Newform orbit 256.8.b.g

• Label: 256.8.b.g
• Level: $256$
• Weight: $8$
• Character orbit: 256.b
• Analytic conductor: $79.971$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 + 22*b * q^3 - 215*b * q^5 + 1224 * q^7 + 251 * q^9 + 1582*b * q^11 + 3059*b * q^13 + 18920 * q^15 - 16270 * q^17 - 2738*b * q^19 + 26928*b * q^21 - 1576 * q^23 - 106775 * q^25 + 53636*b * q^27 + 61419*b * q^29 + 251360 * q^31 - 139216 * q^33 - 263160*b * q^35 + 26169*b * q^37 - 269192 * q^39 + 319398 * q^41 - 355394*b * q^43 - 53965*b * q^45 + 284112 * q^47 + 674633 * q^49 - 357940*b * q^51 - 148031*b * q^53 + 1360520 * q^55 + 240944 * q^57 + 448774*b * q^59 - 442405*b * q^61 + 307224 * q^63 + 2630740 * q^65 + 2329846*b * q^67 - 34672*b * q^69 + 2710792 * q^71 + 5670854 * q^73 - 2349050*b * q^75 + 1936368*b * q^77 - 5124176 * q^79 - 4171031 * q^81 - 781778*b * q^83 + 3498050*b * q^85 - 5404872 * q^87 - 11605674 * q^89 + 3744216*b * q^91 + 5529920*b * q^93 - 2354680 * q^95 + 10931618 * q^97 + 397082*b * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 2448 * q^7 + 502 * q^9 + 37840 * q^15 - 32540 * q^17 - 3152 * q^23 - 213550 * q^25 + 502720 * q^31 - 278432 * q^33 - 538384 * q^39 + 638796 * q^41 + 568224 * q^47 + 1349266 * q^49 + 2721040 * q^55 + 481888 * q^57 + 614448 * q^63 + 5261480 * q^65 + 5421584 * q^71 + 11341708 * q^73 - 10248352 * q^79 - 8342062 * q^81 - 10809744 * q^87 - 23211348 * q^89 - 4709360 * q^95 + 21863236 * q^97

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

−

44.0000i

0

430.000i

0

1224.00

0

251.000

0

129.2

0

44.0000i

0

−

430.000i

0

1224.00

0

251.000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
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	256.8.b.g		2
4.b	odd	2	1		256.8.b.a		2
8.b	even	2	1	inner	256.8.b.g		2
8.d	odd	2	1		256.8.b.a		2
16.e	even	4	1		8.8.a.b	✓	1
16.e	even	4	1		64.8.a.b		1
16.f	odd	4	1		16.8.a.a		1
16.f	odd	4	1		64.8.a.f		1
48.i	odd	4	1		72.8.a.a		1
48.i	odd	4	1		576.8.a.y		1
48.k	even	4	1		144.8.a.a		1
48.k	even	4	1		576.8.a.z		1
80.i	odd	4	1		200.8.c.c		2
80.j	even	4	1		400.8.c.f		2
80.k	odd	4	1		400.8.a.p		1
80.q	even	4	1		200.8.a.b		1
80.s	even	4	1		400.8.c.f		2
80.t	odd	4	1		200.8.c.c		2
112.l	odd	4	1		392.8.a.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.8.a.b	✓	1	16.e	even	4	1
16.8.a.a		1	16.f	odd	4	1
64.8.a.b		1	16.e	even	4	1
64.8.a.f		1	16.f	odd	4	1
72.8.a.a		1	48.i	odd	4	1
144.8.a.a		1	48.k	even	4	1
200.8.a.b		1	80.q	even	4	1
200.8.c.c		2	80.i	odd	4	1
200.8.c.c		2	80.t	odd	4	1
256.8.b.a		2	4.b	odd	2	1
256.8.b.a		2	8.d	odd	2	1
256.8.b.g		2	1.a	even	1	1	trivial
256.8.b.g		2	8.b	even	2	1	inner
392.8.a.b		1	112.l	odd	4	1
400.8.a.p		1	80.k	odd	4	1
400.8.c.f		2	80.j	even	4	1
400.8.c.f		2	80.s	even	4	1
576.8.a.y		1	48.i	odd	4	1
576.8.a.z		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 $S_{8}^{\mathrm{new}}(256, [\chi])$:

$T_{3}^{2} + 1936$

$T_{7} - 1224$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 1936$
$5$	$T^{2} + 184900$
$7$	$(T - 1224)^{2}$
$11$	$T^{2} + 10010896$