Embedded newform 175.2.x.a.47.9

• Label: 175.2.x.a.47.9
• Level: $175$
• Weight: $2$
• Character: 175.47
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $288$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	175.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(18\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			47.9
Character	\(\chi\)	\(=\)	175.47
Dual form			175.2.x.a.108.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.000898858 - 0.0171512i) q^{2} +(0.879015 - 0.711812i) q^{3} +(1.98875 + 0.209026i) q^{4} +(1.43263 - 1.71685i) q^{5} +(-0.0114183 - 0.0157160i) q^{6} +(-2.51481 - 0.822022i) q^{7} +(0.0107461 - 0.0678483i) q^{8} +(-0.357745 + 1.68306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 30 q^{5} - 10 q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.000898858	−	0.0171512i	0.000635588	−	0.0121278i	−0.998293	−	0.0583989i	\(-0.981400\pi\)
							0.998929	+	0.0462711i	\(0.0147338\pi\)
\(3\)	0.879015	−	0.711812i	0.507499	−	0.410965i	−0.341079	−	0.940035i	\(-0.610792\pi\)
							0.848578	+	0.529070i	\(0.177459\pi\)
\(5\)	1.43263	−	1.71685i	0.640690	−	0.767800i
							\(7\)	−2.51481	−	0.822022i	−0.950510	−	0.310695i
\(11\)	−4.96090	+	1.05447i	−1.49577	+	0.317935i								−0.881886	−	0.471463i	\(-0.843726\pi\)
							−0.613882	+	0.789398i	\(0.710393\pi\)
\(13\)	0.145799	−	0.286147i	0.0404375	−	0.0793630i	−0.869904	−	0.493222i	\(-0.835819\pi\)
							0.910341	+	0.413859i	\(0.135819\pi\)
\(17\)	3.48884	−	1.33924i	0.846169	−	0.324814i	0.103629	−	0.994616i	\(-0.466954\pi\)
							0.742540	+	0.669802i	\(0.233621\pi\)
\(19\)	0.698358	+	6.64443i	0.160214	+	1.52434i	0.718991	+	0.695019i	\(0.244604\pi\)
							−0.558777	+	0.829318i	\(0.688729\pi\)
\(23\)	−3.41354	−	0.178896i	−0.711772	−	0.0373024i	−0.306986	−	0.951714i	\(-0.599321\pi\)
							−0.404786	+	0.914412i	\(0.632654\pi\)
\(29\)	1.99592	−	2.74715i	0.370633	−	0.510133i	−0.582440	−	0.812874i	\(-0.697902\pi\)
							0.953073	+	0.302741i	\(0.0979018\pi\)
\(31\)	0.305449	−	0.686049i	0.0548602	−	0.123218i	−0.884036	−	0.467420i	\(-0.845184\pi\)
							0.938896	+	0.344202i	\(0.111850\pi\)
\(37\)	1.79507	+	2.76416i	0.295107	+	0.454426i	0.954860	−	0.297058i	\(-0.0960054\pi\)
							−0.659752	+	0.751483i	\(0.729339\pi\)
\(41\)	−7.88534	+	2.56210i	−1.23148	+	0.400133i	−0.851252	−	0.524758i	\(-0.824156\pi\)
							−0.380232	+	0.924891i	\(0.624156\pi\)
\(43\)	−3.99562	+	3.99562i	−0.609326	+	0.609326i	−0.942770	−	0.333444i	\(-0.891789\pi\)
							0.333444	+	0.942770i	\(0.391789\pi\)
\(47\)	2.25590	−	5.87681i	0.329056	−	0.857221i	−0.664849	−	0.746978i	\(-0.731504\pi\)
							0.993905	−	0.110242i	\(-0.0351627\pi\)