Properties

Label 360.4.q.d
Level $360$
Weight $4$
Character orbit 360.q
Analytic conductor $21.241$
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] = [360,4,Mod(121,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.121");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.q (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: \(21.2406876021\)
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: \( x^{18} - 7 x^{17} + 49 x^{16} - 192 x^{15} + 1434 x^{14} - 9522 x^{13} + 52164 x^{12} - 327240 x^{11} + 1652157 x^{10} - 11483937 x^{9} + 44608239 x^{8} + \cdots + 7625597484987 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{9} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (5 \beta_{2} + 5) q^{5} + ( - \beta_{16} - 2 \beta_{2}) q^{7} + (\beta_{14} + \beta_{7} + 2 \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + (5 \beta_{2} + 5) q^{5} + ( - \beta_{16} - 2 \beta_{2}) q^{7} + (\beta_{14} + \beta_{7} + 2 \beta_{2} - 2) q^{9} + ( - \beta_{9} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 1) q^{11} + ( - \beta_{13} + \beta_{4} - 4 \beta_{2} - 4) q^{13} + ( - 5 \beta_{4} - 5 \beta_{3}) q^{15} + ( - \beta_{15} - \beta_{14} - \beta_{12} - 2 \beta_{5} - 5 \beta_{4} - 2 \beta_1 + 5) q^{17} + ( - \beta_{16} - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{8} + 3 \beta_{4} + \cdots - 7) q^{19}+ \cdots + ( - 2 \beta_{17} - 3 \beta_{16} + \beta_{15} - 19 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + \cdots - 94) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 7 q^{3} + 45 q^{5} + 20 q^{7} - 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 7 q^{3} + 45 q^{5} + 20 q^{7} - 49 q^{9} - 25 q^{11} - 36 q^{13} - 10 q^{15} + 46 q^{17} - 122 q^{19} - 12 q^{21} - 50 q^{23} - 225 q^{25} + 110 q^{27} - 110 q^{29} + 304 q^{31} + 613 q^{33} + 200 q^{35} - 8 q^{37} + 66 q^{39} + 127 q^{41} + 679 q^{43} - 235 q^{45} - 194 q^{47} + 299 q^{49} + 1617 q^{51} - 236 q^{53} - 250 q^{55} - 863 q^{57} - 57 q^{59} + 110 q^{61} - 1422 q^{63} + 180 q^{65} + 1025 q^{67} + 3684 q^{69} + 1664 q^{71} - 1894 q^{73} + 125 q^{75} + 492 q^{77} + 1590 q^{79} - 2761 q^{81} - 948 q^{83} + 115 q^{85} + 2432 q^{87} - 1644 q^{89} - 2172 q^{91} - 664 q^{93} - 305 q^{95} + 1353 q^{97} - 4304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 7 x^{17} + 49 x^{16} - 192 x^{15} + 1434 x^{14} - 9522 x^{13} + 52164 x^{12} - 327240 x^{11} + 1652157 x^{10} - 11483937 x^{9} + 44608239 x^{8} + \cdots + 7625597484987 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3528223223 \nu^{17} + 15222078368 \nu^{16} - 75436636826 \nu^{15} + 528595118826 \nu^{14} - 2994754903602 \nu^{13} + \cdots + 17\!\cdots\!80 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6125546080 \nu^{17} - 52383204461 \nu^{16} + 110844358016 \nu^{15} - 860684346942 \nu^{14} + 5488035129582 \nu^{13} + \cdots - 17\!\cdots\!37 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{17} + 7 \nu^{16} - 49 \nu^{15} + 192 \nu^{14} - 1434 \nu^{13} + 9522 \nu^{12} - 52164 \nu^{11} + 327240 \nu^{10} - 1652157 \nu^{9} + 11483937 \nu^{8} + \cdots + 1977006755367 ) / 282429536481 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 772531 \nu^{17} + 2809246 \nu^{16} + 23704622 \nu^{15} + 25721958 \nu^{14} - 295142718 \nu^{13} + 9050714190 \nu^{12} + \cdots + 24\!\cdots\!50 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11469797150 \nu^{17} + 224656086385 \nu^{16} + 1520239154564 \nu^{15} + 9128092201986 \nu^{14} + 37546434570318 \nu^{13} + \cdots + 77\!\cdots\!09 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 452742797 \nu^{17} - 16336129793 \nu^{16} + 21395912030 \nu^{15} - 297741564132 \nu^{14} + 1158674150508 \nu^{13} + \cdots - 43\!\cdots\!05 ) / 66\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2747535397 \nu^{17} - 26285208439 \nu^{16} + 65022372550 \nu^{15} - 1421305992264 \nu^{14} + 4790756993880 \nu^{13} + \cdots - 50\!\cdots\!83 ) / 29\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3287899388 \nu^{17} + 46217240231 \nu^{16} + 77634394264 \nu^{15} - 38118660438 \nu^{14} - 5188906101642 \nu^{13} + \cdots + 16\!\cdots\!43 ) / 29\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24660563263 \nu^{17} - 22568863315 \nu^{16} + 748142925706 \nu^{15} + 1815822363300 \nu^{14} - 2175824029164 \nu^{13} + \cdots - 40\!\cdots\!95 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24898229356 \nu^{17} - 277116824935 \nu^{16} + 371034583144 \nu^{15} - 5665556475714 \nu^{14} + 10618849215858 \nu^{13} + \cdots - 78\!\cdots\!23 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 43113659821 \nu^{17} + 137120502074 \nu^{16} + 627387305950 \nu^{15} + 704532964398 \nu^{14} + 23092268440170 \nu^{13} + \cdots + 53\!\cdots\!34 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23164090330 \nu^{17} - 8047292075 \nu^{16} - 203405270164 \nu^{15} + 4875059339058 \nu^{14} - 27017581302138 \nu^{13} + \cdots + 71\!\cdots\!09 ) / 89\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 49544083985 \nu^{17} + 550100268827 \nu^{16} - 1403098799222 \nu^{15} + 7114961418648 \nu^{14} - 56066607859776 \nu^{13} + \cdots + 11\!\cdots\!67 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 51187373033 \nu^{17} + 101254334957 \nu^{16} - 2952607211030 \nu^{15} - 1186237047756 \nu^{14} - 46958028337644 \nu^{13} + \cdots - 63\!\cdots\!47 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 51696781307 \nu^{17} + 114423502493 \nu^{16} - 1588397315378 \nu^{15} + 3710358729168 \nu^{14} - 45126437605512 \nu^{13} + \cdots + 54\!\cdots\!25 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 63441813145 \nu^{17} - 360924764572 \nu^{16} + 1792620013606 \nu^{15} - 10397296097670 \nu^{14} + 68110591466046 \nu^{13} + \cdots - 13\!\cdots\!92 ) / 89\!\cdots\!48 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{16} - \beta_{14} + \beta_{12} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} + \beta_{3} - 5\beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{17} - 7 \beta_{16} - 3 \beta_{15} + 4 \beta_{14} - 3 \beta_{13} - \beta_{12} - 2 \beta_{10} - \beta_{9} - 5 \beta_{8} + 6 \beta_{7} - 3 \beta_{6} - 8 \beta_{5} - 21 \beta_{4} - \beta_{3} + 47 \beta_{2} - \beta _1 + 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{17} + 58 \beta_{16} - 15 \beta_{15} - 19 \beta_{14} - 15 \beta_{13} + 7 \beta_{12} - 24 \beta_{11} + 23 \beta_{10} + 16 \beta_{9} + 8 \beta_{8} + 12 \beta_{7} + 3 \beta_{6} - 43 \beta_{5} + 57 \beta_{4} - 2 \beta_{3} + 19 \beta_{2} - 11 \beta _1 - 445 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 48 \beta_{17} - 103 \beta_{16} + 24 \beta_{15} - 8 \beta_{14} + 105 \beta_{13} + 83 \beta_{12} + 3 \beta_{11} - 5 \beta_{10} + 56 \beta_{9} + 190 \beta_{8} + 24 \beta_{7} + 33 \beta_{6} - 290 \beta_{5} + 168 \beta_{4} + 200 \beta_{3} + \cdots + 2395 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 561 \beta_{17} - 779 \beta_{16} - 645 \beta_{15} + 836 \beta_{14} + 246 \beta_{13} - 524 \beta_{12} + 225 \beta_{11} - 409 \beta_{10} - 668 \beta_{9} + 305 \beta_{8} - 645 \beta_{7} - 492 \beta_{6} + 2 \beta_{5} + 4638 \beta_{4} + \cdots + 9047 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3705 \beta_{17} + 4208 \beta_{16} - 1272 \beta_{15} + 5239 \beta_{14} + 510 \beta_{13} - 943 \beta_{12} - 4557 \beta_{11} - 1616 \beta_{10} - 925 \beta_{9} - 4265 \beta_{8} + 7044 \beta_{7} - 588 \beta_{6} + \cdots + 125662 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 9132 \beta_{17} - 28751 \beta_{16} + 327 \beta_{15} + 29051 \beta_{14} + 5106 \beta_{13} + 5569 \beta_{12} - 28830 \beta_{11} + 4019 \beta_{10} - 3881 \beta_{9} - 25444 \beta_{8} + 47442 \beta_{7} + \cdots - 389458 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8241 \beta_{17} + 75731 \beta_{16} - 26031 \beta_{15} - 16442 \beta_{14} - 90588 \beta_{13} + 98750 \beta_{12} + 183960 \beta_{11} + 39343 \beta_{10} + 51212 \beta_{9} - 101825 \beta_{8} + \cdots + 1624174 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 251565 \beta_{17} - 397895 \beta_{16} + 118074 \beta_{15} - 860410 \beta_{14} - 325482 \beta_{13} - 139025 \beta_{12} + 545997 \beta_{11} - 301783 \beta_{10} + 530962 \beta_{9} + \cdots + 26997656 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 879681 \beta_{17} + 26483 \beta_{16} - 109191 \beta_{15} - 5606063 \beta_{14} + 1153032 \beta_{13} + 650504 \beta_{12} - 5513712 \beta_{11} + 17284 \beta_{10} + 2493740 \beta_{9} + \cdots - 29244536 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 3632142 \beta_{17} + 12168958 \beta_{16} - 5053317 \beta_{15} + 11419379 \beta_{14} + 3264978 \beta_{13} + 18267136 \beta_{12} - 525258 \beta_{11} - 2257690 \beta_{10} + \cdots + 209347922 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 44559213 \beta_{17} - 12287434 \beta_{16} - 65356608 \beta_{15} + 135798715 \beta_{14} - 42146382 \beta_{13} - 58624477 \beta_{12} - 112307160 \beta_{11} - 62283029 \beta_{10} + \cdots - 2503714955 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 109827759 \beta_{17} - 9641831 \beta_{16} - 217102353 \beta_{15} + 310620797 \beta_{14} + 110004726 \beta_{13} - 11235638 \beta_{12} - 577882029 \beta_{11} + \cdots + 2489994143 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 173327547 \beta_{17} - 3021058948 \beta_{16} - 1091518284 \beta_{15} + 2498726395 \beta_{14} + 743898462 \beta_{13} + 310773953 \beta_{12} + 848065887 \beta_{11} + \cdots + 81329396866 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 7093227774 \beta_{17} - 4240490315 \beta_{16} + 1547960439 \beta_{15} + 1160297 \beta_{14} - 2927559696 \beta_{13} + 1357031581 \beta_{12} + 3279104904 \beta_{11} + \cdots + 108521645315 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 10249926645 \beta_{17} - 92764885585 \beta_{16} + 23390299365 \beta_{15} - 56023052354 \beta_{14} + 11299301790 \beta_{13} + 33722513912 \beta_{12} + \cdots + 2227797540199 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
5.19319 0.175466i
3.78585 3.55912i
3.56701 + 3.77842i
2.17719 + 4.71804i
1.53879 4.96308i
−1.18421 + 5.05941i
−2.70574 4.43610i
−4.30127 + 2.91531i
−4.57080 2.47139i
5.19319 + 0.175466i
3.78585 + 3.55912i
3.56701 3.77842i
2.17719 4.71804i
1.53879 + 4.96308i
−1.18421 5.05941i
−2.70574 + 4.43610i
−4.30127 2.91531i
−4.57080 + 2.47139i
0 −5.19319 0.175466i 0 2.50000 4.33013i 0 −12.4590 21.5795i 0 26.9384 + 1.82246i 0
121.2 0 −3.78585 3.55912i 0 2.50000 4.33013i 0 8.91967 + 15.4493i 0 1.66528 + 26.9486i 0
121.3 0 −3.56701 + 3.77842i 0 2.50000 4.33013i 0 10.8448 + 18.7837i 0 −1.55286 26.9553i 0
121.4 0 −2.17719 + 4.71804i 0 2.50000 4.33013i 0 −2.33214 4.03939i 0 −17.5197 20.5441i 0
121.5 0 −1.53879 4.96308i 0 2.50000 4.33013i 0 −2.14774 3.71999i 0 −22.2642 + 15.2743i 0
121.6 0 1.18421 + 5.05941i 0 2.50000 4.33013i 0 −3.52039 6.09750i 0 −24.1953 + 11.9828i 0
121.7 0 2.70574 4.43610i 0 2.50000 4.33013i 0 8.01379 + 13.8803i 0 −12.3580 24.0058i 0
121.8 0 4.30127 + 2.91531i 0 2.50000 4.33013i 0 12.6176 + 21.8543i 0 10.0019 + 25.0791i 0
121.9 0 4.57080 2.47139i 0 2.50000 4.33013i 0 −9.93659 17.2107i 0 14.7845 22.5924i 0
241.1 0 −5.19319 + 0.175466i 0 2.50000 + 4.33013i 0 −12.4590 + 21.5795i 0 26.9384 1.82246i 0
241.2 0 −3.78585 + 3.55912i 0 2.50000 + 4.33013i 0 8.91967 15.4493i 0 1.66528 26.9486i 0
241.3 0 −3.56701 3.77842i 0 2.50000 + 4.33013i 0 10.8448 18.7837i 0 −1.55286 + 26.9553i 0
241.4 0 −2.17719 4.71804i 0 2.50000 + 4.33013i 0 −2.33214 + 4.03939i 0 −17.5197 + 20.5441i 0
241.5 0 −1.53879 + 4.96308i 0 2.50000 + 4.33013i 0 −2.14774 + 3.71999i 0 −22.2642 15.2743i 0
241.6 0 1.18421 5.05941i 0 2.50000 + 4.33013i 0 −3.52039 + 6.09750i 0 −24.1953 11.9828i 0
241.7 0 2.70574 + 4.43610i 0 2.50000 + 4.33013i 0 8.01379 13.8803i 0 −12.3580 + 24.0058i 0
241.8 0 4.30127 2.91531i 0 2.50000 + 4.33013i 0 12.6176 21.8543i 0 10.0019 25.0791i 0
241.9 0 4.57080 + 2.47139i 0 2.50000 + 4.33013i 0 −9.93659 + 17.2107i 0 14.7845 + 22.5924i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.9
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 360.4.q.d 18
3.b odd 2 1 1080.4.q.d 18
9.c even 3 1 inner 360.4.q.d 18
9.d odd 6 1 1080.4.q.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.q.d 18 1.a even 1 1 trivial
360.4.q.d 18 9.c even 3 1 inner
1080.4.q.d 18 3.b odd 2 1
1080.4.q.d 18 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{18} - 20 T_{7}^{17} + 1594 T_{7}^{16} - 21068 T_{7}^{15} + 1428898 T_{7}^{14} - 16062644 T_{7}^{13} + 767081524 T_{7}^{12} - 5083391378 T_{7}^{11} + 255944450935 T_{7}^{10} + \cdots + 11\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + 7 T^{17} + \cdots + 7625597484987 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{9} \) Copy content Toggle raw display
$7$ \( T^{18} - 20 T^{17} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{18} + 25 T^{17} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{18} + 36 T^{17} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{9} - 23 T^{8} + \cdots - 10\!\cdots\!88)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} + 61 T^{8} + \cdots + 28\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + 50 T^{17} + \cdots + 75\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{18} + 110 T^{17} + \cdots + 35\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{18} - 304 T^{17} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{9} + 4 T^{8} + \cdots - 13\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} - 127 T^{17} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( T^{18} - 679 T^{17} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{18} + 194 T^{17} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{9} + 118 T^{8} + \cdots - 50\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + 57 T^{17} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{18} - 110 T^{17} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{18} - 1025 T^{17} + \cdots + 30\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( (T^{9} - 832 T^{8} + \cdots - 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} + 947 T^{8} + \cdots - 24\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} - 1590 T^{17} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{18} + 948 T^{17} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{9} + 822 T^{8} + \cdots + 10\!\cdots\!66)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} - 1353 T^{17} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
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