# Properties

 Label 4000.1.g.a Level $4000$ Weight $1$ Character orbit 4000.g Analytic conductor $1.996$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -40 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 4000.g (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: $$\Z[a_1, \ldots, a_{7}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{7} - q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{7} - q^{9} -\beta_{2} q^{11} -\beta_{1} q^{13} + ( -1 - \beta_{2} ) q^{19} + \beta_{1} q^{23} + ( -\beta_{1} - \beta_{3} ) q^{37} + ( -1 - \beta_{2} ) q^{41} -\beta_{1} q^{47} + ( -1 - \beta_{2} ) q^{49} + ( \beta_{1} + \beta_{3} ) q^{53} + ( -1 - \beta_{2} ) q^{59} + ( -\beta_{1} - \beta_{3} ) q^{63} -\beta_{3} q^{77} + q^{81} -\beta_{2} q^{89} + q^{91} + \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 2q^{11} - 2q^{19} - 2q^{41} - 2q^{49} - 2q^{59} + 4q^{81} + 2q^{89} + 4q^{91} - 2q^{99} + O(q^{100})$$

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 $$\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 − 0.618034i − 1.61803i 1.61803i 0.618034i
0 0 0 0 0 1.61803i 0 −1.00000 0
751.2 0 0 0 0 0 0.618034i 0 −1.00000 0
751.3 0 0 0 0 0 0.618034i 0 −1.00000 0
751.4 0 0 0 0 0 1.61803i 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 $$\Q(\sqrt{-10})$$
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 4000.1.g.a 4
4.b odd 2 1 1000.1.g.a 4
5.b even 2 1 inner 4000.1.g.a 4
5.c odd 4 1 4000.1.e.a 2
5.c odd 4 1 4000.1.e.b 2
8.b even 2 1 1000.1.g.a 4
8.d odd 2 1 inner 4000.1.g.a 4
20.d odd 2 1 1000.1.g.a 4
20.e even 4 1 1000.1.e.a 2
20.e even 4 1 1000.1.e.b 2
40.e odd 2 1 CM 4000.1.g.a 4
40.f even 2 1 1000.1.g.a 4
40.i odd 4 1 1000.1.e.a 2
40.i odd 4 1 1000.1.e.b 2
40.k even 4 1 4000.1.e.a 2
40.k even 4 1 4000.1.e.b 2

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(4000, [\chi])$$.

## Hecke characteristic polynomials

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