Properties

Label 8016.2.a.m
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

Related objects

Downloads

Learn more about

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: 3.3.1300.1
Defining polynomial: \(x^{3} - 10 x - 10\)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -\beta_{1} + \beta_{2} ) q^{7} + q^{9} + ( 3 + \beta_{1} - \beta_{2} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( 1 + \beta_{1} + \beta_{2} ) q^{17} + ( 3 - \beta_{1} + \beta_{2} ) q^{19} + ( -\beta_{1} + \beta_{2} ) q^{21} + ( 5 - \beta_{1} + \beta_{2} ) q^{23} + ( 3 - \beta_{1} + \beta_{2} ) q^{25} + q^{27} + ( -3 - \beta_{1} + \beta_{2} ) q^{29} + q^{31} + ( 3 + \beta_{1} - \beta_{2} ) q^{33} + ( 4 - 2 \beta_{1} + 3 \beta_{2} ) q^{35} + ( 5 - \beta_{1} ) q^{37} + ( -1 + \beta_{1} + \beta_{2} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 1 + 3 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -\beta_{1} + \beta_{2} ) q^{47} + ( 2 - 2 \beta_{1} ) q^{49} + ( 1 + \beta_{1} + \beta_{2} ) q^{51} + ( -6 - \beta_{2} ) q^{53} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{55} + ( 3 - \beta_{1} + \beta_{2} ) q^{57} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{59} + ( -4 - 4 \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{2} ) q^{63} + ( -11 - \beta_{1} + \beta_{2} ) q^{65} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{67} + ( 5 - \beta_{1} + \beta_{2} ) q^{69} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{71} + ( -7 + 3 \beta_{1} - \beta_{2} ) q^{73} + ( 3 - \beta_{1} + \beta_{2} ) q^{75} + ( -9 - \beta_{1} + 3 \beta_{2} ) q^{77} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} + ( 12 + 2 \beta_{1} - \beta_{2} ) q^{83} + ( -9 - 3 \beta_{1} + \beta_{2} ) q^{85} + ( -3 - \beta_{1} + \beta_{2} ) q^{87} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{89} + ( 1 + \beta_{1} - 5 \beta_{2} ) q^{91} + q^{93} + ( 7 - 5 \beta_{1} + 3 \beta_{2} ) q^{95} + ( 3 \beta_{1} - 5 \beta_{2} ) q^{97} + ( 3 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 3q^{5} - q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 3q^{5} - q^{7} + 3q^{9} + 10q^{11} - 4q^{13} + 3q^{15} + 2q^{17} + 8q^{19} - q^{21} + 14q^{23} + 8q^{25} + 3q^{27} - 10q^{29} + 3q^{31} + 10q^{33} + 9q^{35} + 15q^{37} - 4q^{39} + 2q^{41} + 6q^{43} + 3q^{45} - q^{47} + 6q^{49} + 2q^{51} - 17q^{53} + 8q^{57} + 5q^{59} - 8q^{61} - q^{63} - 34q^{65} + 7q^{67} + 14q^{69} + 6q^{71} - 20q^{73} + 8q^{75} - 30q^{77} - 4q^{79} + 3q^{81} + 37q^{83} - 28q^{85} - 10q^{87} + 7q^{89} + 8q^{91} + 3q^{93} + 18q^{95} + 5q^{97} + 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 10 x - 10\):

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

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
3.57709
−1.15347
−2.42362
0 1.00000 0 −2.57709 0 −1.35861 0 1.00000 0
1.2 0 1.00000 0 2.15347 0 −3.36258 0 1.00000 0
1.3 0 1.00000 0 3.42362 0 3.72119 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.m 3
4.b odd 2 1 1002.2.a.g 3
12.b even 2 1 3006.2.a.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.g 3 4.b odd 2 1
3006.2.a.q 3 12.b even 2 1
8016.2.a.m 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{3} - 3 T_{5}^{2} - 7 T_{5} + 19 \)
\( T_{7}^{3} + T_{7}^{2} - 13 T_{7} - 17 \)
\( T_{11}^{3} - 10 T_{11}^{2} + 20 T_{11} + 20 \)
\( T_{13}^{3} + 4 T_{13}^{2} - 28 T_{13} - 68 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{3} \)
$5$ \( 1 - 3 T + 8 T^{2} - 11 T^{3} + 40 T^{4} - 75 T^{5} + 125 T^{6} \)
$7$ \( 1 + T + 8 T^{2} - 3 T^{3} + 56 T^{4} + 49 T^{5} + 343 T^{6} \)
$11$ \( 1 - 10 T + 53 T^{2} - 200 T^{3} + 583 T^{4} - 1210 T^{5} + 1331 T^{6} \)
$13$ \( 1 + 4 T + 11 T^{2} + 36 T^{3} + 143 T^{4} + 676 T^{5} + 2197 T^{6} \)
$17$ \( 1 - 2 T + 19 T^{2} - 72 T^{3} + 323 T^{4} - 578 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 8 T + 65 T^{2} - 300 T^{3} + 1235 T^{4} - 2888 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 14 T + 121 T^{2} - 696 T^{3} + 2783 T^{4} - 7406 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 10 T + 107 T^{2} + 560 T^{3} + 3103 T^{4} + 8410 T^{5} + 24389 T^{6} \)
$31$ \( ( 1 - T + 31 T^{2} )^{3} \)
$37$ \( 1 - 15 T + 176 T^{2} - 1175 T^{3} + 6512 T^{4} - 20535 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 2 T + 71 T^{2} - 28 T^{3} + 2911 T^{4} - 3362 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 6 T + 21 T^{2} + 56 T^{3} + 903 T^{4} - 11094 T^{5} + 79507 T^{6} \)
$47$ \( 1 + T + 128 T^{2} + 77 T^{3} + 6016 T^{4} + 2209 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 17 T + 242 T^{2} + 1891 T^{3} + 12826 T^{4} + 47753 T^{5} + 148877 T^{6} \)
$59$ \( 1 - 5 T + 102 T^{2} - 525 T^{3} + 6018 T^{4} - 17405 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 8 T - 9 T^{2} - 688 T^{3} - 549 T^{4} + 29768 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 7 T + 134 T^{2} - 687 T^{3} + 8978 T^{4} - 31423 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 6 T + 125 T^{2} - 1000 T^{3} + 8875 T^{4} - 30246 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 20 T + 279 T^{2} + 2780 T^{3} + 20367 T^{4} + 106580 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 4 T + 189 T^{2} + 664 T^{3} + 14931 T^{4} + 24964 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 37 T + 672 T^{2} - 7551 T^{3} + 55776 T^{4} - 254893 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 7 T + 150 T^{2} - 375 T^{3} + 13350 T^{4} - 55447 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 5 T + 26 T^{2} - 1065 T^{3} + 2522 T^{4} - 47045 T^{5} + 912673 T^{6} \)
show more
show less