# Properties

 Label 800.1.bh.a Level 800 Weight 1 Character orbit 800.bh Analytic conductor 0.399 Analytic rank 0 Dimension 8 Projective image $$A_{5}$$ CM/RM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 800.bh (of order $$10$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.399252010106$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Projective image $$A_{5}$$ Projective field Galois closure of 5.1.25000000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{20}^{7} q^{3} + \zeta_{20}^{4} q^{5} + ( \zeta_{20} + \zeta_{20}^{9} ) q^{7} +O(q^{10})$$ $$q -\zeta_{20}^{7} q^{3} + \zeta_{20}^{4} q^{5} + ( \zeta_{20} + \zeta_{20}^{9} ) q^{7} + ( 1 + \zeta_{20}^{8} ) q^{13} + \zeta_{20} q^{15} + ( -\zeta_{20} - \zeta_{20}^{5} ) q^{19} + ( \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{21} + \zeta_{20} q^{23} + \zeta_{20}^{8} q^{25} + \zeta_{20} q^{27} + ( -1 - \zeta_{20}^{4} ) q^{29} -\zeta_{20}^{3} q^{31} + ( -\zeta_{20}^{3} + \zeta_{20}^{5} ) q^{35} -\zeta_{20}^{4} q^{37} + ( \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{39} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{43} + ( -\zeta_{20} + \zeta_{20}^{3} ) q^{47} + ( -1 + \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{49} + \zeta_{20}^{2} q^{53} + ( -\zeta_{20}^{2} + \zeta_{20}^{8} ) q^{57} + ( -\zeta_{20} + \zeta_{20}^{7} ) q^{59} -\zeta_{20}^{6} q^{61} + ( -\zeta_{20}^{2} + \zeta_{20}^{4} ) q^{65} -\zeta_{20}^{8} q^{69} + ( \zeta_{20} - \zeta_{20}^{3} ) q^{71} -\zeta_{20}^{6} q^{73} + \zeta_{20}^{5} q^{75} + ( \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{79} -\zeta_{20}^{8} q^{81} + \zeta_{20}^{3} q^{83} + ( -\zeta_{20} + \zeta_{20}^{7} ) q^{87} + ( \zeta_{20} - \zeta_{20}^{7} + 2 \zeta_{20}^{9} ) q^{91} - q^{93} + ( -\zeta_{20}^{5} - \zeta_{20}^{9} ) q^{95} + ( \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{5} + O(q^{10})$$ $$8q - 2q^{5} + 6q^{13} + 4q^{21} - 2q^{25} - 6q^{29} + 2q^{37} - 4q^{49} + 2q^{53} - 4q^{57} - 2q^{61} - 4q^{65} + 2q^{69} - 2q^{73} + 2q^{81} - 8q^{93} + 4q^{97} + O(q^{100})$$

## Character Values

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

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{20}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −0.951057 + 0.309017i 0.951057 − 0.309017i 0.587785 − 0.809017i −0.587785 + 0.809017i 0.587785 + 0.809017i −0.587785 − 0.809017i −0.951057 − 0.309017i 0.951057 + 0.309017i
0 −0.587785 0.809017i 0 0.309017 0.951057i 0 0.618034i 0 0 0
31.2 0 0.587785 + 0.809017i 0 0.309017 0.951057i 0 0.618034i 0 0 0
191.1 0 −0.951057 + 0.309017i 0 −0.809017 + 0.587785i 0 1.61803i 0 0 0
191.2 0 0.951057 0.309017i 0 −0.809017 + 0.587785i 0 1.61803i 0 0 0
511.1 0 −0.951057 0.309017i 0 −0.809017 0.587785i 0 1.61803i 0 0 0
511.2 0 0.951057 + 0.309017i 0 −0.809017 0.587785i 0 1.61803i 0 0 0
671.1 0 −0.587785 + 0.809017i 0 0.309017 + 0.951057i 0 0.618034i 0 0 0
671.2 0 0.587785 0.809017i 0 0.309017 + 0.951057i 0 0.618034i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 671.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes
25.d Even 1 yes
100.j Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{1}^{\mathrm{new}}(800, [\chi])$$.