Embedded newform 96.2.o.a.11.5

• Label: 96.2.o.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	96.11
Dual form			96.2.o.a.35.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.816393 - 1.15477i) q^{2} +(-0.266294 - 1.71146i) q^{3} +(-0.667005 + 1.88550i) q^{4} +(-3.14689 - 1.30348i) q^{5} +(-1.75895 + 1.70473i) q^{6} +(0.663471 - 0.663471i) q^{7} +(2.72186 - 0.769068i) q^{8} +(-2.85817 + 0.911503i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.816393	−	1.15477i	−0.577277	−	0.816548i
							\(3\)	−0.266294	−	1.71146i	−0.153745	−	0.988111i
\(5\)	−3.14689	−	1.30348i	−1.40733	−	0.582935i								−0.455687	−	0.890140i	\(-0.650606\pi\)
							−0.951644	+	0.307205i	\(0.900606\pi\)
\(7\)	0.663471	−	0.663471i	0.250769	−	0.250769i	−0.570517	−	0.821286i	\(-0.693257\pi\)
							0.821286	+	0.570517i	\(0.193257\pi\)
\(11\)	1.91929	+	0.794997i	0.578689	+	0.239701i	0.652776	−	0.757551i	\(-0.273604\pi\)
							−0.0740873	+	0.997252i	\(0.523604\pi\)
\(13\)	−2.31672	−	5.59306i	−0.642543	−	1.55123i	−0.823238	−	0.567696i	\(-0.807835\pi\)
							0.180696	−	0.983539i	\(-0.442165\pi\)
\(17\)	2.24987			0.545673			0.272837	−	0.962060i	\(-0.412038\pi\)
							0.272837	+	0.962060i	\(0.412038\pi\)
\(19\)	3.08841	−	1.27926i	0.708530	−	0.293483i	0.000833646	−	1.00000i	\(-0.499735\pi\)
							0.707696	+	0.706517i	\(0.249735\pi\)
\(23\)	4.32171	−	4.32171i	0.901140	−	0.901140i	−0.0943951	−	0.995535i	\(-0.530092\pi\)
							0.995535	+	0.0943951i	\(0.0300917\pi\)
\(29\)	0.546766	+	1.32001i	0.101532	+	0.245120i	0.966480	−	0.256741i	\(-0.0826489\pi\)
							−0.864948	+	0.501861i	\(0.832649\pi\)
\(31\)			2.34273i			0.420767i	0.977619	+	0.210384i	\(0.0674713\pi\)
							−0.977619	+	0.210384i	\(0.932529\pi\)
\(37\)	−0.324682	+	0.783851i	−0.0533774	+	0.128864i	−0.948319	−	0.317319i	\(-0.897217\pi\)
							0.894941	+	0.446184i	\(0.147217\pi\)
\(41\)	−4.73908	−	4.73908i	−0.740120	−	0.740120i	0.232481	−	0.972601i	\(-0.425316\pi\)
							−0.972601	+	0.232481i	\(0.925316\pi\)
\(43\)	0.951664	−	2.29752i	0.145127	−	0.350369i	−0.834555	−	0.550925i	\(-0.814275\pi\)
							0.979682	+	0.200556i	\(0.0642750\pi\)
\(47\)			3.02812i			0.441696i	0.975308	+	0.220848i	\(0.0708825\pi\)
							−0.975308	+	0.220848i	\(0.929117\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.2.o.a.11.5	✓	56
3.2	odd	2	inner	96.2.o.a.11.10	yes	56
4.3	odd	2		384.2.o.a.335.9		56
8.3	odd	2		768.2.o.a.671.6		56
8.5	even	2		768.2.o.b.671.9		56
12.11	even	2		384.2.o.a.335.10		56
24.5	odd	2		768.2.o.b.671.10		56
24.11	even	2		768.2.o.a.671.5		56
32.3	odd	8	inner	96.2.o.a.35.10	yes	56
32.13	even	8		768.2.o.a.95.5		56
32.19	odd	8		768.2.o.b.95.10		56
32.29	even	8		384.2.o.a.47.10		56
96.29	odd	8		384.2.o.a.47.9		56
96.35	even	8	inner	96.2.o.a.35.5	yes	56
96.77	odd	8		768.2.o.a.95.6		56
96.83	even	8		768.2.o.b.95.9		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.o.a.11.5	✓	56	1.1	even	1	trivial
96.2.o.a.11.10	yes	56	3.2	odd	2	inner
96.2.o.a.35.5	yes	56	96.35	even	8	inner
96.2.o.a.35.10	yes	56	32.3	odd	8	inner
384.2.o.a.47.9		56	96.29	odd	8
384.2.o.a.47.10		56	32.29	even	8
384.2.o.a.335.9		56	4.3	odd	2
384.2.o.a.335.10		56	12.11	even	2
768.2.o.a.95.5		56	32.13	even	8
768.2.o.a.95.6		56	96.77	odd	8
768.2.o.a.671.5		56	24.11	even	2
768.2.o.a.671.6		56	8.3	odd	2
768.2.o.b.95.9		56	96.83	even	8
768.2.o.b.95.10		56	32.19	odd	8
768.2.o.b.671.9		56	8.5	even	2
768.2.o.b.671.10		56	24.5	odd	2