# Properties

 Label 4056.2.a.ba Level $4056$ Weight $2$ Character orbit 4056.a Self dual yes Analytic conductor $32.387$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( - \beta_1 + 1) q^{5} + ( - 2 \beta_{2} + \beta_1 - 1) q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (-b1 + 1) * q^5 + (-2*b2 + b1 - 1) * q^7 + q^9 $$q - q^{3} + ( - \beta_1 + 1) q^{5} + ( - 2 \beta_{2} + \beta_1 - 1) q^{7} + q^{9} + \beta_{2} q^{11} + (\beta_1 - 1) q^{15} + ( - \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{2} + \beta_1 - 1) q^{19} + (2 \beta_{2} - \beta_1 + 1) q^{21} + (\beta_1 - 1) q^{23} + (\beta_{2} - 2 \beta_1 - 2) q^{25} - q^{27} + (3 \beta_{2} - 4 \beta_1 + 1) q^{29} + (5 \beta_{2} - 6 \beta_1 + 3) q^{31} - \beta_{2} q^{33} + ( - \beta_{2} + 2 \beta_1 - 1) q^{35} + (3 \beta_{2} + \beta_1 + 4) q^{37} + ( - \beta_{2} + 6 \beta_1 - 2) q^{41} + ( - 2 \beta_{2} + 4 \beta_1 - 5) q^{43} + ( - \beta_1 + 1) q^{45} + (3 \beta_{2} + \beta_1 - 5) q^{47} + ( - 3 \beta_{2} + 2 \beta_1 - 4) q^{49} + (\beta_{2} - \beta_1 + 1) q^{51} + ( - \beta_{2} - 4 \beta_1 + 6) q^{53} - q^{55} + ( - \beta_{2} - \beta_1 + 1) q^{57} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{59} + (7 \beta_{2} - \beta_1 + 1) q^{61} + ( - 2 \beta_{2} + \beta_1 - 1) q^{63} + (4 \beta_{2} - \beta_1 - 6) q^{67} + ( - \beta_1 + 1) q^{69} + ( - \beta_{2} + 4 \beta_1 + 1) q^{71} + (2 \beta_{2} - \beta_1 + 3) q^{73} + ( - \beta_{2} + 2 \beta_1 + 2) q^{75} + (2 \beta_{2} - 2 \beta_1 - 1) q^{77} + ( - 8 \beta_{2} + 5 \beta_1 - 10) q^{79} + q^{81} + (\beta_{2} - \beta_1 - 4) q^{83} + ( - \beta_{2} + 2 \beta_1 - 2) q^{85} + ( - 3 \beta_{2} + 4 \beta_1 - 1) q^{87} + ( - 4 \beta_{2} + 5 \beta_1 + 4) q^{89} + ( - 5 \beta_{2} + 6 \beta_1 - 3) q^{93} + ( - \beta_{2} + 2 \beta_1 - 4) q^{95} + ( - 2 \beta_1 + 13) q^{97} + \beta_{2} q^{99}+O(q^{100})$$ q - q^3 + (-b1 + 1) * q^5 + (-2*b2 + b1 - 1) * q^7 + q^9 + b2 * q^11 + (b1 - 1) * q^15 + (-b2 + b1 - 1) * q^17 + (b2 + b1 - 1) * q^19 + (2*b2 - b1 + 1) * q^21 + (b1 - 1) * q^23 + (b2 - 2*b1 - 2) * q^25 - q^27 + (3*b2 - 4*b1 + 1) * q^29 + (5*b2 - 6*b1 + 3) * q^31 - b2 * q^33 + (-b2 + 2*b1 - 1) * q^35 + (3*b2 + b1 + 4) * q^37 + (-b2 + 6*b1 - 2) * q^41 + (-2*b2 + 4*b1 - 5) * q^43 + (-b1 + 1) * q^45 + (3*b2 + b1 - 5) * q^47 + (-3*b2 + 2*b1 - 4) * q^49 + (b2 - b1 + 1) * q^51 + (-b2 - 4*b1 + 6) * q^53 - q^55 + (-b2 - b1 + 1) * q^57 + (-8*b2 + 2*b1 - 8) * q^59 + (7*b2 - b1 + 1) * q^61 + (-2*b2 + b1 - 1) * q^63 + (4*b2 - b1 - 6) * q^67 + (-b1 + 1) * q^69 + (-b2 + 4*b1 + 1) * q^71 + (2*b2 - b1 + 3) * q^73 + (-b2 + 2*b1 + 2) * q^75 + (2*b2 - 2*b1 - 1) * q^77 + (-8*b2 + 5*b1 - 10) * q^79 + q^81 + (b2 - b1 - 4) * q^83 + (-b2 + 2*b1 - 2) * q^85 + (-3*b2 + 4*b1 - 1) * q^87 + (-4*b2 + 5*b1 + 4) * q^89 + (-5*b2 + 6*b1 - 3) * q^93 + (-b2 + 2*b1 - 4) * q^95 + (-2*b1 + 13) * q^97 + b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 2 q^{5} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 2 * q^5 + 3 * q^9 $$3 q - 3 q^{3} + 2 q^{5} + 3 q^{9} - q^{11} - 2 q^{15} - q^{17} - 3 q^{19} - 2 q^{23} - 9 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} + q^{33} + 10 q^{37} + q^{41} - 9 q^{43} + 2 q^{45} - 17 q^{47} - 7 q^{49} + q^{51} + 15 q^{53} - 3 q^{55} + 3 q^{57} - 14 q^{59} - 5 q^{61} - 23 q^{67} + 2 q^{69} + 8 q^{71} + 6 q^{73} + 9 q^{75} - 7 q^{77} - 17 q^{79} + 3 q^{81} - 14 q^{83} - 3 q^{85} + 4 q^{87} + 21 q^{89} + 2 q^{93} - 9 q^{95} + 37 q^{97} - q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 2 * q^5 + 3 * q^9 - q^11 - 2 * q^15 - q^17 - 3 * q^19 - 2 * q^23 - 9 * q^25 - 3 * q^27 - 4 * q^29 - 2 * q^31 + q^33 + 10 * q^37 + q^41 - 9 * q^43 + 2 * q^45 - 17 * q^47 - 7 * q^49 + q^51 + 15 * q^53 - 3 * q^55 + 3 * q^57 - 14 * q^59 - 5 * q^61 - 23 * q^67 + 2 * q^69 + 8 * q^71 + 6 * q^73 + 9 * q^75 - 7 * q^77 - 17 * q^79 + 3 * q^81 - 14 * q^83 - 3 * q^85 + 4 * q^87 + 21 * q^89 + 2 * q^93 - 9 * q^95 + 37 * q^97 - q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 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
 1.80194 0.445042 −1.24698
0 −1.00000 0 −0.801938 0 −1.69202 0 1.00000 0
1.2 0 −1.00000 0 0.554958 0 3.04892 0 1.00000 0
1.3 0 −1.00000 0 2.24698 0 −1.35690 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.ba yes 3
4.b odd 2 1 8112.2.a.cn 3
13.b even 2 1 4056.2.a.x 3
13.d odd 4 2 4056.2.c.n 6
52.b odd 2 1 8112.2.a.ci 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4056.2.a.x 3 13.b even 2 1
4056.2.a.ba yes 3 1.a even 1 1 trivial
4056.2.c.n 6 13.d odd 4 2
8112.2.a.ci 3 52.b odd 2 1
8112.2.a.cn 3 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4056))$$:

 $$T_{5}^{3} - 2T_{5}^{2} - T_{5} + 1$$ T5^3 - 2*T5^2 - T5 + 1 $$T_{7}^{3} - 7T_{7} - 7$$ T7^3 - 7*T7 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3} - 2T^{2} - T + 1$$
$7$ $$T^{3} - 7T - 7$$
$11$ $$T^{3} + T^{2} - 2T - 1$$
$13$ $$T^{3}$$
$17$ $$T^{3} + T^{2} - 2T - 1$$
$19$ $$T^{3} + 3 T^{2} - 4 T - 13$$
$23$ $$T^{3} + 2T^{2} - T - 1$$
$29$ $$T^{3} + 4 T^{2} - 25 T - 71$$
$31$ $$T^{3} + 2 T^{2} - 71 T - 113$$
$37$ $$T^{3} - 10 T^{2} + 3 T + 13$$
$41$ $$T^{3} - T^{2} - 72 T + 169$$
$43$ $$T^{3} + 9T^{2} - T - 1$$
$47$ $$T^{3} + 17 T^{2} + 66 T - 41$$
$53$ $$T^{3} - 15 T^{2} + 26 T + 169$$
$59$ $$T^{3} + 14 T^{2} - 56 T - 728$$
$61$ $$T^{3} + 5 T^{2} - 92 T - 83$$
$67$ $$T^{3} + 23 T^{2} + 146 T + 251$$
$71$ $$T^{3} - 8 T^{2} - 9 T + 113$$
$73$ $$T^{3} - 6 T^{2} + 5 T + 13$$
$79$ $$T^{3} + 17 T^{2} - 18 T - 923$$
$83$ $$T^{3} + 14 T^{2} + 63 T + 91$$
$89$ $$T^{3} - 21 T^{2} + 98 T + 49$$
$97$ $$T^{3} - 37 T^{2} + 447 T - 1763$$