Embedded newform 221.2.ba.a.148.9

• Label: 221.2.ba.a.148.9
• Level: $221$
• Weight: $2$
• Character: 221.148
• 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			148.9
Character	\(\chi\)	\(=\)	221.148
Dual form			221.2.ba.a.112.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.656637 - 0.271988i) q^{2} +(-0.671616 + 0.448759i) q^{3} +(-1.05702 - 1.05702i) q^{4} +(0.581728 - 0.388698i) q^{5} +(0.563065 - 0.112001i) q^{6} +(-1.99999 - 1.33635i) q^{7} +(0.950557 + 2.29485i) q^{8} +(-0.898367 + 2.16885i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.656637	−	0.271988i	−0.464313	−	0.192325i	0.138248	−	0.990398i	\(-0.455853\pi\)
							−0.602561	+	0.798073i	\(0.705853\pi\)
\(3\)	−0.671616	+	0.448759i	−0.387758	+	0.259091i	−0.734134	−	0.679004i	\(-0.762412\pi\)
							0.346377	+	0.938095i	\(0.387412\pi\)
\(5\)	0.581728	−	0.388698i	0.260157	−	0.173831i	−0.418651	−	0.908147i	\(-0.637497\pi\)
							0.678808	+	0.734316i	\(0.262497\pi\)
\(7\)	−1.99999	−	1.33635i	−0.755924	−	0.505093i	0.116890	−	0.993145i	\(-0.462708\pi\)
							−0.872814	+	0.488052i	\(0.837708\pi\)
\(11\)	−1.77048	+	0.352170i	−0.533819	+	0.106183i	−0.454637	−	0.890677i	\(-0.650231\pi\)
							−0.0791824	+	0.996860i	\(0.525231\pi\)
\(13\)	−3.60022	+	0.196068i	−0.998520	+	0.0543795i
							\(17\)	−3.75802	−	1.69625i	−0.911454	−	0.411402i
\(19\)	−4.14998	−	1.71898i	−0.952071	−	0.394361i								−0.148062	−	0.988978i	\(-0.547304\pi\)
							−0.804009	+	0.594617i	\(0.797304\pi\)
\(23\)	1.62153	−	2.42679i	0.338112	−	0.506021i	−0.622984	−	0.782235i	\(-0.714080\pi\)
							0.961096	+	0.276214i	\(0.0890798\pi\)
\(29\)	−0.192481	+	0.967666i	−0.0357428	+	0.179691i	−0.994533	−	0.104424i	\(-0.966700\pi\)
							0.958790	+	0.284116i	\(0.0916999\pi\)
\(31\)	−0.0645293	−	0.0128357i	−0.0115898	−	0.00230535i	0.189292	−	0.981921i	\(-0.439381\pi\)
							−0.200882	+	0.979615i	\(0.564381\pi\)
\(37\)	7.64730	+	1.52114i	1.25721	+	0.250074i	0.778357	−	0.627822i	\(-0.216053\pi\)
							0.478852	+	0.877896i	\(0.341053\pi\)
\(41\)	−2.55041	+	3.81696i	−0.398307	+	0.596109i	−0.975366	−	0.220594i	\(-0.929200\pi\)
							0.577058	+	0.816703i	\(0.304200\pi\)
\(43\)	4.48766	+	1.85885i	0.684361	+	0.283472i	0.697649	−	0.716440i	\(-0.254230\pi\)
							−0.0132876	+	0.999912i	\(0.504230\pi\)
\(47\)		−	6.47680i		−	0.944738i	−0.881401	−	0.472369i	\(-0.843399\pi\)
							0.881401	−	0.472369i	\(-0.156601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.148.9	yes	152
13.8	odd	4		221.2.z.a.216.11	yes	152
17.10	odd	16		221.2.z.a.44.11	✓	152
221.112	even	16	inner	221.2.ba.a.112.9	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.11	✓	152	17.10	odd	16
221.2.z.a.216.11	yes	152	13.8	odd	4
221.2.ba.a.112.9	yes	152	221.112	even	16	inner
221.2.ba.a.148.9	yes	152	1.1	even	1	trivial