Properties

Label 8016.2.a.n
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 0
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: \(0\)
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 4008)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( 2 - \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 2 - \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{7} + q^{9} + ( 1 + \beta_{1} - \beta_{2} ) q^{11} + 2 q^{13} + ( 2 - \beta_{2} ) q^{15} + 3 \beta_{2} q^{17} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{19} + ( 2 - 2 \beta_{1} ) q^{21} + 8 q^{23} + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{25} + q^{27} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{31} + ( 1 + \beta_{1} - \beta_{2} ) q^{33} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{35} + ( -4 + 2 \beta_{1} ) q^{37} + 2 q^{39} + ( 6 - 6 \beta_{1} - 3 \beta_{2} ) q^{41} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{43} + ( 2 - \beta_{2} ) q^{45} + ( -1 + 3 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} + 4 \beta_{2} ) q^{49} + 3 \beta_{2} q^{51} + ( -4 + 4 \beta_{1} + 3 \beta_{2} ) q^{53} + ( 6 - 4 \beta_{2} ) q^{55} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{57} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -1 + 5 \beta_{1} + 5 \beta_{2} ) q^{61} + ( 2 - 2 \beta_{1} ) q^{63} + ( 4 - 2 \beta_{2} ) q^{65} + ( 8 + 2 \beta_{1} + \beta_{2} ) q^{67} + 8 q^{69} + ( 4 + 4 \beta_{1} + 6 \beta_{2} ) q^{71} + ( 8 - 2 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{75} + ( -4 - 4 \beta_{2} ) q^{77} + ( -4 + 2 \beta_{1} - 5 \beta_{2} ) q^{79} + q^{81} + ( 4 - 8 \beta_{1} - 6 \beta_{2} ) q^{83} + ( -9 + 3 \beta_{1} + 9 \beta_{2} ) q^{85} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{87} + ( -6 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 4 - 4 \beta_{1} ) q^{91} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{93} + ( -8 + 4 \beta_{1} + 8 \beta_{2} ) q^{95} + ( 2 - 6 \beta_{2} ) q^{97} + ( 1 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 6q^{5} + 4q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 6q^{5} + 4q^{7} + 3q^{9} + 4q^{11} + 6q^{13} + 6q^{15} - 4q^{19} + 4q^{21} + 24q^{23} + 5q^{25} + 3q^{27} - 14q^{29} + 4q^{31} + 4q^{33} + 4q^{35} - 10q^{37} + 6q^{39} + 12q^{41} + 8q^{43} + 6q^{45} + 11q^{49} - 8q^{53} + 18q^{55} - 4q^{57} - 4q^{59} + 2q^{61} + 4q^{63} + 12q^{65} + 26q^{67} + 24q^{69} + 16q^{71} + 22q^{73} + 5q^{75} - 12q^{77} - 10q^{79} + 3q^{81} + 4q^{83} - 24q^{85} - 14q^{87} - 6q^{89} + 8q^{91} + 4q^{93} - 20q^{95} + 6q^{97} + 4q^{99} + O(q^{100}) \)

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
−1.48119
2.17009
0.311108
0 1.00000 0 0.324869 0 4.96239 0 1.00000 0
1.2 0 1.00000 0 1.46081 0 −2.34017 0 1.00000 0
1.3 0 1.00000 0 4.21432 0 1.37778 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.n 3
4.b odd 2 1 4008.2.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.e 3 4.b odd 2 1
8016.2.a.n 3 1.a even 1 1 trivial

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{5}^{3} - 6 T_{5}^{2} + 8 T_{5} - 2 \)
\( T_{7}^{3} - 4 T_{7}^{2} - 8 T_{7} + 16 \)
\( T_{11}^{3} - 4 T_{11}^{2} - 4 T_{11} + 20 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{3} \)
$5$ \( 1 - 6 T + 23 T^{2} - 62 T^{3} + 115 T^{4} - 150 T^{5} + 125 T^{6} \)
$7$ \( 1 - 4 T + 13 T^{2} - 40 T^{3} + 91 T^{4} - 196 T^{5} + 343 T^{6} \)
$11$ \( 1 - 4 T + 29 T^{2} - 68 T^{3} + 319 T^{4} - 484 T^{5} + 1331 T^{6} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{3} \)
$17$ \( 1 + 15 T^{2} + 54 T^{3} + 255 T^{4} + 4913 T^{6} \)
$19$ \( 1 + 4 T + 41 T^{2} + 120 T^{3} + 779 T^{4} + 1444 T^{5} + 6859 T^{6} \)
$23$ \( ( 1 - 8 T + 23 T^{2} )^{3} \)
$29$ \( 1 + 14 T + 115 T^{2} + 660 T^{3} + 3335 T^{4} + 11774 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 4 T + 77 T^{2} - 216 T^{3} + 2387 T^{4} - 3844 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 10 T + 131 T^{2} + 732 T^{3} + 4847 T^{4} + 13690 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 12 T + 51 T^{2} - 66 T^{3} + 2091 T^{4} - 20172 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 8 T + 113 T^{2} - 558 T^{3} + 4859 T^{4} - 14792 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 57 T^{2} + 268 T^{3} + 2679 T^{4} + 103823 T^{6} \)
$53$ \( 1 + 8 T + 115 T^{2} + 558 T^{3} + 6095 T^{4} + 22472 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 4 T + 145 T^{2} + 504 T^{3} + 8555 T^{4} + 13924 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 2 T + 51 T^{2} - 248 T^{3} + 3111 T^{4} - 7442 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 26 T + 413 T^{2} - 4038 T^{3} + 27671 T^{4} - 116714 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 16 T + 149 T^{2} - 1232 T^{3} + 10579 T^{4} - 80656 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 22 T + 319 T^{2} - 3316 T^{3} + 23287 T^{4} - 117238 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 10 T + 137 T^{2} + 1302 T^{3} + 10823 T^{4} + 62410 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 4 T - 7 T^{2} + 760 T^{3} - 581 T^{4} - 27556 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 6 T + 119 T^{2} + 148 T^{3} + 10591 T^{4} + 47526 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 6 T + 159 T^{2} - 1316 T^{3} + 15423 T^{4} - 56454 T^{5} + 912673 T^{6} \)
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