Embedded newform 96.2.o.a.35.4

• Label: 96.2.o.a.35.4
• Level: $96$
• Weight: $2$
• Character: 96.35
• 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			35.4
Character	\(\chi\)	\(=\)	96.35
Dual form			96.2.o.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.838298 - 1.13897i) q^{2} +(0.602068 + 1.62404i) q^{3} +(-0.594512 + 1.90960i) q^{4} +(0.378520 - 0.156788i) q^{5} +(1.34503 - 2.04717i) q^{6} +(2.01144 + 2.01144i) q^{7} +(2.67335 - 0.923679i) q^{8} +(-2.27503 + 1.95557i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(37\)	\(65\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.838298	−	1.13897i	−0.592766	−	0.805374i
							\(3\)	0.602068	+	1.62404i	0.347604	+	0.937641i
\(5\)	0.378520	−	0.156788i	0.169279	−	0.0701179i								−0.296434	−	0.955053i	\(-0.595798\pi\)
							0.465714	+	0.884935i	\(0.345798\pi\)
\(7\)	2.01144	+	2.01144i	0.760253	+	0.760253i	0.976368	−	0.216115i	\(-0.0693386\pi\)
							−0.216115	+	0.976368i	\(0.569339\pi\)
\(11\)	0.709852	−	0.294030i	0.214028	−	0.0886534i	−0.273093	−	0.961988i	\(-0.588047\pi\)
							0.487122	+	0.873334i	\(0.338047\pi\)
\(13\)	2.08393	−	5.03104i	0.577977	−	1.39536i	−0.316648	−	0.948543i	\(-0.602557\pi\)
							0.894625	−	0.446817i	\(-0.147443\pi\)
\(17\)	−6.33777			−1.53714			−0.768568	−	0.639768i	\(-0.779030\pi\)
							−0.768568	+	0.639768i	\(0.779030\pi\)
\(19\)	0.646487	+	0.267784i	0.148314	+	0.0614338i	0.455606	−	0.890182i	\(-0.349423\pi\)
							−0.307291	+	0.951615i	\(0.599423\pi\)
\(23\)	1.61798	+	1.61798i	0.337371	+	0.337371i	0.855377	−	0.518006i	\(-0.173325\pi\)
							−0.518006	+	0.855377i	\(0.673325\pi\)
\(29\)	2.04571	−	4.93879i	0.379879	−	0.917110i	−0.612108	−	0.790774i	\(-0.709678\pi\)
							0.991988	−	0.126336i	\(-0.0403217\pi\)
\(31\)		−	5.75464i		−	1.03356i	−0.856117	−	0.516782i	\(-0.827130\pi\)
							0.856117	−	0.516782i	\(-0.172870\pi\)
\(37\)	−2.50232	−	6.04113i	−0.411379	−	0.993156i	−0.984768	−	0.173873i	\(-0.944372\pi\)
							0.573389	−	0.819283i	\(-0.305628\pi\)
\(41\)	5.52228	−	5.52228i	0.862435	−	0.862435i	−0.129185	−	0.991620i	\(-0.541236\pi\)
							0.991620	+	0.129185i	\(0.0412361\pi\)
\(43\)	−0.406593	−	0.981601i	−0.0620048	−	0.149693i	0.889840	−	0.456272i	\(-0.150816\pi\)
							−0.951845	+	0.306579i	\(0.900816\pi\)
\(47\)			10.4826i			1.52904i	0.644597	+	0.764522i	\(0.277025\pi\)
							−0.644597	+	0.764522i	\(0.722975\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.2.o.a.35.4	yes	56
3.2	odd	2	inner	96.2.o.a.35.11	yes	56
4.3	odd	2		384.2.o.a.47.6		56
8.3	odd	2		768.2.o.a.95.9		56
8.5	even	2		768.2.o.b.95.6		56
12.11	even	2		384.2.o.a.47.12		56
24.5	odd	2		768.2.o.b.95.12		56
24.11	even	2		768.2.o.a.95.3		56
32.5	even	8		768.2.o.a.671.3		56
32.11	odd	8	inner	96.2.o.a.11.11	yes	56
32.21	even	8		384.2.o.a.335.12		56
32.27	odd	8		768.2.o.b.671.12		56
96.5	odd	8		768.2.o.a.671.9		56
96.11	even	8	inner	96.2.o.a.11.4	✓	56
96.53	odd	8		384.2.o.a.335.6		56
96.59	even	8		768.2.o.b.671.6		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.o.a.11.4	✓	56	96.11	even	8	inner
96.2.o.a.11.11	yes	56	32.11	odd	8	inner
96.2.o.a.35.4	yes	56	1.1	even	1	trivial
96.2.o.a.35.11	yes	56	3.2	odd	2	inner
384.2.o.a.47.6		56	4.3	odd	2
384.2.o.a.47.12		56	12.11	even	2
384.2.o.a.335.6		56	96.53	odd	8
384.2.o.a.335.12		56	32.21	even	8
768.2.o.a.95.3		56	24.11	even	2
768.2.o.a.95.9		56	8.3	odd	2
768.2.o.a.671.3		56	32.5	even	8
768.2.o.a.671.9		56	96.5	odd	8
768.2.o.b.95.6		56	8.5	even	2
768.2.o.b.95.12		56	24.5	odd	2
768.2.o.b.671.6		56	96.59	even	8
768.2.o.b.671.12		56	32.27	odd	8