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Properties

Label 700.2.k.a
Level $700$
Weight $2$
Character orbit 700.k
Analytic conductor $5.590$
Analytic rank $0$
Dimension $24$
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] = [700,2,Mod(43,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.k (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: \(5.58952814149\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{6} + 28 q^{16} + 56 q^{26} - 44 q^{36} - 80 q^{41} - 80 q^{46} + 4 q^{56} + 64 q^{61} - 72 q^{66} + 8 q^{76} + 8 q^{81} + 64 q^{86} - 200 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.40857 + 0.126161i −0.204473 + 0.204473i 1.96817 − 0.355416i 0 0.262219 − 0.313812i −0.707107 − 0.707107i −2.72747 + 0.748936i 2.91638i 0
43.2 −1.38856 − 0.268135i 1.76019 − 1.76019i 1.85621 + 0.744644i 0 −2.91611 + 1.97217i 0.707107 + 0.707107i −2.37779 − 1.53170i − 3.19656i 0
43.3 −1.15247 + 0.819645i −1.96467 + 1.96467i 0.656365 − 1.88923i 0 0.653887 − 3.87454i 0.707107 + 0.707107i 0.792056 + 2.71526i − 4.71982i 0
43.4 −0.819645 + 1.15247i 1.96467 − 1.96467i −0.656365 − 1.88923i 0 0.653887 + 3.87454i −0.707107 − 0.707107i 2.71526 + 0.792056i − 4.71982i 0
43.5 −0.268135 − 1.38856i 1.76019 − 1.76019i −1.85621 + 0.744644i 0 −2.91611 − 1.97217i 0.707107 + 0.707107i 1.53170 + 2.37779i − 3.19656i 0
43.6 −0.126161 + 1.40857i 0.204473 − 0.204473i −1.96817 − 0.355416i 0 0.262219 + 0.313812i 0.707107 + 0.707107i 0.748936 − 2.72747i 2.91638i 0
43.7 0.126161 − 1.40857i −0.204473 + 0.204473i −1.96817 − 0.355416i 0 0.262219 + 0.313812i −0.707107 − 0.707107i −0.748936 + 2.72747i 2.91638i 0
43.8 0.268135 + 1.38856i −1.76019 + 1.76019i −1.85621 + 0.744644i 0 −2.91611 − 1.97217i −0.707107 − 0.707107i −1.53170 − 2.37779i − 3.19656i 0
43.9 0.819645 − 1.15247i −1.96467 + 1.96467i −0.656365 − 1.88923i 0 0.653887 + 3.87454i 0.707107 + 0.707107i −2.71526 − 0.792056i − 4.71982i 0
43.10 1.15247 − 0.819645i 1.96467 − 1.96467i 0.656365 − 1.88923i 0 0.653887 − 3.87454i −0.707107 − 0.707107i −0.792056 − 2.71526i − 4.71982i 0
43.11 1.38856 + 0.268135i −1.76019 + 1.76019i 1.85621 + 0.744644i 0 −2.91611 + 1.97217i −0.707107 − 0.707107i 2.37779 + 1.53170i − 3.19656i 0
43.12 1.40857 − 0.126161i 0.204473 − 0.204473i 1.96817 − 0.355416i 0 0.262219 − 0.313812i 0.707107 + 0.707107i 2.72747 − 0.748936i 2.91638i 0
407.1 −1.40857 − 0.126161i −0.204473 − 0.204473i 1.96817 + 0.355416i 0 0.262219 + 0.313812i −0.707107 + 0.707107i −2.72747 − 0.748936i − 2.91638i 0
407.2 −1.38856 + 0.268135i 1.76019 + 1.76019i 1.85621 − 0.744644i 0 −2.91611 − 1.97217i 0.707107 − 0.707107i −2.37779 + 1.53170i 3.19656i 0
407.3 −1.15247 − 0.819645i −1.96467 − 1.96467i 0.656365 + 1.88923i 0 0.653887 + 3.87454i 0.707107 − 0.707107i 0.792056 − 2.71526i 4.71982i 0
407.4 −0.819645 − 1.15247i 1.96467 + 1.96467i −0.656365 + 1.88923i 0 0.653887 − 3.87454i −0.707107 + 0.707107i 2.71526 − 0.792056i 4.71982i 0
407.5 −0.268135 + 1.38856i 1.76019 + 1.76019i −1.85621 − 0.744644i 0 −2.91611 + 1.97217i 0.707107 − 0.707107i 1.53170 − 2.37779i 3.19656i 0
407.6 −0.126161 − 1.40857i 0.204473 + 0.204473i −1.96817 + 0.355416i 0 0.262219 − 0.313812i 0.707107 − 0.707107i 0.748936 + 2.72747i − 2.91638i 0
407.7 0.126161 + 1.40857i −0.204473 − 0.204473i −1.96817 + 0.355416i 0 0.262219 − 0.313812i −0.707107 + 0.707107i −0.748936 − 2.72747i − 2.91638i 0
407.8 0.268135 − 1.38856i −1.76019 − 1.76019i −1.85621 − 0.744644i 0 −2.91611 + 1.97217i −0.707107 + 0.707107i −1.53170 + 2.37779i 3.19656i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.k.a ✓ 24
4.b odd 2 1 inner 700.2.k.a ✓ 24
5.b even 2 1 inner 700.2.k.a ✓ 24
5.c odd 4 2 inner 700.2.k.a ✓ 24
20.d odd 2 1 inner 700.2.k.a ✓ 24
20.e even 4 2 inner 700.2.k.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.k.a ✓ 24 1.a even 1 1 trivial
700.2.k.a ✓ 24 4.b odd 2 1 inner
700.2.k.a ✓ 24 5.b even 2 1 inner
700.2.k.a ✓ 24 5.c odd 4 2 inner
700.2.k.a ✓ 24 20.d odd 2 1 inner
700.2.k.a ✓ 24 20.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 98T_{3}^{8} + 2289T_{3}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display