Embedded newform 16.5.f.a.3.3

• Label: 16.5.f.a.3.3
• 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			3.3
Root			\(2.79265 + 0.448449i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.3
Dual form			16.5.f.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.34420 + 3.24110i) q^{2} +(-4.63552 + 4.63552i) q^{3} +(-5.00945 - 15.1956i) q^{4} +(-29.2002 + 29.2002i) q^{5} +(-4.15759 - 25.8908i) q^{6} +59.6196 q^{7} +(60.9935 + 19.3854i) q^{8} +38.0239i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} + 64 q^{6} - 4 q^{7} - 92 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.34420	+	3.24110i	−0.586050	+	0.810275i
							\(3\)	−4.63552	+	4.63552i	−0.515058	+	0.515058i	−0.916072	−	0.401014i	\(-0.868658\pi\)
0.401014	+	0.916072i	\(0.368658\pi\)
\(5\)	−29.2002	+	29.2002i	−1.16801	+	1.16801i	−0.185330	+	0.982676i	\(0.559335\pi\)
							−0.982676	+	0.185330i	\(0.940665\pi\)
\(7\)	59.6196			1.21673			0.608363	−	0.793659i	\(-0.291826\pi\)
							0.608363	+	0.793659i	\(0.291826\pi\)
\(11\)	−18.0837	−	18.0837i	−0.149452	−	0.149452i	0.628421	−	0.777873i	\(-0.283701\pi\)
							−0.777873	+	0.628421i	\(0.783701\pi\)
\(13\)	50.7721	+	50.7721i	0.300427	+	0.300427i	0.841181	−	0.540754i	\(-0.181861\pi\)
							−0.540754	+	0.841181i	\(0.681861\pi\)
\(17\)	−223.769			−0.774289			−0.387144	−	0.922019i	\(-0.626538\pi\)
							−0.387144	+	0.922019i	\(0.626538\pi\)
\(19\)	14.7360	−	14.7360i	0.0408199	−	0.0408199i	−0.686402	−	0.727222i	\(-0.740811\pi\)
							0.727222	+	0.686402i	\(0.240811\pi\)
\(23\)	739.082			1.39713			0.698565	−	0.715547i	\(-0.253822\pi\)
							0.698565	+	0.715547i	\(0.253822\pi\)
\(29\)	938.904	+	938.904i	1.11641	+	1.11641i	0.992263	+	0.124150i	\(0.0396204\pi\)
							0.124150	+	0.992263i	\(0.460380\pi\)
\(31\)			938.741i			0.976838i	0.872609	+	0.488419i	\(0.162426\pi\)
							−0.872609	+	0.488419i	\(0.837574\pi\)
\(37\)	263.837	−	263.837i	0.192722	−	0.192722i	−0.604149	−	0.796871i	\(-0.706487\pi\)
							0.796871	+	0.604149i	\(0.206487\pi\)
\(41\)		−	248.841i		−	0.148031i	−0.997257	−	0.0740157i	\(-0.976418\pi\)
							0.997257	−	0.0740157i	\(-0.0235815\pi\)
\(43\)	1035.00	+	1035.00i	0.559765	+	0.559765i	0.929240	−	0.369476i	\(-0.120463\pi\)
							−0.369476	+	0.929240i	\(0.620463\pi\)
\(47\)		−	2018.46i		−	0.913746i	−0.889532	−	0.456873i	\(-0.848969\pi\)
							0.889532	−	0.456873i	\(-0.151031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.5.f.a.3.3	✓	14
3.2	odd	2		144.5.m.a.19.5		14
4.3	odd	2		64.5.f.a.47.5		14
8.3	odd	2		128.5.f.a.95.3		14
8.5	even	2		128.5.f.b.95.5		14
12.11	even	2		576.5.m.a.559.7		14
16.3	odd	4		128.5.f.b.31.5		14
16.5	even	4		64.5.f.a.15.5		14
16.11	odd	4	inner	16.5.f.a.11.3	yes	14
16.13	even	4		128.5.f.a.31.3		14
48.5	odd	4		576.5.m.a.271.7		14
48.11	even	4		144.5.m.a.91.5		14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.5.f.a.3.3	✓	14	1.1	even	1	trivial
16.5.f.a.11.3	yes	14	16.11	odd	4	inner
64.5.f.a.15.5		14	16.5	even	4
64.5.f.a.47.5		14	4.3	odd	2
128.5.f.a.31.3		14	16.13	even	4
128.5.f.a.95.3		14	8.3	odd	2
128.5.f.b.31.5		14	16.3	odd	4
128.5.f.b.95.5		14	8.5	even	2
144.5.m.a.19.5		14	3.2	odd	2
144.5.m.a.91.5		14	48.11	even	4
576.5.m.a.271.7		14	48.5	odd	4
576.5.m.a.559.7		14	12.11	even	2