Properties

Label 462.4.a.f
Level $462$
Weight $4$
Character orbit 462.a
Self dual yes
Analytic conductor $27.259$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.2588824227\)
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

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + q^{5} - 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9} + O(q^{10}) \) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + q^{5} - 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9} + 2 q^{10} - 11 q^{11} - 12 q^{12} - 43 q^{13} - 14 q^{14} - 3 q^{15} + 16 q^{16} + 100 q^{17} + 18 q^{18} - 87 q^{19} + 4 q^{20} + 21 q^{21} - 22 q^{22} - 58 q^{23} - 24 q^{24} - 124 q^{25} - 86 q^{26} - 27 q^{27} - 28 q^{28} - 223 q^{29} - 6 q^{30} + 88 q^{31} + 32 q^{32} + 33 q^{33} + 200 q^{34} - 7 q^{35} + 36 q^{36} + 37 q^{37} - 174 q^{38} + 129 q^{39} + 8 q^{40} + 128 q^{41} + 42 q^{42} - 458 q^{43} - 44 q^{44} + 9 q^{45} - 116 q^{46} - 341 q^{47} - 48 q^{48} + 49 q^{49} - 248 q^{50} - 300 q^{51} - 172 q^{52} - 342 q^{53} - 54 q^{54} - 11 q^{55} - 56 q^{56} + 261 q^{57} - 446 q^{58} - 105 q^{59} - 12 q^{60} + 190 q^{61} + 176 q^{62} - 63 q^{63} + 64 q^{64} - 43 q^{65} + 66 q^{66} - 579 q^{67} + 400 q^{68} + 174 q^{69} - 14 q^{70} + 128 q^{71} + 72 q^{72} - 161 q^{73} + 74 q^{74} + 372 q^{75} - 348 q^{76} + 77 q^{77} + 258 q^{78} - 396 q^{79} + 16 q^{80} + 81 q^{81} + 256 q^{82} - 420 q^{83} + 84 q^{84} + 100 q^{85} - 916 q^{86} + 669 q^{87} - 88 q^{88} - 798 q^{89} + 18 q^{90} + 301 q^{91} - 232 q^{92} - 264 q^{93} - 682 q^{94} - 87 q^{95} - 96 q^{96} + 1414 q^{97} + 98 q^{98} - 99 q^{99} + O(q^{100}) \)

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
2.00000 −3.00000 4.00000 1.00000 −6.00000 −7.00000 8.00000 9.00000 2.00000
\(n\): 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 462.4.a.f 1
3.b odd 2 1 1386.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.f 1 1.a even 1 1 trivial
1386.4.a.d 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(462))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 3 + T \)
$5$ \( -1 + T \)
$7$ \( 7 + T \)
$11$ \( 11 + T \)
$13$ \( 43 + T \)
$17$ \( -100 + T \)
$19$ \( 87 + T \)
$23$ \( 58 + T \)
$29$ \( 223 + T \)
$31$ \( -88 + T \)
$37$ \( -37 + T \)
$41$ \( -128 + T \)
$43$ \( 458 + T \)
$47$ \( 341 + T \)
$53$ \( 342 + T \)
$59$ \( 105 + T \)
$61$ \( -190 + T \)
$67$ \( 579 + T \)
$71$ \( -128 + T \)
$73$ \( 161 + T \)
$79$ \( 396 + T \)
$83$ \( 420 + T \)
$89$ \( 798 + T \)
$97$ \( -1414 + T \)
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