Properties

Label 2016.2.s.d
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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

Hecke kernels

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

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 5T_{11} + 25 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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