# Properties

 Label 4598.2.a.b Level $4598$ Weight $2$ Character orbit 4598.a Self dual yes Analytic conductor $36.715$ 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] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4598 = 2 \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4598.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: $$36.7152148494$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} + 3 q^{7} - q^{8} - 2 q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 - 2 * q^5 + q^6 + 3 * q^7 - q^8 - 2 * q^9 $$q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} + 3 q^{7} - q^{8} - 2 q^{9} + 2 q^{10} - q^{12} - q^{13} - 3 q^{14} + 2 q^{15} + q^{16} + 7 q^{17} + 2 q^{18} - q^{19} - 2 q^{20} - 3 q^{21} - 5 q^{23} + q^{24} - q^{25} + q^{26} + 5 q^{27} + 3 q^{28} - q^{29} - 2 q^{30} + 10 q^{31} - q^{32} - 7 q^{34} - 6 q^{35} - 2 q^{36} - 6 q^{37} + q^{38} + q^{39} + 2 q^{40} - 6 q^{41} + 3 q^{42} + 4 q^{43} + 4 q^{45} + 5 q^{46} - q^{48} + 2 q^{49} + q^{50} - 7 q^{51} - q^{52} - q^{53} - 5 q^{54} - 3 q^{56} + q^{57} + q^{58} + 3 q^{59} + 2 q^{60} + 12 q^{61} - 10 q^{62} - 6 q^{63} + q^{64} + 2 q^{65} + 3 q^{67} + 7 q^{68} + 5 q^{69} + 6 q^{70} - 10 q^{71} + 2 q^{72} - 3 q^{73} + 6 q^{74} + q^{75} - q^{76} - q^{78} - 8 q^{79} - 2 q^{80} + q^{81} + 6 q^{82} - 8 q^{83} - 3 q^{84} - 14 q^{85} - 4 q^{86} + q^{87} - 8 q^{89} - 4 q^{90} - 3 q^{91} - 5 q^{92} - 10 q^{93} + 2 q^{95} + q^{96} + 8 q^{97} - 2 q^{98}+O(q^{100})$$ q - q^2 - q^3 + q^4 - 2 * q^5 + q^6 + 3 * q^7 - q^8 - 2 * q^9 + 2 * q^10 - q^12 - q^13 - 3 * q^14 + 2 * q^15 + q^16 + 7 * q^17 + 2 * q^18 - q^19 - 2 * q^20 - 3 * q^21 - 5 * q^23 + q^24 - q^25 + q^26 + 5 * q^27 + 3 * q^28 - q^29 - 2 * q^30 + 10 * q^31 - q^32 - 7 * q^34 - 6 * q^35 - 2 * q^36 - 6 * q^37 + q^38 + q^39 + 2 * q^40 - 6 * q^41 + 3 * q^42 + 4 * q^43 + 4 * q^45 + 5 * q^46 - q^48 + 2 * q^49 + q^50 - 7 * q^51 - q^52 - q^53 - 5 * q^54 - 3 * q^56 + q^57 + q^58 + 3 * q^59 + 2 * q^60 + 12 * q^61 - 10 * q^62 - 6 * q^63 + q^64 + 2 * q^65 + 3 * q^67 + 7 * q^68 + 5 * q^69 + 6 * q^70 - 10 * q^71 + 2 * q^72 - 3 * q^73 + 6 * q^74 + q^75 - q^76 - q^78 - 8 * q^79 - 2 * q^80 + q^81 + 6 * q^82 - 8 * q^83 - 3 * q^84 - 14 * q^85 - 4 * q^86 + q^87 - 8 * q^89 - 4 * q^90 - 3 * q^91 - 5 * q^92 - 10 * q^93 + 2 * q^95 + q^96 + 8 * q^97 - 2 * 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 −1.00000 1.00000 −2.00000 1.00000 3.00000 −1.00000 −2.00000 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
$$2$$ $$1$$
$$11$$ $$-1$$
$$19$$ $$1$$

## 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 4598.2.a.b 1
11.b odd 2 1 418.2.a.a 1
33.d even 2 1 3762.2.a.g 1
44.c even 2 1 3344.2.a.h 1
209.d even 2 1 7942.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.a 1 11.b odd 2 1
3344.2.a.h 1 44.c even 2 1
3762.2.a.g 1 33.d even 2 1
4598.2.a.b 1 1.a even 1 1 trivial
7942.2.a.i 1 209.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{5} + 2$$ T5 + 2 $$T_{7} - 3$$ T7 - 3 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

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