# Properties

 Label 4998.2.a.m 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

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

## 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
 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.m 1
7.b odd 2 1 4998.2.a.e 1
7.d odd 6 2 714.2.i.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.i.i 2 7.d odd 6 2
4998.2.a.e 1 7.b odd 2 1
4998.2.a.m 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4998))$$:

 $$T_{5} + 1$$ $$T_{11} + 2$$ $$T_{13} + 1$$ $$T_{23} - 8$$

## Hecke characteristic polynomials

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