Embedded newform 221.2.ba.a.112.13

• Label: 221.2.ba.a.112.13
• Level: $221$
• Weight: $2$
• Character: 221.112
• 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			112.13
Character	\(\chi\)	\(=\)	221.112
Dual form			221.2.ba.a.148.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.739446 - 0.306289i) q^{2} +(-0.471645 - 0.315143i) q^{3} +(-0.961245 + 0.961245i) q^{4} +(-2.95762 - 1.97622i) q^{5} +(-0.445281 - 0.0885718i) q^{6} +(-3.85090 + 2.57309i) q^{7} +(-1.02895 + 2.48410i) q^{8} +(-1.02492 - 2.47437i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.739446	−	0.306289i	0.522868	−	0.216579i	−0.105608	−	0.994408i	\(-0.533679\pi\)
							0.628476	+	0.777829i	\(0.283679\pi\)
\(3\)	−0.471645	−	0.315143i	−0.272304	−	0.181948i	0.411919	−	0.911220i	\(-0.364859\pi\)
							−0.684223	+	0.729273i	\(0.739859\pi\)
\(5\)	−2.95762	−	1.97622i	−1.32269	−	0.883793i	−0.324616	−	0.945846i	\(-0.605235\pi\)
							−0.998073	+	0.0620533i	\(0.980235\pi\)
\(7\)	−3.85090	+	2.57309i	−1.45550	+	0.972535i	−0.459046	+	0.888412i	\(0.651809\pi\)
							−0.996455	+	0.0841226i	\(0.973191\pi\)
\(11\)	0.336691	+	0.0669719i	0.101516	+	0.0201928i	0.245587	−	0.969375i	\(-0.421019\pi\)
							−0.144071	+	0.989567i	\(0.546019\pi\)
\(13\)	2.66147	−	2.43240i	0.738159	−	0.674626i
							\(17\)	2.96019	+	2.87007i	0.717951	+	0.696094i
\(19\)	−2.11728	+	0.877004i	−0.485736	+	0.201199i								−0.612092	−	0.790786i	\(-0.709672\pi\)
							0.126356	+	0.991985i	\(0.459672\pi\)
\(23\)	−2.35817	−	3.52926i	−0.491713	−	0.735901i	0.499767	−	0.866160i	\(-0.333419\pi\)
							−0.991480	+	0.130259i	\(0.958419\pi\)
\(29\)	−0.896391	−	4.50646i	−0.166456	−	0.836829i	−0.970284	−	0.241968i	\(-0.922207\pi\)
							0.803829	−	0.594861i	\(-0.202793\pi\)
\(31\)	−9.39049	+	1.86788i	−1.68658	+	0.335482i	−0.942906	−	0.333059i	\(-0.891919\pi\)
							−0.743676	+	0.668541i	\(0.766919\pi\)
\(37\)	−4.36293	+	0.867840i	−0.717261	+	0.142672i	−0.540211	−	0.841530i	\(-0.681655\pi\)
							−0.177050	+	0.984202i	\(0.556655\pi\)
\(41\)	4.31715	+	6.46108i	0.674226	+	1.00905i	0.998018	+	0.0629226i	\(0.0200421\pi\)
							−0.323792	+	0.946128i	\(0.604958\pi\)
\(43\)	−10.0247	+	4.15236i	−1.52875	+	0.633229i	−0.979321	−	0.202312i	\(-0.935154\pi\)
							−0.549428	+	0.835541i	\(0.685154\pi\)
\(47\)			4.43180i			0.646444i	0.946323	+	0.323222i	\(0.104766\pi\)
							−0.946323	+	0.323222i	\(0.895234\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.112.13	yes	152
13.5	odd	4		221.2.z.a.44.7	✓	152
17.12	odd	16		221.2.z.a.216.7	yes	152
221.148	even	16	inner	221.2.ba.a.148.13	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.7	✓	152	13.5	odd	4
221.2.z.a.216.7	yes	152	17.12	odd	16
221.2.ba.a.112.13	yes	152	1.1	even	1	trivial
221.2.ba.a.148.13	yes	152	221.148	even	16	inner