Properties

Label 5054.2.a.bg
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
Defining polynomial: \( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{5} + \beta_{2} + 1) q^{3} + q^{4} + ( - \beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{2} - 1) q^{6} + q^{7} - q^{8} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_{5} + \beta_{2} + 1) q^{3} + q^{4} + ( - \beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{2} - 1) q^{6} + q^{7} - q^{8} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{9} + (\beta_{6} - \beta_{5} - \beta_{2} + \beta_1) q^{10} + ( - \beta_{7} + 2 \beta_{6} + \beta_{3} + \beta_1 - 1) q^{11} + (\beta_{5} + \beta_{2} + 1) q^{12} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{13} - q^{14} + (\beta_{7} - 4 \beta_{6} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{15} + q^{16} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 1) q^{18} + ( - \beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{20} + (\beta_{5} + \beta_{2} + 1) q^{21} + (\beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_1 + 1) q^{22} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 3) q^{23} + ( - \beta_{5} - \beta_{2} - 1) q^{24} + (\beta_{7} - 6 \beta_{6} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{25} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{26} + (4 \beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{27} + q^{28} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 1) q^{29} + ( - \beta_{7} + 4 \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{30} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{31} - q^{32} + ( - 4 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{33} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{34} + ( - \beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{35} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{36} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{3} + 3 \beta_1 + 5) q^{37} + (\beta_{6} + 2 \beta_{5} + 3 \beta_{2} - 2 \beta_1 + 2) q^{39} + (\beta_{6} - \beta_{5} - \beta_{2} + \beta_1) q^{40} + ( - 4 \beta_{6} + \beta_{5} + 3 \beta_{2} - 1) q^{41} + ( - \beta_{5} - \beta_{2} - 1) q^{42} + (3 \beta_{6} + \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + \beta_1) q^{43} + ( - \beta_{7} + 2 \beta_{6} + \beta_{3} + \beta_1 - 1) q^{44} + (4 \beta_{7} - 4 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{45} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 3) q^{46} + (7 \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{47} + (\beta_{5} + \beta_{2} + 1) q^{48} + q^{49} + ( - \beta_{7} + 6 \beta_{6} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{50} + (4 \beta_{7} + 4 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{51} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{52} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 4) q^{53} + ( - 4 \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{54} + ( - 3 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{2} + 4 \beta_1 - 2) q^{55} - q^{56} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 1) q^{58} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_1 - 1) q^{59} + (\beta_{7} - 4 \beta_{6} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{60} + ( - \beta_{7} + \beta_{5} - \beta_{3} + 2 \beta_{2} + 1) q^{61} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{62} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{63} + q^{64} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 2) q^{65} + (4 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{66} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - \beta_1 + 7) q^{67} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{68} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{69} + (\beta_{6} - \beta_{5} - \beta_{2} + \beta_1) q^{70} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 1) q^{71} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 1) q^{72} + ( - 6 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - \beta_{2} + 3 \beta_1 + 6) q^{73} + (2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{3} - 3 \beta_1 - 5) q^{74} + (6 \beta_{7} - 7 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 6 \beta_{2} - 8 \beta_1 - 3) q^{75} + ( - \beta_{7} + 2 \beta_{6} + \beta_{3} + \beta_1 - 1) q^{77} + ( - \beta_{6} - 2 \beta_{5} - 3 \beta_{2} + 2 \beta_1 - 2) q^{78} + ( - 3 \beta_{6} - \beta_{3} + \beta_{2} + \beta_1 + 6) q^{79} + ( - \beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{80} + (3 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{81} + (4 \beta_{6} - \beta_{5} - 3 \beta_{2} + 1) q^{82} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} - 5) q^{83} + (\beta_{5} + \beta_{2} + 1) q^{84} + (4 \beta_{7} + 4 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{85} + ( - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_1) q^{86} + ( - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 4) q^{87} + (\beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_1 + 1) q^{88} + ( - 2 \beta_{7} + 5 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} + 2) q^{89} + ( - 4 \beta_{7} + 4 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{90} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{91} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 3) q^{92} + (\beta_{7} - 3 \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 2) q^{93} + ( - 7 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{94} + ( - \beta_{5} - \beta_{2} - 1) q^{96} + (\beta_{7} - \beta_{6} - \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - \beta_1 - 1) q^{97} - q^{98} + ( - 5 \beta_{7} + 5 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 2 q^{10} - 12 q^{11} + 4 q^{12} + 10 q^{13} - 8 q^{14} + 24 q^{15} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} + 22 q^{27} + 8 q^{28} + 8 q^{29} - 24 q^{30} + 18 q^{31} - 8 q^{32} - 16 q^{33} + 6 q^{34} - 2 q^{35} + 8 q^{36} + 36 q^{37} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 20 q^{46} - 10 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} + 12 q^{51} + 10 q^{52} + 32 q^{53} - 22 q^{54} - 22 q^{55} - 8 q^{56} - 8 q^{58} - 2 q^{59} + 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} + 44 q^{67} - 6 q^{68} + 2 q^{70} + 8 q^{71} - 8 q^{72} + 30 q^{73} - 36 q^{74} - 16 q^{75} - 12 q^{77} + 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} + 4 q^{84} - 16 q^{85} + 16 q^{86} + 24 q^{87} + 12 q^{88} - 22 q^{89} - 8 q^{90} + 10 q^{91} - 20 q^{92} - 16 q^{93} + 10 q^{94} - 4 q^{96} - 6 q^{97} - 8 q^{98} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 8\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 4\nu^{3} + 4\nu^{2} - 18\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 10\nu^{2} + 16\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 8\nu^{2} + 18\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 12\nu^{5} + 17\nu^{4} + 48\nu^{3} - 32\nu^{2} - 64\nu - 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 5\nu^{6} + 22\nu^{5} - 43\nu^{4} - 84\nu^{3} + 86\nu^{2} + 116\nu + 24 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{5} + 8\beta_{4} + 2\beta_{3} + 4\beta_{2} + 10\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14\beta_{5} + 16\beta_{4} + 12\beta_{3} + 16\beta_{2} + 42\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 4\beta_{6} - 66\beta_{5} + 70\beta_{4} + 30\beta_{3} + 56\beta_{2} + 92\beta _1 + 102 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{7} + 10\beta_{6} - 148\beta_{5} + 180\beta_{4} + 122\beta_{3} + 188\beta_{2} + 326\beta _1 + 188 \) Copy content Toggle raw display

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
2.35877
−0.761439
−2.15124
−1.91631
3.02989
−0.368905
−0.236286
2.04552
−1.00000 −2.35992 1.00000 −4.10066 2.35992 1.00000 −1.00000 2.56920 4.10066
1.2 −1.00000 −1.40760 1.00000 −2.26420 1.40760 1.00000 −1.00000 −1.01865 2.26420
1.3 −1.00000 −1.30521 1.00000 −0.772004 1.30521 1.00000 −1.00000 −1.29642 0.772004
1.4 −1.00000 0.282233 1.00000 2.81658 −0.282233 1.00000 −1.00000 −2.92034 −2.81658
1.5 −1.00000 1.02954 1.00000 −1.38232 −1.02954 1.00000 −1.00000 −1.94005 1.38232
1.6 −1.00000 1.57867 1.00000 0.329540 −1.57867 1.00000 −1.00000 −0.507804 −0.329540
1.7 −1.00000 3.04815 1.00000 3.90247 −3.04815 1.00000 −1.00000 6.29119 −3.90247
1.8 −1.00000 3.13415 1.00000 −0.529405 −3.13415 1.00000 −1.00000 6.82288 0.529405
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(19\) \(1\)

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 5054.2.a.bg 8
19.b odd 2 1 5054.2.a.bh yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.bg 8 1.a even 1 1 trivial
5054.2.a.bh yes 8 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\( T_{3}^{8} - 4T_{3}^{7} - 8T_{3}^{6} + 38T_{3}^{5} + 15T_{3}^{4} - 98T_{3}^{3} + 2T_{3}^{2} + 74T_{3} - 19 \) Copy content Toggle raw display
\( T_{5}^{8} + 2T_{5}^{7} - 22T_{5}^{6} - 46T_{5}^{5} + 90T_{5}^{4} + 254T_{5}^{3} + 138T_{5}^{2} - 18T_{5} - 19 \) Copy content Toggle raw display
\( T_{13}^{8} - 10T_{13}^{7} + 10T_{13}^{6} + 170T_{13}^{5} - 505T_{13}^{4} - 290T_{13}^{3} + 2600T_{13}^{2} - 2960T_{13} + 905 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} - 8 T^{6} + 38 T^{5} + \cdots - 19 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} - 22 T^{6} - 46 T^{5} + \cdots - 19 \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{7} + 28 T^{6} + \cdots - 779 \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + 10 T^{6} + \cdots + 905 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} - 58 T^{6} + \cdots - 24719 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 20 T^{7} + 120 T^{6} + \cdots - 5795 \) Copy content Toggle raw display
$29$ \( T^{8} - 8 T^{7} - 52 T^{6} + \cdots - 2179 \) Copy content Toggle raw display
$31$ \( T^{8} - 18 T^{7} + 78 T^{6} + \cdots + 4541 \) Copy content Toggle raw display
$37$ \( T^{8} - 36 T^{7} + 432 T^{6} + \cdots - 396379 \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} - 148 T^{6} + \cdots - 9559 \) Copy content Toggle raw display
$43$ \( T^{8} + 16 T^{7} - 158 T^{6} + \cdots + 2944801 \) Copy content Toggle raw display
$47$ \( T^{8} + 10 T^{7} - 270 T^{6} + \cdots + 8597405 \) Copy content Toggle raw display
$53$ \( T^{8} - 32 T^{7} + 248 T^{6} + \cdots - 48299 \) Copy content Toggle raw display
$59$ \( T^{8} + 2 T^{7} - 122 T^{6} + \cdots + 29921 \) Copy content Toggle raw display
$61$ \( T^{8} - 2 T^{7} - 82 T^{6} + \cdots + 7421 \) Copy content Toggle raw display
$67$ \( T^{8} - 44 T^{7} + 662 T^{6} + \cdots - 51619 \) Copy content Toggle raw display
$71$ \( T^{8} - 8 T^{7} - 112 T^{6} + \cdots + 17936 \) Copy content Toggle raw display
$73$ \( T^{8} - 30 T^{7} - 10 T^{6} + \cdots + 27334525 \) Copy content Toggle raw display
$79$ \( T^{8} - 60 T^{7} + 1470 T^{6} + \cdots - 6438695 \) Copy content Toggle raw display
$83$ \( T^{8} + 28 T^{7} - 12 T^{6} + \cdots - 23760304 \) Copy content Toggle raw display
$89$ \( T^{8} + 22 T^{7} - 222 T^{6} + \cdots + 580621 \) Copy content Toggle raw display
$97$ \( T^{8} + 6 T^{7} - 378 T^{6} + \cdots - 1404719 \) Copy content Toggle raw display
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