Embedded newform 48.9.i.a.5.14

• Label: 48.9.i.a.5.14
• Level: $48$
• Weight: $9$
• Character: 48.5
• Analytic conductor: $19.554$
• Analytic rank: $0$
• Dimension: $124$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	48.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.5541732829\)
Analytic rank:	\(0\)
Dimension:	\(124\)
Relative dimension:	\(62\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.14
Character	\(\chi\)	\(=\)	48.5
Dual form			48.9.i.a.29.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.8620 + 9.51679i) q^{2} +(-13.6278 + 79.8454i) q^{3} +(74.8613 - 244.810i) q^{4} +(-792.195 + 792.195i) q^{5} +(-584.591 - 1156.66i) q^{6} +2103.96i q^{7} +(1366.94 + 3861.18i) q^{8} +(-6189.56 - 2176.24i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 2 q^{3} - 4 q^{4} - 3236 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.8620	+	9.51679i	−0.803874	+	0.594800i
							\(3\)	−13.6278	+	79.8454i	−0.168245	+	0.985745i
\(5\)	−792.195	+	792.195i	−1.26751	+	1.26751i								−0.320142	+	0.947370i	\(0.603731\pi\)
							−0.947370	+	0.320142i	\(0.896269\pi\)
\(7\)			2103.96i			0.876286i	0.898905	+	0.438143i	\(0.144364\pi\)
							−0.898905	+	0.438143i	\(0.855636\pi\)
\(11\)	−10440.0	+	10440.0i	−0.713065	+	0.713065i	−0.967175	−	0.254111i	\(-0.918217\pi\)
							0.254111	+	0.967175i	\(0.418217\pi\)
\(13\)	−20888.5	+	20888.5i	−0.731366	+	0.731366i	−0.970890	−	0.239524i	\(-0.923008\pi\)
							0.239524	+	0.970890i	\(0.423008\pi\)
\(17\)			149626.i			1.79147i	0.444585	+	0.895737i	\(0.353351\pi\)
							−0.444585	+	0.895737i	\(0.646649\pi\)
\(19\)	−14625.2	+	14625.2i	−0.112224	+	0.112224i	−0.760989	−	0.648765i	\(-0.775286\pi\)
							0.648765	+	0.760989i	\(0.275286\pi\)
\(23\)	363633.			1.29943			0.649714	−	0.760179i	\(-0.274889\pi\)
							0.649714	+	0.760179i	\(0.274889\pi\)
\(29\)	285083.	+	285083.i	0.403068	+	0.403068i	0.879313	−	0.476244i	\(-0.158002\pi\)
							−0.476244	+	0.879313i	\(0.658002\pi\)
\(31\)	509822.			0.552042			0.276021	−	0.961152i	\(-0.410984\pi\)
							0.276021	+	0.961152i	\(0.410984\pi\)
\(37\)	1.08753e6	+	1.08753e6i	0.580274	+	0.580274i	0.934978	−	0.354705i	\(-0.115419\pi\)
							−0.354705	+	0.934978i	\(0.615419\pi\)
\(41\)	805987.			0.285228			0.142614	−	0.989778i	\(-0.454449\pi\)
							0.142614	+	0.989778i	\(0.454449\pi\)
\(43\)	2.25502e6	+	2.25502e6i	0.659595	+	0.659595i	0.955284	−	0.295690i	\(-0.0955494\pi\)
							−0.295690	+	0.955284i	\(0.595549\pi\)
\(47\)		−	171169.i		−	0.0350779i	−0.999846	−	0.0175389i	\(-0.994417\pi\)
							0.999846	−	0.0175389i	\(-0.00558310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.9.i.a.5.14	✓	124
3.2	odd	2	inner	48.9.i.a.5.49	yes	124
16.13	even	4	inner	48.9.i.a.29.49	yes	124
48.29	odd	4	inner	48.9.i.a.29.14	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.9.i.a.5.14	✓	124	1.1	even	1	trivial
48.9.i.a.5.49	yes	124	3.2	odd	2	inner
48.9.i.a.29.14	yes	124	48.29	odd	4	inner
48.9.i.a.29.49	yes	124	16.13	even	4	inner