Embedded newform 169.2.k.a.10.3

• Label: 169.2.k.a.10.3
• Level: $169$
• Weight: $2$
• Character: 169.10
• 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			10.3
Character	\(\chi\)	\(=\)	169.10
Dual form			169.2.k.a.17.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99584 - 0.407454i) q^{2} +(1.34046 - 0.217845i) q^{3} +(1.97740 + 0.842491i) q^{4} +(-0.647035 + 2.62512i) q^{5} +(-2.76410 - 0.111389i) q^{6} +(-3.12689 + 1.48373i) q^{7} +(-0.250449 - 0.172873i) q^{8} +(-1.09624 + 0.365979i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{78}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99584	−	0.407454i	−1.41127	−	0.288113i	−0.566863	−	0.823812i	\(-0.691843\pi\)
							−0.844409	+	0.535699i	\(0.820048\pi\)
\(3\)	1.34046	−	0.217845i	0.773913	−	0.125773i	0.239386	−	0.970924i	\(-0.423054\pi\)
							0.534527	+	0.845151i	\(0.320490\pi\)
\(5\)	−0.647035	+	2.62512i	−0.289363	+	1.17399i	0.628730	+	0.777624i	\(0.283575\pi\)
							−0.918092	+	0.396367i	\(0.870271\pi\)
\(7\)	−3.12689	+	1.48373i	−1.18185	+	0.560796i	−0.915201	−	0.402997i	\(-0.867969\pi\)
							−0.266652	+	0.963793i	\(0.585917\pi\)
\(11\)	−0.457073	+	1.36910i	−0.137813	+	0.412800i	−0.994714	−	0.102680i	\(-0.967258\pi\)
							0.856902	+	0.515480i	\(0.172386\pi\)
\(13\)	−2.59114	−	2.50718i	−0.718654	−	0.695368i
							\(17\)	3.18715	+	6.71677i	0.772997	+	1.62906i	0.777352	+	0.629065i	\(0.216562\pi\)
−0.00435589	+	0.999991i	\(0.501387\pi\)
\(19\)	1.35625	+	0.783031i	0.311145	+	0.179640i	0.647439	−	0.762118i	\(-0.275840\pi\)
							−0.336294	+	0.941757i	\(0.609174\pi\)
\(23\)	2.33811	+	4.04973i	0.487530	+	0.844427i	0.999897	−	0.0143395i	\(-0.00456458\pi\)
							−0.512367	+	0.858767i	\(0.671231\pi\)
\(29\)	1.83251	−	8.97622i	0.340288	−	1.66684i	−0.345024	−	0.938594i	\(-0.612129\pi\)
							0.685313	−	0.728249i	\(-0.259666\pi\)
\(31\)	2.10123	+	4.00356i	0.377393	+	0.719061i	0.997875	−	0.0651589i	\(-0.0207554\pi\)
							−0.620482	+	0.784220i	\(0.713063\pi\)
\(37\)	2.23182	−	3.52934i	0.366909	−	0.580220i	−0.610386	−	0.792104i	\(-0.708986\pi\)
							0.977295	+	0.211884i	\(0.0679601\pi\)
\(41\)	−0.705434	−	4.34071i	−0.110170	−	0.677905i	−0.982112	−	0.188296i	\(-0.939704\pi\)
							0.871942	−	0.489609i	\(-0.162861\pi\)
\(43\)	6.97646	−	4.41165i	1.06390	−	0.672770i	0.116620	−	0.993177i	\(-0.462794\pi\)
							0.947280	+	0.320407i	\(0.103820\pi\)
\(47\)	−4.62870	−	0.562026i	−0.675166	−	0.0819799i	−0.224230	−	0.974536i	\(-0.571987\pi\)
							−0.450936	+	0.892556i	\(0.648910\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.k.a.10.3	✓	360
169.17	even	78	inner	169.2.k.a.17.3	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.k.a.10.3	✓	360	1.1	even	1	trivial
169.2.k.a.17.3	yes	360	169.17	even	78	inner