# Properties

 Label 147.4.e.b Level $147$ Weight $4$ Character orbit 147.e Analytic conductor $8.673$ 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] = [147,4,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, 4, names="a")

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

chi := DirichletCharacter("147.67");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$8.67328077084$$ 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.

 $$f(q)$$ $$=$$ $$q - 4 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (8 \zeta_{6} - 8) q^{4} - 4 \zeta_{6} q^{5} + 12 q^{6} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q - 4*z * q^2 + (3*z - 3) * q^3 + (8*z - 8) * q^4 - 4*z * q^5 + 12 * q^6 - 9*z * q^9 $$q - 4 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (8 \zeta_{6} - 8) q^{4} - 4 \zeta_{6} q^{5} + 12 q^{6} - 9 \zeta_{6} q^{9} + (16 \zeta_{6} - 16) q^{10} + (62 \zeta_{6} - 62) q^{11} - 24 \zeta_{6} q^{12} + 62 q^{13} + 12 q^{15} + 64 \zeta_{6} q^{16} + ( - 84 \zeta_{6} + 84) q^{17} + (36 \zeta_{6} - 36) q^{18} + 100 \zeta_{6} q^{19} + 32 q^{20} + 248 q^{22} + 42 \zeta_{6} q^{23} + ( - 109 \zeta_{6} + 109) q^{25} - 248 \zeta_{6} q^{26} + 27 q^{27} - 10 q^{29} - 48 \zeta_{6} q^{30} + (48 \zeta_{6} - 48) q^{31} + ( - 256 \zeta_{6} + 256) q^{32} - 186 \zeta_{6} q^{33} - 336 q^{34} + 72 q^{36} + 246 \zeta_{6} q^{37} + ( - 400 \zeta_{6} + 400) q^{38} + (186 \zeta_{6} - 186) q^{39} + 248 q^{41} + 68 q^{43} - 496 \zeta_{6} q^{44} + (36 \zeta_{6} - 36) q^{45} + ( - 168 \zeta_{6} + 168) q^{46} + 324 \zeta_{6} q^{47} - 192 q^{48} - 436 q^{50} + 252 \zeta_{6} q^{51} + (496 \zeta_{6} - 496) q^{52} + (258 \zeta_{6} - 258) q^{53} - 108 \zeta_{6} q^{54} + 248 q^{55} - 300 q^{57} + 40 \zeta_{6} q^{58} + ( - 120 \zeta_{6} + 120) q^{59} + (96 \zeta_{6} - 96) q^{60} + 622 \zeta_{6} q^{61} + 192 q^{62} - 512 q^{64} - 248 \zeta_{6} q^{65} + (744 \zeta_{6} - 744) q^{66} + (904 \zeta_{6} - 904) q^{67} + 672 \zeta_{6} q^{68} - 126 q^{69} - 678 q^{71} + (642 \zeta_{6} - 642) q^{73} + ( - 984 \zeta_{6} + 984) q^{74} + 327 \zeta_{6} q^{75} - 800 q^{76} + 744 q^{78} - 740 \zeta_{6} q^{79} + ( - 256 \zeta_{6} + 256) q^{80} + (81 \zeta_{6} - 81) q^{81} - 992 \zeta_{6} q^{82} - 468 q^{83} - 336 q^{85} - 272 \zeta_{6} q^{86} + ( - 30 \zeta_{6} + 30) q^{87} + 200 \zeta_{6} q^{89} + 144 q^{90} - 336 q^{92} - 144 \zeta_{6} q^{93} + ( - 1296 \zeta_{6} + 1296) q^{94} + ( - 400 \zeta_{6} + 400) q^{95} + 768 \zeta_{6} q^{96} + 1266 q^{97} + 558 q^{99} +O(q^{100})$$ q - 4*z * q^2 + (3*z - 3) * q^3 + (8*z - 8) * q^4 - 4*z * q^5 + 12 * q^6 - 9*z * q^9 + (16*z - 16) * q^10 + (62*z - 62) * q^11 - 24*z * q^12 + 62 * q^13 + 12 * q^15 + 64*z * q^16 + (-84*z + 84) * q^17 + (36*z - 36) * q^18 + 100*z * q^19 + 32 * q^20 + 248 * q^22 + 42*z * q^23 + (-109*z + 109) * q^25 - 248*z * q^26 + 27 * q^27 - 10 * q^29 - 48*z * q^30 + (48*z - 48) * q^31 + (-256*z + 256) * q^32 - 186*z * q^33 - 336 * q^34 + 72 * q^36 + 246*z * q^37 + (-400*z + 400) * q^38 + (186*z - 186) * q^39 + 248 * q^41 + 68 * q^43 - 496*z * q^44 + (36*z - 36) * q^45 + (-168*z + 168) * q^46 + 324*z * q^47 - 192 * q^48 - 436 * q^50 + 252*z * q^51 + (496*z - 496) * q^52 + (258*z - 258) * q^53 - 108*z * q^54 + 248 * q^55 - 300 * q^57 + 40*z * q^58 + (-120*z + 120) * q^59 + (96*z - 96) * q^60 + 622*z * q^61 + 192 * q^62 - 512 * q^64 - 248*z * q^65 + (744*z - 744) * q^66 + (904*z - 904) * q^67 + 672*z * q^68 - 126 * q^69 - 678 * q^71 + (642*z - 642) * q^73 + (-984*z + 984) * q^74 + 327*z * q^75 - 800 * q^76 + 744 * q^78 - 740*z * q^79 + (-256*z + 256) * q^80 + (81*z - 81) * q^81 - 992*z * q^82 - 468 * q^83 - 336 * q^85 - 272*z * q^86 + (-30*z + 30) * q^87 + 200*z * q^89 + 144 * q^90 - 336 * q^92 - 144*z * q^93 + (-1296*z + 1296) * q^94 + (-400*z + 400) * q^95 + 768*z * q^96 + 1266 * q^97 + 558 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - 3 q^{3} - 8 q^{4} - 4 q^{5} + 24 q^{6} - 9 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 - 3 * q^3 - 8 * q^4 - 4 * q^5 + 24 * q^6 - 9 * q^9 $$2 q - 4 q^{2} - 3 q^{3} - 8 q^{4} - 4 q^{5} + 24 q^{6} - 9 q^{9} - 16 q^{10} - 62 q^{11} - 24 q^{12} + 124 q^{13} + 24 q^{15} + 64 q^{16} + 84 q^{17} - 36 q^{18} + 100 q^{19} + 64 q^{20} + 496 q^{22} + 42 q^{23} + 109 q^{25} - 248 q^{26} + 54 q^{27} - 20 q^{29} - 48 q^{30} - 48 q^{31} + 256 q^{32} - 186 q^{33} - 672 q^{34} + 144 q^{36} + 246 q^{37} + 400 q^{38} - 186 q^{39} + 496 q^{41} + 136 q^{43} - 496 q^{44} - 36 q^{45} + 168 q^{46} + 324 q^{47} - 384 q^{48} - 872 q^{50} + 252 q^{51} - 496 q^{52} - 258 q^{53} - 108 q^{54} + 496 q^{55} - 600 q^{57} + 40 q^{58} + 120 q^{59} - 96 q^{60} + 622 q^{61} + 384 q^{62} - 1024 q^{64} - 248 q^{65} - 744 q^{66} - 904 q^{67} + 672 q^{68} - 252 q^{69} - 1356 q^{71} - 642 q^{73} + 984 q^{74} + 327 q^{75} - 1600 q^{76} + 1488 q^{78} - 740 q^{79} + 256 q^{80} - 81 q^{81} - 992 q^{82} - 936 q^{83} - 672 q^{85} - 272 q^{86} + 30 q^{87} + 200 q^{89} + 288 q^{90} - 672 q^{92} - 144 q^{93} + 1296 q^{94} + 400 q^{95} + 768 q^{96} + 2532 q^{97} + 1116 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 - 3 * q^3 - 8 * q^4 - 4 * q^5 + 24 * q^6 - 9 * q^9 - 16 * q^10 - 62 * q^11 - 24 * q^12 + 124 * q^13 + 24 * q^15 + 64 * q^16 + 84 * q^17 - 36 * q^18 + 100 * q^19 + 64 * q^20 + 496 * q^22 + 42 * q^23 + 109 * q^25 - 248 * q^26 + 54 * q^27 - 20 * q^29 - 48 * q^30 - 48 * q^31 + 256 * q^32 - 186 * q^33 - 672 * q^34 + 144 * q^36 + 246 * q^37 + 400 * q^38 - 186 * q^39 + 496 * q^41 + 136 * q^43 - 496 * q^44 - 36 * q^45 + 168 * q^46 + 324 * q^47 - 384 * q^48 - 872 * q^50 + 252 * q^51 - 496 * q^52 - 258 * q^53 - 108 * q^54 + 496 * q^55 - 600 * q^57 + 40 * q^58 + 120 * q^59 - 96 * q^60 + 622 * q^61 + 384 * q^62 - 1024 * q^64 - 248 * q^65 - 744 * q^66 - 904 * q^67 + 672 * q^68 - 252 * q^69 - 1356 * q^71 - 642 * q^73 + 984 * q^74 + 327 * q^75 - 1600 * q^76 + 1488 * q^78 - 740 * q^79 + 256 * q^80 - 81 * q^81 - 992 * q^82 - 936 * q^83 - 672 * q^85 - 272 * q^86 + 30 * q^87 + 200 * q^89 + 288 * q^90 - 672 * q^92 - 144 * q^93 + 1296 * q^94 + 400 * q^95 + 768 * q^96 + 2532 * q^97 + 1116 * 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
−2.00000 3.46410i −1.50000 + 2.59808i −4.00000 + 6.92820i −2.00000 3.46410i 12.0000 0 0 −4.50000 7.79423i −8.00000 + 13.8564i
79.1 −2.00000 + 3.46410i −1.50000 2.59808i −4.00000 6.92820i −2.00000 + 3.46410i 12.0000 0 0 −4.50000 + 7.79423i −8.00000 13.8564i
 $$n$$: 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 147.4.e.b 2
3.b odd 2 1 441.4.e.n 2
7.b odd 2 1 147.4.e.c 2
7.c even 3 1 147.4.a.g 1
7.c even 3 1 inner 147.4.e.b 2
7.d odd 6 1 21.4.a.b 1
7.d odd 6 1 147.4.e.c 2
21.c even 2 1 441.4.e.m 2
21.g even 6 1 63.4.a.a 1
21.g even 6 1 441.4.e.m 2
21.h odd 6 1 441.4.a.b 1
21.h odd 6 1 441.4.e.n 2
28.f even 6 1 336.4.a.h 1
28.g odd 6 1 2352.4.a.l 1
35.i odd 6 1 525.4.a.b 1
35.k even 12 2 525.4.d.b 2
56.j odd 6 1 1344.4.a.w 1
56.m even 6 1 1344.4.a.i 1
84.j odd 6 1 1008.4.a.m 1
105.p even 6 1 1575.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 7.d odd 6 1
63.4.a.a 1 21.g even 6 1
147.4.a.g 1 7.c even 3 1
147.4.e.b 2 1.a even 1 1 trivial
147.4.e.b 2 7.c even 3 1 inner
147.4.e.c 2 7.b odd 2 1
147.4.e.c 2 7.d odd 6 1
336.4.a.h 1 28.f even 6 1
441.4.a.b 1 21.h odd 6 1
441.4.e.m 2 21.c even 2 1
441.4.e.m 2 21.g even 6 1
441.4.e.n 2 3.b odd 2 1
441.4.e.n 2 21.h odd 6 1
525.4.a.b 1 35.i odd 6 1
525.4.d.b 2 35.k even 12 2
1008.4.a.m 1 84.j odd 6 1
1344.4.a.i 1 56.m even 6 1
1344.4.a.w 1 56.j odd 6 1
1575.4.a.k 1 105.p even 6 1
2352.4.a.l 1 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{2} + 4T_{2} + 16$$ T2^2 + 4*T2 + 16 $$T_{5}^{2} + 4T_{5} + 16$$ T5^2 + 4*T5 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4T + 16$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} + 4T + 16$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 62T + 3844$$
$13$ $$(T - 62)^{2}$$
$17$ $$T^{2} - 84T + 7056$$
$19$ $$T^{2} - 100T + 10000$$
$23$ $$T^{2} - 42T + 1764$$
$29$ $$(T + 10)^{2}$$
$31$ $$T^{2} + 48T + 2304$$
$37$ $$T^{2} - 246T + 60516$$
$41$ $$(T - 248)^{2}$$
$43$ $$(T - 68)^{2}$$
$47$ $$T^{2} - 324T + 104976$$
$53$ $$T^{2} + 258T + 66564$$
$59$ $$T^{2} - 120T + 14400$$
$61$ $$T^{2} - 622T + 386884$$
$67$ $$T^{2} + 904T + 817216$$
$71$ $$(T + 678)^{2}$$
$73$ $$T^{2} + 642T + 412164$$
$79$ $$T^{2} + 740T + 547600$$
$83$ $$(T + 468)^{2}$$
$89$ $$T^{2} - 200T + 40000$$
$97$ $$(T - 1266)^{2}$$