Properties

Label 7.15.b.a
Level $7$
Weight $15$
Character orbit 7.b
Self dual yes
Analytic conductor $8.703$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.6");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(8.70302777063\)
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)
 
\(f(q)\) \(=\) \( q - 87 q^{2} - 8815 q^{4} - 823543 q^{7} + 2192313 q^{8} + 4782969 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 87 q^{2} - 8815 q^{4} - 823543 q^{7} + 2192313 q^{8} + 4782969 q^{9} + 36437514 q^{11} + 71648241 q^{14} - 46306271 q^{16} - 416118303 q^{18} - 3170063718 q^{22} - 2188914318 q^{23} + 6103515625 q^{25} + 7259531545 q^{28} + 29824366266 q^{29} - 31890210615 q^{32} - 42161871735 q^{36} - 112367216342 q^{37} + 484972531402 q^{43} - 321196685910 q^{44} + 190435545666 q^{46} + 678223072849 q^{49} - 531005859375 q^{50} + 907194972426 q^{53} - 1805464024959 q^{56} - 2594719865142 q^{58} - 3938980639167 q^{63} + 3533130267569 q^{64} + 11528240589818 q^{67} - 4338861915246 q^{71} + 10485765117297 q^{72} + 9775947821754 q^{74} - 30007859592102 q^{77} - 37193960502814 q^{79} + 22876792454961 q^{81} - 42192610231974 q^{86} + 79882435629882 q^{88} + 19295279713170 q^{92} - 59005407337863 q^{98} + 174279499899066 q^{99}+O(q^{100}) \) 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
−87.0000 0 −8815.00 0 0 −823543. 2.19231e6 4.78297e6 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 7.15.b.a 1
3.b odd 2 1 63.15.d.a 1
4.b odd 2 1 112.15.c.a 1
7.b odd 2 1 CM 7.15.b.a 1
7.c even 3 2 49.15.d.a 2
7.d odd 6 2 49.15.d.a 2
21.c even 2 1 63.15.d.a 1
28.d even 2 1 112.15.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.15.b.a 1 1.a even 1 1 trivial
7.15.b.a 1 7.b odd 2 1 CM
49.15.d.a 2 7.c even 3 2
49.15.d.a 2 7.d odd 6 2
63.15.d.a 1 3.b odd 2 1
63.15.d.a 1 21.c even 2 1
112.15.c.a 1 4.b odd 2 1
112.15.c.a 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 87 \) acting on \(S_{15}^{\mathrm{new}}(7, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 87 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 823543 \) Copy content Toggle raw display
$11$ \( T - 36437514 \) 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
$23$ \( T + 2188914318 \) Copy content Toggle raw display
$29$ \( T - 29824366266 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 112367216342 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 484972531402 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 907194972426 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 11528240589818 \) Copy content Toggle raw display
$71$ \( T + 4338861915246 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 37193960502814 \) 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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