Properties

Label 1536.2.a.m
Level $1536$
Weight $2$
Character orbit 1536.a
Self dual yes
Analytic conductor $12.265$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{1} q^{5} -\beta_{3} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{1} q^{5} -\beta_{3} q^{7} + q^{9} + ( -2 - \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{3} ) q^{13} -\beta_{1} q^{15} + ( 2 - \beta_{2} ) q^{17} + \beta_{2} q^{19} + \beta_{3} q^{21} + 2 \beta_{1} q^{23} + ( 5 + 2 \beta_{2} ) q^{25} - q^{27} + ( -\beta_{1} + 2 \beta_{3} ) q^{29} + \beta_{3} q^{31} + ( 2 + \beta_{2} ) q^{33} + ( -2 + 3 \beta_{2} ) q^{35} + ( -\beta_{1} - \beta_{3} ) q^{37} + ( -\beta_{1} + \beta_{3} ) q^{39} + ( 6 - \beta_{2} ) q^{41} + ( 4 - 3 \beta_{2} ) q^{43} + \beta_{1} q^{45} + 2 \beta_{1} q^{47} + ( 7 - 4 \beta_{2} ) q^{49} + ( -2 + \beta_{2} ) q^{51} + ( -\beta_{1} - 2 \beta_{3} ) q^{53} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{55} -\beta_{2} q^{57} + ( -4 - 2 \beta_{2} ) q^{59} + ( -\beta_{1} - \beta_{3} ) q^{61} -\beta_{3} q^{63} + ( 8 + 5 \beta_{2} ) q^{65} + ( 8 + 2 \beta_{2} ) q^{67} -2 \beta_{1} q^{69} + 2 \beta_{3} q^{71} + 4 q^{73} + ( -5 - 2 \beta_{2} ) q^{75} -2 \beta_{1} q^{77} -\beta_{3} q^{79} + q^{81} + ( -6 - \beta_{2} ) q^{83} + 2 \beta_{3} q^{85} + ( \beta_{1} - 2 \beta_{3} ) q^{87} + 10 q^{89} + ( 12 - \beta_{2} ) q^{91} -\beta_{3} q^{93} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{95} + ( 6 + 2 \beta_{2} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{9} - 8q^{11} + 8q^{17} + 20q^{25} - 4q^{27} + 8q^{33} - 8q^{35} + 24q^{41} + 16q^{43} + 28q^{49} - 8q^{51} - 16q^{59} + 32q^{65} + 32q^{67} + 16q^{73} - 20q^{75} + 4q^{81} - 24q^{83} + 40q^{89} + 48q^{91} + 24q^{97} - 8q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu^{2} - 6 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} - 2 \nu^{2} - 8 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 3 \nu^{2} + 2 \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} + 4 \beta_{2} - 5 \beta_{1} + 6\)\()/2\)

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
2.68554
0.334904
−1.74912
−1.27133
0 −1.00000 0 −3.95687 0 −1.63899 0 1.00000 0
1.2 0 −1.00000 0 −2.08402 0 5.03127 0 1.00000 0
1.3 0 −1.00000 0 2.08402 0 −5.03127 0 1.00000 0
1.4 0 −1.00000 0 3.95687 0 1.63899 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1536.2.a.m 4
3.b odd 2 1 4608.2.a.ba 4
4.b odd 2 1 1536.2.a.n yes 4
8.b even 2 1 1536.2.a.n yes 4
8.d odd 2 1 inner 1536.2.a.m 4
12.b even 2 1 4608.2.a.t 4
16.e even 4 2 1536.2.d.g 8
16.f odd 4 2 1536.2.d.g 8
24.f even 2 1 4608.2.a.ba 4
24.h odd 2 1 4608.2.a.t 4
48.i odd 4 2 4608.2.d.p 8
48.k even 4 2 4608.2.d.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.m 4 1.a even 1 1 trivial
1536.2.a.m 4 8.d odd 2 1 inner
1536.2.a.n yes 4 4.b odd 2 1
1536.2.a.n yes 4 8.b even 2 1
1536.2.d.g 8 16.e even 4 2
1536.2.d.g 8 16.f odd 4 2
4608.2.a.t 4 12.b even 2 1
4608.2.a.t 4 24.h odd 2 1
4608.2.a.ba 4 3.b odd 2 1
4608.2.a.ba 4 24.f even 2 1
4608.2.d.p 8 48.i odd 4 2
4608.2.d.p 8 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1536))\):

\( T_{5}^{4} - 20 T_{5}^{2} + 68 \)
\( T_{7}^{4} - 28 T_{7}^{2} + 68 \)
\( T_{11}^{2} + 4 T_{11} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 68 - 20 T^{2} + T^{4} \)
$7$ \( 68 - 28 T^{2} + T^{4} \)
$11$ \( ( -4 + 4 T + T^{2} )^{2} \)
$13$ \( 272 - 40 T^{2} + T^{4} \)
$17$ \( ( -4 - 4 T + T^{2} )^{2} \)
$19$ \( ( -8 + T^{2} )^{2} \)
$23$ \( 1088 - 80 T^{2} + T^{4} \)
$29$ \( 3332 - 116 T^{2} + T^{4} \)
$31$ \( 68 - 28 T^{2} + T^{4} \)
$37$ \( 272 - 56 T^{2} + T^{4} \)
$41$ \( ( 28 - 12 T + T^{2} )^{2} \)
$43$ \( ( -56 - 8 T + T^{2} )^{2} \)
$47$ \( 1088 - 80 T^{2} + T^{4} \)
$53$ \( 68 - 148 T^{2} + T^{4} \)
$59$ \( ( -16 + 8 T + T^{2} )^{2} \)
$61$ \( 272 - 56 T^{2} + T^{4} \)
$67$ \( ( 32 - 16 T + T^{2} )^{2} \)
$71$ \( 1088 - 112 T^{2} + T^{4} \)
$73$ \( ( -4 + T )^{4} \)
$79$ \( 68 - 28 T^{2} + T^{4} \)
$83$ \( ( 28 + 12 T + T^{2} )^{2} \)
$89$ \( ( -10 + T )^{4} \)
$97$ \( ( 4 - 12 T + T^{2} )^{2} \)
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