Properties

Label 192.12.a.q
Level $192$
Weight $12$
Character orbit 192.a
Self dual yes
Analytic conductor $147.522$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 192.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(147.521890667\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 243 q^{3} + 5370 q^{5} - 27760 q^{7} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 243 q^{3} + 5370 q^{5} - 27760 q^{7} + 59049 q^{9} - 637836 q^{11} - 766214 q^{13} + 1304910 q^{15} + 3084354 q^{17} + 19511404 q^{19} - 6745680 q^{21} + 15312360 q^{23} - 19991225 q^{25} + 14348907 q^{27} - 10751262 q^{29} - 50937400 q^{31} - 154994148 q^{33} - 149071200 q^{35} - 664740830 q^{37} - 186190002 q^{39} + 898833450 q^{41} + 957947188 q^{43} + 317093130 q^{45} - 1555741344 q^{47} - 1206709143 q^{49} + 749498022 q^{51} - 3792417030 q^{53} - 3425179320 q^{55} + 4741271172 q^{57} - 555306924 q^{59} - 4950420998 q^{61} - 1639200240 q^{63} - 4114569180 q^{65} - 5292399284 q^{67} + 3720903480 q^{69} - 14831086248 q^{71} + 13971005210 q^{73} - 4857867675 q^{75} + 17706327360 q^{77} + 3720542360 q^{79} + 3486784401 q^{81} - 8768454036 q^{83} + 16562980980 q^{85} - 2612556666 q^{87} - 25472769174 q^{89} + 21270100640 q^{91} - 12377788200 q^{93} + 104776239480 q^{95} - 39092494846 q^{97} - 37663577964 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 243.000 0 5370.00 0 −27760.0 0 59049.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.12.a.q 1
4.b odd 2 1 192.12.a.g 1
8.b even 2 1 3.12.a.a 1
8.d odd 2 1 48.12.a.f 1
24.f even 2 1 144.12.a.l 1
24.h odd 2 1 9.12.a.a 1
40.f even 2 1 75.12.a.a 1
40.i odd 4 2 75.12.b.a 2
56.h odd 2 1 147.12.a.c 1
72.j odd 6 2 81.12.c.e 2
72.n even 6 2 81.12.c.a 2
120.i odd 2 1 225.12.a.f 1
120.w even 4 2 225.12.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 8.b even 2 1
9.12.a.a 1 24.h odd 2 1
48.12.a.f 1 8.d odd 2 1
75.12.a.a 1 40.f even 2 1
75.12.b.a 2 40.i odd 4 2
81.12.c.a 2 72.n even 6 2
81.12.c.e 2 72.j odd 6 2
144.12.a.l 1 24.f even 2 1
147.12.a.c 1 56.h odd 2 1
192.12.a.g 1 4.b odd 2 1
192.12.a.q 1 1.a even 1 1 trivial
225.12.a.f 1 120.i odd 2 1
225.12.b.a 2 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(192))\):

\( T_{5} - 5370 \) Copy content Toggle raw display
\( T_{7} + 27760 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 243 \) Copy content Toggle raw display
$5$ \( T - 5370 \) Copy content Toggle raw display
$7$ \( T + 27760 \) Copy content Toggle raw display
$11$ \( T + 637836 \) Copy content Toggle raw display
$13$ \( T + 766214 \) Copy content Toggle raw display
$17$ \( T - 3084354 \) Copy content Toggle raw display
$19$ \( T - 19511404 \) Copy content Toggle raw display
$23$ \( T - 15312360 \) Copy content Toggle raw display
$29$ \( T + 10751262 \) Copy content Toggle raw display
$31$ \( T + 50937400 \) Copy content Toggle raw display
$37$ \( T + 664740830 \) Copy content Toggle raw display
$41$ \( T - 898833450 \) Copy content Toggle raw display
$43$ \( T - 957947188 \) Copy content Toggle raw display
$47$ \( T + 1555741344 \) Copy content Toggle raw display
$53$ \( T + 3792417030 \) Copy content Toggle raw display
$59$ \( T + 555306924 \) Copy content Toggle raw display
$61$ \( T + 4950420998 \) Copy content Toggle raw display
$67$ \( T + 5292399284 \) Copy content Toggle raw display
$71$ \( T + 14831086248 \) Copy content Toggle raw display
$73$ \( T - 13971005210 \) Copy content Toggle raw display
$79$ \( T - 3720542360 \) Copy content Toggle raw display
$83$ \( T + 8768454036 \) Copy content Toggle raw display
$89$ \( T + 25472769174 \) Copy content Toggle raw display
$97$ \( T + 39092494846 \) Copy content Toggle raw display
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