Properties

Label 60.6.a.a
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

\(f(q)\) \(=\) \( q - 9q^{3} - 25q^{5} + 44q^{7} + 81q^{9} + O(q^{10}) \) \( q - 9q^{3} - 25q^{5} + 44q^{7} + 81q^{9} + 216q^{11} + 770q^{13} + 225q^{15} + 534q^{17} + 1580q^{19} - 396q^{21} + 2904q^{23} + 625q^{25} - 729q^{27} - 4566q^{29} + 2744q^{31} - 1944q^{33} - 1100q^{35} + 1442q^{37} - 6930q^{39} - 13350q^{41} + 17204q^{43} - 2025q^{45} - 10824q^{47} - 14871q^{49} - 4806q^{51} - 9942q^{53} - 5400q^{55} - 14220q^{57} - 15576q^{59} + 39302q^{61} + 3564q^{63} - 19250q^{65} + 55796q^{67} - 26136q^{69} + 57120q^{71} + 50402q^{73} - 5625q^{75} + 9504q^{77} - 10552q^{79} + 6561q^{81} + 108564q^{83} - 13350q^{85} + 41094q^{87} - 116430q^{89} + 33880q^{91} - 24696q^{93} - 39500q^{95} - 2782q^{97} + 17496q^{99} + O(q^{100}) \)

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 44.0000 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.a 1
3.b odd 2 1 180.6.a.d 1
4.b odd 2 1 240.6.a.i 1
5.b even 2 1 300.6.a.d 1
5.c odd 4 2 300.6.d.e 2
8.b even 2 1 960.6.a.z 1
8.d odd 2 1 960.6.a.i 1
12.b even 2 1 720.6.a.p 1
15.d odd 2 1 900.6.a.f 1
15.e even 4 2 900.6.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.a.a 1 1.a even 1 1 trivial
180.6.a.d 1 3.b odd 2 1
240.6.a.i 1 4.b odd 2 1
300.6.a.d 1 5.b even 2 1
300.6.d.e 2 5.c odd 4 2
720.6.a.p 1 12.b even 2 1
900.6.a.f 1 15.d odd 2 1
900.6.d.b 2 15.e even 4 2
960.6.a.i 1 8.d odd 2 1
960.6.a.z 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 44 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(60))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 9 + T \)
$5$ \( 25 + T \)
$7$ \( -44 + T \)
$11$ \( -216 + T \)
$13$ \( -770 + T \)
$17$ \( -534 + T \)
$19$ \( -1580 + T \)
$23$ \( -2904 + T \)
$29$ \( 4566 + T \)
$31$ \( -2744 + T \)
$37$ \( -1442 + T \)
$41$ \( 13350 + T \)
$43$ \( -17204 + T \)
$47$ \( 10824 + T \)
$53$ \( 9942 + T \)
$59$ \( 15576 + T \)
$61$ \( -39302 + T \)
$67$ \( -55796 + T \)
$71$ \( -57120 + T \)
$73$ \( -50402 + T \)
$79$ \( 10552 + T \)
$83$ \( -108564 + T \)
$89$ \( 116430 + T \)
$97$ \( 2782 + T \)
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