Embedded newform 350.2.q.b.11.7

• Label: 350.2.q.b.11.7
• 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			11.7
Character	\(\chi\)	\(=\)	350.11
Dual form			350.2.q.b.191.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.913545 - 0.406737i) q^{2} +(0.880627 + 0.978035i) q^{3} +(0.669131 + 0.743145i) q^{4} +(-1.52041 - 1.63962i) q^{5} +(-0.406690 - 1.25166i) q^{6} +(2.30318 - 1.30207i) q^{7} +(-0.309017 - 0.951057i) q^{8} +(0.132536 - 1.26100i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	0.880627	+	0.978035i	0.508430	+	0.564669i	0.941639	−	0.336625i	\(-0.109285\pi\)
−0.433209	+	0.901294i	\(0.642619\pi\)
\(5\)	−1.52041	−	1.63962i	−0.679946	−	0.733262i
							\(7\)	2.30318	−	1.30207i	0.870519	−	0.492135i
\(11\)	0.0318568	+	0.303097i	0.00960519	+	0.0913873i								0.998274	−	0.0587244i	\(-0.0187033\pi\)
							−0.988669	+	0.150112i	\(0.952037\pi\)
\(13\)	−2.26385	−	1.64478i	−0.627878	−	0.456180i	0.227786	−	0.973711i	\(-0.426851\pi\)
							−0.855664	+	0.517531i	\(0.826851\pi\)
\(17\)	2.06677	+	0.439305i	0.501265	+	0.106547i	0.451603	−	0.892219i	\(-0.350852\pi\)
							0.0496617	+	0.998766i	\(0.484186\pi\)
\(19\)	4.22368	−	4.69087i	0.968978	−	1.07616i	−0.0280879	−	0.999605i	\(-0.508942\pi\)
							0.997065	−	0.0765532i	\(-0.0243915\pi\)
\(23\)	−2.66832	−	1.18801i	−0.556382	−	0.247717i	0.109228	−	0.994017i	\(-0.465162\pi\)
							−0.665610	+	0.746299i	\(0.731829\pi\)
\(29\)	0.488249	−	1.50268i	0.0906655	−	0.279040i	−0.895434	−	0.445194i	\(-0.853135\pi\)
							0.986100	+	0.166154i	\(0.0531348\pi\)
\(31\)	9.59826	+	2.04017i	1.72390	+	0.366426i	0.960236	−	0.279190i	\(-0.0900659\pi\)
							0.763662	+	0.645616i	\(0.223399\pi\)
\(37\)	−0.330193	+	3.14158i	−0.0542834	+	0.516472i	0.933269	+	0.359179i	\(0.116943\pi\)
							−0.987552	+	0.157293i	\(0.949723\pi\)
\(41\)	−0.520460	−	0.378137i	−0.0812823	−	0.0590550i	0.546402	−	0.837523i	\(-0.315997\pi\)
							−0.627684	+	0.778468i	\(0.715997\pi\)
\(43\)	2.37558			0.362273			0.181136	−	0.983458i	\(-0.442022\pi\)
							0.181136	+	0.983458i	\(0.442022\pi\)
\(47\)	−12.5984	+	2.67787i	−1.83766	+	0.390608i	−0.990132	−	0.140137i	\(-0.955246\pi\)
							−0.847532	+	0.530745i	\(0.821912\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.q.b.11.7	✓	72
7.2	even	3	inner	350.2.q.b.261.3	yes	72
25.16	even	5	inner	350.2.q.b.291.3	yes	72
175.16	even	15	inner	350.2.q.b.191.7	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.q.b.11.7	✓	72	1.1	even	1	trivial
350.2.q.b.191.7	yes	72	175.16	even	15	inner
350.2.q.b.261.3	yes	72	7.2	even	3	inner
350.2.q.b.291.3	yes	72	25.16	even	5	inner