# Properties

 Label 112.6.a.c Level $112$ Weight $6$ Character orbit 112.a Self dual yes Analytic conductor $17.963$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 10 q^{3} + 84 q^{5} - 49 q^{7} - 143 q^{9}+O(q^{10})$$ q - 10 * q^3 + 84 * q^5 - 49 * q^7 - 143 * q^9 $$q - 10 q^{3} + 84 q^{5} - 49 q^{7} - 143 q^{9} + 336 q^{11} + 584 q^{13} - 840 q^{15} - 1458 q^{17} - 470 q^{19} + 490 q^{21} + 4200 q^{23} + 3931 q^{25} + 3860 q^{27} + 4866 q^{29} + 7372 q^{31} - 3360 q^{33} - 4116 q^{35} + 14330 q^{37} - 5840 q^{39} + 6222 q^{41} - 3704 q^{43} - 12012 q^{45} + 1812 q^{47} + 2401 q^{49} + 14580 q^{51} - 37242 q^{53} + 28224 q^{55} + 4700 q^{57} - 34302 q^{59} + 24476 q^{61} + 7007 q^{63} + 49056 q^{65} + 17452 q^{67} - 42000 q^{69} - 28224 q^{71} + 3602 q^{73} - 39310 q^{75} - 16464 q^{77} - 42872 q^{79} - 3851 q^{81} + 35202 q^{83} - 122472 q^{85} - 48660 q^{87} + 26730 q^{89} - 28616 q^{91} - 73720 q^{93} - 39480 q^{95} - 16978 q^{97} - 48048 q^{99}+O(q^{100})$$ q - 10 * q^3 + 84 * q^5 - 49 * q^7 - 143 * q^9 + 336 * q^11 + 584 * q^13 - 840 * q^15 - 1458 * q^17 - 470 * q^19 + 490 * q^21 + 4200 * q^23 + 3931 * q^25 + 3860 * q^27 + 4866 * q^29 + 7372 * q^31 - 3360 * q^33 - 4116 * q^35 + 14330 * q^37 - 5840 * q^39 + 6222 * q^41 - 3704 * q^43 - 12012 * q^45 + 1812 * q^47 + 2401 * q^49 + 14580 * q^51 - 37242 * q^53 + 28224 * q^55 + 4700 * q^57 - 34302 * q^59 + 24476 * q^61 + 7007 * q^63 + 49056 * q^65 + 17452 * q^67 - 42000 * q^69 - 28224 * q^71 + 3602 * q^73 - 39310 * q^75 - 16464 * q^77 - 42872 * q^79 - 3851 * q^81 + 35202 * q^83 - 122472 * q^85 - 48660 * q^87 + 26730 * q^89 - 28616 * q^91 - 73720 * q^93 - 39480 * q^95 - 16978 * q^97 - 48048 * 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 −10.0000 0 84.0000 0 −49.0000 0 −143.000 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.c 1
3.b odd 2 1 1008.6.a.b 1
4.b odd 2 1 14.6.a.a 1
7.b odd 2 1 784.6.a.i 1
8.b even 2 1 448.6.a.l 1
8.d odd 2 1 448.6.a.e 1
12.b even 2 1 126.6.a.f 1
20.d odd 2 1 350.6.a.i 1
20.e even 4 2 350.6.c.d 2
28.d even 2 1 98.6.a.a 1
28.f even 6 2 98.6.c.d 2
28.g odd 6 2 98.6.c.c 2
84.h odd 2 1 882.6.a.x 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 10$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(112))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 10$$
$5$ $$T - 84$$
$7$ $$T + 49$$
$11$ $$T - 336$$
$13$ $$T - 584$$
$17$ $$T + 1458$$
$19$ $$T + 470$$
$23$ $$T - 4200$$
$29$ $$T - 4866$$
$31$ $$T - 7372$$
$37$ $$T - 14330$$
$41$ $$T - 6222$$
$43$ $$T + 3704$$
$47$ $$T - 1812$$
$53$ $$T + 37242$$
$59$ $$T + 34302$$
$61$ $$T - 24476$$
$67$ $$T - 17452$$
$71$ $$T + 28224$$
$73$ $$T - 3602$$
$79$ $$T + 42872$$
$83$ $$T - 35202$$
$89$ $$T - 26730$$
$97$ $$T + 16978$$