# Properties

 Label 2007.1.d.a Level 2007 Weight 1 Character orbit 2007.d Self dual yes Analytic conductor 1.002 Analytic rank 0 Dimension 1 Projective image $$D_{2}$$ CM/RM discs -3, -223, 669 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2007 = 3^{2} \cdot 223$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2007.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.00162348035$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-3}, \sqrt{-223})$$ Artin image $D_4$ Artin field Galois closure of 4.0.6021.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{4} - 2q^{7} + O(q^{10})$$ $$q - q^{4} - 2q^{7} + q^{16} + 2q^{19} + q^{25} + 2q^{28} - 2q^{31} + 2q^{37} + 2q^{43} + 3q^{49} - q^{64} - 2q^{73} - 2q^{76} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times$$.

 $$n$$ $$226$$ $$893$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1783.1
 0
0 0 −1.00000 0 0 −2.00000 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
223.b odd 2 1 CM by $$\Q(\sqrt{-223})$$
669.c even 2 1 RM by $$\Q(\sqrt{669})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2007.1.d.a 1
3.b odd 2 1 CM 2007.1.d.a 1
223.b odd 2 1 CM 2007.1.d.a 1
669.c even 2 1 RM 2007.1.d.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2007.1.d.a 1 1.a even 1 1 trivial
2007.1.d.a 1 3.b odd 2 1 CM
2007.1.d.a 1 223.b odd 2 1 CM
2007.1.d.a 1 669.c even 2 1 RM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(2007, [\chi])$$.

## Hecke characteristic polynomials

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