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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [576,3,Mod(353,576)] 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("576.353"); S:= CuspForms(chi, 3); 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, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 18 q^{5} + 2 q^{9} + 30 q^{13} + 66 q^{21} - 74 q^{25} - 54 q^{29} - 120 q^{33} - 216 q^{41} - 366 q^{45} - 86 q^{49} + 144 q^{53} + 302 q^{57} - 42 q^{61} + 306 q^{65} + 54 q^{69} + 196 q^{73} - 414 q^{77}+ \cdots - 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1 0 −2.90962 − 0.730846i 0 3.09427 + 5.35943i 0 0.183090 − 0.317122i 0 7.93173 + 4.25296i 0
353.2 0 −2.81879 + 1.02686i 0 −1.79985 − 3.11744i 0 3.70215 − 6.41232i 0 6.89112 − 5.78899i 0
353.3 0 −2.77167 − 1.14797i 0 1.13094 + 1.95884i 0 2.43128 − 4.21109i 0 6.36431 + 6.36361i 0
353.4 0 −2.72431 + 1.25624i 0 −4.89158 − 8.47246i 0 −3.88386 + 6.72705i 0 5.84373 − 6.84477i 0
353.5 0 −1.32664 − 2.69073i 0 −2.27609 − 3.94230i 0 −4.81066 + 8.33232i 0 −5.48007 + 7.13924i 0
353.6 0 −1.21852 + 2.74139i 0 2.39677 + 4.15133i 0 −1.12793 + 1.95364i 0 −6.03041 − 6.68088i 0
353.7 0 −1.08485 + 2.79698i 0 −3.07518 − 5.32637i 0 −5.35886 + 9.28182i 0 −6.64619 − 6.06862i 0
353.8 0 −0.559363 − 2.94739i 0 0.920717 + 1.59473i 0 −4.56979 + 7.91511i 0 −8.37423 + 3.29733i 0
353.9 0 0.559363 + 2.94739i 0 0.920717 + 1.59473i 0 4.56979 − 7.91511i 0 −8.37423 + 3.29733i 0
353.10 0 1.08485 − 2.79698i 0 −3.07518 − 5.32637i 0 5.35886 − 9.28182i 0 −6.64619 − 6.06862i 0
353.11 0 1.21852 − 2.74139i 0 2.39677 + 4.15133i 0 1.12793 − 1.95364i 0 −6.03041 − 6.68088i 0
353.12 0 1.32664 + 2.69073i 0 −2.27609 − 3.94230i 0 4.81066 − 8.33232i 0 −5.48007 + 7.13924i 0
353.13 0 2.72431 − 1.25624i 0 −4.89158 − 8.47246i 0 3.88386 − 6.72705i 0 5.84373 − 6.84477i 0
353.14 0 2.77167 + 1.14797i 0 1.13094 + 1.95884i 0 −2.43128 + 4.21109i 0 6.36431 + 6.36361i 0
353.15 0 2.81879 − 1.02686i 0 −1.79985 − 3.11744i 0 −3.70215 + 6.41232i 0 6.89112 − 5.78899i 0
353.16 0 2.90962 + 0.730846i 0 3.09427 + 5.35943i 0 −0.183090 + 0.317122i 0 7.93173 + 4.25296i 0
545.1 0 −2.90962 + 0.730846i 0 3.09427 − 5.35943i 0 0.183090 + 0.317122i 0 7.93173 − 4.25296i 0
545.2 0 −2.81879 − 1.02686i 0 −1.79985 + 3.11744i 0 3.70215 + 6.41232i 0 6.89112 + 5.78899i 0
545.3 0 −2.77167 + 1.14797i 0 1.13094 − 1.95884i 0 2.43128 + 4.21109i 0 6.36431 − 6.36361i 0
545.4 0 −2.72431 − 1.25624i 0 −4.89158 + 8.47246i 0 −3.88386 − 6.72705i 0 5.84373 + 6.84477i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 353.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
72.j odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.n.c ✓ 32
3.b odd 2 1 1728.3.n.d 32
4.b odd 2 1 inner 576.3.n.c ✓ 32
8.b even 2 1 576.3.n.d yes 32
8.d odd 2 1 576.3.n.d yes 32
9.c even 3 1 1728.3.n.c 32
9.d odd 6 1 576.3.n.d yes 32
12.b even 2 1 1728.3.n.d 32
24.f even 2 1 1728.3.n.c 32
24.h odd 2 1 1728.3.n.c 32
36.f odd 6 1 1728.3.n.c 32
36.h even 6 1 576.3.n.d yes 32
72.j odd 6 1 inner 576.3.n.c ✓ 32
72.l even 6 1 inner 576.3.n.c ✓ 32
72.n even 6 1 1728.3.n.d 32
72.p odd 6 1 1728.3.n.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.3.n.c ✓ 32 1.a even 1 1 trivial
576.3.n.c ✓ 32 4.b odd 2 1 inner
576.3.n.c ✓ 32 72.j odd 6 1 inner
576.3.n.c ✓ 32 72.l even 6 1 inner
576.3.n.d yes 32 8.b even 2 1
576.3.n.d yes 32 8.d odd 2 1
576.3.n.d yes 32 9.d odd 6 1
576.3.n.d yes 32 36.h even 6 1
1728.3.n.c 32 9.c even 3 1
1728.3.n.c 32 24.f even 2 1
1728.3.n.c 32 24.h odd 2 1
1728.3.n.c 32 36.f odd 6 1
1728.3.n.d 32 3.b odd 2 1
1728.3.n.d 32 12.b even 2 1
1728.3.n.d 32 72.n even 6 1
1728.3.n.d 32 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 9 T_{5}^{15} + 159 T_{5}^{14} + 558 T_{5}^{13} + 9729 T_{5}^{12} + 25515 T_{5}^{11} + \cdots + 14841086976 \) acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display