Properties

Label 192.4.c.a
Level $192$
Weight $4$
Character orbit 192.c
Analytic conductor $11.328$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,4,Mod(191,192)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.191");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 192.c (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: no
Analytic conductor: \(11.3283667211\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 48)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 6 \beta q^{7} - 27 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - 6 \beta q^{7} - 27 q^{9} - 70 q^{13} - 30 \beta q^{19} + 162 q^{21} + 125 q^{25} - 27 \beta q^{27} - 30 \beta q^{31} - 110 q^{37} - 70 \beta q^{39} + 42 \beta q^{43} - 629 q^{49} + 810 q^{57} - 182 q^{61} + 162 \beta q^{63} - 126 \beta q^{67} - 1190 q^{73} + 125 \beta q^{75} + 210 \beta q^{79} + 729 q^{81} + 420 \beta q^{91} + 810 q^{93} + 1330 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 54 q^{9} - 140 q^{13} + 324 q^{21} + 250 q^{25} - 220 q^{37} - 1258 q^{49} + 1620 q^{57} - 364 q^{61} - 2380 q^{73} + 1458 q^{81} + 1620 q^{93} + 2660 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.500000 0.866025i
0.500000 + 0.866025i
0 5.19615i 0 0 0 31.1769i 0 −27.0000 0
191.2 0 5.19615i 0 0 0 31.1769i 0 −27.0000 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 \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.4.c.a 2
3.b odd 2 1 CM 192.4.c.a 2
4.b odd 2 1 inner 192.4.c.a 2
8.b even 2 1 48.4.c.a 2
8.d odd 2 1 48.4.c.a 2
12.b even 2 1 inner 192.4.c.a 2
16.e even 4 2 768.4.f.a 4
16.f odd 4 2 768.4.f.a 4
24.f even 2 1 48.4.c.a 2
24.h odd 2 1 48.4.c.a 2
48.i odd 4 2 768.4.f.a 4
48.k even 4 2 768.4.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.c.a 2 8.b even 2 1
48.4.c.a 2 8.d odd 2 1
48.4.c.a 2 24.f even 2 1
48.4.c.a 2 24.h odd 2 1
192.4.c.a 2 1.a even 1 1 trivial
192.4.c.a 2 3.b odd 2 1 CM
192.4.c.a 2 4.b odd 2 1 inner
192.4.c.a 2 12.b even 2 1 inner
768.4.f.a 4 16.e even 4 2
768.4.f.a 4 16.f odd 4 2
768.4.f.a 4 48.i odd 4 2
768.4.f.a 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{4}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 27 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 972 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 24300 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 24300 \) Copy content Toggle raw display
$37$ \( (T + 110)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 47628 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 182)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 428652 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 1190)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 1190700 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 1330)^{2} \) Copy content Toggle raw display
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