Properties

Label 60.8.a.a
Level $60$
Weight $8$
Character orbit 60.a
Self dual yes
Analytic conductor $18.743$
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: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.7431015290\)
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

\(f(q)\) \(=\) \( q - 27q^{3} - 125q^{5} + 1028q^{7} + 729q^{9} + O(q^{10}) \) \( q - 27q^{3} - 125q^{5} + 1028q^{7} + 729q^{9} + 3096q^{11} - 13030q^{13} + 3375q^{15} + 1878q^{17} - 31180q^{19} - 27756q^{21} - 33288q^{23} + 15625q^{25} - 19683q^{27} - 213054q^{29} - 172696q^{31} - 83592q^{33} - 128500q^{35} + 27434q^{37} + 351810q^{39} + 532650q^{41} - 911908q^{43} - 91125q^{45} - 732648q^{47} + 233241q^{49} - 50706q^{51} + 409074q^{53} - 387000q^{55} + 841860q^{57} + 1508136q^{59} - 302578q^{61} + 749412q^{63} + 1628750q^{65} + 1254332q^{67} + 898776q^{69} + 4781280q^{71} - 502414q^{73} - 421875q^{75} + 3182688q^{77} - 1991368q^{79} + 531441q^{81} - 8099268q^{83} - 234750q^{85} + 5752458q^{87} + 7487970q^{89} - 13394840q^{91} + 4662792q^{93} + 3897500q^{95} - 17172574q^{97} + 2256984q^{99} + O(q^{100}) \)

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 −27.0000 0 −125.000 0 1028.00 0 729.000 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 60.8.a.a 1
3.b odd 2 1 180.8.a.e 1
4.b odd 2 1 240.8.a.i 1
5.b even 2 1 300.8.a.e 1
5.c odd 4 2 300.8.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.8.a.a 1 1.a even 1 1 trivial
180.8.a.e 1 3.b odd 2 1
240.8.a.i 1 4.b odd 2 1
300.8.a.e 1 5.b even 2 1
300.8.d.d 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 1028 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(60))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 27 + T \)
$5$ \( 125 + T \)
$7$ \( -1028 + T \)
$11$ \( -3096 + T \)
$13$ \( 13030 + T \)
$17$ \( -1878 + T \)
$19$ \( 31180 + T \)
$23$ \( 33288 + T \)
$29$ \( 213054 + T \)
$31$ \( 172696 + T \)
$37$ \( -27434 + T \)
$41$ \( -532650 + T \)
$43$ \( 911908 + T \)
$47$ \( 732648 + T \)
$53$ \( -409074 + T \)
$59$ \( -1508136 + T \)
$61$ \( 302578 + T \)
$67$ \( -1254332 + T \)
$71$ \( -4781280 + T \)
$73$ \( 502414 + T \)
$79$ \( 1991368 + T \)
$83$ \( 8099268 + T \)
$89$ \( -7487970 + T \)
$97$ \( 17172574 + T \)
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