Properties

Label 5054.2.a.y
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: \(\Q(\zeta_{20})^+\)
Defining polynomial: \(x^{4} - 5 x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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
−1.90211
−1.17557
1.17557
1.90211
1.00000 −0.902113 1.00000 0.557537 −0.902113 1.00000 1.00000 −2.18619 0.557537
1.2 1.00000 −0.175571 1.00000 −0.284079 −0.175571 1.00000 1.00000 −2.96917 −0.284079
1.3 1.00000 2.17557 1.00000 3.52015 2.17557 1.00000 1.00000 1.73311 3.52015
1.4 1.00000 2.90211 1.00000 −1.79360 2.90211 1.00000 1.00000 5.42226 −1.79360
\(n\): 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.y yes 4
19.b odd 2 1 5054.2.a.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.v 4 19.b odd 2 1
5054.2.a.y yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\( T_{3}^{4} - 4 T_{3}^{3} + T_{3}^{2} + 6 T_{3} + 1 \)
\( T_{5}^{4} - 2 T_{5}^{3} - 6 T_{5}^{2} + 2 T_{5} + 1 \)
\( T_{13}^{4} + 2 T_{13}^{3} - 11 T_{13}^{2} + 8 T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 1 + 6 T + T^{2} - 4 T^{3} + T^{4} \)
$5$ \( 1 + 2 T - 6 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( ( -1 + T )^{4} \)
$11$ \( 61 + 36 T - 14 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( 1 + 8 T - 11 T^{2} + 2 T^{3} + T^{4} \)
$17$ \( -19 + 32 T - 11 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( 61 + 36 T - 14 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( -19 + 12 T + 34 T^{2} - 12 T^{3} + T^{4} \)
$31$ \( -239 + 206 T + 26 T^{2} - 14 T^{3} + T^{4} \)
$37$ \( -1559 - 104 T + 166 T^{2} - 24 T^{3} + T^{4} \)
$41$ \( 205 - 310 T - 95 T^{2} + T^{4} \)
$43$ \( -179 + 66 T + 31 T^{2} - 14 T^{3} + T^{4} \)
$47$ \( -719 - 602 T - 126 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( 905 + 160 T - 90 T^{2} + T^{4} \)
$59$ \( 121 - 198 T - 86 T^{2} - 2 T^{3} + T^{4} \)
$61$ \( 3401 + 162 T - 126 T^{2} - 2 T^{3} + T^{4} \)
$67$ \( -12119 + 2222 T + 19 T^{2} - 22 T^{3} + T^{4} \)
$71$ \( -944 + 816 T - 164 T^{2} - 4 T^{3} + T^{4} \)
$73$ \( 1805 + 760 T - 115 T^{2} - 10 T^{3} + T^{4} \)
$79$ \( 2221 - 658 T - 241 T^{2} - 2 T^{3} + T^{4} \)
$83$ \( 9680 - 200 T^{2} + T^{4} \)
$89$ \( 1205 + 40 T - 115 T^{2} - 10 T^{3} + T^{4} \)
$97$ \( -16739 - 3542 T - 66 T^{2} + 22 T^{3} + T^{4} \)
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