Properties

 Label 841.2.d.j Level $841$ Weight $2$ Character orbit 841.d Analytic conductor $6.715$ Analytic rank $0$ Dimension $12$ CM no Inner twists $6$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$841 = 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 841.d (of order $$7$$, degree $$6$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$6.71541880999$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{7})$$ Coefficient field: 12.0.74049191673856.2 Defining polynomial: $$x^{12} + 2x^{10} + 4x^{8} + 8x^{6} + 16x^{4} + 32x^{2} + 64$$ x^12 + 2*x^10 + 4*x^8 + 8*x^6 + 16*x^4 + 32*x^2 + 64 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 29) Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{9} - \beta_{2}) q^{2} + (\beta_{10} - \beta_{3}) q^{3} + ( - 2 \beta_{11} + \beta_{4}) q^{4} - \beta_{2} q^{5} + (3 \beta_{10} + 3 \beta_{8} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 3 \beta_{2} + \cdots + 3) q^{6}+ \cdots + (2 \beta_{11} + 2 \beta_{9} + 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{9}+O(q^{10})$$ q + (b9 - b2) * q^2 + (b10 - b3) * q^3 + (-2*b11 + b4) * q^4 - b2 * q^5 + (3*b10 + 3*b8 + 3*b6 + 2*b5 + 3*b4 + 3*b2 + 3) * q^6 + 2*b3 * q^7 + (-b11 - b9 - b7 - 3*b6 - b5 - b3 - b1) * q^8 + (2*b11 + 2*b9 + 2*b7 + 2*b5 + 2*b3 + 2*b1) * q^9 $$q + (\beta_{9} - \beta_{2}) q^{2} + (\beta_{10} - \beta_{3}) q^{3} + ( - 2 \beta_{11} + \beta_{4}) q^{4} - \beta_{2} q^{5} + (3 \beta_{10} + 3 \beta_{8} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 3 \beta_{2} + \cdots + 3) q^{6}+ \cdots + ( - 2 \beta_{7} - 4) q^{99}+O(q^{100})$$ q + (b9 - b2) * q^2 + (b10 - b3) * q^3 + (-2*b11 + b4) * q^4 - b2 * q^5 + (3*b10 + 3*b8 + 3*b6 + 2*b5 + 3*b4 + 3*b2 + 3) * q^6 + 2*b3 * q^7 + (-b11 - b9 - b7 - 3*b6 - b5 - b3 - b1) * q^8 + (2*b11 + 2*b9 + 2*b7 + 2*b5 + 2*b3 + 2*b1) * q^9 + (-b11 + b4) * q^10 + (b8 + b1) * q^11 + (-3*b7 + 5) * q^12 + (-b8 + 2*b1) * q^13 + (-4*b10 - 4*b8 - 4*b6 - 2*b5 - 4*b4 - 4*b2 - 4) * q^14 + (b10 + b8 + b6 + b5 + b4 + b2 + 1) * q^15 + 3*b8 * q^16 + (-2*b7 - 2) * q^17 + (-4*b8 + 2*b1) * q^18 + 6*b4 * q^19 + (-2*b11 - 2*b9 - 2*b7 - b6 - 2*b5 - 2*b3 - 2*b1) * q^20 + (-2*b11 - 2*b9 - 2*b7 - 4*b6 - 2*b5 - 2*b3 - 2*b1) * q^21 + b10 * q^22 + (2*b10 + 2*b8 + 2*b6 - 4*b5 + 2*b4 + 2*b2 + 2) * q^23 + (4*b9 - 5*b2) * q^24 - 4*b4 * q^25 + (5*b10 - 3*b3) * q^26 + (b9 + b2) * q^27 + (2*b7 - 8) * q^28 + (-2*b7 + 3) * q^30 + (-5*b9 + 3*b2) * q^31 + (3*b10 + b3) * q^32 - b4 * q^33 - 2*b2 * q^34 - 2*b5 * q^35 + (8*b10 - 2*b3) * q^36 - 4*b6 * q^37 + (-6*b11 - 6*b9 - 6*b7 - 6*b6 - 6*b5 - 6*b3 - 6*b1) * q^38 + (3*b11 - 5*b4) * q^39 + (3*b8 - b1) * q^40 + (6*b7 + 4) * q^41 + (8*b8 - 6*b1) * q^42 + (-5*b10 - 5*b8 - 5*b6 + b5 - 5*b4 - 5*b2 - 5) * q^43 + (3*b10 + 3*b8 + 3*b6 - b5 + 3*b4 + 3*b2 + 3) * q^44 + 2*b1 * q^45 + (2*b7 - 6) * q^46 + (b8 + 3*b1) * q^47 + (-3*b11 + 3*b4) * q^48 + b6 * q^49 + (4*b11 + 4*b9 + 4*b7 + 4*b6 + 4*b5 + 4*b3 + 4*b1) * q^50 + 2*b10 * q^51 + (9*b10 + 9*b8 + 9*b6 + 4*b5 + 9*b4 + 9*b2 + 9) * q^52 + (-6*b9 + b2) * q^53 + b4 * q^54 + (-b10 - b3) * q^55 + (-6*b9 + 4*b2) * q^56 + (-6*b7 + 6) * q^57 + (4*b7 + 2) * q^59 + (3*b9 - 5*b2) * q^60 + (-2*b10 + 2*b3) * q^61 + (8*b11 - 13*b4) * q^62 - 8*b2 * q^63 + (7*b10 + 7*b8 + 7*b6 + 2*b5 + 7*b4 + 7*b2 + 7) * q^64 + (b10 - 2*b3) * q^65 + (b11 + b9 + b7 + b6 + b5 + b3 + b1) * q^66 + (4*b11 + 4*b9 + 4*b7 + 4*b5 + 4*b3 + 4*b1) * q^67 + (2*b11 + 6*b4) * q^68 + (6*b8 - 2*b1) * q^69 + (2*b7 - 4) * q^70 + (-6*b8 + 2*b1) * q^71 + (4*b10 + 4*b8 + 4*b6 + 6*b5 + 4*b4 + 4*b2 + 4) * q^72 + (-4*b10 - 4*b8 - 4*b6 - 4*b4 - 4*b2 - 4) * q^73 + (4*b8 - 4*b1) * q^74 + (4*b7 - 4) * q^75 + (6*b8 - 12*b1) * q^76 + (2*b11 + 4*b4) * q^77 + (8*b11 + 8*b9 + 8*b7 + 11*b6 + 8*b5 + 8*b3 + 8*b1) * q^78 + (-b11 - b9 - b7 - b6 - b5 - b3 - b1) * q^79 - 3*b10 * q^80 + (b10 + b8 + b6 + 6*b5 + b4 + b2 + 1) * q^81 + (-2*b9 + 8*b2) * q^82 + (-4*b11 + 2*b4) * q^83 + (-12*b10 + 10*b3) * q^84 + (2*b9 + 2*b2) * q^85 + (4*b7 - 3) * q^86 + (-2*b7 - 1) * q^88 + (6*b9 - 4*b2) * q^89 + (4*b10 - 2*b3) * q^90 + (-2*b11 + 8*b4) * q^91 + 14*b2 * q^92 + (-13*b10 - 13*b8 - 13*b6 - 8*b5 - 13*b4 - 13*b2 - 13) * q^93 + (5*b10 - 2*b3) * q^94 - 6*b6 * q^95 + (2*b11 + 2*b9 + 2*b7 + b6 + 2*b5 + 2*b3 + 2*b1) * q^96 + (-6*b11 - 4*b4) * q^97 + (-b8 + b1) * q^98 + (-2*b7 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 6 q^{6} + 6 q^{8}+O(q^{10})$$ 12 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^5 + 6 * q^6 + 6 * q^8 $$12 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 6 q^{6} + 6 q^{8} - 2 q^{10} - 2 q^{11} + 60 q^{12} + 2 q^{13} - 8 q^{14} + 2 q^{15} - 6 q^{16} - 24 q^{17} + 8 q^{18} - 12 q^{19} + 2 q^{20} + 8 q^{21} - 2 q^{22} + 4 q^{23} + 10 q^{24} + 8 q^{25} - 10 q^{26} - 2 q^{27} - 96 q^{28} + 36 q^{30} - 6 q^{31} - 6 q^{32} + 2 q^{33} + 4 q^{34} - 16 q^{36} + 8 q^{37} + 12 q^{38} + 10 q^{39} - 6 q^{40} + 48 q^{41} - 16 q^{42} - 10 q^{43} + 6 q^{44} - 72 q^{46} - 2 q^{47} - 6 q^{48} - 2 q^{49} - 8 q^{50} - 4 q^{51} + 18 q^{52} - 2 q^{53} - 2 q^{54} + 2 q^{55} - 8 q^{56} + 72 q^{57} + 24 q^{59} + 10 q^{60} + 4 q^{61} + 26 q^{62} + 16 q^{63} + 14 q^{64} - 2 q^{65} - 2 q^{66} - 12 q^{68} - 12 q^{69} - 48 q^{70} + 12 q^{71} + 8 q^{72} - 8 q^{73} - 8 q^{74} - 48 q^{75} - 12 q^{76} - 8 q^{77} - 22 q^{78} + 2 q^{79} + 6 q^{80} + 2 q^{81} - 16 q^{82} - 4 q^{83} + 24 q^{84} - 4 q^{85} - 36 q^{86} - 12 q^{88} + 8 q^{89} - 8 q^{90} - 16 q^{91} - 28 q^{92} - 26 q^{93} - 10 q^{94} + 12 q^{95} - 2 q^{96} + 8 q^{97} + 2 q^{98} - 48 q^{99}+O(q^{100})$$ 12 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^5 + 6 * q^6 + 6 * q^8 - 2 * q^10 - 2 * q^11 + 60 * q^12 + 2 * q^13 - 8 * q^14 + 2 * q^15 - 6 * q^16 - 24 * q^17 + 8 * q^18 - 12 * q^19 + 2 * q^20 + 8 * q^21 - 2 * q^22 + 4 * q^23 + 10 * q^24 + 8 * q^25 - 10 * q^26 - 2 * q^27 - 96 * q^28 + 36 * q^30 - 6 * q^31 - 6 * q^32 + 2 * q^33 + 4 * q^34 - 16 * q^36 + 8 * q^37 + 12 * q^38 + 10 * q^39 - 6 * q^40 + 48 * q^41 - 16 * q^42 - 10 * q^43 + 6 * q^44 - 72 * q^46 - 2 * q^47 - 6 * q^48 - 2 * q^49 - 8 * q^50 - 4 * q^51 + 18 * q^52 - 2 * q^53 - 2 * q^54 + 2 * q^55 - 8 * q^56 + 72 * q^57 + 24 * q^59 + 10 * q^60 + 4 * q^61 + 26 * q^62 + 16 * q^63 + 14 * q^64 - 2 * q^65 - 2 * q^66 - 12 * q^68 - 12 * q^69 - 48 * q^70 + 12 * q^71 + 8 * q^72 - 8 * q^73 - 8 * q^74 - 48 * q^75 - 12 * q^76 - 8 * q^77 - 22 * q^78 + 2 * q^79 + 6 * q^80 + 2 * q^81 - 16 * q^82 - 4 * q^83 + 24 * q^84 - 4 * q^85 - 36 * q^86 - 12 * q^88 + 8 * q^89 - 8 * q^90 - 16 * q^91 - 28 * q^92 - 26 * q^93 - 10 * q^94 + 12 * q^95 - 2 * q^96 + 8 * q^97 + 2 * q^98 - 48 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 2x^{10} + 4x^{8} + 8x^{6} + 16x^{4} + 32x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} ) / 4$$ (v^4) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{5} ) / 4$$ (v^5) / 4 $$\beta_{6}$$ $$=$$ $$( \nu^{6} ) / 8$$ (v^6) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{7} ) / 8$$ (v^7) / 8 $$\beta_{8}$$ $$=$$ $$( \nu^{8} ) / 16$$ (v^8) / 16 $$\beta_{9}$$ $$=$$ $$( \nu^{9} ) / 16$$ (v^9) / 16 $$\beta_{10}$$ $$=$$ $$( \nu^{10} ) / 32$$ (v^10) / 32 $$\beta_{11}$$ $$=$$ $$( \nu^{11} ) / 32$$ (v^11) / 32
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3 $$\nu^{4}$$ $$=$$ $$4\beta_{4}$$ 4*b4 $$\nu^{5}$$ $$=$$ $$4\beta_{5}$$ 4*b5 $$\nu^{6}$$ $$=$$ $$8\beta_{6}$$ 8*b6 $$\nu^{7}$$ $$=$$ $$8\beta_{7}$$ 8*b7 $$\nu^{8}$$ $$=$$ $$16\beta_{8}$$ 16*b8 $$\nu^{9}$$ $$=$$ $$16\beta_{9}$$ 16*b9 $$\nu^{10}$$ $$=$$ $$32\beta_{10}$$ 32*b10 $$\nu^{11}$$ $$=$$ $$32\beta_{11}$$ 32*b11

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/841\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{4}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
190.1
 −0.314692 + 1.37876i 0.314692 − 1.37876i −0.314692 − 1.37876i 0.314692 + 1.37876i 1.27416 − 0.613604i −1.27416 + 0.613604i 0.881748 + 1.10568i −0.881748 − 1.10568i 0.881748 − 1.10568i −0.881748 + 1.10568i 1.27416 + 0.613604i −1.27416 − 0.613604i
−0.373194 0.179721i −0.258258 + 0.323845i −1.14001 1.42952i 0.900969 + 0.433884i 0.154582 0.0744427i 1.76350 2.21135i 0.352871 + 1.54603i 0.629384 + 2.75751i −0.258258 0.323845i
190.2 2.17513 + 1.04749i 1.50524 1.88751i 2.38699 + 2.99318i 0.900969 + 0.433884i 5.25123 2.52886i −1.76350 + 2.21135i 0.982255 + 4.30354i −0.629384 2.75751i 1.50524 + 1.88751i
571.1 −0.373194 + 0.179721i −0.258258 0.323845i −1.14001 + 1.42952i 0.900969 0.433884i 0.154582 + 0.0744427i 1.76350 + 2.21135i 0.352871 1.54603i 0.629384 2.75751i −0.258258 + 0.323845i
571.2 2.17513 1.04749i 1.50524 + 1.88751i 2.38699 2.99318i 0.900969 0.433884i 5.25123 + 2.52886i −1.76350 2.21135i 0.982255 4.30354i −0.629384 + 2.75751i 1.50524 1.88751i
574.1 −1.50524 + 1.88751i −0.537213 + 2.35368i −0.851905 3.73244i −0.623490 + 0.781831i −3.63396 4.55685i 0.629384 2.75751i 3.97707 + 1.91526i −2.54832 1.22721i −0.537213 2.35368i
574.2 0.258258 0.323845i 0.0921712 0.403828i 0.406863 + 1.78258i −0.623490 + 0.781831i −0.106974 0.134141i −0.629384 + 2.75751i 1.42874 + 0.688047i 2.54832 + 1.22721i 0.0921712 + 0.403828i
605.1 −0.0921712 + 0.403828i 0.373194 0.179721i 1.64736 + 0.793325i 0.222521 0.974928i 0.0381786 + 0.167271i −2.54832 + 1.22721i −0.988722 + 1.23982i −1.76350 + 2.21135i 0.373194 + 0.179721i
605.2 0.537213 2.35368i −2.17513 + 1.04749i −3.44929 1.66109i 0.222521 0.974928i 1.29695 + 5.68230i 2.54832 1.22721i −2.75222 + 3.45117i 1.76350 2.21135i −2.17513 1.04749i
645.1 −0.0921712 0.403828i 0.373194 + 0.179721i 1.64736 0.793325i 0.222521 + 0.974928i 0.0381786 0.167271i −2.54832 1.22721i −0.988722 1.23982i −1.76350 2.21135i 0.373194 0.179721i
645.2 0.537213 + 2.35368i −2.17513 1.04749i −3.44929 + 1.66109i 0.222521 + 0.974928i 1.29695 5.68230i 2.54832 + 1.22721i −2.75222 3.45117i 1.76350 + 2.21135i −2.17513 + 1.04749i
778.1 −1.50524 1.88751i −0.537213 2.35368i −0.851905 + 3.73244i −0.623490 0.781831i −3.63396 + 4.55685i 0.629384 + 2.75751i 3.97707 1.91526i −2.54832 + 1.22721i −0.537213 + 2.35368i
778.2 0.258258 + 0.323845i 0.0921712 + 0.403828i 0.406863 1.78258i −0.623490 0.781831i −0.106974 + 0.134141i −0.629384 2.75751i 1.42874 0.688047i 2.54832 1.22721i 0.0921712 0.403828i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 778.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 5 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.d.j 12
29.b even 2 1 841.2.d.f 12
29.c odd 4 2 841.2.e.k 24
29.d even 7 1 29.2.a.a 2
29.d even 7 5 inner 841.2.d.j 12
29.e even 14 1 841.2.a.d 2
29.e even 14 5 841.2.d.f 12
29.f odd 28 2 841.2.b.a 4
29.f odd 28 10 841.2.e.k 24
87.h odd 14 1 7569.2.a.c 2
87.j odd 14 1 261.2.a.d 2
116.j odd 14 1 464.2.a.h 2
145.n even 14 1 725.2.a.b 2
145.p odd 28 2 725.2.b.b 4
203.n odd 14 1 1421.2.a.j 2
232.p odd 14 1 1856.2.a.w 2
232.s even 14 1 1856.2.a.r 2
319.m odd 14 1 3509.2.a.j 2
348.s even 14 1 4176.2.a.bq 2
377.w even 14 1 4901.2.a.g 2
435.w odd 14 1 6525.2.a.o 2
493.p even 14 1 8381.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 29.d even 7 1
261.2.a.d 2 87.j odd 14 1
464.2.a.h 2 116.j odd 14 1
725.2.a.b 2 145.n even 14 1
725.2.b.b 4 145.p odd 28 2
841.2.a.d 2 29.e even 14 1
841.2.b.a 4 29.f odd 28 2
841.2.d.f 12 29.b even 2 1
841.2.d.f 12 29.e even 14 5
841.2.d.j 12 1.a even 1 1 trivial
841.2.d.j 12 29.d even 7 5 inner
841.2.e.k 24 29.c odd 4 2
841.2.e.k 24 29.f odd 28 10
1421.2.a.j 2 203.n odd 14 1
1856.2.a.r 2 232.s even 14 1
1856.2.a.w 2 232.p odd 14 1
3509.2.a.j 2 319.m odd 14 1
4176.2.a.bq 2 348.s even 14 1
4901.2.a.g 2 377.w even 14 1
6525.2.a.o 2 435.w odd 14 1
7569.2.a.c 2 87.h odd 14 1
8381.2.a.e 2 493.p even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 2 T_{2}^{11} + 5 T_{2}^{10} - 12 T_{2}^{9} + 29 T_{2}^{8} - 70 T_{2}^{7} + 169 T_{2}^{6} + 70 T_{2}^{5} + 29 T_{2}^{4} + 12 T_{2}^{3} + 5 T_{2}^{2} + 2 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(841, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 2 T^{11} + 5 T^{10} - 12 T^{9} + \cdots + 1$$
$3$ $$T^{12} + 2 T^{11} + 5 T^{10} + 12 T^{9} + \cdots + 1$$
$5$ $$(T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$7$ $$T^{12} + 8 T^{10} + 64 T^{8} + \cdots + 262144$$
$11$ $$T^{12} + 2 T^{11} + 5 T^{10} + 12 T^{9} + \cdots + 1$$
$13$ $$T^{12} - 2 T^{11} + 11 T^{10} + \cdots + 117649$$
$17$ $$(T^{2} + 4 T - 4)^{6}$$
$19$ $$(T^{6} + 6 T^{5} + 36 T^{4} + 216 T^{3} + \cdots + 46656)^{2}$$
$23$ $$T^{12} - 4 T^{11} + 44 T^{10} + \cdots + 481890304$$
$29$ $$T^{12}$$
$31$ $$T^{12} + 6 T^{11} + \cdots + 4750104241$$
$37$ $$(T^{6} - 4 T^{5} + 16 T^{4} - 64 T^{3} + \cdots + 4096)^{2}$$
$41$ $$(T^{2} - 8 T - 56)^{6}$$
$43$ $$T^{12} + 10 T^{11} + \cdots + 148035889$$
$47$ $$T^{12} + 2 T^{11} + 21 T^{10} + \cdots + 24137569$$
$53$ $$T^{12} + 2 T^{11} + \cdots + 128100283921$$
$59$ $$(T^{2} - 4 T - 28)^{6}$$
$61$ $$T^{12} - 4 T^{11} + 20 T^{10} + \cdots + 4096$$
$67$ $$T^{12} + 32 T^{10} + \cdots + 1073741824$$
$71$ $$T^{12} - 12 T^{11} + \cdots + 481890304$$
$73$ $$(T^{6} + 4 T^{5} + 16 T^{4} + 64 T^{3} + \cdots + 4096)^{2}$$
$79$ $$T^{12} - 2 T^{11} + 5 T^{10} - 12 T^{9} + \cdots + 1$$
$83$ $$T^{12} + 4 T^{11} + 44 T^{10} + \cdots + 481890304$$
$89$ $$T^{12} - 8 T^{11} + \cdots + 30840979456$$
$97$ $$T^{12} - 8 T^{11} + \cdots + 30840979456$$