# Properties

 Label 4998.2.a.bq 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

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

## 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 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.bq 1
7.b odd 2 1 714.2.a.f 1
21.c even 2 1 2142.2.a.h 1
28.d even 2 1 5712.2.a.o 1

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

 $$T_{5} - 2$$ $$T_{11} - 4$$ $$T_{13} - 2$$ $$T_{23} - 8$$

## Hecke characteristic polynomials

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