Properties

Label 663.4.a.a
Level $663$
Weight $4$
Character orbit 663.a
Self dual yes
Analytic conductor $39.118$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 663.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.1182663338\)
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

\(f(q)\) \(=\) \( q + 4q^{2} + 3q^{3} + 8q^{4} - 10q^{5} + 12q^{6} - 10q^{7} + 9q^{9} + O(q^{10}) \) \( q + 4q^{2} + 3q^{3} + 8q^{4} - 10q^{5} + 12q^{6} - 10q^{7} + 9q^{9} - 40q^{10} + 18q^{11} + 24q^{12} - 13q^{13} - 40q^{14} - 30q^{15} - 64q^{16} - 17q^{17} + 36q^{18} - 74q^{19} - 80q^{20} - 30q^{21} + 72q^{22} - 132q^{23} - 25q^{25} - 52q^{26} + 27q^{27} - 80q^{28} + 210q^{29} - 120q^{30} - 230q^{31} - 256q^{32} + 54q^{33} - 68q^{34} + 100q^{35} + 72q^{36} - 46q^{37} - 296q^{38} - 39q^{39} - 114q^{41} - 120q^{42} + 36q^{43} + 144q^{44} - 90q^{45} - 528q^{46} + 446q^{47} - 192q^{48} - 243q^{49} - 100q^{50} - 51q^{51} - 104q^{52} - 754q^{53} + 108q^{54} - 180q^{55} - 222q^{57} + 840q^{58} - 50q^{59} - 240q^{60} - 226q^{61} - 920q^{62} - 90q^{63} - 512q^{64} + 130q^{65} + 216q^{66} + 582q^{67} - 136q^{68} - 396q^{69} + 400q^{70} - 370q^{71} + 826q^{73} - 184q^{74} - 75q^{75} - 592q^{76} - 180q^{77} - 156q^{78} + 272q^{79} + 640q^{80} + 81q^{81} - 456q^{82} + 162q^{83} - 240q^{84} + 170q^{85} + 144q^{86} + 630q^{87} - 186q^{89} - 360q^{90} + 130q^{91} - 1056q^{92} - 690q^{93} + 1784q^{94} + 740q^{95} - 768q^{96} - 790q^{97} - 972q^{98} + 162q^{99} + O(q^{100}) \)

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
4.00000 3.00000 8.00000 −10.0000 12.0000 −10.0000 0 9.00000 −40.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)
\(17\) \(1\)

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 663.4.a.a 1
3.b odd 2 1 1989.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.4.a.a 1 1.a even 1 1 trivial
1989.4.a.b 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(663))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -3 + T \)
$5$ \( 10 + T \)
$7$ \( 10 + T \)
$11$ \( -18 + T \)
$13$ \( 13 + T \)
$17$ \( 17 + T \)
$19$ \( 74 + T \)
$23$ \( 132 + T \)
$29$ \( -210 + T \)
$31$ \( 230 + T \)
$37$ \( 46 + T \)
$41$ \( 114 + T \)
$43$ \( -36 + T \)
$47$ \( -446 + T \)
$53$ \( 754 + T \)
$59$ \( 50 + T \)
$61$ \( 226 + T \)
$67$ \( -582 + T \)
$71$ \( 370 + T \)
$73$ \( -826 + T \)
$79$ \( -272 + T \)
$83$ \( -162 + T \)
$89$ \( 186 + T \)
$97$ \( 790 + T \)
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