Properties

Label 5390.2.a.bk
Level $5390$
Weight $2$
Character orbit 5390.a
Self dual yes
Analytic conductor $43.039$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.0393666895\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( -1 + \beta ) q^{3} + q^{4} - q^{5} + ( 1 - \beta ) q^{6} - q^{8} + ( 1 - 2 \beta ) q^{9} +O(q^{10})\) \( q - q^{2} + ( -1 + \beta ) q^{3} + q^{4} - q^{5} + ( 1 - \beta ) q^{6} - q^{8} + ( 1 - 2 \beta ) q^{9} + q^{10} + q^{11} + ( -1 + \beta ) q^{12} + ( -2 - 2 \beta ) q^{13} + ( 1 - \beta ) q^{15} + q^{16} + 2 \beta q^{17} + ( -1 + 2 \beta ) q^{18} + ( 1 - \beta ) q^{19} - q^{20} - q^{22} + ( 3 + \beta ) q^{23} + ( 1 - \beta ) q^{24} + q^{25} + ( 2 + 2 \beta ) q^{26} -4 q^{27} + ( -3 + \beta ) q^{29} + ( -1 + \beta ) q^{30} + ( -2 + 4 \beta ) q^{31} - q^{32} + ( -1 + \beta ) q^{33} -2 \beta q^{34} + ( 1 - 2 \beta ) q^{36} + ( 5 + \beta ) q^{37} + ( -1 + \beta ) q^{38} -4 q^{39} + q^{40} + ( 3 - \beta ) q^{41} + ( 2 + 4 \beta ) q^{43} + q^{44} + ( -1 + 2 \beta ) q^{45} + ( -3 - \beta ) q^{46} + ( -1 + \beta ) q^{48} - q^{50} + ( 6 - 2 \beta ) q^{51} + ( -2 - 2 \beta ) q^{52} + ( -3 + \beta ) q^{53} + 4 q^{54} - q^{55} + ( -4 + 2 \beta ) q^{57} + ( 3 - \beta ) q^{58} -8 \beta q^{59} + ( 1 - \beta ) q^{60} -2 q^{61} + ( 2 - 4 \beta ) q^{62} + q^{64} + ( 2 + 2 \beta ) q^{65} + ( 1 - \beta ) q^{66} + ( -4 + 4 \beta ) q^{67} + 2 \beta q^{68} + 2 \beta q^{69} + ( 6 - 2 \beta ) q^{71} + ( -1 + 2 \beta ) q^{72} + ( -8 + 2 \beta ) q^{73} + ( -5 - \beta ) q^{74} + ( -1 + \beta ) q^{75} + ( 1 - \beta ) q^{76} + 4 q^{78} + ( 5 - \beta ) q^{79} - q^{80} + ( 1 + 2 \beta ) q^{81} + ( -3 + \beta ) q^{82} + ( -6 - 6 \beta ) q^{83} -2 \beta q^{85} + ( -2 - 4 \beta ) q^{86} + ( 6 - 4 \beta ) q^{87} - q^{88} + ( 12 - 2 \beta ) q^{89} + ( 1 - 2 \beta ) q^{90} + ( 3 + \beta ) q^{92} + ( 14 - 6 \beta ) q^{93} + ( -1 + \beta ) q^{95} + ( 1 - \beta ) q^{96} + ( -11 - 3 \beta ) q^{97} + ( 1 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 2 q^{16} - 2 q^{18} + 2 q^{19} - 2 q^{20} - 2 q^{22} + 6 q^{23} + 2 q^{24} + 2 q^{25} + 4 q^{26} - 8 q^{27} - 6 q^{29} - 2 q^{30} - 4 q^{31} - 2 q^{32} - 2 q^{33} + 2 q^{36} + 10 q^{37} - 2 q^{38} - 8 q^{39} + 2 q^{40} + 6 q^{41} + 4 q^{43} + 2 q^{44} - 2 q^{45} - 6 q^{46} - 2 q^{48} - 2 q^{50} + 12 q^{51} - 4 q^{52} - 6 q^{53} + 8 q^{54} - 2 q^{55} - 8 q^{57} + 6 q^{58} + 2 q^{60} - 4 q^{61} + 4 q^{62} + 2 q^{64} + 4 q^{65} + 2 q^{66} - 8 q^{67} + 12 q^{71} - 2 q^{72} - 16 q^{73} - 10 q^{74} - 2 q^{75} + 2 q^{76} + 8 q^{78} + 10 q^{79} - 2 q^{80} + 2 q^{81} - 6 q^{82} - 12 q^{83} - 4 q^{86} + 12 q^{87} - 2 q^{88} + 24 q^{89} + 2 q^{90} + 6 q^{92} + 28 q^{93} - 2 q^{95} + 2 q^{96} - 22 q^{97} + 2 q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
−1.00000 −2.73205 1.00000 −1.00000 2.73205 0 −1.00000 4.46410 1.00000
1.2 −1.00000 0.732051 1.00000 −1.00000 −0.732051 0 −1.00000 −2.46410 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 5390.2.a.bk 2
7.b odd 2 1 770.2.a.h 2
21.c even 2 1 6930.2.a.ca 2
28.d even 2 1 6160.2.a.v 2
35.c odd 2 1 3850.2.a.bm 2
35.f even 4 2 3850.2.c.s 4
77.b even 2 1 8470.2.a.ce 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.h 2 7.b odd 2 1
3850.2.a.bm 2 35.c odd 2 1
3850.2.c.s 4 35.f even 4 2
5390.2.a.bk 2 1.a even 1 1 trivial
6160.2.a.v 2 28.d even 2 1
6930.2.a.ca 2 21.c even 2 1
8470.2.a.ce 2 77.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5390))\):

\( T_{3}^{2} + 2 T_{3} - 2 \)
\( T_{13}^{2} + 4 T_{13} - 8 \)
\( T_{17}^{2} - 12 \)
\( T_{19}^{2} - 2 T_{19} - 2 \)
\( T_{31}^{2} + 4 T_{31} - 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -2 + 2 T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( -2 - 2 T + T^{2} \)
$23$ \( 6 - 6 T + T^{2} \)
$29$ \( 6 + 6 T + T^{2} \)
$31$ \( -44 + 4 T + T^{2} \)
$37$ \( 22 - 10 T + T^{2} \)
$41$ \( 6 - 6 T + T^{2} \)
$43$ \( -44 - 4 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 6 + 6 T + T^{2} \)
$59$ \( -192 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -32 + 8 T + T^{2} \)
$71$ \( 24 - 12 T + T^{2} \)
$73$ \( 52 + 16 T + T^{2} \)
$79$ \( 22 - 10 T + T^{2} \)
$83$ \( -72 + 12 T + T^{2} \)
$89$ \( 132 - 24 T + T^{2} \)
$97$ \( 94 + 22 T + T^{2} \)
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