# Properties

 Label 147.6.a.l Level $147$ Weight $6$ Character orbit 147.a Self dual yes Analytic conductor $23.576$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 97x^{2} + 7x + 294$$ x^4 - x^3 - 97*x^2 + 7*x + 294 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} - 9 q^{3} + (\beta_{2} - \beta_1 + 18) q^{4} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{5} + (9 \beta_1 - 9) q^{6} + ( - 2 \beta_{3} + \beta_{2} + \cdots + 38) q^{8}+ \cdots + 81 q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 - 9 * q^3 + (b2 - b1 + 18) * q^4 + (-b3 - b2 + 2*b1 - 1) * q^5 + (9*b1 - 9) * q^6 + (-2*b3 + b2 - 27*b1 + 38) * q^8 + 81 * q^9 $$q + ( - \beta_1 + 1) q^{2} - 9 q^{3} + (\beta_{2} - \beta_1 + 18) q^{4} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{5} + (9 \beta_1 - 9) q^{6} + ( - 2 \beta_{3} + \beta_{2} + \cdots + 38) q^{8}+ \cdots + (243 \beta_{3} + 729 \beta_{2} + \cdots + 8667) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 - 9 * q^3 + (b2 - b1 + 18) * q^4 + (-b3 - b2 + 2*b1 - 1) * q^5 + (9*b1 - 9) * q^6 + (-2*b3 + b2 - 27*b1 + 38) * q^8 + 81 * q^9 + (b2 + 23*b1 - 76) * q^10 + (3*b3 + 9*b2 - 8*b1 + 107) * q^11 + (-9*b2 + 9*b1 - 162) * q^12 + (-9*b3 + b2 - 76*b1 - 96) * q^13 + (9*b3 + 9*b2 - 18*b1 + 9) * q^15 + (-6*b3 + b2 - 85*b1 + 840) * q^16 + (-4*b3 + 8*b2 + 108*b1 - 92) * q^17 + (-81*b1 + 81) * q^18 + (15*b3 + b2 - 220*b1 - 72) * q^19 + (30*b3 + 9*b2 - 29*b1 - 1168) * q^20 + (-12*b3 - b2 - 419*b1 + 448) * q^22 + (20*b3 + 8*b2 - 300*b1 + 1804) * q^23 + (18*b3 - 9*b2 + 243*b1 - 342) * q^24 + (-45*b3 - 95*b2 + 80*b1 + 636) * q^25 + (-20*b3 + 103*b2 - 116*b1 + 3865) * q^26 - 729 * q^27 + (5*b3 - 103*b2 - 254*b1 + 147) * q^29 + (-9*b2 - 207*b1 + 684) * q^30 + (-24*b3 - 94*b2 + 130*b1 + 1523) * q^31 + (50*b3 + 71*b2 - 131*b1 + 3948) * q^32 + (-27*b3 - 81*b2 + 72*b1 - 963) * q^33 + (-24*b3 - 96*b2 - 312*b1 - 5256) * q^34 + (81*b2 - 81*b1 + 1458) * q^36 + (-9*b3 - 95*b2 + 1028*b1 + 3508) * q^37 + (28*b3 + 175*b2 + 316*b1 + 10321) * q^38 + (81*b3 - 9*b2 + 684*b1 + 864) * q^39 + (42*b3 - 93*b2 + 633*b1 + 1932) * q^40 + (-142*b3 + 62*b2 + 328*b1 - 1128) * q^41 + (-33*b3 + 93*b2 + 816*b1 + 7142) * q^43 + (-118*b3 + 167*b2 - 379*b1 + 17864) * q^44 + (-81*b3 - 81*b2 + 162*b1 - 81) * q^45 + (24*b3 + 240*b2 - 1752*b1 + 16008) * q^46 + (28*b3 - 56*b2 - 324*b1 + 3818) * q^47 + (54*b3 - 9*b2 + 765*b1 - 7560) * q^48 + (100*b3 + 55*b2 + 2404*b1 - 2399) * q^50 + (36*b3 - 72*b2 - 972*b1 + 828) * q^51 + (42*b3 + 144*b2 - 6036*b1 + 13450) * q^52 + (-239*b3 + 13*b2 + 1338*b1 + 3095) * q^53 + (729*b1 - 729) * q^54 + (315*b3 + 335*b2 - 506*b1 - 17987) * q^55 + (-135*b3 - 9*b2 + 1980*b1 + 648) * q^57 + (216*b3 + 239*b2 + 4171*b1 + 12154) * q^58 + (71*b3 - 163*b2 + 888*b1 - 9011) * q^59 + (-270*b3 - 81*b2 + 261*b1 + 10512) * q^60 + (-240*b3 + 52*b2 - 796*b1 + 1566) * q^61 + (140*b3 - 58*b2 + 1875*b1 - 4505) * q^62 + (150*b3 - 51*b2 - 3189*b1 - 17600) * q^64 + (246*b3 - 330*b2 + 1660*b1 + 16136) * q^65 + (108*b3 + 9*b2 + 3771*b1 - 4032) * q^66 + (465*b3 - 193*b2 + 2764*b1 - 2286) * q^67 + (272*b3 + 128*b2 + 5280*b1 + 13312) * q^68 + (-180*b3 - 72*b2 + 2700*b1 - 16236) * q^69 + (-180*b3 - 24*b2 + 3660*b1 + 21390) * q^71 + (-162*b3 + 81*b2 - 2187*b1 + 3078) * q^72 + (93*b3 + 143*b2 - 3056*b1 - 14074) * q^73 + (172*b3 - 1001*b2 + 216*b1 - 46915) * q^74 + (405*b3 + 855*b2 - 720*b1 - 5724) * q^75 + (-774*b3 - 432*b2 - 9924*b1 - 3062) * q^76 + (180*b3 - 927*b2 + 1044*b1 - 34785) * q^78 + (-786*b3 - 96*b2 + 2286*b1 - 11635) * q^79 + (-690*b3 - 1047*b2 + 3607*b1 + 6920) * q^80 + 6561 * q^81 + (-408*b3 + 98*b2 - 4112*b1 - 13322) * q^82 + (129*b3 - 1005*b2 + 6432*b1 + 50001) * q^83 + (372*b3 - 192*b2 - 2988*b1 + 9732) * q^85 + (-252*b3 - 717*b2 - 11582*b1 - 31705) * q^86 + (-45*b3 + 927*b2 + 2286*b1 - 1323) * q^87 + (-186*b3 + 765*b2 - 13545*b1 + 25668) * q^88 + (-622*b3 - 1582*b2 - 9508*b1 + 20966) * q^89 + (81*b2 + 1863*b1 - 6156) * q^90 + (-1072*b3 + 1424*b2 - 15792*b1 + 44224) * q^92 + (216*b3 + 846*b2 - 1170*b1 - 13707) * q^93 + (168*b3 + 240*b2 - 990*b1 + 18798) * q^94 + (294*b3 + 1542*b2 + 3820*b1 - 55528) * q^95 + (-450*b3 - 639*b2 + 1179*b1 - 35532) * q^96 + (669*b3 + 863*b2 - 464*b1 + 47705) * q^97 + (243*b3 + 729*b2 - 648*b1 + 8667) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} - 36 q^{3} + 69 q^{4} - 27 q^{6} + 123 q^{8} + 324 q^{9}+O(q^{10})$$ 4 * q + 3 * q^2 - 36 * q^3 + 69 * q^4 - 27 * q^6 + 123 * q^8 + 324 * q^9 $$4 q + 3 q^{2} - 36 q^{3} + 69 q^{4} - 27 q^{6} + 123 q^{8} + 324 q^{9} - 283 q^{10} + 402 q^{11} - 621 q^{12} - 462 q^{13} + 3273 q^{16} - 276 q^{17} + 243 q^{18} - 510 q^{19} - 4719 q^{20} + 1375 q^{22} + 6900 q^{23} - 1107 q^{24} + 2814 q^{25} + 15138 q^{26} - 2916 q^{27} + 540 q^{29} + 2547 q^{30} + 6410 q^{31} + 15519 q^{32} - 3618 q^{33} - 21144 q^{34} + 5589 q^{36} + 15250 q^{37} + 41250 q^{38} + 4158 q^{39} + 8547 q^{40} - 4308 q^{41} + 29198 q^{43} + 70743 q^{44} + 61800 q^{46} + 15060 q^{47} - 29457 q^{48} - 7302 q^{50} + 2484 q^{51} + 47476 q^{52} + 13692 q^{53} - 2187 q^{54} - 73124 q^{55} + 4590 q^{57} + 52309 q^{58} - 34830 q^{59} + 42471 q^{60} + 5364 q^{61} - 16029 q^{62} - 73487 q^{64} + 66864 q^{65} - 12375 q^{66} - 5994 q^{67} + 58272 q^{68} - 62100 q^{69} + 89268 q^{71} + 9963 q^{72} - 59638 q^{73} - 185442 q^{74} - 25326 q^{75} - 21308 q^{76} - 136242 q^{78} - 44062 q^{79} + 33381 q^{80} + 26244 q^{81} - 57596 q^{82} + 208446 q^{83} + 36324 q^{85} - 136968 q^{86} - 4860 q^{87} + 87597 q^{88} + 77520 q^{89} - 22923 q^{90} + 158256 q^{92} - 57690 q^{93} + 73722 q^{94} - 221376 q^{95} - 139671 q^{96} + 188630 q^{97} + 32562 q^{99}+O(q^{100})$$ 4 * q + 3 * q^2 - 36 * q^3 + 69 * q^4 - 27 * q^6 + 123 * q^8 + 324 * q^9 - 283 * q^10 + 402 * q^11 - 621 * q^12 - 462 * q^13 + 3273 * q^16 - 276 * q^17 + 243 * q^18 - 510 * q^19 - 4719 * q^20 + 1375 * q^22 + 6900 * q^23 - 1107 * q^24 + 2814 * q^25 + 15138 * q^26 - 2916 * q^27 + 540 * q^29 + 2547 * q^30 + 6410 * q^31 + 15519 * q^32 - 3618 * q^33 - 21144 * q^34 + 5589 * q^36 + 15250 * q^37 + 41250 * q^38 + 4158 * q^39 + 8547 * q^40 - 4308 * q^41 + 29198 * q^43 + 70743 * q^44 + 61800 * q^46 + 15060 * q^47 - 29457 * q^48 - 7302 * q^50 + 2484 * q^51 + 47476 * q^52 + 13692 * q^53 - 2187 * q^54 - 73124 * q^55 + 4590 * q^57 + 52309 * q^58 - 34830 * q^59 + 42471 * q^60 + 5364 * q^61 - 16029 * q^62 - 73487 * q^64 + 66864 * q^65 - 12375 * q^66 - 5994 * q^67 + 58272 * q^68 - 62100 * q^69 + 89268 * q^71 + 9963 * q^72 - 59638 * q^73 - 185442 * q^74 - 25326 * q^75 - 21308 * q^76 - 136242 * q^78 - 44062 * q^79 + 33381 * q^80 + 26244 * q^81 - 57596 * q^82 + 208446 * q^83 + 36324 * q^85 - 136968 * q^86 - 4860 * q^87 + 87597 * q^88 + 77520 * q^89 - 22923 * q^90 + 158256 * q^92 - 57690 * q^93 + 73722 * q^94 - 221376 * q^95 - 139671 * q^96 + 188630 * q^97 + 32562 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 97x^{2} + 7x + 294$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 49$$ v^2 - v - 49 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 2\nu^{2} - 89\nu + 52 ) / 2$$ (v^3 - 2*v^2 - 89*v + 52) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 49$$ b2 + b1 + 49 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + 2\beta_{2} + 91\beta _1 + 46$$ 2*b3 + 2*b2 + 91*b1 + 46

## 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
 10.1812 1.79080 −1.74818 −9.22385
−9.18123 −9.00000 52.2950 −22.0716 82.6311 0 −186.333 81.0000 202.644
1.2 −0.790805 −9.00000 −31.3746 104.192 7.11724 0 50.1170 81.0000 −82.3953
1.3 2.74818 −9.00000 −24.4475 −58.3673 −24.7336 0 −155.128 81.0000 −160.404
1.4 10.2239 −9.00000 72.5272 −23.7528 −92.0147 0 414.344 81.0000 −242.845
 $$n$$: 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.l 4
3.b odd 2 1 441.6.a.v 4
7.b odd 2 1 147.6.a.m 4
7.c even 3 2 147.6.e.o 8
7.d odd 6 2 21.6.e.c 8
21.c even 2 1 441.6.a.w 4
21.g even 6 2 63.6.e.e 8
28.f even 6 2 336.6.q.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 7.d odd 6 2
63.6.e.e 8 21.g even 6 2
147.6.a.l 4 1.a even 1 1 trivial
147.6.a.m 4 7.b odd 2 1
147.6.e.o 8 7.c even 3 2
336.6.q.j 8 28.f even 6 2
441.6.a.v 4 3.b odd 2 1
441.6.a.w 4 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(147))$$:

 $$T_{2}^{4} - 3T_{2}^{3} - 94T_{2}^{2} + 186T_{2} + 204$$ T2^4 - 3*T2^3 - 94*T2^2 + 186*T2 + 204 $$T_{5}^{4} - 7657T_{5}^{2} - 302700T_{5} - 3188244$$ T5^4 - 7657*T5^2 - 302700*T5 - 3188244

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 3 T^{3} + \cdots + 204$$
$3$ $$(T + 9)^{4}$$
$5$ $$T^{4} - 7657 T^{2} + \cdots - 3188244$$
$7$ $$T^{4}$$
$11$ $$T^{4} + \cdots - 1682132124$$
$13$ $$T^{4} + \cdots + 149501563456$$
$17$ $$T^{4} + \cdots - 50104147968$$
$19$ $$T^{4} + \cdots + 7391138416576$$
$23$ $$T^{4} + \cdots + 3007939608576$$
$29$ $$T^{4} + \cdots + 408027025117872$$
$31$ $$T^{4} + \cdots + 86716089209547$$
$37$ $$T^{4} + \cdots - 50\!\cdots\!56$$
$41$ $$T^{4} + \cdots - 18\!\cdots\!52$$
$43$ $$T^{4} + \cdots - 991662745581932$$
$47$ $$T^{4} + \cdots - 270685655359056$$
$53$ $$T^{4} + \cdots - 85\!\cdots\!72$$
$59$ $$T^{4} + \cdots - 25\!\cdots\!36$$
$61$ $$T^{4} + \cdots + 17\!\cdots\!24$$
$67$ $$T^{4} + \cdots + 55\!\cdots\!84$$
$71$ $$T^{4} + \cdots + 21\!\cdots\!68$$
$73$ $$T^{4} + \cdots - 12\!\cdots\!00$$
$79$ $$T^{4} + \cdots + 16\!\cdots\!59$$
$83$ $$T^{4} + \cdots - 41\!\cdots\!32$$
$89$ $$T^{4} + \cdots - 47\!\cdots\!68$$
$97$ $$T^{4} + \cdots - 11\!\cdots\!44$$