Embedded newform 175.4.x.a.103.30

• Label: 175.4.x.a.103.30
• 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			103.30
Character	\(\chi\)	\(=\)	175.103
Dual form			175.4.x.a.17.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.143226 + 0.00750617i) q^{2} +(-0.396770 - 0.489971i) q^{3} +(-7.93572 - 0.834078i) q^{4} +(-6.20528 + 9.30024i) q^{5} +(-0.0531501 - 0.0731549i) q^{6} +(-18.1483 - 3.69305i) q^{7} +(-2.26360 - 0.358519i) q^{8} +(5.53097 - 26.0212i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 928 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 24 q^{7} + 84 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.143226	+	0.00750617i	0.0506381	+	0.00265383i	0.0776382	−	0.996982i	\(-0.475262\pi\)
							−0.0270001	+	0.999635i	\(0.508595\pi\)
\(3\)	−0.396770	−	0.489971i	−0.0763585	−	0.0942949i	0.737535	−	0.675309i	\(-0.235990\pi\)
							−0.813894	+	0.581014i	\(0.802656\pi\)
\(5\)	−6.20528	+	9.30024i	−0.555017	+	0.831839i
							\(7\)	−18.1483	−	3.69305i	−0.979917	−	0.199406i
\(11\)	62.9213	−	13.3743i	1.72468	−	0.366592i								0.764209	−	0.644969i	\(-0.223130\pi\)
							0.960472	+	0.278377i	\(0.0897963\pi\)
\(13\)	50.0770	+	25.5155i	1.06837	+	0.544364i	0.897539	−	0.440935i	\(-0.145353\pi\)
							0.170836	+	0.985299i	\(0.445353\pi\)
\(17\)	7.31965	+	19.0683i	0.104428	+	0.272044i	0.975832	−	0.218522i	\(-0.0701234\pi\)
							−0.871404	+	0.490566i	\(0.836790\pi\)
\(19\)	7.32981	+	69.7385i	0.0885039	+	0.842058i	0.945256	+	0.326330i	\(0.105812\pi\)
							−0.856752	+	0.515728i	\(0.827521\pi\)
\(23\)	2.91889	−	55.6957i	0.0264622	−	0.504928i	−0.952959	−	0.303100i	\(-0.901978\pi\)
							0.979421	−	0.201828i	\(-0.0646883\pi\)
\(29\)	56.0721	−	77.1766i	0.359046	−	0.494184i	−0.590837	−	0.806791i	\(-0.701202\pi\)
							0.949883	+	0.312607i	\(0.101202\pi\)
\(31\)	30.4126	−	68.3077i	0.176202	−	0.395756i	−0.803756	−	0.594958i	\(-0.797169\pi\)
							0.979958	+	0.199203i	\(0.0638352\pi\)
\(37\)	126.443	−	82.1133i	0.561816	−	0.364847i	−0.232272	−	0.972651i	\(-0.574616\pi\)
							0.794087	+	0.607804i	\(0.207949\pi\)
\(41\)	−214.776	+	69.7848i	−0.818105	+	0.265819i	−0.688027	−	0.725685i	\(-0.741523\pi\)
							−0.130078	+	0.991504i	\(0.541523\pi\)
\(43\)	269.498	+	269.498i	0.955768	+	0.955768i	0.999062	−	0.0432941i	\(-0.0137852\pi\)
							−0.0432941	+	0.999062i	\(0.513785\pi\)
\(47\)	−285.901	−	109.747i	−0.887296	−	0.340601i	−0.128330	−	0.991732i	\(-0.540962\pi\)
							−0.758966	+	0.651131i	\(0.774295\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.x.a.103.30	yes	928
7.3	odd	6	inner	175.4.x.a.3.29	✓	928
25.17	odd	20	inner	175.4.x.a.117.29	yes	928
175.17	even	60	inner	175.4.x.a.17.30	yes	928

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.x.a.3.29	✓	928	7.3	odd	6	inner
175.4.x.a.17.30	yes	928	175.17	even	60	inner
175.4.x.a.103.30	yes	928	1.1	even	1	trivial
175.4.x.a.117.29	yes	928	25.17	odd	20	inner