# Properties

 Label 256.4.b.b 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 128) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} -6 i q^{5} -20 q^{7} + 23 q^{9} +O(q^{10})$$ $$q + 2 i q^{3} -6 i q^{5} -20 q^{7} + 23 q^{9} -14 i q^{11} + 54 i q^{13} + 12 q^{15} -66 q^{17} + 162 i q^{19} -40 i q^{21} -172 q^{23} + 89 q^{25} + 100 i q^{27} -2 i q^{29} -128 q^{31} + 28 q^{33} + 120 i q^{35} -158 i q^{37} -108 q^{39} -202 q^{41} + 298 i q^{43} -138 i q^{45} -408 q^{47} + 57 q^{49} -132 i q^{51} + 690 i q^{53} -84 q^{55} -324 q^{57} + 322 i q^{59} -298 i q^{61} -460 q^{63} + 324 q^{65} + 202 i q^{67} -344 i q^{69} + 700 q^{71} + 418 q^{73} + 178 i q^{75} + 280 i q^{77} + 744 q^{79} + 421 q^{81} -678 i q^{83} + 396 i q^{85} + 4 q^{87} + 82 q^{89} -1080 i q^{91} -256 i q^{93} + 972 q^{95} -1122 q^{97} -322 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 40q^{7} + 46q^{9} + O(q^{10})$$ $$2q - 40q^{7} + 46q^{9} + 24q^{15} - 132q^{17} - 344q^{23} + 178q^{25} - 256q^{31} + 56q^{33} - 216q^{39} - 404q^{41} - 816q^{47} + 114q^{49} - 168q^{55} - 648q^{57} - 920q^{63} + 648q^{65} + 1400q^{71} + 836q^{73} + 1488q^{79} + 842q^{81} + 8q^{87} + 164q^{89} + 1944q^{95} - 2244q^{97} + O(q^{100})$$

## 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 2.00000i 0 6.00000i 0 −20.0000 0 23.0000 0
129.2 0 2.00000i 0 6.00000i 0 −20.0000 0 23.0000 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.b 2
4.b odd 2 1 256.4.b.f 2
8.b even 2 1 inner 256.4.b.b 2
8.d odd 2 1 256.4.b.f 2
16.e even 4 1 128.4.a.a 1
16.e even 4 1 128.4.a.d yes 1
16.f odd 4 1 128.4.a.b yes 1
16.f odd 4 1 128.4.a.c yes 1
48.i odd 4 1 1152.4.a.d 1
48.i odd 4 1 1152.4.a.j 1
48.k even 4 1 1152.4.a.c 1
48.k even 4 1 1152.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.a 1 16.e even 4 1
128.4.a.b yes 1 16.f odd 4 1
128.4.a.c yes 1 16.f odd 4 1
128.4.a.d yes 1 16.e even 4 1
256.4.b.b 2 1.a even 1 1 trivial
256.4.b.b 2 8.b even 2 1 inner
256.4.b.f 2 4.b odd 2 1
256.4.b.f 2 8.d odd 2 1
1152.4.a.c 1 48.k even 4 1
1152.4.a.d 1 48.i odd 4 1
1152.4.a.i 1 48.k even 4 1
1152.4.a.j 1 48.i odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{7} + 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$36 + T^{2}$$
$7$ $$( 20 + T )^{2}$$
$11$ $$196 + T^{2}$$
$13$ $$2916 + T^{2}$$
$17$ $$( 66 + T )^{2}$$
$19$ $$26244 + T^{2}$$
$23$ $$( 172 + T )^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$( 128 + T )^{2}$$
$37$ $$24964 + T^{2}$$
$41$ $$( 202 + T )^{2}$$
$43$ $$88804 + T^{2}$$
$47$ $$( 408 + T )^{2}$$
$53$ $$476100 + T^{2}$$
$59$ $$103684 + T^{2}$$
$61$ $$88804 + T^{2}$$
$67$ $$40804 + T^{2}$$
$71$ $$( -700 + T )^{2}$$
$73$ $$( -418 + T )^{2}$$
$79$ $$( -744 + T )^{2}$$
$83$ $$459684 + T^{2}$$
$89$ $$( -82 + T )^{2}$$
$97$ $$( 1122 + T )^{2}$$