# Properties

 Label 6012.2.a.b Level $6012$ Weight $2$ Character orbit 6012.a Self dual yes Analytic conductor $48.006$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6012 = 2^{2} \cdot 3^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6012.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.0060616952$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + 2 \beta_{2} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + 2 \beta_{2} q^{7} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} + 2 \beta_{1} q^{13} + ( 2 - \beta_{2} ) q^{17} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{19} -4 \beta_{1} q^{23} + ( -2 - \beta_{1} - \beta_{2} ) q^{25} -4 \beta_{2} q^{29} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{31} + ( -6 + 2 \beta_{1} + 2 \beta_{2} ) q^{35} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -4 - 2 \beta_{1} - 3 \beta_{2} ) q^{41} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{43} + ( 5 + \beta_{1} + \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} - 4 \beta_{2} ) q^{49} + ( -4 + 6 \beta_{1} + \beta_{2} ) q^{53} + 2 q^{55} + ( -8 + 2 \beta_{2} ) q^{59} + ( -1 + \beta_{1} + \beta_{2} ) q^{61} + ( 2 - 2 \beta_{1} ) q^{65} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{67} -4 q^{71} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{73} -4 q^{77} + ( 2 - 8 \beta_{1} - \beta_{2} ) q^{79} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{83} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{85} + ( -4 - 4 \beta_{2} ) q^{89} + ( -4 + 4 \beta_{1} ) q^{91} + ( -4 + 4 \beta_{2} ) q^{95} + ( 6 + 6 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + O(q^{10})$$ $$3q - 4q^{11} + 2q^{13} + 6q^{17} - 4q^{19} - 4q^{23} - 7q^{25} + 4q^{31} - 16q^{35} + 10q^{37} - 14q^{41} - 20q^{43} + 16q^{47} + 11q^{49} - 6q^{53} + 6q^{55} - 24q^{59} - 2q^{61} + 4q^{65} + 10q^{67} - 12q^{71} - 10q^{73} - 12q^{77} - 2q^{79} - 4q^{83} + 8q^{85} - 12q^{89} - 8q^{91} - 12q^{95} + 18q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$

## 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
 −1.48119 2.17009 0.311108
0 0 0 −1.67513 0 3.35026 0 0 0
1.2 0 0 0 −0.539189 0 1.07838 0 0 0
1.3 0 0 0 2.21432 0 −4.42864 0 0 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$167$$ $$-1$$

## 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 6012.2.a.b 3
3.b odd 2 1 6012.2.a.c yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6012.2.a.b 3 1.a even 1 1 trivial
6012.2.a.c yes 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - 4 T_{5} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6012))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 11 T^{2} - 2 T^{3} + 55 T^{4} + 125 T^{6}$$
$7$ $$1 + 5 T^{2} + 16 T^{3} + 35 T^{4} + 343 T^{6}$$
$11$ $$1 + 4 T + 33 T^{2} + 84 T^{3} + 363 T^{4} + 484 T^{5} + 1331 T^{6}$$
$13$ $$1 - 2 T + 27 T^{2} - 44 T^{3} + 351 T^{4} - 338 T^{5} + 2197 T^{6}$$
$17$ $$1 - 6 T + 59 T^{2} - 206 T^{3} + 1003 T^{4} - 1734 T^{5} + 4913 T^{6}$$
$19$ $$1 + 4 T + 41 T^{2} + 120 T^{3} + 779 T^{4} + 1444 T^{5} + 6859 T^{6}$$
$23$ $$1 + 4 T + 21 T^{2} + 120 T^{3} + 483 T^{4} + 2116 T^{5} + 12167 T^{6}$$
$29$ $$1 + 23 T^{2} - 128 T^{3} + 667 T^{4} + 24389 T^{6}$$
$31$ $$1 - 4 T + 61 T^{2} - 280 T^{3} + 1891 T^{4} - 3844 T^{5} + 29791 T^{6}$$
$37$ $$1 - 10 T + 91 T^{2} - 748 T^{3} + 3367 T^{4} - 13690 T^{5} + 50653 T^{6}$$
$41$ $$1 + 14 T + 151 T^{2} + 1026 T^{3} + 6191 T^{4} + 23534 T^{5} + 68921 T^{6}$$
$43$ $$1 + 20 T + 241 T^{2} + 1838 T^{3} + 10363 T^{4} + 36980 T^{5} + 79507 T^{6}$$
$47$ $$1 - 16 T + 221 T^{2} - 1628 T^{3} + 10387 T^{4} - 35344 T^{5} + 103823 T^{6}$$
$53$ $$1 + 6 T + 59 T^{2} + 170 T^{3} + 3127 T^{4} + 16854 T^{5} + 148877 T^{6}$$
$59$ $$1 + 24 T + 353 T^{2} + 3232 T^{3} + 20827 T^{4} + 83544 T^{5} + 205379 T^{6}$$
$61$ $$1 + 2 T + 179 T^{2} + 240 T^{3} + 10919 T^{4} + 7442 T^{5} + 226981 T^{6}$$
$67$ $$1 - 10 T + 221 T^{2} - 1314 T^{3} + 14807 T^{4} - 44890 T^{5} + 300763 T^{6}$$
$71$ $$( 1 + 4 T + 71 T^{2} )^{3}$$
$73$ $$1 + 10 T + 231 T^{2} + 1420 T^{3} + 16863 T^{4} + 53290 T^{5} + 389017 T^{6}$$
$79$ $$1 + 2 T + 37 T^{2} + 650 T^{3} + 2923 T^{4} + 12482 T^{5} + 493039 T^{6}$$
$83$ $$1 + 4 T + 161 T^{2} + 680 T^{3} + 13363 T^{4} + 27556 T^{5} + 571787 T^{6}$$
$89$ $$1 + 12 T + 251 T^{2} + 1816 T^{3} + 22339 T^{4} + 95052 T^{5} + 704969 T^{6}$$
$97$ $$1 - 18 T + 255 T^{2} - 2412 T^{3} + 24735 T^{4} - 169362 T^{5} + 912673 T^{6}$$