Embedded newform 221.2.ba.a.31.1

• Label: 221.2.ba.a.31.1
• Level: $221$
• Weight: $2$
• Character: 221.31
• 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			31.1
Character	\(\chi\)	\(=\)	221.31
Dual form			221.2.ba.a.164.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.03706 + 2.50367i) q^{2} +(0.364483 - 1.83238i) q^{3} +(-3.77868 - 3.77868i) q^{4} +(-0.218534 + 1.09865i) q^{5} +(4.20969 + 2.81283i) q^{6} +(0.684640 + 3.44192i) q^{7} +(8.37194 - 3.46777i) q^{8} +(-0.453129 - 0.187692i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{9}{16}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.03706	+	2.50367i	−0.733309	+	1.77036i	−0.102068	+	0.994777i	\(0.532546\pi\)
							−0.631241	+	0.775587i	\(0.717454\pi\)
\(3\)	0.364483	−	1.83238i	0.210434	−	1.05792i	−0.720700	−	0.693248i	\(-0.756179\pi\)
							0.931134	−	0.364677i	\(-0.118821\pi\)
\(5\)	−0.218534	+	1.09865i	−0.0977315	+	0.491329i	0.900654	+	0.434536i	\(0.143088\pi\)
							−0.998386	+	0.0567934i	\(0.981912\pi\)
\(7\)	0.684640	+	3.44192i	0.258769	+	1.30092i	0.863442	+	0.504448i	\(0.168304\pi\)
							−0.604673	+	0.796474i	\(0.706696\pi\)
\(11\)	1.63420	+	1.09194i	0.492729	+	0.329231i	0.776987	−	0.629517i	\(-0.216747\pi\)
							−0.284258	+	0.958748i	\(0.591747\pi\)
\(13\)	−1.36552	+	3.33697i	−0.378727	+	0.925508i
							\(17\)	−3.53240	−	2.12654i	−0.856732	−	0.515762i
\(19\)	−2.66134	+	6.42505i	−0.610554	+	1.47401i								0.251840	+	0.967769i	\(0.418964\pi\)
							−0.862394	+	0.506238i	\(0.831036\pi\)
\(23\)	3.74040	−	0.744011i	0.779927	−	0.155137i	0.210957	−	0.977495i	\(-0.432342\pi\)
							0.568970	+	0.822358i	\(0.307342\pi\)
\(29\)	1.36787	+	2.04716i	0.254007	+	0.380149i	0.936456	−	0.350785i	\(-0.114085\pi\)
							−0.682449	+	0.730933i	\(0.739085\pi\)
\(31\)	−1.79553	+	1.19973i	−0.322486	+	0.215478i	−0.706266	−	0.707946i	\(-0.749622\pi\)
							0.383781	+	0.923424i	\(0.374622\pi\)
\(37\)	0.217383	−	0.145251i	0.0357376	−	0.0238791i	−0.537573	−	0.843217i	\(-0.680659\pi\)
							0.573311	+	0.819338i	\(0.305659\pi\)
\(41\)	2.42511	−	0.482385i	0.378739	−	0.0753358i	−0.00204997	−	0.999998i	\(-0.500653\pi\)
							0.380789	+	0.924662i	\(0.375653\pi\)
\(43\)	0.738277	−	1.78236i	0.112586	−	0.271807i	−0.857536	−	0.514424i	\(-0.828006\pi\)
							0.970122	+	0.242617i	\(0.0780058\pi\)
\(47\)		−	7.16791i		−	1.04555i	−0.852472	−	0.522774i	\(-0.824897\pi\)
							0.852472	−	0.522774i	\(-0.175103\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.31.1	yes	152
13.8	odd	4		221.2.z.a.99.1	yes	152
17.11	odd	16		221.2.z.a.96.1	✓	152
221.164	even	16	inner	221.2.ba.a.164.1	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.96.1	✓	152	17.11	odd	16
221.2.z.a.99.1	yes	152	13.8	odd	4
221.2.ba.a.31.1	yes	152	1.1	even	1	trivial
221.2.ba.a.164.1	yes	152	221.164	even	16	inner