Embedded newform 16.11.f.a.11.1

• Label: 16.11.f.a.11.1
• Level: $16$
• Weight: $11$
• Character: 16.11
• Analytic conductor: $10.166$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1657160428\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	16.11
Dual form			16.11.f.a.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-31.9996 - 0.164143i) q^{2} +(-221.985 - 221.985i) q^{3} +(1023.95 + 10.5050i) q^{4} +(2579.78 + 2579.78i) q^{5} +(7066.97 + 7139.85i) q^{6} +4017.27 q^{7} +(-32764.1 - 504.230i) q^{8} +39505.3i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−31.9996	−	0.164143i	−0.999987	−	0.00512947i
							\(3\)	−221.985	−	221.985i	−0.913517	−	0.913517i	0.0830305	−	0.996547i	\(-0.473540\pi\)
−0.996547	+	0.0830305i	\(0.973540\pi\)
\(5\)	2579.78	+	2579.78i	0.825530	+	0.825530i	0.986895	−	0.161365i	\(-0.0515897\pi\)
							−0.161365	+	0.986895i	\(0.551590\pi\)
\(7\)	4017.27			0.239024			0.119512	−	0.992833i	\(-0.461867\pi\)
							0.119512	+	0.992833i	\(0.461867\pi\)
\(11\)	138039.	−	138039.i	0.857114	−	0.857114i	−0.133883	−	0.990997i	\(-0.542745\pi\)
							0.990997	+	0.133883i	\(0.0427448\pi\)
\(13\)	−18109.0	+	18109.0i	−0.0487727	+	0.0487727i	−0.731073	−	0.682300i	\(-0.760980\pi\)
							0.682300	+	0.731073i	\(0.260980\pi\)
\(17\)	−1.82538e6			−1.28561			−0.642806	−	0.766029i	\(-0.722230\pi\)
							−0.642806	+	0.766029i	\(0.722230\pi\)
\(19\)	−2.84862e6	−	2.84862e6i	−1.15045	−	1.15045i	−0.986463	−	0.163983i	\(-0.947566\pi\)
							−0.163983	−	0.986463i	\(-0.552434\pi\)
\(23\)	7.52016e6			1.16839			0.584195	−	0.811613i	\(-0.301410\pi\)
							0.584195	+	0.811613i	\(0.301410\pi\)
\(29\)	1.29830e7	−	1.29830e7i	0.632973	−	0.632973i	−0.315840	−	0.948813i	\(-0.602286\pi\)
							0.948813	+	0.315840i	\(0.102286\pi\)
\(31\)		−	3.98730e7i		−	1.39274i	−0.717682	−	0.696371i	\(-0.754797\pi\)
							0.717682	−	0.696371i	\(-0.245203\pi\)
\(37\)	1.07957e7	+	1.07957e7i	0.155683	+	0.155683i	0.780651	−	0.624968i	\(-0.214888\pi\)
							−0.624968	+	0.780651i	\(0.714888\pi\)
\(41\)		−	2.03413e8i		−	1.75574i	−0.478899	−	0.877870i	\(-0.658964\pi\)
							0.478899	−	0.877870i	\(-0.341036\pi\)
\(43\)	−5.22721e7	+	5.22721e7i	−0.355572	+	0.355572i	−0.862178	−	0.506606i	\(-0.830900\pi\)
							0.506606	+	0.862178i	\(0.330900\pi\)
\(47\)		−	2.90486e8i		−	1.26659i	−0.773911	−	0.633294i	\(-0.781702\pi\)
							0.773911	−	0.633294i	\(-0.218298\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.11.1	yes	38
4.3	odd	2		64.11.f.a.15.16		38
8.3	odd	2		128.11.f.a.31.4		38
8.5	even	2		128.11.f.b.31.16		38
16.3	odd	4	inner	16.11.f.a.3.1	✓	38
16.5	even	4		128.11.f.a.95.4		38
16.11	odd	4		128.11.f.b.95.16		38
16.13	even	4		64.11.f.a.47.16		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.1	✓	38	16.3	odd	4	inner
16.11.f.a.11.1	yes	38	1.1	even	1	trivial
64.11.f.a.15.16		38	4.3	odd	2
64.11.f.a.47.16		38	16.13	even	4
128.11.f.a.31.4		38	8.3	odd	2
128.11.f.a.95.4		38	16.5	even	4
128.11.f.b.31.16		38	8.5	even	2
128.11.f.b.95.16		38	16.11	odd	4