Properties

Label 546.6.a.h
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

\(f(q)\) \(=\) \( q + 4q^{2} + 9q^{3} + 16q^{4} - 16q^{5} + 36q^{6} - 49q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} + 9q^{3} + 16q^{4} - 16q^{5} + 36q^{6} - 49q^{7} + 64q^{8} + 81q^{9} - 64q^{10} - 114q^{11} + 144q^{12} - 169q^{13} - 196q^{14} - 144q^{15} + 256q^{16} + 538q^{17} + 324q^{18} - 536q^{19} - 256q^{20} - 441q^{21} - 456q^{22} - 4596q^{23} + 576q^{24} - 2869q^{25} - 676q^{26} + 729q^{27} - 784q^{28} + 1594q^{29} - 576q^{30} + 9364q^{31} + 1024q^{32} - 1026q^{33} + 2152q^{34} + 784q^{35} + 1296q^{36} - 12002q^{37} - 2144q^{38} - 1521q^{39} - 1024q^{40} + 4928q^{41} - 1764q^{42} - 14284q^{43} - 1824q^{44} - 1296q^{45} - 18384q^{46} - 22262q^{47} + 2304q^{48} + 2401q^{49} - 11476q^{50} + 4842q^{51} - 2704q^{52} - 474q^{53} + 2916q^{54} + 1824q^{55} - 3136q^{56} - 4824q^{57} + 6376q^{58} - 4182q^{59} - 2304q^{60} - 21830q^{61} + 37456q^{62} - 3969q^{63} + 4096q^{64} + 2704q^{65} - 4104q^{66} + 20780q^{67} + 8608q^{68} - 41364q^{69} + 3136q^{70} + 18682q^{71} + 5184q^{72} - 37866q^{73} - 48008q^{74} - 25821q^{75} - 8576q^{76} + 5586q^{77} - 6084q^{78} - 27840q^{79} - 4096q^{80} + 6561q^{81} + 19712q^{82} - 101914q^{83} - 7056q^{84} - 8608q^{85} - 57136q^{86} + 14346q^{87} - 7296q^{88} - 77644q^{89} - 5184q^{90} + 8281q^{91} - 73536q^{92} + 84276q^{93} - 89048q^{94} + 8576q^{95} + 9216q^{96} - 60050q^{97} + 9604q^{98} - 9234q^{99} + O(q^{100}) \)

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 −16.0000 36.0000 −49.0000 64.0000 81.0000 −64.0000
\(n\): 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.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.6.a.h 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} + 16 \)
\( T_{11} + 114 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -9 + T \)
$5$ \( 16 + T \)
$7$ \( 49 + T \)
$11$ \( 114 + T \)
$13$ \( 169 + T \)
$17$ \( -538 + T \)
$19$ \( 536 + T \)
$23$ \( 4596 + T \)
$29$ \( -1594 + T \)
$31$ \( -9364 + T \)
$37$ \( 12002 + T \)
$41$ \( -4928 + T \)
$43$ \( 14284 + T \)
$47$ \( 22262 + T \)
$53$ \( 474 + T \)
$59$ \( 4182 + T \)
$61$ \( 21830 + T \)
$67$ \( -20780 + T \)
$71$ \( -18682 + T \)
$73$ \( 37866 + T \)
$79$ \( 27840 + T \)
$83$ \( 101914 + T \)
$89$ \( 77644 + T \)
$97$ \( 60050 + T \)
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