Properties

Label 7.7.b.a
Level $7$
Weight $7$
Character orbit 7.b
Self dual yes
Analytic conductor $1.610$
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] = [7,7,Mod(6,7)]
 
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([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.6");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 7.b (of order \(2\), degree \(1\), minimal)

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: \(1.61037858534\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 9 q^{2} + 17 q^{4} - 343 q^{7} - 423 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{2} + 17 q^{4} - 343 q^{7} - 423 q^{8} + 729 q^{9} + 1962 q^{11} - 3087 q^{14} - 4895 q^{16} + 6561 q^{18} + 17658 q^{22} - 22734 q^{23} + 15625 q^{25} - 5831 q^{28} - 21222 q^{29} - 16983 q^{32} + 12393 q^{36} + 101194 q^{37} - 126614 q^{43} + 33354 q^{44} - 204606 q^{46} + 117649 q^{49} + 140625 q^{50} + 50346 q^{53} + 145089 q^{56} - 190998 q^{58} - 250047 q^{63} + 160433 q^{64} - 53926 q^{67} - 242478 q^{71} - 308367 q^{72} + 910746 q^{74} - 672966 q^{77} + 929378 q^{79} + 531441 q^{81} - 1139526 q^{86} - 829926 q^{88} - 386478 q^{92} + 1058841 q^{98} + 1430298 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
0
9.00000 0 17.0000 0 0 −343.000 −423.000 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 7.7.b.a 1
3.b odd 2 1 63.7.d.a 1
4.b odd 2 1 112.7.c.a 1
5.b even 2 1 175.7.d.a 1
5.c odd 4 2 175.7.c.a 2
7.b odd 2 1 CM 7.7.b.a 1
7.c even 3 2 49.7.d.a 2
7.d odd 6 2 49.7.d.a 2
8.b even 2 1 448.7.c.a 1
8.d odd 2 1 448.7.c.b 1
21.c even 2 1 63.7.d.a 1
28.d even 2 1 112.7.c.a 1
35.c odd 2 1 175.7.d.a 1
35.f even 4 2 175.7.c.a 2
56.e even 2 1 448.7.c.b 1
56.h odd 2 1 448.7.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.a 1 1.a even 1 1 trivial
7.7.b.a 1 7.b odd 2 1 CM
49.7.d.a 2 7.c even 3 2
49.7.d.a 2 7.d odd 6 2
63.7.d.a 1 3.b odd 2 1
63.7.d.a 1 21.c even 2 1
112.7.c.a 1 4.b odd 2 1
112.7.c.a 1 28.d even 2 1
175.7.c.a 2 5.c odd 4 2
175.7.c.a 2 35.f even 4 2
175.7.d.a 1 5.b even 2 1
175.7.d.a 1 35.c odd 2 1
448.7.c.a 1 8.b even 2 1
448.7.c.a 1 56.h odd 2 1
448.7.c.b 1 8.d odd 2 1
448.7.c.b 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 9 \) acting on \(S_{7}^{\mathrm{new}}(7, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 9 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 343 \) Copy content Toggle raw display
$11$ \( T - 1962 \) 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
$23$ \( T + 22734 \) Copy content Toggle raw display
$29$ \( T + 21222 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 101194 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 126614 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 50346 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T + 53926 \) Copy content Toggle raw display
$71$ \( T + 242478 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 929378 \) 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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