Properties

Label 2007.1.d.b
Level 2007
Weight 1
Character orbit 2007.d
Self dual Yes
Analytic conductor 1.002
Analytic rank 0
Dimension 3
Projective image \(D_{7}\)
CM disc. -223
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2007 = 3^{2} \cdot 223 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2007.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(1.00162348035\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.11089567.1

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{8} \) \( + ( 1 - \beta_{1} ) q^{14} \) \( + \beta_{1} q^{16} \) \( -\beta_{2} q^{17} \) \( + \beta_{2} q^{19} \) \(+ q^{25}\) \(- q^{28}\) \( + \beta_{1} q^{29} \) \( -\beta_{1} q^{31} \) \(+ q^{32}\) \( + ( -1 - \beta_{2} ) q^{34} \) \( + ( -1 + \beta_{1} - \beta_{2} ) q^{37} \) \( + ( 1 + \beta_{2} ) q^{38} \) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{41} \) \( -\beta_{1} q^{43} \) \( -\beta_{2} q^{47} \) \( + ( 1 - \beta_{1} ) q^{49} \) \( + \beta_{1} q^{50} \) \( -\beta_{2} q^{53} \) \(- q^{56}\) \( + ( 2 + \beta_{2} ) q^{58} \) \( + ( -2 - \beta_{2} ) q^{62} \) \( + ( -1 - \beta_{1} ) q^{68} \) \( -\beta_{1} q^{73} \) \( + ( 1 - \beta_{1} ) q^{74} \) \( + ( 1 + \beta_{1} ) q^{76} \) \( + ( -1 + \beta_{1} ) q^{82} \) \( + \beta_{1} q^{83} \) \( + ( -2 - \beta_{2} ) q^{86} \) \( + ( 1 - \beta_{1} + \beta_{2} ) q^{89} \) \( + ( -1 - \beta_{2} ) q^{94} \) \( + ( -2 + \beta_{1} - \beta_{2} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut q^{16} \) \(\mathstrut +\mathstrut q^{17} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut +\mathstrut 3q^{25} \) \(\mathstrut -\mathstrut 3q^{28} \) \(\mathstrut +\mathstrut q^{29} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut +\mathstrut 3q^{32} \) \(\mathstrut -\mathstrut 2q^{34} \) \(\mathstrut -\mathstrut q^{37} \) \(\mathstrut +\mathstrut 2q^{38} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut -\mathstrut q^{43} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut q^{50} \) \(\mathstrut +\mathstrut q^{53} \) \(\mathstrut -\mathstrut 3q^{56} \) \(\mathstrut +\mathstrut 5q^{58} \) \(\mathstrut -\mathstrut 5q^{62} \) \(\mathstrut -\mathstrut 4q^{68} \) \(\mathstrut -\mathstrut q^{73} \) \(\mathstrut +\mathstrut 2q^{74} \) \(\mathstrut +\mathstrut 4q^{76} \) \(\mathstrut -\mathstrut 2q^{82} \) \(\mathstrut +\mathstrut q^{83} \) \(\mathstrut -\mathstrut 5q^{86} \) \(\mathstrut +\mathstrut q^{89} \) \(\mathstrut -\mathstrut 2q^{94} \) \(\mathstrut -\mathstrut 4q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times\).

\(n\) \(226\) \(893\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1783.1
−1.24698
0.445042
1.80194
−1.24698 0 0.554958 0 0 −1.80194 0.554958 0 0
1783.2 0.445042 0 −0.801938 0 0 1.24698 −0.801938 0 0
1783.3 1.80194 0 2.24698 0 0 −0.445042 2.24698 0 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
223.b Odd 1 CM by \(\Q(\sqrt{-223}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{3} \) \(\mathstrut -\mathstrut T_{2}^{2} \) \(\mathstrut -\mathstrut 2 T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(2007, [\chi])\).