# Properties

 Label 800.1.p.c Level 800 Weight 1 Character orbit 800.p Analytic conductor 0.399 Analytic rank 0 Dimension 2 Projective image $$D_{4}$$ CM discriminant -20 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 800.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.399252010106$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.2.400.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( 1 - i ) q^{3} + ( -1 - i ) q^{7} -i q^{9} +O(q^{10})$$ $$q + ( 1 - i ) q^{3} + ( -1 - i ) q^{7} -i q^{9} -2 q^{21} + ( 1 - i ) q^{23} + 2 i q^{29} + ( -1 + i ) q^{43} + ( 1 + i ) q^{47} + i q^{49} + ( -1 + i ) q^{63} + ( -1 - i ) q^{67} -2 i q^{69} + q^{81} + ( -1 + i ) q^{83} + ( 2 + 2 i ) q^{87} -2 i q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{7} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{7} - 4q^{21} + 2q^{23} - 2q^{43} + 2q^{47} - 2q^{63} - 2q^{67} + 2q^{81} - 2q^{83} + 4q^{87} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 − 1.00000i 1.00000i
0 1.00000 + 1.00000i 0 0 0 −1.00000 + 1.00000i 0 1.00000i 0
257.1 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.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
5.c odd 4 1 inner
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 800.1.p.c yes 2
4.b odd 2 1 800.1.p.a 2
5.b even 2 1 800.1.p.a 2
5.c odd 4 1 800.1.p.a 2
5.c odd 4 1 inner 800.1.p.c yes 2
8.b even 2 1 1600.1.p.a 2
8.d odd 2 1 1600.1.p.c 2
20.d odd 2 1 CM 800.1.p.c yes 2
20.e even 4 1 800.1.p.a 2
20.e even 4 1 inner 800.1.p.c yes 2
40.e odd 2 1 1600.1.p.a 2
40.f even 2 1 1600.1.p.c 2
40.i odd 4 1 1600.1.p.a 2
40.i odd 4 1 1600.1.p.c 2
40.k even 4 1 1600.1.p.a 2
40.k even 4 1 1600.1.p.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.1.p.a 2 4.b odd 2 1
800.1.p.a 2 5.b even 2 1
800.1.p.a 2 5.c odd 4 1
800.1.p.a 2 20.e even 4 1
800.1.p.c yes 2 1.a even 1 1 trivial
800.1.p.c yes 2 5.c odd 4 1 inner
800.1.p.c yes 2 20.d odd 2 1 CM
800.1.p.c yes 2 20.e even 4 1 inner
1600.1.p.a 2 8.b even 2 1
1600.1.p.a 2 40.e odd 2 1
1600.1.p.a 2 40.i odd 4 1
1600.1.p.a 2 40.k even 4 1
1600.1.p.c 2 8.d odd 2 1
1600.1.p.c 2 40.f even 2 1
1600.1.p.c 2 40.i odd 4 1
1600.1.p.c 2 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2 T_{3} + 2$$ acting on $$S_{1}^{\mathrm{new}}(800, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 - T )^{2}( 1 + T^{2} )$$
$5$ 
$7$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$11$ $$( 1 + T^{2} )^{2}$$
$13$ $$1 + T^{4}$$
$17$ $$1 + T^{4}$$
$19$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$23$ $$( 1 - T )^{2}( 1 + T^{2} )$$
$29$ $$( 1 + T^{2} )^{2}$$
$31$ $$( 1 + T^{2} )^{2}$$
$37$ $$1 + T^{4}$$
$41$ $$( 1 + T^{2} )^{2}$$
$43$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$47$ $$( 1 - T )^{2}( 1 + T^{2} )$$
$53$ $$1 + T^{4}$$
$59$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$61$ $$( 1 + T^{2} )^{2}$$
$67$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$71$ $$( 1 + T^{2} )^{2}$$
$73$ $$1 + T^{4}$$
$79$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$83$ $$( 1 + T )^{2}( 1 + T^{2} )$$
$89$ $$( 1 + T^{2} )^{2}$$
$97$ $$1 + T^{4}$$