# Properties

 Label 8112.2.a.cj Level $8112$ Weight $2$ Character orbit 8112.a Self dual yes Analytic conductor $64.775$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1014) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + (\beta_{2} - 3 \beta_1 + 1) q^{5} + (2 \beta_{2} - \beta_1 + 4) q^{7} + q^{9}+O(q^{10})$$ q + q^3 + (b2 - 3*b1 + 1) * q^5 + (2*b2 - b1 + 4) * q^7 + q^9 $$q + q^{3} + (\beta_{2} - 3 \beta_1 + 1) q^{5} + (2 \beta_{2} - \beta_1 + 4) q^{7} + q^{9} + ( - 3 \beta_{2} + 2 \beta_1) q^{11} + (\beta_{2} - 3 \beta_1 + 1) q^{15} + ( - 2 \beta_1 - 2) q^{17} + 4 \beta_{2} q^{19} + (2 \beta_{2} - \beta_1 + 4) q^{21} + (2 \beta_{2} + 2 \beta_1) q^{23} + (4 \beta_{2} - 5 \beta_1 + 9) q^{25} + q^{27} + ( - \beta_{2} - 2 \beta_1 + 4) q^{29} - 5 \beta_{2} q^{31} + ( - 3 \beta_{2} + 2 \beta_1) q^{33} + ( - 11 \beta_1 + 5) q^{35} + (2 \beta_{2} + 4 \beta_1 + 2) q^{37} + ( - 6 \beta_{2} + 4 \beta_1 - 4) q^{41} + (2 \beta_{2} + 2 \beta_1 - 4) q^{43} + (\beta_{2} - 3 \beta_1 + 1) q^{45} - 4 \beta_{2} q^{47} + (9 \beta_{2} - 4 \beta_1 + 11) q^{49} + ( - 2 \beta_1 - 2) q^{51} + ( - 5 \beta_{2} + 3 \beta_1 - 1) q^{53} + (5 \beta_{2} - \beta_1 - 4) q^{55} + 4 \beta_{2} q^{57} + (2 \beta_{2} + 3 \beta_1 - 2) q^{59} + ( - 2 \beta_1 + 8) q^{61} + (2 \beta_{2} - \beta_1 + 4) q^{63} + ( - 2 \beta_{2} + 4 \beta_1) q^{67} + (2 \beta_{2} + 2 \beta_1) q^{69} + (2 \beta_{2} + 2 \beta_1 - 6) q^{71} + ( - 7 \beta_{2} + 3 \beta_1 + 1) q^{73} + (4 \beta_{2} - 5 \beta_1 + 9) q^{75} + ( - \beta_{2} + 2 \beta_1 - 3) q^{77} + (2 \beta_{2} - 5 \beta_1 - 8) q^{79} + q^{81} + (10 \beta_{2} - 7 \beta_1 + 10) q^{83} + (2 \beta_{2} + 4 \beta_1 + 8) q^{85} + ( - \beta_{2} - 2 \beta_1 + 4) q^{87} + (6 \beta_{2} + 2 \beta_1 + 6) q^{89} - 5 \beta_{2} q^{93} + ( - 12 \beta_{2} + 4 \beta_1 - 8) q^{95} + (3 \beta_{2} - 7 \beta_1 + 11) q^{97} + ( - 3 \beta_{2} + 2 \beta_1) q^{99}+O(q^{100})$$ q + q^3 + (b2 - 3*b1 + 1) * q^5 + (2*b2 - b1 + 4) * q^7 + q^9 + (-3*b2 + 2*b1) * q^11 + (b2 - 3*b1 + 1) * q^15 + (-2*b1 - 2) * q^17 + 4*b2 * q^19 + (2*b2 - b1 + 4) * q^21 + (2*b2 + 2*b1) * q^23 + (4*b2 - 5*b1 + 9) * q^25 + q^27 + (-b2 - 2*b1 + 4) * q^29 - 5*b2 * q^31 + (-3*b2 + 2*b1) * q^33 + (-11*b1 + 5) * q^35 + (2*b2 + 4*b1 + 2) * q^37 + (-6*b2 + 4*b1 - 4) * q^41 + (2*b2 + 2*b1 - 4) * q^43 + (b2 - 3*b1 + 1) * q^45 - 4*b2 * q^47 + (9*b2 - 4*b1 + 11) * q^49 + (-2*b1 - 2) * q^51 + (-5*b2 + 3*b1 - 1) * q^53 + (5*b2 - b1 - 4) * q^55 + 4*b2 * q^57 + (2*b2 + 3*b1 - 2) * q^59 + (-2*b1 + 8) * q^61 + (2*b2 - b1 + 4) * q^63 + (-2*b2 + 4*b1) * q^67 + (2*b2 + 2*b1) * q^69 + (2*b2 + 2*b1 - 6) * q^71 + (-7*b2 + 3*b1 + 1) * q^73 + (4*b2 - 5*b1 + 9) * q^75 + (-b2 + 2*b1 - 3) * q^77 + (2*b2 - 5*b1 - 8) * q^79 + q^81 + (10*b2 - 7*b1 + 10) * q^83 + (2*b2 + 4*b1 + 8) * q^85 + (-b2 - 2*b1 + 4) * q^87 + (6*b2 + 2*b1 + 6) * q^89 - 5*b2 * q^93 + (-12*b2 + 4*b1 - 8) * q^95 + (3*b2 - 7*b1 + 11) * q^97 + (-3*b2 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - q^{5} + 9 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - q^5 + 9 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - q^{5} + 9 q^{7} + 3 q^{9} + 5 q^{11} - q^{15} - 8 q^{17} - 4 q^{19} + 9 q^{21} + 18 q^{25} + 3 q^{27} + 11 q^{29} + 5 q^{31} + 5 q^{33} + 4 q^{35} + 8 q^{37} - 2 q^{41} - 12 q^{43} - q^{45} + 4 q^{47} + 20 q^{49} - 8 q^{51} + 5 q^{53} - 18 q^{55} - 4 q^{57} - 5 q^{59} + 22 q^{61} + 9 q^{63} + 6 q^{67} - 18 q^{71} + 13 q^{73} + 18 q^{75} - 6 q^{77} - 31 q^{79} + 3 q^{81} + 13 q^{83} + 26 q^{85} + 11 q^{87} + 14 q^{89} + 5 q^{93} - 8 q^{95} + 23 q^{97} + 5 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - q^5 + 9 * q^7 + 3 * q^9 + 5 * q^11 - q^15 - 8 * q^17 - 4 * q^19 + 9 * q^21 + 18 * q^25 + 3 * q^27 + 11 * q^29 + 5 * q^31 + 5 * q^33 + 4 * q^35 + 8 * q^37 - 2 * q^41 - 12 * q^43 - q^45 + 4 * q^47 + 20 * q^49 - 8 * q^51 + 5 * q^53 - 18 * q^55 - 4 * q^57 - 5 * q^59 + 22 * q^61 + 9 * q^63 + 6 * q^67 - 18 * q^71 + 13 * q^73 + 18 * q^75 - 6 * q^77 - 31 * q^79 + 3 * q^81 + 13 * q^83 + 26 * q^85 + 11 * q^87 + 14 * q^89 + 5 * q^93 - 8 * q^95 + 23 * q^97 + 5 * 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 −3.15883 0 4.69202 0 1.00000 0
1.2 0 1.00000 0 −2.13706 0 −0.0489173 0 1.00000 0
1.3 0 1.00000 0 4.29590 0 4.35690 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.cj 3
4.b odd 2 1 1014.2.a.l 3
12.b even 2 1 3042.2.a.bh 3
13.b even 2 1 8112.2.a.cm 3
52.b odd 2 1 1014.2.a.n yes 3
52.f even 4 2 1014.2.b.f 6
52.i odd 6 2 1014.2.e.l 6
52.j odd 6 2 1014.2.e.n 6
52.l even 12 4 1014.2.i.h 12
156.h even 2 1 3042.2.a.ba 3
156.l odd 4 2 3042.2.b.o 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.l 3 4.b odd 2 1
1014.2.a.n yes 3 52.b odd 2 1
1014.2.b.f 6 52.f even 4 2
1014.2.e.l 6 52.i odd 6 2
1014.2.e.n 6 52.j odd 6 2
1014.2.i.h 12 52.l even 12 4
3042.2.a.ba 3 156.h even 2 1
3042.2.a.bh 3 12.b even 2 1
3042.2.b.o 6 156.l odd 4 2
8112.2.a.cj 3 1.a even 1 1 trivial
8112.2.a.cm 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8112))$$:

 $$T_{5}^{3} + T_{5}^{2} - 16T_{5} - 29$$ T5^3 + T5^2 - 16*T5 - 29 $$T_{7}^{3} - 9T_{7}^{2} + 20T_{7} + 1$$ T7^3 - 9*T7^2 + 20*T7 + 1 $$T_{11}^{3} - 5T_{11}^{2} - 8T_{11} - 1$$ T11^3 - 5*T11^2 - 8*T11 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} + T^{2} - 16 T - 29$$
$7$ $$T^{3} - 9 T^{2} + 20 T + 1$$
$11$ $$T^{3} - 5 T^{2} - 8 T - 1$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 8 T^{2} + 12 T - 8$$
$19$ $$T^{3} + 4 T^{2} - 32 T - 64$$
$23$ $$T^{3} - 28T - 56$$
$29$ $$T^{3} - 11 T^{2} + 24 T + 29$$
$31$ $$T^{3} - 5 T^{2} - 50 T + 125$$
$37$ $$T^{3} - 8 T^{2} - 44 T + 8$$
$41$ $$T^{3} + 2 T^{2} - 64 T - 232$$
$43$ $$T^{3} + 12 T^{2} + 20 T - 104$$
$47$ $$T^{3} - 4 T^{2} - 32 T + 64$$
$53$ $$T^{3} - 5 T^{2} - 36 T - 43$$
$59$ $$T^{3} + 5 T^{2} - 36 T - 167$$
$61$ $$T^{3} - 22 T^{2} + 152 T - 328$$
$67$ $$T^{3} - 6 T^{2} - 16 T + 104$$
$71$ $$T^{3} + 18 T^{2} + 80 T - 8$$
$73$ $$T^{3} - 13 T^{2} - 30 T + 13$$
$79$ $$T^{3} + 31 T^{2} + 276 T + 533$$
$83$ $$T^{3} - 13 T^{2} - 128 T + 1567$$
$89$ $$T^{3} - 14 T^{2} - 56 T + 56$$
$97$ $$T^{3} - 23 T^{2} + 90 T - 97$$