# Properties

 Label 2001.2.a.f Level $2001$ Weight $2$ Character orbit 2001.a Self dual yes Analytic conductor $15.978$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} + ( -1 - \beta ) q^{5} - q^{6} -3 q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} - q^{3} - q^{4} + ( -1 - \beta ) q^{5} - q^{6} -3 q^{8} + q^{9} + ( -1 - \beta ) q^{10} + ( -1 - \beta ) q^{11} + q^{12} + ( -1 - \beta ) q^{13} + ( 1 + \beta ) q^{15} - q^{16} + q^{18} -2 \beta q^{19} + ( 1 + \beta ) q^{20} + ( -1 - \beta ) q^{22} + q^{23} + 3 q^{24} + 3 \beta q^{25} + ( -1 - \beta ) q^{26} - q^{27} - q^{29} + ( 1 + \beta ) q^{30} + ( -5 + \beta ) q^{31} + 5 q^{32} + ( 1 + \beta ) q^{33} - q^{36} + ( 5 - \beta ) q^{37} -2 \beta q^{38} + ( 1 + \beta ) q^{39} + ( 3 + 3 \beta ) q^{40} + ( 5 + \beta ) q^{41} + ( 6 - 4 \beta ) q^{43} + ( 1 + \beta ) q^{44} + ( -1 - \beta ) q^{45} + q^{46} + ( 2 + 2 \beta ) q^{47} + q^{48} -7 q^{49} + 3 \beta q^{50} + ( 1 + \beta ) q^{52} + ( 6 - 4 \beta ) q^{53} - q^{54} + ( 5 + 3 \beta ) q^{55} + 2 \beta q^{57} - q^{58} + ( 7 + \beta ) q^{59} + ( -1 - \beta ) q^{60} + ( 1 - 5 \beta ) q^{61} + ( -5 + \beta ) q^{62} + 7 q^{64} + ( 5 + 3 \beta ) q^{65} + ( 1 + \beta ) q^{66} + ( 9 - \beta ) q^{67} - q^{69} + ( 13 - \beta ) q^{71} -3 q^{72} -2 \beta q^{73} + ( 5 - \beta ) q^{74} -3 \beta q^{75} + 2 \beta q^{76} + ( 1 + \beta ) q^{78} -2 q^{79} + ( 1 + \beta ) q^{80} + q^{81} + ( 5 + \beta ) q^{82} + ( -2 - 2 \beta ) q^{83} + ( 6 - 4 \beta ) q^{86} + q^{87} + ( 3 + 3 \beta ) q^{88} + ( 2 + 6 \beta ) q^{89} + ( -1 - \beta ) q^{90} - q^{92} + ( 5 - \beta ) q^{93} + ( 2 + 2 \beta ) q^{94} + ( 8 + 4 \beta ) q^{95} -5 q^{96} + ( 2 - 6 \beta ) q^{97} -7 q^{98} + ( -1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} - 2q^{4} - 3q^{5} - 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} - 2q^{4} - 3q^{5} - 2q^{6} - 6q^{8} + 2q^{9} - 3q^{10} - 3q^{11} + 2q^{12} - 3q^{13} + 3q^{15} - 2q^{16} + 2q^{18} - 2q^{19} + 3q^{20} - 3q^{22} + 2q^{23} + 6q^{24} + 3q^{25} - 3q^{26} - 2q^{27} - 2q^{29} + 3q^{30} - 9q^{31} + 10q^{32} + 3q^{33} - 2q^{36} + 9q^{37} - 2q^{38} + 3q^{39} + 9q^{40} + 11q^{41} + 8q^{43} + 3q^{44} - 3q^{45} + 2q^{46} + 6q^{47} + 2q^{48} - 14q^{49} + 3q^{50} + 3q^{52} + 8q^{53} - 2q^{54} + 13q^{55} + 2q^{57} - 2q^{58} + 15q^{59} - 3q^{60} - 3q^{61} - 9q^{62} + 14q^{64} + 13q^{65} + 3q^{66} + 17q^{67} - 2q^{69} + 25q^{71} - 6q^{72} - 2q^{73} + 9q^{74} - 3q^{75} + 2q^{76} + 3q^{78} - 4q^{79} + 3q^{80} + 2q^{81} + 11q^{82} - 6q^{83} + 8q^{86} + 2q^{87} + 9q^{88} + 10q^{89} - 3q^{90} - 2q^{92} + 9q^{93} + 6q^{94} + 20q^{95} - 10q^{96} - 2q^{97} - 14q^{98} - 3q^{99} + O(q^{100})$$

## 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
 2.56155 −1.56155
1.00000 −1.00000 −1.00000 −3.56155 −1.00000 0 −3.00000 1.00000 −3.56155
1.2 1.00000 −1.00000 −1.00000 0.561553 −1.00000 0 −3.00000 1.00000 0.561553
 $$n$$: 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.f 2
3.b odd 2 1 6003.2.a.d 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-2 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-2 + 3 T + T^{2}$$
$13$ $$-2 + 3 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-16 + 2 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$16 + 9 T + T^{2}$$
$37$ $$16 - 9 T + T^{2}$$
$41$ $$26 - 11 T + T^{2}$$
$43$ $$-52 - 8 T + T^{2}$$
$47$ $$-8 - 6 T + T^{2}$$
$53$ $$-52 - 8 T + T^{2}$$
$59$ $$52 - 15 T + T^{2}$$
$61$ $$-104 + 3 T + T^{2}$$
$67$ $$68 - 17 T + T^{2}$$
$71$ $$152 - 25 T + T^{2}$$
$73$ $$-16 + 2 T + T^{2}$$
$79$ $$( 2 + T )^{2}$$
$83$ $$-8 + 6 T + T^{2}$$
$89$ $$-128 - 10 T + T^{2}$$
$97$ $$-152 + 2 T + T^{2}$$