# Properties

 Label 147.4.a.d Level $147$ Weight $4$ Character orbit 147.a Self dual yes Analytic conductor $8.673$ 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] = [147,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$8.67328077084$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 3 q^{3} - 7 q^{4} + 12 q^{5} + 3 q^{6} + 15 q^{8} + 9 q^{9}+O(q^{10})$$ q - q^2 - 3 * q^3 - 7 * q^4 + 12 * q^5 + 3 * q^6 + 15 * q^8 + 9 * q^9 $$q - q^{2} - 3 q^{3} - 7 q^{4} + 12 q^{5} + 3 q^{6} + 15 q^{8} + 9 q^{9} - 12 q^{10} + 20 q^{11} + 21 q^{12} - 84 q^{13} - 36 q^{15} + 41 q^{16} - 96 q^{17} - 9 q^{18} + 12 q^{19} - 84 q^{20} - 20 q^{22} - 176 q^{23} - 45 q^{24} + 19 q^{25} + 84 q^{26} - 27 q^{27} + 58 q^{29} + 36 q^{30} - 264 q^{31} - 161 q^{32} - 60 q^{33} + 96 q^{34} - 63 q^{36} + 258 q^{37} - 12 q^{38} + 252 q^{39} + 180 q^{40} + 156 q^{43} - 140 q^{44} + 108 q^{45} + 176 q^{46} - 408 q^{47} - 123 q^{48} - 19 q^{50} + 288 q^{51} + 588 q^{52} - 722 q^{53} + 27 q^{54} + 240 q^{55} - 36 q^{57} - 58 q^{58} + 492 q^{59} + 252 q^{60} - 492 q^{61} + 264 q^{62} - 167 q^{64} - 1008 q^{65} + 60 q^{66} + 412 q^{67} + 672 q^{68} + 528 q^{69} + 296 q^{71} + 135 q^{72} + 240 q^{73} - 258 q^{74} - 57 q^{75} - 84 q^{76} - 252 q^{78} + 776 q^{79} + 492 q^{80} + 81 q^{81} + 924 q^{83} - 1152 q^{85} - 156 q^{86} - 174 q^{87} + 300 q^{88} - 744 q^{89} - 108 q^{90} + 1232 q^{92} + 792 q^{93} + 408 q^{94} + 144 q^{95} + 483 q^{96} - 168 q^{97} + 180 q^{99}+O(q^{100})$$ q - q^2 - 3 * q^3 - 7 * q^4 + 12 * q^5 + 3 * q^6 + 15 * q^8 + 9 * q^9 - 12 * q^10 + 20 * q^11 + 21 * q^12 - 84 * q^13 - 36 * q^15 + 41 * q^16 - 96 * q^17 - 9 * q^18 + 12 * q^19 - 84 * q^20 - 20 * q^22 - 176 * q^23 - 45 * q^24 + 19 * q^25 + 84 * q^26 - 27 * q^27 + 58 * q^29 + 36 * q^30 - 264 * q^31 - 161 * q^32 - 60 * q^33 + 96 * q^34 - 63 * q^36 + 258 * q^37 - 12 * q^38 + 252 * q^39 + 180 * q^40 + 156 * q^43 - 140 * q^44 + 108 * q^45 + 176 * q^46 - 408 * q^47 - 123 * q^48 - 19 * q^50 + 288 * q^51 + 588 * q^52 - 722 * q^53 + 27 * q^54 + 240 * q^55 - 36 * q^57 - 58 * q^58 + 492 * q^59 + 252 * q^60 - 492 * q^61 + 264 * q^62 - 167 * q^64 - 1008 * q^65 + 60 * q^66 + 412 * q^67 + 672 * q^68 + 528 * q^69 + 296 * q^71 + 135 * q^72 + 240 * q^73 - 258 * q^74 - 57 * q^75 - 84 * q^76 - 252 * q^78 + 776 * q^79 + 492 * q^80 + 81 * q^81 + 924 * q^83 - 1152 * q^85 - 156 * q^86 - 174 * q^87 + 300 * q^88 - 744 * q^89 - 108 * q^90 + 1232 * q^92 + 792 * q^93 + 408 * q^94 + 144 * q^95 + 483 * q^96 - 168 * q^97 + 180 * 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
−1.00000 −3.00000 −7.00000 12.0000 3.00000 0 15.0000 9.00000 −12.0000
 $$n$$: 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 147.4.a.d 1
3.b odd 2 1 441.4.a.g 1
4.b odd 2 1 2352.4.a.bi 1
7.b odd 2 1 147.4.a.e yes 1
7.c even 3 2 147.4.e.f 2
7.d odd 6 2 147.4.e.e 2
21.c even 2 1 441.4.a.h 1
21.g even 6 2 441.4.e.f 2
21.h odd 6 2 441.4.e.g 2
28.d even 2 1 2352.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 1.a even 1 1 trivial
147.4.a.e yes 1 7.b odd 2 1
147.4.e.e 2 7.d odd 6 2
147.4.e.f 2 7.c even 3 2
441.4.a.g 1 3.b odd 2 1
441.4.a.h 1 21.c even 2 1
441.4.e.f 2 21.g even 6 2
441.4.e.g 2 21.h odd 6 2
2352.4.a.b 1 28.d even 2 1
2352.4.a.bi 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 12$$ T5 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 3$$
$5$ $$T - 12$$
$7$ $$T$$
$11$ $$T - 20$$
$13$ $$T + 84$$
$17$ $$T + 96$$
$19$ $$T - 12$$
$23$ $$T + 176$$
$29$ $$T - 58$$
$31$ $$T + 264$$
$37$ $$T - 258$$
$41$ $$T$$
$43$ $$T - 156$$
$47$ $$T + 408$$
$53$ $$T + 722$$
$59$ $$T - 492$$
$61$ $$T + 492$$
$67$ $$T - 412$$
$71$ $$T - 296$$
$73$ $$T - 240$$
$79$ $$T - 776$$
$83$ $$T - 924$$
$89$ $$T + 744$$
$97$ $$T + 168$$