Properties

Label 5054.2.a.ba
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.48952000.1
Defining polynomial: \(x^{6} - x^{5} - 13 x^{4} + 44 x^{2} + 36 x + 4\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + q^{4} + ( -1 - \beta_{3} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} ) q^{6} - q^{7} - q^{8} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q - q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + q^{4} + ( -1 - \beta_{3} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} ) q^{6} - q^{7} - q^{8} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} + ( 1 + \beta_{3} ) q^{10} + ( -1 - \beta_{1} - \beta_{3} - \beta_{5} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} ) q^{12} + ( 4 + \beta_{2} - \beta_{5} ) q^{13} + q^{14} + ( -3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{15} + q^{16} + ( -2 \beta_{1} + \beta_{2} - \beta_{5} ) q^{17} + ( -3 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{18} + ( -1 - \beta_{3} ) q^{20} + ( -1 + \beta_{1} - \beta_{2} ) q^{21} + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{22} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} ) q^{24} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{25} + ( -4 - \beta_{2} + \beta_{5} ) q^{26} + ( 3 + 6 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{27} - q^{28} + ( 1 - \beta_{3} - \beta_{4} ) q^{29} + ( 3 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{30} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{31} - q^{32} + ( -1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{33} + ( 2 \beta_{1} - \beta_{2} + \beta_{5} ) q^{34} + ( 1 + \beta_{3} ) q^{35} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{36} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{37} + ( 3 - 5 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{5} ) q^{39} + ( 1 + \beta_{3} ) q^{40} + ( 1 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{41} + ( 1 - \beta_{1} + \beta_{2} ) q^{42} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{43} + ( -1 - \beta_{1} - \beta_{3} - \beta_{5} ) q^{44} + ( -13 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{45} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{46} + ( -1 - 3 \beta_{4} ) q^{47} + ( 1 - \beta_{1} + \beta_{2} ) q^{48} + q^{49} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{50} + ( 7 + \beta_{1} + \beta_{3} - \beta_{5} ) q^{51} + ( 4 + \beta_{2} - \beta_{5} ) q^{52} + ( -3 + \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{53} + ( -3 - 6 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{54} + ( 9 - 2 \beta_{1} + 6 \beta_{2} - 3 \beta_{4} ) q^{55} + q^{56} + ( -1 + \beta_{3} + \beta_{4} ) q^{58} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{59} + ( -3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{60} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{5} ) q^{61} + ( -3 + 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{62} + ( -3 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{63} + q^{64} + ( -2 - \beta_{1} + 5 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{65} + ( 1 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{66} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{67} + ( -2 \beta_{1} + \beta_{2} - \beta_{5} ) q^{68} + ( -1 + \beta_{1} - 7 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{69} + ( -1 - \beta_{3} ) q^{70} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{71} + ( -3 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{72} + ( -3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{73} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{74} + ( 16 - 2 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} ) q^{75} + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{77} + ( -3 + 5 \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{5} ) q^{78} + ( 6 + 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{79} + ( -1 - \beta_{3} ) q^{80} + ( 8 - \beta_{1} + 3 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{81} + ( -1 - 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{82} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{83} + ( -1 + \beta_{1} - \beta_{2} ) q^{84} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{85} + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{86} + ( -1 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{87} + ( 1 + \beta_{1} + \beta_{3} + \beta_{5} ) q^{88} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 13 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{90} + ( -4 - \beta_{2} + \beta_{5} ) q^{91} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{92} + ( 7 - \beta_{1} + 11 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{93} + ( 1 + 3 \beta_{4} ) q^{94} + ( -1 + \beta_{1} - \beta_{2} ) q^{96} + ( 5 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} ) q^{97} - q^{98} + ( -13 + \beta_{1} - 5 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 12 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 12 q^{9} + 5 q^{10} - 4 q^{11} + 2 q^{12} + 23 q^{13} + 6 q^{14} - 6 q^{15} + 6 q^{16} - 3 q^{17} - 12 q^{18} - 5 q^{20} - 2 q^{21} + 4 q^{22} - 4 q^{23} - 2 q^{24} + 13 q^{25} - 23 q^{26} - 4 q^{27} - 6 q^{28} + 5 q^{29} + 6 q^{30} + 12 q^{31} - 6 q^{32} + 10 q^{33} + 3 q^{34} + 5 q^{35} + 12 q^{36} + q^{37} + 4 q^{39} + 5 q^{40} + 19 q^{41} + 2 q^{42} - 2 q^{43} - 4 q^{44} - 55 q^{45} + 4 q^{46} - 12 q^{47} + 2 q^{48} + 6 q^{49} - 13 q^{50} + 44 q^{51} + 23 q^{52} - 11 q^{53} + 4 q^{54} + 28 q^{55} + 6 q^{56} - 5 q^{58} - 4 q^{59} - 6 q^{60} - q^{61} - 12 q^{62} - 12 q^{63} + 6 q^{64} - 24 q^{65} - 10 q^{66} - 14 q^{67} - 3 q^{68} + 10 q^{69} - 5 q^{70} - 8 q^{71} - 12 q^{72} + 5 q^{73} - q^{74} + 56 q^{75} + 4 q^{77} - 4 q^{78} + 30 q^{79} - 5 q^{80} + 30 q^{81} - 19 q^{82} - 12 q^{83} - 2 q^{84} - 16 q^{85} + 2 q^{86} - 14 q^{87} + 4 q^{88} + 11 q^{89} + 55 q^{90} - 23 q^{91} - 4 q^{92} + 10 q^{93} + 12 q^{94} - 2 q^{96} + 25 q^{97} - 6 q^{98} - 50 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 13 x^{4} + 44 x^{2} + 36 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 10 \nu^{3} + 9 \nu^{2} + 28 \nu + 6 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 9 \nu^{3} + 20 \nu^{2} + 22 \nu - 12 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + 3 \nu^{4} + 9 \nu^{3} - 16 \nu^{2} - 30 \nu - 4 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} + 15 \nu^{3} - 6 \nu^{2} - 54 \nu - 16 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 2 \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 9 \beta_{1} + 6\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 13 \beta_{4} + 8 \beta_{3} + 6 \beta_{2} + 25 \beta_{1} + 32\)
\(\nu^{5}\)\(=\)\(12 \beta_{5} + 37 \beta_{4} + 17 \beta_{3} + 36 \beta_{2} + 94 \beta_{1} + 82\)

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
2.54615
3.43145
−0.913852
−2.25033
−0.132439
−1.68097
−1.00000 −3.16418 1.00000 −2.51032 3.16418 −1.00000 −1.00000 7.01204 2.51032
1.2 −1.00000 −1.81341 1.00000 0.208819 1.81341 −1.00000 −1.00000 0.288464 −0.208819
1.3 −1.00000 0.295818 1.00000 1.81581 −0.295818 −1.00000 −1.00000 −2.91249 −1.81581
1.4 −1.00000 1.63229 1.00000 −2.92353 −1.63229 −1.00000 −1.00000 −0.335615 2.92353
1.5 −1.00000 1.75047 1.00000 2.63573 −1.75047 −1.00000 −1.00000 0.0641566 −2.63573
1.6 −1.00000 3.29901 1.00000 −4.22651 −3.29901 −1.00000 −1.00000 7.88345 4.22651
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

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.ba 6
19.b odd 2 1 5054.2.a.bd yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.ba 6 1.a even 1 1 trivial
5054.2.a.bd yes 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\( T_{3}^{6} - 2 T_{3}^{5} - 13 T_{3}^{4} + 26 T_{3}^{3} + 27 T_{3}^{2} - 64 T_{3} + 16 \)
\( T_{5}^{6} + 5 T_{5}^{5} - 9 T_{5}^{4} - 56 T_{5}^{3} + 19 T_{5}^{2} + 147 T_{5} - 31 \)
\( T_{13}^{6} - 23 T_{13}^{5} + 194 T_{13}^{4} - 661 T_{13}^{3} + 212 T_{13}^{2} + 3579 T_{13} - 5571 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( 16 - 64 T + 27 T^{2} + 26 T^{3} - 13 T^{4} - 2 T^{5} + T^{6} \)
$5$ \( -31 + 147 T + 19 T^{2} - 56 T^{3} - 9 T^{4} + 5 T^{5} + T^{6} \)
$7$ \( ( 1 + T )^{6} \)
$11$ \( 656 + 620 T + 9 T^{2} - 128 T^{3} - 26 T^{4} + 4 T^{5} + T^{6} \)
$13$ \( -5571 + 3579 T + 212 T^{2} - 661 T^{3} + 194 T^{4} - 23 T^{5} + T^{6} \)
$17$ \( -3799 + 1305 T + 772 T^{2} - 147 T^{3} - 54 T^{4} + 3 T^{5} + T^{6} \)
$19$ \( T^{6} \)
$23$ \( 944 - 2660 T + 1377 T^{2} - 64 T^{3} - 66 T^{4} + 4 T^{5} + T^{6} \)
$29$ \( 19 - 213 T - 41 T^{2} + 114 T^{3} - 19 T^{4} - 5 T^{5} + T^{6} \)
$31$ \( 80816 - 37308 T + 539 T^{2} + 1340 T^{3} - 92 T^{4} - 12 T^{5} + T^{6} \)
$37$ \( 271 - 2577 T + 1127 T^{2} + 122 T^{3} - 79 T^{4} - T^{5} + T^{6} \)
$41$ \( 17209 - 8521 T - 2142 T^{2} + 825 T^{3} + 36 T^{4} - 19 T^{5} + T^{6} \)
$43$ \( -656 + 240 T + 249 T^{2} - 46 T^{3} - 29 T^{4} + 2 T^{5} + T^{6} \)
$47$ \( 8656 + 10116 T - 2949 T^{2} - 1612 T^{3} - 108 T^{4} + 12 T^{5} + T^{6} \)
$53$ \( 128959 + 49483 T - 1949 T^{2} - 2134 T^{3} - 147 T^{4} + 11 T^{5} + T^{6} \)
$59$ \( -38096 + 500 T + 5067 T^{2} - 84 T^{3} - 156 T^{4} + 4 T^{5} + T^{6} \)
$61$ \( -171 + 303 T + 211 T^{2} - 64 T^{3} - 37 T^{4} + T^{5} + T^{6} \)
$67$ \( -23824 + 45148 T + 3737 T^{2} - 1718 T^{3} - 129 T^{4} + 14 T^{5} + T^{6} \)
$71$ \( -80896 + 27136 T + 6592 T^{2} - 1664 T^{3} - 216 T^{4} + 8 T^{5} + T^{6} \)
$73$ \( -35951 - 22187 T + 6344 T^{2} + 1001 T^{3} - 214 T^{4} - 5 T^{5} + T^{6} \)
$79$ \( -31984 + 24492 T - 4599 T^{2} - 618 T^{3} + 287 T^{4} - 30 T^{5} + T^{6} \)
$83$ \( 2304 - 7680 T - 6736 T^{2} - 1616 T^{3} - 84 T^{4} + 12 T^{5} + T^{6} \)
$89$ \( -392551 - 206605 T + 35912 T^{2} + 3231 T^{3} - 366 T^{4} - 11 T^{5} + T^{6} \)
$97$ \( 2987609 - 459911 T - 36469 T^{2} + 8496 T^{3} - 177 T^{4} - 25 T^{5} + T^{6} \)
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