Embedded newform 40.5.i.a.13.20

• Label: 40.5.i.a.13.20
• Level: $40$
• Weight: $5$
• Character: 40.13
• Analytic conductor: $4.135$
• Analytic rank: $0$
• Dimension: $44$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	40.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.13479852335\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.20
Character	\(\chi\)	\(=\)	40.13
Dual form			40.5.i.a.37.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.46177 - 2.00403i) q^{2} +(-4.26672 + 4.26672i) q^{3} +(7.96776 - 13.8750i) q^{4} +(24.9456 + 1.64905i) q^{5} +(-6.21980 + 23.3211i) q^{6} +(50.3926 - 50.3926i) q^{7} +(-0.223240 - 63.9996i) q^{8} +44.5901i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} + 32 q^{6} - 4 q^{7} - 92 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.46177	−	2.00403i	0.865443	−	0.501007i
							\(3\)	−4.26672	+	4.26672i	−0.474080	+	0.474080i	−0.903232	−	0.429152i	\(-0.858812\pi\)
0.429152	+	0.903232i	\(0.358812\pi\)
\(5\)	24.9456	+	1.64905i	0.997822	+	0.0659621i
							\(7\)	50.3926	−	50.3926i	1.02842	−	1.02842i	0.0288363	−	0.999584i	\(-0.490820\pi\)
0.999584	−	0.0288363i	\(-0.00918016\pi\)
\(11\)		−	40.0246i		−	0.330782i	−0.986228	−	0.165391i	\(-0.947111\pi\)
							0.986228	−	0.165391i	\(-0.0528886\pi\)
\(13\)	−198.123	+	198.123i	−1.17233	+	1.17233i	−0.190671	+	0.981654i	\(0.561066\pi\)
							−0.981654	+	0.190671i	\(0.938934\pi\)
\(17\)	−45.5410	+	45.5410i	−0.157581	+	0.157581i	−0.781494	−	0.623913i	\(-0.785542\pi\)
							0.623913	+	0.781494i	\(0.285542\pi\)
\(19\)	−184.824			−0.511979			−0.255990	−	0.966680i	\(-0.582401\pi\)
							−0.255990	+	0.966680i	\(0.582401\pi\)
\(23\)	409.558	+	409.558i	0.774211	+	0.774211i	0.978840	−	0.204629i	\(-0.0655986\pi\)
							−0.204629	+	0.978840i	\(0.565599\pi\)
\(29\)	−925.786			−1.10082			−0.550408	−	0.834896i	\(-0.685528\pi\)
							−0.550408	+	0.834896i	\(0.685528\pi\)
\(31\)	−1254.72			−1.30564			−0.652821	−	0.757512i	\(-0.726415\pi\)
							−0.652821	+	0.757512i	\(0.726415\pi\)
\(37\)	1473.79	+	1473.79i	1.07655	+	1.07655i	0.996816	+	0.0797322i	\(0.0254065\pi\)
							0.0797322	+	0.996816i	\(0.474593\pi\)
\(41\)	−405.649			−0.241314			−0.120657	−	0.992694i	\(-0.538500\pi\)
							−0.120657	+	0.992694i	\(0.538500\pi\)
\(43\)	1549.51	−	1549.51i	0.838025	−	0.838025i	−0.150574	−	0.988599i	\(-0.548112\pi\)
							0.988599	+	0.150574i	\(0.0481121\pi\)
\(47\)	−402.164	+	402.164i	−0.182057	+	0.182057i	−0.792252	−	0.610195i	\(-0.791091\pi\)
							0.610195	+	0.792252i	\(0.291091\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.5.i.a.13.20	yes	44
4.3	odd	2		160.5.m.a.113.16		44
5.2	odd	4	inner	40.5.i.a.37.15	yes	44
8.3	odd	2		160.5.m.a.113.7		44
8.5	even	2	inner	40.5.i.a.13.15	✓	44
20.7	even	4		160.5.m.a.17.7		44
40.27	even	4		160.5.m.a.17.16		44
40.37	odd	4	inner	40.5.i.a.37.20	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.i.a.13.15	✓	44	8.5	even	2	inner
40.5.i.a.13.20	yes	44	1.1	even	1	trivial
40.5.i.a.37.15	yes	44	5.2	odd	4	inner
40.5.i.a.37.20	yes	44	40.37	odd	4	inner
160.5.m.a.17.7		44	20.7	even	4
160.5.m.a.17.16		44	40.27	even	4
160.5.m.a.113.7		44	8.3	odd	2
160.5.m.a.113.16		44	4.3	odd	2