# Properties

 Label 147.6.a.b Level $147$ Weight $6$ Character orbit 147.a Self dual yes Analytic conductor $23.576$ 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] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, 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("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.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: $$23.5764215125$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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)

 $$f(q)$$ $$=$$ $$q - 6 q^{2} + 9 q^{3} + 4 q^{4} - 78 q^{5} - 54 q^{6} + 168 q^{8} + 81 q^{9}+O(q^{10})$$ q - 6 * q^2 + 9 * q^3 + 4 * q^4 - 78 * q^5 - 54 * q^6 + 168 * q^8 + 81 * q^9 $$q - 6 q^{2} + 9 q^{3} + 4 q^{4} - 78 q^{5} - 54 q^{6} + 168 q^{8} + 81 q^{9} + 468 q^{10} + 444 q^{11} + 36 q^{12} + 442 q^{13} - 702 q^{15} - 1136 q^{16} + 126 q^{17} - 486 q^{18} - 2684 q^{19} - 312 q^{20} - 2664 q^{22} + 4200 q^{23} + 1512 q^{24} + 2959 q^{25} - 2652 q^{26} + 729 q^{27} - 5442 q^{29} + 4212 q^{30} - 80 q^{31} + 1440 q^{32} + 3996 q^{33} - 756 q^{34} + 324 q^{36} - 5434 q^{37} + 16104 q^{38} + 3978 q^{39} - 13104 q^{40} - 7962 q^{41} - 11524 q^{43} + 1776 q^{44} - 6318 q^{45} - 25200 q^{46} + 13920 q^{47} - 10224 q^{48} - 17754 q^{50} + 1134 q^{51} + 1768 q^{52} - 9594 q^{53} - 4374 q^{54} - 34632 q^{55} - 24156 q^{57} + 32652 q^{58} - 27492 q^{59} - 2808 q^{60} - 49478 q^{61} + 480 q^{62} + 27712 q^{64} - 34476 q^{65} - 23976 q^{66} - 59356 q^{67} + 504 q^{68} + 37800 q^{69} + 32040 q^{71} + 13608 q^{72} + 61846 q^{73} + 32604 q^{74} + 26631 q^{75} - 10736 q^{76} - 23868 q^{78} - 65776 q^{79} + 88608 q^{80} + 6561 q^{81} + 47772 q^{82} - 40188 q^{83} - 9828 q^{85} + 69144 q^{86} - 48978 q^{87} + 74592 q^{88} + 7974 q^{89} + 37908 q^{90} + 16800 q^{92} - 720 q^{93} - 83520 q^{94} + 209352 q^{95} + 12960 q^{96} + 143662 q^{97} + 35964 q^{99}+O(q^{100})$$ q - 6 * q^2 + 9 * q^3 + 4 * q^4 - 78 * q^5 - 54 * q^6 + 168 * q^8 + 81 * q^9 + 468 * q^10 + 444 * q^11 + 36 * q^12 + 442 * q^13 - 702 * q^15 - 1136 * q^16 + 126 * q^17 - 486 * q^18 - 2684 * q^19 - 312 * q^20 - 2664 * q^22 + 4200 * q^23 + 1512 * q^24 + 2959 * q^25 - 2652 * q^26 + 729 * q^27 - 5442 * q^29 + 4212 * q^30 - 80 * q^31 + 1440 * q^32 + 3996 * q^33 - 756 * q^34 + 324 * q^36 - 5434 * q^37 + 16104 * q^38 + 3978 * q^39 - 13104 * q^40 - 7962 * q^41 - 11524 * q^43 + 1776 * q^44 - 6318 * q^45 - 25200 * q^46 + 13920 * q^47 - 10224 * q^48 - 17754 * q^50 + 1134 * q^51 + 1768 * q^52 - 9594 * q^53 - 4374 * q^54 - 34632 * q^55 - 24156 * q^57 + 32652 * q^58 - 27492 * q^59 - 2808 * q^60 - 49478 * q^61 + 480 * q^62 + 27712 * q^64 - 34476 * q^65 - 23976 * q^66 - 59356 * q^67 + 504 * q^68 + 37800 * q^69 + 32040 * q^71 + 13608 * q^72 + 61846 * q^73 + 32604 * q^74 + 26631 * q^75 - 10736 * q^76 - 23868 * q^78 - 65776 * q^79 + 88608 * q^80 + 6561 * q^81 + 47772 * q^82 - 40188 * q^83 - 9828 * q^85 + 69144 * q^86 - 48978 * q^87 + 74592 * q^88 + 7974 * q^89 + 37908 * q^90 + 16800 * q^92 - 720 * q^93 - 83520 * q^94 + 209352 * q^95 + 12960 * q^96 + 143662 * q^97 + 35964 * 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
−6.00000 9.00000 4.00000 −78.0000 −54.0000 0 168.000 81.0000 468.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 147.6.a.b 1
3.b odd 2 1 441.6.a.j 1
7.b odd 2 1 21.6.a.a 1
7.c even 3 2 147.6.e.i 2
7.d odd 6 2 147.6.e.j 2
21.c even 2 1 63.6.a.d 1
28.d even 2 1 336.6.a.r 1
35.c odd 2 1 525.6.a.d 1
35.f even 4 2 525.6.d.b 2
84.h odd 2 1 1008.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 7.b odd 2 1
63.6.a.d 1 21.c even 2 1
147.6.a.b 1 1.a even 1 1 trivial
147.6.e.i 2 7.c even 3 2
147.6.e.j 2 7.d odd 6 2
336.6.a.r 1 28.d even 2 1
441.6.a.j 1 3.b odd 2 1
525.6.a.d 1 35.c odd 2 1
525.6.d.b 2 35.f even 4 2
1008.6.a.c 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2} + 6$$ T2 + 6 $$T_{5} + 78$$ T5 + 78

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 6$$
$3$ $$T - 9$$
$5$ $$T + 78$$
$7$ $$T$$
$11$ $$T - 444$$
$13$ $$T - 442$$
$17$ $$T - 126$$
$19$ $$T + 2684$$
$23$ $$T - 4200$$
$29$ $$T + 5442$$
$31$ $$T + 80$$
$37$ $$T + 5434$$
$41$ $$T + 7962$$
$43$ $$T + 11524$$
$47$ $$T - 13920$$
$53$ $$T + 9594$$
$59$ $$T + 27492$$
$61$ $$T + 49478$$
$67$ $$T + 59356$$
$71$ $$T - 32040$$
$73$ $$T - 61846$$
$79$ $$T + 65776$$
$83$ $$T + 40188$$
$89$ $$T - 7974$$
$97$ $$T - 143662$$