# Properties

 Label 2001.2.a.b Level $2001$ Weight $2$ Character orbit 2001.a Self dual yes Analytic conductor $15.978$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.9780654445$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - 2q^{4} + 4q^{5} - 4q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - 2q^{4} + 4q^{5} - 4q^{7} + q^{9} + 4q^{11} + 2q^{12} - 5q^{13} - 4q^{15} + 4q^{16} - 5q^{17} + 5q^{19} - 8q^{20} + 4q^{21} + q^{23} + 11q^{25} - q^{27} + 8q^{28} - q^{29} - 2q^{31} - 4q^{33} - 16q^{35} - 2q^{36} + 5q^{37} + 5q^{39} - 2q^{41} + q^{43} - 8q^{44} + 4q^{45} + 6q^{47} - 4q^{48} + 9q^{49} + 5q^{51} + 10q^{52} + 2q^{53} + 16q^{55} - 5q^{57} + 9q^{59} + 8q^{60} - 10q^{61} - 4q^{63} - 8q^{64} - 20q^{65} + 8q^{67} + 10q^{68} - q^{69} - 3q^{71} + 8q^{73} - 11q^{75} - 10q^{76} - 16q^{77} + 13q^{79} + 16q^{80} + q^{81} - 6q^{83} - 8q^{84} - 20q^{85} + q^{87} - 9q^{89} + 20q^{91} - 2q^{92} + 2q^{93} + 20q^{95} - 6q^{97} + 4q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 −2.00000 4.00000 0 −4.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$23$$ $$-1$$
$$29$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2001.2.a.b 1
3.b odd 2 1 6003.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.b 1 1.a even 1 1 trivial
6003.2.a.a 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2001))$$:

 $$T_{2}$$ $$T_{5} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$-4 + T$$
$7$ $$4 + T$$
$11$ $$-4 + T$$
$13$ $$5 + T$$
$17$ $$5 + T$$
$19$ $$-5 + T$$
$23$ $$-1 + T$$
$29$ $$1 + T$$
$31$ $$2 + T$$
$37$ $$-5 + T$$
$41$ $$2 + T$$
$43$ $$-1 + T$$
$47$ $$-6 + T$$
$53$ $$-2 + T$$
$59$ $$-9 + T$$
$61$ $$10 + T$$
$67$ $$-8 + T$$
$71$ $$3 + T$$
$73$ $$-8 + T$$
$79$ $$-13 + T$$
$83$ $$6 + T$$
$89$ $$9 + T$$
$97$ $$6 + T$$