Newform orbit 60.3.g.a

• Label: 60.3.g.a
• Level: $60$
• Weight: $3$
• Character orbit: 60.g
• Analytic conductor: $1.635$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$1.63488158616$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5})$
Defining polynomial:	$x^{2} + 5$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{-5}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (b + 2) * q^3 + b * q^5 + 2 * q^7 + (4*b - 1) * q^9 - 6*b * q^11 + 8 * q^13 + (2*b - 5) * q^15 - 6*b * q^17 - 34 * q^19 + (2*b + 4) * q^21 - 18*b * q^23 - 5 * q^25 + (7*b - 22) * q^27 + 18*b * q^29 + 14 * q^31 + (-12*b + 30) * q^33 + 2*b * q^35 + 56 * q^37 + (8*b + 16) * q^39 + 12*b * q^41 + 8 * q^43 + (-b - 20) * q^45 + 18*b * q^47 - 45 * q^49 + (-12*b + 30) * q^51 + 18*b * q^53 + 30 * q^55 + (-34*b - 68) * q^57 - 6*b * q^59 - 46 * q^61 + (8*b - 2) * q^63 + 8*b * q^65 + 32 * q^67 + (-36*b + 90) * q^69 - 24*b * q^71 - 106 * q^73 + (-5*b - 10) * q^75 - 12*b * q^77 - 22 * q^79 + (-8*b - 79) * q^81 + 54*b * q^83 + 30 * q^85 + (36*b - 90) * q^87 - 48*b * q^89 + 16 * q^91 + (14*b + 28) * q^93 - 34*b * q^95 + 122 * q^97 + (6*b + 120) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9 + 16 * q^13 - 10 * q^15 - 68 * q^19 + 8 * q^21 - 10 * q^25 - 44 * q^27 + 28 * q^31 + 60 * q^33 + 112 * q^37 + 32 * q^39 + 16 * q^43 - 40 * q^45 - 90 * q^49 + 60 * q^51 + 60 * q^55 - 136 * q^57 - 92 * q^61 - 4 * q^63 + 64 * q^67 + 180 * q^69 - 212 * q^73 - 20 * q^75 - 44 * q^79 - 158 * q^81 + 60 * q^85 - 180 * q^87 + 32 * q^91 + 56 * q^93 + 244 * q^97 + 240 * q^99

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

41.1

	−	2.23607i
		2.23607i

0

2.00000

−

2.23607i

0

−

2.23607i

0

2.00000

0

−1.00000

−

8.94427i

0

41.2

0

2.00000

+

2.23607i

0

2.23607i

0

2.00000

0

−1.00000

+

8.94427i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	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.g.a	✓	2
3.b	odd	2	1	inner	60.3.g.a	✓	2
4.b	odd	2	1		240.3.l.a		2
5.b	even	2	1		300.3.g.d		2
5.c	odd	4	2		300.3.b.c		4
8.b	even	2	1		960.3.l.a		2
8.d	odd	2	1		960.3.l.d		2
9.c	even	3	2		1620.3.o.b		4
9.d	odd	6	2		1620.3.o.b		4
12.b	even	2	1		240.3.l.a		2
15.d	odd	2	1		300.3.g.d		2
15.e	even	4	2		300.3.b.c		4
20.d	odd	2	1		1200.3.l.r		2
20.e	even	4	2		1200.3.c.e		4
24.f	even	2	1		960.3.l.d		2
24.h	odd	2	1		960.3.l.a		2
60.h	even	2	1		1200.3.l.r		2
60.l	odd	4	2		1200.3.c.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.3.g.a	✓	2	1.a	even	1	1	trivial
60.3.g.a	✓	2	3.b	odd	2	1	inner
240.3.l.a		2	4.b	odd	2	1
240.3.l.a		2	12.b	even	2	1
300.3.b.c		4	5.c	odd	4	2
300.3.b.c		4	15.e	even	4	2
300.3.g.d		2	5.b	even	2	1
300.3.g.d		2	15.d	odd	2	1
960.3.l.a		2	8.b	even	2	1
960.3.l.a		2	24.h	odd	2	1
960.3.l.d		2	8.d	odd	2	1
960.3.l.d		2	24.f	even	2	1
1200.3.c.e		4	20.e	even	4	2
1200.3.c.e		4	60.l	odd	4	2
1200.3.l.r		2	20.d	odd	2	1
1200.3.l.r		2	60.h	even	2	1
1620.3.o.b		4	9.c	even	3	2
1620.3.o.b		4	9.d	odd	6	2

Hecke kernels

This newform subspace is the entire newspace $S_{3}^{\mathrm{new}}(60, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} - 4T + 9$
$5$	$T^{2} + 5$
$7$	$(T - 2)^{2}$
$11$	$T^{2} + 180$