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Properties

Label 880.2.bd.i
Level $880$
Weight $2$
Character orbit 880.bd
Analytic conductor $7.027$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,2,Mod(417,880)] 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("880.417"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bd (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,4,0,8,0,0,0,0,0,8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 440)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{3} + 8 q^{5} + 8 q^{11} - 24 q^{15} + 28 q^{23} + 4 q^{25} + 4 q^{27} - 24 q^{31} - 12 q^{33} + 4 q^{37} - 28 q^{45} - 8 q^{47} + 24 q^{53} - 12 q^{55} + 52 q^{67} - 48 q^{71} + 24 q^{75} + 56 q^{77}+ \cdots - 36 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1 0 −2.06525 + 2.06525i 0 1.82253 + 1.29552i 0 −1.52796 + 1.52796i 0 − 5.53050i 0
417.2 0 −2.06525 + 2.06525i 0 1.82253 + 1.29552i 0 1.52796 − 1.52796i 0 − 5.53050i 0
417.3 0 −1.78145 + 1.78145i 0 −0.300869 − 2.21573i 0 −1.52953 + 1.52953i 0 − 3.34712i 0
417.4 0 −1.78145 + 1.78145i 0 −0.300869 − 2.21573i 0 1.52953 − 1.52953i 0 − 3.34712i 0
417.5 0 −0.395428 + 0.395428i 0 −0.0577165 + 2.23532i 0 −2.57694 + 2.57694i 0 2.68727i 0
417.6 0 −0.395428 + 0.395428i 0 −0.0577165 + 2.23532i 0 2.57694 − 2.57694i 0 2.68727i 0
417.7 0 −0.179637 + 0.179637i 0 2.16046 − 0.576544i 0 −1.58807 + 1.58807i 0 2.93546i 0
417.8 0 −0.179637 + 0.179637i 0 2.16046 − 0.576544i 0 1.58807 − 1.58807i 0 2.93546i 0
417.9 0 0.185627 − 0.185627i 0 −2.05027 − 0.892416i 0 −0.202657 + 0.202657i 0 2.93108i 0
417.10 0 0.185627 − 0.185627i 0 −2.05027 − 0.892416i 0 0.202657 − 0.202657i 0 2.93108i 0
417.11 0 1.50516 − 1.50516i 0 1.52592 + 1.63449i 0 −1.92066 + 1.92066i 0 − 1.53101i 0
417.12 0 1.50516 − 1.50516i 0 1.52592 + 1.63449i 0 1.92066 − 1.92066i 0 − 1.53101i 0
417.13 0 1.62830 − 1.62830i 0 1.07475 − 1.96085i 0 −1.24708 + 1.24708i 0 − 2.30275i 0
417.14 0 1.62830 − 1.62830i 0 1.07475 − 1.96085i 0 1.24708 − 1.24708i 0 − 2.30275i 0
417.15 0 2.10267 − 2.10267i 0 −2.17481 − 0.519798i 0 −3.44647 + 3.44647i 0 − 5.84244i 0
417.16 0 2.10267 − 2.10267i 0 −2.17481 − 0.519798i 0 3.44647 − 3.44647i 0 − 5.84244i 0
593.1 0 −2.06525 − 2.06525i 0 1.82253 − 1.29552i 0 −1.52796 − 1.52796i 0 5.53050i 0
593.2 0 −2.06525 − 2.06525i 0 1.82253 − 1.29552i 0 1.52796 + 1.52796i 0 5.53050i 0
593.3 0 −1.78145 − 1.78145i 0 −0.300869 + 2.21573i 0 −1.52953 − 1.52953i 0 3.34712i 0
593.4 0 −1.78145 − 1.78145i 0 −0.300869 + 2.21573i 0 1.52953 + 1.52953i 0 3.34712i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bd.i 32
4.b odd 2 1 440.2.v.b ✓ 32
5.c odd 4 1 inner 880.2.bd.i 32
11.b odd 2 1 inner 880.2.bd.i 32
20.e even 4 1 440.2.v.b ✓ 32
44.c even 2 1 440.2.v.b ✓ 32
55.e even 4 1 inner 880.2.bd.i 32
220.i odd 4 1 440.2.v.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.v.b ✓ 32 4.b odd 2 1
440.2.v.b ✓ 32 20.e even 4 1
440.2.v.b ✓ 32 44.c even 2 1
440.2.v.b ✓ 32 220.i odd 4 1
880.2.bd.i 32 1.a even 1 1 trivial
880.2.bd.i 32 5.c odd 4 1 inner
880.2.bd.i 32 11.b odd 2 1 inner
880.2.bd.i 32 55.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\):

\( T_{3}^{16} - 2 T_{3}^{15} + 2 T_{3}^{14} + 2 T_{3}^{13} + 114 T_{3}^{12} - 226 T_{3}^{11} + 226 T_{3}^{10} + \cdots + 16 \) Copy content Toggle raw display
\( T_{7}^{32} + 874 T_{7}^{28} + 204801 T_{7}^{24} + 18266024 T_{7}^{20} + 764963344 T_{7}^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
\( T_{13}^{32} + 3440 T_{13}^{28} + 4172896 T_{13}^{24} + 2164275456 T_{13}^{20} + 450660724992 T_{13}^{16} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display