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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,9,Mod(2,27)] 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("27.2"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), degree \(6\), 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: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 2.15
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.15

$q$-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+(13.3375 + 2.35175i) q^{2} +(34.9288 + 73.0820i) q^{3} +(-68.2042 - 24.8243i) q^{4} +(-636.805 - 758.914i) q^{5} +(293.990 + 1056.87i) q^{6} +(-2502.00 + 910.653i) q^{7} +(-3853.85 - 2225.02i) q^{8} +(-4120.96 + 5105.33i) q^{9} +(-6708.58 - 11619.6i) q^{10} +(15349.9 - 18293.3i) q^{11} +(-568.081 - 5851.58i) q^{12} +(4780.28 + 27110.3i) q^{13} +(-35511.9 + 6261.71i) q^{14} +(33220.2 - 73046.9i) q^{15} +(-31934.2 - 26796.0i) q^{16} +(-105956. + 61173.5i) q^{17} +(-66969.6 + 58400.7i) q^{18} +(-547.495 + 948.289i) q^{19} +(24593.2 + 67569.4i) q^{20} +(-153944. - 151043. i) q^{21} +(247750. - 207887. i) q^{22} +(50471.0 - 138668. i) q^{23} +(27998.7 - 359365. i) q^{24} +(-102599. + 581870. i) q^{25} +372825. i q^{26} +(-517048. - 122845. i) q^{27} +193253. q^{28} +(-924074. - 162939. i) q^{29} +(614861. - 896135. i) q^{30} +(131238. + 47766.6i) q^{31} +(369368. + 440196. i) q^{32} +(1.87306e6 + 482838. i) q^{33} +(-1.55704e6 + 566717. i) q^{34} +(2.28439e6 + 1.31889e6i) q^{35} +(407803. - 245905. i) q^{36} +(-379045. - 656525. i) q^{37} +(-9532.34 + 11360.2i) q^{38} +(-1.81431e6 + 1.29628e6i) q^{39} +(765550. + 4.34165e6i) q^{40} +(4.60079e6 - 811243. i) q^{41} +(-1.69801e6 - 2.37657e6i) q^{42} +(-1.82392e6 - 1.53045e6i) q^{43} +(-1.50104e6 + 866629. i) q^{44} +(6.49876e6 - 123645. i) q^{45} +(999268. - 1.73078e6i) q^{46} +(425686. + 1.16956e6i) q^{47} +(842880. - 3.26976e6i) q^{48} +(1.01461e6 - 851361. i) q^{49} +(-2.73683e6 + 7.51938e6i) q^{50} +(-8.17158e6 - 5.60673e6i) q^{51} +(346960. - 1.96771e6i) q^{52} -6.96711e6i q^{53} +(-6.60721e6 - 2.85441e6i) q^{54} -2.36579e7 q^{55} +(1.16686e7 + 2.05748e6i) q^{56} +(-88426.2 - 6889.44i) q^{57} +(-1.19416e7 - 4.34639e6i) q^{58} +(4.48304e6 + 5.34268e6i) q^{59} +(-4.07909e6 + 4.15744e6i) q^{60} +(-2.36882e7 + 8.62179e6i) q^{61} +(1.63804e6 + 945725. i) q^{62} +(5.66145e6 - 1.65263e7i) q^{63} +(9.22715e6 + 1.59819e7i) q^{64} +(1.75303e7 - 2.08918e7i) q^{65} +(2.38464e7 + 1.08448e7i) q^{66} +(-3.60777e6 - 2.04607e7i) q^{67} +(8.74520e6 - 1.54201e6i) q^{68} +(1.18970e7 - 1.15498e6i) q^{69} +(2.73663e7 + 2.29630e7i) q^{70} +(-1.06620e7 + 6.15572e6i) q^{71} +(2.72411e7 - 1.05060e7i) q^{72} +(3.50114e6 - 6.06416e6i) q^{73} +(-3.51151e6 - 9.64780e6i) q^{74} +(-4.61079e7 + 1.28258e7i) q^{75} +(60882.1 - 51086.1i) q^{76} +(-2.17466e7 + 5.97482e7i) q^{77} +(-2.72468e7 + 1.30223e7i) q^{78} +(2.79249e6 - 1.58370e7i) q^{79} +4.12991e7i q^{80} +(-9.08211e6 - 4.20777e7i) q^{81} +6.32706e7 q^{82} +(4.48356e6 + 790573. i) q^{83} +(6.75010e6 + 1.41233e7i) q^{84} +(1.13898e8 + 4.14556e7i) q^{85} +(-2.07272e7 - 2.47018e7i) q^{86} +(-2.03689e7 - 7.32244e7i) q^{87} +(-9.98592e7 + 3.63458e7i) q^{88} +(-3.86601e7 - 2.23204e7i) q^{89} +(8.69677e7 + 1.36344e7i) q^{90} +(-3.66484e7 - 6.34768e7i) q^{91} +(-6.88467e6 + 8.20483e6i) q^{92} +(1.09309e6 + 1.12595e7i) q^{93} +(2.92705e6 + 1.66001e7i) q^{94} +(1.06832e6 - 188373. i) q^{95} +(-1.92688e7 + 4.23697e7i) q^{96} +(1.19186e7 + 1.00009e7i) q^{97} +(1.55345e7 - 8.96887e6i) q^{98} +(3.01370e7 + 1.53752e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9} - 3 q^{10} + 28668 q^{11} + 77421 q^{12} - 6 q^{13} - 120975 q^{14} + 105507 q^{15} - 774 q^{16} - 9 q^{17}+ \cdots + 228876057 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13.3375 + 2.35175i 0.833591 + 0.146985i 0.574125 − 0.818768i \(-0.305342\pi\)
0.259466 + 0.965752i \(0.416453\pi\)
\(3\) 34.9288 + 73.0820i 0.431220 + 0.902247i
\(4\) −68.2042 − 24.8243i −0.266423 − 0.0969699i
\(5\) −636.805 − 758.914i −1.01889 − 1.21426i −0.976579 − 0.215158i \(-0.930973\pi\)
−0.0423083 − 0.999105i \(-0.513471\pi\)
\(6\) 293.990 + 1056.87i 0.226844 + 0.815488i
\(7\) −2502.00 + 910.653i −1.04207 + 0.379281i −0.805663 − 0.592375i \(-0.798191\pi\)
−0.236402 + 0.971655i \(0.575968\pi\)
\(8\) −3853.85 − 2225.02i −0.940882 − 0.543219i
\(9\) −4120.96 + 5105.33i −0.628099 + 0.778133i
\(10\) −6708.58 − 11619.6i −0.670858 − 1.16196i
\(11\) 15349.9 − 18293.3i 1.04842 − 1.24946i 0.0808814 − 0.996724i \(-0.474227\pi\)
0.967536 − 0.252732i \(-0.0813290\pi\)
\(12\) −568.081 − 5851.58i −0.0273959 − 0.282194i
\(13\) 4780.28 + 27110.3i 0.167371 + 0.949208i 0.946586 + 0.322451i \(0.104507\pi\)
−0.779215 + 0.626757i \(0.784382\pi\)
\(14\) −35511.9 + 6261.71i −0.924405 + 0.162997i
\(15\) 33220.2 − 73046.9i 0.656201 − 1.44290i
\(16\) −31934.2 − 26796.0i −0.487277 − 0.408874i
\(17\) −105956. + 61173.5i −1.26861 + 0.732432i −0.974725 − 0.223409i \(-0.928281\pi\)
−0.293885 + 0.955841i \(0.594948\pi\)
\(18\) −66969.6 + 58400.7i −0.637952 + 0.556324i
\(19\) −547.495 + 948.289i −0.00420113 + 0.00727656i −0.868118 − 0.496357i \(-0.834671\pi\)
0.863917 + 0.503634i \(0.168004\pi\)
\(20\) 24593.2 + 67569.4i 0.153708 + 0.422309i
\(21\) −153944. − 151043.i −0.791564 − 0.776647i
\(22\) 247750. − 207887.i 1.05760 − 0.887434i
\(23\) 50471.0 − 138668.i 0.180356 − 0.495524i −0.816264 − 0.577680i \(-0.803958\pi\)
0.996619 + 0.0821558i \(0.0261805\pi\)
\(24\) 27998.7 − 359365.i 0.0843905 − 1.08315i
\(25\) −102599. + 581870.i −0.262654 + 1.48959i
\(26\) 372825.i 0.815853i
\(27\) −517048. − 122845.i −0.972917 − 0.231154i
\(28\) 193253. 0.314409
\(29\) −924074. − 162939.i −1.30652 − 0.230374i −0.523313 − 0.852140i \(-0.675304\pi\)
−0.783203 + 0.621766i \(0.786415\pi\)
\(30\) 614861. − 896135.i 0.759088 − 1.10634i
\(31\) 131238. + 47766.6i 0.142106 + 0.0517223i 0.412094 − 0.911141i \(-0.364798\pi\)
−0.269988 + 0.962864i \(0.587020\pi\)
\(32\) 369368. + 440196.i 0.352257 + 0.419804i
\(33\) 1.87306e6 + 482838.i 1.57942 + 0.407142i
\(34\) −1.55704e6 + 566717.i −1.16516 + 0.424083i
\(35\) 2.28439e6 + 1.31889e6i 1.52229 + 0.878897i
\(36\) 407803. − 245905.i 0.242795 − 0.146406i
\(37\) −379045. − 656525.i −0.202248 − 0.350303i 0.747005 − 0.664819i \(-0.231491\pi\)
−0.949252 + 0.314515i \(0.898158\pi\)
\(38\) −9532.34 + 11360.2i −0.00457157 + 0.00544818i
\(39\) −1.81431e6 + 1.29628e6i −0.784247 + 0.560327i
\(40\) 765550. + 4.34165e6i 0.299043 + 1.69596i
\(41\) 4.60079e6 − 811243.i 1.62816 − 0.287088i 0.716361 − 0.697730i \(-0.245806\pi\)
0.911797 + 0.410641i \(0.134695\pi\)
\(42\) −1.69801e6 − 2.37657e6i −0.545685 − 0.763754i
\(43\) −1.82392e6 − 1.53045e6i −0.533498 − 0.447658i 0.335810 − 0.941930i \(-0.390990\pi\)
−0.869307 + 0.494272i \(0.835435\pi\)
\(44\) −1.50104e6 + 866629.i −0.400482 + 0.231218i
\(45\) 6.49876e6 − 123645.i 1.58482 − 0.0301527i
\(46\) 999268. − 1.73078e6i 0.223178 − 0.386555i
\(47\) 425686. + 1.16956e6i 0.0872365 + 0.239680i 0.975638 − 0.219387i \(-0.0704058\pi\)
−0.888401 + 0.459068i \(0.848184\pi\)
\(48\) 842880. − 3.26976e6i 0.158782 − 0.615958i
\(49\) 1.01461e6 − 851361.i 0.176001 − 0.147683i
\(50\) −2.73683e6 + 7.51938e6i −0.437893 + 1.20310i
\(51\) −8.17158e6 − 5.60673e6i −1.20788 − 0.828760i
\(52\) 346960. − 1.96771e6i 0.0474532 − 0.269121i
\(53\) − 6.96711e6i − 0.882977i −0.897267 − 0.441489i \(-0.854451\pi\)
0.897267 − 0.441489i \(-0.145549\pi\)
\(54\) −6.60721e6 − 2.85441e6i −0.777039 − 0.335692i
\(55\) −2.36579e7 −2.58539
\(56\) 1.16686e7 + 2.05748e6i 1.18649 + 0.209211i
\(57\) −88426.2 − 6889.44i −0.00837687 − 0.000652656i
\(58\) −1.19416e7 − 4.34639e6i −1.05524 − 0.384076i
\(59\) 4.48304e6 + 5.34268e6i 0.369968 + 0.440911i 0.918621 − 0.395139i \(-0.129304\pi\)
−0.548653 + 0.836050i \(0.684859\pi\)
\(60\) −4.07909e6 + 4.15744e6i −0.314745 + 0.320790i
\(61\) −2.36882e7 + 8.62179e6i −1.71085 + 0.622699i −0.996986 − 0.0775852i \(-0.975279\pi\)
−0.713865 + 0.700284i \(0.753057\pi\)
\(62\) 1.63804e6 + 945725.i 0.110856 + 0.0640026i
\(63\) 5.66145e6 − 1.65263e7i 0.359389 − 1.04909i
\(64\) 9.22715e6 + 1.59819e7i 0.549981 + 0.952595i
\(65\) 1.75303e7 − 2.08918e7i 0.982056 − 1.17037i
\(66\) 2.38464e7 + 1.08448e7i 1.25674 + 0.571540i
\(67\) −3.60777e6 − 2.04607e7i −0.179036 − 1.01536i −0.933382 − 0.358885i \(-0.883157\pi\)
0.754346 − 0.656477i \(-0.227954\pi\)
\(68\) 8.74520e6 − 1.54201e6i 0.409010 − 0.0721195i
\(69\) 1.18970e7 − 1.15498e6i 0.524858 − 0.0509540i
\(70\) 2.73663e7 + 2.29630e7i 1.13979 + 0.956394i
\(71\) −1.06620e7 + 6.15572e6i −0.419571 + 0.242240i −0.694894 − 0.719112i \(-0.744549\pi\)
0.275323 + 0.961352i \(0.411215\pi\)
\(72\) 2.72411e7 − 1.05060e7i 1.01366 − 0.390937i
\(73\) 3.50114e6 − 6.06416e6i 0.123287 − 0.213540i −0.797775 − 0.602956i \(-0.793990\pi\)
0.921062 + 0.389415i \(0.127323\pi\)
\(74\) −3.51151e6 − 9.64780e6i −0.117103 − 0.321737i
\(75\) −4.61079e7 + 1.28258e7i −1.45724 + 0.405360i
\(76\) 60882.1 − 51086.1i 0.00182488 − 0.00153126i
\(77\) −2.17466e7 + 5.97482e7i −0.618625 + 1.69966i
\(78\) −2.72468e7 + 1.30223e7i −0.736101 + 0.351812i
\(79\) 2.79249e6 − 1.58370e7i 0.0716941 − 0.406598i −0.927748 − 0.373207i \(-0.878258\pi\)
0.999442 − 0.0333910i \(-0.0106306\pi\)
\(80\) 4.12991e7i 1.00828i
\(81\) −9.08211e6 − 4.20777e7i −0.210983 − 0.977490i
\(82\) 6.32706e7 1.39942
\(83\) 4.48356e6 + 790573.i 0.0944737 + 0.0166583i 0.220685 − 0.975345i \(-0.429171\pi\)
−0.126211 + 0.992003i \(0.540282\pi\)
\(84\) 6.75010e6 + 1.41233e7i 0.135579 + 0.283674i
\(85\) 1.13898e8 + 4.14556e7i 2.18194 + 0.794159i
\(86\) −2.07272e7 − 2.47018e7i −0.378920 − 0.451579i
\(87\) −2.03689e7 − 7.32244e7i −0.355541 − 1.27814i
\(88\) −9.98592e7 + 3.63458e7i −1.66517 + 0.606071i
\(89\) −3.86601e7 − 2.23204e7i −0.616174 − 0.355748i 0.159204 − 0.987246i \(-0.449107\pi\)
−0.775378 + 0.631497i \(0.782441\pi\)
\(90\) 8.69677e7 + 1.36344e7i 1.32552 + 0.207809i
\(91\) −3.66484e7 − 6.34768e7i −0.534428 − 0.925656i
\(92\) −6.88467e6 + 8.20483e6i −0.0961018 + 0.114530i
\(93\) 1.09309e6 + 1.12595e7i 0.0146125 + 0.150518i
\(94\) 2.92705e6 + 1.66001e7i 0.0374903 + 0.212618i
\(95\) 1.06832e6 − 188373.i 0.0131161 − 0.00231273i
\(96\) −1.92688e7 + 4.23697e7i −0.226866 + 0.498850i
\(97\) 1.19186e7 + 1.00009e7i 0.134629 + 0.112967i 0.707616 − 0.706598i \(-0.249771\pi\)
−0.572987 + 0.819565i \(0.694215\pi\)
\(98\) 1.55345e7 − 8.96887e6i 0.168420 − 0.0972374i
\(99\) 3.01370e7 + 1.53752e8i 0.313733 + 1.60059i
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 27.9.f.a.2.15 ✓ 138
3.2 odd 2 81.9.f.a.8.9 138
27.13 even 9 81.9.f.a.71.9 138
27.14 odd 18 inner 27.9.f.a.14.15 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.15 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.15 yes 138 27.14 odd 18 inner
81.9.f.a.8.9 138 3.2 odd 2
81.9.f.a.71.9 138 27.13 even 9