Newform orbit 64.9.c.d

• Label: 64.9.c.d
• Level: $64$
• Weight: $9$
• Character orbit: 64.c
• Analytic conductor: $26.072$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	64.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(26.0722310439\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-35}) \)
Defining polynomial:	\( x^{2} - x + 9 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{4}\cdot 3 \)
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 24\sqrt{-35}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{3} + 510 q^{5} - 18 \beta q^{7} - 13599 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1020 q^{5} - 27198 q^{9}+O(q^{10}) \)

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

0.500000	+	2.95804i
0.500000	−	2.95804i

0

−

141.986i

0

510.000

0

−

2555.75i

0

−13599.0

0

63.2

0

141.986i

0

510.000

0

2555.75i

0

−13599.0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	64.9.c.d		2
4.b	odd	2	1	inner	64.9.c.d		2
8.b	even	2	1		16.9.c.a	✓	2
8.d	odd	2	1		16.9.c.a	✓	2
16.e	even	4	2		256.9.d.f		4
16.f	odd	4	2		256.9.d.f		4
24.f	even	2	1		144.9.g.g		2
24.h	odd	2	1		144.9.g.g		2
40.e	odd	2	1		400.9.b.c		2
40.f	even	2	1		400.9.b.c		2
40.i	odd	4	2		400.9.h.b		4
40.k	even	4	2		400.9.h.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
16.9.c.a	✓	2	8.b	even	2	1
16.9.c.a	✓	2	8.d	odd	2	1
64.9.c.d		2	1.a	even	1	1	trivial
64.9.c.d		2	4.b	odd	2	1	inner
144.9.g.g		2	24.f	even	2	1
144.9.g.g		2	24.h	odd	2	1
256.9.d.f		4	16.e	even	4	2
256.9.d.f		4	16.f	odd	4	2
400.9.b.c		2	40.e	odd	2	1
400.9.b.c		2	40.f	even	2	1
400.9.h.b		4	40.i	odd	4	2
400.9.h.b		4	40.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 20160 \) acting on \(S_{9}^{\mathrm{new}}(64, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 20160 \)
$5$	\( (T - 510)^{2} \)
$7$	\( T^{2} + 6531840 \)
$11$	\( T^{2} + 367416000 \)