# Properties

 Label 390.2.bd.a Level $390$ Weight $2$ Character orbit 390.bd Analytic conductor $3.114$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.bd (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{2} q^{2} + \zeta_{24}^{5} q^{3} + \zeta_{24}^{4} q^{4} + ( - \zeta_{24}^{5} + 2 \zeta_{24}^{3} + \zeta_{24}) q^{5} + \zeta_{24}^{7} q^{6} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{6} + 2 \zeta_{24}^{5} - \zeta_{24}^{4} + 2 \zeta_{24}^{3} - \zeta_{24}^{2} - 2 \zeta_{24}) q^{7} + \zeta_{24}^{6} q^{8} + (\zeta_{24}^{6} - \zeta_{24}^{2}) q^{9}+O(q^{10})$$ q + z^2 * q^2 + z^5 * q^3 + z^4 * q^4 + (-z^5 + 2*z^3 + z) * q^5 + z^7 * q^6 + (-2*z^7 - z^6 + 2*z^5 - z^4 + 2*z^3 - z^2 - 2*z) * q^7 + z^6 * q^8 + (z^6 - z^2) * q^9 $$q + \zeta_{24}^{2} q^{2} + \zeta_{24}^{5} q^{3} + \zeta_{24}^{4} q^{4} + ( - \zeta_{24}^{5} + 2 \zeta_{24}^{3} + \zeta_{24}) q^{5} + \zeta_{24}^{7} q^{6} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{6} + 2 \zeta_{24}^{5} - \zeta_{24}^{4} + 2 \zeta_{24}^{3} - \zeta_{24}^{2} - 2 \zeta_{24}) q^{7} + \zeta_{24}^{6} q^{8} + (\zeta_{24}^{6} - \zeta_{24}^{2}) q^{9} + ( - \zeta_{24}^{7} + 2 \zeta_{24}^{5} + \zeta_{24}^{3}) q^{10} + (\zeta_{24}^{7} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{5} + \zeta_{24}^{3} + 2 \zeta_{24}^{2} - 2) q^{11} + (\zeta_{24}^{5} - \zeta_{24}) q^{12} + ( - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 3 \zeta_{24}) q^{13} + (2 \zeta_{24}^{7} - \zeta_{24}^{6} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + 2 \zeta_{24} + 1) q^{14} + (2 \zeta_{24}^{4} + \zeta_{24}^{2} - 2) q^{15} + (\zeta_{24}^{4} - 1) q^{16} + (\zeta_{24}^{6} - \zeta_{24}^{4} + \zeta_{24}^{2} - 4 \zeta_{24} + 2) q^{17} - q^{18} + ( - \zeta_{24}^{7} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{2} - \zeta_{24} - 4) q^{19} + (2 \zeta_{24}^{7} + \zeta_{24}) q^{20} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{5} + 2 \zeta_{24}^{4} + \zeta_{24}^{3} - 2 \zeta_{24}^{2} + \zeta_{24}) q^{21} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{2} - \zeta_{24} + 2) q^{22} + ( - \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{4} + 3 \zeta_{24}^{2} - 4 \zeta_{24} + 3) q^{23} + (\zeta_{24}^{7} - \zeta_{24}^{3}) q^{24} + (3 \zeta_{24}^{6} + 4) q^{25} + ( - 3 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{3}) q^{26} - \zeta_{24}^{3} q^{27} + ( - 2 \zeta_{24}^{6} - \zeta_{24}^{4} + 2 \zeta_{24}^{3} + \zeta_{24}^{2} - 2 \zeta_{24} + 1) q^{28} + ( - 4 \zeta_{24}^{7} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{4} + \zeta_{24}^{3} - 2 \zeta_{24}^{2} - 3 \zeta_{24} - 2) q^{29} + (2 \zeta_{24}^{6} + \zeta_{24}^{4} - 2 \zeta_{24}^{2}) q^{30} + (\zeta_{24}^{6} - 2 \zeta_{24}^{5} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{2} - 2 \zeta_{24} + 1) q^{31} + (\zeta_{24}^{6} - \zeta_{24}^{2}) q^{32} + (2 \zeta_{24}^{6} - 2 \zeta_{24}^{5} + \zeta_{24}^{4} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{2} - 2) q^{33} + ( - \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{2} - 1) q^{34} + ( - \zeta_{24}^{7} + 2 \zeta_{24}^{6} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{2} + \cdots - 2) q^{35} + \cdots + ( - 2 \zeta_{24}^{6} - \zeta_{24}^{5} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{2} - \zeta_{24} - 2) q^{99} +O(q^{100})$$ q + z^2 * q^2 + z^5 * q^3 + z^4 * q^4 + (-z^5 + 2*z^3 + z) * q^5 + z^7 * q^6 + (-2*z^7 - z^6 + 2*z^5 - z^4 + 2*z^3 - z^2 - 2*z) * q^7 + z^6 * q^8 + (z^6 - z^2) * q^9 + (-z^7 + 2*z^5 + z^3) * q^10 + (z^7 - 2*z^6 + 2*z^5 + z^3 + 2*z^2 - 2) * q^11 + (z^5 - z) * q^12 + (-3*z^5 + 2*z^3 + 3*z) * q^13 + (2*z^7 - z^6 - 2*z^4 - 2*z^3 + 2*z + 1) * q^14 + (2*z^4 + z^2 - 2) * q^15 + (z^4 - 1) * q^16 + (z^6 - z^4 + z^2 - 4*z + 2) * q^17 - q^18 + (-z^7 - 2*z^6 + 2*z^4 + 2*z^3 - 2*z^2 - z - 4) * q^19 + (2*z^7 + z) * q^20 + (-2*z^7 - z^5 + 2*z^4 + z^3 - 2*z^2 + z) * q^21 + (2*z^7 + 2*z^5 - 2*z^2 - z + 2) * q^22 + (-z^6 + 2*z^5 - 2*z^4 + 3*z^2 - 4*z + 3) * q^23 + (z^7 - z^3) * q^24 + (3*z^6 + 4) * q^25 + (-3*z^7 + 2*z^5 + 3*z^3) * q^26 - z^3 * q^27 + (-2*z^6 - z^4 + 2*z^3 + z^2 - 2*z + 1) * q^28 + (-4*z^7 + 4*z^5 - 2*z^4 + z^3 - 2*z^2 - 3*z - 2) * q^29 + (2*z^6 + z^4 - 2*z^2) * q^30 + (z^6 - 2*z^5 - 3*z^4 - 3*z^2 - 2*z + 1) * q^31 + (z^6 - z^2) * q^32 + (2*z^6 - 2*z^5 + z^4 + 2*z^3 - 2*z^2 - 2) * q^33 + (-z^6 + 2*z^4 - 4*z^3 + 2*z^2 - 1) * q^34 + (-z^7 + 2*z^6 - 4*z^5 - 2*z^4 - 2*z^3 + 4*z^2 + z - 2) * q^35 - z^2 * q^36 + (2*z^7 + 4*z^6 + 2*z^5 - 2*z^4 - 3*z^3 - 2*z^2 + z + 2) * q^37 + (2*z^6 + z^5 - 4*z^4 - z^3 - 4*z^2 + z + 2) * q^38 + (2*z^4 + 3*z^2 - 2) * q^39 + (2*z^5 + z^3 - 2*z) * q^40 + (-4*z^7 - 4*z^6 + z^4 + 4*z^3 + 3*z^2 + 3) * q^41 + (-z^7 + 2*z^6 - z^5 - 2*z^4 + z^3 + 2*z) * q^42 + (z^7 + 2*z^6 + 4*z^5 + 2*z^4 - 2*z - 2) * q^43 + (2*z^7 + 2*z^5 - 2*z^4 - z^3 + 2*z^2 - 2*z) * q^44 + (z^7 - 2*z) * q^45 + (2*z^7 - 2*z^6 + 2*z^4 - 4*z^3 + 3*z^2 + 1) * q^46 + (-2*z^7 + z^6 + 2*z^3 - 2*z^2 + 2*z + 4) * q^47 - z * q^48 + (-4*z^7 - 4*z^6 - 4*z^5 + 5*z^4 + 4*z^3 + 2*z^2 - 5) * q^49 + (3*z^4 + 4*z^2 - 3) * q^50 + (2*z^7 - 4*z^6 + z^5 - z^3 + z) * q^51 + (2*z^7 + 3*z) * q^52 + (-2*z^6 - 8*z^4 + 8*z^2 + 2) * q^53 - z^5 * q^54 + (-2*z^7 + 4*z^6 + 2*z^5 + 5*z^4 - 4*z^3 + 2*z - 3) * q^55 + (-z^6 + 2*z^5 - z^4 - 2*z^3 + z^2 + 2) * q^56 + (-4*z^7 - z^6 - 2*z^5 + 2*z^4 + 2*z^3 - 2*z - 1) * q^57 + (4*z^7 - 2*z^6 - 3*z^5 - 2*z^4 - 3*z^3 - 2*z^2 + 4*z) * q^58 + (-3*z^7 + 4*z^6 - 4*z^5 + 4*z^4 - 6*z^2 + 2*z + 2) * q^59 + (z^6 - 2) * q^60 + (z^7 - 4*z^6 - 4*z^5 - 2*z^4 - 4*z^3 - 4*z^2 + z) * q^61 + (-2*z^7 - 3*z^6 - 2*z^4 - 2*z^3 + z^2 - 1) * q^62 + (-2*z^7 + 2*z^5 + z^4 + z^2 - 2*z + 1) * q^63 - q^64 + (z^6 + 8) * q^65 + (-2*z^7 + z^6 + 2*z^5 - 2*z^2 - 2) * q^66 + (3*z^7 - 3*z^5 - 3*z^3 - 4*z^2) * q^67 + (2*z^6 - 4*z^5 + z^4 - z^2 + 1) * q^68 + (2*z^7 - 2*z^6 + z^5 + z^3 - 2*z^2 + 2*z) * q^69 + (-4*z^7 - 2*z^6 - 3*z^5 + 6*z^4 + z^3 - 2*z^2 + z - 2) * q^70 + (4*z^7 - 6*z^6 - 4*z^5 - 6*z^4 + 2*z^2 + 2*z + 4) * q^71 - z^4 * q^72 + (2*z^7 + z^6 + 10*z^4 - 2*z^3 + 2*z - 5) * q^73 + (2*z^7 - 2*z^6 - z^5 + 2*z^4 + z^3 + 2*z^2 - 2*z - 4) * q^74 + (3*z^7 + 4*z^5 - 3*z^3) * q^75 + (z^7 - 4*z^6 - z^5 - 2*z^4 + z^3 + 2*z^2 - 2) * q^76 + (2*z^7 + 4*z^6 - 13*z^5 + 2*z^4 - z^3 - 2*z^2 + 13*z - 4) * q^77 + (2*z^6 + 3*z^4 - 2*z^2) * q^78 + (-2*z^7 - 3*z^6 + 2*z^5 - 8*z^4 + 4*z^3 - 4*z + 4) * q^79 + (2*z^7 + z^5 - 2*z^3) * q^80 + (-z^4 + 1) * q^81 + (z^6 - z^4 + 3*z^2 + 4*z + 4) * q^82 + (4*z^7 - 4*z^6 + 3*z^5 - 7*z^3 + 8*z^2 - 7*z - 4) * q^83 + (-z^7 - 2*z^6 + 2*z^4 + 2*z^3 + z - 2) * q^84 + (-3*z^7 + 4*z^6 + 2*z^5 - 8*z^4 + 6*z^3 - 4*z^2 - z) * q^85 + (4*z^7 + 2*z^6 + z^5 + 2*z^4 - 2*z^3 - 2*z^2 - z - 2) * q^86 + (-2*z^7 + z^6 - 4*z^5 + z^4 - 4*z^2 + 2*z + 3) * q^87 + (2*z^7 - 2*z^6 + z^5 + 2*z^4 - 2*z^3 - 2*z) * q^88 + (4*z^7 - 2*z^6 - 2*z^5 + 5*z^4 - 4*z^3 - 3*z^2 + 4*z - 3) * q^89 + (z^5 - 2*z^3 - z) * q^90 + (z^7 + 6*z^6 - 4*z^5 - 6*z^4 - 6*z^3 + 4*z^2 - z + 2) * q^91 + (2*z^6 - 2*z^5 + z^4 + z^2 - 2*z + 2) * q^92 + (-2*z^7 - 4*z^6 - 2*z^5 - z^3 + 2*z^2 + 3*z) * q^93 + (-z^4 + 2*z^3 + 4*z^2 + 2*z - 1) * q^94 + (6*z^7 + 3*z^6 - 4*z^5 - 3*z^4 - 12*z^3 + z^2 + 2*z + 2) * q^95 - z^3 * q^96 + (2*z^7 - z^6 + 4*z^5 - z^4 - 4*z^3 + z^2 - 2*z + 2) * q^97 + (-4*z^7 + 5*z^6 - 2*z^4 - 5*z^2 + 4*z + 4) * q^98 + (-2*z^6 - z^5 + 2*z^4 - 2*z^3 + 2*z^2 - z - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} - 4 q^{7}+O(q^{10})$$ 8 * q + 4 * q^4 - 4 * q^7 $$8 q + 4 q^{4} - 4 q^{7} - 16 q^{11} - 8 q^{15} - 4 q^{16} + 12 q^{17} - 8 q^{18} - 24 q^{19} + 8 q^{21} + 16 q^{22} + 16 q^{23} + 32 q^{25} + 4 q^{28} - 24 q^{29} + 4 q^{30} - 4 q^{31} - 12 q^{33} - 24 q^{35} + 8 q^{37} - 8 q^{39} + 28 q^{41} - 8 q^{42} - 8 q^{43} - 8 q^{44} + 16 q^{46} + 32 q^{47} - 20 q^{49} - 12 q^{50} - 16 q^{53} - 4 q^{55} + 12 q^{56} - 8 q^{58} + 32 q^{59} - 16 q^{60} - 8 q^{61} - 16 q^{62} + 12 q^{63} - 8 q^{64} + 64 q^{65} - 16 q^{66} + 12 q^{68} + 8 q^{70} + 8 q^{71} - 4 q^{72} - 24 q^{74} - 24 q^{76} - 24 q^{77} + 12 q^{78} + 4 q^{81} + 28 q^{82} - 32 q^{83} - 8 q^{84} - 32 q^{85} - 8 q^{86} + 28 q^{87} + 8 q^{88} - 4 q^{89} - 8 q^{91} + 20 q^{92} - 12 q^{94} + 4 q^{95} + 12 q^{97} + 24 q^{98} - 8 q^{99}+O(q^{100})$$ 8 * q + 4 * q^4 - 4 * q^7 - 16 * q^11 - 8 * q^15 - 4 * q^16 + 12 * q^17 - 8 * q^18 - 24 * q^19 + 8 * q^21 + 16 * q^22 + 16 * q^23 + 32 * q^25 + 4 * q^28 - 24 * q^29 + 4 * q^30 - 4 * q^31 - 12 * q^33 - 24 * q^35 + 8 * q^37 - 8 * q^39 + 28 * q^41 - 8 * q^42 - 8 * q^43 - 8 * q^44 + 16 * q^46 + 32 * q^47 - 20 * q^49 - 12 * q^50 - 16 * q^53 - 4 * q^55 + 12 * q^56 - 8 * q^58 + 32 * q^59 - 16 * q^60 - 8 * q^61 - 16 * q^62 + 12 * q^63 - 8 * q^64 + 64 * q^65 - 16 * q^66 + 12 * q^68 + 8 * q^70 + 8 * q^71 - 4 * q^72 - 24 * q^74 - 24 * q^76 - 24 * q^77 + 12 * q^78 + 4 * q^81 + 28 * q^82 - 32 * q^83 - 8 * q^84 - 32 * q^85 - 8 * q^86 + 28 * q^87 + 8 * q^88 - 4 * q^89 - 8 * q^91 + 20 * q^92 - 12 * q^94 + 4 * q^95 + 12 * q^97 + 24 * q^98 - 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$\zeta_{24}^{6}$$ $$\zeta_{24}^{2} - \zeta_{24}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
0.866025 + 0.500000i −0.258819 0.965926i 0.500000 + 0.866025i −2.12132 0.707107i 0.258819 0.965926i −1.88366 3.26260i 1.00000i −0.866025 + 0.500000i −1.48356 1.67303i
7.2 0.866025 + 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i 2.12132 + 0.707107i −0.258819 + 0.965926i −0.848387 1.46945i 1.00000i −0.866025 + 0.500000i 1.48356 + 1.67303i
37.1 −0.866025 + 0.500000i −0.965926 0.258819i 0.500000 0.866025i 2.12132 + 0.707107i 0.965926 0.258819i −1.56583 + 2.71209i 1.00000i 0.866025 + 0.500000i −2.19067 + 0.448288i
37.2 −0.866025 + 0.500000i 0.965926 + 0.258819i 0.500000 0.866025i −2.12132 0.707107i −0.965926 + 0.258819i 2.29788 3.98004i 1.00000i 0.866025 + 0.500000i 2.19067 0.448288i
223.1 0.866025 0.500000i −0.258819 + 0.965926i 0.500000 0.866025i −2.12132 + 0.707107i 0.258819 + 0.965926i −1.88366 + 3.26260i 1.00000i −0.866025 0.500000i −1.48356 + 1.67303i
223.2 0.866025 0.500000i 0.258819 0.965926i 0.500000 0.866025i 2.12132 0.707107i −0.258819 0.965926i −0.848387 + 1.46945i 1.00000i −0.866025 0.500000i 1.48356 1.67303i
253.1 −0.866025 0.500000i −0.965926 + 0.258819i 0.500000 + 0.866025i 2.12132 0.707107i 0.965926 + 0.258819i −1.56583 2.71209i 1.00000i 0.866025 0.500000i −2.19067 0.448288i
253.2 −0.866025 0.500000i 0.965926 0.258819i 0.500000 + 0.866025i −2.12132 + 0.707107i −0.965926 0.258819i 2.29788 + 3.98004i 1.00000i 0.866025 0.500000i 2.19067 + 0.448288i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bd.a 8
5.c odd 4 1 390.2.bn.a yes 8
13.f odd 12 1 390.2.bn.a yes 8
65.t even 12 1 inner 390.2.bd.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bd.a 8 1.a even 1 1 trivial
390.2.bd.a 8 65.t even 12 1 inner
390.2.bn.a yes 8 5.c odd 4 1
390.2.bn.a yes 8 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 4T_{7}^{7} + 32T_{7}^{6} + 112T_{7}^{5} + 700T_{7}^{4} + 2144T_{7}^{3} + 6272T_{7}^{2} + 8096T_{7} + 8464$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$T^{8} - T^{4} + 1$$
$5$ $$(T^{4} - 8 T^{2} + 25)^{2}$$
$7$ $$T^{8} + 4 T^{7} + 32 T^{6} + \cdots + 8464$$
$11$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 36481$$
$13$ $$(T^{4} - 24 T^{2} + 169)^{2}$$
$17$ $$T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 10000$$
$19$ $$T^{8} + 24 T^{7} + 300 T^{6} + \cdots + 150544$$
$23$ $$T^{8} - 16 T^{7} + 98 T^{6} + \cdots + 5041$$
$29$ $$T^{8} + 24 T^{7} + 204 T^{6} + \cdots + 2211169$$
$31$ $$T^{8} + 4 T^{7} + 8 T^{6} - 92 T^{5} + \cdots + 529$$
$37$ $$T^{8} - 8 T^{7} + 92 T^{6} + 112 T^{5} + \cdots + 529$$
$41$ $$T^{8} - 28 T^{7} + 344 T^{6} + \cdots + 8464$$
$43$ $$T^{8} + 8 T^{7} + 80 T^{6} + 216 T^{5} + \cdots + 625$$
$47$ $$(T^{4} - 16 T^{3} + 74 T^{2} - 32 T - 311)^{2}$$
$53$ $$(T^{4} + 8 T^{3} + 32 T^{2} - 704 T + 7744)^{2}$$
$59$ $$T^{8} - 32 T^{7} + 464 T^{6} + \cdots + 3411409$$
$61$ $$T^{8} + 8 T^{7} + 188 T^{6} + \cdots + 234256$$
$67$ $$T^{8} - 68 T^{6} + 4620 T^{4} + \cdots + 16$$
$71$ $$T^{8} - 8 T^{7} + 128 T^{6} + \cdots + 37552384$$
$73$ $$T^{8} + 336 T^{6} + \cdots + 16289296$$
$79$ $$T^{8} + 324 T^{6} + 34470 T^{4} + \cdots + 7834401$$
$83$ $$(T^{4} + 16 T^{3} - 148 T^{2} + \cdots - 15452)^{2}$$
$89$ $$T^{8} + 4 T^{7} + 200 T^{6} + \cdots + 10000$$
$97$ $$T^{8} - 12 T^{7} + 16 T^{6} + \cdots + 150544$$