Properties

Label 8016.2.a.u
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.38569.1
Defining polynomial: \(x^{5} - 5 x^{3} + 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( \beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( \beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + q^{9} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{11} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{13} + ( \beta_{1} + \beta_{4} ) q^{15} + ( 1 + 2 \beta_{2} - 2 \beta_{4} ) q^{17} + ( 4 - \beta_{2} - \beta_{3} ) q^{19} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{21} + ( -3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{23} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{25} + q^{27} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{29} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{33} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{35} + ( -4 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{37} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{39} + ( 1 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{41} + ( 3 - \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{4} ) q^{45} + ( 1 + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{47} + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{49} + ( 1 + 2 \beta_{2} - 2 \beta_{4} ) q^{51} + ( 2 + \beta_{1} - \beta_{2} + 5 \beta_{3} + \beta_{4} ) q^{53} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{55} + ( 4 - \beta_{2} - \beta_{3} ) q^{57} + ( 2 + 5 \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{59} + ( -3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{61} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{63} + ( 5 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{65} + ( 3 + 6 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{67} + ( -3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{69} + ( -1 + 5 \beta_{1} + \beta_{2} + 8 \beta_{3} + 3 \beta_{4} ) q^{71} + ( 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{73} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{75} + ( 5 + \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{77} + ( 1 + 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{79} + q^{81} + ( 1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{83} + ( -2 + 5 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{85} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{87} + ( 4 \beta_{1} + \beta_{2} + 5 \beta_{3} + 4 \beta_{4} ) q^{89} + ( 2 + 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{91} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} ) q^{95} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{97} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} - q^{5} + 4q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} - q^{5} + 4q^{7} + 5q^{9} + 7q^{11} - 8q^{13} - q^{15} + 5q^{17} + 20q^{19} + 4q^{21} - q^{23} - 6q^{25} + 5q^{27} - 5q^{29} + 18q^{31} + 7q^{33} + 6q^{35} - 17q^{37} - 8q^{39} + 6q^{41} + 10q^{43} - q^{45} + 3q^{47} - 13q^{49} + 5q^{51} + 15q^{53} + 9q^{55} + 20q^{57} + 17q^{59} - 2q^{61} + 4q^{63} + 26q^{65} + 24q^{67} - q^{69} - q^{71} + 2q^{73} - 6q^{75} + 26q^{77} + 6q^{79} + 5q^{81} + 7q^{83} - 11q^{85} - 5q^{87} + 15q^{91} + 18q^{93} - q^{95} - 13q^{97} + 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 5 x^{3} + 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 4 \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + 5 \nu^{2} - 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} + \nu^{3} - 5 \nu^{2} - 4 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{3} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(4 \beta_{3} + 5 \beta_{2} + 7\)

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
−1.95408
0.790734
−1.15098
0.275834
2.03850
0 1.00000 0 −3.11102 0 1.66149 0 1.00000 0
1.2 0 1.00000 0 −1.61314 0 −2.77861 0 1.00000 0
1.3 0 1.00000 0 0.0593478 0 1.53510 0 1.00000 0
1.4 0 1.00000 0 1.81885 0 0.619101 0 1.00000 0
1.5 0 1.00000 0 1.84596 0 2.96293 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.u 5
4.b odd 2 1 501.2.a.c 5
12.b even 2 1 1503.2.a.c 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.c 5 4.b odd 2 1
1503.2.a.c 5 12.b even 2 1
8016.2.a.u 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{5} + T_{5}^{4} - 9 T_{5}^{3} - 2 T_{5}^{2} + 17 T_{5} - 1 \)
\( T_{7}^{5} - 4 T_{7}^{4} - 3 T_{7}^{3} + 29 T_{7}^{2} - 37 T_{7} + 13 \)
\( T_{11}^{5} - 7 T_{11}^{4} - 3 T_{11}^{3} + 92 T_{11}^{2} - 101 T_{11} - 121 \)
\( T_{13}^{5} + 8 T_{13}^{4} - T_{13}^{3} - 66 T_{13}^{2} - 48 T_{13} + 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( ( -1 + T )^{5} \)
$5$ \( -1 + 17 T - 2 T^{2} - 9 T^{3} + T^{4} + T^{5} \)
$7$ \( 13 - 37 T + 29 T^{2} - 3 T^{3} - 4 T^{4} + T^{5} \)
$11$ \( -121 - 101 T + 92 T^{2} - 3 T^{3} - 7 T^{4} + T^{5} \)
$13$ \( 17 - 48 T - 66 T^{2} - T^{3} + 8 T^{4} + T^{5} \)
$17$ \( -293 - 255 T + 234 T^{2} - 34 T^{3} - 5 T^{4} + T^{5} \)
$19$ \( -599 + 940 T - 554 T^{2} + 153 T^{3} - 20 T^{4} + T^{5} \)
$23$ \( -1793 + 1301 T - 26 T^{2} - 77 T^{3} + T^{4} + T^{5} \)
$29$ \( 4001 + 1033 T - 330 T^{2} - 71 T^{3} + 5 T^{4} + T^{5} \)
$31$ \( -181 + 444 T - 347 T^{2} + 118 T^{3} - 18 T^{4} + T^{5} \)
$37$ \( -13913 - 7275 T - 1084 T^{2} + 23 T^{3} + 17 T^{4} + T^{5} \)
$41$ \( 1 - 14 T - 55 T^{2} - 40 T^{3} - 6 T^{4} + T^{5} \)
$43$ \( -4159 - 650 T + 689 T^{2} - 52 T^{3} - 10 T^{4} + T^{5} \)
$47$ \( -547 + 208 T + 223 T^{2} - 63 T^{3} - 3 T^{4} + T^{5} \)
$53$ \( 707 - 4314 T + 1417 T^{2} - 65 T^{3} - 15 T^{4} + T^{5} \)
$59$ \( 5737 - 10470 T + 2381 T^{2} - 73 T^{3} - 17 T^{4} + T^{5} \)
$61$ \( -8609 + 2152 T + 474 T^{2} - 147 T^{3} + 2 T^{4} + T^{5} \)
$67$ \( -23449 - 14296 T + 3132 T^{2} - 3 T^{3} - 24 T^{4} + T^{5} \)
$71$ \( 28039 + 15837 T - 207 T^{2} - 292 T^{3} + T^{4} + T^{5} \)
$73$ \( 8743 + 3298 T - 62 T^{2} - 109 T^{3} - 2 T^{4} + T^{5} \)
$79$ \( -31489 + 3112 T + 1471 T^{2} - 186 T^{3} - 6 T^{4} + T^{5} \)
$83$ \( 7703 + 5178 T + 353 T^{2} - 129 T^{3} - 7 T^{4} + T^{5} \)
$89$ \( 17071 + 5064 T - 322 T^{2} - 155 T^{3} + T^{5} \)
$97$ \( -869 - 1989 T - 1201 T^{2} - 112 T^{3} + 13 T^{4} + T^{5} \)
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