# Properties

 Label 4056.2.a.z Level $4056$ Weight $2$ Character orbit 4056.a Self dual yes Analytic conductor $32.387$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.3873230598$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.837.1 Defining polynomial: $$x^{3} - 6x - 1$$ x^3 - 6*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} - \beta_{2} q^{5} + ( - \beta_1 + 1) q^{7} + q^{9}+O(q^{10})$$ q - q^3 - b2 * q^5 + (-b1 + 1) * q^7 + q^9 $$q - q^{3} - \beta_{2} q^{5} + ( - \beta_1 + 1) q^{7} + q^{9} + (\beta_{2} - \beta_1) q^{11} + \beta_{2} q^{15} - \beta_1 q^{17} + (\beta_{2} + \beta_1 + 2) q^{19} + (\beta_1 - 1) q^{21} + (\beta_{2} - \beta_1) q^{23} + (\beta_{2} - 2 \beta_1 + 5) q^{25} - q^{27} + ( - \beta_{2} - 4) q^{29} + ( - \beta_{2} + 1) q^{31} + ( - \beta_{2} + \beta_1) q^{33} + ( - 3 \beta_{2} - \beta_1 + 4) q^{35} + (\beta_1 + 2) q^{37} + ( - 2 \beta_{2} + \beta_1) q^{41} + ( - \beta_1 - 7) q^{43} - \beta_{2} q^{45} + (\beta_{2} - \beta_1 - 4) q^{47} + ( - 3 \beta_{2} + 8) q^{49} + \beta_1 q^{51} + ( - 3 \beta_{2} - 2) q^{53} + ( - 3 \beta_{2} + \beta_1 - 6) q^{55} + ( - \beta_{2} - \beta_1 - 2) q^{57} + (2 \beta_1 + 2) q^{59} + (\beta_{2} + \beta_1 + 1) q^{61} + ( - \beta_1 + 1) q^{63} + ( - \beta_1 + 9) q^{67} + ( - \beta_{2} + \beta_1) q^{69} + (\beta_{2} - \beta_1 + 4) q^{71} + (2 \beta_{2} - 3) q^{73} + ( - \beta_{2} + 2 \beta_1 - 5) q^{75} + (2 \beta_1 + 10) q^{77} + (3 \beta_{2} - 2 \beta_1 + 3) q^{79} + q^{81} + (3 \beta_{2} - \beta_1 - 6) q^{83} + ( - 2 \beta_{2} - \beta_1 + 4) q^{85} + (\beta_{2} + 4) q^{87} + (2 \beta_{2} + 8) q^{89} + (\beta_{2} - 1) q^{93} + ( - \beta_{2} + 3 \beta_1 - 14) q^{95} + ( - \beta_{2} + 7) q^{97} + (\beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q - q^3 - b2 * q^5 + (-b1 + 1) * q^7 + q^9 + (b2 - b1) * q^11 + b2 * q^15 - b1 * q^17 + (b2 + b1 + 2) * q^19 + (b1 - 1) * q^21 + (b2 - b1) * q^23 + (b2 - 2*b1 + 5) * q^25 - q^27 + (-b2 - 4) * q^29 + (-b2 + 1) * q^31 + (-b2 + b1) * q^33 + (-3*b2 - b1 + 4) * q^35 + (b1 + 2) * q^37 + (-2*b2 + b1) * q^41 + (-b1 - 7) * q^43 - b2 * q^45 + (b2 - b1 - 4) * q^47 + (-3*b2 + 8) * q^49 + b1 * q^51 + (-3*b2 - 2) * q^53 + (-3*b2 + b1 - 6) * q^55 + (-b2 - b1 - 2) * q^57 + (2*b1 + 2) * q^59 + (b2 + b1 + 1) * q^61 + (-b1 + 1) * q^63 + (-b1 + 9) * q^67 + (-b2 + b1) * q^69 + (b2 - b1 + 4) * q^71 + (2*b2 - 3) * q^73 + (-b2 + 2*b1 - 5) * q^75 + (2*b1 + 10) * q^77 + (3*b2 - 2*b1 + 3) * q^79 + q^81 + (3*b2 - b1 - 6) * q^83 + (-2*b2 - b1 + 4) * q^85 + (b2 + 4) * q^87 + (2*b2 + 8) * q^89 + (b2 - 1) * q^93 + (-b2 + 3*b1 - 14) * q^95 + (-b2 + 7) * q^97 + (b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 $$3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 6 q^{19} - 3 q^{21} + 15 q^{25} - 3 q^{27} - 12 q^{29} + 3 q^{31} + 12 q^{35} + 6 q^{37} - 21 q^{43} - 12 q^{47} + 24 q^{49} - 6 q^{53} - 18 q^{55} - 6 q^{57} + 6 q^{59} + 3 q^{61} + 3 q^{63} + 27 q^{67} + 12 q^{71} - 9 q^{73} - 15 q^{75} + 30 q^{77} + 9 q^{79} + 3 q^{81} - 18 q^{83} + 12 q^{85} + 12 q^{87} + 24 q^{89} - 3 q^{93} - 42 q^{95} + 21 q^{97}+O(q^{100})$$ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 6 * q^19 - 3 * q^21 + 15 * q^25 - 3 * q^27 - 12 * q^29 + 3 * q^31 + 12 * q^35 + 6 * q^37 - 21 * q^43 - 12 * q^47 + 24 * q^49 - 6 * q^53 - 18 * q^55 - 6 * q^57 + 6 * q^59 + 3 * q^61 + 3 * q^63 + 27 * q^67 + 12 * q^71 - 9 * q^73 - 15 * q^75 + 30 * q^77 + 9 * q^79 + 3 * q^81 - 18 * q^83 + 12 * q^85 + 12 * q^87 + 24 * q^89 - 3 * q^93 - 42 * q^95 + 21 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 4$$ v^2 + v - 4 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 2$$ (-b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 8 ) / 2$$ (b2 + b1 + 8) / 2

## 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
 −2.36147 2.52892 −0.167449
0 −1.00000 0 −3.93800 0 1.78493 0 1.00000 0
1.2 0 −1.00000 0 0.133492 0 −3.92434 0 1.00000 0
1.3 0 −1.00000 0 3.80451 0 5.13941 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$13$$ $$1$$

## 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 4056.2.a.z 3
4.b odd 2 1 8112.2.a.ck 3
13.b even 2 1 4056.2.a.y 3
13.c even 3 2 312.2.q.e 6
13.d odd 4 2 4056.2.c.m 6
39.i odd 6 2 936.2.t.h 6
52.b odd 2 1 8112.2.a.cl 3
52.j odd 6 2 624.2.q.j 6
156.p even 6 2 1872.2.t.u 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.e 6 13.c even 3 2
624.2.q.j 6 52.j odd 6 2
936.2.t.h 6 39.i odd 6 2
1872.2.t.u 6 156.p even 6 2
4056.2.a.y 3 13.b even 2 1
4056.2.a.z 3 1.a even 1 1 trivial
4056.2.c.m 6 13.d odd 4 2
8112.2.a.ck 3 4.b odd 2 1
8112.2.a.cl 3 52.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4056))$$:

 $$T_{5}^{3} - 15T_{5} + 2$$ T5^3 - 15*T5 + 2 $$T_{7}^{3} - 3T_{7}^{2} - 18T_{7} + 36$$ T7^3 - 3*T7^2 - 18*T7 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3} - 15T + 2$$
$7$ $$T^{3} - 3 T^{2} - 18 T + 36$$
$11$ $$T^{3} - 24T + 8$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 21T + 16$$
$19$ $$T^{3} - 6 T^{2} - 36 T + 208$$
$23$ $$T^{3} - 24T + 8$$
$29$ $$T^{3} + 12 T^{2} + 33 T + 6$$
$31$ $$T^{3} - 3 T^{2} - 12 T + 16$$
$37$ $$T^{3} - 6 T^{2} - 9 T + 18$$
$41$ $$T^{3} - 57T + 156$$
$43$ $$T^{3} + 21 T^{2} + 126 T + 212$$
$47$ $$T^{3} + 12 T^{2} + 24 T - 24$$
$53$ $$T^{3} + 6 T^{2} - 123 T - 208$$
$59$ $$T^{3} - 6 T^{2} - 72 T + 32$$
$61$ $$T^{3} - 3 T^{2} - 45 T + 167$$
$67$ $$T^{3} - 27 T^{2} + 222 T - 524$$
$71$ $$T^{3} - 12 T^{2} + 24 T + 40$$
$73$ $$T^{3} + 9 T^{2} - 33 T - 169$$
$79$ $$T^{3} - 9 T^{2} - 120 T - 16$$
$83$ $$T^{3} + 18 T^{2} - 12 T - 992$$
$89$ $$T^{3} - 24 T^{2} + 132 T - 48$$
$97$ $$T^{3} - 21 T^{2} + 132 T - 236$$