Properties

Label 5586.2.a.cc
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.207085568.1
Defining polynomial: \(x^{6} - 2 x^{5} - 13 x^{4} + 28 x^{3} + 10 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + \beta_{5} q^{5} + q^{6} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} + \beta_{5} q^{5} + q^{6} - q^{8} + q^{9} -\beta_{5} q^{10} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{11} - q^{12} + \beta_{4} q^{13} -\beta_{5} q^{15} + q^{16} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} - q^{18} + q^{19} + \beta_{5} q^{20} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} ) q^{22} + ( 3 - \beta_{2} + \beta_{3} + \beta_{5} ) q^{23} + q^{24} + ( 2 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{25} -\beta_{4} q^{26} - q^{27} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + \beta_{5} q^{30} + ( 2 \beta_{3} + \beta_{5} ) q^{31} - q^{32} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} ) q^{33} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + q^{36} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{37} - q^{38} -\beta_{4} q^{39} -\beta_{5} q^{40} + ( -1 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( 1 + \beta_{2} + \beta_{4} - \beta_{5} ) q^{43} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{44} + \beta_{5} q^{45} + ( -3 + \beta_{2} - \beta_{3} - \beta_{5} ) q^{46} + ( -2 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{47} - q^{48} + ( -2 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{50} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + \beta_{4} q^{52} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{53} + q^{54} + ( 6 - 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{5} ) q^{55} - q^{57} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{58} + ( -5 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{59} -\beta_{5} q^{60} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{61} + ( -2 \beta_{3} - \beta_{5} ) q^{62} + q^{64} + ( 3 + \beta_{1} + 5 \beta_{3} - \beta_{4} - \beta_{5} ) q^{65} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{66} + ( -3 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{67} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{68} + ( -3 + \beta_{2} - \beta_{3} - \beta_{5} ) q^{69} + ( 4 - 2 \beta_{3} + 2 \beta_{5} ) q^{71} - q^{72} + ( 3 + \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{73} + ( -\beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{74} + ( -2 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{75} + q^{76} + \beta_{4} q^{78} + ( 7 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{79} + \beta_{5} q^{80} + q^{81} + ( 1 - \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{82} + ( -4 - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{83} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{85} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{86} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} ) q^{88} + ( -4 + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{89} -\beta_{5} q^{90} + ( 3 - \beta_{2} + \beta_{3} + \beta_{5} ) q^{92} + ( -2 \beta_{3} - \beta_{5} ) q^{93} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{94} + \beta_{5} q^{95} + q^{96} + ( -\beta_{1} + \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{97} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} - 6q^{3} + 6q^{4} + 6q^{6} - 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{2} - 6q^{3} + 6q^{4} + 6q^{6} - 6q^{8} + 6q^{9} + 4q^{11} - 6q^{12} + 6q^{16} - 4q^{17} - 6q^{18} + 6q^{19} - 4q^{22} + 16q^{23} + 6q^{24} + 10q^{25} - 6q^{27} + 8q^{29} - 6q^{32} - 4q^{33} + 4q^{34} + 6q^{36} - 6q^{38} - 8q^{41} + 8q^{43} + 4q^{44} - 16q^{46} - 16q^{47} - 6q^{48} - 10q^{50} + 4q^{51} - 4q^{53} + 6q^{54} + 32q^{55} - 6q^{57} - 8q^{58} - 24q^{59} + 6q^{64} + 16q^{65} + 4q^{66} - 20q^{67} - 4q^{68} - 16q^{69} + 24q^{71} - 6q^{72} + 16q^{73} - 10q^{75} + 6q^{76} + 40q^{79} + 6q^{81} + 8q^{82} - 24q^{83} + 8q^{85} - 8q^{86} - 8q^{87} - 4q^{88} - 24q^{89} + 16q^{92} + 16q^{94} + 6q^{96} + 4q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 13 x^{4} + 28 x^{3} + 10 x^{2} - 12 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} + \nu^{3} - 10 \nu^{2} - 4 \nu + 1 \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 13 \nu^{3} + 17 \nu^{2} + 18 \nu - 7 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 14 \nu^{3} + 27 \nu^{2} + 20 \nu - 8 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{5} + 3 \nu^{4} + 27 \nu^{3} - 43 \nu^{2} - 37 \nu + 10 \)
\(\beta_{5}\)\(=\)\( -3 \nu^{5} + 5 \nu^{4} + 40 \nu^{3} - 71 \nu^{2} - 47 \nu + 20 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_{1} + 10\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{5} - 15 \beta_{3} + 9 \beta_{2} - 11 \beta_{1} - 6\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{5} + 20 \beta_{4} + 41 \beta_{3} + 5 \beta_{2} + 19 \beta_{1} + 104\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-24 \beta_{5} - 14 \beta_{4} - 187 \beta_{3} + 89 \beta_{2} - 123 \beta_{1} - 130\)\()/2\)

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
−0.783420
0.341967
2.94362
2.80421
−3.56039
0.254018
−1.00000 −1.00000 1.00000 −3.21939 1.00000 0 −1.00000 1.00000 3.21939
1.2 −1.00000 −1.00000 1.00000 −2.72132 1.00000 0 −1.00000 1.00000 2.72132
1.3 −1.00000 −1.00000 1.00000 −0.933779 1.00000 0 −1.00000 1.00000 0.933779
1.4 −1.00000 −1.00000 1.00000 0.909895 1.00000 0 −1.00000 1.00000 −0.909895
1.5 −1.00000 −1.00000 1.00000 1.81142 1.00000 0 −1.00000 1.00000 −1.81142
1.6 −1.00000 −1.00000 1.00000 4.15317 1.00000 0 −1.00000 1.00000 −4.15317
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 5586.2.a.cc 6
7.b odd 2 1 5586.2.a.cd yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5586.2.a.cc 6 1.a even 1 1 trivial
5586.2.a.cd yes 6 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{6} - 20 T_{5}^{4} - 8 T_{5}^{3} + 82 T_{5}^{2} + 8 T_{5} - 56 \)
\( T_{11}^{6} - 4 T_{11}^{5} - 54 T_{11}^{4} + 240 T_{11}^{3} + 544 T_{11}^{2} - 3712 T_{11} + 4384 \)
\( T_{13}^{6} - 36 T_{13}^{4} + 64 T_{13}^{3} + 82 T_{13}^{2} - 184 T_{13} + 56 \)
\( T_{17}^{6} + 4 T_{17}^{5} - 52 T_{17}^{4} - 224 T_{17}^{3} + 160 T_{17}^{2} + 384 T_{17} + 128 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( ( 1 + T )^{6} \)
$5$ \( -56 + 8 T + 82 T^{2} - 8 T^{3} - 20 T^{4} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 4384 - 3712 T + 544 T^{2} + 240 T^{3} - 54 T^{4} - 4 T^{5} + T^{6} \)
$13$ \( 56 - 184 T + 82 T^{2} + 64 T^{3} - 36 T^{4} + T^{6} \)
$17$ \( 128 + 384 T + 160 T^{2} - 224 T^{3} - 52 T^{4} + 4 T^{5} + T^{6} \)
$19$ \( ( -1 + T )^{6} \)
$23$ \( -4892 + 8040 T - 3934 T^{2} + 616 T^{3} + 34 T^{4} - 16 T^{5} + T^{6} \)
$29$ \( -1792 - 2176 T - 16 T^{2} + 384 T^{3} - 40 T^{4} - 8 T^{5} + T^{6} \)
$31$ \( 56 - 568 T + 418 T^{2} + 40 T^{3} - 44 T^{4} + T^{6} \)
$37$ \( 8072 - 21384 T + 5410 T^{2} + 328 T^{3} - 148 T^{4} + T^{6} \)
$41$ \( 4352 + 4352 T + 448 T^{2} - 448 T^{3} - 66 T^{4} + 8 T^{5} + T^{6} \)
$43$ \( 544 - 1312 T + 164 T^{2} + 320 T^{3} - 52 T^{4} - 8 T^{5} + T^{6} \)
$47$ \( 90808 - 8088 T - 15742 T^{2} - 2848 T^{3} - 84 T^{4} + 16 T^{5} + T^{6} \)
$53$ \( -566848 + 13248 T + 23568 T^{2} - 608 T^{3} - 276 T^{4} + 4 T^{5} + T^{6} \)
$59$ \( -86144 - 71424 T - 19740 T^{2} - 1744 T^{3} + 100 T^{4} + 24 T^{5} + T^{6} \)
$61$ \( -62944 - 5888 T + 9664 T^{2} + 96 T^{3} - 202 T^{4} + T^{6} \)
$67$ \( 1568 + 1920 T - 1824 T^{2} - 304 T^{3} + 90 T^{4} + 20 T^{5} + T^{6} \)
$71$ \( -1792 + 5632 T - 2720 T^{2} + 128 T^{3} + 136 T^{4} - 24 T^{5} + T^{6} \)
$73$ \( 1792 + 2816 T - 3072 T^{2} + 608 T^{3} + 30 T^{4} - 16 T^{5} + T^{6} \)
$79$ \( 18244 - 27576 T + 15634 T^{2} - 4264 T^{3} + 594 T^{4} - 40 T^{5} + T^{6} \)
$83$ \( -34816 - 60416 T - 29312 T^{2} - 3968 T^{3} - 16 T^{4} + 24 T^{5} + T^{6} \)
$89$ \( 59768 + 22240 T - 5876 T^{2} - 1952 T^{3} + 26 T^{4} + 24 T^{5} + T^{6} \)
$97$ \( -255872 + 39296 T + 25504 T^{2} + 416 T^{3} - 308 T^{4} - 4 T^{5} + T^{6} \)
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