Embedded newform 350.2.q.b.11.5

• Label: 350.2.q.b.11.5
• Level: $350$
• Weight: $2$
• Character: 350.11
• 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			11.5
Character	\(\chi\)	\(=\)	350.11
Dual form			350.2.q.b.191.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.913545 - 0.406737i) q^{2} +(-0.318061 - 0.353243i) q^{3} +(0.669131 + 0.743145i) q^{4} +(1.85994 - 1.24123i) q^{5} +(0.146887 + 0.452070i) q^{6} +(-2.60446 - 0.465583i) q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.289968 - 2.75886i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{5}\right)\)

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
							\(3\)	−0.318061	−	0.353243i	−0.183633	−	0.203945i	0.644299	−	0.764774i	\(-0.277149\pi\)
−0.827932	+	0.560829i	\(0.810483\pi\)
\(5\)	1.85994	−	1.24123i	0.831788	−	0.555093i
							\(7\)	−2.60446	−	0.465583i	−0.984395	−	0.175974i
\(11\)	0.0881312	+	0.838513i	0.0265726	+	0.252821i								0.999742	+	0.0227299i	\(0.00723577\pi\)
							−0.973169	+	0.230091i	\(0.926098\pi\)
\(13\)	−1.13989	−	0.828176i	−0.316148	−	0.229695i	0.418382	−	0.908271i	\(-0.362597\pi\)
							−0.734530	+	0.678576i	\(0.762597\pi\)
\(17\)	0.364191	+	0.0774112i	0.0883293	+	0.0187750i	0.251864	−	0.967763i	\(-0.418956\pi\)
							−0.163535	+	0.986538i	\(0.552290\pi\)
\(19\)	3.70653	−	4.11652i	0.850336	−	0.944394i	−0.148674	−	0.988886i	\(-0.547500\pi\)
							0.999010	+	0.0444924i	\(0.0141670\pi\)
\(23\)	−3.95316	−	1.76006i	−0.824290	−	0.366998i	−0.0491534	−	0.998791i	\(-0.515652\pi\)
							−0.775137	+	0.631794i	\(0.782319\pi\)
\(29\)	0.850958	−	2.61898i	0.158019	−	0.486332i	−0.840435	−	0.541912i	\(-0.817701\pi\)
							0.998454	+	0.0555796i	\(0.0177007\pi\)
\(31\)	−6.07277	−	1.29081i	−1.09070	−	0.231836i	−0.372752	−	0.927931i	\(-0.621586\pi\)
							−0.717950	+	0.696095i	\(0.754919\pi\)
\(37\)	0.706362	−	6.72059i	0.116125	−	1.10486i	−0.768918	−	0.639348i	\(-0.779204\pi\)
							0.885043	−	0.465510i	\(-0.154129\pi\)
\(41\)	1.69055	+	1.22826i	0.264020	+	0.191822i	0.711917	−	0.702263i	\(-0.247827\pi\)
							−0.447897	+	0.894085i	\(0.647827\pi\)
\(43\)	−0.698123			−0.106463			−0.0532314	−	0.998582i	\(-0.516952\pi\)
							−0.0532314	+	0.998582i	\(0.516952\pi\)
\(47\)	−0.511396	+	0.108701i	−0.0745948	+	0.0158556i	−0.245058	−	0.969508i	\(-0.578807\pi\)
							0.170463	+	0.985364i	\(0.445474\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.11.5	✓	72
7.2	even	3	inner	350.2.q.b.261.5	yes	72
25.16	even	5	inner	350.2.q.b.291.5	yes	72
175.16	even	15	inner	350.2.q.b.191.5	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.11.5	✓	72	1.1	even	1	trivial
350.2.q.b.191.5	yes	72	175.16	even	15	inner
350.2.q.b.261.5	yes	72	7.2	even	3	inner
350.2.q.b.291.5	yes	72	25.16	even	5	inner