# Properties

 Label 256.5.c.f Level $256$ Weight $5$ Character orbit 256.c Analytic conductor $26.463$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 256.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.4627105495$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-6})$$ Defining polynomial: $$x^{2} + 6$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 $$\beta = 4\sqrt{-6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 8 q^{5} -8 \beta q^{7} -15 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 8 q^{5} -8 \beta q^{7} -15 q^{9} + 11 \beta q^{11} -216 q^{13} + 8 \beta q^{15} -162 q^{17} + 45 \beta q^{19} + 768 q^{21} + 72 \beta q^{23} -561 q^{25} + 66 \beta q^{27} -1304 q^{29} -64 \beta q^{31} -1056 q^{33} -64 \beta q^{35} + 1512 q^{37} -216 \beta q^{39} -1890 q^{41} -297 \beta q^{43} -120 q^{45} + 144 \beta q^{47} -3743 q^{49} -162 \beta q^{51} -1976 q^{53} + 88 \beta q^{55} -4320 q^{57} + 231 \beta q^{59} + 2376 q^{61} + 120 \beta q^{63} -1728 q^{65} -171 \beta q^{67} -6912 q^{69} + 792 \beta q^{71} -2750 q^{73} -561 \beta q^{75} + 8448 q^{77} -816 \beta q^{79} -7551 q^{81} + 953 \beta q^{83} -1296 q^{85} -1304 \beta q^{87} -2430 q^{89} + 1728 \beta q^{91} + 6144 q^{93} + 360 \beta q^{95} + 7454 q^{97} -165 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 16q^{5} - 30q^{9} + O(q^{10})$$ $$2q + 16q^{5} - 30q^{9} - 432q^{13} - 324q^{17} + 1536q^{21} - 1122q^{25} - 2608q^{29} - 2112q^{33} + 3024q^{37} - 3780q^{41} - 240q^{45} - 7486q^{49} - 3952q^{53} - 8640q^{57} + 4752q^{61} - 3456q^{65} - 13824q^{69} - 5500q^{73} + 16896q^{77} - 15102q^{81} - 2592q^{85} - 4860q^{89} + 12288q^{93} + 14908q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
255.1
 − 2.44949i 2.44949i
0 9.79796i 0 8.00000 0 78.3837i 0 −15.0000 0
255.2 0 9.79796i 0 8.00000 0 78.3837i 0 −15.0000 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.5.c.f 2
4.b odd 2 1 inner 256.5.c.f 2
8.b even 2 1 256.5.c.c 2
8.d odd 2 1 256.5.c.c 2
16.e even 4 2 128.5.d.c 4
16.f odd 4 2 128.5.d.c 4
48.i odd 4 2 1152.5.b.i 4
48.k even 4 2 1152.5.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.5.d.c 4 16.e even 4 2
128.5.d.c 4 16.f odd 4 2
256.5.c.c 2 8.b even 2 1
256.5.c.c 2 8.d odd 2 1
256.5.c.f 2 1.a even 1 1 trivial
256.5.c.f 2 4.b odd 2 1 inner
1152.5.b.i 4 48.i odd 4 2
1152.5.b.i 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 $$S_{5}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 96$$ $$T_{5} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 66 T^{2} + 6561 T^{4}$$
$5$ $$( 1 - 8 T + 625 T^{2} )^{2}$$
$7$ $$1 + 1342 T^{2} + 5764801 T^{4}$$
$11$ $$1 - 17666 T^{2} + 214358881 T^{4}$$
$13$ $$( 1 + 216 T + 28561 T^{2} )^{2}$$
$17$ $$( 1 + 162 T + 83521 T^{2} )^{2}$$
$19$ $$1 - 66242 T^{2} + 16983563041 T^{4}$$
$23$ $$1 - 62018 T^{2} + 78310985281 T^{4}$$
$29$ $$( 1 + 1304 T + 707281 T^{2} )^{2}$$
$31$ $$1 - 1453826 T^{2} + 852891037441 T^{4}$$
$37$ $$( 1 - 1512 T + 1874161 T^{2} )^{2}$$
$41$ $$( 1 + 1890 T + 2825761 T^{2} )^{2}$$
$43$ $$1 + 1630462 T^{2} + 11688200277601 T^{4}$$
$47$ $$1 - 7768706 T^{2} + 23811286661761 T^{4}$$
$53$ $$( 1 + 1976 T + 7890481 T^{2} )^{2}$$
$59$ $$1 - 19112066 T^{2} + 146830437604321 T^{4}$$
$61$ $$( 1 - 2376 T + 13845841 T^{2} )^{2}$$
$67$ $$1 - 37495106 T^{2} + 406067677556641 T^{4}$$
$71$ $$1 + 9393982 T^{2} + 645753531245761 T^{4}$$
$73$ $$( 1 + 2750 T + 28398241 T^{2} )^{2}$$
$79$ $$1 - 13977986 T^{2} + 1517108809906561 T^{4}$$
$83$ $$1 - 7728578 T^{2} + 2252292232139041 T^{4}$$
$89$ $$( 1 + 2430 T + 62742241 T^{2} )^{2}$$
$97$ $$( 1 - 7454 T + 88529281 T^{2} )^{2}$$