LMFDB - Embedded newform 117.18.b.c.64.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [117,18,Mod(64,117)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("117.64"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	117.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,-1441794] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$214.369842193$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 2031617 x^{18} + 1715857968816 x^{16} + \cdots + 62\!\cdots\!36 $ x^20 + 2031617x^18 + 1715857968816x^16 + 781461052963404656x^14 + 208332068234930274947584x^12 + 33023860904790497652957818880x^10 + 3016012362020988469355432389902336x^8 + 145208851559261531681366452170941005824x^6 + 3068216631532301212964818397833954540388352x^4 + 24588607789677277492492771752215998914361294848*x^2 + 62374830165048482310808851308621287865317469454336
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	multiple of $ 2^{43}\cdot 3^{33}\cdot 13^{11} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.6
Root			$-354.573i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.18.b.c.64.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-354.573i q^{2} +5349.95 q^{4} +907528. i q^{5} -1.26328e6i q^{7} -4.83715e7i q^{8} +3.21785e8 q^{10} +3.23852e8i q^{11} +(2.76591e9 + 1.00009e9i) q^{13} -4.47925e8 q^{14} -1.64500e10 q^{16} +1.94903e10 q^{17} -5.88040e10i q^{19} +4.85523e9i q^{20} +1.14829e11 q^{22} +1.94361e11 q^{23} -6.06677e10 q^{25} +(3.54606e11 - 9.80715e11i) q^{26} -6.75848e9i q^{28} +3.01254e12 q^{29} +5.92284e12i q^{31} -5.07422e11i q^{32} -6.91073e12i q^{34} +1.14646e12 q^{35} +3.20503e13i q^{37} -2.08503e13 q^{38} +4.38985e13 q^{40} +1.76902e13i q^{41} -1.24868e14 q^{43} +1.73259e12i q^{44} -6.89150e13i q^{46} -2.53458e14i q^{47} +2.31035e14 q^{49} +2.15111e13i q^{50} +(1.47974e13 + 5.35044e12i) q^{52} -4.10308e14 q^{53} -2.93904e14 q^{55} -6.11068e13 q^{56} -1.06817e15i q^{58} -2.19511e15i q^{59} -2.39039e14 q^{61} +2.10008e15 q^{62} -2.33606e15 q^{64} +(-9.07612e14 + 2.51014e15i) q^{65} -4.67250e15i q^{67} +1.04272e14 q^{68} -4.06504e14i q^{70} +2.48245e15i q^{71} +1.32689e16i q^{73} +1.13642e16 q^{74} -3.14599e14i q^{76} +4.09115e14 q^{77} -1.30793e16 q^{79} -1.49289e16i q^{80} +6.27248e15 q^{82} +3.69328e15i q^{83} +1.76880e16i q^{85} +4.42749e16i q^{86} +1.56652e16 q^{88} +7.05606e15i q^{89} +(1.26340e15 - 3.49411e15i) q^{91} +1.03982e15 q^{92} -8.98693e16 q^{94} +5.33663e16 q^{95} +1.11563e17i q^{97} -8.19187e16i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 1441794 q^{4} - 1157719078 q^{10} + 2164505000 q^{13} - 7100918310 q^{14} + 137372157250 q^{16} + 41886537180 q^{17} - 517583912680 q^{22} - 222078810480 q^{23} - 5105437226376 q^{25} + 3236456408130 q^{26}+ \cdots + 46\!\cdots\!04 q^{95}+O(q^{100}) $ 20 * q - 1441794 * q^4 - 1157719078 * q^10 + 2164505000 * q^13 - 7100918310 * q^14 + 137372157250 * q^16 + 41886537180 * q^17 - 517583912680 * q^22 - 222078810480 * q^23 - 5105437226376 * q^25 + 3236456408130 * q^26 - 3521990411568 * q^29 - 47961459509856 * q^35 - 66744999456060 * q^38 + 202362624022474 * q^40 - 201474743080160 * q^43 + 53227142648624 * q^49 - 693691356264060 * q^52 - 2429498493199320 * q^53 + 2118490108324208 * q^55 + 2835869617451466 * q^56 - 140707863052352 * q^61 - 6716044276698600 * q^62 - 5699141619649666 * q^64 - 4675683183703692 * q^65 + 692992269522030 * q^68 - 34108184292603558 * q^74 + 59285142434915520 * q^77 - 31814529658962752 * q^79 - 114501724181741920 * q^82 + 207041488957467040 * q^88 + 89768903072866368 * q^91 - 120908580468048540 * q^92 - 32863399636204954 * q^94 + 462354331213557504 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	354.573i		−	0.979379i	−0.871897	−	0.489689i	$-0.837110\pi$
$2$		−	354.573i		−	0.979379i	0.871897	−	0.489689i	$-0.162890\pi$
$3$		0			0
$3$		0			0
$4$	5349.95			0.0408169
$5$			907528.i			1.03900i	0.854471	+	0.519499i	$0.173882\pi$
$5$			907528.i			1.03900i	−0.854471	+	0.519499i	$0.826118\pi$
$6$		0			0
$7$		−	1.26328e6i		−	0.0828259i	−0.999142	−	0.0414129i	$-0.986814\pi$
$7$		−	1.26328e6i		−	0.0828259i	0.999142	−	0.0414129i	$-0.0131859\pi$
$8$		−	4.83715e7i		−	1.01935i
$9$		0			0
$10$	3.21785e8			1.01757
$11$			3.23852e8i			0.455521i	0.973717	+	0.227760i	$0.0731403\pi$
$11$			3.23852e8i			0.455521i	−0.973717	+	0.227760i	$0.926860\pi$
$12$		0			0
$13$	2.76591e9	+	1.00009e9i	0.940413	+	0.340033i
$13$	2.76591e9	+	1.00009e9i	0.940413	+	0.340033i
$14$	−4.47925e8			−0.0811179
$15$		0			0
$16$	−1.64500e10			−0.957517
$17$	1.94903e10			0.677645			0.338823	−	0.940850i	$-0.389971\pi$
$17$	1.94903e10			0.677645			0.338823	+	0.940850i	$0.389971\pi$
$18$		0			0
$19$		−	5.88040e10i		−	0.794331i	−0.917747	−	0.397166i	$-0.869994\pi$
$19$		−	5.88040e10i		−	0.794331i	0.917747	−	0.397166i	$-0.130006\pi$
$20$			4.85523e9i			0.0424087i
$21$		0			0
$22$	1.14829e11			0.446127
$23$	1.94361e11			0.517513			0.258757	−	0.965943i	$-0.416687\pi$
$23$	1.94361e11			0.517513			0.258757	+	0.965943i	$0.416687\pi$
$24$		0			0
$25$	−6.06677e10			−0.0795183
$26$	3.54606e11	−	9.80715e11i	0.333022	−	0.921021i
$27$		0			0
$28$		−	6.75848e9i		−	0.00338069i
$29$	3.01254e12			1.11828			0.559140	−	0.829073i	$-0.311132\pi$
$29$	3.01254e12			1.11828			0.559140	+	0.829073i	$0.311132\pi$
$30$		0			0
$31$			5.92284e12i			1.24726i	0.781721	+	0.623628i	$0.214342\pi$
$31$			5.92284e12i			1.24726i	−0.781721	+	0.623628i	$0.785658\pi$
$32$		−	5.07422e11i		−	0.0815820i
$33$		0			0
$34$		−	6.91073e12i		−	0.663671i
$35$	1.14646e12			0.0860560
$36$		0			0
$37$			3.20503e13i			1.50009i	0.661386	+	0.750046i	$0.269969\pi$
$37$			3.20503e13i			1.50009i	−0.661386	+	0.750046i	$0.730031\pi$
$38$	−2.08503e13			−0.777951
$39$		0			0
$40$	4.38985e13			1.05911
$41$			1.76902e13i			0.345996i	0.984922	+	0.172998i	$0.0553453\pi$
$41$			1.76902e13i			0.345996i	−0.984922	+	0.172998i	$0.944655\pi$
$42$		0			0
$43$	−1.24868e14			−1.62918			−0.814592	−	0.580035i	$-0.803039\pi$
$43$	−1.24868e14			−1.62918			−0.814592	+	0.580035i	$0.803039\pi$
$44$			1.73259e12i			0.0185929i
$45$		0			0
$46$		−	6.89150e13i		−	0.506842i
$47$		−	2.53458e14i		−	1.55265i	−0.630332	−	0.776326i	$-0.717081\pi$
$47$		−	2.53458e14i		−	1.55265i	0.630332	−	0.776326i	$-0.282919\pi$
$48$		0			0
$49$	2.31035e14			0.993140
$50$			2.15111e13i			0.0778786i
$51$		0			0
$52$	1.47974e13	+	5.35044e12i	0.0383847	+	0.0138791i
$53$	−4.10308e14			−0.905242			−0.452621	−	0.891703i	$-0.649511\pi$
$53$	−4.10308e14			−0.905242			−0.452621	+	0.891703i	$0.649511\pi$
$54$		0			0
$55$	−2.93904e14			−0.473285
$56$	−6.11068e13			−0.0844289
$57$		0			0
$58$		−	1.06817e15i		−	1.09522i
$59$		−	2.19511e15i		−	1.94632i	−0.230123	−	0.973162i	$-0.573913\pi$
$59$		−	2.19511e15i		−	1.94632i	0.230123	−	0.973162i	$-0.426087\pi$
$60$		0			0
$61$	−2.39039e14			−0.159649			−0.0798243	−	0.996809i	$-0.525436\pi$
$61$	−2.39039e14			−0.159649			−0.0798243	+	0.996809i	$0.525436\pi$
$62$	2.10008e15			1.22154
$63$		0			0
$64$	−2.33606e15			−1.03742
$65$	−9.07612e14	+	2.51014e15i	−0.353294	+	0.977088i
$66$		0			0
$67$		−	4.67250e15i		−	1.40577i	−0.711305	−	0.702883i	$-0.751896\pi$
$67$		−	4.67250e15i		−	1.40577i	0.711305	−	0.702883i	$-0.248104\pi$
$68$	1.04272e14			0.0276593
$69$		0			0
$70$		−	4.06504e14i		−	0.0842814i
$71$			2.48245e15i			0.456231i	0.973634	+	0.228115i	$0.0732564\pi$
$71$			2.48245e15i			0.456231i	−0.973634	+	0.228115i	$0.926744\pi$
$72$		0			0
$73$			1.32689e16i			1.92571i	0.270016	+	0.962856i	$0.412971\pi$
$73$			1.32689e16i			1.92571i	−0.270016	+	0.962856i	$0.587029\pi$
$74$	1.13642e16			1.46916
$75$		0			0
$76$		−	3.14599e14i		−	0.0324221i
$77$	4.09115e14			0.0377289
$78$		0			0
$79$	−1.30793e16			−0.969961			−0.484981	−	0.874525i	$-0.661173\pi$
$79$	−1.30793e16			−0.969961			−0.484981	+	0.874525i	$0.661173\pi$
$80$		−	1.49289e16i		−	0.994859i
$81$		0			0
$82$	6.27248e15			0.338861
$83$			3.69328e15i			0.179990i	0.995942	+	0.0899950i	$0.0286851\pi$
$83$			3.69328e15i			0.179990i	−0.995942	+	0.0899950i	$0.971315\pi$
$84$		0			0
$85$			1.76880e16i			0.704073i
$86$			4.42749e16i			1.59559i
$87$		0			0
$88$	1.56652e16			0.464337
$89$			7.05606e15i			0.189997i	0.995477	+	0.0949987i	$0.0302847\pi$
$89$			7.05606e15i			0.189997i	−0.995477	+	0.0949987i	$0.969715\pi$
$90$		0			0
$91$	1.26340e15	−	3.49411e15i	0.0281636	−	0.0778906i
$92$	1.03982e15			0.0211233
$93$		0			0
$94$	−8.98693e16			−1.52063
$95$	5.33663e16			0.825309
$96$		0			0
$97$			1.11563e17i			1.44531i	0.691211	+	0.722653i	$0.257078\pi$
$97$			1.11563e17i			1.44531i	−0.691211	+	0.722653i	$0.742922\pi$
$98$		−	8.19187e16i		−	0.972660i
$99$		0			0
$100$	−3.24569e14			−0.00324569
$101$	7.10749e16			0.653107			0.326554	−	0.945179i	$-0.394113\pi$
$101$	7.10749e16			0.653107			0.326554	+	0.945179i	$0.394113\pi$
$102$		0			0
$103$	9.29172e16			0.722736			0.361368	−	0.932423i	$-0.382310\pi$
$103$	9.29172e16			0.722736			0.361368	+	0.932423i	$0.382310\pi$
$104$	4.83760e16	−	1.33791e17i	0.346614	−	0.958614i
$105$		0			0
$106$			1.45484e17i			0.886575i
$107$	1.79622e17			1.01064			0.505322	−	0.862931i	$-0.331374\pi$
$107$	1.79622e17			1.01064			0.505322	+	0.862931i	$0.331374\pi$
$108$		0			0
$109$			1.02322e16i			0.0491863i	0.999698	+	0.0245931i	$0.00782903\pi$
$109$			1.02322e16i			0.0491863i	−0.999698	+	0.0245931i	$0.992171\pi$
$110$			1.04211e17i			0.463526i
$111$		0			0
$112$			2.07810e16i			0.0793072i
$113$	−2.99075e17			−1.05831			−0.529156	−	0.848525i	$-0.677491\pi$
$113$	−2.99075e17			−1.05831			−0.529156	+	0.848525i	$0.677491\pi$
$114$		0			0
$115$			1.76388e17i			0.537696i
$116$	1.61169e16			0.0456447
$117$		0			0
$118$	−7.78328e17			−1.90619
$119$		−	2.46217e16i		−	0.0561266i
$120$		0			0
$121$	4.00567e17			0.792501
$122$			8.47568e16i			0.156357i
$123$		0			0
$124$			3.16869e16i			0.0509091i
$125$			6.37331e17i			0.956379i
$126$		0			0
$127$	1.46802e17			0.192487			0.0962435	−	0.995358i	$-0.469317\pi$
$127$	1.46802e17			0.192487			0.0962435	+	0.995358i	$0.469317\pi$
$128$			7.61793e17i			0.934442i
$129$		0			0
$130$	8.90027e17	+	3.21815e17i	0.956940	+	0.346009i
$131$	1.14906e18			1.15754			0.578769	−	0.815491i	$-0.303533\pi$
$131$	1.14906e18			1.15754			0.578769	+	0.815491i	$0.303533\pi$
$132$		0			0
$133$	−7.42859e16			−0.0657912
$134$	−1.65674e18			−1.37678
$135$		0			0
$136$		−	9.42775e17i		−	0.690760i
$137$		−	1.05752e18i		−	0.728053i	−0.931389	−	0.364026i	$-0.881402\pi$
$137$		−	1.05752e18i		−	0.728053i	0.931389	−	0.364026i	$-0.118598\pi$
$138$		0			0
$139$	1.56713e18			0.953848			0.476924	−	0.878944i	$-0.341752\pi$
$139$	1.56713e18			0.953848			0.476924	+	0.878944i	$0.341752\pi$
$140$	6.13351e15			0.00351253
$141$		0			0
$142$	8.80210e17			0.446823
$143$	−3.23881e17	+	8.95743e17i	−0.154892	+	0.428378i
$144$		0			0
$145$			2.73397e18i			1.16189i
$146$	4.70480e18			1.88600
$147$		0			0
$148$			1.71468e17i			0.0612290i
$149$			1.66396e18i			0.561123i	0.959836	+	0.280562i	$0.0905207\pi$
$149$			1.66396e18i			0.561123i	−0.959836	+	0.280562i	$0.909479\pi$
$150$		0			0
$151$			2.55772e18i			0.770104i	0.922895	+	0.385052i	$0.125816\pi$
$151$			2.55772e18i			0.770104i	−0.922895	+	0.385052i	$0.874184\pi$
$152$	−2.84444e18			−0.809705
$153$		0			0
$154$		−	1.45061e17i		−	0.0369509i
$155$	−5.37515e18			−1.29590
$156$		0			0
$157$	4.03811e18			0.873033			0.436516	−	0.899696i	$-0.356212\pi$
$157$	4.03811e18			0.873033			0.436516	+	0.899696i	$0.356212\pi$
$158$			4.63757e18i			0.949960i
$159$		0			0
$160$	4.60500e17			0.0847636
$161$		−	2.45532e17i		−	0.0428635i
$162$		0			0
$163$			9.82733e18i			1.54469i	0.635204	+	0.772344i	$0.280916\pi$
$163$			9.82733e18i			1.54469i	−0.635204	+	0.772344i	$0.719084\pi$
$164$			9.46418e16i			0.0141225i
$165$		0			0
$166$	1.30954e18			0.176278
$167$		−	7.83585e17i		−	0.100230i	−0.998743	−	0.0501148i	$-0.984041\pi$
$167$		−	7.83585e17i		−	0.100230i	0.998743	−	0.0501148i	$-0.0159587\pi$
$168$		0			0
$169$	6.65005e18	+	5.53232e18i	0.768755	+	0.639544i
$170$	6.27168e18			0.689554
$171$		0			0
$172$	−6.68038e17			−0.0664982
$173$	−1.35855e17			−0.0128731			−0.00643654	−	0.999979i	$-0.502049\pi$
$173$	−1.35855e17			−0.0128731			−0.00643654	+	0.999979i	$0.502049\pi$
$174$		0			0
$175$			7.66402e16i			0.00658617i
$176$		−	5.32736e18i		−	0.436169i
$177$		0			0
$178$	2.50189e18			0.186079
$179$	1.19942e19			0.850590			0.425295	−	0.905055i	$-0.360170\pi$
$179$	1.19942e19			0.850590			0.425295	+	0.905055i	$0.360170\pi$
$180$		0			0
$181$	4.48126e18			0.289156			0.144578	−	0.989493i	$-0.453818\pi$
$181$	4.48126e18			0.289156			0.144578	+	0.989493i	$0.453818\pi$
$182$	−1.23892e18	−	4.47966e17i	−0.0762844	−	0.0275828i
$183$		0			0
$184$		−	9.40152e18i		−	0.527529i
$185$	−2.90866e19			−1.55859
$186$		0			0
$187$			6.31196e18i			0.308681i
$188$		−	1.35599e18i		−	0.0633744i
$189$		0			0
$190$		−	1.89223e19i		−	0.808290i
$191$	4.34771e19			1.77614			0.888069	−	0.459711i	$-0.152047\pi$
$191$	4.34771e19			1.77614			0.888069	+	0.459711i	$0.152047\pi$
$192$		0			0
$193$		−	2.67073e19i		−	0.998602i	−0.866429	−	0.499301i	$-0.833590\pi$
$193$		−	2.67073e19i		−	0.998602i	0.866429	−	0.499301i	$-0.166410\pi$
$194$	3.95572e19			1.41550
$195$		0			0
$196$	1.23602e18			0.0405368
$197$		−	4.06754e18i		−	0.127752i	−0.997958	−	0.0638762i	$-0.979654\pi$
$197$		−	4.06754e18i		−	0.127752i	0.997958	−	0.0638762i	$-0.0203463\pi$
$198$		0			0
$199$	−6.58440e19			−1.89786			−0.948932	−	0.315480i	$-0.897834\pi$
$199$	−6.58440e19			−1.89786			−0.948932	+	0.315480i	$0.897834\pi$
$200$			2.93459e18i			0.0810573i
$201$		0			0
$202$		−	2.52012e19i		−	0.639639i
$203$		−	3.80568e18i		−	0.0926225i
$204$		0			0
$205$	−1.60544e19			−0.359489
$206$		−	3.29460e19i		−	0.707833i
$207$		0			0
$208$	−4.54992e19	−	1.64515e19i	−0.900462	−	0.325588i
$209$	1.90438e19			0.361834
$210$		0			0
$211$	5.20121e19			0.911389			0.455695	−	0.890136i	$-0.349391\pi$
$211$	5.20121e19			0.911389			0.455695	+	0.890136i	$0.349391\pi$
$212$	−2.19512e18			−0.0369491
$213$		0			0
$214$		−	6.36892e19i		−	0.989803i
$215$		−	1.13321e20i		−	1.69272i
$216$		0			0
$217$	7.48220e18			0.103305
$218$	3.62806e18			0.0481720
$219$		0			0
$220$	−1.57237e18			−0.0193180
$221$	5.39083e19	+	1.94921e19i	0.637267	+	0.230422i
$222$		0			0
$223$		−	1.13813e19i		−	0.124624i	−0.998057	−	0.0623121i	$-0.980153\pi$
$223$		−	1.13813e19i		−	0.124624i	0.998057	−	0.0623121i	$-0.0198474\pi$
$224$	−6.41015e17			−0.00675710
$225$		0			0
$226$			1.06044e20i			1.03649i
$227$			2.41791e19i			0.227625i	0.993502	+	0.113812i	$0.0363063\pi$
$227$			2.41791e19i			0.227625i	−0.993502	+	0.113812i	$0.963694\pi$
$228$		0			0
$229$			1.27743e20i			1.11618i	0.829780	+	0.558091i	$0.188466\pi$
$229$			1.27743e20i			1.11618i	−0.829780	+	0.558091i	$0.811534\pi$
$230$	6.25423e19			0.526608
$231$		0			0
$232$		−	1.45721e20i		−	1.13992i
$233$	1.44577e20			1.09037			0.545184	−	0.838317i	$-0.316460\pi$
$233$	1.44577e20			1.09037			0.545184	+	0.838317i	$0.316460\pi$
$234$		0			0
$235$	2.30020e20			1.61320
$236$		−	1.17437e19i		−	0.0794428i
$237$		0			0
$238$	−8.73018e18			−0.0549692
$239$		−	2.06865e20i		−	1.25691i	−0.777845	−	0.628456i	$-0.783687\pi$
$239$		−	2.06865e20i		−	1.25691i	0.777845	−	0.628456i	$-0.216313\pi$
$240$		0			0
$241$		−	2.60990e19i		−	0.147734i	−0.997268	−	0.0738669i	$-0.976466\pi$
$241$		−	2.60990e19i		−	0.147734i	0.997268	−	0.0738669i	$-0.0235340\pi$
$242$		−	1.42030e20i		−	0.776159i
$243$		0			0
$244$	−1.27885e18			−0.00651636
$245$			2.09670e20i			1.03187i
$246$		0			0
$247$	5.88095e19	−	1.62646e20i	0.270099	−	0.747000i
$248$	2.86497e20			1.27140
$249$		0			0
$250$	2.25981e20			0.936658
$251$	−2.93367e20			−1.17540			−0.587698	−	0.809080i	$-0.699966\pi$
$251$	−2.93367e20			−1.17540			−0.587698	+	0.809080i	$0.699966\pi$
$252$		0			0
$253$			6.29440e19i			0.235738i
$254$		−	5.20521e19i		−	0.188518i
$255$		0			0
$256$	−3.60800e19			−0.122244
$257$	−1.66482e19			−0.0545679			−0.0272839	−	0.999628i	$-0.508686\pi$
$257$	−1.66482e19			−0.0545679			−0.0272839	+	0.999628i	$0.508686\pi$
$258$		0			0
$259$	4.04885e19			0.124246
$260$	−4.85567e18	+	1.34291e19i	−0.0144204	+	0.0398817i
$261$		0			0
$262$		−	4.07425e20i		−	1.13367i
$263$	4.25623e20			1.14657			0.573286	−	0.819355i	$-0.305668\pi$
$263$	4.25623e20			1.14657			0.573286	+	0.819355i	$0.305668\pi$
$264$		0			0
$265$		−	3.72366e20i		−	0.940546i
$266$			2.63398e19i			0.0644345i
$267$		0			0
$268$		−	2.49976e19i		−	0.0573790i
$269$	4.65149e20			1.03442			0.517211	−	0.855858i	$-0.326970\pi$
$269$	4.65149e20			1.03442			0.517211	+	0.855858i	$0.326970\pi$
$270$		0			0
$271$		−	1.76410e20i		−	0.368370i	−0.982892	−	0.184185i	$-0.941035\pi$
$271$		−	1.76410e20i		−	0.368370i	0.982892	−	0.184185i	$-0.0589646\pi$
$272$	−3.20616e20			−0.648857
$273$		0			0
$274$	−3.74967e20			−0.713040
$275$		−	1.96473e19i		−	0.0362222i
$276$		0			0
$277$	6.84726e20			1.18697			0.593484	−	0.804846i	$-0.297752\pi$
$277$	6.84726e20			1.18697			0.593484	+	0.804846i	$0.297752\pi$
$278$		−	5.55662e20i		−	0.934179i
$279$		0			0
$280$		−	5.54561e19i		−	0.0877215i
$281$			7.20394e19i			0.110552i	0.998471	+	0.0552760i	$0.0176039\pi$
$281$			7.20394e19i			0.110552i	−0.998471	+	0.0552760i	$0.982396\pi$
$282$		0			0
$283$	3.01197e20			0.435178			0.217589	−	0.976041i	$-0.430181\pi$
$283$	3.01197e20			0.435178			0.217589	+	0.976041i	$0.430181\pi$
$284$			1.32810e19i			0.0186219i
$285$		0			0
$286$	3.17606e20	+	1.14840e20i	0.419544	+	0.151698i
$287$	2.23477e19			0.0286574
$288$		0			0
$289$	−4.47369e20			−0.540797
$290$	9.69391e20			1.13793
$291$		0			0
$292$			7.09880e19i			0.0786015i
$293$			2.04525e20i			0.219975i	0.993933	+	0.109987i	$0.0350810\pi$
$293$			2.04525e20i			0.219975i	−0.993933	+	0.109987i	$0.964919\pi$
$294$		0			0
$295$	1.99213e21			2.02223
$296$	1.55032e21			1.52912
$297$		0			0
$298$	5.89994e20			0.549552
$299$	5.37583e20	+	1.94378e20i	0.486676	+	0.175972i
$300$		0			0
$301$			1.57743e20i			0.134939i
$302$	9.06899e20			0.754223
$303$		0			0
$304$			9.67328e20i			0.760586i
$305$		−	2.16935e20i		−	0.165875i
$306$		0			0
$307$			1.65655e21i			1.19819i	0.800676	+	0.599097i	$0.204474\pi$
$307$			1.65655e21i			1.19819i	−0.800676	+	0.599097i	$0.795526\pi$
$308$	2.18874e18			0.00153998
$309$		0			0
$310$			1.90588e21i			1.26918i
$311$	−2.95797e20			−0.191660			−0.0958298	−	0.995398i	$-0.530550\pi$
$311$	−2.95797e20			−0.191660			−0.0958298	+	0.995398i	$0.530550\pi$
$312$		0			0
$313$	−8.28602e20			−0.508416			−0.254208	−	0.967150i	$-0.581815\pi$
$313$	−8.28602e20			−0.508416			−0.254208	+	0.967150i	$0.581815\pi$
$314$		−	1.43180e21i		−	0.855030i
$315$		0			0
$316$	−6.99736e19			−0.0395908
$317$			2.94155e21i			1.62021i	0.586283	+	0.810106i	$0.300591\pi$
$317$			2.94155e21i			1.62021i	−0.586283	+	0.810106i	$0.699409\pi$
$318$		0			0
$319$			9.75617e20i			0.509399i
$320$		−	2.12004e21i		−	1.07787i
$321$		0			0
$322$	−8.70589e19			−0.0419796
$323$		−	1.14611e21i		−	0.538275i
$324$		0			0
$325$	−1.67801e20	−	6.06733e19i	−0.0747801	−	0.0270389i
$326$	3.48451e21			1.51284
$327$		0			0
$328$	8.55703e20			0.352692
$329$	−3.20188e20			−0.128600
$330$		0			0
$331$		−	4.55886e20i		−	0.173908i	−0.996212	−	0.0869538i	$-0.972287\pi$
$331$		−	4.55886e20i		−	0.173908i	0.996212	−	0.0869538i	$-0.0277133\pi$
$332$			1.97589e19i			0.00734663i
$333$		0			0
$334$	−2.77838e20			−0.0981628
$335$	4.24042e21			1.46059
$336$		0			0
$337$	−5.90368e20			−0.193316			−0.0966581	−	0.995318i	$-0.530815\pi$
$337$	−5.90368e20			−0.193316			−0.0966581	+	0.995318i	$0.530815\pi$
$338$	1.96161e21	−	2.35793e21i	0.626356	−	0.752902i
$339$		0			0
$340$			9.46298e19i			0.0287380i
$341$	−1.91812e21			−0.568151
$342$		0			0
$343$		−	5.85739e20i		−	0.165084i
$344$			6.04007e21i			1.66072i
$345$		0			0
$346$			4.81704e19i			0.0126076i
$347$	4.50481e21			1.15047			0.575236	−	0.817987i	$-0.304910\pi$
$347$	4.50481e21			1.15047			0.575236	+	0.817987i	$0.304910\pi$
$348$		0			0
$349$		−	4.02398e21i		−	0.978677i	−0.872094	−	0.489339i	$-0.837238\pi$
$349$		−	4.02398e21i		−	0.978677i	0.872094	−	0.489339i	$-0.162762\pi$
$350$	2.71745e19			0.00645036
$351$		0			0
$352$	1.64329e20			0.0371623
$353$		−	3.84005e21i		−	0.847720i	−0.905728	−	0.423860i	$-0.860675\pi$
$353$		−	3.84005e21i		−	0.847720i	0.905728	−	0.423860i	$-0.139325\pi$
$354$		0			0
$355$	−2.25289e21			−0.474023
$356$			3.77496e19i			0.00775509i
$357$		0			0
$358$		−	4.25282e21i		−	0.833050i
$359$			2.75158e21i			0.526356i	0.964747	+	0.263178i	$0.0847706\pi$
$359$			2.75158e21i			0.526356i	−0.964747	+	0.263178i	$0.915229\pi$
$360$		0			0
$361$	2.02247e21			0.369038
$362$		−	1.58893e21i		−	0.283193i
$363$		0			0
$364$	6.75910e18	−	1.86933e19i	0.00114955	−	0.00317925i
$365$	−1.20419e22			−2.00081
$366$		0			0
$367$	5.94290e21			0.942621			0.471311	−	0.881967i	$-0.343781\pi$
$367$	5.94290e21			0.942621			0.471311	+	0.881967i	$0.343781\pi$
$368$	−3.19723e21			−0.495528
$369$		0			0
$370$			1.03133e22i			1.52645i
$371$			5.18333e20i			0.0749775i
$372$		0			0
$373$	5.72899e21			0.791686			0.395843	−	0.918318i	$-0.370452\pi$
$373$	5.72899e21			0.791686			0.395843	+	0.918318i	$0.370452\pi$
$374$	2.23805e21			0.302316
$375$		0			0
$376$	−1.22601e22			−1.58270
$377$	8.33241e21	+	3.01282e21i	1.05164	+	0.380252i
$378$		0			0
$379$		−	9.67569e21i		−	1.16748i	−0.811942	−	0.583739i	$-0.801589\pi$
$379$		−	9.67569e21i		−	1.16748i	0.811942	−	0.583739i	$-0.198411\pi$
$380$	2.85507e20			0.0336865
$381$		0			0
$382$		−	1.54158e22i		−	1.73951i
$383$		−	1.08067e22i		−	1.19262i	−0.802753	−	0.596311i	$-0.796632\pi$
$383$		−	1.08067e22i		−	1.19262i	0.802753	−	0.596311i	$-0.203368\pi$
$384$		0			0
$385$			3.71283e20i			0.0392003i
$386$	−9.46968e21			−0.978009
$387$		0			0
$388$			5.96856e20i			0.0589929i
$389$	−1.50261e22			−1.45303			−0.726515	−	0.687150i	$-0.758861\pi$
$389$	−1.50261e22			−1.45303			−0.726515	+	0.687150i	$0.758861\pi$
$390$		0			0
$391$	3.78814e21			0.350690
$392$		−	1.11755e22i		−	1.01236i
$393$		0			0
$394$	−1.44224e21			−0.125118
$395$		−	1.18698e22i		−	1.00779i
$396$		0			0
$397$		−	1.60199e22i		−	1.30299i	−0.758654	−	0.651493i	$-0.774143\pi$
$397$		−	1.60199e22i		−	1.30299i	0.758654	−	0.651493i	$-0.225857\pi$
$398$			2.33465e22i			1.85873i
$399$		0			0
$400$	9.97984e20			0.0761402
$401$			1.48943e21i			0.111248i	0.998452	+	0.0556241i	$0.0177148\pi$
$401$			1.48943e21i			0.111248i	−0.998452	+	0.0556241i	$0.982285\pi$
$402$		0			0
$403$	−5.92339e21	+	1.63820e22i	−0.424109	+	1.17294i
$404$	3.80247e20			0.0266578
$405$		0			0
$406$	−1.34939e21			−0.0907125
$407$	−1.03795e22			−0.683323
$408$		0			0
$409$		−	2.88555e21i		−	0.182213i	−0.995841	−	0.0911066i	$-0.970960\pi$
$409$		−	2.88555e21i		−	0.182213i	0.995841	−	0.0911066i	$-0.0290404\pi$
$410$			5.69245e21i			0.352076i
$411$		0			0
$412$	4.97102e20			0.0294998
$413$	−2.77304e21			−0.161206
$414$		0			0
$415$	−3.35176e21			−0.187009
$416$	5.07469e20	−	1.40348e21i	0.0277406	−	0.0767208i
$417$		0			0
$418$		−	6.75241e21i		−	0.354373i
$419$	−3.36403e22			−1.72998			−0.864989	−	0.501790i	$-0.832675\pi$
$419$	−3.36403e22			−1.72998			−0.864989	+	0.501790i	$0.832675\pi$
$420$		0			0
$421$			9.19238e21i			0.453973i	0.973898	+	0.226987i	$0.0728873\pi$
$421$			9.19238e21i			0.453973i	−0.973898	+	0.226987i	$0.927113\pi$
$422$		−	1.84421e22i		−	0.892595i
$423$		0			0
$424$			1.98472e22i			0.922762i
$425$	−1.18243e21			−0.0538852
$426$		0			0
$427$			3.01973e20i			0.0132230i
$428$	9.60969e20			0.0412513
$429$		0			0
$430$	−4.01807e22			−1.65781
$431$			4.71393e22i			1.90689i	0.301561	+	0.953447i	$0.402492\pi$
$431$			4.71393e22i			1.90689i	−0.301561	+	0.953447i	$0.597508\pi$
$432$		0			0
$433$	3.63168e22			1.41241			0.706204	−	0.708008i	$-0.250406\pi$
$433$	3.63168e22			1.41241			0.706204	+	0.708008i	$0.250406\pi$
$434$		−	2.65299e21i		−	0.101175i
$435$		0			0
$436$			5.47418e19i			0.00200763i
$437$		−	1.14292e22i		−	0.411077i
$438$		0			0
$439$	−2.78132e22			−0.962282			−0.481141	−	0.876643i	$-0.659777\pi$
$439$	−2.78132e22			−0.962282			−0.481141	+	0.876643i	$0.659777\pi$
$440$			1.42166e22i			0.482445i
$441$		0			0
$442$	6.91137e21	−	1.91144e22i	0.225670	−	0.624126i
$443$	1.40955e21			0.0451491			0.0225745	−	0.999745i	$-0.492814\pi$
$443$	1.40955e21			0.0451491			0.0225745	+	0.999745i	$0.492814\pi$
$444$		0			0
$445$	−6.40358e21			−0.197407
$446$	−4.03552e21			−0.122054
$447$		0			0
$448$			2.95109e21i			0.0859249i
$449$			1.32362e22i			0.378156i	0.981962	+	0.189078i	$0.0605499\pi$
$449$			1.32362e22i			0.378156i	−0.981962	+	0.189078i	$0.939450\pi$
$450$		0			0
$451$	−5.72901e21			−0.157608
$452$	−1.60004e21			−0.0431969
$453$		0			0
$454$	8.57325e21			0.222931
$455$	3.17100e21	+	1.14657e21i	0.0809282	+	0.0292619i
$456$		0			0
$457$			3.83703e22i			0.943425i	0.881752	+	0.471713i	$0.156364\pi$
$457$			3.83703e22i			0.943425i	−0.881752	+	0.471713i	$0.843636\pi$
$458$	4.52941e22			1.09316
$459$		0			0
$460$			9.43665e20i			0.0219470i
$461$			7.82574e22i			1.78677i	0.449296	+	0.893383i	$0.351675\pi$
$461$			7.82574e22i			1.78677i	−0.449296	+	0.893383i	$0.648325\pi$
$462$		0			0
$463$			3.77901e22i			0.831649i	0.909445	+	0.415824i	$0.136507\pi$
$463$			3.77901e22i			0.831649i	−0.909445	+	0.415824i	$0.863493\pi$
$464$	−4.95564e22			−1.07077
$465$		0			0
$466$		−	5.12630e22i		−	1.06788i
$467$	−2.85668e22			−0.584343			−0.292171	−	0.956366i	$-0.594378\pi$
$467$	−2.85668e22			−0.584343			−0.292171	+	0.956366i	$0.594378\pi$
$468$		0			0
$469$	−5.90267e21			−0.116434
$470$		−	8.15589e22i		−	1.57994i
$471$		0			0
$472$	−1.06181e23			−1.98399
$473$		−	4.04388e22i		−	0.742127i
$474$		0			0
$475$			3.56750e21i			0.0631639i
$476$		−	1.31725e20i		−	0.00229091i
$477$		0			0
$478$	−7.33488e22			−1.23099
$479$			4.76485e22i			0.785592i	0.919626	+	0.392796i	$0.128492\pi$
$479$			4.76485e22i			0.785592i	−0.919626	+	0.392796i	$0.871508\pi$
$480$		0			0
$481$	−3.20533e22	+	8.86482e22i	−0.510081	+	1.41071i
$482$	−9.25402e21			−0.144687
$483$		0			0
$484$	2.14301e21			0.0323474
$485$	−1.01246e23			−1.50167
$486$		0			0
$487$			7.92162e22i			1.13453i	0.823534	+	0.567267i	$0.191999\pi$
$487$			7.92162e22i			1.13453i	−0.823534	+	0.567267i	$0.808001\pi$
$488$			1.15627e22i			0.162739i
$489$		0			0
$490$	7.43435e22			1.01059
$491$	2.41059e22			0.322056			0.161028	−	0.986950i	$-0.448519\pi$
$491$	2.41059e22			0.322056			0.161028	+	0.986950i	$0.448519\pi$
$492$		0			0
$493$	5.87153e22			0.757797
$494$	−5.76700e22	−	2.08523e22i	−0.731596	−	0.264529i
$495$		0			0
$496$		−	9.74309e22i		−	1.19427i
$497$	3.13603e21			0.0377877
$498$		0			0
$499$		−	5.95571e22i		−	0.693552i	−0.937948	−	0.346776i	$-0.887276\pi$
$499$		−	5.95571e22i		−	0.693552i	0.937948	−	0.346776i	$-0.112724\pi$
$500$			3.40969e21i			0.0390364i
$501$		0			0
$502$			1.04020e23i			1.15116i
$503$	5.70925e22			0.621228			0.310614	−	0.950536i	$-0.399465\pi$
$503$	5.70925e22			0.621228			0.310614	+	0.950536i	$0.399465\pi$
$504$		0			0
$505$			6.45024e22i			0.678577i
$506$	2.23182e22			0.230877
$507$		0			0
$508$	7.85385e20			0.00785671
$509$		−	9.43917e22i		−	0.928608i	−0.885676	−	0.464304i	$-0.846304\pi$
$509$		−	9.43917e22i		−	0.928608i	0.885676	−	0.464304i	$-0.153696\pi$
$510$		0			0
$511$	1.67624e22			0.159499
$512$			1.12643e23i			1.05417i
$513$		0			0
$514$			5.90302e21i			0.0534426i
$515$			8.43250e22i			0.750922i
$516$		0			0
$517$	8.20827e22			0.707265
$518$		−	1.43561e22i		−	0.121684i
$519$		0			0
$520$	1.21419e23	+	4.39026e22i	0.995999	+	0.360132i
$521$	1.18147e23			0.953463			0.476732	−	0.879049i	$-0.341821\pi$
$521$	1.18147e23			0.953463			0.476732	+	0.879049i	$0.341821\pi$
$522$		0			0
$523$	2.05182e22			0.160279			0.0801394	−	0.996784i	$-0.474463\pi$
$523$	2.05182e22			0.160279			0.0801394	+	0.996784i	$0.474463\pi$
$524$	6.14739e21			0.0472471
$525$		0			0
$526$		−	1.50914e23i		−	1.12293i
$527$			1.15438e23i			0.845197i
$528$		0			0
$529$	−1.03274e23			−0.732180
$530$	−1.32031e23			−0.921151
$531$		0			0
$532$	−3.97426e20			−0.00268539
$533$	−1.76919e22	+	4.89295e22i	−0.117650	+	0.325379i
$534$		0			0
$535$			1.63012e23i			1.05006i
$536$	−2.26016e23			−1.43297
$537$		0			0
$538$		−	1.64929e23i		−	1.01309i
$539$			7.48209e22i			0.452396i
$540$		0			0
$541$			2.29866e23i			1.34679i	0.739285	+	0.673393i	$0.235164\pi$
$541$			2.29866e23i			1.34679i	−0.739285	+	0.673393i	$0.764836\pi$
$542$	−6.25503e22			−0.360774
$543$		0			0
$544$		−	9.88980e21i		−	0.0552837i
$545$	−9.28601e21			−0.0511045
$546$		0			0
$547$	2.87491e23			1.53367			0.766836	−	0.641843i	$-0.221830\pi$
$547$	2.87491e23			1.53367			0.766836	+	0.641843i	$0.221830\pi$
$548$		−	5.65766e21i		−	0.0297168i
$549$		0			0
$550$	−6.96641e21			−0.0354753
$551$		−	1.77150e23i		−	0.888284i
$552$		0			0
$553$			1.65228e22i			0.0803379i
$554$		−	2.42785e23i		−	1.16249i
$555$		0			0
$556$	8.38406e21			0.0389331
$557$			3.70408e23i			1.69399i	0.531601	+	0.846995i	$0.321590\pi$
$557$			3.70408e23i			1.69399i	−0.531601	+	0.846995i	$0.678410\pi$
$558$		0			0
$559$	−3.45374e23	−	1.24880e23i	−1.53211	−	0.553977i
$560$	−1.88593e22			−0.0824001
$561$		0			0
$562$	2.55432e22			0.108272
$563$	3.80961e23			1.59059			0.795296	−	0.606222i	$-0.207316\pi$
$563$	3.80961e23			1.59059			0.795296	+	0.606222i	$0.207316\pi$
$564$		0			0
$565$		−	2.71419e23i		−	1.09958i
$566$		−	1.06796e23i		−	0.426204i
$567$		0			0
$568$	1.20080e23			0.465061
$569$	1.04968e23			0.400499			0.200249	−	0.979745i	$-0.435825\pi$
$569$	1.04968e23			0.400499			0.200249	+	0.979745i	$0.435825\pi$
$570$		0			0
$571$	−1.67680e23			−0.620974			−0.310487	−	0.950578i	$-0.600492\pi$
$571$	−1.67680e23			−0.620974			−0.310487	+	0.950578i	$0.600492\pi$
$572$	−1.73275e21	+	4.79218e21i	−0.00632222	+	0.0174850i
$573$		0			0
$574$		−	7.92389e21i		−	0.0280664i
$575$	−1.17914e22			−0.0411518
$576$		0			0
$577$		−	4.53346e23i		−	1.53615i	−0.640357	−	0.768077i	$-0.721214\pi$
$577$		−	4.53346e23i		−	1.53615i	0.640357	−	0.768077i	$-0.278786\pi$
$578$			1.58625e23i			0.529645i
$579$		0			0
$580$			1.46266e22i			0.0474247i
$581$	4.66564e21			0.0149078
$582$		0			0
$583$		−	1.32879e23i		−	0.412357i
$584$	6.41838e23			1.96298
$585$		0			0
$586$	7.25192e22			0.215439
$587$		−	4.45158e23i		−	1.30344i	−0.758461	−	0.651718i	$-0.774048\pi$
$587$		−	4.45158e23i		−	1.30344i	0.758461	−	0.651718i	$-0.225952\pi$
$588$		0			0
$589$	3.48287e23			0.990735
$590$		−	7.06355e23i		−	1.98053i
$591$		0			0
$592$		−	5.27229e23i		−	1.43636i
$593$		−	4.03490e23i		−	1.08360i	−0.840509	−	0.541798i	$-0.817744\pi$
$593$		−	4.03490e23i		−	1.08360i	0.840509	−	0.541798i	$-0.182256\pi$
$594$		0			0
$595$	2.23449e22			0.0583154
$596$			8.90207e21i			0.0229033i
$597$		0			0
$598$	6.89214e22	−	1.90612e23i	0.172343	−	0.476641i
$599$	−2.43037e23			−0.599162			−0.299581	−	0.954071i	$-0.596847\pi$
$599$	−2.43037e23			−0.599162			−0.299581	+	0.954071i	$0.596847\pi$
$600$		0			0
$601$	−1.90190e21			−0.00455779			−0.00227890	−	0.999997i	$-0.500725\pi$
$601$	−1.90190e21			−0.00455779			−0.00227890	+	0.999997i	$0.500725\pi$
$602$	5.59316e22			0.132156
$603$		0			0
$604$			1.36837e22i			0.0314332i
$605$			3.63526e23i			0.823407i
$606$		0			0
$607$	−4.61810e23			−1.01709			−0.508545	−	0.861035i	$-0.669817\pi$
$607$	−4.61810e23			−1.01709			−0.508545	+	0.861035i	$0.669817\pi$
$608$	−2.98385e22			−0.0648031
$609$		0			0
$610$	−7.69192e22			−0.162454
$611$	2.53481e23	−	7.01040e23i	0.527953	−	1.46013i
$612$		0			0
$613$			6.65144e23i			1.34742i	0.738998	+	0.673708i	$0.235299\pi$
$613$			6.65144e23i			1.34742i	−0.738998	+	0.673708i	$0.764701\pi$
$614$	5.87367e23			1.17349
$615$		0			0
$616$		−	1.97895e22i		−	0.0384591i
$617$		−	5.84913e23i		−	1.12116i	−0.828101	−	0.560580i	$-0.810578\pi$
$617$		−	5.84913e23i		−	1.12116i	0.828101	−	0.560580i	$-0.189422\pi$
$618$		0			0
$619$			4.08278e23i			0.761352i	0.924708	+	0.380676i	$0.124309\pi$
$619$			4.08278e23i			0.761352i	−0.924708	+	0.380676i	$0.875691\pi$
$620$	−2.87567e22			−0.0528945
$621$		0			0
$622$			1.04882e23i			0.187707i
$623$	8.91378e21			0.0157367
$624$		0			0
$625$	−6.24682e23			−1.07320
$626$			2.93800e23i			0.497932i
$627$		0			0
$628$	2.16037e22			0.0356345
$629$			6.24670e23i			1.01653i
$630$		0			0
$631$		−	8.46966e23i		−	1.34158i	−0.741648	−	0.670789i	$-0.765955\pi$
$631$		−	8.46966e23i		−	1.34158i	0.741648	−	0.670789i	$-0.234045\pi$
$632$			6.32666e23i			0.988734i
$633$		0			0
$634$	1.04299e24			1.58680
$635$			1.33227e23i			0.199994i
$636$		0			0
$637$	6.39020e23	+	2.31056e23i	0.933962	+	0.337701i
$638$	3.45927e23			0.498895
$639$		0			0
$640$	−6.91349e23			−0.970884
$641$	8.28678e23			1.14840			0.574199	−	0.818716i	$-0.305314\pi$
$641$	8.28678e23			1.14840			0.574199	+	0.818716i	$0.305314\pi$
$642$		0			0
$643$		−	1.67835e23i		−	0.226510i	−0.993566	−	0.113255i	$-0.963872\pi$
$643$		−	1.67835e23i		−	0.226510i	0.993566	−	0.113255i	$-0.0361278\pi$
$644$		−	1.31358e21i		−	0.00174955i
$645$		0			0
$646$	−4.06379e23			−0.527175
$647$	−1.21885e23			−0.156050			−0.0780252	−	0.996951i	$-0.524861\pi$
$647$	−1.21885e23			−0.156050			−0.0780252	+	0.996951i	$0.524861\pi$
$648$		0			0
$649$	7.10891e23			0.886591
$650$	−2.15131e22	+	5.94977e22i	−0.0264813	+	0.0732381i
$651$		0			0
$652$			5.25757e22i			0.0630493i
$653$	1.44600e24			1.71161			0.855806	−	0.517297i	$-0.173062\pi$
$653$	1.44600e24			1.71161			0.855806	+	0.517297i	$0.173062\pi$
$654$		0			0
$655$			1.04280e24i			1.20268i
$656$		−	2.91005e23i		−	0.331297i
$657$		0			0
$658$			1.13530e23i			0.125948i
$659$	3.49344e23			0.382585			0.191292	−	0.981533i	$-0.438732\pi$
$659$	3.49344e23			0.382585			0.191292	+	0.981533i	$0.438732\pi$
$660$		0			0
$661$			1.12573e24i			1.20149i	0.799440	+	0.600746i	$0.205130\pi$
$661$			1.12573e24i			1.20149i	−0.799440	+	0.600746i	$0.794870\pi$
$662$	−1.61645e23			−0.170321
$663$		0			0
$664$	1.78650e23			0.183474
$665$		−	6.74166e22i		−	0.0683569i
$666$		0			0
$667$	5.85520e23			0.578725
$668$		−	4.19214e21i		−	0.00409106i
$669$		0			0
$670$		−	1.50354e24i		−	1.43047i
$671$		−	7.74132e22i		−	0.0727233i
$672$		0			0
$673$	−1.41075e24			−1.29218			−0.646090	−	0.763261i	$-0.723597\pi$
$673$	−1.41075e24			−1.29218			−0.646090	+	0.763261i	$0.723597\pi$
$674$			2.09328e23i			0.189330i
$675$		0			0
$676$	3.55774e22	+	2.95976e22i	0.0313781	+	0.0261042i
$677$	−1.95672e23			−0.170422			−0.0852108	−	0.996363i	$-0.527156\pi$
$677$	−1.95672e23			−0.170422			−0.0852108	+	0.996363i	$0.527156\pi$
$678$		0			0
$679$	1.40935e23			0.119709
$680$	8.55595e23			0.717699
$681$		0			0
$682$			6.80114e23i			0.556435i
$683$			2.19287e23i			0.177189i	0.996068	+	0.0885945i	$0.0282375\pi$
$683$			2.19287e23i			0.177189i	−0.996068	+	0.0885945i	$0.971762\pi$
$684$		0			0
$685$	9.59727e23			0.756446
$686$	−2.07687e23			−0.161679
$687$		0			0
$688$	2.05408e24			1.55997
$689$	−1.13487e24	−	4.10346e23i	−0.851302	−	0.307813i
$690$		0			0
$691$			9.38357e23i			0.686760i	0.939197	+	0.343380i	$0.111572\pi$
$691$			9.38357e23i			0.686760i	−0.939197	+	0.343380i	$0.888428\pi$
$692$	−7.26815e20			−0.000525439
$693$		0			0
$694$		−	1.59728e24i		−	1.12675i
$695$			1.42221e24i			0.991047i
$696$		0			0
$697$			3.44788e23i			0.234462i
$698$	−1.42679e24			−0.958496
$699$		0			0
$700$			4.10021e20i			0.000268827i
$701$	−1.47903e24			−0.958020			−0.479010	−	0.877809i	$-0.659004\pi$
$701$	−1.47903e24			−0.958020			−0.479010	+	0.877809i	$0.659004\pi$
$702$		0			0
$703$	1.88469e24			1.19157
$704$		−	7.56535e23i		−	0.472565i
$705$		0			0
$706$	−1.36158e24			−0.830239
$707$		−	8.97874e22i		−	0.0540942i
$708$		0			0
$709$			2.72092e22i			0.0160038i	0.999968	+	0.00800189i	$0.00254711\pi$
$709$			2.72092e22i			0.0160038i	−0.999968	+	0.00800189i	$0.997453\pi$
$710$			7.98815e23i			0.464248i
$711$		0			0
$712$	3.41313e23			0.193675
$713$			1.15117e24i			0.645472i
$714$		0			0
$715$	−8.12912e23	−	2.93931e23i	−0.445084	−	0.160933i
$716$	6.41683e22			0.0347184
$717$		0			0
$718$	9.75636e23			0.515502
$719$	−5.01978e23			−0.262113			−0.131057	−	0.991375i	$-0.541837\pi$
$719$	−5.01978e23			−0.262113			−0.131057	+	0.991375i	$0.541837\pi$
$720$		0			0
$721$		−	1.17380e23i		−	0.0598613i
$722$		−	7.17114e23i		−	0.361428i
$723$		0			0
$724$	2.39745e22			0.0118024
$725$	−1.82764e23			−0.0889237
$726$		0			0
$727$	−1.88320e24			−0.895065			−0.447533	−	0.894268i	$-0.647697\pi$
$727$	−1.88320e24			−0.895065			−0.447533	+	0.894268i	$0.647697\pi$
$728$	−1.69016e23	−	6.11124e22i	−0.0793981	−	0.0287086i
$729$		0			0
$730$			4.26974e24i			1.95955i
$731$	−2.43372e24			−1.10401
$732$		0			0
$733$			7.55904e23i			0.335030i	0.985870	+	0.167515i	$0.0535742\pi$
$733$			7.55904e23i			0.335030i	−0.985870	+	0.167515i	$0.946426\pi$
$734$		−	2.10719e24i		−	0.923183i
$735$		0			0
$736$		−	9.86228e22i		−	0.0422198i
$737$	1.51320e24			0.640356
$738$		0			0
$739$			1.98610e22i			0.00821342i	0.999992	+	0.00410671i	$0.00130721\pi$
$739$			1.98610e22i			0.00821342i	−0.999992	+	0.00410671i	$0.998693\pi$
$740$	−1.55612e23			−0.0636169
$741$		0			0
$742$	1.83787e23			0.0734314
$743$			2.93963e24i			1.16115i	0.814208	+	0.580574i	$0.197172\pi$
$743$			2.93963e24i			1.16115i	−0.814208	+	0.580574i	$0.802828\pi$
$744$		0			0
$745$	−1.51009e24			−0.583006
$746$		−	2.03135e24i		−	0.775361i
$747$		0			0
$748$			3.37687e22i			0.0125994i
$749$		−	2.26913e23i		−	0.0837074i
$750$		0			0
$751$	1.54013e24			0.555416			0.277708	−	0.960666i	$-0.410425\pi$
$751$	1.54013e24			0.555416			0.277708	+	0.960666i	$0.410425\pi$
$752$			4.16939e24i			1.48669i
$753$		0			0
$754$	1.06827e24	−	2.95445e24i	0.372411	−	1.02996i
$755$	−2.32120e24			−0.800137
$756$		0			0
$757$	−1.70613e24			−0.575039			−0.287520	−	0.957775i	$-0.592831\pi$
$757$	−1.70613e24			−0.575039			−0.287520	+	0.957775i	$0.592831\pi$
$758$	−3.43074e24			−1.14340
$759$		0			0
$760$		−	2.58141e24i		−	0.841282i
$761$			1.68930e24i			0.544425i	0.962237	+	0.272213i	$0.0877554\pi$
$761$			1.68930e24i			0.544425i	−0.962237	+	0.272213i	$0.912245\pi$
$762$		0			0
$763$	1.29261e22			0.00407390
$764$	2.32600e23			0.0724963
$765$		0			0
$766$	−3.83176e24			−1.16803
$767$	2.19532e24	−	6.07148e24i	0.661815	−	1.83035i
$768$		0			0
$769$			1.90832e24i			0.562700i	0.959605	+	0.281350i	$0.0907822\pi$
$769$			1.90832e24i			0.562700i	−0.959605	+	0.281350i	$0.909218\pi$
$770$	1.31647e23			0.0383919
$771$		0			0
$772$		−	1.42882e23i		−	0.0407598i
$773$			1.49323e24i			0.421309i	0.977561	+	0.210654i	$0.0675595\pi$
$773$			1.49323e24i			0.421309i	−0.977561	+	0.210654i	$0.932441\pi$
$774$		0			0
$775$		−	3.59325e23i		−	0.0991797i
$776$	5.39647e24			1.47328
$777$		0			0
$778$			5.32785e24i			1.42307i
$779$	1.04026e24			0.274835
$780$		0			0
$781$	−8.03945e23			−0.207823
$782$		−	1.34317e24i		−	0.343459i
$783$		0			0
$784$	−3.80052e24			−0.950948
$785$			3.66469e24i			0.907080i
$786$		0			0
$787$		−	5.66871e24i		−	1.37309i	−0.727087	−	0.686545i	$-0.759126\pi$
$787$		−	5.66871e24i		−	1.37309i	0.727087	−	0.686545i	$-0.240874\pi$
$788$		−	2.17611e22i		−	0.00521445i
$789$		0			0
$790$	−4.20872e24			−0.987007
$791$			3.77815e23i			0.0876556i
$792$		0			0
$793$	−6.61159e23	−	2.39061e23i	−0.150136	−	0.0542859i
$794$	−5.68022e24			−1.27612
$795$		0			0
$796$	−3.52262e23			−0.0774649
$797$	−5.06348e24			−1.10167			−0.550837	−	0.834613i	$-0.685692\pi$
$797$	−5.06348e24			−1.10167			−0.550837	+	0.834613i	$0.685692\pi$
$798$		0			0
$799$		−	4.93997e24i		−	1.05215i
$800$			3.07841e22i			0.00648726i
$801$		0			0
$802$	5.28112e23			0.108954
$803$	−4.29716e24			−0.877202
$804$		0			0
$805$	2.22827e23			0.0445351
$806$	5.80862e24	+	2.10027e24i	1.14875	+	0.415363i
$807$		0			0
$808$		−	3.43800e24i		−	0.665747i
$809$	−3.18443e24			−0.610195			−0.305098	−	0.952321i	$-0.598689\pi$
$809$	−3.18443e24			−0.610195			−0.305098	+	0.952321i	$0.598689\pi$
$810$		0			0
$811$		−	6.36497e24i		−	1.19432i	−0.802124	−	0.597158i	$-0.796297\pi$
$811$		−	6.36497e24i		−	1.19432i	0.802124	−	0.597158i	$-0.203703\pi$
$812$		−	2.03602e22i		−	0.00378056i
$813$		0			0
$814$			3.68031e24i			0.669232i
$815$	−8.91858e24			−1.60493
$816$		0			0
$817$			7.34276e24i			1.29411i
$818$	−1.02314e24			−0.178456
$819$		0			0
$820$	−8.58901e22			−0.0146732
$821$		−	1.93799e23i		−	0.0327668i	−0.999866	−	0.0163834i	$-0.994785\pi$
$821$		−	1.93799e23i		−	0.0327668i	0.999866	−	0.0163834i	$-0.00521523\pi$
$822$		0			0
$823$	−7.51783e24			−1.24507			−0.622535	−	0.782592i	$-0.713897\pi$
$823$	−7.51783e24			−1.24507			−0.622535	+	0.782592i	$0.713897\pi$
$824$		−	4.49455e24i		−	0.736724i
$825$		0			0
$826$			9.83246e23i			0.157882i
$827$			1.69926e24i			0.270062i	0.990841	+	0.135031i	$0.0431135\pi$
$827$			1.69926e24i			0.270062i	−0.990841	+	0.135031i	$0.956887\pi$
$828$		0			0
$829$	2.76867e24			0.431080			0.215540	−	0.976495i	$-0.430849\pi$
$829$	2.76867e24			0.431080			0.215540	+	0.976495i	$0.430849\pi$
$830$			1.18844e24i			0.183153i
$831$		0			0
$832$	−6.46131e24	−	2.33627e24i	−0.975601	−	0.352756i
$833$	4.50293e24			0.672997
$834$		0			0
$835$	7.11125e23			0.104138
$836$	1.01883e23			0.0147689
$837$		0			0
$838$			1.19279e25i			1.69430i
$839$		−	1.21805e25i		−	1.71273i	−0.516373	−	0.856364i	$-0.672718\pi$
$839$		−	1.21805e25i		−	1.71273i	0.516373	−	0.856364i	$-0.327282\pi$
$840$		0			0
$841$	1.81827e24			0.250549
$842$	3.25937e24			0.444612
$843$		0			0
$844$	2.78262e23			0.0372000
$845$	−5.02074e24	+	6.03510e24i	−0.664485	+	0.798735i
$846$		0			0
$847$		−	5.06028e23i		−	0.0656396i
$848$	6.74957e24			0.866785
$849$		0			0
$850$			4.19258e23i			0.0527740i
$851$			6.22932e24i			0.776318i
$852$		0			0
$853$			5.78621e24i			0.706850i	0.935463	+	0.353425i	$0.114983\pi$
$853$			5.78621e24i			0.706850i	−0.935463	+	0.353425i	$0.885017\pi$
$854$	1.07072e23			0.0129504
$855$		0			0
$856$		−	8.68860e24i		−	1.03020i
$857$	−1.38024e25			−1.62038			−0.810190	−	0.586168i	$-0.800636\pi$
$857$	−1.38024e25			−1.62038			−0.810190	+	0.586168i	$0.800636\pi$
$858$		0			0
$859$	−4.24691e24			−0.488800			−0.244400	−	0.969675i	$-0.578591\pi$
$859$	−4.24691e24			−0.488800			−0.244400	+	0.969675i	$0.578591\pi$
$860$		−	6.06264e23i		−	0.0690915i
$861$		0			0
$862$	1.67143e25			1.86757
$863$		−	9.77368e23i		−	0.108135i	−0.998537	−	0.0540675i	$-0.982781\pi$
$863$		−	9.77368e23i		−	0.108135i	0.998537	−	0.0540675i	$-0.0172186\pi$
$864$		0			0
$865$		−	1.23292e23i		−	0.0133751i
$866$		−	1.28770e25i		−	1.38328i
$867$		0			0
$868$	4.00294e22			0.00421659
$869$		−	4.23575e24i		−	0.441837i
$870$		0			0
$871$	4.67293e24	−	1.29237e25i	0.478007	−	1.32200i
$872$	4.94948e23			0.0501382
$873$		0			0
$874$	−4.05248e24			−0.402600
$875$	8.05127e23			0.0792129
$876$		0			0
$877$		−	1.39541e25i		−	1.34650i	−0.739415	−	0.673250i	$-0.764898\pi$
$877$		−	1.39541e25i		−	1.34650i	0.739415	−	0.673250i	$-0.235102\pi$
$878$			9.86181e24i			0.942439i
$879$		0			0
$880$	4.83473e24			0.453179
$881$	1.33983e25			1.24381			0.621906	−	0.783092i	$-0.286359\pi$
$881$	1.33983e25			1.24381			0.621906	+	0.783092i	$0.286359\pi$
$882$		0			0
$883$	−4.50243e24			−0.409997			−0.204999	−	0.978762i	$-0.565719\pi$
$883$	−4.50243e24			−0.409997			−0.204999	+	0.978762i	$0.565719\pi$
$884$	2.88406e23	+	1.04282e23i	0.0260112	+	0.00940510i
$885$		0			0
$886$		−	4.99788e23i		−	0.0442180i
$887$	−9.19483e24			−0.805737			−0.402868	−	0.915258i	$-0.631987\pi$
$887$	−9.19483e24			−0.805737			−0.402868	+	0.915258i	$0.631987\pi$
$888$		0			0
$889$		−	1.85452e23i		−	0.0159429i
$890$			2.27054e24i			0.193336i
$891$		0			0
$892$		−	6.08896e22i		−	0.00508676i
$893$	−1.49043e25			−1.23332
$894$		0			0
$895$			1.08851e25i			0.883762i
$896$	9.62358e23			0.0773960
$897$		0			0
$898$	4.69322e24			0.370358
$899$			1.78428e25i			1.39478i
$900$		0			0
$901$	−7.99702e24			−0.613433
$902$			2.03135e24i			0.154358i
$903$		0			0
$904$			1.44667e25i			1.07879i
$905$			4.06687e24i			0.300433i
$906$		0			0
$907$	−4.63861e24			−0.336299			−0.168150	−	0.985761i	$-0.553779\pi$
$907$	−4.63861e24			−0.336299			−0.168150	+	0.985761i	$0.553779\pi$
$908$			1.29357e23i			0.00929094i
$909$		0			0
$910$	4.06542e23	−	1.12435e24i	0.0286585	−	0.0792594i
$911$	−3.94206e24			−0.275307			−0.137653	−	0.990480i	$-0.543956\pi$
$911$	−3.94206e24			−0.275307			−0.137653	+	0.990480i	$0.543956\pi$
$912$		0			0
$913$	−1.19607e24			−0.0819892
$914$	1.36051e25			0.923971
$915$		0			0
$916$			6.83417e23i			0.0455590i
$917$		−	1.45158e24i		−	0.0958741i
$918$		0			0
$919$	2.14434e25			1.39031			0.695155	−	0.718860i	$-0.255336\pi$
$919$	2.14434e25			1.39031			0.695155	+	0.718860i	$0.255336\pi$
$920$	8.53214e24			0.548102
$921$		0			0
$922$	2.77480e25			1.74992
$923$	−2.48268e24	+	6.86622e24i	−0.155134	+	0.429046i
$924$		0			0
$925$		−	1.94442e24i		−	0.119285i
$926$	1.33994e25			0.814499
$927$		0			0
$928$		−	1.52863e24i		−	0.0912315i
$929$		−	1.61203e25i		−	0.953323i	−0.879087	−	0.476662i	$-0.841847\pi$
$929$		−	1.61203e25i		−	0.953323i	0.879087	−	0.476662i	$-0.158153\pi$
$930$		0			0
$931$		−	1.35858e25i		−	0.788882i
$932$	7.73478e23			0.0445054
$933$		0			0
$934$			1.01290e25i			0.572293i
$935$	−5.72828e24			−0.320720
$936$		0			0
$937$	2.01333e25			1.10695			0.553476	−	0.832865i	$-0.313301\pi$
$937$	2.01333e25			1.10695			0.553476	+	0.832865i	$0.313301\pi$
$938$			2.09293e24i			0.114033i
$939$		0			0
$940$	1.23060e24			0.0658459
$941$		−	1.55501e25i		−	0.824560i	−0.911057	−	0.412280i	$-0.864733\pi$
$941$		−	1.55501e25i		−	0.824560i	0.911057	−	0.412280i	$-0.135267\pi$
$942$		0			0
$943$			3.43828e24i			0.179057i
$944$			3.61097e25i			1.86364i
$945$		0			0
$946$	−1.43385e25			−0.726824
$947$		−	1.01550e25i		−	0.510159i	−0.966920	−	0.255080i	$-0.917898\pi$
$947$		−	1.01550e25i		−	0.510159i	0.966920	−	0.255080i	$-0.0821017\pi$
$948$		0			0
$949$	−1.32701e25	+	3.67006e25i	−0.654806	+	1.81097i
$950$	1.26494e24			0.0618614
$951$		0			0
$952$	−1.19099e24			−0.0572128
$953$	4.08461e25			1.94474			0.972368	−	0.233452i	$-0.0750021\pi$
$953$	4.08461e25			1.94474			0.972368	+	0.233452i	$0.0750021\pi$
$954$		0			0
$955$			3.94566e25i			1.84540i
$956$		−	1.10672e24i		−	0.0513032i
$957$		0			0
$958$	1.68949e25			0.769393
$959$	−1.33594e24			−0.0603016
$960$		0			0
$961$	−1.25299e25			−0.555649
$962$	3.14323e25	+	1.13652e25i	1.38162	+	0.499563i
$963$		0			0
$964$		−	1.39628e23i		−	0.00603003i
$965$	2.42376e25			1.03755
$966$		0			0
$967$		−	3.08917e25i		−	1.29932i	−0.760224	−	0.649661i	$-0.774911\pi$
$967$		−	3.08917e25i		−	1.29932i	0.760224	−	0.649661i	$-0.225089\pi$
$968$		−	1.93761e25i		−	0.807839i
$969$		0			0
$970$			3.58993e25i			1.47071i
$971$	−2.68655e24			−0.109102			−0.0545508	−	0.998511i	$-0.517373\pi$
$971$	−2.68655e24			−0.109102			−0.0545508	+	0.998511i	$0.517373\pi$
$972$		0			0
$973$		−	1.97972e24i		−	0.0790033i
$974$	2.80879e25			1.11114
$975$		0			0
$976$	3.93220e24			0.152866
$977$		−	8.93114e24i		−	0.344194i	−0.985080	−	0.172097i	$-0.944946\pi$
$977$		−	8.93114e24i		−	0.344194i	0.985080	−	0.172097i	$-0.0550542\pi$
$978$		0			0
$979$	−2.28512e24			−0.0865477
$980$			1.12173e24i			0.0421177i
$981$		0			0
$982$		−	8.54730e24i		−	0.315414i
$983$			6.47919e24i			0.237037i	0.992952	+	0.118518i	$0.0378145\pi$
$983$			6.47919e24i			0.237037i	−0.992952	+	0.118518i	$0.962186\pi$
$984$		0			0
$985$	3.69141e24			0.132735
$986$		−	2.08189e25i		−	0.742170i
$987$		0			0
$988$	3.14628e23	−	8.70150e23i	0.0110246	−	0.0304902i
$989$	−2.42695e25			−0.843124
$990$		0			0
$991$	2.26655e24			0.0773996			0.0386998	−	0.999251i	$-0.487678\pi$
$991$	2.26655e24			0.0773996			0.0386998	+	0.999251i	$0.487678\pi$
$992$	3.00538e24			0.101754
$993$		0			0
$994$		−	1.11195e24i		−	0.0370085i
$995$		−	5.97553e25i		−	1.97188i
$996$		0			0
$997$	−3.81043e25			−1.23613			−0.618066	−	0.786126i	$-0.712083\pi$
$997$	−3.81043e25			−1.23613			−0.618066	+	0.786126i	$0.712083\pi$
$998$	−2.11174e25			−0.679251
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.18.b.c.64.6		20
3.2	odd	2		13.18.b.a.12.15	yes	20
13.12	even	2	inner	117.18.b.c.64.15		20
39.38	odd	2		13.18.b.a.12.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.18.b.a.12.6	✓	20	39.38	odd	2
13.18.b.a.12.15	yes	20	3.2	odd	2
117.18.b.c.64.6		20	1.1	even	1	trivial
117.18.b.c.64.15		20	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-354.573i q^{2} +5349.95 q^{4} +907528. i q^{5} -1.26328e6i q^{7} -4.83715e7i q^{8} +3.21785e8 q^{10} +3.23852e8i q^{11} +(2.76591e9 + 1.00009e9i) q^{13} -4.47925e8 q^{14} -1.64500e10 q^{16} +1.94903e10 q^{17} -5.88040e10i q^{19} +4.85523e9i q^{20} +1.14829e11 q^{22} +1.94361e11 q^{23} -6.06677e10 q^{25} +(3.54606e11 - 9.80715e11i) q^{26} -6.75848e9i q^{28} +3.01254e12 q^{29} +5.92284e12i q^{31} -5.07422e11i q^{32} -6.91073e12i q^{34} +1.14646e12 q^{35} +3.20503e13i q^{37} -2.08503e13 q^{38} +4.38985e13 q^{40} +1.76902e13i q^{41} -1.24868e14 q^{43} +1.73259e12i q^{44} -6.89150e13i q^{46} -2.53458e14i q^{47} +2.31035e14 q^{49} +2.15111e13i q^{50} +(1.47974e13 + 5.35044e12i) q^{52} -4.10308e14 q^{53} -2.93904e14 q^{55} -6.11068e13 q^{56} -1.06817e15i q^{58} -2.19511e15i q^{59} -2.39039e14 q^{61} +2.10008e15 q^{62} -2.33606e15 q^{64} +(-9.07612e14 + 2.51014e15i) q^{65} -4.67250e15i q^{67} +1.04272e14 q^{68} -4.06504e14i q^{70} +2.48245e15i q^{71} +1.32689e16i q^{73} +1.13642e16 q^{74} -3.14599e14i q^{76} +4.09115e14 q^{77} -1.30793e16 q^{79} -1.49289e16i q^{80} +6.27248e15 q^{82} +3.69328e15i q^{83} +1.76880e16i q^{85} +4.42749e16i q^{86} +1.56652e16 q^{88} +7.05606e15i q^{89} +(1.26340e15 - 3.49411e15i) q^{91} +1.03982e15 q^{92} -8.98693e16 q^{94} +5.33663e16 q^{95} +1.11563e17i q^{97} -8.19187e16i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 1441794 q^{4} - 1157719078 q^{10} + 2164505000 q^{13} - 7100918310 q^{14} + 137372157250 q^{16} + 41886537180 q^{17} - 517583912680 q^{22} - 222078810480 q^{23} - 5105437226376 q^{25} + 3236456408130 q^{26}+ \cdots + 46\!\cdots\!04 q^{95}+O(q^{100}) \) 20 * q - 1441794 * q^4 - 1157719078 * q^10 + 2164505000 * q^13 - 7100918310 * q^14 + 137372157250 * q^16 + 41886537180 * q^17 - 517583912680 * q^22 - 222078810480 * q^23 - 5105437226376 * q^25 + 3236456408130 * q^26 - 3521990411568 * q^29 - 47961459509856 * q^35 - 66744999456060 * q^38 + 202362624022474 * q^40 - 201474743080160 * q^43 + 53227142648624 * q^49 - 693691356264060 * q^52 - 2429498493199320 * q^53 + 2118490108324208 * q^55 + 2835869617451466 * q^56 - 140707863052352 * q^61 - 6716044276698600 * q^62 - 5699141619649666 * q^64 - 4675683183703692 * q^65 + 692992269522030 * q^68 - 34108184292603558 * q^74 + 59285142434915520 * q^77 - 31814529658962752 * q^79 - 114501724181741920 * q^82 + 207041488957467040 * q^88 + 89768903072866368 * q^91 - 120908580468048540 * q^92 - 32863399636204954 * q^94 + 462354331213557504 * q^95

Introduction

L-functions

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Database

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