Properties

Label 192.9.e.b
Level $192$
Weight $9$
Character orbit 192.e
Self dual yes
Analytic conductor $78.217$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,9,Mod(65,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.e (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: \(78.2166931317\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
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)
 
\(f(q)\) \(=\) \( q + 81 q^{3} - 4034 q^{7} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 81 q^{3} - 4034 q^{7} + 6561 q^{9} + 35806 q^{13} - 258526 q^{19} - 326754 q^{21} + 390625 q^{25} + 531441 q^{27} + 1809406 q^{31} - 503522 q^{37} + 2900286 q^{39} + 3492194 q^{43} + 10508355 q^{49} - 20940606 q^{57} + 23826526 q^{61} - 26467074 q^{63} - 5421406 q^{67} + 16169282 q^{73} + 31640625 q^{75} + 18887038 q^{79} + 43046721 q^{81} - 144441404 q^{91} + 146561886 q^{93} + 176908034 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(0\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0
0 81.0000 0 0 0 −4034.00 0 6561.00 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.9.e.b 1
3.b odd 2 1 CM 192.9.e.b 1
4.b odd 2 1 192.9.e.a 1
8.b even 2 1 48.9.e.a 1
8.d odd 2 1 12.9.c.a 1
12.b even 2 1 192.9.e.a 1
24.f even 2 1 12.9.c.a 1
24.h odd 2 1 48.9.e.a 1
40.e odd 2 1 300.9.g.a 1
40.k even 4 2 300.9.b.b 2
72.l even 6 2 324.9.g.a 2
72.p odd 6 2 324.9.g.a 2
120.m even 2 1 300.9.g.a 1
120.q odd 4 2 300.9.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.a 1 8.d odd 2 1
12.9.c.a 1 24.f even 2 1
48.9.e.a 1 8.b even 2 1
48.9.e.a 1 24.h odd 2 1
192.9.e.a 1 4.b odd 2 1
192.9.e.a 1 12.b even 2 1
192.9.e.b 1 1.a even 1 1 trivial
192.9.e.b 1 3.b odd 2 1 CM
300.9.b.b 2 40.k even 4 2
300.9.b.b 2 120.q odd 4 2
300.9.g.a 1 40.e odd 2 1
300.9.g.a 1 120.m even 2 1
324.9.g.a 2 72.l even 6 2
324.9.g.a 2 72.p odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 4034 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 81 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4034 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 35806 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 258526 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 1809406 \) Copy content Toggle raw display
$37$ \( T + 503522 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 3492194 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 23826526 \) Copy content Toggle raw display
$67$ \( T + 5421406 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 16169282 \) Copy content Toggle raw display
$79$ \( T - 18887038 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 176908034 \) Copy content Toggle raw display
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