Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4608,2,Mod(1,4608)] 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("4608.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.28825\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{5} +3.16228 q^{7} -4.47214 q^{11} -4.47214 q^{13} +6.32456 q^{17} +2.82843 q^{19} +4.00000 q^{23} -3.00000 q^{25} -4.24264 q^{29} -3.16228 q^{31} +4.47214 q^{35} -4.47214 q^{37} +6.32456 q^{41} +8.48528 q^{43} +12.0000 q^{47} +3.00000 q^{49} +7.07107 q^{53} -6.32456 q^{55} +13.4164 q^{61} -6.32456 q^{65} +8.00000 q^{71} -4.00000 q^{73} -14.1421 q^{77} +3.16228 q^{79} +4.47214 q^{83} +8.94427 q^{85} -14.1421 q^{91} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{23} - 12 q^{25} + 48 q^{47} + 12 q^{49} + 32 q^{71} - 16 q^{73} + 16 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 1.41421 0.632456 0.316228 − 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 3.16228 1.19523 0.597614 − 0.801784i \(-0.296115\pi\)
0.597614 + 0.801784i \(0.296115\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.47214 −1.34840 −0.674200 − 0.738549i \(-0.735511\pi\)
−0.674200 + 0.738549i \(0.735511\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 − 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.32456 1.53393 0.766965 − 0.641689i \(-0.221766\pi\)
0.766965 + 0.641689i \(0.221766\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.24264 −0.787839 −0.393919 − 0.919145i \(-0.628881\pi\)
−0.393919 + 0.919145i \(0.628881\pi\)
\(30\) 0 0
\(31\) −3.16228 −0.567962 −0.283981 − 0.958830i \(-0.591655\pi\)
−0.283981 + 0.958830i \(0.591655\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.47214 0.755929
\(36\) 0 0
\(37\) −4.47214 −0.735215 −0.367607 − 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.32456 0.987730 0.493865 − 0.869539i \(-0.335584\pi\)
0.493865 + 0.869539i \(0.335584\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 − 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.07107 0.971286 0.485643 − 0.874157i \(-0.338586\pi\)
0.485643 + 0.874157i \(0.338586\pi\)
\(54\) 0 0
\(55\) −6.32456 −0.852803
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 13.4164 1.71780 0.858898 − 0.512148i \(-0.171150\pi\)
0.858898 + 0.512148i \(0.171150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.32456 −0.784465
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.1421 −1.61165
\(78\) 0 0
\(79\) 3.16228 0.355784 0.177892 − 0.984050i \(-0.443072\pi\)
0.177892 + 0.984050i \(0.443072\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.47214 0.490881 0.245440 − 0.969412i \(-0.421067\pi\)
0.245440 + 0.969412i \(0.421067\pi\)
\(84\) 0 0
\(85\) 8.94427 0.970143
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −14.1421 −1.48250
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 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 4608.2.a.y.1.4 yes 4
3.2 odd 2 4608.2.a.x.1.2 yes 4
4.3 odd 2 4608.2.a.x.1.3 yes 4
8.3 odd 2 4608.2.a.x.1.1 ✓ 4
8.5 even 2 inner 4608.2.a.y.1.2 yes 4
12.11 even 2 inner 4608.2.a.y.1.1 yes 4
16.3 odd 4 4608.2.d.l.2305.2 4
16.5 even 4 4608.2.d.i.2305.3 4
16.11 odd 4 4608.2.d.l.2305.4 4
16.13 even 4 4608.2.d.i.2305.1 4
24.5 odd 2 4608.2.a.x.1.4 yes 4
24.11 even 2 inner 4608.2.a.y.1.3 yes 4
48.5 odd 4 4608.2.d.l.2305.1 4
48.11 even 4 4608.2.d.i.2305.2 4
48.29 odd 4 4608.2.d.l.2305.3 4
48.35 even 4 4608.2.d.i.2305.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.x.1.1 ✓ 4 8.3 odd 2
4608.2.a.x.1.2 yes 4 3.2 odd 2
4608.2.a.x.1.3 yes 4 4.3 odd 2
4608.2.a.x.1.4 yes 4 24.5 odd 2
4608.2.a.y.1.1 yes 4 12.11 even 2 inner
4608.2.a.y.1.2 yes 4 8.5 even 2 inner
4608.2.a.y.1.3 yes 4 24.11 even 2 inner
4608.2.a.y.1.4 yes 4 1.1 even 1 trivial
4608.2.d.i.2305.1 4 16.13 even 4
4608.2.d.i.2305.2 4 48.11 even 4
4608.2.d.i.2305.3 4 16.5 even 4
4608.2.d.i.2305.4 4 48.35 even 4
4608.2.d.l.2305.1 4 48.5 odd 4
4608.2.d.l.2305.2 4 16.3 odd 4
4608.2.d.l.2305.3 4 48.29 odd 4
4608.2.d.l.2305.4 4 16.11 odd 4