Properties

Label 8016.2.a.r
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.11256624.1
Defining polynomial: \(x^{5} - x^{4} - 16 x^{3} + 20 x^{2} + 31 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{2} q^{5} + ( -2 + \beta_{4} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{2} q^{5} + ( -2 + \beta_{4} ) q^{7} + q^{9} -\beta_{3} q^{11} + ( 1 + \beta_{1} ) q^{13} -\beta_{2} q^{15} + ( 1 - \beta_{1} ) q^{17} + ( -2 + \beta_{3} ) q^{19} + ( 2 - \beta_{4} ) q^{21} + \beta_{3} q^{23} + ( 3 - 2 \beta_{1} + \beta_{4} ) q^{25} - q^{27} + ( -2 + 2 \beta_{1} - \beta_{3} ) q^{29} + ( -2 - \beta_{3} - \beta_{4} ) q^{31} + \beta_{3} q^{33} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{35} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{37} + ( -1 - \beta_{1} ) q^{39} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{41} + ( -4 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{43} + \beta_{2} q^{45} + ( 2 - 2 \beta_{1} - \beta_{4} ) q^{47} + ( 3 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{49} + ( -1 + \beta_{1} ) q^{51} + ( -3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{53} + ( 2 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( 2 - \beta_{3} ) q^{57} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{59} + ( 4 - 2 \beta_{1} - 2 \beta_{4} ) q^{61} + ( -2 + \beta_{4} ) q^{63} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{65} + ( -2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{67} -\beta_{3} q^{69} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{71} + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{73} + ( -3 + 2 \beta_{1} - \beta_{4} ) q^{75} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{77} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{79} + q^{81} + ( -1 + \beta_{1} + \beta_{2} + 4 \beta_{4} ) q^{83} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{85} + ( 2 - 2 \beta_{1} + \beta_{3} ) q^{87} + ( -2 + 2 \beta_{1} + 3 \beta_{4} ) q^{89} + ( -2 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{91} + ( 2 + \beta_{3} + \beta_{4} ) q^{93} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{95} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{97} -\beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} - q^{5} - 9q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{3} - q^{5} - 9q^{7} + 5q^{9} + 2q^{11} + 6q^{13} + q^{15} + 4q^{17} - 12q^{19} + 9q^{21} - 2q^{23} + 14q^{25} - 5q^{27} - 6q^{29} - 9q^{31} - 2q^{33} - 3q^{35} + 5q^{37} - 6q^{39} + 4q^{41} - 18q^{43} - q^{45} + 7q^{47} + 16q^{49} - 4q^{51} + 3q^{53} + 4q^{55} + 12q^{57} - 13q^{59} + 16q^{61} - 9q^{63} - 10q^{65} - 9q^{67} + 2q^{69} + 10q^{71} + 8q^{73} - 14q^{75} + 6q^{77} - 6q^{79} + 5q^{81} - q^{83} + 8q^{85} + 6q^{87} - 5q^{89} - 12q^{91} + 9q^{93} - 2q^{95} - 7q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 16 x^{3} + 20 x^{2} + 31 x - 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 7 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 11 \nu - 3 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 14 \nu^{2} + 8 \nu + 17 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} - 2 \beta_{2} + 11 \beta_{1} - 4\)
\(\nu^{4}\)\(=\)\(4 \beta_{4} + 28 \beta_{2} - 8 \beta_{1} + 81\)

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
0.308735
−1.14410
2.50259
3.20970
−3.87693
0 −1.00000 0 −3.45234 0 2.53613 0 1.00000 0
1.2 0 −1.00000 0 −2.84552 0 −4.19124 0 1.00000 0
1.3 0 −1.00000 0 −0.368511 0 −4.85901 0 1.00000 0
1.4 0 −1.00000 0 1.65109 0 −0.854515 0 1.00000 0
1.5 0 −1.00000 0 4.01528 0 −1.63136 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.r 5
4.b odd 2 1 1002.2.a.j 5
12.b even 2 1 3006.2.a.t 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.j 5 4.b odd 2 1
3006.2.a.t 5 12.b even 2 1
8016.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{5} + T_{5}^{4} - 19 T_{5}^{3} - 21 T_{5}^{2} + 60 T_{5} + 24 \)
\( T_{7}^{5} + 9 T_{7}^{4} + 15 T_{7}^{3} - 49 T_{7}^{2} - 132 T_{7} - 72 \)
\( T_{11}^{5} - 2 T_{11}^{4} - 28 T_{11}^{3} + 60 T_{11}^{2} + 144 T_{11} - 288 \)
\( T_{13}^{5} - 6 T_{13}^{4} - 2 T_{13}^{3} + 52 T_{13}^{2} - 48 T_{13} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( ( 1 + T )^{5} \)
$5$ \( 24 + 60 T - 21 T^{2} - 19 T^{3} + T^{4} + T^{5} \)
$7$ \( -72 - 132 T - 49 T^{2} + 15 T^{3} + 9 T^{4} + T^{5} \)
$11$ \( -288 + 144 T + 60 T^{2} - 28 T^{3} - 2 T^{4} + T^{5} \)
$13$ \( -8 - 48 T + 52 T^{2} - 2 T^{3} - 6 T^{4} + T^{5} \)
$17$ \( -24 + 24 T + 24 T^{2} - 10 T^{3} - 4 T^{4} + T^{5} \)
$19$ \( 176 - 288 T - 100 T^{2} + 28 T^{3} + 12 T^{4} + T^{5} \)
$23$ \( 288 + 144 T - 60 T^{2} - 28 T^{3} + 2 T^{4} + T^{5} \)
$29$ \( -1728 + 1728 T - 300 T^{2} - 84 T^{3} + 6 T^{4} + T^{5} \)
$31$ \( -256 - 576 T - 229 T^{2} - 5 T^{3} + 9 T^{4} + T^{5} \)
$37$ \( -226 - 344 T + 415 T^{2} - 67 T^{3} - 5 T^{4} + T^{5} \)
$41$ \( 4176 + 1800 T - 132 T^{2} - 110 T^{3} - 4 T^{4} + T^{5} \)
$43$ \( -17424 - 11808 T - 2044 T^{2} - 24 T^{3} + 18 T^{4} + T^{5} \)
$47$ \( 156 + 1104 T + 243 T^{2} - 61 T^{3} - 7 T^{4} + T^{5} \)
$53$ \( -43392 + 10608 T + 717 T^{2} - 209 T^{3} - 3 T^{4} + T^{5} \)
$59$ \( 3306 - 7092 T - 2295 T^{2} - 127 T^{3} + 13 T^{4} + T^{5} \)
$61$ \( -14848 + 128 T + 1024 T^{2} - 28 T^{3} - 16 T^{4} + T^{5} \)
$67$ \( 36476 + 2316 T - 1585 T^{2} - 155 T^{3} + 9 T^{4} + T^{5} \)
$71$ \( -3456 + 4320 T + 1476 T^{2} - 184 T^{3} - 10 T^{4} + T^{5} \)
$73$ \( -2768 + 2128 T + 652 T^{2} - 104 T^{3} - 8 T^{4} + T^{5} \)
$79$ \( 6336 - 96 T - 808 T^{2} - 102 T^{3} + 6 T^{4} + T^{5} \)
$83$ \( -10266 + 16956 T - 165 T^{2} - 281 T^{3} + T^{4} + T^{5} \)
$89$ \( 50268 + 8952 T - 1137 T^{2} - 205 T^{3} + 5 T^{4} + T^{5} \)
$97$ \( 27848 + 5740 T - 1835 T^{2} - 253 T^{3} + 7 T^{4} + T^{5} \)
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