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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 2.17
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.17

$q$-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+(14.2805 + 2.51803i) q^{2} +(-64.8590 - 48.5213i) q^{3} +(-42.9702 - 15.6399i) q^{4} +(501.169 + 597.270i) q^{5} +(-804.038 - 856.223i) q^{6} +(-473.173 + 172.221i) q^{7} +(-3789.11 - 2187.64i) q^{8} +(1852.37 + 6294.08i) q^{9} +(5652.98 + 9791.25i) q^{10} +(-14064.3 + 16761.1i) q^{11} +(2028.14 + 3099.35i) q^{12} +(8434.04 + 47831.8i) q^{13} +(-7190.79 + 1267.93i) q^{14} +(-3525.01 - 63055.7i) q^{15} +(-39634.1 - 33256.9i) q^{16} +(-37714.6 + 21774.5i) q^{17} +(10604.1 + 94546.7i) q^{18} +(26758.6 - 46347.3i) q^{19} +(-12194.1 - 33503.0i) q^{20} +(39045.9 + 11788.9i) q^{21} +(-243049. + 203942. i) q^{22} +(74836.0 - 205610. i) q^{23} +(139610. + 325741. i) q^{24} +(-37729.7 + 213976. i) q^{25} +704297. i q^{26} +(185254. - 498107. i) q^{27} +23025.9 q^{28} +(-1.18416e6 - 208799. i) q^{29} +(108437. - 909340. i) q^{30} +(636861. + 231798. i) q^{31} +(237718. + 283301. i) q^{32} +(1.72546e6 - 404694. i) q^{33} +(-593411. + 215984. i) q^{34} +(-340002. - 196300. i) q^{35} +(18841.8 - 299429. i) q^{36} +(-373921. - 647649. i) q^{37} +(498829. - 594481. i) q^{38} +(1.77384e6 - 3.51155e6i) q^{39} +(-592370. - 3.35950e6i) q^{40} +(686918. - 121122. i) q^{41} +(527909. + 266670. i) q^{42} +(-2.13681e6 - 1.79300e6i) q^{43} +(866486. - 500266. i) q^{44} +(-2.83091e6 + 4.26076e6i) q^{45} +(1.58642e6 - 2.74777e6i) q^{46} +(3.12827e6 + 8.59485e6i) q^{47} +(956957. + 4.08011e6i) q^{48} +(-4.22186e6 + 3.54256e6i) q^{49} +(-1.07760e6 + 2.96067e6i) q^{50} +(3.50266e6 + 417687. i) q^{51} +(385671. - 2.18725e6i) q^{52} +1.07463e7i q^{53} +(3.89976e6 - 6.64672e6i) q^{54} -1.70595e7 q^{55} +(2.16966e6 + 382570. i) q^{56} +(-3.98436e6 + 1.70767e6i) q^{57} +(-1.63846e7 - 5.96350e6i) q^{58} +(-2.12311e6 - 2.53023e6i) q^{59} +(-834712. + 2.76465e6i) q^{60} +(2.02966e7 - 7.38735e6i) q^{61} +(8.51100e6 + 4.91383e6i) q^{62} +(-1.96047e6 - 2.65917e6i) q^{63} +(9.30390e6 + 1.61148e7i) q^{64} +(-2.43416e7 + 2.90092e7i) q^{65} +(2.56595e7 - 1.43444e6i) q^{66} +(-4.71778e6 - 2.67558e7i) q^{67} +(1.96116e6 - 345805. i) q^{68} +(-1.48302e7 + 9.70453e6i) q^{69} +(-4.36110e6 - 3.65940e6i) q^{70} +(1.83188e7 - 1.05764e7i) q^{71} +(6.75035e6 - 2.79013e7i) q^{72} +(-7.57793e6 + 1.31254e7i) q^{73} +(-3.70896e6 - 1.01903e7i) q^{74} +(1.28295e7 - 1.20476e7i) q^{75} +(-1.87469e6 + 1.57305e6i) q^{76} +(3.76821e6 - 1.03531e7i) q^{77} +(3.41734e7 - 4.56800e7i) q^{78} +(-4.40131e6 + 2.49611e7i) q^{79} -4.03396e7i q^{80} +(-3.61842e7 + 2.33180e7i) q^{81} +1.01145e7 q^{82} +(-9.43147e6 - 1.66302e6i) q^{83} +(-1.49343e6 - 1.11724e6i) q^{84} +(-3.19067e7 - 1.16131e7i) q^{85} +(-2.59998e7 - 3.09854e7i) q^{86} +(6.66721e7 + 7.09994e7i) q^{87} +(8.99583e7 - 3.27422e7i) q^{88} +(-4.42841e6 - 2.55674e6i) q^{89} +(-5.11555e7 + 5.37174e7i) q^{90} +(-1.22284e7 - 2.11802e7i) q^{91} +(-6.43143e6 + 7.66468e6i) q^{92} +(-3.00590e7 - 4.59355e7i) q^{93} +(2.30310e7 + 1.30616e8i) q^{94} +(4.10924e7 - 7.24570e6i) q^{95} +(-1.67201e6 - 2.99090e7i) q^{96} +(3.09686e7 + 2.59857e7i) q^{97} +(-6.92104e7 + 3.99586e7i) q^{98} +(-1.31548e8 - 5.74737e7i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.2805 + 2.51803i 0.892529 + 0.157377i 0.601060 − 0.799204i \(-0.294745\pi\)
0.291468 + 0.956580i \(0.405856\pi\)
\(3\) −64.8590 − 48.5213i −0.800728 − 0.599028i
\(4\) −42.9702 − 15.6399i −0.167852 − 0.0610932i
\(5\) 501.169 + 597.270i 0.801870 + 0.955632i 0.999697 − 0.0246050i \(-0.00783282\pi\)
−0.197827 + 0.980237i \(0.563388\pi\)
\(6\) −804.038 − 856.223i −0.620400 − 0.660666i
\(7\) −473.173 + 172.221i −0.197073 + 0.0717289i −0.438671 − 0.898648i \(-0.644551\pi\)
0.241598 + 0.970376i \(0.422329\pi\)
\(8\) −3789.11 − 2187.64i −0.925075 − 0.534092i
\(9\) 1852.37 + 6294.08i 0.282331 + 0.959317i
\(10\) 5652.98 + 9791.25i 0.565298 + 0.979125i
\(11\) −14064.3 + 16761.1i −0.960608 + 1.14481i 0.0287915 + 0.999585i \(0.490834\pi\)
−0.989399 + 0.145222i \(0.953610\pi\)
\(12\) 2028.14 + 3099.35i 0.0978075 + 0.149467i
\(13\) 8434.04 + 47831.8i 0.295299 + 1.67472i 0.665984 + 0.745966i \(0.268012\pi\)
−0.370685 + 0.928759i \(0.620877\pi\)
\(14\) −7190.79 + 1267.93i −0.187182 + 0.0330053i
\(15\) −3525.01 − 63055.7i −0.0696298 − 1.24554i
\(16\) −39634.1 − 33256.9i −0.604768 − 0.507461i
\(17\) −37714.6 + 21774.5i −0.451559 + 0.260707i −0.708488 − 0.705723i \(-0.750622\pi\)
0.256930 + 0.966430i \(0.417289\pi\)
\(18\) 10604.1 + 94546.7i 0.101014 + 0.900651i
\(19\) 26758.6 − 46347.3i 0.205328 − 0.355639i −0.744909 − 0.667166i \(-0.767507\pi\)
0.950237 + 0.311527i \(0.100840\pi\)
\(20\) −12194.1 − 33503.0i −0.0762131 − 0.209394i
\(21\) 39045.9 + 11788.9i 0.200770 + 0.0606172i
\(22\) −243049. + 203942.i −1.03754 + 0.870596i
\(23\) 74836.0 − 205610.i 0.267423 − 0.734739i −0.731194 − 0.682170i \(-0.761037\pi\)
0.998617 − 0.0525697i \(-0.0167412\pi\)
\(24\) 139610. + 325741.i 0.420797 + 0.981809i
\(25\) −37729.7 + 213976.i −0.0965881 + 0.547778i
\(26\) 704297.i 1.54121i
\(27\) 185254. − 498107.i 0.348588 − 0.937276i
\(28\) 23025.9 0.0374614
\(29\) −1.18416e6 − 208799.i −1.67424 − 0.295214i −0.745655 − 0.666332i \(-0.767863\pi\)
−0.928586 + 0.371118i \(0.878975\pi\)
\(30\) 108437. − 909340.i 0.133873 − 1.12264i
\(31\) 636861. + 231798.i 0.689601 + 0.250994i 0.662964 − 0.748651i \(-0.269298\pi\)
0.0266367 + 0.999645i \(0.491520\pi\)
\(32\) 237718. + 283301.i 0.226706 + 0.270177i
\(33\) 1.72546e6 − 404694.i 1.45496 − 0.341249i
\(34\) −593411. + 215984.i −0.444058 + 0.161624i
\(35\) −340002. − 196300.i −0.226574 − 0.130812i
\(36\) 18841.8 − 299429.i 0.0112179 − 0.178272i
\(37\) −373921. − 647649.i −0.199514 − 0.345568i 0.748857 − 0.662731i \(-0.230603\pi\)
−0.948371 + 0.317164i \(0.897270\pi\)
\(38\) 498829. − 594481.i 0.239231 − 0.285104i
\(39\) 1.77384e6 − 3.51155e6i 0.766752 − 1.51789i
\(40\) −592370. − 3.35950e6i −0.231395 − 1.31230i
\(41\) 686918. − 121122.i 0.243091 − 0.0428636i −0.0507748 − 0.998710i \(-0.516169\pi\)
0.293866 + 0.955847i \(0.405058\pi\)
\(42\) 527909. + 266670.i 0.169653 + 0.0856991i
\(43\) −2.13681e6 − 1.79300e6i −0.625017 − 0.524452i 0.274359 − 0.961627i \(-0.411534\pi\)
−0.899376 + 0.437176i \(0.855979\pi\)
\(44\) 866486. − 500266.i 0.231180 − 0.133472i
\(45\) −2.83091e6 + 4.26076e6i −0.690361 + 1.03905i
\(46\) 1.58642e6 − 2.74777e6i 0.354314 − 0.613690i
\(47\) 3.12827e6 + 8.59485e6i 0.641081 + 1.76136i 0.648304 + 0.761382i \(0.275479\pi\)
−0.00722272 + 0.999974i \(0.502299\pi\)
\(48\) 956957. + 4.08011e6i 0.180272 + 0.768611i
\(49\) −4.22186e6 + 3.54256e6i −0.732352 + 0.614516i
\(50\) −1.07760e6 + 2.96067e6i −0.172415 + 0.473707i
\(51\) 3.50266e6 + 417687.i 0.517747 + 0.0617405i
\(52\) 385671. − 2.18725e6i 0.0527477 − 0.299147i
\(53\) 1.07463e7i 1.36193i 0.732317 + 0.680964i \(0.238439\pi\)
−0.732317 + 0.680964i \(0.761561\pi\)
\(54\) 3.89976e6 − 6.64672e6i 0.458630 − 0.781686i
\(55\) −1.70595e7 −1.86430
\(56\) 2.16966e6 + 382570.i 0.220618 + 0.0389008i
\(57\) −3.98436e6 + 1.70767e6i −0.377450 + 0.161773i
\(58\) −1.63846e7 − 5.96350e6i −1.44785 − 0.526974i
\(59\) −2.12311e6 − 2.53023e6i −0.175213 − 0.208810i 0.671290 − 0.741195i \(-0.265740\pi\)
−0.846503 + 0.532385i \(0.821296\pi\)
\(60\) −834712. + 2.76465e6i −0.0644068 + 0.213321i
\(61\) 2.02966e7 − 7.38735e6i 1.46590 − 0.533543i 0.518914 − 0.854826i \(-0.326336\pi\)
0.946983 + 0.321283i \(0.104114\pi\)
\(62\) 8.51100e6 + 4.91383e6i 0.575988 + 0.332547i
\(63\) −1.96047e6 − 2.65917e6i −0.124451 − 0.168805i
\(64\) 9.30390e6 + 1.61148e7i 0.554556 + 0.960519i
\(65\) −2.43416e7 + 2.90092e7i −1.36363 + 1.62511i
\(66\) 2.56595e7 − 1.43444e6i 1.35230 − 0.0755976i
\(67\) −4.71778e6 − 2.67558e7i −0.234120 − 1.32776i −0.844459 − 0.535620i \(-0.820078\pi\)
0.610339 − 0.792140i \(-0.291033\pi\)
\(68\) 1.96116e6 − 345805.i 0.0917226 − 0.0161732i
\(69\) −1.48302e7 + 9.70453e6i −0.654263 + 0.428132i
\(70\) −4.36110e6 − 3.65940e6i −0.181637 − 0.152411i
\(71\) 1.83188e7 − 1.05764e7i 0.720882 − 0.416201i −0.0941955 − 0.995554i \(-0.530028\pi\)
0.815077 + 0.579353i \(0.196695\pi\)
\(72\) 6.75035e6 − 2.79013e7i 0.251187 − 1.03823i
\(73\) −7.57793e6 + 1.31254e7i −0.266845 + 0.462189i −0.968045 − 0.250775i \(-0.919315\pi\)
0.701200 + 0.712964i \(0.252648\pi\)
\(74\) −3.70896e6 − 1.01903e7i −0.123687 − 0.339828i
\(75\) 1.28295e7 − 1.20476e7i 0.405475 − 0.380762i
\(76\) −1.87469e6 + 1.57305e6i −0.0561920 + 0.0471507i
\(77\) 3.76821e6 − 1.03531e7i 0.107194 − 0.294514i
\(78\) 3.41734e7 − 4.56800e7i 0.923230 − 1.23409i
\(79\) −4.40131e6 + 2.49611e7i −0.112999 + 0.640848i 0.874723 + 0.484623i \(0.161043\pi\)
−0.987722 + 0.156224i \(0.950068\pi\)
\(80\) − 4.03396e7i − 0.984853i
\(81\) −3.61842e7 + 2.33180e7i −0.840579 + 0.541690i
\(82\) 1.01145e7 0.223712
\(83\) −9.43147e6 − 1.66302e6i −0.198732 − 0.0350417i 0.0733962 − 0.997303i \(-0.476616\pi\)
−0.272128 + 0.962261i \(0.587727\pi\)
\(84\) −1.49343e6 − 1.11724e6i −0.0299964 − 0.0224404i
\(85\) −3.19067e7 − 1.16131e7i −0.611232 − 0.222470i
\(86\) −2.59998e7 − 3.09854e7i −0.475309 − 0.566451i
\(87\) 6.66721e7 + 7.09994e7i 1.16377 + 1.23930i
\(88\) 8.99583e7 − 3.27422e7i 1.50007 − 0.545980i
\(89\) −4.42841e6 − 2.55674e6i −0.0705810 − 0.0407500i 0.464294 − 0.885681i \(-0.346308\pi\)
−0.534875 + 0.844931i \(0.679641\pi\)
\(90\) −5.11555e7 + 5.37174e7i −0.779690 + 0.818737i
\(91\) −1.22284e7 − 2.11802e7i −0.178322 − 0.308862i
\(92\) −6.43143e6 + 7.66468e6i −0.0897752 + 0.106990i
\(93\) −3.00590e7 − 4.59355e7i −0.401830 − 0.614069i
\(94\) 2.30310e7 + 1.30616e8i 0.294987 + 1.67295i
\(95\) 4.10924e7 − 7.24570e6i 0.504507 − 0.0889582i
\(96\) −1.67201e6 − 2.99090e7i −0.0196858 − 0.352141i
\(97\) 3.09686e7 + 2.59857e7i 0.349811 + 0.293527i 0.800714 − 0.599046i \(-0.204453\pi\)
−0.450903 + 0.892573i \(0.648898\pi\)
\(98\) −6.92104e7 + 3.99586e7i −0.750356 + 0.433218i
\(99\) −1.31548e8 − 5.74737e7i −1.36944 − 0.598313i
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.2.17 ✓ 138
3.2 odd 2 81.9.f.a.8.7 138
27.13 even 9 81.9.f.a.71.7 138
27.14 odd 18 inner 27.9.f.a.14.17 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.17 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.17 yes 138 27.14 odd 18 inner
81.9.f.a.8.7 138 3.2 odd 2
81.9.f.a.71.7 138 27.13 even 9