Properties

 Label 126.6.a.k Level $126$ Weight $6$ Character orbit 126.a Self dual yes Analytic conductor $20.208$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 16 q^{4} + 54 q^{5} + 49 q^{7} + 64 q^{8}+O(q^{10})$$ q + 4 * q^2 + 16 * q^4 + 54 * q^5 + 49 * q^7 + 64 * q^8 $$q + 4 q^{2} + 16 q^{4} + 54 q^{5} + 49 q^{7} + 64 q^{8} + 216 q^{10} - 216 q^{11} + 998 q^{13} + 196 q^{14} + 256 q^{16} - 1302 q^{17} + 884 q^{19} + 864 q^{20} - 864 q^{22} + 2268 q^{23} - 209 q^{25} + 3992 q^{26} + 784 q^{28} + 1482 q^{29} + 8360 q^{31} + 1024 q^{32} - 5208 q^{34} + 2646 q^{35} - 4714 q^{37} + 3536 q^{38} + 3456 q^{40} + 9786 q^{41} + 19436 q^{43} - 3456 q^{44} + 9072 q^{46} - 22200 q^{47} + 2401 q^{49} - 836 q^{50} + 15968 q^{52} - 26790 q^{53} - 11664 q^{55} + 3136 q^{56} + 5928 q^{58} - 28092 q^{59} - 38866 q^{61} + 33440 q^{62} + 4096 q^{64} + 53892 q^{65} + 23948 q^{67} - 20832 q^{68} + 10584 q^{70} + 20628 q^{71} + 290 q^{73} - 18856 q^{74} + 14144 q^{76} - 10584 q^{77} - 99544 q^{79} + 13824 q^{80} + 39144 q^{82} - 19308 q^{83} - 70308 q^{85} + 77744 q^{86} - 13824 q^{88} - 36390 q^{89} + 48902 q^{91} + 36288 q^{92} - 88800 q^{94} + 47736 q^{95} - 79078 q^{97} + 9604 q^{98}+O(q^{100})$$ q + 4 * q^2 + 16 * q^4 + 54 * q^5 + 49 * q^7 + 64 * q^8 + 216 * q^10 - 216 * q^11 + 998 * q^13 + 196 * q^14 + 256 * q^16 - 1302 * q^17 + 884 * q^19 + 864 * q^20 - 864 * q^22 + 2268 * q^23 - 209 * q^25 + 3992 * q^26 + 784 * q^28 + 1482 * q^29 + 8360 * q^31 + 1024 * q^32 - 5208 * q^34 + 2646 * q^35 - 4714 * q^37 + 3536 * q^38 + 3456 * q^40 + 9786 * q^41 + 19436 * q^43 - 3456 * q^44 + 9072 * q^46 - 22200 * q^47 + 2401 * q^49 - 836 * q^50 + 15968 * q^52 - 26790 * q^53 - 11664 * q^55 + 3136 * q^56 + 5928 * q^58 - 28092 * q^59 - 38866 * q^61 + 33440 * q^62 + 4096 * q^64 + 53892 * q^65 + 23948 * q^67 - 20832 * q^68 + 10584 * q^70 + 20628 * q^71 + 290 * q^73 - 18856 * q^74 + 14144 * q^76 - 10584 * q^77 - 99544 * q^79 + 13824 * q^80 + 39144 * q^82 - 19308 * q^83 - 70308 * q^85 + 77744 * q^86 - 13824 * q^88 - 36390 * q^89 + 48902 * q^91 + 36288 * q^92 - 88800 * q^94 + 47736 * q^95 - 79078 * 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 54.0000 0 49.0000 64.0000 0 216.000
 $$n$$: 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.k 1
3.b odd 2 1 42.6.a.a 1
4.b odd 2 1 1008.6.a.x 1
7.b odd 2 1 882.6.a.o 1
12.b even 2 1 336.6.a.j 1
15.d odd 2 1 1050.6.a.n 1
15.e even 4 2 1050.6.g.o 2
21.c even 2 1 294.6.a.h 1
21.g even 6 2 294.6.e.h 2
21.h odd 6 2 294.6.e.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a 1 3.b odd 2 1
126.6.a.k 1 1.a even 1 1 trivial
294.6.a.h 1 21.c even 2 1
294.6.e.h 2 21.g even 6 2
294.6.e.r 2 21.h odd 6 2
336.6.a.j 1 12.b even 2 1
882.6.a.o 1 7.b odd 2 1
1008.6.a.x 1 4.b odd 2 1
1050.6.a.n 1 15.d odd 2 1
1050.6.g.o 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(126))$$:

 $$T_{5} - 54$$ T5 - 54 $$T_{11} + 216$$ T11 + 216

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T$$
$5$ $$T - 54$$
$7$ $$T - 49$$
$11$ $$T + 216$$
$13$ $$T - 998$$
$17$ $$T + 1302$$
$19$ $$T - 884$$
$23$ $$T - 2268$$
$29$ $$T - 1482$$
$31$ $$T - 8360$$
$37$ $$T + 4714$$
$41$ $$T - 9786$$
$43$ $$T - 19436$$
$47$ $$T + 22200$$
$53$ $$T + 26790$$
$59$ $$T + 28092$$
$61$ $$T + 38866$$
$67$ $$T - 23948$$
$71$ $$T - 20628$$
$73$ $$T - 290$$
$79$ $$T + 99544$$
$83$ $$T + 19308$$
$89$ $$T + 36390$$
$97$ $$T + 79078$$