LMFDB - Embedded newform 162.9.b.a.161.3

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3364x^{6} + 4188433x^{4} + 2287495488x^{2} + 462682923264 $ x^8 + 3364x^6 + 4188433x^4 + 2287495488*x^2 + 462682923264
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$30.6613i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.9.b.a.161.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.3137i q^{2} -128.000 q^{4} +654.335i q^{5} +1299.40 q^{7} +1448.15i q^{8} +O(q^{10})$	\(q-11.3137i q^{2} -128.000 q^{4} +654.335i q^{5} +1299.40 q^{7} +1448.15i q^{8} +7402.96 q^{10} +4408.29i q^{11} -246.109 q^{13} -14701.1i q^{14} +16384.0 q^{16} +89030.5i q^{17} +145819. q^{19} -83754.9i q^{20} +49874.1 q^{22} -355568. i q^{23} -37529.5 q^{25} +2784.41i q^{26} -166324. q^{28} -63550.1i q^{29} -972728. q^{31} -185364. i q^{32} +1.00727e6 q^{34} +850245. i q^{35} -408781. q^{37} -1.64975e6i q^{38} -947579. q^{40} +3.59693e6i q^{41} -4.83838e6 q^{43} -564261. i q^{44} -4.02279e6 q^{46} -1.25207e6i q^{47} -4.07635e6 q^{49} +424598. i q^{50} +31502.0 q^{52} +1.23446e7i q^{53} -2.88450e6 q^{55} +1.88174e6i q^{56} -718987. q^{58} +3.30263e6i q^{59} +1.51546e7 q^{61} +1.10052e7i q^{62} -2.09715e6 q^{64} -161038. i q^{65} -1.76819e7 q^{67} -1.13959e7i q^{68} +9.61943e6 q^{70} +2.66372e7i q^{71} -1.28456e7 q^{73} +4.62483e6i q^{74} -1.86648e7 q^{76} +5.72815e6i q^{77} +5.49629e7 q^{79} +1.07206e7i q^{80} +4.06946e7 q^{82} +9.14079e6i q^{83} -5.82558e7 q^{85} +5.47401e7i q^{86} -6.38389e6 q^{88} -5.99482e7i q^{89} -319795. q^{91} +4.55127e7i q^{92} -1.41655e7 q^{94} +9.54142e7i q^{95} +1.07742e8 q^{97} +4.61187e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 1024 q^{4} - 8876 q^{7}+O(q^{10}) $ 8 * q - 1024 * q^4 - 8876 * q^7	$ 8 q - 1024 q^{4} - 8876 q^{7} + 8448 q^{10} - 117380 q^{13} + 131072 q^{16} + 270220 q^{19} - 210816 q^{22} - 1801672 q^{25} + 1136128 q^{28} - 393344 q^{31} - 691968 q^{34} + 1830988 q^{37} - 1081344 q^{40} + 11135236 q^{43} + 5296320 q^{46} - 13586328 q^{49} + 15024640 q^{52} - 1579716 q^{55} - 32988672 q^{58} + 12184204 q^{61} - 16777216 q^{64} - 80355716 q^{67} - 18723264 q^{70} + 197085760 q^{73} - 34588160 q^{76} + 84451852 q^{79} + 144639168 q^{82} - 582634548 q^{85} + 26984448 q^{88} + 373079588 q^{91} - 210121536 q^{94} + 341136928 q^{97}+O(q^{100}) $ 8 * q - 1024 * q^4 - 8876 * q^7 + 8448 * q^10 - 117380 * q^13 + 131072 * q^16 + 270220 * q^19 - 210816 * q^22 - 1801672 * q^25 + 1136128 * q^28 - 393344 * q^31 - 691968 * q^34 + 1830988 * q^37 - 1081344 * q^40 + 11135236 * q^43 + 5296320 * q^46 - 13586328 * q^49 + 15024640 * q^52 - 1579716 * q^55 - 32988672 * q^58 + 12184204 * q^61 - 16777216 * q^64 - 80355716 * q^67 - 18723264 * q^70 + 197085760 * q^73 - 34588160 * q^76 + 84451852 * q^79 + 144639168 * q^82 - 582634548 * q^85 + 26984448 * q^88 + 373079588 * q^91 - 210121536 * q^94 + 341136928 * 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
$5$	654.335i	1.04694i	0.852045	+	0.523468i	$0.175362\pi$
$5$	654.335i	1.04694i	−0.852045	+	0.523468i	$0.824638\pi$
$6$	0	0
$7$	1299.40	0.541192	0.270596	−	0.962693i	$-0.412779\pi$
$7$	1299.40	0.541192	0.270596	+	0.962693i	$0.412779\pi$
$8$	1448.15i	0.353553i
$9$	0	0
$10$	7402.96	0.740296
$11$	4408.29i	0.301092i	0.988603	+	0.150546i	$0.0481032\pi$
$11$	4408.29i	0.301092i	−0.988603	+	0.150546i	$0.951897\pi$
$12$	0	0
$13$	−246.109	−0.00861697	−0.00430849	−	0.999991i	$-0.501371\pi$
$13$	−246.109	−0.00861697	−0.00430849	+	0.999991i	$0.501371\pi$
$14$	− 14701.1i	− 0.382681i
$15$	0	0
$16$	16384.0	0.250000
$17$	89030.5i	1.06597i	0.846126	+	0.532983i	$0.178929\pi$
$17$	89030.5i	1.06597i	−0.846126	+	0.532983i	$0.821071\pi$
$18$	0	0
$19$	145819.	1.11892	0.559459	−	0.828858i	$-0.311009\pi$
$19$	145819.	1.11892	0.559459	+	0.828858i	$0.311009\pi$
$20$	− 83754.9i	− 0.523468i
$21$	0	0
$22$	49874.1	0.212904
$23$	− 355568.i	− 1.27061i	−0.772262	−	0.635304i	$-0.780875\pi$
$23$	− 355568.i	− 1.27061i	0.772262	−	0.635304i	$-0.219125\pi$
$24$	0	0
$25$	−37529.5	−0.0960756
$26$	2784.41i	0.00609312i
$27$	0	0
$28$	−166324.	−0.270596
$29$	− 63550.1i	− 0.0898513i	−0.998990	−	0.0449256i	$-0.985695\pi$
$29$	− 63550.1i	− 0.0898513i	0.998990	−	0.0449256i	$-0.0143051\pi$
$30$	0	0
$31$	−972728.	−1.05328	−0.526641	−	0.850088i	$-0.676549\pi$
$31$	−972728.	−1.05328	−0.526641	+	0.850088i	$0.676549\pi$
$32$	− 185364.i	− 0.176777i
$33$	0	0
$34$	1.00727e6	0.753752
$35$	850245.i	0.566594i
$36$	0	0
$37$	−408781.	−0.218114	−0.109057	−	0.994035i	$-0.534783\pi$
$37$	−408781.	−0.218114	−0.109057	+	0.994035i	$0.534783\pi$
$38$	− 1.64975e6i	− 0.791195i
$39$	0	0
$40$	−947579.	−0.370148
$41$	3.59693e6i	1.27291i	0.771316	+	0.636453i	$0.219599\pi$
$41$	3.59693e6i	1.27291i	−0.771316	+	0.636453i	$0.780401\pi$
$42$	0	0
$43$	−4.83838e6	−1.41523	−0.707614	−	0.706599i	$-0.750228\pi$
$43$	−4.83838e6	−1.41523	−0.707614	+	0.706599i	$0.750228\pi$
$44$	− 564261.i	− 0.150546i
$45$	0	0
$46$	−4.02279e6	−0.898455
$47$	− 1.25207e6i	− 0.256588i	−0.991736	−	0.128294i	$-0.959050\pi$
$47$	− 1.25207e6i	− 0.256588i	0.991736	−	0.128294i	$-0.0409501\pi$
$48$	0	0
$49$	−4.07635e6	−0.707111
$50$	424598.i	0.0679357i
$51$	0	0
$52$	31502.0	0.00430849
$53$	1.23446e7i	1.56450i	0.622965	+	0.782249i	$0.285928\pi$
$53$	1.23446e7i	1.56450i	−0.622965	+	0.782249i	$0.714072\pi$
$54$	0	0
$55$	−2.88450e6	−0.315224
$56$	1.88174e6i	0.191340i
$57$	0	0
$58$	−718987.	−0.0635344
$59$	3.30263e6i	0.272554i	0.990671	+	0.136277i	$0.0435137\pi$
$59$	3.30263e6i	0.272554i	−0.990671	+	0.136277i	$0.956486\pi$
$60$	0	0
$61$	1.51546e7	1.09452	0.547262	−	0.836961i	$-0.315670\pi$
$61$	1.51546e7	1.09452	0.547262	+	0.836961i	$0.315670\pi$
$62$	1.10052e7i	0.744783i
$63$	0	0
$64$	−2.09715e6	−0.125000
$65$	− 161038.i	− 0.00902142i
$66$	0	0
$67$	−1.76819e7	−0.877466	−0.438733	−	0.898617i	$-0.644573\pi$
$67$	−1.76819e7	−0.877466	−0.438733	+	0.898617i	$0.644573\pi$
$68$	− 1.13959e7i	− 0.532983i
$69$	0	0
$70$	9.61943e6	0.400642
$71$	2.66372e7i	1.04823i	0.851649	+	0.524113i	$0.175603\pi$
$71$	2.66372e7i	1.04823i	−0.851649	+	0.524113i	$0.824397\pi$
$72$	0	0
$73$	−1.28456e7	−0.452338	−0.226169	−	0.974088i	$-0.572620\pi$
$73$	−1.28456e7	−0.452338	−0.226169	+	0.974088i	$0.572620\pi$
$74$	4.62483e6i	0.154230i
$75$	0	0
$76$	−1.86648e7	−0.559459
$77$	5.72815e6i	0.162949i
$78$	0	0
$79$	5.49629e7	1.41111	0.705556	−	0.708655i	$-0.250698\pi$
$79$	5.49629e7	1.41111	0.705556	+	0.708655i	$0.250698\pi$
$80$	1.07206e7i	0.261734i
$81$	0	0
$82$	4.06946e7	0.900080
$83$	9.14079e6i	0.192607i	0.995352	+	0.0963034i	$0.0307019\pi$
$83$	9.14079e6i	0.192607i	−0.995352	+	0.0963034i	$0.969298\pi$
$84$	0	0
$85$	−5.82558e7	−1.11600
$86$	5.47401e7i	1.00072i
$87$	0	0
$88$	−6.38389e6	−0.106452
$89$	− 5.99482e7i	− 0.955468i	−0.878505	−	0.477734i	$-0.841458\pi$
$89$	− 5.99482e7i	− 0.955468i	0.878505	−	0.477734i	$-0.158542\pi$
$90$	0	0
$91$	−319795.	−0.00466344
$92$	4.55127e7i	0.635304i
$93$	0	0
$94$	−1.41655e7	−0.181435
$95$	9.54142e7i	1.17144i
$96$	0	0
$97$	1.07742e8	1.21702	0.608512	−	0.793544i	$-0.291767\pi$
$97$	1.07742e8	1.21702	0.608512	+	0.793544i	$0.291767\pi$
$98$	4.61187e7i	0.500003i
$99$	0	0
$100$	4.80378e6	0.0480378
$101$	2.00649e8i	1.92820i	0.265538	+	0.964100i	$0.414450\pi$
$101$	2.00649e8i	1.92820i	−0.265538	+	0.964100i	$0.585550\pi$
$102$	0	0
$103$	−2.24817e8	−1.99747	−0.998733	−	0.0503276i	$-0.983973\pi$
$103$	−2.24817e8	−1.99747	−0.998733	+	0.0503276i	$0.983973\pi$
$104$	− 356404.i	− 0.00304656i
$105$	0	0
$106$	1.39664e8	1.10627
$107$	− 2.03058e7i	− 0.154912i	−0.996996	−	0.0774560i	$-0.975320\pi$
$107$	− 2.03058e7i	− 0.154912i	0.996996	−	0.0774560i	$-0.0246797\pi$
$108$	0	0
$109$	−9.99522e7	−0.708087	−0.354043	−	0.935229i	$-0.615193\pi$
$109$	−9.99522e7	−0.708087	−0.354043	+	0.935229i	$0.615193\pi$
$110$	3.26344e7i	0.222897i
$111$	0	0
$112$	2.12894e7	0.135298
$113$	2.14539e8i	1.31581i	0.753103	+	0.657903i	$0.228556\pi$
$113$	2.14539e8i	1.31581i	−0.753103	+	0.657903i	$0.771444\pi$
$114$	0	0
$115$	2.32661e8	1.33025
$116$	8.13441e6i	0.0449256i
$117$	0	0
$118$	3.73650e7	0.192725
$119$	1.15687e8i	0.576893i
$120$	0	0
$121$	1.94926e8	0.909343
$122$	− 1.71455e8i	− 0.773946i
$123$	0	0
$124$	1.24509e8	0.526641
$125$	2.31043e8i	0.946351i
$126$	0	0
$127$	1.72330e8	0.662440	0.331220	−	0.943553i	$-0.392540\pi$
$127$	1.72330e8	0.662440	0.331220	+	0.943553i	$0.392540\pi$
$128$	2.37266e7i	0.0883883i
$129$	0	0
$130$	−1.82194e6	−0.00637911
$131$	4.54273e7i	0.154252i	0.997021	+	0.0771262i	$0.0245744\pi$
$131$	4.54273e7i	0.154252i	−0.997021	+	0.0771262i	$0.975426\pi$
$132$	0	0
$133$	1.89477e8	0.605550
$134$	2.00048e8i	0.620462i
$135$	0	0
$136$	−1.28930e8	−0.376876
$137$	2.35501e8i	0.668514i	0.942482	+	0.334257i	$0.108485\pi$
$137$	2.35501e8i	0.668514i	−0.942482	+	0.334257i	$0.891515\pi$
$138$	0	0
$139$	−5.78249e8	−1.54902	−0.774508	−	0.632564i	$-0.782002\pi$
$139$	−5.78249e8	−1.54902	−0.774508	+	0.632564i	$0.782002\pi$
$140$	− 1.08831e8i	− 0.283297i
$141$	0	0
$142$	3.01365e8	0.741208
$143$	− 1.08492e6i	− 0.00259450i
$144$	0	0
$145$	4.15831e7	0.0940686
$146$	1.45331e8i	0.319851i
$147$	0	0
$148$	5.23240e7	0.109057
$149$	8.12264e8i	1.64798i	0.566604	+	0.823990i	$0.308257\pi$
$149$	8.12264e8i	1.64798i	−0.566604	+	0.823990i	$0.691743\pi$
$150$	0	0
$151$	−6.85116e8	−1.31782	−0.658911	−	0.752221i	$-0.728983\pi$
$151$	−6.85116e8	−1.31782	−0.658911	+	0.752221i	$0.728983\pi$
$152$	2.11168e8i	0.395598i
$153$	0	0
$154$	6.48066e7	0.115222
$155$	− 6.36490e8i	− 1.10272i
$156$	0	0
$157$	−6.18488e8	−1.01796	−0.508982	−	0.860777i	$-0.669978\pi$
$157$	−6.18488e8	−1.01796	−0.508982	+	0.860777i	$0.669978\pi$
$158$	− 6.21834e8i	− 0.997806i
$159$	0	0
$160$	1.21290e8	0.185074
$161$	− 4.62026e8i	− 0.687643i
$162$	0	0
$163$	−5.97940e8	−0.847047	−0.423523	−	0.905885i	$-0.639207\pi$
$163$	−5.97940e8	−0.847047	−0.423523	+	0.905885i	$0.639207\pi$
$164$	− 4.60407e8i	− 0.636453i
$165$	0	0
$166$	1.03416e8	0.136194
$167$	− 9.51397e7i	− 0.122320i	−0.998128	−	0.0611598i	$-0.980520\pi$
$167$	− 9.51397e7i	− 0.122320i	0.998128	−	0.0611598i	$-0.0194799\pi$
$168$	0	0
$169$	−8.15670e8	−0.999926
$170$	6.59089e8i	0.789130i
$171$	0	0
$172$	6.19313e8	0.707614
$173$	− 5.34126e8i	− 0.596292i	−0.954520	−	0.298146i	$-0.903632\pi$
$173$	− 5.34126e8i	− 0.596292i	0.954520	−	0.298146i	$-0.0963683\pi$
$174$	0	0
$175$	−4.87660e7	−0.0519954
$176$	7.22255e7i	0.0752731i
$177$	0	0
$178$	−6.78237e8	−0.675618
$179$	− 1.22426e9i	− 1.19251i	−0.802794	−	0.596256i	$-0.796654\pi$
$179$	− 1.22426e9i	− 1.19251i	0.802794	−	0.596256i	$-0.203346\pi$
$180$	0	0
$181$	−1.85158e9	−1.72516	−0.862578	−	0.505924i	$-0.831152\pi$
$181$	−1.85158e9	−1.72516	−0.862578	+	0.505924i	$0.831152\pi$
$182$	3.61807e6i	0.00329755i
$183$	0	0
$184$	5.14918e8	0.449228
$185$	− 2.67480e8i	− 0.228352i
$186$	0	0
$187$	−3.92473e8	−0.320954
$188$	1.60265e8i	0.128294i
$189$	0	0
$190$	1.07949e9	0.828331
$191$	− 1.41043e9i	− 1.05979i	−0.848064	−	0.529894i	$-0.822232\pi$
$191$	− 1.41043e9i	− 1.05979i	0.848064	−	0.529894i	$-0.177768\pi$
$192$	0	0
$193$	1.65632e9	1.19376	0.596879	−	0.802332i	$-0.296407\pi$
$193$	1.65632e9	1.19376	0.596879	+	0.802332i	$0.296407\pi$
$194$	− 1.21897e9i	− 0.860566i
$195$	0	0
$196$	5.21773e8	0.353555
$197$	1.67903e8i	0.111479i	0.998445	+	0.0557396i	$0.0177517\pi$
$197$	1.67903e8i	0.111479i	−0.998445	+	0.0557396i	$0.982248\pi$
$198$	0	0
$199$	1.14583e9	0.730646	0.365323	−	0.930881i	$-0.380959\pi$
$199$	1.14583e9	0.730646	0.365323	+	0.930881i	$0.380959\pi$
$200$	− 5.43486e7i	− 0.0339679i
$201$	0	0
$202$	2.27009e9	1.36344
$203$	− 8.25772e7i	− 0.0486268i
$204$	0	0
$205$	−2.35360e9	−1.33265
$206$	2.54351e9i	1.41242i
$207$	0	0
$208$	−4.03226e6	−0.00215424
$209$	6.42811e8i	0.336898i
$210$	0	0
$211$	1.53209e8	0.0772957	0.0386478	−	0.999253i	$-0.487695\pi$
$211$	1.53209e8	0.0772957	0.0386478	+	0.999253i	$0.487695\pi$
$212$	− 1.58011e9i	− 0.782249i
$213$	0	0
$214$	−2.29734e8	−0.109539
$215$	− 3.16592e9i	− 1.48165i
$216$	0	0
$217$	−1.26397e9	−0.570028
$218$	1.13083e9i	0.500693i
$219$	0	0
$220$	3.69216e8	0.157612
$221$	− 2.19112e7i	− 0.00918540i
$222$	0	0
$223$	−2.16048e9	−0.873635	−0.436818	−	0.899550i	$-0.643894\pi$
$223$	−2.16048e9	−0.873635	−0.436818	+	0.899550i	$0.643894\pi$
$224$	− 2.40862e8i	− 0.0956702i
$225$	0	0
$226$	2.42723e9	0.930415
$227$	− 4.94040e8i	− 0.186062i	−0.995663	−	0.0930312i	$-0.970344\pi$
$227$	− 4.94040e8i	− 0.186062i	0.995663	−	0.0930312i	$-0.0296556\pi$
$228$	0	0
$229$	4.22138e9	1.53502	0.767508	−	0.641040i	$-0.221497\pi$
$229$	4.22138e9	1.53502	0.767508	+	0.641040i	$0.221497\pi$
$230$	− 2.63226e9i	− 0.940626i
$231$	0	0
$232$	9.20304e7	0.0317672
$233$	− 5.22591e9i	− 1.77312i	−0.462612	−	0.886561i	$-0.653088\pi$
$233$	− 5.22591e9i	− 1.77312i	0.462612	−	0.886561i	$-0.346912\pi$
$234$	0	0
$235$	8.19272e8	0.268631
$236$	− 4.22737e8i	− 0.136277i
$237$	0	0
$238$	1.30884e9	0.407925
$239$	2.78527e9i	0.853641i	0.904336	+	0.426820i	$0.140366\pi$
$239$	2.78527e9i	0.853641i	−0.904336	+	0.426820i	$0.859634\pi$
$240$	0	0
$241$	−3.49974e9	−1.03745	−0.518726	−	0.854941i	$-0.673593\pi$
$241$	−3.49974e9	−1.03745	−0.518726	+	0.854941i	$0.673593\pi$
$242$	− 2.20533e9i	− 0.643003i
$243$	0	0
$244$	−1.93979e9	−0.547262
$245$	− 2.66730e9i	− 0.740300i
$246$	0	0
$247$	−3.58873e7	−0.00964169
$248$	− 1.40866e9i	− 0.372391i
$249$	0	0
$250$	2.61395e9	0.669171
$251$	− 4.34649e9i	− 1.09508i	−0.836781	−	0.547538i	$-0.815565\pi$
$251$	− 4.34649e9i	− 1.09508i	0.836781	−	0.547538i	$-0.184435\pi$
$252$	0	0
$253$	1.56745e9	0.382570
$254$	− 1.94969e9i	− 0.468416i
$255$	0	0
$256$	2.68435e8	0.0625000
$257$	4.00994e9i	0.919190i	0.888129	+	0.459595i	$0.152005\pi$
$257$	4.00994e9i	0.919190i	−0.888129	+	0.459595i	$0.847995\pi$
$258$	0	0
$259$	−5.31171e8	−0.118042
$260$	2.06129e7i	0.00451071i
$261$	0	0
$262$	5.13951e8	0.109073
$263$	5.15434e9i	1.07733i	0.842519	+	0.538667i	$0.181072\pi$
$263$	5.15434e9i	1.07733i	−0.842519	+	0.538667i	$0.818928\pi$
$264$	0	0
$265$	−8.07754e9	−1.63793
$266$	− 2.14369e9i	− 0.428189i
$267$	0	0
$268$	2.26329e9	0.438733
$269$	− 3.97838e7i	− 0.00759796i	−0.999993	−	0.00379898i	$-0.998791\pi$
$269$	− 3.97838e7i	− 0.00759796i	0.999993	−	0.00379898i	$-0.00120926\pi$
$270$	0	0
$271$	8.93439e8	0.165649	0.0828243	−	0.996564i	$-0.473606\pi$
$271$	8.93439e8	0.165649	0.0828243	+	0.996564i	$0.473606\pi$
$272$	1.45868e9i	0.266491i
$273$	0	0
$274$	2.66439e9	0.472711
$275$	− 1.65441e8i	− 0.0289276i
$276$	0	0
$277$	7.49244e8	0.127264	0.0636318	−	0.997973i	$-0.479732\pi$
$277$	7.49244e8	0.127264	0.0636318	+	0.997973i	$0.479732\pi$
$278$	6.54214e9i	1.09532i
$279$	0	0
$280$	−1.23129e9	−0.200321
$281$	− 1.47805e9i	− 0.237064i	−0.992950	−	0.118532i	$-0.962181\pi$
$281$	− 1.47805e9i	− 0.237064i	0.992950	−	0.118532i	$-0.0378188\pi$
$282$	0	0
$283$	3.17255e9	0.494610	0.247305	−	0.968938i	$-0.420455\pi$
$283$	3.17255e9	0.494610	0.247305	+	0.968938i	$0.420455\pi$
$284$	− 3.40956e9i	− 0.524113i
$285$	0	0
$286$	−1.22745e7	−0.00183459
$287$	4.67386e9i	0.688887i
$288$	0	0
$289$	−9.50675e8	−0.136283
$290$	− 4.70459e8i	− 0.0665165i
$291$	0	0
$292$	1.64424e9	0.226169
$293$	− 2.64033e9i	− 0.358251i	−0.983826	−	0.179126i	$-0.942673\pi$
$293$	− 2.64033e9i	− 0.358251i	0.983826	−	0.179126i	$-0.0573269\pi$
$294$	0	0
$295$	−2.16103e9	−0.285346
$296$	− 5.91978e8i	− 0.0771150i
$297$	0	0
$298$	9.18972e9	1.16530
$299$	8.75087e7i	0.0109488i
$300$	0	0
$301$	−6.28701e9	−0.765911
$302$	7.75121e9i	0.931841i
$303$	0	0
$304$	2.38909e9	0.279730
$305$	9.91620e9i	1.14590i
$306$	0	0
$307$	−1.11920e10	−1.25995	−0.629975	−	0.776615i	$-0.716935\pi$
$307$	−1.11920e10	−1.25995	−0.629975	+	0.776615i	$0.716935\pi$
$308$	− 7.33203e8i	− 0.0814744i
$309$	0	0
$310$	−7.20106e9	−0.779740
$311$	3.70518e9i	0.396066i	0.980195	+	0.198033i	$0.0634554\pi$
$311$	3.70518e9i	0.396066i	−0.980195	+	0.198033i	$0.936545\pi$
$312$	0	0
$313$	1.39092e10	1.44919	0.724596	−	0.689173i	$-0.242026\pi$
$313$	1.39092e10	1.44919	0.724596	+	0.689173i	$0.242026\pi$
$314$	6.99740e9i	0.719810i
$315$	0	0
$316$	−7.03525e9	−0.705556
$317$	7.90702e9i	0.783025i	0.920173	+	0.391512i	$0.128048\pi$
$317$	7.90702e9i	0.783025i	−0.920173	+	0.391512i	$0.871952\pi$
$318$	0	0
$319$	2.80147e8	0.0270535
$320$	− 1.37224e9i	− 0.130867i
$321$	0	0
$322$	−5.22723e9	−0.486237
$323$	1.29823e10i	1.19273i
$324$	0	0
$325$	9.23637e6	0.000827881 0
$326$	6.76492e9i	0.598952i
$327$	0	0
$328$	−5.20891e9	−0.450040
$329$	− 1.62694e9i	− 0.138864i
$330$	0	0
$331$	1.48306e10	1.23551	0.617757	−	0.786369i	$-0.288042\pi$
$331$	1.48306e10	1.23551	0.617757	+	0.786369i	$0.288042\pi$
$332$	− 1.17002e9i	− 0.0963034i
$333$	0	0
$334$	−1.07638e9	−0.0864930
$335$	− 1.15699e10i	− 0.918651i
$336$	0	0
$337$	−1.90954e10	−1.48050	−0.740250	−	0.672332i	$-0.765293\pi$
$337$	−1.90954e10	−1.48050	−0.740250	+	0.672332i	$0.765293\pi$
$338$	9.22825e9i	0.707054i
$339$	0	0
$340$	7.45674e9	0.557999
$341$	− 4.28807e9i	− 0.317135i
$342$	0	0
$343$	−1.27876e10	−0.923875
$344$	− 7.00673e9i	− 0.500359i
$345$	0	0
$346$	−6.04294e9	−0.421642
$347$	2.30111e10i	1.58715i	0.608470	+	0.793577i	$0.291784\pi$
$347$	2.30111e10i	1.58715i	−0.608470	+	0.793577i	$0.708216\pi$
$348$	0	0
$349$	2.75549e8	0.0185736	0.00928682	−	0.999957i	$-0.497044\pi$
$349$	2.75549e8	0.0185736	0.00928682	+	0.999957i	$0.497044\pi$
$350$	5.51724e8i	0.0367663i
$351$	0	0
$352$	8.17138e8	0.0532261
$353$	− 3.80642e9i	− 0.245142i	−0.992460	−	0.122571i	$-0.960886\pi$
$353$	− 3.80642e9i	− 0.245142i	0.992460	−	0.122571i	$-0.0391139\pi$
$354$	0	0
$355$	−1.74297e10	−1.09743
$356$	7.67337e9i	0.477734i
$357$	0	0
$358$	−1.38510e10	−0.843234
$359$	3.37924e9i	0.203442i	0.994813	+	0.101721i	$0.0324349\pi$
$359$	3.37924e9i	0.203442i	−0.994813	+	0.101721i	$0.967565\pi$
$360$	0	0
$361$	4.27950e9	0.251979
$362$	2.09483e10i	1.21987i
$363$	0	0
$364$	4.09338e7	0.00233172
$365$	− 8.40533e9i	− 0.473569i
$366$	0	0
$367$	2.35211e10	1.29656	0.648280	−	0.761402i	$-0.275489\pi$
$367$	2.35211e10	1.29656	0.648280	+	0.761402i	$0.275489\pi$
$368$	− 5.82563e9i	− 0.317652i
$369$	0	0
$370$	−3.02619e9	−0.161469
$371$	1.60407e10i	0.846695i
$372$	0	0
$373$	−5.28403e8	−0.0272979	−0.0136490	−	0.999907i	$-0.504345\pi$
$373$	−5.28403e8	−0.0272979	−0.0136490	+	0.999907i	$0.504345\pi$
$374$	4.44032e9i	0.226949i
$375$	0	0
$376$	1.81319e9	0.0907176
$377$	1.56403e7i	0 0.000774246i
$378$	0	0
$379$	3.35382e10	1.62548	0.812742	−	0.582623i	$-0.197974\pi$
$379$	3.35382e10	1.62548	0.812742	+	0.582623i	$0.197974\pi$
$380$	− 1.22130e10i	− 0.585718i
$381$	0	0
$382$	−1.59572e10	−0.749383
$383$	− 2.11001e10i	− 0.980592i	−0.871556	−	0.490296i	$-0.836889\pi$
$383$	− 2.11001e10i	− 0.980592i	0.871556	−	0.490296i	$-0.163111\pi$
$384$	0	0
$385$	−3.74813e9	−0.170597
$386$	− 1.87392e10i	− 0.844114i
$387$	0	0
$388$	−1.37910e10	−0.608512
$389$	3.10197e10i	1.35469i	0.735667	+	0.677343i	$0.236869\pi$
$389$	3.10197e10i	1.35469i	−0.735667	+	0.677343i	$0.763131\pi$
$390$	0	0
$391$	3.16564e10	1.35442
$392$	− 5.90319e9i	− 0.250001i
$393$	0	0
$394$	1.89961e9	0.0788277
$395$	3.59642e10i	1.47734i
$396$	0	0
$397$	2.51981e10	1.01439	0.507197	−	0.861830i	$-0.330682\pi$
$397$	2.51981e10	1.01439	0.507197	+	0.861830i	$0.330682\pi$
$398$	− 1.29636e10i	− 0.516645i
$399$	0	0
$400$	−6.14884e8	−0.0240189
$401$	− 1.15637e10i	− 0.447217i	−0.974679	−	0.223608i	$-0.928216\pi$
$401$	− 1.15637e10i	− 0.447217i	0.974679	−	0.223608i	$-0.0717837\pi$
$402$	0	0
$403$	2.39397e8	0.00907610
$404$	− 2.56831e10i	− 0.964100i
$405$	0	0
$406$	−9.34254e8	−0.0343844
$407$	− 1.80203e9i	− 0.0656725i
$408$	0	0
$409$	1.80013e10	0.643297	0.321648	−	0.946859i	$-0.395763\pi$
$409$	1.80013e10	0.643297	0.321648	+	0.946859i	$0.395763\pi$
$410$	2.66279e10i	0.942327i
$411$	0	0
$412$	2.87765e10	0.998733
$413$	4.29145e9i	0.147504i
$414$	0	0
$415$	−5.98114e9	−0.201647
$416$	4.56198e7i	0.00152328i
$417$	0	0
$418$	7.27258e9	0.238223
$419$	− 5.43292e9i	− 0.176269i	−0.996109	−	0.0881347i	$-0.971909\pi$
$419$	− 5.43292e9i	− 0.176269i	0.996109	−	0.0881347i	$-0.0280906\pi$
$420$	0	0
$421$	3.52256e10	1.12132	0.560660	−	0.828046i	$-0.310547\pi$
$421$	3.52256e10	1.12132	0.560660	+	0.828046i	$0.310547\pi$
$422$	− 1.73336e9i	− 0.0546563i
$423$	0	0
$424$	−1.78770e10	−0.553134
$425$	− 3.34127e9i	− 0.102413i
$426$	0	0
$427$	1.96919e10	0.592348
$428$	2.59914e9i	0.0774560i
$429$	0	0
$430$	−3.58183e10	−1.04769
$431$	− 1.51218e10i	− 0.438223i	−0.975700	−	0.219112i	$-0.929684\pi$
$431$	− 1.51218e10i	− 0.438223i	0.975700	−	0.219112i	$-0.0703159\pi$
$432$	0	0
$433$	2.23207e10	0.634973	0.317487	−	0.948263i	$-0.397161\pi$
$433$	2.23207e10	0.634973	0.317487	+	0.948263i	$0.397161\pi$
$434$	1.43001e10i	0.403071i
$435$	0	0
$436$	1.27939e10	0.354043
$437$	− 5.18485e10i	− 1.42171i
$438$	0	0
$439$	−3.56756e10	−0.960536	−0.480268	−	0.877122i	$-0.659461\pi$
$439$	−3.56756e10	−0.960536	−0.480268	+	0.877122i	$0.659461\pi$
$440$	− 4.17720e9i	− 0.111449i
$441$	0	0
$442$	−2.47897e8	−0.00649506
$443$	1.32534e10i	0.344121i	0.985086	+	0.172061i	$0.0550425\pi$
$443$	1.32534e10i	0.344121i	−0.985086	+	0.172061i	$0.944958\pi$
$444$	0	0
$445$	3.92262e10	1.00031
$446$	2.44430e10i	0.617753i
$447$	0	0
$448$	−2.72505e9	−0.0676491
$449$	− 3.66405e10i	− 0.901521i	−0.892645	−	0.450761i	$-0.851153\pi$
$449$	− 3.66405e10i	− 0.901521i	0.892645	−	0.450761i	$-0.148847\pi$
$450$	0	0
$451$	−1.58563e10	−0.383262
$452$	− 2.74610e10i	− 0.657903i
$453$	0	0
$454$	−5.58943e9	−0.131566
$455$	− 2.09253e8i	− 0.00488233i
$456$	0	0
$457$	−7.56147e10	−1.73357	−0.866786	−	0.498681i	$-0.833818\pi$
$457$	−7.56147e10	−1.73357	−0.866786	+	0.498681i	$0.833818\pi$
$458$	− 4.77595e10i	− 1.08542i
$459$	0	0
$460$	−2.97806e10	−0.665123
$461$	− 2.22302e10i	− 0.492198i	−0.969245	−	0.246099i	$-0.920851\pi$
$461$	− 2.22302e10i	− 0.492198i	0.969245	−	0.246099i	$-0.0791489\pi$
$462$	0	0
$463$	4.40257e10	0.958038	0.479019	−	0.877805i	$-0.340992\pi$
$463$	4.40257e10	0.958038	0.479019	+	0.877805i	$0.340992\pi$
$464$	− 1.04120e9i	− 0.0224628i
$465$	0	0
$466$	−5.91245e10	−1.25379
$467$	− 1.17066e10i	− 0.246130i	−0.992399	−	0.123065i	$-0.960728\pi$
$467$	− 1.17066e10i	− 0.246130i	0.992399	−	0.123065i	$-0.0392723\pi$
$468$	0	0
$469$	−2.29760e10	−0.474878
$470$	− 9.26901e9i	− 0.189951i
$471$	0	0
$472$	−4.78272e9	−0.0963623
$473$	− 2.13290e10i	− 0.426114i
$474$	0	0
$475$	−5.47251e9	−0.107501
$476$	− 1.48079e10i	− 0.288446i
$477$	0	0
$478$	3.15117e10	0.603615
$479$	− 9.32529e10i	− 1.77141i	−0.464244	−	0.885707i	$-0.653674\pi$
$479$	− 9.32529e10i	− 1.77141i	0.464244	−	0.885707i	$-0.346326\pi$
$480$	0	0
$481$	1.00605e8	0.00187948
$482$	3.95951e10i	0.733589i
$483$	0	0
$484$	−2.49505e10	−0.454672
$485$	7.04996e10i	1.27415i
$486$	0	0
$487$	8.25854e10	1.46821	0.734103	−	0.679038i	$-0.237603\pi$
$487$	8.25854e10	1.46821	0.734103	+	0.679038i	$0.237603\pi$
$488$	2.19462e10i	0.386973i
$489$	0	0
$490$	−3.01771e10	−0.523471
$491$	− 3.72808e10i	− 0.641446i	−0.947173	−	0.320723i	$-0.896074\pi$
$491$	− 3.72808e10i	− 0.641446i	0.947173	−	0.320723i	$-0.103926\pi$
$492$	0	0
$493$	5.65790e9	0.0957784
$494$	4.06019e8i	0.00681771i
$495$	0	0
$496$	−1.59372e10	−0.263320
$497$	3.46124e10i	0.567292i
$498$	0	0
$499$	1.65596e10	0.267085	0.133542	−	0.991043i	$-0.457365\pi$
$499$	1.65596e10	0.267085	0.133542	+	0.991043i	$0.457365\pi$
$500$	− 2.95735e10i	− 0.473176i
$501$	0	0
$502$	−4.91749e10	−0.774335
$503$	− 2.46985e10i	− 0.385832i	−0.981215	−	0.192916i	$-0.938205\pi$
$503$	− 2.46985e10i	− 0.385832i	0.981215	−	0.192916i	$-0.0617945\pi$
$504$	0	0
$505$	−1.31292e11	−2.01870
$506$	− 1.77337e10i	− 0.270518i
$507$	0	0
$508$	−2.20583e10	−0.331220
$509$	− 2.87952e10i	− 0.428991i	−0.976725	−	0.214496i	$-0.931189\pi$
$509$	− 2.87952e10i	− 0.428991i	0.976725	−	0.214496i	$-0.0688107\pi$
$510$	0	0
$511$	−1.66916e10	−0.244802
$512$	− 3.03700e9i	− 0.0441942i
$513$	0	0
$514$	4.53673e10	0.649966
$515$	− 1.47105e11i	− 2.09122i
$516$	0	0
$517$	5.51948e9	0.0772567
$518$	6.00952e9i	0.0834681i
$519$	0	0
$520$	2.33208e8	0.00318955
$521$	1.01030e10i	0.137119i	0.997647	+	0.0685597i	$0.0218404\pi$
$521$	1.01030e10i	0.137119i	−0.997647	+	0.0685597i	$0.978160\pi$
$522$	0	0
$523$	9.96192e10	1.33149	0.665743	−	0.746181i	$-0.268115\pi$
$523$	9.96192e10	1.33149	0.665743	+	0.746181i	$0.268115\pi$
$524$	− 5.81470e9i	− 0.0771262i
$525$	0	0
$526$	5.83147e10	0.761790
$527$	− 8.66025e10i	− 1.12276i
$528$	0	0
$529$	−4.81177e10	−0.614444
$530$	9.13869e10i	1.15819i
$531$	0	0
$532$	−2.42531e10	−0.302775
$533$	− 8.85238e8i	− 0.0109686i
$534$	0	0
$535$	1.32868e10	0.162183
$536$	− 2.56062e10i	− 0.310231i
$537$	0	0
$538$	−4.50102e8	−0.00537257
$539$	− 1.79698e10i	− 0.212906i
$540$	0	0
$541$	−3.18510e10	−0.371821	−0.185911	−	0.982567i	$-0.559524\pi$
$541$	−3.18510e10	−0.371821	−0.185911	+	0.982567i	$0.559524\pi$
$542$	− 1.01081e10i	− 0.117131i
$543$	0	0
$544$	1.65030e10	0.188438
$545$	− 6.54023e10i	− 0.741322i
$546$	0	0
$547$	1.26359e11	1.41142	0.705709	−	0.708502i	$-0.250629\pi$
$547$	1.26359e11	1.41142	0.705709	+	0.708502i	$0.250629\pi$
$548$	− 3.01441e10i	− 0.334257i
$549$	0	0
$550$	−1.87175e9	−0.0204549
$551$	− 9.26679e9i	− 0.100536i
$552$	0	0
$553$	7.14190e10	0.763683
$554$	− 8.47673e9i	− 0.0899889i
$555$	0	0
$556$	7.40159e10	0.774508
$557$	7.08369e10i	0.735934i	0.929839	+	0.367967i	$0.119946\pi$
$557$	7.08369e10i	0.735934i	−0.929839	+	0.367967i	$0.880054\pi$
$558$	0	0
$559$	1.19077e9	0.0121950
$560$	1.39304e10i	0.141649i
$561$	0	0
$562$	−1.67223e10	−0.167629
$563$	1.85667e11i	1.84800i	0.382391	+	0.924001i	$0.375101\pi$
$563$	1.85667e11i	1.84800i	−0.382391	+	0.924001i	$0.624899\pi$
$564$	0	0
$565$	−1.40380e11	−1.37757
$566$	− 3.58934e10i	− 0.349742i
$567$	0	0
$568$	−3.85748e10	−0.370604
$569$	1.25347e11i	1.19581i	0.801565	+	0.597907i	$0.204001\pi$
$569$	1.25347e11i	1.19581i	−0.801565	+	0.597907i	$0.795999\pi$
$570$	0	0
$571$	−1.17675e11	−1.10698	−0.553490	−	0.832856i	$-0.686704\pi$
$571$	−1.17675e11	−1.10698	−0.553490	+	0.832856i	$0.686704\pi$
$572$	1.38870e8i	0.00129725i
$573$	0	0
$574$	5.28787e10	0.487117
$575$	1.33443e10i	0.122074i
$576$	0	0
$577$	6.05429e9	0.0546210	0.0273105	−	0.999627i	$-0.491306\pi$
$577$	6.05429e9	0.0546210	0.0273105	+	0.999627i	$0.491306\pi$
$578$	1.07557e10i	0.0963664i
$579$	0	0
$580$	−5.32263e9	−0.0470343
$581$	1.18776e10i	0.104237i
$582$	0	0
$583$	−5.44188e10	−0.471058
$584$	− 1.86024e10i	− 0.159926i
$585$	0	0
$586$	−2.98719e10	−0.253322
$587$	1.71256e11i	1.44243i	0.692712	+	0.721214i	$0.256416\pi$
$587$	1.71256e11i	1.44243i	−0.692712	+	0.721214i	$0.743584\pi$
$588$	0	0
$589$	−1.41842e11	−1.17854
$590$	2.44493e10i	0.201770i
$591$	0	0
$592$	−6.69747e9	−0.0545286
$593$	− 1.09816e11i	− 0.888070i	−0.896009	−	0.444035i	$-0.853547\pi$
$593$	− 1.09816e11i	− 0.888070i	0.896009	−	0.444035i	$-0.146453\pi$
$594$	0	0
$595$	−7.56978e10	−0.603970
$596$	− 1.03970e11i	− 0.823990i
$597$	0	0
$598$	9.90047e8	0.00774197
$599$	2.07207e11i	1.60952i	0.593598	+	0.804761i	$0.297707\pi$
$599$	2.07207e11i	1.60952i	−0.593598	+	0.804761i	$0.702293\pi$
$600$	0	0
$601$	8.39887e10	0.643758	0.321879	−	0.946781i	$-0.395686\pi$
$601$	8.39887e10	0.643758	0.321879	+	0.946781i	$0.395686\pi$
$602$	7.11294e10i	0.541581i
$603$	0	0
$604$	8.76949e10	0.658911
$605$	1.27547e11i	0.952025i
$606$	0	0
$607$	5.73367e10	0.422355	0.211178	−	0.977448i	$-0.432270\pi$
$607$	5.73367e10	0.422355	0.211178	+	0.977448i	$0.432270\pi$
$608$	− 2.70295e10i	− 0.197799i
$609$	0	0
$610$	1.12189e11	0.810272
$611$	3.08146e8i	0.00221101i
$612$	0	0
$613$	−4.42704e10	−0.313525	−0.156762	−	0.987636i	$-0.550106\pi$
$613$	−4.42704e10	−0.313525	−0.156762	+	0.987636i	$0.550106\pi$
$614$	1.26623e11i	0.890919i
$615$	0	0
$616$	−8.29524e9	−0.0576111
$617$	2.29926e11i	1.58652i	0.608880	+	0.793262i	$0.291619\pi$
$617$	2.29926e11i	1.58652i	−0.608880	+	0.793262i	$0.708381\pi$
$618$	0	0
$619$	2.79506e11	1.90383	0.951917	−	0.306355i	$-0.0991095\pi$
$619$	2.79506e11	1.90383	0.951917	+	0.306355i	$0.0991095\pi$
$620$	8.14707e10i	0.551359i
$621$	0	0
$622$	4.19193e10	0.280061
$623$	− 7.78969e10i	− 0.517092i
$624$	0	0
$625$	−1.65839e11	−1.08685
$626$	− 1.57365e11i	− 1.02473i
$627$	0	0
$628$	7.91665e10	0.508982
$629$	− 3.63940e10i	− 0.232502i
$630$	0	0
$631$	1.51695e11	0.956872	0.478436	−	0.878122i	$-0.341204\pi$
$631$	1.51695e11	0.956872	0.478436	+	0.878122i	$0.341204\pi$
$632$	7.95948e10i	0.498903i
$633$	0	0
$634$	8.94577e10	0.553682
$635$	1.12762e11i	0.693533i
$636$	0	0
$637$	1.00323e9	0.00609315
$638$	− 3.16951e9i	− 0.0191297i
$639$	0	0
$640$	−1.55251e10	−0.0925370
$641$	9.06592e10i	0.537007i	0.963279	+	0.268503i	$0.0865291\pi$
$641$	9.06592e10i	0.537007i	−0.963279	+	0.268503i	$0.913471\pi$
$642$	0	0
$643$	−1.91856e11	−1.12236	−0.561178	−	0.827695i	$-0.689652\pi$
$643$	−1.91856e11	−1.12236	−0.561178	+	0.827695i	$0.689652\pi$
$644$	5.91394e10i	0.343822i
$645$	0	0
$646$	1.46878e11	0.843387
$647$	− 2.69147e11i	− 1.53593i	−0.640489	−	0.767967i	$-0.721268\pi$
$647$	− 2.69147e11i	− 1.53593i	0.640489	−	0.767967i	$-0.278732\pi$
$648$	0	0
$649$	−1.45590e10	−0.0820638
$650$	− 1.04498e8i	0 0.000585400i
$651$	0	0
$652$	7.65363e10	0.423523
$653$	− 1.81639e11i	− 0.998981i	−0.866319	−	0.499490i	$-0.833521\pi$
$653$	− 1.81639e11i	− 0.998981i	0.866319	−	0.499490i	$-0.166479\pi$
$654$	0	0
$655$	−2.97247e10	−0.161492
$656$	5.89321e10i	0.318226i
$657$	0	0
$658$	−1.84067e10	−0.0981914
$659$	− 2.24414e10i	− 0.118989i	−0.998229	−	0.0594947i	$-0.981051\pi$
$659$	− 2.24414e10i	− 0.118989i	0.998229	−	0.0594947i	$-0.0189490\pi$
$660$	0	0
$661$	−8.05608e10	−0.422005	−0.211003	−	0.977485i	$-0.567673\pi$
$661$	−8.05608e10	−0.422005	−0.211003	+	0.977485i	$0.567673\pi$
$662$	− 1.67789e11i	− 0.873640i
$663$	0	0
$664$	−1.32373e10	−0.0680968
$665$	1.23982e11i	0.633973i
$666$	0	0
$667$	−2.25964e10	−0.114166
$668$	1.21779e10i	0.0611598i
$669$	0	0
$670$	−1.30899e11	−0.649585
$671$	6.68060e10i	0.329553i
$672$	0	0
$673$	2.07077e11	1.00942	0.504711	−	0.863289i	$-0.331599\pi$
$673$	2.07077e11	1.00942	0.504711	+	0.863289i	$0.331599\pi$
$674$	2.16039e11i	1.04687i
$675$	0	0
$676$	1.04406e11	0.499963
$677$	1.39552e11i	0.664325i	0.943222	+	0.332162i	$0.107778\pi$
$677$	1.39552e11i	0.664325i	−0.943222	+	0.332162i	$0.892222\pi$
$678$	0	0
$679$	1.40001e11	0.658645
$680$	− 8.43634e10i	− 0.394565i
$681$	0	0
$682$	−4.85140e10	−0.224248
$683$	3.26730e11i	1.50144i	0.660623	+	0.750718i	$0.270292\pi$
$683$	3.26730e11i	1.50144i	−0.660623	+	0.750718i	$0.729708\pi$
$684$	0	0
$685$	−1.54097e11	−0.699892
$686$	1.44675e11i	0.653279i
$687$	0	0
$688$	−7.92721e10	−0.353807
$689$	− 3.03813e9i	− 0.0134812i
$690$	0	0
$691$	2.75937e11	1.21031	0.605156	−	0.796107i	$-0.293111\pi$
$691$	2.75937e11	1.21031	0.605156	+	0.796107i	$0.293111\pi$
$692$	6.83681e10i	0.298146i
$693$	0	0
$694$	2.60341e11	1.12229
$695$	− 3.78369e11i	− 1.62172i
$696$	0	0
$697$	−3.20236e11	−1.35687
$698$	− 3.11748e9i	− 0.0131335i
$699$	0	0
$700$	6.24205e9	0.0259977
$701$	− 3.74663e11i	− 1.55156i	−0.631002	−	0.775781i	$-0.717356\pi$
$701$	− 3.74663e11i	− 1.55156i	0.631002	−	0.775781i	$-0.282644\pi$
$702$	0	0
$703$	−5.96079e10	−0.244052
$704$	− 9.24486e9i	− 0.0376365i
$705$	0	0
$706$	−4.30647e10	−0.173341
$707$	2.60724e11i	1.04353i
$708$	0	0
$709$	−1.08861e11	−0.430813	−0.215406	−	0.976525i	$-0.569108\pi$
$709$	−1.08861e11	−0.430813	−0.215406	+	0.976525i	$0.569108\pi$
$710$	1.97194e11i	0.775997i
$711$	0	0
$712$	8.68143e10	0.337809
$713$	3.45871e11i	1.33831i
$714$	0	0
$715$	7.09903e8	0.00271628
$716$	1.56706e11i	0.596256i
$717$	0	0
$718$	3.82317e10	0.143855
$719$	2.05253e11i	0.768023i	0.923328	+	0.384012i	$0.125458\pi$
$719$	2.05253e11i	0.768023i	−0.923328	+	0.384012i	$0.874542\pi$
$720$	0	0
$721$	−2.92127e11	−1.08101
$722$	− 4.84171e10i	− 0.178176i
$723$	0	0
$724$	2.37002e11	0.862578
$725$	2.38501e9i	0.00863252i
$726$	0	0
$727$	−3.97318e11	−1.42233	−0.711165	−	0.703025i	$-0.751832\pi$
$727$	−3.97318e11	−1.42233	−0.711165	+	0.703025i	$0.751832\pi$
$728$	− 4.63113e8i	− 0.00164878i
$729$	0	0
$730$	−9.50954e10	−0.334864
$731$	− 4.30764e11i	− 1.50858i
$732$	0	0
$733$	−2.13964e11	−0.741180	−0.370590	−	0.928796i	$-0.620845\pi$
$733$	−2.13964e11	−0.741180	−0.370590	+	0.928796i	$0.620845\pi$
$734$	− 2.66111e11i	− 0.916807i
$735$	0	0
$736$	−6.59095e10	−0.224614
$737$	− 7.79471e10i	− 0.264198i
$738$	0	0
$739$	−1.80884e11	−0.606489	−0.303244	−	0.952913i	$-0.598070\pi$
$739$	−1.80884e11	−0.606489	−0.303244	+	0.952913i	$0.598070\pi$
$740$	3.42374e10i	0.114176i
$741$	0	0
$742$	1.81479e11	0.598704
$743$	3.94798e11i	1.29545i	0.761876	+	0.647723i	$0.224278\pi$
$743$	3.94798e11i	1.29545i	−0.761876	+	0.647723i	$0.775722\pi$
$744$	0	0
$745$	−5.31493e11	−1.72533
$746$	5.97820e9i	0.0193026i
$747$	0	0
$748$	5.02365e10	0.160477
$749$	− 2.63854e10i	− 0.0838372i
$750$	0	0
$751$	4.28937e11	1.34845	0.674224	−	0.738527i	$-0.264478\pi$
$751$	4.28937e11	1.34845	0.674224	+	0.738527i	$0.264478\pi$
$752$	− 2.05139e10i	− 0.0641470i
$753$	0	0
$754$	1.76950e8	0.000547475 0
$755$	− 4.48296e11i	− 1.37968i
$756$	0	0
$757$	−2.25950e10	−0.0688063	−0.0344031	−	0.999408i	$-0.510953\pi$
$757$	−2.25950e10	−0.0688063	−0.0344031	+	0.999408i	$0.510953\pi$
$758$	− 3.79441e11i	− 1.14939i
$759$	0	0
$760$	−1.38175e11	−0.414165
$761$	− 5.00151e11i	− 1.49129i	−0.666343	−	0.745645i	$-0.732141\pi$
$761$	− 5.00151e11i	− 1.49129i	0.666343	−	0.745645i	$-0.267859\pi$
$762$	0	0
$763$	−1.29878e11	−0.383211
$764$	1.80535e11i	0.529894i
$765$	0	0
$766$	−2.38720e11	−0.693383
$767$	− 8.12809e8i	− 0.00234859i
$768$	0	0
$769$	−2.65413e11	−0.758956	−0.379478	−	0.925201i	$-0.623896\pi$
$769$	−2.65413e11	−0.758956	−0.379478	+	0.925201i	$0.623896\pi$
$770$	4.24052e10i	0.120630i
$771$	0	0
$772$	−2.12009e11	−0.596879
$773$	8.51838e10i	0.238583i	0.992859	+	0.119291i	$0.0380623\pi$
$773$	8.51838e10i	0.238583i	−0.992859	+	0.119291i	$0.961938\pi$
$774$	0	0
$775$	3.65060e10	0.101195
$776$	1.56028e11i	0.430283i
$777$	0	0
$778$	3.50948e11	0.957908
$779$	5.24499e11i	1.42428i
$780$	0	0
$781$	−1.17425e11	−0.315613
$782$	− 3.58151e11i	− 0.957723i
$783$	0	0
$784$	−6.67870e10	−0.176778
$785$	− 4.04699e11i	− 1.06574i
$786$	0	0
$787$	3.09172e11	0.805936	0.402968	−	0.915214i	$-0.367979\pi$
$787$	3.09172e11	0.805936	0.402968	+	0.915214i	$0.367979\pi$
$788$	− 2.14916e10i	− 0.0557396i
$789$	0	0
$790$	4.06888e11	1.04464
$791$	2.78772e11i	0.712104i
$792$	0	0
$793$	−3.72969e9	−0.00943149
$794$	− 2.85084e11i	− 0.717284i
$795$	0	0
$796$	−1.46666e11	−0.365323
$797$	− 7.58923e11i	− 1.88090i	−0.339938	−	0.940448i	$-0.610406\pi$
$797$	− 7.58923e11i	− 1.88090i	0.339938	−	0.940448i	$-0.389594\pi$
$798$	0	0
$799$	1.11472e11	0.273514
$800$	6.95662e9i	0.0169839i
$801$	0	0
$802$	−1.30828e11	−0.316230
$803$	− 5.66272e10i	− 0.136195i
$804$	0	0
$805$	3.02320e11	0.719919
$806$	− 2.70847e9i	− 0.00641777i
$807$	0	0
$808$	−2.90571e11	−0.681722
$809$	− 4.10334e11i	− 0.957952i	−0.877828	−	0.478976i	$-0.841008\pi$
$809$	− 4.10334e11i	− 0.957952i	0.877828	−	0.478976i	$-0.158992\pi$
$810$	0	0
$811$	2.09073e11	0.483298	0.241649	−	0.970364i	$-0.422312\pi$
$811$	2.09073e11	0.483298	0.241649	+	0.970364i	$0.422312\pi$
$812$	1.05699e10i	0.0243134i
$813$	0	0
$814$	−2.03876e10	−0.0464375
$815$	− 3.91253e11i	− 0.886804i
$816$	0	0
$817$	−7.05526e11	−1.58353
$818$	− 2.03662e11i	− 0.454879i
$819$	0	0
$820$	3.01260e11	0.666326
$821$	6.58897e11i	1.45026i	0.688614	+	0.725128i	$0.258220\pi$
$821$	6.58897e11i	1.45026i	−0.688614	+	0.725128i	$0.741780\pi$
$822$	0	0
$823$	3.39048e11	0.739030	0.369515	−	0.929225i	$-0.379524\pi$
$823$	3.39048e11	0.739030	0.369515	+	0.929225i	$0.379524\pi$
$824$	− 3.25569e11i	− 0.706211i
$825$	0	0
$826$	4.85522e10	0.104301
$827$	6.51031e11i	1.39181i	0.718135	+	0.695904i	$0.244996\pi$
$827$	6.51031e11i	1.39181i	−0.718135	+	0.695904i	$0.755004\pi$
$828$	0	0
$829$	6.46185e11	1.36817	0.684083	−	0.729404i	$-0.260203\pi$
$829$	6.46185e11	1.36817	0.684083	+	0.729404i	$0.260203\pi$
$830$	6.76689e10i	0.142586i
$831$	0	0
$832$	5.16129e8	0.00107712
$833$	− 3.62920e11i	− 0.753756i
$834$	0	0
$835$	6.22532e10	0.128061
$836$	− 8.22798e10i	− 0.168449i
$837$	0	0
$838$	−6.14664e10	−0.124641
$839$	− 6.50212e11i	− 1.31222i	−0.754665	−	0.656111i	$-0.772200\pi$
$839$	− 6.50212e11i	− 1.31222i	0.754665	−	0.656111i	$-0.227800\pi$
$840$	0	0
$841$	4.96208e11	0.991927
$842$	− 3.98532e11i	− 0.792893i
$843$	0	0
$844$	−1.96108e10	−0.0386478
$845$	− 5.33722e11i	− 1.04686i
$846$	0	0
$847$	2.53287e11	0.492130
$848$	2.02255e11i	0.391125i
$849$	0	0
$850$	−3.78022e10	−0.0724171
$851$	1.45350e11i	0.277138i
$852$	0	0
$853$	−1.57184e11	−0.296901	−0.148450	−	0.988920i	$-0.547429\pi$
$853$	−1.57184e11	−0.296901	−0.148450	+	0.988920i	$0.547429\pi$
$854$	− 2.22789e11i	− 0.418854i
$855$	0	0
$856$	2.94060e10	0.0547697
$857$	7.12288e10i	0.132048i	0.997818	+	0.0660241i	$0.0210314\pi$
$857$	7.12288e10i	0.132048i	−0.997818	+	0.0660241i	$0.978969\pi$
$858$	0	0
$859$	5.95995e11	1.09464	0.547318	−	0.836925i	$-0.315649\pi$
$859$	5.95995e11	1.09464	0.547318	+	0.836925i	$0.315649\pi$
$860$	4.05238e11i	0.740827i
$861$	0	0
$862$	−1.71084e11	−0.309871
$863$	− 9.33352e11i	− 1.68268i	−0.540504	−	0.841342i	$-0.681766\pi$
$863$	− 9.33352e11i	− 1.68268i	0.540504	−	0.841342i	$-0.318234\pi$
$864$	0	0
$865$	3.49497e11	0.624280
$866$	− 2.52529e11i	− 0.448994i
$867$	0	0
$868$	1.61788e11	0.285014
$869$	2.42293e11i	0.424875i
$870$	0	0
$871$	4.35169e9	0.00756111
$872$	− 1.44746e11i	− 0.250346i
$873$	0	0
$874$	−5.86598e11	−1.00530
$875$	3.00218e11i	0.512158i
$876$	0	0
$877$	−5.11922e11	−0.865377	−0.432689	−	0.901543i	$-0.642435\pi$
$877$	−5.11922e11	−0.865377	−0.432689	+	0.901543i	$0.642435\pi$
$878$	4.03624e11i	0.679201i
$879$	0	0
$880$	−4.72597e10	−0.0788061
$881$	3.93245e11i	0.652769i	0.945237	+	0.326384i	$0.105830\pi$
$881$	3.93245e11i	0.652769i	−0.945237	+	0.326384i	$0.894170\pi$
$882$	0	0
$883$	8.58653e11	1.41246	0.706228	−	0.707984i	$-0.250395\pi$
$883$	8.58653e11	1.41246	0.706228	+	0.707984i	$0.250395\pi$
$884$	2.80464e9i	0.00459270i
$885$	0	0
$886$	1.49945e11	0.243330
$887$	− 4.38727e10i	− 0.0708762i	−0.999372	−	0.0354381i	$-0.988717\pi$
$887$	− 4.38727e10i	− 0.0708762i	0.999372	−	0.0354381i	$-0.0112827\pi$
$888$	0	0
$889$	2.23927e11	0.358508
$890$	− 4.43794e11i	− 0.707329i
$891$	0	0
$892$	2.76541e11	0.436818
$893$	− 1.82575e11i	− 0.287101i
$894$	0	0
$895$	8.01079e11	1.24849
$896$	3.08304e10i	0.0478351i
$897$	0	0
$898$	−4.14540e11	−0.637472
$899$	6.18169e10i	0.0946387i
$900$	0	0
$901$	−1.09905e12	−1.66770
$902$	1.79394e11i	0.271007i
$903$	0	0
$904$	−3.10685e11	−0.465208
$905$	− 1.21155e12i	− 1.80613i
$906$	0	0
$907$	6.60214e11	0.975563	0.487782	−	0.872966i	$-0.337806\pi$
$907$	6.60214e11	0.975563	0.487782	+	0.872966i	$0.337806\pi$
$908$	6.32371e10i	0.0930312i
$909$	0	0
$910$	−2.36743e9	−0.00345233
$911$	− 6.64022e11i	− 0.964071i	−0.876152	−	0.482035i	$-0.839898\pi$
$911$	− 6.64022e11i	− 0.964071i	0.876152	−	0.482035i	$-0.160102\pi$
$912$	0	0
$913$	−4.02953e10	−0.0579924
$914$	8.55483e11i	1.22582i
$915$	0	0
$916$	−5.40337e11	−0.767508
$917$	5.90284e10i	0.0834802i
$918$	0	0
$919$	−5.25016e11	−0.736056	−0.368028	−	0.929815i	$-0.619967\pi$
$919$	−5.25016e11	−0.736056	−0.368028	+	0.929815i	$0.619967\pi$
$920$	3.36929e11i	0.470313i
$921$	0	0
$922$	−2.51506e11	−0.348037
$923$	− 6.55566e9i	− 0.00903254i
$924$	0	0
$925$	1.53414e10	0.0209555
$926$	− 4.98094e11i	− 0.677435i
$927$	0	0
$928$	−1.17799e10	−0.0158836
$929$	1.35915e12i	1.82476i	0.409347	+	0.912379i	$0.365757\pi$
$929$	1.35915e12i	1.82476i	−0.409347	+	0.912379i	$0.634243\pi$
$930$	0	0
$931$	−5.94408e11	−0.791199
$932$	6.68917e11i	0.886561i
$933$	0	0
$934$	−1.32445e11	−0.174040
$935$	− 2.56809e11i	− 0.336018i
$936$	0	0
$937$	−9.07999e11	−1.17795	−0.588975	−	0.808151i	$-0.700468\pi$
$937$	−9.07999e11	−1.17795	−0.588975	+	0.808151i	$0.700468\pi$
$938$	2.59943e11i	0.335790i
$939$	0	0
$940$	−1.04867e11	−0.134316
$941$	7.94257e11i	1.01298i	0.862245	+	0.506492i	$0.169058\pi$
$941$	7.94257e11i	1.01298i	−0.862245	+	0.506492i	$0.830942\pi$
$942$	0	0
$943$	1.27895e12	1.61736
$944$	5.41103e10i	0.0681385i
$945$	0	0
$946$	−2.41310e11	−0.301308
$947$	− 1.87600e11i	− 0.233256i	−0.993176	−	0.116628i	$-0.962791\pi$
$947$	− 1.87600e11i	− 0.233256i	0.993176	−	0.116628i	$-0.0372086\pi$
$948$	0	0
$949$	3.16142e9	0.00389778
$950$	6.19143e10i	0.0760146i
$951$	0	0
$952$	−1.67532e11	−0.203962
$953$	1.33323e12i	1.61634i	0.588950	+	0.808169i	$0.299541\pi$
$953$	1.33323e12i	1.61634i	−0.588950	+	0.808169i	$0.700459\pi$
$954$	0	0
$955$	9.22895e11	1.10953
$956$	− 3.56514e11i	− 0.426820i
$957$	0	0
$958$	−1.05504e12	−1.25258
$959$	3.06011e11i	0.361795i
$960$	0	0
$961$	9.33083e10	0.109402
$962$	− 1.13821e9i	− 0.00132900i
$963$	0	0
$964$	4.47967e11	0.518726
$965$	1.08379e12i	1.24979i
$966$	0	0
$967$	−3.54726e10	−0.0405684	−0.0202842	−	0.999794i	$-0.506457\pi$
$967$	−3.54726e10	−0.0405684	−0.0202842	+	0.999794i	$0.506457\pi$
$968$	2.82283e11i	0.321501i
$969$	0	0
$970$	7.97612e11	0.900958
$971$	9.73657e10i	0.109529i	0.998499	+	0.0547645i	$0.0174408\pi$
$971$	9.73657e10i	0.109529i	−0.998499	+	0.0547645i	$0.982559\pi$
$972$	0	0
$973$	−7.51379e11	−0.838316
$974$	− 9.34347e11i	− 1.03818i
$975$	0	0
$976$	2.48293e11	0.273631
$977$	− 1.63885e12i	− 1.79871i	−0.437220	−	0.899354i	$-0.644037\pi$
$977$	− 1.63885e12i	− 1.79871i	0.437220	−	0.899354i	$-0.355963\pi$
$978$	0	0
$979$	2.64269e11	0.287684
$980$	3.41415e11i	0.370150i
$981$	0	0
$982$	−4.21785e11	−0.453571
$983$	− 8.99399e11i	− 0.963249i	−0.876378	−	0.481624i	$-0.840047\pi$
$983$	− 8.99399e11i	− 0.963249i	0.876378	−	0.481624i	$-0.159953\pi$
$984$	0	0
$985$	−1.09865e11	−0.116712
$986$	− 6.40118e10i	− 0.0677255i
$987$	0	0
$988$	4.59358e9	0.00482085
$989$	1.72038e12i	1.79820i
$990$	0	0
$991$	1.18507e12	1.22871	0.614357	−	0.789028i	$-0.289415\pi$
$991$	1.18507e12	1.22871	0.614357	+	0.789028i	$0.289415\pi$
$992$	1.80309e11i	0.186196i
$993$	0	0
$994$	3.91595e11	0.401136
$995$	7.49755e11i	0.764940i
$996$	0	0
$997$	4.46164e10	0.0451558	0.0225779	−	0.999745i	$-0.492813\pi$
$997$	4.46164e10	0.0451558	0.0225779	+	0.999745i	$0.492813\pi$
$998$	− 1.87351e11i	− 0.188857i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.9.b.a.161.3	✓	8
3.2	odd	2	inner	162.9.b.a.161.6	yes	8
9.2	odd	6		162.9.d.h.53.7		16
9.4	even	3		162.9.d.h.107.7		16
9.5	odd	6		162.9.d.h.107.2		16
9.7	even	3		162.9.d.h.53.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.9.b.a.161.3	✓	8	1.1	even	1	trivial
162.9.b.a.161.6	yes	8	3.2	odd	2	inner
162.9.d.h.53.2		16	9.7	even	3
162.9.d.h.53.7		16	9.2	odd	6
162.9.d.h.107.2		16	9.5	odd	6
162.9.d.h.107.7		16	9.4	even	3

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)

\(f(q)\)	\(=\)	\(q-11.3137i q^{2} -128.000 q^{4} +654.335i q^{5} +1299.40 q^{7} +1448.15i q^{8} +O(q^{10})\)	\(q-11.3137i q^{2} -128.000 q^{4} +654.335i q^{5} +1299.40 q^{7} +1448.15i q^{8} +7402.96 q^{10} +4408.29i q^{11} -246.109 q^{13} -14701.1i q^{14} +16384.0 q^{16} +89030.5i q^{17} +145819. q^{19} -83754.9i q^{20} +49874.1 q^{22} -355568. i q^{23} -37529.5 q^{25} +2784.41i q^{26} -166324. q^{28} -63550.1i q^{29} -972728. q^{31} -185364. i q^{32} +1.00727e6 q^{34} +850245. i q^{35} -408781. q^{37} -1.64975e6i q^{38} -947579. q^{40} +3.59693e6i q^{41} -4.83838e6 q^{43} -564261. i q^{44} -4.02279e6 q^{46} -1.25207e6i q^{47} -4.07635e6 q^{49} +424598. i q^{50} +31502.0 q^{52} +1.23446e7i q^{53} -2.88450e6 q^{55} +1.88174e6i q^{56} -718987. q^{58} +3.30263e6i q^{59} +1.51546e7 q^{61} +1.10052e7i q^{62} -2.09715e6 q^{64} -161038. i q^{65} -1.76819e7 q^{67} -1.13959e7i q^{68} +9.61943e6 q^{70} +2.66372e7i q^{71} -1.28456e7 q^{73} +4.62483e6i q^{74} -1.86648e7 q^{76} +5.72815e6i q^{77} +5.49629e7 q^{79} +1.07206e7i q^{80} +4.06946e7 q^{82} +9.14079e6i q^{83} -5.82558e7 q^{85} +5.47401e7i q^{86} -6.38389e6 q^{88} -5.99482e7i q^{89} -319795. q^{91} +4.55127e7i q^{92} -1.41655e7 q^{94} +9.54142e7i q^{95} +1.07742e8 q^{97} +4.61187e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 1024 q^{4} - 8876 q^{7}+O(q^{10}) \) 8 * q - 1024 * q^4 - 8876 * q^7	\( 8 q - 1024 q^{4} - 8876 q^{7} + 8448 q^{10} - 117380 q^{13} + 131072 q^{16} + 270220 q^{19} - 210816 q^{22} - 1801672 q^{25} + 1136128 q^{28} - 393344 q^{31} - 691968 q^{34} + 1830988 q^{37} - 1081344 q^{40} + 11135236 q^{43} + 5296320 q^{46} - 13586328 q^{49} + 15024640 q^{52} - 1579716 q^{55} - 32988672 q^{58} + 12184204 q^{61} - 16777216 q^{64} - 80355716 q^{67} - 18723264 q^{70} + 197085760 q^{73} - 34588160 q^{76} + 84451852 q^{79} + 144639168 q^{82} - 582634548 q^{85} + 26984448 q^{88} + 373079588 q^{91} - 210121536 q^{94} + 341136928 q^{97}+O(q^{100}) \) 8 * q - 1024 * q^4 - 8876 * q^7 + 8448 * q^10 - 117380 * q^13 + 131072 * q^16 + 270220 * q^19 - 210816 * q^22 - 1801672 * q^25 + 1136128 * q^28 - 393344 * q^31 - 691968 * q^34 + 1830988 * q^37 - 1081344 * q^40 + 11135236 * q^43 + 5296320 * q^46 - 13586328 * q^49 + 15024640 * q^52 - 1579716 * q^55 - 32988672 * q^58 + 12184204 * q^61 - 16777216 * q^64 - 80355716 * q^67 - 18723264 * q^70 + 197085760 * q^73 - 34588160 * q^76 + 84451852 * q^79 + 144639168 * q^82 - 582634548 * q^85 + 26984448 * q^88 + 373079588 * q^91 - 210121536 * q^94 + 341136928 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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