Properties

Label 11.7.d.a
Level $11$
Weight $7$
Character orbit 11.d
Analytic conductor $2.531$
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] = [11,7,Mod(2,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.2");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 11.d (of order \(10\), 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: \(2.53059491982\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 825 x^{18} + 275175 x^{16} + 47589550 x^{14} + 4569013705 x^{12} + 245564683275 x^{10} + 7342625961605 x^{8} + 117784752305650 x^{6} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 5\cdot 11^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + \beta_{6} - \beta_{4}) q^{2} + (\beta_{11} + 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4}) q^{3} + ( - \beta_{15} - \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 9 \beta_{6} - 16 \beta_{5} + \cdots + 9) q^{4}+ \cdots + ( - 3 \beta_{19} - 3 \beta_{17} - 2 \beta_{16} + 4 \beta_{15} - 6 \beta_{14} + 8 \beta_{13} + \cdots - 30) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} + \beta_{6} - \beta_{4}) q^{2} + (\beta_{11} + 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4}) q^{3} + ( - \beta_{15} - \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 9 \beta_{6} - 16 \beta_{5} + \cdots + 9) q^{4}+ \cdots + ( - 501 \beta_{19} - 1110 \beta_{18} - 2835 \beta_{17} + \cdots - 281502) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{2} - 39 q^{3} + 215 q^{4} + 181 q^{5} - 405 q^{6} - 365 q^{7} + 1595 q^{8} - 704 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{2} - 39 q^{3} + 215 q^{4} + 181 q^{5} - 405 q^{6} - 365 q^{7} + 1595 q^{8} - 704 q^{9} - 3498 q^{11} - 3006 q^{12} - 1805 q^{13} + 7170 q^{14} - 525 q^{15} - 2185 q^{16} + 3635 q^{17} + 11970 q^{18} + 23845 q^{19} + 5144 q^{20} - 2915 q^{22} + 7816 q^{23} - 123775 q^{24} - 30416 q^{25} - 131310 q^{26} - 18687 q^{27} + 226540 q^{28} + 134595 q^{29} + 220420 q^{30} - 71211 q^{31} + 5951 q^{33} - 228190 q^{34} - 377445 q^{35} - 119626 q^{36} - 205731 q^{37} + 127220 q^{38} + 443075 q^{39} + 704340 q^{40} + 490975 q^{41} - 20170 q^{42} - 537812 q^{44} - 805524 q^{45} - 714610 q^{46} + 25329 q^{47} - 935824 q^{48} + 304010 q^{49} + 417855 q^{50} + 1169565 q^{51} + 1468510 q^{52} - 110919 q^{53} - 5511 q^{55} - 862620 q^{56} - 1435995 q^{57} - 667940 q^{58} - 581009 q^{59} + 664640 q^{60} + 892675 q^{61} + 2337360 q^{62} + 900840 q^{63} + 124615 q^{64} + 272910 q^{66} - 960956 q^{67} - 1822680 q^{68} - 624034 q^{69} - 1987140 q^{70} - 288895 q^{71} + 954565 q^{72} - 806585 q^{73} - 404170 q^{74} - 550845 q^{75} + 623095 q^{77} + 1703080 q^{78} + 1662955 q^{79} + 1190956 q^{80} + 1465924 q^{81} - 282095 q^{82} + 14645 q^{83} - 2604390 q^{84} - 33365 q^{85} + 735635 q^{86} - 1860485 q^{88} + 1111620 q^{89} + 4118080 q^{90} + 650935 q^{91} + 4407784 q^{92} - 1556453 q^{93} - 5913080 q^{94} - 4329525 q^{95} - 6429020 q^{96} - 1189281 q^{97} + 252208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 825 x^{18} + 275175 x^{16} + 47589550 x^{14} + 4569013705 x^{12} + 245564683275 x^{10} + 7342625961605 x^{8} + 117784752305650 x^{6} + \cdots + 17\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!58 \nu^{18} + \cdots + 49\!\cdots\!55 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\!\cdots\!93 \nu^{18} + \cdots + 18\!\cdots\!50 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24\!\cdots\!89 \nu^{19} + \cdots - 98\!\cdots\!25 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24\!\cdots\!89 \nu^{19} + \cdots - 98\!\cdots\!25 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 97\!\cdots\!37 \nu^{19} + \cdots + 57\!\cdots\!25 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28\!\cdots\!98 \nu^{19} + \cdots + 12\!\cdots\!25 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28\!\cdots\!98 \nu^{19} + \cdots + 12\!\cdots\!25 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 28\!\cdots\!98 \nu^{19} + \cdots + 49\!\cdots\!25 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 85\!\cdots\!02 \nu^{19} + \cdots + 29\!\cdots\!25 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!26 \nu^{19} + \cdots - 22\!\cdots\!75 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23\!\cdots\!67 \nu^{19} + \cdots - 65\!\cdots\!75 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23\!\cdots\!67 \nu^{19} + \cdots - 65\!\cdots\!75 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16\!\cdots\!54 \nu^{19} + \cdots - 44\!\cdots\!25 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 16\!\cdots\!54 \nu^{19} + \cdots + 44\!\cdots\!25 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 34\!\cdots\!22 \nu^{19} + \cdots - 24\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 22\!\cdots\!11 \nu^{19} + \cdots + 26\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 29\!\cdots\!09 \nu^{19} + \cdots - 20\!\cdots\!50 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 18\!\cdots\!49 \nu^{19} + \cdots + 61\!\cdots\!00 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{8} + 10\beta_{5} + 10\beta_{4} + \beta_{2} - 77 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - 4 \beta_{18} + \beta_{16} + 2 \beta_{14} - 10 \beta_{13} + 2 \beta_{12} - 5 \beta_{11} - \beta_{10} + 11 \beta_{8} - 10 \beta_{7} - 40 \beta_{6} + 23 \beta_{5} - 60 \beta_{4} - 4 \beta_{3} - 150 \beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{19} + 6 \beta_{18} + 6 \beta_{17} + 3 \beta_{16} - 21 \beta_{15} + 15 \beta_{14} + 36 \beta_{13} + 36 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + 416 \beta_{9} + 353 \beta_{8} - 54 \beta_{7} - 2525 \beta_{5} - 2528 \beta_{4} + \cdots + 11581 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 311 \beta_{19} + 913 \beta_{18} - 379 \beta_{16} - 217 \beta_{15} - 440 \beta_{14} + 1894 \beta_{13} - 552 \beta_{12} + 740 \beta_{11} + 311 \beta_{10} - 34 \beta_{9} - 4537 \beta_{8} + 4192 \beta_{7} + 24834 \beta_{6} + \cdots + 12417 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2778 \beta_{19} - 1545 \beta_{18} - 2466 \beta_{17} - 312 \beta_{16} + 8573 \beta_{15} - 7028 \beta_{14} - 7956 \beta_{13} - 7956 \beta_{12} - 1233 \beta_{11} + 312 \beta_{10} - 115496 \beta_{9} - 94164 \beta_{8} + \cdots - 1954786 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 72795 \beta_{19} - 183075 \beta_{18} + 115677 \beta_{16} + 86168 \beta_{15} + 80771 \beta_{14} - 326462 \beta_{13} + 118484 \beta_{12} - 97698 \beta_{11} - 72795 \beta_{10} + 21441 \beta_{9} + \cdots - 3983454 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 724059 \beta_{19} + 357096 \beta_{18} + 733926 \beta_{17} - 9867 \beta_{16} - 2457477 \beta_{15} + 2100381 \beta_{14} + 1494388 \beta_{13} + 1494388 \beta_{12} + 366963 \beta_{11} + \cdots + 340929844 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 16027419 \beta_{19} + 36242477 \beta_{18} - 31149291 \beta_{16} - 25239087 \beta_{15} - 14304854 \beta_{14} + 56139470 \beta_{13} - 23783976 \beta_{12} + 12140436 \beta_{11} + \cdots + 1081812767 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 177433320 \beta_{19} - 80960775 \beta_{18} - 192945090 \beta_{17} + 15511770 \beta_{16} + 620468945 \beta_{15} - 539508170 \beta_{14} - 272508420 \beta_{13} + \cdots - 60752250095 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3458033680 \beta_{19} - 7221323455 \beta_{18} + 7815848902 \beta_{16} + 6559955258 \beta_{15} + 2507396131 \beta_{14} - 9770954160 \beta_{13} + 4668782350 \beta_{12} + \cdots - 271951884092 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 41955882978 \beta_{19} + 18178392552 \beta_{18} + 47554980852 \beta_{17} - 5599097874 \beta_{16} - 147170162846 \beta_{15} + 128991770294 \beta_{14} + \cdots + 11046985263911 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 739136617740 \beta_{19} + 1452420476070 \beta_{18} - 1877678369620 \beta_{16} - 1601707621330 \beta_{15} - 437313110040 \beta_{14} + \cdots + 65355452894190 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9686422297230 \beta_{19} - 4050762433830 \beta_{18} - 11271319726800 \beta_{17} + 1584897429570 \beta_{16} + 33677574684680 \beta_{15} + \cdots - 20\!\cdots\!85 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 157184678046895 \beta_{19} - 294785742200280 \beta_{18} + 438262889376475 \beta_{16} + 376645767821570 \beta_{15} + 75983942598480 \beta_{14} + \cdots - 15\!\cdots\!30 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 21\!\cdots\!25 \beta_{19} + 896902981030500 \beta_{18} + \cdots + 38\!\cdots\!55 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 33\!\cdots\!25 \beta_{19} + \cdots + 34\!\cdots\!05 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 49\!\cdots\!60 \beta_{19} + \cdots - 74\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 70\!\cdots\!75 \beta_{19} + \cdots - 78\!\cdots\!20 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
14.6250i
6.18436i
0.868969i
6.47390i
13.3023i
14.6250i
6.18436i
0.868969i
6.47390i
13.3023i
12.4579i
3.41863i
2.50105i
4.59184i
12.6102i
12.4579i
3.41863i
2.50105i
4.59184i
12.6102i
−12.7912 4.15611i 0.366417 0.266218i 94.5639 + 68.7047i 50.5827 + 155.677i −5.79334 + 1.88237i 50.6754 69.7488i −418.095 575.459i −225.210 + 693.125i 2201.52i
2.2 −4.76365 1.54780i 10.9526 7.95751i −31.4805 22.8719i −55.7063 171.446i −64.4908 + 20.9543i 91.1264 125.425i 302.983 + 417.020i −168.636 + 519.010i 902.932i
2.3 0.291595 + 0.0947451i −33.5534 + 24.3780i −51.7010 37.5630i 29.6861 + 91.3645i −12.0937 + 3.92948i −91.4937 + 125.930i −23.0507 31.7266i 306.271 942.605i 29.4541i
2.4 7.27508 + 2.36382i 23.2263 16.8749i −4.43797 3.22437i 31.2971 + 96.3226i 208.862 67.8633i −91.3973 + 125.798i −312.424 430.015i 29.4247 90.5599i 774.735i
2.5 13.7693 + 4.47392i −20.2452 + 14.7090i 117.801 + 85.5872i −50.2998 154.807i −344.569 + 111.957i 218.726 301.051i 694.490 + 955.883i −31.7606 + 97.7490i 2356.62i
6.1 −12.7912 + 4.15611i 0.366417 + 0.266218i 94.5639 68.7047i 50.5827 155.677i −5.79334 1.88237i 50.6754 + 69.7488i −418.095 + 575.459i −225.210 693.125i 2201.52i
6.2 −4.76365 + 1.54780i 10.9526 + 7.95751i −31.4805 + 22.8719i −55.7063 + 171.446i −64.4908 20.9543i 91.1264 + 125.425i 302.983 417.020i −168.636 519.010i 902.932i
6.3 0.291595 0.0947451i −33.5534 24.3780i −51.7010 + 37.5630i 29.6861 91.3645i −12.0937 3.92948i −91.4937 125.930i −23.0507 + 31.7266i 306.271 + 942.605i 29.4541i
6.4 7.27508 2.36382i 23.2263 + 16.8749i −4.43797 + 3.22437i 31.2971 96.3226i 208.862 + 67.8633i −91.3973 125.798i −312.424 + 430.015i 29.4247 + 90.5599i 774.735i
6.5 13.7693 4.47392i −20.2452 14.7090i 117.801 85.5872i −50.2998 + 154.807i −344.569 111.957i 218.726 + 301.051i 694.490 955.883i −31.7606 97.7490i 2356.62i
7.1 −8.44062 + 11.6175i −3.65047 + 11.2350i −43.9455 135.250i 22.0305 16.0061i −99.7104 137.240i −547.554 + 177.911i 1068.14 + 347.060i 476.874 + 346.469i 391.041i
7.2 −3.12745 + 4.30457i 13.7792 42.4080i 11.0287 + 33.9429i 141.251 102.625i 139.454 + 191.943i 43.6363 14.1783i −504.462 163.910i −1018.80 740.201i 928.982i
7.3 −2.58811 + 3.56223i −1.49018 + 4.58631i 13.7859 + 42.4287i −177.805 + 129.183i −12.4807 17.1783i 410.475 133.371i −454.830 147.783i 570.960 + 414.827i 967.721i
7.4 1.58098 2.17603i −12.6315 + 38.8758i 17.5415 + 53.9871i 114.702 83.3360i 64.6247 + 88.9483i −318.349 + 103.438i 308.927 + 100.377i −761.996 553.623i 381.348i
7.5 6.29405 8.66302i 3.74624 11.5298i −15.6557 48.1834i −15.2392 + 11.0719i −76.3035 105.023i 51.6547 16.7836i 135.823 + 44.1317i 470.872 + 342.109i 201.705i
8.1 −8.44062 11.6175i −3.65047 11.2350i −43.9455 + 135.250i 22.0305 + 16.0061i −99.7104 + 137.240i −547.554 177.911i 1068.14 347.060i 476.874 346.469i 391.041i
8.2 −3.12745 4.30457i 13.7792 + 42.4080i 11.0287 33.9429i 141.251 + 102.625i 139.454 191.943i 43.6363 + 14.1783i −504.462 + 163.910i −1018.80 + 740.201i 928.982i
8.3 −2.58811 3.56223i −1.49018 4.58631i 13.7859 42.4287i −177.805 129.183i −12.4807 + 17.1783i 410.475 + 133.371i −454.830 + 147.783i 570.960 414.827i 967.721i
8.4 1.58098 + 2.17603i −12.6315 38.8758i 17.5415 53.9871i 114.702 + 83.3360i 64.6247 88.9483i −318.349 103.438i 308.927 100.377i −761.996 + 553.623i 381.348i
8.5 6.29405 + 8.66302i 3.74624 + 11.5298i −15.6557 + 48.1834i −15.2392 11.0719i −76.3035 + 105.023i 51.6547 + 16.7836i 135.823 44.1317i 470.872 342.109i 201.705i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.7.d.a 20
3.b odd 2 1 99.7.k.a 20
11.c even 5 1 121.7.b.c 20
11.d odd 10 1 inner 11.7.d.a 20
11.d odd 10 1 121.7.b.c 20
33.f even 10 1 99.7.k.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.7.d.a 20 1.a even 1 1 trivial
11.7.d.a 20 11.d odd 10 1 inner
99.7.k.a 20 3.b odd 2 1
99.7.k.a 20 33.f even 10 1
121.7.b.c 20 11.c even 5 1
121.7.b.c 20 11.d odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(11, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 491270438912000 \) Copy content Toggle raw display
$3$ \( T^{20} + 39 T^{19} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{20} - 181 T^{19} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + 365 T^{19} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + 3498 T^{19} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{20} + 1805 T^{19} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} - 3635 T^{19} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{20} - 23845 T^{19} + \cdots + 94\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{10} - 3908 T^{9} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} - 134595 T^{19} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + 71211 T^{19} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{20} + 205731 T^{19} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} - 490975 T^{19} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + 64938860970 T^{18} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} - 25329 T^{19} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + 110919 T^{19} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + 581009 T^{19} + \cdots + 56\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{20} - 892675 T^{19} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} + 480478 T^{9} + \cdots + 34\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + 288895 T^{19} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{20} + 806585 T^{19} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} - 1662955 T^{19} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} - 14645 T^{19} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{10} - 555810 T^{9} + \cdots + 92\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + 1189281 T^{19} + \cdots + 30\!\cdots\!25 \) Copy content Toggle raw display
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