LMFDB - Embedded newform 252.8.k.e.109.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(37,252)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 89566 x^{14} - 207320 x^{13} + 5161603375 x^{12} - 17143558340 x^{11} + 178819626045814 x^{10} + \cdots + 34\!\cdots\!16 $ x^16 + 89566x^14 - 207320x^13 + 5161603375x^12 - 17143558340x^11 + 178819626045814x^10 - 993183560713460x^9 + 4531523101103784409x^8 - 25038421018919762700x^7 + 77467785793463920028824x^6 - 352418565495722251364640x^5 + 967025470523891143640719680x^4 - 1651641278575894088053178880x^3 + 7200793191683961412674394484736x^2 + 21939035854717669821876654243840x + 34494438285389383231614325978300416
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{13}\cdot 7^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.7
Root			$68.2530 + 118.218i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.8.k.e.37.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(161.248 + 279.290i) q^{5} +(-882.807 - 210.227i) q^{7} +(-43.6956 + 75.6830i) q^{11} -9602.23 q^{13} +(16188.6 - 28039.5i) q^{17} +(-477.164 - 826.473i) q^{19} +(-4780.44 - 8279.97i) q^{23} +(-12939.6 + 22412.0i) q^{25} +210126. q^{29} +(20813.0 - 36049.2i) q^{31} +(-83636.7 - 280458. i) q^{35} +(-13999.4 - 24247.7i) q^{37} -296416. q^{41} +496225. q^{43} +(308439. + 534233. i) q^{47} +(735152. + 371180. i) q^{49} +(-299978. + 519577. i) q^{53} -28183.4 q^{55} +(-1.20550e6 + 2.08799e6i) q^{59} +(-539414. - 934292. i) q^{61} +(-1.54834e6 - 2.68181e6i) q^{65} +(-1.37198e6 + 2.37633e6i) q^{67} +2.21738e6 q^{71} +(1.33265e6 - 2.30822e6i) q^{73} +(54485.4 - 57627.4i) q^{77} +(1.44755e6 + 2.50723e6i) q^{79} +3.60576e6 q^{83} +1.04415e7 q^{85} +(-466027. - 807183. i) q^{89} +(8.47691e6 + 2.01865e6i) q^{91} +(153884. - 266535. i) q^{95} +1.21978e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 1680 q^{7} - 28280 q^{13} + 42224 q^{19} - 80460 q^{25} + 164752 q^{31} - 647980 q^{37} + 1341440 q^{43} + 230104 q^{49} - 323120 q^{55} - 4319336 q^{61} - 3905760 q^{67} + 6471780 q^{73} - 6093104 q^{79}+ \cdots - 27141240 q^{97}+O(q^{100}) $ 16 * q + 1680 * q^7 - 28280 * q^13 + 42224 * q^19 - 80460 * q^25 + 164752 * q^31 - 647980 * q^37 + 1341440 * q^43 + 230104 * q^49 - 323120 * q^55 - 4319336 * q^61 - 3905760 * q^67 + 6471780 * q^73 - 6093104 * q^79 + 456400 * q^85 + 15969856 * q^91 - 27141240 * 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
$5$	161.248	+	279.290i	0.576900	+	0.999220i	0.995832	+	0.0912024i	$0.0290710\pi$
$5$	161.248	+	279.290i	0.576900	+	0.999220i	−0.418933	+	0.908017i	$0.637596\pi$
$6$		0			0
$7$	−882.807	−	210.227i	−0.972797	−	0.231657i
$7$	−882.807	−	210.227i	−0.972797	−	0.231657i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−43.6956	+	75.6830i	−0.00989836	+	0.0171445i	−0.870932	−	0.491403i	$-0.836484\pi$
$11$	−43.6956	+	75.6830i	−0.00989836	+	0.0171445i	0.861034	+	0.508548i	$0.169817\pi$
$12$		0			0
$13$	−9602.23			−1.21219			−0.606095	−	0.795392i	$-0.707265\pi$
$13$	−9602.23			−1.21219			−0.606095	+	0.795392i	$0.707265\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	16188.6	−	28039.5i	0.799168	−	1.38420i	−0.120990	−	0.992654i	$-0.538607\pi$
$17$	16188.6	−	28039.5i	0.799168	−	1.38420i	0.920158	−	0.391546i	$-0.128060\pi$
$18$		0			0
$19$	−477.164	−	826.473i	−0.0159599	−	0.0276434i	0.857935	−	0.513758i	$-0.171747\pi$
$19$	−477.164	−	826.473i	−0.0159599	−	0.0276434i	−0.873895	+	0.486115i	$0.838414\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4780.44	−	8279.97i	−0.0819258	−	0.141900i	0.822151	−	0.569269i	$-0.192774\pi$
$23$	−4780.44	−	8279.97i	−0.0819258	−	0.141900i	−0.904077	+	0.427369i	$0.859440\pi$
$24$		0			0
$25$	−12939.6	+	22412.0i	−0.165627	+	0.286874i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	210126.			1.59988			0.799938	−	0.600083i	$-0.204866\pi$
$29$	210126.			1.59988			0.799938	+	0.600083i	$0.204866\pi$
$30$		0			0
$31$	20813.0	−	36049.2i	0.125478	−	0.217335i	−0.796441	−	0.604716i	$-0.793287\pi$
$31$	20813.0	−	36049.2i	0.125478	−	0.217335i	0.921920	+	0.387381i	$0.126620\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−83636.7	−	280458.i	−0.329730	−	1.10568i
$36$		0			0
$37$	−13999.4	−	24247.7i	−0.0454363	−	0.0786980i	0.842413	−	0.538833i	$-0.181134\pi$
$37$	−13999.4	−	24247.7i	−0.0454363	−	0.0786980i	−0.887849	+	0.460135i	$0.847801\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−296416.			−0.671674			−0.335837	−	0.941920i	$-0.609019\pi$
$41$	−296416.			−0.671674			−0.335837	+	0.941920i	$0.609019\pi$
$42$		0			0
$43$	496225.			0.951784			0.475892	−	0.879504i	$-0.342125\pi$
$43$	496225.			0.951784			0.475892	+	0.879504i	$0.342125\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	308439.	+	534233.i	0.433339	+	0.750564i	0.997158	−	0.0753335i	$-0.0240021\pi$
$47$	308439.	+	534233.i	0.433339	+	0.750564i	−0.563820	+	0.825898i	$0.690669\pi$
$48$		0			0
$49$	735152.	+	371180.i	0.892670	+	0.450711i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−299978.	+	519577.i	−0.276773	+	0.479385i	−0.970581	−	0.240775i	$-0.922598\pi$
$53$	−299978.	+	519577.i	−0.276773	+	0.479385i	0.693808	+	0.720160i	$0.255932\pi$
$54$		0			0
$55$	−28183.4			−0.0228414
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.20550e6	+	2.08799e6i	−0.764161	+	1.32357i	0.176528	+	0.984296i	$0.443513\pi$
$59$	−1.20550e6	+	2.08799e6i	−0.764161	+	1.32357i	−0.940689	+	0.339270i	$0.889820\pi$
$60$		0			0
$61$	−539414.	−	934292.i	−0.304276	−	0.527021i	0.672824	−	0.739803i	$-0.265081\pi$
$61$	−539414.	−	934292.i	−0.304276	−	0.527021i	−0.977100	+	0.212781i	$0.931748\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.54834e6	−	2.68181e6i	−0.699312	−	1.21124i
$66$		0			0
$67$	−1.37198e6	+	2.37633e6i	−0.557294	+	0.965262i	0.440427	+	0.897789i	$0.354827\pi$
$67$	−1.37198e6	+	2.37633e6i	−0.557294	+	0.965262i	−0.997721	+	0.0674736i	$0.978506\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.21738e6			0.735253			0.367626	−	0.929974i	$-0.380171\pi$
$71$	2.21738e6			0.735253			0.367626	+	0.929974i	$0.380171\pi$
$72$		0			0
$73$	1.33265e6	−	2.30822e6i	0.400947	−	0.694461i	−0.592893	−	0.805281i	$-0.702014\pi$
$73$	1.33265e6	−	2.30822e6i	0.400947	−	0.694461i	0.993840	+	0.110820i	$0.0353477\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	54485.4	−	57627.4i	0.0136007	−	0.0143851i
$78$		0			0
$79$	1.44755e6	+	2.50723e6i	0.330323	+	0.572136i	0.982575	−	0.185866i	$-0.0595090\pi$
$79$	1.44755e6	+	2.50723e6i	0.330323	+	0.572136i	−0.652252	+	0.758002i	$0.726176\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.60576e6			0.692188			0.346094	−	0.938200i	$-0.387508\pi$
$83$	3.60576e6			0.692188			0.346094	+	0.938200i	$0.387508\pi$
$84$		0			0
$85$	1.04415e7			1.84416
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−466027.	−	807183.i	−0.0700723	−	0.121369i	0.828860	−	0.559455i	$-0.188990\pi$
$89$	−466027.	−	807183.i	−0.0700723	−	0.121369i	−0.898933	+	0.438087i	$0.855656\pi$
$90$		0			0
$91$	8.47691e6	+	2.01865e6i	1.17921	+	0.280812i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	153884.	−	266535.i	0.0184145	−	0.0318949i
$96$		0			0
$97$	1.21978e7			1.35700			0.678499	−	0.734601i	$-0.262631\pi$
$97$	1.21978e7			1.35700			0.678499	+	0.734601i	$0.262631\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.43499e6	−	1.63419e7i	0.911206	−	1.57826i	0.0988441	−	0.995103i	$-0.468485\pi$
$101$	9.43499e6	−	1.63419e7i	0.911206	−	1.57826i	0.812362	−	0.583153i	$-0.198181\pi$
$102$		0			0
$103$	4.96997e6	+	8.60824e6i	0.448150	+	0.776219i	0.998266	−	0.0588698i	$-0.0187497\pi$
$103$	4.96997e6	+	8.60824e6i	0.448150	+	0.776219i	−0.550116	+	0.835089i	$0.685416\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.52906e6	+	4.38047e6i	0.199580	+	0.345682i	0.948392	−	0.317100i	$-0.102709\pi$
$107$	2.52906e6	+	4.38047e6i	0.199580	+	0.345682i	−0.748812	+	0.662782i	$0.769376\pi$
$108$		0			0
$109$	3.82903e6	−	6.63207e6i	0.283202	−	0.490520i	−0.688970	−	0.724790i	$-0.741937\pi$
$109$	3.82903e6	−	6.63207e6i	0.283202	−	0.490520i	0.972171	+	0.234270i	$0.0752701\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.32872e7			−0.866284			−0.433142	−	0.901326i	$-0.642595\pi$
$113$	−1.32872e7			−0.866284			−0.433142	+	0.901326i	$0.642595\pi$
$114$		0			0
$115$	1.54168e6	−	2.67026e6i	0.0945259	−	0.163724i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.01861e7	+	2.13502e7i	−1.09809	+	1.16141i
$120$		0			0
$121$	9.73977e6	+	1.68698e7i	0.499804	+	0.865686i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.68491e7			0.771599
$126$		0			0
$127$	2.32145e7			1.00565			0.502824	−	0.864389i	$-0.332295\pi$
$127$	2.32145e7			1.00565			0.502824	+	0.864389i	$0.332295\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.08126e6	+	8.80099e6i	0.197479	+	0.342044i	0.947710	−	0.319131i	$-0.103391\pi$
$131$	5.08126e6	+	8.80099e6i	0.197479	+	0.342044i	−0.750231	+	0.661176i	$0.770058\pi$
$132$		0			0
$133$	247497.	+	829928.i	0.00912197	+	0.0305886i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.08111e7	+	3.60459e7i	−0.691470	+	1.19766i	0.279886	+	0.960033i	$0.409703\pi$
$137$	−2.08111e7	+	3.60459e7i	−0.691470	+	1.19766i	−0.971356	+	0.237629i	$0.923630\pi$
$138$		0			0
$139$	4.76834e7			1.50597			0.752983	−	0.658040i	$-0.228614\pi$
$139$	4.76834e7			1.50597			0.752983	+	0.658040i	$0.228614\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	419575.	−	726725.i	0.0119987	−	0.0207823i
$144$		0			0
$145$	3.38824e7	+	5.86861e7i	0.922968	+	1.59863i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.92202e7	−	6.79315e7i	−0.971311	−	1.68236i	−0.691607	−	0.722274i	$-0.743097\pi$
$149$	−3.92202e7	−	6.79315e7i	−0.971311	−	1.68236i	−0.279704	−	0.960086i	$-0.590236\pi$
$150$		0			0
$151$	3.04543e7	−	5.27484e7i	0.719829	−	1.24678i	−0.241239	−	0.970466i	$-0.577554\pi$
$151$	3.04543e7	−	5.27484e7i	0.719829	−	1.24678i	0.961067	−	0.276314i	$-0.0891129\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.34243e7			0.289554
$156$		0			0
$157$	9.46578e6	−	1.63952e7i	0.195213	−	0.338118i	−0.751758	−	0.659440i	$-0.770794\pi$
$157$	9.46578e6	−	1.63952e7i	0.195213	−	0.338118i	0.946970	+	0.321321i	$0.104127\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.47953e6	+	8.31459e6i	0.0468251	+	0.157018i
$162$		0			0
$163$	3.34950e7	+	5.80150e7i	0.605791	+	1.04926i	0.991926	+	0.126819i	$0.0404767\pi$
$163$	3.34950e7	+	5.80150e7i	0.605791	+	1.04926i	−0.386134	+	0.922442i	$0.626190\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.86629e7			1.47311			0.736553	−	0.676380i	$-0.236452\pi$
$167$	8.86629e7			1.47311			0.736553	+	0.676380i	$0.236452\pi$
$168$		0			0
$169$	2.94543e7			0.469403
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.44990e7	+	2.51130e7i	0.212900	+	0.368754i	0.952621	−	0.304160i	$-0.0983758\pi$
$173$	1.44990e7	+	2.51130e7i	0.212900	+	0.368754i	−0.739721	+	0.672914i	$0.765042\pi$
$174$		0			0
$175$	1.61348e7	−	1.70652e7i	0.227578	−	0.240702i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.19272e6	−	5.52995e6i	0.0416079	−	0.0720669i	−0.844472	−	0.535600i	$-0.820085\pi$
$179$	3.19272e6	−	5.52995e6i	0.0416079	−	0.0720669i	0.886079	+	0.463534i	$0.153419\pi$
$180$		0			0
$181$	8.45188e7			1.05944			0.529722	−	0.848171i	$-0.322296\pi$
$181$	8.45188e7			1.05944			0.529722	+	0.848171i	$0.322296\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.51476e6	−	7.81980e6i	0.0524244	−	0.0908018i
$186$		0			0
$187$	1.41474e6	+	2.45040e6i	0.0158209	+	0.0274026i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.05189e7	−	1.39463e8i	−0.836143	−	1.44824i	−0.893096	−	0.449866i	$-0.851472\pi$
$191$	−8.05189e7	−	1.39463e8i	−0.836143	−	1.44824i	0.0569526	−	0.998377i	$-0.481862\pi$
$192$		0			0
$193$	−1.72082e7	+	2.98055e7i	−0.172300	+	0.298433i	−0.939224	−	0.343306i	$-0.888453\pi$
$193$	−1.72082e7	+	2.98055e7i	−0.172300	+	0.298433i	0.766923	+	0.641739i	$0.221787\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.97894e8			1.84417			0.922086	−	0.386984i	$-0.126483\pi$
$197$	1.97894e8			1.84417			0.922086	+	0.386984i	$0.126483\pi$
$198$		0			0
$199$	−2.63954e7	+	4.57181e7i	−0.237434	+	0.411247i	−0.959977	−	0.280078i	$-0.909640\pi$
$199$	−2.63954e7	+	4.57181e7i	−0.237434	+	0.411247i	0.722544	+	0.691325i	$0.242973\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.85500e8	−	4.41741e7i	−1.55636	−	0.370623i
$204$		0			0
$205$	−4.77967e7	−	8.27863e7i	−0.387489	−	0.671150i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	83399.9			0.000631907
$210$		0			0
$211$	−2.95945e7			−0.216881			−0.108441	−	0.994103i	$-0.534586\pi$
$211$	−2.95945e7			−0.216881			−0.108441	+	0.994103i	$0.534586\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00154e7	+	1.38591e8i	0.549084	+	0.951042i
$216$		0			0
$217$	−2.59524e7	+	2.74490e7i	−0.172412	+	0.182355i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.55447e8	+	2.69242e8i	−0.968743	+	1.67791i
$222$		0			0
$223$	−2.98299e8			−1.80130			−0.900649	−	0.434548i	$-0.856908\pi$
$223$	−2.98299e8			−1.80130			−0.900649	+	0.434548i	$0.856908\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.54487e7	−	9.60399e7i	0.314630	−	0.544956i	−0.664728	−	0.747085i	$-0.731453\pi$
$227$	5.54487e7	−	9.60399e7i	0.314630	−	0.544956i	0.979359	+	0.202129i	$0.0647861\pi$
$228$		0			0
$229$	−9.28527e7	−	1.60826e8i	−0.510940	−	0.884975i	−0.999920	−	0.0126792i	$-0.995964\pi$
$229$	−9.28527e7	−	1.60826e8i	−0.510940	−	0.884975i	0.488979	−	0.872295i	$-0.337369\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	697526.	+	1.20815e6i	0.00361256	+	0.00625713i	0.867826	−	0.496868i	$-0.165517\pi$
$233$	697526.	+	1.20815e6i	0.00361256	+	0.00625713i	−0.864213	+	0.503125i	$0.832183\pi$
$234$		0			0
$235$	−9.94707e7	+	1.72288e8i	−0.499986	+	0.866001i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.85224e8			−1.35143			−0.675716	−	0.737162i	$-0.736165\pi$
$239$	−2.85224e8			−1.35143			−0.675716	+	0.737162i	$0.736165\pi$
$240$		0			0
$241$	1.06726e8	−	1.84854e8i	0.491145	−	0.850688i	−0.508803	−	0.860883i	$-0.669912\pi$
$241$	1.06726e8	−	1.84854e8i	0.491145	−	0.850688i	0.999948	+	0.0101950i	$0.00324521\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.48751e7	+	2.65173e8i	0.0646218	+	1.15199i
$246$		0			0
$247$	4.58184e6	+	7.93598e6i	0.0193464	+	0.0335090i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.82490e8			0.728418			0.364209	−	0.931317i	$-0.381339\pi$
$251$	1.82490e8			0.728418			0.364209	+	0.931317i	$0.381339\pi$
$252$		0			0
$253$	835537.			0.00324372
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.34416e8	−	2.32816e8i	−0.493954	−	0.855554i	0.506022	−	0.862521i	$-0.331116\pi$
$257$	−1.34416e8	−	2.32816e8i	−0.493954	−	0.855554i	−0.999976	+	0.00696714i	$0.997782\pi$
$258$		0			0
$259$	7.26124e6	+	2.43491e7i	0.0259694	+	0.0870829i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.36464e8	−	2.36362e8i	0.462564	−	0.801184i	−0.536524	−	0.843885i	$-0.680263\pi$
$263$	1.36464e8	−	2.36362e8i	0.462564	−	0.801184i	0.999088	+	0.0427009i	$0.0135962\pi$
$264$		0			0
$265$	−1.93484e8			−0.638681
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.53736e7	−	9.59099e7i	0.173448	−	0.300421i	−0.766175	−	0.642632i	$-0.777842\pi$
$269$	5.53736e7	−	9.59099e7i	0.173448	−	0.300421i	0.939623	+	0.342211i	$0.111176\pi$
$270$		0			0
$271$	−1.66967e8	−	2.89195e8i	−0.509609	−	0.882669i	−0.999938	−	0.0111314i	$-0.996457\pi$
$271$	−1.66967e8	−	2.89195e8i	−0.509609	−	0.882669i	0.490329	−	0.871537i	$-0.336877\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.13081e6	−	1.95861e6i	−0.00327887	−	0.00567916i
$276$		0			0
$277$	−9.31294e7	+	1.61305e8i	−0.263274	+	0.456003i	−0.967110	−	0.254359i	$-0.918136\pi$
$277$	−9.31294e7	+	1.61305e8i	−0.263274	+	0.456003i	0.703836	+	0.710362i	$0.251469\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.16721e7			−0.219584			−0.109792	−	0.993955i	$-0.535019\pi$
$281$	−8.16721e7			−0.219584			−0.109792	+	0.993955i	$0.535019\pi$
$282$		0			0
$283$	−1.70894e8	+	2.95998e8i	−0.448204	+	0.776311i	−0.998269	−	0.0588101i	$-0.981269\pi$
$283$	−1.70894e8	+	2.95998e8i	−0.448204	+	0.776311i	0.550066	+	0.835121i	$0.314603\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.61678e8	+	6.23148e7i	0.653403	+	0.155598i
$288$		0			0
$289$	−3.18973e8	−	5.52477e8i	−0.777340	−	1.34639i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.40002e8			0.557415			0.278707	−	0.960376i	$-0.410094\pi$
$293$	2.40002e8			0.557415			0.278707	+	0.960376i	$0.410094\pi$
$294$		0			0
$295$	−7.77539e8			−1.76338
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.59029e7	+	7.95062e7i	0.0993096	+	0.172009i
$300$		0			0
$301$	−4.38070e8	−	1.04320e8i	−0.925894	−	0.220488i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.73959e8	−	3.01306e8i	0.351073	−	0.608077i
$306$		0			0
$307$	−8.27000e8			−1.63125			−0.815626	−	0.578579i	$-0.803607\pi$
$307$	−8.27000e8			−1.63125			−0.815626	+	0.578579i	$0.803607\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.02576e7	−	5.24076e7i	0.0570391	−	0.0987946i	−0.836096	−	0.548583i	$-0.815167\pi$
$311$	3.02576e7	−	5.24076e7i	0.0570391	−	0.0987946i	0.893135	+	0.449789i	$0.148501\pi$
$312$		0			0
$313$	−6.22813e7	−	1.07874e8i	−0.114803	−	0.198844i	0.802898	−	0.596116i	$-0.203290\pi$
$313$	−6.22813e7	−	1.07874e8i	−0.114803	−	0.198844i	−0.917701	+	0.397272i	$0.869957\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.75504e7	+	6.50392e7i	0.0662075	+	0.114675i	0.897229	−	0.441566i	$-0.145577\pi$
$317$	3.75504e7	+	6.50392e7i	0.0662075	+	0.114675i	−0.831021	+	0.556240i	$0.812243\pi$
$318$		0			0
$319$	−9.18157e6	+	1.59029e7i	−0.0158361	+	0.0274290i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.08985e7			−0.0510186
$324$		0			0
$325$	1.24249e8	−	2.15206e8i	0.200771	−	0.347746i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.59982e8	−	5.36466e8i	−0.247677	−	0.830533i
$330$		0			0
$331$	3.52636e8	+	6.10784e8i	0.534477	+	0.925741i	0.999188	+	0.0402790i	$0.0128247\pi$
$331$	3.52636e8	+	6.10784e8i	0.534477	+	0.925741i	−0.464712	+	0.885462i	$0.653842\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.84916e8			−1.28601
$336$		0			0
$337$	6.97886e7			0.0993299			0.0496649	−	0.998766i	$-0.484185\pi$
$337$	6.97886e7			0.0993299			0.0496649	+	0.998766i	$0.484185\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.81887e6	+	3.15038e6i	0.00248406	+	0.00430252i
$342$		0			0
$343$	−5.70965e8	−	4.82229e8i	−0.763977	−	0.645244i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.10440e8	+	5.37698e8i	−0.398863	+	0.690852i	−0.993586	−	0.113079i	$-0.963929\pi$
$347$	−3.10440e8	+	5.37698e8i	−0.398863	+	0.690852i	0.594723	+	0.803931i	$0.297262\pi$
$348$		0			0
$349$	8.33992e8			1.05020			0.525101	−	0.851040i	$-0.324028\pi$
$349$	8.33992e8			1.05020			0.525101	+	0.851040i	$0.324028\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.88369e8	+	8.45880e8i	−0.590931	+	1.02352i	0.403177	+	0.915122i	$0.367906\pi$
$353$	−4.88369e8	+	8.45880e8i	−0.590931	+	1.02352i	−0.994107	+	0.108400i	$0.965427\pi$
$354$		0			0
$355$	3.57550e8	+	6.19294e8i	0.424167	+	0.734679i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.02310e8	+	1.38964e9i	0.915191	+	1.58516i	0.806621	+	0.591069i	$0.201294\pi$
$359$	8.02310e8	+	1.38964e9i	0.915191	+	1.58516i	0.108570	+	0.994089i	$0.465373\pi$
$360$		0			0
$361$	4.46480e8	−	7.73327e8i	0.499491	−	0.865143i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.59553e8			0.925226
$366$		0			0
$367$	−4.20508e7	+	7.28341e7i	−0.0444061	+	0.0769136i	−0.887374	−	0.461050i	$-0.847473\pi$
$367$	−4.20508e7	+	7.28341e7i	−0.0444061	+	0.0769136i	0.842968	+	0.537964i	$0.180806\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.74051e8	−	3.95622e8i	0.380297	−	0.402228i
$372$		0			0
$373$	1.20538e8	+	2.08778e8i	0.120266	+	0.208307i	0.919873	−	0.392218i	$-0.128292\pi$
$373$	1.20538e8	+	2.08778e8i	0.120266	+	0.208307i	−0.799607	+	0.600524i	$0.794959\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.01768e9			−1.93935
$378$		0			0
$379$	1.81829e8			0.171564			0.0857818	−	0.996314i	$-0.472661\pi$
$379$	1.81829e8			0.171564			0.0857818	+	0.996314i	$0.472661\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.76485e8	−	4.78886e8i	−0.251464	−	0.435548i	0.712465	−	0.701708i	$-0.247579\pi$
$383$	−2.76485e8	−	4.78886e8i	−0.251464	−	0.435548i	−0.963929	+	0.266159i	$0.914245\pi$
$384$		0			0
$385$	2.48805e7	+	5.92491e6i	0.0222201	+	0.00529138i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.08116e8	−	1.05329e9i	0.523797	−	0.907242i	−0.475820	−	0.879543i	$-0.657849\pi$
$389$	6.08116e8	−	1.05329e9i	0.523797	−	0.907242i	0.999616	−	0.0276994i	$-0.00881811\pi$
$390$		0			0
$391$	−3.09555e8			−0.261890
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.66830e8	+	8.08574e8i	−0.381127	+	0.660131i
$396$		0			0
$397$	4.75231e8	+	8.23124e8i	0.381187	+	0.660235i	0.991232	−	0.132132i	$-0.0421822\pi$
$397$	4.75231e8	+	8.23124e8i	0.381187	+	0.660235i	−0.610046	+	0.792366i	$0.708849\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.92679e8	−	5.06935e8i	−0.226666	−	0.392597i	0.730152	−	0.683285i	$-0.239449\pi$
$401$	−2.92679e8	−	5.06935e8i	−0.226666	−	0.392597i	−0.956818	+	0.290688i	$0.906116\pi$
$402$		0			0
$403$	−1.99851e8	+	3.46153e8i	−0.152104	+	0.263451i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.44685e6			0.00179898
$408$		0			0
$409$	8.82474e8	−	1.52849e9i	0.637779	−	1.10467i	−0.348140	−	0.937443i	$-0.613186\pi$
$409$	8.82474e8	−	1.52849e9i	0.637779	−	1.10467i	0.985919	−	0.167224i	$-0.0534802\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.50317e9	−	1.58986e9i	1.04999	−	1.11054i
$414$		0			0
$415$	5.81424e8	+	1.00706e9i	0.399323	+	0.691648i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.29170e9			0.857849			0.428925	−	0.903340i	$-0.358893\pi$
$419$	1.29170e9			0.857849			0.428925	+	0.903340i	$0.358893\pi$
$420$		0			0
$421$	1.69525e9			1.10725			0.553626	−	0.832765i	$-0.313244\pi$
$421$	1.69525e9			1.10725			0.553626	+	0.832765i	$0.313244\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.18948e8	+	7.25639e8i	0.264727	+	0.458521i
$426$		0			0
$427$	2.79784e8	+	9.38199e8i	0.173911	+	0.583173i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.23276e8	−	1.59916e9i	0.555471	−	0.962103i	−0.442396	−	0.896820i	$-0.645871\pi$
$431$	9.23276e8	−	1.59916e9i	0.555471	−	0.962103i	0.997867	−	0.0652836i	$-0.0207952\pi$
$432$		0			0
$433$	1.12073e9			0.663425			0.331713	−	0.943381i	$-0.392374\pi$
$433$	1.12073e9			0.663425			0.331713	+	0.943381i	$0.392374\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.56211e6	+	7.90181e6i	−0.00261506	+	0.00452941i
$438$		0			0
$439$	1.16071e8	+	2.01041e8i	0.0654783	+	0.113412i	0.896906	−	0.442221i	$-0.145809\pi$
$439$	1.16071e8	+	2.01041e8i	0.0654783	+	0.113412i	−0.831428	+	0.555633i	$0.812476\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.09581e8	+	8.82620e8i	0.278484	+	0.482348i	0.971008	−	0.239046i	$-0.0768348\pi$
$443$	5.09581e8	+	8.82620e8i	0.278484	+	0.482348i	−0.692524	+	0.721395i	$0.743501\pi$
$444$		0			0
$445$	1.50292e8	−	2.60314e8i	0.0808494	−	0.140035i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.68146e9			−1.39801			−0.699004	−	0.715118i	$-0.746373\pi$
$449$	−2.68146e9			−1.39801			−0.699004	+	0.715118i	$0.746373\pi$
$450$		0			0
$451$	1.29521e7	−	2.24337e7i	0.00664847	−	0.0115155i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.03099e8	+	2.69303e9i	0.399696	+	1.34030i
$456$		0			0
$457$	1.16628e9	+	2.02005e9i	0.571605	+	0.990048i	0.996401	+	0.0847596i	$0.0270122\pi$
$457$	1.16628e9	+	2.02005e9i	0.571605	+	0.990048i	−0.424797	+	0.905289i	$0.639654\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.24417e8			0.391917			0.195958	−	0.980612i	$-0.437218\pi$
$461$	8.24417e8			0.391917			0.195958	+	0.980612i	$0.437218\pi$
$462$		0			0
$463$	3.26057e9			1.52672			0.763361	−	0.645972i	$-0.223548\pi$
$463$	3.26057e9			1.52672			0.763361	+	0.645972i	$0.223548\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.09827e9	−	3.63431e9i	−0.953350	−	1.65125i	−0.738100	−	0.674691i	$-0.764277\pi$
$467$	−2.09827e9	−	3.63431e9i	−0.953350	−	1.65125i	−0.215250	−	0.976559i	$-0.569057\pi$
$468$		0			0
$469$	1.71076e9	−	1.80941e9i	0.765744	−	0.809903i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.16828e7	+	3.75557e7i	−0.00942110	+	0.0163178i
$474$		0			0
$475$	2.46972e7			0.0105736
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.32849e9	−	2.30101e9i	0.552310	−	0.956629i	−0.445798	−	0.895134i	$-0.647080\pi$
$479$	1.32849e9	−	2.30101e9i	0.552310	−	0.956629i	0.998107	−	0.0614949i	$-0.0195868\pi$
$480$		0			0
$481$	1.34425e8	+	2.32832e8i	0.0550774	+	0.0953969i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.96687e9	+	3.40672e9i	0.782852	+	1.35594i
$486$		0			0
$487$	−2.31794e9	+	4.01480e9i	−0.909393	+	1.57512i	−0.0944840	+	0.995526i	$0.530120\pi$
$487$	−2.31794e9	+	4.01480e9i	−0.909393	+	1.57512i	−0.814909	+	0.579589i	$0.803213\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.02935e9			−0.392442			−0.196221	−	0.980560i	$-0.562867\pi$
$491$	−1.02935e9			−0.392442			−0.196221	+	0.980560i	$0.562867\pi$
$492$		0			0
$493$	3.40164e9	−	5.89182e9i	1.27857	−	2.21455i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.95752e9	−	4.66154e8i	−0.715252	−	0.170327i
$498$		0			0
$499$	1.39539e9	+	2.41689e9i	0.502741	+	0.870773i	0.999995	+	0.00316767i	$0.00100830\pi$
$499$	1.39539e9	+	2.41689e9i	0.502741	+	0.870773i	−0.497254	+	0.867605i	$0.665658\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.48508e9			−1.57138			−0.785691	−	0.618619i	$-0.787692\pi$
$503$	−4.48508e9			−1.57138			−0.785691	+	0.618619i	$0.787692\pi$
$504$		0			0
$505$	6.08551e9			2.10270
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.02092e9	+	3.50034e9i	0.679262	+	1.17652i	0.975203	+	0.221310i	$0.0710333\pi$
$509$	2.02092e9	+	3.50034e9i	0.679262	+	1.17652i	−0.295941	+	0.955206i	$0.595633\pi$
$510$		0			0
$511$	−1.66173e9	+	1.75756e9i	−0.550917	+	0.582688i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.60280e9	+	2.77613e9i	−0.517075	+	0.895601i
$516$		0			0
$517$	−5.39097e7			−0.0171574
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.08982e9	+	1.88762e9i	−0.337615	+	0.584766i	−0.983984	−	0.178259i	$-0.942954\pi$
$521$	−1.08982e9	+	1.88762e9i	−0.337615	+	0.584766i	0.646369	+	0.763025i	$0.276287\pi$
$522$		0			0
$523$	1.02819e9	+	1.78088e9i	0.314281	+	0.544351i	0.979284	−	0.202489i	$-0.0649032\pi$
$523$	1.02819e9	+	1.78088e9i	0.314281	+	0.544351i	−0.665003	+	0.746841i	$0.731570\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.73867e8	−	1.16717e9i	−0.200557	−	0.347374i
$528$		0			0
$529$	1.65671e9	−	2.86950e9i	0.486576	−	0.842775i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.84626e9			0.814196
$534$		0			0
$535$	−8.15615e8	+	1.41269e9i	−0.230275	+	0.398848i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.02149e7	+	3.94196e7i	−0.0165632	+	0.0108430i
$540$		0			0
$541$	6.39017e8	+	1.10681e9i	0.173509	+	0.300526i	0.939644	−	0.342153i	$-0.111156\pi$
$541$	6.39017e8	+	1.10681e9i	0.173509	+	0.300526i	−0.766135	+	0.642679i	$0.777823\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.46970e9			0.653516
$546$		0			0
$547$	−6.28942e9			−1.64307			−0.821533	−	0.570161i	$-0.806881\pi$
$547$	−6.28942e9			−1.64307			−0.821533	+	0.570161i	$0.806881\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.00265e8	−	1.73663e8i	−0.0255339	−	0.0442260i
$552$		0			0
$553$	−7.50819e8	−	2.51771e9i	−0.188798	−	0.633094i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.33312e7	−	4.04109e7i	0.00572064	−	0.00990844i	−0.863151	−	0.504946i	$-0.831512\pi$
$557$	2.33312e7	−	4.04109e7i	0.00572064	−	0.00990844i	0.868872	+	0.495038i	$0.164846\pi$
$558$		0			0
$559$	−4.76486e9			−1.15374
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.20530e9	−	3.81969e9i	0.520821	−	0.902088i	−0.478886	−	0.877877i	$-0.658959\pi$
$563$	2.20530e9	−	3.81969e9i	0.520821	−	0.902088i	0.999707	−	0.0242113i	$-0.00770745\pi$
$564$		0			0
$565$	−2.14255e9	−	3.71100e9i	−0.499759	−	0.865609i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.77796e9	−	6.54362e9i	−0.859734	−	1.48910i	−0.872182	−	0.489181i	$-0.837296\pi$
$569$	−3.77796e9	−	6.54362e9i	−0.859734	−	1.48910i	0.0124479	−	0.999923i	$-0.496038\pi$
$570$		0			0
$571$	−2.57104e9	+	4.45317e9i	−0.577940	+	1.00102i	0.417776	+	0.908550i	$0.362810\pi$
$571$	−2.57104e9	+	4.45317e9i	−0.577940	+	1.00102i	−0.995715	+	0.0924705i	$0.970524\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.47428e8			0.0542764
$576$		0			0
$577$	1.11111e9	−	1.92450e9i	0.240792	−	0.417065i	−0.720148	−	0.693821i	$-0.755926\pi$
$577$	1.11111e9	−	1.92450e9i	0.240792	−	0.417065i	0.960940	+	0.276756i	$0.0892594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.18319e9	−	7.58030e8i	−0.673359	−	0.160350i
$582$		0			0
$583$	−2.62154e7	−	4.54064e7i	−0.00547919	−	0.00949024i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.10248e9			1.85749			0.928745	−	0.370719i	$-0.120889\pi$
$587$	9.10248e9			1.85749			0.928745	+	0.370719i	$0.120889\pi$
$588$		0			0
$589$	−3.97249e7			−0.00801049
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.33687e7	+	1.44399e8i	0.0164177	+	0.0284363i	0.874118	−	0.485714i	$-0.161441\pi$
$593$	8.33687e7	+	1.44399e8i	0.0164177	+	0.0284363i	−0.857700	+	0.514151i	$0.828107\pi$
$594$		0			0
$595$	−9.21787e9	−	2.19510e9i	−1.79399	−	0.427213i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.87795e8	−	1.19130e9i	0.130757	−	0.226478i	−0.793212	−	0.608946i	$-0.791593\pi$
$599$	6.87795e8	−	1.19130e9i	0.130757	−	0.226478i	0.923969	+	0.382468i	$0.124926\pi$
$600$		0			0
$601$	−5.27171e9			−0.990584			−0.495292	−	0.868727i	$-0.664939\pi$
$601$	−5.27171e9			−0.990584			−0.495292	+	0.868727i	$0.664939\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.14104e9	+	5.44045e9i	−0.576674	+	0.998828i
$606$		0			0
$607$	3.64681e9	+	6.31645e9i	0.661839	+	1.14634i	0.980132	+	0.198346i	$0.0635570\pi$
$607$	3.64681e9	+	6.31645e9i	0.661839	+	1.14634i	−0.318293	+	0.947992i	$0.603110\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.96171e9	−	5.12983e9i	−0.525288	−	0.909826i
$612$		0			0
$613$	2.79988e9	−	4.84953e9i	0.490939	−	0.850331i	−0.509007	−	0.860762i	$-0.669987\pi$
$613$	2.79988e9	−	4.84953e9i	0.490939	−	0.850331i	0.999946	+	0.0104318i	$0.00332060\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.11007e10			1.90263			0.951313	−	0.308226i	$-0.0997352\pi$
$617$	1.11007e10			1.90263			0.951313	+	0.308226i	$0.0997352\pi$
$618$		0			0
$619$	4.29620e9	−	7.44124e9i	0.728060	−	1.26104i	−0.229642	−	0.973275i	$-0.573756\pi$
$619$	4.29620e9	−	7.44124e9i	0.728060	−	1.26104i	0.957702	−	0.287762i	$-0.0929111\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.41720e8	+	8.10558e8i	0.0400502	+	0.134300i
$624$		0			0
$625$	3.72780e9	+	6.45673e9i	0.610762	+	1.05787i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.06523e8			−0.145245
$630$		0			0
$631$	1.71493e8			0.0271734			0.0135867	−	0.999908i	$-0.495675\pi$
$631$	1.71493e8			0.0271734			0.0135867	+	0.999908i	$0.495675\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.74330e9	+	6.48358e9i	0.580158	+	1.00486i
$636$		0			0
$637$	−7.05910e9	−	3.56416e9i	−1.08209	−	0.546347i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.55924e9	+	6.16479e9i	−0.533770	+	0.924517i	0.465452	+	0.885073i	$0.345892\pi$
$641$	−3.55924e9	+	6.16479e9i	−0.533770	+	0.924517i	−0.999222	+	0.0394438i	$0.987441\pi$
$642$		0			0
$643$	−6.98220e9			−1.03575			−0.517874	−	0.855457i	$-0.673276\pi$
$643$	−6.98220e9			−1.03575			−0.517874	+	0.855457i	$0.673276\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	5.19570e8	−	8.99921e8i	0.0754187	−	0.130629i	−0.825850	−	0.563890i	$-0.809304\pi$
$647$	5.19570e8	−	8.99921e8i	0.0754187	−	0.130629i	0.901268	+	0.433262i	$0.142637\pi$
$648$		0			0
$649$	−1.05350e8	−	1.82471e8i	−0.0151279	−	0.0262022i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.96396e9	−	1.03299e10i	−0.838182	−	1.45177i	−0.891413	−	0.453192i	$-0.850285\pi$
$653$	−5.96396e9	−	1.03299e10i	−0.838182	−	1.45177i	0.0532302	−	0.998582i	$-0.483048\pi$
$654$		0			0
$655$	−1.63869e9	+	2.83829e9i	−0.227852	+	0.394650i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.23644e10			−1.68295			−0.841477	−	0.540292i	$-0.818314\pi$
$659$	−1.23644e10			−1.68295			−0.841477	+	0.540292i	$0.818314\pi$
$660$		0			0
$661$	1.48594e9	−	2.57373e9i	0.200123	−	0.346623i	−0.748445	−	0.663197i	$-0.769199\pi$
$661$	1.48594e9	−	2.57373e9i	0.200123	−	0.346623i	0.948568	+	0.316574i	$0.102532\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.91883e8	+	2.02948e8i	−0.0253023	+	0.0267614i
$666$		0			0
$667$	−1.00449e9	−	1.73983e9i	−0.131071	−	0.227022i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	9.42800e7			0.0120473
$672$		0			0
$673$	−8.24543e9			−1.04270			−0.521351	−	0.853342i	$-0.674572\pi$
$673$	−8.24543e9			−1.04270			−0.521351	+	0.853342i	$0.674572\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.34244e9	−	2.32517e9i	−0.166278	−	0.288001i	0.770831	−	0.637040i	$-0.219841\pi$
$677$	−1.34244e9	−	2.32517e9i	−0.166278	−	0.288001i	−0.937108	+	0.349039i	$0.886508\pi$
$678$		0			0
$679$	−1.07683e10	−	2.56430e9i	−1.32008	−	0.314358i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.34998e9	+	5.80234e9i	−0.402319	+	0.696836i	−0.994005	−	0.109332i	$-0.965129\pi$
$683$	−3.34998e9	+	5.80234e9i	−0.402319	+	0.696836i	0.591687	+	0.806168i	$0.298462\pi$
$684$		0			0
$685$	−1.34230e10			−1.59564
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.88046e9	−	4.98910e9i	0.335501	−	0.581105i
$690$		0			0
$691$	−5.57677e9	−	9.65925e9i	−0.642998	−	1.11370i	−0.984760	−	0.173919i	$-0.944357\pi$
$691$	−5.57677e9	−	9.65925e9i	−0.642998	−	1.11370i	0.341762	−	0.939786i	$-0.388976\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.68887e9	+	1.33175e10i	0.868792	+	1.50479i
$696$		0			0
$697$	−4.79857e9	+	8.31136e9i	−0.536781	+	0.929732i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.52110e9			0.715003			0.357502	−	0.933913i	$-0.383629\pi$
$701$	6.52110e9			0.715003			0.357502	+	0.933913i	$0.383629\pi$
$702$		0			0
$703$	−1.33600e7	+	2.31402e7i	−0.00145032	+	0.00251203i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.17648e10	+	1.24432e10i	−1.25203	+	1.32424i
$708$		0			0
$709$	1.09662e9	+	1.89940e9i	0.115556	+	0.200149i	0.918002	−	0.396576i	$-0.129802\pi$
$709$	1.09662e9	+	1.89940e9i	0.115556	+	0.200149i	−0.802446	+	0.596725i	$0.796468\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.97981e8			−0.0411197
$714$		0			0
$715$	2.70623e8			0.0276881
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.05874e8	+	7.02994e8i	0.0407230	+	0.0705343i	0.885669	−	0.464318i	$-0.153701\pi$
$719$	4.05874e8	+	7.02994e8i	0.0407230	+	0.0705343i	−0.844946	+	0.534852i	$0.820367\pi$
$720$		0			0
$721$	−2.57784e9	−	8.64423e9i	−0.256143	−	0.858921i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.71894e9	+	4.70935e9i	−0.264982	+	0.458963i
$726$		0			0
$727$	−1.14066e10			−1.10100			−0.550499	−	0.834836i	$-0.685562\pi$
$727$	−1.14066e10			−1.10100			−0.550499	+	0.834836i	$0.685562\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.03318e9	−	1.39139e10i	0.760636	−	1.31746i
$732$		0			0
$733$	1.76593e9	+	3.05868e9i	0.165619	+	0.286860i	0.936875	−	0.349665i	$-0.113705\pi$
$733$	1.76593e9	+	3.05868e9i	0.165619	+	0.286860i	−0.771256	+	0.636525i	$0.780371\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.19899e8	−	2.07670e8i	−0.0110326	−	0.0191090i
$738$		0			0
$739$	−6.44886e9	+	1.11697e10i	−0.587797	+	1.01809i	0.406724	+	0.913551i	$0.366671\pi$
$739$	−6.44886e9	+	1.11697e10i	−0.587797	+	1.01809i	−0.994520	+	0.104543i	$0.966662\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.49218e10			−1.33463			−0.667316	−	0.744774i	$-0.732557\pi$
$743$	−1.49218e10			−1.33463			−0.667316	+	0.744774i	$0.732557\pi$
$744$		0			0
$745$	1.26484e10	−	2.19077e10i	1.12070	−	1.94111i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.31178e9	−	4.39879e9i	−0.114071	−	0.382513i
$750$		0			0
$751$	8.59883e9	+	1.48936e10i	0.740798	+	1.28310i	0.952132	+	0.305686i	$0.0988857\pi$
$751$	8.59883e9	+	1.48936e10i	0.740798	+	1.28310i	−0.211335	+	0.977414i	$0.567781\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.96428e10			1.66108
$756$		0			0
$757$	1.66346e10			1.39373			0.696863	−	0.717204i	$-0.254578\pi$
$757$	1.66346e10			1.39373			0.696863	+	0.717204i	$0.254578\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.04941e9	−	7.01378e9i	−0.333077	−	0.576907i	0.650036	−	0.759903i	$-0.274754\pi$
$761$	−4.04941e9	−	7.01378e9i	−0.333077	−	0.576907i	−0.983114	+	0.182996i	$0.941420\pi$
$762$		0			0
$763$	−4.77453e9	+	5.04987e9i	−0.389130	+	0.411571i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.15755e10	−	2.00493e10i	0.926308	−	1.60441i
$768$		0			0
$769$	−3.90614e9			−0.309746			−0.154873	−	0.987934i	$-0.549497\pi$
$769$	−3.90614e9			−0.309746			−0.154873	+	0.987934i	$0.549497\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.68336e9	−	2.91567e9i	0.131084	−	0.227044i	−0.793011	−	0.609208i	$-0.791488\pi$
$773$	1.68336e9	−	2.91567e9i	0.131084	−	0.227044i	0.924095	+	0.382164i	$0.124821\pi$
$774$		0			0
$775$	5.38624e8	+	9.32924e8i	0.0415652	+	0.0719930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.41439e8	+	2.44980e8i	0.0107199	+	0.0185673i
$780$		0			0
$781$	−9.68899e7	+	1.67818e8i	−0.00727779	+	0.0126055i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.10537e9			0.450472
$786$		0			0
$787$	9.79288e9	−	1.69618e10i	0.716142	−	1.24039i	−0.246376	−	0.969174i	$-0.579240\pi$
$787$	9.79288e9	−	1.69618e10i	0.716142	−	1.24039i	0.962518	−	0.271220i	$-0.0874269\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.17301e10	+	2.79334e9i	0.842719	+	0.200681i
$792$		0			0
$793$	5.17958e9	+	8.97129e9i	0.368840	+	0.638850i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.32070e10			−0.924059			−0.462029	−	0.886865i	$-0.652879\pi$
$797$	−1.32070e10			−0.924059			−0.462029	+	0.886865i	$0.652879\pi$
$798$		0			0
$799$	1.99728e10			1.38524
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.16462e8	+	2.01718e8i	0.00793744	+	0.0137480i
$804$		0			0
$805$	−1.92236e9	+	2.03322e9i	−0.129882	+	0.137372i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.11284e9	+	1.92749e9i	−0.0738945	+	0.127989i	−0.900605	−	0.434638i	$-0.856876\pi$
$809$	−1.11284e9	+	1.92749e9i	−0.0738945	+	0.127989i	0.826710	+	0.562628i	$0.190209\pi$
$810$		0			0
$811$	−1.14132e10			−0.751337			−0.375668	−	0.926754i	$-0.622587\pi$
$811$	−1.14132e10			−0.751337			−0.375668	+	0.926754i	$0.622587\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.08020e10	+	1.87097e10i	−0.698962	+	1.21064i
$816$		0			0
$817$	−2.36781e8	−	4.10116e8i	−0.0151904	−	0.0263105i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.81684e9	−	6.61096e9i	−0.240715	−	0.416930i	0.720203	−	0.693763i	$-0.244049\pi$
$821$	−3.81684e9	−	6.61096e9i	−0.240715	−	0.416930i	−0.960918	+	0.276833i	$0.910715\pi$
$822$		0			0
$823$	−7.21507e9	+	1.24969e10i	−0.451171	+	0.781451i	−0.998459	−	0.0554930i	$-0.982327\pi$
$823$	−7.21507e9	+	1.24969e10i	−0.451171	+	0.781451i	0.547288	+	0.836944i	$0.315660\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.57532e9			0.404248			0.202124	−	0.979360i	$-0.435216\pi$
$827$	6.57532e9			0.404248			0.202124	+	0.979360i	$0.435216\pi$
$828$		0			0
$829$	1.21968e10	−	2.11255e10i	0.743543	−	1.28785i	−0.207329	−	0.978271i	$-0.566477\pi$
$829$	1.21968e10	−	2.11255e10i	0.743543	−	1.28785i	0.950872	−	0.309583i	$-0.100190\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.23088e10	−	1.46044e10i	1.33727	−	0.875440i
$834$		0			0
$835$	1.42967e10	+	2.47627e10i	0.849835	+	1.47196i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1.33185e10			−0.778555			−0.389278	−	0.921120i	$-0.627275\pi$
$839$	−1.33185e10			−0.778555			−0.389278	+	0.921120i	$0.627275\pi$
$840$		0			0
$841$	2.69030e10			1.55960
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.74947e9	+	8.22632e9i	0.270799	+	0.469037i
$846$		0			0
$847$	−5.05185e9	−	1.69403e10i	−0.285666	−	0.957920i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.33847e8	+	2.31829e8i	−0.00744481	+	0.0128948i
$852$		0			0
$853$	−1.68068e9			−0.0927177			−0.0463589	−	0.998925i	$-0.514762\pi$
$853$	−1.68068e9			−0.0927177			−0.0463589	+	0.998925i	$0.514762\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.58803e10	+	2.75055e10i	−0.861839	+	1.49275i	0.00831350	+	0.999965i	$0.497354\pi$
$857$	−1.58803e10	+	2.75055e10i	−0.861839	+	1.49275i	−0.870152	+	0.492783i	$0.835980\pi$
$858$		0			0
$859$	1.32926e10	+	2.30235e10i	0.715541	+	1.23935i	0.962750	+	0.270392i	$0.0871533\pi$
$859$	1.32926e10	+	2.30235e10i	0.715541	+	1.23935i	−0.247209	+	0.968962i	$0.579513\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	5.81235e9	+	1.00673e10i	0.307832	+	0.533181i	0.977888	−	0.209130i	$-0.0670632\pi$
$863$	5.81235e9	+	1.00673e10i	0.307832	+	0.533181i	−0.670056	+	0.742311i	$0.733730\pi$
$864$		0			0
$865$	−4.67587e9	+	8.09885e9i	−0.245644	+	0.425468i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.53006e8			−0.0130786
$870$		0			0
$871$	1.31740e10	−	2.28181e10i	0.675546	−	1.17008i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.48745e10	−	3.54214e9i	−0.750610	−	0.178746i
$876$		0			0
$877$	1.48291e10	+	2.56847e10i	0.742362	+	1.28581i	0.951417	+	0.307906i	$0.0996281\pi$
$877$	1.48291e10	+	2.56847e10i	0.742362	+	1.28581i	−0.209054	+	0.977904i	$0.567039\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.62547e10			−0.800875			−0.400437	−	0.916324i	$-0.631142\pi$
$881$	−1.62547e10			−0.800875			−0.400437	+	0.916324i	$0.631142\pi$
$882$		0			0
$883$	1.47883e10			0.722864			0.361432	−	0.932398i	$-0.382288\pi$
$883$	1.47883e10			0.722864			0.361432	+	0.932398i	$0.382288\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.54901e10	−	2.68297e10i	−0.745284	−	1.29087i	−0.950062	−	0.312062i	$-0.898980\pi$
$887$	−1.54901e10	−	2.68297e10i	−0.745284	−	1.29087i	0.204778	−	0.978809i	$-0.434353\pi$
$888$		0			0
$889$	−2.04939e10	−	4.88031e9i	−0.978291	−	0.232965i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.94352e8	−	5.09833e8i	0.0138321	−	0.0239579i
$894$		0			0
$895$	2.05928e9			0.0960143
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.37335e9	−	7.57486e9i	0.200750	−	0.347709i
$900$		0			0
$901$	9.71244e9	+	1.68224e10i	0.442376	+	0.766218i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.36285e10	+	2.36053e10i	0.611193	+	1.05862i
$906$		0			0
$907$	1.60810e10	−	2.78531e10i	0.715628	−	1.23950i	−0.247089	−	0.968993i	$-0.579474\pi$
$907$	1.60810e10	−	2.78531e10i	0.715628	−	1.23950i	0.962717	−	0.270511i	$-0.0871926\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.65837e9			−0.204136			−0.102068	−	0.994777i	$-0.532546\pi$
$911$	−4.65837e9			−0.204136			−0.102068	+	0.994777i	$0.532546\pi$
$912$		0			0
$913$	−1.57556e8	+	2.72895e8i	−0.00685152	+	0.0118672i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.63556e9	−	8.83779e9i	−0.112870	−	0.378487i
$918$		0			0
$919$	9.91360e9	+	1.71709e10i	0.421334	+	0.729773i	0.996070	−	0.0885664i	$-0.0282285\pi$
$919$	9.91360e9	+	1.71709e10i	0.421334	+	0.729773i	−0.574736	+	0.818339i	$0.694895\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.12918e10			−0.891266
$924$		0			0
$925$	7.24586e8			0.0301019
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.31456e10	+	4.00894e10i	0.947140	+	1.64050i	0.751408	+	0.659838i	$0.229375\pi$
$929$	2.31456e10	+	4.00894e10i	0.947140	+	1.64050i	0.195733	+	0.980657i	$0.437292\pi$
$930$		0			0
$931$	−4.40182e7	−	7.84697e8i	−0.00178776	−	0.0318697i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.56249e8	+	7.90247e8i	−0.0182542	+	0.0316171i
$936$		0			0
$937$	1.19888e10			0.476088			0.238044	−	0.971254i	$-0.423494\pi$
$937$	1.19888e10			0.476088			0.238044	+	0.971254i	$0.423494\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.65996e9	+	6.33924e9i	−0.143190	+	0.248012i	−0.928696	−	0.370841i	$-0.879069\pi$
$941$	−3.65996e9	+	6.33924e9i	−0.143190	+	0.248012i	0.785506	+	0.618854i	$0.212403\pi$
$942$		0			0
$943$	1.41700e9	+	2.45432e9i	0.0550274	+	0.0953103i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.90645e9	+	1.02303e10i	0.225996	+	0.391437i	0.956618	−	0.291345i	$-0.0941029\pi$
$947$	5.90645e9	+	1.02303e10i	0.225996	+	0.391437i	−0.730621	+	0.682783i	$0.760770\pi$
$948$		0			0
$949$	−1.27964e10	+	2.21641e10i	−0.486024	+	0.841818i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.38534e10			0.892742			0.446371	−	0.894848i	$-0.352716\pi$
$953$	2.38534e10			0.892742			0.446371	+	0.894848i	$0.352716\pi$
$954$		0			0
$955$	2.59671e10	−	4.49763e10i	0.964742	−	1.67098i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.59500e10	−	2.74465e10i	0.950108	−	1.00490i
$960$		0			0
$961$	1.28899e10	+	2.23260e10i	0.468510	+	0.811484i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.10992e10			−0.397600
$966$		0			0
$967$	4.44768e10			1.58176			0.790881	−	0.611970i	$-0.209623\pi$
$967$	4.44768e10			1.58176			0.790881	+	0.611970i	$0.209623\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.06851e10	−	1.85072e10i	−0.374552	−	0.648743i	0.615708	−	0.787975i	$-0.288870\pi$
$971$	−1.06851e10	−	1.85072e10i	−0.374552	−	0.648743i	−0.990260	+	0.139231i	$0.955537\pi$
$972$		0			0
$973$	−4.20952e10	−	1.00243e10i	−1.46500	−	0.348868i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.46697e10	−	2.54086e10i	0.503256	−	0.871665i	−0.496737	−	0.867901i	$-0.665469\pi$
$977$	1.46697e10	−	2.54086e10i	0.503256	−	0.871665i	0.999993	−	0.00376407i	$-0.00119814\pi$
$978$		0			0
$979$	8.14533e7			0.00277440
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.18249e10	−	2.04813e10i	0.397064	−	0.687735i	−0.596298	−	0.802763i	$-0.703363\pi$
$983$	1.18249e10	−	2.04813e10i	0.397064	−	0.687735i	0.993362	+	0.115028i	$0.0366958\pi$
$984$		0			0
$985$	3.19101e10	+	5.52700e10i	1.06390	+	1.84273i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.37217e9	−	4.10872e9i	−0.0779757	−	0.135058i
$990$		0			0
$991$	1.45057e10	−	2.51246e10i	0.473457	−	0.820052i	−0.526081	−	0.850434i	$-0.676339\pi$
$991$	1.45057e10	−	2.51246e10i	0.473457	−	0.820052i	0.999538	+	0.0303824i	$0.00967250\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.70248e10			−0.547901
$996$		0			0
$997$	2.40665e10	−	4.16845e10i	0.769096	−	1.33211i	−0.168958	−	0.985623i	$-0.554040\pi$
$997$	2.40665e10	−	4.16845e10i	0.769096	−	1.33211i	0.938054	−	0.346490i	$-0.112627\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.k.e.109.7	yes	16
3.2	odd	2	inner	252.8.k.e.109.2	yes	16
7.2	even	3	inner	252.8.k.e.37.7	yes	16
21.2	odd	6	inner	252.8.k.e.37.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.k.e.37.2	✓	16	21.2	odd	6	inner
252.8.k.e.37.7	yes	16	7.2	even	3	inner
252.8.k.e.109.2	yes	16	3.2	odd	2	inner
252.8.k.e.109.7	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(161.248 + 279.290i) q^{5} +(-882.807 - 210.227i) q^{7} +(-43.6956 + 75.6830i) q^{11} -9602.23 q^{13} +(16188.6 - 28039.5i) q^{17} +(-477.164 - 826.473i) q^{19} +(-4780.44 - 8279.97i) q^{23} +(-12939.6 + 22412.0i) q^{25} +210126. q^{29} +(20813.0 - 36049.2i) q^{31} +(-83636.7 - 280458. i) q^{35} +(-13999.4 - 24247.7i) q^{37} -296416. q^{41} +496225. q^{43} +(308439. + 534233. i) q^{47} +(735152. + 371180. i) q^{49} +(-299978. + 519577. i) q^{53} -28183.4 q^{55} +(-1.20550e6 + 2.08799e6i) q^{59} +(-539414. - 934292. i) q^{61} +(-1.54834e6 - 2.68181e6i) q^{65} +(-1.37198e6 + 2.37633e6i) q^{67} +2.21738e6 q^{71} +(1.33265e6 - 2.30822e6i) q^{73} +(54485.4 - 57627.4i) q^{77} +(1.44755e6 + 2.50723e6i) q^{79} +3.60576e6 q^{83} +1.04415e7 q^{85} +(-466027. - 807183. i) q^{89} +(8.47691e6 + 2.01865e6i) q^{91} +(153884. - 266535. i) q^{95} +1.21978e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 1680 q^{7} - 28280 q^{13} + 42224 q^{19} - 80460 q^{25} + 164752 q^{31} - 647980 q^{37} + 1341440 q^{43} + 230104 q^{49} - 323120 q^{55} - 4319336 q^{61} - 3905760 q^{67} + 6471780 q^{73} - 6093104 q^{79}+ \cdots - 27141240 q^{97}+O(q^{100}) \) 16 * q + 1680 * q^7 - 28280 * q^13 + 42224 * q^19 - 80460 * q^25 + 164752 * q^31 - 647980 * q^37 + 1341440 * q^43 + 230104 * q^49 - 323120 * q^55 - 4319336 * q^61 - 3905760 * q^67 + 6471780 * q^73 - 6093104 * q^79 + 456400 * q^85 + 15969856 * q^91 - 27141240 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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