# Properties

 Label 8112.2.a.bb 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8112 = 2^{4} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8112.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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 78) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

comment: embeddings in the coefficient field

gp: mfembed(f)

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.bb 1
4.b odd 2 1 1014.2.a.e 1
12.b even 2 1 3042.2.a.d 1
13.b even 2 1 8112.2.a.x 1
13.e even 6 2 624.2.q.b 2
39.h odd 6 2 1872.2.t.i 2
52.b odd 2 1 1014.2.a.a 1
52.f even 4 2 1014.2.b.a 2
52.i odd 6 2 78.2.e.b 2
52.j odd 6 2 1014.2.e.d 2
52.l even 12 4 1014.2.i.e 4
156.h even 2 1 3042.2.a.m 1
156.l odd 4 2 3042.2.b.d 2
156.r even 6 2 234.2.h.b 2
260.w odd 6 2 1950.2.i.b 2
260.bg even 12 4 1950.2.z.b 4

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T - 2$$
$13$ $$T$$
$17$ $$T - 5$$
$19$ $$T + 2$$
$23$ $$T + 6$$
$29$ $$T + 9$$
$31$ $$T + 4$$
$37$ $$T - 11$$
$41$ $$T + 5$$
$43$ $$T + 10$$
$47$ $$T - 2$$
$53$ $$T + 1$$
$59$ $$T + 8$$
$61$ $$T + 11$$
$67$ $$T - 2$$
$71$ $$T + 14$$
$73$ $$T - 13$$
$79$ $$T - 4$$
$83$ $$T - 6$$
$89$ $$T + 2$$
$97$ $$T - 2$$