# 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 disc. -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 $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: Yes Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
40.e Odd 1 CM by $$\Q(\sqrt{-10})$$ yes

## Hecke kernels

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