Properties

Label 693.4.a.a
Level $693$
Weight $4$
Character orbit 693.a
Self dual yes
Analytic conductor $40.888$
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] = [693,4,Mod(1,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, 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("693.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 693.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: \(40.8883236340\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{2} + 17 q^{4} + 6 q^{5} + 7 q^{7} - 45 q^{8} - 30 q^{10} + 11 q^{11} + 70 q^{13} - 35 q^{14} + 89 q^{16} - 126 q^{17} - 80 q^{19} + 102 q^{20} - 55 q^{22} + 200 q^{23} - 89 q^{25} - 350 q^{26}+ \cdots - 245 q^{98}+O(q^{100}) \) 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
−5.00000 0 17.0000 6.00000 0 7.00000 −45.0000 0 −30.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( -1 \)
\(11\) \( -1 \)

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 693.4.a.a 1
3.b odd 2 1 231.4.a.e 1
21.c even 2 1 1617.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.a.e 1 3.b odd 2 1
693.4.a.a 1 1.a even 1 1 trivial
1617.4.a.g 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(693))\):

\( T_{2} + 5 \) Copy content Toggle raw display
\( T_{5} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T - 70 \) Copy content Toggle raw display
$17$ \( T + 126 \) Copy content Toggle raw display
$19$ \( T + 80 \) Copy content Toggle raw display
$23$ \( T - 200 \) Copy content Toggle raw display
$29$ \( T + 134 \) Copy content Toggle raw display
$31$ \( T + 244 \) Copy content Toggle raw display
$37$ \( T + 314 \) Copy content Toggle raw display
$41$ \( T + 278 \) Copy content Toggle raw display
$43$ \( T + 372 \) Copy content Toggle raw display
$47$ \( T - 84 \) Copy content Toggle raw display
$53$ \( T + 182 \) Copy content Toggle raw display
$59$ \( T - 756 \) Copy content Toggle raw display
$61$ \( T - 694 \) Copy content Toggle raw display
$67$ \( T - 820 \) Copy content Toggle raw display
$71$ \( T + 160 \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T - 40 \) Copy content Toggle raw display
$83$ \( T + 760 \) Copy content Toggle raw display
$89$ \( T - 102 \) Copy content Toggle raw display
$97$ \( T + 862 \) Copy content Toggle raw display
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