# Properties

 Label 800.2.o.d Level $800$ Weight $2$ Character orbit 800.o Analytic conductor $6.388$ Analytic rank $0$ Dimension $2$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.o (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 i ) q^{3} -5 i q^{9} +O(q^{10})$$ $$q + ( 2 - 2 i ) q^{3} -5 i q^{9} + 6 q^{11} + ( -4 - 4 i ) q^{17} + 2 i q^{19} + ( -4 - 4 i ) q^{27} + ( 12 - 12 i ) q^{33} -6 q^{41} + ( 6 - 6 i ) q^{43} + 7 i q^{49} -16 q^{51} + ( 4 + 4 i ) q^{57} + 6 i q^{59} + ( -6 - 6 i ) q^{67} + ( -12 + 12 i ) q^{73} - q^{81} + ( 2 - 2 i ) q^{83} + 18 i q^{89} + ( -12 - 12 i ) q^{97} -30 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + O(q^{10})$$ $$2q + 4q^{3} + 12q^{11} - 8q^{17} - 8q^{27} + 24q^{33} - 12q^{41} + 12q^{43} - 32q^{51} + 8q^{57} - 12q^{67} - 24q^{73} - 2q^{81} + 4q^{83} - 24q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 − 1.00000i 1.00000i
0 2.00000 + 2.00000i 0 0 0 0 0 5.00000i 0
207.1 0 2.00000 2.00000i 0 0 0 0 0 5.00000i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
5.c odd 4 1 inner
40.k 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.o.d 2
4.b odd 2 1 200.2.k.a 2
5.b even 2 1 800.2.o.a 2
5.c odd 4 1 800.2.o.a 2
5.c odd 4 1 inner 800.2.o.d 2
8.b even 2 1 200.2.k.a 2
8.d odd 2 1 CM 800.2.o.d 2
20.d odd 2 1 200.2.k.d yes 2
20.e even 4 1 200.2.k.a 2
20.e even 4 1 200.2.k.d yes 2
40.e odd 2 1 800.2.o.a 2
40.f even 2 1 200.2.k.d yes 2
40.i odd 4 1 200.2.k.a 2
40.i odd 4 1 200.2.k.d yes 2
40.k even 4 1 800.2.o.a 2
40.k even 4 1 inner 800.2.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.k.a 2 4.b odd 2 1
200.2.k.a 2 8.b even 2 1
200.2.k.a 2 20.e even 4 1
200.2.k.a 2 40.i odd 4 1
200.2.k.d yes 2 20.d odd 2 1
200.2.k.d yes 2 20.e even 4 1
200.2.k.d yes 2 40.f even 2 1
200.2.k.d yes 2 40.i odd 4 1
800.2.o.a 2 5.b even 2 1
800.2.o.a 2 5.c odd 4 1
800.2.o.a 2 40.e odd 2 1
800.2.o.a 2 40.k even 4 1
800.2.o.d 2 1.a even 1 1 trivial
800.2.o.d 2 5.c odd 4 1 inner
800.2.o.d 2 8.d odd 2 1 CM
800.2.o.d 2 40.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} - 4 T_{3} + 8$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$8 - 4 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$32 + 8 T + T^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$72 - 12 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$72 + 12 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$288 + 24 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$8 - 4 T + T^{2}$$
$89$ $$324 + T^{2}$$
$97$ $$288 + 24 T + T^{2}$$