LMFDB - Embedded newform 2016.2.cp.b.17.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,2,Mod(17,2016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2016, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 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("2016.17"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$28$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.19
Character	$\chi$	$=$	2016.17
Dual form			2016.2.cp.b.593.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00441 - 0.579896i) q^{5} +(-1.24394 - 2.33508i) q^{7} +(-1.41560 + 2.45188i) q^{11} -3.11725 q^{13} +(-0.782206 + 1.35482i) q^{17} +(2.15042 + 3.72463i) q^{19} +(-4.05782 + 2.34278i) q^{23} +(-1.82744 + 3.16522i) q^{25} +4.08861 q^{29} +(-2.40452 - 1.38825i) q^{31} +(-2.60353 - 1.62402i) q^{35} +(-4.96358 + 2.86572i) q^{37} +2.19421 q^{41} +6.52977i q^{43} +(5.34609 + 9.25971i) q^{47} +(-3.90521 + 5.80942i) q^{49} +(5.61902 - 9.73242i) q^{53} +3.28359i q^{55} +(11.9042 + 6.87289i) q^{59} +(5.09458 + 8.82407i) q^{61} +(-3.13100 + 1.80768i) q^{65} +(5.01037 + 2.89274i) q^{67} -4.72781i q^{71} +(-14.0619 - 8.11863i) q^{73} +(7.48627 + 0.255528i) q^{77} +(-2.89324 - 5.01124i) q^{79} +5.93150i q^{83} +1.81439i q^{85} +(-1.33902 - 2.31925i) q^{89} +(3.87769 + 7.27904i) q^{91} +(4.31980 + 2.49404i) q^{95} -11.2024i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 20 q^{7} + 8 q^{25} + 36 q^{31} - 28 q^{49} + 72 q^{73} + 12 q^{79}+O(q^{100}) $ 56 * q + 20 * q^7 + 8 * q^25 + 36 * q^31 - 28 * q^49 + 72 * q^73 + 12 * q^79

Character values

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

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$-1$	$-1$

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
$5$	1.00441	−	0.579896i	0.449185	−	0.259337i	−0.258301	−	0.966065i	$-0.583163\pi$
$5$	1.00441	−	0.579896i	0.449185	−	0.259337i	0.707486	+	0.706727i	$0.249829\pi$
$6$		0			0
$7$	−1.24394	−	2.33508i	−0.470166	−	0.882578i
$7$	−1.24394	−	2.33508i	−0.470166	−	0.882578i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.41560	+	2.45188i	−0.426818	+	0.739271i	−0.996588	−	0.0825332i	$-0.973699\pi$
$11$	−1.41560	+	2.45188i	−0.426818	+	0.739271i	0.569770	+	0.821804i	$0.307032\pi$
$12$		0			0
$13$	−3.11725			−0.864571			−0.432285	−	0.901737i	$-0.642293\pi$
$13$	−3.11725			−0.864571			−0.432285	+	0.901737i	$0.642293\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.782206	+	1.35482i	−0.189713	+	0.328592i	−0.945154	−	0.326624i	$-0.894089\pi$
$17$	−0.782206	+	1.35482i	−0.189713	+	0.328592i	0.755442	+	0.655216i	$0.227422\pi$
$18$		0			0
$19$	2.15042	+	3.72463i	0.493340	+	0.854489i	0.999971	−	0.00767364i	$-0.00244262\pi$
$19$	2.15042	+	3.72463i	0.493340	+	0.854489i	−0.506631	+	0.862163i	$0.669109\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.05782	+	2.34278i	−0.846114	+	0.488504i	−0.859338	−	0.511408i	$-0.829124\pi$
$23$	−4.05782	+	2.34278i	−0.846114	+	0.488504i	0.0132237	+	0.999913i	$0.495791\pi$
$24$		0			0
$25$	−1.82744	+	3.16522i	−0.365488	+	0.633044i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.08861			0.759235			0.379618	−	0.925143i	$-0.376056\pi$
$29$	4.08861			0.759235			0.379618	+	0.925143i	$0.376056\pi$
$30$		0			0
$31$	−2.40452	−	1.38825i	−0.431865	−	0.249337i	0.268276	−	0.963342i	$-0.413546\pi$
$31$	−2.40452	−	1.38825i	−0.431865	−	0.249337i	−0.700141	+	0.714005i	$0.746879\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.60353	−	1.62402i	−0.440077	−	0.274509i
$36$		0			0
$37$	−4.96358	+	2.86572i	−0.816008	+	0.471122i	−0.849038	−	0.528332i	$-0.822818\pi$
$37$	−4.96358	+	2.86572i	−0.816008	+	0.471122i	0.0330302	+	0.999454i	$0.489484\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.19421			0.342678			0.171339	−	0.985212i	$-0.445191\pi$
$41$	2.19421			0.342678			0.171339	+	0.985212i	$0.445191\pi$
$42$		0			0
$43$			6.52977i			0.995781i	0.867240	+	0.497891i	$0.165892\pi$
$43$			6.52977i			0.995781i	−0.867240	+	0.497891i	$0.834108\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.34609	+	9.25971i	0.779808	+	1.35067i	0.932052	+	0.362324i	$0.118017\pi$
$47$	5.34609	+	9.25971i	0.779808	+	1.35067i	−0.152244	+	0.988343i	$0.548650\pi$
$48$		0			0
$49$	−3.90521	+	5.80942i	−0.557887	+	0.829917i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.61902	−	9.73242i	0.771831	−	1.33685i	−0.164727	−	0.986339i	$-0.552674\pi$
$53$	5.61902	−	9.73242i	0.771831	−	1.33685i	0.936558	−	0.350512i	$-0.113992\pi$
$54$		0			0
$55$			3.28359i			0.442760i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	11.9042	+	6.87289i	1.54980	+	0.894775i	0.998157	+	0.0606912i	$0.0193305\pi$
$59$	11.9042	+	6.87289i	1.54980	+	0.894775i	0.551638	+	0.834083i	$0.314003\pi$
$60$		0			0
$61$	5.09458	+	8.82407i	0.652294	+	1.12981i	0.982565	+	0.185920i	$0.0595265\pi$
$61$	5.09458	+	8.82407i	0.652294	+	1.12981i	−0.330271	+	0.943886i	$0.607140\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.13100	+	1.80768i	−0.388352	+	0.224215i
$66$		0			0
$67$	5.01037	+	2.89274i	0.612114	+	0.353404i	0.773793	−	0.633439i	$-0.218357\pi$
$67$	5.01037	+	2.89274i	0.612114	+	0.353404i	−0.161678	+	0.986844i	$0.551691\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.72781i		−	0.561088i	−0.959841	−	0.280544i	$-0.909485\pi$
$71$		−	4.72781i		−	0.561088i	0.959841	−	0.280544i	$-0.0905149\pi$
$72$		0			0
$73$	−14.0619	−	8.11863i	−1.64582	−	0.950213i	−0.978709	−	0.205254i	$-0.934198\pi$
$73$	−14.0619	−	8.11863i	−1.64582	−	0.950213i	−0.667109	−	0.744960i	$-0.732469\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.48627	+	0.255528i	0.853140	+	0.0291201i
$78$		0			0
$79$	−2.89324	−	5.01124i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$79$	−2.89324	−	5.01124i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			5.93150i			0.651066i	0.945531	+	0.325533i	$0.105544\pi$
$83$			5.93150i			0.651066i	−0.945531	+	0.325533i	$0.894456\pi$
$84$		0			0
$85$			1.81439i			0.196798i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.33902	−	2.31925i	−0.141936	−	0.245840i	0.786290	−	0.617858i	$-0.211999\pi$
$89$	−1.33902	−	2.31925i	−0.141936	−	0.245840i	−0.928226	+	0.372018i	$0.878666\pi$
$90$		0			0
$91$	3.87769	+	7.27904i	0.406492	+	0.763051i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.31980	+	2.49404i	0.443202	+	0.255883i
$96$		0			0
$97$		−	11.2024i		−	1.13743i	−0.822534	−	0.568716i	$-0.807440\pi$
$97$		−	11.2024i		−	1.13743i	0.822534	−	0.568716i	$-0.192560\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.473259	−	0.273236i	−0.0470911	−	0.0271880i	0.476270	−	0.879299i	$-0.341989\pi$
$101$	−0.473259	−	0.273236i	−0.0470911	−	0.0271880i	−0.523361	+	0.852111i	$0.675322\pi$
$102$		0			0
$103$	−13.0391	+	7.52810i	−1.28478	+	0.741766i	−0.977718	−	0.209925i	$-0.932678\pi$
$103$	−13.0391	+	7.52810i	−1.28478	+	0.741766i	−0.307059	+	0.951691i	$0.599345\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.42284	−	2.46443i	−0.137551	−	0.238246i	0.789018	−	0.614370i	$-0.210590\pi$
$107$	−1.42284	−	2.46443i	−0.137551	−	0.238246i	−0.926569	+	0.376125i	$0.877256\pi$
$108$		0			0
$109$	−0.806006	−	0.465348i	−0.0772014	−	0.0445722i	0.460902	−	0.887451i	$-0.347526\pi$
$109$	−0.806006	−	0.465348i	−0.0772014	−	0.0445722i	−0.538104	+	0.842879i	$0.680859\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			13.9228i			1.30974i	0.755741	+	0.654871i	$0.227277\pi$
$113$			13.9228i			1.30974i	−0.755741	+	0.654871i	$0.772723\pi$
$114$		0			0
$115$	−2.71714	+	4.70623i	−0.253375	+	0.438858i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.13663	+	0.141195i	0.379205	+	0.0129434i
$120$		0			0
$121$	1.49218	+	2.58452i	0.135652	+	0.234957i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			10.0379i			0.897813i
$126$		0			0
$127$	2.15135			0.190901			0.0954506	−	0.995434i	$-0.469571\pi$
$127$	2.15135			0.190901			0.0954506	+	0.995434i	$0.469571\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.5792	−	6.10788i	0.924306	−	0.533648i	0.0392997	−	0.999227i	$-0.487487\pi$
$131$	10.5792	−	6.10788i	0.924306	−	0.533648i	0.885006	+	0.465579i	$0.154154\pi$
$132$		0			0
$133$	6.02233	−	9.65463i	0.522202	−	0.837163i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.51773	−	4.91771i	−0.727718	−	0.420148i	0.0898684	−	0.995954i	$-0.471355\pi$
$137$	−8.51773	−	4.91771i	−0.727718	−	0.420148i	−0.817587	+	0.575805i	$0.804689\pi$
$138$		0			0
$139$	−17.6445			−1.49658			−0.748292	−	0.663369i	$-0.769126\pi$
$139$	−17.6445			−1.49658			−0.748292	+	0.663369i	$0.769126\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.41277	−	7.64315i	0.369015	−	0.639152i
$144$		0			0
$145$	4.10663	−	2.37097i	0.341037	−	0.196898i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.88010	+	8.45259i	0.399794	+	0.692463i	0.993700	−	0.112071i	$-0.0357484\pi$
$149$	4.88010	+	8.45259i	0.399794	+	0.692463i	−0.593906	+	0.804534i	$0.702415\pi$
$150$		0			0
$151$	−2.13984	+	3.70631i	−0.174138	+	0.301615i	−0.939862	−	0.341553i	$-0.889047\pi$
$151$	−2.13984	+	3.70631i	−0.174138	+	0.301615i	0.765725	+	0.643168i	$0.222380\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.22016			−0.258650
$156$		0			0
$157$	−12.1053	+	20.9671i	−0.966111	+	1.67335i	−0.259512	+	0.965740i	$0.583562\pi$
$157$	−12.1053	+	20.9671i	−0.966111	+	1.67335i	−0.706599	+	0.707614i	$0.749772\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.5183	+	6.56105i	0.828957	+	0.517083i
$162$		0			0
$163$	−7.81820	+	4.51384i	−0.612369	+	0.353551i	−0.773892	−	0.633318i	$-0.781693\pi$
$163$	−7.81820	+	4.51384i	−0.612369	+	0.353551i	0.161523	+	0.986869i	$0.448359\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−16.2972			−1.26112			−0.630559	−	0.776142i	$-0.717174\pi$
$167$	−16.2972			−1.26112			−0.630559	+	0.776142i	$0.717174\pi$
$168$		0			0
$169$	−3.28273			−0.252518
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.0240052	−	0.0138594i	0.00182508	−	0.00105371i	−0.499087	−	0.866552i	$-0.666331\pi$
$173$	0.0240052	−	0.0138594i	0.00182508	−	0.00105371i	0.500912	+	0.865498i	$0.332998\pi$
$174$		0			0
$175$	9.66429	+	0.329870i	0.730551	+	0.0249358i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.39577	−	4.14959i	0.179068	−	0.310155i	−0.762493	−	0.646996i	$-0.776025\pi$
$179$	2.39577	−	4.14959i	0.179068	−	0.310155i	0.941562	+	0.336841i	$0.109358\pi$
$180$		0			0
$181$	7.03270			0.522736			0.261368	−	0.965239i	$-0.415826\pi$
$181$	7.03270			0.522736			0.261368	+	0.965239i	$0.415826\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.32364	+	5.75672i	−0.244359	+	0.423242i
$186$		0			0
$187$	−2.21457	−	3.83576i	−0.161946	−	0.280498i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.9817	+	6.91767i	−0.866969	+	0.500545i	−0.866340	−	0.499455i	$-0.833534\pi$
$191$	−11.9817	+	6.91767i	−0.866969	+	0.500545i	−0.000629171		1.00000i	$0.500200\pi$
$192$		0			0
$193$	−2.10467	+	3.64540i	−0.151498	+	0.262402i	−0.931778	−	0.363028i	$-0.881743\pi$
$193$	−2.10467	+	3.64540i	−0.151498	+	0.262402i	0.780280	+	0.625430i	$0.215076\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−24.3528			−1.73507			−0.867534	−	0.497378i	$-0.834296\pi$
$197$	−24.3528			−1.73507			−0.867534	+	0.497378i	$0.834296\pi$
$198$		0			0
$199$	1.63996	+	0.946831i	0.116254	+	0.0671191i	0.556999	−	0.830513i	$-0.311953\pi$
$199$	1.63996	+	0.946831i	0.116254	+	0.0671191i	−0.440746	+	0.897632i	$0.645286\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.08600	−	9.54723i	−0.356967	−	0.670084i
$204$		0			0
$205$	2.20389	−	1.27241i	0.153926	−	0.0888692i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.1765			−0.842266
$210$		0			0
$211$		−	25.0597i		−	1.72518i	−0.505901	−	0.862592i	$-0.668840\pi$
$211$		−	25.0597i		−	1.72518i	0.505901	−	0.862592i	$-0.331160\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.78659	+	6.55856i	0.258243	+	0.447290i
$216$		0			0
$217$	−0.250592	+	7.34166i	−0.0170113	+	0.498384i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.43833	−	4.22332i	0.164020	−	0.284091i
$222$		0			0
$223$		−	9.42532i		−	0.631166i	−0.948898	−	0.315583i	$-0.897800\pi$
$223$		−	9.42532i		−	0.631166i	0.948898	−	0.315583i	$-0.102200\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.0967	+	9.29346i	1.06838	+	0.616829i	0.927738	−	0.373232i	$-0.121750\pi$
$227$	16.0967	+	9.29346i	1.06838	+	0.616829i	0.140641	+	0.990061i	$0.455084\pi$
$228$		0			0
$229$	13.7016	+	23.7319i	0.905428	+	1.56825i	0.820342	+	0.571874i	$0.193783\pi$
$229$	13.7016	+	23.7319i	0.905428	+	1.56825i	0.0850862	+	0.996374i	$0.472883\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21.0326	−	12.1432i	1.37789	−	0.795526i	0.385987	−	0.922504i	$-0.373861\pi$
$233$	21.0326	−	12.1432i	1.37789	−	0.795526i	0.991906	+	0.126978i	$0.0405278\pi$
$234$		0			0
$235$	10.7393	+	6.20035i	0.700556	+	0.404466i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2.35413i			0.152276i	0.997097	+	0.0761380i	$0.0242590\pi$
$239$			2.35413i			0.152276i	−0.997097	+	0.0761380i	$0.975741\pi$
$240$		0			0
$241$	11.2904	+	6.51850i	0.727277	+	0.419893i	0.817425	−	0.576035i	$-0.195401\pi$
$241$	11.2904	+	6.51850i	0.727277	+	0.419893i	−0.0901483	+	0.995928i	$0.528734\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.553574	+	8.09964i	−0.0353665	+	0.517467i
$246$		0			0
$247$	−6.70340	−	11.6106i	−0.426527	−	0.738767i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	24.0484i		−	1.51792i	−0.651136	−	0.758961i	$-0.725707\pi$
$251$		−	24.0484i		−	1.51792i	0.651136	−	0.758961i	$-0.274293\pi$
$252$		0			0
$253$		−	13.2657i		−	0.834010i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.72517	+	11.6483i	0.419504	+	0.726603i	0.995890	−	0.0905752i	$-0.0288706\pi$
$257$	6.72517	+	11.6483i	0.419504	+	0.726603i	−0.576385	+	0.817178i	$0.695537\pi$
$258$		0			0
$259$	12.8661	+	8.02557i	0.799461	+	0.498684i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.35260	−	0.780923i	−0.0834048	−	0.0481538i	0.457718	−	0.889098i	$-0.348667\pi$
$263$	−1.35260	−	0.780923i	−0.0834048	−	0.0481538i	−0.541122	+	0.840944i	$0.682000\pi$
$264$		0			0
$265$		−	13.0338i		−	0.800658i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.3499	−	8.28490i	−0.874927	−	0.505139i	−0.00594471	−	0.999982i	$-0.501892\pi$
$269$	−14.3499	−	8.28490i	−0.874927	−	0.505139i	−0.868982	+	0.494843i	$0.835226\pi$
$270$		0			0
$271$	−11.9658	+	6.90846i	−0.726871	+	0.419659i	−0.817276	−	0.576246i	$-0.804517\pi$
$271$	−11.9658	+	6.90846i	−0.726871	+	0.419659i	0.0904054	+	0.995905i	$0.471184\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.17384	−	8.96135i	−0.311994	−	0.540390i
$276$		0			0
$277$	−11.6351	−	6.71750i	−0.699083	−	0.403616i	0.107923	−	0.994159i	$-0.465580\pi$
$277$	−11.6351	−	6.71750i	−0.699083	−	0.403616i	−0.807006	+	0.590544i	$0.798913\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	27.9830i		−	1.66933i	−0.550761	−	0.834663i	$-0.685662\pi$
$281$		−	27.9830i		−	1.66933i	0.550761	−	0.834663i	$-0.314338\pi$
$282$		0			0
$283$	9.68568	−	16.7761i	0.575754	−	0.997235i	−0.420205	−	0.907429i	$-0.638042\pi$
$283$	9.68568	−	16.7761i	0.575754	−	0.997235i	0.995959	−	0.0898063i	$-0.0286248\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.72947	−	5.12366i	−0.161116	−	0.302440i
$288$		0			0
$289$	7.27631	+	12.6029i	0.428018	+	0.741349i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			24.4475i			1.42824i	0.700024	+	0.714119i	$0.253173\pi$
$293$			24.4475i			1.42824i	−0.700024	+	0.714119i	$0.746827\pi$
$294$		0			0
$295$	15.9422			0.928193
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.6493	−	7.30305i	0.731525	−	0.422346i
$300$		0			0
$301$	15.2476	−	8.12267i	0.878855	−	0.468183i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.2341	+	5.90865i	0.586001	+	0.338328i
$306$		0			0
$307$	−27.0446			−1.54352			−0.771758	−	0.635917i	$-0.780622\pi$
$307$	−27.0446			−1.54352			−0.771758	+	0.635917i	$0.780622\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.276139	−	0.478286i	0.0156584	−	0.0271211i	−0.858090	−	0.513499i	$-0.828349\pi$
$311$	0.276139	−	0.478286i	0.0156584	−	0.0271211i	0.873748	+	0.486378i	$0.161682\pi$
$312$		0			0
$313$	−13.5478	+	7.82183i	−0.765768	+	0.442116i	−0.831363	−	0.555730i	$-0.812439\pi$
$313$	−13.5478	+	7.82183i	−0.765768	+	0.442116i	0.0655951	+	0.997846i	$0.479105\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.33699	+	10.9760i	0.355921	+	0.616472i	0.987275	−	0.159022i	$-0.0508340\pi$
$317$	6.33699	+	10.9760i	0.355921	+	0.616472i	−0.631355	+	0.775494i	$0.717501\pi$
$318$		0			0
$319$	−5.78782	+	10.0248i	−0.324056	+	0.561281i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.72828			−0.374371
$324$		0			0
$325$	5.69660	−	9.86680i	0.315991	−	0.547312i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	14.9719	−	24.0021i	0.825429	−	1.32328i
$330$		0			0
$331$	2.26793	−	1.30939i	0.124657	−	0.0719706i	−0.436375	−	0.899765i	$-0.643738\pi$
$331$	2.26793	−	1.30939i	0.124657	−	0.0719706i	0.561032	+	0.827794i	$0.310405\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.70995			0.366604
$336$		0			0
$337$	34.9446			1.90356			0.951778	−	0.306788i	$-0.0992542\pi$
$337$	34.9446			1.90356			0.951778	+	0.306788i	$0.0992542\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.80766	−	3.93040i	0.368656	−	0.212843i
$342$		0			0
$343$	18.4233	+	1.89241i	0.994766	+	0.102180i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−16.7205	+	28.9607i	−0.897602	+	1.55469i	−0.0670515	+	0.997750i	$0.521359\pi$
$347$	−16.7205	+	28.9607i	−0.897602	+	1.55469i	−0.830551	+	0.556943i	$0.811974\pi$
$348$		0			0
$349$	−7.32611			−0.392158			−0.196079	−	0.980588i	$-0.562821\pi$
$349$	−7.32611			−0.392158			−0.196079	+	0.980588i	$0.562821\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.9297	+	24.1269i	−0.741402	+	1.28415i	0.210454	+	0.977604i	$0.432506\pi$
$353$	−13.9297	+	24.1269i	−0.741402	+	1.28415i	−0.951857	+	0.306543i	$0.900828\pi$
$354$		0			0
$355$	−2.74164	−	4.74866i	−0.145511	−	0.252033i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.6201	−	9.01825i	0.824396	−	0.475965i	−0.0275343	−	0.999621i	$-0.508766\pi$
$359$	15.6201	−	9.01825i	0.824396	−	0.475965i	0.851930	+	0.523656i	$0.175432\pi$
$360$		0			0
$361$	0.251405	−	0.435447i	0.0132319	−	0.0229182i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−18.8318			−0.985703
$366$		0			0
$367$	−4.16683	−	2.40572i	−0.217507	−	0.125578i	0.387288	−	0.921959i	$-0.373412\pi$
$367$	−4.16683	−	2.40572i	−0.217507	−	0.125578i	−0.604795	+	0.796381i	$0.706745\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−29.7157	−	1.01428i	−1.54276	−	0.0526590i
$372$		0			0
$373$	−0.589575	+	0.340391i	−0.0305270	+	0.0176248i	−0.515186	−	0.857078i	$-0.672277\pi$
$373$	−0.589575	+	0.340391i	−0.0305270	+	0.0176248i	0.484659	+	0.874703i	$0.338944\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.7452			−0.656413
$378$		0			0
$379$		−	23.7081i		−	1.21780i	−0.793245	−	0.608902i	$-0.791610\pi$
$379$		−	23.7081i		−	1.21780i	0.793245	−	0.608902i	$-0.208390\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.7429	−	28.9996i	−0.855522	−	1.48181i	−0.876160	−	0.482021i	$-0.839903\pi$
$383$	−16.7429	−	28.9996i	−0.855522	−	1.48181i	0.0206373	−	0.999787i	$-0.493430\pi$
$384$		0			0
$385$	7.66746	−	4.08460i	0.390770	−	0.208171i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.7772	−	20.3988i	0.597130	−	1.03426i	−0.396113	−	0.918202i	$-0.629641\pi$
$389$	11.7772	−	20.3988i	0.597130	−	1.03426i	0.993243	−	0.116057i	$-0.0370256\pi$
$390$		0			0
$391$		−	7.33015i		−	0.370702i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.81200	−	3.35556i	−0.292433	−	0.168836i
$396$		0			0
$397$	−12.7489	−	22.0817i	−0.639849	−	1.10825i	−0.985466	−	0.169875i	$-0.945664\pi$
$397$	−12.7489	−	22.0817i	−0.639849	−	1.10825i	0.345617	−	0.938376i	$-0.387670\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.2818	+	5.93623i	−0.513451	+	0.296441i	−0.734251	−	0.678878i	$-0.762467\pi$
$401$	−10.2818	+	5.93623i	−0.513451	+	0.296441i	0.220800	+	0.975319i	$0.429133\pi$
$402$		0			0
$403$	7.49550	+	4.32753i	0.373378	+	0.215570i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	16.2268i		−	0.804334i
$408$		0			0
$409$	−4.46727	−	2.57918i	−0.220892	−	0.127532i	0.385471	−	0.922720i	$-0.374039\pi$
$409$	−4.46727	−	2.57918i	−0.220892	−	0.127532i	−0.606363	+	0.795188i	$0.707372\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.24062	−	36.3468i	0.0610469	−	1.78851i
$414$		0			0
$415$	3.43965	+	5.95765i	0.168846	+	0.292449i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			8.63546i			0.421870i	0.977500	+	0.210935i	$0.0676508\pi$
$419$			8.63546i			0.421870i	−0.977500	+	0.210935i	$0.932349\pi$
$420$		0			0
$421$		−	37.2303i		−	1.81449i	−0.420599	−	0.907247i	$-0.638180\pi$
$421$		−	37.2303i		−	1.81449i	0.420599	−	0.907247i	$-0.361820\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.85887	−	4.95171i	−0.138676	−	0.240193i
$426$		0			0
$427$	14.2676	−	22.8729i	0.690455	−	1.10690i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.24374	+	4.18217i	0.348919	+	0.201448i	0.664209	−	0.747547i	$-0.268769\pi$
$431$	7.24374	+	4.18217i	0.348919	+	0.201448i	−0.315290	+	0.948995i	$0.602102\pi$
$432$		0			0
$433$		−	7.63673i		−	0.366998i	−0.983020	−	0.183499i	$-0.941258\pi$
$433$		−	7.63673i		−	0.366998i	0.983020	−	0.183499i	$-0.0587424\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−17.4520	−	10.0759i	−0.834843	−	0.481997i
$438$		0			0
$439$	8.42793	−	4.86587i	0.402243	−	0.232235i	−0.285208	−	0.958466i	$-0.592063\pi$
$439$	8.42793	−	4.86587i	0.402243	−	0.232235i	0.687451	+	0.726230i	$0.258729\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.9352	+	18.9404i	0.519549	+	0.899885i	0.999742	+	0.0227221i	$0.00723330\pi$
$443$	10.9352	+	18.9404i	0.519549	+	0.899885i	−0.480193	+	0.877163i	$0.659433\pi$
$444$		0			0
$445$	−2.68985	−	1.55299i	−0.127511	−	0.0736186i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	16.6555i		−	0.786020i	−0.919534	−	0.393010i	$-0.871434\pi$
$449$		−	16.6555i		−	0.786020i	0.919534	−	0.393010i	$-0.128566\pi$
$450$		0			0
$451$	−3.10612	+	5.37995i	−0.146261	+	0.253332i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.11587	+	5.06248i	0.380478	+	0.237333i
$456$		0			0
$457$	−10.3145	−	17.8652i	−0.482490	−	0.835698i	0.517307	−	0.855800i	$-0.326934\pi$
$457$	−10.3145	−	17.8652i	−0.482490	−	0.835698i	−0.999798	+	0.0201016i	$0.993601\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			31.5292i			1.46846i	0.678901	+	0.734230i	$0.262457\pi$
$461$			31.5292i			1.46846i	−0.678901	+	0.734230i	$0.737543\pi$
$462$		0			0
$463$	40.3843			1.87682			0.938409	−	0.345526i	$-0.112299\pi$
$463$	40.3843			1.87682			0.938409	+	0.345526i	$0.112299\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.1585	+	5.86499i	−0.470077	+	0.271399i	−0.716272	−	0.697821i	$-0.754153\pi$
$467$	−10.1585	+	5.86499i	−0.470077	+	0.271399i	0.246195	+	0.969220i	$0.420820\pi$
$468$		0			0
$469$	0.522166	−	15.2980i	0.0241114	−	0.706398i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−16.0103	−	9.24352i	−0.736152	−	0.425018i
$474$		0			0
$475$	−15.7191			−0.721240
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.299198	+	0.518227i	−0.0136707	+	0.0236784i	−0.872780	−	0.488114i	$-0.837685\pi$
$479$	−0.299198	+	0.518227i	−0.0136707	+	0.0236784i	0.859109	+	0.511792i	$0.171018\pi$
$480$		0			0
$481$	15.4727	−	8.93319i	0.705496	−	0.407318i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.49623	−	11.2518i	−0.294979	−	0.510918i
$486$		0			0
$487$	10.3961	−	18.0065i	0.471090	−	0.815952i	−0.528363	−	0.849019i	$-0.677194\pi$
$487$	10.3961	−	18.0065i	0.471090	−	0.815952i	0.999453	+	0.0330665i	$0.0105273\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−22.4435			−1.01286			−0.506430	−	0.862281i	$-0.669035\pi$
$491$	−22.4435			−1.01286			−0.506430	+	0.862281i	$0.669035\pi$
$492$		0			0
$493$	−3.19813	+	5.53933i	−0.144037	+	0.249479i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.0398	+	5.88113i	−0.495204	+	0.263805i
$498$		0			0
$499$	10.2264	−	5.90424i	0.457799	−	0.264310i	−0.253320	−	0.967383i	$-0.581522\pi$
$499$	10.2264	−	5.90424i	0.457799	−	0.264310i	0.711118	+	0.703072i	$0.248189\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.08281			0.137456			0.0687278	−	0.997635i	$-0.478106\pi$
$503$	3.08281			0.137456			0.0687278	+	0.997635i	$0.478106\pi$
$504$		0			0
$505$	−0.633794			−0.0282035
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.99141	+	2.30444i	−0.176916	+	0.102143i	−0.585843	−	0.810425i	$-0.699236\pi$
$509$	−3.99141	+	2.30444i	−0.176916	+	0.102143i	0.408927	+	0.912567i	$0.365903\pi$
$510$		0			0
$511$	−1.46549	+	42.9347i	−0.0648293	+	1.89932i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.73103	+	15.1226i	−0.384735	+	0.666381i
$516$		0			0
$517$	−30.2716			−1.33135
$518$		0			0
$519$		0			0
$520$		0			0
$521$	19.2542	−	33.3493i	0.843542	−	1.46106i	−0.0433388	−	0.999060i	$-0.513799\pi$
$521$	19.2542	−	33.3493i	0.843542	−	1.46106i	0.886881	−	0.461998i	$-0.152867\pi$
$522$		0			0
$523$	19.3439	+	33.5046i	0.845849	+	1.46505i	0.884882	+	0.465815i	$0.154239\pi$
$523$	19.3439	+	33.5046i	0.845849	+	1.46505i	−0.0390332	+	0.999238i	$0.512428\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.76166	−	2.17180i	0.163860	−	0.0946049i
$528$		0			0
$529$	−0.522727	+	0.905390i	−0.0227273	+	0.0393648i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.83991			−0.296270
$534$		0			0
$535$	−2.85823	−	1.65020i	−0.123572	−	0.0713443i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.71582	−	17.7989i	−0.375417	−	0.766654i
$540$		0			0
$541$	24.9474	−	14.4034i	1.07257	−	0.619249i	0.143688	−	0.989623i	$-0.454104\pi$
$541$	24.9474	−	14.4034i	1.07257	−	0.619249i	0.928883	+	0.370374i	$0.120770\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.07941			−0.0462370
$546$		0			0
$547$		−	26.0874i		−	1.11541i	−0.830038	−	0.557707i	$-0.811681\pi$
$547$		−	26.0874i		−	1.11541i	0.830038	−	0.557707i	$-0.188319\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.79222	+	15.2286i	0.374561	+	0.648759i
$552$		0			0
$553$	−8.10263	+	12.9897i	−0.344559	+	0.552377i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.2896	−	19.5542i	0.478357	−	0.828538i	−0.521335	−	0.853352i	$-0.674566\pi$
$557$	11.2896	−	19.5542i	0.478357	−	0.828538i	0.999692	+	0.0248137i	$0.00789926\pi$
$558$		0			0
$559$		−	20.3550i		−	0.860923i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.9160	+	9.18909i	0.670778	+	0.387274i	0.796371	−	0.604808i	$-0.206750\pi$
$563$	15.9160	+	9.18909i	0.670778	+	0.387274i	−0.125593	+	0.992082i	$0.540083\pi$
$564$		0			0
$565$	8.07374	+	13.9841i	0.339665	+	0.588317i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16.3033	+	9.41272i	−0.683470	+	0.394602i	−0.801161	−	0.598449i	$-0.795784\pi$
$569$	−16.3033	+	9.41272i	−0.683470	+	0.394602i	0.117691	+	0.993050i	$0.462451\pi$
$570$		0			0
$571$	2.34839	+	1.35584i	0.0982770	+	0.0567403i	0.548333	−	0.836260i	$-0.315263\pi$
$571$	2.34839	+	1.35584i	0.0982770	+	0.0567403i	−0.450056	+	0.893000i	$0.648596\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	17.1252i		−	0.714170i
$576$		0			0
$577$	15.8080	+	9.12677i	0.658097	+	0.379952i	0.791551	−	0.611103i	$-0.209274\pi$
$577$	15.8080	+	9.12677i	0.658097	+	0.379952i	−0.133455	+	0.991055i	$0.542607\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	13.8505	−	7.37844i	0.574617	−	0.306109i
$582$		0			0
$583$	15.9085	+	27.5544i	0.658863	+	1.14119i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			32.5679i			1.34422i	0.740451	+	0.672110i	$0.234612\pi$
$587$			32.5679i			1.34422i	−0.740451	+	0.672110i	$0.765388\pi$
$588$		0			0
$589$		−	11.9413i		−	0.492032i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.3152	+	26.5267i	0.628920	+	1.08932i	0.987769	+	0.155925i	$0.0498359\pi$
$593$	15.3152	+	26.5267i	0.628920	+	1.08932i	−0.358849	+	0.933396i	$0.616831\pi$
$594$		0			0
$595$	4.23675	−	2.25700i	0.173690	−	0.0925279i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	28.4250	+	16.4112i	1.16141	+	0.670543i	0.951643	−	0.307207i	$-0.0993944\pi$
$599$	28.4250	+	16.4112i	1.16141	+	0.670543i	0.209772	+	0.977750i	$0.432728\pi$
$600$		0			0
$601$			5.73393i			0.233892i	0.993138	+	0.116946i	$0.0373104\pi$
$601$			5.73393i			0.233892i	−0.993138	+	0.116946i	$0.962690\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.99751	+	1.73061i	0.121866	+	0.0703594i
$606$		0			0
$607$	−4.62054	+	2.66767i	−0.187542	+	0.108277i	−0.590831	−	0.806795i	$-0.701200\pi$
$607$	−4.62054	+	2.66767i	−0.187542	+	0.108277i	0.403289	+	0.915073i	$0.367867\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.6651	−	28.8649i	−0.674199	−	1.16775i
$612$		0			0
$613$	−0.763004	−	0.440521i	−0.0308174	−	0.0177925i	0.484512	−	0.874785i	$-0.338997\pi$
$613$	−0.763004	−	0.440521i	−0.0308174	−	0.0177925i	−0.515330	+	0.856992i	$0.672330\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	33.3940i		−	1.34439i	−0.740373	−	0.672196i	$-0.765351\pi$
$617$		−	33.3940i		−	1.34439i	0.740373	−	0.672196i	$-0.234649\pi$
$618$		0			0
$619$	−0.347094	+	0.601184i	−0.0139509	+	0.0241636i	−0.872917	−	0.487870i	$-0.837774\pi$
$619$	−0.347094	+	0.601184i	−0.0139509	+	0.0241636i	0.858966	+	0.512033i	$0.171108\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.74998	+	6.01174i	−0.150240	+	0.240855i
$624$		0			0
$625$	−3.31630	−	5.74400i	−0.132652	−	0.229760i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	8.96634i		−	0.357511i
$630$		0			0
$631$	−3.67584			−0.146333			−0.0731664	−	0.997320i	$-0.523310\pi$
$631$	−3.67584			−0.146333			−0.0731664	+	0.997320i	$0.523310\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.16083	−	1.24756i	0.0857500	−	0.0495078i
$636$		0			0
$637$	12.1735	−	18.1094i	0.482333	−	0.717522i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20.9407	+	12.0901i	0.827108	+	0.477531i	0.852862	−	0.522137i	$-0.174865\pi$
$641$	20.9407	+	12.0901i	0.827108	+	0.477531i	−0.0257532	+	0.999668i	$0.508198\pi$
$642$		0			0
$643$	7.82797			0.308705			0.154352	−	0.988016i	$-0.450671\pi$
$643$	7.82797			0.308705			0.154352	+	0.988016i	$0.450671\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.30976	+	14.3929i	−0.326690	+	0.565844i	−0.981853	−	0.189644i	$-0.939267\pi$
$647$	−8.30976	+	14.3929i	−0.326690	+	0.565844i	0.655163	+	0.755488i	$0.272600\pi$
$648$		0			0
$649$	−33.7031	+	19.4585i	−1.32296	+	0.763812i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.0211	−	17.3571i	−0.392158	−	0.679237i	0.600576	−	0.799568i	$-0.294938\pi$
$653$	−10.0211	−	17.3571i	−0.392158	−	0.679237i	−0.992734	+	0.120331i	$0.961605\pi$
$654$		0			0
$655$	7.08387	−	12.2696i	0.276790	−	0.479414i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.53973			−0.215797			−0.107899	−	0.994162i	$-0.534412\pi$
$659$	−5.53973			−0.215797			−0.107899	+	0.994162i	$0.534412\pi$
$660$		0			0
$661$	−14.8650	+	25.7470i	−0.578183	+	1.00144i	0.417505	+	0.908675i	$0.362905\pi$
$661$	−14.8650	+	25.7470i	−0.578183	+	1.00144i	−0.995688	+	0.0927674i	$0.970429\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.450196	−	13.1895i	0.0174579	−	0.511468i
$666$		0			0
$667$	−16.5908	+	9.57873i	−0.642400	+	0.370890i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−28.8475			−1.11364
$672$		0			0
$673$	28.0307			1.08050			0.540252	−	0.841503i	$-0.318329\pi$
$673$	28.0307			1.08050			0.540252	+	0.841503i	$0.318329\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.2739	−	5.93164i	0.394858	−	0.227971i	−0.289405	−	0.957207i	$-0.593457\pi$
$677$	10.2739	−	5.93164i	0.394858	−	0.227971i	0.684263	+	0.729235i	$0.260124\pi$
$678$		0			0
$679$	−26.1585	+	13.9352i	−1.00387	+	0.534782i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.62606	−	13.2087i	0.291803	−	0.505418i	−0.682433	−	0.730948i	$-0.739078\pi$
$683$	7.62606	−	13.2087i	0.291803	−	0.505418i	0.974236	+	0.225530i	$0.0724114\pi$
$684$		0			0
$685$	−11.4070			−0.435841
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17.5159	+	30.3384i	−0.667303	+	1.15580i
$690$		0			0
$691$	3.24122	+	5.61395i	0.123302	+	0.213565i	0.921068	−	0.389402i	$-0.127318\pi$
$691$	3.24122	+	5.61395i	0.123302	+	0.213565i	−0.797766	+	0.602967i	$0.793985\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−17.7223	+	10.2320i	−0.672244	+	0.388120i
$696$		0			0
$697$	−1.71632	+	2.97276i	−0.0650104	+	0.112601i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.86157			−0.183619			−0.0918096	−	0.995777i	$-0.529265\pi$
$701$	−4.86157			−0.183619			−0.0918096	+	0.995777i	$0.529265\pi$
$702$		0			0
$703$	−21.3475	−	12.3250i	−0.805138	−	0.464847i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.0493217	+	1.44499i	−0.00185493	+	0.0543444i
$708$		0			0
$709$	−21.3747	+	12.3407i	−0.802744	+	0.463464i	−0.844430	−	0.535667i	$-0.820060\pi$
$709$	−21.3747	+	12.3407i	−0.802744	+	0.463464i	0.0416860	+	0.999131i	$0.486727\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	13.0095			0.487209
$714$		0			0
$715$		−	10.2358i		−	0.382797i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10.2912	+	17.8248i	0.383796	+	0.664754i	0.991601	−	0.129332i	$-0.0412833\pi$
$719$	10.2912	+	17.8248i	0.383796	+	0.664754i	−0.607806	+	0.794086i	$0.707950\pi$
$720$		0			0
$721$	33.7986	+	21.0827i	1.25872	+	0.785162i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.47169	+	12.9414i	−0.277492	+	0.480630i
$726$		0			0
$727$		−	27.0889i		−	1.00467i	−0.864673	−	0.502336i	$-0.832474\pi$
$727$		−	27.0889i		−	1.00467i	0.864673	−	0.502336i	$-0.167526\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.84667	−	5.10763i	−0.327206	−	0.188912i
$732$		0			0
$733$	8.48261	+	14.6923i	0.313312	+	0.542673i	0.979077	−	0.203488i	$-0.0652279\pi$
$733$	8.48261	+	14.6923i	0.313312	+	0.542673i	−0.665765	+	0.746162i	$0.731895\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14.1853	+	8.18990i	−0.522523	+	0.301679i
$738$		0			0
$739$	38.1107	+	22.0032i	1.40193	+	0.809402i	0.994590	−	0.103876i	$-0.0331245\pi$
$739$	38.1107	+	22.0032i	1.40193	+	0.809402i	0.407336	+	0.913278i	$0.366458\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	30.5441i		−	1.12056i	−0.828305	−	0.560278i	$-0.810694\pi$
$743$		−	30.5441i		−	1.12056i	0.828305	−	0.560278i	$-0.189306\pi$
$744$		0			0
$745$	9.80324	+	5.65990i	0.359163	+	0.207363i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.98472	+	6.38806i	−0.145598	+	0.233415i
$750$		0			0
$751$	−8.35120	−	14.4647i	−0.304739	−	0.527824i	0.672464	−	0.740130i	$-0.265236\pi$
$751$	−8.35120	−	14.4647i	−0.304739	−	0.527824i	−0.977203	+	0.212306i	$0.931903\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			4.96353i			0.180641i
$756$		0			0
$757$		−	1.36258i		−	0.0495237i	−0.999693	−	0.0247618i	$-0.992117\pi$
$757$		−	1.36258i		−	0.0495237i	0.999693	−	0.0247618i	$-0.00788275\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.30823	−	12.6582i	−0.264923	−	0.458860i	0.702620	−	0.711565i	$-0.252013\pi$
$761$	−7.30823	−	12.6582i	−0.264923	−	0.458860i	−0.967543	+	0.252705i	$0.918680\pi$
$762$		0			0
$763$	−0.0839995	+	2.46096i	−0.00304099	+	0.0890926i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−37.1084	−	21.4246i	−1.33991	−	0.773596i
$768$		0			0
$769$			1.23072i			0.0443809i	0.999754	+	0.0221904i	$0.00706402\pi$
$769$			1.23072i			0.0443809i	−0.999754	+	0.0221904i	$0.992936\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.4450	+	10.0719i	0.627452	+	0.362259i	0.779765	−	0.626073i	$-0.215339\pi$
$773$	17.4450	+	10.0719i	0.627452	+	0.362259i	−0.152313	+	0.988332i	$0.548672\pi$
$774$		0			0
$775$	8.78824	−	5.07390i	0.315683	−	0.182260i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.71847	+	8.17263i	0.169057	+	0.292815i
$780$		0			0
$781$	11.5921	+	6.69268i	0.414796	+	0.239483i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			28.0793i			1.00219i
$786$		0			0
$787$	0.411250	−	0.712307i	0.0146595	−	0.0253910i	−0.858603	−	0.512642i	$-0.828667\pi$
$787$	0.411250	−	0.712307i	0.0146595	−	0.0253910i	0.873262	+	0.487251i	$0.162000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	32.5108	−	17.3191i	1.15595	−	0.615797i
$792$		0			0
$793$	−15.8811	−	27.5069i	−0.563954	−	0.976797i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			16.0328i			0.567911i	0.958838	+	0.283955i	$0.0916468\pi$
$797$			16.0328i			0.567911i	−0.958838	+	0.283955i	$0.908353\pi$
$798$		0			0
$799$	−16.7270			−0.591758
$800$		0			0
$801$		0			0
$802$		0			0
$803$	39.8119	−	22.9854i	1.40493	−	0.811137i
$804$		0			0
$805$	14.3694	+	0.490469i	0.506454	+	0.0172868i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	19.0060	+	10.9731i	0.668215	+	0.385794i	0.795400	−	0.606085i	$-0.207261\pi$
$809$	19.0060	+	10.9731i	0.668215	+	0.385794i	−0.127185	+	0.991879i	$0.540594\pi$
$810$		0			0
$811$	−40.5865			−1.42518			−0.712592	−	0.701579i	$-0.752479\pi$
$811$	−40.5865			−1.42518			−0.712592	+	0.701579i	$0.752479\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.23511	+	9.06748i	−0.183378	+	0.317620i
$816$		0			0
$817$	−24.3210	+	14.0417i	−0.850885	+	0.491258i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.9753	+	38.0623i	0.766942	+	1.32838i	0.939214	+	0.343333i	$0.111556\pi$
$821$	21.9753	+	38.0623i	0.766942	+	1.32838i	−0.172272	+	0.985049i	$0.555111\pi$
$822$		0			0
$823$	0.112859	−	0.195478i	0.00393403	−	0.00681394i	−0.864052	−	0.503403i	$-0.832081\pi$
$823$	0.112859	−	0.195478i	0.00393403	−	0.00681394i	0.867986	+	0.496589i	$0.165414\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.35761			−0.255849			−0.127925	−	0.991784i	$-0.540832\pi$
$827$	−7.35761			−0.255849			−0.127925	+	0.991784i	$0.540832\pi$
$828$		0			0
$829$	−8.03760	+	13.9215i	−0.279157	+	0.483515i	−0.971176	−	0.238365i	$-0.923388\pi$
$829$	−8.03760	+	13.9215i	−0.279157	+	0.483515i	0.692018	+	0.721880i	$0.256722\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.81603	−	9.83502i	−0.166866	−	0.340763i
$834$		0			0
$835$	−16.3691	+	9.45069i	−0.566475	+	0.327055i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−50.6405			−1.74831			−0.874153	−	0.485651i	$-0.838583\pi$
$839$	−50.6405			−1.74831			−0.874153	+	0.485651i	$0.838583\pi$
$840$		0			0
$841$	−12.2833			−0.423562
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.29720	+	1.90364i	−0.113427	+	0.0654872i
$846$		0			0
$847$	4.17889	−	6.69935i	0.143588	−	0.230192i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	13.4275	−	23.2572i	0.460290	−	0.797246i
$852$		0			0
$853$	−8.65855			−0.296463			−0.148232	−	0.988953i	$-0.547358\pi$
$853$	−8.65855			−0.296463			−0.148232	+	0.988953i	$0.547358\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	17.1294	−	29.6691i	0.585130	−	1.01348i	−0.409729	−	0.912207i	$-0.634377\pi$
$857$	17.1294	−	29.6691i	0.585130	−	1.01348i	0.994859	−	0.101268i	$-0.0322900\pi$
$858$		0			0
$859$	6.28537	+	10.8866i	0.214454	+	0.371445i	0.953104	−	0.302644i	$-0.0978694\pi$
$859$	6.28537	+	10.8866i	0.214454	+	0.371445i	−0.738649	+	0.674090i	$0.764536\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	36.2622	−	20.9360i	1.23438	−	0.712669i	0.266440	−	0.963852i	$-0.414153\pi$
$863$	36.2622	−	20.9360i	1.23438	−	0.712669i	0.967940	+	0.251182i	$0.0808193\pi$
$864$		0			0
$865$	0.0160741	−	0.0278411i	0.000546534	−	0.000946625i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	16.3827			0.555744
$870$		0			0
$871$	−15.6186	−	9.01741i	−0.529216	−	0.305543i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	23.4392	−	12.4865i	0.792390	−	0.422122i
$876$		0			0
$877$	−4.04807	+	2.33716i	−0.136694	+	0.0789201i	−0.566787	−	0.823864i	$-0.691814\pi$
$877$	−4.04807	+	2.33716i	−0.136694	+	0.0789201i	0.430094	+	0.902784i	$0.358481\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13.7595			0.463569			0.231785	−	0.972767i	$-0.425544\pi$
$881$	13.7595			0.463569			0.231785	+	0.972767i	$0.425544\pi$
$882$		0			0
$883$		−	31.7680i		−	1.06908i	−0.845144	−	0.534539i	$-0.820485\pi$
$883$		−	31.7680i		−	1.06908i	0.845144	−	0.534539i	$-0.179515\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.28571	+	16.0833i	0.311784	+	0.540025i	0.978749	−	0.205064i	$-0.0657403\pi$
$887$	9.28571	+	16.0833i	0.311784	+	0.540025i	−0.666965	+	0.745089i	$0.732407\pi$
$888$		0			0
$889$	−2.67615	−	5.02357i	−0.0897553	−	0.168485i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−22.9927	+	39.8245i	−0.769420	+	1.33268i
$894$		0			0
$895$		−	5.55718i		−	0.185756i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.83114	−	5.67601i	−0.327887	−	0.189306i
$900$		0			0
$901$	8.79045	+	15.2255i	0.292852	+	0.507235i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.06371	−	4.07823i	0.234806	−	0.135565i
$906$		0			0
$907$	−18.8054	−	10.8573i	−0.624423	−	0.360511i	0.154166	−	0.988045i	$-0.450731\pi$
$907$	−18.8054	−	10.8573i	−0.624423	−	0.360511i	−0.778589	+	0.627534i	$0.784064\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			34.1200i			1.13044i	0.824939	+	0.565222i	$0.191210\pi$
$911$			34.1200i			1.13044i	−0.824939	+	0.565222i	$0.808790\pi$
$912$		0			0
$913$	−14.5433	−	8.39660i	−0.481314	−	0.277887i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−27.4223	−	17.1054i	−0.905564	−	0.564869i
$918$		0			0
$919$	5.92493	+	10.2623i	0.195445	+	0.338521i	0.947046	−	0.321097i	$-0.104051\pi$
$919$	5.92493	+	10.2623i	0.195445	+	0.338521i	−0.751601	+	0.659618i	$0.770718\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			14.7378i			0.485101i
$924$		0			0
$925$		−	20.9478i		−	0.688759i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	29.2652	+	50.6888i	0.960161	+	1.66305i	0.722090	+	0.691799i	$0.243182\pi$
$929$	29.2652	+	50.6888i	0.960161	+	1.66305i	0.238071	+	0.971248i	$0.423485\pi$
$930$		0			0
$931$	−30.0358	−	2.05281i	−0.984383	−	0.0672781i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.44868	−	2.56844i	−0.145487	−	0.0839971i
$936$		0			0
$937$		−	58.2795i		−	1.90391i	−0.306242	−	0.951954i	$-0.599072\pi$
$937$		−	58.2795i		−	1.90391i	0.306242	−	0.951954i	$-0.400928\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	47.2322	+	27.2695i	1.53973	+	0.888961i	0.998854	+	0.0478564i	$0.0152390\pi$
$941$	47.2322	+	27.2695i	1.53973	+	0.888961i	0.540872	+	0.841105i	$0.318094\pi$
$942$		0			0
$943$	−8.90372	+	5.14056i	−0.289945	+	0.167400i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.3559	+	52.5780i	0.986435	+	1.70856i	0.635379	+	0.772200i	$0.280844\pi$
$947$	30.3559	+	52.5780i	0.986435	+	1.70856i	0.351056	+	0.936355i	$0.385823\pi$
$948$		0			0
$949$	43.8344	+	25.3078i	1.42293	+	0.821527i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	15.3879i		−	0.498463i	−0.968444	−	0.249232i	$-0.919822\pi$
$953$		−	15.3879i		−	0.498463i	0.968444	−	0.249232i	$-0.0801781\pi$
$954$		0			0
$955$	−8.02305	+	13.8963i	−0.259620	+	0.449675i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.887692	+	26.0069i	−0.0286651	+	0.839808i
$960$		0			0
$961$	−11.6455	−	20.1706i	−0.375662	−	0.650666i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			4.88197i			0.157156i
$966$		0			0
$967$	−34.1030			−1.09668			−0.548339	−	0.836256i	$-0.684740\pi$
$967$	−34.1030			−1.09668			−0.548339	+	0.836256i	$0.684740\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	33.1725	−	19.1521i	1.06456	−	0.614621i	0.137866	−	0.990451i	$-0.455976\pi$
$971$	33.1725	−	19.1521i	1.06456	−	0.614621i	0.926689	+	0.375830i	$0.122642\pi$
$972$		0			0
$973$	21.9487	+	41.2013i	0.703644	+	1.32085i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.4340	−	11.2202i	−0.621749	−	0.358967i	0.155801	−	0.987788i	$-0.450204\pi$
$977$	−19.4340	−	11.2202i	−0.621749	−	0.358967i	−0.777550	+	0.628822i	$0.783538\pi$
$978$		0			0
$979$	7.58205			0.242323
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−14.0698	+	24.3695i	−0.448756	+	0.777268i	−0.998305	−	0.0581932i	$-0.981466\pi$
$983$	−14.0698	+	24.3695i	−0.448756	+	0.777268i	0.549550	+	0.835461i	$0.314799\pi$
$984$		0			0
$985$	−24.4602	+	14.1221i	−0.779367	+	0.449968i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−15.2979	−	26.4967i	−0.486443	−	0.842545i
$990$		0			0
$991$	−17.8187	+	30.8629i	−0.566029	+	0.980391i	0.430924	+	0.902388i	$0.358188\pi$
$991$	−17.8187	+	30.8629i	−0.566029	+	0.980391i	−0.996953	+	0.0780029i	$0.975146\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.19625			0.0696259
$996$		0			0
$997$	−12.1376	+	21.0230i	−0.384402	+	0.665804i	−0.991686	−	0.128681i	$-0.958926\pi$
$997$	−12.1376	+	21.0230i	−0.384402	+	0.665804i	0.607284	+	0.794485i	$0.292259\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.cp.b.17.19		56
3.2	odd	2	inner	2016.2.cp.b.17.9		56
4.3	odd	2		504.2.ch.b.269.25	yes	56
7.5	odd	6	inner	2016.2.cp.b.593.20		56
8.3	odd	2		504.2.ch.b.269.16	yes	56
8.5	even	2	inner	2016.2.cp.b.17.10		56
12.11	even	2		504.2.ch.b.269.4	✓	56
21.5	even	6	inner	2016.2.cp.b.593.10		56
24.5	odd	2	inner	2016.2.cp.b.17.20		56
24.11	even	2		504.2.ch.b.269.13	yes	56
28.19	even	6		504.2.ch.b.341.13	yes	56
56.5	odd	6	inner	2016.2.cp.b.593.9		56
56.19	even	6		504.2.ch.b.341.4	yes	56
84.47	odd	6		504.2.ch.b.341.16	yes	56
168.5	even	6	inner	2016.2.cp.b.593.19		56
168.131	odd	6		504.2.ch.b.341.25	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.ch.b.269.4	✓	56	12.11	even	2
504.2.ch.b.269.13	yes	56	24.11	even	2
504.2.ch.b.269.16	yes	56	8.3	odd	2
504.2.ch.b.269.25	yes	56	4.3	odd	2
504.2.ch.b.341.4	yes	56	56.19	even	6
504.2.ch.b.341.13	yes	56	28.19	even	6
504.2.ch.b.341.16	yes	56	84.47	odd	6
504.2.ch.b.341.25	yes	56	168.131	odd	6
2016.2.cp.b.17.9		56	3.2	odd	2	inner
2016.2.cp.b.17.10		56	8.5	even	2	inner
2016.2.cp.b.17.19		56	1.1	even	1	trivial
2016.2.cp.b.17.20		56	24.5	odd	2	inner
2016.2.cp.b.593.9		56	56.5	odd	6	inner
2016.2.cp.b.593.10		56	21.5	even	6	inner
2016.2.cp.b.593.19		56	168.5	even	6	inner
2016.2.cp.b.593.20		56	7.5	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(1.00441 - 0.579896i) q^{5} +(-1.24394 - 2.33508i) q^{7} +(-1.41560 + 2.45188i) q^{11} -3.11725 q^{13} +(-0.782206 + 1.35482i) q^{17} +(2.15042 + 3.72463i) q^{19} +(-4.05782 + 2.34278i) q^{23} +(-1.82744 + 3.16522i) q^{25} +4.08861 q^{29} +(-2.40452 - 1.38825i) q^{31} +(-2.60353 - 1.62402i) q^{35} +(-4.96358 + 2.86572i) q^{37} +2.19421 q^{41} +6.52977i q^{43} +(5.34609 + 9.25971i) q^{47} +(-3.90521 + 5.80942i) q^{49} +(5.61902 - 9.73242i) q^{53} +3.28359i q^{55} +(11.9042 + 6.87289i) q^{59} +(5.09458 + 8.82407i) q^{61} +(-3.13100 + 1.80768i) q^{65} +(5.01037 + 2.89274i) q^{67} -4.72781i q^{71} +(-14.0619 - 8.11863i) q^{73} +(7.48627 + 0.255528i) q^{77} +(-2.89324 - 5.01124i) q^{79} +5.93150i q^{83} +1.81439i q^{85} +(-1.33902 - 2.31925i) q^{89} +(3.87769 + 7.27904i) q^{91} +(4.31980 + 2.49404i) q^{95} -11.2024i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 20 q^{7} + 8 q^{25} + 36 q^{31} - 28 q^{49} + 72 q^{73} + 12 q^{79}+O(q^{100}) \) 56 * q + 20 * q^7 + 8 * q^25 + 36 * q^31 - 28 * q^49 + 72 * q^73 + 12 * q^79

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