Newform orbit 700.2.k.c

• Label: 700.2.k.c
• Level: $700$
• Weight: $2$
• Character orbit: 700.k
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 40 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−1.41091	+	0.0966547i	0.539459	−	0.539459i	1.98132	−	0.272742i		0		−0.708984	+	0.813267i	0.707107	+	0.707107i	−2.76909	+	0.576316i			2.41797i		0
43.2	−1.40452	+	0.165291i	−1.79602	+	1.79602i	1.94536	−	0.464310i		0		2.22568	−	2.81941i	−0.707107	−	0.707107i	−2.65555	+	0.973684i		−	3.45135i		0
43.3	−1.34016	−	0.451632i	−2.23852	+	2.23852i	1.59206	+	1.21052i		0		4.01096	−	1.98898i	0.707107	+	0.707107i	−1.58690	−	2.34131i		−	7.02191i		0
43.4	−1.19057	−	0.763241i	0.421073	−	0.421073i	0.834928	+	1.81739i		0		−0.822698	+	0.179938i	−0.707107	−	0.707107i	0.393061	−	2.80098i			2.64540i		0
43.5	−1.18056	+	0.778639i	−1.22095	+	1.22095i	0.787444	−	1.83846i		0		0.490727	−	2.39209i	−0.707107	−	0.707107i	0.501871	+	2.78355i			0.0185454i		0
43.6	−1.10312	+	0.884938i	0.896840	−	0.896840i	0.433768	−	1.95239i		0		−0.195678	+	1.78297i	−0.707107	−	0.707107i	1.24925	+	2.53759i			1.39136i		0
43.7	−0.884938	+	1.10312i	−0.896840	+	0.896840i	−0.433768	−	1.95239i		0		−0.195678	−	1.78297i	0.707107	+	0.707107i	2.53759	+	1.24925i			1.39136i		0
43.8	−0.778639	+	1.18056i	1.22095	−	1.22095i	−0.787444	−	1.83846i		0		0.490727	+	2.39209i	0.707107	+	0.707107i	2.78355	+	0.501871i			0.0185454i		0
43.9	−0.763241	−	1.19057i	0.421073	−	0.421073i	−0.834928	+	1.81739i		0		−0.822698	−	0.179938i	−0.707107	−	0.707107i	2.80098	−	0.393061i			2.64540i		0
43.10	−0.451632	−	1.34016i	−2.23852	+	2.23852i	−1.59206	+	1.21052i		0		4.01096	+	1.98898i	0.707107	+	0.707107i	2.34131	+	1.58690i		−	7.02191i		0
43.11	−0.165291	+	1.40452i	1.79602	−	1.79602i	−1.94536	−	0.464310i		0		2.22568	+	2.81941i	0.707107	+	0.707107i	0.973684	−	2.65555i		−	3.45135i		0
43.12	−0.0966547	+	1.41091i	−0.539459	+	0.539459i	−1.98132	−	0.272742i		0		−0.708984	−	0.813267i	−0.707107	−	0.707107i	0.576316	−	2.76909i			2.41797i		0
43.13	0.0966547	−	1.41091i	0.539459	−	0.539459i	−1.98132	−	0.272742i		0		−0.708984	−	0.813267i	0.707107	+	0.707107i	−0.576316	+	2.76909i			2.41797i		0
43.14	0.165291	−	1.40452i	−1.79602	+	1.79602i	−1.94536	−	0.464310i		0		2.22568	+	2.81941i	−0.707107	−	0.707107i	−0.973684	+	2.65555i		−	3.45135i		0
43.15	0.451632	+	1.34016i	2.23852	−	2.23852i	−1.59206	+	1.21052i		0		4.01096	+	1.98898i	−0.707107	−	0.707107i	−2.34131	−	1.58690i		−	7.02191i		0
43.16	0.763241	+	1.19057i	−0.421073	+	0.421073i	−0.834928	+	1.81739i		0		−0.822698	−	0.179938i	0.707107	+	0.707107i	−2.80098	+	0.393061i			2.64540i		0
43.17	0.778639	−	1.18056i	−1.22095	+	1.22095i	−0.787444	−	1.83846i		0		0.490727	+	2.39209i	−0.707107	−	0.707107i	−2.78355	−	0.501871i			0.0185454i		0
43.18	0.884938	−	1.10312i	0.896840	−	0.896840i	−0.433768	−	1.95239i		0		−0.195678	−	1.78297i	−0.707107	−	0.707107i	−2.53759	−	1.24925i			1.39136i		0
43.19	1.10312	−	0.884938i	−0.896840	+	0.896840i	0.433768	−	1.95239i		0		−0.195678	+	1.78297i	0.707107	+	0.707107i	−1.24925	−	2.53759i			1.39136i		0
43.20	1.18056	−	0.778639i	1.22095	−	1.22095i	0.787444	−	1.83846i		0		0.490727	−	2.39209i	0.707107	+	0.707107i	−0.501871	−	2.78355i			0.0185454i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
20.d	odd	2	1	inner
20.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.2.k.c	✓	48
4.b	odd	2	1	inner	700.2.k.c	✓	48
5.b	even	2	1	inner	700.2.k.c	✓	48
5.c	odd	4	2	inner	700.2.k.c	✓	48
20.d	odd	2	1	inner	700.2.k.c	✓	48
20.e	even	4	2	inner	700.2.k.c	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.2.k.c	✓	48	1.a	even	1	1	trivial
700.2.k.c	✓	48	4.b	odd	2	1	inner
700.2.k.c	✓	48	5.b	even	2	1	inner
700.2.k.c	✓	48	5.c	odd	4	2	inner
700.2.k.c	✓	48	20.d	odd	2	1	inner
700.2.k.c	✓	48	20.e	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 154T_{3}^{20} + 5905T_{3}^{16} + 53960T_{3}^{12} + 120208T_{3}^{8} + 46848T_{3}^{4} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\).