# Properties

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

# Learn more

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: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338$$ x^6 - 2*x^5 - 59*x^4 + 122*x^3 + 941*x^2 - 1856*x - 2338 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 7^{3}$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - 9 q^{3} + (\beta_{4} - \beta_1 + 25) q^{4} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 16) q^{5} + 9 \beta_{2} q^{6} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \cdots - 28) q^{8}+ \cdots + 81 q^{9}+O(q^{10})$$ q - b2 * q^2 - 9 * q^3 + (b4 - b1 + 25) * q^4 + (b3 + 2*b2 - 2*b1 - 16) * q^5 + 9*b2 * q^6 + (2*b5 - 2*b4 - 3*b3 - 25*b2 + 9*b1 - 28) * q^8 + 81 * q^9 $$q - \beta_{2} q^{2} - 9 q^{3} + (\beta_{4} - \beta_1 + 25) q^{4} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 16) q^{5} + 9 \beta_{2} q^{6} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \cdots - 28) q^{8}+ \cdots + ( - 567 \beta_{5} - 648 \beta_{4} + \cdots + 8667) q^{99}+O(q^{100})$$ q - b2 * q^2 - 9 * q^3 + (b4 - b1 + 25) * q^4 + (b3 + 2*b2 - 2*b1 - 16) * q^5 + 9*b2 * q^6 + (2*b5 - 2*b4 - 3*b3 - 25*b2 + 9*b1 - 28) * q^8 + 81 * q^9 + (-3*b5 - 9*b4 - 3*b3 + 15*b2 + 19*b1 - 138) * q^10 + (-7*b5 - 8*b4 - 3*b3 + 12*b2 + 6*b1 + 107) * q^11 + (-9*b4 + 9*b1 - 225) * q^12 + (3*b5 - 8*b4 - 2*b3 + 34*b2 - 12*b1 - 215) * q^13 + (-9*b3 - 18*b2 + 18*b1 + 144) * q^15 + (12*b5 + 33*b4 + 18*b3 + 126*b2 - 27*b1 + 801) * q^16 + (17*b5 + 8*b4 + 7*b3 - 32*b2 + 31*b1 - 521) * q^17 - 81*b2 * q^18 + (3*b5 + 24*b4 - 12*b3 + 174*b2 + 55*b1 - 231) * q^19 + (-14*b5 + 7*b4 + 5*b3 + 335*b2 - 100*b1 + 41) * q^20 + (-36*b5 - 32*b4 + 18*b3 + 30*b2 - 238*b1 - 664) * q^22 + (51*b5 - 24*b4 + 45*b3 + 16*b2 - 759) * q^23 + (-18*b5 + 18*b4 + 27*b3 + 225*b2 - 81*b1 + 252) * q^24 + (-18*b5 + 16*b4 - 36*b3 + 228*b2 + 290*b1 + 883) * q^25 + (-25*b5 - 7*b4 + 9*b3 + 443*b2 - 45*b1 - 2206) * q^26 - 729 * q^27 + (-63*b5 - 24*b4 - 54*b3 + 86*b2 - 201*b1 - 837) * q^29 + (27*b5 + 81*b4 + 27*b3 - 135*b2 - 171*b1 + 1242) * q^30 + (-63*b5 + 56*b4 - 88*b3 - 94*b2 - 215*b1 - 673) * q^31 + (26*b5 - 128*b4 - 3*b3 - 781*b2 + 687*b1 - 6490) * q^32 + (63*b5 + 72*b4 + 27*b3 - 108*b2 - 54*b1 - 963) * q^33 + (99*b5 + 139*b4 + 25*b3 + 655*b2 + 501*b1 + 2908) * q^34 + (81*b4 - 81*b1 + 2025) * q^36 + (42*b5 + 96*b4 - 54*b3 - 24*b2 - 300*b1 + 3758) * q^37 + (148*b5 - 62*b4 + 22*b3 - 318*b2 + 292*b1 - 8946) * q^38 + (-27*b5 + 72*b4 + 18*b3 - 306*b2 + 108*b1 + 1935) * q^39 + (-30*b5 - 277*b4 - 37*b3 - 1195*b2 - 446*b1 - 17155) * q^40 + (-151*b5 - 136*b4 - 65*b3 - 656*b2 - 23*b1 - 4961) * q^41 + (150*b5 - 32*b4 - 90*b3 - 120*b2 - 208*b1 - 1218) * q^43 + (-236*b5 - 322*b4 - 132*b3 + 80*b2 - 990*b1 - 10718) * q^44 + (81*b3 + 162*b2 - 162*b1 - 1296) * q^45 + (36*b5 + 140*b4 + 54*b3 + 2394*b2 + 1498*b1 + 876) * q^46 + (-47*b5 + 184*b4 + 72*b3 - 206*b2 + 381*b1 - 8645) * q^47 + (-108*b5 - 297*b4 - 162*b3 - 1134*b2 + 243*b1 - 7209) * q^48 + (320*b5 + 100*b4 + 276*b3 - 909*b2 - 312*b1 - 7180) * q^50 + (-153*b5 - 72*b4 - 63*b3 + 288*b2 - 279*b1 + 4689) * q^51 + (-246*b5 - 445*b4 - b3 + 881*b2 + 458*b1 - 19507) * q^52 + (-58*b5 + 208*b4 + 132*b3 + 996*b2 + 1614*b1 + 13832) * q^53 + 729*b2 * q^54 + (417*b5 + 248*b4 + 536*b3 - 2686*b2 - 147*b1 - 2977) * q^55 + (-27*b5 - 216*b4 + 108*b3 - 1566*b2 - 495*b1 + 2079) * q^57 + (-408*b5 - 434*b4 - 162*b3 - 222*b2 - 2452*b1 - 11682) * q^58 + (115*b5 + 280*b4 + 100*b3 - 1794*b2 - 941*b1 - 2115) * q^59 + (126*b5 - 63*b4 - 45*b3 - 3015*b2 + 900*b1 - 369) * q^60 + (-225*b5 + 200*b4 - 328*b3 - 1186*b2 + 74*b1 - 8803) * q^61 + (-148*b5 - 178*b4 - 302*b3 - 2918*b2 - 2492*b1 - 2894) * q^62 + (737*b4 + 396*b3 + 8580*b2 - 125*b1 + 36125) * q^64 + (205*b5 + 392*b4 - 132*b3 - 2942*b2 - 3*b1 - 467) * q^65 + (324*b5 + 288*b4 - 162*b3 - 270*b2 + 2142*b1 + 5976) * q^66 + (354*b5 - 304*b4 + 414*b3 + 2496*b2 + 148*b1 + 1794) * q^67 + (646*b5 + 19*b4 + 73*b3 - 2901*b2 + 3712*b1 - 7407) * q^68 + (-459*b5 + 216*b4 - 405*b3 - 144*b2 + 6831) * q^69 + (-549*b5 - 120*b4 - 243*b3 - 3200*b2 - 2364*b1 - 19359) * q^71 + (162*b5 - 162*b4 - 243*b3 - 2025*b2 + 729*b1 - 2268) * q^72 + (-138*b5 - 32*b4 - 266*b3 - 6584*b2 - 787*b1 - 12854) * q^73 + (168*b5 + 192*b4 - 396*b3 - 7124*b2 + 324*b1 - 7008) * q^74 + (162*b5 - 144*b4 + 324*b3 - 2052*b2 - 2610*b1 - 7947) * q^75 + (432*b5 + 830*b4 + 926*b3 + 8234*b2 + 528*b1 + 35078) * q^76 + (225*b5 + 63*b4 - 81*b3 - 3987*b2 + 405*b1 + 19854) * q^78 + (312*b5 - 272*b4 + 468*b3 + 3240*b2 + 3992*b1 + 19592) * q^79 + (-882*b5 + 777*b4 - 45*b3 + 12525*b2 - 2424*b1 + 55959) * q^80 + 6561 * q^81 + (-819*b5 + 37*b4 + 163*b3 + 6397*b2 - 5841*b1 + 34012) * q^82 + (142*b5 + 16*b4 - 764*b3 - 828*b2 + 622*b1 - 17178) * q^83 + (-888*b5 - 976*b4 - 1134*b3 + 1428*b2 + 514*b1 - 3110) * q^85 + (236*b5 + 1444*b4 + 96*b3 + 3172*b2 + 192*b1 + 1616) * q^86 + (567*b5 + 216*b4 + 486*b3 - 774*b2 + 1809*b1 + 7533) * q^87 + (-1380*b5 - 716*b4 - 1026*b3 + 12162*b2 - 3034*b1 - 11784) * q^88 + (-355*b5 - 1000*b4 + 169*b3 + 4588*b2 + 6481*b1 + 7483) * q^89 + (-243*b5 - 729*b4 - 243*b3 + 1215*b2 + 1539*b1 - 11178) * q^90 + (340*b5 - 286*b4 - 240*b3 - 180*b2 + 6930*b1 - 75050) * q^92 + (567*b5 - 504*b4 + 792*b3 + 846*b2 + 1935*b1 + 6057) * q^93 + (720*b5 - 286*b4 - 106*b3 + 4154*b2 + 2584*b1 + 21314) * q^94 + (338*b5 + 1264*b4 + 366*b3 - 1856*b2 - 9324*b1 - 20578) * q^95 + (-234*b5 + 1152*b4 + 27*b3 + 7029*b2 - 6183*b1 + 58410) * q^96 + (432*b5 - 32*b4 + 22*b3 + 6220*b2 + 6723*b1 - 9896) * q^97 + (-567*b5 - 648*b4 - 243*b3 + 972*b2 + 486*b1 + 8667) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9}+O(q^{10})$$ 6 * q + 2 * q^2 - 54 * q^3 + 150 * q^4 - 100 * q^5 - 18 * q^6 - 114 * q^8 + 486 * q^9 $$6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9} - 864 q^{10} + 604 q^{11} - 1350 q^{12} - 1352 q^{13} + 900 q^{15} + 4578 q^{16} - 3028 q^{17} + 162 q^{18} - 1728 q^{19} - 452 q^{20} - 4116 q^{22} - 4484 q^{23} + 1026 q^{24} + 4806 q^{25} - 14172 q^{26} - 4374 q^{27} - 5320 q^{29} + 7776 q^{30} - 3976 q^{31} - 37326 q^{32} - 5436 q^{33} + 16336 q^{34} + 12150 q^{36} + 22680 q^{37} - 52744 q^{38} + 12168 q^{39} - 100600 q^{40} - 28756 q^{41} - 6768 q^{43} - 64940 q^{44} - 8100 q^{45} + 540 q^{46} - 51552 q^{47} - 41202 q^{48} - 40622 q^{50} + 27252 q^{51} - 119296 q^{52} + 80884 q^{53} - 1458 q^{54} - 11656 q^{55} + 15552 q^{57} - 70464 q^{58} - 8872 q^{59} + 4068 q^{60} - 50896 q^{61} - 11824 q^{62} + 199590 q^{64} + 3492 q^{65} + 37044 q^{66} + 6480 q^{67} - 37348 q^{68} + 40356 q^{69} - 110852 q^{71} - 9234 q^{72} - 64232 q^{73} - 27464 q^{74} - 43254 q^{75} + 194864 q^{76} + 127548 q^{78} + 111696 q^{79} + 308940 q^{80} + 39366 q^{81} + 189640 q^{82} - 101128 q^{83} - 23292 q^{85} + 3824 q^{86} + 47880 q^{87} - 97788 q^{88} + 35012 q^{89} - 69984 q^{90} - 449260 q^{92} + 35784 q^{93} + 121016 q^{94} - 119080 q^{95} + 335934 q^{96} - 70952 q^{97} + 48924 q^{99}+O(q^{100})$$ 6 * q + 2 * q^2 - 54 * q^3 + 150 * q^4 - 100 * q^5 - 18 * q^6 - 114 * q^8 + 486 * q^9 - 864 * q^10 + 604 * q^11 - 1350 * q^12 - 1352 * q^13 + 900 * q^15 + 4578 * q^16 - 3028 * q^17 + 162 * q^18 - 1728 * q^19 - 452 * q^20 - 4116 * q^22 - 4484 * q^23 + 1026 * q^24 + 4806 * q^25 - 14172 * q^26 - 4374 * q^27 - 5320 * q^29 + 7776 * q^30 - 3976 * q^31 - 37326 * q^32 - 5436 * q^33 + 16336 * q^34 + 12150 * q^36 + 22680 * q^37 - 52744 * q^38 + 12168 * q^39 - 100600 * q^40 - 28756 * q^41 - 6768 * q^43 - 64940 * q^44 - 8100 * q^45 + 540 * q^46 - 51552 * q^47 - 41202 * q^48 - 40622 * q^50 + 27252 * q^51 - 119296 * q^52 + 80884 * q^53 - 1458 * q^54 - 11656 * q^55 + 15552 * q^57 - 70464 * q^58 - 8872 * q^59 + 4068 * q^60 - 50896 * q^61 - 11824 * q^62 + 199590 * q^64 + 3492 * q^65 + 37044 * q^66 + 6480 * q^67 - 37348 * q^68 + 40356 * q^69 - 110852 * q^71 - 9234 * q^72 - 64232 * q^73 - 27464 * q^74 - 43254 * q^75 + 194864 * q^76 + 127548 * q^78 + 111696 * q^79 + 308940 * q^80 + 39366 * q^81 + 189640 * q^82 - 101128 * q^83 - 23292 * q^85 + 3824 * q^86 + 47880 * q^87 - 97788 * q^88 + 35012 * q^89 - 69984 * q^90 - 449260 * q^92 + 35784 * q^93 + 121016 * q^94 - 119080 * q^95 + 335934 * q^96 - 70952 * q^97 + 48924 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338$$ :

 $$\beta_{1}$$ $$=$$ $$( 28\nu^{5} - 35\nu^{4} - 546\nu^{3} + 742\nu^{2} - 7063\nu + 11802 ) / 1941$$ (28*v^5 - 35*v^4 - 546*v^3 + 742*v^2 - 7063*v + 11802) / 1941 $$\beta_{2}$$ $$=$$ $$( -87\nu^{5} - 53\nu^{4} + 3961\nu^{3} + 2547\nu^{2} - 42269\nu - 27289 ) / 1941$$ (-87*v^5 - 53*v^4 + 3961*v^3 + 2547*v^2 - 42269*v - 27289) / 1941 $$\beta_{3}$$ $$=$$ $$( 158\nu^{5} - 521\nu^{4} - 6316\nu^{3} + 21656\nu^{2} + 33579\nu - 143678 ) / 1941$$ (158*v^5 - 521*v^4 - 6316*v^3 + 21656*v^2 + 33579*v - 143678) / 1941 $$\beta_{4}$$ $$=$$ $$( 108\nu^{5} + 512\nu^{4} - 5988\nu^{3} - 17195\nu^{2} + 81453\nu + 71402 ) / 647$$ (108*v^5 + 512*v^4 - 5988*v^3 - 17195*v^2 + 81453*v + 71402) / 647 $$\beta_{5}$$ $$=$$ $$( -774\nu^{5} - 1297\nu^{4} + 39032\nu^{3} + 55188\nu^{2} - 447553\nu - 433643 ) / 1941$$ (-774*v^5 - 1297*v^4 + 39032*v^3 + 55188*v^2 - 447553*v - 433643) / 1941
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{3} - 12\beta_{2} - 4\beta _1 + 5 ) / 28$$ (b5 - b3 - 12*b2 - 4*b1 + 5) / 28 $$\nu^{2}$$ $$=$$ $$( 3\beta_{5} + 4\beta_{4} + 5\beta_{3} - 4\beta_{2} - 4\beta _1 + 567 ) / 28$$ (3*b5 + 4*b4 + 5*b3 - 4*b2 - 4*b1 + 567) / 28 $$\nu^{3}$$ $$=$$ $$( 29\beta_{5} + 4\beta_{4} - 28\beta_{3} - 302\beta_{2} - 25\beta _1 - 129 ) / 28$$ (29*b5 + 4*b4 - 28*b3 - 302*b2 - 25*b1 - 129) / 28 $$\nu^{4}$$ $$=$$ $$( 99\beta_{5} + 176\beta_{4} + 155\beta_{3} + 80\beta_{2} + 74\beta _1 + 14843 ) / 28$$ (99*b5 + 176*b4 + 155*b3 + 80*b2 + 74*b1 + 14843) / 28 $$\nu^{5}$$ $$=$$ $$( 862\beta_{5} + 192\beta_{4} - 737\beta_{3} - 8710\beta_{2} + 643\beta _1 - 9528 ) / 28$$ (862*b5 + 192*b4 - 737*b3 - 8710*b2 + 643*b1 - 9528) / 28

## 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
 −5.61145 −0.910122 4.27213 −5.10089 3.75353 5.59680
−11.1881 −9.00000 93.1729 62.3150 100.693 0 −684.407 81.0000 −697.185
1.2 −5.31815 −9.00000 −3.71724 −103.471 47.8634 0 189.950 81.0000 550.272
1.3 −3.09163 −9.00000 −22.4418 −13.7926 27.8246 0 168.314 81.0000 42.6416
1.4 3.38033 −9.00000 −20.5734 54.5253 −30.4230 0 −177.715 81.0000 184.313
1.5 8.20863 −9.00000 35.3816 −29.2259 −73.8777 0 27.7583 81.0000 −239.905
1.6 10.0089 −9.00000 68.1779 −70.3512 −90.0800 0 362.101 81.0000 −704.138
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.n 6
3.b odd 2 1 441.6.a.bb 6
7.b odd 2 1 147.6.a.o yes 6
7.c even 3 2 147.6.e.q 12
7.d odd 6 2 147.6.e.p 12
21.c even 2 1 441.6.a.ba 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.a.n 6 1.a even 1 1 trivial
147.6.a.o yes 6 7.b odd 2 1
147.6.e.p 12 7.d odd 6 2
147.6.e.q 12 7.c even 3 2
441.6.a.ba 6 21.c even 2 1
441.6.a.bb 6 3.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{6} - 2T_{2}^{5} - 169T_{2}^{4} + 336T_{2}^{3} + 6472T_{2}^{2} - 4256T_{2} - 51088$$ T2^6 - 2*T2^5 - 169*T2^4 + 336*T2^3 + 6472*T2^2 - 4256*T2 - 51088 $$T_{5}^{6} + 100T_{5}^{5} - 6778T_{5}^{4} - 651312T_{5}^{3} + 9669292T_{5}^{2} + 959211664T_{5} + 9969962312$$ T5^6 + 100*T5^5 - 6778*T5^4 - 651312*T5^3 + 9669292*T5^2 + 959211664*T5 + 9969962312

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 2 T^{5} + \cdots - 51088$$
$3$ $$(T + 9)^{6}$$
$5$ $$T^{6} + \cdots + 9969962312$$
$7$ $$T^{6}$$
$11$ $$T^{6} + \cdots - 55273527989696$$
$13$ $$T^{6} + \cdots + 88\!\cdots\!04$$
$17$ $$T^{6} + \cdots - 77\!\cdots\!12$$
$19$ $$T^{6} + \cdots - 37\!\cdots\!68$$
$23$ $$T^{6} + \cdots - 10\!\cdots\!48$$
$29$ $$T^{6} + \cdots - 45\!\cdots\!84$$
$31$ $$T^{6} + \cdots - 32\!\cdots\!04$$
$37$ $$T^{6} + \cdots - 79\!\cdots\!96$$
$41$ $$T^{6} + \cdots + 18\!\cdots\!56$$
$43$ $$T^{6} + \cdots - 28\!\cdots\!68$$
$47$ $$T^{6} + \cdots + 13\!\cdots\!28$$
$53$ $$T^{6} + \cdots + 93\!\cdots\!88$$
$59$ $$T^{6} + \cdots - 34\!\cdots\!52$$
$61$ $$T^{6} + \cdots - 18\!\cdots\!36$$
$67$ $$T^{6} + \cdots - 24\!\cdots\!32$$
$71$ $$T^{6} + \cdots - 29\!\cdots\!48$$
$73$ $$T^{6} + \cdots + 32\!\cdots\!84$$
$79$ $$T^{6} + \cdots - 10\!\cdots\!68$$
$83$ $$T^{6} + \cdots - 19\!\cdots\!68$$
$89$ $$T^{6} + \cdots + 48\!\cdots\!72$$
$97$ $$T^{6} + \cdots - 13\!\cdots\!88$$
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