Embedded newform 360.2.bd.b.299.64

• Label: 360.2.bd.b.299.64
• Level: $360$
• Weight: $2$
• Character: 360.299
• Analytic conductor: $2.875$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.87461447277\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			299.64
Character	\(\chi\)	\(=\)	360.299
Dual form			360.2.bd.b.59.64

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41125 + 0.0914777i) q^{2} +(-1.32971 + 1.10989i) q^{3} +(1.98326 + 0.258196i) q^{4} +(-0.451766 - 2.18996i) q^{5} +(-1.97809 + 1.44470i) q^{6} +(2.22153 - 3.84780i) q^{7} +(2.77527 + 0.545804i) q^{8} +(0.536278 - 2.95168i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{4} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\)	\(181\)	\(217\)	\(271\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41125	+	0.0914777i	0.997906	+	0.0646845i
							\(3\)	−1.32971	+	1.10989i	−0.767711	+	0.640797i
\(5\)	−0.451766	−	2.18996i	−0.202036	−	0.979378i
							\(7\)	2.22153	−	3.84780i	0.839659	−	1.45433i	−0.0505211	−	0.998723i	\(-0.516088\pi\)
0.890180	−	0.455609i	\(-0.150578\pi\)
\(11\)	−2.97913	−	1.72000i	−0.898242	−	0.518600i	−0.0216123	−	0.999766i	\(-0.506880\pi\)
							−0.876629	+	0.481166i	\(0.840213\pi\)
\(13\)	3.22299	+	5.58238i	0.893896	+	1.54827i	0.835166	+	0.549999i	\(0.185372\pi\)
							0.0587299	+	0.998274i	\(0.481295\pi\)
\(17\)	2.90120			0.703645			0.351822	−	0.936067i	\(-0.385562\pi\)
							0.351822	+	0.936067i	\(0.385562\pi\)
\(19\)	−1.72505			−0.395754			−0.197877	−	0.980227i	\(-0.563405\pi\)
							−0.197877	+	0.980227i	\(0.563405\pi\)
\(23\)	−4.32408	+	2.49651i	−0.901633	+	0.520558i	−0.877730	−	0.479156i	\(-0.840943\pi\)
							−0.0239032	+	0.999714i	\(0.507609\pi\)
\(29\)	0.523887	−	0.907399i	0.0972834	−	0.168500i	−0.813276	−	0.581878i	\(-0.802318\pi\)
							0.910559	+	0.413378i	\(0.135651\pi\)
\(31\)	1.88690	−	1.08940i	0.338897	−	0.195662i	−0.320887	−	0.947117i	\(-0.603981\pi\)
							0.659784	+	0.751455i	\(0.270648\pi\)
\(37\)	3.02322			0.497015			0.248507	−	0.968630i	\(-0.420060\pi\)
							0.248507	+	0.968630i	\(0.420060\pi\)
\(41\)	−1.94134	+	1.12084i	−0.303187	+	0.175045i	−0.643874	−	0.765132i	\(-0.722674\pi\)
							0.340687	+	0.940177i	\(0.389340\pi\)
\(43\)	−2.07856	−	1.20006i	−0.316978	−	0.183007i	0.333067	−	0.942903i	\(-0.391916\pi\)
							−0.650045	+	0.759896i	\(0.725250\pi\)
\(47\)	0.453489	+	0.261822i	0.0661482	+	0.0381907i	0.532709	−	0.846298i	\(-0.321174\pi\)
							−0.466561	+	0.884489i	\(0.654507\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.bd.b.299.64	yes	128
5.4	even	2	inner	360.2.bd.b.299.1	yes	128
8.3	odd	2	inner	360.2.bd.b.299.20	yes	128
9.5	odd	6	inner	360.2.bd.b.59.45	yes	128
40.19	odd	2	inner	360.2.bd.b.299.45	yes	128
45.14	odd	6	inner	360.2.bd.b.59.20	yes	128
72.59	even	6	inner	360.2.bd.b.59.1	✓	128
360.59	even	6	inner	360.2.bd.b.59.64	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bd.b.59.1	✓	128	72.59	even	6	inner
360.2.bd.b.59.20	yes	128	45.14	odd	6	inner
360.2.bd.b.59.45	yes	128	9.5	odd	6	inner
360.2.bd.b.59.64	yes	128	360.59	even	6	inner
360.2.bd.b.299.1	yes	128	5.4	even	2	inner
360.2.bd.b.299.20	yes	128	8.3	odd	2	inner
360.2.bd.b.299.45	yes	128	40.19	odd	2	inner
360.2.bd.b.299.64	yes	128	1.1	even	1	trivial