Properties

Label 704.2.a.g
Level $704$
Weight $2$
Character orbit 704.a
Self dual yes
Analytic conductor $5.621$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [704,2,Mod(1,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, 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("704.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 704.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: \(5.62146830230\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
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)
 
\(f(q)\) \(=\) \( q + q^{3} - q^{5} - 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} - 4 q^{7} - 2 q^{9} + q^{11} + 2 q^{13} - q^{15} - 2 q^{19} - 4 q^{21} - 9 q^{23} - 4 q^{25} - 5 q^{27} - 4 q^{29} - 5 q^{31} + q^{33} + 4 q^{35} + 9 q^{37} + 2 q^{39} + 2 q^{41} - 6 q^{43} + 2 q^{45} + 4 q^{47} + 9 q^{49} + 6 q^{53} - q^{55} - 2 q^{57} - 5 q^{59} + 8 q^{63} - 2 q^{65} - 13 q^{67} - 9 q^{69} + q^{71} + 14 q^{73} - 4 q^{75} - 4 q^{77} + 10 q^{79} + q^{81} + 14 q^{83} - 4 q^{87} - 13 q^{89} - 8 q^{91} - 5 q^{93} + 2 q^{95} - 19 q^{97} - 2 q^{99}+O(q^{100}) \) 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 1.00000 0 −1.00000 0 −4.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(11\) \( -1 \)

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 704.2.a.g 1
3.b odd 2 1 6336.2.a.bq 1
4.b odd 2 1 704.2.a.d 1
8.b even 2 1 352.2.a.c 1
8.d odd 2 1 352.2.a.e yes 1
11.b odd 2 1 7744.2.a.y 1
12.b even 2 1 6336.2.a.bv 1
16.e even 4 2 2816.2.c.m 2
16.f odd 4 2 2816.2.c.a 2
24.f even 2 1 3168.2.a.j 1
24.h odd 2 1 3168.2.a.g 1
40.e odd 2 1 8800.2.a.i 1
40.f even 2 1 8800.2.a.t 1
44.c even 2 1 7744.2.a.i 1
88.b odd 2 1 3872.2.a.e 1
88.g even 2 1 3872.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.a.c 1 8.b even 2 1
352.2.a.e yes 1 8.d odd 2 1
704.2.a.d 1 4.b odd 2 1
704.2.a.g 1 1.a even 1 1 trivial
2816.2.c.a 2 16.f odd 4 2
2816.2.c.m 2 16.e even 4 2
3168.2.a.g 1 24.h odd 2 1
3168.2.a.j 1 24.f even 2 1
3872.2.a.e 1 88.b odd 2 1
3872.2.a.j 1 88.g even 2 1
6336.2.a.bq 1 3.b odd 2 1
6336.2.a.bv 1 12.b even 2 1
7744.2.a.i 1 44.c even 2 1
7744.2.a.y 1 11.b odd 2 1
8800.2.a.i 1 40.e odd 2 1
8800.2.a.t 1 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(704))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T + 9 \) Copy content Toggle raw display
$29$ \( T + 4 \) Copy content Toggle raw display
$31$ \( T + 5 \) Copy content Toggle raw display
$37$ \( T - 9 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T + 6 \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 5 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 13 \) Copy content Toggle raw display
$71$ \( T - 1 \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T - 10 \) Copy content Toggle raw display
$83$ \( T - 14 \) Copy content Toggle raw display
$89$ \( T + 13 \) Copy content Toggle raw display
$97$ \( T + 19 \) Copy content Toggle raw display
show more
show less