Properties

Label 8016.2.a.o
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.2777.1
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} + x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{7} + q^{9} + ( 2 + \beta_{3} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( 3 + \beta_{1} - \beta_{2} ) q^{17} + 2 \beta_{2} q^{19} + ( -\beta_{2} + \beta_{3} ) q^{21} -2 \beta_{2} q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + q^{27} + ( 4 + 2 \beta_{1} ) q^{29} + ( -\beta_{2} - \beta_{3} ) q^{31} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{35} + ( 2 - 3 \beta_{2} ) q^{37} + ( 2 + \beta_{3} ) q^{39} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{41} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{47} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{49} + ( 3 + \beta_{1} - \beta_{2} ) q^{51} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{53} + 2 \beta_{2} q^{57} + ( 2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{59} -2 \beta_{1} q^{61} + ( -\beta_{2} + \beta_{3} ) q^{63} + ( -4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 3 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{67} -2 \beta_{2} q^{69} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{73} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{75} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( -4 - 4 \beta_{1} - \beta_{2} ) q^{83} + ( -2 - 2 \beta_{1} - 2 \beta_{3} ) q^{85} + ( 4 + 2 \beta_{1} ) q^{87} + ( 4 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{89} + ( 8 - 2 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -\beta_{2} - \beta_{3} ) q^{93} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{95} + ( 2 + 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 5q^{5} - q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 5q^{5} - q^{7} + 4q^{9} + 8q^{13} + 5q^{15} + 10q^{17} + 2q^{19} - q^{21} - 2q^{23} + 5q^{25} + 4q^{27} + 14q^{29} - q^{31} - 5q^{35} + 5q^{37} + 8q^{39} + 14q^{41} - 2q^{43} + 5q^{45} - 7q^{47} + 13q^{49} + 10q^{51} + 3q^{53} + 2q^{57} + 5q^{59} + 2q^{61} - q^{63} + 6q^{65} + 7q^{67} - 2q^{69} - 2q^{71} + 2q^{73} + 5q^{75} + 10q^{79} + 4q^{81} - 13q^{83} - 6q^{85} + 14q^{87} + 13q^{89} + 30q^{91} - q^{93} + 2q^{95} + 5q^{97} + O(q^{100}) \)

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

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

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
2.36234
−0.679643
0.825785
−1.50848
0 1.00000 0 −1.87806 0 3.28324 0 1.00000 0
1.2 0 1.00000 0 0.416566 0 −4.65960 0 1.00000 0
1.3 0 1.00000 0 2.77037 0 −1.86579 0 1.00000 0
1.4 0 1.00000 0 3.69113 0 2.24216 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.o 4
4.b odd 2 1 1002.2.a.i 4
12.b even 2 1 3006.2.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.i 4 4.b odd 2 1
3006.2.a.s 4 12.b even 2 1
8016.2.a.o 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{4} - 5 T_{5}^{3} + 20 T_{5} - 8 \)
\( T_{7}^{4} + T_{7}^{3} - 20 T_{7}^{2} + 64 \)
\( T_{11} \)
\( T_{13}^{4} - 8 T_{13}^{3} + 4 T_{13}^{2} + 56 T_{13} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( -8 + 20 T - 5 T^{3} + T^{4} \)
$7$ \( 64 - 20 T^{2} + T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( -16 + 56 T + 4 T^{2} - 8 T^{3} + T^{4} \)
$17$ \( -16 + 16 T + 20 T^{2} - 10 T^{3} + T^{4} \)
$19$ \( 128 + 32 T - 32 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( 128 - 32 T - 32 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( -32 + 56 T + 36 T^{2} - 14 T^{3} + T^{4} \)
$31$ \( -64 - 96 T - 36 T^{2} + T^{3} + T^{4} \)
$37$ \( 568 + 184 T - 66 T^{2} - 5 T^{3} + T^{4} \)
$41$ \( -1072 + 480 T - 12 T^{2} - 14 T^{3} + T^{4} \)
$43$ \( 32 + 40 T - 32 T^{2} + 2 T^{3} + T^{4} \)
$47$ \( -64 - 208 T - 32 T^{2} + 7 T^{3} + T^{4} \)
$53$ \( 8 + 36 T - 40 T^{2} - 3 T^{3} + T^{4} \)
$59$ \( 808 - 36 T - 116 T^{2} - 5 T^{3} + T^{4} \)
$61$ \( 128 + 72 T - 36 T^{2} - 2 T^{3} + T^{4} \)
$67$ \( -3208 + 1848 T - 186 T^{2} - 7 T^{3} + T^{4} \)
$71$ \( 6784 - 128 T - 168 T^{2} + 2 T^{3} + T^{4} \)
$73$ \( 512 + 72 T - 132 T^{2} - 2 T^{3} + T^{4} \)
$79$ \( 1472 + 376 T - 56 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( 4072 - 772 T - 96 T^{2} + 13 T^{3} + T^{4} \)
$89$ \( -1448 + 516 T - 2 T^{2} - 13 T^{3} + T^{4} \)
$97$ \( 3256 + 628 T - 146 T^{2} - 5 T^{3} + T^{4} \)
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