Newform orbit 64.3.c.a

• Label: 64.3.c.a
• Level: $64$
• Weight: $3$
• Character orbit: 64.c
• Self dual: yes
• Analytic conductor: $1.744$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	64.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.74387369191\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 6 q^{5} + 9 q^{9}+O(q^{10}) \)

Expression as an eta quotient

\(f(z) = \dfrac{\eta(8z)^{18}}{\eta(4z)^{6}\eta(16z)^{6}}=q\prod_{n=1}^\infty(1 - q^{4n})^{-6}(1 - q^{8n})^{18}(1 - q^{16n})^{-6}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(63\)
\(\chi(n)\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

63.1

0

0

0

0

6.00000

0

0

0

9.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	64.3.c.a		1
3.b	odd	2	1		576.3.g.b		1
4.b	odd	2	1	CM	64.3.c.a		1
5.b	even	2	1		1600.3.b.b		1
5.c	odd	4	2		1600.3.h.b		2
8.b	even	2	1		16.3.c.a	✓	1
8.d	odd	2	1		16.3.c.a	✓	1
12.b	even	2	1		576.3.g.b		1
16.e	even	4	2		256.3.d.b		2
16.f	odd	4	2		256.3.d.b		2
20.d	odd	2	1		1600.3.b.b		1
20.e	even	4	2		1600.3.h.b		2
24.f	even	2	1		144.3.g.a		1
24.h	odd	2	1		144.3.g.a		1
40.e	odd	2	1		400.3.b.a		1
40.f	even	2	1		400.3.b.a		1
40.i	odd	4	2		400.3.h.a		2
40.k	even	4	2		400.3.h.a		2
48.i	odd	4	2		2304.3.b.f		2
48.k	even	4	2		2304.3.b.f		2
56.e	even	2	1		784.3.d.b		1
56.h	odd	2	1		784.3.d.b		1
56.j	odd	6	2		784.3.r.d		2
56.k	odd	6	2		784.3.r.e		2
56.m	even	6	2		784.3.r.d		2
56.p	even	6	2		784.3.r.e		2
72.j	odd	6	2		1296.3.o.b		2
72.l	even	6	2		1296.3.o.b		2
72.n	even	6	2		1296.3.o.o		2
72.p	odd	6	2		1296.3.o.o		2
120.i	odd	2	1		3600.3.e.c		1
120.m	even	2	1		3600.3.e.c		1
120.q	odd	4	2		3600.3.j.a		2
120.w	even	4	2		3600.3.j.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
16.3.c.a	✓	1	8.b	even	2	1
16.3.c.a	✓	1	8.d	odd	2	1
64.3.c.a		1	1.a	even	1	1	trivial
64.3.c.a		1	4.b	odd	2	1	CM
144.3.g.a		1	24.f	even	2	1
144.3.g.a		1	24.h	odd	2	1
256.3.d.b		2	16.e	even	4	2
256.3.d.b		2	16.f	odd	4	2
400.3.b.a		1	40.e	odd	2	1
400.3.b.a		1	40.f	even	2	1
400.3.h.a		2	40.i	odd	4	2
400.3.h.a		2	40.k	even	4	2
576.3.g.b		1	3.b	odd	2	1
576.3.g.b		1	12.b	even	2	1
784.3.d.b		1	56.e	even	2	1
784.3.d.b		1	56.h	odd	2	1
784.3.r.d		2	56.j	odd	6	2
784.3.r.d		2	56.m	even	6	2
784.3.r.e		2	56.k	odd	6	2
784.3.r.e		2	56.p	even	6	2
1296.3.o.b		2	72.j	odd	6	2
1296.3.o.b		2	72.l	even	6	2
1296.3.o.o		2	72.n	even	6	2
1296.3.o.o		2	72.p	odd	6	2
1600.3.b.b		1	5.b	even	2	1
1600.3.b.b		1	20.d	odd	2	1
1600.3.h.b		2	5.c	odd	4	2
1600.3.h.b		2	20.e	even	4	2
2304.3.b.f		2	48.i	odd	4	2
2304.3.b.f		2	48.k	even	4	2
3600.3.e.c		1	120.i	odd	2	1
3600.3.e.c		1	120.m	even	2	1
3600.3.j.a		2	120.q	odd	4	2
3600.3.j.a		2	120.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{3}^{\mathrm{new}}(64, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T - 6 \)
$7$	\( T \)
$11$	\( T \)