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Properties

Label 200.3.i.b
Level $200$
Weight $3$
Character orbit 200.i
Analytic conductor $5.450$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [200,3,Mod(93,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("200.93"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 200.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.44960528721\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{21} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_{11} q^{3} - \beta_{15} q^{4} + (\beta_{6} - 1) q^{6} - \beta_{7} q^{7} + (\beta_{18} + \beta_{16} + \cdots + \beta_{6}) q^{8} + (\beta_{19} + 2 \beta_{18} + \cdots + \beta_{4}) q^{9}+ \cdots + (18 \beta_{19} + 3 \beta_{17} + \cdots - 6 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 16 q^{6} + 4 q^{7} - 4 q^{8} + 44 q^{12} - 56 q^{16} + 12 q^{17} - 10 q^{18} - 92 q^{22} + 4 q^{23} + 100 q^{26} - 68 q^{28} - 136 q^{31} - 128 q^{32} - 32 q^{33} + 220 q^{36} + 188 q^{38}+ \cdots - 546 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17 \nu^{18} + 13 \nu^{16} + 71 \nu^{14} + 33 \nu^{12} + 115 \nu^{10} - 140 \nu^{8} + 972 \nu^{6} + \cdots - 2048 ) / 9600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17 \nu^{18} - 13 \nu^{16} - 71 \nu^{14} - 33 \nu^{12} - 115 \nu^{10} + 140 \nu^{8} - 972 \nu^{6} + \cdots + 2048 ) / 9600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2 \nu^{19} + 19 \nu^{17} - 7 \nu^{15} - 93 \nu^{13} - 35 \nu^{11} - 35 \nu^{9} + 132 \nu^{7} + \cdots + 1408 \nu ) / 9600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53 \nu^{18} + 85 \nu^{16} - 145 \nu^{14} - 471 \nu^{12} + 1435 \nu^{10} + 2680 \nu^{8} + \cdots - 17024 ) / 9600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{18} - 13 \nu^{16} + 9 \nu^{14} + 15 \nu^{12} - 35 \nu^{10} - 20 \nu^{8} + 324 \nu^{6} + \cdots + 1600 ) / 320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5 \nu^{19} + 234 \nu^{18} + 215 \nu^{17} + 246 \nu^{16} + 805 \nu^{15} - 1278 \nu^{14} + \cdots - 118272 ) / 38400 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{19} + 234 \nu^{18} - 215 \nu^{17} + 246 \nu^{16} - 805 \nu^{15} - 1278 \nu^{14} + \cdots - 118272 ) / 38400 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15 \nu^{19} + 11 \nu^{18} + 168 \nu^{17} - 175 \nu^{16} - 384 \nu^{15} + 355 \nu^{14} + 12 \nu^{13} + \cdots + 46208 ) / 9600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15 \nu^{19} - 11 \nu^{18} + 168 \nu^{17} + 175 \nu^{16} - 384 \nu^{15} - 355 \nu^{14} + 12 \nu^{13} + \cdots - 46208 ) / 9600 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25 \nu^{19} - 82 \nu^{18} + 45 \nu^{17} + 50 \nu^{16} - 185 \nu^{15} - 10 \nu^{14} - 95 \nu^{13} + \cdots - 16896 ) / 12800 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{18} + \nu^{16} + 3\nu^{14} - 11\nu^{12} - \nu^{10} + 40\nu^{8} - 4\nu^{6} - 176\nu^{4} + 192\nu^{2} + 256 ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11 \nu^{19} - 5 \nu^{17} + 185 \nu^{15} - 177 \nu^{13} - 755 \nu^{11} + 160 \nu^{9} + \cdots + 13312 \nu ) / 4800 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25 \nu^{19} - 82 \nu^{18} - 45 \nu^{17} + 50 \nu^{16} + 185 \nu^{15} - 10 \nu^{14} + 95 \nu^{13} + \cdots - 16896 ) / 12800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17 \nu^{19} - 13 \nu^{17} - 71 \nu^{15} - 33 \nu^{13} - 115 \nu^{11} + 140 \nu^{9} - 972 \nu^{7} + \cdots + 2048 \nu ) / 4800 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 75 \nu^{19} - 16 \nu^{18} + 75 \nu^{17} + 152 \nu^{16} + 225 \nu^{15} - 56 \nu^{14} + \cdots + 11264 ) / 19200 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 75 \nu^{19} - 16 \nu^{18} - 75 \nu^{17} + 152 \nu^{16} - 225 \nu^{15} - 56 \nu^{14} + 825 \nu^{13} + \cdots + 11264 ) / 19200 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 145 \nu^{19} + 301 \nu^{17} - 313 \nu^{15} + 489 \nu^{13} + 1075 \nu^{11} + 1630 \nu^{9} + \cdots - 22784 \nu ) / 19200 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 135 \nu^{19} - 22 \nu^{18} - 153 \nu^{17} + 350 \nu^{16} - 411 \nu^{15} - 710 \nu^{14} + \cdots - 92416 ) / 19200 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{18} - \beta_{17} + \beta_{16} - \beta_{14} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{17} - \beta_{16} - \beta_{14} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{19} + \beta_{18} - \beta_{15} - 2 \beta_{14} + \beta_{13} + 2 \beta_{11} - \beta_{10} + \cdots + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{17} + 2\beta_{16} - \beta_{12} - \beta_{10} + \beta_{9} + 2\beta_{5} - \beta_{3} + \beta_{2} + 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{17} - \beta_{16} - \beta_{15} - \beta_{14} - 2 \beta_{13} + \beta_{11} - \beta_{10} + \cdots - 2 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5 \beta_{17} - 5 \beta_{16} - \beta_{14} - \beta_{11} - 5 \beta_{10} + 5 \beta_{9} - \beta_{8} + \cdots - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2 \beta_{19} - 4 \beta_{18} - \beta_{17} + \beta_{16} + 8 \beta_{15} - \beta_{14} + \cdots - 4 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( \beta_{17} + \beta_{16} - \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + 5 \beta_{8} + \cdots + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 8 \beta_{19} - 2 \beta_{18} - 5 \beta_{17} + 5 \beta_{16} - 12 \beta_{15} + 5 \beta_{14} + \cdots + 5 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 12 \beta_{17} - 12 \beta_{16} + 6 \beta_{14} - 8 \beta_{12} + 6 \beta_{11} - 10 \beta_{10} + \cdots + 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 40 \beta_{19} - 2 \beta_{18} + 21 \beta_{17} - 21 \beta_{16} - 8 \beta_{15} + 17 \beta_{14} + \cdots + 48 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 55 \beta_{17} - 55 \beta_{16} - 71 \beta_{14} + 40 \beta_{12} - 71 \beta_{11} + 23 \beta_{10} + \cdots - 162 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 23 \beta_{19} + 51 \beta_{18} - 30 \beta_{17} + 30 \beta_{16} - 59 \beta_{15} + 27 \beta_{13} + \cdots - 41 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 10 \beta_{17} + 10 \beta_{16} - 36 \beta_{14} - 23 \beta_{12} - 36 \beta_{11} - 15 \beta_{10} + \cdots + 104 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 20 \beta_{18} + 57 \beta_{17} - 57 \beta_{16} - 107 \beta_{15} - 37 \beta_{14} + 46 \beta_{13} + \cdots + 86 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 77 \beta_{17} + 77 \beta_{16} - 279 \beta_{14} - 279 \beta_{11} - 107 \beta_{10} + 107 \beta_{9} + \cdots - 1132 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 130 \beta_{19} + 244 \beta_{18} - 15 \beta_{17} + 15 \beta_{16} + 224 \beta_{15} + 209 \beta_{14} + \cdots - 596 \beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{4}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
1.27574 − 0.610320i
0.541828 − 1.30630i
1.39859 + 0.209644i
−0.541828 − 1.30630i
1.17039 + 0.793843i
−1.27574 − 0.610320i
0.0552378 + 1.41313i
−0.0552378 + 1.41313i
−1.39859 + 0.209644i
−1.17039 + 0.793843i
1.27574 + 0.610320i
0.541828 + 1.30630i
1.39859 − 0.209644i
−0.541828 + 1.30630i
1.17039 − 0.793843i
−1.27574 + 0.610320i
0.0552378 − 1.41313i
−0.0552378 − 1.41313i
−1.39859 − 0.209644i
−1.17039 − 0.793843i
−1.88606 − 0.665418i 3.60765 − 3.60765i 3.11444 + 2.51004i 0 −9.20485 + 4.40365i −1.47907 + 1.47907i −4.20379 − 6.80648i − 17.0303i 0
93.2 −1.84813 + 0.764474i −0.130791 + 0.130791i 2.83116 − 2.82569i 0 0.141733 − 0.341706i 1.59713 − 1.59713i −3.07218 + 7.38659i 8.96579i 0
93.3 −1.18894 − 1.60823i −2.52630 + 2.52630i −1.17282 + 3.82420i 0 7.06650 + 1.05925i 5.20520 − 5.20520i 7.54462 − 2.66059i − 3.76437i 0
93.4 −0.764474 + 1.84813i 0.130791 − 0.130791i −2.83116 − 2.82569i 0 0.141733 + 0.341706i 1.59713 − 1.59713i 7.38659 − 3.07218i 8.96579i 0
93.5 −0.376547 − 1.96423i 0.977390 − 0.977390i −3.71643 + 1.47925i 0 −2.28785 − 1.55179i −8.39950 + 8.39950i 4.30500 + 6.74292i 7.08942i 0
93.6 0.665418 + 1.88606i −3.60765 + 3.60765i −3.11444 + 2.51004i 0 −9.20485 − 4.40365i −1.47907 + 1.47907i −6.80648 − 4.20379i − 17.0303i 0
93.7 1.35790 − 1.46837i −2.57493 + 2.57493i −0.312234 − 3.98780i 0 0.284467 + 7.27744i 4.07624 − 4.07624i −6.27955 − 4.95654i − 4.26050i 0
93.8 1.46837 − 1.35790i 2.57493 − 2.57493i 0.312234 − 3.98780i 0 0.284467 − 7.27744i 4.07624 − 4.07624i −4.95654 − 6.27955i − 4.26050i 0
93.9 1.60823 + 1.18894i 2.52630 − 2.52630i 1.17282 + 3.82420i 0 7.06650 − 1.05925i 5.20520 − 5.20520i −2.66059 + 7.54462i − 3.76437i 0
93.10 1.96423 + 0.376547i −0.977390 + 0.977390i 3.71643 + 1.47925i 0 −2.28785 + 1.55179i −8.39950 + 8.39950i 6.74292 + 4.30500i 7.08942i 0
157.1 −1.88606 + 0.665418i 3.60765 + 3.60765i 3.11444 − 2.51004i 0 −9.20485 − 4.40365i −1.47907 − 1.47907i −4.20379 + 6.80648i 17.0303i 0
157.2 −1.84813 − 0.764474i −0.130791 − 0.130791i 2.83116 + 2.82569i 0 0.141733 + 0.341706i 1.59713 + 1.59713i −3.07218 − 7.38659i − 8.96579i 0
157.3 −1.18894 + 1.60823i −2.52630 − 2.52630i −1.17282 − 3.82420i 0 7.06650 − 1.05925i 5.20520 + 5.20520i 7.54462 + 2.66059i 3.76437i 0
157.4 −0.764474 − 1.84813i 0.130791 + 0.130791i −2.83116 + 2.82569i 0 0.141733 − 0.341706i 1.59713 + 1.59713i 7.38659 + 3.07218i − 8.96579i 0
157.5 −0.376547 + 1.96423i 0.977390 + 0.977390i −3.71643 − 1.47925i 0 −2.28785 + 1.55179i −8.39950 − 8.39950i 4.30500 − 6.74292i − 7.08942i 0
157.6 0.665418 − 1.88606i −3.60765 − 3.60765i −3.11444 − 2.51004i 0 −9.20485 + 4.40365i −1.47907 − 1.47907i −6.80648 + 4.20379i 17.0303i 0
157.7 1.35790 + 1.46837i −2.57493 − 2.57493i −0.312234 + 3.98780i 0 0.284467 − 7.27744i 4.07624 + 4.07624i −6.27955 + 4.95654i 4.26050i 0
157.8 1.46837 + 1.35790i 2.57493 + 2.57493i 0.312234 + 3.98780i 0 0.284467 + 7.27744i 4.07624 + 4.07624i −4.95654 + 6.27955i 4.26050i 0
157.9 1.60823 − 1.18894i 2.52630 + 2.52630i 1.17282 − 3.82420i 0 7.06650 + 1.05925i 5.20520 + 5.20520i −2.66059 − 7.54462i 3.76437i 0
157.10 1.96423 − 0.376547i −0.977390 − 0.977390i 3.71643 − 1.47925i 0 −2.28785 − 1.55179i −8.39950 − 8.39950i 6.74292 − 4.30500i − 7.08942i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 93.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.b even 2 1 inner
40.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.3.i.b 20
4.b odd 2 1 800.3.m.b 20
5.b even 2 1 40.3.i.a ✓ 20
5.c odd 4 1 40.3.i.a ✓ 20
5.c odd 4 1 inner 200.3.i.b 20
8.b even 2 1 inner 200.3.i.b 20
8.d odd 2 1 800.3.m.b 20
15.d odd 2 1 360.3.u.b 20
15.e even 4 1 360.3.u.b 20
20.d odd 2 1 160.3.m.a 20
20.e even 4 1 160.3.m.a 20
20.e even 4 1 800.3.m.b 20
40.e odd 2 1 160.3.m.a 20
40.f even 2 1 40.3.i.a ✓ 20
40.i odd 4 1 40.3.i.a ✓ 20
40.i odd 4 1 inner 200.3.i.b 20
40.k even 4 1 160.3.m.a 20
40.k even 4 1 800.3.m.b 20
120.i odd 2 1 360.3.u.b 20
120.w even 4 1 360.3.u.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.i.a ✓ 20 5.b even 2 1
40.3.i.a ✓ 20 5.c odd 4 1
40.3.i.a ✓ 20 40.f even 2 1
40.3.i.a ✓ 20 40.i odd 4 1
160.3.m.a 20 20.d odd 2 1
160.3.m.a 20 20.e even 4 1
160.3.m.a 20 40.e odd 2 1
160.3.m.a 20 40.k even 4 1
200.3.i.b 20 1.a even 1 1 trivial
200.3.i.b 20 5.c odd 4 1 inner
200.3.i.b 20 8.b even 2 1 inner
200.3.i.b 20 40.i odd 4 1 inner
360.3.u.b 20 15.d odd 2 1
360.3.u.b 20 15.e even 4 1
360.3.u.b 20 120.i odd 2 1
360.3.u.b 20 120.w even 4 1
800.3.m.b 20 4.b odd 2 1
800.3.m.b 20 8.d odd 2 1
800.3.m.b 20 20.e even 4 1
800.3.m.b 20 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 1020T_{3}^{16} + 261904T_{3}^{12} + 20355136T_{3}^{8} + 70885120T_{3}^{4} + 82944 \) acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 2 T^{19} + \cdots + 1048576 \) Copy content Toggle raw display
$3$ \( T^{20} + 1020 T^{16} + \cdots + 82944 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} - 2 T^{9} + \cdots + 5671712)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 552 T^{8} + \cdots + 473497600)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{10} - 6 T^{9} + \cdots + 2344207392)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 384717505536)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 2 T^{9} + \cdots + 415411488)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 2548 T^{8} + \cdots - 31360000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 34 T^{4} + \cdots - 1252704)^{4} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 59\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{5} + 2 T^{4} + \cdots + 74680800)^{4} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 81\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 418650446283552)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 26\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{5} - 62 T^{4} + \cdots + 423110304)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 37\!\cdots\!88)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 90\!\cdots\!32)^{2} \) Copy content Toggle raw display
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