Embedded newform 350.2.u.a.9.12

• Label: 350.2.u.a.9.12
• Level: $350$
• Weight: $2$
• Character: 350.9
• 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			9.12
Character	\(\chi\)	\(=\)	350.9
Dual form			350.2.u.a.39.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.406737 + 0.913545i) q^{2} +(-1.54024 - 1.38684i) q^{3} +(-0.669131 + 0.743145i) q^{4} +(-2.08526 + 0.807273i) q^{5} +(0.640467 - 1.97115i) q^{6} +(2.52231 + 0.798718i) q^{7} +(-0.951057 - 0.309017i) q^{8} +(0.135432 + 1.28855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 20 q^{4} - 2 q^{5} + 8 q^{6} - 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.406737	+	0.913545i	0.287606	+	0.645974i
							\(3\)	−1.54024	−	1.38684i	−0.889257	−	0.800690i	0.0915241	−	0.995803i	\(-0.470826\pi\)
−0.980781	+	0.195112i	\(0.937493\pi\)
\(5\)	−2.08526	+	0.807273i	−0.932557	+	0.361024i
							\(7\)	2.52231	+	0.798718i	0.953344	+	0.301887i
\(11\)	0.554840	−	5.27895i	0.167291	−	1.59166i								−0.512781	−	0.858519i	\(-0.671385\pi\)
							0.680072	−	0.733145i	\(-0.261948\pi\)
\(13\)	−1.24940	−	1.71966i	−0.346522	−	0.476947i	0.599810	−	0.800142i	\(-0.295243\pi\)
							−0.946332	+	0.323196i	\(0.895243\pi\)
\(17\)	−1.18812	−	5.58967i	−0.288162	−	1.35569i	−0.849273	−	0.527954i	\(-0.822959\pi\)
							0.561111	−	0.827741i	\(-0.310374\pi\)
\(19\)	−2.69272	−	2.99057i	−0.617753	−	0.686085i	0.350355	−	0.936617i	\(-0.386061\pi\)
							−0.968108	+	0.250532i	\(0.919394\pi\)
\(23\)	1.55778	+	3.49884i	0.324820	+	0.729559i	0.999967	−	0.00812504i	\(-0.00258631\pi\)
							−0.675147	+	0.737684i	\(0.735920\pi\)
\(29\)	−2.13444	−	6.56912i	−0.396355	−	1.21985i	−0.927902	−	0.372825i	\(-0.878389\pi\)
							0.531547	−	0.847029i	\(-0.321611\pi\)
\(31\)	3.53434	−	0.751247i	0.634786	−	0.134928i	0.120731	−	0.992685i	\(-0.461476\pi\)
							0.514055	+	0.857757i	\(0.328143\pi\)
\(37\)	0.501431	−	0.0527025i	0.0824348	−	0.00866424i	−0.0632213	−	0.998000i	\(-0.520137\pi\)
							0.145656	+	0.989335i	\(0.453471\pi\)
\(41\)	−1.56635	+	1.13802i	−0.244622	+	0.177729i	−0.703340	−	0.710853i	\(-0.748309\pi\)
							0.458718	+	0.888582i	\(0.348309\pi\)
\(43\)			5.89349i			0.898748i	0.893344	+	0.449374i	\(0.148353\pi\)
							−0.893344	+	0.449374i	\(0.851647\pi\)
\(47\)	−0.163102	+	0.767333i	−0.0237908	+	0.111927i	−0.988441	−	0.151608i	\(-0.951555\pi\)
							0.964650	+	0.263535i	\(0.0848883\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.u.a.9.12	✓	160
7.4	even	3	inner	350.2.u.a.109.2	yes	160
25.14	even	10	inner	350.2.u.a.289.2	yes	160
175.39	even	30	inner	350.2.u.a.39.12	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.u.a.9.12	✓	160	1.1	even	1	trivial
350.2.u.a.39.12	yes	160	175.39	even	30	inner
350.2.u.a.109.2	yes	160	7.4	even	3	inner
350.2.u.a.289.2	yes	160	25.14	even	10	inner