# Properties

 Label 4998.2.a.x 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4998,2,Mod(1,4998)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4998, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4998.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4998.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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 102) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + 4 q^{5} - q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + 4 * q^5 - q^6 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + 4 q^{5} - q^{6} - q^{8} + q^{9} - 4 q^{10} + q^{12} + 6 q^{13} + 4 q^{15} + q^{16} + q^{17} - q^{18} - 4 q^{19} + 4 q^{20} + 6 q^{23} - q^{24} + 11 q^{25} - 6 q^{26} + q^{27} - 4 q^{29} - 4 q^{30} + 6 q^{31} - q^{32} - q^{34} + q^{36} - 4 q^{37} + 4 q^{38} + 6 q^{39} - 4 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{45} - 6 q^{46} - 4 q^{47} + q^{48} - 11 q^{50} + q^{51} + 6 q^{52} - 2 q^{53} - q^{54} - 4 q^{57} + 4 q^{58} - 12 q^{59} + 4 q^{60} + 4 q^{61} - 6 q^{62} + q^{64} + 24 q^{65} - 12 q^{67} + q^{68} + 6 q^{69} - 6 q^{71} - q^{72} - 2 q^{73} + 4 q^{74} + 11 q^{75} - 4 q^{76} - 6 q^{78} + 10 q^{79} + 4 q^{80} + q^{81} - 10 q^{82} + 12 q^{83} + 4 q^{85} + 4 q^{86} - 4 q^{87} + 2 q^{89} - 4 q^{90} + 6 q^{92} + 6 q^{93} + 4 q^{94} - 16 q^{95} - q^{96} - 6 q^{97}+O(q^{100})$$ q - q^2 + q^3 + q^4 + 4 * q^5 - q^6 - q^8 + q^9 - 4 * q^10 + q^12 + 6 * q^13 + 4 * q^15 + q^16 + q^17 - q^18 - 4 * q^19 + 4 * q^20 + 6 * q^23 - q^24 + 11 * q^25 - 6 * q^26 + q^27 - 4 * q^29 - 4 * q^30 + 6 * q^31 - q^32 - q^34 + q^36 - 4 * q^37 + 4 * q^38 + 6 * q^39 - 4 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^45 - 6 * q^46 - 4 * q^47 + q^48 - 11 * q^50 + q^51 + 6 * q^52 - 2 * q^53 - q^54 - 4 * q^57 + 4 * q^58 - 12 * q^59 + 4 * q^60 + 4 * q^61 - 6 * q^62 + q^64 + 24 * q^65 - 12 * q^67 + q^68 + 6 * q^69 - 6 * q^71 - q^72 - 2 * q^73 + 4 * q^74 + 11 * q^75 - 4 * q^76 - 6 * q^78 + 10 * q^79 + 4 * q^80 + q^81 - 10 * q^82 + 12 * q^83 + 4 * q^85 + 4 * q^86 - 4 * q^87 + 2 * q^89 - 4 * q^90 + 6 * q^92 + 6 * q^93 + 4 * q^94 - 16 * q^95 - q^96 - 6 * q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 4.00000 −1.00000 0 −1.00000 1.00000 −4.00000
 $$n$$: 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.x 1
7.b odd 2 1 102.2.a.a 1
21.c even 2 1 306.2.a.d 1
28.d even 2 1 816.2.a.h 1
35.c odd 2 1 2550.2.a.be 1
35.f even 4 2 2550.2.d.q 2
56.e even 2 1 3264.2.a.p 1
56.h odd 2 1 3264.2.a.bf 1
84.h odd 2 1 2448.2.a.t 1
105.g even 2 1 7650.2.a.z 1
119.d odd 2 1 1734.2.a.h 1
119.f odd 4 2 1734.2.b.d 2
119.l odd 8 4 1734.2.f.g 4
168.e odd 2 1 9792.2.a.b 1
168.i even 2 1 9792.2.a.a 1
357.c even 2 1 5202.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.a 1 7.b odd 2 1
306.2.a.d 1 21.c even 2 1
816.2.a.h 1 28.d even 2 1
1734.2.a.h 1 119.d odd 2 1
1734.2.b.d 2 119.f odd 4 2
1734.2.f.g 4 119.l odd 8 4
2448.2.a.t 1 84.h odd 2 1
2550.2.a.be 1 35.c odd 2 1
2550.2.d.q 2 35.f even 4 2
3264.2.a.p 1 56.e even 2 1
3264.2.a.bf 1 56.h odd 2 1
4998.2.a.x 1 1.a even 1 1 trivial
5202.2.a.g 1 357.c even 2 1
7650.2.a.z 1 105.g even 2 1
9792.2.a.a 1 168.i even 2 1
9792.2.a.b 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4998))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{11}$$ T11 $$T_{13} - 6$$ T13 - 6 $$T_{23} - 6$$ T23 - 6

## Hecke characteristic polynomials

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