Embedded newform 169.2.k.a.4.13

• Label: 169.2.k.a.4.13
• Level: $169$
• Weight: $2$
• Character: 169.4
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.k (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(15\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	169.4
Dual form			169.2.k.a.127.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.95970 + 0.0789734i) q^{2} +(0.177516 - 0.612858i) q^{3} +(1.84069 + 0.148596i) q^{4} +(0.0713672 + 0.0270660i) q^{5} +(0.396279 - 1.18700i) q^{6} +(-0.605333 - 1.42077i) q^{7} +(-0.298517 - 0.0362466i) q^{8} +(2.19149 + 1.38581i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.95970	+	0.0789734i	1.38572	+	0.0558426i	0.721873	−	0.692026i	\(-0.243281\pi\)
							0.663847	+	0.747868i	\(0.268923\pi\)
\(3\)	0.177516	−	0.612858i	0.102489	−	0.353834i	−0.892840	−	0.450374i	\(-0.851291\pi\)
							0.995329	+	0.0965406i	\(0.0307778\pi\)
\(5\)	0.0713672	+	0.0270660i	0.0319164	+	0.0121043i	0.370511	−	0.928828i	\(-0.379182\pi\)
							−0.338595	+	0.940932i	\(0.609952\pi\)
\(7\)	−0.605333	−	1.42077i	−0.228794	−	0.537000i	0.765529	−	0.643402i	\(-0.222477\pi\)
							−0.994323	+	0.106401i	\(0.966067\pi\)
\(11\)	0.108462	+	0.171519i	0.0327024	+	0.0517148i	0.861146	−	0.508358i	\(-0.169747\pi\)
							−0.828443	+	0.560073i	\(0.810773\pi\)
\(13\)	−1.33034	+	3.35115i	−0.368969	+	0.929442i
							\(17\)	−5.66161	+	2.41219i	−1.37314	+	0.585041i	−0.947472	−	0.319838i	\(-0.896372\pi\)
−0.425669	+	0.904879i	\(0.639961\pi\)
\(19\)	0.558811	−	0.322630i	0.128200	−	0.0740164i	−0.434528	−	0.900658i	\(-0.643085\pi\)
							0.562729	+	0.826642i	\(0.309752\pi\)
\(23\)	2.33752	−	4.04870i	0.487406	−	0.844212i	−0.512489	−	0.858694i	\(-0.671276\pi\)
							0.999895	+	0.0144817i	\(0.00460982\pi\)
\(29\)	0.0968729	−	2.40388i	0.0179888	−	0.446388i	−0.964948	−	0.262443i	\(-0.915472\pi\)
							0.982936	−	0.183946i	\(-0.0588871\pi\)
\(31\)	6.07845	+	6.86115i	1.09172	+	1.23230i	0.970864	+	0.239630i	\(0.0770261\pi\)
							0.120858	+	0.992670i	\(0.461435\pi\)
\(37\)	−1.35802	+	0.277241i	−0.223256	+	0.0455781i	−0.310351	−	0.950622i	\(-0.600447\pi\)
							0.0870951	+	0.996200i	\(0.472242\pi\)
\(41\)	3.60147	+	1.04318i	0.562455	+	0.162917i	0.547253	−	0.836967i	\(-0.315674\pi\)
							0.0152022	+	0.999884i	\(0.495161\pi\)
\(43\)	0.272547	−	1.33502i	0.0415630	−	0.203589i	−0.953707	−	0.300738i	\(-0.902767\pi\)
							0.995270	+	0.0971488i	\(0.0309723\pi\)
\(47\)	−0.916812	+	0.632829i	−0.133731	+	0.0923076i	−0.633052	−	0.774109i	\(-0.718198\pi\)
							0.499321	+	0.866417i	\(0.333583\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.k.a.4.13	✓	360
169.127	even	78	inner	169.2.k.a.127.13	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.k.a.4.13	✓	360	1.1	even	1	trivial
169.2.k.a.127.13	yes	360	169.127	even	78	inner