Embedded newform 360.2.bd.b.59.7

• Label: 360.2.bd.b.59.7
• 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			59.7
Character	\(\chi\)	\(=\)	360.59
Dual form			360.2.bd.b.299.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37227 - 0.341873i) q^{2} +(1.16746 - 1.27946i) q^{3} +(1.76625 + 0.938283i) q^{4} +(-0.996511 + 2.00174i) q^{5} +(-2.03949 + 1.35665i) q^{6} +(-0.0301683 - 0.0522531i) q^{7} +(-2.10299 - 1.89141i) q^{8} +(-0.274059 - 2.98746i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{4} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37227	−	0.341873i	−0.970341	−	0.241741i
							\(3\)	1.16746	−	1.27946i	0.674035	−	0.738699i
\(5\)	−0.996511	+	2.00174i	−0.445653	+	0.895206i
							\(7\)	−0.0301683	−	0.0522531i	−0.0114026	−	0.0197498i	0.860268	−	0.509842i	\(-0.170296\pi\)
−0.871670	+	0.490093i	\(0.836963\pi\)
\(11\)	0.550251	−	0.317688i	0.165907	−	0.0957864i	−0.414747	−	0.909937i	\(-0.636130\pi\)
							0.580654	+	0.814150i	\(0.302797\pi\)
\(13\)	2.83828	−	4.91605i	0.787198	−	1.36347i	−0.140479	−	0.990084i	\(-0.544864\pi\)
							0.927677	−	0.373384i	\(-0.121802\pi\)
\(17\)	3.68210			0.893042			0.446521	−	0.894773i	\(-0.352663\pi\)
							0.446521	+	0.894773i	\(0.352663\pi\)
\(19\)	5.76637			1.32290			0.661448	−	0.749991i	\(-0.269942\pi\)
							0.661448	+	0.749991i	\(0.269942\pi\)
\(23\)	1.92355	+	1.11056i	0.401088	+	0.231568i	0.686953	−	0.726702i	\(-0.258948\pi\)
							−0.285865	+	0.958270i	\(0.592281\pi\)
\(29\)	−2.35784	−	4.08389i	−0.437839	−	0.758359i	0.559684	−	0.828706i	\(-0.310923\pi\)
							−0.997523	+	0.0703470i	\(0.977589\pi\)
\(31\)	6.79485	+	3.92301i	1.22039	+	0.704593i	0.965001	−	0.262245i	\(-0.0844629\pi\)
							0.255390	+	0.966838i	\(0.417796\pi\)
\(37\)	−1.81986			−0.299183			−0.149592	−	0.988748i	\(-0.547796\pi\)
							−0.149592	+	0.988748i	\(0.547796\pi\)
\(41\)	7.85336	+	4.53414i	1.22649	+	0.708113i	0.966294	−	0.257443i	\(-0.0828799\pi\)
							0.260195	+	0.965556i	\(0.416213\pi\)
\(43\)	−8.33102	+	4.80992i	−1.27047	+	0.733505i	−0.975076	−	0.221871i	\(-0.928784\pi\)
							−0.295393	+	0.955376i	\(0.595450\pi\)
\(47\)	−6.57877	+	3.79826i	−0.959613	+	0.554033i	−0.896054	−	0.443945i	\(-0.853578\pi\)
							−0.0635589	+	0.997978i	\(0.520245\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.bd.b.59.7	✓	128
5.4	even	2	inner	360.2.bd.b.59.58	yes	128
8.3	odd	2	inner	360.2.bd.b.59.38	yes	128
9.2	odd	6	inner	360.2.bd.b.299.27	yes	128
40.19	odd	2	inner	360.2.bd.b.59.27	yes	128
45.29	odd	6	inner	360.2.bd.b.299.38	yes	128
72.11	even	6	inner	360.2.bd.b.299.58	yes	128
360.299	even	6	inner	360.2.bd.b.299.7	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bd.b.59.7	✓	128	1.1	even	1	trivial
360.2.bd.b.59.27	yes	128	40.19	odd	2	inner
360.2.bd.b.59.38	yes	128	8.3	odd	2	inner
360.2.bd.b.59.58	yes	128	5.4	even	2	inner
360.2.bd.b.299.7	yes	128	360.299	even	6	inner
360.2.bd.b.299.27	yes	128	9.2	odd	6	inner
360.2.bd.b.299.38	yes	128	45.29	odd	6	inner
360.2.bd.b.299.58	yes	128	72.11	even	6	inner