# Properties

 Label 112.6.a.f 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 56) 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)

 $$f(q)$$ $$=$$ $$q + 6 q^{3} + 4 q^{5} - 49 q^{7} - 207 q^{9}+O(q^{10})$$ q + 6 * q^3 + 4 * q^5 - 49 * q^7 - 207 * q^9 $$q + 6 q^{3} + 4 q^{5} - 49 q^{7} - 207 q^{9} + 240 q^{11} - 744 q^{13} + 24 q^{15} - 1042 q^{17} + 986 q^{19} - 294 q^{21} - 184 q^{23} - 3109 q^{25} - 2700 q^{27} - 734 q^{29} - 5140 q^{31} + 1440 q^{33} - 196 q^{35} - 6054 q^{37} - 4464 q^{39} + 7598 q^{41} - 13016 q^{43} - 828 q^{45} - 14668 q^{47} + 2401 q^{49} - 6252 q^{51} - 14522 q^{53} + 960 q^{55} + 5916 q^{57} + 13362 q^{59} + 9676 q^{61} + 10143 q^{63} - 2976 q^{65} + 62124 q^{67} - 1104 q^{69} + 2112 q^{71} - 28910 q^{73} - 18654 q^{75} - 11760 q^{77} + 101768 q^{79} + 34101 q^{81} + 23922 q^{83} - 4168 q^{85} - 4404 q^{87} + 141674 q^{89} + 36456 q^{91} - 30840 q^{93} + 3944 q^{95} + 99982 q^{97} - 49680 q^{99}+O(q^{100})$$ q + 6 * q^3 + 4 * q^5 - 49 * q^7 - 207 * q^9 + 240 * q^11 - 744 * q^13 + 24 * q^15 - 1042 * q^17 + 986 * q^19 - 294 * q^21 - 184 * q^23 - 3109 * q^25 - 2700 * q^27 - 734 * q^29 - 5140 * q^31 + 1440 * q^33 - 196 * q^35 - 6054 * q^37 - 4464 * q^39 + 7598 * q^41 - 13016 * q^43 - 828 * q^45 - 14668 * q^47 + 2401 * q^49 - 6252 * q^51 - 14522 * q^53 + 960 * q^55 + 5916 * q^57 + 13362 * q^59 + 9676 * q^61 + 10143 * q^63 - 2976 * q^65 + 62124 * q^67 - 1104 * q^69 + 2112 * q^71 - 28910 * q^73 - 18654 * q^75 - 11760 * q^77 + 101768 * q^79 + 34101 * q^81 + 23922 * q^83 - 4168 * q^85 - 4404 * q^87 + 141674 * q^89 + 36456 * q^91 - 30840 * q^93 + 3944 * q^95 + 99982 * q^97 - 49680 * 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 6.00000 0 4.00000 0 −49.0000 0 −207.000 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.f 1
3.b odd 2 1 1008.6.a.p 1
4.b odd 2 1 56.6.a.a 1
7.b odd 2 1 784.6.a.e 1
8.b even 2 1 448.6.a.g 1
8.d odd 2 1 448.6.a.j 1
12.b even 2 1 504.6.a.e 1
28.d even 2 1 392.6.a.c 1
28.f even 6 2 392.6.i.c 2
28.g odd 6 2 392.6.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.a 1 4.b odd 2 1
112.6.a.f 1 1.a even 1 1 trivial
392.6.a.c 1 28.d even 2 1
392.6.i.c 2 28.f even 6 2
392.6.i.d 2 28.g odd 6 2
448.6.a.g 1 8.b even 2 1
448.6.a.j 1 8.d odd 2 1
504.6.a.e 1 12.b even 2 1
784.6.a.e 1 7.b odd 2 1
1008.6.a.p 1 3.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 6$$
$5$ $$T - 4$$
$7$ $$T + 49$$
$11$ $$T - 240$$
$13$ $$T + 744$$
$17$ $$T + 1042$$
$19$ $$T - 986$$
$23$ $$T + 184$$
$29$ $$T + 734$$
$31$ $$T + 5140$$
$37$ $$T + 6054$$
$41$ $$T - 7598$$
$43$ $$T + 13016$$
$47$ $$T + 14668$$
$53$ $$T + 14522$$
$59$ $$T - 13362$$
$61$ $$T - 9676$$
$67$ $$T - 62124$$
$71$ $$T - 2112$$
$73$ $$T + 28910$$
$79$ $$T - 101768$$
$83$ $$T - 23922$$
$89$ $$T - 141674$$
$97$ $$T - 99982$$