Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [576,2,Mod(97,576)] 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("576.97"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 481.1
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 576.481
Dual form 576.2.r.c.97.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.57313 - 0.724745i) q^{3} +(1.94949 + 2.28024i) q^{9} +(-0.476756 + 0.275255i) q^{11} +7.89898 q^{17} -6.34847i q^{19} +(-2.50000 - 4.33013i) q^{25} +(-1.41421 - 5.00000i) q^{27} +(0.949490 - 0.0874863i) q^{33} +(3.39898 - 5.88721i) q^{41} +(10.6941 - 6.17423i) q^{43} +(3.50000 - 6.06218i) q^{49} +(-12.4261 - 5.72474i) q^{51} +(-4.60102 + 9.98698i) q^{57} +(13.2047 + 7.62372i) q^{59} +(-0.301783 - 0.174235i) q^{67} -15.6969 q^{73} +(0.794593 + 8.62372i) q^{75} +(-1.39898 + 8.89060i) q^{81} +(-15.5885 + 9.00000i) q^{83} +18.0000 q^{89} +(-4.84847 - 8.39780i) q^{97} +(-1.55708 - 0.550510i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9} + 24 q^{17} - 20 q^{25} - 12 q^{33} - 12 q^{41} + 28 q^{49} - 76 q^{57} - 8 q^{73} + 28 q^{81} + 144 q^{89} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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
\(3\) −1.57313 − 0.724745i −0.908248 − 0.418432i
\(4\) 0 0
\(5\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) 1.94949 + 2.28024i 0.649830 + 0.760080i
\(10\) 0 0
\(11\) −0.476756 + 0.275255i −0.143747 + 0.0829925i −0.570149 − 0.821541i \(-0.693114\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.89898 1.91578 0.957892 − 0.287129i \(-0.0927008\pi\)
0.957892 + 0.287129i \(0.0927008\pi\)
\(18\) 0 0
\(19\) − 6.34847i − 1.45644i −0.685344 − 0.728219i \(-0.740348\pi\)
0.685344 − 0.728219i \(-0.259652\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −2.50000 − 4.33013i −0.500000 − 0.866025i
\(26\) 0 0
\(27\) −1.41421 − 5.00000i −0.272166 − 0.962250i
\(28\) 0 0
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0 0
\(33\) 0.949490 − 0.0874863i 0.165285 − 0.0152294i
\(34\) 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
\(41\) 3.39898 − 5.88721i 0.530831 − 0.919427i −0.468521 − 0.883452i \(-0.655213\pi\)
0.999353 − 0.0359748i \(-0.0114536\pi\)
\(42\) 0 0
\(43\) 10.6941 − 6.17423i 1.63083 − 0.941562i 0.646997 − 0.762493i \(-0.276025\pi\)
0.983836 − 0.179069i \(-0.0573086\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 3.50000 − 6.06218i 0.500000 − 0.866025i
\(50\) 0 0
\(51\) −12.4261 − 5.72474i −1.74001 − 0.801625i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.60102 + 9.98698i −0.609420 + 1.32281i
\(58\) 0 0
\(59\) 13.2047 + 7.62372i 1.71910 + 0.992524i 0.920575 + 0.390567i \(0.127721\pi\)
0.798528 + 0.601958i \(0.205612\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.301783 − 0.174235i −0.0368687 − 0.0212861i 0.481452 − 0.876472i \(-0.340109\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −15.6969 −1.83719 −0.918594 − 0.395203i \(-0.870674\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) 0.794593 + 8.62372i 0.0917517 + 0.995782i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0 0
\(81\) −1.39898 + 8.89060i −0.155442 + 0.987845i
\(82\) 0 0
\(83\) −15.5885 + 9.00000i −1.71106 + 0.987878i −0.777913 + 0.628372i \(0.783721\pi\)
−0.933143 + 0.359506i \(0.882945\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.84847 − 8.39780i −0.492287 − 0.852667i 0.507673 − 0.861550i \(-0.330506\pi\)
−0.999961 + 0.00888289i \(0.997172\pi\)
\(98\) 0 0
\(99\) −1.55708 − 0.550510i −0.156492 − 0.0553284i
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 576.2.r.c.481.1 yes 8
3.2 odd 2 1728.2.r.c.1441.3 8
4.3 odd 2 inner 576.2.r.c.481.4 yes 8
8.3 odd 2 CM 576.2.r.c.481.1 yes 8
8.5 even 2 inner 576.2.r.c.481.4 yes 8
9.2 odd 6 1728.2.r.c.289.2 8
9.4 even 3 5184.2.d.l.2593.2 4
9.5 odd 6 5184.2.d.e.2593.3 4
9.7 even 3 inner 576.2.r.c.97.4 yes 8
12.11 even 2 1728.2.r.c.1441.2 8
24.5 odd 2 1728.2.r.c.1441.2 8
24.11 even 2 1728.2.r.c.1441.3 8
36.7 odd 6 inner 576.2.r.c.97.1 ✓ 8
36.11 even 6 1728.2.r.c.289.3 8
36.23 even 6 5184.2.d.e.2593.2 4
36.31 odd 6 5184.2.d.l.2593.3 4
72.5 odd 6 5184.2.d.e.2593.2 4
72.11 even 6 1728.2.r.c.289.2 8
72.13 even 6 5184.2.d.l.2593.3 4
72.29 odd 6 1728.2.r.c.289.3 8
72.43 odd 6 inner 576.2.r.c.97.4 yes 8
72.59 even 6 5184.2.d.e.2593.3 4
72.61 even 6 inner 576.2.r.c.97.1 ✓ 8
72.67 odd 6 5184.2.d.l.2593.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.c.97.1 ✓ 8 36.7 odd 6 inner
576.2.r.c.97.1 ✓ 8 72.61 even 6 inner
576.2.r.c.97.4 yes 8 9.7 even 3 inner
576.2.r.c.97.4 yes 8 72.43 odd 6 inner
576.2.r.c.481.1 yes 8 1.1 even 1 trivial
576.2.r.c.481.1 yes 8 8.3 odd 2 CM
576.2.r.c.481.4 yes 8 4.3 odd 2 inner
576.2.r.c.481.4 yes 8 8.5 even 2 inner
1728.2.r.c.289.2 8 9.2 odd 6
1728.2.r.c.289.2 8 72.11 even 6
1728.2.r.c.289.3 8 36.11 even 6
1728.2.r.c.289.3 8 72.29 odd 6
1728.2.r.c.1441.2 8 12.11 even 2
1728.2.r.c.1441.2 8 24.5 odd 2
1728.2.r.c.1441.3 8 3.2 odd 2
1728.2.r.c.1441.3 8 24.11 even 2
5184.2.d.e.2593.2 4 36.23 even 6
5184.2.d.e.2593.2 4 72.5 odd 6
5184.2.d.e.2593.3 4 9.5 odd 6
5184.2.d.e.2593.3 4 72.59 even 6
5184.2.d.l.2593.2 4 9.4 even 3
5184.2.d.l.2593.2 4 72.67 odd 6
5184.2.d.l.2593.3 4 36.31 odd 6
5184.2.d.l.2593.3 4 72.13 even 6