Properties

Label 462.4.a.e
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

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9} - 8 q^{10} - 11 q^{11} - 12 q^{12} + 62 q^{13} - 14 q^{14} + 12 q^{15} + 16 q^{16} - 120 q^{17} + 18 q^{18} + 118 q^{19} - 16 q^{20} + 21 q^{21} - 22 q^{22} - 188 q^{23} - 24 q^{24} - 109 q^{25} + 124 q^{26} - 27 q^{27} - 28 q^{28} + 62 q^{29} + 24 q^{30} - 322 q^{31} + 32 q^{32} + 33 q^{33} - 240 q^{34} + 28 q^{35} + 36 q^{36} - 198 q^{37} + 236 q^{38} - 186 q^{39} - 32 q^{40} + 48 q^{41} + 42 q^{42} + 32 q^{43} - 44 q^{44} - 36 q^{45} - 376 q^{46} - 326 q^{47} - 48 q^{48} + 49 q^{49} - 218 q^{50} + 360 q^{51} + 248 q^{52} - 482 q^{53} - 54 q^{54} + 44 q^{55} - 56 q^{56} - 354 q^{57} + 124 q^{58} + 400 q^{59} + 48 q^{60} + 70 q^{61} - 644 q^{62} - 63 q^{63} + 64 q^{64} - 248 q^{65} + 66 q^{66} - 124 q^{67} - 480 q^{68} + 564 q^{69} + 56 q^{70} - 712 q^{71} + 72 q^{72} + 304 q^{73} - 396 q^{74} + 327 q^{75} + 472 q^{76} + 77 q^{77} - 372 q^{78} - 1016 q^{79} - 64 q^{80} + 81 q^{81} + 96 q^{82} + 430 q^{83} + 84 q^{84} + 480 q^{85} + 64 q^{86} - 186 q^{87} - 88 q^{88} + 442 q^{89} - 72 q^{90} - 434 q^{91} - 752 q^{92} + 966 q^{93} - 652 q^{94} - 472 q^{95} - 96 q^{96} - 966 q^{97} + 98 q^{98} - 99 q^{99}+O(q^{100}) \) 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
2.00000 −3.00000 4.00000 −4.00000 −6.00000 −7.00000 8.00000 9.00000 −8.00000
\(n\): 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.e 1
3.b odd 2 1 1386.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.e 1 1.a even 1 1 trivial
1386.4.a.e 1 3.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T - 62 \) Copy content Toggle raw display
$17$ \( T + 120 \) Copy content Toggle raw display
$19$ \( T - 118 \) Copy content Toggle raw display
$23$ \( T + 188 \) Copy content Toggle raw display
$29$ \( T - 62 \) Copy content Toggle raw display
$31$ \( T + 322 \) Copy content Toggle raw display
$37$ \( T + 198 \) Copy content Toggle raw display
$41$ \( T - 48 \) Copy content Toggle raw display
$43$ \( T - 32 \) Copy content Toggle raw display
$47$ \( T + 326 \) Copy content Toggle raw display
$53$ \( T + 482 \) Copy content Toggle raw display
$59$ \( T - 400 \) Copy content Toggle raw display
$61$ \( T - 70 \) Copy content Toggle raw display
$67$ \( T + 124 \) Copy content Toggle raw display
$71$ \( T + 712 \) Copy content Toggle raw display
$73$ \( T - 304 \) Copy content Toggle raw display
$79$ \( T + 1016 \) Copy content Toggle raw display
$83$ \( T - 430 \) Copy content Toggle raw display
$89$ \( T - 442 \) Copy content Toggle raw display
$97$ \( T + 966 \) Copy content Toggle raw display
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