Embedded newform 350.2.x.a.173.3

• Label: 350.2.x.a.173.3
• Level: $350$
• Weight: $2$
• Character: 350.173
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
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			173.3
Character	\(\chi\)	\(=\)	350.173
Dual form			350.2.x.a.87.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.777146 - 0.629320i) q^{2} +(-0.964634 + 1.48541i) q^{3} +(0.207912 + 0.978148i) q^{4} +(-2.18238 - 0.487036i) q^{5} +(1.68446 - 0.547313i) q^{6} +(1.85882 + 1.88276i) q^{7} +(0.453990 - 0.891007i) q^{8} +(-0.0557023 - 0.125109i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 12 q^{5} - 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.777146	−	0.629320i	−0.549525	−	0.444997i
							\(3\)	−0.964634	+	1.48541i	−0.556932	+	0.857600i	−0.999229	−	0.0392691i	\(-0.987497\pi\)
0.442297	+	0.896869i	\(0.354164\pi\)
\(5\)	−2.18238	−	0.487036i	−0.975991	−	0.217809i
							\(7\)	1.85882	+	1.88276i	0.702567	+	0.711618i
\(11\)	−4.94908	−	2.20347i	−1.49221	−	0.664373i								−0.511396	−	0.859345i	\(-0.670872\pi\)
							−0.980809	+	0.194972i	\(0.937538\pi\)
\(13\)	3.81548	−	0.604312i	1.05822	−	0.167606i	0.397018	−	0.917811i	\(-0.370045\pi\)
							0.661205	+	0.750205i	\(0.270045\pi\)
\(17\)	−5.49481	+	0.287971i	−1.33269	+	0.0698431i	−0.705324	−	0.708885i	\(-0.749199\pi\)
							−0.627362	+	0.778728i	\(0.715865\pi\)
\(19\)	−7.36550	−	1.56559i	−1.68976	−	0.359170i	−0.740112	−	0.672483i	\(-0.765228\pi\)
							−0.949650	+	0.313313i	\(0.898561\pi\)
\(23\)	2.16480	−	2.67331i	0.451392	−	0.557423i	−0.499753	−	0.866168i	\(-0.666576\pi\)
							0.951145	+	0.308745i	\(0.0999091\pi\)
\(29\)	−4.18165	−	1.35870i	−0.776513	−	0.252304i	−0.106162	−	0.994349i	\(-0.533856\pi\)
							−0.670351	+	0.742044i	\(0.733856\pi\)
\(31\)	−4.64255	−	4.18017i	−0.833827	−	0.750781i	0.136998	−	0.990571i	\(-0.456255\pi\)
							−0.970825	+	0.239790i	\(0.922921\pi\)
\(37\)	1.06818	−	2.78271i	0.175608	−	0.457475i	−0.817321	−	0.576183i	\(-0.804542\pi\)
							0.992929	+	0.118707i	\(0.0378751\pi\)
\(41\)	−0.101240	+	0.139345i	−0.0158110	+	0.0217620i	−0.816849	−	0.576851i	\(-0.804281\pi\)
							0.801038	+	0.598613i	\(0.204281\pi\)
\(43\)	−3.80709	−	3.80709i	−0.580576	−	0.580576i	0.354485	−	0.935062i	\(-0.384656\pi\)
							−0.935062	+	0.354485i	\(0.884656\pi\)
\(47\)	−0.706152	+	13.4742i	−0.103003	+	1.96541i	0.126119	+	0.992015i	\(0.459748\pi\)
							−0.229122	+	0.973398i	\(0.573586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.x.a.173.3	yes	320
7.3	odd	6	inner	350.2.x.a.73.13	✓	320
25.12	odd	20	inner	350.2.x.a.187.13	yes	320
175.87	even	60	inner	350.2.x.a.87.3	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.x.a.73.13	✓	320	7.3	odd	6	inner
350.2.x.a.87.3	yes	320	175.87	even	60	inner
350.2.x.a.173.3	yes	320	1.1	even	1	trivial
350.2.x.a.187.13	yes	320	25.12	odd	20	inner