# Properties

 Label 4598.2.a.y Level $4598$ Weight $2$ Character orbit 4598.a Self dual yes Analytic conductor $36.715$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta q^{3} + q^{4} + ( - \beta + 2) q^{5} + \beta q^{6} + ( - \beta + 3) q^{7} - q^{8} + (\beta + 2) q^{9} +O(q^{10})$$ q - q^2 - b * q^3 + q^4 + (-b + 2) * q^5 + b * q^6 + (-b + 3) * q^7 - q^8 + (b + 2) * q^9 $$q - q^{2} - \beta q^{3} + q^{4} + ( - \beta + 2) q^{5} + \beta q^{6} + ( - \beta + 3) q^{7} - q^{8} + (\beta + 2) q^{9} + (\beta - 2) q^{10} - \beta q^{12} + ( - \beta - 3) q^{13} + (\beta - 3) q^{14} + ( - \beta + 5) q^{15} + q^{16} + ( - 2 \beta + 4) q^{17} + ( - \beta - 2) q^{18} - q^{19} + ( - \beta + 2) q^{20} + ( - 2 \beta + 5) q^{21} + (2 \beta + 2) q^{23} + \beta q^{24} + ( - 3 \beta + 4) q^{25} + (\beta + 3) q^{26} - 5 q^{27} + ( - \beta + 3) q^{28} + ( - \beta - 4) q^{29} + (\beta - 5) q^{30} + (\beta - 9) q^{31} - q^{32} + (2 \beta - 4) q^{34} + ( - 4 \beta + 11) q^{35} + (\beta + 2) q^{36} + 8 q^{37} + q^{38} + (4 \beta + 5) q^{39} + (\beta - 2) q^{40} + ( - \beta - 1) q^{41} + (2 \beta - 5) q^{42} + (3 \beta - 2) q^{43} + ( - \beta - 1) q^{45} + ( - 2 \beta - 2) q^{46} + ( - 4 \beta + 2) q^{47} - \beta q^{48} + ( - 5 \beta + 7) q^{49} + (3 \beta - 4) q^{50} + ( - 2 \beta + 10) q^{51} + ( - \beta - 3) q^{52} + (2 \beta + 2) q^{53} + 5 q^{54} + (\beta - 3) q^{56} + \beta q^{57} + (\beta + 4) q^{58} + ( - \beta + 5) q^{60} - 2 q^{61} + ( - \beta + 9) q^{62} + q^{63} + q^{64} + (2 \beta - 1) q^{65} + (\beta - 3) q^{67} + ( - 2 \beta + 4) q^{68} + ( - 4 \beta - 10) q^{69} + (4 \beta - 11) q^{70} + (3 \beta - 6) q^{71} + ( - \beta - 2) q^{72} + (2 \beta - 6) q^{73} - 8 q^{74} + ( - \beta + 15) q^{75} - q^{76} + ( - 4 \beta - 5) q^{78} + ( - 2 \beta + 2) q^{79} + ( - \beta + 2) q^{80} + (2 \beta - 6) q^{81} + (\beta + 1) q^{82} + ( - \beta + 2) q^{83} + ( - 2 \beta + 5) q^{84} + ( - 6 \beta + 18) q^{85} + ( - 3 \beta + 2) q^{86} + (5 \beta + 5) q^{87} + ( - 2 \beta - 8) q^{89} + (\beta + 1) q^{90} + (\beta - 4) q^{91} + (2 \beta + 2) q^{92} + (8 \beta - 5) q^{93} + (4 \beta - 2) q^{94} + (\beta - 2) q^{95} + \beta q^{96} + 8 q^{97} + (5 \beta - 7) q^{98} +O(q^{100})$$ q - q^2 - b * q^3 + q^4 + (-b + 2) * q^5 + b * q^6 + (-b + 3) * q^7 - q^8 + (b + 2) * q^9 + (b - 2) * q^10 - b * q^12 + (-b - 3) * q^13 + (b - 3) * q^14 + (-b + 5) * q^15 + q^16 + (-2*b + 4) * q^17 + (-b - 2) * q^18 - q^19 + (-b + 2) * q^20 + (-2*b + 5) * q^21 + (2*b + 2) * q^23 + b * q^24 + (-3*b + 4) * q^25 + (b + 3) * q^26 - 5 * q^27 + (-b + 3) * q^28 + (-b - 4) * q^29 + (b - 5) * q^30 + (b - 9) * q^31 - q^32 + (2*b - 4) * q^34 + (-4*b + 11) * q^35 + (b + 2) * q^36 + 8 * q^37 + q^38 + (4*b + 5) * q^39 + (b - 2) * q^40 + (-b - 1) * q^41 + (2*b - 5) * q^42 + (3*b - 2) * q^43 + (-b - 1) * q^45 + (-2*b - 2) * q^46 + (-4*b + 2) * q^47 - b * q^48 + (-5*b + 7) * q^49 + (3*b - 4) * q^50 + (-2*b + 10) * q^51 + (-b - 3) * q^52 + (2*b + 2) * q^53 + 5 * q^54 + (b - 3) * q^56 + b * q^57 + (b + 4) * q^58 + (-b + 5) * q^60 - 2 * q^61 + (-b + 9) * q^62 + q^63 + q^64 + (2*b - 1) * q^65 + (b - 3) * q^67 + (-2*b + 4) * q^68 + (-4*b - 10) * q^69 + (4*b - 11) * q^70 + (3*b - 6) * q^71 + (-b - 2) * q^72 + (2*b - 6) * q^73 - 8 * q^74 + (-b + 15) * q^75 - q^76 + (-4*b - 5) * q^78 + (-2*b + 2) * q^79 + (-b + 2) * q^80 + (2*b - 6) * q^81 + (b + 1) * q^82 + (-b + 2) * q^83 + (-2*b + 5) * q^84 + (-6*b + 18) * q^85 + (-3*b + 2) * q^86 + (5*b + 5) * q^87 + (-2*b - 8) * q^89 + (b + 1) * q^90 + (b - 4) * q^91 + (2*b + 2) * q^92 + (8*b - 5) * q^93 + (4*b - 2) * q^94 + (b - 2) * q^95 + b * q^96 + 8 * q^97 + (5*b - 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - q^{3} + 2 q^{4} + 3 q^{5} + q^{6} + 5 q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - q^3 + 2 * q^4 + 3 * q^5 + q^6 + 5 * q^7 - 2 * q^8 + 5 * q^9 $$2 q - 2 q^{2} - q^{3} + 2 q^{4} + 3 q^{5} + q^{6} + 5 q^{7} - 2 q^{8} + 5 q^{9} - 3 q^{10} - q^{12} - 7 q^{13} - 5 q^{14} + 9 q^{15} + 2 q^{16} + 6 q^{17} - 5 q^{18} - 2 q^{19} + 3 q^{20} + 8 q^{21} + 6 q^{23} + q^{24} + 5 q^{25} + 7 q^{26} - 10 q^{27} + 5 q^{28} - 9 q^{29} - 9 q^{30} - 17 q^{31} - 2 q^{32} - 6 q^{34} + 18 q^{35} + 5 q^{36} + 16 q^{37} + 2 q^{38} + 14 q^{39} - 3 q^{40} - 3 q^{41} - 8 q^{42} - q^{43} - 3 q^{45} - 6 q^{46} - q^{48} + 9 q^{49} - 5 q^{50} + 18 q^{51} - 7 q^{52} + 6 q^{53} + 10 q^{54} - 5 q^{56} + q^{57} + 9 q^{58} + 9 q^{60} - 4 q^{61} + 17 q^{62} + 2 q^{63} + 2 q^{64} - 5 q^{67} + 6 q^{68} - 24 q^{69} - 18 q^{70} - 9 q^{71} - 5 q^{72} - 10 q^{73} - 16 q^{74} + 29 q^{75} - 2 q^{76} - 14 q^{78} + 2 q^{79} + 3 q^{80} - 10 q^{81} + 3 q^{82} + 3 q^{83} + 8 q^{84} + 30 q^{85} + q^{86} + 15 q^{87} - 18 q^{89} + 3 q^{90} - 7 q^{91} + 6 q^{92} - 2 q^{93} - 3 q^{95} + q^{96} + 16 q^{97} - 9 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - q^3 + 2 * q^4 + 3 * q^5 + q^6 + 5 * q^7 - 2 * q^8 + 5 * q^9 - 3 * q^10 - q^12 - 7 * q^13 - 5 * q^14 + 9 * q^15 + 2 * q^16 + 6 * q^17 - 5 * q^18 - 2 * q^19 + 3 * q^20 + 8 * q^21 + 6 * q^23 + q^24 + 5 * q^25 + 7 * q^26 - 10 * q^27 + 5 * q^28 - 9 * q^29 - 9 * q^30 - 17 * q^31 - 2 * q^32 - 6 * q^34 + 18 * q^35 + 5 * q^36 + 16 * q^37 + 2 * q^38 + 14 * q^39 - 3 * q^40 - 3 * q^41 - 8 * q^42 - q^43 - 3 * q^45 - 6 * q^46 - q^48 + 9 * q^49 - 5 * q^50 + 18 * q^51 - 7 * q^52 + 6 * q^53 + 10 * q^54 - 5 * q^56 + q^57 + 9 * q^58 + 9 * q^60 - 4 * q^61 + 17 * q^62 + 2 * q^63 + 2 * q^64 - 5 * q^67 + 6 * q^68 - 24 * q^69 - 18 * q^70 - 9 * q^71 - 5 * q^72 - 10 * q^73 - 16 * q^74 + 29 * q^75 - 2 * q^76 - 14 * q^78 + 2 * q^79 + 3 * q^80 - 10 * q^81 + 3 * q^82 + 3 * q^83 + 8 * q^84 + 30 * q^85 + q^86 + 15 * q^87 - 18 * q^89 + 3 * q^90 - 7 * q^91 + 6 * q^92 - 2 * q^93 - 3 * q^95 + q^96 + 16 * q^97 - 9 * 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
 2.79129 −1.79129
−1.00000 −2.79129 1.00000 −0.791288 2.79129 0.208712 −1.00000 4.79129 0.791288
1.2 −1.00000 1.79129 1.00000 3.79129 −1.79129 4.79129 −1.00000 0.208712 −3.79129
 $$n$$: 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.y 2
11.b odd 2 1 418.2.a.f 2
33.d even 2 1 3762.2.a.s 2
44.c even 2 1 3344.2.a.l 2
209.d even 2 1 7942.2.a.w 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + T - 5$$
$5$ $$T^{2} - 3T - 3$$
$7$ $$T^{2} - 5T + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 7T + 7$$
$17$ $$T^{2} - 6T - 12$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 6T - 12$$
$29$ $$T^{2} + 9T + 15$$
$31$ $$T^{2} + 17T + 67$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} + 3T - 3$$
$43$ $$T^{2} + T - 47$$
$47$ $$T^{2} - 84$$
$53$ $$T^{2} - 6T - 12$$
$59$ $$T^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 5T + 1$$
$71$ $$T^{2} + 9T - 27$$
$73$ $$T^{2} + 10T + 4$$
$79$ $$T^{2} - 2T - 20$$
$83$ $$T^{2} - 3T - 3$$
$89$ $$T^{2} + 18T + 60$$
$97$ $$(T - 8)^{2}$$