Properties

Label 49.22.a.b
Level $49$
Weight $22$
Character orbit 49.a
Self dual yes
Analytic conductor $136.944$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,22,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(136.943898701\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q + 2795 q^{2} + 5714873 q^{4} + 10111530195 q^{8} - 10460353203 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2795 q^{2} + 5714873 q^{4} + 10111530195 q^{8} - 10460353203 q^{9} + 147196778612 q^{11} + 16276769553329 q^{16} - 29236687202385 q^{18} + 411414996220540 q^{22} + 380240054265640 q^{23} - 476837158203125 q^{25} - 31\!\cdots\!54 q^{29}+ \cdots - 15\!\cdots\!36 q^{99}+O(q^{100}) \) 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
2795.00 0 5.71487e6 0 0 0 1.01115e10 −1.04604e10 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.22.a.b 1
7.b odd 2 1 CM 49.22.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.22.a.b 1 1.a even 1 1 trivial
49.22.a.b 1 7.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 \(S_{22}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2} - 2795 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2795 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 147196778612 \) 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
$23$ \( T - 380240054265640 \) Copy content Toggle raw display
$29$ \( T + 3189167291874254 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 57\!\cdots\!50 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 21\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 48\!\cdots\!10 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 26\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T - 49\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 62\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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