# Properties

 Label 725.2.a.i 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: 5.5.294577.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 7x^{3} - x^{2} + 7x + 3$$ x^5 - 7*x^3 - x^2 + 7*x + 3 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.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6\nu^{2} + 5\nu + 3$$ v^4 - v^3 - 6*v^2 + 5*v + 3 $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 7\nu^{2} + 5\nu + 6$$ v^4 - v^3 - 7*v^2 + 5*v + 6 $$\beta_{4}$$ $$=$$ $$2\nu^{4} - \nu^{3} - 13\nu^{2} + 5\nu + 9$$ 2*v^4 - v^3 - 13*v^2 + 5*v + 9
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 3$$ -b3 + b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} - \beta_{2} + 5\beta_1$$ b4 - b3 - b2 + 5*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 7\beta_{3} + 6\beta_{2} + 15$$ b4 - 7*b3 + 6*b2 + 15

## 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
 −2.35417 −0.794018 −0.533733 1.22277 2.45914
−2.35417 −1.80364 3.54210 0 4.24606 0.0127981 −3.63036 0.253109 0
1.2 −0.794018 −2.48525 −1.36954 0 1.97334 −3.07654 2.67547 3.17649 0
1.3 −0.533733 0.570435 −1.71513 0 −0.304460 1.47609 1.98289 −2.67460 0
1.4 1.22277 1.05494 −0.504827 0 1.28995 −4.90333 −3.06283 −1.88710 0
1.5 2.45914 −3.33648 4.04739 0 −8.20489 −3.50903 5.03483 8.13211 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.i 5
3.b odd 2 1 6525.2.a.bo 5
5.b even 2 1 725.2.a.j yes 5
5.c odd 4 2 725.2.b.g 10
15.d odd 2 1 6525.2.a.bp 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
725.2.a.i 5 1.a even 1 1 trivial
725.2.a.j yes 5 5.b even 2 1
725.2.b.g 10 5.c odd 4 2
6525.2.a.bo 5 3.b odd 2 1
6525.2.a.bp 5 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(725))$$:

 $$T_{2}^{5} - 7T_{2}^{3} - T_{2}^{2} + 7T_{2} + 3$$ T2^5 - 7*T2^3 - T2^2 + 7*T2 + 3 $$T_{3}^{5} + 6T_{3}^{4} + 7T_{3}^{3} - 11T_{3}^{2} - 13T_{3} + 9$$ T3^5 + 6*T3^4 + 7*T3^3 - 11*T3^2 - 13*T3 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 7 T^{3} - T^{2} + 7 T + 3$$
$3$ $$T^{5} + 6 T^{4} + 7 T^{3} - 11 T^{2} + \cdots + 9$$
$5$ $$T^{5}$$
$7$ $$T^{5} + 10 T^{4} + 26 T^{3} - 11 T^{2} + \cdots + 1$$
$11$ $$T^{5} - 2 T^{4} - 17 T^{3} + 14 T^{2} + \cdots - 27$$
$13$ $$T^{5} + 8 T^{4} - 22 T^{3} - 261 T^{2} + \cdots + 431$$
$17$ $$T^{5} + 3 T^{4} - 41 T^{3} - 161 T^{2} + \cdots + 339$$
$19$ $$T^{5} + 4 T^{4} - 9 T^{3} - 23 T^{2} + \cdots - 17$$
$23$ $$T^{5} + 3 T^{4} - 18 T^{3} - 65 T^{2} + \cdots + 123$$
$29$ $$(T + 1)^{5}$$
$31$ $$T^{5} - 5 T^{4} - 58 T^{3} + 381 T^{2} + \cdots - 251$$
$37$ $$T^{5} + 4 T^{4} - 73 T^{3} + \cdots + 3691$$
$41$ $$T^{5} + 3 T^{4} - 157 T^{3} + \cdots - 19125$$
$43$ $$T^{5} + 12 T^{4} + 3 T^{3} - 252 T^{2} + \cdots - 197$$
$47$ $$T^{5} - T^{4} - 69 T^{3} + 202 T^{2} + \cdots - 2007$$
$53$ $$T^{5} + 12 T^{4} - 31 T^{3} - 167 T^{2} + \cdots - 9$$
$59$ $$T^{5} + 15 T^{4} + 52 T^{3} + \cdots - 375$$
$61$ $$T^{5} + 7 T^{4} - 111 T^{3} - 27 T^{2} + \cdots + 81$$
$67$ $$T^{5} + 17 T^{4} - 61 T^{3} + \cdots + 1679$$
$71$ $$T^{5} + 17 T^{4} - 178 T^{3} + \cdots + 5121$$
$73$ $$T^{5} + 10 T^{4} - 162 T^{3} + \cdots - 13257$$
$79$ $$T^{5} + 2 T^{4} - 228 T^{3} + \cdots + 66293$$
$83$ $$T^{5} + 19 T^{4} - 279 T^{3} + \cdots + 366867$$
$89$ $$T^{5} - 14 T^{4} - 360 T^{3} + \cdots - 278703$$
$97$ $$T^{5} - 9 T^{4} - 263 T^{3} + \cdots + 48079$$