# Properties

 Label 147.6.e.i Level $147$ Weight $6$ Character orbit 147.e Analytic conductor $23.576$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,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, 6, names="a")

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

chi := DirichletCharacter("147.67");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$23.5764215125$$ 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.

 $$f(q)$$ $$=$$ $$q + 6 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (4 \zeta_{6} - 4) q^{4} + 78 \zeta_{6} q^{5} - 54 q^{6} + 168 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10})$$ q + 6*z * q^2 + (9*z - 9) * q^3 + (4*z - 4) * q^4 + 78*z * q^5 - 54 * q^6 + 168 * q^8 - 81*z * q^9 $$q + 6 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (4 \zeta_{6} - 4) q^{4} + 78 \zeta_{6} q^{5} - 54 q^{6} + 168 q^{8} - 81 \zeta_{6} q^{9} + (468 \zeta_{6} - 468) q^{10} + (444 \zeta_{6} - 444) q^{11} - 36 \zeta_{6} q^{12} + 442 q^{13} - 702 q^{15} + 1136 \zeta_{6} q^{16} + (126 \zeta_{6} - 126) q^{17} + ( - 486 \zeta_{6} + 486) q^{18} + 2684 \zeta_{6} q^{19} - 312 q^{20} - 2664 q^{22} - 4200 \zeta_{6} q^{23} + (1512 \zeta_{6} - 1512) q^{24} + (2959 \zeta_{6} - 2959) q^{25} + 2652 \zeta_{6} q^{26} + 729 q^{27} - 5442 q^{29} - 4212 \zeta_{6} q^{30} + ( - 80 \zeta_{6} + 80) q^{31} + (1440 \zeta_{6} - 1440) q^{32} - 3996 \zeta_{6} q^{33} - 756 q^{34} + 324 q^{36} + 5434 \zeta_{6} q^{37} + (16104 \zeta_{6} - 16104) q^{38} + (3978 \zeta_{6} - 3978) q^{39} + 13104 \zeta_{6} q^{40} - 7962 q^{41} - 11524 q^{43} - 1776 \zeta_{6} q^{44} + ( - 6318 \zeta_{6} + 6318) q^{45} + ( - 25200 \zeta_{6} + 25200) q^{46} - 13920 \zeta_{6} q^{47} - 10224 q^{48} - 17754 q^{50} - 1134 \zeta_{6} q^{51} + (1768 \zeta_{6} - 1768) q^{52} + ( - 9594 \zeta_{6} + 9594) q^{53} + 4374 \zeta_{6} q^{54} - 34632 q^{55} - 24156 q^{57} - 32652 \zeta_{6} q^{58} + ( - 27492 \zeta_{6} + 27492) q^{59} + ( - 2808 \zeta_{6} + 2808) q^{60} + 49478 \zeta_{6} q^{61} + 480 q^{62} + 27712 q^{64} + 34476 \zeta_{6} q^{65} + ( - 23976 \zeta_{6} + 23976) q^{66} + ( - 59356 \zeta_{6} + 59356) q^{67} - 504 \zeta_{6} q^{68} + 37800 q^{69} + 32040 q^{71} - 13608 \zeta_{6} q^{72} + (61846 \zeta_{6} - 61846) q^{73} + (32604 \zeta_{6} - 32604) q^{74} - 26631 \zeta_{6} q^{75} - 10736 q^{76} - 23868 q^{78} + 65776 \zeta_{6} q^{79} + (88608 \zeta_{6} - 88608) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 47772 \zeta_{6} q^{82} - 40188 q^{83} - 9828 q^{85} - 69144 \zeta_{6} q^{86} + ( - 48978 \zeta_{6} + 48978) q^{87} + (74592 \zeta_{6} - 74592) q^{88} - 7974 \zeta_{6} q^{89} + 37908 q^{90} + 16800 q^{92} + 720 \zeta_{6} q^{93} + ( - 83520 \zeta_{6} + 83520) q^{94} + (209352 \zeta_{6} - 209352) q^{95} - 12960 \zeta_{6} q^{96} + 143662 q^{97} + 35964 q^{99} +O(q^{100})$$ q + 6*z * q^2 + (9*z - 9) * q^3 + (4*z - 4) * q^4 + 78*z * q^5 - 54 * q^6 + 168 * q^8 - 81*z * q^9 + (468*z - 468) * q^10 + (444*z - 444) * q^11 - 36*z * q^12 + 442 * q^13 - 702 * q^15 + 1136*z * q^16 + (126*z - 126) * q^17 + (-486*z + 486) * q^18 + 2684*z * q^19 - 312 * q^20 - 2664 * q^22 - 4200*z * q^23 + (1512*z - 1512) * q^24 + (2959*z - 2959) * q^25 + 2652*z * q^26 + 729 * q^27 - 5442 * q^29 - 4212*z * q^30 + (-80*z + 80) * q^31 + (1440*z - 1440) * q^32 - 3996*z * q^33 - 756 * q^34 + 324 * q^36 + 5434*z * q^37 + (16104*z - 16104) * q^38 + (3978*z - 3978) * q^39 + 13104*z * q^40 - 7962 * q^41 - 11524 * q^43 - 1776*z * q^44 + (-6318*z + 6318) * q^45 + (-25200*z + 25200) * q^46 - 13920*z * q^47 - 10224 * q^48 - 17754 * q^50 - 1134*z * q^51 + (1768*z - 1768) * q^52 + (-9594*z + 9594) * q^53 + 4374*z * q^54 - 34632 * q^55 - 24156 * q^57 - 32652*z * q^58 + (-27492*z + 27492) * q^59 + (-2808*z + 2808) * q^60 + 49478*z * q^61 + 480 * q^62 + 27712 * q^64 + 34476*z * q^65 + (-23976*z + 23976) * q^66 + (-59356*z + 59356) * q^67 - 504*z * q^68 + 37800 * q^69 + 32040 * q^71 - 13608*z * q^72 + (61846*z - 61846) * q^73 + (32604*z - 32604) * q^74 - 26631*z * q^75 - 10736 * q^76 - 23868 * q^78 + 65776*z * q^79 + (88608*z - 88608) * q^80 + (6561*z - 6561) * q^81 - 47772*z * q^82 - 40188 * q^83 - 9828 * q^85 - 69144*z * q^86 + (-48978*z + 48978) * q^87 + (74592*z - 74592) * q^88 - 7974*z * q^89 + 37908 * q^90 + 16800 * q^92 + 720*z * q^93 + (-83520*z + 83520) * q^94 + (209352*z - 209352) * q^95 - 12960*z * q^96 + 143662 * q^97 + 35964 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} - 9 q^{3} - 4 q^{4} + 78 q^{5} - 108 q^{6} + 336 q^{8} - 81 q^{9}+O(q^{10})$$ 2 * q + 6 * q^2 - 9 * q^3 - 4 * q^4 + 78 * q^5 - 108 * q^6 + 336 * q^8 - 81 * q^9 $$2 q + 6 q^{2} - 9 q^{3} - 4 q^{4} + 78 q^{5} - 108 q^{6} + 336 q^{8} - 81 q^{9} - 468 q^{10} - 444 q^{11} - 36 q^{12} + 884 q^{13} - 1404 q^{15} + 1136 q^{16} - 126 q^{17} + 486 q^{18} + 2684 q^{19} - 624 q^{20} - 5328 q^{22} - 4200 q^{23} - 1512 q^{24} - 2959 q^{25} + 2652 q^{26} + 1458 q^{27} - 10884 q^{29} - 4212 q^{30} + 80 q^{31} - 1440 q^{32} - 3996 q^{33} - 1512 q^{34} + 648 q^{36} + 5434 q^{37} - 16104 q^{38} - 3978 q^{39} + 13104 q^{40} - 15924 q^{41} - 23048 q^{43} - 1776 q^{44} + 6318 q^{45} + 25200 q^{46} - 13920 q^{47} - 20448 q^{48} - 35508 q^{50} - 1134 q^{51} - 1768 q^{52} + 9594 q^{53} + 4374 q^{54} - 69264 q^{55} - 48312 q^{57} - 32652 q^{58} + 27492 q^{59} + 2808 q^{60} + 49478 q^{61} + 960 q^{62} + 55424 q^{64} + 34476 q^{65} + 23976 q^{66} + 59356 q^{67} - 504 q^{68} + 75600 q^{69} + 64080 q^{71} - 13608 q^{72} - 61846 q^{73} - 32604 q^{74} - 26631 q^{75} - 21472 q^{76} - 47736 q^{78} + 65776 q^{79} - 88608 q^{80} - 6561 q^{81} - 47772 q^{82} - 80376 q^{83} - 19656 q^{85} - 69144 q^{86} + 48978 q^{87} - 74592 q^{88} - 7974 q^{89} + 75816 q^{90} + 33600 q^{92} + 720 q^{93} + 83520 q^{94} - 209352 q^{95} - 12960 q^{96} + 287324 q^{97} + 71928 q^{99}+O(q^{100})$$ 2 * q + 6 * q^2 - 9 * q^3 - 4 * q^4 + 78 * q^5 - 108 * q^6 + 336 * q^8 - 81 * q^9 - 468 * q^10 - 444 * q^11 - 36 * q^12 + 884 * q^13 - 1404 * q^15 + 1136 * q^16 - 126 * q^17 + 486 * q^18 + 2684 * q^19 - 624 * q^20 - 5328 * q^22 - 4200 * q^23 - 1512 * q^24 - 2959 * q^25 + 2652 * q^26 + 1458 * q^27 - 10884 * q^29 - 4212 * q^30 + 80 * q^31 - 1440 * q^32 - 3996 * q^33 - 1512 * q^34 + 648 * q^36 + 5434 * q^37 - 16104 * q^38 - 3978 * q^39 + 13104 * q^40 - 15924 * q^41 - 23048 * q^43 - 1776 * q^44 + 6318 * q^45 + 25200 * q^46 - 13920 * q^47 - 20448 * q^48 - 35508 * q^50 - 1134 * q^51 - 1768 * q^52 + 9594 * q^53 + 4374 * q^54 - 69264 * q^55 - 48312 * q^57 - 32652 * q^58 + 27492 * q^59 + 2808 * q^60 + 49478 * q^61 + 960 * q^62 + 55424 * q^64 + 34476 * q^65 + 23976 * q^66 + 59356 * q^67 - 504 * q^68 + 75600 * q^69 + 64080 * q^71 - 13608 * q^72 - 61846 * q^73 - 32604 * q^74 - 26631 * q^75 - 21472 * q^76 - 47736 * q^78 + 65776 * q^79 - 88608 * q^80 - 6561 * q^81 - 47772 * q^82 - 80376 * q^83 - 19656 * q^85 - 69144 * q^86 + 48978 * q^87 - 74592 * q^88 - 7974 * q^89 + 75816 * q^90 + 33600 * q^92 + 720 * q^93 + 83520 * q^94 - 209352 * q^95 - 12960 * q^96 + 287324 * q^97 + 71928 * 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
3.00000 + 5.19615i −4.50000 + 7.79423i −2.00000 + 3.46410i 39.0000 + 67.5500i −54.0000 0 168.000 −40.5000 70.1481i −234.000 + 405.300i
79.1 3.00000 5.19615i −4.50000 7.79423i −2.00000 3.46410i 39.0000 67.5500i −54.0000 0 168.000 −40.5000 + 70.1481i −234.000 405.300i
 $$n$$: 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 147.6.e.i 2
7.b odd 2 1 147.6.e.j 2
7.c even 3 1 147.6.a.b 1
7.c even 3 1 inner 147.6.e.i 2
7.d odd 6 1 21.6.a.a 1
7.d odd 6 1 147.6.e.j 2
21.g even 6 1 63.6.a.d 1
21.h odd 6 1 441.6.a.j 1
28.f even 6 1 336.6.a.r 1
35.i odd 6 1 525.6.a.d 1
35.k even 12 2 525.6.d.b 2
84.j odd 6 1 1008.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 7.d odd 6 1
63.6.a.d 1 21.g even 6 1
147.6.a.b 1 7.c even 3 1
147.6.e.i 2 1.a even 1 1 trivial
147.6.e.i 2 7.c even 3 1 inner
147.6.e.j 2 7.b odd 2 1
147.6.e.j 2 7.d odd 6 1
336.6.a.r 1 28.f even 6 1
441.6.a.j 1 21.h odd 6 1
525.6.a.d 1 35.i odd 6 1
525.6.d.b 2 35.k even 12 2
1008.6.a.c 1 84.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 $$S_{6}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{2} - 6T_{2} + 36$$ T2^2 - 6*T2 + 36 $$T_{5}^{2} - 78T_{5} + 6084$$ T5^2 - 78*T5 + 6084

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 6T + 36$$
$3$ $$T^{2} + 9T + 81$$
$5$ $$T^{2} - 78T + 6084$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 444T + 197136$$
$13$ $$(T - 442)^{2}$$
$17$ $$T^{2} + 126T + 15876$$
$19$ $$T^{2} - 2684 T + 7203856$$
$23$ $$T^{2} + 4200 T + 17640000$$
$29$ $$(T + 5442)^{2}$$
$31$ $$T^{2} - 80T + 6400$$
$37$ $$T^{2} - 5434 T + 29528356$$
$41$ $$(T + 7962)^{2}$$
$43$ $$(T + 11524)^{2}$$
$47$ $$T^{2} + 13920 T + 193766400$$
$53$ $$T^{2} - 9594 T + 92044836$$
$59$ $$T^{2} - 27492 T + 755810064$$
$61$ $$T^{2} + \cdots + 2448072484$$
$67$ $$T^{2} + \cdots + 3523134736$$
$71$ $$(T - 32040)^{2}$$
$73$ $$T^{2} + \cdots + 3824927716$$
$79$ $$T^{2} + \cdots + 4326482176$$
$83$ $$(T + 40188)^{2}$$
$89$ $$T^{2} + 7974 T + 63584676$$
$97$ $$(T - 143662)^{2}$$