Embedded newform 228.2.w.a.67.6

• Label: 228.2.w.a.67.6
• Level: $228$
• Weight: $2$
• Character: 228.67
• Analytic conductor: $1.821$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	228.w (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.82058916609\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			67.6
Character	\(\chi\)	\(=\)	228.67
Dual form			228.2.w.a.211.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.370262 - 1.36488i) q^{2} +(-0.766044 + 0.642788i) q^{3} +(-1.72581 - 1.01073i) q^{4} +(3.29537 + 1.19942i) q^{5} +(0.593693 + 1.28356i) q^{6} +(2.21173 + 1.27694i) q^{7} +(-2.01853 + 1.98130i) q^{8} +(0.173648 - 0.984808i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 3 q^{2} - 3 q^{4} + 3 q^{6} + 3 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/228\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)	\(115\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{17}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.370262	−	1.36488i	0.261815	−	0.965118i
							\(3\)	−0.766044	+	0.642788i	−0.442276	+	0.371114i
\(5\)	3.29537	+	1.19942i	1.47373	+	0.536395i								0.949112	−	0.314940i	\(-0.101984\pi\)
							0.524622	+	0.851335i	\(0.324207\pi\)
\(7\)	2.21173	+	1.27694i	0.835956	+	0.482640i	0.855888	−	0.517162i	\(-0.173011\pi\)
							−0.0199314	+	0.999801i	\(0.506345\pi\)
\(11\)	−0.00324770	+	0.00187506i	−0.000979218	+	0.000565352i	−0.500490	−	0.865743i	\(-0.666847\pi\)
							0.499510	+	0.866308i	\(0.333513\pi\)
\(13\)	0.382681	−	0.456062i	0.106137	−	0.126489i	−0.710361	−	0.703838i	\(-0.751468\pi\)
							0.816498	+	0.577349i	\(0.195913\pi\)
\(17\)	−0.634319	−	3.59740i	−0.153845	−	0.872498i	−0.959834	−	0.280568i	\(-0.909477\pi\)
							0.805989	−	0.591930i	\(-0.201634\pi\)
\(19\)	2.19475	−	3.76604i	0.503509	−	0.863990i
							\(23\)	−0.471769	−	1.29618i	−0.0983707	−	0.270271i	0.880740	−	0.473601i	\(-0.157046\pi\)
−0.979110	+	0.203329i	\(0.934824\pi\)
\(29\)	−4.56231	−	0.804457i	−0.847199	−	0.149384i	−0.266836	−	0.963742i	\(-0.585978\pi\)
							−0.580363	+	0.814358i	\(0.697089\pi\)
\(31\)	−4.50482	+	7.80257i	−0.809089	+	1.40138i	0.104407	+	0.994535i	\(0.466706\pi\)
							−0.913496	+	0.406848i	\(0.866628\pi\)
\(37\)			11.9654i			1.96711i	0.180622	+	0.983553i	\(0.442189\pi\)
							−0.180622	+	0.983553i	\(0.557811\pi\)
\(41\)	−6.76233	−	8.05903i	−1.05610	−	1.25861i	−0.964856	−	0.262780i	\(-0.915361\pi\)
							−0.0912419	−	0.995829i	\(-0.529084\pi\)
\(43\)	−1.12571	+	3.09286i	−0.171669	+	0.471657i	−0.995454	−	0.0952458i	\(-0.969636\pi\)
							0.823785	+	0.566903i	\(0.191859\pi\)
\(47\)	−1.35380	−	0.238711i	−0.197472	−	0.0348196i	0.0740373	−	0.997255i	\(-0.476412\pi\)
							−0.271509	+	0.962436i	\(0.587523\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	228.2.w.a.67.6	✓	60
3.2	odd	2		684.2.cf.c.523.5		60
4.3	odd	2		228.2.w.b.67.8	yes	60
12.11	even	2		684.2.cf.b.523.3		60
19.2	odd	18		228.2.w.b.211.8	yes	60
57.2	even	18		684.2.cf.b.667.3		60
76.59	even	18	inner	228.2.w.a.211.6	yes	60
228.59	odd	18		684.2.cf.c.667.5		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.w.a.67.6	✓	60	1.1	even	1	trivial
228.2.w.a.211.6	yes	60	76.59	even	18	inner
228.2.w.b.67.8	yes	60	4.3	odd	2
228.2.w.b.211.8	yes	60	19.2	odd	18
684.2.cf.b.523.3		60	12.11	even	2
684.2.cf.b.667.3		60	57.2	even	18
684.2.cf.c.523.5		60	3.2	odd	2
684.2.cf.c.667.5		60	228.59	odd	18