LMFDB - Embedded newform 2016.2.cp.b.17.25

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			17.25
Character	$\chi$	$=$	2016.17
Dual form			2016.2.cp.b.593.25

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.16007 - 1.82447i) q^{5} +(1.64838 - 2.06951i) q^{7} +(0.200123 - 0.346624i) q^{11} -0.581419 q^{13} +(-2.27603 + 3.94220i) q^{17} +(2.43222 + 4.21272i) q^{19} +(6.21119 - 3.58603i) q^{23} +(4.15738 - 7.20080i) q^{25} +6.69007 q^{29} +(2.85749 + 1.64978i) q^{31} +(1.43324 - 9.54721i) q^{35} +(-1.93278 + 1.11589i) q^{37} +4.51491 q^{41} -4.09678i q^{43} +(2.86389 + 4.96040i) q^{47} +(-1.56571 - 6.82265i) q^{49} +(-6.74170 + 11.6770i) q^{53} -1.46048i q^{55} +(-7.72158 - 4.45806i) q^{59} +(-6.42386 - 11.1265i) q^{61} +(-1.83733 + 1.06078i) q^{65} +(-8.30513 - 4.79497i) q^{67} -6.42019i q^{71} +(-6.18408 - 3.57038i) q^{73} +(-0.387462 - 0.985523i) q^{77} +(-4.15037 - 7.18866i) q^{79} +10.3852i q^{83} +16.6102i q^{85} +(5.25986 + 9.11034i) q^{89} +(-0.958396 + 1.20325i) q^{91} +(15.3720 + 8.87501i) q^{95} -4.00635i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.16007	−	1.82447i	1.41323	−	0.815928i	0.417537	−	0.908660i	$-0.362893\pi$
$5$	3.16007	−	1.82447i	1.41323	−	0.815928i	0.995691	+	0.0927320i	$0.0295600\pi$
$6$		0			0
$7$	1.64838	−	2.06951i	0.623028	−	0.782200i
$7$	1.64838	−	2.06951i	0.623028	−	0.782200i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.200123	−	0.346624i	0.0603394	−	0.104511i	−0.834278	−	0.551344i	$-0.814115\pi$
$11$	0.200123	−	0.346624i	0.0603394	−	0.104511i	0.894617	+	0.446833i	$0.147448\pi$
$12$		0			0
$13$	−0.581419			−0.161256			−0.0806282	−	0.996744i	$-0.525693\pi$
$13$	−0.581419			−0.161256			−0.0806282	+	0.996744i	$0.525693\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.27603	+	3.94220i	−0.552019	+	0.956124i	0.446110	+	0.894978i	$0.352809\pi$
$17$	−2.27603	+	3.94220i	−0.552019	+	0.956124i	−0.998129	+	0.0611464i	$0.980524\pi$
$18$		0			0
$19$	2.43222	+	4.21272i	0.557989	+	0.966465i	0.997664	+	0.0683081i	$0.0217601\pi$
$19$	2.43222	+	4.21272i	0.557989	+	0.966465i	−0.439676	+	0.898157i	$0.644907\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.21119	−	3.58603i	1.29512	−	0.747739i	0.315565	−	0.948904i	$-0.397806\pi$
$23$	6.21119	−	3.58603i	1.29512	−	0.747739i	0.979557	+	0.201165i	$0.0644728\pi$
$24$		0			0
$25$	4.15738	−	7.20080i	0.831476	−	1.44016i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.69007			1.24232			0.621158	−	0.783686i	$-0.286663\pi$
$29$	6.69007			1.24232			0.621158	+	0.783686i	$0.286663\pi$
$30$		0			0
$31$	2.85749	+	1.64978i	0.513221	+	0.296308i	0.734157	−	0.678980i	$-0.237578\pi$
$31$	2.85749	+	1.64978i	0.513221	+	0.296308i	−0.220936	+	0.975288i	$0.570911\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.43324	−	9.54721i	0.242262	−	1.61377i
$36$		0			0
$37$	−1.93278	+	1.11589i	−0.317747	+	0.183451i	−0.650388	−	0.759602i	$-0.725394\pi$
$37$	−1.93278	+	1.11589i	−0.317747	+	0.183451i	0.332641	+	0.943054i	$0.392060\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.51491			0.705111			0.352556	−	0.935791i	$-0.385313\pi$
$41$	4.51491			0.705111			0.352556	+	0.935791i	$0.385313\pi$
$42$		0			0
$43$		−	4.09678i		−	0.624754i	−0.949958	−	0.312377i	$-0.898875\pi$
$43$		−	4.09678i		−	0.624754i	0.949958	−	0.312377i	$-0.101125\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.86389	+	4.96040i	0.417741	+	0.723549i	0.995712	−	0.0925088i	$-0.0294886\pi$
$47$	2.86389	+	4.96040i	0.417741	+	0.723549i	−0.577971	+	0.816057i	$0.696155\pi$
$48$		0			0
$49$	−1.56571	−	6.82265i	−0.223673	−	0.974664i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.74170	+	11.6770i	−0.926043	+	1.60395i	−0.136169	+	0.990686i	$0.543479\pi$
$53$	−6.74170	+	11.6770i	−0.926043	+	1.60395i	−0.789875	+	0.613268i	$0.789854\pi$
$54$		0			0
$55$		−	1.46048i		−	0.196931i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.72158	−	4.45806i	−1.00526	−	0.580390i	−0.0954627	−	0.995433i	$-0.530433\pi$
$59$	−7.72158	−	4.45806i	−1.00526	−	0.580390i	−0.909802	+	0.415043i	$0.863766\pi$
$60$		0			0
$61$	−6.42386	−	11.1265i	−0.822491	−	1.42460i	−0.903822	−	0.427910i	$-0.859250\pi$
$61$	−6.42386	−	11.1265i	−0.822491	−	1.42460i	0.0813302	−	0.996687i	$-0.474083\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.83733	+	1.06078i	−0.227892	+	0.131574i
$66$		0			0
$67$	−8.30513	−	4.79497i	−1.01463	−	0.585798i	−0.102088	−	0.994775i	$-0.532552\pi$
$67$	−8.30513	−	4.79497i	−1.01463	−	0.585798i	−0.912545	+	0.408977i	$0.865886\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.42019i		−	0.761937i	−0.924588	−	0.380968i	$-0.875591\pi$
$71$		−	6.42019i		−	0.761937i	0.924588	−	0.380968i	$-0.124409\pi$
$72$		0			0
$73$	−6.18408	−	3.57038i	−0.723791	−	0.417881i	0.0923551	−	0.995726i	$-0.470561\pi$
$73$	−6.18408	−	3.57038i	−0.723791	−	0.417881i	−0.816147	+	0.577845i	$0.803894\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.387462	−	0.985523i	−0.0441553	−	0.112311i
$78$		0			0
$79$	−4.15037	−	7.18866i	−0.466953	−	0.808787i	0.532334	−	0.846535i	$-0.321315\pi$
$79$	−4.15037	−	7.18866i	−0.466953	−	0.808787i	−0.999287	+	0.0377474i	$0.987982\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			10.3852i			1.13992i	0.821673	+	0.569959i	$0.193041\pi$
$83$			10.3852i			1.13992i	−0.821673	+	0.569959i	$0.806959\pi$
$84$		0			0
$85$			16.6102i			1.80163i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.25986	+	9.11034i	0.557544	+	0.965694i	0.997701	+	0.0677734i	$0.0215895\pi$
$89$	5.25986	+	9.11034i	0.557544	+	0.965694i	−0.440157	+	0.897921i	$0.645077\pi$
$90$		0			0
$91$	−0.958396	+	1.20325i	−0.100467	+	0.126135i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	15.3720	+	8.87501i	1.57713	+	0.910557i
$96$		0			0
$97$		−	4.00635i		−	0.406783i	−0.979097	−	0.203392i	$-0.934804\pi$
$97$		−	4.00635i		−	0.406783i	0.979097	−	0.203392i	$-0.0651964\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.1877	−	7.03656i	−1.21272	−	0.700164i	−0.249368	−	0.968409i	$-0.580223\pi$
$101$	−12.1877	−	7.03656i	−1.21272	−	0.700164i	−0.963351	+	0.268245i	$0.913556\pi$
$102$		0			0
$103$	17.4850	−	10.0950i	1.72285	−	0.994686i	0.809943	−	0.586508i	$-0.199498\pi$
$103$	17.4850	−	10.0950i	1.72285	−	0.994686i	0.912903	−	0.408177i	$-0.133835\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.09913	+	3.63581i	0.202931	+	0.351487i	0.949472	−	0.313853i	$-0.101620\pi$
$107$	2.09913	+	3.63581i	0.202931	+	0.351487i	−0.746541	+	0.665340i	$0.768287\pi$
$108$		0			0
$109$	−16.0796	−	9.28357i	−1.54015	−	0.889205i	−0.998829	−	0.0483874i	$-0.984592\pi$
$109$	−16.0796	−	9.28357i	−1.54015	−	0.889205i	−0.541319	−	0.840817i	$-0.682075\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	3.44494i		−	0.324073i	−0.986785	−	0.162037i	$-0.948194\pi$
$113$		−	3.44494i		−	0.324073i	0.986785	−	0.162037i	$-0.0518063\pi$
$114$		0			0
$115$	13.0852	−	22.6642i	1.22020	−	2.11345i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.40666	+	11.2085i	0.403958	+	1.02748i
$120$		0			0
$121$	5.41990	+	9.38754i	0.492718	+	0.853413i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.0954i		−	1.08184i
$126$		0			0
$127$	3.82179			0.339129			0.169565	−	0.985519i	$-0.445764\pi$
$127$	3.82179			0.339129			0.169565	+	0.985519i	$0.445764\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.160576	−	0.0927085i	0.0140296	−	0.00809998i	−0.492969	−	0.870047i	$-0.664088\pi$
$131$	0.160576	−	0.0927085i	0.0140296	−	0.00809998i	0.506998	+	0.861947i	$0.330755\pi$
$132$		0			0
$133$	12.7275	+	1.91066i	1.10361	+	0.165676i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.2672	+	7.65983i	1.13350	+	0.654424i	0.944812	−	0.327614i	$-0.106245\pi$
$137$	13.2672	+	7.65983i	1.13350	+	0.654424i	0.188684	+	0.982038i	$0.439578\pi$
$138$		0			0
$139$	−10.5472			−0.894602			−0.447301	−	0.894383i	$-0.647615\pi$
$139$	−10.5472			−0.894602			−0.447301	+	0.894383i	$0.647615\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.116355	+	0.201533i	−0.00973013	+	0.0168531i
$144$		0			0
$145$	21.1411	−	12.2058i	1.75567	−	1.01364i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.59117	+	7.95213i	0.376123	+	0.651464i	0.990494	−	0.137553i	$-0.0439237\pi$
$149$	4.59117	+	7.95213i	0.376123	+	0.651464i	−0.614371	+	0.789017i	$0.710590\pi$
$150$		0			0
$151$	−1.79999	+	3.11768i	−0.146481	+	0.253713i	−0.929925	−	0.367750i	$-0.880128\pi$
$151$	−1.79999	+	3.11768i	−0.146481	+	0.253713i	0.783443	+	0.621463i	$0.213461\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	12.0399			0.967065
$156$		0			0
$157$	−0.359090	+	0.621963i	−0.0286585	+	0.0496380i	−0.879999	−	0.474976i	$-0.842457\pi$
$157$	−0.359090	+	0.621963i	−0.0286585	+	0.0496380i	0.851340	+	0.524614i	$0.175790\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.81706	−	18.7652i	0.222015	−	1.47891i
$162$		0			0
$163$	−19.1125	+	11.0346i	−1.49700	+	0.864296i	−0.999994	−	0.00344810i	$-0.998902\pi$
$163$	−19.1125	+	11.0346i	−1.49700	+	0.864296i	−0.497011	+	0.867744i	$0.665569\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−13.4882			−1.04375			−0.521875	−	0.853022i	$-0.674767\pi$
$167$	−13.4882			−1.04375			−0.521875	+	0.853022i	$0.674767\pi$
$168$		0			0
$169$	−12.6620			−0.973996
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.0098	−	8.08856i	1.06514	−	0.614962i	0.138294	−	0.990391i	$-0.455838\pi$
$173$	14.0098	−	8.08856i	1.06514	−	0.614962i	0.926851	+	0.375430i	$0.122505\pi$
$174$		0			0
$175$	−8.04917	−	20.4733i	−0.608460	−	1.54764i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.872025	−	1.51039i	0.0651782	−	0.112892i	−0.831595	−	0.555383i	$-0.812572\pi$
$179$	0.872025	−	1.51039i	0.0651782	−	0.112892i	0.896773	+	0.442491i	$0.145905\pi$
$180$		0			0
$181$	9.44875			0.702320			0.351160	−	0.936315i	$-0.385787\pi$
$181$	9.44875			0.702320			0.351160	+	0.936315i	$0.385787\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.07182	+	7.05260i	−0.299366	+	0.518517i
$186$		0			0
$187$	0.910974	+	1.57785i	0.0666170	+	0.115384i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.4118	+	9.47537i	−1.18752	+	0.685614i	−0.957742	−	0.287630i	$-0.907133\pi$
$191$	−16.4118	+	9.47537i	−1.18752	+	0.685614i	−0.229776	+	0.973244i	$0.573799\pi$
$192$		0			0
$193$	2.66715	−	4.61965i	0.191986	−	0.332529i	−0.753922	−	0.656964i	$-0.771841\pi$
$193$	2.66715	−	4.61965i	0.191986	−	0.332529i	0.945908	+	0.324434i	$0.105174\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.80474			0.199829			0.0999147	−	0.994996i	$-0.468143\pi$
$197$	2.80474			0.199829			0.0999147	+	0.994996i	$0.468143\pi$
$198$		0			0
$199$	6.44198	+	3.71928i	0.456660	+	0.263653i	0.710639	−	0.703557i	$-0.248406\pi$
$199$	6.44198	+	3.71928i	0.456660	+	0.263653i	−0.253979	+	0.967210i	$0.581739\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	11.0278	−	13.8451i	0.773997	−	0.971739i
$204$		0			0
$205$	14.2675	−	8.23733i	0.996483	−	0.575320i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.94697			0.134675
$210$		0			0
$211$			24.6986i			1.70032i	0.526521	+	0.850162i	$0.323496\pi$
$211$			24.6986i			1.70032i	−0.526521	+	0.850162i	$0.676504\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−7.47446	−	12.9461i	−0.509754	−	0.882920i
$216$		0			0
$217$	8.12445	−	3.19415i	0.551523	−	0.216833i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.32333	−	2.29207i	0.0890166	−	0.154181i
$222$		0			0
$223$			7.98964i			0.535026i	0.963554	+	0.267513i	$0.0862019\pi$
$223$			7.98964i			0.535026i	−0.963554	+	0.267513i	$0.913798\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.04318	+	1.17963i	0.135610	+	0.0782947i	0.566270	−	0.824220i	$-0.308386\pi$
$227$	2.04318	+	1.17963i	0.135610	+	0.0782947i	−0.430660	+	0.902514i	$0.641719\pi$
$228$		0			0
$229$	5.32512	+	9.22338i	0.351894	+	0.609498i	0.986581	−	0.163271i	$-0.0522045\pi$
$229$	5.32512	+	9.22338i	0.351894	+	0.609498i	−0.634687	+	0.772769i	$0.718871\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.6645	−	7.31188i	0.829682	−	0.479017i	−0.0240618	−	0.999710i	$-0.507660\pi$
$233$	12.6645	−	7.31188i	0.829682	−	0.479017i	0.853744	+	0.520693i	$0.174327\pi$
$234$		0			0
$235$	18.1002	+	10.4502i	1.18073	+	0.681693i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	11.5819i		−	0.749168i	−0.927193	−	0.374584i	$-0.877786\pi$
$239$		−	11.5819i		−	0.749168i	0.927193	−	0.374584i	$-0.122214\pi$
$240$		0			0
$241$	1.36690	+	0.789180i	0.0880498	+	0.0508356i	0.543378	−	0.839488i	$-0.317145\pi$
$241$	1.36690	+	0.789180i	0.0880498	+	0.0508356i	−0.455329	+	0.890323i	$0.650478\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−17.3955	−	18.7035i	−1.11136	−	1.19492i
$246$		0			0
$247$	−1.41414	−	2.44935i	−0.0899793	−	0.155849i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			17.9769i			1.13469i	0.823480	+	0.567345i	$0.192029\pi$
$251$			17.9769i			1.13469i	−0.823480	+	0.567345i	$0.807971\pi$
$252$		0			0
$253$		−	2.87059i		−	0.180473i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.71744	+	6.43879i	0.231887	+	0.401640i	0.958363	−	0.285551i	$-0.0921766\pi$
$257$	3.71744	+	6.43879i	0.231887	+	0.401640i	−0.726476	+	0.687192i	$0.758843\pi$
$258$		0			0
$259$	−0.876604	+	5.83931i	−0.0544695	+	0.362837i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.28419	+	4.20553i	0.449162	+	0.259324i	0.707476	−	0.706737i	$-0.249834\pi$
$263$	7.28419	+	4.20553i	0.449162	+	0.259324i	−0.258314	+	0.966061i	$0.583167\pi$
$264$		0			0
$265$			49.2001i			3.02234i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.9259	+	6.88541i	0.727134	+	0.419811i	0.817373	−	0.576109i	$-0.195430\pi$
$269$	11.9259	+	6.88541i	0.727134	+	0.419811i	−0.0902390	+	0.995920i	$0.528763\pi$
$270$		0			0
$271$	−16.0976	+	9.29397i	−0.977861	+	0.564568i	−0.901624	−	0.432521i	$-0.857624\pi$
$271$	−16.0976	+	9.29397i	−0.977861	+	0.564568i	−0.0762374	+	0.997090i	$0.524291\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.66398	−	2.88209i	−0.100342	−	0.173797i
$276$		0			0
$277$	−15.1885	−	8.76910i	−0.912590	−	0.526884i	−0.0313266	−	0.999509i	$-0.509973\pi$
$277$	−15.1885	−	8.76910i	−0.912590	−	0.526884i	−0.881264	+	0.472625i	$0.843307\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	14.8388i		−	0.885208i	−0.896717	−	0.442604i	$-0.854055\pi$
$281$		−	14.8388i		−	0.885208i	0.896717	−	0.442604i	$-0.145945\pi$
$282$		0			0
$283$	−0.293271	+	0.507960i	−0.0174331	+	0.0301951i	−0.874610	−	0.484826i	$-0.838883\pi$
$283$	−0.293271	+	0.507960i	−0.0174331	+	0.0301951i	0.857177	+	0.515022i	$0.172216\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.44228	−	9.34364i	0.439304	−	0.551538i
$288$		0			0
$289$	−1.86064	−	3.22272i	−0.109449	−	0.189572i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			6.07405i			0.354850i	0.984134	+	0.177425i	$0.0567767\pi$
$293$			6.07405i			0.354850i	−0.984134	+	0.177425i	$0.943223\pi$
$294$		0			0
$295$	−32.5344			−1.89422
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.61130	+	2.08498i	−0.208847	+	0.120578i
$300$		0			0
$301$	−8.47832	−	6.75304i	−0.488682	−	0.389239i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−40.5998	−	23.4403i	−2.32474	−	1.34219i
$306$		0			0
$307$	19.0443			1.08691			0.543457	−	0.839437i	$-0.317115\pi$
$307$	19.0443			1.08691			0.543457	+	0.839437i	$0.317115\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.52301	+	7.83409i	−0.256476	+	0.444230i	−0.965295	−	0.261160i	$-0.915895\pi$
$311$	−4.52301	+	7.83409i	−0.256476	+	0.444230i	0.708819	+	0.705390i	$0.249228\pi$
$312$		0			0
$313$	−6.45654	+	3.72768i	−0.364945	+	0.210701i	−0.671248	−	0.741233i	$-0.734241\pi$
$313$	−6.45654	+	3.72768i	−0.364945	+	0.210701i	0.306303	+	0.951934i	$0.400908\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.40732	+	4.16961i	0.135209	+	0.234188i	0.925677	−	0.378314i	$-0.123496\pi$
$317$	2.40732	+	4.16961i	0.135209	+	0.234188i	−0.790468	+	0.612503i	$0.790163\pi$
$318$		0			0
$319$	1.33884	−	2.31894i	0.0749606	−	0.129836i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−22.1432			−1.23208
$324$		0			0
$325$	−2.41718	+	4.18668i	−0.134081	+	0.232235i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	14.9863	+	2.24977i	0.826224	+	0.124034i
$330$		0			0
$331$	−5.43500	+	3.13790i	−0.298735	+	0.172475i	−0.641874	−	0.766810i	$-0.721843\pi$
$331$	−5.43500	+	3.13790i	−0.298735	+	0.172475i	0.343140	+	0.939284i	$0.388510\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−34.9931			−1.91188
$336$		0			0
$337$	−20.1329			−1.09671			−0.548353	−	0.836247i	$-0.684745\pi$
$337$	−20.1329			−1.09671			−0.548353	+	0.836247i	$0.684745\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.14370	−	0.660317i	0.0619350	−	0.0357582i
$342$		0			0
$343$	−16.7004	−	8.00604i	−0.901737	−	0.432285i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.04269	+	15.6624i	−0.485437	+	0.840802i	−0.999860	−	0.0167350i	$-0.994673\pi$
$347$	−9.04269	+	15.6624i	−0.485437	+	0.840802i	0.514423	+	0.857537i	$0.328006\pi$
$348$		0			0
$349$	3.24549			0.173727			0.0868635	−	0.996220i	$-0.472316\pi$
$349$	3.24549			0.173727			0.0868635	+	0.996220i	$0.472316\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.6032	+	25.2934i	−0.777248	+	1.34623i	0.156274	+	0.987714i	$0.450052\pi$
$353$	−14.6032	+	25.2934i	−0.777248	+	1.34623i	−0.933522	+	0.358519i	$0.883282\pi$
$354$		0			0
$355$	−11.7135	−	20.2883i	−0.621685	−	1.07679i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.87600	−	5.12456i	0.468457	−	0.270464i	−0.247136	−	0.968981i	$-0.579490\pi$
$359$	8.87600	−	5.12456i	0.468457	−	0.270464i	0.715594	+	0.698517i	$0.246156\pi$
$360$		0			0
$361$	−2.33135	+	4.03802i	−0.122703	+	0.212528i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−26.0562			−1.36384
$366$		0			0
$367$	26.6034	+	15.3595i	1.38869	+	0.801758i	0.993167	−	0.116700i	$-0.0372317\pi$
$367$	26.6034	+	15.3595i	1.38869	+	0.801758i	0.395518	+	0.918458i	$0.370565\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13.0527	+	33.2000i	0.677662	+	1.72366i
$372$		0			0
$373$	17.5313	−	10.1217i	0.907737	−	0.524082i	0.0280346	−	0.999607i	$-0.491075\pi$
$373$	17.5313	−	10.1217i	0.907737	−	0.524082i	0.879702	+	0.475525i	$0.157742\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.88973			−0.200331
$378$		0			0
$379$		−	15.4020i		−	0.791146i	−0.918435	−	0.395573i	$-0.870546\pi$
$379$		−	15.4020i		−	0.791146i	0.918435	−	0.395573i	$-0.129454\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.96494	+	3.40338i	0.100404	+	0.173905i	0.911851	−	0.410521i	$-0.134653\pi$
$383$	1.96494	+	3.40338i	0.100404	+	0.173905i	−0.811447	+	0.584426i	$0.801320\pi$
$384$		0			0
$385$	−3.02246	−	2.40741i	−0.154039	−	0.122693i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.94490	+	12.0289i	−0.352121	+	0.609891i	−0.986621	−	0.163032i	$-0.947873\pi$
$389$	−6.94490	+	12.0289i	−0.352121	+	0.609891i	0.634500	+	0.772923i	$0.281206\pi$
$390$		0			0
$391$			32.6477i			1.65106i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−26.2310	−	15.1445i	−1.31982	−	0.762001i
$396$		0			0
$397$	4.40738	+	7.63380i	0.221200	+	0.383129i	0.955173	−	0.296050i	$-0.0956693\pi$
$397$	4.40738	+	7.63380i	0.221200	+	0.383129i	−0.733973	+	0.679179i	$0.762336\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.0887	+	7.55674i	−0.653616	+	0.377365i	−0.789840	−	0.613313i	$-0.789837\pi$
$401$	−13.0887	+	7.55674i	−0.653616	+	0.377365i	0.136224	+	0.990678i	$0.456503\pi$
$402$		0			0
$403$	−1.66140	−	0.959210i	−0.0827602	−	0.0477816i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.893263i			0.0442774i
$408$		0			0
$409$	−5.85435	−	3.38001i	−0.289479	−	0.167131i	0.348228	−	0.937410i	$-0.386784\pi$
$409$	−5.85435	−	3.38001i	−0.289479	−	0.167131i	−0.637707	+	0.770279i	$0.720117\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−21.9540	+	8.63131i	−1.08029	+	0.424719i
$414$		0			0
$415$	18.9474	+	32.8179i	0.930091	+	1.61097i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	11.1005i		−	0.542293i	−0.962538	−	0.271147i	$-0.912597\pi$
$419$		−	11.1005i		−	0.542293i	0.962538	−	0.271147i	$-0.0874029\pi$
$420$		0			0
$421$		−	6.72143i		−	0.327582i	−0.986495	−	0.163791i	$-0.947628\pi$
$421$		−	6.72143i		−	0.327582i	0.986495	−	0.163791i	$-0.0523723\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	18.9247	+	32.7785i	0.917981	+	1.58999i
$426$		0			0
$427$	−33.6152	−	5.04636i	−1.62675	−	0.244210i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.32418	+	0.764518i	0.0637837	+	0.0368255i	0.531553	−	0.847025i	$-0.321609\pi$
$431$	1.32418	+	0.764518i	0.0637837	+	0.0368255i	−0.467769	+	0.883851i	$0.654942\pi$
$432$		0			0
$433$			19.9225i			0.957416i	0.877974	+	0.478708i	$0.158895\pi$
$433$			19.9225i			0.957416i	−0.877974	+	0.478708i	$0.841105\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	30.2139	+	17.4440i	1.44533	+	0.834460i
$438$		0			0
$439$	14.9668	−	8.64107i	0.714325	−	0.412416i	−0.0983355	−	0.995153i	$-0.531352\pi$
$439$	14.9668	−	8.64107i	0.714325	−	0.412416i	0.812660	+	0.582738i	$0.198018\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.666057	−	1.15364i	−0.0316453	−	0.0548113i	0.849769	−	0.527155i	$-0.176741\pi$
$443$	−0.666057	−	1.15364i	−0.0316453	−	0.0548113i	−0.881414	+	0.472344i	$0.843408\pi$
$444$		0			0
$445$	33.2431	+	19.1929i	1.57587	+	0.909831i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	38.5884i		−	1.82110i	−0.413397	−	0.910551i	$-0.635658\pi$
$449$		−	38.5884i		−	1.82110i	0.413397	−	0.910551i	$-0.364342\pi$
$450$		0			0
$451$	0.903540	−	1.56498i	0.0425460	−	0.0736919i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.833312	+	5.55092i	−0.0390663	+	0.260231i
$456$		0			0
$457$	7.80133	+	13.5123i	0.364931	+	0.632079i	0.988765	−	0.149478i	$-0.0477593\pi$
$457$	7.80133	+	13.5123i	0.364931	+	0.632079i	−0.623834	+	0.781557i	$0.714426\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			17.6616i			0.822583i	0.911504	+	0.411291i	$0.134922\pi$
$461$			17.6616i			0.822583i	−0.911504	+	0.411291i	$0.865078\pi$
$462$		0			0
$463$	−15.1430			−0.703753			−0.351877	−	0.936046i	$-0.614456\pi$
$463$	−15.1430			−0.703753			−0.351877	+	0.936046i	$0.614456\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0392	+	6.95086i	−0.557110	+	0.321647i	−0.751985	−	0.659181i	$-0.770903\pi$
$467$	−12.0392	+	6.95086i	−0.557110	+	0.321647i	0.194875	+	0.980828i	$0.437570\pi$
$468$		0			0
$469$	−23.6132	+	9.28360i	−1.09036	+	0.428677i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.42004	−	0.819862i	−0.0652936	−	0.0376973i
$474$		0			0
$475$	40.4466			1.85582
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.2782	−	26.4626i	0.698077	−	1.20911i	−0.271055	−	0.962564i	$-0.587373\pi$
$479$	15.2782	−	26.4626i	0.698077	−	1.20911i	0.969132	−	0.246541i	$-0.0792941\pi$
$480$		0			0
$481$	1.12375	−	0.648799i	0.0512388	−	0.0295827i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.30946	−	12.6604i	−0.331906	−	0.574877i
$486$		0			0
$487$	−10.9031	+	18.8848i	−0.494068	+	0.855751i	−0.999977	−	0.00683637i	$-0.997824\pi$
$487$	−10.9031	+	18.8848i	−0.494068	+	0.855751i	0.505909	+	0.862587i	$0.331157\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.38847			0.152919			0.0764597	−	0.997073i	$-0.475638\pi$
$491$	3.38847			0.152919			0.0764597	+	0.997073i	$0.475638\pi$
$492$		0			0
$493$	−15.2268	+	26.3736i	−0.685781	+	1.18781i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−13.2866	−	10.5829i	−0.595987	−	0.474708i
$498$		0			0
$499$	35.9490	−	20.7552i	1.60930	−	0.929128i	0.619770	−	0.784784i	$-0.287226\pi$
$499$	35.9490	−	20.7552i	1.60930	−	0.929128i	0.989528	−	0.144344i	$-0.0461073\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−18.4870			−0.824295			−0.412147	−	0.911117i	$-0.635221\pi$
$503$	−18.4870			−0.824295			−0.412147	+	0.911117i	$0.635221\pi$
$504$		0			0
$505$	−51.3519			−2.28513
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.71745	+	3.30097i	−0.253422	+	0.146313i	−0.621330	−	0.783549i	$-0.713407\pi$
$509$	−5.71745	+	3.30097i	−0.253422	+	0.146313i	0.367908	+	0.929862i	$0.380074\pi$
$510$		0			0
$511$	−17.5826	+	6.91266i	−0.777809	+	0.305798i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	36.8359	−	63.8016i	1.62318	−	2.81144i
$516$		0			0
$517$	2.29252			0.100825
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.54303	−	6.13671i	0.155223	−	0.268854i	−0.777917	−	0.628367i	$-0.783724\pi$
$521$	3.54303	−	6.13671i	0.155223	−	0.268854i	0.933140	+	0.359513i	$0.117057\pi$
$522$		0			0
$523$	−12.3169	−	21.3336i	−0.538583	−	0.932852i	−0.998981	−	0.0451399i	$-0.985627\pi$
$523$	−12.3169	−	21.3336i	−0.538583	−	0.932852i	0.460398	−	0.887713i	$-0.347707\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.0075	+	7.50988i	−0.566615	+	0.327136i
$528$		0			0
$529$	14.2192	−	24.6284i	0.618227	−	1.07080i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.62505			−0.113704
$534$		0			0
$535$	13.2668	+	7.65962i	0.573576	+	0.331154i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.67823	−	0.822658i	−0.115359	−	0.0354344i
$540$		0			0
$541$	−20.3549	+	11.7519i	−0.875125	+	0.505254i	−0.869048	−	0.494728i	$-0.835268\pi$
$541$	−20.3549	+	11.7519i	−0.875125	+	0.505254i	−0.00607698	+	0.999982i	$0.501934\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−67.7504			−2.90211
$546$		0			0
$547$			16.9894i			0.726412i	0.931709	+	0.363206i	$0.118318\pi$
$547$			16.9894i			0.726412i	−0.931709	+	0.363206i	$0.881682\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	16.2717	+	28.1834i	0.693198	+	1.20065i
$552$		0			0
$553$	−21.7183	−	3.26039i	−0.923558	−	0.138646i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.69847	−	4.67389i	0.114338	−	0.198039i	−0.803177	−	0.595740i	$-0.796859\pi$
$557$	2.69847	−	4.67389i	0.114338	−	0.198039i	0.917515	+	0.397701i	$0.130192\pi$
$558$		0			0
$559$			2.38195i			0.100746i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	29.2646	+	16.8959i	1.23336	+	0.712078i	0.967728	−	0.251997i	$-0.0810874\pi$
$563$	29.2646	+	16.8959i	1.23336	+	0.712078i	0.265628	+	0.964076i	$0.414421\pi$
$564$		0			0
$565$	−6.28520	−	10.8863i	−0.264420	−	0.457989i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.2390	+	8.22089i	−0.596930	+	0.344638i	−0.767833	−	0.640650i	$-0.778665\pi$
$569$	−14.2390	+	8.22089i	−0.596930	+	0.344638i	0.170903	+	0.985288i	$0.445332\pi$
$570$		0			0
$571$	33.2026	+	19.1696i	1.38949	+	0.802221i	0.993257	−	0.115931i	$-0.0369850\pi$
$571$	33.2026	+	19.1696i	1.38949	+	0.802221i	0.396230	+	0.918151i	$0.370318\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	59.6340i		−	2.48691i
$576$		0			0
$577$	19.2194	+	11.0963i	0.800113	+	0.461946i	0.843511	−	0.537112i	$-0.180485\pi$
$577$	19.2194	+	11.0963i	0.800113	+	0.461946i	−0.0433975	+	0.999058i	$0.513818\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	21.4921	+	17.1186i	0.891644	+	0.710201i
$582$		0			0
$583$	2.69834	+	4.67366i	0.111754	+	0.193563i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			3.25736i			0.134445i	0.997738	+	0.0672227i	$0.0214138\pi$
$587$			3.25736i			0.134445i	−0.997738	+	0.0672227i	$0.978586\pi$
$588$		0			0
$589$			16.0504i			0.661347i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−19.4634	−	33.7116i	−0.799267	−	1.38437i	−0.920094	−	0.391697i	$-0.871888\pi$
$593$	−19.4634	−	33.7116i	−0.799267	−	1.38437i	0.120827	−	0.992674i	$-0.461445\pi$
$594$		0			0
$595$	34.3749	+	27.3799i	1.40923	+	1.12246i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.92743	−	5.15425i	−0.364765	−	0.210597i	0.306404	−	0.951902i	$-0.400874\pi$
$599$	−8.92743	−	5.15425i	−0.364765	−	0.210597i	−0.671169	+	0.741304i	$0.734207\pi$
$600$		0			0
$601$			16.1379i			0.658277i	0.944282	+	0.329139i	$0.106758\pi$
$601$			16.1379i			0.658277i	−0.944282	+	0.329139i	$0.893242\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	34.2546	+	19.7769i	1.39265	+	0.804045i
$606$		0			0
$607$	31.3010	−	18.0717i	1.27047	−	0.733506i	0.295393	−	0.955376i	$-0.404549\pi$
$607$	31.3010	−	18.0717i	1.27047	−	0.733506i	0.975076	+	0.221870i	$0.0712159\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.66512	−	2.88407i	−0.0673634	−	0.116677i
$612$		0			0
$613$	1.33431	+	0.770363i	0.0538922	+	0.0311147i	0.526704	−	0.850049i	$-0.323428\pi$
$613$	1.33431	+	0.770363i	0.0538922	+	0.0311147i	−0.472812	+	0.881163i	$0.656761\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			40.1412i			1.61602i	0.589165	+	0.808012i	$0.299457\pi$
$617$			40.1412i			1.61602i	−0.589165	+	0.808012i	$0.700543\pi$
$618$		0			0
$619$	−18.4948	+	32.0339i	−0.743368	+	1.28755i	0.207586	+	0.978217i	$0.433439\pi$
$619$	−18.4948	+	32.0339i	−0.743368	+	1.28755i	−0.950953	+	0.309334i	$0.899894\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	27.5241	+	4.13196i	1.10273	+	0.165543i
$624$		0			0
$625$	−1.28074	−	2.21830i	−0.0512295	−	0.0887321i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	10.1592i		−	0.405074i
$630$		0			0
$631$	−31.1044			−1.23825			−0.619123	−	0.785294i	$-0.712512\pi$
$631$	−31.1044			−1.23825			−0.619123	+	0.785294i	$0.712512\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12.0771	−	6.97274i	0.479267	−	0.276705i
$636$		0			0
$637$	0.910334	+	3.96681i	0.0360688	+	0.157171i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.44022	−	1.98621i	−0.135881	−	0.0784507i	0.430519	−	0.902582i	$-0.358331\pi$
$641$	−3.44022	−	1.98621i	−0.135881	−	0.0784507i	−0.566399	+	0.824131i	$0.691664\pi$
$642$		0			0
$643$	−47.5503			−1.87520			−0.937600	−	0.347717i	$-0.886957\pi$
$643$	−47.5503			−1.87520			−0.937600	+	0.347717i	$0.886957\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.9097	−	36.2166i	0.822044	−	1.42382i	−0.0821142	−	0.996623i	$-0.526167\pi$
$647$	20.9097	−	36.2166i	0.822044	−	1.42382i	0.904158	−	0.427199i	$-0.140499\pi$
$648$		0			0
$649$	−3.09054	+	1.78432i	−0.121314	+	0.0700408i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.6456	−	18.4388i	−0.416596	−	0.721565i	0.578999	−	0.815329i	$-0.303444\pi$
$653$	−10.6456	−	18.4388i	−0.416596	−	0.721565i	−0.995595	+	0.0937632i	$0.970110\pi$
$654$		0			0
$655$	0.338288	−	0.585932i	0.0132180	−	0.0228943i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−50.0936			−1.95137			−0.975685	−	0.219179i	$-0.929662\pi$
$659$	−50.0936			−1.95137			−0.975685	+	0.219179i	$0.929662\pi$
$660$		0			0
$661$	11.9756	−	20.7423i	0.465797	−	0.806784i	−0.533440	−	0.845838i	$-0.679101\pi$
$661$	11.9756	−	20.7423i	0.465797	−	0.806784i	0.999237	+	0.0390540i	$0.0124344\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	43.7057	−	17.1830i	1.69483	−	0.666330i
$666$		0			0
$667$	41.5533	−	23.9908i	1.60895	−	0.928927i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.14226			−0.198515
$672$		0			0
$673$	−2.60756			−0.100514			−0.0502570	−	0.998736i	$-0.516004\pi$
$673$	−2.60756			−0.100514			−0.0502570	+	0.998736i	$0.516004\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13.5947	−	7.84890i	0.522486	−	0.301658i	−0.215465	−	0.976512i	$-0.569127\pi$
$677$	13.5947	−	7.84890i	0.522486	−	0.301658i	0.737951	+	0.674854i	$0.235793\pi$
$678$		0			0
$679$	−8.29116	−	6.60397i	−0.318186	−	0.253437i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.7229	−	18.5726i	0.410300	−	0.710661i	−0.584622	−	0.811306i	$-0.698757\pi$
$683$	10.7229	−	18.5726i	0.410300	−	0.710661i	0.994922	+	0.100645i	$0.0320905\pi$
$684$		0			0
$685$	55.9005			2.13585
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.91975	−	6.78920i	0.149330	−	0.258648i
$690$		0			0
$691$	12.6075	+	21.8368i	0.479611	+	0.830710i	0.999727	−	0.0233857i	$-0.00744459\pi$
$691$	12.6075	+	21.8368i	0.479611	+	0.830710i	−0.520116	+	0.854096i	$0.674111\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−33.3300	+	19.2431i	−1.26428	+	0.729931i
$696$		0			0
$697$	−10.2761	+	17.7987i	−0.389235	+	0.674174i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−33.6820			−1.27215			−0.636076	−	0.771626i	$-0.719444\pi$
$701$	−33.6820			−1.27215			−0.636076	+	0.771626i	$0.719444\pi$
$702$		0			0
$703$	−9.40187	−	5.42818i	−0.354598	−	0.204727i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−34.6521	+	13.6236i	−1.30323	+	0.512367i
$708$		0			0
$709$	−33.1687	+	19.1499i	−1.24568	+	0.719191i	−0.970244	−	0.242130i	$-0.922154\pi$
$709$	−33.1687	+	19.1499i	−1.24568	+	0.719191i	−0.275431	+	0.961321i	$0.588821\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	23.6646			0.886245
$714$		0			0
$715$			0.849148i			0.0317563i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.40747	−	5.90191i	−0.127077	−	0.220104i	0.795466	−	0.605998i	$-0.207226\pi$
$719$	−3.40747	−	5.90191i	−0.127077	−	0.220104i	−0.922543	+	0.385894i	$0.873893\pi$
$720$		0			0
$721$	7.93024	−	52.8256i	0.295338	−	1.96733i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	27.8132	−	48.1738i	1.03296	−	1.78913i
$726$		0			0
$727$			44.3315i			1.64416i	0.569369	+	0.822082i	$0.307188\pi$
$727$			44.3315i			1.64416i	−0.569369	+	0.822082i	$0.692812\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.1504	+	9.32441i	0.597342	+	0.344876i
$732$		0			0
$733$	4.98689	+	8.63755i	0.184195	+	0.319035i	0.943305	−	0.331927i	$-0.107699\pi$
$733$	4.98689	+	8.63755i	0.184195	+	0.319035i	−0.759110	+	0.650962i	$0.774366\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.32410	+	1.91917i	−0.122445	+	0.0706935i
$738$		0			0
$739$	−37.1489	−	21.4480i	−1.36655	−	0.788976i	−0.376061	−	0.926595i	$-0.622722\pi$
$739$	−37.1489	−	21.4480i	−1.36655	−	0.788976i	−0.990485	+	0.137619i	$0.956055\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			11.2075i			0.411165i	0.978640	+	0.205582i	$0.0659088\pi$
$743$			11.2075i			0.411165i	−0.978640	+	0.205582i	$0.934091\pi$
$744$		0			0
$745$	29.0169	+	16.7529i	1.06310	+	0.613778i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.9845	+	1.64901i	0.401365	+	0.0602534i
$750$		0			0
$751$	−18.5560	−	32.1399i	−0.677117	−	1.17280i	−0.975845	−	0.218462i	$-0.929896\pi$
$751$	−18.5560	−	32.1399i	−0.677117	−	1.17280i	0.298729	−	0.954338i	$-0.403437\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			13.1361i			0.478073i
$756$		0			0
$757$		−	20.8257i		−	0.756924i	−0.925617	−	0.378462i	$-0.876453\pi$
$757$		−	20.8257i		−	0.756924i	0.925617	−	0.378462i	$-0.123547\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.26978	+	12.5916i	0.263529	+	0.456446i	0.967177	−	0.254103i	$-0.0817801\pi$
$761$	7.26978	+	12.5916i	0.263529	+	0.456446i	−0.703648	+	0.710549i	$0.748447\pi$
$762$		0			0
$763$	−45.7177	+	17.9740i	−1.65509	+	0.650704i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.48947	+	2.59200i	0.162105	+	0.0935916i
$768$		0			0
$769$			2.11604i			0.0763063i	0.999272	+	0.0381532i	$0.0121475\pi$
$769$			2.11604i			0.0763063i	−0.999272	+	0.0381532i	$0.987853\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	18.3194	+	10.5767i	0.658902	+	0.380417i	0.791858	−	0.610705i	$-0.209114\pi$
$773$	18.3194	+	10.5767i	0.658902	+	0.380417i	−0.132957	+	0.991122i	$0.542447\pi$
$774$		0			0
$775$	23.7594	−	13.7175i	0.853462	−	0.492747i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	10.9812	+	19.0201i	0.393444	+	0.681465i
$780$		0			0
$781$	−2.22539	−	1.28483i	−0.0796308	−	0.0459748i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			2.62060i			0.0935331i
$786$		0			0
$787$	15.1516	−	26.2433i	0.540096	−	0.935473i	−0.458802	−	0.888538i	$-0.651721\pi$
$787$	15.1516	−	26.2433i	0.540096	−	0.935473i	0.998898	−	0.0469348i	$-0.0149453\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−7.12933	−	5.67856i	−0.253490	−	0.201906i
$792$		0			0
$793$	3.73495	+	6.46913i	0.132632	+	0.229725i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			12.8964i			0.456812i	0.973566	+	0.228406i	$0.0733514\pi$
$797$			12.8964i			0.456812i	−0.973566	+	0.228406i	$0.926649\pi$
$798$		0			0
$799$	−26.0732			−0.922403
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.47516	+	1.42903i	−0.0873463	+	0.0504294i
$804$		0			0
$805$	−25.3345	−	64.4391i	−0.892922	−	2.27118i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	31.3922	+	18.1243i	1.10369	+	0.637216i	0.937188	−	0.348825i	$-0.113419\pi$
$809$	31.3922	+	18.1243i	1.10369	+	0.637216i	0.166502	+	0.986041i	$0.446753\pi$
$810$		0			0
$811$	−5.48133			−0.192475			−0.0962377	−	0.995358i	$-0.530681\pi$
$811$	−5.48133			−0.192475			−0.0962377	+	0.995358i	$0.530681\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−40.2646	+	69.7403i	−1.41041	+	2.44290i
$816$		0			0
$817$	17.2586	−	9.96427i	0.603803	−	0.348606i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.95824	−	10.3200i	−0.207944	−	0.360169i	0.743123	−	0.669155i	$-0.233344\pi$
$821$	−5.95824	−	10.3200i	−0.207944	−	0.360169i	−0.951067	+	0.308986i	$0.900010\pi$
$822$		0			0
$823$	20.9931	−	36.3612i	0.731775	−	1.26747i	−0.224349	−	0.974509i	$-0.572025\pi$
$823$	20.9931	−	36.3612i	0.731775	−	1.26747i	0.956124	−	0.292963i	$-0.0946412\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.7749			−0.652869			−0.326434	−	0.945220i	$-0.605847\pi$
$827$	−18.7749			−0.652869			−0.326434	+	0.945220i	$0.605847\pi$
$828$		0			0
$829$	7.45588	−	12.9140i	0.258953	−	0.448520i	−0.707009	−	0.707205i	$-0.749956\pi$
$829$	7.45588	−	12.9140i	0.258953	−	0.448520i	0.965962	+	0.258685i	$0.0832891\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	30.4599	+	9.35621i	1.05537	+	0.324173i
$834$		0			0
$835$	−42.6238	+	24.6089i	−1.47506	+	0.851625i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	28.4322			0.981587			0.490794	−	0.871276i	$-0.336707\pi$
$839$	28.4322			0.981587			0.490794	+	0.871276i	$0.336707\pi$
$840$		0			0
$841$	15.7571			0.543347
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−40.0127	+	23.1014i	−1.37648	+	0.794711i
$846$		0			0
$847$	28.3616	+	4.25768i	0.974517	+	0.146296i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.00324	+	13.8620i	−0.274347	+	0.475184i
$852$		0			0
$853$	9.45822			0.323843			0.161922	−	0.986804i	$-0.448231\pi$
$853$	9.45822			0.323843			0.161922	+	0.986804i	$0.448231\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.8068	+	34.3064i	−0.676588	+	1.17188i	0.299415	+	0.954123i	$0.403209\pi$
$857$	−19.8068	+	34.3064i	−0.676588	+	1.17188i	−0.976002	+	0.217761i	$0.930125\pi$
$858$		0			0
$859$	−17.1446	−	29.6953i	−0.584965	−	1.01319i	−0.994880	−	0.101066i	$-0.967775\pi$
$859$	−17.1446	−	29.6953i	−0.584965	−	1.01319i	0.409914	−	0.912124i	$-0.365559\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	30.9385	−	17.8623i	1.05316	−	0.608041i	0.129627	−	0.991563i	$-0.458622\pi$
$863$	30.9385	−	17.8623i	1.05316	−	0.608041i	0.923532	+	0.383522i	$0.125289\pi$
$864$		0			0
$865$	29.5147	−	51.1209i	1.00353	−	1.73816i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.32235			−0.112703
$870$		0			0
$871$	4.82875	+	2.78788i	0.163616	+	0.0944638i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25.0315	−	19.9377i	−0.846218	−	0.674018i
$876$		0			0
$877$	13.3688	−	7.71847i	0.451431	−	0.260634i	−0.257003	−	0.966411i	$-0.582735\pi$
$877$	13.3688	−	7.71847i	0.451431	−	0.260634i	0.708435	+	0.705777i	$0.249402\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.47237			−0.0832962			−0.0416481	−	0.999132i	$-0.513261\pi$
$881$	−2.47237			−0.0832962			−0.0416481	+	0.999132i	$0.513261\pi$
$882$		0			0
$883$		−	22.3515i		−	0.752187i	−0.926582	−	0.376094i	$-0.877267\pi$
$883$		−	22.3515i		−	0.752187i	0.926582	−	0.376094i	$-0.122733\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.2912	+	19.5570i	0.379122	+	0.656658i	0.990935	−	0.134344i	$-0.0428928\pi$
$887$	11.2912	+	19.5570i	0.379122	+	0.656658i	−0.611813	+	0.791003i	$0.709559\pi$
$888$		0			0
$889$	6.29975	−	7.90922i	0.211287	−	0.265267i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13.9312	+	24.1295i	−0.466189	+	0.807464i
$894$		0			0
$895$		−	6.36394i		−	0.212723i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	19.1168	+	11.0371i	0.637582	+	0.368108i
$900$		0			0
$901$	−30.6886	−	53.1543i	−1.02239	−	1.77083i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	29.8588	−	17.2390i	0.992539	−	0.573042i
$906$		0			0
$907$	8.75724	+	5.05599i	0.290779	+	0.167882i	0.638293	−	0.769793i	$-0.279641\pi$
$907$	8.75724	+	5.05599i	0.290779	+	0.167882i	−0.347514	+	0.937675i	$0.612974\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			28.5297i			0.945230i	0.881269	+	0.472615i	$0.156690\pi$
$911$			28.5297i			0.945230i	−0.881269	+	0.472615i	$0.843310\pi$
$912$		0			0
$913$	3.59974	+	2.07831i	0.119134	+	0.0687820i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.0728286	−	0.485131i	0.00240501	−	0.0160204i
$918$		0			0
$919$	−13.4674	−	23.3263i	−0.444249	−	0.769463i	0.553750	−	0.832683i	$-0.313196\pi$
$919$	−13.4674	−	23.3263i	−0.444249	−	0.769463i	−0.998000	+	0.0632203i	$0.979863\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			3.73282i			0.122867i
$924$		0			0
$925$			18.5567i			0.610142i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11.2786	+	19.5350i	0.370038	+	0.640924i	0.989571	−	0.144046i	$-0.0460114\pi$
$929$	11.2786	+	19.5350i	0.370038	+	0.640924i	−0.619533	+	0.784970i	$0.712678\pi$
$930$		0			0
$931$	24.9338	−	23.1901i	0.817171	−	0.760024i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.75749	+	3.32409i	0.188290	+	0.108709i
$936$		0			0
$937$			25.6419i			0.837684i	0.908059	+	0.418842i	$0.137564\pi$
$937$			25.6419i			0.837684i	−0.908059	+	0.418842i	$0.862436\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−49.1787	−	28.3933i	−1.60318	−	0.925596i	−0.990846	−	0.134996i	$-0.956898\pi$
$941$	−49.1787	−	28.3933i	−1.60318	−	0.925596i	−0.612333	−	0.790600i	$-0.709769\pi$
$942$		0			0
$943$	28.0430	−	16.1906i	0.913205	−	0.527239i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.32282	+	9.21939i	0.172968	+	0.299590i	0.939456	−	0.342669i	$-0.111331\pi$
$947$	5.32282	+	9.21939i	0.172968	+	0.299590i	−0.766488	+	0.642259i	$0.777998\pi$
$948$		0			0
$949$	3.59554	+	2.07588i	0.116716	+	0.0673861i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	31.4456i		−	1.01862i	−0.860582	−	0.509312i	$-0.829900\pi$
$953$		−	31.4456i		−	1.01862i	0.860582	−	0.509312i	$-0.170100\pi$
$954$		0			0
$955$	−34.5750	+	59.8857i	−1.11882	+	1.93786i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	37.7214	−	14.8303i	1.21809	−	0.478896i
$960$		0			0
$961$	−10.0565	−	17.4183i	−0.324403	−	0.561882i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	19.4646i		−	0.626587i
$966$		0			0
$967$	−2.45245			−0.0788655			−0.0394327	−	0.999222i	$-0.512555\pi$
$967$	−2.45245			−0.0788655			−0.0394327	+	0.999222i	$0.512555\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28.2257	−	16.2961i	0.905807	−	0.522968i	0.0267274	−	0.999643i	$-0.491491\pi$
$971$	28.2257	−	16.2961i	0.905807	−	0.522968i	0.879080	+	0.476675i	$0.158158\pi$
$972$		0			0
$973$	−17.3858	+	21.8275i	−0.557362	+	0.699758i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	8.83921	+	5.10332i	0.282791	+	0.163270i	0.634686	−	0.772770i	$-0.281129\pi$
$977$	8.83921	+	5.10332i	0.282791	+	0.163270i	−0.351895	+	0.936039i	$0.614463\pi$
$978$		0			0
$979$	4.21048			0.134568
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2.76849	+	4.79516i	−0.0883010	+	0.152942i	−0.906793	−	0.421576i	$-0.861477\pi$
$983$	−2.76849	+	4.79516i	−0.0883010	+	0.152942i	0.818492	+	0.574518i	$0.194810\pi$
$984$		0			0
$985$	8.86318	−	5.11716i	0.282405	−	0.163046i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14.6912	−	25.4459i	−0.467153	−	0.809132i
$990$		0			0
$991$	6.53951	−	11.3268i	0.207734	−	0.359806i	−0.743266	−	0.668996i	$-0.766724\pi$
$991$	6.53951	−	11.3268i	0.207734	−	0.359806i	0.951000	+	0.309189i	$0.100058\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	27.1429			0.860487
$996$		0			0
$997$	−3.01429	+	5.22091i	−0.0954636	+	0.165348i	−0.909802	−	0.415043i	$-0.863767\pi$
$997$	−3.01429	+	5.22091i	−0.0954636	+	0.165348i	0.814338	+	0.580390i	$0.197100\pi$
$998$		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.17.25		56
3.2	odd	2	inner	2016.2.cp.b.17.3		56
4.3	odd	2		504.2.ch.b.269.26	yes	56
7.5	odd	6	inner	2016.2.cp.b.593.26		56
8.3	odd	2		504.2.ch.b.269.21	yes	56
8.5	even	2	inner	2016.2.cp.b.17.4		56
12.11	even	2		504.2.ch.b.269.3	✓	56
21.5	even	6	inner	2016.2.cp.b.593.4		56
24.5	odd	2	inner	2016.2.cp.b.17.26		56
24.11	even	2		504.2.ch.b.269.8	yes	56
28.19	even	6		504.2.ch.b.341.8	yes	56
56.5	odd	6	inner	2016.2.cp.b.593.3		56
56.19	even	6		504.2.ch.b.341.3	yes	56
84.47	odd	6		504.2.ch.b.341.21	yes	56
168.5	even	6	inner	2016.2.cp.b.593.25		56
168.131	odd	6		504.2.ch.b.341.26	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.ch.b.269.3	✓	56	12.11	even	2
504.2.ch.b.269.8	yes	56	24.11	even	2
504.2.ch.b.269.21	yes	56	8.3	odd	2
504.2.ch.b.269.26	yes	56	4.3	odd	2
504.2.ch.b.341.3	yes	56	56.19	even	6
504.2.ch.b.341.8	yes	56	28.19	even	6
504.2.ch.b.341.21	yes	56	84.47	odd	6
504.2.ch.b.341.26	yes	56	168.131	odd	6
2016.2.cp.b.17.3		56	3.2	odd	2	inner
2016.2.cp.b.17.4		56	8.5	even	2	inner
2016.2.cp.b.17.25		56	1.1	even	1	trivial
2016.2.cp.b.17.26		56	24.5	odd	2	inner
2016.2.cp.b.593.3		56	56.5	odd	6	inner
2016.2.cp.b.593.4		56	21.5	even	6	inner
2016.2.cp.b.593.25		56	168.5	even	6	inner
2016.2.cp.b.593.26		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)

\(f(q)\)	\(=\)	\(q+(3.16007 - 1.82447i) q^{5} +(1.64838 - 2.06951i) q^{7} +(0.200123 - 0.346624i) q^{11} -0.581419 q^{13} +(-2.27603 + 3.94220i) q^{17} +(2.43222 + 4.21272i) q^{19} +(6.21119 - 3.58603i) q^{23} +(4.15738 - 7.20080i) q^{25} +6.69007 q^{29} +(2.85749 + 1.64978i) q^{31} +(1.43324 - 9.54721i) q^{35} +(-1.93278 + 1.11589i) q^{37} +4.51491 q^{41} -4.09678i q^{43} +(2.86389 + 4.96040i) q^{47} +(-1.56571 - 6.82265i) q^{49} +(-6.74170 + 11.6770i) q^{53} -1.46048i q^{55} +(-7.72158 - 4.45806i) q^{59} +(-6.42386 - 11.1265i) q^{61} +(-1.83733 + 1.06078i) q^{65} +(-8.30513 - 4.79497i) q^{67} -6.42019i q^{71} +(-6.18408 - 3.57038i) q^{73} +(-0.387462 - 0.985523i) q^{77} +(-4.15037 - 7.18866i) q^{79} +10.3852i q^{83} +16.6102i q^{85} +(5.25986 + 9.11034i) q^{89} +(-0.958396 + 1.20325i) q^{91} +(15.3720 + 8.87501i) q^{95} -4.00635i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 20 q^{7} + 8 q^{25} + 36 q^{31} - 28 q^{49} + 72 q^{73} + 12 q^{79}+O(q^{100}) \) 56 * q + 20 * q^7 + 8 * q^25 + 36 * q^31 - 28 * q^49 + 72 * q^73 + 12 * q^79

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