Properties

Label 462.4.a.h
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

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} - 17 q^{5} + 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} - 17 q^{5} + 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} - 34 q^{10} - 11 q^{11} + 12 q^{12} - 21 q^{13} + 14 q^{14} - 51 q^{15} + 16 q^{16} - 104 q^{17} + 18 q^{18} - 161 q^{19} - 68 q^{20} + 21 q^{21} - 22 q^{22} + 194 q^{23} + 24 q^{24} + 164 q^{25} - 42 q^{26} + 27 q^{27} + 28 q^{28} + 9 q^{29} - 102 q^{30} - 180 q^{31} + 32 q^{32} - 33 q^{33} - 208 q^{34} - 119 q^{35} + 36 q^{36} - 363 q^{37} - 322 q^{38} - 63 q^{39} - 136 q^{40} - 108 q^{41} + 42 q^{42} - 386 q^{43} - 44 q^{44} - 153 q^{45} + 388 q^{46} + 333 q^{47} + 48 q^{48} + 49 q^{49} + 328 q^{50} - 312 q^{51} - 84 q^{52} - 122 q^{53} + 54 q^{54} + 187 q^{55} + 56 q^{56} - 483 q^{57} + 18 q^{58} + 537 q^{59} - 204 q^{60} - 950 q^{61} - 360 q^{62} + 63 q^{63} + 64 q^{64} + 357 q^{65} - 66 q^{66} - 83 q^{67} - 416 q^{68} + 582 q^{69} - 238 q^{70} + 180 q^{71} + 72 q^{72} + 177 q^{73} - 726 q^{74} + 492 q^{75} - 644 q^{76} - 77 q^{77} - 126 q^{78} - 220 q^{79} - 272 q^{80} + 81 q^{81} - 216 q^{82} + 1112 q^{83} + 84 q^{84} + 1768 q^{85} - 772 q^{86} + 27 q^{87} - 88 q^{88} - 394 q^{89} - 306 q^{90} - 147 q^{91} + 776 q^{92} - 540 q^{93} + 666 q^{94} + 2737 q^{95} + 96 q^{96} + 826 q^{97} + 98 q^{98} - 99 q^{99}+O(q^{100}) \) 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 −17.0000 6.00000 7.00000 8.00000 9.00000 −34.0000
\(n\): 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.h 1
3.b odd 2 1 1386.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.4.a.h 1 1.a even 1 1 trivial
1386.4.a.h 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 17 \) 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
$5$ \( T + 17 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 21 \) Copy content Toggle raw display
$17$ \( T + 104 \) Copy content Toggle raw display
$19$ \( T + 161 \) Copy content Toggle raw display
$23$ \( T - 194 \) Copy content Toggle raw display
$29$ \( T - 9 \) Copy content Toggle raw display
$31$ \( T + 180 \) Copy content Toggle raw display
$37$ \( T + 363 \) Copy content Toggle raw display
$41$ \( T + 108 \) Copy content Toggle raw display
$43$ \( T + 386 \) Copy content Toggle raw display
$47$ \( T - 333 \) Copy content Toggle raw display
$53$ \( T + 122 \) Copy content Toggle raw display
$59$ \( T - 537 \) Copy content Toggle raw display
$61$ \( T + 950 \) Copy content Toggle raw display
$67$ \( T + 83 \) Copy content Toggle raw display
$71$ \( T - 180 \) Copy content Toggle raw display
$73$ \( T - 177 \) Copy content Toggle raw display
$79$ \( T + 220 \) Copy content Toggle raw display
$83$ \( T - 1112 \) Copy content Toggle raw display
$89$ \( T + 394 \) Copy content Toggle raw display
$97$ \( T - 826 \) Copy content Toggle raw display
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