Properties

Label 8016.2.a.l
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1002)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 + \beta_{2} ) q^{5} + ( 2 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 + \beta_{2} ) q^{5} + ( 2 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{11} + ( -3 + \beta_{1} - \beta_{2} ) q^{13} + ( -1 + \beta_{2} ) q^{15} + ( -5 - \beta_{1} - \beta_{2} ) q^{17} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{19} + ( 2 - \beta_{1} + \beta_{2} ) q^{21} + ( 1 + \beta_{1} - \beta_{2} ) q^{23} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{25} + q^{27} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{29} + ( 5 - 6 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{33} + ( 2 - \beta_{1} ) q^{35} + ( -1 - 2 \beta_{1} - 3 \beta_{2} ) q^{37} + ( -3 + \beta_{1} - \beta_{2} ) q^{39} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{43} + ( -1 + \beta_{2} ) q^{45} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 4 - 6 \beta_{1} + 4 \beta_{2} ) q^{49} + ( -5 - \beta_{1} - \beta_{2} ) q^{51} + ( -6 + 3 \beta_{1} + 4 \beta_{2} ) q^{53} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{55} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{57} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{59} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} ) q^{63} + ( -1 + \beta_{1} - \beta_{2} ) q^{65} + ( -3 + 6 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 1 + \beta_{1} - \beta_{2} ) q^{69} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{71} + ( 1 + 3 \beta_{1} - 5 \beta_{2} ) q^{73} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{75} + ( -7 + 5 \beta_{1} - 3 \beta_{2} ) q^{77} + ( -10 - 2 \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} + ( -2 + 3 \beta_{1} ) q^{83} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{85} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{87} + ( -8 - 3 \beta_{1} - 5 \beta_{2} ) q^{89} + ( -13 + 7 \beta_{1} - 5 \beta_{2} ) q^{91} + ( 5 - 6 \beta_{1} - 2 \beta_{2} ) q^{93} + ( -9 + 3 \beta_{1} + 7 \beta_{2} ) q^{95} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{97} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - 3q^{5} + 5q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - 3q^{5} + 5q^{7} + 3q^{9} - 8q^{13} - 3q^{15} - 16q^{17} + 2q^{19} + 5q^{21} + 4q^{23} - 4q^{25} + 3q^{27} + 9q^{31} + 5q^{35} - 5q^{37} - 8q^{39} - 14q^{41} + 2q^{43} - 3q^{45} + 3q^{47} + 6q^{49} - 16q^{51} - 15q^{53} + 2q^{55} + 2q^{57} + 5q^{59} - 4q^{61} + 5q^{63} - 2q^{65} - 3q^{67} + 4q^{69} + 4q^{71} + 6q^{73} - 4q^{75} - 16q^{77} - 32q^{79} + 3q^{81} - 3q^{83} + 10q^{85} - 27q^{89} - 32q^{91} + 9q^{93} - 24q^{95} + 5q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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
0.311108
2.17009
−1.48119
0 1.00000 0 −3.21432 0 −0.525428 0 1.00000 0
1.2 0 1.00000 0 −0.460811 0 0.369102 0 1.00000 0
1.3 0 1.00000 0 0.675131 0 5.15633 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(1\)

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 8016.2.a.l 3
4.b odd 2 1 1002.2.a.h 3
12.b even 2 1 3006.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.h 3 4.b odd 2 1
3006.2.a.o 3 12.b even 2 1
8016.2.a.l 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{3} + 3 T_{5}^{2} - T_{5} - 1 \)
\( T_{7}^{3} - 5 T_{7}^{2} - T_{7} + 1 \)
\( T_{11}^{3} - 28 T_{11} - 52 \)
\( T_{13}^{3} + 8 T_{13}^{2} + 12 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -1 - T + 3 T^{2} + T^{3} \)
$7$ \( 1 - T - 5 T^{2} + T^{3} \)
$11$ \( -52 - 28 T + T^{3} \)
$13$ \( 4 + 12 T + 8 T^{2} + T^{3} \)
$17$ \( 124 + 80 T + 16 T^{2} + T^{3} \)
$19$ \( -52 - 32 T - 2 T^{2} + T^{3} \)
$23$ \( 20 - 4 T - 4 T^{2} + T^{3} \)
$29$ \( -52 - 28 T + T^{3} \)
$31$ \( 725 - 85 T - 9 T^{2} + T^{3} \)
$37$ \( -107 - 29 T + 5 T^{2} + T^{3} \)
$41$ \( -152 + 28 T + 14 T^{2} + T^{3} \)
$43$ \( 20 - 44 T - 2 T^{2} + T^{3} \)
$47$ \( 351 - 81 T - 3 T^{2} + T^{3} \)
$53$ \( -139 + 5 T + 15 T^{2} + T^{3} \)
$59$ \( -25 - 57 T - 5 T^{2} + T^{3} \)
$61$ \( -64 - 80 T + 4 T^{2} + T^{3} \)
$67$ \( -621 - 117 T + 3 T^{2} + T^{3} \)
$71$ \( 68 - 40 T - 4 T^{2} + T^{3} \)
$73$ \( 740 - 148 T - 6 T^{2} + T^{3} \)
$79$ \( 992 + 320 T + 32 T^{2} + T^{3} \)
$83$ \( -31 - 27 T + 3 T^{2} + T^{3} \)
$89$ \( -439 + 143 T + 27 T^{2} + T^{3} \)
$97$ \( 61 - 37 T - 5 T^{2} + T^{3} \)
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