Properties

 Label 126.6.a.f 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 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 16 q^{4} - 84 q^{5} + 49 q^{7} + 64 q^{8}+O(q^{10})$$ q + 4 * q^2 + 16 * q^4 - 84 * q^5 + 49 * q^7 + 64 * q^8 $$q + 4 q^{2} + 16 q^{4} - 84 q^{5} + 49 q^{7} + 64 q^{8} - 336 q^{10} + 336 q^{11} + 584 q^{13} + 196 q^{14} + 256 q^{16} + 1458 q^{17} + 470 q^{19} - 1344 q^{20} + 1344 q^{22} + 4200 q^{23} + 3931 q^{25} + 2336 q^{26} + 784 q^{28} - 4866 q^{29} - 7372 q^{31} + 1024 q^{32} + 5832 q^{34} - 4116 q^{35} + 14330 q^{37} + 1880 q^{38} - 5376 q^{40} - 6222 q^{41} + 3704 q^{43} + 5376 q^{44} + 16800 q^{46} + 1812 q^{47} + 2401 q^{49} + 15724 q^{50} + 9344 q^{52} + 37242 q^{53} - 28224 q^{55} + 3136 q^{56} - 19464 q^{58} - 34302 q^{59} + 24476 q^{61} - 29488 q^{62} + 4096 q^{64} - 49056 q^{65} - 17452 q^{67} + 23328 q^{68} - 16464 q^{70} - 28224 q^{71} + 3602 q^{73} + 57320 q^{74} + 7520 q^{76} + 16464 q^{77} + 42872 q^{79} - 21504 q^{80} - 24888 q^{82} + 35202 q^{83} - 122472 q^{85} + 14816 q^{86} + 21504 q^{88} - 26730 q^{89} + 28616 q^{91} + 67200 q^{92} + 7248 q^{94} - 39480 q^{95} - 16978 q^{97} + 9604 q^{98}+O(q^{100})$$ q + 4 * q^2 + 16 * q^4 - 84 * q^5 + 49 * q^7 + 64 * q^8 - 336 * q^10 + 336 * q^11 + 584 * q^13 + 196 * q^14 + 256 * q^16 + 1458 * q^17 + 470 * q^19 - 1344 * q^20 + 1344 * q^22 + 4200 * q^23 + 3931 * q^25 + 2336 * q^26 + 784 * q^28 - 4866 * q^29 - 7372 * q^31 + 1024 * q^32 + 5832 * q^34 - 4116 * q^35 + 14330 * q^37 + 1880 * q^38 - 5376 * q^40 - 6222 * q^41 + 3704 * q^43 + 5376 * q^44 + 16800 * q^46 + 1812 * q^47 + 2401 * q^49 + 15724 * q^50 + 9344 * q^52 + 37242 * q^53 - 28224 * q^55 + 3136 * q^56 - 19464 * q^58 - 34302 * q^59 + 24476 * q^61 - 29488 * q^62 + 4096 * q^64 - 49056 * q^65 - 17452 * q^67 + 23328 * q^68 - 16464 * q^70 - 28224 * q^71 + 3602 * q^73 + 57320 * q^74 + 7520 * q^76 + 16464 * q^77 + 42872 * q^79 - 21504 * q^80 - 24888 * q^82 + 35202 * q^83 - 122472 * q^85 + 14816 * q^86 + 21504 * q^88 - 26730 * q^89 + 28616 * q^91 + 67200 * q^92 + 7248 * q^94 - 39480 * q^95 - 16978 * 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 −84.0000 0 49.0000 64.0000 0 −336.000
 $$n$$: 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.f 1
3.b odd 2 1 14.6.a.a 1
4.b odd 2 1 1008.6.a.b 1
7.b odd 2 1 882.6.a.x 1
12.b even 2 1 112.6.a.c 1
15.d odd 2 1 350.6.a.i 1
15.e even 4 2 350.6.c.d 2
21.c even 2 1 98.6.a.a 1
21.g even 6 2 98.6.c.d 2
21.h odd 6 2 98.6.c.c 2
24.f even 2 1 448.6.a.l 1
24.h odd 2 1 448.6.a.e 1
84.h odd 2 1 784.6.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.a 1 3.b odd 2 1
98.6.a.a 1 21.c even 2 1
98.6.c.c 2 21.h odd 6 2
98.6.c.d 2 21.g even 6 2
112.6.a.c 1 12.b even 2 1
126.6.a.f 1 1.a even 1 1 trivial
350.6.a.i 1 15.d odd 2 1
350.6.c.d 2 15.e even 4 2
448.6.a.e 1 24.h odd 2 1
448.6.a.l 1 24.f even 2 1
784.6.a.i 1 84.h odd 2 1
882.6.a.x 1 7.b odd 2 1
1008.6.a.b 1 4.b odd 2 1

Hecke kernels

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

 $$T_{5} + 84$$ T5 + 84 $$T_{11} - 336$$ T11 - 336

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T$$
$5$ $$T + 84$$
$7$ $$T - 49$$
$11$ $$T - 336$$
$13$ $$T - 584$$
$17$ $$T - 1458$$
$19$ $$T - 470$$
$23$ $$T - 4200$$
$29$ $$T + 4866$$
$31$ $$T + 7372$$
$37$ $$T - 14330$$
$41$ $$T + 6222$$
$43$ $$T - 3704$$
$47$ $$T - 1812$$
$53$ $$T - 37242$$
$59$ $$T + 34302$$
$61$ $$T - 24476$$
$67$ $$T + 17452$$
$71$ $$T + 28224$$
$73$ $$T - 3602$$
$79$ $$T - 42872$$
$83$ $$T - 35202$$
$89$ $$T + 26730$$
$97$ $$T + 16978$$