Newform orbit 64.4.b.a

• Label: 64.4.b.a
• Level: $64$
• Weight: $4$
• Character orbit: 64.b
• Analytic conductor: $3.776$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$64 = 2^{6}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	64.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.77612224037$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 5*b * q^3 - 73 * q^9 + 9*b * q^11 + 90 * q^17 + 53*b * q^19 + 125 * q^25 - 230*b * q^27 - 180 * q^33 + 522 * q^41 + 145*b * q^43 - 343 * q^49 + 450*b * q^51 - 1060 * q^57 - 423*b * q^59 - 35*b * q^67 - 430 * q^73 + 625*b * q^75 + 2629 * q^81 - 675*b * q^83 + 1026 * q^89 - 1910 * q^97 - 657*b * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 146 q^{9} + 180 q^{17} + 250 q^{25} - 360 q^{33} + 1044 q^{41} - 686 q^{49} - 2120 q^{57} - 860 q^{73} + 5258 q^{81} + 2052 q^{89} - 3820 q^{97}+O(q^{100})$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

33.1

	−	1.00000i
		1.00000i

0

−

10.0000i

0

0

0

0

0

−73.0000

0

33.2

0

10.0000i

0

0

0

0

0

−73.0000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	64.4.b.a	✓	2
3.b	odd	2	1		576.4.d.a		2
4.b	odd	2	1	inner	64.4.b.a	✓	2
8.b	even	2	1	inner	64.4.b.a	✓	2
8.d	odd	2	1	CM	64.4.b.a	✓	2
12.b	even	2	1		576.4.d.a		2
16.e	even	4	1		256.4.a.a		1
16.e	even	4	1		256.4.a.h		1
16.f	odd	4	1		256.4.a.a		1
16.f	odd	4	1		256.4.a.h		1
24.f	even	2	1		576.4.d.a		2
24.h	odd	2	1		576.4.d.a		2
48.i	odd	4	1		2304.4.a.h		1
48.i	odd	4	1		2304.4.a.i		1
48.k	even	4	1		2304.4.a.h		1
48.k	even	4	1		2304.4.a.i		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
64.4.b.a	✓	2	1.a	even	1	1	trivial
64.4.b.a	✓	2	4.b	odd	2	1	inner
64.4.b.a	✓	2	8.b	even	2	1	inner
64.4.b.a	✓	2	8.d	odd	2	1	CM
256.4.a.a		1	16.e	even	4	1
256.4.a.a		1	16.f	odd	4	1
256.4.a.h		1	16.e	even	4	1
256.4.a.h		1	16.f	odd	4	1
576.4.d.a		2	3.b	odd	2	1
576.4.d.a		2	12.b	even	2	1
576.4.d.a		2	24.f	even	2	1
576.4.d.a		2	24.h	odd	2	1
2304.4.a.h		1	48.i	odd	4	1
2304.4.a.h		1	48.k	even	4	1
2304.4.a.i		1	48.i	odd	4	1
2304.4.a.i		1	48.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 100$ acting on $S_{4}^{\mathrm{new}}(64, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 100$
$5$	$T^{2}$
$7$	$T^{2}$
$11$	$T^{2} + 324$