Embedded newform 169.4.i.a.3.17

• Label: 169.4.i.a.3.17
• Level: $169$
• Weight: $4$
• Character: 169.3
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.i (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			3.17
Character	\(\chi\)	\(=\)	169.3
Dual form			169.4.i.a.113.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.80419 + 1.35576i) q^{2} +(0.148573 - 0.727759i) q^{3} +(-0.808720 + 2.79203i) q^{4} +(17.9465 + 9.41905i) q^{5} +(0.718615 + 1.51445i) q^{6} +(10.7406 + 1.74552i) q^{7} +(-8.72846 - 23.0150i) q^{8} +(24.3319 + 10.3668i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{31}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.80419	+	1.35576i	−0.637879	+	0.479335i	−0.869803	−	0.493399i	\(-0.835754\pi\)
							0.231924	+	0.972734i	\(0.425498\pi\)
\(3\)	0.148573	−	0.727759i	0.0285929	−	0.140057i	−0.962988	−	0.269543i	\(-0.913127\pi\)
							0.991581	+	0.129485i	\(0.0413326\pi\)
\(5\)	17.9465	+	9.41905i	1.60518	+	0.842465i	0.998884	+	0.0472281i	\(0.0150388\pi\)
							0.606299	+	0.795237i	\(0.292654\pi\)
\(7\)	10.7406	+	1.74552i	0.579939	+	0.0942492i	0.443302	−	0.896372i	\(-0.353807\pi\)
							0.136636	+	0.990621i	\(0.456371\pi\)
\(11\)	10.5595	−	4.49898i	0.289437	−	0.123318i	−0.242368	−	0.970184i	\(-0.577924\pi\)
							0.531805	+	0.846867i	\(0.321514\pi\)
\(13\)	13.2786	−	44.9520i	0.283293	−	0.959033i
							\(17\)	−10.0629	−	1.63538i	−0.143565	−	0.0233316i	0.0882040	−	0.996102i	\(-0.471887\pi\)
−0.231769	+	0.972771i	\(0.574451\pi\)
\(19\)	18.0291	−	31.2273i	0.217692	−	0.377054i	−0.736410	−	0.676536i	\(-0.763480\pi\)
							0.954102	+	0.299481i	\(0.0968137\pi\)
\(23\)	44.7189	+	77.4554i	0.405414	+	0.702198i	0.994370	−	0.105967i	\(-0.0337939\pi\)
							−0.588955	+	0.808166i	\(0.700461\pi\)
\(29\)	−37.2542	+	27.9947i	−0.238549	+	0.179258i	−0.713293	−	0.700866i	\(-0.752797\pi\)
							0.474744	+	0.880124i	\(0.342541\pi\)
\(31\)	−32.1685	+	46.6042i	−0.186375	+	0.270011i	−0.905023	−	0.425363i	\(-0.860146\pi\)
							0.718647	+	0.695375i	\(0.244762\pi\)
\(37\)	−193.252	−	15.6009i	−0.858659	−	0.0693181i	−0.356710	−	0.934215i	\(-0.616101\pi\)
							−0.501949	+	0.864897i	\(0.667384\pi\)
\(41\)	−78.0684	+	382.405i	−0.297372	+	1.45662i	0.506716	+	0.862113i	\(0.330859\pi\)
							−0.804088	+	0.594510i	\(0.797346\pi\)
\(43\)	49.6602	−	4.00898i	0.176119	−	0.0142178i	0.00790742	−	0.999969i	\(-0.497483\pi\)
							0.168211	+	0.985751i	\(0.446201\pi\)
\(47\)	−447.523	−	110.304i	−1.38889	−	0.342331i	−0.527190	−	0.849747i	\(-0.676755\pi\)
							−0.861701	+	0.507416i	\(0.830601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.i.a.3.17	✓	1080
169.113	even	39	inner	169.4.i.a.113.17	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.i.a.3.17	✓	1080	1.1	even	1	trivial
169.4.i.a.113.17	yes	1080	169.113	even	39	inner