LMFDB - Embedded newform 1008.2.cs.o.703.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(271,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))

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

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1008.cs (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$8.04892052375$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			703.2
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	1008.703
Dual form			1008.2.cs.o.271.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.35410 - 1.93649i) q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})$	$q+(3.35410 - 1.93649i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(-3.35410 - 1.93649i) q^{11} -3.46410i q^{13} +(-2.00000 - 3.46410i) q^{19} +(6.70820 - 3.87298i) q^{23} +(5.00000 - 8.66025i) q^{25} -6.70820 q^{29} +(0.500000 - 0.866025i) q^{31} +(-6.70820 + 7.74597i) q^{35} +(-2.00000 - 3.46410i) q^{37} +7.74597i q^{41} -6.92820i q^{43} +(6.70820 + 11.6190i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-3.35410 + 5.80948i) q^{53} -15.0000 q^{55} +(3.35410 - 5.80948i) q^{59} +(9.00000 - 5.19615i) q^{61} +(-6.70820 - 11.6190i) q^{65} +(6.00000 + 3.46410i) q^{67} -7.74597i q^{71} +(-6.00000 - 3.46410i) q^{73} +(10.0623 + 1.93649i) q^{77} +(-10.5000 + 6.06218i) q^{79} -6.70820 q^{83} +(6.70820 - 3.87298i) q^{89} +(3.00000 + 8.66025i) q^{91} +(-13.4164 - 7.74597i) q^{95} +5.19615i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{7}+O(q^{10}) $ 4 * q - 10 * q^7	$ 4 q - 10 q^{7} - 8 q^{19} + 20 q^{25} + 2 q^{31} - 8 q^{37} + 22 q^{49} - 60 q^{55} + 36 q^{61} + 24 q^{67} - 24 q^{73} - 42 q^{79} + 12 q^{91}+O(q^{100}) $ 4 * q - 10 * q^7 - 8 * q^19 + 20 * q^25 + 2 * q^31 - 8 * q^37 + 22 * q^49 - 60 * q^55 + 36 * q^61 + 24 * q^67 - 24 * q^73 - 42 * q^79 + 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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
$5$	3.35410	−	1.93649i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$5$	3.35410	−	1.93649i	1.50000	−	0.866025i	1.00000			$0$
$6$		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
$11$	−3.35410	−	1.93649i	−1.01130	−	0.583874i	−0.0997278	−	0.995015i	$-0.531797\pi$
$11$	−3.35410	−	1.93649i	−1.01130	−	0.583874i	−0.911572	+	0.411141i	$0.865131\pi$
$12$		0			0
$13$		−	3.46410i		−	0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$		−	3.46410i		−	0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.70820	−	3.87298i	1.39876	−	0.807573i	0.404495	−	0.914540i	$-0.367447\pi$
$23$	6.70820	−	3.87298i	1.39876	−	0.807573i	0.994263	+	0.106967i	$0.0341141\pi$
$24$		0			0
$25$	5.00000	−	8.66025i	1.00000	−	1.73205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.70820			−1.24568			−0.622841	−	0.782348i	$-0.714022\pi$
$29$	−6.70820			−1.24568			−0.622841	+	0.782348i	$0.714022\pi$
$30$		0			0
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	−0.817625	−	0.575751i	$-0.804710\pi$
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	0.907428	+	0.420208i	$0.138043\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.70820	+	7.74597i	−1.13389	+	1.30931i
$36$		0			0
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	0.653476	−	0.756948i	$-0.273310\pi$
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	−0.982274	+	0.187453i	$0.939977\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.74597i			1.20972i	0.796333	+	0.604858i	$0.206770\pi$
$41$			7.74597i			1.20972i	−0.796333	+	0.604858i	$0.793230\pi$
$42$		0			0
$43$		−	6.92820i		−	1.05654i	−0.849076	−	0.528271i	$-0.822841\pi$
$43$		−	6.92820i		−	1.05654i	0.849076	−	0.528271i	$-0.177159\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.70820	+	11.6190i	0.978492	+	1.69480i	0.667893	+	0.744257i	$0.267196\pi$
$47$	6.70820	+	11.6190i	0.978492	+	1.69480i	0.310599	+	0.950541i	$0.399470\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.35410	+	5.80948i	−0.460721	+	0.797993i	−0.998997	−	0.0447760i	$-0.985743\pi$
$53$	−3.35410	+	5.80948i	−0.460721	+	0.797993i	0.538276	+	0.842769i	$0.319076\pi$
$54$		0			0
$55$	−15.0000			−2.02260
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.35410	−	5.80948i	0.436667	−	0.756329i	−0.560763	−	0.827976i	$-0.689492\pi$
$59$	3.35410	−	5.80948i	0.436667	−	0.756329i	0.997430	+	0.0716470i	$0.0228255\pi$
$60$		0			0
$61$	9.00000	−	5.19615i	1.15233	−	0.665299i	0.202878	−	0.979204i	$-0.434971\pi$
$61$	9.00000	−	5.19615i	1.15233	−	0.665299i	0.949454	+	0.313905i	$0.101637\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.70820	−	11.6190i	−0.832050	−	1.44115i
$66$		0			0
$67$	6.00000	+	3.46410i	0.733017	+	0.423207i	0.819525	−	0.573044i	$-0.194238\pi$
$67$	6.00000	+	3.46410i	0.733017	+	0.423207i	−0.0865081	+	0.996251i	$0.527571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.74597i		−	0.919277i	−0.888106	−	0.459639i	$-0.847979\pi$
$71$		−	7.74597i		−	0.919277i	0.888106	−	0.459639i	$-0.152021\pi$
$72$		0			0
$73$	−6.00000	−	3.46410i	−0.702247	−	0.405442i	0.105937	−	0.994373i	$-0.466216\pi$
$73$	−6.00000	−	3.46410i	−0.702247	−	0.405442i	−0.808184	+	0.588930i	$0.799549\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.0623	+	1.93649i	1.14671	+	0.220684i
$78$		0			0
$79$	−10.5000	+	6.06218i	−1.18134	+	0.682048i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−10.5000	+	6.06218i	−1.18134	+	0.682048i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.70820			−0.736321			−0.368161	−	0.929762i	$-0.620012\pi$
$83$	−6.70820			−0.736321			−0.368161	+	0.929762i	$0.620012\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.70820	−	3.87298i	0.711068	−	0.410535i	−0.100388	−	0.994948i	$-0.532008\pi$
$89$	6.70820	−	3.87298i	0.711068	−	0.410535i	0.811456	+	0.584413i	$0.198675\pi$
$90$		0			0
$91$	3.00000	+	8.66025i	0.314485	+	0.907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13.4164	−	7.74597i	−1.37649	−	0.794719i
$96$		0			0
$97$			5.19615i			0.527589i	0.964579	+	0.263795i	$0.0849741\pi$
$97$			5.19615i			0.527589i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	0.598858	−	0.800855i	$-0.295621\pi$
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	−0.992990	+	0.118199i	$0.962288\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.0623	−	5.80948i	0.972760	−	0.561623i	0.0726833	−	0.997355i	$-0.476844\pi$
$107$	10.0623	−	5.80948i	0.972760	−	0.561623i	0.900076	+	0.435732i	$0.143510\pi$
$108$		0			0
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	−0.754204	−	0.656640i	$-0.771977\pi$
$109$	2.00000	−	3.46410i	0.191565	−	0.331801i	0.945769	+	0.324840i	$0.105310\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.4164			1.26211			0.631055	−	0.775738i	$-0.282622\pi$
$113$	13.4164			1.26211			0.631055	+	0.775738i	$0.282622\pi$
$114$		0			0
$115$	15.0000	−	25.9808i	1.39876	−	2.42272i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.00000	+	3.46410i	0.181818	+	0.314918i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	19.3649i		−	1.73205i
$126$		0			0
$127$			12.1244i			1.07586i	0.842989	+	0.537931i	$0.180794\pi$
$127$			12.1244i			1.07586i	−0.842989	+	0.537931i	$0.819206\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.35410	+	5.80948i	0.293049	+	0.507576i	0.974529	−	0.224261i	$-0.0719967\pi$
$131$	3.35410	+	5.80948i	0.293049	+	0.507576i	−0.681480	+	0.731837i	$0.738663\pi$
$132$		0			0
$133$	8.00000	+	6.92820i	0.693688	+	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.70820	+	11.6190i	−0.560968	+	0.971625i
$144$		0			0
$145$	−22.5000	+	12.9904i	−1.86852	+	1.07879i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.70820	+	11.6190i	0.549557	+	0.951861i	0.998305	+	0.0582028i	$0.0185370\pi$
$149$	6.70820	+	11.6190i	0.549557	+	0.951861i	−0.448747	+	0.893659i	$0.648130\pi$
$150$		0			0
$151$	13.5000	+	7.79423i	1.09861	+	0.634285i	0.935857	−	0.352381i	$-0.114628\pi$
$151$	13.5000	+	7.79423i	1.09861	+	0.634285i	0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3.87298i		−	0.311086i
$156$		0			0
$157$	−18.0000	−	10.3923i	−1.43656	−	0.829396i	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−18.0000	−	10.3923i	−1.43656	−	0.829396i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13.4164	+	15.4919i	−1.05736	+	1.22094i
$162$		0			0
$163$	9.00000	−	5.19615i	0.704934	−	0.406994i	−0.104248	−	0.994551i	$-0.533244\pi$
$163$	9.00000	−	5.19615i	0.704934	−	0.406994i	0.809183	+	0.587557i	$0.199910\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−13.4164	+	7.74597i	−1.02003	+	0.588915i	−0.914113	−	0.405460i	$-0.867111\pi$
$173$	−13.4164	+	7.74597i	−1.02003	+	0.588915i	−0.105918	+	0.994375i	$0.533778\pi$
$174$		0			0
$175$	−5.00000	+	25.9808i	−0.377964	+	1.96396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	20.1246	+	11.6190i	1.50418	+	0.868441i	0.999988	+	0.00485178i	$0.00154438\pi$
$179$	20.1246	+	11.6190i	1.50418	+	0.868441i	0.504196	+	0.863589i	$0.331789\pi$
$180$		0			0
$181$			24.2487i			1.80239i	0.433411	+	0.901196i	$0.357310\pi$
$181$			24.2487i			1.80239i	−0.433411	+	0.901196i	$0.642690\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−13.4164	−	7.74597i	−0.986394	−	0.569495i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.4164	+	7.74597i	−0.970777	+	0.560478i	−0.899473	−	0.436976i	$-0.856049\pi$
$191$	−13.4164	+	7.74597i	−0.970777	+	0.560478i	−0.0713041	+	0.997455i	$0.522716\pi$
$192$		0			0
$193$	−3.50000	+	6.06218i	−0.251936	+	0.436365i	−0.964059	−	0.265689i	$-0.914400\pi$
$193$	−3.50000	+	6.06218i	−0.251936	+	0.436365i	0.712123	+	0.702055i	$0.247734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.4164			−0.955879			−0.477940	−	0.878393i	$-0.658616\pi$
$197$	−13.4164			−0.955879			−0.477940	+	0.878393i	$0.658616\pi$
$198$		0			0
$199$	4.00000	−	6.92820i	0.283552	−	0.491127i	−0.688705	−	0.725042i	$-0.741820\pi$
$199$	4.00000	−	6.92820i	0.283552	−	0.491127i	0.972257	+	0.233915i	$0.0751537\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	16.7705	−	5.80948i	1.17706	−	0.407745i
$204$		0			0
$205$	15.0000	+	25.9808i	1.04765	+	1.81458i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			15.4919i			1.07160i
$210$		0			0
$211$		−	10.3923i		−	0.715436i	−0.933830	−	0.357718i	$-0.883555\pi$
$211$		−	10.3923i		−	0.715436i	0.933830	−	0.357718i	$-0.116445\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−13.4164	−	23.2379i	−0.914991	−	1.58481i
$216$		0			0
$217$	−0.500000	+	2.59808i	−0.0339422	+	0.176369i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−23.0000			−1.54019			−0.770097	−	0.637927i	$-0.779792\pi$
$223$	−23.0000			−1.54019			−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.35410	−	5.80948i	0.222620	−	0.385588i	−0.732983	−	0.680247i	$-0.761873\pi$
$227$	3.35410	−	5.80948i	0.222620	−	0.385588i	0.955603	+	0.294658i	$0.0952059\pi$
$228$		0			0
$229$	−15.0000	+	8.66025i	−0.991228	+	0.572286i	−0.905641	−	0.424045i	$-0.860610\pi$
$229$	−15.0000	+	8.66025i	−0.991228	+	0.572286i	−0.0855868	+	0.996331i	$0.527276\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$	45.0000	+	25.9808i	2.93548	+	1.69480i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			23.2379i			1.50313i	0.659656	+	0.751567i	$0.270702\pi$
$239$			23.2379i			1.50313i	−0.659656	+	0.751567i	$0.729298\pi$
$240$		0			0
$241$	13.5000	+	7.79423i	0.869611	+	0.502070i	0.867219	−	0.497927i	$-0.165905\pi$
$241$	13.5000	+	7.79423i	0.869611	+	0.502070i	0.00239235	+	0.999997i	$0.499238\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.0623	−	25.1744i	0.642857	−	1.60833i
$246$		0			0
$247$	−12.0000	+	6.92820i	−0.763542	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.70820			0.423418			0.211709	−	0.977333i	$-0.432097\pi$
$251$	6.70820			0.423418			0.211709	+	0.977333i	$0.432097\pi$
$252$		0			0
$253$	−30.0000			−1.88608
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.70820	+	3.87298i	−0.418446	+	0.241590i	−0.694412	−	0.719577i	$-0.744336\pi$
$257$	−6.70820	+	3.87298i	−0.418446	+	0.241590i	0.275966	+	0.961167i	$0.411002\pi$
$258$		0			0
$259$	8.00000	+	6.92820i	0.497096	+	0.430498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	13.4164	+	7.74597i	0.827291	+	0.477637i	0.852924	−	0.522035i	$-0.174827\pi$
$263$	13.4164	+	7.74597i	0.827291	+	0.477637i	−0.0256331	+	0.999671i	$0.508160\pi$
$264$		0			0
$265$			25.9808i			1.59599i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	23.4787	+	13.5554i	1.43152	+	0.826490i	0.997237	−	0.0742854i	$-0.0236676\pi$
$269$	23.4787	+	13.5554i	1.43152	+	0.826490i	0.434285	+	0.900775i	$0.357001\pi$
$270$		0			0
$271$	3.50000	+	6.06218i	0.212610	+	0.368251i	0.952531	−	0.304443i	$-0.0984703\pi$
$271$	3.50000	+	6.06218i	0.212610	+	0.368251i	−0.739921	+	0.672694i	$0.765137\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−33.5410	+	19.3649i	−2.02260	+	1.16775i
$276$		0			0
$277$	10.0000	−	17.3205i	0.600842	−	1.04069i	−0.391852	−	0.920028i	$-0.628166\pi$
$277$	10.0000	−	17.3205i	0.600842	−	1.04069i	0.992694	−	0.120660i	$-0.0385012\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	26.8328			1.60071			0.800356	−	0.599525i	$-0.204644\pi$
$281$	26.8328			1.60071			0.800356	+	0.599525i	$0.204644\pi$
$282$		0			0
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	−0.399188	−	0.916869i	$-0.630708\pi$
$283$	10.0000	−	17.3205i	0.594438	−	1.02960i	0.993626	−	0.112728i	$-0.0359589\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.70820	−	19.3649i	−0.395973	−	1.14307i
$288$		0			0
$289$	−8.50000	−	14.7224i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	11.6190i		−	0.678786i	−0.940645	−	0.339393i	$-0.889778\pi$
$293$		−	11.6190i		−	0.678786i	0.940645	−	0.339393i	$-0.110222\pi$
$294$		0			0
$295$		−	25.9808i		−	1.51266i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−13.4164	−	23.2379i	−0.775891	−	1.34388i
$300$		0			0
$301$	6.00000	+	17.3205i	0.345834	+	0.998337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	20.1246	−	34.8569i	1.15233	−	1.99590i
$306$		0			0
$307$	−14.0000			−0.799022			−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000			−0.799022			−0.399511	+	0.916728i	$0.630820\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.70820	−	11.6190i	0.380387	−	0.658850i	−0.610730	−	0.791839i	$-0.709124\pi$
$311$	6.70820	−	11.6190i	0.380387	−	0.658850i	0.991118	+	0.132989i	$0.0424573\pi$
$312$		0			0
$313$	1.50000	−	0.866025i	0.0847850	−	0.0489506i	−0.457008	−	0.889463i	$-0.651079\pi$
$313$	1.50000	−	0.866025i	0.0847850	−	0.0489506i	0.541793	+	0.840512i	$0.317746\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.0623	+	17.4284i	0.565155	+	0.978878i	0.997035	+	0.0769467i	$0.0245171\pi$
$317$	10.0623	+	17.4284i	0.565155	+	0.978878i	−0.431880	+	0.901931i	$0.642150\pi$
$318$		0			0
$319$	22.5000	+	12.9904i	1.25976	+	0.727322i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−30.0000	−	17.3205i	−1.66410	−	0.960769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−26.8328	−	23.2379i	−1.47934	−	1.28115i
$330$		0			0
$331$	6.00000	−	3.46410i	0.329790	−	0.190404i	−0.325958	−	0.945384i	$-0.605687\pi$
$331$	6.00000	−	3.46410i	0.329790	−	0.190404i	0.655748	+	0.754980i	$0.272353\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	26.8328			1.46603
$336$		0			0
$337$	−1.00000			−0.0544735			−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000			−0.0544735			−0.0272367	+	0.999629i	$0.508671\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.35410	+	1.93649i	−0.181635	+	0.104867i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.70820	−	3.87298i	−0.360115	−	0.207913i	0.309016	−	0.951057i	$-0.400000\pi$
$347$	−6.70820	−	3.87298i	−0.360115	−	0.207913i	−0.669131	+	0.743144i	$0.733334\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.4164	+	7.74597i	0.714083	+	0.412276i	0.812571	−	0.582862i	$-0.198067\pi$
$353$	13.4164	+	7.74597i	0.714083	+	0.412276i	−0.0984878	+	0.995138i	$0.531401\pi$
$354$		0			0
$355$	−15.0000	−	25.9808i	−0.796117	−	1.37892i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−26.8328			−1.40449
$366$		0			0
$367$	5.50000	−	9.52628i	0.287098	−	0.497268i	−0.686018	−	0.727585i	$-0.740643\pi$
$367$	5.50000	−	9.52628i	0.287098	−	0.497268i	0.973116	+	0.230317i	$0.0739762\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.35410	−	17.4284i	0.174136	−	0.904839i
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			23.2379i			1.19681i
$378$		0			0
$379$			17.3205i			0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$			17.3205i			0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.70820	−	11.6190i	−0.342773	−	0.593701i	0.642173	−	0.766559i	$-0.278033\pi$
$383$	−6.70820	−	11.6190i	−0.342773	−	0.593701i	−0.984947	+	0.172859i	$0.944700\pi$
$384$		0			0
$385$	37.5000	−	12.9904i	1.91118	−	0.662051i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.70820	+	11.6190i	−0.340119	+	0.589104i	−0.984455	−	0.175639i	$-0.943801\pi$
$389$	−6.70820	+	11.6190i	−0.340119	+	0.589104i	0.644335	+	0.764743i	$0.277134\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−23.4787	+	40.6663i	−1.18134	+	2.04614i
$396$		0			0
$397$	12.0000	−	6.92820i	0.602263	−	0.347717i	−0.167668	−	0.985843i	$-0.553624\pi$
$397$	12.0000	−	6.92820i	0.602263	−	0.347717i	0.769931	+	0.638127i	$0.220290\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	−3.00000	−	1.73205i	−0.149441	−	0.0862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			15.4919i			0.767907i
$408$		0			0
$409$	−7.50000	−	4.33013i	−0.370851	−	0.214111i	0.302979	−	0.952997i	$-0.402019\pi$
$409$	−7.50000	−	4.33013i	−0.370851	−	0.214111i	−0.673830	+	0.738886i	$0.735352\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.35410	+	17.4284i	−0.165045	+	0.857597i
$414$		0			0
$415$	−22.5000	+	12.9904i	−1.10448	+	0.637673i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−18.0000	+	20.7846i	−0.871081	+	1.00584i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	33.5410	+	19.3649i	1.61561	+	0.932775i	0.988036	+	0.154221i	$0.0492869\pi$
$431$	33.5410	+	19.3649i	1.61561	+	0.932775i	0.627578	+	0.778554i	$0.284046\pi$
$432$		0			0
$433$		−	6.92820i		−	0.332948i	−0.986046	−	0.166474i	$-0.946762\pi$
$433$		−	6.92820i		−	0.332948i	0.986046	−	0.166474i	$-0.0532382\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−26.8328	−	15.4919i	−1.28359	−	0.741080i
$438$		0			0
$439$	−18.5000	−	32.0429i	−0.882957	−	1.52933i	−0.848038	−	0.529936i	$-0.822216\pi$
$439$	−18.5000	−	32.0429i	−0.882957	−	1.52933i	−0.0349192	−	0.999390i	$-0.511117\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.35410	−	1.93649i	0.159358	−	0.0920055i	−0.418200	−	0.908355i	$-0.637339\pi$
$443$	3.35410	−	1.93649i	0.159358	−	0.0920055i	0.577558	+	0.816349i	$0.304006\pi$
$444$		0			0
$445$	15.0000	−	25.9808i	0.711068	−	1.23161i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−13.4164			−0.633159			−0.316580	−	0.948566i	$-0.602534\pi$
$449$	−13.4164			−0.633159			−0.316580	+	0.948566i	$0.602534\pi$
$450$		0			0
$451$	15.0000	−	25.9808i	0.706322	−	1.22339i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	26.8328	+	23.2379i	1.25794	+	1.08941i
$456$		0			0
$457$	−14.5000	−	25.1147i	−0.678281	−	1.17482i	−0.975498	−	0.220008i	$-0.929392\pi$
$457$	−14.5000	−	25.1147i	−0.678281	−	1.17482i	0.297217	−	0.954810i	$-0.403942\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	15.4919i		−	0.721531i	−0.932657	−	0.360766i	$-0.882515\pi$
$461$		−	15.4919i		−	0.721531i	0.932657	−	0.360766i	$-0.117485\pi$
$462$		0			0
$463$			24.2487i			1.12693i	0.826139	+	0.563467i	$0.190533\pi$
$463$			24.2487i			1.12693i	−0.826139	+	0.563467i	$0.809467\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	−18.0000	−	3.46410i	−0.831163	−	0.159957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−13.4164	+	23.2379i	−0.616887	+	1.06848i
$474$		0			0
$475$	−40.0000			−1.83533
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.4164	−	23.2379i	0.613011	−	1.06177i	−0.377719	−	0.925920i	$-0.623291\pi$
$479$	13.4164	−	23.2379i	0.613011	−	1.06177i	0.990730	−	0.135846i	$-0.0433753\pi$
$480$		0			0
$481$	−12.0000	+	6.92820i	−0.547153	+	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.0623	+	17.4284i	0.456906	+	0.791384i
$486$		0			0
$487$	−10.5000	−	6.06218i	−0.475800	−	0.274703i	0.242864	−	0.970060i	$-0.421913\pi$
$487$	−10.5000	−	6.06218i	−0.475800	−	0.274703i	−0.718665	+	0.695357i	$0.755246\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			19.3649i			0.873926i	0.899479	+	0.436963i	$0.143946\pi$
$491$			19.3649i			0.873926i	−0.899479	+	0.436963i	$0.856054\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.70820	+	19.3649i	0.300904	+	0.868635i
$498$		0			0
$499$	−15.0000	+	8.66025i	−0.671492	+	0.387686i	−0.796642	−	0.604452i	$-0.793392\pi$
$499$	−15.0000	+	8.66025i	−0.671492	+	0.387686i	0.125150	+	0.992138i	$0.460059\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−26.8328			−1.19642			−0.598208	−	0.801341i	$-0.704120\pi$
$503$	−26.8328			−1.19642			−0.598208	+	0.801341i	$0.704120\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.0623	−	5.80948i	0.446004	−	0.257500i	−0.260137	−	0.965572i	$-0.583768\pi$
$509$	10.0623	−	5.80948i	0.446004	−	0.257500i	0.706141	+	0.708071i	$0.250434\pi$
$510$		0			0
$511$	18.0000	+	3.46410i	0.796273	+	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−26.8328	−	15.4919i	−1.18240	−	0.682656i
$516$		0			0
$517$		−	51.9615i		−	2.28527i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.4164	−	7.74597i	−0.587784	−	0.339357i	0.176437	−	0.984312i	$-0.443543\pi$
$521$	−13.4164	−	7.74597i	−0.587784	−	0.339357i	−0.764221	+	0.644955i	$0.776876\pi$
$522$		0			0
$523$	11.0000	+	19.0526i	0.480996	+	0.833110i	0.999762	−	0.0218062i	$-0.00694167\pi$
$523$	11.0000	+	19.0526i	0.480996	+	0.833110i	−0.518766	+	0.854916i	$0.673608\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	18.5000	−	32.0429i	0.804348	−	1.39317i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	26.8328			1.16226
$534$		0			0
$535$	22.5000	−	38.9711i	0.972760	−	1.68487i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−26.8328	+	3.87298i	−1.15577	+	0.166821i
$540$		0			0
$541$	16.0000	+	27.7128i	0.687894	+	1.19147i	0.972518	+	0.232828i	$0.0747978\pi$
$541$	16.0000	+	27.7128i	0.687894	+	1.19147i	−0.284624	+	0.958639i	$0.591869\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	15.4919i		−	0.663602i
$546$		0			0
$547$		−	41.5692i		−	1.77737i	−0.458517	−	0.888686i	$-0.651619\pi$
$547$		−	41.5692i		−	1.77737i	0.458517	−	0.888686i	$-0.348381\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.4164	+	23.2379i	0.571558	+	0.989968i
$552$		0			0
$553$	21.0000	−	24.2487i	0.893011	−	1.03116i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.7705	−	29.0474i	0.710589	−	1.23078i	−0.254047	−	0.967192i	$-0.581762\pi$
$557$	16.7705	−	29.0474i	0.710589	−	1.23078i	0.964636	−	0.263585i	$-0.0849049\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.7705	−	29.0474i	0.706793	−	1.22420i	−0.259248	−	0.965811i	$-0.583475\pi$
$563$	16.7705	−	29.0474i	0.706793	−	1.22420i	0.966041	−	0.258390i	$-0.0831920\pi$
$564$		0			0
$565$	45.0000	−	25.9808i	1.89316	−	1.09302i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.4164	−	23.2379i	−0.562445	−	0.974183i	−0.997282	−	0.0736744i	$-0.976527\pi$
$569$	−13.4164	−	23.2379i	−0.562445	−	0.974183i	0.434837	−	0.900509i	$-0.356806\pi$
$570$		0			0
$571$	−15.0000	−	8.66025i	−0.627730	−	0.362420i	0.152142	−	0.988359i	$-0.451383\pi$
$571$	−15.0000	−	8.66025i	−0.627730	−	0.362420i	−0.779873	+	0.625938i	$0.784716\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	77.4597i		−	3.23029i
$576$		0			0
$577$	−19.5000	−	11.2583i	−0.811796	−	0.468690i	0.0357834	−	0.999360i	$-0.488607\pi$
$577$	−19.5000	−	11.2583i	−0.811796	−	0.468690i	−0.847579	+	0.530669i	$0.821941\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.7705	−	5.80948i	0.695758	−	0.241018i
$582$		0			0
$583$	22.5000	−	12.9904i	0.931855	−	0.538007i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.70820			−0.276877			−0.138439	−	0.990371i	$-0.544208\pi$
$587$	−6.70820			−0.276877			−0.138439	+	0.990371i	$0.544208\pi$
$588$		0			0
$589$	−4.00000			−0.164817
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.70820	+	3.87298i	−0.275473	+	0.159044i	−0.631372	−	0.775480i	$-0.717508\pi$
$593$	−6.70820	+	3.87298i	−0.275473	+	0.159044i	0.355899	+	0.934524i	$0.384175\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−26.8328	−	15.4919i	−1.09636	−	0.632983i	−0.161097	−	0.986939i	$-0.551503\pi$
$599$	−26.8328	−	15.4919i	−1.09636	−	0.632983i	−0.935262	+	0.353955i	$0.884836\pi$
$600$		0			0
$601$			43.3013i			1.76630i	0.469095	+	0.883148i	$0.344580\pi$
$601$			43.3013i			1.76630i	−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.4164	+	7.74597i	0.545455	+	0.314918i
$606$		0			0
$607$	5.50000	+	9.52628i	0.223238	+	0.386660i	0.955789	−	0.294052i	$-0.0950039\pi$
$607$	5.50000	+	9.52628i	0.223238	+	0.386660i	−0.732551	+	0.680712i	$0.761671\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	40.2492	−	23.2379i	1.62831	−	0.940105i
$612$		0			0
$613$	5.00000	−	8.66025i	0.201948	−	0.349784i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	5.00000	−	8.66025i	0.201948	−	0.349784i	0.949156	+	0.314806i	$0.101939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	40.2492			1.62037			0.810186	−	0.586172i	$-0.199366\pi$
$617$	40.2492			1.62037			0.810186	+	0.586172i	$0.199366\pi$
$618$		0			0
$619$	−19.0000	+	32.9090i	−0.763674	+	1.32272i	0.177270	+	0.984162i	$0.443273\pi$
$619$	−19.0000	+	32.9090i	−0.763674	+	1.32272i	−0.940945	+	0.338561i	$0.890060\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−13.4164	+	15.4919i	−0.537517	+	0.620671i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	36.3731i		−	1.44799i	−0.689806	−	0.723994i	$-0.742304\pi$
$631$		−	36.3731i		−	1.44799i	0.689806	−	0.723994i	$-0.257696\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.4787	+	40.6663i	0.931724	+	1.61379i
$636$		0			0
$637$	−15.0000	−	19.0526i	−0.594322	−	0.754890i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.70820	+	11.6190i	−0.264958	+	0.458921i	−0.967553	−	0.252669i	$-0.918691\pi$
$641$	−6.70820	+	11.6190i	−0.264958	+	0.458921i	0.702595	+	0.711590i	$0.252025\pi$
$642$		0			0
$643$	−8.00000			−0.315489			−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000			−0.315489			−0.157745	+	0.987480i	$0.550422\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20.1246	+	34.8569i	−0.791180	+	1.37036i	0.134057	+	0.990974i	$0.457200\pi$
$647$	−20.1246	+	34.8569i	−0.791180	+	1.37036i	−0.925237	+	0.379390i	$0.876134\pi$
$648$		0			0
$649$	−22.5000	+	12.9904i	−0.883202	+	0.509917i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−23.4787	−	40.6663i	−0.918793	−	1.59140i	−0.801251	−	0.598328i	$-0.795832\pi$
$653$	−23.4787	−	40.6663i	−0.918793	−	1.59140i	−0.117542	−	0.993068i	$-0.537501\pi$
$654$		0			0
$655$	22.5000	+	12.9904i	0.879148	+	0.507576i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	7.74597i		−	0.301740i	−0.988554	−	0.150870i	$-0.951793\pi$
$659$		−	7.74597i		−	0.301740i	0.988554	−	0.150870i	$-0.0482075\pi$
$660$		0			0
$661$	−6.00000	−	3.46410i	−0.233373	−	0.134738i	0.378754	−	0.925497i	$-0.376353\pi$
$661$	−6.00000	−	3.46410i	−0.233373	−	0.134738i	−0.612127	+	0.790759i	$0.709686\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	40.2492	+	7.74597i	1.56080	+	0.300376i
$666$		0			0
$667$	−45.0000	+	25.9808i	−1.74241	+	1.00598i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−40.2492			−1.55380
$672$		0			0
$673$	13.0000			0.501113			0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000			0.501113			0.250557	+	0.968102i	$0.419386\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.35410	−	1.93649i	0.128909	−	0.0744254i	−0.434159	−	0.900836i	$-0.642954\pi$
$677$	3.35410	−	1.93649i	0.128909	−	0.0744254i	0.563068	+	0.826411i	$0.309621\pi$
$678$		0			0
$679$	−4.50000	−	12.9904i	−0.172694	−	0.498525i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.35410	−	1.93649i	−0.128341	−	0.0740978i	0.434455	−	0.900694i	$-0.356941\pi$
$683$	−3.35410	−	1.93649i	−0.128341	−	0.0740978i	−0.562796	+	0.826596i	$0.690274\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20.1246	+	11.6190i	0.766687	+	0.442647i
$690$		0			0
$691$	−7.00000	−	12.1244i	−0.266293	−	0.461232i	0.701609	−	0.712562i	$-0.252465\pi$
$691$	−7.00000	−	12.1244i	−0.266293	−	0.461232i	−0.967901	+	0.251330i	$0.919132\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.70820	−	3.87298i	0.254457	−	0.146911i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.1246			−0.760096			−0.380048	−	0.924967i	$-0.624093\pi$
$701$	−20.1246			−0.760096			−0.380048	+	0.924967i	$0.624093\pi$
$702$		0			0
$703$	−8.00000	+	13.8564i	−0.301726	+	0.522604i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	0.995228	−	0.0975728i	$-0.0311079\pi$
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	−0.582115	+	0.813107i	$0.697775\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7.74597i		−	0.290089i
$714$		0			0
$715$			51.9615i			1.94325i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.1246	+	34.8569i	0.750521	+	1.29994i	0.947570	+	0.319547i	$0.103531\pi$
$719$	20.1246	+	34.8569i	0.750521	+	1.29994i	−0.197049	+	0.980394i	$0.563136\pi$
$720$		0			0
$721$	16.0000	+	13.8564i	0.595871	+	0.516040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−33.5410	+	58.0948i	−1.24568	+	2.15758i
$726$		0			0
$727$	5.00000			0.185440			0.0927199	−	0.995692i	$-0.470444\pi$
$727$	5.00000			0.185440			0.0927199	+	0.995692i	$0.470444\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	6.00000	−	3.46410i	0.221615	−	0.127950i	−0.385083	−	0.922882i	$-0.625827\pi$
$733$	6.00000	−	3.46410i	0.221615	−	0.127950i	0.606698	+	0.794933i	$0.292494\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13.4164	−	23.2379i	−0.494200	−	0.855979i
$738$		0			0
$739$	3.00000	+	1.73205i	0.110357	+	0.0637145i	0.554162	−	0.832409i	$-0.313039\pi$
$739$	3.00000	+	1.73205i	0.110357	+	0.0637145i	−0.443806	+	0.896123i	$0.646372\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			15.4919i			0.568344i	0.958773	+	0.284172i	$0.0917187\pi$
$743$			15.4919i			0.568344i	−0.958773	+	0.284172i	$0.908281\pi$
$744$		0			0
$745$	45.0000	+	25.9808i	1.64867	+	0.951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−20.1246	+	23.2379i	−0.735337	+	0.849094i
$750$		0			0
$751$	43.5000	−	25.1147i	1.58734	−	0.916450i	0.593594	−	0.804765i	$-0.297709\pi$
$751$	43.5000	−	25.1147i	1.58734	−	0.916450i	0.993744	−	0.111685i	$-0.0356248\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	60.3738			2.19723
$756$		0			0
$757$	−34.0000			−1.23575			−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000			−1.23575			−0.617876	+	0.786276i	$0.712006\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.1246	−	11.6190i	0.729517	−	0.421187i	−0.0887287	−	0.996056i	$-0.528280\pi$
$761$	20.1246	−	11.6190i	0.729517	−	0.421187i	0.818245	+	0.574869i	$0.194947\pi$
$762$		0			0
$763$	−2.00000	+	10.3923i	−0.0724049	+	0.376227i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−20.1246	−	11.6190i	−0.726658	−	0.419536i
$768$		0			0
$769$		−	12.1244i		−	0.437215i	−0.975813	−	0.218608i	$-0.929848\pi$
$769$		−	12.1244i		−	0.437215i	0.975813	−	0.218608i	$-0.0701515\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−13.4164	−	7.74597i	−0.482555	−	0.278603i	0.238926	−	0.971038i	$-0.423205\pi$
$773$	−13.4164	−	7.74597i	−0.482555	−	0.278603i	−0.721480	+	0.692435i	$0.756538\pi$
$774$		0			0
$775$	−5.00000	−	8.66025i	−0.179605	−	0.311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	26.8328	−	15.4919i	0.961385	−	0.555056i
$780$		0			0
$781$	−15.0000	+	25.9808i	−0.536742	+	0.929665i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−80.4984			−2.87311
$786$		0			0
$787$	−7.00000	+	12.1244i	−0.249523	+	0.432187i	−0.963394	−	0.268091i	$-0.913607\pi$
$787$	−7.00000	+	12.1244i	−0.249523	+	0.432187i	0.713871	+	0.700278i	$0.246941\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−33.5410	+	11.6190i	−1.19258	+	0.413122i
$792$		0			0
$793$	−18.0000	−	31.1769i	−0.639199	−	1.10712i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			50.3488i			1.78345i	0.452582	+	0.891723i	$0.350503\pi$
$797$			50.3488i			1.78345i	−0.452582	+	0.891723i	$0.649497\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.4164	+	23.2379i	0.473455	+	0.820048i
$804$		0			0
$805$	−15.0000	+	77.9423i	−0.528681	+	2.74710i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.4164	+	23.2379i	−0.471696	+	0.817001i	−0.999476	−	0.0323801i	$-0.989691\pi$
$809$	−13.4164	+	23.2379i	−0.471696	+	0.817001i	0.527780	+	0.849381i	$0.323025\pi$
$810$		0			0
$811$	44.0000			1.54505			0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000			1.54505			0.772524	+	0.634985i	$0.218994\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	20.1246	−	34.8569i	0.704934	−	1.22098i
$816$		0			0
$817$	−24.0000	+	13.8564i	−0.839654	+	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.35410	−	5.80948i	−0.117059	−	0.202752i	0.801542	−	0.597939i	$-0.204013\pi$
$821$	−3.35410	−	5.80948i	−0.117059	−	0.202752i	−0.918601	+	0.395186i	$0.870680\pi$
$822$		0			0
$823$	9.00000	+	5.19615i	0.313720	+	0.181126i	0.648590	−	0.761138i	$-0.275359\pi$
$823$	9.00000	+	5.19615i	0.313720	+	0.181126i	−0.334870	+	0.942264i	$0.608692\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	3.87298i		−	0.134677i	−0.997730	−	0.0673384i	$-0.978549\pi$
$827$		−	3.87298i		−	0.134677i	0.997730	−	0.0673384i	$-0.0214507\pi$
$828$		0			0
$829$	−12.0000	−	6.92820i	−0.416777	−	0.240626i	0.276920	−	0.960893i	$-0.410686\pi$
$829$	−12.0000	−	6.92820i	−0.416777	−	0.240626i	−0.693698	+	0.720266i	$0.744020\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−26.8328			−0.926372			−0.463186	−	0.886261i	$-0.653294\pi$
$839$	−26.8328			−0.926372			−0.463186	+	0.886261i	$0.653294\pi$
$840$		0			0
$841$	16.0000			0.551724
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.35410	−	1.93649i	0.115385	−	0.0666173i
$846$		0			0
$847$	−8.00000	−	6.92820i	−0.274883	−	0.238056i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−26.8328	−	15.4919i	−0.919817	−	0.531057i
$852$		0			0
$853$			6.92820i			0.237217i	0.992941	+	0.118609i	$0.0378434\pi$
$853$			6.92820i			0.237217i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−46.9574	−	27.1109i	−1.60404	−	0.926090i	−0.990669	−	0.136292i	$-0.956482\pi$
$857$	−46.9574	−	27.1109i	−1.60404	−	0.926090i	−0.613366	−	0.789798i	$-0.710185\pi$
$858$		0			0
$859$	−8.00000	−	13.8564i	−0.272956	−	0.472774i	0.696661	−	0.717400i	$-0.254668\pi$
$859$	−8.00000	−	13.8564i	−0.272956	−	0.472774i	−0.969618	+	0.244626i	$0.921335\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20.1246	−	11.6190i	0.685050	−	0.395514i	−0.116705	−	0.993167i	$-0.537233\pi$
$863$	20.1246	−	11.6190i	0.685050	−	0.395514i	0.801755	+	0.597653i	$0.203900\pi$
$864$		0			0
$865$	−30.0000	+	51.9615i	−1.02003	+	1.76674i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	46.9574			1.59292
$870$		0			0
$871$	12.0000	−	20.7846i	0.406604	−	0.704260i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	16.7705	+	48.4123i	0.566947	+	1.63663i
$876$		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
$881$		−	15.4919i		−	0.521936i	−0.965347	−	0.260968i	$-0.915958\pi$
$881$		−	15.4919i		−	0.521936i	0.965347	−	0.260968i	$-0.0840418\pi$
$882$		0			0
$883$		−	3.46410i		−	0.116576i	−0.998300	−	0.0582882i	$-0.981436\pi$
$883$		−	3.46410i		−	0.116576i	0.998300	−	0.0582882i	$-0.0185642\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.70820	−	11.6190i	−0.225239	−	0.390126i	0.731152	−	0.682215i	$-0.238983\pi$
$887$	−6.70820	−	11.6190i	−0.225239	−	0.390126i	−0.956391	+	0.292089i	$0.905650\pi$
$888$		0			0
$889$	−10.5000	−	30.3109i	−0.352159	−	1.01659i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26.8328	−	46.4758i	0.897926	−	1.55525i
$894$		0			0
$895$	90.0000			3.00837
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.35410	+	5.80948i	−0.111866	+	0.193757i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	46.9574	+	81.3327i	1.56092	+	2.70359i
$906$		0			0
$907$	3.00000	+	1.73205i	0.0996134	+	0.0575118i	0.548979	−	0.835836i	$-0.315017\pi$
$907$	3.00000	+	1.73205i	0.0996134	+	0.0575118i	−0.449366	+	0.893348i	$0.648350\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			23.2379i			0.769906i	0.922936	+	0.384953i	$0.125782\pi$
$911$			23.2379i			0.769906i	−0.922936	+	0.384953i	$0.874218\pi$
$912$		0			0
$913$	22.5000	+	12.9904i	0.744641	+	0.429919i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−13.4164	−	11.6190i	−0.443049	−	0.383692i
$918$		0			0
$919$	33.0000	−	19.0526i	1.08857	−	0.628486i	0.155374	−	0.987856i	$-0.450342\pi$
$919$	33.0000	−	19.0526i	1.08857	−	0.628486i	0.933195	+	0.359370i	$0.117008\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−26.8328			−0.883213
$924$		0			0
$925$	−40.0000			−1.31519
$926$		0			0
$927$		0			0
$928$		0			0
$929$	40.2492	−	23.2379i	1.32053	−	0.762411i	0.336720	−	0.941605i	$-0.390682\pi$
$929$	40.2492	−	23.2379i	1.32053	−	0.762411i	0.983814	+	0.179194i	$0.0573489\pi$
$930$		0			0
$931$	−26.0000	−	10.3923i	−0.852116	−	0.340594i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			12.1244i			0.396085i	0.980193	+	0.198043i	$0.0634585\pi$
$937$			12.1244i			0.396085i	−0.980193	+	0.198043i	$0.936542\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16.7705	+	9.68246i	0.546703	+	0.315639i	0.747791	−	0.663934i	$-0.231114\pi$
$941$	16.7705	+	9.68246i	0.546703	+	0.315639i	−0.201088	+	0.979573i	$0.564448\pi$
$942$		0			0
$943$	30.0000	+	51.9615i	0.976934	+	1.69210i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.70820	−	3.87298i	0.217987	−	0.125855i	−0.387031	−	0.922067i	$-0.626499\pi$
$947$	6.70820	−	3.87298i	0.217987	−	0.125855i	0.605018	+	0.796212i	$0.293166\pi$
$948$		0			0
$949$	−12.0000	+	20.7846i	−0.389536	+	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.8328			−0.869200			−0.434600	−	0.900624i	$-0.643110\pi$
$953$	−26.8328			−0.869200			−0.434600	+	0.900624i	$0.643110\pi$
$954$		0			0
$955$	−30.0000	+	51.9615i	−0.970777	+	1.68144i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			27.1109i			0.872730i
$966$		0			0
$967$			29.4449i			0.946883i	0.880825	+	0.473441i	$0.156988\pi$
$967$			29.4449i			0.946883i	−0.880825	+	0.473441i	$0.843012\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−23.4787	−	40.6663i	−0.753468	−	1.30504i	−0.946132	−	0.323780i	$-0.895046\pi$
$971$	−23.4787	−	40.6663i	−0.753468	−	1.30504i	0.192665	−	0.981265i	$-0.438287\pi$
$972$		0			0
$973$	−5.00000	+	1.73205i	−0.160293	+	0.0555270i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	26.8328	−	46.4758i	0.858458	−	1.48689i	−0.0149418	−	0.999888i	$-0.504756\pi$
$977$	26.8328	−	46.4758i	0.858458	−	1.48689i	0.873400	−	0.487004i	$-0.161910\pi$
$978$		0			0
$979$	−30.0000			−0.958804
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.70820	+	11.6190i	−0.213958	+	0.370587i	−0.952950	−	0.303128i	$-0.901969\pi$
$983$	−6.70820	+	11.6190i	−0.213958	+	0.370587i	0.738991	+	0.673715i	$0.235302\pi$
$984$		0			0
$985$	−45.0000	+	25.9808i	−1.43382	+	0.827816i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26.8328	−	46.4758i	−0.853234	−	1.47784i
$990$		0			0
$991$	1.50000	+	0.866025i	0.0476491	+	0.0275102i	0.523635	−	0.851943i	$-0.324575\pi$
$991$	1.50000	+	0.866025i	0.0476491	+	0.0275102i	−0.475986	+	0.879453i	$0.657909\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	30.9839i		−	0.982255i
$996$		0			0
$997$	−48.0000	−	27.7128i	−1.52018	−	0.877674i	−0.999717	−	0.0237864i	$-0.992428\pi$
$997$	−48.0000	−	27.7128i	−1.52018	−	0.877674i	−0.520458	−	0.853887i	$-0.674239\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.2.cs.o.703.2	yes	4
3.2	odd	2	inner	1008.2.cs.o.703.1	yes	4
4.3	odd	2		1008.2.cs.p.703.2	yes	4
7.3	odd	6		7056.2.b.o.1567.2		4
7.4	even	3		7056.2.b.v.1567.3		4
7.5	odd	6		1008.2.cs.p.271.2	yes	4
12.11	even	2		1008.2.cs.p.703.1	yes	4
21.5	even	6		1008.2.cs.p.271.1	yes	4
21.11	odd	6		7056.2.b.v.1567.1		4
21.17	even	6		7056.2.b.o.1567.4		4
28.3	even	6		7056.2.b.v.1567.2		4
28.11	odd	6		7056.2.b.o.1567.3		4
28.19	even	6	inner	1008.2.cs.o.271.2	yes	4
84.11	even	6		7056.2.b.o.1567.1		4
84.47	odd	6	inner	1008.2.cs.o.271.1	✓	4
84.59	odd	6		7056.2.b.v.1567.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.cs.o.271.1	✓	4	84.47	odd	6	inner
1008.2.cs.o.271.2	yes	4	28.19	even	6	inner
1008.2.cs.o.703.1	yes	4	3.2	odd	2	inner
1008.2.cs.o.703.2	yes	4	1.1	even	1	trivial
1008.2.cs.p.271.1	yes	4	21.5	even	6
1008.2.cs.p.271.2	yes	4	7.5	odd	6
1008.2.cs.p.703.1	yes	4	12.11	even	2
1008.2.cs.p.703.2	yes	4	4.3	odd	2
7056.2.b.o.1567.1		4	84.11	even	6
7056.2.b.o.1567.2		4	7.3	odd	6
7056.2.b.o.1567.3		4	28.11	odd	6
7056.2.b.o.1567.4		4	21.17	even	6
7056.2.b.v.1567.1		4	21.11	odd	6
7056.2.b.v.1567.2		4	28.3	even	6
7056.2.b.v.1567.3		4	7.4	even	3
7056.2.b.v.1567.4		4	84.59	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cs (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.35410 - 1.93649i) q^{5} +(-2.50000 + 0.866025i) q^{7} +O(q^{10})\)	\(q+(3.35410 - 1.93649i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(-3.35410 - 1.93649i) q^{11} -3.46410i q^{13} +(-2.00000 - 3.46410i) q^{19} +(6.70820 - 3.87298i) q^{23} +(5.00000 - 8.66025i) q^{25} -6.70820 q^{29} +(0.500000 - 0.866025i) q^{31} +(-6.70820 + 7.74597i) q^{35} +(-2.00000 - 3.46410i) q^{37} +7.74597i q^{41} -6.92820i q^{43} +(6.70820 + 11.6190i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-3.35410 + 5.80948i) q^{53} -15.0000 q^{55} +(3.35410 - 5.80948i) q^{59} +(9.00000 - 5.19615i) q^{61} +(-6.70820 - 11.6190i) q^{65} +(6.00000 + 3.46410i) q^{67} -7.74597i q^{71} +(-6.00000 - 3.46410i) q^{73} +(10.0623 + 1.93649i) q^{77} +(-10.5000 + 6.06218i) q^{79} -6.70820 q^{83} +(6.70820 - 3.87298i) q^{89} +(3.00000 + 8.66025i) q^{91} +(-13.4164 - 7.74597i) q^{95} +5.19615i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{7}+O(q^{10}) \) 4 * q - 10 * q^7	\( 4 q - 10 q^{7} - 8 q^{19} + 20 q^{25} + 2 q^{31} - 8 q^{37} + 22 q^{49} - 60 q^{55} + 36 q^{61} + 24 q^{67} - 24 q^{73} - 42 q^{79} + 12 q^{91}+O(q^{100}) \) 4 * q - 10 * q^7 - 8 * q^19 + 20 * q^25 + 2 * q^31 - 8 * q^37 + 22 * q^49 - 60 * q^55 + 36 * q^61 + 24 * q^67 - 24 * q^73 - 42 * q^79 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more