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Properties

Label 693.2.bu.e
Level $693$
Weight $2$
Character orbit 693.bu
Analytic conductor $5.534$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(118,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.bu (of order \(10\), degree \(4\), 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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} + 10 q^{7} + 56 q^{16} - 108 q^{22} + 72 q^{25} - 80 q^{28} - 52 q^{37} + 100 q^{46} - 26 q^{49} + 104 q^{58} + 8 q^{64} - 152 q^{67} - 90 q^{70} + 100 q^{79} - 80 q^{85} - 112 q^{88} + 42 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1 −1.12577 − 1.54949i 0 −0.515523 + 1.58662i −0.827784 + 1.13935i 0 0.189188 − 2.63898i −0.604260 + 0.196336i 0 2.69730
118.2 −1.12577 − 1.54949i 0 −0.515523 + 1.58662i 0.827784 − 1.13935i 0 −1.39810 + 2.24618i −0.604260 + 0.196336i 0 −2.69730
118.3 −0.496565 − 0.683463i 0 0.397489 − 1.22335i −1.99713 + 2.74882i 0 1.60220 + 2.10546i −2.64041 + 0.857920i 0 2.87042
118.4 −0.496565 − 0.683463i 0 0.397489 − 1.22335i 1.99713 − 2.74882i 0 2.53376 − 0.761606i −2.64041 + 0.857920i 0 −2.87042
118.5 0.496565 + 0.683463i 0 0.397489 − 1.22335i −1.99713 + 2.74882i 0 2.53376 − 0.761606i 2.64041 − 0.857920i 0 −2.87042
118.6 0.496565 + 0.683463i 0 0.397489 − 1.22335i 1.99713 − 2.74882i 0 1.60220 + 2.10546i 2.64041 − 0.857920i 0 2.87042
118.7 1.12577 + 1.54949i 0 −0.515523 + 1.58662i −0.827784 + 1.13935i 0 −1.39810 + 2.24618i 0.604260 − 0.196336i 0 −2.69730
118.8 1.12577 + 1.54949i 0 −0.515523 + 1.58662i 0.827784 − 1.13935i 0 0.189188 − 2.63898i 0.604260 − 0.196336i 0 2.69730
244.1 −2.43586 − 0.791458i 0 3.68896 + 2.68019i −3.71451 + 1.20692i 0 −2.53985 + 0.741038i −3.85364 − 5.30408i 0 10.0032
244.2 −2.43586 − 0.791458i 0 3.68896 + 2.68019i 3.71451 − 1.20692i 0 0.0800897 + 2.64454i −3.85364 − 5.30408i 0 −10.0032
244.3 −0.229495 − 0.0745674i 0 −1.57093 − 1.14134i −2.55512 + 0.830208i 0 2.63643 − 0.221841i 0.559084 + 0.769513i 0 0.648293
244.4 −0.229495 − 0.0745674i 0 −1.57093 − 1.14134i 2.55512 − 0.830208i 0 −0.603720 − 2.57595i 0.559084 + 0.769513i 0 −0.648293
244.5 0.229495 + 0.0745674i 0 −1.57093 − 1.14134i −2.55512 + 0.830208i 0 −0.603720 − 2.57595i −0.559084 − 0.769513i 0 −0.648293
244.6 0.229495 + 0.0745674i 0 −1.57093 − 1.14134i 2.55512 − 0.830208i 0 2.63643 − 0.221841i −0.559084 − 0.769513i 0 0.648293
244.7 2.43586 + 0.791458i 0 3.68896 + 2.68019i −3.71451 + 1.20692i 0 0.0800897 + 2.64454i 3.85364 + 5.30408i 0 −10.0032
244.8 2.43586 + 0.791458i 0 3.68896 + 2.68019i 3.71451 − 1.20692i 0 −2.53985 + 0.741038i 3.85364 + 5.30408i 0 10.0032
370.1 −1.12577 + 1.54949i 0 −0.515523 − 1.58662i −0.827784 − 1.13935i 0 0.189188 + 2.63898i −0.604260 − 0.196336i 0 2.69730
370.2 −1.12577 + 1.54949i 0 −0.515523 − 1.58662i 0.827784 + 1.13935i 0 −1.39810 − 2.24618i −0.604260 − 0.196336i 0 −2.69730
370.3 −0.496565 + 0.683463i 0 0.397489 + 1.22335i −1.99713 − 2.74882i 0 1.60220 − 2.10546i −2.64041 − 0.857920i 0 2.87042
370.4 −0.496565 + 0.683463i 0 0.397489 + 1.22335i 1.99713 + 2.74882i 0 2.53376 + 0.761606i −2.64041 − 0.857920i 0 −2.87042
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
11.d odd 10 1 inner
21.c even 2 1 inner
33.f even 10 1 inner
77.l even 10 1 inner
231.r odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.bu.e ✓ 32
3.b odd 2 1 inner 693.2.bu.e ✓ 32
7.b odd 2 1 inner 693.2.bu.e ✓ 32
11.d odd 10 1 inner 693.2.bu.e ✓ 32
21.c even 2 1 inner 693.2.bu.e ✓ 32
33.f even 10 1 inner 693.2.bu.e ✓ 32
77.l even 10 1 inner 693.2.bu.e ✓ 32
231.r odd 10 1 inner 693.2.bu.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.bu.e ✓ 32 1.a even 1 1 trivial
693.2.bu.e ✓ 32 3.b odd 2 1 inner
693.2.bu.e ✓ 32 7.b odd 2 1 inner
693.2.bu.e ✓ 32 11.d odd 10 1 inner
693.2.bu.e ✓ 32 21.c even 2 1 inner
693.2.bu.e ✓ 32 33.f even 10 1 inner
693.2.bu.e ✓ 32 77.l even 10 1 inner
693.2.bu.e ✓ 32 231.r odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 8T_{2}^{14} + 30T_{2}^{12} - 38T_{2}^{10} + 579T_{2}^{8} + 178T_{2}^{6} + 275T_{2}^{4} - 27T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display