# Properties

 Label 2008.1.c.a Level 2008 Weight 1 Character orbit 2008.c Self dual yes Analytic conductor 1.002 Analytic rank 0 Dimension 3 Projective image $$D_{7}$$ CM discriminant -2008 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2008 = 2^{3} \cdot 251$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2008.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.00212254537$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{7}$$ Projective field Galois closure of 7.1.8096384512.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} -\beta q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{4} -\beta q^{7} - q^{8} + q^{9} + ( 2 - \beta^{2} ) q^{11} + \beta q^{14} + q^{16} + ( 1 + \beta - \beta^{2} ) q^{17} - q^{18} + ( -1 - \beta + \beta^{2} ) q^{19} + ( -2 + \beta^{2} ) q^{22} + ( -2 + \beta^{2} ) q^{23} + q^{25} -\beta q^{28} + ( -1 - \beta + \beta^{2} ) q^{29} + ( 1 + \beta - \beta^{2} ) q^{31} - q^{32} + ( -1 - \beta + \beta^{2} ) q^{34} + q^{36} + \beta q^{37} + ( 1 + \beta - \beta^{2} ) q^{38} + ( -2 + \beta^{2} ) q^{41} + \beta q^{43} + ( 2 - \beta^{2} ) q^{44} + ( 2 - \beta^{2} ) q^{46} + ( -1 + \beta^{2} ) q^{49} - q^{50} + ( 2 - \beta^{2} ) q^{53} + \beta q^{56} + ( 1 + \beta - \beta^{2} ) q^{58} + \beta q^{59} + ( 2 - \beta^{2} ) q^{61} + ( -1 - \beta + \beta^{2} ) q^{62} -\beta q^{63} + q^{64} + ( 1 + \beta - \beta^{2} ) q^{68} - q^{72} -\beta q^{73} -\beta q^{74} + ( -1 - \beta + \beta^{2} ) q^{76} + ( -1 + \beta^{2} ) q^{77} + ( 1 + \beta - \beta^{2} ) q^{79} + q^{81} + ( 2 - \beta^{2} ) q^{82} -\beta q^{86} + ( -2 + \beta^{2} ) q^{88} -\beta q^{89} + ( -2 + \beta^{2} ) q^{92} + ( 1 - \beta^{2} ) q^{98} + ( 2 - \beta^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 3q^{4} - q^{7} - 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{2} + 3q^{4} - q^{7} - 3q^{8} + 3q^{9} + q^{11} + q^{14} + 3q^{16} - q^{17} - 3q^{18} + q^{19} - q^{22} - q^{23} + 3q^{25} - q^{28} + q^{29} - q^{31} - 3q^{32} + q^{34} + 3q^{36} + q^{37} - q^{38} - q^{41} + q^{43} + q^{44} + q^{46} + 2q^{49} - 3q^{50} + q^{53} + q^{56} - q^{58} + q^{59} + q^{61} + q^{62} - q^{63} + 3q^{64} - q^{68} - 3q^{72} - q^{73} - q^{74} + q^{76} + 2q^{77} - q^{79} + 3q^{81} + q^{82} - q^{86} - q^{88} - q^{89} - q^{92} - 2q^{98} + q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2008\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$503$$ $$1005$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
501.1
 1.80194 0.445042 −1.24698
−1.00000 0 1.00000 0 0 −1.80194 −1.00000 1.00000 0
501.2 −1.00000 0 1.00000 0 0 −0.445042 −1.00000 1.00000 0
501.3 −1.00000 0 1.00000 0 0 1.24698 −1.00000 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
2008.c odd 2 1 CM by $$\Q(\sqrt{-502})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2008.1.c.a 3
8.b even 2 1 2008.1.c.b yes 3
251.b odd 2 1 2008.1.c.b yes 3
2008.c odd 2 1 CM 2008.1.c.a 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.1.c.a 3 1.a even 1 1 trivial
2008.1.c.a 3 2008.c odd 2 1 CM
2008.1.c.b yes 3 8.b even 2 1
2008.1.c.b yes 3 251.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{3} - T_{11}^{2} - 2 T_{11} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2008, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$5$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$7$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$11$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$13$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$17$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$19$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$23$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$29$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$31$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$37$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$41$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$43$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$47$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$53$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$59$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$61$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$67$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$71$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$73$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$79$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$83$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$89$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$97$ $$( 1 - T )^{3}( 1 + T )^{3}$$