Embedded newform 221.2.z.a.216.13

• Label: 221.2.z.a.216.13
• 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			216.13
Character	\(\chi\)	\(=\)	221.216
Dual form			221.2.z.a.44.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.387074 - 0.934480i) q^{2} +(-2.72959 + 1.82386i) q^{3} +(0.690787 + 0.690787i) q^{4} +(-2.09932 - 3.14186i) q^{5} +(0.647801 + 3.25672i) q^{6} +(-0.0152419 + 0.0228111i) q^{7} +(2.78187 - 1.15229i) q^{8} +(2.97618 - 7.18512i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.387074	−	0.934480i	0.273703	−	0.660777i	−0.725933	−	0.687766i	\(-0.758592\pi\)
							0.999636	+	0.0269883i	\(0.00859167\pi\)
\(3\)	−2.72959	+	1.82386i	−1.57593	+	1.05300i	−0.610612	+	0.791930i	\(0.709076\pi\)
							−0.965319	+	0.261073i	\(0.915924\pi\)
\(5\)	−2.09932	−	3.14186i	−0.938846	−	1.40508i	−0.914145	−	0.405387i	\(-0.867137\pi\)
							−0.0247007	−	0.999695i	\(-0.507863\pi\)
\(7\)	−0.0152419	+	0.0228111i	−0.00576089	+	0.00862178i	−0.834339	−	0.551252i	\(-0.814150\pi\)
							0.828578	+	0.559874i	\(0.189150\pi\)
\(11\)	−0.335547	−	1.68691i	−0.101171	−	0.508622i	−0.997827	−	0.0658933i	\(-0.979010\pi\)
							0.896655	−	0.442729i	\(-0.145990\pi\)
\(13\)	−2.06515	−	2.95553i	−0.572768	−	0.819717i
							\(17\)	−3.51397	−	2.15685i	−0.852263	−	0.523114i
\(19\)	0.960614	−	2.31913i	0.220380	−	0.532044i								−0.774562	−	0.632498i	\(-0.782030\pi\)
							0.994942	+	0.100454i	\(0.0320296\pi\)
\(23\)	0.920919	−	1.37825i	0.192025	−	0.287386i	−0.722947	−	0.690904i	\(-0.757213\pi\)
							0.914972	+	0.403518i	\(0.132213\pi\)
\(29\)	0.615794	−	3.09581i	0.114350	−	0.574877i	−0.880545	−	0.473963i	\(-0.842823\pi\)
							0.994895	−	0.100914i	\(-0.0321768\pi\)
\(31\)	−0.374481	+	1.88264i	−0.0672588	+	0.338133i	−0.999733	−	0.0231036i	\(-0.992645\pi\)
							0.932474	+	0.361236i	\(0.117645\pi\)
\(37\)	−1.18991	+	5.98208i	−0.195620	+	0.983447i	0.750805	+	0.660524i	\(0.229666\pi\)
							−0.946425	+	0.322924i	\(0.895334\pi\)
\(41\)	1.32111	+	0.882736i	0.206322	+	0.137860i	0.654439	−	0.756115i	\(-0.272905\pi\)
							−0.448116	+	0.893975i	\(0.647905\pi\)
\(43\)	−6.47602	−	2.68246i	−0.987584	−	0.409071i	−0.170355	−	0.985383i	\(-0.554491\pi\)
							−0.817230	+	0.576312i	\(0.804491\pi\)
\(47\)	−2.82475			−0.412032			−0.206016	−	0.978549i	\(-0.566050\pi\)
							−0.206016	+	0.978549i	\(0.566050\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.44.13	✓	152	221.44	even	16	inner
221.2.z.a.216.13	yes	152	1.1	even	1	trivial
221.2.ba.a.112.7	yes	152	17.10	odd	16
221.2.ba.a.148.7	yes	152	13.5	odd	4