Properties

Label 4864.2.a.r
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \(x^{2} - 11\)
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 \(\beta = \sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{3} + \beta q^{5} -\beta q^{7} + q^{9} +O(q^{10})\) \( q -2 q^{3} + \beta q^{5} -\beta q^{7} + q^{9} + 5 q^{11} -2 \beta q^{15} + 5 q^{17} + q^{19} + 2 \beta q^{21} -2 \beta q^{23} + 6 q^{25} + 4 q^{27} + 2 \beta q^{29} -10 q^{33} -11 q^{35} -2 \beta q^{37} + 6 q^{41} - q^{43} + \beta q^{45} + 3 \beta q^{47} + 4 q^{49} -10 q^{51} -4 \beta q^{53} + 5 \beta q^{55} -2 q^{57} -6 q^{59} + 3 \beta q^{61} -\beta q^{63} -8 q^{67} + 4 \beta q^{69} + 2 \beta q^{71} + 9 q^{73} -12 q^{75} -5 \beta q^{77} -4 \beta q^{79} -11 q^{81} + 4 q^{83} + 5 \beta q^{85} -4 \beta q^{87} + 4 q^{89} + \beta q^{95} -12 q^{97} + 5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} + 2q^{9} + 10q^{11} + 10q^{17} + 2q^{19} + 12q^{25} + 8q^{27} - 20q^{33} - 22q^{35} + 12q^{41} - 2q^{43} + 8q^{49} - 20q^{51} - 4q^{57} - 12q^{59} - 16q^{67} + 18q^{73} - 24q^{75} - 22q^{81} + 8q^{83} + 8q^{89} - 24q^{97} + 10q^{99} + O(q^{100}) \)

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
−3.31662
3.31662
0 −2.00000 0 −3.31662 0 3.31662 0 1.00000 0
1.2 0 −2.00000 0 3.31662 0 −3.31662 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.r 2
4.b odd 2 1 4864.2.a.y 2
8.b even 2 1 4864.2.a.y 2
8.d odd 2 1 inner 4864.2.a.r 2
16.e even 4 2 1216.2.c.f 4
16.f odd 4 2 1216.2.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.f 4 16.e even 4 2
1216.2.c.f 4 16.f odd 4 2
4864.2.a.r 2 1.a even 1 1 trivial
4864.2.a.r 2 8.d odd 2 1 inner
4864.2.a.y 2 4.b odd 2 1
4864.2.a.y 2 8.b even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 2 + T )^{2} \)
$5$ \( -11 + T^{2} \)
$7$ \( -11 + T^{2} \)
$11$ \( ( -5 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -5 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -44 + T^{2} \)
$29$ \( -44 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( -44 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( -99 + T^{2} \)
$53$ \( -176 + T^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( -99 + T^{2} \)
$67$ \( ( 8 + T )^{2} \)
$71$ \( -44 + T^{2} \)
$73$ \( ( -9 + T )^{2} \)
$79$ \( -176 + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( ( -4 + T )^{2} \)
$97$ \( ( 12 + T )^{2} \)
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