Embedded newform 48.8.k.a.11.10

• Label: 48.8.k.a.11.10
• Level: $48$
• Weight: $8$
• Character: 48.11
• Analytic conductor: $14.994$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	48.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9944812232\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(54\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			11.10
Character	\(\chi\)	\(=\)	48.11
Dual form			48.8.k.a.35.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.75422 + 5.73194i) q^{2} +(-37.4127 - 28.0587i) q^{3} +(62.2898 - 111.821i) q^{4} +(-287.248 + 287.248i) q^{5} +(525.763 + 59.2434i) q^{6} +1269.27 q^{7} +(33.3635 + 1447.77i) q^{8} +(612.421 + 2099.50i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{3} - 4 q^{4} + 172 q^{6} - 8 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.75422	+	5.73194i	−0.862160	+	0.506636i
							\(3\)	−37.4127	−	28.0587i	−0.800009	−	0.599988i
\(5\)	−287.248	+	287.248i	−1.02769	+	1.02769i								−0.0280853	+	0.999606i	\(0.508941\pi\)
							−0.999606	+	0.0280853i	\(0.991059\pi\)
\(7\)	1269.27			1.39866			0.699330	−	0.714799i	\(-0.253482\pi\)
							0.699330	+	0.714799i	\(0.253482\pi\)
\(11\)	29.6751	+	29.6751i	0.00672230	+	0.00672230i	0.710460	−	0.703738i	\(-0.248487\pi\)
							−0.703738	+	0.710460i	\(0.748487\pi\)
\(13\)	−3248.46	+	3248.46i	−0.410087	+	0.410087i	−0.881769	−	0.471682i	\(-0.843647\pi\)
							0.471682	+	0.881769i	\(0.343647\pi\)
\(17\)			11581.7i			0.571742i	0.958268	+	0.285871i	\(0.0922828\pi\)
							−0.958268	+	0.285871i	\(0.907717\pi\)
\(19\)	−15034.5	−	15034.5i	−0.502866	−	0.502866i	0.409462	−	0.912327i	\(-0.365717\pi\)
							−0.912327	+	0.409462i	\(0.865717\pi\)
\(23\)		−	88012.9i		−	1.50834i	−0.656680	−	0.754169i	\(-0.728040\pi\)
							0.656680	−	0.754169i	\(-0.271960\pi\)
\(29\)	−102569.	−	102569.i	−0.780948	−	0.780948i	0.199043	−	0.979991i	\(-0.436217\pi\)
							−0.979991	+	0.199043i	\(0.936217\pi\)
\(31\)			292938.i			1.76608i	0.469298	+	0.883040i	\(0.344507\pi\)
							−0.469298	+	0.883040i	\(0.655493\pi\)
\(37\)	−340681.	−	340681.i	−1.10571	−	1.10571i	−0.993708	−	0.112003i	\(-0.964273\pi\)
							−0.112003	−	0.993708i	\(-0.535727\pi\)
\(41\)	163485.			0.370455			0.185228	−	0.982696i	\(-0.440698\pi\)
							0.185228	+	0.982696i	\(0.440698\pi\)
\(43\)	64015.8	−	64015.8i	0.122786	−	0.122786i	−0.643044	−	0.765829i	\(-0.722329\pi\)
							0.765829	+	0.643044i	\(0.222329\pi\)
\(47\)	−1.15527e6			−1.62309			−0.811544	−	0.584291i	\(-0.801372\pi\)
							−0.811544	+	0.584291i	\(0.801372\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.8.k.a.11.10	✓	108
3.2	odd	2	inner	48.8.k.a.11.45	yes	108
16.3	odd	4	inner	48.8.k.a.35.45	yes	108
48.35	even	4	inner	48.8.k.a.35.10	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.8.k.a.11.10	✓	108	1.1	even	1	trivial
48.8.k.a.11.45	yes	108	3.2	odd	2	inner
48.8.k.a.35.10	yes	108	48.35	even	4	inner
48.8.k.a.35.45	yes	108	16.3	odd	4	inner