# Properties

 Label 4000.1.bf.b Level 4000 Weight 1 Character orbit 4000.bf 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.bf (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: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

## Character values

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

 $$n$$ $$1377$$ $$2501$$ $$2751$$ $$\chi(n)$$ $$\zeta_{20}^{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 −0.587785 + 0.809017i 0.587785 − 0.809017i 0.951057 − 0.309017i −0.951057 + 0.309017i 0.951057 + 0.309017i −0.951057 − 0.309017i −0.587785 − 0.809017i 0.587785 + 0.809017i
0 −0.309017 0.951057i 0 0 0 −1.61803 0 0 0
799.2 0 −0.309017 0.951057i 0 0 0 −1.61803 0 0 0
1599.1 0 0.809017 0.587785i 0 0 0 0.618034 0 0 0
1599.2 0 0.809017 0.587785i 0 0 0 0.618034 0 0 0
2399.1 0 0.809017 + 0.587785i 0 0 0 0.618034 0 0 0
2399.2 0 0.809017 + 0.587785i 0 0 0 0.618034 0 0 0
3199.1 0 −0.309017 + 0.951057i 0 0 0 −1.61803 0 0 0
3199.2 0 −0.309017 + 0.951057i 0 0 0 −1.61803 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3199.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner
25.d even 5 1 inner
100.h 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.bf.b 8
4.b odd 2 1 4000.1.bf.a 8
5.b even 2 1 4000.1.bf.a 8
5.c odd 4 1 800.1.bh.a 8
5.c odd 4 1 4000.1.bh.a 8
20.d odd 2 1 inner 4000.1.bf.b 8
20.e even 4 1 800.1.bh.a 8
20.e even 4 1 4000.1.bh.a 8
25.d even 5 1 inner 4000.1.bf.b 8
25.e even 10 1 4000.1.bf.a 8
25.f odd 20 1 800.1.bh.a 8
25.f odd 20 1 4000.1.bh.a 8
40.i odd 4 1 1600.1.bh.b 8
40.k even 4 1 1600.1.bh.b 8
100.h odd 10 1 inner 4000.1.bf.b 8
100.j odd 10 1 4000.1.bf.a 8
100.l even 20 1 800.1.bh.a 8
100.l even 20 1 4000.1.bh.a 8
200.v even 20 1 1600.1.bh.b 8
200.x odd 20 1 1600.1.bh.b 8

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$5$ 1
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$11$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$13$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
$17$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$19$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
$23$ $$( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2}$$
$29$ $$( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$31$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$37$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$41$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$43$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$47$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$53$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$59$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$61$ $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
$67$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$71$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$73$ $$1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16}$$
$79$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )$$
$83$ $$( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2}$$
$89$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$97$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$