# Properties

 Label 4998.2.a.bk Level $4998$ Weight $2$ Character orbit 4998.a Self dual yes Analytic conductor $39.909$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 714) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
1.00000 1.00000 1.00000 −1.00000 1.00000 0 1.00000 1.00000 −1.00000
 $$n$$: 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.bk 1
7.b odd 2 1 4998.2.a.ba 1
7.d odd 6 2 714.2.i.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.i.e 2 7.d odd 6 2
4998.2.a.ba 1 7.b odd 2 1
4998.2.a.bk 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5} + 1$$ $$T_{11} + 6$$ $$T_{13}$$ $$T_{23} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$6 + T$$
$13$ $$T$$
$17$ $$1 + T$$
$19$ $$-3 + T$$
$23$ $$-7 + T$$
$29$ $$-6 + T$$
$31$ $$-8 + T$$
$37$ $$-7 + T$$
$41$ $$T$$
$43$ $$9 + T$$
$47$ $$-6 + T$$
$53$ $$-2 + T$$
$59$ $$-11 + T$$
$61$ $$-6 + T$$
$67$ $$-9 + T$$
$71$ $$9 + T$$
$73$ $$-12 + T$$
$79$ $$8 + T$$
$83$ $$4 + T$$
$89$ $$5 + T$$
$97$ $$-4 + T$$