Embedded newform 221.2.ba.a.148.12

• Label: 221.2.ba.a.148.12
• Level: $221$
• Weight: $2$
• Character: 221.148
• 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.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			148.12
Character	\(\chi\)	\(=\)	221.148
Dual form			221.2.ba.a.112.12

$q$-expansion

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.464472	+	0.192390i	0.328431	+	0.136041i	0.540805	−	0.841148i	\(-0.318120\pi\)
							−0.212374	+	0.977188i	\(0.568120\pi\)
\(3\)	1.59700	−	1.06708i	0.922026	−	0.616078i	−0.00133985	−	0.999999i	\(-0.500426\pi\)
							0.923366	+	0.383921i	\(0.125426\pi\)
\(5\)	0.286663	−	0.191542i	0.128200	−	0.0856603i	−0.489817	−	0.871825i	\(-0.662936\pi\)
							0.618017	+	0.786165i	\(0.287936\pi\)
\(7\)	2.14636	+	1.43415i	0.811248	+	0.542058i	0.890600	−	0.454788i	\(-0.150285\pi\)
							−0.0793520	+	0.996847i	\(0.525285\pi\)
\(11\)	0.312647	−	0.0621893i	0.0942666	−	0.0187508i	−0.147732	−	0.989027i	\(-0.547197\pi\)
							0.241998	+	0.970277i	\(0.422197\pi\)
\(13\)	−0.0245246	−	3.60547i	−0.00680191	−	0.999977i
							\(17\)	3.23071	−	2.56174i	0.783562	−	0.621313i
\(19\)	−4.36515	−	1.80811i	−1.00143	−	0.414808i								−0.179112	−	0.983829i	\(-0.557322\pi\)
							−0.822323	+	0.569021i	\(0.807322\pi\)
\(23\)	−2.41760	+	3.61820i	−0.504105	+	0.754446i	−0.993028	−	0.117881i	\(-0.962390\pi\)
							0.488923	+	0.872327i	\(0.337390\pi\)
\(29\)	−2.00736	+	10.0917i	−0.372757	+	1.87398i	0.103436	+	0.994636i	\(0.467016\pi\)
							−0.476193	+	0.879341i	\(0.657984\pi\)
\(31\)	2.44835	+	0.487006i	0.439736	+	0.0874689i	0.409993	−	0.912089i	\(-0.365531\pi\)
							0.0297429	+	0.999558i	\(0.490531\pi\)
\(37\)	4.34643	+	0.864558i	0.714548	+	0.142132i	0.538960	−	0.842331i	\(-0.318817\pi\)
							0.175588	+	0.984464i	\(0.443817\pi\)
\(41\)	−2.77579	+	4.15426i	−0.433505	+	0.648786i	−0.982331	−	0.187150i	\(-0.940075\pi\)
							0.548826	+	0.835937i	\(0.315075\pi\)
\(43\)	3.22850	+	1.33729i	0.492342	+	0.203935i	0.615019	−	0.788512i	\(-0.289148\pi\)
							−0.122678	+	0.992447i	\(0.539148\pi\)
\(47\)		−	9.32408i		−	1.36006i	−0.733186	−	0.680028i	\(-0.761968\pi\)
							0.733186	−	0.680028i	\(-0.238032\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.8	✓	152	17.10	odd	16
221.2.z.a.216.8	yes	152	13.8	odd	4
221.2.ba.a.112.12	yes	152	221.112	even	16	inner
221.2.ba.a.148.12	yes	152	1.1	even	1	trivial