# Properties

 Label 4598.2.a.bv Level $4598$ Weight $2$ Character orbit 4598.a Self dual yes Analytic conductor $36.715$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.258228.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 12x^{2} + 6x + 24$$ x^4 - 2*x^3 - 12*x^2 + 6*x + 24 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} - \beta_1 q^{5} + ( - \beta_1 + 1) q^{6} + (\beta_{3} + 1) q^{7} + q^{8} + ( - \beta_{3} + \beta_{2} + 4) q^{9}+O(q^{10})$$ q + q^2 + (-b1 + 1) * q^3 + q^4 - b1 * q^5 + (-b1 + 1) * q^6 + (b3 + 1) * q^7 + q^8 + (-b3 + b2 + 4) * q^9 $$q + q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} - \beta_1 q^{5} + ( - \beta_1 + 1) q^{6} + (\beta_{3} + 1) q^{7} + q^{8} + ( - \beta_{3} + \beta_{2} + 4) q^{9} - \beta_1 q^{10} + ( - \beta_1 + 1) q^{12} - \beta_{2} q^{13} + (\beta_{3} + 1) q^{14} + ( - \beta_{3} + \beta_{2} + \beta_1 + 6) q^{15} + q^{16} + ( - \beta_{3} - 2) q^{17} + ( - \beta_{3} + \beta_{2} + 4) q^{18} + q^{19} - \beta_1 q^{20} + (3 \beta_{3} - \beta_1 - 1) q^{21} + (\beta_{2} - 1) q^{23} + ( - \beta_1 + 1) q^{24} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{25} - \beta_{2} q^{26} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{27} + (\beta_{3} + 1) q^{28} + (\beta_{3} - \beta_{2} - \beta_1 - 3) q^{29} + ( - \beta_{3} + \beta_{2} + \beta_1 + 6) q^{30} + ( - 2 \beta_1 + 2) q^{31} + q^{32} + ( - \beta_{3} - 2) q^{34} + (2 \beta_{3} - \beta_1 - 2) q^{35} + ( - \beta_{3} + \beta_{2} + 4) q^{36} + (2 \beta_{3} + \beta_1 + 3) q^{37} + q^{38} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{39} - \beta_1 q^{40} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{41} + (3 \beta_{3} - \beta_1 - 1) q^{42} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{43} + ( - 2 \beta_{2} - 6 \beta_1 + 2) q^{45} + (\beta_{2} - 1) q^{46} + (\beta_{3} - \beta_{2} - 3) q^{47} + ( - \beta_1 + 1) q^{48} + (\beta_{2} - 4 \beta_1 + 8) q^{49} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{50} + ( - 3 \beta_{3} + 2 \beta_1) q^{51} - \beta_{2} q^{52} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{53} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{54} + (\beta_{3} + 1) q^{56} + ( - \beta_1 + 1) q^{57} + (\beta_{3} - \beta_{2} - \beta_1 - 3) q^{58} + ( - \beta_{2} - \beta_1 + 1) q^{59} + ( - \beta_{3} + \beta_{2} + \beta_1 + 6) q^{60} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{61} + ( - 2 \beta_1 + 2) q^{62} + (5 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{63} + q^{64} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{65} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{67} + ( - \beta_{3} - 2) q^{68} + (2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{69} + (2 \beta_{3} - \beta_1 - 2) q^{70} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{71} + ( - \beta_{3} + \beta_{2} + 4) q^{72} + (\beta_{3} - 2) q^{73} + (2 \beta_{3} + \beta_1 + 3) q^{74} + (\beta_{3} - 3 \beta_{2} - 5 \beta_1 - 9) q^{75} + q^{76} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{78} + (\beta_{3} + \beta_{2} - \beta_1 + 6) q^{79} - \beta_1 q^{80} + ( - 5 \beta_{3} + \beta_{2} + 2 \beta_1 + 11) q^{81} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{82} + (\beta_1 - 6) q^{83} + (3 \beta_{3} - \beta_1 - 1) q^{84} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{85} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{86} + (2 \beta_{2} + 6 \beta_1 + 1) q^{87} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{89} + ( - 2 \beta_{2} - 6 \beta_1 + 2) q^{90} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{91} + (\beta_{2} - 1) q^{92} + ( - 2 \beta_{3} + 2 \beta_{2} + 14) q^{93} + (\beta_{3} - \beta_{2} - 3) q^{94} - \beta_1 q^{95} + ( - \beta_1 + 1) q^{96} - 10 q^{97} + (\beta_{2} - 4 \beta_1 + 8) q^{98}+O(q^{100})$$ q + q^2 + (-b1 + 1) * q^3 + q^4 - b1 * q^5 + (-b1 + 1) * q^6 + (b3 + 1) * q^7 + q^8 + (-b3 + b2 + 4) * q^9 - b1 * q^10 + (-b1 + 1) * q^12 - b2 * q^13 + (b3 + 1) * q^14 + (-b3 + b2 + b1 + 6) * q^15 + q^16 + (-b3 - 2) * q^17 + (-b3 + b2 + 4) * q^18 + q^19 - b1 * q^20 + (3*b3 - b1 - 1) * q^21 + (b2 - 1) * q^23 + (-b1 + 1) * q^24 + (-b3 + b2 + 2*b1 + 1) * q^25 - b2 * q^26 + (-b3 - b2 - 3*b1 + 3) * q^27 + (b3 + 1) * q^28 + (b3 - b2 - b1 - 3) * q^29 + (-b3 + b2 + b1 + 6) * q^30 + (-2*b1 + 2) * q^31 + q^32 + (-b3 - 2) * q^34 + (2*b3 - b1 - 2) * q^35 + (-b3 + b2 + 4) * q^36 + (2*b3 + b1 + 3) * q^37 + q^38 + (-2*b3 + b2 + 2*b1) * q^39 - b1 * q^40 + (-b3 - b2 + b1 + 2) * q^41 + (3*b3 - b1 - 1) * q^42 + (-2*b2 - 2*b1 + 4) * q^43 + (-2*b2 - 6*b1 + 2) * q^45 + (b2 - 1) * q^46 + (b3 - b2 - 3) * q^47 + (-b1 + 1) * q^48 + (b2 - 4*b1 + 8) * q^49 + (-b3 + b2 + 2*b1 + 1) * q^50 + (-3*b3 + 2*b1) * q^51 - b2 * q^52 + (-b3 + b2 + b1 - 3) * q^53 + (-b3 - b2 - 3*b1 + 3) * q^54 + (b3 + 1) * q^56 + (-b1 + 1) * q^57 + (b3 - b2 - b1 - 3) * q^58 + (-b2 - b1 + 1) * q^59 + (-b3 + b2 + b1 + 6) * q^60 + (2*b3 - 2*b2 - b1 + 2) * q^61 + (-2*b1 + 2) * q^62 + (5*b3 + b2 + 2*b1 - 4) * q^63 + q^64 + (-2*b3 + 2*b2 + 2*b1) * q^65 + (-2*b3 - b2 + 2*b1 + 2) * q^67 + (-b3 - 2) * q^68 + (2*b3 - b2 - b1 - 1) * q^69 + (2*b3 - b1 - 2) * q^70 + (b3 + b2 - 3*b1 - 2) * q^71 + (-b3 + b2 + 4) * q^72 + (b3 - 2) * q^73 + (2*b3 + b1 + 3) * q^74 + (b3 - 3*b2 - 5*b1 - 9) * q^75 + q^76 + (-2*b3 + b2 + 2*b1) * q^78 + (b3 + b2 - b1 + 6) * q^79 - b1 * q^80 + (-5*b3 + b2 + 2*b1 + 11) * q^81 + (-b3 - b2 + b1 + 2) * q^82 + (b1 - 6) * q^83 + (3*b3 - b1 - 1) * q^84 + (-2*b3 + 2*b1 + 2) * q^85 + (-2*b2 - 2*b1 + 4) * q^86 + (2*b2 + 6*b1 + 1) * q^87 + (b3 + b2 + b1 - 2) * q^89 + (-2*b2 - 6*b1 + 2) * q^90 + (-2*b2 + 2*b1 - 6) * q^91 + (b2 - 1) * q^92 + (-2*b3 + 2*b2 + 14) * q^93 + (b3 - b2 - 3) * q^94 - b1 * q^95 + (-b1 + 1) * q^96 - 10 * q^97 + (b2 - 4*b1 + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} + 4 q^{8} + 16 q^{9}+O(q^{10})$$ 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 + 4 * q^8 + 16 * q^9 $$4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} + 3 q^{7} + 4 q^{8} + 16 q^{9} - 2 q^{10} + 2 q^{12} + q^{13} + 3 q^{14} + 26 q^{15} + 4 q^{16} - 7 q^{17} + 16 q^{18} + 4 q^{19} - 2 q^{20} - 9 q^{21} - 5 q^{23} + 2 q^{24} + 8 q^{25} + q^{26} + 8 q^{27} + 3 q^{28} - 14 q^{29} + 26 q^{30} + 4 q^{31} + 4 q^{32} - 7 q^{34} - 12 q^{35} + 16 q^{36} + 12 q^{37} + 4 q^{38} + 5 q^{39} - 2 q^{40} + 12 q^{41} - 9 q^{42} + 14 q^{43} - 2 q^{45} - 5 q^{46} - 12 q^{47} + 2 q^{48} + 23 q^{49} + 8 q^{50} + 7 q^{51} + q^{52} - 10 q^{53} + 8 q^{54} + 3 q^{56} + 2 q^{57} - 14 q^{58} + 3 q^{59} + 26 q^{60} + 6 q^{61} + 4 q^{62} - 18 q^{63} + 4 q^{64} + 4 q^{65} + 15 q^{67} - 7 q^{68} - 7 q^{69} - 12 q^{70} - 16 q^{71} + 16 q^{72} - 9 q^{73} + 12 q^{74} - 44 q^{75} + 4 q^{76} + 5 q^{78} + 20 q^{79} - 2 q^{80} + 52 q^{81} + 12 q^{82} - 22 q^{83} - 9 q^{84} + 14 q^{85} + 14 q^{86} + 14 q^{87} - 8 q^{89} - 2 q^{90} - 18 q^{91} - 5 q^{92} + 56 q^{93} - 12 q^{94} - 2 q^{95} + 2 q^{96} - 40 q^{97} + 23 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 + 2 * q^6 + 3 * q^7 + 4 * q^8 + 16 * q^9 - 2 * q^10 + 2 * q^12 + q^13 + 3 * q^14 + 26 * q^15 + 4 * q^16 - 7 * q^17 + 16 * q^18 + 4 * q^19 - 2 * q^20 - 9 * q^21 - 5 * q^23 + 2 * q^24 + 8 * q^25 + q^26 + 8 * q^27 + 3 * q^28 - 14 * q^29 + 26 * q^30 + 4 * q^31 + 4 * q^32 - 7 * q^34 - 12 * q^35 + 16 * q^36 + 12 * q^37 + 4 * q^38 + 5 * q^39 - 2 * q^40 + 12 * q^41 - 9 * q^42 + 14 * q^43 - 2 * q^45 - 5 * q^46 - 12 * q^47 + 2 * q^48 + 23 * q^49 + 8 * q^50 + 7 * q^51 + q^52 - 10 * q^53 + 8 * q^54 + 3 * q^56 + 2 * q^57 - 14 * q^58 + 3 * q^59 + 26 * q^60 + 6 * q^61 + 4 * q^62 - 18 * q^63 + 4 * q^64 + 4 * q^65 + 15 * q^67 - 7 * q^68 - 7 * q^69 - 12 * q^70 - 16 * q^71 + 16 * q^72 - 9 * q^73 + 12 * q^74 - 44 * q^75 + 4 * q^76 + 5 * q^78 + 20 * q^79 - 2 * q^80 + 52 * q^81 + 12 * q^82 - 22 * q^83 - 9 * q^84 + 14 * q^85 + 14 * q^86 + 14 * q^87 - 8 * q^89 - 2 * q^90 - 18 * q^91 - 5 * q^92 + 56 * q^93 - 12 * q^94 - 2 * q^95 + 2 * q^96 - 40 * q^97 + 23 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 12x^{2} + 6x + 24$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu^{2} - 8\nu + 2 ) / 2$$ (v^3 - 2*v^2 - 8*v + 2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 4\nu^{2} - 4\nu + 14 ) / 2$$ (v^3 - 4*v^2 - 4*v + 14) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 2\beta _1 + 6$$ -b3 + b2 + 2*b1 + 6 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + 4\beta_{2} + 12\beta _1 + 10$$ -2*b3 + 4*b2 + 12*b1 + 10

## 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
 4.19461 1.63927 −1.48761 −2.34628
1.00000 −3.19461 1.00000 −4.19461 −3.19461 1.32286 1.00000 7.20555 −4.19461
1.2 1.00000 −0.639273 1.00000 −1.63927 −0.639273 1.54956 1.00000 −2.59133 −1.63927
1.3 1.00000 2.48761 1.00000 1.48761 2.48761 4.90325 1.00000 3.18819 1.48761
1.4 1.00000 3.34628 1.00000 2.34628 3.34628 −4.77567 1.00000 8.19759 2.34628
 $$n$$: 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.bv yes 4
11.b odd 2 1 4598.2.a.bs 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.bs 4 11.b odd 2 1
4598.2.a.bv yes 4 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{4} - 2T_{3}^{3} - 12T_{3}^{2} + 20T_{3} + 17$$ T3^4 - 2*T3^3 - 12*T3^2 + 20*T3 + 17 $$T_{5}^{4} + 2T_{5}^{3} - 12T_{5}^{2} - 6T_{5} + 24$$ T5^4 + 2*T5^3 - 12*T5^2 - 6*T5 + 24 $$T_{7}^{4} - 3T_{7}^{3} - 21T_{7}^{2} + 67T_{7} - 48$$ T7^4 - 3*T7^3 - 21*T7^2 + 67*T7 - 48 $$T_{13}^{4} - T_{13}^{3} - 30T_{13}^{2} - 20T_{13} + 104$$ T13^4 - T13^3 - 30*T13^2 - 20*T13 + 104

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{4}$$
$3$ $$T^{4} - 2 T^{3} - 12 T^{2} + 20 T + 17$$
$5$ $$T^{4} + 2 T^{3} - 12 T^{2} - 6 T + 24$$
$7$ $$T^{4} - 3 T^{3} - 21 T^{2} + 67 T - 48$$
$11$ $$T^{4}$$
$13$ $$T^{4} - T^{3} - 30 T^{2} - 20 T + 104$$
$17$ $$T^{4} + 7 T^{3} - 6 T^{2} - 96 T - 132$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} + 5 T^{3} - 21 T^{2} - 33 T + 96$$
$29$ $$T^{4} + 14 T^{3} + 30 T^{2} - 84 T - 69$$
$31$ $$T^{4} - 4 T^{3} - 48 T^{2} + 160 T + 272$$
$37$ $$T^{4} - 12 T^{3} - 78 T^{2} + \cdots - 4569$$
$41$ $$T^{4} - 12 T^{3} - 18 T^{2} + \cdots - 972$$
$43$ $$T^{4} - 14 T^{3} - 120 T^{2} + \cdots - 1376$$
$47$ $$T^{4} + 12 T^{3} + 18 T^{2} + \cdots - 351$$
$53$ $$T^{4} + 10 T^{3} - 6 T^{2} - 204 T - 213$$
$59$ $$T^{4} - 3 T^{3} - 45 T^{2} + 153 T + 108$$
$61$ $$T^{4} - 6 T^{3} - 132 T^{2} + \cdots + 2412$$
$67$ $$T^{4} - 15 T^{3} - 84 T^{2} + \cdots - 7872$$
$71$ $$T^{4} + 16 T^{3} - 54 T^{2} + \cdots - 2916$$
$73$ $$T^{4} + 9 T^{3} + 6 T^{2} - 32 T - 36$$
$79$ $$T^{4} - 20 T^{3} + 78 T^{2} + 34 T - 92$$
$83$ $$T^{4} + 22 T^{3} + 168 T^{2} + \cdots + 492$$
$89$ $$T^{4} + 8 T^{3} - 78 T^{2} + \cdots + 1452$$
$97$ $$(T + 10)^{4}$$