Properties

Label 60.4.a.a
Level $60$
Weight $4$
Character orbit 60.a
Self dual yes
Analytic conductor $3.540$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)

Newform invariants

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

\(f(q)\) \(=\) \( q - 3q^{3} - 5q^{5} - 28q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 5q^{5} - 28q^{7} + 9q^{9} - 24q^{11} - 70q^{13} + 15q^{15} + 102q^{17} + 20q^{19} + 84q^{21} - 72q^{23} + 25q^{25} - 27q^{27} + 306q^{29} - 136q^{31} + 72q^{33} + 140q^{35} - 214q^{37} + 210q^{39} - 150q^{41} - 292q^{43} - 45q^{45} - 72q^{47} + 441q^{49} - 306q^{51} - 414q^{53} + 120q^{55} - 60q^{57} - 744q^{59} - 418q^{61} - 252q^{63} + 350q^{65} + 188q^{67} + 216q^{69} + 480q^{71} + 434q^{73} - 75q^{75} + 672q^{77} + 1352q^{79} + 81q^{81} - 612q^{83} - 510q^{85} - 918q^{87} - 30q^{89} + 1960q^{91} + 408q^{93} - 100q^{95} - 286q^{97} - 216q^{99} + O(q^{100}) \)

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 −28.0000 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.a 1
3.b odd 2 1 180.4.a.d 1
4.b odd 2 1 240.4.a.i 1
5.b even 2 1 300.4.a.i 1
5.c odd 4 2 300.4.d.b 2
8.b even 2 1 960.4.a.bc 1
8.d odd 2 1 960.4.a.r 1
9.c even 3 2 1620.4.i.l 2
9.d odd 6 2 1620.4.i.f 2
12.b even 2 1 720.4.a.bb 1
15.d odd 2 1 900.4.a.q 1
15.e even 4 2 900.4.d.h 2
20.d odd 2 1 1200.4.a.a 1
20.e even 4 2 1200.4.f.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.a 1 1.a even 1 1 trivial
180.4.a.d 1 3.b odd 2 1
240.4.a.i 1 4.b odd 2 1
300.4.a.i 1 5.b even 2 1
300.4.d.b 2 5.c odd 4 2
720.4.a.bb 1 12.b even 2 1
900.4.a.q 1 15.d odd 2 1
900.4.d.h 2 15.e even 4 2
960.4.a.r 1 8.d odd 2 1
960.4.a.bc 1 8.b even 2 1
1200.4.a.a 1 20.d odd 2 1
1200.4.f.n 2 20.e even 4 2
1620.4.i.f 2 9.d odd 6 2
1620.4.i.l 2 9.c even 3 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 28 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(60))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 5 + T \)
$7$ \( 28 + T \)
$11$ \( 24 + T \)
$13$ \( 70 + T \)
$17$ \( -102 + T \)
$19$ \( -20 + T \)
$23$ \( 72 + T \)
$29$ \( -306 + T \)
$31$ \( 136 + T \)
$37$ \( 214 + T \)
$41$ \( 150 + T \)
$43$ \( 292 + T \)
$47$ \( 72 + T \)
$53$ \( 414 + T \)
$59$ \( 744 + T \)
$61$ \( 418 + T \)
$67$ \( -188 + T \)
$71$ \( -480 + T \)
$73$ \( -434 + T \)
$79$ \( -1352 + T \)
$83$ \( 612 + T \)
$89$ \( 30 + T \)
$97$ \( 286 + T \)
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