# Properties

 Label 4056.2.a.w Level $4056$ Weight $2$ Character orbit 4056.a Self dual yes Analytic conductor $32.387$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 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 $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + ( - \beta - 1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^5 + (-b - 1) * q^7 + q^9 $$q + q^{3} + q^{5} + ( - \beta - 1) q^{7} + q^{9} + (\beta + 1) q^{11} + q^{15} + ( - \beta - 2) q^{17} + ( - \beta + 3) q^{19} + ( - \beta - 1) q^{21} + ( - \beta - 1) q^{23} - 4 q^{25} + q^{27} + 3 q^{29} + (2 \beta + 2) q^{31} + (\beta + 1) q^{33} + ( - \beta - 1) q^{35} + (\beta + 6) q^{37} + ( - \beta + 8) q^{41} + (\beta + 1) q^{43} + q^{45} + (\beta + 5) q^{47} + (2 \beta + 7) q^{49} + ( - \beta - 2) q^{51} + 3 q^{53} + (\beta + 1) q^{55} + ( - \beta + 3) q^{57} + (2 \beta - 2) q^{59} + (\beta - 4) q^{61} + ( - \beta - 1) q^{63} + ( - \beta + 7) q^{67} + ( - \beta - 1) q^{69} + (\beta + 13) q^{71} - 7 q^{73} - 4 q^{75} + ( - 2 \beta - 14) q^{77} + 12 q^{79} + q^{81} + ( - 3 \beta + 1) q^{83} + ( - \beta - 2) q^{85} + 3 q^{87} + ( - 4 \beta + 2) q^{89} + (2 \beta + 2) q^{93} + ( - \beta + 3) q^{95} + (2 \beta - 4) q^{97} + (\beta + 1) q^{99}+O(q^{100})$$ q + q^3 + q^5 + (-b - 1) * q^7 + q^9 + (b + 1) * q^11 + q^15 + (-b - 2) * q^17 + (-b + 3) * q^19 + (-b - 1) * q^21 + (-b - 1) * q^23 - 4 * q^25 + q^27 + 3 * q^29 + (2*b + 2) * q^31 + (b + 1) * q^33 + (-b - 1) * q^35 + (b + 6) * q^37 + (-b + 8) * q^41 + (b + 1) * q^43 + q^45 + (b + 5) * q^47 + (2*b + 7) * q^49 + (-b - 2) * q^51 + 3 * q^53 + (b + 1) * q^55 + (-b + 3) * q^57 + (2*b - 2) * q^59 + (b - 4) * q^61 + (-b - 1) * q^63 + (-b + 7) * q^67 + (-b - 1) * q^69 + (b + 13) * q^71 - 7 * q^73 - 4 * q^75 + (-2*b - 14) * q^77 + 12 * q^79 + q^81 + (-3*b + 1) * q^83 + (-b - 2) * q^85 + 3 * q^87 + (-4*b + 2) * q^89 + (2*b + 2) * q^93 + (-b + 3) * q^95 + (2*b - 4) * q^97 + (b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 6 q^{19} - 2 q^{21} - 2 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} + 4 q^{31} + 2 q^{33} - 2 q^{35} + 12 q^{37} + 16 q^{41} + 2 q^{43} + 2 q^{45} + 10 q^{47} + 14 q^{49} - 4 q^{51} + 6 q^{53} + 2 q^{55} + 6 q^{57} - 4 q^{59} - 8 q^{61} - 2 q^{63} + 14 q^{67} - 2 q^{69} + 26 q^{71} - 14 q^{73} - 8 q^{75} - 28 q^{77} + 24 q^{79} + 2 q^{81} + 2 q^{83} - 4 q^{85} + 6 q^{87} + 4 q^{89} + 4 q^{93} + 6 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 + 6 * q^19 - 2 * q^21 - 2 * q^23 - 8 * q^25 + 2 * q^27 + 6 * q^29 + 4 * q^31 + 2 * q^33 - 2 * q^35 + 12 * q^37 + 16 * q^41 + 2 * q^43 + 2 * q^45 + 10 * q^47 + 14 * q^49 - 4 * q^51 + 6 * q^53 + 2 * q^55 + 6 * q^57 - 4 * q^59 - 8 * q^61 - 2 * q^63 + 14 * q^67 - 2 * q^69 + 26 * q^71 - 14 * q^73 - 8 * q^75 - 28 * q^77 + 24 * q^79 + 2 * q^81 + 2 * q^83 - 4 * q^85 + 6 * q^87 + 4 * q^89 + 4 * q^93 + 6 * q^95 - 8 * q^97 + 2 * q^99

## 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
 2.30278 −1.30278
0 1.00000 0 1.00000 0 −4.60555 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 2.60555 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.w 2
4.b odd 2 1 8112.2.a.bn 2
13.b even 2 1 4056.2.a.v 2
13.d odd 4 2 4056.2.c.l 4
13.e even 6 2 312.2.q.d 4
39.h odd 6 2 936.2.t.e 4
52.b odd 2 1 8112.2.a.bl 2
52.i odd 6 2 624.2.q.i 4
156.r even 6 2 1872.2.t.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.d 4 13.e even 6 2
624.2.q.i 4 52.i odd 6 2
936.2.t.e 4 39.h odd 6 2
1872.2.t.q 4 156.r even 6 2
4056.2.a.v 2 13.b even 2 1
4056.2.a.w 2 1.a even 1 1 trivial
4056.2.c.l 4 13.d odd 4 2
8112.2.a.bl 2 52.b odd 2 1
8112.2.a.bn 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4056))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{7}^{2} + 2T_{7} - 12$$ T7^2 + 2*T7 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 2T - 12$$
$11$ $$T^{2} - 2T - 12$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 4T - 9$$
$19$ $$T^{2} - 6T - 4$$
$23$ $$T^{2} + 2T - 12$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} - 4T - 48$$
$37$ $$T^{2} - 12T + 23$$
$41$ $$T^{2} - 16T + 51$$
$43$ $$T^{2} - 2T - 12$$
$47$ $$T^{2} - 10T + 12$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2} + 4T - 48$$
$61$ $$T^{2} + 8T + 3$$
$67$ $$T^{2} - 14T + 36$$
$71$ $$T^{2} - 26T + 156$$
$73$ $$(T + 7)^{2}$$
$79$ $$(T - 12)^{2}$$
$83$ $$T^{2} - 2T - 116$$
$89$ $$T^{2} - 4T - 204$$
$97$ $$T^{2} + 8T - 36$$