# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 2 + \beta ) q^{5} + ( 1 + \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( 2 + \beta ) q^{5} + ( 1 + \beta ) q^{7} + q^{9} + ( -3 + \beta ) q^{11} + ( -2 - \beta ) q^{15} + ( -4 - \beta ) q^{17} + ( -3 - \beta ) q^{19} + ( -1 - \beta ) q^{21} + ( 1 - 3 \beta ) q^{23} + ( 2 + 4 \beta ) q^{25} - q^{27} + ( -1 - 2 \beta ) q^{29} + ( 2 - 2 \beta ) q^{31} + ( 3 - \beta ) q^{33} + ( 5 + 3 \beta ) q^{35} + ( 7 - 2 \beta ) q^{37} + ( 1 - 6 \beta ) q^{41} + ( 1 + 5 \beta ) q^{43} + ( 2 + \beta ) q^{45} + ( 3 - 3 \beta ) q^{47} + ( -3 + 2 \beta ) q^{49} + ( 4 + \beta ) q^{51} + ( -3 - 2 \beta ) q^{53} + ( -3 - \beta ) q^{55} + ( 3 + \beta ) q^{57} -8 q^{59} + ( -4 - 3 \beta ) q^{61} + ( 1 + \beta ) q^{63} + ( -1 - 7 \beta ) q^{67} + ( -1 + 3 \beta ) q^{69} + ( -3 - \beta ) q^{71} + ( -8 + \beta ) q^{73} + ( -2 - 4 \beta ) q^{75} -2 \beta q^{77} + ( 6 - 2 \beta ) q^{79} + q^{81} + ( -5 + 3 \beta ) q^{83} + ( -11 - 6 \beta ) q^{85} + ( 1 + 2 \beta ) q^{87} + ( -6 - 2 \beta ) q^{89} + ( -2 + 2 \beta ) q^{93} + ( -9 - 5 \beta ) q^{95} -6 q^{97} + ( -3 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + 4 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 8 q^{17} - 6 q^{19} - 2 q^{21} + 2 q^{23} + 4 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 6 q^{33} + 10 q^{35} + 14 q^{37} + 2 q^{41} + 2 q^{43} + 4 q^{45} + 6 q^{47} - 6 q^{49} + 8 q^{51} - 6 q^{53} - 6 q^{55} + 6 q^{57} - 16 q^{59} - 8 q^{61} + 2 q^{63} - 2 q^{67} - 2 q^{69} - 6 q^{71} - 16 q^{73} - 4 q^{75} + 12 q^{79} + 2 q^{81} - 10 q^{83} - 22 q^{85} + 2 q^{87} - 12 q^{89} - 4 q^{93} - 18 q^{95} - 12 q^{97} - 6 q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 −1.00000 0 0.267949 0 −0.732051 0 1.00000 0
1.2 0 −1.00000 0 3.73205 0 2.73205 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.bp 2
4.b odd 2 1 1014.2.a.k 2
12.b even 2 1 3042.2.a.p 2
13.b even 2 1 8112.2.a.bj 2
13.f odd 12 2 624.2.bv.e 4
39.k even 12 2 1872.2.by.h 4
52.b odd 2 1 1014.2.a.i 2
52.f even 4 2 1014.2.b.e 4
52.i odd 6 2 1014.2.e.i 4
52.j odd 6 2 1014.2.e.g 4
52.l even 12 2 78.2.i.a 4
52.l even 12 2 1014.2.i.a 4
156.h even 2 1 3042.2.a.y 2
156.l odd 4 2 3042.2.b.i 4
156.v odd 12 2 234.2.l.c 4
260.bc even 12 2 1950.2.bc.d 4
260.be odd 12 2 1950.2.y.b 4
260.bl odd 12 2 1950.2.y.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.a 4 52.l even 12 2
234.2.l.c 4 156.v odd 12 2
624.2.bv.e 4 13.f odd 12 2
1014.2.a.i 2 52.b odd 2 1
1014.2.a.k 2 4.b odd 2 1
1014.2.b.e 4 52.f even 4 2
1014.2.e.g 4 52.j odd 6 2
1014.2.e.i 4 52.i odd 6 2
1014.2.i.a 4 52.l even 12 2
1872.2.by.h 4 39.k even 12 2
1950.2.y.b 4 260.be odd 12 2
1950.2.y.g 4 260.bl odd 12 2
1950.2.bc.d 4 260.bc even 12 2
3042.2.a.p 2 12.b even 2 1
3042.2.a.y 2 156.h even 2 1
3042.2.b.i 4 156.l odd 4 2
8112.2.a.bj 2 13.b even 2 1
8112.2.a.bp 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8112))$$:

 $$T_{5}^{2} - 4 T_{5} + 1$$ $$T_{7}^{2} - 2 T_{7} - 2$$ $$T_{11}^{2} + 6 T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 - 4 T + T^{2}$$
$7$ $$-2 - 2 T + T^{2}$$
$11$ $$6 + 6 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$13 + 8 T + T^{2}$$
$19$ $$6 + 6 T + T^{2}$$
$23$ $$-26 - 2 T + T^{2}$$
$29$ $$-11 + 2 T + T^{2}$$
$31$ $$-8 - 4 T + T^{2}$$
$37$ $$37 - 14 T + T^{2}$$
$41$ $$-107 - 2 T + T^{2}$$
$43$ $$-74 - 2 T + T^{2}$$
$47$ $$-18 - 6 T + T^{2}$$
$53$ $$-3 + 6 T + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$-11 + 8 T + T^{2}$$
$67$ $$-146 + 2 T + T^{2}$$
$71$ $$6 + 6 T + T^{2}$$
$73$ $$61 + 16 T + T^{2}$$
$79$ $$24 - 12 T + T^{2}$$
$83$ $$-2 + 10 T + T^{2}$$
$89$ $$24 + 12 T + T^{2}$$
$97$ $$( 6 + T )^{2}$$
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