Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,9,Mod(8,81)] 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("81.8"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.f (of order \(18\), degree \(6\), 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: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 8.12
Character \(\chi\) \(=\) 81.8
Dual form 81.9.f.a.71.12

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.46646 + 0.434904i) q^{2} +(-234.667 - 85.4118i) q^{4} +(411.949 + 490.942i) q^{5} +(243.334 - 88.5664i) q^{7} +(-1096.91 - 633.301i) q^{8} +(802.546 + 1390.05i) q^{10} +(-4355.96 + 5191.23i) q^{11} +(-733.507 - 4159.93i) q^{13} +(638.693 - 112.619i) q^{14} +(46543.3 + 39054.5i) q^{16} +(27580.9 - 15923.8i) q^{17} +(-31761.5 + 55012.5i) q^{19} +(-54738.7 - 150393. i) q^{20} +(-13001.5 + 10909.6i) q^{22} +(67576.8 - 185666. i) q^{23} +(-3490.62 + 19796.3i) q^{25} -10579.3i q^{26} -64667.1 q^{28} +(-1.01674e6 - 179279. i) q^{29} +(-1.67239e6 - 608702. i) q^{31} +(306236. + 364958. i) q^{32} +(74952.5 - 27280.5i) q^{34} +(143722. + 82978.2i) q^{35} +(-711454. - 1.23227e6i) q^{37} +(-102264. + 121873. i) q^{38} +(-140957. - 799407. i) q^{40} +(-47826.5 + 8433.11i) q^{41} +(670492. + 562610. i) q^{43} +(1.46559e6 - 846160. i) q^{44} +(247423. - 428549. i) q^{46} +(-2.11724e6 - 5.81707e6i) q^{47} +(-4.36473e6 + 3.66244e6i) q^{49} +(-17219.0 + 47308.8i) q^{50} +(-183177. + 1.03885e6i) q^{52} -9.95902e6i q^{53} -4.34303e6 q^{55} +(-323005. - 56954.4i) q^{56} +(-2.42979e6 - 884370. i) q^{58} +(-9.92151e6 - 1.18240e7i) q^{59} +(-2.00776e7 + 7.30764e6i) q^{61} +(-3.86018e6 - 2.22867e6i) q^{62} +(-7.18042e6 - 1.24369e7i) q^{64} +(1.74012e6 - 2.07379e6i) q^{65} +(2.15182e6 + 1.22036e7i) q^{67} +(-7.83240e6 + 1.38106e6i) q^{68} +(318399. + 267168. i) q^{70} +(2.36946e6 - 1.36801e6i) q^{71} +(874977. - 1.51550e6i) q^{73} +(-1.21886e6 - 3.34878e6i) q^{74} +(1.21521e7 - 1.01968e7i) q^{76} +(-600185. + 1.64900e6i) q^{77} +(7.53737e6 - 4.27466e7i) q^{79} +3.89386e7i q^{80} -121630. q^{82} +(5.86063e7 + 1.03339e7i) q^{83} +(1.91796e7 + 6.98080e6i) q^{85} +(1.40906e6 + 1.67926e6i) q^{86} +(8.06570e6 - 2.93568e6i) q^{88} +(3.32007e7 + 1.91684e7i) q^{89} +(-546917. - 947288. i) q^{91} +(-3.17161e7 + 3.77978e7i) q^{92} +(-2.69223e6 - 1.52684e7i) q^{94} +(-4.00921e7 + 7.06932e6i) q^{95} +(1.06250e8 + 8.91546e7i) q^{97} +(-1.23583e7 + 7.13504e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25}+ \cdots - 1293135102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

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

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 0.250193 − 0.968196i \(-0.419506\pi\)
−0.0960385 + 0.995378i \(0.530617\pi\)
\(3\) 0 0
\(4\) −234.667 − 85.4118i −0.916668 − 0.333640i
\(5\) 411.949 + 490.942i 0.659119 + 0.785508i 0.987259 − 0.159122i \(-0.0508662\pi\)
−0.328140 + 0.944629i \(0.606422\pi\)
\(6\) 0 0
\(7\) 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
\(9\) 0 0
\(10\) 802.546 + 1390.05i 0.0802546 + 0.139005i
\(11\) −4355.96 + 5191.23i −0.297518 + 0.354568i −0.894007 − 0.448053i \(-0.852118\pi\)
0.596489 + 0.802621i \(0.296562\pi\)
\(12\) 0 0
\(13\) −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
\(15\) 0 0
\(16\) 46543.3 + 39054.5i 0.710195 + 0.595924i
\(17\) 27580.9 − 15923.8i 0.330227 − 0.190656i −0.325715 − 0.945468i \(-0.605605\pi\)
0.655942 + 0.754812i \(0.272272\pi\)
\(18\) 0 0
\(19\) −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
\(21\) 0 0
\(22\) −13001.5 + 10909.6i −0.0555013 + 0.0465711i
\(23\) 67576.8 − 185666.i 0.241483 − 0.663469i −0.758448 − 0.651733i \(-0.774042\pi\)
0.999931 − 0.0117355i \(-0.00373562\pi\)
\(24\) 0 0
\(25\) −3490.62 + 19796.3i −0.00893599 + 0.0506785i
\(26\) − 10579.3i − 0.0231507i
\(27\) 0 0
\(28\) −64667.1 −0.105209
\(29\) −1.01674e6 − 179279.i −1.43753 − 0.253476i −0.600061 − 0.799954i \(-0.704857\pi\)
−0.837474 + 0.546478i \(0.815968\pi\)
\(30\) 0 0
\(31\) −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
\(33\) 0 0
\(34\) 74952.5 − 27280.5i 0.0560881 − 0.0204144i
\(35\) 143722. + 82978.2i 0.0957750 + 0.0552957i
\(36\) 0 0
\(37\) −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
\(39\) 0 0
\(40\) −140957. − 799407.i −0.0550613 − 0.312268i
\(41\) −47826.5 + 8433.11i −0.0169252 + 0.00298437i −0.182104 − 0.983279i \(-0.558291\pi\)
0.165179 + 0.986264i \(0.447180\pi\)
\(42\) 0 0
\(43\) 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
\(45\) 0 0
\(46\) 247423. − 428549.i 0.0552597 − 0.0957125i
\(47\) −2.11724e6 − 5.81707e6i −0.433889 − 1.19210i −0.943406 − 0.331639i \(-0.892398\pi\)
0.509517 − 0.860460i \(-0.329824\pi\)
\(48\) 0 0
\(49\) −4.36473e6 + 3.66244e6i −0.757134 + 0.635311i
\(50\) −17219.0 + 47308.8i −0.00275504 + 0.00756941i
\(51\) 0 0
\(52\) −183177. + 1.03885e6i −0.0250529 + 0.142082i
\(53\) − 9.95902e6i − 1.26216i −0.775719 − 0.631078i \(-0.782613\pi\)
0.775719 − 0.631078i \(-0.217387\pi\)
\(54\) 0 0
\(55\) −4.34303e6 −0.474616
\(56\) −323005. − 56954.4i −0.0328440 − 0.00579129i
\(57\) 0 0
\(58\) −2.42979e6 − 884370.i −0.214712 − 0.0781487i
\(59\) −9.92151e6 − 1.18240e7i −0.818785 − 0.975790i 0.181186 − 0.983449i \(-0.442006\pi\)
−0.999971 + 0.00765928i \(0.997562\pi\)
\(60\) 0 0
\(61\) −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
\(63\) 0 0
\(64\) −7.18042e6 − 1.24369e7i −0.427987 − 0.741295i
\(65\) 1.74012e6 − 2.07379e6i 0.0974821 − 0.116175i
\(66\) 0 0
\(67\) 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
\(69\) 0 0
\(70\) 318399. + 267168.i 0.0132611 + 0.0111274i
\(71\) 2.36946e6 − 1.36801e6i 0.0932429 − 0.0538338i −0.452654 − 0.891686i \(-0.649523\pi\)
0.545896 + 0.837853i \(0.316189\pi\)
\(72\) 0 0
\(73\) 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
\(75\) 0 0
\(76\) 1.21521e7 − 1.01968e7i 0.364248 − 0.305640i
\(77\) −600185. + 1.64900e6i −0.0170735 + 0.0469091i
\(78\) 0 0
\(79\) 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
\(81\) 0 0
\(82\) −121630. −0.00269020
\(83\) 5.86063e7 + 1.03339e7i 1.23490 + 0.217746i 0.752729 − 0.658330i \(-0.228737\pi\)
0.482172 + 0.876077i \(0.339848\pi\)
\(84\) 0 0
\(85\) 1.91796e7 + 6.98080e6i 0.367421 + 0.133730i
\(86\) 1.40906e6 + 1.67926e6i 0.0257595 + 0.0306990i
\(87\) 0 0
\(88\) 8.06570e6 − 2.93568e6i 0.134497 − 0.0489528i
\(89\) 3.32007e7 + 1.91684e7i 0.529161 + 0.305511i 0.740675 − 0.671864i \(-0.234506\pi\)
−0.211514 + 0.977375i \(0.567839\pi\)
\(90\) 0 0
\(91\) −546917. − 947288.i −0.00797547 − 0.0138139i
\(92\) −3.17161e7 + 3.77978e7i −0.442719 + 0.527612i
\(93\) 0 0
\(94\) −2.69223e6 − 1.52684e7i −0.0344826 − 0.195561i
\(95\) −4.00921e7 + 7.06932e6i −0.492226 + 0.0867927i
\(96\) 0 0
\(97\) 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
\(99\) 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 81.9.f.a.8.12 138
3.2 odd 2 27.9.f.a.2.12 ✓ 138
27.13 even 9 27.9.f.a.14.12 yes 138
27.14 odd 18 inner 81.9.f.a.71.12 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.12 ✓ 138 3.2 odd 2
27.9.f.a.14.12 yes 138 27.13 even 9
81.9.f.a.8.12 138 1.1 even 1 trivial
81.9.f.a.71.12 138 27.14 odd 18 inner