# Properties

 Label 147.6.a.k Level $147$ Weight $6$ Character orbit 147.a Self dual yes Analytic conductor $23.576$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

## 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: yes Analytic conductor: $$23.5764215125$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{249})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 62$$ x^2 - x - 62 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{249})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + 9 q^{3} + (3 \beta + 31) q^{4} + (7 \beta + 13) q^{5} + ( - 9 \beta - 9) q^{6} + ( - 5 \beta - 185) q^{8} + 81 q^{9}+O(q^{10})$$ q + (-b - 1) * q^2 + 9 * q^3 + (3*b + 31) * q^4 + (7*b + 13) * q^5 + (-9*b - 9) * q^6 + (-5*b - 185) * q^8 + 81 * q^9 $$q + ( - \beta - 1) q^{2} + 9 q^{3} + (3 \beta + 31) q^{4} + (7 \beta + 13) q^{5} + ( - 9 \beta - 9) q^{6} + ( - 5 \beta - 185) q^{8} + 81 q^{9} + ( - 27 \beta - 447) q^{10} + (\beta - 569) q^{11} + (27 \beta + 279) q^{12} + ( - 9 \beta - 458) q^{13} + (63 \beta + 117) q^{15} + (99 \beta - 497) q^{16} + ( - 148 \beta + 236) q^{17} + ( - 81 \beta - 81) q^{18} + ( - 27 \beta - 1142) q^{19} + (277 \beta + 1705) q^{20} + (567 \beta + 507) q^{22} + (308 \beta + 644) q^{23} + ( - 45 \beta - 1665) q^{24} + (231 \beta + 82) q^{25} + (476 \beta + 1016) q^{26} + 729 q^{27} + (45 \beta - 1131) q^{29} + ( - 243 \beta - 4023) q^{30} + ( - 768 \beta - 1763) q^{31} + (459 \beta + 279) q^{32} + (9 \beta - 5121) q^{33} + (60 \beta + 8940) q^{34} + (243 \beta + 2511) q^{36} + (855 \beta - 9982) q^{37} + (1196 \beta + 2816) q^{38} + ( - 81 \beta - 4122) q^{39} + ( - 1395 \beta - 4575) q^{40} + ( - 846 \beta + 6852) q^{41} + ( - 2043 \beta - 364) q^{43} + ( - 1673 \beta - 17453) q^{44} + (567 \beta + 1053) q^{45} + ( - 1260 \beta - 19740) q^{46} + (604 \beta + 11278) q^{47} + (891 \beta - 4473) q^{48} + ( - 544 \beta - 14404) q^{50} + ( - 1332 \beta + 2124) q^{51} + ( - 1680 \beta - 15872) q^{52} + ( - 1751 \beta - 14951) q^{53} + ( - 729 \beta - 729) q^{54} + ( - 3963 \beta - 6963) q^{55} + ( - 243 \beta - 10278) q^{57} + (1041 \beta - 1659) q^{58} + (3917 \beta - 22507) q^{59} + (2493 \beta + 15345) q^{60} + (2544 \beta - 22298) q^{61} + (3299 \beta + 49379) q^{62} + ( - 4365 \beta - 12833) q^{64} + ( - 3386 \beta - 9860) q^{65} + (5103 \beta + 4563) q^{66} + ( - 4461 \beta + 17612) q^{67} + ( - 4324 \beta - 20212) q^{68} + (2772 \beta + 5796) q^{69} + (1404 \beta + 50346) q^{71} + ( - 405 \beta - 14985) q^{72} + ( - 5247 \beta + 16912) q^{73} + (8272 \beta - 43028) q^{74} + (2079 \beta + 738) q^{75} + ( - 4344 \beta - 40424) q^{76} + (4284 \beta + 9144) q^{78} + (6834 \beta - 12649) q^{79} + ( - 1499 \beta + 36505) q^{80} + 6561 q^{81} + ( - 5160 \beta + 45600) q^{82} + (1899 \beta - 31539) q^{83} + ( - 1308 \beta - 61164) q^{85} + (4450 \beta + 127030) q^{86} + (405 \beta - 10179) q^{87} + (2655 \beta + 104955) q^{88} + (130 \beta - 14726) q^{89} + ( - 2187 \beta - 36207) q^{90} + (12404 \beta + 77252) q^{92} + ( - 6912 \beta - 15867) q^{93} + ( - 12486 \beta - 48726) q^{94} + ( - 8534 \beta - 26564) q^{95} + (4131 \beta + 2511) q^{96} + (1017 \beta + 4387) q^{97} + (81 \beta - 46089) q^{99}+O(q^{100})$$ q + (-b - 1) * q^2 + 9 * q^3 + (3*b + 31) * q^4 + (7*b + 13) * q^5 + (-9*b - 9) * q^6 + (-5*b - 185) * q^8 + 81 * q^9 + (-27*b - 447) * q^10 + (b - 569) * q^11 + (27*b + 279) * q^12 + (-9*b - 458) * q^13 + (63*b + 117) * q^15 + (99*b - 497) * q^16 + (-148*b + 236) * q^17 + (-81*b - 81) * q^18 + (-27*b - 1142) * q^19 + (277*b + 1705) * q^20 + (567*b + 507) * q^22 + (308*b + 644) * q^23 + (-45*b - 1665) * q^24 + (231*b + 82) * q^25 + (476*b + 1016) * q^26 + 729 * q^27 + (45*b - 1131) * q^29 + (-243*b - 4023) * q^30 + (-768*b - 1763) * q^31 + (459*b + 279) * q^32 + (9*b - 5121) * q^33 + (60*b + 8940) * q^34 + (243*b + 2511) * q^36 + (855*b - 9982) * q^37 + (1196*b + 2816) * q^38 + (-81*b - 4122) * q^39 + (-1395*b - 4575) * q^40 + (-846*b + 6852) * q^41 + (-2043*b - 364) * q^43 + (-1673*b - 17453) * q^44 + (567*b + 1053) * q^45 + (-1260*b - 19740) * q^46 + (604*b + 11278) * q^47 + (891*b - 4473) * q^48 + (-544*b - 14404) * q^50 + (-1332*b + 2124) * q^51 + (-1680*b - 15872) * q^52 + (-1751*b - 14951) * q^53 + (-729*b - 729) * q^54 + (-3963*b - 6963) * q^55 + (-243*b - 10278) * q^57 + (1041*b - 1659) * q^58 + (3917*b - 22507) * q^59 + (2493*b + 15345) * q^60 + (2544*b - 22298) * q^61 + (3299*b + 49379) * q^62 + (-4365*b - 12833) * q^64 + (-3386*b - 9860) * q^65 + (5103*b + 4563) * q^66 + (-4461*b + 17612) * q^67 + (-4324*b - 20212) * q^68 + (2772*b + 5796) * q^69 + (1404*b + 50346) * q^71 + (-405*b - 14985) * q^72 + (-5247*b + 16912) * q^73 + (8272*b - 43028) * q^74 + (2079*b + 738) * q^75 + (-4344*b - 40424) * q^76 + (4284*b + 9144) * q^78 + (6834*b - 12649) * q^79 + (-1499*b + 36505) * q^80 + 6561 * q^81 + (-5160*b + 45600) * q^82 + (1899*b - 31539) * q^83 + (-1308*b - 61164) * q^85 + (4450*b + 127030) * q^86 + (405*b - 10179) * q^87 + (2655*b + 104955) * q^88 + (130*b - 14726) * q^89 + (-2187*b - 36207) * q^90 + (12404*b + 77252) * q^92 + (-6912*b - 15867) * q^93 + (-12486*b - 48726) * q^94 + (-8534*b - 26564) * q^95 + (4131*b + 2511) * q^96 + (1017*b + 4387) * q^97 + (81*b - 46089) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 18 q^{3} + 65 q^{4} + 33 q^{5} - 27 q^{6} - 375 q^{8} + 162 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 18 * q^3 + 65 * q^4 + 33 * q^5 - 27 * q^6 - 375 * q^8 + 162 * q^9 $$2 q - 3 q^{2} + 18 q^{3} + 65 q^{4} + 33 q^{5} - 27 q^{6} - 375 q^{8} + 162 q^{9} - 921 q^{10} - 1137 q^{11} + 585 q^{12} - 925 q^{13} + 297 q^{15} - 895 q^{16} + 324 q^{17} - 243 q^{18} - 2311 q^{19} + 3687 q^{20} + 1581 q^{22} + 1596 q^{23} - 3375 q^{24} + 395 q^{25} + 2508 q^{26} + 1458 q^{27} - 2217 q^{29} - 8289 q^{30} - 4294 q^{31} + 1017 q^{32} - 10233 q^{33} + 17940 q^{34} + 5265 q^{36} - 19109 q^{37} + 6828 q^{38} - 8325 q^{39} - 10545 q^{40} + 12858 q^{41} - 2771 q^{43} - 36579 q^{44} + 2673 q^{45} - 40740 q^{46} + 23160 q^{47} - 8055 q^{48} - 29352 q^{50} + 2916 q^{51} - 33424 q^{52} - 31653 q^{53} - 2187 q^{54} - 17889 q^{55} - 20799 q^{57} - 2277 q^{58} - 41097 q^{59} + 33183 q^{60} - 42052 q^{61} + 102057 q^{62} - 30031 q^{64} - 23106 q^{65} + 14229 q^{66} + 30763 q^{67} - 44748 q^{68} + 14364 q^{69} + 102096 q^{71} - 30375 q^{72} + 28577 q^{73} - 77784 q^{74} + 3555 q^{75} - 85192 q^{76} + 22572 q^{78} - 18464 q^{79} + 71511 q^{80} + 13122 q^{81} + 86040 q^{82} - 61179 q^{83} - 123636 q^{85} + 258510 q^{86} - 19953 q^{87} + 212565 q^{88} - 29322 q^{89} - 74601 q^{90} + 166908 q^{92} - 38646 q^{93} - 109938 q^{94} - 61662 q^{95} + 9153 q^{96} + 9791 q^{97} - 92097 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 18 * q^3 + 65 * q^4 + 33 * q^5 - 27 * q^6 - 375 * q^8 + 162 * q^9 - 921 * q^10 - 1137 * q^11 + 585 * q^12 - 925 * q^13 + 297 * q^15 - 895 * q^16 + 324 * q^17 - 243 * q^18 - 2311 * q^19 + 3687 * q^20 + 1581 * q^22 + 1596 * q^23 - 3375 * q^24 + 395 * q^25 + 2508 * q^26 + 1458 * q^27 - 2217 * q^29 - 8289 * q^30 - 4294 * q^31 + 1017 * q^32 - 10233 * q^33 + 17940 * q^34 + 5265 * q^36 - 19109 * q^37 + 6828 * q^38 - 8325 * q^39 - 10545 * q^40 + 12858 * q^41 - 2771 * q^43 - 36579 * q^44 + 2673 * q^45 - 40740 * q^46 + 23160 * q^47 - 8055 * q^48 - 29352 * q^50 + 2916 * q^51 - 33424 * q^52 - 31653 * q^53 - 2187 * q^54 - 17889 * q^55 - 20799 * q^57 - 2277 * q^58 - 41097 * q^59 + 33183 * q^60 - 42052 * q^61 + 102057 * q^62 - 30031 * q^64 - 23106 * q^65 + 14229 * q^66 + 30763 * q^67 - 44748 * q^68 + 14364 * q^69 + 102096 * q^71 - 30375 * q^72 + 28577 * q^73 - 77784 * q^74 + 3555 * q^75 - 85192 * q^76 + 22572 * q^78 - 18464 * q^79 + 71511 * q^80 + 13122 * q^81 + 86040 * q^82 - 61179 * q^83 - 123636 * q^85 + 258510 * q^86 - 19953 * q^87 + 212565 * q^88 - 29322 * q^89 - 74601 * q^90 + 166908 * q^92 - 38646 * q^93 - 109938 * q^94 - 61662 * q^95 + 9153 * q^96 + 9791 * q^97 - 92097 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 8.38987 −7.38987
−9.38987 9.00000 56.1696 71.7291 −84.5088 0 −226.949 81.0000 −673.526
1.2 6.38987 9.00000 8.83040 −38.7291 57.5088 0 −148.051 81.0000 −247.474
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.a.k 2
3.b odd 2 1 441.6.a.s 2
7.b odd 2 1 147.6.a.i 2
7.c even 3 2 147.6.e.l 4
7.d odd 6 2 21.6.e.b 4
21.c even 2 1 441.6.a.t 2
21.g even 6 2 63.6.e.c 4
28.f even 6 2 336.6.q.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 7.d odd 6 2
63.6.e.c 4 21.g even 6 2
147.6.a.i 2 7.b odd 2 1
147.6.a.k 2 1.a even 1 1 trivial
147.6.e.l 4 7.c even 3 2
336.6.q.e 4 28.f even 6 2
441.6.a.s 2 3.b odd 2 1
441.6.a.t 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2}^{2} + 3T_{2} - 60$$ T2^2 + 3*T2 - 60 $$T_{5}^{2} - 33T_{5} - 2778$$ T5^2 - 33*T5 - 2778

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T - 60$$
$3$ $$(T - 9)^{2}$$
$5$ $$T^{2} - 33T - 2778$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 1137 T + 323130$$
$13$ $$T^{2} + 925T + 208864$$
$17$ $$T^{2} - 324 T - 1337280$$
$19$ $$T^{2} + 2311 T + 1289800$$
$23$ $$T^{2} - 1596 T - 5268480$$
$29$ $$T^{2} + 2217 T + 1102716$$
$31$ $$T^{2} + 4294 T - 32106935$$
$37$ $$T^{2} + 19109 T + 45782164$$
$41$ $$T^{2} - 12858 T - 3221280$$
$43$ $$T^{2} + 2771 T - 257902490$$
$47$ $$T^{2} - 23160 T + 111386604$$
$53$ $$T^{2} + 31653 T + 59619540$$
$59$ $$T^{2} + 41097 T - 532853988$$
$61$ $$T^{2} + 42052 T + 39214660$$
$67$ $$T^{2} + \cdots - 1002216890$$
$71$ $$T^{2} + \cdots + 2483190108$$
$73$ $$T^{2} + \cdots - 1509644078$$
$79$ $$T^{2} + \cdots - 2822066537$$
$83$ $$T^{2} + 61179 T + 711231498$$
$89$ $$T^{2} + 29322 T + 213892896$$
$97$ $$T^{2} - 9791 T - 40418570$$