Embedded newform 350.2.q.b.11.4

• Label: 350.2.q.b.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	350.11
Dual form			350.2.q.b.191.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.913545 - 0.406737i) q^{2} +(-0.848696 - 0.942572i) q^{3} +(0.669131 + 0.743145i) q^{4} +(-2.15057 + 0.612414i) q^{5} +(0.391944 + 1.20628i) q^{6} +(0.891313 + 2.49110i) q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.145428 - 1.38365i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.848696	−	0.942572i	−0.489995	−	0.544194i	0.446542	−	0.894762i	\(-0.352655\pi\)
−0.936537	+	0.350568i	\(0.885989\pi\)
\(5\)	−2.15057	+	0.612414i	−0.961764	+	0.273880i
							\(7\)	0.891313	+	2.49110i	0.336884	+	0.941546i
\(11\)	−0.171476	−	1.63149i	−0.0517020	−	0.491912i								−0.989479	−	0.144674i	\(-0.953787\pi\)
							0.937777	−	0.347237i	\(-0.112880\pi\)
\(13\)	5.23601	+	3.80419i	1.45221	+	1.05509i	0.985309	+	0.170781i	\(0.0546291\pi\)
							0.466900	+	0.884310i	\(0.345371\pi\)
\(17\)	−0.507021	−	0.107771i	−0.122971	−	0.0261382i	0.146015	−	0.989282i	\(-0.453355\pi\)
							−0.268986	+	0.963144i	\(0.586688\pi\)
\(19\)	3.84714	−	4.27269i	0.882595	−	0.980221i	−0.117322	−	0.993094i	\(-0.537431\pi\)
							0.999917	+	0.0128725i	\(0.00409755\pi\)
\(23\)	3.57724	+	1.59269i	0.745905	+	0.332098i	0.744251	−	0.667900i	\(-0.232807\pi\)
							0.00165439	+	0.999999i	\(0.499473\pi\)
\(29\)	2.02623	−	6.23608i	0.376261	−	1.15801i	−0.566364	−	0.824155i	\(-0.691650\pi\)
							0.942624	−	0.333856i	\(-0.108350\pi\)
\(31\)	1.87252	+	0.398017i	0.336315	+	0.0714859i	0.372974	−	0.927842i	\(-0.378338\pi\)
							−0.0366589	+	0.999328i	\(0.511672\pi\)
\(37\)	0.500837	−	4.76514i	0.0823370	−	0.783385i	−0.872971	−	0.487772i	\(-0.837810\pi\)
							0.955308	−	0.295612i	\(-0.0955237\pi\)
\(41\)	9.24206	+	6.71475i	1.44337	+	1.04867i	0.987325	+	0.158709i	\(0.0507332\pi\)
							0.456042	+	0.889958i	\(0.349267\pi\)
\(43\)	−2.81107			−0.428685			−0.214342	−	0.976759i	\(-0.568761\pi\)
							−0.214342	+	0.976759i	\(0.568761\pi\)
\(47\)	6.95263	−	1.47783i	1.01414	−	0.215563i	0.329279	−	0.944233i	\(-0.393194\pi\)
							0.684865	+	0.728670i	\(0.259861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.11.4	✓	72
7.2	even	3	inner	350.2.q.b.261.6	yes	72
25.16	even	5	inner	350.2.q.b.291.6	yes	72
175.16	even	15	inner	350.2.q.b.191.4	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.11.4	✓	72	1.1	even	1	trivial
350.2.q.b.191.4	yes	72	175.16	even	15	inner
350.2.q.b.261.6	yes	72	7.2	even	3	inner
350.2.q.b.291.6	yes	72	25.16	even	5	inner