Embedded newform 48.5.i.a.5.6

• Label: 48.5.i.a.5.6
• Level: $48$
• Weight: $5$
• Character: 48.5
• Analytic conductor: $4.962$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	48.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.96175822802\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	48.5
Dual form			48.5.i.a.29.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.16452 - 2.44659i) q^{2} +(-1.45703 - 8.88128i) q^{3} +(4.02839 + 15.4846i) q^{4} +(30.0724 - 30.0724i) q^{5} +(-17.1180 + 31.6697i) q^{6} -37.1392i q^{7} +(25.1365 - 58.8571i) q^{8} +(-76.7541 + 25.8807i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 2 q^{3} - 4 q^{4} - 68 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.16452	−	2.44659i	−0.791130	−	0.611648i
							\(3\)	−1.45703	−	8.88128i	−0.161893	−	0.986808i
\(5\)	30.0724	−	30.0724i	1.20290	−	1.20290i								0.229616	−	0.973281i	\(-0.426253\pi\)
							0.973281	−	0.229616i	\(-0.0737470\pi\)
\(7\)		−	37.1392i		−	0.757942i	−0.925409	−	0.378971i	\(-0.876278\pi\)
							0.925409	−	0.378971i	\(-0.123722\pi\)
\(11\)	−154.417	+	154.417i	−1.27618	+	1.27618i	−0.333387	+	0.942790i	\(0.608191\pi\)
							−0.942790	+	0.333387i	\(0.891809\pi\)
\(13\)	73.1696	−	73.1696i	0.432956	−	0.432956i	−0.456677	−	0.889633i	\(-0.650960\pi\)
							0.889633	+	0.456677i	\(0.150960\pi\)
\(17\)		−	218.326i		−	0.755453i	−0.925917	−	0.377726i	\(-0.876706\pi\)
							0.925917	−	0.377726i	\(-0.123294\pi\)
\(19\)	86.8686	−	86.8686i	0.240633	−	0.240633i	−0.576479	−	0.817112i	\(-0.695574\pi\)
							0.817112	+	0.576479i	\(0.195574\pi\)
\(23\)	−28.3144			−0.0535243			−0.0267622	−	0.999642i	\(-0.508520\pi\)
							−0.0267622	+	0.999642i	\(0.508520\pi\)
\(29\)	526.593	+	526.593i	0.626151	+	0.626151i	0.947097	−	0.320946i	\(-0.104001\pi\)
							−0.320946	+	0.947097i	\(0.604001\pi\)
\(31\)	−125.958			−0.131070			−0.0655349	−	0.997850i	\(-0.520875\pi\)
							−0.0655349	+	0.997850i	\(0.520875\pi\)
\(37\)	−120.350	−	120.350i	−0.0879111	−	0.0879111i	0.661784	−	0.749695i	\(-0.269800\pi\)
							−0.749695	+	0.661784i	\(0.769800\pi\)
\(41\)	2463.22			1.46533			0.732665	−	0.680589i	\(-0.238276\pi\)
							0.732665	+	0.680589i	\(0.238276\pi\)
\(43\)	−1510.43	−	1510.43i	−0.816888	−	0.816888i	0.168768	−	0.985656i	\(-0.446021\pi\)
							−0.985656	+	0.168768i	\(0.946021\pi\)
\(47\)		−	2215.87i		−	1.00311i	−0.865125	−	0.501556i	\(-0.832761\pi\)
							0.865125	−	0.501556i	\(-0.167239\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.5.i.a.5.6	✓	60
3.2	odd	2	inner	48.5.i.a.5.25	yes	60
4.3	odd	2		192.5.i.a.113.17		60
12.11	even	2		192.5.i.a.113.2		60
16.3	odd	4		192.5.i.a.17.2		60
16.13	even	4	inner	48.5.i.a.29.25	yes	60
48.29	odd	4	inner	48.5.i.a.29.6	yes	60
48.35	even	4		192.5.i.a.17.17		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.i.a.5.6	✓	60	1.1	even	1	trivial
48.5.i.a.5.25	yes	60	3.2	odd	2	inner
48.5.i.a.29.6	yes	60	48.29	odd	4	inner
48.5.i.a.29.25	yes	60	16.13	even	4	inner
192.5.i.a.17.2		60	16.3	odd	4
192.5.i.a.17.17		60	48.35	even	4
192.5.i.a.113.2		60	12.11	even	2
192.5.i.a.113.17		60	4.3	odd	2