Properties

Label 4864.2.a.z
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1216)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{3} -\beta q^{5} + \beta q^{7} + q^{9} +O(q^{10})\) \( q + 2 q^{3} -\beta q^{5} + \beta q^{7} + q^{9} -3 q^{11} -2 \beta q^{15} -3 q^{17} + q^{19} + 2 \beta q^{21} + 2 \beta q^{23} -2 q^{25} -4 q^{27} + 2 \beta q^{29} -4 \beta q^{31} -6 q^{33} -3 q^{35} + 6 \beta q^{37} -6 q^{41} - q^{43} -\beta q^{45} -3 \beta q^{47} -4 q^{49} -6 q^{51} + 3 \beta q^{55} + 2 q^{57} -6 q^{59} -3 \beta q^{61} + \beta q^{63} + 4 q^{67} + 4 \beta q^{69} -2 \beta q^{71} + q^{73} -4 q^{75} -3 \beta q^{77} -11 q^{81} -12 q^{83} + 3 \beta q^{85} + 4 \beta q^{87} -8 \beta q^{93} -\beta q^{95} + 4 q^{97} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{3} + 2q^{9} + O(q^{10}) \) \( 2q + 4q^{3} + 2q^{9} - 6q^{11} - 6q^{17} + 2q^{19} - 4q^{25} - 8q^{27} - 12q^{33} - 6q^{35} - 12q^{41} - 2q^{43} - 8q^{49} - 12q^{51} + 4q^{57} - 12q^{59} + 8q^{67} + 2q^{73} - 8q^{75} - 22q^{81} - 24q^{83} + 8q^{97} - 6q^{99} + O(q^{100}) \)

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
1.73205
−1.73205
0 2.00000 0 −1.73205 0 1.73205 0 1.00000 0
1.2 0 2.00000 0 1.73205 0 −1.73205 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 4864.2.a.z 2
4.b odd 2 1 4864.2.a.q 2
8.b even 2 1 4864.2.a.q 2
8.d odd 2 1 inner 4864.2.a.z 2
16.e even 4 2 1216.2.c.e 4
16.f odd 4 2 1216.2.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.e 4 16.e even 4 2
1216.2.c.e 4 16.f odd 4 2
4864.2.a.q 2 4.b odd 2 1
4864.2.a.q 2 8.b even 2 1
4864.2.a.z 2 1.a even 1 1 trivial
4864.2.a.z 2 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4864))\):

\( T_{3} - 2 \)
\( T_{5}^{2} - 3 \)
\( T_{7}^{2} - 3 \)
\( T_{11} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -2 + T )^{2} \)
$5$ \( -3 + T^{2} \)
$7$ \( -3 + T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( -12 + T^{2} \)
$31$ \( -48 + T^{2} \)
$37$ \( -108 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( -27 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( -27 + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -12 + T^{2} \)
$73$ \( ( -1 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( ( 12 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -4 + T )^{2} \)
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