Embedded newform 16.15.f.a.3.9

• Label: 16.15.f.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-79.3579 - 100.431i) q^{2} +(-417.666 + 417.666i) q^{3} +(-3788.66 + 15939.9i) q^{4} +(-101107. + 101107. i) q^{5} +(75091.6 + 8801.42i) q^{6} -1.56037e6 q^{7} +(1.90152e6 - 884462. i) q^{8} +4.43408e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−79.3579	−	100.431i	−0.619983	−	0.784615i
							\(3\)	−417.666	+	417.666i	−0.190977	+	0.190977i	−0.796118	−	0.605141i	\(-0.793117\pi\)
0.605141	+	0.796118i	\(0.293117\pi\)
\(5\)	−101107.	+	101107.i	−1.29418	+	1.29418i	−0.361997	+	0.932179i	\(0.617905\pi\)
							−0.932179	+	0.361997i	\(0.882095\pi\)
\(7\)	−1.56037e6			−1.89471			−0.947353	−	0.320190i	\(-0.896253\pi\)
							−0.947353	+	0.320190i	\(0.896253\pi\)
\(11\)	−1.05364e7	−	1.05364e7i	−0.540686	−	0.540686i	0.383044	−	0.923730i	\(-0.374876\pi\)
							−0.923730	+	0.383044i	\(0.874876\pi\)
\(13\)	2.86650e7	+	2.86650e7i	0.456823	+	0.456823i	0.897611	−	0.440788i	\(-0.145301\pi\)
							−0.440788	+	0.897611i	\(0.645301\pi\)
\(17\)	2.05708e8			0.501314			0.250657	−	0.968076i	\(-0.419353\pi\)
							0.250657	+	0.968076i	\(0.419353\pi\)
\(19\)	−9.96323e7	+	9.96323e7i	−0.111462	+	0.111462i	−0.760638	−	0.649176i	\(-0.775114\pi\)
							0.649176	+	0.760638i	\(0.275114\pi\)
\(23\)	−3.68677e9			−1.08281			−0.541404	−	0.840762i	\(-0.682107\pi\)
							−0.541404	+	0.840762i	\(0.682107\pi\)
\(29\)	5.20602e9	+	5.20602e9i	0.301800	+	0.301800i	0.841718	−	0.539918i	\(-0.181545\pi\)
							−0.539918	+	0.841718i	\(0.681545\pi\)
\(31\)			2.35259e10i			0.855096i	0.903993	+	0.427548i	\(0.140622\pi\)
							−0.903993	+	0.427548i	\(0.859378\pi\)
\(37\)	−6.36611e10	+	6.36611e10i	−0.670598	+	0.670598i	−0.957854	−	0.287256i	\(-0.907257\pi\)
							0.287256	+	0.957854i	\(0.407257\pi\)
\(41\)			1.61175e11i			0.827579i	0.910373	+	0.413790i	\(0.135795\pi\)
							−0.910373	+	0.413790i	\(0.864205\pi\)
\(43\)	−7.40448e10	−	7.40448e10i	−0.272405	−	0.272405i	0.557663	−	0.830068i	\(-0.311698\pi\)
							−0.830068	+	0.557663i	\(0.811698\pi\)
\(47\)			4.55259e11i			0.898615i	0.893377	+	0.449308i	\(0.148329\pi\)
							−0.893377	+	0.449308i	\(0.851671\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.9	✓	54
4.3	odd	2		64.15.f.a.47.15		54
16.5	even	4		64.15.f.a.15.15		54
16.11	odd	4	inner	16.15.f.a.11.9	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.9	✓	54	1.1	even	1	trivial
16.15.f.a.11.9	yes	54	16.11	odd	4	inner
64.15.f.a.15.15		54	16.5	even	4
64.15.f.a.47.15		54	4.3	odd	2