Properties

Label 546.6.a.f
Level $546$
Weight $6$
Character orbit 546.a
Self dual yes
Analytic conductor $87.570$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \( q + 4q^{2} + 9q^{3} + 16q^{4} - 54q^{5} + 36q^{6} + 49q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} + 9q^{3} + 16q^{4} - 54q^{5} + 36q^{6} + 49q^{7} + 64q^{8} + 81q^{9} - 216q^{10} - 192q^{11} + 144q^{12} + 169q^{13} + 196q^{14} - 486q^{15} + 256q^{16} - 1422q^{17} + 324q^{18} + 1748q^{19} - 864q^{20} + 441q^{21} - 768q^{22} - 3792q^{23} + 576q^{24} - 209q^{25} + 676q^{26} + 729q^{27} + 784q^{28} - 954q^{29} - 1944q^{30} - 568q^{31} + 1024q^{32} - 1728q^{33} - 5688q^{34} - 2646q^{35} + 1296q^{36} + 7886q^{37} + 6992q^{38} + 1521q^{39} - 3456q^{40} - 14802q^{41} + 1764q^{42} + 4964q^{43} - 3072q^{44} - 4374q^{45} - 15168q^{46} - 18948q^{47} + 2304q^{48} + 2401q^{49} - 836q^{50} - 12798q^{51} + 2704q^{52} - 426q^{53} + 2916q^{54} + 10368q^{55} + 3136q^{56} + 15732q^{57} - 3816q^{58} + 34872q^{59} - 7776q^{60} - 25618q^{61} - 2272q^{62} + 3969q^{63} + 4096q^{64} - 9126q^{65} - 6912q^{66} - 67060q^{67} - 22752q^{68} - 34128q^{69} - 10584q^{70} - 28428q^{71} + 5184q^{72} - 22894q^{73} + 31544q^{74} - 1881q^{75} + 27968q^{76} - 9408q^{77} + 6084q^{78} - 1408q^{79} - 13824q^{80} + 6561q^{81} - 59208q^{82} - 17304q^{83} + 7056q^{84} + 76788q^{85} + 19856q^{86} - 8586q^{87} - 12288q^{88} - 93690q^{89} - 17496q^{90} + 8281q^{91} - 60672q^{92} - 5112q^{93} - 75792q^{94} - 94392q^{95} + 9216q^{96} + 16826q^{97} + 9604q^{98} - 15552q^{99} + O(q^{100}) \)

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 −54.0000 36.0000 49.0000 64.0000 81.0000 −216.000
\(n\): 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.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.6.a.f 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} + 54 \)
\( T_{11} + 192 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -9 + T \)
$5$ \( 54 + T \)
$7$ \( -49 + T \)
$11$ \( 192 + T \)
$13$ \( -169 + T \)
$17$ \( 1422 + T \)
$19$ \( -1748 + T \)
$23$ \( 3792 + T \)
$29$ \( 954 + T \)
$31$ \( 568 + T \)
$37$ \( -7886 + T \)
$41$ \( 14802 + T \)
$43$ \( -4964 + T \)
$47$ \( 18948 + T \)
$53$ \( 426 + T \)
$59$ \( -34872 + T \)
$61$ \( 25618 + T \)
$67$ \( 67060 + T \)
$71$ \( 28428 + T \)
$73$ \( 22894 + T \)
$79$ \( 1408 + T \)
$83$ \( 17304 + T \)
$89$ \( 93690 + T \)
$97$ \( -16826 + T \)
show more
show less