Properties

Label 27.2.e.a
Level $27$
Weight $2$
Character orbit 27.e
Analytic conductor $0.216$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.215596085457\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.1952986685049.1
Defining polynomial: \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{3} - 1) q^{2} + ( - \beta_{8} + \beta_{6} - \beta_{2} - 1) q^{3} + (\beta_{9} - \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{4} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{6} + ( - \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{7} + ( - 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} + \beta_{4} - \beta_{2}) q^{8} + (\beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{3} - 1) q^{2} + ( - \beta_{8} + \beta_{6} - \beta_{2} - 1) q^{3} + (\beta_{9} - \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{4} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{6} + ( - \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 1) q^{7} + ( - 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} + \beta_{4} - \beta_{2}) q^{8} + (\beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{2}) q^{9} + (\beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{3} - \beta_{2} - 2) q^{10} + (\beta_{11} + \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{11} + (2 \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2}) q^{12} + ( - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{13} + ( - \beta_{11} + \beta_{8} - \beta_{4} + \beta_{3} + \beta_1 + 3) q^{14} + ( - 3 \beta_{10} + 3 \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1) q^{15}+ \cdots + ( - 3 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + 4 \beta_{8} - 4 \beta_{7} + \beta_{6} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 6 q^{8} - 3 q^{10} + 3 q^{11} + 12 q^{12} - 6 q^{13} + 15 q^{14} + 9 q^{15} + 9 q^{17} + 9 q^{18} - 3 q^{19} - 3 q^{20} - 12 q^{21} + 3 q^{22} - 12 q^{23} - 18 q^{24} + 3 q^{25} - 30 q^{26} - 9 q^{27} - 12 q^{28} - 6 q^{29} - 9 q^{30} + 3 q^{31} + 9 q^{34} + 12 q^{35} + 18 q^{36} - 3 q^{37} + 42 q^{38} + 33 q^{39} + 21 q^{40} + 15 q^{41} + 18 q^{42} + 3 q^{43} + 3 q^{44} - 9 q^{45} - 3 q^{46} - 15 q^{47} - 15 q^{48} + 12 q^{49} - 33 q^{50} - 18 q^{51} + 9 q^{52} - 18 q^{53} - 54 q^{54} - 12 q^{55} - 33 q^{56} - 3 q^{57} + 21 q^{58} - 12 q^{59} + 12 q^{61} - 12 q^{62} + 9 q^{63} + 12 q^{64} + 3 q^{65} - 9 q^{66} - 15 q^{67} + 9 q^{68} + 9 q^{69} - 15 q^{70} + 27 q^{71} + 18 q^{72} + 6 q^{73} + 33 q^{74} + 39 q^{75} - 48 q^{76} + 15 q^{77} + 18 q^{78} - 42 q^{79} + 42 q^{80} + 36 q^{81} - 12 q^{82} + 39 q^{83} + 6 q^{84} - 27 q^{85} + 51 q^{86} + 9 q^{87} - 30 q^{88} + 9 q^{89} + 18 q^{90} + 6 q^{91} - 39 q^{92} - 39 q^{93} - 15 q^{94} - 33 q^{95} + 3 q^{97} - 45 q^{98} - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} - 1030 \nu^{4} + 803 \nu^{3} - 445 \nu^{2} + 155 \nu - 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( - 6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + 1762 \nu^{4} - 1285 \nu^{3} + 649 \nu^{2} - 195 \nu + 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( - 9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + 2918 \nu^{4} - 2170 \nu^{3} + 1118 \nu^{2} - 348 \nu + 49 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} - 3178 \nu^{4} + 2411 \nu^{3} - 1286 \nu^{2} + 421 \nu - 62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} - 3643 \nu^{4} + 2724 \nu^{3} - 1417 \nu^{2} + 442 \nu - 61 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - 16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + 5122 \nu^{4} - 3802 \nu^{3} + 1958 \nu^{2} - 610 \nu + 85 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( - 16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} - 12058 \nu^{4} + 9024 \nu^{3} - 4708 \nu^{2} + 1486 \nu - 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( - 36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( - 42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + 14172 \nu^{4} - 10648 \nu^{3} + 5589 \nu^{2} - 1785 \nu + 257 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 8 \beta _1 - 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{11} + 17 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} - 14 \beta_{5} + 4 \beta_{4} - 14 \beta_{3} - 8 \beta_{2} + \beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19 \beta_{11} - \beta_{10} + 31 \beta_{9} - 32 \beta_{8} + 43 \beta_{7} - 20 \beta_{6} - 41 \beta_{5} - 44 \beta_{4} + 10 \beta_{3} + 25 \beta_{2} + 40 \beta _1 + 87 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 85 \beta_{11} - 97 \beta_{10} + 55 \beta_{9} + 64 \beta_{8} + 10 \beta_{7} - 101 \beta_{6} + 19 \beta_{5} - 62 \beta_{4} + 91 \beta_{3} + 70 \beta_{2} + 31 \beta _1 + 60 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 20 \beta_{11} - 118 \beta_{10} - 134 \beta_{9} + 244 \beta_{8} - 218 \beta_{7} + \beta_{6} + 232 \beta_{5} + 157 \beta_{4} + 52 \beta_{3} - 74 \beta_{2} - 179 \beta _1 - 357 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 503 \beta_{11} + 386 \beta_{10} - 440 \beta_{9} - 47 \beta_{8} - 233 \beta_{7} + 514 \beta_{6} + 163 \beta_{5} + 466 \beta_{4} - 431 \beta_{3} - 461 \beta_{2} - 329 \beta _1 - 639 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 425 \beta_{11} + 1076 \beta_{10} + 319 \beta_{9} - 1313 \beta_{8} + 955 \beta_{7} + 502 \beta_{6} - 1013 \beta_{5} - 332 \beta_{4} - 743 \beta_{3} - 59 \beta_{2} + 631 \beta _1 + 1164 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2299 \beta_{11} - 862 \beta_{10} + 2725 \beta_{9} - 1193 \beta_{8} + 2104 \beta_{7} - 2135 \beta_{6} - 1907 \beta_{5} - 2705 \beta_{4} + 1495 \beta_{3} + 2425 \beta_{2} + 2344 \beta _1 + 4356 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4708 \beta_{11} - 6628 \beta_{10} + 985 \beta_{9} + 5506 \beta_{8} - 3107 \beta_{7} - 4679 \beta_{6} + 3238 \beta_{5} - 992 \beta_{4} + 5476 \beta_{3} + 2770 \beta_{2} - 1043 \beta _1 - 1698 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 1.27297i
0.500000 0.0126039i
0.500000 + 1.27297i
0.500000 + 0.0126039i
0.500000 1.68614i
0.500000 + 1.00210i
0.500000 + 0.258654i
0.500000 2.22827i
0.500000 0.258654i
0.500000 + 2.22827i
0.500000 + 1.68614i
0.500000 1.00210i
−1.57954 0.574906i 1.45446 0.940501i 0.632343 + 0.530599i −0.196143 + 1.11238i −2.83808 + 0.649381i −2.99441 + 2.51261i 0.987144 + 1.70978i 1.23092 2.73584i 0.949332 1.64429i
4.2 0.753189 + 0.274138i −1.68842 0.386327i −1.03995 0.872619i −0.477505 + 2.70806i −1.16579 0.753837i 1.82076 1.52780i −1.34559 2.33062i 2.70150 + 1.30456i −1.10204 + 1.90878i
7.1 −1.57954 + 0.574906i 1.45446 + 0.940501i 0.632343 0.530599i −0.196143 1.11238i −2.83808 0.649381i −2.99441 2.51261i 0.987144 1.70978i 1.23092 + 2.73584i 0.949332 + 1.64429i
7.2 0.753189 0.274138i −1.68842 + 0.386327i −1.03995 + 0.872619i −0.477505 2.70806i −1.16579 + 0.753837i 1.82076 + 1.52780i −1.34559 + 2.33062i 2.70150 1.30456i −1.10204 1.90878i
13.1 −0.417037 2.36514i −0.210069 + 1.71926i −3.54056 + 1.28866i 0.0713060 + 0.0598329i 4.15390 0.220155i 0.544891 + 0.198324i 2.12277 + 3.67675i −2.91174 0.722330i 0.111776 0.193601i
13.2 0.183082 + 1.03831i −1.72962 0.0916693i 0.834822 0.303850i −1.33735 1.12217i −0.221481 1.81266i −2.31094 0.841112i 1.52266 + 2.63732i 2.98319 + 0.317107i 0.920313 1.59403i
16.1 −1.62143 1.36054i −0.986166 1.42389i 0.430663 + 2.44241i 2.52129 + 0.917674i −0.338267 + 3.65046i 0.168844 0.957561i 0.508086 0.880031i −1.05495 + 2.80839i −2.83955 4.91825i
16.2 −0.318266 0.267057i 0.159815 + 1.72466i −0.317323 1.79963i −2.08159 0.757639i 0.409719 0.591580i −0.229151 + 1.29958i −0.795075 + 1.37711i −2.94892 + 0.551252i 0.460168 + 0.797034i
22.1 −1.62143 + 1.36054i −0.986166 + 1.42389i 0.430663 2.44241i 2.52129 0.917674i −0.338267 3.65046i 0.168844 + 0.957561i 0.508086 + 0.880031i −1.05495 2.80839i −2.83955 + 4.91825i
22.2 −0.318266 + 0.267057i 0.159815 1.72466i −0.317323 + 1.79963i −2.08159 + 0.757639i 0.409719 + 0.591580i −0.229151 1.29958i −0.795075 1.37711i −2.94892 0.551252i 0.460168 0.797034i
25.1 −0.417037 + 2.36514i −0.210069 1.71926i −3.54056 1.28866i 0.0713060 0.0598329i 4.15390 + 0.220155i 0.544891 0.198324i 2.12277 3.67675i −2.91174 + 0.722330i 0.111776 + 0.193601i
25.2 0.183082 1.03831i −1.72962 + 0.0916693i 0.834822 + 0.303850i −1.33735 + 1.12217i −0.221481 + 1.81266i −2.31094 + 0.841112i 1.52266 2.63732i 2.98319 0.317107i 0.920313 + 1.59403i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.2
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.2.e.a 12
3.b odd 2 1 81.2.e.a 12
4.b odd 2 1 432.2.u.c 12
5.b even 2 1 675.2.l.c 12
5.c odd 4 2 675.2.u.b 24
9.c even 3 1 243.2.e.c 12
9.c even 3 1 243.2.e.d 12
9.d odd 6 1 243.2.e.a 12
9.d odd 6 1 243.2.e.b 12
27.e even 9 1 inner 27.2.e.a 12
27.e even 9 1 243.2.e.c 12
27.e even 9 1 243.2.e.d 12
27.e even 9 1 729.2.a.a 6
27.e even 9 2 729.2.c.e 12
27.f odd 18 1 81.2.e.a 12
27.f odd 18 1 243.2.e.a 12
27.f odd 18 1 243.2.e.b 12
27.f odd 18 1 729.2.a.d 6
27.f odd 18 2 729.2.c.b 12
108.j odd 18 1 432.2.u.c 12
135.p even 18 1 675.2.l.c 12
135.r odd 36 2 675.2.u.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a 12 1.a even 1 1 trivial
27.2.e.a 12 27.e even 9 1 inner
81.2.e.a 12 3.b odd 2 1
81.2.e.a 12 27.f odd 18 1
243.2.e.a 12 9.d odd 6 1
243.2.e.a 12 27.f odd 18 1
243.2.e.b 12 9.d odd 6 1
243.2.e.b 12 27.f odd 18 1
243.2.e.c 12 9.c even 3 1
243.2.e.c 12 27.e even 9 1
243.2.e.d 12 9.c even 3 1
243.2.e.d 12 27.e even 9 1
432.2.u.c 12 4.b odd 2 1
432.2.u.c 12 108.j odd 18 1
675.2.l.c 12 5.b even 2 1
675.2.l.c 12 135.p even 18 1
675.2.u.b 24 5.c odd 4 2
675.2.u.b 24 135.r odd 36 2
729.2.a.a 6 27.e even 9 1
729.2.a.d 6 27.f odd 18 1
729.2.c.b 12 27.f odd 18 2
729.2.c.e 12 27.e even 9 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 6 T^{11} + 21 T^{10} + 48 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{11} + 18 T^{10} + 39 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} + 3 T^{10} - 12 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + 12 T^{10} - 11 T^{9} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} - 15 T^{10} - 6 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + 48 T^{10} + 214 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} - 9 T^{11} + 72 T^{10} - 189 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{12} + 3 T^{11} + 39 T^{10} - 14 T^{9} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + 48 T^{10} + \cdots + 106929 \) Copy content Toggle raw display
$29$ \( T^{12} + 6 T^{11} + 21 T^{10} + \cdots + 45369 \) Copy content Toggle raw display
$31$ \( T^{12} - 3 T^{11} + 84 T^{10} + \cdots + 26569 \) Copy content Toggle raw display
$37$ \( T^{12} + 3 T^{11} + 66 T^{10} + \cdots + 24334489 \) Copy content Toggle raw display
$41$ \( T^{12} - 15 T^{11} + 93 T^{10} + \cdots + 11229201 \) Copy content Toggle raw display
$43$ \( T^{12} - 3 T^{11} - 60 T^{10} + \cdots + 3308761 \) Copy content Toggle raw display
$47$ \( T^{12} + 15 T^{11} + 111 T^{10} + \cdots + 42732369 \) Copy content Toggle raw display
$53$ \( (T^{6} + 9 T^{5} - 108 T^{4} - 513 T^{3} + \cdots - 12393)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 12 T^{11} + \cdots + 176384961 \) Copy content Toggle raw display
$61$ \( T^{12} - 12 T^{11} + \cdots + 273670849 \) Copy content Toggle raw display
$67$ \( T^{12} + 15 T^{11} + 255 T^{10} + \cdots + 8288641 \) Copy content Toggle raw display
$71$ \( T^{12} - 27 T^{11} + 504 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$73$ \( T^{12} - 6 T^{11} + 210 T^{10} + \cdots + 185761 \) Copy content Toggle raw display
$79$ \( T^{12} + 42 T^{11} + 813 T^{10} + \cdots + 3508129 \) Copy content Toggle raw display
$83$ \( T^{12} - 39 T^{11} + \cdots + 6951057129 \) Copy content Toggle raw display
$89$ \( T^{12} - 9 T^{11} + \cdots + 1062042921 \) Copy content Toggle raw display
$97$ \( T^{12} - 3 T^{11} + 102 T^{10} + \cdots + 66765241 \) Copy content Toggle raw display
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