Embedded newform 16.15.f.a.3.12

• Label: 16.15.f.a.3.12
• Level: $16$
• Weight: $15$
• Character: 16.3
• Analytic conductor: $19.893$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	16.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.8926349043\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) 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			16.15.f.a.11.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-36.7331 - 122.616i) q^{2} +(-1658.66 + 1658.66i) q^{3} +(-13685.4 + 9008.14i) q^{4} +(7093.51 - 7093.51i) q^{5} +(264306. + 142450. i) q^{6} +747019. q^{7} +(1.60725e6 + 1.34715e6i) q^{8} -719331. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−36.7331	−	122.616i	−0.286978	−	0.957937i
							\(3\)	−1658.66	+	1658.66i	−0.758418	+	0.758418i	−0.976034	−	0.217617i	\(-0.930172\pi\)
0.217617	+	0.976034i	\(0.430172\pi\)
\(5\)	7093.51	−	7093.51i	0.0907969	−	0.0907969i	−0.660249	−	0.751046i	\(-0.729549\pi\)
							0.751046	+	0.660249i	\(0.229549\pi\)
\(7\)	747019.			0.907080			0.453540	−	0.891236i	\(-0.350161\pi\)
							0.453540	+	0.891236i	\(0.350161\pi\)
\(11\)	1.32508e6	+	1.32508e6i	0.0679976	+	0.0679976i	0.740288	−	0.672290i	\(-0.234689\pi\)
							−0.672290	+	0.740288i	\(0.734689\pi\)
\(13\)	−2.12896e7	−	2.12896e7i	−0.339285	−	0.339285i	0.516813	−	0.856098i	\(-0.327118\pi\)
							−0.856098	+	0.516813i	\(0.827118\pi\)
\(17\)	−2.24239e8			−0.546472			−0.273236	−	0.961947i	\(-0.588094\pi\)
							−0.273236	+	0.961947i	\(0.588094\pi\)
\(19\)	−4.50730e8	+	4.50730e8i	−0.504245	+	0.504245i	−0.912754	−	0.408509i	\(-0.866049\pi\)
							0.408509	+	0.912754i	\(0.366049\pi\)
\(23\)	−1.08044e9			−0.317325			−0.158663	−	0.987333i	\(-0.550718\pi\)
							−0.158663	+	0.987333i	\(0.550718\pi\)
\(29\)	−1.05455e10	−	1.05455e10i	−0.611340	−	0.611340i	0.331955	−	0.943295i	\(-0.392292\pi\)
							−0.943295	+	0.331955i	\(0.892292\pi\)
\(31\)		−	4.63415e10i		−	1.68437i	−0.539186	−	0.842186i	\(-0.681268\pi\)
							0.539186	−	0.842186i	\(-0.318732\pi\)
\(37\)	−1.76404e10	+	1.76404e10i	−0.185821	+	0.185821i	−0.793887	−	0.608066i	\(-0.791946\pi\)
							0.608066	+	0.793887i	\(0.291946\pi\)
\(41\)			7.50082e10i			0.385143i	0.981283	+	0.192571i	\(0.0616827\pi\)
							−0.981283	+	0.192571i	\(0.938317\pi\)
\(43\)	−1.50371e11	−	1.50371e11i	−0.553205	−	0.553205i	0.374159	−	0.927364i	\(-0.377931\pi\)
							−0.927364	+	0.374159i	\(0.877931\pi\)
\(47\)		−	4.33328e11i		−	0.855327i	−0.903938	−	0.427664i	\(-0.859337\pi\)
							0.903938	−	0.427664i	\(-0.140663\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.12	✓	54
4.3	odd	2		64.15.f.a.47.21		54
16.5	even	4		64.15.f.a.15.21		54
16.11	odd	4	inner	16.15.f.a.11.12	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.12	✓	54	1.1	even	1	trivial
16.15.f.a.11.12	yes	54	16.11	odd	4	inner
64.15.f.a.15.21		54	16.5	even	4
64.15.f.a.47.21		54	4.3	odd	2