# Properties

 Label 3174.2.a.bb Level $3174$ Weight $2$ Character orbit 3174.a Self dual yes Analytic conductor $25.345$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3174 = 2 \cdot 3 \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3174.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.3445176016$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\Q(\zeta_{22})^+$$ Defining polynomial: $$x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1$$ x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 138) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{5} - q^{6} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + (b4 + b2 - 2*b1 + 1) * q^5 - q^6 + (b4 + b2 + b1 - 2) * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{5} - q^{6} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{7} + q^{8} + q^{9} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{10} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{11} - q^{12} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{13} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{14} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{15} + q^{16} + ( - 3 \beta_{3} - \beta_1 + 1) q^{17} + q^{18} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{19} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{20} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{21} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{22} - q^{24} + ( - 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{25} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{26} - q^{27} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{28} + (3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{29} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{30} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} - 4) q^{31} + q^{32} + (3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{33} + ( - 3 \beta_{3} - \beta_1 + 1) q^{34} + ( - 3 \beta_{4} - 4 \beta_{2} + 7 \beta_1 - 5) q^{35} + q^{36} + ( - \beta_{4} + 3 \beta_{3} + \beta_1 - 3) q^{37} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{38} + (\beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{39} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{40} + (\beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 4) q^{41} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{42} + (3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 4) q^{43} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{44} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{45} + ( - \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \beta_1) q^{47} - q^{48} + ( - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{49} + ( - 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{50} + (3 \beta_{3} + \beta_1 - 1) q^{51} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{52} + ( - 4 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 6 \beta_1 - 3) q^{53} - q^{54} + (4 \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 7) q^{55} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{56} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{57} + (3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{58} + ( - 3 \beta_{4} + 7 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 3) q^{59} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{60} + ( - 7 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{61} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} - 4) q^{62} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{63} + q^{64} + (3 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + \beta_1 - 6) q^{65} + (3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{66} + (\beta_{4} - 4 \beta_{3} + 4 \beta_{2} - 5 \beta_1 - 3) q^{67} + ( - 3 \beta_{3} - \beta_1 + 1) q^{68} + ( - 3 \beta_{4} - 4 \beta_{2} + 7 \beta_1 - 5) q^{70} + (5 \beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{71} + q^{72} + ( - \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 1) q^{73} + ( - \beta_{4} + 3 \beta_{3} + \beta_1 - 3) q^{74} + (3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{75} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{76} + (6 \beta_{4} - 8 \beta_{3} + 5 \beta_{2} - 10 \beta_1 + 8) q^{77} + (\beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{78} + (\beta_{4} - 3 \beta_{2} - 2 \beta_1 - 8) q^{79} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{80} + q^{81} + (\beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 4) q^{82} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{83} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{84} + (6 \beta_{4} - \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 1) q^{85} + (3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 4) q^{86} + ( - 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{87} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{88} + (\beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 1) q^{89} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{90} + (6 \beta_{4} - 7 \beta_{3} + 12 \beta_{2} - 8 \beta_1) q^{91} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 4) q^{93} + ( - \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \beta_1) q^{94} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_1 - 1) q^{95} - q^{96} + (3 \beta_{4} + \beta_{2} + 7 \beta_1 - 4) q^{97} + ( - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{98} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + (b4 + b2 - 2*b1 + 1) * q^5 - q^6 + (b4 + b2 + b1 - 2) * q^7 + q^8 + q^9 + (b4 + b2 - 2*b1 + 1) * q^10 + (-3*b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^11 - q^12 + (-b4 + b3 - 3*b2 + 2*b1 + 1) * q^13 + (b4 + b2 + b1 - 2) * q^14 + (-b4 - b2 + 2*b1 - 1) * q^15 + q^16 + (-3*b3 - b1 + 1) * q^17 + q^18 + (b4 - 2*b3 + b2 - b1 - 2) * q^19 + (b4 + b2 - 2*b1 + 1) * q^20 + (-b4 - b2 - b1 + 2) * q^21 + (-3*b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^22 - q^24 + (-3*b4 - 3*b3 + 2*b2 - 2*b1 + 2) * q^25 + (-b4 + b3 - 3*b2 + 2*b1 + 1) * q^26 - q^27 + (b4 + b2 + b1 - 2) * q^28 + (3*b4 - 3*b3 + 2*b2 - b1 + 2) * q^29 + (-b4 - b2 + 2*b1 - 1) * q^30 + (-b4 - b3 - 2*b2 - 4) * q^31 + q^32 + (3*b4 - 2*b3 + 2*b2 - 2*b1 + 4) * q^33 + (-3*b3 - b1 + 1) * q^34 + (-3*b4 - 4*b2 + 7*b1 - 5) * q^35 + q^36 + (-b4 + 3*b3 + b1 - 3) * q^37 + (b4 - 2*b3 + b2 - b1 - 2) * q^38 + (b4 - b3 + 3*b2 - 2*b1 - 1) * q^39 + (b4 + b2 - 2*b1 + 1) * q^40 + (b4 + 3*b3 - 2*b2 - 4) * q^41 + (-b4 - b2 - b1 + 2) * q^42 + (3*b4 + 2*b3 - 3*b2 - 4) * q^43 + (-3*b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^44 + (b4 + b2 - 2*b1 + 1) * q^45 + (-b4 - 2*b3 + 4*b2 + b1) * q^47 - q^48 + (-3*b4 + 3*b3 - b2 - 2*b1 + 3) * q^49 + (-3*b4 - 3*b3 + 2*b2 - 2*b1 + 2) * q^50 + (3*b3 + b1 - 1) * q^51 + (-b4 + b3 - 3*b2 + 2*b1 + 1) * q^52 + (-4*b4 + 2*b3 - 4*b2 + 6*b1 - 3) * q^53 - q^54 + (4*b3 - 4*b2 + 5*b1 - 7) * q^55 + (b4 + b2 + b1 - 2) * q^56 + (-b4 + 2*b3 - b2 + b1 + 2) * q^57 + (3*b4 - 3*b3 + 2*b2 - b1 + 2) * q^58 + (-3*b4 + 7*b3 - 2*b2 + 5*b1 - 3) * q^59 + (-b4 - b2 + 2*b1 - 1) * q^60 + (-7*b4 + b3 - b2 + 2*b1 - 2) * q^61 + (-b4 - b3 - 2*b2 - 4) * q^62 + (b4 + b2 + b1 - 2) * q^63 + q^64 + (3*b4 + 5*b3 - 3*b2 + b1 - 6) * q^65 + (3*b4 - 2*b3 + 2*b2 - 2*b1 + 4) * q^66 + (b4 - 4*b3 + 4*b2 - 5*b1 - 3) * q^67 + (-3*b3 - b1 + 1) * q^68 + (-3*b4 - 4*b2 + 7*b1 - 5) * q^70 + (5*b4 + b3 + 3*b2 + b1 - 1) * q^71 + q^72 + (-b4 + 4*b3 + 3*b2 - 5*b1 - 1) * q^73 + (-b4 + 3*b3 + b1 - 3) * q^74 + (3*b4 + 3*b3 - 2*b2 + 2*b1 - 2) * q^75 + (b4 - 2*b3 + b2 - b1 - 2) * q^76 + (6*b4 - 8*b3 + 5*b2 - 10*b1 + 8) * q^77 + (b4 - b3 + 3*b2 - 2*b1 - 1) * q^78 + (b4 - 3*b2 - 2*b1 - 8) * q^79 + (b4 + b2 - 2*b1 + 1) * q^80 + q^81 + (b4 + 3*b3 - 2*b2 - 4) * q^82 + (-b3 + 2*b2 - 1) * q^83 + (-b4 - b2 - b1 + 2) * q^84 + (6*b4 - b3 + 5*b2 - 6*b1 + 1) * q^85 + (3*b4 + 2*b3 - 3*b2 - 4) * q^86 + (-3*b4 + 3*b3 - 2*b2 + b1 - 2) * q^87 + (-3*b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^88 + (b4 - 4*b3 - 2*b2 + 6*b1 - 1) * q^89 + (b4 + b2 - 2*b1 + 1) * q^90 + (6*b4 - 7*b3 + 12*b2 - 8*b1) * q^91 + (b4 + b3 + 2*b2 + 4) * q^93 + (-b4 - 2*b3 + 4*b2 + b1) * q^94 + (-b4 - 2*b3 + 3*b1 - 1) * q^95 - q^96 + (3*b4 + b2 + 7*b1 - 4) * q^97 + (-3*b4 + 3*b3 - b2 - 2*b1 + 3) * q^98 + (-3*b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} - 5 q^{6} - 11 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10})$$ 5 * q + 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 - 5 * q^6 - 11 * q^7 + 5 * q^8 + 5 * q^9 $$5 q + 5 q^{2} - 5 q^{3} + 5 q^{4} + q^{5} - 5 q^{6} - 11 q^{7} + 5 q^{8} + 5 q^{9} + q^{10} - 11 q^{11} - 5 q^{12} + 12 q^{13} - 11 q^{14} - q^{15} + 5 q^{16} + q^{17} + 5 q^{18} - 15 q^{19} + q^{20} + 11 q^{21} - 11 q^{22} - 5 q^{24} + 6 q^{25} + 12 q^{26} - 5 q^{27} - 11 q^{28} + q^{29} - q^{30} - 18 q^{31} + 5 q^{32} + 11 q^{33} + q^{34} - 11 q^{35} + 5 q^{36} - 10 q^{37} - 15 q^{38} - 12 q^{39} + q^{40} - 16 q^{41} + 11 q^{42} - 18 q^{43} - 11 q^{44} + q^{45} - 4 q^{47} - 5 q^{48} + 20 q^{49} + 6 q^{50} - q^{51} + 12 q^{52} + q^{53} - 5 q^{54} - 22 q^{55} - 11 q^{56} + 15 q^{57} + q^{58} + 2 q^{59} - q^{60} + q^{61} - 18 q^{62} - 11 q^{63} + 5 q^{64} - 24 q^{65} + 11 q^{66} - 29 q^{67} + q^{68} - 11 q^{70} - 11 q^{71} + 5 q^{72} - 8 q^{73} - 10 q^{74} - 6 q^{75} - 15 q^{76} + 11 q^{77} - 12 q^{78} - 40 q^{79} + q^{80} + 5 q^{81} - 16 q^{82} - 8 q^{83} + 11 q^{84} - 13 q^{85} - 18 q^{86} - q^{87} - 11 q^{88} - 2 q^{89} + q^{90} - 33 q^{91} + 18 q^{93} - 4 q^{94} - 3 q^{95} - 5 q^{96} - 17 q^{97} + 20 q^{98} - 11 q^{99}+O(q^{100})$$ 5 * q + 5 * q^2 - 5 * q^3 + 5 * q^4 + q^5 - 5 * q^6 - 11 * q^7 + 5 * q^8 + 5 * q^9 + q^10 - 11 * q^11 - 5 * q^12 + 12 * q^13 - 11 * q^14 - q^15 + 5 * q^16 + q^17 + 5 * q^18 - 15 * q^19 + q^20 + 11 * q^21 - 11 * q^22 - 5 * q^24 + 6 * q^25 + 12 * q^26 - 5 * q^27 - 11 * q^28 + q^29 - q^30 - 18 * q^31 + 5 * q^32 + 11 * q^33 + q^34 - 11 * q^35 + 5 * q^36 - 10 * q^37 - 15 * q^38 - 12 * q^39 + q^40 - 16 * q^41 + 11 * q^42 - 18 * q^43 - 11 * q^44 + q^45 - 4 * q^47 - 5 * q^48 + 20 * q^49 + 6 * q^50 - q^51 + 12 * q^52 + q^53 - 5 * q^54 - 22 * q^55 - 11 * q^56 + 15 * q^57 + q^58 + 2 * q^59 - q^60 + q^61 - 18 * q^62 - 11 * q^63 + 5 * q^64 - 24 * q^65 + 11 * q^66 - 29 * q^67 + q^68 - 11 * q^70 - 11 * q^71 + 5 * q^72 - 8 * q^73 - 10 * q^74 - 6 * q^75 - 15 * q^76 + 11 * q^77 - 12 * q^78 - 40 * q^79 + q^80 + 5 * q^81 - 16 * q^82 - 8 * q^83 + 11 * q^84 - 13 * q^85 - 18 * q^86 - q^87 - 11 * q^88 - 2 * q^89 + q^90 - 33 * q^91 + 18 * q^93 - 4 * q^94 - 3 * q^95 - 5 * q^96 - 17 * q^97 + 20 * q^98 - 11 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{22} + \zeta_{22}^{-1}$$:

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

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.30972 1.91899 0.284630 −0.830830 −1.68251
1.00000 −1.00000 1.00000 −3.82306 −1.00000 −2.89389 1.00000 1.00000 −3.82306
1.2 1.00000 −1.00000 1.00000 −0.324635 −1.00000 2.43232 1.00000 1.00000 −0.324635
1.3 1.00000 −1.00000 1.00000 0.194262 −1.00000 −1.95185 1.00000 1.00000 0.194262
1.4 1.00000 −1.00000 1.00000 1.06731 −1.00000 −4.42518 1.00000 1.00000 1.06731
1.5 1.00000 −1.00000 1.00000 3.88612 −1.00000 −4.16140 1.00000 1.00000 3.88612
 $$n$$: 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$23$$ $$-1$$

## 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 3174.2.a.bb 5
3.b odd 2 1 9522.2.a.br 5
23.b odd 2 1 3174.2.a.ba 5
23.d odd 22 2 138.2.e.b 10
69.c even 2 1 9522.2.a.bs 5
69.g even 22 2 414.2.i.e 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.b 10 23.d odd 22 2
414.2.i.e 10 69.g even 22 2
3174.2.a.ba 5 23.b odd 2 1
3174.2.a.bb 5 1.a even 1 1 trivial
9522.2.a.br 5 3.b odd 2 1
9522.2.a.bs 5 69.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3174))$$:

 $$T_{5}^{5} - T_{5}^{4} - 15T_{5}^{3} + 14T_{5}^{2} + 3T_{5} - 1$$ T5^5 - T5^4 - 15*T5^3 + 14*T5^2 + 3*T5 - 1 $$T_{7}^{5} + 11T_{7}^{4} + 33T_{7}^{3} - 22T_{7}^{2} - 231T_{7} - 253$$ T7^5 + 11*T7^4 + 33*T7^3 - 22*T7^2 - 231*T7 - 253

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{5}$$
$3$ $$(T + 1)^{5}$$
$5$ $$T^{5} - T^{4} - 15 T^{3} + 14 T^{2} + \cdots - 1$$
$7$ $$T^{5} + 11 T^{4} + 33 T^{3} + \cdots - 253$$
$11$ $$T^{5} + 11 T^{4} + 22 T^{3} + \cdots + 253$$
$13$ $$T^{5} - 12 T^{4} + 29 T^{3} + \cdots + 109$$
$17$ $$T^{5} - T^{4} - 37 T^{3} + 47 T^{2} + \cdots - 529$$
$19$ $$T^{5} + 15 T^{4} + 79 T^{3} + 182 T^{2} + \cdots + 67$$
$23$ $$T^{5}$$
$29$ $$T^{5} - T^{4} - 37 T^{3} + 69 T^{2} + \cdots - 23$$
$31$ $$T^{5} + 18 T^{4} + 101 T^{3} + \cdots - 23$$
$37$ $$T^{5} + 10 T^{4} + 7 T^{3} - 118 T^{2} + \cdots + 373$$
$41$ $$T^{5} + 16 T^{4} + 43 T^{3} + \cdots - 659$$
$43$ $$T^{5} + 18 T^{4} + 13 T^{3} + \cdots + 7127$$
$47$ $$T^{5} + 4 T^{4} - 97 T^{3} + \cdots - 4817$$
$53$ $$T^{5} - T^{4} - 114 T^{3} + 146 T^{2} + \cdots - 529$$
$59$ $$T^{5} - 2 T^{4} - 159 T^{3} + \cdots + 11309$$
$61$ $$T^{5} - T^{4} - 169 T^{3} + 102 T^{2} + \cdots - 3389$$
$67$ $$T^{5} + 29 T^{4} + 233 T^{3} + \cdots - 32429$$
$71$ $$T^{5} + 11 T^{4} - 110 T^{3} + \cdots + 5093$$
$73$ $$T^{5} + 8 T^{4} - 245 T^{3} + \cdots + 17621$$
$79$ $$T^{5} + 40 T^{4} + 563 T^{3} + \cdots + 4817$$
$83$ $$T^{5} + 8 T^{4} + 8 T^{3} - 29 T^{2} + \cdots - 1$$
$89$ $$T^{5} + 2 T^{4} - 302 T^{3} + \cdots - 55177$$
$97$ $$T^{5} + 17 T^{4} - 199 T^{3} + \cdots + 3013$$