Embedded newform 221.2.ba.a.5.15

• Label: 221.2.ba.a.5.15
• Level: $221$
• Weight: $2$
• Character: 221.5
• 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			5.15
Character	\(\chi\)	\(=\)	221.5
Dual form			221.2.ba.a.177.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.59516 + 0.660736i) q^{2} +(0.120163 + 0.179836i) q^{3} +(0.693746 + 0.693746i) q^{4} +(1.66060 + 2.48526i) q^{5} +(0.0728543 + 0.366263i) q^{6} +(-2.79811 + 4.18767i) q^{7} +(-0.673221 - 1.62530i) q^{8} +(1.13015 - 2.72842i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} - 16 q^{3} - 8 q^{5} - 8 q^{6} - 8 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(105\)	\(171\)
\(\chi(n)\)	\(e\left(\frac{5}{16}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.59516	+	0.660736i	1.12795	+	0.467211i	0.867083	−	0.498164i	\(-0.165992\pi\)
							0.260865	+	0.965375i	\(0.415992\pi\)
\(3\)	0.120163	+	0.179836i	0.0693759	+	0.103828i	0.864535	−	0.502572i	\(-0.167613\pi\)
							−0.795159	+	0.606400i	\(0.792613\pi\)
\(5\)	1.66060	+	2.48526i	0.742642	+	1.11144i	0.989800	+	0.142467i	\(0.0455034\pi\)
							−0.247158	+	0.968975i	\(0.579497\pi\)
\(7\)	−2.79811	+	4.18767i	−1.05759	+	1.58279i	−0.273775	+	0.961794i	\(0.588272\pi\)
							−0.783813	+	0.620997i	\(0.786728\pi\)
\(11\)	−0.393906	−	1.98030i	−0.118767	−	0.597082i	−0.993628	−	0.112708i	\(-0.964047\pi\)
							0.874861	−	0.484374i	\(-0.160953\pi\)
\(13\)	1.59643	−	3.23287i	0.442769	−	0.896636i
							\(17\)	3.51525	+	2.15476i	0.852574	+	0.522606i
\(19\)	2.80344	+	1.16122i	0.643154	+	0.266403i								0.680330	−	0.732906i	\(-0.261836\pi\)
							−0.0371764	+	0.999309i	\(0.511836\pi\)
\(23\)	−6.91500	−	4.62045i	−1.44188	−	0.963431i	−0.997721	−	0.0674818i	\(-0.978504\pi\)
							−0.444156	−	0.895949i	\(-0.646496\pi\)
\(29\)	0.922259	+	0.183449i	0.171259	+	0.0340656i	0.279975	−	0.960007i	\(-0.409674\pi\)
							−0.108716	+	0.994073i	\(0.534674\pi\)
\(31\)	−0.133927	+	0.673299i	−0.0240541	+	0.120928i	−0.990947	−	0.134255i	\(-0.957136\pi\)
							0.966893	+	0.255183i	\(0.0821358\pi\)
\(37\)	−0.982187	+	4.93779i	−0.161470	+	0.811767i	0.812124	+	0.583484i	\(0.198311\pi\)
							−0.973595	+	0.228283i	\(0.926689\pi\)
\(41\)	−6.80325	−	4.54578i	−1.06249	−	0.709932i	−0.103860	−	0.994592i	\(-0.533119\pi\)
							−0.958629	+	0.284660i	\(0.908119\pi\)
\(43\)	−5.16931	−	2.14120i	−0.788312	−	0.326529i	−0.0480472	−	0.998845i	\(-0.515300\pi\)
							−0.740265	+	0.672316i	\(0.765300\pi\)
\(47\)			3.19931i			0.466667i	0.972397	+	0.233334i	\(0.0749634\pi\)
							−0.972397	+	0.233334i	\(0.925037\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	221.2.ba.a.5.15	yes	152
13.8	odd	4		221.2.z.a.73.5	✓	152
17.7	odd	16		221.2.z.a.109.5	yes	152
221.177	even	16	inner	221.2.ba.a.177.15	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.z.a.73.5	✓	152	13.8	odd	4
221.2.z.a.109.5	yes	152	17.7	odd	16
221.2.ba.a.5.15	yes	152	1.1	even	1	trivial
221.2.ba.a.177.15	yes	152	221.177	even	16	inner