LMFDB - Embedded newform 3024.2.cx.d.559.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3024,2,Mod(559,3024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3024.559"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.cx (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-3,0,5,0,0,0,-6] 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:	$24.1467615712$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			559.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3024.559
Dual form			3024.2.cx.d.2575.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(-3.00000 - 1.73205i) q^{11} +3.46410i q^{17} +5.00000 q^{19} +(-4.50000 + 2.59808i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(1.00000 + 1.73205i) q^{31} +(-3.00000 + 3.46410i) q^{35} -8.00000 q^{37} +(-3.00000 - 1.73205i) q^{43} +(3.00000 - 5.19615i) q^{47} +(5.50000 - 4.33013i) q^{49} +6.00000 q^{53} +6.00000 q^{55} +(6.00000 + 10.3923i) q^{59} +(1.50000 + 0.866025i) q^{61} +(-6.00000 + 3.46410i) q^{67} +12.1244i q^{71} +6.92820i q^{73} +(-9.00000 - 1.73205i) q^{77} +(13.5000 + 7.79423i) q^{79} +(-3.00000 - 5.19615i) q^{85} +13.8564i q^{89} +(-7.50000 + 4.33013i) q^{95} +(-9.00000 - 5.19615i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} + 5 q^{7} - 6 q^{11} + 10 q^{19} - 9 q^{23} - 2 q^{25} + 2 q^{31} - 6 q^{35} - 16 q^{37} - 6 q^{43} + 6 q^{47} + 11 q^{49} + 12 q^{53} + 12 q^{55} + 12 q^{59} + 3 q^{61} - 12 q^{67} - 18 q^{77}+ \cdots - 18 q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 + 5 * q^7 - 6 * q^11 + 10 * q^19 - 9 * q^23 - 2 * q^25 + 2 * q^31 - 6 * q^35 - 16 * q^37 - 6 * q^43 + 6 * q^47 + 11 * q^49 + 12 * q^53 + 12 * q^55 + 12 * q^59 + 3 * q^61 - 12 * q^67 - 18 * q^77 + 27 * q^79 - 6 * q^85 - 15 * q^95 - 18 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.50000	+	0.866025i	−0.670820	+	0.387298i	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−1.50000	+	0.866025i	−0.670820	+	0.387298i	0.125567	+	0.992085i	$0.459925\pi$
$6$		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
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.0258656	−	0.999665i	$-0.508234\pi$
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.878668	+	0.477432i	$0.841568\pi$
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.46410i			0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$			3.46410i			0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$		0			0
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.50000	+	2.59808i	−0.938315	+	0.541736i	−0.889432	−	0.457068i	$-0.848900\pi$
$23$	−4.50000	+	2.59808i	−0.938315	+	0.541736i	−0.0488832	+	0.998805i	$0.515566\pi$
$24$		0			0
$25$	−1.00000	+	1.73205i	−0.200000	+	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	0.941745	−	0.336327i	$-0.109185\pi$
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	−0.762140	+	0.647412i	$0.775851\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000	+	3.46410i	−0.507093	+	0.585540i
$36$		0			0
$37$	−8.00000			−1.31519			−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000			−1.31519			−0.657596	+	0.753371i	$0.728427\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$	−3.00000	−	1.73205i	−0.457496	−	0.264135i	0.253495	−	0.967337i	$-0.418420\pi$
$43$	−3.00000	−	1.73205i	−0.457496	−	0.264135i	−0.710991	+	0.703201i	$0.751753\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	$-0.689164\pi$
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	0.997503	+	0.0706177i	$0.0224970\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	6.00000			0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	0.931282	+	0.364299i	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	1.50000	+	0.866025i	0.192055	+	0.110883i	0.592944	−	0.805243i	$-0.297965\pi$
$61$	1.50000	+	0.866025i	0.192055	+	0.110883i	−0.400889	+	0.916127i	$0.631299\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.00000	+	3.46410i	−0.733017	+	0.423207i	−0.819525	−	0.573044i	$-0.805762\pi$
$67$	−6.00000	+	3.46410i	−0.733017	+	0.423207i	0.0865081	+	0.996251i	$0.472429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.1244i			1.43890i	0.694546	+	0.719448i	$0.255605\pi$
$71$			12.1244i			1.43890i	−0.694546	+	0.719448i	$0.744395\pi$
$72$		0			0
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.00000	−	1.73205i	−1.02565	−	0.197386i
$78$		0			0
$79$	13.5000	+	7.79423i	1.51887	+	0.876919i	0.999753	+	0.0222151i	$0.00707187\pi$
$79$	13.5000	+	7.79423i	1.51887	+	0.876919i	0.519115	+	0.854704i	$0.326261\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$	−3.00000	−	5.19615i	−0.325396	−	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			13.8564i			1.46878i	0.678730	+	0.734388i	$0.262531\pi$
$89$			13.8564i			1.46878i	−0.678730	+	0.734388i	$0.737469\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.50000	+	4.33013i	−0.769484	+	0.444262i
$96$		0			0
$97$	−9.00000	−	5.19615i	−0.913812	−	0.527589i	−0.0321560	−	0.999483i	$-0.510237\pi$
$97$	−9.00000	−	5.19615i	−0.913812	−	0.527589i	−0.881656	+	0.471894i	$0.843571\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.50000	−	0.866025i	−0.149256	−	0.0861727i	0.423512	−	0.905890i	$-0.360797\pi$
$101$	−1.50000	−	0.866025i	−0.149256	−	0.0861727i	−0.572768	+	0.819718i	$0.694130\pi$
$102$		0			0
$103$	7.00000	+	12.1244i	0.689730	+	1.19465i	0.971925	+	0.235291i	$0.0756043\pi$
$103$	7.00000	+	12.1244i	0.689730	+	1.19465i	−0.282194	+	0.959357i	$0.591062\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.92820i		−	0.669775i	−0.942258	−	0.334887i	$-0.891302\pi$
$107$		−	6.92820i		−	0.669775i	0.942258	−	0.334887i	$-0.108698\pi$
$108$		0			0
$109$	−16.0000			−1.53252			−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000			−1.53252			−0.766261	+	0.642529i	$0.777885\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.50000	−	2.59808i	−0.141108	−	0.244406i	0.786806	−	0.617200i	$-0.211733\pi$
$113$	−1.50000	−	2.59808i	−0.141108	−	0.244406i	−0.927914	+	0.372794i	$0.878400\pi$
$114$		0			0
$115$	4.50000	−	7.79423i	0.419627	−	0.726816i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.00000	+	8.66025i	0.275010	+	0.793884i
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.1244i		−	1.08444i
$126$		0			0
$127$			8.66025i			0.768473i	0.923235	+	0.384237i	$0.125535\pi$
$127$			8.66025i			0.768473i	−0.923235	+	0.384237i	$0.874465\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	0.981824	+	0.189794i	$0.0607819\pi$
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	−0.326546	+	0.945181i	$0.605885\pi$
$132$		0			0
$133$	12.5000	−	4.33013i	1.08389	−	0.375470i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	−0.965250	−	0.261329i	$-0.915839\pi$
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	0.708942	+	0.705266i	$0.249173\pi$
$138$		0			0
$139$	−6.50000	−	11.2583i	−0.551323	−	0.954919i	−0.998179	−	0.0603135i	$-0.980790\pi$
$139$	−6.50000	−	11.2583i	−0.551323	−	0.954919i	0.446857	−	0.894606i	$-0.352543\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.0000	−	20.7846i	−0.983078	−	1.70274i	−0.650183	−	0.759778i	$-0.725308\pi$
$149$	−12.0000	−	20.7846i	−0.983078	−	1.70274i	−0.332896	−	0.942964i	$-0.608026\pi$
$150$		0			0
$151$	7.50000	+	4.33013i	0.610341	+	0.352381i	0.773099	−	0.634285i	$-0.218706\pi$
$151$	7.50000	+	4.33013i	0.610341	+	0.352381i	−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.00000	−	1.73205i	−0.240966	−	0.139122i
$156$		0			0
$157$	−7.50000	+	4.33013i	−0.598565	+	0.345582i	−0.768477	−	0.639878i	$-0.778985\pi$
$157$	−7.50000	+	4.33013i	−0.598565	+	0.345582i	0.169912	+	0.985459i	$0.445652\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.00000	+	10.3923i	−0.709299	+	0.819028i
$162$		0			0
$163$			24.2487i			1.89931i	0.313304	+	0.949653i	$0.398564\pi$
$163$			24.2487i			1.89931i	−0.313304	+	0.949653i	$0.601436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	0.999169	−	0.0407502i	$-0.0129748\pi$
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	−0.534875	+	0.844931i	$0.679641\pi$
$168$		0			0
$169$	−6.50000	+	11.2583i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−18.0000	−	10.3923i	−1.36851	−	0.790112i	−0.377776	−	0.925897i	$-0.623311\pi$
$173$	−18.0000	−	10.3923i	−1.36851	−	0.790112i	−0.990738	+	0.135785i	$0.956644\pi$
$174$		0			0
$175$	−1.00000	+	5.19615i	−0.0755929	+	0.392792i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			17.3205i			1.29460i	0.762237	+	0.647298i	$0.224101\pi$
$179$			17.3205i			1.29460i	−0.762237	+	0.647298i	$0.775899\pi$
$180$		0			0
$181$			12.1244i			0.901196i	0.892727	+	0.450598i	$0.148789\pi$
$181$			12.1244i			0.901196i	−0.892727	+	0.450598i	$0.851211\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.0000	−	6.92820i	0.882258	−	0.509372i
$186$		0			0
$187$	6.00000	−	10.3923i	0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.5000	−	11.2583i	−1.41097	−	0.814624i	−0.415491	−	0.909597i	$-0.636390\pi$
$191$	−19.5000	−	11.2583i	−1.41097	−	0.814624i	−0.995480	+	0.0949733i	$0.969723\pi$
$192$		0			0
$193$	11.5000	+	19.9186i	0.827788	+	1.43377i	0.899770	+	0.436365i	$0.143734\pi$
$193$	11.5000	+	19.9186i	0.827788	+	1.43377i	−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−15.0000	−	8.66025i	−1.03757	−	0.599042i
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	6.00000			0.409197
$216$		0			0
$217$	4.00000	+	3.46410i	0.271538	+	0.235159i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	5.00000	−	8.66025i	0.334825	−	0.579934i	−0.648626	−	0.761107i	$-0.724656\pi$
$223$	5.00000	−	8.66025i	0.334825	−	0.579934i	0.983451	+	0.181173i	$0.0579895\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.50000	+	2.59808i	−0.0995585	+	0.172440i	−0.911502	−	0.411296i	$-0.865076\pi$
$227$	−1.50000	+	2.59808i	−0.0995585	+	0.172440i	0.811943	+	0.583736i	$0.198410\pi$
$228$		0			0
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	0.231287	+	0.972886i	$0.425707\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.0000			0.982683			0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000			0.982683			0.491341	+	0.870967i	$0.336507\pi$
$234$		0			0
$235$			10.3923i			0.677919i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.50000	+	0.866025i	−0.0970269	+	0.0560185i	−0.547728	−	0.836656i	$-0.684507\pi$
$239$	−1.50000	+	0.866025i	−0.0970269	+	0.0560185i	0.450701	+	0.892675i	$0.351174\pi$
$240$		0			0
$241$	24.0000	+	13.8564i	1.54598	+	0.892570i	0.998443	+	0.0557856i	$0.0177663\pi$
$241$	24.0000	+	13.8564i	1.54598	+	0.892570i	0.547533	+	0.836784i	$0.315567\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.50000	+	11.2583i	−0.287494	+	0.719268i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.0000			1.32551			0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000			1.32551			0.662754	+	0.748837i	$0.269387\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.0000	−	10.3923i	1.12281	−	0.648254i	0.180693	−	0.983540i	$-0.442166\pi$
$257$	18.0000	−	10.3923i	1.12281	−	0.648254i	0.942117	+	0.335285i	$0.108833\pi$
$258$		0			0
$259$	−20.0000	+	6.92820i	−1.24274	+	0.430498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.5000	−	9.52628i	−1.01743	−	0.587416i	−0.104074	−	0.994570i	$-0.533188\pi$
$263$	−16.5000	−	9.52628i	−1.01743	−	0.587416i	−0.913360	+	0.407154i	$0.866521\pi$
$264$		0			0
$265$	−9.00000	+	5.19615i	−0.552866	+	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			25.9808i			1.58408i	0.610472	+	0.792038i	$0.290980\pi$
$269$			25.9808i			1.58408i	−0.610472	+	0.792038i	$0.709020\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.00000	−	3.46410i	0.361814	−	0.208893i
$276$		0			0
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	0.675810	+	0.737075i	$0.263794\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.5000	+	23.3827i	−0.805342	+	1.39489i	0.110717	+	0.993852i	$0.464685\pi$
$281$	−13.5000	+	23.3827i	−0.805342	+	1.39489i	−0.916060	+	0.401042i	$0.868648\pi$
$282$		0			0
$283$	6.50000	+	11.2583i	0.386385	+	0.669238i	0.991960	−	0.126550i	$-0.0403903\pi$
$283$	6.50000	+	11.2583i	0.386385	+	0.669238i	−0.605575	+	0.795788i	$0.707057\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.00000			0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	13.5000	−	7.79423i	0.788678	−	0.455344i	−0.0508187	−	0.998708i	$-0.516183\pi$
$293$	13.5000	−	7.79423i	0.788678	−	0.455344i	0.839497	+	0.543364i	$0.182850\pi$
$294$		0			0
$295$	−18.0000	−	10.3923i	−1.04800	−	0.605063i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−9.00000	−	1.73205i	−0.518751	−	0.0998337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.00000			−0.171780
$306$		0			0
$307$	23.0000			1.31268			0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000			1.31268			0.656340	+	0.754466i	$0.272104\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.00000	−	15.5885i	−0.510343	−	0.883940i	−0.999928	−	0.0119847i	$-0.996185\pi$
$311$	−9.00000	−	15.5885i	−0.510343	−	0.883940i	0.489585	−	0.871956i	$-0.337148\pi$
$312$		0			0
$313$	−3.00000	−	1.73205i	−0.169570	−	0.0979013i	0.412813	−	0.910816i	$-0.364546\pi$
$313$	−3.00000	−	1.73205i	−0.169570	−	0.0979013i	−0.582383	+	0.812914i	$0.697880\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	−0.494489	−	0.869184i	$-0.664645\pi$
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	0.999980	−	0.00635137i	$-0.00202172\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$			17.3205i			0.963739i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.00000	−	15.5885i	0.165395	−	0.859419i
$330$		0			0
$331$	18.0000	+	10.3923i	0.989369	+	0.571213i	0.905086	−	0.425229i	$-0.139806\pi$
$331$	18.0000	+	10.3923i	0.989369	+	0.571213i	0.0842837	+	0.996442i	$0.473140\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.00000	−	10.3923i	0.327815	−	0.567792i
$336$		0			0
$337$	−7.00000	−	12.1244i	−0.381314	−	0.660456i	0.609936	−	0.792451i	$-0.291195\pi$
$337$	−7.00000	−	12.1244i	−0.381314	−	0.660456i	−0.991250	+	0.131995i	$0.957862\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	6.92820i		−	0.375183i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.0000	−	8.66025i	0.805242	−	0.464907i	−0.0400587	−	0.999197i	$-0.512754\pi$
$347$	15.0000	−	8.66025i	0.805242	−	0.464907i	0.845301	+	0.534291i	$0.179421\pi$
$348$		0			0
$349$	−24.0000	−	13.8564i	−1.28469	−	0.741716i	−0.306988	−	0.951713i	$-0.599321\pi$
$349$	−24.0000	−	13.8564i	−1.28469	−	0.741716i	−0.977702	+	0.209997i	$0.932655\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.00000	+	1.73205i	0.159674	+	0.0921878i	0.577708	−	0.816244i	$-0.303947\pi$
$353$	3.00000	+	1.73205i	0.159674	+	0.0921878i	−0.418034	+	0.908431i	$0.637281\pi$
$354$		0			0
$355$	−10.5000	−	18.1865i	−0.557282	−	0.965241i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	19.0526i		−	1.00556i	−0.864416	−	0.502778i	$-0.832311\pi$
$359$		−	19.0526i		−	1.00556i	0.864416	−	0.502778i	$-0.167689\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.00000	−	10.3923i	−0.314054	−	0.543958i
$366$		0			0
$367$	−13.0000	+	22.5167i	−0.678594	+	1.17536i	0.296810	+	0.954937i	$0.404077\pi$
$367$	−13.0000	+	22.5167i	−0.678594	+	1.17536i	−0.975404	+	0.220423i	$0.929256\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	15.0000	−	5.19615i	0.778761	−	0.269771i
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	27.7128i		−	1.42351i	−0.702427	−	0.711756i	$-0.747900\pi$
$379$		−	27.7128i		−	1.42351i	0.702427	−	0.711756i	$-0.252100\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	0.779143	−	0.626846i	$-0.215654\pi$
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	−0.932436	+	0.361335i	$0.882321\pi$
$384$		0			0
$385$	15.0000	−	5.19615i	0.764471	−	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.00000	−	10.3923i	0.304212	−	0.526911i	−0.672874	−	0.739758i	$-0.734940\pi$
$389$	6.00000	−	10.3923i	0.304212	−	0.526911i	0.977086	+	0.212847i	$0.0682735\pi$
$390$		0			0
$391$	−9.00000	−	15.5885i	−0.455150	−	0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−27.0000			−1.35852
$396$		0			0
$397$		−	6.92820i		−	0.347717i	−0.984771	−	0.173858i	$-0.944377\pi$
$397$		−	6.92820i		−	0.347717i	0.984771	−	0.173858i	$-0.0556235\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	0.826139	−	0.563466i	$-0.190532\pi$
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	−0.901046	+	0.433724i	$0.857199\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	24.0000	+	13.8564i	1.18964	+	0.686837i
$408$		0			0
$409$	30.0000	−	17.3205i	1.48340	−	0.856444i	0.483582	−	0.875299i	$-0.339335\pi$
$409$	30.0000	−	17.3205i	1.48340	−	0.856444i	0.999822	+	0.0188549i	$0.00600205\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	24.0000	+	20.7846i	1.18096	+	1.02274i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.50000	+	7.79423i	0.219839	+	0.380773i	0.954759	−	0.297382i	$-0.0961133\pi$
$419$	4.50000	+	7.79423i	0.219839	+	0.380773i	−0.734919	+	0.678155i	$0.762780\pi$
$420$		0			0
$421$	4.00000	−	6.92820i	0.194948	−	0.337660i	−0.751935	−	0.659237i	$-0.770879\pi$
$421$	4.00000	−	6.92820i	0.194948	−	0.337660i	0.946883	+	0.321577i	$0.104213\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.00000	−	3.46410i	−0.291043	−	0.168034i
$426$		0			0
$427$	4.50000	+	0.866025i	0.217770	+	0.0419099i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			3.46410i			0.166860i	0.996514	+	0.0834300i	$0.0265875\pi$
$431$			3.46410i			0.166860i	−0.996514	+	0.0834300i	$0.973413\pi$
$432$		0			0
$433$		−	34.6410i		−	1.66474i	−0.554220	−	0.832370i	$-0.686983\pi$
$433$		−	34.6410i		−	1.66474i	0.554220	−	0.832370i	$-0.313017\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.5000	+	12.9904i	−1.07632	+	0.621414i
$438$		0			0
$439$	8.00000	−	13.8564i	0.381819	−	0.661330i	−0.609503	−	0.792784i	$-0.708631\pi$
$439$	8.00000	−	13.8564i	0.381819	−	0.661330i	0.991322	+	0.131453i	$0.0419644\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.00000	−	3.46410i	−0.285069	−	0.164584i	0.350647	−	0.936508i	$-0.385962\pi$
$443$	−6.00000	−	3.46410i	−0.285069	−	0.164584i	−0.635716	+	0.771923i	$0.719295\pi$
$444$		0			0
$445$	−12.0000	−	20.7846i	−0.568855	−	0.985285i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	21.0000			0.991051			0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000			0.991051			0.495526	+	0.868593i	$0.334975\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.5000	+	19.9186i	−0.537947	+	0.931752i	0.461067	+	0.887365i	$0.347467\pi$
$457$	−11.5000	+	19.9186i	−0.537947	+	0.931752i	−0.999014	+	0.0443868i	$0.985867\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−34.5000	−	19.9186i	−1.60683	−	0.927701i	−0.990075	−	0.140542i	$-0.955115\pi$
$461$	−34.5000	−	19.9186i	−1.60683	−	0.927701i	−0.616751	−	0.787159i	$-0.711551\pi$
$462$		0			0
$463$	4.50000	−	2.59808i	0.209133	−	0.120743i	−0.391776	−	0.920061i	$-0.628139\pi$
$463$	4.50000	−	2.59808i	0.209133	−	0.120743i	0.600908	+	0.799318i	$0.294806\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$		0			0
$469$	−12.0000	+	13.8564i	−0.554109	+	0.639829i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.00000	+	10.3923i	0.275880	+	0.477839i
$474$		0			0
$475$	−5.00000	+	8.66025i	−0.229416	+	0.397360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	18.0000			0.817338
$486$		0			0
$487$		−	15.5885i		−	0.706380i	−0.935552	−	0.353190i	$-0.885097\pi$
$487$		−	15.5885i		−	0.706380i	0.935552	−	0.353190i	$-0.114903\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18.0000	−	10.3923i	0.812329	−	0.468998i	−0.0354353	−	0.999372i	$-0.511282\pi$
$491$	18.0000	−	10.3923i	0.812329	−	0.468998i	0.847764	+	0.530374i	$0.177948\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.5000	+	30.3109i	0.470989	+	1.35963i
$498$		0			0
$499$	18.0000	−	10.3923i	0.805791	−	0.465223i	−0.0397013	−	0.999212i	$-0.512641\pi$
$499$	18.0000	−	10.3923i	0.805791	−	0.465223i	0.845492	+	0.533988i	$0.179307\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	3.00000			0.133498
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.0000	−	6.92820i	0.531891	−	0.307087i	−0.209895	−	0.977724i	$-0.567312\pi$
$509$	12.0000	−	6.92820i	0.531891	−	0.307087i	0.741786	+	0.670637i	$0.233979\pi$
$510$		0			0
$511$	6.00000	+	17.3205i	0.265424	+	0.766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−21.0000	−	12.1244i	−0.925371	−	0.534263i
$516$		0			0
$517$	−18.0000	+	10.3923i	−0.791639	+	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			38.1051i			1.66942i	0.550693	+	0.834708i	$0.314363\pi$
$521$			38.1051i			1.66942i	−0.550693	+	0.834708i	$0.685637\pi$
$522$		0			0
$523$	−35.0000			−1.53044			−0.765222	−	0.643767i	$-0.777371\pi$
$523$	−35.0000			−1.53044			−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.00000	+	3.46410i	−0.261364	+	0.150899i
$528$		0			0
$529$	2.00000	−	3.46410i	0.0869565	−	0.150613i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	6.00000	+	10.3923i	0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−24.0000	+	3.46410i	−1.03375	+	0.149209i
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	24.0000	−	13.8564i	1.02805	−	0.593543i
$546$		0			0
$547$	−15.0000	−	8.66025i	−0.641354	−	0.370286i	0.143782	−	0.989609i	$-0.454074\pi$
$547$	−15.0000	−	8.66025i	−0.641354	−	0.370286i	−0.785136	+	0.619324i	$0.787407\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	40.5000	+	7.79423i	1.72224	+	0.331444i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	30.0000			1.27114			0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000			1.27114			0.635570	+	0.772043i	$0.280765\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.50000	+	7.79423i	0.189652	+	0.328488i	0.945134	−	0.326682i	$-0.105931\pi$
$563$	4.50000	+	7.79423i	0.189652	+	0.328488i	−0.755482	+	0.655169i	$0.772597\pi$
$564$		0			0
$565$	4.50000	+	2.59808i	0.189316	+	0.109302i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.0000	+	36.3731i	−0.880366	+	1.52484i	−0.0294311	+	0.999567i	$0.509370\pi$
$569$	−21.0000	+	36.3731i	−0.880366	+	1.52484i	−0.850935	+	0.525271i	$0.823964\pi$
$570$		0			0
$571$	6.00000	−	3.46410i	0.251092	−	0.144968i	−0.369172	−	0.929361i	$-0.620359\pi$
$571$	6.00000	−	3.46410i	0.251092	−	0.144968i	0.620264	+	0.784393i	$0.287025\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	10.3923i		−	0.433389i
$576$		0			0
$577$		−	27.7128i		−	1.15370i	−0.816850	−	0.576850i	$-0.804282\pi$
$577$		−	27.7128i		−	1.15370i	0.816850	−	0.576850i	$-0.195718\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−18.0000	−	10.3923i	−0.745484	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	22.5000	−	38.9711i	0.928674	−	1.60851i	0.143132	−	0.989704i	$-0.454283\pi$
$587$	22.5000	−	38.9711i	0.928674	−	1.60851i	0.785543	−	0.618808i	$-0.212384\pi$
$588$		0			0
$589$	5.00000	+	8.66025i	0.206021	+	0.356840i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	24.2487i		−	0.995775i	−0.867242	−	0.497888i	$-0.834109\pi$
$593$		−	24.2487i		−	0.995775i	0.867242	−	0.497888i	$-0.165891\pi$
$594$		0			0
$595$	−12.0000	−	10.3923i	−0.491952	−	0.426043i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	39.0000	−	22.5167i	1.59350	−	0.920006i	0.600796	−	0.799402i	$-0.294850\pi$
$599$	39.0000	−	22.5167i	1.59350	−	0.920006i	0.992701	−	0.120603i	$-0.0384829\pi$
$600$		0			0
$601$	−33.0000	−	19.0526i	−1.34610	−	0.777170i	−0.358404	−	0.933567i	$-0.616679\pi$
$601$	−33.0000	−	19.0526i	−1.34610	−	0.777170i	−0.987694	+	0.156397i	$0.950012\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.50000	−	0.866025i	−0.0609837	−	0.0352089i
$606$		0			0
$607$	2.00000	+	3.46410i	0.0811775	+	0.140604i	0.903756	−	0.428048i	$-0.140799\pi$
$607$	2.00000	+	3.46410i	0.0811775	+	0.140604i	−0.822578	+	0.568652i	$0.807465\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	2.00000			0.0807792			0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000			0.0807792			0.0403896	+	0.999184i	$0.487140\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.0000	+	25.9808i	0.603877	+	1.04595i	0.992228	+	0.124434i	$0.0397116\pi$
$617$	15.0000	+	25.9808i	0.603877	+	1.04595i	−0.388351	+	0.921512i	$0.626955\pi$
$618$		0			0
$619$	−9.50000	+	16.4545i	−0.381837	+	0.661361i	−0.991325	−	0.131434i	$-0.958042\pi$
$619$	−9.50000	+	16.4545i	−0.381837	+	0.661361i	0.609488	+	0.792796i	$0.291375\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.0000	+	34.6410i	0.480770	+	1.38786i
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	27.7128i		−	1.10498i
$630$		0			0
$631$			1.73205i			0.0689519i	0.999406	+	0.0344759i	$0.0109762\pi$
$631$			1.73205i			0.0689519i	−0.999406	+	0.0344759i	$0.989024\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.50000	−	12.9904i	−0.297628	−	0.515508i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7.50000	+	12.9904i	−0.296232	+	0.513089i	−0.975271	−	0.221013i	$-0.929064\pi$
$641$	−7.50000	+	12.9904i	−0.296232	+	0.513089i	0.679039	+	0.734103i	$0.262397\pi$
$642$		0			0
$643$	2.00000	+	3.46410i	0.0788723	+	0.136611i	0.902764	−	0.430137i	$-0.141535\pi$
$643$	2.00000	+	3.46410i	0.0788723	+	0.136611i	−0.823891	+	0.566748i	$0.808201\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−42.0000			−1.65119			−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000			−1.65119			−0.825595	+	0.564263i	$0.809160\pi$
$648$		0			0
$649$		−	41.5692i		−	1.63173i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	0.986634	−	0.162951i	$-0.0521013\pi$
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	−0.634437	+	0.772975i	$0.718768\pi$
$654$		0			0
$655$	−22.5000	−	12.9904i	−0.879148	−	0.507576i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18.0000	−	10.3923i	−0.701180	−	0.404827i	0.106606	−	0.994301i	$-0.466001\pi$
$659$	−18.0000	−	10.3923i	−0.701180	−	0.404827i	−0.807787	+	0.589475i	$0.799335\pi$
$660$		0			0
$661$	28.5000	−	16.4545i	1.10852	−	0.640005i	0.170075	−	0.985431i	$-0.445599\pi$
$661$	28.5000	−	16.4545i	1.10852	−	0.640005i	0.938446	+	0.345426i	$0.112266\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−15.0000	+	17.3205i	−0.581675	+	0.671660i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.00000	−	5.19615i	−0.115814	−	0.200595i
$672$		0			0
$673$	14.5000	−	25.1147i	0.558934	−	0.968102i	−0.438652	−	0.898657i	$-0.644544\pi$
$673$	14.5000	−	25.1147i	0.558934	−	0.968102i	0.997586	−	0.0694449i	$-0.0221228\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.0000	+	6.92820i	0.461197	+	0.266272i	0.712548	−	0.701624i	$-0.247541\pi$
$677$	12.0000	+	6.92820i	0.461197	+	0.266272i	−0.251350	+	0.967896i	$0.580875\pi$
$678$		0			0
$679$	−27.0000	−	5.19615i	−1.03616	−	0.199410i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		−	10.3923i		−	0.397070i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	2.50000	−	4.33013i	0.0951045	−	0.164726i	−0.814548	−	0.580097i	$-0.803015\pi$
$691$	2.50000	−	4.33013i	0.0951045	−	0.164726i	0.909652	+	0.415371i	$0.136348\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	19.5000	+	11.2583i	0.739677	+	0.427053i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$	−40.0000			−1.50863
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.50000	−	0.866025i	−0.169240	−	0.0325702i
$708$		0			0
$709$	−10.0000	+	17.3205i	−0.375558	+	0.650485i	−0.990410	−	0.138157i	$-0.955882\pi$
$709$	−10.0000	+	17.3205i	−0.375558	+	0.650485i	0.614852	+	0.788642i	$0.289216\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.00000	−	5.19615i	−0.337053	−	0.194597i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	28.0000	+	24.2487i	1.04277	+	0.903069i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	10.0000	−	17.3205i	0.370879	−	0.642382i	−0.618822	−	0.785532i	$-0.712390\pi$
$727$	10.0000	−	17.3205i	0.370879	−	0.642382i	0.989701	+	0.143149i	$0.0457230\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.00000	−	10.3923i	0.221918	−	0.384373i
$732$		0			0
$733$	−43.5000	+	25.1147i	−1.60671	+	0.927634i	−0.616609	+	0.787269i	$0.711494\pi$
$733$	−43.5000	+	25.1147i	−1.60671	+	0.927634i	−0.990100	+	0.140365i	$0.955173\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.0000			0.884051
$738$		0			0
$739$		−	27.7128i		−	1.01943i	−0.860343	−	0.509716i	$-0.829750\pi$
$739$		−	27.7128i		−	1.01943i	0.860343	−	0.509716i	$-0.170250\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	21.0000	−	12.1244i	0.770415	−	0.444799i	−0.0626075	−	0.998038i	$-0.519942\pi$
$743$	21.0000	−	12.1244i	0.770415	−	0.444799i	0.833023	+	0.553239i	$0.186608\pi$
$744$		0			0
$745$	36.0000	+	20.7846i	1.31894	+	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.00000	−	17.3205i	−0.219235	−	0.632878i
$750$		0			0
$751$	7.50000	−	4.33013i	0.273679	−	0.158009i	−0.356879	−	0.934150i	$-0.616159\pi$
$751$	7.50000	−	4.33013i	0.273679	−	0.158009i	0.630558	+	0.776142i	$0.282826\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−15.0000			−0.545906
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.00000	+	1.73205i	−0.108750	+	0.0627868i	−0.553388	−	0.832923i	$-0.686665\pi$
$761$	−3.00000	+	1.73205i	−0.108750	+	0.0627868i	0.444639	+	0.895710i	$0.353332\pi$
$762$		0			0
$763$	−40.0000	+	13.8564i	−1.44810	+	0.501636i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−18.0000	+	10.3923i	−0.649097	+	0.374756i	−0.788110	−	0.615534i	$-0.788940\pi$
$769$	−18.0000	+	10.3923i	−0.649097	+	0.374756i	0.139013	+	0.990290i	$0.455607\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			15.5885i			0.560678i	0.959901	+	0.280339i	$0.0904469\pi$
$773$			15.5885i			0.560678i	−0.959901	+	0.280339i	$0.909553\pi$
$774$		0			0
$775$	−4.00000			−0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	21.0000	−	36.3731i	0.751439	−	1.30153i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.50000	−	12.9904i	0.267686	−	0.463647i
$786$		0			0
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	0.828176	−	0.560469i	$-0.189379\pi$
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	−0.899468	+	0.436987i	$0.856046\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.00000	−	5.19615i	−0.213335	−	0.184754i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.5000	−	12.9904i	0.796991	−	0.460143i	−0.0454270	−	0.998968i	$-0.514465\pi$
$797$	22.5000	−	12.9904i	0.796991	−	0.460143i	0.842418	+	0.538825i	$0.181132\pi$
$798$		0			0
$799$	18.0000	+	10.3923i	0.636794	+	0.367653i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.0000	−	20.7846i	0.423471	−	0.733473i
$804$		0			0
$805$	4.50000	−	23.3827i	0.158604	−	0.824131i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−42.0000			−1.47664			−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000			−1.47664			−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−21.0000	−	36.3731i	−0.735598	−	1.27409i
$816$		0			0
$817$	−15.0000	−	8.66025i	−0.524784	−	0.302984i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.00000	+	15.5885i	−0.314102	+	0.544041i	−0.979246	−	0.202674i	$-0.935037\pi$
$821$	−9.00000	+	15.5885i	−0.314102	+	0.544041i	0.665144	+	0.746715i	$0.268370\pi$
$822$		0			0
$823$	9.00000	−	5.19615i	0.313720	−	0.181126i	−0.334870	−	0.942264i	$-0.608692\pi$
$823$	9.00000	−	5.19615i	0.313720	−	0.181126i	0.648590	+	0.761138i	$0.275359\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	3.46410i		−	0.120459i	−0.998185	−	0.0602293i	$-0.980817\pi$
$827$		−	3.46410i		−	0.120459i	0.998185	−	0.0602293i	$-0.0191832\pi$
$828$		0			0
$829$			41.5692i			1.44376i	0.692019	+	0.721879i	$0.256721\pi$
$829$			41.5692i			1.44376i	−0.692019	+	0.721879i	$0.743279\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	15.0000	+	19.0526i	0.519719	+	0.660132i
$834$		0			0
$835$	−18.0000	−	10.3923i	−0.622916	−	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9.00000	−	15.5885i	0.310715	−	0.538173i	−0.667803	−	0.744338i	$-0.732765\pi$
$839$	9.00000	−	15.5885i	0.310715	−	0.538173i	0.978517	+	0.206165i	$0.0660984\pi$
$840$		0			0
$841$	14.5000	+	25.1147i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	22.5167i		−	0.774597i
$846$		0			0
$847$	2.00000	+	1.73205i	0.0687208	+	0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	36.0000	−	20.7846i	1.23406	−	0.712487i
$852$		0			0
$853$	−46.5000	−	26.8468i	−1.59213	−	0.919216i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−46.5000	−	26.8468i	−1.59213	−	0.919216i	−0.599189	−	0.800608i	$-0.704510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	36.0000	+	20.7846i	1.22974	+	0.709989i	0.966975	−	0.254872i	$-0.0820333\pi$
$857$	36.0000	+	20.7846i	1.22974	+	0.709989i	0.262762	+	0.964861i	$0.415367\pi$
$858$		0			0
$859$	20.0000	+	34.6410i	0.682391	+	1.18194i	0.974249	+	0.225475i	$0.0723932\pi$
$859$	20.0000	+	34.6410i	0.682391	+	1.18194i	−0.291858	+	0.956462i	$0.594273\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	46.7654i		−	1.59191i	−0.605355	−	0.795956i	$-0.706969\pi$
$863$		−	46.7654i		−	1.59191i	0.605355	−	0.795956i	$-0.293031\pi$
$864$		0			0
$865$	36.0000			1.22404
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−27.0000	−	46.7654i	−0.915912	−	1.58641i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.5000	−	30.3109i	−0.354965	−	1.02470i
$876$		0			0
$877$	2.00000	+	3.46410i	0.0675352	+	0.116974i	0.897816	−	0.440371i	$-0.145153\pi$
$877$	2.00000	+	3.46410i	0.0675352	+	0.116974i	−0.830281	+	0.557346i	$0.811820\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	55.4256i		−	1.86734i	−0.358139	−	0.933668i	$-0.616589\pi$
$881$		−	55.4256i		−	1.86734i	0.358139	−	0.933668i	$-0.383411\pi$
$882$		0			0
$883$		−	10.3923i		−	0.349729i	−0.984593	−	0.174864i	$-0.944051\pi$
$883$		−	10.3923i		−	0.349729i	0.984593	−	0.174864i	$-0.0559487\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000	+	20.7846i	0.402921	+	0.697879i	0.994077	−	0.108678i	$-0.0346618\pi$
$887$	12.0000	+	20.7846i	0.402921	+	0.697879i	−0.591156	+	0.806557i	$0.701328\pi$
$888$		0			0
$889$	7.50000	+	21.6506i	0.251542	+	0.726139i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	15.0000	−	25.9808i	0.501956	−	0.869413i
$894$		0			0
$895$	−15.0000	−	25.9808i	−0.501395	−	0.868441i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$			20.7846i			0.692436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.5000	−	18.1865i	−0.349032	−	0.604541i
$906$		0			0
$907$	27.0000	+	15.5885i	0.896520	+	0.517606i	0.876070	−	0.482185i	$-0.160157\pi$
$907$	27.0000	+	15.5885i	0.896520	+	0.517606i	0.0204507	+	0.999791i	$0.493490\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.50000	−	2.59808i	−0.149092	−	0.0860781i	0.423598	−	0.905850i	$-0.360767\pi$
$911$	−4.50000	−	2.59808i	−0.149092	−	0.0860781i	−0.572690	+	0.819772i	$0.694100\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	30.0000	+	25.9808i	0.990687	+	0.857960i
$918$		0			0
$919$		−	46.7654i		−	1.54265i	−0.636443	−	0.771324i	$-0.719595\pi$
$919$		−	46.7654i		−	1.54265i	0.636443	−	0.771324i	$-0.280405\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	8.00000	−	13.8564i	0.263038	−	0.455596i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12.0000	+	6.92820i	0.393707	+	0.227307i	0.683765	−	0.729702i	$-0.260341\pi$
$929$	12.0000	+	6.92820i	0.393707	+	0.227307i	−0.290058	+	0.957009i	$0.593675\pi$
$930$		0			0
$931$	27.5000	−	21.6506i	0.901276	−	0.709571i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			20.7846i			0.679729i
$936$		0			0
$937$			6.92820i			0.226335i	0.993576	+	0.113167i	$0.0360996\pi$
$937$			6.92820i			0.226335i	−0.993576	+	0.113167i	$0.963900\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.5000	−	11.2583i	0.635682	−	0.367011i	−0.147267	−	0.989097i	$-0.547048\pi$
$941$	19.5000	−	11.2583i	0.635682	−	0.367011i	0.782949	+	0.622086i	$0.213714\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.0000	+	13.8564i	0.779895	+	0.450273i	0.836393	−	0.548130i	$-0.184660\pi$
$947$	24.0000	+	13.8564i	0.779895	+	0.450273i	−0.0564979	+	0.998403i	$0.517993\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$	39.0000			1.26201
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.00000	+	15.5885i	−0.0968751	+	0.503378i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−34.5000	−	19.9186i	−1.11059	−	0.641202i
$966$		0			0
$967$	−1.50000	+	0.866025i	−0.0482367	+	0.0278495i	−0.523924	−	0.851765i	$-0.675533\pi$
$967$	−1.50000	+	0.866025i	−0.0482367	+	0.0278495i	0.475688	+	0.879614i	$0.342199\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	33.0000			1.05902			0.529510	−	0.848304i	$-0.322376\pi$
$971$	33.0000			1.05902			0.529510	+	0.848304i	$0.322376\pi$
$972$		0			0
$973$	−26.0000	−	22.5167i	−0.833522	−	0.721851i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.0000	−	36.3731i	−0.671850	−	1.16368i	−0.977379	−	0.211495i	$-0.932167\pi$
$977$	−21.0000	−	36.3731i	−0.671850	−	1.16368i	0.305530	−	0.952183i	$-0.401167\pi$
$978$		0			0
$979$	24.0000	−	41.5692i	0.767043	−	1.32856i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.0000	+	36.3731i	−0.669796	+	1.16012i	0.308165	+	0.951333i	$0.400285\pi$
$983$	−21.0000	+	36.3731i	−0.669796	+	1.16012i	−0.977961	+	0.208788i	$0.933048\pi$
$984$		0			0
$985$	9.00000	−	5.19615i	0.286764	−	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	18.0000			0.572367
$990$		0			0
$991$			45.0333i			1.43053i	0.698853	+	0.715265i	$0.253694\pi$
$991$			45.0333i			1.43053i	−0.698853	+	0.715265i	$0.746306\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.00000	−	3.46410i	0.190213	−	0.109819i
$996$		0			0
$997$	31.5000	+	18.1865i	0.997615	+	0.575973i	0.907542	−	0.419962i	$-0.137957\pi$
$997$	31.5000	+	18.1865i	0.997615	+	0.575973i	0.0900732	+	0.995935i	$0.471290\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.cx.d.559.1		2
3.2	odd	2		1008.2.cx.h.223.1	yes	2
4.3	odd	2		3024.2.cx.a.559.1		2
7.6	odd	2		3024.2.cx.g.559.1		2
9.4	even	3		3024.2.cx.f.2575.1		2
9.5	odd	6		1008.2.cx.a.895.1	yes	2
12.11	even	2		1008.2.cx.e.223.1	yes	2
21.20	even	2		1008.2.cx.d.223.1	yes	2
28.27	even	2		3024.2.cx.f.559.1		2
36.23	even	6		1008.2.cx.d.895.1	yes	2
36.31	odd	6		3024.2.cx.g.2575.1		2
63.13	odd	6		3024.2.cx.a.2575.1		2
63.41	even	6		1008.2.cx.e.895.1	yes	2
84.83	odd	2		1008.2.cx.a.223.1	✓	2
252.139	even	6	inner	3024.2.cx.d.2575.1		2
252.167	odd	6		1008.2.cx.h.895.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.cx.a.223.1	✓	2	84.83	odd	2
1008.2.cx.a.895.1	yes	2	9.5	odd	6
1008.2.cx.d.223.1	yes	2	21.20	even	2
1008.2.cx.d.895.1	yes	2	36.23	even	6
1008.2.cx.e.223.1	yes	2	12.11	even	2
1008.2.cx.e.895.1	yes	2	63.41	even	6
1008.2.cx.h.223.1	yes	2	3.2	odd	2
1008.2.cx.h.895.1	yes	2	252.167	odd	6
3024.2.cx.a.559.1		2	4.3	odd	2
3024.2.cx.a.2575.1		2	63.13	odd	6
3024.2.cx.d.559.1		2	1.1	even	1	trivial
3024.2.cx.d.2575.1		2	252.139	even	6	inner
3024.2.cx.f.559.1		2	28.27	even	2
3024.2.cx.f.2575.1		2	9.4	even	3
3024.2.cx.g.559.1		2	7.6	odd	2
3024.2.cx.g.2575.1		2	36.31	odd	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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.cx (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(-3.00000 - 1.73205i) q^{11} +3.46410i q^{17} +5.00000 q^{19} +(-4.50000 + 2.59808i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(1.00000 + 1.73205i) q^{31} +(-3.00000 + 3.46410i) q^{35} -8.00000 q^{37} +(-3.00000 - 1.73205i) q^{43} +(3.00000 - 5.19615i) q^{47} +(5.50000 - 4.33013i) q^{49} +6.00000 q^{53} +6.00000 q^{55} +(6.00000 + 10.3923i) q^{59} +(1.50000 + 0.866025i) q^{61} +(-6.00000 + 3.46410i) q^{67} +12.1244i q^{71} +6.92820i q^{73} +(-9.00000 - 1.73205i) q^{77} +(13.5000 + 7.79423i) q^{79} +(-3.00000 - 5.19615i) q^{85} +13.8564i q^{89} +(-7.50000 + 4.33013i) q^{95} +(-9.00000 - 5.19615i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} + 5 q^{7} - 6 q^{11} + 10 q^{19} - 9 q^{23} - 2 q^{25} + 2 q^{31} - 6 q^{35} - 16 q^{37} - 6 q^{43} + 6 q^{47} + 11 q^{49} + 12 q^{53} + 12 q^{55} + 12 q^{59} + 3 q^{61} - 12 q^{67} - 18 q^{77}+ \cdots - 18 q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 + 5 * q^7 - 6 * q^11 + 10 * q^19 - 9 * q^23 - 2 * q^25 + 2 * q^31 - 6 * q^35 - 16 * q^37 - 6 * q^43 + 6 * q^47 + 11 * q^49 + 12 * q^53 + 12 * q^55 + 12 * q^59 + 3 * q^61 - 12 * q^67 - 18 * q^77 + 27 * q^79 - 6 * q^85 - 15 * q^95 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more