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

## Newform invariants

 Self dual: No Analytic conductor: $$1.99626005053$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ 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 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$3$$ $$x^{2}\mathstrut +\mathstrut$$ $$1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{1}$$

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

## Hecke kernels

There are no other newforms in $$S_{1}^{\mathrm{new}}(4000, [\chi])$$.