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Properties

Label 429.2.n.b
Level $429$
Weight $2$
Character orbit 429.n
Analytic conductor $3.426$
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] = [429,2,Mod(157,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.n (of order \(5\), degree \(4\), 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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 4 x^{18} + 4 x^{17} + 37 x^{16} - 74 x^{15} + 398 x^{14} - 224 x^{13} + 978 x^{12} + \cdots + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{9} q^{3} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_{3}) q^{4}+ \cdots - \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{9} q^{3} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_{3}) q^{4}+ \cdots + ( - \beta_{19} - \beta_{18} + \beta_{17} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{2} + 5 q^{3} + 3 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} - 7 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{2} + 5 q^{3} + 3 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} - 7 q^{8} - 5 q^{9} - 2 q^{10} + 14 q^{11} - 18 q^{12} + 5 q^{13} - q^{14} - 4 q^{15} - 35 q^{16} + 2 q^{17} - 4 q^{18} - 2 q^{19} + 45 q^{20} - 2 q^{21} + 11 q^{22} + 6 q^{23} + 2 q^{24} - 7 q^{25} + 4 q^{26} + 5 q^{27} + 12 q^{28} + 26 q^{29} - 3 q^{30} + 20 q^{31} + 42 q^{32} + q^{33} - 24 q^{34} - 18 q^{35} + 3 q^{36} - 6 q^{37} - 3 q^{38} - 5 q^{39} - 26 q^{41} - 9 q^{42} + 28 q^{43} - 38 q^{44} - 16 q^{45} - 17 q^{46} + 8 q^{47} - 20 q^{48} + 2 q^{49} - 29 q^{50} + 3 q^{51} - 3 q^{52} + q^{53} - 6 q^{54} - 36 q^{56} - 8 q^{57} + 22 q^{58} - 21 q^{59} - 45 q^{60} + 26 q^{61} - 10 q^{62} - 3 q^{63} - 87 q^{64} + 16 q^{65} + 14 q^{66} + 56 q^{67} + 65 q^{68} + 4 q^{69} - 24 q^{70} - 28 q^{71} - 2 q^{72} + 45 q^{73} - 29 q^{74} - 3 q^{75} + 60 q^{76} + 4 q^{77} + 6 q^{78} - 15 q^{79} - 7 q^{80} - 5 q^{81} - 46 q^{82} + 36 q^{83} + 8 q^{84} + 39 q^{86} + 24 q^{87} + 73 q^{88} - 126 q^{89} - 2 q^{90} + 3 q^{91} + 2 q^{92} - 20 q^{93} - 3 q^{94} + 47 q^{95} - 47 q^{96} + 18 q^{97} - 54 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 4 x^{18} + 4 x^{17} + 37 x^{16} - 74 x^{15} + 398 x^{14} - 224 x^{13} + 978 x^{12} + \cdots + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!45 \nu^{19} + \cdots + 21\!\cdots\!64 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!86 \nu^{19} + \cdots + 45\!\cdots\!91 ) / 13\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!34 \nu^{19} + \cdots + 42\!\cdots\!51 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!30 \nu^{19} + \cdots - 10\!\cdots\!15 ) / 13\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!03 \nu^{19} + \cdots + 42\!\cdots\!08 ) / 13\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!89 \nu^{19} + \cdots + 22\!\cdots\!69 ) / 13\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!79 \nu^{19} + \cdots + 79\!\cdots\!82 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29\!\cdots\!05 \nu^{19} + \cdots + 62\!\cdots\!29 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40\!\cdots\!81 \nu^{19} + \cdots + 21\!\cdots\!49 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43\!\cdots\!35 \nu^{19} + \cdots - 48\!\cdots\!89 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 66\!\cdots\!18 \nu^{19} + \cdots - 30\!\cdots\!10 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 81\!\cdots\!57 \nu^{19} + \cdots - 32\!\cdots\!44 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 80\!\cdots\!07 \nu^{19} + \cdots + 61\!\cdots\!13 ) / 13\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 92\!\cdots\!19 \nu^{19} + \cdots + 42\!\cdots\!18 ) / 13\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 10\!\cdots\!52 \nu^{19} + \cdots + 44\!\cdots\!72 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 11\!\cdots\!59 \nu^{19} + \cdots + 24\!\cdots\!71 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 15\!\cdots\!99 \nu^{19} + \cdots + 31\!\cdots\!30 ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 21\!\cdots\!12 \nu^{19} + \cdots - 44\!\cdots\!12 ) / 14\!\cdots\!69 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} + \beta_{12} - \beta_{10} + 3\beta_{9} + \beta_{7} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} - \beta_{16} - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} + \beta_{18} - \beta_{17} + \beta_{16} + 7 \beta_{15} - 7 \beta_{13} + \beta_{11} + \cdots + 9 \beta_{5} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} - 9 \beta_{18} - 2 \beta_{15} - \beta_{14} + 9 \beta_{12} - \beta_{11} + 4 \beta_{10} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 59 \beta_{18} + 9 \beta_{17} - 10 \beta_{16} - 9 \beta_{14} + 49 \beta_{13} - 70 \beta_{12} - 60 \beta_{9} + \cdots - 60 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{19} - 97 \beta_{18} - 2 \beta_{17} + 69 \beta_{16} + 28 \beta_{15} + 10 \beta_{14} - 102 \beta_{13} + \cdots + 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 69 \beta_{19} + 38 \beta_{18} + 69 \beta_{17} - 38 \beta_{16} - 353 \beta_{15} - 3 \beta_{14} + \cdots - 359 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 38 \beta_{19} - 176 \beta_{18} - 81 \beta_{17} + 462 \beta_{16} + 603 \beta_{15} - 603 \beta_{13} + \cdots - 518 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 462 \beta_{19} - 2596 \beta_{18} - 1409 \beta_{15} + 56 \beta_{14} + 2596 \beta_{12} - 462 \beta_{11} + \cdots + 2504 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8119 \beta_{18} + 628 \beta_{17} - 3221 \beta_{16} - 628 \beta_{14} + 4898 \beta_{13} - 7496 \beta_{12} + \cdots - 5271 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3221 \beta_{19} - 15746 \beta_{18} - 3895 \beta_{17} + 4740 \beta_{16} + 11006 \beta_{15} + 3221 \beta_{14} + \cdots + 25771 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4740 \beta_{19} + 29420 \beta_{18} + 4740 \beta_{17} - 29420 \beta_{16} - 39604 \beta_{15} + \cdots - 32914 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 29420 \beta_{19} + 47661 \beta_{18} - 22704 \beta_{17} + 37740 \beta_{16} + 146101 \beta_{15} + \cdots - 43953 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 37740 \beta_{19} - 318589 \beta_{18} - 185820 \beta_{15} + 6213 \beta_{14} + 318589 \beta_{12} + \cdots + 261753 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1406682 \beta_{18} + 162592 \beta_{17} - 295739 \beta_{16} - 162592 \beta_{14} + 1110943 \beta_{13} + \cdots - 1495412 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 295739 \beta_{19} - 3218715 \beta_{18} - 385770 \beta_{17} + 1699794 \beta_{16} + 1518921 \beta_{15} + \cdots + 3165633 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1699794 \beta_{19} + 3271689 \beta_{18} + 1699794 \beta_{17} - 3271689 \beta_{16} - 8500993 \beta_{15} + \cdots - 7372980 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 3271689 \beta_{19} + 3524367 \beta_{18} - 2329951 \beta_{17} + 8742759 \beta_{16} + \cdots - 12993748 \beta_{2} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(1\) \(-1 + \beta_{8} - \beta_{9} + \beta_{10}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−2.26411 − 1.64497i
−0.972448 − 0.706525i
−0.297979 − 0.216494i
1.21086 + 0.879744i
2.01466 + 1.46373i
−0.375422 − 1.15543i
−0.262109 − 0.806688i
0.169480 + 0.521606i
0.566002 + 1.74197i
0.711066 + 2.18844i
−2.26411 + 1.64497i
−0.972448 + 0.706525i
−0.297979 + 0.216494i
1.21086 − 0.879744i
2.01466 − 1.46373i
−0.375422 + 1.15543i
−0.262109 + 0.806688i
0.169480 − 0.521606i
0.566002 − 1.74197i
0.711066 − 2.18844i
−2.26411 + 1.64497i −0.309017 − 0.951057i 1.80222 − 5.54667i 2.68510 + 1.95084i 2.26411 + 1.64497i −0.449002 + 1.38189i 3.31406 + 10.1996i −0.809017 + 0.587785i −9.28845
157.2 −0.972448 + 0.706525i −0.309017 − 0.951057i −0.171556 + 0.527996i −1.69471 − 1.23128i 0.972448 + 0.706525i −0.640431 + 1.97104i −0.949097 − 2.92102i −0.809017 + 0.587785i 2.51795
157.3 −0.297979 + 0.216494i −0.309017 − 0.951057i −0.576112 + 1.77309i −0.859409 − 0.624397i 0.297979 + 0.216494i 1.08879 − 3.35095i −0.439831 − 1.35366i −0.809017 + 0.587785i 0.391264
157.4 1.21086 − 0.879744i −0.309017 − 0.951057i 0.0742075 − 0.228387i 2.08500 + 1.51484i −1.21086 − 0.879744i −0.780732 + 2.40285i 0.813950 + 2.50508i −0.809017 + 0.587785i 3.85732
157.5 2.01466 − 1.46373i −0.309017 − 0.951057i 1.29829 − 3.99572i 1.02009 + 0.741137i −2.01466 − 1.46373i 0.590393 − 1.81704i −1.69400 − 5.21361i −0.809017 + 0.587785i 3.13995
196.1 −0.375422 + 1.15543i 0.809017 − 0.587785i 0.423959 + 0.308024i 0.339998 + 1.04640i 0.375422 + 1.15543i 0.237146 + 0.172297i −2.48080 + 1.80240i 0.309017 − 0.951057i −1.33669
196.2 −0.262109 + 0.806688i 0.809017 − 0.587785i 1.03599 + 0.752690i −1.08398 − 3.33616i 0.262109 + 0.806688i −2.82268 − 2.05079i −2.25115 + 1.63555i 0.309017 − 0.951057i 2.97536
196.3 0.169480 − 0.521606i 0.809017 − 0.587785i 1.37468 + 0.998767i −0.266070 − 0.818878i −0.169480 − 0.521606i 2.33406 + 1.69579i 1.64135 − 1.19251i 0.309017 − 0.951057i −0.472225
196.4 0.566002 − 1.74197i 0.809017 − 0.587785i −1.09608 − 0.796352i 0.725100 + 2.23163i −0.566002 − 1.74197i −1.01039 − 0.734092i 0.956015 − 0.694585i 0.309017 − 0.951057i 4.29785
196.5 0.711066 − 2.18844i 0.809017 − 0.587785i −2.66560 − 1.93667i −0.951111 − 2.92722i −0.711066 − 2.18844i −0.0471527 − 0.0342585i −2.41051 + 1.75134i 0.309017 − 0.951057i −7.08233
235.1 −2.26411 − 1.64497i −0.309017 + 0.951057i 1.80222 + 5.54667i 2.68510 − 1.95084i 2.26411 − 1.64497i −0.449002 − 1.38189i 3.31406 − 10.1996i −0.809017 − 0.587785i −9.28845
235.2 −0.972448 − 0.706525i −0.309017 + 0.951057i −0.171556 − 0.527996i −1.69471 + 1.23128i 0.972448 − 0.706525i −0.640431 − 1.97104i −0.949097 + 2.92102i −0.809017 − 0.587785i 2.51795
235.3 −0.297979 − 0.216494i −0.309017 + 0.951057i −0.576112 − 1.77309i −0.859409 + 0.624397i 0.297979 − 0.216494i 1.08879 + 3.35095i −0.439831 + 1.35366i −0.809017 − 0.587785i 0.391264
235.4 1.21086 + 0.879744i −0.309017 + 0.951057i 0.0742075 + 0.228387i 2.08500 − 1.51484i −1.21086 + 0.879744i −0.780732 − 2.40285i 0.813950 − 2.50508i −0.809017 − 0.587785i 3.85732
235.5 2.01466 + 1.46373i −0.309017 + 0.951057i 1.29829 + 3.99572i 1.02009 − 0.741137i −2.01466 + 1.46373i 0.590393 + 1.81704i −1.69400 + 5.21361i −0.809017 − 0.587785i 3.13995
313.1 −0.375422 − 1.15543i 0.809017 + 0.587785i 0.423959 − 0.308024i 0.339998 − 1.04640i 0.375422 − 1.15543i 0.237146 − 0.172297i −2.48080 − 1.80240i 0.309017 + 0.951057i −1.33669
313.2 −0.262109 − 0.806688i 0.809017 + 0.587785i 1.03599 − 0.752690i −1.08398 + 3.33616i 0.262109 − 0.806688i −2.82268 + 2.05079i −2.25115 − 1.63555i 0.309017 + 0.951057i 2.97536
313.3 0.169480 + 0.521606i 0.809017 + 0.587785i 1.37468 − 0.998767i −0.266070 + 0.818878i −0.169480 + 0.521606i 2.33406 − 1.69579i 1.64135 + 1.19251i 0.309017 + 0.951057i −0.472225
313.4 0.566002 + 1.74197i 0.809017 + 0.587785i −1.09608 + 0.796352i 0.725100 − 2.23163i −0.566002 + 1.74197i −1.01039 + 0.734092i 0.956015 + 0.694585i 0.309017 + 0.951057i 4.29785
313.5 0.711066 + 2.18844i 0.809017 + 0.587785i −2.66560 + 1.93667i −0.951111 + 2.92722i −0.711066 + 2.18844i −0.0471527 + 0.0342585i −2.41051 − 1.75134i 0.309017 + 0.951057i −7.08233
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.n.b ✓ 20
11.c even 5 1 inner 429.2.n.b ✓ 20
11.c even 5 1 4719.2.a.bn 10
11.d odd 10 1 4719.2.a.bi 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.n.b ✓ 20 1.a even 1 1 trivial
429.2.n.b ✓ 20 11.c even 5 1 inner
4719.2.a.bi 10 11.d odd 10 1
4719.2.a.bn 10 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - T_{2}^{19} + 4 T_{2}^{18} + 4 T_{2}^{17} + 37 T_{2}^{16} - 74 T_{2}^{15} + 398 T_{2}^{14} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - T^{19} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} - 4 T^{19} + \cdots + 331776 \) Copy content Toggle raw display
$7$ \( T^{20} + 3 T^{19} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{20} - 2 T^{19} + \cdots + 9554281 \) Copy content Toggle raw display
$19$ \( T^{20} + 2 T^{19} + \cdots + 32364721 \) Copy content Toggle raw display
$23$ \( (T^{10} - 3 T^{9} + \cdots + 3824)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 8008281121 \) Copy content Toggle raw display
$31$ \( T^{20} - 20 T^{19} + \cdots + 41229241 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 1085964073216 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 9398786273536 \) Copy content Toggle raw display
$43$ \( (T^{10} - 14 T^{9} + \cdots + 450224)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 283802574366481 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 2152584874561 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 14896967596281 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 51631683299361 \) Copy content Toggle raw display
$67$ \( (T^{10} - 28 T^{9} + \cdots + 207765424)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 17\!\cdots\!21 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 3035957760000 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 24104358725161 \) Copy content Toggle raw display
$89$ \( (T^{10} + 63 T^{9} + \cdots + 994744784)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
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