# Properties

 Label 2001.2.a.a 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + 3q^{5} + q^{6} + 3q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} - q^{4} + 3q^{5} + q^{6} + 3q^{8} + q^{9} - 3q^{10} + 3q^{11} + q^{12} + 3q^{13} - 3q^{15} - q^{16} - q^{18} + 8q^{19} - 3q^{20} - 3q^{22} - q^{23} - 3q^{24} + 4q^{25} - 3q^{26} - q^{27} + q^{29} + 3q^{30} + 7q^{31} - 5q^{32} - 3q^{33} - q^{36} - 5q^{37} - 8q^{38} - 3q^{39} + 9q^{40} + 3q^{41} + 2q^{43} - 3q^{44} + 3q^{45} + q^{46} - 6q^{47} + q^{48} - 7q^{49} - 4q^{50} - 3q^{52} - 2q^{53} + q^{54} + 9q^{55} - 8q^{57} - q^{58} + q^{59} + 3q^{60} + 7q^{61} - 7q^{62} + 7q^{64} + 9q^{65} + 3q^{66} - 9q^{67} + q^{69} + 3q^{71} + 3q^{72} - 12q^{73} + 5q^{74} - 4q^{75} - 8q^{76} + 3q^{78} - 14q^{79} - 3q^{80} + q^{81} - 3q^{82} + 14q^{83} - 2q^{86} - q^{87} + 9q^{88} - 10q^{89} - 3q^{90} + q^{92} - 7q^{93} + 6q^{94} + 24q^{95} + 5q^{96} + 14q^{97} + 7q^{98} + 3q^{99} + O(q^{100})$$

## 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
−1.00000 −1.00000 −1.00000 3.00000 1.00000 0 3.00000 1.00000 −3.00000
 $$n$$: 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.a 1
3.b odd 2 1 6003.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.a 1 1.a even 1 1 trivial
6003.2.a.c 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2001))$$:

 $$T_{2} + 1$$ $$T_{5} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$-3 + T$$
$7$ $$T$$
$11$ $$-3 + T$$
$13$ $$-3 + T$$
$17$ $$T$$
$19$ $$-8 + T$$
$23$ $$1 + T$$
$29$ $$-1 + T$$
$31$ $$-7 + T$$
$37$ $$5 + T$$
$41$ $$-3 + T$$
$43$ $$-2 + T$$
$47$ $$6 + T$$
$53$ $$2 + T$$
$59$ $$-1 + T$$
$61$ $$-7 + T$$
$67$ $$9 + T$$
$71$ $$-3 + T$$
$73$ $$12 + T$$
$79$ $$14 + T$$
$83$ $$-14 + T$$
$89$ $$10 + T$$
$97$ $$-14 + T$$