LMFDB - Embedded newform 162.9.b.c.161.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,9,Mod(161,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("162.161");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	162.b (of order $2$, degree $1$, 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:	$65.9953348299$
Analytic rank:	$0$
Dimension:	$16$
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} - 8 x^{15} + 5476 x^{14} - 38192 x^{13} + 11414542 x^{12} - 67991120 x^{11} + \cdots + 19\!\cdots\!29 $ x^16 - 8x^15 + 5476x^14 - 38192x^13 + 11414542x^12 - 67991120x^11 + 11330952892x^10 - 56032421816x^9 + 5575009041895x^8 - 21964587980304x^7 + 1366797007396500x^6 - 4023749554164360x^5 + 166750774096111110x^4 - 326820812596206264x^3 + 9443577280676906808x^2 - 9280833855293027160*x + 190324219382219227329
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{36}\cdot 3^{80} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.8
Root			$0.500000 - 11.5235i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.9.b.c.161.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.3137i q^{2} -128.000 q^{4} +1079.93i q^{5} +4112.18 q^{7} +1448.15i q^{8} +O(q^{10})$	\(q-11.3137i q^{2} -128.000 q^{4} +1079.93i q^{5} +4112.18 q^{7} +1448.15i q^{8} +12218.1 q^{10} -9722.03i q^{11} +8232.04 q^{13} -46524.0i q^{14} +16384.0 q^{16} -87809.9i q^{17} +123643. q^{19} -138231. i q^{20} -109992. q^{22} +106266. i q^{23} -775631. q^{25} -93134.9i q^{26} -526359. q^{28} -497543. i q^{29} +1.15595e6 q^{31} -185364. i q^{32} -993456. q^{34} +4.44088e6i q^{35} +1.20145e6 q^{37} -1.39886e6i q^{38} -1.56391e6 q^{40} +1.77769e6i q^{41} +1.05745e6 q^{43} +1.24442e6i q^{44} +1.20227e6 q^{46} +4.39582e6i q^{47} +1.11452e7 q^{49} +8.77527e6i q^{50} -1.05370e6 q^{52} +1.04810e6i q^{53} +1.04991e7 q^{55} +5.95507e6i q^{56} -5.62906e6 q^{58} -1.00465e7i q^{59} +8.31541e6 q^{61} -1.30781e7i q^{62} -2.09715e6 q^{64} +8.89006e6i q^{65} -1.23837e7 q^{67} +1.12397e7i q^{68} +5.02428e7 q^{70} -1.69579e7i q^{71} -2.42736e7 q^{73} -1.35929e7i q^{74} -1.58263e7 q^{76} -3.99787e7i q^{77} -1.27801e7 q^{79} +1.76936e7i q^{80} +2.01122e7 q^{82} +3.22379e7i q^{83} +9.48289e7 q^{85} -1.19637e7i q^{86} +1.40790e7 q^{88} +2.10088e7i q^{89} +3.38516e7 q^{91} -1.36021e7i q^{92} +4.97330e7 q^{94} +1.33526e8i q^{95} +8.93302e7 q^{97} -1.26094e8i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2048 q^{4} + 3692 q^{7}+O(q^{10}) $ 16 * q - 2048 * q^4 + 3692 * q^7	$ 16 q - 2048 q^{4} + 3692 q^{7} + 6740 q^{13} + 262144 q^{16} + 362180 q^{19} + 123648 q^{22} - 1926788 q^{25} - 472576 q^{28} - 1084876 q^{31} - 440832 q^{34} + 3343328 q^{37} - 679024 q^{43} + 7417344 q^{46} + 4729308 q^{49} - 862720 q^{52} - 4584276 q^{55} + 15705600 q^{58} + 1683908 q^{61} - 33554432 q^{64} - 59893288 q^{67} + 68719104 q^{70} - 7547764 q^{73} - 46359040 q^{76} - 67626004 q^{79} - 137346048 q^{82} + 251393544 q^{85} - 15826944 q^{88} + 268578316 q^{91} + 23665152 q^{94} + 178830968 q^{97}+O(q^{100}) $ 16 * q - 2048 * q^4 + 3692 * q^7 + 6740 * q^13 + 262144 * q^16 + 362180 * q^19 + 123648 * q^22 - 1926788 * q^25 - 472576 * q^28 - 1084876 * q^31 - 440832 * q^34 + 3343328 * q^37 - 679024 * q^43 + 7417344 * q^46 + 4729308 * q^49 - 862720 * q^52 - 4584276 * q^55 + 15705600 * q^58 + 1683908 * q^61 - 33554432 * q^64 - 59893288 * q^67 + 68719104 * q^70 - 7547764 * q^73 - 46359040 * q^76 - 67626004 * q^79 - 137346048 * q^82 + 251393544 * q^85 - 15826944 * q^88 + 268578316 * q^91 + 23665152 * q^94 + 178830968 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-1$

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 $

$2$	− 11.3137i	− 0.707107i
$2$	− 11.3137i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−128.000	−0.500000
$5$	1079.93i	1.72789i	0.503583	+	0.863947i	$0.332015\pi$
$5$	1079.93i	1.72789i	−0.503583	+	0.863947i	$0.667985\pi$
$6$	0	0
$7$	4112.18	1.71269	0.856347	−	0.516400i	$-0.172728\pi$
$7$	4112.18	1.71269	0.856347	+	0.516400i	$0.172728\pi$
$8$	1448.15i	0.353553i
$9$	0	0
$10$	12218.1	1.22181
$11$	− 9722.03i	− 0.664028i	−0.943274	−	0.332014i	$-0.892272\pi$
$11$	− 9722.03i	− 0.664028i	0.943274	−	0.332014i	$-0.107728\pi$
$12$	0	0
$13$	8232.04	0.288227	0.144113	−	0.989561i	$-0.453967\pi$
$13$	8232.04	0.288227	0.144113	+	0.989561i	$0.453967\pi$
$14$	− 46524.0i	− 1.21106i
$15$	0	0
$16$	16384.0	0.250000
$17$	− 87809.9i	− 1.05135i	−0.850685	−	0.525676i	$-0.823813\pi$
$17$	− 87809.9i	− 1.05135i	0.850685	−	0.525676i	$-0.176187\pi$
$18$	0	0
$19$	123643.	0.948754	0.474377	−	0.880322i	$-0.342673\pi$
$19$	123643.	0.948754	0.474377	+	0.880322i	$0.342673\pi$
$20$	− 138231.i	− 0.863947i
$21$	0	0
$22$	−109992.	−0.469539
$23$	106266.i	0.379738i	0.981809	+	0.189869i	$0.0608064\pi$
$23$	106266.i	0.379738i	−0.981809	+	0.189869i	$0.939194\pi$
$24$	0	0
$25$	−775631.	−1.98562
$26$	− 93134.9i	− 0.203807i
$27$	0	0
$28$	−526359.	−0.856347
$29$	− 497543.i	− 0.703459i	−0.936102	−	0.351729i	$-0.885594\pi$
$29$	− 497543.i	− 0.703459i	0.936102	−	0.351729i	$-0.114406\pi$
$30$	0	0
$31$	1.15595e6	1.25168	0.625841	−	0.779951i	$-0.284756\pi$
$31$	1.15595e6	1.25168	0.625841	+	0.779951i	$0.284756\pi$
$32$	− 185364.i	− 0.176777i
$33$	0	0
$34$	−993456.	−0.743418
$35$	4.44088e6i	2.95935i
$36$	0	0
$37$	1.20145e6	0.641061	0.320531	−	0.947238i	$-0.396139\pi$
$37$	1.20145e6	0.641061	0.320531	+	0.947238i	$0.396139\pi$
$38$	− 1.39886e6i	− 0.670870i
$39$	0	0
$40$	−1.56391e6	−0.610903
$41$	1.77769e6i	0.629100i	0.949241	+	0.314550i	$0.101854\pi$
$41$	1.77769e6i	0.629100i	−0.949241	+	0.314550i	$0.898146\pi$
$42$	0	0
$43$	1.05745e6	0.309305	0.154652	−	0.987969i	$-0.450574\pi$
$43$	1.05745e6	0.309305	0.154652	+	0.987969i	$0.450574\pi$
$44$	1.24442e6i	0.332014i
$45$	0	0
$46$	1.20227e6	0.268516
$47$	4.39582e6i	0.900841i	0.892817	+	0.450420i	$0.148726\pi$
$47$	4.39582e6i	0.900841i	−0.892817	+	0.450420i	$0.851274\pi$
$48$	0	0
$49$	1.11452e7	1.93332
$50$	8.77527e6i	1.40404i
$51$	0	0
$52$	−1.05370e6	−0.144113
$53$	1.04810e6i	0.132831i	0.997792	+	0.0664157i	$0.0211563\pi$
$53$	1.04810e6i	0.132831i	−0.997792	+	0.0664157i	$0.978844\pi$
$54$	0	0
$55$	1.04991e7	1.14737
$56$	5.95507e6i	0.605529i
$57$	0	0
$58$	−5.62906e6	−0.497421
$59$	− 1.00465e7i	− 0.829096i	−0.910027	−	0.414548i	$-0.863940\pi$
$59$	− 1.00465e7i	− 0.829096i	0.910027	−	0.414548i	$-0.136060\pi$
$60$	0	0
$61$	8.31541e6	0.600571	0.300285	−	0.953849i	$-0.402918\pi$
$61$	8.31541e6	0.600571	0.300285	+	0.953849i	$0.402918\pi$
$62$	− 1.30781e7i	− 0.885073i
$63$	0	0
$64$	−2.09715e6	−0.125000
$65$	8.89006e6i	0.498025i
$66$	0	0
$67$	−1.23837e7	−0.614539	−0.307270	−	0.951622i	$-0.599415\pi$
$67$	−1.23837e7	−0.614539	−0.307270	+	0.951622i	$0.599415\pi$
$68$	1.12397e7i	0.525676i
$69$	0	0
$70$	5.02428e7	2.09258
$71$	− 1.69579e7i	− 0.667328i	−0.942692	−	0.333664i	$-0.891715\pi$
$71$	− 1.69579e7i	− 0.667328i	0.942692	−	0.333664i	$-0.108285\pi$
$72$	0	0
$73$	−2.42736e7	−0.854757	−0.427378	−	0.904073i	$-0.640563\pi$
$73$	−2.42736e7	−0.854757	−0.427378	+	0.904073i	$0.640563\pi$
$74$	− 1.35929e7i	− 0.453299i
$75$	0	0
$76$	−1.58263e7	−0.474377
$77$	− 3.99787e7i	− 1.13728i
$78$	0	0
$79$	−1.27801e7	−0.328114	−0.164057	−	0.986451i	$-0.552458\pi$
$79$	−1.27801e7	−0.328114	−0.164057	+	0.986451i	$0.552458\pi$
$80$	1.76936e7i	0.431973i
$81$	0	0
$82$	2.01122e7	0.444841
$83$	3.22379e7i	0.679289i	0.940554	+	0.339644i	$0.110307\pi$
$83$	3.22379e7i	0.679289i	−0.940554	+	0.339644i	$0.889693\pi$
$84$	0	0
$85$	9.48289e7	1.81662
$86$	− 1.19637e7i	− 0.218712i
$87$	0	0
$88$	1.40790e7	0.234769
$89$	2.10088e7i	0.334843i	0.985885	+	0.167422i	$0.0535441\pi$
$89$	2.10088e7i	0.334843i	−0.985885	+	0.167422i	$0.946456\pi$
$90$	0	0
$91$	3.38516e7	0.493644
$92$	− 1.36021e7i	− 0.189869i
$93$	0	0
$94$	4.97330e7	0.636991
$95$	1.33526e8i	1.63935i
$96$	0	0
$97$	8.93302e7	1.00905	0.504523	−	0.863398i	$-0.331668\pi$
$97$	8.93302e7	1.00905	0.504523	+	0.863398i	$0.331668\pi$
$98$	− 1.26094e8i	− 1.36707i
$99$	0	0
$100$	9.92808e7	0.992808
$101$	3.56144e7i	0.342247i	0.985250	+	0.171124i	$0.0547398\pi$
$101$	3.56144e7i	0.342247i	−0.985250	+	0.171124i	$0.945260\pi$
$102$	0	0
$103$	8.70600e6	0.0773517	0.0386759	−	0.999252i	$-0.487686\pi$
$103$	8.70600e6	0.0773517	0.0386759	+	0.999252i	$0.487686\pi$
$104$	1.19213e7i	0.101904i
$105$	0	0
$106$	1.18579e7	0.0939259
$107$	2.34341e8i	1.78778i	0.448290	+	0.893888i	$0.352033\pi$
$107$	2.34341e8i	1.78778i	−0.448290	+	0.893888i	$0.647967\pi$
$108$	0	0
$109$	2.66492e8	1.88789	0.943947	−	0.330098i	$-0.107082\pi$
$109$	2.66492e8	1.88789	0.943947	+	0.330098i	$0.107082\pi$
$110$	− 1.18784e8i	− 0.811313i
$111$	0	0
$112$	6.73740e7	0.428174
$113$	− 2.85858e8i	− 1.75322i	−0.481202	−	0.876610i	$-0.659800\pi$
$113$	− 2.85858e8i	− 1.75322i	0.481202	−	0.876610i	$-0.340200\pi$
$114$	0	0
$115$	−1.14761e8	−0.656148
$116$	6.36855e7i	0.351729i
$117$	0	0
$118$	−1.13663e8	−0.586259
$119$	− 3.61090e8i	− 1.80064i
$120$	0	0
$121$	1.19841e8	0.559067
$122$	− 9.40781e7i	− 0.424668i
$123$	0	0
$124$	−1.47962e8	−0.625841
$125$	− 4.15781e8i	− 1.70304i
$126$	0	0
$127$	−8.18752e7	−0.314730	−0.157365	−	0.987541i	$-0.550300\pi$
$127$	−8.18752e7	−0.314730	−0.157365	+	0.987541i	$0.550300\pi$
$128$	2.37266e7i	0.0883883i
$129$	0	0
$130$	1.00580e8	0.352157
$131$	3.78511e8i	1.28527i	0.766173	+	0.642634i	$0.222158\pi$
$131$	3.78511e8i	1.28527i	−0.766173	+	0.642634i	$0.777842\pi$
$132$	0	0
$133$	5.08441e8	1.62493
$134$	1.40105e8i	0.434545i
$135$	0	0
$136$	1.27162e8	0.371709
$137$	− 9.66347e7i	− 0.274316i	−0.990549	−	0.137158i	$-0.956203\pi$
$137$	− 9.66347e7i	− 0.274316i	0.990549	−	0.137158i	$-0.0437968\pi$
$138$	0	0
$139$	4.26459e8	1.14240	0.571200	−	0.820811i	$-0.306478\pi$
$139$	4.26459e8	1.14240	0.571200	+	0.820811i	$0.306478\pi$
$140$	− 5.68433e8i	− 1.47968i
$141$	0	0
$142$	−1.91857e8	−0.471872
$143$	− 8.00322e7i	− 0.191391i
$144$	0	0
$145$	5.37313e8	1.21550
$146$	2.74624e8i	0.604404i
$147$	0	0
$148$	−1.53786e8	−0.320531
$149$	4.31206e8i	0.874863i	0.899252	+	0.437431i	$0.144112\pi$
$149$	4.31206e8i	0.874863i	−0.899252	+	0.437431i	$0.855888\pi$
$150$	0	0
$151$	8.40561e8	1.61682	0.808409	−	0.588621i	$-0.200329\pi$
$151$	8.40561e8	1.61682	0.808409	+	0.588621i	$0.200329\pi$
$152$	1.79054e8i	0.335435i
$153$	0	0
$154$	−4.52308e8	−0.804176
$155$	1.24835e9i	2.16277i
$156$	0	0
$157$	−4.21838e8	−0.694301	−0.347150	−	0.937809i	$-0.612851\pi$
$157$	−4.21838e8	−0.694301	−0.347150	+	0.937809i	$0.612851\pi$
$158$	1.44590e8i	0.232012i
$159$	0	0
$160$	2.00181e8	0.305451
$161$	4.36987e8i	0.650376i
$162$	0	0
$163$	−5.50818e8	−0.780293	−0.390146	−	0.920753i	$-0.627576\pi$
$163$	−5.50818e8	−0.780293	−0.390146	+	0.920753i	$0.627576\pi$
$164$	− 2.27544e8i	− 0.314550i
$165$	0	0
$166$	3.64730e8	0.480330
$167$	6.06853e8i	0.780221i	0.920768	+	0.390111i	$0.127563\pi$
$167$	6.06853e8i	0.780221i	−0.920768	+	0.390111i	$0.872437\pi$
$168$	0	0
$169$	−7.47964e8	−0.916925
$170$	− 1.07287e9i	− 1.28455i
$171$	0	0
$172$	−1.35354e8	−0.154652
$173$	9.54495e8i	1.06559i	0.846245	+	0.532794i	$0.178858\pi$
$173$	9.54495e8i	1.06559i	−0.846245	+	0.532794i	$0.821142\pi$
$174$	0	0
$175$	−3.18954e9	−3.40076
$176$	− 1.59286e8i	− 0.166007i
$177$	0	0
$178$	2.37688e8	0.236770
$179$	9.44019e8i	0.919536i	0.888039	+	0.459768i	$0.152067\pi$
$179$	9.44019e8i	0.919536i	−0.888039	+	0.459768i	$0.847933\pi$
$180$	0	0
$181$	−8.68230e8	−0.808948	−0.404474	−	0.914550i	$-0.632545\pi$
$181$	−8.68230e8	−0.808948	−0.404474	+	0.914550i	$0.632545\pi$
$182$	− 3.82988e8i	− 0.349059i
$183$	0	0
$184$	−1.53890e8	−0.134258
$185$	1.29749e9i	1.10769i
$186$	0	0
$187$	−8.53691e8	−0.698126
$188$	− 5.62665e8i	− 0.450420i
$189$	0	0
$190$	1.51067e9	1.15919
$191$	7.04317e7i	0.0529218i	0.999650	+	0.0264609i	$0.00842375\pi$
$191$	7.04317e7i	0.0529218i	−0.999650	+	0.0264609i	$0.991576\pi$
$192$	0	0
$193$	2.73742e9	1.97293	0.986467	−	0.163957i	$-0.0524260\pi$
$193$	2.73742e9	1.97293	0.986467	+	0.163957i	$0.0524260\pi$
$194$	− 1.01066e9i	− 0.713504i
$195$	0	0
$196$	−1.42659e9	−0.966662
$197$	5.47276e8i	0.363364i	0.983357	+	0.181682i	$0.0581541\pi$
$197$	5.47276e8i	0.363364i	−0.983357	+	0.181682i	$0.941846\pi$
$198$	0	0
$199$	−2.66467e9	−1.69915	−0.849574	−	0.527469i	$-0.823141\pi$
$199$	−2.66467e9	−1.69915	−0.849574	+	0.527469i	$0.823141\pi$
$200$	− 1.12323e9i	− 0.702021i
$201$	0	0
$202$	4.02931e8	0.242005
$203$	− 2.04599e9i	− 1.20481i
$204$	0	0
$205$	−1.91978e9	−1.08702
$206$	− 9.84972e7i	− 0.0546959i
$207$	0	0
$208$	1.34874e8	0.0720567
$209$	− 1.20206e9i	− 0.629999i
$210$	0	0
$211$	1.87827e9	0.947609	0.473804	−	0.880630i	$-0.342880\pi$
$211$	1.87827e9	0.947609	0.473804	+	0.880630i	$0.342880\pi$
$212$	− 1.34157e8i	− 0.0664157i
$213$	0	0
$214$	2.65127e9	1.26415
$215$	1.14198e9i	0.534446i
$216$	0	0
$217$	4.75349e9	2.14375
$218$	− 3.01501e9i	− 1.33494i
$219$	0	0
$220$	−1.34389e9	−0.573685
$221$	− 7.22855e8i	− 0.303028i
$222$	0	0
$223$	−2.90104e9	−1.17310	−0.586548	−	0.809914i	$-0.699514\pi$
$223$	−2.90104e9	−1.17310	−0.586548	+	0.809914i	$0.699514\pi$
$224$	− 7.62249e8i	− 0.302765i
$225$	0	0
$226$	−3.23411e9	−1.23971
$227$	− 4.16807e9i	− 1.56975i	−0.619651	−	0.784877i	$-0.712726\pi$
$227$	− 4.16807e9i	− 1.56975i	0.619651	−	0.784877i	$-0.287274\pi$
$228$	0	0
$229$	−3.60037e9	−1.30920	−0.654599	−	0.755977i	$-0.727162\pi$
$229$	−3.60037e9	−1.30920	−0.654599	+	0.755977i	$0.727162\pi$
$230$	1.29837e9i	0.463966i
$231$	0	0
$232$	7.20519e8	0.248710
$233$	2.86365e9i	0.971618i	0.874065	+	0.485809i	$0.161475\pi$
$233$	2.86365e9i	0.971618i	−0.874065	+	0.485809i	$0.838525\pi$
$234$	0	0
$235$	−4.74719e9	−1.55656
$236$	1.28595e9i	0.414548i
$237$	0	0
$238$	−4.08527e9	−1.27325
$239$	1.40296e9i	0.429985i	0.976616	+	0.214992i	$0.0689727\pi$
$239$	1.40296e9i	0.429985i	−0.976616	+	0.214992i	$0.931027\pi$
$240$	0	0
$241$	−4.43615e9	−1.31504	−0.657519	−	0.753438i	$-0.728394\pi$
$241$	−4.43615e9	−1.31504	−0.657519	+	0.753438i	$0.728394\pi$
$242$	− 1.35585e9i	− 0.395320i
$243$	0	0
$244$	−1.06437e9	−0.300285
$245$	1.20361e10i	3.34058i
$246$	0	0
$247$	1.01783e9	0.273456
$248$	1.67400e9i	0.442537i
$249$	0	0
$250$	−4.70403e9	−1.20423
$251$	− 2.42084e9i	− 0.609916i	−0.952366	−	0.304958i	$-0.901357\pi$
$251$	− 2.42084e9i	− 0.609916i	0.952366	−	0.304958i	$-0.0986425\pi$
$252$	0	0
$253$	1.03313e9	0.252157
$254$	9.26312e8i	0.222547i
$255$	0	0
$256$	2.68435e8	0.0625000
$257$	− 2.41234e9i	− 0.552975i	−0.961017	−	0.276488i	$-0.910829\pi$
$257$	− 2.41234e9i	− 0.552975i	0.961017	−	0.276488i	$-0.0891705\pi$
$258$	0	0
$259$	4.94059e9	1.09794
$260$	− 1.13793e9i	− 0.249013i
$261$	0	0
$262$	4.28237e9	0.908822
$263$	− 6.52441e9i	− 1.36370i	−0.731493	−	0.681849i	$-0.761176\pi$
$263$	− 6.52441e9i	− 1.36370i	0.731493	−	0.681849i	$-0.238824\pi$
$264$	0	0
$265$	−1.13188e9	−0.229518
$266$	− 5.75235e9i	− 1.14900i
$267$	0	0
$268$	1.58511e9	0.307270
$269$	− 6.64127e9i	− 1.26836i	−0.773186	−	0.634180i	$-0.781338\pi$
$269$	− 6.64127e9i	− 1.26836i	0.773186	−	0.634180i	$-0.218662\pi$
$270$	0	0
$271$	−2.72229e9	−0.504728	−0.252364	−	0.967632i	$-0.581208\pi$
$271$	−2.72229e9	−0.504728	−0.252364	+	0.967632i	$0.581208\pi$
$272$	− 1.43868e9i	− 0.262838i
$273$	0	0
$274$	−1.09330e9	−0.193971
$275$	7.54071e9i	1.31850i
$276$	0	0
$277$	1.00189e10	1.70177	0.850886	−	0.525350i	$-0.176066\pi$
$277$	1.00189e10	1.70177	0.850886	+	0.525350i	$0.176066\pi$
$278$	− 4.82484e9i	− 0.807799i
$279$	0	0
$280$	−6.43108e9	−1.04629
$281$	7.88683e9i	1.26496i	0.774576	+	0.632480i	$0.217963\pi$
$281$	7.88683e9i	1.26496i	−0.774576	+	0.632480i	$0.782037\pi$
$282$	0	0
$283$	−9.73710e8	−0.151804	−0.0759021	−	0.997115i	$-0.524184\pi$
$283$	−9.73710e8	−0.151804	−0.0759021	+	0.997115i	$0.524184\pi$
$284$	2.17062e9i	0.333664i
$285$	0	0
$286$	−9.05461e8	−0.135334
$287$	7.31016e9i	1.07746i
$288$	0	0
$289$	−7.34822e8	−0.105339
$290$	− 6.07901e9i	− 0.859490i
$291$	0	0
$292$	3.10702e9	0.427378
$293$	− 5.30493e9i	− 0.719795i	−0.932992	−	0.359898i	$-0.882812\pi$
$293$	− 5.30493e9i	− 0.719795i	0.932992	−	0.359898i	$-0.117188\pi$
$294$	0	0
$295$	1.08495e10	1.43259
$296$	1.73989e9i	0.226649i
$297$	0	0
$298$	4.87854e9	0.618621
$299$	8.74789e8i	0.109451i
$300$	0	0
$301$	4.34843e9	0.529745
$302$	− 9.50986e9i	− 1.14326i
$303$	0	0
$304$	2.02576e9	0.237189
$305$	8.98008e9i	1.03772i
$306$	0	0
$307$	1.34241e10	1.51124	0.755618	−	0.655013i	$-0.227337\pi$
$307$	1.34241e10	1.51124	0.755618	+	0.655013i	$0.227337\pi$
$308$	5.11728e9i	0.568639i
$309$	0	0
$310$	1.41235e10	1.52931
$311$	2.28967e9i	0.244754i	0.992484	+	0.122377i	$0.0390518\pi$
$311$	2.28967e9i	0.244754i	−0.992484	+	0.122377i	$0.960948\pi$
$312$	0	0
$313$	−1.78832e10	−1.86324	−0.931619	−	0.363436i	$-0.881603\pi$
$313$	−1.78832e10	−1.86324	−0.931619	+	0.363436i	$0.881603\pi$
$314$	4.77256e9i	0.490945i
$315$	0	0
$316$	1.63585e9	0.164057
$317$	− 6.66729e9i	− 0.660256i	−0.943936	−	0.330128i	$-0.892908\pi$
$317$	− 6.66729e9i	− 0.660256i	0.943936	−	0.330128i	$-0.107092\pi$
$318$	0	0
$319$	−4.83713e9	−0.467116
$320$	− 2.26478e9i	− 0.215987i
$321$	0	0
$322$	4.94394e9	0.459885
$323$	− 1.08570e10i	− 0.997474i
$324$	0	0
$325$	−6.38503e9	−0.572308
$326$	6.23179e9i	0.551750i
$327$	0	0
$328$	−2.57436e9	−0.222420
$329$	1.80764e10i	1.54287i
$330$	0	0
$331$	1.56858e10	1.30676	0.653380	−	0.757030i	$-0.273351\pi$
$331$	1.56858e10	1.30676	0.653380	+	0.757030i	$0.273351\pi$
$332$	− 4.12645e9i	− 0.339644i
$333$	0	0
$334$	6.86576e9	0.551700
$335$	− 1.33735e10i	− 1.06186i
$336$	0	0
$337$	2.58377e9	0.200324	0.100162	−	0.994971i	$-0.468064\pi$
$337$	2.58377e9	0.200324	0.100162	+	0.994971i	$0.468064\pi$
$338$	8.46225e9i	0.648364i
$339$	0	0
$340$	−1.21381e10	−0.908312
$341$	− 1.12382e10i	− 0.831152i
$342$	0	0
$343$	2.21253e10	1.59850
$344$	1.53135e9i	0.109356i
$345$	0	0
$346$	1.07989e10	0.753485
$347$	1.68155e10i	1.15982i	0.814679	+	0.579912i	$0.196913\pi$
$347$	1.68155e10i	1.15982i	−0.814679	+	0.579912i	$0.803087\pi$
$348$	0	0
$349$	−2.44733e10	−1.64964	−0.824822	−	0.565393i	$-0.808725\pi$
$349$	−2.44733e10	−1.64964	−0.824822	+	0.565393i	$0.808725\pi$
$350$	3.60855e10i	2.40470i
$351$	0	0
$352$	−1.80211e9	−0.117385
$353$	1.04511e10i	0.673072i	0.941671	+	0.336536i	$0.109255\pi$
$353$	1.04511e10i	0.673072i	−0.941671	+	0.336536i	$0.890745\pi$
$354$	0	0
$355$	1.83134e10	1.15307
$356$	− 2.68913e9i	− 0.167422i
$357$	0	0
$358$	1.06804e10	0.650210
$359$	2.37497e10i	1.42982i	0.699219	+	0.714908i	$0.253531\pi$
$359$	2.37497e10i	1.42982i	−0.699219	+	0.714908i	$0.746469\pi$
$360$	0	0
$361$	−1.69607e9	−0.0998656
$362$	9.82290e9i	0.572013i
$363$	0	0
$364$	−4.33301e9	−0.246822
$365$	− 2.62139e10i	− 1.47693i
$366$	0	0
$367$	1.03487e10	0.570453	0.285226	−	0.958460i	$-0.407931\pi$
$367$	1.03487e10	0.570453	0.285226	+	0.958460i	$0.407931\pi$
$368$	1.74107e9i	0.0949346i
$369$	0	0
$370$	1.46794e10	0.783252
$371$	4.30999e9i	0.227500i
$372$	0	0
$373$	−8.05344e9	−0.416050	−0.208025	−	0.978123i	$-0.566704\pi$
$373$	−8.05344e9	−0.416050	−0.208025	+	0.978123i	$0.566704\pi$
$374$	9.65841e9i	0.493650i
$375$	0	0
$376$	−6.36582e9	−0.318495
$377$	− 4.09580e9i	− 0.202756i
$378$	0	0
$379$	−2.63218e10	−1.27573	−0.637864	−	0.770149i	$-0.720182\pi$
$379$	−2.63218e10	−1.27573	−0.637864	+	0.770149i	$0.720182\pi$
$380$	− 1.70913e10i	− 0.819673i
$381$	0	0
$382$	7.96844e8	0.0374214
$383$	− 3.05928e10i	− 1.42175i	−0.703317	−	0.710876i	$-0.748299\pi$
$383$	− 3.05928e10i	− 1.42175i	0.703317	−	0.710876i	$-0.251701\pi$
$384$	0	0
$385$	4.31744e10	1.96509
$386$	− 3.09704e10i	− 1.39508i
$387$	0	0
$388$	−1.14343e10	−0.504523
$389$	− 4.35081e10i	− 1.90008i	−0.312133	−	0.950039i	$-0.601043\pi$
$389$	− 4.35081e10i	− 1.90008i	0.312133	−	0.950039i	$-0.398957\pi$
$390$	0	0
$391$	9.33124e9	0.399238
$392$	1.61400e10i	0.683533i
$393$	0	0
$394$	6.19172e9	0.256937
$395$	− 1.38016e10i	− 0.566947i
$396$	0	0
$397$	4.31419e10	1.73675	0.868375	−	0.495908i	$-0.165165\pi$
$397$	4.31419e10	1.73675	0.868375	+	0.495908i	$0.165165\pi$
$398$	3.01473e10i	1.20148i
$399$	0	0
$400$	−1.27079e10	−0.496404
$401$	− 1.81814e10i	− 0.703152i	−0.936159	−	0.351576i	$-0.885646\pi$
$401$	− 1.81814e10i	− 0.703152i	0.936159	−	0.351576i	$-0.114354\pi$
$402$	0	0
$403$	9.51587e9	0.360768
$404$	− 4.55864e9i	− 0.171124i
$405$	0	0
$406$	−2.31477e10	−0.851930
$407$	− 1.16806e10i	− 0.425682i
$408$	0	0
$409$	−6.47448e9	−0.231373	−0.115686	−	0.993286i	$-0.536907\pi$
$409$	−6.47448e9	−0.231373	−0.115686	+	0.993286i	$0.536907\pi$
$410$	2.17199e10i	0.768637i
$411$	0	0
$412$	−1.11437e9	−0.0386759
$413$	− 4.13128e10i	− 1.41999i
$414$	0	0
$415$	−3.48148e10	−1.17374
$416$	− 1.52592e9i	− 0.0509518i
$417$	0	0
$418$	−1.35997e10	−0.445477
$419$	− 3.78853e10i	− 1.22918i	−0.788848	−	0.614589i	$-0.789322\pi$
$419$	− 3.78853e10i	− 1.22918i	0.788848	−	0.614589i	$-0.210678\pi$
$420$	0	0
$421$	−4.77563e10	−1.52020	−0.760102	−	0.649804i	$-0.774851\pi$
$421$	−4.77563e10	−1.52020	−0.760102	+	0.649804i	$0.774851\pi$
$422$	− 2.12502e10i	− 0.670061i
$423$	0	0
$424$	−1.51782e9	−0.0469630
$425$	6.81081e10i	2.08758i
$426$	0	0
$427$	3.41944e10	1.02859
$428$	− 2.99956e10i	− 0.893888i
$429$	0	0
$430$	1.29200e10	0.377910
$431$	− 2.73395e10i	− 0.792285i	−0.918189	−	0.396143i	$-0.870349\pi$
$431$	− 2.73395e10i	− 0.792285i	0.918189	−	0.396143i	$-0.129651\pi$
$432$	0	0
$433$	−6.96958e9	−0.198269	−0.0991345	−	0.995074i	$-0.531607\pi$
$433$	−6.96958e9	−0.198269	−0.0991345	+	0.995074i	$0.531607\pi$
$434$	− 5.37797e10i	− 1.51586i
$435$	0	0
$436$	−3.41109e10	−0.943947
$437$	1.31391e10i	0.360278i
$438$	0	0
$439$	−8.52427e9	−0.229509	−0.114754	−	0.993394i	$-0.536608\pi$
$439$	−8.52427e9	−0.229509	−0.114754	+	0.993394i	$0.536608\pi$
$440$	1.52044e10i	0.405656i
$441$	0	0
$442$	−8.17817e9	−0.214273
$443$	2.04630e10i	0.531317i	0.964067	+	0.265658i	$0.0855894\pi$
$443$	2.04630e10i	0.531317i	−0.964067	+	0.265658i	$0.914411\pi$
$444$	0	0
$445$	−2.26881e10	−0.578574
$446$	3.28215e10i	0.829505i
$447$	0	0
$448$	−8.62387e9	−0.214087
$449$	1.52992e10i	0.376430i	0.982128	+	0.188215i	$0.0602702\pi$
$449$	1.52992e10i	0.376430i	−0.982128	+	0.188215i	$0.939730\pi$
$450$	0	0
$451$	1.72827e10	0.417740
$452$	3.65898e10i	0.876610i
$453$	0	0
$454$	−4.71563e10	−1.10998
$455$	3.65575e10i	0.852965i
$456$	0	0
$457$	2.40211e10	0.550718	0.275359	−	0.961342i	$-0.411203\pi$
$457$	2.40211e10	0.550718	0.275359	+	0.961342i	$0.411203\pi$
$458$	4.07335e10i	0.925742i
$459$	0	0
$460$	1.46894e10	0.328074
$461$	− 5.73650e10i	− 1.27012i	−0.772464	−	0.635058i	$-0.780976\pi$
$461$	− 5.73650e10i	− 1.27012i	0.772464	−	0.635058i	$-0.219024\pi$
$462$	0	0
$463$	−6.06405e10	−1.31959	−0.659795	−	0.751446i	$-0.729357\pi$
$463$	−6.06405e10	−1.31959	−0.659795	+	0.751446i	$0.729357\pi$
$464$	− 8.15175e9i	− 0.175865i
$465$	0	0
$466$	3.23985e10	0.687038
$467$	1.01091e10i	0.212543i	0.994337	+	0.106271i	$0.0338912\pi$
$467$	1.01091e10i	0.212543i	−0.994337	+	0.106271i	$0.966109\pi$
$468$	0	0
$469$	−5.09238e10	−1.05252
$470$	5.37083e10i	1.10065i
$471$	0	0
$472$	1.45488e10	0.293130
$473$	− 1.02806e10i	− 0.205387i
$474$	0	0
$475$	−9.59011e10	−1.88386
$476$	4.62195e10i	0.900322i
$477$	0	0
$478$	1.58727e10	0.304045
$479$	− 2.17051e10i	− 0.412305i	−0.978520	−	0.206153i	$-0.933906\pi$
$479$	− 2.17051e10i	− 0.412305i	0.978520	−	0.206153i	$-0.0660943\pi$
$480$	0	0
$481$	9.89040e9	0.184771
$482$	5.01893e10i	0.929872i
$483$	0	0
$484$	−1.53396e10	−0.279534
$485$	9.64707e10i	1.74353i
$486$	0	0
$487$	−5.82792e9	−0.103609	−0.0518045	−	0.998657i	$-0.516497\pi$
$487$	−5.82792e9	−0.103609	−0.0518045	+	0.998657i	$0.516497\pi$
$488$	1.20420e10i	0.212334i
$489$	0	0
$490$	1.36173e11	2.36215
$491$	− 2.49089e10i	− 0.428577i	−0.976770	−	0.214288i	$-0.931257\pi$
$491$	− 2.49089e10i	− 0.428577i	0.976770	−	0.214288i	$-0.0687433\pi$
$492$	0	0
$493$	−4.36892e10	−0.739582
$494$	− 1.15154e10i	− 0.193363i
$495$	0	0
$496$	1.89392e10	0.312921
$497$	− 6.97341e10i	− 1.14293i
$498$	0	0
$499$	1.63951e10	0.264431	0.132215	−	0.991221i	$-0.457791\pi$
$499$	1.63951e10	0.264431	0.132215	+	0.991221i	$0.457791\pi$
$500$	5.32200e10i	0.851520i
$501$	0	0
$502$	−2.73886e10	−0.431276
$503$	− 4.61285e10i	− 0.720606i	−0.932835	−	0.360303i	$-0.882673\pi$
$503$	− 4.61285e10i	− 0.720606i	0.932835	−	0.360303i	$-0.117327\pi$
$504$	0	0
$505$	−3.84612e10	−0.591367
$506$	− 1.16885e10i	− 0.178302i
$507$	0	0
$508$	1.04800e10	0.157365
$509$	− 7.73007e9i	− 0.115163i	−0.998341	−	0.0575814i	$-0.981661\pi$
$509$	− 7.73007e9i	− 0.115163i	0.998341	−	0.0575814i	$-0.0183389\pi$
$510$	0	0
$511$	−9.98174e10	−1.46394
$512$	− 3.03700e9i	− 0.0441942i
$513$	0	0
$514$	−2.72925e10	−0.391013
$515$	9.40191e9i	0.133656i
$516$	0	0
$517$	4.27363e10	0.598183
$518$	− 5.58964e10i	− 0.776362i
$519$	0	0
$520$	−1.28742e10	−0.176078
$521$	3.97902e10i	0.540039i	0.962855	+	0.270019i	$0.0870301\pi$
$521$	3.97902e10i	0.540039i	−0.962855	+	0.270019i	$0.912970\pi$
$522$	0	0
$523$	−3.49978e10	−0.467772	−0.233886	−	0.972264i	$-0.575144\pi$
$523$	−3.49978e10	−0.467772	−0.233886	+	0.972264i	$0.575144\pi$
$524$	− 4.84495e10i	− 0.642634i
$525$	0	0
$526$	−7.38152e10	−0.964280
$527$	− 1.01504e11i	− 1.31596i
$528$	0	0
$529$	6.70184e10	0.855799
$530$	1.28058e10i	0.162294i
$531$	0	0
$532$	−6.50804e10	−0.812463
$533$	1.46340e10i	0.181323i
$534$	0	0
$535$	−2.53073e11	−3.08909
$536$	− 1.79335e10i	− 0.217272i
$537$	0	0
$538$	−7.51374e10	−0.896865
$539$	− 1.08354e11i	− 1.28378i
$540$	0	0
$541$	2.34001e10	0.273168	0.136584	−	0.990629i	$-0.456388\pi$
$541$	2.34001e10	0.273168	0.136584	+	0.990629i	$0.456388\pi$
$542$	3.07992e10i	0.356896i
$543$	0	0
$544$	−1.62768e10	−0.185854
$545$	2.87793e11i	3.26208i
$546$	0	0
$547$	2.32182e10	0.259346	0.129673	−	0.991557i	$-0.458607\pi$
$547$	2.32182e10	0.259346	0.129673	+	0.991557i	$0.458607\pi$
$548$	1.23692e10i	0.137158i
$549$	0	0
$550$	8.53134e10	0.932324
$551$	− 6.15175e10i	− 0.667409i
$552$	0	0
$553$	−5.25540e10	−0.561960
$554$	− 1.13351e11i	− 1.20333i
$555$	0	0
$556$	−5.45868e10	−0.571200
$557$	− 3.65617e10i	− 0.379844i	−0.981799	−	0.189922i	$-0.939176\pi$
$557$	− 3.65617e10i	− 0.379844i	0.981799	−	0.189922i	$-0.0608236\pi$
$558$	0	0
$559$	8.70499e9	0.0891499
$560$	7.27594e10i	0.739839i
$561$	0	0
$562$	8.92293e10	0.894462
$563$	− 1.33290e11i	− 1.32667i	−0.748321	−	0.663337i	$-0.769140\pi$
$563$	− 1.33290e11i	− 1.32667i	0.748321	−	0.663337i	$-0.230860\pi$
$564$	0	0
$565$	3.08707e11	3.02938
$566$	1.10163e10i	0.107342i
$567$	0	0
$568$	2.45577e10	0.235936
$569$	1.64710e11i	1.57134i	0.618646	+	0.785670i	$0.287682\pi$
$569$	1.64710e11i	1.57134i	−0.618646	+	0.785670i	$0.712318\pi$
$570$	0	0
$571$	1.36919e11	1.28801	0.644007	−	0.765020i	$-0.277271\pi$
$571$	1.36919e11	1.28801	0.644007	+	0.765020i	$0.277271\pi$
$572$	1.02441e10i	0.0956953i
$573$	0	0
$574$	8.27050e10	0.761876
$575$	− 8.24235e10i	− 0.754015i
$576$	0	0
$577$	−1.01555e11	−0.916218	−0.458109	−	0.888896i	$-0.651473\pi$
$577$	−1.01555e11	−0.916218	−0.458109	+	0.888896i	$0.651473\pi$
$578$	8.31356e9i	0.0744862i
$579$	0	0
$580$	−6.87761e10	−0.607751
$581$	1.32568e11i	1.16341i
$582$	0	0
$583$	1.01897e10	0.0882037
$584$	− 3.51519e10i	− 0.302202i
$585$	0	0
$586$	−6.00184e10	−0.508972
$587$	3.76050e10i	0.316733i	0.987380	+	0.158367i	$0.0506228\pi$
$587$	3.76050e10i	0.316733i	−0.987380	+	0.158367i	$0.949377\pi$
$588$	0	0
$589$	1.42925e11	1.18754
$590$	− 1.22748e11i	− 1.01299i
$591$	0	0
$592$	1.96846e10	0.160265
$593$	1.30215e11i	1.05304i	0.850163	+	0.526519i	$0.176503\pi$
$593$	1.30215e11i	1.05304i	−0.850163	+	0.526519i	$0.823497\pi$
$594$	0	0
$595$	3.89953e11	3.11132
$596$	− 5.51944e10i	− 0.437431i
$597$	0	0
$598$	9.89711e9	0.0773934
$599$	1.31461e11i	1.02115i	0.859834	+	0.510575i	$0.170567\pi$
$599$	1.31461e11i	1.02115i	−0.859834	+	0.510575i	$0.829433\pi$
$600$	0	0
$601$	−4.30184e10	−0.329728	−0.164864	−	0.986316i	$-0.552719\pi$
$601$	−4.30184e10	−0.329728	−0.164864	+	0.986316i	$0.552719\pi$
$602$	− 4.91969e10i	− 0.374586i
$603$	0	0
$604$	−1.07592e11	−0.808409
$605$	1.29420e11i	0.966008i
$606$	0	0
$607$	2.59213e10	0.190942	0.0954710	−	0.995432i	$-0.469564\pi$
$607$	2.59213e10	0.190942	0.0954710	+	0.995432i	$0.469564\pi$
$608$	− 2.29189e10i	− 0.167718i
$609$	0	0
$610$	1.01598e11	0.733780
$611$	3.61866e10i	0.259646i
$612$	0	0
$613$	1.28485e11	0.909933	0.454967	−	0.890508i	$-0.349651\pi$
$613$	1.28485e11	0.909933	0.454967	+	0.890508i	$0.349651\pi$
$614$	− 1.51877e11i	− 1.06861i
$615$	0	0
$616$	5.78954e10	0.402088
$617$	− 1.18113e11i	− 0.814997i	−0.913206	−	0.407498i	$-0.866401\pi$
$617$	− 1.18113e11i	− 0.814997i	0.913206	−	0.407498i	$-0.133599\pi$
$618$	0	0
$619$	1.28044e11	0.872159	0.436079	−	0.899908i	$-0.356367\pi$
$619$	1.28044e11	0.872159	0.436079	+	0.899908i	$0.356367\pi$
$620$	− 1.59789e11i	− 1.08139i
$621$	0	0
$622$	2.59046e10	0.173068
$623$	8.63921e10i	0.573484i
$624$	0	0
$625$	1.46035e11	0.957057
$626$	2.02326e11i	1.31751i
$627$	0	0
$628$	5.39953e10	0.347150
$629$	− 1.05499e11i	− 0.673980i
$630$	0	0
$631$	−2.40613e11	−1.51775	−0.758877	−	0.651233i	$-0.774252\pi$
$631$	−2.40613e11	−1.51775	−0.758877	+	0.651233i	$0.774252\pi$
$632$	− 1.85075e10i	− 0.116006i
$633$	0	0
$634$	−7.54317e10	−0.466871
$635$	− 8.84198e10i	− 0.543819i
$636$	0	0
$637$	9.17480e10	0.557236
$638$	5.47259e10i	0.330301i
$639$	0	0
$640$	−2.56231e10	−0.152726
$641$	− 2.16058e11i	− 1.27979i	−0.768464	−	0.639893i	$-0.778979\pi$
$641$	− 2.16058e11i	− 1.27979i	0.768464	−	0.639893i	$-0.221021\pi$
$642$	0	0
$643$	3.71338e10	0.217233	0.108616	−	0.994084i	$-0.465358\pi$
$643$	3.71338e10	0.217233	0.108616	+	0.994084i	$0.465358\pi$
$644$	− 5.59343e10i	− 0.325188i
$645$	0	0
$646$	−1.22833e11	−0.705321
$647$	5.88598e10i	0.335894i	0.985796	+	0.167947i	$0.0537137\pi$
$647$	5.88598e10i	0.335894i	−0.985796	+	0.167947i	$0.946286\pi$
$648$	0	0
$649$	−9.76719e10	−0.550543
$650$	7.22384e10i	0.404683i
$651$	0	0
$652$	7.05047e10	0.390146
$653$	− 1.71979e11i	− 0.945849i	−0.881103	−	0.472925i	$-0.843198\pi$
$653$	− 1.71979e11i	− 0.945849i	0.881103	−	0.472925i	$-0.156802\pi$
$654$	0	0
$655$	−4.08767e11	−2.22081
$656$	2.91256e10i	0.157275i
$657$	0	0
$658$	2.04511e11	1.09097
$659$	− 2.71182e11i	− 1.43787i	−0.695079	−	0.718934i	$-0.744631\pi$
$659$	− 2.71182e11i	− 1.43787i	0.695079	−	0.718934i	$-0.255369\pi$
$660$	0	0
$661$	−2.43194e11	−1.27394	−0.636968	−	0.770890i	$-0.719812\pi$
$661$	−2.43194e11	−1.27394	−0.636968	+	0.770890i	$0.719812\pi$
$662$	− 1.77465e11i	− 0.924018i
$663$	0	0
$664$	−4.66855e10	−0.240165
$665$	5.49082e11i	2.80770i
$666$	0	0
$667$	5.28721e10	0.267130
$668$	− 7.76772e10i	− 0.390111i
$669$	0	0
$670$	−1.51304e11	−0.750847
$671$	− 8.08426e10i	− 0.398796i
$672$	0	0
$673$	−2.89219e10	−0.140983	−0.0704914	−	0.997512i	$-0.522457\pi$
$673$	−2.89219e10	−0.140983	−0.0704914	+	0.997512i	$0.522457\pi$
$674$	− 2.92320e10i	− 0.141651i
$675$	0	0
$676$	9.57394e10	0.458463
$677$	− 1.82531e11i	− 0.868923i	−0.900690	−	0.434462i	$-0.856939\pi$
$677$	− 1.82531e11i	− 0.868923i	0.900690	−	0.434462i	$-0.143061\pi$
$678$	0	0
$679$	3.67342e11	1.72819
$680$	1.37327e11i	0.642273i
$681$	0	0
$682$	−1.27146e11	−0.587713
$683$	− 3.71514e11i	− 1.70723i	−0.520903	−	0.853616i	$-0.674405\pi$
$683$	− 3.71514e11i	− 1.70723i	0.520903	−	0.853616i	$-0.325595\pi$
$684$	0	0
$685$	1.04359e11	0.473989
$686$	− 2.50319e11i	− 1.13031i
$687$	0	0
$688$	1.73253e10	0.0773262
$689$	8.62803e9i	0.0382855i
$690$	0	0
$691$	1.97648e11	0.866921	0.433460	−	0.901173i	$-0.357292\pi$
$691$	1.97648e11	0.866921	0.433460	+	0.901173i	$0.357292\pi$
$692$	− 1.22175e11i	− 0.532794i
$693$	0	0
$694$	1.90246e11	0.820119
$695$	4.60548e11i	1.97395i
$696$	0	0
$697$	1.56098e11	0.661405
$698$	2.76883e11i	1.16647i
$699$	0	0
$700$	4.08261e11	1.70038
$701$	− 6.70096e10i	− 0.277501i	−0.990327	−	0.138751i	$-0.955691\pi$
$701$	− 6.70096e10i	− 0.277501i	0.990327	−	0.138751i	$-0.0443087\pi$
$702$	0	0
$703$	1.48551e11	0.608209
$704$	2.03886e10i	0.0830035i
$705$	0	0
$706$	1.18240e11	0.475933
$707$	1.46453e11i	0.586165i
$708$	0	0
$709$	−1.14058e11	−0.451379	−0.225690	−	0.974199i	$-0.572464\pi$
$709$	−1.14058e11	−0.451379	−0.225690	+	0.974199i	$0.572464\pi$
$710$	− 2.07193e11i	− 0.815345i
$711$	0	0
$712$	−3.04240e10	−0.118385
$713$	1.22839e11i	0.475312i
$714$	0	0
$715$	8.64294e10	0.330703
$716$	− 1.20834e11i	− 0.459768i
$717$	0	0
$718$	2.68697e11	1.01103
$719$	2.31072e11i	0.864633i	0.901722	+	0.432317i	$0.142304\pi$
$719$	2.31072e11i	0.864633i	−0.901722	+	0.432317i	$0.857696\pi$
$720$	0	0
$721$	3.58007e10	0.132480
$722$	1.91889e10i	0.0706156i
$723$	0	0
$724$	1.11133e11	0.404474
$725$	3.85910e11i	1.39680i
$726$	0	0
$727$	−3.41273e11	−1.22170	−0.610850	−	0.791746i	$-0.709172\pi$
$727$	−3.41273e11	−1.22170	−0.610850	+	0.791746i	$0.709172\pi$
$728$	4.90224e10i	0.174530i
$729$	0	0
$730$	−2.96576e11	−1.04435
$731$	− 9.28547e10i	− 0.325188i
$732$	0	0
$733$	−4.14407e11	−1.43553	−0.717763	−	0.696288i	$-0.754834\pi$
$733$	−4.14407e11	−1.43553	−0.717763	+	0.696288i	$0.754834\pi$
$734$	− 1.17082e11i	− 0.403371i
$735$	0	0
$736$	1.96979e10	0.0671289
$737$	1.20394e11i	0.408071i
$738$	0	0
$739$	2.71838e11	0.911449	0.455725	−	0.890121i	$-0.349380\pi$
$739$	2.71838e11	0.911449	0.455725	+	0.890121i	$0.349380\pi$
$740$	− 1.66078e11i	− 0.553843i
$741$	0	0
$742$	4.87620e10	0.160866
$743$	− 3.42572e11i	− 1.12408i	−0.827111	−	0.562039i	$-0.810017\pi$
$743$	− 3.42572e11i	− 1.12408i	0.827111	−	0.562039i	$-0.189983\pi$
$744$	0	0
$745$	−4.65674e11	−1.51167
$746$	9.11142e10i	0.294192i
$747$	0	0
$748$	1.09272e11	0.349063
$749$	9.63652e11i	3.06191i
$750$	0	0
$751$	1.05168e11	0.330615	0.165308	−	0.986242i	$-0.447138\pi$
$751$	1.05168e11	0.330615	0.165308	+	0.986242i	$0.447138\pi$
$752$	7.20211e10i	0.225210i
$753$	0	0
$754$	−4.63386e10	−0.143370
$755$	9.07750e11i	2.79369i
$756$	0	0
$757$	−4.23531e11	−1.28974	−0.644869	−	0.764293i	$-0.723088\pi$
$757$	−4.23531e11	−1.28974	−0.644869	+	0.764293i	$0.723088\pi$
$758$	2.97797e11i	0.902076i
$759$	0	0
$760$	−1.93366e11	−0.579596
$761$	5.33954e11i	1.59208i	0.605243	+	0.796041i	$0.293076\pi$
$761$	5.33954e11i	1.59208i	−0.605243	+	0.796041i	$0.706924\pi$
$762$	0	0
$763$	1.09586e12	3.23339
$764$	− 9.01526e9i	− 0.0264609i
$765$	0	0
$766$	−3.46118e11	−1.00533
$767$	− 8.27028e10i	− 0.238968i
$768$	0	0
$769$	1.13358e11	0.324150	0.162075	−	0.986778i	$-0.448181\pi$
$769$	1.13358e11	0.324150	0.162075	+	0.986778i	$0.448181\pi$
$770$	− 4.88462e11i	− 1.38953i
$771$	0	0
$772$	−3.50390e11	−0.986467
$773$	− 2.83942e11i	− 0.795265i	−0.917545	−	0.397632i	$-0.869832\pi$
$773$	− 2.83942e11i	− 0.795265i	0.917545	−	0.397632i	$-0.130168\pi$
$774$	0	0
$775$	−8.96595e11	−2.48536
$776$	1.29364e11i	0.356752i
$777$	0	0
$778$	−4.92237e11	−1.34356
$779$	2.19798e11i	0.596861i
$780$	0	0
$781$	−1.64866e11	−0.443125
$782$	− 1.05571e11i	− 0.282304i
$783$	0	0
$784$	1.82603e11	0.483331
$785$	− 4.55558e11i	− 1.19968i
$786$	0	0
$787$	−6.51136e11	−1.69736	−0.848678	−	0.528910i	$-0.822601\pi$
$787$	−6.51136e11	−1.69736	−0.848678	+	0.528910i	$0.822601\pi$
$788$	− 7.00513e10i	− 0.181682i
$789$	0	0
$790$	−1.56148e11	−0.400892
$791$	− 1.17550e12i	− 3.00273i
$792$	0	0
$793$	6.84528e10	0.173100
$794$	− 4.88095e11i	− 1.22807i
$795$	0	0
$796$	3.41078e11	0.849574
$797$	4.80896e11i	1.19184i	0.803043	+	0.595921i	$0.203213\pi$
$797$	4.80896e11i	1.19184i	−0.803043	+	0.595921i	$0.796787\pi$
$798$	0	0
$799$	3.85996e11	0.947100
$800$	1.43774e11i	0.351011i
$801$	0	0
$802$	−2.05699e11	−0.497204
$803$	2.35989e11i	0.567582i
$804$	0	0
$805$	−4.71916e11	−1.12378
$806$	− 1.07660e11i	− 0.255102i
$807$	0	0
$808$	−5.15751e10	−0.121003
$809$	1.97505e11i	0.461089i	0.973062	+	0.230545i	$0.0740508\pi$
$809$	1.97505e11i	0.461089i	−0.973062	+	0.230545i	$0.925949\pi$
$810$	0	0
$811$	−1.30697e11	−0.302121	−0.151061	−	0.988524i	$-0.548269\pi$
$811$	−1.30697e11	−0.302121	−0.151061	+	0.988524i	$0.548269\pi$
$812$	2.61886e11i	0.602405i
$813$	0	0
$814$	−1.32150e11	−0.301003
$815$	− 5.94847e11i	− 1.34826i
$816$	0	0
$817$	1.30746e11	0.293454
$818$	7.32504e10i	0.163605i
$819$	0	0
$820$	2.45732e11	0.543509
$821$	− 2.60344e10i	− 0.0573027i	−0.999589	−	0.0286514i	$-0.990879\pi$
$821$	− 2.60344e10i	− 0.0573027i	0.999589	−	0.0286514i	$-0.00912126\pi$
$822$	0	0
$823$	−2.04620e11	−0.446015	−0.223007	−	0.974817i	$-0.571587\pi$
$823$	−2.04620e11	−0.446015	−0.223007	+	0.974817i	$0.571587\pi$
$824$	1.26076e10i	0.0273480i
$825$	0	0
$826$	−4.67401e11	−1.00408
$827$	− 2.31574e11i	− 0.495072i	−0.968879	−	0.247536i	$-0.920379\pi$
$827$	− 2.31574e11i	− 0.495072i	0.968879	−	0.247536i	$-0.0796208\pi$
$828$	0	0
$829$	7.98205e11	1.69004	0.845019	−	0.534736i	$-0.179589\pi$
$829$	7.98205e11	1.69004	0.845019	+	0.534736i	$0.179589\pi$
$830$	3.93884e11i	0.829958i
$831$	0	0
$832$	−1.72638e10	−0.0360283
$833$	− 9.78661e11i	− 2.03260i
$834$	0	0
$835$	−6.55361e11	−1.34814
$836$	1.53863e11i	0.315000i
$837$	0	0
$838$	−4.28623e11	−0.869160
$839$	− 6.42925e11i	− 1.29752i	−0.760995	−	0.648758i	$-0.775289\pi$
$839$	− 6.42925e11i	− 1.29752i	0.760995	−	0.648758i	$-0.224711\pi$
$840$	0	0
$841$	2.52697e11	0.505146
$842$	5.40300e11i	1.07495i
$843$	0	0
$844$	−2.40419e11	−0.473804
$845$	− 8.07752e11i	− 1.58435i
$846$	0	0
$847$	4.92808e11	0.957511
$848$	1.71721e10i	0.0332078i
$849$	0	0
$850$	7.70555e11	1.47614
$851$	1.27674e11i	0.243436i
$852$	0	0
$853$	−1.95077e11	−0.368476	−0.184238	−	0.982882i	$-0.558982\pi$
$853$	−1.95077e11	−0.368476	−0.184238	+	0.982882i	$0.558982\pi$
$854$	− 3.86866e11i	− 0.727326i
$855$	0	0
$856$	−3.39362e11	−0.632074
$857$	4.44448e10i	0.0823945i	0.999151	+	0.0411972i	$0.0131172\pi$
$857$	4.44448e10i	0.0823945i	−0.999151	+	0.0411972i	$0.986883\pi$
$858$	0	0
$859$	2.50588e11	0.460243	0.230122	−	0.973162i	$-0.426088\pi$
$859$	2.50588e11	0.460243	0.230122	+	0.973162i	$0.426088\pi$
$860$	− 1.46173e11i	− 0.267223i
$861$	0	0
$862$	−3.09311e11	−0.560230
$863$	3.91558e11i	0.705915i	0.935639	+	0.352958i	$0.114824\pi$
$863$	3.91558e11i	0.705915i	−0.935639	+	0.352958i	$0.885176\pi$
$864$	0	0
$865$	−1.03079e12	−1.84122
$866$	7.88518e10i	0.140197i
$867$	0	0
$868$	−6.08447e11	−1.07187
$869$	1.24248e11i	0.217877i
$870$	0	0
$871$	−1.01943e11	−0.177127
$872$	3.85921e11i	0.667471i
$873$	0	0
$874$	1.48651e11	0.254755
$875$	− 1.70977e12i	− 2.91679i
$876$	0	0
$877$	−7.98110e10	−0.134916	−0.0674582	−	0.997722i	$-0.521489\pi$
$877$	−7.98110e10	−0.134916	−0.0674582	+	0.997722i	$0.521489\pi$
$878$	9.64411e10i	0.162287i
$879$	0	0
$880$	1.72018e11	0.286842
$881$	5.84453e11i	0.970166i	0.874468	+	0.485083i	$0.161211\pi$
$881$	5.84453e11i	0.970166i	−0.874468	+	0.485083i	$0.838789\pi$
$882$	0	0
$883$	7.52384e10	0.123765	0.0618823	−	0.998083i	$-0.480290\pi$
$883$	7.52384e10	0.123765	0.0618823	+	0.998083i	$0.480290\pi$
$884$	9.25254e10i	0.151514i
$885$	0	0
$886$	2.31512e11	0.375698
$887$	6.99628e11i	1.13025i	0.825007	+	0.565123i	$0.191171\pi$
$887$	6.99628e11i	1.13025i	−0.825007	+	0.565123i	$0.808829\pi$
$888$	0	0
$889$	−3.36686e11	−0.539036
$890$	2.56687e11i	0.409113i
$891$	0	0
$892$	3.71333e11	0.586548
$893$	5.43510e11i	0.854677i
$894$	0	0
$895$	−1.01948e12	−1.58886
$896$	9.75679e10i	0.151382i
$897$	0	0
$898$	1.73091e11	0.266176
$899$	− 5.75137e11i	− 0.880507i
$900$	0	0
$901$	9.20338e10	0.139652
$902$	− 1.95532e11i	− 0.295387i
$903$	0	0
$904$	4.13966e11	0.619857
$905$	− 9.37631e11i	− 1.39778i
$906$	0	0
$907$	1.15989e11	0.171392	0.0856958	−	0.996321i	$-0.472689\pi$
$907$	1.15989e11	0.171392	0.0856958	+	0.996321i	$0.472689\pi$
$908$	5.33513e11i	0.784877i
$909$	0	0
$910$	4.13601e11	0.603137
$911$	− 1.13296e12i	− 1.64491i	−0.568828	−	0.822456i	$-0.692603\pi$
$911$	− 1.13296e12i	− 1.64491i	0.568828	−	0.822456i	$-0.307397\pi$
$912$	0	0
$913$	3.13418e11	0.451066
$914$	− 2.71768e11i	− 0.389416i
$915$	0	0
$916$	4.60847e11	0.654599
$917$	1.55651e12i	2.20127i
$918$	0	0
$919$	−3.06256e11	−0.429361	−0.214680	−	0.976684i	$-0.568871\pi$
$919$	−3.06256e11	−0.429361	−0.214680	+	0.976684i	$0.568871\pi$
$920$	− 1.66191e11i	− 0.231983i
$921$	0	0
$922$	−6.49011e11	−0.898108
$923$	− 1.39598e11i	− 0.192342i
$924$	0	0
$925$	−9.31884e11	−1.27290
$926$	6.86069e11i	0.933090i
$927$	0	0
$928$	−9.22265e10	−0.124355
$929$	1.01050e12i	1.35666i	0.734757	+	0.678331i	$0.237296\pi$
$929$	1.01050e12i	1.35666i	−0.734757	+	0.678331i	$0.762704\pi$
$930$	0	0
$931$	1.37802e12	1.83425
$932$	− 3.66547e11i	− 0.485809i
$933$	0	0
$934$	1.14372e11	0.150290
$935$	− 9.21929e11i	− 1.20629i
$936$	0	0
$937$	−3.80033e11	−0.493018	−0.246509	−	0.969141i	$-0.579283\pi$
$937$	−3.80033e11	−0.493018	−0.246509	+	0.969141i	$0.579283\pi$
$938$	5.76137e11i	0.744243i
$939$	0	0
$940$	6.07640e11	0.778279
$941$	4.85808e11i	0.619593i	0.950803	+	0.309796i	$0.100261\pi$
$941$	4.85808e11i	0.619593i	−0.950803	+	0.309796i	$0.899739\pi$
$942$	0	0
$943$	−1.88908e11	−0.238893
$944$	− 1.64601e11i	− 0.207274i
$945$	0	0
$946$	−1.16311e11	−0.145231
$947$	− 5.36674e9i	− 0.00667284i	−0.999994	−	0.00333642i	$-0.998938\pi$
$947$	− 5.36674e9i	− 0.00667284i	0.999994	−	0.00333642i	$-0.00106202\pi$
$948$	0	0
$949$	−1.99821e11	−0.246364
$950$	1.08500e12i	1.33209i
$951$	0	0
$952$	5.22914e11	0.636624
$953$	− 2.87948e11i	− 0.349094i	−0.984649	−	0.174547i	$-0.944154\pi$
$953$	− 2.87948e11i	− 0.349094i	0.984649	−	0.174547i	$-0.0558462\pi$
$954$	0	0
$955$	−7.60616e10	−0.0914433
$956$	− 1.79579e11i	− 0.214992i
$957$	0	0
$958$	−2.45565e11	−0.291544
$959$	− 3.97379e11i	− 0.469819i
$960$	0	0
$961$	4.83341e11	0.566709
$962$	− 1.11897e11i	− 0.130653i
$963$	0	0
$964$	5.67827e11	0.657519
$965$	2.95624e12i	3.40902i
$966$	0	0
$967$	9.02833e11	1.03253	0.516264	−	0.856430i	$-0.327322\pi$
$967$	9.02833e11	1.03253	0.516264	+	0.856430i	$0.327322\pi$
$968$	1.73548e11i	0.197660i
$969$	0	0
$970$	1.09144e12	1.23286
$971$	− 8.08040e11i	− 0.908983i	−0.890751	−	0.454492i	$-0.849821\pi$
$971$	− 8.08040e11i	− 0.908983i	0.890751	−	0.454492i	$-0.150179\pi$
$972$	0	0
$973$	1.75368e12	1.95658
$974$	6.59354e10i	0.0732627i
$975$	0	0
$976$	1.36240e11	0.150143
$977$	1.61263e12i	1.76994i	0.465652	+	0.884968i	$0.345820\pi$
$977$	1.61263e12i	1.76994i	−0.465652	+	0.884968i	$0.654180\pi$
$978$	0	0
$979$	2.04248e11	0.222345
$980$	− 1.54062e12i	− 1.67029i
$981$	0	0
$982$	−2.81812e11	−0.303050
$983$	− 1.43725e12i	− 1.53928i	−0.638477	−	0.769641i	$-0.720435\pi$
$983$	− 1.43725e12i	− 1.53928i	0.638477	−	0.769641i	$-0.279565\pi$
$984$	0	0
$985$	−5.91022e11	−0.627854
$986$	4.94287e11i	0.522964i
$987$	0	0
$988$	−1.30282e11	−0.136728
$989$	1.12372e11i	0.117455i
$990$	0	0
$991$	−1.11953e12	−1.16075	−0.580377	−	0.814348i	$-0.697095\pi$
$991$	−1.11953e12	−1.16075	−0.580377	+	0.814348i	$0.697095\pi$
$992$	− 2.14272e11i	− 0.221268i
$993$	0	0
$994$	−7.88951e11	−0.808173
$995$	− 2.87767e12i	− 2.93595i
$996$	0	0
$997$	8.91336e11	0.902113	0.451057	−	0.892495i	$-0.351047\pi$
$997$	8.91336e11	0.902113	0.451057	+	0.892495i	$0.351047\pi$
$998$	− 1.85489e11i	− 0.186981i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.9.b.c.161.8		16
3.2	odd	2	inner	162.9.b.c.161.9		16
9.2	odd	6		18.9.d.a.5.6	✓	16
9.4	even	3		18.9.d.a.11.6	yes	16
9.5	odd	6		54.9.d.a.35.1		16
9.7	even	3		54.9.d.a.17.1		16
36.7	odd	6		432.9.q.c.17.1		16
36.11	even	6		144.9.q.b.113.5		16
36.23	even	6		432.9.q.c.305.1		16
36.31	odd	6		144.9.q.b.65.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.9.d.a.5.6	✓	16	9.2	odd	6
18.9.d.a.11.6	yes	16	9.4	even	3
54.9.d.a.17.1		16	9.7	even	3
54.9.d.a.35.1		16	9.5	odd	6
144.9.q.b.65.5		16	36.31	odd	6
144.9.q.b.113.5		16	36.11	even	6
162.9.b.c.161.8		16	1.1	even	1	trivial
162.9.b.c.161.9		16	3.2	odd	2	inner
432.9.q.c.17.1		16	36.7	odd	6
432.9.q.c.305.1		16	36.23	even	6

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	162.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-11.3137i q^{2} -128.000 q^{4} +1079.93i q^{5} +4112.18 q^{7} +1448.15i q^{8} +O(q^{10})\)	\(q-11.3137i q^{2} -128.000 q^{4} +1079.93i q^{5} +4112.18 q^{7} +1448.15i q^{8} +12218.1 q^{10} -9722.03i q^{11} +8232.04 q^{13} -46524.0i q^{14} +16384.0 q^{16} -87809.9i q^{17} +123643. q^{19} -138231. i q^{20} -109992. q^{22} +106266. i q^{23} -775631. q^{25} -93134.9i q^{26} -526359. q^{28} -497543. i q^{29} +1.15595e6 q^{31} -185364. i q^{32} -993456. q^{34} +4.44088e6i q^{35} +1.20145e6 q^{37} -1.39886e6i q^{38} -1.56391e6 q^{40} +1.77769e6i q^{41} +1.05745e6 q^{43} +1.24442e6i q^{44} +1.20227e6 q^{46} +4.39582e6i q^{47} +1.11452e7 q^{49} +8.77527e6i q^{50} -1.05370e6 q^{52} +1.04810e6i q^{53} +1.04991e7 q^{55} +5.95507e6i q^{56} -5.62906e6 q^{58} -1.00465e7i q^{59} +8.31541e6 q^{61} -1.30781e7i q^{62} -2.09715e6 q^{64} +8.89006e6i q^{65} -1.23837e7 q^{67} +1.12397e7i q^{68} +5.02428e7 q^{70} -1.69579e7i q^{71} -2.42736e7 q^{73} -1.35929e7i q^{74} -1.58263e7 q^{76} -3.99787e7i q^{77} -1.27801e7 q^{79} +1.76936e7i q^{80} +2.01122e7 q^{82} +3.22379e7i q^{83} +9.48289e7 q^{85} -1.19637e7i q^{86} +1.40790e7 q^{88} +2.10088e7i q^{89} +3.38516e7 q^{91} -1.36021e7i q^{92} +4.97330e7 q^{94} +1.33526e8i q^{95} +8.93302e7 q^{97} -1.26094e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2048 q^{4} + 3692 q^{7}+O(q^{10}) \) 16 * q - 2048 * q^4 + 3692 * q^7	\( 16 q - 2048 q^{4} + 3692 q^{7} + 6740 q^{13} + 262144 q^{16} + 362180 q^{19} + 123648 q^{22} - 1926788 q^{25} - 472576 q^{28} - 1084876 q^{31} - 440832 q^{34} + 3343328 q^{37} - 679024 q^{43} + 7417344 q^{46} + 4729308 q^{49} - 862720 q^{52} - 4584276 q^{55} + 15705600 q^{58} + 1683908 q^{61} - 33554432 q^{64} - 59893288 q^{67} + 68719104 q^{70} - 7547764 q^{73} - 46359040 q^{76} - 67626004 q^{79} - 137346048 q^{82} + 251393544 q^{85} - 15826944 q^{88} + 268578316 q^{91} + 23665152 q^{94} + 178830968 q^{97}+O(q^{100}) \) 16 * q - 2048 * q^4 + 3692 * q^7 + 6740 * q^13 + 262144 * q^16 + 362180 * q^19 + 123648 * q^22 - 1926788 * q^25 - 472576 * q^28 - 1084876 * q^31 - 440832 * q^34 + 3343328 * q^37 - 679024 * q^43 + 7417344 * q^46 + 4729308 * q^49 - 862720 * q^52 - 4584276 * q^55 + 15705600 * q^58 + 1683908 * q^61 - 33554432 * q^64 - 59893288 * q^67 + 68719104 * q^70 - 7547764 * q^73 - 46359040 * q^76 - 67626004 * q^79 - 137346048 * q^82 + 251393544 * q^85 - 15826944 * q^88 + 268578316 * q^91 + 23665152 * q^94 + 178830968 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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