Properties

Label 546.8.a.a
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
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: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
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

\(f(q)\) \(=\) \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} - 135 q^{5} - 216 q^{6} + 343 q^{7} + 512 q^{8} + 729 q^{9} + O(q^{10}) \) \( q + 8 q^{2} - 27 q^{3} + 64 q^{4} - 135 q^{5} - 216 q^{6} + 343 q^{7} + 512 q^{8} + 729 q^{9} - 1080 q^{10} + 4197 q^{11} - 1728 q^{12} + 2197 q^{13} + 2744 q^{14} + 3645 q^{15} + 4096 q^{16} - 12735 q^{17} + 5832 q^{18} - 28213 q^{19} - 8640 q^{20} - 9261 q^{21} + 33576 q^{22} - 57039 q^{23} - 13824 q^{24} - 59900 q^{25} + 17576 q^{26} - 19683 q^{27} + 21952 q^{28} - 10269 q^{29} + 29160 q^{30} + 98276 q^{31} + 32768 q^{32} - 113319 q^{33} - 101880 q^{34} - 46305 q^{35} + 46656 q^{36} - 352033 q^{37} - 225704 q^{38} - 59319 q^{39} - 69120 q^{40} + 473172 q^{41} - 74088 q^{42} + 891395 q^{43} + 268608 q^{44} - 98415 q^{45} - 456312 q^{46} + 684984 q^{47} - 110592 q^{48} + 117649 q^{49} - 479200 q^{50} + 343845 q^{51} + 140608 q^{52} + 271002 q^{53} - 157464 q^{54} - 566595 q^{55} + 175616 q^{56} + 761751 q^{57} - 82152 q^{58} - 954024 q^{59} + 233280 q^{60} + 3197159 q^{61} + 786208 q^{62} + 250047 q^{63} + 262144 q^{64} - 296595 q^{65} - 906552 q^{66} - 2902018 q^{67} - 815040 q^{68} + 1540053 q^{69} - 370440 q^{70} - 4599768 q^{71} + 373248 q^{72} + 1277111 q^{73} - 2816264 q^{74} + 1617300 q^{75} - 1805632 q^{76} + 1439571 q^{77} - 474552 q^{78} - 1928494 q^{79} - 552960 q^{80} + 531441 q^{81} + 3785376 q^{82} - 355674 q^{83} - 592704 q^{84} + 1719225 q^{85} + 7131160 q^{86} + 277263 q^{87} + 2148864 q^{88} - 217854 q^{89} - 787320 q^{90} + 753571 q^{91} - 3650496 q^{92} - 2653452 q^{93} + 5479872 q^{94} + 3808755 q^{95} - 884736 q^{96} - 4400758 q^{97} + 941192 q^{98} + 3059613 q^{99} + O(q^{100}) \)

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
8.00000 −27.0000 64.0000 −135.000 −216.000 343.000 512.000 729.000 −1080.00
\(n\): 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 546.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 135 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -8 + T \)
$3$ \( 27 + T \)
$5$ \( 135 + T \)
$7$ \( -343 + T \)
$11$ \( -4197 + T \)
$13$ \( -2197 + T \)
$17$ \( 12735 + T \)
$19$ \( 28213 + T \)
$23$ \( 57039 + T \)
$29$ \( 10269 + T \)
$31$ \( -98276 + T \)
$37$ \( 352033 + T \)
$41$ \( -473172 + T \)
$43$ \( -891395 + T \)
$47$ \( -684984 + T \)
$53$ \( -271002 + T \)
$59$ \( 954024 + T \)
$61$ \( -3197159 + T \)
$67$ \( 2902018 + T \)
$71$ \( 4599768 + T \)
$73$ \( -1277111 + T \)
$79$ \( 1928494 + T \)
$83$ \( 355674 + T \)
$89$ \( 217854 + T \)
$97$ \( 4400758 + T \)
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