Embedded newform 350.3.w.a.23.18

• Label: 350.3.w.a.23.18
• 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			23.18
Character	\(\chi\)	\(=\)	350.23
Dual form			350.3.w.a.137.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.09905 + 0.889993i) q^{2} +(4.38001 + 2.84441i) q^{3} +(0.415823 + 1.95630i) q^{4} +(-4.56381 - 2.04246i) q^{5} +(2.28234 + 7.02433i) q^{6} +(-4.20782 + 5.59412i) q^{7} +(-1.28408 + 2.52015i) q^{8} +(7.43318 + 16.6952i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9}+O(q^{10}) \)

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
							\(3\)	4.38001	+	2.84441i	1.46000	+	0.948137i	0.998126	+	0.0611938i	\(0.0194908\pi\)
0.461878	+	0.886944i	\(0.347176\pi\)
\(5\)	−4.56381	−	2.04246i	−0.912762	−	0.408493i
							\(7\)	−4.20782	+	5.59412i	−0.601118	+	0.799160i
\(11\)	6.00270	+	2.67258i	0.545700	+	0.242961i								0.661022	−	0.750367i	\(-0.270123\pi\)
							−0.115321	+	0.993328i	\(0.536790\pi\)
\(13\)	−0.244814	−	1.54570i	−0.0188319	−	0.118900i	0.976482	−	0.215601i	\(-0.0691709\pi\)
							−0.995313	+	0.0967010i	\(0.969171\pi\)
\(17\)	−0.650002	−	12.4028i	−0.0382354	−	0.729575i	−0.948736	−	0.316068i	\(-0.897637\pi\)
							0.910501	−	0.413507i	\(-0.135696\pi\)
\(19\)	−7.08846	+	33.3486i	−0.373077	+	1.75519i	0.245434	+	0.969413i	\(0.421069\pi\)
							−0.618511	+	0.785776i	\(0.712264\pi\)
\(23\)	10.3780	−	12.8157i	0.451216	−	0.557205i	−0.499883	−	0.866093i	\(-0.666624\pi\)
							0.951098	+	0.308888i	\(0.0999569\pi\)
\(29\)	21.5371	+	6.99784i	0.742659	+	0.241305i	0.655820	−	0.754918i	\(-0.272323\pi\)
							0.0868398	+	0.996222i	\(0.472323\pi\)
\(31\)	12.6555	−	14.0554i	0.408243	−	0.453399i	−0.503602	−	0.863936i	\(-0.667992\pi\)
							0.911844	+	0.410537i	\(0.134659\pi\)
\(37\)	16.7667	−	43.6787i	0.453154	−	1.18051i	−0.496731	−	0.867905i	\(-0.665466\pi\)
							0.949884	−	0.312601i	\(-0.101200\pi\)
\(41\)	−42.2106	−	30.6678i	−1.02953	−	0.747995i	−0.0613125	−	0.998119i	\(-0.519529\pi\)
							−0.968214	+	0.250124i	\(0.919529\pi\)
\(43\)	−5.15949	−	5.15949i	−0.119988	−	0.119988i	0.644563	−	0.764551i	\(-0.277039\pi\)
							−0.764551	+	0.644563i	\(0.777039\pi\)
\(47\)	64.6062	+	3.38587i	1.37460	+	0.0720397i	0.725269	−	0.688466i	\(-0.241716\pi\)
							0.649331	+	0.760506i	\(0.275049\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.3.w.a.23.18	✓	320
7.4	even	3	inner	350.3.w.a.123.18	yes	320
25.12	odd	20	inner	350.3.w.a.37.18	yes	320
175.137	odd	60	inner	350.3.w.a.137.18	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.3.w.a.23.18	✓	320	1.1	even	1	trivial
350.3.w.a.37.18	yes	320	25.12	odd	20	inner
350.3.w.a.123.18	yes	320	7.4	even	3	inner
350.3.w.a.137.18	yes	320	175.137	odd	60	inner