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Properties

Label 720.2.bm.g
Level $720$
Weight $2$
Character orbit 720.bm
Analytic conductor $5.749$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(109,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.bm (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} - 24 q^{10} - 64 q^{16} + 32 q^{19} + 32 q^{31} - 72 q^{40} - 32 q^{46} + 128 q^{49} - 32 q^{64} - 104 q^{70} - 32 q^{76} + 224 q^{79} - 48 q^{85} - 32 q^{91} - 48 q^{94}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.29731 − 0.563016i 0 1.36603 + 1.46081i −0.496304 + 2.18029i 0 −0.862231 −0.949697 − 2.66422i 0 1.87140 − 2.54909i
109.2 −1.29731 − 0.563016i 0 1.36603 + 1.46081i 2.18029 − 0.496304i 0 0.862231 −0.949697 − 2.66422i 0 −3.10794 − 0.583681i
109.3 −1.29731 + 0.563016i 0 1.36603 − 1.46081i 0.679792 + 2.13023i 0 −3.29610 −0.949697 + 2.66422i 0 −2.08126 − 2.38084i
109.4 −1.29731 + 0.563016i 0 1.36603 − 1.46081i 2.13023 + 0.679792i 0 3.29610 −0.949697 + 2.66422i 0 −3.14630 + 0.317454i
109.5 −0.903873 − 1.08766i 0 −0.366025 + 1.96622i −2.18690 − 0.466314i 0 −4.79660 2.46943 − 1.37910i 0 1.46949 + 2.80011i
109.6 −0.903873 − 1.08766i 0 −0.366025 + 1.96622i −0.466314 − 2.18690i 0 4.79660 2.46943 − 1.37910i 0 −1.95713 + 2.48388i
109.7 −0.903873 + 1.08766i 0 −0.366025 − 1.96622i −1.80193 + 1.32403i 0 3.06349 2.46943 + 1.37910i 0 0.188608 − 3.15665i
109.8 −0.903873 + 1.08766i 0 −0.366025 − 1.96622i 1.32403 − 1.80193i 0 −3.06349 2.46943 + 1.37910i 0 0.763129 + 3.06882i
109.9 0.903873 − 1.08766i 0 −0.366025 − 1.96622i −1.32403 + 1.80193i 0 −3.06349 −2.46943 − 1.37910i 0 0.763129 + 3.06882i
109.10 0.903873 − 1.08766i 0 −0.366025 − 1.96622i 1.80193 − 1.32403i 0 3.06349 −2.46943 − 1.37910i 0 0.188608 − 3.15665i
109.11 0.903873 + 1.08766i 0 −0.366025 + 1.96622i 0.466314 + 2.18690i 0 4.79660 −2.46943 + 1.37910i 0 −1.95713 + 2.48388i
109.12 0.903873 + 1.08766i 0 −0.366025 + 1.96622i 2.18690 + 0.466314i 0 −4.79660 −2.46943 + 1.37910i 0 1.46949 + 2.80011i
109.13 1.29731 − 0.563016i 0 1.36603 − 1.46081i −2.13023 − 0.679792i 0 3.29610 0.949697 − 2.66422i 0 −3.14630 + 0.317454i
109.14 1.29731 − 0.563016i 0 1.36603 − 1.46081i −0.679792 − 2.13023i 0 −3.29610 0.949697 − 2.66422i 0 −2.08126 − 2.38084i
109.15 1.29731 + 0.563016i 0 1.36603 + 1.46081i −2.18029 + 0.496304i 0 0.862231 0.949697 + 2.66422i 0 −3.10794 − 0.583681i
109.16 1.29731 + 0.563016i 0 1.36603 + 1.46081i 0.496304 − 2.18029i 0 −0.862231 0.949697 + 2.66422i 0 1.87140 − 2.54909i
469.1 −1.29731 − 0.563016i 0 1.36603 + 1.46081i 0.679792 − 2.13023i 0 −3.29610 −0.949697 − 2.66422i 0 −2.08126 + 2.38084i
469.2 −1.29731 − 0.563016i 0 1.36603 + 1.46081i 2.13023 − 0.679792i 0 3.29610 −0.949697 − 2.66422i 0 −3.14630 − 0.317454i
469.3 −1.29731 + 0.563016i 0 1.36603 − 1.46081i −0.496304 − 2.18029i 0 −0.862231 −0.949697 + 2.66422i 0 1.87140 + 2.54909i
469.4 −1.29731 + 0.563016i 0 1.36603 − 1.46081i 2.18029 + 0.496304i 0 0.862231 −0.949697 + 2.66422i 0 −3.10794 + 0.583681i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.16
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
16.e even 4 1 inner
48.i odd 4 1 inner
80.q even 4 1 inner
240.bm odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.bm.g ✓ 32
3.b odd 2 1 inner 720.2.bm.g ✓ 32
5.b even 2 1 inner 720.2.bm.g ✓ 32
15.d odd 2 1 inner 720.2.bm.g ✓ 32
16.e even 4 1 inner 720.2.bm.g ✓ 32
48.i odd 4 1 inner 720.2.bm.g ✓ 32
80.q even 4 1 inner 720.2.bm.g ✓ 32
240.bm odd 4 1 inner 720.2.bm.g ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.bm.g ✓ 32 1.a even 1 1 trivial
720.2.bm.g ✓ 32 3.b odd 2 1 inner
720.2.bm.g ✓ 32 5.b even 2 1 inner
720.2.bm.g ✓ 32 15.d odd 2 1 inner
720.2.bm.g ✓ 32 16.e even 4 1 inner
720.2.bm.g ✓ 32 48.i odd 4 1 inner
720.2.bm.g ✓ 32 80.q even 4 1 inner
720.2.bm.g ✓ 32 240.bm odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{8} - 44T_{7}^{6} + 600T_{7}^{4} - 2768T_{7}^{2} + 1744 \) Copy content Toggle raw display
\( T_{11}^{16} + 1792T_{11}^{12} + 839616T_{11}^{8} + 55447552T_{11}^{4} + 48664576 \) Copy content Toggle raw display
\( T_{13}^{16} + 1696T_{13}^{12} + 86112T_{13}^{8} + 957952T_{13}^{4} + 3041536 \) Copy content Toggle raw display