Embedded newform 350.2.q.b.191.9

• Label: 350.2.q.b.191.9
• Level: $350$
• Weight: $2$
• Character: 350.191
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(9\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			191.9
Character	\(\chi\)	\(=\)	350.191
Dual form			350.2.q.b.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.913545 + 0.406737i) q^{2} +(2.10646 - 2.33946i) q^{3} +(0.669131 - 0.743145i) q^{4} +(-0.760329 + 2.10283i) q^{5} +(-0.972801 + 2.99397i) q^{6} +(-2.25986 - 1.37587i) q^{7} +(-0.309017 + 0.951057i) q^{8} +(-0.722315 - 6.87236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.913545	+	0.406737i	−0.645974	+	0.287606i
							\(3\)	2.10646	−	2.33946i	1.21616	−	1.35069i	0.297959	−	0.954579i	\(-0.403694\pi\)
0.918204	−	0.396108i	\(-0.129639\pi\)
\(5\)	−0.760329	+	2.10283i	−0.340029	+	0.940415i
							\(7\)	−2.25986	−	1.37587i	−0.854147	−	0.520032i
\(11\)	0.277197	−	2.63735i	0.0835781	−	0.795192i								−0.869798	−	0.493408i	\(-0.835751\pi\)
							0.953376	−	0.301784i	\(-0.0975823\pi\)
\(13\)	3.75282	−	2.72658i	1.04084	−	0.756217i	0.0703939	−	0.997519i	\(-0.477574\pi\)
							0.970450	+	0.241302i	\(0.0775744\pi\)
\(17\)	2.61195	−	0.555187i	0.633491	−	0.134653i	0.120037	−	0.992769i	\(-0.461699\pi\)
							0.513455	+	0.858117i	\(0.328365\pi\)
\(19\)	−2.44616	−	2.71674i	−0.561188	−	0.623262i	0.394054	−	0.919087i	\(-0.371072\pi\)
							−0.955242	+	0.295825i	\(0.904405\pi\)
\(23\)	3.94690	−	1.75727i	0.822985	−	0.366416i	0.0483541	−	0.998830i	\(-0.484602\pi\)
							0.774631	+	0.632414i	\(0.217936\pi\)
\(29\)	2.34530	+	7.21809i	0.435511	+	1.34037i	0.892562	+	0.450925i	\(0.148906\pi\)
							−0.457050	+	0.889441i	\(0.651094\pi\)
\(31\)	−4.02406	+	0.855341i	−0.722743	+	0.153624i	−0.554574	−	0.832134i	\(-0.687119\pi\)
							−0.168169	+	0.985758i	\(0.553785\pi\)
\(37\)	0.749420	+	7.13025i	0.123204	+	1.17221i	0.865068	+	0.501655i	\(0.167275\pi\)
							−0.741864	+	0.670551i	\(0.766058\pi\)
\(41\)	−8.00810	+	5.81823i	−1.25066	+	0.908654i	−0.998260	−	0.0589698i	\(-0.981218\pi\)
							−0.252396	+	0.967624i	\(0.581218\pi\)
\(43\)	3.81807			0.582250			0.291125	−	0.956685i	\(-0.405971\pi\)
							0.291125	+	0.956685i	\(0.405971\pi\)
\(47\)	11.9794	+	2.54631i	1.74738	+	0.371417i	0.967176	−	0.254109i	\(-0.0817822\pi\)
							0.780203	+	0.625526i	\(0.215116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.191.9	yes	72
7.4	even	3	inner	350.2.q.b.291.1	yes	72
25.11	even	5	inner	350.2.q.b.261.1	yes	72
175.11	even	15	inner	350.2.q.b.11.9	✓	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.11.9	✓	72	175.11	even	15	inner
350.2.q.b.191.9	yes	72	1.1	even	1	trivial
350.2.q.b.261.1	yes	72	25.11	even	5	inner
350.2.q.b.291.1	yes	72	7.4	even	3	inner