Newform orbit 256.8.a.i

• Label: 256.8.a.i
• Level: $256$
• Weight: $8$
• Character orbit: 256.a
• Self dual: yes
• 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.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{435})$
Defining polynomial:	$x^{2} - 435$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{3}\cdot 3$
Twist minimal:	no (minimal twist has level 64)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 24\sqrt{435}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 30 * q^3 + b * q^5 + 2*b * q^7 - 1287 * q^9 - 4758 * q^11 + 13*b * q^13 + 30*b * q^15 - 13830 * q^17 + 19006 * q^19 + 60*b * q^21 + 182*b * q^23 + 172435 * q^25 - 104220 * q^27 + 65*b * q^29 + 520*b * q^31 - 142740 * q^33 + 501120 * q^35 - 383*b * q^37 + 390*b * q^39 + 454038 * q^41 - 452870 * q^43 - 1287*b * q^45 - 1724*b * q^47 + 178697 * q^49 - 414900 * q^51 - 91*b * q^53 - 4758*b * q^55 + 570180 * q^57 + 2320266 * q^59 + 5265*b * q^61 - 2574*b * q^63 + 3257280 * q^65 - 309010 * q^67 + 5460*b * q^69 + 4930*b * q^71 - 5883410 * q^73 + 5173050 * q^75 - 9516*b * q^77 + 4940*b * q^79 - 311931 * q^81 + 3293550 * q^83 - 13830*b * q^85 + 1950*b * q^87 - 3675906 * q^89 + 6514560 * q^91 + 15600*b * q^93 + 19006*b * q^95 - 11233430 * q^97 + 6123546 * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 60 * q^3 - 2574 * q^9 - 9516 * q^11 - 27660 * q^17 + 38012 * q^19 + 344870 * q^25 - 208440 * q^27 - 285480 * q^33 + 1002240 * q^35 + 908076 * q^41 - 905740 * q^43 + 357394 * q^49 - 829800 * q^51 + 1140360 * q^57 + 4640532 * q^59 + 6514560 * q^65 - 618020 * q^67 - 11766820 * q^73 + 10346100 * q^75 - 623862 * q^81 + 6587100 * q^83 - 7351812 * q^89 + 13029120 * q^91 - 22466860 * q^97 + 12247092 * q^99

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

−20.8567
20.8567

0

30.0000

0

−500.560

0

−1001.12

0

−1287.00

0

1.2

0

30.0000

0

500.560

0

1001.12

0

−1287.00

0

Atkin-Lehner signs

$p$	Sign
$2$	$+1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
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.a.i		2
4.b	odd	2	1		256.8.a.e		2
8.b	even	2	1		256.8.a.e		2
8.d	odd	2	1	inner	256.8.a.i		2
16.e	even	4	2		64.8.b.b	✓	4
16.f	odd	4	2		64.8.b.b	✓	4
48.i	odd	4	2		576.8.d.e		4
48.k	even	4	2		576.8.d.e		4

    

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{8}^{\mathrm{new}}(\Gamma_0(256))$:

$T_{3} - 30$

$T_{5}^{2} - 250560$

$T_{7}^{2} - 1002240$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$(T - 30)^{2}$
$5$	$T^{2} - 250560$
$7$	$T^{2} - 1002240$
$11$	$(T + 4758)^{2}$