Embedded newform 169.4.i.a.3.7

• Label: 169.4.i.a.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	169.3
Dual form			169.4.i.a.113.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.59815 + 2.70384i) q^{2} +(-1.15347 + 5.65006i) q^{3} +(3.41023 - 11.7735i) q^{4} +(1.58751 + 0.833188i) q^{5} +(-11.1265 - 23.4486i) q^{6} +(7.26213 + 1.18021i) q^{7} +(6.79493 + 17.9167i) q^{8} +(-5.75321 - 2.45122i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1080 q - 27 q^{2} - 19 q^{3} + 155 q^{4} - 30 q^{5} - 56 q^{6} + 9 q^{7} + 4 q^{8} + 364 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.59815	+	2.70384i	−1.27214	+	0.955950i	−0.999980	−	0.00630185i	\(-0.997994\pi\)
							−0.272159	+	0.962252i	\(0.587738\pi\)
\(3\)	−1.15347	+	5.65006i	−0.221985	+	1.08735i	0.704118	+	0.710083i	\(0.251343\pi\)
							−0.926103	+	0.377271i	\(0.876863\pi\)
\(5\)	1.58751	+	0.833188i	0.141991	+	0.0745226i	0.534222	−	0.845344i	\(-0.320604\pi\)
							−0.392232	+	0.919867i	\(0.628297\pi\)
\(7\)	7.26213	+	1.18021i	0.392118	+	0.0637254i	0.353274	−	0.935520i	\(-0.385068\pi\)
							0.0388440	+	0.999245i	\(0.487632\pi\)
\(11\)	53.5597	−	22.8196i	1.46808	−	0.625489i	0.497758	−	0.867316i	\(-0.334157\pi\)
							0.970319	+	0.241827i	\(0.0777465\pi\)
\(13\)	11.0811	−	45.5435i	0.236411	−	0.971653i
							\(17\)	96.2693	+	15.6453i	1.37345	+	0.223208i	0.801950	−	0.597391i	\(-0.203796\pi\)
0.571504	+	0.820599i	\(0.306360\pi\)
\(19\)	61.4812	−	106.489i	0.742356	−	1.28580i	−0.209064	−	0.977902i	\(-0.567042\pi\)
							0.951420	−	0.307896i	\(-0.0996249\pi\)
\(23\)	−40.7624	−	70.6026i	−0.369546	−	0.640072i	0.619949	−	0.784642i	\(-0.287153\pi\)
							−0.989495	+	0.144570i	\(0.953820\pi\)
\(29\)	78.4889	−	58.9806i	0.502587	−	0.377670i	−0.318998	−	0.947755i	\(-0.603346\pi\)
							0.821585	+	0.570086i	\(0.193090\pi\)
\(31\)	−164.650	+	238.537i	−0.953936	+	1.38201i	−0.0299656	+	0.999551i	\(0.509540\pi\)
							−0.923970	+	0.382464i	\(0.875076\pi\)
\(37\)	51.6766	+	4.17176i	0.229610	+	0.0185360i	0.194736	−	0.980856i	\(-0.437615\pi\)
							0.0348740	+	0.999392i	\(0.488897\pi\)
\(41\)	59.8685	−	293.255i	0.228046	−	1.11704i	−0.690988	−	0.722866i	\(-0.742824\pi\)
							0.919034	−	0.394177i	\(-0.128970\pi\)
\(43\)	153.029	−	12.3538i	0.542713	−	0.0438123i	0.193929	−	0.981016i	\(-0.437877\pi\)
							0.348784	+	0.937203i	\(0.386595\pi\)
\(47\)	167.870	+	41.3762i	0.520986	+	0.128412i	0.491035	−	0.871140i	\(-0.336619\pi\)
							0.0299510	+	0.999551i	\(0.490465\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.i.a.3.7	✓	1080
169.113	even	39	inner	169.4.i.a.113.7	yes	1080

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.i.a.3.7	✓	1080	1.1	even	1	trivial
169.4.i.a.113.7	yes	1080	169.113	even	39	inner