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Properties

Label 153.2.e.c
Level $153$
Weight $2$
Character orbit 153.e
Analytic conductor $1.222$
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] = [153,2,Mod(52,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.52");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 153.e (of order \(3\), degree \(2\), 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: \(1.22171115093\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 3 x^{18} + 2 x^{17} + 13 x^{16} - 12 x^{15} + 54 x^{14} + 27 x^{13} + 93 x^{12} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{16} q^{2} - \beta_{10} q^{3} + ( - \beta_{5} - \beta_{3} - 1) q^{4} + (\beta_{14} - \beta_{7}) q^{5} + ( - \beta_{18} + \beta_{16} + \cdots - \beta_{3}) q^{6}+ \cdots + ( - \beta_{18} + \beta_{16} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{16} q^{2} - \beta_{10} q^{3} + ( - \beta_{5} - \beta_{3} - 1) q^{4} + (\beta_{14} - \beta_{7}) q^{5} + ( - \beta_{18} + \beta_{16} + \cdots - \beta_{3}) q^{6}+ \cdots + ( - \beta_{19} + 2 \beta_{18} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{3} - 14 q^{4} - q^{5} - 4 q^{6} - 11 q^{7} - 6 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - q^{3} - 14 q^{4} - q^{5} - 4 q^{6} - 11 q^{7} - 6 q^{8} + q^{9} + 12 q^{10} - 7 q^{11} - 7 q^{12} - 9 q^{13} - 2 q^{14} - 22 q^{16} + 20 q^{17} + 31 q^{18} + 18 q^{19} + 20 q^{20} - 14 q^{22} - 4 q^{23} - 36 q^{24} - 19 q^{25} - 20 q^{26} + 14 q^{27} + 48 q^{28} + 3 q^{29} - 16 q^{30} - 19 q^{31} + 11 q^{32} - 34 q^{33} - 10 q^{35} + 76 q^{36} + 32 q^{37} - 16 q^{38} + 20 q^{39} - 13 q^{40} - 6 q^{41} - 33 q^{42} - 26 q^{43} + 54 q^{44} + 6 q^{45} - 6 q^{46} + 4 q^{47} + 36 q^{48} - 11 q^{49} + 2 q^{50} - q^{51} - 5 q^{52} - 40 q^{53} + 47 q^{54} + 46 q^{55} - 12 q^{56} - 3 q^{57} - 2 q^{58} + 12 q^{59} - 95 q^{60} - 5 q^{61} + 54 q^{62} - 6 q^{63} + 10 q^{64} - 11 q^{65} - 20 q^{66} - 30 q^{67} - 14 q^{68} - 36 q^{69} + 10 q^{70} + 14 q^{71} - 12 q^{72} + 36 q^{73} + 4 q^{74} + 48 q^{75} - 2 q^{76} + 18 q^{77} - 90 q^{78} - 28 q^{79} - 134 q^{80} + q^{81} - 20 q^{82} + 15 q^{83} + 60 q^{84} - q^{85} + 19 q^{86} - 9 q^{87} - 31 q^{88} - 28 q^{89} + 55 q^{90} + 48 q^{91} - 22 q^{92} + 7 q^{93} + 21 q^{94} - 44 q^{95} - 36 q^{96} - 8 q^{97} + 118 q^{98} - 29 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 3 x^{18} + 2 x^{17} + 13 x^{16} - 12 x^{15} + 54 x^{14} + 27 x^{13} + 93 x^{12} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 374 \nu^{19} - 6263 \nu^{18} - 2178 \nu^{17} - 7838 \nu^{16} - 28624 \nu^{15} - 64917 \nu^{14} + \cdots - 68477157 ) / 50309748 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 162181 \nu^{19} - 3456124 \nu^{18} - 3311244 \nu^{17} + 20530838 \nu^{16} - 48719141 \nu^{15} + \cdots + 63570657492 ) / 21683501388 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 538 \nu^{19} - 2551 \nu^{18} + 16488 \nu^{17} + 104 \nu^{16} - 51140 \nu^{15} + 62109 \nu^{14} + \cdots + 160711695 ) / 50309748 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 109636 \nu^{19} + 123764 \nu^{18} - 582831 \nu^{17} + 3104843 \nu^{16} - 2102366 \nu^{15} + \cdots + 10404591264 ) / 3613916898 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37454 \nu^{19} - 341995 \nu^{18} + 928802 \nu^{17} - 179921 \nu^{16} - 773861 \nu^{15} + \cdots + 1630998990 ) / 1204638966 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1196 \nu^{19} + 700 \nu^{18} - 9333 \nu^{17} + 10087 \nu^{16} + 35540 \nu^{15} - 29007 \nu^{14} + \cdots - 64284678 ) / 25154874 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{19} + 2 \nu^{18} + 11 \nu^{16} + 19 \nu^{15} + 27 \nu^{14} + 18 \nu^{13} + 189 \nu^{12} + \cdots + 39366 ) / 19683 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1332865 \nu^{19} - 2191763 \nu^{18} - 6262074 \nu^{17} + 5873128 \nu^{16} + \cdots - 41392148337 ) / 21683501388 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 77800 \nu^{19} - 303913 \nu^{18} + 961857 \nu^{17} - 1175878 \nu^{16} + 419125 \nu^{15} + \cdots - 2157965388 ) / 1204638966 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2021131 \nu^{19} - 2555413 \nu^{18} - 21181914 \nu^{17} + 56246276 \nu^{16} + \cdots + 121436374581 ) / 21683501388 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2244395 \nu^{19} + 5526086 \nu^{18} - 16428690 \nu^{17} + 42763478 \nu^{16} + \cdots + 196431379884 ) / 21683501388 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2423198 \nu^{19} - 1251805 \nu^{18} + 10727622 \nu^{17} - 14845306 \nu^{16} + \cdots + 57607948521 ) / 21683501388 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1572781 \nu^{19} - 2419649 \nu^{18} + 128574 \nu^{17} - 22384385 \nu^{16} + \cdots - 132483320550 ) / 10841750694 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 179770 \nu^{19} + 459234 \nu^{18} - 968891 \nu^{17} + 77386 \nu^{16} + 611373 \nu^{15} + \cdots - 2986783713 ) / 1204638966 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 3366157 \nu^{19} - 1719611 \nu^{18} + 6125058 \nu^{17} - 41452910 \nu^{16} + \cdots - 122797454031 ) / 21683501388 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 391159 \nu^{19} - 10925 \nu^{18} - 96908 \nu^{17} + 3502616 \nu^{16} + 1238777 \nu^{15} + \cdots + 11097268839 ) / 2409277932 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 4522507 \nu^{19} + 17331394 \nu^{18} - 25205022 \nu^{17} - 7389914 \nu^{16} + \cdots - 9789773076 ) / 21683501388 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 5495137 \nu^{19} - 19856677 \nu^{18} + 46546218 \nu^{17} - 29477542 \nu^{16} + \cdots - 52593192513 ) / 21683501388 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{18} + \beta_{15} - \beta_{14} + \beta_{13} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} + \beta_{18} - \beta_{17} + 2 \beta_{16} - \beta_{14} + \beta_{11} - \beta_{10} + \beta_{9} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{19} + 2 \beta_{18} - 4 \beta_{17} - 2 \beta_{16} + 3 \beta_{13} - 3 \beta_{12} + \beta_{11} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{19} + 3 \beta_{18} + 2 \beta_{17} - 6 \beta_{16} - 5 \beta_{15} + 4 \beta_{14} + 5 \beta_{13} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10 \beta_{19} - 2 \beta_{18} + 5 \beta_{17} - 10 \beta_{16} - 13 \beta_{15} + 14 \beta_{14} + \cdots + 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 25 \beta_{19} - 31 \beta_{18} - 7 \beta_{17} + 10 \beta_{16} - 6 \beta_{15} - 9 \beta_{14} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 22 \beta_{19} - 15 \beta_{18} + 35 \beta_{17} + 36 \beta_{16} - 8 \beta_{15} + 7 \beta_{14} + \cdots + 68 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 16 \beta_{19} + 22 \beta_{18} + 8 \beta_{17} + 26 \beta_{16} + 17 \beta_{15} - \beta_{14} - 2 \beta_{13} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 83 \beta_{19} + 167 \beta_{18} + 2 \beta_{17} - 107 \beta_{16} + 24 \beta_{15} + 45 \beta_{14} + \cdots - 56 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5 \beta_{19} - 15 \beta_{18} + 179 \beta_{17} - 99 \beta_{16} + 55 \beta_{15} + 61 \beta_{14} + \cdots - 94 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 101 \beta_{19} - 14 \beta_{18} - 280 \beta_{17} + 143 \beta_{16} - 145 \beta_{15} - 334 \beta_{14} + \cdots - 636 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 218 \beta_{19} - 67 \beta_{18} - 214 \beta_{17} + 433 \beta_{16} + 60 \beta_{15} - 243 \beta_{14} + \cdots + 331 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 256 \beta_{19} - 375 \beta_{18} + 566 \beta_{17} + 711 \beta_{16} + 244 \beta_{15} - 506 \beta_{14} + \cdots - 148 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1190 \beta_{19} + 967 \beta_{18} - 487 \beta_{17} - 856 \beta_{16} + 1088 \beta_{15} + 890 \beta_{14} + \cdots + 1326 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1636 \beta_{19} - 3820 \beta_{18} + 3197 \beta_{17} - 647 \beta_{16} + 3084 \beta_{15} + 477 \beta_{14} + \cdots + 4003 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 4 \beta_{19} + 264 \beta_{18} + 1277 \beta_{17} - 4284 \beta_{16} + 334 \beta_{15} + 547 \beta_{14} + \cdots - 328 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 5114 \beta_{19} + 2623 \beta_{18} - 10063 \beta_{17} - 937 \beta_{16} + 1106 \beta_{15} - 1234 \beta_{14} + \cdots - 6756 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\)
\(\chi(n)\) \(1\) \(-1 - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
52.1
1.40050 − 1.01912i
−1.03577 − 1.38823i
−1.01936 + 1.40032i
−1.63674 + 0.566636i
1.60963 + 0.639600i
1.13474 + 1.30857i
−1.46274 − 0.927574i
0.616166 − 1.61875i
0.130006 + 1.72716i
0.763571 − 1.55466i
1.40050 + 1.01912i
−1.03577 + 1.38823i
−1.01936 − 1.40032i
−1.63674 − 0.566636i
1.60963 − 0.639600i
1.13474 − 1.30857i
−1.46274 + 0.927574i
0.616166 + 1.61875i
0.130006 − 1.72716i
0.763571 + 1.55466i
−1.30196 + 2.25507i −0.182333 − 1.72243i −2.39022 − 4.13999i 1.51106 + 2.61722i 4.12158 + 1.83137i −1.47356 + 2.55228i 7.24009 −2.93351 + 0.628109i −7.86936
52.2 −1.14169 + 1.97747i −1.72013 + 0.202892i −1.60693 − 2.78329i −0.889111 − 1.53999i 1.56264 − 3.63314i 0.988673 − 1.71243i 2.77172 2.91767 − 0.698001i 4.06037
52.3 −0.972555 + 1.68451i 0.703036 + 1.58295i −0.891725 − 1.54451i −1.50264 − 2.60265i −3.35025 − 0.355235i −2.26586 + 3.92459i −0.421213 −2.01148 + 2.22575i 5.84560
52.4 −0.375128 + 0.649740i −0.327649 + 1.70078i 0.718558 + 1.24458i 0.764161 + 1.32356i −0.982154 − 0.850896i 1.38856 − 2.40505i −2.57872 −2.78529 − 1.11452i −1.14663
52.5 −0.239792 + 0.415331i 1.35873 − 1.07418i 0.885000 + 1.53286i 0.557032 + 0.964808i 0.120330 + 0.821901i −2.30394 + 3.99054i −1.80803 0.692269 − 2.91903i −0.534287
52.6 0.0269334 − 0.0466500i 1.70063 − 0.328428i 0.998549 + 1.72954i −1.89634 − 3.28456i 0.0304825 − 0.0881800i 1.05082 − 1.82008i 0.215311 2.78427 − 1.11707i −0.204300
52.7 0.672207 − 1.16430i −1.53467 + 0.802982i 0.0962759 + 0.166755i 1.35514 + 2.34718i −0.0967076 + 2.32658i −1.05037 + 1.81929i 2.94770 1.71044 − 2.46463i 3.64375
52.8 0.742286 − 1.28568i −1.09379 − 1.34299i −0.101976 − 0.176628i −1.89642 − 3.28469i −2.53856 + 0.409382i −0.911088 + 1.57805i 2.66636 −0.607237 + 2.93790i −5.63073
52.9 1.27947 − 2.21610i 1.56077 + 0.750994i −2.27407 − 3.93880i 0.0873352 + 0.151269i 3.66123 − 2.49795i −1.25756 + 2.17815i −6.52050 1.87202 + 2.34426i 0.446970
52.10 1.31024 − 2.26940i −0.964587 − 1.43860i −2.43346 − 4.21488i 1.40978 + 2.44182i −4.52861 + 0.304125i 0.334319 − 0.579057i −7.51272 −1.13914 + 2.77531i 7.38862
103.1 −1.30196 − 2.25507i −0.182333 + 1.72243i −2.39022 + 4.13999i 1.51106 − 2.61722i 4.12158 − 1.83137i −1.47356 − 2.55228i 7.24009 −2.93351 − 0.628109i −7.86936
103.2 −1.14169 − 1.97747i −1.72013 − 0.202892i −1.60693 + 2.78329i −0.889111 + 1.53999i 1.56264 + 3.63314i 0.988673 + 1.71243i 2.77172 2.91767 + 0.698001i 4.06037
103.3 −0.972555 − 1.68451i 0.703036 − 1.58295i −0.891725 + 1.54451i −1.50264 + 2.60265i −3.35025 + 0.355235i −2.26586 − 3.92459i −0.421213 −2.01148 − 2.22575i 5.84560
103.4 −0.375128 − 0.649740i −0.327649 − 1.70078i 0.718558 − 1.24458i 0.764161 − 1.32356i −0.982154 + 0.850896i 1.38856 + 2.40505i −2.57872 −2.78529 + 1.11452i −1.14663
103.5 −0.239792 − 0.415331i 1.35873 + 1.07418i 0.885000 − 1.53286i 0.557032 − 0.964808i 0.120330 − 0.821901i −2.30394 − 3.99054i −1.80803 0.692269 + 2.91903i −0.534287
103.6 0.0269334 + 0.0466500i 1.70063 + 0.328428i 0.998549 − 1.72954i −1.89634 + 3.28456i 0.0304825 + 0.0881800i 1.05082 + 1.82008i 0.215311 2.78427 + 1.11707i −0.204300
103.7 0.672207 + 1.16430i −1.53467 − 0.802982i 0.0962759 − 0.166755i 1.35514 − 2.34718i −0.0967076 − 2.32658i −1.05037 − 1.81929i 2.94770 1.71044 + 2.46463i 3.64375
103.8 0.742286 + 1.28568i −1.09379 + 1.34299i −0.101976 + 0.176628i −1.89642 + 3.28469i −2.53856 − 0.409382i −0.911088 − 1.57805i 2.66636 −0.607237 − 2.93790i −5.63073
103.9 1.27947 + 2.21610i 1.56077 − 0.750994i −2.27407 + 3.93880i 0.0873352 − 0.151269i 3.66123 + 2.49795i −1.25756 − 2.17815i −6.52050 1.87202 − 2.34426i 0.446970
103.10 1.31024 + 2.26940i −0.964587 + 1.43860i −2.43346 + 4.21488i 1.40978 − 2.44182i −4.52861 − 0.304125i 0.334319 + 0.579057i −7.51272 −1.13914 − 2.77531i 7.38862
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 52.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.2.e.c ✓ 20
3.b odd 2 1 459.2.e.c 20
9.c even 3 1 inner 153.2.e.c ✓ 20
9.c even 3 1 1377.2.a.l 10
9.d odd 6 1 459.2.e.c 20
9.d odd 6 1 1377.2.a.k 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.2.e.c ✓ 20 1.a even 1 1 trivial
153.2.e.c ✓ 20 9.c even 3 1 inner
459.2.e.c 20 3.b odd 2 1
459.2.e.c 20 9.d odd 6 1
1377.2.a.k 10 9.d odd 6 1
1377.2.a.l 10 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 17 T_{2}^{18} + 2 T_{2}^{17} + 191 T_{2}^{16} + 27 T_{2}^{15} + 1235 T_{2}^{14} + 213 T_{2}^{13} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 17 T^{18} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{20} + T^{19} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} + T^{19} + \cdots + 278784 \) Copy content Toggle raw display
$7$ \( T^{20} + 11 T^{19} + \cdots + 20903184 \) Copy content Toggle raw display
$11$ \( T^{20} + 7 T^{19} + \cdots + 2047761 \) Copy content Toggle raw display
$13$ \( T^{20} + 9 T^{19} + \cdots + 9665881 \) Copy content Toggle raw display
$17$ \( (T - 1)^{20} \) Copy content Toggle raw display
$19$ \( (T^{10} - 9 T^{9} + \cdots - 4608)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 4 T^{19} + \cdots + 8311689 \) Copy content Toggle raw display
$29$ \( T^{20} - 3 T^{19} + \cdots + 18662400 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 199176769 \) Copy content Toggle raw display
$37$ \( (T^{10} - 16 T^{9} + \cdots - 188672)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 1077643305216 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 29240926470144 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 40\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{10} + 20 T^{9} + \cdots + 2324268)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 33867777024 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 78\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{10} - 7 T^{9} + \cdots + 4519104)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 18 T^{9} + \cdots + 14080176576)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 17\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{10} + 14 T^{9} + \cdots + 28566900)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 63\!\cdots\!16 \) Copy content Toggle raw display
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