# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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