# Properties

 Label 1000.1.e.a Level $1000$ Weight $1$ Character orbit 1000.e Self dual yes Analytic conductor $0.499$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -40 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1000 = 2^{3} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1000.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.499065012633$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( 1 - \beta ) q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( 1 - \beta ) q^{7} - q^{8} + q^{9} -\beta q^{11} + \beta q^{13} + ( -1 + \beta ) q^{14} + q^{16} - q^{18} + ( -1 + \beta ) q^{19} + \beta q^{22} + \beta q^{23} -\beta q^{26} + ( 1 - \beta ) q^{28} - q^{32} + q^{36} + ( 1 - \beta ) q^{37} + ( 1 - \beta ) q^{38} + ( -1 + \beta ) q^{41} -\beta q^{44} -\beta q^{46} + \beta q^{47} + ( 1 - \beta ) q^{49} + \beta q^{52} + ( 1 - \beta ) q^{53} + ( -1 + \beta ) q^{56} + ( -1 + \beta ) q^{59} + ( 1 - \beta ) q^{63} + q^{64} - q^{72} + ( -1 + \beta ) q^{74} + ( -1 + \beta ) q^{76} + q^{77} + q^{81} + ( 1 - \beta ) q^{82} + \beta q^{88} -\beta q^{89} - q^{91} + \beta q^{92} -\beta q^{94} + ( -1 + \beta ) q^{98} -\beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} + q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} + q^{7} - 2q^{8} + 2q^{9} - q^{11} + q^{13} - q^{14} + 2q^{16} - 2q^{18} - q^{19} + q^{22} + q^{23} - q^{26} + q^{28} - 2q^{32} + 2q^{36} + q^{37} + q^{38} - q^{41} - q^{44} - q^{46} + q^{47} + q^{49} + q^{52} + q^{53} - q^{56} - q^{59} + q^{63} + 2q^{64} - 2q^{72} - q^{74} - q^{76} + 2q^{77} + 2q^{81} + q^{82} + q^{88} - q^{89} - 2q^{91} + q^{92} - q^{94} - q^{98} - q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times$$.

 $$n$$ $$377$$ $$501$$ $$751$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
499.1
 1.61803 −0.618034
−1.00000 0 1.00000 0 0 −0.618034 −1.00000 1.00000 0
499.2 −1.00000 0 1.00000 0 0 1.61803 −1.00000 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 1000.1.e.a 2
4.b odd 2 1 4000.1.e.a 2
5.b even 2 1 1000.1.e.b 2
5.c odd 4 2 1000.1.g.a 4
8.b even 2 1 4000.1.e.b 2
8.d odd 2 1 1000.1.e.b 2
20.d odd 2 1 4000.1.e.b 2
20.e even 4 2 4000.1.g.a 4
40.e odd 2 1 CM 1000.1.e.a 2
40.f even 2 1 4000.1.e.a 2
40.i odd 4 2 4000.1.g.a 4
40.k even 4 2 1000.1.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.1.e.a 2 1.a even 1 1 trivial
1000.1.e.a 2 40.e odd 2 1 CM
1000.1.e.b 2 5.b even 2 1
1000.1.e.b 2 8.d odd 2 1
1000.1.g.a 4 5.c odd 4 2
1000.1.g.a 4 40.k even 4 2
4000.1.e.a 2 4.b odd 2 1
4000.1.e.a 2 40.f even 2 1
4000.1.e.b 2 8.b even 2 1
4000.1.e.b 2 20.d odd 2 1
4000.1.g.a 4 20.e even 4 2
4000.1.g.a 4 40.i odd 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 $$S_{1}^{\mathrm{new}}(1000, [\chi])$$.

## Hecke characteristic polynomials

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