Embedded newform 360.2.bd.b.59.51

• Label: 360.2.bd.b.59.51
• Level: $360$
• Weight: $2$
• Character: 360.59
• 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			59.51
Character	\(\chi\)	\(=\)	360.59
Dual form			360.2.bd.b.299.51

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.02286 + 0.976600i) q^{2} +(1.56092 + 0.750684i) q^{3} +(0.0925056 + 1.99786i) q^{4} +(1.19110 - 1.89242i) q^{5} +(0.863494 + 2.29224i) q^{6} +(-0.753496 - 1.30509i) q^{7} +(-1.85649 + 2.13388i) q^{8} +(1.87295 + 2.34352i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{4} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(181\)	\(217\)	\(271\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.02286	+	0.976600i	0.723275	+	0.690560i
							\(3\)	1.56092	+	0.750684i	0.901198	+	0.433407i
\(5\)	1.19110	−	1.89242i	0.532678	−	0.846318i
							\(7\)	−0.753496	−	1.30509i	−0.284795	−	0.493279i	0.687765	−	0.725934i	\(-0.258592\pi\)
−0.972559	+	0.232655i	\(0.925259\pi\)
\(11\)	1.76774	−	1.02061i	0.532994	−	0.307724i	−0.209241	−	0.977864i	\(-0.567099\pi\)
							0.742235	+	0.670140i	\(0.233766\pi\)
\(13\)	−0.873834	+	1.51352i	−0.242358	+	0.419776i	−0.961385	−	0.275205i	\(-0.911254\pi\)
							0.719028	+	0.694982i	\(0.244587\pi\)
\(17\)	−5.24049			−1.27101			−0.635503	−	0.772098i	\(-0.719207\pi\)
							−0.635503	+	0.772098i	\(0.719207\pi\)
\(19\)	0.316680			0.0726515			0.0363257	−	0.999340i	\(-0.488435\pi\)
							0.0363257	+	0.999340i	\(0.488435\pi\)
\(23\)	3.53330	+	2.03995i	0.736744	+	0.425359i	0.820884	−	0.571095i	\(-0.193481\pi\)
							−0.0841404	+	0.996454i	\(0.526814\pi\)
\(29\)	−3.95679	−	6.85337i	−0.734758	−	1.27264i	−0.954829	−	0.297155i	\(-0.903962\pi\)
							0.220071	−	0.975484i	\(-0.429371\pi\)
\(31\)	−4.97249	−	2.87087i	−0.893086	−	0.515623i	−0.0181351	−	0.999836i	\(-0.505773\pi\)
							−0.874951	+	0.484212i	\(0.839106\pi\)
\(37\)	−7.74862			−1.27387			−0.636933	−	0.770920i	\(-0.719797\pi\)
							−0.636933	+	0.770920i	\(0.719797\pi\)
\(41\)	3.72948	+	2.15322i	0.582447	+	0.336276i	0.762105	−	0.647453i	\(-0.224166\pi\)
							−0.179658	+	0.983729i	\(0.557499\pi\)
\(43\)	2.74080	−	1.58240i	0.417968	−	0.241314i	−0.276240	−	0.961089i	\(-0.589088\pi\)
							0.694207	+	0.719775i	\(0.255755\pi\)
\(47\)	5.16249	−	2.98057i	0.753027	−	0.434760i	−0.0737596	−	0.997276i	\(-0.523500\pi\)
							0.826787	+	0.562516i	\(0.190166\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.bd.b.59.51	yes	128
5.4	even	2	inner	360.2.bd.b.59.14	✓	128
8.3	odd	2	inner	360.2.bd.b.59.37	yes	128
9.2	odd	6	inner	360.2.bd.b.299.28	yes	128
40.19	odd	2	inner	360.2.bd.b.59.28	yes	128
45.29	odd	6	inner	360.2.bd.b.299.37	yes	128
72.11	even	6	inner	360.2.bd.b.299.14	yes	128
360.299	even	6	inner	360.2.bd.b.299.51	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bd.b.59.14	✓	128	5.4	even	2	inner
360.2.bd.b.59.28	yes	128	40.19	odd	2	inner
360.2.bd.b.59.37	yes	128	8.3	odd	2	inner
360.2.bd.b.59.51	yes	128	1.1	even	1	trivial
360.2.bd.b.299.14	yes	128	72.11	even	6	inner
360.2.bd.b.299.28	yes	128	9.2	odd	6	inner
360.2.bd.b.299.37	yes	128	45.29	odd	6	inner
360.2.bd.b.299.51	yes	128	360.299	even	6	inner