Properties

Label 7.16.c.a
Level $7$
Weight $16$
Character orbit 7.c
Analytic conductor $9.989$
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] = [7,16,Mod(2,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.2");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 7.c (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: \(9.98854535699\)
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: \( x^{18} + 56195 x^{16} - 231748 x^{15} + 2132909286 x^{14} - 7239960392 x^{13} + 44317048303131 x^{12} + 22955841978480 x^{11} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{9}\cdot 5^{6}\cdot 7^{15} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 10 \beta_{2} + \beta_1) q^{2} + ( - \beta_{6} - 486 \beta_{2} - 486) q^{3} + ( - \beta_{8} - \beta_{6} - \beta_{5} - 17283 \beta_{2} - 14 \beta_1 - 17283) q^{4} + (\beta_{10} + 2 \beta_{8} - \beta_{7} - 7 \beta_{6} + 7 \beta_{4} - 44 \beta_{3} + 26010 \beta_{2} + 44 \beta_1) q^{5} + (\beta_{9} + 2 \beta_{5} + 70 \beta_{4} + 972 \beta_{3} - 2284) q^{6} + (\beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} - 112 \beta_{6} + \cdots - 384659) q^{7}+ \cdots + ( - \beta_{17} + 3 \beta_{16} - 4 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{10} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 10 \beta_{2} + \beta_1) q^{2} + ( - \beta_{6} - 486 \beta_{2} - 486) q^{3} + ( - \beta_{8} - \beta_{6} - \beta_{5} - 17283 \beta_{2} - 14 \beta_1 - 17283) q^{4} + (\beta_{10} + 2 \beta_{8} - \beta_{7} - 7 \beta_{6} + 7 \beta_{4} - 44 \beta_{3} + 26010 \beta_{2} + 44 \beta_1) q^{5} + (\beta_{9} + 2 \beta_{5} + 70 \beta_{4} + 972 \beta_{3} - 2284) q^{6} + (\beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} - 112 \beta_{6} + \cdots - 384659) q^{7}+ \cdots + ( - 71043273 \beta_{17} - 144888262 \beta_{16} + \cdots - 260254970147957) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 90 q^{2} - 4375 q^{3} - 155548 q^{4} - 234087 q^{5} - 40964 q^{6} - 5213740 q^{7} + 9633168 q^{8} - 26277934 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 90 q^{2} - 4375 q^{3} - 155548 q^{4} - 234087 q^{5} - 40964 q^{6} - 5213740 q^{7} + 9633168 q^{8} - 26277934 q^{9} - 22514534 q^{10} + 39090045 q^{11} - 288834308 q^{12} + 513884980 q^{13} - 522992946 q^{14} - 1821038062 q^{15} - 994341040 q^{16} + 972374613 q^{17} - 9531746396 q^{18} - 3111766217 q^{19} + 53399044104 q^{20} - 11576465453 q^{21} + 5458693708 q^{22} + 36146955951 q^{23} - 52758967800 q^{24} - 98777758804 q^{25} - 99659957460 q^{26} + 284144943530 q^{27} + 38163694380 q^{28} - 72970512348 q^{29} - 107038110818 q^{30} - 74937071755 q^{31} + 265381674528 q^{32} + 194333417915 q^{33} - 2143875710340 q^{34} + 106671924009 q^{35} + 2042089659472 q^{36} - 483053004979 q^{37} - 817486402194 q^{38} + 1812420982106 q^{39} + 1785313599048 q^{40} - 644745625524 q^{41} - 259221163768 q^{42} + 3557000885544 q^{43} - 4252567613940 q^{44} - 137659135474 q^{45} + 1411771775166 q^{46} - 3161157736821 q^{47} + 13371673883680 q^{48} + 7856431784754 q^{49} - 37218171418224 q^{50} + 2614553875659 q^{51} - 18449613599928 q^{52} - 7481085738279 q^{53} - 42184117534538 q^{54} + 91307835023930 q^{55} + 35462356001184 q^{56} - 16254765652066 q^{57} - 10693074453668 q^{58} - 61004320427463 q^{59} + 23716102438756 q^{60} + 11174039585989 q^{61} + 162863764600908 q^{62} + 174600791207794 q^{63} - 199531737017472 q^{64} - 168824852541702 q^{65} - 251023951343858 q^{66} + 130186921778593 q^{67} + 47805418774092 q^{68} - 43835892593682 q^{69} + 575094592098950 q^{70} + 29890338888768 q^{71} - 734279624880240 q^{72} - 372680541702511 q^{73} + 403717477992882 q^{74} - 53908157287628 q^{75} + 12\!\cdots\!60 q^{76}+ \cdots - 46\!\cdots\!08 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 56195 x^{16} - 231748 x^{15} + 2132909286 x^{14} - 7239960392 x^{13} + 44317048303131 x^{12} + 22955841978480 x^{11} + \cdots + 14\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!14 \nu^{17} + \cdots - 16\!\cdots\!08 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26\!\cdots\!13 \nu^{17} + \cdots - 98\!\cdots\!64 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 35\!\cdots\!99 \nu^{17} + \cdots - 72\!\cdots\!28 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43\!\cdots\!11 \nu^{17} + \cdots - 25\!\cdots\!08 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36\!\cdots\!77 \nu^{17} + \cdots - 15\!\cdots\!56 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!64 \nu^{17} + \cdots - 67\!\cdots\!08 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 68\!\cdots\!38 \nu^{17} + \cdots + 82\!\cdots\!36 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!71 \nu^{17} + \cdots + 69\!\cdots\!12 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!26 \nu^{17} + \cdots - 17\!\cdots\!28 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 34\!\cdots\!08 \nu^{17} + \cdots + 41\!\cdots\!24 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35\!\cdots\!63 \nu^{17} + \cdots + 15\!\cdots\!64 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!51 \nu^{17} + \cdots - 19\!\cdots\!28 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 94\!\cdots\!31 \nu^{17} + \cdots - 30\!\cdots\!68 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27\!\cdots\!78 \nu^{17} + \cdots + 18\!\cdots\!84 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 20\!\cdots\!33 \nu^{17} + \cdots - 22\!\cdots\!76 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 23\!\cdots\!66 \nu^{17} + \cdots + 18\!\cdots\!52 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{6} - \beta_{4} + 6\beta_{3} + 49951\beta_{2} - 6\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{12} - \beta_{9} + 5\beta_{7} + 51\beta_{5} - 310\beta_{4} - 78906\beta_{3} + 309030 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 34 \beta_{17} - 26 \beta_{16} - 20 \beta_{15} + 203 \beta_{14} - 33 \beta_{13} - 88 \beta_{11} - 2649 \beta_{10} + 34 \beta_{9} - 53111 \beta_{8} - 20 \beta_{7} - 62664 \beta_{6} - 53077 \beta_{5} - 34 \beta_{4} + \cdots - 1969663828 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1500 \beta_{17} + 5132 \beta_{16} - 4816 \beta_{15} + 35145 \beta_{14} + 34217 \beta_{13} + 34217 \beta_{12} + 12948 \beta_{11} + 327853 \beta_{10} + 36961 \beta_{9} + 2377187 \beta_{8} - 329353 \beta_{7} + \cdots - 6632 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 633766 \beta_{17} - 2653530 \beta_{16} + 1385998 \beta_{15} + 1267532 \beta_{14} - 1863955 \beta_{12} + 870698 \beta_{11} - 752232 \beta_{10} + 5671505 \beta_{9} + \cdots + 43776610667380 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 143704236 \beta_{17} + 285936300 \beta_{16} + 47410688 \beta_{15} - 1451393323 \beta_{14} - 994378787 \beta_{13} - 239997784 \beta_{11} - 12821606719 \beta_{10} + \cdots - 11\!\cdots\!10 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 21870340596 \beta_{17} + 112549810104 \beta_{16} - 23469394158 \beta_{15} - 284658348033 \beta_{14} + 72237169671 \beta_{13} + 72237169671 \beta_{12} + \cdots - 90679469508 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5866692815172 \beta_{17} - 16545883653384 \beta_{16} + 4812498023040 \beta_{15} + 11733385630344 \beta_{14} - 27646955007451 \beta_{12} + \cdots + 46\!\cdots\!94 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 13\!\cdots\!30 \beta_{17} - 392095957173650 \beta_{16} - 586533438386960 \beta_{15} + \cdots - 26\!\cdots\!36 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 18\!\cdots\!08 \beta_{17} + \cdots - 29\!\cdots\!64 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 24\!\cdots\!22 \beta_{17} + \cdots + 68\!\cdots\!84 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 44\!\cdots\!40 \beta_{17} + \cdots - 52\!\cdots\!26 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 39\!\cdots\!00 \beta_{17} + \cdots - 22\!\cdots\!52 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 13\!\cdots\!04 \beta_{17} + \cdots + 16\!\cdots\!02 ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 31\!\cdots\!26 \beta_{17} + \cdots - 48\!\cdots\!48 ) / 8 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 18\!\cdots\!20 \beta_{17} + \cdots - 78\!\cdots\!96 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
76.4947 + 132.493i
69.4191 + 120.237i
52.9078 + 91.6389i
24.0379 + 41.6349i
−8.49801 14.7190i
−14.7831 25.6051i
−56.6035 98.0401i
−58.5571 101.424i
−84.4177 146.216i
76.4947 132.493i
69.4191 120.237i
52.9078 91.6389i
24.0379 41.6349i
−8.49801 + 14.7190i
−14.7831 + 25.6051i
−56.6035 + 98.0401i
−58.5571 + 101.424i
−84.4177 + 146.216i
−157.989 + 273.646i −3377.85 5850.60i −33537.3 58088.3i −47548.7 + 82356.8i 2.13466e6 −1.20661e6 1.81429e6i 1.08402e7 −1.56452e7 + 2.70983e7i −1.50244e7 2.60230e7i
2.2 −143.838 + 249.135i 2231.37 + 3864.84i −24994.8 43292.3i −135318. + 234377.i −1.28382e6 461443. + 2.12947e6i 4.95426e6 −2.78354e6 + 4.82123e6i −3.89277e7 6.74248e7i
2.3 −110.816 + 191.938i 721.980 + 1250.51i −8176.15 14161.5i 127347. 220572.i −320026. 1.24407e6 1.78881e6i −3.63823e6 6.13194e6 1.06208e7i 2.82241e7 + 4.88856e7i
2.4 −53.0759 + 91.9301i −1023.57 1772.87i 10749.9 + 18619.4i 12633.5 21881.9i 217307. −965276. + 1.95341e6i −5.76062e6 5.07907e6 8.79720e6i 1.34107e6 + 2.32280e6i
2.5 11.9960 20.7777i 2699.32 + 4675.36i 16096.2 + 27879.4i −27229.5 + 47162.9i 129524. −1.83340e6 1.17738e6i 1.55853e6 −7.39818e6 + 1.28140e7i 653291. + 1.13153e6i
2.6 24.5663 42.5500i −1873.73 3245.40i 15177.0 + 26287.3i −88261.2 + 152873.i −184123. 1.92935e6 1.01251e6i 3.10134e6 152696. 264477.i 4.33649e6 + 7.51103e6i
2.7 108.207 187.420i −2791.04 4834.23i −7033.49 12182.4i 139381. 241415.i −1.20804e6 −2.17580e6 + 116027.i 4.04716e6 −8.40541e6 + 1.45586e7i −3.01640e7 5.22455e7i
2.8 112.114 194.188i 1495.64 + 2590.52i −8755.19 15164.4i 59707.2 103416.i 670728. 1.88803e6 + 1.08761e6i 3.42119e6 2.70060e6 4.67758e6i −1.33881e7 2.31888e7i
2.9 163.835 283.771i −269.606 466.971i −37300.1 64605.7i −157755. + 273240.i −176684. −1.94868e6 974795.i −1.37072e7 7.02908e6 1.21747e7i 5.16919e7 + 8.95329e7i
4.1 −157.989 273.646i −3377.85 + 5850.60i −33537.3 + 58088.3i −47548.7 82356.8i 2.13466e6 −1.20661e6 + 1.81429e6i 1.08402e7 −1.56452e7 2.70983e7i −1.50244e7 + 2.60230e7i
4.2 −143.838 249.135i 2231.37 3864.84i −24994.8 + 43292.3i −135318. 234377.i −1.28382e6 461443. 2.12947e6i 4.95426e6 −2.78354e6 4.82123e6i −3.89277e7 + 6.74248e7i
4.3 −110.816 191.938i 721.980 1250.51i −8176.15 + 14161.5i 127347. + 220572.i −320026. 1.24407e6 + 1.78881e6i −3.63823e6 6.13194e6 + 1.06208e7i 2.82241e7 4.88856e7i
4.4 −53.0759 91.9301i −1023.57 + 1772.87i 10749.9 18619.4i 12633.5 + 21881.9i 217307. −965276. 1.95341e6i −5.76062e6 5.07907e6 + 8.79720e6i 1.34107e6 2.32280e6i
4.5 11.9960 + 20.7777i 2699.32 4675.36i 16096.2 27879.4i −27229.5 47162.9i 129524. −1.83340e6 + 1.17738e6i 1.55853e6 −7.39818e6 1.28140e7i 653291. 1.13153e6i
4.6 24.5663 + 42.5500i −1873.73 + 3245.40i 15177.0 26287.3i −88261.2 152873.i −184123. 1.92935e6 + 1.01251e6i 3.10134e6 152696. + 264477.i 4.33649e6 7.51103e6i
4.7 108.207 + 187.420i −2791.04 + 4834.23i −7033.49 + 12182.4i 139381. + 241415.i −1.20804e6 −2.17580e6 116027.i 4.04716e6 −8.40541e6 1.45586e7i −3.01640e7 + 5.22455e7i
4.8 112.114 + 194.188i 1495.64 2590.52i −8755.19 + 15164.4i 59707.2 + 103416.i 670728. 1.88803e6 1.08761e6i 3.42119e6 2.70060e6 + 4.67758e6i −1.33881e7 + 2.31888e7i
4.9 163.835 + 283.771i −269.606 + 466.971i −37300.1 + 64605.7i −157755. 273240.i −176684. −1.94868e6 + 974795.i −1.37072e7 7.02908e6 + 1.21747e7i 5.16919e7 8.95329e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.16.c.a 18
3.b odd 2 1 63.16.e.b 18
7.b odd 2 1 49.16.c.i 18
7.c even 3 1 inner 7.16.c.a 18
7.c even 3 1 49.16.a.g 9
7.d odd 6 1 49.16.a.f 9
7.d odd 6 1 49.16.c.i 18
21.h odd 6 1 63.16.e.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.16.c.a 18 1.a even 1 1 trivial
7.16.c.a 18 7.c even 3 1 inner
49.16.a.f 9 7.d odd 6 1
49.16.a.g 9 7.c even 3 1
49.16.c.i 18 7.b odd 2 1
49.16.c.i 18 7.d odd 6 1
63.16.e.b 18 3.b odd 2 1
63.16.e.b 18 21.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 90 T^{17} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$3$ \( T^{18} + 4375 T^{17} + \cdots + 26\!\cdots\!29 \) Copy content Toggle raw display
$5$ \( T^{18} + 234087 T^{17} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + 5213740 T^{17} + \cdots + 12\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{18} - 39090045 T^{17} + \cdots + 66\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{9} - 256942490 T^{8} + \cdots + 26\!\cdots\!36)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} - 972374613 T^{17} + \cdots + 71\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{18} + 3111766217 T^{17} + \cdots + 77\!\cdots\!29 \) Copy content Toggle raw display
$23$ \( T^{18} - 36146955951 T^{17} + \cdots + 47\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{9} + 36485256174 T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + 74937071755 T^{17} + \cdots + 53\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{18} + 483053004979 T^{17} + \cdots + 96\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{9} + 322372812762 T^{8} + \cdots + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{9} - 1778500442772 T^{8} + \cdots + 82\!\cdots\!08)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + 3161157736821 T^{17} + \cdots + 98\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{18} + 7481085738279 T^{17} + \cdots + 37\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{18} + 61004320427463 T^{17} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{18} - 11174039585989 T^{17} + \cdots + 82\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{18} - 130186921778593 T^{17} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{9} - 14945169444384 T^{8} + \cdots - 28\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + 372680541702511 T^{17} + \cdots + 77\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( T^{18} + 732954179304297 T^{17} + \cdots + 86\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots + 66\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + 983172302764239 T^{17} + \cdots + 93\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots + 91\!\cdots\!88)^{2} \) Copy content Toggle raw display
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