Properties

Label 44.6.a.a
Level $44$
Weight $6$
Character orbit 44.a
Self dual yes
Analytic conductor $7.057$
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] = [44,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("44.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 44 = 2^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(7.05688807177\)
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)
 
\(f(q)\) \(=\) \( q + 7 q^{3} - 79 q^{5} - 50 q^{7} - 194 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 7 q^{3} - 79 q^{5} - 50 q^{7} - 194 q^{9} + 121 q^{11} - 380 q^{13} - 553 q^{15} - 1154 q^{17} - 1824 q^{19} - 350 q^{21} + 3591 q^{23} + 3116 q^{25} - 3059 q^{27} + 8032 q^{29} - 2945 q^{31} + 847 q^{33} + 3950 q^{35} + 6979 q^{37} - 2660 q^{39} - 520 q^{41} - 2486 q^{43} + 15326 q^{45} - 6920 q^{47} - 14307 q^{49} - 8078 q^{51} - 13718 q^{53} - 9559 q^{55} - 12768 q^{57} - 31779 q^{59} + 34156 q^{61} + 9700 q^{63} + 30020 q^{65} - 61503 q^{67} + 25137 q^{69} - 14971 q^{71} - 36444 q^{73} + 21812 q^{75} - 6050 q^{77} - 28538 q^{79} + 25729 q^{81} + 77482 q^{83} + 91166 q^{85} + 56224 q^{87} + 36271 q^{89} + 19000 q^{91} - 20615 q^{93} + 144096 q^{95} - 49799 q^{97} - 23474 q^{99}+O(q^{100}) \) 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 7.00000 0 −79.0000 0 −50.0000 0 −194.000 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 44.6.a.a 1
3.b odd 2 1 396.6.a.e 1
4.b odd 2 1 176.6.a.a 1
5.b even 2 1 1100.6.a.a 1
5.c odd 4 2 1100.6.b.a 2
8.b even 2 1 704.6.a.d 1
8.d odd 2 1 704.6.a.g 1
11.b odd 2 1 484.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.6.a.a 1 1.a even 1 1 trivial
176.6.a.a 1 4.b odd 2 1
396.6.a.e 1 3.b odd 2 1
484.6.a.b 1 11.b odd 2 1
704.6.a.d 1 8.b even 2 1
704.6.a.g 1 8.d odd 2 1
1100.6.a.a 1 5.b even 2 1
1100.6.b.a 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 7 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(44))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 7 \) Copy content Toggle raw display
$5$ \( T + 79 \) Copy content Toggle raw display
$7$ \( T + 50 \) Copy content Toggle raw display
$11$ \( T - 121 \) Copy content Toggle raw display
$13$ \( T + 380 \) Copy content Toggle raw display
$17$ \( T + 1154 \) Copy content Toggle raw display
$19$ \( T + 1824 \) Copy content Toggle raw display
$23$ \( T - 3591 \) Copy content Toggle raw display
$29$ \( T - 8032 \) Copy content Toggle raw display
$31$ \( T + 2945 \) Copy content Toggle raw display
$37$ \( T - 6979 \) Copy content Toggle raw display
$41$ \( T + 520 \) Copy content Toggle raw display
$43$ \( T + 2486 \) Copy content Toggle raw display
$47$ \( T + 6920 \) Copy content Toggle raw display
$53$ \( T + 13718 \) Copy content Toggle raw display
$59$ \( T + 31779 \) Copy content Toggle raw display
$61$ \( T - 34156 \) Copy content Toggle raw display
$67$ \( T + 61503 \) Copy content Toggle raw display
$71$ \( T + 14971 \) Copy content Toggle raw display
$73$ \( T + 36444 \) Copy content Toggle raw display
$79$ \( T + 28538 \) Copy content Toggle raw display
$83$ \( T - 77482 \) Copy content Toggle raw display
$89$ \( T - 36271 \) Copy content Toggle raw display
$97$ \( T + 49799 \) Copy content Toggle raw display
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