# Properties

 Label 2008.1.j.a Level $2008$ Weight $1$ Character orbit 2008.j Analytic conductor $1.002$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2008 = 2^{3} \cdot 251$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2008.j (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.00212254537$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{5}$$ Projective field Galois closure of 5.1.254024064064.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{3} + q^{4} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{6} + q^{8} + ( \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{3} + q^{4} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{6} + q^{8} + ( \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{9} + ( 1 - \zeta_{10}^{3} ) q^{11} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{12} + q^{16} -2 \zeta_{10} q^{17} + ( \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{18} + 2 \zeta_{10}^{2} q^{19} + ( 1 - \zeta_{10}^{3} ) q^{22} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{24} + q^{25} + ( 1 - \zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{27} + q^{32} + ( 1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{33} -2 \zeta_{10} q^{34} + ( \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{36} + 2 \zeta_{10}^{2} q^{38} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{41} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{43} + ( 1 - \zeta_{10}^{3} ) q^{44} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{48} + \zeta_{10}^{4} q^{49} + q^{50} + ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{51} + ( 1 - \zeta_{10} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{54} + ( -2 \zeta_{10}^{3} + 2 \zeta_{10}^{4} ) q^{57} + ( 1 - \zeta_{10}^{3} ) q^{59} + q^{64} + ( 1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{66} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{67} -2 \zeta_{10} q^{68} + ( \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{72} -2 \zeta_{10} q^{73} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{75} + 2 \zeta_{10}^{2} q^{76} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{81} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{82} + ( 1 + \zeta_{10}^{2} ) q^{83} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{86} + ( 1 - \zeta_{10}^{3} ) q^{88} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{89} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{96} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{97} + \zeta_{10}^{4} q^{98} + ( 1 - \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} - 2q^{3} + 4q^{4} - 2q^{6} + 4q^{8} - 3q^{9} + O(q^{10})$$ $$4q + 4q^{2} - 2q^{3} + 4q^{4} - 2q^{6} + 4q^{8} - 3q^{9} + 3q^{11} - 2q^{12} + 4q^{16} - 2q^{17} - 3q^{18} - 2q^{19} + 3q^{22} - 2q^{24} + 4q^{25} + q^{27} + 4q^{32} + q^{33} - 2q^{34} - 3q^{36} - 2q^{38} - 2q^{41} - 2q^{43} + 3q^{44} - 2q^{48} - q^{49} + 4q^{50} - 4q^{51} + q^{54} - 4q^{57} + 3q^{59} + 4q^{64} + q^{66} - 2q^{67} - 2q^{68} - 3q^{72} - 2q^{73} - 2q^{75} - 2q^{76} - 2q^{82} + 3q^{83} - 2q^{86} + 3q^{88} - 2q^{89} - 2q^{96} - 2q^{97} - q^{98} - q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$257$$ $$503$$ $$1005$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
219.1
 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 + 0.951057i −0.309017 − 0.951057i
1.00000 −0.500000 0.363271i 1.00000 0 −0.500000 0.363271i 0 1.00000 −0.190983 0.587785i 0
651.1 1.00000 −0.500000 + 0.363271i 1.00000 0 −0.500000 + 0.363271i 0 1.00000 −0.190983 + 0.587785i 0
1275.1 1.00000 −0.500000 1.53884i 1.00000 0 −0.500000 1.53884i 0 1.00000 −1.30902 + 0.951057i 0
1619.1 1.00000 −0.500000 + 1.53884i 1.00000 0 −0.500000 + 1.53884i 0 1.00000 −1.30902 0.951057i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
251.c even 5 1 inner
2008.j odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2008.1.j.a 4
8.d odd 2 1 CM 2008.1.j.a 4
251.c even 5 1 inner 2008.1.j.a 4
2008.j odd 10 1 inner 2008.1.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.1.j.a 4 1.a even 1 1 trivial
2008.1.j.a 4 8.d odd 2 1 CM
2008.1.j.a 4 251.c even 5 1 inner
2008.1.j.a 4 2008.j odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2008, [\chi])$$.

## Hecke characteristic polynomials

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