Embedded newform 169.2.g.a.27.2

• Label: 169.2.g.a.27.2
• Level: $169$
• Weight: $2$
• Character: 169.27
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			27.2
Character	\(\chi\)	\(=\)	169.27
Dual form			169.2.g.a.144.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.814971 - 2.14890i) q^{2} +(0.722310 - 1.04645i) q^{3} +(-2.45658 + 2.17634i) q^{4} +(-0.370802 - 3.05383i) q^{5} +(-2.83737 - 0.699350i) q^{6} +(0.780099 + 0.409428i) q^{7} +(2.60879 + 1.36920i) q^{8} +(0.490496 + 1.29333i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q - 10 q^{2} - 9 q^{3} - 20 q^{4} - 7 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.814971	−	2.14890i	−0.576272	−	1.51950i	−0.833143	−	0.553058i	\(-0.813461\pi\)
							0.256871	−	0.966446i	\(-0.417308\pi\)
\(3\)	0.722310	−	1.04645i	0.417026	−	0.604166i	−0.557025	−	0.830496i	\(-0.688057\pi\)
							0.974051	+	0.226329i	\(0.0726726\pi\)
\(5\)	−0.370802	−	3.05383i	−0.165828	−	1.36571i	−0.803453	−	0.595368i	\(-0.797006\pi\)
							0.637625	−	0.770347i	\(-0.279917\pi\)
\(7\)	0.780099	+	0.409428i	0.294850	+	0.154749i	0.605662	−	0.795722i	\(-0.292909\pi\)
							−0.310812	+	0.950472i	\(0.600601\pi\)
\(11\)	0.433436	−	1.14288i	0.130686	−	0.344590i	−0.853662	−	0.520828i	\(-0.825623\pi\)
							0.984348	+	0.176237i	\(0.0563927\pi\)
\(13\)	−3.33830	−	1.36225i	−0.925879	−	0.377820i
							\(17\)	5.12941	+	2.69212i	1.24407	+	0.652936i	0.952769	−	0.303697i	\(-0.0982209\pi\)
0.291297	+	0.956633i	\(0.405913\pi\)
\(19\)	0.803292			0.184288			0.0921439	−	0.995746i	\(-0.470628\pi\)
							0.0921439	+	0.995746i	\(0.470628\pi\)
\(23\)	−1.28618			−0.268187			−0.134093	−	0.990969i	\(-0.542812\pi\)
							−0.134093	+	0.990969i	\(0.542812\pi\)
\(29\)	−2.29000	−	6.03824i	−0.425243	−	1.12127i	−0.960914	−	0.276846i	\(-0.910711\pi\)
							0.535672	−	0.844426i	\(-0.320059\pi\)
\(31\)	−4.54890	−	1.12120i	−0.817006	−	0.201374i	−0.191399	−	0.981512i	\(-0.561302\pi\)
							−0.625607	+	0.780138i	\(0.715149\pi\)
\(37\)	8.67654	+	2.13858i	1.42642	+	0.351580i	0.875596	−	0.483045i	\(-0.160469\pi\)
							0.550819	+	0.834624i	\(0.314315\pi\)
\(41\)	4.12633	−	5.97803i	0.644425	−	0.933611i	−0.355574	−	0.934648i	\(-0.615715\pi\)
							0.999999	+	0.00103695i	\(0.000330070\pi\)
\(43\)	3.22950	−	0.796001i	0.492494	−	0.121389i	0.0147560	−	0.999891i	\(-0.495303\pi\)
							0.477738	+	0.878502i	\(0.341457\pi\)
\(47\)	−4.38657	−	3.88617i	−0.639848	−	0.566856i	0.279576	−	0.960123i	\(-0.409806\pi\)
							−0.919424	+	0.393268i	\(0.871344\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.g.a.27.2	✓	156
169.144	even	13	inner	169.2.g.a.144.2	yes	156

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.g.a.27.2	✓	156	1.1	even	1	trivial
169.2.g.a.144.2	yes	156	169.144	even	13	inner