Properties

Label 60.6.a.c
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

\(f(q)\) \(=\) \( q + 9q^{3} - 25q^{5} - 244q^{7} + 81q^{9} + O(q^{10}) \) \( q + 9q^{3} - 25q^{5} - 244q^{7} + 81q^{9} - 144q^{11} + 50q^{13} - 225q^{15} - 1914q^{17} + 140q^{19} - 2196q^{21} - 624q^{23} + 625q^{25} + 729q^{27} - 3126q^{29} - 5176q^{31} - 1296q^{33} + 6100q^{35} + 15698q^{37} + 450q^{39} + 12570q^{41} + 11516q^{43} - 2025q^{45} - 26736q^{47} + 42729q^{49} - 17226q^{51} - 19158q^{53} + 3600q^{55} + 1260q^{57} + 27984q^{59} + 22022q^{61} - 19764q^{63} - 1250q^{65} - 12676q^{67} - 5616q^{69} - 59520q^{71} - 67102q^{73} + 5625q^{75} + 35136q^{77} + 11048q^{79} + 6561q^{81} - 115284q^{83} + 47850q^{85} - 28134q^{87} + 73650q^{89} - 12200q^{91} - 46584q^{93} - 3500q^{95} + 35522q^{97} - 11664q^{99} + O(q^{100}) \)

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 −244.000 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.c 1
3.b odd 2 1 180.6.a.c 1
4.b odd 2 1 240.6.a.d 1
5.b even 2 1 300.6.a.c 1
5.c odd 4 2 300.6.d.c 2
8.b even 2 1 960.6.a.g 1
8.d odd 2 1 960.6.a.bb 1
12.b even 2 1 720.6.a.x 1
15.d odd 2 1 900.6.a.k 1
15.e even 4 2 900.6.d.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.a.c 1 1.a even 1 1 trivial
180.6.a.c 1 3.b odd 2 1
240.6.a.d 1 4.b odd 2 1
300.6.a.c 1 5.b even 2 1
300.6.d.c 2 5.c odd 4 2
720.6.a.x 1 12.b even 2 1
900.6.a.k 1 15.d odd 2 1
900.6.d.f 2 15.e even 4 2
960.6.a.g 1 8.b even 2 1
960.6.a.bb 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 244 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(60))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -9 + T \)
$5$ \( 25 + T \)
$7$ \( 244 + T \)
$11$ \( 144 + T \)
$13$ \( -50 + T \)
$17$ \( 1914 + T \)
$19$ \( -140 + T \)
$23$ \( 624 + T \)
$29$ \( 3126 + T \)
$31$ \( 5176 + T \)
$37$ \( -15698 + T \)
$41$ \( -12570 + T \)
$43$ \( -11516 + T \)
$47$ \( 26736 + T \)
$53$ \( 19158 + T \)
$59$ \( -27984 + T \)
$61$ \( -22022 + T \)
$67$ \( 12676 + T \)
$71$ \( 59520 + T \)
$73$ \( 67102 + T \)
$79$ \( -11048 + T \)
$83$ \( 115284 + T \)
$89$ \( -73650 + T \)
$97$ \( -35522 + T \)
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