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 $4$

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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:

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

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