Newform orbit 704.2.w.c

• Label: 704.2.w.c
• Level: $704$
• Weight: $2$
• Character orbit: 704.w
• Analytic conductor: $5.621$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	704.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.62146830230\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 12 q^{9}+O(q^{10}) \)

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		−1.77223	−	2.43927i		0		−2.80893	+	0.912676i		0		2.83314	+	2.05840i		0		−1.88217	+	5.79272i		0
97.2		0		−1.77223	−	2.43927i		0		2.80893	−	0.912676i		0		−2.83314	−	2.05840i		0		−1.88217	+	5.79272i		0
97.3		0		−1.14226	−	1.57219i		0		−3.23842	+	1.05223i		0		−2.10871	−	1.53207i		0		−0.239964	+	0.738534i		0
97.4		0		−1.14226	−	1.57219i		0		3.23842	−	1.05223i		0		2.10871	+	1.53207i		0		−0.239964	+	0.738534i		0
97.5		0		−0.783643	−	1.07859i		0		−1.90939	+	0.620397i		0		1.15890	+	0.841992i		0		0.377786	−	1.16271i		0
97.6		0		−0.783643	−	1.07859i		0		1.90939	−	0.620397i		0		−1.15890	−	0.841992i		0		0.377786	−	1.16271i		0
97.7		0		−0.741460	−	1.02053i		0		−0.174671	+	0.0567541i		0		−3.83347	−	2.78518i		0		0.435329	−	1.33980i		0
97.8		0		−0.741460	−	1.02053i		0		0.174671	−	0.0567541i		0		3.83347	+	2.78518i		0		0.435329	−	1.33980i		0
97.9		0		0.741460	+	1.02053i		0		−0.174671	+	0.0567541i		0		3.83347	+	2.78518i		0		0.435329	−	1.33980i		0
97.10		0		0.741460	+	1.02053i		0		0.174671	−	0.0567541i		0		−3.83347	−	2.78518i		0		0.435329	−	1.33980i		0
97.11		0		0.783643	+	1.07859i		0		−1.90939	+	0.620397i		0		−1.15890	−	0.841992i		0		0.377786	−	1.16271i		0
97.12		0		0.783643	+	1.07859i		0		1.90939	−	0.620397i		0		1.15890	+	0.841992i		0		0.377786	−	1.16271i		0
97.13		0		1.14226	+	1.57219i		0		−3.23842	+	1.05223i		0		2.10871	+	1.53207i		0		−0.239964	+	0.738534i		0
97.14		0		1.14226	+	1.57219i		0		3.23842	−	1.05223i		0		−2.10871	−	1.53207i		0		−0.239964	+	0.738534i		0
97.15		0		1.77223	+	2.43927i		0		−2.80893	+	0.912676i		0		−2.83314	−	2.05840i		0		−1.88217	+	5.79272i		0
97.16		0		1.77223	+	2.43927i		0		2.80893	−	0.912676i		0		2.83314	+	2.05840i		0		−1.88217	+	5.79272i		0
225.1		0		−1.77223	+	2.43927i		0		−2.80893	−	0.912676i		0		2.83314	−	2.05840i		0		−1.88217	−	5.79272i		0
225.2		0		−1.77223	+	2.43927i		0		2.80893	+	0.912676i		0		−2.83314	+	2.05840i		0		−1.88217	−	5.79272i		0
225.3		0		−1.14226	+	1.57219i		0		−3.23842	−	1.05223i		0		−2.10871	+	1.53207i		0		−0.239964	−	0.738534i		0
225.4		0		−1.14226	+	1.57219i		0		3.23842	+	1.05223i		0		2.10871	−	1.53207i		0		−0.239964	−	0.738534i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
11.c	even	5	1	inner
44.h	odd	10	1	inner
88.l	odd	10	1	inner
88.o	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	704.2.w.c	✓	64
4.b	odd	2	1	inner	704.2.w.c	✓	64
8.b	even	2	1	inner	704.2.w.c	✓	64
8.d	odd	2	1	inner	704.2.w.c	✓	64
11.c	even	5	1	inner	704.2.w.c	✓	64
44.h	odd	10	1	inner	704.2.w.c	✓	64
88.l	odd	10	1	inner	704.2.w.c	✓	64
88.o	even	10	1	inner	704.2.w.c	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
704.2.w.c	✓	64	1.a	even	1	1	trivial
704.2.w.c	✓	64	4.b	odd	2	1	inner
704.2.w.c	✓	64	8.b	even	2	1	inner
704.2.w.c	✓	64	8.d	odd	2	1	inner
704.2.w.c	✓	64	11.c	even	5	1	inner
704.2.w.c	✓	64	44.h	odd	10	1	inner
704.2.w.c	✓	64	88.l	odd	10	1	inner
704.2.w.c	✓	64	88.o	even	10	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^32 - 9*T3^30 + 107*T3^28 - 1104*T3^26 + 8115*T3^24 - 36720*T3^22 + 151231*T3^20 - 550350*T3^18 + 1624916*T3^16 - 2909376*T3^14 + 6788743*T3^12 - 10455015*T3^10 + 13574019*T3^8 - 14653731*T3^6 + 17402180*T3^4 - 6195567*T3^2 + 923521