# Properties

 Label 4998.2.a.bs 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: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} -\beta q^{5} + q^{6} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} -\beta q^{5} + q^{6} - q^{8} + q^{9} + \beta q^{10} + \beta q^{11} - q^{12} + ( -2 + \beta ) q^{13} + \beta q^{15} + q^{16} + q^{17} - q^{18} -6 q^{19} -\beta q^{20} -\beta q^{22} + ( 4 - 2 \beta ) q^{23} + q^{24} + ( 5 + \beta ) q^{25} + ( 2 - \beta ) q^{26} - q^{27} -2 \beta q^{29} -\beta q^{30} - q^{32} -\beta q^{33} - q^{34} + q^{36} + ( -2 + \beta ) q^{37} + 6 q^{38} + ( 2 - \beta ) q^{39} + \beta q^{40} + ( -6 + 2 \beta ) q^{41} + ( 6 - \beta ) q^{43} + \beta q^{44} -\beta q^{45} + ( -4 + 2 \beta ) q^{46} + 8 q^{47} - q^{48} + ( -5 - \beta ) q^{50} - q^{51} + ( -2 + \beta ) q^{52} + ( -4 - \beta ) q^{53} + q^{54} + ( -10 - \beta ) q^{55} + 6 q^{57} + 2 \beta q^{58} + ( 2 - 2 \beta ) q^{59} + \beta q^{60} + 2 q^{61} + q^{64} + ( -10 + \beta ) q^{65} + \beta q^{66} + ( 6 + \beta ) q^{67} + q^{68} + ( -4 + 2 \beta ) q^{69} + ( -4 + 4 \beta ) q^{71} - q^{72} + ( 8 + \beta ) q^{73} + ( 2 - \beta ) q^{74} + ( -5 - \beta ) q^{75} -6 q^{76} + ( -2 + \beta ) q^{78} + ( 10 + \beta ) q^{79} -\beta q^{80} + q^{81} + ( 6 - 2 \beta ) q^{82} + ( -12 + \beta ) q^{83} -\beta q^{85} + ( -6 + \beta ) q^{86} + 2 \beta q^{87} -\beta q^{88} + ( -12 + \beta ) q^{89} + \beta q^{90} + ( 4 - 2 \beta ) q^{92} -8 q^{94} + 6 \beta q^{95} + q^{96} + ( -4 + \beta ) q^{97} + \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - q^{5} + 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - q^{5} + 2q^{6} - 2q^{8} + 2q^{9} + q^{10} + q^{11} - 2q^{12} - 3q^{13} + q^{15} + 2q^{16} + 2q^{17} - 2q^{18} - 12q^{19} - q^{20} - q^{22} + 6q^{23} + 2q^{24} + 11q^{25} + 3q^{26} - 2q^{27} - 2q^{29} - q^{30} - 2q^{32} - q^{33} - 2q^{34} + 2q^{36} - 3q^{37} + 12q^{38} + 3q^{39} + q^{40} - 10q^{41} + 11q^{43} + q^{44} - q^{45} - 6q^{46} + 16q^{47} - 2q^{48} - 11q^{50} - 2q^{51} - 3q^{52} - 9q^{53} + 2q^{54} - 21q^{55} + 12q^{57} + 2q^{58} + 2q^{59} + q^{60} + 4q^{61} + 2q^{64} - 19q^{65} + q^{66} + 13q^{67} + 2q^{68} - 6q^{69} - 4q^{71} - 2q^{72} + 17q^{73} + 3q^{74} - 11q^{75} - 12q^{76} - 3q^{78} + 21q^{79} - q^{80} + 2q^{81} + 10q^{82} - 23q^{83} - q^{85} - 11q^{86} + 2q^{87} - q^{88} - 23q^{89} + q^{90} + 6q^{92} - 16q^{94} + 6q^{95} + 2q^{96} - 7q^{97} + q^{99} + O(q^{100})$$

## 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
 3.70156 −2.70156
−1.00000 −1.00000 1.00000 −3.70156 1.00000 0 −1.00000 1.00000 3.70156
1.2 −1.00000 −1.00000 1.00000 2.70156 1.00000 0 −1.00000 1.00000 −2.70156
 $$n$$: 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.bs 2
7.b odd 2 1 714.2.a.k 2
21.c even 2 1 2142.2.a.y 2
28.d even 2 1 5712.2.a.bi 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.a.k 2 7.b odd 2 1
2142.2.a.y 2 21.c even 2 1
4998.2.a.bs 2 1.a even 1 1 trivial
5712.2.a.bi 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4998))$$:

 $$T_{5}^{2} + T_{5} - 10$$ $$T_{11}^{2} - T_{11} - 10$$ $$T_{13}^{2} + 3 T_{13} - 8$$ $$T_{23}^{2} - 6 T_{23} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-10 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-10 - T + T^{2}$$
$13$ $$-8 + 3 T + T^{2}$$
$17$ $$( -1 + T )^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$-32 - 6 T + T^{2}$$
$29$ $$-40 + 2 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$-8 + 3 T + T^{2}$$
$41$ $$-16 + 10 T + T^{2}$$
$43$ $$20 - 11 T + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$10 + 9 T + T^{2}$$
$59$ $$-40 - 2 T + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$32 - 13 T + T^{2}$$
$71$ $$-160 + 4 T + T^{2}$$
$73$ $$62 - 17 T + T^{2}$$
$79$ $$100 - 21 T + T^{2}$$
$83$ $$122 + 23 T + T^{2}$$
$89$ $$122 + 23 T + T^{2}$$
$97$ $$2 + 7 T + T^{2}$$