Properties

Label 1764.4.a.c
Level $1764$
Weight $4$
Character orbit 1764.a
Self dual yes
Analytic conductor $104.079$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,4,Mod(1,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1");
 
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.a (trivial)

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: yes
Analytic conductor: \(104.079369250\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{5} + 40 q^{11} + 12 q^{13} - 58 q^{17} - 26 q^{19} + 64 q^{23} - 61 q^{25} + 62 q^{29} - 252 q^{31} + 26 q^{37} + 6 q^{41} + 416 q^{43} - 396 q^{47} + 450 q^{53} - 320 q^{55} + 274 q^{59} + 576 q^{61} - 96 q^{65} - 476 q^{67} + 448 q^{71} + 158 q^{73} - 936 q^{79} + 530 q^{83} + 464 q^{85} - 390 q^{89} + 208 q^{95} - 214 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −8.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.a.c 1
3.b odd 2 1 196.4.a.d 1
7.b odd 2 1 252.4.a.d 1
7.c even 3 2 1764.4.k.m 2
7.d odd 6 2 1764.4.k.d 2
12.b even 2 1 784.4.a.a 1
21.c even 2 1 28.4.a.a 1
21.g even 6 2 196.4.e.f 2
21.h odd 6 2 196.4.e.a 2
28.d even 2 1 1008.4.a.o 1
84.h odd 2 1 112.4.a.g 1
105.g even 2 1 700.4.a.n 1
105.k odd 4 2 700.4.e.a 2
168.e odd 2 1 448.4.a.a 1
168.i even 2 1 448.4.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 21.c even 2 1
112.4.a.g 1 84.h odd 2 1
196.4.a.d 1 3.b odd 2 1
196.4.e.a 2 21.h odd 6 2
196.4.e.f 2 21.g even 6 2
252.4.a.d 1 7.b odd 2 1
448.4.a.a 1 168.e odd 2 1
448.4.a.p 1 168.i even 2 1
700.4.a.n 1 105.g even 2 1
700.4.e.a 2 105.k odd 4 2
784.4.a.a 1 12.b even 2 1
1008.4.a.o 1 28.d even 2 1
1764.4.a.c 1 1.a even 1 1 trivial
1764.4.k.d 2 7.d odd 6 2
1764.4.k.m 2 7.c even 3 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1764))\):

\( T_{5} + 8 \) Copy content Toggle raw display
\( T_{11} - 40 \) Copy content Toggle raw display
\( T_{13} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 8 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 40 \) Copy content Toggle raw display
$13$ \( T - 12 \) Copy content Toggle raw display
$17$ \( T + 58 \) Copy content Toggle raw display
$19$ \( T + 26 \) Copy content Toggle raw display
$23$ \( T - 64 \) Copy content Toggle raw display
$29$ \( T - 62 \) Copy content Toggle raw display
$31$ \( T + 252 \) Copy content Toggle raw display
$37$ \( T - 26 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 416 \) Copy content Toggle raw display
$47$ \( T + 396 \) Copy content Toggle raw display
$53$ \( T - 450 \) Copy content Toggle raw display
$59$ \( T - 274 \) Copy content Toggle raw display
$61$ \( T - 576 \) Copy content Toggle raw display
$67$ \( T + 476 \) Copy content Toggle raw display
$71$ \( T - 448 \) Copy content Toggle raw display
$73$ \( T - 158 \) Copy content Toggle raw display
$79$ \( T + 936 \) Copy content Toggle raw display
$83$ \( T - 530 \) Copy content Toggle raw display
$89$ \( T + 390 \) Copy content Toggle raw display
$97$ \( T + 214 \) Copy content Toggle raw display
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