Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [216,5,Mod(125,216)] 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("216.125"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(216, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 216.j (of order \(6\), degree \(2\), 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: \(22.3279120261\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(46\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 125.20
Character \(\chi\) \(=\) 216.125
Dual form 216.5.j.a.197.20

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.02310 - 3.86695i) q^{2} +(-13.9065 + 7.91255i) q^{4} +(-21.1127 + 36.5683i) q^{5} +(-16.9047 - 29.2798i) q^{7} +(44.8252 + 45.6804i) q^{8} +(163.008 + 44.2286i) q^{10} +(28.4273 + 49.2375i) q^{11} +(79.2879 + 45.7769i) q^{13} +(-95.9280 + 95.3256i) q^{14} +(130.783 - 220.072i) q^{16} -122.909i q^{17} +551.287i q^{19} +(4.25597 - 675.594i) q^{20} +(161.315 - 160.302i) q^{22} +(-595.912 - 344.050i) q^{23} +(-578.994 - 1002.85i) q^{25} +(95.8971 - 353.436i) q^{26} +(466.763 + 273.421i) q^{28} +(-665.451 - 1152.59i) q^{29} +(305.568 - 529.260i) q^{31} +(-984.812 - 280.574i) q^{32} +(-475.281 + 125.748i) q^{34} +1427.62 q^{35} -1555.96i q^{37} +(2131.80 - 564.023i) q^{38} +(-2616.84 + 674.743i) q^{40} +(978.349 + 564.850i) q^{41} +(-2635.89 + 1521.83i) q^{43} +(-784.919 - 459.790i) q^{44} +(-720.744 + 2656.36i) q^{46} +(1412.72 - 815.637i) q^{47} +(628.964 - 1089.40i) q^{49} +(-3285.58 + 3264.95i) q^{50} +(-1464.83 - 9.22786i) q^{52} -2783.44 q^{53} -2400.71 q^{55} +(579.757 - 2084.68i) q^{56} +(-3776.20 + 3752.48i) q^{58} +(1220.18 - 2113.42i) q^{59} +(4583.29 - 2646.16i) q^{61} +(-2359.25 - 640.130i) q^{62} +(-77.4039 + 4095.27i) q^{64} +(-3347.97 + 1932.95i) q^{65} +(3539.31 + 2043.42i) q^{67} +(972.521 + 1709.23i) q^{68} +(-1460.59 - 5520.51i) q^{70} -1182.69i q^{71} +585.969 q^{73} +(-6016.83 + 1591.91i) q^{74} +(-4362.09 - 7666.49i) q^{76} +(961.108 - 1664.69i) q^{77} +(-1960.12 - 3395.02i) q^{79} +(5286.48 + 9428.84i) q^{80} +(1183.29 - 4361.12i) q^{82} +(-5665.44 - 9812.83i) q^{83} +(4494.56 + 2594.94i) q^{85} +(8581.63 + 8635.86i) q^{86} +(-974.932 + 3505.65i) q^{88} -6300.52i q^{89} -3095.37i q^{91} +(11009.4 + 69.3548i) q^{92} +(-4599.38 - 4628.45i) q^{94} +(-20159.6 - 11639.2i) q^{95} +(373.253 + 646.493i) q^{97} +(-4856.13 - 1317.60i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 92 q + 3 q^{2} - q^{4} - 2 q^{7} + 28 q^{10} - 852 q^{14} - q^{16} - 1950 q^{20} + 31 q^{22} + 6 q^{23} - 4752 q^{25} + 508 q^{28} - 2 q^{31} - 4947 q^{32} - 387 q^{34} + 5985 q^{38} + 1024 q^{40} + 3318 q^{41}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.02310 − 3.86695i −0.255775 − 0.966736i
\(3\) 0 0
\(4\) −13.9065 + 7.91255i −0.869158 + 0.494535i
\(5\) −21.1127 + 36.5683i −0.844509 + 1.46273i 0.0415382 + 0.999137i \(0.486774\pi\)
−0.886047 + 0.463595i \(0.846559\pi\)
\(6\) 0 0
\(7\) −16.9047 − 29.2798i −0.344993 − 0.597546i 0.640359 − 0.768076i \(-0.278786\pi\)
−0.985353 + 0.170529i \(0.945452\pi\)
\(8\) 44.8252 + 45.6804i 0.700394 + 0.713757i
\(9\) 0 0
\(10\) 163.008 + 44.2286i 1.63008 + 0.442286i
\(11\) 28.4273 + 49.2375i 0.234936 + 0.406922i 0.959254 − 0.282545i \(-0.0911785\pi\)
−0.724318 + 0.689466i \(0.757845\pi\)
\(12\) 0 0
\(13\) 79.2879 + 45.7769i 0.469159 + 0.270869i 0.715888 − 0.698216i \(-0.246022\pi\)
−0.246729 + 0.969085i \(0.579356\pi\)
\(14\) −95.9280 + 95.3256i −0.489429 + 0.486355i
\(15\) 0 0
\(16\) 130.783 − 220.072i 0.510871 − 0.859657i
\(17\) − 122.909i − 0.425289i −0.977130 − 0.212645i \(-0.931792\pi\)
0.977130 − 0.212645i \(-0.0682077\pi\)
\(18\) 0 0
\(19\) 551.287i 1.52711i 0.645742 + 0.763556i \(0.276548\pi\)
−0.645742 + 0.763556i \(0.723452\pi\)
\(20\) 4.25597 − 675.594i 0.0106399 − 1.68898i
\(21\) 0 0
\(22\) 161.315 − 160.302i 0.333295 − 0.331202i
\(23\) −595.912 − 344.050i −1.12649 − 0.650378i −0.183439 − 0.983031i \(-0.558723\pi\)
−0.943049 + 0.332653i \(0.892056\pi\)
\(24\) 0 0
\(25\) −578.994 − 1002.85i −0.926390 − 1.60456i
\(26\) 95.8971 − 353.436i 0.141860 − 0.522835i
\(27\) 0 0
\(28\) 466.763 + 273.421i 0.595361 + 0.348751i
\(29\) −665.451 − 1152.59i −0.791261 − 1.37050i −0.925186 − 0.379513i \(-0.876091\pi\)
0.133925 − 0.990991i \(-0.457242\pi\)
\(30\) 0 0
\(31\) 305.568 − 529.260i 0.317969 − 0.550739i −0.662095 − 0.749420i \(-0.730332\pi\)
0.980064 + 0.198681i \(0.0636658\pi\)
\(32\) −984.812 − 280.574i −0.961730 − 0.273999i
\(33\) 0 0
\(34\) −475.281 + 125.748i −0.411143 + 0.108779i
\(35\) 1427.62 1.16540
\(36\) 0 0
\(37\) − 1555.96i − 1.13657i −0.822832 − 0.568285i \(-0.807607\pi\)
0.822832 − 0.568285i \(-0.192393\pi\)
\(38\) 2131.80 − 564.023i 1.47631 − 0.390598i
\(39\) 0 0
\(40\) −2616.84 + 674.743i −1.63552 + 0.421714i
\(41\) 978.349 + 564.850i 0.582004 + 0.336020i 0.761929 − 0.647660i \(-0.224252\pi\)
−0.179925 + 0.983680i \(0.557586\pi\)
\(42\) 0 0
\(43\) −2635.89 + 1521.83i −1.42558 + 0.823057i −0.996768 − 0.0803375i \(-0.974400\pi\)
−0.428810 + 0.903395i \(0.641067\pi\)
\(44\) −784.919 − 459.790i −0.405433 − 0.237495i
\(45\) 0 0
\(46\) −720.744 + 2656.36i −0.340616 + 1.25537i
\(47\) 1412.72 − 815.637i 0.639531 − 0.369233i −0.144903 − 0.989446i \(-0.546287\pi\)
0.784434 + 0.620212i \(0.212954\pi\)
\(48\) 0 0
\(49\) 628.964 − 1089.40i 0.261959 − 0.453726i
\(50\) −3285.58 + 3264.95i −1.31423 + 1.30598i
\(51\) 0 0
\(52\) −1464.83 − 9.22786i −0.541727 − 0.00341267i
\(53\) −2783.44 −0.990901 −0.495451 − 0.868636i \(-0.664997\pi\)
−0.495451 + 0.868636i \(0.664997\pi\)
\(54\) 0 0
\(55\) −2400.71 −0.793623
\(56\) 579.757 − 2084.68i 0.184871 − 0.664759i
\(57\) 0 0
\(58\) −3776.20 + 3752.48i −1.12253 + 1.11548i
\(59\) 1220.18 − 2113.42i 0.350526 − 0.607129i −0.635816 − 0.771841i \(-0.719336\pi\)
0.986342 + 0.164712i \(0.0526695\pi\)
\(60\) 0 0
\(61\) 4583.29 − 2646.16i 1.23174 − 0.711143i 0.264344 − 0.964429i \(-0.414845\pi\)
0.967391 + 0.253286i \(0.0815113\pi\)
\(62\) −2359.25 − 640.130i −0.613748 − 0.166527i
\(63\) 0 0
\(64\) −77.4039 + 4095.27i −0.0188974 + 0.999821i
\(65\) −3347.97 + 1932.95i −0.792418 + 0.457503i
\(66\) 0 0
\(67\) 3539.31 + 2043.42i 0.788441 + 0.455207i 0.839413 − 0.543493i \(-0.182899\pi\)
−0.0509724 + 0.998700i \(0.516232\pi\)
\(68\) 972.521 + 1709.23i 0.210320 + 0.369644i
\(69\) 0 0
\(70\) −1460.59 − 5520.51i −0.298081 − 1.12663i
\(71\) − 1182.69i − 0.234615i −0.993096 − 0.117307i \(-0.962574\pi\)
0.993096 − 0.117307i \(-0.0374262\pi\)
\(72\) 0 0
\(73\) 585.969 0.109959 0.0549793 − 0.998487i \(-0.482491\pi\)
0.0549793 + 0.998487i \(0.482491\pi\)
\(74\) −6016.83 + 1591.91i −1.09876 + 0.290706i
\(75\) 0 0
\(76\) −4362.09 − 7666.49i −0.755210 − 1.32730i
\(77\) 961.108 − 1664.69i 0.162103 − 0.280771i
\(78\) 0 0
\(79\) −1960.12 − 3395.02i −0.314071 − 0.543986i 0.665169 − 0.746693i \(-0.268360\pi\)
−0.979240 + 0.202707i \(0.935026\pi\)
\(80\) 5286.48 + 9428.84i 0.826013 + 1.47326i
\(81\) 0 0
\(82\) 1183.29 − 4361.12i 0.175981 − 0.648590i
\(83\) −5665.44 − 9812.83i −0.822390 − 1.42442i −0.903898 − 0.427748i \(-0.859307\pi\)
0.0815082 − 0.996673i \(-0.474026\pi\)
\(84\) 0 0
\(85\) 4494.56 + 2594.94i 0.622085 + 0.359161i
\(86\) 8581.63 + 8635.86i 1.16031 + 1.16764i
\(87\) 0 0
\(88\) −974.932 + 3505.65i −0.125895 + 0.452693i
\(89\) − 6300.52i − 0.795419i −0.917511 − 0.397710i \(-0.869805\pi\)
0.917511 − 0.397710i \(-0.130195\pi\)
\(90\) 0 0
\(91\) − 3095.37i − 0.373792i
\(92\) 11009.4 + 69.3548i 1.30073 + 0.00819409i
\(93\) 0 0
\(94\) −4599.38 − 4628.45i −0.520528 − 0.523817i
\(95\) −20159.6 − 11639.2i −2.23376 − 1.28966i
\(96\) 0 0
\(97\) 373.253 + 646.493i 0.0396697 + 0.0687100i 0.885179 − 0.465251i \(-0.154036\pi\)
−0.845509 + 0.533961i \(0.820703\pi\)
\(98\) −4856.13 − 1317.60i −0.505636 − 0.137193i
\(99\) 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 216.5.j.a.125.20 92
3.2 odd 2 72.5.j.a.5.27 yes 92
4.3 odd 2 864.5.n.a.17.2 92
8.3 odd 2 864.5.n.a.17.45 92
8.5 even 2 inner 216.5.j.a.125.35 92
9.2 odd 6 inner 216.5.j.a.197.35 92
9.7 even 3 72.5.j.a.29.12 yes 92
12.11 even 2 288.5.n.a.113.11 92
24.5 odd 2 72.5.j.a.5.12 ✓ 92
24.11 even 2 288.5.n.a.113.36 92
36.7 odd 6 288.5.n.a.209.36 92
36.11 even 6 864.5.n.a.305.45 92
72.11 even 6 864.5.n.a.305.2 92
72.29 odd 6 inner 216.5.j.a.197.20 92
72.43 odd 6 288.5.n.a.209.11 92
72.61 even 6 72.5.j.a.29.27 yes 92
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.5.j.a.5.12 ✓ 92 24.5 odd 2
72.5.j.a.5.27 yes 92 3.2 odd 2
72.5.j.a.29.12 yes 92 9.7 even 3
72.5.j.a.29.27 yes 92 72.61 even 6
216.5.j.a.125.20 92 1.1 even 1 trivial
216.5.j.a.125.35 92 8.5 even 2 inner
216.5.j.a.197.20 92 72.29 odd 6 inner
216.5.j.a.197.35 92 9.2 odd 6 inner
288.5.n.a.113.11 92 12.11 even 2
288.5.n.a.113.36 92 24.11 even 2
288.5.n.a.209.11 92 72.43 odd 6
288.5.n.a.209.36 92 36.7 odd 6
864.5.n.a.17.2 92 4.3 odd 2
864.5.n.a.17.45 92 8.3 odd 2
864.5.n.a.305.2 92 72.11 even 6
864.5.n.a.305.45 92 36.11 even 6