# Properties

 Label 2898.2.a.ba Level $2898$ Weight $2$ Character orbit 2898.a Self dual yes Analytic conductor $23.141$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 966) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} -\beta q^{5} - q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} -\beta q^{5} - q^{7} + q^{8} -\beta q^{10} + ( -2 + 2 \beta ) q^{11} + ( -2 + \beta ) q^{13} - q^{14} + q^{16} + ( -2 - 2 \beta ) q^{17} + ( -2 + 2 \beta ) q^{19} -\beta q^{20} + ( -2 + 2 \beta ) q^{22} - q^{23} + ( -1 + \beta ) q^{25} + ( -2 + \beta ) q^{26} - q^{28} + ( -6 + \beta ) q^{29} -2 \beta q^{31} + q^{32} + ( -2 - 2 \beta ) q^{34} + \beta q^{35} + \beta q^{37} + ( -2 + 2 \beta ) q^{38} -\beta q^{40} + ( -2 + \beta ) q^{41} + ( 2 - 3 \beta ) q^{43} + ( -2 + 2 \beta ) q^{44} - q^{46} + ( -4 - \beta ) q^{47} + q^{49} + ( -1 + \beta ) q^{50} + ( -2 + \beta ) q^{52} + ( -8 + 4 \beta ) q^{53} -8 q^{55} - q^{56} + ( -6 + \beta ) q^{58} + ( -8 - 2 \beta ) q^{59} + ( 4 + 2 \beta ) q^{61} -2 \beta q^{62} + q^{64} + ( -4 + \beta ) q^{65} + ( 2 + 2 \beta ) q^{67} + ( -2 - 2 \beta ) q^{68} + \beta q^{70} -6 \beta q^{71} + ( 6 - 2 \beta ) q^{73} + \beta q^{74} + ( -2 + 2 \beta ) q^{76} + ( 2 - 2 \beta ) q^{77} + ( 8 - 4 \beta ) q^{79} -\beta q^{80} + ( -2 + \beta ) q^{82} + ( 6 - 2 \beta ) q^{83} + ( 8 + 4 \beta ) q^{85} + ( 2 - 3 \beta ) q^{86} + ( -2 + 2 \beta ) q^{88} -14 q^{89} + ( 2 - \beta ) q^{91} - q^{92} + ( -4 - \beta ) q^{94} -8 q^{95} + ( -14 - \beta ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + 2q^{8} - q^{10} - 2q^{11} - 3q^{13} - 2q^{14} + 2q^{16} - 6q^{17} - 2q^{19} - q^{20} - 2q^{22} - 2q^{23} - q^{25} - 3q^{26} - 2q^{28} - 11q^{29} - 2q^{31} + 2q^{32} - 6q^{34} + q^{35} + q^{37} - 2q^{38} - q^{40} - 3q^{41} + q^{43} - 2q^{44} - 2q^{46} - 9q^{47} + 2q^{49} - q^{50} - 3q^{52} - 12q^{53} - 16q^{55} - 2q^{56} - 11q^{58} - 18q^{59} + 10q^{61} - 2q^{62} + 2q^{64} - 7q^{65} + 6q^{67} - 6q^{68} + q^{70} - 6q^{71} + 10q^{73} + q^{74} - 2q^{76} + 2q^{77} + 12q^{79} - q^{80} - 3q^{82} + 10q^{83} + 20q^{85} + q^{86} - 2q^{88} - 28q^{89} + 3q^{91} - 2q^{92} - 9q^{94} - 16q^{95} - 29q^{97} + 2q^{98} + O(q^{100})$$

## 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
 2.56155 −1.56155
1.00000 0 1.00000 −2.56155 0 −1.00000 1.00000 0 −2.56155
1.2 1.00000 0 1.00000 1.56155 0 −1.00000 1.00000 0 1.56155
 $$n$$: 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.ba 2
3.b odd 2 1 966.2.a.l 2
12.b even 2 1 7728.2.a.bm 2
21.c even 2 1 6762.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.l 2 3.b odd 2 1
2898.2.a.ba 2 1.a even 1 1 trivial
6762.2.a.bw 2 21.c even 2 1
7728.2.a.bm 2 12.b even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-4 + T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-16 + 2 T + T^{2}$$
$13$ $$-2 + 3 T + T^{2}$$
$17$ $$-8 + 6 T + T^{2}$$
$19$ $$-16 + 2 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$26 + 11 T + T^{2}$$
$31$ $$-16 + 2 T + T^{2}$$
$37$ $$-4 - T + T^{2}$$
$41$ $$-2 + 3 T + T^{2}$$
$43$ $$-38 - T + T^{2}$$
$47$ $$16 + 9 T + T^{2}$$
$53$ $$-32 + 12 T + T^{2}$$
$59$ $$64 + 18 T + T^{2}$$
$61$ $$8 - 10 T + T^{2}$$
$67$ $$-8 - 6 T + T^{2}$$
$71$ $$-144 + 6 T + T^{2}$$
$73$ $$8 - 10 T + T^{2}$$
$79$ $$-32 - 12 T + T^{2}$$
$83$ $$8 - 10 T + T^{2}$$
$89$ $$( 14 + T )^{2}$$
$97$ $$206 + 29 T + T^{2}$$