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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,7,Mod(17,288)] 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("288.17"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 288.h (of order \(2\), degree \(1\), 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: \(66.2555760825\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.20
Character \(\chi\) \(=\) 288.17
Dual form 288.7.h.a.17.19

$q$-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)
 
\(f(q)\) \(=\) \(q+149.398 q^{5} -498.755 q^{7} +394.875 q^{11} -3962.14i q^{13} +6929.82i q^{17} -1077.01i q^{19} +10605.7i q^{23} +6694.74 q^{25} +13037.3 q^{29} -53775.2 q^{31} -74512.9 q^{35} +40996.0i q^{37} -71003.1i q^{41} +70391.7i q^{43} +13021.7i q^{47} +131107. q^{49} +27017.7 q^{53} +58993.6 q^{55} +40464.9 q^{59} +277921. i q^{61} -591936. i q^{65} -339817. i q^{67} +456044. i q^{71} -303469. q^{73} -196946. q^{77} -471240. q^{79} -1.09854e6 q^{83} +1.03530e6i q^{85} +853707. i q^{89} +1.97614e6i q^{91} -160903. i q^{95} -35614.1 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 75000 q^{25} - 122496 q^{31} + 197448 q^{49} + 232704 q^{55} + 514080 q^{73} + 1193088 q^{79} + 2960832 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 149.398 1.19518 0.597592 − 0.801801i \(-0.296124\pi\)
0.597592 + 0.801801i \(0.296124\pi\)
\(6\) 0 0
\(7\) −498.755 −1.45409 −0.727047 − 0.686587i \(-0.759108\pi\)
−0.727047 + 0.686587i \(0.759108\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 394.875 0.296676 0.148338 − 0.988937i \(-0.452608\pi\)
0.148338 + 0.988937i \(0.452608\pi\)
\(12\) 0 0
\(13\) − 3962.14i − 1.80343i −0.432327 − 0.901717i \(-0.642308\pi\)
0.432327 − 0.901717i \(-0.357692\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6929.82i 1.41051i 0.708955 + 0.705254i \(0.249167\pi\)
−0.708955 + 0.705254i \(0.750833\pi\)
\(18\) 0 0
\(19\) − 1077.01i − 0.157021i −0.996913 − 0.0785105i \(-0.974984\pi\)
0.996913 − 0.0785105i \(-0.0250164\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 10605.7i 0.871675i 0.900025 + 0.435837i \(0.143548\pi\)
−0.900025 + 0.435837i \(0.856452\pi\)
\(24\) 0 0
\(25\) 6694.74 0.428463
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 13037.3 0.534555 0.267277 − 0.963620i \(-0.413876\pi\)
0.267277 + 0.963620i \(0.413876\pi\)
\(30\) 0 0
\(31\) −53775.2 −1.80508 −0.902541 − 0.430604i \(-0.858301\pi\)
−0.902541 + 0.430604i \(0.858301\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −74512.9 −1.73791
\(36\) 0 0
\(37\) 40996.0i 0.809351i 0.914460 + 0.404675i \(0.132615\pi\)
−0.914460 + 0.404675i \(0.867385\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 71003.1i − 1.03021i −0.857127 − 0.515105i \(-0.827753\pi\)
0.857127 − 0.515105i \(-0.172247\pi\)
\(42\) 0 0
\(43\) 70391.7i 0.885352i 0.896682 + 0.442676i \(0.145971\pi\)
−0.896682 + 0.442676i \(0.854029\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13021.7i 0.125422i 0.998032 + 0.0627109i \(0.0199746\pi\)
−0.998032 + 0.0627109i \(0.980025\pi\)
\(48\) 0 0
\(49\) 131107. 1.11439
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 27017.7 0.181477 0.0907383 − 0.995875i \(-0.471077\pi\)
0.0907383 + 0.995875i \(0.471077\pi\)
\(54\) 0 0
\(55\) 58993.6 0.354582
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 40464.9 0.197026 0.0985128 − 0.995136i \(-0.468591\pi\)
0.0985128 + 0.995136i \(0.468591\pi\)
\(60\) 0 0
\(61\) 277921.i 1.22442i 0.790694 + 0.612212i \(0.209720\pi\)
−0.790694 + 0.612212i \(0.790280\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 591936.i − 2.15543i
\(66\) 0 0
\(67\) − 339817.i − 1.12985i −0.825143 − 0.564924i \(-0.808905\pi\)
0.825143 − 0.564924i \(-0.191095\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 456044.i 1.27418i 0.770788 + 0.637091i \(0.219863\pi\)
−0.770788 + 0.637091i \(0.780137\pi\)
\(72\) 0 0
\(73\) −303469. −0.780091 −0.390045 − 0.920796i \(-0.627541\pi\)
−0.390045 + 0.920796i \(0.627541\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −196946. −0.431395
\(78\) 0 0
\(79\) −471240. −0.955786 −0.477893 − 0.878418i \(-0.658599\pi\)
−0.477893 + 0.878418i \(0.658599\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.09854e6 −1.92124 −0.960620 − 0.277866i \(-0.910373\pi\)
−0.960620 + 0.277866i \(0.910373\pi\)
\(84\) 0 0
\(85\) 1.03530e6i 1.68581i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 853707.i 1.21099i 0.795851 + 0.605493i \(0.207024\pi\)
−0.795851 + 0.605493i \(0.792976\pi\)
\(90\) 0 0
\(91\) 1.97614e6i 2.62236i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 160903.i − 0.187669i
\(96\) 0 0
\(97\) −35614.1 −0.0390218 −0.0195109 − 0.999810i \(-0.506211\pi\)
−0.0195109 + 0.999810i \(0.506211\pi\)
\(98\) 0 0
\(99\) 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 288.7.h.a.17.20 24
3.2 odd 2 inner 288.7.h.a.17.6 24
4.3 odd 2 72.7.h.a.53.2 yes 24
8.3 odd 2 72.7.h.a.53.24 yes 24
8.5 even 2 inner 288.7.h.a.17.5 24
12.11 even 2 72.7.h.a.53.23 yes 24
24.5 odd 2 inner 288.7.h.a.17.19 24
24.11 even 2 72.7.h.a.53.1 ✓ 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.7.h.a.53.1 ✓ 24 24.11 even 2
72.7.h.a.53.2 yes 24 4.3 odd 2
72.7.h.a.53.23 yes 24 12.11 even 2
72.7.h.a.53.24 yes 24 8.3 odd 2
288.7.h.a.17.5 24 8.5 even 2 inner
288.7.h.a.17.6 24 3.2 odd 2 inner
288.7.h.a.17.19 24 24.5 odd 2 inner
288.7.h.a.17.20 24 1.1 even 1 trivial