# Properties

 Label 2898.2.a.j Level $2898$ Weight $2$ Character orbit 2898.a Self dual yes Analytic conductor $23.141$ 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] = [2898,2,Mod(1,2898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2898, 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("2898.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2898.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: $$23.1406465058$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 966) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 3 q^{5} + q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + 3 * q^5 + q^7 - q^8 $$q - q^{2} + q^{4} + 3 q^{5} + q^{7} - q^{8} - 3 q^{10} + 5 q^{13} - q^{14} + q^{16} + 8 q^{19} + 3 q^{20} + q^{23} + 4 q^{25} - 5 q^{26} + q^{28} - 3 q^{29} + 2 q^{31} - q^{32} + 3 q^{35} - 7 q^{37} - 8 q^{38} - 3 q^{40} - 9 q^{41} - q^{43} - q^{46} + 3 q^{47} + q^{49} - 4 q^{50} + 5 q^{52} + 12 q^{53} - q^{56} + 3 q^{58} + 6 q^{59} + 14 q^{61} - 2 q^{62} + q^{64} + 15 q^{65} - 4 q^{67} - 3 q^{70} - 6 q^{71} - 4 q^{73} + 7 q^{74} + 8 q^{76} - 16 q^{79} + 3 q^{80} + 9 q^{82} + 12 q^{83} + q^{86} - 6 q^{89} + 5 q^{91} + q^{92} - 3 q^{94} + 24 q^{95} - q^{97} - q^{98}+O(q^{100})$$ q - q^2 + q^4 + 3 * q^5 + q^7 - q^8 - 3 * q^10 + 5 * q^13 - q^14 + q^16 + 8 * q^19 + 3 * q^20 + q^23 + 4 * q^25 - 5 * q^26 + q^28 - 3 * q^29 + 2 * q^31 - q^32 + 3 * q^35 - 7 * q^37 - 8 * q^38 - 3 * q^40 - 9 * q^41 - q^43 - q^46 + 3 * q^47 + q^49 - 4 * q^50 + 5 * q^52 + 12 * q^53 - q^56 + 3 * q^58 + 6 * q^59 + 14 * q^61 - 2 * q^62 + q^64 + 15 * q^65 - 4 * q^67 - 3 * q^70 - 6 * q^71 - 4 * q^73 + 7 * q^74 + 8 * q^76 - 16 * q^79 + 3 * q^80 + 9 * q^82 + 12 * q^83 + q^86 - 6 * q^89 + 5 * q^91 + q^92 - 3 * q^94 + 24 * q^95 - q^97 - q^98

## 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 0 1.00000 3.00000 0 1.00000 −1.00000 0 −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$23$$ $$-1$$

## 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 2898.2.a.j 1
3.b odd 2 1 966.2.a.i 1
12.b even 2 1 7728.2.a.a 1
21.c even 2 1 6762.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.i 1 3.b odd 2 1
2898.2.a.j 1 1.a even 1 1 trivial
6762.2.a.bd 1 21.c even 2 1
7728.2.a.a 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2898))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{11}$$ T11 $$T_{13} - 5$$ T13 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 3$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T - 5$$
$17$ $$T$$
$19$ $$T - 8$$
$23$ $$T - 1$$
$29$ $$T + 3$$
$31$ $$T - 2$$
$37$ $$T + 7$$
$41$ $$T + 9$$
$43$ $$T + 1$$
$47$ $$T - 3$$
$53$ $$T - 12$$
$59$ $$T - 6$$
$61$ $$T - 14$$
$67$ $$T + 4$$
$71$ $$T + 6$$
$73$ $$T + 4$$
$79$ $$T + 16$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T + 1$$