Properties

Label 4864.2.a.bl
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
Defining polynomial: \(x^{4} - 7 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{3} -\beta_{1} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + ( 6 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{3} -\beta_{1} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + ( 6 + \beta_{3} ) q^{9} + \beta_{3} q^{11} + ( 3 \beta_{1} - \beta_{2} ) q^{13} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{15} + 5 q^{17} - q^{19} + ( \beta_{1} + 5 \beta_{2} ) q^{21} + ( \beta_{1} - \beta_{2} ) q^{23} + ( -1 + \beta_{3} ) q^{25} + ( 11 + 3 \beta_{3} ) q^{27} + ( \beta_{1} - 3 \beta_{2} ) q^{29} -2 \beta_{2} q^{31} + 8 q^{33} + ( -6 - \beta_{3} ) q^{35} -4 \beta_{1} q^{37} + ( 3 \beta_{1} + 7 \beta_{2} ) q^{39} + ( -6 + \beta_{3} ) q^{43} + ( -7 \beta_{1} - 2 \beta_{2} ) q^{45} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{47} + 4 q^{49} + ( 5 + 5 \beta_{3} ) q^{51} + ( \beta_{1} - 5 \beta_{2} ) q^{53} + ( -\beta_{1} - 2 \beta_{2} ) q^{55} + ( -1 - \beta_{3} ) q^{57} + ( -1 + \beta_{3} ) q^{59} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{61} + 11 \beta_{1} q^{63} + ( -10 - 2 \beta_{3} ) q^{65} + ( -5 + \beta_{3} ) q^{67} + ( -\beta_{1} + 3 \beta_{2} ) q^{69} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 1 - 4 \beta_{3} ) q^{73} + ( 7 - \beta_{3} ) q^{75} + ( -\beta_{1} + 6 \beta_{2} ) q^{77} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{79} + ( 17 + 8 \beta_{3} ) q^{81} + 8 q^{83} -5 \beta_{1} q^{85} + ( -7 \beta_{1} + 5 \beta_{2} ) q^{87} + ( 6 + 4 \beta_{3} ) q^{89} + ( 17 + \beta_{3} ) q^{91} + ( -6 \beta_{1} + 2 \beta_{2} ) q^{93} + \beta_{1} q^{95} + ( 2 - 2 \beta_{3} ) q^{97} + ( 8 + 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 22q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 22q^{9} - 2q^{11} + 20q^{17} - 4q^{19} - 6q^{25} + 38q^{27} + 32q^{33} - 22q^{35} - 26q^{43} + 16q^{49} + 10q^{51} - 2q^{57} - 6q^{59} - 36q^{65} - 22q^{67} + 12q^{73} + 30q^{75} + 52q^{81} + 32q^{83} + 16q^{89} + 66q^{91} + 12q^{97} + 22q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 7 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 5 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 5 \beta_{1}\)

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
0.792287
−0.792287
2.52434
−2.52434
0 −2.37228 0 −0.792287 0 3.31662 0 2.62772 0
1.2 0 −2.37228 0 0.792287 0 −3.31662 0 2.62772 0
1.3 0 3.37228 0 −2.52434 0 3.31662 0 8.37228 0
1.4 0 3.37228 0 2.52434 0 −3.31662 0 8.37228 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.bl 4
4.b odd 2 1 4864.2.a.bi 4
8.b even 2 1 4864.2.a.bi 4
8.d odd 2 1 inner 4864.2.a.bl 4
16.e even 4 2 1216.2.c.h 8
16.f odd 4 2 1216.2.c.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.h 8 16.e even 4 2
1216.2.c.h 8 16.f odd 4 2
4864.2.a.bi 4 4.b odd 2 1
4864.2.a.bi 4 8.b even 2 1
4864.2.a.bl 4 1.a even 1 1 trivial
4864.2.a.bl 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4864))\):

\( T_{3}^{2} - T_{3} - 8 \)
\( T_{5}^{4} - 7 T_{5}^{2} + 4 \)
\( T_{7}^{2} - 11 \)
\( T_{11}^{2} + T_{11} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -8 - T + T^{2} )^{2} \)
$5$ \( 4 - 7 T^{2} + T^{4} \)
$7$ \( ( -11 + T^{2} )^{2} \)
$11$ \( ( -8 + T + T^{2} )^{2} \)
$13$ \( 576 - 51 T^{2} + T^{4} \)
$17$ \( ( -5 + T )^{4} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( 4 - 7 T^{2} + T^{4} \)
$29$ \( 256 - 43 T^{2} + T^{4} \)
$31$ \( ( -12 + T^{2} )^{2} \)
$37$ \( 1024 - 112 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 34 + 13 T + T^{2} )^{2} \)
$47$ \( 36 - 87 T^{2} + T^{4} \)
$53$ \( 3364 - 127 T^{2} + T^{4} \)
$59$ \( ( -6 + 3 T + T^{2} )^{2} \)
$61$ \( 4356 - 231 T^{2} + T^{4} \)
$67$ \( ( 22 + 11 T + T^{2} )^{2} \)
$71$ \( ( -44 + T^{2} )^{2} \)
$73$ \( ( -123 - 6 T + T^{2} )^{2} \)
$79$ \( 16 - 184 T^{2} + T^{4} \)
$83$ \( ( -8 + T )^{4} \)
$89$ \( ( -116 - 8 T + T^{2} )^{2} \)
$97$ \( ( -24 - 6 T + T^{2} )^{2} \)
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