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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 2.20
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.20

$q$-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+(25.1387 + 4.43262i) q^{2} +(-80.5339 - 8.67746i) q^{3} +(371.742 + 135.303i) q^{4} +(-546.503 - 651.297i) q^{5} +(-1986.05 - 575.116i) q^{6} +(-3663.86 + 1333.53i) q^{7} +(3086.08 + 1781.75i) q^{8} +(6410.40 + 1397.66i) q^{9} +(-10851.4 - 18795.2i) q^{10} +(-7165.64 + 8539.67i) q^{11} +(-28763.8 - 14122.3i) q^{12} +(-6824.34 - 38702.7i) q^{13} +(-98015.5 + 17282.8i) q^{14} +(38360.4 + 57193.8i) q^{15} +(-7898.09 - 6627.28i) q^{16} +(109614. - 63285.8i) q^{17} +(154954. + 63550.2i) q^{18} +(-62328.9 + 107957. i) q^{19} +(-115036. - 316059. i) q^{20} +(306636. - 75601.7i) q^{21} +(-217988. + 182913. i) q^{22} +(-84577.6 + 232375. i) q^{23} +(-233073. - 170270. i) q^{24} +(-57691.0 + 327182. i) q^{25} -1.00318e6i q^{26} +(-504126. - 168185. i) q^{27} -1.54244e6 q^{28} +(-177273. - 31258.0i) q^{29} +(710811. + 1.60781e6i) q^{30} +(557486. + 202908. i) q^{31} +(-755557. - 900438. i) q^{32} +(651179. - 625553. i) q^{33} +(3.03608e6 - 1.10504e6i) q^{34} +(2.87084e6 + 1.65748e6i) q^{35} +(2.19391e6 + 1.38692e6i) q^{36} +(-229672. - 397803. i) q^{37} +(-2.04540e6 + 2.43761e6i) q^{38} +(213749. + 3.17610e6i) q^{39} +(-526104. - 2.98368e6i) q^{40} +(-981871. + 173130. i) q^{41} +(8.04354e6 - 541323. i) q^{42} +(-2.11904e6 - 1.77809e6i) q^{43} +(-3.81922e6 + 2.20503e6i) q^{44} +(-2.59302e6 - 4.93890e6i) q^{45} +(-3.15620e6 + 5.46670e6i) q^{46} +(-1.21462e6 - 3.33714e6i) q^{47} +(578555. + 602256. i) q^{48} +(7.22944e6 - 6.06622e6i) q^{49} +(-2.90055e6 + 7.96919e6i) q^{50} +(-9.37682e6 + 4.14548e6i) q^{51} +(2.69971e6 - 1.53108e7i) q^{52} +6.97006e6i q^{53} +(-1.19276e7 - 6.46254e6i) q^{54} +9.47791e6 q^{55} +(-1.36830e7 - 2.41268e6i) q^{56} +(5.95638e6 - 8.15332e6i) q^{57} +(-4.31784e6 - 1.57157e6i) q^{58} +(-7.66416e6 - 9.13379e6i) q^{59} +(6.52170e6 + 2.64516e7i) q^{60} +(-4.95543e6 + 1.80363e6i) q^{61} +(1.31150e7 + 7.57197e6i) q^{62} +(-2.53506e7 + 3.42767e6i) q^{63} +(-1.36827e7 - 2.36991e7i) q^{64} +(-2.14775e7 + 2.55958e7i) q^{65} +(1.91426e7 - 1.28391e7i) q^{66} +(-5.51669e6 - 3.12867e7i) q^{67} +(4.93110e7 - 8.69486e6i) q^{68} +(8.82779e6 - 1.79801e7i) q^{69} +(6.48220e7 + 5.43921e7i) q^{70} +(-2.14530e7 + 1.23859e7i) q^{71} +(1.72927e7 + 1.57350e7i) q^{72} +(2.29304e6 - 3.97166e6i) q^{73} +(-4.01033e6 - 1.10183e7i) q^{74} +(7.48519e6 - 2.58486e7i) q^{75} +(-3.77772e7 + 3.16988e7i) q^{76} +(1.48659e7 - 4.08438e7i) q^{77} +(-8.70509e6 + 8.07903e7i) q^{78} +(1.68902e6 - 9.57891e6i) q^{79} +8.76584e6i q^{80} +(3.91398e7 + 1.79191e7i) q^{81} -2.54503e7 q^{82} +(-2.39486e7 - 4.22279e6i) q^{83} +(1.24219e8 + 1.33845e7i) q^{84} +(-1.01122e8 - 3.68056e7i) q^{85} +(-4.53882e7 - 5.40916e7i) q^{86} +(1.40052e7 + 4.05560e6i) q^{87} +(-3.73292e7 + 1.35867e7i) q^{88} +(-2.53853e7 - 1.46562e7i) q^{89} +(-4.32926e7 - 1.35651e8i) q^{90} +(7.66148e7 + 1.32701e8i) q^{91} +(-6.28822e7 + 7.49401e7i) q^{92} +(-4.31358e7 - 2.11786e7i) q^{93} +(-1.57416e7 - 8.92752e7i) q^{94} +(1.04375e8 - 1.84041e7i) q^{95} +(5.30344e7 + 7.90720e7i) q^{96} +(8.60715e7 + 7.22226e7i) q^{97} +(2.08628e8 - 1.20451e8i) q^{98} +(-5.78702e7 + 4.47276e7i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 25.1387 + 4.43262i 1.57117 + 0.277039i 0.890302 − 0.455371i \(-0.150493\pi\)
0.680864 + 0.732410i \(0.261604\pi\)
\(3\) −80.5339 − 8.67746i −0.994245 − 0.107129i
\(4\) 371.742 + 135.303i 1.45212 + 0.528528i
\(5\) −546.503 − 651.297i −0.874405 − 1.04208i −0.998757 − 0.0498377i \(-0.984130\pi\)
0.124352 − 0.992238i \(-0.460315\pi\)
\(6\) −1986.05 − 575.116i −1.53244 − 0.443762i
\(7\) −3663.86 + 1333.53i −1.52597 + 0.555408i −0.962631 − 0.270817i \(-0.912706\pi\)
−0.563340 + 0.826225i \(0.690484\pi\)
\(8\) 3086.08 + 1781.75i 0.753437 + 0.434997i
\(9\) 6410.40 + 1397.66i 0.977047 + 0.213025i
\(10\) −10851.4 − 18795.2i −1.08514 − 1.87952i
\(11\) −7165.64 + 8539.67i −0.489423 + 0.583271i −0.953071 − 0.302748i \(-0.902096\pi\)
0.463648 + 0.886020i \(0.346540\pi\)
\(12\) −28763.8 − 14122.3i −1.38714 − 0.681051i
\(13\) −6824.34 − 38702.7i −0.238939 − 1.35509i −0.834158 − 0.551526i \(-0.814046\pi\)
0.595219 − 0.803564i \(-0.297065\pi\)
\(14\) −98015.5 + 17282.8i −2.55142 + 0.449885i
\(15\) 38360.4 + 57193.8i 0.757737 + 1.12975i
\(16\) −7898.09 − 6627.28i −0.120515 − 0.101124i
\(17\) 109614. − 63285.8i 1.31242 − 0.757723i 0.329920 − 0.944009i \(-0.392978\pi\)
0.982496 + 0.186286i \(0.0596450\pi\)
\(18\) 154954. + 63550.2i 1.47609 + 0.605378i
\(19\) −62328.9 + 107957.i −0.478272 + 0.828391i −0.999690 − 0.0249102i \(-0.992070\pi\)
0.521418 + 0.853302i \(0.325403\pi\)
\(20\) −115036. − 316059.i −0.718974 − 1.97537i
\(21\) 306636. − 75601.7i 1.57669 − 0.388736i
\(22\) −217988. + 182913.i −0.930553 + 0.780827i
\(23\) −84577.6 + 232375.i −0.302235 + 0.830383i 0.691876 + 0.722016i \(0.256784\pi\)
−0.994111 + 0.108367i \(0.965438\pi\)
\(24\) −233073. − 170270.i −0.702500 − 0.513209i
\(25\) −57691.0 + 327182.i −0.147689 + 0.837586i
\(26\) − 1.00318e6i − 2.19527i
\(27\) −504126. − 168185.i −0.948603 − 0.316470i
\(28\) −1.54244e6 −2.50944
\(29\) −177273. − 31258.0i −0.250640 − 0.0441946i 0.0469165 − 0.998899i \(-0.485061\pi\)
−0.297556 + 0.954704i \(0.596172\pi\)
\(30\) 710811. + 1.60781e6i 0.877544 + 1.98495i
\(31\) 557486. + 202908.i 0.603653 + 0.219712i 0.625724 − 0.780045i \(-0.284804\pi\)
−0.0220706 + 0.999756i \(0.507026\pi\)
\(32\) −755557. − 900438.i −0.720555 − 0.858724i
\(33\) 651179. − 625553.i 0.549091 − 0.527483i
\(34\) 3.03608e6 − 1.10504e6i 2.27194 − 0.826919i
\(35\) 2.87084e6 + 1.65748e6i 1.91310 + 1.10453i
\(36\) 2.19391e6 + 1.38692e6i 1.30620 + 0.825735i
\(37\) −229672. − 397803.i −0.122546 − 0.212257i 0.798225 − 0.602360i \(-0.205773\pi\)
−0.920771 + 0.390103i \(0.872439\pi\)
\(38\) −2.04540e6 + 2.43761e6i −0.980941 + 1.16904i
\(39\) 213749. + 3.17610e6i 0.0923942 + 1.37289i
\(40\) −526104. − 2.98368e6i −0.205509 − 1.16550i
\(41\) −981871. + 173130.i −0.347471 + 0.0612686i −0.344660 − 0.938727i \(-0.612006\pi\)
−0.00281108 + 0.999996i \(0.500895\pi\)
\(42\) 8.04354e6 − 541323.i 2.58494 − 0.173964i
\(43\) −2.11904e6 − 1.77809e6i −0.619820 − 0.520091i 0.277927 − 0.960602i \(-0.410353\pi\)
−0.897747 + 0.440512i \(0.854797\pi\)
\(44\) −3.81922e6 + 2.20503e6i −1.01898 + 0.588306i
\(45\) −2.59302e6 − 4.93890e6i −0.632346 − 1.20443i
\(46\) −3.15620e6 + 5.46670e6i −0.704909 + 1.22094i
\(47\) −1.21462e6 − 3.33714e6i −0.248914 − 0.683885i −0.999727 − 0.0233703i \(-0.992560\pi\)
0.750813 − 0.660515i \(-0.229662\pi\)
\(48\) 578555. + 602256.i 0.108988 + 0.113453i
\(49\) 7.22944e6 − 6.06622e6i 1.25407 − 1.05229i
\(50\) −2.90055e6 + 7.96919e6i −0.464088 + 1.27507i
\(51\) −9.37682e6 + 4.14548e6i −1.38604 + 0.612765i
\(52\) 2.69971e6 − 1.53108e7i 0.369235 − 2.09404i
\(53\) 6.97006e6i 0.883350i 0.897175 + 0.441675i \(0.145616\pi\)
−0.897175 + 0.441675i \(0.854384\pi\)
\(54\) −1.19276e7 − 6.46254e6i −1.40274 − 0.760026i
\(55\) 9.47791e6 1.03577
\(56\) −1.36830e7 − 2.41268e6i −1.39132 − 0.245328i
\(57\) 5.95638e6 − 8.15332e6i 0.564264 − 0.772387i
\(58\) −4.31784e6 − 1.57157e6i −0.381553 − 0.138874i
\(59\) −7.66416e6 − 9.13379e6i −0.632494 − 0.753777i 0.350671 − 0.936499i \(-0.385954\pi\)
−0.983165 + 0.182722i \(0.941509\pi\)
\(60\) 6.52170e6 + 2.64516e7i 0.503217 + 2.04102i
\(61\) −4.95543e6 + 1.80363e6i −0.357900 + 0.130265i −0.514711 − 0.857364i \(-0.672101\pi\)
0.156811 + 0.987629i \(0.449879\pi\)
\(62\) 1.31150e7 + 7.57197e6i 0.887571 + 0.512439i
\(63\) −2.53506e7 + 3.42767e6i −1.60926 + 0.217589i
\(64\) −1.36827e7 − 2.36991e7i −0.815551 − 1.41258i
\(65\) −2.14775e7 + 2.55958e7i −1.20318 + 1.43389i
\(66\) 1.91426e7 − 1.28391e7i 1.00885 − 0.676644i
\(67\) −5.51669e6 − 3.12867e7i −0.273766 − 1.55260i −0.742854 − 0.669453i \(-0.766528\pi\)
0.469088 − 0.883151i \(-0.344583\pi\)
\(68\) 4.93110e7 − 8.69486e6i 2.30626 − 0.406656i
\(69\) 8.82779e6 − 1.79801e7i 0.389454 − 0.793226i
\(70\) 6.48220e7 + 5.43921e7i 2.69979 + 2.26540i
\(71\) −2.14530e7 + 1.23859e7i −0.844217 + 0.487409i −0.858696 − 0.512486i \(-0.828725\pi\)
0.0144782 + 0.999895i \(0.495391\pi\)
\(72\) 1.72927e7 + 1.57350e7i 0.643477 + 0.585513i
\(73\) 2.29304e6 − 3.97166e6i 0.0807459 − 0.139856i −0.822825 − 0.568295i \(-0.807603\pi\)
0.903571 + 0.428439i \(0.140936\pi\)
\(74\) −4.01033e6 − 1.10183e7i −0.133737 − 0.367441i
\(75\) 7.48519e6 − 2.58486e7i 0.236569 − 0.816944i
\(76\) −3.77772e7 + 3.16988e7i −1.13234 + 0.950143i
\(77\) 1.48659e7 − 4.08438e7i 0.422891 − 1.16188i
\(78\) −8.70509e6 + 8.07903e7i −0.235177 + 2.18263i
\(79\) 1.68902e6 − 9.57891e6i 0.0433637 − 0.245928i −0.955419 − 0.295253i \(-0.904596\pi\)
0.998783 + 0.0493253i \(0.0157071\pi\)
\(80\) 8.76584e6i 0.214010i
\(81\) 3.91398e7 + 1.79191e7i 0.909240 + 0.416271i
\(82\) −2.54503e7 −0.562909
\(83\) −2.39486e7 − 4.22279e6i −0.504625 − 0.0889789i −0.0844600 − 0.996427i \(-0.526917\pi\)
−0.420165 + 0.907448i \(0.638028\pi\)
\(84\) 1.24219e8 + 1.33845e7i 2.49500 + 0.268834i
\(85\) −1.01122e8 − 3.68056e7i −1.93719 − 0.705079i
\(86\) −4.53882e7 − 5.40916e7i −0.829754 − 0.988863i
\(87\) 1.40052e7 + 4.05560e6i 0.244463 + 0.0707911i
\(88\) −3.73292e7 + 1.35867e7i −0.622470 + 0.226561i
\(89\) −2.53853e7 − 1.46562e7i −0.404597 − 0.233594i 0.283869 − 0.958863i \(-0.408382\pi\)
−0.688466 + 0.725269i \(0.741715\pi\)
\(90\) −4.32926e7 − 1.35651e8i −0.659848 − 2.06754i
\(91\) 7.66148e7 + 1.32701e8i 1.11724 + 1.93512i
\(92\) −6.28822e7 + 7.49401e7i −0.877761 + 1.04608i
\(93\) −4.31358e7 − 2.11786e7i −0.576642 − 0.283116i
\(94\) −1.57416e7 − 8.92752e7i −0.201622 − 1.14346i
\(95\) 1.04375e8 − 1.84041e7i 1.28145 − 0.225954i
\(96\) 5.30344e7 + 7.90720e7i 0.624414 + 0.930975i
\(97\) 8.60715e7 + 7.22226e7i 0.972238 + 0.815805i 0.982900 − 0.184139i \(-0.0589495\pi\)
−0.0106623 + 0.999943i \(0.503394\pi\)
\(98\) 2.08628e8 − 1.20451e8i 2.26187 − 1.30589i
\(99\) −5.78702e7 + 4.47276e7i −0.602440 + 0.465624i
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.2.20 ✓ 138
3.2 odd 2 81.9.f.a.8.4 138
27.13 even 9 81.9.f.a.71.4 138
27.14 odd 18 inner 27.9.f.a.14.20 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.20 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.20 yes 138 27.14 odd 18 inner
81.9.f.a.8.4 138 3.2 odd 2
81.9.f.a.71.4 138 27.13 even 9