Embedded newform 169.4.i.a.3.2

• Label: 169.4.i.a.3.2
• Level: $169$
• Weight: $4$
• Character: 169.3
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $1080$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.i (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(1080\)
Relative dimension:	\(45\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			3.2
Character	\(\chi\)	\(=\)	169.3
Dual form			169.4.i.a.113.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.25141 + 3.19473i) q^{2} +(-1.68365 + 8.24708i) q^{3} +(5.64247 - 19.4801i) q^{4} +(-11.4022 - 5.98435i) q^{5} +(-19.1893 - 40.4405i) q^{6} +(-33.3973 - 5.42759i) q^{7} +(23.1588 + 61.0648i) q^{8} +(-40.3401 - 17.1873i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{31}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.25141	+	3.19473i	−1.50310	+	1.12951i	−0.546746	+	0.837298i	\(0.684134\pi\)
							−0.956355	+	0.292209i	\(0.905610\pi\)
\(3\)	−1.68365	+	8.24708i	−0.324019	+	1.58715i	0.412645	+	0.910892i	\(0.364605\pi\)
							−0.736664	+	0.676259i	\(0.763600\pi\)
\(5\)	−11.4022	−	5.98435i	−1.01985	−	0.535257i	−0.129998	−	0.991514i	\(-0.541497\pi\)
							−0.889848	+	0.456258i	\(0.849190\pi\)
\(7\)	−33.3973	−	5.42759i	−1.80329	−	0.293062i	−0.836371	−	0.548163i	\(-0.815327\pi\)
							−0.966914	+	0.255101i	\(0.917891\pi\)
\(11\)	5.57790	−	2.37652i	0.152891	−	0.0651408i	−0.314166	−	0.949368i	\(-0.601725\pi\)
							0.467057	+	0.884227i	\(0.345314\pi\)
\(13\)	−19.9219	+	42.4278i	−0.425026	+	0.905181i
							\(17\)	−51.1398	−	8.31103i	−0.729602	−	0.118572i	−0.215747	−	0.976449i	\(-0.569219\pi\)
−0.513855	+	0.857877i	\(0.671783\pi\)
\(19\)	−7.77469	+	13.4662i	−0.0938756	+	0.162597i	−0.909139	−	0.416493i	\(-0.863259\pi\)
							0.815263	+	0.579091i	\(0.196592\pi\)
\(23\)	83.4953	+	144.618i	0.756956	+	1.31109i	0.944396	+	0.328810i	\(0.106647\pi\)
							−0.187441	+	0.982276i	\(0.560019\pi\)
\(29\)	150.838	−	113.347i	0.965857	−	0.725795i	0.00418358	−	0.999991i	\(-0.498668\pi\)
							0.961674	+	0.274197i	\(0.0884119\pi\)
\(31\)	−103.447	+	149.868i	−0.599341	+	0.868295i	−0.998826	−	0.0484448i	\(-0.984574\pi\)
							0.399485	+	0.916740i	\(0.369189\pi\)
\(37\)	−359.385	−	29.0126i	−1.59683	−	0.128909i	−0.750242	−	0.661164i	\(-0.770063\pi\)
							−0.846584	+	0.532254i	\(0.821345\pi\)
\(41\)	43.7799	−	214.448i	0.166763	−	0.816858i	−0.807222	−	0.590247i	\(-0.799030\pi\)
							0.973985	−	0.226611i	\(-0.0727647\pi\)
\(43\)	−68.1454	+	5.50126i	−0.241676	+	0.0195101i	−0.200711	−	0.979650i	\(-0.564325\pi\)
							−0.0409650	+	0.999161i	\(0.513043\pi\)
\(47\)	93.7551	+	23.1085i	0.290970	+	0.0717176i	0.382099	−	0.924121i	\(-0.375201\pi\)
							−0.0911290	+	0.995839i	\(0.529048\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.i.a.3.2	✓	1080
169.113	even	39	inner	169.4.i.a.113.2	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.i.a.3.2	✓	1080	1.1	even	1	trivial
169.4.i.a.113.2	yes	1080	169.113	even	39	inner