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Properties

Label 2624.2.b.f
Level $2624$
Weight $2$
Character orbit 2624.b
Analytic conductor $20.953$
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] = [2624,2,Mod(1313,2624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2624.1313");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2624 = 2^{6} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2624.b (of order \(2\), degree \(1\), 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: \(20.9527454904\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 28 q^{9} - 8 q^{17} - 36 q^{25} - 64 q^{33} - 28 q^{41} - 20 q^{49} - 80 q^{57} - 8 q^{73} - 20 q^{81} - 72 q^{89} - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1313.1 0 − 3.05092i 0 − 1.75193i 0 2.63946 0 −6.30810 0
1313.2 0 − 3.05092i 0 1.75193i 0 −2.63946 0 −6.30810 0
1313.3 0 − 2.91999i 0 − 1.28166i 0 −3.45556 0 −5.52634 0
1313.4 0 − 2.91999i 0 1.28166i 0 3.45556 0 −5.52634 0
1313.5 0 − 1.76118i 0 − 2.72137i 0 3.26481 0 −0.101746 0
1313.6 0 − 1.76118i 0 2.72137i 0 −3.26481 0 −0.101746 0
1313.7 0 − 1.70996i 0 − 3.02571i 0 −0.654512 0 0.0760526 0
1313.8 0 − 1.70996i 0 3.02571i 0 0.654512 0 0.0760526 0
1313.9 0 − 1.35057i 0 − 4.15097i 0 0.942308 0 1.17597 0
1313.10 0 − 1.35057i 0 4.15097i 0 −0.942308 0 1.17597 0
1313.11 0 − 1.28404i 0 − 0.775188i 0 −3.59076 0 1.35125 0
1313.12 0 − 1.28404i 0 0.775188i 0 3.59076 0 1.35125 0
1313.13 0 − 0.816753i 0 − 2.21266i 0 0.472765 0 2.33292 0
1313.14 0 − 0.816753i 0 2.21266i 0 −0.472765 0 2.33292 0
1313.15 0 0.816753i 0 − 2.21266i 0 −0.472765 0 2.33292 0
1313.16 0 0.816753i 0 2.21266i 0 0.472765 0 2.33292 0
1313.17 0 1.28404i 0 − 0.775188i 0 3.59076 0 1.35125 0
1313.18 0 1.28404i 0 0.775188i 0 −3.59076 0 1.35125 0
1313.19 0 1.35057i 0 − 4.15097i 0 −0.942308 0 1.17597 0
1313.20 0 1.35057i 0 4.15097i 0 0.942308 0 1.17597 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1313.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2624.2.b.f ✓ 28
4.b odd 2 1 inner 2624.2.b.f ✓ 28
8.b even 2 1 inner 2624.2.b.f ✓ 28
8.d odd 2 1 inner 2624.2.b.f ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2624.2.b.f ✓ 28 1.a even 1 1 trivial
2624.2.b.f ✓ 28 4.b odd 2 1 inner
2624.2.b.f ✓ 28 8.b even 2 1 inner
2624.2.b.f ✓ 28 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2624, [\chi])\):

\( T_{3}^{14} + 28T_{3}^{12} + 300T_{3}^{10} + 1580T_{3}^{8} + 4460T_{3}^{6} + 6780T_{3}^{4} + 5116T_{3}^{2} + 1444 \) Copy content Toggle raw display
\( T_{7}^{14} - 44T_{7}^{12} + 732T_{7}^{10} - 5612T_{7}^{8} + 18904T_{7}^{6} - 20736T_{7}^{4} + 8100T_{7}^{2} - 972 \) Copy content Toggle raw display
\( T_{23}^{14} - 180 T_{23}^{12} + 12096 T_{23}^{10} - 380160 T_{23}^{8} + 5676480 T_{23}^{6} + \cdots - 20155392 \) Copy content Toggle raw display