Properties

Label 5054.2.a.u
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)

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
2.17009
−1.48119
0.311108
1.00000 −1.70928 1.00000 −3.24846 −1.70928 −1.00000 1.00000 −0.0783777 −3.24846
1.2 1.00000 0.806063 1.00000 −1.86907 0.806063 −1.00000 1.00000 −2.35026 −1.86907
1.3 1.00000 2.90321 1.00000 4.11753 2.90321 −1.00000 1.00000 5.42864 4.11753
\(n\): e.g. 2-40 or 990-1000
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.u 3
19.b odd 2 1 5054.2.a.q 3
19.c even 3 2 266.2.f.c 6
57.h odd 6 2 2394.2.o.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.f.c 6 19.c even 3 2
2394.2.o.s 6 57.h odd 6 2
5054.2.a.q 3 19.b odd 2 1
5054.2.a.u 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\( T_{3}^{3} - 2 T_{3}^{2} - 4 T_{3} + 4 \)
\( T_{5}^{3} + T_{5}^{2} - 15 T_{5} - 25 \)
\( T_{13}^{3} + T_{13}^{2} - 13 T_{13} - 23 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( 4 - 4 T - 2 T^{2} + T^{3} \)
$5$ \( -25 - 15 T + T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 100 - 16 T - 6 T^{2} + T^{3} \)
$13$ \( -23 - 13 T + T^{2} + T^{3} \)
$17$ \( -148 - 28 T + 6 T^{2} + T^{3} \)
$19$ \( T^{3} \)
$23$ \( -137 - 29 T + 5 T^{2} + T^{3} \)
$29$ \( -148 + 10 T^{2} + T^{3} \)
$31$ \( 52 - 32 T + 2 T^{2} + T^{3} \)
$37$ \( -16 + 8 T + 8 T^{2} + T^{3} \)
$41$ \( -20 - 4 T + 4 T^{2} + T^{3} \)
$43$ \( -100 + 52 T + 16 T^{2} + T^{3} \)
$47$ \( -260 - 96 T + 2 T^{2} + T^{3} \)
$53$ \( -428 - 72 T + 6 T^{2} + T^{3} \)
$59$ \( 27 + 45 T - 15 T^{2} + T^{3} \)
$61$ \( -1069 + 317 T - 31 T^{2} + T^{3} \)
$67$ \( 400 + 176 T + 24 T^{2} + T^{3} \)
$71$ \( -445 + 203 T - 27 T^{2} + T^{3} \)
$73$ \( -632 - 100 T + 6 T^{2} + T^{3} \)
$79$ \( 608 - 160 T + T^{3} \)
$83$ \( -25 - 15 T + T^{2} + T^{3} \)
$89$ \( -1184 - 112 T + 12 T^{2} + T^{3} \)
$97$ \( 148 - 10 T^{2} + T^{3} \)
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