Newform orbit 768.2.o.a

• Label: 768.2.o.a
• Level: $768$
• Weight: $2$
• Character orbit: 768.o
• Analytic conductor: $6.133$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.o (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13251087523\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{8})\)
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{3} + 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
95.1		0		−1.70463	+	0.306981i		0		2.70066	−	1.11865i		0		3.28204	+	3.28204i		0		2.81152	−	1.04658i		0
95.2		0		−1.68370	−	0.406387i		0		−2.81491	+	1.16597i		0		0.543879	+	0.543879i		0		2.66970	+	1.36847i		0
95.3		0		−1.57410	+	0.722645i		0		0.378520	−	0.156788i		0		−2.01144	−	2.01144i		0		1.95557	−	2.27503i		0
95.4		0		−1.29202	−	1.15356i		0		−0.180206	+	0.0746437i		0		0.289055	+	0.289055i		0		0.338620	+	2.98083i		0
95.5		0		−1.02188	+	1.39848i		0		−3.14689	+	1.30348i		0		−0.663471	−	0.663471i		0		−0.911503	−	2.85817i		0
95.6		0		−0.266294	+	1.71146i		0		3.14689	−	1.30348i		0		−0.663471	−	0.663471i		0		−2.85817	−	0.911503i		0
95.7		0		−0.0380372	−	1.73163i		0		−2.18808	+	0.906333i		0		1.93241	+	1.93241i		0		−2.99711	+	0.131733i		0
95.8		0		0.0499748	−	1.73133i		0		1.06973	−	0.443098i		0		−2.37247	−	2.37247i		0		−2.99501	−	0.173046i		0
95.9		0		0.602068	+	1.62404i		0		−0.378520	+	0.156788i		0		−2.01144	−	2.01144i		0		−2.27503	+	1.95557i		0
95.10		0		0.988287	+	1.42242i		0		−2.70066	+	1.11865i		0		3.28204	+	3.28204i		0		−1.04658	+	2.81152i		0
95.11		0		1.18890	−	1.25957i		0		−1.06973	+	0.443098i		0		−2.37247	−	2.37247i		0		−0.173046	−	2.99501i		0
95.12		0		1.25135	−	1.19755i		0		2.18808	−	0.906333i		0		1.93241	+	1.93241i		0		0.131733	−	2.99711i		0
95.13		0		1.47792	+	0.903198i		0		2.81491	−	1.16597i		0		0.543879	+	0.543879i		0		1.36847	+	2.66970i		0
95.14		0		1.72928	+	0.0979076i		0		0.180206	−	0.0746437i		0		0.289055	+	0.289055i		0		2.98083	+	0.338620i		0
287.1		0		−1.71185	+	0.263778i		0		1.20190	−	2.90164i		0		−2.49510	+	2.49510i		0		2.86084	−	0.903094i		0
287.2		0		−1.67029	−	0.458417i		0		0.632159	−	1.52617i		0		2.65940	−	2.65940i		0		2.57971	+	1.53137i		0
287.3		0		−1.54863	−	0.775716i		0		−0.584611	+	1.41138i		0		0.696730	−	0.696730i		0		1.79653	+	2.40260i		0
287.4		0		−1.39698	+	1.02394i		0		−1.20190	+	2.90164i		0		−2.49510	+	2.49510i		0		0.903094	−	2.86084i		0
287.5		0		−0.856920	+	1.50522i		0		−0.632159	+	1.52617i		0		2.65940	−	2.65940i		0		−1.53137	−	2.57971i		0
287.6		0		−0.641495	−	1.60888i		0		−0.296199	+	0.715088i		0		−2.77714	+	2.77714i		0		−2.17697	+	2.06417i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
32.h	odd	8	1	inner
96.o	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	768.2.o.a		56
3.b	odd	2	1	inner	768.2.o.a		56
4.b	odd	2	1		768.2.o.b		56
8.b	even	2	1		384.2.o.a		56
8.d	odd	2	1		96.2.o.a	✓	56
12.b	even	2	1		768.2.o.b		56
24.f	even	2	1		96.2.o.a	✓	56
24.h	odd	2	1		384.2.o.a		56
32.g	even	8	1		96.2.o.a	✓	56
32.g	even	8	1		768.2.o.b		56
32.h	odd	8	1		384.2.o.a		56
32.h	odd	8	1	inner	768.2.o.a		56
96.o	even	8	1		384.2.o.a		56
96.o	even	8	1	inner	768.2.o.a		56
96.p	odd	8	1		96.2.o.a	✓	56
96.p	odd	8	1		768.2.o.b		56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.2.o.a	✓	56	8.d	odd	2	1
96.2.o.a	✓	56	24.f	even	2	1
96.2.o.a	✓	56	32.g	even	8	1
96.2.o.a	✓	56	96.p	odd	8	1
384.2.o.a		56	8.b	even	2	1
384.2.o.a		56	24.h	odd	2	1
384.2.o.a		56	32.h	odd	8	1
384.2.o.a		56	96.o	even	8	1
768.2.o.a		56	1.a	even	1	1	trivial
768.2.o.a		56	3.b	odd	2	1	inner
768.2.o.a		56	32.h	odd	8	1	inner
768.2.o.a		56	96.o	even	8	1	inner
768.2.o.b		56	4.b	odd	2	1
768.2.o.b		56	12.b	even	2	1
768.2.o.b		56	32.g	even	8	1
768.2.o.b		56	96.p	odd	8	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^28 - 4*T7^27 + 8*T7^26 + 636*T7^24 - 2304*T7^23 + 4128*T7^22 - 4048*T7^21 + 137080*T7^20 - 469824*T7^19 + 811456*T7^18 - 1291808*T7^17 + 11973664*T7^16 - 40223872*T7^15 + 73821568*T7^14 - 110905152*T7^13 + 399997328*T7^12 - 1229510080*T7^11 + 2378206336*T7^10 - 2978290304*T7^9 + 2740008384*T7^8 - 2319517184*T7^7 + 2327438336*T7^6 - 2364102656*T7^5 + 1902780160*T7^4 - 1077963776*T7^3 + 409208832*T7^2 - 94291968*T7 + 10863616