Properties

Label 60.3.f.a
Level $60$
Weight $3$
Character orbit 60.f
Analytic conductor $1.635$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 60.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.63488158616\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 4 \zeta_{12}^{2} q^{4} -5 \zeta_{12}^{3} q^{5} + ( 2 + 2 \zeta_{12}^{2} ) q^{6} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{7} + 8 \zeta_{12}^{3} q^{8} + 3 q^{9} +O(q^{10})\) \( q + 2 \zeta_{12} q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 4 \zeta_{12}^{2} q^{4} -5 \zeta_{12}^{3} q^{5} + ( 2 + 2 \zeta_{12}^{2} ) q^{6} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{7} + 8 \zeta_{12}^{3} q^{8} + 3 q^{9} + ( 10 - 10 \zeta_{12}^{2} ) q^{10} + ( -6 + 12 \zeta_{12}^{2} ) q^{11} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{12} -18 \zeta_{12}^{3} q^{13} + ( -12 - 12 \zeta_{12}^{2} ) q^{14} + ( 5 - 10 \zeta_{12}^{2} ) q^{15} + ( -16 + 16 \zeta_{12}^{2} ) q^{16} + 10 \zeta_{12}^{3} q^{17} + 6 \zeta_{12} q^{18} + ( 8 - 16 \zeta_{12}^{2} ) q^{19} + ( 20 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{20} -18 q^{21} + ( -12 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{22} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} + ( -8 + 16 \zeta_{12}^{2} ) q^{24} -25 q^{25} + ( 36 - 36 \zeta_{12}^{2} ) q^{26} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -24 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{28} + 36 q^{29} + ( 10 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{30} + ( 4 - 8 \zeta_{12}^{2} ) q^{31} + ( -32 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{32} + 18 \zeta_{12}^{3} q^{33} + ( -20 + 20 \zeta_{12}^{2} ) q^{34} + ( -30 + 60 \zeta_{12}^{2} ) q^{35} + 12 \zeta_{12}^{2} q^{36} + 54 \zeta_{12}^{3} q^{37} + ( 16 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{38} + ( 18 - 36 \zeta_{12}^{2} ) q^{39} + 40 q^{40} + 18 q^{41} -36 \zeta_{12} q^{42} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{43} + ( -48 + 24 \zeta_{12}^{2} ) q^{44} -15 \zeta_{12}^{3} q^{45} + ( 8 + 8 \zeta_{12}^{2} ) q^{46} + ( -16 \zeta_{12} + 32 \zeta_{12}^{3} ) q^{48} + 59 q^{49} -50 \zeta_{12} q^{50} + ( -10 + 20 \zeta_{12}^{2} ) q^{51} + ( 72 \zeta_{12} - 72 \zeta_{12}^{3} ) q^{52} + 26 \zeta_{12}^{3} q^{53} + ( 6 + 6 \zeta_{12}^{2} ) q^{54} + ( 60 \zeta_{12} - 30 \zeta_{12}^{3} ) q^{55} + ( 48 - 96 \zeta_{12}^{2} ) q^{56} -24 \zeta_{12}^{3} q^{57} + 72 \zeta_{12} q^{58} + ( -18 + 36 \zeta_{12}^{2} ) q^{59} + ( 40 - 20 \zeta_{12}^{2} ) q^{60} -74 q^{61} + ( 8 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{62} + ( -36 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{63} -64 q^{64} -90 q^{65} + ( -36 + 36 \zeta_{12}^{2} ) q^{66} + ( -48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{67} + ( -40 \zeta_{12} + 40 \zeta_{12}^{3} ) q^{68} + 12 q^{69} + ( -60 \zeta_{12} + 120 \zeta_{12}^{3} ) q^{70} + ( 60 - 120 \zeta_{12}^{2} ) q^{71} + 24 \zeta_{12}^{3} q^{72} -36 \zeta_{12}^{3} q^{73} + ( -108 + 108 \zeta_{12}^{2} ) q^{74} + ( -50 \zeta_{12} + 25 \zeta_{12}^{3} ) q^{75} + ( 64 - 32 \zeta_{12}^{2} ) q^{76} -108 \zeta_{12}^{3} q^{77} + ( 36 \zeta_{12} - 72 \zeta_{12}^{3} ) q^{78} + ( 52 - 104 \zeta_{12}^{2} ) q^{79} + 80 \zeta_{12} q^{80} + 9 q^{81} + 36 \zeta_{12} q^{82} + ( -104 \zeta_{12} + 52 \zeta_{12}^{3} ) q^{83} -72 \zeta_{12}^{2} q^{84} + 50 q^{85} + ( 24 + 24 \zeta_{12}^{2} ) q^{86} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{87} + ( -96 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{88} + 18 q^{89} + ( 30 - 30 \zeta_{12}^{2} ) q^{90} + ( -108 + 216 \zeta_{12}^{2} ) q^{91} + ( 16 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{92} -12 \zeta_{12}^{3} q^{93} + ( -80 \zeta_{12} + 40 \zeta_{12}^{3} ) q^{95} + ( -64 + 32 \zeta_{12}^{2} ) q^{96} -72 \zeta_{12}^{3} q^{97} + 118 \zeta_{12} q^{98} + ( -18 + 36 \zeta_{12}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + 12q^{6} + 12q^{9} + O(q^{10}) \) \( 4q + 8q^{4} + 12q^{6} + 12q^{9} + 20q^{10} - 72q^{14} - 32q^{16} - 72q^{21} - 100q^{25} + 72q^{26} + 144q^{29} - 40q^{34} + 24q^{36} + 160q^{40} + 72q^{41} - 144q^{44} + 48q^{46} + 236q^{49} + 36q^{54} + 120q^{60} - 296q^{61} - 256q^{64} - 360q^{65} - 72q^{66} + 48q^{69} - 216q^{74} + 192q^{76} + 36q^{81} - 144q^{84} + 200q^{85} + 144q^{86} + 72q^{89} + 60q^{90} - 192q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−1.73205 1.00000i −1.73205 2.00000 + 3.46410i 5.00000i 3.00000 + 1.73205i 10.3923 8.00000i 3.00000 5.00000 8.66025i
19.2 −1.73205 + 1.00000i −1.73205 2.00000 3.46410i 5.00000i 3.00000 1.73205i 10.3923 8.00000i 3.00000 5.00000 + 8.66025i
19.3 1.73205 1.00000i 1.73205 2.00000 3.46410i 5.00000i 3.00000 1.73205i −10.3923 8.00000i 3.00000 5.00000 + 8.66025i
19.4 1.73205 + 1.00000i 1.73205 2.00000 + 3.46410i 5.00000i 3.00000 + 1.73205i −10.3923 8.00000i 3.00000 5.00000 8.66025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.3.f.a 4
3.b odd 2 1 180.3.f.e 4
4.b odd 2 1 inner 60.3.f.a 4
5.b even 2 1 inner 60.3.f.a 4
5.c odd 4 1 300.3.c.a 2
5.c odd 4 1 300.3.c.c 2
8.b even 2 1 960.3.j.b 4
8.d odd 2 1 960.3.j.b 4
12.b even 2 1 180.3.f.e 4
15.d odd 2 1 180.3.f.e 4
15.e even 4 1 900.3.c.f 2
15.e even 4 1 900.3.c.j 2
20.d odd 2 1 inner 60.3.f.a 4
20.e even 4 1 300.3.c.a 2
20.e even 4 1 300.3.c.c 2
40.e odd 2 1 960.3.j.b 4
40.f even 2 1 960.3.j.b 4
60.h even 2 1 180.3.f.e 4
60.l odd 4 1 900.3.c.f 2
60.l odd 4 1 900.3.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.a 4 1.a even 1 1 trivial
60.3.f.a 4 4.b odd 2 1 inner
60.3.f.a 4 5.b even 2 1 inner
60.3.f.a 4 20.d odd 2 1 inner
180.3.f.e 4 3.b odd 2 1
180.3.f.e 4 12.b even 2 1
180.3.f.e 4 15.d odd 2 1
180.3.f.e 4 60.h even 2 1
300.3.c.a 2 5.c odd 4 1
300.3.c.a 2 20.e even 4 1
300.3.c.c 2 5.c odd 4 1
300.3.c.c 2 20.e even 4 1
900.3.c.f 2 15.e even 4 1
900.3.c.f 2 60.l odd 4 1
900.3.c.j 2 15.e even 4 1
900.3.c.j 2 60.l odd 4 1
960.3.j.b 4 8.b even 2 1
960.3.j.b 4 8.d odd 2 1
960.3.j.b 4 40.e odd 2 1
960.3.j.b 4 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 108 \) acting on \(S_{3}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T^{2} + T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 25 + T^{2} )^{2} \)
$7$ \( ( -108 + T^{2} )^{2} \)
$11$ \( ( 108 + T^{2} )^{2} \)
$13$ \( ( 324 + T^{2} )^{2} \)
$17$ \( ( 100 + T^{2} )^{2} \)
$19$ \( ( 192 + T^{2} )^{2} \)
$23$ \( ( -48 + T^{2} )^{2} \)
$29$ \( ( -36 + T )^{4} \)
$31$ \( ( 48 + T^{2} )^{2} \)
$37$ \( ( 2916 + T^{2} )^{2} \)
$41$ \( ( -18 + T )^{4} \)
$43$ \( ( -432 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( 676 + T^{2} )^{2} \)
$59$ \( ( 972 + T^{2} )^{2} \)
$61$ \( ( 74 + T )^{4} \)
$67$ \( ( -1728 + T^{2} )^{2} \)
$71$ \( ( 10800 + T^{2} )^{2} \)
$73$ \( ( 1296 + T^{2} )^{2} \)
$79$ \( ( 8112 + T^{2} )^{2} \)
$83$ \( ( -8112 + T^{2} )^{2} \)
$89$ \( ( -18 + T )^{4} \)
$97$ \( ( 5184 + T^{2} )^{2} \)
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