Properties

Label 2016.2.s.x
Level $2016$
Weight $2$
Character orbit 2016.s
Analytic conductor $16.098$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1445900625.1
Defining polynomial: \(x^{8} + 9 x^{6} + 77 x^{4} + 36 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{5} + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{7} +O(q^{10})\) \( q + \beta_{7} q^{5} + ( -\beta_{1} + \beta_{4} - \beta_{5} ) q^{7} -\beta_{6} q^{11} + ( 2 - \beta_{2} ) q^{13} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{17} + ( \beta_{1} + \beta_{5} ) q^{19} + ( 2 + 2 \beta_{3} ) q^{23} + ( -\beta_{3} + \beta_{6} ) q^{25} + ( -\beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{29} + ( \beta_{1} - 3 \beta_{4} ) q^{31} + ( -5 + \beta_{2} - 2 \beta_{3} ) q^{35} + ( 2 - \beta_{2} + \beta_{3} + \beta_{6} ) q^{37} + 4 \beta_{5} q^{41} + ( 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} ) q^{43} + ( 4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{47} + ( -1 + \beta_{2} + \beta_{3} - 2 \beta_{6} ) q^{49} + ( -4 \beta_{1} + 5 \beta_{4} ) q^{53} + ( -5 \beta_{4} - 2 \beta_{5} + 5 \beta_{7} ) q^{55} + ( -6 \beta_{3} + \beta_{6} ) q^{59} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{5} + 6 \beta_{7} ) q^{65} + ( \beta_{1} + 4 \beta_{4} ) q^{67} + 2 \beta_{2} q^{71} + ( \beta_{3} + \beta_{6} ) q^{73} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{77} + ( -3 \beta_{1} - 3 \beta_{5} - 5 \beta_{7} ) q^{79} + ( -5 - \beta_{2} ) q^{83} + ( -6 + 2 \beta_{2} ) q^{85} + 6 \beta_{7} q^{89} + ( \beta_{1} + 6 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} ) q^{91} + 2 \beta_{3} q^{95} + ( -5 - \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 2 q^{11} + 12 q^{13} + 8 q^{23} + 2 q^{25} - 28 q^{35} + 6 q^{37} + 12 q^{47} - 4 q^{49} + 22 q^{59} - 12 q^{61} + 8 q^{71} - 6 q^{73} - 44 q^{83} - 40 q^{85} - 8 q^{95} - 44 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9 x^{6} + 77 x^{4} + 36 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 272 \)\()/77\)
\(\beta_{3}\)\(=\)\((\)\( -9 \nu^{6} - 77 \nu^{4} - 693 \nu^{2} - 324 \)\()/308\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 657 \nu \)\()/154\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{7} + 77 \nu^{5} + 693 \nu^{3} + 16 \nu \)\()/308\)
\(\beta_{6}\)\(=\)\((\)\( -41 \nu^{6} - 385 \nu^{4} - 3157 \nu^{2} - 1476 \)\()/308\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 9 \nu^{5} + 77 \nu^{3} + 36 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 5 \beta_{3} - \beta_{2} - 4\)
\(\nu^{3}\)\(=\)\(-2 \beta_{7} + 9 \beta_{5} + 2 \beta_{4}\)
\(\nu^{4}\)\(=\)\(-9 \beta_{6} + 41 \beta_{3}\)
\(\nu^{5}\)\(=\)\(18 \beta_{7} - 77 \beta_{5} - 77 \beta_{1}\)
\(\nu^{6}\)\(=\)\(77 \beta_{2} + 272\)
\(\nu^{7}\)\(=\)\(-154 \beta_{4} + 657 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.342371 + 0.593004i
1.46040 + 2.52950i
−1.46040 2.52950i
−0.342371 0.593004i
0.342371 0.593004i
1.46040 2.52950i
−1.46040 + 2.52950i
−0.342371 + 0.593004i
0 0 0 −1.46040 + 2.52950i 0 1.80278 + 1.93649i 0 0 0
289.2 0 0 0 −0.342371 + 0.593004i 0 1.80278 1.93649i 0 0 0
289.3 0 0 0 0.342371 0.593004i 0 −1.80278 + 1.93649i 0 0 0
289.4 0 0 0 1.46040 2.52950i 0 −1.80278 1.93649i 0 0 0
865.1 0 0 0 −1.46040 2.52950i 0 1.80278 1.93649i 0 0 0
865.2 0 0 0 −0.342371 0.593004i 0 1.80278 + 1.93649i 0 0 0
865.3 0 0 0 0.342371 + 0.593004i 0 −1.80278 1.93649i 0 0 0
865.4 0 0 0 1.46040 + 2.52950i 0 −1.80278 + 1.93649i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 865.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
12.b even 2 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.s.x yes 8
3.b odd 2 1 2016.2.s.w 8
4.b odd 2 1 2016.2.s.w 8
7.c even 3 1 inner 2016.2.s.x yes 8
12.b even 2 1 inner 2016.2.s.x yes 8
21.h odd 6 1 2016.2.s.w 8
28.g odd 6 1 2016.2.s.w 8
84.n even 6 1 inner 2016.2.s.x yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.s.w 8 3.b odd 2 1
2016.2.s.w 8 4.b odd 2 1
2016.2.s.w 8 21.h odd 6 1
2016.2.s.w 8 28.g odd 6 1
2016.2.s.x yes 8 1.a even 1 1 trivial
2016.2.s.x yes 8 7.c even 3 1 inner
2016.2.s.x yes 8 12.b even 2 1 inner
2016.2.s.x yes 8 84.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\):

\( T_{5}^{8} + 9 T_{5}^{6} + 77 T_{5}^{4} + 36 T_{5}^{2} + 16 \)
\( T_{11}^{4} - T_{11}^{3} + 17 T_{11}^{2} + 16 T_{11} + 256 \)
\( T_{13}^{2} - 3 T_{13} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 16 + 36 T^{2} + 77 T^{4} + 9 T^{6} + T^{8} \)
$7$ \( ( 49 + T^{2} + T^{4} )^{2} \)
$11$ \( ( 256 + 16 T + 17 T^{2} - T^{3} + T^{4} )^{2} \)
$13$ \( ( -14 - 3 T + T^{2} )^{4} \)
$17$ \( ( 400 + 20 T^{2} + T^{4} )^{2} \)
$19$ \( 16 + 36 T^{2} + 77 T^{4} + 9 T^{6} + T^{8} \)
$23$ \( ( 4 - 2 T + T^{2} )^{4} \)
$29$ \( ( 784 - 61 T^{2} + T^{4} )^{2} \)
$31$ \( 2401 + 3234 T^{2} + 4307 T^{4} + 66 T^{6} + T^{8} \)
$37$ \( ( 196 + 42 T + 23 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$41$ \( ( 1024 - 144 T^{2} + T^{4} )^{2} \)
$43$ \( ( 784 - 69 T^{2} + T^{4} )^{2} \)
$47$ \( ( 3136 + 336 T + 92 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$53$ \( 92236816 + 2007236 T^{2} + 34077 T^{4} + 209 T^{6} + T^{8} \)
$59$ \( ( 196 - 154 T + 107 T^{2} - 11 T^{3} + T^{4} )^{2} \)
$61$ \( ( 3136 - 336 T + 92 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$67$ \( 24010000 + 906500 T^{2} + 29325 T^{4} + 185 T^{6} + T^{8} \)
$71$ \( ( -64 - 2 T + T^{2} )^{4} \)
$73$ \( ( 196 - 42 T + 23 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$79$ \( 1698181681 + 17555034 T^{2} + 140267 T^{4} + 426 T^{6} + T^{8} \)
$83$ \( ( 14 + 11 T + T^{2} )^{4} \)
$89$ \( 26873856 + 1679616 T^{2} + 99792 T^{4} + 324 T^{6} + T^{8} \)
$97$ \( ( 14 + 11 T + T^{2} )^{4} \)
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