Properties

Label 368.7.f.a
Level $368$
Weight $7$
Character orbit 368.f
Self dual yes
Analytic conductor $84.660$
Analytic rank $0$
Dimension $1$
CM discriminant -23
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [368,7,Mod(321,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.321");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 368.f (of order \(2\), degree \(1\), not 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: \(84.6599027721\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
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)
 
\(f(q)\) \(=\) \( q + 38 q^{3} + 715 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 38 q^{3} + 715 q^{9} + 1082 q^{13} + 12167 q^{23} + 15625 q^{25} - 532 q^{27} + 30746 q^{29} - 58754 q^{31} + 41116 q^{39} + 43634 q^{41} + 205342 q^{47} + 117649 q^{49} + 253942 q^{59} + 462346 q^{69} - 667154 q^{71} + 725042 q^{73} + 593750 q^{75} - 541451 q^{81} + 1168348 q^{87} - 2232652 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\) \(277\)
\(\chi(n)\) \(0\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
0
0 38.0000 0 0 0 0 0 715.000 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
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.7.f.a 1
4.b odd 2 1 23.7.b.a 1
12.b even 2 1 207.7.d.a 1
23.b odd 2 1 CM 368.7.f.a 1
92.b even 2 1 23.7.b.a 1
276.h odd 2 1 207.7.d.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.7.b.a 1 4.b odd 2 1
23.7.b.a 1 92.b even 2 1
207.7.d.a 1 12.b even 2 1
207.7.d.a 1 276.h odd 2 1
368.7.f.a 1 1.a even 1 1 trivial
368.7.f.a 1 23.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 38 \) acting on \(S_{7}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 38 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 1082 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 12167 \) Copy content Toggle raw display
$29$ \( T - 30746 \) Copy content Toggle raw display
$31$ \( T + 58754 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 43634 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 205342 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 253942 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 667154 \) Copy content Toggle raw display
$73$ \( T - 725042 \) Copy content Toggle raw display
$79$ \( T \) 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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