Properties

Label 1764.2.a.l
Level $1764$
Weight $2$
Character orbit 1764.a
Self dual yes
Analytic conductor $14.086$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(14.0856109166\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 196)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} - 4 q^{11} - 3 \beta q^{13} + \beta q^{17} - 2 \beta q^{19} + 4 q^{23} - 3 q^{25} - 8 q^{29} - 8 q^{37} - 5 \beta q^{41} - 4 q^{43} + 4 \beta q^{47} - 10 q^{53} - 4 \beta q^{55} + 10 \beta q^{59} + 5 \beta q^{61} - 6 q^{65} + 5 \beta q^{73} + 8 q^{79} - 10 \beta q^{83} + 2 q^{85} + 5 \beta q^{89} - 4 q^{95} + \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{11} + 8 q^{23} - 6 q^{25} - 16 q^{29} - 16 q^{37} - 8 q^{43} - 20 q^{53} - 12 q^{65} + 16 q^{79} + 4 q^{85} - 8 q^{95}+O(q^{100}) \) 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
−1.41421
1.41421
0 0 0 −1.41421 0 0 0 0 0
1.2 0 0 0 1.41421 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.a.l 2
3.b odd 2 1 196.2.a.c 2
4.b odd 2 1 7056.2.a.cr 2
7.b odd 2 1 inner 1764.2.a.l 2
7.c even 3 2 1764.2.k.l 4
7.d odd 6 2 1764.2.k.l 4
12.b even 2 1 784.2.a.m 2
15.d odd 2 1 4900.2.a.y 2
15.e even 4 2 4900.2.e.p 4
21.c even 2 1 196.2.a.c 2
21.g even 6 2 196.2.e.b 4
21.h odd 6 2 196.2.e.b 4
24.f even 2 1 3136.2.a.bs 2
24.h odd 2 1 3136.2.a.br 2
28.d even 2 1 7056.2.a.cr 2
84.h odd 2 1 784.2.a.m 2
84.j odd 6 2 784.2.i.l 4
84.n even 6 2 784.2.i.l 4
105.g even 2 1 4900.2.a.y 2
105.k odd 4 2 4900.2.e.p 4
168.e odd 2 1 3136.2.a.bs 2
168.i even 2 1 3136.2.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.a.c 2 3.b odd 2 1
196.2.a.c 2 21.c even 2 1
196.2.e.b 4 21.g even 6 2
196.2.e.b 4 21.h odd 6 2
784.2.a.m 2 12.b even 2 1
784.2.a.m 2 84.h odd 2 1
784.2.i.l 4 84.j odd 6 2
784.2.i.l 4 84.n even 6 2
1764.2.a.l 2 1.a even 1 1 trivial
1764.2.a.l 2 7.b odd 2 1 inner
1764.2.k.l 4 7.c even 3 2
1764.2.k.l 4 7.d odd 6 2
3136.2.a.br 2 24.h odd 2 1
3136.2.a.br 2 168.i even 2 1
3136.2.a.bs 2 24.f even 2 1
3136.2.a.bs 2 168.e odd 2 1
4900.2.a.y 2 15.d odd 2 1
4900.2.a.y 2 105.g even 2 1
4900.2.e.p 4 15.e even 4 2
4900.2.e.p 4 105.k odd 4 2
7056.2.a.cr 2 4.b odd 2 1
7056.2.a.cr 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1764))\):

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 18 \) Copy content Toggle raw display
$17$ \( T^{2} - 2 \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 50 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 32 \) Copy content Toggle raw display
$53$ \( (T + 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 200 \) Copy content Toggle raw display
$61$ \( T^{2} - 50 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 50 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 200 \) Copy content Toggle raw display
$89$ \( T^{2} - 50 \) Copy content Toggle raw display
$97$ \( T^{2} - 2 \) Copy content Toggle raw display
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