Properties

Label 192.2.f.a
Level $192$
Weight $2$
Character orbit 192.f
Analytic conductor $1.533$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 192.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.53312771881\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{3} q^{5} + \beta_1 q^{7} - 3 q^{9} + 2 \beta_{2} q^{11} - 3 \beta_1 q^{15} - \beta_{3} q^{21} + 7 q^{25} - 3 \beta_{2} q^{27} + 3 \beta_{3} q^{29} + 5 \beta_1 q^{31} - 6 q^{33} - 4 \beta_{2} q^{35} + 3 \beta_{3} q^{45} + 3 q^{49} - \beta_{3} q^{53} - 6 \beta_1 q^{55} - 6 \beta_{2} q^{59} - 3 \beta_1 q^{63} - 14 q^{73} + 7 \beta_{2} q^{75} - 2 \beta_{3} q^{77} + 5 \beta_1 q^{79} + 9 q^{81} + 10 \beta_{2} q^{83} + 9 \beta_1 q^{87} - 5 \beta_{3} q^{93} + 2 q^{97} - 6 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} + 28 q^{25} - 24 q^{33} + 12 q^{49} - 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 1.73205i 0 −3.46410 0 2.00000i 0 −3.00000 0
95.2 0 1.73205i 0 3.46410 0 2.00000i 0 −3.00000 0
95.3 0 1.73205i 0 −3.46410 0 2.00000i 0 −3.00000 0
95.4 0 1.73205i 0 3.46410 0 2.00000i 0 −3.00000 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.2.f.a 4
3.b odd 2 1 inner 192.2.f.a 4
4.b odd 2 1 inner 192.2.f.a 4
8.b even 2 1 inner 192.2.f.a 4
8.d odd 2 1 inner 192.2.f.a 4
12.b even 2 1 inner 192.2.f.a 4
16.e even 4 2 768.2.c.i 4
16.f odd 4 2 768.2.c.i 4
24.f even 2 1 inner 192.2.f.a 4
24.h odd 2 1 CM 192.2.f.a 4
48.i odd 4 2 768.2.c.i 4
48.k even 4 2 768.2.c.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.f.a 4 1.a even 1 1 trivial
192.2.f.a 4 3.b odd 2 1 inner
192.2.f.a 4 4.b odd 2 1 inner
192.2.f.a 4 8.b even 2 1 inner
192.2.f.a 4 8.d odd 2 1 inner
192.2.f.a 4 12.b even 2 1 inner
192.2.f.a 4 24.f even 2 1 inner
192.2.f.a 4 24.h odd 2 1 CM
768.2.c.i 4 16.e even 4 2
768.2.c.i 4 16.f odd 4 2
768.2.c.i 4 48.i odd 4 2
768.2.c.i 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12 \) acting on \(S_{2}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T + 14)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T - 2)^{4} \) Copy content Toggle raw display
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