Embedded newform 350.2.u.a.179.5

• Label: 350.2.u.a.179.5
• Level: $350$
• Weight: $2$
• Character: 350.179
• 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			179.5
Character	\(\chi\)	\(=\)	350.179
Dual form			350.2.u.a.219.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.207912 - 0.978148i) q^{2} +(0.157719 + 0.354243i) q^{3} +(-0.913545 + 0.406737i) q^{4} +(-2.23582 + 0.0334437i) q^{5} +(0.313711 - 0.227924i) q^{6} +(2.20412 + 1.46351i) q^{7} +(0.587785 + 0.809017i) q^{8} +(1.90678 - 2.11769i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 20 q^{4} - 2 q^{5} + 8 q^{6} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{10}\right)\)

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
							\(3\)	0.157719	+	0.354243i	0.0910593	+	0.204523i	0.953334	−	0.301919i	\(-0.0976272\pi\)
−0.862274	+	0.506442i	\(0.830961\pi\)
\(5\)	−2.23582	+	0.0334437i	−0.999888	+	0.0149565i
							\(7\)	2.20412	+	1.46351i	0.833077	+	0.553156i
\(11\)	1.04697	+	1.16278i	0.315674	+	0.350592i								0.880011	−	0.474953i	\(-0.157535\pi\)
							−0.564337	+	0.825545i	\(0.690868\pi\)
\(13\)	2.33897	−	0.759977i	0.648714	−	0.210780i	0.0338668	−	0.999426i	\(-0.489218\pi\)
							0.614847	+	0.788647i	\(0.289218\pi\)
\(17\)	5.33514	−	0.560746i	1.29396	−	0.136001i	0.567595	−	0.823308i	\(-0.307874\pi\)
							0.726367	+	0.687307i	\(0.241207\pi\)
\(19\)	0.690509	+	0.307435i	0.158414	+	0.0705303i	0.484412	−	0.874840i	\(-0.339034\pi\)
							−0.325998	+	0.945370i	\(0.605700\pi\)
\(23\)	−1.07743	−	5.06892i	−0.224660	−	1.05694i	−0.935428	−	0.353516i	\(-0.884986\pi\)
							0.710768	−	0.703426i	\(-0.248348\pi\)
\(29\)	−1.04137	−	0.756599i	−0.193377	−	0.140497i	0.486884	−	0.873467i	\(-0.338134\pi\)
							−0.680261	+	0.732970i	\(0.738134\pi\)
\(31\)	0.989963	+	9.41887i	0.177803	+	1.69168i	0.611998	+	0.790859i	\(0.290366\pi\)
							−0.434195	+	0.900819i	\(0.642967\pi\)
\(37\)	−2.45669	−	2.21201i	−0.403877	−	0.363652i	0.442014	−	0.897008i	\(-0.354264\pi\)
							−0.845891	+	0.533356i	\(0.820931\pi\)
\(41\)	1.27489	+	3.92370i	0.199104	+	0.612779i	0.999904	+	0.0138473i	\(0.00440787\pi\)
							−0.800800	+	0.598932i	\(0.795592\pi\)
\(43\)			0.876481i			0.133662i	0.997764	+	0.0668311i	\(0.0212889\pi\)
							−0.997764	+	0.0668311i	\(0.978711\pi\)
\(47\)	−1.90230	−	0.199940i	−0.277479	−	0.0291642i	−0.0352331	−	0.999379i	\(-0.511217\pi\)
							−0.242246	+	0.970215i	\(0.577884\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.u.a.179.5	yes	160
7.2	even	3	inner	350.2.u.a.79.6	✓	160
25.19	even	10	inner	350.2.u.a.319.6	yes	160
175.44	even	30	inner	350.2.u.a.219.5	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.u.a.79.6	✓	160	7.2	even	3	inner
350.2.u.a.179.5	yes	160	1.1	even	1	trivial
350.2.u.a.219.5	yes	160	175.44	even	30	inner
350.2.u.a.319.6	yes	160	25.19	even	10	inner