LMFDB - Embedded newform 1600.2.s.e.943.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(207,1600)]

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, 1]))

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

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

chi := DirichletCharacter("1600.207");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.s (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			943.3
Character	$\chi$	$=$	1600.943
Dual form			1600.2.s.e.207.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35800 q^{3} +(-2.66357 + 2.66357i) q^{7} +2.56018 q^{9} +O(q^{10})$	$q-2.35800 q^{3} +(-2.66357 + 2.66357i) q^{7} +2.56018 q^{9} +(2.20666 + 2.20666i) q^{11} +4.16154i q^{13} +(-1.69084 + 1.69084i) q^{17} +(-4.74110 - 4.74110i) q^{19} +(6.28071 - 6.28071i) q^{21} +(3.70658 + 3.70658i) q^{23} +1.03710 q^{27} +(3.65701 - 3.65701i) q^{29} +6.90069i q^{31} +(-5.20331 - 5.20331i) q^{33} +1.10092i q^{37} -9.81293i q^{39} +9.85512i q^{41} -10.0944i q^{43} +(-3.90722 - 3.90722i) q^{47} -7.18923i q^{49} +(3.98700 - 3.98700i) q^{51} -6.19464 q^{53} +(11.1795 + 11.1795i) q^{57} +(-3.42978 + 3.42978i) q^{59} +(-4.57442 - 4.57442i) q^{61} +(-6.81921 + 6.81921i) q^{63} -6.37605i q^{67} +(-8.74012 - 8.74012i) q^{69} +1.03776 q^{71} +(-4.70822 + 4.70822i) q^{73} -11.7552 q^{77} -2.54448 q^{79} -10.1260 q^{81} -7.65615 q^{83} +(-8.62323 + 8.62323i) q^{87} -1.77392 q^{89} +(-11.0846 - 11.0846i) q^{91} -16.2718i q^{93} +(-1.16560 + 1.16560i) q^{97} +(5.64944 + 5.64944i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

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

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
$3$	−2.35800			−1.36139			−0.680697	−	0.732565i	$-0.738323\pi$
$3$	−2.35800			−1.36139			−0.680697	+	0.732565i	$0.738323\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.66357	+	2.66357i	−1.00674	+	1.00674i	−0.00675825	+	0.999977i	$0.502151\pi$
$7$	−2.66357	+	2.66357i	−1.00674	+	1.00674i	−0.999977	+	0.00675825i	$0.997849\pi$
$8$		0			0
$9$	2.56018			0.853392
$10$		0			0
$11$	2.20666	+	2.20666i	0.665333	+	0.665333i	0.956632	−	0.291299i	$-0.0940873\pi$
$11$	2.20666	+	2.20666i	0.665333	+	0.665333i	−0.291299	+	0.956632i	$0.594087\pi$
$12$		0			0
$13$			4.16154i			1.15420i	0.816672	+	0.577102i	$0.195817\pi$
$13$			4.16154i			1.15420i	−0.816672	+	0.577102i	$0.804183\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.69084	+	1.69084i	−0.410088	+	0.410088i	−0.881769	−	0.471681i	$-0.843647\pi$
$17$	−1.69084	+	1.69084i	−0.410088	+	0.410088i	0.471681	+	0.881769i	$0.343647\pi$
$18$		0			0
$19$	−4.74110	−	4.74110i	−1.08768	−	1.08768i	−0.995767	−	0.0919157i	$-0.970701\pi$
$19$	−4.74110	−	4.74110i	−1.08768	−	1.08768i	−0.0919157	−	0.995767i	$-0.529299\pi$
$20$		0			0
$21$	6.28071	−	6.28071i	1.37056	−	1.37056i
$22$		0			0
$23$	3.70658	+	3.70658i	0.772875	+	0.772875i	0.978608	−	0.205733i	$-0.0659580\pi$
$23$	3.70658	+	3.70658i	0.772875	+	0.772875i	−0.205733	+	0.978608i	$0.565958\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	1.03710			0.199591
$28$		0			0
$29$	3.65701	−	3.65701i	0.679089	−	0.679089i	−0.280705	−	0.959794i	$-0.590568\pi$
$29$	3.65701	−	3.65701i	0.679089	−	0.679089i	0.959794	+	0.280705i	$0.0905683\pi$
$30$		0			0
$31$			6.90069i			1.23940i	0.784839	+	0.619700i	$0.212746\pi$
$31$			6.90069i			1.23940i	−0.784839	+	0.619700i	$0.787254\pi$
$32$		0			0
$33$	−5.20331	−	5.20331i	−0.905781	−	0.905781i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			1.10092i			0.180989i	0.995897	+	0.0904947i	$0.0288448\pi$
$37$			1.10092i			0.180989i	−0.995897	+	0.0904947i	$0.971155\pi$
$38$		0			0
$39$		−	9.81293i		−	1.57133i
$40$		0			0
$41$			9.85512i			1.53911i	0.638579	+	0.769556i	$0.279522\pi$
$41$			9.85512i			1.53911i	−0.638579	+	0.769556i	$0.720478\pi$
$42$		0			0
$43$		−	10.0944i		−	1.53938i	−0.638417	−	0.769691i	$-0.720410\pi$
$43$		−	10.0944i		−	1.53938i	0.638417	−	0.769691i	$-0.279590\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.90722	−	3.90722i	−0.569927	−	0.569927i	0.362181	−	0.932108i	$-0.382032\pi$
$47$	−3.90722	−	3.90722i	−0.569927	−	0.569927i	−0.932108	+	0.362181i	$0.882032\pi$
$48$		0			0
$49$		−	7.18923i		−	1.02703i
$50$		0			0
$51$	3.98700	−	3.98700i	0.558291	−	0.558291i
$52$		0			0
$53$	−6.19464			−0.850899			−0.425450	−	0.904982i	$-0.639884\pi$
$53$	−6.19464			−0.850899			−0.425450	+	0.904982i	$0.639884\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	11.1795	+	11.1795i	1.48076	+	1.48076i
$58$		0			0
$59$	−3.42978	+	3.42978i	−0.446519	+	0.446519i	−0.894196	−	0.447677i	$-0.852252\pi$
$59$	−3.42978	+	3.42978i	−0.446519	+	0.446519i	0.447677	+	0.894196i	$0.352252\pi$
$60$		0			0
$61$	−4.57442	−	4.57442i	−0.585694	−	0.585694i	0.350768	−	0.936462i	$-0.385920\pi$
$61$	−4.57442	−	4.57442i	−0.585694	−	0.585694i	−0.936462	+	0.350768i	$0.885920\pi$
$62$		0			0
$63$	−6.81921	+	6.81921i	−0.859140	+	0.859140i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	6.37605i		−	0.778958i	−0.921035	−	0.389479i	$-0.872655\pi$
$67$		−	6.37605i		−	0.778958i	0.921035	−	0.389479i	$-0.127345\pi$
$68$		0			0
$69$	−8.74012	−	8.74012i	−1.05219	−	1.05219i
$70$		0			0
$71$	1.03776			0.123160			0.0615800	−	0.998102i	$-0.480386\pi$
$71$	1.03776			0.123160			0.0615800	+	0.998102i	$0.480386\pi$
$72$		0			0
$73$	−4.70822	+	4.70822i	−0.551056	+	0.551056i	−0.926746	−	0.375690i	$-0.877406\pi$
$73$	−4.70822	+	4.70822i	−0.551056	+	0.551056i	0.375690	+	0.926746i	$0.377406\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−11.7552			−1.33963
$78$		0			0
$79$	−2.54448			−0.286277			−0.143138	−	0.989703i	$-0.545719\pi$
$79$	−2.54448			−0.286277			−0.143138	+	0.989703i	$0.545719\pi$
$80$		0			0
$81$	−10.1260			−1.12511
$82$		0			0
$83$	−7.65615			−0.840371			−0.420186	−	0.907438i	$-0.638035\pi$
$83$	−7.65615			−0.840371			−0.420186	+	0.907438i	$0.638035\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−8.62323	+	8.62323i	−0.924507	+	0.924507i
$88$		0			0
$89$	−1.77392			−0.188036			−0.0940178	−	0.995571i	$-0.529971\pi$
$89$	−1.77392			−0.188036			−0.0940178	+	0.995571i	$0.529971\pi$
$90$		0			0
$91$	−11.0846	−	11.0846i	−1.16198	−	1.16198i
$92$		0			0
$93$		−	16.2718i		−	1.68731i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.16560	+	1.16560i	−0.118349	+	0.118349i	−0.763801	−	0.645452i	$-0.776669\pi$
$97$	−1.16560	+	1.16560i	−0.118349	+	0.118349i	0.645452	+	0.763801i	$0.276669\pi$
$98$		0			0
$99$	5.64944	+	5.64944i	0.567790	+	0.567790i
$100$		0			0
$101$	8.20347	−	8.20347i	0.816275	−	0.816275i	−0.169291	−	0.985566i	$-0.554148\pi$
$101$	8.20347	−	8.20347i	0.816275	−	0.816275i	0.985566	+	0.169291i	$0.0541477\pi$
$102$		0			0
$103$	−9.71614	−	9.71614i	−0.957359	−	0.957359i	0.0417680	−	0.999127i	$-0.486701\pi$
$103$	−9.71614	−	9.71614i	−0.957359	−	0.957359i	−0.999127	+	0.0417680i	$0.986701\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.37605			0.616396			0.308198	−	0.951322i	$-0.400274\pi$
$107$	6.37605			0.616396			0.308198	+	0.951322i	$0.400274\pi$
$108$		0			0
$109$	−0.651659	+	0.651659i	−0.0624176	+	0.0624176i	−0.737627	−	0.675209i	$-0.764053\pi$
$109$	−0.651659	+	0.651659i	−0.0624176	+	0.0624176i	0.675209	+	0.737627i	$0.264053\pi$
$110$		0			0
$111$		−	2.59596i		−	0.246398i
$112$		0			0
$113$	−8.86164	−	8.86164i	−0.833633	−	0.833633i	0.154378	−	0.988012i	$-0.450662\pi$
$113$	−8.86164	−	8.86164i	−0.833633	−	0.833633i	−0.988012	+	0.154378i	$0.950662\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			10.6543i			0.984989i
$118$		0			0
$119$		−	9.00732i		−	0.825700i
$120$		0			0
$121$		−	1.26129i		−	0.114663i
$122$		0			0
$123$		−	23.2384i		−	2.09534i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	5.95966	+	5.95966i	0.528834	+	0.528834i	0.920225	−	0.391391i	$-0.128006\pi$
$127$	5.95966	+	5.95966i	0.528834	+	0.528834i	−0.391391	+	0.920225i	$0.628006\pi$
$128$		0			0
$129$			23.8026i			2.09570i
$130$		0			0
$131$	1.32406	−	1.32406i	0.115683	−	0.115683i	−0.646895	−	0.762579i	$-0.723933\pi$
$131$	1.32406	−	1.32406i	0.115683	−	0.115683i	0.762579	+	0.646895i	$0.223933\pi$
$132$		0			0
$133$	25.2565			2.19002
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−15.4807	−	15.4807i	−1.32260	−	1.32260i	−0.911661	−	0.410943i	$-0.865199\pi$
$137$	−15.4807	−	15.4807i	−1.32260	−	1.32260i	−0.410943	−	0.911661i	$-0.634801\pi$
$138$		0			0
$139$	−5.97722	+	5.97722i	−0.506981	+	0.506981i	−0.913599	−	0.406617i	$-0.866708\pi$
$139$	−5.97722	+	5.97722i	−0.506981	+	0.506981i	0.406617	+	0.913599i	$0.366708\pi$
$140$		0			0
$141$	9.21324	+	9.21324i	0.775895	+	0.775895i
$142$		0			0
$143$	−9.18312	+	9.18312i	−0.767931	+	0.767931i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			16.9522i			1.39820i
$148$		0			0
$149$	−3.16964	−	3.16964i	−0.259667	−	0.259667i	0.565252	−	0.824919i	$-0.308779\pi$
$149$	−3.16964	−	3.16964i	−0.259667	−	0.259667i	−0.824919	+	0.565252i	$0.808779\pi$
$150$		0			0
$151$	11.1344			0.906103			0.453051	−	0.891484i	$-0.350335\pi$
$151$	11.1344			0.906103			0.453051	+	0.891484i	$0.350335\pi$
$152$		0			0
$153$	−4.32884	+	4.32884i	−0.349966	+	0.349966i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	13.4517			1.07356			0.536781	−	0.843722i	$-0.319640\pi$
$157$	13.4517			1.07356			0.536781	+	0.843722i	$0.319640\pi$
$158$		0			0
$159$	14.6070			1.15841
$160$		0			0
$161$	−19.7455			−1.55616
$162$		0			0
$163$	1.01486			0.0794896			0.0397448	−	0.999210i	$-0.487346\pi$
$163$	1.01486			0.0794896			0.0397448	+	0.999210i	$0.487346\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.83926	−	5.83926i	0.451855	−	0.451855i	−0.444115	−	0.895970i	$-0.646482\pi$
$167$	5.83926	−	5.83926i	0.451855	−	0.451855i	0.895970	+	0.444115i	$0.146482\pi$
$168$		0			0
$169$	−4.31844			−0.332188
$170$		0			0
$171$	−12.1380	−	12.1380i	−0.928220	−	0.928220i
$172$		0			0
$173$			6.55812i			0.498605i	0.968426	+	0.249302i	$0.0802013\pi$
$173$			6.55812i			0.498605i	−0.968426	+	0.249302i	$0.919799\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	8.08743	−	8.08743i	0.607888	−	0.607888i
$178$		0			0
$179$	−5.45256	−	5.45256i	−0.407543	−	0.407543i	0.473338	−	0.880881i	$-0.343049\pi$
$179$	−5.45256	−	5.45256i	−0.407543	−	0.407543i	−0.880881	+	0.473338i	$0.843049\pi$
$180$		0			0
$181$	−5.39320	+	5.39320i	−0.400873	+	0.400873i	−0.878541	−	0.477667i	$-0.841482\pi$
$181$	−5.39320	+	5.39320i	−0.400873	+	0.400873i	0.477667	+	0.878541i	$0.341482\pi$
$182$		0			0
$183$	10.7865	+	10.7865i	0.797360	+	0.797360i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−7.46221			−0.545691
$188$		0			0
$189$	−2.76240	+	2.76240i	−0.200935	+	0.200935i
$190$		0			0
$191$			8.78551i			0.635698i	0.948141	+	0.317849i	$0.102960\pi$
$191$			8.78551i			0.635698i	−0.948141	+	0.317849i	$0.897040\pi$
$192$		0			0
$193$	−11.7474	−	11.7474i	−0.845598	−	0.845598i	0.143982	−	0.989580i	$-0.454009\pi$
$193$	−11.7474	−	11.7474i	−0.845598	−	0.845598i	−0.989580	+	0.143982i	$0.954009\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.9577i		−	1.06569i	−0.846212	−	0.532846i	$-0.821123\pi$
$197$		−	14.9577i		−	1.06569i	0.846212	−	0.532846i	$-0.178877\pi$
$198$		0			0
$199$			10.9644i			0.777245i	0.921397	+	0.388623i	$0.127049\pi$
$199$			10.9644i			0.777245i	−0.921397	+	0.388623i	$0.872951\pi$
$200$		0			0
$201$			15.0347i			1.06047i
$202$		0			0
$203$			19.4814i			1.36733i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	9.48949	+	9.48949i	0.659565	+	0.659565i
$208$		0			0
$209$		−	20.9240i		−	1.44734i
$210$		0			0
$211$	−14.1093	+	14.1093i	−0.971326	+	0.971326i	−0.999600	−	0.0282740i	$-0.990999\pi$
$211$	−14.1093	+	14.1093i	−0.971326	+	0.971326i	0.0282740	+	0.999600i	$0.490999\pi$
$212$		0			0
$213$	−2.44705			−0.167669
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−18.3805	−	18.3805i	−1.24775	−	1.24775i
$218$		0			0
$219$	11.1020	−	11.1020i	0.750204	−	0.750204i
$220$		0			0
$221$	−7.03649	−	7.03649i	−0.473325	−	0.473325i
$222$		0			0
$223$	14.1252	−	14.1252i	0.945897	−	0.945897i	−0.0527131	−	0.998610i	$-0.516787\pi$
$223$	14.1252	−	14.1252i	0.945897	−	0.945897i	0.998610	+	0.0527131i	$0.0167869\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			8.40121i			0.557608i	0.960348	+	0.278804i	$0.0899379\pi$
$227$			8.40121i			0.557608i	−0.960348	+	0.278804i	$0.910062\pi$
$228$		0			0
$229$	−15.9755	−	15.9755i	−1.05569	−	1.05569i	−0.998355	−	0.0573343i	$-0.981740\pi$
$229$	−15.9755	−	15.9755i	−1.05569	−	1.05569i	−0.0573343	−	0.998355i	$-0.518260\pi$
$230$		0			0
$231$	27.7188			1.82376
$232$		0			0
$233$	−10.8176	+	10.8176i	−0.708684	+	0.708684i	−0.966258	−	0.257574i	$-0.917077\pi$
$233$	−10.8176	+	10.8176i	−0.708684	+	0.708684i	0.257574	+	0.966258i	$0.417077\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	5.99990			0.389735
$238$		0			0
$239$	12.7033			0.821706			0.410853	−	0.911702i	$-0.365231\pi$
$239$	12.7033			0.821706			0.410853	+	0.911702i	$0.365231\pi$
$240$		0			0
$241$	29.1860			1.88004			0.940019	−	0.341122i	$-0.110807\pi$
$241$	29.1860			1.88004			0.940019	+	0.341122i	$0.110807\pi$
$242$		0			0
$243$	20.7659			1.33213
$244$		0			0
$245$		0			0
$246$		0			0
$247$	19.7303	−	19.7303i	1.25541	−	1.25541i
$248$		0			0
$249$	18.0532			1.14408
$250$		0			0
$251$	−7.59622	−	7.59622i	−0.479469	−	0.479469i	0.425493	−	0.904962i	$-0.360101\pi$
$251$	−7.59622	−	7.59622i	−0.479469	−	0.479469i	−0.904962	+	0.425493i	$0.860101\pi$
$252$		0			0
$253$			16.3583i			1.02844i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.72143	+	4.72143i	−0.294515	+	0.294515i	−0.838861	−	0.544346i	$-0.816778\pi$
$257$	−4.72143	+	4.72143i	−0.294515	+	0.294515i	0.544346	+	0.838861i	$0.316778\pi$
$258$		0			0
$259$	−2.93237	−	2.93237i	−0.182208	−	0.182208i
$260$		0			0
$261$	9.36258	−	9.36258i	0.579529	−	0.579529i
$262$		0			0
$263$	15.6431	+	15.6431i	0.964594	+	0.964594i	0.999394	−	0.0348001i	$-0.0110795\pi$
$263$	15.6431	+	15.6431i	0.964594	+	0.964594i	−0.0348001	+	0.999394i	$0.511079\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	4.18292			0.255991
$268$		0			0
$269$	−11.1216	+	11.1216i	−0.678096	+	0.678096i	−0.959569	−	0.281473i	$-0.909177\pi$
$269$	−11.1216	+	11.1216i	−0.678096	+	0.678096i	0.281473	+	0.959569i	$0.409177\pi$
$270$		0			0
$271$			16.7962i			1.02030i	0.860086	+	0.510148i	$0.170410\pi$
$271$			16.7962i			1.02030i	−0.860086	+	0.510148i	$0.829590\pi$
$272$		0			0
$273$	26.1374	+	26.1374i	1.58191	+	1.58191i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			12.5871i			0.756287i	0.925747	+	0.378144i	$0.123437\pi$
$277$			12.5871i			0.756287i	−0.925747	+	0.378144i	$0.876563\pi$
$278$		0			0
$279$			17.6670i			1.05769i
$280$		0			0
$281$		−	31.3713i		−	1.87146i	−0.352722	−	0.935728i	$-0.614744\pi$
$281$		−	31.3713i		−	1.87146i	0.352722	−	0.935728i	$-0.385256\pi$
$282$		0			0
$283$			7.58242i			0.450728i	0.974275	+	0.225364i	$0.0723571\pi$
$283$			7.58242i			0.450728i	−0.974275	+	0.225364i	$0.927643\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−26.2498	−	26.2498i	−1.54948	−	1.54948i
$288$		0			0
$289$			11.2821i			0.663656i
$290$		0			0
$291$	2.74849	−	2.74849i	0.161119	−	0.161119i
$292$		0			0
$293$	26.6053			1.55430			0.777149	−	0.629317i	$-0.216665\pi$
$293$	26.6053			1.55430			0.777149	+	0.629317i	$0.216665\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.28854	+	2.28854i	0.132794	+	0.132794i
$298$		0			0
$299$	−15.4251	+	15.4251i	−0.892055	+	0.892055i
$300$		0			0
$301$	26.8871	+	26.8871i	1.54975	+	1.54975i
$302$		0			0
$303$	−19.3438	+	19.3438i	−1.11127	+	1.11127i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	2.78617i		−	0.159015i	−0.996834	−	0.0795075i	$-0.974665\pi$
$307$		−	2.78617i		−	0.159015i	0.996834	−	0.0795075i	$-0.0253347\pi$
$308$		0			0
$309$	22.9107	+	22.9107i	1.30334	+	1.30334i
$310$		0			0
$311$	−4.97594			−0.282159			−0.141080	−	0.989998i	$-0.545057\pi$
$311$	−4.97594			−0.282159			−0.141080	+	0.989998i	$0.545057\pi$
$312$		0			0
$313$	6.76254	−	6.76254i	0.382241	−	0.382241i	−0.489668	−	0.871909i	$-0.662882\pi$
$313$	6.76254	−	6.76254i	0.382241	−	0.382241i	0.871909	+	0.489668i	$0.162882\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.81211			0.101778			0.0508892	−	0.998704i	$-0.483794\pi$
$317$	1.81211			0.101778			0.0508892	+	0.998704i	$0.483794\pi$
$318$		0			0
$319$	16.1395			0.903641
$320$		0			0
$321$	−15.0347			−0.839157
$322$		0			0
$323$	16.0328			0.892091
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.53661	−	1.53661i	0.0849749	−	0.0849749i
$328$		0			0
$329$	20.8143			1.14753
$330$		0			0
$331$	1.35992	+	1.35992i	0.0747482	+	0.0747482i	0.743492	−	0.668744i	$-0.233168\pi$
$331$	1.35992	+	1.35992i	0.0747482	+	0.0747482i	−0.668744	+	0.743492i	$0.733168\pi$
$332$		0			0
$333$			2.81854i			0.154455i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.83330	−	3.83330i	0.208813	−	0.208813i	−0.594950	−	0.803763i	$-0.702828\pi$
$337$	3.83330	−	3.83330i	0.208813	−	0.208813i	0.803763	+	0.594950i	$0.202828\pi$
$338$		0			0
$339$	20.8958	+	20.8958i	1.13490	+	1.13490i
$340$		0			0
$341$	−15.2275	+	15.2275i	−0.824614	+	0.824614i
$342$		0			0
$343$	0.504019	+	0.504019i	0.0272144	+	0.0272144i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.4284			−1.15034			−0.575169	−	0.818035i	$-0.695064\pi$
$347$	−21.4284			−1.15034			−0.575169	+	0.818035i	$0.695064\pi$
$348$		0			0
$349$	2.63033	−	2.63033i	0.140798	−	0.140798i	−0.633195	−	0.773993i	$-0.718257\pi$
$349$	2.63033	−	2.63033i	0.140798	−	0.140798i	0.773993	+	0.633195i	$0.218257\pi$
$350$		0			0
$351$			4.31595i			0.230369i
$352$		0			0
$353$	0.754635	+	0.754635i	0.0401651	+	0.0401651i	0.726904	−	0.686739i	$-0.240958\pi$
$353$	0.754635	+	0.754635i	0.0401651	+	0.0401651i	−0.686739	+	0.726904i	$0.740958\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			21.2393i			1.12410i
$358$		0			0
$359$		−	17.0363i		−	0.899140i	−0.893245	−	0.449570i	$-0.851577\pi$
$359$		−	17.0363i		−	0.899140i	0.893245	−	0.449570i	$-0.148423\pi$
$360$		0			0
$361$			25.9560i			1.36611i
$362$		0			0
$363$			2.97412i			0.156101i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.86375	+	4.86375i	0.253886	+	0.253886i	0.822562	−	0.568676i	$-0.192544\pi$
$367$	4.86375	+	4.86375i	0.253886	+	0.253886i	−0.568676	+	0.822562i	$0.692544\pi$
$368$		0			0
$369$			25.2309i			1.31347i
$370$		0			0
$371$	16.4999	−	16.4999i	0.856631	−	0.856631i
$372$		0			0
$373$	15.2209			0.788109			0.394055	−	0.919087i	$-0.371072\pi$
$373$	15.2209			0.788109			0.394055	+	0.919087i	$0.371072\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.2188	+	15.2188i	0.783807	+	0.783807i
$378$		0			0
$379$	−2.75236	+	2.75236i	−0.141379	+	0.141379i	−0.774254	−	0.632875i	$-0.781875\pi$
$379$	−2.75236	+	2.75236i	−0.141379	+	0.141379i	0.632875	+	0.774254i	$0.281875\pi$
$380$		0			0
$381$	−14.0529	−	14.0529i	−0.719951	−	0.719951i
$382$		0			0
$383$	13.6326	−	13.6326i	0.696591	−	0.696591i	−0.267083	−	0.963674i	$-0.586060\pi$
$383$	13.6326	−	13.6326i	0.696591	−	0.696591i	0.963674	+	0.267083i	$0.0860596\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	25.8434i		−	1.31370i
$388$		0			0
$389$	15.9333	+	15.9333i	0.807850	+	0.807850i	0.984308	−	0.176459i	$-0.0564641\pi$
$389$	15.9333	+	15.9333i	0.807850	+	0.807850i	−0.176459	+	0.984308i	$0.556464\pi$
$390$		0			0
$391$	−12.5344			−0.633893
$392$		0			0
$393$	−3.12213	+	3.12213i	−0.157491	+	0.157491i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.4837			0.576353			0.288176	−	0.957577i	$-0.406951\pi$
$397$	11.4837			0.576353			0.288176	+	0.957577i	$0.406951\pi$
$398$		0			0
$399$	−59.5549			−2.98147
$400$		0			0
$401$	−6.56979			−0.328080			−0.164040	−	0.986454i	$-0.552453\pi$
$401$	−6.56979			−0.328080			−0.164040	+	0.986454i	$0.552453\pi$
$402$		0			0
$403$	−28.7175			−1.43052
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.42935	+	2.42935i	−0.120418	+	0.120418i
$408$		0			0
$409$	−21.3172			−1.05407			−0.527034	−	0.849844i	$-0.676696\pi$
$409$	−21.3172			−1.05407			−0.527034	+	0.849844i	$0.676696\pi$
$410$		0			0
$411$	36.5035	+	36.5035i	1.80058	+	1.80058i
$412$		0			0
$413$		−	18.2709i		−	0.899053i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	14.0943	−	14.0943i	0.690201	−	0.690201i
$418$		0			0
$419$	8.25496	+	8.25496i	0.403281	+	0.403281i	0.879388	−	0.476106i	$-0.157952\pi$
$419$	8.25496	+	8.25496i	0.403281	+	0.403281i	−0.476106	+	0.879388i	$0.657952\pi$
$420$		0			0
$421$	3.52333	−	3.52333i	0.171717	−	0.171717i	−0.616017	−	0.787733i	$-0.711255\pi$
$421$	3.52333	−	3.52333i	0.171717	−	0.171717i	0.787733	+	0.616017i	$0.211255\pi$
$422$		0			0
$423$	−10.0032	−	10.0032i	−0.486371	−	0.486371i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	24.3686			1.17928
$428$		0			0
$429$	21.6538	−	21.6538i	1.04546	−	1.04546i
$430$		0			0
$431$		−	33.8672i		−	1.63133i	−0.578526	−	0.815664i	$-0.696372\pi$
$431$		−	33.8672i		−	1.63133i	0.578526	−	0.815664i	$-0.303628\pi$
$432$		0			0
$433$	18.7321	+	18.7321i	0.900206	+	0.900206i	0.995454	−	0.0952478i	$-0.0303643\pi$
$433$	18.7321	+	18.7321i	0.900206	+	0.900206i	−0.0952478	+	0.995454i	$0.530364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	35.1465i		−	1.68128i
$438$		0			0
$439$			21.5595i			1.02898i	0.857497	+	0.514489i	$0.172018\pi$
$439$			21.5595i			1.02898i	−0.857497	+	0.514489i	$0.827982\pi$
$440$		0			0
$441$		−	18.4057i		−	0.876461i
$442$		0			0
$443$			0.517182i			0.0245721i	0.999925	+	0.0122860i	$0.00391086\pi$
$443$			0.517182i			0.0245721i	−0.999925	+	0.0122860i	$0.996089\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	7.47402	+	7.47402i	0.353509	+	0.353509i
$448$		0			0
$449$			5.34582i			0.252285i	0.992012	+	0.126142i	$0.0402597\pi$
$449$			5.34582i			0.252285i	−0.992012	+	0.126142i	$0.959740\pi$
$450$		0			0
$451$	−21.7469	+	21.7469i	−1.02402	+	1.02402i
$452$		0			0
$453$	−26.2549			−1.23356
$454$		0			0
$455$		0			0
$456$		0			0
$457$	20.5862	+	20.5862i	0.962982	+	0.962982i	0.999339	−	0.0363565i	$-0.0115752\pi$
$457$	20.5862	+	20.5862i	0.962982	+	0.962982i	−0.0363565	+	0.999339i	$0.511575\pi$
$458$		0			0
$459$	−1.75357	+	1.75357i	−0.0818498	+	0.0818498i
$460$		0			0
$461$	−20.6538	−	20.6538i	−0.961943	−	0.961943i	0.0373593	−	0.999302i	$-0.488105\pi$
$461$	−20.6538	−	20.6538i	−0.961943	−	0.961943i	−0.999302	+	0.0373593i	$0.988105\pi$
$462$		0			0
$463$	9.94042	−	9.94042i	0.461971	−	0.461971i	−0.437330	−	0.899301i	$-0.644076\pi$
$463$	9.94042	−	9.94042i	0.461971	−	0.461971i	0.899301	+	0.437330i	$0.144076\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	35.1108i		−	1.62473i	−0.583146	−	0.812367i	$-0.698179\pi$
$467$		−	35.1108i		−	1.62473i	0.583146	−	0.812367i	$-0.301821\pi$
$468$		0			0
$469$	16.9831	+	16.9831i	0.784205	+	0.784205i
$470$		0			0
$471$	−31.7191			−1.46154
$472$		0			0
$473$	22.2749	−	22.2749i	1.02420	−	1.02420i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−15.8594			−0.726151
$478$		0			0
$479$	−18.4324			−0.842198			−0.421099	−	0.907015i	$-0.638356\pi$
$479$	−18.4324			−0.842198			−0.421099	+	0.907015i	$0.638356\pi$
$480$		0			0
$481$	−4.58151			−0.208899
$482$		0			0
$483$	46.5599			2.11855
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−24.6160	+	24.6160i	−1.11546	+	1.11546i	−0.123057	+	0.992400i	$0.539270\pi$
$487$	−24.6160	+	24.6160i	−1.11546	+	1.11546i	−0.992400	+	0.123057i	$0.960730\pi$
$488$		0			0
$489$	−2.39303			−0.108217
$490$		0			0
$491$	−19.0271	−	19.0271i	−0.858683	−	0.858683i	0.132500	−	0.991183i	$-0.457700\pi$
$491$	−19.0271	−	19.0271i	−0.858683	−	0.858683i	−0.991183	+	0.132500i	$0.957700\pi$
$492$		0			0
$493$			12.3668i			0.556972i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.76416	+	2.76416i	−0.123989	+	0.123989i
$498$		0			0
$499$	−30.2817	−	30.2817i	−1.35560	−	1.35560i	−0.879268	−	0.476328i	$-0.841968\pi$
$499$	−30.2817	−	30.2817i	−1.35560	−	1.35560i	−0.476328	−	0.879268i	$-0.658032\pi$
$500$		0			0
$501$	−13.7690	+	13.7690i	−0.615153	+	0.615153i
$502$		0			0
$503$	−0.491214	−	0.491214i	−0.0219021	−	0.0219021i	0.696071	−	0.717973i	$-0.254930\pi$
$503$	−0.491214	−	0.491214i	−0.0219021	−	0.0219021i	−0.717973	+	0.696071i	$0.754930\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	10.1829			0.452238
$508$		0			0
$509$	−24.5506	+	24.5506i	−1.08819	+	1.08819i	−0.0924714	+	0.995715i	$0.529477\pi$
$509$	−24.5506	+	24.5506i	−1.08819	+	1.08819i	−0.995715	+	0.0924714i	$0.970523\pi$
$510$		0			0
$511$		−	25.0814i		−	1.10953i
$512$		0			0
$513$	−4.91701	−	4.91701i	−0.217091	−	0.217091i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	17.2438i		−	0.758383i
$518$		0			0
$519$		−	15.4641i		−	0.678797i
$520$		0			0
$521$		−	16.4988i		−	0.722826i	−0.932406	−	0.361413i	$-0.882294\pi$
$521$		−	16.4988i		−	0.722826i	0.932406	−	0.361413i	$-0.117706\pi$
$522$		0			0
$523$			10.9746i			0.479887i	0.970787	+	0.239943i	$0.0771289\pi$
$523$			10.9746i			0.479887i	−0.970787	+	0.239943i	$0.922871\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.6679	−	11.6679i	−0.508263	−	0.508263i
$528$		0			0
$529$			4.47742i			0.194670i
$530$		0			0
$531$	−8.78084	+	8.78084i	−0.381056	+	0.381056i
$532$		0			0
$533$	−41.0125			−1.77645
$534$		0			0
$535$		0			0
$536$		0			0
$537$	12.8571	+	12.8571i	0.554827	+	0.554827i
$538$		0			0
$539$	15.8642	−	15.8642i	0.683319	−	0.683319i
$540$		0			0
$541$	−21.8026	−	21.8026i	−0.937368	−	0.937368i	0.0607833	−	0.998151i	$-0.480640\pi$
$541$	−21.8026	−	21.8026i	−0.937368	−	0.937368i	−0.998151	+	0.0607833i	$0.980640\pi$
$542$		0			0
$543$	12.7172	−	12.7172i	0.545746	−	0.545746i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	19.5619i		−	0.836408i	−0.908353	−	0.418204i	$-0.862660\pi$
$547$		−	19.5619i		−	0.836408i	0.908353	−	0.418204i	$-0.137340\pi$
$548$		0			0
$549$	−11.7113	−	11.7113i	−0.499827	−	0.499827i
$550$		0			0
$551$	−34.6764			−1.47727
$552$		0			0
$553$	6.77741	−	6.77741i	0.288205	−	0.288205i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−23.8550			−1.01077			−0.505385	−	0.862894i	$-0.668649\pi$
$557$	−23.8550			−1.01077			−0.505385	+	0.862894i	$0.668649\pi$
$558$		0			0
$559$	42.0083			1.77676
$560$		0			0
$561$	17.5959			0.742900
$562$		0			0
$563$	−35.6435			−1.50220			−0.751098	−	0.660191i	$-0.770475\pi$
$563$	−35.6435			−1.50220			−0.751098	+	0.660191i	$0.770475\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	26.9714	−	26.9714i	1.13269	−	1.13269i
$568$		0			0
$569$	−21.4354			−0.898619			−0.449310	−	0.893376i	$-0.648330\pi$
$569$	−21.4354			−0.898619			−0.449310	+	0.893376i	$0.648330\pi$
$570$		0			0
$571$	7.26872	+	7.26872i	0.304186	+	0.304186i	0.842649	−	0.538463i	$-0.180995\pi$
$571$	7.26872	+	7.26872i	0.304186	+	0.304186i	−0.538463	+	0.842649i	$0.680995\pi$
$572$		0			0
$573$		−	20.7163i		−	0.865434i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−15.0903	+	15.0903i	−0.628219	+	0.628219i	−0.947620	−	0.319401i	$-0.896518\pi$
$577$	−15.0903	+	15.0903i	−0.628219	+	0.628219i	0.319401	+	0.947620i	$0.396518\pi$
$578$		0			0
$579$	27.7004	+	27.7004i	1.15119	+	1.15119i
$580$		0			0
$581$	20.3927	−	20.3927i	0.846031	−	0.846031i
$582$		0			0
$583$	−13.6695	−	13.6695i	−0.566132	−	0.566132i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0083			−0.495637			−0.247818	−	0.968806i	$-0.579714\pi$
$587$	−12.0083			−0.495637			−0.247818	+	0.968806i	$0.579714\pi$
$588$		0			0
$589$	32.7168	−	32.7168i	1.34807	−	1.34807i
$590$		0			0
$591$			35.2703i			1.45083i
$592$		0			0
$593$	−8.67491	−	8.67491i	−0.356236	−	0.356236i	0.506188	−	0.862423i	$-0.331054\pi$
$593$	−8.67491	−	8.67491i	−0.356236	−	0.356236i	−0.862423	+	0.506188i	$0.831054\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	25.8541i		−	1.05814i
$598$		0			0
$599$			48.1944i			1.96917i	0.174908	+	0.984585i	$0.444037\pi$
$599$			48.1944i			1.96917i	−0.174908	+	0.984585i	$0.555963\pi$
$600$		0			0
$601$			36.1744i			1.47559i	0.675028	+	0.737793i	$0.264132\pi$
$601$			36.1744i			1.47559i	−0.675028	+	0.737793i	$0.735868\pi$
$602$		0			0
$603$		−	16.3238i		−	0.664757i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	4.43441	+	4.43441i	0.179987	+	0.179987i	0.791350	−	0.611363i	$-0.209379\pi$
$607$	4.43441	+	4.43441i	0.179987	+	0.179987i	−0.611363	+	0.791350i	$0.709379\pi$
$608$		0			0
$609$		−	45.9372i		−	1.86147i
$610$		0			0
$611$	16.2601	−	16.2601i	0.657812	−	0.657812i
$612$		0			0
$613$	−36.5740			−1.47721			−0.738605	−	0.674139i	$-0.764515\pi$
$613$	−36.5740			−1.47721			−0.738605	+	0.674139i	$0.764515\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.61726	−	8.61726i	−0.346918	−	0.346918i	0.512042	−	0.858960i	$-0.328889\pi$
$617$	−8.61726	−	8.61726i	−0.346918	−	0.346918i	−0.858960	+	0.512042i	$0.828889\pi$
$618$		0			0
$619$	−16.4732	+	16.4732i	−0.662113	+	0.662113i	−0.955878	−	0.293765i	$-0.905092\pi$
$619$	−16.4732	+	16.4732i	−0.662113	+	0.662113i	0.293765	+	0.955878i	$0.405092\pi$
$620$		0			0
$621$	3.84411	+	3.84411i	0.154259	+	0.154259i
$622$		0			0
$623$	4.72498	−	4.72498i	0.189302	−	0.189302i
$624$		0			0
$625$		0			0
$626$		0			0
$627$			49.3388i			1.97040i
$628$		0			0
$629$	−1.86147	−	1.86147i	−0.0742216	−	0.0742216i
$630$		0			0
$631$	−25.4406			−1.01277			−0.506387	−	0.862306i	$-0.669019\pi$
$631$	−25.4406			−1.01277			−0.506387	+	0.862306i	$0.669019\pi$
$632$		0			0
$633$	33.2698	−	33.2698i	1.32236	−	1.32236i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	29.9183			1.18541
$638$		0			0
$639$	2.65686			0.105104
$640$		0			0
$641$	28.6124			1.13012			0.565060	−	0.825050i	$-0.308853\pi$
$641$	28.6124			1.13012			0.565060	+	0.825050i	$0.308853\pi$
$642$		0			0
$643$	5.55384			0.219022			0.109511	−	0.993986i	$-0.465072\pi$
$643$	5.55384			0.219022			0.109511	+	0.993986i	$0.465072\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−33.4355	+	33.4355i	−1.31448	+	1.31448i	−0.396410	+	0.918073i	$0.629744\pi$
$647$	−33.4355	+	33.4355i	−1.31448	+	1.31448i	−0.918073	+	0.396410i	$0.870256\pi$
$648$		0			0
$649$	−15.1367			−0.594168
$650$		0			0
$651$	43.3412	+	43.3412i	1.69868	+	1.69868i
$652$		0			0
$653$			47.7786i			1.86972i	0.355017	+	0.934860i	$0.384475\pi$
$653$			47.7786i			1.86972i	−0.355017	+	0.934860i	$0.615525\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−12.0539	+	12.0539i	−0.470267	+	0.470267i
$658$		0			0
$659$	−11.3273	−	11.3273i	−0.441249	−	0.441249i	0.451183	−	0.892432i	$-0.351002\pi$
$659$	−11.3273	−	11.3273i	−0.441249	−	0.441249i	−0.892432	+	0.451183i	$0.851002\pi$
$660$		0			0
$661$	−4.24123	+	4.24123i	−0.164965	+	0.164965i	−0.784762	−	0.619797i	$-0.787215\pi$
$661$	−4.24123	+	4.24123i	−0.164965	+	0.164965i	0.619797	+	0.784762i	$0.287215\pi$
$662$		0			0
$663$	16.5921	+	16.5921i	0.644382	+	0.644382i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	27.1099			1.04970
$668$		0			0
$669$	−33.3074	+	33.3074i	−1.28774	+	1.28774i
$670$		0			0
$671$		−	20.1884i		−	0.779364i
$672$		0			0
$673$	−3.48193	−	3.48193i	−0.134218	−	0.134218i	0.636806	−	0.771024i	$-0.280255\pi$
$673$	−3.48193	−	3.48193i	−0.134218	−	0.134218i	−0.771024	+	0.636806i	$0.780255\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			28.3010i			1.08769i	0.839184	+	0.543847i	$0.183033\pi$
$677$			28.3010i			1.08769i	−0.839184	+	0.543847i	$0.816967\pi$
$678$		0			0
$679$		−	6.20932i		−	0.238292i
$680$		0			0
$681$		−	19.8101i		−	0.759123i
$682$		0			0
$683$			26.0075i			0.995151i	0.867421	+	0.497575i	$0.165776\pi$
$683$			26.0075i			0.995151i	−0.867421	+	0.497575i	$0.834224\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	37.6702	+	37.6702i	1.43721	+	1.43721i
$688$		0			0
$689$		−	25.7793i		−	0.982112i
$690$		0			0
$691$	27.2570	−	27.2570i	1.03690	−	1.03690i	0.0376109	−	0.999292i	$-0.488025\pi$
$691$	27.2570	−	27.2570i	1.03690	−	1.03690i	0.999292	−	0.0376109i	$-0.0119747\pi$
$692$		0			0
$693$	−30.0954			−1.14323
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−16.6634	−	16.6634i	−0.631171	−	0.631171i
$698$		0			0
$699$	25.5079	−	25.5079i	0.964798	−	0.964798i
$700$		0			0
$701$	14.0047	+	14.0047i	0.528949	+	0.528949i	0.920259	−	0.391310i	$-0.127978\pi$
$701$	14.0047	+	14.0047i	0.528949	+	0.528949i	−0.391310	+	0.920259i	$0.627978\pi$
$702$		0			0
$703$	5.21955	−	5.21955i	0.196859	−	0.196859i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			43.7010i			1.64355i
$708$		0			0
$709$	24.2686	+	24.2686i	0.911428	+	0.911428i	0.996385	−	0.0849564i	$-0.0270751\pi$
$709$	24.2686	+	24.2686i	0.911428	+	0.911428i	−0.0849564	+	0.996385i	$0.527075\pi$
$710$		0			0
$711$	−6.51432			−0.244306
$712$		0			0
$713$	−25.5779	+	25.5779i	−0.957901	+	0.957901i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−29.9543			−1.11867
$718$		0			0
$719$	−21.2093			−0.790973			−0.395487	−	0.918472i	$-0.629424\pi$
$719$	−21.2093			−0.790973			−0.395487	+	0.918472i	$0.629424\pi$
$720$		0			0
$721$	51.7592			1.92762
$722$		0			0
$723$	−68.8208			−2.55947
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−24.4304	+	24.4304i	−0.906073	+	0.906073i	−0.995953	−	0.0898798i	$-0.971352\pi$
$727$	−24.4304	+	24.4304i	−0.906073	+	0.906073i	0.0898798	+	0.995953i	$0.471352\pi$
$728$		0			0
$729$	−18.5879			−0.688442
$730$		0			0
$731$	17.0680	+	17.0680i	0.631282	+	0.631282i
$732$		0			0
$733$		−	12.5318i		−	0.462872i	−0.972850	−	0.231436i	$-0.925658\pi$
$733$		−	12.5318i		−	0.462872i	0.972850	−	0.231436i	$-0.0743423\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.0698	−	14.0698i	0.518267	−	0.518267i
$738$		0			0
$739$	−8.46558	−	8.46558i	−0.311411	−	0.311411i	0.534045	−	0.845456i	$-0.320671\pi$
$739$	−8.46558	−	8.46558i	−0.311411	−	0.311411i	−0.845456	+	0.534045i	$0.820671\pi$
$740$		0			0
$741$	−46.5241	+	46.5241i	−1.70910	+	1.70910i
$742$		0			0
$743$	−23.7929	−	23.7929i	−0.872876	−	0.872876i	0.119909	−	0.992785i	$-0.461740\pi$
$743$	−23.7929	−	23.7929i	−0.872876	−	0.872876i	−0.992785	+	0.119909i	$0.961740\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−19.6011			−0.717166
$748$		0			0
$749$	−16.9831	+	16.9831i	−0.620548	+	0.620548i
$750$		0			0
$751$		−	35.8257i		−	1.30730i	−0.756797	−	0.653650i	$-0.773237\pi$
$751$		−	35.8257i		−	1.30730i	0.756797	−	0.653650i	$-0.226763\pi$
$752$		0			0
$753$	17.9119	+	17.9119i	0.652746	+	0.652746i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	10.7730i		−	0.391552i	−0.980649	−	0.195776i	$-0.937277\pi$
$757$		−	10.7730i		−	0.391552i	0.980649	−	0.195776i	$-0.0627226\pi$
$758$		0			0
$759$		−	38.5730i		−	1.40011i
$760$		0			0
$761$		−	18.3935i		−	0.666765i	−0.942792	−	0.333383i	$-0.891810\pi$
$761$		−	18.3935i		−	0.666765i	0.942792	−	0.333383i	$-0.108190\pi$
$762$		0			0
$763$		−	3.47148i		−	0.125676i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14.2732	−	14.2732i	−0.515374	−	0.515374i
$768$		0			0
$769$		−	31.2512i		−	1.12695i	−0.826135	−	0.563473i	$-0.809465\pi$
$769$		−	31.2512i		−	1.12695i	0.826135	−	0.563473i	$-0.190535\pi$
$770$		0			0
$771$	11.1331	−	11.1331i	0.400951	−	0.400951i
$772$		0			0
$773$	13.9433			0.501504			0.250752	−	0.968051i	$-0.419322\pi$
$773$	13.9433			0.501504			0.250752	+	0.968051i	$0.419322\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	6.91453	+	6.91453i	0.248057	+	0.248057i
$778$		0			0
$779$	46.7241	−	46.7241i	1.67407	−	1.67407i
$780$		0			0
$781$	2.28999	+	2.28999i	0.0819424	+	0.0819424i
$782$		0			0
$783$	3.79270	−	3.79270i	0.135540	−	0.135540i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			9.23076i			0.329041i	0.986374	+	0.164520i	$0.0526077\pi$
$787$			9.23076i			0.329041i	−0.986374	+	0.164520i	$0.947392\pi$
$788$		0			0
$789$	−36.8864	−	36.8864i	−1.31319	−	1.31319i
$790$		0			0
$791$	47.2072			1.67850
$792$		0			0
$793$	19.0366	−	19.0366i	0.676011	−	0.676011i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−31.2806			−1.10801			−0.554007	−	0.832512i	$-0.686902\pi$
$797$	−31.2806			−1.10801			−0.554007	+	0.832512i	$0.686902\pi$
$798$		0			0
$799$	13.2129			0.467440
$800$		0			0
$801$	−4.54156			−0.160468
$802$		0			0
$803$	−20.7789			−0.733272
$804$		0			0
$805$		0			0
$806$		0			0
$807$	26.2247	−	26.2247i	0.923155	−	0.923155i
$808$		0			0
$809$	43.5905			1.53256			0.766280	−	0.642507i	$-0.222106\pi$
$809$	43.5905			1.53256			0.766280	+	0.642507i	$0.222106\pi$
$810$		0			0
$811$	13.3324	+	13.3324i	0.468164	+	0.468164i	0.901319	−	0.433155i	$-0.142600\pi$
$811$	13.3324	+	13.3324i	0.468164	+	0.468164i	−0.433155	+	0.901319i	$0.642600\pi$
$812$		0			0
$813$		−	39.6055i		−	1.38903i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−47.8585	+	47.8585i	−1.67436	+	1.67436i
$818$		0			0
$819$	−28.3785	−	28.3785i	−0.991623	−	0.991623i
$820$		0			0
$821$	−22.8302	+	22.8302i	−0.796779	+	0.796779i	−0.982586	−	0.185807i	$-0.940510\pi$
$821$	−22.8302	+	22.8302i	−0.796779	+	0.796779i	0.185807	+	0.982586i	$0.440510\pi$
$822$		0			0
$823$	6.70575	+	6.70575i	0.233748	+	0.233748i	0.814255	−	0.580507i	$-0.197146\pi$
$823$	6.70575	+	6.70575i	0.233748	+	0.233748i	−0.580507	+	0.814255i	$0.697146\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−28.9746			−1.00755			−0.503773	−	0.863836i	$-0.668055\pi$
$827$	−28.9746			−1.00755			−0.503773	+	0.863836i	$0.668055\pi$
$828$		0			0
$829$	16.6348	−	16.6348i	0.577749	−	0.577749i	−0.356534	−	0.934282i	$-0.616041\pi$
$829$	16.6348	−	16.6348i	0.577749	−	0.577749i	0.934282	+	0.356534i	$0.116041\pi$
$830$		0			0
$831$		−	29.6805i		−	1.02960i
$832$		0			0
$833$	12.1558	+	12.1558i	0.421174	+	0.421174i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			7.15673i			0.247373i
$838$		0			0
$839$		−	56.4946i		−	1.95041i	−0.221300	−	0.975206i	$-0.571030\pi$
$839$		−	56.4946i		−	1.95041i	0.221300	−	0.975206i	$-0.428970\pi$
$840$		0			0
$841$			2.25262i			0.0776766i
$842$		0			0
$843$			73.9737i			2.54779i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	3.35953	+	3.35953i	0.115435	+	0.115435i
$848$		0			0
$849$		−	17.8794i		−	0.613618i
$850$		0			0
$851$	−4.08063	+	4.08063i	−0.139882	+	0.139882i
$852$		0			0
$853$	−5.13309			−0.175754			−0.0878768	−	0.996131i	$-0.528008\pi$
$853$	−5.13309			−0.175754			−0.0878768	+	0.996131i	$0.528008\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.91389	−	9.91389i	−0.338652	−	0.338652i	0.517208	−	0.855860i	$-0.326971\pi$
$857$	−9.91389	−	9.91389i	−0.338652	−	0.338652i	−0.855860	+	0.517208i	$0.826971\pi$
$858$		0			0
$859$	21.1068	−	21.1068i	0.720154	−	0.720154i	−0.248482	−	0.968636i	$-0.579932\pi$
$859$	21.1068	−	21.1068i	0.720154	−	0.720154i	0.968636	+	0.248482i	$0.0799317\pi$
$860$		0			0
$861$	61.8972	+	61.8972i	2.10945	+	2.10945i
$862$		0			0
$863$	−4.72724	+	4.72724i	−0.160917	+	0.160917i	−0.782973	−	0.622056i	$-0.786298\pi$
$863$	−4.72724	+	4.72724i	−0.160917	+	0.160917i	0.622056	+	0.782973i	$0.286298\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	26.6033i		−	0.903497i
$868$		0			0
$869$	−5.61481	−	5.61481i	−0.190469	−	0.190469i
$870$		0			0
$871$	26.5342			0.899077
$872$		0			0
$873$	−2.98414	+	2.98414i	−0.100998	+	0.100998i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	52.8569			1.78485			0.892425	−	0.451195i	$-0.149002\pi$
$877$	52.8569			1.78485			0.892425	+	0.451195i	$0.149002\pi$
$878$		0			0
$879$	−62.7353			−2.11601
$880$		0			0
$881$	−31.1635			−1.04993			−0.524963	−	0.851125i	$-0.675921\pi$
$881$	−31.1635			−1.04993			−0.524963	+	0.851125i	$0.675921\pi$
$882$		0			0
$883$	37.8593			1.27407			0.637033	−	0.770837i	$-0.280162\pi$
$883$	37.8593			1.27407			0.637033	+	0.770837i	$0.280162\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	34.4782	−	34.4782i	1.15766	−	1.15766i	0.172688	−	0.984976i	$-0.444755\pi$
$887$	34.4782	−	34.4782i	1.15766	−	1.15766i	0.984976	−	0.172688i	$-0.0552454\pi$
$888$		0			0
$889$	−31.7479			−1.06479
$890$		0			0
$891$	−22.3447	−	22.3447i	−0.748576	−	0.748576i
$892$		0			0
$893$			37.0491i			1.23980i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	36.3724	−	36.3724i	1.21444	−	1.21444i
$898$		0			0
$899$	25.2359	+	25.2359i	0.841663	+	0.841663i
$900$		0			0
$901$	10.4741	−	10.4741i	0.348944	−	0.348944i
$902$		0			0
$903$	−63.4000	−	63.4000i	−2.10982	−	2.10982i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	6.44928			0.214145			0.107072	−	0.994251i	$-0.465852\pi$
$907$	6.44928			0.214145			0.107072	+	0.994251i	$0.465852\pi$
$908$		0			0
$909$	21.0023	−	21.0023i	0.696603	−	0.696603i
$910$		0			0
$911$		−	1.53662i		−	0.0509105i	−0.999676	−	0.0254553i	$-0.991896\pi$
$911$		−	1.53662i		−	0.0509105i	0.999676	−	0.0254553i	$-0.00810353\pi$
$912$		0			0
$913$	−16.8945	−	16.8945i	−0.559127	−	0.559127i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			7.05344i			0.232925i
$918$		0			0
$919$			55.6116i			1.83446i	0.398361	+	0.917229i	$0.369579\pi$
$919$			55.6116i			1.83446i	−0.398361	+	0.917229i	$0.630421\pi$
$920$		0			0
$921$			6.56979i			0.216482i
$922$		0			0
$923$			4.31870i			0.142152i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−24.8750	−	24.8750i	−0.817003	−	0.817003i
$928$		0			0
$929$			7.19045i			0.235911i	0.993019	+	0.117955i	$0.0376340\pi$
$929$			7.19045i			0.235911i	−0.993019	+	0.117955i	$0.962366\pi$
$930$		0			0
$931$	−34.0848	+	34.0848i	−1.11709	+	1.11709i
$932$		0			0
$933$	11.7333			0.384130
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.71397	+	4.71397i	0.153999	+	0.153999i	0.779901	−	0.625903i	$-0.215269\pi$
$937$	4.71397	+	4.71397i	0.153999	+	0.153999i	−0.625903	+	0.779901i	$0.715269\pi$
$938$		0			0
$939$	−15.9461	+	15.9461i	−0.520381	+	0.520381i
$940$		0			0
$941$	−5.36011	−	5.36011i	−0.174735	−	0.174735i	0.614321	−	0.789056i	$-0.289430\pi$
$941$	−5.36011	−	5.36011i	−0.174735	−	0.174735i	−0.789056	+	0.614321i	$0.789430\pi$
$942$		0			0
$943$	−36.5288	+	36.5288i	−1.18954	+	1.18954i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			21.9071i			0.711886i	0.934508	+	0.355943i	$0.115840\pi$
$947$			21.9071i			0.711886i	−0.934508	+	0.355943i	$0.884160\pi$
$948$		0			0
$949$	−19.5935	−	19.5935i	−0.636031	−	0.636031i
$950$		0			0
$951$	−4.27297			−0.138560
$952$		0			0
$953$	−43.3690	+	43.3690i	−1.40486	+	1.40486i	−0.621241	+	0.783620i	$0.713371\pi$
$953$	−43.3690	+	43.3690i	−1.40486	+	1.40486i	−0.783620	+	0.621241i	$0.786629\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−38.0571			−1.23021
$958$		0			0
$959$	82.4678			2.66302
$960$		0			0
$961$	−16.6195			−0.536113
$962$		0			0
$963$	16.3238			0.526028
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−11.5130	+	11.5130i	−0.370232	+	0.370232i	−0.867562	−	0.497330i	$-0.834314\pi$
$967$	−11.5130	+	11.5130i	−0.370232	+	0.370232i	0.497330	+	0.867562i	$0.334314\pi$
$968$		0			0
$969$	−37.8055			−1.21449
$970$		0			0
$971$	−25.2567	−	25.2567i	−0.810524	−	0.810524i	0.174188	−	0.984712i	$-0.444270\pi$
$971$	−25.2567	−	25.2567i	−0.810524	−	0.810524i	−0.984712	+	0.174188i	$0.944270\pi$
$972$		0			0
$973$		−	31.8415i		−	1.02079i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.59728	−	5.59728i	0.179073	−	0.179073i	−0.611879	−	0.790952i	$-0.709586\pi$
$977$	5.59728	−	5.59728i	0.179073	−	0.179073i	0.790952	+	0.611879i	$0.209586\pi$
$978$		0			0
$979$	−3.91445	−	3.91445i	−0.125106	−	0.125106i
$980$		0			0
$981$	−1.66836	+	1.66836i	−0.0532667	+	0.0532667i
$982$		0			0
$983$	−20.9203	−	20.9203i	−0.667255	−	0.667255i	0.289825	−	0.957080i	$-0.406403\pi$
$983$	−20.9203	−	20.9203i	−0.667255	−	0.667255i	−0.957080	+	0.289825i	$0.906403\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−49.0803			−1.56224
$988$		0			0
$989$	37.4157	−	37.4157i	1.18975	−	1.18975i
$990$		0			0
$991$			35.6265i			1.13171i	0.824503	+	0.565857i	$0.191455\pi$
$991$			35.6265i			1.13171i	−0.824503	+	0.565857i	$0.808545\pi$
$992$		0			0
$993$	−3.20671	−	3.20671i	−0.101762	−	0.101762i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			44.8208i			1.41949i	0.704459	+	0.709745i	$0.251190\pi$
$997$			44.8208i			1.41949i	−0.704459	+	0.709745i	$0.748810\pi$
$998$		0			0
$999$			1.14176i			0.0361238i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.s.e.943.3		24
4.3	odd	2		400.2.s.e.243.7	yes	24
5.2	odd	4		1600.2.j.e.1007.10		24
5.3	odd	4		1600.2.j.e.1007.3		24
5.4	even	2	inner	1600.2.s.e.943.10		24
16.5	even	4		400.2.j.e.43.12	yes	24
16.11	odd	4		1600.2.j.e.143.3		24
20.3	even	4		400.2.j.e.307.1	yes	24
20.7	even	4		400.2.j.e.307.12	yes	24
20.19	odd	2		400.2.s.e.243.6	yes	24
80.27	even	4	inner	1600.2.s.e.207.3		24
80.37	odd	4		400.2.s.e.107.7	yes	24
80.43	even	4	inner	1600.2.s.e.207.10		24
80.53	odd	4		400.2.s.e.107.6	yes	24
80.59	odd	4		1600.2.j.e.143.10		24
80.69	even	4		400.2.j.e.43.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.j.e.43.1	✓	24	80.69	even	4
400.2.j.e.43.12	yes	24	16.5	even	4
400.2.j.e.307.1	yes	24	20.3	even	4
400.2.j.e.307.12	yes	24	20.7	even	4
400.2.s.e.107.6	yes	24	80.53	odd	4
400.2.s.e.107.7	yes	24	80.37	odd	4
400.2.s.e.243.6	yes	24	20.19	odd	2
400.2.s.e.243.7	yes	24	4.3	odd	2
1600.2.j.e.143.3		24	16.11	odd	4
1600.2.j.e.143.10		24	80.59	odd	4
1600.2.j.e.1007.3		24	5.3	odd	4
1600.2.j.e.1007.10		24	5.2	odd	4
1600.2.s.e.207.3		24	80.27	even	4	inner
1600.2.s.e.207.10		24	80.43	even	4	inner
1600.2.s.e.943.3		24	1.1	even	1	trivial
1600.2.s.e.943.10		24	5.4	even	2	inner

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.s (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.35800 q^{3} +(-2.66357 + 2.66357i) q^{7} +2.56018 q^{9} +O(q^{10})\)	\(q-2.35800 q^{3} +(-2.66357 + 2.66357i) q^{7} +2.56018 q^{9} +(2.20666 + 2.20666i) q^{11} +4.16154i q^{13} +(-1.69084 + 1.69084i) q^{17} +(-4.74110 - 4.74110i) q^{19} +(6.28071 - 6.28071i) q^{21} +(3.70658 + 3.70658i) q^{23} +1.03710 q^{27} +(3.65701 - 3.65701i) q^{29} +6.90069i q^{31} +(-5.20331 - 5.20331i) q^{33} +1.10092i q^{37} -9.81293i q^{39} +9.85512i q^{41} -10.0944i q^{43} +(-3.90722 - 3.90722i) q^{47} -7.18923i q^{49} +(3.98700 - 3.98700i) q^{51} -6.19464 q^{53} +(11.1795 + 11.1795i) q^{57} +(-3.42978 + 3.42978i) q^{59} +(-4.57442 - 4.57442i) q^{61} +(-6.81921 + 6.81921i) q^{63} -6.37605i q^{67} +(-8.74012 - 8.74012i) q^{69} +1.03776 q^{71} +(-4.70822 + 4.70822i) q^{73} -11.7552 q^{77} -2.54448 q^{79} -10.1260 q^{81} -7.65615 q^{83} +(-8.62323 + 8.62323i) q^{87} -1.77392 q^{89} +(-11.0846 - 11.0846i) q^{91} -16.2718i q^{93} +(-1.16560 + 1.16560i) q^{97} +(5.64944 + 5.64944i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 40 q^{9}+O(q^{10}) \) 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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