# Properties

 Label 4598.2.a.t 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.

 $$f(q)$$ $$=$$ $$q - q^{2} - 2 q^{3} + q^{4} + ( - \beta + 1) q^{5} + 2 q^{6} + ( - 3 \beta + 3) q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - 2 * q^3 + q^4 + (-b + 1) * q^5 + 2 * q^6 + (-3*b + 3) * q^7 - q^8 + q^9 $$q - q^{2} - 2 q^{3} + q^{4} + ( - \beta + 1) q^{5} + 2 q^{6} + ( - 3 \beta + 3) q^{7} - q^{8} + q^{9} + (\beta - 1) q^{10} - 2 q^{12} + 4 \beta q^{13} + (3 \beta - 3) q^{14} + (2 \beta - 2) q^{15} + q^{16} + ( - \beta - 4) q^{17} - q^{18} - q^{19} + ( - \beta + 1) q^{20} + (6 \beta - 6) q^{21} + (3 \beta - 6) q^{23} + 2 q^{24} + ( - \beta - 3) q^{25} - 4 \beta q^{26} + 4 q^{27} + ( - 3 \beta + 3) q^{28} + (4 \beta - 8) q^{29} + ( - 2 \beta + 2) q^{30} + 2 q^{31} - q^{32} + (\beta + 4) q^{34} + ( - 3 \beta + 6) q^{35} + q^{36} + 4 q^{37} + q^{38} - 8 \beta q^{39} + (\beta - 1) q^{40} + (2 \beta + 6) q^{41} + ( - 6 \beta + 6) q^{42} + ( - 5 \beta + 6) q^{43} + ( - \beta + 1) q^{45} + ( - 3 \beta + 6) q^{46} + (\beta + 2) q^{47} - 2 q^{48} + ( - 9 \beta + 11) q^{49} + (\beta + 3) q^{50} + (2 \beta + 8) q^{51} + 4 \beta q^{52} + ( - 2 \beta - 6) q^{53} - 4 q^{54} + (3 \beta - 3) q^{56} + 2 q^{57} + ( - 4 \beta + 8) q^{58} + ( - 4 \beta - 6) q^{59} + (2 \beta - 2) q^{60} + ( - 9 \beta + 4) q^{61} - 2 q^{62} + ( - 3 \beta + 3) q^{63} + q^{64} - 4 q^{65} + (2 \beta + 4) q^{67} + ( - \beta - 4) q^{68} + ( - 6 \beta + 12) q^{69} + (3 \beta - 6) q^{70} - 8 \beta q^{71} - q^{72} + 14 q^{73} - 4 q^{74} + (2 \beta + 6) q^{75} - q^{76} + 8 \beta q^{78} + 6 \beta q^{79} + ( - \beta + 1) q^{80} - 11 q^{81} + ( - 2 \beta - 6) q^{82} + ( - 11 \beta + 7) q^{83} + (6 \beta - 6) q^{84} + (4 \beta - 3) q^{85} + (5 \beta - 6) q^{86} + ( - 8 \beta + 16) q^{87} + ( - 2 \beta + 10) q^{89} + (\beta - 1) q^{90} - 12 q^{91} + (3 \beta - 6) q^{92} - 4 q^{93} + ( - \beta - 2) q^{94} + (\beta - 1) q^{95} + 2 q^{96} + (2 \beta - 8) q^{97} + (9 \beta - 11) q^{98} +O(q^{100})$$ q - q^2 - 2 * q^3 + q^4 + (-b + 1) * q^5 + 2 * q^6 + (-3*b + 3) * q^7 - q^8 + q^9 + (b - 1) * q^10 - 2 * q^12 + 4*b * q^13 + (3*b - 3) * q^14 + (2*b - 2) * q^15 + q^16 + (-b - 4) * q^17 - q^18 - q^19 + (-b + 1) * q^20 + (6*b - 6) * q^21 + (3*b - 6) * q^23 + 2 * q^24 + (-b - 3) * q^25 - 4*b * q^26 + 4 * q^27 + (-3*b + 3) * q^28 + (4*b - 8) * q^29 + (-2*b + 2) * q^30 + 2 * q^31 - q^32 + (b + 4) * q^34 + (-3*b + 6) * q^35 + q^36 + 4 * q^37 + q^38 - 8*b * q^39 + (b - 1) * q^40 + (2*b + 6) * q^41 + (-6*b + 6) * q^42 + (-5*b + 6) * q^43 + (-b + 1) * q^45 + (-3*b + 6) * q^46 + (b + 2) * q^47 - 2 * q^48 + (-9*b + 11) * q^49 + (b + 3) * q^50 + (2*b + 8) * q^51 + 4*b * q^52 + (-2*b - 6) * q^53 - 4 * q^54 + (3*b - 3) * q^56 + 2 * q^57 + (-4*b + 8) * q^58 + (-4*b - 6) * q^59 + (2*b - 2) * q^60 + (-9*b + 4) * q^61 - 2 * q^62 + (-3*b + 3) * q^63 + q^64 - 4 * q^65 + (2*b + 4) * q^67 + (-b - 4) * q^68 + (-6*b + 12) * q^69 + (3*b - 6) * q^70 - 8*b * q^71 - q^72 + 14 * q^73 - 4 * q^74 + (2*b + 6) * q^75 - q^76 + 8*b * q^78 + 6*b * q^79 + (-b + 1) * q^80 - 11 * q^81 + (-2*b - 6) * q^82 + (-11*b + 7) * q^83 + (6*b - 6) * q^84 + (4*b - 3) * q^85 + (5*b - 6) * q^86 + (-8*b + 16) * q^87 + (-2*b + 10) * q^89 + (b - 1) * q^90 - 12 * q^91 + (3*b - 6) * q^92 - 4 * q^93 + (-b - 2) * q^94 + (b - 1) * q^95 + 2 * q^96 + (2*b - 8) * q^97 + (9*b - 11) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + q^{5} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + q^5 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + q^{5} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} - q^{10} - 4 q^{12} + 4 q^{13} - 3 q^{14} - 2 q^{15} + 2 q^{16} - 9 q^{17} - 2 q^{18} - 2 q^{19} + q^{20} - 6 q^{21} - 9 q^{23} + 4 q^{24} - 7 q^{25} - 4 q^{26} + 8 q^{27} + 3 q^{28} - 12 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} + 9 q^{34} + 9 q^{35} + 2 q^{36} + 8 q^{37} + 2 q^{38} - 8 q^{39} - q^{40} + 14 q^{41} + 6 q^{42} + 7 q^{43} + q^{45} + 9 q^{46} + 5 q^{47} - 4 q^{48} + 13 q^{49} + 7 q^{50} + 18 q^{51} + 4 q^{52} - 14 q^{53} - 8 q^{54} - 3 q^{56} + 4 q^{57} + 12 q^{58} - 16 q^{59} - 2 q^{60} - q^{61} - 4 q^{62} + 3 q^{63} + 2 q^{64} - 8 q^{65} + 10 q^{67} - 9 q^{68} + 18 q^{69} - 9 q^{70} - 8 q^{71} - 2 q^{72} + 28 q^{73} - 8 q^{74} + 14 q^{75} - 2 q^{76} + 8 q^{78} + 6 q^{79} + q^{80} - 22 q^{81} - 14 q^{82} + 3 q^{83} - 6 q^{84} - 2 q^{85} - 7 q^{86} + 24 q^{87} + 18 q^{89} - q^{90} - 24 q^{91} - 9 q^{92} - 8 q^{93} - 5 q^{94} - q^{95} + 4 q^{96} - 14 q^{97} - 13 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + q^5 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 - q^10 - 4 * q^12 + 4 * q^13 - 3 * q^14 - 2 * q^15 + 2 * q^16 - 9 * q^17 - 2 * q^18 - 2 * q^19 + q^20 - 6 * q^21 - 9 * q^23 + 4 * q^24 - 7 * q^25 - 4 * q^26 + 8 * q^27 + 3 * q^28 - 12 * q^29 + 2 * q^30 + 4 * q^31 - 2 * q^32 + 9 * q^34 + 9 * q^35 + 2 * q^36 + 8 * q^37 + 2 * q^38 - 8 * q^39 - q^40 + 14 * q^41 + 6 * q^42 + 7 * q^43 + q^45 + 9 * q^46 + 5 * q^47 - 4 * q^48 + 13 * q^49 + 7 * q^50 + 18 * q^51 + 4 * q^52 - 14 * q^53 - 8 * q^54 - 3 * q^56 + 4 * q^57 + 12 * q^58 - 16 * q^59 - 2 * q^60 - q^61 - 4 * q^62 + 3 * q^63 + 2 * q^64 - 8 * q^65 + 10 * q^67 - 9 * q^68 + 18 * q^69 - 9 * q^70 - 8 * q^71 - 2 * q^72 + 28 * q^73 - 8 * q^74 + 14 * q^75 - 2 * q^76 + 8 * q^78 + 6 * q^79 + q^80 - 22 * q^81 - 14 * q^82 + 3 * q^83 - 6 * q^84 - 2 * q^85 - 7 * q^86 + 24 * q^87 + 18 * q^89 - q^90 - 24 * q^91 - 9 * q^92 - 8 * q^93 - 5 * q^94 - q^95 + 4 * q^96 - 14 * q^97 - 13 * 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 −2.00000 1.00000 −0.618034 2.00000 −1.85410 −1.00000 1.00000 0.618034
1.2 −1.00000 −2.00000 1.00000 1.61803 2.00000 4.85410 −1.00000 1.00000 −1.61803
 $$n$$: 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.t 2
11.b odd 2 1 4598.2.a.bb 2
11.d odd 10 2 418.2.f.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.d 4 11.d odd 10 2
4598.2.a.t 2 1.a even 1 1 trivial
4598.2.a.bb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{5}^{2} - T_{5} - 1$$ T5^2 - T5 - 1 $$T_{7}^{2} - 3T_{7} - 9$$ T7^2 - 3*T7 - 9 $$T_{13}^{2} - 4T_{13} - 16$$ T13^2 - 4*T13 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 2)^{2}$$
$5$ $$T^{2} - T - 1$$
$7$ $$T^{2} - 3T - 9$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T - 16$$
$17$ $$T^{2} + 9T + 19$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 9T + 9$$
$29$ $$T^{2} + 12T + 16$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} - 14T + 44$$
$43$ $$T^{2} - 7T - 19$$
$47$ $$T^{2} - 5T + 5$$
$53$ $$T^{2} + 14T + 44$$
$59$ $$T^{2} + 16T + 44$$
$61$ $$T^{2} + T - 101$$
$67$ $$T^{2} - 10T + 20$$
$71$ $$T^{2} + 8T - 64$$
$73$ $$(T - 14)^{2}$$
$79$ $$T^{2} - 6T - 36$$
$83$ $$T^{2} - 3T - 149$$
$89$ $$T^{2} - 18T + 76$$
$97$ $$T^{2} + 14T + 44$$