Properties

Label 2016.2.s.o
Level $2016$
Weight $2$
Character orbit 2016.s
Analytic conductor $16.098$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_{2} q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q + 3 \beta_{2} q^{5} + \beta_{3} q^{7} -\beta_{1} q^{11} -4 q^{13} + ( 1 + \beta_{2} ) q^{17} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{3} ) q^{23} + ( -4 - 4 \beta_{2} ) q^{25} + 4 q^{29} + \beta_{1} q^{31} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{35} -5 \beta_{2} q^{37} -8 q^{41} -4 \beta_{3} q^{43} + ( -\beta_{1} - \beta_{3} ) q^{47} + 7 q^{49} + ( 7 + 7 \beta_{2} ) q^{53} -3 \beta_{3} q^{55} + \beta_{1} q^{59} -5 \beta_{2} q^{61} -12 \beta_{2} q^{65} -\beta_{1} q^{67} + ( 9 + 9 \beta_{2} ) q^{73} + ( 7 + 7 \beta_{2} ) q^{77} + ( -\beta_{1} - \beta_{3} ) q^{79} -4 \beta_{3} q^{83} -3 q^{85} + 9 \beta_{2} q^{89} -4 \beta_{3} q^{91} + 9 \beta_{1} q^{95} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} + O(q^{10}) \) \( 4 q - 6 q^{5} - 16 q^{13} + 2 q^{17} - 8 q^{25} + 16 q^{29} + 10 q^{37} - 32 q^{41} + 28 q^{49} + 14 q^{53} + 10 q^{61} + 24 q^{65} + 18 q^{73} + 14 q^{77} - 12 q^{85} - 18 q^{89} - 32 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\) \(1\)

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
1.32288 + 2.29129i
−1.32288 2.29129i
1.32288 2.29129i
−1.32288 + 2.29129i
0 0 0 −1.50000 + 2.59808i 0 −2.64575 0 0 0
289.2 0 0 0 −1.50000 + 2.59808i 0 2.64575 0 0 0
865.1 0 0 0 −1.50000 2.59808i 0 −2.64575 0 0 0
865.2 0 0 0 −1.50000 2.59808i 0 2.64575 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
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 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.o 4
3.b odd 2 1 224.2.i.c 4
4.b odd 2 1 inner 2016.2.s.o 4
7.c even 3 1 inner 2016.2.s.o 4
12.b even 2 1 224.2.i.c 4
21.c even 2 1 1568.2.i.p 4
21.g even 6 1 1568.2.a.t 2
21.g even 6 1 1568.2.i.p 4
21.h odd 6 1 224.2.i.c 4
21.h odd 6 1 1568.2.a.m 2
24.f even 2 1 448.2.i.h 4
24.h odd 2 1 448.2.i.h 4
28.g odd 6 1 inner 2016.2.s.o 4
84.h odd 2 1 1568.2.i.p 4
84.j odd 6 1 1568.2.a.t 2
84.j odd 6 1 1568.2.i.p 4
84.n even 6 1 224.2.i.c 4
84.n even 6 1 1568.2.a.m 2
168.s odd 6 1 448.2.i.h 4
168.s odd 6 1 3136.2.a.bv 2
168.v even 6 1 448.2.i.h 4
168.v even 6 1 3136.2.a.bv 2
168.ba even 6 1 3136.2.a.bg 2
168.be odd 6 1 3136.2.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.c 4 3.b odd 2 1
224.2.i.c 4 12.b even 2 1
224.2.i.c 4 21.h odd 6 1
224.2.i.c 4 84.n even 6 1
448.2.i.h 4 24.f even 2 1
448.2.i.h 4 24.h odd 2 1
448.2.i.h 4 168.s odd 6 1
448.2.i.h 4 168.v even 6 1
1568.2.a.m 2 21.h odd 6 1
1568.2.a.m 2 84.n even 6 1
1568.2.a.t 2 21.g even 6 1
1568.2.a.t 2 84.j odd 6 1
1568.2.i.p 4 21.c even 2 1
1568.2.i.p 4 21.g even 6 1
1568.2.i.p 4 84.h odd 2 1
1568.2.i.p 4 84.j odd 6 1
2016.2.s.o 4 1.a even 1 1 trivial
2016.2.s.o 4 4.b odd 2 1 inner
2016.2.s.o 4 7.c even 3 1 inner
2016.2.s.o 4 28.g odd 6 1 inner
3136.2.a.bg 2 168.ba even 6 1
3136.2.a.bg 2 168.be odd 6 1
3136.2.a.bv 2 168.s odd 6 1
3136.2.a.bv 2 168.v even 6 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11}^{4} + 7 T_{11}^{2} + 49 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 9 + 3 T + T^{2} )^{2} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 49 + 7 T^{2} + T^{4} \)
$13$ \( ( 4 + T )^{4} \)
$17$ \( ( 1 - T + T^{2} )^{2} \)
$19$ \( 3969 + 63 T^{2} + T^{4} \)
$23$ \( 49 + 7 T^{2} + T^{4} \)
$29$ \( ( -4 + T )^{4} \)
$31$ \( 49 + 7 T^{2} + T^{4} \)
$37$ \( ( 25 - 5 T + T^{2} )^{2} \)
$41$ \( ( 8 + T )^{4} \)
$43$ \( ( -112 + T^{2} )^{2} \)
$47$ \( 49 + 7 T^{2} + T^{4} \)
$53$ \( ( 49 - 7 T + T^{2} )^{2} \)
$59$ \( 49 + 7 T^{2} + T^{4} \)
$61$ \( ( 25 - 5 T + T^{2} )^{2} \)
$67$ \( 49 + 7 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 81 - 9 T + T^{2} )^{2} \)
$79$ \( 49 + 7 T^{2} + T^{4} \)
$83$ \( ( -112 + T^{2} )^{2} \)
$89$ \( ( 81 + 9 T + T^{2} )^{2} \)
$97$ \( ( 8 + T )^{4} \)
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