Newform orbit 960.2.cp.a

• Label: 960.2.cp.a
• Level: $960$
• Weight: $2$
• Character orbit: 960.cp
• Analytic conductor: $7.666$
• Analytic rank: $0$
• Dimension: $32$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.cp (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.66563859404\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

$q$-expansion

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

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.06750	−	0.927598i	−0.962276	+	1.44015i	0.279124	+	1.98043i	−2.19310	−	0.436235i	2.36311	−	0.644756i		0		1.53908	−	2.37303i	−1.14805	−	2.77164i	1.93649	+	2.50000i
59.2	−0.631267	+	1.26550i	−0.962276	+	1.44015i	−1.20300	−	1.59774i	2.19310	+	0.436235i	−1.21506	−	2.12688i		0		2.78137	−	0.513803i	−1.14805	−	2.77164i	−1.93649	+	2.50000i
59.3	0.631267	−	1.26550i	0.962276	−	1.44015i	−1.20300	−	1.59774i	−2.19310	−	0.436235i	−1.21506	−	2.12688i		0		−2.78137	+	0.513803i	−1.14805	−	2.77164i	−1.93649	+	2.50000i
59.4	1.06750	+	0.927598i	0.962276	−	1.44015i	0.279124	+	1.98043i	2.19310	+	0.436235i	2.36311	−	0.644756i		0		−1.53908	+	2.37303i	−1.14805	−	2.77164i	1.93649	+	2.50000i
179.1	−1.06750	+	0.927598i	−0.962276	−	1.44015i	0.279124	−	1.98043i	−2.19310	+	0.436235i	2.36311	+	0.644756i		0		1.53908	+	2.37303i	−1.14805	+	2.77164i	1.93649	−	2.50000i
179.2	−0.631267	−	1.26550i	−0.962276	−	1.44015i	−1.20300	+	1.59774i	2.19310	−	0.436235i	−1.21506	+	2.12688i		0		2.78137	+	0.513803i	−1.14805	+	2.77164i	−1.93649	−	2.50000i
179.3	0.631267	+	1.26550i	0.962276	+	1.44015i	−1.20300	+	1.59774i	−2.19310	+	0.436235i	−1.21506	+	2.12688i		0		−2.78137	−	0.513803i	−1.14805	+	2.77164i	−1.93649	−	2.50000i
179.4	1.06750	−	0.927598i	0.962276	+	1.44015i	0.279124	−	1.98043i	2.19310	−	0.436235i	2.36311	+	0.644756i		0		−1.53908	−	2.37303i	−1.14805	+	2.77164i	1.93649	−	2.50000i
299.1	−1.34122	+	0.448473i	1.69877	−	0.337906i	1.59774	−	1.20300i	1.85922	−	1.24229i	−2.12688	+	1.21506i		0		−1.60341	+	2.33004i	2.77164	−	1.14805i	−1.93649	+	2.50000i
299.2	−0.0989274	−	1.41075i	1.69877	−	0.337906i	−1.98043	+	0.279124i	−1.85922	+	1.24229i	−0.644756	−	2.36311i		0		0.589692	+	2.76627i	2.77164	−	1.14805i	1.93649	+	2.50000i
299.3	0.0989274	+	1.41075i	−1.69877	+	0.337906i	−1.98043	+	0.279124i	1.85922	−	1.24229i	−0.644756	−	2.36311i		0		−0.589692	−	2.76627i	2.77164	−	1.14805i	1.93649	+	2.50000i
299.4	1.34122	−	0.448473i	−1.69877	+	0.337906i	1.59774	−	1.20300i	−1.85922	+	1.24229i	−2.12688	+	1.21506i		0		1.60341	−	2.33004i	2.77164	−	1.14805i	−1.93649	+	2.50000i
419.1	−1.41075	−	0.0989274i	−0.337906	+	1.69877i	1.98043	+	0.279124i	−1.24229	+	1.85922i	0.644756	−	2.36311i		0		−2.76627	−	0.589692i	−2.77164	−	1.14805i	1.93649	−	2.50000i
419.2	−0.448473	+	1.34122i	0.337906	−	1.69877i	−1.59774	−	1.20300i	−1.24229	+	1.85922i	2.12688	+	1.21506i		0		2.33004	−	1.60341i	−2.77164	−	1.14805i	−1.93649	−	2.50000i
419.3	0.448473	−	1.34122i	−0.337906	+	1.69877i	−1.59774	−	1.20300i	1.24229	−	1.85922i	2.12688	+	1.21506i		0		−2.33004	+	1.60341i	−2.77164	−	1.14805i	−1.93649	−	2.50000i
419.4	1.41075	+	0.0989274i	0.337906	−	1.69877i	1.98043	+	0.279124i	1.24229	−	1.85922i	0.644756	−	2.36311i		0		2.76627	+	0.589692i	−2.77164	−	1.14805i	1.93649	−	2.50000i
539.1	−1.26550	−	0.631267i	−1.44015	−	0.962276i	1.20300	+	1.59774i	0.436235	−	2.19310i	1.21506	+	2.12688i		0		−0.513803	−	2.78137i	1.14805	+	2.77164i	−1.93649	+	2.50000i
539.2	−0.927598	+	1.06750i	1.44015	+	0.962276i	−0.279124	−	1.98043i	0.436235	−	2.19310i	−2.36311	+	0.644756i		0		2.37303	+	1.53908i	1.14805	+	2.77164i	1.93649	+	2.50000i
539.3	0.927598	−	1.06750i	−1.44015	−	0.962276i	−0.279124	−	1.98043i	−0.436235	+	2.19310i	−2.36311	+	0.644756i		0		−2.37303	−	1.53908i	1.14805	+	2.77164i	1.93649	+	2.50000i
539.4	1.26550	+	0.631267i	1.44015	+	0.962276i	1.20300	+	1.59774i	−0.436235	+	2.19310i	1.21506	+	2.12688i		0		0.513803	+	2.78137i	1.14805	+	2.77164i	−1.93649	+	2.50000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by \(\Q(\sqrt{-15}) \)
3.b	odd	2	1	inner
5.b	even	2	1	inner
64.j	odd	16	1	inner
192.s	even	16	1	inner
320.bh	odd	16	1	inner
960.cp	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	960.2.cp.a	✓	32
3.b	odd	2	1	inner	960.2.cp.a	✓	32
5.b	even	2	1	inner	960.2.cp.a	✓	32
15.d	odd	2	1	CM	960.2.cp.a	✓	32
64.j	odd	16	1	inner	960.2.cp.a	✓	32
192.s	even	16	1	inner	960.2.cp.a	✓	32
320.bh	odd	16	1	inner	960.2.cp.a	✓	32
960.cp	even	16	1	inner	960.2.cp.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
960.2.cp.a	✓	32	1.a	even	1	1	trivial
960.2.cp.a	✓	32	3.b	odd	2	1	inner
960.2.cp.a	✓	32	5.b	even	2	1	inner
960.2.cp.a	✓	32	15.d	odd	2	1	CM
960.2.cp.a	✓	32	64.j	odd	16	1	inner
960.2.cp.a	✓	32	192.s	even	16	1	inner
960.2.cp.a	✓	32	320.bh	odd	16	1	inner
960.2.cp.a	✓	32	960.cp	even	16	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\).