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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,5,Mod(79,288)] 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("288.79"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 288.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.7705493681\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 79.2
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 288.79
Dual form 288.5.t.a.175.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.39898 + 8.89060i) q^{3} +(-77.0857 + 24.8755i) q^{9} +(114.379 - 198.109i) q^{11} +228.212 q^{17} +716.696 q^{19} +(-312.500 + 541.266i) q^{25} +(-329.000 - 650.538i) q^{27} +(1921.33 + 739.744i) q^{33} +(1040.62 + 1802.40i) q^{41} +(361.107 - 625.456i) q^{43} +(-1200.50 - 2079.33i) q^{49} +(319.264 + 2028.94i) q^{51} +(1002.64 + 6371.86i) q^{57} +(3072.37 + 5321.51i) q^{59} +(4472.74 + 7747.01i) q^{67} -579.069 q^{73} +(-5249.36 - 2021.09i) q^{75} +(5323.41 - 3835.10i) q^{81} +(5593.00 - 9687.36i) q^{83} +5474.00 q^{89} +(4412.06 - 7641.91i) q^{97} +(-3888.87 + 18116.6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} - 34 q^{9} + 46 q^{11} + 1148 q^{17} + 868 q^{19} - 1250 q^{25} - 1316 q^{27} + 4354 q^{33} - 1246 q^{41} + 3502 q^{43} - 4802 q^{49} - 5170 q^{51} + 6754 q^{57} + 238 q^{59} + 5134 q^{67}+ \cdots - 1564 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\)

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
\(3\) 1.39898 + 8.89060i 0.155442 + 0.987845i
\(4\) 0 0
\(5\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0 0
\(7\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 0 0
\(9\) −77.0857 + 24.8755i −0.951675 + 0.307106i
\(10\) 0 0
\(11\) 114.379 − 198.109i 0.945277 − 1.63727i 0.190083 − 0.981768i \(-0.439124\pi\)
0.755195 − 0.655500i \(-0.227542\pi\)
\(12\) 0 0
\(13\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 228.212 0.789662 0.394831 − 0.918754i \(-0.370803\pi\)
0.394831 + 0.918754i \(0.370803\pi\)
\(18\) 0 0
\(19\) 716.696 1.98531 0.992654 − 0.120991i \(-0.0386072\pi\)
0.992654 + 0.120991i \(0.0386072\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −312.500 + 541.266i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) −329.000 − 650.538i −0.451303 − 0.892371i
\(28\) 0 0
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0 0
\(33\) 1921.33 + 739.744i 1.76430 + 0.679287i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1040.62 + 1802.40i 0.619047 + 1.07222i 0.989660 + 0.143434i \(0.0458144\pi\)
−0.370613 + 0.928787i \(0.620852\pi\)
\(42\) 0 0
\(43\) 361.107 − 625.456i 0.195299 − 0.338267i −0.751700 − 0.659505i \(-0.770766\pi\)
0.946998 + 0.321238i \(0.104099\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) −1200.50 − 2079.33i −0.500000 − 0.866025i
\(50\) 0 0
\(51\) 319.264 + 2028.94i 0.122747 + 0.780063i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1002.64 + 6371.86i 0.308600 + 1.96118i
\(58\) 0 0
\(59\) 3072.37 + 5321.51i 0.882612 + 1.52873i 0.848426 + 0.529313i \(0.177550\pi\)
0.0341856 + 0.999416i \(0.489116\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4472.74 + 7747.01i 0.996377 + 1.72578i 0.571842 + 0.820364i \(0.306229\pi\)
0.424535 + 0.905412i \(0.360438\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −579.069 −0.108664 −0.0543319 − 0.998523i \(-0.517303\pi\)
−0.0543319 + 0.998523i \(0.517303\pi\)
\(74\) 0 0
\(75\) −5249.36 − 2021.09i −0.933220 − 0.359306i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) 5323.41 − 3835.10i 0.811372 − 0.584530i
\(82\) 0 0
\(83\) 5593.00 − 9687.36i 0.811874 − 1.40621i −0.0996769 − 0.995020i \(-0.531781\pi\)
0.911551 − 0.411187i \(-0.134886\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5474.00 0.691074 0.345537 − 0.938405i \(-0.387697\pi\)
0.345537 + 0.938405i \(0.387697\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4412.06 − 7641.91i 0.468919 − 0.812192i −0.530450 − 0.847716i \(-0.677977\pi\)
0.999369 + 0.0355246i \(0.0113102\pi\)
\(98\) 0 0
\(99\) −3888.87 + 18116.6i −0.396783 + 1.84845i
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 288.5.t.a.79.2 4
3.2 odd 2 864.5.t.a.559.1 4
4.3 odd 2 72.5.p.a.43.1 ✓ 4
8.3 odd 2 CM 288.5.t.a.79.2 4
8.5 even 2 72.5.p.a.43.1 ✓ 4
9.4 even 3 inner 288.5.t.a.175.2 4
9.5 odd 6 864.5.t.a.847.1 4
12.11 even 2 216.5.p.a.19.2 4
24.5 odd 2 216.5.p.a.19.2 4
24.11 even 2 864.5.t.a.559.1 4
36.23 even 6 216.5.p.a.91.2 4
36.31 odd 6 72.5.p.a.67.1 yes 4
72.5 odd 6 216.5.p.a.91.2 4
72.13 even 6 72.5.p.a.67.1 yes 4
72.59 even 6 864.5.t.a.847.1 4
72.67 odd 6 inner 288.5.t.a.175.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.5.p.a.43.1 ✓ 4 4.3 odd 2
72.5.p.a.43.1 ✓ 4 8.5 even 2
72.5.p.a.67.1 yes 4 36.31 odd 6
72.5.p.a.67.1 yes 4 72.13 even 6
216.5.p.a.19.2 4 12.11 even 2
216.5.p.a.19.2 4 24.5 odd 2
216.5.p.a.91.2 4 36.23 even 6
216.5.p.a.91.2 4 72.5 odd 6
288.5.t.a.79.2 4 1.1 even 1 trivial
288.5.t.a.79.2 4 8.3 odd 2 CM
288.5.t.a.175.2 4 9.4 even 3 inner
288.5.t.a.175.2 4 72.67 odd 6 inner
864.5.t.a.559.1 4 3.2 odd 2
864.5.t.a.559.1 4 24.11 even 2
864.5.t.a.847.1 4 9.5 odd 6
864.5.t.a.847.1 4 72.59 even 6