# Properties

 Label 2898.2.a.w 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.00000 0 1.00000 −2.73205 0 1.00000 −1.00000 0 2.73205
1.2 −1.00000 0 1.00000 0.732051 0 1.00000 −1.00000 0 −0.732051
 $$n$$: 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.w 2
3.b odd 2 1 322.2.a.f 2
12.b even 2 1 2576.2.a.u 2
15.d odd 2 1 8050.2.a.x 2
21.c even 2 1 2254.2.a.o 2
69.c even 2 1 7406.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.f 2 3.b odd 2 1
2254.2.a.o 2 21.c even 2 1
2576.2.a.u 2 12.b even 2 1
2898.2.a.w 2 1.a even 1 1 trivial
7406.2.a.s 2 69.c even 2 1
8050.2.a.x 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2898))$$:

 $$T_{5}^{2} + 2 T_{5} - 2$$ $$T_{11}^{2} - 12$$ $$T_{13}^{2} - 4 T_{13} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + 2 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$-8 - 4 T + T^{2}$$
$17$ $$-2 + 2 T + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$-2 + 10 T + T^{2}$$
$37$ $$4 - 8 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$-18 - 6 T + T^{2}$$
$53$ $$-12 + T^{2}$$
$59$ $$-66 - 6 T + T^{2}$$
$61$ $$-18 - 6 T + T^{2}$$
$67$ $$52 + 16 T + T^{2}$$
$71$ $$-104 - 4 T + T^{2}$$
$73$ $$4 + 8 T + T^{2}$$
$79$ $$-12 + 12 T + T^{2}$$
$83$ $$-92 + 8 T + T^{2}$$
$89$ $$54 + 18 T + T^{2}$$
$97$ $$46 + 14 T + T^{2}$$