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Properties

Label 8820.2.f.a
Level $8820$
Weight $2$
Character orbit 8820.f
Analytic conductor $70.428$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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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] = [8820,2,Mod(4409,8820)] 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("8820.4409"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8820, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(70.4280545828\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 1260)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{25} - 64 q^{79} + 32 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4409.1 0 0 0 −2.19078 − 0.447731i 0 0 0 0 0
4409.2 0 0 0 −2.19078 − 0.447731i 0 0 0 0 0
4409.3 0 0 0 −2.19078 + 0.447731i 0 0 0 0 0
4409.4 0 0 0 −2.19078 + 0.447731i 0 0 0 0 0
4409.5 0 0 0 −2.01406 − 0.971367i 0 0 0 0 0
4409.6 0 0 0 −2.01406 − 0.971367i 0 0 0 0 0
4409.7 0 0 0 −2.01406 + 0.971367i 0 0 0 0 0
4409.8 0 0 0 −2.01406 + 0.971367i 0 0 0 0 0
4409.9 0 0 0 −1.61232 − 1.54934i 0 0 0 0 0
4409.10 0 0 0 −1.61232 − 1.54934i 0 0 0 0 0
4409.11 0 0 0 −1.61232 + 1.54934i 0 0 0 0 0
4409.12 0 0 0 −1.61232 + 1.54934i 0 0 0 0 0
4409.13 0 0 0 −0.210848 − 2.22610i 0 0 0 0 0
4409.14 0 0 0 −0.210848 − 2.22610i 0 0 0 0 0
4409.15 0 0 0 −0.210848 + 2.22610i 0 0 0 0 0
4409.16 0 0 0 −0.210848 + 2.22610i 0 0 0 0 0
4409.17 0 0 0 0.210848 − 2.22610i 0 0 0 0 0
4409.18 0 0 0 0.210848 − 2.22610i 0 0 0 0 0
4409.19 0 0 0 0.210848 + 2.22610i 0 0 0 0 0
4409.20 0 0 0 0.210848 + 2.22610i 0 0 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4409.32
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
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8820.2.f.a 32
3.b odd 2 1 inner 8820.2.f.a 32
5.b even 2 1 inner 8820.2.f.a 32
7.b odd 2 1 inner 8820.2.f.a 32
7.c even 3 1 1260.2.dc.a ✓ 32
7.d odd 6 1 1260.2.dc.a ✓ 32
15.d odd 2 1 inner 8820.2.f.a 32
21.c even 2 1 inner 8820.2.f.a 32
21.g even 6 1 1260.2.dc.a ✓ 32
21.h odd 6 1 1260.2.dc.a ✓ 32
35.c odd 2 1 inner 8820.2.f.a 32
35.i odd 6 1 1260.2.dc.a ✓ 32
35.j even 6 1 1260.2.dc.a ✓ 32
35.k even 12 2 6300.2.ch.f 32
35.l odd 12 2 6300.2.ch.f 32
105.g even 2 1 inner 8820.2.f.a 32
105.o odd 6 1 1260.2.dc.a ✓ 32
105.p even 6 1 1260.2.dc.a ✓ 32
105.w odd 12 2 6300.2.ch.f 32
105.x even 12 2 6300.2.ch.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.dc.a ✓ 32 7.c even 3 1
1260.2.dc.a ✓ 32 7.d odd 6 1
1260.2.dc.a ✓ 32 21.g even 6 1
1260.2.dc.a ✓ 32 21.h odd 6 1
1260.2.dc.a ✓ 32 35.i odd 6 1
1260.2.dc.a ✓ 32 35.j even 6 1
1260.2.dc.a ✓ 32 105.o odd 6 1
1260.2.dc.a ✓ 32 105.p even 6 1
6300.2.ch.f 32 35.k even 12 2
6300.2.ch.f 32 35.l odd 12 2
6300.2.ch.f 32 105.w odd 12 2
6300.2.ch.f 32 105.x even 12 2
8820.2.f.a 32 1.a even 1 1 trivial
8820.2.f.a 32 3.b odd 2 1 inner
8820.2.f.a 32 5.b even 2 1 inner
8820.2.f.a 32 7.b odd 2 1 inner
8820.2.f.a 32 15.d odd 2 1 inner
8820.2.f.a 32 21.c even 2 1 inner
8820.2.f.a 32 35.c odd 2 1 inner
8820.2.f.a 32 105.g even 2 1 inner