# Properties

 Label 800.4.c.i Level $800$ Weight $4$ Character orbit 800.c 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.2015280046$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 2 \beta_1) q^{3} + ( - 5 \beta_{2} - 2 \beta_1) q^{7} + (4 \beta_{3} - 13) q^{9}+O(q^{10})$$ q + (-b2 + 2*b1) * q^3 + (-5*b2 - 2*b1) * q^7 + (4*b3 - 13) * q^9 $$q + ( - \beta_{2} + 2 \beta_1) q^{3} + ( - 5 \beta_{2} - 2 \beta_1) q^{7} + (4 \beta_{3} - 13) q^{9} + (3 \beta_{3} - 32) q^{11} + (8 \beta_{2} - 3 \beta_1) q^{13} + (24 \beta_{2} + \beta_1) q^{17} + (4 \beta_{3} + 104) q^{19} + (8 \beta_{3} - 104) q^{21} + (11 \beta_{2} + 30 \beta_1) q^{23} + (18 \beta_{2} - 68 \beta_1) q^{27} + ( - 8 \beta_{3} + 146) q^{29} + ( - 3 \beta_{3} - 88) q^{31} + (56 \beta_{2} - 136 \beta_1) q^{33} + ( - 16 \beta_{2} - 89 \beta_1) q^{37} + ( - 19 \beta_{3} + 216) q^{39} + (20 \beta_{3} + 50) q^{41} + ( - 37 \beta_{2} - 94 \beta_1) q^{43} + ( - 13 \beta_{2} + 70 \beta_1) q^{47} + ( - 20 \beta_{3} - 273) q^{49} + ( - 47 \beta_{3} + 568) q^{51} + (24 \beta_{2} - 79 \beta_1) q^{53} + ( - 72 \beta_{2} + 112 \beta_1) q^{57} + (10 \beta_{3} + 360) q^{59} + (16 \beta_{3} - 634) q^{61} + (33 \beta_{2} - 454 \beta_1) q^{63} + (47 \beta_{2} + 186 \beta_1) q^{67} + (8 \beta_{3} + 24) q^{69} + (109 \beta_{3} + 24) q^{71} + ( - 120 \beta_{2} + 235 \beta_1) q^{73} + (136 \beta_{2} - 296 \beta_1) q^{77} + (86 \beta_{3} + 16) q^{79} + (4 \beta_{3} + 625) q^{81} + (47 \beta_{2} - 398 \beta_1) q^{83} + ( - 210 \beta_{2} + 484 \beta_1) q^{87} + (24 \beta_{3} + 390) q^{89} + (\beta_{3} + 936) q^{91} + (64 \beta_{2} - 104 \beta_1) q^{93} + (40 \beta_{2} + 305 \beta_1) q^{97} + ( - 167 \beta_{3} + 1568) q^{99}+O(q^{100})$$ q + (-b2 + 2*b1) * q^3 + (-5*b2 - 2*b1) * q^7 + (4*b3 - 13) * q^9 + (3*b3 - 32) * q^11 + (8*b2 - 3*b1) * q^13 + (24*b2 + b1) * q^17 + (4*b3 + 104) * q^19 + (8*b3 - 104) * q^21 + (11*b2 + 30*b1) * q^23 + (18*b2 - 68*b1) * q^27 + (-8*b3 + 146) * q^29 + (-3*b3 - 88) * q^31 + (56*b2 - 136*b1) * q^33 + (-16*b2 - 89*b1) * q^37 + (-19*b3 + 216) * q^39 + (20*b3 + 50) * q^41 + (-37*b2 - 94*b1) * q^43 + (-13*b2 + 70*b1) * q^47 + (-20*b3 - 273) * q^49 + (-47*b3 + 568) * q^51 + (24*b2 - 79*b1) * q^53 + (-72*b2 + 112*b1) * q^57 + (10*b3 + 360) * q^59 + (16*b3 - 634) * q^61 + (33*b2 - 454*b1) * q^63 + (47*b2 + 186*b1) * q^67 + (8*b3 + 24) * q^69 + (109*b3 + 24) * q^71 + (-120*b2 + 235*b1) * q^73 + (136*b2 - 296*b1) * q^77 + (86*b3 + 16) * q^79 + (4*b3 + 625) * q^81 + (47*b2 - 398*b1) * q^83 + (-210*b2 + 484*b1) * q^87 + (24*b3 + 390) * q^89 + (b3 + 936) * q^91 + (64*b2 - 104*b1) * q^93 + (40*b2 + 305*b1) * q^97 + (-167*b3 + 1568) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 52 q^{9}+O(q^{10})$$ 4 * q - 52 * q^9 $$4 q - 52 q^{9} - 128 q^{11} + 416 q^{19} - 416 q^{21} + 584 q^{29} - 352 q^{31} + 864 q^{39} + 200 q^{41} - 1092 q^{49} + 2272 q^{51} + 1440 q^{59} - 2536 q^{61} + 96 q^{69} + 96 q^{71} + 64 q^{79} + 2500 q^{81} + 1560 q^{89} + 3744 q^{91} + 6272 q^{99}+O(q^{100})$$ 4 * q - 52 * q^9 - 128 * q^11 + 416 * q^19 - 416 * q^21 + 584 * q^29 - 352 * q^31 + 864 * q^39 + 200 * q^41 - 1092 * q^49 + 2272 * q^51 + 1440 * q^59 - 2536 * q^61 + 96 * q^69 + 96 * q^71 + 64 * q^79 + 2500 * q^81 + 1560 * q^89 + 3744 * q^91 + 6272 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{2} ) / 3$$ (2*v^2) / 3 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} + 6\nu ) / 3$$ (2*v^3 + 6*v) / 3 $$\beta_{3}$$ $$=$$ $$( -4\nu^{3} + 12\nu ) / 3$$ (-4*v^3 + 12*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_{2} ) / 8$$ (b3 + 2*b2) / 8 $$\nu^{2}$$ $$=$$ $$( 3\beta_1 ) / 2$$ (3*b1) / 2 $$\nu^{3}$$ $$=$$ $$( -3\beta_{3} + 6\beta_{2} ) / 8$$ (-3*b3 + 6*b2) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −1.22474 + 1.22474i 1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i
0 8.89898i 0 0 0 20.4949i 0 −52.1918 0
449.2 0 0.898979i 0 0 0 28.4949i 0 26.1918 0
449.3 0 0.898979i 0 0 0 28.4949i 0 26.1918 0
449.4 0 8.89898i 0 0 0 20.4949i 0 −52.1918 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.c.i 4
4.b odd 2 1 800.4.c.k 4
5.b even 2 1 inner 800.4.c.i 4
5.c odd 4 1 160.4.a.c 2
5.c odd 4 1 800.4.a.s 2
15.e even 4 1 1440.4.a.t 2
20.d odd 2 1 800.4.c.k 4
20.e even 4 1 160.4.a.g yes 2
20.e even 4 1 800.4.a.m 2
40.i odd 4 1 320.4.a.s 2
40.i odd 4 1 1600.4.a.cd 2
40.k even 4 1 320.4.a.o 2
40.k even 4 1 1600.4.a.cn 2
60.l odd 4 1 1440.4.a.x 2
80.i odd 4 1 1280.4.d.x 4
80.j even 4 1 1280.4.d.q 4
80.s even 4 1 1280.4.d.q 4
80.t odd 4 1 1280.4.d.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.c 2 5.c odd 4 1
160.4.a.g yes 2 20.e even 4 1
320.4.a.o 2 40.k even 4 1
320.4.a.s 2 40.i odd 4 1
800.4.a.m 2 20.e even 4 1
800.4.a.s 2 5.c odd 4 1
800.4.c.i 4 1.a even 1 1 trivial
800.4.c.i 4 5.b even 2 1 inner
800.4.c.k 4 4.b odd 2 1
800.4.c.k 4 20.d odd 2 1
1280.4.d.q 4 80.j even 4 1
1280.4.d.q 4 80.s even 4 1
1280.4.d.x 4 80.i odd 4 1
1280.4.d.x 4 80.t odd 4 1
1440.4.a.t 2 15.e even 4 1
1440.4.a.x 2 60.l odd 4 1
1600.4.a.cd 2 40.i odd 4 1
1600.4.a.cn 2 40.k even 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 80T_{3}^{2} + 64$$ T3^4 + 80*T3^2 + 64 $$T_{11}^{2} + 64T_{11} + 160$$ T11^2 + 64*T11 + 160

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 80T^{2} + 64$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 1232 T^{2} + 341056$$
$11$ $$(T^{2} + 64 T + 160)^{2}$$
$13$ $$T^{4} + 3144 T^{2} + \cdots + 2250000$$
$17$ $$T^{4} + 27656 T^{2} + \cdots + 190992400$$
$19$ $$(T^{2} - 208 T + 9280)^{2}$$
$23$ $$T^{4} + 13008 T^{2} + \cdots + 484416$$
$29$ $$(T^{2} - 292 T + 15172)^{2}$$
$31$ $$(T^{2} + 176 T + 6880)^{2}$$
$37$ $$T^{4} + 75656 T^{2} + \cdots + 652291600$$
$41$ $$(T^{2} - 100 T - 35900)^{2}$$
$43$ $$T^{4} + 136400 T^{2} + \cdots + 6190144$$
$47$ $$T^{4} + 47312 T^{2} + \cdots + 241615936$$
$53$ $$T^{4} + 77576 T^{2} + \cdots + 124099600$$
$59$ $$(T^{2} - 720 T + 120000)^{2}$$
$61$ $$(T^{2} + 1268 T + 377380)^{2}$$
$67$ $$T^{4} + 382800 T^{2} + \cdots + 7287695424$$
$71$ $$(T^{2} - 48 T - 1140000)^{2}$$
$73$ $$T^{4} + 1133000 T^{2} + \cdots + 15550090000$$
$79$ $$(T^{2} - 32 T - 709760)^{2}$$
$83$ $$T^{4} + 1373264 T^{2} + \cdots + 337096360000$$
$89$ $$(T^{2} - 780 T + 96804)^{2}$$
$97$ $$T^{4} + 821000 T^{2} + \cdots + 111355690000$$