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(2305,4608)] 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("4608.2305"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.d (of order \(2\), degree \(1\), not minimal)

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: no
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: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2305.2
Root \(1.58114 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2305
Dual form 4608.2.d.l.2305.4

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) − 1.41421i − 0.632456i −0.948683 − 0.316228i \(-0.897584\pi\)
0.948683 − 0.316228i \(-0.102416\pi\)
\(6\) 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
\(11\) − 4.47214i − 1.34840i −0.738549 − 0.674200i \(-0.764489\pi\)
0.738549 − 0.674200i \(-0.235511\pi\)
\(12\) 0 0
\(13\) − 4.47214i − 1.24035i −0.784465 − 0.620174i \(-0.787062\pi\)
0.784465 − 0.620174i \(-0.212938\pi\)
\(14\) 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
\(19\) − 2.82843i − 0.648886i −0.945905 − 0.324443i \(-0.894823\pi\)
0.945905 − 0.324443i \(-0.105177\pi\)
\(20\) 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
\(29\) − 4.24264i − 0.787839i −0.919145 − 0.393919i \(-0.871119\pi\)
0.919145 − 0.393919i \(-0.128881\pi\)
\(30\) 0 0
\(31\) 3.16228 0.567962 0.283981 − 0.958830i \(-0.408345\pi\)
0.283981 + 0.958830i \(0.408345\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.47214i − 0.755929i
\(36\) 0 0
\(37\) 4.47214i 0.735215i 0.929981 + 0.367607i \(0.119823\pi\)
−0.929981 + 0.367607i \(0.880177\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.32456 −0.987730 −0.493865 − 0.869539i \(-0.664416\pi\)
−0.493865 + 0.869539i \(0.664416\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 7.07107i − 0.971286i −0.874157 − 0.485643i \(-0.838586\pi\)
0.874157 − 0.485643i \(-0.161414\pi\)
\(54\) 0 0
\(55\) −6.32456 −0.852803
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 13.4164i 1.71780i 0.512148 + 0.858898i \(0.328850\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.32456 −0.784465
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 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 \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 14.1421i − 1.61165i
\(78\) 0 0
\(79\) −3.16228 −0.355784 −0.177892 − 0.984050i \(-0.556928\pi\)
−0.177892 + 0.984050i \(0.556928\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 4.47214i − 0.490881i −0.969412 − 0.245440i \(-0.921067\pi\)
0.969412 − 0.245440i \(-0.0789325\pi\)
\(84\) 0 0
\(85\) − 8.94427i − 0.970143i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) − 14.1421i − 1.48250i
\(92\) 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.d.l.2305.2 4
3.2 odd 2 4608.2.d.i.2305.4 4
4.3 odd 2 4608.2.d.i.2305.1 4
8.3 odd 2 4608.2.d.i.2305.3 4
8.5 even 2 inner 4608.2.d.l.2305.4 4
12.11 even 2 inner 4608.2.d.l.2305.3 4
16.3 odd 4 4608.2.a.y.1.2 yes 4
16.5 even 4 4608.2.a.x.1.3 yes 4
16.11 odd 4 4608.2.a.y.1.4 yes 4
16.13 even 4 4608.2.a.x.1.1 ✓ 4
24.5 odd 2 4608.2.d.i.2305.2 4
24.11 even 2 inner 4608.2.d.l.2305.1 4
48.5 odd 4 4608.2.a.y.1.1 yes 4
48.11 even 4 4608.2.a.x.1.2 yes 4
48.29 odd 4 4608.2.a.y.1.3 yes 4
48.35 even 4 4608.2.a.x.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.x.1.1 ✓ 4 16.13 even 4
4608.2.a.x.1.2 yes 4 48.11 even 4
4608.2.a.x.1.3 yes 4 16.5 even 4
4608.2.a.x.1.4 yes 4 48.35 even 4
4608.2.a.y.1.1 yes 4 48.5 odd 4
4608.2.a.y.1.2 yes 4 16.3 odd 4
4608.2.a.y.1.3 yes 4 48.29 odd 4
4608.2.a.y.1.4 yes 4 16.11 odd 4
4608.2.d.i.2305.1 4 4.3 odd 2
4608.2.d.i.2305.2 4 24.5 odd 2
4608.2.d.i.2305.3 4 8.3 odd 2
4608.2.d.i.2305.4 4 3.2 odd 2
4608.2.d.l.2305.1 4 24.11 even 2 inner
4608.2.d.l.2305.2 4 1.1 even 1 trivial
4608.2.d.l.2305.3 4 12.11 even 2 inner
4608.2.d.l.2305.4 4 8.5 even 2 inner