Properties

Label 882.2.u.j
Level $882$
Weight $2$
Character orbit 882.u
Analytic conductor $7.043$
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] = [882,2,Mod(127,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.u (of order \(7\), degree \(6\), 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: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 15 x^{16} - 23 x^{15} + 72 x^{14} - 85 x^{13} + 432 x^{12} - 282 x^{11} + 1786 x^{10} - 1092 x^{9} + 7272 x^{8} - 10168 x^{7} + 25378 x^{6} - 43359 x^{5} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 98)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} + (\beta_{14} + \beta_{13} + \beta_{10} - \beta_{9} + \beta_{8} - 1) q^{4} + ( - \beta_{17} + \beta_{14} + \beta_{11}) q^{5} + (\beta_{12} + \beta_{10} + \beta_1 - 1) q^{7} + \beta_{14} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{2} + (\beta_{14} + \beta_{13} + \beta_{10} - \beta_{9} + \beta_{8} - 1) q^{4} + ( - \beta_{17} + \beta_{14} + \beta_{11}) q^{5} + (\beta_{12} + \beta_{10} + \beta_1 - 1) q^{7} + \beta_{14} q^{8} + ( - \beta_{16} - \beta_{8} + \beta_{7}) q^{10} + (\beta_{17} + \beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{7} - 2 \beta_{5}) q^{11} + (\beta_{15} + \beta_{14} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3}) q^{13} + ( - \beta_{17} - \beta_{13} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{14} - \beta_{8} q^{16} + (\beta_{16} - \beta_{15} - \beta_{14} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{17} + ( - \beta_{13} - \beta_{10} + \beta_{6} + 2 \beta_{3} - 2 \beta_{2} + 3) q^{19} + (\beta_{17} - \beta_{16} + \beta_{15} - \beta_{11} - \beta_{9} + \beta_{7}) q^{20} + (\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{3} - 1) q^{22} + ( - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{5} + \cdots - 1) q^{23}+ \cdots + (\beta_{17} - \beta_{16} + 2 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{2} - 3 q^{4} + 6 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{2} - 3 q^{4} + 6 q^{5} - 7 q^{7} + 3 q^{8} - 6 q^{10} + q^{11} + 7 q^{14} - 3 q^{16} + 11 q^{17} + 36 q^{19} - q^{20} - q^{22} - 5 q^{23} - 23 q^{25} + 7 q^{26} + 13 q^{29} - 4 q^{31} + 3 q^{32} - 11 q^{34} + 7 q^{35} + 33 q^{37} - 15 q^{38} + q^{40} + 28 q^{41} - 20 q^{43} - 6 q^{44} + 5 q^{46} - 36 q^{47} + 49 q^{49} - 26 q^{50} - 48 q^{53} + 47 q^{55} - 7 q^{56} + 36 q^{58} + 25 q^{59} + q^{61} + 11 q^{62} - 3 q^{64} + 56 q^{65} + 34 q^{67} - 38 q^{68} - 14 q^{70} + 6 q^{71} - 39 q^{73} + 23 q^{74} - 20 q^{76} + 28 q^{77} - 2 q^{79} - 8 q^{80} - 28 q^{82} - 35 q^{83} + 33 q^{85} - 36 q^{86} + 6 q^{88} + 6 q^{89} - 35 q^{91} - 5 q^{92} - 13 q^{94} + 17 q^{95} + 56 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} + 15 x^{16} - 23 x^{15} + 72 x^{14} - 85 x^{13} + 432 x^{12} - 282 x^{11} + 1786 x^{10} - 1092 x^{9} + 7272 x^{8} - 10168 x^{7} + 25378 x^{6} - 43359 x^{5} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!35 \nu^{17} + \cdots + 71\!\cdots\!94 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!55 \nu^{17} + \cdots - 25\!\cdots\!02 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 62\!\cdots\!27 \nu^{17} + \cdots + 29\!\cdots\!45 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 79\!\cdots\!31 \nu^{17} + \cdots + 18\!\cdots\!12 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 98\!\cdots\!25 \nu^{17} + \cdots + 15\!\cdots\!71 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!90 \nu^{17} + \cdots + 29\!\cdots\!71 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 65\!\cdots\!97 \nu^{17} + \cdots - 53\!\cdots\!57 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76\!\cdots\!84 \nu^{17} + \cdots - 13\!\cdots\!25 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 34\!\cdots\!38 \nu^{17} + \cdots + 15\!\cdots\!98 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37\!\cdots\!36 \nu^{17} + \cdots - 82\!\cdots\!95 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!95 \nu^{17} + \cdots - 60\!\cdots\!57 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!06 \nu^{17} + \cdots - 31\!\cdots\!28 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41\!\cdots\!05 \nu^{17} + \cdots + 39\!\cdots\!56 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 59\!\cdots\!48 \nu^{17} + \cdots - 32\!\cdots\!74 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 21\!\cdots\!01 \nu^{17} + \cdots - 90\!\cdots\!23 ) / 35\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 10\!\cdots\!83 \nu^{17} + \cdots - 28\!\cdots\!02 ) / 16\!\cdots\!91 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} + \beta_{16} - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - 5\beta_{10} + 2\beta_{9} - \beta_{8} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} - 6 \beta_{10} + 6 \beta_{9} - 4 \beta_{8} + \beta_{7} + \beta_{5} + 5 \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} - 12 \beta_{14} + 6 \beta_{11} - 29 \beta_{10} + 45 \beta_{9} - 29 \beta_{8} + 8 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 4 \beta_{3} + \beta_{2} + 2 \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32 \beta_{17} - 17 \beta_{15} - 55 \beta_{14} + 23 \beta_{13} + 20 \beta_{12} + 17 \beta_{11} - 55 \beta_{10} + 98 \beta_{9} - 98 \beta_{8} + 20 \beta_{7} + 21 \beta_{6} + 41 \beta_{5} - 6 \beta_{4} + 21 \beta_{3} + 6 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 162 \beta_{17} - 78 \beta_{16} - 35 \beta_{15} - 242 \beta_{14} + 180 \beta_{13} + 162 \beta_{12} + 242 \beta_{9} - 380 \beta_{8} + 78 \beta_{7} + 101 \beta_{6} + 101 \beta_{5} - 63 \beta_{4} + 63 \beta_{3} - 20 \beta_{2} + 20 \beta _1 - 180 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 581 \beta_{17} - 581 \beta_{16} - 695 \beta_{14} + 867 \beta_{13} + 738 \beta_{12} - 260 \beta_{11} + 867 \beta_{10} - 695 \beta_{8} + 260 \beta_{7} + 482 \beta_{6} + 338 \beta_{5} - 338 \beta_{4} + 131 \beta_{3} - 131 \beta_{2} + \cdots - 1248 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1788 \beta_{17} - 2877 \beta_{16} + 548 \beta_{15} - 1672 \beta_{14} + 3498 \beta_{13} + 2877 \beta_{12} - 1788 \beta_{11} + 6220 \beta_{10} - 3498 \beta_{9} + 548 \beta_{7} + 1302 \beta_{6} + 710 \beta_{5} - 1302 \beta_{4} + \cdots - 6220 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3907 \beta_{17} - 11007 \beta_{16} + 3907 \beta_{15} + 10554 \beta_{13} + 8778 \beta_{12} - 8778 \beta_{11} + 29522 \beta_{10} - 23691 \beta_{9} + 10554 \beta_{8} + 2950 \beta_{6} - 4471 \beta_{4} + \cdots - 23691 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 34258 \beta_{16} + 18780 \beta_{15} + 23343 \beta_{14} + 23343 \beta_{13} + 18780 \beta_{12} - 34258 \beta_{11} + 115569 \beta_{10} - 115569 \beta_{9} + 74768 \beta_{8} - 8317 \beta_{7} + \cdots - 74768 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 58350 \beta_{17} - 72822 \beta_{16} + 72822 \beta_{15} + 157800 \beta_{14} - 105493 \beta_{11} + 354367 \beta_{10} - 441790 \beta_{9} + 354367 \beta_{8} - 58350 \beta_{7} - 43108 \beta_{6} + \cdots - 157800 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 405403 \beta_{17} + 226992 \beta_{15} + 761719 \beta_{14} - 340865 \beta_{13} - 280742 \beta_{12} - 226992 \beta_{11} + 761719 \beta_{10} - 1370373 \beta_{9} + 1370373 \beta_{8} + \cdots - 132121 \beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1953735 \beta_{17} + 1084493 \beta_{16} + 483774 \beta_{15} + 2929197 \beta_{14} - 2350098 \beta_{13} - 1953735 \beta_{12} - 2929197 \beta_{9} + 4232543 \beta_{8} - 1084493 \beta_{7} + \cdots + 2350098 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7543291 \beta_{17} + 7543291 \beta_{16} + 9062260 \beta_{14} - 11313804 \beta_{13} - 9402630 \beta_{12} + 3361682 \beta_{11} - 11313804 \beta_{10} + 9062260 \beta_{8} + \cdots + 16356758 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 23320604 \beta_{17} + 36256752 \beta_{16} - 7185651 \beta_{15} + 19399238 \beta_{14} - 43593021 \beta_{13} - 36256752 \beta_{12} + 23320604 \beta_{11} - 78558012 \beta_{10} + \cdots + 78558012 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 49934244 \beta_{17} + 139836396 \beta_{16} - 49934244 \beta_{15} - 134834431 \beta_{13} - 112154680 \beta_{12} + 112154680 \beta_{11} - 377923739 \beta_{10} + \cdots + 303034222 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 432441179 \beta_{16} - 239996980 \beta_{15} - 288598279 \beta_{14} - 288598279 \beta_{13} - 239996980 \beta_{12} + 432441179 \beta_{11} - 1457176683 \beta_{10} + \cdots + 937092022 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\beta_{10}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.432251 + 1.89382i
−0.392347 1.71899i
0.281648 + 1.23398i
0.937338 0.451398i
−1.88755 + 0.908997i
3.47467 1.67331i
0.101116 + 0.126795i
−1.30651 1.63831i
1.35938 + 1.70461i
0.101116 0.126795i
−1.30651 + 1.63831i
1.35938 1.70461i
0.937338 + 0.451398i
−1.88755 0.908997i
3.47467 + 1.67331i
0.432251 1.89382i
−0.392347 + 1.71899i
0.281648 1.23398i
0.900969 + 0.433884i 0 0.623490 + 0.781831i −0.482195 2.11264i 0 −2.20649 + 1.45993i 0.222521 + 0.974928i 0 0.482195 2.11264i
127.2 0.900969 + 0.433884i 0 0.623490 + 0.781831i −0.0678680 0.297349i 0 −1.53790 2.15287i 0.222521 + 0.974928i 0 0.0678680 0.297349i
127.3 0.900969 + 0.433884i 0 0.623490 + 0.781831i 0.871615 + 3.81880i 0 2.52187 + 0.800098i 0.222521 + 0.974928i 0 −0.871615 + 3.81880i
253.1 −0.623490 0.781831i 0 −0.222521 + 0.974928i −0.426714 + 0.205495i 0 2.62504 + 0.330433i 0.900969 0.433884i 0 0.426714 + 0.205495i
253.2 −0.623490 0.781831i 0 −0.222521 + 0.974928i 0.309110 0.148859i 0 −2.03497 + 1.69083i 0.900969 0.433884i 0 −0.309110 0.148859i
253.3 −0.623490 0.781831i 0 −0.222521 + 0.974928i 2.64206 1.27235i 0 −2.49104 0.891482i 0.900969 0.433884i 0 −2.64206 1.27235i
379.1 0.222521 + 0.974928i 0 −0.900969 + 0.433884i −2.67921 3.35962i 0 −2.50613 0.848133i −0.623490 0.781831i 0 2.67921 3.35962i
379.2 0.222521 + 0.974928i 0 −0.900969 + 0.433884i 1.25326 + 1.57153i 0 −0.413525 2.61324i −0.623490 0.781831i 0 −1.25326 + 1.57153i
379.3 0.222521 + 0.974928i 0 −0.900969 + 0.433884i 1.57994 + 1.98119i 0 2.54314 + 0.729681i −0.623490 0.781831i 0 −1.57994 + 1.98119i
505.1 0.222521 0.974928i 0 −0.900969 0.433884i −2.67921 + 3.35962i 0 −2.50613 + 0.848133i −0.623490 + 0.781831i 0 2.67921 + 3.35962i
505.2 0.222521 0.974928i 0 −0.900969 0.433884i 1.25326 1.57153i 0 −0.413525 + 2.61324i −0.623490 + 0.781831i 0 −1.25326 1.57153i
505.3 0.222521 0.974928i 0 −0.900969 0.433884i 1.57994 1.98119i 0 2.54314 0.729681i −0.623490 + 0.781831i 0 −1.57994 1.98119i
631.1 −0.623490 + 0.781831i 0 −0.222521 0.974928i −0.426714 0.205495i 0 2.62504 0.330433i 0.900969 + 0.433884i 0 0.426714 0.205495i
631.2 −0.623490 + 0.781831i 0 −0.222521 0.974928i 0.309110 + 0.148859i 0 −2.03497 1.69083i 0.900969 + 0.433884i 0 −0.309110 + 0.148859i
631.3 −0.623490 + 0.781831i 0 −0.222521 0.974928i 2.64206 + 1.27235i 0 −2.49104 + 0.891482i 0.900969 + 0.433884i 0 −2.64206 + 1.27235i
757.1 0.900969 0.433884i 0 0.623490 0.781831i −0.482195 + 2.11264i 0 −2.20649 1.45993i 0.222521 0.974928i 0 0.482195 + 2.11264i
757.2 0.900969 0.433884i 0 0.623490 0.781831i −0.0678680 + 0.297349i 0 −1.53790 + 2.15287i 0.222521 0.974928i 0 0.0678680 + 0.297349i
757.3 0.900969 0.433884i 0 0.623490 0.781831i 0.871615 3.81880i 0 2.52187 0.800098i 0.222521 0.974928i 0 −0.871615 3.81880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.u.j 18
3.b odd 2 1 98.2.e.a 18
12.b even 2 1 784.2.u.c 18
21.c even 2 1 686.2.e.a 18
21.g even 6 2 686.2.g.j 36
21.h odd 6 2 686.2.g.i 36
49.e even 7 1 inner 882.2.u.j 18
147.k even 14 1 686.2.e.a 18
147.k even 14 1 4802.2.a.e 9
147.l odd 14 1 98.2.e.a 18
147.l odd 14 1 4802.2.a.h 9
147.n odd 42 2 686.2.g.i 36
147.o even 42 2 686.2.g.j 36
588.u even 14 1 784.2.u.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.e.a 18 3.b odd 2 1
98.2.e.a 18 147.l odd 14 1
686.2.e.a 18 21.c even 2 1
686.2.e.a 18 147.k even 14 1
686.2.g.i 36 21.h odd 6 2
686.2.g.i 36 147.n odd 42 2
686.2.g.j 36 21.g even 6 2
686.2.g.j 36 147.o even 42 2
784.2.u.c 18 12.b even 2 1
784.2.u.c 18 588.u even 14 1
882.2.u.j 18 1.a even 1 1 trivial
882.2.u.j 18 49.e even 7 1 inner
4802.2.a.e 9 147.k even 14 1
4802.2.a.h 9 147.l odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 6 T_{5}^{17} + 37 T_{5}^{16} - 166 T_{5}^{15} + 780 T_{5}^{14} - 3448 T_{5}^{13} + 13353 T_{5}^{12} - 39036 T_{5}^{11} + 92243 T_{5}^{10} - 170517 T_{5}^{9} + 267705 T_{5}^{8} - 260614 T_{5}^{7} + 136735 T_{5}^{6} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 6 T^{17} + 37 T^{16} - 166 T^{15} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{18} + 7 T^{17} - 154 T^{15} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{18} - T^{17} + 38 T^{16} + \cdots + 6126662529 \) Copy content Toggle raw display
$13$ \( T^{18} + 14 T^{16} + 721 T^{14} + \cdots + 19882681 \) Copy content Toggle raw display
$17$ \( T^{18} - 11 T^{17} + \cdots + 438986304 \) Copy content Toggle raw display
$19$ \( (T^{9} - 18 T^{8} + 49 T^{7} + \cdots + 500387)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + 5 T^{17} + 89 T^{16} + \cdots + 1225449 \) Copy content Toggle raw display
$29$ \( T^{18} - 13 T^{17} + 101 T^{16} + \cdots + 1347921 \) Copy content Toggle raw display
$31$ \( (T^{9} + 2 T^{8} - 42 T^{7} - 62 T^{6} + \cdots - 13)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} - 33 T^{17} + \cdots + 1703523546481 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 661949588881161 \) Copy content Toggle raw display
$43$ \( T^{18} + 20 T^{17} + \cdots + 63645702961 \) Copy content Toggle raw display
$47$ \( T^{18} + 36 T^{17} + 625 T^{16} + \cdots + 9308601 \) Copy content Toggle raw display
$53$ \( T^{18} + 48 T^{17} + \cdots + 97031627001 \) Copy content Toggle raw display
$59$ \( T^{18} - 25 T^{17} + \cdots + 83822409441 \) Copy content Toggle raw display
$61$ \( T^{18} - T^{17} + \cdots + 4441105256449 \) Copy content Toggle raw display
$67$ \( (T^{9} - 17 T^{8} - 297 T^{7} + \cdots - 149884139)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} - 6 T^{17} + 3 T^{16} + \cdots + 14085009 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 351446147128609 \) Copy content Toggle raw display
$79$ \( (T^{9} + T^{8} - 243 T^{7} - 742 T^{6} + \cdots + 5529329)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + 35 T^{17} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{18} - 6 T^{17} + \cdots + 10124977628529 \) Copy content Toggle raw display
$97$ \( (T^{9} - 28 T^{8} + 238 T^{7} - 224 T^{6} + \cdots + 8869)^{2} \) Copy content Toggle raw display
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