Properties

Label 5054.2.a.x
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.151572.1
Defining polynomial: \(x^{4} - x^{3} - 10 x^{2} + 8 x + 4\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 10 x^{2} + 8 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 10 \nu - 4 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 10 \nu + 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 10 \beta_{1} - 1\)

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
−3.02917
−0.352271
1.16566
3.21578
1.00000 −3.02917 1.00000 −3.36893 −3.02917 1.00000 1.00000 6.17589 −3.36893
1.2 1.00000 −0.352271 1.00000 4.32518 −0.352271 1.00000 1.00000 −2.87591 4.32518
1.3 1.00000 1.16566 1.00000 −1.55010 1.16566 1.00000 1.00000 −1.64123 −1.55010
1.4 1.00000 3.21578 1.00000 1.59385 3.21578 1.00000 1.00000 7.34125 1.59385
\(n\): 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.x 4
19.b odd 2 1 5054.2.a.w 4
19.d odd 6 2 266.2.f.d 8
57.f even 6 2 2394.2.o.v 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.f.d 8 19.d odd 6 2
2394.2.o.v 8 57.f even 6 2
5054.2.a.w 4 19.b odd 2 1
5054.2.a.x 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\( T_{3}^{4} - T_{3}^{3} - 10 T_{3}^{2} + 8 T_{3} + 4 \)
\( T_{5}^{4} - T_{5}^{3} - 17 T_{5}^{2} + 3 T_{5} + 36 \)
\( T_{13}^{4} - 5 T_{13}^{3} - 25 T_{13}^{2} + 145 T_{13} - 98 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 4 + 8 T - 10 T^{2} - T^{3} + T^{4} \)
$5$ \( 36 + 3 T - 17 T^{2} - T^{3} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( -12 + 12 T + 8 T^{2} - 7 T^{3} + T^{4} \)
$13$ \( -98 + 145 T - 25 T^{2} - 5 T^{3} + T^{4} \)
$17$ \( 24 + 156 T - 52 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( 462 + 177 T - 59 T^{2} - 5 T^{3} + T^{4} \)
$29$ \( 48 - 396 T - 80 T^{2} + 4 T^{3} + T^{4} \)
$31$ \( -328 + 292 T - 40 T^{2} - 6 T^{3} + T^{4} \)
$37$ \( 288 - 240 T - 32 T^{2} + 10 T^{3} + T^{4} \)
$41$ \( 132 - 216 T + 98 T^{2} - 17 T^{3} + T^{4} \)
$43$ \( -16 - 28 T + 68 T^{2} - 18 T^{3} + T^{4} \)
$47$ \( 48 - 396 T - 80 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 72 + 60 T - 16 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( -321 + 198 T + 70 T^{2} - 20 T^{3} + T^{4} \)
$61$ \( 2552 + 695 T - 121 T^{2} - 9 T^{3} + T^{4} \)
$67$ \( -16 - 72 T - 68 T^{2} - 7 T^{3} + T^{4} \)
$71$ \( 66 + 159 T + 109 T^{2} + 21 T^{3} + T^{4} \)
$73$ \( -8392 + 1796 T + 14 T^{2} - 21 T^{3} + T^{4} \)
$79$ \( 5504 - 288 T - 176 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( -11061 - 3126 T - 182 T^{2} + 12 T^{3} + T^{4} \)
$89$ \( 2112 + 1728 T + 392 T^{2} + 34 T^{3} + T^{4} \)
$97$ \( 388 - 148 T - 84 T^{2} + 5 T^{3} + T^{4} \)
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