LMFDB - Embedded newform 252.8.k.e.109.4

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			109.4
Root			$-50.1630 - 86.8849i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.8.k.e.37.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-117.526 - 203.561i) q^{5} +(881.619 - 215.154i) q^{7} +(3934.09 - 6814.05i) q^{11} -5762.92 q^{13} +(3577.84 - 6196.99i) q^{17} +(-19965.0 - 34580.3i) q^{19} +(28266.6 + 48959.2i) q^{23} +(11437.7 - 19810.8i) q^{25} +130038. q^{29} +(-25781.5 + 44654.8i) q^{31} +(-147410. - 154177. i) q^{35} +(139818. + 242173. i) q^{37} +345556. q^{41} -111709. q^{43} +(-425409. - 736830. i) q^{47} +(730961. - 379368. i) q^{49} +(-599912. + 1.03908e6i) q^{53} -1.84943e6 q^{55} +(-856228. + 1.48303e6i) q^{59} +(-1.22726e6 - 2.12568e6i) q^{61} +(677294. + 1.17311e6i) q^{65} +(-313531. + 543052. i) q^{67} -2.97409e6 q^{71} +(476382. - 825118. i) q^{73} +(2.00230e6 - 6.85383e6i) q^{77} +(-2.95297e6 - 5.11469e6i) q^{79} +9.78669e6 q^{83} -1.68196e6 q^{85} +(-2.59171e6 - 4.48897e6i) q^{89} +(-5.08070e6 + 1.23992e6i) q^{91} +(-4.69281e6 + 8.12818e6i) q^{95} -4.87158e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−117.526	−	203.561i	−0.420474	−	0.728282i	0.575512	−	0.817793i	$-0.304803\pi$
$5$	−117.526	−	203.561i	−0.420474	−	0.728282i	−0.995986	+	0.0895111i	$0.971470\pi$
$6$		0			0
$7$	881.619	−	215.154i	0.971489	−	0.237086i
$7$	881.619	−	215.154i	0.971489	−	0.237086i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3934.09	−	6814.05i	0.891189	−	1.54359i	0.0527383	−	0.998608i	$-0.483205\pi$
$11$	3934.09	−	6814.05i	0.891189	−	1.54359i	0.838451	−	0.544977i	$-0.183462\pi$
$12$		0			0
$13$	−5762.92			−0.727514			−0.363757	−	0.931494i	$-0.618506\pi$
$13$	−5762.92			−0.727514			−0.363757	+	0.931494i	$0.618506\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3577.84	−	6196.99i	0.176624	−	0.305921i	−0.764098	−	0.645100i	$-0.776816\pi$
$17$	3577.84	−	6196.99i	0.176624	−	0.305921i	0.940722	+	0.339179i	$0.110149\pi$
$18$		0			0
$19$	−19965.0	−	34580.3i	−0.667776	−	1.15662i	−0.978525	−	0.206130i	$-0.933913\pi$
$19$	−19965.0	−	34580.3i	−0.667776	−	1.15662i	0.310749	−	0.950492i	$-0.399420\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	28266.6	+	48959.2i	0.484425	+	0.839048i	0.999840	−	0.0178925i	$-0.00569567\pi$
$23$	28266.6	+	48959.2i	0.484425	+	0.839048i	−0.515415	+	0.856940i	$0.672362\pi$
$24$		0			0
$25$	11437.7	−	19810.8i	0.146403	−	0.253578i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	130038.			0.990096			0.495048	−	0.868866i	$-0.335150\pi$
$29$	130038.			0.990096			0.495048	+	0.868866i	$0.335150\pi$
$30$		0			0
$31$	−25781.5	+	44654.8i	−0.155432	+	0.269217i	−0.933216	−	0.359315i	$-0.883010\pi$
$31$	−25781.5	+	44654.8i	−0.155432	+	0.269217i	0.777784	+	0.628532i	$0.216344\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−147410.	−	154177.i	−0.581151	−	0.607830i
$36$		0			0
$37$	139818.	+	242173.i	0.453794	+	0.785994i	0.998618	−	0.0525566i	$-0.0167370\pi$
$37$	139818.	+	242173.i	0.453794	+	0.785994i	−0.544824	+	0.838550i	$0.683404\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	345556.			0.783024			0.391512	−	0.920173i	$-0.371952\pi$
$41$	345556.			0.783024			0.391512	+	0.920173i	$0.371952\pi$
$42$		0			0
$43$	−111709.			−0.214264			−0.107132	−	0.994245i	$-0.534167\pi$
$43$	−111709.			−0.214264			−0.107132	+	0.994245i	$0.534167\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−425409.	−	736830.i	−0.597674	−	1.03520i	−0.993164	−	0.116731i	$-0.962758\pi$
$47$	−425409.	−	736830.i	−0.597674	−	1.03520i	0.395489	−	0.918471i	$-0.370575\pi$
$48$		0			0
$49$	730961.	−	379368.i	0.887580	−	0.460653i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−599912.	+	1.03908e6i	−0.553506	+	0.958700i	0.444512	+	0.895773i	$0.353377\pi$
$53$	−599912.	+	1.03908e6i	−0.553506	+	0.958700i	−0.998018	+	0.0629273i	$0.979956\pi$
$54$		0			0
$55$	−1.84943e6			−1.49889
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−856228.	+	1.48303e6i	−0.542759	+	0.940086i	0.455985	+	0.889987i	$0.349287\pi$
$59$	−856228.	+	1.48303e6i	−0.542759	+	0.940086i	−0.998744	+	0.0500990i	$0.984046\pi$
$60$		0			0
$61$	−1.22726e6	−	2.12568e6i	−0.692282	−	1.19907i	−0.971088	−	0.238721i	$-0.923272\pi$
$61$	−1.22726e6	−	2.12568e6i	−0.692282	−	1.19907i	0.278806	−	0.960347i	$-0.410061\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	677294.	+	1.17311e6i	0.305901	+	0.529835i
$66$		0			0
$67$	−313531.	+	543052.i	−0.127356	+	0.220587i	−0.922651	−	0.385635i	$-0.873982\pi$
$67$	−313531.	+	543052.i	−0.127356	+	0.220587i	0.795296	+	0.606222i	$0.207316\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.97409e6			−0.986165			−0.493082	−	0.869983i	$-0.664130\pi$
$71$	−2.97409e6			−0.986165			−0.493082	+	0.869983i	$0.664130\pi$
$72$		0			0
$73$	476382.	−	825118.i	0.143326	−	0.248248i	−0.785421	−	0.618962i	$-0.787554\pi$
$73$	476382.	−	825118.i	0.143326	−	0.248248i	0.928747	+	0.370714i	$0.120887\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.00230e6	−	6.85383e6i	0.499818	−	1.71086i
$78$		0			0
$79$	−2.95297e6	−	5.11469e6i	−0.673852	−	1.16715i	−0.976803	−	0.214139i	$-0.931305\pi$
$79$	−2.95297e6	−	5.11469e6i	−0.673852	−	1.16715i	0.302952	−	0.953006i	$-0.402028\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.78669e6			1.87872			0.939361	−	0.342930i	$-0.111419\pi$
$83$	9.78669e6			1.87872			0.939361	+	0.342930i	$0.111419\pi$
$84$		0			0
$85$	−1.68196e6			−0.297063
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.59171e6	−	4.48897e6i	−0.389692	−	0.674966i	0.602716	−	0.797956i	$-0.294085\pi$
$89$	−2.59171e6	−	4.48897e6i	−0.389692	−	0.674966i	−0.992408	+	0.122989i	$0.960752\pi$
$90$		0			0
$91$	−5.08070e6	+	1.23992e6i	−0.706771	+	0.172483i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.69281e6	+	8.12818e6i	−0.561565	+	0.972659i
$96$		0			0
$97$	−4.87158e6			−0.541962			−0.270981	−	0.962585i	$-0.587348\pi$
$97$	−4.87158e6			−0.541962			−0.270981	+	0.962585i	$0.587348\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.03637e6	−	3.52710e6i	0.196667	−	0.340638i	−0.750778	−	0.660554i	$-0.770321\pi$
$101$	2.03637e6	−	3.52710e6i	0.196667	−	0.340638i	0.947446	+	0.319916i	$0.103655\pi$
$102$		0			0
$103$	−7.15804e6	−	1.23981e7i	−0.645452	−	1.11795i	−0.984197	−	0.177077i	$-0.943336\pi$
$103$	−7.15804e6	−	1.23981e7i	−0.645452	−	1.11795i	0.338745	−	0.940878i	$-0.389997\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.47668e6	+	1.64141e7i	0.747847	+	1.29531i	0.948853	+	0.315720i	$0.102246\pi$
$107$	9.47668e6	+	1.64141e7i	0.747847	+	1.29531i	−0.201005	+	0.979590i	$0.564421\pi$
$108$		0			0
$109$	−2.83878e6	+	4.91690e6i	−0.209961	+	0.363663i	−0.951702	−	0.307023i	$-0.900667\pi$
$109$	−2.83878e6	+	4.91690e6i	−0.209961	+	0.363663i	0.741741	+	0.670686i	$0.234000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.34292e6			−0.413537			−0.206769	−	0.978390i	$-0.566295\pi$
$113$	−6.34292e6			−0.413537			−0.206769	+	0.978390i	$0.566295\pi$
$114$		0			0
$115$	6.64412e6	−	1.15080e7i	0.407376	−	0.705596i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.82098e6	−	6.23317e6i	0.0990583	−	0.339074i
$120$		0			0
$121$	−2.12106e7	−	3.67378e7i	−1.08844	−	1.88523i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.37404e7			−1.08718
$126$		0			0
$127$	−4.12014e7			−1.78484			−0.892420	−	0.451205i	$-0.850994\pi$
$127$	−4.12014e7			−1.78484			−0.892420	+	0.451205i	$0.850994\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.09350e7	−	1.89400e7i	−0.424982	−	0.736091i	0.571436	−	0.820646i	$-0.306386\pi$
$131$	−1.09350e7	−	1.89400e7i	−0.424982	−	0.736091i	−0.996419	+	0.0845554i	$0.973053\pi$
$132$		0			0
$133$	−2.50416e7	−	2.61911e7i	−0.922956	−	0.965324i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.87470e7	−	3.24708e7i	0.622889	−	1.07888i	−0.366056	−	0.930593i	$-0.619292\pi$
$137$	1.87470e7	−	3.24708e7i	0.622889	−	1.07888i	0.988945	−	0.148283i	$-0.0473746\pi$
$138$		0			0
$139$	−3.15043e7			−0.994989			−0.497494	−	0.867467i	$-0.665746\pi$
$139$	−3.15043e7			−0.994989			−0.497494	+	0.867467i	$0.665746\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.26719e7	+	3.92688e7i	−0.648352	+	1.12298i
$144$		0			0
$145$	−1.52829e7	−	2.64707e7i	−0.416310	−	0.721070i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.28779e6	+	1.43549e7i	0.205252	+	0.355506i	0.950213	−	0.311601i	$-0.100865\pi$
$149$	8.28779e6	+	1.43549e7i	0.205252	+	0.355506i	−0.744961	+	0.667108i	$0.767532\pi$
$150$		0			0
$151$	3.11762e7	−	5.39987e7i	0.736891	−	1.27633i	−0.216997	−	0.976172i	$-0.569626\pi$
$151$	3.11762e7	−	5.39987e7i	0.736891	−	1.27633i	0.953888	−	0.300161i	$-0.0970405\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.21200e7			0.261421
$156$		0			0
$157$	−1.99021e7	+	3.44715e7i	−0.410441	+	0.710905i	−0.994938	−	0.100491i	$-0.967959\pi$
$157$	−1.99021e7	+	3.44715e7i	−0.410441	+	0.710905i	0.584497	+	0.811396i	$0.301292\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.54541e7	+	3.70817e7i	0.669540	+	0.700275i
$162$		0			0
$163$	1.67496e7	+	2.90111e7i	0.302933	+	0.524695i	0.976799	−	0.214158i	$-0.0687008\pi$
$163$	1.67496e7	+	2.90111e7i	0.302933	+	0.524695i	−0.673866	+	0.738854i	$0.735367\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.31239e7			−1.38108			−0.690539	−	0.723295i	$-0.742627\pi$
$167$	−8.31239e7			−1.38108			−0.690539	+	0.723295i	$0.742627\pi$
$168$		0			0
$169$	−2.95372e7			−0.470724
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.73005e7	+	6.46063e7i	0.547713	+	0.948666i	0.998431	+	0.0560000i	$0.0178347\pi$
$173$	3.73005e7	+	6.46063e7i	0.547713	+	0.948666i	−0.450718	+	0.892666i	$0.648832\pi$
$174$		0			0
$175$	5.82137e6	−	1.99264e7i	0.0821093	−	0.281058i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.29744e7	−	1.26395e8i	0.951010	−	1.64720i	0.207765	−	0.978179i	$-0.433381\pi$
$179$	7.29744e7	−	1.26395e8i	0.951010	−	1.64720i	0.743245	−	0.669019i	$-0.233286\pi$
$180$		0			0
$181$	−8.25291e7			−1.03450			−0.517252	−	0.855833i	$-0.673045\pi$
$181$	−8.25291e7			−1.03450			−0.517252	+	0.855833i	$0.673045\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.28646e7	−	5.69232e7i	0.381617	−	0.660980i
$186$		0			0
$187$	−2.81511e7	−	4.87591e7i	−0.314810	−	0.545268i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.07989e6	−	1.22627e7i	−0.0735207	−	0.127342i	0.826921	−	0.562318i	$-0.190090\pi$
$191$	−7.07989e6	−	1.22627e7i	−0.0735207	−	0.127342i	−0.900442	+	0.434976i	$0.856757\pi$
$192$		0			0
$193$	−5.70040e6	+	9.87338e6i	−0.0570761	+	0.0988588i	−0.893152	−	0.449756i	$-0.851511\pi$
$193$	−5.70040e6	+	9.87338e6i	−0.0570761	+	0.0988588i	0.836076	+	0.548614i	$0.184844\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.08549e7			0.567106			0.283553	−	0.958957i	$-0.408487\pi$
$197$	6.08549e7			0.567106			0.283553	+	0.958957i	$0.408487\pi$
$198$		0			0
$199$	4.04759e7	−	7.01063e7i	0.364092	−	0.630625i	−0.624538	−	0.780994i	$-0.714713\pi$
$199$	4.04759e7	−	7.01063e7i	0.364092	−	0.630625i	0.988630	+	0.150369i	$0.0480461\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.14644e8	−	2.79782e7i	0.961867	−	0.234738i
$204$		0			0
$205$	−4.06119e7	−	7.03418e7i	−0.329241	−	0.570263i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.14176e8			−2.38046
$210$		0			0
$211$	5.64186e6			0.0413460			0.0206730	−	0.999786i	$-0.493419\pi$
$211$	5.64186e6			0.0413460			0.0206730	+	0.999786i	$0.493419\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.31288e7	+	2.27397e7i	0.0900926	+	0.156045i
$216$		0			0
$217$	−1.31218e7	+	4.49155e7i	−0.0871732	+	0.298392i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.06188e7	+	3.57128e7i	−0.128496	+	0.222562i
$222$		0			0
$223$	1.69180e8			1.02160			0.510801	−	0.859699i	$-0.329349\pi$
$223$	1.69180e8			1.02160			0.510801	+	0.859699i	$0.329349\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.58940e7	+	1.31452e8i	−0.430643	+	0.745895i	−0.996929	−	0.0783137i	$-0.975046\pi$
$227$	−7.58940e7	+	1.31452e8i	−0.430643	+	0.745895i	0.566286	+	0.824209i	$0.308380\pi$
$228$		0			0
$229$	1.01394e8	+	1.75620e8i	0.557942	+	0.966384i	0.997668	+	0.0682522i	$0.0217423\pi$
$229$	1.01394e8	+	1.75620e8i	0.557942	+	0.966384i	−0.439726	+	0.898132i	$0.644924\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.53553e8	+	2.65962e8i	0.795266	+	1.37744i	0.922670	+	0.385591i	$0.126002\pi$
$233$	1.53553e8	+	2.65962e8i	0.795266	+	1.37744i	−0.127403	+	0.991851i	$0.540664\pi$
$234$		0			0
$235$	−9.99934e7	+	1.73194e8i	−0.502613	+	0.870551i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.38306e8			−1.12913			−0.564564	−	0.825389i	$-0.690956\pi$
$239$	−2.38306e8			−1.12913			−0.564564	+	0.825389i	$0.690956\pi$
$240$		0			0
$241$	−695427.	+	1.20451e6i	−0.00320031	+	0.00554309i	−0.867621	−	0.497226i	$-0.834352\pi$
$241$	−695427.	+	1.20451e6i	−0.00320031	+	0.00554309i	0.864421	+	0.502769i	$0.167685\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.63131e8	−	1.04210e8i	−0.708690	−	0.452717i
$246$		0			0
$247$	1.15056e8	+	1.99284e8i	0.485816	+	0.841458i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3.96244e8			1.58163			0.790815	−	0.612055i	$-0.209657\pi$
$251$	3.96244e8			1.58163			0.790815	+	0.612055i	$0.209657\pi$
$252$		0			0
$253$	4.44814e8			1.72686
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.72046e8	−	2.97993e8i	−0.632237	−	1.09507i	−0.987093	−	0.160146i	$-0.948804\pi$
$257$	−1.72046e8	−	2.97993e8i	−0.632237	−	1.09507i	0.354857	−	0.934921i	$-0.384530\pi$
$258$		0			0
$259$	1.75371e8	+	1.83421e8i	0.627203	+	0.655996i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.01986e6	−	1.56229e7i	0.0305742	−	0.0529560i	−0.850333	−	0.526244i	$-0.823600\pi$
$263$	9.01986e6	−	1.56229e7i	0.0305742	−	0.0529560i	0.880908	+	0.473288i	$0.156933\pi$
$264$		0			0
$265$	2.82021e8			0.930939
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.13538e7	−	3.69859e7i	0.0668871	−	0.115852i	−0.830642	−	0.556806i	$-0.812027\pi$
$269$	2.13538e7	−	3.69859e7i	0.0668871	−	0.115852i	0.897530	+	0.440954i	$0.145360\pi$
$270$		0			0
$271$	7.46844e7	+	1.29357e8i	0.227949	+	0.394819i	0.957200	−	0.289427i	$-0.0934648\pi$
$271$	7.46844e7	+	1.29357e8i	0.227949	+	0.394819i	−0.729251	+	0.684246i	$0.760131\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.99943e7	−	1.55875e8i	−0.260946	−	0.451972i
$276$		0			0
$277$	−1.80069e8	+	3.11889e8i	−0.509050	+	0.881701i	0.490895	+	0.871219i	$0.336670\pi$
$277$	−1.80069e8	+	3.11889e8i	−0.509050	+	0.881701i	−0.999945	+	0.0104821i	$0.996663\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.84959e8			0.497282			0.248641	−	0.968596i	$-0.420016\pi$
$281$	1.84959e8			0.497282			0.248641	+	0.968596i	$0.420016\pi$
$282$		0			0
$283$	2.14592e7	−	3.71684e7i	0.0562809	−	0.0974813i	−0.836512	−	0.547948i	$-0.815409\pi$
$283$	2.14592e7	−	3.71684e7i	0.0562809	−	0.0974813i	0.892793	+	0.450467i	$0.148742\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.04649e8	−	7.43478e7i	0.760699	−	0.185644i
$288$		0			0
$289$	1.79568e8	+	3.11020e8i	0.437608	+	0.757959i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.88345e8			1.36646			0.683228	−	0.730205i	$-0.260576\pi$
$293$	5.88345e8			1.36646			0.683228	+	0.730205i	$0.260576\pi$
$294$		0			0
$295$	4.02516e8			0.912865
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.62898e8	−	2.82148e8i	−0.352425	−	0.610419i
$300$		0			0
$301$	−9.84850e7	+	2.40347e7i	−0.208155	+	0.0507991i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.88471e8	+	4.99646e8i	−0.582173	+	1.00835i
$306$		0			0
$307$	−4.17824e7			−0.0824155			−0.0412077	−	0.999151i	$-0.513121\pi$
$307$	−4.17824e7			−0.0824155			−0.0412077	+	0.999151i	$0.513121\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.03292e8	+	3.52113e8i	−0.383230	+	0.663774i	−0.991522	−	0.129939i	$-0.958522\pi$
$311$	−2.03292e8	+	3.52113e8i	−0.383230	+	0.663774i	0.608292	+	0.793714i	$0.291855\pi$
$312$		0			0
$313$	2.29934e8	+	3.98258e8i	0.423837	+	0.734107i	0.996311	−	0.0858160i	$-0.0273497\pi$
$313$	2.29934e8	+	3.98258e8i	0.423837	+	0.734107i	−0.572474	+	0.819923i	$0.694016\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.06886e8	+	5.31543e8i	0.541091	+	0.937197i	0.998842	+	0.0481166i	$0.0153219\pi$
$317$	3.06886e8	+	5.31543e8i	0.541091	+	0.937197i	−0.457751	+	0.889081i	$0.651345\pi$
$318$		0			0
$319$	5.11581e8	−	8.86085e8i	0.882363	−	1.52830i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.85725e8			−0.471780
$324$		0			0
$325$	−6.59149e7	+	1.14168e8i	−0.106510	+	0.184481i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.33581e8	−	5.58075e8i	−0.826066	−	0.863987i
$330$		0			0
$331$	5.71651e8	+	9.90129e8i	0.866429	+	1.50070i	0.865621	+	0.500700i	$0.166924\pi$
$331$	5.71651e8	+	9.90129e8i	0.866429	+	1.50070i	0.000808163		1.00000i	$0.499743\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.47392e8			0.214199
$336$		0			0
$337$	−8.58803e8			−1.22233			−0.611166	−	0.791503i	$-0.709299\pi$
$337$	−8.58803e8			−1.22233			−0.611166	+	0.791503i	$0.709299\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.02853e8	+	3.51352e8i	0.277039	+	0.479846i
$342$		0			0
$343$	5.62806e8	−	4.91727e8i	0.753060	−	0.657952i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.59150e8	+	4.48860e8i	−0.332964	+	0.576711i	−0.983092	−	0.183114i	$-0.941382\pi$
$347$	−2.59150e8	+	4.48860e8i	−0.332964	+	0.576711i	0.650128	+	0.759825i	$0.274715\pi$
$348$		0			0
$349$	−4.10903e8			−0.517428			−0.258714	−	0.965954i	$-0.583299\pi$
$349$	−4.10903e8			−0.517428			−0.258714	+	0.965954i	$0.583299\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.77472e8	−	1.34662e9i	0.940749	−	1.62942i	0.176701	−	0.984265i	$-0.443458\pi$
$353$	7.77472e8	−	1.34662e9i	0.940749	−	1.62942i	0.764048	−	0.645160i	$-0.223209\pi$
$354$		0			0
$355$	3.49533e8	+	6.05409e8i	0.414657	+	0.718206i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.73553e8	−	1.51304e9i	−0.996457	−	1.72591i	−0.571062	−	0.820907i	$-0.693469\pi$
$359$	−8.73553e8	−	1.51304e9i	−0.996457	−	1.72591i	−0.425395	−	0.905008i	$-0.639865\pi$
$360$		0			0
$361$	−3.50263e8	+	6.06673e8i	−0.391849	+	0.678703i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.23949e8			−0.241060
$366$		0			0
$367$	1.44227e8	−	2.49808e8i	0.152305	−	0.263800i	−0.779769	−	0.626067i	$-0.784664\pi$
$367$	1.44227e8	−	2.49808e8i	0.152305	−	0.263800i	0.932074	+	0.362267i	$0.117997\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.05332e8	+	1.04514e9i	−0.310430	+	1.06259i
$372$		0			0
$373$	−9.04946e7	−	1.56741e8i	−0.0902904	−	0.156388i	0.817343	−	0.576152i	$-0.195446\pi$
$373$	−9.04946e7	−	1.56741e8i	−0.0902904	−	0.156388i	−0.907633	+	0.419764i	$0.862113\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.49399e8			−0.720308
$378$		0			0
$379$	1.42653e9			1.34599			0.672997	−	0.739645i	$-0.265007\pi$
$379$	1.42653e9			1.34599			0.672997	+	0.739645i	$0.265007\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.99160e8	+	1.55739e9i	0.817789	+	1.41645i	0.907308	+	0.420467i	$0.138134\pi$
$383$	8.99160e8	+	1.55739e9i	0.817789	+	1.41645i	−0.0895186	+	0.995985i	$0.528533\pi$
$384$		0			0
$385$	−1.63049e9	+	3.97913e8i	−1.45615	+	0.355366i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.57575e8	−	2.72928e8i	0.135726	−	0.235085i	−0.790148	−	0.612916i	$-0.789997\pi$
$389$	1.57575e8	−	2.72928e8i	0.135726	−	0.235085i	0.925875	+	0.377831i	$0.123330\pi$
$390$		0			0
$391$	4.04533e8			0.342244
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.94102e8	+	1.20222e9i	−0.566674	+	0.981509i
$396$		0			0
$397$	−8.25550e8	−	1.42989e9i	−0.662181	−	1.14693i	−0.980041	−	0.198793i	$-0.936298\pi$
$397$	−8.25550e8	−	1.42989e9i	−0.662181	−	1.14693i	0.317861	−	0.948137i	$-0.397036\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.17565e8	−	8.96449e8i	−0.400830	−	0.694257i	0.592997	−	0.805205i	$-0.297945\pi$
$401$	−5.17565e8	−	8.96449e8i	−0.400830	−	0.694257i	−0.993826	+	0.110948i	$0.964611\pi$
$402$		0			0
$403$	1.48577e8	−	2.57342e8i	0.113079	−	0.195859i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.20023e9			1.61766
$408$		0			0
$409$	4.39925e8	−	7.61973e8i	0.317941	−	0.550691i	−0.662117	−	0.749401i	$-0.730342\pi$
$409$	4.39925e8	−	7.61973e8i	0.317941	−	0.550691i	0.980058	+	0.198710i	$0.0636751\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.35787e8	+	1.49169e9i	−0.304403	+	1.04196i
$414$		0			0
$415$	−1.15019e9	−	1.99219e9i	−0.789954	−	1.36824i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.88360e8			0.125095			0.0625473	−	0.998042i	$-0.480078\pi$
$419$	1.88360e8			0.125095			0.0625473	+	0.998042i	$0.480078\pi$
$420$		0			0
$421$	−2.55915e9			−1.67151			−0.835753	−	0.549106i	$-0.814968\pi$
$421$	−2.55915e9			−1.67151			−0.835753	+	0.549106i	$0.814968\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.18448e7	−	1.41759e8i	−0.0517166	−	0.0895757i
$426$		0			0
$427$	−1.53933e9	−	1.60999e9i	−0.956827	−	1.00075i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.09832e9	−	1.90235e9i	0.660785	−	1.14451i	−0.319625	−	0.947544i	$-0.603557\pi$
$431$	1.09832e9	−	1.90235e9i	0.660785	−	1.14451i	0.980410	−	0.196969i	$-0.0631097\pi$
$432$		0			0
$433$	−2.66829e9			−1.57952			−0.789761	−	0.613414i	$-0.789796\pi$
$433$	−2.66829e9			−1.57952			−0.789761	+	0.613414i	$0.789796\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.12868e9	−	1.95494e9i	0.646974	−	1.12059i
$438$		0			0
$439$	−4.10653e8	−	7.11272e8i	−0.231659	−	0.401245i	0.726637	−	0.687021i	$-0.241082\pi$
$439$	−4.10653e8	−	7.11272e8i	−0.231659	−	0.401245i	−0.958296	+	0.285776i	$0.907749\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.27229e8	−	1.08639e9i	−0.342778	−	0.593709i	0.642169	−	0.766563i	$-0.278035\pi$
$443$	−6.27229e8	−	1.08639e9i	−0.342778	−	0.593709i	−0.984948	+	0.172854i	$0.944701\pi$
$444$		0			0
$445$	−6.09187e8	+	1.05514e9i	−0.327711	+	0.567612i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8.34122e7			0.0434878			0.0217439	−	0.999764i	$-0.493078\pi$
$449$	8.34122e7			0.0434878			0.0217439	+	0.999764i	$0.493078\pi$
$450$		0			0
$451$	1.35945e9	−	2.35464e9i	0.697823	−	1.20866i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.49513e8	+	8.88511e8i	0.422796	+	0.442204i
$456$		0			0
$457$	−2.00176e8	−	3.46714e8i	−0.0981080	−	0.169928i	0.812794	−	0.582552i	$-0.197946\pi$
$457$	−2.00176e8	−	3.46714e8i	−0.0981080	−	0.169928i	−0.910901	+	0.412624i	$0.864612\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.55230e7			−0.0216410			−0.0108205	−	0.999941i	$-0.503444\pi$
$461$	−4.55230e7			−0.0216410			−0.0108205	+	0.999941i	$0.503444\pi$
$462$		0			0
$463$	1.41715e9			0.663565			0.331782	−	0.943356i	$-0.392350\pi$
$463$	1.41715e9			0.663565			0.331782	+	0.943356i	$0.392350\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.76416e8	−	1.34479e9i	−0.352765	−	0.611007i	0.633968	−	0.773359i	$-0.281425\pi$
$467$	−7.76416e8	−	1.34479e9i	−0.352765	−	0.611007i	−0.986733	+	0.162352i	$0.948092\pi$
$468$		0			0
$469$	−1.59575e8	+	5.46222e8i	−0.0714267	+	0.244492i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.39475e8	+	7.61192e8i	−0.190950	+	0.330735i
$474$		0			0
$475$	−9.13416e8			−0.391058
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.98584e8	+	1.03678e9i	−0.248857	+	0.431034i	−0.963209	−	0.268753i	$-0.913388\pi$
$479$	−5.98584e8	+	1.03678e9i	−0.248857	+	0.431034i	0.714352	+	0.699787i	$0.246722\pi$
$480$		0			0
$481$	−8.05763e8	−	1.39562e9i	−0.330141	−	0.571821i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.72538e8	+	9.91664e8i	0.227881	+	0.394701i
$486$		0			0
$487$	−8.14788e8	+	1.41125e9i	−0.319664	+	0.553674i	−0.980418	−	0.196929i	$-0.936903\pi$
$487$	−8.14788e8	+	1.41125e9i	−0.319664	+	0.553674i	0.660754	+	0.750602i	$0.270237\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.66094e9			−0.633240			−0.316620	−	0.948553i	$-0.602548\pi$
$491$	−1.66094e9			−0.633240			−0.316620	+	0.948553i	$0.602548\pi$
$492$		0			0
$493$	4.65255e8	−	8.05845e8i	0.174875	−	0.302892i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.62201e9	+	6.39887e8i	−0.958048	+	0.233806i
$498$		0			0
$499$	4.21787e8	+	7.30556e8i	0.151964	+	0.263210i	0.931949	−	0.362588i	$-0.118107\pi$
$499$	4.21787e8	+	7.30556e8i	0.151964	+	0.263210i	−0.779985	+	0.625798i	$0.784774\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.39033e7			−0.00837472			−0.00418736	−	0.999991i	$-0.501333\pi$
$503$	−2.39033e7			−0.00837472			−0.00418736	+	0.999991i	$0.501333\pi$
$504$		0			0
$505$	−9.57307e8			−0.330774
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.04440e9	+	3.54101e9i	0.687153	+	1.19018i	0.972755	+	0.231836i	$0.0744733\pi$
$509$	2.04440e9	+	3.54101e9i	0.687153	+	1.19018i	−0.285601	+	0.958348i	$0.592193\pi$
$510$		0			0
$511$	2.42460e8	−	8.29935e8i	0.0803836	−	0.275151i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.68251e9	+	2.91420e9i	−0.542791	+	0.940142i
$516$		0			0
$517$	−6.69440e9			−2.13056
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.40519e7	+	1.28262e8i	−0.0229406	+	0.0397342i	−0.877268	−	0.480001i	$-0.840636\pi$
$521$	−7.40519e7	+	1.28262e8i	−0.0229406	+	0.0397342i	0.854327	+	0.519736i	$0.173970\pi$
$522$		0			0
$523$	−8.06974e8	−	1.39772e9i	−0.246663	−	0.427232i	0.715935	−	0.698167i	$-0.246001\pi$
$523$	−8.06974e8	−	1.39772e9i	−0.246663	−	0.427232i	−0.962598	+	0.270934i	$0.912667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.84484e8	+	3.19535e8i	0.0549061	+	0.0951002i
$528$		0			0
$529$	1.04411e8	−	1.80846e8i	0.0306657	−	0.0531145i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.99141e9			−0.569661
$534$		0			0
$535$	2.22751e9	−	3.85817e9i	0.628901	−	1.08929i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.90638e8	−	6.47326e9i	0.0799450	−	1.78058i
$540$		0			0
$541$	1.89435e9	+	3.28111e9i	0.514362	+	0.890902i	0.999861	+	0.0166643i	$0.00530466\pi$
$541$	1.89435e9	+	3.28111e9i	0.514362	+	0.890902i	−0.485499	+	0.874237i	$0.661362\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.33452e9			0.353132
$546$		0			0
$547$	2.21090e9			0.577582			0.288791	−	0.957392i	$-0.406747\pi$
$547$	2.21090e9			0.577582			0.288791	+	0.957392i	$0.406747\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.59620e9	−	4.49676e9i	−0.661162	−	1.14517i
$552$		0			0
$553$	−3.70384e9	−	3.87387e9i	−0.931353	−	0.974107i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.70947e7	−	1.68173e8i	0.0238069	−	0.0412347i	−0.853877	−	0.520476i	$-0.825755\pi$
$557$	9.70947e7	−	1.68173e8i	0.0238069	−	0.0412347i	0.877683	+	0.479241i	$0.159088\pi$
$558$		0			0
$559$	6.43772e8			0.155880
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.43470e9	−	2.48497e9i	0.338830	−	0.586871i	−0.645383	−	0.763859i	$-0.723302\pi$
$563$	1.43470e9	−	2.48497e9i	0.338830	−	0.586871i	0.984213	+	0.176989i	$0.0566356\pi$
$564$		0			0
$565$	7.45458e8	+	1.29117e9i	0.173882	+	0.301172i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.75350e9	+	4.76921e9i	0.626604	+	1.08531i	0.988228	+	0.152986i	$0.0488888\pi$
$569$	2.75350e9	+	4.76921e9i	0.626604	+	1.08531i	−0.361625	+	0.932324i	$0.617778\pi$
$570$		0			0
$571$	2.55175e9	−	4.41975e9i	0.573603	−	0.993509i	−0.422589	−	0.906321i	$-0.638879\pi$
$571$	2.55175e9	−	4.41975e9i	0.573603	−	0.993509i	0.996192	−	0.0871876i	$-0.0277880\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.29322e9			0.283685
$576$		0			0
$577$	−1.72918e9	+	2.99503e9i	−0.374736	+	0.649062i	−0.990287	−	0.139035i	$-0.955600\pi$
$577$	−1.72918e9	+	2.99503e9i	−0.374736	+	0.649062i	0.615552	+	0.788097i	$0.288933\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.62813e9	−	2.10564e9i	1.82516	−	0.445419i
$582$		0			0
$583$	4.72022e9	+	8.17566e9i	0.986557	+	1.70877i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.14732e9			0.846318			0.423159	−	0.906055i	$-0.360921\pi$
$587$	4.14732e9			0.846318			0.423159	+	0.906055i	$0.360921\pi$
$588$		0			0
$589$	2.05890e9			0.415176
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.97134e9	−	6.87856e9i	−0.782069	−	1.35458i	−0.930734	−	0.365696i	$-0.880831\pi$
$593$	−3.97134e9	−	6.87856e9i	−0.782069	−	1.35458i	0.148665	−	0.988888i	$-0.452502\pi$
$594$		0			0
$595$	−1.48284e9	+	3.61879e8i	−0.288593	+	0.0704295i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.80065e9	+	3.11882e9i	−0.342323	+	0.592921i	−0.984864	−	0.173331i	$-0.944547\pi$
$599$	−1.80065e9	+	3.11882e9i	−0.342323	+	0.592921i	0.642541	+	0.766252i	$0.277880\pi$
$600$		0			0
$601$	−7.09831e8			−0.133381			−0.0666906	−	0.997774i	$-0.521244\pi$
$601$	−7.09831e8			−0.133381			−0.0666906	+	0.997774i	$0.521244\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.98559e9	+	8.63529e9i	−0.915319	+	1.58538i
$606$		0			0
$607$	3.94285e9	+	6.82922e9i	0.715566	+	1.23940i	0.962741	+	0.270426i	$0.0871646\pi$
$607$	3.94285e9	+	6.82922e9i	0.715566	+	1.23940i	−0.247174	+	0.968971i	$0.579502\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.45160e9	+	4.24630e9i	0.434816	+	0.753123i
$612$		0			0
$613$	−5.08263e9	+	8.80338e9i	−0.891204	+	1.54361i	−0.0527708	+	0.998607i	$0.516805\pi$
$613$	−5.08263e9	+	8.80338e9i	−0.891204	+	1.54361i	−0.838433	+	0.545004i	$0.816528\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.64250e9			−0.967105			−0.483552	−	0.875315i	$-0.660654\pi$
$617$	−5.64250e9			−0.967105			−0.483552	+	0.875315i	$0.660654\pi$
$618$		0			0
$619$	5.82276e9	−	1.00853e10i	0.986760	−	1.70912i	0.352922	−	0.935653i	$-0.385188\pi$
$619$	5.82276e9	−	1.00853e10i	0.986760	−	1.70912i	0.633838	−	0.773466i	$-0.281479\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.25072e9	−	3.39995e9i	−0.538607	−	0.563332i
$624$		0			0
$625$	1.89654e9	+	3.28490e9i	0.310729	+	0.538199i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.00099e9			0.320603
$630$		0			0
$631$	1.07600e10			1.70494			0.852469	−	0.522777i	$-0.175104\pi$
$631$	1.07600e10			1.70494			0.852469	+	0.522777i	$0.175104\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.84224e9	+	8.38701e9i	0.750479	+	1.29987i
$636$		0			0
$637$	−4.21247e9	+	2.18627e9i	−0.645727	+	0.335131i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.13595e9	−	8.89572e9i	0.770225	−	1.33407i	−0.167215	−	0.985921i	$-0.553477\pi$
$641$	5.13595e9	−	8.89572e9i	0.770225	−	1.33407i	0.937439	−	0.348148i	$-0.113189\pi$
$642$		0			0
$643$	1.16555e10			1.72899			0.864497	−	0.502638i	$-0.167637\pi$
$643$	1.16555e10			1.72899			0.864497	+	0.502638i	$0.167637\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.47741e9	+	2.55895e9i	−0.214455	+	0.371447i	−0.953104	−	0.302644i	$-0.902131\pi$
$647$	−1.47741e9	+	2.55895e9i	−0.214455	+	0.371447i	0.738649	+	0.674090i	$0.235464\pi$
$648$		0			0
$649$	6.73695e9	+	1.16687e10i	0.967402	+	1.67559i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.44343e9	−	4.23214e9i	−0.343403	−	0.594791i	0.641660	−	0.766990i	$-0.278246\pi$
$653$	−2.44343e9	−	4.23214e9i	−0.343403	−	0.594791i	−0.985062	+	0.172199i	$0.944913\pi$
$654$		0			0
$655$	−2.57030e9	+	4.45190e9i	−0.357388	+	0.619014i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.12839e10			1.53589			0.767947	−	0.640513i	$-0.221278\pi$
$659$	1.12839e10			1.53589			0.767947	+	0.640513i	$0.221278\pi$
$660$		0			0
$661$	3.19740e9	−	5.53807e9i	0.430618	−	0.745853i	−0.566308	−	0.824193i	$-0.691629\pi$
$661$	3.19740e9	−	5.53807e9i	0.430618	−	0.745853i	0.996927	+	0.0783406i	$0.0249622\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.38846e9	+	8.17563e9i	−0.314950	+	1.07807i
$666$		0			0
$667$	3.67573e9	+	6.36656e9i	0.479627	+	0.830738i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.93127e10			−2.46782
$672$		0			0
$673$	2.86561e8			0.0362380			0.0181190	−	0.999836i	$-0.494232\pi$
$673$	2.86561e8			0.0362380			0.0181190	+	0.999836i	$0.494232\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.77078e9	−	1.17273e10i	−0.838646	−	1.45258i	−0.891027	−	0.453950i	$-0.850015\pi$
$677$	−6.77078e9	−	1.17273e10i	−0.838646	−	1.45258i	0.0523818	−	0.998627i	$-0.483319\pi$
$678$		0			0
$679$	−4.29488e9	+	1.04814e9i	−0.526510	+	0.128492i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.56295e9	−	2.70711e9i	0.187704	−	0.325112i	−0.756781	−	0.653669i	$-0.773229\pi$
$683$	1.56295e9	−	2.70711e9i	0.187704	−	0.325112i	0.944484	+	0.328557i	$0.106562\pi$
$684$		0			0
$685$	−8.81307e9			−1.04763
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.45725e9	−	5.98813e9i	0.402683	−	0.697467i
$690$		0			0
$691$	3.72246e9	+	6.44749e9i	0.429197	+	0.743392i	0.996802	−	0.0799095i	$-0.0254631\pi$
$691$	3.72246e9	+	6.44749e9i	0.429197	+	0.743392i	−0.567605	+	0.823301i	$0.692130\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.70258e9	+	6.41305e9i	0.418367	+	0.724633i
$696$		0			0
$697$	1.23634e9	−	2.14141e9i	0.138301	−	0.239544i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−8.02247e9			−0.879620			−0.439810	−	0.898091i	$-0.644954\pi$
$701$	−8.02247e9			−0.879620			−0.439810	+	0.898091i	$0.644954\pi$
$702$		0			0
$703$	5.58294e9	−	9.66993e9i	0.606065	−	1.04974i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.03643e9	−	3.54769e9i	0.110300	−	0.377553i
$708$		0			0
$709$	9.07598e9	+	1.57201e10i	0.956383	+	1.65650i	0.731172	+	0.682193i	$0.238974\pi$
$709$	9.07598e9	+	1.57201e10i	0.956383	+	1.65650i	0.225211	+	0.974310i	$0.427693\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.91502e9			−0.301181
$714$		0			0
$715$	1.06581e10			1.09046
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.98920e9	+	1.21056e10i	0.701255	+	1.21461i	0.968026	+	0.250850i	$0.0807100\pi$
$719$	6.98920e9	+	1.21056e10i	0.701255	+	1.21461i	−0.266771	+	0.963760i	$0.585957\pi$
$720$		0			0
$721$	−8.97816e9	−	9.39030e9i	−0.892100	−	0.933053i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.48734e9	−	2.57615e9i	0.144953	−	0.251066i
$726$		0			0
$727$	−6.81996e9			−0.658281			−0.329140	−	0.944281i	$-0.606759\pi$
$727$	−6.81996e9			−0.658281			−0.329140	+	0.944281i	$0.606759\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.99678e8	+	6.92262e8i	−0.0378442	+	0.0655480i
$732$		0			0
$733$	−5.74796e9	−	9.95576e9i	−0.539076	−	0.933706i	−0.998954	−	0.0457245i	$-0.985440\pi$
$733$	−5.74796e9	−	9.95576e9i	−0.539076	−	0.933706i	0.459879	−	0.887982i	$-0.347893\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.46692e9	+	4.27283e9i	0.226996	+	0.393169i
$738$		0			0
$739$	−6.74312e8	+	1.16794e9i	−0.0614618	+	0.106455i	−0.895119	−	0.445827i	$-0.852910\pi$
$739$	−6.74312e8	+	1.16794e9i	−0.0614618	+	0.106455i	0.833657	+	0.552282i	$0.186243\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.00361e9			−0.179206			−0.0896029	−	0.995978i	$-0.528560\pi$
$743$	−2.00361e9			−0.179206			−0.0896029	+	0.995978i	$0.528560\pi$
$744$		0			0
$745$	1.94806e9	−	3.37414e9i	0.172606	−	0.298962i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.18864e10	+	1.24320e10i	1.03363	+	1.08107i
$750$		0			0
$751$	3.69709e9	+	6.40354e9i	0.318508	+	0.551671i	0.980177	−	0.198125i	$-0.0634850\pi$
$751$	3.69709e9	+	6.40354e9i	0.318508	+	0.551671i	−0.661669	+	0.749796i	$0.730152\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.46561e10			−1.23937
$756$		0			0
$757$	1.79000e10			1.49975			0.749874	−	0.661581i	$-0.230114\pi$
$757$	1.79000e10			1.49975			0.749874	+	0.661581i	$0.230114\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.51905e9	+	1.12913e10i	0.536214	+	0.928749i	0.999104	+	0.0423334i	$0.0134792\pi$
$761$	6.51905e9	+	1.12913e10i	0.536214	+	0.928749i	−0.462890	+	0.886416i	$0.653188\pi$
$762$		0			0
$763$	−1.44483e9	+	4.94561e9i	−0.117755	+	0.403073i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.93437e9	−	8.54658e9i	0.394865	−	0.683926i
$768$		0			0
$769$	−1.72914e10			−1.37116			−0.685580	−	0.727997i	$-0.740451\pi$
$769$	−1.72914e10			−1.37116			−0.685580	+	0.727997i	$0.740451\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.33950e9	−	9.24829e9i	0.415789	−	0.720167i	−0.579722	−	0.814814i	$-0.696839\pi$
$773$	5.33950e9	−	9.24829e9i	0.415789	−	0.720167i	0.995511	+	0.0946470i	$0.0301722\pi$
$774$		0			0
$775$	5.89764e8	+	1.02150e9i	0.0455116	+	0.0788284i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.89902e9	−	1.19494e10i	−0.522885	−	0.905663i
$780$		0			0
$781$	−1.17003e10	+	2.02656e10i	−0.878860	+	1.52223i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	9.35608e9			0.690319
$786$		0			0
$787$	1.13691e10	−	1.96918e10i	0.831407	−	1.44004i	−0.0655160	−	0.997852i	$-0.520869\pi$
$787$	1.13691e10	−	1.96918e10i	0.831407	−	1.44004i	0.896923	−	0.442187i	$-0.145797\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5.59204e9	+	1.36470e9i	−0.401747	+	0.0980440i
$792$		0			0
$793$	7.07262e9	+	1.22501e10i	0.503645	+	0.872338i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.15349e10			0.807067			0.403534	−	0.914965i	$-0.367782\pi$
$797$	1.15349e10			0.807067			0.403534	+	0.914965i	$0.367782\pi$
$798$		0			0
$799$	−6.08818e9			−0.422254
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.74826e9	−	6.49218e9i	−0.255462	−	0.442472i
$804$		0			0
$805$	3.38160e9	−	1.15751e10i	0.228474	−	0.782061i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.85450e8	−	1.18723e9i	0.0455152	−	0.0788346i	−0.842370	−	0.538899i	$-0.818840\pi$
$809$	6.85450e8	−	1.18723e9i	0.0455152	−	0.0788346i	0.887885	+	0.460065i	$0.152174\pi$
$810$		0			0
$811$	−2.38277e10			−1.56859			−0.784293	−	0.620390i	$-0.786974\pi$
$811$	−2.38277e10			−1.56859			−0.784293	+	0.620390i	$0.786974\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.93702e9	−	6.81912e9i	0.254751	−	0.441242i
$816$		0			0
$817$	2.23027e9	+	3.86294e9i	0.143081	+	0.247823i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.09835e10	−	1.90240e10i	−0.692691	−	1.19978i	−0.970953	−	0.239271i	$-0.923092\pi$
$821$	−1.09835e10	−	1.90240e10i	−0.692691	−	1.19978i	0.278262	−	0.960505i	$-0.410242\pi$
$822$		0			0
$823$	−4.92559e8	+	8.53137e8i	−0.0308006	+	0.0533481i	−0.881015	−	0.473089i	$-0.843139\pi$
$823$	−4.92559e8	+	8.53137e8i	−0.0308006	+	0.0533481i	0.850214	+	0.526437i	$0.176472\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.53940e10			0.946415			0.473208	−	0.880951i	$-0.343096\pi$
$827$	1.53940e10			0.946415			0.473208	+	0.880951i	$0.343096\pi$
$828$		0			0
$829$	1.18113e10	−	2.04578e10i	0.720040	−	1.24715i	−0.240944	−	0.970539i	$-0.577457\pi$
$829$	1.18113e10	−	2.04578e10i	0.720040	−	1.24715i	0.960983	−	0.276606i	$-0.0892097\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.64319e8	−	5.88707e9i	0.0158442	−	0.352892i
$834$		0			0
$835$	9.76923e9	+	1.69208e10i	0.580708	+	1.00582i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.00471e10			0.587319			0.293660	−	0.955910i	$-0.405127\pi$
$839$	1.00471e10			0.587319			0.293660	+	0.955910i	$0.405127\pi$
$840$		0			0
$841$	−3.39991e8			−0.0197098
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.47140e9	+	6.01263e9i	0.197927	+	0.342820i
$846$		0			0
$847$	−2.66039e10	−	2.78252e10i	−1.50437	−	1.57342i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.90438e9	+	1.36908e10i	−0.439658	+	0.761509i
$852$		0			0
$853$	−1.86048e10			−1.02637			−0.513185	−	0.858278i	$-0.671534\pi$
$853$	−1.86048e10			−1.02637			−0.513185	+	0.858278i	$0.671534\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.41716e10	+	2.45459e10i	−0.769105	+	1.33213i	0.168944	+	0.985626i	$0.445964\pi$
$857$	−1.41716e10	+	2.45459e10i	−0.769105	+	1.33213i	−0.938049	+	0.346503i	$0.887369\pi$
$858$		0			0
$859$	−1.72295e10	−	2.98424e10i	−0.927463	−	1.60641i	−0.787551	−	0.616250i	$-0.788651\pi$
$859$	−1.72295e10	−	2.98424e10i	−0.927463	−	1.60641i	−0.139913	−	0.990164i	$-0.544682\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.59668e9	+	4.49758e9i	0.137525	+	0.238199i	0.926559	−	0.376149i	$-0.122752\pi$
$863$	2.59668e9	+	4.49758e9i	0.137525	+	0.238199i	−0.789034	+	0.614349i	$0.789419\pi$
$864$		0			0
$865$	8.76755e9	−	1.51858e10i	0.460598	−	0.797779i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.64690e10			−2.40212
$870$		0			0
$871$	1.80686e9	−	3.12957e9i	0.0926531	−	0.160480i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.09300e10	+	5.10784e9i	−1.05619	+	0.257756i
$876$		0			0
$877$	−1.25283e10	−	2.16997e10i	−0.627184	−	1.08632i	−0.988114	−	0.153722i	$-0.950874\pi$
$877$	−1.25283e10	−	2.16997e10i	−0.627184	−	1.08632i	0.360930	−	0.932593i	$-0.382459\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.93660e10			1.44687			0.723434	−	0.690394i	$-0.242563\pi$
$881$	2.93660e10			1.44687			0.723434	+	0.690394i	$0.242563\pi$
$882$		0			0
$883$	−1.74989e10			−0.855360			−0.427680	−	0.903930i	$-0.640669\pi$
$883$	−1.74989e10			−0.855360			−0.427680	+	0.903930i	$0.640669\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.08503e9	−	5.34343e9i	−0.148432	−	0.257091i	0.782216	−	0.623007i	$-0.214089\pi$
$887$	−3.08503e9	−	5.34343e9i	−0.148432	−	0.257091i	−0.930648	+	0.365916i	$0.880756\pi$
$888$		0			0
$889$	−3.63240e10	+	8.86465e9i	−1.73395	+	0.423161i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.69866e10	+	2.94216e10i	−0.798225	+	1.38257i
$894$		0			0
$895$	−3.43056e10			−1.59950
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.35257e9	+	5.80682e9i	−0.153893	+	0.266551i
$900$		0			0
$901$	4.29277e9	+	7.43530e9i	0.195525	+	0.338658i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.69932e9	+	1.67997e10i	0.434982	+	0.753411i
$906$		0			0
$907$	−1.53133e10	+	2.65234e10i	−0.681464	+	1.18033i	0.293069	+	0.956091i	$0.405323\pi$
$907$	−1.53133e10	+	2.65234e10i	−0.681464	+	1.18033i	−0.974534	+	0.224240i	$0.928010\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.76610e10			1.21214			0.606071	−	0.795411i	$-0.292745\pi$
$911$	2.76610e10			1.21214			0.606071	+	0.795411i	$0.292745\pi$
$912$		0			0
$913$	3.85017e10	−	6.66869e10i	1.67430	−	2.89997i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.37156e10	−	1.43452e10i	−0.587382	−	0.614347i
$918$		0			0
$919$	5.88796e8	+	1.01982e9i	0.0250242	+	0.0433432i	0.878266	−	0.478172i	$-0.158700\pi$
$919$	5.88796e8	+	1.01982e9i	0.0250242	+	0.0433432i	−0.853242	+	0.521515i	$0.825367\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.71394e10			0.717448
$924$		0			0
$925$	6.39683e9			0.265747
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.36178e10	−	4.09073e10i	−0.966463	−	1.67396i	−0.705633	−	0.708578i	$-0.749337\pi$
$929$	−2.36178e10	−	4.09073e10i	−0.966463	−	1.67396i	−0.260830	−	0.965385i	$-0.583996\pi$
$930$		0			0
$931$	−2.77122e10	−	1.77028e10i	−1.12551	−	0.718982i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.61697e9	+	1.14609e10i	−0.264739	+	0.458542i
$936$		0			0
$937$	1.50207e10			0.596489			0.298245	−	0.954489i	$-0.403599\pi$
$937$	1.50207e10			0.596489			0.298245	+	0.954489i	$0.403599\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.66955e10	−	2.89175e10i	0.653185	−	1.13135i	−0.329161	−	0.944274i	$-0.606766\pi$
$941$	1.66955e10	−	2.89175e10i	0.653185	−	1.13135i	0.982346	−	0.187075i	$-0.0599008\pi$
$942$		0			0
$943$	9.76770e9	+	1.69182e10i	0.379316	+	0.656995i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.60668e9	−	1.14431e10i	−0.252789	−	0.437843i	0.711504	−	0.702682i	$-0.248015\pi$
$947$	−6.60668e9	−	1.14431e10i	−0.252789	−	0.437843i	−0.964293	+	0.264839i	$0.914681\pi$
$948$		0			0
$949$	−2.74535e9	+	4.75509e9i	−0.104272	+	0.180604i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.62784e10			0.983499			0.491750	−	0.870737i	$-0.336358\pi$
$953$	2.62784e10			0.983499			0.491750	+	0.870737i	$0.336358\pi$
$954$		0			0
$955$	−1.66414e9	+	2.88238e9i	−0.0618271	+	0.107088i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.54152e9	−	3.26604e10i	0.349343	−	1.19579i
$960$		0			0
$961$	1.24269e10	+	2.15241e10i	0.451682	+	0.782335i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.67978e9			0.0959961
$966$		0			0
$967$	4.66478e10			1.65897			0.829486	−	0.558528i	$-0.188634\pi$
$967$	4.66478e10			1.65897			0.829486	+	0.558528i	$0.188634\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.09381e9	+	7.09069e9i	0.143503	+	0.248554i	0.928813	−	0.370548i	$-0.120830\pi$
$971$	4.09381e9	+	7.09069e9i	0.143503	+	0.248554i	−0.785310	+	0.619102i	$0.787497\pi$
$972$		0			0
$973$	−2.77748e10	+	6.77828e9i	−0.966620	+	0.235898i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.36224e9	+	1.10197e10i	−0.218262	+	0.378042i	−0.954277	−	0.298924i	$-0.903372\pi$
$977$	−6.36224e9	+	1.10197e10i	−0.218262	+	0.378042i	0.736014	+	0.676966i	$0.236706\pi$
$978$		0			0
$979$	−4.07841e10			−1.38916
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.99730e9	−	8.65558e9i	0.167802	−	0.290642i	−0.769844	−	0.638232i	$-0.779666\pi$
$983$	4.99730e9	−	8.65558e9i	0.167802	−	0.290642i	0.937647	+	0.347589i	$0.113000\pi$
$984$		0			0
$985$	−7.15204e9	−	1.23877e10i	−0.238453	−	0.413013i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.15764e9	−	5.46920e9i	−0.103795	−	0.179778i
$990$		0			0
$991$	6.83437e8	−	1.18375e9i	0.0223070	−	0.0386368i	−0.854656	−	0.519194i	$-0.826232\pi$
$991$	6.83437e8	−	1.18375e9i	0.0223070	−	0.0386368i	0.876963	+	0.480557i	$0.159566\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.90279e10			−0.612364
$996$		0			0
$997$	7.10973e9	−	1.23144e10i	0.227206	−	0.393533i	−0.729773	−	0.683690i	$-0.760374\pi$
$997$	7.10973e9	−	1.23144e10i	0.227206	−	0.393533i	0.956979	+	0.290157i	$0.0937075\pi$
$998$		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.109.4	yes	16
3.2	odd	2	inner	252.8.k.e.109.5	yes	16
7.2	even	3	inner	252.8.k.e.37.4	✓	16
21.2	odd	6	inner	252.8.k.e.37.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.k.e.37.4	✓	16	7.2	even	3	inner
252.8.k.e.37.5	yes	16	21.2	odd	6	inner
252.8.k.e.109.4	yes	16	1.1	even	1	trivial
252.8.k.e.109.5	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-117.526 - 203.561i) q^{5} +(881.619 - 215.154i) q^{7} +(3934.09 - 6814.05i) q^{11} -5762.92 q^{13} +(3577.84 - 6196.99i) q^{17} +(-19965.0 - 34580.3i) q^{19} +(28266.6 + 48959.2i) q^{23} +(11437.7 - 19810.8i) q^{25} +130038. q^{29} +(-25781.5 + 44654.8i) q^{31} +(-147410. - 154177. i) q^{35} +(139818. + 242173. i) q^{37} +345556. q^{41} -111709. q^{43} +(-425409. - 736830. i) q^{47} +(730961. - 379368. i) q^{49} +(-599912. + 1.03908e6i) q^{53} -1.84943e6 q^{55} +(-856228. + 1.48303e6i) q^{59} +(-1.22726e6 - 2.12568e6i) q^{61} +(677294. + 1.17311e6i) q^{65} +(-313531. + 543052. i) q^{67} -2.97409e6 q^{71} +(476382. - 825118. i) q^{73} +(2.00230e6 - 6.85383e6i) q^{77} +(-2.95297e6 - 5.11469e6i) q^{79} +9.78669e6 q^{83} -1.68196e6 q^{85} +(-2.59171e6 - 4.48897e6i) q^{89} +(-5.08070e6 + 1.23992e6i) q^{91} +(-4.69281e6 + 8.12818e6i) q^{95} -4.87158e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 1680 q^{7} - 28280 q^{13} + 42224 q^{19} - 80460 q^{25} + 164752 q^{31} - 647980 q^{37} + 1341440 q^{43} + 230104 q^{49} - 323120 q^{55} - 4319336 q^{61} - 3905760 q^{67} + 6471780 q^{73} - 6093104 q^{79}+ \cdots - 27141240 q^{97}+O(q^{100}) \) 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

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Representations

Groups

Database

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