Properties

Label 546.2.a.e
Level $546$
Weight $2$
Character orbit 546.a
Self dual yes
Analytic conductor $4.360$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + 3q^{10} + q^{11} - q^{12} - q^{13} - q^{14} - 3q^{15} + q^{16} + 7q^{17} + q^{18} + q^{19} + 3q^{20} + q^{21} + q^{22} - 7q^{23} - q^{24} + 4q^{25} - q^{26} - q^{27} - q^{28} + 3q^{29} - 3q^{30} + q^{32} - q^{33} + 7q^{34} - 3q^{35} + q^{36} - 5q^{37} + q^{38} + q^{39} + 3q^{40} + 4q^{41} + q^{42} + 11q^{43} + q^{44} + 3q^{45} - 7q^{46} - q^{48} + q^{49} + 4q^{50} - 7q^{51} - q^{52} - 14q^{53} - q^{54} + 3q^{55} - q^{56} - q^{57} + 3q^{58} + 4q^{59} - 3q^{60} + q^{61} - q^{63} + q^{64} - 3q^{65} - q^{66} - 6q^{67} + 7q^{68} + 7q^{69} - 3q^{70} - 12q^{71} + q^{72} + 5q^{73} - 5q^{74} - 4q^{75} + q^{76} - q^{77} + q^{78} - 10q^{79} + 3q^{80} + q^{81} + 4q^{82} - 14q^{83} + q^{84} + 21q^{85} + 11q^{86} - 3q^{87} + q^{88} - 6q^{89} + 3q^{90} + q^{91} - 7q^{92} + 3q^{95} - q^{96} + 6q^{97} + q^{98} + q^{99} + O(q^{100}) \)

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
1.00000 −1.00000 1.00000 3.00000 −1.00000 −1.00000 1.00000 1.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(13\) \(1\)

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 546.2.a.e 1
3.b odd 2 1 1638.2.a.a 1
4.b odd 2 1 4368.2.a.z 1
7.b odd 2 1 3822.2.a.bc 1
13.b even 2 1 7098.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.e 1 1.a even 1 1 trivial
1638.2.a.a 1 3.b odd 2 1
3822.2.a.bc 1 7.b odd 2 1
4368.2.a.z 1 4.b odd 2 1
7098.2.a.b 1 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} - 3 \)
\( T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( -3 + T \)
$7$ \( 1 + T \)
$11$ \( -1 + T \)
$13$ \( 1 + T \)
$17$ \( -7 + T \)
$19$ \( -1 + T \)
$23$ \( 7 + T \)
$29$ \( -3 + T \)
$31$ \( T \)
$37$ \( 5 + T \)
$41$ \( -4 + T \)
$43$ \( -11 + T \)
$47$ \( T \)
$53$ \( 14 + T \)
$59$ \( -4 + T \)
$61$ \( -1 + T \)
$67$ \( 6 + T \)
$71$ \( 12 + T \)
$73$ \( -5 + T \)
$79$ \( 10 + T \)
$83$ \( 14 + T \)
$89$ \( 6 + T \)
$97$ \( -6 + T \)
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