Properties

Label 855.2.bs.b
Level $855$
Weight $2$
Character orbit 855.bs
Analytic conductor $6.827$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(226,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.bs (of order \(9\), 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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 12 x^{16} - 8 x^{15} + 96 x^{14} - 75 x^{13} + 448 x^{12} - 405 x^{11} + 1521 x^{10} - 1294 x^{9} + 3333 x^{8} - 2616 x^{7} + 5113 x^{6} - 3126 x^{5} + 4032 x^{4} + \cdots + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 12 x^{16} - 8 x^{15} + 96 x^{14} - 75 x^{13} + 448 x^{12} - 405 x^{11} + 1521 x^{10} - 1294 x^{9} + 3333 x^{8} - 2616 x^{7} + 5113 x^{6} - 3126 x^{5} + 4032 x^{4} + \cdots + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 136164871502623 \nu^{17} + \cdots - 12\!\cdots\!58 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 305669388244104 \nu^{17} + 493873766181026 \nu^{16} + \cdots + 47\!\cdots\!16 ) / 78\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!19 \nu^{17} + \cdots + 40\!\cdots\!81 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!95 \nu^{17} + \cdots + 49\!\cdots\!59 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!82 \nu^{17} + \cdots + 73\!\cdots\!88 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!56 \nu^{17} + 305669388244104 \nu^{16} + \cdots - 15\!\cdots\!37 ) / 78\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 42\!\cdots\!70 \nu^{17} + \cdots - 79\!\cdots\!20 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 116934812256397 \nu^{17} - 400329673137974 \nu^{16} + \cdots - 19\!\cdots\!38 ) / 63\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47\!\cdots\!03 \nu^{17} + \cdots + 52\!\cdots\!54 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 61\!\cdots\!77 \nu^{17} + 828266836359098 \nu^{16} + \cdots + 60\!\cdots\!85 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 84\!\cdots\!25 \nu^{17} + \cdots - 58\!\cdots\!05 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 84\!\cdots\!44 \nu^{17} + \cdots - 15\!\cdots\!70 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 87\!\cdots\!64 \nu^{17} + \cdots + 47\!\cdots\!08 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 92\!\cdots\!56 \nu^{17} + \cdots + 30\!\cdots\!25 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 316478299376213 \nu^{17} - 588887179277519 \nu^{16} + \cdots - 38\!\cdots\!43 ) / 63\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 17\!\cdots\!19 \nu^{17} + \cdots - 11\!\cdots\!95 ) / 23\!\cdots\!13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{16} + 3\beta_{7} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{12} - 2\beta_{10} + \beta_{9} - \beta_{4} - 3\beta_{3} - 2\beta_{2} - 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} + 6 \beta_{16} + \beta_{14} - \beta_{13} + 2 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} - 5 \beta_{9} - 6 \beta_{8} - 11 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - 7 \beta_{11} + 9 \beta_{10} + 10 \beta_{8} + 9 \beta_{7} + 7 \beta_{6} + \beta_{5} + \beta_{4} + 11 \beta_{3} + 16 \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{17} - 8 \beta_{16} + 10 \beta_{15} - 9 \beta_{14} + 10 \beta_{13} - 12 \beta_{12} - 33 \beta_{11} - 58 \beta_{10} + 36 \beta_{9} + 24 \beta_{8} + 8 \beta_{6} + \beta_{5} - 22 \beta_{4} - 9 \beta_{3} - 39 \beta_{2} - 9 \beta _1 + 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 46 \beta_{17} + 69 \beta_{16} - 10 \beta_{15} + 22 \beta_{14} - 12 \beta_{13} + 77 \beta_{12} + 69 \beta_{11} + 79 \beta_{10} - 65 \beta_{9} - 79 \beta_{8} - 66 \beta_{7} - 46 \beta_{6} - 22 \beta_{5} + 63 \beta_{4} + 49 \beta _1 - 112 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 52 \beta_{17} - 176 \beta_{16} - 63 \beta_{15} - 14 \beta_{14} - 14 \beta_{13} - 76 \beta_{12} - 39 \beta_{11} + 157 \beta_{10} - 52 \beta_{9} + 129 \beta_{8} + 258 \beta_{7} + 39 \beta_{6} + 63 \beta_{5} + 76 \beta_{4} + 66 \beta_{3} + 267 \beta_{2} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 252 \beta_{17} - 53 \beta_{16} + 181 \beta_{15} - 105 \beta_{14} + 181 \beta_{13} - 417 \beta_{12} - 249 \beta_{11} - 1051 \beta_{10} + 501 \beta_{9} + 84 \beta_{8} + 53 \beta_{6} + 76 \beta_{5} - 550 \beta_{4} - 258 \beta_{3} + \cdots + 518 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 682 \beta_{17} + 1497 \beta_{16} - 133 \beta_{15} + 550 \beta_{14} - 417 \beta_{13} + 1349 \beta_{12} + 1497 \beta_{11} + 1702 \beta_{10} - 1169 \beta_{9} - 1702 \beta_{8} - 1527 \beta_{7} - 682 \beta_{6} + \cdots - 2209 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 487 \beta_{17} - 3061 \beta_{16} - 816 \beta_{15} - 533 \beta_{14} - 533 \beta_{13} - 1081 \beta_{12} - 1560 \beta_{11} + 3235 \beta_{10} - 487 \beta_{9} + 3342 \beta_{8} + 3246 \beta_{7} + 1560 \beta_{6} + 816 \beta_{5} + \cdots + 1560 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2804 \beta_{17} - 2093 \beta_{16} + 3829 \beta_{15} - 2748 \beta_{14} + 3829 \beta_{13} - 5996 \beta_{12} - 7227 \beta_{11} - 19673 \beta_{10} + 10031 \beta_{9} + 4035 \beta_{8} + 2093 \beta_{6} + 1081 \beta_{5} + \cdots + 11782 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 13860 \beta_{17} + 24807 \beta_{16} - 3646 \beta_{15} + 9642 \beta_{14} - 5996 \beta_{13} + 26415 \beta_{12} + 24807 \beta_{11} + 29062 \beta_{10} - 20922 \beta_{9} - 29062 \beta_{8} - 22573 \beta_{7} + \cdots - 36433 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13621 \beta_{17} - 54352 \beta_{16} - 18275 \beta_{15} - 8140 \beta_{14} - 8140 \beta_{13} - 24798 \beta_{12} - 20828 \beta_{11} + 56416 \beta_{10} - 13621 \beta_{9} + 53972 \beta_{8} + 63849 \beta_{7} + \cdots + 20828 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 65420 \beta_{17} - 28968 \beta_{16} + 67593 \beta_{15} - 42795 \beta_{14} + 67593 \beta_{13} - 122798 \beta_{12} - 106818 \beta_{11} - 353905 \beta_{10} + 172238 \beta_{9} + 49440 \beta_{8} + \cdots + 185488 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 239831 \beta_{17} + 463488 \beta_{16} - 58869 \beta_{15} + 181667 \beta_{14} - 122798 \beta_{13} + 469405 \beta_{12} + 463488 \beta_{11} + 542052 \beta_{10} - 373270 \beta_{9} + \cdots - 669036 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 208482 \beta_{17} - 982637 \beta_{16} - 300623 \beta_{15} - 168782 \beta_{14} - 168782 \beta_{13} - 416438 \beta_{12} - 436673 \beta_{11} + 1040364 \beta_{10} - 208482 \beta_{9} + 1039838 \beta_{8} + \cdots + 436673 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4} + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
−1.31112 + 2.27092i
−0.359728 + 0.623068i
0.731154 1.26640i
1.01081 1.75077i
0.597039 1.03410i
−0.841804 + 1.45805i
−1.31112 2.27092i
−0.359728 0.623068i
0.731154 + 1.26640i
−0.908512 + 1.57359i
0.296923 0.514286i
0.785237 1.36007i
1.01081 + 1.75077i
0.597039 + 1.03410i
−0.841804 1.45805i
−0.908512 1.57359i
0.296923 + 0.514286i
0.785237 + 1.36007i
−0.455347 + 2.58240i 0 −4.58206 1.66773i −0.939693 + 0.342020i 0 1.91579 + 3.31824i 3.77094 6.53146i 0 −0.455347 2.58240i
226.2 −0.124932 + 0.708527i 0 1.39298 + 0.507004i −0.939693 + 0.342020i 0 −0.645970 1.11885i −1.25271 + 2.16976i 0 −0.124932 0.708527i
226.3 0.253927 1.44009i 0 −0.130002 0.0473169i −0.939693 + 0.342020i 0 −2.03586 3.52622i 1.36116 2.35759i 0 0.253927 + 1.44009i
271.1 −1.89970 + 0.691434i 0 1.59869 1.34146i 0.766044 + 0.642788i 0 −1.54609 2.67790i −0.0878797 + 0.152212i 0 −1.89970 0.691434i
271.2 −1.12207 + 0.408399i 0 −0.439845 + 0.369074i 0.766044 + 0.642788i 0 1.09825 + 1.90222i 1.53688 2.66196i 0 −1.12207 0.408399i
271.3 1.58207 0.575828i 0 0.639290 0.536428i 0.766044 + 0.642788i 0 0.274194 + 0.474919i −0.981094 + 1.69930i 0 1.58207 + 0.575828i
541.1 −0.455347 2.58240i 0 −4.58206 + 1.66773i −0.939693 0.342020i 0 1.91579 3.31824i 3.77094 + 6.53146i 0 −0.455347 + 2.58240i
541.2 −0.124932 0.708527i 0 1.39298 0.507004i −0.939693 0.342020i 0 −0.645970 + 1.11885i −1.25271 2.16976i 0 −0.124932 + 0.708527i
541.3 0.253927 + 1.44009i 0 −0.130002 + 0.0473169i −0.939693 0.342020i 0 −2.03586 + 3.52622i 1.36116 + 2.35759i 0 0.253927 1.44009i
586.1 −1.39192 1.16796i 0 0.226016 + 1.28180i 0.173648 0.984808i 0 −0.536732 0.929646i −0.634528 + 1.09903i 0 −1.39192 + 1.16796i
586.2 0.454913 + 0.381717i 0 −0.286059 1.62232i 0.173648 0.984808i 0 −0.530259 0.918436i 1.08298 1.87578i 0 0.454913 0.381717i
586.3 1.20305 + 1.00948i 0 0.0809872 + 0.459301i 0.173648 0.984808i 0 2.00668 + 3.47568i 1.20425 2.08582i 0 1.20305 1.00948i
631.1 −1.89970 0.691434i 0 1.59869 + 1.34146i 0.766044 0.642788i 0 −1.54609 + 2.67790i −0.0878797 0.152212i 0 −1.89970 + 0.691434i
631.2 −1.12207 0.408399i 0 −0.439845 0.369074i 0.766044 0.642788i 0 1.09825 1.90222i 1.53688 + 2.66196i 0 −1.12207 + 0.408399i
631.3 1.58207 + 0.575828i 0 0.639290 + 0.536428i 0.766044 0.642788i 0 0.274194 0.474919i −0.981094 1.69930i 0 1.58207 0.575828i
766.1 −1.39192 + 1.16796i 0 0.226016 1.28180i 0.173648 + 0.984808i 0 −0.536732 + 0.929646i −0.634528 1.09903i 0 −1.39192 1.16796i
766.2 0.454913 0.381717i 0 −0.286059 + 1.62232i 0.173648 + 0.984808i 0 −0.530259 + 0.918436i 1.08298 + 1.87578i 0 0.454913 + 0.381717i
766.3 1.20305 1.00948i 0 0.0809872 0.459301i 0.173648 + 0.984808i 0 2.00668 3.47568i 1.20425 + 2.08582i 0 1.20305 + 1.00948i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.bs.b 18
3.b odd 2 1 95.2.k.b 18
15.d odd 2 1 475.2.l.b 18
15.e even 4 2 475.2.u.c 36
19.e even 9 1 inner 855.2.bs.b 18
57.j even 18 1 1805.2.a.u 9
57.l odd 18 1 95.2.k.b 18
57.l odd 18 1 1805.2.a.t 9
285.bd odd 18 1 475.2.l.b 18
285.bd odd 18 1 9025.2.a.ce 9
285.bf even 18 1 9025.2.a.cd 9
285.bi even 36 2 475.2.u.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.b 18 3.b odd 2 1
95.2.k.b 18 57.l odd 18 1
475.2.l.b 18 15.d odd 2 1
475.2.l.b 18 285.bd odd 18 1
475.2.u.c 36 15.e even 4 2
475.2.u.c 36 285.bi even 36 2
855.2.bs.b 18 1.a even 1 1 trivial
855.2.bs.b 18 19.e even 9 1 inner
1805.2.a.t 9 57.l odd 18 1
1805.2.a.u 9 57.j even 18 1
9025.2.a.cd 9 285.bf even 18 1
9025.2.a.ce 9 285.bd odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 3 T_{2}^{17} + 6 T_{2}^{16} + 5 T_{2}^{15} - 15 T_{2}^{14} - 24 T_{2}^{13} + 70 T_{2}^{12} + 96 T_{2}^{11} - 30 T_{2}^{10} + 52 T_{2}^{9} + 282 T_{2}^{8} - 357 T_{2}^{7} + 553 T_{2}^{6} + 1890 T_{2}^{5} + 750 T_{2}^{4} + 36 T_{2}^{3} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 3 T^{17} + 6 T^{16} + 5 T^{15} + \cdots + 361 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} + 33 T^{16} + 20 T^{15} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{18} + 36 T^{16} + 80 T^{15} + \cdots + 361 \) Copy content Toggle raw display
$13$ \( T^{18} + 3 T^{17} + 12 T^{16} + \cdots + 98743969 \) Copy content Toggle raw display
$17$ \( T^{18} - 24 T^{17} + 306 T^{16} - 2641 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{18} + 12 T^{17} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + 21 T^{17} + \cdots + 3993860809 \) Copy content Toggle raw display
$29$ \( T^{18} - 9 T^{17} + \cdots + 492805404001 \) Copy content Toggle raw display
$31$ \( T^{18} - 30 T^{17} + \cdots + 993257404129 \) Copy content Toggle raw display
$37$ \( (T^{9} + 30 T^{8} + 267 T^{7} + \cdots + 27721)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} - 6 T^{17} + \cdots + 132479256529 \) Copy content Toggle raw display
$43$ \( T^{18} + 6 T^{17} + 186 T^{16} + \cdots + 101989801 \) Copy content Toggle raw display
$47$ \( T^{18} + 33 T^{17} + \cdots + 32737816559809 \) Copy content Toggle raw display
$53$ \( T^{18} + 24 T^{17} + \cdots + 19250147475001 \) Copy content Toggle raw display
$59$ \( T^{18} + 18 T^{17} + \cdots + 333749999521 \) Copy content Toggle raw display
$61$ \( T^{18} - 6 T^{17} + \cdots + 3468418292161 \) Copy content Toggle raw display
$67$ \( T^{18} + 24 T^{17} + \cdots + 36\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{18} + 24 T^{17} + \cdots + 8590138489 \) Copy content Toggle raw display
$73$ \( T^{18} - 6 T^{17} + \cdots + 1047660743809 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 316957384070209 \) Copy content Toggle raw display
$83$ \( T^{18} + 171 T^{16} + \cdots + 96609241 \) Copy content Toggle raw display
$89$ \( T^{18} - 6 T^{17} - 63 T^{16} + \cdots + 143304841 \) Copy content Toggle raw display
$97$ \( T^{18} + 87 T^{17} + \cdots + 36\!\cdots\!09 \) Copy content Toggle raw display
show more
show less