Embedded newform 221.2.z.a.216.8

• Label: 221.2.z.a.216.8
• Level: $221$
• Weight: $2$
• Character: 221.216
• 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.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			216.8
Character	\(\chi\)	\(=\)	221.216
Dual form			221.2.z.a.44.8

$q$-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 \(\left(\mathbb{Z}/221\mathbb{Z}\right)^\times\).

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{13}{16}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.192390	+	0.464472i	−0.136041	+	0.328431i	−0.977188	−	0.212374i	\(-0.931880\pi\)
							0.841148	+	0.540805i	\(0.181880\pi\)
\(3\)	1.59700	−	1.06708i	0.922026	−	0.616078i	−0.00133985	−	0.999999i	\(-0.500426\pi\)
							0.923366	+	0.383921i	\(0.125426\pi\)
\(5\)	0.191542	+	0.286663i	0.0856603	+	0.128200i	0.871825	−	0.489817i	\(-0.162936\pi\)
							−0.786165	+	0.618017i	\(0.787936\pi\)
\(7\)	1.43415	−	2.14636i	0.542058	−	0.811248i	−0.454788	−	0.890600i	\(-0.650285\pi\)
							0.996847	+	0.0793520i	\(0.0252851\pi\)
\(11\)	−0.0621893	−	0.312647i	−0.0187508	−	0.0942666i	0.970277	−	0.241998i	\(-0.0778028\pi\)
							−0.989027	+	0.147732i	\(0.952803\pi\)
\(13\)	−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								−0.569021	−	0.822323i	\(-0.692678\pi\)
							0.983829	−	0.179112i	\(-0.0573223\pi\)
\(23\)	2.41760	−	3.61820i	0.504105	−	0.754446i	−0.488923	−	0.872327i	\(-0.662610\pi\)
							0.993028	+	0.117881i	\(0.0376101\pi\)
\(29\)	−2.00736	+	10.0917i	−0.372757	+	1.87398i	0.103436	+	0.994636i	\(0.467016\pi\)
							−0.476193	+	0.879341i	\(0.657984\pi\)
\(31\)	−0.487006	+	2.44835i	−0.0874689	+	0.439736i	0.912089	+	0.409993i	\(0.134469\pi\)
							−0.999558	+	0.0297429i	\(0.990531\pi\)
\(37\)	0.864558	−	4.34643i	0.142132	−	0.714548i	−0.842331	−	0.538960i	\(-0.818817\pi\)
							0.984464	−	0.175588i	\(-0.0561826\pi\)
\(41\)	−4.15426	−	2.77579i	−0.648786	−	0.433505i	0.187150	−	0.982331i	\(-0.440075\pi\)
							−0.835937	+	0.548826i	\(0.815075\pi\)
\(43\)	−3.22850	−	1.33729i	−0.492342	−	0.203935i	0.122678	−	0.992447i	\(-0.460852\pi\)
							−0.615019	+	0.788512i	\(0.710852\pi\)
\(47\)	−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.216.8	yes	152
13.5	odd	4		221.2.ba.a.148.12	yes	152
17.10	odd	16		221.2.ba.a.112.12	yes	152
221.44	even	16	inner	221.2.z.a.44.8	✓	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.8	✓	152	221.44	even	16	inner
221.2.z.a.216.8	yes	152	1.1	even	1	trivial
221.2.ba.a.112.12	yes	152	17.10	odd	16
221.2.ba.a.148.12	yes	152	13.5	odd	4