# Properties

 Label 2001.2.a.g Level $2001$ Weight $2$ Character orbit 2001.a Self dual yes Analytic conductor $15.978$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.5744.1 Defining polynomial: $$x^{4} - 5 x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} -2 q^{4} -\beta_{1} q^{5} + ( -1 - \beta_{2} + \beta_{3} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} -2 q^{4} -\beta_{1} q^{5} + ( -1 - \beta_{2} + \beta_{3} ) q^{7} + q^{9} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + 2 q^{12} + ( 1 + 2 \beta_{2} ) q^{13} + \beta_{1} q^{15} + 4 q^{16} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{17} + ( 2 + 2 \beta_{2} + \beta_{3} ) q^{19} + 2 \beta_{1} q^{20} + ( 1 + \beta_{2} - \beta_{3} ) q^{21} + q^{23} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} - q^{27} + ( 2 + 2 \beta_{2} - 2 \beta_{3} ) q^{28} + q^{29} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{33} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} -2 q^{36} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{37} + ( -1 - 2 \beta_{2} ) q^{39} + ( 2 - \beta_{1} - 2 \beta_{3} ) q^{41} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{44} -\beta_{1} q^{45} + \beta_{1} q^{47} -4 q^{48} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{49} + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 - 4 \beta_{2} ) q^{52} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -2 + 2 \beta_{3} ) q^{55} + ( -2 - 2 \beta_{2} - \beta_{3} ) q^{57} + ( -6 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{59} -2 \beta_{1} q^{60} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -1 - \beta_{2} + \beta_{3} ) q^{63} -8 q^{64} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 2 + 2 \beta_{2} - 2 \beta_{3} ) q^{67} + ( -2 - 2 \beta_{2} + 4 \beta_{3} ) q^{68} - q^{69} + ( -5 - \beta_{1} + 2 \beta_{3} ) q^{71} + ( -5 + \beta_{2} - \beta_{3} ) q^{73} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{75} + ( -4 - 4 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 2 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{77} + ( -6 + 2 \beta_{1} + 3 \beta_{3} ) q^{79} -4 \beta_{1} q^{80} + q^{81} + ( -6 + 3 \beta_{1} - 4 \beta_{2} ) q^{83} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{84} + ( -1 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{85} - q^{87} + ( 1 - 6 \beta_{1} + 3 \beta_{2} ) q^{89} + ( -11 + 6 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{91} -2 q^{92} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( -2 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{95} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 8q^{4} - 2q^{5} - 2q^{7} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 8q^{4} - 2q^{5} - 2q^{7} + 4q^{9} + 8q^{12} + 2q^{15} + 16q^{16} + 2q^{17} + 4q^{19} + 4q^{20} + 2q^{21} + 4q^{23} - 4q^{25} - 4q^{27} + 4q^{28} + 4q^{29} + 2q^{31} + 4q^{35} - 8q^{36} + 4q^{37} + 6q^{41} - 2q^{45} + 2q^{47} - 16q^{48} + 4q^{49} - 2q^{51} - 8q^{53} - 8q^{55} - 4q^{57} - 22q^{59} - 4q^{60} - 16q^{61} - 2q^{63} - 32q^{64} - 10q^{65} + 4q^{67} - 4q^{68} - 4q^{69} - 22q^{71} - 22q^{73} + 4q^{75} - 8q^{76} - 8q^{77} - 20q^{79} - 8q^{80} + 4q^{81} - 10q^{83} - 4q^{84} - 2q^{85} - 4q^{87} - 14q^{89} - 26q^{91} - 8q^{92} - 2q^{93} - 14q^{95} - 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 4 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 5 \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + \beta_{1} - 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} - 2 \beta_{2} + 3 \beta_{1} - 1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.37988 −0.751024 −1.92022 0.291367
0 −1.00000 −2.00000 −2.95969 0 1.80007 0 1.00000 0
1.2 0 −1.00000 −2.00000 −1.58049 0 −3.08254 0 1.00000 0
1.3 0 −1.00000 −2.00000 0.399447 0 −3.44099 0 1.00000 0
1.4 0 −1.00000 −2.00000 2.14073 0 2.72347 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.g 4
3.b odd 2 1 6003.2.a.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.g 4 1.a even 1 1 trivial
6003.2.a.g 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2001))$$:

 $$T_{2}$$ $$T_{5}^{4} + 2 T_{5}^{3} - 6 T_{5}^{2} - 8 T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$4 - 8 T - 6 T^{2} + 2 T^{3} + T^{4}$$
$7$ $$52 - 16 T - 14 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$-16 - 48 T - 28 T^{2} + T^{4}$$
$13$ $$125 - 24 T - 38 T^{2} + T^{4}$$
$17$ $$355 + 22 T - 44 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$835 + 116 T - 56 T^{2} - 4 T^{3} + T^{4}$$
$23$ $$( -1 + T )^{4}$$
$29$ $$( -1 + T )^{4}$$
$31$ $$4 - 48 T - 38 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$835 + 108 T - 56 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$-364 + 288 T - 46 T^{2} - 6 T^{3} + T^{4}$$
$43$ $$-73 - 216 T - 92 T^{2} + T^{4}$$
$47$ $$4 + 8 T - 6 T^{2} - 2 T^{3} + T^{4}$$
$53$ $$-304 - 416 T - 56 T^{2} + 8 T^{3} + T^{4}$$
$59$ $$133 + 262 T + 130 T^{2} + 22 T^{3} + T^{4}$$
$61$ $$-2240 - 736 T + 8 T^{2} + 16 T^{3} + T^{4}$$
$67$ $$832 + 128 T - 56 T^{2} - 4 T^{3} + T^{4}$$
$71$ $$197 + 278 T + 130 T^{2} + 22 T^{3} + T^{4}$$
$73$ $$508 + 496 T + 166 T^{2} + 22 T^{3} + T^{4}$$
$79$ $$-7133 - 1676 T + 20 T^{3} + T^{4}$$
$83$ $$-68 + 216 T - 122 T^{2} + 10 T^{3} + T^{4}$$
$89$ $$10207 - 1930 T - 192 T^{2} + 14 T^{3} + T^{4}$$
$97$ $$64 - 32 T - 24 T^{2} + 8 T^{3} + T^{4}$$