# Properties

 Label 8112.2.a.bv Level $8112$ Weight $2$ Character orbit 8112.a Self dual yes Analytic conductor $64.775$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta q^{7} + q^{9} +O(q^{10})$$ q + q^3 - b * q^7 + q^9 $$q + q^{3} - \beta q^{7} + q^{9} - \beta q^{11} + 6 q^{17} - \beta q^{19} - \beta q^{21} - 5 q^{25} + q^{27} + 6 q^{29} + \beta q^{31} - \beta q^{33} - 2 \beta q^{37} - 2 \beta q^{41} - 4 q^{43} + \beta q^{47} + 5 q^{49} + 6 q^{51} + 6 q^{53} - \beta q^{57} + 3 \beta q^{59} - 2 q^{61} - \beta q^{63} + 3 \beta q^{67} - \beta q^{71} - 5 q^{75} + 12 q^{77} + 8 q^{79} + q^{81} + \beta q^{83} + 6 q^{87} + 2 \beta q^{89} + \beta q^{93} - 4 \beta q^{97} - \beta q^{99} +O(q^{100})$$ q + q^3 - b * q^7 + q^9 - b * q^11 + 6 * q^17 - b * q^19 - b * q^21 - 5 * q^25 + q^27 + 6 * q^29 + b * q^31 - b * q^33 - 2*b * q^37 - 2*b * q^41 - 4 * q^43 + b * q^47 + 5 * q^49 + 6 * q^51 + 6 * q^53 - b * q^57 + 3*b * q^59 - 2 * q^61 - b * q^63 + 3*b * q^67 - b * q^71 - 5 * q^75 + 12 * q^77 + 8 * q^79 + q^81 + b * q^83 + 6 * q^87 + 2*b * q^89 + b * q^93 - 4*b * q^97 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{9} + 12 q^{17} - 10 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{43} + 10 q^{49} + 12 q^{51} + 12 q^{53} - 4 q^{61} - 10 q^{75} + 24 q^{77} + 16 q^{79} + 2 q^{81} + 12 q^{87}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^9 + 12 * q^17 - 10 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^43 + 10 * q^49 + 12 * q^51 + 12 * q^53 - 4 * q^61 - 10 * q^75 + 24 * q^77 + 16 * q^79 + 2 * q^81 + 12 * q^87

## 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 0 −3.46410 0 1.00000 0
1.2 0 1.00000 0 0 0 3.46410 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8112.2.a.bv 2
4.b odd 2 1 507.2.a.f 2
12.b even 2 1 1521.2.a.l 2
13.b even 2 1 inner 8112.2.a.bv 2
13.d odd 4 2 624.2.c.e 2
39.f even 4 2 1872.2.c.e 2
52.b odd 2 1 507.2.a.f 2
52.f even 4 2 39.2.b.a 2
52.i odd 6 2 507.2.e.e 4
52.j odd 6 2 507.2.e.e 4
52.l even 12 2 507.2.j.a 2
52.l even 12 2 507.2.j.c 2
104.j odd 4 2 2496.2.c.d 2
104.m even 4 2 2496.2.c.k 2
156.h even 2 1 1521.2.a.l 2
156.l odd 4 2 117.2.b.a 2
260.l odd 4 2 975.2.h.f 4
260.s odd 4 2 975.2.h.f 4
260.u even 4 2 975.2.b.d 2
364.p odd 4 2 1911.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 52.f even 4 2
117.2.b.a 2 156.l odd 4 2
507.2.a.f 2 4.b odd 2 1
507.2.a.f 2 52.b odd 2 1
507.2.e.e 4 52.i odd 6 2
507.2.e.e 4 52.j odd 6 2
507.2.j.a 2 52.l even 12 2
507.2.j.c 2 52.l even 12 2
624.2.c.e 2 13.d odd 4 2
975.2.b.d 2 260.u even 4 2
975.2.h.f 4 260.l odd 4 2
975.2.h.f 4 260.s odd 4 2
1521.2.a.l 2 12.b even 2 1
1521.2.a.l 2 156.h even 2 1
1872.2.c.e 2 39.f even 4 2
1911.2.c.d 2 364.p odd 4 2
2496.2.c.d 2 104.j odd 4 2
2496.2.c.k 2 104.m even 4 2
8112.2.a.bv 2 1.a even 1 1 trivial
8112.2.a.bv 2 13.b even 2 1 inner

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{11}^{2} - 12$$ T11^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2} - 12$$
$13$ $$T^{2}$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} - 12$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 12$$
$37$ $$T^{2} - 48$$
$41$ $$T^{2} - 48$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 12$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 108$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} - 108$$
$71$ $$T^{2} - 12$$
$73$ $$T^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 12$$
$89$ $$T^{2} - 48$$
$97$ $$T^{2} - 192$$