Properties

Label 845.2.l.e
Level $845$
Weight $2$
Character orbit 845.l
Analytic conductor $6.747$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.89539436150784.1
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{7} - \beta_1 + 1) q^{2} + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{9} + 2 \beta_{7}) q^{4} + (\beta_{8} + \beta_{2}) q^{5} + ( - \beta_{6} - \beta_{4} - \beta_{3}) q^{6} + (\beta_{10} - \beta_{9} + \beta_{7}) q^{7} + ( - 3 \beta_{2} - 4) q^{8} + (\beta_{10} - 3 \beta_{9}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_{7} - \beta_1 + 1) q^{2} + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{9} + 2 \beta_{7}) q^{4} + (\beta_{8} + \beta_{2}) q^{5} + ( - \beta_{6} - \beta_{4} - \beta_{3}) q^{6} + (\beta_{10} - \beta_{9} + \beta_{7}) q^{7} + ( - 3 \beta_{2} - 4) q^{8} + (\beta_{10} - 3 \beta_{9}) q^{9} + (2 \beta_{11} - 2 \beta_{9} + \beta_{7} + \beta_{5} + 2 \beta_{2} + 1) q^{10} + ( - \beta_{11} - \beta_{8} - \beta_{5} + \beta_{4}) q^{11} + (\beta_{11} - \beta_{6}) q^{12} + ( - \beta_{2} + \beta_1 + 1) q^{14} + (\beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{15} + (2 \beta_{10} + 4 \beta_{9} - 3 \beta_{7} - 4 \beta_{2} + 2 \beta_1 - 3) q^{16} + ( - \beta_{6} - 2 \beta_{4} + \beta_{3}) q^{17} - 5 \beta_{2} q^{18} - \beta_{4} q^{19} + ( - \beta_{10} - 2 \beta_{9} + 3 \beta_{7} + 3 \beta_{6} - \beta_{4} - \beta_{3}) q^{20} + ( - \beta_{11} + 2 \beta_{8} + \beta_{6} + \beta_{5} + \beta_{3}) q^{21} + ( - 3 \beta_{6} + 2 \beta_{4} + 2 \beta_{3}) q^{22} + ( - 3 \beta_{11} - 2 \beta_{5}) q^{23} + (3 \beta_{8} + 2 \beta_{5} - 3 \beta_{4}) q^{24} + (2 \beta_{11} - 2 \beta_{6} - 2 \beta_{2} - 2 \beta_1 - 1) q^{25} + ( - \beta_{11} + 4 \beta_{8} + \beta_{6} + 3 \beta_{5} + 3 \beta_{3}) q^{27} + (\beta_{10} + \beta_{9} - \beta_{7} - \beta_{2} + \beta_1 - 1) q^{28} + (3 \beta_{10} - 3 \beta_{9} + 3 \beta_{7} + 3 \beta_{2} + 3 \beta_1 + 3) q^{29} + ( - \beta_{10} - \beta_{9} - \beta_{7} + 2 \beta_{4} + 3 \beta_{3}) q^{30} + ( - 3 \beta_{11} + \beta_{8} + 3 \beta_{6} - \beta_{5} - \beta_{3}) q^{31} + (3 \beta_{10} + 4 \beta_{9} - 5 \beta_{7}) q^{32} + ( - \beta_{10} + 3 \beta_{9} + \beta_{7}) q^{33} + ( - 3 \beta_{11} + 2 \beta_{8} + 3 \beta_{6} - \beta_{5} - \beta_{3}) q^{34} + ( - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} - 2 \beta_{4} + \beta_{3}) q^{35} + (2 \beta_{10} + 4 \beta_{9} - 5 \beta_{7} - 4 \beta_{2} + 2 \beta_1 - 5) q^{36} + ( - \beta_{10} + 3 \beta_{9} - \beta_{7} - 3 \beta_{2} - \beta_1 - 1) q^{37} + ( - 2 \beta_{11} + 2 \beta_{6} - \beta_{5} - \beta_{3}) q^{38} + ( - 3 \beta_{11} - 4 \beta_{8} + 3 \beta_{6} - \beta_{2} + 3 \beta_1 - 6) q^{40} + (2 \beta_{11} + 2 \beta_{8} - 2 \beta_{4}) q^{41} + (\beta_{11} + 3 \beta_{5}) q^{42} + ( - \beta_{6} + 2 \beta_{4} - 2 \beta_{3}) q^{43} + (6 \beta_{11} + 3 \beta_{8} - 6 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{44} + ( - 3 \beta_{10} + 4 \beta_{9} + 5 \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3}) q^{45} + ( - 4 \beta_{6} + 5 \beta_{4} + 2 \beta_{3}) q^{46} + ( - \beta_{2} - \beta_1 + 3) q^{47} + (2 \beta_{6} - 2 \beta_{4} - 5 \beta_{3}) q^{48} + ( - 2 \beta_{10} + 3 \beta_{7} - 2 \beta_1 + 3) q^{49} + (4 \beta_{11} + \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} - 2 \beta_{4} - 2 \beta_{2} + \cdots + 3) q^{50}+ \cdots + (4 \beta_{11} - \beta_{8} - 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} - 10 q^{4} - 4 q^{5} - 4 q^{7} - 36 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} - 10 q^{4} - 4 q^{5} - 4 q^{7} - 36 q^{8} + 6 q^{9} + 2 q^{10} + 16 q^{14} + 4 q^{15} - 10 q^{16} + 20 q^{18} - 14 q^{20} - 4 q^{25} - 4 q^{28} + 12 q^{29} + 8 q^{30} + 22 q^{32} - 12 q^{33} - 8 q^{35} - 22 q^{36} - 68 q^{40} - 38 q^{45} + 40 q^{47} + 18 q^{49} + 22 q^{50} - 48 q^{51} + 16 q^{55} + 40 q^{56} - 8 q^{57} + 24 q^{58} - 8 q^{60} - 12 q^{61} - 36 q^{63} + 132 q^{64} - 32 q^{66} + 20 q^{67} - 24 q^{69} - 40 q^{70} + 90 q^{72} + 96 q^{73} - 4 q^{74} + 16 q^{75} + 64 q^{79} - 58 q^{80} - 46 q^{81} + 88 q^{83} + 32 q^{85} - 140 q^{90} + 4 q^{93} + 32 q^{94} + 16 q^{95} - 28 q^{97} - 50 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - \nu^{10} + 4 \nu^{9} - 28 \nu^{8} + 18 \nu^{7} - 22 \nu^{6} + 94 \nu^{5} + 146 \nu^{4} - 144 \nu^{3} + 48 \nu^{2} - 48 \nu - 748 ) / 460 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} - 454 \nu^{4} + 444 \nu^{3} - 148 \nu^{2} + 148 \nu + 612 ) / 460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10 \nu^{11} + 32 \nu^{10} - 32 \nu^{9} + 82 \nu^{8} - 123 \nu^{7} - 197 \nu^{6} + 322 \nu^{5} - 84 \nu^{4} + 34 \nu^{3} + 306 \nu^{2} - 10 \nu + 10 ) / 230 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + 110 \nu^{4} - 50 \nu^{3} - 1140 \nu^{2} + 12 \nu - 12 ) / 230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 39 \nu^{11} - 83 \nu^{10} + 94 \nu^{9} - 328 \nu^{8} + 220 \nu^{7} + 482 \nu^{6} - 330 \nu^{5} + 734 \nu^{4} + 1060 \nu^{3} - 436 \nu^{2} - 156 \nu - 156 ) / 460 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22 \nu^{11} + 75 \nu^{10} - 75 \nu^{9} + 185 \nu^{8} - 289 \nu^{7} - 461 \nu^{6} + 736 \nu^{5} - 194 \nu^{4} + 84 \nu^{3} + 1676 \nu^{2} - 22 \nu + 22 ) / 230 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 32 ) / 20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} - 1774 \nu^{4} - 3856 \nu^{3} - 524 \nu^{2} + 524 \nu + 476 ) / 460 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 117 \nu^{11} + 195 \nu^{10} - 195 \nu^{9} + 941 \nu^{8} - 250 \nu^{7} - 1700 \nu^{6} + 92 \nu^{5} - 2510 \nu^{4} - 2790 \nu^{3} + 1294 \nu^{2} - 1060 \nu + 1060 ) / 460 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} - 1424 \nu^{4} - 1636 \nu^{3} + 732 \nu^{2} - 612 \nu + 612 ) / 230 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 187 \nu^{11} + 293 \nu^{10} - 216 \nu^{9} + 1338 \nu^{8} - 116 \nu^{7} - 3250 \nu^{6} + 174 \nu^{5} - 3050 \nu^{4} - 5552 \nu^{3} + 2560 \nu^{2} + 748 \nu + 748 ) / 460 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{11} + 4\beta_{8} + 3\beta_{6} - \beta_{5} - \beta_{3} + 2\beta_{2} + 4\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{10} - \beta_{9} + 7\beta_{7} + \beta_{2} + 5\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{10} - 3\beta_{9} + 9\beta_{7} + 6\beta_{6} + 8\beta_{4} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{11} + 22\beta_{8} + 17\beta_{6} - 11\beta_{5} - 11\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 25 \beta_{11} + 33 \beta_{10} - 11 \beta_{9} + 33 \beta_{8} + 39 \beta_{7} - 14 \beta_{5} - 33 \beta_{4} + 11 \beta_{2} + 33 \beta _1 + 39 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94\beta_{10} - 28\beta_{9} + 116\beta_{7} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -105\beta_{11} + 138\beta_{8} + 105\beta_{6} - 61\beta_{5} - 61\beta_{3} - 44\beta_{2} - 138\beta _1 - 166 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -304\beta_{11} + 398\beta_{8} - 182\beta_{5} - 398\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 580\beta_{10} - 182\beta_{9} + 702\beta_{7} - 442\beta_{6} - 580\beta_{4} + 260\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-\beta_{7}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
654.1
−0.531325 1.98293i
1.98293 0.531325i
0.312819 + 1.16746i
−1.16746 + 0.312819i
0.550552 0.147520i
−0.147520 0.550552i
−0.531325 + 1.98293i
1.98293 + 0.531325i
0.312819 1.16746i
−1.16746 0.312819i
0.550552 + 0.147520i
−0.147520 + 0.550552i
−0.607160 + 1.05163i −1.13545 0.655554i 0.262714 + 0.455034i −0.311108 + 2.21432i 1.37880 0.796052i −1.45161 2.51426i −3.06668 −0.640498 1.10938i −2.13976 1.67162i
654.2 −0.607160 + 1.05163i 1.13545 + 0.655554i 0.262714 + 0.455034i −0.311108 2.21432i −1.37880 + 0.796052i −1.45161 2.51426i −3.06668 −0.640498 1.10938i 2.51754 + 1.01728i
654.3 0.769594 1.33298i −2.74538 1.58504i −0.184551 0.319652i −2.17009 0.539189i −4.22565 + 2.43968i 0.854638 + 1.48028i 2.51026 3.52472 + 6.10500i −2.38881 + 2.47772i
654.4 0.769594 1.33298i 2.74538 + 1.58504i −0.184551 0.319652i −2.17009 + 0.539189i 4.22565 2.43968i 0.854638 + 1.48028i 2.51026 3.52472 + 6.10500i −0.951360 + 3.30763i
654.5 1.33757 2.31673i −0.416726 0.240597i −2.57816 4.46551i 1.48119 + 1.67513i −1.11480 + 0.643629i −0.403032 0.698071i −8.44358 −1.38423 2.39755i 5.86202 1.19093i
654.6 1.33757 2.31673i 0.416726 + 0.240597i −2.57816 4.46551i 1.48119 1.67513i 1.11480 0.643629i −0.403032 0.698071i −8.44358 −1.38423 2.39755i −1.89963 5.67213i
699.1 −0.607160 1.05163i −1.13545 + 0.655554i 0.262714 0.455034i −0.311108 2.21432i 1.37880 + 0.796052i −1.45161 + 2.51426i −3.06668 −0.640498 + 1.10938i −2.13976 + 1.67162i
699.2 −0.607160 1.05163i 1.13545 0.655554i 0.262714 0.455034i −0.311108 + 2.21432i −1.37880 0.796052i −1.45161 + 2.51426i −3.06668 −0.640498 + 1.10938i 2.51754 1.01728i
699.3 0.769594 + 1.33298i −2.74538 + 1.58504i −0.184551 + 0.319652i −2.17009 + 0.539189i −4.22565 2.43968i 0.854638 1.48028i 2.51026 3.52472 6.10500i −2.38881 2.47772i
699.4 0.769594 + 1.33298i 2.74538 1.58504i −0.184551 + 0.319652i −2.17009 0.539189i 4.22565 + 2.43968i 0.854638 1.48028i 2.51026 3.52472 6.10500i −0.951360 3.30763i
699.5 1.33757 + 2.31673i −0.416726 + 0.240597i −2.57816 + 4.46551i 1.48119 1.67513i −1.11480 0.643629i −0.403032 + 0.698071i −8.44358 −1.38423 + 2.39755i 5.86202 + 1.19093i
699.6 1.33757 + 2.31673i 0.416726 0.240597i −2.57816 + 4.46551i 1.48119 + 1.67513i 1.11480 + 0.643629i −0.403032 + 0.698071i −8.44358 −1.38423 + 2.39755i −1.89963 + 5.67213i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 699.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
65.d even 2 1 inner
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.l.e 12
5.b even 2 1 845.2.l.d 12
13.b even 2 1 845.2.l.d 12
13.c even 3 1 845.2.d.a 6
13.c even 3 1 inner 845.2.l.e 12
13.d odd 4 1 845.2.n.f 12
13.d odd 4 1 845.2.n.g 12
13.e even 6 1 845.2.d.b 6
13.e even 6 1 845.2.l.d 12
13.f odd 12 1 65.2.b.a 6
13.f odd 12 1 845.2.b.c 6
13.f odd 12 1 845.2.n.f 12
13.f odd 12 1 845.2.n.g 12
39.k even 12 1 585.2.c.b 6
52.l even 12 1 1040.2.d.c 6
65.d even 2 1 inner 845.2.l.e 12
65.g odd 4 1 845.2.n.f 12
65.g odd 4 1 845.2.n.g 12
65.l even 6 1 845.2.d.a 6
65.l even 6 1 inner 845.2.l.e 12
65.n even 6 1 845.2.d.b 6
65.n even 6 1 845.2.l.d 12
65.o even 12 1 325.2.a.j 3
65.o even 12 1 4225.2.a.ba 3
65.s odd 12 1 65.2.b.a 6
65.s odd 12 1 845.2.b.c 6
65.s odd 12 1 845.2.n.f 12
65.s odd 12 1 845.2.n.g 12
65.t even 12 1 325.2.a.k 3
65.t even 12 1 4225.2.a.bh 3
195.bc odd 12 1 2925.2.a.bf 3
195.bh even 12 1 585.2.c.b 6
195.bn odd 12 1 2925.2.a.bj 3
260.bc even 12 1 1040.2.d.c 6
260.be odd 12 1 5200.2.a.cj 3
260.bl odd 12 1 5200.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 13.f odd 12 1
65.2.b.a 6 65.s odd 12 1
325.2.a.j 3 65.o even 12 1
325.2.a.k 3 65.t even 12 1
585.2.c.b 6 39.k even 12 1
585.2.c.b 6 195.bh even 12 1
845.2.b.c 6 13.f odd 12 1
845.2.b.c 6 65.s odd 12 1
845.2.d.a 6 13.c even 3 1
845.2.d.a 6 65.l even 6 1
845.2.d.b 6 13.e even 6 1
845.2.d.b 6 65.n even 6 1
845.2.l.d 12 5.b even 2 1
845.2.l.d 12 13.b even 2 1
845.2.l.d 12 13.e even 6 1
845.2.l.d 12 65.n even 6 1
845.2.l.e 12 1.a even 1 1 trivial
845.2.l.e 12 13.c even 3 1 inner
845.2.l.e 12 65.d even 2 1 inner
845.2.l.e 12 65.l even 6 1 inner
845.2.n.f 12 13.d odd 4 1
845.2.n.f 12 13.f odd 12 1
845.2.n.f 12 65.g odd 4 1
845.2.n.f 12 65.s odd 12 1
845.2.n.g 12 13.d odd 4 1
845.2.n.g 12 13.f odd 12 1
845.2.n.g 12 65.g odd 4 1
845.2.n.g 12 65.s odd 12 1
1040.2.d.c 6 52.l even 12 1
1040.2.d.c 6 260.bc even 12 1
2925.2.a.bf 3 195.bc odd 12 1
2925.2.a.bj 3 195.bn odd 12 1
4225.2.a.ba 3 65.o even 12 1
4225.2.a.bh 3 65.t even 12 1
5200.2.a.cb 3 260.bl odd 12 1
5200.2.a.cj 3 260.be odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3T_{2}^{5} + 10T_{2}^{4} - 7T_{2}^{3} + 16T_{2}^{2} - 5T_{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 3 T^{5} + 10 T^{4} - 7 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 12 T^{10} + 124 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 2 T^{5} + 8 T^{4} + 24 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 20 T^{10} + 312 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 44 T^{10} + 1824 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{10} + 48 T^{8} - 120 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{12} - 72 T^{10} + 3748 T^{8} + \cdots + 54700816 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + 72 T^{4} + 1944 T^{2} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 60 T^{4} + 920 T^{2} + 676)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 28 T^{4} + 104 T^{3} + 784 T^{2} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 80 T^{10} + 5632 T^{8} + \cdots + 1048576 \) Copy content Toggle raw display
$43$ \( T^{12} - 128 T^{10} + \cdots + 5972816656 \) Copy content Toggle raw display
$47$ \( (T^{3} - 10 T^{2} + 28 T - 20)^{4} \) Copy content Toggle raw display
$53$ \( (T^{6} + 144 T^{4} + 6464 T^{2} + \cdots + 92416)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 144 T^{10} + \cdots + 4711998736 \) Copy content Toggle raw display
$61$ \( (T^{6} + 6 T^{5} + 52 T^{4} - 104 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 10 T^{5} + 160 T^{4} + \cdots + 364816)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 320 T^{10} + \cdots + 323210442256 \) Copy content Toggle raw display
$73$ \( (T^{3} - 24 T^{2} + 164 T - 236)^{4} \) Copy content Toggle raw display
$79$ \( (T^{3} - 16 T^{2} + 24 T + 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{3} - 22 T^{2} + 152 T - 316)^{4} \) Copy content Toggle raw display
$89$ \( T^{12} - 204 T^{10} + \cdots + 1600000000 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + 280 T^{4} + \cdots + 40000)^{2} \) Copy content Toggle raw display
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