Properties

Label 192.3.h.a
Level $192$
Weight $3$
Character orbit 192.h
Analytic conductor $5.232$
Analytic rank $0$
Dimension $4$
CM discriminant -3
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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \zeta_{12}^{3} q^{3} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 \zeta_{12}^{3} q^{3} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{7} -9 q^{9} + ( 8 - 16 \zeta_{12}^{2} ) q^{13} + 26 \zeta_{12}^{3} q^{19} + ( 24 - 48 \zeta_{12}^{2} ) q^{21} -25 q^{25} -27 \zeta_{12}^{3} q^{27} + ( -48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{31} + ( -40 + 80 \zeta_{12}^{2} ) q^{37} + ( 48 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{39} + 22 \zeta_{12}^{3} q^{43} + 143 q^{49} -78 q^{57} + ( -56 + 112 \zeta_{12}^{2} ) q^{61} + ( 144 \zeta_{12} - 72 \zeta_{12}^{3} ) q^{63} -122 \zeta_{12}^{3} q^{67} -46 q^{73} -75 \zeta_{12}^{3} q^{75} + ( 80 \zeta_{12} - 40 \zeta_{12}^{3} ) q^{79} + 81 q^{81} + 192 \zeta_{12}^{3} q^{91} + ( 72 - 144 \zeta_{12}^{2} ) q^{93} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 36q^{9} + O(q^{10}) \) \( 4q - 36q^{9} - 100q^{25} + 572q^{49} - 312q^{57} - 184q^{73} + 324q^{81} - 8q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 3.00000i 0 0 0 −13.8564 0 −9.00000 0
161.2 0 3.00000i 0 0 0 13.8564 0 −9.00000 0
161.3 0 3.00000i 0 0 0 −13.8564 0 −9.00000 0
161.4 0 3.00000i 0 0 0 13.8564 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
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.a 4
3.b odd 2 1 CM 192.3.h.a 4
4.b odd 2 1 inner 192.3.h.a 4
8.b even 2 1 inner 192.3.h.a 4
8.d odd 2 1 inner 192.3.h.a 4
12.b even 2 1 inner 192.3.h.a 4
16.e even 4 1 768.3.e.a 2
16.e even 4 1 768.3.e.f 2
16.f odd 4 1 768.3.e.a 2
16.f odd 4 1 768.3.e.f 2
24.f even 2 1 inner 192.3.h.a 4
24.h odd 2 1 inner 192.3.h.a 4
48.i odd 4 1 768.3.e.a 2
48.i odd 4 1 768.3.e.f 2
48.k even 4 1 768.3.e.a 2
48.k even 4 1 768.3.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.3.h.a 4 1.a even 1 1 trivial
192.3.h.a 4 3.b odd 2 1 CM
192.3.h.a 4 4.b odd 2 1 inner
192.3.h.a 4 8.b even 2 1 inner
192.3.h.a 4 8.d odd 2 1 inner
192.3.h.a 4 12.b even 2 1 inner
192.3.h.a 4 24.f even 2 1 inner
192.3.h.a 4 24.h odd 2 1 inner
768.3.e.a 2 16.e even 4 1
768.3.e.a 2 16.f odd 4 1
768.3.e.a 2 48.i odd 4 1
768.3.e.a 2 48.k even 4 1
768.3.e.f 2 16.e even 4 1
768.3.e.f 2 16.f odd 4 1
768.3.e.f 2 48.i odd 4 1
768.3.e.f 2 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 \(S_{3}^{\mathrm{new}}(192, [\chi])\):

\( T_{5} \)
\( T_{7}^{2} - 192 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 9 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( -192 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 192 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 676 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( -1728 + T^{2} )^{2} \)
$37$ \( ( 4800 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( 484 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 9408 + T^{2} )^{2} \)
$67$ \( ( 14884 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( ( 46 + T )^{4} \)
$79$ \( ( -4800 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 2 + T )^{4} \)
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