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Properties

Label 936.2.w.h
Level $936$
Weight $2$
Character orbit 936.w
Analytic conductor $7.474$
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] = [936,2,Mod(307,936)] 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("936.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,-2,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.47399762919\)
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: \( x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 104)
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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} - \beta_{13} q^{4} + (\beta_{16} - \beta_{14} + \cdots - \beta_{3}) q^{5} + ( - \beta_{18} + \beta_{14} + \beta_{12} + \cdots - 1) q^{7} + ( - \beta_{18} + \beta_{12} + \cdots + \beta_{4}) q^{8}+ \cdots + ( - 3 \beta_{18} - 3 \beta_{17} + \cdots - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 4 q^{8} + 8 q^{11} + 20 q^{14} + 20 q^{16} - 12 q^{19} - 8 q^{20} - 8 q^{22} + 34 q^{26} + 12 q^{28} + 8 q^{32} + 4 q^{34} - 44 q^{35} + 44 q^{40} + 24 q^{41} - 32 q^{44} + 20 q^{46} - 6 q^{50}+ \cdots + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3 \nu^{19} - 20 \nu^{18} - 58 \nu^{17} + 316 \nu^{16} - 125 \nu^{15} + 124 \nu^{14} - 42 \nu^{13} + \cdots - 38912 ) / 31232 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 100 \nu^{19} - 49 \nu^{18} - 28 \nu^{17} + 80 \nu^{16} - 1268 \nu^{15} + 1783 \nu^{14} + \cdots - 85504 ) / 62464 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{19} - 73 \nu^{18} + 64 \nu^{17} - 104 \nu^{16} - 481 \nu^{15} + 719 \nu^{14} + 272 \nu^{13} + \cdots + 13824 ) / 62464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28 \nu^{19} - 21 \nu^{18} + 23 \nu^{17} + 40 \nu^{16} + 56 \nu^{15} - 93 \nu^{14} - 257 \nu^{13} + \cdots - 7680 ) / 15616 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{19} - 42 \nu^{18} - 42 \nu^{17} - 24 \nu^{16} - 147 \nu^{15} + 774 \nu^{14} - 186 \nu^{13} + \cdots - 24064 ) / 31232 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 62 \nu^{19} - 61 \nu^{18} + 288 \nu^{17} - 192 \nu^{16} + 282 \nu^{15} - 53 \nu^{14} + \cdots + 82432 ) / 62464 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 66 \nu^{19} + 81 \nu^{18} + 72 \nu^{17} - 118 \nu^{15} + 361 \nu^{14} - 232 \nu^{13} - 912 \nu^{12} + \cdots - 27136 ) / 62464 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9 \nu^{19} + 17 \nu^{18} - 38 \nu^{17} - 64 \nu^{16} + 73 \nu^{15} - 27 \nu^{14} + 106 \nu^{13} + \cdots + 8576 ) / 7808 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53 \nu^{19} - 54 \nu^{18} - 46 \nu^{17} + 80 \nu^{16} - 171 \nu^{15} + 122 \nu^{14} + 162 \nu^{13} + \cdots + 7168 ) / 15616 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 53 \nu^{19} - 54 \nu^{18} - 46 \nu^{17} + 80 \nu^{16} - 171 \nu^{15} + 122 \nu^{14} + 162 \nu^{13} + \cdots + 7168 ) / 15616 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 35 \nu^{19} + 56 \nu^{18} + 64 \nu^{17} - 136 \nu^{16} + 153 \nu^{15} - 232 \nu^{14} + \cdots + 9216 ) / 15616 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{19} + 2 \nu^{18} - 2 \nu^{17} + 7 \nu^{15} - 14 \nu^{14} + 14 \nu^{13} - 8 \nu^{12} + \cdots + 1024 ) / 512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - \nu^{19} + \nu^{18} - 2 \nu^{16} + 7 \nu^{15} - 7 \nu^{14} + 6 \nu^{12} - 24 \nu^{11} + 24 \nu^{10} + \cdots + 512 ) / 256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 7 \nu^{19} + 67 \nu^{18} - 68 \nu^{17} - 46 \nu^{16} + 129 \nu^{15} - 269 \nu^{14} + 220 \nu^{13} + \cdots + 17920 ) / 15616 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 227 \nu^{19} - 333 \nu^{18} - 100 \nu^{17} + 520 \nu^{16} - 1053 \nu^{15} + 2283 \nu^{14} + \cdots + 12800 ) / 62464 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 41 \nu^{19} + 165 \nu^{18} - 362 \nu^{17} + 68 \nu^{16} + 383 \nu^{15} - 771 \nu^{14} + \cdots + 91136 ) / 31232 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 302 \nu^{19} - 423 \nu^{18} - 200 \nu^{17} + 696 \nu^{16} - 1482 \nu^{15} + 1617 \nu^{14} + \cdots - 62976 ) / 62464 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 37 \nu^{19} + 138 \nu^{18} - 127 \nu^{17} - 64 \nu^{16} + 207 \nu^{15} - 774 \nu^{14} + \cdots + 28672 ) / 15616 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 7 \nu^{19} + 271 \nu^{18} - 164 \nu^{17} + 44 \nu^{16} - 15 \nu^{15} - 1193 \nu^{14} + \cdots + 91136 ) / 31232 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{19} - \beta_{18} + \beta_{17} + \beta_{16} - \beta_{15} - \beta_{14} + 2 \beta_{13} - \beta_{12} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{17} + 2\beta_{14} - \beta_{10} - \beta_{9} + 2\beta_{6} - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} - 3 \beta_{15} - \beta_{14} - \beta_{12} + \cdots + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{19} + 4 \beta_{18} + 2 \beta_{16} + 4 \beta_{15} - 4 \beta_{14} - 2 \beta_{13} + \cdots - 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3 \beta_{19} + 5 \beta_{18} - 3 \beta_{17} + \beta_{16} + 5 \beta_{15} - \beta_{14} - 3 \beta_{12} + \cdots + 6 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10 \beta_{18} + 6 \beta_{17} - 2 \beta_{14} - 2 \beta_{13} - 16 \beta_{12} - \beta_{10} - \beta_{9} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - \beta_{19} + \beta_{18} + \beta_{17} + 11 \beta_{16} - 9 \beta_{15} - 11 \beta_{14} + \cdots - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6 \beta_{19} + 20 \beta_{18} - 10 \beta_{16} + 20 \beta_{15} - 12 \beta_{14} + 2 \beta_{13} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 13 \beta_{19} + 27 \beta_{18} - 13 \beta_{17} - 17 \beta_{16} + 19 \beta_{15} + \beta_{14} + 11 \beta_{12} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 14 \beta_{18} + 10 \beta_{17} - 46 \beta_{14} - 22 \beta_{13} - 15 \beta_{10} - 15 \beta_{9} + 22 \beta_{8} + \cdots + 46 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 9 \beta_{19} - 9 \beta_{18} - 9 \beta_{17} + 13 \beta_{16} - 15 \beta_{15} - 13 \beta_{14} - 41 \beta_{12} + \cdots + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 70 \beta_{19} - 52 \beta_{18} + 10 \beta_{16} - 36 \beta_{15} - 100 \beta_{14} + 46 \beta_{13} + \cdots - 136 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 21 \beta_{19} - 83 \beta_{18} + 21 \beta_{17} + 25 \beta_{16} + 5 \beta_{15} + 87 \beta_{14} + 32 \beta_{13} + \cdots - 88 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 110 \beta_{18} - 74 \beta_{17} - 2 \beta_{14} - 90 \beta_{13} + 176 \beta_{12} - 25 \beta_{10} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 47 \beta_{19} - 47 \beta_{18} - 47 \beta_{17} + 107 \beta_{16} + 55 \beta_{15} - 107 \beta_{14} + \cdots + 96 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 6 \beta_{19} + 20 \beta_{18} + 22 \beta_{16} + 276 \beta_{15} - 140 \beta_{14} + 2 \beta_{13} + \cdots - 184 ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 195 \beta_{19} - 85 \beta_{18} + 195 \beta_{17} - 129 \beta_{16} + 3 \beta_{15} + 753 \beta_{14} + \cdots - 88 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 114 \beta_{18} + 202 \beta_{17} + 370 \beta_{14} - 118 \beta_{13} + 32 \beta_{12} + \beta_{10} + \cdots - 370 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(\beta_{14}\) \(1\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.209223 + 1.39865i
0.549370 + 1.30315i
−0.813947 + 1.15650i
1.30315 + 0.549370i
−1.33837 + 0.456912i
−1.41123 − 0.0918109i
1.39865 − 0.209223i
1.15650 − 0.813947i
0.456912 − 1.33837i
−0.0918109 − 1.41123i
−0.209223 − 1.39865i
0.549370 − 1.30315i
−0.813947 − 1.15650i
1.30315 − 0.549370i
−1.33837 − 0.456912i
−1.41123 + 0.0918109i
1.39865 + 0.209223i
1.15650 + 0.813947i
0.456912 + 1.33837i
−0.0918109 + 1.41123i
−1.39865 + 0.209223i 0 1.91245 − 0.585261i −1.55756 + 1.55756i 0 1.24165 + 1.24165i −2.55240 + 1.21871i 0 1.85261 − 2.50437i
307.2 −1.30315 − 0.549370i 0 1.39639 + 1.43182i 0.328612 − 0.328612i 0 −2.68064 − 2.68064i −1.03310 − 2.63300i 0 −0.608760 + 0.247700i
307.3 −1.15650 + 0.813947i 0 0.674982 − 1.88266i 2.40872 − 2.40872i 0 −0.127019 − 0.127019i 0.751766 + 2.72669i 0 −0.825113 + 4.74625i
307.4 −0.549370 − 1.30315i 0 −1.39639 + 1.43182i −0.328612 + 0.328612i 0 2.68064 + 2.68064i 2.63300 + 1.03310i 0 0.608760 + 0.247700i
307.5 −0.456912 + 1.33837i 0 −1.58246 − 1.22303i 1.04334 − 1.04334i 0 −2.37322 − 2.37322i 2.35992 − 1.55910i 0 0.919655 + 1.87308i
307.6 0.0918109 + 1.41123i 0 −1.98314 + 0.259133i −0.274863 + 0.274863i 0 −0.352378 − 0.352378i −0.547770 − 2.77488i 0 −0.413130 − 0.362660i
307.7 0.209223 − 1.39865i 0 −1.91245 − 0.585261i 1.55756 − 1.55756i 0 −1.24165 − 1.24165i −1.21871 + 2.55240i 0 −1.85261 − 2.50437i
307.8 0.813947 − 1.15650i 0 −0.674982 − 1.88266i −2.40872 + 2.40872i 0 0.127019 + 0.127019i −2.72669 − 0.751766i 0 0.825113 + 4.74625i
307.9 1.33837 − 0.456912i 0 1.58246 − 1.22303i −1.04334 + 1.04334i 0 2.37322 + 2.37322i 1.55910 − 2.35992i 0 −0.919655 + 1.87308i
307.10 1.41123 + 0.0918109i 0 1.98314 + 0.259133i 0.274863 − 0.274863i 0 0.352378 + 0.352378i 2.77488 + 0.547770i 0 0.413130 − 0.362660i
811.1 −1.39865 − 0.209223i 0 1.91245 + 0.585261i −1.55756 − 1.55756i 0 1.24165 − 1.24165i −2.55240 − 1.21871i 0 1.85261 + 2.50437i
811.2 −1.30315 + 0.549370i 0 1.39639 − 1.43182i 0.328612 + 0.328612i 0 −2.68064 + 2.68064i −1.03310 + 2.63300i 0 −0.608760 − 0.247700i
811.3 −1.15650 − 0.813947i 0 0.674982 + 1.88266i 2.40872 + 2.40872i 0 −0.127019 + 0.127019i 0.751766 − 2.72669i 0 −0.825113 − 4.74625i
811.4 −0.549370 + 1.30315i 0 −1.39639 − 1.43182i −0.328612 − 0.328612i 0 2.68064 − 2.68064i 2.63300 − 1.03310i 0 0.608760 − 0.247700i
811.5 −0.456912 − 1.33837i 0 −1.58246 + 1.22303i 1.04334 + 1.04334i 0 −2.37322 + 2.37322i 2.35992 + 1.55910i 0 0.919655 − 1.87308i
811.6 0.0918109 − 1.41123i 0 −1.98314 − 0.259133i −0.274863 − 0.274863i 0 −0.352378 + 0.352378i −0.547770 + 2.77488i 0 −0.413130 + 0.362660i
811.7 0.209223 + 1.39865i 0 −1.91245 + 0.585261i 1.55756 + 1.55756i 0 −1.24165 + 1.24165i −1.21871 − 2.55240i 0 −1.85261 + 2.50437i
811.8 0.813947 + 1.15650i 0 −0.674982 + 1.88266i −2.40872 − 2.40872i 0 0.127019 − 0.127019i −2.72669 + 0.751766i 0 0.825113 − 4.74625i
811.9 1.33837 + 0.456912i 0 1.58246 + 1.22303i −1.04334 − 1.04334i 0 2.37322 − 2.37322i 1.55910 + 2.35992i 0 −0.919655 − 1.87308i
811.10 1.41123 − 0.0918109i 0 1.98314 − 0.259133i 0.274863 + 0.274863i 0 0.352378 − 0.352378i 2.77488 − 0.547770i 0 0.413130 + 0.362660i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
13.d odd 4 1 inner
104.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.w.h 20
3.b odd 2 1 104.2.m.b ✓ 20
8.d odd 2 1 inner 936.2.w.h 20
12.b even 2 1 416.2.u.b 20
13.d odd 4 1 inner 936.2.w.h 20
24.f even 2 1 104.2.m.b ✓ 20
24.h odd 2 1 416.2.u.b 20
39.f even 4 1 104.2.m.b ✓ 20
104.m even 4 1 inner 936.2.w.h 20
156.l odd 4 1 416.2.u.b 20
312.w odd 4 1 104.2.m.b ✓ 20
312.y even 4 1 416.2.u.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.m.b ✓ 20 3.b odd 2 1
104.2.m.b ✓ 20 24.f even 2 1
104.2.m.b ✓ 20 39.f even 4 1
104.2.m.b ✓ 20 312.w odd 4 1
416.2.u.b 20 12.b even 2 1
416.2.u.b 20 24.h odd 2 1
416.2.u.b 20 156.l odd 4 1
416.2.u.b 20 312.y even 4 1
936.2.w.h 20 1.a even 1 1 trivial
936.2.w.h 20 8.d odd 2 1 inner
936.2.w.h 20 13.d odd 4 1 inner
936.2.w.h 20 104.m even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\):

\( T_{5}^{20} + 163T_{5}^{16} + 3931T_{5}^{12} + 15297T_{5}^{8} + 1048T_{5}^{4} + 16 \) Copy content Toggle raw display
\( T_{7}^{20} + 343T_{7}^{16} + 29399T_{7}^{12} + 251005T_{7}^{8} + 15628T_{7}^{4} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 2 T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 163 T^{16} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{20} + 343 T^{16} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{10} - 4 T^{9} + \cdots + 5408)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( (T^{10} + 71 T^{8} + \cdots + 355216)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 6 T^{9} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 94 T^{8} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 186 T^{8} + \cdots + 21632)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 92829679353856 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 911076029249296 \) Copy content Toggle raw display
$41$ \( (T^{10} - 12 T^{9} + \cdots + 512)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 111 T^{8} + \cdots + 795664)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 1416468496 \) Copy content Toggle raw display
$53$ \( (T^{10} + 230 T^{8} + \cdots + 3442688)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 10 T^{9} + \cdots + 1384448)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 392 T^{8} + \cdots + 8388608)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 8 T^{9} + \cdots + 78575648)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 97459622042896 \) Copy content Toggle raw display
$73$ \( (T^{10} - 12 T^{9} + \cdots + 512)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 442 T^{8} + \cdots + 336338048)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 8 T^{9} + \cdots + 414950432)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 117546549248)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} - 6 T^{9} + \cdots + 7270250528)^{2} \) Copy content Toggle raw display
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