Properties

 Label 342.2.u.a Level $342$ Weight $2$ Character orbit 342.u Analytic conductor $2.731$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 342.u (of order $$9$$, degree $$6$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.73088374913$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{5} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{7} - \zeta_{18}^{3} q^{8} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 + (z^5 - z^3 + z^2 - 1) * q^5 + (z^5 + z^4 - z^3 - z^2 + 1) * q^7 - z^3 * q^8 $$q - \zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{5} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{7} - \zeta_{18}^{3} q^{8} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18} + 1) q^{10} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{11} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{13} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18} + 1) q^{14} + \zeta_{18}^{4} q^{16} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} - \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{17} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \zeta_{18} - 2) q^{19} + ( - \zeta_{18}^{5} + 2 \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18}) q^{20} + ( - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{22} + (2 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18} + 2) q^{23} + ( - \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}) q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18} + 2) q^{26} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} - 1) q^{28} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{29} + ( - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18} + 3) q^{31} - \zeta_{18}^{5} q^{32} + (3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{34} + ( - 2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 4 \zeta_{18}^{2} - 4) q^{35} + ( - 2 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 6 \zeta_{18}^{2} + 6 \zeta_{18} - 3) q^{37} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{38} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{40} + (4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{41} + (\zeta_{18}^{5} - 5 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 5 \zeta_{18} - 1) q^{43} + (\zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{44} + ( - 2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 2 \zeta_{18}) q^{46} + ( - \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} + \zeta_{18}^{2}) q^{47} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18}) q^{49} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 1) q^{50} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{52} + (2 \zeta_{18}^{4} - 8 \zeta_{18}^{3} - \zeta_{18}^{2} - 8 \zeta_{18} + 2) q^{53} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - 2) q^{55} + ( - \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 1) q^{56} + (3 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{58} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 4 \zeta_{18} - 6) q^{59} + (\zeta_{18}^{4} - \zeta_{18}^{3} - 11 \zeta_{18}^{2} - \zeta_{18} + 1) q^{61} + (3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18}) q^{62} + (\zeta_{18}^{3} - 1) q^{64} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}) q^{65} + ( - 4 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 4 \zeta_{18} + 4) q^{67} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18}) q^{68} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 4 \zeta_{18} - 2) q^{70} + (7 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} + 5 \zeta_{18} - 7) q^{71} + (5 \zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 7) q^{73} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 3 \zeta_{18} - 2) q^{74} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{76} - q^{77} + ( - 10 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - 5) q^{79} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} - 2) q^{80} + ( - 3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - \zeta_{18} + 4) q^{82} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} - 9 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18} + 9) q^{83} + ( - 9 \zeta_{18}^{5} - \zeta_{18}^{4} + 8 \zeta_{18}^{3} + 9 \zeta_{18}^{2} - 7 \zeta_{18} - 7) q^{85} + (5 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 5 \zeta_{18}^{2} + \zeta_{18} + 1) q^{86} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{88} + (5 \zeta_{18}^{5} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 3 \zeta_{18} + 7) q^{89} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{2} - 2) q^{91} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{92} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 1) q^{94} + ( - 10 \zeta_{18}^{5} - \zeta_{18}^{4} + 6 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - \zeta_{18} - 3) q^{95} + ( - \zeta_{18}^{2} - 4 \zeta_{18} - 1) q^{97} + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3) q^{98} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 + (z^5 - z^3 + z^2 - 1) * q^5 + (z^5 + z^4 - z^3 - z^2 + 1) * q^7 - z^3 * q^8 + (z^4 - 2*z^3 + z + 1) * q^10 + (z^4 + z^2) * q^11 + (2*z^5 + z^4 + z^3 - 1) * q^13 + (-z^5 + z^4 - z + 1) * q^14 + z^4 * q^16 + (3*z^5 - 3*z^3 - z^2 + 2*z + 2) * q^17 + (2*z^5 + 2*z^4 - 2*z^3 - z - 2) * q^19 + (-z^5 + 2*z^4 - z^2 - z) * q^20 + (-z^5 - z^3) * q^22 + (2*z^4 + 3*z^3 - 2*z^2 + 3*z + 2) * q^23 + (-z^5 + 3*z^4 + 3*z^3 - 3*z) * q^25 + (-z^5 - z^4 - 2*z^3 + z + 2) * q^26 + (-z^5 + z^3 + z^2 - z - 1) * q^28 + (2*z^5 - 3*z^4 + 5*z^3 - 2*z^2 - 2*z - 2) * q^29 + (-3*z^3 + 3*z^2 - 3*z + 3) * q^31 - z^5 * q^32 + (3*z^4 - 2*z^3 - 2*z^2 - 2*z + 3) * q^34 + (-2*z^5 - 3*z^4 + 2*z^3 + 4*z^2 - 4) * q^35 + (-2*z^5 - 4*z^4 + 6*z^2 + 6*z - 3) * q^37 + (-2*z^5 + 2*z^4 - 2*z^3 + z^2 + 2*z + 2) * q^38 + (-2*z^5 + 2*z^3 + z^2 - 1) * q^40 + (4*z^5 + 3*z^4 + 3*z^3 - z^2 + 1) * q^41 + (z^5 - 5*z^4 + 3*z^3 - 3*z^2 + 5*z - 1) * q^43 + (z^4 + z^3 - 1) * q^44 + (-2*z^5 - 3*z^4 + 2*z^3 - 3*z^2 - 2*z) * q^46 + (-z^5 - 3*z^4 + 3*z^3 + z^2) * q^47 + (3*z^5 - z^4 + 4*z^3 - z^2 + 3*z) * q^49 + (-3*z^5 - 3*z^4 + z^3 + 3*z^2 - 1) * q^50 + (z^5 + 2*z^4 + z^3 - z^2 - 2*z - 1) * q^52 + (2*z^4 - 8*z^3 - z^2 - 8*z + 2) * q^53 + (-z^5 + z^3 - z^2 - 2) * q^55 + (-z^4 + z^2 + z - 1) * q^56 + (3*z^5 - 5*z^4 + 2*z^2 + 2*z + 2) * q^58 + (-3*z^5 + 3*z^3 - 3*z^2 + 4*z - 6) * q^59 + (z^4 - z^3 - 11*z^2 - z + 1) * q^61 + (3*z^4 - 3*z^3 + 3*z^2 - 3*z) * q^62 + (z^3 - 1) * q^64 + (-3*z^5 - 3*z) * q^65 + (-4*z^5 - 3*z^4 - z^3 + 4*z^2 + 4*z + 4) * q^67 + (-3*z^5 + 2*z^4 + 2*z^3 + 2*z^2 - 3*z) * q^68 + (3*z^5 - 2*z^4 - 2*z^3 + 4*z - 2) * q^70 + (7*z^5 - 5*z^4 + z^3 - z^2 + 5*z - 7) * q^71 + (5*z^5 + z^4 - 2*z^3 - 7*z^2 + 7) * q^73 + (4*z^5 - 4*z^3 - 6*z^2 + 3*z - 2) * q^74 + (-2*z^5 + 2*z^4 + z^3 - 2*z^2 - 2*z - 2) * q^76 - q^77 + (-10*z^5 - 5*z^4 - 5*z^3 + 5*z^2 - 5) * q^79 + (-2*z^4 + z^3 + z - 2) * q^80 + (-3*z^5 - 3*z^4 - 3*z^3 - z + 4) * q^82 + (-z^5 - z^4 - 9*z^3 - 2*z^2 + 3*z + 9) * q^83 + (-9*z^5 - z^4 + 8*z^3 + 9*z^2 - 7*z - 7) * q^85 + (5*z^5 - 3*z^4 + 2*z^3 - 5*z^2 + z + 1) * q^86 + (-z^5 - z^4 + z) * q^88 + (5*z^5 - 4*z^4 - 4*z^3 - 3*z + 7) * q^89 + (-2*z^4 + z^2 - 2) * q^91 + (3*z^5 - 2*z^4 + 5*z^3 + 2*z^2 - 2) * q^92 + (3*z^5 - 3*z^4 - 1) * q^94 + (-10*z^5 - z^4 + 6*z^3 + 2*z^2 - z - 3) * q^95 + (-z^2 - 4*z - 1) * q^97 + (z^5 - 4*z^4 - 2*z^3 - 3*z^2 + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 9 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10})$$ 6 * q - 9 * q^5 + 3 * q^7 - 3 * q^8 $$6 q - 9 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{13} + 6 q^{14} + 3 q^{17} - 18 q^{19} - 3 q^{22} + 21 q^{23} + 9 q^{25} + 6 q^{26} - 3 q^{28} + 3 q^{29} + 9 q^{31} + 12 q^{34} - 18 q^{35} - 18 q^{37} + 6 q^{38} + 15 q^{41} + 3 q^{43} - 3 q^{44} + 6 q^{46} + 9 q^{47} + 12 q^{49} - 3 q^{50} - 3 q^{52} - 12 q^{53} - 9 q^{55} - 6 q^{56} + 12 q^{58} - 27 q^{59} + 3 q^{61} - 9 q^{62} - 3 q^{64} + 21 q^{67} + 6 q^{68} - 18 q^{70} - 39 q^{71} + 36 q^{73} - 24 q^{74} - 9 q^{76} - 6 q^{77} - 45 q^{79} - 9 q^{80} + 15 q^{82} + 27 q^{83} - 18 q^{85} + 12 q^{86} + 30 q^{89} - 12 q^{91} + 3 q^{92} - 6 q^{94} - 6 q^{97} + 12 q^{98}+O(q^{100})$$ 6 * q - 9 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^13 + 6 * q^14 + 3 * q^17 - 18 * q^19 - 3 * q^22 + 21 * q^23 + 9 * q^25 + 6 * q^26 - 3 * q^28 + 3 * q^29 + 9 * q^31 + 12 * q^34 - 18 * q^35 - 18 * q^37 + 6 * q^38 + 15 * q^41 + 3 * q^43 - 3 * q^44 + 6 * q^46 + 9 * q^47 + 12 * q^49 - 3 * q^50 - 3 * q^52 - 12 * q^53 - 9 * q^55 - 6 * q^56 + 12 * q^58 - 27 * q^59 + 3 * q^61 - 9 * q^62 - 3 * q^64 + 21 * q^67 + 6 * q^68 - 18 * q^70 - 39 * q^71 + 36 * q^73 - 24 * q^74 - 9 * q^76 - 6 * q^77 - 45 * q^79 - 9 * q^80 + 15 * q^82 + 27 * q^83 - 18 * q^85 + 12 * q^86 + 30 * q^89 - 12 * q^91 + 3 * q^92 - 6 * q^94 - 6 * q^97 + 12 * q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −0.173648 + 0.984808i −0.766044 − 0.642788i −0.173648 − 0.984808i −0.766044 + 0.642788i 0.939693 − 0.342020i 0.939693 + 0.342020i
0.173648 0.984808i 0 −0.939693 0.342020i −3.20574 + 1.16679i 0 1.43969 + 2.49362i −0.500000 + 0.866025i 0 0.592396 + 3.35965i
73.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.386659 + 2.19285i 0 0.326352 + 0.565258i −0.500000 + 0.866025i 0 −1.70574 + 1.43128i
199.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i −3.20574 1.16679i 0 1.43969 2.49362i −0.500000 0.866025i 0 0.592396 3.35965i
253.1 0.766044 0.642788i 0 0.173648 0.984808i −0.386659 2.19285i 0 0.326352 0.565258i −0.500000 0.866025i 0 −1.70574 1.43128i
271.1 −0.939693 + 0.342020i 0 0.766044 0.642788i −0.907604 0.761570i 0 −0.266044 0.460802i −0.500000 + 0.866025i 0 1.11334 + 0.405223i
289.1 −0.939693 0.342020i 0 0.766044 + 0.642788i −0.907604 + 0.761570i 0 −0.266044 + 0.460802i −0.500000 0.866025i 0 1.11334 0.405223i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.u.a 6
3.b odd 2 1 114.2.i.d 6
12.b even 2 1 912.2.bo.f 6
19.e even 9 1 inner 342.2.u.a 6
19.e even 9 1 6498.2.a.bs 3
19.f odd 18 1 6498.2.a.bn 3
57.j even 18 1 2166.2.a.u 3
57.l odd 18 1 114.2.i.d 6
57.l odd 18 1 2166.2.a.o 3
228.v even 18 1 912.2.bo.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.d 6 3.b odd 2 1
114.2.i.d 6 57.l odd 18 1
342.2.u.a 6 1.a even 1 1 trivial
342.2.u.a 6 19.e even 9 1 inner
912.2.bo.f 6 12.b even 2 1
912.2.bo.f 6 228.v even 18 1
2166.2.a.o 3 57.l odd 18 1
2166.2.a.u 3 57.j even 18 1
6498.2.a.bn 3 19.f odd 18 1
6498.2.a.bs 3 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 9T_{5}^{5} + 36T_{5}^{4} + 90T_{5}^{3} + 162T_{5}^{2} + 162T_{5} + 81$$ acting on $$S_{2}^{\mathrm{new}}(342, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 9 T^{5} + 36 T^{4} + 90 T^{3} + \cdots + 81$$
$7$ $$T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1$$
$11$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$13$ $$T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9$$
$17$ $$T^{6} - 3 T^{5} + 48 T^{4} - 244 T^{3} + \cdots + 289$$
$19$ $$T^{6} + 18 T^{5} + 162 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 21 T^{5} + 210 T^{4} + \cdots + 72361$$
$29$ $$T^{6} - 3 T^{5} + 18 T^{4} + \cdots + 45369$$
$31$ $$T^{6} - 9 T^{5} + 81 T^{4} - 54 T^{3} + \cdots + 729$$
$37$ $$(T^{3} + 9 T^{2} - 57 T - 361)^{2}$$
$41$ $$T^{6} - 15 T^{5} + 177 T^{4} + \cdots + 11881$$
$43$ $$T^{6} - 3 T^{5} + 99 T^{4} + \cdots + 3249$$
$47$ $$T^{6} - 9 T^{5} + 63 T^{4} + \cdots + 2809$$
$53$ $$T^{6} + 12 T^{5} + 174 T^{4} + \cdots + 94249$$
$59$ $$T^{6} + 27 T^{5} + 324 T^{4} + \cdots + 289$$
$61$ $$T^{6} - 3 T^{5} - 60 T^{4} + \cdots + 1682209$$
$67$ $$T^{6} - 21 T^{5} + 126 T^{4} - 24 T^{3} + \cdots + 9$$
$71$ $$T^{6} + 39 T^{5} + 561 T^{4} + \cdots + 201601$$
$73$ $$T^{6} - 36 T^{5} + 558 T^{4} + \cdots + 1369$$
$79$ $$T^{6} + 45 T^{5} + 1125 T^{4} + \cdots + 4515625$$
$83$ $$T^{6} - 27 T^{5} + 507 T^{4} + \cdots + 253009$$
$89$ $$T^{6} - 30 T^{5} + 246 T^{4} + \cdots + 11449$$
$97$ $$T^{6} + 6 T^{5} + 3 T^{4} + 35 T^{3} + \cdots + 2809$$