Embedded newform 16.15.f.a.3.20

• Label: 16.15.f.a.3.20
• 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			3.20
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(83.7629 + 96.7872i) q^{2} +(-2050.98 + 2050.98i) q^{3} +(-2351.54 + 16214.4i) q^{4} +(-70340.6 + 70340.6i) q^{5} +(-370305. - 26712.6i) q^{6} -283091. q^{7} +(-1.76632e6 + 1.13056e6i) q^{8} -3.63010e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	83.7629	+	96.7872i	0.654398	+	0.756150i
							\(3\)	−2050.98	+	2050.98i	−0.937807	+	0.937807i	−0.998176	−	0.0603693i	\(-0.980772\pi\)
0.0603693	+	0.998176i	\(0.480772\pi\)
\(5\)	−70340.6	+	70340.6i	−0.900360	+	0.900360i	−0.995467	−	0.0951072i	\(-0.969681\pi\)
							0.0951072	+	0.995467i	\(0.469681\pi\)
\(7\)	−283091.			−0.343748			−0.171874	−	0.985119i	\(-0.554982\pi\)
							−0.171874	+	0.985119i	\(0.554982\pi\)
\(11\)	2.38907e7	+	2.38907e7i	1.22597	+	1.22597i	0.965475	+	0.260494i	\(0.0838854\pi\)
							0.260494	+	0.965475i	\(0.416115\pi\)
\(13\)	−1.65227e7	−	1.65227e7i	−0.263316	−	0.263316i	0.563084	−	0.826400i	\(-0.309615\pi\)
							−0.826400	+	0.563084i	\(0.809615\pi\)
\(17\)	5.92212e8			1.44323			0.721614	−	0.692296i	\(-0.243401\pi\)
							0.721614	+	0.692296i	\(0.243401\pi\)
\(19\)	−6.30412e8	+	6.30412e8i	−0.705260	+	0.705260i	−0.965535	−	0.260275i	\(-0.916187\pi\)
							0.260275	+	0.965535i	\(0.416187\pi\)
\(23\)	3.15943e9			0.927928			0.463964	−	0.885854i	\(-0.346427\pi\)
							0.463964	+	0.885854i	\(0.346427\pi\)
\(29\)	−3.01339e8	−	3.01339e8i	−0.0174690	−	0.0174690i	0.698318	−	0.715787i	\(-0.253932\pi\)
							−0.715787	+	0.698318i	\(0.753932\pi\)
\(31\)		−	3.91587e10i		−	1.42330i	−0.702535	−	0.711649i	\(-0.747949\pi\)
							0.702535	−	0.711649i	\(-0.252051\pi\)
\(37\)	−8.99794e10	+	8.99794e10i	−0.947831	+	0.947831i	−0.998705	−	0.0508743i	\(-0.983799\pi\)
							0.0508743	+	0.998705i	\(0.483799\pi\)
\(41\)			1.67053e11i			0.857763i	0.903361	+	0.428881i	\(0.141092\pi\)
							−0.903361	+	0.428881i	\(0.858908\pi\)
\(43\)	2.59334e11	+	2.59334e11i	0.954070	+	0.954070i	0.998991	−	0.0449205i	\(-0.0143034\pi\)
							−0.0449205	+	0.998991i	\(0.514303\pi\)
\(47\)		−	1.99009e11i		−	0.392815i	−0.980522	−	0.196408i	\(-0.937072\pi\)
							0.980522	−	0.196408i	\(-0.0629276\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.20	✓	54
4.3	odd	2		64.15.f.a.47.23		54
16.5	even	4		64.15.f.a.15.23		54
16.11	odd	4	inner	16.15.f.a.11.20	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.20	✓	54	1.1	even	1	trivial
16.15.f.a.11.20	yes	54	16.11	odd	4	inner
64.15.f.a.15.23		54	16.5	even	4
64.15.f.a.47.23		54	4.3	odd	2