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Properties

Label 252.12.k.e
Level $252$
Weight $12$
Character orbit 252.k
Analytic conductor $193.622$
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] = [252,12,Mod(37,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 252.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: \(193.622481501\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 85666 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 85666 q^{7} + 2794700 q^{13} + 37414826 q^{19} - 135469250 q^{25} + 71052098 q^{31} - 395898874 q^{37} + 980170268 q^{43} - 1253333270 q^{49} - 8844657360 q^{55} + 16692685100 q^{61} + 10934009870 q^{67} + 25578690650 q^{73} - 113974174090 q^{79} + 111789733440 q^{85} - 267181531078 q^{91} + 279658007384 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 0 0 0 −5969.07 + 10338.7i 0 −19562.4 + 39932.9i 0 0 0
37.2 0 0 0 −5383.85 + 9325.11i 0 38734.1 − 21840.2i 0 0 0
37.3 0 0 0 −4415.92 + 7648.60i 0 −36565.0 − 25304.7i 0 0 0
37.4 0 0 0 −2661.82 + 4610.40i 0 −34772.1 + 27717.0i 0 0 0
37.5 0 0 0 −2320.27 + 4018.82i 0 9109.27 − 43524.1i 0 0 0
37.6 0 0 0 −1749.37 + 3029.99i 0 22921.7 + 38104.1i 0 0 0
37.7 0 0 0 −1654.60 + 2865.85i 0 41550.9 + 15838.2i 0 0 0
37.8 0 0 0 1654.60 − 2865.85i 0 41550.9 + 15838.2i 0 0 0
37.9 0 0 0 1749.37 − 3029.99i 0 22921.7 + 38104.1i 0 0 0
37.10 0 0 0 2320.27 − 4018.82i 0 9109.27 − 43524.1i 0 0 0
37.11 0 0 0 2661.82 − 4610.40i 0 −34772.1 + 27717.0i 0 0 0
37.12 0 0 0 4415.92 − 7648.60i 0 −36565.0 − 25304.7i 0 0 0
37.13 0 0 0 5383.85 − 9325.11i 0 38734.1 − 21840.2i 0 0 0
37.14 0 0 0 5969.07 − 10338.7i 0 −19562.4 + 39932.9i 0 0 0
109.1 0 0 0 −5969.07 − 10338.7i 0 −19562.4 − 39932.9i 0 0 0
109.2 0 0 0 −5383.85 − 9325.11i 0 38734.1 + 21840.2i 0 0 0
109.3 0 0 0 −4415.92 − 7648.60i 0 −36565.0 + 25304.7i 0 0 0
109.4 0 0 0 −2661.82 − 4610.40i 0 −34772.1 − 27717.0i 0 0 0
109.5 0 0 0 −2320.27 − 4018.82i 0 9109.27 + 43524.1i 0 0 0
109.6 0 0 0 −1749.37 − 3029.99i 0 22921.7 − 38104.1i 0 0 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.12.k.e ✓ 28
3.b odd 2 1 inner 252.12.k.e ✓ 28
7.c even 3 1 inner 252.12.k.e ✓ 28
21.h odd 6 1 inner 252.12.k.e ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.12.k.e ✓ 28 1.a even 1 1 trivial
252.12.k.e ✓ 28 3.b odd 2 1 inner
252.12.k.e ✓ 28 7.c even 3 1 inner
252.12.k.e ✓ 28 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{28} + 409531500 T_{5}^{26} + \cdots + 11\!\cdots\!00 \) acting on \(S_{12}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display