# Properties

 Label 800.4.c.h Level $800$ Weight $4$ Character orbit 800.c Analytic conductor $47.202$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + \beta_{2} q^{7} - 25 q^{9}+O(q^{10})$$ q - b2 * q^3 + b2 * q^7 - 25 * q^9 $$q - \beta_{2} q^{3} + \beta_{2} q^{7} - 25 q^{9} + 3 \beta_{3} q^{11} - 17 \beta_1 q^{13} + 57 \beta_1 q^{17} + 52 q^{21} + 29 \beta_{2} q^{23} - 2 \beta_{2} q^{27} + 26 q^{29} + 7 \beta_{3} q^{31} - 156 \beta_1 q^{33} - 75 \beta_1 q^{37} - 17 \beta_{3} q^{39} + 342 q^{41} + 63 \beta_{2} q^{43} + 81 \beta_{2} q^{47} + 291 q^{49} + 57 \beta_{3} q^{51} + 131 \beta_1 q^{53} + 34 \beta_{3} q^{59} - 262 q^{61} - 25 \beta_{2} q^{63} - 69 \beta_{2} q^{67} + 1508 q^{69} - 73 \beta_{3} q^{71} - 341 \beta_1 q^{73} + 156 \beta_1 q^{77} + 14 \beta_{3} q^{79} - 779 q^{81} - 21 \beta_{2} q^{83} - 26 \beta_{2} q^{87} + 630 q^{89} + 17 \beta_{3} q^{91} - 364 \beta_1 q^{93} - 483 \beta_1 q^{97} - 75 \beta_{3} q^{99}+O(q^{100})$$ q - b2 * q^3 + b2 * q^7 - 25 * q^9 + 3*b3 * q^11 - 17*b1 * q^13 + 57*b1 * q^17 + 52 * q^21 + 29*b2 * q^23 - 2*b2 * q^27 + 26 * q^29 + 7*b3 * q^31 - 156*b1 * q^33 - 75*b1 * q^37 - 17*b3 * q^39 + 342 * q^41 + 63*b2 * q^43 + 81*b2 * q^47 + 291 * q^49 + 57*b3 * q^51 + 131*b1 * q^53 + 34*b3 * q^59 - 262 * q^61 - 25*b2 * q^63 - 69*b2 * q^67 + 1508 * q^69 - 73*b3 * q^71 - 341*b1 * q^73 + 156*b1 * q^77 + 14*b3 * q^79 - 779 * q^81 - 21*b2 * q^83 - 26*b2 * q^87 + 630 * q^89 + 17*b3 * q^91 - 364*b1 * q^93 - 483*b1 * q^97 - 75*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 100 q^{9}+O(q^{10})$$ 4 * q - 100 * q^9 $$4 q - 100 q^{9} + 208 q^{21} + 104 q^{29} + 1368 q^{41} + 1164 q^{49} - 1048 q^{61} + 6032 q^{69} - 3116 q^{81} + 2520 q^{89}+O(q^{100})$$ 4 * q - 100 * q^9 + 208 * q^21 + 104 * q^29 + 1368 * q^41 + 1164 * q^49 - 1048 * q^61 + 6032 * q^69 - 3116 * q^81 + 2520 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} + 8\nu ) / 3$$ (2*v^3 + 8*v) / 3 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} + 20\nu ) / 3$$ (2*v^3 + 20*v) / 3 $$\beta_{3}$$ $$=$$ $$8\nu^{2} + 28$$ 8*v^2 + 28
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 4$$ (b2 - b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 28 ) / 8$$ (b3 - 28) / 8 $$\nu^{3}$$ $$=$$ $$( -2\beta_{2} + 5\beta_1 ) / 2$$ (-2*b2 + 5*b1) / 2

## 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
 2.30278i 1.30278i − 2.30278i − 1.30278i
0 7.21110i 0 0 0 7.21110i 0 −25.0000 0
449.2 0 7.21110i 0 0 0 7.21110i 0 −25.0000 0
449.3 0 7.21110i 0 0 0 7.21110i 0 −25.0000 0
449.4 0 7.21110i 0 0 0 7.21110i 0 −25.0000 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
5.b even 2 1 inner
20.d 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.c.h 4
4.b odd 2 1 inner 800.4.c.h 4
5.b even 2 1 inner 800.4.c.h 4
5.c odd 4 1 160.4.a.e 2
5.c odd 4 1 800.4.a.q 2
15.e even 4 1 1440.4.a.bd 2
20.d odd 2 1 inner 800.4.c.h 4
20.e even 4 1 160.4.a.e 2
20.e even 4 1 800.4.a.q 2
40.i odd 4 1 320.4.a.r 2
40.i odd 4 1 1600.4.a.ci 2
40.k even 4 1 320.4.a.r 2
40.k even 4 1 1600.4.a.ci 2
60.l odd 4 1 1440.4.a.bd 2
80.i odd 4 1 1280.4.d.t 4
80.j even 4 1 1280.4.d.t 4
80.s even 4 1 1280.4.d.t 4
80.t odd 4 1 1280.4.d.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.e 2 5.c odd 4 1
160.4.a.e 2 20.e even 4 1
320.4.a.r 2 40.i odd 4 1
320.4.a.r 2 40.k even 4 1
800.4.a.q 2 5.c odd 4 1
800.4.a.q 2 20.e even 4 1
800.4.c.h 4 1.a even 1 1 trivial
800.4.c.h 4 4.b odd 2 1 inner
800.4.c.h 4 5.b even 2 1 inner
800.4.c.h 4 20.d odd 2 1 inner
1280.4.d.t 4 80.i odd 4 1
1280.4.d.t 4 80.j even 4 1
1280.4.d.t 4 80.s even 4 1
1280.4.d.t 4 80.t odd 4 1
1440.4.a.bd 2 15.e even 4 1
1440.4.a.bd 2 60.l odd 4 1
1600.4.a.ci 2 40.i odd 4 1
1600.4.a.ci 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 $$S_{4}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} + 52$$ T3^2 + 52 $$T_{11}^{2} - 1872$$ T11^2 - 1872

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 52)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 52)^{2}$$
$11$ $$(T^{2} - 1872)^{2}$$
$13$ $$(T^{2} + 1156)^{2}$$
$17$ $$(T^{2} + 12996)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 43732)^{2}$$
$29$ $$(T - 26)^{4}$$
$31$ $$(T^{2} - 10192)^{2}$$
$37$ $$(T^{2} + 22500)^{2}$$
$41$ $$(T - 342)^{4}$$
$43$ $$(T^{2} + 206388)^{2}$$
$47$ $$(T^{2} + 341172)^{2}$$
$53$ $$(T^{2} + 68644)^{2}$$
$59$ $$(T^{2} - 240448)^{2}$$
$61$ $$(T + 262)^{4}$$
$67$ $$(T^{2} + 247572)^{2}$$
$71$ $$(T^{2} - 1108432)^{2}$$
$73$ $$(T^{2} + 465124)^{2}$$
$79$ $$(T^{2} - 40768)^{2}$$
$83$ $$(T^{2} + 22932)^{2}$$
$89$ $$(T - 630)^{4}$$
$97$ $$(T^{2} + 933156)^{2}$$