# Properties

 Label 4851.2.a.bi Level $4851$ Weight $2$ Character orbit 4851.a Self dual yes Analytic conductor $38.735$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{2} + ( 2 + \beta_{1} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( -1 - 2 \beta_{1} ) q^{8} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{2} + ( 2 + \beta_{1} ) q^{4} + ( 1 - \beta_{1} - \beta_{2} ) q^{5} + ( -1 - 2 \beta_{1} ) q^{8} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{10} + q^{11} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{13} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{16} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 3 - \beta_{1} + \beta_{2} ) q^{19} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{20} + ( -1 - \beta_{2} ) q^{22} + ( -4 - 2 \beta_{2} ) q^{23} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{25} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{26} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{29} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{31} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{32} + ( -6 \beta_{1} - 4 \beta_{2} ) q^{34} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{37} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{38} + ( 7 + \beta_{1} + 3 \beta_{2} ) q^{40} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 2 + \beta_{1} ) q^{44} + ( 10 + 2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 1 + \beta_{1} + 3 \beta_{2} ) q^{47} + ( 4 + \beta_{1} - 7 \beta_{2} ) q^{50} + ( 10 + 6 \beta_{1} + \beta_{2} ) q^{52} + 2 \beta_{1} q^{53} + ( 1 - \beta_{1} - \beta_{2} ) q^{55} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{59} + 2 q^{61} + ( -12 + 2 \beta_{2} ) q^{62} + ( -1 - 4 \beta_{1} + 2 \beta_{2} ) q^{64} + ( -7 + \beta_{1} - 7 \beta_{2} ) q^{65} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{67} + ( 14 + 8 \beta_{1} ) q^{68} + ( 4 - 4 \beta_{1} ) q^{71} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{73} + ( -9 - \beta_{1} + 2 \beta_{2} ) q^{74} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{76} + ( 4 - 4 \beta_{1} ) q^{79} + ( -13 - \beta_{1} + 2 \beta_{2} ) q^{80} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{82} + ( 2 - 6 \beta_{1} ) q^{83} + ( -6 + 2 \beta_{1} - 12 \beta_{2} ) q^{85} + ( -6 - 10 \beta_{1} + 6 \beta_{2} ) q^{86} + ( -1 - 2 \beta_{1} ) q^{88} + ( 10 + 4 \beta_{2} ) q^{89} + ( -10 - 6 \beta_{1} - 4 \beta_{2} ) q^{92} + ( -11 - 5 \beta_{1} + 2 \beta_{2} ) q^{94} + ( 3 - 5 \beta_{1} + \beta_{2} ) q^{95} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 6 q^{4} + 4 q^{5} - 3 q^{8} + O(q^{10})$$ $$3 q - 2 q^{2} + 6 q^{4} + 4 q^{5} - 3 q^{8} + 11 q^{10} + 3 q^{11} + 4 q^{13} - 4 q^{16} + 8 q^{17} + 8 q^{19} - 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} - q^{26} + 4 q^{29} + 2 q^{31} - 8 q^{32} + 4 q^{34} - 13 q^{38} + 18 q^{40} + 14 q^{41} - 14 q^{43} + 6 q^{44} + 28 q^{46} + 19 q^{50} + 29 q^{52} + 4 q^{55} - q^{58} + 6 q^{61} - 38 q^{62} - 5 q^{64} - 14 q^{65} - 4 q^{67} + 42 q^{68} + 12 q^{71} + 20 q^{73} - 29 q^{74} + 11 q^{76} + 12 q^{79} - 41 q^{80} + 6 q^{82} + 6 q^{83} - 6 q^{85} - 24 q^{86} - 3 q^{88} + 26 q^{89} - 26 q^{92} - 35 q^{94} + 8 q^{95} + 4 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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 −1.86081 −0.254102
−2.47283 0 4.11491 −2.58774 0 0 −5.22982 0 6.39905
1.2 −1.46260 0 0.139194 2.39821 0 0 2.72161 0 −3.50761
1.3 1.93543 0 1.74590 4.18953 0 0 −0.491797 0 8.10856
 $$n$$: 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.bi 3
3.b odd 2 1 1617.2.a.t 3
7.b odd 2 1 693.2.a.l 3
21.c even 2 1 231.2.a.e 3
77.b even 2 1 7623.2.a.cd 3
84.h odd 2 1 3696.2.a.bo 3
105.g even 2 1 5775.2.a.bp 3
231.h odd 2 1 2541.2.a.bg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 21.c even 2 1
693.2.a.l 3 7.b odd 2 1
1617.2.a.t 3 3.b odd 2 1
2541.2.a.bg 3 231.h odd 2 1
3696.2.a.bo 3 84.h odd 2 1
4851.2.a.bi 3 1.a even 1 1 trivial
5775.2.a.bp 3 105.g even 2 1
7623.2.a.cd 3 77.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4851))$$:

 $$T_{2}^{3} + 2 T_{2}^{2} - 4 T_{2} - 7$$ $$T_{5}^{3} - 4 T_{5}^{2} - 7 T_{5} + 26$$ $$T_{13}^{3} - 4 T_{13}^{2} - 27 T_{13} + 94$$ $$T_{17}^{3} - 8 T_{17}^{2} - 40 T_{17} + 328$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-7 - 4 T + 2 T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$26 - 7 T - 4 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$94 - 27 T - 4 T^{2} + T^{3}$$
$17$ $$328 - 40 T - 8 T^{2} + T^{3}$$
$19$ $$-4 + 15 T - 8 T^{2} + T^{3}$$
$23$ $$-64 + 12 T + 10 T^{2} + T^{3}$$
$29$ $$94 - 27 T - 4 T^{2} + T^{3}$$
$31$ $$256 - 76 T - 2 T^{2} + T^{3}$$
$37$ $$106 - 43 T + T^{3}$$
$41$ $$32 + 40 T - 14 T^{2} + T^{3}$$
$43$ $$-848 - 44 T + 14 T^{2} + T^{3}$$
$47$ $$32 - 61 T + T^{3}$$
$53$ $$-8 - 16 T + T^{3}$$
$59$ $$-52 - 57 T + T^{3}$$
$61$ $$( -2 + T )^{3}$$
$67$ $$-236 - 85 T + 4 T^{2} + T^{3}$$
$71$ $$256 - 16 T - 12 T^{2} + T^{3}$$
$73$ $$-134 + 101 T - 20 T^{2} + T^{3}$$
$79$ $$256 - 16 T - 12 T^{2} + T^{3}$$
$83$ $$496 - 132 T - 6 T^{2} + T^{3}$$
$89$ $$328 + 140 T - 26 T^{2} + T^{3}$$
$97$ $$232 - 120 T - 4 T^{2} + T^{3}$$
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