Properties

Label 169.3.d.e
Level $169$
Weight $3$
Character orbit 169.d
Analytic conductor $4.605$
Analytic rank $0$
Dimension $24$
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] = [169,3,Mod(70,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.70");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 169.d (of order \(4\), degree \(2\), 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: \(4.60491646769\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{3} + 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{3} + 84 q^{9} + 188 q^{14} + 188 q^{16} - 136 q^{22} + 60 q^{27} + 84 q^{29} + 176 q^{35} - 524 q^{40} - 368 q^{42} - 368 q^{48} - 44 q^{53} - 704 q^{55} - 8 q^{61} - 740 q^{66} - 168 q^{68} + 744 q^{74} - 300 q^{79} + 240 q^{81} - 952 q^{87} + 1736 q^{92} + 1132 q^{94}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1 −2.28029 + 2.28029i −2.04607 6.39942i −5.50472 + 5.50472i 4.66563 4.66563i −2.31365 2.31365i 5.47136 + 5.47136i −4.81360 25.1047i
70.2 −2.02149 + 2.02149i 4.64845 4.17281i 2.43155 2.43155i −9.39676 + 9.39676i 0.759383 + 0.759383i 0.349324 + 0.349324i 12.6080 9.83067i
70.3 −1.66446 + 1.66446i −5.02350 1.54085i 1.10964 1.10964i 8.36141 8.36141i −4.64797 4.64797i −4.09315 4.09315i 16.2355 3.69391i
70.4 −1.40788 + 1.40788i −0.599528 0.0357421i 5.65111 5.65111i 0.844064 0.844064i −1.43861 1.43861i −5.68184 5.68184i −8.64057 15.9122i
70.5 −1.17461 + 1.17461i 1.35405 1.24058i −1.57408 + 1.57408i −1.59048 + 1.59048i −8.90433 8.90433i −6.15564 6.15564i −7.16655 3.69787i
70.6 −0.285703 + 0.285703i 4.66660 3.83675i 4.14024 4.14024i −1.33326 + 1.33326i 1.61555 + 1.61555i −2.23898 2.23898i 12.7772 2.36576i
70.7 0.285703 0.285703i 4.66660 3.83675i −4.14024 + 4.14024i 1.33326 1.33326i −1.61555 1.61555i 2.23898 + 2.23898i 12.7772 2.36576i
70.8 1.17461 1.17461i 1.35405 1.24058i 1.57408 1.57408i 1.59048 1.59048i 8.90433 + 8.90433i 6.15564 + 6.15564i −7.16655 3.69787i
70.9 1.40788 1.40788i −0.599528 0.0357421i −5.65111 + 5.65111i −0.844064 + 0.844064i 1.43861 + 1.43861i 5.68184 + 5.68184i −8.64057 15.9122i
70.10 1.66446 1.66446i −5.02350 1.54085i −1.10964 + 1.10964i −8.36141 + 8.36141i 4.64797 + 4.64797i 4.09315 + 4.09315i 16.2355 3.69391i
70.11 2.02149 2.02149i 4.64845 4.17281i −2.43155 + 2.43155i 9.39676 9.39676i −0.759383 0.759383i −0.349324 0.349324i 12.6080 9.83067i
70.12 2.28029 2.28029i −2.04607 6.39942i 5.50472 5.50472i −4.66563 + 4.66563i 2.31365 + 2.31365i −5.47136 5.47136i −4.81360 25.1047i
99.1 −2.28029 2.28029i −2.04607 6.39942i −5.50472 5.50472i 4.66563 + 4.66563i −2.31365 + 2.31365i 5.47136 5.47136i −4.81360 25.1047i
99.2 −2.02149 2.02149i 4.64845 4.17281i 2.43155 + 2.43155i −9.39676 9.39676i 0.759383 0.759383i 0.349324 0.349324i 12.6080 9.83067i
99.3 −1.66446 1.66446i −5.02350 1.54085i 1.10964 + 1.10964i 8.36141 + 8.36141i −4.64797 + 4.64797i −4.09315 + 4.09315i 16.2355 3.69391i
99.4 −1.40788 1.40788i −0.599528 0.0357421i 5.65111 + 5.65111i 0.844064 + 0.844064i −1.43861 + 1.43861i −5.68184 + 5.68184i −8.64057 15.9122i
99.5 −1.17461 1.17461i 1.35405 1.24058i −1.57408 1.57408i −1.59048 1.59048i −8.90433 + 8.90433i −6.15564 + 6.15564i −7.16655 3.69787i
99.6 −0.285703 0.285703i 4.66660 3.83675i 4.14024 + 4.14024i −1.33326 1.33326i 1.61555 1.61555i −2.23898 + 2.23898i 12.7772 2.36576i
99.7 0.285703 + 0.285703i 4.66660 3.83675i −4.14024 4.14024i 1.33326 + 1.33326i −1.61555 + 1.61555i 2.23898 2.23898i 12.7772 2.36576i
99.8 1.17461 + 1.17461i 1.35405 1.24058i 1.57408 + 1.57408i 1.59048 + 1.59048i 8.90433 8.90433i 6.15564 6.15564i −7.16655 3.69787i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.d odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.3.d.e 24
13.b even 2 1 inner 169.3.d.e 24
13.c even 3 2 169.3.f.g 48
13.d odd 4 2 inner 169.3.d.e 24
13.e even 6 2 169.3.f.g 48
13.f odd 12 4 169.3.f.g 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.3.d.e 24 1.a even 1 1 trivial
169.3.d.e 24 13.b even 2 1 inner
169.3.d.e 24 13.d odd 4 2 inner
169.3.f.g 48 13.c even 3 2
169.3.f.g 48 13.e even 6 2
169.3.f.g 48 13.f odd 12 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 229T_{2}^{20} + 17518T_{2}^{16} + 540679T_{2}^{12} + 6695460T_{2}^{8} + 26716280T_{2}^{4} + 707281 \) acting on \(S_{3}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display