Properties

Label 8016.2.a.k
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
0 1.00000 0 −3.41421 0 −2.82843 0 1.00000 0
1.2 0 1.00000 0 −0.585786 0 2.82843 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.k 2
4.b odd 2 1 1002.2.a.f 2
12.b even 2 1 3006.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.f 2 4.b odd 2 1
3006.2.a.n 2 12.b even 2 1
8016.2.a.k 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4} \)
$7$ \( 1 + 6 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{2} \)
$13$ \( 1 + 18 T^{2} + 169 T^{4} \)
$17$ \( 1 + 8 T + 48 T^{2} + 136 T^{3} + 289 T^{4} \)
$19$ \( 1 + 8 T + 46 T^{2} + 152 T^{3} + 361 T^{4} \)
$23$ \( 1 + 14 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 8 T + 70 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 4 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 8 T + 96 T^{2} + 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 8 T + 84 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 20 T + 186 T^{2} - 940 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 4 T + 92 T^{2} + 212 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 20 T + 210 T^{2} - 1180 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 78 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 132 T^{2} + 4489 T^{4} \)
$71$ \( 1 - 8 T + 86 T^{2} - 568 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 4 T - 50 T^{2} + 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 12 T + 176 T^{2} - 948 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 12 T + 170 T^{2} - 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 4 T + 54 T^{2} + 356 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 12 T + 102 T^{2} + 1164 T^{3} + 9409 T^{4} \)
show more
show less