Embedded newform 48.7.l.a.19.4

• Label: 48.7.l.a.19.4
• Level: $48$
• Weight: $7$
• Character: 48.19
• Analytic conductor: $11.043$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	48.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0425960138\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			19.4
Character	\(\chi\)	\(=\)	48.19
Dual form			48.7.l.a.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.29112 + 3.29235i) q^{2} +(11.0227 - 11.0227i) q^{3} +(42.3208 - 48.0099i) q^{4} +(-73.3870 + 73.3870i) q^{5} +(-44.0772 + 116.658i) q^{6} +291.401 q^{7} +(-150.501 + 489.381i) q^{8} -243.000i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 180 q^{4} - 1932 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.29112	+	3.29235i	−0.911390	+	0.411544i
							\(3\)	11.0227	−	11.0227i	0.408248	−	0.408248i
\(5\)	−73.3870	+	73.3870i	−0.587096	+	0.587096i								−0.936844	−	0.349748i	\(-0.886267\pi\)
							0.349748	+	0.936844i	\(0.386267\pi\)
\(7\)	291.401			0.849565			0.424782	−	0.905296i	\(-0.360351\pi\)
							0.424782	+	0.905296i	\(0.360351\pi\)
\(11\)	−1594.09	−	1594.09i	−1.19767	−	1.19767i	−0.974864	−	0.222802i	\(-0.928480\pi\)
							−0.222802	−	0.974864i	\(-0.571520\pi\)
\(13\)	1768.57	+	1768.57i	0.804994	+	0.804994i	0.983871	−	0.178878i	\(-0.0572466\pi\)
							−0.178878	+	0.983871i	\(0.557247\pi\)
\(17\)	6756.86			1.37530			0.687652	−	0.726041i	\(-0.258642\pi\)
							0.687652	+	0.726041i	\(0.258642\pi\)
\(19\)	8599.42	−	8599.42i	1.25374	−	1.25374i	0.299713	−	0.954029i	\(-0.403109\pi\)
							0.954029	−	0.299713i	\(-0.0968909\pi\)
\(23\)	16174.5			1.32937			0.664687	−	0.747122i	\(-0.268565\pi\)
							0.664687	+	0.747122i	\(0.268565\pi\)
\(29\)	−7778.56	−	7778.56i	−0.318937	−	0.318937i	0.529422	−	0.848359i	\(-0.322409\pi\)
							−0.848359	+	0.529422i	\(0.822409\pi\)
\(31\)		−	18422.8i		−	0.618400i	−0.950997	−	0.309200i	\(-0.899939\pi\)
							0.950997	−	0.309200i	\(-0.100061\pi\)
\(37\)	52107.5	−	52107.5i	1.02872	−	1.02872i	0.0291404	−	0.999575i	\(-0.490723\pi\)
							0.999575	−	0.0291404i	\(-0.00927700\pi\)
\(41\)		−	39387.8i		−	0.571491i	−0.958305	−	0.285746i	\(-0.907759\pi\)
							0.958305	−	0.285746i	\(-0.0922413\pi\)
\(43\)	4815.39	+	4815.39i	0.0605656	+	0.0605656i	0.736741	−	0.676175i	\(-0.236364\pi\)
							−0.676175	+	0.736741i	\(0.736364\pi\)
\(47\)			11143.5i			0.107332i	0.998559	+	0.0536658i	\(0.0170906\pi\)
							−0.998559	+	0.0536658i	\(0.982909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.7.l.a.19.4	✓	48
4.3	odd	2		192.7.l.a.175.3		48
8.3	odd	2		384.7.l.a.223.22		48
8.5	even	2		384.7.l.b.223.3		48
16.3	odd	4		384.7.l.b.31.3		48
16.5	even	4		192.7.l.a.79.3		48
16.11	odd	4	inner	48.7.l.a.43.4	yes	48
16.13	even	4		384.7.l.a.31.22		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.l.a.19.4	✓	48	1.1	even	1	trivial
48.7.l.a.43.4	yes	48	16.11	odd	4	inner
192.7.l.a.79.3		48	16.5	even	4
192.7.l.a.175.3		48	4.3	odd	2
384.7.l.a.31.22		48	16.13	even	4
384.7.l.a.223.22		48	8.3	odd	2
384.7.l.b.31.3		48	16.3	odd	4
384.7.l.b.223.3		48	8.5	even	2