LMFDB - Embedded newform 1008.2.cs.e.271.1

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:	$2$
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 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1008.271
Dual form			1008.2.cs.e.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 0.866025i) q^{5} +(0.500000 - 2.59808i) q^{7} +O(q^{10})$	$q+(-1.50000 - 0.866025i) q^{5} +(0.500000 - 2.59808i) q^{7} +(-1.50000 + 0.866025i) q^{11} +(-3.00000 + 1.73205i) q^{17} +(1.00000 - 1.73205i) q^{19} +(-1.00000 - 1.73205i) q^{25} -9.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-3.00000 + 3.46410i) q^{35} +(-5.00000 + 8.66025i) q^{37} +10.3923i q^{41} +3.46410i q^{43} +(6.00000 - 10.3923i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-4.50000 - 7.79423i) q^{53} +3.00000 q^{55} +(-4.50000 - 7.79423i) q^{59} +(-12.0000 + 6.92820i) q^{67} -13.8564i q^{71} +(-6.00000 + 3.46410i) q^{73} +(1.50000 + 4.33013i) q^{77} +(4.50000 + 2.59808i) q^{79} -3.00000 q^{83} +6.00000 q^{85} +(3.00000 + 1.73205i) q^{89} +(-3.00000 + 1.73205i) q^{95} -19.0526i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} + q^{7}+O(q^{10}) $ 2 * q - 3 * q^5 + q^7	$ 2 q - 3 q^{5} + q^{7} - 3 q^{11} - 6 q^{17} + 2 q^{19} - 2 q^{25} - 18 q^{29} - 5 q^{31} - 6 q^{35} - 10 q^{37} + 12 q^{47} - 13 q^{49} - 9 q^{53} + 6 q^{55} - 9 q^{59} - 24 q^{67} - 12 q^{73} + 3 q^{77} + 9 q^{79} - 6 q^{83} + 12 q^{85} + 6 q^{89} - 6 q^{95}+O(q^{100}) $ 2 * q - 3 * q^5 + q^7 - 3 * q^11 - 6 * q^17 + 2 * q^19 - 2 * q^25 - 18 * q^29 - 5 * q^31 - 6 * q^35 - 10 * q^37 + 12 * q^47 - 13 * q^49 - 9 * q^53 + 6 * q^55 - 9 * q^59 - 24 * q^67 - 12 * q^73 + 3 * q^77 + 9 * q^79 - 6 * q^83 + 12 * q^85 + 6 * q^89 - 6 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	0.125567	−	0.992085i	$-0.459925\pi$
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	−0.796387	+	0.604787i	$0.793258\pi$
$6$		0			0
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.50000	+	0.866025i	−0.452267	+	0.261116i	−0.708787	−	0.705422i	$-0.750757\pi$
$11$	−1.50000	+	0.866025i	−0.452267	+	0.261116i	0.256520	+	0.966539i	$0.417424\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000	+	1.73205i	−0.727607	+	0.420084i	−0.817546	−	0.575863i	$-0.804666\pi$
$17$	−3.00000	+	1.73205i	−0.727607	+	0.420084i	0.0899392	+	0.995947i	$0.471333\pi$
$18$		0			0
$19$	1.00000	−	1.73205i	0.229416	−	0.397360i	−0.728219	−	0.685344i	$-0.759652\pi$
$19$	1.00000	−	1.73205i	0.229416	−	0.397360i	0.957635	+	0.287984i	$0.0929851\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−1.00000	−	1.73205i	−0.200000	−	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	0.549309	−	0.835619i	$-0.314891\pi$
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	−0.998322	+	0.0579057i	$0.981558\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000	+	3.46410i	−0.507093	+	0.585540i
$36$		0			0
$37$	−5.00000	+	8.66025i	−0.821995	+	1.42374i	0.0821995	+	0.996616i	$0.473806\pi$
$37$	−5.00000	+	8.66025i	−0.821995	+	1.42374i	−0.904194	+	0.427121i	$0.859528\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.3923i			1.62301i	0.584349	+	0.811503i	$0.301350\pi$
$41$			10.3923i			1.62301i	−0.584349	+	0.811503i	$0.698650\pi$
$42$		0			0
$43$			3.46410i			0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$			3.46410i			0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000	−	10.3923i	0.875190	−	1.51587i	0.0186297	−	0.999826i	$-0.494070\pi$
$47$	6.00000	−	10.3923i	0.875190	−	1.51587i	0.856560	−	0.516047i	$-0.172597\pi$
$48$		0			0
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	−0.989828	−	0.142269i	$-0.954560\pi$
$53$	−4.50000	−	7.79423i	−0.618123	−	1.07062i	0.371706	−	0.928351i	$-0.378773\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	−0.994769	−	0.102151i	$-0.967427\pi$
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	0.408919	−	0.912571i	$-0.365906\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−12.0000	+	6.92820i	−1.46603	+	0.846415i	−0.999279	−	0.0379722i	$-0.987910\pi$
$67$	−12.0000	+	6.92820i	−1.46603	+	0.846415i	−0.466755	+	0.884387i	$0.654577\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	13.8564i		−	1.64445i	−0.569160	−	0.822226i	$-0.692732\pi$
$71$		−	13.8564i		−	1.64445i	0.569160	−	0.822226i	$-0.307268\pi$
$72$		0			0
$73$	−6.00000	+	3.46410i	−0.702247	+	0.405442i	−0.808184	−	0.588930i	$-0.799549\pi$
$73$	−6.00000	+	3.46410i	−0.702247	+	0.405442i	0.105937	+	0.994373i	$0.466216\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.50000	+	4.33013i	0.170941	+	0.493464i
$78$		0			0
$79$	4.50000	+	2.59808i	0.506290	+	0.292306i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	4.50000	+	2.59808i	0.506290	+	0.292306i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.00000			−0.329293			−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000			−0.329293			−0.164646	+	0.986353i	$0.552648\pi$
$84$		0			0
$85$	6.00000			0.650791
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000	+	1.73205i	0.317999	+	0.183597i	0.650500	−	0.759506i	$-0.274559\pi$
$89$	3.00000	+	1.73205i	0.317999	+	0.183597i	−0.332501	+	0.943103i	$0.607893\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.00000	+	1.73205i	−0.307794	+	0.177705i
$96$		0			0
$97$		−	19.0526i		−	1.93449i	−0.253837	−	0.967247i	$-0.581693\pi$
$97$		−	19.0526i		−	1.93449i	0.253837	−	0.967247i	$-0.418307\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.0000	−	6.92820i	1.19404	−	0.689382i	0.234823	−	0.972038i	$-0.424549\pi$
$101$	12.0000	−	6.92820i	1.19404	−	0.689382i	0.959221	+	0.282656i	$0.0912155\pi$
$102$		0			0
$103$	2.00000	−	3.46410i	0.197066	−	0.341328i	−0.750510	−	0.660859i	$-0.770192\pi$
$103$	2.00000	−	3.46410i	0.197066	−	0.341328i	0.947576	+	0.319531i	$0.103525\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.5000	+	6.06218i	1.01507	+	0.586053i	0.912673	−	0.408690i	$-0.134014\pi$
$107$	10.5000	+	6.06218i	1.01507	+	0.586053i	0.102400	+	0.994743i	$0.467348\pi$
$108$		0			0
$109$	2.00000	+	3.46410i	0.191565	+	0.331801i	0.945769	−	0.324840i	$-0.105310\pi$
$109$	2.00000	+	3.46410i	0.191565	+	0.331801i	−0.754204	+	0.656640i	$0.771977\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.00000	+	8.66025i	0.275010	+	0.793884i
$120$		0			0
$121$	−4.00000	+	6.92820i	−0.363636	+	0.629837i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.1244i			1.08444i
$126$		0			0
$127$			5.19615i			0.461084i	0.973062	+	0.230542i	$0.0740499\pi$
$127$			5.19615i			0.461084i	−0.973062	+	0.230542i	$0.925950\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.50000	+	7.79423i	−0.393167	+	0.680985i	−0.992865	−	0.119241i	$-0.961954\pi$
$131$	−4.50000	+	7.79423i	−0.393167	+	0.680985i	0.599699	+	0.800226i	$0.295287\pi$
$132$		0			0
$133$	−4.00000	−	3.46410i	−0.346844	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	−0.999893	−	0.0146279i	$-0.995344\pi$
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	0.487278	−	0.873247i	$-0.337990\pi$
$138$		0			0
$139$	14.0000			1.18746			0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000			1.18746			0.593732	+	0.804663i	$0.297654\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	13.5000	+	7.79423i	1.12111	+	0.647275i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	4.50000	−	2.59808i	0.366205	−	0.211428i	−0.305594	−	0.952162i	$-0.598855\pi$
$151$	4.50000	−	2.59808i	0.366205	−	0.211428i	0.671799	+	0.740733i	$0.265522\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			8.66025i			0.695608i
$156$		0			0
$157$	6.00000	−	3.46410i	0.478852	−	0.276465i	−0.241086	−	0.970504i	$-0.577504\pi$
$157$	6.00000	−	3.46410i	0.478852	−	0.276465i	0.719938	+	0.694038i	$0.244170\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−12.0000	−	6.92820i	−0.939913	−	0.542659i	−0.0499796	−	0.998750i	$-0.515916\pi$
$163$	−12.0000	−	6.92820i	−0.939913	−	0.542659i	−0.889933	+	0.456091i	$0.849249\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	−5.00000	+	1.73205i	−0.377964	+	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−21.0000	+	12.1244i	−1.56961	+	0.906217i	−0.573400	+	0.819275i	$0.694376\pi$
$179$	−21.0000	+	12.1244i	−1.56961	+	0.906217i	−0.996213	+	0.0869415i	$0.972291\pi$
$180$		0			0
$181$			10.3923i			0.772454i	0.922404	+	0.386227i	$0.126222\pi$
$181$			10.3923i			0.772454i	−0.922404	+	0.386227i	$0.873778\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.0000	−	8.66025i	1.10282	−	0.636715i
$186$		0			0
$187$	3.00000	−	5.19615i	0.219382	−	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.0000	+	6.92820i	0.868290	+	0.501307i	0.866779	−	0.498692i	$-0.166186\pi$
$191$	12.0000	+	6.92820i	0.868290	+	0.501307i	0.00151007	+	0.999999i	$0.499519\pi$
$192$		0			0
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	0.941865	−	0.335993i	$-0.109072\pi$
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	−0.761911	+	0.647682i	$0.775738\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	−0.996850	−	0.0793045i	$-0.974730\pi$
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	0.429745	−	0.902950i	$-0.358603\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.50000	+	23.3827i	−0.315838	+	1.64114i
$204$		0			0
$205$	9.00000	−	15.5885i	0.628587	−	1.08875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3.46410i			0.239617i
$210$		0			0
$211$		−	6.92820i		−	0.476957i	−0.971148	−	0.238479i	$-0.923351\pi$
$211$		−	6.92820i		−	0.476957i	0.971148	−	0.238479i	$-0.0766487\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.00000	−	5.19615i	0.204598	−	0.354375i
$216$		0			0
$217$	−12.5000	+	4.33013i	−0.848555	+	0.293948i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	19.0000			1.27233			0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000			1.27233			0.636167	+	0.771551i	$0.280519\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.5000	−	18.1865i	−0.696909	−	1.20708i	−0.969533	−	0.244962i	$-0.921225\pi$
$227$	−10.5000	−	18.1865i	−0.696909	−	1.20708i	0.272623	−	0.962121i	$-0.412109\pi$
$228$		0			0
$229$	−3.00000	−	1.73205i	−0.198246	−	0.114457i	0.397591	−	0.917563i	$-0.369846\pi$
$229$	−3.00000	−	1.73205i	−0.198246	−	0.114457i	−0.595837	+	0.803105i	$0.703180\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	−0.404674	−	0.914461i	$-0.632615\pi$
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	0.994283	−	0.106773i	$-0.0340517\pi$
$234$		0			0
$235$	−18.0000	+	10.3923i	−1.17419	+	0.677919i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.92820i		−	0.448148i	−0.974572	−	0.224074i	$-0.928064\pi$
$239$		−	6.92820i		−	0.448148i	0.974572	−	0.224074i	$-0.0719358\pi$
$240$		0			0
$241$	19.5000	−	11.2583i	1.25611	−	0.725213i	0.283790	−	0.958886i	$-0.408408\pi$
$241$	19.5000	−	11.2583i	1.25611	−	0.725213i	0.972315	+	0.233674i	$0.0750747\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.50000	+	9.52628i	0.479157	+	0.608612i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.0000			0.946792			0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000			0.946792			0.473396	+	0.880850i	$0.343028\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	+	5.19615i	0.561405	+	0.324127i	0.753709	−	0.657208i	$-0.228263\pi$
$257$	9.00000	+	5.19615i	0.561405	+	0.324127i	−0.192304	+	0.981335i	$0.561596\pi$
$258$		0			0
$259$	20.0000	+	17.3205i	1.24274	+	1.07624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−15.0000	+	8.66025i	−0.924940	+	0.534014i	−0.885208	−	0.465196i	$-0.845984\pi$
$263$	−15.0000	+	8.66025i	−0.924940	+	0.534014i	−0.0397320	+	0.999210i	$0.512650\pi$
$264$		0			0
$265$			15.5885i			0.957591i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.5000	−	7.79423i	0.823110	−	0.475223i	−0.0283781	−	0.999597i	$-0.509034\pi$
$269$	13.5000	−	7.79423i	0.823110	−	0.475223i	0.851488	+	0.524375i	$0.175701\pi$
$270$		0			0
$271$	−5.50000	+	9.52628i	−0.334101	+	0.578680i	−0.983312	−	0.181928i	$-0.941766\pi$
$271$	−5.50000	+	9.52628i	−0.334101	+	0.578680i	0.649211	+	0.760609i	$0.275099\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.00000	+	1.73205i	0.180907	+	0.104447i
$276$		0			0
$277$	−14.0000	−	24.2487i	−0.841178	−	1.45696i	−0.888899	−	0.458103i	$-0.848529\pi$
$277$	−14.0000	−	24.2487i	−0.841178	−	1.45696i	0.0477206	−	0.998861i	$-0.484804\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	24.0000			1.43172			0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000			1.43172			0.715860	+	0.698244i	$0.246035\pi$
$282$		0			0
$283$	7.00000	+	12.1244i	0.416107	+	0.720718i	0.995544	−	0.0942988i	$-0.0300609\pi$
$283$	7.00000	+	12.1244i	0.416107	+	0.720718i	−0.579437	+	0.815017i	$0.696728\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	27.0000	+	5.19615i	1.59376	+	0.306719i
$288$		0			0
$289$	−2.50000	+	4.33013i	−0.147059	+	0.254713i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	5.19615i		−	0.303562i	−0.988414	−	0.151781i	$-0.951499\pi$
$293$		−	5.19615i		−	0.303562i	0.988414	−	0.151781i	$-0.0485009\pi$
$294$		0			0
$295$			15.5885i			0.907595i
$296$		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
$305$		0			0
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		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
$313$	−4.50000	−	2.59808i	−0.254355	−	0.146852i	0.367402	−	0.930062i	$-0.380247\pi$
$313$	−4.50000	−	2.59808i	−0.254355	−	0.146852i	−0.621757	+	0.783210i	$0.713581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.5000	+	18.1865i	−0.589739	+	1.02146i	0.404528	+	0.914526i	$0.367436\pi$
$317$	−10.5000	+	18.1865i	−0.589739	+	1.02146i	−0.994266	+	0.106932i	$0.965897\pi$
$318$		0			0
$319$	13.5000	−	7.79423i	0.755855	−	0.436393i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6.92820i			0.385496i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−24.0000	−	20.7846i	−1.32316	−	1.14589i
$330$		0			0
$331$	12.0000	+	6.92820i	0.659580	+	0.380808i	0.792117	−	0.610370i	$-0.208979\pi$
$331$	12.0000	+	6.92820i	0.659580	+	0.380808i	−0.132537	+	0.991178i	$0.542312\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	24.0000			1.31126
$336$		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
$341$	7.50000	+	4.33013i	0.406148	+	0.234490i
$342$		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
$349$		−	6.92820i		−	0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$		−	6.92820i		−	0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000	+	3.46410i	−0.319348	+	0.184376i	−0.651102	−	0.758990i	$-0.725693\pi$
$353$	−6.00000	+	3.46410i	−0.319348	+	0.184376i	0.331754	+	0.943366i	$0.392360\pi$
$354$		0			0
$355$	−12.0000	+	20.7846i	−0.636894	+	1.10313i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.0000	−	13.8564i	−1.26667	−	0.731313i	−0.292315	−	0.956322i	$-0.594426\pi$
$359$	−24.0000	−	13.8564i	−1.26667	−	0.731313i	−0.974357	+	0.225009i	$0.927759\pi$
$360$		0			0
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.0000			0.628109
$366$		0			0
$367$	−9.50000	−	16.4545i	−0.495896	−	0.858917i	0.504093	−	0.863649i	$-0.331827\pi$
$367$	−9.50000	−	16.4545i	−0.495896	−	0.858917i	−0.999989	+	0.00473247i	$0.998494\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−22.5000	+	7.79423i	−1.16814	+	0.404656i
$372$		0			0
$373$	−1.00000	+	1.73205i	−0.0517780	+	0.0896822i	−0.890753	−	0.454488i	$-0.849822\pi$
$373$	−1.00000	+	1.73205i	−0.0517780	+	0.0896822i	0.838975	+	0.544170i	$0.183156\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	24.2487i		−	1.24557i	−0.782392	−	0.622786i	$-0.786001\pi$
$379$		−	24.2487i		−	1.24557i	0.782392	−	0.622786i	$-0.213999\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.00000	−	15.5885i	0.459879	−	0.796533i	−0.539076	−	0.842257i	$-0.681226\pi$
$383$	9.00000	−	15.5885i	0.459879	−	0.796533i	0.998954	+	0.0457244i	$0.0145596\pi$
$384$		0			0
$385$	1.50000	−	7.79423i	0.0764471	−	0.397231i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	0.779895	−	0.625910i	$-0.215272\pi$
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	−0.932002	+	0.362454i	$0.881939\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.50000	−	7.79423i	−0.226420	−	0.392170i
$396$		0			0
$397$	−18.0000	−	10.3923i	−0.903394	−	0.521575i	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−18.0000	−	10.3923i	−0.903394	−	0.521575i	−0.878300	+	0.478110i	$0.841322\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.00000	+	10.3923i	−0.299626	+	0.518967i	−0.976050	−	0.217545i	$-0.930195\pi$
$401$	−6.00000	+	10.3923i	−0.299626	+	0.518967i	0.676425	+	0.736512i	$0.263528\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	17.3205i		−	0.858546i
$408$		0			0
$409$	−19.5000	+	11.2583i	−0.964213	+	0.556689i	−0.897467	−	0.441081i	$-0.854595\pi$
$409$	−19.5000	+	11.2583i	−0.964213	+	0.556689i	−0.0667458	+	0.997770i	$0.521262\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−22.5000	+	7.79423i	−1.10715	+	0.383529i
$414$		0			0
$415$	4.50000	+	2.59808i	0.220896	+	0.127535i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−20.0000			−0.974740			−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000			−0.974740			−0.487370	+	0.873195i	$0.662044\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.00000	+	3.46410i	0.291043	+	0.168034i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−33.0000	+	19.0526i	−1.58955	+	0.917729i	−0.596174	+	0.802855i	$0.703313\pi$
$431$	−33.0000	+	19.0526i	−1.58955	+	0.917729i	−0.993380	+	0.114874i	$0.963353\pi$
$432$		0			0
$433$			34.6410i			1.66474i	0.554220	+	0.832370i	$0.313017\pi$
$433$			34.6410i			1.66474i	−0.554220	+	0.832370i	$0.686983\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	14.5000	−	25.1147i	0.692047	−	1.19866i	−0.279119	−	0.960257i	$-0.590042\pi$
$439$	14.5000	−	25.1147i	0.692047	−	1.19866i	0.971166	−	0.238404i	$-0.0766244\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.5000	−	6.06218i	−0.498870	−	0.288023i	0.229377	−	0.973338i	$-0.426331\pi$
$443$	−10.5000	−	6.06218i	−0.498870	−	0.288023i	−0.728247	+	0.685315i	$0.759665\pi$
$444$		0			0
$445$	−3.00000	−	5.19615i	−0.142214	−	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$	−9.00000	−	15.5885i	−0.423793	−	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	15.5000	−	26.8468i	0.725059	−	1.25584i	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	15.5000	−	26.8468i	0.725059	−	1.25584i	0.958950	−	0.283577i	$-0.0915211\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			6.92820i			0.322679i	0.986899	+	0.161339i	$0.0515813\pi$
$461$			6.92820i			0.322679i	−0.986899	+	0.161339i	$0.948419\pi$
$462$		0			0
$463$		−	38.1051i		−	1.77090i	−0.464739	−	0.885448i	$-0.653852\pi$
$463$		−	38.1051i		−	1.77090i	0.464739	−	0.885448i	$-0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.00000	−	10.3923i	0.277647	−	0.480899i	−0.693153	−	0.720791i	$-0.743779\pi$
$467$	6.00000	−	10.3923i	0.277647	−	0.480899i	0.970799	+	0.239892i	$0.0771121\pi$
$468$		0			0
$469$	12.0000	+	34.6410i	0.554109	+	1.59957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.00000	−	5.19615i	−0.137940	−	0.238919i
$474$		0			0
$475$	−4.00000			−0.183533
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.00000	+	5.19615i	0.137073	+	0.237418i	0.926388	−	0.376571i	$-0.122897\pi$
$479$	3.00000	+	5.19615i	0.137073	+	0.237418i	−0.789314	+	0.613990i	$0.789564\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−16.5000	+	28.5788i	−0.749226	+	1.29770i
$486$		0			0
$487$	−7.50000	+	4.33013i	−0.339857	+	0.196217i	−0.660209	−	0.751082i	$-0.729532\pi$
$487$	−7.50000	+	4.33013i	−0.339857	+	0.196217i	0.320352	+	0.947299i	$0.396199\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			25.9808i			1.17250i	0.810132	+	0.586248i	$0.199395\pi$
$491$			25.9808i			1.17250i	−0.810132	+	0.586248i	$0.800605\pi$
$492$		0			0
$493$	27.0000	−	15.5885i	1.21602	−	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−36.0000	−	6.92820i	−1.61482	−	0.310772i
$498$		0			0
$499$	3.00000	+	1.73205i	0.134298	+	0.0775372i	0.565644	−	0.824650i	$-0.308628\pi$
$499$	3.00000	+	1.73205i	0.134298	+	0.0775372i	−0.431346	+	0.902187i	$0.641961\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6.00000			−0.267527			−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000			−0.267527			−0.133763	+	0.991013i	$0.542706\pi$
$504$		0			0
$505$	−24.0000			−1.06799
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−28.5000	−	16.4545i	−1.26324	−	0.729332i	−0.289540	−	0.957166i	$-0.593502\pi$
$509$	−28.5000	−	16.4545i	−1.26324	−	0.729332i	−0.973700	+	0.227834i	$0.926836\pi$
$510$		0			0
$511$	6.00000	+	17.3205i	0.265424	+	0.766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.00000	+	3.46410i	−0.264392	+	0.152647i
$516$		0			0
$517$			20.7846i			0.914106i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.0000	+	19.0526i	−1.44576	+	0.834708i	−0.998225	−	0.0595604i	$-0.981030\pi$
$521$	−33.0000	+	19.0526i	−1.44576	+	0.834708i	−0.447532	+	0.894268i	$0.647697\pi$
$522$		0			0
$523$	14.0000	−	24.2487i	0.612177	−	1.06032i	−0.378695	−	0.925521i	$-0.623627\pi$
$523$	14.0000	−	24.2487i	0.612177	−	1.06032i	0.990873	−	0.134801i	$-0.0430394\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	15.0000	+	8.66025i	0.653410	+	0.377247i
$528$		0			0
$529$	−11.5000	−	19.9186i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−10.5000	−	18.1865i	−0.453955	−	0.786272i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12.0000	−	1.73205i	0.516877	−	0.0746047i
$540$		0			0
$541$	−11.0000	+	19.0526i	−0.472927	+	0.819133i	−0.999520	−	0.0309841i	$-0.990136\pi$
$541$	−11.0000	+	19.0526i	−0.472927	+	0.819133i	0.526593	+	0.850118i	$0.323469\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	6.92820i		−	0.296772i
$546$		0			0
$547$		−	10.3923i		−	0.444343i	−0.975008	−	0.222171i	$-0.928686\pi$
$547$		−	10.3923i		−	0.444343i	0.975008	−	0.222171i	$-0.0713145\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9.00000	+	15.5885i	−0.383413	+	0.664091i
$552$		0			0
$553$	9.00000	−	10.3923i	0.382719	−	0.441926i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.5000	−	23.3827i	−0.572013	−	0.990756i	−0.996359	−	0.0852559i	$-0.972829\pi$
$557$	−13.5000	−	23.3827i	−0.572013	−	0.990756i	0.424346	−	0.905500i	$-0.360504\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22.5000	−	38.9711i	−0.948262	−	1.64244i	−0.749085	−	0.662474i	$-0.769506\pi$
$563$	−22.5000	−	38.9711i	−0.948262	−	1.64244i	−0.199177	−	0.979963i	$-0.563827\pi$
$564$		0			0
$565$	9.00000	+	5.19615i	0.378633	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−18.0000	+	10.3923i	−0.753277	+	0.434904i	−0.826877	−	0.562383i	$-0.809885\pi$
$571$	−18.0000	+	10.3923i	−0.753277	+	0.434904i	0.0736000	+	0.997288i	$0.476551\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.50000	+	0.866025i	−0.0624458	+	0.0360531i	−0.530898	−	0.847436i	$-0.678145\pi$
$577$	−1.50000	+	0.866025i	−0.0624458	+	0.0360531i	0.468452	+	0.883489i	$0.344812\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.50000	+	7.79423i	−0.0622305	+	0.323359i
$582$		0			0
$583$	13.5000	+	7.79423i	0.559113	+	0.322804i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	33.0000			1.36206			0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000			1.36206			0.681028	+	0.732257i	$0.261533\pi$
$588$		0			0
$589$	−10.0000			−0.412043
$590$		0			0
$591$		0			0
$592$		0			0
$593$	24.0000	+	13.8564i	0.985562	+	0.569014i	0.903945	−	0.427649i	$-0.140658\pi$
$593$	24.0000	+	13.8564i	0.985562	+	0.569014i	0.0816172	+	0.996664i	$0.473992\pi$
$594$		0			0
$595$	3.00000	−	15.5885i	0.122988	−	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−24.0000	+	13.8564i	−0.980613	+	0.566157i	−0.902455	−	0.430784i	$-0.858237\pi$
$599$	−24.0000	+	13.8564i	−0.980613	+	0.566157i	−0.0781581	+	0.996941i	$0.524904\pi$
$600$		0			0
$601$		−	15.5885i		−	0.635866i	−0.948113	−	0.317933i	$-0.897011\pi$
$601$		−	15.5885i		−	0.635866i	0.948113	−	0.317933i	$-0.102989\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12.0000	−	6.92820i	0.487869	−	0.281672i
$606$		0			0
$607$	−9.50000	+	16.4545i	−0.385593	+	0.667867i	−0.991851	−	0.127401i	$-0.959336\pi$
$607$	−9.50000	+	16.4545i	−0.385593	+	0.667867i	0.606258	+	0.795268i	$0.292670\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	2.00000	+	3.46410i	0.0807792	+	0.139914i	0.903585	−	0.428409i	$-0.140926\pi$
$613$	2.00000	+	3.46410i	0.0807792	+	0.139914i	−0.822806	+	0.568323i	$0.807592\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−12.0000			−0.483102			−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000			−0.483102			−0.241551	+	0.970388i	$0.577656\pi$
$618$		0			0
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	0.997259	+	0.0739910i	$0.0235736\pi$
$619$	14.0000	+	24.2487i	0.562708	+	0.974638i	−0.434551	+	0.900647i	$0.643093\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.00000	−	6.92820i	0.240385	−	0.277573i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	34.6410i		−	1.38123i
$630$		0			0
$631$			32.9090i			1.31009i	0.755592	+	0.655043i	$0.227349\pi$
$631$			32.9090i			1.31009i	−0.755592	+	0.655043i	$0.772651\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.50000	−	7.79423i	0.178577	−	0.309305i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.0000	+	31.1769i	0.710957	+	1.23141i	0.964498	+	0.264089i	$0.0850714\pi$
$641$	18.0000	+	31.1769i	0.710957	+	1.23141i	−0.253541	+	0.967325i	$0.581595\pi$
$642$		0			0
$643$	16.0000			0.630978			0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000			0.630978			0.315489	+	0.948929i	$0.397831\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$	13.5000	+	7.79423i	0.529921	+	0.305950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13.5000	+	23.3827i	−0.528296	+	0.915035i	0.471160	+	0.882048i	$0.343835\pi$
$653$	−13.5000	+	23.3827i	−0.528296	+	0.915035i	−0.999456	+	0.0329874i	$0.989498\pi$
$654$		0			0
$655$	13.5000	−	7.79423i	0.527489	−	0.304546i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	10.3923i		−	0.404827i	−0.979300	−	0.202413i	$-0.935122\pi$
$659$		−	10.3923i		−	0.404827i	0.979300	−	0.202413i	$-0.0648785\pi$
$660$		0			0
$661$	3.00000	−	1.73205i	0.116686	−	0.0673690i	−0.440521	−	0.897742i	$-0.645206\pi$
$661$	3.00000	−	1.73205i	0.116686	−	0.0673690i	0.557207	+	0.830373i	$0.311873\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.00000	+	8.66025i	0.116335	+	0.335830i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−35.0000			−1.34915			−0.674575	−	0.738206i	$-0.735673\pi$
$673$	−35.0000			−1.34915			−0.674575	+	0.738206i	$0.735673\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−25.5000	−	14.7224i	−0.980045	−	0.565829i	−0.0777610	−	0.996972i	$-0.524777\pi$
$677$	−25.5000	−	14.7224i	−0.980045	−	0.565829i	−0.902284	+	0.431143i	$0.858110\pi$
$678$		0			0
$679$	−49.5000	−	9.52628i	−1.89964	−	0.365585i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22.5000	−	12.9904i	0.860939	−	0.497063i	−0.00338791	−	0.999994i	$-0.501078\pi$
$683$	22.5000	−	12.9904i	0.860939	−	0.497063i	0.864326	+	0.502931i	$0.167745\pi$
$684$		0			0
$685$			20.7846i			0.794139i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	11.0000	−	19.0526i	0.418460	−	0.724793i	−0.577325	−	0.816514i	$-0.695903\pi$
$691$	11.0000	−	19.0526i	0.418460	−	0.724793i	0.995785	+	0.0917209i	$0.0292368\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−21.0000	−	12.1244i	−0.796575	−	0.459903i
$696$		0			0
$697$	−18.0000	−	31.1769i	−0.681799	−	1.18091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	45.0000			1.69963			0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000			1.69963			0.849813	+	0.527084i	$0.176715\pi$
$702$		0			0
$703$	10.0000	+	17.3205i	0.377157	+	0.653255i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.0000	−	34.6410i	−0.451306	−	1.30281i
$708$		0			0
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	−0.756730	−	0.653727i	$-0.773204\pi$
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	0.944509	+	0.328484i	$0.106538\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.00000	−	15.5885i	0.335643	−	0.581351i	−0.647965	−	0.761670i	$-0.724380\pi$
$719$	9.00000	−	15.5885i	0.335643	−	0.581351i	0.983608	+	0.180319i	$0.0577130\pi$
$720$		0			0
$721$	−8.00000	−	6.92820i	−0.297936	−	0.258020i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.00000	+	15.5885i	0.334252	+	0.578941i
$726$		0			0
$727$	−37.0000			−1.37225			−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000			−1.37225			−0.686127	+	0.727482i	$0.740691\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−6.00000	−	10.3923i	−0.221918	−	0.384373i
$732$		0			0
$733$	−9.00000	−	5.19615i	−0.332423	−	0.191924i	0.324494	−	0.945888i	$-0.394806\pi$
$733$	−9.00000	−	5.19615i	−0.332423	−	0.191924i	−0.656916	+	0.753964i	$0.728139\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.0000	−	20.7846i	0.442026	−	0.765611i
$738$		0			0
$739$	9.00000	−	5.19615i	0.331070	−	0.191144i	−0.325246	−	0.945629i	$-0.605447\pi$
$739$	9.00000	−	5.19615i	0.331070	−	0.191144i	0.656316	+	0.754486i	$0.272114\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			13.8564i			0.508342i	0.967159	+	0.254171i	$0.0818026\pi$
$743$			13.8564i			0.508342i	−0.967159	+	0.254171i	$0.918197\pi$
$744$		0			0
$745$	−9.00000	+	5.19615i	−0.329734	+	0.190372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	21.0000	−	24.2487i	0.767323	−	0.886029i
$750$		0			0
$751$	4.50000	+	2.59808i	0.164207	+	0.0948051i	0.579852	−	0.814722i	$-0.303111\pi$
$751$	4.50000	+	2.59808i	0.164207	+	0.0948051i	−0.415644	+	0.909527i	$0.636444\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.00000			−0.327544
$756$		0			0
$757$	−4.00000			−0.145382			−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000			−0.145382			−0.0726912	+	0.997354i	$0.523159\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.0000	+	24.2487i	1.52250	+	0.879015i	0.999646	+	0.0265919i	$0.00846546\pi$
$761$	42.0000	+	24.2487i	1.52250	+	0.879015i	0.522852	+	0.852423i	$0.324868\pi$
$762$		0			0
$763$	10.0000	−	3.46410i	0.362024	−	0.125409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			19.0526i			0.687053i	0.939143	+	0.343526i	$0.111621\pi$
$769$			19.0526i			0.687053i	−0.939143	+	0.343526i	$0.888379\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−5.00000	+	8.66025i	−0.179605	+	0.311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18.0000	+	10.3923i	0.644917	+	0.372343i
$780$		0			0
$781$	12.0000	+	20.7846i	0.429394	+	0.743732i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−12.0000			−0.428298
$786$		0			0
$787$	14.0000	+	24.2487i	0.499046	+	0.864373i	0.999999	−	0.00110111i	$-0.000350496\pi$
$787$	14.0000	+	24.2487i	0.499046	+	0.864373i	−0.500953	+	0.865474i	$0.667017\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.00000	+	15.5885i	−0.106668	+	0.554262i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	5.19615i		−	0.184057i	−0.995756	−	0.0920286i	$-0.970665\pi$
$797$		−	5.19615i		−	0.184057i	0.995756	−	0.0920286i	$-0.0293351\pi$
$798$		0			0
$799$			41.5692i			1.47061i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.00000	−	10.3923i	0.211735	−	0.366736i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0000	+	20.7846i	0.420342	+	0.728053i
$816$		0			0
$817$	6.00000	+	3.46410i	0.209913	+	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.50000	−	2.59808i	0.0523504	−	0.0906735i	−0.838663	−	0.544651i	$-0.816662\pi$
$821$	1.50000	−	2.59808i	0.0523504	−	0.0906735i	0.891013	+	0.453978i	$0.149995\pi$
$822$		0			0
$823$	−33.0000	+	19.0526i	−1.15031	+	0.664130i	−0.948962	−	0.315390i	$-0.897865\pi$
$823$	−33.0000	+	19.0526i	−1.15031	+	0.664130i	−0.201345	+	0.979520i	$0.564531\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	39.8372i		−	1.38527i	−0.721286	−	0.692637i	$-0.756449\pi$
$827$		−	39.8372i		−	1.38527i	0.721286	−	0.692637i	$-0.243551\pi$
$828$		0			0
$829$	−15.0000	+	8.66025i	−0.520972	+	0.300783i	−0.737332	−	0.675530i	$-0.763915\pi$
$829$	−15.0000	+	8.66025i	−0.520972	+	0.300783i	0.216361	+	0.976314i	$0.430581\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	24.0000	−	3.46410i	0.831551	−	0.120024i
$834$		0			0
$835$	−18.0000	−	10.3923i	−0.622916	−	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−30.0000			−1.03572			−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000			−1.03572			−0.517858	+	0.855467i	$0.673270\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.5000	−	11.2583i	−0.670820	−	0.387298i
$846$		0			0
$847$	16.0000	+	13.8564i	0.549767	+	0.476112i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	48.4974i		−	1.66052i	−0.557376	−	0.830260i	$-0.688192\pi$
$853$		−	48.4974i		−	1.66052i	0.557376	−	0.830260i	$-0.311808\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	7.00000	−	12.1244i	0.238837	−	0.413678i	−0.721544	−	0.692369i	$-0.756567\pi$
$859$	7.00000	−	12.1244i	0.238837	−	0.413678i	0.960381	+	0.278691i	$0.0899005\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27.0000	+	15.5885i	0.919091	+	0.530637i	0.883345	−	0.468724i	$-0.155286\pi$
$863$	27.0000	+	15.5885i	0.919091	+	0.530637i	0.0357458	+	0.999361i	$0.488619\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.00000			−0.305304
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	31.5000	+	6.06218i	1.06489	+	0.204939i
$876$		0			0
$877$	−16.0000	+	27.7128i	−0.540282	+	0.935795i	0.458606	+	0.888640i	$0.348349\pi$
$877$	−16.0000	+	27.7128i	−0.540282	+	0.935795i	−0.998888	+	0.0471555i	$0.984984\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			13.8564i			0.466834i	0.972377	+	0.233417i	$0.0749907\pi$
$881$			13.8564i			0.466834i	−0.972377	+	0.233417i	$0.925009\pi$
$882$		0			0
$883$		−	24.2487i		−	0.816034i	−0.912974	−	0.408017i	$-0.866220\pi$
$883$		−	24.2487i		−	0.816034i	0.912974	−	0.408017i	$-0.133780\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.0000	+	41.5692i	−0.805841	+	1.39576i	0.109881	+	0.993945i	$0.464953\pi$
$887$	−24.0000	+	41.5692i	−0.805841	+	1.39576i	−0.915722	+	0.401813i	$0.868380\pi$
$888$		0			0
$889$	13.5000	+	2.59808i	0.452775	+	0.0871367i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.0000	−	20.7846i	−0.401565	−	0.695530i
$894$		0			0
$895$	42.0000			1.40391
$896$		0			0
$897$		0			0
$898$		0			0
$899$	22.5000	+	38.9711i	0.750417	+	1.29976i
$900$		0			0
$901$	27.0000	+	15.5885i	0.899500	+	0.519327i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.00000	−	15.5885i	0.299170	−	0.518178i
$906$		0			0
$907$	18.0000	−	10.3923i	0.597680	−	0.345071i	−0.170448	−	0.985367i	$-0.554522\pi$
$907$	18.0000	−	10.3923i	0.597680	−	0.345071i	0.768128	+	0.640296i	$0.221188\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			41.5692i			1.37725i	0.725118	+	0.688625i	$0.241785\pi$
$911$			41.5692i			1.37725i	−0.725118	+	0.688625i	$0.758215\pi$
$912$		0			0
$913$	4.50000	−	2.59808i	0.148928	−	0.0859838i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.0000	+	15.5885i	0.594412	+	0.514776i
$918$		0			0
$919$	15.0000	+	8.66025i	0.494804	+	0.285675i	0.726565	−	0.687097i	$-0.241115\pi$
$919$	15.0000	+	8.66025i	0.494804	+	0.285675i	−0.231761	+	0.972773i	$0.574449\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	20.0000			0.657596
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.0000	+	8.66025i	0.492134	+	0.284134i	0.725459	−	0.688265i	$-0.241627\pi$
$929$	15.0000	+	8.66025i	0.492134	+	0.284134i	−0.233325	+	0.972399i	$0.574961\pi$
$930$		0			0
$931$	−11.0000	+	8.66025i	−0.360510	+	0.283828i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9.00000	+	5.19615i	−0.294331	+	0.169932i
$936$		0			0
$937$		−	19.0526i		−	0.622420i	−0.950341	−	0.311210i	$-0.899266\pi$
$937$		−	19.0526i		−	0.622420i	0.950341	−	0.311210i	$-0.100734\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	34.5000	−	19.9186i	1.12467	−	0.649327i	0.182079	−	0.983284i	$-0.441717\pi$
$941$	34.5000	−	19.9186i	1.12467	−	0.649327i	0.942588	+	0.333957i	$0.108384\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.0000	+	19.0526i	1.07236	+	0.619125i	0.928824	−	0.370521i	$-0.120821\pi$
$947$	33.0000	+	19.0526i	1.07236	+	0.619125i	0.143532	+	0.989646i	$0.454154\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.0000			0.583077			0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000			0.583077			0.291539	+	0.956559i	$0.405833\pi$
$954$		0			0
$955$	−12.0000	−	20.7846i	−0.388311	−	0.672574i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−30.0000	+	10.3923i	−0.968751	+	0.335585i
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	8.66025i		−	0.278783i
$966$		0			0
$967$		−	39.8372i		−	1.28108i	−0.767926	−	0.640538i	$-0.778711\pi$
$967$		−	39.8372i		−	1.28108i	0.767926	−	0.640538i	$-0.221289\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16.5000	+	28.5788i	−0.529510	+	0.917139i	0.469897	+	0.882721i	$0.344291\pi$
$971$	−16.5000	+	28.5788i	−0.529510	+	0.917139i	−0.999408	+	0.0344175i	$0.989042\pi$
$972$		0			0
$973$	7.00000	−	36.3731i	0.224410	−	1.16607i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.00000	−	15.5885i	−0.287936	−	0.498719i	0.685381	−	0.728184i	$-0.259636\pi$
$977$	−9.00000	−	15.5885i	−0.287936	−	0.498719i	−0.973317	+	0.229465i	$0.926302\pi$
$978$		0			0
$979$	−6.00000			−0.191761
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.00000	+	15.5885i	0.287055	+	0.497195i	0.973106	−	0.230360i	$-0.0739903\pi$
$983$	9.00000	+	15.5885i	0.287055	+	0.497195i	−0.686050	+	0.727554i	$0.740657\pi$
$984$		0			0
$985$	−9.00000	−	5.19615i	−0.286764	−	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	10.5000	−	6.06218i	0.333543	−	0.192571i	−0.323870	−	0.946102i	$-0.604984\pi$
$991$	10.5000	−	6.06218i	0.333543	−	0.192571i	0.657413	+	0.753530i	$0.271651\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			27.7128i			0.878555i
$996$		0			0
$997$	24.0000	−	13.8564i	0.760088	−	0.438837i	−0.0692396	−	0.997600i	$-0.522057\pi$
$997$	24.0000	−	13.8564i	0.760088	−	0.438837i	0.829327	+	0.558763i	$0.188724\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.2.cs.e.271.1		2
3.2	odd	2		336.2.bl.c.271.1	yes	2
4.3	odd	2		1008.2.cs.d.271.1		2
7.2	even	3		7056.2.b.e.1567.2		2
7.3	odd	6		1008.2.cs.d.703.1		2
7.5	odd	6		7056.2.b.i.1567.1		2
12.11	even	2		336.2.bl.g.271.1	yes	2
21.2	odd	6		2352.2.b.g.1567.1		2
21.5	even	6		2352.2.b.c.1567.2		2
21.11	odd	6		2352.2.bl.b.31.1		2
21.17	even	6		336.2.bl.g.31.1	yes	2
21.20	even	2		2352.2.bl.h.607.1		2
24.5	odd	2		1344.2.bl.f.1279.1		2
24.11	even	2		1344.2.bl.b.1279.1		2
28.3	even	6	inner	1008.2.cs.e.703.1		2
28.19	even	6		7056.2.b.e.1567.1		2
28.23	odd	6		7056.2.b.i.1567.2		2
84.11	even	6		2352.2.bl.h.31.1		2
84.23	even	6		2352.2.b.c.1567.1		2
84.47	odd	6		2352.2.b.g.1567.2		2
84.59	odd	6		336.2.bl.c.31.1	✓	2
84.83	odd	2		2352.2.bl.b.607.1		2
168.59	odd	6		1344.2.bl.f.703.1		2
168.101	even	6		1344.2.bl.b.703.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bl.c.31.1	✓	2	84.59	odd	6
336.2.bl.c.271.1	yes	2	3.2	odd	2
336.2.bl.g.31.1	yes	2	21.17	even	6
336.2.bl.g.271.1	yes	2	12.11	even	2
1008.2.cs.d.271.1		2	4.3	odd	2
1008.2.cs.d.703.1		2	7.3	odd	6
1008.2.cs.e.271.1		2	1.1	even	1	trivial
1008.2.cs.e.703.1		2	28.3	even	6	inner
1344.2.bl.b.703.1		2	168.101	even	6
1344.2.bl.b.1279.1		2	24.11	even	2
1344.2.bl.f.703.1		2	168.59	odd	6
1344.2.bl.f.1279.1		2	24.5	odd	2
2352.2.b.c.1567.1		2	84.23	even	6
2352.2.b.c.1567.2		2	21.5	even	6
2352.2.b.g.1567.1		2	21.2	odd	6
2352.2.b.g.1567.2		2	84.47	odd	6
2352.2.bl.b.31.1		2	21.11	odd	6
2352.2.bl.b.607.1		2	84.83	odd	2
2352.2.bl.h.31.1		2	84.11	even	6
2352.2.bl.h.607.1		2	21.20	even	2
7056.2.b.e.1567.1		2	28.19	even	6
7056.2.b.e.1567.2		2	7.2	even	3
7056.2.b.i.1567.1		2	7.5	odd	6
7056.2.b.i.1567.2		2	28.23	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)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 0.866025i) q^{5} +(0.500000 - 2.59808i) q^{7} +O(q^{10})\)	\(q+(-1.50000 - 0.866025i) q^{5} +(0.500000 - 2.59808i) q^{7} +(-1.50000 + 0.866025i) q^{11} +(-3.00000 + 1.73205i) q^{17} +(1.00000 - 1.73205i) q^{19} +(-1.00000 - 1.73205i) q^{25} -9.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-3.00000 + 3.46410i) q^{35} +(-5.00000 + 8.66025i) q^{37} +10.3923i q^{41} +3.46410i q^{43} +(6.00000 - 10.3923i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-4.50000 - 7.79423i) q^{53} +3.00000 q^{55} +(-4.50000 - 7.79423i) q^{59} +(-12.0000 + 6.92820i) q^{67} -13.8564i q^{71} +(-6.00000 + 3.46410i) q^{73} +(1.50000 + 4.33013i) q^{77} +(4.50000 + 2.59808i) q^{79} -3.00000 q^{83} +6.00000 q^{85} +(3.00000 + 1.73205i) q^{89} +(-3.00000 + 1.73205i) q^{95} -19.0526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} + q^{7}+O(q^{10}) \) 2 * q - 3 * q^5 + q^7	\( 2 q - 3 q^{5} + q^{7} - 3 q^{11} - 6 q^{17} + 2 q^{19} - 2 q^{25} - 18 q^{29} - 5 q^{31} - 6 q^{35} - 10 q^{37} + 12 q^{47} - 13 q^{49} - 9 q^{53} + 6 q^{55} - 9 q^{59} - 24 q^{67} - 12 q^{73} + 3 q^{77} + 9 q^{79} - 6 q^{83} + 12 q^{85} + 6 q^{89} - 6 q^{95}+O(q^{100}) \) 2 * q - 3 * q^5 + q^7 - 3 * q^11 - 6 * q^17 + 2 * q^19 - 2 * q^25 - 18 * q^29 - 5 * q^31 - 6 * q^35 - 10 * q^37 + 12 * q^47 - 13 * q^49 - 9 * q^53 + 6 * q^55 - 9 * q^59 - 24 * q^67 - 12 * q^73 + 3 * q^77 + 9 * q^79 - 6 * q^83 + 12 * q^85 + 6 * q^89 - 6 * q^95

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