LMFDB - Embedded newform 2016.2.cp.b.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(17,2016)]

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([0, 3, 3, 1]))

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

//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: 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:	$56$
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			17.2
Character	$\chi$	$=$	2016.17
Dual form			2016.2.cp.b.593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.18706 + 1.84005i) q^{5} +(0.998380 + 2.45015i) q^{7} +O(q^{10})$	$q+(-3.18706 + 1.84005i) q^{5} +(0.998380 + 2.45015i) q^{7} +(0.568599 - 0.984843i) q^{11} +3.62005 q^{13} +(2.84947 - 4.93542i) q^{17} +(2.63386 + 4.56198i) q^{19} +(3.19122 - 1.84245i) q^{23} +(4.27158 - 7.39859i) q^{25} -1.82838 q^{29} +(5.52097 + 3.18753i) q^{31} +(-7.69030 - 5.97171i) q^{35} +(-8.63700 + 4.98657i) q^{37} -3.46900 q^{41} +7.00429i q^{43} +(3.98886 + 6.90891i) q^{47} +(-5.00648 + 4.89236i) q^{49} +(-2.84057 + 4.92001i) q^{53} +4.18501i q^{55} +(-0.813177 - 0.469488i) q^{59} +(-1.98482 - 3.43782i) q^{61} +(-11.5373 + 6.66109i) q^{65} +(-2.18293 - 1.26031i) q^{67} +14.0093i q^{71} +(6.72069 + 3.88019i) q^{73} +(2.98069 + 0.409906i) q^{77} +(-7.68289 - 13.3072i) q^{79} +2.89078i q^{83} +20.9727i q^{85} +(1.04281 + 1.80621i) q^{89} +(3.61419 + 8.86968i) q^{91} +(-16.7886 - 9.69289i) q^{95} -10.3900i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 20 q^{7}+O(q^{10}) $ 56 * q + 20 * q^7	$ 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

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−3.18706	+	1.84005i	−1.42530	+	0.822896i	−0.996745	−	0.0806207i	$-0.974310\pi$
$5$	−3.18706	+	1.84005i	−1.42530	+	0.822896i	−0.428553	+	0.903517i	$0.640976\pi$
$6$		0			0
$7$	0.998380	+	2.45015i	0.377352	+	0.926070i
$7$	0.998380	+	2.45015i	0.377352	+	0.926070i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.568599	−	0.984843i	0.171439	−	0.296941i	−0.767484	−	0.641068i	$-0.778492\pi$
$11$	0.568599	−	0.984843i	0.171439	−	0.296941i	0.938923	+	0.344127i	$0.111825\pi$
$12$		0			0
$13$	3.62005			1.00402			0.502011	−	0.864861i	$-0.332594\pi$
$13$	3.62005			1.00402			0.502011	+	0.864861i	$0.332594\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.84947	−	4.93542i	0.691097	−	1.19702i	−0.280381	−	0.959889i	$-0.590461\pi$
$17$	2.84947	−	4.93542i	0.691097	−	1.19702i	0.971479	−	0.237127i	$-0.0762057\pi$
$18$		0			0
$19$	2.63386	+	4.56198i	0.604250	+	1.04659i	0.992170	+	0.124898i	$0.0398603\pi$
$19$	2.63386	+	4.56198i	0.604250	+	1.04659i	−0.387920	+	0.921693i	$0.626806\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.19122	−	1.84245i	0.665416	−	0.384178i	−0.128922	−	0.991655i	$-0.541152\pi$
$23$	3.19122	−	1.84245i	0.665416	−	0.384178i	0.794337	+	0.607477i	$0.207818\pi$
$24$		0			0
$25$	4.27158	−	7.39859i	0.854316	−	1.47972i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.82838			−0.339522			−0.169761	−	0.985485i	$-0.554300\pi$
$29$	−1.82838			−0.339522			−0.169761	+	0.985485i	$0.554300\pi$
$30$		0			0
$31$	5.52097	+	3.18753i	0.991595	+	0.572498i	0.905751	−	0.423811i	$-0.139308\pi$
$31$	5.52097	+	3.18753i	0.991595	+	0.572498i	0.0858442	+	0.996309i	$0.472641\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.69030	−	5.97171i	−1.29990	−	1.00940i
$36$		0			0
$37$	−8.63700	+	4.98657i	−1.41991	+	0.819788i	−0.996291	−	0.0860510i	$-0.972575\pi$
$37$	−8.63700	+	4.98657i	−1.41991	+	0.819788i	−0.423623	+	0.905839i	$0.639242\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.46900			−0.541767			−0.270884	−	0.962612i	$-0.587316\pi$
$41$	−3.46900			−0.541767			−0.270884	+	0.962612i	$0.587316\pi$
$42$		0			0
$43$			7.00429i			1.06814i	0.845439	+	0.534072i	$0.179339\pi$
$43$			7.00429i			1.06814i	−0.845439	+	0.534072i	$0.820661\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.98886	+	6.90891i	0.581835	+	1.00777i	0.995262	+	0.0972305i	$0.0309984\pi$
$47$	3.98886	+	6.90891i	0.581835	+	1.00777i	−0.413427	+	0.910537i	$0.635668\pi$
$48$		0			0
$49$	−5.00648	+	4.89236i	−0.715211	+	0.698909i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.84057	+	4.92001i	−0.390182	+	0.675815i	−0.992473	−	0.122461i	$-0.960921\pi$
$53$	−2.84057	+	4.92001i	−0.390182	+	0.675815i	0.602291	+	0.798276i	$0.294255\pi$
$54$		0			0
$55$			4.18501i			0.564306i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.813177	−	0.469488i	−0.105867	−	0.0611221i	0.446132	−	0.894967i	$-0.352801\pi$
$59$	−0.813177	−	0.469488i	−0.105867	−	0.0611221i	−0.551998	+	0.833845i	$0.686135\pi$
$60$		0			0
$61$	−1.98482	−	3.43782i	−0.254131	−	0.440167i	0.710528	−	0.703669i	$-0.248456\pi$
$61$	−1.98482	−	3.43782i	−0.254131	−	0.440167i	−0.964659	+	0.263501i	$0.915123\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−11.5373	+	6.66109i	−1.43103	+	0.826206i
$66$		0			0
$67$	−2.18293	−	1.26031i	−0.266687	−	0.153972i	0.360694	−	0.932684i	$-0.382540\pi$
$67$	−2.18293	−	1.26031i	−0.266687	−	0.153972i	−0.627381	+	0.778712i	$0.715873\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			14.0093i			1.66260i	0.555821	+	0.831302i	$0.312404\pi$
$71$			14.0093i			1.66260i	−0.555821	+	0.831302i	$0.687596\pi$
$72$		0			0
$73$	6.72069	+	3.88019i	0.786597	+	0.454142i	0.838763	−	0.544496i	$-0.183279\pi$
$73$	6.72069	+	3.88019i	0.786597	+	0.454142i	−0.0521661	+	0.998638i	$0.516613\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.98069	+	0.409906i	0.339681	+	0.0467132i
$78$		0			0
$79$	−7.68289	−	13.3072i	−0.864393	−	1.49717i	−0.867649	−	0.497177i	$-0.834370\pi$
$79$	−7.68289	−	13.3072i	−0.864393	−	1.49717i	0.00325650	−	0.999995i	$-0.498963\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			2.89078i			0.317304i	0.987335	+	0.158652i	$0.0507148\pi$
$83$			2.89078i			0.317304i	−0.987335	+	0.158652i	$0.949285\pi$
$84$		0			0
$85$			20.9727i			2.27480i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.04281	+	1.80621i	0.110538	+	0.191458i	0.915987	−	0.401207i	$-0.131409\pi$
$89$	1.04281	+	1.80621i	0.110538	+	0.191458i	−0.805449	+	0.592665i	$0.798076\pi$
$90$		0			0
$91$	3.61419	+	8.86968i	0.378870	+	0.929795i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−16.7886	−	9.69289i	−1.72247	−	0.994469i
$96$		0			0
$97$		−	10.3900i		−	1.05494i	−0.849573	−	0.527471i	$-0.823140\pi$
$97$		−	10.3900i		−	1.05494i	0.849573	−	0.527471i	$-0.176860\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−17.0193	−	9.82609i	−1.69348	−	0.977732i	−0.951669	−	0.307126i	$-0.900633\pi$
$101$	−17.0193	−	9.82609i	−1.69348	−	0.977732i	−0.741813	−	0.670606i	$-0.766034\pi$
$102$		0			0
$103$	−7.80827	+	4.50811i	−0.769372	+	0.444197i	−0.832650	−	0.553799i	$-0.813178\pi$
$103$	−7.80827	+	4.50811i	−0.769372	+	0.444197i	0.0632787	+	0.997996i	$0.479844\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.22682	+	5.58902i	0.311949	+	0.540311i	0.978784	−	0.204894i	$-0.0656850\pi$
$107$	3.22682	+	5.58902i	0.311949	+	0.540311i	−0.666836	+	0.745205i	$0.732352\pi$
$108$		0			0
$109$	6.00494	+	3.46695i	0.575169	+	0.332074i	0.759211	−	0.650844i	$-0.225585\pi$
$109$	6.00494	+	3.46695i	0.575169	+	0.332074i	−0.184042	+	0.982918i	$0.558918\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.9073i			1.68458i	0.539028	+	0.842288i	$0.318792\pi$
$113$			17.9073i			1.68458i	−0.539028	+	0.842288i	$0.681208\pi$
$114$		0			0
$115$	−6.78042	+	11.7440i	−0.632277	+	1.09514i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.9374	+	2.05420i	1.36931	+	0.188308i
$120$		0			0
$121$	4.85339	+	8.40632i	0.441217	+	0.764211i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			13.0392i			1.16626i
$126$		0			0
$127$	1.16038			0.102967			0.0514837	−	0.998674i	$-0.483605\pi$
$127$	1.16038			0.102967			0.0514837	+	0.998674i	$0.483605\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.2555	−	5.92101i	0.896026	−	0.517321i	0.0201174	−	0.999798i	$-0.493596\pi$
$131$	10.2555	−	5.92101i	0.896026	−	0.517321i	0.875909	+	0.482477i	$0.160263\pi$
$132$		0			0
$133$	−8.54795	+	11.0080i	−0.741201	+	0.954511i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.36092	−	1.94043i	−0.287143	−	0.165782i	0.349510	−	0.936933i	$-0.386348\pi$
$137$	−3.36092	−	1.94043i	−0.287143	−	0.165782i	−0.636653	+	0.771151i	$0.719681\pi$
$138$		0			0
$139$	7.63508			0.647599			0.323799	−	0.946126i	$-0.395040\pi$
$139$	7.63508			0.647599			0.323799	+	0.946126i	$0.395040\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.05836	−	3.56518i	0.172129	−	0.298136i
$144$		0			0
$145$	5.82716	−	3.36432i	0.483920	−	0.279391i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.50369	−	6.06857i	−0.287034	−	0.497157i	0.686067	−	0.727539i	$-0.259336\pi$
$149$	−3.50369	−	6.06857i	−0.287034	−	0.497157i	−0.973100	+	0.230382i	$0.926002\pi$
$150$		0			0
$151$	9.11816	−	15.7931i	0.742025	−	1.28523i	−0.209547	−	0.977799i	$-0.567199\pi$
$151$	9.11816	−	15.7931i	0.742025	−	1.28523i	0.951572	−	0.307427i	$-0.0994679\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−23.4609			−1.88442
$156$		0			0
$157$	6.98079	−	12.0911i	0.557128	−	0.964974i	−0.440607	−	0.897700i	$-0.645237\pi$
$157$	6.98079	−	12.0911i	0.557128	−	0.964974i	0.997735	−	0.0672736i	$-0.0214300\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.70034	+	5.97951i	0.606872	+	0.471251i
$162$		0			0
$163$	8.75280	−	5.05343i	0.685572	−	0.395815i	−0.116379	−	0.993205i	$-0.537129\pi$
$163$	8.75280	−	5.05343i	0.685572	−	0.395815i	0.801951	+	0.597390i	$0.203795\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.21398			0.403470			0.201735	−	0.979440i	$-0.435342\pi$
$167$	5.21398			0.403470			0.201735	+	0.979440i	$0.435342\pi$
$168$		0			0
$169$	0.104796			0.00806124
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.22284	−	1.86071i	0.245028	−	0.141467i	−0.372457	−	0.928049i	$-0.621485\pi$
$173$	3.22284	−	1.86071i	0.245028	−	0.141467i	0.617486	+	0.786582i	$0.288151\pi$
$174$		0			0
$175$	22.3923	+	3.07941i	1.69270	+	0.232781i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.1792	+	22.8270i	−0.985057	+	1.70617i	−0.343377	+	0.939198i	$0.611571\pi$
$179$	−13.1792	+	22.8270i	−0.985057	+	1.70617i	−0.641681	+	0.766972i	$0.721763\pi$
$180$		0			0
$181$	4.91329			0.365202			0.182601	−	0.983187i	$-0.441548\pi$
$181$	4.91329			0.365202			0.182601	+	0.983187i	$0.441548\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	18.3511	−	31.7850i	1.34920	−	2.33688i
$186$		0			0
$187$	−3.24041	−	5.61255i	−0.236962	−	0.410431i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.29844	+	3.05906i	−0.383382	+	0.221345i	−0.679289	−	0.733871i	$-0.737712\pi$
$191$	−5.29844	+	3.05906i	−0.383382	+	0.221345i	0.295907	+	0.955217i	$0.404378\pi$
$192$		0			0
$193$	−9.25188	+	16.0247i	−0.665965	+	1.15349i	0.313058	+	0.949734i	$0.398647\pi$
$193$	−9.25188	+	16.0247i	−0.665965	+	1.15349i	−0.979023	+	0.203751i	$0.934687\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.99827			−0.213618			−0.106809	−	0.994280i	$-0.534063\pi$
$197$	−2.99827			−0.213618			−0.106809	+	0.994280i	$0.534063\pi$
$198$		0			0
$199$	1.27718	+	0.737381i	0.0905370	+	0.0522715i	0.544585	−	0.838706i	$-0.316687\pi$
$199$	1.27718	+	0.737381i	0.0905370	+	0.0522715i	−0.454048	+	0.890977i	$0.650021\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.82542	−	4.47981i	−0.128119	−	0.314421i
$204$		0			0
$205$	11.0559	−	6.38315i	0.772180	−	0.445818i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.99045			0.414368
$210$		0			0
$211$			17.8952i			1.23196i	0.787764	+	0.615978i	$0.211239\pi$
$211$			17.8952i			1.23196i	−0.787764	+	0.615978i	$0.788761\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.8882	−	22.3231i	−0.878971	−	1.52242i
$216$		0			0
$217$	−2.29791	+	16.7096i	−0.155992	+	1.13432i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.3152	−	17.8665i	0.693877	−	1.20183i
$222$		0			0
$223$		−	1.26160i		−	0.0844828i	−0.999107	−	0.0422414i	$-0.986550\pi$
$223$		−	1.26160i		−	0.0844828i	0.999107	−	0.0422414i	$-0.0134499\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.642193	+	0.370770i	0.0426238	+	0.0246089i	0.521160	−	0.853459i	$-0.325499\pi$
$227$	0.642193	+	0.370770i	0.0426238	+	0.0246089i	−0.478537	+	0.878068i	$0.658833\pi$
$228$		0			0
$229$	4.62136	+	8.00442i	0.305388	+	0.528947i	0.977348	−	0.211640i	$-0.0678806\pi$
$229$	4.62136	+	8.00442i	0.305388	+	0.528947i	−0.671960	+	0.740588i	$0.734547\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.423385	−	0.244441i	0.0277369	−	0.0160139i	−0.486067	−	0.873921i	$-0.661569\pi$
$233$	0.423385	−	0.244441i	0.0277369	−	0.0160139i	0.513804	+	0.857907i	$0.328236\pi$
$234$		0			0
$235$	−25.4255	−	14.6794i	−1.65858	−	0.957579i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1.36060i			0.0880098i	0.999031	+	0.0440049i	$0.0140117\pi$
$239$			1.36060i			0.0880098i	−0.999031	+	0.0440049i	$0.985988\pi$
$240$		0			0
$241$	0.282523	+	0.163115i	0.0181989	+	0.0105071i	0.509072	−	0.860724i	$-0.329989\pi$
$241$	0.282523	+	0.163115i	0.0181989	+	0.0105071i	−0.490873	+	0.871231i	$0.663322\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.95375	−	24.8044i	0.444259	−	1.58470i
$246$		0			0
$247$	9.53473	+	16.5146i	0.606680	+	1.05080i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	0.972392i		−	0.0613768i	−0.999529	−	0.0306884i	$-0.990230\pi$
$251$		−	0.972392i		−	0.0613768i	0.999529	−	0.0306884i	$-0.00976996\pi$
$252$		0			0
$253$		−	4.19047i		−	0.263453i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.08066	+	12.2641i	0.441680	+	0.765012i	0.997814	−	0.0660806i	$-0.0210494\pi$
$257$	7.08066	+	12.2641i	0.441680	+	0.765012i	−0.556135	+	0.831092i	$0.687716\pi$
$258$		0			0
$259$	−20.8409	−	16.1835i	−1.29499	−	1.00559i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−15.9754	−	9.22341i	−0.985086	−	0.568740i	−0.0812844	−	0.996691i	$-0.525902\pi$
$263$	−15.9754	−	9.22341i	−0.985086	−	0.568740i	−0.903802	+	0.427951i	$0.859236\pi$
$264$		0			0
$265$		−	20.9072i		−	1.28432i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.9445	+	9.78291i	1.03312	+	0.596475i	0.917878	−	0.396862i	$-0.129901\pi$
$269$	16.9445	+	9.78291i	1.03312	+	0.596475i	0.115246	+	0.993337i	$0.463234\pi$
$270$		0			0
$271$	−17.4255	+	10.0606i	−1.05853	+	0.611141i	−0.925024	−	0.379908i	$-0.875956\pi$
$271$	−17.4255	+	10.0606i	−1.05853	+	0.611141i	−0.133502	+	0.991049i	$0.542622\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.85763	−	8.41367i	−0.292926	−	0.507363i
$276$		0			0
$277$	−6.82065	−	3.93790i	−0.409813	−	0.236606i	0.280896	−	0.959738i	$-0.409368\pi$
$277$	−6.82065	−	3.93790i	−0.409813	−	0.236606i	−0.690709	+	0.723132i	$0.742702\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	19.6351i		−	1.17133i	−0.810552	−	0.585666i	$-0.800833\pi$
$281$		−	19.6351i		−	1.17133i	0.810552	−	0.585666i	$-0.199167\pi$
$282$		0			0
$283$	11.0872	−	19.2035i	0.659063	−	1.14153i	−0.321796	−	0.946809i	$-0.604286\pi$
$283$	11.0872	−	19.2035i	0.659063	−	1.14153i	0.980859	−	0.194721i	$-0.0623803\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.46338	−	8.49958i	−0.204437	−	0.501715i
$288$		0			0
$289$	−7.73892	−	13.4042i	−0.455231	−	0.788483i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			18.6531i			1.08973i	0.838525	+	0.544863i	$0.183418\pi$
$293$			18.6531i			1.08973i	−0.838525	+	0.544863i	$0.816582\pi$
$294$		0			0
$295$	3.45553			0.201189
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.5524	−	6.66978i	0.668092	−	0.385723i
$300$		0			0
$301$	−17.1616	+	6.99294i	−0.989176	+	0.403066i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.6515	+	7.30436i	0.724424	+	0.418246i
$306$		0			0
$307$	26.7926			1.52913			0.764566	−	0.644546i	$-0.222953\pi$
$307$	26.7926			1.52913			0.764566	+	0.644546i	$0.222953\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.8790	+	22.3071i	−0.730302	+	1.26492i	0.226452	+	0.974022i	$0.427287\pi$
$311$	−12.8790	+	22.3071i	−0.730302	+	1.26492i	−0.956754	+	0.290898i	$0.906046\pi$
$312$		0			0
$313$	11.2421	−	6.49065i	0.635443	−	0.366873i	−0.147414	−	0.989075i	$-0.547095\pi$
$313$	11.2421	−	6.49065i	0.635443	−	0.366873i	0.782857	+	0.622202i	$0.213762\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.5090	+	18.2021i	0.590243	+	1.02233i	0.994199	+	0.107553i	$0.0343015\pi$
$317$	10.5090	+	18.2021i	0.590243	+	1.02233i	−0.403956	+	0.914778i	$0.632365\pi$
$318$		0			0
$319$	−1.03962	+	1.80067i	−0.0582073	+	0.100818i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	30.0204			1.67038
$324$		0			0
$325$	15.4633	−	26.7833i	0.857752	−	1.48567i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.9455	+	16.6710i	−0.713707	+	0.919103i
$330$		0			0
$331$	−11.6548	+	6.72892i	−0.640607	+	0.369855i	−0.784848	−	0.619688i	$-0.787259\pi$
$331$	−11.6548	+	6.72892i	−0.640607	+	0.369855i	0.144241	+	0.989543i	$0.453926\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	9.27617			0.506811
$336$		0			0
$337$	−24.1178			−1.31378			−0.656891	−	0.753985i	$-0.728129\pi$
$337$	−24.1178			−1.31378			−0.656891	+	0.753985i	$0.728129\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.27843	−	3.62486i	0.339996	−	0.196297i
$342$		0			0
$343$	−16.9854	−	7.38218i	−0.917125	−	0.398600i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.10821	−	3.65153i	0.113175	−	0.196025i	−0.803874	−	0.594800i	$-0.797231\pi$
$347$	2.10821	−	3.65153i	0.113175	−	0.196025i	0.917049	+	0.398775i	$0.130565\pi$
$348$		0			0
$349$	9.32452			0.499130			0.249565	−	0.968358i	$-0.419712\pi$
$349$	9.32452			0.499130			0.249565	+	0.968358i	$0.419712\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−10.5650	+	18.2992i	−0.562320	+	0.973967i	0.434974	+	0.900443i	$0.356758\pi$
$353$	−10.5650	+	18.2992i	−0.562320	+	0.973967i	−0.997293	+	0.0735234i	$0.976576\pi$
$354$		0			0
$355$	−25.7779	−	44.6487i	−1.36815	−	2.36970i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.51917	−	5.49590i	0.502403	−	0.290062i	−0.227303	−	0.973824i	$-0.572991\pi$
$359$	9.51917	−	5.49590i	0.502403	−	0.290062i	0.729705	+	0.683762i	$0.239657\pi$
$360$		0			0
$361$	−4.37446	+	7.57680i	−0.230235	+	0.398779i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−28.5590			−1.49485
$366$		0			0
$367$	−16.8547	−	9.73104i	−0.879806	−	0.507956i	−0.00921137	−	0.999958i	$-0.502932\pi$
$367$	−16.8547	−	9.73104i	−0.879806	−	0.507956i	−0.870594	+	0.492002i	$0.836265\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−14.8907	−	2.04778i	−0.773088	−	0.106316i
$372$		0			0
$373$	0.130569	−	0.0753842i	0.00676062	−	0.00390325i	−0.496616	−	0.867970i	$-0.665424\pi$
$373$	0.130569	−	0.0753842i	0.00676062	−	0.00390325i	0.503377	+	0.864067i	$0.332091\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.61884			−0.340888
$378$		0			0
$379$			32.9127i			1.69061i	0.534283	+	0.845306i	$0.320582\pi$
$379$			32.9127i			1.69061i	−0.534283	+	0.845306i	$0.679418\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.38834	+	14.5290i	0.428624	+	0.742398i	0.996751	−	0.0805423i	$-0.0256652\pi$
$383$	8.38834	+	14.5290i	0.428624	+	0.742398i	−0.568127	+	0.822941i	$0.692332\pi$
$384$		0			0
$385$	−10.2539	+	4.17823i	−0.522587	+	0.212942i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.6509	−	21.9121i	0.641428	−	1.11099i	−0.343686	−	0.939085i	$-0.611676\pi$
$389$	12.6509	−	21.9121i	0.641428	−	1.11099i	0.985114	−	0.171902i	$-0.0549912\pi$
$390$		0			0
$391$		−	21.0000i		−	1.06202i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	48.9717	+	28.2738i	2.46403	+	1.42261i
$396$		0			0
$397$	−17.7452	−	30.7356i	−0.890606	−	1.54257i	−0.839151	−	0.543899i	$-0.816947\pi$
$397$	−17.7452	−	30.7356i	−0.890606	−	1.54257i	−0.0514552	−	0.998675i	$-0.516386\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.98000	−	1.72050i	0.148814	−	0.0859179i	−0.423744	−	0.905782i	$-0.639284\pi$
$401$	2.98000	−	1.72050i	0.148814	−	0.0859179i	0.572558	+	0.819864i	$0.305951\pi$
$402$		0			0
$403$	19.9862	+	11.5390i	0.995584	+	0.574800i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			11.3414i			0.562175i
$408$		0			0
$409$	−5.30301	−	3.06169i	−0.262217	−	0.151391i	0.363129	−	0.931739i	$-0.381709\pi$
$409$	−5.30301	−	3.06169i	−0.262217	−	0.151391i	−0.625345	+	0.780348i	$0.715042\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.338457	−	2.46113i	0.0166544	−	0.121104i
$414$		0			0
$415$	−5.31918	−	9.21309i	−0.261108	−	0.452253i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			11.9546i			0.584022i	0.956415	+	0.292011i	$0.0943244\pi$
$419$			11.9546i			0.584022i	−0.956415	+	0.292011i	$0.905676\pi$
$420$		0			0
$421$		−	12.1616i		−	0.592720i	−0.955076	−	0.296360i	$-0.904227\pi$
$421$		−	12.1616i		−	0.592720i	0.955076	−	0.296360i	$-0.0957728\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.3434	−	42.1641i	−1.18083	−	2.04526i
$426$		0			0
$427$	6.44156	−	8.29536i	0.311729	−	0.401441i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	19.6396	+	11.3389i	0.946006	+	0.546177i	0.891838	−	0.452355i	$-0.149416\pi$
$431$	19.6396	+	11.3389i	0.946006	+	0.546177i	0.0541683	+	0.998532i	$0.482749\pi$
$432$		0			0
$433$			0.754762i			0.0362716i	0.999836	+	0.0181358i	$0.00577312\pi$
$433$			0.754762i			0.0362716i	−0.999836	+	0.0181358i	$0.994227\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.8105	+	9.70554i	0.804154	+	0.464279i
$438$		0			0
$439$	−16.2969	+	9.40902i	−0.777809	+	0.449068i	−0.835653	−	0.549258i	$-0.814911\pi$
$439$	−16.2969	+	9.40902i	−0.777809	+	0.449068i	0.0578444	+	0.998326i	$0.481577\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.46180	−	5.99600i	−0.164475	−	0.284879i	0.771994	−	0.635630i	$-0.219260\pi$
$443$	−3.46180	−	5.99600i	−0.164475	−	0.284879i	−0.936469	+	0.350751i	$0.885926\pi$
$444$		0			0
$445$	−6.64703	−	3.83766i	−0.315099	−	0.181923i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	0.270365i		−	0.0127593i	−0.999980	−	0.00637965i	$-0.997969\pi$
$449$		−	0.270365i		−	0.0127593i	0.999980	−	0.00637965i	$-0.00203072\pi$
$450$		0			0
$451$	−1.97247	+	3.41642i	−0.0928801	+	0.160873i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−27.8393	−	21.6179i	−1.30513	−	1.01346i
$456$		0			0
$457$	−1.91551	−	3.31776i	−0.0896037	−	0.155198i	0.817740	−	0.575588i	$-0.195227\pi$
$457$	−1.91551	−	3.31776i	−0.0896037	−	0.155198i	−0.907344	+	0.420390i	$0.861893\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			15.4863i			0.721267i	0.932708	+	0.360634i	$0.117439\pi$
$461$			15.4863i			0.721267i	−0.932708	+	0.360634i	$0.882561\pi$
$462$		0			0
$463$	6.51872			0.302950			0.151475	−	0.988461i	$-0.451598\pi$
$463$	6.51872			0.302950			0.151475	+	0.988461i	$0.451598\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	19.1086	−	11.0323i	0.884240	−	0.510516i	0.0121856	−	0.999926i	$-0.496121\pi$
$467$	19.1086	−	11.0323i	0.884240	−	0.510516i	0.872054	+	0.489410i	$0.162788\pi$
$468$		0			0
$469$	0.908568	−	6.60677i	0.0419538	−	0.305072i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.89812	+	3.98263i	0.317176	+	0.183122i
$474$		0			0
$475$	45.0030			2.06488
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.97672	+	10.3520i	−0.273083	+	0.472994i	−0.969650	−	0.244498i	$-0.921377\pi$
$479$	−5.97672	+	10.3520i	−0.273083	+	0.472994i	0.696567	+	0.717492i	$0.254710\pi$
$480$		0			0
$481$	−31.2664	+	18.0517i	−1.42563	+	0.823085i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	19.1181	+	33.1135i	0.868107	+	1.50361i
$486$		0			0
$487$	−1.13421	+	1.96451i	−0.0513959	+	0.0890203i	−0.890579	−	0.454829i	$-0.849700\pi$
$487$	−1.13421	+	1.96451i	−0.0513959	+	0.0890203i	0.839183	+	0.543849i	$0.183034\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.1155			−0.682153			−0.341077	−	0.940036i	$-0.610792\pi$
$491$	−15.1155			−0.682153			−0.341077	+	0.940036i	$0.610792\pi$
$492$		0			0
$493$	−5.20991	+	9.02383i	−0.234643	+	0.406413i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−34.3250	+	13.9867i	−1.53969	+	0.627387i
$498$		0			0
$499$	37.1204	−	21.4315i	1.66174	−	0.959404i	0.689851	−	0.723952i	$-0.257676\pi$
$499$	37.1204	−	21.4315i	1.66174	−	0.959404i	0.971886	−	0.235452i	$-0.0756571\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	40.5210			1.80674			0.903372	−	0.428858i	$-0.141084\pi$
$503$	40.5210			1.80674			0.903372	+	0.428858i	$0.141084\pi$
$504$		0			0
$505$	72.3220			3.21829
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.81718	−	5.66795i	0.435139	−	0.251227i	−0.266395	−	0.963864i	$-0.585832\pi$
$509$	9.81718	−	5.66795i	0.435139	−	0.251227i	0.701533	+	0.712637i	$0.252499\pi$
$510$		0			0
$511$	−2.79725	+	20.3406i	−0.123743	+	0.899815i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16.5903	−	28.7352i	0.731056	−	1.26623i
$516$		0			0
$517$	9.07225			0.398997
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4.86146	−	8.42029i	0.212984	−	0.368900i	−0.739663	−	0.672978i	$-0.765015\pi$
$521$	4.86146	−	8.42029i	0.212984	−	0.368900i	0.952647	+	0.304078i	$0.0983483\pi$
$522$		0			0
$523$	−9.28564	−	16.0832i	−0.406033	−	0.703269i	0.588408	−	0.808564i	$-0.299755\pi$
$523$	−9.28564	−	16.0832i	−0.406033	−	0.703269i	−0.994441	+	0.105295i	$0.966421\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	31.4636	−	18.1655i	1.37058	−	0.791303i
$528$		0			0
$529$	−4.71073	+	8.15923i	−0.204815	+	0.354749i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.5580			−0.543947
$534$		0			0
$535$	−20.5682	−	11.8750i	−0.889239	−	0.513402i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.97153	+	7.71238i	0.0849198	+	0.332196i
$540$		0			0
$541$	0.795518	−	0.459292i	0.0342020	−	0.0197465i	−0.482802	−	0.875730i	$-0.660381\pi$
$541$	0.795518	−	0.459292i	0.0342020	−	0.0197465i	0.517004	+	0.855983i	$0.327047\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−25.5175			−1.09305
$546$		0			0
$547$		−	28.4217i		−	1.21523i	−0.794233	−	0.607613i	$-0.792127\pi$
$547$		−	28.4217i		−	1.21523i	0.794233	−	0.607613i	$-0.207873\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.81570	−	8.34105i	−0.205156	−	0.355340i
$552$		0			0
$553$	24.9341	−	32.1098i	1.06031	−	1.36545i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12.7308	−	22.0504i	0.539420	−	0.934304i	−0.459515	−	0.888170i	$-0.651977\pi$
$557$	12.7308	−	22.0504i	0.539420	−	0.934304i	0.998935	−	0.0461335i	$-0.0146900\pi$
$558$		0			0
$559$			25.3559i			1.07244i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.6292	−	9.02355i	−0.658694	−	0.380297i	0.133085	−	0.991105i	$-0.457512\pi$
$563$	−15.6292	−	9.02355i	−0.658694	−	0.380297i	−0.791779	+	0.610807i	$0.790845\pi$
$564$		0			0
$565$	−32.9503	−	57.0716i	−1.38623	−	2.40102i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	26.9348	−	15.5508i	1.12917	−	0.651925i	0.185442	−	0.982655i	$-0.440628\pi$
$569$	26.9348	−	15.5508i	1.12917	−	0.651925i	0.943725	+	0.330730i	$0.107295\pi$
$570$		0			0
$571$	−38.0094	−	21.9447i	−1.59064	−	0.918359i	−0.993197	−	0.116450i	$-0.962848\pi$
$571$	−38.0094	−	21.9447i	−1.59064	−	0.918359i	−0.597447	−	0.801908i	$-0.703818\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	31.4807i		−	1.31284i
$576$		0			0
$577$	−25.7081	−	14.8426i	−1.07024	−	0.617906i	−0.141996	−	0.989867i	$-0.545352\pi$
$577$	−25.7081	−	14.8426i	−1.07024	−	0.617906i	−0.928248	+	0.371962i	$0.878685\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.08284	+	2.88610i	−0.293846	+	0.119735i
$582$		0			0
$583$	3.23029	+	5.59503i	0.133785	+	0.231722i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	39.8767i		−	1.64589i	−0.568122	−	0.822944i	$-0.692330\pi$
$587$		−	39.8767i		−	1.64589i	0.568122	−	0.822944i	$-0.307670\pi$
$588$		0			0
$589$			33.5821i			1.38373i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.90705	−	5.03515i	−0.119378	−	0.206769i	0.800143	−	0.599809i	$-0.204757\pi$
$593$	−2.90705	−	5.03515i	−0.119378	−	0.206769i	−0.919521	+	0.393040i	$0.871423\pi$
$594$		0			0
$595$	−51.3862	+	20.9387i	−2.10663	+	0.858402i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.85117	+	4.53288i	0.320790	+	0.185208i	0.651745	−	0.758438i	$-0.274037\pi$
$599$	7.85117	+	4.53288i	0.320790	+	0.185208i	−0.330955	+	0.943647i	$0.607371\pi$
$600$		0			0
$601$			24.2715i			0.990056i	0.868877	+	0.495028i	$0.164842\pi$
$601$			24.2715i			0.990056i	−0.868877	+	0.495028i	$0.835158\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−30.9361	−	17.8610i	−1.25773	−	0.726152i
$606$		0			0
$607$	21.9254	−	12.6586i	0.889923	−	0.513797i	0.0160059	−	0.999872i	$-0.494905\pi$
$607$	21.9254	−	12.6586i	0.889923	−	0.513797i	0.873917	+	0.486074i	$0.161572\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	14.4399	+	25.0106i	0.584175	+	1.01182i
$612$		0			0
$613$	−0.971418	−	0.560849i	−0.0392352	−	0.0226525i	0.480254	−	0.877129i	$-0.340544\pi$
$613$	−0.971418	−	0.560849i	−0.0392352	−	0.0226525i	−0.519489	+	0.854477i	$0.673878\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	24.9684i		−	1.00519i	−0.864522	−	0.502596i	$-0.832378\pi$
$617$		−	24.9684i		−	1.00519i	0.864522	−	0.502596i	$-0.167622\pi$
$618$		0			0
$619$	16.5995	−	28.7513i	0.667192	−	1.15561i	−0.311494	−	0.950248i	$-0.600829\pi$
$619$	16.5995	−	28.7513i	0.667192	−	1.15561i	0.978686	−	0.205362i	$-0.0658373\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.38435	+	4.35833i	−0.135591	+	0.174613i
$624$		0			0
$625$	−2.63487	−	4.56373i	−0.105395	−	0.182549i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			56.8363i			2.26621i
$630$		0			0
$631$	15.5394			0.618613			0.309307	−	0.950962i	$-0.399903\pi$
$631$	15.5394			0.618613			0.309307	+	0.950962i	$0.399903\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.69821	+	2.13516i	−0.146759	+	0.0847314i
$636$		0			0
$637$	−18.1237	+	17.7106i	−0.718088	+	0.701720i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	33.1136	+	19.1182i	1.30791	+	0.755122i	0.981747	−	0.190191i	$-0.0609108\pi$
$641$	33.1136	+	19.1182i	1.30791	+	0.755122i	0.326163	+	0.945314i	$0.394244\pi$
$642$		0			0
$643$	3.17969			0.125395			0.0626974	−	0.998033i	$-0.480030\pi$
$643$	3.17969			0.125395			0.0626974	+	0.998033i	$0.480030\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.13828	−	15.8280i	0.359263	−	0.622262i	−0.628575	−	0.777749i	$-0.716361\pi$
$647$	9.13828	−	15.8280i	0.359263	−	0.622262i	0.987838	+	0.155487i	$0.0496948\pi$
$648$		0			0
$649$	−0.924744	+	0.533901i	−0.0362994	+	0.0209574i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−22.2442	−	38.5282i	−0.870485	−	1.50772i	−0.861496	−	0.507764i	$-0.830472\pi$
$653$	−22.2442	−	38.5282i	−0.870485	−	1.50772i	−0.00898848	−	0.999960i	$-0.502861\pi$
$654$		0			0
$655$	−21.7899	+	37.7413i	−0.851403	+	1.47467i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−24.7165			−0.962818			−0.481409	−	0.876496i	$-0.659875\pi$
$659$	−24.7165			−0.962818			−0.481409	+	0.876496i	$0.659875\pi$
$660$		0			0
$661$	9.39973	−	16.2808i	0.365607	−	0.633250i	−0.623266	−	0.782010i	$-0.714195\pi$
$661$	9.39973	−	16.2808i	0.365607	−	0.633250i	0.988873	+	0.148760i	$0.0475281\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.98766	−	50.8117i	0.270970	−	1.97039i
$666$		0			0
$667$	−5.83477	+	3.36871i	−0.225923	+	0.130437i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.51428			−0.174272
$672$		0			0
$673$	22.7132			0.875531			0.437766	−	0.899089i	$-0.355770\pi$
$673$	22.7132			0.875531			0.437766	+	0.899089i	$0.355770\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.0283	−	17.9142i	1.19251	−	0.688498i	0.233638	−	0.972324i	$-0.424937\pi$
$677$	31.0283	−	17.9142i	1.19251	−	0.688498i	0.958876	+	0.283825i	$0.0916036\pi$
$678$		0			0
$679$	25.4570	−	10.3731i	0.976950	−	0.398084i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.91001	−	10.2364i	0.226140	−	0.391686i	−0.730521	−	0.682891i	$-0.760723\pi$
$683$	5.91001	−	10.2364i	0.226140	−	0.391686i	0.956661	+	0.291204i	$0.0940559\pi$
$684$		0			0
$685$	14.2820			0.545686
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10.2830	+	17.8107i	−0.391752	+	0.678534i
$690$		0			0
$691$	16.8555	+	29.1945i	0.641213	+	1.11061i	0.985162	+	0.171625i	$0.0549016\pi$
$691$	16.8555	+	29.1945i	0.641213	+	1.11061i	−0.343950	+	0.938988i	$0.611765\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−24.3335	+	14.0489i	−0.923021	+	0.532906i
$696$		0			0
$697$	−9.88481	+	17.1210i	−0.374414	+	0.648504i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.35428			−0.277767			−0.138884	−	0.990309i	$-0.544351\pi$
$701$	−7.35428			−0.277767			−0.138884	+	0.990309i	$0.544351\pi$
$702$		0			0
$703$	−45.4973	−	26.2679i	−1.71596	−	0.990713i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.08368	−	51.5100i	0.266409	−	1.93723i
$708$		0			0
$709$	32.1899	−	18.5848i	1.20892	−	0.697968i	0.246394	−	0.969170i	$-0.420754\pi$
$709$	32.1899	−	18.5848i	1.20892	−	0.697968i	0.962523	+	0.271202i	$0.0874210\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	23.4915			0.879764
$714$		0			0
$715$			15.1500i			0.566576i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.83186	+	17.0293i	0.366667	+	0.635085i	0.989042	−	0.147634i	$-0.0471657\pi$
$719$	9.83186	+	17.0293i	0.366667	+	0.635085i	−0.622376	+	0.782719i	$0.713832\pi$
$720$		0			0
$721$	−18.8412	−	14.6306i	−0.701682	−	0.544873i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.81007	+	13.5274i	−0.290059	+	0.502397i
$726$		0			0
$727$			24.1400i			0.895302i	0.894208	+	0.447651i	$0.147739\pi$
$727$			24.1400i			0.895302i	−0.894208	+	0.447651i	$0.852261\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	34.5691	+	19.9585i	1.27858	+	0.738191i
$732$		0			0
$733$	−0.741275	−	1.28393i	−0.0273796	−	0.0474229i	0.852011	−	0.523524i	$-0.175383\pi$
$733$	−0.741275	−	1.28393i	−0.0273796	−	0.0474229i	−0.879391	+	0.476101i	$0.842050\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.48242	+	1.43323i	−0.0914412	+	0.0527936i
$738$		0			0
$739$	20.3252	+	11.7348i	0.747676	+	0.431671i	0.824854	−	0.565346i	$-0.191257\pi$
$739$	20.3252	+	11.7348i	0.747676	+	0.431671i	−0.0771774	+	0.997017i	$0.524591\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	5.90858i		−	0.216765i	−0.994109	−	0.108382i	$-0.965433\pi$
$743$		−	5.90858i		−	0.216765i	0.994109	−	0.108382i	$-0.0345671\pi$
$744$		0			0
$745$	22.3330	+	12.8939i	0.818217	+	0.472398i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−10.4723	+	13.4862i	−0.382651	+	0.492774i
$750$		0			0
$751$	16.1457	+	27.9652i	0.589166	+	1.02047i	0.994342	+	0.106227i	$0.0338771\pi$
$751$	16.1457	+	27.9652i	0.589166	+	1.02047i	−0.405175	+	0.914239i	$0.632790\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			67.1115i			2.44244i
$756$		0			0
$757$			7.01154i			0.254839i	0.991849	+	0.127419i	$0.0406694\pi$
$757$			7.01154i			0.254839i	−0.991849	+	0.127419i	$0.959331\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.4606	−	30.2426i	−0.632945	−	1.09629i	−0.986946	−	0.161049i	$-0.948512\pi$
$761$	−17.4606	−	30.2426i	−0.632945	−	1.09629i	0.354001	−	0.935245i	$-0.384821\pi$
$762$		0			0
$763$	−2.49935	+	18.1743i	−0.0904824	+	0.657955i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.94375	−	1.69957i	−0.106292	−	0.0613680i
$768$		0			0
$769$		−	40.6649i		−	1.46642i	−0.680005	−	0.733208i	$-0.738022\pi$
$769$		−	40.6649i		−	1.46642i	0.680005	−	0.733208i	$-0.261978\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	35.3357	+	20.4011i	1.27094	+	0.733777i	0.975165	−	0.221481i	$-0.0710892\pi$
$773$	35.3357	+	20.4011i	1.27094	+	0.733777i	0.295774	+	0.955258i	$0.404423\pi$
$774$		0			0
$775$	47.1665	−	27.2316i	1.69427	−	0.978187i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.13688	−	15.8255i	−0.327363	−	0.567009i
$780$		0			0
$781$	13.7970	+	7.96570i	0.493696	+	0.285035i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			51.3801i			1.83383i
$786$		0			0
$787$	−15.9930	+	27.7007i	−0.570089	+	0.987423i	0.426467	+	0.904503i	$0.359758\pi$
$787$	−15.9930	+	27.7007i	−0.570089	+	0.987423i	−0.996556	+	0.0829199i	$0.973575\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−43.8755	+	17.8783i	−1.56003	+	0.635678i
$792$		0			0
$793$	−7.18517	−	12.4451i	−0.255153	−	0.441938i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	27.1417i		−	0.961410i	−0.876883	−	0.480705i	$-0.840381\pi$
$797$		−	27.1417i		−	0.961410i	0.876883	−	0.480705i	$-0.159619\pi$
$798$		0			0
$799$	45.4645			1.60842
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.64276	−	4.41255i	0.269707	−	0.155715i
$804$		0			0
$805$	−35.5441	−	4.88804i	−1.25276	−	0.172281i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10.8693	−	6.27540i	−0.382144	−	0.220631i	0.296606	−	0.955000i	$-0.404145\pi$
$809$	−10.8693	−	6.27540i	−0.382144	−	0.220631i	−0.678751	+	0.734369i	$0.737478\pi$
$810$		0			0
$811$	12.0857			0.424388			0.212194	−	0.977228i	$-0.431939\pi$
$811$	12.0857			0.424388			0.212194	+	0.977228i	$0.431939\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.5971	+	32.2112i	−0.651430	+	1.12831i
$816$		0			0
$817$	−31.9534	+	18.4483i	−1.11791	+	0.645425i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.2725	−	26.4527i	−0.533012	−	0.923204i	−0.999257	−	0.0385483i	$-0.987727\pi$
$821$	−15.2725	−	26.4527i	−0.533012	−	0.923204i	0.466245	−	0.884656i	$-0.345607\pi$
$822$		0			0
$823$	−10.4306	+	18.0664i	−0.363589	+	0.629754i	−0.988549	−	0.150903i	$-0.951782\pi$
$823$	−10.4306	+	18.0664i	−0.363589	+	0.629754i	0.624960	+	0.780657i	$0.285115\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−50.1969			−1.74552			−0.872759	−	0.488151i	$-0.837672\pi$
$827$	−50.1969			−1.74552			−0.872759	+	0.488151i	$0.837672\pi$
$828$		0			0
$829$	−9.59915	+	16.6262i	−0.333392	+	0.577453i	−0.983175	−	0.182668i	$-0.941527\pi$
$829$	−9.59915	+	16.6262i	−0.333392	+	0.577453i	0.649782	+	0.760120i	$0.274860\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.88008	+	38.6497i	0.342325	+	1.33913i
$834$		0			0
$835$	−16.6173	+	9.59400i	−0.575065	+	0.332014i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.77418			−0.337442			−0.168721	−	0.985664i	$-0.553964\pi$
$839$	−9.77418			−0.337442			−0.168721	+	0.985664i	$0.553964\pi$
$840$		0			0
$841$	−25.6570			−0.884725
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.333992	+	0.192830i	−0.0114897	+	0.00663357i
$846$		0			0
$847$	−15.7512	+	20.2842i	−0.541218	+	0.696975i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18.3751	+	31.8265i	−0.629889	+	1.09100i
$852$		0			0
$853$	52.3052			1.79090			0.895449	−	0.445165i	$-0.146855\pi$
$853$	52.3052			1.79090			0.895449	+	0.445165i	$0.146855\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.5757	+	33.9062i	−0.668695	+	1.15821i	0.309575	+	0.950875i	$0.399813\pi$
$857$	−19.5757	+	33.9062i	−0.668695	+	1.15821i	−0.978269	+	0.207338i	$0.933520\pi$
$858$		0			0
$859$	23.9471	+	41.4775i	0.817063	+	1.41520i	0.907837	+	0.419324i	$0.137733\pi$
$859$	23.9471	+	41.4775i	0.817063	+	1.41520i	−0.0907734	+	0.995872i	$0.528934\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−19.8433	+	11.4565i	−0.675473	+	0.389985i	−0.798147	−	0.602462i	$-0.794186\pi$
$863$	−19.8433	+	11.4565i	−0.675473	+	0.389985i	0.122674	+	0.992447i	$0.460853\pi$
$864$		0			0
$865$	−6.84760	+	11.8604i	−0.232825	+	0.403265i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−17.4739			−0.592763
$870$		0			0
$871$	−7.90232	−	4.56241i	−0.267760	−	0.154591i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−31.9480	+	13.0181i	−1.08004	+	0.440091i
$876$		0			0
$877$	−20.9012	+	12.0673i	−0.705783	+	0.407484i	−0.809498	−	0.587123i	$-0.800260\pi$
$877$	−20.9012	+	12.0673i	−0.705783	+	0.407484i	0.103715	+	0.994607i	$0.466927\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	49.8730			1.68026			0.840131	−	0.542383i	$-0.182478\pi$
$881$	49.8730			1.68026			0.840131	+	0.542383i	$0.182478\pi$
$882$		0			0
$883$			10.1798i			0.342578i	0.985221	+	0.171289i	$0.0547932\pi$
$883$			10.1798i			0.342578i	−0.985221	+	0.171289i	$0.945207\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.2756	+	17.7979i	0.345022	+	0.597596i	0.985358	−	0.170499i	$-0.0545381\pi$
$887$	10.2756	+	17.7979i	0.345022	+	0.597596i	−0.640336	+	0.768095i	$0.721205\pi$
$888$		0			0
$889$	1.15850	+	2.84311i	0.0388549	+	0.0953549i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−21.0122	+	36.3942i	−0.703147	+	1.21789i
$894$		0			0
$895$		−	97.0014i		−	3.24240i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−10.0944	−	5.82802i	−0.336668	−	0.194375i
$900$		0			0
$901$	16.1882	+	28.0388i	0.539307	+	0.934108i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.6589	+	9.04070i	−0.520521	+	0.300523i
$906$		0			0
$907$	−32.1180	−	18.5433i	−1.06646	−	0.615721i	−0.139248	−	0.990258i	$-0.544468\pi$
$907$	−32.1180	−	18.5433i	−1.06646	−	0.615721i	−0.927212	+	0.374537i	$0.877802\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			9.53166i			0.315798i	0.987455	+	0.157899i	$0.0504720\pi$
$911$			9.53166i			0.315798i	−0.987455	+	0.157899i	$0.949528\pi$
$912$		0			0
$913$	2.84696	+	1.64369i	0.0942207	+	0.0543984i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.7462	+	19.2161i	0.817193	+	0.634571i
$918$		0			0
$919$	−13.0199	−	22.5511i	−0.429487	−	0.743894i	0.567340	−	0.823483i	$-0.307972\pi$
$919$	−13.0199	−	22.5511i	−0.429487	−	0.743894i	−0.996828	+	0.0795896i	$0.974639\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			50.7146i			1.66929i
$924$		0			0
$925$			85.2022i			2.80143i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16.7273	+	28.9725i	0.548804	+	0.950556i	0.998357	+	0.0573025i	$0.0182499\pi$
$929$	16.7273	+	28.9725i	0.548804	+	0.950556i	−0.449553	+	0.893254i	$0.648417\pi$
$930$		0			0
$931$	−35.5052	−	9.95365i	−1.16364	−	0.326218i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	20.6548	+	11.9250i	0.675483	+	0.389991i
$936$		0			0
$937$		−	18.9751i		−	0.619890i	−0.950754	−	0.309945i	$-0.899689\pi$
$937$		−	18.9751i		−	0.619890i	0.950754	−	0.309945i	$-0.100311\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.97034	+	2.29228i	0.129429	+	0.0747261i	0.563317	−	0.826241i	$-0.309525\pi$
$941$	3.97034	+	2.29228i	0.129429	+	0.0747261i	−0.433887	+	0.900967i	$0.642858\pi$
$942$		0			0
$943$	−11.0704	+	6.39148i	−0.360501	+	0.208135i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.53547	−	13.0518i	−0.244870	−	0.424127i	0.717225	−	0.696842i	$-0.245412\pi$
$947$	−7.53547	−	13.0518i	−0.244870	−	0.424127i	−0.962095	+	0.272714i	$0.912079\pi$
$948$		0			0
$949$	24.3293	+	14.0465i	0.789761	+	0.455969i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.61355i			0.0522679i	0.999658	+	0.0261339i	$0.00831964\pi$
$953$			1.61355i			0.0522679i	−0.999658	+	0.0261339i	$0.991680\pi$
$954$		0			0
$955$	11.2576	−	19.4988i	0.364289	−	0.630966i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.39887	−	10.1720i	0.0451718	−	0.328473i
$960$		0			0
$961$	4.82071	+	8.34972i	0.155507	+	0.269346i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	68.0957i		−	2.19208i
$966$		0			0
$967$	−50.2361			−1.61548			−0.807742	−	0.589536i	$-0.799311\pi$
$967$	−50.2361			−1.61548			−0.807742	+	0.589536i	$0.799311\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19.2190	+	11.0961i	−0.616767	+	0.356091i	−0.775609	−	0.631213i	$-0.782557\pi$
$971$	−19.2190	+	11.0961i	−0.616767	+	0.356091i	0.158842	+	0.987304i	$0.449224\pi$
$972$		0			0
$973$	7.62271	+	18.7071i	0.244373	+	0.599722i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.72372	−	0.995188i	−0.0551466	−	0.0318389i	0.472173	−	0.881506i	$-0.343470\pi$
$977$	−1.72372	−	0.995188i	−0.0551466	−	0.0318389i	−0.527320	+	0.849667i	$0.676803\pi$
$978$		0			0
$979$	2.37177			0.0758022
$980$		0			0
$981$		0			0
$982$		0			0
$983$	28.8788	−	50.0195i	0.921089	−	1.59537i	0.123356	−	0.992362i	$-0.460634\pi$
$983$	28.8788	−	50.0195i	0.921089	−	1.59537i	0.797733	−	0.603011i	$-0.206032\pi$
$984$		0			0
$985$	9.55568	−	5.51698i	0.304469	−	0.175785i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.9051	+	22.3522i	0.410357	+	0.710760i
$990$		0			0
$991$	28.6512	−	49.6252i	0.910134	−	1.57640i	0.0962601	−	0.995356i	$-0.469312\pi$
$991$	28.6512	−	49.6252i	0.910134	−	1.57640i	0.813874	−	0.581042i	$-0.197355\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.42727			−0.172056
$996$		0			0
$997$	−10.3607	+	17.9452i	−0.328125	+	0.568330i	−0.982140	−	0.188152i	$-0.939750\pi$
$997$	−10.3607	+	17.9452i	−0.328125	+	0.568330i	0.654015	+	0.756482i	$0.273083\pi$
$998$		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.17.2		56
3.2	odd	2	inner	2016.2.cp.b.17.28		56
4.3	odd	2		504.2.ch.b.269.17	yes	56
7.5	odd	6	inner	2016.2.cp.b.593.1		56
8.3	odd	2		504.2.ch.b.269.6	✓	56
8.5	even	2	inner	2016.2.cp.b.17.27		56
12.11	even	2		504.2.ch.b.269.12	yes	56
21.5	even	6	inner	2016.2.cp.b.593.27		56
24.5	odd	2	inner	2016.2.cp.b.17.1		56
24.11	even	2		504.2.ch.b.269.23	yes	56
28.19	even	6		504.2.ch.b.341.23	yes	56
56.5	odd	6	inner	2016.2.cp.b.593.28		56
56.19	even	6		504.2.ch.b.341.12	yes	56
84.47	odd	6		504.2.ch.b.341.6	yes	56
168.5	even	6	inner	2016.2.cp.b.593.2		56
168.131	odd	6		504.2.ch.b.341.17	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.ch.b.269.6	✓	56	8.3	odd	2
504.2.ch.b.269.12	yes	56	12.11	even	2
504.2.ch.b.269.17	yes	56	4.3	odd	2
504.2.ch.b.269.23	yes	56	24.11	even	2
504.2.ch.b.341.6	yes	56	84.47	odd	6
504.2.ch.b.341.12	yes	56	56.19	even	6
504.2.ch.b.341.17	yes	56	168.131	odd	6
504.2.ch.b.341.23	yes	56	28.19	even	6
2016.2.cp.b.17.1		56	24.5	odd	2	inner
2016.2.cp.b.17.2		56	1.1	even	1	trivial
2016.2.cp.b.17.27		56	8.5	even	2	inner
2016.2.cp.b.17.28		56	3.2	odd	2	inner
2016.2.cp.b.593.1		56	7.5	odd	6	inner
2016.2.cp.b.593.2		56	168.5	even	6	inner
2016.2.cp.b.593.27		56	21.5	even	6	inner
2016.2.cp.b.593.28		56	56.5	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+(-3.18706 + 1.84005i) q^{5} +(0.998380 + 2.45015i) q^{7} +O(q^{10})\)	\(q+(-3.18706 + 1.84005i) q^{5} +(0.998380 + 2.45015i) q^{7} +(0.568599 - 0.984843i) q^{11} +3.62005 q^{13} +(2.84947 - 4.93542i) q^{17} +(2.63386 + 4.56198i) q^{19} +(3.19122 - 1.84245i) q^{23} +(4.27158 - 7.39859i) q^{25} -1.82838 q^{29} +(5.52097 + 3.18753i) q^{31} +(-7.69030 - 5.97171i) q^{35} +(-8.63700 + 4.98657i) q^{37} -3.46900 q^{41} +7.00429i q^{43} +(3.98886 + 6.90891i) q^{47} +(-5.00648 + 4.89236i) q^{49} +(-2.84057 + 4.92001i) q^{53} +4.18501i q^{55} +(-0.813177 - 0.469488i) q^{59} +(-1.98482 - 3.43782i) q^{61} +(-11.5373 + 6.66109i) q^{65} +(-2.18293 - 1.26031i) q^{67} +14.0093i q^{71} +(6.72069 + 3.88019i) q^{73} +(2.98069 + 0.409906i) q^{77} +(-7.68289 - 13.3072i) q^{79} +2.89078i q^{83} +20.9727i q^{85} +(1.04281 + 1.80621i) q^{89} +(3.61419 + 8.86968i) q^{91} +(-16.7886 - 9.69289i) q^{95} -10.3900i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 20 q^{7}+O(q^{10}) \) 56 * q + 20 * q^7	\( 56 q + 20 q^{7} + 8 q^{25} + 36 q^{31} - 28 q^{49} + 72 q^{73} + 12 q^{79}+O(q^{100}) \) 56 * q + 20 * q^7 + 8 * q^25 + 36 * q^31 - 28 * q^49 + 72 * q^73 + 12 * q^79

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