Embedded newform 16.15.f.a.3.4

• Label: 16.15.f.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-118.943 + 47.2917i) q^{2} +(-1108.18 + 1108.18i) q^{3} +(11911.0 - 11250.1i) q^{4} +(-55052.8 + 55052.8i) q^{5} +(79403.1 - 184219. i) q^{6} +519580. q^{7} +(-884698. + 1.90141e6i) q^{8} +2.32682e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−118.943	+	47.2917i	−0.929244	+	0.369466i
							\(3\)	−1108.18	+	1108.18i	−0.506714	+	0.506714i	−0.913516	−	0.406802i	\(-0.866644\pi\)
0.406802	+	0.913516i	\(0.366644\pi\)
\(5\)	−55052.8	+	55052.8i	−0.704676	+	0.704676i	−0.965411	−	0.260734i	\(-0.916035\pi\)
							0.260734	+	0.965411i	\(0.416035\pi\)
\(7\)	519580.			0.630908			0.315454	−	0.948941i	\(-0.397843\pi\)
							0.315454	+	0.948941i	\(0.397843\pi\)
\(11\)	1.34844e7	+	1.34844e7i	0.691963	+	0.691963i	0.962664	−	0.270701i	\(-0.0872554\pi\)
							−0.270701	+	0.962664i	\(0.587255\pi\)
\(13\)	8.37297e7	+	8.37297e7i	1.33437	+	1.33437i	0.901416	+	0.432953i	\(0.142529\pi\)
							0.432953	+	0.901416i	\(0.357471\pi\)
\(17\)	−3.52150e8			−0.858194			−0.429097	−	0.903258i	\(-0.641168\pi\)
							−0.429097	+	0.903258i	\(0.641168\pi\)
\(19\)	1.52967e8	−	1.52967e8i	0.171129	−	0.171129i	−0.616347	−	0.787475i	\(-0.711388\pi\)
							0.787475	+	0.616347i	\(0.211388\pi\)
\(23\)	−2.03600e9			−0.597974			−0.298987	−	0.954257i	\(-0.596649\pi\)
							−0.298987	+	0.954257i	\(0.596649\pi\)
\(29\)	−1.70313e10	−	1.70313e10i	−0.987329	−	0.987329i	0.0125922	−	0.999921i	\(-0.495992\pi\)
							−0.999921	+	0.0125922i	\(0.995992\pi\)
\(31\)			2.49816e10i			0.908004i	0.891001	+	0.454002i	\(0.150004\pi\)
							−0.891001	+	0.454002i	\(0.849996\pi\)
\(37\)	9.94297e10	−	9.94297e10i	1.04738	−	1.04738i	0.0485587	−	0.998820i	\(-0.484537\pi\)
							0.998820	−	0.0485587i	\(-0.0154628\pi\)
\(41\)			1.06387e11i			0.546262i	0.961977	+	0.273131i	\(0.0880593\pi\)
							−0.961977	+	0.273131i	\(0.911941\pi\)
\(43\)	1.70983e11	+	1.70983e11i	0.629032	+	0.629032i	0.947825	−	0.318792i	\(-0.103277\pi\)
							−0.318792	+	0.947825i	\(0.603277\pi\)
\(47\)			2.13266e11i			0.420956i	0.977599	+	0.210478i	\(0.0675020\pi\)
							−0.977599	+	0.210478i	\(0.932498\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.4	✓	54
4.3	odd	2		64.15.f.a.47.19		54
16.5	even	4		64.15.f.a.15.19		54
16.11	odd	4	inner	16.15.f.a.11.4	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.4	✓	54	1.1	even	1	trivial
16.15.f.a.11.4	yes	54	16.11	odd	4	inner
64.15.f.a.15.19		54	16.5	even	4
64.15.f.a.47.19		54	4.3	odd	2