Properties

Label 462.8.a.a
Level $462$
Weight $8$
Character orbit 462.a
Self dual yes
Analytic conductor $144.322$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(144.321881774\)
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

\(f(q)\) \(=\) \( q - 8 q^{2} - 27 q^{3} + 64 q^{4} + 182 q^{5} + 216 q^{6} + 343 q^{7} - 512 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} - 27 q^{3} + 64 q^{4} + 182 q^{5} + 216 q^{6} + 343 q^{7} - 512 q^{8} + 729 q^{9} - 1456 q^{10} + 1331 q^{11} - 1728 q^{12} + 13758 q^{13} - 2744 q^{14} - 4914 q^{15} + 4096 q^{16} + 14738 q^{17} - 5832 q^{18} + 34516 q^{19} + 11648 q^{20} - 9261 q^{21} - 10648 q^{22} + 49816 q^{23} + 13824 q^{24} - 45001 q^{25} - 110064 q^{26} - 19683 q^{27} + 21952 q^{28} + 141174 q^{29} + 39312 q^{30} - 86112 q^{31} - 32768 q^{32} - 35937 q^{33} - 117904 q^{34} + 62426 q^{35} + 46656 q^{36} + 535038 q^{37} - 276128 q^{38} - 371466 q^{39} - 93184 q^{40} + 708282 q^{41} + 74088 q^{42} + 162292 q^{43} + 85184 q^{44} + 132678 q^{45} - 398528 q^{46} + 397296 q^{47} - 110592 q^{48} + 117649 q^{49} + 360008 q^{50} - 397926 q^{51} + 880512 q^{52} + 224798 q^{53} + 157464 q^{54} + 242242 q^{55} - 175616 q^{56} - 931932 q^{57} - 1129392 q^{58} + 694668 q^{59} - 314496 q^{60} + 50110 q^{61} + 688896 q^{62} + 250047 q^{63} + 262144 q^{64} + 2503956 q^{65} + 287496 q^{66} - 3228404 q^{67} + 943232 q^{68} - 1345032 q^{69} - 499408 q^{70} - 1871496 q^{71} - 373248 q^{72} - 937542 q^{73} - 4280304 q^{74} + 1215027 q^{75} + 2209024 q^{76} + 456533 q^{77} + 2971728 q^{78} + 1266032 q^{79} + 745472 q^{80} + 531441 q^{81} - 5666256 q^{82} - 2545004 q^{83} - 592704 q^{84} + 2682316 q^{85} - 1298336 q^{86} - 3811698 q^{87} - 681472 q^{88} - 9909830 q^{89} - 1061424 q^{90} + 4718994 q^{91} + 3188224 q^{92} + 2325024 q^{93} - 3178368 q^{94} + 6281912 q^{95} + 884736 q^{96} - 14010830 q^{97} - 941192 q^{98} + 970299 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 182.000 216.000 343.000 −512.000 729.000 −1456.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 462.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.8.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 182 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(462))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T + 27 \) Copy content Toggle raw display
$5$ \( T - 182 \) Copy content Toggle raw display
$7$ \( T - 343 \) Copy content Toggle raw display
$11$ \( T - 1331 \) Copy content Toggle raw display
$13$ \( T - 13758 \) Copy content Toggle raw display
$17$ \( T - 14738 \) Copy content Toggle raw display
$19$ \( T - 34516 \) Copy content Toggle raw display
$23$ \( T - 49816 \) Copy content Toggle raw display
$29$ \( T - 141174 \) Copy content Toggle raw display
$31$ \( T + 86112 \) Copy content Toggle raw display
$37$ \( T - 535038 \) Copy content Toggle raw display
$41$ \( T - 708282 \) Copy content Toggle raw display
$43$ \( T - 162292 \) Copy content Toggle raw display
$47$ \( T - 397296 \) Copy content Toggle raw display
$53$ \( T - 224798 \) Copy content Toggle raw display
$59$ \( T - 694668 \) Copy content Toggle raw display
$61$ \( T - 50110 \) Copy content Toggle raw display
$67$ \( T + 3228404 \) Copy content Toggle raw display
$71$ \( T + 1871496 \) Copy content Toggle raw display
$73$ \( T + 937542 \) Copy content Toggle raw display
$79$ \( T - 1266032 \) Copy content Toggle raw display
$83$ \( T + 2545004 \) Copy content Toggle raw display
$89$ \( T + 9909830 \) Copy content Toggle raw display
$97$ \( T + 14010830 \) Copy content Toggle raw display
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