LMFDB - Embedded newform 9.15.b.a.8.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,15,Mod(8,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 15, names="a")

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

chi := DirichletCharacter("9.8");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	9.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:	$11.1896071337$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3745})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 1867x^{2} + 1868x + 879846 $ x^4 - 2x^3 - 1867x^2 + 1868*x + 879846
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			8.2
Root			$-30.0982 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	9.8
Dual form			9.15.b.a.8.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-110.725i q^{2} +4123.91 q^{4} -35180.3i q^{5} +883527. q^{7} -2.27074e6i q^{8} +O(q^{10})$	\(q-110.725i q^{2} +4123.91 q^{4} -35180.3i q^{5} +883527. q^{7} -2.27074e6i q^{8} -3.89535e6 q^{10} -1.16632e7i q^{11} -1.07693e8 q^{13} -9.78288e7i q^{14} -1.83863e8 q^{16} -6.94152e8i q^{17} +4.57205e8 q^{19} -1.45080e8i q^{20} -1.29142e9 q^{22} +3.31786e9i q^{23} +4.86586e9 q^{25} +1.19243e10i q^{26} +3.64358e9 q^{28} -5.59359e9i q^{29} +2.96133e10 q^{31} -1.68456e10i q^{32} -7.68602e10 q^{34} -3.10827e10i q^{35} -6.69873e10 q^{37} -5.06242e10i q^{38} -7.98854e10 q^{40} -4.19037e10i q^{41} +2.68989e11 q^{43} -4.80981e10i q^{44} +3.67371e11 q^{46} +8.88014e11i q^{47} +1.02397e11 q^{49} -5.38774e11i q^{50} -4.44116e11 q^{52} +2.06534e12i q^{53} -4.10316e11 q^{55} -2.00626e12i q^{56} -6.19352e11 q^{58} -3.12449e12i q^{59} +1.70805e12 q^{61} -3.27895e12i q^{62} -4.87764e12 q^{64} +3.78866e12i q^{65} +2.82893e12 q^{67} -2.86262e12i q^{68} -3.44164e12 q^{70} +1.53177e13i q^{71} +9.14210e12 q^{73} +7.41719e12i q^{74} +1.88547e12 q^{76} -1.03048e13i q^{77} +2.06959e13 q^{79} +6.46834e12i q^{80} -4.63980e12 q^{82} -8.35164e12i q^{83} -2.44205e13 q^{85} -2.97839e13i q^{86} -2.64842e13 q^{88} -1.57894e13i q^{89} -9.51496e13 q^{91} +1.36825e13i q^{92} +9.83256e13 q^{94} -1.60846e13i q^{95} +3.41627e13 q^{97} -1.13379e13i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3332 q^{4} - 1065904 q^{7}+O(q^{10}) $ 4 * q - 3332 * q^4 - 1065904 * q^7	$ 4 q - 3332 q^{4} - 1065904 q^{7} - 20201220 q^{10} - 193633024 q^{13} - 1027273976 q^{16} - 1738250560 q^{19} + 5179999824 q^{22} + 18465731740 q^{25} + 23689734992 q^{28} - 22530472816 q^{31} - 118458902412 q^{34} - 135170189128 q^{37} - 291249051480 q^{40} + 567822762272 q^{43} + 95457780144 q^{46} + 2861171717628 q^{49} - 1014177699808 q^{52} + 1351760277600 q^{55} - 9671683881852 q^{58} - 66091393528 q^{61} - 13634169987344 q^{64} + 15196895050208 q^{67} + 10695968312400 q^{70} + 20468321135936 q^{73} + 19129607873600 q^{76} + 19688660582384 q^{79} - 68290319744532 q^{82} - 38973146130360 q^{85} + 29271666534624 q^{88} - 221111416338176 q^{91} + 73435199966064 q^{94} + 104819071127168 q^{97}+O(q^{100}) $ 4 * q - 3332 * q^4 - 1065904 * q^7 - 20201220 * q^10 - 193633024 * q^13 - 1027273976 * q^16 - 1738250560 * q^19 + 5179999824 * q^22 + 18465731740 * q^25 + 23689734992 * q^28 - 22530472816 * q^31 - 118458902412 * q^34 - 135170189128 * q^37 - 291249051480 * q^40 + 567822762272 * q^43 + 95457780144 * q^46 + 2861171717628 * q^49 - 1014177699808 * q^52 + 1351760277600 * q^55 - 9671683881852 * q^58 - 66091393528 * q^61 - 13634169987344 * q^64 + 15196895050208 * q^67 + 10695968312400 * q^70 + 20468321135936 * q^73 + 19129607873600 * q^76 + 19688660582384 * q^79 - 68290319744532 * q^82 - 38973146130360 * q^85 + 29271666534624 * q^88 - 221111416338176 * q^91 + 73435199966064 * q^94 + 104819071127168 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 110.725i	− 0.865041i	−0.901624	−	0.432521i	$-0.857624\pi$
$2$	− 110.725i	− 0.865041i	0.901624	−	0.432521i	$-0.142376\pi$
$3$	0	0
$3$	0	0
$4$	4123.91	0.251703
$5$	− 35180.3i	− 0.450307i	−0.974323	−	0.225154i	$-0.927712\pi$
$5$	− 35180.3i	− 0.450307i	0.974323	−	0.225154i	$-0.0722884\pi$
$6$	0	0
$7$	883527.	1.07284	0.536418	−	0.843952i	$-0.319777\pi$
$7$	883527.	1.07284	0.536418	+	0.843952i	$0.319777\pi$
$8$	− 2.27074e6i	− 1.08278i
$9$	0	0
$10$	−3.89535e6	−0.389535
$11$	− 1.16632e7i	− 0.598509i	−0.954173	−	0.299254i	$-0.903262\pi$
$11$	− 1.16632e7i	− 0.598509i	0.954173	−	0.299254i	$-0.0967379\pi$
$12$	0	0
$13$	−1.07693e8	−1.71626	−0.858131	−	0.513431i	$-0.828374\pi$
$13$	−1.07693e8	−1.71626	−0.858131	+	0.513431i	$0.828374\pi$
$14$	− 9.78288e7i	− 0.928048i
$15$	0	0
$16$	−1.83863e8	−0.684942
$17$	− 6.94152e8i	− 1.69166i	−0.533455	−	0.845828i	$-0.679107\pi$
$17$	− 6.94152e8i	− 1.69166i	0.533455	−	0.845828i	$-0.320893\pi$
$18$	0	0
$19$	4.57205e8	0.511488	0.255744	−	0.966744i	$-0.417680\pi$
$19$	4.57205e8	0.511488	0.255744	+	0.966744i	$0.417680\pi$
$20$	− 1.45080e8i	− 0.113344i
$21$	0	0
$22$	−1.29142e9	−0.517735
$23$	3.31786e9i	0.974457i	0.873274	+	0.487229i	$0.161992\pi$
$23$	3.31786e9i	0.974457i	−0.873274	+	0.487229i	$0.838008\pi$
$24$	0	0
$25$	4.86586e9	0.797223
$26$	1.19243e10i	1.48464i
$27$	0	0
$28$	3.64358e9	0.270037
$29$	− 5.59359e9i	− 0.324268i	−0.986769	−	0.162134i	$-0.948162\pi$
$29$	− 5.59359e9i	− 0.324268i	0.986769	−	0.162134i	$-0.0518377\pi$
$30$	0	0
$31$	2.96133e10	1.07636	0.538178	−	0.842831i	$-0.319113\pi$
$31$	2.96133e10	1.07636	0.538178	+	0.842831i	$0.319113\pi$
$32$	− 1.68456e10i	− 0.490272i
$33$	0	0
$34$	−7.68602e10	−1.46335
$35$	− 3.10827e10i	− 0.483106i
$36$	0	0
$37$	−6.69873e10	−0.705636	−0.352818	−	0.935692i	$-0.614777\pi$
$37$	−6.69873e10	−0.705636	−0.352818	+	0.935692i	$0.614777\pi$
$38$	− 5.06242e10i	− 0.442459i
$39$	0	0
$40$	−7.98854e10	−0.487582
$41$	− 4.19037e10i	− 0.215162i	−0.994196	−	0.107581i	$-0.965690\pi$
$41$	− 4.19037e10i	− 0.215162i	0.994196	−	0.107581i	$-0.0343105\pi$
$42$	0	0
$43$	2.68989e11	0.989591	0.494796	−	0.869009i	$-0.335243\pi$
$43$	2.68989e11	0.989591	0.494796	+	0.869009i	$0.335243\pi$
$44$	− 4.80981e10i	− 0.150647i
$45$	0	0
$46$	3.67371e11	0.842946
$47$	8.88014e11i	1.75281i	0.481574	+	0.876405i	$0.340065\pi$
$47$	8.88014e11i	1.75281i	−0.481574	+	0.876405i	$0.659935\pi$
$48$	0	0
$49$	1.02397e11	0.150978
$50$	− 5.38774e11i	− 0.689631i
$51$	0	0
$52$	−4.44116e11	−0.431989
$53$	2.06534e12i	1.75816i	0.476670	+	0.879082i	$0.341844\pi$
$53$	2.06534e12i	1.75816i	−0.476670	+	0.879082i	$0.658156\pi$
$54$	0	0
$55$	−4.10316e11	−0.269513
$56$	− 2.00626e12i	− 1.16164i
$57$	0	0
$58$	−6.19352e11	−0.280506
$59$	− 3.12449e12i	− 1.25550i	−0.778417	−	0.627748i	$-0.783977\pi$
$59$	− 3.12449e12i	− 1.25550i	0.778417	−	0.627748i	$-0.216023\pi$
$60$	0	0
$61$	1.70805e12	0.543491	0.271745	−	0.962369i	$-0.412399\pi$
$61$	1.70805e12	0.543491	0.271745	+	0.962369i	$0.412399\pi$
$62$	− 3.27895e12i	− 0.931092i
$63$	0	0
$64$	−4.87764e12	−1.10905
$65$	3.78866e12i	0.772845i
$66$	0	0
$67$	2.82893e12	0.466765	0.233383	−	0.972385i	$-0.425021\pi$
$67$	2.82893e12	0.466765	0.233383	+	0.972385i	$0.425021\pi$
$68$	− 2.86262e12i	− 0.425796i
$69$	0	0
$70$	−3.44164e12	−0.417907
$71$	1.53177e13i	1.68417i	0.539343	+	0.842086i	$0.318673\pi$
$71$	1.53177e13i	1.68417i	−0.539343	+	0.842086i	$0.681327\pi$
$72$	0	0
$73$	9.14210e12	0.827534	0.413767	−	0.910383i	$-0.364213\pi$
$73$	9.14210e12	0.827534	0.413767	+	0.910383i	$0.364213\pi$
$74$	7.41719e12i	0.610404i
$75$	0	0
$76$	1.88547e12	0.128743
$77$	− 1.03048e13i	− 0.642102i
$78$	0	0
$79$	2.06959e13	1.07769	0.538846	−	0.842404i	$-0.318860\pi$
$79$	2.06959e13	1.07769	0.538846	+	0.842404i	$0.318860\pi$
$80$	6.46834e12i	0.308434i
$81$	0	0
$82$	−4.63980e12	−0.186124
$83$	− 8.35164e12i	− 0.307769i	−0.988089	−	0.153885i	$-0.950822\pi$
$83$	− 8.35164e12i	− 0.307769i	0.988089	−	0.153885i	$-0.0491784\pi$
$84$	0	0
$85$	−2.44205e13	−0.761765
$86$	− 2.97839e13i	− 0.856038i
$87$	0	0
$88$	−2.64842e13	−0.648050
$89$	− 1.57894e13i	− 0.356973i	−0.983942	−	0.178487i	$-0.942880\pi$
$89$	− 1.57894e13i	− 0.356973i	0.983942	−	0.178487i	$-0.0571202\pi$
$90$	0	0
$91$	−9.51496e13	−1.84127
$92$	1.36825e13i	0.245274i
$93$	0	0
$94$	9.83256e13	1.51625
$95$	− 1.60846e13i	− 0.230327i
$96$	0	0
$97$	3.41627e13	0.422814	0.211407	−	0.977398i	$-0.432195\pi$
$97$	3.41627e13	0.422814	0.211407	+	0.977398i	$0.432195\pi$
$98$	− 1.13379e13i	− 0.130602i
$99$	0	0
$100$	2.00664e13	0.200664
$101$	− 7.98246e12i	− 0.0744539i	−0.999307	−	0.0372269i	$-0.988148\pi$
$101$	− 7.98246e12i	− 0.0744539i	0.999307	−	0.0372269i	$-0.0118524\pi$
$102$	0	0
$103$	2.33697e13	0.190017	0.0950085	−	0.995476i	$-0.469712\pi$
$103$	2.33697e13	0.190017	0.0950085	+	0.995476i	$0.469712\pi$
$104$	2.44543e14i	1.85833i
$105$	0	0
$106$	2.28685e14	1.52088
$107$	− 4.37651e13i	− 0.272547i	−0.990671	−	0.136274i	$-0.956487\pi$
$107$	− 4.37651e13i	− 0.272547i	0.990671	−	0.136274i	$-0.0435126\pi$
$108$	0	0
$109$	−2.01167e14	−1.10045	−0.550227	−	0.835015i	$-0.685459\pi$
$109$	−2.01167e14	−1.10045	−0.550227	+	0.835015i	$0.685459\pi$
$110$	4.54324e13i	0.233140i
$111$	0	0
$112$	−1.62448e14	−0.734831
$113$	9.91777e13i	0.421565i	0.977533	+	0.210783i	$0.0676013\pi$
$113$	9.91777e13i	0.421565i	−0.977533	+	0.210783i	$0.932399\pi$
$114$	0	0
$115$	1.16723e14	0.438805
$116$	− 2.30675e13i	− 0.0816195i
$117$	0	0
$118$	−3.45960e14	−1.08606
$119$	− 6.13302e14i	− 1.81487i
$120$	0	0
$121$	2.43719e14	0.641787
$122$	− 1.89124e14i	− 0.470142i
$123$	0	0
$124$	1.22123e14	0.270922
$125$	− 3.85906e14i	− 0.809303i
$126$	0	0
$127$	−5.29363e14	−0.993407	−0.496704	−	0.867920i	$-0.665456\pi$
$127$	−5.29363e14	−0.993407	−0.496704	+	0.867920i	$0.665456\pi$
$128$	2.64080e14i	0.469100i
$129$	0	0
$130$	4.19501e14	0.668543
$131$	7.20146e14i	1.08773i	0.839173	+	0.543865i	$0.183040\pi$
$131$	7.20146e14i	1.08773i	−0.839173	+	0.543865i	$0.816960\pi$
$132$	0	0
$133$	4.03953e14	0.548743
$134$	− 3.13234e14i	− 0.403771i
$135$	0	0
$136$	−1.57624e15	−1.83168
$137$	1.37360e15i	1.51641i	0.652016	+	0.758206i	$0.273924\pi$
$137$	1.37360e15i	1.51641i	−0.652016	+	0.758206i	$0.726076\pi$
$138$	0	0
$139$	1.14888e15	1.14596	0.572980	−	0.819570i	$-0.305787\pi$
$139$	1.14888e15	1.14596	0.572980	+	0.819570i	$0.305787\pi$
$140$	− 1.28182e14i	− 0.121599i
$141$	0	0
$142$	1.69606e15	1.45688
$143$	1.25605e15i	1.02720i
$144$	0	0
$145$	−1.96784e14	−0.146020
$146$	− 1.01226e15i	− 0.715852i
$147$	0	0
$148$	−2.76250e14	−0.177611
$149$	− 6.67540e14i	− 0.409424i	−0.978822	−	0.204712i	$-0.934374\pi$
$149$	− 6.67540e14i	− 0.409424i	0.978822	−	0.204712i	$-0.0656258\pi$
$150$	0	0
$151$	1.45481e14	0.0812772	0.0406386	−	0.999174i	$-0.487061\pi$
$151$	1.45481e14	0.0812772	0.0406386	+	0.999174i	$0.487061\pi$
$152$	− 1.03820e15i	− 0.553827i
$153$	0	0
$154$	−1.14100e15	−0.555445
$155$	− 1.04181e15i	− 0.484691i
$156$	0	0
$157$	−5.28795e14	−0.224900	−0.112450	−	0.993657i	$-0.535870\pi$
$157$	−5.28795e14	−0.224900	−0.112450	+	0.993657i	$0.535870\pi$
$158$	− 2.29156e15i	− 0.932249i
$159$	0	0
$160$	−5.92633e14	−0.220773
$161$	2.93142e15i	1.04543i
$162$	0	0
$163$	−5.79477e15	−1.89549	−0.947747	−	0.319022i	$-0.896646\pi$
$163$	−5.79477e15	−1.89549	−0.947747	+	0.319022i	$0.896646\pi$
$164$	− 1.72807e14i	− 0.0541570i
$165$	0	0
$166$	−9.24738e14	−0.266233
$167$	− 3.18234e15i	− 0.878480i	−0.898370	−	0.439240i	$-0.855248\pi$
$167$	− 3.18234e15i	− 0.878480i	0.898370	−	0.439240i	$-0.144752\pi$
$168$	0	0
$169$	7.66038e15	1.94555
$170$	2.70396e15i	0.658959i
$171$	0	0
$172$	1.10929e15	0.249084
$173$	2.07646e15i	0.447715i	0.974622	+	0.223857i	$0.0718650\pi$
$173$	2.07646e15i	0.447715i	−0.974622	+	0.223857i	$0.928135\pi$
$174$	0	0
$175$	4.29912e15	0.855290
$176$	2.14444e15i	0.409944i
$177$	0	0
$178$	−1.74829e15	−0.308797
$179$	− 5.08283e15i	− 0.863245i	−0.902054	−	0.431623i	$-0.857941\pi$
$179$	− 5.08283e15i	− 0.863245i	0.902054	−	0.431623i	$-0.142059\pi$
$180$	0	0
$181$	1.92902e15	0.303101	0.151550	−	0.988450i	$-0.451573\pi$
$181$	1.92902e15	0.303101	0.151550	+	0.988450i	$0.451573\pi$
$182$	1.05355e16i	1.59277i
$183$	0	0
$184$	7.53401e15	1.05512
$185$	2.35663e15i	0.317753i
$186$	0	0
$187$	−8.09606e15	−1.01247
$188$	3.66209e15i	0.441188i
$189$	0	0
$190$	−1.78097e15	−0.199242
$191$	− 9.36150e15i	− 1.00951i	−0.863262	−	0.504756i	$-0.831582\pi$
$191$	− 9.36150e15i	− 1.00951i	0.863262	−	0.504756i	$-0.168418\pi$
$192$	0	0
$193$	−1.39399e16	−1.39752	−0.698762	−	0.715354i	$-0.746266\pi$
$193$	−1.39399e16	−1.39752	−0.698762	+	0.715354i	$0.746266\pi$
$194$	− 3.78267e15i	− 0.365752i
$195$	0	0
$196$	4.22274e14	0.0380016
$197$	− 3.18295e15i	− 0.276418i	−0.990403	−	0.138209i	$-0.955865\pi$
$197$	− 3.18295e15i	− 0.276418i	0.990403	−	0.138209i	$-0.0441346\pi$
$198$	0	0
$199$	1.71134e15	0.138473	0.0692365	−	0.997600i	$-0.477944\pi$
$199$	1.71134e15	0.138473	0.0692365	+	0.997600i	$0.477944\pi$
$200$	− 1.10491e16i	− 0.863214i
$201$	0	0
$202$	−8.83860e14	−0.0644057
$203$	− 4.94209e15i	− 0.347887i
$204$	0	0
$205$	−1.47418e15	−0.0968890
$206$	− 2.58762e15i	− 0.164373i
$207$	0	0
$208$	1.98007e16	1.17554
$209$	− 5.33249e15i	− 0.306130i
$210$	0	0
$211$	−1.45864e16	−0.783375	−0.391688	−	0.920098i	$-0.628109\pi$
$211$	−1.45864e16	−0.783375	−0.391688	+	0.920098i	$0.628109\pi$
$212$	8.51725e15i	0.442536i
$213$	0	0
$214$	−4.84591e15	−0.235765
$215$	− 9.46312e15i	− 0.445620i
$216$	0	0
$217$	2.61642e16	1.15475
$218$	2.22743e16i	0.951939i
$219$	0	0
$220$	−1.69211e15	−0.0678373
$221$	7.47552e16i	2.90333i
$222$	0	0
$223$	−3.55539e16	−1.29644	−0.648220	−	0.761453i	$-0.724486\pi$
$223$	−3.55539e16	−1.29644	−0.648220	+	0.761453i	$0.724486\pi$
$224$	− 1.48836e16i	− 0.525982i
$225$	0	0
$226$	1.09815e16	0.364672
$227$	− 1.64473e16i	− 0.529559i	−0.964309	−	0.264780i	$-0.914701\pi$
$227$	− 1.64473e16i	− 0.529559i	0.964309	−	0.264780i	$-0.0852992\pi$
$228$	0	0
$229$	5.36604e16	1.62482	0.812410	−	0.583086i	$-0.198155\pi$
$229$	5.36604e16	1.62482	0.812410	+	0.583086i	$0.198155\pi$
$230$	− 1.29242e16i	− 0.379585i
$231$	0	0
$232$	−1.27016e16	−0.351110
$233$	1.80740e16i	0.484800i	0.970176	+	0.242400i	$0.0779347\pi$
$233$	1.80740e16i	0.484800i	−0.970176	+	0.242400i	$0.922065\pi$
$234$	0	0
$235$	3.12406e16	0.789303
$236$	− 1.28851e16i	− 0.316013i
$237$	0	0
$238$	−6.79080e16	−1.56994
$239$	− 4.85761e16i	− 1.09053i	−0.838264	−	0.545265i	$-0.816429\pi$
$239$	− 4.85761e16i	− 1.09053i	0.838264	−	0.545265i	$-0.183571\pi$
$240$	0	0
$241$	1.53621e16	0.325335	0.162667	−	0.986681i	$-0.447990\pi$
$241$	1.53621e16	0.325335	0.162667	+	0.986681i	$0.447990\pi$
$242$	− 2.69858e16i	− 0.555173i
$243$	0	0
$244$	7.04385e15	0.136798
$245$	− 3.60234e15i	− 0.0679864i
$246$	0	0
$247$	−4.92377e16	−0.877848
$248$	− 6.72443e16i	− 1.16545i
$249$	0	0
$250$	−4.27295e16	−0.700081
$251$	1.44706e16i	0.230553i	0.993333	+	0.115276i	$0.0367754\pi$
$251$	1.44706e16i	0.230553i	−0.993333	+	0.115276i	$0.963225\pi$
$252$	0	0
$253$	3.86970e16	0.583221
$254$	5.86139e16i	0.859339i
$255$	0	0
$256$	−5.06750e16	−0.703257
$257$	5.74778e16i	0.776190i	0.921619	+	0.388095i	$0.126867\pi$
$257$	5.74778e16i	0.776190i	−0.921619	+	0.388095i	$0.873133\pi$
$258$	0	0
$259$	−5.91851e16	−0.757032
$260$	1.56241e16i	0.194528i
$261$	0	0
$262$	7.97383e16	0.940932
$263$	− 7.67487e16i	− 0.881821i	−0.897551	−	0.440911i	$-0.854656\pi$
$263$	− 7.67487e16i	− 0.881821i	0.897551	−	0.440911i	$-0.145344\pi$
$264$	0	0
$265$	7.26590e16	0.791714
$266$	− 4.47278e16i	− 0.474686i
$267$	0	0
$268$	1.16662e16	0.117486
$269$	1.74215e17i	1.70930i	0.519203	+	0.854651i	$0.326229\pi$
$269$	1.74215e17i	1.70930i	−0.519203	+	0.854651i	$0.673771\pi$
$270$	0	0
$271$	9.30219e16	0.866563	0.433282	−	0.901259i	$-0.357356\pi$
$271$	9.30219e16	0.866563	0.433282	+	0.901259i	$0.357356\pi$
$272$	1.27629e17i	1.15869i
$273$	0	0
$274$	1.52092e17	1.31176
$275$	− 5.67518e16i	− 0.477145i
$276$	0	0
$277$	−7.02731e14	−0.00561605	−0.00280802	−	0.999996i	$-0.500894\pi$
$277$	−7.02731e14	−0.00561605	−0.00280802	+	0.999996i	$0.500894\pi$
$278$	− 1.27210e17i	− 0.991302i
$279$	0	0
$280$	−7.05809e16	−0.523095
$281$	4.78678e16i	0.346019i	0.984920	+	0.173009i	$0.0553491\pi$
$281$	4.78678e16i	0.346019i	−0.984920	+	0.173009i	$0.944651\pi$
$282$	0	0
$283$	−1.93594e17	−1.33164	−0.665818	−	0.746114i	$-0.731917\pi$
$283$	−1.93594e17	−1.33164	−0.665818	+	0.746114i	$0.731917\pi$
$284$	6.31690e16i	0.423912i
$285$	0	0
$286$	1.39076e17	0.888569
$287$	− 3.70230e16i	− 0.230833i
$288$	0	0
$289$	−3.13469e17	−1.86170
$290$	2.17890e16i	0.126314i
$291$	0	0
$292$	3.77012e16	0.208293
$293$	− 2.27930e17i	− 1.22950i	−0.788722	−	0.614750i	$-0.789257\pi$
$293$	− 2.27930e17i	− 1.22950i	0.788722	−	0.614750i	$-0.210743\pi$
$294$	0	0
$295$	−1.09920e17	−0.565359
$296$	1.52111e17i	0.764045i
$297$	0	0
$298$	−7.39136e16	−0.354169
$299$	− 3.57310e17i	− 1.67242i
$300$	0	0
$301$	2.37659e17	1.06167
$302$	− 1.61085e16i	− 0.0703081i
$303$	0	0
$304$	−8.40630e16	−0.350340
$305$	− 6.00897e16i	− 0.244738i
$306$	0	0
$307$	−4.36377e15	−0.0169783	−0.00848913	−	0.999964i	$-0.502702\pi$
$307$	−4.36377e15	−0.0169783	−0.00848913	+	0.999964i	$0.502702\pi$
$308$	− 4.24960e16i	− 0.161619i
$309$	0	0
$310$	−1.15354e17	−0.419277
$311$	1.95266e17i	0.693912i	0.937881	+	0.346956i	$0.112785\pi$
$311$	1.95266e17i	0.693912i	−0.937881	+	0.346956i	$0.887215\pi$
$312$	0	0
$313$	4.74136e17	1.61099	0.805494	−	0.592604i	$-0.201900\pi$
$313$	4.74136e17	1.61099	0.805494	+	0.592604i	$0.201900\pi$
$314$	5.85510e16i	0.194548i
$315$	0	0
$316$	8.53481e16	0.271259
$317$	1.42514e17i	0.443040i	0.975156	+	0.221520i	$0.0711018\pi$
$317$	1.42514e17i	0.443040i	−0.975156	+	0.221520i	$0.928898\pi$
$318$	0	0
$319$	−6.52394e16	−0.194077
$320$	1.71597e17i	0.499412i
$321$	0	0
$322$	3.24582e17	0.904343
$323$	− 3.17370e17i	− 0.865263i
$324$	0	0
$325$	−5.24019e17	−1.36824
$326$	6.41627e17i	1.63968i
$327$	0	0
$328$	−9.51526e16	−0.232972
$329$	7.84584e17i	1.88048i
$330$	0	0
$331$	3.38601e17	0.777843	0.388922	−	0.921271i	$-0.372848\pi$
$331$	3.38601e17	0.777843	0.388922	+	0.921271i	$0.372848\pi$
$332$	− 3.44414e16i	− 0.0774665i
$333$	0	0
$334$	−3.52366e17	−0.759921
$335$	− 9.95224e16i	− 0.210188i
$336$	0	0
$337$	5.63776e17	1.14208	0.571041	−	0.820922i	$-0.306540\pi$
$337$	5.63776e17	1.14208	0.571041	+	0.820922i	$0.306540\pi$
$338$	− 8.48198e17i	− 1.68299i
$339$	0	0
$340$	−1.00708e17	−0.191739
$341$	− 3.45388e17i	− 0.644208i
$342$	0	0
$343$	−5.08758e17	−0.910862
$344$	− 6.10806e17i	− 1.07151i
$345$	0	0
$346$	2.29917e17	0.387292
$347$	6.66164e17i	1.09970i	0.835263	+	0.549851i	$0.185316\pi$
$347$	6.66164e17i	1.09970i	−0.835263	+	0.549851i	$0.814684\pi$
$348$	0	0
$349$	−9.80546e16	−0.155485	−0.0777427	−	0.996973i	$-0.524771\pi$
$349$	−9.80546e16	−0.155485	−0.0777427	+	0.996973i	$0.524771\pi$
$350$	− 4.76022e17i	− 0.739861i
$351$	0	0
$352$	−1.96475e17	−0.293432
$353$	− 9.85542e17i	− 1.44295i	−0.692439	−	0.721476i	$-0.743464\pi$
$353$	− 9.85542e17i	− 1.44295i	0.692439	−	0.721476i	$-0.256536\pi$
$354$	0	0
$355$	5.38882e17	0.758395
$356$	− 6.51141e16i	− 0.0898514i
$357$	0	0
$358$	−5.62798e17	−0.746743
$359$	1.08574e18i	1.41275i	0.707840	+	0.706373i	$0.249670\pi$
$359$	1.08574e18i	1.41275i	−0.707840	+	0.706373i	$0.750330\pi$
$360$	0	0
$361$	−5.89970e17	−0.738380
$362$	− 2.13591e17i	− 0.262195i
$363$	0	0
$364$	−3.92388e17	−0.463453
$365$	− 3.21622e17i	− 0.372645i
$366$	0	0
$367$	−2.62517e15	−0.00292749	−0.00146374	−	0.999999i	$-0.500466\pi$
$367$	−2.62517e15	−0.00292749	−0.00146374	+	0.999999i	$0.500466\pi$
$368$	− 6.10030e17i	− 0.667447i
$369$	0	0
$370$	2.60939e17	0.274870
$371$	1.82478e18i	1.88622i
$372$	0	0
$373$	−6.80263e17	−0.677197	−0.338598	−	0.940931i	$-0.609953\pi$
$373$	−6.80263e17	−0.677197	−0.338598	+	0.940931i	$0.609953\pi$
$374$	8.96439e17i	0.875830i
$375$	0	0
$376$	2.01645e18	1.89790
$377$	6.02390e17i	0.556529i
$378$	0	0
$379$	1.49973e18	1.33518	0.667589	−	0.744530i	$-0.267326\pi$
$379$	1.49973e18	1.33518	0.667589	+	0.744530i	$0.267326\pi$
$380$	− 6.63314e16i	− 0.0579741i
$381$	0	0
$382$	−1.03655e18	−0.873270
$383$	− 1.51578e18i	− 1.25384i	−0.779082	−	0.626922i	$-0.784315\pi$
$383$	− 1.51578e18i	− 1.25384i	0.779082	−	0.626922i	$-0.215685\pi$
$384$	0	0
$385$	−3.62525e17	−0.289143
$386$	1.54350e18i	1.20892i
$387$	0	0
$388$	1.40884e17	0.106424
$389$	1.81414e17i	0.134593i	0.997733	+	0.0672966i	$0.0214374\pi$
$389$	1.81414e17i	0.134593i	−0.997733	+	0.0672966i	$0.978563\pi$
$390$	0	0
$391$	2.30310e18	1.64845
$392$	− 2.32517e17i	− 0.163475i
$393$	0	0
$394$	−3.52434e17	−0.239113
$395$	− 7.28088e17i	− 0.485293i
$396$	0	0
$397$	6.62747e17	0.426397	0.213199	−	0.977009i	$-0.431612\pi$
$397$	6.62747e17	0.426397	0.213199	+	0.977009i	$0.431612\pi$
$398$	− 1.89489e17i	− 0.119785i
$399$	0	0
$400$	−8.94651e17	−0.546052
$401$	− 1.51330e18i	− 0.907644i	−0.891092	−	0.453822i	$-0.850060\pi$
$401$	− 1.51330e18i	− 0.907644i	0.891092	−	0.453822i	$-0.149940\pi$
$402$	0	0
$403$	−3.18915e18	−1.84731
$404$	− 3.29189e16i	− 0.0187403i
$405$	0	0
$406$	−5.47214e17	−0.300937
$407$	7.81290e17i	0.422329i
$408$	0	0
$409$	−1.84141e18	−0.961807	−0.480904	−	0.876773i	$-0.659691\pi$
$409$	−1.84141e18	−0.961807	−0.480904	+	0.876773i	$0.659691\pi$
$410$	1.63229e17i	0.0838130i
$411$	0	0
$412$	9.63745e16	0.0478279
$413$	− 2.76057e18i	− 1.34694i
$414$	0	0
$415$	−2.93813e17	−0.138591
$416$	1.81415e18i	0.841435i
$417$	0	0
$418$	−5.90442e17	−0.264815
$419$	− 1.85544e18i	− 0.818369i	−0.912452	−	0.409184i	$-0.865813\pi$
$419$	− 1.85544e18i	− 0.818369i	0.912452	−	0.409184i	$-0.134187\pi$
$420$	0	0
$421$	−6.91577e17	−0.295030	−0.147515	−	0.989060i	$-0.547127\pi$
$421$	−6.91577e17	−0.295030	−0.147515	+	0.989060i	$0.547127\pi$
$422$	1.61508e18i	0.677652i
$423$	0	0
$424$	4.68985e18	1.90370
$425$	− 3.37765e18i	− 1.34863i
$426$	0	0
$427$	1.50911e18	0.583076
$428$	− 1.80483e17i	− 0.0686011i
$429$	0	0
$430$	−1.04781e18	−0.385480
$431$	4.62095e18i	1.67259i	0.548279	+	0.836296i	$0.315283\pi$
$431$	4.62095e18i	1.67259i	−0.548279	+	0.836296i	$0.684717\pi$
$432$	0	0
$433$	−1.41466e18	−0.495720	−0.247860	−	0.968796i	$-0.579727\pi$
$433$	−1.41466e18	−0.495720	−0.247860	+	0.968796i	$0.579727\pi$
$434$	− 2.89704e18i	− 0.998909i
$435$	0	0
$436$	−8.29596e17	−0.276988
$437$	1.51694e18i	0.498424i
$438$	0	0
$439$	2.29678e18	0.730917	0.365459	−	0.930828i	$-0.380912\pi$
$439$	2.29678e18	0.730917	0.365459	+	0.930828i	$0.380912\pi$
$440$	9.31723e17i	0.291822i
$441$	0	0
$442$	8.27730e18	2.51150
$443$	4.28200e18i	1.27885i	0.768853	+	0.639426i	$0.220828\pi$
$443$	4.28200e18i	1.27885i	−0.768853	+	0.639426i	$0.779172\pi$
$444$	0	0
$445$	−5.55476e17	−0.160748
$446$	3.93671e18i	1.12147i
$447$	0	0
$448$	−4.30953e18	−1.18983
$449$	− 2.12941e18i	− 0.578808i	−0.957207	−	0.289404i	$-0.906543\pi$
$449$	− 2.12941e18i	− 0.578808i	0.957207	−	0.289404i	$-0.0934571\pi$
$450$	0	0
$451$	−4.88733e17	−0.128776
$452$	4.09000e17i	0.106109i
$453$	0	0
$454$	−1.82114e18	−0.458091
$455$	3.34739e18i	0.829137i
$456$	0	0
$457$	−4.65543e18	−1.11827	−0.559134	−	0.829077i	$-0.688866\pi$
$457$	−4.65543e18	−1.11827	−0.559134	+	0.829077i	$0.688866\pi$
$458$	− 5.94156e18i	− 1.40554i
$459$	0	0
$460$	4.81355e17	0.110449
$461$	4.67445e17i	0.105639i	0.998604	+	0.0528195i	$0.0168208\pi$
$461$	4.67445e17i	0.105639i	−0.998604	+	0.0528195i	$0.983179\pi$
$462$	0	0
$463$	9.15959e17	0.200821	0.100410	−	0.994946i	$-0.467984\pi$
$463$	9.15959e17	0.200821	0.100410	+	0.994946i	$0.467984\pi$
$464$	1.02845e18i	0.222105i
$465$	0	0
$466$	2.00125e18	0.419372
$467$	− 2.90169e18i	− 0.599007i	−0.954095	−	0.299504i	$-0.903179\pi$
$467$	− 2.90169e18i	− 0.599007i	0.954095	−	0.299504i	$-0.0968211\pi$
$468$	0	0
$469$	2.49943e18	0.500762
$470$	− 3.45912e18i	− 0.682780i
$471$	0	0
$472$	−7.09492e18	−1.35942
$473$	− 3.13729e18i	− 0.592279i
$474$	0	0
$475$	2.22470e18	0.407770
$476$	− 2.52920e18i	− 0.456809i
$477$	0	0
$478$	−5.37861e18	−0.943354
$479$	− 3.64102e18i	− 0.629325i	−0.949204	−	0.314662i	$-0.898109\pi$
$479$	− 3.64102e18i	− 0.629325i	0.949204	−	0.314662i	$-0.101891\pi$
$480$	0	0
$481$	7.21406e18	1.21106
$482$	− 1.70097e18i	− 0.281428i
$483$	0	0
$484$	1.00507e18	0.161540
$485$	− 1.20185e18i	− 0.190396i
$486$	0	0
$487$	−8.79353e17	−0.135351	−0.0676754	−	0.997707i	$-0.521558\pi$
$487$	−8.79353e17	−0.135351	−0.0676754	+	0.997707i	$0.521558\pi$
$488$	− 3.87855e18i	− 0.588478i
$489$	0	0
$490$	−3.98870e17	−0.0588111
$491$	6.00110e18i	0.872290i	0.899876	+	0.436145i	$0.143656\pi$
$491$	6.00110e18i	0.872290i	−0.899876	+	0.436145i	$0.856344\pi$
$492$	0	0
$493$	−3.88280e18	−0.548551
$494$	5.45186e18i	0.759375i
$495$	0	0
$496$	−5.44479e18	−0.737241
$497$	1.35336e19i	1.80684i
$498$	0	0
$499$	4.69782e18	0.609807	0.304903	−	0.952383i	$-0.401376\pi$
$499$	4.69782e18	0.609807	0.304903	+	0.952383i	$0.401376\pi$
$500$	− 1.59144e18i	− 0.203704i
$501$	0	0
$502$	1.60226e18	0.199438
$503$	2.79016e18i	0.342494i	0.985228	+	0.171247i	$0.0547797\pi$
$503$	2.79016e18i	0.342494i	−0.985228	+	0.171247i	$0.945220\pi$
$504$	0	0
$505$	−2.80825e17	−0.0335271
$506$	− 4.28473e18i	− 0.504510i
$507$	0	0
$508$	−2.18304e18	−0.250044
$509$	2.04248e18i	0.230745i	0.993322	+	0.115373i	$0.0368062\pi$
$509$	2.04248e18i	0.230745i	−0.993322	+	0.115373i	$0.963194\pi$
$510$	0	0
$511$	8.07729e18	0.887809
$512$	9.93769e18i	1.07745i
$513$	0	0
$514$	6.36425e18	0.671437
$515$	− 8.22152e17i	− 0.0855660i
$516$	0	0
$517$	1.03571e19	1.04907
$518$	6.55329e18i	0.654864i
$519$	0	0
$520$	8.60309e18	0.836818
$521$	− 4.24170e18i	− 0.407077i	−0.979067	−	0.203538i	$-0.934756\pi$
$521$	− 4.24170e18i	− 0.407077i	0.979067	−	0.203538i	$-0.0652441\pi$
$522$	0	0
$523$	−1.52348e19	−1.42340	−0.711698	−	0.702486i	$-0.752073\pi$
$523$	−1.52348e19	−1.42340	−0.711698	+	0.702486i	$0.752073\pi$
$524$	2.96981e18i	0.273785i
$525$	0	0
$526$	−8.49802e18	−0.762812
$527$	− 2.05562e19i	− 1.82082i
$528$	0	0
$529$	5.84660e17	0.0504329
$530$	− 8.04519e18i	− 0.684866i
$531$	0	0
$532$	1.66587e18	0.138121
$533$	4.51273e18i	0.369274i
$534$	0	0
$535$	−1.53967e18	−0.122730
$536$	− 6.42377e18i	− 0.505402i
$537$	0	0
$538$	1.92900e19	1.47862
$539$	− 1.19428e18i	− 0.0903615i
$540$	0	0
$541$	−2.21223e19	−1.63099	−0.815493	−	0.578767i	$-0.803534\pi$
$541$	−2.21223e19	−1.63099	−0.815493	+	0.578767i	$0.803534\pi$
$542$	− 1.02999e19i	− 0.749613i
$543$	0	0
$544$	−1.16934e19	−0.829372
$545$	7.07712e18i	0.495543i
$546$	0	0
$547$	−2.02110e19	−1.37936	−0.689679	−	0.724115i	$-0.742248\pi$
$547$	−2.02110e19	−1.37936	−0.689679	+	0.724115i	$0.742248\pi$
$548$	5.66461e18i	0.381686i
$549$	0	0
$550$	−6.28386e18	−0.412750
$551$	− 2.55742e18i	− 0.165860i
$552$	0	0
$553$	1.82854e19	1.15619
$554$	7.78101e16i	0.00485811i
$555$	0	0
$556$	4.73786e18	0.288442
$557$	− 2.04747e18i	− 0.123093i	−0.998104	−	0.0615463i	$-0.980397\pi$
$557$	− 2.04747e18i	− 0.123093i	0.998104	−	0.0615463i	$-0.0196032\pi$
$558$	0	0
$559$	−2.89682e19	−1.69840
$560$	5.71495e18i	0.330900i
$561$	0	0
$562$	5.30018e18	0.299320
$563$	− 2.55905e19i	− 1.42732i	−0.700493	−	0.713659i	$-0.747037\pi$
$563$	− 2.55905e19i	− 1.42732i	0.700493	−	0.713659i	$-0.252963\pi$
$564$	0	0
$565$	3.48910e18	0.189834
$566$	2.14357e19i	1.15192i
$567$	0	0
$568$	3.47827e19	1.82358
$569$	− 3.28939e18i	− 0.170345i	−0.996366	−	0.0851724i	$-0.972856\pi$
$569$	− 3.28939e18i	− 0.170345i	0.996366	−	0.0851724i	$-0.0271441\pi$
$570$	0	0
$571$	2.96783e19	1.49964	0.749819	−	0.661643i	$-0.230141\pi$
$571$	2.96783e19	1.49964	0.749819	+	0.661643i	$0.230141\pi$
$572$	5.17983e18i	0.258549i
$573$	0	0
$574$	−4.09939e18	−0.199680
$575$	1.61442e19i	0.776860i
$576$	0	0
$577$	−1.39821e17	−0.00656661	−0.00328331	−	0.999995i	$-0.501045\pi$
$577$	−1.39821e17	−0.00656661	−0.00328331	+	0.999995i	$0.501045\pi$
$578$	3.47090e19i	1.61045i
$579$	0	0
$580$	−8.11519e17	−0.0367538
$581$	− 7.37890e18i	− 0.330186i
$582$	0	0
$583$	2.40885e19	1.05228
$584$	− 2.07594e19i	− 0.896034i
$585$	0	0
$586$	−2.52376e19	−1.06357
$587$	− 1.46302e19i	− 0.609234i	−0.952475	−	0.304617i	$-0.901471\pi$
$587$	− 1.46302e19i	− 0.609234i	0.952475	−	0.304617i	$-0.0985285\pi$
$588$	0	0
$589$	1.35394e19	0.550543
$590$	1.21710e19i	0.489059i
$591$	0	0
$592$	1.23165e19	0.483320
$593$	3.26597e19i	1.26657i	0.773918	+	0.633286i	$0.218294\pi$
$593$	3.26597e19i	1.26657i	−0.773918	+	0.633286i	$0.781706\pi$
$594$	0	0
$595$	−2.15761e19	−0.817250
$596$	− 2.75287e18i	− 0.103053i
$597$	0	0
$598$	−3.95632e19	−1.44672
$599$	3.23807e19i	1.17030i	0.810924	+	0.585151i	$0.198965\pi$
$599$	3.23807e19i	1.17030i	−0.810924	+	0.585151i	$0.801035\pi$
$600$	0	0
$601$	6.07127e18	0.214367	0.107184	−	0.994239i	$-0.465817\pi$
$601$	6.07127e18	0.214367	0.107184	+	0.994239i	$0.465817\pi$
$602$	− 2.63149e19i	− 0.918388i
$603$	0	0
$604$	5.99952e17	0.0204577
$605$	− 8.57409e18i	− 0.289002i
$606$	0	0
$607$	−3.25128e19	−1.07086	−0.535430	−	0.844579i	$-0.679851\pi$
$607$	−3.25128e19	−1.07086	−0.535430	+	0.844579i	$0.679851\pi$
$608$	− 7.70190e18i	− 0.250769i
$609$	0	0
$610$	−6.65345e18	−0.211708
$611$	− 9.56328e19i	− 3.00828i
$612$	0	0
$613$	5.00591e19	1.53907	0.769537	−	0.638602i	$-0.220487\pi$
$613$	5.00591e19	1.53907	0.769537	+	0.638602i	$0.220487\pi$
$614$	4.83180e17i	0.0146869i
$615$	0	0
$616$	−2.33995e19	−0.695252
$617$	4.85392e19i	1.42593i	0.701202	+	0.712963i	$0.252647\pi$
$617$	4.85392e19i	1.42593i	−0.701202	+	0.712963i	$0.747353\pi$
$618$	0	0
$619$	−4.94298e19	−1.41956	−0.709782	−	0.704422i	$-0.751206\pi$
$619$	−4.94298e19	−1.41956	−0.709782	+	0.704422i	$0.751206\pi$
$620$	− 4.29631e18i	− 0.121998i
$621$	0	0
$622$	2.16209e19	0.600263
$623$	− 1.39504e19i	− 0.382974i
$624$	0	0
$625$	1.61226e19	0.432788
$626$	− 5.24988e19i	− 1.39357i
$627$	0	0
$628$	−2.18070e18	−0.0566081
$629$	4.64994e19i	1.19369i
$630$	0	0
$631$	2.28715e19	0.574235	0.287117	−	0.957895i	$-0.407303\pi$
$631$	2.28715e19	0.574235	0.287117	+	0.957895i	$0.407303\pi$
$632$	− 4.69951e19i	− 1.16690i
$633$	0	0
$634$	1.57799e19	0.383248
$635$	1.86231e19i	0.447339i
$636$	0	0
$637$	−1.10274e19	−0.259117
$638$	7.22365e18i	0.167885i
$639$	0	0
$640$	9.29040e18	0.211239
$641$	− 1.67328e19i	− 0.376325i	−0.982138	−	0.188162i	$-0.939747\pi$
$641$	− 1.67328e19i	− 0.376325i	0.982138	−	0.188162i	$-0.0602532\pi$
$642$	0	0
$643$	4.23021e19	0.930861	0.465430	−	0.885084i	$-0.345900\pi$
$643$	4.23021e19	0.930861	0.465430	+	0.885084i	$0.345900\pi$
$644$	1.20889e19i	0.263139i
$645$	0	0
$646$	−3.51409e19	−0.748488
$647$	3.78080e19i	0.796624i	0.917250	+	0.398312i	$0.130404\pi$
$647$	3.78080e19i	0.796624i	−0.917250	+	0.398312i	$0.869596\pi$
$648$	0	0
$649$	−3.64417e19	−0.751425
$650$	5.80222e19i	1.18359i
$651$	0	0
$652$	−2.38971e19	−0.477103
$653$	8.97706e19i	1.77313i	0.462600	+	0.886567i	$0.346917\pi$
$653$	8.97706e19i	1.77313i	−0.462600	+	0.886567i	$0.653083\pi$
$654$	0	0
$655$	2.53349e19	0.489813
$656$	7.70453e18i	0.147373i
$657$	0	0
$658$	8.68733e19	1.62669
$659$	4.52112e19i	0.837623i	0.908073	+	0.418811i	$0.137553\pi$
$659$	4.52112e19i	0.837623i	−0.908073	+	0.418811i	$0.862447\pi$
$660$	0	0
$661$	−2.48686e19	−0.451067	−0.225533	−	0.974235i	$-0.572412\pi$
$661$	−2.48686e19	−0.451067	−0.225533	+	0.974235i	$0.572412\pi$
$662$	− 3.74917e19i	− 0.672867i
$663$	0	0
$664$	−1.89644e19	−0.333245
$665$	− 1.42112e19i	− 0.247103i
$666$	0	0
$667$	1.85587e19	0.315986
$668$	− 1.31237e19i	− 0.221116i
$669$	0	0
$670$	−1.10197e19	−0.181821
$671$	− 1.99214e19i	− 0.325284i
$672$	0	0
$673$	−5.56277e18	−0.0889582	−0.0444791	−	0.999010i	$-0.514163\pi$
$673$	−5.56277e18	−0.0889582	−0.0444791	+	0.999010i	$0.514163\pi$
$674$	− 6.24242e19i	− 0.987948i
$675$	0	0
$676$	3.15907e19	0.489703
$677$	− 7.32519e19i	− 1.12382i	−0.827197	−	0.561912i	$-0.810066\pi$
$677$	− 7.32519e19i	− 1.12382i	0.827197	−	0.561912i	$-0.189934\pi$
$678$	0	0
$679$	3.01836e19	0.453611
$680$	5.54526e19i	0.824821i
$681$	0	0
$682$	−3.82431e19	−0.557267
$683$	2.87786e19i	0.415074i	0.978227	+	0.207537i	$0.0665447\pi$
$683$	2.87786e19i	0.415074i	−0.978227	+	0.207537i	$0.933455\pi$
$684$	0	0
$685$	4.83237e19	0.682851
$686$	5.63324e19i	0.787933i
$687$	0	0
$688$	−4.94571e19	−0.677813
$689$	− 2.22422e20i	− 3.01747i
$690$	0	0
$691$	−1.23571e20	−1.64275	−0.821374	−	0.570390i	$-0.806792\pi$
$691$	−1.23571e20	−1.64275	−0.821374	+	0.570390i	$0.806792\pi$
$692$	8.56314e18i	0.112691i
$693$	0	0
$694$	7.37612e19	0.951288
$695$	− 4.04177e19i	− 0.516034i
$696$	0	0
$697$	−2.90875e19	−0.363980
$698$	1.08571e19i	0.134501i
$699$	0	0
$700$	1.77292e19	0.215279
$701$	4.08830e18i	0.0491492i	0.999698	+	0.0245746i	$0.00782313\pi$
$701$	4.08830e18i	0.0491492i	−0.999698	+	0.0245746i	$0.992177\pi$
$702$	0	0
$703$	−3.06270e19	−0.360925
$704$	5.68891e19i	0.663775i
$705$	0	0
$706$	−1.09124e20	−1.24821
$707$	− 7.05272e18i	− 0.0798768i
$708$	0	0
$709$	−4.42555e19	−0.491410	−0.245705	−	0.969345i	$-0.579019\pi$
$709$	−4.42555e19	−0.491410	−0.245705	+	0.969345i	$0.579019\pi$
$710$	− 5.96679e19i	− 0.656043i
$711$	0	0
$712$	−3.58537e19	−0.386522
$713$	9.82528e19i	1.04886i
$714$	0	0
$715$	4.41881e19	0.462555
$716$	− 2.09611e19i	− 0.217282i
$717$	0	0
$718$	1.20219e20	1.22208
$719$	5.39816e19i	0.543430i	0.962378	+	0.271715i	$0.0875907\pi$
$719$	5.39816e19i	0.543430i	−0.962378	+	0.271715i	$0.912409\pi$
$720$	0	0
$721$	2.06477e19	0.203857
$722$	6.53246e19i	0.638729i
$723$	0	0
$724$	7.95510e18	0.0762915
$725$	− 2.72176e19i	− 0.258514i
$726$	0	0
$727$	8.15992e19	0.760230	0.380115	−	0.924939i	$-0.375884\pi$
$727$	8.15992e19	0.760230	0.380115	+	0.924939i	$0.375884\pi$
$728$	2.16060e20i	1.99368i
$729$	0	0
$730$	−3.56116e19	−0.322353
$731$	− 1.86720e20i	− 1.67405i
$732$	0	0
$733$	4.81616e18	0.0423616	0.0211808	−	0.999776i	$-0.493257\pi$
$733$	4.81616e18	0.0423616	0.0211808	+	0.999776i	$0.493257\pi$
$734$	2.90673e17i	0.00253240i
$735$	0	0
$736$	5.58914e19	0.477749
$737$	− 3.29945e19i	− 0.279363i
$738$	0	0
$739$	6.74961e19	0.560748	0.280374	−	0.959891i	$-0.409542\pi$
$739$	6.74961e19	0.560748	0.280374	+	0.959891i	$0.409542\pi$
$740$	9.71854e18i	0.0799795i
$741$	0	0
$742$	2.02049e20	1.63166
$743$	2.22177e20i	1.77737i	0.458518	+	0.888685i	$0.348381\pi$
$743$	2.22177e20i	1.77737i	−0.458518	+	0.888685i	$0.651619\pi$
$744$	0	0
$745$	−2.34842e19	−0.184367
$746$	7.53223e19i	0.585803i
$747$	0	0
$748$	−3.33874e19	−0.254842
$749$	− 3.86677e19i	− 0.292399i
$750$	0	0
$751$	4.90998e19	0.364418	0.182209	−	0.983260i	$-0.441675\pi$
$751$	4.90998e19	0.364418	0.182209	+	0.983260i	$0.441675\pi$
$752$	− 1.63273e20i	− 1.20057i
$753$	0	0
$754$	6.66998e19	0.481421
$755$	− 5.11807e18i	− 0.0365997i
$756$	0	0
$757$	−2.77057e20	−1.94490	−0.972452	−	0.233101i	$-0.925113\pi$
$757$	−2.77057e20	−1.94490	−0.972452	+	0.233101i	$0.925113\pi$
$758$	− 1.66058e20i	− 1.15498i
$759$	0	0
$760$	−3.65240e19	−0.249392
$761$	− 1.28035e19i	− 0.0866235i	−0.999062	−	0.0433118i	$-0.986209\pi$
$761$	− 1.28035e19i	− 0.0866235i	0.999062	−	0.0433118i	$-0.0137909\pi$
$762$	0	0
$763$	−1.77737e20	−1.18061
$764$	− 3.86060e19i	− 0.254098i
$765$	0	0
$766$	−1.67835e20	−1.08463
$767$	3.36486e20i	2.15476i
$768$	0	0
$769$	−4.25064e19	−0.267282	−0.133641	−	0.991030i	$-0.542667\pi$
$769$	−4.25064e19	−0.267282	−0.133641	+	0.991030i	$0.542667\pi$
$770$	4.01407e19i	0.250121i
$771$	0	0
$772$	−5.74870e19	−0.351762
$773$	− 2.65437e20i	− 1.60955i	−0.593581	−	0.804774i	$-0.702286\pi$
$773$	− 2.65437e20i	− 1.60955i	0.593581	−	0.804774i	$-0.297714\pi$
$774$	0	0
$775$	1.44095e20	0.858095
$776$	− 7.75747e19i	− 0.457813i
$777$	0	0
$778$	2.00871e19	0.116429
$779$	− 1.91586e19i	− 0.110053i
$780$	0	0
$781$	1.78655e20	1.00799
$782$	− 2.55011e20i	− 1.42597i
$783$	0	0
$784$	−1.88269e19	−0.103411
$785$	1.86031e19i	0.101274i
$786$	0	0
$787$	−1.39036e20	−0.743539	−0.371769	−	0.928325i	$-0.621249\pi$
$787$	−1.39036e20	−0.743539	−0.371769	+	0.928325i	$0.621249\pi$
$788$	− 1.31262e19i	− 0.0695754i
$789$	0	0
$790$	−8.06177e19	−0.419799
$791$	8.76262e19i	0.452271i
$792$	0	0
$793$	−1.83945e20	−0.932772
$794$	− 7.33829e19i	− 0.368851i
$795$	0	0
$796$	7.05742e18	0.0348541
$797$	2.45859e19i	0.120359i	0.998188	+	0.0601793i	$0.0191673\pi$
$797$	2.45859e19i	0.120359i	−0.998188	+	0.0601793i	$0.980833\pi$
$798$	0	0
$799$	6.16417e20	2.96515
$800$	− 8.19685e19i	− 0.390856i
$801$	0	0
$802$	−1.67561e20	−0.785150
$803$	− 1.06627e20i	− 0.495287i
$804$	0	0
$805$	1.03128e20	0.470766
$806$	3.53119e20i	1.59800i
$807$	0	0
$808$	−1.81261e19	−0.0806168
$809$	1.17284e20i	0.517128i	0.965994	+	0.258564i	$0.0832492\pi$
$809$	1.17284e20i	0.517128i	−0.965994	+	0.258564i	$0.916751\pi$
$810$	0	0
$811$	1.13339e20	0.491171	0.245586	−	0.969375i	$-0.421020\pi$
$811$	1.13339e20	0.491171	0.245586	+	0.969375i	$0.421020\pi$
$812$	− 2.03807e19i	− 0.0875643i
$813$	0	0
$814$	8.65085e19	0.365332
$815$	2.03861e20i	0.853555i
$816$	0	0
$817$	1.22983e20	0.506165
$818$	2.03891e20i	0.832003i
$819$	0	0
$820$	−6.07940e18	−0.0243873
$821$	6.14223e19i	0.244300i	0.992512	+	0.122150i	$0.0389789\pi$
$821$	6.14223e19i	0.244300i	−0.992512	+	0.122150i	$0.961021\pi$
$822$	0	0
$823$	−4.05684e19	−0.158631	−0.0793157	−	0.996850i	$-0.525274\pi$
$823$	−4.05684e19	−0.158631	−0.0793157	+	0.996850i	$0.525274\pi$
$824$	− 5.30666e19i	− 0.205746i
$825$	0	0
$826$	−3.05665e20	−1.16516
$827$	1.59384e20i	0.602431i	0.953556	+	0.301216i	$0.0973924\pi$
$827$	1.59384e20i	0.602431i	−0.953556	+	0.301216i	$0.902608\pi$
$828$	0	0
$829$	−3.87834e20	−1.44133	−0.720666	−	0.693282i	$-0.756164\pi$
$829$	−3.87834e20	−1.44133	−0.720666	+	0.693282i	$0.756164\pi$
$830$	3.25325e19i	0.119887i
$831$	0	0
$832$	5.25287e20	1.90342
$833$	− 7.10788e19i	− 0.255403i
$834$	0	0
$835$	−1.11956e20	−0.395586
$836$	− 2.19907e19i	− 0.0770540i
$837$	0	0
$838$	−2.05444e20	−0.707923
$839$	2.49688e20i	0.853225i	0.904435	+	0.426612i	$0.140293\pi$
$839$	2.49688e20i	0.853225i	−0.904435	+	0.426612i	$0.859707\pi$
$840$	0	0
$841$	2.66270e20	0.894850
$842$	7.65750e19i	0.255213i
$843$	0	0
$844$	−6.01528e19	−0.197178
$845$	− 2.69494e20i	− 0.876098i
$846$	0	0
$847$	2.15332e20	0.688533
$848$	− 3.79738e20i	− 1.20424i
$849$	0	0
$850$	−3.73991e20	−1.16662
$851$	− 2.22254e20i	− 0.687612i
$852$	0	0
$853$	1.59506e20	0.485439	0.242720	−	0.970096i	$-0.421960\pi$
$853$	1.59506e20	0.485439	0.242720	+	0.970096i	$0.421960\pi$
$854$	− 1.67097e20i	− 0.504385i
$855$	0	0
$856$	−9.93794e19	−0.295107
$857$	− 1.79666e20i	− 0.529177i	−0.964361	−	0.264589i	$-0.914764\pi$
$857$	− 1.79666e20i	− 0.529177i	0.964361	−	0.264589i	$-0.0852361\pi$
$858$	0	0
$859$	−4.72567e20	−1.36934	−0.684669	−	0.728854i	$-0.740053\pi$
$859$	−4.72567e20	−1.36934	−0.684669	+	0.728854i	$0.740053\pi$
$860$	− 3.90250e19i	− 0.112164i
$861$	0	0
$862$	5.11656e20	1.44686
$863$	− 3.89191e20i	− 1.09166i	−0.837896	−	0.545830i	$-0.816214\pi$
$863$	− 3.89191e20i	− 1.09166i	0.837896	−	0.545830i	$-0.183786\pi$
$864$	0	0
$865$	7.30505e19	0.201609
$866$	1.56638e20i	0.428818i
$867$	0	0
$868$	1.07899e20	0.290655
$869$	− 2.41381e20i	− 0.645009i
$870$	0	0
$871$	−3.04655e20	−0.801091
$872$	4.56800e20i	1.19154i
$873$	0	0
$874$	1.67964e20	0.431157
$875$	− 3.40958e20i	− 0.868250i
$876$	0	0
$877$	−5.95465e20	−1.49231	−0.746155	−	0.665772i	$-0.768102\pi$
$877$	−5.95465e20	−1.49231	−0.746155	+	0.665772i	$0.768102\pi$
$878$	− 2.54312e20i	− 0.632273i
$879$	0	0
$880$	7.54418e19	0.184601
$881$	4.64234e20i	1.12695i	0.826133	+	0.563476i	$0.190536\pi$
$881$	4.64234e20i	1.12695i	−0.826133	+	0.563476i	$0.809464\pi$
$882$	0	0
$883$	−1.90591e20	−0.455385	−0.227692	−	0.973733i	$-0.573118\pi$
$883$	−1.90591e20	−0.455385	−0.227692	+	0.973733i	$0.573118\pi$
$884$	3.08284e20i	0.730777i
$885$	0	0
$886$	4.74126e20	1.10626
$887$	− 1.42329e19i	− 0.0329479i	−0.999864	−	0.0164739i	$-0.994756\pi$
$887$	− 1.42329e19i	− 0.0329479i	0.999864	−	0.0164739i	$-0.00524406\pi$
$888$	0	0
$889$	−4.67706e20	−1.06576
$890$	6.15052e19i	0.139053i
$891$	0	0
$892$	−1.46621e20	−0.326319
$893$	4.06005e20i	0.896542i
$894$	0	0
$895$	−1.78815e20	−0.388726
$896$	2.33322e20i	0.503267i
$897$	0	0
$898$	−2.35779e20	−0.500693
$899$	− 1.65645e20i	− 0.349028i
$900$	0	0
$901$	1.43366e21	2.97421
$902$	5.41151e19i	0.111397i
$903$	0	0
$904$	2.25207e20	0.456461
$905$	− 6.78634e19i	− 0.136488i
$906$	0	0
$907$	−8.12017e19	−0.160810	−0.0804052	−	0.996762i	$-0.525621\pi$
$907$	−8.12017e19	−0.160810	−0.0804052	+	0.996762i	$0.525621\pi$
$908$	− 6.78273e19i	− 0.133292i
$909$	0	0
$910$	3.70640e20	0.717237
$911$	8.31841e20i	1.59739i	0.601735	+	0.798696i	$0.294476\pi$
$911$	8.31841e20i	1.59739i	−0.601735	+	0.798696i	$0.705524\pi$
$912$	0	0
$913$	−9.74072e19	−0.184202
$914$	5.15473e20i	0.967348i
$915$	0	0
$916$	2.21291e20	0.408973
$917$	6.36268e20i	1.16696i
$918$	0	0
$919$	6.70003e19	0.121023	0.0605115	−	0.998167i	$-0.480727\pi$
$919$	6.70003e19	0.121023	0.0605115	+	0.998167i	$0.480727\pi$
$920$	− 2.65048e20i	− 0.475128i
$921$	0	0
$922$	5.17580e19	0.0913821
$923$	− 1.64961e21i	− 2.89048i
$924$	0	0
$925$	−3.25951e20	−0.562549
$926$	− 1.01420e20i	− 0.173718i
$927$	0	0
$928$	−9.42275e19	−0.158980
$929$	− 3.74247e20i	− 0.626684i	−0.949640	−	0.313342i	$-0.898551\pi$
$929$	− 3.74247e20i	− 0.626684i	0.949640	−	0.313342i	$-0.101449\pi$
$930$	0	0
$931$	4.68163e19	0.0772234
$932$	7.45356e19i	0.122026i
$933$	0	0
$934$	−3.21290e20	−0.518166
$935$	2.84822e20i	0.455923i
$936$	0	0
$937$	6.49972e20	1.02498	0.512492	−	0.858692i	$-0.328722\pi$
$937$	6.49972e20	1.02498	0.512492	+	0.858692i	$0.328722\pi$
$938$	− 2.76751e20i	− 0.433180i
$939$	0	0
$940$	1.28833e20	0.198670
$941$	6.03849e19i	0.0924274i	0.998932	+	0.0462137i	$0.0147155\pi$
$941$	6.03849e19i	0.0924274i	−0.998932	+	0.0462137i	$0.985284\pi$
$942$	0	0
$943$	1.39030e20	0.209666
$944$	5.74478e20i	0.859942i
$945$	0	0
$946$	−3.47377e20	−0.512346
$947$	− 4.27652e20i	− 0.626096i	−0.949737	−	0.313048i	$-0.898650\pi$
$947$	− 4.27652e20i	− 0.626096i	0.949737	−	0.313048i	$-0.101350\pi$
$948$	0	0
$949$	−9.84539e20	−1.42027
$950$	− 2.46330e20i	− 0.352738i
$951$	0	0
$952$	−1.39265e21	−1.96510
$953$	− 7.13135e20i	− 0.998898i	−0.866343	−	0.499449i	$-0.833536\pi$
$953$	− 7.13135e20i	− 0.998898i	0.866343	−	0.499449i	$-0.166464\pi$
$954$	0	0
$955$	−3.29340e20	−0.454591
$956$	− 2.00324e20i	− 0.274490i
$957$	0	0
$958$	−4.03153e20	−0.544392
$959$	1.21361e21i	1.62686i
$960$	0	0
$961$	1.20006e20	0.158540
$962$	− 7.98779e20i	− 1.04761i
$963$	0	0
$964$	6.33518e19	0.0818879
$965$	4.90411e20i	0.629316i
$966$	0	0
$967$	7.77128e20	0.982896	0.491448	−	0.870907i	$-0.336468\pi$
$967$	7.77128e20	0.982896	0.491448	+	0.870907i	$0.336468\pi$
$968$	− 5.53423e20i	− 0.694911i
$969$	0	0
$970$	−1.33075e20	−0.164701
$971$	− 1.36648e21i	− 1.67907i	−0.543308	−	0.839534i	$-0.682828\pi$
$971$	− 1.36648e21i	− 1.67907i	0.543308	−	0.839534i	$-0.317172\pi$
$972$	0	0
$973$	1.01506e21	1.22943
$974$	9.73666e19i	0.117084i
$975$	0	0
$976$	−3.14047e20	−0.372259
$977$	1.36093e21i	1.60167i	0.598883	+	0.800837i	$0.295612\pi$
$977$	1.36093e21i	1.60167i	−0.598883	+	0.800837i	$0.704388\pi$
$978$	0	0
$979$	−1.84156e20	−0.213652
$980$	− 1.48557e19i	− 0.0171124i
$981$	0	0
$982$	6.64474e20	0.754567
$983$	− 2.58784e20i	− 0.291785i	−0.989300	−	0.145892i	$-0.953395\pi$
$983$	− 2.58784e20i	− 0.291785i	0.989300	−	0.145892i	$-0.0466053\pi$
$984$	0	0
$985$	−1.11977e20	−0.124473
$986$	4.29924e20i	0.474519i
$987$	0	0
$988$	−2.03052e20	−0.220957
$989$	8.92468e20i	0.964315i
$990$	0	0
$991$	7.02597e20	0.748498	0.374249	−	0.927328i	$-0.377901\pi$
$991$	7.02597e20	0.748498	0.374249	+	0.927328i	$0.377901\pi$
$992$	− 4.98855e20i	− 0.527707i
$993$	0	0
$994$	1.49852e21	1.56299
$995$	− 6.02055e19i	− 0.0623554i
$996$	0	0
$997$	1.76275e21	1.80022	0.900108	−	0.435666i	$-0.143487\pi$
$997$	1.76275e21	1.80022	0.900108	+	0.435666i	$0.143487\pi$
$998$	− 5.20167e20i	− 0.527508i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.15.b.a.8.2	✓	4
3.2	odd	2	inner	9.15.b.a.8.3	yes	4
4.3	odd	2		144.15.e.d.17.2		4
12.11	even	2		144.15.e.d.17.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.15.b.a.8.2	✓	4	1.1	even	1	trivial
9.15.b.a.8.3	yes	4	3.2	odd	2	inner
144.15.e.d.17.2		4	4.3	odd	2
144.15.e.d.17.3		4	12.11	even	2

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:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	9.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-110.725i q^{2} +4123.91 q^{4} -35180.3i q^{5} +883527. q^{7} -2.27074e6i q^{8} +O(q^{10})\)	\(q-110.725i q^{2} +4123.91 q^{4} -35180.3i q^{5} +883527. q^{7} -2.27074e6i q^{8} -3.89535e6 q^{10} -1.16632e7i q^{11} -1.07693e8 q^{13} -9.78288e7i q^{14} -1.83863e8 q^{16} -6.94152e8i q^{17} +4.57205e8 q^{19} -1.45080e8i q^{20} -1.29142e9 q^{22} +3.31786e9i q^{23} +4.86586e9 q^{25} +1.19243e10i q^{26} +3.64358e9 q^{28} -5.59359e9i q^{29} +2.96133e10 q^{31} -1.68456e10i q^{32} -7.68602e10 q^{34} -3.10827e10i q^{35} -6.69873e10 q^{37} -5.06242e10i q^{38} -7.98854e10 q^{40} -4.19037e10i q^{41} +2.68989e11 q^{43} -4.80981e10i q^{44} +3.67371e11 q^{46} +8.88014e11i q^{47} +1.02397e11 q^{49} -5.38774e11i q^{50} -4.44116e11 q^{52} +2.06534e12i q^{53} -4.10316e11 q^{55} -2.00626e12i q^{56} -6.19352e11 q^{58} -3.12449e12i q^{59} +1.70805e12 q^{61} -3.27895e12i q^{62} -4.87764e12 q^{64} +3.78866e12i q^{65} +2.82893e12 q^{67} -2.86262e12i q^{68} -3.44164e12 q^{70} +1.53177e13i q^{71} +9.14210e12 q^{73} +7.41719e12i q^{74} +1.88547e12 q^{76} -1.03048e13i q^{77} +2.06959e13 q^{79} +6.46834e12i q^{80} -4.63980e12 q^{82} -8.35164e12i q^{83} -2.44205e13 q^{85} -2.97839e13i q^{86} -2.64842e13 q^{88} -1.57894e13i q^{89} -9.51496e13 q^{91} +1.36825e13i q^{92} +9.83256e13 q^{94} -1.60846e13i q^{95} +3.41627e13 q^{97} -1.13379e13i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3332 q^{4} - 1065904 q^{7}+O(q^{10}) \) 4 * q - 3332 * q^4 - 1065904 * q^7	\( 4 q - 3332 q^{4} - 1065904 q^{7} - 20201220 q^{10} - 193633024 q^{13} - 1027273976 q^{16} - 1738250560 q^{19} + 5179999824 q^{22} + 18465731740 q^{25} + 23689734992 q^{28} - 22530472816 q^{31} - 118458902412 q^{34} - 135170189128 q^{37} - 291249051480 q^{40} + 567822762272 q^{43} + 95457780144 q^{46} + 2861171717628 q^{49} - 1014177699808 q^{52} + 1351760277600 q^{55} - 9671683881852 q^{58} - 66091393528 q^{61} - 13634169987344 q^{64} + 15196895050208 q^{67} + 10695968312400 q^{70} + 20468321135936 q^{73} + 19129607873600 q^{76} + 19688660582384 q^{79} - 68290319744532 q^{82} - 38973146130360 q^{85} + 29271666534624 q^{88} - 221111416338176 q^{91} + 73435199966064 q^{94} + 104819071127168 q^{97}+O(q^{100}) \) 4 * q - 3332 * q^4 - 1065904 * q^7 - 20201220 * q^10 - 193633024 * q^13 - 1027273976 * q^16 - 1738250560 * q^19 + 5179999824 * q^22 + 18465731740 * q^25 + 23689734992 * q^28 - 22530472816 * q^31 - 118458902412 * q^34 - 135170189128 * q^37 - 291249051480 * q^40 + 567822762272 * q^43 + 95457780144 * q^46 + 2861171717628 * q^49 - 1014177699808 * q^52 + 1351760277600 * q^55 - 9671683881852 * q^58 - 66091393528 * q^61 - 13634169987344 * q^64 + 15196895050208 * q^67 + 10695968312400 * q^70 + 20468321135936 * q^73 + 19129607873600 * q^76 + 19688660582384 * q^79 - 68290319744532 * q^82 - 38973146130360 * q^85 + 29271666534624 * q^88 - 221111416338176 * q^91 + 73435199966064 * q^94 + 104819071127168 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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