Embedded newform 289.2.g.a.135.21

• Label: 289.2.g.a.135.21
• Level: $289$
• Weight: $2$
• Character: 289.135
• 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			135.21
Character	\(\chi\)	\(=\)	289.135
Dual form			289.2.g.a.152.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.07129 + 0.387192i) q^{2} +(1.49935 - 0.746589i) q^{3} +(2.27540 + 0.881494i) q^{4} +(-1.56941 - 2.53468i) q^{5} +(3.39467 - 0.965868i) q^{6} +(1.27355 + 0.961742i) q^{7} +(0.788602 + 0.488282i) q^{8} +(-0.117239 + 0.155250i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 17 q^{2} - 17 q^{3} - 41 q^{4} + 17 q^{5} - 17 q^{6} - 17 q^{7} - 17 q^{8} + 7 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{34}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.07129	+	0.387192i	1.46463	+	0.273786i	0.854886	−	0.518815i	\(-0.173627\pi\)
							0.609740	+	0.792602i	\(0.291274\pi\)
\(3\)	1.49935	−	0.746589i	0.865652	−	0.431043i	0.0426554	−	0.999090i	\(-0.486418\pi\)
							0.822997	+	0.568046i	\(0.192301\pi\)
\(5\)	−1.56941	−	2.53468i	−0.701862	−	1.13355i	−0.984680	−	0.174372i	\(-0.944211\pi\)
							0.282818	−	0.959173i	\(-0.408731\pi\)
\(7\)	1.27355	+	0.961742i	0.481358	+	0.363504i	0.815252	−	0.579106i	\(-0.196598\pi\)
							−0.333895	+	0.942610i	\(0.608363\pi\)
\(11\)	−1.94110	+	5.01057i	−0.585265	+	1.51074i	0.254101	+	0.967178i	\(0.418220\pi\)
							−0.839367	+	0.543566i	\(0.817074\pi\)
\(13\)	0.486100	+	0.300981i	0.134820	+	0.0834770i	0.592185	−	0.805802i	\(-0.298266\pi\)
							−0.457365	+	0.889279i	\(0.651207\pi\)
\(17\)	2.07195	−	3.56469i	0.502523	−	0.864564i
							\(19\)	−2.41551	+	0.451537i	−0.554156	+	0.103590i	−0.453376	−	0.891319i	\(-0.649781\pi\)
−0.100779	+	0.994909i	\(0.532134\pi\)
\(23\)	4.65667	+	3.51656i	0.970984	+	0.733253i	0.963774	−	0.266721i	\(-0.0859403\pi\)
							0.00720981	+	0.999974i	\(0.497705\pi\)
\(29\)	1.08201	−	2.79298i	0.200923	−	0.518643i	−0.795226	−	0.606313i	\(-0.792648\pi\)
							0.996150	+	0.0876698i	\(0.0279420\pi\)
\(31\)	−1.72340	+	2.78339i	−0.309532	+	0.499912i	−0.967411	−	0.253212i	\(-0.918513\pi\)
							0.657878	+	0.753124i	\(0.271454\pi\)
\(37\)	−0.168197	−	0.0155857i	−0.0276514	−	0.00256228i	0.0784338	−	0.996919i	\(-0.475008\pi\)
							−0.106085	+	0.994357i	\(0.533832\pi\)
\(41\)	0.851874	−	0.424183i	0.133040	−	0.0662463i	−0.377986	−	0.925811i	\(-0.623383\pi\)
							0.511026	+	0.859565i	\(0.329266\pi\)
\(43\)	−7.82744	+	7.13566i	−1.19367	+	1.08818i	−0.199555	+	0.979887i	\(0.563950\pi\)
							−0.994119	+	0.108291i	\(0.965462\pi\)
\(47\)	−6.08023	−	8.05152i	−0.886892	−	1.17444i	−0.983823	−	0.179144i	\(-0.942667\pi\)
							0.0969304	−	0.995291i	\(-0.469098\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.g.a.135.21	✓	384
289.152	even	34	inner	289.2.g.a.152.21	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.g.a.135.21	✓	384	1.1	even	1	trivial
289.2.g.a.152.21	yes	384	289.152	even	34	inner