Properties

Label 192.3.e.f
Level $192$
Weight $3$
Character orbit 192.e
Analytic conductor $5.232$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 96)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( \beta_{1} + 2 \beta_{3} ) q^{7} + ( 5 - \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( \beta_{1} + 2 \beta_{3} ) q^{7} + ( 5 - \beta_{2} ) q^{9} -5 \beta_{1} q^{11} -10 q^{13} + ( -5 \beta_{1} + 4 \beta_{3} ) q^{15} + ( 5 \beta_{1} + 10 \beta_{3} ) q^{19} + ( 14 - \beta_{2} ) q^{21} + 6 \beta_{1} q^{23} -31 q^{25} + ( 10 \beta_{1} + \beta_{3} ) q^{27} -5 \beta_{2} q^{29} + ( 5 \beta_{1} + 10 \beta_{3} ) q^{31} + ( 20 + 5 \beta_{2} ) q^{33} -14 \beta_{1} q^{35} -10 q^{37} + ( -10 \beta_{1} - 10 \beta_{3} ) q^{39} + 2 \beta_{2} q^{41} + ( -11 \beta_{1} - 22 \beta_{3} ) q^{43} + ( 56 + 5 \beta_{2} ) q^{45} + 4 \beta_{1} q^{47} -21 q^{49} + 5 \beta_{2} q^{53} + ( -20 \beta_{1} - 40 \beta_{3} ) q^{55} + ( 70 - 5 \beta_{2} ) q^{57} + 35 \beta_{1} q^{59} -90 q^{61} + ( 19 \beta_{1} + 10 \beta_{3} ) q^{63} -10 \beta_{2} q^{65} + ( \beta_{1} + 2 \beta_{3} ) q^{67} + ( -24 - 6 \beta_{2} ) q^{69} + 10 \beta_{1} q^{71} -30 q^{73} + ( -31 \beta_{1} - 31 \beta_{3} ) q^{75} + 10 \beta_{2} q^{77} + ( 5 \beta_{1} + 10 \beta_{3} ) q^{79} + ( -31 - 10 \beta_{2} ) q^{81} + 9 \beta_{1} q^{83} + ( 25 \beta_{1} - 20 \beta_{3} ) q^{87} + 10 \beta_{2} q^{89} + ( -10 \beta_{1} - 20 \beta_{3} ) q^{91} + ( 70 - 5 \beta_{2} ) q^{93} -70 \beta_{1} q^{95} + 10 q^{97} + ( -5 \beta_{1} + 40 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 20q^{9} + O(q^{10}) \) \( 4q + 20q^{9} - 40q^{13} + 56q^{21} - 124q^{25} + 80q^{33} - 40q^{37} + 224q^{45} - 84q^{49} + 280q^{57} - 360q^{61} - 96q^{69} - 120q^{73} - 124q^{81} + 280q^{93} + 40q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} - 10 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} + 22 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 3 \nu^{2} + 5 \nu + 12 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{1} - 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} - 11 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
2.57794i
2.57794i
1.16372i
1.16372i
0 −2.64575 1.41421i 0 7.48331i 0 −5.29150 0 5.00000 + 7.48331i 0
65.2 0 −2.64575 + 1.41421i 0 7.48331i 0 −5.29150 0 5.00000 7.48331i 0
65.3 0 2.64575 1.41421i 0 7.48331i 0 5.29150 0 5.00000 7.48331i 0
65.4 0 2.64575 + 1.41421i 0 7.48331i 0 5.29150 0 5.00000 + 7.48331i 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 inner
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 192.3.e.f 4
3.b odd 2 1 inner 192.3.e.f 4
4.b odd 2 1 inner 192.3.e.f 4
8.b even 2 1 96.3.e.b 4
8.d odd 2 1 96.3.e.b 4
12.b even 2 1 inner 192.3.e.f 4
16.e even 4 2 768.3.h.e 8
16.f odd 4 2 768.3.h.e 8
24.f even 2 1 96.3.e.b 4
24.h odd 2 1 96.3.e.b 4
48.i odd 4 2 768.3.h.e 8
48.k even 4 2 768.3.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.e.b 4 8.b even 2 1
96.3.e.b 4 8.d odd 2 1
96.3.e.b 4 24.f even 2 1
96.3.e.b 4 24.h odd 2 1
192.3.e.f 4 1.a even 1 1 trivial
192.3.e.f 4 3.b odd 2 1 inner
192.3.e.f 4 4.b odd 2 1 inner
192.3.e.f 4 12.b even 2 1 inner
768.3.h.e 8 16.e even 4 2
768.3.h.e 8 16.f odd 4 2
768.3.h.e 8 48.i odd 4 2
768.3.h.e 8 48.k even 4 2

Hecke kernels

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

\( T_{5}^{2} + 56 \)
\( T_{7}^{2} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 10 T^{2} + T^{4} \)
$5$ \( ( 56 + T^{2} )^{2} \)
$7$ \( ( -28 + T^{2} )^{2} \)
$11$ \( ( 200 + T^{2} )^{2} \)
$13$ \( ( 10 + T )^{4} \)
$17$ \( T^{4} \)
$19$ \( ( -700 + T^{2} )^{2} \)
$23$ \( ( 288 + T^{2} )^{2} \)
$29$ \( ( 1400 + T^{2} )^{2} \)
$31$ \( ( -700 + T^{2} )^{2} \)
$37$ \( ( 10 + T )^{4} \)
$41$ \( ( 224 + T^{2} )^{2} \)
$43$ \( ( -3388 + T^{2} )^{2} \)
$47$ \( ( 128 + T^{2} )^{2} \)
$53$ \( ( 1400 + T^{2} )^{2} \)
$59$ \( ( 9800 + T^{2} )^{2} \)
$61$ \( ( 90 + T )^{4} \)
$67$ \( ( -28 + T^{2} )^{2} \)
$71$ \( ( 800 + T^{2} )^{2} \)
$73$ \( ( 30 + T )^{4} \)
$79$ \( ( -700 + T^{2} )^{2} \)
$83$ \( ( 648 + T^{2} )^{2} \)
$89$ \( ( 5600 + T^{2} )^{2} \)
$97$ \( ( -10 + T )^{4} \)
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