# Properties

 Label 112.6.a.g Level $112$ Weight $6$ Character orbit 112.a Self dual yes Analytic conductor $17.963$ 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] = [112,6,Mod(1,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("112.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 112.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: $$17.9629878191$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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)

 $$f(q)$$ $$=$$ $$q + 14 q^{3} - 56 q^{5} + 49 q^{7} - 47 q^{9}+O(q^{10})$$ q + 14 * q^3 - 56 * q^5 + 49 * q^7 - 47 * q^9 $$q + 14 q^{3} - 56 q^{5} + 49 q^{7} - 47 q^{9} - 232 q^{11} - 140 q^{13} - 784 q^{15} - 1722 q^{17} + 98 q^{19} + 686 q^{21} - 1824 q^{23} + 11 q^{25} - 4060 q^{27} + 3418 q^{29} + 7644 q^{31} - 3248 q^{33} - 2744 q^{35} - 10398 q^{37} - 1960 q^{39} - 17962 q^{41} - 10880 q^{43} + 2632 q^{45} - 9324 q^{47} + 2401 q^{49} - 24108 q^{51} + 2262 q^{53} + 12992 q^{55} + 1372 q^{57} + 2730 q^{59} + 25648 q^{61} - 2303 q^{63} + 7840 q^{65} + 48404 q^{67} - 25536 q^{69} + 58560 q^{71} + 68082 q^{73} + 154 q^{75} - 11368 q^{77} - 31784 q^{79} - 45419 q^{81} + 20538 q^{83} + 96432 q^{85} + 47852 q^{87} - 50582 q^{89} - 6860 q^{91} + 107016 q^{93} - 5488 q^{95} - 58506 q^{97} + 10904 q^{99}+O(q^{100})$$ q + 14 * q^3 - 56 * q^5 + 49 * q^7 - 47 * q^9 - 232 * q^11 - 140 * q^13 - 784 * q^15 - 1722 * q^17 + 98 * q^19 + 686 * q^21 - 1824 * q^23 + 11 * q^25 - 4060 * q^27 + 3418 * q^29 + 7644 * q^31 - 3248 * q^33 - 2744 * q^35 - 10398 * q^37 - 1960 * q^39 - 17962 * q^41 - 10880 * q^43 + 2632 * q^45 - 9324 * q^47 + 2401 * q^49 - 24108 * q^51 + 2262 * q^53 + 12992 * q^55 + 1372 * q^57 + 2730 * q^59 + 25648 * q^61 - 2303 * q^63 + 7840 * q^65 + 48404 * q^67 - 25536 * q^69 + 58560 * q^71 + 68082 * q^73 + 154 * q^75 - 11368 * q^77 - 31784 * q^79 - 45419 * q^81 + 20538 * q^83 + 96432 * q^85 + 47852 * q^87 - 50582 * q^89 - 6860 * q^91 + 107016 * q^93 - 5488 * q^95 - 58506 * q^97 + 10904 * 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 14.0000 0 −56.0000 0 49.0000 0 −47.0000 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
$$2$$ $$-1$$
$$7$$ $$-1$$

## 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 112.6.a.g 1
3.b odd 2 1 1008.6.a.y 1
4.b odd 2 1 7.6.a.a 1
7.b odd 2 1 784.6.a.c 1
8.b even 2 1 448.6.a.c 1
8.d odd 2 1 448.6.a.m 1
12.b even 2 1 63.6.a.e 1
20.d odd 2 1 175.6.a.b 1
20.e even 4 2 175.6.b.a 2
28.d even 2 1 49.6.a.a 1
28.f even 6 2 49.6.c.b 2
28.g odd 6 2 49.6.c.c 2
44.c even 2 1 847.6.a.b 1
84.h odd 2 1 441.6.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 4.b odd 2 1
49.6.a.a 1 28.d even 2 1
49.6.c.b 2 28.f even 6 2
49.6.c.c 2 28.g odd 6 2
63.6.a.e 1 12.b even 2 1
112.6.a.g 1 1.a even 1 1 trivial
175.6.a.b 1 20.d odd 2 1
175.6.b.a 2 20.e even 4 2
441.6.a.k 1 84.h odd 2 1
448.6.a.c 1 8.b even 2 1
448.6.a.m 1 8.d odd 2 1
784.6.a.c 1 7.b odd 2 1
847.6.a.b 1 44.c even 2 1
1008.6.a.y 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 14$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(112))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 14$$
$5$ $$T + 56$$
$7$ $$T - 49$$
$11$ $$T + 232$$
$13$ $$T + 140$$
$17$ $$T + 1722$$
$19$ $$T - 98$$
$23$ $$T + 1824$$
$29$ $$T - 3418$$
$31$ $$T - 7644$$
$37$ $$T + 10398$$
$41$ $$T + 17962$$
$43$ $$T + 10880$$
$47$ $$T + 9324$$
$53$ $$T - 2262$$
$59$ $$T - 2730$$
$61$ $$T - 25648$$
$67$ $$T - 48404$$
$71$ $$T - 58560$$
$73$ $$T - 68082$$
$79$ $$T + 31784$$
$83$ $$T - 20538$$
$89$ $$T + 50582$$
$97$ $$T + 58506$$