Embedded newform 221.2.ba.a.5.19

• Label: 221.2.ba.a.5.19
• 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			5.19
Character	\(\chi\)	\(=\)	221.5
Dual form			221.2.ba.a.177.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.51774 + 1.04288i) q^{2} +(-0.310064 - 0.464044i) q^{3} +(3.83722 + 3.83722i) q^{4} +(-1.13843 - 1.70379i) q^{5} +(-0.296719 - 1.49170i) q^{6} +(-1.65822 + 2.48170i) q^{7} +(3.57359 + 8.62741i) q^{8} +(1.02885 - 2.48387i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.51774	+	1.04288i	1.78031	+	0.737430i	0.992606	+	0.121382i	\(0.0387325\pi\)
							0.787708	+	0.616048i	\(0.211267\pi\)
\(3\)	−0.310064	−	0.464044i	−0.179016	−	0.267916i	0.731100	−	0.682270i	\(-0.239007\pi\)
							−0.910116	+	0.414355i	\(0.864007\pi\)
\(5\)	−1.13843	−	1.70379i	−0.509123	−	0.761957i	0.484490	−	0.874797i	\(-0.339005\pi\)
							−0.993613	+	0.112840i	\(0.964005\pi\)
\(7\)	−1.65822	+	2.48170i	−0.626748	+	0.937995i	0.373200	+	0.927751i	\(0.378261\pi\)
							−0.999948	+	0.0102436i	\(0.996739\pi\)
\(11\)	−0.615741	−	3.09554i	−0.185653	−	0.933340i	−0.955473	−	0.295079i	\(-0.904654\pi\)
							0.769820	−	0.638261i	\(-0.220346\pi\)
\(13\)	0.809088	+	3.51360i	0.224401	+	0.974497i
							\(17\)	−0.324598	−	4.11031i	−0.0787266	−	0.996896i
\(19\)	−7.19419	−	2.97993i	−1.65046	−	0.683643i								−0.653170	−	0.757211i	\(-0.726561\pi\)
							−0.997290	+	0.0735680i	\(0.976561\pi\)
\(23\)	−0.612434	−	0.409215i	−0.127701	−	0.0853273i	0.490079	−	0.871678i	\(-0.336968\pi\)
							−0.617780	+	0.786351i	\(0.711968\pi\)
\(29\)	3.14811	+	0.626198i	0.584589	+	0.116282i	0.478520	−	0.878077i	\(-0.341173\pi\)
							0.106069	+	0.994359i	\(0.466173\pi\)
\(31\)	−0.402862	+	2.02532i	−0.0723561	+	0.363759i	−0.999952	−	0.00982721i	\(-0.996872\pi\)
							0.927596	+	0.373586i	\(0.121872\pi\)
\(37\)	−2.02403	+	10.1755i	−0.332749	+	1.67284i	0.345818	+	0.938302i	\(0.387601\pi\)
							−0.678567	+	0.734539i	\(0.737399\pi\)
\(41\)	2.80667	+	1.87536i	0.438328	+	0.292881i	0.755087	−	0.655625i	\(-0.227595\pi\)
							−0.316759	+	0.948506i	\(0.602595\pi\)
\(43\)	4.78797	+	1.98324i	0.730158	+	0.302441i	0.716617	−	0.697467i	\(-0.245690\pi\)
							0.0135412	+	0.999908i	\(0.495690\pi\)
\(47\)			0.606308i			0.0884392i	0.999022	+	0.0442196i	\(0.0140801\pi\)
							−0.999022	+	0.0442196i	\(0.985920\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.5.19	yes	152
13.8	odd	4		221.2.z.a.73.1	✓	152
17.7	odd	16		221.2.z.a.109.1	yes	152
221.177	even	16	inner	221.2.ba.a.177.19	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.73.1	✓	152	13.8	odd	4
221.2.z.a.109.1	yes	152	17.7	odd	16
221.2.ba.a.5.19	yes	152	1.1	even	1	trivial
221.2.ba.a.177.19	yes	152	221.177	even	16	inner