Properties

Label 150.6.g.a
Level $150$
Weight $6$
Character orbit 150.g
Analytic conductor $24.058$
Analytic rank $0$
Dimension $20$
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] = [150,6,Mod(31,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.31");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.g (of order \(5\), degree \(4\), minimal)

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: no
Analytic conductor: \(24.0575729719\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} + 67300 x^{18} - 605415 x^{17} + 1505002471 x^{16} - 12026298320 x^{15} + 15161344295580 x^{14} - 105918997871590 x^{13} + \cdots + 45\!\cdots\!05 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{6} q^{2} - 9 \beta_{7} q^{3} + 16 \beta_{7} q^{4} + ( - \beta_{18} + \beta_{16} - \beta_{13} + 3 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} - 5) q^{5} + ( - 36 \beta_{7} - 36 \beta_{6} + 36 \beta_{5} - 36) q^{6} + ( - \beta_{19} - 2 \beta_{17} - \beta_{16} + \beta_{13} - \beta_{12} + \beta_{9} + 22 \beta_{6} + \cdots + 20) q^{7}+ \cdots - 81 \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{6} q^{2} - 9 \beta_{7} q^{3} + 16 \beta_{7} q^{4} + ( - \beta_{18} + \beta_{16} - \beta_{13} + 3 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} - 5) q^{5} + ( - 36 \beta_{7} - 36 \beta_{6} + 36 \beta_{5} - 36) q^{6} + ( - \beta_{19} - 2 \beta_{17} - \beta_{16} + \beta_{13} - \beta_{12} + \beta_{9} + 22 \beta_{6} + \cdots + 20) q^{7}+ \cdots + ( - 162 \beta_{19} + 162 \beta_{15} + 162 \beta_{14} - 162 \beta_{11} + \cdots - 4212) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} + 45 q^{3} - 80 q^{4} - 55 q^{5} - 180 q^{6} + 180 q^{7} + 320 q^{8} - 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} + 45 q^{3} - 80 q^{4} - 55 q^{5} - 180 q^{6} + 180 q^{7} + 320 q^{8} - 405 q^{9} + 220 q^{10} + 380 q^{11} + 720 q^{12} - 230 q^{13} - 920 q^{14} - 630 q^{15} - 1280 q^{16} + 2870 q^{17} - 6480 q^{18} + 1560 q^{19} + 1120 q^{20} + 2880 q^{21} - 720 q^{22} + 760 q^{23} + 11520 q^{24} + 1725 q^{25} + 7320 q^{26} + 3645 q^{27} - 5120 q^{28} + 12000 q^{29} + 5220 q^{30} - 1710 q^{31} - 20480 q^{32} - 1620 q^{33} + 20620 q^{34} - 22440 q^{35} - 6480 q^{36} - 5595 q^{37} - 5240 q^{38} + 2070 q^{39} - 9280 q^{40} - 2820 q^{41} - 11520 q^{42} + 66020 q^{43} + 2880 q^{44} - 8505 q^{45} + 21960 q^{46} - 19630 q^{47} + 11520 q^{48} - 61780 q^{49} + 100 q^{50} - 41130 q^{51} - 3680 q^{52} + 62735 q^{53} - 14580 q^{54} + 66590 q^{55} + 20480 q^{56} + 51660 q^{57} - 48000 q^{58} + 105100 q^{59} + 15120 q^{60} + 34790 q^{61} - 28560 q^{62} + 18630 q^{63} - 20480 q^{64} - 44345 q^{65} + 13680 q^{66} - 19470 q^{67} + 73120 q^{68} + 49410 q^{69} - 81240 q^{70} - 111720 q^{71} + 25920 q^{72} - 198830 q^{73} - 104120 q^{74} + 60975 q^{75} - 91840 q^{76} + 144740 q^{77} + 41220 q^{78} + 71210 q^{79} - 26880 q^{80} - 32805 q^{81} + 166680 q^{82} - 288690 q^{83} - 33120 q^{84} + 338735 q^{85} + 39920 q^{86} + 108225 q^{87} - 24320 q^{88} + 114225 q^{89} + 17820 q^{90} - 155800 q^{91} - 87840 q^{92} + 97740 q^{93} + 78520 q^{94} - 416900 q^{95} - 46080 q^{96} - 446970 q^{97} - 123380 q^{98} - 90720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} + 67300 x^{18} - 605415 x^{17} + 1505002471 x^{16} - 12026298320 x^{15} + 15161344295580 x^{14} - 105918997871590 x^{13} + \cdots + 45\!\cdots\!05 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 53\!\cdots\!68 \nu^{18} + \cdots + 20\!\cdots\!50 ) / 39\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15\!\cdots\!66 \nu^{18} + \cdots + 71\!\cdots\!00 ) / 39\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19\!\cdots\!69 \nu^{18} + \cdots - 67\!\cdots\!70 ) / 39\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39\!\cdots\!96 \nu^{18} + \cdots + 14\!\cdots\!30 ) / 39\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 40\!\cdots\!33 \nu^{19} + \cdots + 43\!\cdots\!80 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40\!\cdots\!33 \nu^{19} + \cdots + 12\!\cdots\!80 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 74\!\cdots\!49 \nu^{19} + \cdots - 19\!\cdots\!15 ) / 16\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37\!\cdots\!51 \nu^{19} + \cdots + 41\!\cdots\!10 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 85\!\cdots\!28 \nu^{19} + \cdots + 11\!\cdots\!80 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 60\!\cdots\!27 \nu^{19} + \cdots + 94\!\cdots\!40 ) / 24\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23\!\cdots\!21 \nu^{19} + \cdots - 34\!\cdots\!65 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!78 \nu^{19} + \cdots - 25\!\cdots\!70 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 30\!\cdots\!78 \nu^{19} + \cdots + 39\!\cdots\!30 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!18 \nu^{19} + \cdots + 11\!\cdots\!05 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 41\!\cdots\!39 \nu^{19} + \cdots + 19\!\cdots\!15 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 55\!\cdots\!36 \nu^{19} + \cdots - 12\!\cdots\!90 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 55\!\cdots\!36 \nu^{19} + \cdots - 23\!\cdots\!10 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 62\!\cdots\!14 \nu^{19} + \cdots + 21\!\cdots\!85 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 64\!\cdots\!64 \nu^{19} + \cdots + 14\!\cdots\!15 ) / 37\!\cdots\!25 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - \beta_{19} - 6 \beta_{18} - 2 \beta_{17} - 2 \beta_{16} + \beta_{15} + \beta_{14} - 5 \beta_{13} - 5 \beta_{12} + \beta_{11} + 4 \beta_{10} - 5 \beta_{9} - 8 \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta _1 + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 479 \beta_{19} - 6 \beta_{18} + 715 \beta_{17} + 164 \beta_{16} + 72 \beta_{15} + 72 \beta_{14} - 95 \beta_{13} + 85 \beta_{12} - 70 \beta_{11} + 4 \beta_{10} - 485 \beta_{9} - 8 \beta_{8} + \beta_{7} - 15121 \beta_{6} + 15124 \beta_{5} + \cdots - 41186 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 73144 \beta_{19} + 127678 \beta_{18} + 94620 \beta_{17} + 30914 \beta_{16} + 5273 \beta_{15} - 2547 \beta_{14} + 101173 \beta_{13} + 101443 \beta_{12} - 2760 \beta_{11} - 46619 \beta_{10} + 118323 \beta_{9} + \cdots + 1366978 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12782616 \beta_{19} + 255362 \beta_{18} - 19400826 \beta_{17} - 7643713 \beta_{16} + 1054925 \beta_{15} + 1039285 \beta_{14} + 13577353 \beta_{13} - 13172111 \beta_{12} + \cdots + 1038724838 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2875933971 \beta_{19} - 3770461473 \beta_{18} - 3925309999 \beta_{17} - 637850009 \beta_{16} - 824345837 \beta_{15} - 112290440 \beta_{14} - 2777741908 \beta_{13} + \cdots - 51800034605 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 407271823029 \beta_{19} - 11312022827 \beta_{18} + 596082402258 \beta_{17} + 266865262882 \beta_{16} - 79293216662 \beta_{15} - 77157011371 \beta_{14} + \cdots - 33167410765564 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 103942207979079 \beta_{19} + 125492989445627 \beta_{18} + 145648312616615 \beta_{17} + 18571609580247 \beta_{16} + 36697714996121 \beta_{15} + \cdots + 18\!\cdots\!26 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 13\!\cdots\!46 \beta_{19} + 502024747818216 \beta_{18} + \cdots + 11\!\cdots\!25 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 36\!\cdots\!61 \beta_{19} + \cdots - 62\!\cdots\!23 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 47\!\cdots\!54 \beta_{19} + \cdots - 39\!\cdots\!40 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13\!\cdots\!14 \beta_{19} + \cdots + 21\!\cdots\!75 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 16\!\cdots\!46 \beta_{19} + \cdots + 13\!\cdots\!30 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 46\!\cdots\!81 \beta_{19} + \cdots - 75\!\cdots\!77 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 58\!\cdots\!39 \beta_{19} + \cdots - 49\!\cdots\!97 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 16\!\cdots\!29 \beta_{19} + \cdots + 26\!\cdots\!25 ) / 5 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 20\!\cdots\!41 \beta_{19} + \cdots + 17\!\cdots\!43 ) / 5 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 57\!\cdots\!41 \beta_{19} + \cdots - 90\!\cdots\!64 ) / 5 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 71\!\cdots\!69 \beta_{19} + \cdots - 61\!\cdots\!68 ) / 5 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 20\!\cdots\!04 \beta_{19} + \cdots + 31\!\cdots\!88 ) / 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 105.104i
0.500000 187.698i
0.500000 + 19.6223i
0.500000 + 4.33815i
0.500000 + 62.0735i
0.500000 22.2192i
0.500000 3.30996i
0.500000 + 100.713i
0.500000 + 3.46508i
0.500000 77.8364i
0.500000 + 22.2192i
0.500000 + 3.30996i
0.500000 100.713i
0.500000 3.46508i
0.500000 + 77.8364i
0.500000 105.104i
0.500000 + 187.698i
0.500000 19.6223i
0.500000 4.33815i
0.500000 62.0735i
−1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i −55.7220 + 4.47850i −29.1246 21.1603i −89.1519 51.7771 + 37.6183i 25.0304 77.0356i 85.9134 + 206.443i
31.2 −1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i −49.2840 26.3835i −29.1246 21.1603i 255.057 51.7771 + 37.6183i 25.0304 77.0356i −39.4504 + 220.099i
31.3 −1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i −13.4674 + 54.2552i −29.1246 21.1603i 11.3383 51.7771 + 37.6183i 25.0304 77.0356i 223.046 15.8303i
31.4 −1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i 11.2821 54.7514i −29.1246 21.1603i 29.3060 51.7771 + 37.6183i 25.0304 77.0356i −222.232 + 24.7567i
31.5 −1.23607 3.80423i 7.28115 5.29007i −12.9443 + 9.40456i 51.5150 + 21.7074i −29.1246 21.1603i −38.5661 51.7771 + 37.6183i 25.0304 77.0356i 18.9037 222.806i
61.1 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i −41.6409 37.2966i 11.1246 + 34.2380i −58.1692 −19.7771 60.8676i −65.5304 47.6106i −222.442 22.7907i
61.2 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i −38.1312 + 40.8780i 11.1246 + 34.2380i −22.2017 −19.7771 60.8676i −65.5304 47.6106i −27.2854 + 221.936i
61.3 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i 22.2851 51.2677i 11.1246 + 34.2380i 175.661 −19.7771 60.8676i −65.5304 47.6106i −48.4213 218.301i
61.4 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i 32.6761 + 45.3572i 11.1246 + 34.2380i −9.31479 −19.7771 60.8676i −65.5304 47.6106i 212.383 + 69.9528i
61.5 3.23607 2.35114i −2.78115 + 8.55951i 4.94427 15.2169i 52.9872 17.8146i 11.1246 + 34.2380i −163.959 −19.7771 60.8676i −65.5304 47.6106i 129.585 182.230i
91.1 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i −41.6409 + 37.2966i 11.1246 34.2380i −58.1692 −19.7771 + 60.8676i −65.5304 + 47.6106i −222.442 + 22.7907i
91.2 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i −38.1312 40.8780i 11.1246 34.2380i −22.2017 −19.7771 + 60.8676i −65.5304 + 47.6106i −27.2854 221.936i
91.3 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i 22.2851 + 51.2677i 11.1246 34.2380i 175.661 −19.7771 + 60.8676i −65.5304 + 47.6106i −48.4213 + 218.301i
91.4 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i 32.6761 45.3572i 11.1246 34.2380i −9.31479 −19.7771 + 60.8676i −65.5304 + 47.6106i 212.383 69.9528i
91.5 3.23607 + 2.35114i −2.78115 8.55951i 4.94427 + 15.2169i 52.9872 + 17.8146i 11.1246 34.2380i −163.959 −19.7771 + 60.8676i −65.5304 + 47.6106i 129.585 + 182.230i
121.1 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i −55.7220 4.47850i −29.1246 + 21.1603i −89.1519 51.7771 37.6183i 25.0304 + 77.0356i 85.9134 206.443i
121.2 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i −49.2840 + 26.3835i −29.1246 + 21.1603i 255.057 51.7771 37.6183i 25.0304 + 77.0356i −39.4504 220.099i
121.3 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i −13.4674 54.2552i −29.1246 + 21.1603i 11.3383 51.7771 37.6183i 25.0304 + 77.0356i 223.046 + 15.8303i
121.4 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i 11.2821 + 54.7514i −29.1246 + 21.1603i 29.3060 51.7771 37.6183i 25.0304 + 77.0356i −222.232 24.7567i
121.5 −1.23607 + 3.80423i 7.28115 + 5.29007i −12.9443 9.40456i 51.5150 21.7074i −29.1246 + 21.1603i −38.5661 51.7771 37.6183i 25.0304 + 77.0356i 18.9037 + 222.806i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.6.g.a 20
25.d even 5 1 inner 150.6.g.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.g.a 20 1.a even 1 1 trivial
150.6.g.a 20 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} - 90 T_{7}^{9} - 64540 T_{7}^{8} + 499140 T_{7}^{7} + 1080778560 T_{7}^{6} + 65226626294 T_{7}^{5} + 17548207095 T_{7}^{4} - 66684308302480 T_{7}^{3} + \cdots + 10\!\cdots\!84 \) acting on \(S_{6}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256)^{5} \) Copy content Toggle raw display
$3$ \( (T^{4} - 9 T^{3} + 81 T^{2} - 729 T + 6561)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} + 55 T^{19} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{10} - 90 T^{9} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} - 380 T^{19} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{20} + 230 T^{19} + \cdots + 69\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{20} - 2870 T^{19} + \cdots + 43\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{20} - 1560 T^{19} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} - 760 T^{19} + \cdots + 51\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{20} - 12000 T^{19} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{20} + 1710 T^{19} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{20} + 5595 T^{19} + \cdots + 94\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{20} + 2820 T^{19} + \cdots + 56\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( (T^{10} - 33010 T^{9} + \cdots - 92\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + 19630 T^{19} + \cdots + 80\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{20} - 62735 T^{19} + \cdots + 37\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{20} - 105100 T^{19} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{20} - 34790 T^{19} + \cdots + 50\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{20} + 19470 T^{19} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{20} + 111720 T^{19} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{20} + 198830 T^{19} + \cdots + 32\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{20} - 71210 T^{19} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + 288690 T^{19} + \cdots + 78\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} - 114225 T^{19} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{20} + 446970 T^{19} + \cdots + 63\!\cdots\!21 \) Copy content Toggle raw display
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