Embedded newform 40.5.i.a.13.10

• Label: 40.5.i.a.13.10
• 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			13.10
Character	\(\chi\)	\(=\)	40.13
Dual form			40.5.i.a.37.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.401499 - 3.97980i) q^{2} +(7.77558 - 7.77558i) q^{3} +(-15.6776 + 3.19577i) q^{4} +(13.6749 - 20.9284i) q^{5} +(-34.0671 - 27.8233i) q^{6} +(-5.91736 + 5.91736i) q^{7} +(19.0131 + 61.1106i) q^{8} -39.9192i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 2 q^{2} + 32 q^{6} - 4 q^{7} - 92 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.401499	−	3.97980i	−0.100375	−	0.994950i
							\(3\)	7.77558	−	7.77558i	0.863953	−	0.863953i	−0.127841	−	0.991795i	\(-0.540805\pi\)
0.991795	+	0.127841i	\(0.0408049\pi\)
\(5\)	13.6749	−	20.9284i	0.546997	−	0.837135i
							\(7\)	−5.91736	+	5.91736i	−0.120762	+	0.120762i	−0.764905	−	0.644143i	\(-0.777214\pi\)
0.644143	+	0.764905i	\(0.277214\pi\)
\(11\)		−	102.177i		−	0.844441i	−0.906493	−	0.422221i	\(-0.861251\pi\)
							0.906493	−	0.422221i	\(-0.138749\pi\)
\(13\)	−181.739	+	181.739i	−1.07538	+	1.07538i	−0.0784635	+	0.996917i	\(0.525001\pi\)
							−0.996917	+	0.0784635i	\(0.974999\pi\)
\(17\)	276.818	−	276.818i	0.957848	−	0.957848i	−0.0412990	−	0.999147i	\(-0.513150\pi\)
							0.999147	+	0.0412990i	\(0.0131496\pi\)
\(19\)	359.569			0.996035			0.498018	−	0.867167i	\(-0.334061\pi\)
							0.498018	+	0.867167i	\(0.334061\pi\)
\(23\)	516.386	+	516.386i	0.976155	+	0.976155i	0.999722	−	0.0235669i	\(-0.00750228\pi\)
							−0.0235669	+	0.999722i	\(0.507502\pi\)
\(29\)	122.460			0.145612			0.0728061	−	0.997346i	\(-0.476805\pi\)
							0.0728061	+	0.997346i	\(0.476805\pi\)
\(31\)	546.305			0.568476			0.284238	−	0.958754i	\(-0.408259\pi\)
							0.284238	+	0.958754i	\(0.408259\pi\)
\(37\)	−858.571	−	858.571i	−0.627152	−	0.627152i	0.320198	−	0.947350i	\(-0.396250\pi\)
							−0.947350	+	0.320198i	\(0.896250\pi\)
\(41\)	−1326.64			−0.789197			−0.394599	−	0.918854i	\(-0.629116\pi\)
							−0.394599	+	0.918854i	\(0.629116\pi\)
\(43\)	30.9254	−	30.9254i	0.0167255	−	0.0167255i	−0.698695	−	0.715420i	\(-0.746235\pi\)
							0.715420	+	0.698695i	\(0.246235\pi\)
\(47\)	−1055.53	+	1055.53i	−0.477831	+	0.477831i	−0.904437	−	0.426606i	\(-0.859709\pi\)
							0.426606	+	0.904437i	\(0.359709\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.5.i.a.13.10	✓	44
4.3	odd	2		160.5.m.a.113.5		44
5.2	odd	4	inner	40.5.i.a.37.21	yes	44
8.3	odd	2		160.5.m.a.113.18		44
8.5	even	2	inner	40.5.i.a.13.21	yes	44
20.7	even	4		160.5.m.a.17.18		44
40.27	even	4		160.5.m.a.17.5		44
40.37	odd	4	inner	40.5.i.a.37.10	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.i.a.13.10	✓	44	1.1	even	1	trivial
40.5.i.a.13.21	yes	44	8.5	even	2	inner
40.5.i.a.37.10	yes	44	40.37	odd	4	inner
40.5.i.a.37.21	yes	44	5.2	odd	4	inner
160.5.m.a.17.5		44	40.27	even	4
160.5.m.a.17.18		44	20.7	even	4
160.5.m.a.113.5		44	4.3	odd	2
160.5.m.a.113.18		44	8.3	odd	2