# Properties

 Label 8112.2.a.by 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: $$x^{3} - x^{2} - 2x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( - \beta_1 - 1) q^{5} + (\beta_1 + 3) q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (-b1 - 1) * q^5 + (b1 + 3) * q^7 + q^9 $$q - q^{3} + ( - \beta_1 - 1) q^{5} + (\beta_1 + 3) q^{7} + q^{9} + (3 \beta_{2} - 4 \beta_1 + 2) q^{11} + (\beta_1 + 1) q^{15} + ( - \beta_{2} + \beta_1 - 3) q^{17} + ( - 3 \beta_{2} - \beta_1 + 3) q^{19} + ( - \beta_1 - 3) q^{21} + ( - 2 \beta_{2} + 5 \beta_1 - 3) q^{23} + (\beta_{2} + 2 \beta_1 - 2) q^{25} - q^{27} + (\beta_{2} + 2 \beta_1 - 3) q^{29} + (5 \beta_{2} - 2 \beta_1 + 5) q^{31} + ( - 3 \beta_{2} + 4 \beta_1 - 2) q^{33} + ( - \beta_{2} - 4 \beta_1 - 5) q^{35} + (\beta_{2} - \beta_1 - 4) q^{37} + \beta_{2} q^{41} + ( - 4 \beta_{2} + 2 \beta_1 - 1) q^{43} + ( - \beta_1 - 1) q^{45} + ( - 9 \beta_{2} + 3 \beta_1 - 7) q^{47} + (\beta_{2} + 6 \beta_1 + 4) q^{49} + (\beta_{2} - \beta_1 + 3) q^{51} + ( - \beta_{2} - 2 \beta_1 - 4) q^{53} + ( - 2 \beta_{2} + 2 \beta_1 + 3) q^{55} + (3 \beta_{2} + \beta_1 - 3) q^{57} + ( - 4 \beta_{2} + 6 \beta_1 - 8) q^{59} + ( - 5 \beta_{2} + 3 \beta_1 - 7) q^{61} + (\beta_1 + 3) q^{63} + (4 \beta_{2} - 3 \beta_1 + 4) q^{67} + (2 \beta_{2} - 5 \beta_1 + 3) q^{69} + (5 \beta_{2} + 2 \beta_1 - 1) q^{71} + ( - 2 \beta_{2} + \beta_1 - 7) q^{73} + ( - \beta_{2} - 2 \beta_1 + 2) q^{75} + (8 \beta_{2} - 10 \beta_1 + 1) q^{77} + ( - 4 \beta_{2} + 5 \beta_1) q^{79} + q^{81} + (\beta_{2} + 3 \beta_1 - 6) q^{83} + (\beta_{2} + 2 \beta_1 + 2) q^{85} + ( - \beta_{2} - 2 \beta_1 + 3) q^{87} + (2 \beta_{2} + \beta_1 + 2) q^{89} + ( - 5 \beta_{2} + 2 \beta_1 - 5) q^{93} + (7 \beta_{2} - 2 \beta_1 + 2) q^{95} + (4 \beta_{2} - 10 \beta_1 + 3) q^{97} + (3 \beta_{2} - 4 \beta_1 + 2) q^{99}+O(q^{100})$$ q - q^3 + (-b1 - 1) * q^5 + (b1 + 3) * q^7 + q^9 + (3*b2 - 4*b1 + 2) * q^11 + (b1 + 1) * q^15 + (-b2 + b1 - 3) * q^17 + (-3*b2 - b1 + 3) * q^19 + (-b1 - 3) * q^21 + (-2*b2 + 5*b1 - 3) * q^23 + (b2 + 2*b1 - 2) * q^25 - q^27 + (b2 + 2*b1 - 3) * q^29 + (5*b2 - 2*b1 + 5) * q^31 + (-3*b2 + 4*b1 - 2) * q^33 + (-b2 - 4*b1 - 5) * q^35 + (b2 - b1 - 4) * q^37 + b2 * q^41 + (-4*b2 + 2*b1 - 1) * q^43 + (-b1 - 1) * q^45 + (-9*b2 + 3*b1 - 7) * q^47 + (b2 + 6*b1 + 4) * q^49 + (b2 - b1 + 3) * q^51 + (-b2 - 2*b1 - 4) * q^53 + (-2*b2 + 2*b1 + 3) * q^55 + (3*b2 + b1 - 3) * q^57 + (-4*b2 + 6*b1 - 8) * q^59 + (-5*b2 + 3*b1 - 7) * q^61 + (b1 + 3) * q^63 + (4*b2 - 3*b1 + 4) * q^67 + (2*b2 - 5*b1 + 3) * q^69 + (5*b2 + 2*b1 - 1) * q^71 + (-2*b2 + b1 - 7) * q^73 + (-b2 - 2*b1 + 2) * q^75 + (8*b2 - 10*b1 + 1) * q^77 + (-4*b2 + 5*b1) * q^79 + q^81 + (b2 + 3*b1 - 6) * q^83 + (b2 + 2*b1 + 2) * q^85 + (-b2 - 2*b1 + 3) * q^87 + (2*b2 + b1 + 2) * q^89 + (-5*b2 + 2*b1 - 5) * q^93 + (7*b2 - 2*b1 + 2) * q^95 + (4*b2 - 10*b1 + 3) * q^97 + (3*b2 - 4*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9 $$3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9} - q^{11} + 4 q^{15} - 7 q^{17} + 11 q^{19} - 10 q^{21} - 2 q^{23} - 5 q^{25} - 3 q^{27} - 8 q^{29} + 8 q^{31} + q^{33} - 18 q^{35} - 14 q^{37} - q^{41} + 3 q^{43} - 4 q^{45} - 9 q^{47} + 17 q^{49} + 7 q^{51} - 13 q^{53} + 13 q^{55} - 11 q^{57} - 14 q^{59} - 13 q^{61} + 10 q^{63} + 5 q^{67} + 2 q^{69} - 6 q^{71} - 18 q^{73} + 5 q^{75} - 15 q^{77} + 9 q^{79} + 3 q^{81} - 16 q^{83} + 7 q^{85} + 8 q^{87} + 5 q^{89} - 8 q^{93} - 3 q^{95} - 5 q^{97} - q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9 - q^11 + 4 * q^15 - 7 * q^17 + 11 * q^19 - 10 * q^21 - 2 * q^23 - 5 * q^25 - 3 * q^27 - 8 * q^29 + 8 * q^31 + q^33 - 18 * q^35 - 14 * q^37 - q^41 + 3 * q^43 - 4 * q^45 - 9 * q^47 + 17 * q^49 + 7 * q^51 - 13 * q^53 + 13 * q^55 - 11 * q^57 - 14 * q^59 - 13 * q^61 + 10 * q^63 + 5 * q^67 + 2 * q^69 - 6 * q^71 - 18 * q^73 + 5 * q^75 - 15 * q^77 + 9 * q^79 + 3 * q^81 - 16 * q^83 + 7 * q^85 + 8 * q^87 + 5 * q^89 - 8 * q^93 - 3 * q^95 - 5 * q^97 - q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 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
 1.80194 0.445042 −1.24698
0 −1.00000 0 −2.80194 0 4.80194 0 1.00000 0
1.2 0 −1.00000 0 −1.44504 0 3.44504 0 1.00000 0
1.3 0 −1.00000 0 0.246980 0 1.75302 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.by 3
4.b odd 2 1 507.2.a.j 3
12.b even 2 1 1521.2.a.q 3
13.b even 2 1 8112.2.a.cf 3
52.b odd 2 1 507.2.a.k yes 3
52.f even 4 2 507.2.b.g 6
52.i odd 6 2 507.2.e.j 6
52.j odd 6 2 507.2.e.k 6
52.l even 12 4 507.2.j.h 12
156.h even 2 1 1521.2.a.p 3
156.l odd 4 2 1521.2.b.m 6

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

 $$T_{5}^{3} + 4T_{5}^{2} + 3T_{5} - 1$$ T5^3 + 4*T5^2 + 3*T5 - 1 $$T_{7}^{3} - 10T_{7}^{2} + 31T_{7} - 29$$ T7^3 - 10*T7^2 + 31*T7 - 29 $$T_{11}^{3} + T_{11}^{2} - 30T_{11} - 43$$ T11^3 + T11^2 - 30*T11 - 43

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3} + 4 T^{2} + 3 T - 1$$
$7$ $$T^{3} - 10 T^{2} + 31 T - 29$$
$11$ $$T^{3} + T^{2} - 30 T - 43$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 7 T^{2} + 14 T + 7$$
$19$ $$T^{3} - 11 T^{2} + 10 T + 113$$
$23$ $$T^{3} + 2 T^{2} - 43 T + 83$$
$29$ $$T^{3} + 8 T^{2} + 5 T - 43$$
$31$ $$T^{3} - 8 T^{2} - 23 T + 197$$
$37$ $$T^{3} + 14 T^{2} + 63 T + 91$$
$41$ $$T^{3} + T^{2} - 2T - 1$$
$43$ $$T^{3} - 3 T^{2} - 25 T - 29$$
$47$ $$T^{3} + 9 T^{2} - 120 T - 911$$
$53$ $$T^{3} + 13 T^{2} + 40 T + 29$$
$59$ $$T^{3} + 14T^{2} - 56$$
$61$ $$T^{3} + 13 T^{2} + 12 T - 223$$
$67$ $$T^{3} - 5 T^{2} - 22 T + 97$$
$71$ $$T^{3} + 6 T^{2} - 79 T - 461$$
$73$ $$T^{3} + 18 T^{2} + 101 T + 167$$
$79$ $$T^{3} - 9 T^{2} - 22 T + 169$$
$83$ $$T^{3} + 16 T^{2} + 55 T - 43$$
$89$ $$T^{3} - 5 T^{2} - 8 T - 1$$
$97$ $$T^{3} + 5 T^{2} - 169 T - 1189$$