Properties

Label 5586.2.a.ce
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 22 x^{6} + 60 x^{5} + 87 x^{4} - 176 x^{3} - 40 x^{2} + 64 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 22 x^{6} + 60 x^{5} + 87 x^{4} - 176 x^{3} - 40 x^{2} + 64 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 9 \nu^{7} + 56 \nu^{6} - 502 \nu^{5} - 1524 \nu^{4} + 4267 \nu^{3} + 6100 \nu^{2} - 6148 \nu - 4644 \)\()/1972\)
\(\beta_{2}\)\(=\)\((\)\( 268 \nu^{7} - 1893 \nu^{6} - 3226 \nu^{5} + 37114 \nu^{4} - 14484 \nu^{3} - 159895 \nu^{2} + 87638 \nu + 96380 \)\()/13804\)
\(\beta_{3}\)\(=\)\((\)\( -675 \nu^{7} + 1716 \nu^{6} + 18916 \nu^{5} - 19796 \nu^{4} - 116909 \nu^{3} + 47332 \nu^{2} + 136706 \nu - 24408 \)\()/13804\)
\(\beta_{4}\)\(=\)\((\)\( -1983 \nu^{7} + 7710 \nu^{6} + 42902 \nu^{5} - 107912 \nu^{4} - 146761 \nu^{3} + 237182 \nu^{2} - 71804 \nu + 43144 \)\()/27608\)
\(\beta_{5}\)\(=\)\((\)\( -167 \nu^{7} + 659 \nu^{6} + 3618 \nu^{5} - 9518 \nu^{4} - 13005 \nu^{3} + 25125 \nu^{2} + 4524 \nu - 6512 \)\()/1972\)
\(\beta_{6}\)\(=\)\((\)\( -167 \nu^{7} + 659 \nu^{6} + 3618 \nu^{5} - 9518 \nu^{4} - 13005 \nu^{3} + 25125 \nu^{2} + 580 \nu - 4540 \)\()/1972\)
\(\beta_{7}\)\(=\)\((\)\( 2859 \nu^{7} - 13434 \nu^{6} - 55610 \nu^{5} + 215936 \nu^{4} + 149277 \nu^{3} - 653770 \nu^{2} + 140592 \nu + 139824 \)\()/27608\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 4 \beta_{1} + 14\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(8 \beta_{7} - 21 \beta_{6} + 17 \beta_{5} + 12 \beta_{4} + 4 \beta_{3} - 8 \beta_{2} + 10 \beta_{1} + 35\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(75 \beta_{7} - 27 \beta_{6} + 51 \beta_{5} + 53 \beta_{4} + 31 \beta_{3} - 48 \beta_{2} + 114 \beta_{1} + 302\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(298 \beta_{7} - 545 \beta_{6} + 445 \beta_{5} + 390 \beta_{4} + 170 \beta_{3} - 260 \beta_{2} + 448 \beta_{1} + 1267\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(2113 \beta_{7} - 1757 \beta_{6} + 1951 \beta_{5} + 1939 \beta_{4} + 1023 \beta_{3} - 1466 \beta_{2} + 3368 \beta_{1} + 8642\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(10348 \beta_{7} - 15443 \beta_{6} + 13263 \beta_{5} + 12296 \beta_{4} + 5792 \beta_{3} - 8360 \beta_{2} + 16322 \beta_{1} + 43669\)\()/2\)

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
2.60141
5.71612
−3.71976
−0.261561
−1.97842
0.804035
1.35228
−0.514116
1.00000 −1.00000 1.00000 −4.15767 −1.00000 0 1.00000 1.00000 −4.15767
1.2 1.00000 −1.00000 1.00000 −3.15280 −1.00000 0 1.00000 1.00000 −3.15280
1.3 1.00000 −1.00000 1.00000 −2.63704 −1.00000 0 1.00000 1.00000 −2.63704
1.4 1.00000 −1.00000 1.00000 −2.05470 −1.00000 0 1.00000 1.00000 −2.05470
1.5 1.00000 −1.00000 1.00000 0.153106 −1.00000 0 1.00000 1.00000 0.153106
1.6 1.00000 −1.00000 1.00000 0.856159 −1.00000 0 1.00000 1.00000 0.856159
1.7 1.00000 −1.00000 1.00000 3.05440 −1.00000 0 1.00000 1.00000 3.05440
1.8 1.00000 −1.00000 1.00000 3.93854 −1.00000 0 1.00000 1.00000 3.93854
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.ce 8
7.b odd 2 1 5586.2.a.cf yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5586.2.a.ce 8 1.a even 1 1 trivial
5586.2.a.cf yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\(T_{5}^{8} + \cdots\)
\(T_{11}^{8} - \cdots\)
\(T_{13}^{8} - \cdots\)
\(T_{17}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( ( 1 + T )^{8} \)
$5$ \( 112 - 768 T + 128 T^{2} + 704 T^{3} + 122 T^{4} - 104 T^{5} - 24 T^{6} + 4 T^{7} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 448 - 5760 T + 7584 T^{2} - 2464 T^{3} - 532 T^{4} + 368 T^{5} - 28 T^{6} - 8 T^{7} + T^{8} \)
$13$ \( 59248 + 20544 T - 20800 T^{2} - 4928 T^{3} + 2074 T^{4} + 272 T^{5} - 80 T^{6} - 4 T^{7} + T^{8} \)
$17$ \( 256 - 17920 T + 1920 T^{2} + 5312 T^{3} + 24 T^{4} - 480 T^{5} - 48 T^{6} + 8 T^{7} + T^{8} \)
$19$ \( ( 1 + T )^{8} \)
$23$ \( -8 - 32 T + 280 T^{2} + 1008 T^{3} + 750 T^{4} + 80 T^{5} - 52 T^{6} - 4 T^{7} + T^{8} \)
$29$ \( 31744 - 352256 T + 236288 T^{2} - 36864 T^{3} - 5584 T^{4} + 1728 T^{5} - 40 T^{6} - 16 T^{7} + T^{8} \)
$31$ \( -149536 - 129472 T + 6416 T^{2} + 25072 T^{3} + 4034 T^{4} - 616 T^{5} - 128 T^{6} + 4 T^{7} + T^{8} \)
$37$ \( 109856 + 267456 T + 38192 T^{2} - 36208 T^{3} - 1822 T^{4} + 1800 T^{5} - 112 T^{6} - 12 T^{7} + T^{8} \)
$41$ \( 691712 + 428544 T - 324736 T^{2} - 7424 T^{3} + 15972 T^{4} + 32 T^{5} - 228 T^{6} + T^{8} \)
$43$ \( -13555456 + 1425408 T + 1490624 T^{2} - 322432 T^{3} - 1148 T^{4} + 4480 T^{5} - 204 T^{6} - 16 T^{7} + T^{8} \)
$47$ \( 2896352 - 434496 T - 477840 T^{2} + 78064 T^{3} + 14850 T^{4} - 1936 T^{5} - 200 T^{6} + 12 T^{7} + T^{8} \)
$53$ \( -2142464 + 774656 T + 658432 T^{2} - 79744 T^{3} - 29376 T^{4} + 4896 T^{5} - 48 T^{6} - 24 T^{7} + T^{8} \)
$59$ \( 260096 - 8192 T - 139264 T^{2} - 17024 T^{3} + 9956 T^{4} + 816 T^{5} - 188 T^{6} - 8 T^{7} + T^{8} \)
$61$ \( -2787904 + 1870848 T - 206688 T^{2} - 83392 T^{3} + 16500 T^{4} + 640 T^{5} - 236 T^{6} + T^{8} \)
$67$ \( -387136 + 134272 T + 87840 T^{2} - 36320 T^{3} - 2676 T^{4} + 2288 T^{5} - 204 T^{6} - 8 T^{7} + T^{8} \)
$71$ \( 14336 + 327680 T + 207360 T^{2} - 38912 T^{3} - 23008 T^{4} + 4160 T^{5} - 32 T^{6} - 24 T^{7} + T^{8} \)
$73$ \( 3523072 + 1732096 T - 445184 T^{2} - 146240 T^{3} + 18372 T^{4} + 3136 T^{5} - 228 T^{6} - 16 T^{7} + T^{8} \)
$79$ \( -1016968 + 203936 T + 421560 T^{2} - 87024 T^{3} - 13826 T^{4} + 3824 T^{5} - 100 T^{6} - 20 T^{7} + T^{8} \)
$83$ \( -974848 + 303104 T + 567296 T^{2} + 114688 T^{3} - 11072 T^{4} - 4096 T^{5} - 176 T^{6} + 16 T^{7} + T^{8} \)
$89$ \( 185104 + 582784 T - 297952 T^{2} - 133760 T^{3} + 40088 T^{4} + 544 T^{5} - 440 T^{6} + T^{8} \)
$97$ \( -44800 - 327680 T - 416896 T^{2} + 19776 T^{3} + 33496 T^{4} + 1024 T^{5} - 352 T^{6} - 8 T^{7} + T^{8} \)
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