Properties

 Label 800.1.e.a Level 800 Weight 1 Character orbit 800.e Analytic conductor 0.399 Analytic rank 0 Dimension 2 Projective image $$D_{3}$$ CM discriminant -8 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 800.e (of order $$2$$, degree $$1$$, not 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: no (minimal twist has level 200) Projective image $$D_{3}$$ Projective field Galois closure of 3.1.200.1

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q -i q^{3} +O(q^{10})$$ $$q -i q^{3} + q^{11} -i q^{17} - q^{19} -i q^{27} -i q^{33} - q^{41} + 2 i q^{43} - q^{49} - q^{51} + i q^{57} + 2 q^{59} + i q^{67} + i q^{73} - q^{81} -i q^{83} + q^{89} + 2 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 2q^{11} - 2q^{19} - 2q^{41} - 2q^{49} - 2q^{51} + 4q^{59} - 2q^{81} + 2q^{89} + O(q^{100})$$

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
399.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 0 0 0 0
399.2 0 1.00000i 0 0 0 0 0 0 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.b even 2 1 inner
40.e odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.1.e.a 2
4.b odd 2 1 200.1.e.a 2
5.b even 2 1 inner 800.1.e.a 2
5.c odd 4 1 800.1.g.a 1
5.c odd 4 1 800.1.g.b 1
8.b even 2 1 200.1.e.a 2
8.d odd 2 1 CM 800.1.e.a 2
12.b even 2 1 1800.1.p.a 2
20.d odd 2 1 200.1.e.a 2
20.e even 4 1 200.1.g.a 1
20.e even 4 1 200.1.g.b yes 1
24.h odd 2 1 1800.1.p.a 2
40.e odd 2 1 inner 800.1.e.a 2
40.f even 2 1 200.1.e.a 2
40.i odd 4 1 200.1.g.a 1
40.i odd 4 1 200.1.g.b yes 1
40.k even 4 1 800.1.g.a 1
40.k even 4 1 800.1.g.b 1
60.h even 2 1 1800.1.p.a 2
60.l odd 4 1 1800.1.g.a 1
60.l odd 4 1 1800.1.g.b 1
120.i odd 2 1 1800.1.p.a 2
120.w even 4 1 1800.1.g.a 1
120.w even 4 1 1800.1.g.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.1.e.a 2 4.b odd 2 1
200.1.e.a 2 8.b even 2 1
200.1.e.a 2 20.d odd 2 1
200.1.e.a 2 40.f even 2 1
200.1.g.a 1 20.e even 4 1
200.1.g.a 1 40.i odd 4 1
200.1.g.b yes 1 20.e even 4 1
200.1.g.b yes 1 40.i odd 4 1
800.1.e.a 2 1.a even 1 1 trivial
800.1.e.a 2 5.b even 2 1 inner
800.1.e.a 2 8.d odd 2 1 CM
800.1.e.a 2 40.e odd 2 1 inner
800.1.g.a 1 5.c odd 4 1
800.1.g.a 1 40.k even 4 1
800.1.g.b 1 5.c odd 4 1
800.1.g.b 1 40.k even 4 1
1800.1.g.a 1 60.l odd 4 1
1800.1.g.a 1 120.w even 4 1
1800.1.g.b 1 60.l odd 4 1
1800.1.g.b 1 120.w even 4 1
1800.1.p.a 2 12.b even 2 1
1800.1.p.a 2 24.h odd 2 1
1800.1.p.a 2 60.h even 2 1
1800.1.p.a 2 120.i odd 2 1

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(800, [\chi])$$.

Hecke Characteristic Polynomials

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