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Properties

Label 688.2.bb.b
Level $688$
Weight $2$
Character orbit 688.bb
Analytic conductor $5.494$
Analytic rank $0$
Dimension $96$
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] = [688,2,Mod(223,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.bb (of order \(14\), degree \(6\), 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: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 12 q^{9} - 34 q^{13} + 14 q^{17} + 32 q^{25} + 28 q^{29} + 12 q^{41} + 14 q^{45} + 112 q^{49} + 14 q^{53} + 80 q^{57} + 14 q^{73} + 14 q^{77} - 140 q^{81} - 42 q^{89} - 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1 0 −0.710701 − 3.11379i 0 0.908788 + 0.724734i 0 −2.79930 0 −6.48766 + 3.12429i 0
223.2 0 −0.606744 − 2.65832i 0 1.21126 + 0.965947i 0 −0.238951 0 −3.99561 + 1.92419i 0
223.3 0 −0.543685 − 2.38204i 0 −3.15515 − 2.51615i 0 −3.48134 0 −2.67561 + 1.28851i 0
223.4 0 −0.381990 − 1.67361i 0 −2.98293 − 2.37881i 0 0.0286281 0 0.0478660 − 0.0230510i 0
223.5 0 −0.378079 − 1.65647i 0 −0.557847 − 0.444868i 0 4.45565 0 0.101956 − 0.0490993i 0
223.6 0 −0.246870 − 1.08161i 0 −0.836233 − 0.666874i 0 −0.304137 0 1.59397 − 0.767616i 0
223.7 0 −0.203002 − 0.889408i 0 2.99103 + 2.38527i 0 −1.21112 0 1.95307 − 0.940549i 0
223.8 0 −0.0140553 − 0.0615802i 0 0.896625 + 0.715034i 0 −4.35237 0 2.69931 − 1.29992i 0
223.9 0 0.0140553 + 0.0615802i 0 0.896625 + 0.715034i 0 4.35237 0 2.69931 − 1.29992i 0
223.10 0 0.203002 + 0.889408i 0 2.99103 + 2.38527i 0 1.21112 0 1.95307 − 0.940549i 0
223.11 0 0.246870 + 1.08161i 0 −0.836233 − 0.666874i 0 0.304137 0 1.59397 − 0.767616i 0
223.12 0 0.378079 + 1.65647i 0 −0.557847 − 0.444868i 0 −4.45565 0 0.101956 − 0.0490993i 0
223.13 0 0.381990 + 1.67361i 0 −2.98293 − 2.37881i 0 −0.0286281 0 0.0478660 − 0.0230510i 0
223.14 0 0.543685 + 2.38204i 0 −3.15515 − 2.51615i 0 3.48134 0 −2.67561 + 1.28851i 0
223.15 0 0.606744 + 2.65832i 0 1.21126 + 0.965947i 0 0.238951 0 −3.99561 + 1.92419i 0
223.16 0 0.710701 + 3.11379i 0 0.908788 + 0.724734i 0 2.79930 0 −6.48766 + 3.12429i 0
303.1 0 −2.92701 − 1.40957i 0 −1.31990 − 0.301258i 0 −0.894145 0 4.71000 + 5.90616i 0
303.2 0 −2.69405 − 1.29739i 0 −3.12296 − 0.712795i 0 4.26097 0 3.70423 + 4.64495i 0
303.3 0 −2.06483 − 0.994370i 0 1.93655 + 0.442006i 0 0.602140 0 1.40429 + 1.76092i 0
303.4 0 −1.98988 − 0.958274i 0 4.01778 + 0.917033i 0 2.60731 0 1.17085 + 1.46820i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
43.f odd 14 1 inner
172.j even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.bb.b ✓ 96
4.b odd 2 1 inner 688.2.bb.b ✓ 96
43.f odd 14 1 inner 688.2.bb.b ✓ 96
172.j even 14 1 inner 688.2.bb.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
688.2.bb.b ✓ 96 1.a even 1 1 trivial
688.2.bb.b ✓ 96 4.b odd 2 1 inner
688.2.bb.b ✓ 96 43.f odd 14 1 inner
688.2.bb.b ✓ 96 172.j even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{96} + 30 T_{3}^{94} + 647 T_{3}^{92} + 10802 T_{3}^{90} + 155319 T_{3}^{88} + \cdots + 3616238492881 \) acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\). Copy content Toggle raw display