Properties

Label 5054.2.a.bc
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: 6.6.1528713.1
Defining polynomial: \(x^{6} - 3 x^{5} - 3 x^{4} + 7 x^{3} + 3 x^{2} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 3 x^{4} + 7 x^{3} + 3 x^{2} - 3 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 2 \nu^{2} + 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 2 \nu^{3} + 4 \nu^{2} + \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 2 \nu^{3} + 5 \nu^{2} - \nu - 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 4 \nu^{4} + 9 \nu^{2} - 2 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + 2 \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 7 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-3 \beta_{5} + 11 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} + 21 \beta_{1} + 12\)
\(\nu^{5}\)\(=\)\(-11 \beta_{5} + 35 \beta_{4} - 23 \beta_{3} - 8 \beta_{2} + 68 \beta_{1} + 33\)

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
0.768740
−0.725554
−0.313223
3.19261
1.37826
−1.30083
1.00000 −2.44476 1.00000 −1.45177 −2.44476 −1.00000 1.00000 2.97685 −1.45177
1.2 1.00000 −2.11161 1.00000 −3.59028 −2.11161 −1.00000 1.00000 1.45891 −3.59028
1.3 1.00000 −1.10878 1.00000 3.89136 −1.10878 −1.00000 1.00000 −1.77060 3.89136
1.4 1.00000 0.108781 1.00000 −1.47989 0.108781 −1.00000 1.00000 −2.98817 −1.47989
1.5 1.00000 1.11161 1.00000 0.363595 1.11161 −1.00000 1.00000 −1.76432 0.363595
1.6 1.00000 1.44476 1.00000 −0.733019 1.44476 −1.00000 1.00000 −0.912670 −0.733019
\(n\): 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.bc 6
19.b odd 2 1 5054.2.a.bb 6
19.f odd 18 2 266.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.u.b 12 19.f odd 18 2
5054.2.a.bb 6 19.b odd 2 1
5054.2.a.bc 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\( T_{3}^{6} + 3 T_{3}^{5} - 3 T_{3}^{4} - 11 T_{3}^{3} + 3 T_{3}^{2} + 9 T_{3} - 1 \)
\( T_{5}^{6} + 3 T_{5}^{5} - 12 T_{5}^{4} - 47 T_{5}^{3} - 42 T_{5}^{2} + 8 \)
\( T_{13}^{6} + 6 T_{13}^{5} + 3 T_{13}^{4} - 17 T_{13}^{3} - 12 T_{13}^{2} + 12 T_{13} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{6} \)
$3$ \( -1 + 9 T + 3 T^{2} - 11 T^{3} - 3 T^{4} + 3 T^{5} + T^{6} \)
$5$ \( 8 - 42 T^{2} - 47 T^{3} - 12 T^{4} + 3 T^{5} + T^{6} \)
$7$ \( ( 1 + T )^{6} \)
$11$ \( 1 + 15 T + 33 T^{2} + 15 T^{3} - 9 T^{4} - 3 T^{5} + T^{6} \)
$13$ \( 8 + 12 T - 12 T^{2} - 17 T^{3} + 3 T^{4} + 6 T^{5} + T^{6} \)
$17$ \( 1 - 348 T - 738 T^{2} - 349 T^{3} - 24 T^{4} + 9 T^{5} + T^{6} \)
$19$ \( T^{6} \)
$23$ \( 1216 - 1728 T + 480 T^{2} + 144 T^{3} - 60 T^{4} + T^{6} \)
$29$ \( 296 - 156 T - 558 T^{2} - 265 T^{3} - 27 T^{4} + 6 T^{5} + T^{6} \)
$31$ \( 2368 + 4320 T + 1164 T^{2} - 223 T^{3} - 72 T^{4} + 3 T^{5} + T^{6} \)
$37$ \( -296 - 624 T + 1710 T^{2} + 33 T^{3} - 102 T^{4} - 3 T^{5} + T^{6} \)
$41$ \( 9 + 297 T - 117 T^{2} - 153 T^{3} - 9 T^{4} + 9 T^{5} + T^{6} \)
$43$ \( -43919 + 17430 T + 4239 T^{2} - 732 T^{3} - 129 T^{4} + 6 T^{5} + T^{6} \)
$47$ \( 1864 + 2700 T - 3924 T^{2} + 1359 T^{3} - 120 T^{4} - 9 T^{5} + T^{6} \)
$53$ \( 49832 + 33828 T + 2784 T^{2} - 1127 T^{3} - 147 T^{4} + 6 T^{5} + T^{6} \)
$59$ \( 41887 + 9981 T - 4425 T^{2} - 1415 T^{3} - 45 T^{4} + 15 T^{5} + T^{6} \)
$61$ \( 35648 - 26208 T - 16944 T^{2} - 1352 T^{3} + 204 T^{4} + 30 T^{5} + T^{6} \)
$67$ \( -47843 + 40623 T + 3093 T^{2} - 1573 T^{3} - 93 T^{4} + 15 T^{5} + T^{6} \)
$71$ \( 9784 + 22692 T - 10248 T^{2} - 2589 T^{3} - 24 T^{4} + 21 T^{5} + T^{6} \)
$73$ \( 73511 - 36609 T - 27399 T^{2} - 2393 T^{3} + 219 T^{4} + 33 T^{5} + T^{6} \)
$79$ \( 17704 - 10668 T - 7692 T^{2} + 1185 T^{3} + 183 T^{4} - 30 T^{5} + T^{6} \)
$83$ \( 9343 - 7815 T - 19371 T^{2} - 2113 T^{3} + 213 T^{4} + 33 T^{5} + T^{6} \)
$89$ \( -5013 + 3348 T + 5787 T^{2} + 696 T^{3} - 135 T^{4} - 12 T^{5} + T^{6} \)
$97$ \( 14912 + 33552 T + 18456 T^{2} + 1997 T^{3} - 177 T^{4} - 18 T^{5} + T^{6} \)
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