Embedded newform 221.2.z.a.44.8

• Label: 221.2.z.a.44.8
• Level: $221$
• Weight: $2$
• Character: 221.44
• Analytic conductor: $1.765$
• Analytic rank: $0$
• Dimension: $152$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 221 = 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	221.z (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			44.8
Character	\(\chi\)	\(=\)	221.44
Dual form			221.2.z.a.216.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.192390 - 0.464472i) q^{2} +(1.59700 + 1.06708i) q^{3} +(1.23549 - 1.23549i) q^{4} +(0.191542 - 0.286663i) q^{5} +(0.188381 - 0.947055i) q^{6} +(1.43415 + 2.14636i) q^{7} +(-1.74049 - 0.720935i) q^{8} +(0.263689 + 0.636602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.192390	−	0.464472i	−0.136041	−	0.328431i	0.841148	−	0.540805i	\(-0.181880\pi\)
							−0.977188	+	0.212374i	\(0.931880\pi\)
\(3\)	1.59700	+	1.06708i	0.922026	+	0.616078i	0.923366	−	0.383921i	\(-0.125426\pi\)
							−0.00133985	+	0.999999i	\(0.500426\pi\)
\(5\)	0.191542	−	0.286663i	0.0856603	−	0.128200i	−0.786165	−	0.618017i	\(-0.787936\pi\)
							0.871825	+	0.489817i	\(0.162936\pi\)
\(7\)	1.43415	+	2.14636i	0.542058	+	0.811248i	0.996847	−	0.0793520i	\(-0.0252851\pi\)
							−0.454788	+	0.890600i	\(0.650285\pi\)
\(11\)	−0.0621893	+	0.312647i	−0.0187508	+	0.0942666i	−0.989027	−	0.147732i	\(-0.952803\pi\)
							0.970277	+	0.241998i	\(0.0778028\pi\)
\(13\)	−3.60547	+	0.0245246i	−0.999977	+	0.00680191i
							\(17\)	−3.23071	−	2.56174i	−0.783562	−	0.621313i
\(19\)	1.80811	+	4.36515i	0.414808	+	1.00143i								0.983829	+	0.179112i	\(0.0573223\pi\)
							−0.569021	+	0.822323i	\(0.692678\pi\)
\(23\)	2.41760	+	3.61820i	0.504105	+	0.754446i	0.993028	−	0.117881i	\(-0.0376101\pi\)
							−0.488923	+	0.872327i	\(0.662610\pi\)
\(29\)	−2.00736	−	10.0917i	−0.372757	−	1.87398i	−0.476193	−	0.879341i	\(-0.657984\pi\)
							0.103436	−	0.994636i	\(-0.467016\pi\)
\(31\)	−0.487006	−	2.44835i	−0.0874689	−	0.439736i	−0.999558	−	0.0297429i	\(-0.990531\pi\)
							0.912089	−	0.409993i	\(-0.134469\pi\)
\(37\)	0.864558	+	4.34643i	0.142132	+	0.714548i	0.984464	+	0.175588i	\(0.0561826\pi\)
							−0.842331	+	0.538960i	\(0.818817\pi\)
\(41\)	−4.15426	+	2.77579i	−0.648786	+	0.433505i	−0.835937	−	0.548826i	\(-0.815075\pi\)
							0.187150	+	0.982331i	\(0.440075\pi\)
\(43\)	−3.22850	+	1.33729i	−0.492342	+	0.203935i	−0.615019	−	0.788512i	\(-0.710852\pi\)
							0.122678	+	0.992447i	\(0.460852\pi\)
\(47\)	−9.32408			−1.36006			−0.680028	−	0.733186i	\(-0.738032\pi\)
							−0.680028	+	0.733186i	\(0.738032\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.z.a.44.8	✓	152
13.8	odd	4		221.2.ba.a.112.12	yes	152
17.12	odd	16		221.2.ba.a.148.12	yes	152
221.216	even	16	inner	221.2.z.a.216.8	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.8	✓	152	1.1	even	1	trivial
221.2.z.a.216.8	yes	152	221.216	even	16	inner
221.2.ba.a.112.12	yes	152	13.8	odd	4
221.2.ba.a.148.12	yes	152	17.12	odd	16