Embedded newform 968.2.k.e.475.3

• Label: 968.2.k.e.475.3
• Level: $968$
• Weight: $2$
• Character: 968.475
• Analytic conductor: $7.730$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.k (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:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			475.3
Character	\(\chi\)	\(=\)	968.475
Dual form			968.2.k.e.699.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.918861 + 1.07503i) q^{2} +(0.795382 - 0.577879i) q^{3} +(-0.311390 - 1.97561i) q^{4} +(0.644597 - 0.209442i) q^{5} +(-0.109607 + 1.38605i) q^{6} +(3.67554 + 2.67043i) q^{7} +(2.40997 + 1.48056i) q^{8} +(-0.628362 + 1.93390i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 5 q^{2} - 2 q^{3} - 5 q^{4} + 10 q^{6} - 5 q^{8} + 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(485\)	\(727\)	\(849\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.918861	+	1.07503i	−0.649733	+	0.760163i
							\(3\)	0.795382	−	0.577879i	0.459214	−	0.333639i	−0.334009	−	0.942570i	\(-0.608402\pi\)
0.793223	+	0.608931i	\(0.208402\pi\)
\(5\)	0.644597	−	0.209442i	0.288273	−	0.0936655i	−0.161311	−	0.986904i	\(-0.551572\pi\)
							0.449584	+	0.893238i	\(0.351572\pi\)
\(7\)	3.67554	+	2.67043i	1.38922	+	1.00933i	0.995950	+	0.0899076i	\(0.0286572\pi\)
							0.393273	+	0.919422i	\(0.371343\pi\)
\(11\)		0			0
							\(13\)	−0.727443	+	2.23884i	−0.201756	+	0.620942i	0.798075	+	0.602559i	\(0.205852\pi\)
−0.999831	+	0.0183835i	\(0.994148\pi\)
\(17\)	−3.35652	+	1.09060i	−0.814077	+	0.264510i	−0.686323	−	0.727296i	\(-0.740777\pi\)
							−0.127753	+	0.991806i	\(0.540777\pi\)
\(19\)	2.71305	+	3.73419i	0.622415	+	0.856681i	0.997526	−	0.0702987i	\(-0.0223952\pi\)
							−0.375110	+	0.926980i	\(0.622395\pi\)
\(23\)		−	1.11065i		−	0.231588i	−0.993273	−	0.115794i	\(-0.963059\pi\)
							0.993273	−	0.115794i	\(-0.0369412\pi\)
\(29\)	−0.993277	−	0.721658i	−0.184447	−	0.134008i	0.491730	−	0.870748i	\(-0.336365\pi\)
							−0.676177	+	0.736739i	\(0.736365\pi\)
\(31\)	−5.60016	−	1.81960i	−1.00582	−	0.326810i	−0.240630	−	0.970617i	\(-0.577354\pi\)
							−0.765188	+	0.643807i	\(0.777354\pi\)
\(37\)	5.43052	−	7.47447i	0.892772	−	1.22880i	−0.0799443	−	0.996799i	\(-0.525474\pi\)
							0.972717	−	0.231996i	\(-0.0745258\pi\)
\(41\)	−1.16524	−	1.60381i	−0.181979	−	0.250473i	0.708275	−	0.705936i	\(-0.249474\pi\)
							−0.890255	+	0.455463i	\(0.849474\pi\)
\(43\)			2.02637i			0.309019i	0.987991	+	0.154509i	\(0.0493797\pi\)
							−0.987991	+	0.154509i	\(0.950620\pi\)
\(47\)	−2.77280	−	3.81644i	−0.404455	−	0.556684i	0.557400	−	0.830244i	\(-0.311799\pi\)
							−0.961855	+	0.273560i	\(0.911799\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.2.k.e.475.3		32
8.3	odd	2	inner	968.2.k.e.475.5		32
11.2	odd	10		88.2.k.b.19.1	✓	32
11.3	even	5		88.2.k.b.51.6	yes	32
11.4	even	5		968.2.g.e.483.7		32
11.5	even	5		968.2.k.i.699.4		32
11.6	odd	10	inner	968.2.k.e.699.5		32
11.7	odd	10		968.2.g.e.483.26		32
11.8	odd	10		968.2.k.h.403.3		32
11.9	even	5		968.2.k.h.723.8		32
11.10	odd	2		968.2.k.i.475.6		32
33.2	even	10		792.2.bp.b.19.8		32
33.14	odd	10		792.2.bp.b.667.3		32
44.3	odd	10		352.2.s.b.271.6		32
44.7	even	10		3872.2.g.d.1935.23		32
44.15	odd	10		3872.2.g.d.1935.24		32
44.35	even	10		352.2.s.b.239.5		32
88.3	odd	10		88.2.k.b.51.1	yes	32
88.13	odd	10		352.2.s.b.239.6		32
88.19	even	10		968.2.k.h.403.8		32
88.27	odd	10		968.2.k.i.699.6		32
88.29	odd	10		3872.2.g.d.1935.21		32
88.35	even	10		88.2.k.b.19.6	yes	32
88.37	even	10		3872.2.g.d.1935.22		32
88.43	even	2		968.2.k.i.475.4		32
88.51	even	10		968.2.g.e.483.8		32
88.59	odd	10		968.2.g.e.483.25		32
88.69	even	10		352.2.s.b.271.5		32
88.75	odd	10		968.2.k.h.723.3		32
88.83	even	10	inner	968.2.k.e.699.3		32
264.35	odd	10		792.2.bp.b.19.3		32
264.179	even	10		792.2.bp.b.667.8		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.k.b.19.1	✓	32	11.2	odd	10
88.2.k.b.19.6	yes	32	88.35	even	10
88.2.k.b.51.1	yes	32	88.3	odd	10
88.2.k.b.51.6	yes	32	11.3	even	5
352.2.s.b.239.5		32	44.35	even	10
352.2.s.b.239.6		32	88.13	odd	10
352.2.s.b.271.5		32	88.69	even	10
352.2.s.b.271.6		32	44.3	odd	10
792.2.bp.b.19.3		32	264.35	odd	10
792.2.bp.b.19.8		32	33.2	even	10
792.2.bp.b.667.3		32	33.14	odd	10
792.2.bp.b.667.8		32	264.179	even	10
968.2.g.e.483.7		32	11.4	even	5
968.2.g.e.483.8		32	88.51	even	10
968.2.g.e.483.25		32	88.59	odd	10
968.2.g.e.483.26		32	11.7	odd	10
968.2.k.e.475.3		32	1.1	even	1	trivial
968.2.k.e.475.5		32	8.3	odd	2	inner
968.2.k.e.699.3		32	88.83	even	10	inner
968.2.k.e.699.5		32	11.6	odd	10	inner
968.2.k.h.403.3		32	11.8	odd	10
968.2.k.h.403.8		32	88.19	even	10
968.2.k.h.723.3		32	88.75	odd	10
968.2.k.h.723.8		32	11.9	even	5
968.2.k.i.475.4		32	88.43	even	2
968.2.k.i.475.6		32	11.10	odd	2
968.2.k.i.699.4		32	11.5	even	5
968.2.k.i.699.6		32	88.27	odd	10
3872.2.g.d.1935.21		32	88.29	odd	10
3872.2.g.d.1935.22		32	88.37	even	10
3872.2.g.d.1935.23		32	44.7	even	10
3872.2.g.d.1935.24		32	44.15	odd	10