Embedded newform 32.2.g.b.5.2

• Label: 32.2.g.b.5.2
• Level: $32$
• Weight: $2$
• Character: 32.5
• Analytic conductor: $0.256$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.255521286468\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{8})\)
Coefficient field:	8.0.18939904.2
Defining polynomial:	\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			5.2
Root			\(0.500000 - 0.691860i\) of defining polynomial
Character	\(\chi\)	\(=\)	32.5
Dual form			32.2.g.b.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.443806 + 1.34277i) q^{2} +(0.0794708 + 0.191860i) q^{3} +(-1.60607 - 1.19186i) q^{4} +(0.707107 + 0.292893i) q^{5} +(-0.292893 + 0.0215628i) q^{6} +(-2.27133 - 2.27133i) q^{7} +(2.31318 - 1.62764i) q^{8} +(2.09083 - 2.09083i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{6} - 8 q^{7} - 4 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.443806	+	1.34277i	−0.313818	+	0.949483i
							\(3\)	0.0794708	+	0.191860i	0.0458825	+	0.110770i	0.945159	−	0.326610i	\(-0.105906\pi\)
−0.899277	+	0.437380i	\(0.855906\pi\)
\(5\)	0.707107	+	0.292893i	0.316228	+	0.130986i	0.535151	−	0.844756i	\(-0.320255\pi\)
							−0.218924	+	0.975742i	\(0.570255\pi\)
\(7\)	−2.27133	−	2.27133i	−0.858482	−	0.858482i	0.132677	−	0.991159i	\(-0.457643\pi\)
							−0.991159	+	0.132677i	\(0.957643\pi\)
\(11\)	−1.49368	+	3.60607i	−0.450363	+	1.08727i	0.521821	+	0.853055i	\(0.325253\pi\)
							−0.972184	+	0.234217i	\(0.924747\pi\)
\(13\)	−4.50504	+	1.86605i	−1.24947	+	0.517549i	−0.906663	−	0.421856i	\(-0.861379\pi\)
							−0.342810	+	0.939405i	\(0.611379\pi\)
\(17\)			3.05320i			0.740511i	0.928930	+	0.370255i	\(0.120730\pi\)
							−0.928930	+	0.370255i	\(0.879270\pi\)
\(19\)	3.87740	−	1.60607i	0.889537	−	0.368458i	0.109349	−	0.994003i	\(-0.465123\pi\)
							0.780188	+	0.625545i	\(0.215123\pi\)
\(23\)	0.271330	−	0.271330i	0.0565763	−	0.0565763i	−0.678253	−	0.734829i	\(-0.737262\pi\)
							0.734829	+	0.678253i	\(0.237262\pi\)
\(29\)	−0.931884	−	2.24977i	−0.173047	−	0.417771i	0.813432	−	0.581660i	\(-0.197596\pi\)
							−0.986479	+	0.163888i	\(0.947596\pi\)
\(31\)	6.82843			1.22642			0.613211	−	0.789919i	\(-0.289878\pi\)
							0.613211	+	0.789919i	\(0.289878\pi\)
\(37\)	3.63349	+	1.50504i	0.597342	+	0.247427i	0.660806	−	0.750557i	\(-0.270215\pi\)
							−0.0634640	+	0.997984i	\(0.520215\pi\)
\(41\)	−1.54266	+	1.54266i	−0.240923	+	0.240923i	−0.817232	−	0.576309i	\(-0.804493\pi\)
							0.576309	+	0.817232i	\(0.304493\pi\)
\(43\)	0.748956	−	1.80814i	0.114215	−	0.275739i	−0.856427	−	0.516268i	\(-0.827321\pi\)
							0.970642	+	0.240529i	\(0.0773209\pi\)
\(47\)			7.37109i			1.07518i	0.843205	+	0.537592i	\(0.180666\pi\)
							−0.843205	+	0.537592i	\(0.819334\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.2.g.b.5.2	✓	8
3.2	odd	2		288.2.v.b.37.1		8
4.3	odd	2		128.2.g.b.113.1		8
5.2	odd	4		800.2.ba.c.549.1		8
5.3	odd	4		800.2.ba.d.549.2		8
5.4	even	2		800.2.y.b.101.1		8
8.3	odd	2		256.2.g.c.225.2		8
8.5	even	2		256.2.g.d.225.1		8
12.11	even	2		1152.2.v.b.1009.2		8
16.3	odd	4		512.2.g.f.193.2		8
16.5	even	4		512.2.g.e.193.2		8
16.11	odd	4		512.2.g.g.193.1		8
16.13	even	4		512.2.g.h.193.1		8
32.3	odd	8		256.2.g.c.33.2		8
32.5	even	8		512.2.g.h.321.1		8
32.11	odd	8		512.2.g.g.321.1		8
32.13	even	8	inner	32.2.g.b.13.2	yes	8
32.19	odd	8		128.2.g.b.17.1		8
32.21	even	8		512.2.g.e.321.2		8
32.27	odd	8		512.2.g.f.321.2		8
32.29	even	8		256.2.g.d.33.1		8
64.13	even	16		4096.2.a.k.1.5		8
64.19	odd	16		4096.2.a.q.1.5		8
64.45	even	16		4096.2.a.k.1.4		8
64.51	odd	16		4096.2.a.q.1.4		8
96.77	odd	8		288.2.v.b.109.1		8
96.83	even	8		1152.2.v.b.145.2		8
160.13	odd	8		800.2.ba.c.749.1		8
160.77	odd	8		800.2.ba.d.749.2		8
160.109	even	8		800.2.y.b.301.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.2.g.b.5.2	✓	8	1.1	even	1	trivial
32.2.g.b.13.2	yes	8	32.13	even	8	inner
128.2.g.b.17.1		8	32.19	odd	8
128.2.g.b.113.1		8	4.3	odd	2
256.2.g.c.33.2		8	32.3	odd	8
256.2.g.c.225.2		8	8.3	odd	2
256.2.g.d.33.1		8	32.29	even	8
256.2.g.d.225.1		8	8.5	even	2
288.2.v.b.37.1		8	3.2	odd	2
288.2.v.b.109.1		8	96.77	odd	8
512.2.g.e.193.2		8	16.5	even	4
512.2.g.e.321.2		8	32.21	even	8
512.2.g.f.193.2		8	16.3	odd	4
512.2.g.f.321.2		8	32.27	odd	8
512.2.g.g.193.1		8	16.11	odd	4
512.2.g.g.321.1		8	32.11	odd	8
512.2.g.h.193.1		8	16.13	even	4
512.2.g.h.321.1		8	32.5	even	8
800.2.y.b.101.1		8	5.4	even	2
800.2.y.b.301.1		8	160.109	even	8
800.2.ba.c.549.1		8	5.2	odd	4
800.2.ba.c.749.1		8	160.13	odd	8
800.2.ba.d.549.2		8	5.3	odd	4
800.2.ba.d.749.2		8	160.77	odd	8
1152.2.v.b.145.2		8	96.83	even	8
1152.2.v.b.1009.2		8	12.11	even	2
4096.2.a.k.1.4		8	64.45	even	16
4096.2.a.k.1.5		8	64.13	even	16
4096.2.a.q.1.4		8	64.51	odd	16
4096.2.a.q.1.5		8	64.19	odd	16