Embedded newform 16.13.f.a.3.12

• Label: 16.13.f.a.3.12
• Level: $16$
• Weight: $13$
• Character: 16.3
• Analytic conductor: $14.624$
• Analytic rank: $0$
• Dimension: $46$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.6239010764\)
Analytic rank:	\(0\)
Dimension:	\(46\)
Relative dimension:	\(23\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.12
Character	\(\chi\)	\(=\)	16.3
Dual form			16.13.f.a.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.36222 + 63.9116i) q^{2} +(837.232 - 837.232i) q^{3} +(-4073.39 + 429.769i) q^{4} +(5671.68 - 5671.68i) q^{5} +(56323.8 + 50693.9i) q^{6} -4585.41 q^{7} +(-41162.9 - 258892. i) q^{8} -870475. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 2 q^{2} - 2 q^{3} - 4232 q^{4} - 2 q^{5} + 72928 q^{6} - 4 q^{7} - 478172 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.36222	+	63.9116i	0.0525346	+	0.998619i
							\(3\)	837.232	−	837.232i	1.14847	−	1.14847i	0.161613	−	0.986854i	\(-0.448330\pi\)
0.986854	−	0.161613i	\(-0.0516695\pi\)
\(5\)	5671.68	−	5671.68i	0.362987	−	0.362987i	−0.501924	−	0.864912i	\(-0.667374\pi\)
							0.864912	+	0.501924i	\(0.167374\pi\)
\(7\)	−4585.41			−0.0389753			−0.0194877	−	0.999810i	\(-0.506204\pi\)
							−0.0194877	+	0.999810i	\(0.506204\pi\)
\(11\)	−1.20751e6	−	1.20751e6i	−0.681610	−	0.681610i	0.278753	−	0.960363i	\(-0.410079\pi\)
							−0.960363	+	0.278753i	\(0.910079\pi\)
\(13\)	−3.66470e6	−	3.66470e6i	−0.759238	−	0.759238i	0.216946	−	0.976184i	\(-0.430391\pi\)
							−0.976184	+	0.216946i	\(0.930391\pi\)
\(17\)	2.38723e7			0.989011			0.494505	−	0.869175i	\(-0.335349\pi\)
							0.494505	+	0.869175i	\(0.335349\pi\)
\(19\)	4.96293e7	−	4.96293e7i	1.05491	−	1.05491i	0.0565114	−	0.998402i	\(-0.482002\pi\)
							0.998402	−	0.0565114i	\(-0.0179977\pi\)
\(23\)	−1.13469e8			−0.766495			−0.383247	−	0.923646i	\(-0.625194\pi\)
							−0.383247	+	0.923646i	\(0.625194\pi\)
\(29\)	5.30250e8	+	5.30250e8i	0.891441	+	0.891441i	0.994659	−	0.103218i	\(-0.0329140\pi\)
							−0.103218	+	0.994659i	\(0.532914\pi\)
\(31\)		−	1.47187e9i		−	1.65844i	−0.558923	−	0.829220i	\(-0.688785\pi\)
							0.558923	−	0.829220i	\(-0.311215\pi\)
\(37\)	−1.24658e9	+	1.24658e9i	−0.485859	+	0.485859i	−0.906997	−	0.421138i	\(-0.861631\pi\)
							0.421138	+	0.906997i	\(0.361631\pi\)
\(41\)		−	7.97177e9i		−	1.67823i	−0.543954	−	0.839115i	\(-0.683074\pi\)
							0.543954	−	0.839115i	\(-0.316926\pi\)
\(43\)	3.00483e9	+	3.00483e9i	0.475345	+	0.475345i	0.903639	−	0.428294i	\(-0.140885\pi\)
							−0.428294	+	0.903639i	\(0.640885\pi\)
\(47\)			1.72206e10i			1.59758i	0.601612	+	0.798789i	\(0.294526\pi\)
							−0.601612	+	0.798789i	\(0.705474\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.13.f.a.3.12	✓	46
4.3	odd	2		64.13.f.a.47.2		46
8.3	odd	2		128.13.f.a.95.22		46
8.5	even	2		128.13.f.b.95.2		46
16.3	odd	4		128.13.f.b.31.2		46
16.5	even	4		64.13.f.a.15.2		46
16.11	odd	4	inner	16.13.f.a.11.12	yes	46
16.13	even	4		128.13.f.a.31.22		46

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.13.f.a.3.12	✓	46	1.1	even	1	trivial
16.13.f.a.11.12	yes	46	16.11	odd	4	inner
64.13.f.a.15.2		46	16.5	even	4
64.13.f.a.47.2		46	4.3	odd	2
128.13.f.a.31.22		46	16.13	even	4
128.13.f.a.95.22		46	8.3	odd	2
128.13.f.b.31.2		46	16.3	odd	4
128.13.f.b.95.2		46	8.5	even	2