# Properties

 Label 725.2.a.h Level $725$ Weight $2$ Character orbit 725.a Self dual yes Analytic conductor $5.789$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.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: $$5.78915414654$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.240881.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 5x - 7$$ x^5 - 2*x^4 - 5*x^3 + 9*x^2 + 5*x - 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{6} + (\beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{8} + ( - 2 \beta_{4} - 2 \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b4 - 1) * q^3 + (b2 + 1) * q^4 + (-b4 - b3 + 2*b1 - 1) * q^6 + (b3 - b1 - 1) * q^7 + (-b3 - b2 + b1 - 1) * q^8 + (-2*b4 - 2*b2 + b1 + 1) * q^9 $$q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{6} + (\beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{8} + ( - 2 \beta_{4} - 2 \beta_{2} + \beta_1 + 1) q^{9} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{11} + (\beta_{3} - \beta_1 - 1) q^{12} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 2) q^{13} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 + 1) q^{14} + (\beta_{4} + \beta_{3} - 2) q^{16} + ( - \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{17} + (2 \beta_{4} + 4 \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{18} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{19} + ( - 2 \beta_{4} - 3 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{21} + ( - 3 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 3) q^{22} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{23} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{24} + (\beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{26} + (3 \beta_{4} - \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 3) q^{27} + (\beta_{4} - \beta_{2} - 2) q^{28} + q^{29} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{31} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{32} + ( - 3 \beta_{4} - 4 \beta_{3} + 4 \beta_1) q^{33} + (3 \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{34} + ( - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 4) q^{36} + (2 \beta_{2} - \beta_1 - 2) q^{37} + (2 \beta_{4} - \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{38} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 1) q^{39} + (\beta_{2} - 2 \beta_1 + 2) q^{41} + (5 \beta_{4} + \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 2) q^{42} + (\beta_{4} + \beta_{3} + 3 \beta_{2} - 5) q^{43} + (3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 2) q^{44} + (3 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{46} + (\beta_{4} - 2 \beta_{2} + 2 \beta_1 - 4) q^{47} + ( - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{48} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{49} + (5 \beta_{3} - 5 \beta_1 + 2) q^{51} + ( - \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 3) q^{52} + (\beta_{4} + 3 \beta_{3} - 2 \beta_1 + 2) q^{53} + ( - 2 \beta_{4} - 7 \beta_{3} + 5 \beta_1 + 1) q^{54} + (\beta_{4} + 3 \beta_{2} - \beta_1 - 2) q^{56} + (5 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 1) q^{57} - \beta_1 q^{58} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{59} + (\beta_{4} - 4 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{61} + (2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{62} + (3 \beta_{4} + 7 \beta_{3} + \beta_{2} - 9 \beta_1 + 4) q^{63} + ( - \beta_{4} - 4 \beta_{2} - 2) q^{64} + (7 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 7 \beta_1 - 1) q^{66} + (2 \beta_{4} - 3 \beta_1 - 3) q^{67} + ( - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 1) q^{68} + (2 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 8 \beta_1 - 2) q^{69} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{71} + (\beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{72} + ( - \beta_{4} - 3 \beta_{2} - 4) q^{73} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{74} + ( - \beta_{4} - \beta_{3} - 2 \beta_1 + 4) q^{76} + ( - 4 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 4) q^{77} + ( - 3 \beta_{4} - 4 \beta_{3} + 2 \beta_1 + 6) q^{78} + (2 \beta_{4} - \beta_{3} - \beta_{2} - 4 \beta_1) q^{79} + ( - 6 \beta_{4} + 4 \beta_{3} - \beta_{2} - \beta_1 + 8) q^{81} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{82} + ( - 5 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 4) q^{83} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 5) q^{84} + ( - 2 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} + 7 \beta_1 - 6) q^{86} + (\beta_{4} - 1) q^{87} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 6) q^{88} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - 4) q^{89} + (3 \beta_{4} + \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 3) q^{91} + ( - \beta_{2} + 3 \beta_1 + 4) q^{92} + (3 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{93} + ( - \beta_{4} + \beta_{3} + 5 \beta_1 - 5) q^{94} + (3 \beta_{4} + 5 \beta_{2} - 5 \beta_1 - 4) q^{96} + (\beta_{4} + 2 \beta_{3} - 6 \beta_{2} - 6) q^{97} + (3 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - \beta_1) q^{98} + (4 \beta_{4} + 6 \beta_{3} - \beta_{2} - 12 \beta_1 - 1) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b4 - 1) * q^3 + (b2 + 1) * q^4 + (-b4 - b3 + 2*b1 - 1) * q^6 + (b3 - b1 - 1) * q^7 + (-b3 - b2 + b1 - 1) * q^8 + (-2*b4 - 2*b2 + b1 + 1) * q^9 + (b4 + 2*b3 + b2 - b1) * q^11 + (b3 - b1 - 1) * q^12 + (-b4 - b2 + 2*b1 - 2) * q^13 + (-b4 - b2 + 2*b1 + 1) * q^14 + (b4 + b3 - 2) * q^16 + (-b4 - 2*b3 + b1 - 2) * q^17 + (2*b4 + 4*b3 + b2 - 3*b1 + 1) * q^18 + (-2*b3 + b2 + 1) * q^19 + (-2*b4 - 3*b3 + b2 + 3*b1 - 1) * q^21 + (-3*b4 - 2*b3 - 4*b2 + 3*b1 - 3) * q^22 + (-2*b4 - b3 + b2 + 2*b1 - 1) * q^23 + (b4 + 2*b3 - b2 - 2*b1 + 3) * q^24 + (b4 + 2*b3 - b2 + b1 - 4) * q^26 + (3*b4 - b3 + 4*b2 - 2*b1 - 3) * q^27 + (b4 - b2 - 2) * q^28 + q^29 + (-b4 - b3 - 2*b2 + b1 - 1) * q^31 + (-2*b4 + b3 + 2*b1 - 1) * q^32 + (-3*b4 - 4*b3 + 4*b1) * q^33 + (3*b4 + b3 + 3*b2 - b1 + 2) * q^34 + (-2*b4 - 3*b3 - 2*b2 + 3*b1 - 4) * q^36 + (2*b2 - b1 - 2) * q^37 + (2*b4 - b3 + 3*b2 - 3*b1 + 3) * q^38 + (2*b4 + b3 + 2*b2 - 4*b1 + 1) * q^39 + (b2 - 2*b1 + 2) * q^41 + (5*b4 + b3 + 2*b2 - 4*b1 - 2) * q^42 + (b4 + b3 + 3*b2 - 5) * q^43 + (3*b4 + 3*b3 + 3*b2 + 2) * q^44 + (3*b4 + b3 - b2 - 2*b1 - 3) * q^46 + (b4 - 2*b2 + 2*b1 - 4) * q^47 + (-3*b4 - 2*b3 - b2 + 2*b1 + 4) * q^48 + (-2*b4 - b3 - 2*b2 + 2*b1 - 2) * q^49 + (5*b3 - 5*b1 + 2) * q^51 + (-b4 - 2*b2 + 3*b1 - 3) * q^52 + (b4 + 3*b3 - 2*b1 + 2) * q^53 + (-2*b4 - 7*b3 + 5*b1 + 1) * q^54 + (b4 + 3*b2 - b1 - 2) * q^56 + (5*b3 - 2*b2 - 3*b1 + 1) * q^57 - b1 * q^58 + (-b3 - 2*b2 + 4*b1 - 4) * q^59 + (b4 - 4*b3 - 3*b2 + b1 - 1) * q^61 + (2*b4 + 3*b3 + 3*b2 - b1 + 2) * q^62 + (3*b4 + 7*b3 + b2 - 9*b1 + 4) * q^63 + (-b4 - 4*b2 - 2) * q^64 + (7*b4 + 3*b3 + 4*b2 - 7*b1 - 1) * q^66 + (2*b4 - 3*b1 - 3) * q^67 + (-2*b4 - 2*b3 - 4*b2 - 1) * q^68 + (2*b4 + 5*b3 + 3*b2 - 8*b1 - 2) * q^69 + (-3*b4 - 2*b3 + b2 - 1) * q^71 + (b4 - 4*b3 + 3*b2 + 5*b1 - 1) * q^72 + (-b4 - 3*b2 - 4) * q^73 + (-2*b3 - b2 + 2*b1 + 1) * q^74 + (-b4 - b3 - 2*b1 + 4) * q^76 + (-4*b4 - 3*b3 - 3*b2 + 3*b1 + 4) * q^77 + (-3*b4 - 4*b3 + 2*b1 + 6) * q^78 + (2*b4 - b3 - b2 - 4*b1) * q^79 + (-6*b4 + 4*b3 - b2 - b1 + 8) * q^81 + (-b3 + b2 - 2*b1 + 5) * q^82 + (-5*b4 - 2*b3 - 2*b2 + 3*b1 - 4) * q^83 + (-2*b4 - b3 - 2*b2 + 2*b1 + 5) * q^84 + (-2*b4 - 4*b3 - 5*b2 + 7*b1 - 6) * q^86 + (b4 - 1) * q^87 + (-2*b3 - b2 - 2*b1 - 6) * q^88 + (-2*b4 - b3 - b2 - 4) * q^89 + (3*b4 + b3 + 2*b2 - 5*b1 + 3) * q^91 + (-b2 + 3*b1 + 4) * q^92 + (3*b4 + b3 + b2 - 2*b1) * q^93 + (-b4 + b3 + 5*b1 - 5) * q^94 + (3*b4 + 5*b2 - 5*b1 - 4) * q^96 + (b4 + 2*b3 - 6*b2 - 6) * q^97 + (3*b4 + 4*b3 + 2*b2 - b1) * q^98 + (4*b4 + 6*b3 - b2 - 12*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{2} - 6 q^{3} + 4 q^{4} - q^{6} - 6 q^{7} - 3 q^{8} + 11 q^{9}+O(q^{10})$$ 5 * q - 2 * q^2 - 6 * q^3 + 4 * q^4 - q^6 - 6 * q^7 - 3 * q^8 + 11 * q^9 $$5 q - 2 q^{2} - 6 q^{3} + 4 q^{4} - q^{6} - 6 q^{7} - 3 q^{8} + 11 q^{9} - 2 q^{11} - 6 q^{12} - 4 q^{13} + 11 q^{14} - 10 q^{16} - 9 q^{17} + 2 q^{19} - q^{21} - 4 q^{22} - q^{23} + 13 q^{24} - 16 q^{26} - 27 q^{27} - 10 q^{28} + 5 q^{29} - q^{31} + 2 q^{32} + 7 q^{33} + 3 q^{34} - 13 q^{36} - 14 q^{37} + 3 q^{38} - 6 q^{39} + 5 q^{41} - 24 q^{42} - 28 q^{43} + 7 q^{44} - 20 q^{46} - 15 q^{47} + 26 q^{48} - 3 q^{49} + 5 q^{51} - 6 q^{52} + 8 q^{53} + 10 q^{54} - 16 q^{56} + 6 q^{57} - 2 q^{58} - 11 q^{59} - 5 q^{61} + 6 q^{62} + 5 q^{63} - 5 q^{64} - 27 q^{66} - 23 q^{67} - q^{68} - 26 q^{69} - 5 q^{71} - 3 q^{72} - 16 q^{73} + 8 q^{74} + 16 q^{76} + 30 q^{77} + 33 q^{78} - 10 q^{79} + 49 q^{81} + 19 q^{82} - 9 q^{83} + 32 q^{84} - 13 q^{86} - 6 q^{87} - 35 q^{88} - 18 q^{89} + q^{91} + 27 q^{92} - 7 q^{93} - 13 q^{94} - 38 q^{96} - 23 q^{97} - 3 q^{98} - 26 q^{99}+O(q^{100})$$ 5 * q - 2 * q^2 - 6 * q^3 + 4 * q^4 - q^6 - 6 * q^7 - 3 * q^8 + 11 * q^9 - 2 * q^11 - 6 * q^12 - 4 * q^13 + 11 * q^14 - 10 * q^16 - 9 * q^17 + 2 * q^19 - q^21 - 4 * q^22 - q^23 + 13 * q^24 - 16 * q^26 - 27 * q^27 - 10 * q^28 + 5 * q^29 - q^31 + 2 * q^32 + 7 * q^33 + 3 * q^34 - 13 * q^36 - 14 * q^37 + 3 * q^38 - 6 * q^39 + 5 * q^41 - 24 * q^42 - 28 * q^43 + 7 * q^44 - 20 * q^46 - 15 * q^47 + 26 * q^48 - 3 * q^49 + 5 * q^51 - 6 * q^52 + 8 * q^53 + 10 * q^54 - 16 * q^56 + 6 * q^57 - 2 * q^58 - 11 * q^59 - 5 * q^61 + 6 * q^62 + 5 * q^63 - 5 * q^64 - 27 * q^66 - 23 * q^67 - q^68 - 26 * q^69 - 5 * q^71 - 3 * q^72 - 16 * q^73 + 8 * q^74 + 16 * q^76 + 30 * q^77 + 33 * q^78 - 10 * q^79 + 49 * q^81 + 19 * q^82 - 9 * q^83 + 32 * q^84 - 13 * q^86 - 6 * q^87 - 35 * q^88 - 18 * q^89 + q^91 + 27 * q^92 - 7 * q^93 - 13 * q^94 - 38 * q^96 - 23 * q^97 - 3 * q^98 - 26 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 5x - 7$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 2$$ v^3 - v^2 - 3*v + 2 $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 4$$ v^4 - v^3 - 5*v^2 + 3*v + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 3\beta _1 + 1$$ b3 + b2 + 3*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 6\beta_{2} + 12$$ b4 + b3 + 6*b2 + 12

## 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
 2.33090 1.78154 0.838718 −1.06634 −1.88481
−2.33090 −0.318289 3.43308 0 0.741899 −1.09271 −3.34037 −2.89869 0
1.2 −1.78154 −3.10566 1.17387 0 5.53284 −3.64565 1.47177 6.64510 0
1.3 −0.838718 1.90376 −1.29655 0 −1.59672 −2.46832 2.76488 0.624302 0
1.4 1.06634 −3.37897 −0.862915 0 −3.60314 2.91576 −3.05285 8.41742 0
1.5 1.88481 −1.10085 1.55251 0 −2.07489 −1.70908 −0.843434 −1.78814 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$29$$ $$-1$$

## 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 725.2.a.h 5
3.b odd 2 1 6525.2.a.bq 5
5.b even 2 1 725.2.a.k yes 5
5.c odd 4 2 725.2.b.f 10
15.d odd 2 1 6525.2.a.bm 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.a.h 5 1.a even 1 1 trivial
725.2.a.k yes 5 5.b even 2 1
725.2.b.f 10 5.c odd 4 2
6525.2.a.bm 5 15.d odd 2 1
6525.2.a.bq 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(725))$$:

 $$T_{2}^{5} + 2T_{2}^{4} - 5T_{2}^{3} - 9T_{2}^{2} + 5T_{2} + 7$$ T2^5 + 2*T2^4 - 5*T2^3 - 9*T2^2 + 5*T2 + 7 $$T_{3}^{5} + 6T_{3}^{4} + 5T_{3}^{3} - 21T_{3}^{2} - 29T_{3} - 7$$ T3^5 + 6*T3^4 + 5*T3^3 - 21*T3^2 - 29*T3 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} + 2 T^{4} - 5 T^{3} - 9 T^{2} + \cdots + 7$$
$3$ $$T^{5} + 6 T^{4} + 5 T^{3} - 21 T^{2} + \cdots - 7$$
$5$ $$T^{5}$$
$7$ $$T^{5} + 6 T^{4} + 2 T^{3} - 45 T^{2} + \cdots - 49$$
$11$ $$T^{5} + 2 T^{4} - 35 T^{3} - 64 T^{2} + \cdots + 307$$
$13$ $$T^{5} + 4 T^{4} - 18 T^{3} - 27 T^{2} + \cdots + 1$$
$17$ $$T^{5} + 9 T^{4} - 3 T^{3} - 155 T^{2} + \cdots + 439$$
$19$ $$T^{5} - 2 T^{4} - 47 T^{3} + 81 T^{2} + \cdots + 5$$
$23$ $$T^{5} + T^{4} - 62 T^{3} - 111 T^{2} + \cdots + 581$$
$29$ $$(T - 1)^{5}$$
$31$ $$T^{5} + T^{4} - 30 T^{3} + 43 T^{2} + \cdots - 73$$
$37$ $$T^{5} + 14 T^{4} + 53 T^{3} + 11 T^{2} + \cdots + 63$$
$41$ $$T^{5} - 5 T^{4} - 13 T^{3} + 22 T^{2} + \cdots + 9$$
$43$ $$T^{5} + 28 T^{4} + 235 T^{3} + \cdots - 22833$$
$47$ $$T^{5} + 15 T^{4} + 37 T^{3} + \cdots - 2263$$
$53$ $$T^{5} - 8 T^{4} - 55 T^{3} + 295 T^{2} + \cdots - 863$$
$59$ $$T^{5} + 11 T^{4} - 36 T^{3} + \cdots + 2205$$
$61$ $$T^{5} + 5 T^{4} - 233 T^{3} + \cdots - 16381$$
$67$ $$T^{5} + 23 T^{4} + 135 T^{3} + \cdots + 353$$
$71$ $$T^{5} + 5 T^{4} - 98 T^{3} + \cdots + 2837$$
$73$ $$T^{5} + 16 T^{4} + 30 T^{3} - 271 T^{2} + \cdots - 9$$
$79$ $$T^{5} + 10 T^{4} - 146 T^{3} + \cdots + 10035$$
$83$ $$T^{5} + 9 T^{4} - 163 T^{3} + \cdots + 18113$$
$89$ $$T^{5} + 18 T^{4} + 86 T^{3} + 51 T^{2} + \cdots - 25$$
$97$ $$T^{5} + 23 T^{4} - 101 T^{3} + \cdots + 196403$$