919i	$0.936582\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	50.4397			2.75581
$336$		0			0
$337$	2.81073			0.153110			0.0765550	−	0.997065i	$-0.475608\pi$
$337$	2.81073			0.153110			0.0765550	+	0.997065i	$0.475608\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.65872	−	8.06914i	−0.252284	−	0.436968i
$342$		0			0
$343$	−2.30639	+	18.3761i	−0.124533	+	0.992215i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.7883	+	10.8474i	1.00861	+	0.582320i	0.910784	−	0.412884i	$-0.135478\pi$
$347$	18.7883	+	10.8474i	1.00861	+	0.582320i	0.0978242	+	0.995204i	$0.468812\pi$
$348$		0			0
$349$			7.44465i			0.398503i	0.979948	+	0.199251i	$0.0638511\pi$
$349$			7.44465i			0.398503i	−0.979948	+	0.199251i	$0.936149\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.0365	+	7.52665i	0.693865	+	0.400603i	0.805058	−	0.593196i	$-0.202134\pi$
$353$	13.0365	+	7.52665i	0.693865	+	0.400603i	−0.111193	+	0.993799i	$0.535467\pi$
$354$		0			0
$355$	−17.7063	−	30.6682i	−0.939754	−	1.62770i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.01876	+	15.6210i	0.475992	+	0.824442i	0.999622	−	0.0275035i	$-0.00875575\pi$
$359$	9.01876	+	15.6210i	0.475992	+	0.824442i	−0.523630	+	0.851946i	$0.675422\pi$
$360$		0			0
$361$	6.71162	−	11.6249i	0.353243	−	0.611835i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.20816			0.429635
$366$		0			0
$367$	3.38897	+	1.95662i	0.176903	+	0.102135i	0.585837	−	0.810429i	$-0.300766\pi$
$367$	3.38897	+	1.95662i	0.176903	+	0.102135i	−0.408934	+	0.912564i	$0.634099\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.29236	−	1.19920i	0.119013	−	0.0622594i
$372$		0			0
$373$	15.8513	−	9.15178i	0.820752	−	0.473861i	−0.0299239	−	0.999552i	$-0.509527\pi$
$373$	15.8513	−	9.15178i	0.820752	−	0.473861i	0.850676	+	0.525691i	$0.176193\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	38.4203i		−	1.97875i
$378$		0			0
$379$	−25.8073			−1.32563			−0.662816	−	0.748783i	$-0.730639\pi$
$379$	−25.8073			−1.32563			−0.662816	+	0.748783i	$0.730639\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.37665	+	12.7767i	0.376929	+	0.652860i	0.990614	−	0.136691i	$-0.0436467\pi$
$383$	7.37665	+	12.7767i	0.376929	+	0.652860i	−0.613685	+	0.789551i	$0.710313\pi$
$384$		0			0
$385$	9.06159	−	14.2873i	0.461822	−	0.728147i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.70264	−	15.0734i	0.441242	−	0.764253i	−0.556540	−	0.830821i	$-0.687872\pi$
$389$	8.70264	−	15.0734i	0.441242	−	0.764253i	0.997782	+	0.0665677i	$0.0212049\pi$
$390$		0			0
$391$		−	3.41290i		−	0.172598i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−56.8186	−	32.8042i	−2.85886	−	1.65056i
$396$		0			0
$397$	10.0273	−	5.78926i	0.503255	−	0.290554i	−0.226802	−	0.973941i	$-0.572827\pi$
$397$	10.0273	−	5.78926i	0.503255	−	0.290554i	0.730057	+	0.683387i	$0.239494\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−31.1453	+	17.9818i	−1.55532	+	0.897967i	−0.557630	+	0.830089i	$0.688289\pi$
$401$	−31.1453	+	17.9818i	−1.55532	+	0.897967i	−0.997694	+	0.0678773i	$0.978377\pi$
$402$		0			0
$403$	15.4768	−	26.8065i	0.770953	−	1.33533i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.2000			−0.505595
$408$		0			0
$409$	12.1615	−	21.0644i	0.601349	−	1.04157i	−0.391268	−	0.920277i	$-0.627964\pi$
$409$	12.1615	−	21.0644i	0.601349	−	1.04157i	0.992617	−	0.121290i	$-0.0387031\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.45111	+	18.0665i	0.465059	+	0.888993i
$414$		0			0
$415$	38.7580	−	22.3769i	1.90256	−	1.09844i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			6.92954i			0.338530i	0.985571	+	0.169265i	$0.0541394\pi$
$419$			6.92954i			0.338530i	−0.985571	+	0.169265i	$0.945861\pi$
$420$		0			0
$421$			12.6192i			0.615024i	0.951544	+	0.307512i	$0.0994965\pi$
$421$			12.6192i			0.615024i	−0.951544	+	0.307512i	$0.900504\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−20.7391	+	11.9737i	−1.00600	+	0.580812i
$426$		0			0
$427$	23.3352	+	0.971750i	1.12927	+	0.0470263i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11.1759	−	19.3571i	0.538322	−	0.932401i	−0.460673	−	0.887570i	$-0.652392\pi$
$431$	11.1759	−	19.3571i	0.538322	−	0.932401i	0.998995	−	0.0448306i	$-0.0142748\pi$
$432$		0			0
$433$	−32.2813			−1.55134			−0.775671	−	0.631138i	$-0.782588\pi$
$433$	−32.2813			−1.55134			−0.775671	+	0.631138i	$0.782588\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.54413	−	2.67451i	0.0738658	−	0.127939i
$438$		0			0
$439$	2.42798	−	1.40180i	0.115881	−	0.0669041i	−0.440939	−	0.897537i	$-0.645355\pi$
$439$	2.42798	−	1.40180i	0.115881	−	0.0669041i	0.556820	+	0.830633i	$0.312021\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.2066	+	9.35689i	−0.769999	+	0.444559i	−0.832874	−	0.553462i	$-0.813306\pi$
$443$	−16.2066	+	9.35689i	−0.769999	+	0.444559i	0.0628752	+	0.998021i	$0.479973\pi$
$444$		0			0
$445$	36.0940	+	20.8389i	1.71102	+	0.987859i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	13.8950i		−	0.655747i	−0.944722	−	0.327874i	$-0.893668\pi$
$449$		−	13.8950i		−	0.655747i	0.944722	−	0.327874i	$-0.106332\pi$
$450$		0			0
$451$	−5.18395	+	8.97886i	−0.244103	+	0.422798i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	56.1567	+	2.33854i	2.63267	+	0.109633i
$456$		0			0
$457$	2.88922	+	5.00428i	0.135152	+	0.234090i	0.925656	−	0.378367i	$-0.123514\pi$
$457$	2.88922	+	5.00428i	0.135152	+	0.234090i	−0.790503	+	0.612458i	$0.790181\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.4222			−0.718285			−0.359142	−	0.933283i	$-0.616931\pi$
$461$	−15.4222			−0.718285			−0.359142	+	0.933283i	$0.616931\pi$
$462$		0			0
$463$		−	11.5902i		−	0.538644i	−0.963050	−	0.269322i	$-0.913200\pi$
$463$		−	11.5902i		−	0.538644i	0.963050	−	0.269322i	$-0.0867996\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.98153	−	1.14404i	0.0916943	−	0.0529397i	−0.453452	−	0.891281i	$-0.649808\pi$
$467$	1.98153	−	1.14404i	0.0916943	−	0.0529397i	0.545146	+	0.838341i	$0.316474\pi$
$468$		0			0
$469$	−31.4061	+	16.4295i	−1.45020	+	0.758643i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.55761	+	5.51809i	0.439459	+	0.253722i
$474$		0			0
$475$	−21.6696			−0.994270
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.42948	+	16.3323i	−0.430844	+	0.746244i	−0.996946	−	0.0780912i	$-0.975117\pi$
$479$	−9.42948	+	16.3323i	−0.430844	+	0.746244i	0.566102	+	0.824335i	$0.308451\pi$
$480$		0			0
$481$	−16.9427	−	29.3457i	−0.772522	−	1.33805i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−28.3573	−	49.1162i	−1.28764	−	2.23025i
$486$		0			0
$487$	26.9019	+	15.5318i	1.21904	+	0.703813i	0.964713	−	0.263305i	$-0.0848126\pi$
$487$	26.9019	+	15.5318i	1.21904	+	0.703813i	0.254327	+	0.967118i	$0.418146\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7.52948i		−	0.339801i	−0.985461	−	0.169900i	$-0.945655\pi$
$491$		−	7.52948i		−	0.339801i	0.985461	−	0.169900i	$-0.0543446\pi$
$492$		0			0
$493$	−15.3901	−	8.88549i	−0.693136	−	0.400182i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	21.0142	+	13.3281i	0.942617	+	0.597848i
$498$		0			0
$499$	0.883277	+	1.52988i	0.0395409	+	0.0684869i	0.885119	−	0.465366i	$-0.154077\pi$
$499$	0.883277	+	1.52988i	0.0395409	+	0.0684869i	−0.845578	+	0.533852i	$0.820744\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.9584			−0.756136			−0.378068	−	0.925778i	$-0.623411\pi$
$503$	−16.9584			−0.756136			−0.378068	+	0.925778i	$0.623411\pi$
$504$		0			0
$505$	5.85777			0.260667
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.13114	+	8.88739i	0.227434	+	0.393927i	0.957047	−	0.289934i	$-0.0936332\pi$
$509$	5.13114	+	8.88739i	0.227434	+	0.393927i	−0.729613	+	0.683860i	$0.760300\pi$
$510$		0			0
$511$	−5.11079	+	2.67361i	−0.226088	+	0.118273i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	26.1324	+	15.0876i	1.15153	+	0.664837i
$516$		0			0
$517$			12.0028i			0.527884i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−32.7368	−	18.9006i	−1.43423	−	0.828051i	−0.436787	−	0.899565i	$-0.643884\pi$
$521$	−32.7368	−	18.9006i	−1.43423	−	0.828051i	−0.997440	+	0.0715138i	$0.977217\pi$
$522$		0			0
$523$	−19.6691	−	34.0679i	−0.860070	−	1.48968i	−0.871861	−	0.489754i	$-0.837087\pi$
$523$	−19.6691	−	34.0679i	−0.860070	−	1.48968i	0.0117911	−	0.999930i	$-0.496247\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.15864	−	12.3991i	−0.311835	−	0.540114i
$528$		0			0
$529$	10.6449	−	18.4375i	0.462822	−	0.801631i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−34.4433			−1.49190
$534$		0			0
$535$	−47.5372	−	27.4456i	−2.05521	−	1.18658i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.988451	+	11.8475i	−0.0425756	+	0.510309i
$540$		0			0
$541$	12.8385	−	7.41231i	0.551970	−	0.318680i	−0.197946	−	0.980213i	$-0.563427\pi$
$541$	12.8385	−	7.41231i	0.551970	−	0.318680i	0.749916	+	0.661533i	$0.230094\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.84849i			0.122016i
$546$		0			0
$547$	16.2017			0.692736			0.346368	−	0.938099i	$-0.387415\pi$
$547$	16.2017			0.692736			0.346368	+	0.938099i	$0.387415\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−8.04030	−	13.9262i	−0.342528	−	0.593277i
$552$		0			0
$553$	46.0632	+	1.91822i	1.95881	+	0.0815708i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.17534	+	2.03574i	−0.0498006	+	0.0862572i	−0.889851	−	0.456251i	$-0.849192\pi$
$557$	−1.17534	+	2.03574i	−0.0498006	+	0.0862572i	0.840051	+	0.542508i	$0.182525\pi$
$558$		0			0
$559$			36.6634i			1.55070i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.40537	+	3.69814i	0.269954	+	0.155858i	0.628867	−	0.777513i	$-0.283519\pi$
$563$	6.40537	+	3.69814i	0.269954	+	0.155858i	−0.358913	+	0.933371i	$0.616852\pi$
$564$		0			0
$565$	−32.1957	+	18.5882i	−1.35449	+	0.782012i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−30.2952	+	17.4909i	−1.27004	+	0.733258i	−0.974995	−	0.222226i	$-0.928668\pi$
$569$	−30.2952	+	17.4909i	−1.27004	+	0.733258i	−0.295045	+	0.955483i	$0.595334\pi$
$570$		0			0
$571$	−16.4081	+	28.4197i	−0.686659	+	1.18933i	0.286254	+	0.958154i	$0.407590\pi$
$571$	−16.4081	+	28.4197i	−0.686659	+	1.18933i	−0.972912	+	0.231174i	$0.925743\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.0001			0.500437
$576$		0			0
$577$	4.58139	−	7.93520i	0.190726	−	0.330347i	−0.754765	−	0.655995i	$-0.772249\pi$
$577$	4.58139	−	7.93520i	0.190726	−	0.330347i	0.945491	+	0.325648i	$0.105583\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16.8438	+	26.5574i	−0.698800	+	1.10179i
$582$		0			0
$583$	1.43823	−	0.830362i	0.0595653	−	0.0343901i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	17.4528i		−	0.720354i	−0.932884	−	0.360177i	$-0.882716\pi$
$587$		−	17.4528i		−	0.720354i	0.932884	−	0.360177i	$-0.117284\pi$
$588$		0			0
$589$		−	12.9554i		−	0.533819i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.0385	+	9.83721i	−0.699689	+	0.403966i	−0.807232	−	0.590235i	$-0.799035\pi$
$593$	−17.0385	+	9.83721i	−0.699689	+	0.403966i	0.107542	+	0.994200i	$0.465702\pi$
$594$		0			0
$595$	13.9241	−	21.9540i	0.570834	−	0.900025i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.68048	−	4.64273i	0.109521	−	0.189697i	−0.806055	−	0.591841i	$-0.798402\pi$
$599$	2.68048	−	4.64273i	0.109521	−	0.189697i	0.915576	+	0.402144i	$0.131735\pi$
$600$		0			0
$601$	12.4609			0.508291			0.254145	−	0.967166i	$-0.418206\pi$
$601$	12.4609			0.508291			0.254145	+	0.967166i	$0.418206\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.2779	+	26.4621i	−0.621135	+	1.07584i
$606$		0			0
$607$	−10.6807	+	6.16649i	−0.433515	+	0.250290i	−0.700843	−	0.713315i	$-0.747193\pi$
$607$	−10.6807	+	6.16649i	−0.433515	+	0.250290i	0.267328	+	0.963606i	$0.413859\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−34.5325	+	19.9373i	−1.39704	+	0.806579i
$612$		0			0
$613$	3.55744	+	2.05389i	0.143683	+	0.0829557i	0.570118	−	0.821563i	$-0.306897\pi$
$613$	3.55744	+	2.05389i	0.143683	+	0.0829557i	−0.426435	+	0.904518i	$0.640231\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	36.1255i		−	1.45436i	−0.686447	−	0.727180i	$-0.740831\pi$
$617$		−	36.1255i		−	1.45436i	0.686447	−	0.727180i	$-0.259169\pi$
$618$		0			0
$619$	0.802674	−	1.39027i	0.0322622	−	0.0558798i	−0.849443	−	0.527680i	$-0.823062\pi$
$619$	0.802674	−	1.39027i	0.0322622	−	0.0558798i	0.881706	+	0.471800i	$0.156396\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−29.2616	−	1.21855i	−1.17234	−	0.0488201i
$624$		0			0
$625$	−6.66043	−	11.5362i	−0.266417	−	0.461448i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.6734			−0.624940
$630$		0			0
$631$		−	8.98270i		−	0.357596i	−0.983886	−	0.178798i	$-0.942779\pi$
$631$		−	8.98270i		−	0.357596i	0.983886	−	0.178798i	$-0.0572208\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.7007	−	8.48748i	0.583381	−	0.336815i
$636$		0			0
$637$	−35.7276	+	16.8356i	−1.41558	+	0.667050i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	34.1214	+	19.7000i	1.34772	+	0.778104i	0.987926	−	0.154929i	$-0.0495149\pi$
$641$	34.1214	+	19.7000i	1.34772	+	0.778104i	0.359790	+	0.933033i	$0.382848\pi$
$642$		0			0
$643$	−40.0002			−1.57745			−0.788726	−	0.614745i	$-0.789259\pi$
$643$	−40.0002			−1.57745			−0.788726	+	0.614745i	$0.789259\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−19.8854	+	34.4425i	−0.781775	+	1.35407i	0.149132	+	0.988817i	$0.452352\pi$
$647$	−19.8854	+	34.4425i	−0.781775	+	1.35407i	−0.930907	+	0.365257i	$0.880981\pi$
$648$		0			0
$649$	6.54422	+	11.3349i	0.256883	+	0.444935i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.64003	−	2.84061i	−0.0641792	−	0.111162i	0.832150	−	0.554550i	$-0.187110\pi$
$653$	−1.64003	−	2.84061i	−0.0641792	−	0.111162i	−0.896330	+	0.443388i	$0.853776\pi$
$654$		0			0
$655$	−52.9651	−	30.5794i	−2.06952	−	1.19484i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			16.7484i			0.652425i	0.945297	+	0.326212i	$0.105772\pi$
$659$			16.7484i			0.652425i	−0.945297	+	0.326212i	$0.894228\pi$
$660$		0			0
$661$	4.50164	+	2.59902i	0.175093	+	0.101090i	0.584985	−	0.811044i	$-0.301100\pi$
$661$	4.50164	+	2.59902i	0.175093	+	0.101090i	−0.409892	+	0.912134i	$0.634434\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20.8445	−	10.9044i	0.808315	−	0.422854i
$666$		0			0
$667$	4.45251	+	7.71197i	0.172402	+	0.298609i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.9925			0.578779
$672$		0			0
$673$	−29.9571			−1.15476			−0.577380	−	0.816475i	$-0.695925\pi$
$673$	−29.9571			−1.15476			−0.577380	+	0.816475i	$0.695925\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.2704	−	26.4492i	−0.586891	−	1.01652i	−0.994637	−	0.103429i	$-0.967018\pi$
$677$	−15.2704	−	26.4492i	−0.586891	−	1.01652i	0.407746	−	0.913095i	$-0.366315\pi$
$678$		0			0
$679$	33.6550	+	21.3454i	1.29156	+	0.819162i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.1942	−	6.46296i	−0.428333	−	0.247298i	0.270303	−	0.962775i	$-0.412876\pi$
$683$	−11.1942	−	6.46296i	−0.428333	−	0.247298i	−0.698636	+	0.715477i	$0.746209\pi$
$684$		0			0
$685$			15.8022i			0.603770i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.77795	+	2.75855i	0.182025	+	0.105092i
$690$		0			0
$691$	8.33676	+	14.4397i	0.317145	+	0.549312i	0.979891	−	0.199533i	$-0.0639424\pi$
$691$	8.33676	+	14.4397i	0.317145	+	0.549312i	−0.662746	+	0.748844i	$0.730609\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.7358	−	18.5949i	−0.407231	−	0.705344i
$696$		0			0
$697$	−7.96571	+	13.7970i	−0.301723	+	0.522599i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−24.5914			−0.928805			−0.464403	−	0.885624i	$-0.653731\pi$
$701$	−24.5914			−0.928805			−0.464403	+	0.885624i	$0.653731\pi$
$702$		0			0
$703$	−12.2825	−	7.09128i	−0.463242	−	0.267453i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.64733	+	1.90803i	−0.137172	+	0.0717587i
$708$		0			0
$709$	−4.59843	+	2.65490i	−0.172698	+	0.0997070i	−0.583857	−	0.811856i	$-0.698457\pi$
$709$	−4.59843	+	2.65490i	−0.172698	+	0.0997070i	0.411160	+	0.911563i	$0.365124\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			7.17437i			0.268682i
$714$		0			0
$715$	36.0799			1.34931
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.16456	+	8.94528i	0.192606	+	0.333603i	0.946113	−	0.323837i	$-0.104973\pi$
$719$	5.16456	+	8.94528i	0.192606	+	0.333603i	−0.753507	+	0.657439i	$0.771640\pi$
$720$		0			0
$721$	−21.1857	−	0.882240i	−0.788997	−	0.0328563i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	31.2422	−	54.1131i	1.16031	−	2.00971i
$726$		0			0
$727$			28.6727i			1.06341i	0.846929	+	0.531706i	$0.178449\pi$
$727$			28.6727i			1.06341i	−0.846929	+	0.531706i	$0.821551\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	14.6863	+	8.47916i	0.543193	+	0.313613i
$732$		0			0
$733$	33.6091	−	19.4042i	1.24138	−	0.716710i	0.272004	−	0.962296i	$-0.412314\pi$
$733$	33.6091	−	19.4042i	1.24138	−	0.716710i	0.969375	+	0.245586i	$0.0789803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19.7042	+	11.3763i	−0.725815	+	0.419050i
$738$		0			0
$739$	5.59773	−	9.69556i	0.205916	−	0.356657i	−0.744508	−	0.667613i	$-0.767316\pi$
$739$	5.59773	−	9.69556i	0.205916	−	0.356657i	0.950424	+	0.310956i	$0.100649\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.68201			0.245139			0.122570	−	0.992460i	$-0.460887\pi$
$743$	6.68201			0.245139			0.122570	+	0.992460i	$0.460887\pi$
$744$		0			0
$745$	−25.3843	+	43.9668i	−0.930007	+	1.61082i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	38.5387	+	1.60487i	1.40817	+	0.0586407i
$750$		0			0
$751$	43.2883	−	24.9925i	1.57961	−	0.911990i	0.584701	−	0.811249i	$-0.301212\pi$
$751$	43.2883	−	24.9925i	1.57961	−	0.911990i	0.994913	−	0.100741i	$-0.0321215\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	80.3117i		−	2.92284i
$756$		0			0
$757$		−	32.8378i		−	1.19351i	−0.802424	−	0.596755i	$-0.796457\pi$
$757$		−	32.8378i		−	1.19351i	0.802424	−	0.596755i	$-0.203543\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	28.7157	−	16.5790i	1.04094	−	0.600988i	0.120842	−	0.992672i	$-0.461440\pi$
$761$	28.7157	−	16.5790i	1.04094	−	0.600988i	0.920100	+	0.391683i	$0.128107\pi$
$762$		0			0
$763$	−0.927825	−	1.77360i	−0.0335895	−	0.0642088i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−21.7406	+	37.6559i	−0.785008	+	1.35967i
$768$		0			0
$769$	−48.1183			−1.73519			−0.867596	−	0.497270i	$-0.834336\pi$
$769$	−48.1183			−1.73519			−0.867596	+	0.497270i	$0.834336\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	14.2293	−	24.6458i	0.511791	−	0.886449i	−0.488115	−	0.872779i	$-0.662315\pi$
$773$	14.2293	−	24.6458i	0.511791	−	0.886449i	0.999907	−	0.0136695i	$-0.00435128\pi$
$774$		0			0
$775$	43.5965	−	25.1704i	1.56603	−	0.904148i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.4847	+	7.20802i	−0.447309	+	0.258254i
$780$		0			0
$781$	13.8339	+	7.98703i	0.495017	+	0.285798i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			34.6612i			1.23711i
$786$		0			0
$787$	11.0597	−	19.1559i	0.394234	−	0.682834i	−0.598769	−	0.800922i	$-0.704343\pi$
$787$	11.0597	−	19.1559i	0.394234	−	0.682834i	0.993003	+	0.118088i	$0.0376766\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	13.9920	−	22.0609i	0.497497	−	0.784395i
$792$		0			0
$793$	24.9033	+	43.1339i	0.884344	+	1.53173i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	25.7820			0.913247			0.456623	−	0.889660i	$-0.349059\pi$
$797$	25.7820			0.913247			0.456623	+	0.889660i	$0.349059\pi$
$798$		0			0
$799$			18.4437i			0.652490i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.20652	+	1.85128i	−0.113156	+	0.0653304i
$804$		0			0
$805$	−11.5431	+	6.03856i	−0.406842	+	0.212831i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.56573	+	1.48132i	0.0902062	+	0.0520806i	0.544425	−	0.838810i	$-0.316748\pi$
$809$	2.56573	+	1.48132i	0.0902062	+	0.0520806i	−0.454218	+	0.890890i	$0.650081\pi$
$810$		0			0
$811$	51.9174			1.82307			0.911533	−	0.411228i	$-0.134900\pi$
$811$	51.9174			1.82307			0.911533	+	0.411228i	$0.134900\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.226030	+	0.391495i	−0.00791747	+	0.0137135i
$816$		0			0
$817$	7.67262	+	13.2894i	0.268431	+	0.464936i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.7741	+	18.6612i	0.376017	+	0.651281i	0.990479	−	0.137666i	$-0.0439601\pi$
$821$	10.7741	+	18.6612i	0.376017	+	0.651281i	−0.614462	+	0.788947i	$0.710627\pi$
$822$		0			0
$823$	30.0162	+	17.3299i	1.04630	+	0.604082i	0.921611	−	0.388114i	$-0.126873\pi$
$823$	30.0162	+	17.3299i	1.04630	+	0.604082i	0.124689	+	0.992196i	$0.460207\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	40.6029i		−	1.41190i	−0.708260	−	0.705951i	$-0.750520\pi$
$827$		−	40.6029i		−	1.41190i	0.708260	−	0.705951i	$-0.249480\pi$
$828$		0			0
$829$	28.2837	+	16.3296i	0.982334	+	0.567151i	0.902974	−	0.429695i	$-0.141379\pi$
$829$	28.2837	+	16.3296i	0.982334	+	0.567151i	0.0793604	+	0.996846i	$0.474712\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.51886	+	18.2050i	−0.0526256	+	0.630767i
$834$		0			0
$835$	24.4376	+	42.3271i	0.845697	+	1.46479i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	35.4704			1.22458			0.612288	−	0.790635i	$-0.290249\pi$
$839$	35.4704			1.22458			0.612288	+	0.790635i	$0.290249\pi$
$840$		0			0
$841$	17.3685			0.598913
$842$		0			0
$843$		0			0
$844$		0			0
$845$	35.4573	+	61.4138i	1.21977	+	2.11270i
$846$		0			0
$847$	0.893370	−	21.4530i	0.0306965	−	0.737133i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.80170	+	3.92697i	0.233159	+	0.134615i
$852$		0			0
$853$			2.87597i			0.0984714i	0.998787	+	0.0492357i	$0.0156786\pi$
$853$			2.87597i			0.0984714i	−0.998787	+	0.0492357i	$0.984321\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.76468	+	3.32824i	0.196918	+	0.113691i	0.595217	−	0.803565i	$-0.297066\pi$
$857$	5.76468	+	3.32824i	0.196918	+	0.113691i	−0.398299	+	0.917256i	$0.630399\pi$
$858$		0			0
$859$	−10.2685	−	17.7855i	−0.350355	−	0.606833i	0.635956	−	0.771725i	$-0.280606\pi$
$859$	−10.2685	−	17.7855i	−0.350355	−	0.606833i	−0.986312	+	0.164892i	$0.947272\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−15.7546	−	27.2877i	−0.536292	−	0.928886i	−0.999100	−	0.0424266i	$-0.986491\pi$
$863$	−15.7546	−	27.2877i	−0.536292	−	0.928886i	0.462807	−	0.886459i	$-0.346842\pi$
$864$		0			0
$865$	12.9692	−	22.4634i	0.440968	−	0.763778i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	29.5949			1.00394
$870$		0			0
$871$	−65.4596	−	37.7931i	−2.21801	−	1.28057i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	35.1308	+	22.2815i	1.18764	+	0.753251i
$876$		0			0
$877$	48.3067	−	27.8899i	1.63120	−	0.941774i	0.647477	−	0.762085i	$-0.275824\pi$
$877$	48.3067	−	27.8899i	1.63120	−	0.941774i	0.983723	−	0.179689i	$-0.0575091\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	21.9485i		−	0.739464i	−0.929138	−	0.369732i	$-0.879450\pi$
$881$		−	21.9485i		−	0.739464i	0.929138	−	0.369732i	$-0.120550\pi$
$882$		0			0
$883$	−33.4287			−1.12497			−0.562483	−	0.826809i	$-0.690154\pi$
$883$	−33.4287			−1.12497			−0.562483	+	0.826809i	$0.690154\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.18309	−	10.7094i	−0.207608	−	0.359587i	0.743353	−	0.668900i	$-0.233234\pi$
$887$	−6.18309	−	10.7094i	−0.207608	−	0.359587i	−0.950960	+	0.309313i	$0.899901\pi$
$888$		0			0
$889$	−6.38880	+	10.0731i	−0.214273	+	0.337841i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.34466	+	14.4534i	−0.279243	+	0.483664i
$894$		0			0
$895$			59.1460i			1.97703i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	32.3521	+	18.6785i	1.07900	+	0.622963i
$900$		0			0
$901$	2.21000	−	1.27594i	0.0736257	−	0.0425078i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	51.0996	−	29.5024i	1.69861	−	0.980692i
$906$		0			0
$907$	6.85914	−	11.8804i	0.227754	−	0.394481i	−0.729388	−	0.684100i	$-0.760195\pi$
$907$	6.85914	−	11.8804i	0.227754	−	0.394481i	0.957142	+	0.289619i	$0.0935285\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.02706			0.299080			0.149540	−	0.988756i	$-0.452221\pi$
$911$	9.02706			0.299080			0.149540	+	0.988756i	$0.452221\pi$
$912$		0			0
$913$	−10.0939	+	17.4831i	−0.334058	+	0.578606i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	42.9391	+	1.78812i	1.41797	+	0.0590489i
$918$		0			0
$919$	−37.8104	+	21.8298i	−1.24725	+	0.720100i	−0.970560	−	0.240860i	$-0.922571\pi$
$919$	−37.8104	+	21.8298i	−1.24725	+	0.720100i	−0.276689	+	0.960959i	$0.589237\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			53.0675i			1.74674i
$924$		0			0
$925$		−	55.1092i		−	1.81198i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	43.0944	−	24.8806i	1.41388	−	0.816304i	0.418129	−	0.908387i	$-0.362686\pi$
$929$	43.0944	−	24.8806i	1.41388	−	0.816304i	0.995751	+	0.0920830i	$0.0293525\pi$
$930$		0			0
$931$	−9.42695	+	13.5792i	−0.308956	+	0.445039i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.34421	−	14.4526i	0.272885	−	0.472650i
$936$		0			0
$937$	40.5415			1.32443			0.662216	−	0.749313i	$-0.269616\pi$
$937$	40.5415			1.32443			0.662216	+	0.749313i	$0.269616\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−27.1914	+	47.0968i	−0.886414	+	1.53531i	−0.0423286	+	0.999104i	$0.513478\pi$
$941$	−27.1914	+	47.0968i	−0.886414	+	1.53531i	−0.844085	+	0.536209i	$0.819856\pi$
$942$		0			0
$943$	6.91367	−	3.99161i	0.225140	−	0.129985i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.3501	−	7.13032i	0.401323	−	0.231704i	−0.285732	−	0.958310i	$-0.592237\pi$
$947$	12.3501	−	7.13032i	0.401323	−	0.231704i	0.687055	+	0.726606i	$0.258903\pi$
$948$		0			0
$949$	−10.6524	−	6.15016i	−0.345791	−	0.199643i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			3.31610i			0.107419i	0.998557	+	0.0537095i	$0.0171045\pi$
$953$			3.31610i			0.107419i	−0.998557	+	0.0537095i	$0.982896\pi$
$954$		0			0
$955$	−45.4200	+	78.6697i	−1.46976	+	2.54569i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5.14717	−	9.83919i	−0.166211	−	0.317724i
$960$		0			0
$961$	−0.451578	−	0.782157i	−0.0145670	−	0.0252309i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	36.1682			1.16430
$966$		0			0
$967$		−	18.6637i		−	0.600185i	−0.953910	−	0.300092i	$-0.902982\pi$
$967$		−	18.6637i		−	0.600185i	0.953910	−	0.300092i	$-0.0970175\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.93811	+	2.85102i	−0.158472	+	0.0914936i	−0.577139	−	0.816646i	$-0.695831\pi$
$971$	−4.93811	+	2.85102i	−0.158472	+	0.0914936i	0.418667	+	0.908140i	$0.362497\pi$
$972$		0			0
$973$	12.7414	+	8.08115i	0.408471	+	0.259070i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.52730	−	2.03649i	−0.112848	−	0.0651530i	0.442514	−	0.896762i	$-0.354087\pi$
$977$	−3.52730	−	2.03649i	−0.112848	−	0.0651530i	−0.555362	+	0.831609i	$0.687420\pi$
$978$		0			0
$979$	−18.8002			−0.600856
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13.6318	+	23.6109i	−0.434786	+	0.753071i	−0.997278	−	0.0737313i	$-0.976509\pi$
$983$	−13.6318	+	23.6109i	−0.434786	+	0.753071i	0.562492	+	0.826803i	$0.309843\pi$
$984$		0			0
$985$	3.86500	+	6.69438i	0.123149	+	0.213301i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.24890	−	7.35930i	−0.135107	−	0.234012i
$990$		0			0
$991$	−7.91978	−	4.57249i	−0.251580	−	0.145250i	0.368907	−	0.929466i	$-0.379732\pi$
$991$	−7.91978	−	4.57249i	−0.251580	−	0.145250i	−0.620488	+	0.784216i	$0.713065\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			37.4807i			1.18822i
$996$		0			0
$997$	−40.5025	−	23.3841i	−1.28273	−	0.740583i	−0.305381	−	0.952230i	$-0.598784\pi$
$997$	−40.5025	−	23.3841i	−1.28273	−	0.740583i	−0.977346	+	0.211647i	$0.932117\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.bu.c.431.1		48
3.2	odd	2	inner	2016.2.bu.c.431.24		48
4.3	odd	2		504.2.bm.c.179.10	yes	48
7.2	even	3	inner	2016.2.bu.c.1871.2		48
8.3	odd	2	inner	2016.2.bu.c.431.23		48
8.5	even	2		504.2.bm.c.179.17	yes	48
12.11	even	2		504.2.bm.c.179.15	yes	48
21.2	odd	6	inner	2016.2.bu.c.1871.23		48
24.5	odd	2		504.2.bm.c.179.8	yes	48
24.11	even	2	inner	2016.2.bu.c.431.2		48
28.23	odd	6		504.2.bm.c.107.8	✓	48
56.37	even	6		504.2.bm.c.107.15	yes	48
56.51	odd	6	inner	2016.2.bu.c.1871.24		48
84.23	even	6		504.2.bm.c.107.17	yes	48
168.107	even	6	inner	2016.2.bu.c.1871.1		48
168.149	odd	6		504.2.bm.c.107.10	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bm.c.107.8	✓	48	28.23	odd	6
504.2.bm.c.107.10	yes	48	168.149	odd	6
504.2.bm.c.107.15	yes	48	56.37	even	6
504.2.bm.c.107.17	yes	48	84.23	even	6
504.2.bm.c.179.8	yes	48	24.5	odd	2
504.2.bm.c.179.10	yes	48	4.3	odd	2
504.2.bm.c.179.15	yes	48	12.11	even	2
504.2.bm.c.179.17	yes	48	8.5	even	2
2016.2.bu.c.431.1		48	1.1	even	1	trivial
2016.2.bu.c.431.2		48	24.11	even	2	inner
2016.2.bu.c.431.23		48	8.3	odd	2	inner
2016.2.bu.c.431.24		48	3.2	odd	2	inner
2016.2.bu.c.1871.1		48	168.107	even	6	inner
2016.2.bu.c.1871.2		48	7.2	even	3	inner
2016.2.bu.c.1871.23		48	21.2	odd	6	inner
2016.2.bu.c.1871.24		48	56.51	odd	6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.bu (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.88256 - 3.26069i) q^{5} +(2.23426 + 1.41706i) q^{7} +O(q^{10})\)	\(q+(-1.88256 - 3.26069i) q^{5} +(2.23426 + 1.41706i) q^{7} +(1.47084 + 0.849192i) q^{11} +5.64222i q^{13} +(2.26012 + 1.30488i) q^{17} +(1.18076 + 2.04513i) q^{19} +(-0.653873 - 1.13254i) q^{23} +(-4.58807 + 7.94677i) q^{25} -6.80944 q^{29} +(-4.75107 - 2.74303i) q^{31} +(0.414472 - 9.95295i) q^{35} +(-5.20109 + 3.00285i) q^{37} +6.10456i q^{41} +6.49804 q^{43} +(3.53360 + 6.12038i) q^{47} +(2.98386 + 6.33219i) q^{49} +(0.488913 - 0.846821i) q^{53} -6.39462i q^{55} +(6.67395 + 3.85320i) q^{59} +(7.64484 - 4.41375i) q^{61} +(18.3975 - 10.6218i) q^{65} +(-6.69828 + 11.6018i) q^{67} +9.40544 q^{71} +(-1.09003 + 1.88798i) q^{73} +(2.08289 + 3.98160i) q^{77} +(15.0908 - 8.71267i) q^{79} +11.8864i q^{83} -9.82605i q^{85} +(-9.58642 + 5.53472i) q^{89} +(-7.99538 + 12.6062i) q^{91} +(4.44570 - 7.70018i) q^{95} +15.0631 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 32 q^{19} + 160 q^{43} + 56 q^{49} + 16 q^{73} + 32 q^{91} + 240 q^{97}+O(q^{100}) \) 48 * q + 32 * q^19 + 160 * q^43 + 56 * q^49 + 16 * q^73 + 32 * q^91 + 240 * q^97

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