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 discriminant -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 \(6\), degree \(2\), not minimal)

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: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Projective image \(D_{7}\)
Projective field Galois closure of 7.1.23639903.1
Artin image $C_6\times D_7$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{42} + \cdots)\)

$q$-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.

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

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
287.d odd 2 1 CM by \(\Q(\sqrt{-287}) \)
7.c even 3 1 inner
287.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2009.1.i.a 6
7.b odd 2 1 2009.1.i.b 6
7.c even 3 1 287.1.d.b yes 3
7.c even 3 1 inner 2009.1.i.a 6
7.d odd 6 1 287.1.d.a 3
7.d odd 6 1 2009.1.i.b 6
21.g even 6 1 2583.1.f.b 3
21.h odd 6 1 2583.1.f.a 3
41.b even 2 1 2009.1.i.b 6
287.d odd 2 1 CM 2009.1.i.a 6
287.i odd 6 1 287.1.d.b yes 3
287.i odd 6 1 inner 2009.1.i.a 6
287.j even 6 1 287.1.d.a 3
287.j even 6 1 2009.1.i.b 6
861.r even 6 1 2583.1.f.a 3
861.t odd 6 1 2583.1.f.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.1.d.a 3 7.d odd 6 1
287.1.d.a 3 287.j even 6 1
287.1.d.b yes 3 7.c even 3 1
287.1.d.b yes 3 287.i odd 6 1
2009.1.i.a 6 1.a even 1 1 trivial
2009.1.i.a 6 7.c even 3 1 inner
2009.1.i.a 6 287.d odd 2 1 CM
2009.1.i.a 6 287.i odd 6 1 inner
2009.1.i.b 6 7.b odd 2 1
2009.1.i.b 6 7.d odd 6 1
2009.1.i.b 6 41.b even 2 1
2009.1.i.b 6 287.j even 6 1
2583.1.f.a 3 21.h odd 6 1
2583.1.f.a 3 861.r even 6 1
2583.1.f.b 3 21.g even 6 1
2583.1.f.b 3 861.t odd 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
$3$ \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
$5$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$7$ 1
$11$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$13$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
$17$ \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
$19$ \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
$23$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
$29$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$31$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$37$ \( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \)
$41$ \( ( 1 + T )^{6} \)
$43$ \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \)
$47$ \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
$53$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$59$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$61$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$67$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$71$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$73$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$79$ \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \)
$83$ \( ( 1 - T )^{6}( 1 + T )^{6} \)
$89$ \( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \)
$97$ \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \)
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