Properties

Label 1936.4.a.a
Level $1936$
Weight $4$
Character orbit 1936.a
Self dual yes
Analytic conductor $114.228$
Analytic rank $1$
Dimension $1$
CM discriminant -11
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1936,4,Mod(1,1936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1936, 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("1936.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1936 = 2^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1936.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: \(114.227697771\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 121)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 8 q^{3} + 18 q^{5} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{3} + 18 q^{5} + 37 q^{9} - 144 q^{15} + 108 q^{23} + 199 q^{25} - 80 q^{27} - 340 q^{31} - 434 q^{37} + 666 q^{45} + 36 q^{47} - 343 q^{49} - 738 q^{53} + 720 q^{59} + 416 q^{67} - 864 q^{69} - 612 q^{71} - 1592 q^{75} - 359 q^{81} + 1674 q^{89} + 2720 q^{93} - 34 q^{97}+O(q^{100}) \) 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 −8.00000 0 18.0000 0 0 0 37.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1936.4.a.a 1
4.b odd 2 1 121.4.a.a 1
11.b odd 2 1 CM 1936.4.a.a 1
12.b even 2 1 1089.4.a.f 1
44.c even 2 1 121.4.a.a 1
44.g even 10 4 121.4.c.a 4
44.h odd 10 4 121.4.c.a 4
132.d odd 2 1 1089.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.a 1 4.b odd 2 1
121.4.a.a 1 44.c even 2 1
121.4.c.a 4 44.g even 10 4
121.4.c.a 4 44.h odd 10 4
1089.4.a.f 1 12.b even 2 1
1089.4.a.f 1 132.d odd 2 1
1936.4.a.a 1 1.a even 1 1 trivial
1936.4.a.a 1 11.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1936))\):

\( T_{3} + 8 \) Copy content Toggle raw display
\( T_{5} - 18 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 8 \) Copy content Toggle raw display
$5$ \( T - 18 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 108 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 340 \) Copy content Toggle raw display
$37$ \( T + 434 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 36 \) Copy content Toggle raw display
$53$ \( T + 738 \) Copy content Toggle raw display
$59$ \( T - 720 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 416 \) Copy content Toggle raw display
$71$ \( T + 612 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 1674 \) Copy content Toggle raw display
$97$ \( T + 34 \) Copy content Toggle raw display
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