Properties

Label 1536.2.a.h
Level $1536$
Weight $2$
Character orbit 1536.a
Self dual yes
Analytic conductor $12.265$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
0 1.00000 0 −3.41421 0 1.41421 0 1.00000 0
1.2 0 1.00000 0 −0.585786 0 −1.41421 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

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 1536.2.a.h yes 2
3.b odd 2 1 4608.2.a.q 2
4.b odd 2 1 1536.2.a.a 2
8.b even 2 1 1536.2.a.f yes 2
8.d odd 2 1 1536.2.a.k yes 2
12.b even 2 1 4608.2.a.o 2
16.e even 4 2 1536.2.d.b 4
16.f odd 4 2 1536.2.d.e 4
24.f even 2 1 4608.2.a.d 2
24.h odd 2 1 4608.2.a.b 2
48.i odd 4 2 4608.2.d.f 4
48.k even 4 2 4608.2.d.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.a 2 4.b odd 2 1
1536.2.a.f yes 2 8.b even 2 1
1536.2.a.h yes 2 1.a even 1 1 trivial
1536.2.a.k yes 2 8.d odd 2 1
1536.2.d.b 4 16.e even 4 2
1536.2.d.e 4 16.f odd 4 2
4608.2.a.b 2 24.h odd 2 1
4608.2.a.d 2 24.f even 2 1
4608.2.a.o 2 12.b even 2 1
4608.2.a.q 2 3.b odd 2 1
4608.2.d.f 4 48.i odd 4 2
4608.2.d.h 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1536))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 2 + 4 T + T^{2} \)
$7$ \( -2 + T^{2} \)
$11$ \( -4 + 4 T + T^{2} \)
$13$ \( -4 + 4 T + T^{2} \)
$17$ \( -4 - 4 T + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( 8 + 8 T + T^{2} \)
$29$ \( 34 + 12 T + T^{2} \)
$31$ \( -50 + T^{2} \)
$37$ \( 4 + 12 T + T^{2} \)
$41$ \( -68 + 4 T + T^{2} \)
$43$ \( 8 - 8 T + T^{2} \)
$47$ \( -56 + 8 T + T^{2} \)
$53$ \( 18 + 12 T + T^{2} \)
$59$ \( -16 + 8 T + T^{2} \)
$61$ \( 4 + 12 T + T^{2} \)
$67$ \( -32 + T^{2} \)
$71$ \( -72 + T^{2} \)
$73$ \( -128 + T^{2} \)
$79$ \( 254 - 32 T + T^{2} \)
$83$ \( 28 + 12 T + T^{2} \)
$89$ \( -92 + 12 T + T^{2} \)
$97$ \( -28 + 4 T + T^{2} \)
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