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,-16,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.517638\) 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)
 
\(f(q)\) \(=\) \(q-2.44949 q^{5} +1.41421 q^{7} +3.46410 q^{11} +4.89898 q^{13} -4.00000 q^{17} -6.92820 q^{19} -5.65685 q^{23} +1.00000 q^{25} -2.44949 q^{29} -1.41421 q^{31} -3.46410 q^{35} +4.89898 q^{37} -4.00000 q^{41} +6.92820 q^{43} +5.65685 q^{47} -5.00000 q^{49} +7.34847 q^{53} -8.48528 q^{55} -13.8564 q^{59} +4.89898 q^{61} -12.0000 q^{65} -11.3137 q^{71} -4.00000 q^{73} +4.89898 q^{77} +7.07107 q^{79} +10.3923 q^{83} +9.79796 q^{85} -16.0000 q^{89} +6.92820 q^{91} +16.9706 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{17} + 4 q^{25} - 16 q^{41} - 20 q^{49} - 48 q^{65} - 16 q^{73} - 64 q^{89} + 24 q^{97}+O(q^{100}) \) 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
\(5\) −2.44949 −1.09545 −0.547723 − 0.836660i \(-0.684505\pi\)
−0.547723 + 0.836660i \(0.684505\pi\)
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 − 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 4.89898 1.35873 0.679366 − 0.733799i \(-0.262255\pi\)
0.679366 + 0.733799i \(0.262255\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −6.92820 −1.58944 −0.794719 − 0.606977i \(-0.792382\pi\)
−0.794719 + 0.606977i \(0.792382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.65685 −1.17954 −0.589768 − 0.807573i \(-0.700781\pi\)
−0.589768 + 0.807573i \(0.700781\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.44949 −0.454859 −0.227429 − 0.973795i \(-0.573032\pi\)
−0.227429 + 0.973795i \(0.573032\pi\)
\(30\) 0 0
\(31\) −1.41421 −0.254000 −0.127000 − 0.991903i \(-0.540535\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) 4.89898 0.805387 0.402694 − 0.915335i \(-0.368074\pi\)
0.402694 + 0.915335i \(0.368074\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 6.92820 1.05654 0.528271 − 0.849076i \(-0.322841\pi\)
0.528271 + 0.849076i \(0.322841\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.65685 0.825137 0.412568 − 0.910927i \(-0.364632\pi\)
0.412568 + 0.910927i \(0.364632\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.34847 1.00939 0.504695 − 0.863298i \(-0.331605\pi\)
0.504695 + 0.863298i \(0.331605\pi\)
\(54\) 0 0
\(55\) −8.48528 −1.14416
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.8564 −1.80395 −0.901975 − 0.431788i \(-0.857883\pi\)
−0.901975 + 0.431788i \(0.857883\pi\)
\(60\) 0 0
\(61\) 4.89898 0.627250 0.313625 − 0.949547i \(-0.398457\pi\)
0.313625 + 0.949547i \(0.398457\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 − 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 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
\(77\) 4.89898 0.558291
\(78\) 0 0
\(79\) 7.07107 0.795557 0.397779 − 0.917481i \(-0.369781\pi\)
0.397779 + 0.917481i \(0.369781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.3923 1.14070 0.570352 − 0.821401i \(-0.306807\pi\)
0.570352 + 0.821401i \(0.306807\pi\)
\(84\) 0 0
\(85\) 9.79796 1.06274
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −16.0000 −1.69600 −0.847998 − 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 6.92820 0.726273
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.9706 1.74114
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 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.v.1.2 yes 4
3.2 odd 2 4608.2.a.z.1.4 yes 4
4.3 odd 2 inner 4608.2.a.v.1.1 ✓ 4
8.3 odd 2 inner 4608.2.a.v.1.3 yes 4
8.5 even 2 inner 4608.2.a.v.1.4 yes 4
12.11 even 2 4608.2.a.z.1.3 yes 4
16.3 odd 4 4608.2.d.e.2305.4 4
16.5 even 4 4608.2.d.e.2305.1 4
16.11 odd 4 4608.2.d.e.2305.2 4
16.13 even 4 4608.2.d.e.2305.3 4
24.5 odd 2 4608.2.a.z.1.2 yes 4
24.11 even 2 4608.2.a.z.1.1 yes 4
48.5 odd 4 4608.2.d.m.2305.3 4
48.11 even 4 4608.2.d.m.2305.4 4
48.29 odd 4 4608.2.d.m.2305.1 4
48.35 even 4 4608.2.d.m.2305.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.v.1.1 ✓ 4 4.3 odd 2 inner
4608.2.a.v.1.2 yes 4 1.1 even 1 trivial
4608.2.a.v.1.3 yes 4 8.3 odd 2 inner
4608.2.a.v.1.4 yes 4 8.5 even 2 inner
4608.2.a.z.1.1 yes 4 24.11 even 2
4608.2.a.z.1.2 yes 4 24.5 odd 2
4608.2.a.z.1.3 yes 4 12.11 even 2
4608.2.a.z.1.4 yes 4 3.2 odd 2
4608.2.d.e.2305.1 4 16.5 even 4
4608.2.d.e.2305.2 4 16.11 odd 4
4608.2.d.e.2305.3 4 16.13 even 4
4608.2.d.e.2305.4 4 16.3 odd 4
4608.2.d.m.2305.1 4 48.29 odd 4
4608.2.d.m.2305.2 4 48.35 even 4
4608.2.d.m.2305.3 4 48.5 odd 4
4608.2.d.m.2305.4 4 48.11 even 4