Properties

Label 7.6.a.a
Level $7$
Weight $6$
Character orbit 7.a
Self dual yes
Analytic conductor $1.123$
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] = [7,6,Mod(1,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 7.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: \(1.12268673869\)
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)
 
\(f(q)\) \(=\) \( q - 10 q^{2} - 14 q^{3} + 68 q^{4} - 56 q^{5} + 140 q^{6} - 49 q^{7} - 360 q^{8} - 47 q^{9} + 560 q^{10} + 232 q^{11} - 952 q^{12} - 140 q^{13} + 490 q^{14} + 784 q^{15} + 1424 q^{16} - 1722 q^{17} + 470 q^{18}+ \cdots - 10904 q^{99}+O(q^{100}) \) 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
−10.0000 −14.0000 68.0000 −56.0000 140.000 −49.0000 −360.000 −47.0000 560.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 7.6.a.a 1
3.b odd 2 1 63.6.a.e 1
4.b odd 2 1 112.6.a.g 1
5.b even 2 1 175.6.a.b 1
5.c odd 4 2 175.6.b.a 2
7.b odd 2 1 49.6.a.a 1
7.c even 3 2 49.6.c.c 2
7.d odd 6 2 49.6.c.b 2
8.b even 2 1 448.6.a.m 1
8.d odd 2 1 448.6.a.c 1
11.b odd 2 1 847.6.a.b 1
12.b even 2 1 1008.6.a.y 1
21.c even 2 1 441.6.a.k 1
28.d even 2 1 784.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 1.a even 1 1 trivial
49.6.a.a 1 7.b odd 2 1
49.6.c.b 2 7.d odd 6 2
49.6.c.c 2 7.c even 3 2
63.6.a.e 1 3.b odd 2 1
112.6.a.g 1 4.b odd 2 1
175.6.a.b 1 5.b even 2 1
175.6.b.a 2 5.c odd 4 2
441.6.a.k 1 21.c even 2 1
448.6.a.c 1 8.d odd 2 1
448.6.a.m 1 8.b even 2 1
784.6.a.c 1 28.d even 2 1
847.6.a.b 1 11.b odd 2 1
1008.6.a.y 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 10 \) Copy content Toggle raw display
$3$ \( T + 14 \) Copy content Toggle raw display
$5$ \( T + 56 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T - 232 \) Copy content Toggle raw display
$13$ \( T + 140 \) Copy content Toggle raw display
$17$ \( T + 1722 \) Copy content Toggle raw display
$19$ \( T + 98 \) Copy content Toggle raw display
$23$ \( T - 1824 \) Copy content Toggle raw display
$29$ \( T - 3418 \) Copy content Toggle raw display
$31$ \( T + 7644 \) Copy content Toggle raw display
$37$ \( T + 10398 \) Copy content Toggle raw display
$41$ \( T + 17962 \) Copy content Toggle raw display
$43$ \( T - 10880 \) Copy content Toggle raw display
$47$ \( T - 9324 \) Copy content Toggle raw display
$53$ \( T - 2262 \) Copy content Toggle raw display
$59$ \( T + 2730 \) Copy content Toggle raw display
$61$ \( T - 25648 \) Copy content Toggle raw display
$67$ \( T + 48404 \) Copy content Toggle raw display
$71$ \( T + 58560 \) Copy content Toggle raw display
$73$ \( T - 68082 \) Copy content Toggle raw display
$79$ \( T - 31784 \) Copy content Toggle raw display
$83$ \( T + 20538 \) Copy content Toggle raw display
$89$ \( T + 50582 \) Copy content Toggle raw display
$97$ \( T + 58506 \) Copy content Toggle raw display
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