# Properties

 Label 4998.2.a.bx Level $4998$ Weight $2$ Character orbit 4998.a Self dual yes Analytic conductor $39.909$ Analytic rank $1$ Dimension $2$ 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: $$1$$ 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 714) 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.

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

## 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 −1.00000 1.00000 −3.56155 −1.00000 0 1.00000 1.00000 −3.56155
1.2 1.00000 −1.00000 1.00000 0.561553 −1.00000 0 1.00000 1.00000 0.561553
 $$n$$: 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.bx 2
7.b odd 2 1 714.2.a.l 2
21.c even 2 1 2142.2.a.w 2
28.d even 2 1 5712.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.a.l 2 7.b odd 2 1
2142.2.a.w 2 21.c even 2 1
4998.2.a.bx 2 1.a even 1 1 trivial
5712.2.a.bl 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4998))$$:

 $$T_{5}^{2} + 3 T_{5} - 2$$ $$T_{11}^{2} - T_{11} - 4$$ $$T_{13}^{2} + 5 T_{13} + 2$$ $$T_{23}^{2} + 2 T_{23} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-2 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-4 - T + T^{2}$$
$13$ $$2 + 5 T + T^{2}$$
$17$ $$( 1 + T )^{2}$$
$19$ $$-8 - 6 T + T^{2}$$
$23$ $$-16 + 2 T + T^{2}$$
$29$ $$-68 + T^{2}$$
$31$ $$-64 + 4 T + T^{2}$$
$37$ $$-2 - 3 T + T^{2}$$
$41$ $$-16 + 2 T + T^{2}$$
$43$ $$68 + 17 T + T^{2}$$
$47$ $$-64 - 4 T + T^{2}$$
$53$ $$18 + 15 T + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$-68 + T^{2}$$
$67$ $$4 - 13 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$-94 + 7 T + T^{2}$$
$79$ $$-32 - 5 T + T^{2}$$
$83$ $$-196 + 7 T + T^{2}$$
$89$ $$-26 - 7 T + T^{2}$$
$97$ $$178 + 27 T + T^{2}$$