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Properties

Label 208.4.k.c
Level $208$
Weight $4$
Character orbit 208.k
Analytic conductor $12.272$
Analytic rank $0$
Dimension $28$
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] = [208,4,Mod(31,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.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: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{5} - 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{5} - 252 q^{9} + 80 q^{13} + 176 q^{21} + 864 q^{29} + 928 q^{33} - 548 q^{37} - 1852 q^{41} + 1460 q^{45} - 2152 q^{53} + 2288 q^{57} - 936 q^{61} - 1948 q^{65} + 540 q^{73} + 732 q^{81} + 2056 q^{85} - 2444 q^{89} - 912 q^{93} + 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 − 9.86664i 0 −5.55393 + 5.55393i 0 4.04572 − 4.04572i 0 −70.3507 0
31.2 0 − 7.62963i 0 5.44390 − 5.44390i 0 −22.3158 + 22.3158i 0 −31.2112 0
31.3 0 − 6.16186i 0 −15.3461 + 15.3461i 0 −3.13430 + 3.13430i 0 −10.9686 0
31.4 0 − 5.13423i 0 6.69756 − 6.69756i 0 12.7815 − 12.7815i 0 0.639662 0
31.5 0 − 4.33333i 0 13.0206 − 13.0206i 0 19.0375 − 19.0375i 0 8.22222 0
31.6 0 − 3.40333i 0 0.265199 − 0.265199i 0 −13.0952 + 13.0952i 0 15.4174 0
31.7 0 − 1.32242i 0 −5.52719 + 5.52719i 0 1.59213 − 1.59213i 0 25.2512 0
31.8 0 1.32242i 0 −5.52719 + 5.52719i 0 −1.59213 + 1.59213i 0 25.2512 0
31.9 0 3.40333i 0 0.265199 − 0.265199i 0 13.0952 − 13.0952i 0 15.4174 0
31.10 0 4.33333i 0 13.0206 − 13.0206i 0 −19.0375 + 19.0375i 0 8.22222 0
31.11 0 5.13423i 0 6.69756 − 6.69756i 0 −12.7815 + 12.7815i 0 0.639662 0
31.12 0 6.16186i 0 −15.3461 + 15.3461i 0 3.13430 − 3.13430i 0 −10.9686 0
31.13 0 7.62963i 0 5.44390 − 5.44390i 0 22.3158 − 22.3158i 0 −31.2112 0
31.14 0 9.86664i 0 −5.55393 + 5.55393i 0 −4.04572 + 4.04572i 0 −70.3507 0
47.1 0 − 9.86664i 0 −5.55393 − 5.55393i 0 −4.04572 − 4.04572i 0 −70.3507 0
47.2 0 − 7.62963i 0 5.44390 + 5.44390i 0 22.3158 + 22.3158i 0 −31.2112 0
47.3 0 − 6.16186i 0 −15.3461 − 15.3461i 0 3.13430 + 3.13430i 0 −10.9686 0
47.4 0 − 5.13423i 0 6.69756 + 6.69756i 0 −12.7815 − 12.7815i 0 0.639662 0
47.5 0 − 4.33333i 0 13.0206 + 13.0206i 0 −19.0375 − 19.0375i 0 8.22222 0
47.6 0 − 3.40333i 0 0.265199 + 0.265199i 0 13.0952 + 13.0952i 0 15.4174 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.k.c ✓ 28
4.b odd 2 1 inner 208.4.k.c ✓ 28
13.d odd 4 1 inner 208.4.k.c ✓ 28
52.f even 4 1 inner 208.4.k.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
208.4.k.c ✓ 28 1.a even 1 1 trivial
208.4.k.c ✓ 28 4.b odd 2 1 inner
208.4.k.c ✓ 28 13.d odd 4 1 inner
208.4.k.c ✓ 28 52.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 252 T_{3}^{12} + 24006 T_{3}^{10} + 1115540 T_{3}^{8} + 26972049 T_{3}^{6} + \cdots + 2157293568 \) acting on \(S_{4}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display