Properties

Label 546.8.a.b
Level $546$
Weight $8$
Character orbit 546.a
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + 8q^{2} + 27q^{3} + 64q^{4} - 390q^{5} + 216q^{6} - 343q^{7} + 512q^{8} + 729q^{9} + O(q^{10}) \) \( q + 8q^{2} + 27q^{3} + 64q^{4} - 390q^{5} + 216q^{6} - 343q^{7} + 512q^{8} + 729q^{9} - 3120q^{10} - 388q^{11} + 1728q^{12} - 2197q^{13} - 2744q^{14} - 10530q^{15} + 4096q^{16} - 37294q^{17} + 5832q^{18} + 35164q^{19} - 24960q^{20} - 9261q^{21} - 3104q^{22} - 102980q^{23} + 13824q^{24} + 73975q^{25} - 17576q^{26} + 19683q^{27} - 21952q^{28} + 224826q^{29} - 84240q^{30} - 150552q^{31} + 32768q^{32} - 10476q^{33} - 298352q^{34} + 133770q^{35} + 46656q^{36} + 306058q^{37} + 281312q^{38} - 59319q^{39} - 199680q^{40} + 784994q^{41} - 74088q^{42} - 771532q^{43} - 24832q^{44} - 284310q^{45} - 823840q^{46} + 653976q^{47} + 110592q^{48} + 117649q^{49} + 591800q^{50} - 1006938q^{51} - 140608q^{52} - 6646q^{53} + 157464q^{54} + 151320q^{55} - 175616q^{56} + 949428q^{57} + 1798608q^{58} + 1376600q^{59} - 673920q^{60} - 1215494q^{61} - 1204416q^{62} - 250047q^{63} + 262144q^{64} + 856830q^{65} - 83808q^{66} - 3041808q^{67} - 2386816q^{68} - 2780460q^{69} + 1070160q^{70} + 611256q^{71} + 373248q^{72} + 3531686q^{73} + 2448464q^{74} + 1997325q^{75} + 2250496q^{76} + 133084q^{77} - 474552q^{78} - 1351792q^{79} - 1597440q^{80} + 531441q^{81} + 6279952q^{82} - 4882216q^{83} - 592704q^{84} + 14544660q^{85} - 6172256q^{86} + 6070302q^{87} - 198656q^{88} + 6893754q^{89} - 2274480q^{90} + 753571q^{91} - 6590720q^{92} - 4064904q^{93} + 5231808q^{94} - 13713960q^{95} + 884736q^{96} + 3768126q^{97} + 941192q^{98} - 282852q^{99} + O(q^{100}) \)

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 −390.000 216.000 −343.000 512.000 729.000 −3120.00
\(n\): 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.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.b 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 390 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -8 + T \)
$3$ \( -27 + T \)
$5$ \( 390 + T \)
$7$ \( 343 + T \)
$11$ \( 388 + T \)
$13$ \( 2197 + T \)
$17$ \( 37294 + T \)
$19$ \( -35164 + T \)
$23$ \( 102980 + T \)
$29$ \( -224826 + T \)
$31$ \( 150552 + T \)
$37$ \( -306058 + T \)
$41$ \( -784994 + T \)
$43$ \( 771532 + T \)
$47$ \( -653976 + T \)
$53$ \( 6646 + T \)
$59$ \( -1376600 + T \)
$61$ \( 1215494 + T \)
$67$ \( 3041808 + T \)
$71$ \( -611256 + T \)
$73$ \( -3531686 + T \)
$79$ \( 1351792 + T \)
$83$ \( 4882216 + T \)
$89$ \( -6893754 + T \)
$97$ \( -3768126 + T \)
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