Properties

Label 45.6.e.a
Level $45$
Weight $6$
Character orbit 45.e
Analytic conductor $7.217$
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] = [45,6,Mod(16,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.16");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(7.21727189158\)
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: \( x^{18} - 4 x^{17} + 208 x^{16} - 308 x^{15} + 27209 x^{14} - 27878 x^{13} + 1986588 x^{12} + 177366 x^{11} + 100470813 x^{10} + 10889662 x^{9} + \cdots + 247382885376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{17} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{9} + \beta_{3} + \beta_{2} + 1) q^{3} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} + 11 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 11) q^{4} + ( - 25 \beta_{3} + 25) q^{5} + (\beta_{12} + \beta_{8} + \beta_{7} - 16 \beta_{3} + 3 \beta_{2} - 7 \beta_1 - 12) q^{6} + (\beta_{15} + \beta_{14} - \beta_{10} - 3 \beta_{9} - \beta_{8} + 18 \beta_{3} + 2 \beta_{2} + \beta_1) q^{7} + ( - \beta_{17} + \beta_{16} + \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 51) q^{8}+ \cdots + ( - \beta_{17} - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{9} + \beta_{3} + \beta_{2} + 1) q^{3} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} + 11 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 11) q^{4} + ( - 25 \beta_{3} + 25) q^{5} + (\beta_{12} + \beta_{8} + \beta_{7} - 16 \beta_{3} + 3 \beta_{2} - 7 \beta_1 - 12) q^{6} + (\beta_{15} + \beta_{14} - \beta_{10} - 3 \beta_{9} - \beta_{8} + 18 \beta_{3} + 2 \beta_{2} + \beta_1) q^{7} + ( - \beta_{17} + \beta_{16} + \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 51) q^{8}+ \cdots + (90 \beta_{17} - 566 \beta_{16} - 75 \beta_{15} + 525 \beta_{14} + \cdots + 39118) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{2} + 33 q^{3} - 112 q^{4} + 225 q^{5} - 369 q^{6} + 167 q^{7} + 996 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 4 q^{2} + 33 q^{3} - 112 q^{4} + 225 q^{5} - 369 q^{6} + 167 q^{7} + 996 q^{8} + 3 q^{9} - 200 q^{10} + 134 q^{11} - 2937 q^{12} + 418 q^{13} + 207 q^{14} + 825 q^{15} - 1252 q^{16} + 4172 q^{17} + 2487 q^{18} - 7448 q^{19} + 2800 q^{20} - 6840 q^{21} + 5183 q^{22} - 3207 q^{23} + 10368 q^{24} - 5625 q^{25} + 17276 q^{26} + 5940 q^{27} - 29446 q^{28} - 4121 q^{29} - 10950 q^{30} + 13568 q^{31} - 15050 q^{32} + 20610 q^{33} + 14033 q^{34} + 8350 q^{35} + 1653 q^{36} - 26352 q^{37} + 10547 q^{38} - 15846 q^{39} + 12450 q^{40} + 9595 q^{41} + 27549 q^{42} + 41632 q^{43} - 31958 q^{44} - 2100 q^{45} - 95574 q^{46} - 32263 q^{47} - 13545 q^{48} - 24296 q^{49} - 2500 q^{50} - 52320 q^{51} + 73242 q^{52} + 16844 q^{53} + 107748 q^{54} + 6700 q^{55} + 108171 q^{56} + 13686 q^{57} + 46789 q^{58} + 19850 q^{59} + 10125 q^{60} - 19001 q^{61} - 324060 q^{62} + 60039 q^{63} - 75240 q^{64} - 10450 q^{65} - 122982 q^{66} + 99107 q^{67} + 33121 q^{68} - 41445 q^{69} - 5175 q^{70} + 18052 q^{71} + 498654 q^{72} - 292868 q^{73} + 138124 q^{74} + 195615 q^{76} - 12654 q^{77} - 209766 q^{78} + 135110 q^{79} - 62600 q^{80} + 317295 q^{81} - 173480 q^{82} - 124401 q^{83} - 820305 q^{84} + 52150 q^{85} + 202961 q^{86} - 87300 q^{87} + 237261 q^{88} + 453654 q^{89} + 53700 q^{90} - 626036 q^{91} - 147153 q^{92} - 338580 q^{93} + 78857 q^{94} - 93100 q^{95} - 460497 q^{96} + 112332 q^{97} + 907606 q^{98} + 622290 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} + 208 x^{16} - 308 x^{15} + 27209 x^{14} - 27878 x^{13} + 1986588 x^{12} + 177366 x^{11} + 100470813 x^{10} + 10889662 x^{9} + \cdots + 247382885376 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 59\!\cdots\!21 \nu^{17} + \cdots + 35\!\cdots\!68 ) / 43\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24\!\cdots\!11 \nu^{17} + \cdots + 11\!\cdots\!56 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!43 \nu^{17} + \cdots + 80\!\cdots\!12 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24\!\cdots\!55 \nu^{17} + \cdots - 14\!\cdots\!16 ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 82\!\cdots\!23 \nu^{17} + \cdots + 94\!\cdots\!20 ) / 72\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 48\!\cdots\!07 \nu^{17} + \cdots - 12\!\cdots\!92 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 48\!\cdots\!03 \nu^{17} + \cdots + 22\!\cdots\!08 ) / 24\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!09 \nu^{17} + \cdots + 29\!\cdots\!36 ) / 72\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 41\!\cdots\!21 \nu^{17} + \cdots + 49\!\cdots\!92 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 42\!\cdots\!44 \nu^{17} + \cdots - 36\!\cdots\!68 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 89\!\cdots\!91 \nu^{17} + \cdots - 15\!\cdots\!80 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28\!\cdots\!17 \nu^{17} + \cdots + 18\!\cdots\!20 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10\!\cdots\!87 \nu^{17} + \cdots + 29\!\cdots\!36 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18\!\cdots\!53 \nu^{17} + \cdots + 12\!\cdots\!92 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 75\!\cdots\!01 \nu^{17} + \cdots - 63\!\cdots\!04 ) / 72\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 10\!\cdots\!11 \nu^{17} + \cdots - 14\!\cdots\!24 ) / 72\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} + 43\beta_{3} - 2\beta_{2} + 2\beta _1 - 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{16} - \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} - 74 \beta_{2} - \beta _1 - 51 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} - 3 \beta_{16} - 5 \beta_{15} - 10 \beta_{14} + \beta_{13} + 8 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} + 113 \beta_{9} + \beta_{8} + \beta_{7} - 91 \beta_{4} - 3215 \beta_{3} - 8 \beta_{2} - 273 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 18 \beta_{17} - 103 \beta_{16} - 132 \beta_{15} - 42 \beta_{14} + 142 \beta_{13} + 275 \beta_{12} + 275 \beta_{11} - 145 \beta_{10} + 75 \beta_{9} - 54 \beta_{8} - 18 \beta_{7} + 132 \beta_{6} - 42 \beta_{5} - 178 \beta_{4} + \cdots + 8175 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 286 \beta_{17} + 1070 \beta_{16} - 280 \beta_{13} - 538 \beta_{12} + 1292 \beta_{11} - 2362 \beta_{10} + 406 \beta_{9} - 13281 \beta_{8} - 8663 \beta_{7} + 862 \beta_{6} - 1608 \beta_{5} - 286 \beta_{4} + \cdots + 278287 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 15995 \beta_{17} + 31544 \beta_{16} + 15575 \beta_{15} + 8788 \beta_{14} - 18879 \beta_{13} - 17397 \beta_{12} - 14147 \beta_{11} - 22935 \beta_{10} - 33120 \beta_{9} + 7926 \beta_{8} - 18879 \beta_{7} + \cdots + 2884 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 32670 \beta_{17} - 40366 \beta_{16} + 115623 \beta_{15} + 204342 \beta_{14} - 19757 \beta_{13} - 124393 \beta_{12} - 124393 \beta_{11} + 163976 \beta_{10} - 1361228 \beta_{9} + 1383416 \beta_{8} + \cdots - 26306975 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2057350 \beta_{17} - 2798106 \beta_{16} + 359318 \beta_{13} - 1464060 \beta_{12} - 1988145 \beta_{11} + 4786251 \beta_{10} + 640375 \beta_{9} + 4236061 \beta_{8} + 3353508 \beta_{7} + \cdots - 130813685 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4217396 \beta_{17} - 17498056 \beta_{16} - 14331776 \beta_{15} - 24124084 \beta_{14} + 7906088 \beta_{13} + 19408348 \beta_{12} - 1910292 \beta_{11} + 22213792 \beta_{10} + \cdots - 3688692 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 41592456 \beta_{17} - 45899661 \beta_{16} - 199222437 \beta_{15} - 178432008 \beta_{14} + 178122525 \beta_{13} + 374040498 \beta_{12} + 374040498 \beta_{11} + \cdots + 15754263391 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1077118309 \beta_{17} + 2799621625 \beta_{16} - 415696890 \beta_{13} + 32477143 \beta_{12} + 2242319368 \beta_{11} - 5041940993 \beta_{10} + 1011669429 \beta_{9} + \cdots + 274469931945 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 18765266930 \beta_{17} + 40619715831 \beta_{16} + 22319540960 \beta_{15} + 22398084082 \beta_{14} - 23469273196 \beta_{13} - 25266320993 \beta_{12} + \cdots + 4704006266 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 47018189520 \beta_{17} - 56857339604 \beta_{16} + 202055977122 \beta_{15} + 313641865248 \beta_{14} - 91324743778 \beta_{13} - 282833912702 \beta_{12} + \cdots - 29194792731351 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2525181332587 \beta_{17} - 4302374416211 \beta_{16} + 528989769292 \beta_{13} - 1589101979927 \beta_{12} - 2849315104277 \beta_{11} + \cdots - 222844017192427 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 11790027911657 \beta_{17} - 34214349236405 \beta_{16} - 23528268202379 \beta_{15} - 35385805287886 \beta_{14} + 17125403203839 \beta_{13} + \cdots - 5335375292182 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 59504970822750 \beta_{17} - 159300985835 \beta_{16} - 280983049719822 \beta_{15} - 321740831401926 \beta_{14} + 214658669760404 \beta_{13} + \cdots + 26\!\cdots\!39 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
5.33109 9.23372i
4.15211 7.19167i
3.32238 5.75454i
0.596411 1.03301i
0.516068 0.893855i
−0.917202 + 1.58864i
−3.05744 + 5.29565i
−3.28578 + 5.69114i
−4.65764 + 8.06726i
5.33109 + 9.23372i
4.15211 + 7.19167i
3.32238 + 5.75454i
0.596411 + 1.03301i
0.516068 + 0.893855i
−0.917202 1.58864i
−3.05744 5.29565i
−3.28578 5.69114i
−4.65764 8.06726i
−5.33109 + 9.23372i 15.1576 3.63953i −40.8410 70.7387i 12.5000 + 21.6506i −47.2003 + 159.364i 120.579 208.849i 529.719 216.508 110.333i −266.554
16.2 −4.15211 + 7.19167i 5.05797 + 14.7451i −18.4801 32.0085i 12.5000 + 21.6506i −127.043 24.8480i −87.6462 + 151.808i 41.1908 −191.834 + 149.160i −207.606
16.3 −3.32238 + 5.75454i −0.746006 15.5706i −6.07646 10.5247i 12.5000 + 21.6506i 92.0801 + 47.4386i −12.2857 + 21.2794i −131.879 −241.887 + 23.2315i −166.119
16.4 −0.596411 + 1.03301i 14.8460 4.75362i 15.2886 + 26.4806i 12.5000 + 21.6506i −3.94375 + 18.1712i −41.8376 + 72.4649i −74.6434 197.806 141.144i −29.8205
16.5 −0.516068 + 0.893855i −5.64574 + 14.5302i 15.4673 + 26.7902i 12.5000 + 21.6506i −10.0743 12.5450i 13.9409 24.1464i −64.9571 −179.251 164.067i −25.8034
16.6 0.917202 1.58864i −12.5080 9.30322i 14.3175 + 24.7986i 12.5000 + 21.6506i −26.2518 + 11.3378i 92.5869 160.365i 111.229 69.9002 + 232.729i 45.8601
16.7 3.05744 5.29565i −15.3601 + 2.65868i −2.69593 4.66949i 12.5000 + 21.6506i −32.8831 + 89.4702i −85.5735 + 148.218i 162.706 228.863 81.6749i 152.872
16.8 3.28578 5.69114i 13.7855 + 7.27744i −5.59275 9.68692i 12.5000 + 21.6506i 86.7130 54.5429i 32.5978 56.4610i 136.784 137.078 + 200.646i 164.289
16.9 4.65764 8.06726i 1.91277 15.4707i −27.3871 47.4359i 12.5000 + 21.6506i −115.897 87.4875i 51.1380 88.5737i −212.149 −235.683 59.1837i 232.882
31.1 −5.33109 9.23372i 15.1576 + 3.63953i −40.8410 + 70.7387i 12.5000 21.6506i −47.2003 159.364i 120.579 + 208.849i 529.719 216.508 + 110.333i −266.554
31.2 −4.15211 7.19167i 5.05797 14.7451i −18.4801 + 32.0085i 12.5000 21.6506i −127.043 + 24.8480i −87.6462 151.808i 41.1908 −191.834 149.160i −207.606
31.3 −3.32238 5.75454i −0.746006 + 15.5706i −6.07646 + 10.5247i 12.5000 21.6506i 92.0801 47.4386i −12.2857 21.2794i −131.879 −241.887 23.2315i −166.119
31.4 −0.596411 1.03301i 14.8460 + 4.75362i 15.2886 26.4806i 12.5000 21.6506i −3.94375 18.1712i −41.8376 72.4649i −74.6434 197.806 + 141.144i −29.8205
31.5 −0.516068 0.893855i −5.64574 14.5302i 15.4673 26.7902i 12.5000 21.6506i −10.0743 + 12.5450i 13.9409 + 24.1464i −64.9571 −179.251 + 164.067i −25.8034
31.6 0.917202 + 1.58864i −12.5080 + 9.30322i 14.3175 24.7986i 12.5000 21.6506i −26.2518 11.3378i 92.5869 + 160.365i 111.229 69.9002 232.729i 45.8601
31.7 3.05744 + 5.29565i −15.3601 2.65868i −2.69593 + 4.66949i 12.5000 21.6506i −32.8831 89.4702i −85.5735 148.218i 162.706 228.863 + 81.6749i 152.872
31.8 3.28578 + 5.69114i 13.7855 7.27744i −5.59275 + 9.68692i 12.5000 21.6506i 86.7130 + 54.5429i 32.5978 + 56.4610i 136.784 137.078 200.646i 164.289
31.9 4.65764 + 8.06726i 1.91277 + 15.4707i −27.3871 + 47.4359i 12.5000 21.6506i −115.897 + 87.4875i 51.1380 + 88.5737i −212.149 −235.683 + 59.1837i 232.882
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.9
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 45.6.e.a 18
3.b odd 2 1 135.6.e.a 18
9.c even 3 1 inner 45.6.e.a 18
9.c even 3 1 405.6.a.f 9
9.d odd 6 1 135.6.e.a 18
9.d odd 6 1 405.6.a.e 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.6.e.a 18 1.a even 1 1 trivial
45.6.e.a 18 9.c even 3 1 inner
135.6.e.a 18 3.b odd 2 1
135.6.e.a 18 9.d odd 6 1
405.6.a.e 9 9.d odd 6 1
405.6.a.f 9 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 4 T_{2}^{17} + 208 T_{2}^{16} + 308 T_{2}^{15} + 27209 T_{2}^{14} + 27878 T_{2}^{13} + 1986588 T_{2}^{12} - 177366 T_{2}^{11} + 100470813 T_{2}^{10} - 10889662 T_{2}^{9} + \cdots + 247382885376 \) acting on \(S_{6}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 4 T^{17} + \cdots + 247382885376 \) Copy content Toggle raw display
$3$ \( T^{18} - 33 T^{17} + \cdots + 29\!\cdots\!43 \) Copy content Toggle raw display
$5$ \( (T^{2} - 25 T + 625)^{9} \) Copy content Toggle raw display
$7$ \( T^{18} - 167 T^{17} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{18} - 134 T^{17} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{18} - 418 T^{17} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( (T^{9} - 2086 T^{8} + \cdots - 59\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} + 3724 T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + 3207 T^{17} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{18} + 4121 T^{17} + \cdots + 82\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{18} - 13568 T^{17} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{9} + 13176 T^{8} + \cdots + 63\!\cdots\!24)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} - 9595 T^{17} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{18} - 41632 T^{17} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{18} + 32263 T^{17} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{9} - 8422 T^{8} + \cdots - 31\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} - 19850 T^{17} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{18} + 19001 T^{17} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{18} - 99107 T^{17} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{9} - 9026 T^{8} + \cdots - 18\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} + 146434 T^{8} + \cdots + 23\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} - 135110 T^{17} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{18} + 124401 T^{17} + \cdots + 86\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{9} - 226827 T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} - 112332 T^{17} + \cdots + 71\!\cdots\!24 \) Copy content Toggle raw display
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