Embedded newform 180.2.x.a.103.15

• Label: 180.2.x.a.103.15
• Level: $180$
• Weight: $2$
• Character: 180.103
• Analytic conductor: $1.437$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	180.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.43730723638\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(32\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			103.15
Character	\(\chi\)	\(=\)	180.103
Dual form			180.2.x.a.7.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.241682 - 1.39341i) q^{2} +(-1.73134 - 0.0494428i) q^{3} +(-1.88318 + 0.673524i) q^{4} +(-1.26760 - 1.84206i) q^{5} +(0.349541 + 2.42442i) q^{6} +(-1.09131 + 4.07282i) q^{7} +(1.39362 + 2.46126i) q^{8} +(2.99511 + 0.171205i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 2 q^{2} - 4 q^{5} - 8 q^{6} - 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(91\)	\(101\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.241682	−	1.39341i	−0.170895	−	0.985289i
							\(3\)	−1.73134	−	0.0494428i	−0.999592	−	0.0285458i
\(5\)	−1.26760	−	1.84206i	−0.566889	−	0.823794i
							\(7\)	−1.09131	+	4.07282i	−0.412476	+	1.53938i	0.377361	+	0.926066i	\(0.376832\pi\)
−0.789837	+	0.613317i	\(0.789835\pi\)
\(11\)	−0.447973	+	0.258637i	−0.135069	+	0.0779821i	−0.566012	−	0.824397i	\(-0.691514\pi\)
							0.430943	+	0.902379i	\(0.358181\pi\)
\(13\)	−1.88061	+	0.503908i	−0.521588	+	0.139759i	−0.510001	−	0.860174i	\(-0.670355\pi\)
							−0.0115869	+	0.999933i	\(0.503688\pi\)
\(17\)	−4.37652	+	4.37652i	−1.06146	+	1.06146i	−0.0634784	+	0.997983i	\(0.520219\pi\)
							−0.997983	+	0.0634784i	\(0.979781\pi\)
\(19\)	−5.33020			−1.22283			−0.611416	−	0.791309i	\(-0.709400\pi\)
							−0.611416	+	0.791309i	\(0.709400\pi\)
\(23\)	−0.996062	−	3.71735i	−0.207693	−	0.775122i	−0.988612	−	0.150489i	\(-0.951915\pi\)
							0.780918	−	0.624633i	\(-0.214751\pi\)
\(29\)	3.49078	−	2.01540i	0.648222	−	0.374251i	−0.139553	−	0.990215i	\(-0.544566\pi\)
							0.787775	+	0.615964i	\(0.211233\pi\)
\(31\)	−1.01483	−	0.585914i	−0.182269	−	0.105233i	0.406089	−	0.913833i	\(-0.366892\pi\)
							−0.588358	+	0.808600i	\(0.700226\pi\)
\(37\)	−1.89681	+	1.89681i	−0.311833	+	0.311833i	−0.845619	−	0.533786i	\(-0.820769\pi\)
							0.533786	+	0.845619i	\(0.320769\pi\)
\(41\)	−0.924526	+	1.60133i	−0.144387	+	0.250085i	−0.929144	−	0.369718i	\(-0.879454\pi\)
							0.784757	+	0.619803i	\(0.212788\pi\)
\(43\)	−1.13787	−	0.304891i	−0.173524	−	0.0464955i	0.171011	−	0.985269i	\(-0.445297\pi\)
							−0.344535	+	0.938774i	\(0.611963\pi\)
\(47\)	0.627826	−	2.34308i	0.0915779	−	0.341773i	−0.904901	−	0.425623i	\(-0.860055\pi\)
							0.996478	+	0.0838499i	\(0.0267216\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.2.x.a.103.15	yes	128
3.2	odd	2		540.2.y.a.523.18		128
4.3	odd	2	inner	180.2.x.a.103.10	yes	128
5.2	odd	4	inner	180.2.x.a.67.30	yes	128
5.3	odd	4		900.2.bf.e.607.3		128
5.4	even	2		900.2.bf.e.643.18		128
9.2	odd	6		540.2.y.a.343.24		128
9.7	even	3	inner	180.2.x.a.43.9	yes	128
12.11	even	2		540.2.y.a.523.23		128
15.2	even	4		540.2.y.a.307.3		128
20.3	even	4		900.2.bf.e.607.24		128
20.7	even	4	inner	180.2.x.a.67.9	yes	128
20.19	odd	2		900.2.bf.e.643.23		128
36.7	odd	6	inner	180.2.x.a.43.30	yes	128
36.11	even	6		540.2.y.a.343.3		128
45.2	even	12		540.2.y.a.127.23		128
45.7	odd	12	inner	180.2.x.a.7.10	✓	128
45.34	even	6		900.2.bf.e.43.24		128
45.43	odd	12		900.2.bf.e.7.23		128
60.47	odd	4		540.2.y.a.307.24		128
180.7	even	12	inner	180.2.x.a.7.15	yes	128
180.43	even	12		900.2.bf.e.7.18		128
180.47	odd	12		540.2.y.a.127.18		128
180.79	odd	6		900.2.bf.e.43.3		128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.x.a.7.10	✓	128	45.7	odd	12	inner
180.2.x.a.7.15	yes	128	180.7	even	12	inner
180.2.x.a.43.9	yes	128	9.7	even	3	inner
180.2.x.a.43.30	yes	128	36.7	odd	6	inner
180.2.x.a.67.9	yes	128	20.7	even	4	inner
180.2.x.a.67.30	yes	128	5.2	odd	4	inner
180.2.x.a.103.10	yes	128	4.3	odd	2	inner
180.2.x.a.103.15	yes	128	1.1	even	1	trivial
540.2.y.a.127.18		128	180.47	odd	12
540.2.y.a.127.23		128	45.2	even	12
540.2.y.a.307.3		128	15.2	even	4
540.2.y.a.307.24		128	60.47	odd	4
540.2.y.a.343.3		128	36.11	even	6
540.2.y.a.343.24		128	9.2	odd	6
540.2.y.a.523.18		128	3.2	odd	2
540.2.y.a.523.23		128	12.11	even	2
900.2.bf.e.7.18		128	180.43	even	12
900.2.bf.e.7.23		128	45.43	odd	12
900.2.bf.e.43.3		128	180.79	odd	6
900.2.bf.e.43.24		128	45.34	even	6
900.2.bf.e.607.3		128	5.3	odd	4
900.2.bf.e.607.24		128	20.3	even	4
900.2.bf.e.643.18		128	5.4	even	2
900.2.bf.e.643.23		128	20.19	odd	2