Embedded newform 175.3.w.a.163.37

• Label: 175.3.w.a.163.37
• Level: $175$
• Weight: $3$
• Character: 175.163
• Analytic conductor: $4.768$
• Analytic rank: $0$
• Dimension: $608$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	175.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.76840462631\)
Analytic rank:	\(0\)
Dimension:	\(608\)
Relative dimension:	\(38\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			163.37
Character	\(\chi\)	\(=\)	175.163
Dual form			175.3.w.a.102.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.25970 - 3.28163i) q^{2} +(4.62972 - 0.242633i) q^{3} +(-6.20970 - 5.59124i) q^{4} +(-4.99868 + 0.114674i) q^{5} +(5.03583 - 15.4987i) q^{6} +(-1.41942 - 6.85458i) q^{7} +(-13.6428 + 6.95137i) q^{8} +(12.4247 - 1.30589i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 608 q - 8 q^{2} - 8 q^{3} - 10 q^{4} - 6 q^{5} - 24 q^{6} - 14 q^{7} - 4 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.25970	−	3.28163i	0.629851	−	1.64082i	−0.130610	−	0.991434i	\(-0.541694\pi\)
							0.760461	−	0.649384i	\(-0.224973\pi\)
\(3\)	4.62972	−	0.242633i	1.54324	−	0.0808777i	0.738396	−	0.674367i	\(-0.235583\pi\)
							0.804842	+	0.593489i	\(0.202250\pi\)
\(5\)	−4.99868	+	0.114674i	−0.999737	+	0.0229348i
							\(7\)	−1.41942	−	6.85458i	−0.202774	−	0.979225i
\(11\)	−1.35734	+	12.9142i	−0.123394	+	1.17402i								0.741106	+	0.671388i	\(0.234301\pi\)
							−0.864501	+	0.502632i	\(0.832365\pi\)
\(13\)	19.1286	+	3.02967i	1.47143	+	0.233051i	0.840086	−	0.542454i	\(-0.182505\pi\)
							0.631342	+	0.775505i	\(0.282505\pi\)
\(17\)	12.3899	−	19.0788i	0.728818	−	1.12228i	−0.258881	−	0.965909i	\(-0.583354\pi\)
							0.987698	−	0.156371i	\(-0.0499797\pi\)
\(19\)	−11.2882	+	10.1640i	−0.594118	+	0.534946i	−0.910406	−	0.413717i	\(-0.864230\pi\)
							0.316287	+	0.948663i	\(0.397564\pi\)
\(23\)	16.9746	+	6.51595i	0.738027	+	0.283302i	0.698198	−	0.715904i	\(-0.253985\pi\)
							0.0398289	+	0.999207i	\(0.487319\pi\)
\(29\)	8.58100	−	2.78814i	0.295897	−	0.0961426i	−0.157307	−	0.987550i	\(-0.550281\pi\)
							0.453203	+	0.891407i	\(0.350281\pi\)
\(31\)	−8.84588	+	1.88025i	−0.285351	+	0.0606532i	−0.348364	−	0.937360i	\(-0.613263\pi\)
							0.0630125	+	0.998013i	\(0.479929\pi\)
\(37\)	19.2079	+	15.5542i	0.519132	+	0.420385i	0.852747	−	0.522325i	\(-0.174935\pi\)
							−0.333615	+	0.942709i	\(0.608268\pi\)
\(41\)	−59.3313	+	43.1067i	−1.44710	+	1.05138i	−0.460606	+	0.887604i	\(0.652368\pi\)
							−0.986497	+	0.163778i	\(0.947632\pi\)
\(43\)	8.13714	+	8.13714i	0.189236	+	0.189236i	0.795366	−	0.606130i	\(-0.207279\pi\)
							−0.606130	+	0.795366i	\(0.707279\pi\)
\(47\)	9.34361	−	6.06781i	0.198800	−	0.129102i	−0.441400	−	0.897310i	\(-0.645518\pi\)
							0.640200	+	0.768208i	\(0.278851\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.3.w.a.163.37	yes	608
7.4	even	3	inner	175.3.w.a.88.37	yes	608
25.2	odd	20	inner	175.3.w.a.2.37	✓	608
175.102	odd	60	inner	175.3.w.a.102.37	yes	608

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.3.w.a.2.37	✓	608	25.2	odd	20	inner
175.3.w.a.88.37	yes	608	7.4	even	3	inner
175.3.w.a.102.37	yes	608	175.102	odd	60	inner
175.3.w.a.163.37	yes	608	1.1	even	1	trivial