# Properties

 Label 4998.2.a.bw Level $4998$ Weight $2$ Character orbit 4998.a Self dual yes Analytic conductor $39.909$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −2.44949 2.44949
1.00000 −1.00000 1.00000 −2.00000 −1.00000 0 1.00000 1.00000 −2.00000
1.2 1.00000 −1.00000 1.00000 −2.00000 −1.00000 0 1.00000 1.00000 −2.00000
 $$n$$: 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.bw 2
7.b odd 2 1 714.2.a.m 2
21.c even 2 1 2142.2.a.v 2
28.d even 2 1 5712.2.a.bm 2

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

 $$T_{5} + 2$$ $$T_{11}^{2} - 24$$ $$T_{13}^{2} - 4 T_{13} - 20$$ $$T_{23} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 2 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-24 + T^{2}$$
$13$ $$-20 - 4 T + T^{2}$$
$17$ $$( -1 + T )^{2}$$
$19$ $$-24 + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$-20 - 4 T + T^{2}$$
$31$ $$-96 + T^{2}$$
$37$ $$-20 + 4 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$-8 - 8 T + T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$-80 + 8 T + T^{2}$$
$71$ $$-80 - 8 T + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$-96 + T^{2}$$
$83$ $$120 - 24 T + T^{2}$$
$89$ $$-92 - 4 T + T^{2}$$
$97$ $$( 6 + T )^{2}$$