Properties

Label 1536.2.a.d
Level $1536$
Weight $2$
Character orbit 1536.a
Self dual yes
Analytic conductor $12.265$
Analytic rank $0$
Dimension $2$
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: \(2\)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta q^{5} + 3 \beta q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta q^{5} + 3 \beta q^{7} + q^{9} + 6 q^{11} + 4 \beta q^{13} -\beta q^{15} -6 q^{17} + 4 q^{19} -3 \beta q^{21} -2 \beta q^{23} -3 q^{25} - q^{27} + \beta q^{29} + \beta q^{31} -6 q^{33} + 6 q^{35} -6 \beta q^{37} -4 \beta q^{39} -2 q^{41} + \beta q^{45} -2 \beta q^{47} + 11 q^{49} + 6 q^{51} -7 \beta q^{53} + 6 \beta q^{55} -4 q^{57} + 4 q^{59} -6 \beta q^{61} + 3 \beta q^{63} + 8 q^{65} + 8 q^{67} + 2 \beta q^{69} + 2 \beta q^{71} + 8 q^{73} + 3 q^{75} + 18 \beta q^{77} -9 \beta q^{79} + q^{81} + 2 q^{83} -6 \beta q^{85} -\beta q^{87} + 2 q^{89} + 24 q^{91} -\beta q^{93} + 4 \beta q^{95} + 2 q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 12q^{11} - 12q^{17} + 8q^{19} - 6q^{25} - 2q^{27} - 12q^{33} + 12q^{35} - 4q^{41} + 22q^{49} + 12q^{51} - 8q^{57} + 8q^{59} + 16q^{65} + 16q^{67} + 16q^{73} + 6q^{75} + 2q^{81} + 4q^{83} + 4q^{89} + 48q^{91} + 4q^{97} + 12q^{99} + O(q^{100}) \)

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 −1.41421 0 −4.24264 0 1.00000 0
1.2 0 −1.00000 0 1.41421 0 4.24264 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.d 2
3.b odd 2 1 4608.2.a.g 2
4.b odd 2 1 1536.2.a.i yes 2
8.b even 2 1 1536.2.a.i yes 2
8.d odd 2 1 inner 1536.2.a.d 2
12.b even 2 1 4608.2.a.l 2
16.e even 4 2 1536.2.d.c 4
16.f odd 4 2 1536.2.d.c 4
24.f even 2 1 4608.2.a.g 2
24.h odd 2 1 4608.2.a.l 2
48.i odd 4 2 4608.2.d.n 4
48.k even 4 2 4608.2.d.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.d 2 1.a even 1 1 trivial
1536.2.a.d 2 8.d odd 2 1 inner
1536.2.a.i yes 2 4.b odd 2 1
1536.2.a.i yes 2 8.b even 2 1
1536.2.d.c 4 16.e even 4 2
1536.2.d.c 4 16.f odd 4 2
4608.2.a.g 2 3.b odd 2 1
4608.2.a.g 2 24.f even 2 1
4608.2.a.l 2 12.b even 2 1
4608.2.a.l 2 24.h odd 2 1
4608.2.d.n 4 48.i odd 4 2
4608.2.d.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1536))\):

\( T_{5}^{2} - 2 \)
\( T_{7}^{2} - 18 \)
\( T_{11} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -2 + T^{2} \)
$7$ \( -18 + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( -8 + T^{2} \)
$29$ \( -2 + T^{2} \)
$31$ \( -2 + T^{2} \)
$37$ \( -72 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( -98 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( -72 + T^{2} \)
$67$ \( ( -8 + T )^{2} \)
$71$ \( -8 + T^{2} \)
$73$ \( ( -8 + T )^{2} \)
$79$ \( -162 + T^{2} \)
$83$ \( ( -2 + T )^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( ( -2 + T )^{2} \)
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