Embedded newform 289.2.g.a.101.5

• Label: 289.2.g.a.101.5
• Level: $289$
• Weight: $2$
• Character: 289.101
• 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			101.5
Character	\(\chi\)	\(=\)	289.101
Dual form			289.2.g.a.186.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18359 - 1.56733i) q^{2} +(1.30829 + 1.43512i) q^{3} +(-0.508305 + 1.78651i) q^{4} +(-1.37561 + 3.55086i) q^{5} +(0.700831 - 3.74912i) q^{6} +(-0.287579 - 0.0266482i) q^{7} +(-0.261137 + 0.101165i) q^{8} +(-0.0711567 + 0.767903i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 17 q^{2} - 17 q^{3} - 41 q^{4} + 17 q^{5} - 17 q^{6} - 17 q^{7} - 17 q^{8} + 7 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{34}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18359	−	1.56733i	−0.836926	−	1.10827i	−0.992716	−	0.120479i	\(-0.961557\pi\)
							0.155790	−	0.987790i	\(-0.450208\pi\)
\(3\)	1.30829	+	1.43512i	0.755340	+	0.828569i	0.989636	−	0.143597i	\(-0.0458668\pi\)
							−0.234296	+	0.972165i	\(0.575279\pi\)
\(5\)	−1.37561	+	3.55086i	−0.615191	+	1.58799i	0.181357	+	0.983417i	\(0.441951\pi\)
							−0.796549	+	0.604574i	\(0.793343\pi\)
\(7\)	−0.287579	−	0.0266482i	−0.108695	−	0.0100721i	0.0377834	−	0.999286i	\(-0.487970\pi\)
							−0.146478	+	0.989214i	\(0.546794\pi\)
\(11\)	−4.70847	−	1.33968i	−1.41966	−	0.403928i	−0.525162	−	0.851002i	\(-0.675995\pi\)
							−0.894497	+	0.447075i	\(0.852466\pi\)
\(13\)	−4.36236	+	1.68999i	−1.20990	+	0.468718i	−0.879853	−	0.475246i	\(-0.842359\pi\)
							−0.330048	+	0.943964i	\(0.607065\pi\)
\(17\)	−0.136386	+	4.12085i	−0.0330785	+	0.999453i
							\(19\)	1.52862	−	2.02423i	0.350690	−	0.464389i	−0.588145	−	0.808756i	\(-0.700142\pi\)
0.938835	+	0.344366i	\(0.111906\pi\)
\(23\)	−5.38846	−	0.499315i	−1.12357	−	0.104114i	−0.485605	−	0.874178i	\(-0.661401\pi\)
							−0.637967	+	0.770064i	\(0.720224\pi\)
\(29\)	7.58544	+	2.15824i	1.40858	+	0.400776i	0.890667	−	0.454657i	\(-0.150238\pi\)
							0.517914	+	0.855433i	\(0.326709\pi\)
\(31\)	1.86495	+	4.81398i	0.334954	+	0.864617i	0.993682	+	0.112235i	\(0.0358010\pi\)
							−0.658727	+	0.752382i	\(0.728905\pi\)
\(37\)	7.62654	+	3.79757i	1.25380	+	0.624316i	0.945080	−	0.326838i	\(-0.105983\pi\)
							0.308715	+	0.951154i	\(0.400101\pi\)
\(41\)	6.32183	+	6.93472i	0.987304	+	1.08302i	0.996544	+	0.0830635i	\(0.0264704\pi\)
							−0.00924031	+	0.999957i	\(0.502941\pi\)
\(43\)	3.98216	−	2.46565i	0.607274	−	0.376008i	−0.188079	−	0.982154i	\(-0.560226\pi\)
							0.795354	+	0.606145i	\(0.207285\pi\)
\(47\)	0.466615	+	5.03558i	0.0680628	+	0.734514i	0.959563	+	0.281494i	\(0.0908299\pi\)
							−0.891500	+	0.453020i	\(0.850347\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.g.a.101.5	✓	384
289.186	even	34	inner	289.2.g.a.186.5	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.g.a.101.5	✓	384	1.1	even	1	trivial
289.2.g.a.186.5	yes	384	289.186	even	34	inner