Properties

Label 2016.2.s.f
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: no (minimal twist has level 672)
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.

\(f(q)\) \(=\) \( q + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + ( -2 + 3 \zeta_{6} ) q^{7} + ( -2 + 2 \zeta_{6} ) q^{11} + 5 q^{13} + ( -2 + 2 \zeta_{6} ) q^{17} + 3 \zeta_{6} q^{19} -2 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} -8 q^{29} + ( -1 + \zeta_{6} ) q^{31} + 5 \zeta_{6} q^{37} -2 q^{41} -7 q^{43} + 8 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + ( -2 + 2 \zeta_{6} ) q^{53} + ( -10 + 10 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( -11 + 11 \zeta_{6} ) q^{67} + 12 q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} + ( -2 - 4 \zeta_{6} ) q^{77} + 17 \zeta_{6} q^{79} -16 q^{83} + 12 \zeta_{6} q^{89} + ( -10 + 15 \zeta_{6} ) q^{91} -14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{7} + O(q^{10}) \) \( 2q - q^{7} - 2q^{11} + 10q^{13} - 2q^{17} + 3q^{19} - 2q^{23} + 5q^{25} - 16q^{29} - q^{31} + 5q^{37} - 4q^{41} - 14q^{43} + 8q^{47} - 13q^{49} - 2q^{53} - 10q^{59} + 2q^{61} - 11q^{67} + 24q^{71} + 3q^{73} - 8q^{77} + 17q^{79} - 32q^{83} + 12q^{89} - 5q^{91} - 28q^{97} + O(q^{100}) \)

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 0 0 −0.500000 2.59808i 0 0 0
865.1 0 0 0 0 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.f 2
3.b odd 2 1 672.2.q.i yes 2
4.b odd 2 1 2016.2.s.g 2
7.c even 3 1 inner 2016.2.s.f 2
12.b even 2 1 672.2.q.c 2
21.g even 6 1 4704.2.a.x 1
21.h odd 6 1 672.2.q.i yes 2
21.h odd 6 1 4704.2.a.h 1
24.f even 2 1 1344.2.q.p 2
24.h odd 2 1 1344.2.q.f 2
28.g odd 6 1 2016.2.s.g 2
84.j odd 6 1 4704.2.a.i 1
84.n even 6 1 672.2.q.c 2
84.n even 6 1 4704.2.a.bb 1
168.s odd 6 1 1344.2.q.f 2
168.s odd 6 1 9408.2.a.cm 1
168.v even 6 1 1344.2.q.p 2
168.v even 6 1 9408.2.a.s 1
168.ba even 6 1 9408.2.a.y 1
168.be odd 6 1 9408.2.a.cj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.c 2 12.b even 2 1
672.2.q.c 2 84.n even 6 1
672.2.q.i yes 2 3.b odd 2 1
672.2.q.i yes 2 21.h odd 6 1
1344.2.q.f 2 24.h odd 2 1
1344.2.q.f 2 168.s odd 6 1
1344.2.q.p 2 24.f even 2 1
1344.2.q.p 2 168.v even 6 1
2016.2.s.f 2 1.a even 1 1 trivial
2016.2.s.f 2 7.c even 3 1 inner
2016.2.s.g 2 4.b odd 2 1
2016.2.s.g 2 28.g odd 6 1
4704.2.a.h 1 21.h odd 6 1
4704.2.a.i 1 84.j odd 6 1
4704.2.a.x 1 21.g even 6 1
4704.2.a.bb 1 84.n even 6 1
9408.2.a.s 1 168.v even 6 1
9408.2.a.y 1 168.ba even 6 1
9408.2.a.cj 1 168.be odd 6 1
9408.2.a.cm 1 168.s odd 6 1

Hecke kernels

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

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