Newform orbit 256.8.b.d

• Label: 256.8.b.d
• Level: $256$
• Weight: $8$
• Character orbit: 256.b
• Analytic conductor: $79.971$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -4
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$256 = 2^{8}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 32)
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 + 29*b * q^5 + 2187 * q^9 - 4449*b * q^13 - 40094 * q^17 + 74761 * q^25 + 116615*b * q^29 - 281987*b * q^37 - 9530 * q^41 + 63423*b * q^45 - 823543 * q^49 + 399301*b * q^53 - 1752665*b * q^61 + 516084 * q^65 - 3917418 * q^73 + 4782969 * q^81 - 1162726*b * q^85 - 9246170 * q^89 - 17567406 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4374 q^{9} - 80188 q^{17} + 149522 q^{25} - 19060 q^{41} - 1647086 q^{49} + 1032168 q^{65} - 7834836 q^{73} + 9565938 q^{81} - 18492340 q^{89} - 35134812 q^{97}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

129.1

	−	1.00000i
		1.00000i

0

0

0

−

58.0000i

0

0

0

2187.00

0

129.2

0

0

0

58.0000i

0

0

0

2187.00

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	256.8.b.d		2
4.b	odd	2	1	CM	256.8.b.d		2
8.b	even	2	1	inner	256.8.b.d		2
8.d	odd	2	1	inner	256.8.b.d		2
16.e	even	4	1		32.8.a.a	✓	1
16.e	even	4	1		64.8.a.d		1
16.f	odd	4	1		32.8.a.a	✓	1
16.f	odd	4	1		64.8.a.d		1
48.i	odd	4	1		288.8.a.b		1
48.i	odd	4	1		576.8.a.n		1
48.k	even	4	1		288.8.a.b		1
48.k	even	4	1		576.8.a.n		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.8.a.a	✓	1	16.e	even	4	1
32.8.a.a	✓	1	16.f	odd	4	1
64.8.a.d		1	16.e	even	4	1
64.8.a.d		1	16.f	odd	4	1
256.8.b.d		2	1.a	even	1	1	trivial
256.8.b.d		2	4.b	odd	2	1	CM
256.8.b.d		2	8.b	even	2	1	inner
256.8.b.d		2	8.d	odd	2	1	inner
288.8.a.b		1	48.i	odd	4	1
288.8.a.b		1	48.k	even	4	1
576.8.a.n		1	48.i	odd	4	1
576.8.a.n		1	48.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{8}^{\mathrm{new}}(256, [\chi])$:

$T_{3}$

$T_{7}$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$T^{2} + 3364$
$7$	$T^{2}$
$11$	$T^{2}$