# Properties

 Label 4998.2.a.ci Level $4998$ Weight $2$ Character orbit 4998.a Self dual yes Analytic conductor $39.909$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4998.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.9092309302$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.469.1 Defining polynomial: $$x^{3} - x^{2} - 5 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} -\beta_{2} q^{5} - q^{6} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} -\beta_{2} q^{5} - q^{6} + q^{8} + q^{9} -\beta_{2} q^{10} + ( 1 + \beta_{1} ) q^{11} - q^{12} + ( -1 + \beta_{1} ) q^{13} + \beta_{2} q^{15} + q^{16} + q^{17} + q^{18} + ( 1 - \beta_{1} - \beta_{2} ) q^{19} -\beta_{2} q^{20} + ( 1 + \beta_{1} ) q^{22} + 2 q^{23} - q^{24} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{25} + ( -1 + \beta_{1} ) q^{26} - q^{27} + ( 1 + \beta_{1} - \beta_{2} ) q^{29} + \beta_{2} q^{30} + ( -2 + 2 \beta_{1} ) q^{31} + q^{32} + ( -1 - \beta_{1} ) q^{33} + q^{34} + q^{36} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{37} + ( 1 - \beta_{1} - \beta_{2} ) q^{38} + ( 1 - \beta_{1} ) q^{39} -\beta_{2} q^{40} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 - \beta_{2} ) q^{43} + ( 1 + \beta_{1} ) q^{44} -\beta_{2} q^{45} + 2 q^{46} -2 \beta_{2} q^{47} - q^{48} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{50} - q^{51} + ( -1 + \beta_{1} ) q^{52} + ( 4 - \beta_{2} ) q^{53} - q^{54} + ( -2 + \beta_{2} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} ) q^{57} + ( 1 + \beta_{1} - \beta_{2} ) q^{58} + ( 3 - \beta_{1} + \beta_{2} ) q^{59} + \beta_{2} q^{60} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -2 + 2 \beta_{1} ) q^{62} + q^{64} + ( -2 + 3 \beta_{2} ) q^{65} + ( -1 - \beta_{1} ) q^{66} + ( 8 + 2 \beta_{1} - \beta_{2} ) q^{67} + q^{68} -2 q^{69} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + q^{72} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{73} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{74} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} ) q^{76} + ( 1 - \beta_{1} ) q^{78} + ( 2 - 4 \beta_{1} - \beta_{2} ) q^{79} -\beta_{2} q^{80} + q^{81} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{83} -\beta_{2} q^{85} + ( 2 - \beta_{2} ) q^{86} + ( -1 - \beta_{1} + \beta_{2} ) q^{87} + ( 1 + \beta_{1} ) q^{88} + ( -4 + 3 \beta_{2} ) q^{89} -\beta_{2} q^{90} + 2 q^{92} + ( 2 - 2 \beta_{1} ) q^{93} -2 \beta_{2} q^{94} + ( 10 - 2 \beta_{1} - 2 \beta_{2} ) q^{95} - q^{96} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{97} + ( 1 + \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - q^{5} - 3q^{6} + 3q^{8} + 3q^{9} - q^{10} + 3q^{11} - 3q^{12} - 3q^{13} + q^{15} + 3q^{16} + 3q^{17} + 3q^{18} + 2q^{19} - q^{20} + 3q^{22} + 6q^{23} - 3q^{24} + 10q^{25} - 3q^{26} - 3q^{27} + 2q^{29} + q^{30} - 6q^{31} + 3q^{32} - 3q^{33} + 3q^{34} + 3q^{36} + q^{37} + 2q^{38} + 3q^{39} - q^{40} - 10q^{41} + 5q^{43} + 3q^{44} - q^{45} + 6q^{46} - 2q^{47} - 3q^{48} + 10q^{50} - 3q^{51} - 3q^{52} + 11q^{53} - 3q^{54} - 5q^{55} - 2q^{57} + 2q^{58} + 10q^{59} + q^{60} - 2q^{61} - 6q^{62} + 3q^{64} - 3q^{65} - 3q^{66} + 23q^{67} + 3q^{68} - 6q^{69} + 10q^{71} + 3q^{72} + 7q^{73} + q^{74} - 10q^{75} + 2q^{76} + 3q^{78} + 5q^{79} - q^{80} + 3q^{81} - 10q^{82} - q^{83} - q^{85} + 5q^{86} - 2q^{87} + 3q^{88} - 9q^{89} - q^{90} + 6q^{92} + 6q^{93} - 2q^{94} + 28q^{95} - 3q^{96} - 5q^{97} + 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 4$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 7$$$$)/2$$

## 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
 −2.16425 2.39138 0.772866
1.00000 −1.00000 1.00000 −3.84822 −1.00000 0 1.00000 1.00000 −3.84822
1.2 1.00000 −1.00000 1.00000 −0.327327 −1.00000 0 1.00000 1.00000 −0.327327
1.3 1.00000 −1.00000 1.00000 3.17554 −1.00000 0 1.00000 1.00000 3.17554
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$17$$ $$-1$$

## 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 4998.2.a.ci 3
7.b odd 2 1 4998.2.a.cj yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4998.2.a.ci 3 1.a even 1 1 trivial
4998.2.a.cj yes 3 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4998))$$:

 $$T_{5}^{3} + T_{5}^{2} - 12 T_{5} - 4$$ $$T_{11}^{3} - 3 T_{11}^{2} - 10 T_{11} - 4$$ $$T_{13}^{3} + 3 T_{13}^{2} - 10 T_{13} - 28$$ $$T_{23} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$-4 - 12 T + T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$-4 - 10 T - 3 T^{2} + T^{3}$$
$13$ $$-28 - 10 T + 3 T^{2} + T^{3}$$
$17$ $$( -1 + T )^{3}$$
$19$ $$-32 - 28 T - 2 T^{2} + T^{3}$$
$23$ $$( -2 + T )^{3}$$
$29$ $$32 - 20 T - 2 T^{2} + T^{3}$$
$31$ $$-224 - 40 T + 6 T^{2} + T^{3}$$
$37$ $$172 - 54 T - T^{2} + T^{3}$$
$41$ $$-392 - 52 T + 10 T^{2} + T^{3}$$
$43$ $$16 - 4 T - 5 T^{2} + T^{3}$$
$47$ $$-32 - 48 T + 2 T^{2} + T^{3}$$
$53$ $$-4 + 28 T - 11 T^{2} + T^{3}$$
$59$ $$16 + 12 T - 10 T^{2} + T^{3}$$
$61$ $$88 - 84 T + 2 T^{2} + T^{3}$$
$67$ $$-112 + 120 T - 23 T^{2} + T^{3}$$
$71$ $$-56 - 84 T - 10 T^{2} + T^{3}$$
$73$ $$212 - 40 T - 7 T^{2} + T^{3}$$
$79$ $$944 - 228 T - 5 T^{2} + T^{3}$$
$83$ $$76 - 142 T + T^{2} + T^{3}$$
$89$ $$-308 - 84 T + 9 T^{2} + T^{3}$$
$97$ $$-76 - 64 T + 5 T^{2} + T^{3}$$