Properties

Label 462.6.a.c
Level $462$
Weight $6$
Character orbit 462.a
Self dual yes
Analytic conductor $74.097$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(74.0973247536\)
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

\(f(q)\) \(=\) \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} + 106 q^{5} + 36 q^{6} - 49 q^{7} - 64 q^{8} + 81 q^{9} + O(q^{10}) \) \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} + 106 q^{5} + 36 q^{6} - 49 q^{7} - 64 q^{8} + 81 q^{9} - 424 q^{10} + 121 q^{11} - 144 q^{12} - 166 q^{13} + 196 q^{14} - 954 q^{15} + 256 q^{16} - 842 q^{17} - 324 q^{18} + 1200 q^{19} + 1696 q^{20} + 441 q^{21} - 484 q^{22} + 1724 q^{23} + 576 q^{24} + 8111 q^{25} + 664 q^{26} - 729 q^{27} - 784 q^{28} + 1010 q^{29} + 3816 q^{30} - 1328 q^{31} - 1024 q^{32} - 1089 q^{33} + 3368 q^{34} - 5194 q^{35} + 1296 q^{36} + 15718 q^{37} - 4800 q^{38} + 1494 q^{39} - 6784 q^{40} - 7498 q^{41} - 1764 q^{42} - 2456 q^{43} + 1936 q^{44} + 8586 q^{45} - 6896 q^{46} - 2252 q^{47} - 2304 q^{48} + 2401 q^{49} - 32444 q^{50} + 7578 q^{51} - 2656 q^{52} + 15714 q^{53} + 2916 q^{54} + 12826 q^{55} + 3136 q^{56} - 10800 q^{57} - 4040 q^{58} - 5340 q^{59} - 15264 q^{60} - 25718 q^{61} + 5312 q^{62} - 3969 q^{63} + 4096 q^{64} - 17596 q^{65} + 4356 q^{66} - 4572 q^{67} - 13472 q^{68} - 15516 q^{69} + 20776 q^{70} - 67708 q^{71} - 5184 q^{72} - 7406 q^{73} - 62872 q^{74} - 72999 q^{75} + 19200 q^{76} - 5929 q^{77} - 5976 q^{78} + 25560 q^{79} + 27136 q^{80} + 6561 q^{81} + 29992 q^{82} + 22404 q^{83} + 7056 q^{84} - 89252 q^{85} + 9824 q^{86} - 9090 q^{87} - 7744 q^{88} + 61530 q^{89} - 34344 q^{90} + 8134 q^{91} + 27584 q^{92} + 11952 q^{93} + 9008 q^{94} + 127200 q^{95} + 9216 q^{96} - 72782 q^{97} - 9604 q^{98} + 9801 q^{99} + O(q^{100}) \)

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
−4.00000 −9.00000 16.0000 106.000 36.0000 −49.0000 −64.0000 81.0000 −424.000
\(n\): 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 462.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.6.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 106 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(462))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( 9 + T \)
$5$ \( -106 + T \)
$7$ \( 49 + T \)
$11$ \( -121 + T \)
$13$ \( 166 + T \)
$17$ \( 842 + T \)
$19$ \( -1200 + T \)
$23$ \( -1724 + T \)
$29$ \( -1010 + T \)
$31$ \( 1328 + T \)
$37$ \( -15718 + T \)
$41$ \( 7498 + T \)
$43$ \( 2456 + T \)
$47$ \( 2252 + T \)
$53$ \( -15714 + T \)
$59$ \( 5340 + T \)
$61$ \( 25718 + T \)
$67$ \( 4572 + T \)
$71$ \( 67708 + T \)
$73$ \( 7406 + T \)
$79$ \( -25560 + T \)
$83$ \( -22404 + T \)
$89$ \( -61530 + T \)
$97$ \( 72782 + T \)
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