# Properties

 Label 2898.2.a.bg Level $2898$ Weight $2$ Character orbit 2898.a Self dual yes Analytic conductor $23.141$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2898.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.1406465058$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} -\beta_{2} q^{5} - q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} -\beta_{2} q^{5} - q^{7} + q^{8} -\beta_{2} q^{10} + 2 \beta_{1} q^{11} + ( 2 - \beta_{1} ) q^{13} - q^{14} + q^{16} + ( \beta_{1} - \beta_{2} ) q^{17} + ( 2 + \beta_{1} + \beta_{2} ) q^{19} -\beta_{2} q^{20} + 2 \beta_{1} q^{22} - q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{25} + ( 2 - \beta_{1} ) q^{26} - q^{28} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{29} + ( 2 - \beta_{1} + \beta_{2} ) q^{31} + q^{32} + ( \beta_{1} - \beta_{2} ) q^{34} + \beta_{2} q^{35} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( 2 + \beta_{1} + \beta_{2} ) q^{38} -\beta_{2} q^{40} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{43} + 2 \beta_{1} q^{44} - q^{46} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{50} + ( 2 - \beta_{1} ) q^{52} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 4 + 2 \beta_{2} ) q^{55} - q^{56} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{58} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} ) q^{62} + q^{64} + ( -2 - 3 \beta_{2} ) q^{65} + 4 \beta_{1} q^{67} + ( \beta_{1} - \beta_{2} ) q^{68} + \beta_{2} q^{70} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{71} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{74} + ( 2 + \beta_{1} + \beta_{2} ) q^{76} -2 \beta_{1} q^{77} + ( 2 - 2 \beta_{1} ) q^{79} -\beta_{2} q^{80} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{82} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{83} + ( 8 - 2 \beta_{1} ) q^{85} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{86} + 2 \beta_{1} q^{88} + ( 5 \beta_{1} + 5 \beta_{2} ) q^{89} + ( -2 + \beta_{1} ) q^{91} - q^{92} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{94} + ( -4 + 2 \beta_{1} ) q^{95} + ( 2 + \beta_{1} ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 3q^{4} + q^{5} - 3q^{7} + 3q^{8} + O(q^{10})$$ $$3q + 3q^{2} + 3q^{4} + q^{5} - 3q^{7} + 3q^{8} + q^{10} + 2q^{11} + 5q^{13} - 3q^{14} + 3q^{16} + 2q^{17} + 6q^{19} + q^{20} + 2q^{22} - 3q^{23} + 2q^{25} + 5q^{26} - 3q^{28} + 11q^{29} + 4q^{31} + 3q^{32} + 2q^{34} - q^{35} + 5q^{37} + 6q^{38} + q^{40} + 9q^{41} + 5q^{43} + 2q^{44} - 3q^{46} + 11q^{47} + 3q^{49} + 2q^{50} + 5q^{52} - 2q^{53} + 10q^{55} - 3q^{56} + 11q^{58} + 10q^{61} + 4q^{62} + 3q^{64} - 3q^{65} + 4q^{67} + 2q^{68} - q^{70} - 10q^{71} + 18q^{73} + 5q^{74} + 6q^{76} - 2q^{77} + 4q^{79} + q^{80} + 9q^{82} + 4q^{83} + 22q^{85} + 5q^{86} + 2q^{88} - 5q^{91} - 3q^{92} + 11q^{94} - 10q^{95} + 7q^{97} + 3q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

## 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
 −1.76156 3.12489 −0.363328
1.00000 0 1.00000 −2.62620 0 −1.00000 1.00000 0 −2.62620
1.2 1.00000 0 1.00000 0.484862 0 −1.00000 1.00000 0 0.484862
1.3 1.00000 0 1.00000 3.14134 0 −1.00000 1.00000 0 3.14134
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$
$$23$$ $$1$$

## 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 2898.2.a.bg yes 3
3.b odd 2 1 2898.2.a.bf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2898.2.a.bf 3 3.b odd 2 1
2898.2.a.bg yes 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2898))$$:

 $$T_{5}^{3} - T_{5}^{2} - 8 T_{5} + 4$$ $$T_{11}^{3} - 2 T_{11}^{2} - 24 T_{11} - 16$$ $$T_{13}^{3} - 5 T_{13}^{2} + 2 T_{13} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$T^{3}$$
$5$ $$4 - 8 T - T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-16 - 24 T - 2 T^{2} + T^{3}$$
$13$ $$10 + 2 T - 5 T^{2} + T^{3}$$
$17$ $$44 - 18 T - 2 T^{2} + T^{3}$$
$19$ $$20 + 2 T - 6 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$68 + 16 T - 11 T^{2} + T^{3}$$
$31$ $$-8 - 14 T - 4 T^{2} + T^{3}$$
$37$ $$64 - 16 T - 5 T^{2} + T^{3}$$
$41$ $$44 - 16 T - 9 T^{2} + T^{3}$$
$43$ $$352 - 64 T - 5 T^{2} + T^{3}$$
$47$ $$122 + 10 T - 11 T^{2} + T^{3}$$
$53$ $$160 - 96 T + 2 T^{2} + T^{3}$$
$59$ $$T^{3}$$
$61$ $$472 - 44 T - 10 T^{2} + T^{3}$$
$67$ $$-128 - 96 T - 4 T^{2} + T^{3}$$
$71$ $$-848 - 88 T + 10 T^{2} + T^{3}$$
$73$ $$88 + 68 T - 18 T^{2} + T^{3}$$
$79$ $$64 - 20 T - 4 T^{2} + T^{3}$$
$83$ $$232 - 62 T - 4 T^{2} + T^{3}$$
$89$ $$1000 - 250 T + T^{3}$$
$97$ $$-2 + 10 T - 7 T^{2} + T^{3}$$