# Properties

 Label 4140.2.f.a Level $4140$ Weight $2$ Character orbit 4140.f Analytic conductor $33.058$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4140.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.0580664368$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + \beta_1) q^{5} - \beta_{3} q^{7}+O(q^{10})$$ q + (-b4 + b1) * q^5 - b3 * q^7 $$q + ( - \beta_{4} + \beta_1) q^{5} - \beta_{3} q^{7} + \beta_1 q^{11} + (2 \beta_{5} - \beta_{4} + \beta_{3}) q^{13} + ( - 2 \beta_{5} + \beta_{4}) q^{17} + (\beta_{2} - 2 \beta_1 + 1) q^{19} + \beta_{3} q^{23} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{25} + (3 \beta_{2} - 2) q^{29} + (\beta_{2} - \beta_1 - 4) q^{31} + (\beta_{5} + \beta_{2}) q^{35} + (\beta_{5} - 3 \beta_{4}) q^{37} + (4 \beta_{2} + \beta_1 + 3) q^{41} + ( - \beta_{5} - 3 \beta_{4} + \beta_{3}) q^{43} + ( - \beta_{5} + 2 \beta_{4} + 4 \beta_{3}) q^{47} + 6 q^{49} + (\beta_{5} + 2 \beta_{4} + \beta_{3}) q^{53} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 3) q^{55} + (\beta_{2} + 2) q^{59} + ( - \beta_{2} + 4 \beta_1 - 7) q^{61} + ( - 3 \beta_{5} + \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - \beta_1) q^{65} + ( - \beta_{5} - 5 \beta_{4} + 2 \beta_{3}) q^{67} + ( - 4 \beta_{2} + 3 \beta_1 + 1) q^{71} + ( - 2 \beta_{5} - \beta_{4} - 7 \beta_{3}) q^{73} + \beta_{5} q^{77} + (2 \beta_{2} - 6 \beta_1 + 2) q^{79} + ( - 2 \beta_{5} + 3 \beta_{4} - 8 \beta_{3}) q^{83} + (2 \beta_{5} - \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + \beta_1) q^{85} + (4 \beta_{2} + 2) q^{89} + (\beta_{2} - 2 \beta_1 + 1) q^{91} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 3 \beta_1 - 5) q^{95} + ( - 5 \beta_{5} + 7 \beta_{4} - 3 \beta_{3}) q^{97}+O(q^{100})$$ q + (-b4 + b1) * q^5 - b3 * q^7 + b1 * q^11 + (2*b5 - b4 + b3) * q^13 + (-2*b5 + b4) * q^17 + (b2 - 2*b1 + 1) * q^19 + b3 * q^23 + (-2*b4 + 2*b3 + 2*b2 - 2*b1 + 1) * q^25 + (3*b2 - 2) * q^29 + (b2 - b1 - 4) * q^31 + (b5 + b2) * q^35 + (b5 - 3*b4) * q^37 + (4*b2 + b1 + 3) * q^41 + (-b5 - 3*b4 + b3) * q^43 + (-b5 + 2*b4 + 4*b3) * q^47 + 6 * q^49 + (b5 + 2*b4 + b3) * q^53 + (-b4 + b3 + b2 - b1 + 3) * q^55 + (b2 + 2) * q^59 + (-b2 + 4*b1 - 7) * q^61 + (-3*b5 + b4 - 5*b3 + 2*b2 - b1) * q^65 + (-b5 - 5*b4 + 2*b3) * q^67 + (-4*b2 + 3*b1 + 1) * q^71 + (-2*b5 - b4 - 7*b3) * q^73 + b5 * q^77 + (2*b2 - 6*b1 + 2) * q^79 + (-2*b5 + 3*b4 - 8*b3) * q^83 + (2*b5 - b4 + 5*b3 - 3*b2 + b1) * q^85 + (4*b2 + 2) * q^89 + (b2 - 2*b1 + 1) * q^91 + (-b5 + 2*b4 - b2 + 3*b1 - 5) * q^95 + (-5*b5 + 7*b4 - 3*b3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q + 4 q^{19} + 2 q^{25} - 18 q^{29} - 26 q^{31} - 2 q^{35} + 10 q^{41} + 36 q^{49} + 16 q^{55} + 10 q^{59} - 40 q^{61} - 4 q^{65} + 14 q^{71} + 8 q^{79} + 6 q^{85} + 4 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100})$$ 6 * q + 4 * q^19 + 2 * q^25 - 18 * q^29 - 26 * q^31 - 2 * q^35 + 10 * q^41 + 36 * q^49 + 16 * q^55 + 10 * q^59 - 40 * q^61 - 4 * q^65 + 14 * q^71 + 8 * q^79 + 6 * q^85 + 4 * q^89 + 4 * q^91 - 28 * q^95

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

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23$$ (-v^5 + 8*v^4 - 4*v^3 - v^2 + 2*v + 38) / 23 $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23$$ (-5*v^5 + 17*v^4 - 20*v^3 - 5*v^2 + 10*v + 29) / 23 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{4}$$ $$=$$ $$( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23$$ (-11*v^5 + 19*v^4 - 21*v^3 - 11*v^2 - 70*v + 27) / 23 $$\beta_{5}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 37*v^2 - 64*v + 26) / 23
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2$$ (b5 - b4 + b3 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{3}$$ b5 + 2*b3 $$\nu^{3}$$ $$=$$ $$2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2$$ 2*b5 - b4 + 2*b3 - b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5\beta _1 - 7$$ -b2 + 5*b1 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9$$ -8*b5 + 3*b4 - 9*b3 - 3*b2 + 8*b1 - 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$$.

 $$n$$ $$461$$ $$1657$$ $$2071$$ $$3961$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
829.1
 1.45161 + 1.45161i 1.45161 − 1.45161i −0.854638 − 0.854638i −0.854638 + 0.854638i 0.403032 − 0.403032i 0.403032 + 0.403032i
0 0 0 −2.21432 0.311108i 0 1.00000i 0 0 0
829.2 0 0 0 −2.21432 + 0.311108i 0 1.00000i 0 0 0
829.3 0 0 0 0.539189 2.17009i 0 1.00000i 0 0 0
829.4 0 0 0 0.539189 + 2.17009i 0 1.00000i 0 0 0
829.5 0 0 0 1.67513 1.48119i 0 1.00000i 0 0 0
829.6 0 0 0 1.67513 + 1.48119i 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 829.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.2.f.a 6
3.b odd 2 1 1380.2.f.a 6
5.b even 2 1 inner 4140.2.f.a 6
15.d odd 2 1 1380.2.f.a 6
15.e even 4 1 6900.2.a.y 3
15.e even 4 1 6900.2.a.z 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.a 6 3.b odd 2 1
1380.2.f.a 6 15.d odd 2 1
4140.2.f.a 6 1.a even 1 1 trivial
4140.2.f.a 6 5.b even 2 1 inner
6900.2.a.y 3 15.e even 4 1
6900.2.a.z 3 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(4140, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{4} + 16 T^{3} - 5 T^{2} + \cdots + 125$$
$7$ $$(T^{2} + 1)^{3}$$
$11$ $$(T^{3} - 4 T + 2)^{2}$$
$13$ $$T^{6} + 32 T^{4} + 156 T^{2} + \cdots + 100$$
$17$ $$T^{6} + 31 T^{4} + 275 T^{2} + \cdots + 625$$
$19$ $$(T^{3} - 2 T^{2} - 14 T - 10)^{2}$$
$23$ $$(T^{2} + 1)^{3}$$
$29$ $$(T^{3} + 9 T^{2} - 3 T - 61)^{2}$$
$31$ $$(T^{3} + 13 T^{2} + 51 T + 59)^{2}$$
$37$ $$T^{6} + 59 T^{4} + 475 T^{2} + \cdots + 625$$
$41$ $$(T^{3} - 5 T^{2} - 57 T - 25)^{2}$$
$43$ $$T^{6} + 80 T^{4} + 1600 T^{2} + \cdots + 5776$$
$47$ $$T^{6} + 92 T^{4} + 1080 T^{2} + \cdots + 3364$$
$53$ $$T^{6} + 51 T^{4} + 839 T^{2} + \cdots + 4489$$
$59$ $$(T^{3} - 5 T^{2} + 5 T + 1)^{2}$$
$61$ $$(T^{3} + 20 T^{2} + 74 T + 74)^{2}$$
$67$ $$T^{6} + 195 T^{4} + 9083 T^{2} + \cdots + 26569$$
$71$ $$(T^{3} - 7 T^{2} - 49 T + 5)^{2}$$
$73$ $$T^{6} + 208 T^{4} + 9980 T^{2} + \cdots + 42436$$
$79$ $$(T^{3} - 4 T^{2} - 128 T - 416)^{2}$$
$83$ $$T^{6} + 215 T^{4} + 10963 T^{2} + \cdots + 1849$$
$89$ $$(T^{3} - 2 T^{2} - 52 T + 40)^{2}$$
$97$ $$T^{6} + 388 T^{4} + 40400 T^{2} + \cdots + 781456$$