# Properties

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

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## 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + (\beta_{4} - \beta_{3} - 1) q^{5} + q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + (b4 - b3 - 1) * q^5 + q^6 + (-b4 + b3 - b2 - 2) * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + (\beta_{4} - \beta_{3} - 1) q^{5} + q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{7} + q^{8} + q^{9} + (\beta_{4} - \beta_{3} - 1) q^{10} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{11} + q^{12} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{13} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{14} + (\beta_{4} - \beta_{3} - 1) q^{15} + q^{16} + (\beta_{4} + \beta_{3} + \beta_1 - 2) q^{17} + q^{18} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{19} + (\beta_{4} - \beta_{3} - 1) q^{20} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{21} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{22} + q^{24} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{25} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{26} + q^{27} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{28} + (5 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{29} + (\beta_{4} - \beta_{3} - 1) q^{30} + (\beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 1) q^{31} + q^{32} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{33} + (\beta_{4} + \beta_{3} + \beta_1 - 2) q^{34} + ( - \beta_{3} + 3 \beta_{2} + 1) q^{35} + q^{36} + (\beta_{4} - \beta_1 - 2) q^{37} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{38} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{39} + (\beta_{4} - \beta_{3} - 1) q^{40} + (\beta_{4} - 3 \beta_{2} + 3 \beta_1 - 3) q^{41} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{42} + ( - 3 \beta_{4} - \beta_{3} - 4 \beta_{2} - 2) q^{43} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{44} + (\beta_{4} - \beta_{3} - 1) q^{45} + ( - 4 \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_1 - 6) q^{47} + q^{48} + (2 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{49} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{50} + (\beta_{4} + \beta_{3} + \beta_1 - 2) q^{51} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{52} + ( - 2 \beta_{4} + 4 \beta_{3} - 8 \beta_{2} + 2 \beta_1 - 5) q^{53} + q^{54} + ( - 6 \beta_{4} + 8 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 1) q^{55} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{56} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{57} + (5 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{58} + (\beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \beta_1 - 4) q^{59} + (\beta_{4} - \beta_{3} - 1) q^{60} + ( - 2 \beta_{4} + 8 \beta_{3} - 3 \beta_{2} + 7 \beta_1 - 5) q^{61} + (\beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 1) q^{62} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{63} + q^{64} + ( - 2 \beta_{3} + 4 \beta_{2} - \beta_1 - 1) q^{65} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{66} + ( - 6 \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_1 - 5) q^{67} + (\beta_{4} + \beta_{3} + \beta_1 - 2) q^{68} + ( - \beta_{3} + 3 \beta_{2} + 1) q^{70} + (4 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 7 \beta_1) q^{71} + q^{72} + ( - \beta_{4} - 3 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 1) q^{73} + (\beta_{4} - \beta_1 - 2) q^{74} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{75} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{76} + (5 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{77} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{78} + (3 \beta_{4} - 5 \beta_{3} + 4 \beta_1) q^{79} + (\beta_{4} - \beta_{3} - 1) q^{80} + q^{81} + (\beta_{4} - 3 \beta_{2} + 3 \beta_1 - 3) q^{82} + (5 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} - 7 \beta_1) q^{83} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{84} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{85} + ( - 3 \beta_{4} - \beta_{3} - 4 \beta_{2} - 2) q^{86} + (5 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{87} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{88} + ( - 4 \beta_{4} - 3 \beta_{3} - 1) q^{89} + (\beta_{4} - \beta_{3} - 1) q^{90} + (3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 5) q^{91} + (\beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 1) q^{93} + ( - 4 \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_1 - 6) q^{94} + ( - 4 \beta_{4} + 3 \beta_{3} - \beta_{2} + 6 \beta_1 - 5) q^{95} + q^{96} + (11 \beta_{4} - 5 \beta_{3} + 7 \beta_{2} - 8 \beta_1 + 8) q^{97} + (2 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{98} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + (b4 - b3 - 1) * q^5 + q^6 + (-b4 + b3 - b2 - 2) * q^7 + q^8 + q^9 + (b4 - b3 - 1) * q^10 + (-b3 + 2*b2 - 2) * q^11 + q^12 + (-2*b4 + 2*b3 - b2 + b1 - 2) * q^13 + (-b4 + b3 - b2 - 2) * q^14 + (b4 - b3 - 1) * q^15 + q^16 + (b4 + b3 + b1 - 2) * q^17 + q^18 + (-b4 + b3 + b2 - 2*b1 - 2) * q^19 + (b4 - b3 - 1) * q^20 + (-b4 + b3 - b2 - 2) * q^21 + (-b3 + 2*b2 - 2) * q^22 + q^24 + (-b4 + 2*b3 - b2 - b1 - 1) * q^25 + (-2*b4 + 2*b3 - b2 + b1 - 2) * q^26 + q^27 + (-b4 + b3 - b2 - 2) * q^28 + (5*b4 - 2*b3 + 4*b2 - b1 + 1) * q^29 + (b4 - b3 - 1) * q^30 + (b4 - 4*b3 + 3*b2 - 5*b1 + 1) * q^31 + q^32 + (-b3 + 2*b2 - 2) * q^33 + (b4 + b3 + b1 - 2) * q^34 + (-b3 + 3*b2 + 1) * q^35 + q^36 + (b4 - b1 - 2) * q^37 + (-b4 + b3 + b2 - 2*b1 - 2) * q^38 + (-2*b4 + 2*b3 - b2 + b1 - 2) * q^39 + (b4 - b3 - 1) * q^40 + (b4 - 3*b2 + 3*b1 - 3) * q^41 + (-b4 + b3 - b2 - 2) * q^42 + (-3*b4 - b3 - 4*b2 - 2) * q^43 + (-b3 + 2*b2 - 2) * q^44 + (b4 - b3 - 1) * q^45 + (-4*b4 - b3 + b2 + 4*b1 - 6) * q^47 + q^48 + (2*b4 + b2 + b1 - 2) * q^49 + (-b4 + 2*b3 - b2 - b1 - 1) * q^50 + (b4 + b3 + b1 - 2) * q^51 + (-2*b4 + 2*b3 - b2 + b1 - 2) * q^52 + (-2*b4 + 4*b3 - 8*b2 + 2*b1 - 5) * q^53 + q^54 + (-6*b4 + 8*b3 - 5*b2 + 2*b1 - 1) * q^55 + (-b4 + b3 - b2 - 2) * q^56 + (-b4 + b3 + b2 - 2*b1 - 2) * q^57 + (5*b4 - 2*b3 + 4*b2 - b1 + 1) * q^58 + (b4 + 2*b3 - 4*b2 + b1 - 4) * q^59 + (b4 - b3 - 1) * q^60 + (-2*b4 + 8*b3 - 3*b2 + 7*b1 - 5) * q^61 + (b4 - 4*b3 + 3*b2 - 5*b1 + 1) * q^62 + (-b4 + b3 - b2 - 2) * q^63 + q^64 + (-2*b3 + 4*b2 - b1 - 1) * q^65 + (-b3 + 2*b2 - 2) * q^66 + (-6*b4 + b3 + b2 + 6*b1 - 5) * q^67 + (b4 + b3 + b1 - 2) * q^68 + (-b3 + 3*b2 + 1) * q^70 + (4*b4 - 6*b3 + 2*b2 - 7*b1) * q^71 + q^72 + (-b4 - 3*b3 - 7*b2 + 4*b1 - 1) * q^73 + (b4 - b1 - 2) * q^74 + (-b4 + 2*b3 - b2 - b1 - 1) * q^75 + (-b4 + b3 + b2 - 2*b1 - 2) * q^76 + (5*b4 - 6*b3 + 2*b2 - 2*b1 + 4) * q^77 + (-2*b4 + 2*b3 - b2 + b1 - 2) * q^78 + (3*b4 - 5*b3 + 4*b1) * q^79 + (b4 - b3 - 1) * q^80 + q^81 + (b4 - 3*b2 + 3*b1 - 3) * q^82 + (5*b4 - 7*b3 + 5*b2 - 7*b1) * q^83 + (-b4 + b3 - b2 - 2) * q^84 + (-2*b4 - b3 + b2 - 3*b1 + 4) * q^85 + (-3*b4 - b3 - 4*b2 - 2) * q^86 + (5*b4 - 2*b3 + 4*b2 - b1 + 1) * q^87 + (-b3 + 2*b2 - 2) * q^88 + (-4*b4 - 3*b3 - 1) * q^89 + (b4 - b3 - 1) * q^90 + (3*b4 - b3 - b2 - b1 + 5) * q^91 + (b4 - 4*b3 + 3*b2 - 5*b1 + 1) * q^93 + (-4*b4 - b3 + b2 + 4*b1 - 6) * q^94 + (-4*b4 + 3*b3 - b2 + 6*b1 - 5) * q^95 + q^96 + (11*b4 - 5*b3 + 7*b2 - 8*b1 + 8) * q^97 + (2*b4 + b2 + b1 - 2) * q^98 + (-b3 + 2*b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 7 q^{5} + 5 q^{6} - 7 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10})$$ 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 - 7 * q^5 + 5 * q^6 - 7 * q^7 + 5 * q^8 + 5 * q^9 $$5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 7 q^{5} + 5 q^{6} - 7 q^{7} + 5 q^{8} + 5 q^{9} - 7 q^{10} - 13 q^{11} + 5 q^{12} - 4 q^{13} - 7 q^{14} - 7 q^{15} + 5 q^{16} - 9 q^{17} + 5 q^{18} - 11 q^{19} - 7 q^{20} - 7 q^{21} - 13 q^{22} + 5 q^{24} - 2 q^{25} - 4 q^{26} + 5 q^{27} - 7 q^{28} - 7 q^{29} - 7 q^{30} - 8 q^{31} + 5 q^{32} - 13 q^{33} - 9 q^{34} + q^{35} + 5 q^{36} - 12 q^{37} - 11 q^{38} - 4 q^{39} - 7 q^{40} - 10 q^{41} - 7 q^{42} - 4 q^{43} - 13 q^{44} - 7 q^{45} - 24 q^{47} + 5 q^{48} - 12 q^{49} - 2 q^{50} - 9 q^{51} - 4 q^{52} - 9 q^{53} + 5 q^{54} + 16 q^{55} - 7 q^{56} - 11 q^{57} - 7 q^{58} - 14 q^{59} - 7 q^{60} - 5 q^{61} - 8 q^{62} - 7 q^{63} + 5 q^{64} - 12 q^{65} - 13 q^{66} - 13 q^{67} - 9 q^{68} + q^{70} - 19 q^{71} + 5 q^{72} + 4 q^{73} - 12 q^{74} - 2 q^{75} - 11 q^{76} + 5 q^{77} - 4 q^{78} - 4 q^{79} - 7 q^{80} + 5 q^{81} - 10 q^{82} - 24 q^{83} - 7 q^{84} + 17 q^{85} - 4 q^{86} - 7 q^{87} - 13 q^{88} - 4 q^{89} - 7 q^{90} + 21 q^{91} - 8 q^{93} - 24 q^{94} - 11 q^{95} + 5 q^{96} + 9 q^{97} - 12 q^{98} - 13 q^{99}+O(q^{100})$$ 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 - 7 * q^5 + 5 * q^6 - 7 * q^7 + 5 * q^8 + 5 * q^9 - 7 * q^10 - 13 * q^11 + 5 * q^12 - 4 * q^13 - 7 * q^14 - 7 * q^15 + 5 * q^16 - 9 * q^17 + 5 * q^18 - 11 * q^19 - 7 * q^20 - 7 * q^21 - 13 * q^22 + 5 * q^24 - 2 * q^25 - 4 * q^26 + 5 * q^27 - 7 * q^28 - 7 * q^29 - 7 * q^30 - 8 * q^31 + 5 * q^32 - 13 * q^33 - 9 * q^34 + q^35 + 5 * q^36 - 12 * q^37 - 11 * q^38 - 4 * q^39 - 7 * q^40 - 10 * q^41 - 7 * q^42 - 4 * q^43 - 13 * q^44 - 7 * q^45 - 24 * q^47 + 5 * q^48 - 12 * q^49 - 2 * q^50 - 9 * q^51 - 4 * q^52 - 9 * q^53 + 5 * q^54 + 16 * q^55 - 7 * q^56 - 11 * q^57 - 7 * q^58 - 14 * q^59 - 7 * q^60 - 5 * q^61 - 8 * q^62 - 7 * q^63 + 5 * q^64 - 12 * q^65 - 13 * q^66 - 13 * q^67 - 9 * q^68 + q^70 - 19 * q^71 + 5 * q^72 + 4 * q^73 - 12 * q^74 - 2 * q^75 - 11 * q^76 + 5 * q^77 - 4 * q^78 - 4 * q^79 - 7 * q^80 + 5 * q^81 - 10 * q^82 - 24 * q^83 - 7 * q^84 + 17 * q^85 - 4 * q^86 - 7 * q^87 - 13 * q^88 - 4 * q^89 - 7 * q^90 + 21 * q^91 - 8 * q^93 - 24 * q^94 - 11 * q^95 + 5 * q^96 + 9 * q^97 - 12 * q^98 - 13 * 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
 −0.830830 −1.68251 1.91899 1.30972 0.284630
1.00000 1.00000 1.00000 −3.20362 1.00000 1.51334 1.00000 1.00000 −3.20362
1.2 1.00000 1.00000 1.00000 −2.59435 1.00000 −1.23648 1.00000 1.00000 −2.59435
1.3 1.00000 1.00000 1.00000 −1.47889 1.00000 −3.20362 1.00000 1.00000 −1.47889
1.4 1.00000 1.00000 1.00000 −1.23648 1.00000 −1.47889 1.00000 1.00000 −1.23648
1.5 1.00000 1.00000 1.00000 1.51334 1.00000 −2.59435 1.00000 1.00000 1.51334
 $$n$$: 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.bc 5
3.b odd 2 1 9522.2.a.bt 5
23.b odd 2 1 3174.2.a.bd 5
23.c even 11 2 138.2.e.a 10
69.c even 2 1 9522.2.a.bq 5
69.h odd 22 2 414.2.i.d 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.a 10 23.c even 11 2
414.2.i.d 10 69.h odd 22 2
3174.2.a.bc 5 1.a even 1 1 trivial
3174.2.a.bd 5 23.b odd 2 1
9522.2.a.bq 5 69.c even 2 1
9522.2.a.bt 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3174))$$:

 $$T_{5}^{5} + 7T_{5}^{4} + 13T_{5}^{3} - 6T_{5}^{2} - 35T_{5} - 23$$ T5^5 + 7*T5^4 + 13*T5^3 - 6*T5^2 - 35*T5 - 23 $$T_{7}^{5} + 7T_{7}^{4} + 13T_{7}^{3} - 6T_{7}^{2} - 35T_{7} - 23$$ T7^5 + 7*T7^4 + 13*T7^3 - 6*T7^2 - 35*T7 - 23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{5}$$
$3$ $$(T - 1)^{5}$$
$5$ $$T^{5} + 7 T^{4} + 13 T^{3} - 6 T^{2} + \cdots - 23$$
$7$ $$T^{5} + 7 T^{4} + 13 T^{3} - 6 T^{2} + \cdots - 23$$
$11$ $$T^{5} + 13 T^{4} + 50 T^{3} + 53 T^{2} + \cdots - 1$$
$13$ $$T^{5} + 4 T^{4} - 9 T^{3} - 27 T^{2} + \cdots + 1$$
$17$ $$T^{5} + 9 T^{4} + 17 T^{3} - 31 T^{2} + \cdots - 43$$
$19$ $$T^{5} + 11 T^{4} + 11 T^{3} + \cdots + 253$$
$23$ $$T^{5}$$
$29$ $$T^{5} + 7 T^{4} - 75 T^{3} + \cdots + 1693$$
$31$ $$T^{5} + 8 T^{4} - 69 T^{3} - 733 T^{2} + \cdots - 947$$
$37$ $$T^{5} + 12 T^{4} + 51 T^{3} + 96 T^{2} + \cdots + 23$$
$41$ $$T^{5} + 10 T^{4} - 37 T^{3} - 195 T^{2} + \cdots - 23$$
$43$ $$T^{5} + 4 T^{4} - 97 T^{3} + 149 T^{2} + \cdots - 439$$
$47$ $$T^{5} + 24 T^{4} + 83 T^{3} + \cdots - 10649$$
$53$ $$T^{5} + 9 T^{4} - 170 T^{3} + \cdots + 19009$$
$59$ $$T^{5} + 14 T^{4} - 3 T^{3} + \cdots + 5633$$
$61$ $$T^{5} + 5 T^{4} - 243 T^{3} + \cdots + 52933$$
$67$ $$T^{5} + 13 T^{4} - 181 T^{3} + \cdots + 42481$$
$71$ $$T^{5} + 19 T^{4} - 36 T^{3} + \cdots + 38609$$
$73$ $$T^{5} - 4 T^{4} - 317 T^{3} + \cdots + 15377$$
$79$ $$T^{5} + 4 T^{4} - 251 T^{3} + \cdots - 49169$$
$83$ $$T^{5} + 24 T^{4} + 50 T^{3} + \cdots + 25673$$
$89$ $$T^{5} + 4 T^{4} - 130 T^{3} + \cdots + 9637$$
$97$ $$T^{5} - 9 T^{4} - 335 T^{3} + \cdots - 149381$$
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