Embedded newform 968.2.o.c.245.3

• Label: 968.2.o.c.245.3
• Level: $968$
• Weight: $2$
• Character: 968.245
• Analytic conductor: $7.730$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.o (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.72951891566\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			245.3
Character	\(\chi\)	\(=\)	968.245
Dual form			968.2.o.c.565.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.876503 + 1.10984i) q^{2} +(-1.40150 + 0.455375i) q^{3} +(-0.463486 - 1.94555i) q^{4} +(2.09113 - 2.87820i) q^{5} +(0.723026 - 1.95458i) q^{6} +(1.18995 - 3.66230i) q^{7} +(2.56550 + 1.19089i) q^{8} +(-0.670212 + 0.486937i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{4} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\)	\(485\)	\(727\)	\(849\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.876503	+	1.10984i	−0.619781	+	0.784775i
							\(3\)	−1.40150	+	0.455375i	−0.809157	+	0.262911i	−0.684241	−	0.729256i	\(-0.739866\pi\)
−0.124916	+	0.992167i	\(0.539866\pi\)
\(5\)	2.09113	−	2.87820i	0.935183	−	1.28717i	−0.0226203	−	0.999744i	\(-0.507201\pi\)
							0.957803	−	0.287425i	\(-0.0927991\pi\)
\(7\)	1.18995	−	3.66230i	0.449760	−	1.38422i	−0.427418	−	0.904054i	\(-0.640577\pi\)
							0.877178	−	0.480165i	\(-0.159423\pi\)
\(11\)		0			0
							\(13\)	2.80839	+	3.86541i	0.778906	+	1.07207i	0.995402	+	0.0957885i	\(0.0305372\pi\)
−0.216496	+	0.976284i	\(0.569463\pi\)
\(17\)	1.29042	+	0.937542i	0.312972	+	0.227387i	0.733171	−	0.680045i	\(-0.238040\pi\)
							−0.420199	+	0.907432i	\(0.638040\pi\)
\(19\)	3.94104	−	1.28052i	0.904138	−	0.293772i	0.180194	−	0.983631i	\(-0.442327\pi\)
							0.723944	+	0.689859i	\(0.242327\pi\)
\(23\)	−3.24264			−0.676137			−0.338069	−	0.941121i	\(-0.609774\pi\)
							−0.338069	+	0.941121i	\(0.609774\pi\)
\(29\)	7.02929	+	2.28396i	1.30531	+	0.424120i	0.877425	−	0.479714i	\(-0.159260\pi\)
							0.427882	+	0.903834i	\(0.359260\pi\)
\(31\)	2.90096	−	2.10767i	0.521028	−	0.378549i	−0.295963	−	0.955199i	\(-0.595640\pi\)
							0.816991	+	0.576650i	\(0.195640\pi\)
\(37\)	−0.580521	−	0.188623i	−0.0954370	−	0.0310094i	0.260909	−	0.965363i	\(-0.415978\pi\)
							−0.356346	+	0.934354i	\(0.615978\pi\)
\(41\)	−2.87280	−	8.84157i	−0.448656	−	1.38082i	−0.878424	−	0.477882i	\(-0.841405\pi\)
							0.429768	−	0.902939i	\(-0.358595\pi\)
\(43\)		−	3.06147i		−	0.466869i	−0.972372	−	0.233435i	\(-0.925003\pi\)
							0.972372	−	0.233435i	\(-0.0749965\pi\)
\(47\)	−1.49207	−	4.59211i	−0.217640	−	0.669828i	−0.998956	−	0.0456913i	\(-0.985451\pi\)
							0.781315	−	0.624136i	\(-0.214549\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.2.o.c.245.3		32
8.5	even	2	inner	968.2.o.c.245.2		32
11.2	odd	10		968.2.c.c.485.2	yes	8
11.3	even	5	inner	968.2.o.c.269.7		32
11.4	even	5	inner	968.2.o.c.565.2		32
11.5	even	5	inner	968.2.o.c.493.5		32
11.6	odd	10	inner	968.2.o.c.493.4		32
11.7	odd	10	inner	968.2.o.c.565.7		32
11.8	odd	10	inner	968.2.o.c.269.2		32
11.9	even	5		968.2.c.c.485.7	yes	8
11.10	odd	2	inner	968.2.o.c.245.6		32
44.31	odd	10		3872.2.c.c.1937.3		8
44.35	even	10		3872.2.c.c.1937.4		8
88.5	even	10	inner	968.2.o.c.493.7		32
88.13	odd	10		968.2.c.c.485.1	✓	8
88.21	odd	2	inner	968.2.o.c.245.7		32
88.29	odd	10	inner	968.2.o.c.565.6		32
88.35	even	10		3872.2.c.c.1937.6		8
88.37	even	10	inner	968.2.o.c.565.3		32
88.53	even	10		968.2.c.c.485.8	yes	8
88.61	odd	10	inner	968.2.o.c.493.2		32
88.69	even	10	inner	968.2.o.c.269.5		32
88.75	odd	10		3872.2.c.c.1937.5		8
88.85	odd	10	inner	968.2.o.c.269.4		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
968.2.c.c.485.1	✓	8	88.13	odd	10
968.2.c.c.485.2	yes	8	11.2	odd	10
968.2.c.c.485.7	yes	8	11.9	even	5
968.2.c.c.485.8	yes	8	88.53	even	10
968.2.o.c.245.2		32	8.5	even	2	inner
968.2.o.c.245.3		32	1.1	even	1	trivial
968.2.o.c.245.6		32	11.10	odd	2	inner
968.2.o.c.245.7		32	88.21	odd	2	inner
968.2.o.c.269.2		32	11.8	odd	10	inner
968.2.o.c.269.4		32	88.85	odd	10	inner
968.2.o.c.269.5		32	88.69	even	10	inner
968.2.o.c.269.7		32	11.3	even	5	inner
968.2.o.c.493.2		32	88.61	odd	10	inner
968.2.o.c.493.4		32	11.6	odd	10	inner
968.2.o.c.493.5		32	11.5	even	5	inner
968.2.o.c.493.7		32	88.5	even	10	inner
968.2.o.c.565.2		32	11.4	even	5	inner
968.2.o.c.565.3		32	88.37	even	10	inner
968.2.o.c.565.6		32	88.29	odd	10	inner
968.2.o.c.565.7		32	11.7	odd	10	inner
3872.2.c.c.1937.3		8	44.31	odd	10
3872.2.c.c.1937.4		8	44.35	even	10
3872.2.c.c.1937.5		8	88.75	odd	10
3872.2.c.c.1937.6		8	88.35	even	10