↑

Properties

Label 1332.2.p.a
Level $1332$
Weight $2$
Character orbit 1332.p
Analytic conductor $10.636$
Analytic rank $0$
Dimension $28$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1332,2,Mod(413,1332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1332, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1332.413");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1332 = 2^{2} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1332.p (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: \(10.6360735492\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{13} - 8 q^{19} - 24 q^{37} - 16 q^{43} + 28 q^{49} + 16 q^{55} - 4 q^{61} - 40 q^{79} - 8 q^{91} + 52 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
413.1 0 0 0 −2.57235 + 2.57235i 0 −4.47360 0 0 0
413.2 0 0 0 −2.32864 + 2.32864i 0 −0.751489 0 0 0
413.3 0 0 0 −1.79775 + 1.79775i 0 3.98763 0 0 0
413.4 0 0 0 −1.60363 + 1.60363i 0 −0.770063 0 0 0
413.5 0 0 0 −1.03246 + 1.03246i 0 0.562775 0 0 0
413.6 0 0 0 −0.806150 + 0.806150i 0 −2.24138 0 0 0
413.7 0 0 0 −0.664147 + 0.664147i 0 3.68613 0 0 0
413.8 0 0 0 0.664147 − 0.664147i 0 3.68613 0 0 0
413.9 0 0 0 0.806150 − 0.806150i 0 −2.24138 0 0 0
413.10 0 0 0 1.03246 − 1.03246i 0 0.562775 0 0 0
413.11 0 0 0 1.60363 − 1.60363i 0 −0.770063 0 0 0
413.12 0 0 0 1.79775 − 1.79775i 0 3.98763 0 0 0
413.13 0 0 0 2.32864 − 2.32864i 0 −0.751489 0 0 0
413.14 0 0 0 2.57235 − 2.57235i 0 −4.47360 0 0 0
845.1 0 0 0 −2.57235 − 2.57235i 0 −4.47360 0 0 0
845.2 0 0 0 −2.32864 − 2.32864i 0 −0.751489 0 0 0
845.3 0 0 0 −1.79775 − 1.79775i 0 3.98763 0 0 0
845.4 0 0 0 −1.60363 − 1.60363i 0 −0.770063 0 0 0
845.5 0 0 0 −1.03246 − 1.03246i 0 0.562775 0 0 0
845.6 0 0 0 −0.806150 − 0.806150i 0 −2.24138 0 0 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 413.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.d odd 4 1 inner
111.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1332.2.p.a ✓ 28
3.b odd 2 1 inner 1332.2.p.a ✓ 28
37.d odd 4 1 inner 1332.2.p.a ✓ 28
111.g even 4 1 inner 1332.2.p.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1332.2.p.a ✓ 28 1.a even 1 1 trivial
1332.2.p.a ✓ 28 3.b odd 2 1 inner
1332.2.p.a ✓ 28 37.d odd 4 1 inner
1332.2.p.a ✓ 28 111.g even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1332, [\chi])\).