Embedded newform 48.6.k.a.11.6

• Label: 48.6.k.a.11.6
• 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			11.6
Character	\(\chi\)	\(=\)	48.11
Dual form			48.6.k.a.35.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.17653 + 2.28112i) q^{2} +(14.1218 - 6.60105i) q^{3} +(21.5930 - 23.6166i) q^{4} +(62.8699 - 62.8699i) q^{5} +(-58.0444 + 66.3841i) q^{6} -197.388 q^{7} +(-57.9049 + 171.508i) q^{8} +(155.852 - 186.438i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 2 q^{3} - 4 q^{4} - 116 q^{6} - 8 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.17653	+	2.28112i	−0.915091	+	0.403248i
							\(3\)	14.1218	−	6.60105i	0.905916	−	0.423457i
\(5\)	62.8699	−	62.8699i	1.12465	−	1.12465i								0.133619	−	0.991033i	\(-0.457340\pi\)
							0.991033	−	0.133619i	\(-0.0426597\pi\)
\(7\)	−197.388			−1.52256			−0.761281	−	0.648422i	\(-0.775429\pi\)
							−0.761281	+	0.648422i	\(0.775429\pi\)
\(11\)	−231.401	−	231.401i	−0.576611	−	0.576611i	0.357357	−	0.933968i	\(-0.383678\pi\)
							−0.933968	+	0.357357i	\(0.883678\pi\)
\(13\)	−85.9906	+	85.9906i	−0.141121	+	0.141121i	−0.774138	−	0.633017i	\(-0.781816\pi\)
							0.633017	+	0.774138i	\(0.281816\pi\)
\(17\)		−	1243.86i		−	1.04388i	−0.852982	−	0.521940i	\(-0.825208\pi\)
							0.852982	−	0.521940i	\(-0.174792\pi\)
\(19\)	1215.72	+	1215.72i	0.772591	+	0.772591i	0.978559	−	0.205967i	\(-0.0660341\pi\)
							−0.205967	+	0.978559i	\(0.566034\pi\)
\(23\)		−	1841.73i		−	0.725950i	−0.931799	−	0.362975i	\(-0.881761\pi\)
							0.931799	−	0.362975i	\(-0.118239\pi\)
\(29\)	2684.59	+	2684.59i	0.592766	+	0.592766i	0.938378	−	0.345612i	\(-0.112329\pi\)
							−0.345612	+	0.938378i	\(0.612329\pi\)
\(31\)			2123.28i			0.396829i	0.980118	+	0.198414i	\(0.0635792\pi\)
							−0.980118	+	0.198414i	\(0.936421\pi\)
\(37\)	−1246.55	−	1246.55i	−0.149695	−	0.149695i	0.628287	−	0.777982i	\(-0.283756\pi\)
							−0.777982	+	0.628287i	\(0.783756\pi\)
\(41\)	15712.5			1.45978			0.729889	−	0.683566i	\(-0.239572\pi\)
							0.729889	+	0.683566i	\(0.239572\pi\)
\(43\)	4085.67	−	4085.67i	0.336971	−	0.336971i	−0.518255	−	0.855226i	\(-0.673418\pi\)
							0.855226	+	0.518255i	\(0.173418\pi\)
\(47\)	−15463.6			−1.02110			−0.510548	−	0.859849i	\(-0.670557\pi\)
							−0.510548	+	0.859849i	\(0.670557\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.6.k.a.11.6	✓	76
3.2	odd	2	inner	48.6.k.a.11.33	yes	76
4.3	odd	2		192.6.k.a.143.5		76
12.11	even	2		192.6.k.a.143.24		76
16.3	odd	4	inner	48.6.k.a.35.33	yes	76
16.13	even	4		192.6.k.a.47.24		76
48.29	odd	4		192.6.k.a.47.5		76
48.35	even	4	inner	48.6.k.a.35.6	yes	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.6.k.a.11.6	✓	76	1.1	even	1	trivial
48.6.k.a.11.33	yes	76	3.2	odd	2	inner
48.6.k.a.35.6	yes	76	48.35	even	4	inner
48.6.k.a.35.33	yes	76	16.3	odd	4	inner
192.6.k.a.47.5		76	48.29	odd	4
192.6.k.a.47.24		76	16.13	even	4
192.6.k.a.143.5		76	4.3	odd	2
192.6.k.a.143.24		76	12.11	even	2