Properties

Label 104.2.m.b
Level $104$
Weight $2$
Character orbit 104.m
Analytic conductor $0.830$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [104,2,Mod(83,104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(104, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("104.83");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 104.m (of order \(4\), 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: \(0.830444181021\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{14} q^{2} - \beta_{5} q^{3} - \beta_{2} q^{4} + \beta_{10} q^{5} + ( - \beta_{17} - \beta_{16} + \beta_{15} + \cdots - 1) q^{6}+ \cdots + (\beta_{17} + \beta_{14} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{14} q^{2} - \beta_{5} q^{3} - \beta_{2} q^{4} + \beta_{10} q^{5} + ( - \beta_{17} - \beta_{16} + \beta_{15} + \cdots - 1) q^{6}+ \cdots + (\beta_{18} + \beta_{17} + 3 \beta_{15} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 4 q^{3} - 4 q^{6} - 4 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} - 4 q^{3} - 4 q^{6} - 4 q^{8} + 16 q^{9} - 8 q^{11} - 20 q^{14} + 20 q^{16} - 26 q^{18} - 12 q^{19} + 8 q^{20} - 8 q^{22} + 4 q^{24} - 34 q^{26} - 52 q^{27} + 12 q^{28} - 8 q^{32} - 12 q^{33} + 4 q^{34} + 44 q^{35} + 44 q^{40} - 24 q^{41} + 28 q^{42} + 32 q^{44} + 20 q^{46} - 20 q^{48} + 6 q^{50} + 36 q^{52} + 24 q^{54} + 8 q^{57} + 12 q^{58} + 20 q^{59} - 56 q^{60} - 8 q^{65} + 56 q^{66} - 16 q^{67} - 28 q^{68} + 8 q^{70} - 44 q^{72} + 24 q^{73} + 40 q^{74} + 44 q^{76} + 68 q^{78} - 12 q^{80} - 44 q^{81} + 16 q^{83} - 60 q^{84} - 36 q^{86} - 16 q^{89} + 32 q^{91} - 88 q^{92} - 44 q^{94} - 84 q^{96} + 12 q^{97} - 26 q^{98} + 20 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9 \nu^{19} + 10 \nu^{18} - 250 \nu^{17} + 272 \nu^{16} + 121 \nu^{15} - 390 \nu^{14} + 950 \nu^{13} + \cdots + 44032 ) / 31232 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3 \nu^{19} + 20 \nu^{18} + 58 \nu^{17} - 316 \nu^{16} + 125 \nu^{15} - 124 \nu^{14} + \cdots + 38912 ) / 31232 \) 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
\(\beta_{7}\)\(=\) \( ( 71 \nu^{19} - 12 \nu^{18} - 176 \nu^{17} + 300 \nu^{16} - 289 \nu^{15} + 692 \nu^{14} + 368 \nu^{13} + \cdots - 18432 ) / 31232 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8 \nu^{19} + 47 \nu^{18} - 44 \nu^{17} + 12 \nu^{16} - 208 \nu^{15} - 729 \nu^{14} + 804 \nu^{13} + \cdots + 51712 ) / 31232 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41 \nu^{19} - 64 \nu^{18} - 152 \nu^{17} + 208 \nu^{16} - 239 \nu^{15} + 48 \nu^{14} + 1160 \nu^{13} + \cdots + 85504 ) / 31232 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 66 \nu^{19} + 81 \nu^{18} + 72 \nu^{17} - 118 \nu^{15} + 361 \nu^{14} - 232 \nu^{13} - 912 \nu^{12} + \cdots - 27136 ) / 62464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 85 \nu^{19} + \nu^{18} + 20 \nu^{17} - 76 \nu^{16} - 91 \nu^{15} + 313 \nu^{14} + 260 \nu^{13} + \cdots + 52224 ) / 31232 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 161 \nu^{19} + 111 \nu^{18} - 116 \nu^{17} - 128 \nu^{16} + 991 \nu^{15} - 729 \nu^{14} + \cdots + 54784 ) / 62464 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 206 \nu^{19} - 193 \nu^{18} - 304 \nu^{17} + 504 \nu^{16} - 666 \nu^{15} + 1415 \nu^{14} + \cdots - 152064 ) / 62464 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 53 \nu^{19} - 54 \nu^{18} - 46 \nu^{17} + 80 \nu^{16} - 171 \nu^{15} + 122 \nu^{14} + 162 \nu^{13} + \cdots + 7168 ) / 15616 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7 \nu^{19} + 67 \nu^{18} - 68 \nu^{17} - 46 \nu^{16} + 129 \nu^{15} - 269 \nu^{14} + 220 \nu^{13} + \cdots + 17920 ) / 15616 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 222 \nu^{19} + 293 \nu^{18} - 216 \nu^{17} - 80 \nu^{16} + 2122 \nu^{15} - 3491 \nu^{14} + \cdots + 210432 ) / 62464 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 131 \nu^{19} - 234 \nu^{18} - 6 \nu^{17} + 272 \nu^{16} - 733 \nu^{15} + 1318 \nu^{14} + \cdots - 80896 ) / 31232 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 177 \nu^{19} - 453 \nu^{18} + 348 \nu^{17} + 96 \nu^{16} - 927 \nu^{15} + 3571 \nu^{14} + \cdots - 237056 ) / 62464 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 13 \nu^{19} - 38 \nu^{18} + 20 \nu^{17} + 50 \nu^{16} - 155 \nu^{15} + 226 \nu^{14} - 188 \nu^{13} + \cdots - 13568 ) / 3904 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{10} - \beta_{9} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{18} + \beta_{17} + \beta_{13} + \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + 2\beta_{6} + \beta_{5} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - 3 \beta_{18} + \beta_{17} - 2 \beta_{15} + \beta_{12} + 2 \beta_{11} - 2 \beta_{9} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{18} - 2 \beta_{16} + 2 \beta_{15} + \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{19} + 3 \beta_{17} + 2 \beta_{16} + 7 \beta_{14} - 9 \beta_{13} - 3 \beta_{10} + \cdots - \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{18} + 3 \beta_{17} - 2 \beta_{16} + 2 \beta_{15} + 4 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - \beta_{19} - \beta_{18} - \beta_{17} + 6 \beta_{15} - 13 \beta_{12} - 6 \beta_{11} + 6 \beta_{9} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5 \beta_{18} - 6 \beta_{16} + 6 \beta_{15} - 8 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 6 \beta_{11} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 11 \beta_{19} + \beta_{17} - 2 \beta_{16} - 32 \beta_{15} + 17 \beta_{14} - 23 \beta_{13} - 13 \beta_{10} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 8 \beta_{19} + 3 \beta_{18} + 13 \beta_{17} - 6 \beta_{16} + 6 \beta_{15} - 12 \beta_{14} - 5 \beta_{13} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 23 \beta_{19} + 9 \beta_{18} - 15 \beta_{17} - 22 \beta_{15} - 43 \beta_{12} - 26 \beta_{11} + \cdots - 22 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11 \beta_{18} - 10 \beta_{16} + 10 \beta_{15} - 48 \beta_{14} + 13 \beta_{13} - 3 \beta_{12} + 10 \beta_{11} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 45 \beta_{19} + 15 \beta_{17} + 18 \beta_{16} - 64 \beta_{15} - 9 \beta_{14} + 47 \beta_{13} + \cdots + 64 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 56 \beta_{19} - 99 \beta_{18} + 75 \beta_{17} + 70 \beta_{16} - 70 \beta_{15} - 140 \beta_{14} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - \beta_{19} - 17 \beta_{18} - 97 \beta_{17} - 90 \beta_{15} - 93 \beta_{12} + 42 \beta_{11} + \cdots - 90 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 85 \beta_{18} + 74 \beta_{16} + 342 \beta_{15} - 56 \beta_{14} - 77 \beta_{13} - 173 \beta_{12} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 59 \beta_{19} + 97 \beta_{17} + 94 \beta_{16} - 32 \beta_{15} + 321 \beta_{14} - 103 \beta_{13} + \cdots + 32 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(53\) \(79\)
\(\chi(n)\) \(-\beta_{15}\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1
−0.0918109 1.41123i
0.456912 1.33837i
1.15650 0.813947i
1.39865 0.209223i
−1.41123 0.0918109i
−1.33837 + 0.456912i
1.30315 + 0.549370i
−0.813947 + 1.15650i
0.549370 + 1.30315i
−0.209223 + 1.39865i
−0.0918109 + 1.41123i
0.456912 + 1.33837i
1.15650 + 0.813947i
1.39865 + 0.209223i
−1.41123 + 0.0918109i
−1.33837 0.456912i
1.30315 0.549370i
−0.813947 1.15650i
0.549370 1.30315i
−0.209223 1.39865i
−1.41123 + 0.0918109i 2.19371 1.98314 0.259133i −0.274863 0.274863i −3.09583 + 0.201407i 0.352378 0.352378i −2.77488 + 0.547770i 1.81237 0.413130 + 0.362660i
83.2 −1.33837 0.456912i −2.95212 1.58246 + 1.22303i 1.04334 + 1.04334i 3.95103 + 1.34886i 2.37322 2.37322i −1.55910 2.35992i 5.71504 −0.919655 1.87308i
83.3 −0.813947 1.15650i 1.38809 −0.674982 + 1.88266i 2.40872 + 2.40872i −1.12983 1.60533i 0.127019 0.127019i 2.72669 0.751766i −1.07320 0.825113 4.74625i
83.4 −0.209223 1.39865i −1.86790 −1.91245 + 0.585261i −1.55756 1.55756i 0.390808 + 2.61254i −1.24165 + 1.24165i 1.21871 + 2.55240i 0.489042 −1.85261 + 2.50437i
83.5 −0.0918109 + 1.41123i 2.19371 −1.98314 0.259133i 0.274863 + 0.274863i −0.201407 + 3.09583i −0.352378 + 0.352378i 0.547770 2.77488i 1.81237 −0.413130 + 0.362660i
83.6 0.456912 + 1.33837i −2.95212 −1.58246 + 1.22303i −1.04334 1.04334i −1.34886 3.95103i −2.37322 + 2.37322i −2.35992 1.55910i 5.71504 0.919655 1.87308i
83.7 0.549370 1.30315i 0.238218 −1.39639 1.43182i 0.328612 + 0.328612i 0.130870 0.310433i 2.68064 2.68064i −2.63300 + 1.03310i −2.94325 0.608760 0.247700i
83.8 1.15650 + 0.813947i 1.38809 0.674982 + 1.88266i −2.40872 2.40872i 1.60533 + 1.12983i −0.127019 + 0.127019i −0.751766 + 2.72669i −1.07320 −0.825113 4.74625i
83.9 1.30315 0.549370i 0.238218 1.39639 1.43182i −0.328612 0.328612i 0.310433 0.130870i −2.68064 + 2.68064i 1.03310 2.63300i −2.94325 −0.608760 0.247700i
83.10 1.39865 + 0.209223i −1.86790 1.91245 + 0.585261i 1.55756 + 1.55756i −2.61254 0.390808i 1.24165 1.24165i 2.55240 + 1.21871i 0.489042 1.85261 + 2.50437i
99.1 −1.41123 0.0918109i 2.19371 1.98314 + 0.259133i −0.274863 + 0.274863i −3.09583 0.201407i 0.352378 + 0.352378i −2.77488 0.547770i 1.81237 0.413130 0.362660i
99.2 −1.33837 + 0.456912i −2.95212 1.58246 1.22303i 1.04334 1.04334i 3.95103 1.34886i 2.37322 + 2.37322i −1.55910 + 2.35992i 5.71504 −0.919655 + 1.87308i
99.3 −0.813947 + 1.15650i 1.38809 −0.674982 1.88266i 2.40872 2.40872i −1.12983 + 1.60533i 0.127019 + 0.127019i 2.72669 + 0.751766i −1.07320 0.825113 + 4.74625i
99.4 −0.209223 + 1.39865i −1.86790 −1.91245 0.585261i −1.55756 + 1.55756i 0.390808 2.61254i −1.24165 1.24165i 1.21871 2.55240i 0.489042 −1.85261 2.50437i
99.5 −0.0918109 1.41123i 2.19371 −1.98314 + 0.259133i 0.274863 0.274863i −0.201407 3.09583i −0.352378 0.352378i 0.547770 + 2.77488i 1.81237 −0.413130 0.362660i
99.6 0.456912 1.33837i −2.95212 −1.58246 1.22303i −1.04334 + 1.04334i −1.34886 + 3.95103i −2.37322 2.37322i −2.35992 + 1.55910i 5.71504 0.919655 + 1.87308i
99.7 0.549370 + 1.30315i 0.238218 −1.39639 + 1.43182i 0.328612 0.328612i 0.130870 + 0.310433i 2.68064 + 2.68064i −2.63300 1.03310i −2.94325 0.608760 + 0.247700i
99.8 1.15650 0.813947i 1.38809 0.674982 1.88266i −2.40872 + 2.40872i 1.60533 1.12983i −0.127019 0.127019i −0.751766 2.72669i −1.07320 −0.825113 + 4.74625i
99.9 1.30315 + 0.549370i 0.238218 1.39639 + 1.43182i −0.328612 + 0.328612i 0.310433 + 0.130870i −2.68064 2.68064i 1.03310 + 2.63300i −2.94325 −0.608760 + 0.247700i
99.10 1.39865 0.209223i −1.86790 1.91245 0.585261i 1.55756 1.55756i −2.61254 + 0.390808i 1.24165 + 1.24165i 2.55240 1.21871i 0.489042 1.85261 2.50437i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.10
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 104.2.m.b 20
3.b odd 2 1 936.2.w.h 20
4.b odd 2 1 416.2.u.b 20
8.b even 2 1 416.2.u.b 20
8.d odd 2 1 inner 104.2.m.b 20
13.d odd 4 1 inner 104.2.m.b 20
24.f even 2 1 936.2.w.h 20
39.f even 4 1 936.2.w.h 20
52.f even 4 1 416.2.u.b 20
104.j odd 4 1 416.2.u.b 20
104.m even 4 1 inner 104.2.m.b 20
312.w odd 4 1 936.2.w.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.m.b 20 1.a even 1 1 trivial
104.2.m.b 20 8.d odd 2 1 inner
104.2.m.b 20 13.d odd 4 1 inner
104.2.m.b 20 104.m even 4 1 inner
416.2.u.b 20 4.b odd 2 1
416.2.u.b 20 8.b even 2 1
416.2.u.b 20 52.f even 4 1
416.2.u.b 20 104.j odd 4 1
936.2.w.h 20 3.b odd 2 1
936.2.w.h 20 24.f even 2 1
936.2.w.h 20 39.f even 4 1
936.2.w.h 20 312.w odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + T_{3}^{4} - 9T_{3}^{3} - 3T_{3}^{2} + 18T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(104, [\chi])\). 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
$3$ \( (T^{5} + T^{4} - 9 T^{3} + \cdots - 4)^{4} \) 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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