# Properties

 Label 800.4.a.z Level $800$ Weight $4$ Character orbit 800.a Self dual yes Analytic conductor $47.202$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$47.2015280046$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.37485.2 Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + 12x - 3$$ x^4 - x^3 - 8*x^2 + 12*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + ( - 3 \beta_{2} + 19) q^{9}+O(q^{10})$$ q - b1 * q^3 + (b3 + b1) * q^7 + (-3*b2 + 19) * q^9 $$q - \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + ( - 3 \beta_{2} + 19) q^{9} + 2 \beta_{3} q^{11} + ( - 2 \beta_{2} + 52) q^{13} - 8 \beta_{2} q^{17} + (2 \beta_{3} - 16 \beta_1) q^{19} + (17 \beta_{2} - 34) q^{21} + (\beta_{3} + 11 \beta_1) q^{23} + ( - 6 \beta_{3} - 10 \beta_1) q^{27} + ( - 16 \beta_{2} - 78) q^{29} - 16 \beta_1 q^{31} + (28 \beta_{2} + 24) q^{33} + (14 \beta_{2} + 68) q^{37} + ( - 4 \beta_{3} - 64 \beta_1) q^{39} + ( - 13 \beta_{2} - 24) q^{41} + ( - 6 \beta_{3} - 49 \beta_1) q^{43} + (9 \beta_{3} + 37 \beta_1) q^{47} + (17 \beta_{2} + 303) q^{49} + ( - 16 \beta_{3} - 48 \beta_1) q^{51} + ( - 30 \beta_{2} + 12) q^{53} + ( - 20 \beta_{2} + 760) q^{57} + ( - 10 \beta_{3} - 48 \beta_1) q^{59} + ( - 27 \beta_{2} + 264) q^{61} + (7 \beta_{3} + 109 \beta_1) q^{63} + (10 \beta_{3} - 51 \beta_1) q^{67} + (47 \beta_{2} - 494) q^{69} + ( - 4 \beta_{3} + 16 \beta_1) q^{71} + ( - 44 \beta_{2} + 360) q^{73} + (68 \beta_{2} + 1224) q^{77} + (24 \beta_{3} + 80 \beta_1) q^{79} + ( - 33 \beta_{2} - 125) q^{81} + (22 \beta_{3} - 117 \beta_1) q^{83} + ( - 32 \beta_{3} - 18 \beta_1) q^{87} + (80 \beta_{2} - 10) q^{89} + (44 \beta_{3} + 112 \beta_1) q^{91} + ( - 48 \beta_{2} + 736) q^{93} + 1136 q^{97} + (2 \beta_{3} + 144 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^3 + (b3 + b1) * q^7 + (-3*b2 + 19) * q^9 + 2*b3 * q^11 + (-2*b2 + 52) * q^13 - 8*b2 * q^17 + (2*b3 - 16*b1) * q^19 + (17*b2 - 34) * q^21 + (b3 + 11*b1) * q^23 + (-6*b3 - 10*b1) * q^27 + (-16*b2 - 78) * q^29 - 16*b1 * q^31 + (28*b2 + 24) * q^33 + (14*b2 + 68) * q^37 + (-4*b3 - 64*b1) * q^39 + (-13*b2 - 24) * q^41 + (-6*b3 - 49*b1) * q^43 + (9*b3 + 37*b1) * q^47 + (17*b2 + 303) * q^49 + (-16*b3 - 48*b1) * q^51 + (-30*b2 + 12) * q^53 + (-20*b2 + 760) * q^57 + (-10*b3 - 48*b1) * q^59 + (-27*b2 + 264) * q^61 + (7*b3 + 109*b1) * q^63 + (10*b3 - 51*b1) * q^67 + (47*b2 - 494) * q^69 + (-4*b3 + 16*b1) * q^71 + (-44*b2 + 360) * q^73 + (68*b2 + 1224) * q^77 + (24*b3 + 80*b1) * q^79 + (-33*b2 - 125) * q^81 + (22*b3 - 117*b1) * q^83 + (-32*b3 - 18*b1) * q^87 + (80*b2 - 10) * q^89 + (44*b3 + 112*b1) * q^91 + (-48*b2 + 736) * q^93 + 1136 * q^97 + (2*b3 + 144*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 76 q^{9}+O(q^{10})$$ 4 * q + 76 * q^9 $$4 q + 76 q^{9} + 208 q^{13} - 136 q^{21} - 312 q^{29} + 96 q^{33} + 272 q^{37} - 96 q^{41} + 1212 q^{49} + 48 q^{53} + 3040 q^{57} + 1056 q^{61} - 1976 q^{69} + 1440 q^{73} + 4896 q^{77} - 500 q^{81} - 40 q^{89} + 2944 q^{93} + 4544 q^{97}+O(q^{100})$$ 4 * q + 76 * q^9 + 208 * q^13 - 136 * q^21 - 312 * q^29 + 96 * q^33 + 272 * q^37 - 96 * q^41 + 1212 * q^49 + 48 * q^53 + 3040 * q^57 + 1056 * q^61 - 1976 * q^69 + 1440 * q^73 + 4896 * q^77 - 500 * q^81 - 40 * q^89 + 2944 * q^93 + 4544 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8x^{2} + 12x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{3} - 18\nu + 10$$ 2*v^3 - 18*v + 10 $$\beta_{2}$$ $$=$$ $$4\nu^{3} - 28\nu + 18$$ 4*v^3 - 28*v + 18 $$\beta_{3}$$ $$=$$ $$8\nu^{2} + 8\nu - 36$$ 8*v^2 + 8*v - 36
 $$\nu$$ $$=$$ $$( \beta_{2} - 2\beta _1 + 2 ) / 8$$ (b2 - 2*b1 + 2) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - \beta_{2} + 2\beta _1 + 34 ) / 8$$ (b3 - b2 + 2*b1 + 34) / 8 $$\nu^{3}$$ $$=$$ $$( 9\beta_{2} - 14\beta _1 - 22 ) / 8$$ (9*b2 - 14*b1 - 22) / 8

## 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
 −3.03888 0.320221 2.47107 1.24759
0 −8.57295 0 0 0 22.1403 0 46.4955 0
1.2 0 −4.30169 0 0 0 −28.3162 0 −8.49545 0
1.3 0 4.30169 0 0 0 28.3162 0 −8.49545 0
1.4 0 8.57295 0 0 0 −22.1403 0 46.4955 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
$$2$$ $$-1$$
$$5$$ $$-1$$

## 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 800.4.a.z 4
4.b odd 2 1 inner 800.4.a.z 4
5.b even 2 1 800.4.a.y 4
5.c odd 4 2 160.4.c.d 8
8.b even 2 1 1600.4.a.cu 4
8.d odd 2 1 1600.4.a.cu 4
15.e even 4 2 1440.4.f.k 8
20.d odd 2 1 800.4.a.y 4
20.e even 4 2 160.4.c.d 8
40.e odd 2 1 1600.4.a.cv 4
40.f even 2 1 1600.4.a.cv 4
40.i odd 4 2 320.4.c.j 8
40.k even 4 2 320.4.c.j 8
60.l odd 4 2 1440.4.f.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.d 8 5.c odd 4 2
160.4.c.d 8 20.e even 4 2
320.4.c.j 8 40.i odd 4 2
320.4.c.j 8 40.k even 4 2
800.4.a.y 4 5.b even 2 1
800.4.a.y 4 20.d odd 2 1
800.4.a.z 4 1.a even 1 1 trivial
800.4.a.z 4 4.b odd 2 1 inner
1440.4.f.k 8 15.e even 4 2
1440.4.f.k 8 60.l odd 4 2
1600.4.a.cu 4 8.b even 2 1
1600.4.a.cu 4 8.d odd 2 1
1600.4.a.cv 4 40.e odd 2 1
1600.4.a.cv 4 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(800))$$:

 $$T_{3}^{4} - 92T_{3}^{2} + 1360$$ T3^4 - 92*T3^2 + 1360 $$T_{11}^{4} - 4992T_{11}^{2} + 3133440$$ T11^4 - 4992*T11^2 + 3133440 $$T_{13}^{2} - 104T_{13} + 2368$$ T13^2 - 104*T13 + 2368

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 92T^{2} + 1360$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 1292 T^{2} + 393040$$
$11$ $$T^{4} - 4992 T^{2} + \cdots + 3133440$$
$13$ $$(T^{2} - 104 T + 2368)^{2}$$
$17$ $$(T^{2} - 5376)^{2}$$
$19$ $$T^{4} - 30080 T^{2} + \cdots + 217600000$$
$23$ $$T^{4} - 11852 T^{2} + \cdots + 2514640$$
$29$ $$(T^{2} + 156 T - 15420)^{2}$$
$31$ $$T^{4} - 23552 T^{2} + \cdots + 89128960$$
$37$ $$(T^{2} - 136 T - 11840)^{2}$$
$41$ $$(T^{2} + 48 T - 13620)^{2}$$
$43$ $$T^{4} - 251708 T^{2} + \cdots + 57154000$$
$47$ $$T^{4} - 211052 T^{2} + \cdots + 3485953360$$
$53$ $$(T^{2} - 24 T - 75456)^{2}$$
$59$ $$T^{4} - 313728 T^{2} + \cdots + 4289679360$$
$61$ $$(T^{2} - 528 T + 8460)^{2}$$
$67$ $$T^{4} - 388572 T^{2} + \cdots + 27064708560$$
$71$ $$T^{4} - 46592 T^{2} + \cdots + 272957440$$
$73$ $$(T^{2} - 720 T - 33024)^{2}$$
$79$ $$T^{4} - 1215488 T^{2} + \cdots + 196885872640$$
$83$ $$T^{4} - 1986972 T^{2} + \cdots + 739915650000$$
$89$ $$(T^{2} + 20 T - 537500)^{2}$$
$97$ $$(T - 1136)^{4}$$