# Properties

 Label 8112.2.a.bk Level $8112$ Weight $2$ Character orbit 8112.a Self dual yes Analytic conductor $64.775$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( - \beta - 1) q^{5} + ( - \beta + 2) q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (-b - 1) * q^5 + (-b + 2) * q^7 + q^9 $$q - q^{3} + ( - \beta - 1) q^{5} + ( - \beta + 2) q^{7} + q^{9} + 2 q^{11} + (\beta + 1) q^{15} + ( - \beta + 1) q^{17} + ( - 2 \beta - 2) q^{19} + (\beta - 2) q^{21} - 2 q^{23} + 3 \beta q^{25} - q^{27} + (3 \beta - 1) q^{29} - \beta q^{31} - 2 q^{33} + 2 q^{35} + (\beta + 5) q^{37} + ( - \beta + 1) q^{41} + ( - \beta - 2) q^{43} + ( - \beta - 1) q^{45} + ( - 4 \beta + 2) q^{47} + ( - 3 \beta + 1) q^{49} + (\beta - 1) q^{51} + ( - 3 \beta + 7) q^{53} + ( - 2 \beta - 2) q^{55} + (2 \beta + 2) q^{57} + ( - 2 \beta + 8) q^{59} + ( - 2 \beta + 9) q^{61} + ( - \beta + 2) q^{63} + ( - \beta - 2) q^{67} + 2 q^{69} - 14 q^{71} + ( - 2 \beta - 5) q^{73} - 3 \beta q^{75} + ( - 2 \beta + 4) q^{77} + (\beta - 8) q^{79} + q^{81} + ( - 2 \beta + 6) q^{83} + (\beta + 3) q^{85} + ( - 3 \beta + 1) q^{87} + ( - 2 \beta + 10) q^{89} + \beta q^{93} + (6 \beta + 10) q^{95} + ( - \beta - 6) q^{97} + 2 q^{99} +O(q^{100})$$ q - q^3 + (-b - 1) * q^5 + (-b + 2) * q^7 + q^9 + 2 * q^11 + (b + 1) * q^15 + (-b + 1) * q^17 + (-2*b - 2) * q^19 + (b - 2) * q^21 - 2 * q^23 + 3*b * q^25 - q^27 + (3*b - 1) * q^29 - b * q^31 - 2 * q^33 + 2 * q^35 + (b + 5) * q^37 + (-b + 1) * q^41 + (-b - 2) * q^43 + (-b - 1) * q^45 + (-4*b + 2) * q^47 + (-3*b + 1) * q^49 + (b - 1) * q^51 + (-3*b + 7) * q^53 + (-2*b - 2) * q^55 + (2*b + 2) * q^57 + (-2*b + 8) * q^59 + (-2*b + 9) * q^61 + (-b + 2) * q^63 + (-b - 2) * q^67 + 2 * q^69 - 14 * q^71 + (-2*b - 5) * q^73 - 3*b * q^75 + (-2*b + 4) * q^77 + (b - 8) * q^79 + q^81 + (-2*b + 6) * q^83 + (b + 3) * q^85 + (-3*b + 1) * q^87 + (-2*b + 10) * q^89 + b * q^93 + (6*b + 10) * q^95 + (-b - 6) * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 2 * q^9 $$2 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 2 q^{9} + 4 q^{11} + 3 q^{15} + q^{17} - 6 q^{19} - 3 q^{21} - 4 q^{23} + 3 q^{25} - 2 q^{27} + q^{29} - q^{31} - 4 q^{33} + 4 q^{35} + 11 q^{37} + q^{41} - 5 q^{43} - 3 q^{45} - q^{49} - q^{51} + 11 q^{53} - 6 q^{55} + 6 q^{57} + 14 q^{59} + 16 q^{61} + 3 q^{63} - 5 q^{67} + 4 q^{69} - 28 q^{71} - 12 q^{73} - 3 q^{75} + 6 q^{77} - 15 q^{79} + 2 q^{81} + 10 q^{83} + 7 q^{85} - q^{87} + 18 q^{89} + q^{93} + 26 q^{95} - 13 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 3 * q^5 + 3 * q^7 + 2 * q^9 + 4 * q^11 + 3 * q^15 + q^17 - 6 * q^19 - 3 * q^21 - 4 * q^23 + 3 * q^25 - 2 * q^27 + q^29 - q^31 - 4 * q^33 + 4 * q^35 + 11 * q^37 + q^41 - 5 * q^43 - 3 * q^45 - q^49 - q^51 + 11 * q^53 - 6 * q^55 + 6 * q^57 + 14 * q^59 + 16 * q^61 + 3 * q^63 - 5 * q^67 + 4 * q^69 - 28 * q^71 - 12 * q^73 - 3 * q^75 + 6 * q^77 - 15 * q^79 + 2 * q^81 + 10 * q^83 + 7 * q^85 - q^87 + 18 * q^89 + q^93 + 26 * q^95 - 13 * q^97 + 4 * q^99

## 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
 2.56155 −1.56155
0 −1.00000 0 −3.56155 0 −0.561553 0 1.00000 0
1.2 0 −1.00000 0 0.561553 0 3.56155 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.bk 2
4.b odd 2 1 507.2.a.g 2
12.b even 2 1 1521.2.a.g 2
13.b even 2 1 8112.2.a.bo 2
13.c even 3 2 624.2.q.h 4
39.i odd 6 2 1872.2.t.r 4
52.b odd 2 1 507.2.a.d 2
52.f even 4 2 507.2.b.d 4
52.i odd 6 2 507.2.e.g 4
52.j odd 6 2 39.2.e.b 4
52.l even 12 4 507.2.j.g 8
156.h even 2 1 1521.2.a.m 2
156.l odd 4 2 1521.2.b.h 4
156.p even 6 2 117.2.g.c 4
260.v odd 6 2 975.2.i.k 4
260.bj even 12 4 975.2.bb.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 52.j odd 6 2
117.2.g.c 4 156.p even 6 2
507.2.a.d 2 52.b odd 2 1
507.2.a.g 2 4.b odd 2 1
507.2.b.d 4 52.f even 4 2
507.2.e.g 4 52.i odd 6 2
507.2.j.g 8 52.l even 12 4
624.2.q.h 4 13.c even 3 2
975.2.i.k 4 260.v odd 6 2
975.2.bb.i 8 260.bj even 12 4
1521.2.a.g 2 12.b even 2 1
1521.2.a.m 2 156.h even 2 1
1521.2.b.h 4 156.l odd 4 2
1872.2.t.r 4 39.i odd 6 2
8112.2.a.bk 2 1.a even 1 1 trivial
8112.2.a.bo 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8112))$$:

 $$T_{5}^{2} + 3T_{5} - 2$$ T5^2 + 3*T5 - 2 $$T_{7}^{2} - 3T_{7} - 2$$ T7^2 - 3*T7 - 2 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 3T - 2$$
$7$ $$T^{2} - 3T - 2$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - T - 4$$
$19$ $$T^{2} + 6T - 8$$
$23$ $$(T + 2)^{2}$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} + T - 4$$
$37$ $$T^{2} - 11T + 26$$
$41$ $$T^{2} - T - 4$$
$43$ $$T^{2} + 5T + 2$$
$47$ $$T^{2} - 68$$
$53$ $$T^{2} - 11T - 8$$
$59$ $$T^{2} - 14T + 32$$
$61$ $$T^{2} - 16T + 47$$
$67$ $$T^{2} + 5T + 2$$
$71$ $$(T + 14)^{2}$$
$73$ $$T^{2} + 12T + 19$$
$79$ $$T^{2} + 15T + 52$$
$83$ $$T^{2} - 10T + 8$$
$89$ $$T^{2} - 18T + 64$$
$97$ $$T^{2} + 13T + 38$$