Properties

Label 2016.2.a.r
Level $2016$
Weight $2$
Character orbit 2016.a
Self dual yes
Analytic conductor $16.098$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \beta q^{5} + q^{7} +O(q^{10})\) \( q -2 \beta q^{5} + q^{7} + 4 \beta q^{11} + ( 4 - 2 \beta ) q^{13} + ( 2 - 4 \beta ) q^{17} + ( 2 - 2 \beta ) q^{19} -4 q^{23} + ( -1 + 4 \beta ) q^{25} + ( -2 + 4 \beta ) q^{29} + ( 4 - 4 \beta ) q^{31} -2 \beta q^{35} + ( 2 - 4 \beta ) q^{37} + ( 2 + 4 \beta ) q^{41} + 4 \beta q^{43} + ( -4 - 4 \beta ) q^{47} + q^{49} + 10 q^{53} + ( -8 - 8 \beta ) q^{55} + ( 6 + 2 \beta ) q^{59} + ( 8 + 2 \beta ) q^{61} + ( 4 - 4 \beta ) q^{65} + 4 q^{67} + ( -8 + 8 \beta ) q^{71} + ( 10 - 8 \beta ) q^{73} + 4 \beta q^{77} + 8 \beta q^{79} + ( 6 + 2 \beta ) q^{83} + ( 8 + 4 \beta ) q^{85} + 6 q^{89} + ( 4 - 2 \beta ) q^{91} + 4 q^{95} + ( 6 + 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 2q^{7} + O(q^{10}) \) \( 2q - 2q^{5} + 2q^{7} + 4q^{11} + 6q^{13} + 2q^{19} - 8q^{23} + 2q^{25} + 4q^{31} - 2q^{35} + 8q^{41} + 4q^{43} - 12q^{47} + 2q^{49} + 20q^{53} - 24q^{55} + 14q^{59} + 18q^{61} + 4q^{65} + 8q^{67} - 8q^{71} + 12q^{73} + 4q^{77} + 8q^{79} + 14q^{83} + 20q^{85} + 12q^{89} + 6q^{91} + 8q^{95} + 16q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
0 0 0 −3.23607 0 1.00000 0 0 0
1.2 0 0 0 1.23607 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.a.r 2
3.b odd 2 1 224.2.a.c 2
4.b odd 2 1 2016.2.a.o 2
8.b even 2 1 4032.2.a.bw 2
8.d odd 2 1 4032.2.a.bv 2
12.b even 2 1 224.2.a.d yes 2
15.d odd 2 1 5600.2.a.bk 2
21.c even 2 1 1568.2.a.v 2
21.g even 6 2 1568.2.i.n 4
21.h odd 6 2 1568.2.i.v 4
24.f even 2 1 448.2.a.i 2
24.h odd 2 1 448.2.a.j 2
48.i odd 4 2 1792.2.b.k 4
48.k even 4 2 1792.2.b.m 4
60.h even 2 1 5600.2.a.z 2
84.h odd 2 1 1568.2.a.k 2
84.j odd 6 2 1568.2.i.w 4
84.n even 6 2 1568.2.i.m 4
168.e odd 2 1 3136.2.a.by 2
168.i even 2 1 3136.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 3.b odd 2 1
224.2.a.d yes 2 12.b even 2 1
448.2.a.i 2 24.f even 2 1
448.2.a.j 2 24.h odd 2 1
1568.2.a.k 2 84.h odd 2 1
1568.2.a.v 2 21.c even 2 1
1568.2.i.m 4 84.n even 6 2
1568.2.i.n 4 21.g even 6 2
1568.2.i.v 4 21.h odd 6 2
1568.2.i.w 4 84.j odd 6 2
1792.2.b.k 4 48.i odd 4 2
1792.2.b.m 4 48.k even 4 2
2016.2.a.o 2 4.b odd 2 1
2016.2.a.r 2 1.a even 1 1 trivial
3136.2.a.bf 2 168.i even 2 1
3136.2.a.by 2 168.e odd 2 1
4032.2.a.bv 2 8.d odd 2 1
4032.2.a.bw 2 8.b even 2 1
5600.2.a.z 2 60.h even 2 1
5600.2.a.bk 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2016))\):

\( T_{5}^{2} + 2 T_{5} - 4 \)
\( T_{11}^{2} - 4 T_{11} - 16 \)
\( T_{13}^{2} - 6 T_{13} + 4 \)
\( T_{17}^{2} - 20 \)
\( T_{19}^{2} - 2 T_{19} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -4 + 2 T + T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( -16 - 4 T + T^{2} \)
$13$ \( 4 - 6 T + T^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( -4 - 2 T + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( -20 + T^{2} \)
$31$ \( -16 - 4 T + T^{2} \)
$37$ \( -20 + T^{2} \)
$41$ \( -4 - 8 T + T^{2} \)
$43$ \( -16 - 4 T + T^{2} \)
$47$ \( 16 + 12 T + T^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( 44 - 14 T + T^{2} \)
$61$ \( 76 - 18 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -64 + 8 T + T^{2} \)
$73$ \( -44 - 12 T + T^{2} \)
$79$ \( -64 - 8 T + T^{2} \)
$83$ \( 44 - 14 T + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 44 - 16 T + T^{2} \)
show more
show less