Embedded newform 16.5.f.a.3.1

• Label: 16.5.f.a.3.1
• Level: $16$
• Weight: $5$
• Character: 16.3
• Analytic conductor: $1.654$
• Analytic rank: $0$
• Dimension: $14$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.65391940934\)
Analytic rank:	\(0\)
Dimension:	\(14\)
Relative dimension:	\(7\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial:	x^14 - 4*x^13 + 15*x^12 - 34*x^11 + 62*x^10 - 312*x^9 + 1432*x^8 - 4960*x^7 + 11456*x^6 - 19968*x^5 + 31744*x^4 - 139264*x^3 + 491520*x^2 - 1048576*x + 2097152
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{21} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.1
Root			\(2.24452 - 1.72109i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.3
Dual form			16.5.f.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.96560 + 0.523430i) q^{2} +(5.54016 - 5.54016i) q^{3} +(15.4520 - 4.15143i) q^{4} +(21.7374 - 21.7374i) q^{5} +(-19.0702 + 24.8700i) q^{6} -6.62054 q^{7} +(-59.1037 + 24.5510i) q^{8} +19.6133i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} + 64 q^{6} - 4 q^{7} - 92 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.96560	+	0.523430i	−0.991401	+	0.130858i
							\(3\)	5.54016	−	5.54016i	0.615573	−	0.615573i	−0.328820	−	0.944393i	\(-0.606651\pi\)
0.944393	+	0.328820i	\(0.106651\pi\)
\(5\)	21.7374	−	21.7374i	0.869495	−	0.869495i	−0.122921	−	0.992416i	\(-0.539226\pi\)
							0.992416	+	0.122921i	\(0.0392262\pi\)
\(7\)	−6.62054			−0.135113			−0.0675565	−	0.997715i	\(-0.521520\pi\)
							−0.0675565	+	0.997715i	\(0.521520\pi\)
\(11\)	−90.9986	−	90.9986i	−0.752054	−	0.752054i	0.222808	−	0.974862i	\(-0.428478\pi\)
							−0.974862	+	0.222808i	\(0.928478\pi\)
\(13\)	221.402	+	221.402i	1.31007	+	1.31007i	0.921362	+	0.388707i	\(0.127078\pi\)
							0.388707	+	0.921362i	\(0.372922\pi\)
\(17\)	−132.575			−0.458738			−0.229369	−	0.973339i	\(-0.573666\pi\)
							−0.229369	+	0.973339i	\(0.573666\pi\)
\(19\)	−402.520	+	402.520i	−1.11501	+	1.11501i	−0.122552	+	0.992462i	\(0.539108\pi\)
							−0.992462	+	0.122552i	\(0.960892\pi\)
\(23\)	27.5037			0.0519918			0.0259959	−	0.999662i	\(-0.491724\pi\)
							0.0259959	+	0.999662i	\(0.491724\pi\)
\(29\)	174.909	+	174.909i	0.207978	+	0.207978i	0.803407	−	0.595430i	\(-0.203018\pi\)
							−0.595430	+	0.803407i	\(0.703018\pi\)
\(31\)		−	1083.96i		−	1.12795i	−0.825791	−	0.563976i	\(-0.809271\pi\)
							0.825791	−	0.563976i	\(-0.190729\pi\)
\(37\)	553.474	−	553.474i	0.404291	−	0.404291i	−0.475451	−	0.879742i	\(-0.657715\pi\)
							0.879742	+	0.475451i	\(0.157715\pi\)
\(41\)		−	1803.47i		−	1.07285i	−0.843947	−	0.536427i	\(-0.819774\pi\)
							0.843947	−	0.536427i	\(-0.180226\pi\)
\(43\)	17.8633	+	17.8633i	0.00966108	+	0.00966108i	0.711921	−	0.702260i	\(-0.247825\pi\)
							−0.702260	+	0.711921i	\(0.747825\pi\)
\(47\)			2268.26i			1.02683i	0.858141	+	0.513414i	\(0.171620\pi\)
							−0.858141	+	0.513414i	\(0.828380\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.5.f.a.3.1	✓	14
3.2	odd	2		144.5.m.a.19.7		14
4.3	odd	2		64.5.f.a.47.2		14
8.3	odd	2		128.5.f.a.95.6		14
8.5	even	2		128.5.f.b.95.2		14
12.11	even	2		576.5.m.a.559.2		14
16.3	odd	4		128.5.f.b.31.2		14
16.5	even	4		64.5.f.a.15.2		14
16.11	odd	4	inner	16.5.f.a.11.1	yes	14
16.13	even	4		128.5.f.a.31.6		14
48.5	odd	4		576.5.m.a.271.2		14
48.11	even	4		144.5.m.a.91.7		14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.5.f.a.3.1	✓	14	1.1	even	1	trivial
16.5.f.a.11.1	yes	14	16.11	odd	4	inner
64.5.f.a.15.2		14	16.5	even	4
64.5.f.a.47.2		14	4.3	odd	2
128.5.f.a.31.6		14	16.13	even	4
128.5.f.a.95.6		14	8.3	odd	2
128.5.f.b.31.2		14	16.3	odd	4
128.5.f.b.95.2		14	8.5	even	2
144.5.m.a.19.7		14	3.2	odd	2
144.5.m.a.91.7		14	48.11	even	4
576.5.m.a.271.2		14	48.5	odd	4
576.5.m.a.559.2		14	12.11	even	2