Embedded newform 16.15.f.a.3.13

• Label: 16.15.f.a.3.13
• 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			3.13
Character	\(\chi\)	\(=\)	16.3
Dual form			16.15.f.a.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.9550 - 127.440i) q^{2} +(1873.37 - 1873.37i) q^{3} +(-16098.2 + 3047.10i) q^{4} +(56067.7 - 56067.7i) q^{5} +(-261139. - 216347. i) q^{6} -548166. q^{7} +(580777. + 2.01513e6i) q^{8} -2.23606e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 2 q^{2} - 2 q^{3} + 15832 q^{4} - 2 q^{5} - 182528 q^{6} - 4 q^{7} + 2748484 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.9550	−	127.440i	−0.0933983	−	0.995629i
							\(3\)	1873.37	−	1873.37i	0.856593	−	0.856593i	−0.134342	−	0.990935i	\(-0.542892\pi\)
0.990935	+	0.134342i	\(0.0428919\pi\)
\(5\)	56067.7	−	56067.7i	0.717667	−	0.717667i	−0.250460	−	0.968127i	\(-0.580582\pi\)
							0.968127	+	0.250460i	\(0.0805819\pi\)
\(7\)	−548166.			−0.665619			−0.332809	−	0.942994i	\(-0.607997\pi\)
							−0.332809	+	0.942994i	\(0.607997\pi\)
\(11\)	−2.20667e7	−	2.20667e7i	−1.13237	−	1.13237i	−0.989782	−	0.142590i	\(-0.954457\pi\)
							−0.142590	−	0.989782i	\(-0.545543\pi\)
\(13\)	−2.11499e7	−	2.11499e7i	−0.337059	−	0.337059i	0.518200	−	0.855259i	\(-0.326602\pi\)
							−0.855259	+	0.518200i	\(0.826602\pi\)
\(17\)	1.22910e8			0.299534			0.149767	−	0.988721i	\(-0.452148\pi\)
							0.149767	+	0.988721i	\(0.452148\pi\)
\(19\)	−9.16566e7	+	9.16566e7i	−0.102539	+	0.102539i	−0.756515	−	0.653976i	\(-0.773100\pi\)
							0.653976	+	0.756515i	\(0.273100\pi\)
\(23\)	−3.35020e9			−0.983957			−0.491979	−	0.870607i	\(-0.663726\pi\)
							−0.491979	+	0.870607i	\(0.663726\pi\)
\(29\)	−1.34212e9	−	1.34212e9i	−0.0778049	−	0.0778049i	0.667133	−	0.744938i	\(-0.267521\pi\)
							−0.744938	+	0.667133i	\(0.767521\pi\)
\(31\)			3.91148e10i			1.42171i	0.703341	+	0.710853i	\(0.251691\pi\)
							−0.703341	+	0.710853i	\(0.748309\pi\)
\(37\)	9.77502e10	−	9.77502e10i	1.02969	−	1.02969i	0.0301417	−	0.999546i	\(-0.490404\pi\)
							0.999546	−	0.0301417i	\(-0.00959586\pi\)
\(41\)		−	3.32567e11i		−	1.70762i	−0.520581	−	0.853812i	\(-0.674285\pi\)
							0.520581	−	0.853812i	\(-0.325715\pi\)
\(43\)	1.53730e11	+	1.53730e11i	0.565561	+	0.565561i	0.930882	−	0.365320i	\(-0.119041\pi\)
							−0.365320	+	0.930882i	\(0.619041\pi\)
\(47\)		−	9.80194e11i		−	1.93476i	−0.253329	−	0.967380i	\(-0.581526\pi\)
							0.253329	−	0.967380i	\(-0.418474\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.f.a.3.13	✓	54
4.3	odd	2		64.15.f.a.47.7		54
16.5	even	4		64.15.f.a.15.7		54
16.11	odd	4	inner	16.15.f.a.11.13	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.f.a.3.13	✓	54	1.1	even	1	trivial
16.15.f.a.11.13	yes	54	16.11	odd	4	inner
64.15.f.a.15.7		54	16.5	even	4
64.15.f.a.47.7		54	4.3	odd	2