Embedded newform 175.4.n.a.169.12

• Label: 175.4.n.a.169.12
• Level: $175$
• Weight: $4$
• Character: 175.169
• Analytic conductor: $10.325$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3253342510\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			169.12
Character	\(\chi\)	\(=\)	175.169
Dual form			175.4.n.a.29.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.14628 + 1.02229i) q^{2} +(5.97409 + 8.22263i) q^{3} +(2.38186 - 1.73053i) q^{4} +(9.51619 - 5.86874i) q^{5} +(-27.2020 - 19.7634i) q^{6} -7.00000i q^{7} +(9.83115 - 13.5314i) q^{8} +(-23.5784 + 72.5669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q + 192 q^{4} - 46 q^{5} + 386 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.14628	+	1.02229i	−1.11238	+	0.361433i	−0.806854	−	0.590751i	\(-0.798832\pi\)
							−0.305524	+	0.952184i	\(0.598832\pi\)
\(3\)	5.97409	+	8.22263i	1.14971	+	1.58245i	0.743603	+	0.668621i	\(0.233115\pi\)
							0.406111	+	0.913824i	\(0.366885\pi\)
\(5\)	9.51619	−	5.86874i	0.851154	−	0.524916i
							\(7\)		−	7.00000i		−	0.377964i
\(11\)	9.04891	+	27.8497i	0.248032	+	0.763363i								0.995123	+	0.0986415i	\(0.0314497\pi\)
							−0.747091	+	0.664721i	\(0.768550\pi\)
\(13\)	63.2889	+	20.5638i	1.35024	+	0.438721i	0.892774	−	0.450504i	\(-0.148756\pi\)
							0.457470	+	0.889225i	\(0.348756\pi\)
\(17\)	−58.0070	+	79.8398i	−0.827575	+	1.13906i	0.160795	+	0.986988i	\(0.448594\pi\)
							−0.988370	+	0.152071i	\(0.951406\pi\)
\(19\)	−54.7786	−	39.7990i	−0.661425	−	0.480553i	0.205719	−	0.978611i	\(-0.434047\pi\)
							−0.867144	+	0.498058i	\(0.834047\pi\)
\(23\)	13.5376	−	4.39862i	0.122729	−	0.0398772i	−0.247008	−	0.969013i	\(-0.579448\pi\)
							0.369738	+	0.929136i	\(0.379448\pi\)
\(29\)	56.0150	−	40.6973i	0.358680	−	0.260596i	−0.393821	−	0.919187i	\(-0.628847\pi\)
							0.752501	+	0.658591i	\(0.228847\pi\)
\(31\)	129.176	+	93.8515i	0.748407	+	0.543749i	0.895333	−	0.445398i	\(-0.146938\pi\)
							−0.146926	+	0.989148i	\(0.546938\pi\)
\(37\)	−189.138	−	61.4548i	−0.840382	−	0.273057i	−0.142970	−	0.989727i	\(-0.545665\pi\)
							−0.697412	+	0.716670i	\(0.745665\pi\)
\(41\)	56.8400	−	174.936i	0.216510	−	0.666350i	−0.782533	−	0.622609i	\(-0.786073\pi\)
							0.999043	−	0.0437403i	\(-0.0139274\pi\)
\(43\)			231.141i			0.819735i	0.912145	+	0.409868i	\(0.134425\pi\)
							−0.912145	+	0.409868i	\(0.865575\pi\)
\(47\)	0.0431285	+	0.0593612i	0.000133850	+	0.000184228i	0.809084	−	0.587693i	\(-0.199964\pi\)
							−0.808950	+	0.587877i	\(0.799964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.n.a.169.12	yes	184
25.4	even	10	inner	175.4.n.a.29.12	✓	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.n.a.29.12	✓	184	25.4	even	10	inner
175.4.n.a.169.12	yes	184	1.1	even	1	trivial