Properties

Label 462.6.a.a
Level $462$
Weight $6$
Character orbit 462.a
Self dual yes
Analytic conductor $74.097$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 78 q^{5} + 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 78 q^{5} + 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9} + 312 q^{10} + 121 q^{11} - 144 q^{12} - 502 q^{13} - 196 q^{14} + 702 q^{15} + 256 q^{16} - 642 q^{17} - 324 q^{18} - 520 q^{19} - 1248 q^{20} - 441 q^{21} - 484 q^{22} + 1020 q^{23} + 576 q^{24} + 2959 q^{25} + 2008 q^{26} - 729 q^{27} + 784 q^{28} + 4818 q^{29} - 2808 q^{30} + 1784 q^{31} - 1024 q^{32} - 1089 q^{33} + 2568 q^{34} - 3822 q^{35} + 1296 q^{36} + 7958 q^{37} + 2080 q^{38} + 4518 q^{39} + 4992 q^{40} + 2430 q^{41} + 1764 q^{42} + 22904 q^{43} + 1936 q^{44} - 6318 q^{45} - 4080 q^{46} - 11316 q^{47} - 2304 q^{48} + 2401 q^{49} - 11836 q^{50} + 5778 q^{51} - 8032 q^{52} - 12222 q^{53} + 2916 q^{54} - 9438 q^{55} - 3136 q^{56} + 4680 q^{57} - 19272 q^{58} - 15852 q^{59} + 11232 q^{60} + 46298 q^{61} - 7136 q^{62} + 3969 q^{63} + 4096 q^{64} + 39156 q^{65} + 4356 q^{66} + 19412 q^{67} - 10272 q^{68} - 9180 q^{69} + 15288 q^{70} - 17292 q^{71} - 5184 q^{72} - 30214 q^{73} - 31832 q^{74} - 26631 q^{75} - 8320 q^{76} + 5929 q^{77} - 18072 q^{78} + 35672 q^{79} - 19968 q^{80} + 6561 q^{81} - 9720 q^{82} - 43428 q^{83} - 7056 q^{84} + 50076 q^{85} - 91616 q^{86} - 43362 q^{87} - 7744 q^{88} - 14934 q^{89} + 25272 q^{90} - 24598 q^{91} + 16320 q^{92} - 16056 q^{93} + 45264 q^{94} + 40560 q^{95} + 9216 q^{96} + 85106 q^{97} - 9604 q^{98} + 9801 q^{99}+O(q^{100}) \) 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
−4.00000 −9.00000 16.0000 −78.0000 36.0000 49.0000 −64.0000 81.0000 312.000
\(n\): 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.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.6.a.a 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 78 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(462))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T + 78 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 121 \) Copy content Toggle raw display
$13$ \( T + 502 \) Copy content Toggle raw display
$17$ \( T + 642 \) Copy content Toggle raw display
$19$ \( T + 520 \) Copy content Toggle raw display
$23$ \( T - 1020 \) Copy content Toggle raw display
$29$ \( T - 4818 \) Copy content Toggle raw display
$31$ \( T - 1784 \) Copy content Toggle raw display
$37$ \( T - 7958 \) Copy content Toggle raw display
$41$ \( T - 2430 \) Copy content Toggle raw display
$43$ \( T - 22904 \) Copy content Toggle raw display
$47$ \( T + 11316 \) Copy content Toggle raw display
$53$ \( T + 12222 \) Copy content Toggle raw display
$59$ \( T + 15852 \) Copy content Toggle raw display
$61$ \( T - 46298 \) Copy content Toggle raw display
$67$ \( T - 19412 \) Copy content Toggle raw display
$71$ \( T + 17292 \) Copy content Toggle raw display
$73$ \( T + 30214 \) Copy content Toggle raw display
$79$ \( T - 35672 \) Copy content Toggle raw display
$83$ \( T + 43428 \) Copy content Toggle raw display
$89$ \( T + 14934 \) Copy content Toggle raw display
$97$ \( T - 85106 \) Copy content Toggle raw display
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