Embedded newform 16.5.f.a.3.6

• Label: 16.5.f.a.3.6
• 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			3.6
Root			\(0.336831 + 2.80830i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.3
Dual form			16.5.f.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.47147 + 3.14513i) q^{2} +(-7.86839 + 7.86839i) q^{3} +(-3.78368 + 15.5462i) q^{4} +(27.2309 - 27.2309i) q^{5} +(-44.1936 - 5.30063i) q^{6} +50.3097 q^{7} +(-58.2460 + 26.5217i) q^{8} -42.8233i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14q - 2q^{2} - 2q^{3} - 8q^{4} - 2q^{5} + 64q^{6} - 4q^{7} - 92q^{8} + O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.47147	+	3.14513i	0.617867	+	0.786282i
							\(3\)	−7.86839	+	7.86839i	−0.874266	+	0.874266i	−0.992934	−	0.118668i	\(-0.962138\pi\)
0.118668	+	0.992934i	\(0.462138\pi\)
\(5\)	27.2309	−	27.2309i	1.08924	−	1.08924i	0.0936308	−	0.995607i	\(-0.470153\pi\)
							0.995607	−	0.0936308i	\(-0.0298473\pi\)
\(7\)	50.3097			1.02673			0.513365	−	0.858171i	\(-0.328399\pi\)
							0.513365	+	0.858171i	\(0.328399\pi\)
\(11\)	−53.1047	−	53.1047i	−0.438881	−	0.438881i	0.452754	−	0.891635i	\(-0.350442\pi\)
							−0.891635	+	0.452754i	\(0.850442\pi\)
\(13\)	−125.128	−	125.128i	−0.740404	−	0.740404i	0.232252	−	0.972656i	\(-0.425391\pi\)
							−0.972656	+	0.232252i	\(0.925391\pi\)
\(17\)	286.271			0.990557			0.495279	−	0.868734i	\(-0.335066\pi\)
							0.495279	+	0.868734i	\(0.335066\pi\)
\(19\)	−99.5010	+	99.5010i	−0.275626	+	0.275626i	−0.831360	−	0.555734i	\(-0.812437\pi\)
							0.555734	+	0.831360i	\(0.312437\pi\)
\(23\)	−100.505			−0.189991			−0.0949957	−	0.995478i	\(-0.530284\pi\)
							−0.0949957	+	0.995478i	\(0.530284\pi\)
\(29\)	343.872	+	343.872i	0.408885	+	0.408885i	0.881350	−	0.472465i	\(-0.156636\pi\)
							−0.472465	+	0.881350i	\(0.656636\pi\)
\(31\)		−	208.400i		−	0.216858i	−0.994104	−	0.108429i	\(-0.965418\pi\)
							0.994104	−	0.108429i	\(-0.0345820\pi\)
\(37\)	−1159.47	+	1159.47i	−0.846946	+	0.846946i	−0.989751	−	0.142805i	\(-0.954388\pi\)
							0.142805	+	0.989751i	\(0.454388\pi\)
\(41\)			2335.63i			1.38943i	0.719286	+	0.694714i	\(0.244469\pi\)
							−0.719286	+	0.694714i	\(0.755531\pi\)
\(43\)	−2079.41	−	2079.41i	−1.12461	−	1.12461i	−0.991039	−	0.133575i	\(-0.957354\pi\)
							−0.133575	−	0.991039i	\(-0.542646\pi\)
\(47\)			1054.04i			0.477159i	0.971123	+	0.238580i	\(0.0766818\pi\)
							−0.971123	+	0.238580i	\(0.923318\pi\)