Properties

Label 546.8.a.c
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

\(f(q)\) \(=\) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} - 10 q^{5} + 216 q^{6} - 343 q^{7} + 512 q^{8} + 729 q^{9} + O(q^{10}) \) \( q + 8 q^{2} + 27 q^{3} + 64 q^{4} - 10 q^{5} + 216 q^{6} - 343 q^{7} + 512 q^{8} + 729 q^{9} - 80 q^{10} + 1508 q^{11} + 1728 q^{12} + 2197 q^{13} - 2744 q^{14} - 270 q^{15} + 4096 q^{16} - 1042 q^{17} + 5832 q^{18} - 9068 q^{19} - 640 q^{20} - 9261 q^{21} + 12064 q^{22} - 98988 q^{23} + 13824 q^{24} - 78025 q^{25} + 17576 q^{26} + 19683 q^{27} - 21952 q^{28} - 213642 q^{29} - 2160 q^{30} - 22048 q^{31} + 32768 q^{32} + 40716 q^{33} - 8336 q^{34} + 3430 q^{35} + 46656 q^{36} + 418246 q^{37} - 72544 q^{38} + 59319 q^{39} - 5120 q^{40} - 76414 q^{41} - 74088 q^{42} - 177524 q^{43} + 96512 q^{44} - 7290 q^{45} - 791904 q^{46} + 631916 q^{47} + 110592 q^{48} + 117649 q^{49} - 624200 q^{50} - 28134 q^{51} + 140608 q^{52} - 982354 q^{53} + 157464 q^{54} - 15080 q^{55} - 175616 q^{56} - 244836 q^{57} - 1709136 q^{58} - 596384 q^{59} - 17280 q^{60} + 1863406 q^{61} - 176384 q^{62} - 250047 q^{63} + 262144 q^{64} - 21970 q^{65} + 325728 q^{66} - 1845652 q^{67} - 66688 q^{68} - 2672676 q^{69} + 27440 q^{70} + 1632280 q^{71} + 373248 q^{72} + 216650 q^{73} + 3345968 q^{74} - 2106675 q^{75} - 580352 q^{76} - 517244 q^{77} + 474552 q^{78} - 6460168 q^{79} - 40960 q^{80} + 531441 q^{81} - 611312 q^{82} + 3592152 q^{83} - 592704 q^{84} + 10420 q^{85} - 1420192 q^{86} - 5768334 q^{87} + 772096 q^{88} - 9482662 q^{89} - 58320 q^{90} - 753571 q^{91} - 6335232 q^{92} - 595296 q^{93} + 5055328 q^{94} + 90680 q^{95} + 884736 q^{96} + 840226 q^{97} + 941192 q^{98} + 1099332 q^{99} + O(q^{100}) \)

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 −10.0000 216.000 −343.000 512.000 729.000 −80.0000
\(n\): 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 546.8.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 10 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -8 + T \)
$3$ \( -27 + T \)
$5$ \( 10 + T \)
$7$ \( 343 + T \)
$11$ \( -1508 + T \)
$13$ \( -2197 + T \)
$17$ \( 1042 + T \)
$19$ \( 9068 + T \)
$23$ \( 98988 + T \)
$29$ \( 213642 + T \)
$31$ \( 22048 + T \)
$37$ \( -418246 + T \)
$41$ \( 76414 + T \)
$43$ \( 177524 + T \)
$47$ \( -631916 + T \)
$53$ \( 982354 + T \)
$59$ \( 596384 + T \)
$61$ \( -1863406 + T \)
$67$ \( 1845652 + T \)
$71$ \( -1632280 + T \)
$73$ \( -216650 + T \)
$79$ \( 6460168 + T \)
$83$ \( -3592152 + T \)
$89$ \( 9482662 + T \)
$97$ \( -840226 + T \)
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