Properties

Label 60.6.a.d
Level $60$
Weight $6$
Character orbit 60.a
Self dual yes
Analytic conductor $9.623$
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)

Newform invariants

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

\(f(q)\) \(=\) \( q + 9q^{3} + 25q^{5} + 56q^{7} + 81q^{9} + O(q^{10}) \) \( q + 9q^{3} + 25q^{5} + 56q^{7} + 81q^{9} + 156q^{11} + 350q^{13} + 225q^{15} + 786q^{17} + 740q^{19} + 504q^{21} + 2376q^{23} + 625q^{25} + 729q^{27} + 2574q^{29} - 4576q^{31} + 1404q^{33} + 1400q^{35} - 12202q^{37} + 3150q^{39} - 10230q^{41} - 16084q^{43} + 2025q^{45} + 864q^{47} - 13671q^{49} + 7074q^{51} - 17658q^{53} + 3900q^{55} + 6660q^{57} + 48684q^{59} - 33778q^{61} + 4536q^{63} + 8750q^{65} + 3524q^{67} + 21384q^{69} + 38280q^{71} - 79702q^{73} + 5625q^{75} + 8736q^{77} + 99248q^{79} + 6561q^{81} - 22284q^{83} + 19650q^{85} + 23166q^{87} + 94650q^{89} + 19600q^{91} - 41184q^{93} + 18500q^{95} + 9122q^{97} + 12636q^{99} + O(q^{100}) \)

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 56.0000 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.d 1
3.b odd 2 1 180.6.a.b 1
4.b odd 2 1 240.6.a.e 1
5.b even 2 1 300.6.a.b 1
5.c odd 4 2 300.6.d.d 2
8.b even 2 1 960.6.a.e 1
8.d odd 2 1 960.6.a.p 1
12.b even 2 1 720.6.a.d 1
15.d odd 2 1 900.6.a.e 1
15.e even 4 2 900.6.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.a.d 1 1.a even 1 1 trivial
180.6.a.b 1 3.b odd 2 1
240.6.a.e 1 4.b odd 2 1
300.6.a.b 1 5.b even 2 1
300.6.d.d 2 5.c odd 4 2
720.6.a.d 1 12.b even 2 1
900.6.a.e 1 15.d odd 2 1
900.6.d.c 2 15.e even 4 2
960.6.a.e 1 8.b even 2 1
960.6.a.p 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 56 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(60))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -9 + T \)
$5$ \( -25 + T \)
$7$ \( -56 + T \)
$11$ \( -156 + T \)
$13$ \( -350 + T \)
$17$ \( -786 + T \)
$19$ \( -740 + T \)
$23$ \( -2376 + T \)
$29$ \( -2574 + T \)
$31$ \( 4576 + T \)
$37$ \( 12202 + T \)
$41$ \( 10230 + T \)
$43$ \( 16084 + T \)
$47$ \( -864 + T \)
$53$ \( 17658 + T \)
$59$ \( -48684 + T \)
$61$ \( 33778 + T \)
$67$ \( -3524 + T \)
$71$ \( -38280 + T \)
$73$ \( 79702 + T \)
$79$ \( -99248 + T \)
$83$ \( 22284 + T \)
$89$ \( -94650 + T \)
$97$ \( -9122 + T \)
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