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Properties

Label 532.2.k.b
Level $532$
Weight $2$
Character orbit 532.k
Analytic conductor $4.248$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [532,2,Mod(121,532)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(532, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("532.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 532 = 2^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 532.k (of order \(3\), 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: \(4.24804138753\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{3} + 12 q^{5} - 6 q^{7} - 15 q^{9} + q^{11} - 7 q^{13} - 2 q^{15} + 3 q^{17} - 19 q^{19} + 12 q^{21} + 8 q^{23} + 16 q^{25} + 20 q^{27} - 22 q^{29} - 7 q^{31} - 14 q^{33} + 18 q^{35} + 9 q^{37}+ \cdots - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0 −1.66499 + 2.88385i 0 2.17686 0 −2.16805 − 1.51643i 0 −4.04441 − 7.00512i 0
121.2 0 −1.49182 + 2.58391i 0 0.446078 0 1.18643 + 2.36482i 0 −2.95105 − 5.11136i 0
121.3 0 −1.00455 + 1.73993i 0 −0.208844 0 0.452777 − 2.60672i 0 −0.518227 − 0.897596i 0
121.4 0 −0.771712 + 1.33665i 0 −2.99584 0 −2.55868 + 0.673181i 0 0.308920 + 0.535065i 0
121.5 0 −0.520631 + 0.901759i 0 3.47279 0 2.62969 + 0.291052i 0 0.957887 + 1.65911i 0
121.6 0 −0.0989313 + 0.171354i 0 2.23136 0 −1.27819 − 2.31651i 0 1.48043 + 2.56417i 0
121.7 0 0.0144506 − 0.0250292i 0 −0.520300 0 −2.55086 + 0.702209i 0 1.49958 + 2.59735i 0
121.8 0 0.536703 − 0.929597i 0 3.52186 0 −1.25747 + 2.32783i 0 0.923900 + 1.60024i 0
121.9 0 0.609809 − 1.05622i 0 −3.23495 0 1.15617 + 2.37976i 0 0.756266 + 1.30989i 0
121.10 0 1.02695 − 1.77872i 0 −2.32366 0 −0.804015 − 2.52063i 0 −0.609239 − 1.05523i 0
121.11 0 1.27630 − 2.21062i 0 0.584291 0 0.854338 + 2.50402i 0 −1.75791 − 3.04478i 0
121.12 0 1.58842 − 2.75122i 0 2.85036 0 1.33785 − 2.28258i 0 −3.54616 − 6.14212i 0
277.1 0 −1.66499 − 2.88385i 0 2.17686 0 −2.16805 + 1.51643i 0 −4.04441 + 7.00512i 0
277.2 0 −1.49182 − 2.58391i 0 0.446078 0 1.18643 − 2.36482i 0 −2.95105 + 5.11136i 0
277.3 0 −1.00455 − 1.73993i 0 −0.208844 0 0.452777 + 2.60672i 0 −0.518227 + 0.897596i 0
277.4 0 −0.771712 − 1.33665i 0 −2.99584 0 −2.55868 − 0.673181i 0 0.308920 − 0.535065i 0
277.5 0 −0.520631 − 0.901759i 0 3.47279 0 2.62969 − 0.291052i 0 0.957887 − 1.65911i 0
277.6 0 −0.0989313 − 0.171354i 0 2.23136 0 −1.27819 + 2.31651i 0 1.48043 − 2.56417i 0
277.7 0 0.0144506 + 0.0250292i 0 −0.520300 0 −2.55086 − 0.702209i 0 1.49958 − 2.59735i 0
277.8 0 0.536703 + 0.929597i 0 3.52186 0 −1.25747 − 2.32783i 0 0.923900 − 1.60024i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 532.2.k.b ✓ 24
7.c even 3 1 532.2.l.b yes 24
19.c even 3 1 532.2.l.b yes 24
133.h even 3 1 inner 532.2.k.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.k.b ✓ 24 1.a even 1 1 trivial
532.2.k.b ✓ 24 133.h even 3 1 inner
532.2.l.b yes 24 7.c even 3 1
532.2.l.b yes 24 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + T_{3}^{23} + 26 T_{3}^{22} + 15 T_{3}^{21} + 425 T_{3}^{20} + 193 T_{3}^{19} + 4117 T_{3}^{18} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(532, [\chi])\). Copy content Toggle raw display