Properties

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

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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: 5.5.135076.1
Defining polynomial: \(x^{5} - x^{4} - 5 x^{3} + 4 x^{2} + 4 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 3192)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} - q^{7} + q^{9} + \beta_{4} q^{11} + ( 2 - \beta_{3} ) q^{13} + \beta_{1} q^{15} -\beta_{2} q^{17} - q^{19} - q^{21} -\beta_{3} q^{23} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{25} + q^{27} + ( \beta_{1} + \beta_{3} ) q^{29} + ( \beta_{3} - \beta_{4} ) q^{31} + \beta_{4} q^{33} -\beta_{1} q^{35} + ( 2 + \beta_{3} + \beta_{4} ) q^{37} + ( 2 - \beta_{3} ) q^{39} + ( -2 + 2 \beta_{1} ) q^{41} + \beta_{1} q^{45} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{47} + q^{49} -\beta_{2} q^{51} + ( 4 - \beta_{1} - \beta_{3} ) q^{53} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{55} - q^{57} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{59} + 6 q^{61} - q^{63} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{65} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{67} -\beta_{3} q^{69} + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{71} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{73} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{75} -\beta_{4} q^{77} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{79} + q^{81} + ( 2 - \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{83} + ( 4 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{85} + ( \beta_{1} + \beta_{3} ) q^{87} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{89} + ( -2 + \beta_{3} ) q^{91} + ( \beta_{3} - \beta_{4} ) q^{93} -\beta_{1} q^{95} + ( 2 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{97} + \beta_{4} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} + O(q^{10}) \) \( 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} - 5 q^{21} - 2 q^{23} + 19 q^{25} + 5 q^{27} + 4 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} + 2 q^{45} + 2 q^{47} + 5 q^{49} - 2 q^{51} + 16 q^{53} + 16 q^{55} - 5 q^{57} + 12 q^{59} + 30 q^{61} - 5 q^{63} + 4 q^{65} - 18 q^{67} - 2 q^{69} + 10 q^{71} + 14 q^{73} + 19 q^{75} + 2 q^{77} + 2 q^{79} + 5 q^{81} + 6 q^{83} + 12 q^{85} + 4 q^{87} + 10 q^{89} - 8 q^{91} + 4 q^{93} - 2 q^{95} + 8 q^{97} - 2 q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 2 \nu^{2} - 8 \nu - 4 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{4} + 2 \nu^{3} + 8 \nu^{2} - 6 \nu - 2 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{4} - 10 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 8\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{4} + \beta_{3} + \beta_{2} + 7 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_{1} + 28\)\()/4\)

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
−1.94486
−0.895793
0.420632
1.26848
2.15154
0 1.00000 0 −3.88971 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −1.79159 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 0.841263 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 2.53696 0 −1.00000 0 1.00000 0
1.5 0 1.00000 0 4.30308 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.ce 5
4.b odd 2 1 3192.2.a.ba 5
12.b even 2 1 9576.2.a.co 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.ba 5 4.b odd 2 1
6384.2.a.ce 5 1.a even 1 1 trivial
9576.2.a.co 5 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6384))\):

\( T_{5}^{5} - 2 T_{5}^{4} - 20 T_{5}^{3} + 32 T_{5}^{2} + 64 T_{5} - 64 \)
\( T_{11}^{5} + 2 T_{11}^{4} - 28 T_{11}^{3} - 40 T_{11}^{2} + 160 T_{11} + 128 \)
\( T_{13}^{5} - 8 T_{13}^{4} - 12 T_{13}^{3} + 208 T_{13}^{2} - 352 T_{13} - 64 \)
\( T_{17}^{5} + 2 T_{17}^{4} - 92 T_{17}^{3} - 192 T_{17}^{2} + 1984 T_{17} + 5504 \)
\( T_{23}^{5} + 2 T_{23}^{4} - 36 T_{23}^{3} + 24 T_{23}^{2} + 160 T_{23} - 128 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( ( -1 + T )^{5} \)
$5$ \( -64 + 64 T + 32 T^{2} - 20 T^{3} - 2 T^{4} + T^{5} \)
$7$ \( ( 1 + T )^{5} \)
$11$ \( 128 + 160 T - 40 T^{2} - 28 T^{3} + 2 T^{4} + T^{5} \)
$13$ \( -64 - 352 T + 208 T^{2} - 12 T^{3} - 8 T^{4} + T^{5} \)
$17$ \( 5504 + 1984 T - 192 T^{2} - 92 T^{3} + 2 T^{4} + T^{5} \)
$19$ \( ( 1 + T )^{5} \)
$23$ \( -128 + 160 T + 24 T^{2} - 36 T^{3} + 2 T^{4} + T^{5} \)
$29$ \( -1376 + 128 T + 264 T^{2} - 52 T^{3} - 4 T^{4} + T^{5} \)
$31$ \( 256 + 1408 T + 208 T^{2} - 80 T^{3} - 4 T^{4} + T^{5} \)
$37$ \( -352 - 304 T + 224 T^{2} - 8 T^{3} - 10 T^{4} + T^{5} \)
$41$ \( 352 + 1040 T - 240 T^{2} - 72 T^{3} + 6 T^{4} + T^{5} \)
$43$ \( T^{5} \)
$47$ \( 17728 + 6624 T - 80 T^{2} - 180 T^{3} - 2 T^{4} + T^{5} \)
$53$ \( -32 + 104 T^{2} + 44 T^{3} - 16 T^{4} + T^{5} \)
$59$ \( -105472 + 11008 T + 2944 T^{2} - 256 T^{3} - 12 T^{4} + T^{5} \)
$61$ \( ( -6 + T )^{5} \)
$67$ \( -10624 - 10784 T - 2984 T^{2} - 108 T^{3} + 18 T^{4} + T^{5} \)
$71$ \( -44288 + 2880 T + 1840 T^{2} - 172 T^{3} - 10 T^{4} + T^{5} \)
$73$ \( 8992 + 5776 T + 752 T^{2} - 104 T^{3} - 14 T^{4} + T^{5} \)
$79$ \( 512 - 128 T - 504 T^{2} - 156 T^{3} - 2 T^{4} + T^{5} \)
$83$ \( -4096 + 1536 T + 448 T^{2} - 92 T^{3} - 6 T^{4} + T^{5} \)
$89$ \( -6304 + 2448 T + 464 T^{2} - 168 T^{3} - 10 T^{4} + T^{5} \)
$97$ \( -128 - 704 T - 656 T^{2} - 172 T^{3} - 8 T^{4} + T^{5} \)
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