LMFDB - Embedded newform 2016.2.s.h.289.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,2,Mod(289,2016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2016.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2016, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.s (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,5,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2016.289
Dual form			2016.2.s.h.865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(2.50000 - 4.33013i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +8.00000 q^{29} +(-1.50000 - 2.59808i) q^{31} +(4.50000 - 7.79423i) q^{37} -2.00000 q^{41} -1.00000 q^{43} +(-4.00000 + 6.92820i) q^{47} +(5.50000 - 4.33013i) q^{49} +(3.00000 + 5.19615i) q^{53} +(-3.00000 - 5.19615i) q^{59} +(1.00000 - 1.73205i) q^{61} +(-2.50000 - 4.33013i) q^{67} +4.00000 q^{71} +(5.50000 + 9.52628i) q^{73} +(4.00000 + 3.46410i) q^{77} +(-2.50000 + 4.33013i) q^{79} +(6.00000 - 10.3923i) q^{89} +(2.50000 - 0.866025i) q^{91} +18.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{7} + 2 q^{11} + 2 q^{13} - 2 q^{17} + 5 q^{19} - 6 q^{23} + 5 q^{25} + 16 q^{29} - 3 q^{31} + 9 q^{37} - 4 q^{41} - 2 q^{43} - 8 q^{47} + 11 q^{49} + 6 q^{53} - 6 q^{59} + 2 q^{61} - 5 q^{67}+ \cdots + 36 q^{97}+O(q^{100}) $ 2 * q + 5 * q^7 + 2 * q^11 + 2 * q^13 - 2 * q^17 + 5 * q^19 - 6 * q^23 + 5 * q^25 + 16 * q^29 - 3 * q^31 + 9 * q^37 - 4 * q^41 - 2 * q^43 - 8 * q^47 + 11 * q^49 + 6 * q^53 - 6 * q^59 + 2 * q^61 - 5 * q^67 + 8 * q^71 + 11 * q^73 + 8 * q^77 - 5 * q^79 + 12 * q^89 + 5 * q^91 + 36 * q^97

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	2.50000	−	0.866025i	0.944911	−	0.327327i
$7$	2.50000	−	0.866025i	0.944911	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		0			0
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	0.718900	−	0.695113i	$-0.244646\pi$
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	−0.961436	+	0.275029i	$0.911312\pi$
$18$		0			0
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	−0.422659	−	0.906289i	$-0.638903\pi$
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	0.996199	−	0.0871106i	$-0.0277634\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	8.00000			1.48556			0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000			1.48556			0.742781	+	0.669534i	$0.233506\pi$
$30$		0			0
$31$	−1.50000	−	2.59808i	−0.269408	−	0.466628i	0.699301	−	0.714827i	$-0.253495\pi$
$31$	−1.50000	−	2.59808i	−0.269408	−	0.466628i	−0.968709	+	0.248199i	$0.920161\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.50000	−	7.79423i	0.739795	−	1.28136i	−0.212792	−	0.977098i	$-0.568256\pi$
$37$	4.50000	−	7.79423i	0.739795	−	1.28136i	0.952587	−	0.304266i	$-0.0984111\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	0.995117	−	0.0987002i	$-0.0314685\pi$
$53$	3.00000	+	5.19615i	0.412082	+	0.713746i	−0.583036	+	0.812447i	$0.698135\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	0.601958	−	0.798528i	$-0.294388\pi$
$59$	−3.00000	−	5.19615i	−0.390567	−	0.676481i	−0.992524	+	0.122047i	$0.961054\pi$
$60$		0			0
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	−0.794879	−	0.606768i	$-0.792466\pi$
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	0.922916	+	0.385002i	$0.125799\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	−2.50000	−	4.33013i	−0.305424	−	0.529009i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$	5.50000	+	9.52628i	0.643726	+	1.11497i	0.984594	+	0.174855i	$0.0559458\pi$
$73$	5.50000	+	9.52628i	0.643726	+	1.11497i	−0.340868	+	0.940111i	$0.610721\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000	+	3.46410i	0.455842	+	0.394771i
$78$		0			0
$79$	−2.50000	+	4.33013i	−0.281272	+	0.487177i	−0.971698	−	0.236225i	$-0.924090\pi$
$79$	−2.50000	+	4.33013i	−0.281272	+	0.487177i	0.690426	+	0.723403i	$0.257423\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000	−	10.3923i	0.635999	−	1.10158i	−0.350304	−	0.936636i	$-0.613922\pi$
$89$	6.00000	−	10.3923i	0.635999	−	1.10158i	0.986303	−	0.164946i	$-0.0527450\pi$
$90$		0			0
$91$	2.50000	−	0.866025i	0.262071	−	0.0907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	18.0000			1.82762			0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000			1.82762			0.913812	+	0.406138i	$0.133125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	0.677284	−	0.735721i	$-0.263157\pi$
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	−0.975796	+	0.218685i	$0.929823\pi$
$102$		0			0
$103$	−5.50000	+	9.52628i	−0.541931	+	0.938652i	0.456862	+	0.889538i	$0.348973\pi$
$103$	−5.50000	+	9.52628i	−0.541931	+	0.938652i	−0.998793	+	0.0491146i	$0.984360\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.00813215	−	0.999967i	$-0.497411\pi$
$107$	9.00000	−	15.5885i	0.870063	−	1.50699i	0.861931	−	0.507026i	$-0.169255\pi$
$108$		0			0
$109$	1.50000	+	2.59808i	0.143674	+	0.248851i	0.928877	−	0.370387i	$-0.120775\pi$
$109$	1.50000	+	2.59808i	0.143674	+	0.248851i	−0.785203	+	0.619238i	$0.787442\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.0000			1.12887			0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000			1.12887			0.564433	+	0.825479i	$0.309095\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.00000	−	3.46410i	−0.366679	−	0.317554i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−9.00000			−0.798621			−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000			−0.798621			−0.399310	+	0.916816i	$0.630750\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.00000	−	8.66025i	0.436852	−	0.756650i	−0.560593	−	0.828092i	$-0.689427\pi$
$131$	5.00000	−	8.66025i	0.436852	−	0.756650i	0.997445	+	0.0714417i	$0.0227600\pi$
$132$		0			0
$133$	2.50000	−	12.9904i	0.216777	−	1.12641i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	−0.999893	−	0.0146279i	$-0.995344\pi$
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	0.487278	−	0.873247i	$-0.337990\pi$
$138$		0			0
$139$	15.0000			1.27228			0.636142	−	0.771572i	$-0.280529\pi$
$139$	15.0000			1.27228			0.636142	+	0.771572i	$0.280529\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.00000	+	1.73205i	0.0836242	+	0.144841i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0000	+	17.3205i	−0.819232	+	1.41895i	0.0870170	+	0.996207i	$0.472267\pi$
$149$	−10.0000	+	17.3205i	−0.819232	+	1.41895i	−0.906249	+	0.422744i	$0.861067\pi$
$150$		0			0
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	0.982873	+	0.184284i	$0.0589965\pi$
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	−0.331842	+	0.943335i	$0.607670\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	11.0000	+	19.0526i	0.877896	+	1.52056i	0.853646	+	0.520854i	$0.174386\pi$
$157$	11.0000	+	19.0526i	0.877896	+	1.52056i	0.0242497	+	0.999706i	$0.492280\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.00000	+	15.5885i	−0.236433	+	1.22854i
$162$		0			0
$163$	−10.0000	+	17.3205i	−0.783260	+	1.35665i	0.146772	+	0.989170i	$0.453112\pi$
$163$	−10.0000	+	17.3205i	−0.783260	+	1.35665i	−0.930033	+	0.367477i	$0.880222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.00000	−	15.5885i	0.684257	−	1.18517i	−0.289412	−	0.957205i	$-0.593460\pi$
$173$	9.00000	−	15.5885i	0.684257	−	1.18517i	0.973670	−	0.227964i	$-0.0732068\pi$
$174$		0			0
$175$	10.0000	+	8.66025i	0.755929	+	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2.00000	−	3.46410i	−0.149487	−	0.258919i	0.781551	−	0.623841i	$-0.214429\pi$
$179$	−2.00000	−	3.46410i	−0.149487	−	0.258919i	−0.931038	+	0.364922i	$0.881096\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	2.00000	−	3.46410i	0.146254	−	0.253320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.0000	+	17.3205i	−0.723575	+	1.25327i	0.235983	+	0.971757i	$0.424169\pi$
$191$	−10.0000	+	17.3205i	−0.723575	+	1.25327i	−0.959558	+	0.281511i	$0.909164\pi$
$192$		0			0
$193$	−3.50000	−	6.06218i	−0.251936	−	0.436365i	0.712123	−	0.702055i	$-0.247734\pi$
$193$	−3.50000	−	6.06218i	−0.251936	−	0.436365i	−0.964059	+	0.265689i	$0.914400\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.0000			0.712470			0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000			0.712470			0.356235	+	0.934396i	$0.384060\pi$
$198$		0			0
$199$	−6.00000	−	10.3923i	−0.425329	−	0.736691i	0.571122	−	0.820865i	$-0.306508\pi$
$199$	−6.00000	−	10.3923i	−0.425329	−	0.736691i	−0.996451	+	0.0841740i	$0.973175\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	20.0000	−	6.92820i	1.40372	−	0.486265i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.0000			0.691714
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6.00000	−	5.19615i	−0.407307	−	0.352738i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.00000	−	1.73205i	−0.0672673	−	0.116510i
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.0000	−	19.0526i	−0.730096	−	1.26456i	−0.956842	−	0.290609i	$-0.906142\pi$
$227$	−11.0000	−	19.0526i	−0.730096	−	1.26456i	0.226746	−	0.973954i	$-0.427191\pi$
$228$		0			0
$229$	−0.500000	+	0.866025i	−0.0330409	+	0.0572286i	−0.882073	−	0.471113i	$-0.843853\pi$
$229$	−0.500000	+	0.866025i	−0.0330409	+	0.0572286i	0.849032	+	0.528341i	$0.177186\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.00000	−	3.46410i	0.131024	−	0.226941i	−0.793047	−	0.609160i	$-0.791507\pi$
$233$	2.00000	−	3.46410i	0.131024	−	0.226941i	0.924072	+	0.382219i	$0.124840\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−22.0000			−1.42306			−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000			−1.42306			−0.711531	+	0.702655i	$0.751998\pi$
$240$		0			0
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	0.658838	−	0.752285i	$-0.271048\pi$
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	−0.980917	+	0.194429i	$0.937715\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.50000	−	4.33013i	0.159071	−	0.275519i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.00000			0.378717			0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000			0.378717			0.189358	+	0.981908i	$0.439359\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.0000	+	27.7128i	−0.998053	+	1.72868i	−0.445005	+	0.895528i	$0.646798\pi$
$257$	−16.0000	+	27.7128i	−0.998053	+	1.72868i	−0.553047	+	0.833150i	$0.686535\pi$
$258$		0			0
$259$	4.50000	−	23.3827i	0.279616	−	1.45293i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	0.962594	−	0.270947i	$-0.0873367\pi$
$263$	4.00000	+	6.92820i	0.246651	+	0.427211i	−0.715944	+	0.698158i	$0.754003\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.0000	−	22.5167i	−0.792624	−	1.37287i	−0.924337	−	0.381577i	$-0.875381\pi$
$269$	−13.0000	−	22.5167i	−0.792624	−	1.37287i	0.131713	−	0.991288i	$-0.457952\pi$
$270$		0			0
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	−0.961563	−	0.274586i	$-0.911459\pi$
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	0.718580	+	0.695444i	$0.244792\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.00000	+	8.66025i	−0.301511	+	0.522233i
$276$		0			0
$277$	5.50000	+	9.52628i	0.330463	+	0.572379i	0.982603	−	0.185720i	$-0.0594618\pi$
$277$	5.50000	+	9.52628i	0.330463	+	0.572379i	−0.652140	+	0.758099i	$0.726128\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.0000			0.835170			0.417585	−	0.908638i	$-0.362877\pi$
$281$	14.0000			0.835170			0.417585	+	0.908638i	$0.362877\pi$
$282$		0			0
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.797283	−	0.603606i	$-0.793730\pi$
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.124096	−	0.992270i	$-0.539603\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.00000	+	1.73205i	−0.295141	+	0.102240i
$288$		0			0
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−28.0000			−1.63578			−0.817889	−	0.575376i	$-0.804856\pi$
$293$	−28.0000			−1.63578			−0.817889	+	0.575376i	$0.804856\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	−2.50000	+	0.866025i	−0.144098	+	0.0499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	25.0000			1.42683			0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000			1.42683			0.713413	+	0.700744i	$0.247149\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.00000	+	8.66025i	0.283524	+	0.491078i	0.972250	−	0.233944i	$-0.0751631\pi$
$311$	5.00000	+	8.66025i	0.283524	+	0.491078i	−0.688726	+	0.725022i	$0.741830\pi$
$312$		0			0
$313$	−15.5000	+	26.8468i	−0.876112	+	1.51747i	−0.0205381	+	0.999789i	$0.506538\pi$
$313$	−15.5000	+	26.8468i	−0.876112	+	1.51747i	−0.855574	+	0.517681i	$0.826795\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.00000	−	3.46410i	0.112331	−	0.194563i	−0.804379	−	0.594117i	$-0.797502\pi$
$317$	2.00000	−	3.46410i	0.112331	−	0.194563i	0.916710	+	0.399554i	$0.130835\pi$
$318$		0			0
$319$	8.00000	+	13.8564i	0.447914	+	0.775810i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.0000			−0.556415
$324$		0			0
$325$	2.50000	+	4.33013i	0.138675	+	0.240192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.00000	+	20.7846i	−0.220527	+	1.14589i
$330$		0			0
$331$	−8.50000	+	14.7224i	−0.467202	+	0.809218i	−0.999298	−	0.0374662i	$-0.988071\pi$
$331$	−8.50000	+	14.7224i	−0.467202	+	0.809218i	0.532096	+	0.846684i	$0.321405\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−31.0000			−1.68868			−0.844339	−	0.535810i	$-0.820006\pi$
$337$	−31.0000			−1.68868			−0.844339	+	0.535810i	$0.820006\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.00000	−	5.19615i	0.162459	−	0.281387i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.00000	+	15.5885i	0.483145	+	0.836832i	0.999813	−	0.0193540i	$-0.00616095\pi$
$347$	9.00000	+	15.5885i	0.483145	+	0.836832i	−0.516667	+	0.856186i	$0.672828\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.0000	−	20.7846i	−0.638696	−	1.10625i	−0.985719	−	0.168397i	$-0.946141\pi$
$353$	−12.0000	−	20.7846i	−0.638696	−	1.10625i	0.347024	−	0.937856i	$-0.387192\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.0000	+	27.7128i	−0.844448	+	1.46263i	0.0416523	+	0.999132i	$0.486738\pi$
$359$	−16.0000	+	27.7128i	−0.844448	+	1.46263i	−0.886100	+	0.463494i	$0.846596\pi$
$360$		0			0
$361$	−3.00000	−	5.19615i	−0.157895	−	0.273482i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−18.5000	−	32.0429i	−0.965692	−	1.67263i	−0.707744	−	0.706469i	$-0.750287\pi$
$367$	−18.5000	−	32.0429i	−0.965692	−	1.67263i	−0.257948	−	0.966159i	$-0.583046\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.0000	+	10.3923i	0.623009	+	0.539542i
$372$		0			0
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	−0.852791	−	0.522253i	$-0.825092\pi$
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	0.878680	+	0.477412i	$0.158425\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.00000			0.412021
$378$		0			0
$379$	25.0000			1.28416			0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000			1.28416			0.642082	+	0.766636i	$0.278071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.00000	−	1.73205i	0.0510976	−	0.0885037i	−0.839345	−	0.543599i	$-0.817061\pi$
$383$	1.00000	−	1.73205i	0.0510976	−	0.0885037i	0.890443	+	0.455095i	$0.150395\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.00000	−	6.92820i	−0.202808	−	0.351274i	0.746624	−	0.665246i	$-0.231673\pi$
$389$	−4.00000	−	6.92820i	−0.202808	−	0.351274i	−0.949432	+	0.313972i	$0.898340\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.50000	+	12.9904i	−0.376414	+	0.651969i	−0.990538	−	0.137241i	$-0.956176\pi$
$397$	−7.50000	+	12.9904i	−0.376414	+	0.651969i	0.614123	+	0.789210i	$0.289510\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.0000	+	19.0526i	−0.549314	+	0.951439i	0.449008	+	0.893528i	$0.351777\pi$
$401$	−11.0000	+	19.0526i	−0.549314	+	0.951439i	−0.998322	+	0.0579116i	$0.981556\pi$
$402$		0			0
$403$	−1.50000	−	2.59808i	−0.0747203	−	0.129419i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.0000			0.892227
$408$		0			0
$409$	−2.50000	−	4.33013i	−0.123617	−	0.214111i	0.797574	−	0.603220i	$-0.206116\pi$
$409$	−2.50000	−	4.33013i	−0.123617	−	0.214111i	−0.921192	+	0.389109i	$0.872783\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.0000	−	10.3923i	−0.590481	−	0.511372i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−34.0000			−1.66101			−0.830504	−	0.557012i	$-0.811948\pi$
$419$	−34.0000			−1.66101			−0.830504	+	0.557012i	$0.811948\pi$
$420$		0			0
$421$	−11.0000			−0.536107			−0.268054	−	0.963404i	$-0.586380\pi$
$421$	−11.0000			−0.536107			−0.268054	+	0.963404i	$0.586380\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.00000	−	8.66025i	0.242536	−	0.420084i
$426$		0			0
$427$	1.00000	−	5.19615i	0.0483934	−	0.251459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	0.684564	−	0.728953i	$-0.259993\pi$
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	−0.973574	+	0.228373i	$0.926659\pi$
$432$		0			0
$433$	−25.0000			−1.20142			−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000			−1.20142			−0.600712	+	0.799466i	$0.705116\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.0000	+	25.9808i	0.717547	+	1.24283i
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.0000	−	19.0526i	0.522626	−	0.905214i	−0.477028	−	0.878888i	$-0.658286\pi$
$443$	11.0000	−	19.0526i	0.522626	−	0.905214i	0.999653	−	0.0263261i	$-0.00838082\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−24.0000			−1.13263			−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000			−1.13263			−0.566315	+	0.824189i	$0.691631\pi$
$450$		0			0
$451$	−2.00000	−	3.46410i	−0.0941763	−	0.163118i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.500000	+	0.866025i	−0.0233890	+	0.0405110i	−0.877483	−	0.479608i	$-0.840779\pi$
$457$	−0.500000	+	0.866025i	−0.0233890	+	0.0405110i	0.854094	+	0.520119i	$0.174112\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$	−11.0000			−0.511213			−0.255607	−	0.966781i	$-0.582275\pi$
$463$	−11.0000			−0.511213			−0.255607	+	0.966781i	$0.582275\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.0000	+	17.3205i	−0.462745	+	0.801498i	−0.999097	−	0.0424970i	$-0.986469\pi$
$467$	−10.0000	+	17.3205i	−0.462745	+	0.801498i	0.536352	+	0.843995i	$0.319802\pi$
$468$		0			0
$469$	−10.0000	−	8.66025i	−0.461757	−	0.399893i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.00000	−	1.73205i	−0.0459800	−	0.0796398i
$474$		0			0
$475$	25.0000			1.14708
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.00000	−	5.19615i	−0.137073	−	0.237418i	0.789314	−	0.613990i	$-0.210436\pi$
$479$	−3.00000	−	5.19615i	−0.137073	−	0.237418i	−0.926388	+	0.376571i	$0.877103\pi$
$480$		0			0
$481$	4.50000	−	7.79423i	0.205182	−	0.355386i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	3.50000	+	6.06218i	0.158600	+	0.274703i	0.934364	−	0.356320i	$-0.115969\pi$
$487$	3.50000	+	6.06218i	0.158600	+	0.274703i	−0.775764	+	0.631023i	$0.782635\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	−8.00000	−	13.8564i	−0.360302	−	0.624061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.0000	−	3.46410i	0.448561	−	0.155386i
$498$		0			0
$499$	−11.5000	+	19.9186i	−0.514811	+	0.891678i	0.485042	+	0.874491i	$0.338804\pi$
$499$	−11.5000	+	19.9186i	−0.514811	+	0.891678i	−0.999852	+	0.0171872i	$0.994529\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	42.0000			1.87269			0.936344	−	0.351085i	$-0.114187\pi$
$503$	42.0000			1.87269			0.936344	+	0.351085i	$0.114187\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10.0000	+	17.3205i	−0.443242	+	0.767718i	−0.997928	−	0.0643419i	$-0.979505\pi$
$509$	−10.0000	+	17.3205i	−0.443242	+	0.767718i	0.554686	+	0.832060i	$0.312839\pi$
$510$		0			0
$511$	22.0000	+	19.0526i	0.973223	+	0.842836i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−16.0000			−0.703679
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	−6.50000	+	11.2583i	−0.284225	+	0.492292i	−0.972421	−	0.233233i	$-0.925070\pi$
$523$	−6.50000	+	11.2583i	−0.284225	+	0.492292i	0.688196	+	0.725525i	$0.258403\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.00000	+	5.19615i	−0.130682	+	0.226348i
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.00000			−0.0866296
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13.0000	+	5.19615i	0.559950	+	0.223814i
$540$		0			0
$541$	−12.5000	+	21.6506i	−0.537417	+	0.930834i	0.461625	+	0.887075i	$0.347267\pi$
$541$	−12.5000	+	21.6506i	−0.537417	+	0.930834i	−0.999042	+	0.0437584i	$0.986067\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	20.0000	−	34.6410i	0.852029	−	1.47576i
$552$		0			0
$553$	−2.50000	+	12.9904i	−0.106311	+	0.552407i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13.0000	+	22.5167i	0.550828	+	0.954062i	0.998215	+	0.0597213i	$0.0190212\pi$
$557$	13.0000	+	22.5167i	0.550828	+	0.954062i	−0.447387	+	0.894340i	$0.647645\pi$
$558$		0			0
$559$	−1.00000			−0.0422955
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.0000	−	17.3205i	−0.421450	−	0.729972i	0.574632	−	0.818412i	$-0.305145\pi$
$563$	−10.0000	−	17.3205i	−0.421450	−	0.729972i	−0.996082	+	0.0884397i	$0.971812\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.0000	−	22.5167i	0.544988	−	0.943948i	−0.453619	−	0.891196i	$-0.649867\pi$
$569$	13.0000	−	22.5167i	0.544988	−	0.943948i	0.998608	−	0.0527519i	$-0.0167993\pi$
$570$		0			0
$571$	−3.50000	−	6.06218i	−0.146470	−	0.253694i	0.783450	−	0.621455i	$-0.213458\pi$
$571$	−3.50000	−	6.06218i	−0.146470	−	0.253694i	−0.929921	+	0.367760i	$0.880125\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−30.0000			−1.25109
$576$		0			0
$577$	8.50000	+	14.7224i	0.353860	+	0.612903i	0.986922	−	0.161198i	$-0.0515357\pi$
$577$	8.50000	+	14.7224i	0.353860	+	0.612903i	−0.633062	+	0.774101i	$0.718202\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6.00000	+	10.3923i	−0.248495	+	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24.0000			0.990586			0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000			0.990586			0.495293	+	0.868726i	$0.335061\pi$
$588$		0			0
$589$	−15.0000			−0.618064
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7.00000	−	12.1244i	0.287456	−	0.497888i	−0.685746	−	0.727841i	$-0.740524\pi$
$593$	7.00000	−	12.1244i	0.287456	−	0.497888i	0.973202	+	0.229953i	$0.0738573\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.0000	−	17.3205i	−0.408589	−	0.707697i	0.586143	−	0.810208i	$-0.300646\pi$
$599$	−10.0000	−	17.3205i	−0.408589	−	0.707697i	−0.994732	+	0.102511i	$0.967312\pi$
$600$		0			0
$601$	3.00000			0.122373			0.0611863	−	0.998126i	$-0.480512\pi$
$601$	3.00000			0.122373			0.0611863	+	0.998126i	$0.480512\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−6.50000	+	11.2583i	−0.263827	+	0.456962i	−0.967256	−	0.253804i	$-0.918318\pi$
$607$	−6.50000	+	11.2583i	−0.263827	+	0.456962i	0.703429	+	0.710766i	$0.251651\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.00000	+	6.92820i	−0.161823	+	0.280285i
$612$		0			0
$613$	−5.00000	−	8.66025i	−0.201948	−	0.349784i	0.747208	−	0.664590i	$-0.231394\pi$
$613$	−5.00000	−	8.66025i	−0.201948	−	0.349784i	−0.949156	+	0.314806i	$0.898061\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.0000			−1.36879			−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000			−1.36879			−0.684394	+	0.729112i	$0.739933\pi$
$618$		0			0
$619$	−2.50000	−	4.33013i	−0.100483	−	0.174042i	0.811400	−	0.584491i	$-0.198706\pi$
$619$	−2.50000	−	4.33013i	−0.100483	−	0.174042i	−0.911884	+	0.410448i	$0.865372\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.00000	−	31.1769i	0.240385	−	1.24908i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−18.0000			−0.717707
$630$		0			0
$631$	−44.0000			−1.75161			−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000			−1.75161			−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	5.50000	−	4.33013i	0.217918	−	0.171566i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−19.0000	−	32.9090i	−0.750455	−	1.29983i	−0.947602	−	0.319452i	$-0.896501\pi$
$641$	−19.0000	−	32.9090i	−0.750455	−	1.29983i	0.197148	−	0.980374i	$-0.436832\pi$
$642$		0			0
$643$	−13.0000			−0.512670			−0.256335	−	0.966588i	$-0.582515\pi$
$643$	−13.0000			−0.512670			−0.256335	+	0.966588i	$0.582515\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.0000	−	36.3731i	−0.825595	−	1.42997i	−0.901464	−	0.432855i	$-0.857506\pi$
$647$	−21.0000	−	36.3731i	−0.825595	−	1.42997i	0.0758684	−	0.997118i	$-0.475827\pi$
$648$		0			0
$649$	6.00000	−	10.3923i	0.235521	−	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.0000	+	32.9090i	−0.743527	+	1.28783i	0.207352	+	0.978266i	$0.433515\pi$
$653$	−19.0000	+	32.9090i	−0.743527	+	1.28783i	−0.950880	+	0.309561i	$0.899818\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	−7.50000	−	12.9904i	−0.291716	−	0.505267i	0.682499	−	0.730886i	$-0.260893\pi$
$661$	−7.50000	−	12.9904i	−0.291716	−	0.505267i	−0.974216	+	0.225619i	$0.927560\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−24.0000	+	41.5692i	−0.929284	+	1.60957i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.00000			0.154418
$672$		0			0
$673$	35.0000			1.34915			0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000			1.34915			0.674575	+	0.738206i	$0.264327\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	−0.279453	−	0.960159i	$-0.590153\pi$
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	0.971249	−	0.238067i	$-0.0765137\pi$
$678$		0			0
$679$	45.0000	−	15.5885i	1.72694	−	0.598230i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.00000	+	15.5885i	0.344375	+	0.596476i	0.985240	−	0.171178i	$-0.0547574\pi$
$683$	9.00000	+	15.5885i	0.344375	+	0.596476i	−0.640865	+	0.767654i	$0.721424\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.00000	+	5.19615i	0.114291	+	0.197958i
$690$		0			0
$691$	−14.5000	+	25.1147i	−0.551606	+	0.955410i	0.446553	+	0.894757i	$0.352651\pi$
$691$	−14.5000	+	25.1147i	−0.551606	+	0.955410i	−0.998159	+	0.0606524i	$0.980682\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.00000	+	3.46410i	0.0757554	+	0.131212i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14.0000			0.528773			0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000			0.528773			0.264386	+	0.964417i	$0.414831\pi$
$702$		0			0
$703$	−22.5000	−	38.9711i	−0.848604	−	1.46982i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.0000	−	10.3923i	−0.451306	−	0.390843i
$708$		0			0
$709$	−5.00000	+	8.66025i	−0.187779	+	0.325243i	−0.944509	−	0.328484i	$-0.893462\pi$
$709$	−5.00000	+	8.66025i	−0.187779	+	0.325243i	0.756730	+	0.653727i	$0.226796\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.0000			0.674105
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.0000	+	25.9808i	−0.559406	+	0.968919i	0.438141	+	0.898906i	$0.355637\pi$
$719$	−15.0000	+	25.9808i	−0.559406	+	0.968919i	−0.997546	+	0.0700124i	$0.977696\pi$
$720$		0			0
$721$	−5.50000	+	28.5788i	−0.204831	+	1.06433i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	20.0000	+	34.6410i	0.742781	+	1.28654i
$726$		0			0
$727$	17.0000			0.630495			0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000			0.630495			0.315248	+	0.949009i	$0.397912\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.00000	+	1.73205i	0.0369863	+	0.0640622i
$732$		0			0
$733$	19.5000	−	33.7750i	0.720249	−	1.24751i	−0.240651	−	0.970612i	$-0.577361\pi$
$733$	19.5000	−	33.7750i	0.720249	−	1.24751i	0.960900	−	0.276896i	$-0.0893058\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.00000	−	8.66025i	0.184177	−	0.319005i
$738$		0			0
$739$	16.5000	+	28.5788i	0.606962	+	1.05129i	0.991738	+	0.128279i	$0.0409454\pi$
$739$	16.5000	+	28.5788i	0.606962	+	1.05129i	−0.384776	+	0.923010i	$0.625721\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	54.0000			1.98107			0.990534	−	0.137268i	$-0.0438322\pi$
$743$	54.0000			1.98107			0.990534	+	0.137268i	$0.0438322\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.00000	−	46.7654i	0.328853	−	1.70877i
$750$		0			0
$751$	−15.5000	+	26.8468i	−0.565603	+	0.979653i	0.431390	+	0.902165i	$0.358023\pi$
$751$	−15.5000	+	26.8468i	−0.565603	+	0.979653i	−0.996993	+	0.0774878i	$0.975310\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.0000	+	43.3013i	−0.906249	+	1.56967i	−0.0870179	+	0.996207i	$0.527734\pi$
$761$	−25.0000	+	43.3013i	−0.906249	+	1.56967i	−0.819231	+	0.573463i	$0.805600\pi$
$762$		0			0
$763$	6.00000	+	5.19615i	0.217215	+	0.188113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.00000	−	5.19615i	−0.108324	−	0.187622i
$768$		0			0
$769$	23.0000			0.829401			0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000			0.829401			0.414701	+	0.909958i	$0.363886\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21.0000	+	36.3731i	0.755318	+	1.30825i	0.945216	+	0.326445i	$0.105851\pi$
$773$	21.0000	+	36.3731i	0.755318	+	1.30825i	−0.189899	+	0.981804i	$0.560816\pi$
$774$		0			0
$775$	7.50000	−	12.9904i	0.269408	−	0.466628i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.00000	+	8.66025i	−0.179144	+	0.310286i
$780$		0			0
$781$	4.00000	+	6.92820i	0.143131	+	0.247911i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−4.00000	−	6.92820i	−0.142585	−	0.246964i	0.785885	−	0.618373i	$-0.212208\pi$
$787$	−4.00000	−	6.92820i	−0.142585	−	0.246964i	−0.928469	+	0.371409i	$0.878875\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	30.0000	−	10.3923i	1.06668	−	0.369508i
$792$		0			0
$793$	1.00000	−	1.73205i	0.0355110	−	0.0615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.00000			−0.212531			−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000			−0.212531			−0.106265	+	0.994338i	$0.533889\pi$
$798$		0			0
$799$	16.0000			0.566039
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.0000	+	19.0526i	−0.388182	+	0.672350i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−14.0000	−	24.2487i	−0.492214	−	0.852539i	0.507746	−	0.861507i	$-0.330479\pi$
$809$	−14.0000	−	24.2487i	−0.492214	−	0.852539i	−0.999960	+	0.00896753i	$0.997146\pi$
$810$		0			0
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.50000	+	4.33013i	−0.0874639	+	0.151492i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	24.0000	−	41.5692i	0.837606	−	1.45078i	−0.0542853	−	0.998525i	$-0.517288\pi$
$821$	24.0000	−	41.5692i	0.837606	−	1.45078i	0.891891	−	0.452250i	$-0.149379\pi$
$822$		0			0
$823$	16.0000	+	27.7128i	0.557725	+	0.966008i	0.997686	+	0.0679910i	$0.0216589\pi$
$823$	16.0000	+	27.7128i	0.557725	+	0.966008i	−0.439961	+	0.898017i	$0.645008\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.0000			−0.834562			−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000			−0.834562			−0.417281	+	0.908778i	$0.637017\pi$
$828$		0			0
$829$	−24.5000	−	42.4352i	−0.850920	−	1.47384i	−0.880379	−	0.474271i	$-0.842712\pi$
$829$	−24.5000	−	42.4352i	−0.850920	−	1.47384i	0.0294587	−	0.999566i	$-0.490622\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.0000	−	5.19615i	−0.450423	−	0.180036i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4.00000			−0.138095			−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000			−0.138095			−0.0690477	+	0.997613i	$0.521996\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	3.50000	−	18.1865i	0.120261	−	0.624897i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	27.0000	+	46.7654i	0.925548	+	1.60310i
$852$		0			0
$853$	−21.0000			−0.719026			−0.359513	−	0.933140i	$-0.617057\pi$
$853$	−21.0000			−0.719026			−0.359513	+	0.933140i	$0.617057\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.00000	+	6.92820i	0.136637	+	0.236663i	0.926222	−	0.376979i	$-0.123037\pi$
$857$	4.00000	+	6.92820i	0.136637	+	0.236663i	−0.789584	+	0.613642i	$0.789704\pi$
$858$		0			0
$859$	22.0000	−	38.1051i	0.750630	−	1.30013i	−0.196887	−	0.980426i	$-0.563083\pi$
$859$	22.0000	−	38.1051i	0.750630	−	1.30013i	0.947518	−	0.319704i	$-0.103583\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.00000	−	15.5885i	0.306364	−	0.530637i	−0.671200	−	0.741276i	$-0.734221\pi$
$863$	9.00000	−	15.5885i	0.306364	−	0.530637i	0.977564	+	0.210639i	$0.0675543\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−10.0000			−0.339227
$870$		0			0
$871$	−2.50000	−	4.33013i	−0.0847093	−	0.146721i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5.00000	+	8.66025i	−0.168838	+	0.292436i	−0.938012	−	0.346604i	$-0.887335\pi$
$877$	−5.00000	+	8.66025i	−0.168838	+	0.292436i	0.769174	+	0.639040i	$0.220668\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	54.0000			1.81931			0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000			1.81931			0.909653	+	0.415369i	$0.136347\pi$
$882$		0			0
$883$	−19.0000			−0.639401			−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000			−0.639401			−0.319700	+	0.947519i	$0.603582\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.00000	−	5.19615i	0.100730	−	0.174470i	−0.811256	−	0.584692i	$-0.801215\pi$
$887$	3.00000	−	5.19615i	0.100730	−	0.174470i	0.911986	+	0.410222i	$0.134549\pi$
$888$		0			0
$889$	−22.5000	+	7.79423i	−0.754626	+	0.261410i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	20.0000	+	34.6410i	0.669274	+	1.15922i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.0000	−	20.7846i	−0.400222	−	0.693206i
$900$		0			0
$901$	6.00000	−	10.3923i	0.199889	−	0.346218i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.500000	−	0.866025i	−0.0166022	−	0.0287559i	0.857605	−	0.514309i	$-0.171952\pi$
$907$	−0.500000	−	0.866025i	−0.0166022	−	0.0287559i	−0.874207	+	0.485553i	$0.838618\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−22.0000			−0.728893			−0.364446	−	0.931224i	$-0.618742\pi$
$911$	−22.0000			−0.728893			−0.364446	+	0.931224i	$0.618742\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.00000	−	25.9808i	0.165115	−	0.857960i
$918$		0			0
$919$	17.5000	−	30.3109i	0.577272	−	0.999864i	−0.418519	−	0.908208i	$-0.637451\pi$
$919$	17.5000	−	30.3109i	0.577272	−	0.999864i	0.995791	−	0.0916559i	$-0.0292160\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.00000			0.131662
$924$		0			0
$925$	45.0000			1.47959
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.00000	+	13.8564i	−0.262471	+	0.454614i	−0.966898	−	0.255163i	$-0.917871\pi$
$929$	−8.00000	+	13.8564i	−0.262471	+	0.454614i	0.704427	+	0.709777i	$0.251204\pi$
$930$		0			0
$931$	−5.00000	−	34.6410i	−0.163868	−	1.13531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	51.0000			1.66610			0.833049	−	0.553200i	$-0.186593\pi$
$937$	51.0000			1.66610			0.833049	+	0.553200i	$0.186593\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.0000	+	22.5167i	0.423788	+	0.734022i	0.996306	−	0.0858697i	$-0.0273669\pi$
$941$	13.0000	+	22.5167i	0.423788	+	0.734022i	−0.572518	+	0.819892i	$0.694034\pi$
$942$		0			0
$943$	6.00000	−	10.3923i	0.195387	−	0.338420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−25.0000	+	43.3013i	−0.812391	+	1.40710i	0.0987955	+	0.995108i	$0.468501\pi$
$947$	−25.0000	+	43.3013i	−0.812391	+	1.40710i	−0.911186	+	0.411994i	$0.864832\pi$
$948$		0			0
$949$	5.50000	+	9.52628i	0.178538	+	0.309236i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−14.0000			−0.453504			−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000			−0.453504			−0.226752	+	0.973952i	$0.572811\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−24.0000	−	20.7846i	−0.775000	−	0.671170i
$960$		0			0
$961$	11.0000	−	19.0526i	0.354839	−	0.614599i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	13.0000			0.418052			0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000			0.418052			0.209026	+	0.977910i	$0.432971\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.0000	+	22.5167i	−0.417190	+	0.722594i	−0.995656	−	0.0931127i	$-0.970318\pi$
$971$	−13.0000	+	22.5167i	−0.417190	+	0.722594i	0.578466	+	0.815707i	$0.303652\pi$
$972$		0			0
$973$	37.5000	−	12.9904i	1.20219	−	0.416452i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.0000	+	25.9808i	0.479893	+	0.831198i	0.999734	−	0.0230645i	$-0.00734232\pi$
$977$	15.0000	+	25.9808i	0.479893	+	0.831198i	−0.519841	+	0.854263i	$0.674009\pi$
$978$		0			0
$979$	24.0000			0.767043
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.00000	+	1.73205i	0.0318950	+	0.0552438i	0.881532	−	0.472124i	$-0.156512\pi$
$983$	1.00000	+	1.73205i	0.0318950	+	0.0552438i	−0.849637	+	0.527368i	$0.823179\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.00000	−	5.19615i	0.0953945	−	0.165228i
$990$		0			0
$991$	−27.5000	−	47.6314i	−0.873566	−	1.51306i	−0.858282	−	0.513178i	$-0.828468\pi$
$991$	−27.5000	−	47.6314i	−0.873566	−	1.51306i	−0.0152841	−	0.999883i	$-0.504865\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	20.5000	+	35.5070i	0.649242	+	1.12452i	0.983304	+	0.181968i	$0.0582469\pi$
$997$	20.5000	+	35.5070i	0.649242	+	1.12452i	−0.334063	+	0.942551i	$0.608420\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.s.h.289.1		2
3.2	odd	2		672.2.q.d.289.1	yes	2
4.3	odd	2		2016.2.s.e.289.1		2
7.4	even	3	inner	2016.2.s.h.865.1		2
12.11	even	2		672.2.q.h.289.1	yes	2
21.2	odd	6		4704.2.a.ba.1.1		1
21.5	even	6		4704.2.a.j.1.1		1
21.11	odd	6		672.2.q.d.193.1	✓	2
24.5	odd	2		1344.2.q.q.961.1		2
24.11	even	2		1344.2.q.e.961.1		2
28.11	odd	6		2016.2.s.e.865.1		2
84.11	even	6		672.2.q.h.193.1	yes	2
84.23	even	6		4704.2.a.g.1.1		1
84.47	odd	6		4704.2.a.y.1.1		1
168.5	even	6		9408.2.a.ci.1.1		1
168.11	even	6		1344.2.q.e.193.1		2
168.53	odd	6		1344.2.q.q.193.1		2
168.107	even	6		9408.2.a.cn.1.1		1
168.131	odd	6		9408.2.a.x.1.1		1
168.149	odd	6		9408.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.q.d.193.1	✓	2	21.11	odd	6
672.2.q.d.289.1	yes	2	3.2	odd	2
672.2.q.h.193.1	yes	2	84.11	even	6
672.2.q.h.289.1	yes	2	12.11	even	2
1344.2.q.e.193.1		2	168.11	even	6
1344.2.q.e.961.1		2	24.11	even	2
1344.2.q.q.193.1		2	168.53	odd	6
1344.2.q.q.961.1		2	24.5	odd	2
2016.2.s.e.289.1		2	4.3	odd	2
2016.2.s.e.865.1		2	28.11	odd	6
2016.2.s.h.289.1		2	1.1	even	1	trivial
2016.2.s.h.865.1		2	7.4	even	3	inner
4704.2.a.g.1.1		1	84.23	even	6
4704.2.a.j.1.1		1	21.5	even	6
4704.2.a.y.1.1		1	84.47	odd	6
4704.2.a.ba.1.1		1	21.2	odd	6
9408.2.a.t.1.1		1	168.149	odd	6
9408.2.a.x.1.1		1	168.131	odd	6
9408.2.a.ci.1.1		1	168.5	even	6
9408.2.a.cn.1.1		1	168.107	even	6

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.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{11} +1.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(2.50000 - 4.33013i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +8.00000 q^{29} +(-1.50000 - 2.59808i) q^{31} +(4.50000 - 7.79423i) q^{37} -2.00000 q^{41} -1.00000 q^{43} +(-4.00000 + 6.92820i) q^{47} +(5.50000 - 4.33013i) q^{49} +(3.00000 + 5.19615i) q^{53} +(-3.00000 - 5.19615i) q^{59} +(1.00000 - 1.73205i) q^{61} +(-2.50000 - 4.33013i) q^{67} +4.00000 q^{71} +(5.50000 + 9.52628i) q^{73} +(4.00000 + 3.46410i) q^{77} +(-2.50000 + 4.33013i) q^{79} +(6.00000 - 10.3923i) q^{89} +(2.50000 - 0.866025i) q^{91} +18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{7} + 2 q^{11} + 2 q^{13} - 2 q^{17} + 5 q^{19} - 6 q^{23} + 5 q^{25} + 16 q^{29} - 3 q^{31} + 9 q^{37} - 4 q^{41} - 2 q^{43} - 8 q^{47} + 11 q^{49} + 6 q^{53} - 6 q^{59} + 2 q^{61} - 5 q^{67}+ \cdots + 36 q^{97}+O(q^{100}) \) 2 * q + 5 * q^7 + 2 * q^11 + 2 * q^13 - 2 * q^17 + 5 * q^19 - 6 * q^23 + 5 * q^25 + 16 * q^29 - 3 * q^31 + 9 * q^37 - 4 * q^41 - 2 * q^43 - 8 * q^47 + 11 * q^49 + 6 * q^53 - 6 * q^59 + 2 * q^61 - 5 * q^67 + 8 * q^71 + 11 * q^73 + 8 * q^77 - 5 * q^79 + 12 * q^89 + 5 * q^91 + 36 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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