# Properties

 Label 8112.2.a.s 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

 $$f(q)$$ $$=$$ $$q + q^{3} - 2 q^{5} - 4 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - 2 * q^5 - 4 * q^7 + q^9 $$q + q^{3} - 2 q^{5} - 4 q^{7} + q^{9} + 4 q^{11} - 2 q^{15} + 2 q^{17} - 4 q^{21} - q^{25} + q^{27} - 10 q^{29} + 4 q^{31} + 4 q^{33} + 8 q^{35} + 2 q^{37} - 6 q^{41} + 12 q^{43} - 2 q^{45} + 9 q^{49} + 2 q^{51} + 6 q^{53} - 8 q^{55} + 12 q^{59} - 2 q^{61} - 4 q^{63} - 8 q^{67} - 2 q^{73} - q^{75} - 16 q^{77} - 8 q^{79} + q^{81} + 4 q^{83} - 4 q^{85} - 10 q^{87} + 2 q^{89} + 4 q^{93} - 10 q^{97} + 4 q^{99}+O(q^{100})$$ q + q^3 - 2 * q^5 - 4 * q^7 + q^9 + 4 * q^11 - 2 * q^15 + 2 * q^17 - 4 * q^21 - q^25 + q^27 - 10 * q^29 + 4 * q^31 + 4 * q^33 + 8 * q^35 + 2 * q^37 - 6 * q^41 + 12 * q^43 - 2 * q^45 + 9 * q^49 + 2 * q^51 + 6 * q^53 - 8 * q^55 + 12 * q^59 - 2 * q^61 - 4 * q^63 - 8 * q^67 - 2 * q^73 - q^75 - 16 * q^77 - 8 * q^79 + q^81 + 4 * q^83 - 4 * q^85 - 10 * q^87 + 2 * q^89 + 4 * q^93 - 10 * 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
 0
0 1.00000 0 −2.00000 0 −4.00000 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.s 1
4.b odd 2 1 507.2.a.a 1
12.b even 2 1 1521.2.a.e 1
13.b even 2 1 624.2.a.i 1
39.d odd 2 1 1872.2.a.h 1
52.b odd 2 1 39.2.a.a 1
52.f even 4 2 507.2.b.a 2
52.i odd 6 2 507.2.e.a 2
52.j odd 6 2 507.2.e.b 2
52.l even 12 4 507.2.j.e 4
104.e even 2 1 2496.2.a.e 1
104.h odd 2 1 2496.2.a.q 1
156.h even 2 1 117.2.a.a 1
156.l odd 4 2 1521.2.b.b 2
260.g odd 2 1 975.2.a.f 1
260.p even 4 2 975.2.c.f 2
312.b odd 2 1 7488.2.a.by 1
312.h even 2 1 7488.2.a.bl 1
364.h even 2 1 1911.2.a.f 1
468.x even 6 2 1053.2.e.d 2
468.bg odd 6 2 1053.2.e.b 2
572.b even 2 1 4719.2.a.c 1
780.d even 2 1 2925.2.a.p 1
780.w odd 4 2 2925.2.c.e 2
1092.d odd 2 1 5733.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 52.b odd 2 1
117.2.a.a 1 156.h even 2 1
507.2.a.a 1 4.b odd 2 1
507.2.b.a 2 52.f even 4 2
507.2.e.a 2 52.i odd 6 2
507.2.e.b 2 52.j odd 6 2
507.2.j.e 4 52.l even 12 4
624.2.a.i 1 13.b even 2 1
975.2.a.f 1 260.g odd 2 1
975.2.c.f 2 260.p even 4 2
1053.2.e.b 2 468.bg odd 6 2
1053.2.e.d 2 468.x even 6 2
1521.2.a.e 1 12.b even 2 1
1521.2.b.b 2 156.l odd 4 2
1872.2.a.h 1 39.d odd 2 1
1911.2.a.f 1 364.h even 2 1
2496.2.a.e 1 104.e even 2 1
2496.2.a.q 1 104.h odd 2 1
2925.2.a.p 1 780.d even 2 1
2925.2.c.e 2 780.w odd 4 2
4719.2.a.c 1 572.b even 2 1
5733.2.a.e 1 1092.d odd 2 1
7488.2.a.bl 1 312.h even 2 1
7488.2.a.by 1 312.b odd 2 1
8112.2.a.s 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8112))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 4$$ T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T + 4$$
$11$ $$T - 4$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 10$$
$31$ $$T - 4$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T - 12$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T + 2$$
$67$ $$T + 8$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T + 8$$
$83$ $$T - 4$$
$89$ $$T - 2$$
$97$ $$T + 10$$