# Properties

 Label 147.6.a.g Level $147$ Weight $6$ Character orbit 147.a Self dual yes Analytic conductor $23.576$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 10 q^{2} - 9 q^{3} + 68 q^{4} + 106 q^{5} - 90 q^{6} + 360 q^{8} + 81 q^{9}+O(q^{10})$$ q + 10 * q^2 - 9 * q^3 + 68 * q^4 + 106 * q^5 - 90 * q^6 + 360 * q^8 + 81 * q^9 $$q + 10 q^{2} - 9 q^{3} + 68 q^{4} + 106 q^{5} - 90 q^{6} + 360 q^{8} + 81 q^{9} + 1060 q^{10} + 92 q^{11} - 612 q^{12} - 670 q^{13} - 954 q^{15} + 1424 q^{16} + 222 q^{17} + 810 q^{18} + 908 q^{19} + 7208 q^{20} + 920 q^{22} - 1176 q^{23} - 3240 q^{24} + 8111 q^{25} - 6700 q^{26} - 729 q^{27} + 1118 q^{29} - 9540 q^{30} - 3696 q^{31} + 2720 q^{32} - 828 q^{33} + 2220 q^{34} + 5508 q^{36} + 4182 q^{37} + 9080 q^{38} + 6030 q^{39} + 38160 q^{40} + 6662 q^{41} - 3700 q^{43} + 6256 q^{44} + 8586 q^{45} - 11760 q^{46} + 7056 q^{47} - 12816 q^{48} + 81110 q^{50} - 1998 q^{51} - 45560 q^{52} - 37578 q^{53} - 7290 q^{54} + 9752 q^{55} - 8172 q^{57} + 11180 q^{58} - 32700 q^{59} - 64872 q^{60} + 10802 q^{61} - 36960 q^{62} - 18368 q^{64} - 71020 q^{65} - 8280 q^{66} + 64996 q^{67} + 15096 q^{68} + 10584 q^{69} - 61320 q^{71} + 29160 q^{72} - 38922 q^{73} + 41820 q^{74} - 72999 q^{75} + 61744 q^{76} + 60300 q^{78} - 88096 q^{79} + 150944 q^{80} + 6561 q^{81} + 66620 q^{82} - 71892 q^{83} + 23532 q^{85} - 37000 q^{86} - 10062 q^{87} + 33120 q^{88} - 111818 q^{89} + 85860 q^{90} - 79968 q^{92} + 33264 q^{93} + 70560 q^{94} + 96248 q^{95} - 24480 q^{96} + 150846 q^{97} + 7452 q^{99}+O(q^{100})$$ q + 10 * q^2 - 9 * q^3 + 68 * q^4 + 106 * q^5 - 90 * q^6 + 360 * q^8 + 81 * q^9 + 1060 * q^10 + 92 * q^11 - 612 * q^12 - 670 * q^13 - 954 * q^15 + 1424 * q^16 + 222 * q^17 + 810 * q^18 + 908 * q^19 + 7208 * q^20 + 920 * q^22 - 1176 * q^23 - 3240 * q^24 + 8111 * q^25 - 6700 * q^26 - 729 * q^27 + 1118 * q^29 - 9540 * q^30 - 3696 * q^31 + 2720 * q^32 - 828 * q^33 + 2220 * q^34 + 5508 * q^36 + 4182 * q^37 + 9080 * q^38 + 6030 * q^39 + 38160 * q^40 + 6662 * q^41 - 3700 * q^43 + 6256 * q^44 + 8586 * q^45 - 11760 * q^46 + 7056 * q^47 - 12816 * q^48 + 81110 * q^50 - 1998 * q^51 - 45560 * q^52 - 37578 * q^53 - 7290 * q^54 + 9752 * q^55 - 8172 * q^57 + 11180 * q^58 - 32700 * q^59 - 64872 * q^60 + 10802 * q^61 - 36960 * q^62 - 18368 * q^64 - 71020 * q^65 - 8280 * q^66 + 64996 * q^67 + 15096 * q^68 + 10584 * q^69 - 61320 * q^71 + 29160 * q^72 - 38922 * q^73 + 41820 * q^74 - 72999 * q^75 + 61744 * q^76 + 60300 * q^78 - 88096 * q^79 + 150944 * q^80 + 6561 * q^81 + 66620 * q^82 - 71892 * q^83 + 23532 * q^85 - 37000 * q^86 - 10062 * q^87 + 33120 * q^88 - 111818 * q^89 + 85860 * q^90 - 79968 * q^92 + 33264 * q^93 + 70560 * q^94 + 96248 * q^95 - 24480 * q^96 + 150846 * q^97 + 7452 * 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
10.0000 −9.00000 68.0000 106.000 −90.0000 0 360.000 81.0000 1060.00
 $$n$$: 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.g 1
3.b odd 2 1 441.6.a.b 1
7.b odd 2 1 21.6.a.d 1
7.c even 3 2 147.6.e.b 2
7.d odd 6 2 147.6.e.a 2
21.c even 2 1 63.6.a.a 1
28.d even 2 1 336.6.a.a 1
35.c odd 2 1 525.6.a.a 1
35.f even 4 2 525.6.d.a 2
84.h odd 2 1 1008.6.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.d 1 7.b odd 2 1
63.6.a.a 1 21.c even 2 1
147.6.a.g 1 1.a even 1 1 trivial
147.6.e.a 2 7.d odd 6 2
147.6.e.b 2 7.c even 3 2
336.6.a.a 1 28.d even 2 1
441.6.a.b 1 3.b odd 2 1
525.6.a.a 1 35.c odd 2 1
525.6.d.a 2 35.f even 4 2
1008.6.a.bc 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2} - 10$$ T2 - 10 $$T_{5} - 106$$ T5 - 106

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 10$$
$3$ $$T + 9$$
$5$ $$T - 106$$
$7$ $$T$$
$11$ $$T - 92$$
$13$ $$T + 670$$
$17$ $$T - 222$$
$19$ $$T - 908$$
$23$ $$T + 1176$$
$29$ $$T - 1118$$
$31$ $$T + 3696$$
$37$ $$T - 4182$$
$41$ $$T - 6662$$
$43$ $$T + 3700$$
$47$ $$T - 7056$$
$53$ $$T + 37578$$
$59$ $$T + 32700$$
$61$ $$T - 10802$$
$67$ $$T - 64996$$
$71$ $$T + 61320$$
$73$ $$T + 38922$$
$79$ $$T + 88096$$
$83$ $$T + 71892$$
$89$ $$T + 111818$$
$97$ $$T - 150846$$