# Properties

 Label 147.2.e.b 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + ( - \zeta_{6} + 1) q^{4} + 2 \zeta_{6} q^{5} - q^{6} + 3 q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + (z - 1) * q^3 + (-z + 1) * q^4 + 2*z * q^5 - q^6 + 3 * q^8 - z * q^9 $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + ( - \zeta_{6} + 1) q^{4} + 2 \zeta_{6} q^{5} - q^{6} + 3 q^{8} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{10} + (4 \zeta_{6} - 4) q^{11} + \zeta_{6} q^{12} - 2 q^{13} - 2 q^{15} + \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + ( - \zeta_{6} + 1) q^{18} - 4 \zeta_{6} q^{19} + 2 q^{20} - 4 q^{22} + (3 \zeta_{6} - 3) q^{24} + ( - \zeta_{6} + 1) q^{25} - 2 \zeta_{6} q^{26} + q^{27} - 2 q^{29} - 2 \zeta_{6} q^{30} + ( - 5 \zeta_{6} + 5) q^{32} - 4 \zeta_{6} q^{33} + 6 q^{34} - q^{36} - 6 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + ( - 2 \zeta_{6} + 2) q^{39} + 6 \zeta_{6} q^{40} + 2 q^{41} - 4 q^{43} + 4 \zeta_{6} q^{44} + ( - 2 \zeta_{6} + 2) q^{45} - q^{48} + q^{50} + 6 \zeta_{6} q^{51} + (2 \zeta_{6} - 2) q^{52} + (6 \zeta_{6} - 6) q^{53} + \zeta_{6} q^{54} - 8 q^{55} + 4 q^{57} - 2 \zeta_{6} q^{58} + (12 \zeta_{6} - 12) q^{59} + (2 \zeta_{6} - 2) q^{60} + 2 \zeta_{6} q^{61} + 7 q^{64} - 4 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{66} + (4 \zeta_{6} - 4) q^{67} - 6 \zeta_{6} q^{68} - 3 \zeta_{6} q^{72} + ( - 6 \zeta_{6} + 6) q^{73} + ( - 6 \zeta_{6} + 6) q^{74} + \zeta_{6} q^{75} - 4 q^{76} + 2 q^{78} + 16 \zeta_{6} q^{79} + (2 \zeta_{6} - 2) q^{80} + (\zeta_{6} - 1) q^{81} + 2 \zeta_{6} q^{82} - 12 q^{83} + 12 q^{85} - 4 \zeta_{6} q^{86} + ( - 2 \zeta_{6} + 2) q^{87} + (12 \zeta_{6} - 12) q^{88} + 14 \zeta_{6} q^{89} + 2 q^{90} + ( - 8 \zeta_{6} + 8) q^{95} + 5 \zeta_{6} q^{96} + 18 q^{97} + 4 q^{99} +O(q^{100})$$ q + z * q^2 + (z - 1) * q^3 + (-z + 1) * q^4 + 2*z * q^5 - q^6 + 3 * q^8 - z * q^9 + (2*z - 2) * q^10 + (4*z - 4) * q^11 + z * q^12 - 2 * q^13 - 2 * q^15 + z * q^16 + (-6*z + 6) * q^17 + (-z + 1) * q^18 - 4*z * q^19 + 2 * q^20 - 4 * q^22 + (3*z - 3) * q^24 + (-z + 1) * q^25 - 2*z * q^26 + q^27 - 2 * q^29 - 2*z * q^30 + (-5*z + 5) * q^32 - 4*z * q^33 + 6 * q^34 - q^36 - 6*z * q^37 + (-4*z + 4) * q^38 + (-2*z + 2) * q^39 + 6*z * q^40 + 2 * q^41 - 4 * q^43 + 4*z * q^44 + (-2*z + 2) * q^45 - q^48 + q^50 + 6*z * q^51 + (2*z - 2) * q^52 + (6*z - 6) * q^53 + z * q^54 - 8 * q^55 + 4 * q^57 - 2*z * q^58 + (12*z - 12) * q^59 + (2*z - 2) * q^60 + 2*z * q^61 + 7 * q^64 - 4*z * q^65 + (-4*z + 4) * q^66 + (4*z - 4) * q^67 - 6*z * q^68 - 3*z * q^72 + (-6*z + 6) * q^73 + (-6*z + 6) * q^74 + z * q^75 - 4 * q^76 + 2 * q^78 + 16*z * q^79 + (2*z - 2) * q^80 + (z - 1) * q^81 + 2*z * q^82 - 12 * q^83 + 12 * q^85 - 4*z * q^86 + (-2*z + 2) * q^87 + (12*z - 12) * q^88 + 14*z * q^89 + 2 * q^90 + (-8*z + 8) * q^95 + 5*z * q^96 + 18 * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{3} + q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 - q^3 + q^4 + 2 * q^5 - 2 * q^6 + 6 * q^8 - q^9 $$2 q + q^{2} - q^{3} + q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8} - q^{9} - 2 q^{10} - 4 q^{11} + q^{12} - 4 q^{13} - 4 q^{15} + q^{16} + 6 q^{17} + q^{18} - 4 q^{19} + 4 q^{20} - 8 q^{22} - 3 q^{24} + q^{25} - 2 q^{26} + 2 q^{27} - 4 q^{29} - 2 q^{30} + 5 q^{32} - 4 q^{33} + 12 q^{34} - 2 q^{36} - 6 q^{37} + 4 q^{38} + 2 q^{39} + 6 q^{40} + 4 q^{41} - 8 q^{43} + 4 q^{44} + 2 q^{45} - 2 q^{48} + 2 q^{50} + 6 q^{51} - 2 q^{52} - 6 q^{53} + q^{54} - 16 q^{55} + 8 q^{57} - 2 q^{58} - 12 q^{59} - 2 q^{60} + 2 q^{61} + 14 q^{64} - 4 q^{65} + 4 q^{66} - 4 q^{67} - 6 q^{68} - 3 q^{72} + 6 q^{73} + 6 q^{74} + q^{75} - 8 q^{76} + 4 q^{78} + 16 q^{79} - 2 q^{80} - q^{81} + 2 q^{82} - 24 q^{83} + 24 q^{85} - 4 q^{86} + 2 q^{87} - 12 q^{88} + 14 q^{89} + 4 q^{90} + 8 q^{95} + 5 q^{96} + 36 q^{97} + 8 q^{99}+O(q^{100})$$ 2 * q + q^2 - q^3 + q^4 + 2 * q^5 - 2 * q^6 + 6 * q^8 - q^9 - 2 * q^10 - 4 * q^11 + q^12 - 4 * q^13 - 4 * q^15 + q^16 + 6 * q^17 + q^18 - 4 * q^19 + 4 * q^20 - 8 * q^22 - 3 * q^24 + q^25 - 2 * q^26 + 2 * q^27 - 4 * q^29 - 2 * q^30 + 5 * q^32 - 4 * q^33 + 12 * q^34 - 2 * q^36 - 6 * q^37 + 4 * q^38 + 2 * q^39 + 6 * q^40 + 4 * q^41 - 8 * q^43 + 4 * q^44 + 2 * q^45 - 2 * q^48 + 2 * q^50 + 6 * q^51 - 2 * q^52 - 6 * q^53 + q^54 - 16 * q^55 + 8 * q^57 - 2 * q^58 - 12 * q^59 - 2 * q^60 + 2 * q^61 + 14 * q^64 - 4 * q^65 + 4 * q^66 - 4 * q^67 - 6 * q^68 - 3 * q^72 + 6 * q^73 + 6 * q^74 + q^75 - 8 * q^76 + 4 * q^78 + 16 * q^79 - 2 * q^80 - q^81 + 2 * q^82 - 24 * q^83 + 24 * q^85 - 4 * q^86 + 2 * q^87 - 12 * q^88 + 14 * q^89 + 4 * q^90 + 8 * q^95 + 5 * q^96 + 36 * q^97 + 8 * 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
0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i 1.00000 + 1.73205i −1.00000 0 3.00000 −0.500000 0.866025i −1.00000 + 1.73205i
79.1 0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 1.00000 1.73205i −1.00000 0 3.00000 −0.500000 + 0.866025i −1.00000 1.73205i
 $$n$$: 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.b 2
3.b odd 2 1 441.2.e.a 2
4.b odd 2 1 2352.2.q.x 2
7.b odd 2 1 147.2.e.c 2
7.c even 3 1 21.2.a.a 1
7.c even 3 1 inner 147.2.e.b 2
7.d odd 6 1 147.2.a.a 1
7.d odd 6 1 147.2.e.c 2
21.c even 2 1 441.2.e.b 2
21.g even 6 1 441.2.a.f 1
21.g even 6 1 441.2.e.b 2
21.h odd 6 1 63.2.a.a 1
21.h odd 6 1 441.2.e.a 2
28.d even 2 1 2352.2.q.e 2
28.f even 6 1 2352.2.a.v 1
28.f even 6 1 2352.2.q.e 2
28.g odd 6 1 336.2.a.a 1
28.g odd 6 1 2352.2.q.x 2
35.i odd 6 1 3675.2.a.n 1
35.j even 6 1 525.2.a.d 1
35.l odd 12 2 525.2.d.a 2
56.j odd 6 1 9408.2.a.bv 1
56.k odd 6 1 1344.2.a.s 1
56.m even 6 1 9408.2.a.m 1
56.p even 6 1 1344.2.a.g 1
63.g even 3 1 567.2.f.g 2
63.h even 3 1 567.2.f.g 2
63.j odd 6 1 567.2.f.b 2
63.n odd 6 1 567.2.f.b 2
77.h odd 6 1 2541.2.a.j 1
84.j odd 6 1 7056.2.a.p 1
84.n even 6 1 1008.2.a.l 1
91.r even 6 1 3549.2.a.c 1
105.o odd 6 1 1575.2.a.c 1
105.x even 12 2 1575.2.d.a 2
112.u odd 12 2 5376.2.c.l 2
112.w even 12 2 5376.2.c.r 2
119.j even 6 1 6069.2.a.b 1
133.r odd 6 1 7581.2.a.d 1
140.p odd 6 1 8400.2.a.bn 1
168.s odd 6 1 4032.2.a.h 1
168.v even 6 1 4032.2.a.k 1
231.l even 6 1 7623.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 7.c even 3 1
63.2.a.a 1 21.h odd 6 1
147.2.a.a 1 7.d odd 6 1
147.2.e.b 2 1.a even 1 1 trivial
147.2.e.b 2 7.c even 3 1 inner
147.2.e.c 2 7.b odd 2 1
147.2.e.c 2 7.d odd 6 1
336.2.a.a 1 28.g odd 6 1
441.2.a.f 1 21.g even 6 1
441.2.e.a 2 3.b odd 2 1
441.2.e.a 2 21.h odd 6 1
441.2.e.b 2 21.c even 2 1
441.2.e.b 2 21.g even 6 1
525.2.a.d 1 35.j even 6 1
525.2.d.a 2 35.l odd 12 2
567.2.f.b 2 63.j odd 6 1
567.2.f.b 2 63.n odd 6 1
567.2.f.g 2 63.g even 3 1
567.2.f.g 2 63.h even 3 1
1008.2.a.l 1 84.n even 6 1
1344.2.a.g 1 56.p even 6 1
1344.2.a.s 1 56.k odd 6 1
1575.2.a.c 1 105.o odd 6 1
1575.2.d.a 2 105.x even 12 2
2352.2.a.v 1 28.f even 6 1
2352.2.q.e 2 28.d even 2 1
2352.2.q.e 2 28.f even 6 1
2352.2.q.x 2 4.b odd 2 1
2352.2.q.x 2 28.g odd 6 1
2541.2.a.j 1 77.h odd 6 1
3549.2.a.c 1 91.r even 6 1
3675.2.a.n 1 35.i odd 6 1
4032.2.a.h 1 168.s odd 6 1
4032.2.a.k 1 168.v even 6 1
5376.2.c.l 2 112.u odd 12 2
5376.2.c.r 2 112.w even 12 2
6069.2.a.b 1 119.j even 6 1
7056.2.a.p 1 84.j odd 6 1
7581.2.a.d 1 133.r odd 6 1
7623.2.a.g 1 231.l even 6 1
8400.2.a.bn 1 140.p odd 6 1
9408.2.a.m 1 56.m even 6 1
9408.2.a.bv 1 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 $$S_{2}^{\mathrm{new}}(147, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2}$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 6T + 36$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 6T + 36$$
$79$ $$T^{2} - 16T + 256$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} - 14T + 196$$
$97$ $$(T - 18)^{2}$$