Properties

Label 8496.2.a.bl
Level $8496$
Weight $2$
Character orbit 8496.a
Self dual yes
Analytic conductor $67.841$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8496 = 2^{4} \cdot 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8496.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.8409015573\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( -3 - \beta_{1} ) q^{7} +O(q^{10})\) \( q + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + ( -3 - \beta_{1} ) q^{7} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} ) q^{13} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{17} + ( -2 + \beta_{2} ) q^{19} + ( -2 \beta_{1} - \beta_{2} ) q^{23} + ( -3 \beta_{1} + \beta_{2} ) q^{25} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{29} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{31} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{35} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -1 - 5 \beta_{1} + 3 \beta_{2} ) q^{43} + ( 5 - \beta_{1} + 4 \beta_{2} ) q^{47} + ( 5 + 6 \beta_{1} + \beta_{2} ) q^{49} + ( -1 + 5 \beta_{1} - \beta_{2} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} ) q^{55} + q^{59} + ( 1 + \beta_{1} + 4 \beta_{2} ) q^{61} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{65} + ( -5 - \beta_{1} - 5 \beta_{2} ) q^{67} + ( 9 - 3 \beta_{1} + \beta_{2} ) q^{71} + ( 1 - 3 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 7 + 5 \beta_{1} + 4 \beta_{2} ) q^{77} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{79} + ( 1 - 5 \beta_{1} + 6 \beta_{2} ) q^{83} + ( -12 + 6 \beta_{1} - \beta_{2} ) q^{85} + ( 9 - 3 \beta_{1} + 4 \beta_{2} ) q^{89} + ( 1 + 3 \beta_{1} + 4 \beta_{2} ) q^{91} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{95} + ( 5 - 5 \beta_{1} + \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2q^{5} - 9q^{7} + O(q^{10}) \) \( 3q + 2q^{5} - 9q^{7} - 2q^{11} + 4q^{13} - 3q^{17} - 7q^{19} + q^{23} - q^{25} + 11q^{29} - 13q^{31} - q^{35} - 5q^{37} + q^{41} - 6q^{43} + 11q^{47} + 14q^{49} - 2q^{53} - 4q^{55} + 3q^{59} - q^{61} - 10q^{67} + 26q^{71} + 7q^{73} + 17q^{77} - 2q^{79} - 3q^{83} - 35q^{85} + 23q^{89} - q^{91} + 3q^{95} + 14q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 4 x - 1\):

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

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
−0.254102
2.11491
−1.86081
0 0 0 −1.68133 0 −2.74590 0 0 0
1.2 0 0 0 0.357926 0 −5.11491 0 0 0
1.3 0 0 0 3.32340 0 −1.13919 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(59\) \(-1\)

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 8496.2.a.bl 3
3.b odd 2 1 2832.2.a.t 3
4.b odd 2 1 531.2.a.d 3
12.b even 2 1 177.2.a.d 3
60.h even 2 1 4425.2.a.w 3
84.h odd 2 1 8673.2.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.d 3 12.b even 2 1
531.2.a.d 3 4.b odd 2 1
2832.2.a.t 3 3.b odd 2 1
4425.2.a.w 3 60.h even 2 1
8496.2.a.bl 3 1.a even 1 1 trivial
8673.2.a.s 3 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8496))\):

\( T_{5}^{3} - 2 T_{5}^{2} - 5 T_{5} + 2 \)
\( T_{7}^{3} + 9 T_{7}^{2} + 23 T_{7} + 16 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 11 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( 2 - 5 T - 2 T^{2} + T^{3} \)
$7$ \( 16 + 23 T + 9 T^{2} + T^{3} \)
$11$ \( 4 - 11 T + 2 T^{2} + T^{3} \)
$13$ \( 26 - 7 T - 4 T^{2} + T^{3} \)
$17$ \( -98 - 43 T + 3 T^{2} + T^{3} \)
$19$ \( 4 + 11 T + 7 T^{2} + T^{3} \)
$23$ \( 64 - 27 T - T^{2} + T^{3} \)
$29$ \( 74 + 9 T - 11 T^{2} + T^{3} \)
$31$ \( -28 + 37 T + 13 T^{2} + T^{3} \)
$37$ \( 14 - 19 T + 5 T^{2} + T^{3} \)
$41$ \( -74 - 39 T - T^{2} + T^{3} \)
$43$ \( -592 - 91 T + 6 T^{2} + T^{3} \)
$47$ \( 496 - 37 T - 11 T^{2} + T^{3} \)
$53$ \( 58 - 89 T + 2 T^{2} + T^{3} \)
$59$ \( ( -1 + T )^{3} \)
$61$ \( 98 - 101 T + T^{2} + T^{3} \)
$67$ \( -784 - 119 T + 10 T^{2} + T^{3} \)
$71$ \( -424 + 193 T - 26 T^{2} + T^{3} \)
$73$ \( 718 - 141 T - 7 T^{2} + T^{3} \)
$79$ \( 32 - 31 T + 2 T^{2} + T^{3} \)
$83$ \( -148 - 199 T + 3 T^{2} + T^{3} \)
$89$ \( 278 + 91 T - 23 T^{2} + T^{3} \)
$97$ \( 202 - 25 T - 14 T^{2} + T^{3} \)
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