Properties

Label 6012.2.a.e
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0060616952\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.149169.1
Defining polynomial: \(x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{3} + \beta_{4} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{13} + ( 3 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{17} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{19} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{25} + ( -\beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{29} + ( -3 + 2 \beta_{2} + \beta_{3} ) q^{31} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{35} + ( -4 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{37} + ( 2 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 2 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{43} + ( -4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + ( -4 + \beta_{2} - \beta_{3} ) q^{49} + ( -1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{53} + ( -4 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{59} + ( -4 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{61} + ( 5 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{65} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{67} + ( -5 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} ) q^{71} + ( -4 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{73} + ( 1 + \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{77} + ( -6 - 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{79} + ( 1 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{83} + ( 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{85} + ( 7 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{89} + ( -3 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{91} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{95} + ( -4 - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 3q^{5} - 2q^{7} + O(q^{10}) \) \( 5q + 3q^{5} - 2q^{7} - 3q^{11} - 4q^{13} + 7q^{17} - 2q^{19} - 13q^{23} - 2q^{25} + 3q^{29} - 12q^{31} - 10q^{35} - 7q^{37} + 16q^{41} - q^{47} - 17q^{49} - 3q^{53} - 23q^{55} - q^{59} - 22q^{61} + 20q^{65} + 2q^{67} - 9q^{71} - 28q^{73} + 10q^{77} - 28q^{79} - 7q^{83} - 11q^{85} + 30q^{89} - 13q^{91} - 3q^{95} - 33q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-5 \beta_{4} + 6 \beta_{3} + \beta_{2} + 3 \beta_{1} + 14\)

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
−1.81853
2.54970
−1.25464
−0.228573
0.752046
0 0 0 −2.12560 0 0.221696 0 0 0
1.2 0 0 0 −0.951283 0 −1.28171 0 0 0
1.3 0 0 0 0.171230 0 2.92708 0 0 0
1.4 0 0 0 2.71918 0 −2.29630 0 0 0
1.5 0 0 0 3.18647 0 −1.57076 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(1\)

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 6012.2.a.e 5
3.b odd 2 1 2004.2.a.a 5
12.b even 2 1 8016.2.a.t 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.a 5 3.b odd 2 1
6012.2.a.e 5 1.a even 1 1 trivial
8016.2.a.t 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 3 T_{5}^{4} - 7 T_{5}^{3} + 16 T_{5}^{2} + 15 T_{5} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( T^{5} \)
$5$ \( -3 + 15 T + 16 T^{2} - 7 T^{3} - 3 T^{4} + T^{5} \)
$7$ \( 3 - 9 T - 19 T^{2} - 7 T^{3} + 2 T^{4} + T^{5} \)
$11$ \( 3 + 15 T - 16 T^{2} - 7 T^{3} + 3 T^{4} + T^{5} \)
$13$ \( -11 - 94 T - 78 T^{2} - 13 T^{3} + 4 T^{4} + T^{5} \)
$17$ \( -627 - 171 T + 218 T^{2} - 22 T^{3} - 7 T^{4} + T^{5} \)
$19$ \( 459 + 216 T - 86 T^{2} - 41 T^{3} + 2 T^{4} + T^{5} \)
$23$ \( 779 - 57 T - 232 T^{2} + 13 T^{3} + 13 T^{4} + T^{5} \)
$29$ \( -7689 + 3075 T + 310 T^{2} - 121 T^{3} - 3 T^{4} + T^{5} \)
$31$ \( 9 - 30 T - 17 T^{2} + 26 T^{3} + 12 T^{4} + T^{5} \)
$37$ \( 25111 + 4239 T - 874 T^{2} - 135 T^{3} + 7 T^{4} + T^{5} \)
$41$ \( 16439 - 9684 T + 1657 T^{2} - 24 T^{3} - 16 T^{4} + T^{5} \)
$43$ \( -17441 + 4820 T + 337 T^{2} - 146 T^{3} + T^{5} \)
$47$ \( 513 + 108 T - 327 T^{2} - 125 T^{3} + T^{4} + T^{5} \)
$53$ \( -579 + 360 T + 115 T^{2} - 73 T^{3} + 3 T^{4} + T^{5} \)
$59$ \( 3189 + 1500 T - 295 T^{2} - 105 T^{3} + T^{4} + T^{5} \)
$61$ \( -25497 - 13596 T - 1810 T^{2} + 49 T^{3} + 22 T^{4} + T^{5} \)
$67$ \( 189 + 990 T + 486 T^{2} - 125 T^{3} - 2 T^{4} + T^{5} \)
$71$ \( -1393 - 1961 T - 873 T^{2} - 106 T^{3} + 9 T^{4} + T^{5} \)
$73$ \( -9697 - 20684 T - 2702 T^{2} + 105 T^{3} + 28 T^{4} + T^{5} \)
$79$ \( -10659 - 10146 T - 1457 T^{2} + 140 T^{3} + 28 T^{4} + T^{5} \)
$83$ \( -116041 + 31754 T - 523 T^{2} - 329 T^{3} + 7 T^{4} + T^{5} \)
$89$ \( 5409 - 9024 T + 976 T^{2} + 197 T^{3} - 30 T^{4} + T^{5} \)
$97$ \( -7169 - 7769 T - 249 T^{2} + 284 T^{3} + 33 T^{4} + T^{5} \)
show more
show less