Embedded newform 169.2.k.a.10.5

• Label: 169.2.k.a.10.5
• 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			10.5
Character	\(\chi\)	\(=\)	169.10
Dual form			169.2.k.a.17.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44039 - 0.294058i) q^{2} +(-2.46995 + 0.401407i) q^{3} +(0.148295 + 0.0631826i) q^{4} +(0.720721 - 2.92408i) q^{5} +(3.67574 + 0.148127i) q^{6} +(-2.39377 + 1.13586i) q^{7} +(2.22471 + 1.53561i) q^{8} +(3.09394 - 1.03291i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - 23 q^{2} - 24 q^{3} - 41 q^{4} - 26 q^{5} - 32 q^{6} - 26 q^{7} - 26 q^{8} - 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44039	−	0.294058i	−1.01851	−	0.207930i	−0.338370	−	0.941013i	\(-0.609876\pi\)
							−0.680139	+	0.733083i	\(0.738081\pi\)
\(3\)	−2.46995	+	0.401407i	−1.42603	+	0.231752i	−0.823940	−	0.566677i	\(-0.808229\pi\)
							−0.602089	+	0.798429i	\(0.705665\pi\)
\(5\)	0.720721	−	2.92408i	0.322316	−	1.30769i	−0.556627	−	0.830762i	\(-0.687905\pi\)
							0.878944	−	0.476926i	\(-0.158249\pi\)
\(7\)	−2.39377	+	1.13586i	−0.904761	+	0.429314i	−0.823459	−	0.567375i	\(-0.807959\pi\)
							−0.0813016	+	0.996690i	\(0.525908\pi\)
\(11\)	−0.728042	+	2.18075i	−0.219513	+	0.657522i	0.779984	+	0.625800i	\(0.215227\pi\)
							−0.999497	+	0.0317220i	\(0.989901\pi\)
\(13\)	1.92669	+	3.04760i	0.534368	+	0.845252i
							\(17\)	1.97711	+	4.16666i	0.479519	+	1.01056i	0.988700	+	0.149905i	\(0.0478969\pi\)
−0.509182	+	0.860659i	\(0.670052\pi\)
\(19\)	−1.79523	−	1.03648i	−0.411855	−	0.237785i	0.279732	−	0.960078i	\(-0.409755\pi\)
							−0.691586	+	0.722294i	\(0.743088\pi\)
\(23\)	3.57304	+	6.18869i	0.745031	+	1.29043i	0.950180	+	0.311701i	\(0.100899\pi\)
							−0.205149	+	0.978731i	\(0.565768\pi\)
\(29\)	0.932811	−	4.56921i	0.173219	−	0.848481i	−0.796572	−	0.604543i	\(-0.793356\pi\)
							0.969791	−	0.243938i	\(-0.0784393\pi\)
\(31\)	−1.28190	−	2.44246i	−0.230237	−	0.438679i	0.742741	−	0.669579i	\(-0.233525\pi\)
							−0.972978	+	0.230900i	\(0.925833\pi\)
\(37\)	−1.42045	+	2.24625i	−0.233520	+	0.369282i	−0.941535	−	0.336915i	\(-0.890616\pi\)
							0.708015	+	0.706197i	\(0.249591\pi\)
\(41\)	1.11879	+	6.88422i	0.174726	+	1.07513i	0.915185	+	0.403033i	\(0.132044\pi\)
							−0.740459	+	0.672101i	\(0.765392\pi\)
\(43\)	−7.73330	+	4.89024i	−1.17932	+	0.745755i	−0.972563	−	0.232639i	\(-0.925264\pi\)
							−0.206753	+	0.978393i	\(0.566290\pi\)
\(47\)	−1.71533	−	0.208278i	−0.250206	−	0.0303805i	−0.00552730	−	0.999985i	\(-0.501759\pi\)
							−0.244679	+	0.969604i	\(0.578682\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.k.a.10.5	✓	360
169.17	even	78	inner	169.2.k.a.17.5	yes	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.k.a.10.5	✓	360	1.1	even	1	trivial
169.2.k.a.17.5	yes	360	169.17	even	78	inner