Properties

Label 546.6.a.g
Level $546$
Weight $6$
Character orbit 546.a
Self dual yes
Analytic conductor $87.570$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(87.5695656179\)
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

\(f(q)\) \(=\) \( q + 4q^{2} + 9q^{3} + 16q^{4} - 21q^{5} + 36q^{6} + 49q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} + 9q^{3} + 16q^{4} - 21q^{5} + 36q^{6} + 49q^{7} + 64q^{8} + 81q^{9} - 84q^{10} - 654q^{11} + 144q^{12} + 169q^{13} + 196q^{14} - 189q^{15} + 256q^{16} + 756q^{17} + 324q^{18} - 2905q^{19} - 336q^{20} + 441q^{21} - 2616q^{22} + 3039q^{23} + 576q^{24} - 2684q^{25} + 676q^{26} + 729q^{27} + 784q^{28} - 3957q^{29} - 756q^{30} - 3241q^{31} + 1024q^{32} - 5886q^{33} + 3024q^{34} - 1029q^{35} + 1296q^{36} - 628q^{37} - 11620q^{38} + 1521q^{39} - 1344q^{40} + 3678q^{41} + 1764q^{42} - 8797q^{43} - 10464q^{44} - 1701q^{45} + 12156q^{46} - 14163q^{47} + 2304q^{48} + 2401q^{49} - 10736q^{50} + 6804q^{51} + 2704q^{52} - 16497q^{53} + 2916q^{54} + 13734q^{55} + 3136q^{56} - 26145q^{57} - 15828q^{58} - 24528q^{59} - 3024q^{60} + 3686q^{61} - 12964q^{62} + 3969q^{63} + 4096q^{64} - 3549q^{65} - 23544q^{66} + 63818q^{67} + 12096q^{68} + 27351q^{69} - 4116q^{70} - 14040q^{71} + 5184q^{72} - 4579q^{73} - 2512q^{74} - 24156q^{75} - 46480q^{76} - 32046q^{77} + 6084q^{78} - 20713q^{79} - 5376q^{80} + 6561q^{81} + 14712q^{82} + 59421q^{83} + 7056q^{84} - 15876q^{85} - 35188q^{86} - 35613q^{87} - 41856q^{88} - 112995q^{89} - 6804q^{90} + 8281q^{91} + 48624q^{92} - 29169q^{93} - 56652q^{94} + 61005q^{95} + 9216q^{96} - 78511q^{97} + 9604q^{98} - 52974q^{99} + O(q^{100}) \)

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
4.00000 9.00000 16.0000 −21.0000 36.0000 49.0000 64.0000 81.0000 −84.0000
\(n\): 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 546.6.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.6.a.g 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} + 21 \)
\( T_{11} + 654 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -9 + T \)
$5$ \( 21 + T \)
$7$ \( -49 + T \)
$11$ \( 654 + T \)
$13$ \( -169 + T \)
$17$ \( -756 + T \)
$19$ \( 2905 + T \)
$23$ \( -3039 + T \)
$29$ \( 3957 + T \)
$31$ \( 3241 + T \)
$37$ \( 628 + T \)
$41$ \( -3678 + T \)
$43$ \( 8797 + T \)
$47$ \( 14163 + T \)
$53$ \( 16497 + T \)
$59$ \( 24528 + T \)
$61$ \( -3686 + T \)
$67$ \( -63818 + T \)
$71$ \( 14040 + T \)
$73$ \( 4579 + T \)
$79$ \( 20713 + T \)
$83$ \( -59421 + T \)
$89$ \( 112995 + T \)
$97$ \( 78511 + T \)
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