Embedded newform 175.4.x.a.103.17

• Label: 175.4.x.a.103.17
• 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			103.17
Character	\(\chi\)	\(=\)	175.103
Dual form			175.4.x.a.17.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.70791 - 0.141916i) q^{2} +(-2.31104 - 2.85390i) q^{3} +(-0.643524 - 0.0676371i) q^{4} +(8.55160 + 7.20209i) q^{5} +(5.85308 + 8.05607i) q^{6} +(-16.6592 - 8.09134i) q^{7} +(23.1590 + 3.66802i) q^{8} +(2.80980 - 13.2191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 928 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 24 q^{7} + 84 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.70791	−	0.141916i	−0.957392	−	0.0501748i	−0.432765	−	0.901507i	\(-0.642462\pi\)
							−0.524627	+	0.851332i	\(0.675795\pi\)
\(3\)	−2.31104	−	2.85390i	−0.444760	−	0.549233i	0.504629	−	0.863336i	\(-0.331629\pi\)
							−0.949389	+	0.314104i	\(0.898296\pi\)
\(5\)	8.55160	+	7.20209i	0.764879	+	0.644174i
							\(7\)	−16.6592	−	8.09134i	−0.899514	−	0.436891i
\(11\)	30.3217	−	6.44509i	0.831123	−	0.176661i								0.227346	−	0.973814i	\(-0.426995\pi\)
							0.603777	+	0.797153i	\(0.293662\pi\)
\(13\)	−54.8773	−	27.9614i	−1.17079	−	0.596545i	−0.243135	−	0.969993i	\(-0.578176\pi\)
							−0.927651	+	0.373447i	\(0.878176\pi\)
\(17\)	20.8135	+	54.2210i	0.296942	+	0.773560i	0.998161	+	0.0606146i	\(0.0193060\pi\)
							−0.701219	+	0.712946i	\(0.747361\pi\)
\(19\)	−12.2739	−	116.778i	−0.148201	−	1.41004i	−0.775543	−	0.631295i	\(-0.782524\pi\)
							0.627342	−	0.778744i	\(-0.284143\pi\)
\(23\)	−7.28080	+	138.926i	−0.0660066	+	1.25948i	0.742037	+	0.670359i	\(0.233860\pi\)
							−0.808044	+	0.589122i	\(0.799474\pi\)
\(29\)	−38.9574	+	53.6202i	−0.249455	+	0.343346i	−0.915320	−	0.402726i	\(-0.868063\pi\)
							0.665865	+	0.746072i	\(0.268063\pi\)
\(31\)	−95.5571	+	214.625i	−0.553631	+	1.24348i	0.392518	+	0.919744i	\(0.371604\pi\)
							−0.946149	+	0.323731i	\(0.895063\pi\)
\(37\)	−158.913	+	103.199i	−0.706085	+	0.458537i	−0.847075	−	0.531474i	\(-0.821638\pi\)
							0.140989	+	0.990011i	\(0.454972\pi\)
\(41\)	−184.619	+	59.9863i	−0.703235	+	0.228495i	−0.638739	−	0.769423i	\(-0.720544\pi\)
							−0.0644954	+	0.997918i	\(0.520544\pi\)
\(43\)	−361.069	−	361.069i	−1.28052	−	1.28052i	−0.940369	−	0.340156i	\(-0.889520\pi\)
							−0.340156	−	0.940369i	\(-0.610480\pi\)
\(47\)	−142.507	−	54.7034i	−0.442273	−	0.169773i	0.127033	−	0.991898i	\(-0.459455\pi\)
							−0.569306	+	0.822126i	\(0.692788\pi\)