Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + 10 \zeta_{12}^{3} q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + 10 \zeta_{12}^{3} q^{7} + 3 q^{9} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{11} + ( 4 - 8 \zeta_{12}^{2} ) q^{13} + 6 \zeta_{12}^{3} q^{15} + 30 q^{17} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{19} + ( 10 - 20 \zeta_{12}^{2} ) q^{21} + 12 \zeta_{12}^{3} q^{23} + 13 q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -30 + 60 \zeta_{12}^{2} ) q^{29} + 14 \zeta_{12}^{3} q^{31} -24 q^{33} + ( 40 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{35} + ( 32 - 64 \zeta_{12}^{2} ) q^{37} + 12 \zeta_{12}^{3} q^{39} -6 q^{41} + ( -72 \zeta_{12} + 36 \zeta_{12}^{3} ) q^{43} + ( 6 - 12 \zeta_{12}^{2} ) q^{45} -84 \zeta_{12}^{3} q^{47} -51 q^{49} + ( -60 \zeta_{12} + 30 \zeta_{12}^{3} ) q^{51} + ( -10 + 20 \zeta_{12}^{2} ) q^{53} -48 \zeta_{12}^{3} q^{55} -12 q^{57} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{59} + ( -56 + 112 \zeta_{12}^{2} ) q^{61} + 30 \zeta_{12}^{3} q^{63} -24 q^{65} + ( -56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{67} + ( 12 - 24 \zeta_{12}^{2} ) q^{69} -60 \zeta_{12}^{3} q^{71} -86 q^{73} + ( -26 \zeta_{12} + 13 \zeta_{12}^{3} ) q^{75} + ( -80 + 160 \zeta_{12}^{2} ) q^{77} + 38 \zeta_{12}^{3} q^{79} + 9 q^{81} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{83} + ( 60 - 120 \zeta_{12}^{2} ) q^{85} -90 \zeta_{12}^{3} q^{87} + 78 q^{89} + ( 80 \zeta_{12} - 40 \zeta_{12}^{3} ) q^{91} + ( 14 - 28 \zeta_{12}^{2} ) q^{93} -24 \zeta_{12}^{3} q^{95} + 62 q^{97} + ( 48 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 120q^{17} + 52q^{25} - 96q^{33} - 24q^{41} - 204q^{49} - 48q^{57} - 96q^{65} - 344q^{73} + 36q^{81} + 312q^{89} + 248q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0 −1.73205 0 3.46410i 0 10.0000i 0 3.00000 0
31.2 0 −1.73205 0 3.46410i 0 10.0000i 0 3.00000 0
31.3 0 1.73205 0 3.46410i 0 10.0000i 0 3.00000 0
31.4 0 1.73205 0 3.46410i 0 10.0000i 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
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 192.3.b.b 4
3.b odd 2 1 576.3.b.a 4
4.b odd 2 1 inner 192.3.b.b 4
8.b even 2 1 inner 192.3.b.b 4
8.d odd 2 1 inner 192.3.b.b 4
12.b even 2 1 576.3.b.a 4
16.e even 4 2 768.3.g.f 4
16.f odd 4 2 768.3.g.f 4
24.f even 2 1 576.3.b.a 4
24.h odd 2 1 576.3.b.a 4
48.i odd 4 2 2304.3.g.q 4
48.k even 4 2 2304.3.g.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.3.b.b 4 1.a even 1 1 trivial
192.3.b.b 4 4.b odd 2 1 inner
192.3.b.b 4 8.b even 2 1 inner
192.3.b.b 4 8.d odd 2 1 inner
576.3.b.a 4 3.b odd 2 1
576.3.b.a 4 12.b even 2 1
576.3.b.a 4 24.f even 2 1
576.3.b.a 4 24.h odd 2 1
768.3.g.f 4 16.e even 4 2
768.3.g.f 4 16.f odd 4 2
2304.3.g.q 4 48.i odd 4 2
2304.3.g.q 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 100 \) acting on \(S_{3}^{\mathrm{new}}(192, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 12 + T^{2} )^{2} \)
$7$ \( ( 100 + T^{2} )^{2} \)
$11$ \( ( -192 + T^{2} )^{2} \)
$13$ \( ( 48 + T^{2} )^{2} \)
$17$ \( ( -30 + T )^{4} \)
$19$ \( ( -48 + T^{2} )^{2} \)
$23$ \( ( 144 + T^{2} )^{2} \)
$29$ \( ( 2700 + T^{2} )^{2} \)
$31$ \( ( 196 + T^{2} )^{2} \)
$37$ \( ( 3072 + T^{2} )^{2} \)
$41$ \( ( 6 + T )^{4} \)
$43$ \( ( -3888 + T^{2} )^{2} \)
$47$ \( ( 7056 + T^{2} )^{2} \)
$53$ \( ( 300 + T^{2} )^{2} \)
$59$ \( ( -3888 + T^{2} )^{2} \)
$61$ \( ( 9408 + T^{2} )^{2} \)
$67$ \( ( -2352 + T^{2} )^{2} \)
$71$ \( ( 3600 + T^{2} )^{2} \)
$73$ \( ( 86 + T )^{4} \)
$79$ \( ( 1444 + T^{2} )^{2} \)
$83$ \( ( -192 + T^{2} )^{2} \)
$89$ \( ( -78 + T )^{4} \)
$97$ \( ( -62 + T )^{4} \)
show more
show less