# Properties

 Label 800.2.n.h Level $800$ Weight $2$ Character orbit 800.n Analytic conductor $6.388$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.n (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ 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: $$1$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{3} + ( - i + 1) q^{7} - i q^{9} +O(q^{10})$$ q + (i + 1) * q^3 + (-i + 1) * q^7 - i * q^9 $$q + (i + 1) q^{3} + ( - i + 1) q^{7} - i q^{9} - 6 i q^{11} + ( - i + 1) q^{13} + ( - i - 1) q^{17} + 4 q^{19} + 2 q^{21} + ( - 5 i - 5) q^{23} + ( - 4 i + 4) q^{27} + 8 i q^{29} + 2 i q^{31} + ( - 6 i + 6) q^{33} + (5 i + 5) q^{37} + 2 q^{39} + 6 q^{41} + ( - 3 i - 3) q^{43} + (7 i - 7) q^{47} + 5 i q^{49} - 2 i q^{51} + ( - i + 1) q^{53} + (4 i + 4) q^{57} + 4 q^{59} + 2 q^{61} + ( - i - 1) q^{63} + ( - 7 i + 7) q^{67} - 10 i q^{69} - 6 i q^{71} + (9 i - 9) q^{73} + ( - 6 i - 6) q^{77} - 8 q^{79} + 5 q^{81} + (5 i + 5) q^{83} + (8 i - 8) q^{87} - 2 i q^{91} + (2 i - 2) q^{93} + (3 i + 3) q^{97} - 6 q^{99} +O(q^{100})$$ q + (i + 1) * q^3 + (-i + 1) * q^7 - i * q^9 - 6*i * q^11 + (-i + 1) * q^13 + (-i - 1) * q^17 + 4 * q^19 + 2 * q^21 + (-5*i - 5) * q^23 + (-4*i + 4) * q^27 + 8*i * q^29 + 2*i * q^31 + (-6*i + 6) * q^33 + (5*i + 5) * q^37 + 2 * q^39 + 6 * q^41 + (-3*i - 3) * q^43 + (7*i - 7) * q^47 + 5*i * q^49 - 2*i * q^51 + (-i + 1) * q^53 + (4*i + 4) * q^57 + 4 * q^59 + 2 * q^61 + (-i - 1) * q^63 + (-7*i + 7) * q^67 - 10*i * q^69 - 6*i * q^71 + (9*i - 9) * q^73 + (-6*i - 6) * q^77 - 8 * q^79 + 5 * q^81 + (5*i + 5) * q^83 + (8*i - 8) * q^87 - 2*i * q^91 + (2*i - 2) * q^93 + (3*i + 3) * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^7 $$2 q + 2 q^{3} + 2 q^{7} + 2 q^{13} - 2 q^{17} + 8 q^{19} + 4 q^{21} - 10 q^{23} + 8 q^{27} + 12 q^{33} + 10 q^{37} + 4 q^{39} + 12 q^{41} - 6 q^{43} - 14 q^{47} + 2 q^{53} + 8 q^{57} + 8 q^{59} + 4 q^{61} - 2 q^{63} + 14 q^{67} - 18 q^{73} - 12 q^{77} - 16 q^{79} + 10 q^{81} + 10 q^{83} - 16 q^{87} - 4 q^{93} + 6 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^13 - 2 * q^17 + 8 * q^19 + 4 * q^21 - 10 * q^23 + 8 * q^27 + 12 * q^33 + 10 * q^37 + 4 * q^39 + 12 * q^41 - 6 * q^43 - 14 * q^47 + 2 * q^53 + 8 * q^57 + 8 * q^59 + 4 * q^61 - 2 * q^63 + 14 * q^67 - 18 * q^73 - 12 * q^77 - 16 * q^79 + 10 * q^81 + 10 * q^83 - 16 * q^87 - 4 * q^93 + 6 * q^97 - 12 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
543.1
 − 1.00000i 1.00000i
0 1.00000 1.00000i 0 0 0 1.00000 + 1.00000i 0 1.00000i 0
607.1 0 1.00000 + 1.00000i 0 0 0 1.00000 1.00000i 0 1.00000i 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
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.n.h 2
4.b odd 2 1 800.2.n.c 2
5.b even 2 1 160.2.n.b 2
5.c odd 4 1 160.2.n.e yes 2
5.c odd 4 1 800.2.n.c 2
8.b even 2 1 1600.2.n.e 2
8.d odd 2 1 1600.2.n.j 2
15.d odd 2 1 1440.2.x.b 2
15.e even 4 1 1440.2.x.e 2
20.d odd 2 1 160.2.n.e yes 2
20.e even 4 1 160.2.n.b 2
20.e even 4 1 inner 800.2.n.h 2
40.e odd 2 1 320.2.n.c 2
40.f even 2 1 320.2.n.f 2
40.i odd 4 1 320.2.n.c 2
40.i odd 4 1 1600.2.n.j 2
40.k even 4 1 320.2.n.f 2
40.k even 4 1 1600.2.n.e 2
60.h even 2 1 1440.2.x.e 2
60.l odd 4 1 1440.2.x.b 2
80.i odd 4 1 1280.2.o.d 2
80.j even 4 1 1280.2.o.e 2
80.k odd 4 1 1280.2.o.d 2
80.k odd 4 1 1280.2.o.n 2
80.q even 4 1 1280.2.o.e 2
80.q even 4 1 1280.2.o.k 2
80.s even 4 1 1280.2.o.k 2
80.t odd 4 1 1280.2.o.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.b 2 5.b even 2 1
160.2.n.b 2 20.e even 4 1
160.2.n.e yes 2 5.c odd 4 1
160.2.n.e yes 2 20.d odd 2 1
320.2.n.c 2 40.e odd 2 1
320.2.n.c 2 40.i odd 4 1
320.2.n.f 2 40.f even 2 1
320.2.n.f 2 40.k even 4 1
800.2.n.c 2 4.b odd 2 1
800.2.n.c 2 5.c odd 4 1
800.2.n.h 2 1.a even 1 1 trivial
800.2.n.h 2 20.e even 4 1 inner
1280.2.o.d 2 80.i odd 4 1
1280.2.o.d 2 80.k odd 4 1
1280.2.o.e 2 80.j even 4 1
1280.2.o.e 2 80.q even 4 1
1280.2.o.k 2 80.q even 4 1
1280.2.o.k 2 80.s even 4 1
1280.2.o.n 2 80.k odd 4 1
1280.2.o.n 2 80.t odd 4 1
1440.2.x.b 2 15.d odd 2 1
1440.2.x.b 2 60.l odd 4 1
1440.2.x.e 2 15.e even 4 1
1440.2.x.e 2 60.h even 2 1
1600.2.n.e 2 8.b even 2 1
1600.2.n.e 2 40.k even 4 1
1600.2.n.j 2 8.d odd 2 1
1600.2.n.j 2 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 $$S_{2}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} - 2T_{3} + 2$$ T3^2 - 2*T3 + 2 $$T_{7}^{2} - 2T_{7} + 2$$ T7^2 - 2*T7 + 2 $$T_{11}^{2} + 36$$ T11^2 + 36 $$T_{13}^{2} - 2T_{13} + 2$$ T13^2 - 2*T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 2T + 2$$
$11$ $$T^{2} + 36$$
$13$ $$T^{2} - 2T + 2$$
$17$ $$T^{2} + 2T + 2$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 10T + 50$$
$29$ $$T^{2} + 64$$
$31$ $$T^{2} + 4$$
$37$ $$T^{2} - 10T + 50$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 6T + 18$$
$47$ $$T^{2} + 14T + 98$$
$53$ $$T^{2} - 2T + 2$$
$59$ $$(T - 4)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} - 14T + 98$$
$71$ $$T^{2} + 36$$
$73$ $$T^{2} + 18T + 162$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} - 10T + 50$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 6T + 18$$