Properties

Label 5390.2.a.bq
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{33}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} -2 q^{3} + q^{4} - q^{5} -2 q^{6} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} -2 q^{3} + q^{4} - q^{5} -2 q^{6} + q^{8} + q^{9} - q^{10} - q^{11} -2 q^{12} + ( -1 - \beta ) q^{13} + 2 q^{15} + q^{16} + ( 1 + \beta ) q^{17} + q^{18} + ( 1 + \beta ) q^{19} - q^{20} - q^{22} + ( -1 - \beta ) q^{23} -2 q^{24} + q^{25} + ( -1 - \beta ) q^{26} + 4 q^{27} + ( 3 + \beta ) q^{29} + 2 q^{30} + ( 1 - \beta ) q^{31} + q^{32} + 2 q^{33} + ( 1 + \beta ) q^{34} + q^{36} + ( -5 + \beta ) q^{37} + ( 1 + \beta ) q^{38} + ( 2 + 2 \beta ) q^{39} - q^{40} + 4 q^{41} + 4 q^{43} - q^{44} - q^{45} + ( -1 - \beta ) q^{46} + ( 1 - \beta ) q^{47} -2 q^{48} + q^{50} + ( -2 - 2 \beta ) q^{51} + ( -1 - \beta ) q^{52} + ( -7 - \beta ) q^{53} + 4 q^{54} + q^{55} + ( -2 - 2 \beta ) q^{57} + ( 3 + \beta ) q^{58} + ( 3 + \beta ) q^{59} + 2 q^{60} + ( -7 + \beta ) q^{61} + ( 1 - \beta ) q^{62} + q^{64} + ( 1 + \beta ) q^{65} + 2 q^{66} -4 q^{67} + ( 1 + \beta ) q^{68} + ( 2 + 2 \beta ) q^{69} -4 q^{71} + q^{72} + ( -5 - \beta ) q^{73} + ( -5 + \beta ) q^{74} -2 q^{75} + ( 1 + \beta ) q^{76} + ( 2 + 2 \beta ) q^{78} + ( 1 + \beta ) q^{79} - q^{80} -11 q^{81} + 4 q^{82} -8 q^{83} + ( -1 - \beta ) q^{85} + 4 q^{86} + ( -6 - 2 \beta ) q^{87} - q^{88} + ( -4 - 2 \beta ) q^{89} - q^{90} + ( -1 - \beta ) q^{92} + ( -2 + 2 \beta ) q^{93} + ( 1 - \beta ) q^{94} + ( -1 - \beta ) q^{95} -2 q^{96} + ( 11 + \beta ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} - 4q^{6} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} - 4q^{6} + 2q^{8} + 2q^{9} - 2q^{10} - 2q^{11} - 4q^{12} - 2q^{13} + 4q^{15} + 2q^{16} + 2q^{17} + 2q^{18} + 2q^{19} - 2q^{20} - 2q^{22} - 2q^{23} - 4q^{24} + 2q^{25} - 2q^{26} + 8q^{27} + 6q^{29} + 4q^{30} + 2q^{31} + 2q^{32} + 4q^{33} + 2q^{34} + 2q^{36} - 10q^{37} + 2q^{38} + 4q^{39} - 2q^{40} + 8q^{41} + 8q^{43} - 2q^{44} - 2q^{45} - 2q^{46} + 2q^{47} - 4q^{48} + 2q^{50} - 4q^{51} - 2q^{52} - 14q^{53} + 8q^{54} + 2q^{55} - 4q^{57} + 6q^{58} + 6q^{59} + 4q^{60} - 14q^{61} + 2q^{62} + 2q^{64} + 2q^{65} + 4q^{66} - 8q^{67} + 2q^{68} + 4q^{69} - 8q^{71} + 2q^{72} - 10q^{73} - 10q^{74} - 4q^{75} + 2q^{76} + 4q^{78} + 2q^{79} - 2q^{80} - 22q^{81} + 8q^{82} - 16q^{83} - 2q^{85} + 8q^{86} - 12q^{87} - 2q^{88} - 8q^{89} - 2q^{90} - 2q^{92} - 4q^{93} + 2q^{94} - 2q^{95} - 4q^{96} + 22q^{97} - 2q^{99} + O(q^{100}) \)

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
3.37228
−2.37228
1.00000 −2.00000 1.00000 −1.00000 −2.00000 0 1.00000 1.00000 −1.00000
1.2 1.00000 −2.00000 1.00000 −1.00000 −2.00000 0 1.00000 1.00000 −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.bq 2
7.b odd 2 1 770.2.a.k 2
21.c even 2 1 6930.2.a.bo 2
28.d even 2 1 6160.2.a.r 2
35.c odd 2 1 3850.2.a.bc 2
35.f even 4 2 3850.2.c.y 4
77.b even 2 1 8470.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.k 2 7.b odd 2 1
3850.2.a.bc 2 35.c odd 2 1
3850.2.c.y 4 35.f even 4 2
5390.2.a.bq 2 1.a even 1 1 trivial
6160.2.a.r 2 28.d even 2 1
6930.2.a.bo 2 21.c even 2 1
8470.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5390))\):

\( T_{3} + 2 \)
\( T_{13}^{2} + 2 T_{13} - 32 \)
\( T_{17}^{2} - 2 T_{17} - 32 \)
\( T_{19}^{2} - 2 T_{19} - 32 \)
\( T_{31}^{2} - 2 T_{31} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 2 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -32 + 2 T + T^{2} \)
$17$ \( -32 - 2 T + T^{2} \)
$19$ \( -32 - 2 T + T^{2} \)
$23$ \( -32 + 2 T + T^{2} \)
$29$ \( -24 - 6 T + T^{2} \)
$31$ \( -32 - 2 T + T^{2} \)
$37$ \( -8 + 10 T + T^{2} \)
$41$ \( ( -4 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -32 - 2 T + T^{2} \)
$53$ \( 16 + 14 T + T^{2} \)
$59$ \( -24 - 6 T + T^{2} \)
$61$ \( 16 + 14 T + T^{2} \)
$67$ \( ( 4 + T )^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( -8 + 10 T + T^{2} \)
$79$ \( -32 - 2 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( -116 + 8 T + T^{2} \)
$97$ \( 88 - 22 T + T^{2} \)
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