# Properties

 Label 4998.2.a.be 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 102) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} + 2q^{5} - q^{6} + q^{8} + q^{9} + 2q^{10} - 4q^{11} - q^{12} + 2q^{13} - 2q^{15} + q^{16} - q^{17} + q^{18} - 4q^{19} + 2q^{20} - 4q^{22} - q^{24} - q^{25} + 2q^{26} - q^{27} - 10q^{29} - 2q^{30} - 8q^{31} + q^{32} + 4q^{33} - q^{34} + q^{36} - 2q^{37} - 4q^{38} - 2q^{39} + 2q^{40} - 10q^{41} + 12q^{43} - 4q^{44} + 2q^{45} - q^{48} - q^{50} + q^{51} + 2q^{52} + 6q^{53} - q^{54} - 8q^{55} + 4q^{57} - 10q^{58} - 12q^{59} - 2q^{60} + 10q^{61} - 8q^{62} + q^{64} + 4q^{65} + 4q^{66} - 12q^{67} - q^{68} + q^{72} - 10q^{73} - 2q^{74} + q^{75} - 4q^{76} - 2q^{78} - 8q^{79} + 2q^{80} + q^{81} - 10q^{82} - 4q^{83} - 2q^{85} + 12q^{86} + 10q^{87} - 4q^{88} + 6q^{89} + 2q^{90} + 8q^{93} - 8q^{95} - q^{96} + 14q^{97} - 4q^{99} + O(q^{100})$$

## 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.be 1
7.b odd 2 1 102.2.a.c 1
21.c even 2 1 306.2.a.b 1
28.d even 2 1 816.2.a.b 1
35.c odd 2 1 2550.2.a.c 1
35.f even 4 2 2550.2.d.m 2
56.e even 2 1 3264.2.a.bc 1
56.h odd 2 1 3264.2.a.m 1
84.h odd 2 1 2448.2.a.p 1
105.g even 2 1 7650.2.a.ca 1
119.d odd 2 1 1734.2.a.j 1
119.f odd 4 2 1734.2.b.b 2
119.l odd 8 4 1734.2.f.e 4
168.e odd 2 1 9792.2.a.l 1
168.i even 2 1 9792.2.a.k 1
357.c even 2 1 5202.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 7.b odd 2 1
306.2.a.b 1 21.c even 2 1
816.2.a.b 1 28.d even 2 1
1734.2.a.j 1 119.d odd 2 1
1734.2.b.b 2 119.f odd 4 2
1734.2.f.e 4 119.l odd 8 4
2448.2.a.p 1 84.h odd 2 1
2550.2.a.c 1 35.c odd 2 1
2550.2.d.m 2 35.f even 4 2
3264.2.a.m 1 56.h odd 2 1
3264.2.a.bc 1 56.e even 2 1
4998.2.a.be 1 1.a even 1 1 trivial
5202.2.a.c 1 357.c even 2 1
7650.2.a.ca 1 105.g even 2 1
9792.2.a.k 1 168.i even 2 1
9792.2.a.l 1 168.e odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ $$1 + T$$
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ 1
$11$ $$1 + 4 T + 11 T^{2}$$
$13$ $$1 - 2 T + 13 T^{2}$$
$17$ $$1 + T$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 10 T + 29 T^{2}$$
$31$ $$1 + 8 T + 31 T^{2}$$
$37$ $$1 + 2 T + 37 T^{2}$$
$41$ $$1 + 10 T + 41 T^{2}$$
$43$ $$1 - 12 T + 43 T^{2}$$
$47$ $$1 + 47 T^{2}$$
$53$ $$1 - 6 T + 53 T^{2}$$
$59$ $$1 + 12 T + 59 T^{2}$$
$61$ $$1 - 10 T + 61 T^{2}$$
$67$ $$1 + 12 T + 67 T^{2}$$
$71$ $$1 + 71 T^{2}$$
$73$ $$1 + 10 T + 73 T^{2}$$
$79$ $$1 + 8 T + 79 T^{2}$$
$83$ $$1 + 4 T + 83 T^{2}$$
$89$ $$1 - 6 T + 89 T^{2}$$
$97$ $$1 - 14 T + 97 T^{2}$$