# Properties

 Label 256.4.a.l Level $256$ Weight $4$ Character orbit 256.a Self dual yes 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.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.1044889615$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 2 \beta q^{5} + 8 q^{7} + q^{9}+O(q^{10})$$ q + b * q^3 + 2*b * q^5 + 8 * q^7 + q^9 $$q + \beta q^{3} + 2 \beta q^{5} + 8 q^{7} + q^{9} + 3 \beta q^{11} + 10 \beta q^{13} + 56 q^{15} - 14 q^{17} - 7 \beta q^{19} + 8 \beta q^{21} + 152 q^{23} - 13 q^{25} - 26 \beta q^{27} - 30 \beta q^{29} + 224 q^{31} + 84 q^{33} + 16 \beta q^{35} - 46 \beta q^{37} + 280 q^{39} + 70 q^{41} + 83 \beta q^{43} + 2 \beta q^{45} + 336 q^{47} - 279 q^{49} - 14 \beta q^{51} - 6 \beta q^{53} + 168 q^{55} - 196 q^{57} - 101 \beta q^{59} + 18 \beta q^{61} + 8 q^{63} + 560 q^{65} + 33 \beta q^{67} + 152 \beta q^{69} + 72 q^{71} + 294 q^{73} - 13 \beta q^{75} + 24 \beta q^{77} - 464 q^{79} - 755 q^{81} - 103 \beta q^{83} - 28 \beta q^{85} - 840 q^{87} - 266 q^{89} + 80 \beta q^{91} + 224 \beta q^{93} - 392 q^{95} + 994 q^{97} + 3 \beta q^{99} +O(q^{100})$$ q + b * q^3 + 2*b * q^5 + 8 * q^7 + q^9 + 3*b * q^11 + 10*b * q^13 + 56 * q^15 - 14 * q^17 - 7*b * q^19 + 8*b * q^21 + 152 * q^23 - 13 * q^25 - 26*b * q^27 - 30*b * q^29 + 224 * q^31 + 84 * q^33 + 16*b * q^35 - 46*b * q^37 + 280 * q^39 + 70 * q^41 + 83*b * q^43 + 2*b * q^45 + 336 * q^47 - 279 * q^49 - 14*b * q^51 - 6*b * q^53 + 168 * q^55 - 196 * q^57 - 101*b * q^59 + 18*b * q^61 + 8 * q^63 + 560 * q^65 + 33*b * q^67 + 152*b * q^69 + 72 * q^71 + 294 * q^73 - 13*b * q^75 + 24*b * q^77 - 464 * q^79 - 755 * q^81 - 103*b * q^83 - 28*b * q^85 - 840 * q^87 - 266 * q^89 + 80*b * q^91 + 224*b * q^93 - 392 * q^95 + 994 * q^97 + 3*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 16 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 16 * q^7 + 2 * q^9 $$2 q + 16 q^{7} + 2 q^{9} + 112 q^{15} - 28 q^{17} + 304 q^{23} - 26 q^{25} + 448 q^{31} + 168 q^{33} + 560 q^{39} + 140 q^{41} + 672 q^{47} - 558 q^{49} + 336 q^{55} - 392 q^{57} + 16 q^{63} + 1120 q^{65} + 144 q^{71} + 588 q^{73} - 928 q^{79} - 1510 q^{81} - 1680 q^{87} - 532 q^{89} - 784 q^{95} + 1988 q^{97}+O(q^{100})$$ 2 * q + 16 * q^7 + 2 * q^9 + 112 * q^15 - 28 * q^17 + 304 * q^23 - 26 * q^25 + 448 * q^31 + 168 * q^33 + 560 * q^39 + 140 * q^41 + 672 * q^47 - 558 * q^49 + 336 * q^55 - 392 * q^57 + 16 * q^63 + 1120 * q^65 + 144 * q^71 + 588 * q^73 - 928 * q^79 - 1510 * q^81 - 1680 * q^87 - 532 * q^89 - 784 * q^95 + 1988 * q^97

## 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
 −2.64575 2.64575
0 −5.29150 0 −10.5830 0 8.00000 0 1.00000 0
1.2 0 5.29150 0 10.5830 0 8.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$

## 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.a.l 2
3.b odd 2 1 2304.4.a.bn 2
4.b odd 2 1 256.4.a.j 2
8.b even 2 1 inner 256.4.a.l 2
8.d odd 2 1 256.4.a.j 2
12.b even 2 1 2304.4.a.v 2
16.e even 4 2 8.4.b.a 2
16.f odd 4 2 32.4.b.a 2
24.f even 2 1 2304.4.a.v 2
24.h odd 2 1 2304.4.a.bn 2
48.i odd 4 2 72.4.d.b 2
48.k even 4 2 288.4.d.a 2
80.i odd 4 2 200.4.f.a 4
80.j even 4 2 800.4.f.a 4
80.k odd 4 2 800.4.d.a 2
80.q even 4 2 200.4.d.a 2
80.s even 4 2 800.4.f.a 4
80.t odd 4 2 200.4.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 16.e even 4 2
32.4.b.a 2 16.f odd 4 2
72.4.d.b 2 48.i odd 4 2
200.4.d.a 2 80.q even 4 2
200.4.f.a 4 80.i odd 4 2
200.4.f.a 4 80.t odd 4 2
256.4.a.j 2 4.b odd 2 1
256.4.a.j 2 8.d odd 2 1
256.4.a.l 2 1.a even 1 1 trivial
256.4.a.l 2 8.b even 2 1 inner
288.4.d.a 2 48.k even 4 2
800.4.d.a 2 80.k odd 4 2
800.4.f.a 4 80.j even 4 2
800.4.f.a 4 80.s even 4 2
2304.4.a.v 2 12.b even 2 1
2304.4.a.v 2 24.f even 2 1
2304.4.a.bn 2 3.b odd 2 1
2304.4.a.bn 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(256))$$:

 $$T_{3}^{2} - 28$$ T3^2 - 28 $$T_{5}^{2} - 112$$ T5^2 - 112 $$T_{7} - 8$$ T7 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 28$$
$5$ $$T^{2} - 112$$
$7$ $$(T - 8)^{2}$$
$11$ $$T^{2} - 252$$
$13$ $$T^{2} - 2800$$
$17$ $$(T + 14)^{2}$$
$19$ $$T^{2} - 1372$$
$23$ $$(T - 152)^{2}$$
$29$ $$T^{2} - 25200$$
$31$ $$(T - 224)^{2}$$
$37$ $$T^{2} - 59248$$
$41$ $$(T - 70)^{2}$$
$43$ $$T^{2} - 192892$$
$47$ $$(T - 336)^{2}$$
$53$ $$T^{2} - 1008$$
$59$ $$T^{2} - 285628$$
$61$ $$T^{2} - 9072$$
$67$ $$T^{2} - 30492$$
$71$ $$(T - 72)^{2}$$
$73$ $$(T - 294)^{2}$$
$79$ $$(T + 464)^{2}$$
$83$ $$T^{2} - 297052$$
$89$ $$(T + 266)^{2}$$
$97$ $$(T - 994)^{2}$$