Newform orbit 702.2.j.b

• Label: 702.2.j.b
• Level: $702$
• Weight: $2$
• Character orbit: 702.j
• Analytic conductor: $5.605$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	702.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.60549822189\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{16} - 24 q^{19} + 24 q^{34} + 24 q^{37} + 24 q^{46} + 48 q^{70} - 24 q^{73} - 24 q^{76} - 24 q^{79} - 24 q^{91} + 72 q^{94} + 96 q^{97}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
161.1	−0.707107	−	0.707107i		0				1.00000i	−2.82990	−	2.82990i		0		−1.76787	−	1.76787i	0.707107	−	0.707107i		0				4.00208i
161.2	−0.707107	−	0.707107i		0				1.00000i	−2.08925	−	2.08925i		0		−0.0160989	−	0.0160989i	0.707107	−	0.707107i		0				2.95464i
161.3	−0.707107	−	0.707107i		0				1.00000i	−0.136488	−	0.136488i		0		3.40994	+	3.40994i	0.707107	−	0.707107i		0				0.193023i
161.4	−0.707107	−	0.707107i		0				1.00000i	0.730953	+	0.730953i		0		−2.49696	−	2.49696i	0.707107	−	0.707107i		0			−	1.03372i
161.5	−0.707107	−	0.707107i		0				1.00000i	1.23433	+	1.23433i		0		−1.64208	−	1.64208i	0.707107	−	0.707107i		0			−	1.74561i
161.6	−0.707107	−	0.707107i		0				1.00000i	3.09035	+	3.09035i		0		2.51306	+	2.51306i	0.707107	−	0.707107i		0			−	4.37041i
161.7	0.707107	+	0.707107i		0				1.00000i	−3.09035	−	3.09035i		0		2.51306	+	2.51306i	−0.707107	+	0.707107i		0			−	4.37041i
161.8	0.707107	+	0.707107i		0				1.00000i	−1.23433	−	1.23433i		0		−1.64208	−	1.64208i	−0.707107	+	0.707107i		0			−	1.74561i
161.9	0.707107	+	0.707107i		0				1.00000i	−0.730953	−	0.730953i		0		−2.49696	−	2.49696i	−0.707107	+	0.707107i		0			−	1.03372i
161.10	0.707107	+	0.707107i		0				1.00000i	0.136488	+	0.136488i		0		3.40994	+	3.40994i	−0.707107	+	0.707107i		0				0.193023i
161.11	0.707107	+	0.707107i		0				1.00000i	2.08925	+	2.08925i		0		−0.0160989	−	0.0160989i	−0.707107	+	0.707107i		0				2.95464i
161.12	0.707107	+	0.707107i		0				1.00000i	2.82990	+	2.82990i		0		−1.76787	−	1.76787i	−0.707107	+	0.707107i		0				4.00208i
593.1	−0.707107	+	0.707107i		0			−	1.00000i	−2.82990	+	2.82990i		0		−1.76787	+	1.76787i	0.707107	+	0.707107i		0			−	4.00208i
593.2	−0.707107	+	0.707107i		0			−	1.00000i	−2.08925	+	2.08925i		0		−0.0160989	+	0.0160989i	0.707107	+	0.707107i		0			−	2.95464i
593.3	−0.707107	+	0.707107i		0			−	1.00000i	−0.136488	+	0.136488i		0		3.40994	−	3.40994i	0.707107	+	0.707107i		0			−	0.193023i
593.4	−0.707107	+	0.707107i		0			−	1.00000i	0.730953	−	0.730953i		0		−2.49696	+	2.49696i	0.707107	+	0.707107i		0				1.03372i
593.5	−0.707107	+	0.707107i		0			−	1.00000i	1.23433	−	1.23433i		0		−1.64208	+	1.64208i	0.707107	+	0.707107i		0				1.74561i
593.6	−0.707107	+	0.707107i		0			−	1.00000i	3.09035	−	3.09035i		0		2.51306	−	2.51306i	0.707107	+	0.707107i		0				4.37041i
593.7	0.707107	−	0.707107i		0			−	1.00000i	−3.09035	+	3.09035i		0		2.51306	−	2.51306i	−0.707107	−	0.707107i		0				4.37041i
593.8	0.707107	−	0.707107i		0			−	1.00000i	−1.23433	+	1.23433i		0		−1.64208	+	1.64208i	−0.707107	−	0.707107i		0				1.74561i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
13.d	odd	4	1	inner
39.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	702.2.j.b	✓	24
3.b	odd	2	1	inner	702.2.j.b	✓	24
13.d	odd	4	1	inner	702.2.j.b	✓	24
39.f	even	4	1	inner	702.2.j.b	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
702.2.j.b	✓	24	1.a	even	1	1	trivial
702.2.j.b	✓	24	3.b	odd	2	1	inner
702.2.j.b	✓	24	13.d	odd	4	1	inner
702.2.j.b	✓	24	39.f	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 708T_{5}^{20} + 148230T_{5}^{16} + 8609868T_{5}^{12} + 75878289T_{5}^{8} + 75728520T_{5}^{4} + 104976 \) acting on \(S_{2}^{\mathrm{new}}(702, [\chi])\).