# Properties

 Label 2001.1.i.d Level 2001 Weight 1 Character orbit 2001.i Analytic conductor 0.999 Analytic rank 0 Dimension 4 Projective image $$D_{12}$$ CM discriminant -23 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2001.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.998629090279$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{12}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{2} -\zeta_{12}^{2} q^{3} + ( \zeta_{12}^{2} - \zeta_{12}^{3} + \zeta_{12}^{4} ) q^{4} + ( \zeta_{12}^{3} - \zeta_{12}^{4} ) q^{6} + ( -1 - \zeta_{12}^{3} + \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{8} + \zeta_{12}^{4} q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{2} -\zeta_{12}^{2} q^{3} + ( \zeta_{12}^{2} - \zeta_{12}^{3} + \zeta_{12}^{4} ) q^{4} + ( \zeta_{12}^{3} - \zeta_{12}^{4} ) q^{6} + ( -1 - \zeta_{12}^{3} + \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{8} + \zeta_{12}^{4} q^{9} + ( 1 - \zeta_{12}^{4} + \zeta_{12}^{5} ) q^{12} -\zeta_{12}^{3} q^{13} + ( -1 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{16} + ( -1 - \zeta_{12}^{5} ) q^{18} -\zeta_{12}^{3} q^{23} + ( 1 - \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{24} + q^{25} + ( \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{26} + q^{27} -\zeta_{12}^{5} q^{29} + ( \zeta_{12} + \zeta_{12}^{2} ) q^{31} + ( -1 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} - \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{32} + ( -1 + \zeta_{12} - \zeta_{12}^{2} ) q^{36} + \zeta_{12}^{5} q^{39} + ( -\zeta_{12} - \zeta_{12}^{2} ) q^{41} + ( \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{46} + ( -\zeta_{12}^{4} + \zeta_{12}^{5} ) q^{47} + ( 1 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} + \zeta_{12}^{4} ) q^{48} + q^{49} + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{50} + ( -1 + \zeta_{12} - \zeta_{12}^{5} ) q^{52} + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{54} + ( -1 + \zeta_{12} ) q^{58} + 2 \zeta_{12}^{3} q^{59} + ( -\zeta_{12}^{2} + \zeta_{12}^{4} ) q^{62} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} - \zeta_{12}^{4} + \zeta_{12}^{5} ) q^{64} + \zeta_{12}^{5} q^{69} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{71} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} - \zeta_{12}^{4} ) q^{72} + ( -\zeta_{12}^{4} - \zeta_{12}^{5} ) q^{73} -\zeta_{12}^{2} q^{75} + ( 1 - \zeta_{12} ) q^{78} -\zeta_{12}^{2} q^{81} + ( \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{82} -\zeta_{12} q^{87} + ( -1 + \zeta_{12} - \zeta_{12}^{5} ) q^{92} + ( -\zeta_{12}^{3} - \zeta_{12}^{4} ) q^{93} + ( 2 - \zeta_{12} + \zeta_{12}^{5} ) q^{94} + ( -1 - \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} + \zeta_{12}^{4} - \zeta_{12}^{5} ) q^{96} + ( -\zeta_{12} + \zeta_{12}^{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{3} + 2q^{6} - 6q^{8} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{3} + 2q^{6} - 6q^{8} - 2q^{9} + 6q^{12} - 8q^{16} - 4q^{18} + 6q^{24} + 4q^{25} - 2q^{26} + 4q^{27} + 2q^{31} - 4q^{32} - 6q^{36} - 2q^{41} - 2q^{46} + 2q^{47} + 4q^{48} + 4q^{49} + 2q^{50} - 4q^{52} + 2q^{54} - 4q^{58} - 4q^{62} + 2q^{73} - 2q^{75} + 4q^{78} - 2q^{81} + 4q^{82} - 4q^{92} + 2q^{93} + 8q^{94} - 4q^{96} + 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times$$.

 $$n$$ $$553$$ $$668$$ $$1132$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1172.1
 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
−0.366025 + 0.366025i −0.500000 0.866025i 0.732051i 0 0.500000 + 0.133975i 0 −0.633975 0.633975i −0.500000 + 0.866025i 0
1172.2 1.36603 1.36603i −0.500000 + 0.866025i 2.73205i 0 0.500000 + 1.86603i 0 −2.36603 2.36603i −0.500000 0.866025i 0
1931.1 −0.366025 0.366025i −0.500000 + 0.866025i 0.732051i 0 0.500000 0.133975i 0 −0.633975 + 0.633975i −0.500000 0.866025i 0
1931.2 1.36603 + 1.36603i −0.500000 0.866025i 2.73205i 0 0.500000 1.86603i 0 −2.36603 + 2.36603i −0.500000 + 0.866025i 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
23.b odd 2 1 CM by $$\Q(\sqrt{-23})$$
87.f even 4 1 inner
2001.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2001.1.i.d yes 4
3.b odd 2 1 2001.1.i.c 4
23.b odd 2 1 CM 2001.1.i.d yes 4
29.c odd 4 1 2001.1.i.c 4
69.c even 2 1 2001.1.i.c 4
87.f even 4 1 inner 2001.1.i.d yes 4
667.f even 4 1 2001.1.i.c 4
2001.i odd 4 1 inner 2001.1.i.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.1.i.c 4 3.b odd 2 1
2001.1.i.c 4 29.c odd 4 1
2001.1.i.c 4 69.c even 2 1
2001.1.i.c 4 667.f even 4 1
2001.1.i.d yes 4 1.a even 1 1 trivial
2001.1.i.d yes 4 23.b odd 2 1 CM
2001.1.i.d yes 4 87.f even 4 1 inner
2001.1.i.d yes 4 2001.i odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2 T_{2}^{3} + 2 T_{2}^{2} + 2 T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2001, [\chi])$$.

## Hecke characteristic polynomials

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