Embedded newform 169.2.h.a.25.2

• Label: 169.2.h.a.25.2
• Level: $169$
• Weight: $2$
• Character: 169.25
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			25.2
Character	\(\chi\)	\(=\)	169.25
Dual form			169.2.h.a.142.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.16507 - 0.821103i) q^{2} +(1.73033 + 2.50681i) q^{3} +(2.51629 + 2.22924i) q^{4} +(3.34407 + 0.406044i) q^{5} +(-1.68793 - 6.84821i) q^{6} +(-0.858950 - 1.63659i) q^{7} +(-1.46534 - 2.79198i) q^{8} +(-2.22627 + 5.87018i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 13 q^{2} - 13 q^{3} + q^{4} - 13 q^{5} - 13 q^{6} - 13 q^{7} - 13 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{26}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.16507	−	0.821103i	−1.53094	−	0.580607i	−0.561830	−	0.827253i	\(-0.689902\pi\)
							−0.969106	+	0.246646i	\(0.920672\pi\)
\(3\)	1.73033	+	2.50681i	0.999006	+	1.44731i	0.891454	+	0.453111i	\(0.149686\pi\)
							0.107552	+	0.994199i	\(0.465699\pi\)
\(5\)	3.34407	+	0.406044i	1.49551	+	0.181588i	0.827038	−	0.562146i	\(-0.190024\pi\)
							0.668476	+	0.743734i	\(0.266947\pi\)
\(7\)	−0.858950	−	1.63659i	−0.324653	−	0.618574i	0.667357	−	0.744738i	\(-0.267426\pi\)
							−0.992009	+	0.126164i	\(0.959733\pi\)
\(11\)	−1.93228	+	0.732816i	−0.582604	+	0.220952i	−0.628258	−	0.778005i	\(-0.716232\pi\)
							0.0456547	+	0.998957i	\(0.485463\pi\)
\(13\)	2.73847	−	2.34537i	0.759516	−	0.650489i
							\(17\)	−3.92016	+	2.05746i	−0.950778	+	0.499007i	−0.867455	−	0.497516i	\(-0.834246\pi\)
−0.0833228	+	0.996523i	\(0.526553\pi\)
\(19\)		−	4.57866i		−	1.05042i	−0.850973	−	0.525209i	\(-0.823987\pi\)
							0.850973	−	0.525209i	\(-0.176013\pi\)
\(23\)	1.39775			0.291451			0.145726	−	0.989325i	\(-0.453448\pi\)
							0.145726	+	0.989325i	\(0.453448\pi\)
\(29\)	−0.913692	+	2.40921i	−0.169668	+	0.447379i	−0.992699	−	0.120618i	\(-0.961512\pi\)
							0.823031	+	0.567997i	\(0.192282\pi\)
\(31\)	2.22313	+	9.01957i	0.399285	+	1.61996i	0.735047	+	0.678016i	\(0.237160\pi\)
							−0.335762	+	0.941947i	\(0.608994\pi\)
\(37\)	−2.00301	−	8.12654i	−0.329293	−	1.33600i	−0.869279	−	0.494321i	\(-0.835417\pi\)
							0.539986	−	0.841674i	\(-0.318429\pi\)
\(41\)	0.588837	−	0.406445i	0.0919609	−	0.0634760i	−0.521197	−	0.853436i	\(-0.674514\pi\)
							0.613158	+	0.789960i	\(0.289899\pi\)
\(43\)	−6.47343	−	1.59556i	−0.987189	−	0.243320i	−0.287510	−	0.957778i	\(-0.592827\pi\)
							−0.699679	+	0.714458i	\(0.746674\pi\)
\(47\)	−7.62554	−	8.60746i	−1.11230	−	1.25553i	−0.963803	−	0.266614i	\(-0.914095\pi\)
							−0.148497	−	0.988913i	\(-0.547443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.h.a.25.2	✓	168
169.142	even	26	inner	169.2.h.a.142.2	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.h.a.25.2	✓	168	1.1	even	1	trivial
169.2.h.a.142.2	yes	168	169.142	even	26	inner