Embedded newform 96.2.o.a.59.2

• Label: 96.2.o.a.59.2
• Level: $96$
• Weight: $2$
• Character: 96.59
• 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			59.2
Character	\(\chi\)	\(=\)	96.59
Dual form			96.2.o.a.83.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27175 + 0.618595i) q^{2} +(1.44423 + 0.956131i) q^{3} +(1.23468 - 1.57339i) q^{4} +(1.56013 - 3.76650i) q^{5} +(-2.42816 - 0.322561i) q^{6} +(-0.838552 + 0.838552i) q^{7} +(-0.596910 + 2.76472i) q^{8} +(1.17163 + 2.76175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27175	+	0.618595i	−0.899261	+	0.437413i
							\(3\)	1.44423	+	0.956131i	0.833829	+	0.552022i
\(5\)	1.56013	−	3.76650i	0.697713	−	1.68443i								−0.0309186	−	0.999522i	\(-0.509843\pi\)
							0.728631	−	0.684906i	\(-0.240157\pi\)
\(7\)	−0.838552	+	0.838552i	−0.316943	+	0.316943i	−0.847592	−	0.530649i	\(-0.821948\pi\)
							0.530649	+	0.847592i	\(0.321948\pi\)
\(11\)	−0.249049	+	0.601256i	−0.0750910	+	0.181286i	−0.956968	−	0.290194i	\(-0.906280\pi\)
							0.881877	+	0.471480i	\(0.156280\pi\)
\(13\)	−2.05771	+	0.852332i	−0.570706	+	0.236394i	−0.649326	−	0.760510i	\(-0.724949\pi\)
							0.0786194	+	0.996905i	\(0.474949\pi\)
\(17\)	−3.23677			−0.785032			−0.392516	−	0.919745i	\(-0.628395\pi\)
							−0.392516	+	0.919745i	\(0.628395\pi\)
\(19\)	1.47818	+	3.56865i	0.339119	+	0.818705i	0.997801	+	0.0662838i	\(0.0211143\pi\)
							−0.658682	+	0.752421i	\(0.728886\pi\)
\(23\)	−2.58369	+	2.58369i	−0.538737	+	0.538737i	−0.923158	−	0.384421i	\(-0.874401\pi\)
							0.384421	+	0.923158i	\(0.374401\pi\)
\(29\)	3.52027	−	1.45815i	0.653698	−	0.270771i	−0.0310857	−	0.999517i	\(-0.509896\pi\)
							0.684784	+	0.728746i	\(0.259896\pi\)
\(31\)		−	7.63408i		−	1.37112i	−0.728015	−	0.685561i	\(-0.759557\pi\)
							0.728015	−	0.685561i	\(-0.240443\pi\)
\(37\)	0.579146	+	0.239890i	0.0952110	+	0.0394377i	0.429781	−	0.902933i	\(-0.358591\pi\)
							−0.334570	+	0.942371i	\(0.608591\pi\)
\(41\)	−3.54554	−	3.54554i	−0.553720	−	0.553720i	0.373792	−	0.927512i	\(-0.378057\pi\)
							−0.927512	+	0.373792i	\(0.878057\pi\)
\(43\)	−3.19595	−	1.32381i	−0.487378	−	0.201879i	0.125442	−	0.992101i	\(-0.459965\pi\)
							−0.612820	+	0.790222i	\(0.709965\pi\)
\(47\)			5.96658i			0.870315i	0.900354	+	0.435158i	\(0.143307\pi\)
							−0.900354	+	0.435158i	\(0.856693\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.2.o.a.59.2	✓	56
3.2	odd	2	inner	96.2.o.a.59.13	yes	56
4.3	odd	2		384.2.o.a.143.3		56
8.3	odd	2		768.2.o.a.287.12		56
8.5	even	2		768.2.o.b.287.3		56
12.11	even	2		384.2.o.a.143.6		56
24.5	odd	2		768.2.o.b.287.6		56
24.11	even	2		768.2.o.a.287.9		56
32.3	odd	8		768.2.o.b.479.6		56
32.13	even	8		384.2.o.a.239.6		56
32.19	odd	8	inner	96.2.o.a.83.13	yes	56
32.29	even	8		768.2.o.a.479.9		56
96.29	odd	8		768.2.o.a.479.12		56
96.35	even	8		768.2.o.b.479.3		56
96.77	odd	8		384.2.o.a.239.3		56
96.83	even	8	inner	96.2.o.a.83.2	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.o.a.59.2	✓	56	1.1	even	1	trivial
96.2.o.a.59.13	yes	56	3.2	odd	2	inner
96.2.o.a.83.2	yes	56	96.83	even	8	inner
96.2.o.a.83.13	yes	56	32.19	odd	8	inner
384.2.o.a.143.3		56	4.3	odd	2
384.2.o.a.143.6		56	12.11	even	2
384.2.o.a.239.3		56	96.77	odd	8
384.2.o.a.239.6		56	32.13	even	8
768.2.o.a.287.9		56	24.11	even	2
768.2.o.a.287.12		56	8.3	odd	2
768.2.o.a.479.9		56	32.29	even	8
768.2.o.a.479.12		56	96.29	odd	8
768.2.o.b.287.3		56	8.5	even	2
768.2.o.b.287.6		56	24.5	odd	2
768.2.o.b.479.3		56	96.35	even	8
768.2.o.b.479.6		56	32.3	odd	8