# Properties

 Label 4598.2.a.v 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: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - 2 \beta q^{3} + q^{4} + (3 \beta - 1) q^{5} + 2 \beta q^{6} + (3 \beta - 1) q^{7} - q^{8} + (4 \beta + 1) q^{9} +O(q^{10})$$ q - q^2 - 2*b * q^3 + q^4 + (3*b - 1) * q^5 + 2*b * q^6 + (3*b - 1) * q^7 - q^8 + (4*b + 1) * q^9 $$q - q^{2} - 2 \beta q^{3} + q^{4} + (3 \beta - 1) q^{5} + 2 \beta q^{6} + (3 \beta - 1) q^{7} - q^{8} + (4 \beta + 1) q^{9} + ( - 3 \beta + 1) q^{10} - 2 \beta q^{12} + 4 q^{13} + ( - 3 \beta + 1) q^{14} + ( - 4 \beta - 6) q^{15} + q^{16} - \beta q^{17} + ( - 4 \beta - 1) q^{18} - q^{19} + (3 \beta - 1) q^{20} + ( - 4 \beta - 6) q^{21} + (3 \beta - 4) q^{23} + 2 \beta q^{24} + (3 \beta + 5) q^{25} - 4 q^{26} + ( - 4 \beta - 8) q^{27} + (3 \beta - 1) q^{28} + (6 \beta - 4) q^{29} + (4 \beta + 6) q^{30} + (2 \beta - 6) q^{31} - q^{32} + \beta q^{34} + (3 \beta + 10) q^{35} + (4 \beta + 1) q^{36} - 2 \beta q^{37} + q^{38} - 8 \beta q^{39} + ( - 3 \beta + 1) q^{40} + ( - 4 \beta + 6) q^{41} + (4 \beta + 6) q^{42} + (\beta + 6) q^{43} + (11 \beta + 11) q^{45} + ( - 3 \beta + 4) q^{46} + ( - 3 \beta + 4) q^{47} - 2 \beta q^{48} + (3 \beta + 3) q^{49} + ( - 3 \beta - 5) q^{50} + (2 \beta + 2) q^{51} + 4 q^{52} + (2 \beta + 8) q^{53} + (4 \beta + 8) q^{54} + ( - 3 \beta + 1) q^{56} + 2 \beta q^{57} + ( - 6 \beta + 4) q^{58} + (8 \beta - 6) q^{59} + ( - 4 \beta - 6) q^{60} + ( - 3 \beta + 6) q^{61} + ( - 2 \beta + 6) q^{62} + (11 \beta + 11) q^{63} + q^{64} + (12 \beta - 4) q^{65} + ( - 2 \beta - 6) q^{67} - \beta q^{68} + (2 \beta - 6) q^{69} + ( - 3 \beta - 10) q^{70} + (2 \beta - 6) q^{71} + ( - 4 \beta - 1) q^{72} + (8 \beta - 2) q^{73} + 2 \beta q^{74} + ( - 16 \beta - 6) q^{75} - q^{76} + 8 \beta q^{78} + (2 \beta + 6) q^{79} + (3 \beta - 1) q^{80} + (12 \beta + 5) q^{81} + (4 \beta - 6) q^{82} + (3 \beta - 7) q^{83} + ( - 4 \beta - 6) q^{84} + ( - 2 \beta - 3) q^{85} + ( - \beta - 6) q^{86} + ( - 4 \beta - 12) q^{87} + ( - 10 \beta + 2) q^{89} + ( - 11 \beta - 11) q^{90} + (12 \beta - 4) q^{91} + (3 \beta - 4) q^{92} + (8 \beta - 4) q^{93} + (3 \beta - 4) q^{94} + ( - 3 \beta + 1) q^{95} + 2 \beta q^{96} + (8 \beta + 4) q^{97} + ( - 3 \beta - 3) q^{98} +O(q^{100})$$ q - q^2 - 2*b * q^3 + q^4 + (3*b - 1) * q^5 + 2*b * q^6 + (3*b - 1) * q^7 - q^8 + (4*b + 1) * q^9 + (-3*b + 1) * q^10 - 2*b * q^12 + 4 * q^13 + (-3*b + 1) * q^14 + (-4*b - 6) * q^15 + q^16 - b * q^17 + (-4*b - 1) * q^18 - q^19 + (3*b - 1) * q^20 + (-4*b - 6) * q^21 + (3*b - 4) * q^23 + 2*b * q^24 + (3*b + 5) * q^25 - 4 * q^26 + (-4*b - 8) * q^27 + (3*b - 1) * q^28 + (6*b - 4) * q^29 + (4*b + 6) * q^30 + (2*b - 6) * q^31 - q^32 + b * q^34 + (3*b + 10) * q^35 + (4*b + 1) * q^36 - 2*b * q^37 + q^38 - 8*b * q^39 + (-3*b + 1) * q^40 + (-4*b + 6) * q^41 + (4*b + 6) * q^42 + (b + 6) * q^43 + (11*b + 11) * q^45 + (-3*b + 4) * q^46 + (-3*b + 4) * q^47 - 2*b * q^48 + (3*b + 3) * q^49 + (-3*b - 5) * q^50 + (2*b + 2) * q^51 + 4 * q^52 + (2*b + 8) * q^53 + (4*b + 8) * q^54 + (-3*b + 1) * q^56 + 2*b * q^57 + (-6*b + 4) * q^58 + (8*b - 6) * q^59 + (-4*b - 6) * q^60 + (-3*b + 6) * q^61 + (-2*b + 6) * q^62 + (11*b + 11) * q^63 + q^64 + (12*b - 4) * q^65 + (-2*b - 6) * q^67 - b * q^68 + (2*b - 6) * q^69 + (-3*b - 10) * q^70 + (2*b - 6) * q^71 + (-4*b - 1) * q^72 + (8*b - 2) * q^73 + 2*b * q^74 + (-16*b - 6) * q^75 - q^76 + 8*b * q^78 + (2*b + 6) * q^79 + (3*b - 1) * q^80 + (12*b + 5) * q^81 + (4*b - 6) * q^82 + (3*b - 7) * q^83 + (-4*b - 6) * q^84 + (-2*b - 3) * q^85 + (-b - 6) * q^86 + (-4*b - 12) * q^87 + (-10*b + 2) * q^89 + (-11*b - 11) * q^90 + (12*b - 4) * q^91 + (3*b - 4) * q^92 + (8*b - 4) * q^93 + (3*b - 4) * q^94 + (-3*b + 1) * q^95 + 2*b * q^96 + (8*b + 4) * q^97 + (-3*b - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 + q^7 - 2 * q^8 + 6 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + q^{7} - 2 q^{8} + 6 q^{9} - q^{10} - 2 q^{12} + 8 q^{13} - q^{14} - 16 q^{15} + 2 q^{16} - q^{17} - 6 q^{18} - 2 q^{19} + q^{20} - 16 q^{21} - 5 q^{23} + 2 q^{24} + 13 q^{25} - 8 q^{26} - 20 q^{27} + q^{28} - 2 q^{29} + 16 q^{30} - 10 q^{31} - 2 q^{32} + q^{34} + 23 q^{35} + 6 q^{36} - 2 q^{37} + 2 q^{38} - 8 q^{39} - q^{40} + 8 q^{41} + 16 q^{42} + 13 q^{43} + 33 q^{45} + 5 q^{46} + 5 q^{47} - 2 q^{48} + 9 q^{49} - 13 q^{50} + 6 q^{51} + 8 q^{52} + 18 q^{53} + 20 q^{54} - q^{56} + 2 q^{57} + 2 q^{58} - 4 q^{59} - 16 q^{60} + 9 q^{61} + 10 q^{62} + 33 q^{63} + 2 q^{64} + 4 q^{65} - 14 q^{67} - q^{68} - 10 q^{69} - 23 q^{70} - 10 q^{71} - 6 q^{72} + 4 q^{73} + 2 q^{74} - 28 q^{75} - 2 q^{76} + 8 q^{78} + 14 q^{79} + q^{80} + 22 q^{81} - 8 q^{82} - 11 q^{83} - 16 q^{84} - 8 q^{85} - 13 q^{86} - 28 q^{87} - 6 q^{89} - 33 q^{90} + 4 q^{91} - 5 q^{92} - 5 q^{94} - q^{95} + 2 q^{96} + 16 q^{97} - 9 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + q^5 + 2 * q^6 + q^7 - 2 * q^8 + 6 * q^9 - q^10 - 2 * q^12 + 8 * q^13 - q^14 - 16 * q^15 + 2 * q^16 - q^17 - 6 * q^18 - 2 * q^19 + q^20 - 16 * q^21 - 5 * q^23 + 2 * q^24 + 13 * q^25 - 8 * q^26 - 20 * q^27 + q^28 - 2 * q^29 + 16 * q^30 - 10 * q^31 - 2 * q^32 + q^34 + 23 * q^35 + 6 * q^36 - 2 * q^37 + 2 * q^38 - 8 * q^39 - q^40 + 8 * q^41 + 16 * q^42 + 13 * q^43 + 33 * q^45 + 5 * q^46 + 5 * q^47 - 2 * q^48 + 9 * q^49 - 13 * q^50 + 6 * q^51 + 8 * q^52 + 18 * q^53 + 20 * q^54 - q^56 + 2 * q^57 + 2 * q^58 - 4 * q^59 - 16 * q^60 + 9 * q^61 + 10 * q^62 + 33 * q^63 + 2 * q^64 + 4 * q^65 - 14 * q^67 - q^68 - 10 * q^69 - 23 * q^70 - 10 * q^71 - 6 * q^72 + 4 * q^73 + 2 * q^74 - 28 * q^75 - 2 * q^76 + 8 * q^78 + 14 * q^79 + q^80 + 22 * q^81 - 8 * q^82 - 11 * q^83 - 16 * q^84 - 8 * q^85 - 13 * q^86 - 28 * q^87 - 6 * q^89 - 33 * q^90 + 4 * q^91 - 5 * q^92 - 5 * q^94 - q^95 + 2 * 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
 1.61803 −0.618034
−1.00000 −3.23607 1.00000 3.85410 3.23607 3.85410 −1.00000 7.47214 −3.85410
1.2 −1.00000 1.23607 1.00000 −2.85410 −1.23607 −2.85410 −1.00000 −1.47214 2.85410
 $$n$$: 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.v 2
11.b odd 2 1 4598.2.a.be 2
11.d odd 10 2 418.2.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.a 4 11.d odd 10 2
4598.2.a.v 2 1.a even 1 1 trivial
4598.2.a.be 2 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3}^{2} + 2T_{3} - 4$$ T3^2 + 2*T3 - 4 $$T_{5}^{2} - T_{5} - 11$$ T5^2 - T5 - 11 $$T_{7}^{2} - T_{7} - 11$$ T7^2 - T7 - 11 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 2T - 4$$
$5$ $$T^{2} - T - 11$$
$7$ $$T^{2} - T - 11$$
$11$ $$T^{2}$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} + T - 1$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 5T - 5$$
$29$ $$T^{2} + 2T - 44$$
$31$ $$T^{2} + 10T + 20$$
$37$ $$T^{2} + 2T - 4$$
$41$ $$T^{2} - 8T - 4$$
$43$ $$T^{2} - 13T + 41$$
$47$ $$T^{2} - 5T - 5$$
$53$ $$T^{2} - 18T + 76$$
$59$ $$T^{2} + 4T - 76$$
$61$ $$T^{2} - 9T + 9$$
$67$ $$T^{2} + 14T + 44$$
$71$ $$T^{2} + 10T + 20$$
$73$ $$T^{2} - 4T - 76$$
$79$ $$T^{2} - 14T + 44$$
$83$ $$T^{2} + 11T + 19$$
$89$ $$T^{2} + 6T - 116$$
$97$ $$T^{2} - 16T - 16$$