Properties

Label 60.6.a.b
Level $60$
Weight $6$
Character orbit 60.a
Self dual yes
Analytic conductor $9.623$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.62302918878\)
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

\(f(q)\) \(=\) \( q - 9q^{3} + 25q^{5} - 16q^{7} + 81q^{9} + O(q^{10}) \) \( q - 9q^{3} + 25q^{5} - 16q^{7} + 81q^{9} - 564q^{11} - 370q^{13} - 225q^{15} - 1086q^{17} - 2860q^{19} + 144q^{21} + 1584q^{23} + 625q^{25} - 729q^{27} + 1134q^{29} - 6016q^{31} + 5076q^{33} - 400q^{35} - 538q^{37} + 3330q^{39} + 11370q^{41} + 5444q^{43} + 2025q^{45} + 10296q^{47} - 16551q^{49} + 9774q^{51} + 34758q^{53} - 14100q^{55} + 25740q^{57} - 26196q^{59} + 9422q^{61} - 1296q^{63} - 9250q^{65} - 51124q^{67} - 14256q^{69} + 14520q^{71} - 22678q^{73} - 5625q^{75} + 9024q^{77} - 97312q^{79} + 6561q^{81} - 7956q^{83} - 27150q^{85} - 10206q^{87} - 47910q^{89} + 5920q^{91} + 54144q^{93} - 71500q^{95} + 140738q^{97} - 45684q^{99} + O(q^{100}) \)

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
0 −9.00000 0 25.0000 0 −16.0000 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)

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 60.6.a.b 1
3.b odd 2 1 180.6.a.a 1
4.b odd 2 1 240.6.a.m 1
5.b even 2 1 300.6.a.e 1
5.c odd 4 2 300.6.d.a 2
8.b even 2 1 960.6.a.r 1
8.d odd 2 1 960.6.a.c 1
12.b even 2 1 720.6.a.f 1
15.d odd 2 1 900.6.a.g 1
15.e even 4 2 900.6.d.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.a.b 1 1.a even 1 1 trivial
180.6.a.a 1 3.b odd 2 1
240.6.a.m 1 4.b odd 2 1
300.6.a.e 1 5.b even 2 1
300.6.d.a 2 5.c odd 4 2
720.6.a.f 1 12.b even 2 1
900.6.a.g 1 15.d odd 2 1
900.6.d.i 2 15.e even 4 2
960.6.a.c 1 8.d odd 2 1
960.6.a.r 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 16 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(60))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 9 + T \)
$5$ \( -25 + T \)
$7$ \( 16 + T \)
$11$ \( 564 + T \)
$13$ \( 370 + T \)
$17$ \( 1086 + T \)
$19$ \( 2860 + T \)
$23$ \( -1584 + T \)
$29$ \( -1134 + T \)
$31$ \( 6016 + T \)
$37$ \( 538 + T \)
$41$ \( -11370 + T \)
$43$ \( -5444 + T \)
$47$ \( -10296 + T \)
$53$ \( -34758 + T \)
$59$ \( 26196 + T \)
$61$ \( -9422 + T \)
$67$ \( 51124 + T \)
$71$ \( -14520 + T \)
$73$ \( 22678 + T \)
$79$ \( 97312 + T \)
$83$ \( 7956 + T \)
$89$ \( 47910 + T \)
$97$ \( -140738 + T \)
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