Properties

Label 1408.2.i.a
Level $1408$
Weight $2$
Character orbit 1408.i
Analytic conductor $11.243$
Analytic rank $0$
Dimension $44$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1408,2,Mod(351,1408)] 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("1408.351"); 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([2, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.i (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [44,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2429366046\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{3} + 4 q^{5} - 6 q^{11} + 24 q^{23} + 8 q^{27} - 4 q^{33} + 20 q^{37} - 28 q^{45} - 28 q^{49} - 12 q^{53} + 36 q^{55} - 20 q^{59} + 36 q^{67} + 16 q^{69} + 40 q^{71} + 60 q^{75} - 4 q^{77} - 20 q^{81}+ \cdots - 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1 0 −2.13264 + 2.13264i 0 −0.987102 + 0.987102i 0 3.85199i 0 − 6.09629i 0
351.2 0 −2.13264 + 2.13264i 0 −0.987102 + 0.987102i 0 − 3.85199i 0 − 6.09629i 0
351.3 0 −1.82463 + 1.82463i 0 2.57632 − 2.57632i 0 − 2.30642i 0 − 3.65855i 0
351.4 0 −1.82463 + 1.82463i 0 2.57632 − 2.57632i 0 2.30642i 0 − 3.65855i 0
351.5 0 −1.64229 + 1.64229i 0 −1.97065 + 1.97065i 0 1.30445i 0 − 2.39422i 0
351.6 0 −1.64229 + 1.64229i 0 −1.97065 + 1.97065i 0 − 1.30445i 0 − 2.39422i 0
351.7 0 −0.766690 + 0.766690i 0 0.946416 − 0.946416i 0 − 0.840189i 0 1.82437i 0
351.8 0 −0.766690 + 0.766690i 0 0.946416 − 0.946416i 0 0.840189i 0 1.82437i 0
351.9 0 −0.429854 + 0.429854i 0 −0.703721 + 0.703721i 0 − 3.48768i 0 2.63045i 0
351.10 0 −0.429854 + 0.429854i 0 −0.703721 + 0.703721i 0 3.48768i 0 2.63045i 0
351.11 0 −0.259076 + 0.259076i 0 1.45525 − 1.45525i 0 − 0.413174i 0 2.86576i 0
351.12 0 −0.259076 + 0.259076i 0 1.45525 − 1.45525i 0 0.413174i 0 2.86576i 0
351.13 0 0.108192 − 0.108192i 0 −2.69023 + 2.69023i 0 − 3.17010i 0 2.97659i 0
351.14 0 0.108192 − 0.108192i 0 −2.69023 + 2.69023i 0 3.17010i 0 2.97659i 0
351.15 0 1.15889 − 1.15889i 0 2.51141 − 2.51141i 0 − 4.37614i 0 0.313934i 0
351.16 0 1.15889 − 1.15889i 0 2.51141 − 2.51141i 0 4.37614i 0 0.313934i 0
351.17 0 1.22947 − 1.22947i 0 0.282901 − 0.282901i 0 − 4.20092i 0 − 0.0232059i 0
351.18 0 1.22947 − 1.22947i 0 0.282901 − 0.282901i 0 4.20092i 0 − 0.0232059i 0
351.19 0 1.33912 − 1.33912i 0 −1.27873 + 1.27873i 0 0.148220i 0 − 0.586494i 0
351.20 0 1.33912 − 1.33912i 0 −1.27873 + 1.27873i 0 − 0.148220i 0 − 0.586494i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 351.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
16.f odd 4 1 inner
176.i 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.i.a 44
4.b odd 2 1 1408.2.i.b 44
8.b even 2 1 704.2.i.a 44
8.d odd 2 1 176.2.i.a ✓ 44
11.b odd 2 1 inner 1408.2.i.a 44
16.e even 4 1 176.2.i.a ✓ 44
16.e even 4 1 1408.2.i.b 44
16.f odd 4 1 704.2.i.a 44
16.f odd 4 1 inner 1408.2.i.a 44
44.c even 2 1 1408.2.i.b 44
88.b odd 2 1 704.2.i.a 44
88.g even 2 1 176.2.i.a ✓ 44
176.i even 4 1 704.2.i.a 44
176.i even 4 1 inner 1408.2.i.a 44
176.l odd 4 1 176.2.i.a ✓ 44
176.l odd 4 1 1408.2.i.b 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.i.a ✓ 44 8.d odd 2 1
176.2.i.a ✓ 44 16.e even 4 1
176.2.i.a ✓ 44 88.g even 2 1
176.2.i.a ✓ 44 176.l odd 4 1
704.2.i.a 44 8.b even 2 1
704.2.i.a 44 16.f odd 4 1
704.2.i.a 44 88.b odd 2 1
704.2.i.a 44 176.i even 4 1
1408.2.i.a 44 1.a even 1 1 trivial
1408.2.i.a 44 11.b odd 2 1 inner
1408.2.i.a 44 16.f odd 4 1 inner
1408.2.i.a 44 176.i even 4 1 inner
1408.2.i.b 44 4.b odd 2 1
1408.2.i.b 44 16.e even 4 1
1408.2.i.b 44 44.c even 2 1
1408.2.i.b 44 176.l odd 4 1