Embedded newform 221.2.ba.a.148.7

• Label: 221.2.ba.a.148.7
• Level: $221$
• Weight: $2$
• Character: 221.148
• 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			148.7
Character	\(\chi\)	\(=\)	221.148
Dual form			221.2.ba.a.112.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.934480 - 0.387074i) q^{2} +(-2.72959 + 1.82386i) q^{3} +(-0.690787 - 0.690787i) q^{4} +(-3.14186 + 2.09932i) q^{5} +(3.25672 - 0.647801i) q^{6} +(-0.0228111 - 0.0152419i) q^{7} +(1.15229 + 2.78187i) q^{8} +(2.97618 - 7.18512i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.934480	−	0.387074i	−0.660777	−	0.273703i	0.0269883	−	0.999636i	\(-0.491408\pi\)
							−0.687766	+	0.725933i	\(0.741408\pi\)
\(3\)	−2.72959	+	1.82386i	−1.57593	+	1.05300i	−0.610612	+	0.791930i	\(0.709076\pi\)
							−0.965319	+	0.261073i	\(0.915924\pi\)
\(5\)	−3.14186	+	2.09932i	−1.40508	+	0.938846i	−0.405387	+	0.914145i	\(0.632863\pi\)
							−0.999695	+	0.0247007i	\(0.992137\pi\)
\(7\)	−0.0228111	−	0.0152419i	−0.00862178	−	0.00576089i	0.551252	−	0.834339i	\(-0.314150\pi\)
							−0.559874	+	0.828578i	\(0.689150\pi\)
\(11\)	1.68691	−	0.335547i	0.508622	−	0.101171i	0.0658933	−	0.997827i	\(-0.479010\pi\)
							0.442729	+	0.896655i	\(0.354010\pi\)
\(13\)	−2.95553	−	2.06515i	−0.819717	−	0.572768i
							\(17\)	3.51397	+	2.15685i	0.852263	+	0.523114i
\(19\)	−2.31913	−	0.960614i	−0.532044	−	0.220380i								0.100454	−	0.994942i	\(-0.467970\pi\)
							−0.632498	+	0.774562i	\(0.717970\pi\)
\(23\)	−0.920919	+	1.37825i	−0.192025	+	0.287386i	−0.914972	−	0.403518i	\(-0.867787\pi\)
							0.722947	+	0.690904i	\(0.242787\pi\)
\(29\)	0.615794	−	3.09581i	0.114350	−	0.574877i	−0.880545	−	0.473963i	\(-0.842823\pi\)
							0.994895	−	0.100914i	\(-0.0321768\pi\)
\(31\)	1.88264	+	0.374481i	0.338133	+	0.0672588i	0.361236	−	0.932474i	\(-0.382355\pi\)
							−0.0231036	+	0.999733i	\(0.507355\pi\)
\(37\)	−5.98208	−	1.18991i	−0.983447	−	0.195620i	−0.322924	−	0.946425i	\(-0.604666\pi\)
							−0.660524	+	0.750805i	\(0.729666\pi\)
\(41\)	0.882736	−	1.32111i	0.137860	−	0.206322i	−0.756115	−	0.654439i	\(-0.772905\pi\)
							0.893975	+	0.448116i	\(0.147905\pi\)
\(43\)	6.47602	+	2.68246i	0.987584	+	0.409071i	0.817230	−	0.576312i	\(-0.195509\pi\)
							0.170355	+	0.985383i	\(0.445509\pi\)
\(47\)		−	2.82475i		−	0.412032i	−0.978549	−	0.206016i	\(-0.933950\pi\)
							0.978549	−	0.206016i	\(-0.0660498\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.148.7	yes	152
13.8	odd	4		221.2.z.a.216.13	yes	152
17.10	odd	16		221.2.z.a.44.13	✓	152
221.112	even	16	inner	221.2.ba.a.112.7	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.13	✓	152	17.10	odd	16
221.2.z.a.216.13	yes	152	13.8	odd	4
221.2.ba.a.112.7	yes	152	221.112	even	16	inner
221.2.ba.a.148.7	yes	152	1.1	even	1	trivial