Properties

Label 192.3.e.a
Level $192$
Weight $3$
Character orbit 192.e
Self dual yes
Analytic conductor $5.232$
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] = [192,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(5.23162107572\)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 2 q^{7} + 9 q^{9} + 22 q^{13} + 26 q^{19} + 6 q^{21} + 25 q^{25} - 27 q^{27} + 46 q^{31} - 26 q^{37} - 66 q^{39} - 22 q^{43} - 45 q^{49} - 78 q^{57} - 74 q^{61} - 18 q^{63} + 122 q^{67} - 46 q^{73} - 75 q^{75} + 142 q^{79} + 81 q^{81} - 44 q^{91} - 138 q^{93} + 2 q^{97}+O(q^{100}) \) 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 −3.00000 0 0 0 −2.00000 0 9.00000 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 192.3.e.a 1
3.b odd 2 1 CM 192.3.e.a 1
4.b odd 2 1 192.3.e.b 1
8.b even 2 1 48.3.e.a 1
8.d odd 2 1 12.3.c.a 1
12.b even 2 1 192.3.e.b 1
16.e even 4 2 768.3.h.b 2
16.f odd 4 2 768.3.h.a 2
24.f even 2 1 12.3.c.a 1
24.h odd 2 1 48.3.e.a 1
40.e odd 2 1 300.3.g.b 1
40.f even 2 1 1200.3.l.b 1
40.i odd 4 2 1200.3.c.c 2
40.k even 4 2 300.3.b.a 2
48.i odd 4 2 768.3.h.b 2
48.k even 4 2 768.3.h.a 2
56.e even 2 1 588.3.c.c 1
56.k odd 6 2 588.3.p.c 2
56.m even 6 2 588.3.p.b 2
72.j odd 6 2 1296.3.q.b 2
72.l even 6 2 324.3.g.b 2
72.n even 6 2 1296.3.q.b 2
72.p odd 6 2 324.3.g.b 2
88.g even 2 1 1452.3.e.b 1
120.i odd 2 1 1200.3.l.b 1
120.m even 2 1 300.3.g.b 1
120.q odd 4 2 300.3.b.a 2
120.w even 4 2 1200.3.c.c 2
168.e odd 2 1 588.3.c.c 1
168.v even 6 2 588.3.p.c 2
168.be odd 6 2 588.3.p.b 2
264.p odd 2 1 1452.3.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 8.d odd 2 1
12.3.c.a 1 24.f even 2 1
48.3.e.a 1 8.b even 2 1
48.3.e.a 1 24.h odd 2 1
192.3.e.a 1 1.a even 1 1 trivial
192.3.e.a 1 3.b odd 2 1 CM
192.3.e.b 1 4.b odd 2 1
192.3.e.b 1 12.b even 2 1
300.3.b.a 2 40.k even 4 2
300.3.b.a 2 120.q odd 4 2
300.3.g.b 1 40.e odd 2 1
300.3.g.b 1 120.m even 2 1
324.3.g.b 2 72.l even 6 2
324.3.g.b 2 72.p odd 6 2
588.3.c.c 1 56.e even 2 1
588.3.c.c 1 168.e odd 2 1
588.3.p.b 2 56.m even 6 2
588.3.p.b 2 168.be odd 6 2
588.3.p.c 2 56.k odd 6 2
588.3.p.c 2 168.v even 6 2
768.3.h.a 2 16.f odd 4 2
768.3.h.a 2 48.k even 4 2
768.3.h.b 2 16.e even 4 2
768.3.h.b 2 48.i odd 4 2
1200.3.c.c 2 40.i odd 4 2
1200.3.c.c 2 120.w even 4 2
1200.3.l.b 1 40.f even 2 1
1200.3.l.b 1 120.i odd 2 1
1296.3.q.b 2 72.j odd 6 2
1296.3.q.b 2 72.n even 6 2
1452.3.e.b 1 88.g even 2 1
1452.3.e.b 1 264.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(192, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 22 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 26 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 46 \) Copy content Toggle raw display
$37$ \( T + 26 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 22 \) 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
$61$ \( T + 74 \) Copy content Toggle raw display
$67$ \( T - 122 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 46 \) Copy content Toggle raw display
$79$ \( T - 142 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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