LMFDB - Embedded newform 1200.2.v.l.593.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,2,Mod(257,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("1200.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1200.v (of order $4$, degree $2$, not 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:	$9.58204824255$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.4
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1200.593
Dual form			1200.2.v.l.257.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.67303 - 0.448288i) q^{3} +(-2.44949 + 2.44949i) q^{7} +(2.59808 - 1.50000i) q^{9} +5.19615i q^{11} +(2.12132 + 2.12132i) q^{17} -1.00000i q^{19} +(-3.00000 + 5.19615i) q^{21} +(-4.24264 + 4.24264i) q^{23} +(3.67423 - 3.67423i) q^{27} +2.00000 q^{31} +(2.32937 + 8.69333i) q^{33} +(-2.44949 + 2.44949i) q^{37} +5.19615i q^{41} +(2.44949 + 2.44949i) q^{43} -5.00000i q^{49} +(4.50000 + 2.59808i) q^{51} +(-4.24264 + 4.24264i) q^{53} +(-0.448288 - 1.67303i) q^{57} +10.3923 q^{59} +14.0000 q^{61} +(-2.68973 + 10.0382i) q^{63} +(-3.67423 + 3.67423i) q^{67} +(-5.19615 + 9.00000i) q^{69} +(-6.12372 - 6.12372i) q^{73} +(-12.7279 - 12.7279i) q^{77} -14.0000i q^{79} +(4.50000 - 7.79423i) q^{81} +(2.12132 - 2.12132i) q^{83} +15.5885 q^{89} +(3.34607 - 0.896575i) q^{93} +(4.89898 - 4.89898i) q^{97} +(7.79423 + 13.5000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{21} + 16 q^{31} + 36 q^{51} + 112 q^{61} + 36 q^{81}+O(q^{100}) $ 8 * q - 24 * q^21 + 16 * q^31 + 36 * q^51 + 112 * q^61 + 36 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	1.67303	−	0.448288i	0.965926	−	0.258819i
$3$	1.67303	−	0.448288i	0.965926	−	0.258819i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.44949	+	2.44949i	−0.925820	+	0.925820i	−0.997433	−	0.0716124i	$-0.977186\pi$
$7$	−2.44949	+	2.44949i	−0.925820	+	0.925820i	0.0716124	+	0.997433i	$0.477186\pi$
$8$		0			0
$9$	2.59808	−	1.50000i	0.866025	−	0.500000i
$10$		0			0
$11$			5.19615i			1.56670i	0.621582	+	0.783349i	$0.286490\pi$
$11$			5.19615i			1.56670i	−0.621582	+	0.783349i	$0.713510\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.12132	+	2.12132i	0.514496	+	0.514496i	0.915901	−	0.401405i	$-0.131478\pi$
$17$	2.12132	+	2.12132i	0.514496	+	0.514496i	−0.401405	+	0.915901i	$0.631478\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		0			0
$21$	−3.00000	+	5.19615i	−0.654654	+	1.13389i
$22$		0			0
$23$	−4.24264	+	4.24264i	−0.884652	+	0.884652i	−0.994003	−	0.109351i	$-0.965123\pi$
$23$	−4.24264	+	4.24264i	−0.884652	+	0.884652i	0.109351	+	0.994003i	$0.465123\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.67423	−	3.67423i	0.707107	−	0.707107i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$	2.32937	+	8.69333i	0.405492	+	1.51331i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.44949	+	2.44949i	−0.402694	+	0.402694i	−0.879181	−	0.476488i	$-0.841910\pi$
$37$	−2.44949	+	2.44949i	−0.402694	+	0.402694i	0.476488	+	0.879181i	$0.341910\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.19615i			0.811503i	0.913984	+	0.405751i	$0.132990\pi$
$41$			5.19615i			0.811503i	−0.913984	+	0.405751i	$0.867010\pi$
$42$		0			0
$43$	2.44949	+	2.44949i	0.373544	+	0.373544i	0.868766	−	0.495222i	$-0.164913\pi$
$43$	2.44949	+	2.44949i	0.373544	+	0.373544i	−0.495222	+	0.868766i	$0.664913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$	4.50000	+	2.59808i	0.630126	+	0.363803i
$52$		0			0
$53$	−4.24264	+	4.24264i	−0.582772	+	0.582772i	−0.935664	−	0.352892i	$-0.885198\pi$
$53$	−4.24264	+	4.24264i	−0.582772	+	0.582772i	0.352892	+	0.935664i	$0.385198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.448288	−	1.67303i	−0.0593772	−	0.221599i
$58$		0			0
$59$	10.3923			1.35296			0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923			1.35296			0.676481	+	0.736460i	$0.263504\pi$
$60$		0			0
$61$	14.0000			1.79252			0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000			1.79252			0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$	−2.68973	+	10.0382i	−0.338874	+	1.26469i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.67423	+	3.67423i	−0.448879	+	0.448879i	−0.894982	−	0.446103i	$-0.852812\pi$
$67$	−3.67423	+	3.67423i	−0.448879	+	0.448879i	0.446103	+	0.894982i	$0.352812\pi$
$68$		0			0
$69$	−5.19615	+	9.00000i	−0.625543	+	1.08347i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−6.12372	−	6.12372i	−0.716728	−	0.716728i	0.251206	−	0.967934i	$-0.419173\pi$
$73$	−6.12372	−	6.12372i	−0.716728	−	0.716728i	−0.967934	+	0.251206i	$0.919173\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.7279	−	12.7279i	−1.45048	−	1.45048i
$78$		0			0
$79$		−	14.0000i		−	1.57512i	−0.616236	−	0.787562i	$-0.711343\pi$
$79$		−	14.0000i		−	1.57512i	0.616236	−	0.787562i	$-0.288657\pi$
$80$		0			0
$81$	4.50000	−	7.79423i	0.500000	−	0.866025i
$82$		0			0
$83$	2.12132	−	2.12132i	0.232845	−	0.232845i	−0.581034	−	0.813879i	$-0.697352\pi$
$83$	2.12132	−	2.12132i	0.232845	−	0.232845i	0.813879	+	0.581034i	$0.197352\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	15.5885			1.65237			0.826187	−	0.563397i	$-0.190506\pi$
$89$	15.5885			1.65237			0.826187	+	0.563397i	$0.190506\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	3.34607	−	0.896575i	0.346971	−	0.0929705i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.89898	−	4.89898i	0.497416	−	0.497416i	−0.413217	−	0.910633i	$-0.635595\pi$
$97$	4.89898	−	4.89898i	0.497416	−	0.497416i	0.910633	+	0.413217i	$0.135595\pi$
$98$		0			0
$99$	7.79423	+	13.5000i	0.783349	+	1.35680i
$100$		0			0
$101$			10.3923i			1.03407i	0.855963	+	0.517036i	$0.172965\pi$
$101$			10.3923i			1.03407i	−0.855963	+	0.517036i	$0.827035\pi$
$102$		0			0
$103$	−9.79796	−	9.79796i	−0.965422	−	0.965422i	0.0340002	−	0.999422i	$-0.489175\pi$
$103$	−9.79796	−	9.79796i	−0.965422	−	0.965422i	−0.999422	+	0.0340002i	$0.989175\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.36396	−	6.36396i	−0.615227	−	0.615227i	0.329076	−	0.944303i	$-0.393263\pi$
$107$	−6.36396	−	6.36396i	−0.615227	−	0.615227i	−0.944303	+	0.329076i	$0.893263\pi$
$108$		0			0
$109$		−	10.0000i		−	0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$		−	10.0000i		−	0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$		0			0
$111$	−3.00000	+	5.19615i	−0.284747	+	0.493197i
$112$		0			0
$113$	−6.36396	+	6.36396i	−0.598671	+	0.598671i	−0.939959	−	0.341288i	$-0.889137\pi$
$113$	−6.36396	+	6.36396i	−0.598671	+	0.598671i	0.341288	+	0.939959i	$0.389137\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.3923			−0.952661
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$	2.32937	+	8.69333i	0.210032	+	0.783851i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7.34847	−	7.34847i	0.652071	−	0.652071i	−0.301420	−	0.953491i	$-0.597461\pi$
$127$	7.34847	−	7.34847i	0.652071	−	0.652071i	0.953491	+	0.301420i	$0.0974607\pi$
$128$		0			0
$129$	5.19615	+	3.00000i	0.457496	+	0.264135i
$130$		0			0
$131$		−	10.3923i		−	0.907980i	−0.891007	−	0.453990i	$-0.850000\pi$
$131$		−	10.3923i		−	0.907980i	0.891007	−	0.453990i	$-0.150000\pi$
$132$		0			0
$133$	2.44949	+	2.44949i	0.212398	+	0.212398i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.8492	+	14.8492i	1.26866	+	1.26866i	0.946783	+	0.321874i	$0.104313\pi$
$137$	14.8492	+	14.8492i	1.26866	+	1.26866i	0.321874	+	0.946783i	$0.395687\pi$
$138$		0			0
$139$		−	7.00000i		−	0.593732i	−0.954919	−	0.296866i	$-0.904058\pi$
$139$		−	7.00000i		−	0.593732i	0.954919	−	0.296866i	$-0.0959415\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.24144	−	8.36516i	−0.184871	−	0.689947i
$148$		0			0
$149$	−20.7846			−1.70274			−0.851371	−	0.524564i	$-0.824228\pi$
$149$	−20.7846			−1.70274			−0.851371	+	0.524564i	$0.824228\pi$
$150$		0			0
$151$	14.0000			1.13930			0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000			1.13930			0.569652	+	0.821886i	$0.307078\pi$
$152$		0			0
$153$	8.69333	+	2.32937i	0.702814	+	0.188319i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−12.2474	+	12.2474i	−0.977453	+	0.977453i	−0.999751	−	0.0222985i	$-0.992902\pi$
$157$	−12.2474	+	12.2474i	−0.977453	+	0.977453i	0.0222985	+	0.999751i	$0.492902\pi$
$158$		0			0
$159$	−5.19615	+	9.00000i	−0.412082	+	0.713746i
$160$		0			0
$161$		−	20.7846i		−	1.63806i
$162$		0			0
$163$	−3.67423	−	3.67423i	−0.287788	−	0.287788i	0.548417	−	0.836205i	$-0.315231\pi$
$163$	−3.67423	−	3.67423i	−0.287788	−	0.287788i	−0.836205	+	0.548417i	$0.815231\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.48528	+	8.48528i	0.656611	+	0.656611i	0.954577	−	0.297966i	$-0.0963081\pi$
$167$	8.48528	+	8.48528i	0.656611	+	0.656611i	−0.297966	+	0.954577i	$0.596308\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	−1.50000	−	2.59808i	−0.114708	−	0.198680i
$172$		0			0
$173$	8.48528	−	8.48528i	0.645124	−	0.645124i	−0.306687	−	0.951811i	$-0.599220\pi$
$173$	8.48528	−	8.48528i	0.645124	−	0.645124i	0.951811	+	0.306687i	$0.0992203\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	17.3867	−	4.65874i	1.30686	−	0.350173i
$178$		0			0
$179$	−15.5885			−1.16514			−0.582568	−	0.812782i	$-0.697952\pi$
$179$	−15.5885			−1.16514			−0.582568	+	0.812782i	$0.697952\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	23.4225	−	6.27603i	1.73144	−	0.463937i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−11.0227	+	11.0227i	−0.806060	+	0.806060i
$188$		0			0
$189$			18.0000i			1.30931i
$190$		0			0
$191$			10.3923i			0.751961i	0.926628	+	0.375980i	$0.122694\pi$
$191$			10.3923i			0.751961i	−0.926628	+	0.375980i	$0.877306\pi$
$192$		0			0
$193$	−6.12372	−	6.12372i	−0.440795	−	0.440795i	0.451484	−	0.892279i	$-0.350895\pi$
$193$	−6.12372	−	6.12372i	−0.440795	−	0.440795i	−0.892279	+	0.451484i	$0.850895\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.24264	+	4.24264i	0.302276	+	0.302276i	0.841904	−	0.539628i	$-0.181435\pi$
$197$	4.24264	+	4.24264i	0.302276	+	0.302276i	−0.539628	+	0.841904i	$0.681435\pi$
$198$		0			0
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$		0			0
$201$	−4.50000	+	7.79423i	−0.317406	+	0.549762i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−4.65874	+	17.3867i	−0.323805	+	1.20846i
$208$		0			0
$209$	5.19615			0.359425
$210$		0			0
$211$	−23.0000			−1.58339			−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000			−1.58339			−0.791693	+	0.610920i	$0.790800\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−4.89898	+	4.89898i	−0.332564	+	0.332564i
$218$		0			0
$219$	−12.9904	−	7.50000i	−0.877809	−	0.506803i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−9.79796	−	9.79796i	−0.656120	−	0.656120i	0.298340	−	0.954460i	$-0.403567\pi$
$223$	−9.79796	−	9.79796i	−0.656120	−	0.656120i	−0.954460	+	0.298340i	$0.903567\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.48528	+	8.48528i	0.563188	+	0.563188i	0.930212	−	0.367024i	$-0.119623\pi$
$227$	8.48528	+	8.48528i	0.563188	+	0.563188i	−0.367024	+	0.930212i	$0.619623\pi$
$228$		0			0
$229$			16.0000i			1.05731i	0.848837	+	0.528655i	$0.177303\pi$
$229$			16.0000i			1.05731i	−0.848837	+	0.528655i	$0.822697\pi$
$230$		0			0
$231$	−27.0000	−	15.5885i	−1.77647	−	1.02565i
$232$		0			0
$233$	12.7279	−	12.7279i	0.833834	−	0.833834i	−0.154205	−	0.988039i	$-0.549282\pi$
$233$	12.7279	−	12.7279i	0.833834	−	0.833834i	0.988039	+	0.154205i	$0.0492816\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−6.27603	−	23.4225i	−0.407672	−	1.52145i
$238$		0			0
$239$	20.7846			1.34444			0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846			1.34444			0.672222	+	0.740349i	$0.265340\pi$
$240$		0			0
$241$	−1.00000			−0.0644157			−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000			−0.0644157			−0.0322078	+	0.999481i	$0.510254\pi$
$242$		0			0
$243$	4.03459	−	15.0573i	0.258819	−	0.965926i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	2.59808	−	4.50000i	0.164646	−	0.285176i
$250$		0			0
$251$			5.19615i			0.327978i	0.986462	+	0.163989i	$0.0524362\pi$
$251$			5.19615i			0.327978i	−0.986462	+	0.163989i	$0.947564\pi$
$252$		0			0
$253$	−22.0454	−	22.0454i	−1.38598	−	1.38598i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.24264	−	4.24264i	−0.264649	−	0.264649i	0.562291	−	0.826940i	$-0.309920\pi$
$257$	−4.24264	−	4.24264i	−0.264649	−	0.264649i	−0.826940	+	0.562291i	$0.809920\pi$
$258$		0			0
$259$		−	12.0000i		−	0.745644i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.7279	−	12.7279i	0.784837	−	0.784837i	−0.195805	−	0.980643i	$-0.562732\pi$
$263$	12.7279	−	12.7279i	0.784837	−	0.784837i	0.980643	+	0.195805i	$0.0627321\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	26.0800	−	6.98811i	1.59607	−	0.427666i
$268$		0			0
$269$	−20.7846			−1.26726			−0.633630	−	0.773636i	$-0.718436\pi$
$269$	−20.7846			−1.26726			−0.633630	+	0.773636i	$0.718436\pi$
$270$		0			0
$271$	10.0000			0.607457			0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000			0.607457			0.303728	+	0.952759i	$0.401768\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.79796	+	9.79796i	−0.588702	+	0.588702i	−0.937280	−	0.348578i	$-0.886665\pi$
$277$	−9.79796	+	9.79796i	−0.588702	+	0.588702i	0.348578	+	0.937280i	$0.386665\pi$
$278$		0			0
$279$	5.19615	−	3.00000i	0.311086	−	0.179605i
$280$		0			0
$281$		−	20.7846i		−	1.23991i	−0.784639	−	0.619953i	$-0.787152\pi$
$281$		−	20.7846i		−	1.23991i	0.784639	−	0.619953i	$-0.212848\pi$
$282$		0			0
$283$	−6.12372	−	6.12372i	−0.364018	−	0.364018i	0.501272	−	0.865290i	$-0.332866\pi$
$283$	−6.12372	−	6.12372i	−0.364018	−	0.364018i	−0.865290	+	0.501272i	$0.832866\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.7279	−	12.7279i	−0.751305	−	0.751305i
$288$		0			0
$289$		−	8.00000i		−	0.470588i
$290$		0			0
$291$	6.00000	−	10.3923i	0.351726	−	0.609208i
$292$		0			0
$293$	21.2132	−	21.2132i	1.23929	−	1.23929i	0.278996	−	0.960292i	$-0.409998\pi$
$293$	21.2132	−	21.2132i	1.23929	−	1.23929i	0.960292	−	0.278996i	$-0.0900018\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	19.0919	+	19.0919i	1.10782	+	1.10782i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−12.0000			−0.691669
$302$		0			0
$303$	4.65874	+	17.3867i	0.267638	+	0.998838i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.22474	−	1.22474i	0.0698999	−	0.0698999i	−0.671293	−	0.741192i	$-0.734261\pi$
$307$	1.22474	−	1.22474i	0.0698999	−	0.0698999i	0.741192	+	0.671293i	$0.234261\pi$
$308$		0			0
$309$	−20.7846	−	12.0000i	−1.18240	−	0.682656i
$310$		0			0
$311$		−	31.1769i		−	1.76788i	−0.467600	−	0.883940i	$-0.654881\pi$
$311$		−	31.1769i		−	1.76788i	0.467600	−	0.883940i	$-0.345119\pi$
$312$		0			0
$313$	14.6969	+	14.6969i	0.830720	+	0.830720i	0.987615	−	0.156895i	$-0.0501485\pi$
$313$	14.6969	+	14.6969i	0.830720	+	0.830720i	−0.156895	+	0.987615i	$0.550148\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.48528	−	8.48528i	−0.476581	−	0.476581i	0.427456	−	0.904036i	$-0.359410\pi$
$317$	−8.48528	−	8.48528i	−0.476581	−	0.476581i	−0.904036	+	0.427456i	$0.859410\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−13.5000	−	7.79423i	−0.753497	−	0.435031i
$322$		0			0
$323$	2.12132	−	2.12132i	0.118033	−	0.118033i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−4.48288	−	16.7303i	−0.247904	−	0.925189i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	13.0000			0.714545			0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000			0.714545			0.357272	+	0.934000i	$0.383707\pi$
$332$		0			0
$333$	−2.68973	+	10.0382i	−0.147396	+	0.550090i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.67423	−	3.67423i	0.200148	−	0.200148i	−0.599915	−	0.800064i	$-0.704799\pi$
$337$	3.67423	−	3.67423i	0.200148	−	0.200148i	0.800064	+	0.599915i	$0.204799\pi$
$338$		0			0
$339$	−7.79423	+	13.5000i	−0.423324	+	0.733219i
$340$		0			0
$341$			10.3923i			0.562775i
$342$		0			0
$343$	−4.89898	−	4.89898i	−0.264520	−	0.264520i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.36396	+	6.36396i	0.341635	+	0.341635i	0.856982	−	0.515347i	$-0.172337\pi$
$347$	6.36396	+	6.36396i	0.341635	+	0.341635i	−0.515347	+	0.856982i	$0.672337\pi$
$348$		0			0
$349$			22.0000i			1.17763i	0.808267	+	0.588817i	$0.200406\pi$
$349$			22.0000i			1.17763i	−0.808267	+	0.588817i	$0.799594\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.7279	+	12.7279i	−0.677439	+	0.677439i	−0.959420	−	0.281981i	$-0.909008\pi$
$353$	−12.7279	+	12.7279i	−0.677439	+	0.677439i	0.281981	+	0.959420i	$0.409008\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−17.3867	+	4.65874i	−0.920200	+	0.246567i
$358$		0			0
$359$	−10.3923			−0.548485			−0.274242	−	0.961661i	$-0.588427\pi$
$359$	−10.3923			−0.548485			−0.274242	+	0.961661i	$0.588427\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$	−26.7685	+	7.17260i	−1.40498	+	0.376464i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	14.6969	−	14.6969i	0.767174	−	0.767174i	−0.210434	−	0.977608i	$-0.567488\pi$
$367$	14.6969	−	14.6969i	0.767174	−	0.767174i	0.977608	+	0.210434i	$0.0674877\pi$
$368$		0			0
$369$	7.79423	+	13.5000i	0.405751	+	0.702782i
$370$		0			0
$371$		−	20.7846i		−	1.07908i
$372$		0			0
$373$	2.44949	+	2.44949i	0.126830	+	0.126830i	0.767672	−	0.640843i	$-0.221415\pi$
$373$	2.44949	+	2.44949i	0.126830	+	0.126830i	−0.640843	+	0.767672i	$0.721415\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			11.0000i			0.565032i	0.959263	+	0.282516i	$0.0911690\pi$
$379$			11.0000i			0.565032i	−0.959263	+	0.282516i	$0.908831\pi$
$380$		0			0
$381$	9.00000	−	15.5885i	0.461084	−	0.798621i
$382$		0			0
$383$	4.24264	−	4.24264i	0.216789	−	0.216789i	−0.590355	−	0.807144i	$-0.701012\pi$
$383$	4.24264	−	4.24264i	0.216789	−	0.216789i	0.807144	+	0.590355i	$0.201012\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	10.0382	+	2.68973i	0.510270	+	0.136726i
$388$		0			0
$389$	10.3923			0.526911			0.263455	−	0.964672i	$-0.415138\pi$
$389$	10.3923			0.526911			0.263455	+	0.964672i	$0.415138\pi$
$390$		0			0
$391$	−18.0000			−0.910299
$392$		0			0
$393$	−4.65874	−	17.3867i	−0.235002	−	0.877041i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	19.5959	−	19.5959i	0.983491	−	0.983491i	−0.0163750	−	0.999866i	$-0.505213\pi$
$397$	19.5959	−	19.5959i	0.983491	−	0.983491i	0.999866	+	0.0163750i	$0.00521255\pi$
$398$		0			0
$399$	5.19615	+	3.00000i	0.260133	+	0.150188i
$400$		0			0
$401$		−	5.19615i		−	0.259483i	−0.991548	−	0.129742i	$-0.958585\pi$
$401$		−	5.19615i		−	0.259483i	0.991548	−	0.129742i	$-0.0414148\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.7279	−	12.7279i	−0.630900	−	0.630900i
$408$		0			0
$409$		−	5.00000i		−	0.247234i	−0.992330	−	0.123617i	$-0.960551\pi$
$409$		−	5.00000i		−	0.247234i	0.992330	−	0.123617i	$-0.0394494\pi$
$410$		0			0
$411$	31.5000	+	18.1865i	1.55378	+	0.897076i
$412$		0			0
$413$	−25.4558	+	25.4558i	−1.25260	+	1.25260i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.13801	−	11.7112i	−0.153669	−	0.573501i
$418$		0			0
$419$	−25.9808			−1.26924			−0.634622	−	0.772823i	$-0.718844\pi$
$419$	−25.9808			−1.26924			−0.634622	+	0.772823i	$0.718844\pi$
$420$		0			0
$421$	4.00000			0.194948			0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000			0.194948			0.0974740	+	0.995238i	$0.468924\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−34.2929	+	34.2929i	−1.65955	+	1.65955i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			10.3923i			0.500580i	0.968171	+	0.250290i	$0.0805259\pi$
$431$			10.3923i			0.500580i	−0.968171	+	0.250290i	$0.919474\pi$
$432$		0			0
$433$	−15.9217	−	15.9217i	−0.765147	−	0.765147i	0.212101	−	0.977248i	$-0.431970\pi$
$433$	−15.9217	−	15.9217i	−0.765147	−	0.765147i	−0.977248	+	0.212101i	$0.931970\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.24264	+	4.24264i	0.202953	+	0.202953i
$438$		0			0
$439$			4.00000i			0.190910i	0.995434	+	0.0954548i	$0.0304305\pi$
$439$			4.00000i			0.190910i	−0.995434	+	0.0954548i	$0.969569\pi$
$440$		0			0
$441$	−7.50000	−	12.9904i	−0.357143	−	0.618590i
$442$		0			0
$443$	−14.8492	+	14.8492i	−0.705509	+	0.705509i	−0.965587	−	0.260079i	$-0.916252\pi$
$443$	−14.8492	+	14.8492i	−0.705509	+	0.705509i	0.260079	+	0.965587i	$0.416252\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−34.7733	+	9.31749i	−1.64472	+	0.440702i
$448$		0			0
$449$	25.9808			1.22611			0.613054	−	0.790041i	$-0.289941\pi$
$449$	25.9808			1.22611			0.613054	+	0.790041i	$0.289941\pi$
$450$		0			0
$451$	−27.0000			−1.27138
$452$		0			0
$453$	23.4225	−	6.27603i	1.10048	−	0.294874i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	18.3712	−	18.3712i	0.859367	−	0.859367i	−0.131896	−	0.991264i	$-0.542107\pi$
$457$	18.3712	−	18.3712i	0.859367	−	0.859367i	0.991264	+	0.131896i	$0.0421066\pi$
$458$		0			0
$459$	15.5885			0.727607
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	24.4949	+	24.4949i	1.13837	+	1.13837i	0.988742	+	0.149633i	$0.0478091\pi$
$463$	24.4949	+	24.4949i	1.13837	+	1.13837i	0.149633	+	0.988742i	$0.452191\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.48528	+	8.48528i	0.392652	+	0.392652i	0.875632	−	0.482980i	$-0.160445\pi$
$467$	8.48528	+	8.48528i	0.392652	+	0.392652i	−0.482980	+	0.875632i	$0.660445\pi$
$468$		0			0
$469$		−	18.0000i		−	0.831163i
$470$		0			0
$471$	−15.0000	+	25.9808i	−0.691164	+	1.19713i
$472$		0			0
$473$	−12.7279	+	12.7279i	−0.585230	+	0.585230i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−4.65874	+	17.3867i	−0.213309	+	0.796081i
$478$		0			0
$479$	10.3923			0.474837			0.237418	−	0.971408i	$-0.423699\pi$
$479$	10.3923			0.474837			0.237418	+	0.971408i	$0.423699\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−9.31749	−	34.7733i	−0.423960	−	1.58224i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	22.0454	−	22.0454i	0.998973	−	0.998973i	−0.00102669	−	0.999999i	$-0.500327\pi$
$487$	22.0454	−	22.0454i	0.998973	−	0.998973i	0.999999	+	0.00102669i	$0.000326807\pi$
$488$		0			0
$489$	−7.79423	−	4.50000i	−0.352467	−	0.203497i
$490$		0			0
$491$			31.1769i			1.40699i	0.710698	+	0.703497i	$0.248379\pi$
$491$			31.1769i			1.40699i	−0.710698	+	0.703497i	$0.751621\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$		0			0
$501$	18.0000	+	10.3923i	0.804181	+	0.464294i
$502$		0			0
$503$	8.48528	−	8.48528i	0.378340	−	0.378340i	−0.492163	−	0.870503i	$-0.663794\pi$
$503$	8.48528	−	8.48528i	0.378340	−	0.378340i	0.870503	+	0.492163i	$0.163794\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5.82774	−	21.7494i	−0.258819	−	0.965926i
$508$		0			0
$509$	10.3923			0.460631			0.230315	−	0.973116i	$-0.426024\pi$
$509$	10.3923			0.460631			0.230315	+	0.973116i	$0.426024\pi$
$510$		0			0
$511$	30.0000			1.32712
$512$		0			0
$513$	−3.67423	−	3.67423i	−0.162221	−	0.162221i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	10.3923	−	18.0000i	0.456172	−	0.790112i
$520$		0			0
$521$			36.3731i			1.59353i	0.604287	+	0.796766i	$0.293458\pi$
$521$			36.3731i			1.59353i	−0.604287	+	0.796766i	$0.706542\pi$
$522$		0			0
$523$	30.6186	+	30.6186i	1.33886	+	1.33886i	0.897167	+	0.441692i	$0.145622\pi$
$523$	30.6186	+	30.6186i	1.33886	+	1.33886i	0.441692	+	0.897167i	$0.354378\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.24264	+	4.24264i	0.184812	+	0.184812i
$528$		0			0
$529$		−	13.0000i		−	0.565217i
$530$		0			0
$531$	27.0000	−	15.5885i	1.17170	−	0.676481i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−26.0800	+	6.98811i	−1.12543	+	0.301559i
$538$		0			0
$539$	25.9808			1.11907
$540$		0			0
$541$	8.00000			0.343947			0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000			0.343947			0.171973	+	0.985102i	$0.444986\pi$
$542$		0			0
$543$	−3.34607	+	0.896575i	−0.143593	+	0.0384757i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.57321	−	8.57321i	0.366564	−	0.366564i	−0.499658	−	0.866223i	$-0.666541\pi$
$547$	8.57321	−	8.57321i	0.366564	−	0.366564i	0.866223	+	0.499658i	$0.166541\pi$
$548$		0			0
$549$	36.3731	−	21.0000i	1.55236	−	0.896258i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	34.2929	+	34.2929i	1.45828	+	1.45828i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−13.5000	+	23.3827i	−0.569970	+	0.987218i
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	8.06918	+	30.1146i	0.338874	+	1.26469i
$568$		0			0
$569$	−25.9808			−1.08917			−0.544585	−	0.838706i	$-0.683313\pi$
$569$	−25.9808			−1.08917			−0.544585	+	0.838706i	$0.683313\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$		0			0
$573$	4.65874	+	17.3867i	0.194622	+	0.726338i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−13.4722	+	13.4722i	−0.560855	+	0.560855i	−0.929550	−	0.368695i	$-0.879805\pi$
$577$	−13.4722	+	13.4722i	−0.560855	+	0.560855i	0.368695	+	0.929550i	$0.379805\pi$
$578$		0			0
$579$	−12.9904	−	7.50000i	−0.539862	−	0.311689i
$580$		0			0
$581$			10.3923i			0.431145i
$582$		0			0
$583$	−22.0454	−	22.0454i	−0.913027	−	0.913027i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.8492	−	14.8492i	−0.612894	−	0.612894i	0.330805	−	0.943699i	$-0.392680\pi$
$587$	−14.8492	−	14.8492i	−0.612894	−	0.612894i	−0.943699	+	0.330805i	$0.892680\pi$
$588$		0			0
$589$		−	2.00000i		−	0.0824086i
$590$		0			0
$591$	9.00000	+	5.19615i	0.370211	+	0.213741i
$592$		0			0
$593$	23.3345	−	23.3345i	0.958234	−	0.958234i	−0.0409281	−	0.999162i	$-0.513031\pi$
$593$	23.3345	−	23.3345i	0.958234	−	0.958234i	0.999162	+	0.0409281i	$0.0130314\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	7.17260	+	26.7685i	0.293555	+	1.09556i
$598$		0			0
$599$	−10.3923			−0.424618			−0.212309	−	0.977203i	$-0.568098\pi$
$599$	−10.3923			−0.424618			−0.212309	+	0.977203i	$0.568098\pi$
$600$		0			0
$601$	−5.00000			−0.203954			−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000			−0.203954			−0.101977	+	0.994787i	$0.532517\pi$
$602$		0			0
$603$	−4.03459	+	15.0573i	−0.164301	+	0.613180i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.89898	+	4.89898i	−0.198843	+	0.198843i	−0.799504	−	0.600661i	$-0.794904\pi$
$607$	−4.89898	+	4.89898i	−0.198843	+	0.198843i	0.600661	+	0.799504i	$0.294904\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	17.1464	+	17.1464i	0.692538	+	0.692538i	0.962790	−	0.270252i	$-0.0871070\pi$
$613$	17.1464	+	17.1464i	0.692538	+	0.692538i	−0.270252	+	0.962790i	$0.587107\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.2132	+	21.2132i	0.854011	+	0.854011i	0.990624	−	0.136613i	$-0.0436217\pi$
$617$	21.2132	+	21.2132i	0.854011	+	0.854011i	−0.136613	+	0.990624i	$0.543622\pi$
$618$		0			0
$619$			4.00000i			0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$			4.00000i			0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$		0			0
$621$			31.1769i			1.25109i
$622$		0			0
$623$	−38.1838	+	38.1838i	−1.52980	+	1.52980i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	8.69333	−	2.32937i	0.347178	−	0.0930261i
$628$		0			0
$629$	−10.3923			−0.414368
$630$		0			0
$631$	34.0000			1.35352			0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000			1.35352			0.676759	+	0.736204i	$0.263384\pi$
$632$		0			0
$633$	−38.4797	+	10.3106i	−1.52943	+	0.409810i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			20.7846i			0.820943i	0.911873	+	0.410471i	$0.134636\pi$
$641$			20.7846i			0.820943i	−0.911873	+	0.410471i	$0.865364\pi$
$642$		0			0
$643$	22.0454	+	22.0454i	0.869386	+	0.869386i	0.992404	−	0.123018i	$-0.0392574\pi$
$643$	22.0454	+	22.0454i	0.869386	+	0.869386i	−0.123018	+	0.992404i	$0.539257\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	33.9411	+	33.9411i	1.33436	+	1.33436i	0.901422	+	0.432941i	$0.142524\pi$
$647$	33.9411	+	33.9411i	1.33436	+	1.33436i	0.432941	+	0.901422i	$0.357476\pi$
$648$		0			0
$649$			54.0000i			2.11969i
$650$		0			0
$651$	−6.00000	+	10.3923i	−0.235159	+	0.407307i
$652$		0			0
$653$	−4.24264	+	4.24264i	−0.166027	+	0.166027i	−0.785231	−	0.619203i	$-0.787456\pi$
$653$	−4.24264	+	4.24264i	−0.166027	+	0.166027i	0.619203	+	0.785231i	$0.287456\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−25.0955	−	6.72432i	−0.979068	−	0.262341i
$658$		0			0
$659$	25.9808			1.01207			0.506033	−	0.862514i	$-0.331111\pi$
$659$	25.9808			1.01207			0.506033	+	0.862514i	$0.331111\pi$
$660$		0			0
$661$	−20.0000			−0.777910			−0.388955	−	0.921257i	$-0.627164\pi$
$661$	−20.0000			−0.777910			−0.388955	+	0.921257i	$0.627164\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−20.7846	−	12.0000i	−0.803579	−	0.463947i
$670$		0			0
$671$			72.7461i			2.80833i
$672$		0			0
$673$	4.89898	+	4.89898i	0.188842	+	0.188842i	0.795195	−	0.606353i	$-0.207368\pi$
$673$	4.89898	+	4.89898i	0.188842	+	0.188842i	−0.606353	+	0.795195i	$0.707368\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−25.4558	−	25.4558i	−0.978348	−	0.978348i	0.0214229	−	0.999771i	$-0.493180\pi$
$677$	−25.4558	−	25.4558i	−0.978348	−	0.978348i	−0.999771	+	0.0214229i	$0.993180\pi$
$678$		0			0
$679$			24.0000i			0.921035i
$680$		0			0
$681$	18.0000	+	10.3923i	0.689761	+	0.398234i
$682$		0			0
$683$	−14.8492	+	14.8492i	−0.568190	+	0.568190i	−0.931621	−	0.363431i	$-0.881605\pi$
$683$	−14.8492	+	14.8492i	−0.568190	+	0.568190i	0.363431	+	0.931621i	$0.381605\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	7.17260	+	26.7685i	0.273652	+	1.02128i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−37.0000			−1.40755			−0.703773	−	0.710425i	$-0.748503\pi$
$691$	−37.0000			−1.40755			−0.703773	+	0.710425i	$0.748503\pi$
$692$		0			0
$693$	−52.1600	−	13.9762i	−1.98139	−	0.530913i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−11.0227	+	11.0227i	−0.417515	+	0.417515i
$698$		0			0
$699$	15.5885	−	27.0000i	0.589610	−	1.02123i
$700$		0			0
$701$		−	20.7846i		−	0.785024i	−0.919747	−	0.392512i	$-0.871606\pi$
$701$		−	20.7846i		−	0.785024i	0.919747	−	0.392512i	$-0.128394\pi$
$702$		0			0
$703$	2.44949	+	2.44949i	0.0923843	+	0.0923843i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−25.4558	−	25.4558i	−0.957366	−	0.957366i
$708$		0			0
$709$			40.0000i			1.50223i	0.660171	+	0.751116i	$0.270484\pi$
$709$			40.0000i			1.50223i	−0.660171	+	0.751116i	$0.729516\pi$
$710$		0			0
$711$	−21.0000	−	36.3731i	−0.787562	−	1.36410i
$712$		0			0
$713$	−8.48528	+	8.48528i	−0.317776	+	0.317776i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	34.7733	−	9.31749i	1.29863	−	0.347968i
$718$		0			0
$719$	31.1769			1.16270			0.581351	−	0.813653i	$-0.302524\pi$
$719$	31.1769			1.16270			0.581351	+	0.813653i	$0.302524\pi$
$720$		0			0
$721$	48.0000			1.78761
$722$		0			0
$723$	−1.67303	+	0.448288i	−0.0622208	+	0.0166720i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	4.89898	−	4.89898i	0.181693	−	0.181693i	−0.610400	−	0.792093i	$-0.708991\pi$
$727$	4.89898	−	4.89898i	0.181693	−	0.181693i	0.792093	+	0.610400i	$0.208991\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$			10.3923i			0.384373i
$732$		0			0
$733$	14.6969	+	14.6969i	0.542844	+	0.542844i	0.924362	−	0.381518i	$-0.124598\pi$
$733$	14.6969	+	14.6969i	0.542844	+	0.542844i	−0.381518	+	0.924362i	$0.624598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19.0919	−	19.0919i	−0.703259	−	0.703259i
$738$		0			0
$739$			20.0000i			0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$			20.0000i			0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.7279	−	12.7279i	0.466942	−	0.466942i	−0.433980	−	0.900922i	$-0.642891\pi$
$743$	12.7279	−	12.7279i	0.466942	−	0.466942i	0.900922	+	0.433980i	$0.142891\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	2.32937	−	8.69333i	0.0852272	−	0.318072i
$748$		0			0
$749$	31.1769			1.13918
$750$		0			0
$751$	20.0000			0.729810			0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000			0.729810			0.364905	+	0.931045i	$0.381101\pi$
$752$		0			0
$753$	2.32937	+	8.69333i	0.0848870	+	0.316803i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	24.4949	−	24.4949i	0.890282	−	0.890282i	−0.104267	−	0.994549i	$-0.533250\pi$
$757$	24.4949	−	24.4949i	0.890282	−	0.890282i	0.994549	+	0.104267i	$0.0332497\pi$
$758$		0			0
$759$	−46.7654	−	27.0000i	−1.69748	−	0.980038i
$760$		0			0
$761$			5.19615i			0.188360i	0.995555	+	0.0941802i	$0.0300230\pi$
$761$			5.19615i			0.188360i	−0.995555	+	0.0941802i	$0.969977\pi$
$762$		0			0
$763$	24.4949	+	24.4949i	0.886775	+	0.886775i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	13.0000i		−	0.468792i	−0.972141	−	0.234396i	$-0.924689\pi$
$769$		−	13.0000i		−	0.468792i	0.972141	−	0.234396i	$-0.0753112\pi$
$770$		0			0
$771$	−9.00000	−	5.19615i	−0.324127	−	0.187135i
$772$		0			0
$773$	−8.48528	+	8.48528i	−0.305194	+	0.305194i	−0.843042	−	0.537848i	$-0.819238\pi$
$773$	−8.48528	+	8.48528i	−0.305194	+	0.305194i	0.537848	+	0.843042i	$0.319238\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−5.37945	−	20.0764i	−0.192987	−	0.720237i
$778$		0			0
$779$	5.19615			0.186171
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−17.1464	+	17.1464i	−0.611204	+	0.611204i	−0.943260	−	0.332056i	$-0.892258\pi$
$787$	−17.1464	+	17.1464i	−0.611204	+	0.611204i	0.332056	+	0.943260i	$0.392258\pi$
$788$		0			0
$789$	15.5885	−	27.0000i	0.554964	−	0.961225i
$790$		0			0
$791$		−	31.1769i		−	1.10852i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	29.6985	+	29.6985i	1.05197	+	1.05197i	0.998573	+	0.0534012i	$0.0170062\pi$
$797$	29.6985	+	29.6985i	1.05197	+	1.05197i	0.0534012	+	0.998573i	$0.482994\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	40.5000	−	23.3827i	1.43100	−	0.826187i
$802$		0			0
$803$	31.8198	−	31.8198i	1.12290	−	1.12290i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−34.7733	+	9.31749i	−1.22408	+	0.327991i
$808$		0			0
$809$	−20.7846			−0.730748			−0.365374	−	0.930861i	$-0.619059\pi$
$809$	−20.7846			−0.730748			−0.365374	+	0.930861i	$0.619059\pi$
$810$		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
$813$	16.7303	−	4.48288i	0.586758	−	0.157221i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.44949	−	2.44949i	0.0856968	−	0.0856968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	31.1769i		−	1.08808i	−0.839059	−	0.544041i	$-0.816894\pi$
$821$		−	31.1769i		−	1.08808i	0.839059	−	0.544041i	$-0.183106\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.12132	−	2.12132i	−0.0737655	−	0.0737655i	0.669261	−	0.743027i	$-0.266611\pi$
$827$	−2.12132	−	2.12132i	−0.0737655	−	0.0737655i	−0.743027	+	0.669261i	$0.766611\pi$
$828$		0			0
$829$		−	44.0000i		−	1.52818i	−0.645108	−	0.764092i	$-0.723188\pi$
$829$		−	44.0000i		−	1.52818i	0.645108	−	0.764092i	$-0.276812\pi$
$830$		0			0
$831$	−12.0000	+	20.7846i	−0.416275	+	0.721010i
$832$		0			0
$833$	10.6066	−	10.6066i	0.367497	−	0.367497i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	7.34847	−	7.34847i	0.254000	−	0.254000i
$838$		0			0
$839$	−20.7846			−0.717564			−0.358782	−	0.933421i	$-0.616808\pi$
$839$	−20.7846			−0.717564			−0.358782	+	0.933421i	$0.616808\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−9.31749	−	34.7733i	−0.320911	−	1.19766i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	39.1918	−	39.1918i	1.34665	−	1.34665i
$848$		0			0
$849$	−12.9904	−	7.50000i	−0.445829	−	0.257399i
$850$		0			0
$851$		−	20.7846i		−	0.712487i
$852$		0			0
$853$	−7.34847	−	7.34847i	−0.251607	−	0.251607i	0.570022	−	0.821629i	$-0.306935\pi$
$853$	−7.34847	−	7.34847i	−0.251607	−	0.251607i	−0.821629	+	0.570022i	$0.806935\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−36.0624	−	36.0624i	−1.23187	−	1.23187i	−0.963245	−	0.268625i	$-0.913431\pi$
$857$	−36.0624	−	36.0624i	−1.23187	−	1.23187i	−0.268625	−	0.963245i	$-0.586569\pi$
$858$		0			0
$859$			7.00000i			0.238837i	0.992844	+	0.119418i	$0.0381030\pi$
$859$			7.00000i			0.238837i	−0.992844	+	0.119418i	$0.961897\pi$
$860$		0			0
$861$	−27.0000	−	15.5885i	−0.920158	−	0.531253i
$862$		0			0
$863$	−33.9411	+	33.9411i	−1.15537	+	1.15537i	−0.169910	+	0.985460i	$0.554348\pi$
$863$	−33.9411	+	33.9411i	−1.15537	+	1.15537i	−0.985460	+	0.169910i	$0.945652\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−3.58630	−	13.3843i	−0.121797	−	0.454553i
$868$		0			0
$869$	72.7461			2.46774
$870$		0			0
$871$		0			0
$872$		0			0
$873$	5.37945	−	20.0764i	0.182067	−	0.679483i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−34.2929	+	34.2929i	−1.15799	+	1.15799i	−0.173080	+	0.984908i	$0.555372\pi$
$877$	−34.2929	+	34.2929i	−1.15799	+	1.15799i	−0.984908	+	0.173080i	$0.944628\pi$
$878$		0			0
$879$	25.9808	−	45.0000i	0.876309	−	1.51781i
$880$		0			0
$881$			20.7846i			0.700251i	0.936703	+	0.350126i	$0.113861\pi$
$881$			20.7846i			0.700251i	−0.936703	+	0.350126i	$0.886139\pi$
$882$		0			0
$883$	−20.8207	−	20.8207i	−0.700671	−	0.700671i	0.263883	−	0.964555i	$-0.414997\pi$
$883$	−20.8207	−	20.8207i	−0.700671	−	0.700671i	−0.964555	+	0.263883i	$0.914997\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−33.9411	−	33.9411i	−1.13963	−	1.13963i	−0.988516	−	0.151115i	$-0.951714\pi$
$887$	−33.9411	−	33.9411i	−1.13963	−	1.13963i	−0.151115	−	0.988516i	$-0.548286\pi$
$888$		0			0
$889$			36.0000i			1.20740i
$890$		0			0
$891$	40.5000	+	23.3827i	1.35680	+	0.783349i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−18.0000			−0.599667
$902$		0			0
$903$	−20.0764	+	5.37945i	−0.668100	+	0.179017i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	17.1464	−	17.1464i	0.569338	−	0.569338i	−0.362605	−	0.931943i	$-0.618113\pi$
$907$	17.1464	−	17.1464i	0.569338	−	0.569338i	0.931943	+	0.362605i	$0.118113\pi$
$908$		0			0
$909$	15.5885	+	27.0000i	0.517036	+	0.895533i
$910$		0			0
$911$		−	41.5692i		−	1.37725i	−0.725118	−	0.688625i	$-0.758215\pi$
$911$		−	41.5692i		−	1.37725i	0.725118	−	0.688625i	$-0.241785\pi$
$912$		0			0
$913$	11.0227	+	11.0227i	0.364798	+	0.364798i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	25.4558	+	25.4558i	0.840626	+	0.840626i
$918$		0			0
$919$			26.0000i			0.857661i	0.903385	+	0.428830i	$0.141074\pi$
$919$			26.0000i			0.857661i	−0.903385	+	0.428830i	$0.858926\pi$
$920$		0			0
$921$	1.50000	−	2.59808i	0.0494267	−	0.0856095i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−40.1528	−	10.7589i	−1.31879	−	0.353369i
$928$		0			0
$929$	−41.5692			−1.36384			−0.681921	−	0.731426i	$-0.738855\pi$
$929$	−41.5692			−1.36384			−0.681921	+	0.731426i	$0.738855\pi$
$930$		0			0
$931$	−5.00000			−0.163868
$932$		0			0
$933$	−13.9762	−	52.1600i	−0.457561	−	1.70764i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	6.12372	−	6.12372i	0.200053	−	0.200053i	−0.599970	−	0.800023i	$-0.704821\pi$
$937$	6.12372	−	6.12372i	0.200053	−	0.200053i	0.800023	+	0.599970i	$0.204821\pi$
$938$		0			0
$939$	31.1769	+	18.0000i	1.01742	+	0.587408i
$940$		0			0
$941$		−	20.7846i		−	0.677559i	−0.940866	−	0.338779i	$-0.889986\pi$
$941$		−	20.7846i		−	0.677559i	0.940866	−	0.338779i	$-0.110014\pi$
$942$		0			0
$943$	−22.0454	−	22.0454i	−0.717897	−	0.717897i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.9706	+	16.9706i	0.551469	+	0.551469i	0.926865	−	0.375396i	$-0.122493\pi$
$947$	16.9706	+	16.9706i	0.551469	+	0.551469i	−0.375396	+	0.926865i	$0.622493\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−18.0000	−	10.3923i	−0.583690	−	0.336994i
$952$		0			0
$953$	−6.36396	+	6.36396i	−0.206149	+	0.206149i	−0.802628	−	0.596479i	$-0.796566\pi$
$953$	−6.36396	+	6.36396i	−0.206149	+	0.206149i	0.596479	+	0.802628i	$0.296566\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−72.7461			−2.34910
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$	−26.0800	−	6.98811i	−0.840416	−	0.225189i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	9.79796	−	9.79796i	0.315081	−	0.315081i	−0.531793	−	0.846874i	$-0.678482\pi$
$967$	9.79796	−	9.79796i	0.315081	−	0.315081i	0.846874	+	0.531793i	$0.178482\pi$
$968$		0			0
$969$	2.59808	−	4.50000i	0.0834622	−	0.144561i
$970$		0			0
$971$		−	5.19615i		−	0.166752i	−0.996518	−	0.0833762i	$-0.973430\pi$
$971$		−	5.19615i		−	0.166752i	0.996518	−	0.0833762i	$-0.0265703\pi$
$972$		0			0
$973$	17.1464	+	17.1464i	0.549689	+	0.549689i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6.36396	+	6.36396i	0.203601	+	0.203601i	0.801541	−	0.597940i	$-0.204014\pi$
$977$	6.36396	+	6.36396i	0.203601	+	0.203601i	−0.597940	+	0.801541i	$0.704014\pi$
$978$		0			0
$979$			81.0000i			2.58877i
$980$		0			0
$981$	−15.0000	−	25.9808i	−0.478913	−	0.829502i
$982$		0			0
$983$	21.2132	−	21.2132i	0.676596	−	0.676596i	−0.282632	−	0.959228i	$-0.591208\pi$
$983$	21.2132	−	21.2132i	0.676596	−	0.676596i	0.959228	+	0.282632i	$0.0912076\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−20.7846			−0.660912
$990$		0			0
$991$	10.0000			0.317660			0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000			0.317660			0.158830	+	0.987306i	$0.449228\pi$
$992$		0			0
$993$	21.7494	−	5.82774i	0.690197	−	0.184938i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	44.0908	−	44.0908i	1.39637	−	1.39637i	0.586214	−	0.810157i	$-0.300618\pi$
$997$	44.0908	−	44.0908i	1.39637	−	1.39637i	0.810157	−	0.586214i	$-0.199382\pi$
$998$		0			0
$999$			18.0000i			0.569495i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.2.v.l.593.4		8
3.2	odd	2	inner	1200.2.v.l.593.2		8
4.3	odd	2		150.2.e.b.143.1	yes	8
5.2	odd	4	inner	1200.2.v.l.257.2		8
5.3	odd	4	inner	1200.2.v.l.257.3		8
5.4	even	2	inner	1200.2.v.l.593.1		8
12.11	even	2		150.2.e.b.143.3	yes	8
15.2	even	4	inner	1200.2.v.l.257.4		8
15.8	even	4	inner	1200.2.v.l.257.1		8
15.14	odd	2	inner	1200.2.v.l.593.3		8
20.3	even	4		150.2.e.b.107.2	yes	8
20.7	even	4		150.2.e.b.107.3	yes	8
20.19	odd	2		150.2.e.b.143.4	yes	8
60.23	odd	4		150.2.e.b.107.4	yes	8
60.47	odd	4		150.2.e.b.107.1	✓	8
60.59	even	2		150.2.e.b.143.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.e.b.107.1	✓	8	60.47	odd	4
150.2.e.b.107.2	yes	8	20.3	even	4
150.2.e.b.107.3	yes	8	20.7	even	4
150.2.e.b.107.4	yes	8	60.23	odd	4
150.2.e.b.143.1	yes	8	4.3	odd	2
150.2.e.b.143.2	yes	8	60.59	even	2
150.2.e.b.143.3	yes	8	12.11	even	2
150.2.e.b.143.4	yes	8	20.19	odd	2
1200.2.v.l.257.1		8	15.8	even	4	inner
1200.2.v.l.257.2		8	5.2	odd	4	inner
1200.2.v.l.257.3		8	5.3	odd	4	inner
1200.2.v.l.257.4		8	15.2	even	4	inner
1200.2.v.l.593.1		8	5.4	even	2	inner
1200.2.v.l.593.2		8	3.2	odd	2	inner
1200.2.v.l.593.3		8	15.14	odd	2	inner
1200.2.v.l.593.4		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.67303 - 0.448288i) q^{3} +(-2.44949 + 2.44949i) q^{7} +(2.59808 - 1.50000i) q^{9} +5.19615i q^{11} +(2.12132 + 2.12132i) q^{17} -1.00000i q^{19} +(-3.00000 + 5.19615i) q^{21} +(-4.24264 + 4.24264i) q^{23} +(3.67423 - 3.67423i) q^{27} +2.00000 q^{31} +(2.32937 + 8.69333i) q^{33} +(-2.44949 + 2.44949i) q^{37} +5.19615i q^{41} +(2.44949 + 2.44949i) q^{43} -5.00000i q^{49} +(4.50000 + 2.59808i) q^{51} +(-4.24264 + 4.24264i) q^{53} +(-0.448288 - 1.67303i) q^{57} +10.3923 q^{59} +14.0000 q^{61} +(-2.68973 + 10.0382i) q^{63} +(-3.67423 + 3.67423i) q^{67} +(-5.19615 + 9.00000i) q^{69} +(-6.12372 - 6.12372i) q^{73} +(-12.7279 - 12.7279i) q^{77} -14.0000i q^{79} +(4.50000 - 7.79423i) q^{81} +(2.12132 - 2.12132i) q^{83} +15.5885 q^{89} +(3.34607 - 0.896575i) q^{93} +(4.89898 - 4.89898i) q^{97} +(7.79423 + 13.5000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{21} + 16 q^{31} + 36 q^{51} + 112 q^{61} + 36 q^{81}+O(q^{100}) \) 8 * q - 24 * q^21 + 16 * q^31 + 36 * q^51 + 112 * q^61 + 36 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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