Properties

Label 60.4.a.b
Level $60$
Weight $4$
Character orbit 60.a
Self dual yes
Analytic conductor $3.540$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.54011460034\)
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

\(f(q)\) \(=\) \( q - 3q^{3} + 5q^{5} + 32q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} + 5q^{5} + 32q^{7} + 9q^{9} + 36q^{11} - 10q^{13} - 15q^{15} - 78q^{17} + 140q^{19} - 96q^{21} - 192q^{23} + 25q^{25} - 27q^{27} + 6q^{29} - 16q^{31} - 108q^{33} + 160q^{35} - 34q^{37} + 30q^{39} - 390q^{41} - 52q^{43} + 45q^{45} + 408q^{47} + 681q^{49} + 234q^{51} - 114q^{53} + 180q^{55} - 420q^{57} + 516q^{59} - 58q^{61} + 288q^{63} - 50q^{65} - 892q^{67} + 576q^{69} - 120q^{71} - 646q^{73} - 75q^{75} + 1152q^{77} - 1168q^{79} + 81q^{81} - 732q^{83} - 390q^{85} - 18q^{87} - 1590q^{89} - 320q^{91} + 48q^{93} + 700q^{95} + 194q^{97} + 324q^{99} + O(q^{100}) \)

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 −3.00000 0 5.00000 0 32.0000 0 9.00000 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 60.4.a.b 1
3.b odd 2 1 180.4.a.c 1
4.b odd 2 1 240.4.a.j 1
5.b even 2 1 300.4.a.e 1
5.c odd 4 2 300.4.d.d 2
8.b even 2 1 960.4.a.bb 1
8.d odd 2 1 960.4.a.a 1
9.c even 3 2 1620.4.i.a 2
9.d odd 6 2 1620.4.i.g 2
12.b even 2 1 720.4.a.c 1
15.d odd 2 1 900.4.a.b 1
15.e even 4 2 900.4.d.b 2
20.d odd 2 1 1200.4.a.s 1
20.e even 4 2 1200.4.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.b 1 1.a even 1 1 trivial
180.4.a.c 1 3.b odd 2 1
240.4.a.j 1 4.b odd 2 1
300.4.a.e 1 5.b even 2 1
300.4.d.d 2 5.c odd 4 2
720.4.a.c 1 12.b even 2 1
900.4.a.b 1 15.d odd 2 1
900.4.d.b 2 15.e even 4 2
960.4.a.a 1 8.d odd 2 1
960.4.a.bb 1 8.b even 2 1
1200.4.a.s 1 20.d odd 2 1
1200.4.f.e 2 20.e even 4 2
1620.4.i.a 2 9.c even 3 2
1620.4.i.g 2 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 32 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(60))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( -5 + T \)
$7$ \( -32 + T \)
$11$ \( -36 + T \)
$13$ \( 10 + T \)
$17$ \( 78 + T \)
$19$ \( -140 + T \)
$23$ \( 192 + T \)
$29$ \( -6 + T \)
$31$ \( 16 + T \)
$37$ \( 34 + T \)
$41$ \( 390 + T \)
$43$ \( 52 + T \)
$47$ \( -408 + T \)
$53$ \( 114 + T \)
$59$ \( -516 + T \)
$61$ \( 58 + T \)
$67$ \( 892 + T \)
$71$ \( 120 + T \)
$73$ \( 646 + T \)
$79$ \( 1168 + T \)
$83$ \( 732 + T \)
$89$ \( 1590 + T \)
$97$ \( -194 + T \)
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