# Properties

 Label 2898.2.a.bh Level $2898$ Weight $2$ Character orbit 2898.a Self dual yes Analytic conductor $23.141$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.271296.1 Defining polynomial: $$x^{4} - 18 x^{2} - 8 x + 24$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta_{1} q^{5} + q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + \beta_{1} q^{5} + q^{7} - q^{8} -\beta_{1} q^{10} + ( -2 + \beta_{1} ) q^{11} + \beta_{2} q^{13} - q^{14} + q^{16} + \beta_{2} q^{17} + ( \beta_{1} + \beta_{2} ) q^{19} + \beta_{1} q^{20} + ( 2 - \beta_{1} ) q^{22} - q^{23} + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} -\beta_{2} q^{26} + q^{28} -2 q^{29} + ( 1 - \beta_{1} - \beta_{3} ) q^{31} - q^{32} -\beta_{2} q^{34} + \beta_{1} q^{35} + ( 1 - \beta_{2} - \beta_{3} ) q^{37} + ( -\beta_{1} - \beta_{2} ) q^{38} -\beta_{1} q^{40} + 2 q^{41} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{43} + ( -2 + \beta_{1} ) q^{44} + q^{46} + ( 3 + \beta_{1} + \beta_{3} ) q^{47} + q^{49} + ( -4 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{50} + \beta_{2} q^{52} + ( -3 - \beta_{2} - \beta_{3} ) q^{53} + ( 9 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{55} - q^{56} + 2 q^{58} + ( 4 - 2 \beta_{1} ) q^{59} + 3 \beta_{1} q^{61} + ( -1 + \beta_{1} + \beta_{3} ) q^{62} + q^{64} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{65} + ( 6 + 3 \beta_{1} + 2 \beta_{2} ) q^{67} + \beta_{2} q^{68} -\beta_{1} q^{70} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{71} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{73} + ( -1 + \beta_{2} + \beta_{3} ) q^{74} + ( \beta_{1} + \beta_{2} ) q^{76} + ( -2 + \beta_{1} ) q^{77} + ( 7 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{79} + \beta_{1} q^{80} -2 q^{82} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{86} + ( 2 - \beta_{1} ) q^{88} + ( -2 - \beta_{2} - 2 \beta_{3} ) q^{89} + \beta_{2} q^{91} - q^{92} + ( -3 - \beta_{1} - \beta_{3} ) q^{94} + ( 9 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{95} + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 4q^{4} + 4q^{7} - 4q^{8} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{4} + 4q^{7} - 4q^{8} - 8q^{11} - 4q^{14} + 4q^{16} + 8q^{22} - 4q^{23} + 16q^{25} + 4q^{28} - 8q^{29} + 4q^{31} - 4q^{32} + 4q^{37} + 8q^{41} + 8q^{43} - 8q^{44} + 4q^{46} + 12q^{47} + 4q^{49} - 16q^{50} - 12q^{53} + 36q^{55} - 4q^{56} + 8q^{58} + 16q^{59} - 4q^{62} + 4q^{64} + 24q^{67} + 4q^{71} + 12q^{73} - 4q^{74} - 8q^{77} + 28q^{79} - 8q^{82} - 8q^{83} - 8q^{86} + 8q^{88} - 8q^{89} - 4q^{92} - 12q^{94} + 36q^{95} - 8q^{97} - 4q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 18 x^{2} - 8 x + 24$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 2 \nu^{2} - 14 \nu + 12$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 18 \nu - 24$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - \beta_{2} + \beta_{1} + 9$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 2 \beta_{2} + 16 \beta_{1} + 6$$

## 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
 −3.76613 −1.51727 0.974808 4.30859
−1.00000 0 1.00000 −3.76613 0 1.00000 −1.00000 0 3.76613
1.2 −1.00000 0 1.00000 −1.51727 0 1.00000 −1.00000 0 1.51727
1.3 −1.00000 0 1.00000 0.974808 0 1.00000 −1.00000 0 −0.974808
1.4 −1.00000 0 1.00000 4.30859 0 1.00000 −1.00000 0 −4.30859
 $$n$$: 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.bh 4
3.b odd 2 1 2898.2.a.bi yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2898.2.a.bh 4 1.a even 1 1 trivial
2898.2.a.bi yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2898))$$:

 $$T_{5}^{4} - 18 T_{5}^{2} - 8 T_{5} + 24$$ $$T_{11}^{4} + 8 T_{11}^{3} + 6 T_{11}^{2} - 48 T_{11} - 48$$ $$T_{13}^{4} - 30 T_{13}^{2} - 56 T_{13} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$T^{4}$$
$5$ $$24 - 8 T - 18 T^{2} + T^{4}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$-48 - 48 T + 6 T^{2} + 8 T^{3} + T^{4}$$
$13$ $$-24 - 56 T - 30 T^{2} + T^{4}$$
$17$ $$-24 - 56 T - 30 T^{2} + T^{4}$$
$19$ $$-36 + 128 T - 48 T^{2} + T^{4}$$
$23$ $$( 1 + T )^{4}$$
$29$ $$( 2 + T )^{4}$$
$31$ $$-8 + 104 T - 66 T^{2} - 4 T^{3} + T^{4}$$
$37$ $$64 - 64 T - 78 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$( -2 + T )^{4}$$
$43$ $$-3168 + 1296 T - 114 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$-648 + 360 T - 18 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$-1440 - 624 T - 30 T^{2} + 12 T^{3} + T^{4}$$
$59$ $$-768 + 384 T + 24 T^{2} - 16 T^{3} + T^{4}$$
$61$ $$1944 - 216 T - 162 T^{2} + T^{4}$$
$67$ $$-23904 + 4592 T - 66 T^{2} - 24 T^{3} + T^{4}$$
$71$ $$416 + 448 T - 108 T^{2} - 4 T^{3} + T^{4}$$
$73$ $$192 + 544 T - 36 T^{2} - 12 T^{3} + T^{4}$$
$79$ $$-1536 + 204 T^{2} - 28 T^{3} + T^{4}$$
$83$ $$10364 - 2000 T - 288 T^{2} + 8 T^{3} + T^{4}$$
$89$ $$-11304 - 3384 T - 246 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$872 - 1928 T - 270 T^{2} + 8 T^{3} + T^{4}$$