LMFDB - Embedded newform 162.9.b.a.161.7

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			161.7
Root			$25.8318i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.9.b.a.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137i q^{2} -128.000 q^{4} +618.080i q^{5} -549.440 q^{7} -1448.15i q^{8} +O(q^{10})$	\(q+11.3137i q^{2} -128.000 q^{4} +618.080i q^{5} -549.440 q^{7} -1448.15i q^{8} -6992.78 q^{10} -19275.3i q^{11} +2083.78 q^{13} -6216.21i q^{14} +16384.0 q^{16} +58190.7i q^{17} +202758. q^{19} -79114.2i q^{20} +218075. q^{22} -69096.5i q^{23} +8602.17 q^{25} +23575.3i q^{26} +70328.3 q^{28} -362035. i q^{29} +54553.0 q^{31} +185364. i q^{32} -658352. q^{34} -339598. i q^{35} +632925. q^{37} +2.29395e6i q^{38} +895075. q^{40} -5.05168e6i q^{41} +1.99703e6 q^{43} +2.46724e6i q^{44} +781737. q^{46} +7.04440e6i q^{47} -5.46292e6 q^{49} +97322.5i q^{50} -266724. q^{52} +4.75974e6i q^{53} +1.19137e7 q^{55} +795674. i q^{56} +4.09595e6 q^{58} +6.52082e6i q^{59} +9.32433e6 q^{61} +617197. i q^{62} -2.09715e6 q^{64} +1.28794e6i q^{65} +1.17800e7 q^{67} -7.44840e6i q^{68} +3.84211e6 q^{70} +3.81048e7i q^{71} +5.36350e7 q^{73} +7.16073e6i q^{74} -2.59531e7 q^{76} +1.05906e7i q^{77} -5.15683e7 q^{79} +1.01266e7i q^{80} +5.71532e7 q^{82} +3.30371e7i q^{83} -3.59665e7 q^{85} +2.25938e7i q^{86} -2.79136e7 q^{88} -1.36524e7i q^{89} -1.14491e6 q^{91} +8.84435e6i q^{92} -7.96983e7 q^{94} +1.25321e8i q^{95} -1.10521e8 q^{97} -6.18058e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	618.080i	0.988928i	0.869198	+	0.494464i	$0.164636\pi$
$5$	618.080i	0.988928i	−0.869198	+	0.494464i	$0.835364\pi$
$6$	0	0
$7$	−549.440	−0.228838	−0.114419	−	0.993433i	$-0.536501\pi$
$7$	−549.440	−0.228838	−0.114419	+	0.993433i	$0.536501\pi$
$8$	− 1448.15i	− 0.353553i
$9$	0	0
$10$	−6992.78	−0.699278
$11$	− 19275.3i	− 1.31653i	−0.752787	−	0.658264i	$-0.771291\pi$
$11$	− 19275.3i	− 1.31653i	0.752787	−	0.658264i	$-0.228709\pi$
$12$	0	0
$13$	2083.78	0.0729589	0.0364795	−	0.999334i	$-0.488386\pi$
$13$	2083.78	0.0729589	0.0364795	+	0.999334i	$0.488386\pi$
$14$	− 6216.21i	− 0.161813i
$15$	0	0
$16$	16384.0	0.250000
$17$	58190.7i	0.696719i	0.937361	+	0.348359i	$0.113261\pi$
$17$	58190.7i	0.696719i	−0.937361	+	0.348359i	$0.886739\pi$
$18$	0	0
$19$	202758.	1.55584	0.777919	−	0.628364i	$-0.216275\pi$
$19$	202758.	1.55584	0.777919	+	0.628364i	$0.216275\pi$
$20$	− 79114.2i	− 0.494464i
$21$	0	0
$22$	218075.	0.930927
$23$	− 69096.5i	− 0.246913i	−0.992350	−	0.123457i	$-0.960602\pi$
$23$	− 69096.5i	− 0.246913i	0.992350	−	0.123457i	$-0.0393980\pi$
$24$	0	0
$25$	8602.17	0.0220216
$26$	23575.3i	0.0515898i
$27$	0	0
$28$	70328.3	0.114419
$29$	− 362035.i	− 0.511868i	−0.966694	−	0.255934i	$-0.917617\pi$
$29$	− 362035.i	− 0.511868i	0.966694	−	0.255934i	$-0.0823830\pi$
$30$	0	0
$31$	54553.0	0.0590707	0.0295353	−	0.999564i	$-0.490597\pi$
$31$	54553.0	0.0590707	0.0295353	+	0.999564i	$0.490597\pi$
$32$	185364.i	0.176777i
$33$	0	0
$34$	−658352.	−0.492655
$35$	− 339598.i	− 0.226304i
$36$	0	0
$37$	632925.	0.337711	0.168856	−	0.985641i	$-0.445993\pi$
$37$	632925.	0.337711	0.168856	+	0.985641i	$0.445993\pi$
$38$	2.29395e6i	1.10014i
$39$	0	0
$40$	895075.	0.349639
$41$	− 5.05168e6i	− 1.78772i	−0.448344	−	0.893861i	$-0.647986\pi$
$41$	− 5.05168e6i	− 1.78772i	0.448344	−	0.893861i	$-0.352014\pi$
$42$	0	0
$43$	1.99703e6	0.584131	0.292065	−	0.956398i	$-0.405658\pi$
$43$	1.99703e6	0.584131	0.292065	+	0.956398i	$0.405658\pi$
$44$	2.46724e6i	0.658264i
$45$	0	0
$46$	781737.	0.174594
$47$	7.04440e6i	1.44362i	0.692092	+	0.721810i	$0.256689\pi$
$47$	7.04440e6i	1.44362i	−0.692092	+	0.721810i	$0.743311\pi$
$48$	0	0
$49$	−5.46292e6	−0.947633
$50$	97322.5i	0.0155716i
$51$	0	0
$52$	−266724.	−0.0364795
$53$	4.75974e6i	0.603225i	0.953431	+	0.301613i	$0.0975249\pi$
$53$	4.75974e6i	0.603225i	−0.953431	+	0.301613i	$0.902475\pi$
$54$	0	0
$55$	1.19137e7	1.30195
$56$	795674.i	0.0809065i
$57$	0	0
$58$	4.09595e6	0.361945
$59$	6.52082e6i	0.538139i	0.963121	+	0.269069i	$0.0867161\pi$
$59$	6.52082e6i	0.538139i	−0.963121	+	0.269069i	$0.913284\pi$
$60$	0	0
$61$	9.32433e6	0.673439	0.336720	−	0.941605i	$-0.390683\pi$
$61$	9.32433e6	0.673439	0.336720	+	0.941605i	$0.390683\pi$
$62$	617197.i	0.0417693i
$63$	0	0
$64$	−2.09715e6	−0.125000
$65$	1.28794e6i	0.0721511i
$66$	0	0
$67$	1.17800e7	0.584583	0.292292	−	0.956329i	$-0.405582\pi$
$67$	1.17800e7	0.584583	0.292292	+	0.956329i	$0.405582\pi$
$68$	− 7.44840e6i	− 0.348359i
$69$	0	0
$70$	3.84211e6	0.160021
$71$	3.81048e7i	1.49950i	0.661721	+	0.749750i	$0.269826\pi$
$71$	3.81048e7i	1.49950i	−0.661721	+	0.749750i	$0.730174\pi$
$72$	0	0
$73$	5.36350e7	1.88867	0.944336	−	0.328983i	$-0.106706\pi$
$73$	5.36350e7	1.88867	0.944336	+	0.328983i	$0.106706\pi$
$74$	7.16073e6i	0.238798i
$75$	0	0
$76$	−2.59531e7	−0.777919
$77$	1.05906e7i	0.301272i
$78$	0	0
$79$	−5.15683e7	−1.32396	−0.661979	−	0.749522i	$-0.730283\pi$
$79$	−5.15683e7	−1.32396	−0.661979	+	0.749522i	$0.730283\pi$
$80$	1.01266e7i	0.247232i
$81$	0	0
$82$	5.71532e7	1.26411
$83$	3.30371e7i	0.696129i	0.937471	+	0.348065i	$0.113161\pi$
$83$	3.30371e7i	0.696129i	−0.937471	+	0.348065i	$0.886839\pi$
$84$	0	0
$85$	−3.59665e7	−0.689005
$86$	2.25938e7i	0.413043i
$87$	0	0
$88$	−2.79136e7	−0.465463
$89$	− 1.36524e7i	− 0.217595i	−0.994064	−	0.108798i	$-0.965300\pi$
$89$	− 1.36524e7i	− 0.217595i	0.994064	−	0.108798i	$-0.0347001\pi$
$90$	0	0
$91$	−1.14491e6	−0.0166958
$92$	8.84435e6i	0.123457i
$93$	0	0
$94$	−7.96983e7	−1.02079
$95$	1.25321e8i	1.53861i
$96$	0	0
$97$	−1.10521e8	−1.24841	−0.624206	−	0.781260i	$-0.714577\pi$
$97$	−1.10521e8	−1.24841	−0.624206	+	0.781260i	$0.714577\pi$
$98$	− 6.18058e7i	− 0.670078i
$99$	0	0
$100$	−1.10108e6	−0.0110108
$101$	4.09854e7i	0.393862i	0.980417	+	0.196931i	$0.0630975\pi$
$101$	4.09854e7i	0.393862i	−0.980417	+	0.196931i	$0.936903\pi$
$102$	0	0
$103$	−1.25291e8	−1.11320	−0.556599	−	0.830781i	$-0.687894\pi$
$103$	−1.25291e8	−1.11320	−0.556599	+	0.830781i	$0.687894\pi$
$104$	− 3.01764e6i	− 0.0257949i
$105$	0	0
$106$	−5.38503e7	−0.426545
$107$	1.39381e8i	1.06333i	0.846955	+	0.531665i	$0.178434\pi$
$107$	1.39381e8i	1.06333i	−0.846955	+	0.531665i	$0.821566\pi$
$108$	0	0
$109$	2.70252e8	1.91453	0.957267	−	0.289205i	$-0.0933909\pi$
$109$	2.70252e8	1.91453	0.957267	+	0.289205i	$0.0933909\pi$
$110$	1.34788e8i	0.920619i
$111$	0	0
$112$	−9.00203e6	−0.0572095
$113$	2.18515e8i	1.34019i	0.742274	+	0.670096i	$0.233747\pi$
$113$	2.18515e8i	1.34019i	−0.742274	+	0.670096i	$0.766253\pi$
$114$	0	0
$115$	4.27071e7	0.244179
$116$	4.63404e7i	0.255934i
$117$	0	0
$118$	−7.37747e7	−0.380522
$119$	− 3.19723e7i	− 0.159436i
$120$	0	0
$121$	−1.57178e8	−0.733248
$122$	1.05493e8i	0.476194i
$123$	0	0
$124$	−6.98279e6	−0.0295353
$125$	2.46754e8i	1.01071i
$126$	0	0
$127$	3.19550e8	1.22835	0.614177	−	0.789168i	$-0.289488\pi$
$127$	3.19550e8	1.22835	0.614177	+	0.789168i	$0.289488\pi$
$128$	− 2.37266e7i	− 0.0883883i
$129$	0	0
$130$	−1.45714e7	−0.0510186
$131$	3.79019e8i	1.28699i	0.765449	+	0.643496i	$0.222517\pi$
$131$	3.79019e8i	1.28699i	−0.765449	+	0.643496i	$0.777483\pi$
$132$	0	0
$133$	−1.11404e8	−0.356035
$134$	1.33276e8i	0.413363i
$135$	0	0
$136$	8.42691e7	0.246327
$137$	− 3.40543e8i	− 0.966696i	−0.875428	−	0.483348i	$-0.839421\pi$
$137$	− 3.40543e8i	− 0.966696i	0.875428	−	0.483348i	$-0.160579\pi$
$138$	0	0
$139$	−2.61367e8	−0.700150	−0.350075	−	0.936722i	$-0.613844\pi$
$139$	−2.61367e8	−0.700150	−0.350075	+	0.936722i	$0.613844\pi$
$140$	4.34685e7i	0.113152i
$141$	0	0
$142$	−4.31107e8	−1.06031
$143$	− 4.01655e7i	− 0.0960526i
$144$	0	0
$145$	2.23766e8	0.506201
$146$	6.06810e8i	1.33549i
$147$	0	0
$148$	−8.10144e7	−0.168856
$149$	− 6.17740e8i	− 1.25332i	−0.779294	−	0.626658i	$-0.784422\pi$
$149$	− 6.17740e8i	− 1.25332i	0.779294	−	0.626658i	$-0.215578\pi$
$150$	0	0
$151$	8.53219e8	1.64117	0.820583	−	0.571527i	$-0.193649\pi$
$151$	8.53219e8	1.64117	0.820583	+	0.571527i	$0.193649\pi$
$152$	− 2.93626e8i	− 0.550072i
$153$	0	0
$154$	−1.19819e8	−0.213031
$155$	3.37181e7i	0.0584167i
$156$	0	0
$157$	1.09170e9	1.79682	0.898410	−	0.439157i	$-0.144723\pi$
$157$	1.09170e9	1.79682	0.898410	+	0.439157i	$0.144723\pi$
$158$	− 5.83429e8i	− 0.936180i
$159$	0	0
$160$	−1.14570e8	−0.174819
$161$	3.79644e7i	0.0565032i
$162$	0	0
$163$	2.61300e8	0.370159	0.185080	−	0.982724i	$-0.440746\pi$
$163$	2.61300e8	0.370159	0.185080	+	0.982724i	$0.440746\pi$
$164$	6.46614e8i	0.893861i
$165$	0	0
$166$	−3.73772e8	−0.492238
$167$	− 4.78413e8i	− 0.615088i	−0.951534	−	0.307544i	$-0.900493\pi$
$167$	− 4.78413e8i	− 0.615088i	0.951534	−	0.307544i	$-0.0995070\pi$
$168$	0	0
$169$	−8.11389e8	−0.994677
$170$	− 4.06914e8i	− 0.487200i
$171$	0	0
$172$	−2.55619e8	−0.292065
$173$	− 5.57282e7i	− 0.0622144i	−0.999516	−	0.0311072i	$-0.990097\pi$
$173$	− 5.57282e7i	− 0.0622144i	0.999516	−	0.0311072i	$-0.00990332\pi$
$174$	0	0
$175$	−4.72638e6	−0.00503937
$176$	− 3.15807e8i	− 0.329132i
$177$	0	0
$178$	1.54459e8	0.153863
$179$	9.47029e8i	0.922468i	0.887279	+	0.461234i	$0.152593\pi$
$179$	9.47029e8i	0.922468i	−0.887279	+	0.461234i	$0.847407\pi$
$180$	0	0
$181$	−9.11578e7	−0.0849336	−0.0424668	−	0.999098i	$-0.513522\pi$
$181$	−9.11578e7	−0.0849336	−0.0424668	+	0.999098i	$0.513522\pi$
$182$	− 1.29532e7i	− 0.0118057i
$183$	0	0
$184$	−1.00062e8	−0.0872970
$185$	3.91198e8i	0.333972i
$186$	0	0
$187$	1.12164e9	0.917250
$188$	− 9.01683e8i	− 0.721810i
$189$	0	0
$190$	−1.41784e9	−1.08796
$191$	− 1.44530e9i	− 1.08599i	−0.839737	−	0.542993i	$-0.817291\pi$
$191$	− 1.44530e9i	− 1.08599i	0.839737	−	0.542993i	$-0.182709\pi$
$192$	0	0
$193$	2.13739e9	1.54048	0.770238	−	0.637757i	$-0.220138\pi$
$193$	2.13739e9	1.54048	0.770238	+	0.637757i	$0.220138\pi$
$194$	− 1.25040e9i	− 0.882760i
$195$	0	0
$196$	6.99253e8	0.473817
$197$	− 1.87499e7i	− 0.0124490i	−0.999981	−	0.00622450i	$-0.998019\pi$
$197$	− 1.87499e7i	− 0.0124490i	0.999981	−	0.00622450i	$-0.00198133\pi$
$198$	0	0
$199$	−2.25293e9	−1.43660	−0.718298	−	0.695736i	$-0.755078\pi$
$199$	−2.25293e9	−1.43660	−0.718298	+	0.695736i	$0.755078\pi$
$200$	− 1.24573e7i	− 0.00778580i
$201$	0	0
$202$	−4.63697e8	−0.278502
$203$	1.98916e8i	0.117135i
$204$	0	0
$205$	3.12234e9	1.76793
$206$	− 1.41751e9i	− 0.787150i
$207$	0	0
$208$	3.41407e7	0.0182397
$209$	− 3.90823e9i	− 2.04831i
$210$	0	0
$211$	3.05079e9	1.53916	0.769578	−	0.638553i	$-0.220467\pi$
$211$	3.05079e9	1.53916	0.769578	+	0.638553i	$0.220467\pi$
$212$	− 6.09246e8i	− 0.301613i
$213$	0	0
$214$	−1.57691e9	−0.751888
$215$	1.23432e9i	0.577663i
$216$	0	0
$217$	−2.99736e7	−0.0135176
$218$	3.05755e9i	1.35378i
$219$	0	0
$220$	−1.52495e9	−0.650976
$221$	1.21257e8i	0.0508319i
$222$	0	0
$223$	1.83878e9	0.743551	0.371776	−	0.928323i	$-0.378749\pi$
$223$	1.83878e9	0.743551	0.371776	+	0.928323i	$0.378749\pi$
$224$	− 1.01846e8i	− 0.0404532i
$225$	0	0
$226$	−2.47221e9	−0.947659
$227$	− 2.46535e9i	− 0.928484i	−0.885708	−	0.464242i	$-0.846327\pi$
$227$	− 2.46535e9i	− 0.928484i	0.885708	−	0.464242i	$-0.153673\pi$
$228$	0	0
$229$	−8.15259e8	−0.296452	−0.148226	−	0.988954i	$-0.547356\pi$
$229$	−8.15259e8	−0.296452	−0.148226	+	0.988954i	$0.547356\pi$
$230$	4.83176e8i	0.172661i
$231$	0	0
$232$	−5.24282e8	−0.180973
$233$	− 4.78516e9i	− 1.62358i	−0.583951	−	0.811789i	$-0.698494\pi$
$233$	− 4.78516e9i	− 1.62358i	0.583951	−	0.811789i	$-0.301506\pi$
$234$	0	0
$235$	−4.35400e9	−1.42764
$236$	− 8.34665e8i	− 0.269069i
$237$	0	0
$238$	3.61725e8	0.112738
$239$	1.78320e9i	0.546524i	0.961940	+	0.273262i	$0.0881025\pi$
$239$	1.78320e9i	0.546524i	−0.961940	+	0.273262i	$0.911897\pi$
$240$	0	0
$241$	−3.23819e9	−0.959918	−0.479959	−	0.877291i	$-0.659348\pi$
$241$	−3.23819e9	−0.959918	−0.479959	+	0.877291i	$0.659348\pi$
$242$	− 1.77827e9i	− 0.518485i
$243$	0	0
$244$	−1.19351e9	−0.336720
$245$	− 3.37652e9i	− 0.937141i
$246$	0	0
$247$	4.22504e8	0.113512
$248$	− 7.90012e7i	− 0.0208846i
$249$	0	0
$250$	−2.79171e9	−0.714677
$251$	− 2.70554e9i	− 0.681646i	−0.940127	−	0.340823i	$-0.889294\pi$
$251$	− 2.70554e9i	− 0.681646i	0.940127	−	0.340823i	$-0.110706\pi$
$252$	0	0
$253$	−1.33185e9	−0.325068
$254$	3.61529e9i	0.868578i
$255$	0	0
$256$	2.68435e8	0.0625000
$257$	− 1.98000e9i	− 0.453872i	−0.973910	−	0.226936i	$-0.927129\pi$
$257$	− 1.98000e9i	− 0.453872i	0.973910	−	0.226936i	$-0.0728708\pi$
$258$	0	0
$259$	−3.47755e8	−0.0772812
$260$	− 1.64857e8i	− 0.0360756i
$261$	0	0
$262$	−4.28811e9	−0.910041
$263$	− 9.63237e8i	− 0.201331i	−0.994920	−	0.100665i	$-0.967903\pi$
$263$	− 9.63237e8i	− 0.201331i	0.994920	−	0.100665i	$-0.0320972\pi$
$264$	0	0
$265$	−2.94190e9	−0.596546
$266$	− 1.26039e9i	− 0.251755i
$267$	0	0
$268$	−1.50784e9	−0.292292
$269$	6.15308e9i	1.17512i	0.809179	+	0.587562i	$0.199912\pi$
$269$	6.15308e9i	1.17512i	−0.809179	+	0.587562i	$0.800088\pi$
$270$	0	0
$271$	8.15716e9	1.51238	0.756191	−	0.654350i	$-0.227058\pi$
$271$	8.15716e9	1.51238	0.756191	+	0.654350i	$0.227058\pi$
$272$	9.53396e8i	0.174180i
$273$	0	0
$274$	3.85281e9	0.683558
$275$	− 1.65809e8i	− 0.0289920i
$276$	0	0
$277$	7.58355e9	1.28811	0.644056	−	0.764978i	$-0.277250\pi$
$277$	7.58355e9	1.28811	0.644056	+	0.764978i	$0.277250\pi$
$278$	− 2.95703e9i	− 0.495081i
$279$	0	0
$280$	−4.91790e8	−0.0800107
$281$	− 2.77360e9i	− 0.444855i	−0.974949	−	0.222427i	$-0.928602\pi$
$281$	− 2.77360e9i	− 0.444855i	0.974949	−	0.222427i	$-0.0713980\pi$
$282$	0	0
$283$	4.14403e9	0.646067	0.323033	−	0.946388i	$-0.395297\pi$
$283$	4.14403e9	0.646067	0.323033	+	0.946388i	$0.395297\pi$
$284$	− 4.87742e9i	− 0.749750i
$285$	0	0
$286$	4.54421e8	0.0679194
$287$	2.77559e9i	0.409099i
$288$	0	0
$289$	3.58961e9	0.514583
$290$	2.53163e9i	0.357938i
$291$	0	0
$292$	−6.86527e9	−0.944336
$293$	8.18497e9i	1.11057i	0.831659	+	0.555286i	$0.187391\pi$
$293$	8.18497e9i	1.11057i	−0.831659	+	0.555286i	$0.812609\pi$
$294$	0	0
$295$	−4.03039e9	−0.532181
$296$	− 9.16574e8i	− 0.119399i
$297$	0	0
$298$	6.98893e9	0.886229
$299$	− 1.43982e8i	− 0.0180145i
$300$	0	0
$301$	−1.09725e9	−0.133671
$302$	9.65307e9i	1.16048i
$303$	0	0
$304$	3.32199e9	0.388960
$305$	5.76318e9i	0.665983i
$306$	0	0
$307$	8.92948e9	1.00525	0.502623	−	0.864506i	$-0.332368\pi$
$307$	8.92948e9	1.00525	0.502623	+	0.864506i	$0.332368\pi$
$308$	− 1.35560e9i	− 0.150636i
$309$	0	0
$310$	−3.81477e8	−0.0413068
$311$	− 1.16159e9i	− 0.124169i	−0.998071	−	0.0620844i	$-0.980225\pi$
$311$	− 1.16159e9i	− 0.124169i	0.998071	−	0.0620844i	$-0.0197748\pi$
$312$	0	0
$313$	1.52821e9	0.159223	0.0796117	−	0.996826i	$-0.474632\pi$
$313$	1.52821e9	0.159223	0.0796117	+	0.996826i	$0.474632\pi$
$314$	1.23512e10i	1.27054i
$315$	0	0
$316$	6.60074e9	0.661979
$317$	1.68572e10i	1.66935i	0.550740	+	0.834677i	$0.314346\pi$
$317$	1.68572e10i	1.66935i	−0.550740	+	0.834677i	$0.685654\pi$
$318$	0	0
$319$	−6.97833e9	−0.673889
$320$	− 1.29621e9i	− 0.123616i
$321$	0	0
$322$	−4.29518e8	−0.0399538
$323$	1.17986e10i	1.08398i
$324$	0	0
$325$	1.79250e7	0.00160667
$326$	2.95627e9i	0.261742i
$327$	0	0
$328$	−7.31561e9	−0.632055
$329$	− 3.87048e9i	− 0.330355i
$330$	0	0
$331$	−6.36850e9	−0.530549	−0.265274	−	0.964173i	$-0.585463\pi$
$331$	−6.36850e9	−0.530549	−0.265274	+	0.964173i	$0.585463\pi$
$332$	− 4.22875e9i	− 0.348065i
$333$	0	0
$334$	5.41262e9	0.434933
$335$	7.28099e9i	0.578111i
$336$	0	0
$337$	1.52117e10	1.17939	0.589695	−	0.807626i	$-0.299248\pi$
$337$	1.52117e10	1.17939	0.589695	+	0.807626i	$0.299248\pi$
$338$	− 9.17981e9i	− 0.703343i
$339$	0	0
$340$	4.60371e9	0.344502
$341$	− 1.05153e9i	− 0.0777683i
$342$	0	0
$343$	6.16896e9	0.445693
$344$	− 2.89200e9i	− 0.206521i
$345$	0	0
$346$	6.30493e8	0.0439922
$347$	− 1.09643e10i	− 0.756249i	−0.925755	−	0.378125i	$-0.876569\pi$
$347$	− 1.09643e10i	− 0.756249i	0.925755	−	0.378125i	$-0.123431\pi$
$348$	0	0
$349$	−5.02924e9	−0.339001	−0.169501	−	0.985530i	$-0.554215\pi$
$349$	−5.02924e9	−0.339001	−0.169501	+	0.985530i	$0.554215\pi$
$350$	− 5.34729e7i	− 0.00356337i
$351$	0	0
$352$	3.57294e9	0.232732
$353$	2.99280e10i	1.92743i	0.266929	+	0.963716i	$0.413991\pi$
$353$	2.99280e10i	1.92743i	−0.266929	+	0.963716i	$0.586009\pi$
$354$	0	0
$355$	−2.35518e10	−1.48290
$356$	1.74751e9i	0.108798i
$357$	0	0
$358$	−1.07144e10	−0.652283
$359$	− 2.07140e9i	− 0.124705i	−0.998054	−	0.0623527i	$-0.980140\pi$
$359$	− 2.07140e9i	− 0.124705i	0.998054	−	0.0623527i	$-0.0198604\pi$
$360$	0	0
$361$	2.41274e10	1.42063
$362$	− 1.03133e9i	− 0.0600572i
$363$	0	0
$364$	1.46549e8	0.00834789
$365$	3.31507e10i	1.86776i
$366$	0	0
$367$	−2.69882e10	−1.48768	−0.743840	−	0.668357i	$-0.766998\pi$
$367$	−2.69882e10	−1.48768	−0.743840	+	0.668357i	$0.766998\pi$
$368$	− 1.13208e9i	− 0.0617283i
$369$	0	0
$370$	−4.42590e9	−0.236154
$371$	− 2.61519e9i	− 0.138041i
$372$	0	0
$373$	−2.69087e10	−1.39014	−0.695068	−	0.718944i	$-0.744626\pi$
$373$	−2.69087e10	−1.39014	−0.695068	+	0.718944i	$0.744626\pi$
$374$	1.26899e10i	0.648594i
$375$	0	0
$376$	1.02014e10	0.510397
$377$	− 7.54401e8i	− 0.0373454i
$378$	0	0
$379$	6.42912e9	0.311598	0.155799	−	0.987789i	$-0.450205\pi$
$379$	6.42912e9	0.311598	0.155799	+	0.987789i	$0.450205\pi$
$380$	− 1.60411e10i	− 0.769306i
$381$	0	0
$382$	1.63517e10	0.767908
$383$	− 3.43138e9i	− 0.159468i	−0.996816	−	0.0797340i	$-0.974593\pi$
$383$	− 3.43138e9i	− 0.159468i	0.996816	−	0.0797340i	$-0.0254071\pi$
$384$	0	0
$385$	−6.54585e9	−0.297936
$386$	2.41818e10i	1.08928i
$387$	0	0
$388$	1.41467e10	0.624206
$389$	− 7.34763e9i	− 0.320885i	−0.987045	−	0.160442i	$-0.948708\pi$
$389$	− 7.34763e9i	− 0.320885i	0.987045	−	0.160442i	$-0.0512920\pi$
$390$	0	0
$391$	4.02077e9	0.172029
$392$	7.91115e9i	0.335039i
$393$	0	0
$394$	2.12131e8	0.00880277
$395$	− 3.18733e10i	− 1.30930i
$396$	0	0
$397$	−2.54242e10	−1.02349	−0.511746	−	0.859137i	$-0.671001\pi$
$397$	−2.54242e10	−1.02349	−0.511746	+	0.859137i	$0.671001\pi$
$398$	− 2.54889e10i	− 1.01583i
$399$	0	0
$400$	1.40938e8	0.00550539
$401$	− 1.63772e10i	− 0.633376i	−0.948530	−	0.316688i	$-0.897429\pi$
$401$	− 1.63772e10i	− 0.633376i	0.948530	−	0.316688i	$-0.102571\pi$
$402$	0	0
$403$	1.13677e8	0.00430974
$404$	− 5.24613e9i	− 0.196931i
$405$	0	0
$406$	−2.25048e9	−0.0828269
$407$	− 1.21998e10i	− 0.444607i
$408$	0	0
$409$	−1.08754e10	−0.388645	−0.194323	−	0.980938i	$-0.562251\pi$
$409$	−1.08754e10	−0.388645	−0.194323	+	0.980938i	$0.562251\pi$
$410$	3.53252e10i	1.25011i
$411$	0	0
$412$	1.60373e10	0.556599
$413$	− 3.58280e9i	− 0.123147i
$414$	0	0
$415$	−2.04196e10	−0.688422
$416$	3.86257e8i	0.0128974i
$417$	0	0
$418$	4.42166e10	1.44837
$419$	2.62072e9i	0.0850286i	0.999096	+	0.0425143i	$0.0135368\pi$
$419$	2.62072e9i	0.0850286i	−0.999096	+	0.0425143i	$0.986463\pi$
$420$	0	0
$421$	3.07039e10	0.977385	0.488693	−	0.872456i	$-0.337474\pi$
$421$	3.07039e10	0.977385	0.488693	+	0.872456i	$0.337474\pi$
$422$	3.45158e10i	1.08835i
$423$	0	0
$424$	6.89284e9	0.213272
$425$	5.00566e8i	0.0153428i
$426$	0	0
$427$	−5.12316e9	−0.154109
$428$	− 1.78408e10i	− 0.531665i
$429$	0	0
$430$	−1.39648e10	−0.408469
$431$	1.11735e10i	0.323804i	0.986807	+	0.161902i	$0.0517628\pi$
$431$	1.11735e10i	0.323804i	−0.986807	+	0.161902i	$0.948237\pi$
$432$	0	0
$433$	−3.08778e10	−0.878406	−0.439203	−	0.898388i	$-0.644739\pi$
$433$	−3.08778e10	−0.878406	−0.439203	+	0.898388i	$0.644739\pi$
$434$	− 3.39113e8i	− 0.00955840i
$435$	0	0
$436$	−3.45923e10	−0.957267
$437$	− 1.40099e10i	− 0.384157i
$438$	0	0
$439$	−3.27137e9	−0.0880788	−0.0440394	−	0.999030i	$-0.514023\pi$
$439$	−3.27137e9	−0.0880788	−0.0440394	+	0.999030i	$0.514023\pi$
$440$	− 1.72528e10i	− 0.460310i
$441$	0	0
$442$	−1.37186e9	−0.0359436
$443$	3.06880e9i	0.0796808i	0.999206	+	0.0398404i	$0.0126849\pi$
$443$	3.06880e9i	0.0796808i	−0.999206	+	0.0398404i	$0.987315\pi$
$444$	0	0
$445$	8.43829e9	0.215186
$446$	2.08034e10i	0.525770i
$447$	0	0
$448$	1.15226e9	0.0286048
$449$	− 6.75557e10i	− 1.66218i	−0.556141	−	0.831088i	$-0.687719\pi$
$449$	− 6.75557e10i	− 1.66218i	0.556141	−	0.831088i	$-0.312281\pi$
$450$	0	0
$451$	−9.73725e10	−2.35359
$452$	− 2.79699e10i	− 0.670096i
$453$	0	0
$454$	2.78922e10	0.656537
$455$	− 7.07648e8i	− 0.0165109i
$456$	0	0
$457$	−2.39103e10	−0.548177	−0.274089	−	0.961704i	$-0.588376\pi$
$457$	−2.39103e10	−0.548177	−0.274089	+	0.961704i	$0.588376\pi$
$458$	− 9.22360e9i	− 0.209623i
$459$	0	0
$460$	−5.46651e9	−0.122090
$461$	1.47480e10i	0.326534i	0.986582	+	0.163267i	$0.0522033\pi$
$461$	1.47480e10i	0.326534i	−0.986582	+	0.163267i	$0.947797\pi$
$462$	0	0
$463$	−8.03448e9	−0.174837	−0.0874186	−	0.996172i	$-0.527862\pi$
$463$	−8.03448e9	−0.174837	−0.0874186	+	0.996172i	$0.527862\pi$
$464$	− 5.93158e9i	− 0.127967i
$465$	0	0
$466$	5.41379e10	1.14804
$467$	− 1.58552e10i	− 0.333352i	−0.986012	−	0.166676i	$-0.946697\pi$
$467$	− 1.58552e10i	− 0.333352i	0.986012	−	0.166676i	$-0.0533035\pi$
$468$	0	0
$469$	−6.47241e9	−0.133775
$470$	− 4.92599e10i	− 1.00949i
$471$	0	0
$472$	9.44316e9	0.190261
$473$	− 3.84933e10i	− 0.769025i
$474$	0	0
$475$	1.74416e9	0.0342620
$476$	4.09245e9i	0.0797179i
$477$	0	0
$478$	−2.01746e10	−0.386450
$479$	9.78801e10i	1.85931i	0.368429	+	0.929656i	$0.379896\pi$
$479$	9.78801e10i	1.85931i	−0.368429	+	0.929656i	$0.620104\pi$
$480$	0	0
$481$	1.31888e9	0.0246391
$482$	− 3.66359e10i	− 0.678765i
$483$	0	0
$484$	2.01188e10	0.366624
$485$	− 6.83108e10i	− 1.23459i
$486$	0	0
$487$	1.43016e10	0.254255	0.127127	−	0.991886i	$-0.459424\pi$
$487$	1.43016e10	0.254255	0.127127	+	0.991886i	$0.459424\pi$
$488$	− 1.35031e10i	− 0.238097i
$489$	0	0
$490$	3.82010e10	0.662659
$491$	− 8.13457e10i	− 1.39962i	−0.714331	−	0.699808i	$-0.753269\pi$
$491$	− 8.13457e10i	− 1.39962i	0.714331	−	0.699808i	$-0.246731\pi$
$492$	0	0
$493$	2.10670e10	0.356628
$494$	4.78009e9i	0.0802654i
$495$	0	0
$496$	8.93797e8	0.0147677
$497$	− 2.09363e10i	− 0.343143i
$498$	0	0
$499$	1.00357e11	1.61863	0.809314	−	0.587377i	$-0.199839\pi$
$499$	1.00357e11	1.61863	0.809314	+	0.587377i	$0.199839\pi$
$500$	− 3.15846e10i	− 0.505353i
$501$	0	0
$502$	3.06097e10	0.481997
$503$	8.91388e10i	1.39250i	0.717800	+	0.696249i	$0.245149\pi$
$503$	8.91388e10i	1.39250i	−0.717800	+	0.696249i	$0.754851\pi$
$504$	0	0
$505$	−2.53323e10	−0.389501
$506$	− 1.50682e10i	− 0.229858i
$507$	0	0
$508$	−4.09024e10	−0.614177
$509$	− 7.94775e10i	− 1.18406i	−0.805916	−	0.592029i	$-0.798327\pi$
$509$	− 7.94775e10i	− 1.18406i	0.805916	−	0.592029i	$-0.201673\pi$
$510$	0	0
$511$	−2.94692e10	−0.432200
$512$	3.03700e9i	0.0441942i
$513$	0	0
$514$	2.24012e10	0.320936
$515$	− 7.74401e10i	− 1.10087i
$516$	0	0
$517$	1.35783e11	1.90057
$518$	− 3.93439e9i	− 0.0546460i
$519$	0	0
$520$	1.86514e9	0.0255093
$521$	− 5.94692e10i	− 0.807126i	−0.914952	−	0.403563i	$-0.867772\pi$
$521$	− 5.94692e10i	− 0.807126i	0.914952	−	0.403563i	$-0.132228\pi$
$522$	0	0
$523$	−1.09394e11	−1.46213	−0.731063	−	0.682310i	$-0.760976\pi$
$523$	−1.09394e11	−1.46213	−0.731063	+	0.682310i	$0.760976\pi$
$524$	− 4.85145e10i	− 0.643496i
$525$	0	0
$526$	1.08978e10	0.142362
$527$	3.17448e9i	0.0411557i
$528$	0	0
$529$	7.35367e10	0.939034
$530$	− 3.32838e10i	− 0.421822i
$531$	0	0
$532$	1.42597e10	0.178018
$533$	− 1.05266e10i	− 0.130430i
$534$	0	0
$535$	−8.61485e10	−1.05156
$536$	− 1.70593e10i	− 0.206681i
$537$	0	0
$538$	−6.96142e10	−0.830938
$539$	1.05299e11i	1.24759i
$540$	0	0
$541$	1.26322e11	1.47465	0.737325	−	0.675538i	$-0.236089\pi$
$541$	1.26322e11	1.47465	0.737325	+	0.675538i	$0.236089\pi$
$542$	9.22877e10i	1.06942i
$543$	0	0
$544$	−1.07864e10	−0.123164
$545$	1.67037e11i	1.89334i
$546$	0	0
$547$	8.53542e10	0.953400	0.476700	−	0.879066i	$-0.341833\pi$
$547$	8.53542e10	0.953400	0.476700	+	0.879066i	$0.341833\pi$
$548$	4.35895e10i	0.483348i
$549$	0	0
$550$	1.87592e9	0.0205005
$551$	− 7.34056e10i	− 0.796384i
$552$	0	0
$553$	2.83337e10	0.302972
$554$	8.57981e10i	0.910833i
$555$	0	0
$556$	3.34549e10	0.350075
$557$	− 7.02941e10i	− 0.730295i	−0.930950	−	0.365148i	$-0.881019\pi$
$557$	− 7.02941e10i	− 0.730295i	0.930950	−	0.365148i	$-0.118981\pi$
$558$	0	0
$559$	4.16136e9	0.0426176
$560$	− 5.56397e9i	− 0.0565761i
$561$	0	0
$562$	3.13797e10	0.314560
$563$	− 1.10285e11i	− 1.09770i	−0.835921	−	0.548849i	$-0.815066\pi$
$563$	− 1.10285e11i	− 1.09770i	0.835921	−	0.548849i	$-0.184934\pi$
$564$	0	0
$565$	−1.35060e11	−1.32535
$566$	4.68844e10i	0.456838i
$567$	0	0
$568$	5.51817e10	0.530153
$569$	1.27112e11i	1.21266i	0.795213	+	0.606330i	$0.207359\pi$
$569$	1.27112e11i	1.21266i	−0.795213	+	0.606330i	$0.792641\pi$
$570$	0	0
$571$	−1.33049e11	−1.25161	−0.625803	−	0.779981i	$-0.715228\pi$
$571$	−1.33049e11	−1.25161	−0.625803	+	0.779981i	$0.715228\pi$
$572$	5.14118e9i	0.0480263i
$573$	0	0
$574$	−3.14023e10	−0.289277
$575$	− 5.94380e8i	− 0.00543742i
$576$	0	0
$577$	4.19953e9	0.0378877	0.0189438	−	0.999821i	$-0.493970\pi$
$577$	4.19953e9	0.0378877	0.0189438	+	0.999821i	$0.493970\pi$
$578$	4.06118e10i	0.363865i
$579$	0	0
$580$	−2.86421e10	−0.253100
$581$	− 1.81519e10i	− 0.159301i
$582$	0	0
$583$	9.17454e10	0.794164
$584$	− 7.76717e10i	− 0.667746i
$585$	0	0
$586$	−9.26024e10	−0.785293
$587$	3.28670e10i	0.276826i	0.990375	+	0.138413i	$0.0442002\pi$
$587$	3.28670e10i	0.276826i	−0.990375	+	0.138413i	$0.955800\pi$
$588$	0	0
$589$	1.10611e10	0.0919045
$590$	− 4.55987e10i	− 0.376308i
$591$	0	0
$592$	1.03698e10	0.0844278
$593$	6.35989e10i	0.514317i	0.966369	+	0.257159i	$0.0827863\pi$
$593$	6.35989e10i	0.514317i	−0.966369	+	0.257159i	$0.917214\pi$
$594$	0	0
$595$	1.97614e10	0.157671
$596$	7.90707e10i	0.626658i
$597$	0	0
$598$	1.62897e9	0.0127382
$599$	1.02201e11i	0.793864i	0.917848	+	0.396932i	$0.129925\pi$
$599$	1.02201e11i	0.793864i	−0.917848	+	0.396932i	$0.870075\pi$
$600$	0	0
$601$	−3.68415e10	−0.282384	−0.141192	−	0.989982i	$-0.545093\pi$
$601$	−3.68415e10	−0.282384	−0.141192	+	0.989982i	$0.545093\pi$
$602$	− 1.24139e10i	− 0.0945199i
$603$	0	0
$604$	−1.09212e11	−0.820583
$605$	− 9.71487e10i	− 0.725130i
$606$	0	0
$607$	−2.16701e11	−1.59627	−0.798134	−	0.602480i	$-0.794179\pi$
$607$	−2.16701e11	−1.59627	−0.798134	+	0.602480i	$0.794179\pi$
$608$	3.75841e10i	0.275036i
$609$	0	0
$610$	−6.52030e10	−0.470921
$611$	1.46790e10i	0.105325i
$612$	0	0
$613$	1.77119e11	1.25436	0.627180	−	0.778874i	$-0.284209\pi$
$613$	1.77119e11	1.25436	0.627180	+	0.778874i	$0.284209\pi$
$614$	1.01025e11i	0.710817i
$615$	0	0
$616$	1.53369e10	0.106516
$617$	2.00900e11i	1.38624i	0.720822	+	0.693120i	$0.243764\pi$
$617$	2.00900e11i	1.38624i	−0.720822	+	0.693120i	$0.756236\pi$
$618$	0	0
$619$	−1.09856e11	−0.748272	−0.374136	−	0.927374i	$-0.622061\pi$
$619$	−1.09856e11	−0.748272	−0.374136	+	0.927374i	$0.622061\pi$
$620$	− 4.31592e9i	− 0.0292083i
$621$	0	0
$622$	1.31419e10	0.0878006
$623$	7.50119e9i	0.0497941i
$624$	0	0
$625$	−1.49154e11	−0.977493
$626$	1.72898e10i	0.112588i
$627$	0	0
$628$	−1.39738e11	−0.898410
$629$	3.68303e10i	0.235290i
$630$	0	0
$631$	2.34271e11	1.47775	0.738875	−	0.673842i	$-0.235357\pi$
$631$	2.34271e11	1.47775	0.738875	+	0.673842i	$0.235357\pi$
$632$	7.46789e10i	0.468090i
$633$	0	0
$634$	−1.90717e11	−1.18041
$635$	1.97507e11i	1.21475i
$636$	0	0
$637$	−1.13835e10	−0.0691383
$638$	− 7.89507e10i	− 0.476512i
$639$	0	0
$640$	1.46649e10	0.0874097
$641$	− 8.19659e10i	− 0.485514i	−0.970087	−	0.242757i	$-0.921948\pi$
$641$	− 8.19659e10i	− 0.485514i	0.970087	−	0.242757i	$-0.0780517\pi$
$642$	0	0
$643$	7.63003e10	0.446357	0.223179	−	0.974778i	$-0.428357\pi$
$643$	7.63003e10	0.446357	0.223179	+	0.974778i	$0.428357\pi$
$644$	− 4.85944e9i	− 0.0282516i
$645$	0	0
$646$	−1.33486e11	−0.766491
$647$	3.07177e11i	1.75296i	0.481440	+	0.876479i	$0.340114\pi$
$647$	3.07177e11i	1.75296i	−0.481440	+	0.876479i	$0.659886\pi$
$648$	0	0
$649$	1.25691e11	0.708475
$650$	2.02799e8i	0.00113609i
$651$	0	0
$652$	−3.34464e10	−0.185080
$653$	2.54588e11i	1.40018i	0.714053	+	0.700092i	$0.246858\pi$
$653$	2.54588e11i	1.40018i	−0.714053	+	0.700092i	$0.753142\pi$
$654$	0	0
$655$	−2.34264e11	−1.27274
$656$	− 8.27666e10i	− 0.446930i
$657$	0	0
$658$	4.37895e10	0.233596
$659$	− 2.79368e11i	− 1.48127i	−0.671905	−	0.740637i	$-0.734524\pi$
$659$	− 2.79368e11i	− 1.48127i	0.671905	−	0.740637i	$-0.265476\pi$
$660$	0	0
$661$	1.52502e11	0.798860	0.399430	−	0.916764i	$-0.369208\pi$
$661$	1.52502e11	0.798860	0.399430	+	0.916764i	$0.369208\pi$
$662$	− 7.20514e10i	− 0.375155i
$663$	0	0
$664$	4.78429e10	0.246119
$665$	− 6.88564e10i	− 0.352093i
$666$	0	0
$667$	−2.50153e10	−0.126387
$668$	6.12368e10i	0.307544i
$669$	0	0
$670$	−8.23750e10	−0.408786
$671$	− 1.79729e11i	− 0.886602i
$672$	0	0
$673$	2.83359e11	1.38126	0.690632	−	0.723207i	$-0.257333\pi$
$673$	2.83359e11	1.38126	0.690632	+	0.723207i	$0.257333\pi$
$674$	1.72101e11i	0.833955i
$675$	0	0
$676$	1.03858e11	0.497338
$677$	2.24639e11i	1.06937i	0.845050	+	0.534687i	$0.179571\pi$
$677$	2.24639e11i	1.06937i	−0.845050	+	0.534687i	$0.820429\pi$
$678$	0	0
$679$	6.07247e10	0.285684
$680$	5.20850e10i	0.243600i
$681$	0	0
$682$	1.18967e10	0.0549905
$683$	1.17660e11i	0.540685i	0.962764	+	0.270343i	$0.0871370\pi$
$683$	1.17660e11i	0.540685i	−0.962764	+	0.270343i	$0.912863\pi$
$684$	0	0
$685$	2.10483e11	0.955993
$686$	6.97938e10i	0.315152i
$687$	0	0
$688$	3.27193e10	0.146033
$689$	9.91825e9i	0.0440107i
$690$	0	0
$691$	−1.92479e10	−0.0844249	−0.0422124	−	0.999109i	$-0.513441\pi$
$691$	−1.92479e10	−0.0844249	−0.0422124	+	0.999109i	$0.513441\pi$
$692$	7.13321e9i	0.0311072i
$693$	0	0
$694$	1.24047e11	0.534749
$695$	− 1.61546e11i	− 0.692398i
$696$	0	0
$697$	2.93960e11	1.24554
$698$	− 5.68994e10i	− 0.239710i
$699$	0	0
$700$	6.04977e8	0.00251969
$701$	− 7.50876e10i	− 0.310954i	−0.987840	−	0.155477i	$-0.950309\pi$
$701$	− 7.50876e10i	− 0.310954i	0.987840	−	0.155477i	$-0.0496915\pi$
$702$	0	0
$703$	1.28331e11	0.525424
$704$	4.04232e10i	0.164566i
$705$	0	0
$706$	−3.38597e11	−1.36290
$707$	− 2.25190e10i	− 0.0901306i
$708$	0	0
$709$	1.42979e11	0.565832	0.282916	−	0.959145i	$-0.408698\pi$
$709$	1.42979e11	0.565832	0.282916	+	0.959145i	$0.408698\pi$
$710$	− 2.66458e11i	− 1.04857i
$711$	0	0
$712$	−1.97708e10	−0.0769316
$713$	− 3.76942e9i	− 0.0145853i
$714$	0	0
$715$	2.48255e10	0.0949891
$716$	− 1.21220e11i	− 0.461234i
$717$	0	0
$718$	2.34352e10	0.0881801
$719$	3.97995e11i	1.48923i	0.667493	+	0.744616i	$0.267368\pi$
$719$	3.97995e11i	1.48923i	−0.667493	+	0.744616i	$0.732632\pi$
$720$	0	0
$721$	6.88401e10	0.254742
$722$	2.72971e11i	1.00454i
$723$	0	0
$724$	1.16682e10	0.0424668
$725$	− 3.11428e9i	− 0.0112721i
$726$	0	0
$727$	−2.04083e11	−0.730583	−0.365291	−	0.930893i	$-0.619031\pi$
$727$	−2.04083e11	−0.730583	−0.365291	+	0.930893i	$0.619031\pi$
$728$	1.65801e9i	0.00590285i
$729$	0	0
$730$	−3.75057e11	−1.32071
$731$	1.16208e11i	0.406975i
$732$	0	0
$733$	1.25071e10	0.0433252	0.0216626	−	0.999765i	$-0.493104\pi$
$733$	1.25071e10	0.0433252	0.0216626	+	0.999765i	$0.493104\pi$
$734$	− 3.05337e11i	− 1.05195i
$735$	0	0
$736$	1.28080e10	0.0436485
$737$	− 2.27063e11i	− 0.769621i
$738$	0	0
$739$	−4.31386e11	−1.44640	−0.723200	−	0.690639i	$-0.757329\pi$
$739$	−4.31386e11	−1.44640	−0.723200	+	0.690639i	$0.757329\pi$
$740$	− 5.00734e10i	− 0.166986i
$741$	0	0
$742$	2.95875e10	0.0976097
$743$	− 2.05331e10i	− 0.0673752i	−0.999432	−	0.0336876i	$-0.989275\pi$
$743$	− 2.05331e10i	− 0.0673752i	0.999432	−	0.0336876i	$-0.0107251\pi$
$744$	0	0
$745$	3.81813e11	1.23944
$746$	− 3.04437e11i	− 0.982975i
$747$	0	0
$748$	−1.43570e11	−0.458625
$749$	− 7.65815e10i	− 0.243330i
$750$	0	0
$751$	−2.80002e11	−0.880240	−0.440120	−	0.897939i	$-0.645064\pi$
$751$	−2.80002e11	−0.880240	−0.440120	+	0.897939i	$0.645064\pi$
$752$	1.15415e11i	0.360905i
$753$	0	0
$754$	8.53507e9	0.0264072
$755$	5.27358e11i	1.62300i
$756$	0	0
$757$	4.02781e10	0.122655	0.0613275	−	0.998118i	$-0.480467\pi$
$757$	4.02781e10	0.122655	0.0613275	+	0.998118i	$0.480467\pi$
$758$	7.27372e10i	0.220333i
$759$	0	0
$760$	1.81484e11	0.543982
$761$	1.10593e11i	0.329754i	0.986314	+	0.164877i	$0.0527227\pi$
$761$	1.10593e11i	0.329754i	−0.986314	+	0.164877i	$0.947277\pi$
$762$	0	0
$763$	−1.48487e11	−0.438118
$764$	1.84998e11i	0.542993i
$765$	0	0
$766$	3.88216e10	0.112761
$767$	1.35880e10i	0.0392620i
$768$	0	0
$769$	2.93433e10	0.0839082	0.0419541	−	0.999120i	$-0.486642\pi$
$769$	2.93433e10	0.0839082	0.0419541	+	0.999120i	$0.486642\pi$
$770$	− 7.40579e10i	− 0.210673i
$771$	0	0
$772$	−2.73586e11	−0.770238
$773$	− 5.67214e11i	− 1.58865i	−0.607492	−	0.794326i	$-0.707824\pi$
$773$	− 5.67214e11i	− 1.58865i	0.607492	−	0.794326i	$-0.292176\pi$
$774$	0	0
$775$	4.69275e8	0.00130083
$776$	1.60051e11i	0.441380i
$777$	0	0
$778$	8.31289e10	0.226900
$779$	− 1.02427e12i	− 2.78141i
$780$	0	0
$781$	7.34482e11	1.97414
$782$	4.54898e10i	0.121643i
$783$	0	0
$784$	−8.95044e10	−0.236908
$785$	6.74758e11i	1.77693i
$786$	0	0
$787$	6.30035e11	1.64235	0.821175	−	0.570676i	$-0.193319\pi$
$787$	6.30035e11	1.64235	0.821175	+	0.570676i	$0.193319\pi$
$788$	2.39999e9i	0.00622450i
$789$	0	0
$790$	3.60606e11	0.925815
$791$	− 1.20061e11i	− 0.306687i
$792$	0	0
$793$	1.94299e10	0.0491334
$794$	− 2.87642e11i	− 0.723718i
$795$	0	0
$796$	2.88374e11	0.718298
$797$	1.08992e11i	0.270124i	0.990837	+	0.135062i	$0.0431233\pi$
$797$	1.08992e11i	0.270124i	−0.990837	+	0.135062i	$0.956877\pi$
$798$	0	0
$799$	−4.09918e11	−1.00580
$800$	1.59453e9i	0.00389290i
$801$	0	0
$802$	1.85287e11	0.447864
$803$	− 1.03383e12i	− 2.48649i
$804$	0	0
$805$	−2.34650e10	−0.0558776
$806$	1.28610e9i	0.00304744i
$807$	0	0
$808$	5.93532e10	0.139251
$809$	5.44516e10i	0.127121i	0.997978	+	0.0635604i	$0.0202455\pi$
$809$	5.44516e10i	0.127121i	−0.997978	+	0.0635604i	$0.979754\pi$
$810$	0	0
$811$	−7.27746e11	−1.68227	−0.841137	−	0.540822i	$-0.818113\pi$
$811$	−7.27746e11	−1.68227	−0.841137	+	0.540822i	$0.818113\pi$
$812$	− 2.54613e10i	− 0.0585675i
$813$	0	0
$814$	1.38025e11	0.314384
$815$	1.61504e11i	0.366061i
$816$	0	0
$817$	4.04914e11	0.908813
$818$	− 1.23041e11i	− 0.274814i
$819$	0	0
$820$	−3.99659e11	−0.883964
$821$	− 7.22729e11i	− 1.59075i	−0.606116	−	0.795376i	$-0.707273\pi$
$821$	− 7.22729e11i	− 1.59075i	0.606116	−	0.795376i	$-0.292727\pi$
$822$	0	0
$823$	1.33800e11	0.291647	0.145824	−	0.989311i	$-0.453417\pi$
$823$	1.33800e11	0.291647	0.145824	+	0.989311i	$0.453417\pi$
$824$	1.81441e11i	0.393575i
$825$	0	0
$826$	4.05348e10	0.0870778
$827$	− 8.49721e11i	− 1.81658i	−0.418343	−	0.908289i	$-0.637389\pi$
$827$	− 8.49721e11i	− 1.81658i	0.418343	−	0.908289i	$-0.362611\pi$
$828$	0	0
$829$	−6.24241e11	−1.32170	−0.660852	−	0.750517i	$-0.729805\pi$
$829$	−6.24241e11	−1.32170	−0.660852	+	0.750517i	$0.729805\pi$
$830$	− 2.31021e11i	− 0.486788i
$831$	0	0
$832$	−4.37000e9	−0.00911987
$833$	− 3.17891e11i	− 0.660234i
$834$	0	0
$835$	2.95697e11	0.608277
$836$	5.00253e11i	1.02415i
$837$	0	0
$838$	−2.96501e10	−0.0601243
$839$	− 2.86199e11i	− 0.577590i	−0.957391	−	0.288795i	$-0.906745\pi$
$839$	− 2.86199e11i	− 0.577590i	0.957391	−	0.288795i	$-0.0932547\pi$
$840$	0	0
$841$	3.69177e11	0.737991
$842$	3.47375e11i	0.691116i
$843$	0	0
$844$	−3.90501e11	−0.769578
$845$	− 5.01503e11i	− 0.983664i
$846$	0	0
$847$	8.63601e10	0.167795
$848$	7.79836e10i	0.150806i
$849$	0	0
$850$	−5.66326e9	−0.0108490
$851$	− 4.37329e10i	− 0.0833854i
$852$	0	0
$853$	−6.86295e11	−1.29633	−0.648163	−	0.761501i	$-0.724463\pi$
$853$	−6.86295e11	−1.29633	−0.648163	+	0.761501i	$0.724463\pi$
$854$	− 5.79620e10i	− 0.108971i
$855$	0	0
$856$	2.01845e11	0.375944
$857$	− 2.02706e11i	− 0.375788i	−0.982189	−	0.187894i	$-0.939834\pi$
$857$	− 2.02706e11i	− 0.375788i	0.982189	−	0.187894i	$-0.0601661\pi$
$858$	0	0
$859$	−6.99529e11	−1.28479	−0.642396	−	0.766373i	$-0.722060\pi$
$859$	−6.99529e11	−1.28479	−0.642396	+	0.766373i	$0.722060\pi$
$860$	− 1.57993e11i	− 0.288832i
$861$	0	0
$862$	−1.26414e11	−0.228964
$863$	5.41586e11i	0.976392i	0.872734	+	0.488196i	$0.162345\pi$
$863$	5.41586e11i	0.976392i	−0.872734	+	0.488196i	$0.837655\pi$
$864$	0	0
$865$	3.44445e10	0.0615255
$866$	− 3.49343e11i	− 0.621127i
$867$	0	0
$868$	3.83662e9	0.00675881
$869$	9.93994e11i	1.74303i
$870$	0	0
$871$	2.45470e10	0.0426506
$872$	− 3.91367e11i	− 0.676890i
$873$	0	0
$874$	1.58504e11	0.271640
$875$	− 1.35577e11i	− 0.231288i
$876$	0	0
$877$	1.42507e11	0.240901	0.120451	−	0.992719i	$-0.461566\pi$
$877$	1.42507e11	0.240901	0.120451	+	0.992719i	$0.461566\pi$
$878$	− 3.70113e10i	− 0.0622811i
$879$	0	0
$880$	1.95194e11	0.325488
$881$	− 5.82792e11i	− 0.967409i	−0.875231	−	0.483705i	$-0.839291\pi$
$881$	− 5.82792e11i	− 0.967409i	0.875231	−	0.483705i	$-0.160709\pi$
$882$	0	0
$883$	−4.46455e11	−0.734404	−0.367202	−	0.930141i	$-0.619684\pi$
$883$	−4.46455e11	−0.734404	−0.367202	+	0.930141i	$0.619684\pi$
$884$	− 1.55208e10i	− 0.0254159i
$885$	0	0
$886$	−3.47195e10	−0.0563428
$887$	− 7.82518e11i	− 1.26415i	−0.774906	−	0.632077i	$-0.782203\pi$
$887$	− 7.82518e11i	− 1.26415i	0.774906	−	0.632077i	$-0.217797\pi$
$888$	0	0
$889$	−1.75574e11	−0.281094
$890$	9.54683e10i	0.152160i
$891$	0	0
$892$	−2.35364e11	−0.371776
$893$	1.42831e12i	2.24604i
$894$	0	0
$895$	−5.85340e11	−0.912254
$896$	1.30363e10i	0.0202266i
$897$	0	0
$898$	7.64306e11	1.17534
$899$	− 1.97501e10i	− 0.0302364i
$900$	0	0
$901$	−2.76972e11	−0.420278
$902$	− 1.10164e12i	− 1.66424i
$903$	0	0
$904$	3.16443e11	0.473830
$905$	− 5.63428e10i	− 0.0839933i
$906$	0	0
$907$	−5.34694e11	−0.790089	−0.395044	−	0.918662i	$-0.629271\pi$
$907$	−5.34694e11	−0.790089	−0.395044	+	0.918662i	$0.629271\pi$
$908$	3.15564e11i	0.464242i
$909$	0	0
$910$	8.00612e9	0.0116750
$911$	1.09432e12i	1.58880i	0.607394	+	0.794400i	$0.292215\pi$
$911$	1.09432e12i	1.58880i	−0.607394	+	0.794400i	$0.707785\pi$
$912$	0	0
$913$	6.36801e11	0.916474
$914$	− 2.70515e11i	− 0.387620i
$915$	0	0
$916$	1.04353e11	0.148226
$917$	− 2.08248e11i	− 0.294513i
$918$	0	0
$919$	−1.14588e12	−1.60648	−0.803242	−	0.595653i	$-0.796893\pi$
$919$	−1.14588e12	−1.60648	−0.803242	+	0.595653i	$0.796893\pi$
$920$	− 6.18465e10i	− 0.0863305i
$921$	0	0
$922$	−1.66854e11	−0.230895
$923$	7.94021e10i	0.109402i
$924$	0	0
$925$	5.44453e9	0.00743693
$926$	− 9.08998e10i	− 0.123629i
$927$	0	0
$928$	6.71081e10	0.0904864
$929$	− 4.39923e10i	− 0.0590627i	−0.999564	−	0.0295314i	$-0.990599\pi$
$929$	− 4.39923e10i	− 0.0590627i	0.999564	−	0.0295314i	$-0.00940149\pi$
$930$	0	0
$931$	−1.10765e12	−1.47436
$932$	6.12501e11i	0.811789i
$933$	0	0
$934$	1.79381e11	0.235716
$935$	6.93265e11i	0.907095i
$936$	0	0
$937$	5.55029e11	0.720041	0.360021	−	0.932944i	$-0.382770\pi$
$937$	5.55029e11	0.720041	0.360021	+	0.932944i	$0.382770\pi$
$938$	− 7.32270e10i	− 0.0945932i
$939$	0	0
$940$	5.57312e11	0.713818
$941$	1.91990e11i	0.244862i	0.992477	+	0.122431i	$0.0390689\pi$
$941$	1.91990e11i	0.244862i	−0.992477	+	0.122431i	$0.960931\pi$
$942$	0	0
$943$	−3.49053e11	−0.441412
$944$	1.06837e11i	0.134535i
$945$	0	0
$946$	4.35502e11	0.543783
$947$	6.16187e11i	0.766147i	0.923718	+	0.383074i	$0.125134\pi$
$947$	6.16187e11i	0.766147i	−0.923718	+	0.383074i	$0.874866\pi$
$948$	0	0
$949$	1.11763e11	0.137796
$950$	1.97330e10i	0.0242269i
$951$	0	0
$952$	−4.63008e10	−0.0563691
$953$	− 1.34665e12i	− 1.63261i	−0.577618	−	0.816307i	$-0.696018\pi$
$953$	− 1.34665e12i	− 1.63261i	0.577618	−	0.816307i	$-0.303982\pi$
$954$	0	0
$955$	8.93310e11	1.07396
$956$	− 2.28250e11i	− 0.273262i
$957$	0	0
$958$	−1.10739e12	−1.31473
$959$	1.87108e11i	0.221217i
$960$	0	0
$961$	−8.49915e11	−0.996511
$962$	1.49214e10i	0.0174224i
$963$	0	0
$964$	4.14488e11	0.479959
$965$	1.32108e12i	1.52342i
$966$	0	0
$967$	−1.39724e11	−0.159796	−0.0798978	−	0.996803i	$-0.525459\pi$
$967$	−1.39724e11	−0.159796	−0.0798978	+	0.996803i	$0.525459\pi$
$968$	2.27618e11i	0.259242i
$969$	0	0
$970$	7.72848e11	0.872986
$971$	1.72539e12i	1.94093i	0.241242	+	0.970465i	$0.422445\pi$
$971$	1.72539e12i	1.94093i	−0.241242	+	0.970465i	$0.577555\pi$
$972$	0	0
$973$	1.43605e11	0.160221
$974$	1.61804e11i	0.179785i
$975$	0	0
$976$	1.52770e11	0.168360
$977$	1.11223e12i	1.22072i	0.792125	+	0.610358i	$0.208975\pi$
$977$	1.11223e12i	1.22072i	−0.792125	+	0.610358i	$0.791025\pi$
$978$	0	0
$979$	−2.63154e11	−0.286471
$980$	4.32194e11i	0.468570i
$981$	0	0
$982$	9.20322e11	0.989678
$983$	− 1.23691e12i	− 1.32472i	−0.749186	−	0.662360i	$-0.769555\pi$
$983$	− 1.23691e12i	− 1.32472i	0.749186	−	0.662360i	$-0.230445\pi$
$984$	0	0
$985$	1.15889e10	0.0123112
$986$	2.38346e11i	0.252174i
$987$	0	0
$988$	−5.40805e10	−0.0567562
$989$	− 1.37987e11i	− 0.144230i
$990$	0	0
$991$	4.57888e11	0.474749	0.237375	−	0.971418i	$-0.423713\pi$
$991$	4.57888e11	0.474749	0.237375	+	0.971418i	$0.423713\pi$
$992$	1.01122e10i	0.0104423i
$993$	0	0
$994$	2.36867e11	0.242639
$995$	− 1.39249e12i	− 1.42069i
$996$	0	0
$997$	1.76730e12	1.78867	0.894333	−	0.447402i	$-0.147651\pi$
$997$	1.76730e12	1.78867	0.894333	+	0.447402i	$0.147651\pi$
$998$	1.13541e12i	1.14454i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.9.b.a.161.7	yes	8
3.2	odd	2	inner	162.9.b.a.161.2	✓	8
9.2	odd	6		162.9.d.h.53.3		16
9.4	even	3		162.9.d.h.107.3		16
9.5	odd	6		162.9.d.h.107.6		16
9.7	even	3		162.9.d.h.53.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.9.b.a.161.2	✓	8	3.2	odd	2	inner
162.9.b.a.161.7	yes	8	1.1	even	1	trivial
162.9.d.h.53.3		16	9.2	odd	6
162.9.d.h.53.6		16	9.7	even	3
162.9.d.h.107.3		16	9.4	even	3
162.9.d.h.107.6		16	9.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+11.3137i q^{2} -128.000 q^{4} +618.080i q^{5} -549.440 q^{7} -1448.15i q^{8} +O(q^{10})\)	\(q+11.3137i q^{2} -128.000 q^{4} +618.080i q^{5} -549.440 q^{7} -1448.15i q^{8} -6992.78 q^{10} -19275.3i q^{11} +2083.78 q^{13} -6216.21i q^{14} +16384.0 q^{16} +58190.7i q^{17} +202758. q^{19} -79114.2i q^{20} +218075. q^{22} -69096.5i q^{23} +8602.17 q^{25} +23575.3i q^{26} +70328.3 q^{28} -362035. i q^{29} +54553.0 q^{31} +185364. i q^{32} -658352. q^{34} -339598. i q^{35} +632925. q^{37} +2.29395e6i q^{38} +895075. q^{40} -5.05168e6i q^{41} +1.99703e6 q^{43} +2.46724e6i q^{44} +781737. q^{46} +7.04440e6i q^{47} -5.46292e6 q^{49} +97322.5i q^{50} -266724. q^{52} +4.75974e6i q^{53} +1.19137e7 q^{55} +795674. i q^{56} +4.09595e6 q^{58} +6.52082e6i q^{59} +9.32433e6 q^{61} +617197. i q^{62} -2.09715e6 q^{64} +1.28794e6i q^{65} +1.17800e7 q^{67} -7.44840e6i q^{68} +3.84211e6 q^{70} +3.81048e7i q^{71} +5.36350e7 q^{73} +7.16073e6i q^{74} -2.59531e7 q^{76} +1.05906e7i q^{77} -5.15683e7 q^{79} +1.01266e7i q^{80} +5.71532e7 q^{82} +3.30371e7i q^{83} -3.59665e7 q^{85} +2.25938e7i q^{86} -2.79136e7 q^{88} -1.36524e7i q^{89} -1.14491e6 q^{91} +8.84435e6i q^{92} -7.96983e7 q^{94} +1.25321e8i q^{95} -1.10521e8 q^{97} -6.18058e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 1024 q^{4} - 8876 q^{7}+O(q^{10}) \) 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

Related objects

Downloads

Learn more