# Properties

 Label 112.6.a.e 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 28) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - 96 q^{5} - 49 q^{7} - 239 q^{9}+O(q^{10})$$ q + 2 * q^3 - 96 * q^5 - 49 * q^7 - 239 * q^9 $$q + 2 q^{3} - 96 q^{5} - 49 q^{7} - 239 q^{9} + 720 q^{11} + 572 q^{13} - 192 q^{15} + 1254 q^{17} + 94 q^{19} - 98 q^{21} - 96 q^{23} + 6091 q^{25} - 964 q^{27} - 4374 q^{29} + 6244 q^{31} + 1440 q^{33} + 4704 q^{35} - 10798 q^{37} + 1144 q^{39} + 12006 q^{41} + 9160 q^{43} + 22944 q^{45} + 25836 q^{47} + 2401 q^{49} + 2508 q^{51} + 1014 q^{53} - 69120 q^{55} + 188 q^{57} - 1242 q^{59} + 7592 q^{61} + 11711 q^{63} - 54912 q^{65} - 41132 q^{67} - 192 q^{69} + 37632 q^{71} - 13438 q^{73} + 12182 q^{75} - 35280 q^{77} - 6248 q^{79} + 56149 q^{81} + 25254 q^{83} - 120384 q^{85} - 8748 q^{87} - 45126 q^{89} - 28028 q^{91} + 12488 q^{93} - 9024 q^{95} + 107222 q^{97} - 172080 q^{99}+O(q^{100})$$ q + 2 * q^3 - 96 * q^5 - 49 * q^7 - 239 * q^9 + 720 * q^11 + 572 * q^13 - 192 * q^15 + 1254 * q^17 + 94 * q^19 - 98 * q^21 - 96 * q^23 + 6091 * q^25 - 964 * q^27 - 4374 * q^29 + 6244 * q^31 + 1440 * q^33 + 4704 * q^35 - 10798 * q^37 + 1144 * q^39 + 12006 * q^41 + 9160 * q^43 + 22944 * q^45 + 25836 * q^47 + 2401 * q^49 + 2508 * q^51 + 1014 * q^53 - 69120 * q^55 + 188 * q^57 - 1242 * q^59 + 7592 * q^61 + 11711 * q^63 - 54912 * q^65 - 41132 * q^67 - 192 * q^69 + 37632 * q^71 - 13438 * q^73 + 12182 * q^75 - 35280 * q^77 - 6248 * q^79 + 56149 * q^81 + 25254 * q^83 - 120384 * q^85 - 8748 * q^87 - 45126 * q^89 - 28028 * q^91 + 12488 * q^93 - 9024 * q^95 + 107222 * q^97 - 172080 * 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 2.00000 0 −96.0000 0 −49.0000 0 −239.000 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.e 1
3.b odd 2 1 1008.6.a.bb 1
4.b odd 2 1 28.6.a.a 1
7.b odd 2 1 784.6.a.f 1
8.b even 2 1 448.6.a.h 1
8.d odd 2 1 448.6.a.i 1
12.b even 2 1 252.6.a.d 1
20.d odd 2 1 700.6.a.d 1
20.e even 4 2 700.6.e.d 2
28.d even 2 1 196.6.a.d 1
28.f even 6 2 196.6.e.e 2
28.g odd 6 2 196.6.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.a 1 4.b odd 2 1
112.6.a.e 1 1.a even 1 1 trivial
196.6.a.d 1 28.d even 2 1
196.6.e.e 2 28.f even 6 2
196.6.e.f 2 28.g odd 6 2
252.6.a.d 1 12.b even 2 1
448.6.a.h 1 8.b even 2 1
448.6.a.i 1 8.d odd 2 1
700.6.a.d 1 20.d odd 2 1
700.6.e.d 2 20.e even 4 2
784.6.a.f 1 7.b odd 2 1
1008.6.a.bb 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 2$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(112))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T + 96$$
$7$ $$T + 49$$
$11$ $$T - 720$$
$13$ $$T - 572$$
$17$ $$T - 1254$$
$19$ $$T - 94$$
$23$ $$T + 96$$
$29$ $$T + 4374$$
$31$ $$T - 6244$$
$37$ $$T + 10798$$
$41$ $$T - 12006$$
$43$ $$T - 9160$$
$47$ $$T - 25836$$
$53$ $$T - 1014$$
$59$ $$T + 1242$$
$61$ $$T - 7592$$
$67$ $$T + 41132$$
$71$ $$T - 37632$$
$73$ $$T + 13438$$
$79$ $$T + 6248$$
$83$ $$T - 25254$$
$89$ $$T + 45126$$
$97$ $$T - 107222$$