# Properties

 Label 8112.2.a.m Level $8112$ Weight $2$ Character orbit 8112.a Self dual yes Analytic conductor $64.775$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + 3 q^{5} - 2 q^{7} + q^{9}+O(q^{10})$$ q - q^3 + 3 * q^5 - 2 * q^7 + q^9 $$q - q^{3} + 3 q^{5} - 2 q^{7} + q^{9} - 6 q^{11} - 3 q^{15} - 3 q^{17} - 2 q^{19} + 2 q^{21} + 6 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} + 4 q^{31} + 6 q^{33} - 6 q^{35} - 7 q^{37} - 3 q^{41} + 10 q^{43} + 3 q^{45} - 6 q^{47} - 3 q^{49} + 3 q^{51} + 3 q^{53} - 18 q^{55} + 2 q^{57} - 7 q^{61} - 2 q^{63} + 10 q^{67} - 6 q^{69} - 6 q^{71} - 13 q^{73} - 4 q^{75} + 12 q^{77} + 4 q^{79} + q^{81} + 6 q^{83} - 9 q^{85} - 3 q^{87} + 18 q^{89} - 4 q^{93} - 6 q^{95} + 14 q^{97} - 6 q^{99}+O(q^{100})$$ q - q^3 + 3 * q^5 - 2 * q^7 + q^9 - 6 * q^11 - 3 * q^15 - 3 * q^17 - 2 * q^19 + 2 * q^21 + 6 * q^23 + 4 * q^25 - q^27 + 3 * q^29 + 4 * q^31 + 6 * q^33 - 6 * q^35 - 7 * q^37 - 3 * q^41 + 10 * q^43 + 3 * q^45 - 6 * q^47 - 3 * q^49 + 3 * q^51 + 3 * q^53 - 18 * q^55 + 2 * q^57 - 7 * q^61 - 2 * q^63 + 10 * q^67 - 6 * q^69 - 6 * q^71 - 13 * q^73 - 4 * q^75 + 12 * q^77 + 4 * q^79 + q^81 + 6 * q^83 - 9 * q^85 - 3 * q^87 + 18 * q^89 - 4 * q^93 - 6 * q^95 + 14 * q^97 - 6 * 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
 0
0 −1.00000 0 3.00000 0 −2.00000 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.m 1
4.b odd 2 1 1014.2.a.c 1
12.b even 2 1 3042.2.a.i 1
13.b even 2 1 8112.2.a.c 1
13.c even 3 2 624.2.q.g 2
39.i odd 6 2 1872.2.t.c 2
52.b odd 2 1 1014.2.a.f 1
52.f even 4 2 1014.2.b.c 2
52.i odd 6 2 1014.2.e.a 2
52.j odd 6 2 78.2.e.a 2
52.l even 12 4 1014.2.i.b 4
156.h even 2 1 3042.2.a.h 1
156.l odd 4 2 3042.2.b.h 2
156.p even 6 2 234.2.h.a 2
260.v odd 6 2 1950.2.i.m 2
260.bj even 12 4 1950.2.z.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 52.j odd 6 2
234.2.h.a 2 156.p even 6 2
624.2.q.g 2 13.c even 3 2
1014.2.a.c 1 4.b odd 2 1
1014.2.a.f 1 52.b odd 2 1
1014.2.b.c 2 52.f even 4 2
1014.2.e.a 2 52.i odd 6 2
1014.2.i.b 4 52.l even 12 4
1872.2.t.c 2 39.i odd 6 2
1950.2.i.m 2 260.v odd 6 2
1950.2.z.g 4 260.bj even 12 4
3042.2.a.h 1 156.h even 2 1
3042.2.a.i 1 12.b even 2 1
3042.2.b.h 2 156.l odd 4 2
8112.2.a.c 1 13.b even 2 1
8112.2.a.m 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8112))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 6$$ T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 3$$
$7$ $$T + 2$$
$11$ $$T + 6$$
$13$ $$T$$
$17$ $$T + 3$$
$19$ $$T + 2$$
$23$ $$T - 6$$
$29$ $$T - 3$$
$31$ $$T - 4$$
$37$ $$T + 7$$
$41$ $$T + 3$$
$43$ $$T - 10$$
$47$ $$T + 6$$
$53$ $$T - 3$$
$59$ $$T$$
$61$ $$T + 7$$
$67$ $$T - 10$$
$71$ $$T + 6$$
$73$ $$T + 13$$
$79$ $$T - 4$$
$83$ $$T - 6$$
$89$ $$T - 18$$
$97$ $$T - 14$$