Properties

Label 5054.2.a.z
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.36538000.1
Defining polynomial: \(x^{6} - 13 x^{4} - 4 x^{3} + 41 x^{2} + 16 x - 16\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 13 x^{4} - 4 x^{3} + 41 x^{2} + 16 x - 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 4 \nu^{4} + 3 \nu^{3} + 24 \nu^{2} - 55 \nu - 44 \)\()/40\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} + 7 \nu^{3} - 24 \nu^{2} - 15 \nu + 4 \)\()/10\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{5} + 12 \nu^{4} + 31 \nu^{3} - 112 \nu^{2} - 75 \nu + 172 \)\()/40\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{5} - 2 \nu^{4} - 31 \nu^{3} + 2 \nu^{2} + 65 \nu + 18 \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{2} + 7 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(\beta_{5} - 7 \beta_{4} + 11 \beta_{3} + 11 \beta_{2} + 12 \beta_{1} + 36\)
\(\nu^{5}\)\(=\)\(4 \beta_{5} - 4 \beta_{4} + 17 \beta_{3} + 48 \beta_{2} + 58 \beta_{1} + 56\)

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
−2.52231
−2.14489
−0.969024
0.479869
2.04245
3.11392
−1.00000 −2.52231 1.00000 1.42123 2.52231 1.00000 −1.00000 3.36207 −1.42123
1.2 −1.00000 −2.14489 1.00000 2.45305 2.14489 1.00000 −1.00000 1.60057 −2.45305
1.3 −1.00000 −0.969024 1.00000 −3.11111 0.969024 1.00000 −1.00000 −2.06099 3.11111
1.4 −1.00000 0.479869 1.00000 −2.85512 −0.479869 1.00000 −1.00000 −2.76973 2.85512
1.5 −1.00000 2.04245 1.00000 2.05192 −2.04245 1.00000 −1.00000 1.17159 −2.05192
1.6 −1.00000 3.11392 1.00000 −0.959972 −3.11392 1.00000 −1.00000 6.69649 0.959972
\(n\): 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.z 6
19.b odd 2 1 5054.2.a.be yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.z 6 1.a even 1 1 trivial
5054.2.a.be 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\( T_{3}^{6} - 13 T_{3}^{4} - 4 T_{3}^{3} + 41 T_{3}^{2} + 16 T_{3} - 16 \)
\( T_{5}^{6} + T_{5}^{5} - 15 T_{5}^{4} - 6 T_{5}^{3} + 67 T_{5}^{2} - 7 T_{5} - 61 \)
\( T_{13}^{6} + 15 T_{13}^{5} + 70 T_{13}^{4} + 73 T_{13}^{3} - 250 T_{13}^{2} - 525 T_{13} - 109 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{6} \)
$3$ \( -16 + 16 T + 41 T^{2} - 4 T^{3} - 13 T^{4} + T^{6} \)
$5$ \( -61 - 7 T + 67 T^{2} - 6 T^{3} - 15 T^{4} + T^{5} + T^{6} \)
$7$ \( ( -1 + T )^{6} \)
$11$ \( 16 - 356 T + 213 T^{2} + 96 T^{3} - 30 T^{4} - 4 T^{5} + T^{6} \)
$13$ \( -109 - 525 T - 250 T^{2} + 73 T^{3} + 70 T^{4} + 15 T^{5} + T^{6} \)
$17$ \( 4975 + 3525 T - 170 T^{2} - 365 T^{3} - 26 T^{4} + 9 T^{5} + T^{6} \)
$19$ \( T^{6} \)
$23$ \( -16 - 20 T + 153 T^{2} + 56 T^{3} - 26 T^{4} - 4 T^{5} + T^{6} \)
$29$ \( -12625 + 5875 T + 1451 T^{2} - 418 T^{3} - 67 T^{4} + 7 T^{5} + T^{6} \)
$31$ \( -14384 + 236 T + 2533 T^{2} - 176 T^{3} - 90 T^{4} + 4 T^{5} + T^{6} \)
$37$ \( -41 + 19 T + 155 T^{2} - 102 T^{3} - 51 T^{4} + 3 T^{5} + T^{6} \)
$41$ \( -11449 + 10807 T + 510 T^{2} - 783 T^{3} - 58 T^{4} + 11 T^{5} + T^{6} \)
$43$ \( 2224 + 7904 T - 159 T^{2} - 826 T^{3} - 73 T^{4} + 10 T^{5} + T^{6} \)
$47$ \( 11056 + 12364 T - 9643 T^{2} + 1968 T^{3} - 90 T^{4} - 12 T^{5} + T^{6} \)
$53$ \( 2111 + 7131 T + 3139 T^{2} + 122 T^{3} - 99 T^{4} - 5 T^{5} + T^{6} \)
$59$ \( -59120 - 11460 T + 9101 T^{2} - 224 T^{3} - 178 T^{4} + 4 T^{5} + T^{6} \)
$61$ \( 1625711 + 398589 T - 5245 T^{2} - 5806 T^{3} - 187 T^{4} + 21 T^{5} + T^{6} \)
$67$ \( -390224 + 90828 T + 11745 T^{2} - 2242 T^{3} - 173 T^{4} + 14 T^{5} + T^{6} \)
$71$ \( 256 - 2944 T - 880 T^{2} + 416 T^{3} + 188 T^{4} + 24 T^{5} + T^{6} \)
$73$ \( -57301 - 54095 T - 17422 T^{2} - 2009 T^{3} + 34 T^{4} + 21 T^{5} + T^{6} \)
$79$ \( -407920 + 38220 T + 64181 T^{2} + 13858 T^{3} + 1303 T^{4} + 58 T^{5} + T^{6} \)
$83$ \( 649984 - 68096 T - 35504 T^{2} + 6464 T^{3} - 188 T^{4} - 20 T^{5} + T^{6} \)
$89$ \( 150355 - 152645 T + 41766 T^{2} - 783 T^{3} - 402 T^{4} + 7 T^{5} + T^{6} \)
$97$ \( 3254519 + 931427 T + 53583 T^{2} - 5430 T^{3} - 503 T^{4} + 7 T^{5} + T^{6} \)
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