Properties

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

Newform invariants

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

\(f(q)\) \(=\) \( q - 4q^{2} + 9q^{3} + 16q^{4} + 81q^{5} - 36q^{6} - 49q^{7} - 64q^{8} + 81q^{9} + O(q^{10}) \) \( q - 4q^{2} + 9q^{3} + 16q^{4} + 81q^{5} - 36q^{6} - 49q^{7} - 64q^{8} + 81q^{9} - 324q^{10} + 191q^{11} + 144q^{12} - 169q^{13} + 196q^{14} + 729q^{15} + 256q^{16} - 871q^{17} - 324q^{18} - 479q^{19} + 1296q^{20} - 441q^{21} - 764q^{22} + 1387q^{23} - 576q^{24} + 3436q^{25} + 676q^{26} + 729q^{27} - 784q^{28} - 5295q^{29} - 2916q^{30} + 5940q^{31} - 1024q^{32} + 1719q^{33} + 3484q^{34} - 3969q^{35} + 1296q^{36} + 13543q^{37} + 1916q^{38} - 1521q^{39} - 5184q^{40} + 9464q^{41} + 1764q^{42} + 17387q^{43} + 3056q^{44} + 6561q^{45} - 5548q^{46} - 8112q^{47} + 2304q^{48} + 2401q^{49} - 13744q^{50} - 7839q^{51} - 2704q^{52} + 18038q^{53} - 2916q^{54} + 15471q^{55} + 3136q^{56} - 4311q^{57} + 21180q^{58} + 28784q^{59} + 11664q^{60} + 14773q^{61} - 23760q^{62} - 3969q^{63} + 4096q^{64} - 13689q^{65} - 6876q^{66} - 54354q^{67} - 13936q^{68} + 12483q^{69} + 15876q^{70} + 64608q^{71} - 5184q^{72} + 39461q^{73} - 54172q^{74} + 30924q^{75} - 7664q^{76} - 9359q^{77} + 6084q^{78} - 95554q^{79} + 20736q^{80} + 6561q^{81} - 37856q^{82} - 69634q^{83} - 7056q^{84} - 70551q^{85} - 69548q^{86} - 47655q^{87} - 12224q^{88} - 51906q^{89} - 26244q^{90} + 8281q^{91} + 22192q^{92} + 53460q^{93} + 32448q^{94} - 38799q^{95} - 9216q^{96} + 162654q^{97} - 9604q^{98} + 15471q^{99} + O(q^{100}) \)

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 81.0000 −36.0000 −49.0000 −64.0000 81.0000 −324.000
\(n\): 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.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.6.a.a 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} - 81 \)
\( T_{11} - 191 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( -9 + T \)
$5$ \( -81 + T \)
$7$ \( 49 + T \)
$11$ \( -191 + T \)
$13$ \( 169 + T \)
$17$ \( 871 + T \)
$19$ \( 479 + T \)
$23$ \( -1387 + T \)
$29$ \( 5295 + T \)
$31$ \( -5940 + T \)
$37$ \( -13543 + T \)
$41$ \( -9464 + T \)
$43$ \( -17387 + T \)
$47$ \( 8112 + T \)
$53$ \( -18038 + T \)
$59$ \( -28784 + T \)
$61$ \( -14773 + T \)
$67$ \( 54354 + T \)
$71$ \( -64608 + T \)
$73$ \( -39461 + T \)
$79$ \( 95554 + T \)
$83$ \( 69634 + T \)
$89$ \( 51906 + T \)
$97$ \( -162654 + T \)
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