Properties

Label 546.6.a.d
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

\(f(q)\) \(=\) \( q + 4q^{2} - 9q^{3} + 16q^{4} + 33q^{5} - 36q^{6} + 49q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} - 9q^{3} + 16q^{4} + 33q^{5} - 36q^{6} + 49q^{7} + 64q^{8} + 81q^{9} + 132q^{10} - 94q^{11} - 144q^{12} - 169q^{13} + 196q^{14} - 297q^{15} + 256q^{16} - 1112q^{17} + 324q^{18} - 1909q^{19} + 528q^{20} - 441q^{21} - 376q^{22} + 961q^{23} - 576q^{24} - 2036q^{25} - 676q^{26} - 729q^{27} + 784q^{28} - 6337q^{29} - 1188q^{30} + 7015q^{31} + 1024q^{32} + 846q^{33} - 4448q^{34} + 1617q^{35} + 1296q^{36} + 928q^{37} - 7636q^{38} + 1521q^{39} + 2112q^{40} + 11442q^{41} - 1764q^{42} - 12711q^{43} - 1504q^{44} + 2673q^{45} + 3844q^{46} - 15107q^{47} - 2304q^{48} + 2401q^{49} - 8144q^{50} + 10008q^{51} - 2704q^{52} + 18691q^{53} - 2916q^{54} - 3102q^{55} + 3136q^{56} + 17181q^{57} - 25348q^{58} - 12360q^{59} - 4752q^{60} + 14110q^{61} + 28060q^{62} + 3969q^{63} + 4096q^{64} - 5577q^{65} + 3384q^{66} - 53746q^{67} - 17792q^{68} - 8649q^{69} + 6468q^{70} - 47748q^{71} + 5184q^{72} - 25301q^{73} + 3712q^{74} + 18324q^{75} - 30544q^{76} - 4606q^{77} + 6084q^{78} - 5447q^{79} + 8448q^{80} + 6561q^{81} + 45768q^{82} + 29393q^{83} - 7056q^{84} - 36696q^{85} - 50844q^{86} + 57033q^{87} - 6016q^{88} - 77621q^{89} + 10692q^{90} - 8281q^{91} + 15376q^{92} - 63135q^{93} - 60428q^{94} - 62997q^{95} - 9216q^{96} + 73607q^{97} + 9604q^{98} - 7614q^{99} + O(q^{100}) \)

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 33.0000 −36.0000 49.0000 64.0000 81.0000 132.000
\(n\): 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.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.6.a.d 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\):

\( T_{5} - 33 \)
\( T_{11} + 94 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( 9 + T \)
$5$ \( -33 + T \)
$7$ \( -49 + T \)
$11$ \( 94 + T \)
$13$ \( 169 + T \)
$17$ \( 1112 + T \)
$19$ \( 1909 + T \)
$23$ \( -961 + T \)
$29$ \( 6337 + T \)
$31$ \( -7015 + T \)
$37$ \( -928 + T \)
$41$ \( -11442 + T \)
$43$ \( 12711 + T \)
$47$ \( 15107 + T \)
$53$ \( -18691 + T \)
$59$ \( 12360 + T \)
$61$ \( -14110 + T \)
$67$ \( 53746 + T \)
$71$ \( 47748 + T \)
$73$ \( 25301 + T \)
$79$ \( 5447 + T \)
$83$ \( -29393 + T \)
$89$ \( 77621 + T \)
$97$ \( -73607 + T \)
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