Embedded newform 40.5.i.a.13.19

• Label: 40.5.i.a.13.19
• 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			13.19
Character	\(\chi\)	\(=\)	40.13
Dual form			40.5.i.a.37.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.32460 + 2.22419i) q^{2} +(9.32929 - 9.32929i) q^{3} +(6.10592 + 14.7891i) q^{4} +(-13.0913 + 21.2983i) q^{5} +(51.7663 - 10.2660i) q^{6} +(43.0150 - 43.0150i) q^{7} +(-12.5941 + 62.7486i) q^{8} -93.0715i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} + 32 q^{6} - 4 q^{7} - 92 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.32460	+	2.22419i	0.831150	+	0.556048i
							\(3\)	9.32929	−	9.32929i	1.03659	−	1.03659i	0.0372835	−	0.999305i	\(-0.488130\pi\)
0.999305	−	0.0372835i	\(-0.0118705\pi\)
\(5\)	−13.0913	+	21.2983i	−0.523650	+	0.851933i
							\(7\)	43.0150	−	43.0150i	0.877856	−	0.877856i	−0.115456	−	0.993313i	\(-0.536833\pi\)
0.993313	+	0.115456i	\(0.0368331\pi\)
\(11\)		−	128.598i		−	1.06280i	−0.847122	−	0.531398i	\(-0.821667\pi\)
							0.847122	−	0.531398i	\(-0.178333\pi\)
\(13\)	−193.990	+	193.990i	−1.14787	+	1.14787i	−0.160899	+	0.986971i	\(0.551439\pi\)
							−0.986971	+	0.160899i	\(0.948561\pi\)
\(17\)	41.5464	−	41.5464i	0.143759	−	0.143759i	−0.631564	−	0.775324i	\(-0.717587\pi\)
							0.775324	+	0.631564i	\(0.217587\pi\)
\(19\)	−480.667			−1.33149			−0.665744	−	0.746180i	\(-0.731886\pi\)
							−0.665744	+	0.746180i	\(0.731886\pi\)
\(23\)	−89.7712	−	89.7712i	−0.169700	−	0.169700i	0.617148	−	0.786847i	\(-0.288288\pi\)
							−0.786847	+	0.617148i	\(0.788288\pi\)
\(29\)	449.595			0.534596			0.267298	−	0.963614i	\(-0.413869\pi\)
							0.267298	+	0.963614i	\(0.413869\pi\)
\(31\)	264.763			0.275508			0.137754	−	0.990466i	\(-0.456012\pi\)
							0.137754	+	0.990466i	\(0.456012\pi\)
\(37\)	46.3379	+	46.3379i	0.0338480	+	0.0338480i	0.723828	−	0.689980i	\(-0.242381\pi\)
							−0.689980	+	0.723828i	\(0.742381\pi\)
\(41\)	2330.15			1.38617			0.693085	−	0.720856i	\(-0.256251\pi\)
							0.693085	+	0.720856i	\(0.256251\pi\)
\(43\)	−38.4885	+	38.4885i	−0.0208158	+	0.0208158i	−0.717438	−	0.696622i	\(-0.754685\pi\)
							0.696622	+	0.717438i	\(0.254685\pi\)
\(47\)	781.056	−	781.056i	0.353579	−	0.353579i	−0.507860	−	0.861439i	\(-0.669563\pi\)
							0.861439	+	0.507860i	\(0.169563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.5.i.a.13.19	yes	44
4.3	odd	2		160.5.m.a.113.3		44
5.2	odd	4	inner	40.5.i.a.37.8	yes	44
8.3	odd	2		160.5.m.a.113.20		44
8.5	even	2	inner	40.5.i.a.13.8	✓	44
20.7	even	4		160.5.m.a.17.20		44
40.27	even	4		160.5.m.a.17.3		44
40.37	odd	4	inner	40.5.i.a.37.19	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.i.a.13.8	✓	44	8.5	even	2	inner
40.5.i.a.13.19	yes	44	1.1	even	1	trivial
40.5.i.a.37.8	yes	44	5.2	odd	4	inner
40.5.i.a.37.19	yes	44	40.37	odd	4	inner
160.5.m.a.17.3		44	40.27	even	4
160.5.m.a.17.20		44	20.7	even	4
160.5.m.a.113.3		44	4.3	odd	2
160.5.m.a.113.20		44	8.3	odd	2