Embedded newform 102.5.e.a.47.3

• Label: 102.5.e.a.47.3
• Level: $102$
• Weight: $5$
• Character: 102.47
• Analytic conductor: $10.544$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 102 = 2 \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	102.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.5437362346\)
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			47.3
Character	\(\chi\)	\(=\)	102.47
Dual form			102.5.e.a.89.3

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.82843 q^{2} +(-7.71559 + 4.63353i) q^{3} +8.00000 q^{4} +(-1.85553 + 1.85553i) q^{5} +(21.8230 - 13.1056i) q^{6} +(25.3003 - 25.3003i) q^{7} -22.6274 q^{8} +(38.0607 - 71.5009i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{3} + 384 q^{4} - 32 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(37\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.82843			−0.707107
							\(3\)	−7.71559	+	4.63353i	−0.857288	+	0.514837i
\(5\)	−1.85553	+	1.85553i	−0.0742213	+	0.0742213i								−0.743243	−	0.669022i	\(-0.766713\pi\)
							0.669022	+	0.743243i	\(0.266713\pi\)
\(7\)	25.3003	−	25.3003i	0.516332	−	0.516332i	−0.400127	−	0.916460i	\(-0.631034\pi\)
							0.916460	+	0.400127i	\(0.131034\pi\)
\(11\)	−14.3492	−	14.3492i	−0.118589	−	0.118589i	0.645322	−	0.763911i	\(-0.276723\pi\)
							−0.763911	+	0.645322i	\(0.776723\pi\)
\(13\)	33.5484			0.198511			0.0992557	−	0.995062i	\(-0.468354\pi\)
							0.0992557	+	0.995062i	\(0.468354\pi\)
\(17\)	−1.22203	+	288.997i	−0.00422847	+	0.999991i
							\(19\)			149.755i			0.414835i	0.978253	+	0.207417i	\(0.0665058\pi\)
−0.978253	+	0.207417i	\(0.933494\pi\)
\(23\)	219.486	+	219.486i	0.414907	+	0.414907i	0.883444	−	0.468537i	\(-0.155219\pi\)
							−0.468537	+	0.883444i	\(0.655219\pi\)
\(29\)	−482.493	+	482.493i	−0.573713	+	0.573713i	−0.933164	−	0.359451i	\(-0.882964\pi\)
							0.359451	+	0.933164i	\(0.382964\pi\)
\(31\)	360.066	+	360.066i	0.374678	+	0.374678i	0.869178	−	0.494500i	\(-0.164649\pi\)
							−0.494500	+	0.869178i	\(0.664649\pi\)
\(37\)	979.187	+	979.187i	0.715257	+	0.715257i	0.967630	−	0.252373i	\(-0.0812109\pi\)
							−0.252373	+	0.967630i	\(0.581211\pi\)
\(41\)	−733.956	−	733.956i	−0.436619	−	0.436619i	0.454254	−	0.890872i	\(-0.349906\pi\)
							−0.890872	+	0.454254i	\(0.849906\pi\)
\(43\)			904.929i			0.489415i	0.969597	+	0.244708i	\(0.0786920\pi\)
							−0.969597	+	0.244708i	\(0.921308\pi\)
\(47\)		−	2298.28i		−	1.04042i	−0.854040	−	0.520208i	\(-0.825854\pi\)
							0.854040	−	0.520208i	\(-0.174146\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	102.5.e.a.47.3	✓	48
3.2	odd	2	inner	102.5.e.a.47.21	yes	48
17.4	even	4	inner	102.5.e.a.89.21	yes	48
51.38	odd	4	inner	102.5.e.a.89.3	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.5.e.a.47.3	✓	48	1.1	even	1	trivial
102.5.e.a.47.21	yes	48	3.2	odd	2	inner
102.5.e.a.89.3	yes	48	51.38	odd	4	inner
102.5.e.a.89.21	yes	48	17.4	even	4	inner