LMFDB - Embedded newform 6300.2.ch.e.1601.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6300,2,Mod(1601,6300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6300.1601"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6300, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.ch (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 32 x^{18} - 124 x^{16} + 5094 x^{14} + 61094 x^{12} + 245850 x^{10} + 420152 x^{8} + \cdots + 15876 $ x^20 - 32x^18 - 124x^16 + 5094x^14 + 61094x^12 + 245850x^10 + 420152x^8 + 466066x^6 + 1736173x^4 - 23778*x^2 + 15876
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.1
Root			$4.99853 + 0.314967i$ of defining polynomial
Character	$\chi$	$=$	6300.1601
Dual form			6300.2.ch.e.4301.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.37152 - 1.17299i) q^{7} +(-0.545538 + 0.314967i) q^{11} -3.17384i q^{13} +(-0.0398515 - 0.0690248i) q^{17} +(-3.42210 - 1.97575i) q^{19} +(-7.83251 - 4.52210i) q^{23} +0.634956i q^{29} +(-1.15101 + 0.664538i) q^{31} +(0.996014 - 1.72515i) q^{37} +3.23801 q^{41} +5.39522 q^{43} +(-3.33967 + 5.78448i) q^{47} +(4.24819 + 5.56353i) q^{49} +(1.64531 - 0.949922i) q^{53} +(7.24278 + 12.5449i) q^{59} +(-6.68049 - 3.85698i) q^{61} +(3.47711 + 6.02252i) q^{67} +5.78534i q^{71} +(5.24659 - 3.02912i) q^{73} +(1.66321 - 0.107039i) q^{77} +(-5.62370 + 9.74053i) q^{79} -3.23801 q^{83} +(4.49284 - 7.78183i) q^{89} +(-3.72288 + 7.52683i) q^{91} +3.13513i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{7} + 6 q^{31} + 2 q^{37} - 20 q^{43} + 4 q^{49} + 6 q^{61} + 28 q^{67} - 6 q^{73} + 2 q^{79} - 40 q^{91}+O(q^{100}) $ 20 * q + 4 * q^7 + 6 * q^31 + 2 * q^37 - 20 * q^43 + 4 * q^49 + 6 * q^61 + 28 * q^67 - 6 * q^73 + 2 * q^79 - 40 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.37152	−	1.17299i	−0.896350	−	0.443348i
$7$	−2.37152	−	1.17299i	−0.896350	−	0.443348i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.545538	+	0.314967i	−0.164486	+	0.0949660i	−0.579983	−	0.814628i	$-0.696941\pi$
$11$	−0.545538	+	0.314967i	−0.164486	+	0.0949660i	0.415497	+	0.909594i	$0.363608\pi$
$12$		0			0
$13$		−	3.17384i		−	0.880266i	−0.897933	−	0.440133i	$-0.854931\pi$
$13$		−	3.17384i		−	0.880266i	0.897933	−	0.440133i	$-0.145069\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.0398515	−	0.0690248i	−0.00966541	−	0.0167410i	0.861152	−	0.508347i	$-0.169743\pi$
$17$	−0.0398515	−	0.0690248i	−0.00966541	−	0.0167410i	−0.870818	+	0.491606i	$0.836410\pi$
$18$		0			0
$19$	−3.42210	−	1.97575i	−0.785084	−	0.453268i	0.0531451	−	0.998587i	$-0.483075\pi$
$19$	−3.42210	−	1.97575i	−0.785084	−	0.453268i	−0.838229	+	0.545318i	$0.816409\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.83251	−	4.52210i	−1.63319	−	0.942924i	−0.983100	−	0.183069i	$-0.941397\pi$
$23$	−7.83251	−	4.52210i	−1.63319	−	0.942924i	−0.650092	−	0.759855i	$-0.725270\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			0.634956i			0.117908i	0.998261	+	0.0589541i	$0.0187766\pi$
$29$			0.634956i			0.117908i	−0.998261	+	0.0589541i	$0.981223\pi$
$30$		0			0
$31$	−1.15101	+	0.664538i	−0.206728	+	0.119355i	−0.599790	−	0.800158i	$-0.704749\pi$
$31$	−1.15101	+	0.664538i	−0.206728	+	0.119355i	0.393062	+	0.919512i	$0.371416\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.996014	−	1.72515i	0.163744	−	0.283612i	−0.772465	−	0.635058i	$-0.780976\pi$
$37$	0.996014	−	1.72515i	0.163744	−	0.283612i	0.936208	+	0.351445i	$0.114310\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.23801			0.505692			0.252846	−	0.967507i	$-0.418633\pi$
$41$	3.23801			0.505692			0.252846	+	0.967507i	$0.418633\pi$
$42$		0			0
$43$	5.39522			0.822764			0.411382	−	0.911463i	$-0.365046\pi$
$43$	5.39522			0.822764			0.411382	+	0.911463i	$0.365046\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.33967	+	5.78448i	−0.487141	+	0.843753i	−0.999891	−	0.0147849i	$-0.995294\pi$
$47$	−3.33967	+	5.78448i	−0.487141	+	0.843753i	0.512749	+	0.858538i	$0.328627\pi$
$48$		0			0
$49$	4.24819	+	5.56353i	0.606885	+	0.794790i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.64531	−	0.949922i	0.226001	−	0.130482i	−0.382725	−	0.923862i	$-0.625014\pi$
$53$	1.64531	−	0.949922i	0.226001	−	0.130482i	0.608726	+	0.793381i	$0.291681\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.24278	+	12.5449i	0.942929	+	1.63320i	0.759845	+	0.650104i	$0.225275\pi$
$59$	7.24278	+	12.5449i	0.942929	+	1.63320i	0.183084	+	0.983097i	$0.441392\pi$
$60$		0			0
$61$	−6.68049	−	3.85698i	−0.855350	−	0.493836i	0.00710270	−	0.999975i	$-0.497739\pi$
$61$	−6.68049	−	3.85698i	−0.855350	−	0.493836i	−0.862452	+	0.506139i	$0.831072\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.47711	+	6.02252i	0.424796	+	0.735769i	0.996401	−	0.0847603i	$-0.0270124\pi$
$67$	3.47711	+	6.02252i	0.424796	+	0.735769i	−0.571605	+	0.820529i	$0.693679\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.78534i			0.686593i	0.939227	+	0.343297i	$0.111544\pi$
$71$			5.78534i			0.686593i	−0.939227	+	0.343297i	$0.888456\pi$
$72$		0			0
$73$	5.24659	−	3.02912i	0.614067	−	0.354531i	−0.160489	−	0.987038i	$-0.551307\pi$
$73$	5.24659	−	3.02912i	0.614067	−	0.354531i	0.774555	+	0.632506i	$0.217974\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.66321	−	0.107039i	0.189540	−	0.0121982i
$78$		0			0
$79$	−5.62370	+	9.74053i	−0.632716	+	1.09590i	0.354279	+	0.935140i	$0.384726\pi$
$79$	−5.62370	+	9.74053i	−0.632716	+	1.09590i	−0.986994	+	0.160756i	$0.948607\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.23801			−0.355418			−0.177709	−	0.984083i	$-0.556869\pi$
$83$	−3.23801			−0.355418			−0.177709	+	0.984083i	$0.556869\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.49284	−	7.78183i	0.476240	−	0.824872i	−0.523389	−	0.852094i	$-0.675333\pi$
$89$	4.49284	−	7.78183i	0.476240	−	0.824872i	0.999629	+	0.0272213i	$0.00866589\pi$
$90$		0			0
$91$	−3.72288	+	7.52683i	−0.390264	+	0.789026i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			3.13513i			0.318324i	0.987252	+	0.159162i	$0.0508793\pi$
$97$			3.13513i			0.318324i	−0.987252	+	0.159162i	$0.949121\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.28297	+	10.8824i	0.625179	+	1.08284i	0.988506	+	0.151180i	$0.0483074\pi$
$101$	6.28297	+	10.8824i	0.625179	+	1.08284i	−0.363327	+	0.931662i	$0.618359\pi$
$102$		0			0
$103$	−6.33444	−	3.65719i	−0.624151	−	0.360354i	0.154332	−	0.988019i	$-0.450677\pi$
$103$	−6.33444	−	3.65719i	−0.624151	−	0.360354i	−0.778483	+	0.627665i	$0.784011\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.7715	+	6.21891i	1.04132	+	0.601205i	0.920206	−	0.391434i	$-0.128021\pi$
$107$	10.7715	+	6.21891i	1.04132	+	0.601205i	0.121111	+	0.992639i	$0.461354\pi$
$108$		0			0
$109$	−2.24464	−	3.88784i	−0.214998	−	0.372387i	0.738274	−	0.674501i	$-0.235641\pi$
$109$	−2.24464	−	3.88784i	−0.214998	−	0.372387i	−0.953272	+	0.302114i	$0.902308\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	14.1946i		−	1.33531i	−0.744469	−	0.667657i	$-0.767297\pi$
$113$		−	14.1946i		−	1.33531i	0.744469	−	0.667657i	$-0.232703\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.0135432	+	0.210439i	0.00124150	+	0.0192909i
$120$		0			0
$121$	−5.30159	+	9.18263i	−0.481963	+	0.834784i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−14.0140			−1.24355			−0.621773	−	0.783198i	$-0.713587\pi$
$127$	−14.0140			−1.24355			−0.621773	+	0.783198i	$0.713587\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.47529	+	6.01939i	−0.303638	+	0.525916i	−0.976957	−	0.213436i	$-0.931535\pi$
$131$	−3.47529	+	6.01939i	−0.303638	+	0.525916i	0.673319	+	0.739352i	$0.264868\pi$
$132$		0			0
$133$	5.79804	+	8.69962i	0.502754	+	0.754352i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.3492	−	6.55245i	0.969625	−	0.559813i	0.0705030	−	0.997512i	$-0.477540\pi$
$137$	11.3492	−	6.55245i	0.969625	−	0.559813i	0.899122	+	0.437698i	$0.144206\pi$
$138$		0			0
$139$		−	11.8932i		−	1.00877i	−0.863479	−	0.504385i	$-0.831719\pi$
$139$		−	11.8932i		−	1.00877i	0.863479	−	0.504385i	$-0.168281\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.999655	+	1.73145i	0.0835953	+	0.144791i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.24390	−	1.87287i	−0.265751	−	0.153431i	0.361204	−	0.932487i	$-0.382366\pi$
$149$	−3.24390	−	1.87287i	−0.265751	−	0.153431i	−0.626955	+	0.779055i	$0.715699\pi$
$150$		0			0
$151$	−2.55679	−	4.42850i	−0.208069	−	0.360386i	0.743037	−	0.669250i	$-0.233385\pi$
$151$	−2.55679	−	4.42850i	−0.208069	−	0.360386i	−0.951106	+	0.308864i	$0.900051\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−13.6027	+	7.85355i	−1.08562	+	0.626782i	−0.932406	−	0.361412i	$-0.882295\pi$
$157$	−13.6027	+	7.85355i	−1.08562	+	0.626782i	−0.153211	+	0.988193i	$0.548962\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	13.2706	+	19.9117i	1.04587	+	1.56926i
$162$		0			0
$163$	5.07710	−	8.79380i	0.397669	−	0.688783i	−0.595769	−	0.803156i	$-0.703152\pi$
$163$	5.07710	−	8.79380i	0.397669	−	0.688783i	0.993438	+	0.114373i	$0.0364858\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.38941			0.107516			0.0537578	−	0.998554i	$-0.482880\pi$
$167$	1.38941			0.107516			0.0537578	+	0.998554i	$0.482880\pi$
$168$		0			0
$169$	2.92672			0.225132
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.37552	−	4.11452i	0.180607	−	0.312821i	−0.761480	−	0.648188i	$-0.775527\pi$
$173$	2.37552	−	4.11452i	0.180607	−	0.312821i	0.942088	+	0.335367i	$0.108860\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.94883	+	5.74396i	−0.743610	+	0.429324i	−0.823380	−	0.567490i	$-0.807915\pi$
$179$	−9.94883	+	5.74396i	−0.743610	+	0.429324i	0.0797703	+	0.996813i	$0.474581\pi$
$180$		0			0
$181$			17.5511i			1.30456i	0.757978	+	0.652280i	$0.226187\pi$
$181$			17.5511i			1.30456i	−0.757978	+	0.652280i	$0.773813\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.0434810	+	0.0251038i	0.00317965	+	0.00183577i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	15.0829	+	8.70814i	1.09136	+	0.630099i	0.933939	−	0.357432i	$-0.116348\pi$
$191$	15.0829	+	8.70814i	1.09136	+	0.630099i	0.157424	+	0.987531i	$0.449681\pi$
$192$		0			0
$193$	0.0326560	+	0.0565619i	0.00235063	+	0.00407141i	0.867198	−	0.497963i	$-0.165918\pi$
$193$	0.0326560	+	0.0565619i	0.00235063	+	0.00407141i	−0.864848	+	0.502034i	$0.832585\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.2109i			1.51122i	0.655023	+	0.755609i	$0.272659\pi$
$197$			21.2109i			1.51122i	−0.655023	+	0.755609i	$0.727341\pi$
$198$		0			0
$199$	−11.5823	+	6.68706i	−0.821050	+	0.474033i	−0.850778	−	0.525525i	$-0.823869\pi$
$199$	−11.5823	+	6.68706i	−0.821050	+	0.474033i	0.0297286	+	0.999558i	$0.490536\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.744796	−	1.50581i	0.0522744	−	0.105687i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.48918			0.172180
$210$		0			0
$211$	−18.8346			−1.29663			−0.648315	−	0.761372i	$-0.724526\pi$
$211$	−18.8346			−1.29663			−0.648315	+	0.761372i	$0.724526\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	3.50914	−	0.225838i	0.238216	−	0.0153309i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.219074	+	0.126482i	−0.0147365	+	0.00850813i
$222$		0			0
$223$		−	0.904185i		−	0.0605487i	−0.999542	−	0.0302743i	$-0.990362\pi$
$223$		−	0.904185i		−	0.0605487i	0.999542	−	0.0302743i	$-0.00963809\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.0680	+	22.6345i	0.867354	+	1.50230i	0.864690	+	0.502305i	$0.167515\pi$
$227$	13.0680	+	22.6345i	0.867354	+	1.50230i	0.00266398	+	0.999996i	$0.499152\pi$
$228$		0			0
$229$	4.13905	+	2.38968i	0.273516	+	0.157915i	0.630485	−	0.776202i	$-0.282856\pi$
$229$	4.13905	+	2.38968i	0.273516	+	0.157915i	−0.356968	+	0.934117i	$0.616190\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.80032	−	4.50351i	−0.511016	−	0.295035i	0.222235	−	0.974993i	$-0.428665\pi$
$233$	−7.80032	−	4.50351i	−0.511016	−	0.295035i	−0.733251	+	0.679958i	$0.761998\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			3.75749i			0.243052i	0.992588	+	0.121526i	$0.0387787\pi$
$239$			3.75749i			0.243052i	−0.992588	+	0.121526i	$0.961221\pi$
$240$		0			0
$241$	−16.6168	+	9.59371i	−1.07038	+	0.617985i	−0.928286	−	0.371868i	$-0.878717\pi$
$241$	−16.6168	+	9.59371i	−1.07038	+	0.617985i	−0.142096	+	0.989853i	$0.545384\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.27073	+	10.8612i	−0.398997	+	0.691083i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.3871			1.72866			0.864328	−	0.502928i	$-0.167744\pi$
$251$	27.3871			1.72866			0.864328	+	0.502928i	$0.167744\pi$
$252$		0			0
$253$	5.69725			0.358183
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.30972	+	16.1249i	−0.580725	+	1.00584i	0.414669	+	0.909972i	$0.363897\pi$
$257$	−9.30972	+	16.1249i	−0.580725	+	1.00584i	−0.995394	+	0.0958722i	$0.969436\pi$
$258$		0			0
$259$	−4.38564	+	2.92290i	−0.272511	+	0.181620i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.42624	+	0.823440i	−0.0879457	+	0.0507755i	−0.543328	−	0.839521i	$-0.682836\pi$
$263$	−1.42624	+	0.823440i	−0.0879457	+	0.0507755i	0.455382	+	0.890296i	$0.349503\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.72546	+	8.18474i	0.288116	+	0.499032i	0.973360	−	0.229282i	$-0.0736377\pi$
$269$	4.72546	+	8.18474i	0.288116	+	0.499032i	−0.685244	+	0.728314i	$0.740304\pi$
$270$		0			0
$271$	6.56613	+	3.79096i	0.398864	+	0.230284i	0.685994	−	0.727607i	$-0.259368\pi$
$271$	6.56613	+	3.79096i	0.398864	+	0.230284i	−0.287130	+	0.957892i	$0.592701\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3.50161	+	6.06496i	0.210391	+	0.364408i	0.951837	−	0.306605i	$-0.0991929\pi$
$277$	3.50161	+	6.06496i	0.210391	+	0.364408i	−0.741446	+	0.671013i	$0.765860\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			15.1496i			0.903747i	0.892082	+	0.451873i	$0.149244\pi$
$281$			15.1496i			0.903747i	−0.892082	+	0.451873i	$0.850756\pi$
$282$		0			0
$283$	−4.39336	+	2.53651i	−0.261158	+	0.150780i	−0.624863	−	0.780735i	$-0.714845\pi$
$283$	−4.39336	+	2.53651i	−0.261158	+	0.150780i	0.363705	+	0.931514i	$0.381512\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.67900	−	3.79815i	−0.453277	−	0.224198i
$288$		0			0
$289$	8.49682	−	14.7169i	0.499813	−	0.865702i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.38425			−0.431393			−0.215696	−	0.976460i	$-0.569202\pi$
$293$	−7.38425			−0.431393			−0.215696	+	0.976460i	$0.569202\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−14.3525	+	24.8592i	−0.830024	+	1.43764i
$300$		0			0
$301$	−12.7949	−	6.32853i	−0.737484	−	0.364771i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	12.3870i		−	0.706963i	−0.935441	−	0.353482i	$-0.884998\pi$
$307$		−	12.3870i		−	0.706963i	0.935441	−	0.353482i	$-0.115002\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.77181	+	16.9253i	0.554109	+	0.959744i	0.997972	+	0.0636498i	$0.0202741\pi$
$311$	9.77181	+	16.9253i	0.554109	+	0.959744i	−0.443864	+	0.896094i	$0.646393\pi$
$312$		0			0
$313$	21.0291	+	12.1412i	1.18864	+	0.686260i	0.957997	−	0.286779i	$-0.0925847\pi$
$313$	21.0291	+	12.1412i	1.18864	+	0.686260i	0.230640	+	0.973039i	$0.425918\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.64538	−	3.83671i	−0.373242	−	0.215491i	0.301632	−	0.953424i	$-0.402469\pi$
$317$	−6.64538	−	3.83671i	−0.373242	−	0.215491i	−0.674874	+	0.737933i	$0.735802\pi$
$318$		0			0
$319$	−0.199990	−	0.346393i	−0.0111973	−	0.0193943i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.314947i			0.0175241i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	14.7052	−	9.80060i	0.810725	−	0.540325i
$330$		0			0
$331$	−8.07835	+	13.9921i	−0.444026	+	0.769076i	−0.997984	−	0.0634691i	$-0.979784\pi$
$331$	−8.07835	+	13.9921i	−0.444026	+	0.769076i	0.553958	+	0.832545i	$0.313117\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.77851			−0.205828			−0.102914	−	0.994690i	$-0.532817\pi$
$337$	−3.77851			−0.205828			−0.102914	+	0.994690i	$0.532817\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.418615	−	0.725062i	0.0226693	−	0.0392643i
$342$		0			0
$343$	−3.54871	−	18.1771i	−0.191612	−	0.981471i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.7689	−	11.9909i	1.11493	−	0.643708i	0.174832	−	0.984598i	$-0.444062\pi$
$347$	20.7689	−	11.9909i	1.11493	−	0.643708i	0.940103	+	0.340890i	$0.110729\pi$
$348$		0			0
$349$			27.1667i			1.45420i	0.686530	+	0.727101i	$0.259133\pi$
$349$			27.1667i			1.45420i	−0.686530	+	0.727101i	$0.740867\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.8712	−	20.5616i	−0.631842	−	1.09438i	−0.987175	−	0.159643i	$-0.948966\pi$
$353$	−11.8712	−	20.5616i	−0.631842	−	1.09438i	0.355332	−	0.934740i	$-0.384368\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.9963	+	10.9675i	1.00259	+	0.578843i	0.909012	−	0.416769i	$-0.136838\pi$
$359$	18.9963	+	10.9675i	1.00259	+	0.578843i	0.0935733	+	0.995612i	$0.470171\pi$
$360$		0			0
$361$	−1.69281	−	2.93204i	−0.0890955	−	0.154318i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−6.45856	+	3.72885i	−0.337134	+	0.194644i	−0.659004	−	0.752139i	$-0.729022\pi$
$367$	−6.45856	+	3.72885i	−0.337134	+	0.194644i	0.321870	+	0.946784i	$0.395689\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.01614	+	0.322823i	−0.260425	+	0.0167601i
$372$		0			0
$373$	7.09682	−	12.2921i	0.367459	−	0.636458i	−0.621708	−	0.783249i	$-0.713561\pi$
$373$	7.09682	−	12.2921i	0.367459	−	0.636458i	0.989168	+	0.146791i	$0.0468944\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.01525			0.103791
$378$		0			0
$379$	−24.5439			−1.26074			−0.630369	−	0.776296i	$-0.717096\pi$
$379$	−24.5439			−1.26074			−0.630369	+	0.776296i	$0.717096\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.14447	+	1.98228i	−0.0584797	+	0.101290i	−0.893783	−	0.448499i	$-0.851959\pi$
$383$	−1.14447	+	1.98228i	−0.0584797	+	0.101290i	0.835303	+	0.549789i	$0.185292\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	31.2676	−	18.0524i	1.58533	−	0.915292i	0.591271	−	0.806473i	$-0.298626\pi$
$389$	31.2676	−	18.0524i	1.58533	−	0.915292i	0.994062	−	0.108819i	$-0.0347070\pi$
$390$		0			0
$391$			0.720851i			0.0364550i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−31.4630	−	18.1652i	−1.57908	−	0.911685i	−0.994987	−	0.100001i	$-0.968115\pi$
$397$	−31.4630	−	18.1652i	−1.57908	−	0.911685i	−0.584097	−	0.811684i	$-0.698551\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	24.1148	+	13.9227i	1.20423	+	0.695264i	0.961494	−	0.274827i	$-0.0886206\pi$
$401$	24.1148	+	13.9227i	1.20423	+	0.695264i	0.242740	+	0.970091i	$0.421954\pi$
$402$		0			0
$403$	2.10914	+	3.65314i	0.105064	+	0.181976i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			1.25484i			0.0622003i
$408$		0			0
$409$	−20.2326	+	11.6813i	−1.00044	+	0.577604i	−0.908378	−	0.418150i	$-0.862679\pi$
$409$	−20.2326	+	11.6813i	−1.00044	+	0.577604i	−0.0920608	+	0.995753i	$0.529345\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.46140	−	38.2460i	−0.121117	−	1.88196i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−17.2387			−0.842167			−0.421084	−	0.907022i	$-0.638350\pi$
$419$	−17.2387			−0.842167			−0.421084	+	0.907022i	$0.638350\pi$
$420$		0			0
$421$	14.5092			0.707135			0.353567	−	0.935409i	$-0.384968\pi$
$421$	14.5092			0.707135			0.353567	+	0.935409i	$0.384968\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	11.3187	+	16.9831i	0.547751	+	0.821867i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−35.3429	+	20.4052i	−1.70241	+	0.982886i	−0.759097	+	0.650977i	$0.774359\pi$
$431$	−35.3429	+	20.4052i	−1.70241	+	0.982886i	−0.943311	+	0.331909i	$0.892307\pi$
$432$		0			0
$433$		−	24.3665i		−	1.17098i	−0.810679	−	0.585491i	$-0.800902\pi$
$433$		−	24.3665i		−	1.17098i	0.810679	−	0.585491i	$-0.199098\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	17.8691	+	30.9502i	0.854795	+	1.48055i
$438$		0			0
$439$	0.748822	+	0.432332i	0.0357393	+	0.0206341i	0.517763	−	0.855524i	$-0.326765\pi$
$439$	0.748822	+	0.432332i	0.0357393	+	0.0206341i	−0.482024	+	0.876158i	$0.660098\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.18260	−	2.99217i	−0.246233	−	0.142162i	0.371805	−	0.928311i	$-0.378739\pi$
$443$	−5.18260	−	2.99217i	−0.246233	−	0.142162i	−0.618038	+	0.786148i	$0.712072\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	0.597775i		−	0.0282107i	−0.999901	−	0.0141054i	$-0.995510\pi$
$449$		−	0.597775i		−	0.0282107i	0.999901	−	0.0141054i	$-0.00449003\pi$
$450$		0			0
$451$	−1.76646	+	1.01987i	−0.0831793	+	0.0480236i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.71155	+	6.42860i	−0.173619	+	0.300717i	−0.939683	−	0.342048i	$-0.888879\pi$
$457$	−3.71155	+	6.42860i	−0.173619	+	0.300717i	0.766063	+	0.642765i	$0.222213\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.92784			−0.229513			−0.114756	−	0.993394i	$-0.536609\pi$
$461$	−4.92784			−0.229513			−0.114756	+	0.993394i	$0.536609\pi$
$462$		0			0
$463$	4.92546			0.228906			0.114453	−	0.993429i	$-0.463489\pi$
$463$	4.92546			0.228906			0.114453	+	0.993429i	$0.463489\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.7758	−	18.6643i	0.498645	−	0.863679i	−0.501354	−	0.865243i	$-0.667164\pi$
$467$	10.7758	−	18.6643i	0.498645	−	0.863679i	0.999999	+	0.00156365i	$0.000497725\pi$
$468$		0			0
$469$	−1.18167	−	18.3611i	−0.0545642	−	0.847838i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.94330	+	1.69931i	−0.135333	+	0.0781346i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.41091	+	11.1040i	0.292922	+	0.507356i	0.974499	−	0.224390i	$-0.0720391\pi$
$479$	6.41091	+	11.1040i	0.292922	+	0.507356i	−0.681577	+	0.731746i	$0.738706\pi$
$480$		0			0
$481$	−5.47534	−	3.16119i	−0.249654	−	0.144138i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	9.06050	+	15.6932i	0.410570	+	0.711129i	0.994952	−	0.100350i	$-0.0319963\pi$
$487$	9.06050	+	15.6932i	0.410570	+	0.711129i	−0.584382	+	0.811479i	$0.698663\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			16.4664i			0.743117i	0.928409	+	0.371559i	$0.121177\pi$
$491$			16.4664i			0.743117i	−0.928409	+	0.371559i	$0.878823\pi$
$492$		0			0
$493$	0.0438277	−	0.0253039i	0.00197390	−	0.00113963i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.78613	−	13.7200i	0.304400	−	0.615427i
$498$		0			0
$499$	10.3994	−	18.0123i	0.465541	−	0.806340i	−0.533685	−	0.845683i	$-0.679193\pi$
$499$	10.3994	−	18.0123i	0.465541	−	0.806340i	0.999226	+	0.0393432i	$0.0125266\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−35.7828			−1.59547			−0.797737	−	0.603005i	$-0.793970\pi$
$503$	−35.7828			−1.59547			−0.797737	+	0.603005i	$0.793970\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.20179	+	10.7418i	−0.274890	+	0.476123i	−0.970107	−	0.242677i	$-0.921975\pi$
$509$	−6.20179	+	10.7418i	−0.274890	+	0.476123i	0.695218	+	0.718799i	$0.255308\pi$
$510$		0			0
$511$	−15.9955	+	1.02942i	−0.707599	+	0.0455389i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	4.20754i		−	0.185047i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20.1236	−	34.8551i	−0.881631	−	1.52703i	−0.849527	−	0.527545i	$-0.823113\pi$
$521$	−20.1236	−	34.8551i	−0.881631	−	1.52703i	−0.0321033	−	0.999485i	$-0.510221\pi$
$522$		0			0
$523$	2.23019	+	1.28760i	0.0975196	+	0.0563029i	0.547967	−	0.836500i	$-0.315402\pi$
$523$	2.23019	+	1.28760i	0.0975196	+	0.0563029i	−0.450447	+	0.892803i	$0.648735\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.0917392	+	0.0529657i	0.00399622	+	0.00230722i
$528$		0			0
$529$	29.3989	+	50.9203i	1.27821	+	2.21393i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	10.2769i		−	0.445144i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.06988	−	1.69708i	−0.175302	−	0.0730983i
$540$		0			0
$541$	4.04711	−	7.00979i	0.173999	−	0.301374i	−0.765816	−	0.643060i	$-0.777664\pi$
$541$	4.04711	−	7.00979i	0.173999	−	0.301374i	0.939814	+	0.341686i	$0.110998\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.64318			0.369556			0.184778	−	0.982780i	$-0.440843\pi$
$547$	8.64318			0.369556			0.184778	+	0.982780i	$0.440843\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.25451	−	2.17288i	0.0534441	−	0.0925679i
$552$		0			0
$553$	24.7622	−	16.5033i	1.05300	−	0.701792i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.7450	+	7.93569i	−0.582395	+	0.336246i	−0.762085	−	0.647478i	$-0.775824\pi$
$557$	−13.7450	+	7.93569i	−0.582395	+	0.336246i	0.179690	+	0.983723i	$0.442491\pi$
$558$		0			0
$559$		−	17.1236i		−	0.724251i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.31444	+	4.00872i	0.0975419	+	0.168948i	0.910667	−	0.413142i	$-0.135569\pi$
$563$	2.31444	+	4.00872i	0.0975419	+	0.168948i	−0.813125	+	0.582089i	$0.802235\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.43907	−	3.14025i	−0.228018	−	0.131646i	0.381640	−	0.924311i	$-0.375360\pi$
$569$	−5.43907	−	3.14025i	−0.228018	−	0.131646i	−0.609657	+	0.792665i	$0.708693\pi$
$570$		0			0
$571$	−6.72730	−	11.6520i	−0.281529	−	0.487622i	0.690233	−	0.723588i	$-0.257508\pi$
$571$	−6.72730	−	11.6520i	−0.281529	−	0.487622i	−0.971762	+	0.235965i	$0.924175\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−8.97939	+	5.18425i	−0.373817	+	0.215823i	−0.675125	−	0.737704i	$-0.735910\pi$
$577$	−8.97939	+	5.18425i	−0.373817	+	0.215823i	0.301308	+	0.953527i	$0.402577\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.67900	+	3.79815i	0.318579	+	0.157574i
$582$		0			0
$583$	−0.598388	+	1.03644i	−0.0247827	+	0.0429249i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−29.7632			−1.22846			−0.614230	−	0.789127i	$-0.710533\pi$
$587$	−29.7632			−1.22846			−0.614230	+	0.789127i	$0.710533\pi$
$588$		0			0
$589$	5.25185			0.216399
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.989732	−	1.71427i	0.0406434	−	0.0703965i	−0.844988	−	0.534785i	$-0.820393\pi$
$593$	0.989732	−	1.71427i	0.0406434	−	0.0703965i	0.885632	+	0.464389i	$0.153726\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	14.8980	−	8.60138i	0.608717	−	0.351443i	−0.163746	−	0.986502i	$-0.552358\pi$
$599$	14.8980	−	8.60138i	0.608717	−	0.351443i	0.772463	+	0.635060i	$0.219025\pi$
$600$		0			0
$601$			16.8733i			0.688274i	0.938919	+	0.344137i	$0.111829\pi$
$601$			16.8733i			0.688274i	−0.938919	+	0.344137i	$0.888171\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	17.5293	+	10.1206i	0.711493	+	0.410781i	0.811614	−	0.584194i	$-0.198589\pi$
$607$	17.5293	+	10.1206i	0.711493	+	0.410781i	−0.100120	+	0.994975i	$0.531923\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	18.3590	+	10.5996i	0.742727	+	0.428814i
$612$		0			0
$613$	7.14230	+	12.3708i	0.288475	+	0.499653i	0.973446	−	0.228917i	$-0.0735185\pi$
$613$	7.14230	+	12.3708i	0.288475	+	0.499653i	−0.684971	+	0.728570i	$0.740185\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			18.4895i			0.744358i	0.928161	+	0.372179i	$0.121389\pi$
$617$			18.4895i			0.744358i	−0.928161	+	0.372179i	$0.878611\pi$
$618$		0			0
$619$	−14.7219	+	8.49969i	−0.591723	+	0.341631i	−0.765778	−	0.643104i	$-0.777646\pi$
$619$	−14.7219	+	8.49969i	−0.591723	+	0.341631i	0.174056	+	0.984736i	$0.444313\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−19.7829	+	13.1847i	−0.792583	+	0.528234i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.158771			−0.00633060
$630$		0			0
$631$	−35.9321			−1.43043			−0.715216	−	0.698903i	$-0.753672\pi$
$631$	−35.9321			−1.43043			−0.715216	+	0.698903i	$0.753672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	17.6578	−	13.4831i	0.699626	−	0.534220i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.4525	−	11.2309i	0.768328	−	0.443594i	−0.0639497	−	0.997953i	$-0.520370\pi$
$641$	19.4525	−	11.2309i	0.768328	−	0.443594i	0.832278	+	0.554359i	$0.187036\pi$
$642$		0			0
$643$			8.27095i			0.326175i	0.986612	+	0.163087i	$0.0521452\pi$
$643$			8.27095i			0.326175i	−0.986612	+	0.163087i	$0.947855\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.06882	+	15.7077i	0.356532	+	0.617532i	0.987379	−	0.158376i	$-0.0506257\pi$
$647$	9.06882	+	15.7077i	0.356532	+	0.617532i	−0.630847	+	0.775907i	$0.717292\pi$
$648$		0			0
$649$	−7.90242	−	4.56247i	−0.310197	−	0.179092i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	24.3365	+	14.0507i	0.952359	+	0.549845i	0.893813	−	0.448440i	$-0.148020\pi$
$653$	24.3365	+	14.0507i	0.952359	+	0.549845i	0.0585463	+	0.998285i	$0.481353\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			30.1912i			1.17608i	0.808832	+	0.588040i	$0.200100\pi$
$659$			30.1912i			1.17608i	−0.808832	+	0.588040i	$0.799900\pi$
$660$		0			0
$661$	27.7832	−	16.0407i	1.08064	−	0.623909i	0.149573	−	0.988751i	$-0.452210\pi$
$661$	27.7832	−	16.0407i	1.08064	−	0.623909i	0.931070	+	0.364841i	$0.118877\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.87134	−	4.97330i	0.111179	−	0.192567i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.85929			0.187591
$672$		0			0
$673$	−28.5578			−1.10082			−0.550412	−	0.834893i	$-0.685529\pi$
$673$	−28.5578			−1.10082			−0.550412	+	0.834893i	$0.685529\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12.9988	+	22.5146i	−0.499586	+	0.865308i	−1.00000		0.000478375i	$-0.999848\pi$
$677$	−12.9988	+	22.5146i	−0.499586	+	0.865308i	0.500414	+	0.865786i	$0.333181\pi$
$678$		0			0
$679$	3.67747	−	7.43502i	0.141128	−	0.285330i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	28.2820	−	16.3286i	1.08218	−	0.624797i	0.150696	−	0.988580i	$-0.451849\pi$
$683$	28.2820	−	16.3286i	1.08218	−	0.624797i	0.931484	+	0.363783i	$0.118515\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.01490	−	5.22197i	−0.114859	−	0.198941i
$690$		0			0
$691$	20.3090	+	11.7254i	0.772593	+	0.446057i	0.833799	−	0.552069i	$-0.186161\pi$
$691$	20.3090	+	11.7254i	0.772593	+	0.446057i	−0.0612061	+	0.998125i	$0.519495\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.129040	−	0.223503i	−0.00488772	−	0.00846578i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	23.4918i		−	0.887272i	−0.896207	−	0.443636i	$-0.853688\pi$
$701$		−	23.4918i		−	0.887272i	0.896207	−	0.443636i	$-0.146312\pi$
$702$		0			0
$703$	−6.81692	+	3.93575i	−0.257105	+	0.148440i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.13522	−	33.1777i	−0.0803030	−	1.24778i
$708$		0			0
$709$	−14.1045	+	24.4296i	−0.529704	+	0.917474i	0.469696	+	0.882828i	$0.344364\pi$
$709$	−14.1045	+	24.4296i	−0.529704	+	0.917474i	−0.999400	+	0.0346459i	$0.988970\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12.0204			0.450169
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.58682	+	16.6049i	−0.357528	+	0.619257i	−0.987547	−	0.157323i	$-0.949714\pi$
$719$	−9.58682	+	16.6049i	−0.357528	+	0.619257i	0.630019	+	0.776580i	$0.283047\pi$
$720$		0			0
$721$	10.7324	+	16.1033i	0.399695	+	0.599719i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	33.2243i		−	1.23222i	−0.787659	−	0.616111i	$-0.788707\pi$
$727$		−	33.2243i		−	1.23222i	0.787659	−	0.616111i	$-0.211293\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.215008	−	0.372404i	−0.00795235	−	0.0137739i
$732$		0			0
$733$	−11.0897	−	6.40264i	−0.409608	−	0.236487i	0.281014	−	0.959704i	$-0.409329\pi$
$733$	−11.0897	−	6.40264i	−0.409608	−	0.236487i	−0.690621	+	0.723217i	$0.742663\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.79379	−	2.19035i	−0.139746	−	0.0806824i
$738$		0			0
$739$	−2.76955	−	4.79701i	−0.101880	−	0.176461i	0.810579	−	0.585629i	$-0.199152\pi$
$739$	−2.76955	−	4.79701i	−0.101880	−	0.176461i	−0.912459	+	0.409168i	$0.865819\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			27.3484i			1.00332i	0.865066	+	0.501658i	$0.167276\pi$
$743$			27.3484i			1.00332i	−0.865066	+	0.501658i	$0.832724\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.2500	−	27.3831i	−0.666841	−	1.00056i
$750$		0			0
$751$	−3.11938	+	5.40292i	−0.113828	+	0.197155i	−0.917310	−	0.398173i	$-0.869644\pi$
$751$	−3.11938	+	5.40292i	−0.113828	+	0.197155i	0.803483	+	0.595328i	$0.202978\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−28.2206			−1.02570			−0.512848	−	0.858479i	$-0.671410\pi$
$757$	−28.2206			−1.02570			−0.512848	+	0.858479i	$0.671410\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.29393	−	5.70525i	0.119405	−	0.206815i	−0.800127	−	0.599830i	$-0.795235\pi$
$761$	3.29393	−	5.70525i	0.119405	−	0.206815i	0.919532	+	0.393015i	$0.128568\pi$
$762$		0			0
$763$	0.762823	+	11.8530i	0.0276161	+	0.429108i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	39.8154	−	22.9874i	1.43765	−	0.830028i
$768$		0			0
$769$		−	39.0076i		−	1.40665i	−0.710868	−	0.703325i	$-0.751698\pi$
$769$		−	39.0076i		−	1.40665i	0.710868	−	0.703325i	$-0.248302\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.3147	+	29.9899i	0.622766	+	1.07866i	0.988968	+	0.148127i	$0.0473246\pi$
$773$	17.3147	+	29.9899i	0.622766	+	1.07866i	−0.366202	+	0.930535i	$0.619342\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.0808	−	6.39750i	−0.397011	−	0.229214i
$780$		0			0
$781$	−1.82219	−	3.15612i	−0.0652030	−	0.112935i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−42.2269	+	24.3797i	−1.50523	+	0.869044i	−0.505247	+	0.862975i	$0.668598\pi$
$787$	−42.2269	+	24.3797i	−1.50523	+	0.869044i	−0.999982	+	0.00606887i	$0.998068\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−16.6501	+	33.6627i	−0.592009	+	1.19691i
$792$		0			0
$793$	−12.2415	+	21.2028i	−0.434707	+	0.752935i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−54.3306			−1.92449			−0.962244	−	0.272189i	$-0.912252\pi$
$797$	−54.3306			−1.92449			−0.962244	+	0.272189i	$0.912252\pi$
$798$		0			0
$799$	0.532364			0.0188337
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.90814	+	3.30500i	−0.0673369	+	0.116631i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	31.3176	−	18.0812i	1.10107	−	0.635701i	0.164566	−	0.986366i	$-0.447378\pi$
$809$	31.3176	−	18.0812i	1.10107	−	0.635701i	0.936501	+	0.350665i	$0.114044\pi$
$810$		0			0
$811$			10.0168i			0.351738i	0.984414	+	0.175869i	$0.0562735\pi$
$811$			10.0168i			0.351738i	−0.984414	+	0.175869i	$0.943726\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−18.4630	−	10.6596i	−0.645938	−	0.372933i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−25.2767	−	14.5935i	−0.882163	−	0.509317i	−0.0107921	−	0.999942i	$-0.503435\pi$
$821$	−25.2767	−	14.5935i	−0.882163	−	0.509317i	−0.871371	+	0.490625i	$0.836769\pi$
$822$		0			0
$823$	−6.80415	−	11.7851i	−0.237178	−	0.410804i	0.722725	−	0.691135i	$-0.242889\pi$
$823$	−6.80415	−	11.7851i	−0.237178	−	0.410804i	−0.959903	+	0.280331i	$0.909556\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	36.3414i		−	1.26372i	−0.775084	−	0.631858i	$-0.782293\pi$
$827$		−	36.3414i		−	1.26372i	0.775084	−	0.631858i	$-0.217707\pi$
$828$		0			0
$829$	43.2073	−	24.9457i	1.50065	−	0.866401i	0.500651	−	0.865649i	$-0.333094\pi$
$829$	43.2073	−	24.9457i	1.50065	−	0.866401i	1.00000		0.000751649i	$-0.000239257\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.214725	−	0.514946i	0.00743977	−	0.0178418i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−31.6185			−1.09159			−0.545796	−	0.837918i	$-0.683773\pi$
$839$	−31.6185			−1.09159			−0.545796	+	0.837918i	$0.683773\pi$
$840$		0			0
$841$	28.5968			0.986098
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	23.3439	−	15.5581i	0.802107	−	0.534581i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−15.6026	+	9.00816i	−0.534850	+	0.308796i
$852$		0			0
$853$			30.7487i			1.05281i	0.850232	+	0.526407i	$0.176461\pi$
$853$			30.7487i			1.05281i	−0.850232	+	0.526407i	$0.823539\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.9682	+	36.3180i	0.716260	+	1.24060i	0.962472	+	0.271382i	$0.0874808\pi$
$857$	20.9682	+	36.3180i	0.716260	+	1.24060i	−0.246212	+	0.969216i	$0.579186\pi$
$858$		0			0
$859$	−39.3503	−	22.7189i	−1.34262	−	0.775160i	−0.355425	−	0.934705i	$-0.615664\pi$
$859$	−39.3503	−	22.7189i	−1.34262	−	0.775160i	−0.987191	+	0.159545i	$0.948997\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9.60951	−	5.54805i	−0.327112	−	0.188858i	0.327446	−	0.944870i	$-0.393812\pi$
$863$	−9.60951	−	5.54805i	−0.327112	−	0.188858i	−0.654558	+	0.756012i	$0.727145\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	7.08511i		−	0.240346i
$870$		0			0
$871$	19.1146	−	11.0358i	0.647672	−	0.373934i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	13.9542	−	24.1693i	0.471198	−	0.816139i	−0.528259	−	0.849083i	$-0.677155\pi$
$877$	13.9542	−	24.1693i	0.471198	−	0.816139i	0.999457	+	0.0329440i	$0.0104883\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	52.9244			1.78307			0.891534	−	0.452953i	$-0.149630\pi$
$881$	52.9244			1.78307			0.891534	+	0.452953i	$0.149630\pi$
$882$		0			0
$883$	−14.8744			−0.500562			−0.250281	−	0.968173i	$-0.580523\pi$
$883$	−14.8744			−0.500562			−0.250281	+	0.968173i	$0.580523\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.571118	−	0.989205i	0.0191763	−	0.0332142i	−0.856278	−	0.516515i	$-0.827229\pi$
$887$	0.571118	−	0.989205i	0.0191763	−	0.0332142i	0.875454	+	0.483301i	$0.160562\pi$
$888$		0			0
$889$	33.2346	+	16.4383i	1.11465	+	0.551323i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	22.8574	−	13.1967i	0.764894	−	0.441612i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.421952	−	0.730842i	−0.0140729	−	0.0243750i
$900$		0			0
$901$	−0.131136	−	0.0757117i	−0.00436879	−	0.00252232i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	10.6487	+	18.4442i	0.353586	+	0.612428i	0.986875	−	0.161487i	$-0.0516289\pi$
$907$	10.6487	+	18.4442i	0.353586	+	0.612428i	−0.633289	+	0.773915i	$0.718296\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	3.29445i		−	0.109150i	−0.998510	−	0.0545749i	$-0.982620\pi$
$911$		−	3.29445i		−	0.109150i	0.998510	−	0.0545749i	$-0.0173804\pi$
$912$		0			0
$913$	1.76646	−	1.01987i	0.0584612	−	0.0337526i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15.3024	−	10.1986i	0.505330	−	0.336788i
$918$		0			0
$919$	−10.5108	+	18.2052i	−0.346718	+	0.600533i	−0.985664	−	0.168718i	$-0.946037\pi$
$919$	−10.5108	+	18.2052i	−0.346718	+	0.600533i	0.638946	+	0.769251i	$0.279371\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	18.3618			0.604384
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.9527	−	36.2911i	0.687436	−	1.19067i	−0.285229	−	0.958459i	$-0.592070\pi$
$929$	20.9527	−	36.2911i	0.687436	−	1.19067i	0.972665	−	0.232214i	$-0.0745970\pi$
$930$		0			0
$931$	−3.54560	−	27.4323i	−0.116202	−	0.899058i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			5.94712i			0.194284i	0.995271	+	0.0971419i	$0.0309701\pi$
$937$			5.94712i			0.194284i	−0.995271	+	0.0971419i	$0.969030\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23.9262	−	41.4414i	−0.779973	−	1.35095i	−0.931957	−	0.362570i	$-0.881900\pi$
$941$	−23.9262	−	41.4414i	−0.779973	−	1.35095i	0.151984	−	0.988383i	$-0.451434\pi$
$942$		0			0
$943$	−25.3618	−	14.6426i	−0.825893	−	0.476829i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.4399	−	15.2651i	−0.859182	−	0.496049i	0.00455619	−	0.999990i	$-0.498550\pi$
$947$	−26.4399	−	15.2651i	−0.859182	−	0.496049i	−0.863738	+	0.503941i	$0.831883\pi$
$948$		0			0
$949$	−9.61395	−	16.6518i	−0.312082	−	0.540542i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			30.5262i			0.988842i	0.869223	+	0.494421i	$0.164620\pi$
$953$			30.5262i			0.988842i	−0.869223	+	0.494421i	$0.835380\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−34.6007	+	2.22679i	−1.11731	+	0.0719069i
$960$		0			0
$961$	−14.6168	+	25.3170i	−0.471509	+	0.816678i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	55.8252			1.79522			0.897609	−	0.440793i	$-0.145303\pi$
$967$	55.8252			1.79522			0.897609	+	0.440793i	$0.145303\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.8910	−	32.7202i	0.606242	−	1.05004i	−0.385612	−	0.922661i	$-0.626010\pi$
$971$	18.8910	−	32.7202i	0.606242	−	1.05004i	0.991854	−	0.127381i	$-0.0406572\pi$
$972$		0			0
$973$	−13.9506	+	28.2050i	−0.447236	+	0.904210i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	43.8820	−	25.3353i	1.40391	−	0.810547i	0.409117	−	0.912482i	$-0.365837\pi$
$977$	43.8820	−	25.3353i	1.40391	−	0.810547i	0.994791	+	0.101935i	$0.0325034\pi$
$978$		0			0
$979$			5.66038i			0.180907i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20.9873	+	36.3510i	0.669390	+	1.15942i	0.978075	+	0.208254i	$0.0667779\pi$
$983$	20.9873	+	36.3510i	0.669390	+	1.15942i	−0.308685	+	0.951164i	$0.599889\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−42.2582	−	24.3978i	−1.34373	−	0.775803i
$990$		0			0
$991$	−20.7463	−	35.9336i	−0.659027	−	1.14147i	−0.980868	−	0.194674i	$-0.937635\pi$
$991$	−20.7463	−	35.9336i	−0.659027	−	1.14147i	0.321841	−	0.946794i	$-0.395698\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−24.7630	+	14.2969i	−0.784253	+	0.452788i	−0.837935	−	0.545770i	$-0.816237\pi$
$997$	−24.7630	+	14.2969i	−0.784253	+	0.452788i	0.0536827	+	0.998558i	$0.482904\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.ch.e.1601.1	yes	20
3.2	odd	2	inner	6300.2.ch.e.1601.2	yes	20
5.2	odd	4		6300.2.dd.d.1349.13		40
5.3	odd	4		6300.2.dd.d.1349.7		40
5.4	even	2		6300.2.ch.d.1601.9	✓	20
7.3	odd	6	inner	6300.2.ch.e.4301.2	yes	20
15.2	even	4		6300.2.dd.d.1349.14		40
15.8	even	4		6300.2.dd.d.1349.8		40
15.14	odd	2		6300.2.ch.d.1601.10	yes	20
21.17	even	6	inner	6300.2.ch.e.4301.1	yes	20
35.3	even	12		6300.2.dd.d.4049.14		40
35.17	even	12		6300.2.dd.d.4049.8		40
35.24	odd	6		6300.2.ch.d.4301.10	yes	20
105.17	odd	12		6300.2.dd.d.4049.7		40
105.38	odd	12		6300.2.dd.d.4049.13		40
105.59	even	6		6300.2.ch.d.4301.9	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6300.2.ch.d.1601.9	✓	20	5.4	even	2
6300.2.ch.d.1601.10	yes	20	15.14	odd	2
6300.2.ch.d.4301.9	yes	20	105.59	even	6
6300.2.ch.d.4301.10	yes	20	35.24	odd	6
6300.2.ch.e.1601.1	yes	20	1.1	even	1	trivial
6300.2.ch.e.1601.2	yes	20	3.2	odd	2	inner
6300.2.ch.e.4301.1	yes	20	21.17	even	6	inner
6300.2.ch.e.4301.2	yes	20	7.3	odd	6	inner
6300.2.dd.d.1349.7		40	5.3	odd	4
6300.2.dd.d.1349.8		40	15.8	even	4
6300.2.dd.d.1349.13		40	5.2	odd	4
6300.2.dd.d.1349.14		40	15.2	even	4
6300.2.dd.d.4049.7		40	105.17	odd	12
6300.2.dd.d.4049.8		40	35.17	even	12
6300.2.dd.d.4049.13		40	105.38	odd	12
6300.2.dd.d.4049.14		40	35.3	even	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.ch (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.37152 - 1.17299i) q^{7} +(-0.545538 + 0.314967i) q^{11} -3.17384i q^{13} +(-0.0398515 - 0.0690248i) q^{17} +(-3.42210 - 1.97575i) q^{19} +(-7.83251 - 4.52210i) q^{23} +0.634956i q^{29} +(-1.15101 + 0.664538i) q^{31} +(0.996014 - 1.72515i) q^{37} +3.23801 q^{41} +5.39522 q^{43} +(-3.33967 + 5.78448i) q^{47} +(4.24819 + 5.56353i) q^{49} +(1.64531 - 0.949922i) q^{53} +(7.24278 + 12.5449i) q^{59} +(-6.68049 - 3.85698i) q^{61} +(3.47711 + 6.02252i) q^{67} +5.78534i q^{71} +(5.24659 - 3.02912i) q^{73} +(1.66321 - 0.107039i) q^{77} +(-5.62370 + 9.74053i) q^{79} -3.23801 q^{83} +(4.49284 - 7.78183i) q^{89} +(-3.72288 + 7.52683i) q^{91} +3.13513i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{7} + 6 q^{31} + 2 q^{37} - 20 q^{43} + 4 q^{49} + 6 q^{61} + 28 q^{67} - 6 q^{73} + 2 q^{79} - 40 q^{91}+O(q^{100}) \) 20 * q + 4 * q^7 + 6 * q^31 + 2 * q^37 - 20 * q^43 + 4 * q^49 + 6 * q^61 + 28 * q^67 - 6 * q^73 + 2 * q^79 - 40 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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