Embedded newform 360.2.bd.b.59.16

• Label: 360.2.bd.b.59.16
• 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			59.16
Character	\(\chi\)	\(=\)	360.59
Dual form			360.2.bd.b.299.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.933002 + 1.06278i) q^{2} +(1.70706 + 0.293175i) q^{3} +(-0.259016 - 1.98316i) q^{4} +(1.32418 + 1.80181i) q^{5} +(-1.90427 + 1.54070i) q^{6} +(0.193132 + 0.334514i) q^{7} +(2.34933 + 1.57501i) q^{8} +(2.82810 + 1.00093i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{4} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.933002	+	1.06278i	−0.659732	+	0.751501i
							\(3\)	1.70706	+	0.293175i	0.985571	+	0.169265i
\(5\)	1.32418	+	1.80181i	0.592193	+	0.805796i
							\(7\)	0.193132	+	0.334514i	0.0729970	+	0.126435i	0.900213	−	0.435449i	\(-0.143410\pi\)
−0.827216	+	0.561883i	\(0.810077\pi\)
\(11\)	−2.78273	+	1.60661i	−0.839026	+	0.484412i	−0.856933	−	0.515428i	\(-0.827633\pi\)
							0.0179071	+	0.999840i	\(0.494300\pi\)
\(13\)	1.73079	−	2.99781i	0.480034	−	0.831444i	−0.519703	−	0.854347i	\(-0.673958\pi\)
							0.999738	+	0.0229029i	\(0.00729087\pi\)
\(17\)	0.211121			0.0512044			0.0256022	−	0.999672i	\(-0.491850\pi\)
							0.0256022	+	0.999672i	\(0.491850\pi\)
\(19\)	0.235635			0.0540584			0.0270292	−	0.999635i	\(-0.491395\pi\)
							0.0270292	+	0.999635i	\(0.491395\pi\)
\(23\)	5.94857	+	3.43441i	1.24036	+	0.716123i	0.969168	−	0.246402i	\(-0.0792482\pi\)
							0.271194	+	0.962525i	\(0.412582\pi\)
\(29\)	−1.62707	−	2.81817i	−0.302140	−	0.523322i	0.674480	−	0.738293i	\(-0.264368\pi\)
							−0.976620	+	0.214971i	\(0.931034\pi\)
\(31\)	−6.89939	−	3.98336i	−1.23917	−	0.715433i	−0.270243	−	0.962792i	\(-0.587104\pi\)
							−0.968924	+	0.247359i	\(0.920437\pi\)
\(37\)	0.625495			0.102831			0.0514154	−	0.998677i	\(-0.483627\pi\)
							0.0514154	+	0.998677i	\(0.483627\pi\)
\(41\)	2.40884	+	1.39075i	0.376198	+	0.217198i	0.676163	−	0.736752i	\(-0.263642\pi\)
							−0.299965	+	0.953950i	\(0.596975\pi\)
\(43\)	−9.64632	+	5.56931i	−1.47105	+	0.849311i	−0.999471	−	0.0325125i	\(-0.989649\pi\)
							−0.471579	+	0.881824i	\(0.656316\pi\)
\(47\)	3.32214	−	1.91804i	0.484584	−	0.279774i	−0.237741	−	0.971329i	\(-0.576407\pi\)
							0.722325	+	0.691554i	\(0.243074\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.2.bd.b.59.16	yes	128
5.4	even	2	inner	360.2.bd.b.59.49	yes	128
8.3	odd	2	inner	360.2.bd.b.59.60	yes	128
9.2	odd	6	inner	360.2.bd.b.299.5	yes	128
40.19	odd	2	inner	360.2.bd.b.59.5	✓	128
45.29	odd	6	inner	360.2.bd.b.299.60	yes	128
72.11	even	6	inner	360.2.bd.b.299.49	yes	128
360.299	even	6	inner	360.2.bd.b.299.16	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.bd.b.59.5	✓	128	40.19	odd	2	inner
360.2.bd.b.59.16	yes	128	1.1	even	1	trivial
360.2.bd.b.59.49	yes	128	5.4	even	2	inner
360.2.bd.b.59.60	yes	128	8.3	odd	2	inner
360.2.bd.b.299.5	yes	128	9.2	odd	6	inner
360.2.bd.b.299.16	yes	128	360.299	even	6	inner
360.2.bd.b.299.49	yes	128	72.11	even	6	inner
360.2.bd.b.299.60	yes	128	45.29	odd	6	inner