Properties

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

Newform invariants

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

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 3 q^{5} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + O(q^{10}) \) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + 3 q^{5} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + 6 q^{10} - 11 q^{11} - 12 q^{12} + 41 q^{13} + 14 q^{14} - 9 q^{15} + 16 q^{16} + 6 q^{17} + 18 q^{18} - 43 q^{19} + 12 q^{20} - 21 q^{21} - 22 q^{22} + 120 q^{23} - 24 q^{24} - 116 q^{25} + 82 q^{26} - 27 q^{27} + 28 q^{28} + 111 q^{29} - 18 q^{30} + 266 q^{31} + 32 q^{32} + 33 q^{33} + 12 q^{34} + 21 q^{35} + 36 q^{36} - 79 q^{37} - 86 q^{38} - 123 q^{39} + 24 q^{40} + 216 q^{41} - 42 q^{42} + 284 q^{43} - 44 q^{44} + 27 q^{45} + 240 q^{46} + 213 q^{47} - 48 q^{48} + 49 q^{49} - 232 q^{50} - 18 q^{51} + 164 q^{52} - 216 q^{53} - 54 q^{54} - 33 q^{55} + 56 q^{56} + 129 q^{57} + 222 q^{58} + 393 q^{59} - 36 q^{60} + 350 q^{61} + 532 q^{62} + 63 q^{63} + 64 q^{64} + 123 q^{65} + 66 q^{66} + 821 q^{67} + 24 q^{68} - 360 q^{69} + 42 q^{70} - 264 q^{71} + 72 q^{72} - 865 q^{73} - 158 q^{74} + 348 q^{75} - 172 q^{76} - 77 q^{77} - 246 q^{78} - 484 q^{79} + 48 q^{80} + 81 q^{81} + 432 q^{82} + 1158 q^{83} - 84 q^{84} + 18 q^{85} + 568 q^{86} - 333 q^{87} - 88 q^{88} + 330 q^{89} + 54 q^{90} + 287 q^{91} + 480 q^{92} - 798 q^{93} + 426 q^{94} - 129 q^{95} - 96 q^{96} + 980 q^{97} + 98 q^{98} - 99 q^{99} + O(q^{100}) \)

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(462))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( 3 + T \)
$5$ \( -3 + T \)
$7$ \( -7 + T \)
$11$ \( 11 + T \)
$13$ \( -41 + T \)
$17$ \( -6 + T \)
$19$ \( 43 + T \)
$23$ \( -120 + T \)
$29$ \( -111 + T \)
$31$ \( -266 + T \)
$37$ \( 79 + T \)
$41$ \( -216 + T \)
$43$ \( -284 + T \)
$47$ \( -213 + T \)
$53$ \( 216 + T \)
$59$ \( -393 + T \)
$61$ \( -350 + T \)
$67$ \( -821 + T \)
$71$ \( 264 + T \)
$73$ \( 865 + T \)
$79$ \( 484 + T \)
$83$ \( -1158 + T \)
$89$ \( -330 + T \)
$97$ \( -980 + T \)
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