# Properties

 Label 126.6.a.i Level $126$ Weight $6$ Character orbit 126.a Self dual yes Analytic conductor $20.208$ 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] = [126,6,Mod(1,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("126.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 126.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: $$20.2083612964$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 16 q^{4} - 26 q^{5} - 49 q^{7} + 64 q^{8}+O(q^{10})$$ q + 4 * q^2 + 16 * q^4 - 26 * q^5 - 49 * q^7 + 64 * q^8 $$q + 4 q^{2} + 16 q^{4} - 26 q^{5} - 49 q^{7} + 64 q^{8} - 104 q^{10} - 664 q^{11} + 318 q^{13} - 196 q^{14} + 256 q^{16} - 1582 q^{17} + 236 q^{19} - 416 q^{20} - 2656 q^{22} - 2212 q^{23} - 2449 q^{25} + 1272 q^{26} - 784 q^{28} + 4954 q^{29} - 7128 q^{31} + 1024 q^{32} - 6328 q^{34} + 1274 q^{35} + 4358 q^{37} + 944 q^{38} - 1664 q^{40} - 10542 q^{41} - 8452 q^{43} - 10624 q^{44} - 8848 q^{46} - 5352 q^{47} + 2401 q^{49} - 9796 q^{50} + 5088 q^{52} + 33354 q^{53} + 17264 q^{55} - 3136 q^{56} + 19816 q^{58} + 15436 q^{59} - 36762 q^{61} - 28512 q^{62} + 4096 q^{64} - 8268 q^{65} + 40972 q^{67} - 25312 q^{68} + 5096 q^{70} + 9092 q^{71} - 73454 q^{73} + 17432 q^{74} + 3776 q^{76} + 32536 q^{77} + 89400 q^{79} - 6656 q^{80} - 42168 q^{82} + 6428 q^{83} + 41132 q^{85} - 33808 q^{86} - 42496 q^{88} + 122658 q^{89} - 15582 q^{91} - 35392 q^{92} - 21408 q^{94} - 6136 q^{95} + 21370 q^{97} + 9604 q^{98}+O(q^{100})$$ q + 4 * q^2 + 16 * q^4 - 26 * q^5 - 49 * q^7 + 64 * q^8 - 104 * q^10 - 664 * q^11 + 318 * q^13 - 196 * q^14 + 256 * q^16 - 1582 * q^17 + 236 * q^19 - 416 * q^20 - 2656 * q^22 - 2212 * q^23 - 2449 * q^25 + 1272 * q^26 - 784 * q^28 + 4954 * q^29 - 7128 * q^31 + 1024 * q^32 - 6328 * q^34 + 1274 * q^35 + 4358 * q^37 + 944 * q^38 - 1664 * q^40 - 10542 * q^41 - 8452 * q^43 - 10624 * q^44 - 8848 * q^46 - 5352 * q^47 + 2401 * q^49 - 9796 * q^50 + 5088 * q^52 + 33354 * q^53 + 17264 * q^55 - 3136 * q^56 + 19816 * q^58 + 15436 * q^59 - 36762 * q^61 - 28512 * q^62 + 4096 * q^64 - 8268 * q^65 + 40972 * q^67 - 25312 * q^68 + 5096 * q^70 + 9092 * q^71 - 73454 * q^73 + 17432 * q^74 + 3776 * q^76 + 32536 * q^77 + 89400 * q^79 - 6656 * q^80 - 42168 * q^82 + 6428 * q^83 + 41132 * q^85 - 33808 * q^86 - 42496 * q^88 + 122658 * q^89 - 15582 * q^91 - 35392 * q^92 - 21408 * q^94 - 6136 * q^95 + 21370 * q^97 + 9604 * q^98

## 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
4.00000 0 16.0000 −26.0000 0 −49.0000 64.0000 0 −104.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$+1$$

## 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 126.6.a.i 1
3.b odd 2 1 42.6.a.d 1
4.b odd 2 1 1008.6.a.j 1
7.b odd 2 1 882.6.a.s 1
12.b even 2 1 336.6.a.h 1
15.d odd 2 1 1050.6.a.k 1
15.e even 4 2 1050.6.g.i 2
21.c even 2 1 294.6.a.b 1
21.g even 6 2 294.6.e.p 2
21.h odd 6 2 294.6.e.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.d 1 3.b odd 2 1
126.6.a.i 1 1.a even 1 1 trivial
294.6.a.b 1 21.c even 2 1
294.6.e.i 2 21.h odd 6 2
294.6.e.p 2 21.g even 6 2
336.6.a.h 1 12.b even 2 1
882.6.a.s 1 7.b odd 2 1
1008.6.a.j 1 4.b odd 2 1
1050.6.a.k 1 15.d odd 2 1
1050.6.g.i 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(126))$$:

 $$T_{5} + 26$$ T5 + 26 $$T_{11} + 664$$ T11 + 664

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T$$
$5$ $$T + 26$$
$7$ $$T + 49$$
$11$ $$T + 664$$
$13$ $$T - 318$$
$17$ $$T + 1582$$
$19$ $$T - 236$$
$23$ $$T + 2212$$
$29$ $$T - 4954$$
$31$ $$T + 7128$$
$37$ $$T - 4358$$
$41$ $$T + 10542$$
$43$ $$T + 8452$$
$47$ $$T + 5352$$
$53$ $$T - 33354$$
$59$ $$T - 15436$$
$61$ $$T + 36762$$
$67$ $$T - 40972$$
$71$ $$T - 9092$$
$73$ $$T + 73454$$
$79$ $$T - 89400$$
$83$ $$T - 6428$$
$89$ $$T - 122658$$
$97$ $$T - 21370$$