LMFDB - Embedded newform 252.8.k.c.109.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,8,Mod(37,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.37"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,-249] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 $ x^10 - x^9 + 342x^8 + 2165x^7 + 113605x^6 + 319380x^5 + 1438128x^4 + 1705752x^3 + 7544016x^2 + 8573040x + 23619600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8}\cdot 7^{5} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.3
Root			$10.0194 + 17.3541i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.8.k.c.37.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-58.1649 - 100.745i) q^{5} +(-276.518 + 864.338i) q^{7} +(-3407.25 + 5901.54i) q^{11} -11924.5 q^{13} +(3301.82 - 5718.92i) q^{17} +(7387.57 + 12795.6i) q^{19} +(-16134.9 - 27946.4i) q^{23} +(32296.2 - 55938.6i) q^{25} +126792. q^{29} +(-94753.8 + 164118. i) q^{31} +(103161. - 22416.5i) q^{35} +(76077.0 + 131769. i) q^{37} +355426. q^{41} -650136. q^{43} +(-50866.2 - 88102.8i) q^{47} +(-670618. - 478010. i) q^{49} +(1.02218e6 - 1.77047e6i) q^{53} +792730. q^{55} +(-100356. + 173822. i) q^{59} +(-453981. - 786317. i) q^{61} +(693586. + 1.20133e6i) q^{65} +(-1.11616e6 + 1.93324e6i) q^{67} -4.16441e6 q^{71} +(1.61704e6 - 2.80080e6i) q^{73} +(-4.15876e6 - 4.57690e6i) q^{77} +(684840. + 1.18618e6i) q^{79} +8.14674e6 q^{83} -768201. q^{85} +(-1.77793e6 - 3.07947e6i) q^{89} +(3.29733e6 - 1.03068e7i) q^{91} +(859394. - 1.48851e6i) q^{95} +1.48623e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 249 q^{5} + 332 q^{7} - 6399 q^{11} - 26988 q^{13} - 3609 q^{17} - 12403 q^{19} + 13959 q^{23} - 162364 q^{25} - 26148 q^{29} - 20181 q^{31} - 791715 q^{35} - 54763 q^{37} + 1824468 q^{41} - 1938424 q^{43}+ \cdots + 285580 q^{97}+O(q^{100}) $ 10 * q - 249 * q^5 + 332 * q^7 - 6399 * q^11 - 26988 * q^13 - 3609 * q^17 - 12403 * q^19 + 13959 * q^23 - 162364 * q^25 - 26148 * q^29 - 20181 * q^31 - 791715 * q^35 - 54763 * q^37 + 1824468 * q^41 - 1938424 * q^43 - 1217253 * q^47 + 1722058 * q^49 + 2349159 * q^53 + 3123494 * q^55 - 2188611 * q^59 + 6107445 * q^61 + 3022014 * q^65 - 3171581 * q^67 + 18938976 * q^71 + 8034853 * q^73 - 24100191 * q^77 - 7589559 * q^79 + 13452408 * q^83 - 48717670 * q^85 - 29864757 * q^89 + 34267512 * q^91 + 25812009 * q^95 + 285580 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−58.1649	−	100.745i	−0.208097	−	0.360435i	0.743018	−	0.669271i	$-0.233394\pi$
$5$	−58.1649	−	100.745i	−0.208097	−	0.360435i	−0.951115	+	0.308837i	$0.900060\pi$
$6$		0			0
$7$	−276.518	+	864.338i	−0.304706	+	0.952447i
$7$	−276.518	+	864.338i	−0.304706	+	0.952447i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3407.25	+	5901.54i	−0.771845	+	1.33687i	0.164706	+	0.986343i	$0.447332\pi$
$11$	−3407.25	+	5901.54i	−0.771845	+	1.33687i	−0.936551	+	0.350532i	$0.886001\pi$
$12$		0			0
$13$	−11924.5			−1.50535			−0.752675	−	0.658392i	$-0.771237\pi$
$13$	−11924.5			−1.50535			−0.752675	+	0.658392i	$0.771237\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3301.82	−	5718.92i	0.162998	−	0.282321i	−0.772944	−	0.634474i	$-0.781217\pi$
$17$	3301.82	−	5718.92i	0.162998	−	0.282321i	0.935943	+	0.352153i	$0.114550\pi$
$18$		0			0
$19$	7387.57	+	12795.6i	0.247095	+	0.427981i	0.962719	−	0.270505i	$-0.0871907\pi$
$19$	7387.57	+	12795.6i	0.247095	+	0.427981i	−0.715624	+	0.698486i	$0.753857\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−16134.9	−	27946.4i	−0.276515	−	0.478938i	0.694001	−	0.719974i	$-0.255846\pi$
$23$	−16134.9	−	27946.4i	−0.276515	−	0.478938i	−0.970516	+	0.241036i	$0.922513\pi$
$24$		0			0
$25$	32296.2	−	55938.6i	0.413391	−	0.716015i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	126792.			0.965382			0.482691	−	0.875791i	$-0.339659\pi$
$29$	126792.			0.965382			0.482691	+	0.875791i	$0.339659\pi$
$30$		0			0
$31$	−94753.8	+	164118.i	−0.571256	+	0.989444i	0.425182	+	0.905108i	$0.360210\pi$
$31$	−94753.8	+	164118.i	−0.571256	+	0.989444i	−0.996437	+	0.0843359i	$0.973123\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	103161.	−	22416.5i	0.406703	−	0.0883749i
$36$		0			0
$37$	76077.0	+	131769.i	0.246915	+	0.427669i	0.962668	−	0.270684i	$-0.0872499\pi$
$37$	76077.0	+	131769.i	0.246915	+	0.427669i	−0.715753	+	0.698353i	$0.753917\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	355426.			0.805389			0.402694	−	0.915334i	$-0.368074\pi$
$41$	355426.			0.805389			0.402694	+	0.915334i	$0.368074\pi$
$42$		0			0
$43$	−650136.			−1.24699			−0.623497	−	0.781825i	$-0.714289\pi$
$43$	−650136.			−1.24699			−0.623497	+	0.781825i	$0.714289\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−50866.2	−	88102.8i	−0.0714638	−	0.123779i	0.828079	−	0.560611i	$-0.189434\pi$
$47$	−50866.2	−	88102.8i	−0.0714638	−	0.123779i	−0.899543	+	0.436832i	$0.856100\pi$
$48$		0			0
$49$	−670618.	−	478010.i	−0.814309	−	0.580432i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.02218e6	−	1.77047e6i	0.943111	−	1.63352i	0.183620	−	0.982997i	$-0.441218\pi$
$53$	1.02218e6	−	1.77047e6i	0.943111	−	1.63352i	0.759490	−	0.650519i	$-0.225448\pi$
$54$		0			0
$55$	792730.			0.642475
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−100356.	+	173822.i	−0.0636153	+	0.110185i	−0.896079	−	0.443895i	$-0.853596\pi$
$59$	−100356.	+	173822.i	−0.0636153	+	0.110185i	0.832464	+	0.554080i	$0.186930\pi$
$60$		0			0
$61$	−453981.	−	786317.i	−0.256084	−	0.443551i	0.709105	−	0.705103i	$-0.249099\pi$
$61$	−453981.	−	786317.i	−0.256084	−	0.443551i	−0.965189	+	0.261552i	$0.915766\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	693586.	+	1.20133e6i	0.313259	+	0.542580i
$66$		0			0
$67$	−1.11616e6	+	1.93324e6i	−0.453382	+	0.785280i	−0.998594	−	0.0530181i	$-0.983116\pi$
$67$	−1.11616e6	+	1.93324e6i	−0.453382	+	0.785280i	0.545212	+	0.838298i	$0.316449\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.16441e6			−1.38086			−0.690430	−	0.723399i	$-0.742579\pi$
$71$	−4.16441e6			−1.38086			−0.690430	+	0.723399i	$0.742579\pi$
$72$		0			0
$73$	1.61704e6	−	2.80080e6i	0.486510	−	0.842660i	−0.513370	−	0.858168i	$-0.671603\pi$
$73$	1.61704e6	−	2.80080e6i	0.486510	−	0.842660i	0.999880	+	0.0155074i	$0.00493635\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.15876e6	−	4.57690e6i	−1.03812	−	1.14249i
$78$		0			0
$79$	684840.	+	1.18618e6i	0.156277	+	0.270679i	0.933523	−	0.358517i	$-0.116717\pi$
$79$	684840.	+	1.18618e6i	0.156277	+	0.270679i	−0.777246	+	0.629196i	$0.783384\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.14674e6			1.56391			0.781953	−	0.623337i	$-0.214224\pi$
$83$	8.14674e6			1.56391			0.781953	+	0.623337i	$0.214224\pi$
$84$		0			0
$85$	−768201.			−0.135678
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.77793e6	−	3.07947e6i	−0.267331	−	0.463031i	0.700840	−	0.713318i	$-0.252808\pi$
$89$	−1.77793e6	−	3.07947e6i	−0.267331	−	0.463031i	−0.968172	+	0.250287i	$0.919475\pi$
$90$		0			0
$91$	3.29733e6	−	1.03068e7i	0.458689	−	1.43377i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	859394.	−	1.48851e6i	0.102839	−	0.178123i
$96$		0			0
$97$	1.48623e7			1.65343			0.826715	−	0.562622i	$-0.190207\pi$
$97$	1.48623e7			1.65343			0.826715	+	0.562622i	$0.190207\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.42329e6	−	2.46521e6i	0.137458	−	0.238084i	−0.789076	−	0.614296i	$-0.789440\pi$
$101$	1.42329e6	−	2.46521e6i	0.137458	−	0.238084i	0.926534	+	0.376212i	$0.122774\pi$
$102$		0			0
$103$	297848.	+	515888.i	0.0268574	+	0.0465184i	0.879142	−	0.476560i	$-0.158117\pi$
$103$	297848.	+	515888.i	0.0268574	+	0.0465184i	−0.852284	+	0.523079i	$0.824783\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.93320e6	−	1.72048e7i	−0.783873	−	1.35771i	−0.929670	−	0.368393i	$-0.879908\pi$
$107$	−9.93320e6	−	1.72048e7i	−0.783873	−	1.35771i	0.145797	−	0.989315i	$-0.453425\pi$
$108$		0			0
$109$	3.92925e6	−	6.80565e6i	0.290614	−	0.503358i	−0.683341	−	0.730099i	$-0.739474\pi$
$109$	3.92925e6	−	6.80565e6i	0.290614	−	0.503358i	0.973955	+	0.226741i	$0.0728071\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.16317e7			0.758347			0.379174	−	0.925325i	$-0.376208\pi$
$113$	1.16317e7			0.758347			0.379174	+	0.925325i	$0.376208\pi$
$114$		0			0
$115$	−1.87697e6	+	3.25100e6i	−0.115084	+	0.199331i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.03007e6	+	4.43528e6i	0.219229	+	0.241272i
$120$		0			0
$121$	−1.34752e7	−	2.33397e7i	−0.691489	−	1.19769i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.66023e7			−0.760296
$126$		0			0
$127$	1.02118e6			0.0442372			0.0221186	−	0.999755i	$-0.492959\pi$
$127$	1.02118e6			0.0442372			0.0221186	+	0.999755i	$0.492959\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.41713e7	−	4.18660e7i	−0.939401	−	1.62709i	−0.766591	−	0.642135i	$-0.778049\pi$
$131$	−2.41713e7	−	4.18660e7i	−0.939401	−	1.62709i	−0.172810	−	0.984955i	$-0.555285\pi$
$132$		0			0
$133$	−1.31026e7	+	2.84713e6i	−0.482920	+	0.104937i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.23915e7	−	2.14626e7i	0.411719	−	0.713118i	−0.583359	−	0.812214i	$-0.698262\pi$
$137$	1.23915e7	−	2.14626e7i	0.411719	−	0.713118i	0.995078	+	0.0990967i	$0.0315953\pi$
$138$		0			0
$139$	−1.93687e7			−0.611713			−0.305857	−	0.952078i	$-0.598943\pi$
$139$	−1.93687e7			−0.611713			−0.305857	+	0.952078i	$0.598943\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.06297e7	−	7.03727e7i	1.16190	−	2.01246i
$144$		0			0
$145$	−7.37485e6	−	1.27736e7i	−0.200893	−	0.347957i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.53357e6	+	1.47806e7i	0.211339	+	0.366049i	0.952134	−	0.305682i	$-0.0988844\pi$
$149$	8.53357e6	+	1.47806e7i	0.211339	+	0.366049i	−0.740795	+	0.671731i	$0.765551\pi$
$150$		0			0
$151$	−1.31225e7	+	2.27288e7i	−0.310167	+	0.537225i	−0.978398	−	0.206728i	$-0.933718\pi$
$151$	−1.31225e7	+	2.27288e7i	−0.310167	+	0.537225i	0.668231	+	0.743954i	$0.267052\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.20454e7			0.475507
$156$		0			0
$157$	2.04968e7	−	3.55016e7i	0.422706	−	0.732148i	−0.573498	−	0.819207i	$-0.694414\pi$
$157$	2.04968e7	−	3.55016e7i	0.422706	−	0.732148i	0.996203	+	0.0870598i	$0.0277471\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.86168e7	−	6.21830e6i	0.540418	−	0.117431i
$162$		0			0
$163$	−2.61227e7	−	4.52458e7i	−0.472455	−	0.818317i	0.527048	−	0.849836i	$-0.323299\pi$
$163$	−2.61227e7	−	4.52458e7i	−0.472455	−	0.818317i	−0.999503	+	0.0315189i	$0.989966\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.67268e7			0.942498			0.471249	−	0.882000i	$-0.343803\pi$
$167$	5.67268e7			0.942498			0.471249	+	0.882000i	$0.343803\pi$
$168$		0			0
$169$	7.94446e7			1.26608
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−774366.	−	1.34124e6i	−0.0113706	−	0.0196945i	0.860284	−	0.509815i	$-0.170286\pi$
$173$	−774366.	−	1.34124e6i	−0.0113706	−	0.0196945i	−0.871655	+	0.490120i	$0.836953\pi$
$174$		0			0
$175$	3.94194e7	+	4.33829e7i	0.556003	+	0.611907i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.87023e7	−	3.23933e7i	0.243730	−	0.422153i	−0.718044	−	0.695998i	$-0.754962\pi$
$179$	1.87023e7	−	3.23933e7i	0.243730	−	0.422153i	0.961774	+	0.273845i	$0.0882955\pi$
$180$		0			0
$181$	−1.08072e8			−1.35469			−0.677343	−	0.735667i	$-0.736869\pi$
$181$	−1.08072e8			−1.35469			−0.677343	+	0.735667i	$0.736869\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.85002e6	−	1.53287e7i	0.102765	−	0.177993i
$186$		0			0
$187$	2.25003e7	+	3.89716e7i	0.251618	+	0.435816i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.76760e7	−	9.98978e7i	−0.598933	−	1.03738i	−0.992979	−	0.118292i	$-0.962258\pi$
$191$	−5.76760e7	−	9.98978e7i	−0.598933	−	1.03738i	0.394046	−	0.919091i	$-0.371075\pi$
$192$		0			0
$193$	4.10645e7	−	7.11257e7i	0.411164	−	0.712158i	−0.583853	−	0.811859i	$-0.698456\pi$
$193$	4.10645e7	−	7.11257e7i	0.411164	−	0.712158i	0.995017	+	0.0997017i	$0.0317888\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.32327e8			−1.23315			−0.616576	−	0.787296i	$-0.711481\pi$
$197$	−1.32327e8			−1.23315			−0.616576	+	0.787296i	$0.711481\pi$
$198$		0			0
$199$	−3.70915e7	+	6.42444e7i	−0.333648	+	0.577896i	−0.983224	−	0.182401i	$-0.941613\pi$
$199$	−3.70915e7	+	6.42444e7i	−0.333648	+	0.577896i	0.649576	+	0.760297i	$0.274946\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.50603e7	+	1.09591e8i	−0.294157	+	0.919475i
$204$		0			0
$205$	−2.06733e7	−	3.58072e7i	−0.167599	−	0.290290i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.00685e8			−0.762876
$210$		0			0
$211$	5.84260e7			0.428171			0.214086	−	0.976815i	$-0.431323\pi$
$211$	5.84260e7			0.428171			0.214086	+	0.976815i	$0.431323\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.78151e7	+	6.54977e7i	0.259496	+	0.449460i
$216$		0			0
$217$	−1.15653e8	−	1.27281e8i	−0.768328	−	0.845580i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.93725e7	+	6.81952e7i	−0.245369	+	0.424992i
$222$		0			0
$223$	2.61801e8			1.58090			0.790449	−	0.612528i	$-0.209847\pi$
$223$	2.61801e8			1.58090			0.790449	+	0.612528i	$0.209847\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.89678e7	+	5.01737e7i	−0.164371	+	0.284699i	−0.936432	−	0.350850i	$-0.885893\pi$
$227$	−2.89678e7	+	5.01737e7i	−0.164371	+	0.284699i	0.772061	+	0.635549i	$0.219226\pi$
$228$		0			0
$229$	8.34771e7	+	1.44587e8i	0.459349	+	0.795616i	0.998927	−	0.0463198i	$-0.0147493\pi$
$229$	8.34771e7	+	1.44587e8i	0.459349	+	0.795616i	−0.539577	+	0.841936i	$0.681416\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.08941e8	+	1.88691e8i	0.564217	+	0.977252i	0.997122	+	0.0758125i	$0.0241550\pi$
$233$	1.08941e8	+	1.88691e8i	0.564217	+	0.977252i	−0.432906	+	0.901439i	$0.642512\pi$
$234$		0			0
$235$	−5.91725e6	+	1.02490e7i	−0.0297428	+	0.0515161i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.23970e8			−1.06120			−0.530600	−	0.847622i	$-0.678033\pi$
$239$	−2.23970e8			−1.06120			−0.530600	+	0.847622i	$0.678033\pi$
$240$		0			0
$241$	−4.93506e7	+	8.54778e7i	−0.227108	+	0.393363i	−0.956950	−	0.290253i	$-0.906260\pi$
$241$	−4.93506e7	+	8.54778e7i	−0.227108	+	0.393363i	0.729842	+	0.683616i	$0.239594\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.15049e6	+	9.53646e7i	−0.0397524	+	0.414291i
$246$		0			0
$247$	−8.80929e7	−	1.52581e8i	−0.371964	−	0.644261i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.65751e8			1.06076			0.530379	−	0.847760i	$-0.322050\pi$
$251$	2.65751e8			1.06076			0.530379	+	0.847760i	$0.322050\pi$
$252$		0			0
$253$	2.19903e8			0.853706
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.52841e7	+	2.64728e7i	0.0561660	+	0.0972824i	0.892741	−	0.450570i	$-0.148779\pi$
$257$	1.52841e7	+	2.64728e7i	0.0561660	+	0.0972824i	−0.836575	+	0.547852i	$0.815446\pi$
$258$		0			0
$259$	−1.34930e8	+	2.93197e7i	−0.482568	+	0.104860i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.61770e6	+	6.26604e6i	−0.0122627	+	0.0212397i	−0.872092	−	0.489343i	$-0.837237\pi$
$263$	−3.61770e6	+	6.26604e6i	−0.0122627	+	0.0212397i	0.859829	+	0.510582i	$0.170570\pi$
$264$		0			0
$265$	−2.37820e8			−0.785034
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.42076e7	−	5.92494e7i	0.107150	−	0.185588i	−0.807465	−	0.589916i	$-0.799161\pi$
$269$	3.42076e7	−	5.92494e7i	0.107150	−	0.185588i	0.914614	+	0.404327i	$0.132494\pi$
$270$		0			0
$271$	4.88220e7	+	8.45622e7i	0.149013	+	0.258097i	0.930863	−	0.365369i	$-0.119057\pi$
$271$	4.88220e7	+	8.45622e7i	0.149013	+	0.258097i	−0.781850	+	0.623466i	$0.785724\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.20083e8	+	3.81194e8i	0.638148	+	1.10530i
$276$		0			0
$277$	−1.93980e8	+	3.35983e8i	−0.548375	+	0.949814i	0.450011	+	0.893023i	$0.351420\pi$
$277$	−1.93980e8	+	3.35983e8i	−0.548375	+	0.949814i	−0.998386	+	0.0567907i	$0.981913\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.44469e8			−1.19500			−0.597502	−	0.801868i	$-0.703840\pi$
$281$	−4.44469e8			−1.19500			−0.597502	+	0.801868i	$0.703840\pi$
$282$		0			0
$283$	−3.11056e8	+	5.38765e8i	−0.815805	+	1.41302i	0.0929429	+	0.995671i	$0.470373\pi$
$283$	−3.11056e8	+	5.38765e8i	−0.815805	+	1.41302i	−0.908748	+	0.417345i	$0.862961\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.82817e7	+	3.07208e8i	−0.245407	+	0.767090i
$288$		0			0
$289$	1.83365e8	+	3.17598e8i	0.446863	+	0.773990i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.26197e8			0.757606			0.378803	−	0.925477i	$-0.376336\pi$
$293$	3.26197e8			0.757606			0.378803	+	0.925477i	$0.376336\pi$
$294$		0			0
$295$	2.33488e7			0.0529526
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.92400e8	+	3.33247e8i	0.416252	+	0.720969i
$300$		0			0
$301$	1.79774e8	−	5.61937e8i	0.379966	−	1.18770i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.28115e7	+	9.14721e7i	−0.106581	+	0.184603i
$306$		0			0
$307$	2.17327e8			0.428675			0.214338	−	0.976760i	$-0.431241\pi$
$307$	2.17327e8			0.428675			0.214338	+	0.976760i	$0.431241\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.26348e8	+	2.18842e8i	−0.238182	+	0.412543i	−0.960193	−	0.279339i	$-0.909885\pi$
$311$	−1.26348e8	+	2.18842e8i	−0.238182	+	0.412543i	0.722011	+	0.691882i	$0.243218\pi$
$312$		0			0
$313$	−1.71496e8	−	2.97039e8i	−0.316117	−	0.547531i	0.663557	−	0.748125i	$-0.269046\pi$
$313$	−1.71496e8	−	2.97039e8i	−0.316117	−	0.547531i	−0.979674	+	0.200595i	$0.935713\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.70616e8	+	2.95516e8i	0.300824	+	0.521042i	0.976323	−	0.216319i	$-0.0694050\pi$
$317$	1.70616e8	+	2.95516e8i	0.300824	+	0.521042i	−0.675499	+	0.737361i	$0.736072\pi$
$318$		0			0
$319$	−4.32013e8	+	7.48268e8i	−0.745125	+	1.29059i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9.75697e7			0.161104
$324$		0			0
$325$	−3.85115e8	+	6.67039e8i	−0.622299	+	1.07785i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.02160e7	−	1.96036e7i	0.139668	−	0.0303493i
$330$		0			0
$331$	−2.70854e8	−	4.69133e8i	−0.410523	−	0.711046i	0.584424	−	0.811448i	$-0.301320\pi$
$331$	−2.70854e8	−	4.69133e8i	−0.410523	−	0.711046i	−0.994947	+	0.100402i	$0.967987\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.59685e8			0.377390
$336$		0			0
$337$	5.08120e7			0.0723205			0.0361602	−	0.999346i	$-0.488487\pi$
$337$	5.08120e7			0.0723205			0.0361602	+	0.999346i	$0.488487\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.45700e8	−	1.11839e9i	−0.881842	−	1.52739i
$342$		0			0
$343$	5.98601e8	−	4.47463e8i	0.800955	−	0.598725i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.39930e8	+	5.88776e8i	−0.436753	+	0.756479i	−0.997437	−	0.0715511i	$-0.977205\pi$
$347$	−3.39930e8	+	5.88776e8i	−0.436753	+	0.756479i	0.560684	+	0.828030i	$0.310538\pi$
$348$		0			0
$349$	−4.75862e8			−0.599228			−0.299614	−	0.954060i	$-0.596858\pi$
$349$	−4.75862e8			−0.599228			−0.299614	+	0.954060i	$0.596858\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.08969e8	−	3.61945e8i	0.252855	−	0.437957i	−0.711456	−	0.702731i	$-0.751964\pi$
$353$	2.08969e8	−	3.61945e8i	0.252855	−	0.437957i	0.964311	+	0.264774i	$0.0852972\pi$
$354$		0			0
$355$	2.42223e8	+	4.19542e8i	0.287353	+	0.497710i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.89840e8	+	3.28812e8i	0.216549	+	0.375074i	0.953751	−	0.300599i	$-0.0971865\pi$
$359$	1.89840e8	+	3.28812e8i	0.216549	+	0.375074i	−0.737201	+	0.675673i	$0.763853\pi$
$360$		0			0
$361$	3.37784e8	−	5.85058e8i	0.377888	−	0.654521i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.76221e8			−0.404965
$366$		0			0
$367$	2.24517e8	−	3.88875e8i	0.237093	−	0.410656i	−0.722786	−	0.691072i	$-0.757139\pi$
$367$	2.24517e8	−	3.88875e8i	0.237093	−	0.410656i	0.959879	+	0.280415i	$0.0904722\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.24763e9	+	1.37308e9i	1.26847	+	1.39600i
$372$		0			0
$373$	−1.92208e8	−	3.32913e8i	−0.191774	−	0.332162i	0.754064	−	0.656801i	$-0.228091\pi$
$373$	−1.92208e8	−	3.32913e8i	−0.191774	−	0.332162i	−0.945838	+	0.324638i	$0.894757\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.51193e9			−1.45324
$378$		0			0
$379$	1.06012e9			1.00027			0.500137	−	0.865946i	$-0.333283\pi$
$379$	1.06012e9			1.00027			0.500137	+	0.865946i	$0.333283\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.71522e8	−	8.16701e8i	−0.428851	−	0.742792i	0.567920	−	0.823084i	$-0.307748\pi$
$383$	−4.71522e8	−	8.16701e8i	−0.428851	−	0.742792i	−0.996771	+	0.0802915i	$0.974415\pi$
$384$		0			0
$385$	−2.19204e8	+	6.85187e8i	−0.195766	+	0.611923i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.74414e8	−	4.75300e8i	0.236365	−	0.409396i	−0.723303	−	0.690530i	$-0.757377\pi$
$389$	2.74414e8	−	4.75300e8i	0.236365	−	0.409396i	0.959669	+	0.281134i	$0.0907106\pi$
$390$		0			0
$391$	−2.13098e8			−0.180286
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.96673e7	−	1.37988e8i	0.0650415	−	0.112655i
$396$		0			0
$397$	7.09112e8	+	1.22822e9i	0.568785	+	0.985164i	0.996687	+	0.0813387i	$0.0259196\pi$
$397$	7.09112e8	+	1.22822e9i	0.568785	+	0.985164i	−0.427902	+	0.903825i	$0.640747\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.03119e8	−	8.71428e8i	−0.389642	−	0.674880i	0.602759	−	0.797923i	$-0.294068\pi$
$401$	−5.03119e8	−	8.71428e8i	−0.389642	−	0.674880i	−0.992401	+	0.123043i	$0.960735\pi$
$402$		0			0
$403$	1.12989e9	−	1.95703e9i	0.859940	−	1.48946i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.03685e9			−0.762320
$408$		0			0
$409$	4.83763e8	−	8.37902e8i	0.349624	−	0.605566i	−0.636559	−	0.771228i	$-0.719643\pi$
$409$	4.83763e8	−	8.37902e8i	0.349624	−	0.605566i	0.986183	+	0.165662i	$0.0529761\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.22491e8	−	1.34806e8i	−0.0855613	−	0.0941641i
$414$		0			0
$415$	−4.73854e8	−	8.20740e8i	−0.325444	−	0.563686i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.33931e9			0.889473			0.444736	−	0.895662i	$-0.353297\pi$
$419$	1.33931e9			0.889473			0.444736	+	0.895662i	$0.353297\pi$
$420$		0			0
$421$	4.59395e8			0.300054			0.150027	−	0.988682i	$-0.452064\pi$
$421$	4.59395e8			0.300054			0.150027	+	0.988682i	$0.452064\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.13273e8	−	3.69399e8i	−0.134764	−	0.233418i
$426$		0			0
$427$	8.05178e8	−	1.74962e8i	0.500489	−	0.108754i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.25854e9	−	2.17985e9i	0.757173	−	1.31146i	−0.187114	−	0.982338i	$-0.559913\pi$
$431$	1.25854e9	−	2.17985e9i	0.757173	−	1.31146i	0.944287	−	0.329123i	$-0.106753\pi$
$432$		0			0
$433$	−1.94780e9			−1.15302			−0.576511	−	0.817090i	$-0.695586\pi$
$433$	−1.94780e9			−1.15302			−0.576511	+	0.817090i	$0.695586\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.38395e8	−	4.12913e8i	0.136651	−	0.236686i
$438$		0			0
$439$	−6.98148e8	−	1.20923e9i	−0.393842	−	0.682154i	0.599111	−	0.800666i	$-0.295521\pi$
$439$	−6.98148e8	−	1.20923e9i	−0.393842	−	0.682154i	−0.992953	+	0.118512i	$0.962188\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.85364e8	−	1.70670e9i	−0.538498	−	0.932705i	−0.998985	−	0.0450392i	$-0.985659\pi$
$443$	−9.85364e8	−	1.70670e9i	−0.538498	−	0.932705i	0.460488	−	0.887666i	$-0.347675\pi$
$444$		0			0
$445$	−2.06826e8	+	3.58234e8i	−0.111262	+	0.192711i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8.24242e8			−0.429727			−0.214864	−	0.976644i	$-0.568931\pi$
$449$	−8.24242e8			−0.429727			−0.214864	+	0.976644i	$0.568931\pi$
$450$		0			0
$451$	−1.21103e9	+	2.09756e9i	−0.621635	+	1.07670i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.23014e9	+	2.67304e8i	−0.612231	+	0.133035i
$456$		0			0
$457$	3.05260e8	+	5.28726e8i	0.149611	+	0.259134i	0.931084	−	0.364806i	$-0.118865\pi$
$457$	3.05260e8	+	5.28726e8i	0.149611	+	0.259134i	−0.781473	+	0.623939i	$0.785531\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.46019e9			0.694155			0.347078	−	0.937836i	$-0.387174\pi$
$461$	1.46019e9			0.694155			0.347078	+	0.937836i	$0.387174\pi$
$462$		0			0
$463$	−6.99630e7			−0.0327593			−0.0163797	−	0.999866i	$-0.505214\pi$
$463$	−6.99630e7			−0.0327593			−0.0163797	+	0.999866i	$0.505214\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.79060e9	+	3.10141e9i	0.813559	+	1.40913i	0.910358	+	0.413822i	$0.135806\pi$
$467$	1.79060e9	+	3.10141e9i	0.813559	+	1.40913i	−0.0967986	+	0.995304i	$0.530860\pi$
$468$		0			0
$469$	−1.36234e9	−	1.49932e9i	−0.609790	−	0.671101i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.21518e9	−	3.83680e9i	0.962486	−	1.66708i
$474$		0			0
$475$	9.54361e8			0.408588
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.65832e7	−	1.67287e8i	0.0401539	−	0.0695485i	−0.845250	−	0.534371i	$-0.820549\pi$
$479$	9.65832e7	−	1.67287e8i	0.0401539	−	0.0695485i	0.885404	+	0.464822i	$0.153882\pi$
$480$		0			0
$481$	−9.07178e8	−	1.57128e9i	−0.371693	−	0.643792i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.64465e8	−	1.49730e9i	−0.344074	−	0.595953i
$486$		0			0
$487$	1.79836e8	−	3.11486e8i	0.0705547	−	0.122204i	−0.828590	−	0.559856i	$-0.810856\pi$
$487$	1.79836e8	−	3.11486e8i	0.0705547	−	0.122204i	0.899145	+	0.437652i	$0.144190\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.79812e9			1.06680			0.533398	−	0.845865i	$-0.320915\pi$
$491$	2.79812e9			1.06680			0.533398	+	0.845865i	$0.320915\pi$
$492$		0			0
$493$	4.18645e8	−	7.25114e8i	0.157355	−	0.272548i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.15154e9	−	3.59946e9i	0.420756	−	1.31520i
$498$		0			0
$499$	−1.02578e8	−	1.77670e8i	−0.0369573	−	0.0640120i	0.846955	−	0.531664i	$-0.178433\pi$
$499$	−1.02578e8	−	1.77670e8i	−0.0369573	−	0.0640120i	−0.883912	+	0.467652i	$0.845100\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.67473e9			0.937114			0.468557	−	0.883433i	$-0.344774\pi$
$503$	2.67473e9			0.937114			0.468557	+	0.883433i	$0.344774\pi$
$504$		0			0
$505$	−3.31143e8			−0.114418
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.49115e9	−	4.31481e9i	−0.837314	−	1.45027i	−0.892132	−	0.451774i	$-0.850791\pi$
$509$	−2.49115e9	−	4.31481e9i	−0.837314	−	1.45027i	0.0548181	−	0.998496i	$-0.482542\pi$
$510$		0			0
$511$	1.97370e9	+	2.17215e9i	0.654346	+	0.720138i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.46486e7	−	6.00131e7i	0.0111779	−	0.0193607i
$516$		0			0
$517$	6.93255e8			0.220636
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.19617e8	+	1.59282e9i	−0.284888	+	0.493441i	−0.972582	−	0.232560i	$-0.925290\pi$
$521$	−9.19617e8	+	1.59282e9i	−0.284888	+	0.493441i	0.687694	+	0.726001i	$0.258623\pi$
$522$		0			0
$523$	2.73502e8	+	4.73720e8i	0.0835997	+	0.144799i	0.904794	−	0.425850i	$-0.140025\pi$
$523$	2.73502e8	+	4.73720e8i	0.0835997	+	0.144799i	−0.821194	+	0.570649i	$0.806692\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.25720e8	+	1.08378e9i	0.186227	+	0.322555i
$528$		0			0
$529$	1.18174e9	−	2.04684e9i	0.347079	−	0.601158i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.23827e9			−1.21239
$534$		0			0
$535$	−1.15553e9	+	2.00143e9i	−0.326243	+	0.565070i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.10596e9	−	2.32898e9i	1.40448	−	0.640626i
$540$		0			0
$541$	−3.54288e9	−	6.13644e9i	−0.961979	−	1.66620i	−0.717522	−	0.696536i	$-0.754724\pi$
$541$	−3.54288e9	−	6.13644e9i	−0.961979	−	1.66620i	−0.244457	−	0.969660i	$-0.578610\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.14177e8			−0.241904
$546$		0			0
$547$	−6.74238e9			−1.76140			−0.880700	−	0.473675i	$-0.842927\pi$
$547$	−6.74238e9			−1.76140			−0.880700	+	0.473675i	$0.842927\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9.36685e8	+	1.62239e9i	0.238541	+	0.413165i
$552$		0			0
$553$	−1.21463e9	+	2.63934e8i	−0.305426	+	0.0663678i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.68771e9	+	6.38730e9i	−0.904198	+	1.56612i	−0.0822077	+	0.996615i	$0.526197\pi$
$557$	−3.68771e9	+	6.38730e9i	−0.904198	+	1.56612i	−0.821990	+	0.569502i	$0.807136\pi$
$558$		0			0
$559$	7.75253e9			1.87716
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.19279e9	+	5.53008e9i	−0.754034	+	1.30603i	0.191818	+	0.981430i	$0.438562\pi$
$563$	−3.19279e9	+	5.53008e9i	−0.754034	+	1.30603i	−0.945853	+	0.324596i	$0.894772\pi$
$564$		0			0
$565$	−6.76556e8	−	1.17183e9i	−0.157810	−	0.273335i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00015e9	−	5.19642e9i	−0.682733	−	1.18253i	−0.974144	−	0.225929i	$-0.927458\pi$
$569$	−3.00015e9	−	5.19642e9i	−0.682733	−	1.18253i	0.291411	−	0.956598i	$-0.405875\pi$
$570$		0			0
$571$	−1.47621e9	+	2.55687e9i	−0.331835	+	0.574755i	−0.982872	−	0.184291i	$-0.941001\pi$
$571$	−1.47621e9	+	2.55687e9i	−0.331835	+	0.574755i	0.651037	+	0.759046i	$0.274334\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.08438e9			−0.457235
$576$		0			0
$577$	−4.88420e8	+	8.45969e8i	−0.105847	+	0.183332i	−0.914084	−	0.405525i	$-0.867089\pi$
$577$	−4.88420e8	+	8.45969e8i	−0.105847	+	0.183332i	0.808237	+	0.588857i	$0.200422\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.25272e9	+	7.04154e9i	−0.476531	+	1.48954i
$582$		0			0
$583$	6.96567e9	+	1.20649e10i	1.45587	+	2.52164i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.30141e9			−0.265572			−0.132786	−	0.991145i	$-0.542392\pi$
$587$	−1.30141e9			−0.265572			−0.132786	+	0.991145i	$0.542392\pi$
$588$		0			0
$589$	−2.80000e9			−0.564618
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.46216e8	+	2.53254e8i	0.0287942	+	0.0498730i	0.880063	−	0.474856i	$-0.157500\pi$
$593$	1.46216e8	+	2.53254e8i	0.0287942	+	0.0498730i	−0.851269	+	0.524729i	$0.824167\pi$
$594$		0			0
$595$	2.12421e8	−	6.63985e8i	0.0413417	−	0.129226i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.08228e8	−	1.05348e9i	0.115631	−	0.200278i	−0.802401	−	0.596785i	$-0.796444\pi$
$599$	6.08228e8	−	1.05348e9i	0.115631	−	0.200278i	0.918032	+	0.396507i	$0.129778\pi$
$600$		0			0
$601$	8.70073e9			1.63491			0.817457	−	0.575990i	$-0.195383\pi$
$601$	8.70073e9			1.63491			0.817457	+	0.575990i	$0.195383\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.56756e9	+	2.71510e9i	−0.287794	+	0.498473i
$606$		0			0
$607$	−1.73761e9	−	3.00963e9i	−0.315350	−	0.546202i	0.664162	−	0.747589i	$-0.268788\pi$
$607$	−1.73761e9	−	3.00963e9i	−0.315350	−	0.546202i	−0.979512	+	0.201387i	$0.935455\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.06552e8	+	1.05058e9i	0.107578	+	0.186331i
$612$		0			0
$613$	−9.58454e8	+	1.66009e9i	−0.168058	+	0.291085i	−0.937737	−	0.347346i	$-0.887083\pi$
$613$	−9.58454e8	+	1.66009e9i	−0.168058	+	0.291085i	0.769679	+	0.638431i	$0.220416\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.94935e9			1.70528			0.852642	−	0.522496i	$-0.174999\pi$
$617$	9.94935e9			1.70528			0.852642	+	0.522496i	$0.174999\pi$
$618$		0			0
$619$	5.68330e9	−	9.84377e9i	0.963127	−	1.66818i	0.248562	−	0.968616i	$-0.420042\pi$
$619$	5.68330e9	−	9.84377e9i	0.963127	−	1.66818i	0.714565	−	0.699569i	$-0.246625\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.15333e9	−	6.85206e8i	0.522470	−	0.113531i
$624$		0			0
$625$	−1.55747e9	−	2.69762e9i	−0.255176	−	0.441977i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.00477e9			0.160987
$630$		0			0
$631$	7.79638e9			1.23535			0.617676	−	0.786433i	$-0.288075\pi$
$631$	7.79638e9			1.23535			0.617676	+	0.786433i	$0.288075\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.93966e7	−	1.02878e8i	−0.00920563	−	0.0159446i
$636$		0			0
$637$	7.99677e9	+	5.70002e9i	1.22582	+	0.873753i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.82020e9	+	1.00809e10i	−0.872841	+	1.51181i	−0.0137961	+	0.999905i	$0.504392\pi$
$641$	−5.82020e9	+	1.00809e10i	−0.872841	+	1.51181i	−0.859045	+	0.511900i	$0.828942\pi$
$642$		0			0
$643$	−9.21864e9			−1.36750			−0.683752	−	0.729714i	$-0.739653\pi$
$643$	−9.21864e9			−1.36750			−0.683752	+	0.729714i	$0.739653\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.05598e9	+	8.75721e9i	−0.733905	+	1.27116i	0.221297	+	0.975206i	$0.428971\pi$
$647$	−5.05598e9	+	8.75721e9i	−0.733905	+	1.27116i	−0.955202	+	0.295955i	$0.904362\pi$
$648$		0			0
$649$	−6.83877e8	−	1.18451e9i	−0.0982023	−	0.170091i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.45809e9	−	1.11857e10i	−0.907629	−	1.57206i	−0.817349	−	0.576142i	$-0.804557\pi$
$653$	−6.45809e9	−	1.11857e10i	−0.907629	−	1.57206i	−0.0902794	−	0.995916i	$-0.528776\pi$
$654$		0			0
$655$	−2.81185e9	+	4.87026e9i	−0.390973	+	0.677186i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.30898e9			0.178170			0.0890848	−	0.996024i	$-0.471606\pi$
$659$	1.30898e9			0.178170			0.0890848	+	0.996024i	$0.471606\pi$
$660$		0			0
$661$	−5.09754e9	+	8.82919e9i	−0.686523	+	1.18909i	0.286432	+	0.958100i	$0.407531\pi$
$661$	−5.09754e9	+	8.82919e9i	−0.686523	+	1.18909i	−0.972956	+	0.230993i	$0.925803\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.04894e9	+	1.15441e9i	0.138317	+	0.152224i
$666$		0			0
$667$	−2.04578e9	−	3.54339e9i	−0.266943	−	0.462358i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.18731e9			0.790629
$672$		0			0
$673$	−1.13920e10			−1.44061			−0.720305	−	0.693658i	$-0.755998\pi$
$673$	−1.13920e10			−1.44061			−0.720305	+	0.693658i	$0.755998\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.89269e9	+	8.47439e9i	0.606021	+	1.04966i	0.991889	+	0.127105i	$0.0405686\pi$
$677$	4.89269e9	+	8.47439e9i	0.606021	+	1.04966i	−0.385868	+	0.922554i	$0.626098\pi$
$678$		0			0
$679$	−4.10970e9	+	1.28461e10i	−0.503809	+	1.57480i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3.21469e9	−	5.56801e9i	0.386071	−	0.668694i	−0.605847	−	0.795582i	$-0.707165\pi$
$683$	3.21469e9	−	5.56801e9i	0.386071	−	0.668694i	0.991917	+	0.126888i	$0.0404988\pi$
$684$		0			0
$685$	−2.88299e9			−0.342710
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.21890e10	+	2.11119e10i	−1.41971	+	2.45901i
$690$		0			0
$691$	8.28687e7	+	1.43533e8i	0.00955471	+	0.0165492i	0.870763	−	0.491703i	$-0.163625\pi$
$691$	8.28687e7	+	1.43533e8i	0.00955471	+	0.0165492i	−0.861209	+	0.508252i	$0.830292\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.12658e9	+	1.95129e9i	0.127296	+	0.220483i
$696$		0			0
$697$	1.17355e9	−	2.03265e9i	0.131277	−	0.227378i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.81761e9			−0.308935			−0.154468	−	0.987998i	$-0.549366\pi$
$701$	−2.81761e9			−0.308935			−0.154468	+	0.987998i	$0.549366\pi$
$702$		0			0
$703$	−1.12405e9	+	1.94691e9i	−0.122023	+	0.211350i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.73721e9	+	1.91188e9i	0.184878	+	0.203467i
$708$		0			0
$709$	−3.67090e9	−	6.35819e9i	−0.386822	−	0.669995i	0.605198	−	0.796075i	$-0.293094\pi$
$709$	−3.67090e9	−	6.35819e9i	−0.386822	−	0.669995i	−0.992020	+	0.126080i	$0.959761\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.11537e9			0.631843
$714$		0			0
$715$	−9.45289e9			−0.967149
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.73646e8	−	1.68640e9i	−0.0976899	−	0.169204i	0.813038	−	0.582210i	$-0.197812\pi$
$719$	−9.73646e8	−	1.68640e9i	−0.0976899	−	0.169204i	−0.910728	+	0.413006i	$0.864479\pi$
$720$		0			0
$721$	−5.28262e8	+	1.14789e8i	−0.0524899	+	0.0114058i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.09490e9	−	7.09258e9i	0.399080	−	0.691228i
$726$		0			0
$727$	8.55516e9			0.825768			0.412884	−	0.910784i	$-0.364522\pi$
$727$	8.55516e9			0.825768			0.412884	+	0.910784i	$0.364522\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.14663e9	+	3.71808e9i	−0.203258	+	0.352053i
$732$		0			0
$733$	5.23062e9	+	9.05970e9i	0.490557	+	0.849669i	0.999941	−	0.0108699i	$-0.00346006\pi$
$733$	5.23062e9	+	9.05970e9i	0.490557	+	0.849669i	−0.509384	+	0.860539i	$0.670127\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.60607e9	−	1.31741e10i	−0.699881	−	1.21223i
$738$		0			0
$739$	6.64787e8	−	1.15144e9i	0.0605936	−	0.104951i	−0.834137	−	0.551557i	$-0.814034\pi$
$739$	6.64787e8	−	1.15144e9i	0.0605936	−	0.104951i	0.894731	+	0.446606i	$0.147367\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.68408e9			0.150627			0.0753133	−	0.997160i	$-0.476004\pi$
$743$	1.68408e9			0.150627			0.0753133	+	0.997160i	$0.476004\pi$
$744$		0			0
$745$	9.92708e8	−	1.71942e9i	0.0879579	−	0.152348i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.76175e10	−	3.82820e9i	1.53199	−	0.332896i
$750$		0			0
$751$	−2.22383e9	−	3.85178e9i	−0.191585	−	0.331835i	0.754191	−	0.656655i	$-0.228029\pi$
$751$	−2.22383e9	−	3.85178e9i	−0.191585	−	0.331835i	−0.945776	+	0.324821i	$0.894696\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.05307e9			0.258179
$756$		0			0
$757$	−7.99647e9			−0.669981			−0.334991	−	0.942221i	$-0.608733\pi$
$757$	−7.99647e9			−0.669981			−0.334991	+	0.942221i	$0.608733\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.22630e9	−	7.32016e9i	−0.347627	−	0.602108i	0.638200	−	0.769870i	$-0.279679\pi$
$761$	−4.22630e9	−	7.32016e9i	−0.347627	−	0.602108i	−0.985827	+	0.167762i	$0.946346\pi$
$762$		0			0
$763$	4.79588e9	+	5.27808e9i	0.390870	+	0.430170i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.19669e9	−	2.07273e9i	0.0957633	−	0.165867i
$768$		0			0
$769$	4.93932e9			0.391674			0.195837	−	0.980636i	$-0.437258\pi$
$769$	4.93932e9			0.391674			0.195837	+	0.980636i	$0.437258\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.76862e9	+	4.79539e9i	−0.215593	+	0.373418i	−0.953456	−	0.301532i	$-0.902502\pi$
$773$	−2.76862e9	+	4.79539e9i	−0.215593	+	0.373418i	0.737863	+	0.674951i	$0.235835\pi$
$774$		0			0
$775$	6.12037e9	+	1.06008e10i	0.472304	+	0.818055i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.62573e9	+	4.54790e9i	0.199008	+	0.344691i
$780$		0			0
$781$	1.41892e10	−	2.45764e10i	1.06581	−	1.84604i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.76879e9			−0.351855
$786$		0			0
$787$	−2.39855e9	+	4.15440e9i	−0.175403	+	0.303806i	−0.940301	−	0.340345i	$-0.889456\pi$
$787$	−2.39855e9	+	4.15440e9i	−0.175403	+	0.303806i	0.764898	+	0.644152i	$0.222789\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.21637e9	+	1.00537e10i	−0.231073	+	0.722285i
$792$		0			0
$793$	5.41348e9	+	9.37642e9i	0.385496	+	0.667699i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.15558e9			−0.0808526			−0.0404263	−	0.999183i	$-0.512872\pi$
$797$	−1.15558e9			−0.0808526			−0.0404263	+	0.999183i	$0.512872\pi$
$798$		0			0
$799$	−6.71804e8			−0.0465939
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.10194e10	+	1.90861e10i	0.751021	+	1.30081i
$804$		0			0
$805$	−2.29095e9	−	2.52130e9i	−0.154786	−	0.170349i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.19483e10	−	2.06951e10i	0.793390	−	1.37419i	−0.130467	−	0.991453i	$-0.541648\pi$
$809$	1.19483e10	−	2.06951e10i	0.793390	−	1.37419i	0.923857	−	0.382739i	$-0.125019\pi$
$810$		0			0
$811$	−1.09421e10			−0.720326			−0.360163	−	0.932889i	$-0.617279\pi$
$811$	−1.09421e10			−0.720326			−0.360163	+	0.932889i	$0.617279\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.03884e9	+	5.26343e9i	−0.196633	+	0.340579i
$816$		0			0
$817$	−4.80292e9	−	8.31891e9i	−0.308126	−	0.533690i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.11318e10	−	1.92808e10i	−0.702043	−	1.21597i	−0.967748	−	0.251920i	$-0.918938\pi$
$821$	−1.11318e10	−	1.92808e10i	−0.702043	−	1.21597i	0.265705	−	0.964054i	$-0.414395\pi$
$822$		0			0
$823$	1.83529e9	−	3.17881e9i	0.114764	−	0.198777i	−0.802921	−	0.596085i	$-0.796722\pi$
$823$	1.83529e9	−	3.17881e9i	0.114764	−	0.198777i	0.917685	+	0.397308i	$0.130056\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8.88318e9			−0.546134			−0.273067	−	0.961995i	$-0.588038\pi$
$827$	−8.88318e9			−0.546134			−0.273067	+	0.961995i	$0.588038\pi$
$828$		0			0
$829$	−6.36202e9	+	1.10193e10i	−0.387842	+	0.671761i	−0.992159	−	0.124982i	$-0.960113\pi$
$829$	−6.36202e9	+	1.10193e10i	−0.387842	+	0.671761i	0.604317	+	0.796744i	$0.293446\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.94797e9	+	2.25691e9i	−0.296599	+	0.135287i
$834$		0			0
$835$	−3.29951e9	−	5.71491e9i	−0.196131	−	0.339709i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.40186e9			−0.140404			−0.0702022	−	0.997533i	$-0.522364\pi$
$839$	−2.40186e9			−0.140404			−0.0702022	+	0.997533i	$0.522364\pi$
$840$		0			0
$841$	−1.17364e9			−0.0680375
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.62089e9	−	8.00361e9i	−0.263467	−	0.456339i
$846$		0			0
$847$	2.38995e10	−	5.19326e9i	1.35144	−	0.293662i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.45499e9	−	4.25216e9i	0.136551	−	0.236514i
$852$		0			0
$853$	2.38086e10			1.31344			0.656722	−	0.754133i	$-0.271942\pi$
$853$	2.38086e10			1.31344			0.656722	+	0.754133i	$0.271942\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.45036e8	+	1.63685e9i	−0.0512880	+	0.0888334i	−0.890530	−	0.454925i	$-0.849666\pi$
$857$	−9.45036e8	+	1.63685e9i	−0.0512880	+	0.0888334i	0.839242	+	0.543759i	$0.182999\pi$
$858$		0			0
$859$	−1.62617e10	−	2.81661e10i	−0.875368	−	1.51618i	−0.856371	−	0.516362i	$-0.827286\pi$
$859$	−1.62617e10	−	2.81661e10i	−0.875368	−	1.51618i	−0.0189969	−	0.999820i	$-0.506047\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.57375e10	−	2.72581e10i	−0.833483	−	1.44364i	−0.895259	−	0.445545i	$-0.853010\pi$
$863$	−1.57375e10	−	2.72581e10i	−0.833483	−	1.44364i	0.0617761	−	0.998090i	$-0.480324\pi$
$864$		0			0
$865$	−9.00818e7	+	1.56026e8i	−0.00473239	+	0.00819675i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.33370e9			−0.482486
$870$		0			0
$871$	1.33096e10	−	2.30529e10i	0.682498	−	1.18212i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.59083e9	−	1.43500e10i	0.231666	−	0.724142i
$876$		0			0
$877$	−4.66652e9	−	8.08264e9i	−0.233612	−	0.404627i	0.725257	−	0.688479i	$-0.241721\pi$
$877$	−4.66652e9	−	8.08264e9i	−0.233612	−	0.404627i	−0.958868	+	0.283851i	$0.908388\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.08746e10			1.02850			0.514248	−	0.857641i	$-0.328071\pi$
$881$	2.08746e10			1.02850			0.514248	+	0.857641i	$0.328071\pi$
$882$		0			0
$883$	2.12927e10			1.04080			0.520402	−	0.853922i	$-0.325782\pi$
$883$	2.12927e10			1.04080			0.520402	+	0.853922i	$0.325782\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.74469e10	−	3.02188e10i	−0.839430	−	1.45394i	−0.890372	−	0.455234i	$-0.849556\pi$
$887$	−1.74469e10	−	3.02188e10i	−0.839430	−	1.45394i	0.0509420	−	0.998702i	$-0.483778\pi$
$888$		0			0
$889$	−2.82374e8	+	8.82642e8i	−0.0134793	+	0.0421336i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.51554e8	−	1.30173e9i	0.0353167	−	0.0611703i
$894$		0			0
$895$	−4.35127e9			−0.202878
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.20140e10	+	2.08089e10i	−0.551480	+	0.955191i
$900$		0			0
$901$	−6.75013e9	−	1.16916e10i	−0.307450	−	0.532520i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.28601e9	+	1.08877e10i	0.281906	+	0.488276i
$906$		0			0
$907$	1.66490e9	−	2.88370e9i	0.0740907	−	0.128329i	−0.826600	−	0.562790i	$-0.809728\pi$
$907$	1.66490e9	−	2.88370e9i	0.0740907	−	0.128329i	0.900691	+	0.434461i	$0.143061\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.05647e10			−1.77760			−0.888799	−	0.458297i	$-0.848460\pi$
$911$	−4.05647e10			−1.77760			−0.888799	+	0.458297i	$0.848460\pi$
$912$		0			0
$913$	−2.77580e10	+	4.80783e10i	−1.20709	+	2.09075i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.28702e10	−	9.31551e9i	1.83596	−	0.398946i
$918$		0			0
$919$	−1.14032e10	−	1.97509e10i	−0.484644	−	0.839428i	0.515200	−	0.857070i	$-0.327717\pi$
$919$	−1.14032e10	−	1.97509e10i	−0.484644	−	0.839428i	−0.999844	+	0.0176418i	$0.994384\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.96585e10			2.07868
$924$		0			0
$925$	9.82798e9			0.408290
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.60242e10	−	2.77547e10i	−0.655724	−	1.13575i	−0.981712	−	0.190374i	$-0.939030\pi$
$929$	−1.60242e10	−	2.77547e10i	−0.655724	−	1.13575i	0.325987	−	0.945374i	$-0.394303\pi$
$930$		0			0
$931$	1.16221e9	−	1.21123e10i	0.0472021	−	0.491931i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.61745e9	−	4.53356e9i	0.104722	−	0.181384i
$936$		0			0
$937$	2.73774e10			1.08718			0.543592	−	0.839350i	$-0.317064\pi$
$937$	2.73774e10			1.08718			0.543592	+	0.839350i	$0.317064\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.29951e9	+	7.44697e9i	−0.168211	+	0.291350i	−0.937791	−	0.347200i	$-0.887132\pi$
$941$	−4.29951e9	+	7.44697e9i	−0.168211	+	0.291350i	0.769580	+	0.638551i	$0.220466\pi$
$942$		0			0
$943$	−5.73476e9	−	9.93289e9i	−0.222702	−	0.385731i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.99474e10	−	3.45500e10i	−0.763242	−	1.32197i	−0.941171	−	0.337930i	$-0.890273\pi$
$947$	−1.99474e10	−	3.45500e10i	−0.763242	−	1.32197i	0.177930	−	0.984043i	$-0.443060\pi$
$948$		0			0
$949$	−1.92824e10	+	3.33981e10i	−0.732368	+	1.26850i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.66469e9			0.286860			0.143430	−	0.989660i	$-0.454187\pi$
$953$	7.66469e9			0.286860			0.143430	+	0.989660i	$0.454187\pi$
$954$		0			0
$955$	−6.70944e9	+	1.16211e10i	−0.249272	+	0.431753i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.51245e10	+	1.66452e10i	0.553753	+	0.609431i
$960$		0			0
$961$	−4.20025e9	−	7.27504e9i	−0.152666	−	0.264426i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.55404e9			−0.342249
$966$		0			0
$967$	−5.31576e10			−1.89048			−0.945241	−	0.326373i	$-0.894174\pi$
$967$	−5.31576e10			−1.89048			−0.945241	+	0.326373i	$0.894174\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.20926e9	+	7.29066e9i	0.147550	+	0.255564i	0.930321	−	0.366745i	$-0.119528\pi$
$971$	4.20926e9	+	7.29066e9i	0.147550	+	0.255564i	−0.782772	+	0.622309i	$0.786195\pi$
$972$		0			0
$973$	5.35579e9	−	1.67411e10i	0.186393	−	0.582624i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.60872e9	−	1.31787e10i	0.261024	−	0.452107i	−0.705490	−	0.708720i	$-0.749273\pi$
$977$	7.60872e9	−	1.31787e10i	0.261024	−	0.452107i	0.966514	+	0.256613i	$0.0826065\pi$
$978$		0			0
$979$	2.42314e10			0.825353
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.96526e10	−	3.40393e10i	0.659907	−	1.14299i	−0.320733	−	0.947170i	$-0.603929\pi$
$983$	1.96526e10	−	3.40393e10i	0.659907	−	1.14299i	0.980639	−	0.195822i	$-0.0627375\pi$
$984$		0			0
$985$	7.69678e9	+	1.33312e10i	0.256615	+	0.444470i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.04899e10	+	1.81690e10i	0.344813	+	0.597233i
$990$		0			0
$991$	4.22919e9	−	7.32516e9i	0.138038	−	0.239089i	−0.788716	−	0.614758i	$-0.789254\pi$
$991$	4.22919e9	−	7.32516e9i	0.138038	−	0.239089i	0.926754	+	0.375669i	$0.122587\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.62970e9			0.277725
$996$		0			0
$997$	−4.10207e8	+	7.10500e8i	−0.0131090	+	0.0227055i	−0.872505	−	0.488604i	$-0.837506\pi$
$997$	−4.10207e8	+	7.10500e8i	−0.0131090	+	0.0227055i	0.859396	+	0.511310i	$0.170840\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.k.c.109.3		10
3.2	odd	2		28.8.e.a.25.1	yes	10
7.2	even	3	inner	252.8.k.c.37.3		10
12.11	even	2		112.8.i.d.81.5		10
21.2	odd	6		28.8.e.a.9.1	✓	10
21.5	even	6		196.8.e.f.177.5		10
21.11	odd	6		196.8.a.d.1.5		5
21.17	even	6		196.8.a.e.1.1		5
21.20	even	2		196.8.e.f.165.5		10
84.23	even	6		112.8.i.d.65.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.e.a.9.1	✓	10	21.2	odd	6
28.8.e.a.25.1	yes	10	3.2	odd	2
112.8.i.d.65.5		10	84.23	even	6
112.8.i.d.81.5		10	12.11	even	2
196.8.a.d.1.5		5	21.11	odd	6
196.8.a.e.1.1		5	21.17	even	6
196.8.e.f.165.5		10	21.20	even	2
196.8.e.f.177.5		10	21.5	even	6
252.8.k.c.37.3		10	7.2	even	3	inner
252.8.k.c.109.3		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-58.1649 - 100.745i) q^{5} +(-276.518 + 864.338i) q^{7} +(-3407.25 + 5901.54i) q^{11} -11924.5 q^{13} +(3301.82 - 5718.92i) q^{17} +(7387.57 + 12795.6i) q^{19} +(-16134.9 - 27946.4i) q^{23} +(32296.2 - 55938.6i) q^{25} +126792. q^{29} +(-94753.8 + 164118. i) q^{31} +(103161. - 22416.5i) q^{35} +(76077.0 + 131769. i) q^{37} +355426. q^{41} -650136. q^{43} +(-50866.2 - 88102.8i) q^{47} +(-670618. - 478010. i) q^{49} +(1.02218e6 - 1.77047e6i) q^{53} +792730. q^{55} +(-100356. + 173822. i) q^{59} +(-453981. - 786317. i) q^{61} +(693586. + 1.20133e6i) q^{65} +(-1.11616e6 + 1.93324e6i) q^{67} -4.16441e6 q^{71} +(1.61704e6 - 2.80080e6i) q^{73} +(-4.15876e6 - 4.57690e6i) q^{77} +(684840. + 1.18618e6i) q^{79} +8.14674e6 q^{83} -768201. q^{85} +(-1.77793e6 - 3.07947e6i) q^{89} +(3.29733e6 - 1.03068e7i) q^{91} +(859394. - 1.48851e6i) q^{95} +1.48623e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 249 q^{5} + 332 q^{7} - 6399 q^{11} - 26988 q^{13} - 3609 q^{17} - 12403 q^{19} + 13959 q^{23} - 162364 q^{25} - 26148 q^{29} - 20181 q^{31} - 791715 q^{35} - 54763 q^{37} + 1824468 q^{41} - 1938424 q^{43}+ \cdots + 285580 q^{97}+O(q^{100}) \) 10 * q - 249 * q^5 + 332 * q^7 - 6399 * q^11 - 26988 * q^13 - 3609 * q^17 - 12403 * q^19 + 13959 * q^23 - 162364 * q^25 - 26148 * q^29 - 20181 * q^31 - 791715 * q^35 - 54763 * q^37 + 1824468 * q^41 - 1938424 * q^43 - 1217253 * q^47 + 1722058 * q^49 + 2349159 * q^53 + 3123494 * q^55 - 2188611 * q^59 + 6107445 * q^61 + 3022014 * q^65 - 3171581 * q^67 + 18938976 * q^71 + 8034853 * q^73 - 24100191 * q^77 - 7589559 * q^79 + 13452408 * q^83 - 48717670 * q^85 - 29864757 * q^89 + 34267512 * q^91 + 25812009 * q^95 + 285580 * q^97

Introduction

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