# Properties

 Label 147.4.e.g 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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (\zeta_{6} - 1) q^{4} - 18 \zeta_{6} q^{5} - 9 q^{6} + 21 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + 3*z * q^2 + (3*z - 3) * q^3 + (z - 1) * q^4 - 18*z * q^5 - 9 * q^6 + 21 * q^8 - 9*z * q^9 $$q + 3 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (\zeta_{6} - 1) q^{4} - 18 \zeta_{6} q^{5} - 9 q^{6} + 21 q^{8} - 9 \zeta_{6} q^{9} + ( - 54 \zeta_{6} + 54) q^{10} + ( - 36 \zeta_{6} + 36) q^{11} - 3 \zeta_{6} q^{12} + 34 q^{13} + 54 q^{15} + 71 \zeta_{6} q^{16} + ( - 42 \zeta_{6} + 42) q^{17} + ( - 27 \zeta_{6} + 27) q^{18} - 124 \zeta_{6} q^{19} + 18 q^{20} + 108 q^{22} + (63 \zeta_{6} - 63) q^{24} + (199 \zeta_{6} - 199) q^{25} + 102 \zeta_{6} q^{26} + 27 q^{27} + 102 q^{29} + 162 \zeta_{6} q^{30} + (160 \zeta_{6} - 160) q^{31} + (45 \zeta_{6} - 45) q^{32} + 108 \zeta_{6} q^{33} + 126 q^{34} + 9 q^{36} - 398 \zeta_{6} q^{37} + ( - 372 \zeta_{6} + 372) q^{38} + (102 \zeta_{6} - 102) q^{39} - 378 \zeta_{6} q^{40} + 318 q^{41} - 268 q^{43} + 36 \zeta_{6} q^{44} + (162 \zeta_{6} - 162) q^{45} + 240 \zeta_{6} q^{47} - 213 q^{48} - 597 q^{50} + 126 \zeta_{6} q^{51} + (34 \zeta_{6} - 34) q^{52} + ( - 498 \zeta_{6} + 498) q^{53} + 81 \zeta_{6} q^{54} - 648 q^{55} + 372 q^{57} + 306 \zeta_{6} q^{58} + (132 \zeta_{6} - 132) q^{59} + (54 \zeta_{6} - 54) q^{60} + 398 \zeta_{6} q^{61} - 480 q^{62} + 433 q^{64} - 612 \zeta_{6} q^{65} + (324 \zeta_{6} - 324) q^{66} + (92 \zeta_{6} - 92) q^{67} + 42 \zeta_{6} q^{68} - 720 q^{71} - 189 \zeta_{6} q^{72} + (502 \zeta_{6} - 502) q^{73} + ( - 1194 \zeta_{6} + 1194) q^{74} - 597 \zeta_{6} q^{75} + 124 q^{76} - 306 q^{78} + 1024 \zeta_{6} q^{79} + ( - 1278 \zeta_{6} + 1278) q^{80} + (81 \zeta_{6} - 81) q^{81} + 954 \zeta_{6} q^{82} + 204 q^{83} - 756 q^{85} - 804 \zeta_{6} q^{86} + (306 \zeta_{6} - 306) q^{87} + ( - 756 \zeta_{6} + 756) q^{88} + 354 \zeta_{6} q^{89} - 486 q^{90} - 480 \zeta_{6} q^{93} + (720 \zeta_{6} - 720) q^{94} + (2232 \zeta_{6} - 2232) q^{95} - 135 \zeta_{6} q^{96} + 286 q^{97} - 324 q^{99} +O(q^{100})$$ q + 3*z * q^2 + (3*z - 3) * q^3 + (z - 1) * q^4 - 18*z * q^5 - 9 * q^6 + 21 * q^8 - 9*z * q^9 + (-54*z + 54) * q^10 + (-36*z + 36) * q^11 - 3*z * q^12 + 34 * q^13 + 54 * q^15 + 71*z * q^16 + (-42*z + 42) * q^17 + (-27*z + 27) * q^18 - 124*z * q^19 + 18 * q^20 + 108 * q^22 + (63*z - 63) * q^24 + (199*z - 199) * q^25 + 102*z * q^26 + 27 * q^27 + 102 * q^29 + 162*z * q^30 + (160*z - 160) * q^31 + (45*z - 45) * q^32 + 108*z * q^33 + 126 * q^34 + 9 * q^36 - 398*z * q^37 + (-372*z + 372) * q^38 + (102*z - 102) * q^39 - 378*z * q^40 + 318 * q^41 - 268 * q^43 + 36*z * q^44 + (162*z - 162) * q^45 + 240*z * q^47 - 213 * q^48 - 597 * q^50 + 126*z * q^51 + (34*z - 34) * q^52 + (-498*z + 498) * q^53 + 81*z * q^54 - 648 * q^55 + 372 * q^57 + 306*z * q^58 + (132*z - 132) * q^59 + (54*z - 54) * q^60 + 398*z * q^61 - 480 * q^62 + 433 * q^64 - 612*z * q^65 + (324*z - 324) * q^66 + (92*z - 92) * q^67 + 42*z * q^68 - 720 * q^71 - 189*z * q^72 + (502*z - 502) * q^73 + (-1194*z + 1194) * q^74 - 597*z * q^75 + 124 * q^76 - 306 * q^78 + 1024*z * q^79 + (-1278*z + 1278) * q^80 + (81*z - 81) * q^81 + 954*z * q^82 + 204 * q^83 - 756 * q^85 - 804*z * q^86 + (306*z - 306) * q^87 + (-756*z + 756) * q^88 + 354*z * q^89 - 486 * q^90 - 480*z * q^93 + (720*z - 720) * q^94 + (2232*z - 2232) * q^95 - 135*z * q^96 + 286 * q^97 - 324 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - 3 q^{3} - q^{4} - 18 q^{5} - 18 q^{6} + 42 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q + 3 * q^2 - 3 * q^3 - q^4 - 18 * q^5 - 18 * q^6 + 42 * q^8 - 9 * q^9 $$2 q + 3 q^{2} - 3 q^{3} - q^{4} - 18 q^{5} - 18 q^{6} + 42 q^{8} - 9 q^{9} + 54 q^{10} + 36 q^{11} - 3 q^{12} + 68 q^{13} + 108 q^{15} + 71 q^{16} + 42 q^{17} + 27 q^{18} - 124 q^{19} + 36 q^{20} + 216 q^{22} - 63 q^{24} - 199 q^{25} + 102 q^{26} + 54 q^{27} + 204 q^{29} + 162 q^{30} - 160 q^{31} - 45 q^{32} + 108 q^{33} + 252 q^{34} + 18 q^{36} - 398 q^{37} + 372 q^{38} - 102 q^{39} - 378 q^{40} + 636 q^{41} - 536 q^{43} + 36 q^{44} - 162 q^{45} + 240 q^{47} - 426 q^{48} - 1194 q^{50} + 126 q^{51} - 34 q^{52} + 498 q^{53} + 81 q^{54} - 1296 q^{55} + 744 q^{57} + 306 q^{58} - 132 q^{59} - 54 q^{60} + 398 q^{61} - 960 q^{62} + 866 q^{64} - 612 q^{65} - 324 q^{66} - 92 q^{67} + 42 q^{68} - 1440 q^{71} - 189 q^{72} - 502 q^{73} + 1194 q^{74} - 597 q^{75} + 248 q^{76} - 612 q^{78} + 1024 q^{79} + 1278 q^{80} - 81 q^{81} + 954 q^{82} + 408 q^{83} - 1512 q^{85} - 804 q^{86} - 306 q^{87} + 756 q^{88} + 354 q^{89} - 972 q^{90} - 480 q^{93} - 720 q^{94} - 2232 q^{95} - 135 q^{96} + 572 q^{97} - 648 q^{99}+O(q^{100})$$ 2 * q + 3 * q^2 - 3 * q^3 - q^4 - 18 * q^5 - 18 * q^6 + 42 * q^8 - 9 * q^9 + 54 * q^10 + 36 * q^11 - 3 * q^12 + 68 * q^13 + 108 * q^15 + 71 * q^16 + 42 * q^17 + 27 * q^18 - 124 * q^19 + 36 * q^20 + 216 * q^22 - 63 * q^24 - 199 * q^25 + 102 * q^26 + 54 * q^27 + 204 * q^29 + 162 * q^30 - 160 * q^31 - 45 * q^32 + 108 * q^33 + 252 * q^34 + 18 * q^36 - 398 * q^37 + 372 * q^38 - 102 * q^39 - 378 * q^40 + 636 * q^41 - 536 * q^43 + 36 * q^44 - 162 * q^45 + 240 * q^47 - 426 * q^48 - 1194 * q^50 + 126 * q^51 - 34 * q^52 + 498 * q^53 + 81 * q^54 - 1296 * q^55 + 744 * q^57 + 306 * q^58 - 132 * q^59 - 54 * q^60 + 398 * q^61 - 960 * q^62 + 866 * q^64 - 612 * q^65 - 324 * q^66 - 92 * q^67 + 42 * q^68 - 1440 * q^71 - 189 * q^72 - 502 * q^73 + 1194 * q^74 - 597 * q^75 + 248 * q^76 - 612 * q^78 + 1024 * q^79 + 1278 * q^80 - 81 * q^81 + 954 * q^82 + 408 * q^83 - 1512 * q^85 - 804 * q^86 - 306 * q^87 + 756 * q^88 + 354 * q^89 - 972 * q^90 - 480 * q^93 - 720 * q^94 - 2232 * q^95 - 135 * q^96 + 572 * q^97 - 648 * 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
1.50000 + 2.59808i −1.50000 + 2.59808i −0.500000 + 0.866025i −9.00000 15.5885i −9.00000 0 21.0000 −4.50000 7.79423i 27.0000 46.7654i
79.1 1.50000 2.59808i −1.50000 2.59808i −0.500000 0.866025i −9.00000 + 15.5885i −9.00000 0 21.0000 −4.50000 + 7.79423i 27.0000 + 46.7654i
 $$n$$: 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.g 2
3.b odd 2 1 441.4.e.d 2
7.b odd 2 1 147.4.e.i 2
7.c even 3 1 147.4.a.c 1
7.c even 3 1 inner 147.4.e.g 2
7.d odd 6 1 21.4.a.a 1
7.d odd 6 1 147.4.e.i 2
21.c even 2 1 441.4.e.b 2
21.g even 6 1 63.4.a.c 1
21.g even 6 1 441.4.e.b 2
21.h odd 6 1 441.4.a.j 1
21.h odd 6 1 441.4.e.d 2
28.f even 6 1 336.4.a.f 1
28.g odd 6 1 2352.4.a.r 1
35.i odd 6 1 525.4.a.g 1
35.k even 12 2 525.4.d.c 2
56.j odd 6 1 1344.4.a.ba 1
56.m even 6 1 1344.4.a.n 1
84.j odd 6 1 1008.4.a.v 1
105.p even 6 1 1575.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 7.d odd 6 1
63.4.a.c 1 21.g even 6 1
147.4.a.c 1 7.c even 3 1
147.4.e.g 2 1.a even 1 1 trivial
147.4.e.g 2 7.c even 3 1 inner
147.4.e.i 2 7.b odd 2 1
147.4.e.i 2 7.d odd 6 1
336.4.a.f 1 28.f even 6 1
441.4.a.j 1 21.h odd 6 1
441.4.e.b 2 21.c even 2 1
441.4.e.b 2 21.g even 6 1
441.4.e.d 2 3.b odd 2 1
441.4.e.d 2 21.h odd 6 1
525.4.a.g 1 35.i odd 6 1
525.4.d.c 2 35.k even 12 2
1008.4.a.v 1 84.j odd 6 1
1344.4.a.n 1 56.m even 6 1
1344.4.a.ba 1 56.j odd 6 1
1575.4.a.b 1 105.p even 6 1
2352.4.a.r 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 $$S_{4}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{2} - 3T_{2} + 9$$ T2^2 - 3*T2 + 9 $$T_{5}^{2} + 18T_{5} + 324$$ T5^2 + 18*T5 + 324

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 9$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} + 18T + 324$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 36T + 1296$$
$13$ $$(T - 34)^{2}$$
$17$ $$T^{2} - 42T + 1764$$
$19$ $$T^{2} + 124T + 15376$$
$23$ $$T^{2}$$
$29$ $$(T - 102)^{2}$$
$31$ $$T^{2} + 160T + 25600$$
$37$ $$T^{2} + 398T + 158404$$
$41$ $$(T - 318)^{2}$$
$43$ $$(T + 268)^{2}$$
$47$ $$T^{2} - 240T + 57600$$
$53$ $$T^{2} - 498T + 248004$$
$59$ $$T^{2} + 132T + 17424$$
$61$ $$T^{2} - 398T + 158404$$
$67$ $$T^{2} + 92T + 8464$$
$71$ $$(T + 720)^{2}$$
$73$ $$T^{2} + 502T + 252004$$
$79$ $$T^{2} - 1024 T + 1048576$$
$83$ $$(T - 204)^{2}$$
$89$ $$T^{2} - 354T + 125316$$
$97$ $$(T - 286)^{2}$$