Properties

Label 192.4.d.a
Level $192$
Weight $4$
Character orbit 192.d
Analytic conductor $11.328$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 192.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.3283667211\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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.

\(f(q)\) \(=\) \( q - 3 \beta_1 q^{3} - 3 \beta_{2} q^{5} - \beta_{3} q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_1 q^{3} - 3 \beta_{2} q^{5} - \beta_{3} q^{7} - 9 q^{9} - 16 \beta_{2} q^{13} - 9 \beta_{3} q^{15} - 90 q^{17} + 116 \beta_1 q^{19} + 3 \beta_{2} q^{21} - 30 \beta_{3} q^{23} + 17 q^{25} + 27 \beta_1 q^{27} - 75 \beta_{2} q^{29} - 87 \beta_{3} q^{31} + 36 \beta_1 q^{35} - 10 \beta_{2} q^{37} - 48 \beta_{3} q^{39} - 54 q^{41} - 20 \beta_1 q^{43} + 27 \beta_{2} q^{45} + 114 \beta_{3} q^{47} - 331 q^{49} + 270 \beta_1 q^{51} - 141 \beta_{2} q^{53} + 348 q^{57} - 324 \beta_1 q^{59} + 166 \beta_{2} q^{61} + 9 \beta_{3} q^{63} - 576 q^{65} - 116 \beta_1 q^{67} + 90 \beta_{2} q^{69} + 318 \beta_{3} q^{71} + 1106 q^{73} - 51 \beta_1 q^{75} - 43 \beta_{3} q^{79} + 81 q^{81} - 1152 \beta_1 q^{83} + 270 \beta_{2} q^{85} - 225 \beta_{3} q^{87} + 918 q^{89} + 192 \beta_1 q^{91} + 261 \beta_{2} q^{93} + 348 \beta_{3} q^{95} + 190 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{9} - 360 q^{17} + 68 q^{25} - 216 q^{41} - 1324 q^{49} + 1392 q^{57} - 2304 q^{65} + 4424 q^{73} + 324 q^{81} + 3672 q^{89} + 760 q^{97}+O(q^{100}) \) 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 3.00000i 0 10.3923i 0 −3.46410 0 −9.00000 0
97.2 0 3.00000i 0 10.3923i 0 3.46410 0 −9.00000 0
97.3 0 3.00000i 0 10.3923i 0 3.46410 0 −9.00000 0
97.4 0 3.00000i 0 10.3923i 0 −3.46410 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
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 192.4.d.a 4
3.b odd 2 1 576.4.d.g 4
4.b odd 2 1 inner 192.4.d.a 4
8.b even 2 1 inner 192.4.d.a 4
8.d odd 2 1 inner 192.4.d.a 4
12.b even 2 1 576.4.d.g 4
16.e even 4 1 768.4.a.g 2
16.e even 4 1 768.4.a.n 2
16.f odd 4 1 768.4.a.g 2
16.f odd 4 1 768.4.a.n 2
24.f even 2 1 576.4.d.g 4
24.h odd 2 1 576.4.d.g 4
48.i odd 4 1 2304.4.a.be 2
48.i odd 4 1 2304.4.a.bg 2
48.k even 4 1 2304.4.a.be 2
48.k even 4 1 2304.4.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.4.d.a 4 1.a even 1 1 trivial
192.4.d.a 4 4.b odd 2 1 inner
192.4.d.a 4 8.b even 2 1 inner
192.4.d.a 4 8.d odd 2 1 inner
576.4.d.g 4 3.b odd 2 1
576.4.d.g 4 12.b even 2 1
576.4.d.g 4 24.f even 2 1
576.4.d.g 4 24.h odd 2 1
768.4.a.g 2 16.e even 4 1
768.4.a.g 2 16.f odd 4 1
768.4.a.n 2 16.e even 4 1
768.4.a.n 2 16.f odd 4 1
2304.4.a.be 2 48.i odd 4 1
2304.4.a.be 2 48.k even 4 1
2304.4.a.bg 2 48.i odd 4 1
2304.4.a.bg 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 108 \) acting on \(S_{4}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3072)^{2} \) Copy content Toggle raw display
$17$ \( (T + 90)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 13456)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 10800)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 67500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 90828)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1200)^{2} \) Copy content Toggle raw display
$41$ \( (T + 54)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 155952)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 238572)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 104976)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 330672)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 13456)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1213488)^{2} \) Copy content Toggle raw display
$73$ \( (T - 1106)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 22188)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1327104)^{2} \) Copy content Toggle raw display
$89$ \( (T - 918)^{4} \) Copy content Toggle raw display
$97$ \( (T - 190)^{4} \) Copy content Toggle raw display
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