Embedded newform 96.2.o.a.11.8

• Label: 96.2.o.a.11.8
• Level: $96$
• Weight: $2$
• Character: 96.11
• Analytic conductor: $0.767$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	96.o (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.766563859404\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			11.8
Character	\(\chi\)	\(=\)	96.11
Dual form			96.2.o.a.35.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.310967 - 1.37960i) q^{2} +(0.988287 - 1.42242i) q^{3} +(-1.80660 - 0.858022i) q^{4} +(2.70066 + 1.11865i) q^{5} +(-1.65505 - 1.80577i) q^{6} +(-3.28204 + 3.28204i) q^{7} +(-1.74552 + 2.22557i) q^{8} +(-1.04658 - 2.81152i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(37\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.310967	−	1.37960i	0.219887	−	0.975525i
							\(3\)	0.988287	−	1.42242i	0.570588	−	0.821237i
\(5\)	2.70066	+	1.11865i	1.20777	+	0.500275i								0.893502	−	0.449059i	\(-0.148241\pi\)
							0.314269	+	0.949334i	\(0.398241\pi\)
\(7\)	−3.28204	+	3.28204i	−1.24049	+	1.24049i	−0.280697	+	0.959797i	\(0.590565\pi\)
							−0.959797	+	0.280697i	\(0.909435\pi\)
\(11\)	−0.276697	−	0.114612i	−0.0834272	−	0.0345567i	0.340580	−	0.940216i	\(-0.389377\pi\)
							−0.424007	+	0.905659i	\(0.639377\pi\)
\(13\)	−0.155281	−	0.374881i	−0.0430671	−	0.103973i	0.900882	−	0.434064i	\(-0.142921\pi\)
							−0.943949	+	0.330091i	\(0.892921\pi\)
\(17\)	1.41723			0.343728			0.171864	−	0.985121i	\(-0.445021\pi\)
							0.171864	+	0.985121i	\(0.445021\pi\)
\(19\)	−2.51972	+	1.04370i	−0.578062	+	0.239441i	−0.652506	−	0.757784i	\(-0.726282\pi\)
							0.0744431	+	0.997225i	\(0.476282\pi\)
\(23\)	3.30034	−	3.30034i	0.688169	−	0.688169i	−0.273658	−	0.961827i	\(-0.588234\pi\)
							0.961827	+	0.273658i	\(0.0882337\pi\)
\(29\)	0.650676	+	1.57087i	0.120827	+	0.291703i	0.972708	−	0.232033i	\(-0.0745377\pi\)
							−0.851880	+	0.523736i	\(0.824538\pi\)
\(31\)		−	4.16702i		−	0.748419i	−0.927344	−	0.374209i	\(-0.877914\pi\)
							0.927344	−	0.374209i	\(-0.122086\pi\)
\(37\)	−2.82757	+	6.82635i	−0.464849	+	1.12225i	0.501534	+	0.865138i	\(0.332769\pi\)
							−0.966383	+	0.257107i	\(0.917231\pi\)
\(41\)	−3.81949	−	3.81949i	−0.596504	−	0.596504i	0.342877	−	0.939380i	\(-0.388599\pi\)
							−0.939380	+	0.342877i	\(0.888599\pi\)
\(43\)	2.07136	−	5.00071i	0.315880	−	0.762602i	−0.683584	−	0.729872i	\(-0.739580\pi\)
							0.999464	−	0.0327301i	\(-0.0104202\pi\)
\(47\)			5.44085i			0.793629i	0.917899	+	0.396815i	\(0.129884\pi\)
							−0.917899	+	0.396815i	\(0.870116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.2.o.a.11.8	yes	56
3.2	odd	2	inner	96.2.o.a.11.7	✓	56
4.3	odd	2		384.2.o.a.335.5		56
8.3	odd	2		768.2.o.a.671.10		56
8.5	even	2		768.2.o.b.671.5		56
12.11	even	2		384.2.o.a.335.14		56
24.5	odd	2		768.2.o.b.671.14		56
24.11	even	2		768.2.o.a.671.1		56
32.3	odd	8	inner	96.2.o.a.35.7	yes	56
32.13	even	8		768.2.o.a.95.1		56
32.19	odd	8		768.2.o.b.95.14		56
32.29	even	8		384.2.o.a.47.14		56
96.29	odd	8		384.2.o.a.47.5		56
96.35	even	8	inner	96.2.o.a.35.8	yes	56
96.77	odd	8		768.2.o.a.95.10		56
96.83	even	8		768.2.o.b.95.5		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.o.a.11.7	✓	56	3.2	odd	2	inner
96.2.o.a.11.8	yes	56	1.1	even	1	trivial
96.2.o.a.35.7	yes	56	32.3	odd	8	inner
96.2.o.a.35.8	yes	56	96.35	even	8	inner
384.2.o.a.47.5		56	96.29	odd	8
384.2.o.a.47.14		56	32.29	even	8
384.2.o.a.335.5		56	4.3	odd	2
384.2.o.a.335.14		56	12.11	even	2
768.2.o.a.95.1		56	32.13	even	8
768.2.o.a.95.10		56	96.77	odd	8
768.2.o.a.671.1		56	24.11	even	2
768.2.o.a.671.10		56	8.3	odd	2
768.2.o.b.95.5		56	96.83	even	8
768.2.o.b.95.14		56	32.19	odd	8
768.2.o.b.671.5		56	8.5	even	2
768.2.o.b.671.14		56	24.5	odd	2