# Properties

 Label 4000.1.bh.a Level 4000 Weight 1 Character orbit 4000.bh Analytic conductor 1.996 Analytic rank 0 Dimension 8 Projective image $$A_{5}$$ CM/RM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 4000.bh (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.99626005053$$ 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: $$1$$ Twist minimal: no (minimal twist has level 800) Projective image $$A_{5}$$ Projective field Galois closure of 5.1.25000000.2

## $q$-expansion

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

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

## Character values

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

 $$n$$ $$1377$$ $$2501$$ $$2751$$ $$\chi(n)$$ $$-\zeta_{20}^{6}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
351.1
 0.587785 − 0.809017i −0.587785 + 0.809017i 0.587785 + 0.809017i −0.587785 − 0.809017i 0.951057 − 0.309017i −0.951057 + 0.309017i 0.951057 + 0.309017i −0.951057 − 0.309017i
0 −0.587785 + 0.809017i 0 0 0 0.618034i 0 0 0
351.2 0 0.587785 0.809017i 0 0 0 0.618034i 0 0 0
1151.1 0 −0.587785 0.809017i 0 0 0 0.618034i 0 0 0
1151.2 0 0.587785 + 0.809017i 0 0 0 0.618034i 0 0 0
1951.1 0 −0.951057 + 0.309017i 0 0 0 1.61803i 0 0 0
1951.2 0 0.951057 0.309017i 0 0 0 1.61803i 0 0 0
3551.1 0 −0.951057 0.309017i 0 0 0 1.61803i 0 0 0
3551.2 0 0.951057 + 0.309017i 0 0 0 1.61803i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3551.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.d even 5 1 inner
100.j odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.1.bh.a 8
4.b odd 2 1 inner 4000.1.bh.a 8
5.b even 2 1 800.1.bh.a 8
5.c odd 4 1 4000.1.bf.a 8
5.c odd 4 1 4000.1.bf.b 8
20.d odd 2 1 800.1.bh.a 8
20.e even 4 1 4000.1.bf.a 8
20.e even 4 1 4000.1.bf.b 8
25.d even 5 1 inner 4000.1.bh.a 8
25.e even 10 1 800.1.bh.a 8
25.f odd 20 1 4000.1.bf.a 8
25.f odd 20 1 4000.1.bf.b 8
40.e odd 2 1 1600.1.bh.b 8
40.f even 2 1 1600.1.bh.b 8
100.h odd 10 1 800.1.bh.a 8
100.j odd 10 1 inner 4000.1.bh.a 8
100.l even 20 1 4000.1.bf.a 8
100.l even 20 1 4000.1.bf.b 8
200.o even 10 1 1600.1.bh.b 8
200.s odd 10 1 1600.1.bh.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.1.bh.a 8 5.b even 2 1
800.1.bh.a 8 20.d odd 2 1
800.1.bh.a 8 25.e even 10 1
800.1.bh.a 8 100.h odd 10 1
1600.1.bh.b 8 40.e odd 2 1
1600.1.bh.b 8 40.f even 2 1
1600.1.bh.b 8 200.o even 10 1
1600.1.bh.b 8 200.s odd 10 1
4000.1.bf.a 8 5.c odd 4 1
4000.1.bf.a 8 20.e even 4 1
4000.1.bf.a 8 25.f odd 20 1
4000.1.bf.a 8 100.l even 20 1
4000.1.bf.b 8 5.c odd 4 1
4000.1.bf.b 8 20.e even 4 1
4000.1.bf.b 8 25.f odd 20 1
4000.1.bf.b 8 100.l even 20 1
4000.1.bh.a 8 1.a even 1 1 trivial
4000.1.bh.a 8 4.b odd 2 1 inner
4000.1.bh.a 8 25.d even 5 1 inner
4000.1.bh.a 8 100.j odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(4000, [\chi])$$.