Embedded newform 350.2.u.a.219.7

• Label: 350.2.u.a.219.7
• Level: $350$
• Weight: $2$
• Character: 350.219
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.u (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(20\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			219.7
Character	\(\chi\)	\(=\)	350.219
Dual form			350.2.u.a.179.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.207912 + 0.978148i) q^{2} +(0.355990 - 0.799566i) q^{3} +(-0.913545 - 0.406737i) q^{4} +(1.13322 - 1.92764i) q^{5} +(0.708079 + 0.514449i) q^{6} +(-1.42020 + 2.23227i) q^{7} +(0.587785 - 0.809017i) q^{8} +(1.49481 + 1.66016i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 20 q^{4} - 2 q^{5} + 8 q^{6} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.207912	+	0.978148i	−0.147016	+	0.691655i
							\(3\)	0.355990	−	0.799566i	0.205531	−	0.461630i	−0.781141	−	0.624355i	\(-0.785362\pi\)
0.986671	+	0.162726i	\(0.0520285\pi\)
\(5\)	1.13322	−	1.92764i	0.506791	−	0.862069i
							\(7\)	−1.42020	+	2.23227i	−0.536785	+	0.843719i
\(11\)	3.39981	−	3.77587i	1.02508	−	1.13847i								0.0347977	−	0.999394i	\(-0.488921\pi\)
							0.990282	−	0.139072i	\(-0.0444120\pi\)
\(13\)	−2.88279	−	0.936675i	−0.799542	−	0.259787i	−0.119380	−	0.992849i	\(-0.538091\pi\)
							−0.680162	+	0.733062i	\(0.738091\pi\)
\(17\)	4.96201	+	0.521528i	1.20346	+	0.126489i	0.684956	−	0.728584i	\(-0.259821\pi\)
							0.518507	+	0.855073i	\(0.326488\pi\)
\(19\)	6.19088	−	2.75636i	1.42029	−	0.632352i	0.454276	−	0.890861i	\(-0.349898\pi\)
							0.966009	+	0.258509i	\(0.0832312\pi\)
\(23\)	1.34003	−	6.30436i	0.279416	−	1.31455i	−0.584697	−	0.811252i	\(-0.698787\pi\)
							0.864113	−	0.503298i	\(-0.167880\pi\)
\(29\)	−6.25297	+	4.54305i	−1.16115	+	0.843623i	−0.989923	−	0.141609i	\(-0.954772\pi\)
							−0.171224	+	0.985232i	\(0.554772\pi\)
\(31\)	−0.225116	+	2.14184i	−0.0404321	+	0.384686i	0.955529	+	0.294898i	\(0.0952856\pi\)
							−0.995961	+	0.0897880i	\(0.971381\pi\)
\(37\)	−5.08052	+	4.57452i	−0.835232	+	0.752046i	−0.971098	−	0.238681i	\(-0.923285\pi\)
							0.135866	+	0.990727i	\(0.456618\pi\)
\(41\)	−0.670143	+	2.06249i	−0.104659	+	0.322107i	−0.989650	−	0.143501i	\(-0.954164\pi\)
							0.884991	+	0.465607i	\(0.154164\pi\)
\(43\)			7.21837i			1.10079i	0.834904	+	0.550395i	\(0.185523\pi\)
							−0.834904	+	0.550395i	\(0.814477\pi\)
\(47\)	−3.80447	+	0.399866i	−0.554939	+	0.0583264i	−0.377847	−	0.925868i	\(-0.623336\pi\)
							−0.177091	+	0.984194i	\(0.556669\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.u.a.219.7	yes	160
7.4	even	3	inner	350.2.u.a.319.4	yes	160
25.4	even	10	inner	350.2.u.a.79.4	✓	160
175.4	even	30	inner	350.2.u.a.179.7	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.u.a.79.4	✓	160	25.4	even	10	inner
350.2.u.a.179.7	yes	160	175.4	even	30	inner
350.2.u.a.219.7	yes	160	1.1	even	1	trivial
350.2.u.a.319.4	yes	160	7.4	even	3	inner