Properties

Label 462.4.a.b
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

\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 14 q^{5} + 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9} + O(q^{10}) \) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 14 q^{5} + 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9} + 28 q^{10} + 11 q^{11} - 12 q^{12} + 38 q^{13} + 14 q^{14} + 42 q^{15} + 16 q^{16} + 54 q^{17} - 18 q^{18} + 40 q^{19} - 56 q^{20} + 21 q^{21} - 22 q^{22} + 8 q^{23} + 24 q^{24} + 71 q^{25} - 76 q^{26} - 27 q^{27} - 28 q^{28} - 170 q^{29} - 84 q^{30} + 92 q^{31} - 32 q^{32} - 33 q^{33} - 108 q^{34} + 98 q^{35} + 36 q^{36} + 294 q^{37} - 80 q^{38} - 114 q^{39} + 112 q^{40} - 258 q^{41} - 42 q^{42} - 52 q^{43} + 44 q^{44} - 126 q^{45} - 16 q^{46} - 76 q^{47} - 48 q^{48} + 49 q^{49} - 142 q^{50} - 162 q^{51} + 152 q^{52} - 322 q^{53} + 54 q^{54} - 154 q^{55} + 56 q^{56} - 120 q^{57} + 340 q^{58} + 260 q^{59} + 168 q^{60} + 22 q^{61} - 184 q^{62} - 63 q^{63} + 64 q^{64} - 532 q^{65} + 66 q^{66} - 436 q^{67} + 216 q^{68} - 24 q^{69} - 196 q^{70} - 368 q^{71} - 72 q^{72} - 2 q^{73} - 588 q^{74} - 213 q^{75} + 160 q^{76} - 77 q^{77} + 228 q^{78} - 200 q^{79} - 224 q^{80} + 81 q^{81} + 516 q^{82} - 952 q^{83} + 84 q^{84} - 756 q^{85} + 104 q^{86} + 510 q^{87} - 88 q^{88} - 70 q^{89} + 252 q^{90} - 266 q^{91} + 32 q^{92} - 276 q^{93} + 152 q^{94} - 560 q^{95} + 96 q^{96} - 1086 q^{97} - 98 q^{98} + 99 q^{99} + O(q^{100}) \)

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 −14.0000 6.00000 −7.00000 −8.00000 9.00000 28.0000
\(n\): 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.b 1
3.b odd 2 1 1386.4.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.b 1 1.a even 1 1 trivial
1386.4.a.m 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(462))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( 3 + T \)
$5$ \( 14 + T \)
$7$ \( 7 + T \)
$11$ \( -11 + T \)
$13$ \( -38 + T \)
$17$ \( -54 + T \)
$19$ \( -40 + T \)
$23$ \( -8 + T \)
$29$ \( 170 + T \)
$31$ \( -92 + T \)
$37$ \( -294 + T \)
$41$ \( 258 + T \)
$43$ \( 52 + T \)
$47$ \( 76 + T \)
$53$ \( 322 + T \)
$59$ \( -260 + T \)
$61$ \( -22 + T \)
$67$ \( 436 + T \)
$71$ \( 368 + T \)
$73$ \( 2 + T \)
$79$ \( 200 + T \)
$83$ \( 952 + T \)
$89$ \( 70 + T \)
$97$ \( 1086 + T \)
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