# Properties

 Label 4851.2.a.bl Level $4851$ Weight $2$ Character orbit 4851.a Self dual yes Analytic conductor $38.735$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4851 = 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4851.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.7354300205$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1617) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} - q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + 1) * q^4 + (b2 - b1 - 1) * q^5 - q^8 $$q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} - q^{8} + (\beta_{2} + 2) q^{10} - q^{11} + (\beta_{2} + \beta_1 + 1) q^{13} + ( - 2 \beta_{2} + \beta_1 - 2) q^{16} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} + ( - 2 \beta_{2} - \beta_1 + 1) q^{20} + \beta_1 q^{22} + ( - 3 \beta_{2} + \beta_1) q^{25} + ( - \beta_{2} - 2 \beta_1 - 4) q^{26} + (3 \beta_{2} + \beta_1 - 1) q^{29} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{31} + ( - \beta_{2} + 4 \beta_1 + 1) q^{32} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{34} + ( - \beta_{2} - \beta_1 - 3) q^{37} + ( - \beta_{2} - 2) q^{38} + ( - \beta_{2} + \beta_1 + 1) q^{40} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_{2} + 2 \beta_1 - 2) q^{43} + ( - \beta_{2} - 1) q^{44} + (\beta_{2} - 3 \beta_1 - 3) q^{47} + ( - \beta_{2} + 3 \beta_1) q^{50} + (3 \beta_1 + 5) q^{52} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{53} + ( - \beta_{2} + \beta_1 + 1) q^{55} + ( - \beta_{2} - 2 \beta_1 - 6) q^{58} + ( - 3 \beta_{2} + 5 \beta_1 - 3) q^{59} + (2 \beta_{2} + 2 \beta_1 + 6) q^{61} + (2 \beta_{2} + 8) q^{62} + ( - 2 \beta_1 - 7) q^{64} + ( - 3 \beta_{2} - \beta_1 - 1) q^{65} + ( - 5 \beta_{2} + 5 \beta_1 + 1) q^{67} + (2 \beta_1 - 6) q^{68} + ( - 6 \beta_{2} + 6 \beta_1 - 2) q^{71} + ( - 5 \beta_{2} - \beta_1 - 1) q^{73} + (\beta_{2} + 4 \beta_1 + 4) q^{74} + (2 \beta_{2} + \beta_1 - 1) q^{76} + ( - 4 \beta_1 - 4) q^{79} + (3 \beta_{2} + 2 \beta_1 - 4) q^{80} + (2 \beta_{2} + 8) q^{82} + (2 \beta_{2} + 2 \beta_1 + 2) q^{83} + (2 \beta_{2} + 2 \beta_1 - 6) q^{85} + ( - 2 \beta_{2} - 8) q^{86} + q^{88} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{89} + (3 \beta_{2} + 2 \beta_1 + 8) q^{94} + (3 \beta_{2} - \beta_1 - 5) q^{95} + ( - 2 \beta_{2} - 6 \beta_1 + 2) q^{97}+O(q^{100})$$ q - b1 * q^2 + (b2 + 1) * q^4 + (b2 - b1 - 1) * q^5 - q^8 + (b2 + 2) * q^10 - q^11 + (b2 + b1 + 1) * q^13 + (-2*b2 + b1 - 2) * q^16 + (-2*b2 + 2*b1 - 2) * q^17 + (-b2 + b1 + 1) * q^19 + (-2*b2 - b1 + 1) * q^20 + b1 * q^22 + (-3*b2 + b1) * q^25 + (-b2 - 2*b1 - 4) * q^26 + (3*b2 + b1 - 1) * q^29 + (-2*b2 - 2*b1 + 2) * q^31 + (-b2 + 4*b1 + 1) * q^32 + (-2*b2 + 4*b1 - 4) * q^34 + (-b2 - b1 - 3) * q^37 + (-b2 - 2) * q^38 + (-b2 + b1 + 1) * q^40 + (-2*b2 - 2*b1 + 2) * q^41 + (2*b2 + 2*b1 - 2) * q^43 + (-b2 - 1) * q^44 + (b2 - 3*b1 - 3) * q^47 + (-b2 + 3*b1) * q^50 + (3*b1 + 5) * q^52 + (-2*b2 + 2*b1 - 4) * q^53 + (-b2 + b1 + 1) * q^55 + (-b2 - 2*b1 - 6) * q^58 + (-3*b2 + 5*b1 - 3) * q^59 + (2*b2 + 2*b1 + 6) * q^61 + (2*b2 + 8) * q^62 + (-2*b1 - 7) * q^64 + (-3*b2 - b1 - 1) * q^65 + (-5*b2 + 5*b1 + 1) * q^67 + (2*b1 - 6) * q^68 + (-6*b2 + 6*b1 - 2) * q^71 + (-5*b2 - b1 - 1) * q^73 + (b2 + 4*b1 + 4) * q^74 + (2*b2 + b1 - 1) * q^76 + (-4*b1 - 4) * q^79 + (3*b2 + 2*b1 - 4) * q^80 + (2*b2 + 8) * q^82 + (2*b2 + 2*b1 + 2) * q^83 + (2*b2 + 2*b1 - 6) * q^85 + (-2*b2 - 8) * q^86 + q^88 + (-2*b2 + 2*b1 - 6) * q^89 + (3*b2 + 2*b1 + 8) * q^94 + (3*b2 - b1 - 5) * q^95 + (-2*b2 - 6*b1 + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{4} - 4 q^{5} - 3 q^{8}+O(q^{10})$$ 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8 $$3 q + 2 q^{4} - 4 q^{5} - 3 q^{8} + 5 q^{10} - 3 q^{11} + 2 q^{13} - 4 q^{16} - 4 q^{17} + 4 q^{19} + 5 q^{20} + 3 q^{25} - 11 q^{26} - 6 q^{29} + 8 q^{31} + 4 q^{32} - 10 q^{34} - 8 q^{37} - 5 q^{38} + 4 q^{40} + 8 q^{41} - 8 q^{43} - 2 q^{44} - 10 q^{47} + q^{50} + 15 q^{52} - 10 q^{53} + 4 q^{55} - 17 q^{58} - 6 q^{59} + 16 q^{61} + 22 q^{62} - 21 q^{64} + 8 q^{67} - 18 q^{68} + 2 q^{73} + 11 q^{74} - 5 q^{76} - 12 q^{79} - 15 q^{80} + 22 q^{82} + 4 q^{83} - 20 q^{85} - 22 q^{86} + 3 q^{88} - 16 q^{89} + 21 q^{94} - 18 q^{95} + 8 q^{97}+O(q^{100})$$ 3 * q + 2 * q^4 - 4 * q^5 - 3 * q^8 + 5 * q^10 - 3 * q^11 + 2 * q^13 - 4 * q^16 - 4 * q^17 + 4 * q^19 + 5 * q^20 + 3 * q^25 - 11 * q^26 - 6 * q^29 + 8 * q^31 + 4 * q^32 - 10 * q^34 - 8 * q^37 - 5 * q^38 + 4 * q^40 + 8 * q^41 - 8 * q^43 - 2 * q^44 - 10 * q^47 + q^50 + 15 * q^52 - 10 * q^53 + 4 * q^55 - 17 * q^58 - 6 * q^59 + 16 * q^61 + 22 * q^62 - 21 * q^64 + 8 * q^67 - 18 * q^68 + 2 * q^73 + 11 * q^74 - 5 * q^76 - 12 * q^79 - 15 * q^80 + 22 * q^82 + 4 * q^83 - 20 * q^85 - 22 * q^86 + 3 * q^88 - 16 * q^89 + 21 * q^94 - 18 * q^95 + 8 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.11491 −0.254102 −1.86081
−2.11491 0 2.47283 −1.64207 0 0 −1.00000 0 3.47283
1.2 0.254102 0 −1.93543 −3.68133 0 0 −1.00000 0 −0.935432
1.3 1.86081 0 1.46260 1.32340 0 0 −1.00000 0 2.46260
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 4851.2.a.bl 3
3.b odd 2 1 1617.2.a.r yes 3
7.b odd 2 1 4851.2.a.bm 3
21.c even 2 1 1617.2.a.q 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.2.a.q 3 21.c even 2 1
1617.2.a.r yes 3 3.b odd 2 1
4851.2.a.bl 3 1.a even 1 1 trivial
4851.2.a.bm 3 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4851))$$:

 $$T_{2}^{3} - 4T_{2} + 1$$ T2^3 - 4*T2 + 1 $$T_{5}^{3} + 4T_{5}^{2} - T_{5} - 8$$ T5^3 + 4*T5^2 - T5 - 8 $$T_{13}^{3} - 2T_{13}^{2} - 11T_{13} - 4$$ T13^3 - 2*T13^2 - 11*T13 - 4 $$T_{17}^{3} + 4T_{17}^{2} - 20T_{17} - 16$$ T17^3 + 4*T17^2 - 20*T17 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 4T + 1$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 4T^{2} - T - 8$$
$7$ $$T^{3}$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3} - 2 T^{2} - 11 T - 4$$
$17$ $$T^{3} + 4 T^{2} - 20 T - 16$$
$19$ $$T^{3} - 4T^{2} - T + 8$$
$23$ $$T^{3}$$
$29$ $$T^{3} + 6 T^{2} - 49 T - 82$$
$31$ $$T^{3} - 8 T^{2} - 28 T + 208$$
$37$ $$T^{3} + 8 T^{2} + 9 T - 2$$
$41$ $$T^{3} - 8 T^{2} - 28 T + 208$$
$43$ $$T^{3} + 8 T^{2} - 28 T - 208$$
$47$ $$T^{3} + 10 T^{2} + T - 124$$
$53$ $$T^{3} + 10 T^{2} + 8 T - 32$$
$59$ $$T^{3} + 6 T^{2} - 91 T + 196$$
$61$ $$T^{3} - 16 T^{2} + 36 T + 16$$
$67$ $$T^{3} - 8 T^{2} - 137 T + 644$$
$71$ $$T^{3} - 228T + 416$$
$73$ $$T^{3} - 2 T^{2} - 151 T - 212$$
$79$ $$T^{3} + 12 T^{2} - 16 T - 128$$
$83$ $$T^{3} - 4 T^{2} - 44 T - 32$$
$89$ $$T^{3} + 16 T^{2} + 60 T + 32$$
$97$ $$T^{3} - 8 T^{2} - 180 T + 1568$$