Properties

 Label 2898.2.a.bd Level $2898$ Weight $2$ Character orbit 2898.a Self dual yes Analytic conductor $23.141$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( 1 + \beta ) q^{5} - q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} + ( 1 + \beta ) q^{5} - q^{7} + q^{8} + ( 1 + \beta ) q^{10} + ( -2 - 2 \beta ) q^{13} - q^{14} + q^{16} + ( 5 - \beta ) q^{17} + 2 \beta q^{19} + ( 1 + \beta ) q^{20} + q^{23} + ( 1 + 2 \beta ) q^{25} + ( -2 - 2 \beta ) q^{26} - q^{28} + 2 q^{29} + ( 1 + \beta ) q^{31} + q^{32} + ( 5 - \beta ) q^{34} + ( -1 - \beta ) q^{35} + 6 q^{37} + 2 \beta q^{38} + ( 1 + \beta ) q^{40} + 10 q^{41} + ( -2 + 2 \beta ) q^{43} + q^{46} + ( 9 + \beta ) q^{47} + q^{49} + ( 1 + 2 \beta ) q^{50} + ( -2 - 2 \beta ) q^{52} + 6 q^{53} - q^{56} + 2 q^{58} + ( -9 + \beta ) q^{59} + ( 5 - 3 \beta ) q^{61} + ( 1 + \beta ) q^{62} + q^{64} + ( -12 - 4 \beta ) q^{65} + 4 q^{67} + ( 5 - \beta ) q^{68} + ( -1 - \beta ) q^{70} + ( -2 - 2 \beta ) q^{71} + 6 \beta q^{73} + 6 q^{74} + 2 \beta q^{76} -4 \beta q^{79} + ( 1 + \beta ) q^{80} + 10 q^{82} + ( -2 - 4 \beta ) q^{83} + 4 \beta q^{85} + ( -2 + 2 \beta ) q^{86} + ( 5 - 5 \beta ) q^{89} + ( 2 + 2 \beta ) q^{91} + q^{92} + ( 9 + \beta ) q^{94} + ( 10 + 2 \beta ) q^{95} + ( -3 - 5 \beta ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{4} + 2q^{5} - 2q^{7} + 2q^{8} + 2q^{10} - 4q^{13} - 2q^{14} + 2q^{16} + 10q^{17} + 2q^{20} + 2q^{23} + 2q^{25} - 4q^{26} - 2q^{28} + 4q^{29} + 2q^{31} + 2q^{32} + 10q^{34} - 2q^{35} + 12q^{37} + 2q^{40} + 20q^{41} - 4q^{43} + 2q^{46} + 18q^{47} + 2q^{49} + 2q^{50} - 4q^{52} + 12q^{53} - 2q^{56} + 4q^{58} - 18q^{59} + 10q^{61} + 2q^{62} + 2q^{64} - 24q^{65} + 8q^{67} + 10q^{68} - 2q^{70} - 4q^{71} + 12q^{74} + 2q^{80} + 20q^{82} - 4q^{83} - 4q^{86} + 10q^{89} + 4q^{91} + 2q^{92} + 18q^{94} + 20q^{95} - 6q^{97} + 2q^{98} + O(q^{100})$$

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
 −0.618034 1.61803
1.00000 0 1.00000 −1.23607 0 −1.00000 1.00000 0 −1.23607
1.2 1.00000 0 1.00000 3.23607 0 −1.00000 1.00000 0 3.23607
 $$n$$: 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.bd 2
3.b odd 2 1 322.2.a.e 2
12.b even 2 1 2576.2.a.t 2
15.d odd 2 1 8050.2.a.bf 2
21.c even 2 1 2254.2.a.k 2
69.c even 2 1 7406.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.a.e 2 3.b odd 2 1
2254.2.a.k 2 21.c even 2 1
2576.2.a.t 2 12.b even 2 1
2898.2.a.bd 2 1.a even 1 1 trivial
7406.2.a.j 2 69.c even 2 1
8050.2.a.bf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2898))$$:

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-4 - 2 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-16 + 4 T + T^{2}$$
$17$ $$20 - 10 T + T^{2}$$
$19$ $$-20 + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$-4 - 2 T + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$-16 + 4 T + T^{2}$$
$47$ $$76 - 18 T + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$76 + 18 T + T^{2}$$
$61$ $$-20 - 10 T + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$-16 + 4 T + T^{2}$$
$73$ $$-180 + T^{2}$$
$79$ $$-80 + T^{2}$$
$83$ $$-76 + 4 T + T^{2}$$
$89$ $$-100 - 10 T + T^{2}$$
$97$ $$-116 + 6 T + T^{2}$$