# Properties

 Label 8112.2.a.cg Level $8112$ Weight $2$ Character orbit 8112.a Self dual yes Analytic conductor $64.775$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8112 = 2^{4} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.7746461197$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 507) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -2 + \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -2 + \beta_{1} + \beta_{2} ) q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} + q^{9} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -2 + \beta_{1} + \beta_{2} ) q^{15} + ( -2 \beta_{1} - \beta_{2} ) q^{17} + ( -2 + \beta_{2} ) q^{19} + ( \beta_{1} - \beta_{2} ) q^{21} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{23} + ( 4 - 3 \beta_{1} - 2 \beta_{2} ) q^{25} + q^{27} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -3 - 5 \beta_{1} + 2 \beta_{2} ) q^{31} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{33} + ( 1 - 3 \beta_{1} + 4 \beta_{2} ) q^{35} + ( -7 + \beta_{2} ) q^{37} + ( -2 - 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -2 + \beta_{1} + \beta_{2} ) q^{45} + ( 2 - \beta_{2} ) q^{47} + ( -6 + \beta_{1} - 2 \beta_{2} ) q^{49} + ( -2 \beta_{1} - \beta_{2} ) q^{51} + ( -8 + 3 \beta_{1} - 4 \beta_{2} ) q^{53} + ( 5 - 8 \beta_{1} + 10 \beta_{2} ) q^{55} + ( -2 + \beta_{2} ) q^{57} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -8 + 6 \beta_{1} - 5 \beta_{2} ) q^{61} + ( \beta_{1} - \beta_{2} ) q^{63} + ( 7 - 7 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{69} + ( 1 - 5 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 2 - 9 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 4 - 3 \beta_{1} - 2 \beta_{2} ) q^{75} + ( 3 + 2 \beta_{1} - 4 \beta_{2} ) q^{77} + ( -3 + 3 \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{83} + ( -8 + 3 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 1 - 3 \beta_{1} + 2 \beta_{2} ) q^{87} + ( -3 + \beta_{1} - 7 \beta_{2} ) q^{89} + ( -3 - 5 \beta_{1} + 2 \beta_{2} ) q^{93} + ( 6 - \beta_{1} - 4 \beta_{2} ) q^{95} + ( -1 - 2 \beta_{1} - 10 \beta_{2} ) q^{97} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} - 6q^{5} + 2q^{7} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} - 6q^{5} + 2q^{7} + 3q^{9} + 5q^{11} - 6q^{15} - q^{17} - 7q^{19} + 2q^{21} + 11q^{25} + 3q^{27} - 2q^{29} - 16q^{31} + 5q^{33} - 4q^{35} - 22q^{37} - 11q^{41} + 15q^{43} - 6q^{45} + 7q^{47} - 15q^{49} - q^{51} - 17q^{53} - 3q^{55} - 7q^{57} - 6q^{59} - 13q^{61} + 2q^{63} + 11q^{67} - 6q^{73} + 11q^{75} + 15q^{77} - 3q^{79} + 3q^{81} + 12q^{83} - 19q^{85} - 2q^{87} - q^{89} - 16q^{93} + 21q^{95} + 5q^{97} + 5q^{99} + O(q^{100})$$

## 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
 −1.24698 0.445042 1.80194
0 1.00000 0 −3.69202 0 −0.801938 0 1.00000 0
1.2 0 1.00000 0 −3.35690 0 2.24698 0 1.00000 0
1.3 0 1.00000 0 1.04892 0 0.554958 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 8112.2.a.cg 3
4.b odd 2 1 507.2.a.i 3
12.b even 2 1 1521.2.a.s 3
13.b even 2 1 8112.2.a.cp 3
52.b odd 2 1 507.2.a.l yes 3
52.f even 4 2 507.2.b.f 6
52.i odd 6 2 507.2.e.i 6
52.j odd 6 2 507.2.e.l 6
52.l even 12 4 507.2.j.i 12
156.h even 2 1 1521.2.a.n 3
156.l odd 4 2 1521.2.b.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 4.b odd 2 1
507.2.a.l yes 3 52.b odd 2 1
507.2.b.f 6 52.f even 4 2
507.2.e.i 6 52.i odd 6 2
507.2.e.l 6 52.j odd 6 2
507.2.j.i 12 52.l even 12 4
1521.2.a.n 3 156.h even 2 1
1521.2.a.s 3 12.b even 2 1
1521.2.b.k 6 156.l odd 4 2
8112.2.a.cg 3 1.a even 1 1 trivial
8112.2.a.cp 3 13.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8112))$$:

 $$T_{5}^{3} + 6 T_{5}^{2} + 5 T_{5} - 13$$ $$T_{7}^{3} - 2 T_{7}^{2} - T_{7} + 1$$ $$T_{11}^{3} - 5 T_{11}^{2} - 8 T_{11} + 41$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-13 + 5 T + 6 T^{2} + T^{3}$$
$7$ $$1 - T - 2 T^{2} + T^{3}$$
$11$ $$41 - 8 T - 5 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$13 - 16 T + T^{2} + T^{3}$$
$19$ $$7 + 14 T + 7 T^{2} + T^{3}$$
$23$ $$-91 - 49 T + T^{3}$$
$29$ $$-29 - 15 T + 2 T^{2} + T^{3}$$
$31$ $$-197 + 41 T + 16 T^{2} + T^{3}$$
$37$ $$377 + 159 T + 22 T^{2} + T^{3}$$
$41$ $$-29 + 24 T + 11 T^{2} + T^{3}$$
$43$ $$-41 + 47 T - 15 T^{2} + T^{3}$$
$47$ $$-7 + 14 T - 7 T^{2} + T^{3}$$
$53$ $$-41 + 66 T + 17 T^{2} + T^{3}$$
$59$ $$-104 - 16 T + 6 T^{2} + T^{3}$$
$61$ $$-167 - 16 T + 13 T^{2} + T^{3}$$
$67$ $$-41 - 46 T - 11 T^{2} + T^{3}$$
$71$ $$203 - 91 T + T^{3}$$
$73$ $$-923 - 135 T + 6 T^{2} + T^{3}$$
$79$ $$-27 - 18 T + 3 T^{2} + T^{3}$$
$83$ $$-43 + 41 T - 12 T^{2} + T^{3}$$
$89$ $$-113 - 100 T + T^{2} + T^{3}$$
$97$ $$1637 - 281 T - 5 T^{2} + T^{3}$$