Embedded newform 16.11.f.a.3.10

• Label: 16.11.f.a.3.10
• Level: $16$
• Weight: $11$
• Character: 16.3
• 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			3.10
Character	\(\chi\)	\(=\)	16.3
Dual form			16.11.f.a.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21444 - 31.9769i) q^{2} +(287.146 - 287.146i) q^{3} +(-1021.05 + 77.6681i) q^{4} +(-3974.49 + 3974.49i) q^{5} +(-9530.78 - 8833.33i) q^{6} -11755.9 q^{7} +(3723.59 + 32555.7i) q^{8} -105857. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{2} - 2 q^{3} - 1256 q^{4} - 2 q^{5} - 17216 q^{6} - 4 q^{7} + 69124 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21444	−	31.9769i	−0.0379512	−	0.999280i
							\(3\)	287.146	−	287.146i	1.18167	−	1.18167i	0.202360	−	0.979311i	\(-0.435139\pi\)
0.979311	−	0.202360i	\(-0.0648610\pi\)
\(5\)	−3974.49	+	3974.49i	−1.27184	+	1.27184i	−0.326712	+	0.945124i	\(0.605941\pi\)
							−0.945124	+	0.326712i	\(0.894059\pi\)
\(7\)	−11755.9			−0.699463			−0.349732	−	0.936850i	\(-0.613727\pi\)
							−0.349732	+	0.936850i	\(0.613727\pi\)
\(11\)	−2670.51	−	2670.51i	−0.0165818	−	0.0165818i	0.698767	−	0.715349i	\(-0.253732\pi\)
							−0.715349	+	0.698767i	\(0.753732\pi\)
\(13\)	−443886.	−	443886.i	−1.19551	−	1.19551i	−0.975496	−	0.220018i	\(-0.929388\pi\)
							−0.220018	−	0.975496i	\(-0.570612\pi\)
\(17\)	−13922.7			−0.00980572			−0.00490286	−	0.999988i	\(-0.501561\pi\)
							−0.00490286	+	0.999988i	\(0.501561\pi\)
\(19\)	814378.	−	814378.i	0.328896	−	0.328896i	−0.523271	−	0.852166i	\(-0.675288\pi\)
							0.852166	+	0.523271i	\(0.175288\pi\)
\(23\)	−4.54668e6			−0.706407			−0.353203	−	0.935547i	\(-0.614908\pi\)
							−0.353203	+	0.935547i	\(0.614908\pi\)
\(29\)	178026.	+	178026.i	0.00867948	+	0.00867948i	0.711433	−	0.702754i	\(-0.248046\pi\)
							−0.702754	+	0.711433i	\(0.748046\pi\)
\(31\)		−	270764.i		−	0.00945765i	−0.999989	−	0.00472882i	\(-0.998495\pi\)
							0.999989	−	0.00472882i	\(-0.00150524\pi\)
\(37\)	−4.38637e7	+	4.38637e7i	−0.632553	+	0.632553i	−0.948708	−	0.316155i	\(-0.897608\pi\)
							0.316155	+	0.948708i	\(0.397608\pi\)
\(41\)		−	4.64100e6i		−	0.0400583i	−0.999799	−	0.0200292i	\(-0.993624\pi\)
							0.999799	−	0.0200292i	\(-0.00637590\pi\)
\(43\)	8.47711e6	+	8.47711e6i	0.0576641	+	0.0576641i	0.735351	−	0.677687i	\(-0.237017\pi\)
							−0.677687	+	0.735351i	\(0.737017\pi\)
\(47\)		−	1.88545e8i		−	0.822104i	−0.911612	−	0.411052i	\(-0.865161\pi\)
							0.911612	−	0.411052i	\(-0.134839\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.f.a.3.10	✓	38
4.3	odd	2		64.11.f.a.47.2		38
8.3	odd	2		128.11.f.a.95.18		38
8.5	even	2		128.11.f.b.95.2		38
16.3	odd	4		128.11.f.b.31.2		38
16.5	even	4		64.11.f.a.15.2		38
16.11	odd	4	inner	16.11.f.a.11.10	yes	38
16.13	even	4		128.11.f.a.31.18		38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.f.a.3.10	✓	38	1.1	even	1	trivial
16.11.f.a.11.10	yes	38	16.11	odd	4	inner
64.11.f.a.15.2		38	16.5	even	4
64.11.f.a.47.2		38	4.3	odd	2
128.11.f.a.31.18		38	16.13	even	4
128.11.f.a.95.18		38	8.3	odd	2
128.11.f.b.31.2		38	16.3	odd	4
128.11.f.b.95.2		38	8.5	even	2