# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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