# Properties

 Label 4000.1.e.b Level $4000$ Weight $1$ Character orbit 4000.e Self dual yes Analytic conductor $1.996$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -40 Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1000) Projective image $$D_{5}$$ Projective field Galois closure of 5.1.1000000.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{7} + q^{9} +O(q^{10})$$ $$q + \beta q^{7} + q^{9} + ( 1 - \beta ) q^{11} + ( -1 + \beta ) q^{13} + \beta q^{19} + ( 1 - \beta ) q^{23} -\beta q^{37} -\beta q^{41} + ( 1 - \beta ) q^{47} + \beta q^{49} -\beta q^{53} + \beta q^{59} + \beta q^{63} - q^{77} + q^{81} + ( -1 + \beta ) q^{89} + q^{91} + ( 1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{7} + 2q^{9} + O(q^{10})$$ $$2q + q^{7} + 2q^{9} + q^{11} - q^{13} + q^{19} + q^{23} - q^{37} - q^{41} + q^{47} + q^{49} - q^{53} + q^{59} + q^{63} - 2q^{77} + 2q^{81} - q^{89} + 2q^{91} + q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1999.1
 −0.618034 1.61803
0 0 0 0 0 −0.618034 0 1.00000 0
1999.2 0 0 0 0 0 1.61803 0 1.00000 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.1.e.b 2
4.b odd 2 1 1000.1.e.b 2
5.b even 2 1 4000.1.e.a 2
5.c odd 4 2 4000.1.g.a 4
8.b even 2 1 1000.1.e.a 2
8.d odd 2 1 4000.1.e.a 2
20.d odd 2 1 1000.1.e.a 2
20.e even 4 2 1000.1.g.a 4
40.e odd 2 1 CM 4000.1.e.b 2
40.f even 2 1 1000.1.e.b 2
40.i odd 4 2 1000.1.g.a 4
40.k even 4 2 4000.1.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.1.e.a 2 8.b even 2 1
1000.1.e.a 2 20.d odd 2 1
1000.1.e.b 2 4.b odd 2 1
1000.1.e.b 2 40.f even 2 1
1000.1.g.a 4 20.e even 4 2
1000.1.g.a 4 40.i odd 4 2
4000.1.e.a 2 5.b even 2 1
4000.1.e.a 2 8.d odd 2 1
4000.1.e.b 2 1.a even 1 1 trivial
4000.1.e.b 2 40.e odd 2 1 CM
4000.1.g.a 4 5.c odd 4 2
4000.1.g.a 4 40.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - T_{7} - 1$$ acting on $$S_{1}^{\mathrm{new}}(4000, [\chi])$$.

## Hecke characteristic polynomials

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