# Properties

 Label 147.6.e.l Level $147$ Weight $6$ Character orbit 147.e Analytic conductor $23.576$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-83})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 20x^{2} - 21x + 441$$ x^4 - x^3 - 20*x^2 - 21*x + 441 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1 + 1) q^{2} + 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 3 \beta_{2} + 31 \beta_1) q^{4} + (7 \beta_{3} - 13 \beta_1 - 13) q^{5} + (9 \beta_{2} - 9) q^{6} + (5 \beta_{2} - 185) q^{8} + ( - 81 \beta_1 - 81) q^{9}+O(q^{10})$$ q + (-b3 + b1 + 1) * q^2 + 9*b1 * q^3 + (-3*b3 + 3*b2 + 31*b1) * q^4 + (7*b3 - 13*b1 - 13) * q^5 + (9*b2 - 9) * q^6 + (5*b2 - 185) * q^8 + (-81*b1 - 81) * q^9 $$q + ( - \beta_{3} + \beta_1 + 1) q^{2} + 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 3 \beta_{2} + 31 \beta_1) q^{4} + (7 \beta_{3} - 13 \beta_1 - 13) q^{5} + (9 \beta_{2} - 9) q^{6} + (5 \beta_{2} - 185) q^{8} + ( - 81 \beta_1 - 81) q^{9} + (27 \beta_{3} - 27 \beta_{2} - 447 \beta_1) q^{10} + ( - \beta_{3} + \beta_{2} - 569 \beta_1) q^{11} + (27 \beta_{3} - 279 \beta_1 - 279) q^{12} + (9 \beta_{2} - 458) q^{13} + ( - 63 \beta_{2} + 117) q^{15} + (99 \beta_{3} + 497 \beta_1 + 497) q^{16} + (148 \beta_{3} - 148 \beta_{2} + 236 \beta_1) q^{17} + (81 \beta_{3} - 81 \beta_{2} - 81 \beta_1) q^{18} + ( - 27 \beta_{3} + 1142 \beta_1 + 1142) q^{19} + ( - 277 \beta_{2} + 1705) q^{20} + ( - 567 \beta_{2} + 507) q^{22} + (308 \beta_{3} - 644 \beta_1 - 644) q^{23} + (45 \beta_{3} - 45 \beta_{2} - 1665 \beta_1) q^{24} + ( - 231 \beta_{3} + 231 \beta_{2} + 82 \beta_1) q^{25} + (476 \beta_{3} - 1016 \beta_1 - 1016) q^{26} + 729 q^{27} + ( - 45 \beta_{2} - 1131) q^{29} + ( - 243 \beta_{3} + 4023 \beta_1 + 4023) q^{30} + (768 \beta_{3} - 768 \beta_{2} - 1763 \beta_1) q^{31} + ( - 459 \beta_{3} + 459 \beta_{2} + 279 \beta_1) q^{32} + (9 \beta_{3} + 5121 \beta_1 + 5121) q^{33} + ( - 60 \beta_{2} + 8940) q^{34} + ( - 243 \beta_{2} + 2511) q^{36} + (855 \beta_{3} + 9982 \beta_1 + 9982) q^{37} + ( - 1196 \beta_{3} + 1196 \beta_{2} + 2816 \beta_1) q^{38} + (81 \beta_{3} - 81 \beta_{2} - 4122 \beta_1) q^{39} + ( - 1395 \beta_{3} + 4575 \beta_1 + 4575) q^{40} + (846 \beta_{2} + 6852) q^{41} + (2043 \beta_{2} - 364) q^{43} + ( - 1673 \beta_{3} + 17453 \beta_1 + 17453) q^{44} + ( - 567 \beta_{3} + 567 \beta_{2} + 1053 \beta_1) q^{45} + (1260 \beta_{3} - 1260 \beta_{2} - 19740 \beta_1) q^{46} + (604 \beta_{3} - 11278 \beta_1 - 11278) q^{47} + ( - 891 \beta_{2} - 4473) q^{48} + (544 \beta_{2} - 14404) q^{50} + ( - 1332 \beta_{3} - 2124 \beta_1 - 2124) q^{51} + (1680 \beta_{3} - 1680 \beta_{2} - 15872 \beta_1) q^{52} + (1751 \beta_{3} - 1751 \beta_{2} - 14951 \beta_1) q^{53} + ( - 729 \beta_{3} + 729 \beta_1 + 729) q^{54} + (3963 \beta_{2} - 6963) q^{55} + (243 \beta_{2} - 10278) q^{57} + (1041 \beta_{3} + 1659 \beta_1 + 1659) q^{58} + ( - 3917 \beta_{3} + 3917 \beta_{2} - 22507 \beta_1) q^{59} + ( - 2493 \beta_{3} + 2493 \beta_{2} + 15345 \beta_1) q^{60} + (2544 \beta_{3} + 22298 \beta_1 + 22298) q^{61} + ( - 3299 \beta_{2} + 49379) q^{62} + (4365 \beta_{2} - 12833) q^{64} + ( - 3386 \beta_{3} + 9860 \beta_1 + 9860) q^{65} + ( - 5103 \beta_{3} + 5103 \beta_{2} + 4563 \beta_1) q^{66} + (4461 \beta_{3} - 4461 \beta_{2} + 17612 \beta_1) q^{67} + ( - 4324 \beta_{3} + 20212 \beta_1 + 20212) q^{68} + ( - 2772 \beta_{2} + 5796) q^{69} + ( - 1404 \beta_{2} + 50346) q^{71} + ( - 405 \beta_{3} + 14985 \beta_1 + 14985) q^{72} + (5247 \beta_{3} - 5247 \beta_{2} + 16912 \beta_1) q^{73} + ( - 8272 \beta_{3} + 8272 \beta_{2} - 43028 \beta_1) q^{74} + (2079 \beta_{3} - 738 \beta_1 - 738) q^{75} + (4344 \beta_{2} - 40424) q^{76} + ( - 4284 \beta_{2} + 9144) q^{78} + (6834 \beta_{3} + 12649 \beta_1 + 12649) q^{79} + (1499 \beta_{3} - 1499 \beta_{2} + 36505 \beta_1) q^{80} + 6561 \beta_1 q^{81} + ( - 5160 \beta_{3} - 45600 \beta_1 - 45600) q^{82} + ( - 1899 \beta_{2} - 31539) q^{83} + (1308 \beta_{2} - 61164) q^{85} + (4450 \beta_{3} - 127030 \beta_1 - 127030) q^{86} + ( - 405 \beta_{3} + 405 \beta_{2} - 10179 \beta_1) q^{87} + ( - 2655 \beta_{3} + 2655 \beta_{2} + 104955 \beta_1) q^{88} + (130 \beta_{3} + 14726 \beta_1 + 14726) q^{89} + (2187 \beta_{2} - 36207) q^{90} + ( - 12404 \beta_{2} + 77252) q^{92} + ( - 6912 \beta_{3} + 15867 \beta_1 + 15867) q^{93} + (12486 \beta_{3} - 12486 \beta_{2} - 48726 \beta_1) q^{94} + (8534 \beta_{3} - 8534 \beta_{2} - 26564 \beta_1) q^{95} + (4131 \beta_{3} - 2511 \beta_1 - 2511) q^{96} + ( - 1017 \beta_{2} + 4387) q^{97} + ( - 81 \beta_{2} - 46089) q^{99}+O(q^{100})$$ q + (-b3 + b1 + 1) * q^2 + 9*b1 * q^3 + (-3*b3 + 3*b2 + 31*b1) * q^4 + (7*b3 - 13*b1 - 13) * q^5 + (9*b2 - 9) * q^6 + (5*b2 - 185) * q^8 + (-81*b1 - 81) * q^9 + (27*b3 - 27*b2 - 447*b1) * q^10 + (-b3 + b2 - 569*b1) * q^11 + (27*b3 - 279*b1 - 279) * q^12 + (9*b2 - 458) * q^13 + (-63*b2 + 117) * q^15 + (99*b3 + 497*b1 + 497) * q^16 + (148*b3 - 148*b2 + 236*b1) * q^17 + (81*b3 - 81*b2 - 81*b1) * q^18 + (-27*b3 + 1142*b1 + 1142) * q^19 + (-277*b2 + 1705) * q^20 + (-567*b2 + 507) * q^22 + (308*b3 - 644*b1 - 644) * q^23 + (45*b3 - 45*b2 - 1665*b1) * q^24 + (-231*b3 + 231*b2 + 82*b1) * q^25 + (476*b3 - 1016*b1 - 1016) * q^26 + 729 * q^27 + (-45*b2 - 1131) * q^29 + (-243*b3 + 4023*b1 + 4023) * q^30 + (768*b3 - 768*b2 - 1763*b1) * q^31 + (-459*b3 + 459*b2 + 279*b1) * q^32 + (9*b3 + 5121*b1 + 5121) * q^33 + (-60*b2 + 8940) * q^34 + (-243*b2 + 2511) * q^36 + (855*b3 + 9982*b1 + 9982) * q^37 + (-1196*b3 + 1196*b2 + 2816*b1) * q^38 + (81*b3 - 81*b2 - 4122*b1) * q^39 + (-1395*b3 + 4575*b1 + 4575) * q^40 + (846*b2 + 6852) * q^41 + (2043*b2 - 364) * q^43 + (-1673*b3 + 17453*b1 + 17453) * q^44 + (-567*b3 + 567*b2 + 1053*b1) * q^45 + (1260*b3 - 1260*b2 - 19740*b1) * q^46 + (604*b3 - 11278*b1 - 11278) * q^47 + (-891*b2 - 4473) * q^48 + (544*b2 - 14404) * q^50 + (-1332*b3 - 2124*b1 - 2124) * q^51 + (1680*b3 - 1680*b2 - 15872*b1) * q^52 + (1751*b3 - 1751*b2 - 14951*b1) * q^53 + (-729*b3 + 729*b1 + 729) * q^54 + (3963*b2 - 6963) * q^55 + (243*b2 - 10278) * q^57 + (1041*b3 + 1659*b1 + 1659) * q^58 + (-3917*b3 + 3917*b2 - 22507*b1) * q^59 + (-2493*b3 + 2493*b2 + 15345*b1) * q^60 + (2544*b3 + 22298*b1 + 22298) * q^61 + (-3299*b2 + 49379) * q^62 + (4365*b2 - 12833) * q^64 + (-3386*b3 + 9860*b1 + 9860) * q^65 + (-5103*b3 + 5103*b2 + 4563*b1) * q^66 + (4461*b3 - 4461*b2 + 17612*b1) * q^67 + (-4324*b3 + 20212*b1 + 20212) * q^68 + (-2772*b2 + 5796) * q^69 + (-1404*b2 + 50346) * q^71 + (-405*b3 + 14985*b1 + 14985) * q^72 + (5247*b3 - 5247*b2 + 16912*b1) * q^73 + (-8272*b3 + 8272*b2 - 43028*b1) * q^74 + (2079*b3 - 738*b1 - 738) * q^75 + (4344*b2 - 40424) * q^76 + (-4284*b2 + 9144) * q^78 + (6834*b3 + 12649*b1 + 12649) * q^79 + (1499*b3 - 1499*b2 + 36505*b1) * q^80 + 6561*b1 * q^81 + (-5160*b3 - 45600*b1 - 45600) * q^82 + (-1899*b2 - 31539) * q^83 + (1308*b2 - 61164) * q^85 + (4450*b3 - 127030*b1 - 127030) * q^86 + (-405*b3 + 405*b2 - 10179*b1) * q^87 + (-2655*b3 + 2655*b2 + 104955*b1) * q^88 + (130*b3 + 14726*b1 + 14726) * q^89 + (2187*b2 - 36207) * q^90 + (-12404*b2 + 77252) * q^92 + (-6912*b3 + 15867*b1 + 15867) * q^93 + (12486*b3 - 12486*b2 - 48726*b1) * q^94 + (8534*b3 - 8534*b2 - 26564*b1) * q^95 + (4131*b3 - 2511*b1 - 2511) * q^96 + (-1017*b2 + 4387) * q^97 + (-81*b2 - 46089) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{2} - 18 q^{3} - 65 q^{4} - 33 q^{5} - 54 q^{6} - 750 q^{8} - 162 q^{9}+O(q^{10})$$ 4 * q + 3 * q^2 - 18 * q^3 - 65 * q^4 - 33 * q^5 - 54 * q^6 - 750 * q^8 - 162 * q^9 $$4 q + 3 q^{2} - 18 q^{3} - 65 q^{4} - 33 q^{5} - 54 q^{6} - 750 q^{8} - 162 q^{9} + 921 q^{10} + 1137 q^{11} - 585 q^{12} - 1850 q^{13} + 594 q^{15} + 895 q^{16} - 324 q^{17} + 243 q^{18} + 2311 q^{19} + 7374 q^{20} + 3162 q^{22} - 1596 q^{23} + 3375 q^{24} - 395 q^{25} - 2508 q^{26} + 2916 q^{27} - 4434 q^{29} + 8289 q^{30} + 4294 q^{31} - 1017 q^{32} + 10233 q^{33} + 35880 q^{34} + 10530 q^{36} + 19109 q^{37} - 6828 q^{38} + 8325 q^{39} + 10545 q^{40} + 25716 q^{41} - 5542 q^{43} + 36579 q^{44} - 2673 q^{45} + 40740 q^{46} - 23160 q^{47} - 16110 q^{48} - 58704 q^{50} - 2916 q^{51} + 33424 q^{52} + 31653 q^{53} + 2187 q^{54} - 35778 q^{55} - 41598 q^{57} + 2277 q^{58} + 41097 q^{59} - 33183 q^{60} + 42052 q^{61} + 204114 q^{62} - 60062 q^{64} + 23106 q^{65} - 14229 q^{66} - 30763 q^{67} + 44748 q^{68} + 28728 q^{69} + 204192 q^{71} + 30375 q^{72} - 28577 q^{73} + 77784 q^{74} - 3555 q^{75} - 170384 q^{76} + 45144 q^{78} + 18464 q^{79} - 71511 q^{80} - 13122 q^{81} - 86040 q^{82} - 122358 q^{83} - 247272 q^{85} - 258510 q^{86} + 19953 q^{87} - 212565 q^{88} + 29322 q^{89} - 149202 q^{90} + 333816 q^{92} + 38646 q^{93} + 109938 q^{94} + 61662 q^{95} - 9153 q^{96} + 19582 q^{97} - 184194 q^{99}+O(q^{100})$$ 4 * q + 3 * q^2 - 18 * q^3 - 65 * q^4 - 33 * q^5 - 54 * q^6 - 750 * q^8 - 162 * q^9 + 921 * q^10 + 1137 * q^11 - 585 * q^12 - 1850 * q^13 + 594 * q^15 + 895 * q^16 - 324 * q^17 + 243 * q^18 + 2311 * q^19 + 7374 * q^20 + 3162 * q^22 - 1596 * q^23 + 3375 * q^24 - 395 * q^25 - 2508 * q^26 + 2916 * q^27 - 4434 * q^29 + 8289 * q^30 + 4294 * q^31 - 1017 * q^32 + 10233 * q^33 + 35880 * q^34 + 10530 * q^36 + 19109 * q^37 - 6828 * q^38 + 8325 * q^39 + 10545 * q^40 + 25716 * q^41 - 5542 * q^43 + 36579 * q^44 - 2673 * q^45 + 40740 * q^46 - 23160 * q^47 - 16110 * q^48 - 58704 * q^50 - 2916 * q^51 + 33424 * q^52 + 31653 * q^53 + 2187 * q^54 - 35778 * q^55 - 41598 * q^57 + 2277 * q^58 + 41097 * q^59 - 33183 * q^60 + 42052 * q^61 + 204114 * q^62 - 60062 * q^64 + 23106 * q^65 - 14229 * q^66 - 30763 * q^67 + 44748 * q^68 + 28728 * q^69 + 204192 * q^71 + 30375 * q^72 - 28577 * q^73 + 77784 * q^74 - 3555 * q^75 - 170384 * q^76 + 45144 * q^78 + 18464 * q^79 - 71511 * q^80 - 13122 * q^81 - 86040 * q^82 - 122358 * q^83 - 247272 * q^85 - 258510 * q^86 + 19953 * q^87 - 212565 * q^88 + 29322 * q^89 - 149202 * q^90 + 333816 * q^92 + 38646 * q^93 + 109938 * q^94 + 61662 * q^95 - 9153 * q^96 + 19582 * q^97 - 184194 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 20x^{2} - 21x + 441$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420$$ (v^3 + 20*v^2 - 20*v - 441) / 420 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 41\nu ) / 21$$ (-v^3 + v^2 + 41*v) / 21 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 20\nu - 41 ) / 20$$ (v^3 + 20*v - 41) / 20
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3$$ (b3 + b2 - b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3$$ (-b3 + 2*b2 + 61*b1 + 62) / 3 $$\nu^{3}$$ $$=$$ $$( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3$$ (40*b3 - 20*b2 + 20*b1 + 103) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{1}$$

## 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
 4.19493 + 1.84460i −3.69493 − 2.71062i 4.19493 − 1.84460i −3.69493 + 2.71062i
−3.19493 5.53379i −4.50000 + 7.79423i −4.41520 + 7.64735i 19.3645 + 33.5404i 57.5088 0 −148.051 −40.5000 70.1481i 123.737 214.318i
67.2 4.69493 + 8.13186i −4.50000 + 7.79423i −28.0848 + 48.6443i −35.8645 62.1192i −84.5088 0 −226.949 −40.5000 70.1481i 336.763 583.291i
79.1 −3.19493 + 5.53379i −4.50000 7.79423i −4.41520 7.64735i 19.3645 33.5404i 57.5088 0 −148.051 −40.5000 + 70.1481i 123.737 + 214.318i
79.2 4.69493 8.13186i −4.50000 7.79423i −28.0848 48.6443i −35.8645 + 62.1192i −84.5088 0 −226.949 −40.5000 + 70.1481i 336.763 + 583.291i
 $$n$$: 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.l 4
7.b odd 2 1 21.6.e.b 4
7.c even 3 1 147.6.a.k 2
7.c even 3 1 inner 147.6.e.l 4
7.d odd 6 1 21.6.e.b 4
7.d odd 6 1 147.6.a.i 2
21.c even 2 1 63.6.e.c 4
21.g even 6 1 63.6.e.c 4
21.g even 6 1 441.6.a.t 2
21.h odd 6 1 441.6.a.s 2
28.d even 2 1 336.6.q.e 4
28.f even 6 1 336.6.q.e 4

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

## Hecke kernels

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

 $$T_{2}^{4} - 3T_{2}^{3} + 69T_{2}^{2} + 180T_{2} + 3600$$ T2^4 - 3*T2^3 + 69*T2^2 + 180*T2 + 3600 $$T_{5}^{4} + 33T_{5}^{3} + 3867T_{5}^{2} - 91674T_{5} + 7717284$$ T5^4 + 33*T5^3 + 3867*T5^2 - 91674*T5 + 7717284

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 3 T^{3} + 69 T^{2} + \cdots + 3600$$
$3$ $$(T^{2} + 9 T + 81)^{2}$$
$5$ $$T^{4} + 33 T^{3} + 3867 T^{2} + \cdots + 7717284$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 1137 T^{3} + \cdots + 104412996900$$
$13$ $$(T^{2} + 925 T + 208864)^{2}$$
$17$ $$T^{4} + 324 T^{3} + \cdots + 1788317798400$$
$19$ $$T^{4} - 2311 T^{3} + \cdots + 1663584040000$$
$23$ $$T^{4} + 1596 T^{3} + \cdots + 27756881510400$$
$29$ $$(T^{2} + 2217 T + 1102716)^{2}$$
$31$ $$T^{4} - 4294 T^{3} + \cdots + 10\!\cdots\!25$$
$37$ $$T^{4} - 19109 T^{3} + \cdots + 20\!\cdots\!96$$
$41$ $$(T^{2} - 12858 T - 3221280)^{2}$$
$43$ $$(T^{2} + 2771 T - 257902490)^{2}$$
$47$ $$T^{4} + 23160 T^{3} + \cdots + 12\!\cdots\!16$$
$53$ $$T^{4} - 31653 T^{3} + \cdots + 35\!\cdots\!00$$
$59$ $$T^{4} - 41097 T^{3} + \cdots + 28\!\cdots\!44$$
$61$ $$T^{4} - 42052 T^{3} + \cdots + 15\!\cdots\!00$$
$67$ $$T^{4} + 30763 T^{3} + \cdots + 10\!\cdots\!00$$
$71$ $$(T^{2} - 102096 T + 2483190108)^{2}$$
$73$ $$T^{4} + 28577 T^{3} + \cdots + 22\!\cdots\!84$$
$79$ $$T^{4} - 18464 T^{3} + \cdots + 79\!\cdots\!69$$
$83$ $$(T^{2} + 61179 T + 711231498)^{2}$$
$89$ $$T^{4} - 29322 T^{3} + \cdots + 45\!\cdots\!16$$
$97$ $$(T^{2} - 9791 T - 40418570)^{2}$$