Properties

Label 1764.4.t.c
Level $1764$
Weight $4$
Character orbit 1764.t
Analytic conductor $104.079$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,4,Mod(521,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.521");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.t (of order \(6\), degree \(2\), 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: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 888 q^{25} - 864 q^{37} - 2496 q^{43} - 1056 q^{67} - 16128 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1 0 0 0 −10.7460 18.6127i 0 0 0 0 0
521.2 0 0 0 −10.7460 18.6127i 0 0 0 0 0
521.3 0 0 0 −7.13585 12.3596i 0 0 0 0 0
521.4 0 0 0 −7.13585 12.3596i 0 0 0 0 0
521.5 0 0 0 −6.78515 11.7522i 0 0 0 0 0
521.6 0 0 0 −6.78515 11.7522i 0 0 0 0 0
521.7 0 0 0 −4.41277 7.64315i 0 0 0 0 0
521.8 0 0 0 −4.41277 7.64315i 0 0 0 0 0
521.9 0 0 0 −3.27916 5.67967i 0 0 0 0 0
521.10 0 0 0 −3.27916 5.67967i 0 0 0 0 0
521.11 0 0 0 −0.582251 1.00849i 0 0 0 0 0
521.12 0 0 0 −0.582251 1.00849i 0 0 0 0 0
521.13 0 0 0 0.582251 + 1.00849i 0 0 0 0 0
521.14 0 0 0 0.582251 + 1.00849i 0 0 0 0 0
521.15 0 0 0 3.27916 + 5.67967i 0 0 0 0 0
521.16 0 0 0 3.27916 + 5.67967i 0 0 0 0 0
521.17 0 0 0 4.41277 + 7.64315i 0 0 0 0 0
521.18 0 0 0 4.41277 + 7.64315i 0 0 0 0 0
521.19 0 0 0 6.78515 + 11.7522i 0 0 0 0 0
521.20 0 0 0 6.78515 + 11.7522i 0 0 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 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 1764.4.t.c 48
3.b odd 2 1 inner 1764.4.t.c 48
7.b odd 2 1 inner 1764.4.t.c 48
7.c even 3 1 1764.4.f.b 24
7.c even 3 1 inner 1764.4.t.c 48
7.d odd 6 1 1764.4.f.b 24
7.d odd 6 1 inner 1764.4.t.c 48
21.c even 2 1 inner 1764.4.t.c 48
21.g even 6 1 1764.4.f.b 24
21.g even 6 1 inner 1764.4.t.c 48
21.h odd 6 1 1764.4.f.b 24
21.h odd 6 1 inner 1764.4.t.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.4.f.b 24 7.c even 3 1
1764.4.f.b 24 7.d odd 6 1
1764.4.f.b 24 21.g even 6 1
1764.4.f.b 24 21.h odd 6 1
1764.4.t.c 48 1.a even 1 1 trivial
1764.4.t.c 48 3.b odd 2 1 inner
1764.4.t.c 48 7.b odd 2 1 inner
1764.4.t.c 48 7.c even 3 1 inner
1764.4.t.c 48 7.d odd 6 1 inner
1764.4.t.c 48 21.c even 2 1 inner
1764.4.t.c 48 21.g even 6 1 inner
1764.4.t.c 48 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 972 T_{5}^{22} + 620730 T_{5}^{20} + 221373352 T_{5}^{18} + 56634544824 T_{5}^{16} + 9623457152688 T_{5}^{14} + \cdots + 61\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display