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Properties

Label 209.2.e.a
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: \( x^{18} - x^{17} + 14 x^{16} - 15 x^{15} + 132 x^{14} - 132 x^{13} + 671 x^{12} - 520 x^{11} + 2312 x^{10} - 1413 x^{9} + 4605 x^{8} - 985 x^{7} + 5289 x^{6} - 427 x^{5} + 4276 x^{4} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{13} - \beta_{9} - \beta_{4} - 1) q^{4} + ( - \beta_{17} + \beta_{10}) q^{5} + (\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{6} + (\beta_{8} + \beta_{7} - \beta_{6} - 1) q^{7} + ( - \beta_{16} - \beta_{15} + \beta_{12} - \beta_{11} + \beta_{6} - \beta_{4} + \beta_{3}) q^{8} + ( - \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} + 2 \beta_{6} - \beta_{4} - \beta_{2} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{13} - \beta_{9} - \beta_{4} - 1) q^{4} + ( - \beta_{17} + \beta_{10}) q^{5} + (\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{6} + (\beta_{8} + \beta_{7} - \beta_{6} - 1) q^{7} + ( - \beta_{16} - \beta_{15} + \beta_{12} - \beta_{11} + \beta_{6} - \beta_{4} + \beta_{3}) q^{8} + ( - \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} + 2 \beta_{6} - \beta_{4} - \beta_{2} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{8} + 2 \beta_{6} - \beta_{4} - \beta_{2} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - q^{2} - 4 q^{3} - 9 q^{4} + 7 q^{6} - 6 q^{7} - 12 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - q^{2} - 4 q^{3} - 9 q^{4} + 7 q^{6} - 6 q^{7} - 12 q^{8} - 11 q^{9} - 11 q^{10} + 18 q^{11} + 8 q^{12} - q^{13} - 3 q^{14} - q^{15} - 5 q^{16} + 2 q^{17} + 12 q^{18} + 6 q^{19} - 44 q^{20} - 24 q^{21} - q^{22} + 11 q^{23} + 17 q^{24} - 15 q^{25} - 20 q^{26} + 14 q^{27} + 8 q^{28} - 13 q^{29} + 104 q^{30} + 16 q^{31} + 36 q^{32} - 4 q^{33} - 4 q^{34} + 6 q^{35} - 22 q^{36} + 28 q^{37} + 20 q^{38} - 4 q^{39} - 44 q^{40} - 16 q^{41} - 9 q^{42} + 15 q^{43} - 9 q^{44} - 34 q^{45} + 6 q^{46} - 2 q^{47} - 71 q^{48} + 12 q^{49} - 38 q^{50} + 26 q^{51} - 29 q^{52} - 16 q^{53} + 49 q^{54} - 10 q^{56} - 43 q^{57} - 37 q^{59} + 17 q^{60} - 15 q^{61} + q^{62} - 24 q^{63} + 32 q^{64} + 40 q^{65} + 7 q^{66} - 14 q^{67} + 92 q^{68} + 48 q^{69} + 26 q^{70} - 8 q^{71} + 33 q^{72} + 18 q^{73} + 7 q^{74} + 50 q^{75} - 56 q^{76} - 6 q^{77} - 9 q^{78} - q^{79} + 23 q^{80} - 53 q^{81} + 13 q^{82} - 16 q^{83} - 2 q^{84} + 31 q^{85} - 19 q^{86} + 20 q^{87} - 12 q^{88} - 3 q^{89} - 25 q^{90} - 28 q^{91} + 75 q^{92} - 8 q^{93} - 26 q^{94} - 5 q^{95} - 100 q^{96} - 44 q^{97} - 44 q^{98} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 14 x^{16} - 15 x^{15} + 132 x^{14} - 132 x^{13} + 671 x^{12} - 520 x^{11} + 2312 x^{10} - 1413 x^{9} + 4605 x^{8} - 985 x^{7} + 5289 x^{6} - 427 x^{5} + 4276 x^{4} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!36 \nu^{17} + \cdots - 22\!\cdots\!24 ) / 18\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21\!\cdots\!69 \nu^{17} + \cdots + 15\!\cdots\!64 ) / 61\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!22 \nu^{17} + \cdots - 61\!\cdots\!43 ) / 20\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 50\!\cdots\!84 \nu^{17} + \cdots + 16\!\cdots\!00 ) / 61\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!57 \nu^{17} + \cdots - 49\!\cdots\!02 ) / 92\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 90\!\cdots\!82 \nu^{17} + \cdots + 48\!\cdots\!31 ) / 61\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 58\!\cdots\!25 \nu^{17} + \cdots + 63\!\cdots\!60 ) / 18\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 58\!\cdots\!32 \nu^{17} + \cdots - 57\!\cdots\!94 ) / 18\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 55\!\cdots\!54 \nu^{17} + \cdots + 15\!\cdots\!19 ) / 92\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39\!\cdots\!59 \nu^{17} + \cdots - 24\!\cdots\!51 ) / 61\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 44\!\cdots\!05 \nu^{17} + \cdots - 32\!\cdots\!82 ) / 61\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 55\!\cdots\!66 \nu^{17} + \cdots - 55\!\cdots\!05 ) / 61\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15\!\cdots\!87 \nu^{17} + \cdots - 62\!\cdots\!10 ) / 92\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 66\!\cdots\!60 \nu^{17} + \cdots + 17\!\cdots\!10 ) / 30\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 45\!\cdots\!69 \nu^{17} + \cdots + 43\!\cdots\!90 ) / 18\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 50\!\cdots\!80 \nu^{17} + \cdots + 23\!\cdots\!86 ) / 18\!\cdots\!18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - 3\beta_{9} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} + \beta_{15} - \beta_{12} + \beta_{11} - \beta_{6} + \beta_{4} - 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} - \beta_{16} + \beta_{14} - 8 \beta_{13} - \beta_{12} + \beta_{10} + 13 \beta_{9} - 2 \beta_{7} - \beta_{5} + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} + \beta_{16} - 2 \beta_{15} - 3 \beta_{14} + 11 \beta_{13} + 9 \beta_{12} - 9 \beta_{9} - 2 \beta_{8} + 2 \beta_{6} + 10 \beta_{5} - 11 \beta_{4} + 30 \beta_{3} + 8 \beta_{2} - 30 \beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{16} + 12 \beta_{15} - \beta_{12} + 3 \beta_{11} - 11 \beta_{10} + 21 \beta_{8} + 21 \beta_{7} - 10 \beta_{6} - 2 \beta_{5} + 58 \beta_{4} - 25 \beta_{3} + 68 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14 \beta_{17} - 81 \beta_{16} - 44 \beta_{15} + 37 \beta_{14} - 99 \beta_{13} - 26 \beta_{12} - 55 \beta_{11} + 14 \beta_{10} + 69 \beta_{9} - 28 \beta_{7} + 44 \beta_{6} - 70 \beta_{5} - 55 \beta_{2} + 194 \beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 92 \beta_{17} + 26 \beta_{16} - 94 \beta_{15} - 97 \beta_{14} + 417 \beta_{13} + 112 \beta_{12} - 387 \beta_{9} - 174 \beta_{8} + 71 \beta_{6} + 138 \beta_{5} - 417 \beta_{4} + 236 \beta_{3} + 44 \beta_{2} - 236 \beta _1 - 413 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 523 \beta_{16} + 523 \beta_{15} - 279 \beta_{12} + 373 \beta_{11} - 141 \beta_{10} + 278 \beta_{8} + 278 \beta_{7} - 518 \beta_{6} - 94 \beta_{5} + 823 \beta_{4} - 1309 \beta_{3} + 337 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 706 \beta_{17} - 1195 \beta_{16} - 206 \beta_{15} + 797 \beta_{14} - 3015 \beta_{13} - 745 \beta_{12} - 450 \beta_{11} + 706 \beta_{10} + 2425 \beta_{9} - 1337 \beta_{7} + 206 \beta_{6} - 951 \beta_{5} - 450 \beta_{2} + \cdots + 206 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1247 \beta_{17} + 745 \beta_{16} - 2031 \beta_{15} - 2685 \beta_{14} + 6563 \beta_{13} + 3851 \beta_{12} - 3978 \beta_{9} - 2429 \beta_{8} + 1940 \beta_{6} + 4596 \beta_{5} - 6563 \beta_{4} + 9085 \beta_{3} + \cdots - 4723 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7706 \beta_{16} + 7706 \beta_{15} - 1967 \beta_{12} + 3998 \beta_{11} - 5250 \beta_{10} + 9976 \beta_{8} + 9976 \beta_{7} - 6268 \beta_{6} - 2031 \beta_{5} + 21960 \beta_{4} - 16427 \beta_{3} + 16162 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10330 \beta_{17} - 33967 \beta_{16} - 12223 \beta_{15} + 20662 \beta_{14} - 51109 \beta_{13} - 16005 \beta_{12} - 17962 \beta_{11} + 10330 \beta_{10} + 30448 \beta_{9} - 19963 \beta_{7} + \cdots + 12223 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 38624 \beta_{17} + 16005 \beta_{16} - 43664 \beta_{15} - 49337 \beta_{14} + 160985 \beta_{13} + 60806 \beta_{12} - 111173 \beta_{9} - 73609 \beta_{8} + 33332 \beta_{6} + 76811 \beta_{5} + \cdots - 127178 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 206935 \beta_{16} + 206935 \beta_{15} - 84174 \beta_{12} + 127838 \beta_{11} - 82484 \beta_{10} + 158662 \beta_{8} + 158662 \beta_{7} - 196222 \beta_{6} - 43664 \beta_{5} + 392404 \beta_{4} + \cdots + 192501 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 283550 \beta_{17} - 594730 \beta_{16} - 141805 \beta_{15} + 379697 \beta_{14} - 1186209 \beta_{13} - 330164 \beta_{12} - 264566 \beta_{11} + 283550 \beta_{10} + 788264 \beta_{9} + \cdots + 141805 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 644263 \beta_{17} + 330164 \beta_{16} - 928802 \beta_{15} - 1162819 \beta_{14} + 2986476 \beta_{13} + 1520281 \beta_{12} - 1778402 \beta_{9} - 1236074 \beta_{8} + 832655 \beta_{6} + \cdots - 2108566 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(78\) \(134\)
\(\chi(n)\) \(-1 - \beta_{9}\) \(1\)

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
1.17624 + 2.03730i
1.01915 + 1.76522i
0.831273 + 1.43981i
0.605399 + 1.04858i
0.111589 + 0.193278i
−0.424456 − 0.735179i
−0.448336 − 0.776541i
−1.00350 − 1.73811i
−1.36735 − 2.36833i
1.17624 − 2.03730i
1.01915 − 1.76522i
0.831273 − 1.43981i
0.605399 − 1.04858i
0.111589 − 0.193278i
−0.424456 + 0.735179i
−0.448336 + 0.776541i
−1.00350 + 1.73811i
−1.36735 + 2.36833i
−1.17624 − 2.03730i −0.948876 − 1.64350i −1.76706 + 3.06064i −1.45451 − 2.51928i −2.23220 + 3.86629i −0.513749 3.60897 −0.300733 + 0.520885i −3.42169 + 5.92654i
45.2 −1.01915 − 1.76522i 1.62111 + 2.80784i −1.07732 + 1.86598i 0.953652 + 1.65177i 3.30430 − 5.72321i −1.44214 0.315214 −3.75599 + 6.50557i 1.94382 − 3.36680i
45.3 −0.831273 − 1.43981i −0.205287 − 0.355568i −0.382031 + 0.661697i 1.09220 + 1.89174i −0.341299 + 0.591148i 4.21687 −2.05481 1.41571 − 2.45209i 1.81583 − 3.14511i
45.4 −0.605399 − 1.04858i 0.391191 + 0.677562i 0.266984 − 0.462430i −0.857641 − 1.48548i 0.473653 − 0.820391i −5.17559 −3.06812 1.19394 − 2.06796i −1.03843 + 1.79861i
45.5 −0.111589 − 0.193278i −1.70486 − 2.95291i 0.975096 − 1.68892i −0.909261 − 1.57489i −0.380488 + 0.659025i 3.46341 −0.881596 −4.31312 + 7.47054i −0.202927 + 0.351480i
45.6 0.424456 + 0.735179i 0.287689 + 0.498292i 0.639674 − 1.10795i −2.13253 − 3.69366i −0.244223 + 0.423006i 0.190562 2.78388 1.33447 − 2.31137i 1.81033 − 3.13559i
45.7 0.448336 + 0.776541i −0.820353 − 1.42089i 0.597989 − 1.03575i 1.31548 + 2.27848i 0.735588 − 1.27408i −1.27368 2.86575 0.154042 − 0.266809i −1.17956 + 2.04305i
45.8 1.00350 + 1.73811i 0.669895 + 1.16029i −1.01402 + 1.75633i 0.304819 + 0.527963i −1.34448 + 2.32870i −2.87509 −0.0562672 0.602481 − 1.04353i −0.611771 + 1.05962i
45.9 1.36735 + 2.36833i −1.29050 − 2.23522i −2.73931 + 4.74463i 1.68779 + 2.92334i 3.52915 − 6.11267i 0.409409 −9.51302 −1.83080 + 3.17105i −4.61562 + 7.99448i
144.1 −1.17624 + 2.03730i −0.948876 + 1.64350i −1.76706 − 3.06064i −1.45451 + 2.51928i −2.23220 − 3.86629i −0.513749 3.60897 −0.300733 − 0.520885i −3.42169 − 5.92654i
144.2 −1.01915 + 1.76522i 1.62111 − 2.80784i −1.07732 − 1.86598i 0.953652 − 1.65177i 3.30430 + 5.72321i −1.44214 0.315214 −3.75599 − 6.50557i 1.94382 + 3.36680i
144.3 −0.831273 + 1.43981i −0.205287 + 0.355568i −0.382031 − 0.661697i 1.09220 − 1.89174i −0.341299 − 0.591148i 4.21687 −2.05481 1.41571 + 2.45209i 1.81583 + 3.14511i
144.4 −0.605399 + 1.04858i 0.391191 − 0.677562i 0.266984 + 0.462430i −0.857641 + 1.48548i 0.473653 + 0.820391i −5.17559 −3.06812 1.19394 + 2.06796i −1.03843 − 1.79861i
144.5 −0.111589 + 0.193278i −1.70486 + 2.95291i 0.975096 + 1.68892i −0.909261 + 1.57489i −0.380488 − 0.659025i 3.46341 −0.881596 −4.31312 − 7.47054i −0.202927 − 0.351480i
144.6 0.424456 − 0.735179i 0.287689 − 0.498292i 0.639674 + 1.10795i −2.13253 + 3.69366i −0.244223 − 0.423006i 0.190562 2.78388 1.33447 + 2.31137i 1.81033 + 3.13559i
144.7 0.448336 − 0.776541i −0.820353 + 1.42089i 0.597989 + 1.03575i 1.31548 − 2.27848i 0.735588 + 1.27408i −1.27368 2.86575 0.154042 + 0.266809i −1.17956 − 2.04305i
144.8 1.00350 − 1.73811i 0.669895 − 1.16029i −1.01402 − 1.75633i 0.304819 − 0.527963i −1.34448 − 2.32870i −2.87509 −0.0562672 0.602481 + 1.04353i −0.611771 − 1.05962i
144.9 1.36735 − 2.36833i −1.29050 + 2.23522i −2.73931 − 4.74463i 1.68779 − 2.92334i 3.52915 + 6.11267i 0.409409 −9.51302 −1.83080 − 3.17105i −4.61562 − 7.99448i
\(n\): 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.a ✓ 18
19.c even 3 1 inner 209.2.e.a ✓ 18
19.c even 3 1 3971.2.a.m 9
19.d odd 6 1 3971.2.a.j 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.e.a ✓ 18 1.a even 1 1 trivial
209.2.e.a ✓ 18 19.c even 3 1 inner
3971.2.a.j 9 19.d odd 6 1
3971.2.a.m 9 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + T_{2}^{17} + 14 T_{2}^{16} + 15 T_{2}^{15} + 132 T_{2}^{14} + 132 T_{2}^{13} + 671 T_{2}^{12} + 520 T_{2}^{11} + 2312 T_{2}^{10} + 1413 T_{2}^{9} + 4605 T_{2}^{8} + 985 T_{2}^{7} + 5289 T_{2}^{6} + 427 T_{2}^{5} + 4276 T_{2}^{4} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + T^{17} + 14 T^{16} + 15 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{18} + 4 T^{17} + 27 T^{16} + 66 T^{15} + \cdots + 484 \) Copy content Toggle raw display
$5$ \( T^{18} + 30 T^{16} - 26 T^{15} + \cdots + 762129 \) Copy content Toggle raw display
$7$ \( (T^{9} + 3 T^{8} - 30 T^{7} - 82 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{18} \) Copy content Toggle raw display
$13$ \( T^{18} + T^{17} + 45 T^{16} + 150 T^{15} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{18} - 2 T^{17} + 70 T^{16} + \cdots + 18974736 \) Copy content Toggle raw display
$19$ \( T^{18} - 6 T^{17} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} - 11 T^{17} + \cdots + 19115274564 \) Copy content Toggle raw display
$29$ \( T^{18} + 13 T^{17} + \cdots + 38699545284 \) Copy content Toggle raw display
$31$ \( (T^{9} - 8 T^{8} - 35 T^{7} + 485 T^{6} + \cdots + 3872)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} - 14 T^{8} - 66 T^{7} + \cdots - 471948)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + 16 T^{17} + \cdots + 77720518656 \) Copy content Toggle raw display
$43$ \( T^{18} - 15 T^{17} + \cdots + 37122499584 \) Copy content Toggle raw display
$47$ \( T^{18} + 2 T^{17} + \cdots + 361636255044 \) Copy content Toggle raw display
$53$ \( T^{18} + 16 T^{17} + \cdots + 49461315201 \) Copy content Toggle raw display
$59$ \( T^{18} + 37 T^{17} + \cdots + 1647254170116 \) Copy content Toggle raw display
$61$ \( T^{18} + 15 T^{17} + \cdots + 8564391936 \) Copy content Toggle raw display
$67$ \( T^{18} + 14 T^{17} + \cdots + 36309590753536 \) Copy content Toggle raw display
$71$ \( T^{18} + 8 T^{17} + \cdots + 343531965456 \) Copy content Toggle raw display
$73$ \( T^{18} - 18 T^{17} + \cdots + 981997757764 \) Copy content Toggle raw display
$79$ \( T^{18} + T^{17} + \cdots + 432502207201 \) Copy content Toggle raw display
$83$ \( (T^{9} + 8 T^{8} - 567 T^{7} + \cdots + 141072192)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 111035618574336 \) Copy content Toggle raw display
$97$ \( T^{18} + 44 T^{17} + \cdots + 17\!\cdots\!49 \) Copy content Toggle raw display
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