Embedded newform 221.2.ba.a.112.16

• Label: 221.2.ba.a.112.16
• Level: $221$
• Weight: $2$
• Character: 221.112
• 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			112.16
Character	\(\chi\)	\(=\)	221.112
Dual form			221.2.ba.a.148.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.75759 - 0.728016i) q^{2} +(-0.779718 - 0.520991i) q^{3} +(1.14489 - 1.14489i) q^{4} +(-2.64697 - 1.76865i) q^{5} +(-1.74971 - 0.348039i) q^{6} +(3.96063 - 2.64641i) q^{7} +(-0.277288 + 0.669433i) q^{8} +(-0.811522 - 1.95919i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.75759	−	0.728016i	1.24280	−	0.514785i	0.338212	−	0.941070i	\(-0.390178\pi\)
							0.904589	+	0.426285i	\(0.140178\pi\)
\(3\)	−0.779718	−	0.520991i	−0.450170	−	0.300794i	0.309741	−	0.950821i	\(-0.399758\pi\)
							−0.759911	+	0.650027i	\(0.774758\pi\)
\(5\)	−2.64697	−	1.76865i	−1.18376	−	0.790964i	−0.201685	−	0.979450i	\(-0.564642\pi\)
							−0.982076	+	0.188487i	\(0.939642\pi\)
\(7\)	3.96063	−	2.64641i	1.49698	−	1.00025i	0.506553	−	0.862209i	\(-0.330920\pi\)
							0.990427	−	0.138040i	\(-0.0440804\pi\)
\(11\)	4.22578	+	0.840561i	1.27412	+	0.253439i	0.785393	−	0.618997i	\(-0.212461\pi\)
							0.488729	+	0.872436i	\(0.337461\pi\)
\(13\)	−3.60019	+	0.196515i	−0.998514	+	0.0545034i
							\(17\)	0.283954	+	4.11332i	0.0688688	+	0.997626i
\(19\)	2.25899	−	0.935706i	0.518249	−	0.214666i								−0.108199	−	0.994129i	\(-0.534508\pi\)
							0.626448	+	0.779464i	\(0.284508\pi\)
\(23\)	0.411512	+	0.615871i	0.0858061	+	0.128418i	0.871890	−	0.489702i	\(-0.162895\pi\)
							−0.786084	+	0.618120i	\(0.787895\pi\)
\(29\)	1.26782	+	6.37378i	0.235429	+	1.18358i	0.899842	+	0.436216i	\(0.143682\pi\)
							−0.664413	+	0.747366i	\(0.731318\pi\)
\(31\)	2.26831	−	0.451196i	0.407401	−	0.0810371i	0.0128643	−	0.999917i	\(-0.495905\pi\)
							0.394537	+	0.918880i	\(0.370905\pi\)
\(37\)	5.28200	−	1.05065i	0.868355	−	0.172727i	0.259244	−	0.965812i	\(-0.416527\pi\)
							0.609111	+	0.793085i	\(0.291527\pi\)
\(41\)	−3.45102	−	5.16482i	−0.538959	−	0.806609i	0.457629	−	0.889143i	\(-0.348699\pi\)
							−0.996588	+	0.0825339i	\(0.973699\pi\)
\(43\)	−2.32276	+	0.962120i	−0.354218	+	0.146722i	−0.552695	−	0.833384i	\(-0.686401\pi\)
							0.198477	+	0.980106i	\(0.436401\pi\)
\(47\)		−	8.28997i		−	1.20922i	−0.796523	−	0.604608i	\(-0.793330\pi\)
							0.796523	−	0.604608i	\(-0.206670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.112.16	yes	152
13.5	odd	4		221.2.z.a.44.4	✓	152
17.12	odd	16		221.2.z.a.216.4	yes	152
221.148	even	16	inner	221.2.ba.a.148.16	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.4	✓	152	13.5	odd	4
221.2.z.a.216.4	yes	152	17.12	odd	16
221.2.ba.a.112.16	yes	152	1.1	even	1	trivial
221.2.ba.a.148.16	yes	152	221.148	even	16	inner