Embedded newform 48.5.i.a.29.6

• Label: 48.5.i.a.29.6
• Level: $48$
• Weight: $5$
• Character: 48.29
• 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			29.6
Character	\(\chi\)	\(=\)	48.29
Dual form			48.5.i.a.5.6

$q$-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 \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(31\)	\(37\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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								0.973281	+	0.229616i	\(0.0737470\pi\)
							0.229616	+	0.973281i	\(0.426253\pi\)
\(7\)			37.1392i			0.757942i	0.925409	+	0.378971i	\(0.123722\pi\)
							−0.925409	+	0.378971i	\(0.876278\pi\)
\(11\)	−154.417	−	154.417i	−1.27618	−	1.27618i	−0.942790	−	0.333387i	\(-0.891809\pi\)
							−0.333387	−	0.942790i	\(-0.608191\pi\)
\(13\)	73.1696	+	73.1696i	0.432956	+	0.432956i	0.889633	−	0.456677i	\(-0.150960\pi\)
							−0.456677	+	0.889633i	\(0.650960\pi\)
\(17\)			218.326i			0.755453i	0.925917	+	0.377726i	\(0.123294\pi\)
							−0.925917	+	0.377726i	\(0.876706\pi\)
\(19\)	86.8686	+	86.8686i	0.240633	+	0.240633i	0.817112	−	0.576479i	\(-0.195574\pi\)
							−0.576479	+	0.817112i	\(0.695574\pi\)
\(23\)	−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	−0.320946	−	0.947097i	\(-0.604001\pi\)
							0.947097	+	0.320946i	\(0.104001\pi\)
\(31\)	−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	−0.749695	−	0.661784i	\(-0.769800\pi\)
							0.661784	+	0.749695i	\(0.269800\pi\)
\(41\)	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	−0.985656	−	0.168768i	\(-0.946021\pi\)
							0.168768	+	0.985656i	\(0.446021\pi\)
\(47\)			2215.87i			1.00311i	0.865125	+	0.501556i	\(0.167239\pi\)
							−0.865125	+	0.501556i	\(0.832761\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.i.a.5.6	✓	60	48.5	odd	4	inner
48.5.i.a.5.25	yes	60	16.5	even	4	inner
48.5.i.a.29.6	yes	60	1.1	even	1	trivial
48.5.i.a.29.25	yes	60	3.2	odd	2	inner
192.5.i.a.17.2		60	12.11	even	2
192.5.i.a.17.17		60	4.3	odd	2
192.5.i.a.113.2		60	16.11	odd	4
192.5.i.a.113.17		60	48.11	even	4