# 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 discriminant -223 Inner twists 4

# Related objects

## Newspace parameters

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

## 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: $$1$$ Twist minimal: yes 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 root $$\beta$$ of the polynomial $$x^{6} - 7 x^{4} + 14 x^{2} - 7$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( -1 + \beta^{2} ) q^{4} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{7} + ( 2 \beta - \beta^{3} ) q^{8} +O(q^{10})$$ $$q -\beta q^{2} + ( -1 + \beta^{2} ) q^{4} + ( -2 + 4 \beta^{2} - \beta^{4} ) q^{7} + ( 2 \beta - \beta^{3} ) q^{8} + ( 2 \beta - 4 \beta^{3} + \beta^{5} ) q^{14} + ( 1 - 3 \beta^{2} + \beta^{4} ) q^{16} + ( -5 \beta + 5 \beta^{3} - \beta^{5} ) q^{17} + ( 2 - \beta^{2} ) q^{19} + q^{25} + ( -5 + 8 \beta^{2} - 2 \beta^{4} ) q^{28} + \beta q^{29} + ( 5 - 5 \beta^{2} + \beta^{4} ) q^{31} + ( -3 \beta + 4 \beta^{3} - \beta^{5} ) q^{32} + ( 7 - 9 \beta^{2} + 2 \beta^{4} ) q^{34} + ( 2 - 4 \beta^{2} + \beta^{4} ) q^{37} + ( -2 \beta + \beta^{3} ) q^{38} + ( -3 \beta + \beta^{3} ) q^{41} + ( -5 + 5 \beta^{2} - \beta^{4} ) q^{43} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{47} + ( -4 + 5 \beta^{2} - \beta^{4} ) q^{49} -\beta q^{50} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{53} + ( 3 \beta - 4 \beta^{3} + \beta^{5} ) q^{56} -\beta^{2} q^{58} + ( -5 \beta + 5 \beta^{3} - \beta^{5} ) q^{62} + ( 6 - 8 \beta^{2} + 2 \beta^{4} ) q^{64} + ( -2 \beta + 4 \beta^{3} - \beta^{5} ) q^{68} + ( 5 - 5 \beta^{2} + \beta^{4} ) q^{73} + ( -2 \beta + 4 \beta^{3} - \beta^{5} ) q^{74} + ( -2 + 3 \beta^{2} - \beta^{4} ) q^{76} + ( 3 \beta^{2} - \beta^{4} ) q^{82} + \beta q^{83} + ( 5 \beta - 5 \beta^{3} + \beta^{5} ) q^{86} + ( 3 \beta - \beta^{3} ) q^{89} + ( -7 + 9 \beta^{2} - 2 \beta^{4} ) q^{94} + ( 4 \beta - 5 \beta^{3} + \beta^{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 8q^{4} + 2q^{7} + O(q^{10})$$ $$6q + 8q^{4} + 2q^{7} + 6q^{16} - 2q^{19} + 6q^{25} - 2q^{28} + 2q^{31} - 2q^{37} - 2q^{43} + 4q^{49} - 14q^{58} + 8q^{64} + 2q^{73} - 12q^{76} + O(q^{100})$$

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
223.b odd 2 1 CM by $$\Q(\sqrt{-223})$$
3.b odd 2 1 inner
669.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2007.1.d.c 6
3.b odd 2 1 inner 2007.1.d.c 6
223.b odd 2 1 CM 2007.1.d.c 6
669.c even 2 1 inner 2007.1.d.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2007.1.d.c 6 1.a even 1 1 trivial
2007.1.d.c 6 3.b odd 2 1 inner
2007.1.d.c 6 223.b odd 2 1 CM
2007.1.d.c 6 669.c even 2 1 inner

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$3$ 1
$5$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}$$
$11$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$13$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$17$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$19$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$23$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$29$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$31$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}$$
$37$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$41$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$43$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$47$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$53$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$59$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$61$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$67$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$71$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$73$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}$$
$79$ $$( 1 - T )^{6}( 1 + T )^{6}$$
$83$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$89$ $$1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12}$$
$97$ $$( 1 - T )^{6}( 1 + T )^{6}$$