Properties

Label 4864.2.a.v
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -2 q^{5} + ( -1 + \beta ) q^{7} -2 q^{9} +O(q^{10})\) \( q + q^{3} -2 q^{5} + ( -1 + \beta ) q^{7} -2 q^{9} -4 q^{11} + ( 1 - \beta ) q^{13} -2 q^{15} + ( -1 + 2 \beta ) q^{17} + q^{19} + ( -1 + \beta ) q^{21} + ( 3 - \beta ) q^{23} - q^{25} -5 q^{27} + ( 1 + \beta ) q^{29} + ( 2 + 2 \beta ) q^{31} -4 q^{33} + ( 2 - 2 \beta ) q^{35} -2 q^{37} + ( 1 - \beta ) q^{39} + 2 \beta q^{41} + ( -2 - 2 \beta ) q^{43} + 4 q^{45} -2 \beta q^{47} + ( 6 - 2 \beta ) q^{49} + ( -1 + 2 \beta ) q^{51} + ( 9 - \beta ) q^{53} + 8 q^{55} + q^{57} + ( -3 + 2 \beta ) q^{59} + ( -2 + 2 \beta ) q^{61} + ( 2 - 2 \beta ) q^{63} + ( -2 + 2 \beta ) q^{65} + ( 1 + 2 \beta ) q^{67} + ( 3 - \beta ) q^{69} + 14 q^{71} -7 q^{73} - q^{75} + ( 4 - 4 \beta ) q^{77} -2 q^{79} + q^{81} + ( -4 - 2 \beta ) q^{83} + ( 2 - 4 \beta ) q^{85} + ( 1 + \beta ) q^{87} + ( 6 + 2 \beta ) q^{89} + ( -13 + 2 \beta ) q^{91} + ( 2 + 2 \beta ) q^{93} -2 q^{95} + ( 6 - 2 \beta ) q^{97} + 8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{5} - 2q^{7} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{5} - 2q^{7} - 4q^{9} - 8q^{11} + 2q^{13} - 4q^{15} - 2q^{17} + 2q^{19} - 2q^{21} + 6q^{23} - 2q^{25} - 10q^{27} + 2q^{29} + 4q^{31} - 8q^{33} + 4q^{35} - 4q^{37} + 2q^{39} - 4q^{43} + 8q^{45} + 12q^{49} - 2q^{51} + 18q^{53} + 16q^{55} + 2q^{57} - 6q^{59} - 4q^{61} + 4q^{63} - 4q^{65} + 2q^{67} + 6q^{69} + 28q^{71} - 14q^{73} - 2q^{75} + 8q^{77} - 4q^{79} + 2q^{81} - 8q^{83} + 4q^{85} + 2q^{87} + 12q^{89} - 26q^{91} + 4q^{93} - 4q^{95} + 12q^{97} + 16q^{99} + O(q^{100}) \)

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 1.00000 0 −2.00000 0 −4.46410 0 −2.00000 0
1.2 0 1.00000 0 −2.00000 0 2.46410 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4864.2.a.v 2
4.b odd 2 1 4864.2.a.s 2
8.b even 2 1 4864.2.a.u 2
8.d odd 2 1 4864.2.a.x 2
16.e even 4 2 2432.2.c.d yes 4
16.f odd 4 2 2432.2.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2432.2.c.c 4 16.f odd 4 2
2432.2.c.d yes 4 16.e even 4 2
4864.2.a.s 2 4.b odd 2 1
4864.2.a.u 2 8.b even 2 1
4864.2.a.v 2 1.a even 1 1 trivial
4864.2.a.x 2 8.d odd 2 1

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{5} + 2 \)
\( T_{7}^{2} + 2 T_{7} - 11 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( -11 + 2 T + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -11 - 2 T + T^{2} \)
$17$ \( -47 + 2 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -3 - 6 T + T^{2} \)
$29$ \( -11 - 2 T + T^{2} \)
$31$ \( -44 - 4 T + T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( -48 + T^{2} \)
$43$ \( -44 + 4 T + T^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( 69 - 18 T + T^{2} \)
$59$ \( -39 + 6 T + T^{2} \)
$61$ \( -44 + 4 T + T^{2} \)
$67$ \( -47 - 2 T + T^{2} \)
$71$ \( ( -14 + T )^{2} \)
$73$ \( ( 7 + T )^{2} \)
$79$ \( ( 2 + T )^{2} \)
$83$ \( -32 + 8 T + T^{2} \)
$89$ \( -12 - 12 T + T^{2} \)
$97$ \( -12 - 12 T + T^{2} \)
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