Embedded newform 175.4.x.a.103.36

• Label: 175.4.x.a.103.36
• 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			103.36
Character	\(\chi\)	\(=\)	175.103
Dual form			175.4.x.a.17.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.48373 + 0.0777588i) q^{2} +(2.51040 + 3.10009i) q^{3} +(-5.76078 - 0.605482i) q^{4} +(4.25151 + 10.3404i) q^{5} +(3.48370 + 4.79490i) q^{6} +(-12.4178 + 13.7404i) q^{7} +(-20.2401 - 3.20572i) q^{8} +(2.30518 - 10.8450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 928 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 24 q^{7} + 84 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.48373	+	0.0777588i	0.524577	+	0.0274919i	0.312787	−	0.949823i	\(-0.398737\pi\)
							0.211790	+	0.977315i	\(0.432071\pi\)
\(3\)	2.51040	+	3.10009i	0.483128	+	0.596613i	0.959177	−	0.282806i	\(-0.0912651\pi\)
							−0.476050	+	0.879418i	\(0.657932\pi\)
\(5\)	4.25151	+	10.3404i	0.380266	+	0.924877i
							\(7\)	−12.4178	+	13.7404i	−0.670498	+	0.741912i
\(11\)	−33.2598	+	7.06959i	−0.911655	+	0.193778i								−0.639784	−	0.768555i	\(-0.720976\pi\)
							−0.271872	+	0.962333i	\(0.587643\pi\)
\(13\)	−13.9029	−	7.08390i	−0.296614	−	0.151132i	0.299351	−	0.954143i	\(-0.403230\pi\)
							−0.595964	+	0.803011i	\(0.703230\pi\)
\(17\)	−27.0101	−	70.3638i	−0.385348	−	1.00387i	−0.979843	−	0.199768i	\(-0.935981\pi\)
							0.594495	−	0.804099i	\(-0.297352\pi\)
\(19\)	3.94372	+	37.5220i	0.0476185	+	0.453059i	0.992188	+	0.124749i	\(0.0398125\pi\)
							−0.944570	+	0.328310i	\(0.893521\pi\)
\(23\)	−10.3465	+	197.422i	−0.0937993	+	1.78980i	0.396290	+	0.918126i	\(0.370298\pi\)
							−0.490089	+	0.871672i	\(0.663036\pi\)
\(29\)	−33.8927	+	46.6493i	−0.217025	+	0.298709i	−0.903623	−	0.428328i	\(-0.859103\pi\)
							0.686599	+	0.727036i	\(0.259103\pi\)
\(31\)	−69.5763	+	156.271i	−0.403105	+	0.905390i	0.591941	+	0.805981i	\(0.298362\pi\)
							−0.995047	+	0.0994085i	\(0.968305\pi\)
\(37\)	48.0518	−	31.2052i	0.213504	−	0.138651i	−0.433457	−	0.901174i	\(-0.642706\pi\)
							0.646961	+	0.762523i	\(0.276040\pi\)
\(41\)	−17.8048	+	5.78515i	−0.0678207	+	0.0220363i	−0.342731	−	0.939434i	\(-0.611352\pi\)
							0.274910	+	0.961470i	\(0.411352\pi\)
\(43\)	1.93480	+	1.93480i	0.00686173	+	0.00686173i	0.710529	−	0.703668i	\(-0.248456\pi\)
							−0.703668	+	0.710529i	\(0.748456\pi\)
\(47\)	535.324	+	205.492i	1.66138	+	0.637746i	0.994239	−	0.107181i	\(-0.0341826\pi\)
							0.667146	+	0.744927i	\(0.267516\pi\)