Properties

Label 1408.2.j.b
Level $1408$
Weight $2$
Character orbit 1408.j
Analytic conductor $11.243$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1408,2,Mod(353,1408)] 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("1408.353"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1408, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,0,0,0,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2429366046\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 176)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 16 q^{15} - 16 q^{19} + 24 q^{27} - 24 q^{31} + 24 q^{35} + 40 q^{47} - 40 q^{49} - 24 q^{51} - 8 q^{59} - 32 q^{61} - 16 q^{65} - 24 q^{67} - 32 q^{69} - 56 q^{75} - 32 q^{79} - 40 q^{81} + 40 q^{83}+ \cdots + 48 q^{95}+O(q^{100}) \) 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 −2.27066 + 2.27066i 0 −1.24014 − 1.24014i 0 − 3.02382i 0 − 7.31182i 0
353.2 0 −2.24228 + 2.24228i 0 2.71114 + 2.71114i 0 1.46050i 0 − 7.05562i 0
353.3 0 −1.97112 + 1.97112i 0 −2.29075 − 2.29075i 0 3.79929i 0 − 4.77065i 0
353.4 0 −1.51913 + 1.51913i 0 1.06305 + 1.06305i 0 − 3.12852i 0 − 1.61549i 0
353.5 0 −1.27582 + 1.27582i 0 0.795196 + 0.795196i 0 4.67475i 0 − 0.255441i 0
353.6 0 −1.15765 + 1.15765i 0 −2.69017 − 2.69017i 0 − 0.0889173i 0 0.319711i 0
353.7 0 −1.15526 + 1.15526i 0 −0.731846 − 0.731846i 0 − 2.61186i 0 0.330747i 0
353.8 0 −0.273979 + 0.273979i 0 2.48024 + 2.48024i 0 − 4.65464i 0 2.84987i 0
353.9 0 −0.248483 + 0.248483i 0 0.0434775 + 0.0434775i 0 3.27975i 0 2.87651i 0
353.10 0 −0.151436 + 0.151436i 0 −1.85628 − 1.85628i 0 − 0.615059i 0 2.95413i 0
353.11 0 0.190424 − 0.190424i 0 −0.291418 − 0.291418i 0 − 2.64639i 0 2.92748i 0
353.12 0 0.529792 − 0.529792i 0 −1.79526 − 1.79526i 0 1.13150i 0 2.43864i 0
353.13 0 0.586507 − 0.586507i 0 1.32116 + 1.32116i 0 − 1.42100i 0 2.31202i 0
353.14 0 0.791470 − 0.791470i 0 2.78615 + 2.78615i 0 − 0.0391973i 0 1.74715i 0
353.15 0 1.28225 − 1.28225i 0 2.06176 + 2.06176i 0 3.54041i 0 − 0.288318i 0
353.16 0 1.34498 − 1.34498i 0 −2.40794 − 2.40794i 0 4.88654i 0 − 0.617917i 0
353.17 0 1.60387 − 1.60387i 0 0.215519 + 0.215519i 0 1.18021i 0 − 2.14479i 0
353.18 0 1.72812 − 1.72812i 0 −0.893733 − 0.893733i 0 − 2.98840i 0 − 2.97277i 0
353.19 0 2.04787 − 2.04787i 0 −0.814963 − 0.814963i 0 − 2.42498i 0 − 5.38755i 0
353.20 0 2.16054 − 2.16054i 0 1.53482 + 1.53482i 0 − 0.310169i 0 − 6.33589i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 353.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1408.2.j.b 40
4.b odd 2 1 1408.2.j.a 40
8.b even 2 1 704.2.j.a 40
8.d odd 2 1 176.2.j.a ✓ 40
16.e even 4 1 704.2.j.a 40
16.e even 4 1 inner 1408.2.j.b 40
16.f odd 4 1 176.2.j.a ✓ 40
16.f odd 4 1 1408.2.j.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.j.a ✓ 40 8.d odd 2 1
176.2.j.a ✓ 40 16.f odd 4 1
704.2.j.a 40 8.b even 2 1
704.2.j.a 40 16.e even 4 1
1408.2.j.a 40 4.b odd 2 1
1408.2.j.a 40 16.f odd 4 1
1408.2.j.b 40 1.a even 1 1 trivial
1408.2.j.b 40 16.e even 4 1 inner