Embedded newform 175.4.x.a.103.27

• Label: 175.4.x.a.103.27
• Level: $175$
• Weight: $4$
• Character: 175.103
• Analytic conductor: $10.325$
• Analytic rank: $0$
• Dimension: $928$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3253342510\)
Analytic rank:	\(0\)
Dimension:	\(928\)
Relative dimension:	\(58\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			103.27
Character	\(\chi\)	\(=\)	175.103
Dual form			175.4.x.a.17.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.570569 - 0.0299022i) q^{2} +(1.41700 + 1.74985i) q^{3} +(-7.63152 - 0.802105i) q^{4} +(-10.8302 - 2.77628i) q^{5} +(-0.756170 - 1.04078i) q^{6} +(18.4956 - 0.955975i) q^{7} +(8.84486 + 1.40089i) q^{8} +(4.55954 - 21.4509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 928 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 24 q^{7} + 84 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.570569	−	0.0299022i	−0.201727	−	0.0105720i	−0.0487949	−	0.998809i	\(-0.515538\pi\)
							−0.152932	+	0.988237i	\(0.548871\pi\)
\(3\)	1.41700	+	1.74985i	0.272701	+	0.336758i	0.895037	−	0.445993i	\(-0.147149\pi\)
							−0.622335	+	0.782751i	\(0.713816\pi\)
\(5\)	−10.8302	−	2.77628i	−0.968679	−	0.248318i
							\(7\)	18.4956	−	0.955975i	0.998667	−	0.0516178i
\(11\)	−42.9973	+	9.13936i	−1.17856	+	0.250511i								−0.755228	−	0.655462i	\(-0.772474\pi\)
							−0.423334	+	0.905974i	\(0.639140\pi\)
\(13\)	63.7457	+	32.4801i	1.35999	+	0.692950i	0.973360	−	0.229284i	\(-0.0736385\pi\)
							0.386632	+	0.922234i	\(0.373639\pi\)
\(17\)	44.2617	+	115.306i	0.631473	+	1.64504i	0.757358	+	0.653000i	\(0.226490\pi\)
							−0.125885	+	0.992045i	\(0.540177\pi\)
\(19\)	−2.03508	−	19.3625i	−0.0245726	−	0.233792i	−0.999914	−	0.0130923i	\(-0.995832\pi\)
							0.975342	−	0.220700i	\(-0.0708342\pi\)
\(23\)	−5.56252	+	106.139i	−0.0504289	+	0.962241i	0.849876	+	0.526982i	\(0.176677\pi\)
							−0.900305	+	0.435259i	\(0.856657\pi\)
\(29\)	−0.465140	+	0.640210i	−0.00297842	+	0.00409945i	−0.810504	−	0.585734i	\(-0.800806\pi\)
							0.807525	+	0.589833i	\(0.200806\pi\)
\(31\)	2.69447	−	6.05188i	0.0156110	−	0.0350629i	−0.905572	−	0.424193i	\(-0.860558\pi\)
							0.921183	+	0.389130i	\(0.127224\pi\)
\(37\)	305.530	−	198.414i	1.35754	−	0.881594i	0.358878	−	0.933384i	\(-0.383159\pi\)
							0.998658	+	0.0517902i	\(0.0164927\pi\)
\(41\)	−113.102	+	36.7490i	−0.430818	+	0.139981i	−0.516395	−	0.856350i	\(-0.672727\pi\)
							0.0855774	+	0.996332i	\(0.472727\pi\)
\(43\)	18.9499	+	18.9499i	0.0672053	+	0.0672053i	0.739911	−	0.672705i	\(-0.234868\pi\)
							−0.672705	+	0.739911i	\(0.734868\pi\)
\(47\)	525.773	+	201.825i	1.63174	+	0.626367i	0.990065	−	0.140608i	\(-0.0449056\pi\)
							0.641677	+	0.766975i	\(0.278239\pi\)