# Properties

 Label 8112.2.a.w Level $8112$ Weight $2$ Character orbit 8112.a Self dual yes Analytic conductor $64.775$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} - 2 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - q^5 - 2 * q^7 + q^9 $$q + q^{3} - q^{5} - 2 q^{7} + q^{9} + 2 q^{11} - q^{15} - 7 q^{17} + 6 q^{19} - 2 q^{21} + 6 q^{23} - 4 q^{25} + q^{27} - q^{29} - 4 q^{31} + 2 q^{33} + 2 q^{35} + q^{37} + 9 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} - 3 q^{49} - 7 q^{51} - 9 q^{53} - 2 q^{55} + 6 q^{57} + q^{61} - 2 q^{63} + 2 q^{67} + 6 q^{69} - 6 q^{71} + 11 q^{73} - 4 q^{75} - 4 q^{77} + 4 q^{79} + q^{81} + 14 q^{83} + 7 q^{85} - q^{87} - 14 q^{89} - 4 q^{93} - 6 q^{95} - 2 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^3 - q^5 - 2 * q^7 + q^9 + 2 * q^11 - q^15 - 7 * q^17 + 6 * q^19 - 2 * q^21 + 6 * q^23 - 4 * q^25 + q^27 - q^29 - 4 * q^31 + 2 * q^33 + 2 * q^35 + q^37 + 9 * q^41 - 6 * q^43 - q^45 - 6 * q^47 - 3 * q^49 - 7 * q^51 - 9 * q^53 - 2 * q^55 + 6 * q^57 + q^61 - 2 * q^63 + 2 * q^67 + 6 * q^69 - 6 * q^71 + 11 * q^73 - 4 * q^75 - 4 * q^77 + 4 * q^79 + q^81 + 14 * q^83 + 7 * q^85 - q^87 - 14 * q^89 - 4 * q^93 - 6 * q^95 - 2 * q^97 + 2 * 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 −1.00000 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.w 1
4.b odd 2 1 507.2.a.c 1
12.b even 2 1 1521.2.a.a 1
13.b even 2 1 8112.2.a.bc 1
13.c even 3 2 624.2.q.c 2
39.i odd 6 2 1872.2.t.j 2
52.b odd 2 1 507.2.a.b 1
52.f even 4 2 507.2.b.b 2
52.i odd 6 2 507.2.e.c 2
52.j odd 6 2 39.2.e.a 2
52.l even 12 4 507.2.j.d 4
156.h even 2 1 1521.2.a.d 1
156.l odd 4 2 1521.2.b.c 2
156.p even 6 2 117.2.g.b 2
260.v odd 6 2 975.2.i.f 2
260.bj even 12 4 975.2.bb.d 4

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

 $$T_{5} + 1$$ T5 + 1 $$T_{7} + 2$$ T7 + 2 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T + 2$$
$11$ $$T - 2$$
$13$ $$T$$
$17$ $$T + 7$$
$19$ $$T - 6$$
$23$ $$T - 6$$
$29$ $$T + 1$$
$31$ $$T + 4$$
$37$ $$T - 1$$
$41$ $$T - 9$$
$43$ $$T + 6$$
$47$ $$T + 6$$
$53$ $$T + 9$$
$59$ $$T$$
$61$ $$T - 1$$
$67$ $$T - 2$$
$71$ $$T + 6$$
$73$ $$T - 11$$
$79$ $$T - 4$$
$83$ $$T - 14$$
$89$ $$T + 14$$
$97$ $$T + 2$$