Embedded newform 350.3.w.a.23.1

• Label: 350.3.w.a.23.1
• Level: $350$
• Weight: $3$
• Character: 350.23
• Analytic conductor: $9.537$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	350.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.53680925261\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(20\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			23.1
Character	\(\chi\)	\(=\)	350.23
Dual form			350.3.w.a.137.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09905 + 0.889993i) q^{2} +(-4.16727 - 2.70626i) q^{3} +(0.415823 + 1.95630i) q^{4} +(3.21899 - 3.82598i) q^{5} +(-2.17149 - 6.68315i) q^{6} +(-0.176682 + 6.99777i) q^{7} +(-1.28408 + 2.52015i) q^{8} +(6.38168 + 14.3335i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.09905	+	0.889993i	0.549525	+	0.444997i
							\(3\)	−4.16727	−	2.70626i	−1.38909	−	0.902085i	−0.389271	−	0.921123i	\(-0.627273\pi\)
−0.999819	+	0.0190379i	\(0.993940\pi\)
\(5\)	3.21899	−	3.82598i	0.643798	−	0.765196i
							\(7\)	−0.176682	+	6.99777i	−0.0252403	+	0.999681i
\(11\)	11.5074	+	5.12343i	1.04613	+	0.465766i								0.856531	−	0.516096i	\(-0.172615\pi\)
							0.189597	+	0.981862i	\(0.439282\pi\)
\(13\)	−3.42705	−	21.6376i	−0.263619	−	1.66443i	−0.663739	−	0.747965i	\(-0.731031\pi\)
							0.400119	−	0.916463i	\(-0.368969\pi\)
\(17\)	−0.384054	−	7.32818i	−0.0225914	−	0.431070i	−0.986388	−	0.164432i	\(-0.947421\pi\)
							0.963797	−	0.266637i	\(-0.0859126\pi\)
\(19\)	−0.369390	+	1.73784i	−0.0194416	+	0.0914653i	−0.986799	−	0.161952i	\(-0.948221\pi\)
							0.967357	+	0.253417i	\(0.0815545\pi\)
\(23\)	24.2377	−	29.9311i	1.05381	−	1.30135i	0.101963	−	0.994788i	\(-0.467488\pi\)
							0.951850	−	0.306563i	\(-0.0991789\pi\)
\(29\)	−6.18664	−	2.01016i	−0.213332	−	0.0693159i	0.200401	−	0.979714i	\(-0.435775\pi\)
							−0.413733	+	0.910398i	\(0.635775\pi\)
\(31\)	29.4938	−	32.7562i	0.951414	−	1.05665i	−0.0469167	−	0.998899i	\(-0.514940\pi\)
							0.998331	−	0.0577537i	\(-0.0183938\pi\)
\(37\)	8.05158	−	20.9751i	0.217610	−	0.566894i	−0.780706	−	0.624898i	\(-0.785140\pi\)
							0.998316	+	0.0580044i	\(0.0184737\pi\)
\(41\)	−22.3240	−	16.2193i	−0.544487	−	0.395593i	0.281262	−	0.959631i	\(-0.409247\pi\)
							−0.825749	+	0.564038i	\(0.809247\pi\)
\(43\)	16.1797	+	16.1797i	0.376272	+	0.376272i	0.869755	−	0.493483i	\(-0.164277\pi\)
							−0.493483	+	0.869755i	\(0.664277\pi\)
\(47\)	−1.46869	−	0.0769708i	−0.0312487	−	0.00163768i	0.0367052	−	0.999326i	\(-0.488314\pi\)
							−0.0679539	+	0.997688i	\(0.521647\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.3.w.a.23.1	✓	320
7.4	even	3	inner	350.3.w.a.123.1	yes	320
25.12	odd	20	inner	350.3.w.a.37.1	yes	320
175.137	odd	60	inner	350.3.w.a.137.1	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.3.w.a.23.1	✓	320	1.1	even	1	trivial
350.3.w.a.37.1	yes	320	25.12	odd	20	inner
350.3.w.a.123.1	yes	320	7.4	even	3	inner
350.3.w.a.137.1	yes	320	175.137	odd	60	inner