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Properties

Label 1680.2.bl.d
Level $1680$
Weight $2$
Character orbit 1680.bl
Analytic conductor $13.415$
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] = [1680,2,Mod(127,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.bl (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: \(13.4148675396\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{13} - 8 q^{17} - 24 q^{21} + 8 q^{25} - 24 q^{33} + 24 q^{37} + 32 q^{41} + 8 q^{45} - 16 q^{53} - 8 q^{57} - 32 q^{61} + 32 q^{65} - 48 q^{73} - 24 q^{77} - 24 q^{81} + 72 q^{85} - 40 q^{93} - 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 0 −0.707107 − 0.707107i 0 2.14388 − 0.635431i 0 0.707107 − 0.707107i 0 1.00000i 0
127.2 0 −0.707107 − 0.707107i 0 1.85671 + 1.24605i 0 0.707107 − 0.707107i 0 1.00000i 0
127.3 0 −0.707107 − 0.707107i 0 −2.06542 − 0.856762i 0 0.707107 − 0.707107i 0 1.00000i 0
127.4 0 −0.707107 − 0.707107i 0 −0.970109 + 2.01467i 0 0.707107 − 0.707107i 0 1.00000i 0
127.5 0 −0.707107 − 0.707107i 0 −1.55110 − 1.61062i 0 0.707107 − 0.707107i 0 1.00000i 0
127.6 0 −0.707107 − 0.707107i 0 0.586039 − 2.15791i 0 0.707107 − 0.707107i 0 1.00000i 0
127.7 0 0.707107 + 0.707107i 0 1.85671 + 1.24605i 0 −0.707107 + 0.707107i 0 1.00000i 0
127.8 0 0.707107 + 0.707107i 0 2.14388 − 0.635431i 0 −0.707107 + 0.707107i 0 1.00000i 0
127.9 0 0.707107 + 0.707107i 0 0.586039 − 2.15791i 0 −0.707107 + 0.707107i 0 1.00000i 0
127.10 0 0.707107 + 0.707107i 0 −1.55110 − 1.61062i 0 −0.707107 + 0.707107i 0 1.00000i 0
127.11 0 0.707107 + 0.707107i 0 −0.970109 + 2.01467i 0 −0.707107 + 0.707107i 0 1.00000i 0
127.12 0 0.707107 + 0.707107i 0 −2.06542 − 0.856762i 0 −0.707107 + 0.707107i 0 1.00000i 0
463.1 0 −0.707107 + 0.707107i 0 2.14388 + 0.635431i 0 0.707107 + 0.707107i 0 − 1.00000i 0
463.2 0 −0.707107 + 0.707107i 0 1.85671 − 1.24605i 0 0.707107 + 0.707107i 0 − 1.00000i 0
463.3 0 −0.707107 + 0.707107i 0 −2.06542 + 0.856762i 0 0.707107 + 0.707107i 0 − 1.00000i 0
463.4 0 −0.707107 + 0.707107i 0 −0.970109 − 2.01467i 0 0.707107 + 0.707107i 0 − 1.00000i 0
463.5 0 −0.707107 + 0.707107i 0 −1.55110 + 1.61062i 0 0.707107 + 0.707107i 0 − 1.00000i 0
463.6 0 −0.707107 + 0.707107i 0 0.586039 + 2.15791i 0 0.707107 + 0.707107i 0 − 1.00000i 0
463.7 0 0.707107 − 0.707107i 0 1.85671 − 1.24605i 0 −0.707107 − 0.707107i 0 − 1.00000i 0
463.8 0 0.707107 − 0.707107i 0 2.14388 + 0.635431i 0 −0.707107 − 0.707107i 0 − 1.00000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.bl.d ✓ 24
4.b odd 2 1 inner 1680.2.bl.d ✓ 24
5.c odd 4 1 inner 1680.2.bl.d ✓ 24
20.e even 4 1 inner 1680.2.bl.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1680.2.bl.d ✓ 24 1.a even 1 1 trivial
1680.2.bl.d ✓ 24 4.b odd 2 1 inner
1680.2.bl.d ✓ 24 5.c odd 4 1 inner
1680.2.bl.d ✓ 24 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

\( T_{11}^{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{12} + 8 T_{13}^{11} + 32 T_{13}^{10} + 32 T_{13}^{9} + 1520 T_{13}^{8} + 11360 T_{13}^{7} + \cdots + 4000000 \) Copy content Toggle raw display