LMFDB - Embedded newform 162.9.b.a.161.4

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			161.4
Root			$26.3494i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.9.b.a.161.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.3137i q^{2} -128.000 q^{4} +1061.95i q^{5} -3233.60 q^{7} +1448.15i q^{8} +O(q^{10})$	\(q-11.3137i q^{2} -128.000 q^{4} +1061.95i q^{5} -3233.60 q^{7} +1448.15i q^{8} +12014.6 q^{10} -7892.01i q^{11} -54494.5 q^{13} +36584.0i q^{14} +16384.0 q^{16} +85389.0i q^{17} -153936. q^{19} -135930. i q^{20} -89287.9 q^{22} +352061. i q^{23} -737118. q^{25} +616535. i q^{26} +413901. q^{28} -880286. i q^{29} +1.43554e6 q^{31} -185364. i q^{32} +966067. q^{34} -3.43393e6i q^{35} +1.22625e6 q^{37} +1.74158e6i q^{38} -1.53787e6 q^{40} -291589. i q^{41} +2.30050e6 q^{43} +1.01018e6i q^{44} +3.98311e6 q^{46} +1.84797e6i q^{47} +4.69138e6 q^{49} +8.33954e6i q^{50} +6.97530e6 q^{52} -3.72595e6i q^{53} +8.38094e6 q^{55} -4.68276e6i q^{56} -9.95929e6 q^{58} -3.50160e6i q^{59} -47311.8 q^{61} -1.62413e7i q^{62} -2.09715e6 q^{64} -5.78706e7i q^{65} -8.18647e6 q^{67} -1.09298e7i q^{68} -3.88505e7 q^{70} +6.23977e6i q^{71} +3.16653e7 q^{73} -1.38734e7i q^{74} +1.97038e7 q^{76} +2.55196e7i q^{77} -2.77912e7 q^{79} +1.73990e7i q^{80} -3.29895e6 q^{82} +8.77450e7i q^{83} -9.06791e7 q^{85} -2.60272e7i q^{86} +1.14289e7 q^{88} -8.43269e7i q^{89} +1.76214e8 q^{91} -4.50637e7i q^{92} +2.09074e7 q^{94} -1.63472e8i q^{95} +2.33853e7 q^{97} -5.30769e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1061.95i	1.69912i	0.527489	+	0.849562i	$0.323134\pi$
$5$	1061.95i	1.69912i	−0.527489	+	0.849562i	$0.676866\pi$
$6$	0	0
$7$	−3233.60	−1.34677	−0.673386	−	0.739291i	$-0.735161\pi$
$7$	−3233.60	−1.34677	−0.673386	+	0.739291i	$0.735161\pi$
$8$	1448.15i	0.353553i
$9$	0	0
$10$	12014.6	1.20146
$11$	− 7892.01i	− 0.539035i	−0.962995	−	0.269517i	$-0.913136\pi$
$11$	− 7892.01i	− 0.539035i	0.962995	−	0.269517i	$-0.0868642\pi$
$12$	0	0
$13$	−54494.5	−1.90800	−0.954002	−	0.299800i	$-0.903080\pi$
$13$	−54494.5	−1.90800	−0.954002	+	0.299800i	$0.903080\pi$
$14$	36584.0i	0.952312i
$15$	0	0
$16$	16384.0	0.250000
$17$	85389.0i	1.02237i	0.859472	+	0.511183i	$0.170793\pi$
$17$	85389.0i	1.02237i	−0.859472	+	0.511183i	$0.829207\pi$
$18$	0	0
$19$	−153936.	−1.18120	−0.590602	−	0.806963i	$-0.701110\pi$
$19$	−153936.	−1.18120	−0.590602	+	0.806963i	$0.701110\pi$
$20$	− 135930.i	− 0.849562i
$21$	0	0
$22$	−89287.9	−0.381155
$23$	352061.i	1.25807i	0.777376	+	0.629037i	$0.216550\pi$
$23$	352061.i	1.25807i	−0.777376	+	0.629037i	$0.783450\pi$
$24$	0	0
$25$	−737118.	−1.88702
$26$	616535.i	1.34916i
$27$	0	0
$28$	413901.	0.673386
$29$	− 880286.i	− 1.24461i	−0.782777	−	0.622303i	$-0.786197\pi$
$29$	− 880286.i	− 1.24461i	0.782777	−	0.622303i	$-0.213803\pi$
$30$	0	0
$31$	1.43554e6	1.55442	0.777212	−	0.629239i	$-0.216633\pi$
$31$	1.43554e6	1.55442	0.777212	+	0.629239i	$0.216633\pi$
$32$	− 185364.i	− 0.176777i
$33$	0	0
$34$	966067.	0.722922
$35$	− 3.43393e6i	− 2.28833i
$36$	0	0
$37$	1.22625e6	0.654293	0.327146	−	0.944974i	$-0.393913\pi$
$37$	1.22625e6	0.654293	0.327146	+	0.944974i	$0.393913\pi$
$38$	1.74158e6i	0.835237i
$39$	0	0
$40$	−1.53787e6	−0.600731
$41$	− 291589.i	− 0.103189i	−0.998668	−	0.0515947i	$-0.983570\pi$
$41$	− 291589.i	− 0.103189i	0.998668	−	0.0515947i	$-0.0164304\pi$
$42$	0	0
$43$	2.30050e6	0.672897	0.336448	−	0.941702i	$-0.390774\pi$
$43$	2.30050e6	0.672897	0.336448	+	0.941702i	$0.390774\pi$
$44$	1.01018e6i	0.269517i
$45$	0	0
$46$	3.98311e6	0.889592
$47$	1.84797e6i	0.378706i	0.981909	+	0.189353i	$0.0606391\pi$
$47$	1.84797e6i	0.378706i	−0.981909	+	0.189353i	$0.939361\pi$
$48$	0	0
$49$	4.69138e6	0.813797
$50$	8.33954e6i	1.33433i
$51$	0	0
$52$	6.97530e6	0.954002
$53$	− 3.72595e6i	− 0.472208i	−0.971728	−	0.236104i	$-0.924129\pi$
$53$	− 3.72595e6i	− 0.472208i	0.971728	−	0.236104i	$-0.0758706\pi$
$54$	0	0
$55$	8.38094e6	0.915887
$56$	− 4.68276e6i	− 0.476156i
$57$	0	0
$58$	−9.95929e6	−0.880069
$59$	− 3.50160e6i	− 0.288974i	−0.989507	−	0.144487i	$-0.953847\pi$
$59$	− 3.50160e6i	− 0.288974i	0.989507	−	0.144487i	$-0.0461532\pi$
$60$	0	0
$61$	−47311.8	−0.00341704	−0.00170852	−	0.999999i	$-0.500544\pi$
$61$	−47311.8	−0.00341704	−0.00170852	+	0.999999i	$0.500544\pi$
$62$	− 1.62413e7i	− 1.09914i
$63$	0	0
$64$	−2.09715e6	−0.125000
$65$	− 5.78706e7i	− 3.24194i
$66$	0	0
$67$	−8.18647e6	−0.406254	−0.203127	−	0.979152i	$-0.565110\pi$
$67$	−8.18647e6	−0.406254	−0.203127	+	0.979152i	$0.565110\pi$
$68$	− 1.09298e7i	− 0.511183i
$69$	0	0
$70$	−3.88505e7	−1.61810
$71$	6.23977e6i	0.245547i	0.992435	+	0.122774i	$0.0391789\pi$
$71$	6.23977e6i	0.245547i	−0.992435	+	0.122774i	$0.960821\pi$
$72$	0	0
$73$	3.16653e7	1.11505	0.557523	−	0.830162i	$-0.311752\pi$
$73$	3.16653e7	1.11505	0.557523	+	0.830162i	$0.311752\pi$
$74$	− 1.38734e7i	− 0.462655i
$75$	0	0
$76$	1.97038e7	0.590602
$77$	2.55196e7i	0.725958i
$78$	0	0
$79$	−2.77912e7	−0.713509	−0.356755	−	0.934198i	$-0.616117\pi$
$79$	−2.77912e7	−0.713509	−0.356755	+	0.934198i	$0.616117\pi$
$80$	1.73990e7i	0.424781i
$81$	0	0
$82$	−3.29895e6	−0.0729660
$83$	8.77450e7i	1.84889i	0.381321	+	0.924443i	$0.375469\pi$
$83$	8.77450e7i	1.84889i	−0.381321	+	0.924443i	$0.624531\pi$
$84$	0	0
$85$	−9.06791e7	−1.73713
$86$	− 2.60272e7i	− 0.475810i
$87$	0	0
$88$	1.14289e7	0.190578
$89$	− 8.43269e7i	− 1.34402i	−0.740541	−	0.672011i	$-0.765431\pi$
$89$	− 8.43269e7i	− 1.34402i	0.740541	−	0.672011i	$-0.234569\pi$
$90$	0	0
$91$	1.76214e8	2.56965
$92$	− 4.50637e7i	− 0.629037i
$93$	0	0
$94$	2.09074e7	0.267786
$95$	− 1.63472e8i	− 2.00701i
$96$	0	0
$97$	2.33853e7	0.264153	0.132077	−	0.991239i	$-0.457835\pi$
$97$	2.33853e7	0.264153	0.132077	+	0.991239i	$0.457835\pi$
$98$	− 5.30769e7i	− 0.575441i
$99$	0	0
$100$	9.43511e7	0.943511
$101$	7.12639e7i	0.684833i	0.939548	+	0.342416i	$0.111245\pi$
$101$	7.12639e7i	0.684833i	−0.939548	+	0.342416i	$0.888755\pi$
$102$	0	0
$103$	−1.75219e8	−1.55679	−0.778397	−	0.627772i	$-0.783967\pi$
$103$	−1.75219e8	−1.55679	−0.778397	+	0.627772i	$0.783967\pi$
$104$	− 7.89165e7i	− 0.674581i
$105$	0	0
$106$	−4.21543e7	−0.333901
$107$	− 3.54632e7i	− 0.270547i	−0.990808	−	0.135273i	$-0.956809\pi$
$107$	− 3.54632e7i	− 0.270547i	0.990808	−	0.135273i	$-0.0431913\pi$
$108$	0	0
$109$	5.93417e7	0.420392	0.210196	−	0.977659i	$-0.432590\pi$
$109$	5.93417e7	0.420392	0.210196	+	0.977659i	$0.432590\pi$
$110$	− 9.48195e7i	− 0.647630i
$111$	0	0
$112$	−5.29793e7	−0.336693
$113$	− 2.32353e8i	− 1.42506i	−0.701640	−	0.712532i	$-0.747548\pi$
$113$	− 2.32353e8i	− 1.42506i	0.701640	−	0.712532i	$-0.252452\pi$
$114$	0	0
$115$	−3.73872e8	−2.13762
$116$	1.12677e8i	0.622303i
$117$	0	0
$118$	−3.96161e7	−0.204335
$119$	− 2.76114e8i	− 1.37689i
$120$	0	0
$121$	1.52075e8	0.709441
$122$	535272.i	0.00241621i
$123$	0	0
$124$	−1.83750e8	−0.777212
$125$	− 3.67959e8i	− 1.50716i
$126$	0	0
$127$	−4.40554e7	−0.169350	−0.0846749	−	0.996409i	$-0.526985\pi$
$127$	−4.40554e7	−0.169350	−0.0846749	+	0.996409i	$0.526985\pi$
$128$	2.37266e7i	0.0883883i
$129$	0	0
$130$	−6.54731e8	−2.29239
$131$	− 1.32203e8i	− 0.448906i	−0.974485	−	0.224453i	$-0.927940\pi$
$131$	− 1.32203e8i	− 0.448906i	0.974485	−	0.224453i	$-0.0720596\pi$
$132$	0	0
$133$	4.97766e8	1.59081
$134$	9.26194e7i	0.287265i
$135$	0	0
$136$	−1.23657e8	−0.361461
$137$	− 5.07826e8i	− 1.44156i	−0.693164	−	0.720780i	$-0.743784\pi$
$137$	− 5.07826e8i	− 1.44156i	0.693164	−	0.720780i	$-0.256216\pi$
$138$	0	0
$139$	1.39585e7	0.0373920	0.0186960	−	0.999825i	$-0.494049\pi$
$139$	1.39585e7	0.0373920	0.0186960	+	0.999825i	$0.494049\pi$
$140$	4.39543e8i	1.14417i
$141$	0	0
$142$	7.05950e7	0.173628
$143$	4.30071e8i	1.02848i
$144$	0	0
$145$	9.34821e8	2.11474
$146$	− 3.58252e8i	− 0.788456i
$147$	0	0
$148$	−1.56960e8	−0.327146
$149$	− 2.72813e8i	− 0.553504i	−0.960941	−	0.276752i	$-0.910742\pi$
$149$	− 2.72813e8i	− 0.553504i	0.960941	−	0.276752i	$-0.0892580\pi$
$150$	0	0
$151$	−9.29048e8	−1.78702	−0.893512	−	0.449039i	$-0.851767\pi$
$151$	−9.29048e8	−1.78702	−0.893512	+	0.449039i	$0.851767\pi$
$152$	− 2.22923e8i	− 0.417618i
$153$	0	0
$154$	2.88722e8	0.513330
$155$	1.52448e9i	2.64116i
$156$	0	0
$157$	1.59766e8	0.262958	0.131479	−	0.991319i	$-0.458027\pi$
$157$	1.59766e8	0.262958	0.131479	+	0.991319i	$0.458027\pi$
$158$	3.14422e8i	0.504527i
$159$	0	0
$160$	1.96848e8	0.300366
$161$	− 1.13842e9i	− 1.69434i
$162$	0	0
$163$	4.67512e8	0.662281	0.331140	−	0.943581i	$-0.392567\pi$
$163$	4.67512e8	0.662281	0.331140	+	0.943581i	$0.392567\pi$
$164$	3.73234e7i	0.0515947i
$165$	0	0
$166$	9.92721e8	1.30736
$167$	3.25225e8i	0.418136i	0.977901	+	0.209068i	$0.0670431\pi$
$167$	3.25225e8i	0.418136i	−0.977901	+	0.209068i	$0.932957\pi$
$168$	0	0
$169$	2.15392e9	2.64048
$170$	1.02592e9i	1.22833i
$171$	0	0
$172$	−2.94464e8	−0.336448
$173$	5.39504e8i	0.602297i	0.953577	+	0.301148i	$0.0973699\pi$
$173$	5.39504e8i	0.602297i	−0.953577	+	0.301148i	$0.902630\pi$
$174$	0	0
$175$	2.38355e9	2.54139
$176$	− 1.29303e8i	− 0.134759i
$177$	0	0
$178$	−9.54050e8	−0.950367
$179$	− 7.15170e8i	− 0.696622i	−0.937379	−	0.348311i	$-0.886755\pi$
$179$	− 7.15170e8i	− 0.696622i	0.937379	−	0.348311i	$-0.113245\pi$
$180$	0	0
$181$	−2.01588e9	−1.87824	−0.939118	−	0.343594i	$-0.888356\pi$
$181$	−2.01588e9	−1.87824	−0.939118	+	0.343594i	$0.888356\pi$
$182$	− 1.99363e9i	− 1.81702i
$183$	0	0
$184$	−5.09838e8	−0.444796
$185$	1.30222e9i	1.11172i
$186$	0	0
$187$	6.73891e8	0.551091
$188$	− 2.36540e8i	− 0.189353i
$189$	0	0
$190$	−1.84948e9	−1.41917
$191$	1.02979e9i	0.773775i	0.922127	+	0.386888i	$0.126450\pi$
$191$	1.02979e9i	0.773775i	−0.922127	+	0.386888i	$0.873550\pi$
$192$	0	0
$193$	−1.82400e9	−1.31460	−0.657301	−	0.753628i	$-0.728302\pi$
$193$	−1.82400e9	−1.31460	−0.657301	+	0.753628i	$0.728302\pi$
$194$	− 2.64575e8i	− 0.186785i
$195$	0	0
$196$	−6.00496e8	−0.406899
$197$	1.21693e9i	0.807981i	0.914763	+	0.403991i	$0.132377\pi$
$197$	1.21693e9i	0.807981i	−0.914763	+	0.403991i	$0.867623\pi$
$198$	0	0
$199$	1.71825e9	1.09566	0.547829	−	0.836591i	$-0.315455\pi$
$199$	1.71825e9	1.09566	0.547829	+	0.836591i	$0.315455\pi$
$200$	− 1.06746e9i	− 0.667163i
$201$	0	0
$202$	8.06260e8	0.484250
$203$	2.84649e9i	1.67620i
$204$	0	0
$205$	3.09653e8	0.175332
$206$	1.98237e9i	1.10082i
$207$	0	0
$208$	−8.92838e8	−0.477001
$209$	1.21486e9i	0.636710i
$210$	0	0
$211$	−6.18692e8	−0.312136	−0.156068	−	0.987746i	$-0.549882\pi$
$211$	−6.18692e8	−0.312136	−0.156068	+	0.987746i	$0.549882\pi$
$212$	4.76921e8i	0.236104i
$213$	0	0
$214$	−4.01220e8	−0.191305
$215$	2.44302e9i	1.14333i
$216$	0	0
$217$	−4.64198e9	−2.09346
$218$	− 6.71375e8i	− 0.297262i
$219$	0	0
$220$	−1.07276e9	−0.457944
$221$	− 4.65323e9i	− 1.95068i
$222$	0	0
$223$	3.75384e9	1.51795	0.758973	−	0.651122i	$-0.225701\pi$
$223$	3.75384e9	1.51795	0.758973	+	0.651122i	$0.225701\pi$
$224$	5.99393e8i	0.238078i
$225$	0	0
$226$	−2.62877e9	−1.00767
$227$	4.30166e9i	1.62007i	0.586384	+	0.810033i	$0.300551\pi$
$227$	4.30166e9i	1.62007i	−0.586384	+	0.810033i	$0.699449\pi$
$228$	0	0
$229$	4.03628e9	1.46771	0.733854	−	0.679308i	$-0.237720\pi$
$229$	4.03628e9	1.46771	0.733854	+	0.679308i	$0.237720\pi$
$230$	4.22987e9i	1.51153i
$231$	0	0
$232$	1.27479e9	0.440034
$233$	1.15028e9i	0.390282i	0.980775	+	0.195141i	$0.0625164\pi$
$233$	1.15028e9i	0.390282i	−0.980775	+	0.195141i	$0.937484\pi$
$234$	0	0
$235$	−1.96245e9	−0.643469
$236$	4.48205e8i	0.144487i
$237$	0	0
$238$	−3.12387e9	−0.973612
$239$	− 3.60021e9i	− 1.10341i	−0.834040	−	0.551705i	$-0.813978\pi$
$239$	− 3.60021e9i	− 1.10341i	0.834040	−	0.551705i	$-0.186022\pi$
$240$	0	0
$241$	−3.55755e8	−0.105459	−0.0527294	−	0.998609i	$-0.516792\pi$
$241$	−3.55755e8	−0.105459	−0.0527294	+	0.998609i	$0.516792\pi$
$242$	− 1.72053e9i	− 0.501651i
$243$	0	0
$244$	6.05591e6	0.00170852
$245$	4.98202e9i	1.38274i
$246$	0	0
$247$	8.38864e9	2.25374
$248$	2.07889e9i	0.549572i
$249$	0	0
$250$	−4.16299e9	−1.06572
$251$	− 5.98696e9i	− 1.50838i	−0.656654	−	0.754192i	$-0.728029\pi$
$251$	− 5.98696e9i	− 1.50838i	0.656654	−	0.754192i	$-0.271971\pi$
$252$	0	0
$253$	2.77847e9	0.678146
$254$	4.98430e8i	0.119748i
$255$	0	0
$256$	2.68435e8	0.0625000
$257$	− 2.89949e9i	− 0.664643i	−0.943166	−	0.332322i	$-0.892168\pi$
$257$	− 2.89949e9i	− 0.664643i	0.943166	−	0.332322i	$-0.107832\pi$
$258$	0	0
$259$	−3.96520e9	−0.881184
$260$	7.40743e9i	1.62097i
$261$	0	0
$262$	−1.49570e9	−0.317425
$263$	2.66911e9i	0.557882i	0.960308	+	0.278941i	$0.0899835\pi$
$263$	2.66911e9i	0.557882i	−0.960308	+	0.278941i	$0.910017\pi$
$264$	0	0
$265$	3.95678e9	0.802339
$266$	− 5.63158e9i	− 1.12487i
$267$	0	0
$268$	1.04787e9	0.203127
$269$	− 1.26012e9i	− 0.240660i	−0.992734	−	0.120330i	$-0.961605\pi$
$269$	− 1.26012e9i	− 0.240660i	0.992734	−	0.120330i	$-0.0383953\pi$
$270$	0	0
$271$	1.48125e9	0.274632	0.137316	−	0.990527i	$-0.456152\pi$
$271$	1.48125e9	0.274632	0.137316	+	0.990527i	$0.456152\pi$
$272$	1.39901e9i	0.255591i
$273$	0	0
$274$	−5.74540e9	−1.01934
$275$	5.81735e9i	1.01717i
$276$	0	0
$277$	6.41212e8	0.108914	0.0544568	−	0.998516i	$-0.482657\pi$
$277$	6.41212e8	0.108914	0.0544568	+	0.998516i	$0.482657\pi$
$278$	− 1.57922e8i	− 0.0264401i
$279$	0	0
$280$	4.97286e9	0.809048
$281$	− 1.27667e9i	− 0.204763i	−0.994745	−	0.102382i	$-0.967354\pi$
$281$	− 1.27667e9i	− 0.204763i	0.994745	−	0.102382i	$-0.0326463\pi$
$282$	0	0
$283$	−1.12894e10	−1.76006	−0.880029	−	0.474921i	$-0.842477\pi$
$283$	−1.12894e10	−1.76006	−0.880029	+	0.474921i	$0.842477\pi$
$284$	− 7.98691e8i	− 0.122774i
$285$	0	0
$286$	4.86570e9	0.727246
$287$	9.42882e8i	0.138973i
$288$	0	0
$289$	−3.15528e8	−0.0452321
$290$	− 1.05763e10i	− 1.49535i
$291$	0	0
$292$	−4.05316e9	−0.557523
$293$	1.64745e9i	0.223533i	0.993735	+	0.111766i	$0.0356508\pi$
$293$	1.64745e9i	0.223533i	−0.993735	+	0.111766i	$0.964349\pi$
$294$	0	0
$295$	3.71854e9	0.491003
$296$	1.77580e9i	0.231327i
$297$	0	0
$298$	−3.08653e9	−0.391386
$299$	− 1.91854e10i	− 2.40041i
$300$	0	0
$301$	−7.43890e9	−0.906239
$302$	1.05110e10i	1.26362i
$303$	0	0
$304$	−2.52208e9	−0.295301
$305$	− 5.02429e7i	− 0.00580598i
$306$	0	0
$307$	−4.02739e9	−0.453388	−0.226694	−	0.973966i	$-0.572792\pi$
$307$	−4.02739e9	−0.453388	−0.226694	+	0.973966i	$0.572792\pi$
$308$	− 3.26651e9i	− 0.362979i
$309$	0	0
$310$	1.72475e10	1.86758
$311$	2.03690e9i	0.217735i	0.994056	+	0.108868i	$0.0347224\pi$
$311$	2.03690e9i	0.217735i	−0.994056	+	0.108868i	$0.965278\pi$
$312$	0	0
$313$	−5.27110e9	−0.549192	−0.274596	−	0.961560i	$-0.588544\pi$
$313$	−5.27110e9	−0.549192	−0.274596	+	0.961560i	$0.588544\pi$
$314$	− 1.80755e9i	− 0.185939i
$315$	0	0
$316$	3.55728e9	0.356755
$317$	1.11628e10i	1.10545i	0.833365	+	0.552723i	$0.186411\pi$
$317$	1.11628e10i	1.10545i	−0.833365	+	0.552723i	$0.813589\pi$
$318$	0	0
$319$	−6.94722e9	−0.670886
$320$	− 2.22708e9i	− 0.212391i
$321$	0	0
$322$	−1.28798e10	−1.19808
$323$	− 1.31444e10i	− 1.20762i
$324$	0	0
$325$	4.01689e10	3.60045
$326$	− 5.28929e9i	− 0.468303i
$327$	0	0
$328$	4.22266e8	0.0364830
$329$	− 5.97559e9i	− 0.510031i
$330$	0	0
$331$	7.64734e9	0.637086	0.318543	−	0.947908i	$-0.396806\pi$
$331$	7.64734e9	0.637086	0.318543	+	0.947908i	$0.396806\pi$
$332$	− 1.12314e10i	− 0.924443i
$333$	0	0
$334$	3.67950e9	0.295667
$335$	− 8.69365e9i	− 0.690276i
$336$	0	0
$337$	−1.17075e8	−0.00907703	−0.00453852	−	0.999990i	$-0.501445\pi$
$337$	−1.17075e8	−0.00907703	−0.00453852	+	0.999990i	$0.501445\pi$
$338$	− 2.43688e10i	− 1.86710i
$339$	0	0
$340$	1.16069e10	0.868563
$341$	− 1.13293e10i	− 0.837889i
$342$	0	0
$343$	3.47102e9	0.250773
$344$	3.33148e9i	0.237905i
$345$	0	0
$346$	6.10379e9	0.425888
$347$	− 1.54670e10i	− 1.06681i	−0.845859	−	0.533407i	$-0.820912\pi$
$347$	− 1.54670e10i	− 1.06681i	0.845859	−	0.533407i	$-0.179088\pi$
$348$	0	0
$349$	−1.67148e10	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$349$	−1.67148e10	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$350$	− 2.69668e10i	− 1.79703i
$351$	0	0
$352$	−1.46289e9	−0.0952888
$353$	− 1.76650e10i	− 1.13767i	−0.822452	−	0.568834i	$-0.807395\pi$
$353$	− 1.76650e10i	− 1.13767i	0.822452	−	0.568834i	$-0.192605\pi$
$354$	0	0
$355$	−6.62634e9	−0.417215
$356$	1.07938e10i	0.672011i
$357$	0	0
$358$	−8.09123e9	−0.492586
$359$	4.69625e9i	0.282731i	0.989957	+	0.141365i	$0.0451493\pi$
$359$	4.69625e9i	0.282731i	−0.989957	+	0.141365i	$0.954851\pi$
$360$	0	0
$361$	6.71260e9	0.395241
$362$	2.28071e10i	1.32811i
$363$	0	0
$364$	−2.25553e10	−1.28482
$365$	3.36271e10i	1.89460i
$366$	0	0
$367$	2.64435e10	1.45766	0.728829	−	0.684696i	$-0.240065\pi$
$367$	2.64435e10	1.45766	0.728829	+	0.684696i	$0.240065\pi$
$368$	5.76816e9i	0.314518i
$369$	0	0
$370$	1.47329e10	0.786108
$371$	1.20482e10i	0.635957i
$372$	0	0
$373$	1.53246e10	0.791685	0.395843	−	0.918318i	$-0.370453\pi$
$373$	1.53246e10	0.791685	0.395843	+	0.918318i	$0.370453\pi$
$374$	− 7.62421e9i	− 0.389680i
$375$	0	0
$376$	−2.67614e9	−0.133893
$377$	4.79707e10i	2.37471i
$378$	0	0
$379$	1.88826e10	0.915175	0.457588	−	0.889165i	$-0.348714\pi$
$379$	1.88826e10	0.915175	0.457588	+	0.889165i	$0.348714\pi$
$380$	2.09245e10i	1.00351i
$381$	0	0
$382$	1.16507e10	0.547142
$383$	− 4.19990e10i	− 1.95184i	−0.218128	−	0.975920i	$-0.569995\pi$
$383$	− 4.19990e10i	− 1.95184i	0.218128	−	0.975920i	$-0.430005\pi$
$384$	0	0
$385$	−2.71006e10	−1.23349
$386$	2.06362e10i	0.929564i
$387$	0	0
$388$	−2.99332e9	−0.132077
$389$	− 2.28908e10i	− 0.999684i	−0.866117	−	0.499842i	$-0.833391\pi$
$389$	− 2.28908e10i	− 0.999684i	0.866117	−	0.499842i	$-0.166609\pi$
$390$	0	0
$391$	−3.00621e10	−1.28621
$392$	6.79384e9i	0.287721i
$393$	0	0
$394$	1.37680e10	0.571329
$395$	− 2.95130e10i	− 1.21234i
$396$	0	0
$397$	−3.22176e10	−1.29697	−0.648487	−	0.761225i	$-0.724598\pi$
$397$	−3.22176e10	−1.29697	−0.648487	+	0.761225i	$0.724598\pi$
$398$	− 1.94398e10i	− 0.774747i
$399$	0	0
$400$	−1.20769e10	−0.471756
$401$	3.38757e10i	1.31012i	0.755577	+	0.655059i	$0.227356\pi$
$401$	3.38757e10i	1.31012i	−0.755577	+	0.655059i	$0.772644\pi$
$402$	0	0
$403$	−7.82292e10	−2.96585
$404$	− 9.12179e9i	− 0.342416i
$405$	0	0
$406$	3.22044e10	1.18525
$407$	− 9.67758e9i	− 0.352687i
$408$	0	0
$409$	−3.28412e10	−1.17362	−0.586808	−	0.809726i	$-0.699616\pi$
$409$	−3.28412e10	−1.17362	−0.586808	+	0.809726i	$0.699616\pi$
$410$	− 3.50333e9i	− 0.123978i
$411$	0	0
$412$	2.24280e10	0.778397
$413$	1.13228e10i	0.389182i
$414$	0	0
$415$	−9.31810e10	−3.14149
$416$	1.01013e10i	0.337291i
$417$	0	0
$418$	1.37446e10	0.450222
$419$	− 4.20810e10i	− 1.36531i	−0.730743	−	0.682653i	$-0.760826\pi$
$419$	− 4.20810e10i	− 1.36531i	0.730743	−	0.682653i	$-0.239174\pi$
$420$	0	0
$421$	2.16574e10	0.689410	0.344705	−	0.938711i	$-0.387979\pi$
$421$	2.16574e10	0.689410	0.344705	+	0.938711i	$0.387979\pi$
$422$	6.99970e9i	0.220714i
$423$	0	0
$424$	5.39575e9	0.166951
$425$	− 6.29418e10i	− 1.92923i
$426$	0	0
$427$	1.52988e8	0.00460198
$428$	4.53928e9i	0.135273i
$429$	0	0
$430$	2.76396e10	0.808460
$431$	5.79188e9i	0.167846i	0.996472	+	0.0839229i	$0.0267450\pi$
$431$	5.79188e9i	0.167846i	−0.996472	+	0.0839229i	$0.973255\pi$
$432$	0	0
$433$	−2.54361e10	−0.723601	−0.361801	−	0.932255i	$-0.617838\pi$
$433$	−2.54361e10	−0.723601	−0.361801	+	0.932255i	$0.617838\pi$
$434$	5.25180e10i	1.48030i
$435$	0	0
$436$	−7.59574e9	−0.210196
$437$	− 5.41946e10i	− 1.48604i
$438$	0	0
$439$	−1.90617e10	−0.513219	−0.256610	−	0.966515i	$-0.582605\pi$
$439$	−1.90617e10	−0.513219	−0.256610	+	0.966515i	$0.582605\pi$
$440$	1.21369e10i	0.323815i
$441$	0	0
$442$	−5.26453e10	−1.37934
$443$	1.92985e10i	0.501083i	0.968106	+	0.250541i	$0.0806086\pi$
$443$	1.92985e10i	0.501083i	−0.968106	+	0.250541i	$0.919391\pi$
$444$	0	0
$445$	8.95512e10	2.28366
$446$	− 4.24699e10i	− 1.07335i
$447$	0	0
$448$	6.78135e9	0.168347
$449$	2.56335e9i	0.0630700i	0.999503	+	0.0315350i	$0.0100396\pi$
$449$	2.56335e9i	0.0630700i	−0.999503	+	0.0315350i	$0.989960\pi$
$450$	0	0
$451$	−2.30122e9	−0.0556227
$452$	2.97412e10i	0.712532i
$453$	0	0
$454$	4.86677e10	1.14556
$455$	1.87130e11i	4.36615i
$456$	0	0
$457$	3.22717e10	0.739873	0.369937	−	0.929057i	$-0.379379\pi$
$457$	3.22717e10	0.739873	0.369937	+	0.929057i	$0.379379\pi$
$458$	− 4.56653e10i	− 1.03783i
$459$	0	0
$460$	4.78556e10	1.06881
$461$	6.87557e10i	1.52232i	0.648567	+	0.761158i	$0.275369\pi$
$461$	6.87557e10i	1.52232i	−0.648567	+	0.761158i	$0.724631\pi$
$462$	0	0
$463$	9.26696e9	0.201657	0.100829	−	0.994904i	$-0.467851\pi$
$463$	9.26696e9	0.201657	0.100829	+	0.994904i	$0.467851\pi$
$464$	− 1.44226e10i	− 0.311151i
$465$	0	0
$466$	1.30139e10	0.275971
$467$	5.14063e10i	1.08081i	0.841406	+	0.540404i	$0.181729\pi$
$467$	5.14063e10i	1.08081i	−0.841406	+	0.540404i	$0.818271\pi$
$468$	0	0
$469$	2.64718e10	0.547132
$470$	2.22026e10i	0.455001i
$471$	0	0
$472$	5.07086e9	0.102168
$473$	− 1.81556e10i	− 0.362715i
$474$	0	0
$475$	1.13469e11	2.22896
$476$	3.53426e10i	0.688447i
$477$	0	0
$478$	−4.07318e10	−0.780228
$479$	7.06306e10i	1.34169i	0.741600	+	0.670843i	$0.234067\pi$
$479$	7.06306e10i	1.34169i	−0.741600	+	0.670843i	$0.765933\pi$
$480$	0	0
$481$	−6.68239e10	−1.24839
$482$	4.02491e9i	0.0745706i
$483$	0	0
$484$	−1.94656e10	−0.354721
$485$	2.48341e10i	0.448829i
$486$	0	0
$487$	−4.61712e10	−0.820834	−0.410417	−	0.911898i	$-0.634617\pi$
$487$	−4.61712e10	−0.820834	−0.410417	+	0.911898i	$0.634617\pi$
$488$	− 6.85148e7i	− 0.00120811i
$489$	0	0
$490$	5.63651e10	0.977747
$491$	3.86564e10i	0.665112i	0.943083	+	0.332556i	$0.107911\pi$
$491$	3.86564e10i	0.665112i	−0.943083	+	0.332556i	$0.892089\pi$
$492$	0	0
$493$	7.51667e10	1.27244
$494$	− 9.49067e10i	− 1.59364i
$495$	0	0
$496$	2.35199e10	0.388606
$497$	− 2.01769e10i	− 0.330697i
$498$	0	0
$499$	1.18884e10	0.191744	0.0958721	−	0.995394i	$-0.469436\pi$
$499$	1.18884e10	0.191744	0.0958721	+	0.995394i	$0.469436\pi$
$500$	4.70988e10i	0.753581i
$501$	0	0
$502$	−6.77348e10	−1.06659
$503$	7.35514e10i	1.14900i	0.818506	+	0.574499i	$0.194803\pi$
$503$	7.35514e10i	1.14900i	−0.818506	+	0.574499i	$0.805197\pi$
$504$	0	0
$505$	−7.56789e10	−1.16362
$506$	− 3.14347e10i	− 0.479521i
$507$	0	0
$508$	5.63909e9	0.0846749
$509$	− 2.05011e10i	− 0.305427i	−0.988271	−	0.152713i	$-0.951199\pi$
$509$	− 2.05011e10i	− 0.305427i	0.988271	−	0.152713i	$-0.0488011\pi$
$510$	0	0
$511$	−1.02393e11	−1.50171
$512$	− 3.03700e9i	− 0.0441942i
$513$	0	0
$514$	−3.28039e10	−0.469974
$515$	− 1.86074e11i	− 2.64519i
$516$	0	0
$517$	1.45842e10	0.204136
$518$	4.48612e10i	0.623091i
$519$	0	0
$520$	8.38055e10	1.14620
$521$	7.15997e10i	0.971763i	0.874025	+	0.485881i	$0.161501\pi$
$521$	7.15997e10i	0.971763i	−0.874025	+	0.485881i	$0.838499\pi$
$522$	0	0
$523$	1.01958e11	1.36275	0.681376	−	0.731934i	$-0.261382\pi$
$523$	1.01958e11	1.36275	0.681376	+	0.731934i	$0.261382\pi$
$524$	1.69220e10i	0.224453i
$525$	0	0
$526$	3.01975e10	0.394482
$527$	1.22580e11i	1.58919i
$528$	0	0
$529$	−4.56356e10	−0.582749
$530$	− 4.47658e10i	− 0.567340i
$531$	0	0
$532$	−6.37141e10	−0.795406
$533$	1.58900e10i	0.196886i
$534$	0	0
$535$	3.76602e10	0.459692
$536$	− 1.18553e10i	− 0.143632i
$537$	0	0
$538$	−1.42567e10	−0.170172
$539$	− 3.70244e10i	− 0.438665i
$540$	0	0
$541$	−7.17904e10	−0.838064	−0.419032	−	0.907972i	$-0.637630\pi$
$541$	−7.17904e10	−0.838064	−0.419032	+	0.907972i	$0.637630\pi$
$542$	− 1.67584e10i	− 0.194194i
$543$	0	0
$544$	1.58280e10	0.180730
$545$	6.30181e10i	0.714298i
$546$	0	0
$547$	−1.28661e11	−1.43714	−0.718568	−	0.695456i	$-0.755202\pi$
$547$	−1.28661e11	−1.43714	−0.718568	+	0.695456i	$0.755202\pi$
$548$	6.50017e10i	0.720780i
$549$	0	0
$550$	6.58157e10	0.719249
$551$	1.35507e11i	1.47013i
$552$	0	0
$553$	8.98658e10	0.960935
$554$	− 7.25448e9i	− 0.0770136i
$555$	0	0
$556$	−1.78669e9	−0.0186960
$557$	− 1.41085e10i	− 0.146575i	−0.997311	−	0.0732874i	$-0.976651\pi$
$557$	− 1.41085e10i	− 0.146575i	0.997311	−	0.0732874i	$-0.0233490\pi$
$558$	0	0
$559$	−1.25365e11	−1.28389
$560$	− 5.62615e10i	− 0.572084i
$561$	0	0
$562$	−1.44438e10	−0.144790
$563$	− 2.80691e10i	− 0.279380i	−0.990195	−	0.139690i	$-0.955389\pi$
$563$	− 2.80691e10i	− 0.279380i	0.990195	−	0.139690i	$-0.0446105\pi$
$564$	0	0
$565$	2.46748e11	2.42136
$566$	1.27725e11i	1.24455i
$567$	0	0
$568$	−9.03616e9	−0.0868141
$569$	− 1.43549e11i	− 1.36946i	−0.728796	−	0.684731i	$-0.759920\pi$
$569$	− 1.43549e11i	− 1.36946i	0.728796	−	0.684731i	$-0.240080\pi$
$570$	0	0
$571$	−6.83821e10	−0.643277	−0.321639	−	0.946863i	$-0.604234\pi$
$571$	−6.83821e10	−0.643277	−0.321639	+	0.946863i	$0.604234\pi$
$572$	− 5.50491e10i	− 0.514240i
$573$	0	0
$574$	1.06675e10	0.0982686
$575$	− 2.59510e11i	− 2.37401i
$576$	0	0
$577$	9.83443e10	0.887250	0.443625	−	0.896213i	$-0.353692\pi$
$577$	9.83443e10	0.887250	0.443625	+	0.896213i	$0.353692\pi$
$578$	3.56979e9i	0.0319839i
$579$	0	0
$580$	−1.19657e11	−1.05737
$581$	− 2.83732e11i	− 2.49003i
$582$	0	0
$583$	−2.94052e10	−0.254536
$584$	4.58563e10i	0.394228i
$585$	0	0
$586$	1.86387e10	0.158061
$587$	3.62220e10i	0.305085i	0.988297	+	0.152542i	$0.0487460\pi$
$587$	3.62220e10i	0.305085i	−0.988297	+	0.152542i	$0.951254\pi$
$588$	0	0
$589$	−2.20981e11	−1.83609
$590$	− 4.20704e10i	− 0.347191i
$591$	0	0
$592$	2.00909e10	0.163573
$593$	− 8.30807e10i	− 0.671864i	−0.941886	−	0.335932i	$-0.890949\pi$
$593$	− 8.30807e10i	− 0.671864i	0.941886	−	0.335932i	$-0.109051\pi$
$594$	0	0
$595$	2.93220e11	2.33951
$596$	3.49201e10i	0.276752i
$597$	0	0
$598$	−2.17058e11	−1.69735
$599$	− 3.03534e10i	− 0.235776i	−0.993027	−	0.117888i	$-0.962388\pi$
$599$	− 3.03534e10i	− 0.235776i	0.993027	−	0.117888i	$-0.0376124\pi$
$600$	0	0
$601$	4.92836e10	0.377750	0.188875	−	0.982001i	$-0.439516\pi$
$601$	4.92836e10	0.377750	0.188875	+	0.982001i	$0.439516\pi$
$602$	8.41615e10i	0.640808i
$603$	0	0
$604$	1.18918e11	0.893512
$605$	1.61496e11i	1.20543i
$606$	0	0
$607$	2.37886e11	1.75233	0.876163	−	0.482015i	$-0.160095\pi$
$607$	2.37886e11	1.75233	0.876163	+	0.482015i	$0.160095\pi$
$608$	2.85341e10i	0.208809i
$609$	0	0
$610$	−5.68433e8	−0.00410545
$611$	− 1.00704e11i	− 0.722573i
$612$	0	0
$613$	1.03907e11	0.735875	0.367938	−	0.929850i	$-0.380064\pi$
$613$	1.03907e11	0.735875	0.367938	+	0.929850i	$0.380064\pi$
$614$	4.55647e10i	0.320594i
$615$	0	0
$616$	−3.69564e10	−0.256665
$617$	− 2.25154e11i	− 1.55360i	−0.629748	−	0.776800i	$-0.716842\pi$
$617$	− 2.25154e11i	− 1.55360i	0.629748	−	0.776800i	$-0.283158\pi$
$618$	0	0
$619$	−2.27847e11	−1.55196	−0.775981	−	0.630756i	$-0.782745\pi$
$619$	−2.27847e11	−1.55196	−0.775981	+	0.630756i	$0.782745\pi$
$620$	− 1.95133e11i	− 1.32058i
$621$	0	0
$622$	2.30449e10	0.153962
$623$	2.72680e11i	1.81009i
$624$	0	0
$625$	1.02819e11	0.673832
$626$	5.96357e10i	0.388337i
$627$	0	0
$628$	−2.04501e10	−0.131479
$629$	1.04708e11i	0.668927i
$630$	0	0
$631$	−1.35217e11	−0.852931	−0.426466	−	0.904504i	$-0.640242\pi$
$631$	−1.35217e11	−0.852931	−0.426466	+	0.904504i	$0.640242\pi$
$632$	− 4.02460e10i	− 0.252264i
$633$	0	0
$634$	1.26293e11	0.781668
$635$	− 4.67848e10i	− 0.287746i
$636$	0	0
$637$	−2.55654e11	−1.55273
$638$	7.85989e10i	0.474388i
$639$	0	0
$640$	−2.51965e10	−0.150183
$641$	− 2.40867e11i	− 1.42674i	−0.700788	−	0.713370i	$-0.747168\pi$
$641$	− 2.40867e11i	− 1.42674i	0.700788	−	0.713370i	$-0.252832\pi$
$642$	0	0
$643$	−3.23996e11	−1.89538	−0.947690	−	0.319193i	$-0.896588\pi$
$643$	−3.23996e11	−1.89538	−0.947690	+	0.319193i	$0.896588\pi$
$644$	1.45718e11i	0.847170i
$645$	0	0
$646$	−1.48712e11	−0.853918
$647$	− 2.57799e11i	− 1.47118i	−0.677429	−	0.735588i	$-0.736906\pi$
$647$	− 2.57799e11i	− 1.47118i	0.677429	−	0.735588i	$-0.263094\pi$
$648$	0	0
$649$	−2.76347e10	−0.155767
$650$	− 4.54459e11i	− 2.54590i
$651$	0	0
$652$	−5.98415e10	−0.331140
$653$	− 2.12821e11i	− 1.17048i	−0.810861	−	0.585238i	$-0.801001\pi$
$653$	− 2.12821e11i	− 1.17048i	0.810861	−	0.585238i	$-0.198999\pi$
$654$	0	0
$655$	1.40393e11	0.762747
$656$	− 4.77739e9i	− 0.0257974i
$657$	0	0
$658$	−6.76060e10	−0.360647
$659$	7.65162e9i	0.0405707i	0.999794	+	0.0202853i	$0.00645746\pi$
$659$	7.65162e9i	0.0405707i	−0.999794	+	0.0202853i	$0.993543\pi$
$660$	0	0
$661$	5.80465e10	0.304068	0.152034	−	0.988375i	$-0.451418\pi$
$661$	5.80465e10	0.304068	0.152034	+	0.988375i	$0.451418\pi$
$662$	− 8.65197e10i	− 0.450488i
$663$	0	0
$664$	−1.27068e11	−0.653680
$665$	5.28604e11i	2.70299i
$666$	0	0
$667$	3.09914e11	1.56580
$668$	− 4.16288e10i	− 0.209068i
$669$	0	0
$670$	−9.83574e10	−0.488099
$671$	3.73385e8i	0.00184190i
$672$	0	0
$673$	2.03908e10	0.0993973	0.0496986	−	0.998764i	$-0.484174\pi$
$673$	2.03908e10	0.0993973	0.0496986	+	0.998764i	$0.484174\pi$
$674$	1.32455e9i	0.00641843i
$675$	0	0
$676$	−2.75702e11	−1.32024
$677$	− 3.03449e11i	− 1.44454i	−0.691609	−	0.722272i	$-0.743098\pi$
$677$	− 3.03449e11i	− 1.44454i	0.691609	−	0.722272i	$-0.256902\pi$
$678$	0	0
$679$	−7.56188e10	−0.355755
$680$	− 1.31317e11i	− 0.614167i
$681$	0	0
$682$	−1.28177e11	−0.592477
$683$	− 2.08506e11i	− 0.958156i	−0.877772	−	0.479078i	$-0.840971\pi$
$683$	− 2.08506e11i	− 0.958156i	0.877772	−	0.479078i	$-0.159029\pi$
$684$	0	0
$685$	5.39287e11	2.44939
$686$	− 3.92701e10i	− 0.177323i
$687$	0	0
$688$	3.76914e10	0.168224
$689$	2.03044e11i	0.900974i
$690$	0	0
$691$	1.91880e10	0.0841624	0.0420812	−	0.999114i	$-0.486601\pi$
$691$	1.91880e10	0.0841624	0.0420812	+	0.999114i	$0.486601\pi$
$692$	− 6.90565e10i	− 0.301148i
$693$	0	0
$694$	−1.74989e11	−0.754351
$695$	1.48232e10i	0.0635337i
$696$	0	0
$697$	2.48985e10	0.105497
$698$	1.89106e11i	0.796679i
$699$	0	0
$700$	−3.05094e11	−1.27070
$701$	− 3.44220e11i	− 1.42549i	−0.701423	−	0.712745i	$-0.747452\pi$
$701$	− 3.44220e11i	− 1.42549i	0.701423	−	0.712745i	$-0.252548\pi$
$702$	0	0
$703$	−1.88764e11	−0.772853
$704$	1.65507e10i	0.0673794i
$705$	0	0
$706$	−1.99857e11	−0.804453
$707$	− 2.30439e11i	− 0.922314i
$708$	0	0
$709$	2.20130e11	0.871153	0.435576	−	0.900152i	$-0.356545\pi$
$709$	2.20130e11	0.871153	0.435576	+	0.900152i	$0.356545\pi$
$710$	7.49685e10i	0.295016i
$711$	0	0
$712$	1.22118e11	0.475183
$713$	5.05398e11i	1.95558i
$714$	0	0
$715$	−4.56715e11	−1.74752
$716$	9.15418e10i	0.348311i
$717$	0	0
$718$	5.31320e10	0.199921
$719$	2.06177e11i	0.771479i	0.922608	+	0.385740i	$0.126054\pi$
$719$	2.06177e11i	0.771479i	−0.922608	+	0.385740i	$0.873946\pi$
$720$	0	0
$721$	5.66587e11	2.09665
$722$	− 7.59444e10i	− 0.279478i
$723$	0	0
$724$	2.58033e11	0.939118
$725$	6.48875e11i	2.34860i
$726$	0	0
$727$	4.32232e11	1.54732	0.773659	−	0.633602i	$-0.218424\pi$
$727$	4.32232e11	1.54732	0.773659	+	0.633602i	$0.218424\pi$
$728$	2.55184e11i	0.908508i
$729$	0	0
$730$	3.80447e11	1.33969
$731$	1.96437e11i	0.687947i
$732$	0	0
$733$	4.82047e11	1.66984	0.834918	−	0.550374i	$-0.185515\pi$
$733$	4.82047e11	1.66984	0.834918	+	0.550374i	$0.185515\pi$
$734$	− 2.99175e11i	− 1.03072i
$735$	0	0
$736$	6.52593e10	0.222398
$737$	6.46077e10i	0.218985i
$738$	0	0
$739$	−5.20187e11	−1.74414	−0.872070	−	0.489380i	$-0.837223\pi$
$739$	−5.20187e11	−1.74414	−0.872070	+	0.489380i	$0.837223\pi$
$740$	− 1.66684e11i	− 0.555862i
$741$	0	0
$742$	1.36310e11	0.449689
$743$	1.34453e11i	0.441178i	0.975367	+	0.220589i	$0.0707980\pi$
$743$	1.34453e11i	0.441178i	−0.975367	+	0.220589i	$0.929202\pi$
$744$	0	0
$745$	2.89715e11	0.940472
$746$	− 1.73378e11i	− 0.559806i
$747$	0	0
$748$	−8.62581e10	−0.275545
$749$	1.14674e11i	0.364365i
$750$	0	0
$751$	1.80147e11	0.566328	0.283164	−	0.959071i	$-0.408616\pi$
$751$	1.80147e11	0.566328	0.283164	+	0.959071i	$0.408616\pi$
$752$	3.02771e10i	0.0946766i
$753$	0	0
$754$	5.42727e11	1.67917
$755$	− 9.86605e11i	− 3.03638i
$756$	0	0
$757$	5.77282e10	0.175794	0.0878971	−	0.996130i	$-0.471985\pi$
$757$	5.77282e10	0.175794	0.0878971	+	0.996130i	$0.471985\pi$
$758$	− 2.13632e11i	− 0.647127i
$759$	0	0
$760$	2.36733e11	0.709586
$761$	− 2.25519e11i	− 0.672425i	−0.941786	−	0.336212i	$-0.890854\pi$
$761$	− 2.25519e11i	− 0.672425i	0.941786	−	0.336212i	$-0.109146\pi$
$762$	0	0
$763$	−1.91888e11	−0.566172
$764$	− 1.31813e11i	− 0.386888i
$765$	0	0
$766$	−4.75165e11	−1.38016
$767$	1.90818e11i	0.551364i
$768$	0	0
$769$	4.70600e11	1.34569	0.672847	−	0.739782i	$-0.265071\pi$
$769$	4.70600e11	1.34569	0.672847	+	0.739782i	$0.265071\pi$
$770$	3.06609e11i	0.872211i
$771$	0	0
$772$	2.33471e11	0.657301
$773$	5.56707e10i	0.155923i	0.996956	+	0.0779613i	$0.0248411\pi$
$773$	5.56707e10i	0.155923i	−0.996956	+	0.0779613i	$0.975159\pi$
$774$	0	0
$775$	−1.05817e12	−2.93323
$776$	3.38655e10i	0.0933923i
$777$	0	0
$778$	−2.58980e11	−0.706883
$779$	4.48859e10i	0.121888i
$780$	0	0
$781$	4.92443e10	0.132359
$782$	3.40114e11i	0.909489i
$783$	0	0
$784$	7.68635e10	0.203449
$785$	1.69664e11i	0.446798i
$786$	0	0
$787$	−1.83736e11	−0.478956	−0.239478	−	0.970902i	$-0.576976\pi$
$787$	−1.83736e11	−0.478956	−0.239478	+	0.970902i	$0.576976\pi$
$788$	− 1.55767e11i	− 0.403991i
$789$	0	0
$790$	−3.33901e11	−0.857254
$791$	7.51337e11i	1.91924i
$792$	0	0
$793$	2.57823e9	0.00651973
$794$	3.64501e11i	0.917100i
$795$	0	0
$796$	−2.19936e11	−0.547829
$797$	− 5.78800e11i	− 1.43448i	−0.696826	−	0.717241i	$-0.745405\pi$
$797$	− 5.78800e11i	− 1.43448i	0.696826	−	0.717241i	$-0.254595\pi$
$798$	0	0
$799$	−1.57796e11	−0.387176
$800$	1.36635e11i	0.333582i
$801$	0	0
$802$	3.83260e11	0.926394
$803$	− 2.49903e11i	− 0.601049i
$804$	0	0
$805$	1.20895e12	2.87889
$806$	8.85063e11i	2.09717i
$807$	0	0
$808$	−1.03201e11	−0.242125
$809$	2.72068e10i	0.0635161i	0.999496	+	0.0317581i	$0.0101106\pi$
$809$	2.72068e10i	0.0635161i	−0.999496	+	0.0317581i	$0.989889\pi$
$810$	0	0
$811$	2.76712e11	0.639653	0.319827	−	0.947476i	$-0.396375\pi$
$811$	2.76712e11	0.639653	0.319827	+	0.947476i	$0.396375\pi$
$812$	− 3.64351e11i	− 0.838100i
$813$	0	0
$814$	−1.09489e11	−0.249387
$815$	4.96476e11i	1.12530i
$816$	0	0
$817$	−3.54129e11	−0.794828
$818$	3.71556e11i	0.829872i
$819$	0	0
$820$	−3.96356e10	−0.0876658
$821$	− 3.61935e11i	− 0.796632i	−0.917248	−	0.398316i	$-0.869595\pi$
$821$	− 3.61935e11i	− 0.796632i	0.917248	−	0.398316i	$-0.130405\pi$
$822$	0	0
$823$	−3.97144e11	−0.865663	−0.432831	−	0.901475i	$-0.642485\pi$
$823$	−3.97144e11	−0.865663	−0.432831	+	0.901475i	$0.642485\pi$
$824$	− 2.53744e11i	− 0.550410i
$825$	0	0
$826$	1.28103e11	0.275193
$827$	− 4.91891e11i	− 1.05159i	−0.850611	−	0.525796i	$-0.823768\pi$
$827$	− 4.91891e11i	− 1.05159i	0.850611	−	0.525796i	$-0.176232\pi$
$828$	0	0
$829$	−6.55696e11	−1.38830	−0.694152	−	0.719828i	$-0.744220\pi$
$829$	−6.55696e11	−1.38830	−0.694152	+	0.719828i	$0.744220\pi$
$830$	1.05422e12i	2.22137i
$831$	0	0
$832$	1.14283e11	0.238500
$833$	4.00592e11i	0.831998i
$834$	0	0
$835$	−3.45373e11	−0.710466
$836$	− 1.55502e11i	− 0.318355i
$837$	0	0
$838$	−4.76092e11	−0.965417
$839$	− 6.10795e11i	− 1.23267i	−0.787483	−	0.616336i	$-0.788616\pi$
$839$	− 6.10795e11i	− 1.23267i	0.787483	−	0.616336i	$-0.211384\pi$
$840$	0	0
$841$	−2.74656e11	−0.549042
$842$	− 2.45025e11i	− 0.487487i
$843$	0	0
$844$	7.91925e10	0.156068
$845$	2.28736e12i	4.48650i
$846$	0	0
$847$	−4.91750e11	−0.955456
$848$	− 6.10459e10i	− 0.118052i
$849$	0	0
$850$	−7.12105e11	−1.36417
$851$	4.31714e11i	0.823148i
$852$	0	0
$853$	4.05108e11	0.765199	0.382599	−	0.923914i	$-0.375029\pi$
$853$	4.05108e11	0.765199	0.382599	+	0.923914i	$0.375029\pi$
$854$	− 1.73086e9i	− 0.00325409i
$855$	0	0
$856$	5.13561e10	0.0956527
$857$	− 2.10370e10i	− 0.0389996i	−0.999810	−	0.0194998i	$-0.993793\pi$
$857$	− 2.10370e10i	− 0.0389996i	0.999810	−	0.0194998i	$-0.00620737\pi$
$858$	0	0
$859$	2.43542e10	0.0447302	0.0223651	−	0.999750i	$-0.492880\pi$
$859$	2.43542e10	0.0447302	0.0223651	+	0.999750i	$0.492880\pi$
$860$	− 3.12707e11i	− 0.571667i
$861$	0	0
$862$	6.55277e10	0.118685
$863$	− 6.16048e11i	− 1.11064i	−0.831638	−	0.555318i	$-0.812597\pi$
$863$	− 6.16048e11i	− 1.11064i	0.831638	−	0.555318i	$-0.187403\pi$
$864$	0	0
$865$	−5.72928e11	−1.02338
$866$	2.87777e11i	0.511663i
$867$	0	0
$868$	5.94173e11	1.04673
$869$	2.19329e11i	0.384606i
$870$	0	0
$871$	4.46118e11	0.775134
$872$	8.59360e10i	0.148631i
$873$	0	0
$874$	−6.13142e11	−1.05079
$875$	1.18983e12i	2.02980i
$876$	0	0
$877$	−1.01983e12	−1.72397	−0.861987	−	0.506930i	$-0.830780\pi$
$877$	−1.01983e12	−1.72397	−0.861987	+	0.506930i	$0.830780\pi$
$878$	2.15658e11i	0.362901i
$879$	0	0
$880$	1.37313e11	0.228972
$881$	1.04617e12i	1.73659i	0.496050	+	0.868294i	$0.334783\pi$
$881$	1.04617e12i	1.73659i	−0.496050	+	0.868294i	$0.665217\pi$
$882$	0	0
$883$	−1.11194e11	−0.182911	−0.0914554	−	0.995809i	$-0.529152\pi$
$883$	−1.11194e11	−0.182911	−0.0914554	+	0.995809i	$0.529152\pi$
$884$	5.95614e11i	0.975339i
$885$	0	0
$886$	2.18338e11	0.354319
$887$	7.81949e10i	0.126324i	0.998003	+	0.0631618i	$0.0201184\pi$
$887$	7.81949e10i	0.126324i	−0.998003	+	0.0631618i	$0.979882\pi$
$888$	0	0
$889$	1.42458e11	0.228076
$890$	− 1.01316e12i	− 1.61479i
$891$	0	0
$892$	−4.80492e11	−0.758973
$893$	− 2.84468e11i	− 0.447329i
$894$	0	0
$895$	7.59477e11	1.18365
$896$	− 7.67223e10i	− 0.119039i
$897$	0	0
$898$	2.90010e10	0.0445973
$899$	− 1.26369e12i	− 1.93464i
$900$	0	0
$901$	3.18155e11	0.482769
$902$	2.60353e10i	0.0393312i
$903$	0	0
$904$	3.36483e11	0.503836
$905$	− 2.14077e12i	− 3.19136i
$906$	0	0
$907$	−4.43060e11	−0.654686	−0.327343	−	0.944906i	$-0.606153\pi$
$907$	−4.43060e11	−0.654686	−0.327343	+	0.944906i	$0.606153\pi$
$908$	− 5.50613e11i	− 0.810033i
$909$	0	0
$910$	2.11714e12	3.08733
$911$	1.14326e12i	1.65986i	0.557866	+	0.829931i	$0.311620\pi$
$911$	1.14326e12i	1.65986i	−0.557866	+	0.829931i	$0.688380\pi$
$912$	0	0
$913$	6.92484e11	0.996614
$914$	− 3.65113e11i	− 0.523170i
$915$	0	0
$916$	−5.16644e11	−0.733854
$917$	4.27491e11i	0.604575i
$918$	0	0
$919$	1.85438e11	0.259978	0.129989	−	0.991515i	$-0.458506\pi$
$919$	1.85438e11	0.259978	0.129989	+	0.991515i	$0.458506\pi$
$920$	− 5.41424e11i	− 0.755764i
$921$	0	0
$922$	7.77882e11	1.07644
$923$	− 3.40033e11i	− 0.468505i
$924$	0	0
$925$	−9.03891e11	−1.23467
$926$	− 1.04844e11i	− 0.142593i
$927$	0	0
$928$	−1.63173e11	−0.220017
$929$	6.47121e10i	0.0868806i	0.999056	+	0.0434403i	$0.0138318\pi$
$929$	6.47121e10i	0.0868806i	−0.999056	+	0.0434403i	$0.986168\pi$
$930$	0	0
$931$	−7.22170e11	−0.961260
$932$	− 1.47235e11i	− 0.195141i
$933$	0	0
$934$	5.81595e11	0.764247
$935$	7.15640e11i	0.936372i
$936$	0	0
$937$	8.18788e11	1.06222	0.531108	−	0.847304i	$-0.321776\pi$
$937$	8.18788e11	1.06222	0.531108	+	0.847304i	$0.321776\pi$
$938$	− 2.99494e11i	− 0.386881i
$939$	0	0
$940$	2.51194e11	0.321735
$941$	8.70441e11i	1.11015i	0.831801	+	0.555074i	$0.187310\pi$
$941$	8.70441e11i	1.11015i	−0.831801	+	0.555074i	$0.812690\pi$
$942$	0	0
$943$	1.02657e11	0.129820
$944$	− 5.73703e10i	− 0.0722435i
$945$	0	0
$946$	−2.05407e11	−0.256478
$947$	− 1.48793e12i	− 1.85005i	−0.379906	−	0.925025i	$-0.624044\pi$
$947$	− 1.48793e12i	− 1.85005i	0.379906	−	0.925025i	$-0.375956\pi$
$948$	0	0
$949$	−1.72559e12	−2.12751
$950$	− 1.28375e12i	− 1.57611i
$951$	0	0
$952$	3.99856e11	0.486806
$953$	− 1.64141e12i	− 1.98997i	−0.100019	−	0.994986i	$-0.531890\pi$
$953$	− 1.64141e12i	− 1.98997i	0.100019	−	0.994986i	$-0.468110\pi$
$954$	0	0
$955$	−1.09359e12	−1.31474
$956$	4.60827e11i	0.551705i
$957$	0	0
$958$	7.99094e11	0.948715
$959$	1.64211e12i	1.94145i
$960$	0	0
$961$	1.20789e12	1.41624
$962$	7.56026e11i	0.882747i
$963$	0	0
$964$	4.55366e10	0.0527294
$965$	− 1.93700e12i	− 2.23367i
$966$	0	0
$967$	−7.74230e11	−0.885450	−0.442725	−	0.896657i	$-0.645988\pi$
$967$	−7.74230e11	−0.885450	−0.442725	+	0.896657i	$0.645988\pi$
$968$	2.20228e11i	0.250825i
$969$	0	0
$970$	2.80966e11	0.317370
$971$	− 6.61550e11i	− 0.744194i	−0.928194	−	0.372097i	$-0.878639\pi$
$971$	− 6.61550e11i	− 0.744194i	0.928194	−	0.372097i	$-0.121361\pi$
$972$	0	0
$973$	−4.51362e10	−0.0503585
$974$	5.22368e11i	0.580418i
$975$	0	0
$976$	−7.75157e8	−0.000854260 0
$977$	− 3.68827e11i	− 0.404803i	−0.979303	−	0.202402i	$-0.935125\pi$
$977$	− 3.68827e11i	− 0.404803i	0.979303	−	0.202402i	$-0.0648746\pi$
$978$	0	0
$979$	−6.65509e11	−0.724475
$980$	− 6.37699e11i	− 0.691371i
$981$	0	0
$982$	4.37347e11	0.470305
$983$	− 6.01419e9i	− 0.00644114i	−0.999995	−	0.00322057i	$-0.998975\pi$
$983$	− 6.01419e9i	− 0.00644114i	0.999995	−	0.00322057i	$-0.00102514\pi$
$984$	0	0
$985$	−1.29232e12	−1.37286
$986$	− 8.50414e11i	− 0.899752i
$987$	0	0
$988$	−1.07375e12	−1.12687
$989$	8.09915e11i	0.846553i
$990$	0	0
$991$	1.32877e12	1.37770	0.688851	−	0.724903i	$-0.258115\pi$
$991$	1.32877e12	1.37770	0.688851	+	0.724903i	$0.258115\pi$
$992$	− 2.66098e11i	− 0.274786i
$993$	0	0
$994$	−2.28276e11	−0.233838
$995$	1.82470e12i	1.86166i
$996$	0	0
$997$	5.76153e11	0.583119	0.291559	−	0.956553i	$-0.405826\pi$
$997$	5.76153e11	0.583119	0.291559	+	0.956553i	$0.405826\pi$
$998$	− 1.34502e11i	− 0.135584i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.9.b.a.161.4	✓	8
3.2	odd	2	inner	162.9.b.a.161.5	yes	8
9.2	odd	6		162.9.d.h.53.8		16
9.4	even	3		162.9.d.h.107.8		16
9.5	odd	6		162.9.d.h.107.1		16
9.7	even	3		162.9.d.h.53.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.9.b.a.161.4	✓	8	1.1	even	1	trivial
162.9.b.a.161.5	yes	8	3.2	odd	2	inner
162.9.d.h.53.1		16	9.7	even	3
162.9.d.h.53.8		16	9.2	odd	6
162.9.d.h.107.1		16	9.5	odd	6
162.9.d.h.107.8		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)

\(f(q)\)	\(=\)	\(q-11.3137i q^{2} -128.000 q^{4} +1061.95i q^{5} -3233.60 q^{7} +1448.15i q^{8} +O(q^{10})\)	\(q-11.3137i q^{2} -128.000 q^{4} +1061.95i q^{5} -3233.60 q^{7} +1448.15i q^{8} +12014.6 q^{10} -7892.01i q^{11} -54494.5 q^{13} +36584.0i q^{14} +16384.0 q^{16} +85389.0i q^{17} -153936. q^{19} -135930. i q^{20} -89287.9 q^{22} +352061. i q^{23} -737118. q^{25} +616535. i q^{26} +413901. q^{28} -880286. i q^{29} +1.43554e6 q^{31} -185364. i q^{32} +966067. q^{34} -3.43393e6i q^{35} +1.22625e6 q^{37} +1.74158e6i q^{38} -1.53787e6 q^{40} -291589. i q^{41} +2.30050e6 q^{43} +1.01018e6i q^{44} +3.98311e6 q^{46} +1.84797e6i q^{47} +4.69138e6 q^{49} +8.33954e6i q^{50} +6.97530e6 q^{52} -3.72595e6i q^{53} +8.38094e6 q^{55} -4.68276e6i q^{56} -9.95929e6 q^{58} -3.50160e6i q^{59} -47311.8 q^{61} -1.62413e7i q^{62} -2.09715e6 q^{64} -5.78706e7i q^{65} -8.18647e6 q^{67} -1.09298e7i q^{68} -3.88505e7 q^{70} +6.23977e6i q^{71} +3.16653e7 q^{73} -1.38734e7i q^{74} +1.97038e7 q^{76} +2.55196e7i q^{77} -2.77912e7 q^{79} +1.73990e7i q^{80} -3.29895e6 q^{82} +8.77450e7i q^{83} -9.06791e7 q^{85} -2.60272e7i q^{86} +1.14289e7 q^{88} -8.43269e7i q^{89} +1.76214e8 q^{91} -4.50637e7i q^{92} +2.09074e7 q^{94} -1.63472e8i q^{95} +2.33853e7 q^{97} -5.30769e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 1024 q^{4} - 8876 q^{7}+O(q^{10}) \) 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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