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Properties

Label 1408.2.i.b
Level $1408$
Weight $2$
Character orbit 1408.i
Analytic conductor $11.243$
Analytic rank $0$
Dimension $44$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1408,2,Mod(351,1408)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.i (of order \(4\), degree \(2\), not 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: \(11.2429366046\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 4 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 4 q^{3} + 4 q^{5} + 6 q^{11} - 24 q^{23} - 8 q^{27} - 4 q^{33} + 20 q^{37} - 28 q^{45} - 28 q^{49} - 12 q^{53} - 36 q^{55} + 20 q^{59} - 36 q^{67} + 16 q^{69} - 40 q^{71} - 60 q^{75} - 4 q^{77} - 20 q^{81} - 56 q^{91} - 8 q^{93} - 8 q^{97} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1 0 −2.21950 + 2.21950i 0 0.858137 − 0.858137i 0 1.49353i 0 − 6.85234i 0
351.2 0 −2.21950 + 2.21950i 0 0.858137 − 0.858137i 0 − 1.49353i 0 − 6.85234i 0
351.3 0 −1.33912 + 1.33912i 0 −1.27873 + 1.27873i 0 − 0.148220i 0 − 0.586494i 0
351.4 0 −1.33912 + 1.33912i 0 −1.27873 + 1.27873i 0 0.148220i 0 − 0.586494i 0
351.5 0 −1.22947 + 1.22947i 0 0.282901 − 0.282901i 0 − 4.20092i 0 − 0.0232059i 0
351.6 0 −1.22947 + 1.22947i 0 0.282901 − 0.282901i 0 4.20092i 0 − 0.0232059i 0
351.7 0 −1.15889 + 1.15889i 0 2.51141 − 2.51141i 0 − 4.37614i 0 0.313934i 0
351.8 0 −1.15889 + 1.15889i 0 2.51141 − 2.51141i 0 4.37614i 0 0.313934i 0
351.9 0 −0.108192 + 0.108192i 0 −2.69023 + 2.69023i 0 3.17010i 0 2.97659i 0
351.10 0 −0.108192 + 0.108192i 0 −2.69023 + 2.69023i 0 − 3.17010i 0 2.97659i 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 0.429854 − 0.429854i 0 −0.703721 + 0.703721i 0 3.48768i 0 2.63045i 0
351.14 0 0.429854 − 0.429854i 0 −0.703721 + 0.703721i 0 − 3.48768i 0 2.63045i 0
351.15 0 0.766690 − 0.766690i 0 0.946416 − 0.946416i 0 0.840189i 0 1.82437i 0
351.16 0 0.766690 − 0.766690i 0 0.946416 − 0.946416i 0 − 0.840189i 0 1.82437i 0
351.17 0 1.64229 − 1.64229i 0 −1.97065 + 1.97065i 0 − 1.30445i 0 − 2.39422i 0
351.18 0 1.64229 − 1.64229i 0 −1.97065 + 1.97065i 0 1.30445i 0 − 2.39422i 0
351.19 0 1.82463 − 1.82463i 0 2.57632 − 2.57632i 0 − 2.30642i 0 − 3.65855i 0
351.20 0 1.82463 − 1.82463i 0 2.57632 − 2.57632i 0 2.30642i 0 − 3.65855i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
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.b 44
4.b odd 2 1 1408.2.i.a 44
8.b even 2 1 176.2.i.a ✓ 44
8.d odd 2 1 704.2.i.a 44
11.b odd 2 1 inner 1408.2.i.b 44
16.e even 4 1 704.2.i.a 44
16.e even 4 1 1408.2.i.a 44
16.f odd 4 1 176.2.i.a ✓ 44
16.f odd 4 1 inner 1408.2.i.b 44
44.c even 2 1 1408.2.i.a 44
88.b odd 2 1 176.2.i.a ✓ 44
88.g even 2 1 704.2.i.a 44
176.i even 4 1 176.2.i.a ✓ 44
176.i even 4 1 inner 1408.2.i.b 44
176.l odd 4 1 704.2.i.a 44
176.l odd 4 1 1408.2.i.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.i.a ✓ 44 8.b even 2 1
176.2.i.a ✓ 44 16.f odd 4 1
176.2.i.a ✓ 44 88.b odd 2 1
176.2.i.a ✓ 44 176.i even 4 1
704.2.i.a 44 8.d odd 2 1
704.2.i.a 44 16.e even 4 1
704.2.i.a 44 88.g even 2 1
704.2.i.a 44 176.l odd 4 1
1408.2.i.a 44 4.b odd 2 1
1408.2.i.a 44 16.e even 4 1
1408.2.i.a 44 44.c even 2 1
1408.2.i.a 44 176.l odd 4 1
1408.2.i.b 44 1.a even 1 1 trivial
1408.2.i.b 44 11.b odd 2 1 inner
1408.2.i.b 44 16.f odd 4 1 inner
1408.2.i.b 44 176.i even 4 1 inner