# Properties

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

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

## 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
 0
−1.00000 1.00000 1.00000 −3.00000 −1.00000 0 −1.00000 1.00000 3.00000
 $$n$$: 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.k 1
7.b odd 2 1 4998.2.a.j 1
7.c even 3 2 714.2.i.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.i.h 2 7.c even 3 2
4998.2.a.j 1 7.b odd 2 1
4998.2.a.k 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4998))$$:

 $$T_{5} + 3$$ $$T_{11}$$ $$T_{13} + 1$$ $$T_{23} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$3 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$1 + T$$
$17$ $$-1 + T$$
$19$ $$4 + T$$
$23$ $$-6 + T$$
$29$ $$-3 + T$$
$31$ $$1 + T$$
$37$ $$4 + T$$
$41$ $$-3 + T$$
$43$ $$4 + T$$
$47$ $$-3 + T$$
$53$ $$-12 + T$$
$59$ $$-9 + T$$
$61$ $$10 + T$$
$67$ $$-2 + T$$
$71$ $$6 + T$$
$73$ $$16 + T$$
$79$ $$4 + T$$
$83$ $$9 + T$$
$89$ $$12 + T$$
$97$ $$-8 + T$$