Newform orbit 768.2.r.a

• Label: 768.2.r.a
• Level: $768$
• Weight: $2$
• Character orbit: 768.r
• Analytic conductor: $6.133$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.r (of order \(16\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.13251087523\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{16})\)
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
49.1		0		−0.980785	+	0.195090i		0		−1.84004	−	2.75382i		0		−0.915702	−	0.379296i		0		0.923880	−	0.382683i		0
49.2		0		−0.980785	+	0.195090i		0		−1.65192	−	2.47227i		0		2.43319	+	1.00786i		0		0.923880	−	0.382683i		0
49.3		0		−0.980785	+	0.195090i		0		−1.12099	−	1.67768i		0		−3.63169	−	1.50430i		0		0.923880	−	0.382683i		0
49.4		0		−0.980785	+	0.195090i		0		−1.10999	−	1.66121i		0		1.97014	+	0.816058i		0		0.923880	−	0.382683i		0
49.5		0		−0.980785	+	0.195090i		0		0.264413	+	0.395723i		0		−2.00633	−	0.831051i		0		0.923880	−	0.382683i		0
49.6		0		−0.980785	+	0.195090i		0		0.988484	+	1.47937i		0		−0.0771711	−	0.0319653i		0		0.923880	−	0.382683i		0
49.7		0		−0.980785	+	0.195090i		0		1.62504	+	2.43205i		0		−0.294182	−	0.121854i		0		0.923880	−	0.382683i		0
49.8		0		−0.980785	+	0.195090i		0		1.73386	+	2.59490i		0		4.36951	+	1.80991i		0		0.923880	−	0.382683i		0
49.9		0		0.980785	−	0.195090i		0		−1.96590	−	2.94217i		0		−2.86576	−	1.18704i		0		0.923880	−	0.382683i		0
49.10		0		0.980785	−	0.195090i		0		−0.620781	−	0.929065i		0		3.49586	+	1.44803i		0		0.923880	−	0.382683i		0
49.11		0		0.980785	−	0.195090i		0		−0.605864	−	0.906740i		0		−1.87610	−	0.777107i		0		0.923880	−	0.382683i		0
49.12		0		0.980785	−	0.195090i		0		−0.546564	−	0.817990i		0		3.46555	+	1.43548i		0		0.923880	−	0.382683i		0
49.13		0		0.980785	−	0.195090i		0		−0.0578655	−	0.0866018i		0		−2.40918	−	0.997914i		0		0.923880	−	0.382683i		0
49.14		0		0.980785	−	0.195090i		0		0.683874	+	1.02349i		0		0.296793	+	0.122936i		0		0.923880	−	0.382683i		0
49.15		0		0.980785	−	0.195090i		0		2.00072	+	2.99429i		0		3.93986	+	1.63194i		0		0.923880	−	0.382683i		0
49.16		0		0.980785	−	0.195090i		0		2.22352	+	3.32773i		0		−2.19925	−	0.910960i		0		0.923880	−	0.382683i		0
145.1		0		−0.831470	+	0.555570i		0		−4.01045	+	0.797728i		0		1.73589	+	4.19080i		0		0.382683	−	0.923880i		0
145.2		0		−0.831470	+	0.555570i		0		−1.24882	+	0.248406i		0		0.409753	+	0.989232i		0		0.382683	−	0.923880i		0
145.3		0		−0.831470	+	0.555570i		0		−1.15816	+	0.230372i		0		−1.66636	−	4.02294i		0		0.382683	−	0.923880i		0
145.4		0		−0.831470	+	0.555570i		0		−0.668642	+	0.133001i		0		−0.440731	−	1.06402i		0		0.382683	−	0.923880i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
64.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	768.2.r.a		128
4.b	odd	2	1		192.2.r.a	✓	128
12.b	even	2	1		576.2.bd.b		128
64.i	even	16	1	inner	768.2.r.a		128
64.j	odd	16	1		192.2.r.a	✓	128
192.s	even	16	1		576.2.bd.b		128

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
192.2.r.a	✓	128	4.b	odd	2	1
192.2.r.a	✓	128	64.j	odd	16	1
576.2.bd.b		128	12.b	even	2	1
576.2.bd.b		128	192.s	even	16	1
768.2.r.a		128	1.a	even	1	1	trivial
768.2.r.a		128	64.i	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(768, [\chi])\).