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 disc. -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 \(2\) and degree \(1\))

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: \( 1 \)
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.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\)

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
2008.c Odd 1 CM by \(\Q(\sqrt{-502}) \) yes

Hecke kernels

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