# Properties

 Label 147.6.a.d 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 9 q^{3} - 28 q^{4} - 11 q^{5} - 18 q^{6} + 120 q^{8} + 81 q^{9}+O(q^{10})$$ q - 2 * q^2 + 9 * q^3 - 28 * q^4 - 11 * q^5 - 18 * q^6 + 120 * q^8 + 81 * q^9 $$q - 2 q^{2} + 9 q^{3} - 28 q^{4} - 11 q^{5} - 18 q^{6} + 120 q^{8} + 81 q^{9} + 22 q^{10} + 269 q^{11} - 252 q^{12} + 308 q^{13} - 99 q^{15} + 656 q^{16} - 1896 q^{17} - 162 q^{18} + 164 q^{19} + 308 q^{20} - 538 q^{22} - 3264 q^{23} + 1080 q^{24} - 3004 q^{25} - 616 q^{26} + 729 q^{27} + 2417 q^{29} + 198 q^{30} - 2841 q^{31} - 5152 q^{32} + 2421 q^{33} + 3792 q^{34} - 2268 q^{36} - 11328 q^{37} - 328 q^{38} + 2772 q^{39} - 1320 q^{40} + 16856 q^{41} - 7894 q^{43} - 7532 q^{44} - 891 q^{45} + 6528 q^{46} - 21102 q^{47} + 5904 q^{48} + 6008 q^{50} - 17064 q^{51} - 8624 q^{52} - 29691 q^{53} - 1458 q^{54} - 2959 q^{55} + 1476 q^{57} - 4834 q^{58} + 8163 q^{59} + 2772 q^{60} - 15166 q^{61} + 5682 q^{62} - 10688 q^{64} - 3388 q^{65} - 4842 q^{66} - 32078 q^{67} + 53088 q^{68} - 29376 q^{69} - 38274 q^{71} + 9720 q^{72} - 34866 q^{73} + 22656 q^{74} - 27036 q^{75} - 4592 q^{76} - 5544 q^{78} + 13529 q^{79} - 7216 q^{80} + 6561 q^{81} - 33712 q^{82} + 68103 q^{83} + 20856 q^{85} + 15788 q^{86} + 21753 q^{87} + 32280 q^{88} + 114922 q^{89} + 1782 q^{90} + 91392 q^{92} - 25569 q^{93} + 42204 q^{94} - 1804 q^{95} - 46368 q^{96} - 154959 q^{97} + 21789 q^{99}+O(q^{100})$$ q - 2 * q^2 + 9 * q^3 - 28 * q^4 - 11 * q^5 - 18 * q^6 + 120 * q^8 + 81 * q^9 + 22 * q^10 + 269 * q^11 - 252 * q^12 + 308 * q^13 - 99 * q^15 + 656 * q^16 - 1896 * q^17 - 162 * q^18 + 164 * q^19 + 308 * q^20 - 538 * q^22 - 3264 * q^23 + 1080 * q^24 - 3004 * q^25 - 616 * q^26 + 729 * q^27 + 2417 * q^29 + 198 * q^30 - 2841 * q^31 - 5152 * q^32 + 2421 * q^33 + 3792 * q^34 - 2268 * q^36 - 11328 * q^37 - 328 * q^38 + 2772 * q^39 - 1320 * q^40 + 16856 * q^41 - 7894 * q^43 - 7532 * q^44 - 891 * q^45 + 6528 * q^46 - 21102 * q^47 + 5904 * q^48 + 6008 * q^50 - 17064 * q^51 - 8624 * q^52 - 29691 * q^53 - 1458 * q^54 - 2959 * q^55 + 1476 * q^57 - 4834 * q^58 + 8163 * q^59 + 2772 * q^60 - 15166 * q^61 + 5682 * q^62 - 10688 * q^64 - 3388 * q^65 - 4842 * q^66 - 32078 * q^67 + 53088 * q^68 - 29376 * q^69 - 38274 * q^71 + 9720 * q^72 - 34866 * q^73 + 22656 * q^74 - 27036 * q^75 - 4592 * q^76 - 5544 * q^78 + 13529 * q^79 - 7216 * q^80 + 6561 * q^81 - 33712 * q^82 + 68103 * q^83 + 20856 * q^85 + 15788 * q^86 + 21753 * q^87 + 32280 * q^88 + 114922 * q^89 + 1782 * q^90 + 91392 * q^92 - 25569 * q^93 + 42204 * q^94 - 1804 * q^95 - 46368 * q^96 - 154959 * q^97 + 21789 * 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
−2.00000 9.00000 −28.0000 −11.0000 −18.0000 0 120.000 81.0000 22.0000
 $$n$$: 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.d 1
3.b odd 2 1 441.6.a.h 1
7.b odd 2 1 147.6.a.c 1
7.c even 3 2 147.6.e.g 2
7.d odd 6 2 21.6.e.a 2
21.c even 2 1 441.6.a.g 1
21.g even 6 2 63.6.e.a 2
28.f even 6 2 336.6.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.a 2 7.d odd 6 2
63.6.e.a 2 21.g even 6 2
147.6.a.c 1 7.b odd 2 1
147.6.a.d 1 1.a even 1 1 trivial
147.6.e.g 2 7.c even 3 2
336.6.q.b 2 28.f even 6 2
441.6.a.g 1 21.c even 2 1
441.6.a.h 1 3.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 $$S_{6}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{5} + 11$$ T5 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 9$$
$5$ $$T + 11$$
$7$ $$T$$
$11$ $$T - 269$$
$13$ $$T - 308$$
$17$ $$T + 1896$$
$19$ $$T - 164$$
$23$ $$T + 3264$$
$29$ $$T - 2417$$
$31$ $$T + 2841$$
$37$ $$T + 11328$$
$41$ $$T - 16856$$
$43$ $$T + 7894$$
$47$ $$T + 21102$$
$53$ $$T + 29691$$
$59$ $$T - 8163$$
$61$ $$T + 15166$$
$67$ $$T + 32078$$
$71$ $$T + 38274$$
$73$ $$T + 34866$$
$79$ $$T - 13529$$
$83$ $$T - 68103$$
$89$ $$T - 114922$$
$97$ $$T + 154959$$