# Properties

 Label 4056.2.a.bh Level $4056$ Weight $2$ Character orbit 4056.a Self dual yes Analytic conductor $32.387$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.27700337.1 Defining polynomial: $$x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127$$ x^6 - x^5 - 19*x^4 + 17*x^3 + 103*x^2 - 71*x - 127 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( - \beta_{3} + \beta_1) q^{5} + ( - \beta_{4} - \beta_{2} + 1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 + (-b3 + b1) * q^5 + (-b4 - b2 + 1) * q^7 + q^9 $$q + q^{3} + ( - \beta_{3} + \beta_1) q^{5} + ( - \beta_{4} - \beta_{2} + 1) q^{7} + q^{9} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{11} + ( - \beta_{3} + \beta_1) q^{15} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{17} + ( - 3 \beta_{3} - \beta_{2} + \beta_1) q^{19} + ( - \beta_{4} - \beta_{2} + 1) q^{21} + (\beta_{4} - 2 \beta_{2} - \beta_1 + 3) q^{23} + ( - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 3) q^{25} + q^{27} + ( - 2 \beta_{4} - \beta_1 + 1) q^{29} + (2 \beta_{2} + \beta_1 + 1) q^{31} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{33} + (\beta_{4} + 3 \beta_{3} + 6 \beta_{2} + \beta_1 - 2) q^{35} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 1) q^{37} + (2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{41} + (3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{43} + ( - \beta_{3} + \beta_1) q^{45} + (2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + \beta_1 - 2) q^{47} + ( - 2 \beta_{4} + 6 \beta_{3} - \beta_1) q^{49} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{51} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{53} + (\beta_{4} - \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 2) q^{55} + ( - 3 \beta_{3} - \beta_{2} + \beta_1) q^{57} + (2 \beta_{5} + 4 \beta_{3} - \beta_{2} - \beta_1) q^{59} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{61} + ( - \beta_{4} - \beta_{2} + 1) q^{63} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 2) q^{67} + (\beta_{4} - 2 \beta_{2} - \beta_1 + 3) q^{69} + ( - \beta_{5} - 6 \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{71} + ( - 2 \beta_{5} - \beta_{3} + 6 \beta_{2} + \beta_1 - 4) q^{73} + ( - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 3) q^{75} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 10) q^{77} + ( - 3 \beta_{5} - 7 \beta_{3} - 9 \beta_{2} + \beta_1 + 8) q^{79} + q^{81} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{83} + (\beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 7) q^{85} + ( - 2 \beta_{4} - \beta_1 + 1) q^{87} + (\beta_{5} + 9 \beta_{3} + 4 \beta_{2} - 10) q^{89} + (2 \beta_{2} + \beta_1 + 1) q^{93} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 9) q^{95} + (3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_1 - 8) q^{97} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{99}+O(q^{100})$$ q + q^3 + (-b3 + b1) * q^5 + (-b4 - b2 + 1) * q^7 + q^9 + (-b5 - b4 - b3 - 2*b2 + 2) * q^11 + (-b3 + b1) * q^15 + (-b3 - b2 + b1 + 2) * q^17 + (-3*b3 - b2 + b1) * q^19 + (-b4 - b2 + 1) * q^21 + (b4 - 2*b2 - b1 + 3) * q^23 + (-b5 - 2*b3 - 2*b2 + 3) * q^25 + q^27 + (-2*b4 - b1 + 1) * q^29 + (2*b2 + b1 + 1) * q^31 + (-b5 - b4 - b3 - 2*b2 + 2) * q^33 + (b4 + 3*b3 + 6*b2 + b1 - 2) * q^35 + (-b5 + b4 + 2*b3 - b2 - 1) * q^37 + (2*b5 - b4 + b3 + b2 - 2*b1 - 3) * q^41 + (3*b5 + b4 + b3 + 2*b2 + b1 + 1) * q^43 + (-b3 + b1) * q^45 + (2*b5 + 2*b4 + 5*b3 + 5*b2 + b1 - 2) * q^47 + (-2*b4 + 6*b3 - b1) * q^49 + (-b3 - b2 + b1 + 2) * q^51 + (-2*b3 + b2 + 4) * q^53 + (b4 - b3 + 8*b2 + 2*b1 - 2) * q^55 + (-3*b3 - b2 + b1) * q^57 + (2*b5 + 4*b3 - b2 - b1) * q^59 + (3*b3 - b2 - 3*b1 + 4) * q^61 + (-b4 - b2 + 1) * q^63 + (-b5 + 2*b4 + 3*b3 + 2*b2 - 2) * q^67 + (b4 - 2*b2 - b1 + 3) * q^69 + (-b5 - 6*b3 - 3*b2 - b1 + 2) * q^71 + (-2*b5 - b3 + 6*b2 + b1 - 4) * q^73 + (-b5 - 2*b3 - 2*b2 + 3) * q^75 + (-b5 - 2*b4 - 2*b3 - 5*b2 - 2*b1 + 10) * q^77 + (-3*b5 - 7*b3 - 9*b2 + b1 + 8) * q^79 + q^81 + (-b5 + b4 - 2*b3 + 2*b2 - 3*b1 + 3) * q^83 + (b4 - 3*b3 - b2 + 2*b1 + 7) * q^85 + (-2*b4 - b1 + 1) * q^87 + (b5 + 9*b3 + 4*b2 - 10) * q^89 + (2*b2 + b1 + 1) * q^93 + (-2*b5 + b4 - b3 - b2 + 9) * q^95 + (3*b5 + 2*b4 - 2*b3 + b1 - 8) * q^97 + (-b5 - b4 - b3 - 2*b2 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10})$$ 6 * q + 6 * q^3 - q^5 + 5 * q^7 + 6 * q^9 $$6 q + 6 q^{3} - q^{5} + 5 q^{7} + 6 q^{9} + 6 q^{11} - q^{15} + 9 q^{17} - 7 q^{19} + 5 q^{21} + 12 q^{23} + 9 q^{25} + 6 q^{27} + 7 q^{29} + 11 q^{31} + 6 q^{33} + 6 q^{35} - 6 q^{37} - 13 q^{41} + 15 q^{43} - q^{45} + 9 q^{47} + 13 q^{49} + 9 q^{51} + 22 q^{53} + 3 q^{55} - 7 q^{57} + 7 q^{59} + 25 q^{61} + 5 q^{63} - 5 q^{67} + 12 q^{69} - 8 q^{71} - 15 q^{73} + 9 q^{75} + 45 q^{77} + 14 q^{79} + 6 q^{81} + 13 q^{83} + 35 q^{85} + 7 q^{87} - 33 q^{89} + 11 q^{93} + 47 q^{95} - 50 q^{97} + 6 q^{99}+O(q^{100})$$ 6 * q + 6 * q^3 - q^5 + 5 * q^7 + 6 * q^9 + 6 * q^11 - q^15 + 9 * q^17 - 7 * q^19 + 5 * q^21 + 12 * q^23 + 9 * q^25 + 6 * q^27 + 7 * q^29 + 11 * q^31 + 6 * q^33 + 6 * q^35 - 6 * q^37 - 13 * q^41 + 15 * q^43 - q^45 + 9 * q^47 + 13 * q^49 + 9 * q^51 + 22 * q^53 + 3 * q^55 - 7 * q^57 + 7 * q^59 + 25 * q^61 + 5 * q^63 - 5 * q^67 + 12 * q^69 - 8 * q^71 - 15 * q^73 + 9 * q^75 + 45 * q^77 + 14 * q^79 + 6 * q^81 + 13 * q^83 + 35 * q^85 + 7 * q^87 - 33 * q^89 + 11 * q^93 + 47 * q^95 - 50 * q^97 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -4\nu^{5} + 11\nu^{4} + 15\nu^{3} - 136\nu^{2} + 160\nu + 338 ) / 167$$ (-4*v^5 + 11*v^4 + 15*v^3 - 136*v^2 + 160*v + 338) / 167 $$\beta_{3}$$ $$=$$ $$( -9\nu^{5} - 17\nu^{4} + 159\nu^{3} + 195\nu^{2} - 642\nu - 325 ) / 167$$ (-9*v^5 - 17*v^4 + 159*v^3 + 195*v^2 - 642*v - 325) / 167 $$\beta_{4}$$ $$=$$ $$( 10\nu^{5} + 56\nu^{4} - 121\nu^{3} - 662\nu^{2} + 268\nu + 1326 ) / 167$$ (10*v^5 + 56*v^4 - 121*v^3 - 662*v^2 + 268*v + 1326) / 167 $$\beta_{5}$$ $$=$$ $$( -13\nu^{5} - 6\nu^{4} + 174\nu^{3} + 226\nu^{2} - 482\nu - 1156 ) / 167$$ (-13*v^5 - 6*v^4 + 174*v^3 + 226*v^2 - 482*v - 1156) / 167
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{3} - \beta_{2} + 7$$ b5 - b3 - b2 + 7 $$\nu^{3}$$ $$=$$ $$\beta_{4} + 2\beta_{3} - 2\beta_{2} + 8\beta_1$$ b4 + 2*b3 - 2*b2 + 8*b1 $$\nu^{4}$$ $$=$$ $$12\beta_{5} + 3\beta_{4} - 10\beta_{3} - 9\beta_{2} + 58$$ 12*b5 + 3*b4 - 10*b3 - 9*b2 + 58 $$\nu^{5}$$ $$=$$ $$-\beta_{5} + 12\beta_{4} + 14\beta_{3} - 40\beta_{2} + 70\beta _1 + 6$$ -b5 + 12*b4 + 14*b3 - 40*b2 + 70*b1 + 6

## 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
 −2.71914 −0.920510 −3.12925 2.72245 3.16419 1.88227
0 1.00000 0 −3.16419 0 3.98252 0 1.00000 0
1.2 0 1.00000 0 −2.72245 0 −3.31535 0 1.00000 0
1.3 0 1.00000 0 −1.88227 0 −1.29156 0 1.00000 0
1.4 0 1.00000 0 0.920510 0 4.87031 0 1.00000 0
1.5 0 1.00000 0 2.71914 0 −0.735542 0 1.00000 0
1.6 0 1.00000 0 3.12925 0 1.48962 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.bh 6
4.b odd 2 1 8112.2.a.ct 6
13.b even 2 1 4056.2.a.bi yes 6
13.d odd 4 2 4056.2.c.r 12
52.b odd 2 1 8112.2.a.cu 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4056.2.a.bh 6 1.a even 1 1 trivial
4056.2.a.bi yes 6 13.b even 2 1
4056.2.c.r 12 13.d odd 4 2
8112.2.a.ct 6 4.b odd 2 1
8112.2.a.cu 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4056))$$:

 $$T_{5}^{6} + T_{5}^{5} - 19T_{5}^{4} - 17T_{5}^{3} + 103T_{5}^{2} + 71T_{5} - 127$$ T5^6 + T5^5 - 19*T5^4 - 17*T5^3 + 103*T5^2 + 71*T5 - 127 $$T_{7}^{6} - 5T_{7}^{5} - 15T_{7}^{4} + 69T_{7}^{3} + 63T_{7}^{2} - 119T_{7} - 91$$ T7^6 - 5*T7^5 - 15*T7^4 + 69*T7^3 + 63*T7^2 - 119*T7 - 91

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T - 1)^{6}$$
$5$ $$T^{6} + T^{5} - 19 T^{4} - 17 T^{3} + \cdots - 127$$
$7$ $$T^{6} - 5 T^{5} - 15 T^{4} + 69 T^{3} + \cdots - 91$$
$11$ $$T^{6} - 6 T^{5} - 23 T^{4} + 118 T^{3} + \cdots - 91$$
$13$ $$T^{6}$$
$17$ $$T^{6} - 9 T^{5} + 12 T^{4} + 49 T^{3} + \cdots + 8$$
$19$ $$T^{6} + 7 T^{5} - 20 T^{4} - 161 T^{3} + \cdots + 832$$
$23$ $$T^{6} - 12 T^{5} - 15 T^{4} + \cdots + 16856$$
$29$ $$T^{6} - 7 T^{5} - 87 T^{4} + \cdots + 27397$$
$31$ $$T^{6} - 11 T^{5} + 17 T^{4} + \cdots - 301$$
$37$ $$T^{6} + 6 T^{5} - 89 T^{4} - 683 T^{3} + \cdots + 664$$
$41$ $$T^{6} + 13 T^{5} - 116 T^{4} + \cdots - 78728$$
$43$ $$T^{6} - 15 T^{5} - 117 T^{4} + \cdots - 249992$$
$47$ $$T^{6} - 9 T^{5} - 188 T^{4} + \cdots + 287552$$
$53$ $$(T^{3} - 11 T^{2} + 24 T - 13)^{2}$$
$59$ $$T^{6} - 7 T^{5} - 150 T^{4} + \cdots + 1112$$
$61$ $$T^{6} - 25 T^{5} + 74 T^{4} + \cdots + 22504$$
$67$ $$T^{6} + 5 T^{5} - 176 T^{4} + \cdots + 7112$$
$71$ $$T^{6} + 8 T^{5} - 145 T^{4} + \cdots - 23192$$
$73$ $$T^{6} + 15 T^{5} - 219 T^{4} + \cdots + 109067$$
$79$ $$T^{6} - 14 T^{5} - 329 T^{4} + \cdots + 784147$$
$83$ $$T^{6} - 13 T^{5} - 277 T^{4} + \cdots + 598949$$
$89$ $$T^{6} + 33 T^{5} + 176 T^{4} + \cdots + 786344$$
$97$ $$T^{6} + 50 T^{5} + 766 T^{4} + \cdots + 711061$$