Properties

Label 2009.1.i.a
Level 2009
Weight 1
Character orbit 2009.i
Analytic conductor 1.003
Analytic rank 0
Dimension 6
Projective image \(D_{7}\)
CM disc. -287
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2009.i (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00262161038\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.64827.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.23639903.1

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{3} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} \) \( + ( \beta_{2} - \beta_{3} ) q^{6} \) \( + ( -\beta_{2} + \beta_{3} ) q^{8} \) \( + ( -1 + \beta_{4} + \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{3} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} \) \( + ( \beta_{2} - \beta_{3} ) q^{6} \) \( + ( -\beta_{2} + \beta_{3} ) q^{8} \) \( + ( -1 + \beta_{4} + \beta_{5} ) q^{9} \) \( + ( 1 + \beta_{1} - \beta_{5} ) q^{12} \) \( + \beta_{2} q^{13} \) \( -\beta_{1} q^{16} \) \( + ( \beta_{3} + \beta_{4} ) q^{17} \) \( + \beta_{5} q^{18} \) \( -\beta_{4} q^{19} \) \( + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{23} \) \( + ( \beta_{1} - \beta_{2} - \beta_{5} ) q^{24} \) \( -\beta_{5} q^{25} \) \( + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{26} \) \( + ( 1 + \beta_{3} ) q^{27} \) \( + \beta_{5} q^{32} \) \( + ( 1 - \beta_{2} ) q^{34} \) \( + ( -1 + \beta_{2} - \beta_{3} ) q^{36} \) \( + \beta_{1} q^{37} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{38} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{39} \) \(- q^{41}\) \( + \beta_{3} q^{43} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{46} \) \( -\beta_{1} q^{47} \) \( + ( -\beta_{2} + \beta_{3} ) q^{48} \) \( -\beta_{2} q^{50} \) \( + ( 1 - \beta_{4} - \beta_{5} ) q^{51} \) \( + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{52} \) \( + ( 1 - \beta_{5} ) q^{54} \) \( + ( -1 - \beta_{3} ) q^{57} \) \( + ( -1 + \beta_{5} ) q^{68} \) \( + ( -2 - \beta_{3} ) q^{69} \) \( + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{72} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{74} \) \( + ( -1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{75} \) \(+ q^{76}\) \( + ( 2 \beta_{2} - \beta_{3} ) q^{78} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{81} \) \( -\beta_{1} q^{82} \) \( + ( 1 - \beta_{1} - \beta_{5} ) q^{86} \) \( -\beta_{1} q^{89} \) \( + ( 1 + \beta_{2} ) q^{92} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{94} \) \( + ( 1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{96} \) \( + ( 1 - \beta_{2} + \beta_{3} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 4q^{12} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut q^{16} \) \(\mathstrut -\mathstrut q^{17} \) \(\mathstrut +\mathstrut 3q^{18} \) \(\mathstrut -\mathstrut q^{19} \) \(\mathstrut +\mathstrut q^{23} \) \(\mathstrut -\mathstrut 4q^{24} \) \(\mathstrut -\mathstrut 3q^{25} \) \(\mathstrut +\mathstrut 5q^{26} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut +\mathstrut 3q^{32} \) \(\mathstrut +\mathstrut 4q^{34} \) \(\mathstrut -\mathstrut 2q^{36} \) \(\mathstrut +\mathstrut q^{37} \) \(\mathstrut -\mathstrut 2q^{38} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 2q^{46} \) \(\mathstrut -\mathstrut q^{47} \) \(\mathstrut -\mathstrut 4q^{48} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut +\mathstrut 2q^{51} \) \(\mathstrut -\mathstrut 3q^{52} \) \(\mathstrut +\mathstrut 3q^{54} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 3q^{68} \) \(\mathstrut -\mathstrut 10q^{69} \) \(\mathstrut -\mathstrut q^{72} \) \(\mathstrut -\mathstrut 5q^{74} \) \(\mathstrut -\mathstrut q^{75} \) \(\mathstrut +\mathstrut 6q^{76} \) \(\mathstrut +\mathstrut 6q^{78} \) \(\mathstrut -\mathstrut q^{81} \) \(\mathstrut -\mathstrut q^{82} \) \(\mathstrut +\mathstrut 2q^{86} \) \(\mathstrut -\mathstrut q^{89} \) \(\mathstrut +\mathstrut 8q^{92} \) \(\mathstrut +\mathstrut 5q^{94} \) \(\mathstrut +\mathstrut q^{96} \) \(\mathstrut +\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut +\mathstrut \) \(3\) \(x^{4}\mathstrut +\mathstrut \) \(5\) \(x^{2}\mathstrut -\mathstrut \) \(2\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6 \)\()/13\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18 \)\()/13\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2 \)\()/13\)
\(\beta_{5}\)\(=\)\((\)\( -6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10 \)\()/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{5}\)\(=\)\(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut -\mathstrut \) \(9\) \(\beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2009\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(785\)
\(\chi(n)\) \(1 - \beta_{5}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
−0.623490 + 1.07992i
0.222521 0.385418i
0.900969 1.56052i
−0.623490 1.07992i
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.623490 + 1.07992i −0.222521 0.385418i −0.277479 0.480608i 0 0.554958 0 −0.554958 0.400969 0.694498i 0
901.2 0.222521 0.385418i −0.900969 1.56052i 0.400969 + 0.694498i 0 −0.801938 0 0.801938 −1.12349 + 1.94594i 0
901.3 0.900969 1.56052i 0.623490 + 1.07992i −1.12349 1.94594i 0 2.24698 0 −2.24698 −0.277479 + 0.480608i 0
1844.1 −0.623490 1.07992i −0.222521 + 0.385418i −0.277479 + 0.480608i 0 0.554958 0 −0.554958 0.400969 + 0.694498i 0
1844.2 0.222521 + 0.385418i −0.900969 + 1.56052i 0.400969 0.694498i 0 −0.801938 0 0.801938 −1.12349 1.94594i 0
1844.3 0.900969 + 1.56052i 0.623490 1.07992i −1.12349 + 1.94594i 0 2.24698 0 −2.24698 −0.277479 0.480608i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1844.3
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
287.d Odd 1 CM by \(\Q(\sqrt{-287}) \) yes
7.c Even 1 yes
287.i Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{6} \) \(\mathstrut +\mathstrut T_{3}^{5} \) \(\mathstrut +\mathstrut 3 T_{3}^{4} \) \(\mathstrut +\mathstrut 5 T_{3}^{2} \) \(\mathstrut +\mathstrut 2 T_{3} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(2009, [\chi])\).