Properties

Label 5054.2.a.s
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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
1.87939
−0.347296
−1.53209
−1.00000 −0.879385 1.00000 0.532089 0.879385 1.00000 −1.00000 −2.22668 −0.532089
1.2 −1.00000 1.34730 1.00000 −2.87939 −1.34730 1.00000 −1.00000 −1.18479 2.87939
1.3 −1.00000 2.53209 1.00000 −0.652704 −2.53209 1.00000 −1.00000 3.41147 0.652704
\(n\): 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.s 3
19.b odd 2 1 5054.2.a.t 3
19.f odd 18 2 266.2.u.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.u.a 6 19.f odd 18 2
5054.2.a.s 3 1.a even 1 1 trivial
5054.2.a.t 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\( T_{3}^{3} - 3 T_{3}^{2} + 3 \)
\( T_{5}^{3} + 3 T_{5}^{2} - 1 \)
\( T_{13}^{3} - 3 T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( 3 - 3 T^{2} + T^{3} \)
$5$ \( -1 + 3 T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( -1 + 3 T^{2} + T^{3} \)
$13$ \( 1 - 3 T + T^{3} \)
$17$ \( -27 - 27 T + T^{3} \)
$19$ \( T^{3} \)
$23$ \( 8 - 24 T + 6 T^{2} + T^{3} \)
$29$ \( -171 + 99 T - 18 T^{2} + T^{3} \)
$31$ \( -53 - 24 T + 3 T^{2} + T^{3} \)
$37$ \( -307 - 30 T + 9 T^{2} + T^{3} \)
$41$ \( -233 - 66 T + 9 T^{2} + T^{3} \)
$43$ \( -127 + 39 T + 15 T^{2} + T^{3} \)
$47$ \( 9 + 18 T - 9 T^{2} + T^{3} \)
$53$ \( 73 + 9 T - 12 T^{2} + T^{3} \)
$59$ \( -199 - 72 T + 3 T^{2} + T^{3} \)
$61$ \( 136 + 96 T + 18 T^{2} + T^{3} \)
$67$ \( -489 - 144 T + 3 T^{2} + T^{3} \)
$71$ \( 27 - 54 T + 9 T^{2} + T^{3} \)
$73$ \( -19 + 108 T + 21 T^{2} + T^{3} \)
$79$ \( 17 - 63 T - 6 T^{2} + T^{3} \)
$83$ \( -17 + 48 T - 15 T^{2} + T^{3} \)
$89$ \( 2319 - 153 T - 15 T^{2} + T^{3} \)
$97$ \( -192 + 12 T^{2} + T^{3} \)
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