Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,8,Mod(10,27)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27.10"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 10.4
Root \(0.500000 - 2.70685i\) of defining polynomial
Character \(\chi\) \(=\) 27.10
Dual form 27.8.c.a.19.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.09420 + 5.35931i) q^{2} +(44.8519 - 77.6857i) q^{4} +(-167.952 + 290.901i) q^{5} +(442.025 + 765.610i) q^{7} +1347.24 q^{8} -2078.70 q^{10} +(2106.95 + 3649.35i) q^{11} +(-6257.63 + 10838.5i) q^{13} +(-2735.43 + 4737.90i) q^{14} +(-1572.42 - 2723.51i) q^{16} +742.627 q^{17} +9111.12 q^{19} +(15065.9 + 26094.9i) q^{20} +(-13038.7 + 22583.6i) q^{22} +(22651.2 - 39233.1i) q^{23} +(-17352.9 - 30056.2i) q^{25} -77449.4 q^{26} +79302.6 q^{28} +(-17291.8 - 29950.3i) q^{29} +(138773. - 240361. i) q^{31} +(95953.9 - 166197. i) q^{32} +(2297.84 + 3979.97i) q^{34} -296955. q^{35} -209817. q^{37} +(28191.6 + 48829.3i) q^{38} +(-226271. + 391912. i) q^{40} +(53466.0 - 92605.8i) q^{41} +(-8512.90 - 14744.8i) q^{43} +378003. q^{44} +280350. q^{46} +(675738. + 1.17041e6i) q^{47} +(20999.2 - 36371.6i) q^{49} +(107387. - 186000. i) q^{50} +(561333. + 972257. i) q^{52} -1.83419e6 q^{53} -1.41546e6 q^{55} +(595513. + 1.03146e6i) q^{56} +(107009. - 185345. i) q^{58} +(435574. - 754437. i) q^{59} +(-487289. - 844009. i) q^{61} +1.71756e6 q^{62} +785063. q^{64} +(-2.10196e6 - 3.64070e6i) q^{65} +(143227. - 248076. i) q^{67} +(33308.2 - 57691.5i) q^{68} +(-918838. - 1.59147e6i) q^{70} -967923. q^{71} +4.50531e6 q^{73} +(-649216. - 1.12448e6i) q^{74} +(408651. - 707804. i) q^{76} +(-1.86265e6 + 3.22621e6i) q^{77} +(-1.22765e6 - 2.12634e6i) q^{79} +1.05636e6 q^{80} +661738. q^{82} +(695940. + 1.20540e6i) q^{83} +(-124725. + 216031. i) q^{85} +(52681.2 - 91246.5i) q^{86} +(2.83856e6 + 4.91654e6i) q^{88} +7.88308e6 q^{89} -1.10641e7 q^{91} +(-2.03190e6 - 3.51936e6i) q^{92} +(-4.18174e6 + 7.24298e6i) q^{94} +(-1.53023e6 + 2.65043e6i) q^{95} +(3.43723e6 + 5.95346e6i) q^{97} +259902. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 9 q^{2} - 321 q^{4} + 180 q^{5} - 84 q^{7} - 5922 q^{8} + 252 q^{10} + 8460 q^{11} - 1848 q^{13} + 16272 q^{14} - 12417 q^{16} - 30564 q^{17} + 24432 q^{19} + 40788 q^{20} - 35001 q^{22} + 51588 q^{23}+ \cdots + 95833314 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.09420 + 5.35931i 0.273491 + 0.473701i 0.969753 − 0.244087i \(-0.0784882\pi\)
−0.696262 + 0.717788i \(0.745155\pi\)
\(3\) 0 0
\(4\) 44.8519 − 77.6857i 0.350405 − 0.606920i
\(5\) −167.952 + 290.901i −0.600882 + 1.04076i 0.391806 + 0.920048i \(0.371850\pi\)
−0.992688 + 0.120710i \(0.961483\pi\)
\(6\) 0 0
\(7\) 442.025 + 765.610i 0.487084 + 0.843654i 0.999890 − 0.0148506i \(-0.00472726\pi\)
−0.512806 + 0.858505i \(0.671394\pi\)
\(8\) 1347.24 0.930313
\(9\) 0 0
\(10\) −2078.70 −0.657343
\(11\) 2106.95 + 3649.35i 0.477288 + 0.826687i 0.999661 − 0.0260303i \(-0.00828663\pi\)
−0.522373 + 0.852717i \(0.674953\pi\)
\(12\) 0 0
\(13\) −6257.63 + 10838.5i −0.789966 + 1.36826i 0.136021 + 0.990706i \(0.456568\pi\)
−0.925987 + 0.377555i \(0.876765\pi\)
\(14\) −2735.43 + 4737.90i −0.266426 + 0.461464i
\(15\) 0 0
\(16\) −1572.42 − 2723.51i −0.0959728 − 0.166230i
\(17\) 742.627 0.0366606 0.0183303 − 0.999832i \(-0.494165\pi\)
0.0183303 + 0.999832i \(0.494165\pi\)
\(18\) 0 0
\(19\) 9111.12 0.304743 0.152372 − 0.988323i \(-0.451309\pi\)
0.152372 + 0.988323i \(0.451309\pi\)
\(20\) 15065.9 + 26094.9i 0.421104 + 0.729374i
\(21\) 0 0
\(22\) −13038.7 + 22583.6i −0.261068 + 0.452183i
\(23\) 22651.2 − 39233.1i 0.388190 − 0.672365i −0.604016 − 0.796972i \(-0.706434\pi\)
0.992206 + 0.124607i \(0.0397670\pi\)
\(24\) 0 0
\(25\) −17352.9 − 30056.2i −0.222118 − 0.384719i
\(26\) −77449.4 −0.864195
\(27\) 0 0
\(28\) 79302.6 0.682707
\(29\) −17291.8 − 29950.3i −0.131658 − 0.228039i 0.792658 − 0.609667i \(-0.208697\pi\)
−0.924316 + 0.381628i \(0.875363\pi\)
\(30\) 0 0
\(31\) 138773. − 240361.i 0.836639 − 1.44910i −0.0560492 − 0.998428i \(-0.517850\pi\)
0.892689 − 0.450674i \(-0.148816\pi\)
\(32\) 95953.9 − 166197.i 0.517652 − 0.896600i
\(33\) 0 0
\(34\) 2297.84 + 3979.97i 0.0100263 + 0.0173661i
\(35\) −296955. −1.17072
\(36\) 0 0
\(37\) −209817. −0.680981 −0.340491 − 0.940248i \(-0.610593\pi\)
−0.340491 + 0.940248i \(0.610593\pi\)
\(38\) 28191.6 + 48829.3i 0.0833446 + 0.144357i
\(39\) 0 0
\(40\) −226271. + 391912.i −0.559008 + 0.968231i
\(41\) 53466.0 − 92605.8i 0.121153 − 0.209843i −0.799070 − 0.601239i \(-0.794674\pi\)
0.920223 + 0.391395i \(0.128007\pi\)
\(42\) 0 0
\(43\) −8512.90 − 14744.8i −0.0163282 − 0.0282812i 0.857746 − 0.514074i \(-0.171864\pi\)
−0.874074 + 0.485793i \(0.838531\pi\)
\(44\) 378003. 0.668976
\(45\) 0 0
\(46\) 280350. 0.424666
\(47\) 675738. + 1.17041e6i 0.949371 + 1.64436i 0.746754 + 0.665100i \(0.231611\pi\)
0.202616 + 0.979258i \(0.435056\pi\)
\(48\) 0 0
\(49\) 20999.2 − 36371.6i 0.0254986 − 0.0441648i
\(50\) 107387. − 186000.i 0.121494 − 0.210435i
\(51\) 0 0
\(52\) 561333. + 972257.i 0.553616 + 0.958892i
\(53\) −1.83419e6 −1.69230 −0.846152 − 0.532942i \(-0.821086\pi\)
−0.846152 + 0.532942i \(0.821086\pi\)
\(54\) 0 0
\(55\) −1.41546e6 −1.14717
\(56\) 595513. + 1.03146e6i 0.453141 + 0.784862i
\(57\) 0 0
\(58\) 107009. − 185345.i 0.0720147 − 0.124733i
\(59\) 435574. − 754437.i 0.276109 − 0.478235i −0.694305 − 0.719680i \(-0.744288\pi\)
0.970414 + 0.241446i \(0.0776216\pi\)
\(60\) 0 0
\(61\) −487289. − 844009.i −0.274873 − 0.476094i 0.695230 − 0.718787i \(-0.255302\pi\)
−0.970103 + 0.242693i \(0.921969\pi\)
\(62\) 1.71756e6 0.915254
\(63\) 0 0
\(64\) 785063. 0.374347
\(65\) −2.10196e6 − 3.64070e6i −0.949352 − 1.64433i
\(66\) 0 0
\(67\) 143227. − 248076.i 0.0581785 − 0.100768i −0.835469 − 0.549537i \(-0.814804\pi\)
0.893648 + 0.448769i \(0.148137\pi\)
\(68\) 33308.2 − 57691.5i 0.0128461 − 0.0222500i
\(69\) 0 0
\(70\) −918838. − 1.59147e6i −0.320181 − 0.554570i
\(71\) −967923. −0.320950 −0.160475 − 0.987040i \(-0.551303\pi\)
−0.160475 + 0.987040i \(0.551303\pi\)
\(72\) 0 0
\(73\) 4.50531e6 1.35548 0.677742 − 0.735299i \(-0.262958\pi\)
0.677742 + 0.735299i \(0.262958\pi\)
\(74\) −649216. − 1.12448e6i −0.186242 − 0.322581i
\(75\) 0 0
\(76\) 408651. − 707804.i 0.106784 − 0.184955i
\(77\) −1.86265e6 + 3.22621e6i −0.464958 + 0.805331i
\(78\) 0 0
\(79\) −1.22765e6 − 2.12634e6i −0.280142 − 0.485220i 0.691278 − 0.722589i \(-0.257048\pi\)
−0.971419 + 0.237369i \(0.923715\pi\)
\(80\) 1.05636e6 0.230673
\(81\) 0 0
\(82\) 661738. 0.132537
\(83\) 695940. + 1.20540e6i 0.133598 + 0.231398i 0.925061 − 0.379819i \(-0.124014\pi\)
−0.791463 + 0.611217i \(0.790680\pi\)
\(84\) 0 0
\(85\) −124725. + 216031.i −0.0220287 + 0.0381548i
\(86\) 52681.2 − 91246.5i 0.00893123 − 0.0154693i
\(87\) 0 0
\(88\) 2.83856e6 + 4.91654e6i 0.444027 + 0.769077i
\(89\) 7.88308e6 1.18531 0.592653 − 0.805458i \(-0.298080\pi\)
0.592653 + 0.805458i \(0.298080\pi\)
\(90\) 0 0
\(91\) −1.10641e7 −1.53912
\(92\) −2.03190e6 − 3.51936e6i −0.272048 − 0.471201i
\(93\) 0 0
\(94\) −4.18174e6 + 7.24298e6i −0.519289 + 0.899435i
\(95\) −1.53023e6 + 2.65043e6i −0.183115 + 0.317164i
\(96\) 0 0
\(97\) 3.43723e6 + 5.95346e6i 0.382391 + 0.662321i 0.991404 − 0.130840i \(-0.0417673\pi\)
−0.609012 + 0.793161i \(0.708434\pi\)
\(98\) 259902. 0.0278945
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 27.8.c.a.10.4 12
3.2 odd 2 9.8.c.a.4.3 ✓ 12
4.3 odd 2 432.8.i.c.145.1 12
9.2 odd 6 9.8.c.a.7.3 yes 12
9.4 even 3 81.8.a.c.1.3 6
9.5 odd 6 81.8.a.e.1.4 6
9.7 even 3 inner 27.8.c.a.19.4 12
12.11 even 2 144.8.i.c.49.2 12
36.7 odd 6 432.8.i.c.289.1 12
36.11 even 6 144.8.i.c.97.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.3 ✓ 12 3.2 odd 2
9.8.c.a.7.3 yes 12 9.2 odd 6
27.8.c.a.10.4 12 1.1 even 1 trivial
27.8.c.a.19.4 12 9.7 even 3 inner
81.8.a.c.1.3 6 9.4 even 3
81.8.a.e.1.4 6 9.5 odd 6
144.8.i.c.49.2 12 12.11 even 2
144.8.i.c.97.2 12 36.11 even 6
432.8.i.c.145.1 12 4.3 odd 2
432.8.i.c.289.1 12 36.7 odd 6