Embedded newform 16.5.f.a.3.7

• Label: 16.5.f.a.3.7
• 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			3.7
Root			\(-2.40693 + 1.48549i\) of defining polynomial
Character	\(\chi\)	\(=\)	16.3
Dual form			16.5.f.a.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.89242 - 0.921438i) q^{2} +(-0.0461995 + 0.0461995i) q^{3} +(14.3019 - 7.17325i) q^{4} +(-8.04297 + 8.04297i) q^{5} +(-0.137258 + 0.222398i) q^{6} -49.8797 q^{7} +(49.0594 - 41.0996i) q^{8} +80.9957i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{2} - 2 q^{3} - 8 q^{4} - 2 q^{5} + 64 q^{6} - 4 q^{7} - 92 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.89242	−	0.921438i	0.973106	−	0.230359i
							\(3\)	−0.0461995	+	0.0461995i	−0.00513328	+	0.00513328i	−0.709669	−	0.704535i	\(-0.751155\pi\)
0.704535	+	0.709669i	\(0.251155\pi\)
\(5\)	−8.04297	+	8.04297i	−0.321719	+	0.321719i	−0.849426	−	0.527707i	\(-0.823052\pi\)
							0.527707	+	0.849426i	\(0.323052\pi\)
\(7\)	−49.8797			−1.01795			−0.508976	−	0.860781i	\(-0.669976\pi\)
							−0.508976	+	0.860781i	\(0.669976\pi\)
\(11\)	−84.2573	−	84.2573i	−0.696341	−	0.696341i	0.267278	−	0.963619i	\(-0.413876\pi\)
							−0.963619	+	0.267278i	\(0.913876\pi\)
\(13\)	19.4838	+	19.4838i	0.115289	+	0.115289i	0.762398	−	0.647109i	\(-0.224022\pi\)
							−0.647109	+	0.762398i	\(0.724022\pi\)
\(17\)	437.855			1.51507			0.757535	−	0.652795i	\(-0.226404\pi\)
							0.757535	+	0.652795i	\(0.226404\pi\)
\(19\)	349.021	−	349.021i	0.966817	−	0.966817i	−0.0326494	−	0.999467i	\(-0.510394\pi\)
							0.999467	+	0.0326494i	\(0.0103945\pi\)
\(23\)	−404.840			−0.765293			−0.382647	−	0.923895i	\(-0.624987\pi\)
							−0.382647	+	0.923895i	\(0.624987\pi\)
\(29\)	−1031.65	−	1031.65i	−1.22670	−	1.22670i	−0.965206	−	0.261490i	\(-0.915786\pi\)
							−0.261490	−	0.965206i	\(-0.584214\pi\)
\(31\)			1506.15i			1.56728i	0.621217	+	0.783638i	\(0.286638\pi\)
							−0.621217	+	0.783638i	\(0.713362\pi\)
\(37\)	−434.262	+	434.262i	−0.317211	+	0.317211i	−0.847695	−	0.530484i	\(-0.822010\pi\)
							0.530484	+	0.847695i	\(0.322010\pi\)
\(41\)		−	696.847i		−	0.414543i	−0.978283	−	0.207272i	\(-0.933542\pi\)
							0.978283	−	0.207272i	\(-0.0664584\pi\)
\(43\)	917.612	+	917.612i	0.496275	+	0.496275i	0.910276	−	0.414002i	\(-0.135869\pi\)
							−0.414002	+	0.910276i	\(0.635869\pi\)
\(47\)		−	111.917i		−	0.0506641i	−0.999679	−	0.0253321i	\(-0.991936\pi\)
							0.999679	−	0.0253321i	\(-0.00806431\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.5.f.a.3.7	✓	14
3.2	odd	2		144.5.m.a.19.1		14
4.3	odd	2		64.5.f.a.47.4		14
8.3	odd	2		128.5.f.a.95.4		14
8.5	even	2		128.5.f.b.95.4		14
12.11	even	2		576.5.m.a.559.5		14
16.3	odd	4		128.5.f.b.31.4		14
16.5	even	4		64.5.f.a.15.4		14
16.11	odd	4	inner	16.5.f.a.11.7	yes	14
16.13	even	4		128.5.f.a.31.4		14
48.5	odd	4		576.5.m.a.271.5		14
48.11	even	4		144.5.m.a.91.1		14

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.5.f.a.3.7	✓	14	1.1	even	1	trivial
16.5.f.a.11.7	yes	14	16.11	odd	4	inner
64.5.f.a.15.4		14	16.5	even	4
64.5.f.a.47.4		14	4.3	odd	2
128.5.f.a.31.4		14	16.13	even	4
128.5.f.a.95.4		14	8.3	odd	2
128.5.f.b.31.4		14	16.3	odd	4
128.5.f.b.95.4		14	8.5	even	2
144.5.m.a.19.1		14	3.2	odd	2
144.5.m.a.91.1		14	48.11	even	4
576.5.m.a.271.5		14	48.5	odd	4
576.5.m.a.559.5		14	12.11	even	2