Embedded newform 32.8.g.a.13.15

• Label: 32.8.g.a.13.15
• Level: $32$
• Weight: $8$
• Character: 32.13
• Analytic conductor: $9.996$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.99632081549\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(27\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			13.15
Character	\(\chi\)	\(=\)	32.13
Dual form			32.8.g.a.5.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.61553 + 11.1978i) q^{2} +(-17.7725 + 42.9066i) q^{3} +(-122.780 + 36.1807i) q^{4} +(97.0019 - 40.1795i) q^{5} +(-509.170 - 129.695i) q^{6} +(-891.771 + 891.771i) q^{7} +(-603.498 - 1316.41i) q^{8} +(21.3266 + 21.3266i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.61553	+	11.1978i	0.142794	+	0.989752i
							\(3\)	−17.7725	+	42.9066i	−0.380036	+	0.917487i	0.611922	+	0.790918i	\(0.290397\pi\)
−0.991958	+	0.126569i	\(0.959603\pi\)
\(5\)	97.0019	−	40.1795i	0.347045	−	0.143751i	−0.202351	−	0.979313i	\(-0.564858\pi\)
							0.549395	+	0.835563i	\(0.314858\pi\)
\(7\)	−891.771	+	891.771i	−0.982676	+	0.982676i	−0.999852	−	0.0171765i	\(-0.994532\pi\)
							0.0171765	+	0.999852i	\(0.494532\pi\)
\(11\)	−2879.18	−	6950.94i	−0.652219	−	1.57460i	−0.809550	−	0.587051i	\(-0.800289\pi\)
							0.157331	−	0.987546i	\(-0.449711\pi\)
\(13\)	8855.65	+	3668.13i	1.11794	+	0.463066i	0.863665	−	0.504066i	\(-0.168163\pi\)
							0.254275	+	0.967132i	\(0.418163\pi\)
\(17\)			2200.77i			0.108643i	0.998523	+	0.0543216i	\(0.0172996\pi\)
							−0.998523	+	0.0543216i	\(0.982700\pi\)
\(19\)	−13605.9	−	5635.76i	−0.455083	−	0.188501i	0.143354	−	0.989671i	\(-0.454211\pi\)
							−0.598437	+	0.801170i	\(0.704211\pi\)
\(23\)	−41289.8	−	41289.8i	−0.707613	−	0.707613i	0.258420	−	0.966033i	\(-0.416798\pi\)
							−0.966033	+	0.258420i	\(0.916798\pi\)
\(29\)	−60946.2	+	147137.i	−0.464038	+	1.12029i	0.502686	+	0.864469i	\(0.332345\pi\)
							−0.966725	+	0.255819i	\(0.917655\pi\)
\(31\)	−89272.2			−0.538208			−0.269104	−	0.963111i	\(-0.586728\pi\)
							−0.269104	+	0.963111i	\(0.586728\pi\)
\(37\)	−200353.	+	82988.9i	−0.650264	+	0.269348i	−0.683335	−	0.730105i	\(-0.739471\pi\)
							0.0330713	+	0.999453i	\(0.489471\pi\)
\(41\)	−139647.	−	139647.i	−0.316438	−	0.316438i	0.530959	−	0.847397i	\(-0.321832\pi\)
							−0.847397	+	0.530959i	\(0.821832\pi\)
\(43\)	380738.	+	919182.i	0.730275	+	1.76304i	0.641681	+	0.766972i	\(0.278237\pi\)
							0.0885942	+	0.996068i	\(0.471763\pi\)
\(47\)		−	701493.i		−	0.985556i	−0.870155	−	0.492778i	\(-0.835982\pi\)
							0.870155	−	0.492778i	\(-0.164018\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.g.a.13.15	yes	108
32.5	even	8	inner	32.8.g.a.5.15	✓	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.g.a.5.15	✓	108	32.5	even	8	inner
32.8.g.a.13.15	yes	108	1.1	even	1	trivial