# Properties

 Label 4000.1.h.a Level 4000 Weight 1 Character orbit 4000.h Analytic conductor 1.996 Analytic rank 0 Dimension 4 Projective image $$A_{5}$$ CM/RM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$A_{5}$$ Projective field Galois closure of 5.1.1000000.2

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{3} + q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{3} + q^{7} + ( 1 + \beta_{2} ) q^{9} + \beta_{3} q^{11} + ( -\beta_{1} - \beta_{3} ) q^{13} + \beta_{3} q^{17} + ( \beta_{1} + \beta_{3} ) q^{19} + ( -1 - \beta_{2} ) q^{21} - q^{27} + q^{29} -\beta_{1} q^{31} + ( -\beta_{1} - \beta_{3} ) q^{33} + ( \beta_{1} + 2 \beta_{3} ) q^{39} + q^{41} - q^{43} -\beta_{2} q^{47} + ( -\beta_{1} - \beta_{3} ) q^{51} + \beta_{1} q^{53} + ( -\beta_{1} - 2 \beta_{3} ) q^{57} -\beta_{1} q^{59} + ( -1 - \beta_{2} ) q^{61} + ( 1 + \beta_{2} ) q^{63} + \beta_{2} q^{67} + \beta_{3} q^{71} + \beta_{1} q^{73} + \beta_{3} q^{77} -\beta_{3} q^{79} + ( -1 - \beta_{2} ) q^{87} + ( -\beta_{1} - \beta_{3} ) q^{91} + \beta_{3} q^{93} + \beta_{1} q^{97} + ( \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$4q - 2q^{3} + 4q^{7} + 2q^{9} - 2q^{21} - 4q^{27} + 4q^{29} + 4q^{41} - 4q^{43} + 2q^{47} - 2q^{61} + 2q^{63} - 2q^{67} - 2q^{87} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2 \beta_{1}$$

## Character values

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

 $$n$$ $$1377$$ $$2501$$ $$2751$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3999.1
 − 0.618034i 0.618034i 1.61803i − 1.61803i
0 −1.61803 0 0 0 1.00000 0 1.61803 0
3999.2 0 −1.61803 0 0 0 1.00000 0 1.61803 0
3999.3 0 0.618034 0 0 0 1.00000 0 −0.618034 0
3999.4 0 0.618034 0 0 0 1.00000 0 −0.618034 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
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.1.h.a 4
4.b odd 2 1 4000.1.h.b 4
5.b even 2 1 4000.1.h.b 4
5.c odd 4 1 4000.1.b.a 4
5.c odd 4 1 4000.1.b.b yes 4
20.d odd 2 1 inner 4000.1.h.a 4
20.e even 4 1 4000.1.b.a 4
20.e even 4 1 4000.1.b.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.1.b.a 4 5.c odd 4 1
4000.1.b.a 4 20.e even 4 1
4000.1.b.b yes 4 5.c odd 4 1
4000.1.b.b yes 4 20.e even 4 1
4000.1.h.a 4 1.a even 1 1 trivial
4000.1.h.a 4 20.d odd 2 1 inner
4000.1.h.b 4 4.b odd 2 1
4000.1.h.b 4 5.b even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

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