# Properties

 Label 130.2.l.a Level $130$ Weight $2$ Character orbit 130.l Analytic conductor $1.038$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 130.l (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.03805522628$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/130\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i −0.366025 0.633975i 0.500000 0.866025i 1.00000i 0.633975 + 0.366025i 2.59808 + 1.50000i 1.00000i 1.23205 2.13397i −0.500000 0.866025i
101.2 0.866025 0.500000i 1.36603 + 2.36603i 0.500000 0.866025i 1.00000i 2.36603 + 1.36603i −2.59808 1.50000i 1.00000i −2.23205 + 3.86603i −0.500000 0.866025i
121.1 −0.866025 0.500000i −0.366025 + 0.633975i 0.500000 + 0.866025i 1.00000i 0.633975 0.366025i 2.59808 1.50000i 1.00000i 1.23205 + 2.13397i −0.500000 + 0.866025i
121.2 0.866025 + 0.500000i 1.36603 2.36603i 0.500000 + 0.866025i 1.00000i 2.36603 1.36603i −2.59808 + 1.50000i 1.00000i −2.23205 3.86603i −0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.l.a 4
3.b odd 2 1 1170.2.bs.c 4
4.b odd 2 1 1040.2.da.a 4
5.b even 2 1 650.2.m.a 4
5.c odd 4 1 650.2.n.a 4
5.c odd 4 1 650.2.n.b 4
13.b even 2 1 1690.2.l.g 4
13.c even 3 1 1690.2.d.f 4
13.c even 3 1 1690.2.l.g 4
13.d odd 4 1 1690.2.e.l 4
13.d odd 4 1 1690.2.e.n 4
13.e even 6 1 inner 130.2.l.a 4
13.e even 6 1 1690.2.d.f 4
13.f odd 12 1 1690.2.a.j 2
13.f odd 12 1 1690.2.a.m 2
13.f odd 12 1 1690.2.e.l 4
13.f odd 12 1 1690.2.e.n 4
39.h odd 6 1 1170.2.bs.c 4
52.i odd 6 1 1040.2.da.a 4
65.l even 6 1 650.2.m.a 4
65.r odd 12 1 650.2.n.a 4
65.r odd 12 1 650.2.n.b 4
65.s odd 12 1 8450.2.a.bf 2
65.s odd 12 1 8450.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 1.a even 1 1 trivial
130.2.l.a 4 13.e even 6 1 inner
650.2.m.a 4 5.b even 2 1
650.2.m.a 4 65.l even 6 1
650.2.n.a 4 5.c odd 4 1
650.2.n.a 4 65.r odd 12 1
650.2.n.b 4 5.c odd 4 1
650.2.n.b 4 65.r odd 12 1
1040.2.da.a 4 4.b odd 2 1
1040.2.da.a 4 52.i odd 6 1
1170.2.bs.c 4 3.b odd 2 1
1170.2.bs.c 4 39.h odd 6 1
1690.2.a.j 2 13.f odd 12 1
1690.2.a.m 2 13.f odd 12 1
1690.2.d.f 4 13.c even 3 1
1690.2.d.f 4 13.e even 6 1
1690.2.e.l 4 13.d odd 4 1
1690.2.e.l 4 13.f odd 12 1
1690.2.e.n 4 13.d odd 4 1
1690.2.e.n 4 13.f odd 12 1
1690.2.l.g 4 13.b even 2 1
1690.2.l.g 4 13.c even 3 1
8450.2.a.bf 2 65.s odd 12 1
8450.2.a.bm 2 65.s odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 2T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(130, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} - 9T^{2} + 81$$
$11$ $$T^{4} - 9T^{2} + 81$$
$13$ $$(T^{2} - 7 T + 13)^{2}$$
$17$ $$T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324$$
$19$ $$T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9$$
$23$ $$T^{4} + 12 T^{3} + 120 T^{2} + \cdots + 576$$
$29$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$31$ $$T^{4} + 24T^{2} + 36$$
$37$ $$T^{4} - 18 T^{3} + 99 T^{2} + 162 T + 81$$
$41$ $$(T^{2} + 18 T + 108)^{2}$$
$43$ $$(T^{2} + 2 T + 4)^{2}$$
$47$ $$(T^{2} + 9)^{2}$$
$53$ $$(T^{2} + 6 T - 3)^{2}$$
$59$ $$(T^{2} - 18 T + 108)^{2}$$
$61$ $$T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676$$
$67$ $$T^{4}$$
$71$ $$T^{4} - 36T^{2} + 1296$$
$73$ $$T^{4} + 168T^{2} + 4356$$
$79$ $$(T^{2} - 2 T - 26)^{2}$$
$83$ $$T^{4} + 72T^{2} + 324$$
$89$ $$T^{4} + 18 T^{3} - 9 T^{2} + \cdots + 13689$$
$97$ $$T^{4} - 42 T^{3} + 726 T^{2} + \cdots + 19044$$