Embedded newform 60.4.h.c.59.19

• Label: 60.4.h.c.59.19
• Level: $60$
• Weight: $4$
• Character: 60.59
• Analytic conductor: $3.540$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.54011460034\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			59.19
Character	\(\chi\)	\(=\)	60.59
Dual form			60.4.h.c.59.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.40208 + 1.49331i) q^{2} +(-3.96106 - 3.36304i) q^{3} +(3.54002 + 7.17414i) q^{4} +(9.10892 + 6.48287i) q^{5} +(-4.49272 - 13.9934i) q^{6} +26.5236 q^{7} +(-2.20980 + 22.5193i) q^{8} +(4.37993 + 26.6424i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 56 q^{4} + 12 q^{6} + 192 q^{9}+O(q^{10}) \)

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.40208	+	1.49331i	0.849265	+	0.527966i
							\(3\)	−3.96106	−	3.36304i	−0.762306	−	0.647217i
\(5\)	9.10892	+	6.48287i	0.814726	+	0.579846i
							\(7\)	26.5236			1.43214			0.716071	−	0.698028i	\(-0.245939\pi\)
0.716071	+	0.698028i	\(0.245939\pi\)
\(11\)	−17.2030			−0.471536			−0.235768	−	0.971809i	\(-0.575761\pi\)
							−0.235768	+	0.971809i	\(0.575761\pi\)
\(13\)		−	66.7249i		−	1.42355i	−0.702407	−	0.711775i	\(-0.747891\pi\)
							0.702407	−	0.711775i	\(-0.252109\pi\)
\(17\)	−87.8981			−1.25403			−0.627013	−	0.779009i	\(-0.715723\pi\)
							−0.627013	+	0.779009i	\(0.715723\pi\)
\(19\)		−	22.3542i		−	0.269916i	−0.990851	−	0.134958i	\(-0.956910\pi\)
							0.990851	−	0.134958i	\(-0.0430900\pi\)
\(23\)		−	30.5698i		−	0.277141i	−0.990353	−	0.138571i	\(-0.955749\pi\)
							0.990353	−	0.138571i	\(-0.0442508\pi\)
\(29\)			54.3226i			0.347843i	0.984759	+	0.173922i	\(0.0556440\pi\)
							−0.984759	+	0.173922i	\(0.944356\pi\)
\(31\)		−	143.859i		−	0.833477i	−0.909026	−	0.416738i	\(-0.863173\pi\)
							0.909026	−	0.416738i	\(-0.136827\pi\)
\(37\)		−	285.192i		−	1.26717i	−0.773674	−	0.633584i	\(-0.781583\pi\)
							0.773674	−	0.633584i	\(-0.218417\pi\)
\(41\)			57.8269i			0.220269i	0.993917	+	0.110135i	\(0.0351282\pi\)
							−0.993917	+	0.110135i	\(0.964872\pi\)
\(43\)	−284.756			−1.00988			−0.504941	−	0.863154i	\(-0.668486\pi\)
							−0.504941	+	0.863154i	\(0.668486\pi\)
\(47\)			111.229i			0.345199i	0.984992	+	0.172600i	\(0.0552167\pi\)
							−0.984992	+	0.172600i	\(0.944783\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.h.c.59.19	yes	24
3.2	odd	2	inner	60.4.h.c.59.5	✓	24
4.3	odd	2	inner	60.4.h.c.59.18	yes	24
5.4	even	2	inner	60.4.h.c.59.6	yes	24
12.11	even	2	inner	60.4.h.c.59.8	yes	24
15.14	odd	2	inner	60.4.h.c.59.20	yes	24
20.19	odd	2	inner	60.4.h.c.59.7	yes	24
60.59	even	2	inner	60.4.h.c.59.17	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.h.c.59.5	✓	24	3.2	odd	2	inner
60.4.h.c.59.6	yes	24	5.4	even	2	inner
60.4.h.c.59.7	yes	24	20.19	odd	2	inner
60.4.h.c.59.8	yes	24	12.11	even	2	inner
60.4.h.c.59.17	yes	24	60.59	even	2	inner
60.4.h.c.59.18	yes	24	4.3	odd	2	inner
60.4.h.c.59.19	yes	24	1.1	even	1	trivial
60.4.h.c.59.20	yes	24	15.14	odd	2	inner