Properties

Label 224.2.a.a
Level $224$
Weight $2$
Character orbit 224.a
Self dual yes
Analytic conductor $1.789$
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] = [224,2,Mod(1,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, 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("224.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.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: \(1.78864900528\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{3} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} - q^{7} + q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{17} - 6 q^{19} + 2 q^{21} + 8 q^{23} - 5 q^{25} + 4 q^{27} + 2 q^{29} - 4 q^{31} + 8 q^{33} + 10 q^{37} + 8 q^{39} - 10 q^{41} + 4 q^{43} + 4 q^{47} + q^{49} + 4 q^{51} - 2 q^{53} + 12 q^{57} + 10 q^{59} - 8 q^{61} - q^{63} - 8 q^{67} - 16 q^{69} - 6 q^{73} + 10 q^{75} + 4 q^{77} - 16 q^{79} - 11 q^{81} + 2 q^{83} - 4 q^{87} + 18 q^{89} + 4 q^{91} + 8 q^{93} - 2 q^{97} - 4 q^{99}+O(q^{100}) \) 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 −2.00000 0 0 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( +1 \)

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 224.2.a.a 1
3.b odd 2 1 2016.2.a.e 1
4.b odd 2 1 224.2.a.b yes 1
5.b even 2 1 5600.2.a.t 1
7.b odd 2 1 1568.2.a.h 1
7.c even 3 2 1568.2.i.k 2
7.d odd 6 2 1568.2.i.c 2
8.b even 2 1 448.2.a.f 1
8.d odd 2 1 448.2.a.b 1
12.b even 2 1 2016.2.a.g 1
16.e even 4 2 1792.2.b.f 2
16.f odd 4 2 1792.2.b.b 2
20.d odd 2 1 5600.2.a.c 1
24.f even 2 1 4032.2.a.z 1
24.h odd 2 1 4032.2.a.p 1
28.d even 2 1 1568.2.a.b 1
28.f even 6 2 1568.2.i.j 2
28.g odd 6 2 1568.2.i.b 2
56.e even 2 1 3136.2.a.y 1
56.h odd 2 1 3136.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 1.a even 1 1 trivial
224.2.a.b yes 1 4.b odd 2 1
448.2.a.b 1 8.d odd 2 1
448.2.a.f 1 8.b even 2 1
1568.2.a.b 1 28.d even 2 1
1568.2.a.h 1 7.b odd 2 1
1568.2.i.b 2 28.g odd 6 2
1568.2.i.c 2 7.d odd 6 2
1568.2.i.j 2 28.f even 6 2
1568.2.i.k 2 7.c even 3 2
1792.2.b.b 2 16.f odd 4 2
1792.2.b.f 2 16.e even 4 2
2016.2.a.e 1 3.b odd 2 1
2016.2.a.g 1 12.b even 2 1
3136.2.a.f 1 56.h odd 2 1
3136.2.a.y 1 56.e even 2 1
4032.2.a.p 1 24.h odd 2 1
4032.2.a.z 1 24.f even 2 1
5600.2.a.c 1 20.d odd 2 1
5600.2.a.t 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(224))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T - 10 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T + 8 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T - 2 \) Copy content Toggle raw display
$89$ \( T - 18 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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