Properties

Label 192.2.d.a
Level $192$
Weight $2$
Character orbit 192.d
Analytic conductor $1.533$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 192.d (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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} - q^{9} + \beta_{3} q^{15} + 6 q^{17} + 4 \beta_1 q^{19} + \beta_{2} q^{21} - 2 \beta_{3} q^{23} - 7 q^{25} - \beta_1 q^{27} - \beta_{2} q^{29} - \beta_{3} q^{31} - 12 \beta_1 q^{35} + 2 \beta_{2} q^{37} - 6 q^{41} - 4 \beta_1 q^{43} + \beta_{2} q^{45} - 2 \beta_{3} q^{47} + 5 q^{49} + 6 \beta_1 q^{51} + \beta_{2} q^{53} - 4 q^{57} + 12 \beta_1 q^{59} + 2 \beta_{2} q^{61} - \beta_{3} q^{63} - 4 \beta_1 q^{67} - 2 \beta_{2} q^{69} + 2 \beta_{3} q^{71} + 2 q^{73} - 7 \beta_1 q^{75} + 3 \beta_{3} q^{79} + q^{81} - 6 \beta_{2} q^{85} + \beta_{3} q^{87} + 6 q^{89} - \beta_{2} q^{93} + 4 \beta_{3} q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 24 q^{17} - 28 q^{25} - 24 q^{41} + 20 q^{49} - 16 q^{57} + 8 q^{73} + 4 q^{81} + 24 q^{89} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{12}^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 1.00000i 0 3.46410i 0 −3.46410 0 −1.00000 0
97.2 0 1.00000i 0 3.46410i 0 3.46410 0 −1.00000 0
97.3 0 1.00000i 0 3.46410i 0 3.46410 0 −1.00000 0
97.4 0 1.00000i 0 3.46410i 0 −3.46410 0 −1.00000 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.2.d.a 4
3.b odd 2 1 576.2.d.b 4
4.b odd 2 1 inner 192.2.d.a 4
5.b even 2 1 4800.2.k.j 4
5.c odd 4 1 4800.2.d.j 4
5.c odd 4 1 4800.2.d.o 4
8.b even 2 1 inner 192.2.d.a 4
8.d odd 2 1 inner 192.2.d.a 4
12.b even 2 1 576.2.d.b 4
16.e even 4 1 768.2.a.j 2
16.e even 4 1 768.2.a.k 2
16.f odd 4 1 768.2.a.j 2
16.f odd 4 1 768.2.a.k 2
20.d odd 2 1 4800.2.k.j 4
20.e even 4 1 4800.2.d.j 4
20.e even 4 1 4800.2.d.o 4
24.f even 2 1 576.2.d.b 4
24.h odd 2 1 576.2.d.b 4
40.e odd 2 1 4800.2.k.j 4
40.f even 2 1 4800.2.k.j 4
40.i odd 4 1 4800.2.d.j 4
40.i odd 4 1 4800.2.d.o 4
40.k even 4 1 4800.2.d.j 4
40.k even 4 1 4800.2.d.o 4
48.i odd 4 1 2304.2.a.s 2
48.i odd 4 1 2304.2.a.u 2
48.k even 4 1 2304.2.a.s 2
48.k even 4 1 2304.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.d.a 4 1.a even 1 1 trivial
192.2.d.a 4 4.b odd 2 1 inner
192.2.d.a 4 8.b even 2 1 inner
192.2.d.a 4 8.d odd 2 1 inner
576.2.d.b 4 3.b odd 2 1
576.2.d.b 4 12.b even 2 1
576.2.d.b 4 24.f even 2 1
576.2.d.b 4 24.h odd 2 1
768.2.a.j 2 16.e even 4 1
768.2.a.j 2 16.f odd 4 1
768.2.a.k 2 16.e even 4 1
768.2.a.k 2 16.f odd 4 1
2304.2.a.s 2 48.i odd 4 1
2304.2.a.s 2 48.k even 4 1
2304.2.a.u 2 48.i odd 4 1
2304.2.a.u 2 48.k even 4 1
4800.2.d.j 4 5.c odd 4 1
4800.2.d.j 4 20.e even 4 1
4800.2.d.j 4 40.i odd 4 1
4800.2.d.j 4 40.k even 4 1
4800.2.d.o 4 5.c odd 4 1
4800.2.d.o 4 20.e even 4 1
4800.2.d.o 4 40.i odd 4 1
4800.2.d.o 4 40.k even 4 1
4800.2.k.j 4 5.b even 2 1
4800.2.k.j 4 20.d odd 2 1
4800.2.k.j 4 40.e odd 2 1
4800.2.k.j 4 40.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(192, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T - 6)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( (T + 2)^{4} \) Copy content Toggle raw display
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