Properties

Label 192.3.g.a
Level $192$
Weight $3$
Character orbit 192.g
Analytic conductor $5.232$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} -6 q^{5} + ( -4 + 8 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} -6 q^{5} + ( -4 + 8 \zeta_{6} ) q^{7} -3 q^{9} + ( -12 + 24 \zeta_{6} ) q^{11} + 14 q^{13} + ( -6 + 12 \zeta_{6} ) q^{15} -6 q^{17} + ( -4 + 8 \zeta_{6} ) q^{19} + 12 q^{21} + 11 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} -30 q^{29} + ( -12 + 24 \zeta_{6} ) q^{31} + 36 q^{33} + ( 24 - 48 \zeta_{6} ) q^{35} -26 q^{37} + ( 14 - 28 \zeta_{6} ) q^{39} -54 q^{41} + ( -12 + 24 \zeta_{6} ) q^{43} + 18 q^{45} + ( -24 + 48 \zeta_{6} ) q^{47} + q^{49} + ( -6 + 12 \zeta_{6} ) q^{51} + 18 q^{53} + ( 72 - 144 \zeta_{6} ) q^{55} + 12 q^{57} + ( 12 - 24 \zeta_{6} ) q^{59} + 70 q^{61} + ( 12 - 24 \zeta_{6} ) q^{63} -84 q^{65} + ( 68 - 136 \zeta_{6} ) q^{67} + ( 48 - 96 \zeta_{6} ) q^{71} + 82 q^{73} + ( 11 - 22 \zeta_{6} ) q^{75} -144 q^{77} + ( -44 + 88 \zeta_{6} ) q^{79} + 9 q^{81} + ( 12 - 24 \zeta_{6} ) q^{83} + 36 q^{85} + ( -30 + 60 \zeta_{6} ) q^{87} + 114 q^{89} + ( -56 + 112 \zeta_{6} ) q^{91} + 36 q^{93} + ( 24 - 48 \zeta_{6} ) q^{95} + 34 q^{97} + ( 36 - 72 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 12q^{5} - 6q^{9} + O(q^{10}) \) \( 2q - 12q^{5} - 6q^{9} + 28q^{13} - 12q^{17} + 24q^{21} + 22q^{25} - 60q^{29} + 72q^{33} - 52q^{37} - 108q^{41} + 36q^{45} + 2q^{49} + 36q^{53} + 24q^{57} + 140q^{61} - 168q^{65} + 164q^{73} - 288q^{77} + 18q^{81} + 72q^{85} + 228q^{89} + 72q^{93} + 68q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 −6.00000 0 6.92820i 0 −3.00000 0
127.2 0 1.73205i 0 −6.00000 0 6.92820i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.3.g.a 2
3.b odd 2 1 576.3.g.i 2
4.b odd 2 1 inner 192.3.g.a 2
8.b even 2 1 48.3.g.a 2
8.d odd 2 1 48.3.g.a 2
12.b even 2 1 576.3.g.i 2
16.e even 4 2 768.3.b.b 4
16.f odd 4 2 768.3.b.b 4
24.f even 2 1 144.3.g.b 2
24.h odd 2 1 144.3.g.b 2
40.e odd 2 1 1200.3.e.h 2
40.f even 2 1 1200.3.e.h 2
40.i odd 4 2 1200.3.j.a 4
40.k even 4 2 1200.3.j.a 4
48.i odd 4 2 2304.3.b.n 4
48.k even 4 2 2304.3.b.n 4
56.e even 2 1 2352.3.m.a 2
56.h odd 2 1 2352.3.m.a 2
72.j odd 6 1 1296.3.o.n 2
72.j odd 6 1 1296.3.o.p 2
72.l even 6 1 1296.3.o.n 2
72.l even 6 1 1296.3.o.p 2
72.n even 6 1 1296.3.o.a 2
72.n even 6 1 1296.3.o.c 2
72.p odd 6 1 1296.3.o.a 2
72.p odd 6 1 1296.3.o.c 2
120.i odd 2 1 3600.3.e.t 2
120.m even 2 1 3600.3.e.t 2
120.q odd 4 2 3600.3.j.i 4
120.w even 4 2 3600.3.j.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.g.a 2 8.b even 2 1
48.3.g.a 2 8.d odd 2 1
144.3.g.b 2 24.f even 2 1
144.3.g.b 2 24.h odd 2 1
192.3.g.a 2 1.a even 1 1 trivial
192.3.g.a 2 4.b odd 2 1 inner
576.3.g.i 2 3.b odd 2 1
576.3.g.i 2 12.b even 2 1
768.3.b.b 4 16.e even 4 2
768.3.b.b 4 16.f odd 4 2
1200.3.e.h 2 40.e odd 2 1
1200.3.e.h 2 40.f even 2 1
1200.3.j.a 4 40.i odd 4 2
1200.3.j.a 4 40.k even 4 2
1296.3.o.a 2 72.n even 6 1
1296.3.o.a 2 72.p odd 6 1
1296.3.o.c 2 72.n even 6 1
1296.3.o.c 2 72.p odd 6 1
1296.3.o.n 2 72.j odd 6 1
1296.3.o.n 2 72.l even 6 1
1296.3.o.p 2 72.j odd 6 1
1296.3.o.p 2 72.l even 6 1
2304.3.b.n 4 48.i odd 4 2
2304.3.b.n 4 48.k even 4 2
2352.3.m.a 2 56.e even 2 1
2352.3.m.a 2 56.h odd 2 1
3600.3.e.t 2 120.i odd 2 1
3600.3.e.t 2 120.m even 2 1
3600.3.j.i 4 120.q odd 4 2
3600.3.j.i 4 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 6 \) acting on \(S_{3}^{\mathrm{new}}(192, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( ( 6 + T )^{2} \)
$7$ \( 48 + T^{2} \)
$11$ \( 432 + T^{2} \)
$13$ \( ( -14 + T )^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( 48 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 30 + T )^{2} \)
$31$ \( 432 + T^{2} \)
$37$ \( ( 26 + T )^{2} \)
$41$ \( ( 54 + T )^{2} \)
$43$ \( 432 + T^{2} \)
$47$ \( 1728 + T^{2} \)
$53$ \( ( -18 + T )^{2} \)
$59$ \( 432 + T^{2} \)
$61$ \( ( -70 + T )^{2} \)
$67$ \( 13872 + T^{2} \)
$71$ \( 6912 + T^{2} \)
$73$ \( ( -82 + T )^{2} \)
$79$ \( 5808 + T^{2} \)
$83$ \( 432 + T^{2} \)
$89$ \( ( -114 + T )^{2} \)
$97$ \( ( -34 + T )^{2} \)
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