# Properties

 Label 4000.1.bo.b Level 4000 Weight 1 Character orbit 4000.bo Analytic conductor 1.996 Analytic rank 0 Dimension 8 Projective image $$D_{20}$$ CM disc. -4 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 4000.bo (of order $$20$$ and degree $$8$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{20})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Projective image $$D_{20}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{20} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$-\zeta_{20}^{9} q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\zeta_{20}^{9} q^{9}$$ $$+ ( \zeta_{20} - \zeta_{20}^{2} ) q^{13}$$ $$+ ( \zeta_{20}^{4} + \zeta_{20}^{7} ) q^{17}$$ $$+ ( -\zeta_{20}^{6} - \zeta_{20}^{8} ) q^{29}$$ $$+ ( \zeta_{20}^{5} - \zeta_{20}^{8} ) q^{37}$$ $$+ ( \zeta_{20} + \zeta_{20}^{7} ) q^{41}$$ $$+ \zeta_{20}^{5} q^{49}$$ $$+ ( -\zeta_{20}^{4} - \zeta_{20}^{5} ) q^{53}$$ $$+ ( \zeta_{20}^{3} + \zeta_{20}^{9} ) q^{61}$$ $$+ ( -\zeta_{20}^{8} - \zeta_{20}^{9} ) q^{73}$$ $$-\zeta_{20}^{8} q^{81}$$ $$+ ( -1 - \zeta_{20}^{2} ) q^{89}$$ $$+ ( -\zeta_{20}^{3} + \zeta_{20}^{6} ) q^{97}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 2q^{13}$$ $$\mathstrut -\mathstrut 2q^{17}$$ $$\mathstrut +\mathstrut 2q^{37}$$ $$\mathstrut +\mathstrut 2q^{53}$$ $$\mathstrut +\mathstrut 2q^{73}$$ $$\mathstrut +\mathstrut 2q^{81}$$ $$\mathstrut -\mathstrut 10q^{89}$$ $$\mathstrut +\mathstrut 2q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$$.

 $$n$$ $$1377$$ $$2501$$ $$2751$$ $$\chi(n)$$ $$\zeta_{20}^{7}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −0.951057 − 0.309017i 0.951057 + 0.309017i 0.587785 − 0.809017i 0.951057 − 0.309017i −0.587785 − 0.809017i 0.587785 + 0.809017i −0.951057 + 0.309017i −0.587785 + 0.809017i
0 0 0 0 0 0 0 −0.951057 + 0.309017i 0
993.1 0 0 0 0 0 0 0 0.951057 0.309017i 0
1793.1 0 0 0 0 0 0 0 0.587785 + 0.809017i 0
1857.1 0 0 0 0 0 0 0 0.951057 + 0.309017i 0
2593.1 0 0 0 0 0 0 0 −0.587785 + 0.809017i 0
2657.1 0 0 0 0 0 0 0 0.587785 0.809017i 0
3393.1 0 0 0 0 0 0 0 −0.951057 0.309017i 0
3457.1 0 0 0 0 0 0 0 −0.587785 0.809017i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3457.1 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 CM by $$\Q(\sqrt{-1})$$ yes
25.f Odd 1 yes
100.l Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{13}^{8}$$ $$\mathstrut +\mathstrut 2 T_{13}^{7}$$ $$\mathstrut +\mathstrut 2 T_{13}^{6}$$ $$\mathstrut -\mathstrut 4 T_{13}^{4}$$ $$\mathstrut -\mathstrut 10 T_{13}^{3}$$ $$\mathstrut +\mathstrut 13 T_{13}^{2}$$ $$\mathstrut -\mathstrut 4 T_{13}$$ $$\mathstrut +\mathstrut 1$$ acting on $$S_{1}^{\mathrm{new}}(4000, [\chi])$$.