# Properties

 Label 4235.2.a.ba Level $4235$ Weight $2$ Character orbit 4235.a Self dual yes Analytic conductor $33.817$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4235 = 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4235.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.8166452560$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.580017.1 Defining polynomial: $$x^{5} - 7x^{3} - 4x^{2} + 6x + 3$$ x^5 - 7*x^3 - 4*x^2 + 6*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

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{3} q^{3} + ( - \beta_{4} + \beta_{3}) q^{4} - q^{5} + (\beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} - q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{8} + ( - 2 \beta_{4} + \beta_{3} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 - b3 * q^3 + (-b4 + b3) * q^4 - q^5 + (b4 - b2 - b1 - 1) * q^6 - q^7 + (-b4 + b3 + b2 + 2) * q^8 + (-2*b4 + b3 - b1 + 1) * q^9 $$q + \beta_1 q^{2} - \beta_{3} q^{3} + ( - \beta_{4} + \beta_{3}) q^{4} - q^{5} + (\beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} - q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{8} + ( - 2 \beta_{4} + \beta_{3} - \beta_1 + 1) q^{9} - \beta_1 q^{10} + (2 \beta_{4} - 2 \beta_{3} - \beta_1 - 3) q^{12} + ( - \beta_{4} + \beta_{2} + 2) q^{13} - \beta_1 q^{14} + \beta_{3} q^{15} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{16} + ( - \beta_{2} - \beta_1) q^{17} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{18} + ( - 2 \beta_{4} - \beta_{3} - 1) q^{19} + (\beta_{4} - \beta_{3}) q^{20} + \beta_{3} q^{21} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{4} - 3 \beta_{3} - 5 \beta_1 - 4) q^{24} + q^{25} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 1) q^{26} + (\beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{27} + (\beta_{4} - \beta_{3}) q^{28} + ( - 2 \beta_{4} - \beta_{2} - \beta_1 - 1) q^{29} + ( - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{30} + (4 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{31} + ( - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{32} + (2 \beta_{4} - 3 \beta_{3} + \beta_1 - 2) q^{34} + q^{35} + ( - 2 \beta_{4} + 4 \beta_{3} + \beta_{2} + 3 \beta_1 + 7) q^{36} + ( - 4 \beta_{4} + \beta_{3} - 5 \beta_1 - 2) q^{37} + (\beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{38} + ( - \beta_{4} - 2 \beta_{3} - 6 \beta_1) q^{39} + (\beta_{4} - \beta_{3} - \beta_{2} - 2) q^{40} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{41} + ( - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{42} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{43} + (2 \beta_{4} - \beta_{3} + \beta_1 - 1) q^{45} + ( - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 4) q^{46} + (\beta_{4} - \beta_{3} + 4 \beta_1 + 4) q^{47} + (4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 6 \beta_1 - 8) q^{48} + q^{49} + \beta_1 q^{50} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 + 2) q^{51} + ( - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + 5 \beta_1 + 4) q^{52} + ( - 3 \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 2) q^{53} + (2 \beta_{4} - \beta_{3} - \beta_{2} - 4 \beta_1 - 6) q^{54} + (\beta_{4} - \beta_{3} - \beta_{2} - 2) q^{56} + ( - 2 \beta_{4} - 5 \beta_1 + 6) q^{57} + (2 \beta_{4} - \beta_{3} + 2 \beta_1) q^{58} + (\beta_{2} + \beta_1 - 6) q^{59} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_1 + 3) q^{60} + (\beta_{3} + \beta_{2} - \beta_1 + 8) q^{61} + ( - 2 \beta_{4} - \beta_{2} - 6 \beta_1 - 1) q^{62} + (2 \beta_{4} - \beta_{3} + \beta_1 - 1) q^{63} + ( - 4 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_1 + 6) q^{64} + (\beta_{4} - \beta_{2} - 2) q^{65} + ( - 3 \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{67} + (2 \beta_{4} - \beta_{3} - \beta_{2} - 5 \beta_1 - 3) q^{68} + (5 \beta_{4} - 3 \beta_{3} - \beta_{2} + 6 \beta_1 - 2) q^{69} + \beta_1 q^{70} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{71} + ( - 8 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 10) q^{72} + ( - 4 \beta_{4} + 3 \beta_{3} - \beta_{2} - 1) q^{73} + (4 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{74} - \beta_{3} q^{75} + (3 \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 3) q^{76} + (8 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - \beta_1 - 13) q^{78} + ( - \beta_{3} + 5 \beta_1 + 3) q^{79} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{80} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{81} + (2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{82} + (3 \beta_{4} - \beta_{2} + 5 \beta_1 + 6) q^{83} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_1 + 3) q^{84} + (\beta_{2} + \beta_1) q^{85} + (\beta_{2} + 2 \beta_1 - 3) q^{86} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{87} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{89} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{90} + (\beta_{4} - \beta_{2} - 2) q^{91} + (3 \beta_{3} + 2 \beta_{2} - \beta_1 + 6) q^{92} + ( - \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{93} + ( - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 6) q^{94} + (2 \beta_{4} + \beta_{3} + 1) q^{95} + (9 \beta_{4} - 10 \beta_{3} - 2 \beta_{2} - \beta_1 - 10) q^{96} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{97} + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 - b3 * q^3 + (-b4 + b3) * q^4 - q^5 + (b4 - b2 - b1 - 1) * q^6 - q^7 + (-b4 + b3 + b2 + 2) * q^8 + (-2*b4 + b3 - b1 + 1) * q^9 - b1 * q^10 + (2*b4 - 2*b3 - b1 - 3) * q^12 + (-b4 + b2 + 2) * q^13 - b1 * q^14 + b3 * q^15 + (b3 + b2 + 3*b1 + 2) * q^16 + (-b2 - b1) * q^17 + (b3 + b2 + 4*b1 + 1) * q^18 + (-2*b4 - b3 - 1) * q^19 + (b4 - b3) * q^20 + b3 * q^21 + (-b4 + b3 - 2*b2 + b1 - 1) * q^23 + (b4 - 3*b3 - 5*b1 - 4) * q^24 + q^25 + (-b4 + 3*b3 + 2*b1 + 1) * q^26 + (b4 - b3 + b2 - 2*b1 - 1) * q^27 + (b4 - b3) * q^28 + (-2*b4 - b2 - b1 - 1) * q^29 + (-b4 + b2 + b1 + 1) * q^30 + (4*b4 - b3 + b2 + 2*b1) * q^31 + (-3*b4 + 3*b3 - b2 + 2*b1 + 3) * q^32 + (2*b4 - 3*b3 + b1 - 2) * q^34 + q^35 + (-2*b4 + 4*b3 + b2 + 3*b1 + 7) * q^36 + (-4*b4 + b3 - 5*b1 - 2) * q^37 + (b4 + 2*b3 - b2 + 1) * q^38 + (-b4 - 2*b3 - 6*b1) * q^39 + (b4 - b3 - b2 - 2) * q^40 + (-b4 - b3 - 2*b2 + b1 - 3) * q^41 + (-b4 + b2 + b1 + 1) * q^42 + (b3 + b2 - 2*b1 + 2) * q^43 + (2*b4 - b3 + b1 - 1) * q^45 + (-2*b3 + b2 + 3*b1 + 4) * q^46 + (b4 - b3 + 4*b1 + 4) * q^47 + (4*b4 - 2*b3 - 3*b2 - 6*b1 - 8) * q^48 + q^49 + b1 * q^50 + (-b3 + b2 + 5*b1 + 2) * q^51 + (-3*b4 + 3*b3 + b2 + 5*b1 + 4) * q^52 + (-3*b4 + b3 - 3*b2 - 2*b1 - 2) * q^53 + (2*b4 - b3 - b2 - 4*b1 - 6) * q^54 + (b4 - b3 - b2 - 2) * q^56 + (-2*b4 - 5*b1 + 6) * q^57 + (2*b4 - b3 + 2*b1) * q^58 + (b2 + b1 - 6) * q^59 + (-2*b4 + 2*b3 + b1 + 3) * q^60 + (b3 + b2 - b1 + 8) * q^61 + (-2*b4 - b2 - 6*b1 - 1) * q^62 + (2*b4 - b3 + b1 - 1) * q^63 + (-4*b4 + b3 + b2 + 4*b1 + 6) * q^64 + (b4 - b2 - 2) * q^65 + (-3*b4 - b3 + b2 + 1) * q^67 + (2*b4 - b3 - b2 - 5*b1 - 3) * q^68 + (5*b4 - 3*b3 - b2 + 6*b1 - 2) * q^69 + b1 * q^70 + (-2*b4 + b3 - 2*b2) * q^71 + (-8*b4 + 5*b3 + 2*b2 + 4*b1 + 10) * q^72 + (-4*b4 + 3*b3 - b2 - 1) * q^73 + (4*b4 - b3 + b2 + 3*b1 - 5) * q^74 - b3 * q^75 + (3*b4 - b3 + 2*b2 + 3*b1 + 3) * q^76 + (8*b4 - 5*b3 - 2*b2 - b1 - 13) * q^78 + (-b3 + 5*b1 + 3) * q^79 + (-b3 - b2 - 3*b1 - 2) * q^80 + (b4 + b3 + 2*b2 + 2*b1 + 1) * q^81 + (2*b4 - 2*b3 - b2 - b1 + 2) * q^82 + (3*b4 - b2 + 5*b1 + 6) * q^83 + (-2*b4 + 2*b3 + b1 + 3) * q^84 + (b2 + b1) * q^85 + (b2 + 2*b1 - 3) * q^86 + (-2*b3 + b2 + b1 + 4) * q^87 + (-2*b4 - b3 + b2 + b1 + 1) * q^89 + (-b3 - b2 - 4*b1 - 1) * q^90 + (b4 - b2 - 2) * q^91 + (3*b3 + 2*b2 - b1 + 6) * q^92 + (-b4 + 6*b3 - 2*b2 + b1 - 3) * q^93 + (-3*b4 + 3*b3 - b2 + 2*b1 + 6) * q^94 + (2*b4 + b3 + 1) * q^95 + (9*b4 - 10*b3 - 2*b2 - b1 - 10) * q^96 + (-2*b4 + 2*b3 - b2 + 2*b1 - 3) * q^97 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} - 5 q^{7} + 12 q^{8} + 11 q^{9}+O(q^{10})$$ 5 * q - 2 * q^3 + 4 * q^4 - 5 * q^5 - 5 * q^6 - 5 * q^7 + 12 * q^8 + 11 * q^9 $$5 q - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} - 5 q^{7} + 12 q^{8} + 11 q^{9} - 23 q^{12} + 10 q^{13} + 2 q^{15} + 10 q^{16} + 2 q^{17} + 5 q^{18} - 3 q^{19} - 4 q^{20} + 2 q^{21} + 3 q^{23} - 28 q^{24} + 5 q^{25} + 13 q^{26} - 11 q^{27} - 4 q^{28} + q^{29} + 5 q^{30} - 12 q^{31} + 29 q^{32} - 20 q^{34} + 5 q^{35} + 45 q^{36} + 9 q^{38} - 2 q^{39} - 12 q^{40} - 11 q^{41} + 5 q^{42} + 10 q^{43} - 11 q^{45} + 14 q^{46} + 16 q^{47} - 46 q^{48} + 5 q^{49} + 6 q^{51} + 30 q^{52} + 4 q^{53} - 34 q^{54} - 12 q^{56} + 34 q^{57} - 6 q^{58} - 32 q^{59} + 23 q^{60} + 40 q^{61} + q^{62} - 11 q^{63} + 38 q^{64} - 10 q^{65} + 7 q^{67} - 19 q^{68} - 24 q^{69} + 10 q^{71} + 72 q^{72} + 11 q^{73} - 37 q^{74} - 2 q^{75} + 3 q^{76} - 87 q^{78} + 13 q^{79} - 10 q^{80} + q^{81} + 4 q^{82} + 26 q^{83} + 23 q^{84} - 2 q^{85} - 17 q^{86} + 14 q^{87} + 5 q^{89} - 5 q^{90} - 10 q^{91} + 32 q^{92} + 3 q^{93} + 44 q^{94} + 3 q^{95} - 84 q^{96} - 5 q^{97}+O(q^{100})$$ 5 * q - 2 * q^3 + 4 * q^4 - 5 * q^5 - 5 * q^6 - 5 * q^7 + 12 * q^8 + 11 * q^9 - 23 * q^12 + 10 * q^13 + 2 * q^15 + 10 * q^16 + 2 * q^17 + 5 * q^18 - 3 * q^19 - 4 * q^20 + 2 * q^21 + 3 * q^23 - 28 * q^24 + 5 * q^25 + 13 * q^26 - 11 * q^27 - 4 * q^28 + q^29 + 5 * q^30 - 12 * q^31 + 29 * q^32 - 20 * q^34 + 5 * q^35 + 45 * q^36 + 9 * q^38 - 2 * q^39 - 12 * q^40 - 11 * q^41 + 5 * q^42 + 10 * q^43 - 11 * q^45 + 14 * q^46 + 16 * q^47 - 46 * q^48 + 5 * q^49 + 6 * q^51 + 30 * q^52 + 4 * q^53 - 34 * q^54 - 12 * q^56 + 34 * q^57 - 6 * q^58 - 32 * q^59 + 23 * q^60 + 40 * q^61 + q^62 - 11 * q^63 + 38 * q^64 - 10 * q^65 + 7 * q^67 - 19 * q^68 - 24 * q^69 + 10 * q^71 + 72 * q^72 + 11 * q^73 - 37 * q^74 - 2 * q^75 + 3 * q^76 - 87 * q^78 + 13 * q^79 - 10 * q^80 + q^81 + 4 * q^82 + 26 * q^83 + 23 * q^84 - 2 * q^85 - 17 * q^86 + 14 * q^87 + 5 * q^89 - 5 * q^90 - 10 * q^91 + 32 * q^92 + 3 * q^93 + 44 * q^94 + 3 * q^95 - 84 * q^96 - 5 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 7x^{3} - 4x^{2} + 6x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu$$ v^3 - v^2 - 4*v $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 2$$ v^4 - v^3 - 5*v^2 + v + 2 $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6\nu^{2} + \nu + 4$$ v^4 - v^3 - 6*v^2 + v + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + 2$$ -b4 + b3 + 2 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 2$$ -b4 + b3 + b2 + 4*b1 + 2 $$\nu^{4}$$ $$=$$ $$-6\beta_{4} + 7\beta_{3} + \beta_{2} + 3\beta _1 + 10$$ -6*b4 + 7*b3 + b2 + 3*b1 + 10

## 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
 −1.93511 −1.27779 −0.470042 0.941569 2.74137
−1.93511 −2.61037 1.74465 −1.00000 5.05136 −1.00000 0.494123 3.81405 1.93511
1.2 −1.27779 2.68935 −0.367260 −1.00000 −3.43642 −1.00000 3.02485 4.23262 1.27779
1.3 −0.470042 −0.577925 −1.77906 −1.00000 0.271649 −1.00000 1.77632 −2.66600 0.470042
1.4 0.941569 1.53997 −1.11345 −1.00000 1.44999 −1.00000 −2.93153 −0.628495 −0.941569
1.5 2.74137 −3.04102 5.51511 −1.00000 −8.33658 −1.00000 9.63623 6.24783 −2.74137
 $$n$$: 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
$$5$$ $$1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## 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 4235.2.a.ba 5
11.b odd 2 1 4235.2.a.bb yes 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4235.2.a.ba 5 1.a even 1 1 trivial
4235.2.a.bb yes 5 11.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4235))$$:

 $$T_{2}^{5} - 7T_{2}^{3} - 4T_{2}^{2} + 6T_{2} + 3$$ T2^5 - 7*T2^3 - 4*T2^2 + 6*T2 + 3 $$T_{3}^{5} + 2T_{3}^{4} - 11T_{3}^{3} - 17T_{3}^{2} + 27T_{3} + 19$$ T3^5 + 2*T3^4 - 11*T3^3 - 17*T3^2 + 27*T3 + 19 $$T_{13}^{5} - 10T_{13}^{4} + 8T_{13}^{3} + 99T_{13}^{2} - 4T_{13} - 133$$ T13^5 - 10*T13^4 + 8*T13^3 + 99*T13^2 - 4*T13 - 133

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 7 T^{3} - 4 T^{2} + 6 T + 3$$
$3$ $$T^{5} + 2 T^{4} - 11 T^{3} - 17 T^{2} + \cdots + 19$$
$5$ $$(T + 1)^{5}$$
$7$ $$(T + 1)^{5}$$
$11$ $$T^{5}$$
$13$ $$T^{5} - 10 T^{4} + 8 T^{3} + 99 T^{2} + \cdots - 133$$
$17$ $$T^{5} - 2 T^{4} - 28 T^{3} + 43 T^{2} + \cdots + 9$$
$19$ $$T^{5} + 3 T^{4} - 51 T^{3} - 114 T^{2} + \cdots - 269$$
$23$ $$T^{5} - 3 T^{4} - 70 T^{3} + \cdots - 3573$$
$29$ $$T^{5} - T^{4} - 36 T^{3} + 115 T^{2} + \cdots - 147$$
$31$ $$T^{5} + 12 T^{4} - 39 T^{3} + \cdots - 5161$$
$37$ $$T^{5} - 145 T^{3} - 243 T^{2} + \cdots + 11545$$
$41$ $$T^{5} + 11 T^{4} - 30 T^{3} + \cdots - 1425$$
$43$ $$T^{5} - 10 T^{4} + 3 T^{3} + 247 T^{2} + \cdots + 761$$
$47$ $$T^{5} - 16 T^{4} + 21 T^{3} + \cdots - 1335$$
$53$ $$T^{5} - 4 T^{4} - 167 T^{3} + \cdots - 2307$$
$59$ $$T^{5} + 32 T^{4} + 380 T^{3} + \cdots + 3267$$
$61$ $$T^{5} - 40 T^{4} + 613 T^{3} + \cdots - 13937$$
$67$ $$T^{5} - 7 T^{4} - 117 T^{3} + 392 T^{2} + \cdots - 5$$
$71$ $$T^{5} - 10 T^{4} - 35 T^{3} + \cdots - 1155$$
$73$ $$T^{5} - 11 T^{4} - 141 T^{3} + \cdots + 121$$
$79$ $$T^{5} - 13 T^{4} - 95 T^{3} + 1120 T^{2} + \cdots - 59$$
$83$ $$T^{5} - 26 T^{4} + 116 T^{3} + \cdots + 23445$$
$89$ $$T^{5} - 5 T^{4} - 95 T^{3} + 377 T^{2} + \cdots - 927$$
$97$ $$T^{5} + 5 T^{4} - 123 T^{3} - 751 T^{2} + \cdots + 49$$