# Properties

 Label 162.10.c.t Level $162$ Weight $10$ Character orbit 162.c Analytic conductor $83.436$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,10,Mod(55,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("162.55");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 162.c (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: $$83.4358054585$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 1910x^{6} + 1303429x^{4} + 373918128x^{2} + 38299272804$$ x^8 + 1910*x^6 + 1303429*x^4 + 373918128*x^2 + 38299272804 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3^{18}$$ Twist minimal: yes 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 16 \beta_{2} q^{2} + (256 \beta_{2} - 256) q^{4} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 492 \beta_{2} + \beta_1 - 492) q^{5} + ( - 18 \beta_{7} + \beta_{6} - 1124 \beta_{2}) q^{7} - 4096 q^{8}+O(q^{10})$$ q + 16*b2 * q^2 + (256*b2 - 256) * q^4 + (b7 - b6 + b5 - b4 + 492*b2 + b1 - 492) * q^5 + (-18*b7 + b6 - 1124*b2) * q^7 - 4096 * q^8 $$q + 16 \beta_{2} q^{2} + (256 \beta_{2} - 256) q^{4} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 492 \beta_{2} + \beta_1 - 492) q^{5} + ( - 18 \beta_{7} + \beta_{6} - 1124 \beta_{2}) q^{7} - 4096 q^{8} + ( - 16 \beta_{4} - 16 \beta_{3} + 16 \beta_1 - 7872) q^{10} + ( - 28 \beta_{7} + 15 \beta_{6} + 61 \beta_{5} + 61 \beta_{3} + 2196 \beta_{2}) q^{11} + ( - 166 \beta_{7} - 93 \beta_{6} - 52 \beta_{5} - 93 \beta_{4} + \cdots + 40639) q^{13}+ \cdots + (41472 \beta_{4} + 15488 \beta_{3} + 546048 \beta_1 + 509697072) q^{98}+O(q^{100})$$ q + 16*b2 * q^2 + (256*b2 - 256) * q^4 + (b7 - b6 + b5 - b4 + 492*b2 + b1 - 492) * q^5 + (-18*b7 + b6 - 1124*b2) * q^7 - 4096 * q^8 + (-16*b4 - 16*b3 + 16*b1 - 7872) * q^10 + (-28*b7 + 15*b6 + 61*b5 + 61*b3 + 2196*b2) * q^11 + (-166*b7 - 93*b6 - 52*b5 - 93*b4 - 40639*b2 - 166*b1 + 40639) * q^13 + (-288*b7 + 16*b6 + 16*b4 - 17984*b2 - 288*b1 + 17984) * q^14 - 65536*b2 * q^16 + (-254*b4 + 138*b3 - 1607*b1 + 134520) * q^17 + (355*b4 + 400*b3 - 1766*b1 - 2056) * q^19 + (-256*b7 + 256*b6 - 256*b5 - 256*b3 - 125952*b2) * q^20 + (-448*b7 + 240*b6 + 976*b5 + 240*b4 + 35136*b2 - 448*b1 - 35136) * q^22 + (1832*b7 - 795*b6 - 449*b5 - 795*b4 + 648684*b2 + 1832*b1 - 648684) * q^23 + (2196*b7 + 494*b6 - 1264*b5 - 1264*b3 - 468298*b2) * q^25 + (-1488*b4 + 832*b3 - 2656*b1 + 650224) * q^26 + (256*b4 - 4608*b1 + 287744) * q^28 + (15545*b7 + 2875*b6 + 1421*b5 + 1421*b3 - 808164*b2) * q^29 + (-980*b7 - 2710*b6 - 448*b5 - 2710*b4 + 870644*b2 - 980*b1 - 870644) * q^31 + (-1048576*b2 + 1048576) * q^32 + (25712*b7 + 4064*b6 + 2208*b5 + 2208*b3 + 2152320*b2) * q^34 + (-1947*b4 + 3025*b3 - 680*b1 + 1992324) * q^35 + (1221*b4 - 908*b3 - 48570*b1 - 621973) * q^37 + (28256*b7 - 5680*b6 + 6400*b5 + 6400*b3 - 32896*b2) * q^38 + (-4096*b7 + 4096*b6 - 4096*b5 + 4096*b4 - 2015232*b2 - 4096*b1 + 2015232) * q^40 + (54748*b7 + 3864*b6 + 5456*b5 + 3864*b4 + 8664288*b2 + 54748*b1 - 8664288) * q^41 + (-4378*b7 - 18963*b6 - 1824*b5 - 1824*b3 + 10352500*b2) * q^43 + (3840*b4 - 15616*b3 - 7168*b1 - 562176) * q^44 + (-12720*b4 + 7184*b3 + 29312*b1 - 10378944) * q^46 + (119964*b7 - 10208*b6 - 33440*b5 - 33440*b3 - 10139712*b2) * q^47 + (34128*b7 + 2592*b6 - 968*b5 + 2592*b4 - 31856067*b2 + 34128*b1 + 31856067) * q^49 + (35136*b7 + 7904*b6 - 20224*b5 + 7904*b4 - 7492768*b2 + 35136*b1 + 7492768) * q^50 + (42496*b7 + 23808*b6 + 13312*b5 + 13312*b3 + 10403584*b2) * q^52 + (70438*b4 - 6314*b3 - 129292*b1 + 2105256) * q^53 + (6017*b4 - 42944*b3 + 432174*b1 - 57267000) * q^55 + (73728*b7 - 4096*b6 + 4603904*b2) * q^56 + (248720*b7 + 46000*b6 + 22736*b5 + 46000*b4 - 12930624*b2 + 248720*b1 + 12930624) * q^58 + (555248*b7 - 31300*b6 + 43092*b5 - 31300*b4 + 6710832*b2 + 555248*b1 - 6710832) * q^59 + (-256278*b7 - 43861*b6 - 6004*b5 - 6004*b3 + 27876121*b2) * q^61 + (-43360*b4 + 7168*b3 - 15680*b1 - 13930304) * q^62 + 16777216 * q^64 + (1158355*b7 + 28798*b6 + 99330*b5 + 99330*b3 - 5876904*b2) * q^65 + (679386*b7 - 104445*b6 - 104016*b5 - 104445*b4 - 52016128*b2 + 679386*b1 + 52016128) * q^67 + (411392*b7 + 65024*b6 + 35328*b5 + 65024*b4 + 34437120*b2 + 411392*b1 - 34437120) * q^68 + (10880*b7 + 31152*b6 + 48400*b5 + 48400*b3 + 31877184*b2) * q^70 + (-212315*b4 + 57537*b3 - 601268*b1 + 89002068) * q^71 + (29032*b4 + 192008*b3 + 926688*b1 - 9281527) * q^73 + (777120*b7 - 19536*b6 - 14528*b5 - 14528*b3 - 9951568*b2) * q^74 + (452096*b7 - 90880*b6 + 102400*b5 - 90880*b4 - 526336*b2 + 452096*b1 + 526336) * q^76 + (77708*b7 - 36832*b6 - 223560*b5 - 36832*b4 + 8252640*b2 + 77708*b1 - 8252640) * q^77 + (1559314*b7 + 146143*b6 - 206656*b5 - 206656*b3 + 161283712*b2) * q^79 + (65536*b4 + 65536*b3 - 65536*b1 + 32243712) * q^80 + (61824*b4 - 87296*b3 + 875968*b1 - 138628608) * q^82 + (1163580*b7 + 189046*b6 - 179630*b5 - 179630*b3 - 254759064*b2) * q^83 + (3671574*b7 + 272597*b6 + 416564*b5 + 272597*b4 + 20238885*b2 + 3671574*b1 - 20238885) * q^85 + (-70048*b7 - 303408*b6 - 29184*b5 - 303408*b4 + 165640000*b2 - 70048*b1 - 165640000) * q^86 + (114688*b7 - 61440*b6 - 249856*b5 - 249856*b3 - 8994816*b2) * q^88 + (-10280*b4 - 59736*b3 - 1246781*b1 + 387048192) * q^89 + (-84891*b4 - 224112*b3 + 1353206*b1 - 32694152) * q^91 + (-468992*b7 + 203520*b6 + 114944*b5 + 114944*b3 - 166063104*b2) * q^92 + (1919424*b7 - 163328*b6 - 535040*b5 - 163328*b4 - 162235392*b2 + 1919424*b1 + 162235392) * q^94 + (-104188*b7 + 274325*b6 + 456207*b5 + 274325*b4 + 861483468*b2 - 104188*b1 - 861483468) * q^95 + (3580696*b7 + 127068*b6 + 958072*b5 + 958072*b3 - 278003642*b2) * q^97 + (41472*b4 + 15488*b3 + 546048*b1 + 509697072) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 64 q^{2} - 1024 q^{4} - 1968 q^{5} - 4496 q^{7} - 32768 q^{8}+O(q^{10})$$ 8 * q + 64 * q^2 - 1024 * q^4 - 1968 * q^5 - 4496 * q^7 - 32768 * q^8 $$8 q + 64 q^{2} - 1024 q^{4} - 1968 q^{5} - 4496 q^{7} - 32768 q^{8} - 62976 q^{10} + 8784 q^{11} + 162556 q^{13} + 71936 q^{14} - 262144 q^{16} + 1076160 q^{17} - 16448 q^{19} - 503808 q^{20} - 140544 q^{22} - 2594736 q^{23} - 1873192 q^{25} + 5201792 q^{26} + 2301952 q^{28} - 3232656 q^{29} - 3482576 q^{31} + 4194304 q^{32} + 8609280 q^{34} + 15938592 q^{35} - 4975784 q^{37} - 131584 q^{38} + 8060928 q^{40} - 34657152 q^{41} + 41410000 q^{43} - 4497408 q^{44} - 83031552 q^{46} - 40558848 q^{47} + 127424268 q^{49} + 29971072 q^{50} + 41614336 q^{52} + 16842048 q^{53} - 458136000 q^{55} + 18415616 q^{56} + 51722496 q^{58} - 26843328 q^{59} + 111504484 q^{61} - 111442432 q^{62} + 134217728 q^{64} - 23507616 q^{65} + 208064512 q^{67} - 137748480 q^{68} + 127508736 q^{70} + 712016544 q^{71} - 74252216 q^{73} - 39806272 q^{74} + 2105344 q^{76} - 33010560 q^{77} + 645134848 q^{79} + 257949696 q^{80} - 1109028864 q^{82} - 1019036256 q^{83} - 80955540 q^{85} - 662560000 q^{86} - 35979264 q^{88} + 3096385536 q^{89} - 261553216 q^{91} - 664252416 q^{92} + 648941568 q^{94} - 3445933872 q^{95} - 1112014568 q^{97} + 4077576576 q^{98}+O(q^{100})$$ 8 * q + 64 * q^2 - 1024 * q^4 - 1968 * q^5 - 4496 * q^7 - 32768 * q^8 - 62976 * q^10 + 8784 * q^11 + 162556 * q^13 + 71936 * q^14 - 262144 * q^16 + 1076160 * q^17 - 16448 * q^19 - 503808 * q^20 - 140544 * q^22 - 2594736 * q^23 - 1873192 * q^25 + 5201792 * q^26 + 2301952 * q^28 - 3232656 * q^29 - 3482576 * q^31 + 4194304 * q^32 + 8609280 * q^34 + 15938592 * q^35 - 4975784 * q^37 - 131584 * q^38 + 8060928 * q^40 - 34657152 * q^41 + 41410000 * q^43 - 4497408 * q^44 - 83031552 * q^46 - 40558848 * q^47 + 127424268 * q^49 + 29971072 * q^50 + 41614336 * q^52 + 16842048 * q^53 - 458136000 * q^55 + 18415616 * q^56 + 51722496 * q^58 - 26843328 * q^59 + 111504484 * q^61 - 111442432 * q^62 + 134217728 * q^64 - 23507616 * q^65 + 208064512 * q^67 - 137748480 * q^68 + 127508736 * q^70 + 712016544 * q^71 - 74252216 * q^73 - 39806272 * q^74 + 2105344 * q^76 - 33010560 * q^77 + 645134848 * q^79 + 257949696 * q^80 - 1109028864 * q^82 - 1019036256 * q^83 - 80955540 * q^85 - 662560000 * q^86 - 35979264 * q^88 + 3096385536 * q^89 - 261553216 * q^91 - 664252416 * q^92 + 648941568 * q^94 - 3445933872 * q^95 - 1112014568 * q^97 + 4077576576 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 1910x^{6} + 1303429x^{4} + 373918128x^{2} + 38299272804$$ :

 $$\beta_{1}$$ $$=$$ $$( -27\nu^{6} - 38691\nu^{4} - 15890526\nu^{2} - 1707648804 ) / 2152700$$ (-27*v^6 - 38691*v^4 - 15890526*v^2 - 1707648804) / 2152700 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - 1910\nu^{5} - 1107727\nu^{3} - 187022718\nu + 40314612 ) / 80629224$$ (-v^7 - 1910*v^5 - 1107727*v^3 - 187022718*v + 40314612) / 80629224 $$\beta_{3}$$ $$=$$ $$( -307\nu^{6} - 408581\nu^{4} - 170116216\nu^{2} - 22532569914 ) / 538175$$ (-307*v^6 - 408581*v^4 - 170116216*v^2 - 22532569914) / 538175 $$\beta_{4}$$ $$=$$ $$( -617\nu^{6} - 1009561\nu^{4} - 521633546\nu^{2} - 82066424784 ) / 1076350$$ (-617*v^6 - 1009561*v^4 - 521633546*v^2 - 82066424784) / 1076350 $$\beta_{5}$$ $$=$$ $$( - 81971 \nu^{7} + 20026838 \nu^{6} - 103529368 \nu^{5} + 26653372954 \nu^{4} - 27776247923 \nu^{3} + 11097361234544 \nu^{2} + \cdots + 14\!\cdots\!76 ) / 70214615900$$ (-81971*v^7 + 20026838*v^6 - 103529368*v^5 + 26653372954*v^4 - 27776247923*v^3 + 11097361234544*v^2 - 299191320942*v + 1469889665769876) / 70214615900 $$\beta_{6}$$ $$=$$ $$( 155513 \nu^{7} + 20124689 \nu^{6} + 213758629 \nu^{5} + 32928851137 \nu^{4} + 80304894194 \nu^{3} + 17014121369882 \nu^{2} + \cdots + 26\!\cdots\!28 ) / 70214615900$$ (155513*v^7 + 20124689*v^6 + 213758629*v^5 + 32928851137*v^4 + 80304894194*v^3 + 17014121369882*v^2 + 4315483564476*v + 2676760577179728) / 70214615900 $$\beta_{7}$$ $$=$$ $$( 661878 \nu^{7} + 880659 \nu^{6} + 992063349 \nu^{5} + 1261984347 \nu^{4} + 472757816439 \nu^{3} + 518301286542 \nu^{2} + \cdots + 55698381040068 ) / 140429231800$$ (661878*v^7 + 880659*v^6 + 992063349*v^5 + 1261984347*v^4 + 472757816439*v^3 + 518301286542*v^2 + 70270933519206*v + 55698381040068) / 140429231800
 $$\nu$$ $$=$$ $$( 4\beta_{7} - 18\beta_{6} - 18\beta_{5} - 9\beta_{4} - 9\beta_{3} + 2\beta_1 ) / 972$$ (4*b7 - 18*b6 - 18*b5 - 9*b4 - 9*b3 + 2*b1) / 972 $$\nu^{2}$$ $$=$$ $$( -3\beta_{4} - 6\beta_{3} + 410\beta _1 - 154710 ) / 324$$ (-3*b4 - 6*b3 + 410*b1 - 154710) / 324 $$\nu^{3}$$ $$=$$ $$( 668 \beta_{7} + 6750 \beta_{6} + 12312 \beta_{5} + 3375 \beta_{4} + 6156 \beta_{3} + 300348 \beta_{2} + 334 \beta _1 - 150174 ) / 972$$ (668*b7 + 6750*b6 + 12312*b5 + 3375*b4 + 6156*b3 + 300348*b2 + 334*b1 - 150174) / 972 $$\nu^{4}$$ $$=$$ $$( 1011\beta_{4} + 7584\beta_{3} - 391138\beta _1 + 84340602 ) / 324$$ (1011*b4 + 7584*b3 - 391138*b1 + 84340602) / 324 $$\nu^{5}$$ $$=$$ $$( - 1927508 \beta_{7} - 2919906 \beta_{6} - 8242740 \beta_{5} - 1459953 \beta_{4} - 4121370 \beta_{3} - 478053900 \beta_{2} - 963754 \beta _1 + 239026950 ) / 972$$ (-1927508*b7 - 2919906*b6 - 8242740*b5 - 1459953*b4 - 4121370*b3 - 478053900*b2 - 963754*b1 + 239026950) / 972 $$\nu^{6}$$ $$=$$ $$( 105617\beta_{4} - 2445548\beta_{3} + 97789258\beta _1 - 16766384778 ) / 108$$ (105617*b4 - 2445548*b3 + 97789258*b1 - 16766384778) / 108 $$\nu^{7}$$ $$=$$ $$( 2193487772 \beta_{7} + 1466272134 \beta_{6} + 5471707500 \beta_{5} + 733136067 \beta_{4} + 2735853750 \beta_{3} + 502007754276 \beta_{2} + \cdots - 251003877138 ) / 972$$ (2193487772*b7 + 1466272134*b6 + 5471707500*b5 + 733136067*b4 + 2735853750*b3 + 502007754276*b2 + 1096743886*b1 - 251003877138) / 972

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 26.1057i − 17.7872i − 25.1057i 16.7872i − 26.1057i 17.7872i 25.1057i − 16.7872i
8.00000 13.8564i 0 −128.000 221.703i −992.635 1719.29i 0 −1748.17 + 3027.91i −4096.00 0 −31764.3
55.2 8.00000 13.8564i 0 −128.000 221.703i −957.062 1657.68i 0 24.9899 43.2838i −4096.00 0 −30626.0
55.3 8.00000 13.8564i 0 −128.000 221.703i 318.770 + 552.126i 0 −1890.77 + 3274.91i −4096.00 0 10200.6
55.4 8.00000 13.8564i 0 −128.000 221.703i 646.928 + 1120.51i 0 1365.95 2365.89i −4096.00 0 20701.7
109.1 8.00000 + 13.8564i 0 −128.000 + 221.703i −992.635 + 1719.29i 0 −1748.17 3027.91i −4096.00 0 −31764.3
109.2 8.00000 + 13.8564i 0 −128.000 + 221.703i −957.062 + 1657.68i 0 24.9899 + 43.2838i −4096.00 0 −30626.0
109.3 8.00000 + 13.8564i 0 −128.000 + 221.703i 318.770 552.126i 0 −1890.77 3274.91i −4096.00 0 10200.6
109.4 8.00000 + 13.8564i 0 −128.000 + 221.703i 646.928 1120.51i 0 1365.95 + 2365.89i −4096.00 0 20701.7
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.10.c.t 8
3.b odd 2 1 162.10.c.s 8
9.c even 3 1 162.10.a.f 4
9.c even 3 1 inner 162.10.c.t 8
9.d odd 6 1 162.10.a.g yes 4
9.d odd 6 1 162.10.c.s 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.10.a.f 4 9.c even 3 1
162.10.a.g yes 4 9.d odd 6 1
162.10.c.s 8 3.b odd 2 1
162.10.c.s 8 9.d odd 6 1
162.10.c.t 8 1.a even 1 1 trivial
162.10.c.t 8 9.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 1968 T_{5}^{7} + 6779358 T_{5}^{6} + 2526052608 T_{5}^{5} + 13425956722611 T_{5}^{4} - 355408098312960 T_{5}^{3} + \cdots + 98\!\cdots\!25$$ acting on $$S_{10}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 16 T + 256)^{4}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 1968 T^{7} + \cdots + 98\!\cdots\!25$$
$7$ $$T^{8} + 4496 T^{7} + \cdots + 32\!\cdots\!96$$
$11$ $$T^{8} - 8784 T^{7} + \cdots + 51\!\cdots\!04$$
$13$ $$T^{8} - 162556 T^{7} + \cdots + 16\!\cdots\!89$$
$17$ $$(T^{4} - 538080 T^{3} + \cdots - 66\!\cdots\!59)^{2}$$
$19$ $$(T^{4} + 8224 T^{3} + \cdots + 56\!\cdots\!24)^{2}$$
$23$ $$T^{8} + 2594736 T^{7} + \cdots + 16\!\cdots\!76$$
$29$ $$T^{8} + 3232656 T^{7} + \cdots + 10\!\cdots\!09$$
$31$ $$T^{8} + 3482576 T^{7} + \cdots + 47\!\cdots\!24$$
$37$ $$(T^{4} + 2487892 T^{3} + \cdots + 18\!\cdots\!21)^{2}$$
$41$ $$T^{8} + 34657152 T^{7} + \cdots + 78\!\cdots\!04$$
$43$ $$T^{8} - 41410000 T^{7} + \cdots + 90\!\cdots\!44$$
$47$ $$T^{8} + 40558848 T^{7} + \cdots + 22\!\cdots\!16$$
$53$ $$(T^{4} - 8421024 T^{3} + \cdots + 13\!\cdots\!84)^{2}$$
$59$ $$T^{8} + 26843328 T^{7} + \cdots + 10\!\cdots\!00$$
$61$ $$T^{8} - 111504484 T^{7} + \cdots + 17\!\cdots\!09$$
$67$ $$T^{8} - 208064512 T^{7} + \cdots + 35\!\cdots\!96$$
$71$ $$(T^{4} - 356008272 T^{3} + \cdots - 13\!\cdots\!48)^{2}$$
$73$ $$(T^{4} + 37126108 T^{3} + \cdots - 25\!\cdots\!99)^{2}$$
$79$ $$T^{8} - 645134848 T^{7} + \cdots + 12\!\cdots\!00$$
$83$ $$T^{8} + 1019036256 T^{7} + \cdots + 12\!\cdots\!44$$
$89$ $$(T^{4} - 1548192768 T^{3} + \cdots + 13\!\cdots\!01)^{2}$$
$97$ $$T^{8} + 1112014568 T^{7} + \cdots + 17\!\cdots\!24$$
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