# Properties

 Label 147.2.e.a Level $147$ Weight $2$ Character orbit 147.e Analytic conductor $1.174$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,2,Mod(67,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("147.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 147.e (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$1.17380090971$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 2) q^{4} - 2 \zeta_{6} q^{5} - 2 q^{6} - \zeta_{6} q^{9} +O(q^{10})$$ q - 2*z * q^2 + (-z + 1) * q^3 + (2*z - 2) * q^4 - 2*z * q^5 - 2 * q^6 - z * q^9 $$q - 2 \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 2) q^{4} - 2 \zeta_{6} q^{5} - 2 q^{6} - \zeta_{6} q^{9} + (4 \zeta_{6} - 4) q^{10} + ( - 2 \zeta_{6} + 2) q^{11} + 2 \zeta_{6} q^{12} - q^{13} - 2 q^{15} + 4 \zeta_{6} q^{16} + (2 \zeta_{6} - 2) q^{18} + \zeta_{6} q^{19} + 4 q^{20} - 4 q^{22} + ( - \zeta_{6} + 1) q^{25} + 2 \zeta_{6} q^{26} - q^{27} + 4 q^{29} + 4 \zeta_{6} q^{30} + ( - 9 \zeta_{6} + 9) q^{31} + ( - 8 \zeta_{6} + 8) q^{32} - 2 \zeta_{6} q^{33} + 2 q^{36} - 3 \zeta_{6} q^{37} + ( - 2 \zeta_{6} + 2) q^{38} + (\zeta_{6} - 1) q^{39} + 10 q^{41} + 5 q^{43} + 4 \zeta_{6} q^{44} + (2 \zeta_{6} - 2) q^{45} - 6 \zeta_{6} q^{47} + 4 q^{48} - 2 q^{50} + ( - 2 \zeta_{6} + 2) q^{52} + (12 \zeta_{6} - 12) q^{53} + 2 \zeta_{6} q^{54} - 4 q^{55} + q^{57} - 8 \zeta_{6} q^{58} + (12 \zeta_{6} - 12) q^{59} + ( - 4 \zeta_{6} + 4) q^{60} + 10 \zeta_{6} q^{61} - 18 q^{62} - 8 q^{64} + 2 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{66} + ( - 5 \zeta_{6} + 5) q^{67} - 6 q^{71} + (3 \zeta_{6} - 3) q^{73} + (6 \zeta_{6} - 6) q^{74} - \zeta_{6} q^{75} - 2 q^{76} + 2 q^{78} + \zeta_{6} q^{79} + ( - 8 \zeta_{6} + 8) q^{80} + (\zeta_{6} - 1) q^{81} - 20 \zeta_{6} q^{82} - 6 q^{83} - 10 \zeta_{6} q^{86} + ( - 4 \zeta_{6} + 4) q^{87} + 16 \zeta_{6} q^{89} + 4 q^{90} - 9 \zeta_{6} q^{93} + (12 \zeta_{6} - 12) q^{94} + ( - 2 \zeta_{6} + 2) q^{95} - 8 \zeta_{6} q^{96} + 6 q^{97} - 2 q^{99} +O(q^{100})$$ q - 2*z * q^2 + (-z + 1) * q^3 + (2*z - 2) * q^4 - 2*z * q^5 - 2 * q^6 - z * q^9 + (4*z - 4) * q^10 + (-2*z + 2) * q^11 + 2*z * q^12 - q^13 - 2 * q^15 + 4*z * q^16 + (2*z - 2) * q^18 + z * q^19 + 4 * q^20 - 4 * q^22 + (-z + 1) * q^25 + 2*z * q^26 - q^27 + 4 * q^29 + 4*z * q^30 + (-9*z + 9) * q^31 + (-8*z + 8) * q^32 - 2*z * q^33 + 2 * q^36 - 3*z * q^37 + (-2*z + 2) * q^38 + (z - 1) * q^39 + 10 * q^41 + 5 * q^43 + 4*z * q^44 + (2*z - 2) * q^45 - 6*z * q^47 + 4 * q^48 - 2 * q^50 + (-2*z + 2) * q^52 + (12*z - 12) * q^53 + 2*z * q^54 - 4 * q^55 + q^57 - 8*z * q^58 + (12*z - 12) * q^59 + (-4*z + 4) * q^60 + 10*z * q^61 - 18 * q^62 - 8 * q^64 + 2*z * q^65 + (4*z - 4) * q^66 + (-5*z + 5) * q^67 - 6 * q^71 + (3*z - 3) * q^73 + (6*z - 6) * q^74 - z * q^75 - 2 * q^76 + 2 * q^78 + z * q^79 + (-8*z + 8) * q^80 + (z - 1) * q^81 - 20*z * q^82 - 6 * q^83 - 10*z * q^86 + (-4*z + 4) * q^87 + 16*z * q^89 + 4 * q^90 - 9*z * q^93 + (12*z - 12) * q^94 + (-2*z + 2) * q^95 - 8*z * q^96 + 6 * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + q^3 - 2 * q^4 - 2 * q^5 - 4 * q^6 - q^9 $$2 q - 2 q^{2} + q^{3} - 2 q^{4} - 2 q^{5} - 4 q^{6} - q^{9} - 4 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} - 4 q^{15} + 4 q^{16} - 2 q^{18} + q^{19} + 8 q^{20} - 8 q^{22} + q^{25} + 2 q^{26} - 2 q^{27} + 8 q^{29} + 4 q^{30} + 9 q^{31} + 8 q^{32} - 2 q^{33} + 4 q^{36} - 3 q^{37} + 2 q^{38} - q^{39} + 20 q^{41} + 10 q^{43} + 4 q^{44} - 2 q^{45} - 6 q^{47} + 8 q^{48} - 4 q^{50} + 2 q^{52} - 12 q^{53} + 2 q^{54} - 8 q^{55} + 2 q^{57} - 8 q^{58} - 12 q^{59} + 4 q^{60} + 10 q^{61} - 36 q^{62} - 16 q^{64} + 2 q^{65} - 4 q^{66} + 5 q^{67} - 12 q^{71} - 3 q^{73} - 6 q^{74} - q^{75} - 4 q^{76} + 4 q^{78} + q^{79} + 8 q^{80} - q^{81} - 20 q^{82} - 12 q^{83} - 10 q^{86} + 4 q^{87} + 16 q^{89} + 8 q^{90} - 9 q^{93} - 12 q^{94} + 2 q^{95} - 8 q^{96} + 12 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + q^3 - 2 * q^4 - 2 * q^5 - 4 * q^6 - q^9 - 4 * q^10 + 2 * q^11 + 2 * q^12 - 2 * q^13 - 4 * q^15 + 4 * q^16 - 2 * q^18 + q^19 + 8 * q^20 - 8 * q^22 + q^25 + 2 * q^26 - 2 * q^27 + 8 * q^29 + 4 * q^30 + 9 * q^31 + 8 * q^32 - 2 * q^33 + 4 * q^36 - 3 * q^37 + 2 * q^38 - q^39 + 20 * q^41 + 10 * q^43 + 4 * q^44 - 2 * q^45 - 6 * q^47 + 8 * q^48 - 4 * q^50 + 2 * q^52 - 12 * q^53 + 2 * q^54 - 8 * q^55 + 2 * q^57 - 8 * q^58 - 12 * q^59 + 4 * q^60 + 10 * q^61 - 36 * q^62 - 16 * q^64 + 2 * q^65 - 4 * q^66 + 5 * q^67 - 12 * q^71 - 3 * q^73 - 6 * q^74 - q^75 - 4 * q^76 + 4 * q^78 + q^79 + 8 * q^80 - q^81 - 20 * q^82 - 12 * q^83 - 10 * q^86 + 4 * q^87 + 16 * q^89 + 8 * q^90 - 9 * q^93 - 12 * q^94 + 2 * q^95 - 8 * q^96 + 12 * q^97 - 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 0.500000 0.866025i −1.00000 + 1.73205i −1.00000 1.73205i −2.00000 0 0 −0.500000 0.866025i −2.00000 + 3.46410i
79.1 −1.00000 + 1.73205i 0.500000 + 0.866025i −1.00000 1.73205i −1.00000 + 1.73205i −2.00000 0 0 −0.500000 + 0.866025i −2.00000 3.46410i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.2.e.a 2
3.b odd 2 1 441.2.e.e 2
4.b odd 2 1 2352.2.q.c 2
7.b odd 2 1 21.2.e.a 2
7.c even 3 1 147.2.a.b 1
7.c even 3 1 inner 147.2.e.a 2
7.d odd 6 1 21.2.e.a 2
7.d odd 6 1 147.2.a.c 1
21.c even 2 1 63.2.e.b 2
21.g even 6 1 63.2.e.b 2
21.g even 6 1 441.2.a.b 1
21.h odd 6 1 441.2.a.a 1
21.h odd 6 1 441.2.e.e 2
28.d even 2 1 336.2.q.f 2
28.f even 6 1 336.2.q.f 2
28.f even 6 1 2352.2.a.d 1
28.g odd 6 1 2352.2.a.w 1
28.g odd 6 1 2352.2.q.c 2
35.c odd 2 1 525.2.i.e 2
35.f even 4 2 525.2.r.e 4
35.i odd 6 1 525.2.i.e 2
35.i odd 6 1 3675.2.a.a 1
35.j even 6 1 3675.2.a.c 1
35.k even 12 2 525.2.r.e 4
56.e even 2 1 1344.2.q.c 2
56.h odd 2 1 1344.2.q.m 2
56.j odd 6 1 1344.2.q.m 2
56.j odd 6 1 9408.2.a.bg 1
56.k odd 6 1 9408.2.a.k 1
56.m even 6 1 1344.2.q.c 2
56.m even 6 1 9408.2.a.cv 1
56.p even 6 1 9408.2.a.bz 1
63.i even 6 1 567.2.g.f 2
63.k odd 6 1 567.2.h.f 2
63.l odd 6 1 567.2.g.a 2
63.l odd 6 1 567.2.h.f 2
63.o even 6 1 567.2.g.f 2
63.o even 6 1 567.2.h.a 2
63.s even 6 1 567.2.h.a 2
63.t odd 6 1 567.2.g.a 2
84.h odd 2 1 1008.2.s.d 2
84.j odd 6 1 1008.2.s.d 2
84.j odd 6 1 7056.2.a.bp 1
84.n even 6 1 7056.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 7.b odd 2 1
21.2.e.a 2 7.d odd 6 1
63.2.e.b 2 21.c even 2 1
63.2.e.b 2 21.g even 6 1
147.2.a.b 1 7.c even 3 1
147.2.a.c 1 7.d odd 6 1
147.2.e.a 2 1.a even 1 1 trivial
147.2.e.a 2 7.c even 3 1 inner
336.2.q.f 2 28.d even 2 1
336.2.q.f 2 28.f even 6 1
441.2.a.a 1 21.h odd 6 1
441.2.a.b 1 21.g even 6 1
441.2.e.e 2 3.b odd 2 1
441.2.e.e 2 21.h odd 6 1
525.2.i.e 2 35.c odd 2 1
525.2.i.e 2 35.i odd 6 1
525.2.r.e 4 35.f even 4 2
525.2.r.e 4 35.k even 12 2
567.2.g.a 2 63.l odd 6 1
567.2.g.a 2 63.t odd 6 1
567.2.g.f 2 63.i even 6 1
567.2.g.f 2 63.o even 6 1
567.2.h.a 2 63.o even 6 1
567.2.h.a 2 63.s even 6 1
567.2.h.f 2 63.k odd 6 1
567.2.h.f 2 63.l odd 6 1
1008.2.s.d 2 84.h odd 2 1
1008.2.s.d 2 84.j odd 6 1
1344.2.q.c 2 56.e even 2 1
1344.2.q.c 2 56.m even 6 1
1344.2.q.m 2 56.h odd 2 1
1344.2.q.m 2 56.j odd 6 1
2352.2.a.d 1 28.f even 6 1
2352.2.a.w 1 28.g odd 6 1
2352.2.q.c 2 4.b odd 2 1
2352.2.q.c 2 28.g odd 6 1
3675.2.a.a 1 35.i odd 6 1
3675.2.a.c 1 35.j even 6 1
7056.2.a.m 1 84.n even 6 1
7056.2.a.bp 1 84.j odd 6 1
9408.2.a.k 1 56.k odd 6 1
9408.2.a.bg 1 56.j odd 6 1
9408.2.a.bz 1 56.p even 6 1
9408.2.a.cv 1 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{2} + 2T_{2} + 4$$ T2^2 + 2*T2 + 4 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2}$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} - 9T + 81$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$(T - 10)^{2}$$
$43$ $$(T - 5)^{2}$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - 10T + 100$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 3T + 9$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 16T + 256$$
$97$ $$(T - 6)^{2}$$