Properties

 Label 126.6.a.e 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: yes 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 $$q - 4 q^{2} + 16 q^{4} + 54 q^{5} + 49 q^{7} - 64 q^{8} - 216 q^{10} + 594 q^{11} + 26 q^{13} - 196 q^{14} + 256 q^{16} - 534 q^{17} - 3004 q^{19} + 864 q^{20} - 2376 q^{22} + 3510 q^{23} - 209 q^{25} - 104 q^{26} + 784 q^{28} + 4296 q^{29} + 8036 q^{31} - 1024 q^{32} + 2136 q^{34} + 2646 q^{35} - 502 q^{37} + 12016 q^{38} - 3456 q^{40} + 9870 q^{41} + 9068 q^{43} + 9504 q^{44} - 14040 q^{46} + 1140 q^{47} + 2401 q^{49} + 836 q^{50} + 416 q^{52} + 28356 q^{53} + 32076 q^{55} - 3136 q^{56} - 17184 q^{58} - 8196 q^{59} + 29822 q^{61} - 32144 q^{62} + 4096 q^{64} + 1404 q^{65} - 62884 q^{67} - 8544 q^{68} - 10584 q^{70} - 34398 q^{71} + 56990 q^{73} + 2008 q^{74} - 48064 q^{76} + 29106 q^{77} + 49496 q^{79} + 13824 q^{80} - 39480 q^{82} - 52512 q^{83} - 28836 q^{85} - 36272 q^{86} - 38016 q^{88} - 48282 q^{89} + 1274 q^{91} + 56160 q^{92} - 4560 q^{94} - 162216 q^{95} - 83938 q^{97} - 9604 q^{98}+O(q^{100})$$ q - 4 * q^2 + 16 * q^4 + 54 * q^5 + 49 * q^7 - 64 * q^8 - 216 * q^10 + 594 * q^11 + 26 * q^13 - 196 * q^14 + 256 * q^16 - 534 * q^17 - 3004 * q^19 + 864 * q^20 - 2376 * q^22 + 3510 * q^23 - 209 * q^25 - 104 * q^26 + 784 * q^28 + 4296 * q^29 + 8036 * q^31 - 1024 * q^32 + 2136 * q^34 + 2646 * q^35 - 502 * q^37 + 12016 * q^38 - 3456 * q^40 + 9870 * q^41 + 9068 * q^43 + 9504 * q^44 - 14040 * q^46 + 1140 * q^47 + 2401 * q^49 + 836 * q^50 + 416 * q^52 + 28356 * q^53 + 32076 * q^55 - 3136 * q^56 - 17184 * q^58 - 8196 * q^59 + 29822 * q^61 - 32144 * q^62 + 4096 * q^64 + 1404 * q^65 - 62884 * q^67 - 8544 * q^68 - 10584 * q^70 - 34398 * q^71 + 56990 * q^73 + 2008 * q^74 - 48064 * q^76 + 29106 * q^77 + 49496 * q^79 + 13824 * q^80 - 39480 * q^82 - 52512 * q^83 - 28836 * q^85 - 36272 * q^86 - 38016 * q^88 - 48282 * q^89 + 1274 * q^91 + 56160 * q^92 - 4560 * q^94 - 162216 * q^95 - 83938 * 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.e 1
3.b odd 2 1 126.6.a.g yes 1
4.b odd 2 1 1008.6.a.w 1
7.b odd 2 1 882.6.a.b 1
12.b even 2 1 1008.6.a.f 1
21.c even 2 1 882.6.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.6.a.e 1 1.a even 1 1 trivial
126.6.a.g yes 1 3.b odd 2 1
882.6.a.b 1 7.b odd 2 1
882.6.a.w 1 21.c even 2 1
1008.6.a.f 1 12.b even 2 1
1008.6.a.w 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(126))$$:

 $$T_{5} - 54$$ T5 - 54 $$T_{11} - 594$$ T11 - 594

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T$$
$5$ $$T - 54$$
$7$ $$T - 49$$
$11$ $$T - 594$$
$13$ $$T - 26$$
$17$ $$T + 534$$
$19$ $$T + 3004$$
$23$ $$T - 3510$$
$29$ $$T - 4296$$
$31$ $$T - 8036$$
$37$ $$T + 502$$
$41$ $$T - 9870$$
$43$ $$T - 9068$$
$47$ $$T - 1140$$
$53$ $$T - 28356$$
$59$ $$T + 8196$$
$61$ $$T - 29822$$
$67$ $$T + 62884$$
$71$ $$T + 34398$$
$73$ $$T - 56990$$
$79$ $$T - 49496$$
$83$ $$T + 52512$$
$89$ $$T + 48282$$
$97$ $$T + 83938$$