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Properties

Label 289.4.a.g
Level $289$
Weight $4$
Character orbit 289.a
Self dual yes
Analytic conductor $17.052$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,4,Mod(1,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.0515519917\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + 38476 x^{4} - 32336 x^{3} - 57024 x^{2} - 15776 x + 1156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + (\beta_{5} - 1) q^{2} + \beta_{7} q^{3} + (\beta_{10} - \beta_{5} + 2 \beta_1 + 2) q^{4} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} + 2 \beta_{3}) q^{5} + ( - \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{3} + 2 \beta_{2}) q^{6} + (2 \beta_{9} + 2 \beta_{8} - 2 \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{11} - 3 \beta_{10} + 2 \beta_{6} - 3 \beta_{5} - 5 \beta_1 - 9) q^{8} + (2 \beta_{11} - \beta_{10} - 2 \beta_{6} - 5 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 1) q^{2} + \beta_{7} q^{3} + (\beta_{10} - \beta_{5} + 2 \beta_1 + 2) q^{4} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} + 2 \beta_{3}) q^{5} + ( - \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{3} + 2 \beta_{2}) q^{6} + (2 \beta_{9} + 2 \beta_{8} - 2 \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{11} - 3 \beta_{10} + 2 \beta_{6} - 3 \beta_{5} - 5 \beta_1 - 9) q^{8} + (2 \beta_{11} - \beta_{10} - 2 \beta_{6} - 5 \beta_1 - 2) q^{9} + (3 \beta_{9} - 4 \beta_{8} + 3 \beta_{7} - 6 \beta_{4} - 6 \beta_{3} - 4 \beta_{2}) q^{10} + (\beta_{9} - 7 \beta_{8} - 5 \beta_{7} - \beta_{4} - 3 \beta_{3} - \beta_{2}) q^{11} + (3 \beta_{9} - \beta_{8} - \beta_{7} + 6 \beta_{4} - 22 \beta_{3} - 4 \beta_{2}) q^{12} + ( - 4 \beta_{11} - 3 \beta_{10} - 6 \beta_{6} - 6 \beta_{5} + 8 \beta_1 + 1) q^{13} + ( - \beta_{9} + 4 \beta_{8} - 6 \beta_{7} - 15 \beta_{4} + 8 \beta_{3} + 2 \beta_{2}) q^{14} + ( - 2 \beta_{10} + 5 \beta_{6} + 2 \beta_{5} - 22 \beta_1 - 19) q^{15} + (5 \beta_{11} - 2 \beta_{10} - 8 \beta_{6} - 5 \beta_{5} + 7 \beta_1 - 10) q^{16} + ( - \beta_{11} - 2 \beta_{10} + \beta_{6} - 13 \beta_{5} - 18 \beta_1 + 3) q^{18} + (3 \beta_{11} + 3 \beta_{10} + 19 \beta_{6} - 14 \beta_{5} - 11 \beta_1 - 29) q^{19} + ( - 6 \beta_{9} + 8 \beta_{8} - 9 \beta_{7} + 21 \beta_{4} + 44 \beta_{3} + 12 \beta_{2}) q^{20} + ( - 5 \beta_{11} + \beta_{10} + 5 \beta_{6} - 16 \beta_{5} + 41 \beta_1 - 19) q^{21} + ( - 4 \beta_{9} + 16 \beta_{8} + \beta_{7} + 49 \beta_{4} + 17 \beta_{3} - 7 \beta_{2}) q^{22} + ( - 9 \beta_{8} - 15 \beta_{4} - 20 \beta_{3} + 24 \beta_{2}) q^{23} + (\beta_{9} + 13 \beta_{8} - 5 \beta_{7} + 8 \beta_{4} + 46 \beta_{3} + 4 \beta_{2}) q^{24} + (3 \beta_{11} + 10 \beta_{10} - 7 \beta_{6} - 22 \beta_{5} + 37 \beta_1 - 27) q^{25} + ( - 3 \beta_{11} - 2 \beta_{10} - 4 \beta_{6} - 2 \beta_{5} - 59 \beta_1 - 65) q^{26} + ( - 11 \beta_{9} - 18 \beta_{7} - 60 \beta_{4} + 37 \beta_{3} - 4 \beta_{2}) q^{27} + ( - 3 \beta_{9} - 2 \beta_{8} + 12 \beta_{7} - 29 \beta_{4} - 32 \beta_{3} - 12 \beta_{2}) q^{28} + ( - 14 \beta_{9} - 3 \beta_{8} - 9 \beta_{7} + 54 \beta_{4} - 74 \beta_{3} - 10 \beta_{2}) q^{29} + (7 \beta_{11} + 11 \beta_{10} - 22 \beta_{6} - 20 \beta_{5} + 73 \beta_1 + 82) q^{30} + (6 \beta_{9} - 3 \beta_{8} + 14 \beta_{7} - 71 \beta_{4} - 12 \beta_{3} + 4 \beta_{2}) q^{31} + (2 \beta_{11} + 10 \beta_{10} + 6 \beta_{6} - 15 \beta_{5} - 62 \beta_1 + 9) q^{32} + ( - 11 \beta_{11} + 16 \beta_{10} - 12 \beta_{6} + 6 \beta_{5} + 72 \beta_1 - 113) q^{33} + ( - 6 \beta_{11} - 2 \beta_{10} + 11 \beta_{6} + 12 \beta_{5} - 50 \beta_1 - 63) q^{35} + ( - 13 \beta_{11} + \beta_{10} - 5 \beta_{6} + \beta_{5} + 46 \beta_1 - 74) q^{36} + (12 \beta_{9} - 26 \beta_{8} + 9 \beta_{7} + 99 \beta_{4} + 86 \beta_{3} + 15 \beta_{2}) q^{37} + (16 \beta_{11} - 4 \beta_{10} - 2 \beta_{6} - 10 \beta_{5} + 142 \beta_1 - 44) q^{38} + (26 \beta_{9} - 14 \beta_{8} + 4 \beta_{7} + 100 \beta_{4} + 6 \beta_{3} - 4 \beta_{2}) q^{39} + (11 \beta_{9} - 6 \beta_{8} + 21 \beta_{7} - 14 \beta_{4} - 150 \beta_{3} - 30 \beta_{2}) q^{40} + ( - 23 \beta_{9} + 7 \beta_{8} + 12 \beta_{7} - 6 \beta_{4} + 67 \beta_{3} - 21 \beta_{2}) q^{41} + (4 \beta_{11} - 8 \beta_{10} + 26 \beta_{6} + 4 \beta_{5} - 14 \beta_1 - 150) q^{42} + ( - \beta_{11} + 28 \beta_{10} - 6 \beta_{6} - 18 \beta_{5} - 108 \beta_1 - 83) q^{43} + (4 \beta_{9} - 22 \beta_{8} + 33 \beta_{7} - 93 \beta_{4} + 47 \beta_{3} - 7 \beta_{2}) q^{44} + (3 \beta_{9} - 10 \beta_{8} + 9 \beta_{7} - 44 \beta_{4} + 8 \beta_{3} + 13 \beta_{2}) q^{45} + (15 \beta_{9} + 24 \beta_{8} + 48 \beta_{7} + 39 \beta_{4} - 202 \beta_{3} + 20 \beta_{2}) q^{46} + (22 \beta_{11} - 6 \beta_{10} + 52 \beta_{6} + 30 \beta_{5} - 6 \beta_1 - 148) q^{47} + ( - 3 \beta_{9} - 5 \beta_{8} + 19 \beta_{7} - 144 \beta_{4} + 54 \beta_{3} - 24 \beta_{2}) q^{48} + (\beta_{11} - 5 \beta_{10} - 43 \beta_{6} + 34 \beta_{5} - 63 \beta_1 - 86) q^{49} + ( - 17 \beta_{11} - 52 \beta_{10} + 46 \beta_{6} - 13 \beta_{5} - 163 \beta_1 - 272) q^{50} + (30 \beta_{11} + 25 \beta_{10} - 20 \beta_{6} - 18 \beta_{5} - 34 \beta_1 + 97) q^{52} + (13 \beta_{11} - 40 \beta_{10} - 39 \beta_{6} + 58 \beta_{5} - 11 \beta_1 - 224) q^{53} + (25 \beta_{9} + 45 \beta_{8} + 32 \beta_{7} + 15 \beta_{4} - 33 \beta_{3} - 73 \beta_{2}) q^{54} + ( - 6 \beta_{11} - 22 \beta_{10} - 15 \beta_{6} + 66 \beta_{5} + 100 \beta_1 - 145) q^{55} + ( - 15 \beta_{9} - 22 \beta_{8} + 18 \beta_{7} + 149 \beta_{4} + \cdots + 40 \beta_{2}) q^{56}+ \cdots + (72 \beta_{9} + 93 \beta_{8} - 47 \beta_{7} + 443 \beta_{4} - 344 \beta_{3} + 17 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36} - 624 q^{38} - 1664 q^{42} - 1200 q^{43} - 1512 q^{47} - 1052 q^{49} - 2856 q^{50} + 792 q^{52} - 2504 q^{53} - 1424 q^{55} - 3408 q^{59} - 2808 q^{60} + 272 q^{64} + 272 q^{66} - 1080 q^{67} - 344 q^{69} + 2600 q^{70} + 248 q^{72} + 896 q^{76} + 848 q^{77} - 2404 q^{81} - 2960 q^{83} + 4768 q^{84} - 1200 q^{86} - 160 q^{87} - 2144 q^{89} + 3800 q^{93} + 5984 q^{94} + 3464 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + 38476 x^{4} - 32336 x^{3} - 57024 x^{2} - 15776 x + 1156 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 231283 \nu^{11} + 841213 \nu^{10} + 14752558 \nu^{9} - 44008343 \nu^{8} - 354341075 \nu^{7} + 777504271 \nu^{6} + 3924198436 \nu^{5} + \cdots + 4105733002 ) / 4091924922 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11149490123 \nu^{11} - 51020278232 \nu^{10} - 611532188522 \nu^{9} + 2624844361002 \nu^{8} + 11433844746415 \nu^{7} + \cdots + 31493930441268 ) / 10761762544860 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17902986718 \nu^{11} - 83369711285 \nu^{10} - 985140561222 \nu^{9} + 4302540706534 \nu^{8} + 18581870878046 \nu^{7} + \cdots + 51299241770360 ) / 10761762544860 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 47174762280 \nu^{11} + 217474977641 \nu^{10} + 2599094071560 \nu^{9} - 11199287270444 \nu^{8} - 49100090474212 \nu^{7} + \cdots - 77253066815524 ) / 10761762544860 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47174762280 \nu^{11} - 217474977641 \nu^{10} - 2599094071560 \nu^{9} + 11199287270444 \nu^{8} + 49100090474212 \nu^{7} + \cdots + 77253066815524 ) / 10761762544860 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32649227984 \nu^{11} + 152182003823 \nu^{10} + 1796290322926 \nu^{9} - 7854645162419 \nu^{8} - 33869660532884 \nu^{7} + \cdots - 58543690480892 ) / 5380881272430 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 66713717634 \nu^{11} - 305126955302 \nu^{10} - 3685257193021 \nu^{9} + 15720173879810 \nu^{8} + 69874539344772 \nu^{7} + \cdots + 171477393597988 ) / 10761762544860 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25114138837 \nu^{11} + 117434909377 \nu^{10} + 1378643536448 \nu^{9} - 6056198909874 \nu^{8} - 25909524049613 \nu^{7} + \cdots - 40353881572248 ) / 3587254181620 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 84442594199 \nu^{11} + 384554299885 \nu^{10} + 4672664055666 \nu^{9} - 19793560640642 \nu^{8} - 88897390259683 \nu^{7} + \cdots - 79447320838000 ) / 10761762544860 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 102901796452 \nu^{11} + 484922088431 \nu^{10} + 5633967919588 \nu^{9} - 25022164246710 \nu^{8} + \cdots - 294047688406684 ) / 10761762544860 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 256162049158 \nu^{11} - 1189777436801 \nu^{10} - 14112515646972 \nu^{9} + 61398425862658 \nu^{8} + 266793921860618 \nu^{7} + \cdots + 432454605018044 ) / 10761762544860 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 2\beta_{8} + \beta_{5} + 2\beta_{4} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - 3\beta_{9} - 3\beta_{8} - \beta_{6} + 19\beta_{5} + 27\beta_{4} + 5\beta_{3} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{11} + 22 \beta_{10} - 4 \beta_{9} - 56 \beta_{8} + 8 \beta_{7} - 6 \beta_{6} + 33 \beta_{5} + 76 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} + 13 \beta _1 + 219 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 45 \beta_{11} + 14 \beta_{10} - 95 \beta_{9} - 135 \beta_{8} + 10 \beta_{7} - 60 \beta_{6} + 439 \beta_{5} + 720 \beta_{4} + 186 \beta_{3} + 10 \beta_{2} - 7 \beta _1 + 267 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 224 \beta_{11} + 505 \beta_{10} - 194 \beta_{9} - 1514 \beta_{8} + 300 \beta_{7} - 347 \beta_{6} + 1093 \beta_{5} + 2508 \beta_{4} + 228 \beta_{3} - 96 \beta_{2} + 60 \beta _1 + 5072 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1570 \beta_{11} + 659 \beta_{10} - 2639 \beta_{9} - 4830 \beta_{8} + 644 \beta_{7} - 2377 \beta_{6} + 11033 \beta_{5} + 19725 \beta_{4} + 4872 \beta_{3} + 623 \beta_{2} - 1536 \beta _1 + 10046 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7889 \beta_{11} + 12206 \beta_{10} - 7080 \beta_{9} - 41696 \beta_{8} + 9664 \beta_{7} - 14076 \beta_{6} + 35049 \beta_{5} + 79736 \beta_{4} + 4752 \beta_{3} - 584 \beta_{2} - 5299 \beta _1 + 127235 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 50915 \beta_{11} + 23736 \beta_{10} - 72039 \beta_{9} - 159765 \beta_{8} + 28770 \beta_{7} - 84154 \beta_{6} + 291885 \beta_{5} + 554744 \beta_{4} + 111150 \beta_{3} + 28068 \beta_{2} + \cdots + 345105 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 259614 \beta_{11} + 306949 \beta_{10} - 235118 \beta_{9} - 1175670 \beta_{8} + 302044 \beta_{7} - 503145 \beta_{6} + 1096319 \beta_{5} + 2483388 \beta_{4} + 68892 \beta_{3} + \cdots + 3357908 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1606870 \beta_{11} + 780957 \beta_{10} - 1985665 \beta_{9} - 5090250 \beta_{8} + 1106688 \beta_{7} - 2844705 \beta_{6} + 8002111 \beta_{5} + 15931861 \beta_{4} + \cdots + 11265364 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.09630
−4.62703
−4.34747
−0.651949
−1.47083
0.0598990
3.07564
−0.619881
1.83828
5.53380
4.91828
3.38755
−4.86166 −5.06554 15.6358 18.5634 24.6269 −14.0210 −37.1225 −1.34032 −90.2489
1.2 −4.86166 5.06554 15.6358 −18.5634 −24.6269 14.0210 −37.1225 −1.34032 90.2489
1.3 −3.49971 −2.82539 4.24796 −8.71933 9.88806 −6.85501 13.1311 −19.0171 30.5151
1.4 −3.49971 2.82539 4.24796 8.71933 −9.88806 6.85501 13.1311 −19.0171 −30.5151
1.5 −1.70547 −4.44379 −5.09138 6.80983 7.57875 −13.8760 22.3269 −7.25269 −11.6140
1.6 −1.70547 4.44379 −5.09138 −6.80983 −7.57875 13.8760 22.3269 −7.25269 11.6140
1.7 0.227878 −8.23916 −7.94807 −2.75144 −1.87752 21.5220 −3.63421 40.8838 −0.626993
1.8 0.227878 8.23916 −7.94807 2.75144 1.87752 −21.5220 −3.63421 40.8838 0.626993
1.9 2.68604 −4.33137 −0.785167 2.08666 −11.6343 24.9985 −23.5973 −8.23920 5.60485
1.10 2.68604 4.33137 −0.785167 −2.08666 11.6343 −24.9985 −23.5973 −8.23920 −5.60485
1.11 3.15292 −1.99138 1.94089 5.00761 −6.27866 −2.78516 −19.1039 −23.0344 15.7886
1.12 3.15292 1.99138 1.94089 −5.00761 6.27866 2.78516 −19.1039 −23.0344 −15.7886
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.4.a.g 12
17.b even 2 1 inner 289.4.a.g 12
17.c even 4 2 289.4.b.e 12
17.e odd 16 2 17.4.d.a ✓ 12
51.i even 16 2 153.4.l.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.d.a ✓ 12 17.e odd 16 2
153.4.l.a 12 51.i even 16 2
289.4.a.g 12 1.a even 1 1 trivial
289.4.a.g 12 17.b even 2 1 inner
289.4.b.e 12 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(289))\):

\( T_{2}^{6} + 4T_{2}^{5} - 20T_{2}^{4} - 64T_{2}^{3} + 111T_{2}^{2} + 224T_{2} - 56 \) Copy content Toggle raw display
\( T_{3}^{12} - 144T_{3}^{10} + 7324T_{3}^{8} - 174192T_{3}^{6} + 2041764T_{3}^{4} - 10931104T_{3}^{2} + 20428832 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 4 T^{5} - 20 T^{4} - 64 T^{3} + \cdots - 56)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 144 T^{10} + \cdots + 20428832 \) Copy content Toggle raw display
$5$ \( T^{12} - 504 T^{10} + \cdots + 1004236928 \) Copy content Toggle raw display
$7$ \( T^{12} - 1532 T^{10} + \cdots + 3993906708992 \) Copy content Toggle raw display
$11$ \( T^{12} - 8880 T^{10} + \cdots + 44\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( (T^{6} + 4 T^{5} - 5950 T^{4} + \cdots - 587761088)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + 176 T^{5} + \cdots + 37116511456)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 77052 T^{10} + \cdots + 78\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{12} - 158712 T^{10} + \cdots + 73\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( T^{12} - 98188 T^{10} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{12} - 281576 T^{10} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{12} - 292140 T^{10} + \cdots + 36\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( (T^{6} + 600 T^{5} + \cdots - 25553688086896)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 756 T^{5} + \cdots + 426029287281152)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 1252 T^{5} + \cdots + 309016062416752)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 1704 T^{5} + \cdots - 847356213991824)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} - 1655176 T^{10} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( (T^{6} + 540 T^{5} + \cdots + 61\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 838348 T^{10} + \cdots + 32\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{12} - 1240236 T^{10} + \cdots + 99\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{12} - 1494012 T^{10} + \cdots + 93\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( (T^{6} + 1480 T^{5} + \cdots + 124020435364336)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 1072 T^{5} + \cdots + 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 5358828 T^{10} + \cdots + 16\!\cdots\!52 \) Copy content Toggle raw display
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