Embedded newform 48.7.i.a.5.16

• Label: 48.7.i.a.5.16
• Level: $48$
• Weight: $7$
• Character: 48.5
• Analytic conductor: $11.043$
• Analytic rank: $0$
• Dimension: $92$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	48.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0425960138\)
Analytic rank:	\(0\)
Dimension:	\(92\)
Relative dimension:	\(46\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.16
Character	\(\chi\)	\(=\)	48.5
Dual form			48.7.i.a.29.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.66778 + 7.10967i) q^{2} +(-22.4096 - 15.0602i) q^{3} +(-37.0947 - 52.1535i) q^{4} +(-13.2524 + 13.2524i) q^{5} +(189.267 - 104.087i) q^{6} -437.375i q^{7} +(506.849 - 72.4434i) q^{8} +(275.382 + 674.986i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q - 2 q^{3} - 4 q^{4} + 508 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(31\)	\(37\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.66778	+	7.10967i	−0.458473	+	0.888708i
							\(3\)	−22.4096	−	15.0602i	−0.829986	−	0.557785i
\(5\)	−13.2524	+	13.2524i	−0.106019	+	0.106019i								−0.758127	−	0.652107i	\(-0.773885\pi\)
							0.652107	+	0.758127i	\(0.273885\pi\)
\(7\)		−	437.375i		−	1.27514i	−0.770390	−	0.637572i	\(-0.779939\pi\)
							0.770390	−	0.637572i	\(-0.220061\pi\)
\(11\)	−1727.31	+	1727.31i	−1.29775	+	1.29775i	−0.367876	+	0.929875i	\(0.619915\pi\)
							−0.929875	+	0.367876i	\(0.880085\pi\)
\(13\)	2220.94	−	2220.94i	1.01090	−	1.01090i	0.0109555	−	0.999940i	\(-0.496513\pi\)
							0.999940	−	0.0109555i	\(-0.00348730\pi\)
\(17\)			3704.29i			0.753977i	0.926218	+	0.376989i	\(0.123040\pi\)
							−0.926218	+	0.376989i	\(0.876960\pi\)
\(19\)	−5370.90	+	5370.90i	−0.783045	+	0.783045i	−0.980343	−	0.197299i	\(-0.936783\pi\)
							0.197299	+	0.980343i	\(0.436783\pi\)
\(23\)	2921.45			0.240112			0.120056	−	0.992767i	\(-0.461693\pi\)
							0.120056	+	0.992767i	\(0.461693\pi\)
\(29\)	17194.6	+	17194.6i	0.705013	+	0.705013i	0.965482	−	0.260469i	\(-0.0838772\pi\)
							−0.260469	+	0.965482i	\(0.583877\pi\)
\(31\)	27777.2			0.932404			0.466202	−	0.884678i	\(-0.345622\pi\)
							0.466202	+	0.884678i	\(0.345622\pi\)
\(37\)	24150.7	+	24150.7i	0.476787	+	0.476787i	0.904102	−	0.427316i	\(-0.140541\pi\)
							−0.427316	+	0.904102i	\(0.640541\pi\)
\(41\)	40529.4			0.588056			0.294028	−	0.955797i	\(-0.405004\pi\)
							0.294028	+	0.955797i	\(0.405004\pi\)
\(43\)	68279.4	+	68279.4i	0.858784	+	0.858784i	0.991195	−	0.132410i	\(-0.0422717\pi\)
							−0.132410	+	0.991195i	\(0.542272\pi\)
\(47\)		−	143294.i		−	1.38018i	−0.723726	−	0.690088i	\(-0.757572\pi\)
							0.723726	−	0.690088i	\(-0.242428\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.7.i.a.5.16	✓	92
3.2	odd	2	inner	48.7.i.a.5.31	yes	92
4.3	odd	2		192.7.i.a.113.37		92
12.11	even	2		192.7.i.a.113.15		92
16.3	odd	4		192.7.i.a.17.15		92
16.13	even	4	inner	48.7.i.a.29.31	yes	92
48.29	odd	4	inner	48.7.i.a.29.16	yes	92
48.35	even	4		192.7.i.a.17.37		92

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.i.a.5.16	✓	92	1.1	even	1	trivial
48.7.i.a.5.31	yes	92	3.2	odd	2	inner
48.7.i.a.29.16	yes	92	48.29	odd	4	inner
48.7.i.a.29.31	yes	92	16.13	even	4	inner
192.7.i.a.17.15		92	16.3	odd	4
192.7.i.a.17.37		92	48.35	even	4
192.7.i.a.113.15		92	12.11	even	2
192.7.i.a.113.37		92	4.3	odd	2