Properties

Label 546.6.a.c
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

\(f(q)\) \(=\) \( q + 4q^{2} - 9q^{3} + 16q^{4} - 61q^{5} - 36q^{6} - 49q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} - 9q^{3} + 16q^{4} - 61q^{5} - 36q^{6} - 49q^{7} + 64q^{8} + 81q^{9} - 244q^{10} + 181q^{11} - 144q^{12} - 169q^{13} - 196q^{14} + 549q^{15} + 256q^{16} - 1965q^{17} + 324q^{18} + 2193q^{19} - 976q^{20} + 441q^{21} + 724q^{22} - 4015q^{23} - 576q^{24} + 596q^{25} - 676q^{26} - 729q^{27} - 784q^{28} - 6841q^{29} + 2196q^{30} + 8992q^{31} + 1024q^{32} - 1629q^{33} - 7860q^{34} + 2989q^{35} + 1296q^{36} - 9753q^{37} + 8772q^{38} + 1521q^{39} - 3904q^{40} - 9900q^{41} + 1764q^{42} + 13975q^{43} + 2896q^{44} - 4941q^{45} - 16060q^{46} + 18808q^{47} - 2304q^{48} + 2401q^{49} + 2384q^{50} + 17685q^{51} - 2704q^{52} + 2082q^{53} - 2916q^{54} - 11041q^{55} - 3136q^{56} - 19737q^{57} - 27364q^{58} + 31700q^{59} + 8784q^{60} + 21577q^{61} + 35968q^{62} - 3969q^{63} + 4096q^{64} + 10309q^{65} - 6516q^{66} - 49694q^{67} - 31440q^{68} + 36135q^{69} + 11956q^{70} + 53500q^{71} + 5184q^{72} + 60137q^{73} - 39012q^{74} - 5364q^{75} + 35088q^{76} - 8869q^{77} + 6084q^{78} + 17678q^{79} - 15616q^{80} + 6561q^{81} - 39600q^{82} + 80658q^{83} + 7056q^{84} + 119865q^{85} + 55900q^{86} + 61569q^{87} + 11584q^{88} + 81690q^{89} - 19764q^{90} + 8281q^{91} - 64240q^{92} - 80928q^{93} + 75232q^{94} - 133773q^{95} - 9216q^{96} + 38054q^{97} + 9604q^{98} + 14661q^{99} + O(q^{100}) \)

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 −61.0000 −36.0000 −49.0000 64.0000 81.0000 −244.000
\(n\): 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.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.6.a.c 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} + 61 \)
\( T_{11} - 181 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( 9 + T \)
$5$ \( 61 + T \)
$7$ \( 49 + T \)
$11$ \( -181 + T \)
$13$ \( 169 + T \)
$17$ \( 1965 + T \)
$19$ \( -2193 + T \)
$23$ \( 4015 + T \)
$29$ \( 6841 + T \)
$31$ \( -8992 + T \)
$37$ \( 9753 + T \)
$41$ \( 9900 + T \)
$43$ \( -13975 + T \)
$47$ \( -18808 + T \)
$53$ \( -2082 + T \)
$59$ \( -31700 + T \)
$61$ \( -21577 + T \)
$67$ \( 49694 + T \)
$71$ \( -53500 + T \)
$73$ \( -60137 + T \)
$79$ \( -17678 + T \)
$83$ \( -80658 + T \)
$89$ \( -81690 + T \)
$97$ \( -38054 + T \)
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