Embedded newform 16.15.f.a.3.10

• Label: 16.15.f.a.3.10
• Level: $16$
• Weight: $15$
• Character: 16.3
• Analytic conductor: $19.893$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.8926349043\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.10
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-72.6989 + 105.351i) q^{2} +(692.853 - 692.853i) q^{3} +(-5813.74 - 15317.8i) q^{4} +(-62322.7 + 62322.7i) q^{5} +(22623.2 + 123363. i) q^{6} -300008. q^{7} +(2.03640e6 + 501105. i) q^{8} +3.82288e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−72.6989	+	105.351i	−0.567960	+	0.823056i
							\(3\)	692.853	−	692.853i	0.316805	−	0.316805i	−0.530733	−	0.847539i	\(-0.678083\pi\)
0.847539	+	0.530733i	\(0.178083\pi\)
\(5\)	−62322.7	+	62322.7i	−0.797730	+	0.797730i	−0.982737	−	0.185007i	\(-0.940769\pi\)
							0.185007	+	0.982737i	\(0.440769\pi\)
\(7\)	−300008.			−0.364290			−0.182145	−	0.983272i	\(-0.558304\pi\)
							−0.182145	+	0.983272i	\(0.558304\pi\)
\(11\)	−702232.	−	702232.i	−0.0360356	−	0.0360356i	0.688859	−	0.724895i	\(-0.258112\pi\)
							−0.724895	+	0.688859i	\(0.758112\pi\)
\(13\)	−8.38874e7	−	8.38874e7i	−1.33688	−	1.33688i	−0.899062	−	0.437820i	\(-0.855751\pi\)
							−0.437820	−	0.899062i	\(-0.644249\pi\)
\(17\)	3.11542e8			0.759230			0.379615	−	0.925144i	\(-0.376056\pi\)
							0.379615	+	0.925144i	\(0.376056\pi\)
\(19\)	1.02517e9	−	1.02517e9i	1.14689	−	1.14689i	0.159731	−	0.987161i	\(-0.448937\pi\)
							0.987161	−	0.159731i	\(-0.0510627\pi\)
\(23\)	3.69692e9			1.08579			0.542894	−	0.839801i	\(-0.317328\pi\)
							0.542894	+	0.839801i	\(0.317328\pi\)
\(29\)	−5.00693e9	−	5.00693e9i	−0.290259	−	0.290259i	0.546924	−	0.837183i	\(-0.315799\pi\)
							−0.837183	+	0.546924i	\(0.815799\pi\)
\(31\)		−	2.82491e10i		−	1.02677i	−0.858159	−	0.513385i	\(-0.828391\pi\)
							0.858159	−	0.513385i	\(-0.171609\pi\)
\(37\)	8.06331e10	−	8.06331e10i	0.849379	−	0.849379i	−0.140677	−	0.990056i	\(-0.544928\pi\)
							0.990056	+	0.140677i	\(0.0449278\pi\)
\(41\)			7.87272e10i			0.404238i	0.979361	+	0.202119i	\(0.0647828\pi\)
							−0.979361	+	0.202119i	\(0.935217\pi\)
\(43\)	1.27929e11	+	1.27929e11i	0.470640	+	0.470640i	0.902122	−	0.431482i	\(-0.142009\pi\)
							−0.431482	+	0.902122i	\(0.642009\pi\)
\(47\)		−	5.95749e11i		−	1.17592i	−0.808889	−	0.587961i	\(-0.799931\pi\)
							0.808889	−	0.587961i	\(-0.200069\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.10	✓	54
4.3	odd	2		64.15.f.a.47.11		54
16.5	even	4		64.15.f.a.15.11		54
16.11	odd	4	inner	16.15.f.a.11.10	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.10	✓	54	1.1	even	1	trivial
16.15.f.a.11.10	yes	54	16.11	odd	4	inner
64.15.f.a.15.11		54	16.5	even	4
64.15.f.a.47.11		54	4.3	odd	2