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Properties

Label 483.2.i.h
Level $483$
Weight $2$
Character orbit 483.i
Analytic conductor $3.857$
Analytic rank $0$
Dimension $20$
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] = [483,2,Mod(277,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.i (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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} + 22 x^{18} - 43 x^{17} + 245 x^{16} - 416 x^{15} + 1707 x^{14} - 2021 x^{13} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_1) q^{2} - \beta_{9} q^{3} + ( - \beta_{17} + \beta_{9} + \cdots - \beta_1) q^{4}+ \cdots + ( - \beta_{9} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_1) q^{2} - \beta_{9} q^{3} + ( - \beta_{17} + \beta_{9} + \cdots - \beta_1) q^{4}+ \cdots + (\beta_{7} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 3 q^{2} + 10 q^{3} - 15 q^{4} + 5 q^{5} - 6 q^{6} + 18 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 3 q^{2} + 10 q^{3} - 15 q^{4} + 5 q^{5} - 6 q^{6} + 18 q^{8} - 10 q^{9} - 11 q^{10} - 8 q^{11} + 15 q^{12} + 11 q^{14} + 10 q^{15} - 37 q^{16} + 11 q^{17} - 3 q^{18} - q^{19} - 30 q^{20} + 12 q^{22} - 10 q^{23} + 9 q^{24} - 21 q^{25} - q^{26} - 20 q^{27} + 44 q^{28} + 44 q^{29} + 11 q^{30} + 3 q^{31} - 11 q^{32} + 8 q^{33} - 6 q^{34} - 9 q^{35} + 30 q^{36} + 3 q^{37} - 16 q^{38} - 39 q^{40} - 52 q^{41} + 7 q^{42} + 54 q^{43} - 16 q^{44} + 5 q^{45} - 3 q^{46} - 11 q^{47} - 74 q^{48} + 22 q^{49} + 4 q^{50} - 11 q^{51} - 29 q^{52} - 5 q^{53} + 3 q^{54} - 36 q^{55} + 43 q^{56} - 2 q^{57} - 16 q^{58} + 10 q^{59} - 15 q^{60} - 22 q^{61} - 64 q^{62} + 138 q^{64} + 11 q^{65} + 6 q^{66} + 2 q^{67} + 21 q^{68} - 20 q^{69} + 84 q^{70} + 54 q^{71} - 9 q^{72} + 8 q^{73} - 14 q^{74} + 21 q^{75} - 44 q^{76} + 8 q^{77} - 2 q^{78} - 21 q^{79} + 53 q^{80} - 10 q^{81} - 36 q^{82} - 24 q^{83} + 25 q^{84} + 46 q^{85} - 18 q^{86} + 22 q^{87} + 10 q^{88} - 6 q^{89} + 22 q^{90} - 62 q^{91} + 30 q^{92} - 3 q^{93} - 35 q^{94} - 44 q^{95} + 11 q^{96} + 12 q^{97} + 2 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{19} + 22 x^{18} - 43 x^{17} + 245 x^{16} - 416 x^{15} + 1707 x^{14} - 2021 x^{13} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22\!\cdots\!01 \nu^{19} + \cdots - 20\!\cdots\!14 ) / 69\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22\!\cdots\!60 \nu^{19} + \cdots - 13\!\cdots\!22 ) / 36\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 62\!\cdots\!25 \nu^{19} + \cdots + 18\!\cdots\!90 ) / 69\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!73 \nu^{19} + \cdots - 89\!\cdots\!04 ) / 36\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!92 \nu^{19} + \cdots + 13\!\cdots\!44 ) / 36\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 51\!\cdots\!36 \nu^{19} + \cdots - 12\!\cdots\!86 ) / 36\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!44 \nu^{19} + \cdots + 92\!\cdots\!32 ) / 36\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 90\!\cdots\!45 \nu^{19} + \cdots - 45\!\cdots\!36 ) / 13\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 74\!\cdots\!85 \nu^{19} + \cdots + 40\!\cdots\!04 ) / 73\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 77\!\cdots\!09 \nu^{19} + \cdots + 28\!\cdots\!96 ) / 73\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!99 \nu^{19} + \cdots - 47\!\cdots\!12 ) / 73\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 55\!\cdots\!40 \nu^{19} + \cdots + 11\!\cdots\!72 ) / 36\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!71 \nu^{19} + \cdots - 21\!\cdots\!28 ) / 73\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 71\!\cdots\!56 \nu^{19} + \cdots - 83\!\cdots\!36 ) / 36\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 73\!\cdots\!06 \nu^{19} + \cdots - 22\!\cdots\!76 ) / 36\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 17\!\cdots\!37 \nu^{19} + \cdots - 93\!\cdots\!88 ) / 73\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 18\!\cdots\!67 \nu^{19} + \cdots + 14\!\cdots\!12 ) / 73\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 18\!\cdots\!75 \nu^{19} + \cdots - 11\!\cdots\!52 ) / 73\!\cdots\!28 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} - 3\beta_{9} + \beta_{8} - \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{7} - \beta_{5} + 7\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - 8 \beta_{17} - \beta_{16} - \beta_{14} - \beta_{10} + 17 \beta_{9} - 7 \beta_{8} + \cdots - 7 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{19} - 10 \beta_{18} + 2 \beta_{17} - \beta_{16} - 2 \beta_{15} + 11 \beta_{14} + \cdots - 29 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13 \beta_{16} - 13 \beta_{15} + \beta_{14} + \beta_{12} + 12 \beta_{11} + 12 \beta_{10} - 14 \beta_{7} + \cdots + 112 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 40 \beta_{19} + 83 \beta_{18} - 28 \beta_{17} + 28 \beta_{16} + 12 \beta_{15} - 26 \beta_{14} + \cdots + 179 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 65 \beta_{19} + 444 \beta_{17} + 2 \beta_{16} + 131 \beta_{15} + 81 \beta_{14} - 20 \beta_{13} + \cdots - 779 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 177 \beta_{16} + 177 \beta_{15} - 539 \beta_{14} - 539 \beta_{12} - 581 \beta_{11} - 581 \beta_{10} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 433 \beta_{19} + 4 \beta_{18} - 3278 \beta_{17} - 1195 \beta_{16} - 44 \beta_{15} - 673 \beta_{14} + \cdots - 2026 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3452 \beta_{19} - 5001 \beta_{18} + 2464 \beta_{17} - 1026 \beta_{16} - 2722 \beta_{15} + 6345 \beta_{14} + \cdots - 56 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 9737 \beta_{16} - 9737 \beta_{15} + 332 \beta_{14} + 332 \beta_{12} + 8135 \beta_{11} + 8135 \beta_{10} + \cdots + 40791 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 28749 \beta_{19} + 37910 \beta_{18} - 20356 \beta_{17} + 23956 \beta_{16} + 8829 \beta_{15} + \cdots + 52927 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 16645 \beta_{19} - 1004 \beta_{18} + 179652 \beta_{17} + 6992 \beta_{16} + 86941 \beta_{15} + \cdots - 302152 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 129544 \beta_{16} + 129544 \beta_{15} - 218511 \beta_{14} - 218511 \beta_{12} - 282629 \beta_{11} + \cdots + 252 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 95613 \beta_{19} + 10008 \beta_{18} - 1336036 \beta_{17} - 715923 \beta_{16} - 71110 \beta_{15} + \cdots - 682063 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1851193 \beta_{19} - 2149200 \beta_{18} + 1263312 \beta_{17} - 613931 \beta_{16} - 1695944 \beta_{15} + \cdots + 36976 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 5142647 \beta_{16} - 5142647 \beta_{15} - 145569 \beta_{14} - 145569 \beta_{12} + 4254666 \beta_{11} + \cdots + 17011682 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 14578856 \beta_{19} + 16161297 \beta_{18} - 9686600 \beta_{17} + 13906594 \beta_{16} + \cdots + 19667087 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(-1 - \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
1.39981 − 2.42454i
1.25760 − 2.17823i
1.13751 − 1.97023i
0.599601 − 1.03854i
0.432430 − 0.748990i
−0.0131282 + 0.0227388i
−0.384545 + 0.666052i
−0.526755 + 0.912367i
−1.03253 + 1.78839i
−1.37000 + 2.37290i
1.39981 + 2.42454i
1.25760 + 2.17823i
1.13751 + 1.97023i
0.599601 + 1.03854i
0.432430 + 0.748990i
−0.0131282 − 0.0227388i
−0.384545 − 0.666052i
−0.526755 − 0.912367i
−1.03253 − 1.78839i
−1.37000 − 2.37290i
−1.39981 − 2.42454i 0.500000 − 0.866025i −2.91893 + 5.05574i 0.973538 + 1.68622i −2.79962 −2.64333 + 0.113245i 10.7446 −0.500000 − 0.866025i 2.72553 − 4.72076i
277.2 −1.25760 − 2.17823i 0.500000 − 0.866025i −2.16313 + 3.74664i −2.15868 − 3.73894i −2.51520 1.42694 − 2.22797i 5.85100 −0.500000 − 0.866025i −5.42951 + 9.40419i
277.3 −1.13751 − 1.97023i 0.500000 − 0.866025i −1.58786 + 2.75026i 0.619347 + 1.07274i −2.27502 1.84753 + 1.89384i 2.67481 −0.500000 − 0.866025i 1.40903 − 2.44051i
277.4 −0.599601 − 1.03854i 0.500000 − 0.866025i 0.280958 − 0.486633i 0.286327 + 0.495932i −1.19920 −1.87398 − 1.86767i −3.07225 −0.500000 − 0.866025i 0.343363 − 0.594723i
277.5 −0.432430 − 0.748990i 0.500000 − 0.866025i 0.626009 − 1.08428i 1.24779 + 2.16123i −0.864859 −0.0271380 + 2.64561i −2.81254 −0.500000 − 0.866025i 1.07916 − 1.86916i
277.6 0.0131282 + 0.0227388i 0.500000 − 0.866025i 0.999655 − 1.73145i −1.38914 − 2.40606i 0.0262565 −2.58821 + 0.548771i 0.105008 −0.500000 − 0.866025i 0.0364740 − 0.0631748i
277.7 0.384545 + 0.666052i 0.500000 − 0.866025i 0.704250 − 1.21980i 2.01355 + 3.48756i 0.769091 2.63574 − 0.229917i 2.62145 −0.500000 − 0.866025i −1.54860 + 2.68225i
277.8 0.526755 + 0.912367i 0.500000 − 0.866025i 0.445058 − 0.770863i −1.08806 − 1.88457i 1.05351 1.36439 + 2.26681i 3.04477 −0.500000 − 0.866025i 1.14628 − 1.98542i
277.9 1.03253 + 1.78839i 0.500000 − 0.866025i −1.13223 + 1.96108i 0.304427 + 0.527282i 2.06506 2.05425 − 1.66735i −0.546135 −0.500000 − 0.866025i −0.628658 + 1.08887i
277.10 1.37000 + 2.37290i 0.500000 − 0.866025i −2.75378 + 4.76968i 1.69091 + 2.92874i 2.73999 −2.19620 − 1.47537i −9.61066 −0.500000 − 0.866025i −4.63307 + 8.02471i
415.1 −1.39981 + 2.42454i 0.500000 + 0.866025i −2.91893 − 5.05574i 0.973538 − 1.68622i −2.79962 −2.64333 − 0.113245i 10.7446 −0.500000 + 0.866025i 2.72553 + 4.72076i
415.2 −1.25760 + 2.17823i 0.500000 + 0.866025i −2.16313 − 3.74664i −2.15868 + 3.73894i −2.51520 1.42694 + 2.22797i 5.85100 −0.500000 + 0.866025i −5.42951 − 9.40419i
415.3 −1.13751 + 1.97023i 0.500000 + 0.866025i −1.58786 − 2.75026i 0.619347 − 1.07274i −2.27502 1.84753 − 1.89384i 2.67481 −0.500000 + 0.866025i 1.40903 + 2.44051i
415.4 −0.599601 + 1.03854i 0.500000 + 0.866025i 0.280958 + 0.486633i 0.286327 − 0.495932i −1.19920 −1.87398 + 1.86767i −3.07225 −0.500000 + 0.866025i 0.343363 + 0.594723i
415.5 −0.432430 + 0.748990i 0.500000 + 0.866025i 0.626009 + 1.08428i 1.24779 − 2.16123i −0.864859 −0.0271380 − 2.64561i −2.81254 −0.500000 + 0.866025i 1.07916 + 1.86916i
415.6 0.0131282 − 0.0227388i 0.500000 + 0.866025i 0.999655 + 1.73145i −1.38914 + 2.40606i 0.0262565 −2.58821 − 0.548771i 0.105008 −0.500000 + 0.866025i 0.0364740 + 0.0631748i
415.7 0.384545 − 0.666052i 0.500000 + 0.866025i 0.704250 + 1.21980i 2.01355 − 3.48756i 0.769091 2.63574 + 0.229917i 2.62145 −0.500000 + 0.866025i −1.54860 − 2.68225i
415.8 0.526755 − 0.912367i 0.500000 + 0.866025i 0.445058 + 0.770863i −1.08806 + 1.88457i 1.05351 1.36439 − 2.26681i 3.04477 −0.500000 + 0.866025i 1.14628 + 1.98542i
415.9 1.03253 − 1.78839i 0.500000 + 0.866025i −1.13223 − 1.96108i 0.304427 − 0.527282i 2.06506 2.05425 + 1.66735i −0.546135 −0.500000 + 0.866025i −0.628658 − 1.08887i
415.10 1.37000 − 2.37290i 0.500000 + 0.866025i −2.75378 − 4.76968i 1.69091 − 2.92874i 2.73999 −2.19620 + 1.47537i −9.61066 −0.500000 + 0.866025i −4.63307 − 8.02471i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.10
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 483.2.i.h ✓ 20
7.c even 3 1 inner 483.2.i.h ✓ 20
7.c even 3 1 3381.2.a.bi 10
7.d odd 6 1 3381.2.a.bj 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.h ✓ 20 1.a even 1 1 trivial
483.2.i.h ✓ 20 7.c even 3 1 inner
3381.2.a.bi 10 7.c even 3 1
3381.2.a.bj 10 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\( T_{2}^{20} + 3 T_{2}^{19} + 22 T_{2}^{18} + 43 T_{2}^{17} + 245 T_{2}^{16} + 416 T_{2}^{15} + 1707 T_{2}^{14} + \cdots + 4 \) Copy content Toggle raw display
\( T_{5}^{20} - 5 T_{5}^{19} + 48 T_{5}^{18} - 187 T_{5}^{17} + 1263 T_{5}^{16} - 4454 T_{5}^{15} + \cdots + 556516 \) Copy content Toggle raw display
\( T_{11}^{20} + 8 T_{11}^{19} + 88 T_{11}^{18} + 320 T_{11}^{17} + 2504 T_{11}^{16} + 6744 T_{11}^{15} + \cdots + 31719424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 3 T^{19} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} - 5 T^{19} + \cdots + 556516 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} + 8 T^{19} + \cdots + 31719424 \) Copy content Toggle raw display
$13$ \( (T^{10} - 57 T^{8} + \cdots + 30667)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 53029799524 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 13176284944 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$29$ \( (T^{10} - 22 T^{9} + \cdots + 198784)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 3370484748544 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 345393601613824 \) Copy content Toggle raw display
$41$ \( (T^{10} + 26 T^{9} + \cdots - 30634016)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 27 T^{9} + \cdots + 59884)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + 11 T^{19} + \cdots + 5161984 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 7340588259904 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 119638508544 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 155728101376 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 3442498027609 \) Copy content Toggle raw display
$71$ \( (T^{10} - 27 T^{9} + \cdots + 5367896)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 60216094569409 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{10} + 12 T^{9} + \cdots - 947876608)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 175748762176 \) Copy content Toggle raw display
$97$ \( (T^{10} - 6 T^{9} + \cdots - 21133952)^{2} \) Copy content Toggle raw display
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