Embedded newform 16.5.f.a.3.4

• Label: 16.5.f.a.3.4
• 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			3.4
Root			\(-2.15805 - 1.82834i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.3
Dual form			16.5.f.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.329715 - 3.98639i) q^{2} +(3.91498 - 3.91498i) q^{3} +(-15.7826 - 2.62875i) q^{4} +(4.72348 - 4.72348i) q^{5} +(-14.3158 - 16.8975i) q^{6} +45.3712 q^{7} +(-15.6830 + 62.0487i) q^{8} +50.3458i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 64q^{6} - 4q^{7} - 92q^{8} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.329715	−	3.98639i	0.0824289	−	0.996597i
							\(3\)	3.91498	−	3.91498i	0.434998	−	0.434998i	−0.455327	−	0.890324i	\(-0.650477\pi\)
0.890324	+	0.455327i	\(0.150477\pi\)
\(5\)	4.72348	−	4.72348i	0.188939	−	0.188939i	−0.606298	−	0.795237i	\(-0.707346\pi\)
							0.795237	+	0.606298i	\(0.207346\pi\)
\(7\)	45.3712			0.925943			0.462971	−	0.886373i	\(-0.346783\pi\)
							0.462971	+	0.886373i	\(0.346783\pi\)
\(11\)	110.228	+	110.228i	0.910974	+	0.910974i	0.996349	−	0.0853752i	\(-0.0272089\pi\)
							−0.0853752	+	0.996349i	\(0.527209\pi\)
\(13\)	−157.128	−	157.128i	−0.929754	−	0.929754i	0.0679357	−	0.997690i	\(-0.478359\pi\)
							−0.997690	+	0.0679357i	\(0.978359\pi\)
\(17\)	−378.592			−1.31001			−0.655003	−	0.755626i	\(-0.727333\pi\)
							−0.655003	+	0.755626i	\(0.727333\pi\)
\(19\)	203.127	−	203.127i	0.562680	−	0.562680i	−0.367388	−	0.930068i	\(-0.619748\pi\)
							0.930068	+	0.367388i	\(0.119748\pi\)
\(23\)	−740.444			−1.39971			−0.699853	−	0.714287i	\(-0.746751\pi\)
							−0.699853	+	0.714287i	\(0.746751\pi\)
\(29\)	82.6244	+	82.6244i	0.0982455	+	0.0982455i	0.754521	−	0.656276i	\(-0.227869\pi\)
							−0.656276	+	0.754521i	\(0.727869\pi\)
\(31\)		−	286.217i		−	0.297833i	−0.988850	−	0.148916i	\(-0.952421\pi\)
							0.988850	−	0.148916i	\(-0.0475785\pi\)
\(37\)	1470.39	−	1470.39i	1.07406	−	1.07406i	0.0770352	−	0.997028i	\(-0.475455\pi\)
							0.997028	−	0.0770352i	\(-0.0245454\pi\)
\(41\)			1301.21i			0.774070i	0.922065	+	0.387035i	\(0.126501\pi\)
							−0.922065	+	0.387035i	\(0.873499\pi\)
\(43\)	−366.979	−	366.979i	−0.198474	−	0.198474i	0.600871	−	0.799346i	\(-0.294820\pi\)
							−0.799346	+	0.600871i	\(0.794820\pi\)
\(47\)		−	751.307i		−	0.340112i	−0.985434	−	0.170056i	\(-0.945605\pi\)
							0.985434	−	0.170056i	\(-0.0543948\pi\)