# Properties

 Label 2001.2.a.d 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} -2 q^{5} + q^{6} + 3 q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} - q^{4} -2 q^{5} + q^{6} + 3 q^{8} + q^{9} + 2 q^{10} + ( -2 - \beta ) q^{11} + q^{12} -2 q^{13} + 2 q^{15} - q^{16} + \beta q^{17} - q^{18} + ( -2 - \beta ) q^{19} + 2 q^{20} + ( 2 + \beta ) q^{22} - q^{23} -3 q^{24} - q^{25} + 2 q^{26} - q^{27} + q^{29} -2 q^{30} -8 q^{31} -5 q^{32} + ( 2 + \beta ) q^{33} -\beta q^{34} - q^{36} -\beta q^{37} + ( 2 + \beta ) q^{38} + 2 q^{39} -6 q^{40} + ( -2 - 2 \beta ) q^{41} + ( 2 + \beta ) q^{43} + ( 2 + \beta ) q^{44} -2 q^{45} + q^{46} + ( 4 + 2 \beta ) q^{47} + q^{48} -7 q^{49} + q^{50} -\beta q^{51} + 2 q^{52} -2 q^{53} + q^{54} + ( 4 + 2 \beta ) q^{55} + ( 2 + \beta ) q^{57} - q^{58} -4 q^{59} -2 q^{60} + ( -8 - \beta ) q^{61} + 8 q^{62} + 7 q^{64} + 4 q^{65} + ( -2 - \beta ) q^{66} -4 q^{67} -\beta q^{68} + q^{69} + 8 q^{71} + 3 q^{72} + ( -2 - 2 \beta ) q^{73} + \beta q^{74} + q^{75} + ( 2 + \beta ) q^{76} -2 q^{78} + ( 6 + \beta ) q^{79} + 2 q^{80} + q^{81} + ( 2 + 2 \beta ) q^{82} + 4 q^{83} -2 \beta q^{85} + ( -2 - \beta ) q^{86} - q^{87} + ( -6 - 3 \beta ) q^{88} -3 \beta q^{89} + 2 q^{90} + q^{92} + 8 q^{93} + ( -4 - 2 \beta ) q^{94} + ( 4 + 2 \beta ) q^{95} + 5 q^{96} + ( 4 + 3 \beta ) q^{97} + 7 q^{98} + ( -2 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} - 2q^{4} - 4q^{5} + 2q^{6} + 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} - 2q^{4} - 4q^{5} + 2q^{6} + 6q^{8} + 2q^{9} + 4q^{10} - 4q^{11} + 2q^{12} - 4q^{13} + 4q^{15} - 2q^{16} - 2q^{18} - 4q^{19} + 4q^{20} + 4q^{22} - 2q^{23} - 6q^{24} - 2q^{25} + 4q^{26} - 2q^{27} + 2q^{29} - 4q^{30} - 16q^{31} - 10q^{32} + 4q^{33} - 2q^{36} + 4q^{38} + 4q^{39} - 12q^{40} - 4q^{41} + 4q^{43} + 4q^{44} - 4q^{45} + 2q^{46} + 8q^{47} + 2q^{48} - 14q^{49} + 2q^{50} + 4q^{52} - 4q^{53} + 2q^{54} + 8q^{55} + 4q^{57} - 2q^{58} - 8q^{59} - 4q^{60} - 16q^{61} + 16q^{62} + 14q^{64} + 8q^{65} - 4q^{66} - 8q^{67} + 2q^{69} + 16q^{71} + 6q^{72} - 4q^{73} + 2q^{75} + 4q^{76} - 4q^{78} + 12q^{79} + 4q^{80} + 2q^{81} + 4q^{82} + 8q^{83} - 4q^{86} - 2q^{87} - 12q^{88} + 4q^{90} + 2q^{92} + 16q^{93} - 8q^{94} + 8q^{95} + 10q^{96} + 8q^{97} + 14q^{98} - 4q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
−1.00000 −1.00000 −1.00000 −2.00000 1.00000 0 3.00000 1.00000 2.00000
1.2 −1.00000 −1.00000 −1.00000 −2.00000 1.00000 0 3.00000 1.00000 2.00000
 $$n$$: 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.d 2
3.b odd 2 1 6003.2.a.f 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 2 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-16 + 4 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$-20 + T^{2}$$
$19$ $$-16 + 4 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$-20 + T^{2}$$
$41$ $$-76 + 4 T + T^{2}$$
$43$ $$-16 - 4 T + T^{2}$$
$47$ $$-64 - 8 T + T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$44 + 16 T + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$-76 + 4 T + T^{2}$$
$79$ $$16 - 12 T + T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$-180 + T^{2}$$
$97$ $$-164 - 8 T + T^{2}$$