Properties

Label 2008.1.c.b
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

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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: \(\Q(\zeta_{14})^+\)
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
Artin image $D_7$
Artin field Galois closure of 7.1.8096384512.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a root \(\beta\) of the polynomial \(x^{3} - x^{2} - 2 x + 1\). 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:

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

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

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