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Properties

Label 3360.2.z.d
Level $3360$
Weight $2$
Character orbit 3360.z
Analytic conductor $26.830$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3360,2,Mod(1231,3360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3360.1231");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.z (of order \(2\), degree \(1\), not 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: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 840)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{5} - 4 q^{7} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{5} - 4 q^{7} - 28 q^{9} - 8 q^{13} + 28 q^{25} - 24 q^{31} - 4 q^{35} - 24 q^{43} - 28 q^{45} + 24 q^{47} + 28 q^{49} + 8 q^{57} - 16 q^{61} + 4 q^{63} - 8 q^{65} + 8 q^{67} - 24 q^{69} + 16 q^{77} + 28 q^{81} - 16 q^{87} - 48 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1231.1 0 − 1.00000i 0 1.00000 0 −2.61024 − 0.432017i 0 −1.00000 0
1231.2 0 1.00000i 0 1.00000 0 −2.61024 + 0.432017i 0 −1.00000 0
1231.3 0 − 1.00000i 0 1.00000 0 2.64060 − 0.165018i 0 −1.00000 0
1231.4 0 1.00000i 0 1.00000 0 2.64060 + 0.165018i 0 −1.00000 0
1231.5 0 − 1.00000i 0 1.00000 0 −0.658347 − 2.56253i 0 −1.00000 0
1231.6 0 1.00000i 0 1.00000 0 −0.658347 + 2.56253i 0 −1.00000 0
1231.7 0 − 1.00000i 0 1.00000 0 −1.57556 + 2.12547i 0 −1.00000 0
1231.8 0 1.00000i 0 1.00000 0 −1.57556 − 2.12547i 0 −1.00000 0
1231.9 0 − 1.00000i 0 1.00000 0 −2.48628 − 0.904667i 0 −1.00000 0
1231.10 0 1.00000i 0 1.00000 0 −2.48628 + 0.904667i 0 −1.00000 0
1231.11 0 − 1.00000i 0 1.00000 0 −2.37591 − 1.16407i 0 −1.00000 0
1231.12 0 1.00000i 0 1.00000 0 −2.37591 + 1.16407i 0 −1.00000 0
1231.13 0 − 1.00000i 0 1.00000 0 −1.92209 + 1.81812i 0 −1.00000 0
1231.14 0 1.00000i 0 1.00000 0 −1.92209 − 1.81812i 0 −1.00000 0
1231.15 0 − 1.00000i 0 1.00000 0 1.33527 − 2.28409i 0 −1.00000 0
1231.16 0 1.00000i 0 1.00000 0 1.33527 + 2.28409i 0 −1.00000 0
1231.17 0 − 1.00000i 0 1.00000 0 1.64136 + 2.07507i 0 −1.00000 0
1231.18 0 1.00000i 0 1.00000 0 1.64136 − 2.07507i 0 −1.00000 0
1231.19 0 − 1.00000i 0 1.00000 0 0.864047 + 2.50068i 0 −1.00000 0
1231.20 0 1.00000i 0 1.00000 0 0.864047 − 2.50068i 0 −1.00000 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1231.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.2.z.d 28
4.b odd 2 1 840.2.z.d yes 28
7.b odd 2 1 3360.2.z.c 28
8.b even 2 1 840.2.z.c ✓ 28
8.d odd 2 1 3360.2.z.c 28
28.d even 2 1 840.2.z.c ✓ 28
56.e even 2 1 inner 3360.2.z.d 28
56.h odd 2 1 840.2.z.d yes 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.z.c ✓ 28 8.b even 2 1
840.2.z.c ✓ 28 28.d even 2 1
840.2.z.d yes 28 4.b odd 2 1
840.2.z.d yes 28 56.h odd 2 1
3360.2.z.c 28 7.b odd 2 1
3360.2.z.c 28 8.d odd 2 1
3360.2.z.d 28 1.a even 1 1 trivial
3360.2.z.d 28 56.e even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11}^{14} - 84 T_{11}^{12} + 40 T_{11}^{11} + 2516 T_{11}^{10} - 2512 T_{11}^{9} - 33200 T_{11}^{8} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{13}^{14} + 4 T_{13}^{13} - 68 T_{13}^{12} - 176 T_{13}^{11} + 1812 T_{13}^{10} + 2160 T_{13}^{9} + \cdots - 32768 \) Copy content Toggle raw display