# Properties

 Label 2583.1.f.b Level 2583 Weight 1 Character orbit 2583.f Self dual yes Analytic conductor 1.289 Analytic rank 0 Dimension 3 Projective image $$D_{7}$$ CM discriminant -287 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2583 = 3^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2583.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.28908492763$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $D_{14}$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{14} + \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( -1 + \beta^{2} ) q^{4} + q^{7} + ( -1 + \beta^{2} ) q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( -1 + \beta^{2} ) q^{4} + q^{7} + ( -1 + \beta^{2} ) q^{8} -\beta q^{13} + \beta q^{14} + \beta q^{16} + ( -1 - \beta + \beta^{2} ) q^{17} + ( 1 + \beta - \beta^{2} ) q^{19} + ( 2 - \beta^{2} ) q^{23} + q^{25} -\beta^{2} q^{26} + ( -1 + \beta^{2} ) q^{28} + q^{32} + ( -1 + \beta ) q^{34} -\beta q^{37} + ( 1 - \beta ) q^{38} - q^{41} + ( 1 + \beta - \beta^{2} ) q^{43} + ( 1 - \beta^{2} ) q^{46} + \beta q^{47} + q^{49} + \beta q^{50} + ( 1 - \beta - \beta^{2} ) q^{52} + ( -1 + \beta^{2} ) q^{56} + q^{68} -\beta^{2} q^{74} - q^{76} -\beta q^{82} + ( 1 - \beta ) q^{86} + \beta q^{89} -\beta q^{91} + ( -1 - \beta ) q^{92} + \beta^{2} q^{94} + ( -2 + \beta^{2} ) q^{97} + \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} + 2q^{4} + 3q^{7} + 2q^{8} + O(q^{10})$$ $$3q + q^{2} + 2q^{4} + 3q^{7} + 2q^{8} - q^{13} + q^{14} + q^{16} + q^{17} - q^{19} + q^{23} + 3q^{25} - 5q^{26} + 2q^{28} + 3q^{32} - 2q^{34} - q^{37} + 2q^{38} - 3q^{41} - q^{43} - 2q^{46} + q^{47} + 3q^{49} + q^{50} - 3q^{52} + 2q^{56} + 3q^{68} - 5q^{74} - 3q^{76} - q^{82} + 2q^{86} + q^{89} - q^{91} - 4q^{92} + 5q^{94} - q^{97} + q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2583\mathbb{Z}\right)^\times$$.

 $$n$$ $$1072$$ $$2215$$ $$2297$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2008.1
 −1.24698 0.445042 1.80194
−1.24698 0 0.554958 0 0 1.00000 0.554958 0 0
2008.2 0.445042 0 −0.801938 0 0 1.00000 −0.801938 0 0
2008.3 1.80194 0 2.24698 0 0 1.00000 2.24698 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 $$\Q(\sqrt{-287})$$

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.1.d.a 3 3.b odd 2 1
287.1.d.a 3 861.e even 2 1
287.1.d.b yes 3 21.c even 2 1
287.1.d.b yes 3 123.b odd 2 1
2009.1.i.a 6 21.g even 6 2
2009.1.i.a 6 861.t odd 6 2
2009.1.i.b 6 21.h odd 6 2
2009.1.i.b 6 861.r even 6 2
2583.1.f.a 3 7.b odd 2 1
2583.1.f.a 3 41.b even 2 1
2583.1.f.b 3 1.a even 1 1 trivial
2583.1.f.b 3 287.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{3} + T_{13}^{2} - 2 T_{13} - 1$$ acting on $$S_{1}^{\mathrm{new}}(2583, [\chi])$$.

## Hecke characteristic polynomials

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