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Properties

Label 209.2.e.b
Level $209$
Weight $2$
Character orbit 209.e
Analytic conductor $1.669$
Analytic rank $0$
Dimension $18$
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] = [209,2,Mod(45,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.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: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + 14 x^{16} - 11 x^{15} + 130 x^{14} - 92 x^{13} + 629 x^{12} - 276 x^{11} + 2060 x^{10} - 681 x^{9} + 3955 x^{8} - 115 x^{7} + 4735 x^{6} + 105 x^{5} + 2904 x^{4} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{12} + \beta_{4}) q^{3} + (\beta_{13} - \beta_{8} + \beta_{3} - 1) q^{4} + ( - \beta_{16} - \beta_{9} - \beta_1) q^{5} + ( - \beta_{14} - \beta_{10} - \beta_{7} - \beta_{2} - \beta_1) q^{6} + (\beta_{5} + \beta_{4}) q^{7} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}) q^{8} + (\beta_{16} - \beta_{15} + \beta_{12} - \beta_{10} - \beta_{8} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{12} + \beta_{4}) q^{3} + (\beta_{13} - \beta_{8} + \beta_{3} - 1) q^{4} + ( - \beta_{16} - \beta_{9} - \beta_1) q^{5} + ( - \beta_{14} - \beta_{10} - \beta_{7} - \beta_{2} - \beta_1) q^{6} + (\beta_{5} + \beta_{4}) q^{7} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}) q^{8} + (\beta_{16} - \beta_{15} + \beta_{12} - \beta_{10} - \beta_{8} - 1) q^{9} + (\beta_{15} - \beta_{13} - \beta_{12} + 2 \beta_{8} - \beta_{3} + 2) q^{10} - q^{11} + ( - \beta_{7} - \beta_{5} - \beta_{4} + 1) q^{12} + ( - \beta_{16} + \beta_{15} - \beta_{13} + \beta_{8} - \beta_{3} + 1) q^{13} + ( - \beta_{16} + \beta_{15} - \beta_{14} - \beta_{10} - \beta_{9} - \beta_{6} + \beta_{5} - \beta_1) q^{14} + ( - \beta_{16} - \beta_{15} + \beta_{14} + 2 \beta_{10} + \beta_{8} + \beta_{7} + 2 \beta_{2} + 2 \beta_1 + 1) q^{15} + (\beta_{16} + \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - 2 \beta_{4} + \cdots + \beta_1) q^{16}+ \cdots + ( - \beta_{16} + \beta_{15} - \beta_{12} + \beta_{10} + \beta_{8} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + q^{2} - 9 q^{4} - 3 q^{6} - 6 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + q^{2} - 9 q^{4} - 3 q^{6} - 6 q^{7} - 11 q^{9} + 21 q^{10} - 18 q^{11} + 16 q^{12} + 11 q^{13} + q^{14} + 7 q^{15} - 13 q^{16} + 6 q^{17} - 8 q^{18} + 4 q^{19} + 4 q^{20} + 24 q^{21} - q^{22} - q^{23} - 3 q^{24} - 7 q^{25} - 4 q^{26} + 6 q^{27} - 2 q^{28} + 13 q^{29} - 64 q^{30} - 32 q^{31} - 14 q^{32} + 8 q^{34} - 12 q^{35} + 10 q^{36} - 36 q^{37} - 20 q^{38} + 20 q^{39} + 32 q^{40} + 4 q^{41} + 11 q^{42} + q^{43} + 9 q^{44} + 22 q^{45} - 14 q^{46} - 14 q^{47} + 37 q^{48} - 12 q^{49} + 30 q^{50} - 16 q^{51} + 57 q^{52} - 4 q^{53} - 33 q^{54} - 34 q^{56} - 7 q^{57} + 8 q^{58} + 31 q^{59} + 9 q^{60} + 3 q^{61} + 5 q^{62} + 50 q^{63} + 64 q^{64} - 56 q^{65} + 3 q^{66} - 2 q^{67} - 96 q^{68} - 58 q^{70} - 12 q^{71} + 5 q^{72} - 26 q^{73} + 35 q^{74} + 50 q^{75} - 36 q^{76} + 6 q^{77} + 3 q^{78} + 31 q^{79} + 31 q^{80} - 21 q^{81} + 25 q^{82} + 36 q^{83} - 102 q^{84} - 11 q^{85} + 41 q^{86} - 124 q^{87} - 11 q^{89} + 51 q^{90} - 20 q^{91} - 33 q^{92} + 20 q^{93} - 86 q^{94} + 17 q^{95} + 60 q^{96} + 16 q^{97} + 4 q^{98} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 14 x^{16} - 11 x^{15} + 130 x^{14} - 92 x^{13} + 629 x^{12} - 276 x^{11} + 2060 x^{10} - 681 x^{9} + 3955 x^{8} - 115 x^{7} + 4735 x^{6} + 105 x^{5} + 2904 x^{4} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\!\cdots\!19 \nu^{17} + \cdots + 65\!\cdots\!64 ) / 57\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 43\!\cdots\!56 \nu^{17} + \cdots + 17\!\cdots\!35 ) / 57\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27\!\cdots\!03 \nu^{17} + \cdots + 74\!\cdots\!51 ) / 41\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!99 \nu^{17} + \cdots - 69\!\cdots\!97 ) / 20\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\!\cdots\!49 \nu^{17} + \cdots - 23\!\cdots\!00 ) / 41\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49\!\cdots\!66 \nu^{17} + \cdots - 40\!\cdots\!41 ) / 41\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21\!\cdots\!88 \nu^{17} + \cdots + 24\!\cdots\!52 ) / 17\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!49 \nu^{17} + \cdots - 33\!\cdots\!94 ) / 88\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 33\!\cdots\!75 \nu^{17} + \cdots + 12\!\cdots\!04 ) / 15\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 92\!\cdots\!95 \nu^{17} + \cdots - 21\!\cdots\!58 ) / 41\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 40\!\cdots\!23 \nu^{17} + \cdots + 27\!\cdots\!00 ) / 12\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 22\!\cdots\!44 \nu^{17} + \cdots + 24\!\cdots\!23 ) / 57\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 49\!\cdots\!08 \nu^{17} + \cdots - 42\!\cdots\!13 ) / 12\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 22\!\cdots\!36 \nu^{17} + \cdots + 16\!\cdots\!02 ) / 41\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 25\!\cdots\!33 \nu^{17} + \cdots + 18\!\cdots\!08 ) / 41\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 35\!\cdots\!91 \nu^{17} + \cdots - 25\!\cdots\!96 ) / 41\!\cdots\!46 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - 3\beta_{8} + \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + \beta_{14} - 7\beta_{13} + 2\beta_{12} - \beta_{10} + \beta_{9} + 15\beta_{8} - \beta_{6} - 2\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{17} - 9 \beta_{16} - 9 \beta_{15} + 10 \beta_{14} + 9 \beta_{13} + 11 \beta_{10} - 2 \beta_{8} + 10 \beta_{7} + 9 \beta_{3} - 29 \beta_{2} - 29 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} - 10\beta_{9} + 10\beta_{7} + 10\beta_{6} + \beta_{5} + 23\beta_{4} - 49\beta_{3} + 12\beta_{2} + 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 68 \beta_{17} + 67 \beta_{16} + 70 \beta_{15} - 81 \beta_{14} - 71 \beta_{13} - 68 \beta_{11} - 92 \beta_{10} + 67 \beta_{9} + 25 \beta_{8} - 92 \beta_{6} + 70 \beta_{5} + 184 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{17} - 84 \beta_{16} + 13 \beta_{15} - 78 \beta_{14} + 347 \beta_{13} - 198 \beta_{12} + 82 \beta_{10} - 590 \beta_{8} - 78 \beta_{7} + 347 \beta_{3} - 110 \beta_{2} - 110 \beta _1 - 590 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 493 \beta_{11} - 480 \beta_{9} - 609 \beta_{7} + 705 \beta_{6} - 522 \beta_{5} + 2 \beta_{4} - 539 \beta_{3} + 1232 \beta_{2} + 236 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 142 \beta_{17} + 664 \beta_{16} - 112 \beta_{15} + 561 \beta_{14} - 2476 \beta_{13} + 1539 \beta_{12} - 142 \beta_{11} - 637 \beta_{10} + 664 \beta_{9} + 4037 \beta_{8} - 637 \beta_{6} - 112 \beta_{5} - 1539 \beta_{4} + \cdots + 915 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3532 \beta_{17} - 3420 \beta_{16} - 3813 \beta_{15} + 4434 \beta_{14} + 4028 \beta_{13} - 49 \beta_{12} + 5213 \beta_{10} - 2015 \beta_{8} + 4434 \beta_{7} + 4028 \beta_{3} - 8506 \beta_{2} - 8506 \beta _1 - 2015 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1275 \beta_{11} - 5088 \beta_{9} + 3889 \beta_{7} + 4856 \beta_{6} + 799 \beta_{5} + 11440 \beta_{4} - 17747 \beta_{3} + 7269 \beta_{2} + 28213 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 25217 \beta_{17} + 24418 \beta_{16} + 27523 \beta_{15} - 31801 \beta_{14} - 29872 \beta_{13} + 735 \beta_{12} - 25217 \beta_{11} - 37932 \beta_{10} + 24418 \beta_{9} + 16411 \beta_{8} - 37932 \beta_{6} + \cdots + 59737 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 10786 \beta_{17} - 38309 \beta_{16} + 5034 \beta_{15} - 26411 \beta_{14} + 127541 \beta_{13} - 83247 \beta_{12} + 36738 \beta_{10} - 199309 \beta_{8} - 26411 \beta_{7} + 127541 \beta_{3} + \cdots - 199309 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 179904 \beta_{11} - 174870 \beta_{9} - 226413 \beta_{7} + 273937 \beta_{6} - 197295 \beta_{5} + 8756 \beta_{4} - 220613 \beta_{3} + 423507 \beta_{2} + 130234 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 88233 \beta_{17} + 285528 \beta_{16} - 28225 \beta_{15} + 176948 \beta_{14} - 918057 \beta_{13} + 599417 \beta_{12} - 88233 \beta_{11} - 276893 \beta_{10} + 285528 \beta_{9} + 1416097 \beta_{8} + \cdots + 430324 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1283665 \beta_{17} - 1255440 \beta_{16} - 1408758 \beta_{15} + 1606189 \beta_{14} + 1625274 \beta_{13} - 91310 \beta_{12} + 1971315 \beta_{10} - 1017156 \beta_{8} + 1606189 \beta_{7} + \cdots - 1017156 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(134\)
\(\chi(n)\) \(-1 - \beta_{8}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
−1.34782 − 2.33450i
−0.918695 − 1.59123i
−0.615930 − 1.06682i
−0.198213 − 0.343315i
−0.0694302 − 0.120257i
0.527190 + 0.913120i
0.797832 + 1.38189i
1.00688 + 1.74398i
1.31818 + 2.28316i
−1.34782 + 2.33450i
−0.918695 + 1.59123i
−0.615930 + 1.06682i
−0.198213 + 0.343315i
−0.0694302 + 0.120257i
0.527190 − 0.913120i
0.797832 − 1.38189i
1.00688 − 1.74398i
1.31818 − 2.28316i
−1.34782 − 2.33450i 0.0468836 + 0.0812047i −2.63325 + 4.56093i 1.12231 + 1.94390i 0.126382 − 0.218899i 0.215810 8.80535 1.49560 − 2.59046i 3.02535 − 5.24006i
45.2 −0.918695 − 1.59123i −1.56355 − 2.70815i −0.688002 + 1.19165i 1.63491 + 2.83176i −2.87286 + 4.97594i −3.63195 −1.14653 −3.38940 + 5.87061i 3.00398 − 5.20304i
45.3 −0.615930 − 1.06682i 1.01385 + 1.75603i 0.241261 − 0.417876i −1.31548 − 2.27848i 1.24892 − 2.16319i 3.75846 −3.05812 −0.555768 + 0.962618i −1.62049 + 2.80677i
45.4 −0.198213 − 0.343315i −0.634005 − 1.09813i 0.921423 − 1.59595i −0.220960 − 0.382715i −0.251336 + 0.435327i −0.667657 −1.52341 0.696076 − 1.20564i −0.0875945 + 0.151718i
45.5 −0.0694302 − 0.120257i 0.748393 + 1.29625i 0.990359 − 1.71535i 1.75825 + 3.04538i 0.103922 − 0.179998i −1.46902 −0.552764 0.379817 − 0.657862i 0.244151 − 0.422882i
45.6 0.527190 + 0.913120i 1.63062 + 2.82432i 0.444141 − 0.769275i −0.650102 − 1.12601i −1.71929 + 2.97791i −1.25919 3.04535 −3.81785 + 6.61271i 0.685455 − 1.18724i
45.7 0.797832 + 1.38189i −0.583506 − 1.01066i −0.273072 + 0.472974i 0.206639 + 0.357909i 0.931080 − 1.61268i 2.20560 2.31987 0.819041 − 1.41862i −0.329727 + 0.571103i
45.8 1.00688 + 1.74398i −1.29029 − 2.23484i −1.02763 + 1.77991i −1.77613 − 3.07635i 2.59834 − 4.50045i −4.09817 −0.111292 −1.82967 + 3.16909i 3.57672 − 6.19505i
45.9 1.31818 + 2.28316i 0.631607 + 1.09397i −2.47522 + 4.28721i −0.759439 − 1.31539i −1.66515 + 2.88412i 1.94612 −7.77846 0.702146 − 1.21615i 2.00216 − 3.46785i
144.1 −1.34782 + 2.33450i 0.0468836 − 0.0812047i −2.63325 − 4.56093i 1.12231 − 1.94390i 0.126382 + 0.218899i 0.215810 8.80535 1.49560 + 2.59046i 3.02535 + 5.24006i
144.2 −0.918695 + 1.59123i −1.56355 + 2.70815i −0.688002 − 1.19165i 1.63491 − 2.83176i −2.87286 − 4.97594i −3.63195 −1.14653 −3.38940 − 5.87061i 3.00398 + 5.20304i
144.3 −0.615930 + 1.06682i 1.01385 − 1.75603i 0.241261 + 0.417876i −1.31548 + 2.27848i 1.24892 + 2.16319i 3.75846 −3.05812 −0.555768 − 0.962618i −1.62049 − 2.80677i
144.4 −0.198213 + 0.343315i −0.634005 + 1.09813i 0.921423 + 1.59595i −0.220960 + 0.382715i −0.251336 − 0.435327i −0.667657 −1.52341 0.696076 + 1.20564i −0.0875945 − 0.151718i
144.5 −0.0694302 + 0.120257i 0.748393 − 1.29625i 0.990359 + 1.71535i 1.75825 − 3.04538i 0.103922 + 0.179998i −1.46902 −0.552764 0.379817 + 0.657862i 0.244151 + 0.422882i
144.6 0.527190 − 0.913120i 1.63062 − 2.82432i 0.444141 + 0.769275i −0.650102 + 1.12601i −1.71929 − 2.97791i −1.25919 3.04535 −3.81785 − 6.61271i 0.685455 + 1.18724i
144.7 0.797832 − 1.38189i −0.583506 + 1.01066i −0.273072 − 0.472974i 0.206639 − 0.357909i 0.931080 + 1.61268i 2.20560 2.31987 0.819041 + 1.41862i −0.329727 − 0.571103i
144.8 1.00688 − 1.74398i −1.29029 + 2.23484i −1.02763 − 1.77991i −1.77613 + 3.07635i 2.59834 + 4.50045i −4.09817 −0.111292 −1.82967 − 3.16909i 3.57672 + 6.19505i
144.9 1.31818 − 2.28316i 0.631607 − 1.09397i −2.47522 − 4.28721i −0.759439 + 1.31539i −1.66515 − 2.88412i 1.94612 −7.77846 0.702146 + 1.21615i 2.00216 + 3.46785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.e.b ✓ 18
19.c even 3 1 inner 209.2.e.b ✓ 18
19.c even 3 1 3971.2.a.k 9
19.d odd 6 1 3971.2.a.l 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.e.b ✓ 18 1.a even 1 1 trivial
209.2.e.b ✓ 18 19.c even 3 1 inner
3971.2.a.k 9 19.c even 3 1
3971.2.a.l 9 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - T_{2}^{17} + 14 T_{2}^{16} - 11 T_{2}^{15} + 130 T_{2}^{14} - 92 T_{2}^{13} + 629 T_{2}^{12} - 276 T_{2}^{11} + 2060 T_{2}^{10} - 681 T_{2}^{9} + 3955 T_{2}^{8} - 115 T_{2}^{7} + 4735 T_{2}^{6} + 105 T_{2}^{5} + 2904 T_{2}^{4} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - T^{17} + 14 T^{16} - 11 T^{15} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{18} + 19 T^{16} - 2 T^{15} + 251 T^{14} + \cdots + 196 \) Copy content Toggle raw display
$5$ \( T^{18} + 26 T^{16} + 14 T^{15} + \cdots + 7569 \) Copy content Toggle raw display
$7$ \( (T^{9} + 3 T^{8} - 24 T^{7} - 64 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{18} \) Copy content Toggle raw display
$13$ \( T^{18} - 11 T^{17} + 119 T^{16} + \cdots + 23020804 \) Copy content Toggle raw display
$17$ \( T^{18} - 6 T^{17} + 78 T^{16} + \cdots + 13571856 \) Copy content Toggle raw display
$19$ \( T^{18} - 4 T^{17} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + T^{17} + 61 T^{16} + 168 T^{15} + \cdots + 72900 \) Copy content Toggle raw display
$29$ \( T^{18} - 13 T^{17} + \cdots + 463344321636 \) Copy content Toggle raw display
$31$ \( (T^{9} + 16 T^{8} - 75 T^{7} + \cdots - 331264)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} + 18 T^{8} - 78 T^{7} + \cdots - 1086460)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 127125805400064 \) Copy content Toggle raw display
$43$ \( T^{18} - T^{17} + 143 T^{16} + \cdots + 41783296 \) Copy content Toggle raw display
$47$ \( T^{18} + 14 T^{17} + 230 T^{16} + \cdots + 101124 \) Copy content Toggle raw display
$53$ \( T^{18} + 4 T^{17} + 231 T^{16} + \cdots + 267813225 \) Copy content Toggle raw display
$59$ \( T^{18} - 31 T^{17} + \cdots + 76585986800964 \) Copy content Toggle raw display
$61$ \( T^{18} - 3 T^{17} + \cdots + 16374647833600 \) Copy content Toggle raw display
$67$ \( T^{18} + 2 T^{17} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{18} + 12 T^{17} + \cdots + 10122094962576 \) Copy content Toggle raw display
$73$ \( T^{18} + 26 T^{17} + \cdots + 120629793124 \) Copy content Toggle raw display
$79$ \( T^{18} - 31 T^{17} + \cdots + 51046624225 \) Copy content Toggle raw display
$83$ \( (T^{9} - 18 T^{8} - 155 T^{7} + \cdots - 3773520)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + 11 T^{17} + \cdots + 3020601600 \) Copy content Toggle raw display
$97$ \( T^{18} - 16 T^{17} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
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