# Properties

 Label 112.6.a.d 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 14) 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)

 $$f(q)$$ $$=$$ $$q - 8 q^{3} + 10 q^{5} + 49 q^{7} - 179 q^{9}+O(q^{10})$$ q - 8 * q^3 + 10 * q^5 + 49 * q^7 - 179 * q^9 $$q - 8 q^{3} + 10 q^{5} + 49 q^{7} - 179 q^{9} + 340 q^{11} - 294 q^{13} - 80 q^{15} + 1226 q^{17} - 2432 q^{19} - 392 q^{21} - 2000 q^{23} - 3025 q^{25} + 3376 q^{27} - 6746 q^{29} - 8856 q^{31} - 2720 q^{33} + 490 q^{35} + 9182 q^{37} + 2352 q^{39} - 14574 q^{41} - 8108 q^{43} - 1790 q^{45} + 312 q^{47} + 2401 q^{49} - 9808 q^{51} - 14634 q^{53} + 3400 q^{55} + 19456 q^{57} + 27656 q^{59} + 34338 q^{61} - 8771 q^{63} - 2940 q^{65} - 12316 q^{67} + 16000 q^{69} - 36920 q^{71} - 61718 q^{73} + 24200 q^{75} + 16660 q^{77} + 64752 q^{79} + 16489 q^{81} + 77056 q^{83} + 12260 q^{85} + 53968 q^{87} - 8166 q^{89} - 14406 q^{91} + 70848 q^{93} - 24320 q^{95} + 20650 q^{97} - 60860 q^{99}+O(q^{100})$$ q - 8 * q^3 + 10 * q^5 + 49 * q^7 - 179 * q^9 + 340 * q^11 - 294 * q^13 - 80 * q^15 + 1226 * q^17 - 2432 * q^19 - 392 * q^21 - 2000 * q^23 - 3025 * q^25 + 3376 * q^27 - 6746 * q^29 - 8856 * q^31 - 2720 * q^33 + 490 * q^35 + 9182 * q^37 + 2352 * q^39 - 14574 * q^41 - 8108 * q^43 - 1790 * q^45 + 312 * q^47 + 2401 * q^49 - 9808 * q^51 - 14634 * q^53 + 3400 * q^55 + 19456 * q^57 + 27656 * q^59 + 34338 * q^61 - 8771 * q^63 - 2940 * q^65 - 12316 * q^67 + 16000 * q^69 - 36920 * q^71 - 61718 * q^73 + 24200 * q^75 + 16660 * q^77 + 64752 * q^79 + 16489 * q^81 + 77056 * q^83 + 12260 * q^85 + 53968 * q^87 - 8166 * q^89 - 14406 * q^91 + 70848 * q^93 - 24320 * q^95 + 20650 * q^97 - 60860 * 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 −8.00000 0 10.0000 0 49.0000 0 −179.000 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.d 1
3.b odd 2 1 1008.6.a.n 1
4.b odd 2 1 14.6.a.b 1
7.b odd 2 1 784.6.a.h 1
8.b even 2 1 448.6.a.k 1
8.d odd 2 1 448.6.a.f 1
12.b even 2 1 126.6.a.c 1
20.d odd 2 1 350.6.a.b 1
20.e even 4 2 350.6.c.f 2
28.d even 2 1 98.6.a.b 1
28.f even 6 2 98.6.c.b 2
28.g odd 6 2 98.6.c.a 2
84.h odd 2 1 882.6.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 4.b odd 2 1
98.6.a.b 1 28.d even 2 1
98.6.c.a 2 28.g odd 6 2
98.6.c.b 2 28.f even 6 2
112.6.a.d 1 1.a even 1 1 trivial
126.6.a.c 1 12.b even 2 1
350.6.a.b 1 20.d odd 2 1
350.6.c.f 2 20.e even 4 2
448.6.a.f 1 8.d odd 2 1
448.6.a.k 1 8.b even 2 1
784.6.a.h 1 7.b odd 2 1
882.6.a.g 1 84.h odd 2 1
1008.6.a.n 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 8$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(112))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 8$$
$5$ $$T - 10$$
$7$ $$T - 49$$
$11$ $$T - 340$$
$13$ $$T + 294$$
$17$ $$T - 1226$$
$19$ $$T + 2432$$
$23$ $$T + 2000$$
$29$ $$T + 6746$$
$31$ $$T + 8856$$
$37$ $$T - 9182$$
$41$ $$T + 14574$$
$43$ $$T + 8108$$
$47$ $$T - 312$$
$53$ $$T + 14634$$
$59$ $$T - 27656$$
$61$ $$T - 34338$$
$67$ $$T + 12316$$
$71$ $$T + 36920$$
$73$ $$T + 61718$$
$79$ $$T - 64752$$
$83$ $$T - 77056$$
$89$ $$T + 8166$$
$97$ $$T - 20650$$