Embedded newform 48.6.k.a.11.13

• Label: 48.6.k.a.11.13
• Level: $48$
• Weight: $6$
• Character: 48.11
• Analytic conductor: $7.698$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	48.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.69842335102\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			11.13
Character	\(\chi\)	\(=\)	48.11
Dual form			48.6.k.a.35.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.92159 - 4.84400i) q^{2} +(15.5263 - 1.39064i) q^{3} +(-14.9286 + 28.3043i) q^{4} +(15.8062 - 15.8062i) q^{5} +(-52.0977 - 71.1465i) q^{6} +95.8790 q^{7} +(180.722 - 10.3793i) q^{8} +(239.132 - 43.1829i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 2 q^{3} - 4 q^{4} - 116 q^{6} - 8 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.92159	−	4.84400i	−0.516469	−	0.856306i
							\(3\)	15.5263	−	1.39064i	0.996013	−	0.0892093i
\(5\)	15.8062	−	15.8062i	0.282749	−	0.282749i								−0.551455	−	0.834205i	\(-0.685927\pi\)
							0.834205	+	0.551455i	\(0.185927\pi\)
\(7\)	95.8790			0.739569			0.369784	−	0.929118i	\(-0.379432\pi\)
							0.369784	+	0.929118i	\(0.379432\pi\)
\(11\)	21.3053	+	21.3053i	0.0530892	+	0.0530892i	0.733153	−	0.680064i	\(-0.238048\pi\)
							−0.680064	+	0.733153i	\(0.738048\pi\)
\(13\)	230.726	−	230.726i	0.378651	−	0.378651i	−0.491964	−	0.870615i	\(-0.663721\pi\)
							0.870615	+	0.491964i	\(0.163721\pi\)
\(17\)		−	1585.39i		−	1.33049i	−0.746624	−	0.665246i	\(-0.768327\pi\)
							0.746624	−	0.665246i	\(-0.231673\pi\)
\(19\)	−370.178	−	370.178i	−0.235248	−	0.235248i	0.579631	−	0.814879i	\(-0.303197\pi\)
							−0.814879	+	0.579631i	\(0.803197\pi\)
\(23\)		−	1131.29i		−	0.445917i	−0.974828	−	0.222958i	\(-0.928429\pi\)
							0.974828	−	0.222958i	\(-0.0715714\pi\)
\(29\)	−471.553	−	471.553i	−0.104120	−	0.104120i	0.653128	−	0.757248i	\(-0.273457\pi\)
							−0.757248	+	0.653128i	\(0.773457\pi\)
\(31\)			6217.54i			1.16202i	0.813896	+	0.581011i	\(0.197343\pi\)
							−0.813896	+	0.581011i	\(0.802657\pi\)
\(37\)	10076.2	+	10076.2i	1.21002	+	1.21002i	0.971021	+	0.238995i	\(0.0768179\pi\)
							0.238995	+	0.971021i	\(0.423182\pi\)
\(41\)	−13666.2			−1.26966			−0.634831	−	0.772651i	\(-0.718930\pi\)
							−0.634831	+	0.772651i	\(0.718930\pi\)
\(43\)	9568.00	−	9568.00i	0.789133	−	0.789133i	−0.192219	−	0.981352i	\(-0.561568\pi\)
							0.981352	+	0.192219i	\(0.0615684\pi\)
\(47\)	18199.6			1.20176			0.600879	−	0.799340i	\(-0.294817\pi\)
							0.600879	+	0.799340i	\(0.294817\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.6.k.a.11.13	✓	76
3.2	odd	2	inner	48.6.k.a.11.26	yes	76
4.3	odd	2		192.6.k.a.143.2		76
12.11	even	2		192.6.k.a.143.21		76
16.3	odd	4	inner	48.6.k.a.35.26	yes	76
16.13	even	4		192.6.k.a.47.21		76
48.29	odd	4		192.6.k.a.47.2		76
48.35	even	4	inner	48.6.k.a.35.13	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.6.k.a.11.13	✓	76	1.1	even	1	trivial
48.6.k.a.11.26	yes	76	3.2	odd	2	inner
48.6.k.a.35.13	yes	76	48.35	even	4	inner
48.6.k.a.35.26	yes	76	16.3	odd	4	inner
192.6.k.a.47.2		76	48.29	odd	4
192.6.k.a.47.21		76	16.13	even	4
192.6.k.a.143.2		76	4.3	odd	2
192.6.k.a.143.21		76	12.11	even	2