# Properties

 Label 147.2.e.d Level $147$ Weight $2$ Character orbit 147.e Analytic conductor $1.174$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{2} + \beta_{2} q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + ( - 2 \beta_{2} + \beta_1 - 2) q^{5} + (\beta_{3} - 1) q^{6} + (\beta_{3} - 3) q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^2 + b2 * q^3 + (2*b3 + b2 + 2*b1) * q^4 + (-2*b2 + b1 - 2) * q^5 + (b3 - 1) * q^6 + (b3 - 3) * q^8 + (-b2 - 1) * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{2} + \beta_{2} q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + ( - 2 \beta_{2} + \beta_1 - 2) q^{5} + (\beta_{3} - 1) q^{6} + (\beta_{3} - 3) q^{8} + ( - \beta_{2} - 1) q^{9} + ( - \beta_{3} - \beta_1) q^{10} - 2 \beta_{2} q^{11} + ( - \beta_{2} - 2 \beta_1 - 1) q^{12} + ( - \beta_{3} + 4) q^{13} + (\beta_{3} + 2) q^{15} + ( - 3 \beta_{2} - 3) q^{16} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{18} - 2 \beta_1 q^{19} + ( - 3 \beta_{3} - 2) q^{20} + ( - 2 \beta_{3} + 2) q^{22} + (2 \beta_{2} - 4 \beta_1 + 2) q^{23} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{24} + ( - 4 \beta_{3} + \beta_{2} - 4 \beta_1) q^{25} + (6 \beta_{2} + 5 \beta_1 + 6) q^{26} + q^{27} + ( - 2 \beta_{3} - 4) q^{29} + \beta_1 q^{30} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{31} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{32} + (2 \beta_{2} + 2) q^{33} + (5 \beta_{3} - 8) q^{34} + ( - 2 \beta_{3} + 1) q^{36} + (4 \beta_{2} + 4) q^{37} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{38} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{39} + (4 \beta_{2} - \beta_1 + 4) q^{40} + (3 \beta_{3} + 2) q^{41} + 4 \beta_{3} q^{43} + (2 \beta_{2} + 4 \beta_1 + 2) q^{44} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{45} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{46} - 2 \beta_1 q^{47} + 3 q^{48} + ( - 3 \beta_{3} + 7) q^{50} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{51} + (9 \beta_{3} + 8 \beta_{2} + 9 \beta_1) q^{52} - 2 \beta_{2} q^{53} + (\beta_{2} + \beta_1 + 1) q^{54} + ( - 2 \beta_{3} - 4) q^{55} - 2 \beta_{3} q^{57} - 2 \beta_1 q^{58} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{59} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{60} + ( - 8 \beta_{2} + 3 \beta_1 - 8) q^{61} + ( - 6 \beta_{3} + 8) q^{62} + (2 \beta_{3} - 7) q^{64} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{65} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{66} + (4 \beta_{3} + 4 \beta_1) q^{67} + ( - 14 \beta_{2} - 7 \beta_1 - 14) q^{68} + ( - 4 \beta_{3} - 2) q^{69} + (8 \beta_{3} - 2) q^{71} + (3 \beta_{2} + \beta_1 + 3) q^{72} + ( - 7 \beta_{3} + 4 \beta_{2} - 7 \beta_1) q^{73} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{74} + ( - \beta_{2} + 4 \beta_1 - 1) q^{75} + ( - 2 \beta_{3} + 8) q^{76} + (5 \beta_{3} - 6) q^{78} + ( - 8 \beta_{2} + 4 \beta_1 - 8) q^{79} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1) q^{80} + \beta_{2} q^{81} + ( - 4 \beta_{2} - \beta_1 - 4) q^{82} + (8 \beta_{3} - 4) q^{83} + ( - 4 \beta_{3} - 2) q^{85} + ( - 8 \beta_{2} - 4 \beta_1 - 8) q^{86} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{87} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{88} + (10 \beta_{2} - 3 \beta_1 + 10) q^{89} + \beta_{3} q^{90} + 14 q^{92} + (4 \beta_{2} + 2 \beta_1 + 4) q^{93} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{94} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{95} + ( - 3 \beta_{2} + \beta_1 - 3) q^{96} + ( - \beta_{3} + 4) q^{97} - 2 q^{99}+O(q^{100})$$ q + (b2 + b1 + 1) * q^2 + b2 * q^3 + (2*b3 + b2 + 2*b1) * q^4 + (-2*b2 + b1 - 2) * q^5 + (b3 - 1) * q^6 + (b3 - 3) * q^8 + (-b2 - 1) * q^9 + (-b3 - b1) * q^10 - 2*b2 * q^11 + (-b2 - 2*b1 - 1) * q^12 + (-b3 + 4) * q^13 + (b3 + 2) * q^15 + (-3*b2 - 3) * q^16 + (3*b3 + 2*b2 + 3*b1) * q^17 + (-b3 - b2 - b1) * q^18 - 2*b1 * q^19 + (-3*b3 - 2) * q^20 + (-2*b3 + 2) * q^22 + (2*b2 - 4*b1 + 2) * q^23 + (-b3 - 3*b2 - b1) * q^24 + (-4*b3 + b2 - 4*b1) * q^25 + (6*b2 + 5*b1 + 6) * q^26 + q^27 + (-2*b3 - 4) * q^29 + b1 * q^30 + (-2*b3 - 4*b2 - 2*b1) * q^31 + (-b3 + 3*b2 - b1) * q^32 + (2*b2 + 2) * q^33 + (5*b3 - 8) * q^34 + (-2*b3 + 1) * q^36 + (4*b2 + 4) * q^37 + (-2*b3 - 4*b2 - 2*b1) * q^38 + (b3 + 4*b2 + b1) * q^39 + (4*b2 - b1 + 4) * q^40 + (3*b3 + 2) * q^41 + 4*b3 * q^43 + (2*b2 + 4*b1 + 2) * q^44 + (-b3 + 2*b2 - b1) * q^45 + (-2*b3 - 6*b2 - 2*b1) * q^46 - 2*b1 * q^47 + 3 * q^48 + (-3*b3 + 7) * q^50 + (-2*b2 - 3*b1 - 2) * q^51 + (9*b3 + 8*b2 + 9*b1) * q^52 - 2*b2 * q^53 + (b2 + b1 + 1) * q^54 + (-2*b3 - 4) * q^55 - 2*b3 * q^57 - 2*b1 * q^58 + (-2*b3 - 4*b2 - 2*b1) * q^59 + (3*b3 - 2*b2 + 3*b1) * q^60 + (-8*b2 + 3*b1 - 8) * q^61 + (-6*b3 + 8) * q^62 + (2*b3 - 7) * q^64 + (-6*b2 + 2*b1 - 6) * q^65 + (2*b3 + 2*b2 + 2*b1) * q^66 + (4*b3 + 4*b1) * q^67 + (-14*b2 - 7*b1 - 14) * q^68 + (-4*b3 - 2) * q^69 + (8*b3 - 2) * q^71 + (3*b2 + b1 + 3) * q^72 + (-7*b3 + 4*b2 - 7*b1) * q^73 + (4*b3 + 4*b2 + 4*b1) * q^74 + (-b2 + 4*b1 - 1) * q^75 + (-2*b3 + 8) * q^76 + (5*b3 - 6) * q^78 + (-8*b2 + 4*b1 - 8) * q^79 + (-3*b3 + 6*b2 - 3*b1) * q^80 + b2 * q^81 + (-4*b2 - b1 - 4) * q^82 + (8*b3 - 4) * q^83 + (-4*b3 - 2) * q^85 + (-8*b2 - 4*b1 - 8) * q^86 + (2*b3 - 4*b2 + 2*b1) * q^87 + (2*b3 + 6*b2 + 2*b1) * q^88 + (10*b2 - 3*b1 + 10) * q^89 + b3 * q^90 + 14 * q^92 + (4*b2 + 2*b1 + 4) * q^93 + (-2*b3 - 4*b2 - 2*b1) * q^94 + (4*b3 - 4*b2 + 4*b1) * q^95 + (-3*b2 + b1 - 3) * q^96 + (-b3 + 4) * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 - 2 * q^9 $$4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 4 q^{5} - 4 q^{6} - 12 q^{8} - 2 q^{9} + 4 q^{11} - 2 q^{12} + 16 q^{13} + 8 q^{15} - 6 q^{16} - 4 q^{17} + 2 q^{18} - 8 q^{20} + 8 q^{22} + 4 q^{23} + 6 q^{24} - 2 q^{25} + 12 q^{26} + 4 q^{27} - 16 q^{29} + 8 q^{31} - 6 q^{32} + 4 q^{33} - 32 q^{34} + 4 q^{36} + 8 q^{37} + 8 q^{38} - 8 q^{39} + 8 q^{40} + 8 q^{41} + 4 q^{44} - 4 q^{45} + 12 q^{46} + 12 q^{48} + 28 q^{50} - 4 q^{51} - 16 q^{52} + 4 q^{53} + 2 q^{54} - 16 q^{55} + 8 q^{59} + 4 q^{60} - 16 q^{61} + 32 q^{62} - 28 q^{64} - 12 q^{65} - 4 q^{66} - 28 q^{68} - 8 q^{69} - 8 q^{71} + 6 q^{72} - 8 q^{73} - 8 q^{74} - 2 q^{75} + 32 q^{76} - 24 q^{78} - 16 q^{79} - 12 q^{80} - 2 q^{81} - 8 q^{82} - 16 q^{83} - 8 q^{85} - 16 q^{86} + 8 q^{87} - 12 q^{88} + 20 q^{89} + 56 q^{92} + 8 q^{93} + 8 q^{94} + 8 q^{95} - 6 q^{96} + 16 q^{97} - 8 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 - 4 * q^5 - 4 * q^6 - 12 * q^8 - 2 * q^9 + 4 * q^11 - 2 * q^12 + 16 * q^13 + 8 * q^15 - 6 * q^16 - 4 * q^17 + 2 * q^18 - 8 * q^20 + 8 * q^22 + 4 * q^23 + 6 * q^24 - 2 * q^25 + 12 * q^26 + 4 * q^27 - 16 * q^29 + 8 * q^31 - 6 * q^32 + 4 * q^33 - 32 * q^34 + 4 * q^36 + 8 * q^37 + 8 * q^38 - 8 * q^39 + 8 * q^40 + 8 * q^41 + 4 * q^44 - 4 * q^45 + 12 * q^46 + 12 * q^48 + 28 * q^50 - 4 * q^51 - 16 * q^52 + 4 * q^53 + 2 * q^54 - 16 * q^55 + 8 * q^59 + 4 * q^60 - 16 * q^61 + 32 * q^62 - 28 * q^64 - 12 * q^65 - 4 * q^66 - 28 * q^68 - 8 * q^69 - 8 * q^71 + 6 * q^72 - 8 * q^73 - 8 * q^74 - 2 * q^75 + 32 * q^76 - 24 * q^78 - 16 * q^79 - 12 * q^80 - 2 * q^81 - 8 * q^82 - 16 * q^83 - 8 * q^85 - 16 * q^86 + 8 * q^87 - 12 * q^88 + 20 * q^89 + 56 * q^92 + 8 * q^93 + 8 * q^94 + 8 * q^95 - 6 * q^96 + 16 * q^97 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.207107 0.358719i −0.500000 + 0.866025i 0.914214 1.58346i −1.70711 2.95680i 0.414214 0 −1.58579 −0.500000 0.866025i −0.707107 + 1.22474i
67.2 1.20711 + 2.09077i −0.500000 + 0.866025i −1.91421 + 3.31552i −0.292893 0.507306i −2.41421 0 −4.41421 −0.500000 0.866025i 0.707107 1.22474i
79.1 −0.207107 + 0.358719i −0.500000 0.866025i 0.914214 + 1.58346i −1.70711 + 2.95680i 0.414214 0 −1.58579 −0.500000 + 0.866025i −0.707107 1.22474i
79.2 1.20711 2.09077i −0.500000 0.866025i −1.91421 3.31552i −0.292893 + 0.507306i −2.41421 0 −4.41421 −0.500000 + 0.866025i 0.707107 + 1.22474i
 $$n$$: 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.d 4
3.b odd 2 1 441.2.e.g 4
4.b odd 2 1 2352.2.q.bd 4
7.b odd 2 1 147.2.e.e 4
7.c even 3 1 147.2.a.e yes 2
7.c even 3 1 inner 147.2.e.d 4
7.d odd 6 1 147.2.a.d 2
7.d odd 6 1 147.2.e.e 4
21.c even 2 1 441.2.e.f 4
21.g even 6 1 441.2.a.j 2
21.g even 6 1 441.2.e.f 4
21.h odd 6 1 441.2.a.i 2
21.h odd 6 1 441.2.e.g 4
28.d even 2 1 2352.2.q.bb 4
28.f even 6 1 2352.2.a.be 2
28.f even 6 1 2352.2.q.bb 4
28.g odd 6 1 2352.2.a.bc 2
28.g odd 6 1 2352.2.q.bd 4
35.i odd 6 1 3675.2.a.bf 2
35.j even 6 1 3675.2.a.bd 2
56.j odd 6 1 9408.2.a.ef 2
56.k odd 6 1 9408.2.a.dt 2
56.m even 6 1 9408.2.a.dq 2
56.p even 6 1 9408.2.a.di 2
84.j odd 6 1 7056.2.a.cv 2
84.n even 6 1 7056.2.a.cf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 7.d odd 6 1
147.2.a.e yes 2 7.c even 3 1
147.2.e.d 4 1.a even 1 1 trivial
147.2.e.d 4 7.c even 3 1 inner
147.2.e.e 4 7.b odd 2 1
147.2.e.e 4 7.d odd 6 1
441.2.a.i 2 21.h odd 6 1
441.2.a.j 2 21.g even 6 1
441.2.e.f 4 21.c even 2 1
441.2.e.f 4 21.g even 6 1
441.2.e.g 4 3.b odd 2 1
441.2.e.g 4 21.h odd 6 1
2352.2.a.bc 2 28.g odd 6 1
2352.2.a.be 2 28.f even 6 1
2352.2.q.bb 4 28.d even 2 1
2352.2.q.bb 4 28.f even 6 1
2352.2.q.bd 4 4.b odd 2 1
2352.2.q.bd 4 28.g odd 6 1
3675.2.a.bd 2 35.j even 6 1
3675.2.a.bf 2 35.i odd 6 1
7056.2.a.cf 2 84.n even 6 1
7056.2.a.cv 2 84.j odd 6 1
9408.2.a.di 2 56.p even 6 1
9408.2.a.dq 2 56.m even 6 1
9408.2.a.dt 2 56.k odd 6 1
9408.2.a.ef 2 56.j odd 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1$$ T2^4 - 2*T2^3 + 5*T2^2 + 2*T2 + 1 $$T_{5}^{4} + 4T_{5}^{3} + 14T_{5}^{2} + 8T_{5} + 4$$ T5^4 + 4*T5^3 + 14*T5^2 + 8*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + \cdots + 1$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$T^{4} + 4 T^{3} + \cdots + 4$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$(T^{2} - 8 T + 14)^{2}$$
$17$ $$T^{4} + 4 T^{3} + \cdots + 196$$
$19$ $$T^{4} + 8T^{2} + 64$$
$23$ $$T^{4} - 4 T^{3} + \cdots + 784$$
$29$ $$(T^{2} + 8 T + 8)^{2}$$
$31$ $$T^{4} - 8 T^{3} + \cdots + 64$$
$37$ $$(T^{2} - 4 T + 16)^{2}$$
$41$ $$(T^{2} - 4 T - 14)^{2}$$
$43$ $$(T^{2} - 32)^{2}$$
$47$ $$T^{4} + 8T^{2} + 64$$
$53$ $$(T^{2} - 2 T + 4)^{2}$$
$59$ $$T^{4} - 8 T^{3} + \cdots + 64$$
$61$ $$T^{4} + 16 T^{3} + \cdots + 2116$$
$67$ $$T^{4} + 32T^{2} + 1024$$
$71$ $$(T^{2} + 4 T - 124)^{2}$$
$73$ $$T^{4} + 8 T^{3} + \cdots + 6724$$
$79$ $$T^{4} + 16 T^{3} + \cdots + 1024$$
$83$ $$(T^{2} + 8 T - 112)^{2}$$
$89$ $$T^{4} - 20 T^{3} + \cdots + 6724$$
$97$ $$(T^{2} - 8 T + 14)^{2}$$