Show commands: Magma / Pari/GP / SageMath

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 2.12
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.12

$q$-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+(-2.46646 - 0.434904i) q^{2} +(-52.6972 - 61.5143i) q^{3} +(-234.667 - 85.4118i) q^{4} +(-411.949 - 490.942i) q^{5} +(103.223 + 174.641i) q^{6} +(243.334 - 88.5664i) q^{7} +(1096.91 + 633.301i) q^{8} +(-1007.01 + 6483.26i) q^{9} +(802.546 + 1390.05i) q^{10} +(4355.96 - 5191.23i) q^{11} +(7112.24 + 18936.3i) q^{12} +(-733.507 - 4159.93i) q^{13} +(-638.693 + 112.619i) q^{14} +(-8491.39 + 51212.1i) q^{15} +(46543.3 + 39054.5i) q^{16} +(-27580.9 + 15923.8i) q^{17} +(5303.36 - 15552.8i) q^{18} +(-31761.5 + 55012.5i) q^{19} +(54738.7 + 150393. i) q^{20} +(-18271.1 - 10301.3i) q^{21} +(-13001.5 + 10909.6i) q^{22} +(-67576.8 + 185666. i) q^{23} +(-18847.0 - 100849. i) q^{24} +(-3490.62 + 19796.3i) q^{25} +10579.3i q^{26} +(451880. - 279704. i) q^{27} -64667.1 q^{28} +(1.01674e6 + 179279. i) q^{29} +(43216.0 - 122620. i) q^{30} +(-1.67239e6 - 608702. i) q^{31} +(-306236. - 364958. i) q^{32} +(-548882. + 5609.45i) q^{33} +(74952.5 - 27280.5i) q^{34} +(-143722. - 82978.2i) q^{35} +(790060. - 1.43540e6i) q^{36} +(-711454. - 1.23227e6i) q^{37} +(102264. - 121873. i) q^{38} +(-217241. + 264338. i) q^{39} +(-140957. - 799407. i) q^{40} +(47826.5 - 8433.11i) q^{41} +(40585.0 + 33354.1i) q^{42} +(670492. + 562610. i) q^{43} +(-1.46559e6 + 846160. i) q^{44} +(3.59774e6 - 2.17639e6i) q^{45} +(247423. - 428549. i) q^{46} +(2.11724e6 + 5.81707e6i) q^{47} +(-50293.0 - 4.92114e6i) q^{48} +(-4.36473e6 + 3.66244e6i) q^{49} +(17219.0 - 47308.8i) q^{50} +(2.43298e6 + 857477. i) q^{51} +(-183177. + 1.03885e6i) q^{52} +9.95902e6i q^{53} +(-1.23619e6 + 493355. i) q^{54} -4.34303e6 q^{55} +(323005. + 56954.4i) q^{56} +(5.05780e6 - 945219. i) q^{57} +(-2.42979e6 - 884370. i) q^{58} +(9.92151e6 + 1.18240e7i) q^{59} +(6.36676e6 - 1.12925e7i) q^{60} +(-2.00776e7 + 7.30764e6i) q^{61} +(3.86018e6 + 2.22867e6i) q^{62} +(329158. + 1.66679e6i) q^{63} +(-7.18042e6 - 1.24369e7i) q^{64} +(-1.74012e6 + 2.07379e6i) q^{65} +(1.35624e6 + 224875. i) q^{66} +(2.15182e6 + 1.22036e7i) q^{67} +(7.83240e6 - 1.38106e6i) q^{68} +(1.49822e7 - 5.62712e6i) q^{69} +(318399. + 267168. i) q^{70} +(-2.36946e6 + 1.36801e6i) q^{71} +(-5.21046e6 + 6.47380e6i) q^{72} +(874977. - 1.51550e6i) q^{73} +(1.21886e6 + 3.34878e6i) q^{74} +(1.40170e6 - 828486. i) q^{75} +(1.21521e7 - 1.01968e7i) q^{76} +(600185. - 1.64900e6i) q^{77} +(650779. - 557500. i) q^{78} +(7.53737e6 - 4.27466e7i) q^{79} -3.89386e7i q^{80} +(-4.10186e7 - 1.30575e7i) q^{81} -121630. q^{82} +(-5.86063e7 - 1.03339e7i) q^{83} +(3.40778e6 + 3.97795e6i) q^{84} +(1.91796e7 + 6.98080e6i) q^{85} +(-1.40906e6 - 1.67926e6i) q^{86} +(-4.25512e7 - 7.19916e7i) q^{87} +(8.06570e6 - 2.93568e6i) q^{88} +(-3.32007e7 - 1.91684e7i) q^{89} +(-9.82023e6 + 3.80331e6i) q^{90} +(-546917. - 947288. i) q^{91} +(3.17161e7 - 3.77978e7i) q^{92} +(5.06866e7 + 1.34953e8i) q^{93} +(-2.69223e6 - 1.52684e7i) q^{94} +(4.00921e7 - 7.06932e6i) q^{95} +(-6.31236e6 + 3.80702e7i) q^{96} +(1.06250e8 + 8.91546e7i) q^{97} +(1.23583e7 - 7.13504e6i) q^{98} +(2.92696e7 + 3.34685e7i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.46646 − 0.434904i −0.154154 − 0.0271815i 0.0960385 − 0.995378i \(-0.469383\pi\)
−0.250193 + 0.968196i \(0.580494\pi\)
\(3\) −52.6972 − 61.5143i −0.650582 − 0.759436i
\(4\) −234.667 − 85.4118i −0.916668 − 0.333640i
\(5\) −411.949 − 490.942i −0.659119 − 0.785508i 0.328140 − 0.944629i \(-0.393578\pi\)
−0.987259 + 0.159122i \(0.949134\pi\)
\(6\) 103.223 + 174.641i 0.0796473 + 0.134754i
\(7\) 243.334 − 88.5664i 0.101347 − 0.0368873i −0.290849 − 0.956769i \(-0.593938\pi\)
0.392196 + 0.919882i \(0.371715\pi\)
\(8\) 1096.91 + 633.301i 0.267800 + 0.154614i
\(9\) −1007.01 + 6483.26i −0.153485 + 0.988151i
\(10\) 802.546 + 1390.05i 0.0802546 + 0.139005i
\(11\) 4355.96 − 5191.23i 0.297518 − 0.354568i −0.596489 − 0.802621i \(-0.703438\pi\)
0.894007 + 0.448053i \(0.147882\pi\)
\(12\) 7112.24 + 18936.3i 0.342990 + 0.913211i
\(13\) −733.507 − 4159.93i −0.0256821 − 0.145651i 0.969270 − 0.245998i \(-0.0791157\pi\)
−0.994952 + 0.100347i \(0.968005\pi\)
\(14\) −638.693 + 112.619i −0.0166257 + 0.00293156i
\(15\) −8491.39 + 51212.1i −0.167731 + 1.01160i
\(16\) 46543.3 + 39054.5i 0.710195 + 0.595924i
\(17\) −27580.9 + 15923.8i −0.330227 + 0.190656i −0.655942 − 0.754812i \(-0.727728\pi\)
0.325715 + 0.945468i \(0.394395\pi\)
\(18\) 5303.36 − 15552.8i 0.0505198 − 0.148156i
\(19\) −31761.5 + 55012.5i −0.243717 + 0.422131i −0.961770 − 0.273858i \(-0.911700\pi\)
0.718053 + 0.695989i \(0.245034\pi\)
\(20\) 54738.7 + 150393.i 0.342117 + 0.939958i
\(21\) −18271.1 − 10301.3i −0.0939481 − 0.0529683i
\(22\) −13001.5 + 10909.6i −0.0555013 + 0.0465711i
\(23\) −67576.8 + 185666.i −0.241483 + 0.663469i 0.758448 + 0.651733i \(0.225958\pi\)
−0.999931 + 0.0117355i \(0.996264\pi\)
\(24\) −18847.0 − 100849.i −0.0568063 − 0.303966i
\(25\) −3490.62 + 19796.3i −0.00893599 + 0.0506785i
\(26\) 10579.3i 0.0231507i
\(27\) 451880. − 279704.i 0.850292 − 0.526312i
\(28\) −64667.1 −0.105209
\(29\) 1.01674e6 + 179279.i 1.43753 + 0.253476i 0.837474 − 0.546478i \(-0.184032\pi\)
0.600061 + 0.799954i \(0.295143\pi\)
\(30\) 43216.0 − 122620.i 0.0533531 − 0.151382i
\(31\) −1.67239e6 − 608702.i −1.81089 − 0.659110i −0.996940 − 0.0781752i \(-0.975091\pi\)
−0.813950 − 0.580935i \(-0.802687\pi\)
\(32\) −306236. − 364958.i −0.292050 − 0.348051i
\(33\) −548882. + 5609.45i −0.462832 + 0.00473004i
\(34\) 74952.5 − 27280.5i 0.0560881 − 0.0204144i
\(35\) −143722. − 82978.2i −0.0957750 − 0.0552957i
\(36\) 790060. − 1.43540e6i 0.470381 − 0.854598i
\(37\) −711454. − 1.23227e6i −0.379612 − 0.657507i 0.611394 − 0.791327i \(-0.290609\pi\)
−0.991006 + 0.133819i \(0.957276\pi\)
\(38\) 102264. − 121873.i 0.0490442 − 0.0584486i
\(39\) −217241. + 264338.i −0.0939039 + 0.114262i
\(40\) −140957. − 799407.i −0.0550613 − 0.312268i
\(41\) 47826.5 − 8433.11i 0.0169252 − 0.00298437i −0.165179 − 0.986264i \(-0.552820\pi\)
0.182104 + 0.983279i \(0.441709\pi\)
\(42\) 40585.0 + 33354.1i 0.0130427 + 0.0107189i
\(43\) 670492. + 562610.i 0.196119 + 0.164564i 0.735559 − 0.677461i \(-0.236920\pi\)
−0.539440 + 0.842024i \(0.681364\pi\)
\(44\) −1.46559e6 + 846160.i −0.391023 + 0.225757i
\(45\) 3.59774e6 − 2.17639e6i 0.877365 − 0.530746i
\(46\) 247423. − 428549.i 0.0552597 − 0.0957125i
\(47\) 2.11724e6 + 5.81707e6i 0.433889 + 1.19210i 0.943406 + 0.331639i \(0.107602\pi\)
−0.509517 + 0.860460i \(0.670176\pi\)
\(48\) −50293.0 − 4.92114e6i −0.00947420 − 0.927045i
\(49\) −4.36473e6 + 3.66244e6i −0.757134 + 0.635311i
\(50\) 17219.0 − 47308.8i 0.00275504 − 0.00756941i
\(51\) 2.43298e6 + 857477.i 0.359631 + 0.126748i
\(52\) −183177. + 1.03885e6i −0.0250529 + 0.142082i
\(53\) 9.95902e6i 1.26216i 0.775719 + 0.631078i \(0.217387\pi\)
−0.775719 + 0.631078i \(0.782613\pi\)
\(54\) −1.23619e6 + 493355.i −0.145382 + 0.0580209i
\(55\) −4.34303e6 −0.474616
\(56\) 323005. + 56954.4i 0.0328440 + 0.00579129i
\(57\) 5.05780e6 − 945219.i 0.479140 − 0.0895433i
\(58\) −2.42979e6 − 884370.i −0.214712 − 0.0781487i
\(59\) 9.92151e6 + 1.18240e7i 0.818785 + 0.975790i 0.999971 − 0.00765928i \(-0.00243805\pi\)
−0.181186 + 0.983449i \(0.557994\pi\)
\(60\) 6.36676e6 − 1.12925e7i 0.491263 − 0.871336i
\(61\) −2.00776e7 + 7.30764e6i −1.45008 + 0.527786i −0.942613 − 0.333886i \(-0.891640\pi\)
−0.507466 + 0.861672i \(0.669418\pi\)
\(62\) 3.86018e6 + 2.22867e6i 0.261240 + 0.150827i
\(63\) 329158. + 1.66679e6i 0.0208950 + 0.105808i
\(64\) −7.18042e6 − 1.24369e7i −0.427987 − 0.741295i
\(65\) −1.74012e6 + 2.07379e6i −0.0974821 + 0.116175i
\(66\) 1.35624e6 + 224875.i 0.0714759 + 0.0118513i
\(67\) 2.15182e6 + 1.22036e7i 0.106784 + 0.605602i 0.990493 + 0.137565i \(0.0439276\pi\)
−0.883709 + 0.468037i \(0.844961\pi\)
\(68\) 7.83240e6 − 1.38106e6i 0.366319 − 0.0645919i
\(69\) 1.49822e7 − 5.62712e6i 0.660966 − 0.248250i
\(70\) 318399. + 267168.i 0.0132611 + 0.0111274i
\(71\) −2.36946e6 + 1.36801e6i −0.0932429 + 0.0538338i −0.545896 − 0.837853i \(-0.683811\pi\)
0.452654 + 0.891686i \(0.350477\pi\)
\(72\) −5.21046e6 + 6.47380e6i −0.193886 + 0.240896i
\(73\) 874977. − 1.51550e6i 0.0308109 − 0.0533661i −0.850209 − 0.526446i \(-0.823524\pi\)
0.881020 + 0.473080i \(0.156858\pi\)
\(74\) 1.21886e6 + 3.34878e6i 0.0406467 + 0.111676i
\(75\) 1.40170e6 − 828486.i 0.0443007 − 0.0261843i
\(76\) 1.21521e7 − 1.01968e7i 0.364248 − 0.305640i
\(77\) 600185. − 1.64900e6i 0.0170735 − 0.0469091i
\(78\) 650779. − 557500.i 0.0175815 − 0.0150614i
\(79\) 7.53737e6 − 4.27466e7i 0.193514 − 1.09747i −0.721006 − 0.692929i \(-0.756320\pi\)
0.914520 − 0.404542i \(-0.132569\pi\)
\(80\) − 3.89386e7i − 0.950648i
\(81\) −4.10186e7 − 1.30575e7i −0.952885 − 0.303333i
\(82\) −121630. −0.00269020
\(83\) −5.86063e7 − 1.03339e7i −1.23490 − 0.217746i −0.482172 − 0.876077i \(-0.660152\pi\)
−0.752729 + 0.658330i \(0.771263\pi\)
\(84\) 3.40778e6 + 3.97795e6i 0.0684469 + 0.0798992i
\(85\) 1.91796e7 + 6.98080e6i 0.367421 + 0.133730i
\(86\) −1.40906e6 − 1.67926e6i −0.0257595 − 0.0306990i
\(87\) −4.25512e7 − 7.19916e7i −0.742736 − 1.25662i
\(88\) 8.06570e6 − 2.93568e6i 0.134497 − 0.0489528i
\(89\) −3.32007e7 − 1.91684e7i −0.529161 − 0.305511i 0.211514 − 0.977375i \(-0.432161\pi\)
−0.740675 + 0.671864i \(0.765494\pi\)
\(90\) −9.82023e6 + 3.80331e6i −0.149676 + 0.0579685i
\(91\) −546917. − 947288.i −0.00797547 − 0.0138139i
\(92\) 3.17161e7 − 3.77978e7i 0.442719 − 0.527612i
\(93\) 5.06866e7 + 1.34953e8i 0.677581 + 1.80406i
\(94\) −2.69223e6 − 1.52684e7i −0.0344826 − 0.195561i
\(95\) 4.00921e7 − 7.06932e6i 0.492226 − 0.0867927i
\(96\) −6.31236e6 + 3.80702e7i −0.0743201 + 0.448229i
\(97\) 1.06250e8 + 8.91546e7i 1.20017 + 1.00706i 0.999625 + 0.0273787i \(0.00871601\pi\)
0.200546 + 0.979684i \(0.435728\pi\)
\(98\) 1.23583e7 − 7.13504e6i 0.133984 − 0.0773557i
\(99\) 2.92696e7 + 3.34685e7i 0.304702 + 0.348413i
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.2.12 ✓ 138
3.2 odd 2 81.9.f.a.8.12 138
27.13 even 9 81.9.f.a.71.12 138
27.14 odd 18 inner 27.9.f.a.14.12 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.12 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.12 yes 138 27.14 odd 18 inner
81.9.f.a.8.12 138 3.2 odd 2
81.9.f.a.71.12 138 27.13 even 9