# 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

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

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4998))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 2$$ T13 - 2 $$T_{23}$$ T23

## Hecke characteristic polynomials

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