# Properties

 Label 2112.2.a.bf Level $2112$ Weight $2$ Character orbit 2112.a Self dual yes Analytic conductor $16.864$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
0 1.00000 0 −3.23607 0 1.23607 0 1.00000 0
1.2 0 1.00000 0 1.23607 0 −3.23607 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.bf 2
3.b odd 2 1 6336.2.a.cs 2
4.b odd 2 1 2112.2.a.be 2
8.b even 2 1 1056.2.a.k 2
8.d odd 2 1 1056.2.a.l yes 2
12.b even 2 1 6336.2.a.ct 2
24.f even 2 1 3168.2.a.bf 2
24.h odd 2 1 3168.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1056.2.a.k 2 8.b even 2 1
1056.2.a.l yes 2 8.d odd 2 1
2112.2.a.be 2 4.b odd 2 1
2112.2.a.bf 2 1.a even 1 1 trivial
3168.2.a.be 2 24.h odd 2 1
3168.2.a.bf 2 24.f even 2 1
6336.2.a.cs 2 3.b odd 2 1
6336.2.a.ct 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2112))$$:

 $$T_{5}^{2} + 2 T_{5} - 4$$ $$T_{7}^{2} + 2 T_{7} - 4$$ $$T_{13}^{2} + 6 T_{13} + 4$$ $$T_{17}^{2} - 20$$ $$T_{19}^{2} - 4 T_{19} - 16$$ $$T_{23}^{2} + 2 T_{23} - 44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T )^{2}$$
$5$ $$1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$1 + 6 T + 30 T^{2} + 78 T^{3} + 169 T^{4}$$
$17$ $$1 + 14 T^{2} + 289 T^{4}$$
$19$ $$1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 + 8 T + 54 T^{2} + 232 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 + 6 T + 37 T^{2} )^{2}$$
$41$ $$1 - 4 T + 6 T^{2} - 164 T^{3} + 1681 T^{4}$$
$43$ $$1 + 8 T + 22 T^{2} + 344 T^{3} + 1849 T^{4}$$
$47$ $$1 - 2 T + 50 T^{2} - 94 T^{3} + 2209 T^{4}$$
$53$ $$1 + 18 T + 182 T^{2} + 954 T^{3} + 2809 T^{4}$$
$59$ $$1 + 20 T + 198 T^{2} + 1180 T^{3} + 3481 T^{4}$$
$61$ $$1 + 10 T + 102 T^{2} + 610 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 8 T + 67 T^{2} )^{2}$$
$71$ $$1 + 2 T + 98 T^{2} + 142 T^{3} + 5041 T^{4}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{2}$$
$79$ $$1 - 10 T + 138 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$1 + 8 T + 102 T^{2} + 664 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{2}$$
$97$ $$1 + 8 T + 190 T^{2} + 776 T^{3} + 9409 T^{4}$$