Properties

Label 2007.1.d.c
Level 2007
Weight 1
Character orbit 2007.d
Self dual Yes
Analytic conductor 1.002
Analytic rank 0
Dimension 6
Projective image \(D_{14}\)
CM disc. -223
Inner twists 4

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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: \(6\)
Coefficient field: \(\Q(\zeta_{28})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{14}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( 1 - \beta_{2} - \beta_{4} ) q^{7} \) \( + ( -\beta_{1} - \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( 1 - \beta_{2} - \beta_{4} ) q^{7} \) \( + ( -\beta_{1} - \beta_{3} ) q^{8} \) \( + ( \beta_{3} + \beta_{5} ) q^{14} \) \( + ( 2 \beta_{2} + \beta_{4} ) q^{16} \) \( -\beta_{5} q^{17} \) \( -\beta_{2} q^{19} \) \(+ q^{25}\) \( + ( 1 - 2 \beta_{2} - 2 \beta_{4} ) q^{28} \) \( + \beta_{1} q^{29} \) \( + \beta_{4} q^{31} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{32} \) \( + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{34} \) \( + ( -1 + \beta_{2} + \beta_{4} ) q^{37} \) \( + ( \beta_{1} + \beta_{3} ) q^{38} \) \( + \beta_{3} q^{41} \) \( -\beta_{4} q^{43} \) \( + \beta_{5} q^{47} \) \( + ( 1 - \beta_{4} ) q^{49} \) \( -\beta_{1} q^{50} \) \( + \beta_{5} q^{53} \) \( + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{56} \) \( + ( -2 - \beta_{2} ) q^{58} \) \( -\beta_{5} q^{62} \) \( + ( 2 \beta_{2} + 2 \beta_{4} ) q^{64} \) \( + ( -\beta_{3} - \beta_{5} ) q^{68} \) \( + \beta_{4} q^{73} \) \( + ( -\beta_{3} - \beta_{5} ) q^{74} \) \( + ( -1 - 2 \beta_{2} - \beta_{4} ) q^{76} \) \( + ( 1 - 2 \beta_{2} - \beta_{4} ) q^{82} \) \( + \beta_{1} q^{83} \) \( + \beta_{5} q^{86} \) \( -\beta_{3} q^{89} \) \( + ( 1 - \beta_{2} - 2 \beta_{4} ) q^{94} \) \( + ( -\beta_{1} + \beta_{5} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 6q^{16} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut -\mathstrut 2q^{28} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 14q^{58} \) \(\mathstrut +\mathstrut 8q^{64} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 12q^{76} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{28} + \zeta_{28}^{-1}\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 5 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 5 \nu^{3} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{5}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\)

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
1.94986
1.56366
0.867767
−0.867767
−1.56366
−1.94986
−1.94986 0 2.80194 0 0 −1.24698 −3.51352 0 0
1783.2 −1.56366 0 1.44504 0 0 1.80194 −0.695895 0 0
1783.3 −0.867767 0 −0.246980 0 0 0.445042 1.08209 0 0
1783.4 0.867767 0 −0.246980 0 0 0.445042 −1.08209 0 0
1783.5 1.56366 0 1.44504 0 0 1.80194 0.695895 0 0
1783.6 1.94986 0 2.80194 0 0 −1.24698 3.51352 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1783.6
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
3.b Odd 1 yes
669.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{6} \) \(\mathstrut -\mathstrut 7 T_{2}^{4} \) \(\mathstrut +\mathstrut 14 T_{2}^{2} \) \(\mathstrut -\mathstrut 7 \) acting on \(S_{1}^{\mathrm{new}}(2007, [\chi])\).