LMFDB - Embedded newform 1600.2.j.e.143.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(143,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 3]))

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

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

chi := DirichletCharacter("1600.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.j (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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.2
Character	$\chi$	$=$	1600.143
Dual form			1600.2.j.e.1007.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70780i q^{3} +(-1.60911 + 1.60911i) q^{7} -4.33218 q^{9} +O(q^{10})$	$q-2.70780i q^{3} +(-1.60911 + 1.60911i) q^{7} -4.33218 q^{9} +(3.90559 - 3.90559i) q^{11} +1.48572 q^{13} +(4.26356 - 4.26356i) q^{17} +(0.162635 - 0.162635i) q^{19} +(4.35714 + 4.35714i) q^{21} +(-5.87429 - 5.87429i) q^{23} +3.60728i q^{27} +(1.48745 + 1.48745i) q^{29} +6.78608i q^{31} +(-10.5756 - 10.5756i) q^{33} -6.13260 q^{37} -4.02305i q^{39} -2.75628i q^{41} +3.39392 q^{43} +(-9.44332 - 9.44332i) q^{47} +1.82155i q^{49} +(-11.5449 - 11.5449i) q^{51} -1.54070i q^{53} +(-0.440383 - 0.440383i) q^{57} +(-2.53072 - 2.53072i) q^{59} +(0.600858 - 0.600858i) q^{61} +(6.97095 - 6.97095i) q^{63} -8.14130 q^{67} +(-15.9064 + 15.9064i) q^{69} +4.55096 q^{71} +(4.84446 - 4.84446i) q^{73} +12.5690i q^{77} +0.455580 q^{79} -3.22874 q^{81} +2.84390i q^{83} +(4.02772 - 4.02772i) q^{87} -1.91001 q^{89} +(-2.39069 + 2.39069i) q^{91} +18.3754 q^{93} +(-1.73249 + 1.73249i) q^{97} +(-16.9197 + 16.9197i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 40 q^{9}+O(q^{10}) $ 24 * q - 40 * q^9	$ 24 q - 40 q^{9} + 20 q^{11} + 12 q^{19} + 8 q^{29} - 20 q^{51} - 8 q^{59} - 48 q^{61} - 64 q^{69} + 16 q^{71} + 104 q^{79} + 48 q^{81} + 96 q^{89} - 64 q^{91} - 128 q^{99}+O(q^{100}) $ 24 * q - 40 * q^9 + 20 * q^11 + 12 * q^19 + 8 * q^29 - 20 * q^51 - 8 * q^59 - 48 * q^61 - 64 * q^69 + 16 * q^71 + 104 * q^79 + 48 * q^81 + 96 * q^89 - 64 * q^91 - 128 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{1}{4}\right)$	$-1$

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
$3$		−	2.70780i		−	1.56335i	−0.623686	−	0.781675i	$-0.714366\pi$
$3$		−	2.70780i		−	1.56335i	0.623686	−	0.781675i	$-0.285634\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.60911	+	1.60911i	−0.608185	+	0.608185i	−0.942472	−	0.334286i	$-0.891505\pi$
$7$	−1.60911	+	1.60911i	−0.608185	+	0.608185i	0.334286	+	0.942472i	$0.391505\pi$
$8$		0			0
$9$	−4.33218			−1.44406
$10$		0			0
$11$	3.90559	−	3.90559i	1.17758	−	1.17758i	0.197221	−	0.980359i	$-0.436808\pi$
$11$	3.90559	−	3.90559i	1.17758	−	1.17758i	0.980359	−	0.197221i	$-0.0631916\pi$
$12$		0			0
$13$	1.48572			0.412066			0.206033	−	0.978545i	$-0.433945\pi$
$13$	1.48572			0.412066			0.206033	+	0.978545i	$0.433945\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.26356	−	4.26356i	1.03406	−	1.03406i	0.0346652	−	0.999399i	$-0.488964\pi$
$17$	4.26356	−	4.26356i	1.03406	−	1.03406i	0.999399	−	0.0346652i	$-0.0110365\pi$
$18$		0			0
$19$	0.162635	−	0.162635i	0.0373110	−	0.0373110i	−0.688205	−	0.725516i	$-0.741601\pi$
$19$	0.162635	−	0.162635i	0.0373110	−	0.0373110i	0.725516	+	0.688205i	$0.241601\pi$
$20$		0			0
$21$	4.35714	+	4.35714i	0.950806	+	0.950806i
$22$		0			0
$23$	−5.87429	−	5.87429i	−1.22487	−	1.22487i	−0.965879	−	0.258996i	$-0.916609\pi$
$23$	−5.87429	−	5.87429i	−1.22487	−	1.22487i	−0.258996	−	0.965879i	$-0.583391\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			3.60728i			0.694222i
$28$		0			0
$29$	1.48745	+	1.48745i	0.276213	+	0.276213i	0.831595	−	0.555382i	$-0.187428\pi$
$29$	1.48745	+	1.48745i	0.276213	+	0.276213i	−0.555382	+	0.831595i	$0.687428\pi$
$30$		0			0
$31$			6.78608i			1.21882i	0.792857	+	0.609408i	$0.208593\pi$
$31$			6.78608i			1.21882i	−0.792857	+	0.609408i	$0.791407\pi$
$32$		0			0
$33$	−10.5756	−	10.5756i	−1.84097	−	1.84097i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.13260			−1.00819			−0.504097	−	0.863647i	$-0.668174\pi$
$37$	−6.13260			−1.00819			−0.504097	+	0.863647i	$0.668174\pi$
$38$		0			0
$39$		−	4.02305i		−	0.644203i
$40$		0			0
$41$		−	2.75628i		−	0.430459i	−0.976563	−	0.215230i	$-0.930950\pi$
$41$		−	2.75628i		−	0.430459i	0.976563	−	0.215230i	$-0.0690500\pi$
$42$		0			0
$43$	3.39392			0.517569			0.258784	−	0.965935i	$-0.416678\pi$
$43$	3.39392			0.517569			0.258784	+	0.965935i	$0.416678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.44332	−	9.44332i	−1.37745	−	1.37745i	−0.848906	−	0.528544i	$-0.822738\pi$
$47$	−9.44332	−	9.44332i	−1.37745	−	1.37745i	−0.528544	−	0.848906i	$-0.677262\pi$
$48$		0			0
$49$			1.82155i			0.260221i
$50$		0			0
$51$	−11.5449	−	11.5449i	−1.61660	−	1.61660i
$52$		0			0
$53$		−	1.54070i		−	0.211632i	−0.994386	−	0.105816i	$-0.966255\pi$
$53$		−	1.54070i		−	0.211632i	0.994386	−	0.105816i	$-0.0337454\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.440383	−	0.440383i	−0.0583302	−	0.0583302i
$58$		0			0
$59$	−2.53072	−	2.53072i	−0.329471	−	0.329471i	0.522914	−	0.852385i	$-0.324845\pi$
$59$	−2.53072	−	2.53072i	−0.329471	−	0.329471i	−0.852385	+	0.522914i	$0.824845\pi$
$60$		0			0
$61$	0.600858	−	0.600858i	0.0769320	−	0.0769320i	−0.667594	−	0.744526i	$-0.732676\pi$
$61$	0.600858	−	0.600858i	0.0769320	−	0.0769320i	0.744526	+	0.667594i	$0.232676\pi$
$62$		0			0
$63$	6.97095	−	6.97095i	0.878257	−	0.878257i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−8.14130			−0.994618			−0.497309	−	0.867574i	$-0.665678\pi$
$67$	−8.14130			−0.994618			−0.497309	+	0.867574i	$0.665678\pi$
$68$		0			0
$69$	−15.9064	+	15.9064i	−1.91491	+	1.91491i
$70$		0			0
$71$	4.55096			0.540099			0.270050	−	0.962846i	$-0.412960\pi$
$71$	4.55096			0.540099			0.270050	+	0.962846i	$0.412960\pi$
$72$		0			0
$73$	4.84446	−	4.84446i	0.567001	−	0.567001i	−0.364286	−	0.931287i	$-0.618687\pi$
$73$	4.84446	−	4.84446i	0.567001	−	0.567001i	0.931287	+	0.364286i	$0.118687\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			12.5690i			1.43237i
$78$		0			0
$79$	0.455580			0.0512568			0.0256284	−	0.999672i	$-0.491841\pi$
$79$	0.455580			0.0512568			0.0256284	+	0.999672i	$0.491841\pi$
$80$		0			0
$81$	−3.22874			−0.358749
$82$		0			0
$83$			2.84390i			0.312158i	0.987745	+	0.156079i	$0.0498855\pi$
$83$			2.84390i			0.312158i	−0.987745	+	0.156079i	$0.950115\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	4.02772	−	4.02772i	0.431817	−	0.431817i
$88$		0			0
$89$	−1.91001			−0.202461			−0.101230	−	0.994863i	$-0.532278\pi$
$89$	−1.91001			−0.202461			−0.101230	+	0.994863i	$0.532278\pi$
$90$		0			0
$91$	−2.39069	+	2.39069i	−0.250612	+	0.250612i
$92$		0			0
$93$	18.3754			1.90544
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.73249	+	1.73249i	−0.175908	+	0.175908i	−0.789569	−	0.613662i	$-0.789696\pi$
$97$	−1.73249	+	1.73249i	−0.175908	+	0.175908i	0.613662	+	0.789569i	$0.289696\pi$
$98$		0			0
$99$	−16.9197	+	16.9197i	−1.70050	+	1.70050i
$100$		0			0
$101$	−7.75459	−	7.75459i	−0.771610	−	0.771610i	0.206778	−	0.978388i	$-0.433702\pi$
$101$	−7.75459	−	7.75459i	−0.771610	−	0.771610i	−0.978388	+	0.206778i	$0.933702\pi$
$102$		0			0
$103$	−0.377727	−	0.377727i	−0.0372186	−	0.0372186i	0.688253	−	0.725471i	$-0.258378\pi$
$103$	−0.377727	−	0.377727i	−0.0372186	−	0.0372186i	−0.725471	+	0.688253i	$0.758378\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.14130i			0.787049i	0.919314	+	0.393524i	$0.128744\pi$
$107$			8.14130i			0.787049i	−0.919314	+	0.393524i	$0.871256\pi$
$108$		0			0
$109$	9.51087	+	9.51087i	0.910976	+	0.910976i	0.996349	−	0.0853729i	$-0.0272082\pi$
$109$	9.51087	+	9.51087i	0.910976	+	0.910976i	−0.0853729	+	0.996349i	$0.527208\pi$
$110$		0			0
$111$			16.6059i			1.57616i
$112$		0			0
$113$	3.89633	+	3.89633i	0.366536	+	0.366536i	0.866212	−	0.499676i	$-0.166548\pi$
$113$	3.89633	+	3.89633i	0.366536	+	0.366536i	−0.499676	+	0.866212i	$0.666548\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−6.43643			−0.595048
$118$		0			0
$119$			13.7210i			1.25781i
$120$		0			0
$121$		−	19.5073i		−	1.77339i
$122$		0			0
$123$	−7.46346			−0.672958
$124$		0			0
$125$		0			0
$126$		0			0
$127$	5.63683	+	5.63683i	0.500188	+	0.500188i	0.911496	−	0.411309i	$-0.134928\pi$
$127$	5.63683	+	5.63683i	0.500188	+	0.500188i	−0.411309	+	0.911496i	$0.634928\pi$
$128$		0			0
$129$		−	9.19007i		−	0.809140i
$130$		0			0
$131$	−4.77637	−	4.77637i	−0.417313	−	0.417313i	0.466963	−	0.884277i	$-0.345348\pi$
$131$	−4.77637	−	4.77637i	−0.417313	−	0.417313i	−0.884277	+	0.466963i	$0.845348\pi$
$132$		0			0
$133$			0.523394i			0.0453841i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.13294	−	3.13294i	−0.267666	−	0.267666i	0.560493	−	0.828159i	$-0.310612\pi$
$137$	−3.13294	−	3.13294i	−0.267666	−	0.267666i	−0.828159	+	0.560493i	$0.810612\pi$
$138$		0			0
$139$	9.27118	+	9.27118i	0.786372	+	0.786372i	0.980897	−	0.194526i	$-0.0623168\pi$
$139$	9.27118	+	9.27118i	0.786372	+	0.786372i	−0.194526	+	0.980897i	$0.562317\pi$
$140$		0			0
$141$	−25.5706	+	25.5706i	−2.15344	+	2.15344i
$142$		0			0
$143$	5.80263	−	5.80263i	0.485240	−	0.485240i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.93239			0.406816
$148$		0			0
$149$	15.1098	−	15.1098i	1.23784	−	1.23784i	0.276964	−	0.960880i	$-0.410672\pi$
$149$	15.1098	−	15.1098i	1.23784	−	1.23784i	0.960880	−	0.276964i	$-0.0893283\pi$
$150$		0			0
$151$	−8.69622			−0.707688			−0.353844	−	0.935304i	$-0.615126\pi$
$151$	−8.69622			−0.707688			−0.353844	+	0.935304i	$0.615126\pi$
$152$		0			0
$153$	−18.4705	+	18.4705i	−1.49325	+	1.49325i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	15.3620i		−	1.22602i	−0.790076	−	0.613009i	$-0.789959\pi$
$157$		−	15.3620i		−	1.22602i	0.790076	−	0.613009i	$-0.210041\pi$
$158$		0			0
$159$	−4.17191			−0.330854
$160$		0			0
$161$	18.9047			1.48990
$162$		0			0
$163$			17.1904i			1.34645i	0.739436	+	0.673227i	$0.235092\pi$
$163$			17.1904i			1.34645i	−0.739436	+	0.673227i	$0.764908\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.32562	+	5.32562i	−0.412109	+	0.412109i	−0.882473	−	0.470364i	$-0.844123\pi$
$167$	−5.32562	+	5.32562i	−0.412109	+	0.412109i	0.470364	+	0.882473i	$0.344123\pi$
$168$		0			0
$169$	−10.7926			−0.830202
$170$		0			0
$171$	−0.704565	+	0.704565i	−0.0538794	+	0.0538794i
$172$		0			0
$173$	14.0580			1.06881			0.534405	−	0.845228i	$-0.320536\pi$
$173$	14.0580			1.06881			0.534405	+	0.845228i	$0.320536\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−6.85268	+	6.85268i	−0.515079	+	0.515079i
$178$		0			0
$179$	−3.80190	+	3.80190i	−0.284168	+	0.284168i	−0.834769	−	0.550601i	$-0.814399\pi$
$179$	−3.80190	+	3.80190i	−0.284168	+	0.284168i	0.550601	+	0.834769i	$0.314399\pi$
$180$		0			0
$181$	−9.75230	−	9.75230i	−0.724883	−	0.724883i	0.244713	−	0.969596i	$-0.421306\pi$
$181$	−9.75230	−	9.75230i	−0.724883	−	0.724883i	−0.969596	+	0.244713i	$0.921306\pi$
$182$		0			0
$183$	−1.62700	−	1.62700i	−0.120272	−	0.120272i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	33.3034i		−	2.43539i
$188$		0			0
$189$	−5.80451	−	5.80451i	−0.422216	−	0.422216i
$190$		0			0
$191$		−	17.3470i		−	1.25519i	−0.778542	−	0.627593i	$-0.784040\pi$
$191$		−	17.3470i		−	1.25519i	0.778542	−	0.627593i	$-0.215960\pi$
$192$		0			0
$193$	−11.9979	−	11.9979i	−0.863627	−	0.863627i	0.128130	−	0.991757i	$-0.459102\pi$
$193$	−11.9979	−	11.9979i	−0.863627	−	0.863627i	−0.991757	+	0.128130i	$0.959102\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.87712			0.347480			0.173740	−	0.984792i	$-0.444415\pi$
$197$	4.87712			0.347480			0.173740	+	0.984792i	$0.444415\pi$
$198$		0			0
$199$		−	7.34946i		−	0.520989i	−0.965475	−	0.260495i	$-0.916114\pi$
$199$		−	7.34946i		−	0.520989i	0.965475	−	0.260495i	$-0.0838857\pi$
$200$		0			0
$201$			22.0450i			1.55493i
$202$		0			0
$203$	−4.78694			−0.335977
$204$		0			0
$205$		0			0
$206$		0			0
$207$	25.4485	+	25.4485i	1.76879	+	1.76879i
$208$		0			0
$209$		−	1.27037i		−	0.0878735i
$210$		0			0
$211$	6.60321	+	6.60321i	0.454584	+	0.454584i	0.896873	−	0.442289i	$-0.145833\pi$
$211$	6.60321	+	6.60321i	0.454584	+	0.454584i	−0.442289	+	0.896873i	$0.645833\pi$
$212$		0			0
$213$		−	12.3231i		−	0.844364i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−10.9195	−	10.9195i	−0.741267	−	0.741267i
$218$		0			0
$219$	−13.1178	−	13.1178i	−0.886421	−	0.886421i
$220$		0			0
$221$	6.33447	−	6.33447i	0.426103	−	0.426103i
$222$		0			0
$223$	16.7751	−	16.7751i	1.12334	−	1.12334i	0.132106	−	0.991236i	$-0.457826\pi$
$223$	16.7751	−	16.7751i	1.12334	−	1.12334i	0.991236	−	0.132106i	$-0.0421738\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.19897			−0.411440			−0.205720	−	0.978611i	$-0.565954\pi$
$227$	−6.19897			−0.411440			−0.205720	+	0.978611i	$0.565954\pi$
$228$		0			0
$229$	12.4206	−	12.4206i	0.820779	−	0.820779i	−0.165440	−	0.986220i	$-0.552905\pi$
$229$	12.4206	−	12.4206i	0.820779	−	0.820779i	0.986220	+	0.165440i	$0.0529045\pi$
$230$		0			0
$231$	34.0344			2.23930
$232$		0			0
$233$	12.7462	−	12.7462i	0.835030	−	0.835030i	−0.153170	−	0.988200i	$-0.548948\pi$
$233$	12.7462	−	12.7462i	0.835030	−	0.835030i	0.988200	+	0.153170i	$0.0489483\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	1.23362i		−	0.0801323i
$238$		0			0
$239$	24.6380			1.59370			0.796849	−	0.604179i	$-0.206499\pi$
$239$	24.6380			1.59370			0.796849	+	0.604179i	$0.206499\pi$
$240$		0			0
$241$	19.3783			1.24826			0.624132	−	0.781319i	$-0.285453\pi$
$241$	19.3783			1.24826			0.624132	+	0.781319i	$0.285453\pi$
$242$		0			0
$243$			19.5646i			1.25507i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.241631	−	0.241631i	0.0153746	−	0.0153746i
$248$		0			0
$249$	7.70070			0.488012
$250$		0			0
$251$	4.08108	−	4.08108i	0.257596	−	0.257596i	−0.566480	−	0.824076i	$-0.691695\pi$
$251$	4.08108	−	4.08108i	0.257596	−	0.257596i	0.824076	+	0.566480i	$0.191695\pi$
$252$		0			0
$253$	−45.8851			−2.88477
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.3161	+	11.3161i	−0.705878	+	0.705878i	−0.965666	−	0.259788i	$-0.916347\pi$
$257$	−11.3161	+	11.3161i	−0.705878	+	0.705878i	0.259788	+	0.965666i	$0.416347\pi$
$258$		0			0
$259$	9.86801	−	9.86801i	0.613168	−	0.613168i
$260$		0			0
$261$	−6.44391	−	6.44391i	−0.398868	−	0.398868i
$262$		0			0
$263$	−0.625808	−	0.625808i	−0.0385890	−	0.0385890i	0.687549	−	0.726138i	$-0.258687\pi$
$263$	−0.625808	−	0.625808i	−0.0385890	−	0.0385890i	−0.726138	+	0.687549i	$0.758687\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			5.17193i			0.316517i
$268$		0			0
$269$	9.04646	+	9.04646i	0.551573	+	0.551573i	0.926895	−	0.375322i	$-0.122468\pi$
$269$	9.04646	+	9.04646i	0.551573	+	0.551573i	−0.375322	+	0.926895i	$0.622468\pi$
$270$		0			0
$271$			2.64963i			0.160954i	0.996756	+	0.0804769i	$0.0256443\pi$
$271$			2.64963i			0.160954i	−0.996756	+	0.0804769i	$0.974356\pi$
$272$		0			0
$273$	6.47351	+	6.47351i	0.391795	+	0.391795i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	19.5983			1.17755			0.588774	−	0.808298i	$-0.299611\pi$
$277$	19.5983			1.17755			0.588774	+	0.808298i	$0.299611\pi$
$278$		0			0
$279$		−	29.3986i		−	1.76005i
$280$		0			0
$281$		−	18.8586i		−	1.12501i	−0.826793	−	0.562506i	$-0.809837\pi$
$281$		−	18.8586i		−	1.12501i	0.826793	−	0.562506i	$-0.190163\pi$
$282$		0			0
$283$	12.9828			0.771745			0.385873	−	0.922552i	$-0.373900\pi$
$283$	12.9828			0.771745			0.385873	+	0.922552i	$0.373900\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.43516	+	4.43516i	0.261799	+	0.261799i
$288$		0			0
$289$		−	19.3558i		−	1.13858i
$290$		0			0
$291$	4.69124	+	4.69124i	0.275005	+	0.275005i
$292$		0			0
$293$			27.3139i			1.59570i	0.602858	+	0.797848i	$0.294028\pi$
$293$			27.3139i			1.59570i	−0.602858	+	0.797848i	$0.705972\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	14.0886	+	14.0886i	0.817502	+	0.817502i
$298$		0			0
$299$	−8.72758	−	8.72758i	−0.504729	−	0.504729i
$300$		0			0
$301$	−5.46119	+	5.46119i	−0.314778	+	0.314778i
$302$		0			0
$303$	−20.9979	+	20.9979i	−1.20630	+	1.20630i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.8250			0.617815			0.308908	−	0.951092i	$-0.400037\pi$
$307$	10.8250			0.617815			0.308908	+	0.951092i	$0.400037\pi$
$308$		0			0
$309$	−1.02281	+	1.02281i	−0.0581856	+	0.0581856i
$310$		0			0
$311$	25.4742			1.44451			0.722256	−	0.691626i	$-0.243105\pi$
$311$	25.4742			1.44451			0.722256	+	0.691626i	$0.243105\pi$
$312$		0			0
$313$	−13.0982	+	13.0982i	−0.740354	+	0.740354i	−0.972646	−	0.232292i	$-0.925378\pi$
$313$	−13.0982	+	13.0982i	−0.740354	+	0.740354i	0.232292	+	0.972646i	$0.425378\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			5.34769i			0.300356i	0.988659	+	0.150178i	$0.0479847\pi$
$317$			5.34769i			0.300356i	−0.988659	+	0.150178i	$0.952015\pi$
$318$		0			0
$319$	11.6188			0.650525
$320$		0			0
$321$	22.0450			1.23043
$322$		0			0
$323$		−	1.38681i		−	0.0771640i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	25.7535	−	25.7535i	1.42417	−	1.42417i
$328$		0			0
$329$	30.3906			1.67549
$330$		0			0
$331$	−12.1138	+	12.1138i	−0.665837	+	0.665837i	−0.956750	−	0.290913i	$-0.906041\pi$
$331$	−12.1138	+	12.1138i	−0.665837	+	0.665837i	0.290913	+	0.956750i	$0.406041\pi$
$332$		0			0
$333$	26.5675			1.45589
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.80287	−	9.80287i	0.533996	−	0.533996i	−0.387763	−	0.921759i	$-0.626752\pi$
$337$	9.80287	−	9.80287i	0.533996	−	0.533996i	0.921759	+	0.387763i	$0.126752\pi$
$338$		0			0
$339$	10.5505	−	10.5505i	0.573024	−	0.573024i
$340$		0			0
$341$	26.5037	+	26.5037i	1.43525	+	1.43525i
$342$		0			0
$343$	−14.1948	−	14.1948i	−0.766448	−	0.766448i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6.32581i			0.339587i	0.985480	+	0.169794i	$0.0543101\pi$
$347$			6.32581i			0.339587i	−0.985480	+	0.169794i	$0.945690\pi$
$348$		0			0
$349$	−8.73170	−	8.73170i	−0.467397	−	0.467397i	0.433673	−	0.901070i	$-0.357217\pi$
$349$	−8.73170	−	8.73170i	−0.467397	−	0.467397i	−0.901070	+	0.433673i	$0.857217\pi$
$350$		0			0
$351$			5.35943i			0.286065i
$352$		0			0
$353$	10.1374	+	10.1374i	0.539561	+	0.539561i	0.923400	−	0.383839i	$-0.125398\pi$
$353$	10.1374	+	10.1374i	0.539561	+	0.539561i	−0.383839	+	0.923400i	$0.625398\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	37.1538			1.96639
$358$		0			0
$359$			6.95941i			0.367303i	0.982991	+	0.183652i	$0.0587918\pi$
$359$			6.95941i			0.367303i	−0.982991	+	0.183652i	$0.941208\pi$
$360$		0			0
$361$			18.9471i			0.997216i
$362$		0			0
$363$	−52.8218			−2.77243
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−10.5831	−	10.5831i	−0.552431	−	0.552431i	0.374710	−	0.927142i	$-0.377742\pi$
$367$	−10.5831	−	10.5831i	−0.552431	−	0.552431i	−0.927142	+	0.374710i	$0.877742\pi$
$368$		0			0
$369$			11.9407i			0.621609i
$370$		0			0
$371$	2.47916	+	2.47916i	0.128711	+	0.128711i
$372$		0			0
$373$		−	8.69529i		−	0.450225i	−0.974333	−	0.225113i	$-0.927725\pi$
$373$		−	8.69529i		−	0.450225i	0.974333	−	0.225113i	$-0.0722750\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.20994	+	2.20994i	0.113818	+	0.113818i
$378$		0			0
$379$	6.85696	+	6.85696i	0.352218	+	0.352218i	0.860934	−	0.508716i	$-0.169880\pi$
$379$	6.85696	+	6.85696i	0.352218	+	0.352218i	−0.508716	+	0.860934i	$0.669880\pi$
$380$		0			0
$381$	15.2634	−	15.2634i	0.781968	−	0.781968i
$382$		0			0
$383$	0.555884	−	0.555884i	0.0284043	−	0.0284043i	−0.692762	−	0.721166i	$-0.743606\pi$
$383$	0.555884	−	0.555884i	0.0284043	−	0.0284043i	0.721166	+	0.692762i	$0.243606\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−14.7031			−0.747400
$388$		0			0
$389$	−13.6874	+	13.6874i	−0.693979	+	0.693979i	−0.963105	−	0.269126i	$-0.913265\pi$
$389$	−13.6874	+	13.6874i	−0.693979	+	0.693979i	0.269126	+	0.963105i	$0.413265\pi$
$390$		0			0
$391$	−50.0907			−2.53320
$392$		0			0
$393$	−12.9334	+	12.9334i	−0.652406	+	0.652406i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	7.41556i		−	0.372176i	−0.982533	−	0.186088i	$-0.940419\pi$
$397$		−	7.41556i		−	0.372176i	0.982533	−	0.186088i	$-0.0595810\pi$
$398$		0			0
$399$	1.41725			0.0709511
$400$		0			0
$401$	−29.3119			−1.46377			−0.731884	−	0.681429i	$-0.761359\pi$
$401$	−29.3119			−1.46377			−0.731884	+	0.681429i	$0.761359\pi$
$402$		0			0
$403$			10.0823i			0.502233i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−23.9514	+	23.9514i	−1.18723	+	1.18723i
$408$		0			0
$409$	−37.0355			−1.83129			−0.915644	−	0.401990i	$-0.868319\pi$
$409$	−37.0355			−1.83129			−0.915644	+	0.401990i	$0.868319\pi$
$410$		0			0
$411$	−8.48339	+	8.48339i	−0.418455	+	0.418455i
$412$		0			0
$413$	8.14439			0.400759
$414$		0			0
$415$		0			0
$416$		0			0
$417$	25.1045	−	25.1045i	1.22937	−	1.22937i
$418$		0			0
$419$	18.5780	−	18.5780i	0.907594	−	0.907594i	−0.0884832	−	0.996078i	$-0.528202\pi$
$419$	18.5780	−	18.5780i	0.907594	−	0.907594i	0.996078	+	0.0884832i	$0.0282020\pi$
$420$		0			0
$421$	−7.39938	−	7.39938i	−0.360624	−	0.360624i	0.503419	−	0.864042i	$-0.332075\pi$
$421$	−7.39938	−	7.39938i	−0.360624	−	0.360624i	−0.864042	+	0.503419i	$0.832075\pi$
$422$		0			0
$423$	40.9102	+	40.9102i	1.98912	+	1.98912i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			1.93369i			0.0935778i
$428$		0			0
$429$	−15.7124	−	15.7124i	−0.758600	−	0.758600i
$430$		0			0
$431$		−	12.9340i		−	0.623011i	−0.950244	−	0.311506i	$-0.899167\pi$
$431$		−	12.9340i		−	0.623011i	0.950244	−	0.311506i	$-0.100833\pi$
$432$		0			0
$433$	18.7932	+	18.7932i	0.903145	+	0.903145i	0.995707	−	0.0925619i	$-0.0295056\pi$
$433$	18.7932	+	18.7932i	0.903145	+	0.903145i	−0.0925619	+	0.995707i	$0.529506\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.91073			−0.0914026
$438$		0			0
$439$			17.0988i			0.816082i	0.912964	+	0.408041i	$0.133788\pi$
$439$			17.0988i			0.816082i	−0.912964	+	0.408041i	$0.866212\pi$
$440$		0			0
$441$		−	7.89127i		−	0.375775i
$442$		0			0
$443$	36.1064			1.71547			0.857733	−	0.514096i	$-0.171872\pi$
$443$	36.1064			1.71547			0.857733	+	0.514096i	$0.171872\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−40.9144	−	40.9144i	−1.93518	−	1.93518i
$448$		0			0
$449$		−	14.7784i		−	0.697436i	−0.937228	−	0.348718i	$-0.886617\pi$
$449$		−	14.7784i		−	0.697436i	0.937228	−	0.348718i	$-0.113383\pi$
$450$		0			0
$451$	−10.7649	−	10.7649i	−0.506900	−	0.506900i
$452$		0			0
$453$			23.5476i			1.10636i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	15.4771	+	15.4771i	0.723987	+	0.723987i	0.969415	−	0.245428i	$-0.0789285\pi$
$457$	15.4771	+	15.4771i	0.723987	+	0.723987i	−0.245428	+	0.969415i	$0.578928\pi$
$458$		0			0
$459$	15.3799	+	15.3799i	0.717870	+	0.717870i
$460$		0			0
$461$	1.98835	−	1.98835i	0.0926068	−	0.0926068i	−0.659286	−	0.751893i	$-0.729141\pi$
$461$	1.98835	−	1.98835i	0.0926068	−	0.0926068i	0.751893	+	0.659286i	$0.229141\pi$
$462$		0			0
$463$	−15.6906	+	15.6906i	−0.729202	+	0.729202i	−0.970461	−	0.241258i	$-0.922440\pi$
$463$	−15.6906	+	15.6906i	−0.729202	+	0.729202i	0.241258	+	0.970461i	$0.422440\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.2593			0.521018			0.260509	−	0.965471i	$-0.416110\pi$
$467$	11.2593			0.521018			0.260509	+	0.965471i	$0.416110\pi$
$468$		0			0
$469$	13.1002	−	13.1002i	0.604912	−	0.604912i
$470$		0			0
$471$	−41.5971			−1.91669
$472$		0			0
$473$	13.2553	−	13.2553i	0.609478	−	0.609478i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			6.67460i			0.305609i
$478$		0			0
$479$	−28.4486			−1.29985			−0.649925	−	0.759998i	$-0.725200\pi$
$479$	−28.4486			−1.29985			−0.649925	+	0.759998i	$0.725200\pi$
$480$		0			0
$481$	−9.11135			−0.415442
$482$		0			0
$483$		−	51.1902i		−	2.32924i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−7.65129	+	7.65129i	−0.346713	+	0.346713i	−0.858884	−	0.512171i	$-0.828841\pi$
$487$	−7.65129	+	7.65129i	−0.346713	+	0.346713i	0.512171	+	0.858884i	$0.328841\pi$
$488$		0			0
$489$	46.5481			2.10498
$490$		0			0
$491$	18.3385	−	18.3385i	0.827607	−	0.827607i	−0.159578	−	0.987185i	$-0.551013\pi$
$491$	18.3385	−	18.3385i	0.827607	−	0.827607i	0.987185	+	0.159578i	$0.0510135\pi$
$492$		0			0
$493$	12.6837			0.571244
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.32298	+	7.32298i	−0.328481	+	0.328481i
$498$		0			0
$499$	16.4254	−	16.4254i	0.735301	−	0.735301i	−0.236363	−	0.971665i	$-0.575956\pi$
$499$	16.4254	−	16.4254i	0.735301	−	0.735301i	0.971665	+	0.236363i	$0.0759556\pi$
$500$		0			0
$501$	14.4207	+	14.4207i	0.644271	+	0.644271i
$502$		0			0
$503$	11.6628	+	11.6628i	0.520017	+	0.520017i	0.917577	−	0.397559i	$-0.130143\pi$
$503$	11.6628	+	11.6628i	0.520017	+	0.520017i	−0.397559	+	0.917577i	$0.630143\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			29.2243i			1.29790i
$508$		0			0
$509$	29.7342	+	29.7342i	1.31795	+	1.31795i	0.915399	+	0.402549i	$0.131876\pi$
$509$	29.7342	+	29.7342i	1.31795	+	1.31795i	0.402549	+	0.915399i	$0.368124\pi$
$510$		0			0
$511$			15.5905i			0.689684i
$512$		0			0
$513$	0.586671	+	0.586671i	0.0259021	+	0.0259021i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−73.7635			−3.24412
$518$		0			0
$519$		−	38.0663i		−	1.67092i
$520$		0			0
$521$			2.02127i			0.0885534i	0.999019	+	0.0442767i	$0.0140983\pi$
$521$			2.02127i			0.0885534i	−0.999019	+	0.0442767i	$0.985902\pi$
$522$		0			0
$523$	30.3472			1.32699			0.663495	−	0.748181i	$-0.269073\pi$
$523$	30.3472			1.32699			0.663495	+	0.748181i	$0.269073\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	28.9329	+	28.9329i	1.26033	+	1.26033i
$528$		0			0
$529$			46.0146i			2.00063i
$530$		0			0
$531$	10.9635	+	10.9635i	0.475777	+	0.475777i
$532$		0			0
$533$		−	4.09508i		−	0.177378i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	10.2948	+	10.2948i	0.444253	+	0.444253i
$538$		0			0
$539$	7.11422	+	7.11422i	0.306431	+	0.306431i
$540$		0			0
$541$	11.1901	−	11.1901i	0.481099	−	0.481099i	−0.424384	−	0.905482i	$-0.639509\pi$
$541$	11.1901	−	11.1901i	0.481099	−	0.481099i	0.905482	+	0.424384i	$0.139509\pi$
$542$		0			0
$543$	−26.4073	+	26.4073i	−1.13324	+	1.13324i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	24.9664			1.06749			0.533743	−	0.845647i	$-0.320785\pi$
$547$	24.9664			1.06749			0.533743	+	0.845647i	$0.320785\pi$
$548$		0			0
$549$	−2.60303	+	2.60303i	−0.111094	+	0.111094i
$550$		0			0
$551$	0.483823			0.0206116
$552$		0			0
$553$	−0.733078	+	0.733078i	−0.0311736	+	0.0311736i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			24.1273i			1.02230i	0.859490	+	0.511152i	$0.170781\pi$
$557$			24.1273i			1.02230i	−0.859490	+	0.511152i	$0.829219\pi$
$558$		0			0
$559$	5.04244			0.213272
$560$		0			0
$561$	−90.1790			−3.80736
$562$		0			0
$563$			30.9864i			1.30592i	0.757392	+	0.652961i	$0.226473\pi$
$563$			30.9864i			1.30592i	−0.757392	+	0.652961i	$0.773527\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	5.19540	−	5.19540i	0.218186	−	0.218186i
$568$		0			0
$569$	−24.3753			−1.02186			−0.510932	−	0.859621i	$-0.670700\pi$
$569$	−24.3753			−1.02186			−0.510932	+	0.859621i	$0.670700\pi$
$570$		0			0
$571$	26.5820	−	26.5820i	1.11242	−	1.11242i	0.119602	−	0.992822i	$-0.461838\pi$
$571$	26.5820	−	26.5820i	1.11242	−	1.11242i	0.992822	−	0.119602i	$-0.0381618\pi$
$572$		0			0
$573$	−46.9722			−1.96229
$574$		0			0
$575$		0			0
$576$		0			0
$577$	30.9055	−	30.9055i	1.28661	−	1.28661i	0.349782	−	0.936831i	$-0.386256\pi$
$577$	30.9055	−	30.9055i	1.28661	−	1.28661i	0.936831	−	0.349782i	$-0.113744\pi$
$578$		0			0
$579$	−32.4879	+	32.4879i	−1.35015	+	1.35015i
$580$		0			0
$581$	−4.57613	−	4.57613i	−0.189850	−	0.189850i
$582$		0			0
$583$	−6.01735	−	6.01735i	−0.249213	−	0.249213i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			16.7155i			0.689923i	0.938617	+	0.344962i	$0.112108\pi$
$587$			16.7155i			0.689923i	−0.938617	+	0.344962i	$0.887892\pi$
$588$		0			0
$589$	1.10366	+	1.10366i	0.0454753	+	0.0454753i
$590$		0			0
$591$		−	13.2063i		−	0.543233i
$592$		0			0
$593$	2.47626	+	2.47626i	0.101688	+	0.101688i	0.756120	−	0.654433i	$-0.227092\pi$
$593$	2.47626	+	2.47626i	0.101688	+	0.101688i	−0.654433	+	0.756120i	$0.727092\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−19.9009			−0.814488
$598$		0			0
$599$			25.4533i			1.03999i	0.854168	+	0.519997i	$0.174067\pi$
$599$			25.4533i			1.03999i	−0.854168	+	0.519997i	$0.825933\pi$
$600$		0			0
$601$			34.4210i			1.40406i	0.712147	+	0.702031i	$0.247723\pi$
$601$			34.4210i			1.40406i	−0.712147	+	0.702031i	$0.752277\pi$
$602$		0			0
$603$	35.2696			1.43629
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−5.02033	−	5.02033i	−0.203769	−	0.203769i	0.597844	−	0.801613i	$-0.296024\pi$
$607$	−5.02033	−	5.02033i	−0.203769	−	0.203769i	−0.801613	+	0.597844i	$0.796024\pi$
$608$		0			0
$609$			12.9621i			0.525250i
$610$		0			0
$611$	−14.0302	−	14.0302i	−0.567600	−	0.567600i
$612$		0			0
$613$			37.4177i			1.51129i	0.654983	+	0.755644i	$0.272676\pi$
$613$			37.4177i			1.51129i	−0.654983	+	0.755644i	$0.727324\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.1640	−	16.1640i	−0.650740	−	0.650740i	0.302431	−	0.953171i	$-0.402202\pi$
$617$	−16.1640	−	16.1640i	−0.650740	−	0.650740i	−0.953171	+	0.302431i	$0.902202\pi$
$618$		0			0
$619$	12.9228	+	12.9228i	0.519412	+	0.519412i	0.917393	−	0.397982i	$-0.130289\pi$
$619$	12.9228	+	12.9228i	0.519412	+	0.519412i	−0.397982	+	0.917393i	$0.630289\pi$
$620$		0			0
$621$	21.1902	−	21.1902i	0.850334	−	0.850334i
$622$		0			0
$623$	3.07342	−	3.07342i	0.123134	−	0.123134i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−3.43991			−0.137377
$628$		0			0
$629$	−26.1467	+	26.1467i	−1.04254	+	1.04254i
$630$		0			0
$631$	28.2978			1.12652			0.563259	−	0.826281i	$-0.309547\pi$
$631$	28.2978			1.12652			0.563259	+	0.826281i	$0.309547\pi$
$632$		0			0
$633$	17.8802	−	17.8802i	0.710673	−	0.710673i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			2.70632i			0.107228i
$638$		0			0
$639$	−19.7156			−0.779936
$640$		0			0
$641$	−5.16460			−0.203989			−0.101995	−	0.994785i	$-0.532522\pi$
$641$	−5.16460			−0.203989			−0.101995	+	0.994785i	$0.532522\pi$
$642$		0			0
$643$		−	36.9131i		−	1.45571i	−0.685730	−	0.727856i	$-0.740517\pi$
$643$		−	36.9131i		−	1.45571i	0.685730	−	0.727856i	$-0.259483\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.9996	−	18.9996i	0.746953	−	0.746953i	−0.226953	−	0.973906i	$-0.572876\pi$
$647$	18.9996	−	18.9996i	0.746953	−	0.746953i	0.973906	+	0.226953i	$0.0728762\pi$
$648$		0			0
$649$	−19.7679			−0.775958
$650$		0			0
$651$	−29.5679	+	29.5679i	−1.15886	+	1.15886i
$652$		0			0
$653$	−29.1069			−1.13904			−0.569520	−	0.821977i	$-0.692871\pi$
$653$	−29.1069			−1.13904			−0.569520	+	0.821977i	$0.692871\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−20.9871	+	20.9871i	−0.818784	+	0.818784i
$658$		0			0
$659$	29.4708	−	29.4708i	1.14802	−	1.14802i	0.161080	−	0.986941i	$-0.448502\pi$
$659$	29.4708	−	29.4708i	1.14802	−	1.14802i	0.986941	−	0.161080i	$-0.0514977\pi$
$660$		0			0
$661$	8.20363	+	8.20363i	0.319084	+	0.319084i	0.848415	−	0.529331i	$-0.177557\pi$
$661$	8.20363	+	8.20363i	0.319084	+	0.319084i	−0.529331	+	0.848415i	$0.677557\pi$
$662$		0			0
$663$	−17.1525	−	17.1525i	−0.666147	−	0.666147i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	17.4754i		−	0.676652i
$668$		0			0
$669$	−45.4235	−	45.4235i	−1.75617	−	1.75617i
$670$		0			0
$671$		−	4.69341i		−	0.181187i
$672$		0			0
$673$	−17.5626	−	17.5626i	−0.676988	−	0.676988i	0.282329	−	0.959318i	$-0.408893\pi$
$673$	−17.5626	−	17.5626i	−0.676988	−	0.676988i	−0.959318	+	0.282329i	$0.908893\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	33.7993			1.29901			0.649507	−	0.760356i	$-0.274975\pi$
$677$	33.7993			1.29901			0.649507	+	0.760356i	$0.274975\pi$
$678$		0			0
$679$		−	5.57552i		−	0.213969i
$680$		0			0
$681$			16.7856i			0.643224i
$682$		0			0
$683$	2.75261			0.105326			0.0526628	−	0.998612i	$-0.483229\pi$
$683$	2.75261			0.105326			0.0526628	+	0.998612i	$0.483229\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−33.6326	−	33.6326i	−1.28316	−	1.28316i
$688$		0			0
$689$		−	2.28906i		−	0.0872062i
$690$		0			0
$691$	25.4193	+	25.4193i	0.966995	+	0.966995i	0.999472	−	0.0324773i	$-0.0103397\pi$
$691$	25.4193	+	25.4193i	0.966995	+	0.966995i	−0.0324773	+	0.999472i	$0.510340\pi$
$692$		0			0
$693$		−	54.4513i		−	2.06843i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−11.7516	−	11.7516i	−0.445122	−	0.445122i
$698$		0			0
$699$	−34.5141	−	34.5141i	−1.30544	−	1.30544i
$700$		0			0
$701$	10.4690	−	10.4690i	0.395410	−	0.395410i	−0.481201	−	0.876610i	$-0.659799\pi$
$701$	10.4690	−	10.4690i	0.395410	−	0.395410i	0.876610	+	0.481201i	$0.159799\pi$
$702$		0			0
$703$	−0.997376	+	0.997376i	−0.0376167	+	0.0376167i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.9559			0.938564
$708$		0			0
$709$	28.5665	−	28.5665i	1.07284	−	1.07284i	0.0757076	−	0.997130i	$-0.475878\pi$
$709$	28.5665	−	28.5665i	1.07284	−	1.07284i	0.997130	−	0.0757076i	$-0.0241216\pi$
$710$		0			0
$711$	−1.97366			−0.0740179
$712$		0			0
$713$	39.8634	−	39.8634i	1.49290	−	1.49290i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	66.7147i		−	2.49151i
$718$		0			0
$719$	−18.9471			−0.706607			−0.353303	−	0.935509i	$-0.614942\pi$
$719$	−18.9471			−0.706607			−0.353303	+	0.935509i	$0.614942\pi$
$720$		0			0
$721$	1.21561			0.0452716
$722$		0			0
$723$		−	52.4725i		−	1.95147i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−31.1761	+	31.1761i	−1.15626	+	1.15626i	−0.170982	+	0.985274i	$0.554694\pi$
$727$	−31.1761	+	31.1761i	−1.15626	+	1.15626i	−0.985274	+	0.170982i	$0.945306\pi$
$728$		0			0
$729$	43.2909			1.60337
$730$		0			0
$731$	14.4702	−	14.4702i	0.535199	−	0.535199i
$732$		0			0
$733$	−10.1960			−0.376599			−0.188300	−	0.982112i	$-0.560298\pi$
$733$	−10.1960			−0.376599			−0.188300	+	0.982112i	$0.560298\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−31.7966	+	31.7966i	−1.17124	+	1.17124i
$738$		0			0
$739$	−34.1178	+	34.1178i	−1.25504	+	1.25504i	−0.301615	+	0.953430i	$0.597526\pi$
$739$	−34.1178	+	34.1178i	−1.25504	+	1.25504i	−0.953430	+	0.301615i	$0.902474\pi$
$740$		0			0
$741$	−0.654288	−	0.654288i	−0.0240359	−	0.0240359i
$742$		0			0
$743$	−11.3933	−	11.3933i	−0.417981	−	0.417981i	0.466526	−	0.884507i	$-0.345505\pi$
$743$	−11.3933	−	11.3933i	−0.417981	−	0.417981i	−0.884507	+	0.466526i	$0.845505\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	12.3203i		−	0.450775i
$748$		0			0
$749$	−13.1002	−	13.1002i	−0.478672	−	0.478672i
$750$		0			0
$751$			2.94217i			0.107361i	0.998558	+	0.0536807i	$0.0170953\pi$
$751$			2.94217i			0.107361i	−0.998558	+	0.0536807i	$0.982905\pi$
$752$		0			0
$753$	−11.0508	−	11.0508i	−0.402712	−	0.402712i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.83398			0.103003			0.0515014	−	0.998673i	$-0.483599\pi$
$757$	2.83398			0.103003			0.0515014	+	0.998673i	$0.483599\pi$
$758$		0			0
$759$			124.248i			4.50991i
$760$		0			0
$761$		−	22.5458i		−	0.817284i	−0.912695	−	0.408642i	$-0.866002\pi$
$761$		−	22.5458i		−	0.817284i	0.912695	−	0.408642i	$-0.133998\pi$
$762$		0			0
$763$	−30.6080			−1.10808
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.75995	−	3.75995i	−0.135764	−	0.135764i
$768$		0			0
$769$		−	10.9238i		−	0.393923i	−0.980411	−	0.196961i	$-0.936893\pi$
$769$		−	10.9238i		−	0.393923i	0.980411	−	0.196961i	$-0.0631074\pi$
$770$		0			0
$771$	30.6417	+	30.6417i	1.10353	+	1.10353i
$772$		0			0
$773$		−	28.7653i		−	1.03461i	−0.855800	−	0.517307i	$-0.826934\pi$
$773$		−	28.7653i		−	1.03461i	0.855800	−	0.517307i	$-0.173066\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−26.7206	−	26.7206i	−0.958596	−	0.958596i
$778$		0			0
$779$	−0.448268	−	0.448268i	−0.0160609	−	0.0160609i
$780$		0			0
$781$	17.7742	−	17.7742i	0.636010	−	0.636010i
$782$		0			0
$783$	−5.36566	+	5.36566i	−0.191753	+	0.191753i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	7.84805			0.279753			0.139876	−	0.990169i	$-0.455329\pi$
$787$	7.84805			0.279753			0.139876	+	0.990169i	$0.455329\pi$
$788$		0			0
$789$	−1.69456	+	1.69456i	−0.0603280	+	0.0603280i
$790$		0			0
$791$	−12.5392			−0.445844
$792$		0			0
$793$	0.892709	−	0.892709i	0.0317010	−	0.0317010i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			33.1600i			1.17459i	0.809374	+	0.587294i	$0.199807\pi$
$797$			33.1600i			1.17459i	−0.809374	+	0.587294i	$0.800193\pi$
$798$		0			0
$799$	−80.5243			−2.84874
$800$		0			0
$801$	8.27452			0.292366
$802$		0			0
$803$		−	37.8410i		−	1.33538i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	24.4960	−	24.4960i	0.862301	−	0.862301i
$808$		0			0
$809$	1.71735			0.0603787			0.0301893	−	0.999544i	$-0.490389\pi$
$809$	1.71735			0.0603787			0.0301893	+	0.999544i	$0.490389\pi$
$810$		0			0
$811$	−5.70133	+	5.70133i	−0.200201	+	0.200201i	−0.800086	−	0.599885i	$-0.795213\pi$
$811$	−5.70133	+	5.70133i	−0.200201	+	0.200201i	0.599885	+	0.800086i	$0.295213\pi$
$812$		0			0
$813$	7.17468			0.251627
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.551971	−	0.551971i	0.0193110	−	0.0193110i
$818$		0			0
$819$	10.3569	−	10.3569i	0.361900	−	0.361900i
$820$		0			0
$821$	14.8755	+	14.8755i	0.519160	+	0.519160i	0.917317	−	0.398157i	$-0.130350\pi$
$821$	14.8755	+	14.8755i	0.519160	+	0.519160i	−0.398157	+	0.917317i	$0.630350\pi$
$822$		0			0
$823$	−9.74767	−	9.74767i	−0.339782	−	0.339782i	0.516503	−	0.856285i	$-0.327233\pi$
$823$	−9.74767	−	9.74767i	−0.339782	−	0.339782i	−0.856285	+	0.516503i	$0.827233\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			38.3314i			1.33291i	0.745544	+	0.666456i	$0.232190\pi$
$827$			38.3314i			1.33291i	−0.745544	+	0.666456i	$0.767810\pi$
$828$		0			0
$829$	−12.1114	−	12.1114i	−0.420645	−	0.420645i	0.464781	−	0.885426i	$-0.346133\pi$
$829$	−12.1114	−	12.1114i	−0.420645	−	0.420645i	−0.885426	+	0.464781i	$0.846133\pi$
$830$		0			0
$831$		−	53.0683i		−	1.84092i
$832$		0			0
$833$	7.76627	+	7.76627i	0.269085	+	0.269085i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−24.4793			−0.846129
$838$		0			0
$839$		−	11.1976i		−	0.386586i	−0.981141	−	0.193293i	$-0.938083\pi$
$839$		−	11.1976i		−	0.386586i	0.981141	−	0.193293i	$-0.0619167\pi$
$840$		0			0
$841$		−	24.5750i		−	0.847413i
$842$		0			0
$843$	−51.0654			−1.75879
$844$		0			0
$845$		0			0
$846$		0			0
$847$	31.3893	+	31.3893i	1.07855	+	1.07855i
$848$		0			0
$849$		−	35.1547i		−	1.20651i
$850$		0			0
$851$	36.0247	+	36.0247i	1.23491	+	1.23491i
$852$		0			0
$853$		−	16.4185i		−	0.562160i	−0.959684	−	0.281080i	$-0.909307\pi$
$853$		−	16.4185i		−	0.562160i	0.959684	−	0.281080i	$-0.0906927\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.0907	+	10.0907i	0.344691	+	0.344691i	0.858128	−	0.513436i	$-0.171628\pi$
$857$	10.0907	+	10.0907i	0.344691	+	0.344691i	−0.513436	+	0.858128i	$0.671628\pi$
$858$		0			0
$859$	−6.14854	−	6.14854i	−0.209785	−	0.209785i	0.594391	−	0.804176i	$-0.297393\pi$
$859$	−6.14854	−	6.14854i	−0.209785	−	0.209785i	−0.804176	+	0.594391i	$0.797393\pi$
$860$		0			0
$861$	12.0095	−	12.0095i	0.409283	−	0.409283i
$862$		0			0
$863$	9.80142	−	9.80142i	0.333644	−	0.333644i	−0.520325	−	0.853969i	$-0.674189\pi$
$863$	9.80142	−	9.80142i	0.333644	−	0.333644i	0.853969	+	0.520325i	$0.174189\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−52.4117			−1.77999
$868$		0			0
$869$	1.77931	−	1.77931i	0.0603590	−	0.0603590i
$870$		0			0
$871$	−12.0957			−0.409848
$872$		0			0
$873$	7.50546	−	7.50546i	0.254021	−	0.254021i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			26.9934i			0.911503i	0.890107	+	0.455751i	$0.150629\pi$
$877$			26.9934i			0.911503i	−0.890107	+	0.455751i	$0.849371\pi$
$878$		0			0
$879$	73.9606			2.49463
$880$		0			0
$881$	44.8712			1.51175			0.755874	−	0.654717i	$-0.227212\pi$
$881$	44.8712			1.51175			0.755874	+	0.654717i	$0.227212\pi$
$882$		0			0
$883$		−	1.92019i		−	0.0646195i	−0.999478	−	0.0323098i	$-0.989714\pi$
$883$		−	1.92019i		−	0.0646195i	0.999478	−	0.0323098i	$-0.0102863\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.90523	−	9.90523i	0.332585	−	0.332585i	−0.520982	−	0.853567i	$-0.674434\pi$
$887$	9.90523	−	9.90523i	0.332585	−	0.332585i	0.853567	+	0.520982i	$0.174434\pi$
$888$		0			0
$889$	−18.1405			−0.608414
$890$		0			0
$891$	−12.6102	+	12.6102i	−0.422456	+	0.422456i
$892$		0			0
$893$	−3.07163			−0.102788
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−23.6325	+	23.6325i	−0.789067	+	0.789067i
$898$		0			0
$899$	−10.0940	+	10.0940i	−0.336653	+	0.336653i
$900$		0			0
$901$	−6.56887	−	6.56887i	−0.218841	−	0.218841i
$902$		0			0
$903$	14.7878	+	14.7878i	0.492107	+	0.492107i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	33.3923i		−	1.10877i	−0.832259	−	0.554387i	$-0.812953\pi$
$907$		−	33.3923i		−	1.10877i	0.832259	−	0.554387i	$-0.187047\pi$
$908$		0			0
$909$	33.5943	+	33.5943i	1.11425	+	1.11425i
$910$		0			0
$911$		−	23.8525i		−	0.790270i	−0.918623	−	0.395135i	$-0.870698\pi$
$911$		−	23.8525i		−	0.790270i	0.918623	−	0.395135i	$-0.129302\pi$
$912$		0			0
$913$	11.1071	+	11.1071i	0.367591	+	0.367591i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15.3714			0.507607
$918$		0			0
$919$		−	54.1050i		−	1.78476i	−0.451286	−	0.892379i	$-0.649035\pi$
$919$		−	54.1050i		−	1.78476i	0.451286	−	0.892379i	$-0.350965\pi$
$920$		0			0
$921$		−	29.3119i		−	0.965861i
$922$		0			0
$923$	6.76147			0.222556
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.63638	+	1.63638i	0.0537459	+	0.0537459i
$928$		0			0
$929$			41.4213i			1.35899i	0.733680	+	0.679495i	$0.237801\pi$
$929$			41.4213i			1.35899i	−0.733680	+	0.679495i	$0.762199\pi$
$930$		0			0
$931$	0.296247	+	0.296247i	0.00970912	+	0.00970912i
$932$		0			0
$933$		−	68.9792i		−	2.25828i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	18.9679	+	18.9679i	0.619655	+	0.619655i	0.945443	−	0.325788i	$-0.105630\pi$
$937$	18.9679	+	18.9679i	0.619655	+	0.619655i	−0.325788	+	0.945443i	$0.605630\pi$
$938$		0			0
$939$	35.4673	+	35.4673i	1.15743	+	1.15743i
$940$		0			0
$941$	−2.67037	+	2.67037i	−0.0870515	+	0.0870515i	−0.749292	−	0.662240i	$-0.769606\pi$
$941$	−2.67037	+	2.67037i	−0.0870515	+	0.0870515i	0.662240	+	0.749292i	$0.269606\pi$
$942$		0			0
$943$	−16.1912	+	16.1912i	−0.527258	+	0.527258i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−30.2062			−0.981571			−0.490785	−	0.871280i	$-0.663290\pi$
$947$	−30.2062			−0.981571			−0.490785	+	0.871280i	$0.663290\pi$
$948$		0			0
$949$	7.19753	−	7.19753i	0.233642	−	0.233642i
$950$		0			0
$951$	14.4805			0.469562
$952$		0			0
$953$	−25.5333	+	25.5333i	−0.827104	+	0.827104i	−0.987115	−	0.160011i	$-0.948847\pi$
$953$	−25.5333	+	25.5333i	−0.827104	+	0.827104i	0.160011	+	0.987115i	$0.448847\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	31.4613i		−	1.01700i
$958$		0			0
$959$	10.0825			0.325581
$960$		0			0
$961$	−15.0509			−0.485514
$962$		0			0
$963$		−	35.2696i		−	1.13655i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−33.1842	+	33.1842i	−1.06713	+	1.06713i	−0.0695536	+	0.997578i	$0.522157\pi$
$967$	−33.1842	+	33.1842i	−1.06713	+	1.06713i	−0.997578	+	0.0695536i	$0.977843\pi$
$968$		0			0
$969$	−3.75520			−0.120634
$970$		0			0
$971$	−9.74290	+	9.74290i	−0.312664	+	0.312664i	−0.845941	−	0.533277i	$-0.820961\pi$
$971$	−9.74290	+	9.74290i	−0.312664	+	0.312664i	0.533277	+	0.845941i	$0.320961\pi$
$972$		0			0
$973$	−29.8367			−0.956519
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−31.1381	+	31.1381i	−0.996197	+	0.996197i	−0.999993	−	0.00379623i	$-0.998792\pi$
$977$	−31.1381	+	31.1381i	−0.996197	+	0.996197i	0.00379623	+	0.999993i	$0.498792\pi$
$978$		0			0
$979$	−7.45973	+	7.45973i	−0.238414	+	0.238414i
$980$		0			0
$981$	−41.2028	−	41.2028i	−1.31550	−	1.31550i
$982$		0			0
$983$	9.29644	+	9.29644i	0.296510	+	0.296510i	0.839645	−	0.543135i	$-0.182763\pi$
$983$	9.29644	+	9.29644i	0.296510	+	0.296510i	−0.543135	+	0.839645i	$0.682763\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	82.2918i		−	2.61938i
$988$		0			0
$989$	−19.9369	−	19.9369i	−0.633956	−	0.633956i
$990$		0			0
$991$			18.2628i			0.580136i	0.957006	+	0.290068i	$0.0936780\pi$
$991$			18.2628i			0.580136i	−0.957006	+	0.290068i	$0.906322\pi$
$992$		0			0
$993$	32.8019	+	32.8019i	1.04094	+	1.04094i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.29522			0.167701			0.0838506	−	0.996478i	$-0.473278\pi$
$997$	5.29522			0.167701			0.0838506	+	0.996478i	$0.473278\pi$
$998$		0			0
$999$		−	22.1220i		−	0.699910i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.j.e.143.2		24
4.3	odd	2		400.2.j.e.43.9	yes	24
5.2	odd	4		1600.2.s.e.207.2		24
5.3	odd	4		1600.2.s.e.207.11		24
5.4	even	2	inner	1600.2.j.e.143.11		24
16.3	odd	4		1600.2.s.e.943.2		24
16.13	even	4		400.2.s.e.243.3	yes	24
20.3	even	4		400.2.s.e.107.10	yes	24
20.7	even	4		400.2.s.e.107.3	yes	24
20.19	odd	2		400.2.j.e.43.4	✓	24
80.3	even	4	inner	1600.2.j.e.1007.2		24
80.13	odd	4		400.2.j.e.307.4	yes	24
80.19	odd	4		1600.2.s.e.943.11		24
80.29	even	4		400.2.s.e.243.10	yes	24
80.67	even	4	inner	1600.2.j.e.1007.11		24
80.77	odd	4		400.2.j.e.307.9	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.j.e.43.4	✓	24	20.19	odd	2
400.2.j.e.43.9	yes	24	4.3	odd	2
400.2.j.e.307.4	yes	24	80.13	odd	4
400.2.j.e.307.9	yes	24	80.77	odd	4
400.2.s.e.107.3	yes	24	20.7	even	4
400.2.s.e.107.10	yes	24	20.3	even	4
400.2.s.e.243.3	yes	24	16.13	even	4
400.2.s.e.243.10	yes	24	80.29	even	4
1600.2.j.e.143.2		24	1.1	even	1	trivial
1600.2.j.e.143.11		24	5.4	even	2	inner
1600.2.j.e.1007.2		24	80.3	even	4	inner
1600.2.j.e.1007.11		24	80.67	even	4	inner
1600.2.s.e.207.2		24	5.2	odd	4
1600.2.s.e.207.11		24	5.3	odd	4
1600.2.s.e.943.2		24	16.3	odd	4
1600.2.s.e.943.11		24	80.19	odd	4

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.70780i q^{3} +(-1.60911 + 1.60911i) q^{7} -4.33218 q^{9} +O(q^{10})\)	\(q-2.70780i q^{3} +(-1.60911 + 1.60911i) q^{7} -4.33218 q^{9} +(3.90559 - 3.90559i) q^{11} +1.48572 q^{13} +(4.26356 - 4.26356i) q^{17} +(0.162635 - 0.162635i) q^{19} +(4.35714 + 4.35714i) q^{21} +(-5.87429 - 5.87429i) q^{23} +3.60728i q^{27} +(1.48745 + 1.48745i) q^{29} +6.78608i q^{31} +(-10.5756 - 10.5756i) q^{33} -6.13260 q^{37} -4.02305i q^{39} -2.75628i q^{41} +3.39392 q^{43} +(-9.44332 - 9.44332i) q^{47} +1.82155i q^{49} +(-11.5449 - 11.5449i) q^{51} -1.54070i q^{53} +(-0.440383 - 0.440383i) q^{57} +(-2.53072 - 2.53072i) q^{59} +(0.600858 - 0.600858i) q^{61} +(6.97095 - 6.97095i) q^{63} -8.14130 q^{67} +(-15.9064 + 15.9064i) q^{69} +4.55096 q^{71} +(4.84446 - 4.84446i) q^{73} +12.5690i q^{77} +0.455580 q^{79} -3.22874 q^{81} +2.84390i q^{83} +(4.02772 - 4.02772i) q^{87} -1.91001 q^{89} +(-2.39069 + 2.39069i) q^{91} +18.3754 q^{93} +(-1.73249 + 1.73249i) q^{97} +(-16.9197 + 16.9197i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 40 q^{9}+O(q^{10}) \) 24 * q - 40 * q^9	\( 24 q - 40 q^{9} + 20 q^{11} + 12 q^{19} + 8 q^{29} - 20 q^{51} - 8 q^{59} - 48 q^{61} - 64 q^{69} + 16 q^{71} + 104 q^{79} + 48 q^{81} + 96 q^{89} - 64 q^{91} - 128 q^{99}+O(q^{100}) \) 24 * q - 40 * q^9 + 20 * q^11 + 12 * q^19 + 8 * q^29 - 20 * q^51 - 8 * q^59 - 48 * q^61 - 64 * q^69 + 16 * q^71 + 104 * q^79 + 48 * q^81 + 96 * q^89 - 64 * q^91 - 128 * q^99

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