Embedded newform 169.4.h.a.12.4

• Label: 169.4.h.a.12.4
• 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			12.4
Character	\(\chi\)	\(=\)	169.12
Dual form			169.4.h.a.155.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.99097 + 2.75477i) q^{2} +(-3.64670 - 1.91393i) q^{3} +(5.50225 - 14.5082i) q^{4} +(6.25474 + 7.06014i) q^{5} +(19.8263 - 2.40735i) q^{6} +(-8.42904 - 34.1979i) q^{7} +(8.72323 + 35.3915i) q^{8} +(-5.70250 - 8.26149i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 13 q^{2} - 11 q^{3} + 157 q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 339 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.99097	+	2.75477i	−1.41102	+	0.973957i	−0.413022	+	0.910721i	\(0.635527\pi\)
							−0.997999	+	0.0632358i	\(0.979858\pi\)
\(3\)	−3.64670	−	1.91393i	−0.701807	−	0.368337i	0.0757692	−	0.997125i	\(-0.475859\pi\)
							−0.777576	+	0.628789i	\(0.783551\pi\)
\(5\)	6.25474	+	7.06014i	0.559441	+	0.631478i	0.958379	−	0.285500i	\(-0.0921596\pi\)
							−0.398938	+	0.916978i	\(0.630621\pi\)
\(7\)	−8.42904	−	34.1979i	−0.455125	−	1.84652i	−0.528101	−	0.849182i	\(-0.677096\pi\)
							0.0729759	−	0.997334i	\(-0.476750\pi\)
\(11\)	51.9373	+	35.8498i	1.42361	+	0.982646i	0.996934	+	0.0782410i	\(0.0249304\pi\)
							0.426674	+	0.904405i	\(0.359685\pi\)
\(13\)	−5.53082	+	46.5447i	−0.117998	+	0.993014i
							\(17\)	−93.2392	+	22.9814i	−1.33023	+	0.327871i	−0.839460	−	0.543421i	\(-0.817129\pi\)
−0.490765	+	0.871292i	\(0.663283\pi\)
\(19\)		−	89.3500i		−	1.07886i	−0.842031	−	0.539429i	\(-0.818640\pi\)
							0.842031	−	0.539429i	\(-0.181360\pi\)
\(23\)	41.7218			0.378243			0.189122	−	0.981954i	\(-0.439436\pi\)
							0.189122	+	0.981954i	\(0.439436\pi\)
\(29\)	−64.5244	−	93.4798i	−0.413168	−	0.598578i	0.560050	−	0.828459i	\(-0.310782\pi\)
							−0.973219	+	0.229881i	\(0.926166\pi\)
\(31\)	−157.147	+	19.0811i	−0.910466	+	0.110551i	−0.562348	−	0.826900i	\(-0.690102\pi\)
							−0.348117	+	0.937451i	\(0.613179\pi\)
\(37\)	8.97585	−	1.08987i	0.0398816	−	0.00484251i	−0.100572	−	0.994930i	\(-0.532067\pi\)
							0.140453	+	0.990087i	\(0.455144\pi\)
\(41\)	11.5005	−	21.9124i	0.0438068	−	0.0834668i	−0.862569	−	0.505940i	\(-0.831146\pi\)
							0.906375	+	0.422473i	\(0.138838\pi\)
\(43\)	14.2950	−	117.730i	0.0506968	−	0.417526i	−0.945161	−	0.326604i	\(-0.894096\pi\)
							0.995858	−	0.0909216i	\(-0.0289813\pi\)
\(47\)	−212.834	+	80.7174i	−0.660533	+	0.250507i	−0.662014	−	0.749492i	\(-0.730298\pi\)
							0.00148046	+	0.999999i	\(0.499529\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.h.a.12.4	✓	528
169.155	even	26	inner	169.4.h.a.155.4	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.h.a.12.4	✓	528	1.1	even	1	trivial
169.4.h.a.155.4	yes	528	169.155	even	26	inner