Embedded newform 16.15.f.a.3.16

• Label: 16.15.f.a.3.16
• 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			3.16
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(42.3109 - 120.805i) q^{2} +(859.991 - 859.991i) q^{3} +(-12803.6 - 10222.7i) q^{4} +(-55370.8 + 55370.8i) q^{5} +(-67504.0 - 140278. i) q^{6} +175518. q^{7} +(-1.77668e6 + 1.11420e6i) q^{8} +3.30380e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	42.3109	−	120.805i	0.330554	−	0.943787i
							\(3\)	859.991	−	859.991i	0.393228	−	0.393228i	−0.482608	−	0.875836i	\(-0.660310\pi\)
0.875836	+	0.482608i	\(0.160310\pi\)
\(5\)	−55370.8	+	55370.8i	−0.708746	+	0.708746i	−0.966271	−	0.257526i	\(-0.917093\pi\)
							0.257526	+	0.966271i	\(0.417093\pi\)
\(7\)	175518.			0.213126			0.106563	−	0.994306i	\(-0.466015\pi\)
							0.106563	+	0.994306i	\(0.466015\pi\)
\(11\)	1.27142e7	+	1.27142e7i	0.652442	+	0.652442i	0.953580	−	0.301139i	\(-0.0973667\pi\)
							−0.301139	+	0.953580i	\(0.597367\pi\)
\(13\)	−6.39406e6	−	6.39406e6i	−0.101900	−	0.101900i	0.654319	−	0.756219i	\(-0.272955\pi\)
							−0.756219	+	0.654319i	\(0.772955\pi\)
\(17\)	−4.26614e8			−1.03966			−0.519832	−	0.854269i	\(-0.674005\pi\)
							−0.519832	+	0.854269i	\(0.674005\pi\)
\(19\)	−1.17940e8	+	1.17940e8i	−0.131942	+	0.131942i	−0.769994	−	0.638051i	\(-0.779741\pi\)
							0.638051	+	0.769994i	\(0.279741\pi\)
\(23\)	4.32243e9			1.26950			0.634750	−	0.772718i	\(-0.281103\pi\)
							0.634750	+	0.772718i	\(0.281103\pi\)
\(29\)	2.12197e9	+	2.12197e9i	0.123014	+	0.123014i	0.765934	−	0.642920i	\(-0.222277\pi\)
							−0.642920	+	0.765934i	\(0.722277\pi\)
\(31\)			1.85131e10i			0.672894i	0.941702	+	0.336447i	\(0.109225\pi\)
							−0.941702	+	0.336447i	\(0.890775\pi\)
\(37\)	−1.03365e11	+	1.03365e11i	−1.08883	+	1.08883i	−0.0931807	+	0.995649i	\(0.529703\pi\)
							−0.995649	+	0.0931807i	\(0.970297\pi\)
\(41\)		−	1.05263e11i		−	0.540492i	−0.962791	−	0.270246i	\(-0.912895\pi\)
							0.962791	−	0.270246i	\(-0.0871051\pi\)
\(43\)	−2.75073e11	−	2.75073e11i	−1.01197	−	1.01197i	−0.999927	−	0.0120465i	\(-0.996165\pi\)
							−0.0120465	−	0.999927i	\(-0.503835\pi\)
\(47\)			6.67502e11i			1.31755i	0.752340	+	0.658775i	\(0.228925\pi\)
							−0.752340	+	0.658775i	\(0.771075\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.16	✓	54
4.3	odd	2		64.15.f.a.47.10		54
16.5	even	4		64.15.f.a.15.10		54
16.11	odd	4	inner	16.15.f.a.11.16	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.16	✓	54	1.1	even	1	trivial
16.15.f.a.11.16	yes	54	16.11	odd	4	inner
64.15.f.a.15.10		54	16.5	even	4
64.15.f.a.47.10		54	4.3	odd	2