Properties

Label 546.2.a.g
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + 2q^{10} - 4q^{11} + q^{12} - q^{13} + q^{14} + 2q^{15} + q^{16} + 6q^{17} + q^{18} - 4q^{19} + 2q^{20} + q^{21} - 4q^{22} + q^{24} - q^{25} - q^{26} + q^{27} + q^{28} - 6q^{29} + 2q^{30} - 8q^{31} + q^{32} - 4q^{33} + 6q^{34} + 2q^{35} + q^{36} + 10q^{37} - 4q^{38} - q^{39} + 2q^{40} - 6q^{41} + q^{42} + 4q^{43} - 4q^{44} + 2q^{45} + 4q^{47} + q^{48} + q^{49} - q^{50} + 6q^{51} - q^{52} + 10q^{53} + q^{54} - 8q^{55} + q^{56} - 4q^{57} - 6q^{58} + 4q^{59} + 2q^{60} - 6q^{61} - 8q^{62} + q^{63} + q^{64} - 2q^{65} - 4q^{66} - 8q^{67} + 6q^{68} + 2q^{70} + q^{72} - 10q^{73} + 10q^{74} - q^{75} - 4q^{76} - 4q^{77} - q^{78} - 8q^{79} + 2q^{80} + q^{81} - 6q^{82} + 4q^{83} + q^{84} + 12q^{85} + 4q^{86} - 6q^{87} - 4q^{88} - 6q^{89} + 2q^{90} - q^{91} - 8q^{93} + 4q^{94} - 8q^{95} + q^{96} - 2q^{97} + q^{98} - 4q^{99} + O(q^{100}) \)

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 2.00000 1.00000 1.00000 1.00000 1.00000 2.00000
\(n\): 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.g 1
3.b odd 2 1 1638.2.a.d 1
4.b odd 2 1 4368.2.a.k 1
7.b odd 2 1 3822.2.a.t 1
13.b even 2 1 7098.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.g 1 1.a even 1 1 trivial
1638.2.a.d 1 3.b odd 2 1
3822.2.a.t 1 7.b odd 2 1
4368.2.a.k 1 4.b odd 2 1
7098.2.a.j 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} - 2 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( -2 + T \)
$7$ \( -1 + T \)
$11$ \( 4 + T \)
$13$ \( 1 + T \)
$17$ \( -6 + T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( 8 + T \)
$37$ \( -10 + T \)
$41$ \( 6 + T \)
$43$ \( -4 + T \)
$47$ \( -4 + T \)
$53$ \( -10 + T \)
$59$ \( -4 + T \)
$61$ \( 6 + T \)
$67$ \( 8 + T \)
$71$ \( T \)
$73$ \( 10 + T \)
$79$ \( 8 + T \)
$83$ \( -4 + T \)
$89$ \( 6 + T \)
$97$ \( 2 + T \)
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