# Properties

 Label 256.4.b.a Level $256$ Weight $4$ Character orbit 256.b Analytic conductor $15.104$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.1044889615$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{3} -2 i q^{5} -24 q^{7} + 11 q^{9} +O(q^{10})$$ $$q + 4 i q^{3} -2 i q^{5} -24 q^{7} + 11 q^{9} -44 i q^{11} -22 i q^{13} + 8 q^{15} + 50 q^{17} -44 i q^{19} -96 i q^{21} + 56 q^{23} + 121 q^{25} + 152 i q^{27} -198 i q^{29} -160 q^{31} + 176 q^{33} + 48 i q^{35} -162 i q^{37} + 88 q^{39} + 198 q^{41} + 52 i q^{43} -22 i q^{45} + 528 q^{47} + 233 q^{49} + 200 i q^{51} -242 i q^{53} -88 q^{55} + 176 q^{57} -668 i q^{59} -550 i q^{61} -264 q^{63} -44 q^{65} -188 i q^{67} + 224 i q^{69} -728 q^{71} -154 q^{73} + 484 i q^{75} + 1056 i q^{77} -656 q^{79} -311 q^{81} -236 i q^{83} -100 i q^{85} + 792 q^{87} -714 q^{89} + 528 i q^{91} -640 i q^{93} -88 q^{95} -478 q^{97} -484 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 48q^{7} + 22q^{9} + O(q^{10})$$ $$2q - 48q^{7} + 22q^{9} + 16q^{15} + 100q^{17} + 112q^{23} + 242q^{25} - 320q^{31} + 352q^{33} + 176q^{39} + 396q^{41} + 1056q^{47} + 466q^{49} - 176q^{55} + 352q^{57} - 528q^{63} - 88q^{65} - 1456q^{71} - 308q^{73} - 1312q^{79} - 622q^{81} + 1584q^{87} - 1428q^{89} - 176q^{95} - 956q^{97} + O(q^{100})$$

## 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 4.00000i 0 2.00000i 0 −24.0000 0 11.0000 0
129.2 0 4.00000i 0 2.00000i 0 −24.0000 0 11.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 256.4.b.a 2
4.b odd 2 1 256.4.b.g 2
8.b even 2 1 inner 256.4.b.a 2
8.d odd 2 1 256.4.b.g 2
16.e even 4 1 8.4.a.a 1
16.e even 4 1 64.4.a.d 1
16.f odd 4 1 16.4.a.a 1
16.f odd 4 1 64.4.a.b 1
48.i odd 4 1 72.4.a.c 1
48.i odd 4 1 576.4.a.k 1
48.k even 4 1 144.4.a.e 1
48.k even 4 1 576.4.a.j 1
80.i odd 4 1 200.4.c.e 2
80.j even 4 1 400.4.c.i 2
80.k odd 4 1 400.4.a.g 1
80.k odd 4 1 1600.4.a.bm 1
80.q even 4 1 200.4.a.g 1
80.q even 4 1 1600.4.a.o 1
80.s even 4 1 400.4.c.i 2
80.t odd 4 1 200.4.c.e 2
112.j even 4 1 784.4.a.e 1
112.l odd 4 1 392.4.a.e 1
112.w even 12 2 392.4.i.g 2
112.x odd 12 2 392.4.i.b 2
144.w odd 12 2 648.4.i.e 2
144.x even 12 2 648.4.i.h 2
176.i even 4 1 1936.4.a.l 1
176.l odd 4 1 968.4.a.a 1
208.p even 4 1 1352.4.a.a 1
240.bb even 4 1 1800.4.f.u 2
240.bf even 4 1 1800.4.f.u 2
240.bm odd 4 1 1800.4.a.d 1
272.r even 4 1 2312.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 16.e even 4 1
16.4.a.a 1 16.f odd 4 1
64.4.a.b 1 16.f odd 4 1
64.4.a.d 1 16.e even 4 1
72.4.a.c 1 48.i odd 4 1
144.4.a.e 1 48.k even 4 1
200.4.a.g 1 80.q even 4 1
200.4.c.e 2 80.i odd 4 1
200.4.c.e 2 80.t odd 4 1
256.4.b.a 2 1.a even 1 1 trivial
256.4.b.a 2 8.b even 2 1 inner
256.4.b.g 2 4.b odd 2 1
256.4.b.g 2 8.d odd 2 1
392.4.a.e 1 112.l odd 4 1
392.4.i.b 2 112.x odd 12 2
392.4.i.g 2 112.w even 12 2
400.4.a.g 1 80.k odd 4 1
400.4.c.i 2 80.j even 4 1
400.4.c.i 2 80.s even 4 1
576.4.a.j 1 48.k even 4 1
576.4.a.k 1 48.i odd 4 1
648.4.i.e 2 144.w odd 12 2
648.4.i.h 2 144.x even 12 2
784.4.a.e 1 112.j even 4 1
968.4.a.a 1 176.l odd 4 1
1352.4.a.a 1 208.p even 4 1
1600.4.a.o 1 80.q even 4 1
1600.4.a.bm 1 80.k odd 4 1
1800.4.a.d 1 240.bm odd 4 1
1800.4.f.u 2 240.bb even 4 1
1800.4.f.u 2 240.bf even 4 1
1936.4.a.l 1 176.i even 4 1
2312.4.a.a 1 272.r even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 16$$ $$T_{7} + 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 38 T^{2} + 729 T^{4}$$
$5$ $$1 - 246 T^{2} + 15625 T^{4}$$
$7$ $$( 1 + 24 T + 343 T^{2} )^{2}$$
$11$ $$1 - 726 T^{2} + 1771561 T^{4}$$
$13$ $$1 - 3910 T^{2} + 4826809 T^{4}$$
$17$ $$( 1 - 50 T + 4913 T^{2} )^{2}$$
$19$ $$1 - 11782 T^{2} + 47045881 T^{4}$$
$23$ $$( 1 - 56 T + 12167 T^{2} )^{2}$$
$29$ $$1 - 9574 T^{2} + 594823321 T^{4}$$
$31$ $$( 1 + 160 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 75062 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 - 198 T + 68921 T^{2} )^{2}$$
$43$ $$1 - 156310 T^{2} + 6321363049 T^{4}$$
$47$ $$( 1 - 528 T + 103823 T^{2} )^{2}$$
$53$ $$1 - 239190 T^{2} + 22164361129 T^{4}$$
$59$ $$1 + 35466 T^{2} + 42180533641 T^{4}$$
$61$ $$1 - 151462 T^{2} + 51520374361 T^{4}$$
$67$ $$1 - 566182 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 + 728 T + 357911 T^{2} )^{2}$$
$73$ $$( 1 + 154 T + 389017 T^{2} )^{2}$$
$79$ $$( 1 + 656 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 1087878 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 + 714 T + 704969 T^{2} )^{2}$$
$97$ $$( 1 + 478 T + 912673 T^{2} )^{2}$$