# Properties

 Label 2001.1.i.a Level 2001 Weight 1 Character orbit 2001.i Analytic conductor 0.999 Analytic rank 0 Dimension 2 Projective image $$D_{4}$$ 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: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.2.5048523.1

## $q$-expansion

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

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

## Character values

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

 $$n$$ $$553$$ $$668$$ $$1132$$ $$\chi(n)$$ $$i$$ $$-1$$ $$-1$$

## 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
 1.00000i − 1.00000i
−1.00000 + 1.00000i 1.00000 1.00000i 0 −1.00000 + 1.00000i 0 0 1.00000 0
1931.1 −1.00000 1.00000i 1.00000 1.00000i 0 −1.00000 1.00000i 0 0 1.00000 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.a 2
3.b odd 2 1 2001.1.i.b yes 2
23.b odd 2 1 CM 2001.1.i.a 2
29.c odd 4 1 2001.1.i.b yes 2
69.c even 2 1 2001.1.i.b yes 2
87.f even 4 1 inner 2001.1.i.a 2
667.f even 4 1 2001.1.i.b yes 2
2001.i odd 4 1 inner 2001.1.i.a 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

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