LMFDB - Embedded newform 27.9.b.d.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,9,Mod(26,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, 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("27.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	27.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:	$10.9992224717$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.6171673600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 40x^{3} + 225x^{2} + 150x + 50 $ x^6 - 2x^5 + 2x^4 + 40x^3 + 225x^2 + 150*x + 50
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{21} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$-2.05620 + 2.05620i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.9.b.d.26.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-29.0422i q^{2} -587.450 q^{4} -1141.55i q^{5} -618.310 q^{7} +9626.03i q^{8} +O(q^{10})$	\(q-29.0422i q^{2} -587.450 q^{4} -1141.55i q^{5} -618.310 q^{7} +9626.03i q^{8} -33153.0 q^{10} +5408.54i q^{11} +24561.4 q^{13} +17957.1i q^{14} +129174. q^{16} +11332.1i q^{17} +38797.9 q^{19} +670601. i q^{20} +157076. q^{22} -347537. i q^{23} -912504. q^{25} -713316. i q^{26} +363226. q^{28} +513264. i q^{29} -1.02526e6 q^{31} -1.28723e6i q^{32} +329110. q^{34} +705830. i q^{35} -2.38722e6 q^{37} -1.12678e6i q^{38} +1.09886e7 q^{40} -4.26661e6i q^{41} +3.81535e6 q^{43} -3.17724e6i q^{44} -1.00932e7 q^{46} -3.71085e6i q^{47} -5.38249e6 q^{49} +2.65011e7i q^{50} -1.44286e7 q^{52} +6.10781e6i q^{53} +6.17410e6 q^{55} -5.95187e6i q^{56} +1.49063e7 q^{58} -1.02804e7i q^{59} -1.28725e7 q^{61} +2.97757e7i q^{62} -4.31555e6 q^{64} -2.80379e7i q^{65} +3.05616e7 q^{67} -6.65705e6i q^{68} +2.04989e7 q^{70} -7.37721e6i q^{71} +1.85229e7 q^{73} +6.93302e7i q^{74} -2.27918e7 q^{76} -3.34415e6i q^{77} +3.10689e7 q^{79} -1.47458e8i q^{80} -1.23912e8 q^{82} -2.33350e7i q^{83} +1.29361e7 q^{85} -1.10806e8i q^{86} -5.20627e7 q^{88} +3.49841e7i q^{89} -1.51865e7 q^{91} +2.04160e8i q^{92} -1.07771e8 q^{94} -4.42896e7i q^{95} +2.06846e7 q^{97} +1.56319e8i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 786 q^{4} - 1698 q^{7}+O(q^{10}) $ 6 * q - 786 * q^4 - 1698 * q^7	$ 6 q - 786 q^{4} - 1698 q^{7} - 46278 q^{10} + 41844 q^{13} + 170466 q^{16} - 36384 q^{19} + 647298 q^{22} - 1646688 q^{25} - 838986 q^{28} + 2058474 q^{31} - 1320300 q^{34} - 9395880 q^{37} + 29560086 q^{40} - 2737284 q^{43} - 31514616 q^{46} + 28900656 q^{49} - 34081692 q^{52} - 26674542 q^{55} + 75244572 q^{58} - 40180776 q^{61} - 48541458 q^{64} + 111355284 q^{67} + 17727282 q^{70} + 12821718 q^{73} - 38623776 q^{76} + 21820404 q^{79} - 138785832 q^{82} - 83844396 q^{85} + 57552174 q^{88} + 156632964 q^{91} - 502013916 q^{94} + 53735106 q^{97}+O(q^{100}) $ 6 * q - 786 * q^4 - 1698 * q^7 - 46278 * q^10 + 41844 * q^13 + 170466 * q^16 - 36384 * q^19 + 647298 * q^22 - 1646688 * q^25 - 838986 * q^28 + 2058474 * q^31 - 1320300 * q^34 - 9395880 * q^37 + 29560086 * q^40 - 2737284 * q^43 - 31514616 * q^46 + 28900656 * q^49 - 34081692 * q^52 - 26674542 * q^55 + 75244572 * q^58 - 40180776 * q^61 - 48541458 * q^64 + 111355284 * q^67 + 17727282 * q^70 + 12821718 * q^73 - 38623776 * q^76 + 21820404 * q^79 - 138785832 * q^82 - 83844396 * q^85 + 57552174 * q^88 + 156632964 * q^91 - 502013916 * q^94 + 53735106 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$.

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

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 $

$2$	− 29.0422i	− 1.81514i	−0.419903	−	0.907569i	$-0.637936\pi$
$2$	− 29.0422i	− 1.81514i	0.419903	−	0.907569i	$-0.362064\pi$
$3$	0	0
$3$	0	0
$4$	−587.450	−2.29472
$5$	− 1141.55i	− 1.82648i	−0.407428	−	0.913238i	$-0.633574\pi$
$5$	− 1141.55i	− 1.82648i	0.407428	−	0.913238i	$-0.366426\pi$
$6$	0	0
$7$	−618.310	−0.257522	−0.128761	−	0.991676i	$-0.541100\pi$
$7$	−618.310	−0.257522	−0.128761	+	0.991676i	$0.541100\pi$
$8$	9626.03i	2.35010i
$9$	0	0
$10$	−33153.0	−3.31530
$11$	5408.54i	0.369410i	0.982794	+	0.184705i	$0.0591330\pi$
$11$	5408.54i	0.369410i	−0.982794	+	0.184705i	$0.940867\pi$
$12$	0	0
$13$	24561.4	0.859962	0.429981	−	0.902838i	$-0.358520\pi$
$13$	24561.4	0.859962	0.429981	+	0.902838i	$0.358520\pi$
$14$	17957.1i	0.467438i
$15$	0	0
$16$	129174.	1.97104
$17$	11332.1i	0.135680i	0.997696	+	0.0678400i	$0.0216107\pi$
$17$	11332.1i	0.135680i	−0.997696	+	0.0678400i	$0.978389\pi$
$18$	0	0
$19$	38797.9	0.297710	0.148855	−	0.988859i	$-0.452441\pi$
$19$	38797.9	0.297710	0.148855	+	0.988859i	$0.452441\pi$
$20$	670601.i	4.19126i
$21$	0	0
$22$	157076.	0.670531
$23$	− 347537.i	− 1.24191i	−0.783847	−	0.620954i	$-0.786745\pi$
$23$	− 347537.i	− 1.24191i	0.783847	−	0.620954i	$-0.213255\pi$
$24$	0	0
$25$	−912504.	−2.33601
$26$	− 713316.i	− 1.56095i
$27$	0	0
$28$	363226.	0.590942
$29$	513264.i	0.725687i	0.931850	+	0.362843i	$0.118194\pi$
$29$	513264.i	0.725687i	−0.931850	+	0.362843i	$0.881806\pi$
$30$	0	0
$31$	−1.02526e6	−1.11016	−0.555080	−	0.831797i	$-0.687312\pi$
$31$	−1.02526e6	−1.11016	−0.555080	+	0.831797i	$0.687312\pi$
$32$	− 1.28723e6i	− 1.22760i
$33$	0	0
$34$	329110.	0.246278
$35$	705830.i	0.470357i
$36$	0	0
$37$	−2.38722e6	−1.27375	−0.636877	−	0.770965i	$-0.719774\pi$
$37$	−2.38722e6	−1.27375	−0.636877	+	0.770965i	$0.719774\pi$
$38$	− 1.12678e6i	− 0.540385i
$39$	0	0
$40$	1.09886e7	4.29241
$41$	− 4.26661e6i	− 1.50990i	−0.655785	−	0.754948i	$-0.727662\pi$
$41$	− 4.26661e6i	− 1.50990i	0.655785	−	0.754948i	$-0.272338\pi$
$42$	0	0
$43$	3.81535e6	1.11599	0.557996	−	0.829844i	$-0.311571\pi$
$43$	3.81535e6	1.11599	0.557996	+	0.829844i	$0.311571\pi$
$44$	− 3.17724e6i	− 0.847695i
$45$	0	0
$46$	−1.00932e7	−2.25424
$47$	− 3.71085e6i	− 0.760470i	−0.924890	−	0.380235i	$-0.875843\pi$
$47$	− 3.71085e6i	− 0.760470i	0.924890	−	0.380235i	$-0.124157\pi$
$48$	0	0
$49$	−5.38249e6	−0.933682
$50$	2.65011e7i	4.24018i
$51$	0	0
$52$	−1.44286e7	−1.97338
$53$	6.10781e6i	0.774073i	0.922064	+	0.387036i	$0.126501\pi$
$53$	6.10781e6i	0.774073i	−0.922064	+	0.387036i	$0.873499\pi$
$54$	0	0
$55$	6.17410e6	0.674719
$56$	− 5.95187e6i	− 0.605203i
$57$	0	0
$58$	1.49063e7	1.31722
$59$	− 1.02804e7i	− 0.848401i	−0.905568	−	0.424200i	$-0.860555\pi$
$59$	− 1.02804e7i	− 0.848401i	0.905568	−	0.424200i	$-0.139445\pi$
$60$	0	0
$61$	−1.28725e7	−0.929699	−0.464849	−	0.885390i	$-0.653892\pi$
$61$	−1.28725e7	−0.929699	−0.464849	+	0.885390i	$0.653892\pi$
$62$	2.97757e7i	2.01509i
$63$	0	0
$64$	−4.31555e6	−0.257227
$65$	− 2.80379e7i	− 1.57070i
$66$	0	0
$67$	3.05616e7	1.51662	0.758311	−	0.651893i	$-0.226025\pi$
$67$	3.05616e7	1.51662	0.758311	+	0.651893i	$0.226025\pi$
$68$	− 6.65705e6i	− 0.311348i
$69$	0	0
$70$	2.04989e7	0.853763
$71$	− 7.37721e6i	− 0.290308i	−0.989409	−	0.145154i	$-0.953632\pi$
$71$	− 7.37721e6i	− 0.290308i	0.989409	−	0.145154i	$-0.0463677\pi$
$72$	0	0
$73$	1.85229e7	0.652255	0.326127	−	0.945326i	$-0.394256\pi$
$73$	1.85229e7	0.652255	0.326127	+	0.945326i	$0.394256\pi$
$74$	6.93302e7i	2.31204i
$75$	0	0
$76$	−2.27918e7	−0.683163
$77$	− 3.34415e6i	− 0.0951312i
$78$	0	0
$79$	3.10689e7	0.797659	0.398830	−	0.917025i	$-0.369416\pi$
$79$	3.10689e7	0.797659	0.398830	+	0.917025i	$0.369416\pi$
$80$	− 1.47458e8i	− 3.60005i
$81$	0	0
$82$	−1.23912e8	−2.74067
$83$	− 2.33350e7i	− 0.491694i	−0.969309	−	0.245847i	$-0.920934\pi$
$83$	− 2.33350e7i	− 0.491694i	0.969309	−	0.245847i	$-0.0790661\pi$
$84$	0	0
$85$	1.29361e7	0.247816
$86$	− 1.10806e8i	− 2.02568i
$87$	0	0
$88$	−5.20627e7	−0.868153
$89$	3.49841e7i	0.557584i	0.960351	+	0.278792i	$0.0899340\pi$
$89$	3.49841e7i	0.557584i	−0.960351	+	0.278792i	$0.910066\pi$
$90$	0	0
$91$	−1.51865e7	−0.221459
$92$	2.04160e8i	2.84984i
$93$	0	0
$94$	−1.07771e8	−1.38036
$95$	− 4.42896e7i	− 0.543760i
$96$	0	0
$97$	2.06846e7	0.233647	0.116824	−	0.993153i	$-0.462729\pi$
$97$	2.06846e7	0.233647	0.116824	+	0.993153i	$0.462729\pi$
$98$	1.56319e8i	1.69476i
$99$	0	0
$100$	5.36050e8	5.36050
$101$	− 1.58855e8i	− 1.52657i	−0.646064	−	0.763283i	$-0.723586\pi$
$101$	− 1.58855e8i	− 1.52657i	0.646064	−	0.763283i	$-0.276414\pi$
$102$	0	0
$103$	−1.52284e8	−1.35302	−0.676511	−	0.736433i	$-0.736509\pi$
$103$	−1.52284e8	−1.35302	−0.676511	+	0.736433i	$0.736509\pi$
$104$	2.36428e8i	2.02100i
$105$	0	0
$106$	1.77384e8	1.40505
$107$	1.23325e8i	0.940843i	0.882442	+	0.470421i	$0.155898\pi$
$107$	1.23325e8i	0.940843i	−0.882442	+	0.470421i	$0.844102\pi$
$108$	0	0
$109$	−4.87939e7	−0.345669	−0.172834	−	0.984951i	$-0.555293\pi$
$109$	−4.87939e7	−0.345669	−0.172834	+	0.984951i	$0.555293\pi$
$110$	− 1.79309e8i	− 1.22471i
$111$	0	0
$112$	−7.98695e7	−0.507585
$113$	6.23131e7i	0.382178i	0.981573	+	0.191089i	$0.0612019\pi$
$113$	6.23131e7i	0.382178i	−0.981573	+	0.191089i	$0.938798\pi$
$114$	0	0
$115$	−3.96730e8	−2.26832
$116$	− 3.01517e8i	− 1.66525i
$117$	0	0
$118$	−2.98565e8	−1.53996
$119$	− 7.00676e6i	− 0.0349405i
$120$	0	0
$121$	1.85107e8	0.863536
$122$	3.73845e8i	1.68753i
$123$	0	0
$124$	6.02286e8	2.54751
$125$	5.95750e8i	2.44019i
$126$	0	0
$127$	−1.46369e8	−0.562646	−0.281323	−	0.959613i	$-0.590773\pi$
$127$	−1.46369e8	−0.562646	−0.281323	+	0.959613i	$0.590773\pi$
$128$	− 2.04199e8i	− 0.760699i
$129$	0	0
$130$	−8.14284e8	−2.85103
$131$	− 4.08362e8i	− 1.38663i	−0.720636	−	0.693314i	$-0.756150\pi$
$131$	− 4.08362e8i	− 1.38663i	0.720636	−	0.693314i	$-0.243850\pi$
$132$	0	0
$133$	−2.39891e7	−0.0766669
$134$	− 8.87577e8i	− 2.75288i
$135$	0	0
$136$	−1.09083e8	−0.318862
$137$	− 1.27898e8i	− 0.363062i	−0.983385	−	0.181531i	$-0.941895\pi$
$137$	− 1.27898e8i	− 0.363062i	0.983385	−	0.181531i	$-0.0581053\pi$
$138$	0	0
$139$	6.05634e8	1.62237	0.811187	−	0.584787i	$-0.198822\pi$
$139$	6.05634e8	1.62237	0.811187	+	0.584787i	$0.198822\pi$
$140$	− 4.14639e8i	− 1.07934i
$141$	0	0
$142$	−2.14250e8	−0.526949
$143$	1.32841e8i	0.317679i
$144$	0	0
$145$	5.85915e8	1.32545
$146$	− 5.37945e8i	− 1.18393i
$147$	0	0
$148$	1.40237e9	2.92292
$149$	5.22005e8i	1.05908i	0.848284	+	0.529541i	$0.177636\pi$
$149$	5.22005e8i	1.05908i	−0.848284	+	0.529541i	$0.822364\pi$
$150$	0	0
$151$	1.64223e8	0.315884	0.157942	−	0.987448i	$-0.449514\pi$
$151$	1.64223e8	0.315884	0.157942	+	0.987448i	$0.449514\pi$
$152$	3.73470e8i	0.699650i
$153$	0	0
$154$	−9.71216e7	−0.172676
$155$	1.17038e9i	2.02768i
$156$	0	0
$157$	−6.93059e8	−1.14070	−0.570350	−	0.821402i	$-0.693192\pi$
$157$	−6.93059e8	−1.14070	−0.570350	+	0.821402i	$0.693192\pi$
$158$	− 9.02309e8i	− 1.44786i
$159$	0	0
$160$	−1.46944e9	−2.24218
$161$	2.14886e8i	0.319819i
$162$	0	0
$163$	5.78961e8	0.820161	0.410080	−	0.912049i	$-0.365501\pi$
$163$	5.78961e8	0.820161	0.410080	+	0.912049i	$0.365501\pi$
$164$	2.50642e9i	3.46480i
$165$	0	0
$166$	−6.77699e8	−0.892492
$167$	− 8.55075e8i	− 1.09936i	−0.835377	−	0.549678i	$-0.814751\pi$
$167$	− 8.55075e8i	− 1.09936i	0.835377	−	0.549678i	$-0.185249\pi$
$168$	0	0
$169$	−2.12470e8	−0.260466
$170$	− 3.75694e8i	− 0.449820i
$171$	0	0
$172$	−2.24133e9	−2.56089
$173$	− 9.56430e8i	− 1.06775i	−0.845564	−	0.533874i	$-0.820736\pi$
$173$	− 9.56430e8i	− 1.06775i	0.845564	−	0.533874i	$-0.179264\pi$
$174$	0	0
$175$	5.64211e8	0.601574
$176$	6.98642e8i	0.728122i
$177$	0	0
$178$	1.01601e9	1.01209
$179$	− 1.79135e9i	− 1.74489i	−0.488714	−	0.872444i	$-0.662534\pi$
$179$	− 1.79135e9i	− 1.74489i	0.488714	−	0.872444i	$-0.337466\pi$
$180$	0	0
$181$	−1.11171e9	−1.03580	−0.517901	−	0.855441i	$-0.673286\pi$
$181$	−1.11171e9	−1.03580	−0.517901	+	0.855441i	$0.673286\pi$
$182$	4.41050e8i	0.401978i
$183$	0	0
$184$	3.34540e9	2.91861
$185$	2.72512e9i	2.32648i
$186$	0	0
$187$	−6.12902e7	−0.0501216
$188$	2.17994e9i	1.74507i
$189$	0	0
$190$	−1.28627e9	−0.987000
$191$	6.64690e8i	0.499443i	0.968318	+	0.249721i	$0.0803390\pi$
$191$	6.64690e8i	0.499443i	−0.968318	+	0.249721i	$0.919661\pi$
$192$	0	0
$193$	2.25896e9	1.62810	0.814048	−	0.580797i	$-0.197259\pi$
$193$	2.25896e9	1.62810	0.814048	+	0.580797i	$0.197259\pi$
$194$	− 6.00728e8i	− 0.424102i
$195$	0	0
$196$	3.16194e9	2.14254
$197$	− 6.71494e8i	− 0.445838i	−0.974837	−	0.222919i	$-0.928441\pi$
$197$	− 6.71494e8i	− 0.445838i	0.974837	−	0.222919i	$-0.0715585\pi$
$198$	0	0
$199$	1.70981e9	1.09027	0.545137	−	0.838347i	$-0.316478\pi$
$199$	1.70981e9	1.09027	0.545137	+	0.838347i	$0.316478\pi$
$200$	− 8.78379e9i	− 5.48987i
$201$	0	0
$202$	−4.61350e9	−2.77093
$203$	− 3.17357e8i	− 0.186880i
$204$	0	0
$205$	−4.87053e9	−2.75779
$206$	4.42266e9i	2.45592i
$207$	0	0
$208$	3.17269e9	1.69502
$209$	2.09840e8i	0.109977i
$210$	0	0
$211$	−1.05211e9	−0.530802	−0.265401	−	0.964138i	$-0.585504\pi$
$211$	−1.05211e9	−0.530802	−0.265401	+	0.964138i	$0.585504\pi$
$212$	− 3.58803e9i	− 1.77628i
$213$	0	0
$214$	3.58164e9	1.70776
$215$	− 4.35540e9i	− 2.03833i
$216$	0	0
$217$	6.33926e8	0.285890
$218$	1.41708e9i	0.627436i
$219$	0	0
$220$	−3.62697e9	−1.54829
$221$	2.78332e8i	0.116680i
$222$	0	0
$223$	−3.67967e9	−1.48795	−0.743977	−	0.668205i	$-0.767063\pi$
$223$	−3.67967e9	−1.48795	−0.743977	+	0.668205i	$0.767063\pi$
$224$	7.95909e8i	0.316134i
$225$	0	0
$226$	1.80971e9	0.693706
$227$	− 1.78206e9i	− 0.671150i	−0.942013	−	0.335575i	$-0.891069\pi$
$227$	− 1.78206e9i	− 0.671150i	0.942013	−	0.335575i	$-0.108931\pi$
$228$	0	0
$229$	−6.48600e8	−0.235850	−0.117925	−	0.993023i	$-0.537624\pi$
$229$	−6.48600e8	−0.235850	−0.117925	+	0.993023i	$0.537624\pi$
$230$	1.15219e10i	4.11730i
$231$	0	0
$232$	−4.94070e9	−1.70544
$233$	− 692043.i	0 0.000234806i	−1.00000		0.000117403i	$-0.999963\pi$
$233$	− 692043.i	0 0.000234806i	1.00000		0.000117403i	$-3.73706e-5\pi$
$234$	0	0
$235$	−4.23611e9	−1.38898
$236$	6.03920e9i	1.94685i
$237$	0	0
$238$	−2.03492e8	−0.0634219
$239$	− 3.41159e9i	− 1.04560i	−0.852456	−	0.522799i	$-0.824888\pi$
$239$	− 3.41159e9i	− 1.04560i	0.852456	−	0.522799i	$-0.175112\pi$
$240$	0	0
$241$	4.95411e9	1.46858	0.734289	−	0.678837i	$-0.237516\pi$
$241$	4.95411e9	1.46858	0.734289	+	0.678837i	$0.237516\pi$
$242$	− 5.37590e9i	− 1.56744i
$243$	0	0
$244$	7.56192e9	2.13340
$245$	6.14437e9i	1.70535i
$246$	0	0
$247$	9.52929e8	0.256019
$248$	− 9.86914e9i	− 2.60899i
$249$	0	0
$250$	1.73019e10	4.42928
$251$	− 2.53734e9i	− 0.639268i	−0.947541	−	0.319634i	$-0.896440\pi$
$251$	− 2.53734e9i	− 0.639268i	0.947541	−	0.319634i	$-0.103560\pi$
$252$	0	0
$253$	1.87967e9	0.458774
$254$	4.25089e9i	1.02128i
$255$	0	0
$256$	−7.03516e9	−1.63800
$257$	− 1.33068e9i	− 0.305028i	−0.988301	−	0.152514i	$-0.951263\pi$
$257$	− 1.33068e9i	− 0.305028i	0.988301	−	0.152514i	$-0.0487369\pi$
$258$	0	0
$259$	1.47604e9	0.328020
$260$	1.64709e10i	3.60432i
$261$	0	0
$262$	−1.18597e10	−2.51692
$263$	− 7.77170e8i	− 0.162440i	−0.996696	−	0.0812200i	$-0.974118\pi$
$263$	− 7.77170e8i	− 0.162440i	0.996696	−	0.0812200i	$-0.0258816\pi$
$264$	0	0
$265$	6.97235e9	1.41382
$266$	6.96697e8i	0.139161i
$267$	0	0
$268$	−1.79534e10	−3.48023
$269$	3.50474e9i	0.669339i	0.942336	+	0.334670i	$0.108625\pi$
$269$	3.50474e9i	0.669339i	−0.942336	+	0.334670i	$0.891375\pi$
$270$	0	0
$271$	5.23655e9	0.970886	0.485443	−	0.874268i	$-0.338658\pi$
$271$	5.23655e9	0.970886	0.485443	+	0.874268i	$0.338658\pi$
$272$	1.46381e9i	0.267430i
$273$	0	0
$274$	−3.71444e9	−0.659008
$275$	− 4.93531e9i	− 0.862947i
$276$	0	0
$277$	−1.55490e9	−0.264109	−0.132054	−	0.991242i	$-0.542157\pi$
$277$	−1.55490e9	−0.264109	−0.132054	+	0.991242i	$0.542157\pi$
$278$	− 1.75889e10i	− 2.94483i
$279$	0	0
$280$	−6.79434e9	−1.10539
$281$	7.17780e7i	0.0115124i	0.999983	+	0.00575620i	$0.00183227\pi$
$281$	7.17780e7i	0.0115124i	−0.999983	+	0.00575620i	$0.998168\pi$
$282$	0	0
$283$	3.79519e9	0.591681	0.295841	−	0.955237i	$-0.404400\pi$
$283$	3.79519e9	0.591681	0.295841	+	0.955237i	$0.404400\pi$
$284$	4.33374e9i	0.666176i
$285$	0	0
$286$	3.85800e9	0.576631
$287$	2.63808e9i	0.388831i
$288$	0	0
$289$	6.84734e9	0.981591
$290$	− 1.70163e10i	− 2.40587i
$291$	0	0
$292$	−1.08813e10	−1.49674
$293$	7.72213e9i	1.04777i	0.851789	+	0.523886i	$0.175518\pi$
$293$	7.72213e9i	1.04777i	−0.851789	+	0.523886i	$0.824482\pi$
$294$	0	0
$295$	−1.17355e10	−1.54958
$296$	− 2.29795e10i	− 2.99346i
$297$	0	0
$298$	1.51602e10	1.92238
$299$	− 8.53598e9i	− 1.06799i
$300$	0	0
$301$	−2.35907e9	−0.287392
$302$	− 4.76941e9i	− 0.573372i
$303$	0	0
$304$	5.01168e9	0.586798
$305$	1.46945e10i	1.69807i
$306$	0	0
$307$	1.79909e9	0.202535	0.101267	−	0.994859i	$-0.467710\pi$
$307$	1.79909e9	0.202535	0.101267	+	0.994859i	$0.467710\pi$
$308$	1.96452e9i	0.218300i
$309$	0	0
$310$	3.39903e10	3.68052
$311$	7.28196e9i	0.778407i	0.921152	+	0.389204i	$0.127250\pi$
$311$	7.28196e9i	0.778407i	−0.921152	+	0.389204i	$0.872750\pi$
$312$	0	0
$313$	−1.61533e9	−0.168300	−0.0841499	−	0.996453i	$-0.526817\pi$
$313$	−1.61533e9	−0.168300	−0.0841499	+	0.996453i	$0.526817\pi$
$314$	2.01280e10i	2.07053i
$315$	0	0
$316$	−1.82514e10	−1.83041
$317$	1.20595e10i	1.19424i	0.802153	+	0.597118i	$0.203688\pi$
$317$	1.20595e10i	1.19424i	−0.802153	+	0.597118i	$0.796312\pi$
$318$	0	0
$319$	−2.77601e9	−0.268076
$320$	4.92640e9i	0.469818i
$321$	0	0
$322$	6.24075e9	0.580515
$323$	4.39663e8i	0.0403933i
$324$	0	0
$325$	−2.24124e10	−2.00888
$326$	− 1.68143e10i	− 1.48870i
$327$	0	0
$328$	4.10705e10	3.54841
$329$	2.29446e9i	0.195838i
$330$	0	0
$331$	1.21361e10	1.01104	0.505520	−	0.862815i	$-0.331301\pi$
$331$	1.21361e10	1.01104	0.505520	+	0.862815i	$0.331301\pi$
$332$	1.37081e10i	1.12830i
$333$	0	0
$334$	−2.48333e10	−1.99548
$335$	− 3.48875e10i	− 2.77007i
$336$	0	0
$337$	−1.74659e10	−1.35416	−0.677081	−	0.735909i	$-0.736755\pi$
$337$	−1.74659e10	−1.35416	−0.677081	+	0.735909i	$0.736755\pi$
$338$	6.17060e9i	0.472782i
$339$	0	0
$340$	−7.59934e9	−0.568670
$341$	− 5.54513e9i	− 0.410104i
$342$	0	0
$343$	6.89248e9	0.497966
$344$	3.67267e10i	2.62270i
$345$	0	0
$346$	−2.77768e10	−1.93811
$347$	1.19555e10i	0.824612i	0.911046	+	0.412306i	$0.135276\pi$
$347$	1.19555e10i	0.824612i	−0.911046	+	0.412306i	$0.864724\pi$
$348$	0	0
$349$	3.18133e9	0.214441	0.107220	−	0.994235i	$-0.465805\pi$
$349$	3.18133e9	0.214441	0.107220	+	0.994235i	$0.465805\pi$
$350$	− 1.63859e10i	− 1.09194i
$351$	0	0
$352$	6.96204e9	0.453488
$353$	2.05597e10i	1.32409i	0.749464	+	0.662045i	$0.230311\pi$
$353$	2.05597e10i	1.32409i	−0.749464	+	0.662045i	$0.769689\pi$
$354$	0	0
$355$	−8.42143e9	−0.530240
$356$	− 2.05514e10i	− 1.27950i
$357$	0	0
$358$	−5.20246e10	−3.16721
$359$	3.13212e10i	1.88565i	0.333292	+	0.942824i	$0.391841\pi$
$359$	3.13212e10i	1.88565i	−0.333292	+	0.942824i	$0.608159\pi$
$360$	0	0
$361$	−1.54783e10	−0.911369
$362$	3.22865e10i	1.88012i
$363$	0	0
$364$	8.92132e9	0.508187
$365$	− 2.11447e10i	− 1.19133i
$366$	0	0
$367$	1.94574e10	1.07256	0.536279	−	0.844041i	$-0.319829\pi$
$367$	1.94574e10	1.07256	0.536279	+	0.844041i	$0.319829\pi$
$368$	− 4.48927e10i	− 2.44785i
$369$	0	0
$370$	7.91436e10	4.22288
$371$	− 3.77652e9i	− 0.199341i
$372$	0	0
$373$	8.72469e9	0.450728	0.225364	−	0.974275i	$-0.427643\pi$
$373$	8.72469e9	0.450728	0.225364	+	0.974275i	$0.427643\pi$
$374$	1.78000e9i	0.0909775i
$375$	0	0
$376$	3.57208e10	1.78718
$377$	1.26065e10i	0.624063i
$378$	0	0
$379$	1.25483e8	0.00608172	0.00304086	−	0.999995i	$-0.499032\pi$
$379$	1.25483e8	0.00608172	0.00304086	+	0.999995i	$0.499032\pi$
$380$	2.60179e10i	1.24778i
$381$	0	0
$382$	1.93041e10	0.906557
$383$	− 2.72346e10i	− 1.26568i	−0.774281	−	0.632842i	$-0.781888\pi$
$383$	− 2.72346e10i	− 1.26568i	0.774281	−	0.632842i	$-0.218112\pi$
$384$	0	0
$385$	−3.81751e9	−0.173755
$386$	− 6.56053e10i	− 2.95522i
$387$	0	0
$388$	−1.21512e10	−0.536157
$389$	4.88881e9i	0.213504i	0.994286	+	0.106752i	$0.0340450\pi$
$389$	4.88881e9i	0.213504i	−0.994286	+	0.106752i	$0.965955\pi$
$390$	0	0
$391$	3.93833e9	0.168502
$392$	− 5.18120e10i	− 2.19425i
$393$	0	0
$394$	−1.95017e10	−0.809257
$395$	− 3.54666e10i	− 1.45690i
$396$	0	0
$397$	3.91261e10	1.57509	0.787544	−	0.616258i	$-0.211352\pi$
$397$	3.91261e10	1.57509	0.787544	+	0.616258i	$0.211352\pi$
$398$	− 4.96567e10i	− 1.97900i
$399$	0	0
$400$	−1.17872e11	−4.60437
$401$	2.52756e9i	0.0977515i	0.998805	+	0.0488758i	$0.0155638\pi$
$401$	2.52756e9i	0.0977515i	−0.998805	+	0.0488758i	$0.984436\pi$
$402$	0	0
$403$	−2.51817e10	−0.954695
$404$	9.33194e10i	3.50305i
$405$	0	0
$406$	−9.21673e9	−0.339213
$407$	− 1.29114e10i	− 0.470538i
$408$	0	0
$409$	−7.35880e9	−0.262974	−0.131487	−	0.991318i	$-0.541975\pi$
$409$	−7.35880e9	−0.262974	−0.131487	+	0.991318i	$0.541975\pi$
$410$	1.41451e11i	5.00576i
$411$	0	0
$412$	8.94590e10	3.10481
$413$	6.35646e9i	0.218482i
$414$	0	0
$415$	−2.66380e10	−0.898067
$416$	− 3.16162e10i	− 1.05569i
$417$	0	0
$418$	6.09421e9	0.199624
$419$	− 8.57360e9i	− 0.278168i	−0.990281	−	0.139084i	$-0.955584\pi$
$419$	− 8.57360e9i	− 0.278168i	0.990281	−	0.139084i	$-0.0444158\pi$
$420$	0	0
$421$	5.61642e10	1.78785	0.893926	−	0.448215i	$-0.147940\pi$
$421$	5.61642e10	1.78785	0.893926	+	0.448215i	$0.147940\pi$
$422$	3.05557e10i	0.963479i
$423$	0	0
$424$	−5.87939e10	−1.81915
$425$	− 1.03406e10i	− 0.316950i
$426$	0	0
$427$	7.95917e9	0.239418
$428$	− 7.24474e10i	− 2.15898i
$429$	0	0
$430$	−1.26491e11	−3.69985
$431$	1.13012e10i	0.327502i	0.986502	+	0.163751i	$0.0523594\pi$
$431$	1.13012e10i	0.327502i	−0.986502	+	0.163751i	$0.947641\pi$
$432$	0	0
$433$	−1.39390e10	−0.396535	−0.198267	−	0.980148i	$-0.563531\pi$
$433$	−1.39390e10	−0.396535	−0.198267	+	0.980148i	$0.563531\pi$
$434$	− 1.84106e10i	− 0.518930i
$435$	0	0
$436$	2.86640e10	0.793214
$437$	− 1.34837e10i	− 0.369729i
$438$	0	0
$439$	3.56214e10	0.959075	0.479537	−	0.877521i	$-0.340804\pi$
$439$	3.56214e10	0.959075	0.479537	+	0.877521i	$0.340804\pi$
$440$	5.94320e10i	1.58566i
$441$	0	0
$442$	8.08339e9	0.211789
$443$	2.89753e10i	0.752339i	0.926551	+	0.376170i	$0.122759\pi$
$443$	2.89753e10i	0.752339i	−0.926551	+	0.376170i	$0.877241\pi$
$444$	0	0
$445$	3.99360e10	1.01841
$446$	1.06866e11i	2.70084i
$447$	0	0
$448$	2.66835e9	0.0662415
$449$	− 7.12282e10i	− 1.75253i	−0.481825	−	0.876267i	$-0.660026\pi$
$449$	− 7.12282e10i	− 1.75253i	0.481825	−	0.876267i	$-0.339974\pi$
$450$	0	0
$451$	2.30761e10	0.557771
$452$	− 3.66058e10i	− 0.876993i
$453$	0	0
$454$	−5.17551e10	−1.21823
$455$	1.73361e10i	0.404489i
$456$	0	0
$457$	4.94096e10	1.13278	0.566392	−	0.824136i	$-0.308339\pi$
$457$	4.94096e10	1.13278	0.566392	+	0.824136i	$0.308339\pi$
$458$	1.88368e10i	0.428100i
$459$	0	0
$460$	2.33059e11	5.20516
$461$	− 2.92993e10i	− 0.648715i	−0.945935	−	0.324358i	$-0.894852\pi$
$461$	− 2.92993e10i	− 0.648715i	0.945935	−	0.324358i	$-0.105148\pi$
$462$	0	0
$463$	−9.00438e10	−1.95943	−0.979715	−	0.200397i	$-0.935777\pi$
$463$	−9.00438e10	−1.95943	−0.979715	+	0.200397i	$0.935777\pi$
$464$	6.63004e10i	1.43036i
$465$	0	0
$466$	−2.00985e7	−0.000426206 0
$467$	− 5.43476e10i	− 1.14265i	−0.820724	−	0.571325i	$-0.806430\pi$
$467$	− 5.43476e10i	− 1.14265i	0.820724	−	0.571325i	$-0.193570\pi$
$468$	0	0
$469$	−1.88966e10	−0.390563
$470$	1.23026e11i	2.52119i
$471$	0	0
$472$	9.89592e10	1.99383
$473$	2.06355e10i	0.412259i
$474$	0	0
$475$	−3.54033e10	−0.695455
$476$	4.11612e9i	0.0801789i
$477$	0	0
$478$	−9.90800e10	−1.89790
$479$	3.96744e10i	0.753647i	0.926285	+	0.376824i	$0.122984\pi$
$479$	3.96744e10i	0.753647i	−0.926285	+	0.376824i	$0.877016\pi$
$480$	0	0
$481$	−5.86334e10	−1.09538
$482$	− 1.43878e11i	− 2.66567i
$483$	0	0
$484$	−1.08741e11	−1.98158
$485$	− 2.36125e10i	− 0.426751i
$486$	0	0
$487$	9.50722e8	0.0169020	0.00845099	−	0.999964i	$-0.497310\pi$
$487$	9.50722e8	0.0169020	0.00845099	+	0.999964i	$0.497310\pi$
$488$	− 1.23911e11i	− 2.18489i
$489$	0	0
$490$	1.78446e11	3.09544
$491$	7.83335e10i	1.34779i	0.738828	+	0.673894i	$0.235380\pi$
$491$	7.83335e10i	1.34779i	−0.738828	+	0.673894i	$0.764620\pi$
$492$	0	0
$493$	−5.81638e9	−0.0984611
$494$	− 2.76752e10i	− 0.464711i
$495$	0	0
$496$	−1.32436e11	−2.18817
$497$	4.56140e9i	0.0747606i
$498$	0	0
$499$	−5.43384e10	−0.876404	−0.438202	−	0.898876i	$-0.644385\pi$
$499$	−5.43384e10	−0.876404	−0.438202	+	0.898876i	$0.644385\pi$
$500$	− 3.49973e11i	− 5.59957i
$501$	0	0
$502$	−7.36898e10	−1.16036
$503$	− 2.22062e10i	− 0.346898i	−0.984843	−	0.173449i	$-0.944509\pi$
$503$	− 2.22062e10i	− 0.346898i	0.984843	−	0.173449i	$-0.0554913\pi$
$504$	0	0
$505$	−1.81341e11	−2.78823
$506$	− 5.45897e10i	− 0.832738i
$507$	0	0
$508$	8.59847e10	1.29112
$509$	1.78138e10i	0.265390i	0.991157	+	0.132695i	$0.0423631\pi$
$509$	1.78138e10i	0.265390i	−0.991157	+	0.132695i	$0.957637\pi$
$510$	0	0
$511$	−1.14529e10	−0.167970
$512$	1.52042e11i	2.21250i
$513$	0	0
$514$	−3.86457e10	−0.553668
$515$	1.73839e11i	2.47126i
$516$	0	0
$517$	2.00703e10	0.280925
$518$	− 4.28675e10i	− 0.595401i
$519$	0	0
$520$	2.69894e11	3.69130
$521$	− 4.80679e9i	− 0.0652385i	−0.999468	−	0.0326193i	$-0.989615\pi$
$521$	− 4.80679e9i	− 0.0652385i	0.999468	−	0.0326193i	$-0.0103849\pi$
$522$	0	0
$523$	1.04562e11	1.39755	0.698777	−	0.715339i	$-0.253728\pi$
$523$	1.04562e11	1.39755	0.698777	+	0.715339i	$0.253728\pi$
$524$	2.39892e11i	3.18193i
$525$	0	0
$526$	−2.25707e10	−0.294851
$527$	− 1.16183e10i	− 0.150626i
$528$	0	0
$529$	−4.24710e10	−0.542337
$530$	− 2.02492e11i	− 2.56629i
$531$	0	0
$532$	1.40924e10	0.175929
$533$	− 1.04794e11i	− 1.29845i
$534$	0	0
$535$	1.40782e11	1.71843
$536$	2.94187e11i	3.56422i
$537$	0	0
$538$	1.01785e11	1.21494
$539$	− 2.91114e10i	− 0.344912i
$540$	0	0
$541$	6.34784e10	0.741032	0.370516	−	0.928826i	$-0.379181\pi$
$541$	6.34784e10	0.741032	0.370516	+	0.928826i	$0.379181\pi$
$542$	− 1.52081e11i	− 1.76229i
$543$	0	0
$544$	1.45871e10	0.166561
$545$	5.57006e10i	0.631355i
$546$	0	0
$547$	−1.68089e10	−0.187755	−0.0938773	−	0.995584i	$-0.529926\pi$
$547$	−1.68089e10	−0.187755	−0.0938773	+	0.995584i	$0.529926\pi$
$548$	7.51336e10i	0.833128i
$549$	0	0
$550$	−1.43332e11	−1.56637
$551$	1.99136e10i	0.216044i
$552$	0	0
$553$	−1.92102e10	−0.205415
$554$	4.51577e10i	0.479394i
$555$	0	0
$556$	−3.55779e11	−3.72290
$557$	− 2.79408e10i	− 0.290281i	−0.989411	−	0.145141i	$-0.953637\pi$
$557$	− 2.79408e10i	− 0.290281i	0.989411	−	0.145141i	$-0.0463634\pi$
$558$	0	0
$559$	9.37103e10	0.959710
$560$	9.11748e10i	0.927092i
$561$	0	0
$562$	2.08459e9	0.0208966
$563$	− 1.90449e10i	− 0.189559i	−0.995498	−	0.0947795i	$-0.969785\pi$
$563$	− 1.90449e10i	− 0.189559i	0.995498	−	0.0947795i	$-0.0302146\pi$
$564$	0	0
$565$	7.11333e10	0.698038
$566$	− 1.10221e11i	− 1.07398i
$567$	0	0
$568$	7.10132e10	0.682253
$569$	− 4.41514e10i	− 0.421207i	−0.977572	−	0.210603i	$-0.932457\pi$
$569$	− 4.41514e10i	− 0.421207i	0.977572	−	0.210603i	$-0.0675429\pi$
$570$	0	0
$571$	6.17813e10	0.581182	0.290591	−	0.956847i	$-0.406148\pi$
$571$	6.17813e10	0.581182	0.290591	+	0.956847i	$0.406148\pi$
$572$	− 7.80374e10i	− 0.728985i
$573$	0	0
$574$	7.66158e10	0.705782
$575$	3.17129e11i	2.90111i
$576$	0	0
$577$	5.67064e10	0.511598	0.255799	−	0.966730i	$-0.417661\pi$
$577$	5.67064e10	0.511598	0.255799	+	0.966730i	$0.417661\pi$
$578$	− 1.98862e11i	− 1.78172i
$579$	0	0
$580$	−3.44196e11	−3.04154
$581$	1.44282e10i	0.126622i
$582$	0	0
$583$	−3.30343e10	−0.285951
$584$	1.78302e11i	1.53287i
$585$	0	0
$586$	2.24268e11	1.90185
$587$	1.27993e11i	1.07803i	0.842295	+	0.539017i	$0.181204\pi$
$587$	1.27993e11i	1.07803i	−0.842295	+	0.539017i	$0.818796\pi$
$588$	0	0
$589$	−3.97778e10	−0.330506
$590$	3.40826e11i	2.81271i
$591$	0	0
$592$	−3.08367e11	−2.51062
$593$	1.46325e11i	1.18332i	0.806189	+	0.591658i	$0.201526\pi$
$593$	1.46325e11i	1.18332i	−0.806189	+	0.591658i	$0.798474\pi$
$594$	0	0
$595$	−7.99855e9	−0.0638180
$596$	− 3.06652e11i	− 2.43030i
$597$	0	0
$598$	−2.47904e11	−1.93856
$599$	− 1.55799e11i	− 1.21020i	−0.796150	−	0.605099i	$-0.793133\pi$
$599$	− 1.55799e11i	− 1.21020i	0.796150	−	0.605099i	$-0.206867\pi$
$600$	0	0
$601$	8.14475e10	0.624281	0.312141	−	0.950036i	$-0.398954\pi$
$601$	8.14475e10	0.624281	0.312141	+	0.950036i	$0.398954\pi$
$602$	6.85126e10i	0.521656i
$603$	0	0
$604$	−9.64729e10	−0.724866
$605$	− 2.11308e11i	− 1.57723i
$606$	0	0
$607$	−8.45502e10	−0.622816	−0.311408	−	0.950276i	$-0.600801\pi$
$607$	−8.45502e10	−0.622816	−0.311408	+	0.950276i	$0.600801\pi$
$608$	− 4.99419e10i	− 0.365469i
$609$	0	0
$610$	4.26761e11	3.08223
$611$	− 9.11436e10i	− 0.653975i
$612$	0	0
$613$	−2.66389e11	−1.88658	−0.943288	−	0.331974i	$-0.892285\pi$
$613$	−2.66389e11	−1.88658	−0.943288	+	0.331974i	$0.892285\pi$
$614$	− 5.22496e10i	− 0.367629i
$615$	0	0
$616$	3.21909e10	0.223568
$617$	− 2.02535e11i	− 1.39752i	−0.715354	−	0.698762i	$-0.753735\pi$
$617$	− 2.02535e11i	− 1.39752i	0.715354	−	0.698762i	$-0.246265\pi$
$618$	0	0
$619$	6.91745e10	0.471176	0.235588	−	0.971853i	$-0.424298\pi$
$619$	6.91745e10	0.471176	0.235588	+	0.971853i	$0.424298\pi$
$620$	− 6.87538e11i	− 4.65297i
$621$	0	0
$622$	2.11484e11	1.41292
$623$	− 2.16310e10i	− 0.143590i
$624$	0	0
$625$	3.23629e11	2.12094
$626$	4.69127e10i	0.305487i
$627$	0	0
$628$	4.07137e11	2.61759
$629$	− 2.70523e10i	− 0.172823i
$630$	0	0
$631$	1.30412e11	0.822623	0.411311	−	0.911495i	$-0.365071\pi$
$631$	1.30412e11	0.822623	0.411311	+	0.911495i	$0.365071\pi$
$632$	2.99070e11i	1.87458i
$633$	0	0
$634$	3.50233e11	2.16770
$635$	1.67088e11i	1.02766i
$636$	0	0
$637$	−1.32201e11	−0.802931
$638$	8.06214e10i	0.486595i
$639$	0	0
$640$	−2.33102e11	−1.38940
$641$	− 1.61521e11i	− 0.956744i	−0.878157	−	0.478372i	$-0.841227\pi$
$641$	− 1.61521e11i	− 0.956744i	0.878157	−	0.478372i	$-0.158773\pi$
$642$	0	0
$643$	−2.49573e11	−1.46000	−0.730000	−	0.683447i	$-0.760480\pi$
$643$	−2.49573e11	−1.46000	−0.730000	+	0.683447i	$0.760480\pi$
$644$	− 1.26234e11i	− 0.733896i
$645$	0	0
$646$	1.27688e10	0.0733194
$647$	3.14220e11i	1.79315i	0.442892	+	0.896575i	$0.353952\pi$
$647$	3.14220e11i	1.79315i	−0.442892	+	0.896575i	$0.646048\pi$
$648$	0	0
$649$	5.56018e10	0.313408
$650$	6.50904e11i	3.64639i
$651$	0	0
$652$	−3.40110e11	−1.88204
$653$	− 1.44558e11i	− 0.795038i	−0.917594	−	0.397519i	$-0.869871\pi$
$653$	− 1.44558e11i	− 0.795038i	0.917594	−	0.397519i	$-0.130129\pi$
$654$	0	0
$655$	−4.66164e11	−2.53264
$656$	− 5.51134e11i	− 2.97606i
$657$	0	0
$658$	6.66361e10	0.355472
$659$	9.51659e10i	0.504591i	0.967650	+	0.252296i	$0.0811856\pi$
$659$	9.51659e10i	0.504591i	−0.967650	+	0.252296i	$0.918814\pi$
$660$	0	0
$661$	2.42049e11	1.26793	0.633967	−	0.773360i	$-0.281426\pi$
$661$	2.42049e11	1.26793	0.633967	+	0.773360i	$0.281426\pi$
$662$	− 3.52460e11i	− 1.83518i
$663$	0	0
$664$	2.24623e11	1.15553
$665$	2.73847e10i	0.140030i
$666$	0	0
$667$	1.78378e11	0.901237
$668$	5.02313e11i	2.52272i
$669$	0	0
$670$	−1.01321e12	−5.02806
$671$	− 6.96212e10i	− 0.343440i
$672$	0	0
$673$	−3.24434e10	−0.158149	−0.0790745	−	0.996869i	$-0.525196\pi$
$673$	−3.24434e10	−0.158149	−0.0790745	+	0.996869i	$0.525196\pi$
$674$	5.07247e11i	2.45799i
$675$	0	0
$676$	1.24815e11	0.597698
$677$	5.23991e10i	0.249442i	0.992192	+	0.124721i	$0.0398035\pi$
$677$	5.23991e10i	0.249442i	−0.992192	+	0.124721i	$0.960196\pi$
$678$	0	0
$679$	−1.27895e10	−0.0601693
$680$	1.24524e11i	0.582393i
$681$	0	0
$682$	−1.61043e11	−0.744396
$683$	− 3.63011e11i	− 1.66816i	−0.551646	−	0.834078i	$-0.686000\pi$
$683$	− 3.63011e11i	− 1.66816i	0.551646	−	0.834078i	$-0.314000\pi$
$684$	0	0
$685$	−1.46001e11	−0.663124
$686$	− 2.00173e11i	− 0.903876i
$687$	0	0
$688$	4.92844e11	2.19966
$689$	1.50016e11i	0.665673i
$690$	0	0
$691$	−3.35488e11	−1.47152	−0.735758	−	0.677245i	$-0.763174\pi$
$691$	−3.35488e11	−1.47152	−0.735758	+	0.677245i	$0.763174\pi$
$692$	5.61854e11i	2.45019i
$693$	0	0
$694$	3.47214e11	1.49678
$695$	− 6.91359e11i	− 2.96322i
$696$	0	0
$697$	4.83497e10	0.204863
$698$	− 9.23928e10i	− 0.389239i
$699$	0	0
$700$	−3.31445e11	−1.38045
$701$	− 3.62513e11i	− 1.50124i	−0.660732	−	0.750622i	$-0.729754\pi$
$701$	− 3.62513e11i	− 1.50124i	0.660732	−	0.750622i	$-0.270246\pi$
$702$	0	0
$703$	−9.26192e10	−0.379210
$704$	− 2.33408e10i	− 0.0950222i
$705$	0	0
$706$	5.97099e11	2.40341
$707$	9.82217e10i	0.393124i
$708$	0	0
$709$	−1.60913e10	−0.0636805	−0.0318402	−	0.999493i	$-0.510137\pi$
$709$	−1.60913e10	−0.0636805	−0.0318402	+	0.999493i	$0.510137\pi$
$710$	2.44577e11i	0.962458i
$711$	0	0
$712$	−3.36758e11	−1.31038
$713$	3.56314e11i	1.37872i
$714$	0	0
$715$	1.51644e11	0.580232
$716$	1.05233e12i	4.00404i
$717$	0	0
$718$	9.09636e11	3.42271
$719$	3.17146e11i	1.18671i	0.804942	+	0.593353i	$0.202196\pi$
$719$	3.17146e11i	1.18671i	−0.804942	+	0.593353i	$0.797804\pi$
$720$	0	0
$721$	9.41586e10	0.348433
$722$	4.49524e11i	1.65426i
$723$	0	0
$724$	6.53073e11	2.37688
$725$	− 4.68356e11i	− 1.69521i
$726$	0	0
$727$	−1.89203e10	−0.0677314	−0.0338657	−	0.999426i	$-0.510782\pi$
$727$	−1.89203e10	−0.0677314	−0.0338657	+	0.999426i	$0.510782\pi$
$728$	− 1.46186e11i	− 0.520452i
$729$	0	0
$730$	−6.14090e11	−2.16242
$731$	4.32360e10i	0.151418i
$732$	0	0
$733$	−4.47743e11	−1.55100	−0.775502	−	0.631346i	$-0.782503\pi$
$733$	−4.47743e11	−1.55100	−0.775502	+	0.631346i	$0.782503\pi$
$734$	− 5.65086e11i	− 1.94684i
$735$	0	0
$736$	−4.47361e11	−1.52457
$737$	1.65294e11i	0.560256i
$738$	0	0
$739$	6.72795e10	0.225582	0.112791	−	0.993619i	$-0.464021\pi$
$739$	6.72795e10	0.225582	0.112791	+	0.993619i	$0.464021\pi$
$740$	− 1.60087e12i	− 5.33863i
$741$	0	0
$742$	−1.09678e11	−0.361831
$743$	5.93373e11i	1.94703i	0.228626	+	0.973514i	$0.426577\pi$
$743$	5.93373e11i	1.94703i	−0.228626	+	0.973514i	$0.573423\pi$
$744$	0	0
$745$	5.95893e11	1.93439
$746$	− 2.53384e11i	− 0.818134i
$747$	0	0
$748$	3.60049e10	0.115015
$749$	− 7.62533e10i	− 0.242288i
$750$	0	0
$751$	1.58553e11	0.498444	0.249222	−	0.968446i	$-0.419825\pi$
$751$	1.58553e11	0.498444	0.249222	+	0.968446i	$0.419825\pi$
$752$	− 4.79345e11i	− 1.49892i
$753$	0	0
$754$	3.66120e11	1.13276
$755$	− 1.87469e11i	− 0.576953i
$756$	0	0
$757$	−4.60322e11	−1.40177	−0.700887	−	0.713273i	$-0.747212\pi$
$757$	−4.60322e11	−1.40177	−0.700887	+	0.713273i	$0.747212\pi$
$758$	− 3.64429e9i	− 0.0110392i
$759$	0	0
$760$	4.26333e11	1.27789
$761$	2.30446e11i	0.687116i	0.939132	+	0.343558i	$0.111632\pi$
$761$	2.30446e11i	0.687116i	−0.939132	+	0.343558i	$0.888368\pi$
$762$	0	0
$763$	3.01698e10	0.0890172
$764$	− 3.90472e11i	− 1.14608i
$765$	0	0
$766$	−7.90952e11	−2.29739
$767$	− 2.52500e11i	− 0.729592i
$768$	0	0
$769$	8.20544e10	0.234637	0.117319	−	0.993094i	$-0.462570\pi$
$769$	8.20544e10	0.234637	0.117319	+	0.993094i	$0.462570\pi$
$770$	1.10869e11i	0.315389i
$771$	0	0
$772$	−1.32703e12	−3.73603
$773$	− 2.14108e11i	− 0.599674i	−0.953990	−	0.299837i	$-0.903068\pi$
$773$	− 2.14108e11i	− 0.599674i	0.953990	−	0.299837i	$-0.0969323\pi$
$774$	0	0
$775$	9.35550e11	2.59335
$776$	1.99111e11i	0.549096i
$777$	0	0
$778$	1.41982e11	0.387538
$779$	− 1.65535e11i	− 0.449512i
$780$	0	0
$781$	3.98999e10	0.107243
$782$	− 1.14378e11i	− 0.305854i
$783$	0	0
$784$	−6.95278e11	−1.84032
$785$	7.91159e11i	2.08346i
$786$	0	0
$787$	2.38400e11	0.621452	0.310726	−	0.950500i	$-0.399428\pi$
$787$	2.38400e11	0.621452	0.310726	+	0.950500i	$0.399428\pi$
$788$	3.94469e11i	1.02308i
$789$	0	0
$790$	−1.03003e12	−2.64448
$791$	− 3.85288e10i	− 0.0984192i
$792$	0	0
$793$	−3.16165e11	−0.799505
$794$	− 1.13631e12i	− 2.85900i
$795$	0	0
$796$	−1.00443e12	−2.50188
$797$	1.42021e11i	0.351981i	0.984392	+	0.175991i	$0.0563128\pi$
$797$	1.42021e11i	0.351981i	−0.984392	+	0.175991i	$0.943687\pi$
$798$	0	0
$799$	4.20518e10	0.103181
$800$	1.17461e12i	2.86769i
$801$	0	0
$802$	7.34058e10	0.177433
$803$	1.00182e11i	0.240950i
$804$	0	0
$805$	2.45302e11	0.584141
$806$	7.31332e11i	1.73290i
$807$	0	0
$808$	1.52914e12	3.58759
$809$	1.70718e11i	0.398552i	0.979943	+	0.199276i	$0.0638591\pi$
$809$	1.70718e11i	0.398552i	−0.979943	+	0.199276i	$0.936141\pi$
$810$	0	0
$811$	7.94459e11	1.83649	0.918244	−	0.396015i	$-0.129607\pi$
$811$	7.94459e11	1.83649	0.918244	+	0.396015i	$0.129607\pi$
$812$	1.86431e11i	0.428839i
$813$	0	0
$814$	−3.74975e11	−0.854091
$815$	− 6.60911e11i	− 1.49800i
$816$	0	0
$817$	1.48028e11	0.332242
$818$	2.13716e11i	0.477335i
$819$	0	0
$820$	2.86119e12	6.32836
$821$	6.63787e11i	1.46102i	0.682902	+	0.730510i	$0.260718\pi$
$821$	6.63787e11i	1.46102i	−0.682902	+	0.730510i	$0.739282\pi$
$822$	0	0
$823$	−1.41192e11	−0.307758	−0.153879	−	0.988090i	$-0.549177\pi$
$823$	−1.41192e11	−0.307758	−0.153879	+	0.988090i	$0.549177\pi$
$824$	− 1.46589e12i	− 3.17974i
$825$	0	0
$826$	1.84606e11	0.396574
$827$	− 5.29237e11i	− 1.13143i	−0.824600	−	0.565716i	$-0.808600\pi$
$827$	− 5.29237e11i	− 1.13143i	0.824600	−	0.565716i	$-0.191400\pi$
$828$	0	0
$829$	−2.76078e11	−0.584540	−0.292270	−	0.956336i	$-0.594411\pi$
$829$	−2.76078e11	−0.584540	−0.292270	+	0.956336i	$0.594411\pi$
$830$	7.73625e11i	1.63011i
$831$	0	0
$832$	−1.05996e11	−0.221205
$833$	− 6.09951e10i	− 0.126682i
$834$	0	0
$835$	−9.76108e11	−2.00795
$836$	− 1.23270e11i	− 0.252368i
$837$	0	0
$838$	−2.48996e11	−0.504913
$839$	4.80367e11i	0.969450i	0.874667	+	0.484725i	$0.161080\pi$
$839$	4.80367e11i	0.969450i	−0.874667	+	0.484725i	$0.838920\pi$
$840$	0	0
$841$	2.36806e11	0.473379
$842$	− 1.63113e12i	− 3.24520i
$843$	0	0
$844$	6.18064e11	1.21805
$845$	2.42545e11i	0.475735i
$846$	0	0
$847$	−1.14453e11	−0.222379
$848$	7.88969e11i	1.52573i
$849$	0	0
$850$	−3.00314e11	−0.575308
$851$	8.29648e11i	1.58189i
$852$	0	0
$853$	−3.99406e11	−0.754428	−0.377214	−	0.926126i	$-0.623118\pi$
$853$	−3.99406e11	−0.754428	−0.377214	+	0.926126i	$0.623118\pi$
$854$	− 2.31152e11i	− 0.434576i
$855$	0	0
$856$	−1.18713e12	−2.21108
$857$	3.45379e11i	0.640284i	0.947370	+	0.320142i	$0.103731\pi$
$857$	3.45379e11i	0.640284i	−0.947370	+	0.320142i	$0.896269\pi$
$858$	0	0
$859$	−1.49233e11	−0.274090	−0.137045	−	0.990565i	$-0.543760\pi$
$859$	−1.49233e11	−0.274090	−0.137045	+	0.990565i	$0.543760\pi$
$860$	2.55858e12i	4.67741i
$861$	0	0
$862$	3.28211e11	0.594461
$863$	6.94428e10i	0.125194i	0.998039	+	0.0625971i	$0.0199383\pi$
$863$	6.94428e10i	0.125194i	−0.998039	+	0.0625971i	$0.980062\pi$
$864$	0	0
$865$	−1.09181e12	−1.95021
$866$	4.04820e11i	0.719765i
$867$	0	0
$868$	−3.72400e11	−0.656040
$869$	1.68037e11i	0.294664i
$870$	0	0
$871$	7.50635e11	1.30424
$872$	− 4.69692e11i	− 0.812357i
$873$	0	0
$874$	−3.91597e11	−0.671109
$875$	− 3.68358e11i	− 0.628402i
$876$	0	0
$877$	3.28637e10	0.0555544	0.0277772	−	0.999614i	$-0.491157\pi$
$877$	3.28637e10	0.0555544	0.0277772	+	0.999614i	$0.491157\pi$
$878$	− 1.03452e12i	− 1.74085i
$879$	0	0
$880$	7.97533e11	1.32990
$881$	− 4.25529e11i	− 0.706359i	−0.935556	−	0.353180i	$-0.885100\pi$
$881$	− 4.25529e11i	− 0.706359i	0.935556	−	0.353180i	$-0.114900\pi$
$882$	0	0
$883$	−9.79606e11	−1.61142	−0.805710	−	0.592311i	$-0.798216\pi$
$883$	−9.79606e11	−1.61142	−0.805710	+	0.592311i	$0.798216\pi$
$884$	− 1.63506e11i	− 0.267747i
$885$	0	0
$886$	8.41508e11	1.36560
$887$	− 6.35999e11i	− 1.02745i	−0.857954	−	0.513727i	$-0.828265\pi$
$887$	− 6.35999e11i	− 1.02745i	0.857954	−	0.513727i	$-0.171735\pi$
$888$	0	0
$889$	9.05017e10	0.144894
$890$	− 1.15983e12i	− 1.84856i
$891$	0	0
$892$	2.16162e12	3.41445
$893$	− 1.43973e11i	− 0.226400i
$894$	0	0
$895$	−2.04491e12	−3.18699
$896$	1.26258e11i	0.195897i
$897$	0	0
$898$	−2.06862e12	−3.18109
$899$	− 5.26227e11i	− 0.805628i
$900$	0	0
$901$	−6.92144e10	−0.105026
$902$	− 6.70181e11i	− 1.01243i
$903$	0	0
$904$	−5.99828e11	−0.898158
$905$	1.26907e12i	1.89187i
$906$	0	0
$907$	6.40236e11	0.946043	0.473021	−	0.881051i	$-0.343163\pi$
$907$	6.40236e11	0.946043	0.473021	+	0.881051i	$0.343163\pi$
$908$	1.04687e12i	1.54011i
$909$	0	0
$910$	5.03480e11	0.734204
$911$	− 9.37983e11i	− 1.36183i	−0.732364	−	0.680913i	$-0.761583\pi$
$911$	− 9.37983e11i	− 1.36183i	0.732364	−	0.680913i	$-0.238417\pi$
$912$	0	0
$913$	1.26208e11	0.181637
$914$	− 1.43496e12i	− 2.05616i
$915$	0	0
$916$	3.81020e11	0.541210
$917$	2.52494e11i	0.357087i
$918$	0	0
$919$	5.87760e11	0.824021	0.412010	−	0.911179i	$-0.364827\pi$
$919$	5.87760e11	0.824021	0.412010	+	0.911179i	$0.364827\pi$
$920$	− 3.81893e12i	− 5.33078i
$921$	0	0
$922$	−8.50918e11	−1.17751
$923$	− 1.81194e11i	− 0.249654i
$924$	0	0
$925$	2.17835e12	2.97550
$926$	2.61507e12i	3.55663i
$927$	0	0
$928$	6.60691e11	0.890854
$929$	4.82213e11i	0.647405i	0.946159	+	0.323703i	$0.104928\pi$
$929$	4.82213e11i	0.647405i	−0.946159	+	0.323703i	$0.895072\pi$
$930$	0	0
$931$	−2.08829e11	−0.277967
$932$	4.06541e8i	0 0.000538816i
$933$	0	0
$934$	−1.57838e12	−2.07407
$935$	6.99656e10i	0.0915458i
$936$	0	0
$937$	5.57423e11	0.723147	0.361573	−	0.932344i	$-0.382240\pi$
$937$	5.57423e11	0.723147	0.361573	+	0.932344i	$0.382240\pi$
$938$	5.48798e11i	0.708926i
$939$	0	0
$940$	2.48850e12	3.18733
$941$	− 3.08524e11i	− 0.393487i	−0.980455	−	0.196743i	$-0.936963\pi$
$941$	− 3.08524e11i	− 0.393487i	0.980455	−	0.196743i	$-0.0630366\pi$
$942$	0	0
$943$	−1.48280e12	−1.87515
$944$	− 1.32796e12i	− 1.67223i
$945$	0	0
$946$	5.99300e11	0.748306
$947$	3.89670e11i	0.484503i	0.970213	+	0.242252i	$0.0778859\pi$
$947$	3.89670e11i	0.484503i	−0.970213	+	0.242252i	$0.922114\pi$
$948$	0	0
$949$	4.54947e11	0.560914
$950$	1.02819e12i	1.26235i
$951$	0	0
$952$	6.74473e10	0.0821139
$953$	1.03386e12i	1.25341i	0.779259	+	0.626703i	$0.215596\pi$
$953$	1.03386e12i	1.25341i	−0.779259	+	0.626703i	$0.784404\pi$
$954$	0	0
$955$	7.58775e11	0.912220
$956$	2.00414e12i	2.39936i
$957$	0	0
$958$	1.15223e12	1.36797
$959$	7.90806e10i	0.0934965i
$960$	0	0
$961$	1.98259e11	0.232455
$962$	1.70284e12i	1.98827i
$963$	0	0
$964$	−2.91029e12	−3.36998
$965$	− 2.57871e12i	− 2.97368i
$966$	0	0
$967$	7.48439e11	0.855954	0.427977	−	0.903790i	$-0.359226\pi$
$967$	7.48439e11	0.855954	0.427977	+	0.903790i	$0.359226\pi$
$968$	1.78184e12i	2.02940i
$969$	0	0
$970$	−6.85759e11	−0.774612
$971$	− 5.21049e11i	− 0.586140i	−0.956091	−	0.293070i	$-0.905323\pi$
$971$	− 5.21049e11i	− 0.586140i	0.956091	−	0.293070i	$-0.0946769\pi$
$972$	0	0
$973$	−3.74469e11	−0.417797
$974$	− 2.76111e10i	− 0.0306794i
$975$	0	0
$976$	−1.66279e12	−1.83247
$977$	1.26211e12i	1.38522i	0.721313	+	0.692610i	$0.243539\pi$
$977$	1.26211e12i	1.38522i	−0.721313	+	0.692610i	$0.756461\pi$
$978$	0	0
$979$	−1.89213e11	−0.205977
$980$	− 3.60951e12i	− 3.91330i
$981$	0	0
$982$	2.27498e12	2.44642
$983$	− 5.61877e11i	− 0.601765i	−0.953661	−	0.300882i	$-0.902719\pi$
$983$	− 5.61877e11i	− 0.601765i	0.953661	−	0.300882i	$-0.0972812\pi$
$984$	0	0
$985$	−7.66542e11	−0.814312
$986$	1.68920e11i	0.178720i
$987$	0	0
$988$	−5.59798e11	−0.587494
$989$	− 1.32598e12i	− 1.38596i
$990$	0	0
$991$	2.81006e11	0.291354	0.145677	−	0.989332i	$-0.453464\pi$
$991$	2.81006e11	0.291354	0.145677	+	0.989332i	$0.453464\pi$
$992$	1.31974e12i	1.36283i
$993$	0	0
$994$	1.32473e11	0.135701
$995$	− 1.95183e12i	− 1.99136i
$996$	0	0
$997$	1.38386e12	1.40059	0.700294	−	0.713855i	$-0.253052\pi$
$997$	1.38386e12	1.40059	0.700294	+	0.713855i	$0.253052\pi$
$998$	1.57811e12i	1.59079i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.9.b.d.26.1	✓	6
3.2	odd	2	inner	27.9.b.d.26.6	yes	6
4.3	odd	2		432.9.e.k.161.1		6
9.2	odd	6		81.9.d.f.53.6		12
9.4	even	3		81.9.d.f.26.6		12
9.5	odd	6		81.9.d.f.26.1		12
9.7	even	3		81.9.d.f.53.1		12
12.11	even	2		432.9.e.k.161.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.9.b.d.26.1	✓	6	1.1	even	1	trivial
27.9.b.d.26.6	yes	6	3.2	odd	2	inner
81.9.d.f.26.1		12	9.5	odd	6
81.9.d.f.26.6		12	9.4	even	3
81.9.d.f.53.1		12	9.7	even	3
81.9.d.f.53.6		12	9.2	odd	6
432.9.e.k.161.1		6	4.3	odd	2
432.9.e.k.161.6		6	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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	27.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-29.0422i q^{2} -587.450 q^{4} -1141.55i q^{5} -618.310 q^{7} +9626.03i q^{8} +O(q^{10})\)	\(q-29.0422i q^{2} -587.450 q^{4} -1141.55i q^{5} -618.310 q^{7} +9626.03i q^{8} -33153.0 q^{10} +5408.54i q^{11} +24561.4 q^{13} +17957.1i q^{14} +129174. q^{16} +11332.1i q^{17} +38797.9 q^{19} +670601. i q^{20} +157076. q^{22} -347537. i q^{23} -912504. q^{25} -713316. i q^{26} +363226. q^{28} +513264. i q^{29} -1.02526e6 q^{31} -1.28723e6i q^{32} +329110. q^{34} +705830. i q^{35} -2.38722e6 q^{37} -1.12678e6i q^{38} +1.09886e7 q^{40} -4.26661e6i q^{41} +3.81535e6 q^{43} -3.17724e6i q^{44} -1.00932e7 q^{46} -3.71085e6i q^{47} -5.38249e6 q^{49} +2.65011e7i q^{50} -1.44286e7 q^{52} +6.10781e6i q^{53} +6.17410e6 q^{55} -5.95187e6i q^{56} +1.49063e7 q^{58} -1.02804e7i q^{59} -1.28725e7 q^{61} +2.97757e7i q^{62} -4.31555e6 q^{64} -2.80379e7i q^{65} +3.05616e7 q^{67} -6.65705e6i q^{68} +2.04989e7 q^{70} -7.37721e6i q^{71} +1.85229e7 q^{73} +6.93302e7i q^{74} -2.27918e7 q^{76} -3.34415e6i q^{77} +3.10689e7 q^{79} -1.47458e8i q^{80} -1.23912e8 q^{82} -2.33350e7i q^{83} +1.29361e7 q^{85} -1.10806e8i q^{86} -5.20627e7 q^{88} +3.49841e7i q^{89} -1.51865e7 q^{91} +2.04160e8i q^{92} -1.07771e8 q^{94} -4.42896e7i q^{95} +2.06846e7 q^{97} +1.56319e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 786 q^{4} - 1698 q^{7}+O(q^{10}) \) 6 * q - 786 * q^4 - 1698 * q^7	\( 6 q - 786 q^{4} - 1698 q^{7} - 46278 q^{10} + 41844 q^{13} + 170466 q^{16} - 36384 q^{19} + 647298 q^{22} - 1646688 q^{25} - 838986 q^{28} + 2058474 q^{31} - 1320300 q^{34} - 9395880 q^{37} + 29560086 q^{40} - 2737284 q^{43} - 31514616 q^{46} + 28900656 q^{49} - 34081692 q^{52} - 26674542 q^{55} + 75244572 q^{58} - 40180776 q^{61} - 48541458 q^{64} + 111355284 q^{67} + 17727282 q^{70} + 12821718 q^{73} - 38623776 q^{76} + 21820404 q^{79} - 138785832 q^{82} - 83844396 q^{85} + 57552174 q^{88} + 156632964 q^{91} - 502013916 q^{94} + 53735106 q^{97}+O(q^{100}) \) 6 * q - 786 * q^4 - 1698 * q^7 - 46278 * q^10 + 41844 * q^13 + 170466 * q^16 - 36384 * q^19 + 647298 * q^22 - 1646688 * q^25 - 838986 * q^28 + 2058474 * q^31 - 1320300 * q^34 - 9395880 * q^37 + 29560086 * q^40 - 2737284 * q^43 - 31514616 * q^46 + 28900656 * q^49 - 34081692 * q^52 - 26674542 * q^55 + 75244572 * q^58 - 40180776 * q^61 - 48541458 * q^64 + 111355284 * q^67 + 17727282 * q^70 + 12821718 * q^73 - 38623776 * q^76 + 21820404 * q^79 - 138785832 * q^82 - 83844396 * q^85 + 57552174 * q^88 + 156632964 * q^91 - 502013916 * q^94 + 53735106 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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