Properties

Label 546.4.a.a
Level $546$
Weight $4$
Character orbit 546.a
Self dual yes
Analytic conductor $32.215$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.2150428631\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - 3q^{3} + 4q^{4} + 12q^{5} + 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} - 3q^{3} + 4q^{4} + 12q^{5} + 6q^{6} + 7q^{7} - 8q^{8} + 9q^{9} - 24q^{10} - 22q^{11} - 12q^{12} - 13q^{13} - 14q^{14} - 36q^{15} + 16q^{16} - 2q^{17} - 18q^{18} - 88q^{19} + 48q^{20} - 21q^{21} + 44q^{22} - 80q^{23} + 24q^{24} + 19q^{25} + 26q^{26} - 27q^{27} + 28q^{28} - 22q^{29} + 72q^{30} - 92q^{31} - 32q^{32} + 66q^{33} + 4q^{34} + 84q^{35} + 36q^{36} + 118q^{37} + 176q^{38} + 39q^{39} - 96q^{40} + 324q^{41} + 42q^{42} + 84q^{43} - 88q^{44} + 108q^{45} + 160q^{46} - 134q^{47} - 48q^{48} + 49q^{49} - 38q^{50} + 6q^{51} - 52q^{52} - 194q^{53} + 54q^{54} - 264q^{55} - 56q^{56} + 264q^{57} + 44q^{58} + 210q^{59} - 144q^{60} - 470q^{61} + 184q^{62} + 63q^{63} + 64q^{64} - 156q^{65} - 132q^{66} - 292q^{67} - 8q^{68} + 240q^{69} - 168q^{70} - 66q^{71} - 72q^{72} - 506q^{73} - 236q^{74} - 57q^{75} - 352q^{76} - 154q^{77} - 78q^{78} - 776q^{79} + 192q^{80} + 81q^{81} - 648q^{82} - 778q^{83} - 84q^{84} - 24q^{85} - 168q^{86} + 66q^{87} + 176q^{88} - 920q^{89} - 216q^{90} - 91q^{91} - 320q^{92} + 276q^{93} + 268q^{94} - 1056q^{95} + 96q^{96} - 490q^{97} - 98q^{98} - 198q^{99} + O(q^{100}) \)

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
−2.00000 −3.00000 4.00000 12.0000 6.00000 7.00000 −8.00000 9.00000 −24.0000
\(n\): 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 546.4.a.a 1
3.b odd 2 1 1638.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.a.a 1 1.a even 1 1 trivial
1638.4.a.i 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 12 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 3 + T \)
$5$ \( -12 + T \)
$7$ \( -7 + T \)
$11$ \( 22 + T \)
$13$ \( 13 + T \)
$17$ \( 2 + T \)
$19$ \( 88 + T \)
$23$ \( 80 + T \)
$29$ \( 22 + T \)
$31$ \( 92 + T \)
$37$ \( -118 + T \)
$41$ \( -324 + T \)
$43$ \( -84 + T \)
$47$ \( 134 + T \)
$53$ \( 194 + T \)
$59$ \( -210 + T \)
$61$ \( 470 + T \)
$67$ \( 292 + T \)
$71$ \( 66 + T \)
$73$ \( 506 + T \)
$79$ \( 776 + T \)
$83$ \( 778 + T \)
$89$ \( 920 + T \)
$97$ \( 490 + T \)
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