Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 3 \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + ( -5 + 5 \zeta_{6} ) q^{11} + 2 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} + 6 \zeta_{6} q^{19} -2 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} - q^{29} + ( 1 - \zeta_{6} ) q^{31} + ( -9 + 6 \zeta_{6} ) q^{35} -10 \zeta_{6} q^{37} -4 q^{41} + 4 q^{43} + 8 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 5 - 5 \zeta_{6} ) q^{53} -15 q^{55} + ( -13 + 13 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + ( 14 - 14 \zeta_{6} ) q^{67} -12 q^{71} + ( 6 - 6 \zeta_{6} ) q^{73} + ( -10 - 5 \zeta_{6} ) q^{77} -11 \zeta_{6} q^{79} -7 q^{83} + 6 q^{85} -6 \zeta_{6} q^{89} + ( -2 + 6 \zeta_{6} ) q^{91} + ( -18 + 18 \zeta_{6} ) q^{95} + 19 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} + q^{7} + O(q^{10}) \) \( 2 q + 3 q^{5} + q^{7} - 5 q^{11} + 4 q^{13} + 2 q^{17} + 6 q^{19} - 2 q^{23} - 4 q^{25} - 2 q^{29} + q^{31} - 12 q^{35} - 10 q^{37} - 8 q^{41} + 8 q^{43} + 8 q^{47} - 13 q^{49} + 5 q^{53} - 30 q^{55} - 13 q^{59} + 8 q^{61} + 6 q^{65} + 14 q^{67} - 24 q^{71} + 6 q^{73} - 25 q^{77} - 11 q^{79} - 14 q^{83} + 12 q^{85} - 6 q^{89} + 2 q^{91} - 18 q^{95} + 38 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 1.50000 2.59808i 0 0.500000 2.59808i 0 0 0
865.1 0 0 0 1.50000 + 2.59808i 0 0.500000 + 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

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

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 9 \)
\( T_{11}^{2} + 5 T_{11} + 25 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 - 3 T + T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 25 + 5 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 4 - 2 T + T^{2} \)
$19$ \( 36 - 6 T + T^{2} \)
$23$ \( 4 + 2 T + T^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( 1 - T + T^{2} \)
$37$ \( 100 + 10 T + T^{2} \)
$41$ \( ( 4 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 64 - 8 T + T^{2} \)
$53$ \( 25 - 5 T + T^{2} \)
$59$ \( 169 + 13 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 196 - 14 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 36 - 6 T + T^{2} \)
$79$ \( 121 + 11 T + T^{2} \)
$83$ \( ( 7 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( -19 + T )^{2} \)
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