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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(464,465)] 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("465.464"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 40 q^{4} - 16 q^{9} - 20 q^{10} + 56 q^{16} - 16 q^{19} + 8 q^{31} + 24 q^{36} - 64 q^{39} - 48 q^{40} + 28 q^{45} - 160 q^{49} - 112 q^{51} - 32 q^{64} - 168 q^{66} + 160 q^{69} + 188 q^{70} + 24 q^{76}+ \cdots + 208 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
464.1 −2.57161 −0.894699 − 1.48308i 4.61316 1.24654 − 1.85638i 2.30081 + 3.81389i 0.730671i −6.72001 −1.39903 + 2.65381i −3.20561 + 4.77387i
464.2 −2.57161 −0.894699 + 1.48308i 4.61316 1.24654 + 1.85638i 2.30081 − 3.81389i − 0.730671i −6.72001 −1.39903 − 2.65381i −3.20561 − 4.77387i
464.3 −2.57161 0.894699 − 1.48308i 4.61316 1.24654 + 1.85638i −2.30081 + 3.81389i − 0.730671i −6.72001 −1.39903 − 2.65381i −3.20561 − 4.77387i
464.4 −2.57161 0.894699 + 1.48308i 4.61316 1.24654 − 1.85638i −2.30081 − 3.81389i 0.730671i −6.72001 −1.39903 + 2.65381i −3.20561 + 4.77387i
464.5 −2.20453 −1.64942 − 0.528597i 2.85997 −0.685489 + 2.12840i 3.63620 + 1.16531i 4.42299i −1.89584 2.44117 + 1.74376i 1.51118 − 4.69214i
464.6 −2.20453 −1.64942 + 0.528597i 2.85997 −0.685489 − 2.12840i 3.63620 − 1.16531i − 4.42299i −1.89584 2.44117 − 1.74376i 1.51118 + 4.69214i
464.7 −2.20453 1.64942 − 0.528597i 2.85997 −0.685489 − 2.12840i −3.63620 + 1.16531i − 4.42299i −1.89584 2.44117 − 1.74376i 1.51118 + 4.69214i
464.8 −2.20453 1.64942 + 0.528597i 2.85997 −0.685489 + 2.12840i −3.63620 − 1.16531i 4.42299i −1.89584 2.44117 + 1.74376i 1.51118 − 4.69214i
464.9 −1.11979 −0.286684 − 1.70816i −0.746077 1.75264 − 1.38861i 0.321025 + 1.91278i − 3.60916i 3.07502 −2.83562 + 0.979406i −1.96259 + 1.55495i
464.10 −1.11979 −0.286684 + 1.70816i −0.746077 1.75264 + 1.38861i 0.321025 − 1.91278i 3.60916i 3.07502 −2.83562 − 0.979406i −1.96259 − 1.55495i
464.11 −1.11979 0.286684 − 1.70816i −0.746077 1.75264 + 1.38861i −0.321025 + 1.91278i 3.60916i 3.07502 −2.83562 − 0.979406i −1.96259 − 1.55495i
464.12 −1.11979 0.286684 + 1.70816i −0.746077 1.75264 − 1.38861i −0.321025 − 1.91278i − 3.60916i 3.07502 −2.83562 + 0.979406i −1.96259 + 1.55495i
464.13 −0.522444 −1.18184 − 1.26620i −1.72705 −2.21461 − 0.309042i 0.617444 + 0.661518i − 3.85710i 1.94718 −0.206518 + 2.99288i 1.15701 + 0.161457i
464.14 −0.522444 −1.18184 + 1.26620i −1.72705 −2.21461 + 0.309042i 0.617444 − 0.661518i 3.85710i 1.94718 −0.206518 − 2.99288i 1.15701 − 0.161457i
464.15 −0.522444 1.18184 − 1.26620i −1.72705 −2.21461 + 0.309042i −0.617444 + 0.661518i 3.85710i 1.94718 −0.206518 − 2.99288i 1.15701 − 0.161457i
464.16 −0.522444 1.18184 + 1.26620i −1.72705 −2.21461 − 0.309042i −0.617444 − 0.661518i − 3.85710i 1.94718 −0.206518 + 2.99288i 1.15701 + 0.161457i
464.17 0.522444 −1.18184 − 1.26620i −1.72705 2.21461 − 0.309042i −0.617444 − 0.661518i 3.85710i −1.94718 −0.206518 + 2.99288i 1.15701 − 0.161457i
464.18 0.522444 −1.18184 + 1.26620i −1.72705 2.21461 + 0.309042i −0.617444 + 0.661518i − 3.85710i −1.94718 −0.206518 − 2.99288i 1.15701 + 0.161457i
464.19 0.522444 1.18184 − 1.26620i −1.72705 2.21461 + 0.309042i 0.617444 − 0.661518i − 3.85710i −1.94718 −0.206518 − 2.99288i 1.15701 + 0.161457i
464.20 0.522444 1.18184 + 1.26620i −1.72705 2.21461 − 0.309042i 0.617444 + 0.661518i 3.85710i −1.94718 −0.206518 + 2.99288i 1.15701 − 0.161457i
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 464.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner
155.c odd 2 1 inner
465.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.g.f ✓ 32
3.b odd 2 1 inner 465.2.g.f ✓ 32
5.b even 2 1 inner 465.2.g.f ✓ 32
15.d odd 2 1 inner 465.2.g.f ✓ 32
31.b odd 2 1 inner 465.2.g.f ✓ 32
93.c even 2 1 inner 465.2.g.f ✓ 32
155.c odd 2 1 inner 465.2.g.f ✓ 32
465.g even 2 1 inner 465.2.g.f ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.g.f ✓ 32 1.a even 1 1 trivial
465.2.g.f ✓ 32 3.b odd 2 1 inner
465.2.g.f ✓ 32 5.b even 2 1 inner
465.2.g.f ✓ 32 15.d odd 2 1 inner
465.2.g.f ✓ 32 31.b odd 2 1 inner
465.2.g.f ✓ 32 93.c even 2 1 inner
465.2.g.f ✓ 32 155.c odd 2 1 inner
465.2.g.f ✓ 32 465.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\):

\( T_{2}^{8} - 13T_{2}^{6} + 50T_{2}^{4} - 53T_{2}^{2} + 11 \) Copy content Toggle raw display
\( T_{37}^{8} - 118T_{37}^{6} + 3568T_{37}^{4} - 5208T_{37}^{2} + 1936 \) Copy content Toggle raw display