# Properties

 Label 147.6.a.e 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + 9 q^{3} - 31 q^{4} + 34 q^{5} + 9 q^{6} - 63 q^{8} + 81 q^{9}+O(q^{10})$$ q + q^2 + 9 * q^3 - 31 * q^4 + 34 * q^5 + 9 * q^6 - 63 * q^8 + 81 * q^9 $$q + q^{2} + 9 q^{3} - 31 q^{4} + 34 q^{5} + 9 q^{6} - 63 q^{8} + 81 q^{9} + 34 q^{10} - 340 q^{11} - 279 q^{12} - 454 q^{13} + 306 q^{15} + 929 q^{16} + 798 q^{17} + 81 q^{18} - 892 q^{19} - 1054 q^{20} - 340 q^{22} - 3192 q^{23} - 567 q^{24} - 1969 q^{25} - 454 q^{26} + 729 q^{27} - 8242 q^{29} + 306 q^{30} + 2496 q^{31} + 2945 q^{32} - 3060 q^{33} + 798 q^{34} - 2511 q^{36} + 9798 q^{37} - 892 q^{38} - 4086 q^{39} - 2142 q^{40} - 19834 q^{41} - 17236 q^{43} + 10540 q^{44} + 2754 q^{45} - 3192 q^{46} - 8928 q^{47} + 8361 q^{48} - 1969 q^{50} + 7182 q^{51} + 14074 q^{52} + 150 q^{53} + 729 q^{54} - 11560 q^{55} - 8028 q^{57} - 8242 q^{58} + 42396 q^{59} - 9486 q^{60} - 14758 q^{61} + 2496 q^{62} - 26783 q^{64} - 15436 q^{65} - 3060 q^{66} - 1676 q^{67} - 24738 q^{68} - 28728 q^{69} + 14568 q^{71} - 5103 q^{72} - 78378 q^{73} + 9798 q^{74} - 17721 q^{75} + 27652 q^{76} - 4086 q^{78} - 2272 q^{79} + 31586 q^{80} + 6561 q^{81} - 19834 q^{82} + 37764 q^{83} + 27132 q^{85} - 17236 q^{86} - 74178 q^{87} + 21420 q^{88} + 117286 q^{89} + 2754 q^{90} + 98952 q^{92} + 22464 q^{93} - 8928 q^{94} - 30328 q^{95} + 26505 q^{96} - 10002 q^{97} - 27540 q^{99}+O(q^{100})$$ q + q^2 + 9 * q^3 - 31 * q^4 + 34 * q^5 + 9 * q^6 - 63 * q^8 + 81 * q^9 + 34 * q^10 - 340 * q^11 - 279 * q^12 - 454 * q^13 + 306 * q^15 + 929 * q^16 + 798 * q^17 + 81 * q^18 - 892 * q^19 - 1054 * q^20 - 340 * q^22 - 3192 * q^23 - 567 * q^24 - 1969 * q^25 - 454 * q^26 + 729 * q^27 - 8242 * q^29 + 306 * q^30 + 2496 * q^31 + 2945 * q^32 - 3060 * q^33 + 798 * q^34 - 2511 * q^36 + 9798 * q^37 - 892 * q^38 - 4086 * q^39 - 2142 * q^40 - 19834 * q^41 - 17236 * q^43 + 10540 * q^44 + 2754 * q^45 - 3192 * q^46 - 8928 * q^47 + 8361 * q^48 - 1969 * q^50 + 7182 * q^51 + 14074 * q^52 + 150 * q^53 + 729 * q^54 - 11560 * q^55 - 8028 * q^57 - 8242 * q^58 + 42396 * q^59 - 9486 * q^60 - 14758 * q^61 + 2496 * q^62 - 26783 * q^64 - 15436 * q^65 - 3060 * q^66 - 1676 * q^67 - 24738 * q^68 - 28728 * q^69 + 14568 * q^71 - 5103 * q^72 - 78378 * q^73 + 9798 * q^74 - 17721 * q^75 + 27652 * q^76 - 4086 * q^78 - 2272 * q^79 + 31586 * q^80 + 6561 * q^81 - 19834 * q^82 + 37764 * q^83 + 27132 * q^85 - 17236 * q^86 - 74178 * q^87 + 21420 * q^88 + 117286 * q^89 + 2754 * q^90 + 98952 * q^92 + 22464 * q^93 - 8928 * q^94 - 30328 * q^95 + 26505 * q^96 - 10002 * q^97 - 27540 * 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 9.00000 −31.0000 34.0000 9.00000 0 −63.0000 81.0000 34.0000
 $$n$$: 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.e 1
3.b odd 2 1 441.6.a.d 1
7.b odd 2 1 21.6.a.b 1
7.c even 3 2 147.6.e.e 2
7.d odd 6 2 147.6.e.f 2
21.c even 2 1 63.6.a.c 1
28.d even 2 1 336.6.a.l 1
35.c odd 2 1 525.6.a.c 1
35.f even 4 2 525.6.d.d 2
84.h odd 2 1 1008.6.a.t 1

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

 $$T_{2} - 1$$ T2 - 1 $$T_{5} - 34$$ T5 - 34

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 9$$
$5$ $$T - 34$$
$7$ $$T$$
$11$ $$T + 340$$
$13$ $$T + 454$$
$17$ $$T - 798$$
$19$ $$T + 892$$
$23$ $$T + 3192$$
$29$ $$T + 8242$$
$31$ $$T - 2496$$
$37$ $$T - 9798$$
$41$ $$T + 19834$$
$43$ $$T + 17236$$
$47$ $$T + 8928$$
$53$ $$T - 150$$
$59$ $$T - 42396$$
$61$ $$T + 14758$$
$67$ $$T + 1676$$
$71$ $$T - 14568$$
$73$ $$T + 78378$$
$79$ $$T + 2272$$
$83$ $$T - 37764$$
$89$ $$T - 117286$$
$97$ $$T + 10002$$