LMFDB - Embedded newform 81.10.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,10,Mod(1,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	81.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.7179027293$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2930 x^{6} - 1276 x^{5} + 2487472 x^{4} + 3423248 x^{3} - 586568096 x^{2} + \cdots + 965565184 $ x^8 - x^7 - 2930x^6 - 1276x^5 + 2487472x^4 + 3423248x^3 - 586568096x^2 - 2825363008x + 965565184
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{18}\cdot 17 $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-32.8660$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-34.8660 q^{2} +703.639 q^{4} +2006.52 q^{5} +4867.88 q^{7} -6681.68 q^{8} -69959.4 q^{10} -73917.1 q^{11} +26492.1 q^{13} -169724. q^{14} -127299. q^{16} -300444. q^{17} +180460. q^{19} +1.41187e6 q^{20} +2.57720e6 q^{22} -2.13285e6 q^{23} +2.07300e6 q^{25} -923673. q^{26} +3.42523e6 q^{28} -1.89635e6 q^{29} +5.07398e6 q^{31} +7.85945e6 q^{32} +1.04753e7 q^{34} +9.76750e6 q^{35} -2.17155e6 q^{37} -6.29194e6 q^{38} -1.34069e7 q^{40} -7.63351e6 q^{41} -1.37351e7 q^{43} -5.20110e7 q^{44} +7.43639e7 q^{46} -5.06676e7 q^{47} -1.66573e7 q^{49} -7.22772e7 q^{50} +1.86409e7 q^{52} +2.60958e7 q^{53} -1.48316e8 q^{55} -3.25256e7 q^{56} +6.61182e7 q^{58} +4.22136e7 q^{59} -4.87874e7 q^{61} -1.76910e8 q^{62} -2.08850e8 q^{64} +5.31569e7 q^{65} -1.71276e8 q^{67} -2.11404e8 q^{68} -3.40554e8 q^{70} +8.48941e7 q^{71} +1.62304e8 q^{73} +7.57135e7 q^{74} +1.26979e8 q^{76} -3.59820e8 q^{77} +1.90082e8 q^{79} -2.55429e8 q^{80} +2.66150e8 q^{82} -4.43914e8 q^{83} -6.02848e8 q^{85} +4.78889e8 q^{86} +4.93891e8 q^{88} -4.26398e8 q^{89} +1.28960e8 q^{91} -1.50076e9 q^{92} +1.76658e9 q^{94} +3.62098e8 q^{95} +1.61351e9 q^{97} +5.80775e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 15 q^{2} + 1793 q^{4} - 453 q^{5} + 343 q^{7} - 7239 q^{8} + 510 q^{10} - 99150 q^{11} - 32435 q^{13} - 394824 q^{14} + 328193 q^{16} - 415539 q^{17} - 85277 q^{19} - 1855164 q^{20} - 529359 q^{22}+ \cdots + 2413650159 q^{98}+O(q^{100}) $ 8 * q - 15 * q^2 + 1793 * q^4 - 453 * q^5 + 343 * q^7 - 7239 * q^8 + 510 * q^10 - 99150 * q^11 - 32435 * q^13 - 394824 * q^14 + 328193 * q^16 - 415539 * q^17 - 85277 * q^19 - 1855164 * q^20 - 529359 * q^22 - 1064559 * q^23 + 2293229 * q^25 + 1218156 * q^26 + 612862 * q^28 + 1309053 * q^29 + 2359819 * q^31 - 5760063 * q^32 - 981801 * q^34 - 15533277 * q^35 + 8195758 * q^37 - 39490203 * q^38 + 16760496 * q^40 - 54747318 * q^41 - 15249608 * q^43 - 166254963 * q^44 + 1195260 * q^46 - 156295545 * q^47 - 15239583 * q^49 - 315590163 * q^50 + 19773358 * q^52 - 262758114 * q^53 - 3789885 * q^55 - 470339790 * q^56 - 55408560 * q^58 - 307774074 * q^59 - 69192125 * q^61 - 457218462 * q^62 - 201794239 * q^64 - 482470359 * q^65 - 14328044 * q^67 - 915409575 * q^68 + 229271934 * q^70 - 619800696 * q^71 + 299306599 * q^73 - 1022736000 * q^74 - 119954093 * q^76 - 717995541 * q^77 - 30257531 * q^79 - 1463913264 * q^80 - 101188011 * q^82 - 1176168291 * q^83 - 4818366 * q^85 - 1426944009 * q^86 - 911312427 * q^88 - 1658520648 * q^89 - 369615061 * q^91 + 76813998 * q^92 + 1954316784 * q^94 + 391400652 * q^95 + 267311278 * q^97 + 2413650159 * q^98

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 $

$2$	−34.8660	−1.54087	−0.770437	−	0.637516i	$-0.779962\pi$
$2$	−34.8660	−1.54087	−0.770437	+	0.637516i	$0.779962\pi$
$3$	0	0
$3$	0	0
$4$	703.639	1.37429
$5$	2006.52	1.43575	0.717874	−	0.696173i	$-0.245115\pi$
$5$	2006.52	1.43575	0.717874	+	0.696173i	$0.245115\pi$
$6$	0	0
$7$	4867.88	0.766300	0.383150	−	0.923686i	$-0.374839\pi$
$7$	4867.88	0.766300	0.383150	+	0.923686i	$0.374839\pi$
$8$	−6681.68	−0.576741
$9$	0	0
$10$	−69959.4	−2.21231
$11$	−73917.1	−1.52222	−0.761111	−	0.648621i	$-0.775346\pi$
$11$	−73917.1	−1.52222	−0.761111	+	0.648621i	$0.775346\pi$
$12$	0	0
$13$	26492.1	0.257259	0.128630	−	0.991693i	$-0.458942\pi$
$13$	26492.1	0.257259	0.128630	+	0.991693i	$0.458942\pi$
$14$	−169724.	−1.18077
$15$	0	0
$16$	−127299.	−0.485609
$17$	−300444.	−0.872457	−0.436228	−	0.899836i	$-0.643686\pi$
$17$	−300444.	−0.872457	−0.436228	+	0.899836i	$0.643686\pi$
$18$	0	0
$19$	180460.	0.317681	0.158840	−	0.987304i	$-0.449224\pi$
$19$	180460.	0.317681	0.158840	+	0.987304i	$0.449224\pi$
$20$	1.41187e6	1.97314
$21$	0	0
$22$	2.57720e6	2.34555
$23$	−2.13285e6	−1.58922	−0.794611	−	0.607119i	$-0.792325\pi$
$23$	−2.13285e6	−1.58922	−0.794611	+	0.607119i	$0.792325\pi$
$24$	0	0
$25$	2.07300e6	1.06138
$26$	−923673.	−0.396404
$27$	0	0
$28$	3.42523e6	1.05312
$29$	−1.89635e6	−0.497884	−0.248942	−	0.968518i	$-0.580083\pi$
$29$	−1.89635e6	−0.497884	−0.248942	+	0.968518i	$0.580083\pi$
$30$	0	0
$31$	5.07398e6	0.986782	0.493391	−	0.869808i	$-0.335757\pi$
$31$	5.07398e6	0.986782	0.493391	+	0.869808i	$0.335757\pi$
$32$	7.85945e6	1.32500
$33$	0	0
$34$	1.04753e7	1.34435
$35$	9.76750e6	1.10021
$36$	0	0
$37$	−2.17155e6	−0.190486	−0.0952430	−	0.995454i	$-0.530363\pi$
$37$	−2.17155e6	−0.190486	−0.0952430	+	0.995454i	$0.530363\pi$
$38$	−6.29194e6	−0.489506
$39$	0	0
$40$	−1.34069e7	−0.828056
$41$	−7.63351e6	−0.421888	−0.210944	−	0.977498i	$-0.567654\pi$
$41$	−7.63351e6	−0.421888	−0.210944	+	0.977498i	$0.567654\pi$
$42$	0	0
$43$	−1.37351e7	−0.612667	−0.306333	−	0.951924i	$-0.599102\pi$
$43$	−1.37351e7	−0.612667	−0.306333	+	0.951924i	$0.599102\pi$
$44$	−5.20110e7	−2.09198
$45$	0	0
$46$	7.43639e7	2.44879
$47$	−5.06676e7	−1.51457	−0.757286	−	0.653083i	$-0.773475\pi$
$47$	−5.06676e7	−1.51457	−0.757286	+	0.653083i	$0.773475\pi$
$48$	0	0
$49$	−1.66573e7	−0.412784
$50$	−7.22772e7	−1.63545
$51$	0	0
$52$	1.86409e7	0.353550
$53$	2.60958e7	0.454286	0.227143	−	0.973861i	$-0.427062\pi$
$53$	2.60958e7	0.454286	0.227143	+	0.973861i	$0.427062\pi$
$54$	0	0
$55$	−1.48316e8	−2.18553
$56$	−3.25256e7	−0.441957
$57$	0	0
$58$	6.61182e7	0.767176
$59$	4.22136e7	0.453543	0.226771	−	0.973948i	$-0.427183\pi$
$59$	4.22136e7	0.453543	0.226771	+	0.973948i	$0.427183\pi$
$60$	0	0
$61$	−4.87874e7	−0.451153	−0.225576	−	0.974225i	$-0.572427\pi$
$61$	−4.87874e7	−0.451153	−0.225576	+	0.974225i	$0.572427\pi$
$62$	−1.76910e8	−1.52051
$63$	0	0
$64$	−2.08850e8	−1.55606
$65$	5.31569e7	0.369360
$66$	0	0
$67$	−1.71276e8	−1.03839	−0.519194	−	0.854656i	$-0.673768\pi$
$67$	−1.71276e8	−1.03839	−0.519194	+	0.854656i	$0.673768\pi$
$68$	−2.11404e8	−1.19901
$69$	0	0
$70$	−3.40554e8	−1.69529
$71$	8.48941e7	0.396474	0.198237	−	0.980154i	$-0.436478\pi$
$71$	8.48941e7	0.396474	0.198237	+	0.980154i	$0.436478\pi$
$72$	0	0
$73$	1.62304e8	0.668923	0.334462	−	0.942409i	$-0.391445\pi$
$73$	1.62304e8	0.668923	0.334462	+	0.942409i	$0.391445\pi$
$74$	7.57135e7	0.293515
$75$	0	0
$76$	1.26979e8	0.436587
$77$	−3.59820e8	−1.16648
$78$	0	0
$79$	1.90082e8	0.549059	0.274530	−	0.961579i	$-0.411478\pi$
$79$	1.90082e8	0.549059	0.274530	+	0.961579i	$0.411478\pi$
$80$	−2.55429e8	−0.697212
$81$	0	0
$82$	2.66150e8	0.650076
$83$	−4.43914e8	−1.02671	−0.513354	−	0.858177i	$-0.671597\pi$
$83$	−4.43914e8	−1.02671	−0.513354	+	0.858177i	$0.671597\pi$
$84$	0	0
$85$	−6.02848e8	−1.25263
$86$	4.78889e8	0.944043
$87$	0	0
$88$	4.93891e8	0.877928
$89$	−4.26398e8	−0.720377	−0.360189	−	0.932879i	$-0.617288\pi$
$89$	−4.26398e8	−0.720377	−0.360189	+	0.932879i	$0.617288\pi$
$90$	0	0
$91$	1.28960e8	0.197138
$92$	−1.50076e9	−2.18406
$93$	0	0
$94$	1.76658e9	2.33377
$95$	3.62098e8	0.456110
$96$	0	0
$97$	1.61351e9	1.85055	0.925273	−	0.379301i	$-0.123836\pi$
$97$	1.61351e9	1.85055	0.925273	+	0.379301i	$0.123836\pi$
$98$	5.80775e8	0.636049
$99$	0	0
$100$	1.45864e9	1.45864
$101$	6.30985e8	0.603355	0.301677	−	0.953410i	$-0.402453\pi$
$101$	6.30985e8	0.603355	0.301677	+	0.953410i	$0.402453\pi$
$102$	0	0
$103$	−9.06864e8	−0.793916	−0.396958	−	0.917837i	$-0.629934\pi$
$103$	−9.06864e8	−0.793916	−0.396958	+	0.917837i	$0.629934\pi$
$104$	−1.77012e8	−0.148372
$105$	0	0
$106$	−9.09856e8	−0.699997
$107$	1.81912e9	1.34163	0.670817	−	0.741623i	$-0.265944\pi$
$107$	1.81912e9	1.34163	0.670817	+	0.741623i	$0.265944\pi$
$108$	0	0
$109$	−8.90034e8	−0.603931	−0.301966	−	0.953319i	$-0.597643\pi$
$109$	−8.90034e8	−0.603931	−0.301966	+	0.953319i	$0.597643\pi$
$110$	5.17120e9	3.36763
$111$	0	0
$112$	−6.19679e8	−0.372122
$113$	6.76258e7	0.0390175	0.0195088	−	0.999810i	$-0.493790\pi$
$113$	6.76258e7	0.0390175	0.0195088	+	0.999810i	$0.493790\pi$
$114$	0	0
$115$	−4.27960e9	−2.28172
$116$	−1.33435e9	−0.684239
$117$	0	0
$118$	−1.47182e9	−0.698853
$119$	−1.46253e9	−0.668564
$120$	0	0
$121$	3.10580e9	1.31716
$122$	1.70102e9	0.695170
$123$	0	0
$124$	3.57025e9	1.35613
$125$	2.40528e8	0.0881191
$126$	0	0
$127$	−3.43570e9	−1.17192	−0.585960	−	0.810340i	$-0.699283\pi$
$127$	−3.43570e9	−1.17192	−0.585960	+	0.810340i	$0.699283\pi$
$128$	3.25774e9	1.07268
$129$	0	0
$130$	−1.85337e9	−0.569137
$131$	−1.42075e9	−0.421500	−0.210750	−	0.977540i	$-0.567591\pi$
$131$	−1.42075e9	−0.421500	−0.210750	+	0.977540i	$0.567591\pi$
$132$	0	0
$133$	8.78460e8	0.243439
$134$	5.97171e9	1.60003
$135$	0	0
$136$	2.00747e9	0.503182
$137$	−4.08435e9	−0.990558	−0.495279	−	0.868734i	$-0.664934\pi$
$137$	−4.08435e9	−0.990558	−0.495279	+	0.868734i	$0.664934\pi$
$138$	0	0
$139$	−4.29846e9	−0.976667	−0.488333	−	0.872657i	$-0.662395\pi$
$139$	−4.29846e9	−0.976667	−0.488333	+	0.872657i	$0.662395\pi$
$140$	6.87279e9	1.51202
$141$	0	0
$142$	−2.95992e9	−0.610917
$143$	−1.95822e9	−0.391606
$144$	0	0
$145$	−3.80507e9	−0.714836
$146$	−5.65889e9	−1.03073
$147$	0	0
$148$	−1.52799e9	−0.261784
$149$	−1.90031e8	−0.0315855	−0.0157927	−	0.999875i	$-0.505027\pi$
$149$	−1.90031e8	−0.0315855	−0.0157927	+	0.999875i	$0.505027\pi$
$150$	0	0
$151$	−9.42107e9	−1.47470	−0.737351	−	0.675510i	$-0.763923\pi$
$151$	−9.42107e9	−1.47470	−0.737351	+	0.675510i	$0.763923\pi$
$152$	−1.20578e9	−0.183220
$153$	0	0
$154$	1.25455e10	1.79740
$155$	1.01810e10	1.41677
$156$	0	0
$157$	−1.18095e10	−1.55125	−0.775625	−	0.631194i	$-0.782565\pi$
$157$	−1.18095e10	−1.55125	−0.775625	+	0.631194i	$0.782565\pi$
$158$	−6.62740e9	−0.846031
$159$	0	0
$160$	1.57701e10	1.90237
$161$	−1.03825e10	−1.21782
$162$	0	0
$163$	−1.93114e9	−0.214274	−0.107137	−	0.994244i	$-0.534168\pi$
$163$	−1.93114e9	−0.214274	−0.107137	+	0.994244i	$0.534168\pi$
$164$	−5.37123e9	−0.579798
$165$	0	0
$166$	1.54775e10	1.58203
$167$	8.70120e8	0.0865675	0.0432838	−	0.999063i	$-0.486218\pi$
$167$	8.70120e8	0.0865675	0.0432838	+	0.999063i	$0.486218\pi$
$168$	0	0
$169$	−9.90267e9	−0.933818
$170$	2.10189e10	1.93014
$171$	0	0
$172$	−9.66456e9	−0.841985
$173$	1.60328e9	0.136083	0.0680413	−	0.997683i	$-0.478325\pi$
$173$	1.60328e9	0.136083	0.0680413	+	0.997683i	$0.478325\pi$
$174$	0	0
$175$	1.00911e10	0.813332
$176$	9.40961e9	0.739205
$177$	0	0
$178$	1.48668e10	1.11001
$179$	1.20800e9	0.0879486	0.0439743	−	0.999033i	$-0.485998\pi$
$179$	1.20800e9	0.0879486	0.0439743	+	0.999033i	$0.485998\pi$
$180$	0	0
$181$	2.02866e10	1.40494	0.702468	−	0.711715i	$-0.252081\pi$
$181$	2.02866e10	1.40494	0.702468	+	0.711715i	$0.252081\pi$
$182$	−4.49633e9	−0.303765
$183$	0	0
$184$	1.42510e10	0.916570
$185$	−4.35727e9	−0.273490
$186$	0	0
$187$	2.22080e10	1.32807
$188$	−3.56517e10	−2.08147
$189$	0	0
$190$	−1.26249e10	−0.702808
$191$	−3.50273e9	−0.190439	−0.0952197	−	0.995456i	$-0.530355\pi$
$191$	−3.50273e9	−0.190439	−0.0952197	+	0.995456i	$0.530355\pi$
$192$	0	0
$193$	2.43071e10	1.26103	0.630514	−	0.776178i	$-0.282844\pi$
$193$	2.43071e10	1.26103	0.630514	+	0.776178i	$0.282844\pi$
$194$	−5.62568e10	−2.85146
$195$	0	0
$196$	−1.17207e10	−0.567287
$197$	−3.50660e10	−1.65878	−0.829390	−	0.558670i	$-0.811312\pi$
$197$	−3.50660e10	−1.65878	−0.829390	+	0.558670i	$0.811312\pi$
$198$	0	0
$199$	−3.04182e9	−0.137498	−0.0687488	−	0.997634i	$-0.521901\pi$
$199$	−3.04182e9	−0.137498	−0.0687488	+	0.997634i	$0.521901\pi$
$200$	−1.38511e10	−0.612139
$201$	0	0
$202$	−2.19999e10	−0.929694
$203$	−9.23121e9	−0.381528
$204$	0	0
$205$	−1.53168e10	−0.605725
$206$	3.16187e10	1.22332
$207$	0	0
$208$	−3.37243e9	−0.124927
$209$	−1.33391e10	−0.483581
$210$	0	0
$211$	−2.02783e10	−0.704303	−0.352151	−	0.935943i	$-0.614550\pi$
$211$	−2.02783e10	−0.704303	−0.352151	+	0.935943i	$0.614550\pi$
$212$	1.83620e10	0.624322
$213$	0	0
$214$	−6.34254e10	−2.06729
$215$	−2.75598e10	−0.879636
$216$	0	0
$217$	2.46995e10	0.756171
$218$	3.10320e10	0.930582
$219$	0	0
$220$	−1.04361e11	−3.00356
$221$	−7.95940e9	−0.224448
$222$	0	0
$223$	3.13424e10	0.848712	0.424356	−	0.905495i	$-0.360500\pi$
$223$	3.13424e10	0.848712	0.424356	+	0.905495i	$0.360500\pi$
$224$	3.82589e10	1.01535
$225$	0	0
$226$	−2.35784e9	−0.0601211
$227$	−7.23264e10	−1.80792	−0.903962	−	0.427612i	$-0.859355\pi$
$227$	−7.23264e10	−1.80792	−0.903962	+	0.427612i	$0.859355\pi$
$228$	0	0
$229$	2.34943e10	0.564550	0.282275	−	0.959334i	$-0.408911\pi$
$229$	2.34943e10	0.564550	0.282275	+	0.959334i	$0.408911\pi$
$230$	1.49213e11	3.51585
$231$	0	0
$232$	1.26708e10	0.287150
$233$	8.06827e10	1.79341	0.896703	−	0.442633i	$-0.145955\pi$
$233$	8.06827e10	1.79341	0.896703	+	0.442633i	$0.145955\pi$
$234$	0	0
$235$	−1.01666e11	−2.17455
$236$	2.97031e10	0.623301
$237$	0	0
$238$	5.09925e10	1.03017
$239$	−7.38972e10	−1.46500	−0.732500	−	0.680767i	$-0.761647\pi$
$239$	−7.38972e10	−1.46500	−0.732500	+	0.680767i	$0.761647\pi$
$240$	0	0
$241$	2.76671e10	0.528307	0.264154	−	0.964481i	$-0.414907\pi$
$241$	2.76671e10	0.528307	0.264154	+	0.964481i	$0.414907\pi$
$242$	−1.08287e11	−2.02958
$243$	0	0
$244$	−3.43287e10	−0.620017
$245$	−3.34233e10	−0.592655
$246$	0	0
$247$	4.78077e9	0.0817263
$248$	−3.39027e10	−0.569118
$249$	0	0
$250$	−8.38625e9	−0.135781
$251$	6.64513e10	1.05675	0.528375	−	0.849011i	$-0.322802\pi$
$251$	6.64513e10	1.05675	0.528375	+	0.849011i	$0.322802\pi$
$252$	0	0
$253$	1.57654e11	2.41915
$254$	1.19789e11	1.80578
$255$	0	0
$256$	−6.65304e9	−0.0968145
$257$	6.64345e10	0.949936	0.474968	−	0.880003i	$-0.342460\pi$
$257$	6.64345e10	0.949936	0.474968	+	0.880003i	$0.342460\pi$
$258$	0	0
$259$	−1.05709e10	−0.145969
$260$	3.74033e10	0.507609
$261$	0	0
$262$	4.95359e10	0.649478
$263$	1.35863e11	1.75105	0.875526	−	0.483172i	$-0.160515\pi$
$263$	1.35863e11	1.75105	0.875526	+	0.483172i	$0.160515\pi$
$264$	0	0
$265$	5.23617e10	0.652240
$266$	−3.06284e10	−0.375109
$267$	0	0
$268$	−1.20516e11	−1.42705
$269$	−9.96606e10	−1.16048	−0.580241	−	0.814445i	$-0.697042\pi$
$269$	−9.96606e10	−1.16048	−0.580241	+	0.814445i	$0.697042\pi$
$270$	0	0
$271$	−4.00569e10	−0.451145	−0.225572	−	0.974226i	$-0.572425\pi$
$271$	−4.00569e10	−0.451145	−0.225572	+	0.974226i	$0.572425\pi$
$272$	3.82464e10	0.423673
$273$	0	0
$274$	1.42405e11	1.52633
$275$	−1.53230e11	−1.61565
$276$	0	0
$277$	1.65483e11	1.68886	0.844432	−	0.535662i	$-0.179938\pi$
$277$	1.65483e11	1.68886	0.844432	+	0.535662i	$0.179938\pi$
$278$	1.49870e11	1.50492
$279$	0	0
$280$	−6.52634e10	−0.634539
$281$	−1.83536e11	−1.75608	−0.878038	−	0.478591i	$-0.841148\pi$
$281$	−1.83536e11	−1.75608	−0.878038	+	0.478591i	$0.841148\pi$
$282$	0	0
$283$	−7.83487e10	−0.726094	−0.363047	−	0.931771i	$-0.618264\pi$
$283$	−7.83487e10	−0.726094	−0.363047	+	0.931771i	$0.618264\pi$
$284$	5.97348e10	0.544872
$285$	0	0
$286$	6.82753e10	0.603415
$287$	−3.71590e10	−0.323293
$288$	0	0
$289$	−2.83210e10	−0.238819
$290$	1.32668e11	1.10147
$291$	0	0
$292$	1.14203e11	0.919298
$293$	1.84598e11	1.46326	0.731632	−	0.681700i	$-0.238759\pi$
$293$	1.84598e11	1.46326	0.731632	+	0.681700i	$0.238759\pi$
$294$	0	0
$295$	8.47024e10	0.651174
$296$	1.45096e10	0.109861
$297$	0	0
$298$	6.62564e9	0.0486693
$299$	−5.65036e10	−0.408842
$300$	0	0
$301$	−6.68609e10	−0.469487
$302$	3.28475e11	2.27233
$303$	0	0
$304$	−2.29725e10	−0.154269
$305$	−9.78930e10	−0.647742
$306$	0	0
$307$	1.19307e11	0.766554	0.383277	−	0.923633i	$-0.374795\pi$
$307$	1.19307e11	0.766554	0.383277	+	0.923633i	$0.374795\pi$
$308$	−2.53183e11	−1.60309
$309$	0	0
$310$	−3.54973e11	−2.18307
$311$	−5.79966e9	−0.0351545	−0.0175773	−	0.999846i	$-0.505595\pi$
$311$	−5.79966e9	−0.0351545	−0.0175773	+	0.999846i	$0.505595\pi$
$312$	0	0
$313$	−3.08296e11	−1.81559	−0.907796	−	0.419412i	$-0.862236\pi$
$313$	−3.08296e11	−1.81559	−0.907796	+	0.419412i	$0.862236\pi$
$314$	4.11749e11	2.39028
$315$	0	0
$316$	1.33749e11	0.754569
$317$	2.29270e11	1.27521	0.637603	−	0.770365i	$-0.279926\pi$
$317$	2.29270e11	1.27521	0.637603	+	0.770365i	$0.279926\pi$
$318$	0	0
$319$	1.40173e11	0.757890
$320$	−4.19062e11	−2.23411
$321$	0	0
$322$	3.61995e11	1.87651
$323$	−5.42183e10	−0.277163
$324$	0	0
$325$	5.49180e10	0.273049
$326$	6.73312e10	0.330169
$327$	0	0
$328$	5.10047e10	0.243320
$329$	−2.46644e11	−1.16062
$330$	0	0
$331$	−1.45906e11	−0.668110	−0.334055	−	0.942554i	$-0.608417\pi$
$331$	−1.45906e11	−0.668110	−0.334055	+	0.942554i	$0.608417\pi$
$332$	−3.12355e11	−1.41100
$333$	0	0
$334$	−3.03376e10	−0.133390
$335$	−3.43669e11	−1.49087
$336$	0	0
$337$	−2.82273e11	−1.19216	−0.596081	−	0.802924i	$-0.703276\pi$
$337$	−2.82273e11	−1.19216	−0.596081	+	0.802924i	$0.703276\pi$
$338$	3.45267e11	1.43890
$339$	0	0
$340$	−4.24187e11	−1.72148
$341$	−3.75054e11	−1.50210
$342$	0	0
$343$	−2.77523e11	−1.08262
$344$	9.17737e10	0.353350
$345$	0	0
$346$	−5.59001e10	−0.209686
$347$	−3.18802e11	−1.18042	−0.590212	−	0.807248i	$-0.700956\pi$
$347$	−3.18802e11	−1.18042	−0.590212	+	0.807248i	$0.700956\pi$
$348$	0	0
$349$	−1.39860e11	−0.504638	−0.252319	−	0.967644i	$-0.581193\pi$
$349$	−1.39860e11	−0.504638	−0.252319	+	0.967644i	$0.581193\pi$
$350$	−3.51837e11	−1.25324
$351$	0	0
$352$	−5.80948e11	−2.01695
$353$	1.61673e11	0.554181	0.277090	−	0.960844i	$-0.410630\pi$
$353$	1.61673e11	0.554181	0.277090	+	0.960844i	$0.410630\pi$
$354$	0	0
$355$	1.70342e11	0.569237
$356$	−3.00030e11	−0.990011
$357$	0	0
$358$	−4.21182e10	−0.135518
$359$	3.30259e11	1.04937	0.524687	−	0.851295i	$-0.324182\pi$
$359$	3.30259e11	1.04937	0.524687	+	0.851295i	$0.324182\pi$
$360$	0	0
$361$	−2.90122e11	−0.899079
$362$	−7.07314e11	−2.16483
$363$	0	0
$364$	9.07415e10	0.270925
$365$	3.25666e11	0.960406
$366$	0	0
$367$	4.26074e11	1.22599	0.612996	−	0.790086i	$-0.289964\pi$
$367$	4.26074e11	1.22599	0.612996	+	0.790086i	$0.289964\pi$
$368$	2.71510e11	0.771740
$369$	0	0
$370$	1.51921e11	0.421414
$371$	1.27031e11	0.348119
$372$	0	0
$373$	2.01040e11	0.537766	0.268883	−	0.963173i	$-0.413345\pi$
$373$	2.01040e11	0.537766	0.268883	+	0.963173i	$0.413345\pi$
$374$	−7.74304e11	−2.04639
$375$	0	0
$376$	3.38545e11	0.873517
$377$	−5.02383e10	−0.128085
$378$	0	0
$379$	7.50298e10	0.186792	0.0933959	−	0.995629i	$-0.470228\pi$
$379$	7.50298e10	0.186792	0.0933959	+	0.995629i	$0.470228\pi$
$380$	2.54786e11	0.626829
$381$	0	0
$382$	1.22126e11	0.293443
$383$	1.64739e11	0.391204	0.195602	−	0.980683i	$-0.437334\pi$
$383$	1.64739e11	0.391204	0.195602	+	0.980683i	$0.437334\pi$
$384$	0	0
$385$	−7.21986e11	−1.67477
$386$	−8.47490e11	−1.94308
$387$	0	0
$388$	1.13533e12	2.54320
$389$	1.03184e11	0.228474	0.114237	−	0.993454i	$-0.463558\pi$
$389$	1.03184e11	0.228474	0.114237	+	0.993454i	$0.463558\pi$
$390$	0	0
$391$	6.40802e11	1.38653
$392$	1.11299e11	0.238070
$393$	0	0
$394$	1.22261e12	2.55597
$395$	3.81403e11	0.788311
$396$	0	0
$397$	7.76925e11	1.56972	0.784860	−	0.619673i	$-0.212735\pi$
$397$	7.76925e11	1.56972	0.784860	+	0.619673i	$0.212735\pi$
$398$	1.06056e11	0.211867
$399$	0	0
$400$	−2.63891e11	−0.515413
$401$	5.66184e11	1.09347	0.546736	−	0.837305i	$-0.315870\pi$
$401$	5.66184e11	1.09347	0.546736	+	0.837305i	$0.315870\pi$
$402$	0	0
$403$	1.34420e11	0.253859
$404$	4.43986e11	0.829188
$405$	0	0
$406$	3.21856e11	0.587887
$407$	1.60515e11	0.289962
$408$	0	0
$409$	−7.56762e11	−1.33722	−0.668612	−	0.743611i	$-0.733111\pi$
$409$	−7.56762e11	−1.33722	−0.668612	+	0.743611i	$0.733111\pi$
$410$	5.34035e11	0.933346
$411$	0	0
$412$	−6.38104e11	−1.09107
$413$	2.05491e11	0.347550
$414$	0	0
$415$	−8.90722e11	−1.47410
$416$	2.08213e11	0.340869
$417$	0	0
$418$	4.65082e11	0.745137
$419$	−4.02426e11	−0.637856	−0.318928	−	0.947779i	$-0.603323\pi$
$419$	−4.02426e11	−0.637856	−0.318928	+	0.947779i	$0.603323\pi$
$420$	0	0
$421$	−4.00580e11	−0.621470	−0.310735	−	0.950497i	$-0.600575\pi$
$421$	−4.00580e11	−0.621470	−0.310735	+	0.950497i	$0.600575\pi$
$422$	7.07022e11	1.08524
$423$	0	0
$424$	−1.74364e11	−0.262005
$425$	−6.22821e11	−0.926004
$426$	0	0
$427$	−2.37491e11	−0.345718
$428$	1.28000e12	1.84380
$429$	0	0
$430$	9.60900e11	1.35541
$431$	−5.54723e11	−0.774335	−0.387167	−	0.922009i	$-0.626546\pi$
$431$	−5.54723e11	−0.774335	−0.387167	+	0.922009i	$0.626546\pi$
$432$	0	0
$433$	−4.39147e11	−0.600364	−0.300182	−	0.953882i	$-0.597047\pi$
$433$	−4.39147e11	−0.600364	−0.300182	+	0.953882i	$0.597047\pi$
$434$	−8.61175e11	−1.16517
$435$	0	0
$436$	−6.26263e11	−0.829980
$437$	−3.84895e11	−0.504865
$438$	0	0
$439$	5.91760e11	0.760424	0.380212	−	0.924899i	$-0.375851\pi$
$439$	5.91760e11	0.760424	0.380212	+	0.924899i	$0.375851\pi$
$440$	9.91002e11	1.26048
$441$	0	0
$442$	2.77512e11	0.345846
$443$	5.70684e11	0.704010	0.352005	−	0.935998i	$-0.385500\pi$
$443$	5.70684e11	0.704010	0.352005	+	0.935998i	$0.385500\pi$
$444$	0	0
$445$	−8.55576e11	−1.03428
$446$	−1.09278e12	−1.30776
$447$	0	0
$448$	−1.01666e12	−1.19241
$449$	−1.06817e12	−1.24031	−0.620156	−	0.784479i	$-0.712931\pi$
$449$	−1.06817e12	−1.24031	−0.620156	+	0.784479i	$0.712931\pi$
$450$	0	0
$451$	5.64247e11	0.642207
$452$	4.75841e10	0.0536215
$453$	0	0
$454$	2.52173e12	2.78579
$455$	2.58761e11	0.283040
$456$	0	0
$457$	−2.83903e11	−0.304471	−0.152236	−	0.988344i	$-0.548647\pi$
$457$	−2.83903e11	−0.304471	−0.152236	+	0.988344i	$0.548647\pi$
$458$	−8.19152e11	−0.869901
$459$	0	0
$460$	−3.01130e12	−3.13576
$461$	−2.67602e11	−0.275953	−0.137977	−	0.990435i	$-0.544060\pi$
$461$	−2.67602e11	−0.275953	−0.137977	+	0.990435i	$0.544060\pi$
$462$	0	0
$463$	−1.12929e12	−1.14206	−0.571032	−	0.820928i	$-0.693457\pi$
$463$	−1.12929e12	−1.14206	−0.571032	+	0.820928i	$0.693457\pi$
$464$	2.41404e11	0.241777
$465$	0	0
$466$	−2.81308e12	−2.76341
$467$	1.35767e12	1.32089	0.660446	−	0.750873i	$-0.270367\pi$
$467$	1.35767e12	1.32089	0.660446	+	0.750873i	$0.270367\pi$
$468$	0	0
$469$	−8.33752e11	−0.795717
$470$	3.54467e12	3.35070
$471$	0	0
$472$	−2.82058e11	−0.261577
$473$	1.01526e12	0.932615
$474$	0	0
$475$	3.74094e11	0.337178
$476$	−1.02909e12	−0.918804
$477$	0	0
$478$	2.57650e12	2.25738
$479$	1.73815e12	1.50861	0.754304	−	0.656525i	$-0.227974\pi$
$479$	1.73815e12	1.50861	0.754304	+	0.656525i	$0.227974\pi$
$480$	0	0
$481$	−5.75290e10	−0.0490043
$482$	−9.64640e11	−0.814055
$483$	0	0
$484$	2.18536e12	1.81017
$485$	3.23755e12	2.65692
$486$	0	0
$487$	3.33116e11	0.268358	0.134179	−	0.990957i	$-0.457160\pi$
$487$	3.33116e11	0.268358	0.134179	+	0.990957i	$0.457160\pi$
$488$	3.25982e11	0.260198
$489$	0	0
$490$	1.16534e12	0.913206
$491$	−3.75694e11	−0.291721	−0.145860	−	0.989305i	$-0.546595\pi$
$491$	−3.75694e11	−0.291721	−0.145860	+	0.989305i	$0.546595\pi$
$492$	0	0
$493$	5.69748e11	0.434382
$494$	−1.66687e11	−0.125930
$495$	0	0
$496$	−6.45915e11	−0.479190
$497$	4.13254e11	0.303818
$498$	0	0
$499$	−6.39178e11	−0.461497	−0.230749	−	0.973013i	$-0.574117\pi$
$499$	−6.39178e11	−0.461497	−0.230749	+	0.973013i	$0.574117\pi$
$500$	1.69245e11	0.121102
$501$	0	0
$502$	−2.31689e12	−1.62832
$503$	−6.06590e11	−0.422513	−0.211256	−	0.977431i	$-0.567755\pi$
$503$	−6.06590e11	−0.422513	−0.211256	+	0.977431i	$0.567755\pi$
$504$	0	0
$505$	1.26608e12	0.866266
$506$	−5.49677e12	−3.72761
$507$	0	0
$508$	−2.41749e12	−1.61056
$509$	2.03325e12	1.34264	0.671321	−	0.741167i	$-0.265727\pi$
$509$	2.03325e12	1.34264	0.671321	+	0.741167i	$0.265727\pi$
$510$	0	0
$511$	7.90077e11	0.512596
$512$	−1.43600e12	−0.923504
$513$	0	0
$514$	−2.31630e12	−1.46373
$515$	−1.81964e12	−1.13986
$516$	0	0
$517$	3.74521e12	2.30552
$518$	3.68564e11	0.224921
$519$	0	0
$520$	−3.55178e11	−0.213025
$521$	4.96377e9	0.00295150	0.00147575	−	0.999999i	$-0.499530\pi$
$521$	4.96377e9	0.00295150	0.00147575	+	0.999999i	$0.499530\pi$
$522$	0	0
$523$	5.29813e11	0.309646	0.154823	−	0.987942i	$-0.450519\pi$
$523$	5.29813e11	0.309646	0.154823	+	0.987942i	$0.450519\pi$
$524$	−9.99696e11	−0.579265
$525$	0	0
$526$	−4.73698e12	−2.69815
$527$	−1.52445e12	−0.860925
$528$	0	0
$529$	2.74789e12	1.52563
$530$	−1.82564e12	−1.00502
$531$	0	0
$532$	6.18119e11	0.334557
$533$	−2.02228e11	−0.108535
$534$	0	0
$535$	3.65010e12	1.92625
$536$	1.14441e12	0.598882
$537$	0	0
$538$	3.47477e12	1.78816
$539$	1.23126e12	0.628349
$540$	0	0
$541$	2.56439e11	0.128705	0.0643526	−	0.997927i	$-0.479502\pi$
$541$	2.56439e11	0.128705	0.0643526	+	0.997927i	$0.479502\pi$
$542$	1.39663e12	0.695157
$543$	0	0
$544$	−2.36133e12	−1.15601
$545$	−1.78587e12	−0.867094
$546$	0	0
$547$	−1.27191e12	−0.607456	−0.303728	−	0.952759i	$-0.598231\pi$
$547$	−1.27191e12	−0.607456	−0.303728	+	0.952759i	$0.598231\pi$
$548$	−2.87390e12	−1.36132
$549$	0	0
$550$	5.34252e12	2.48951
$551$	−3.42216e11	−0.158168
$552$	0	0
$553$	9.25297e11	0.420744
$554$	−5.76974e12	−2.60233
$555$	0	0
$556$	−3.02456e12	−1.34223
$557$	2.20000e11	0.0968446	0.0484223	−	0.998827i	$-0.484581\pi$
$557$	2.20000e11	0.0968446	0.0484223	+	0.998827i	$0.484581\pi$
$558$	0	0
$559$	−3.63872e11	−0.157614
$560$	−1.24340e12	−0.534274
$561$	0	0
$562$	6.39918e12	2.70589
$563$	3.60793e12	1.51346	0.756730	−	0.653728i	$-0.226796\pi$
$563$	3.60793e12	1.51346	0.756730	+	0.653728i	$0.226796\pi$
$564$	0	0
$565$	1.35693e11	0.0560193
$566$	2.73171e12	1.11882
$567$	0	0
$568$	−5.67235e11	−0.228663
$569$	1.16911e12	0.467573	0.233786	−	0.972288i	$-0.424888\pi$
$569$	1.16911e12	0.467573	0.233786	+	0.972288i	$0.424888\pi$
$570$	0	0
$571$	−1.99352e12	−0.784797	−0.392398	−	0.919795i	$-0.628355\pi$
$571$	−1.99352e12	−0.784797	−0.392398	+	0.919795i	$0.628355\pi$
$572$	−1.37788e12	−0.538182
$573$	0	0
$574$	1.29559e12	0.498153
$575$	−4.42139e12	−1.68676
$576$	0	0
$577$	4.79261e12	1.80003	0.900017	−	0.435854i	$-0.143554\pi$
$577$	4.79261e12	1.80003	0.900017	+	0.435854i	$0.143554\pi$
$578$	9.87441e11	0.367990
$579$	0	0
$580$	−2.67739e12	−0.982395
$581$	−2.16092e12	−0.786767
$582$	0	0
$583$	−1.92893e12	−0.691524
$584$	−1.08446e12	−0.385796
$585$	0	0
$586$	−6.43620e12	−2.25471
$587$	−7.78336e11	−0.270580	−0.135290	−	0.990806i	$-0.543197\pi$
$587$	−7.78336e11	−0.270580	−0.135290	+	0.990806i	$0.543197\pi$
$588$	0	0
$589$	9.15653e11	0.313482
$590$	−2.95324e12	−1.00338
$591$	0	0
$592$	2.76438e11	0.0925017
$593$	3.16841e11	0.105219	0.0526097	−	0.998615i	$-0.483246\pi$
$593$	3.16841e11	0.105219	0.0526097	+	0.998615i	$0.483246\pi$
$594$	0	0
$595$	−2.93459e12	−0.959890
$596$	−1.33714e11	−0.0434077
$597$	0	0
$598$	1.97006e12	0.629975
$599$	4.31656e12	1.36999	0.684994	−	0.728548i	$-0.259805\pi$
$599$	4.31656e12	1.36999	0.684994	+	0.728548i	$0.259805\pi$
$600$	0	0
$601$	3.25888e11	0.101891	0.0509453	−	0.998701i	$-0.483777\pi$
$601$	3.25888e11	0.101891	0.0509453	+	0.998701i	$0.483777\pi$
$602$	2.33117e12	0.723420
$603$	0	0
$604$	−6.62903e12	−2.02667
$605$	6.23184e12	1.89111
$606$	0	0
$607$	−9.34391e11	−0.279370	−0.139685	−	0.990196i	$-0.544609\pi$
$607$	−9.34391e11	−0.279370	−0.139685	+	0.990196i	$0.544609\pi$
$608$	1.41832e12	0.420928
$609$	0	0
$610$	3.41314e12	0.998089
$611$	−1.34229e12	−0.389638
$612$	0	0
$613$	−5.38098e12	−1.53918	−0.769590	−	0.638538i	$-0.779540\pi$
$613$	−5.38098e12	−1.53918	−0.769590	+	0.638538i	$0.779540\pi$
$614$	−4.15976e12	−1.18116
$615$	0	0
$616$	2.40420e12	0.672756
$617$	3.76946e11	0.104712	0.0523560	−	0.998628i	$-0.483327\pi$
$617$	3.76946e11	0.104712	0.0523560	+	0.998628i	$0.483327\pi$
$618$	0	0
$619$	−3.68366e12	−1.00849	−0.504245	−	0.863561i	$-0.668229\pi$
$619$	−3.68366e12	−1.00849	−0.504245	+	0.863561i	$0.668229\pi$
$620$	7.16378e12	1.94706
$621$	0	0
$622$	2.02211e11	0.0541687
$623$	−2.07565e12	−0.552025
$624$	0	0
$625$	−3.56620e12	−0.934858
$626$	1.07490e13	2.79760
$627$	0	0
$628$	−8.30960e12	−2.13188
$629$	6.52432e11	0.166191
$630$	0	0
$631$	6.53753e12	1.64165	0.820827	−	0.571178i	$-0.193513\pi$
$631$	6.53753e12	1.64165	0.820827	+	0.571178i	$0.193513\pi$
$632$	−1.27007e12	−0.316665
$633$	0	0
$634$	−7.99373e12	−1.96493
$635$	−6.89380e12	−1.68258
$636$	0	0
$637$	−4.41288e11	−0.106193
$638$	−4.88727e12	−1.16781
$639$	0	0
$640$	6.53672e12	1.54010
$641$	6.40714e12	1.49900	0.749502	−	0.662002i	$-0.230293\pi$
$641$	6.40714e12	1.49900	0.749502	+	0.662002i	$0.230293\pi$
$642$	0	0
$643$	−5.48032e12	−1.26432	−0.632159	−	0.774839i	$-0.717831\pi$
$643$	−5.48032e12	−1.26432	−0.632159	+	0.774839i	$0.717831\pi$
$644$	−7.30550e12	−1.67365
$645$	0	0
$646$	1.89038e12	0.427073
$647$	−2.31980e12	−0.520454	−0.260227	−	0.965548i	$-0.583797\pi$
$647$	−2.31980e12	−0.520454	−0.260227	+	0.965548i	$0.583797\pi$
$648$	0	0
$649$	−3.12031e12	−0.690393
$650$	−1.91477e12	−0.420734
$651$	0	0
$652$	−1.35883e12	−0.294476
$653$	5.42695e11	0.116801	0.0584005	−	0.998293i	$-0.481400\pi$
$653$	5.42695e11	0.116801	0.0584005	+	0.998293i	$0.481400\pi$
$654$	0	0
$655$	−2.85077e12	−0.605168
$656$	9.71741e11	0.204872
$657$	0	0
$658$	8.59949e12	1.78837
$659$	−2.93198e12	−0.605587	−0.302793	−	0.953056i	$-0.597919\pi$
$659$	−2.93198e12	−0.605587	−0.302793	+	0.953056i	$0.597919\pi$
$660$	0	0
$661$	−1.79719e12	−0.366173	−0.183087	−	0.983097i	$-0.558609\pi$
$661$	−1.79719e12	−0.366173	−0.183087	+	0.983097i	$0.558609\pi$
$662$	5.08717e12	1.02947
$663$	0	0
$664$	2.96609e12	0.592145
$665$	1.76265e12	0.349517
$666$	0	0
$667$	4.04463e12	0.791248
$668$	6.12250e11	0.118969
$669$	0	0
$670$	1.19824e13	2.29724
$671$	3.60623e12	0.686755
$672$	0	0
$673$	6.97900e12	1.31137	0.655686	−	0.755034i	$-0.272380\pi$
$673$	6.97900e12	1.31137	0.655686	+	0.755034i	$0.272380\pi$
$674$	9.84174e12	1.83697
$675$	0	0
$676$	−6.96790e12	−1.28334
$677$	−2.44120e12	−0.446636	−0.223318	−	0.974746i	$-0.571689\pi$
$677$	−2.44120e12	−0.446636	−0.223318	+	0.974746i	$0.571689\pi$
$678$	0	0
$679$	7.85440e12	1.41807
$680$	4.02804e12	0.722443
$681$	0	0
$682$	1.30766e13	2.31455
$683$	−3.80854e12	−0.669676	−0.334838	−	0.942276i	$-0.608682\pi$
$683$	−3.80854e12	−0.669676	−0.334838	+	0.942276i	$0.608682\pi$
$684$	0	0
$685$	−8.19532e12	−1.42219
$686$	9.67610e12	1.66818
$687$	0	0
$688$	1.74847e12	0.297516
$689$	6.91332e11	0.116869
$690$	0	0
$691$	−9.58272e12	−1.59896	−0.799480	−	0.600693i	$-0.794891\pi$
$691$	−9.58272e12	−1.59896	−0.799480	+	0.600693i	$0.794891\pi$
$692$	1.12813e12	0.187018
$693$	0	0
$694$	1.11153e13	1.81888
$695$	−8.62495e12	−1.40225
$696$	0	0
$697$	2.29344e12	0.368079
$698$	4.87637e12	0.777583
$699$	0	0
$700$	7.10050e12	1.11776
$701$	4.17769e12	0.653439	0.326720	−	0.945121i	$-0.394057\pi$
$701$	4.17769e12	0.653439	0.326720	+	0.945121i	$0.394057\pi$
$702$	0	0
$703$	−3.91880e11	−0.0605137
$704$	1.54376e13	2.36866
$705$	0	0
$706$	−5.63689e12	−0.853923
$707$	3.07156e12	0.462351
$708$	0	0
$709$	2.71135e12	0.402975	0.201487	−	0.979491i	$-0.435423\pi$
$709$	2.71135e12	0.402975	0.201487	+	0.979491i	$0.435423\pi$
$710$	−5.93913e12	−0.877123
$711$	0	0
$712$	2.84906e12	0.415471
$713$	−1.08220e13	−1.56822
$714$	0	0
$715$	−3.92921e12	−0.562248
$716$	8.49997e11	0.120867
$717$	0	0
$718$	−1.15148e13	−1.61695
$719$	4.34037e12	0.605684	0.302842	−	0.953041i	$-0.402064\pi$
$719$	4.34037e12	0.605684	0.302842	+	0.953041i	$0.402064\pi$
$720$	0	0
$721$	−4.41450e12	−0.608378
$722$	1.01154e13	1.38537
$723$	0	0
$724$	1.42745e13	1.93080
$725$	−3.93113e12	−0.528441
$726$	0	0
$727$	8.81788e12	1.17074	0.585368	−	0.810768i	$-0.300950\pi$
$727$	8.81788e12	1.17074	0.585368	+	0.810768i	$0.300950\pi$
$728$	−8.61672e11	−0.113697
$729$	0	0
$730$	−1.13547e13	−1.47987
$731$	4.12664e12	0.534525
$732$	0	0
$733$	4.56782e12	0.584442	0.292221	−	0.956351i	$-0.405606\pi$
$733$	4.56782e12	0.584442	0.292221	+	0.956351i	$0.405606\pi$
$734$	−1.48555e13	−1.88910
$735$	0	0
$736$	−1.67630e13	−2.10573
$737$	1.26602e13	1.58066
$738$	0	0
$739$	−5.19636e12	−0.640913	−0.320456	−	0.947263i	$-0.603836\pi$
$739$	−5.19636e12	−0.640913	−0.320456	+	0.947263i	$0.603836\pi$
$740$	−3.06594e12	−0.375856
$741$	0	0
$742$	−4.42907e12	−0.536408
$743$	−1.10644e13	−1.33192	−0.665962	−	0.745986i	$-0.731979\pi$
$743$	−1.10644e13	−1.33192	−0.665962	+	0.745986i	$0.731979\pi$
$744$	0	0
$745$	−3.81302e11	−0.0453488
$746$	−7.00948e12	−0.828630
$747$	0	0
$748$	1.56264e13	1.82516
$749$	8.85525e12	1.02809
$750$	0	0
$751$	−1.12230e13	−1.28745	−0.643724	−	0.765258i	$-0.722612\pi$
$751$	−1.12230e13	−1.28745	−0.643724	+	0.765258i	$0.722612\pi$
$752$	6.44996e12	0.735490
$753$	0	0
$754$	1.75161e12	0.197363
$755$	−1.89036e13	−2.11730
$756$	0	0
$757$	6.89383e11	0.0763008	0.0381504	−	0.999272i	$-0.487853\pi$
$757$	6.89383e11	0.0763008	0.0381504	+	0.999272i	$0.487853\pi$
$758$	−2.61599e12	−0.287823
$759$	0	0
$760$	−2.41942e12	−0.263057
$761$	−9.56410e12	−1.03374	−0.516872	−	0.856062i	$-0.672904\pi$
$761$	−9.56410e12	−1.03374	−0.516872	+	0.856062i	$0.672904\pi$
$762$	0	0
$763$	−4.33258e12	−0.462793
$764$	−2.46466e12	−0.261720
$765$	0	0
$766$	−5.74381e12	−0.602796
$767$	1.11833e12	0.116678
$768$	0	0
$769$	7.19521e12	0.741951	0.370975	−	0.928643i	$-0.379023\pi$
$769$	7.19521e12	0.741951	0.370975	+	0.928643i	$0.379023\pi$
$770$	2.51728e13	2.58061
$771$	0	0
$772$	1.71034e13	1.73302
$773$	6.24748e12	0.629357	0.314678	−	0.949198i	$-0.398103\pi$
$773$	6.24748e12	0.629357	0.314678	+	0.949198i	$0.398103\pi$
$774$	0	0
$775$	1.05184e13	1.04735
$776$	−1.07810e13	−1.06729
$777$	0	0
$778$	−3.59760e12	−0.352050
$779$	−1.37755e12	−0.134026
$780$	0	0
$781$	−6.27513e12	−0.603522
$782$	−2.23422e13	−2.13647
$783$	0	0
$784$	2.12047e12	0.200452
$785$	−2.36959e13	−2.22721
$786$	0	0
$787$	−6.65966e12	−0.618822	−0.309411	−	0.950928i	$-0.600132\pi$
$787$	−6.65966e12	−0.618822	−0.309411	+	0.950928i	$0.600132\pi$
$788$	−2.46738e13	−2.27965
$789$	0	0
$790$	−1.32980e13	−1.21469
$791$	3.29194e11	0.0298991
$792$	0	0
$793$	−1.29248e12	−0.116063
$794$	−2.70883e13	−2.41874
$795$	0	0
$796$	−2.14035e12	−0.188962
$797$	9.51973e12	0.835723	0.417861	−	0.908511i	$-0.362780\pi$
$797$	9.51973e12	0.835723	0.417861	+	0.908511i	$0.362780\pi$
$798$	0	0
$799$	1.52228e13	1.32140
$800$	1.62926e13	1.40633
$801$	0	0
$802$	−1.97406e13	−1.68490
$803$	−1.19971e13	−1.01825
$804$	0	0
$805$	−2.08326e13	−1.74849
$806$	−4.68670e12	−0.391165
$807$	0	0
$808$	−4.21604e12	−0.347980
$809$	−1.32179e13	−1.08491	−0.542454	−	0.840085i	$-0.682505\pi$
$809$	−1.32179e13	−1.08491	−0.542454	+	0.840085i	$0.682505\pi$
$810$	0	0
$811$	−1.21897e13	−0.989459	−0.494729	−	0.869047i	$-0.664733\pi$
$811$	−1.21897e13	−0.989459	−0.494729	+	0.869047i	$0.664733\pi$
$812$	−6.49544e12	−0.524332
$813$	0	0
$814$	−5.59652e12	−0.446795
$815$	−3.87487e12	−0.307644
$816$	0	0
$817$	−2.47865e12	−0.194632
$818$	2.63853e13	2.06050
$819$	0	0
$820$	−1.07775e13	−0.832444
$821$	2.20788e13	1.69602	0.848010	−	0.529981i	$-0.177801\pi$
$821$	2.20788e13	1.69602	0.848010	+	0.529981i	$0.177801\pi$
$822$	0	0
$823$	9.75977e12	0.741550	0.370775	−	0.928723i	$-0.379092\pi$
$823$	9.75977e12	0.741550	0.370775	+	0.928723i	$0.379092\pi$
$824$	6.05938e12	0.457884
$825$	0	0
$826$	−7.16464e12	−0.535531
$827$	1.29365e12	0.0961706	0.0480853	−	0.998843i	$-0.484688\pi$
$827$	1.29365e12	0.0961706	0.0480853	+	0.998843i	$0.484688\pi$
$828$	0	0
$829$	1.67176e13	1.22936	0.614681	−	0.788776i	$-0.289285\pi$
$829$	1.67176e13	1.22936	0.614681	+	0.788776i	$0.289285\pi$
$830$	3.10559e13	2.27140
$831$	0	0
$832$	−5.53288e12	−0.400310
$833$	5.00460e12	0.360137
$834$	0	0
$835$	1.74591e12	0.124289
$836$	−9.38593e12	−0.664582
$837$	0	0
$838$	1.40310e13	0.982857
$839$	2.39369e12	0.166778	0.0833889	−	0.996517i	$-0.473426\pi$
$839$	2.39369e12	0.166778	0.0833889	+	0.996517i	$0.473426\pi$
$840$	0	0
$841$	−1.09110e13	−0.752112
$842$	1.39666e13	0.957607
$843$	0	0
$844$	−1.42686e13	−0.967920
$845$	−1.98699e13	−1.34073
$846$	0	0
$847$	1.51186e13	1.00934
$848$	−3.32198e12	−0.220605
$849$	0	0
$850$	2.17153e13	1.42686
$851$	4.63160e12	0.302725
$852$	0	0
$853$	−2.04942e12	−0.132544	−0.0662719	−	0.997802i	$-0.521110\pi$
$853$	−2.04942e12	−0.132544	−0.0662719	+	0.997802i	$0.521110\pi$
$854$	8.28038e12	0.532709
$855$	0	0
$856$	−1.21548e13	−0.773775
$857$	1.46083e13	0.925094	0.462547	−	0.886595i	$-0.346936\pi$
$857$	1.46083e13	0.925094	0.462547	+	0.886595i	$0.346936\pi$
$858$	0	0
$859$	2.65526e13	1.66394	0.831970	−	0.554820i	$-0.187213\pi$
$859$	2.65526e13	1.66394	0.831970	+	0.554820i	$0.187213\pi$
$860$	−1.93921e13	−1.20888
$861$	0	0
$862$	1.93410e13	1.19315
$863$	−2.27251e13	−1.39463	−0.697313	−	0.716766i	$-0.745621\pi$
$863$	−2.27251e13	−1.39463	−0.697313	+	0.716766i	$0.745621\pi$
$864$	0	0
$865$	3.21702e12	0.195381
$866$	1.53113e13	0.925085
$867$	0	0
$868$	1.73796e13	1.03920
$869$	−1.40503e13	−0.835790
$870$	0	0
$871$	−4.53746e12	−0.267135
$872$	5.94693e12	0.348312
$873$	0	0
$874$	1.34197e13	0.777934
$875$	1.17086e12	0.0675257
$876$	0	0
$877$	1.36504e13	0.779194	0.389597	−	0.920985i	$-0.372614\pi$
$877$	1.36504e13	0.779194	0.389597	+	0.920985i	$0.372614\pi$
$878$	−2.06323e13	−1.17172
$879$	0	0
$880$	1.88806e13	1.06131
$881$	2.64554e13	1.47953	0.739763	−	0.672867i	$-0.234938\pi$
$881$	2.64554e13	1.47953	0.739763	+	0.672867i	$0.234938\pi$
$882$	0	0
$883$	−2.86010e13	−1.58328	−0.791641	−	0.610987i	$-0.790773\pi$
$883$	−2.86010e13	−1.58328	−0.791641	+	0.610987i	$0.790773\pi$
$884$	−5.60054e12	−0.308457
$885$	0	0
$886$	−1.98975e13	−1.08479
$887$	−2.34532e13	−1.27217	−0.636085	−	0.771619i	$-0.719447\pi$
$887$	−2.34532e13	−1.27217	−0.636085	+	0.771619i	$0.719447\pi$
$888$	0	0
$889$	−1.67246e13	−0.898043
$890$	2.98305e13	1.59370
$891$	0	0
$892$	2.20537e13	1.16638
$893$	−9.14350e12	−0.481151
$894$	0	0
$895$	2.42388e12	0.126272
$896$	1.58583e13	0.821997
$897$	0	0
$898$	3.72427e13	1.91116
$899$	−9.62205e12	−0.491303
$900$	0	0
$901$	−7.84033e12	−0.396345
$902$	−1.96730e13	−0.989560
$903$	0	0
$904$	−4.51854e11	−0.0225030
$905$	4.07056e13	2.01714
$906$	0	0
$907$	8.68701e12	0.426224	0.213112	−	0.977028i	$-0.431640\pi$
$907$	8.68701e12	0.426224	0.213112	+	0.977028i	$0.431640\pi$
$908$	−5.08916e13	−2.48462
$909$	0	0
$910$	−9.02198e12	−0.436130
$911$	−2.67001e13	−1.28434	−0.642169	−	0.766563i	$-0.721965\pi$
$911$	−2.67001e13	−1.28434	−0.642169	+	0.766563i	$0.721965\pi$
$912$	0	0
$913$	3.28128e13	1.56288
$914$	9.89855e12	0.469152
$915$	0	0
$916$	1.65315e13	0.775858
$917$	−6.91605e12	−0.322995
$918$	0	0
$919$	3.64071e13	1.68371	0.841853	−	0.539707i	$-0.181465\pi$
$919$	3.64071e13	1.68371	0.841853	+	0.539707i	$0.181465\pi$
$920$	2.85950e13	1.31596
$921$	0	0
$922$	9.33022e12	0.425209
$923$	2.24902e12	0.101997
$924$	0	0
$925$	−4.50163e12	−0.202177
$926$	3.93738e13	1.75978
$927$	0	0
$928$	−1.49043e13	−0.659698
$929$	7.69628e12	0.339008	0.169504	−	0.985529i	$-0.445783\pi$
$929$	7.69628e12	0.339008	0.169504	+	0.985529i	$0.445783\pi$
$930$	0	0
$931$	−3.00599e12	−0.131134
$932$	5.67715e13	2.46467
$933$	0	0
$934$	−4.73364e13	−2.03533
$935$	4.45608e13	1.90678
$936$	0	0
$937$	2.28334e13	0.967702	0.483851	−	0.875150i	$-0.339238\pi$
$937$	2.28334e13	0.967702	0.483851	+	0.875150i	$0.339238\pi$
$938$	2.90696e13	1.22610
$939$	0	0
$940$	−7.15359e13	−2.98847
$941$	4.41700e12	0.183643	0.0918214	−	0.995775i	$-0.470731\pi$
$941$	4.41700e12	0.183643	0.0918214	+	0.995775i	$0.470731\pi$
$942$	0	0
$943$	1.62811e13	0.670473
$944$	−5.37377e12	−0.220244
$945$	0	0
$946$	−3.53981e13	−1.43704
$947$	1.20960e13	0.488727	0.244363	−	0.969684i	$-0.421421\pi$
$947$	1.20960e13	0.488727	0.244363	+	0.969684i	$0.421421\pi$
$948$	0	0
$949$	4.29977e12	0.172087
$950$	−1.30432e13	−0.519550
$951$	0	0
$952$	9.77215e12	0.385588
$953$	1.37598e13	0.540372	0.270186	−	0.962808i	$-0.412915\pi$
$953$	1.37598e13	0.540372	0.270186	+	0.962808i	$0.412915\pi$
$954$	0	0
$955$	−7.02830e12	−0.273423
$956$	−5.19970e13	−2.01334
$957$	0	0
$958$	−6.06022e13	−2.32458
$959$	−1.98821e13	−0.759065
$960$	0	0
$961$	−6.94330e11	−0.0262610
$962$	2.00581e12	0.0755095
$963$	0	0
$964$	1.94676e13	0.726050
$965$	4.87726e13	1.81052
$966$	0	0
$967$	3.43806e13	1.26443	0.632213	−	0.774794i	$-0.282147\pi$
$967$	3.43806e13	1.26443	0.632213	+	0.774794i	$0.282147\pi$
$968$	−2.07519e13	−0.759661
$969$	0	0
$970$	−1.12880e14	−4.09398
$971$	−2.24449e13	−0.810271	−0.405135	−	0.914257i	$-0.632776\pi$
$971$	−2.24449e13	−0.810271	−0.405135	+	0.914257i	$0.632776\pi$
$972$	0	0
$973$	−2.09244e13	−0.748420
$974$	−1.16144e13	−0.413506
$975$	0	0
$976$	6.21061e12	0.219084
$977$	3.36242e13	1.18066	0.590332	−	0.807160i	$-0.298997\pi$
$977$	3.36242e13	1.18066	0.590332	+	0.807160i	$0.298997\pi$
$978$	0	0
$979$	3.15181e13	1.09657
$980$	−2.35179e13	−0.814482
$981$	0	0
$982$	1.30990e13	0.449505
$983$	−5.23017e11	−0.0178659	−0.00893296	−	0.999960i	$-0.502843\pi$
$983$	−5.23017e11	−0.0178659	−0.00893296	+	0.999960i	$0.502843\pi$
$984$	0	0
$985$	−7.03607e13	−2.38159
$986$	−1.98648e13	−0.669328
$987$	0	0
$988$	3.36394e12	0.112316
$989$	2.92949e13	0.973664
$990$	0	0
$991$	3.32760e12	0.109597	0.0547985	−	0.998497i	$-0.482548\pi$
$991$	3.32760e12	0.109597	0.0547985	+	0.998497i	$0.482548\pi$
$992$	3.98787e13	1.30749
$993$	0	0
$994$	−1.44085e13	−0.468146
$995$	−6.10348e12	−0.197412
$996$	0	0
$997$	−1.15804e13	−0.371188	−0.185594	−	0.982626i	$-0.559421\pi$
$997$	−1.15804e13	−0.371188	−0.185594	+	0.982626i	$0.559421\pi$
$998$	2.22856e13	0.711109
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.10.a.c.1.2		8
3.2	odd	2		81.10.a.d.1.7		8
9.2	odd	6		27.10.c.a.10.2		16
9.4	even	3		9.10.c.a.7.7	yes	16
9.5	odd	6		27.10.c.a.19.2		16
9.7	even	3		9.10.c.a.4.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.10.c.a.4.7	✓	16	9.7	even	3
9.10.c.a.7.7	yes	16	9.4	even	3
27.10.c.a.10.2		16	9.2	odd	6
27.10.c.a.19.2		16	9.5	odd	6
81.10.a.c.1.2		8	1.1	even	1	trivial
81.10.a.d.1.7		8	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-34.8660 q^{2} +703.639 q^{4} +2006.52 q^{5} +4867.88 q^{7} -6681.68 q^{8} -69959.4 q^{10} -73917.1 q^{11} +26492.1 q^{13} -169724. q^{14} -127299. q^{16} -300444. q^{17} +180460. q^{19} +1.41187e6 q^{20} +2.57720e6 q^{22} -2.13285e6 q^{23} +2.07300e6 q^{25} -923673. q^{26} +3.42523e6 q^{28} -1.89635e6 q^{29} +5.07398e6 q^{31} +7.85945e6 q^{32} +1.04753e7 q^{34} +9.76750e6 q^{35} -2.17155e6 q^{37} -6.29194e6 q^{38} -1.34069e7 q^{40} -7.63351e6 q^{41} -1.37351e7 q^{43} -5.20110e7 q^{44} +7.43639e7 q^{46} -5.06676e7 q^{47} -1.66573e7 q^{49} -7.22772e7 q^{50} +1.86409e7 q^{52} +2.60958e7 q^{53} -1.48316e8 q^{55} -3.25256e7 q^{56} +6.61182e7 q^{58} +4.22136e7 q^{59} -4.87874e7 q^{61} -1.76910e8 q^{62} -2.08850e8 q^{64} +5.31569e7 q^{65} -1.71276e8 q^{67} -2.11404e8 q^{68} -3.40554e8 q^{70} +8.48941e7 q^{71} +1.62304e8 q^{73} +7.57135e7 q^{74} +1.26979e8 q^{76} -3.59820e8 q^{77} +1.90082e8 q^{79} -2.55429e8 q^{80} +2.66150e8 q^{82} -4.43914e8 q^{83} -6.02848e8 q^{85} +4.78889e8 q^{86} +4.93891e8 q^{88} -4.26398e8 q^{89} +1.28960e8 q^{91} -1.50076e9 q^{92} +1.76658e9 q^{94} +3.62098e8 q^{95} +1.61351e9 q^{97} +5.80775e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 15 q^{2} + 1793 q^{4} - 453 q^{5} + 343 q^{7} - 7239 q^{8} + 510 q^{10} - 99150 q^{11} - 32435 q^{13} - 394824 q^{14} + 328193 q^{16} - 415539 q^{17} - 85277 q^{19} - 1855164 q^{20} - 529359 q^{22}+ \cdots + 2413650159 q^{98}+O(q^{100}) \) 8 * q - 15 * q^2 + 1793 * q^4 - 453 * q^5 + 343 * q^7 - 7239 * q^8 + 510 * q^10 - 99150 * q^11 - 32435 * q^13 - 394824 * q^14 + 328193 * q^16 - 415539 * q^17 - 85277 * q^19 - 1855164 * q^20 - 529359 * q^22 - 1064559 * q^23 + 2293229 * q^25 + 1218156 * q^26 + 612862 * q^28 + 1309053 * q^29 + 2359819 * q^31 - 5760063 * q^32 - 981801 * q^34 - 15533277 * q^35 + 8195758 * q^37 - 39490203 * q^38 + 16760496 * q^40 - 54747318 * q^41 - 15249608 * q^43 - 166254963 * q^44 + 1195260 * q^46 - 156295545 * q^47 - 15239583 * q^49 - 315590163 * q^50 + 19773358 * q^52 - 262758114 * q^53 - 3789885 * q^55 - 470339790 * q^56 - 55408560 * q^58 - 307774074 * q^59 - 69192125 * q^61 - 457218462 * q^62 - 201794239 * q^64 - 482470359 * q^65 - 14328044 * q^67 - 915409575 * q^68 + 229271934 * q^70 - 619800696 * q^71 + 299306599 * q^73 - 1022736000 * q^74 - 119954093 * q^76 - 717995541 * q^77 - 30257531 * q^79 - 1463913264 * q^80 - 101188011 * q^82 - 1176168291 * q^83 - 4818366 * q^85 - 1426944009 * q^86 - 911312427 * q^88 - 1658520648 * q^89 - 369615061 * q^91 + 76813998 * q^92 + 1954316784 * q^94 + 391400652 * q^95 + 267311278 * q^97 + 2413650159 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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