Newform orbit 768.2.n.a

• Label: 768.2.n.a
• Level: $768$
• Weight: $2$
• Character orbit: 768.n
• Analytic conductor: $6.133$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13251087523\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
97.1		0		−0.923880	+	0.382683i		0		−0.750897	+	1.81283i		0		0.638460	+	0.638460i		0		0.707107	−	0.707107i		0
97.2		0		−0.923880	+	0.382683i		0		0.00259461	−	0.00626394i		0		−2.41880	−	2.41880i		0		0.707107	−	0.707107i		0
97.3		0		−0.923880	+	0.382683i		0		0.155637	−	0.375742i		0		0.709092	+	0.709092i		0		0.707107	−	0.707107i		0
97.4		0		−0.923880	+	0.382683i		0		1.35803	−	3.27858i		0		2.48546	+	2.48546i		0		0.707107	−	0.707107i		0
97.5		0		0.923880	−	0.382683i		0		−1.36206	+	3.28830i		0		−2.73097	−	2.73097i		0		0.707107	−	0.707107i		0
97.6		0		0.923880	−	0.382683i		0		−0.705805	+	1.70396i		0		3.24150	+	3.24150i		0		0.707107	−	0.707107i		0
97.7		0		0.923880	−	0.382683i		0		−0.184062	+	0.444366i		0		−0.134531	−	0.134531i		0		0.707107	−	0.707107i		0
97.8		0		0.923880	−	0.382683i		0		1.48656	−	3.58888i		0		1.03821	+	1.03821i		0		0.707107	−	0.707107i		0
289.1		0		−0.382683	+	0.923880i		0		−3.68816	+	1.52768i		0		−1.63704	+	1.63704i		0		−0.707107	−	0.707107i		0
289.2		0		−0.382683	+	0.923880i		0		−0.825824	+	0.342068i		0		1.17750	−	1.17750i		0		−0.707107	−	0.707107i		0
289.3		0		−0.382683	+	0.923880i		0		1.20409	−	0.498752i		0		2.59422	−	2.59422i		0		−0.707107	−	0.707107i		0
289.4		0		−0.382683	+	0.923880i		0		1.46213	−	0.605634i		0		−3.54889	+	3.54889i		0		−0.707107	−	0.707107i		0
289.5		0		0.382683	−	0.923880i		0		−2.14986	+	0.890503i		0		−1.10001	+	1.10001i		0		−0.707107	−	0.707107i		0
289.6		0		0.382683	−	0.923880i		0		−1.60930	+	0.666593i		0		−0.589445	+	0.589445i		0		−0.707107	−	0.707107i		0
289.7		0		0.382683	−	0.923880i		0		2.51374	−	1.04122i		0		2.01027	−	2.01027i		0		−0.707107	−	0.707107i		0
289.8		0		0.382683	−	0.923880i		0		3.09318	−	1.28124i		0		−1.73503	+	1.73503i		0		−0.707107	−	0.707107i		0
481.1		0		−0.382683	−	0.923880i		0		−3.68816	−	1.52768i		0		−1.63704	−	1.63704i		0		−0.707107	+	0.707107i		0
481.2		0		−0.382683	−	0.923880i		0		−0.825824	−	0.342068i		0		1.17750	+	1.17750i		0		−0.707107	+	0.707107i		0
481.3		0		−0.382683	−	0.923880i		0		1.20409	+	0.498752i		0		2.59422	+	2.59422i		0		−0.707107	+	0.707107i		0
481.4		0		−0.382683	−	0.923880i		0		1.46213	+	0.605634i		0		−3.54889	−	3.54889i		0		−0.707107	+	0.707107i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
32.g	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.n.a		32
4.b	odd	2	1		768.2.n.b		32
8.b	even	2	1		96.2.n.a	✓	32
8.d	odd	2	1		384.2.n.a		32
24.f	even	2	1		1152.2.v.c		32
24.h	odd	2	1		288.2.v.d		32
32.g	even	8	1		96.2.n.a	✓	32
32.g	even	8	1	inner	768.2.n.a		32
32.h	odd	8	1		384.2.n.a		32
32.h	odd	8	1		768.2.n.b		32
96.o	even	8	1		1152.2.v.c		32
96.p	odd	8	1		288.2.v.d		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.2.n.a	✓	32	8.b	even	2	1
96.2.n.a	✓	32	32.g	even	8	1
288.2.v.d		32	24.h	odd	2	1
288.2.v.d		32	96.p	odd	8	1
384.2.n.a		32	8.d	odd	2	1
384.2.n.a		32	32.h	odd	8	1
768.2.n.a		32	1.a	even	1	1	trivial
768.2.n.a		32	32.g	even	8	1	inner
768.2.n.b		32	4.b	odd	2	1
768.2.n.b		32	32.h	odd	8	1
1152.2.v.c		32	24.f	even	2	1
1152.2.v.c		32	96.o	even	8	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^32 - 16*T7^29 + 960*T7^28 - 352*T7^27 + 128*T7^26 - 7328*T7^25 + 284304*T7^24 - 154368*T7^23 + 56320*T7^22 - 1036992*T7^21 + 33081600*T7^20 - 20145792*T7^19 + 7321088*T7^18 - 63516544*T7^17 + 1458295392*T7^16 - 918034432*T7^15 + 346710016*T7^14 - 2310927104*T7^13 + 21542736896*T7^12 - 19222802944*T7^11 + 9040271360*T7^10 - 10587135488*T7^9 + 88705161472*T7^8 - 92169613312*T7^7 + 46372995072*T7^6 + 8425397248*T7^5 + 39292022784*T7^4 - 40369174528*T7^3 + 20840652800*T7^2 + 7059036160*T7 + 1195499776