Properties

Label 6384.2.a.cc
Level $6384$
Weight $2$
Character orbit 6384.a
Self dual yes
Analytic conductor $50.976$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.9764966504\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1240016.1
Defining polynomial: \(x^{5} - x^{4} - 8 x^{3} + 2 x^{2} + 16 x + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 399)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{2} q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{2} q^{5} - q^{7} + q^{9} + \beta_{1} q^{11} + ( 2 - \beta_{3} ) q^{13} -\beta_{2} q^{15} + ( -1 - \beta_{2} - \beta_{4} ) q^{17} - q^{19} + q^{21} -\beta_{1} q^{23} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{25} - q^{27} + ( \beta_{1} + \beta_{2} ) q^{29} + ( -2 + \beta_{1} ) q^{31} -\beta_{1} q^{33} -\beta_{2} q^{35} + ( 1 + 2 \beta_{2} - \beta_{4} ) q^{37} + ( -2 + \beta_{3} ) q^{39} + ( 1 - \beta_{3} + \beta_{4} ) q^{41} + ( -3 - \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{43} + \beta_{2} q^{45} + ( 6 - \beta_{1} + \beta_{2} ) q^{47} + q^{49} + ( 1 + \beta_{2} + \beta_{4} ) q^{51} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{53} + ( 1 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{55} + q^{57} + 2 \beta_{1} q^{59} + ( 3 + 2 \beta_{2} + \beta_{4} ) q^{61} - q^{63} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{65} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{67} + \beta_{1} q^{69} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{71} + ( 1 + 2 \beta_{3} - \beta_{4} ) q^{73} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{75} -\beta_{1} q^{77} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{79} + q^{81} + ( 5 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{83} + ( -6 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -\beta_{1} - \beta_{2} ) q^{87} + ( 1 - 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{89} + ( -2 + \beta_{3} ) q^{91} + ( 2 - \beta_{1} ) q^{93} -\beta_{2} q^{95} + ( -5 + 3 \beta_{1} + \beta_{3} - \beta_{4} ) q^{97} + \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9} + O(q^{10}) \) \( 5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9} + 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} + 5 q^{21} - 2 q^{23} + 11 q^{25} - 5 q^{27} - 8 q^{31} - 2 q^{33} + 2 q^{35} + 2 q^{37} - 8 q^{39} + 2 q^{41} - 20 q^{43} - 2 q^{45} + 26 q^{47} + 5 q^{49} + 2 q^{51} + 4 q^{53} + 4 q^{55} + 5 q^{57} + 4 q^{59} + 10 q^{61} - 5 q^{63} - 4 q^{65} - 10 q^{67} + 2 q^{69} - 10 q^{71} + 10 q^{73} - 11 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 34 q^{83} - 36 q^{85} + 10 q^{89} - 8 q^{91} + 8 q^{93} + 2 q^{95} - 16 q^{97} + 2 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 8 x^{3} + 2 x^{2} + 16 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 2 \nu - 6 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{4} - 4 \nu^{3} - 10 \nu^{2} + 12 \nu + 11 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{1} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 6\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{4} + 7 \beta_{3} + 4 \beta_{2} + 9 \beta_{1} + 31\)\()/2\)

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
1.91889
−1.89281
−1.36162
−0.437507
2.77304
0 −1.00000 0 −4.29208 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 −2.79287 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 1.06804 0 −1.00000 0 1.00000 0
1.4 0 −1.00000 0 1.47487 0 −1.00000 0 1.00000 0
1.5 0 −1.00000 0 2.54204 0 −1.00000 0 1.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(19\) \(1\)

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 6384.2.a.cc 5
4.b odd 2 1 399.2.a.f 5
12.b even 2 1 1197.2.a.p 5
20.d odd 2 1 9975.2.a.bq 5
28.d even 2 1 2793.2.a.be 5
76.d even 2 1 7581.2.a.x 5
84.h odd 2 1 8379.2.a.ce 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.a.f 5 4.b odd 2 1
1197.2.a.p 5 12.b even 2 1
2793.2.a.be 5 28.d even 2 1
6384.2.a.cc 5 1.a even 1 1 trivial
7581.2.a.x 5 76.d even 2 1
8379.2.a.ce 5 84.h odd 2 1
9975.2.a.bq 5 20.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6384))\):

\( T_{5}^{5} + 2 T_{5}^{4} - 16 T_{5}^{3} - 8 T_{5}^{2} + 68 T_{5} - 48 \)
\( T_{11}^{5} - 2 T_{11}^{4} - 32 T_{11}^{3} + 16 T_{11}^{2} + 256 T_{11} + 192 \)
\( T_{13}^{5} - 8 T_{13}^{4} - 8 T_{13}^{3} + 112 T_{13}^{2} + 32 T_{13} - 256 \)
\( T_{17}^{5} + 2 T_{17}^{4} - 72 T_{17}^{3} - 176 T_{17}^{2} + 1108 T_{17} + 3168 \)
\( T_{23}^{5} + 2 T_{23}^{4} - 32 T_{23}^{3} - 16 T_{23}^{2} + 256 T_{23} - 192 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( ( 1 + T )^{5} \)
$5$ \( -48 + 68 T - 8 T^{2} - 16 T^{3} + 2 T^{4} + T^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( 192 + 256 T + 16 T^{2} - 32 T^{3} - 2 T^{4} + T^{5} \)
$13$ \( -256 + 32 T + 112 T^{2} - 8 T^{3} - 8 T^{4} + T^{5} \)
$17$ \( 3168 + 1108 T - 176 T^{2} - 72 T^{3} + 2 T^{4} + T^{5} \)
$19$ \( ( 1 + T )^{5} \)
$23$ \( -192 + 256 T - 16 T^{2} - 32 T^{3} + 2 T^{4} + T^{5} \)
$29$ \( 24 + 28 T - 80 T^{2} - 56 T^{3} + T^{5} \)
$31$ \( 512 - 48 T - 144 T^{2} - 8 T^{3} + 8 T^{4} + T^{5} \)
$37$ \( 608 + 2128 T + 176 T^{2} - 120 T^{3} - 2 T^{4} + T^{5} \)
$41$ \( 96 + 176 T - 96 T^{2} - 72 T^{3} - 2 T^{4} + T^{5} \)
$43$ \( -13184 - 8272 T - 1376 T^{2} + 32 T^{3} + 20 T^{4} + T^{5} \)
$47$ \( 3648 - 836 T - 592 T^{2} + 224 T^{3} - 26 T^{4} + T^{5} \)
$53$ \( 20376 + 8812 T + 248 T^{2} - 192 T^{3} - 4 T^{4} + T^{5} \)
$59$ \( 6144 + 4096 T + 128 T^{2} - 128 T^{3} - 4 T^{4} + T^{5} \)
$61$ \( -3872 - 304 T + 944 T^{2} - 88 T^{3} - 10 T^{4} + T^{5} \)
$67$ \( 40064 + 3520 T - 1408 T^{2} - 136 T^{3} + 10 T^{4} + T^{5} \)
$71$ \( 3888 - 236 T - 896 T^{2} - 84 T^{3} + 10 T^{4} + T^{5} \)
$73$ \( -32 - 5232 T + 1744 T^{2} - 120 T^{3} - 10 T^{4} + T^{5} \)
$79$ \( 4864 + 1024 T - 512 T^{2} - 72 T^{3} + 14 T^{4} + T^{5} \)
$83$ \( 159888 - 45476 T + 2872 T^{2} + 248 T^{3} - 34 T^{4} + T^{5} \)
$89$ \( -114336 + 24176 T + 2304 T^{2} - 312 T^{3} - 10 T^{4} + T^{5} \)
$97$ \( 194816 + 19536 T - 4544 T^{2} - 312 T^{3} + 16 T^{4} + T^{5} \)
show more
show less