Properties

Label 5054.2.a.o
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
1.00000 −0.618034 1.00000 2.85410 −0.618034 −1.00000 1.00000 −2.61803 2.85410
1.2 1.00000 1.61803 1.00000 −3.85410 1.61803 −1.00000 1.00000 −0.381966 −3.85410
\(n\): 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.o yes 2
19.b odd 2 1 5054.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.d 2 19.b odd 2 1
5054.2.a.o yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -1 - T + T^{2} \)
$5$ \( -11 + T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -1 - T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( -4 + 2 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( -11 - 9 T + T^{2} \)
$31$ \( -4 - 8 T + T^{2} \)
$37$ \( -19 - 7 T + T^{2} \)
$41$ \( 19 + 11 T + T^{2} \)
$43$ \( 5 - 5 T + T^{2} \)
$47$ \( 89 - 19 T + T^{2} \)
$53$ \( -31 + 11 T + T^{2} \)
$59$ \( 109 - 21 T + T^{2} \)
$61$ \( -95 - 5 T + T^{2} \)
$67$ \( 20 - 10 T + T^{2} \)
$71$ \( -95 + 5 T + T^{2} \)
$73$ \( -64 - 8 T + T^{2} \)
$79$ \( -1 - 11 T + T^{2} \)
$83$ \( 4 - 14 T + T^{2} \)
$89$ \( -49 - 7 T + T^{2} \)
$97$ \( 19 - 9 T + T^{2} \)
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