Embedded newform 169.4.h.a.12.5

• Label: 169.4.h.a.12.5
• Level: $169$
• Weight: $4$
• Character: 169.12
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $528$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.h (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(44\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			12.5
Character	\(\chi\)	\(=\)	169.12
Dual form			169.4.h.a.155.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.98439 + 2.75022i) q^{2} +(8.05272 + 4.22640i) q^{3} +(5.47477 - 14.4358i) q^{4} +(14.3051 + 16.1471i) q^{5} +(-43.7087 + 5.30720i) q^{6} +(-2.97876 - 12.0853i) q^{7} +(8.61907 + 34.9689i) q^{8} +(31.6462 + 45.8474i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 13 q^{2} - 11 q^{3} + 157 q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 339 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{21}{26}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.98439	+	2.75022i	−1.40869	+	0.972351i	−0.410526	+	0.911849i	\(0.634655\pi\)
							−0.998168	+	0.0605019i	\(0.980730\pi\)
\(3\)	8.05272	+	4.22640i	1.54975	+	0.813370i	0.999762	−	0.0218041i	\(-0.00694102\pi\)
							0.549985	+	0.835174i	\(0.314633\pi\)
\(5\)	14.3051	+	16.1471i	1.27948	+	1.44424i	0.836749	+	0.547586i	\(0.184453\pi\)
							0.442735	+	0.896653i	\(0.354008\pi\)
\(7\)	−2.97876	−	12.0853i	−0.160838	−	0.652545i	−0.994899	−	0.100874i	\(-0.967836\pi\)
							0.834061	−	0.551672i	\(-0.186010\pi\)
\(11\)	−33.4559	−	23.0929i	−0.917030	−	0.632980i	0.0132176	−	0.999913i	\(-0.495793\pi\)
							−0.930248	+	0.366933i	\(0.880408\pi\)
\(13\)	−28.3948	+	37.2926i	−0.605792	+	0.795623i
							\(17\)	58.2449	−	14.3561i	0.830968	−	0.204815i	0.199187	−	0.979962i	\(-0.436170\pi\)
0.631782	+	0.775146i	\(0.282324\pi\)
\(19\)			55.8406i			0.674248i	0.941460	+	0.337124i	\(0.109454\pi\)
							−0.941460	+	0.337124i	\(0.890546\pi\)
\(23\)	108.617			0.984709			0.492355	−	0.870395i	\(-0.336136\pi\)
							0.492355	+	0.870395i	\(0.336136\pi\)
\(29\)	−76.9716	−	111.513i	−0.492871	−	0.714047i	0.495074	−	0.868851i	\(-0.335141\pi\)
							−0.987945	+	0.154804i	\(0.950526\pi\)
\(31\)	−13.0387	+	1.58318i	−0.0755426	+	0.00917253i	−0.158221	−	0.987404i	\(-0.550576\pi\)
							0.0826781	+	0.996576i	\(0.473653\pi\)
\(37\)	−156.294	+	18.9775i	−0.694448	+	0.0843212i	−0.460147	−	0.887843i	\(-0.652203\pi\)
							−0.234301	+	0.972164i	\(0.575280\pi\)
\(41\)	−129.043	+	245.871i	−0.491540	+	0.936552i	0.505729	+	0.862693i	\(0.331224\pi\)
							−0.997269	+	0.0738589i	\(0.976469\pi\)
\(43\)	58.5150	−	481.914i	0.207522	−	1.70910i	−0.401428	−	0.915891i	\(-0.631486\pi\)
							0.608950	−	0.793208i	\(-0.291591\pi\)
\(47\)	288.723	−	109.498i	0.896055	−	0.339829i	0.136782	−	0.990601i	\(-0.456324\pi\)
							0.759273	+	0.650772i	\(0.225555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.h.a.12.5	✓	528
169.155	even	26	inner	169.4.h.a.155.5	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.h.a.12.5	✓	528	1.1	even	1	trivial
169.4.h.a.155.5	yes	528	169.155	even	26	inner