Embedded newform 32.5.h.a.11.2

• Label: 32.5.h.a.11.2
• Level: $32$
• Weight: $5$
• Character: 32.11
• Analytic conductor: $3.308$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	32.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.30783881868\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(15\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	32.11
Dual form			32.5.h.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.85182 + 1.07865i) q^{2} +(4.50708 + 10.8811i) q^{3} +(13.6730 - 8.30956i) q^{4} +(-0.960017 + 2.31769i) q^{5} +(-29.0974 - 37.0503i) q^{6} +(-43.4088 + 43.4088i) q^{7} +(-43.7028 + 46.7554i) q^{8} +(-40.8081 + 40.8081i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.85182	+	1.07865i	−0.962955	+	0.269664i
							\(3\)	4.50708	+	10.8811i	0.500787	+	1.20901i	0.949056	+	0.315108i	\(0.102041\pi\)
−0.448269	+	0.893899i	\(0.647959\pi\)
\(5\)	−0.960017	+	2.31769i	−0.0384007	+	0.0927074i	−0.941916	−	0.335848i	\(-0.890977\pi\)
							0.903516	+	0.428555i	\(0.140977\pi\)
\(7\)	−43.4088	+	43.4088i	−0.885895	+	0.885895i	−0.994126	−	0.108231i	\(-0.965481\pi\)
							0.108231	+	0.994126i	\(0.465481\pi\)
\(11\)	−45.8774	+	110.758i	−0.379152	+	0.915354i	0.612973	+	0.790104i	\(0.289973\pi\)
							−0.992125	+	0.125251i	\(0.960027\pi\)
\(13\)	−50.7165	−	122.441i	−0.300098	−	0.724500i	−0.999948	−	0.0102106i	\(-0.996750\pi\)
							0.699850	−	0.714290i	\(-0.253250\pi\)
\(17\)			203.288i			0.703417i	0.936109	+	0.351709i	\(0.114399\pi\)
							−0.936109	+	0.351709i	\(0.885601\pi\)
\(19\)	499.925	−	207.076i	1.38483	−	0.573617i	0.439064	−	0.898456i	\(-0.355310\pi\)
							0.945769	+	0.324839i	\(0.105310\pi\)
\(23\)	483.112	+	483.112i	0.913255	+	0.913255i	0.996527	−	0.0832723i	\(-0.0265371\pi\)
							−0.0832723	+	0.996527i	\(0.526537\pi\)
\(29\)	−774.667	+	320.877i	−0.921126	+	0.381543i	−0.792305	−	0.610125i	\(-0.791119\pi\)
							−0.128821	+	0.991668i	\(0.541119\pi\)
\(31\)		−	1177.63i		−	1.22542i	−0.790309	−	0.612708i	\(-0.790080\pi\)
							0.790309	−	0.612708i	\(-0.209920\pi\)
\(37\)	633.612	−	1529.68i	0.462829	−	1.11737i	−0.504402	−	0.863469i	\(-0.668287\pi\)
							0.967231	−	0.253898i	\(-0.0817128\pi\)
\(41\)	1208.09	−	1208.09i	0.718674	−	0.718674i	−0.249659	−	0.968334i	\(-0.580319\pi\)
							0.968334	+	0.249659i	\(0.0803186\pi\)
\(43\)	−1254.64	+	3028.96i	−0.678548	+	1.63816i	0.0881151	+	0.996110i	\(0.471916\pi\)
							−0.766663	+	0.642050i	\(0.778084\pi\)
\(47\)	2999.77			1.35797			0.678987	−	0.734150i	\(-0.262419\pi\)
							0.678987	+	0.734150i	\(0.262419\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.5.h.a.11.2	yes	60
4.3	odd	2		128.5.h.a.79.3		60
32.3	odd	8	inner	32.5.h.a.3.2	✓	60
32.29	even	8		128.5.h.a.47.3		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.5.h.a.3.2	✓	60	32.3	odd	8	inner
32.5.h.a.11.2	yes	60	1.1	even	1	trivial
128.5.h.a.47.3		60	32.29	even	8
128.5.h.a.79.3		60	4.3	odd	2