# Properties

 Label 800.4.c.e Level $800$ Weight $4$ Character orbit 800.c Analytic conductor $47.202$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 3 \beta q^{7} + 23 q^{9}+O(q^{10})$$ q + b * q^3 + 3*b * q^7 + 23 * q^9 $$q + \beta q^{3} + 3 \beta q^{7} + 23 q^{9} - 60 q^{11} + 25 \beta q^{13} + 15 \beta q^{17} + 40 q^{19} - 12 q^{21} - 89 \beta q^{23} + 50 \beta q^{27} - 166 q^{29} - 20 q^{31} - 60 \beta q^{33} - 5 \beta q^{37} - 100 q^{39} - 250 q^{41} - 71 \beta q^{43} + 107 \beta q^{47} + 307 q^{49} - 60 q^{51} + 245 \beta q^{53} + 40 \beta q^{57} - 800 q^{59} + 250 q^{61} + 69 \beta q^{63} - 387 \beta q^{67} + 356 q^{69} - 100 q^{71} - 115 \beta q^{73} - 180 \beta q^{77} - 1320 q^{79} + 421 q^{81} - 491 \beta q^{83} - 166 \beta q^{87} - 874 q^{89} - 300 q^{91} - 20 \beta q^{93} + 155 \beta q^{97} - 1380 q^{99} +O(q^{100})$$ q + b * q^3 + 3*b * q^7 + 23 * q^9 - 60 * q^11 + 25*b * q^13 + 15*b * q^17 + 40 * q^19 - 12 * q^21 - 89*b * q^23 + 50*b * q^27 - 166 * q^29 - 20 * q^31 - 60*b * q^33 - 5*b * q^37 - 100 * q^39 - 250 * q^41 - 71*b * q^43 + 107*b * q^47 + 307 * q^49 - 60 * q^51 + 245*b * q^53 + 40*b * q^57 - 800 * q^59 + 250 * q^61 + 69*b * q^63 - 387*b * q^67 + 356 * q^69 - 100 * q^71 - 115*b * q^73 - 180*b * q^77 - 1320 * q^79 + 421 * q^81 - 491*b * q^83 - 166*b * q^87 - 874 * q^89 - 300 * q^91 - 20*b * q^93 + 155*b * q^97 - 1380 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 46 q^{9}+O(q^{10})$$ 2 * q + 46 * q^9 $$2 q + 46 q^{9} - 120 q^{11} + 80 q^{19} - 24 q^{21} - 332 q^{29} - 40 q^{31} - 200 q^{39} - 500 q^{41} + 614 q^{49} - 120 q^{51} - 1600 q^{59} + 500 q^{61} + 712 q^{69} - 200 q^{71} - 2640 q^{79} + 842 q^{81} - 1748 q^{89} - 600 q^{91} - 2760 q^{99}+O(q^{100})$$ 2 * q + 46 * q^9 - 120 * q^11 + 80 * q^19 - 24 * q^21 - 332 * q^29 - 40 * q^31 - 200 * q^39 - 500 * q^41 + 614 * q^49 - 120 * q^51 - 1600 * q^59 + 500 * q^61 + 712 * q^69 - 200 * q^71 - 2640 * q^79 + 842 * q^81 - 1748 * q^89 - 600 * q^91 - 2760 * q^99

## 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
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 6.00000i 0 23.0000 0
449.2 0 2.00000i 0 0 0 6.00000i 0 23.0000 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
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.e 2
4.b odd 2 1 800.4.c.f 2
5.b even 2 1 inner 800.4.c.e 2
5.c odd 4 1 160.4.a.b yes 1
5.c odd 4 1 800.4.a.d 1
15.e even 4 1 1440.4.a.n 1
20.d odd 2 1 800.4.c.f 2
20.e even 4 1 160.4.a.a 1
20.e even 4 1 800.4.a.h 1
40.i odd 4 1 320.4.a.f 1
40.i odd 4 1 1600.4.a.bj 1
40.k even 4 1 320.4.a.i 1
40.k even 4 1 1600.4.a.r 1
60.l odd 4 1 1440.4.a.o 1
80.i odd 4 1 1280.4.d.k 2
80.j even 4 1 1280.4.d.f 2
80.s even 4 1 1280.4.d.f 2
80.t odd 4 1 1280.4.d.k 2

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

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{11} + 60$$ T11 + 60

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 36$$
$11$ $$(T + 60)^{2}$$
$13$ $$T^{2} + 2500$$
$17$ $$T^{2} + 900$$
$19$ $$(T - 40)^{2}$$
$23$ $$T^{2} + 31684$$
$29$ $$(T + 166)^{2}$$
$31$ $$(T + 20)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 250)^{2}$$
$43$ $$T^{2} + 20164$$
$47$ $$T^{2} + 45796$$
$53$ $$T^{2} + 240100$$
$59$ $$(T + 800)^{2}$$
$61$ $$(T - 250)^{2}$$
$67$ $$T^{2} + 599076$$
$71$ $$(T + 100)^{2}$$
$73$ $$T^{2} + 52900$$
$79$ $$(T + 1320)^{2}$$
$83$ $$T^{2} + 964324$$
$89$ $$(T + 874)^{2}$$
$97$ $$T^{2} + 96100$$