Properties

Label 192.3.h.c
Level $192$
Weight $3$
Character orbit 192.h
Analytic conductor $5.232$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
Defining polynomial: \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} -\beta_{4} q^{5} -\beta_{1} q^{7} + ( 1 - \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} -\beta_{4} q^{5} -\beta_{1} q^{7} + ( 1 - \beta_{5} ) q^{9} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{11} + 3 \beta_{6} q^{13} + ( -5 \beta_{1} + \beta_{7} ) q^{15} + ( \beta_{2} - \beta_{3} ) q^{19} + ( -\beta_{4} - \beta_{6} ) q^{21} + 2 \beta_{7} q^{23} + 35 q^{25} + ( 2 \beta_{2} - 9 \beta_{3} ) q^{27} -\beta_{4} q^{29} + 7 \beta_{1} q^{31} + ( -30 + 3 \beta_{5} ) q^{33} + ( 6 \beta_{2} + 6 \beta_{3} ) q^{35} + 5 \beta_{6} q^{37} + ( -12 \beta_{1} - 3 \beta_{7} ) q^{39} -6 \beta_{5} q^{41} + ( -13 \beta_{2} + 13 \beta_{3} ) q^{43} + ( -\beta_{4} - 10 \beta_{6} ) q^{45} -4 \beta_{7} q^{47} -37 q^{49} + 7 \beta_{4} q^{53} + 30 \beta_{1} q^{55} + ( -8 - \beta_{5} ) q^{57} + ( 9 \beta_{2} + 9 \beta_{3} ) q^{59} -\beta_{6} q^{61} + ( -\beta_{1} + 2 \beta_{7} ) q^{63} + 18 \beta_{5} q^{65} + ( -7 \beta_{2} + 7 \beta_{3} ) q^{67} + ( 8 \beta_{4} - 10 \beta_{6} ) q^{69} -2 \beta_{7} q^{71} + 74 q^{73} + 35 \beta_{2} q^{75} + 6 \beta_{4} q^{77} + 15 \beta_{1} q^{79} + ( -79 - 2 \beta_{5} ) q^{81} + ( -27 \beta_{2} - 27 \beta_{3} ) q^{83} + ( -5 \beta_{1} + \beta_{7} ) q^{87} -6 \beta_{5} q^{89} + ( -18 \beta_{2} + 18 \beta_{3} ) q^{91} + ( 7 \beta_{4} + 7 \beta_{6} ) q^{93} + 2 \beta_{7} q^{95} -62 q^{97} + ( -33 \beta_{2} + 27 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{9} + 280q^{25} - 240q^{33} - 296q^{49} - 64q^{57} + 592q^{73} - 632q^{81} - 496q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{7} - 4 \nu^{5} + 10 \nu^{3} - 7 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 6 \nu^{7} - \nu^{6} - 16 \nu^{5} + 40 \nu^{3} - 2 \nu - 9 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -6 \nu^{7} - \nu^{6} + 16 \nu^{5} - 40 \nu^{3} + 2 \nu - 9 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} + 16 \nu^{5} - 44 \nu^{3} + 31 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\( 6 \nu^{7} - 16 \nu^{5} + 44 \nu^{3} - 2 \nu \)
\(\beta_{6}\)\(=\)\( -3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 5 \)
\(\beta_{7}\)\(=\)\( -8 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_{1}\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 3 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} + 12\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + 2 \beta_{3} - 2 \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{7} - 7 \beta_{6} - 6 \beta_{3} - 6 \beta_{2} - 28\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{5} - 10 \beta_{4} + 11 \beta_{3} - 11 \beta_{2} - 22 \beta_{1}\)\()/16\)
\(\nu^{6}\)\(=\)\(-2 \beta_{3} - 2 \beta_{2} - 9\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{5} - 26 \beta_{4} - 29 \beta_{3} + 29 \beta_{2} - 58 \beta_{1}\)\()/16\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.535233 0.309017i
−0.535233 0.309017i
0.535233 + 0.309017i
−0.535233 + 0.309017i
1.40126 + 0.809017i
−1.40126 + 0.809017i
1.40126 0.809017i
−1.40126 0.809017i
0 −2.23607 2.00000i 0 −7.74597 0 3.46410 0 1.00000 + 8.94427i 0
161.2 0 −2.23607 2.00000i 0 7.74597 0 −3.46410 0 1.00000 + 8.94427i 0
161.3 0 −2.23607 + 2.00000i 0 −7.74597 0 3.46410 0 1.00000 8.94427i 0
161.4 0 −2.23607 + 2.00000i 0 7.74597 0 −3.46410 0 1.00000 8.94427i 0
161.5 0 2.23607 2.00000i 0 −7.74597 0 −3.46410 0 1.00000 8.94427i 0
161.6 0 2.23607 2.00000i 0 7.74597 0 3.46410 0 1.00000 8.94427i 0
161.7 0 2.23607 + 2.00000i 0 −7.74597 0 −3.46410 0 1.00000 + 8.94427i 0
161.8 0 2.23607 + 2.00000i 0 7.74597 0 3.46410 0 1.00000 + 8.94427i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.3.h.c 8
3.b odd 2 1 inner 192.3.h.c 8
4.b odd 2 1 inner 192.3.h.c 8
8.b even 2 1 inner 192.3.h.c 8
8.d odd 2 1 inner 192.3.h.c 8
12.b even 2 1 inner 192.3.h.c 8
16.e even 4 1 768.3.e.g 4
16.e even 4 1 768.3.e.n 4
16.f odd 4 1 768.3.e.g 4
16.f odd 4 1 768.3.e.n 4
24.f even 2 1 inner 192.3.h.c 8
24.h odd 2 1 inner 192.3.h.c 8
48.i odd 4 1 768.3.e.g 4
48.i odd 4 1 768.3.e.n 4
48.k even 4 1 768.3.e.g 4
48.k even 4 1 768.3.e.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.3.h.c 8 1.a even 1 1 trivial
192.3.h.c 8 3.b odd 2 1 inner
192.3.h.c 8 4.b odd 2 1 inner
192.3.h.c 8 8.b even 2 1 inner
192.3.h.c 8 8.d odd 2 1 inner
192.3.h.c 8 12.b even 2 1 inner
192.3.h.c 8 24.f even 2 1 inner
192.3.h.c 8 24.h odd 2 1 inner
768.3.e.g 4 16.e even 4 1
768.3.e.g 4 16.f odd 4 1
768.3.e.g 4 48.i odd 4 1
768.3.e.g 4 48.k even 4 1
768.3.e.n 4 16.e even 4 1
768.3.e.n 4 16.f odd 4 1
768.3.e.n 4 48.i odd 4 1
768.3.e.n 4 48.k even 4 1

Hecke kernels

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

\( T_{5}^{2} - 60 \)
\( T_{7}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 81 - 2 T^{2} + T^{4} )^{2} \)
$5$ \( ( -60 + T^{2} )^{4} \)
$7$ \( ( -12 + T^{2} )^{4} \)
$11$ \( ( -180 + T^{2} )^{4} \)
$13$ \( ( 432 + T^{2} )^{4} \)
$17$ \( T^{8} \)
$19$ \( ( 16 + T^{2} )^{4} \)
$23$ \( ( 960 + T^{2} )^{4} \)
$29$ \( ( -60 + T^{2} )^{4} \)
$31$ \( ( -588 + T^{2} )^{4} \)
$37$ \( ( 1200 + T^{2} )^{4} \)
$41$ \( ( 2880 + T^{2} )^{4} \)
$43$ \( ( 2704 + T^{2} )^{4} \)
$47$ \( ( 3840 + T^{2} )^{4} \)
$53$ \( ( -2940 + T^{2} )^{4} \)
$59$ \( ( -1620 + T^{2} )^{4} \)
$61$ \( ( 48 + T^{2} )^{4} \)
$67$ \( ( 784 + T^{2} )^{4} \)
$71$ \( ( 960 + T^{2} )^{4} \)
$73$ \( ( -74 + T )^{8} \)
$79$ \( ( -2700 + T^{2} )^{4} \)
$83$ \( ( -14580 + T^{2} )^{4} \)
$89$ \( ( 2880 + T^{2} )^{4} \)
$97$ \( ( 62 + T )^{8} \)
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