# Properties

 Label 1000.1.g.a Level $1000$ Weight $1$ Character orbit 1000.g Analytic conductor $0.499$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -40 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1000 = 2^{3} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1000.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.499065012633$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} - q^{4} -\beta_{1} q^{7} + \beta_{3} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} - q^{4} -\beta_{1} q^{7} + \beta_{3} q^{8} - q^{9} + ( -1 - \beta_{2} ) q^{11} + ( -\beta_{1} - \beta_{3} ) q^{13} -\beta_{2} q^{14} + q^{16} + \beta_{3} q^{18} -\beta_{2} q^{19} + ( \beta_{1} + \beta_{3} ) q^{22} + ( -\beta_{1} - \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{26} + \beta_{1} q^{28} -\beta_{3} q^{32} + q^{36} -\beta_{1} q^{37} + \beta_{1} q^{38} + \beta_{2} q^{41} + ( 1 + \beta_{2} ) q^{44} + ( -1 - \beta_{2} ) q^{46} + ( \beta_{1} + \beta_{3} ) q^{47} + \beta_{2} q^{49} + ( \beta_{1} + \beta_{3} ) q^{52} + \beta_{1} q^{53} + \beta_{2} q^{56} -\beta_{2} q^{59} + \beta_{1} q^{63} - q^{64} -\beta_{3} q^{72} -\beta_{2} q^{74} + \beta_{2} q^{76} + \beta_{3} q^{77} + q^{81} -\beta_{1} q^{82} + ( -\beta_{1} - \beta_{3} ) q^{88} + ( 1 + \beta_{2} ) q^{89} - q^{91} + ( \beta_{1} + \beta_{3} ) q^{92} + ( 1 + \beta_{2} ) q^{94} -\beta_{1} q^{98} + ( 1 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{9} - 2q^{11} + 2q^{14} + 4q^{16} + 2q^{19} - 2q^{26} + 4q^{36} - 2q^{41} + 2q^{44} - 2q^{46} - 2q^{49} - 2q^{56} + 2q^{59} - 4q^{64} + 2q^{74} - 2q^{76} + 4q^{81} + 2q^{89} - 4q^{91} + 2q^{94} + 2q^{99} + O(q^{100})$$

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 $$\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 0.618034i − 1.61803i 1.61803i − 0.618034i
1.00000i 0 −1.00000 0 0 0.618034i 1.00000i −1.00000 0
251.2 1.00000i 0 −1.00000 0 0 1.61803i 1.00000i −1.00000 0
251.3 1.00000i 0 −1.00000 0 0 1.61803i 1.00000i −1.00000 0
251.4 1.00000i 0 −1.00000 0 0 0.618034i 1.00000i −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 $$\Q(\sqrt{-10})$$
5.b even 2 1 inner
8.d odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1000, [\chi])$$.

## Hecke characteristic polynomials

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