Embedded newform 289.2.g.a.118.3

• Label: 289.2.g.a.118.3
• Level: $289$
• Weight: $2$
• Character: 289.118
• Analytic conductor: $2.308$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	289.g (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30767661842\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(24\) over \(\Q(\zeta_{34})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{34}]$

Embedding invariants

Embedding label			118.3
Character	\(\chi\)	\(=\)	289.118
Dual form			289.2.g.a.169.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93346 - 1.19715i) q^{2} +(-0.263727 + 1.41082i) q^{3} +(1.41364 + 2.83897i) q^{4} +(3.28580 - 0.304475i) q^{5} +(2.19887 - 2.41204i) q^{6} +(0.242364 - 0.625614i) q^{7} +(0.245802 - 2.65262i) q^{8} +(0.876569 + 0.339585i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 17 q^{2} - 17 q^{3} - 41 q^{4} + 17 q^{5} - 17 q^{6} - 17 q^{7} - 17 q^{8} + 7 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{19}{34}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93346	−	1.19715i	−1.36717	−	0.846513i	−0.370558	−	0.928809i	\(-0.620834\pi\)
							−0.996608	+	0.0822960i	\(0.973775\pi\)
\(3\)	−0.263727	+	1.41082i	−0.152263	+	0.814535i	0.818502	+	0.574503i	\(0.194805\pi\)
							−0.970765	+	0.240031i	\(0.922842\pi\)
\(5\)	3.28580	−	0.304475i	1.46946	−	0.136165i	0.672453	−	0.740140i	\(-0.265241\pi\)
							0.797003	+	0.603975i	\(0.206417\pi\)
\(7\)	0.242364	−	0.625614i	0.0916051	−	0.236460i	−0.879144	−	0.476556i	\(-0.841885\pi\)
							0.970749	+	0.240096i	\(0.0771790\pi\)
\(11\)	−3.96670	+	1.97518i	−1.19601	+	0.595540i	−0.929689	−	0.368345i	\(-0.879924\pi\)
							−0.266317	+	0.963886i	\(0.585807\pi\)
\(13\)	−0.0517138	+	0.558081i	−0.0143428	+	0.154784i	−0.999893	−	0.0145973i	\(-0.995353\pi\)
							0.985551	+	0.169381i	\(0.0541769\pi\)
\(17\)	1.05591	+	3.98561i	0.256096	+	0.966651i
							\(19\)	2.85486	−	1.76765i	0.654949	−	0.405527i	−0.158342	−	0.987384i	\(-0.550615\pi\)
0.813291	+	0.581857i	\(0.197674\pi\)
\(23\)	0.0390231	−	0.100730i	0.00813689	−	0.0210037i	−0.928345	−	0.371721i	\(-0.878768\pi\)
							0.936482	+	0.350717i	\(0.114062\pi\)
\(29\)	2.07217	−	1.03182i	0.384793	−	0.191604i	−0.242925	−	0.970045i	\(-0.578107\pi\)
							0.627718	+	0.778441i	\(0.283989\pi\)
\(31\)	4.72717	+	0.438037i	0.849026	+	0.0786738i	0.507976	−	0.861371i	\(-0.330394\pi\)
							0.341050	+	0.940045i	\(0.389217\pi\)
\(37\)	5.29892	+	1.50767i	0.871137	+	0.247860i	0.679562	−	0.733618i	\(-0.262170\pi\)
							0.191575	+	0.981478i	\(0.438640\pi\)
\(41\)	0.482024	−	2.57860i	0.0752795	−	0.402710i	−0.924489	−	0.381208i	\(-0.875508\pi\)
							0.999769	−	0.0215017i	\(-0.00684474\pi\)
\(43\)	2.03256	+	2.69154i	0.309962	+	0.410456i	0.926157	−	0.377137i	\(-0.123092\pi\)
							−0.616195	+	0.787593i	\(0.711327\pi\)
\(47\)	−6.92830	+	2.68404i	−1.01060	+	0.391507i	−0.808903	−	0.587942i	\(-0.799938\pi\)
							−0.201693	+	0.979449i	\(0.564644\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.g.a.118.3	✓	384
289.169	even	34	inner	289.2.g.a.169.3	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.g.a.118.3	✓	384	1.1	even	1	trivial
289.2.g.a.169.3	yes	384	289.169	even	34	inner