# Properties

 Label 2112.2.a.bg Level $2112$ Weight $2$ Character orbit 2112.a Self dual yes Analytic conductor $16.864$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2112 = 2^{6} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.8644049069$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1056) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta_{1} q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta_{1} q^{5} + ( 1 + \beta_{2} ) q^{7} + q^{9} - q^{11} + \beta_{1} q^{13} -\beta_{1} q^{15} + ( 3 + \beta_{1} - \beta_{2} ) q^{17} + ( -1 - \beta_{1} + \beta_{2} ) q^{19} + ( -1 - \beta_{2} ) q^{21} + ( -2 + \beta_{1} ) q^{23} + ( 5 + 2 \beta_{2} ) q^{25} - q^{27} + ( -3 - \beta_{1} + \beta_{2} ) q^{29} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{31} + q^{33} + ( 4 + 4 \beta_{1} ) q^{35} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{37} -\beta_{1} q^{39} + ( 3 - \beta_{1} - \beta_{2} ) q^{41} + ( -3 - \beta_{1} - \beta_{2} ) q^{43} + \beta_{1} q^{45} + ( -6 - \beta_{1} ) q^{47} + ( 9 + 2 \beta_{1} ) q^{49} + ( -3 - \beta_{1} + \beta_{2} ) q^{51} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{53} -\beta_{1} q^{55} + ( 1 + \beta_{1} - \beta_{2} ) q^{57} -2 \beta_{1} q^{59} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{61} + ( 1 + \beta_{2} ) q^{63} + ( 10 + 2 \beta_{2} ) q^{65} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 2 - \beta_{1} ) q^{69} + ( -10 + \beta_{1} ) q^{71} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -5 - 2 \beta_{2} ) q^{75} + ( -1 - \beta_{2} ) q^{77} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{79} + q^{81} + 8 q^{83} + ( 6 + 2 \beta_{2} ) q^{85} + ( 3 + \beta_{1} - \beta_{2} ) q^{87} + 2 q^{89} + ( 4 + 4 \beta_{1} ) q^{91} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( 2 - 2 \beta_{1} ) q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 4q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 4q^{7} + 3q^{9} - 3q^{11} + 8q^{17} - 2q^{19} - 4q^{21} - 6q^{23} + 17q^{25} - 3q^{27} - 8q^{29} + 4q^{31} + 3q^{33} + 12q^{35} + 2q^{37} + 8q^{41} - 10q^{43} - 18q^{47} + 27q^{49} - 8q^{51} + 4q^{53} + 2q^{57} - 8q^{61} + 4q^{63} + 32q^{65} + 4q^{67} + 6q^{69} - 30q^{71} + 10q^{73} - 17q^{75} - 4q^{77} + 16q^{79} + 3q^{81} + 24q^{83} + 20q^{85} + 8q^{87} + 6q^{89} + 12q^{91} - 4q^{93} - 20q^{95} + 6q^{97} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 5$$$$)/2$$

## 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
 −1.86081 −0.254102 2.11491
0 −1.00000 0 −3.72161 0 2.92520 0 1.00000 0
1.2 0 −1.00000 0 −0.508203 0 −3.87086 0 1.00000 0
1.3 0 −1.00000 0 4.22982 0 4.94567 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$11$$ $$1$$

## 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 2112.2.a.bg 3
3.b odd 2 1 6336.2.a.cz 3
4.b odd 2 1 2112.2.a.bh 3
8.b even 2 1 1056.2.a.n yes 3
8.d odd 2 1 1056.2.a.m 3
12.b even 2 1 6336.2.a.cy 3
24.f even 2 1 3168.2.a.bg 3
24.h odd 2 1 3168.2.a.bh 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1056.2.a.m 3 8.d odd 2 1
1056.2.a.n yes 3 8.b even 2 1
2112.2.a.bg 3 1.a even 1 1 trivial
2112.2.a.bh 3 4.b odd 2 1
3168.2.a.bg 3 24.f even 2 1
3168.2.a.bh 3 24.h odd 2 1
6336.2.a.cy 3 12.b even 2 1
6336.2.a.cz 3 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2112))$$:

 $$T_{5}^{3} - 16 T_{5} - 8$$ $$T_{7}^{3} - 4 T_{7}^{2} - 16 T_{7} + 56$$ $$T_{13}^{3} - 16 T_{13} - 8$$ $$T_{17}^{3} - 8 T_{17}^{2} - 4 T_{17} + 64$$ $$T_{19}^{3} + 2 T_{19}^{2} - 24 T_{19} - 32$$ $$T_{23}^{3} + 6 T_{23}^{2} - 4 T_{23} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T )^{3}$$
$5$ $$1 - T^{2} - 8 T^{3} - 5 T^{4} + 125 T^{6}$$
$7$ $$1 - 4 T + 5 T^{2} + 35 T^{4} - 196 T^{5} + 343 T^{6}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$1 + 23 T^{2} - 8 T^{3} + 299 T^{4} + 2197 T^{6}$$
$17$ $$1 - 8 T + 47 T^{2} - 208 T^{3} + 799 T^{4} - 2312 T^{5} + 4913 T^{6}$$
$19$ $$1 + 2 T + 33 T^{2} + 44 T^{3} + 627 T^{4} + 722 T^{5} + 6859 T^{6}$$
$23$ $$1 + 6 T + 65 T^{2} + 244 T^{3} + 1495 T^{4} + 3174 T^{5} + 12167 T^{6}$$
$29$ $$1 + 8 T + 83 T^{2} + 400 T^{3} + 2407 T^{4} + 6728 T^{5} + 24389 T^{6}$$
$31$ $$1 - 4 T - 3 T^{2} + 8 T^{3} - 93 T^{4} - 3844 T^{5} + 29791 T^{6}$$
$37$ $$1 - 2 T + 11 T^{2} - 204 T^{3} + 407 T^{4} - 2738 T^{5} + 50653 T^{6}$$
$41$ $$1 - 8 T + 95 T^{2} - 448 T^{3} + 3895 T^{4} - 13448 T^{5} + 68921 T^{6}$$
$43$ $$1 + 10 T + 113 T^{2} + 828 T^{3} + 4859 T^{4} + 18490 T^{5} + 79507 T^{6}$$
$47$ $$1 + 18 T + 233 T^{2} + 1820 T^{3} + 10951 T^{4} + 39762 T^{5} + 103823 T^{6}$$
$53$ $$1 - 4 T + 39 T^{2} - 192 T^{3} + 2067 T^{4} - 11236 T^{5} + 148877 T^{6}$$
$59$ $$1 + 113 T^{2} + 64 T^{3} + 6667 T^{4} + 205379 T^{6}$$
$61$ $$1 + 8 T + 127 T^{2} + 584 T^{3} + 7747 T^{4} + 29768 T^{5} + 226981 T^{6}$$
$67$ $$1 - 4 T + 9 T^{2} + 488 T^{3} + 603 T^{4} - 17956 T^{5} + 300763 T^{6}$$
$71$ $$1 + 30 T + 497 T^{2} + 5092 T^{3} + 35287 T^{4} + 151230 T^{5} + 357911 T^{6}$$
$73$ $$1 - 10 T + 55 T^{2} - 76 T^{3} + 4015 T^{4} - 53290 T^{5} + 389017 T^{6}$$
$79$ $$1 - 16 T + 261 T^{2} - 2536 T^{3} + 20619 T^{4} - 99856 T^{5} + 493039 T^{6}$$
$83$ $$( 1 - 8 T + 83 T^{2} )^{3}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{3}$$
$97$ $$1 - 6 T + 239 T^{2} - 980 T^{3} + 23183 T^{4} - 56454 T^{5} + 912673 T^{6}$$