# Properties

 Label 2898.2.a.be Level $2898$ Weight $2$ Character orbit 2898.a Self dual yes Analytic conductor $23.141$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 322) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -1 + \beta_{2} ) q^{5} - q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -1 + \beta_{2} ) q^{5} - q^{7} - q^{8} + ( 1 - \beta_{2} ) q^{10} + ( 2 - \beta_{1} + \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} + q^{14} + q^{16} + ( -3 + \beta_{1} ) q^{17} + ( -\beta_{1} - \beta_{2} ) q^{19} + ( -1 + \beta_{2} ) q^{20} + ( -2 + \beta_{1} - \beta_{2} ) q^{22} + q^{23} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{25} + ( -\beta_{1} + \beta_{2} ) q^{26} - q^{28} + ( 2 + \beta_{1} - 3 \beta_{2} ) q^{29} + ( -1 + \beta_{1} ) q^{31} - q^{32} + ( 3 - \beta_{1} ) q^{34} + ( 1 - \beta_{2} ) q^{35} + ( -4 + 2 \beta_{1} ) q^{37} + ( \beta_{1} + \beta_{2} ) q^{38} + ( 1 - \beta_{2} ) q^{40} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 2 - \beta_{1} + \beta_{2} ) q^{44} - q^{46} + ( -1 + \beta_{1} ) q^{47} + q^{49} + ( -1 + \beta_{1} + 3 \beta_{2} ) q^{50} + ( \beta_{1} - \beta_{2} ) q^{52} + 2 \beta_{1} q^{53} + ( 2 + 2 \beta_{2} ) q^{55} + q^{56} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{58} + ( -3 - 3 \beta_{2} ) q^{59} + ( -1 - 2 \beta_{1} + 3 \beta_{2} ) q^{61} + ( 1 - \beta_{1} ) q^{62} + q^{64} -4 q^{65} + ( 2 - \beta_{1} + \beta_{2} ) q^{67} + ( -3 + \beta_{1} ) q^{68} + ( -1 + \beta_{2} ) q^{70} + ( 10 - 2 \beta_{1} ) q^{71} + ( -4 + 4 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 4 - 2 \beta_{1} ) q^{74} + ( -\beta_{1} - \beta_{2} ) q^{76} + ( -2 + \beta_{1} - \beta_{2} ) q^{77} + ( -8 - \beta_{1} - 5 \beta_{2} ) q^{79} + ( -1 + \beta_{2} ) q^{80} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -10 + \beta_{1} - \beta_{2} ) q^{83} + ( 4 - \beta_{1} - 5 \beta_{2} ) q^{85} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{86} + ( -2 + \beta_{1} - \beta_{2} ) q^{88} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{89} + ( -\beta_{1} + \beta_{2} ) q^{91} + q^{92} + ( 1 - \beta_{1} ) q^{94} + ( -6 + 2 \beta_{1} + 4 \beta_{2} ) q^{95} + ( 7 - \beta_{1} + 4 \beta_{2} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 3q^{4} - 4q^{5} - 3q^{7} - 3q^{8} + O(q^{10})$$ $$3q - 3q^{2} + 3q^{4} - 4q^{5} - 3q^{7} - 3q^{8} + 4q^{10} + 4q^{11} + 2q^{13} + 3q^{14} + 3q^{16} - 8q^{17} - 4q^{20} - 4q^{22} + 3q^{23} + 5q^{25} - 2q^{26} - 3q^{28} + 10q^{29} - 2q^{31} - 3q^{32} + 8q^{34} + 4q^{35} - 10q^{37} + 4q^{40} - 6q^{41} - 4q^{43} + 4q^{44} - 3q^{46} - 2q^{47} + 3q^{49} - 5q^{50} + 2q^{52} + 2q^{53} + 4q^{55} + 3q^{56} - 10q^{58} - 6q^{59} - 8q^{61} + 2q^{62} + 3q^{64} - 12q^{65} + 4q^{67} - 8q^{68} - 4q^{70} + 28q^{71} - 6q^{73} + 10q^{74} - 4q^{77} - 20q^{79} - 4q^{80} + 6q^{82} - 28q^{83} + 16q^{85} + 4q^{86} - 4q^{88} - 2q^{91} + 3q^{92} + 2q^{94} - 20q^{95} + 16q^{97} - 3q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 6$$$$)/2$$

## 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
 0.470683 2.34292 −1.81361
−1.00000 0 1.00000 −4.24914 0 −1.00000 −1.00000 0 4.24914
1.2 −1.00000 0 1.00000 −0.853635 0 −1.00000 −1.00000 0 0.853635
1.3 −1.00000 0 1.00000 1.10278 0 −1.00000 −1.00000 0 −1.10278
 $$n$$: 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.be 3
3.b odd 2 1 322.2.a.g 3
12.b even 2 1 2576.2.a.w 3
15.d odd 2 1 8050.2.a.bh 3
21.c even 2 1 2254.2.a.p 3
69.c even 2 1 7406.2.a.x 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.g 3 3.b odd 2 1
2254.2.a.p 3 21.c even 2 1
2576.2.a.w 3 12.b even 2 1
2898.2.a.be 3 1.a even 1 1 trivial
7406.2.a.x 3 69.c even 2 1
8050.2.a.bh 3 15.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{3} + 4 T_{5}^{2} - 2 T_{5} - 4$$ $$T_{11}^{3} - 4 T_{11}^{2} - 12 T_{11} + 16$$ $$T_{13}^{3} - 2 T_{13}^{2} - 16 T_{13} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$T^{3}$$
$5$ $$-4 - 2 T + 4 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$16 - 12 T - 4 T^{2} + T^{3}$$
$13$ $$16 - 16 T - 2 T^{2} + T^{3}$$
$17$ $$-44 + 6 T + 8 T^{2} + T^{3}$$
$19$ $$-16 - 28 T + T^{3}$$
$23$ $$( -1 + T )^{3}$$
$29$ $$352 - 32 T - 10 T^{2} + T^{3}$$
$31$ $$-32 - 14 T + 2 T^{2} + T^{3}$$
$37$ $$-344 - 28 T + 10 T^{2} + T^{3}$$
$41$ $$-344 - 100 T + 6 T^{2} + T^{3}$$
$43$ $$-128 - 64 T + 4 T^{2} + T^{3}$$
$47$ $$-32 - 14 T + 2 T^{2} + T^{3}$$
$53$ $$-136 - 60 T - 2 T^{2} + T^{3}$$
$59$ $$-216 - 54 T + 6 T^{2} + T^{3}$$
$61$ $$-524 - 74 T + 8 T^{2} + T^{3}$$
$67$ $$16 - 12 T - 4 T^{2} + T^{3}$$
$71$ $$-64 + 200 T - 28 T^{2} + T^{3}$$
$73$ $$-1448 - 220 T + 6 T^{2} + T^{3}$$
$79$ $$-2432 - 92 T + 20 T^{2} + T^{3}$$
$83$ $$656 + 244 T + 28 T^{2} + T^{3}$$
$89$ $$-76 - 34 T + T^{3}$$
$97$ $$172 - 26 T - 16 T^{2} + T^{3}$$