Properties

Label 44.4.a.a
Level $44$
Weight $4$
Character orbit 44.a
Self dual yes
Analytic conductor $2.596$
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] = [44,4,Mod(1,44)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(44, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("44.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 44 = 2^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 44.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: \(2.59608404025\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 5 q^{3} - 7 q^{5} - 26 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{3} - 7 q^{5} - 26 q^{7} - 2 q^{9} - 11 q^{11} + 52 q^{13} + 35 q^{15} + 46 q^{17} - 96 q^{19} + 130 q^{21} + 27 q^{23} - 76 q^{25} + 145 q^{27} + 16 q^{29} - 293 q^{31} + 55 q^{33} + 182 q^{35} - 29 q^{37} - 260 q^{39} - 472 q^{41} - 110 q^{43} + 14 q^{45} - 224 q^{47} + 333 q^{49} - 230 q^{51} + 754 q^{53} + 77 q^{55} + 480 q^{57} + 825 q^{59} - 548 q^{61} + 52 q^{63} - 364 q^{65} - 123 q^{67} - 135 q^{69} + 1001 q^{71} - 1020 q^{73} + 380 q^{75} + 286 q^{77} + 526 q^{79} - 671 q^{81} - 158 q^{83} - 322 q^{85} - 80 q^{87} - 1217 q^{89} - 1352 q^{91} + 1465 q^{93} + 672 q^{95} - 263 q^{97} + 22 q^{99}+O(q^{100}) \) 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 −5.00000 0 −7.00000 0 −26.0000 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 44.4.a.a 1
3.b odd 2 1 396.4.a.e 1
4.b odd 2 1 176.4.a.e 1
5.b even 2 1 1100.4.a.d 1
5.c odd 4 2 1100.4.b.c 2
7.b odd 2 1 2156.4.a.b 1
8.b even 2 1 704.4.a.j 1
8.d odd 2 1 704.4.a.c 1
11.b odd 2 1 484.4.a.a 1
12.b even 2 1 1584.4.a.p 1
44.c even 2 1 1936.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.4.a.a 1 1.a even 1 1 trivial
176.4.a.e 1 4.b odd 2 1
396.4.a.e 1 3.b odd 2 1
484.4.a.a 1 11.b odd 2 1
704.4.a.c 1 8.d odd 2 1
704.4.a.j 1 8.b even 2 1
1100.4.a.d 1 5.b even 2 1
1100.4.b.c 2 5.c odd 4 2
1584.4.a.p 1 12.b even 2 1
1936.4.a.m 1 44.c even 2 1
2156.4.a.b 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(44))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 5 \) Copy content Toggle raw display
$5$ \( T + 7 \) Copy content Toggle raw display
$7$ \( T + 26 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T - 52 \) Copy content Toggle raw display
$17$ \( T - 46 \) Copy content Toggle raw display
$19$ \( T + 96 \) Copy content Toggle raw display
$23$ \( T - 27 \) Copy content Toggle raw display
$29$ \( T - 16 \) Copy content Toggle raw display
$31$ \( T + 293 \) Copy content Toggle raw display
$37$ \( T + 29 \) Copy content Toggle raw display
$41$ \( T + 472 \) Copy content Toggle raw display
$43$ \( T + 110 \) Copy content Toggle raw display
$47$ \( T + 224 \) Copy content Toggle raw display
$53$ \( T - 754 \) Copy content Toggle raw display
$59$ \( T - 825 \) Copy content Toggle raw display
$61$ \( T + 548 \) Copy content Toggle raw display
$67$ \( T + 123 \) Copy content Toggle raw display
$71$ \( T - 1001 \) Copy content Toggle raw display
$73$ \( T + 1020 \) Copy content Toggle raw display
$79$ \( T - 526 \) Copy content Toggle raw display
$83$ \( T + 158 \) Copy content Toggle raw display
$89$ \( T + 1217 \) Copy content Toggle raw display
$97$ \( T + 263 \) Copy content Toggle raw display
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