Embedded newform 221.2.ba.a.5.16

• Label: 221.2.ba.a.5.16
• Level: $221$
• Weight: $2$
• Character: 221.5
• Analytic conductor: $1.765$
• Analytic rank: $0$
• Dimension: $152$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 221 = 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	221.ba (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.76469388467\)
Analytic rank:	\(0\)
Dimension:	\(152\)
Relative dimension:	\(19\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			5.16
Character	\(\chi\)	\(=\)	221.5
Dual form			221.2.ba.a.177.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.74086 + 0.721088i) q^{2} +(-1.81248 - 2.71257i) q^{3} +(1.09641 + 1.09641i) q^{4} +(-0.443834 - 0.664244i) q^{5} +(-1.19927 - 6.02916i) q^{6} +(0.330817 - 0.495102i) q^{7} +(-0.324086 - 0.782412i) q^{8} +(-2.92489 + 7.06132i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/221\mathbb{Z}\right)^\times\).

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.74086	+	0.721088i	1.23097	+	0.509886i	0.900883	−	0.434063i	\(-0.142920\pi\)
							0.330091	+	0.943949i	\(0.392920\pi\)
\(3\)	−1.81248	−	2.71257i	−1.04644	−	1.56610i	−0.802812	−	0.596233i	\(-0.796663\pi\)
							−0.243625	−	0.969870i	\(-0.578337\pi\)
\(5\)	−0.443834	−	0.664244i	−0.198488	−	0.297059i	0.718852	−	0.695164i	\(-0.244668\pi\)
							−0.917340	+	0.398105i	\(0.869668\pi\)
\(7\)	0.330817	−	0.495102i	0.125037	−	0.187131i	−0.763670	−	0.645607i	\(-0.776604\pi\)
							0.888707	+	0.458476i	\(0.151604\pi\)
\(11\)	−1.13221	−	5.69202i	−0.341375	−	1.71621i	−0.645657	−	0.763627i	\(-0.723417\pi\)
							0.304282	−	0.952582i	\(-0.401583\pi\)
\(13\)	3.40207	+	1.19412i	0.943564	+	0.331189i
							\(17\)	−0.330194	+	4.10986i	−0.0800839	+	0.996788i
\(19\)	4.72753	+	1.95821i	1.08457	+	0.449244i								0.852110	−	0.523362i	\(-0.175322\pi\)
							0.232460	+	0.972606i	\(0.425322\pi\)
\(23\)	4.22506	+	2.82309i	0.880985	+	0.588655i	0.911693	−	0.410873i	\(-0.134776\pi\)
							−0.0307075	+	0.999528i	\(0.509776\pi\)
\(29\)	−3.13263	−	0.623120i	−0.581716	−	0.115710i	−0.104544	−	0.994520i	\(-0.533338\pi\)
							−0.477172	+	0.878810i	\(0.658338\pi\)
\(31\)	−0.999876	+	5.02672i	−0.179583	+	0.902825i	0.780936	+	0.624611i	\(0.214742\pi\)
							−0.960519	+	0.278214i	\(0.910258\pi\)
\(37\)	−0.660199	+	3.31905i	−0.108536	+	0.545648i	0.887808	+	0.460214i	\(0.152227\pi\)
							−0.996344	+	0.0854334i	\(0.972773\pi\)
\(41\)	0.255756	+	0.170891i	0.0399424	+	0.0266887i	0.575381	−	0.817885i	\(-0.304854\pi\)
							−0.535439	+	0.844574i	\(0.679854\pi\)
\(43\)	−1.98251	−	0.821184i	−0.302330	−	0.125229i	0.226361	−	0.974043i	\(-0.427317\pi\)
							−0.528691	+	0.848814i	\(0.677317\pi\)
\(47\)		−	2.31805i		−	0.338122i	−0.985606	−	0.169061i	\(-0.945926\pi\)
							0.985606	−	0.169061i	\(-0.0540735\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.5.16	yes	152
13.8	odd	4		221.2.z.a.73.4	✓	152
17.7	odd	16		221.2.z.a.109.4	yes	152
221.177	even	16	inner	221.2.ba.a.177.16	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.73.4	✓	152	13.8	odd	4
221.2.z.a.109.4	yes	152	17.7	odd	16
221.2.ba.a.5.16	yes	152	1.1	even	1	trivial
221.2.ba.a.177.16	yes	152	221.177	even	16	inner