Properties

Label 390.2.l.c
Level $390$
Weight $2$
Character orbit 390.l
Analytic conductor $3.114$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [390,2,Mod(53,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.l (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: \(3.11416567883\)
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: \( x^{20} + 2 x^{18} - 12 x^{17} + 6 x^{16} - 24 x^{15} + 72 x^{14} - 112 x^{13} + 189 x^{12} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + \beta_1 q^{3} - \beta_{11} q^{4} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{5} + \beta_{6} q^{6} + (\beta_{15} - \beta_{4} + \beta_{2}) q^{7} + \beta_{4} q^{8} + ( - \beta_{19} + \beta_{15} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + \beta_1 q^{3} - \beta_{11} q^{4} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{5} + \beta_{6} q^{6} + (\beta_{15} - \beta_{4} + \beta_{2}) q^{7} + \beta_{4} q^{8} + ( - \beta_{19} + \beta_{15} + \cdots + \beta_{2}) q^{9}+ \cdots + (\beta_{18} - 2 \beta_{15} - \beta_{13} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{7} - 4 q^{9} - 8 q^{10} - 4 q^{12} + 16 q^{14} + 4 q^{15} - 20 q^{16} - 12 q^{17} + 8 q^{18} - 16 q^{21} + 4 q^{22} - 16 q^{23} - 4 q^{24} + 32 q^{25} + 36 q^{27} - 4 q^{28} + 24 q^{29} - 4 q^{30} - 8 q^{31} + 36 q^{33} + 20 q^{35} - 4 q^{36} - 16 q^{37} - 24 q^{38} - 8 q^{40} + 4 q^{42} - 20 q^{43} - 16 q^{44} + 24 q^{45} - 40 q^{46} - 16 q^{47} + 12 q^{51} - 32 q^{53} - 28 q^{54} - 4 q^{55} + 16 q^{57} + 28 q^{58} - 16 q^{60} + 24 q^{61} - 40 q^{62} + 36 q^{63} + 8 q^{65} + 12 q^{68} - 72 q^{69} + 20 q^{70} + 8 q^{72} + 28 q^{73} + 48 q^{74} - 52 q^{75} + 16 q^{76} - 8 q^{77} + 4 q^{78} - 16 q^{81} - 112 q^{83} - 8 q^{85} + 4 q^{87} + 4 q^{88} + 88 q^{89} - 4 q^{90} - 16 q^{91} - 16 q^{92} + 40 q^{93} + 8 q^{95} + 36 q^{97} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 2 x^{18} - 12 x^{17} + 6 x^{16} - 24 x^{15} + 72 x^{14} - 112 x^{13} + 189 x^{12} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 625 \nu^{19} - 9513 \nu^{18} + 6131 \nu^{17} - 15987 \nu^{16} + 58143 \nu^{15} - 86973 \nu^{14} + \cdots - 148035843 ) / 17530992 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{19} + 2 \nu^{17} - 12 \nu^{16} + 6 \nu^{15} - 24 \nu^{14} + 72 \nu^{13} - 112 \nu^{12} + \cdots + 13122 \nu ) / 19683 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1501 \nu^{19} + 5969 \nu^{18} - 3497 \nu^{17} + 21805 \nu^{16} - 56055 \nu^{15} + \cdots - 42981111 ) / 17530992 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29 \nu^{19} + 30 \nu^{18} + 13 \nu^{17} - 288 \nu^{16} + 129 \nu^{15} - 462 \nu^{14} + \cdots + 1614006 ) / 314928 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1627 \nu^{19} + 3513 \nu^{18} - 19921 \nu^{17} - 4965 \nu^{16} - 58089 \nu^{15} + \cdots + 187244379 ) / 17530992 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 864 \nu^{19} - 5383 \nu^{18} + 7827 \nu^{17} - 15239 \nu^{16} + 62700 \nu^{15} - 105765 \nu^{14} + \cdots - 110992437 ) / 8765496 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6551 \nu^{19} + 13509 \nu^{18} - 66823 \nu^{17} + 110085 \nu^{16} - 235551 \nu^{15} + \cdots + 378366309 ) / 52592976 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3171 \nu^{19} + 1627 \nu^{18} - 2829 \nu^{17} + 18131 \nu^{16} - 23991 \nu^{15} + \cdots - 12301875 ) / 17530992 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3271 \nu^{19} + 5948 \nu^{18} - 6335 \nu^{17} + 30916 \nu^{16} - 44877 \nu^{15} + \cdots + 45087192 ) / 17530992 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5383 \nu^{19} + 6099 \nu^{18} - 4871 \nu^{17} + 57516 \nu^{16} - 85029 \nu^{15} + \cdots - 51018336 ) / 26296488 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2033 \nu^{19} + 1965 \nu^{18} - 2360 \nu^{17} - 17577 \nu^{16} - 2931 \nu^{15} + 447 \nu^{14} + \cdots + 105953589 ) / 8765496 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4141 \nu^{19} - 4446 \nu^{18} - 3343 \nu^{17} - 26832 \nu^{16} + 35103 \nu^{15} - 17178 \nu^{14} + \cdots + 72171000 ) / 17530992 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12785 \nu^{19} - 2760 \nu^{18} + 13229 \nu^{17} + 55452 \nu^{16} + 123351 \nu^{15} + \cdots - 478178802 ) / 52592976 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13582 \nu^{19} + 23283 \nu^{18} - 26278 \nu^{17} - 55587 \nu^{16} - 185556 \nu^{15} + \cdots + 817100379 ) / 52592976 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 33 \nu^{19} - 56 \nu^{18} + 51 \nu^{17} - 526 \nu^{16} + 759 \nu^{15} - 984 \nu^{14} + \cdots + 1246590 ) / 104976 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 5969 \nu^{19} - 495 \nu^{18} + 3793 \nu^{17} - 47049 \nu^{16} + 19065 \nu^{15} - 55911 \nu^{14} + \cdots + 88632549 ) / 17530992 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 33799 \nu^{19} - 51414 \nu^{18} + 68795 \nu^{17} - 359052 \nu^{16} + 654405 \nu^{15} + \cdots + 32437584 ) / 52592976 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 38233 \nu^{19} - 31014 \nu^{18} + 12809 \nu^{17} - 316272 \nu^{16} + 274263 \nu^{15} + \cdots + 722208636 ) / 52592976 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} + \beta_{15} - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} + \beta_{18} + \beta_{17} + \beta_{15} + \beta_{13} + \beta_{11} + 2\beta_{9} - 2\beta_{5} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} + 3 \beta_{17} + \beta_{14} + \beta_{13} + 3 \beta_{11} + 2 \beta_{10} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{18} - 2 \beta_{17} + 2 \beta_{16} + 3 \beta_{15} + \beta_{14} + 2 \beta_{13} + 6 \beta_{11} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{19} - \beta_{18} + \beta_{17} + 4 \beta_{16} - 4 \beta_{14} + \beta_{13} + 6 \beta_{12} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 8 \beta_{18} + 6 \beta_{17} - 4 \beta_{16} + 3 \beta_{15} - \beta_{14} - 4 \beta_{13} + 12 \beta_{12} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5 \beta_{19} + 9 \beta_{18} - 23 \beta_{17} - 8 \beta_{16} + 9 \beta_{15} - 12 \beta_{14} - 13 \beta_{13} + \cdots - 29 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 33 \beta_{19} + 35 \beta_{18} - 9 \beta_{17} - 2 \beta_{16} + 2 \beta_{15} - 13 \beta_{14} + \cdots - 46 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 15 \beta_{19} + 44 \beta_{18} + 6 \beta_{17} - 8 \beta_{16} + 8 \beta_{15} - 7 \beta_{14} + 76 \beta_{13} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 52 \beta_{19} - 14 \beta_{18} - 60 \beta_{17} + 22 \beta_{16} - 8 \beta_{15} - 126 \beta_{14} + \cdots - 108 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 180 \beta_{19} + 2 \beta_{18} - 48 \beta_{17} + 96 \beta_{16} + 44 \beta_{15} - 72 \beta_{14} + \cdots + 103 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 88 \beta_{19} - 22 \beta_{18} + 128 \beta_{17} + 38 \beta_{16} - 10 \beta_{15} - 8 \beta_{14} + \cdots - 588 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 199 \beta_{19} - 416 \beta_{18} - 214 \beta_{17} - 168 \beta_{16} + 101 \beta_{15} - 196 \beta_{14} + \cdots - 856 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 535 \beta_{19} + 871 \beta_{18} - 687 \beta_{17} - 122 \beta_{16} - 559 \beta_{15} - 42 \beta_{14} + \cdots - 498 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 717 \beta_{19} + 649 \beta_{18} + 3 \beta_{17} - 312 \beta_{16} - 2096 \beta_{15} + 205 \beta_{14} + \cdots - 3133 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 612 \beta_{19} - 1100 \beta_{18} - 4534 \beta_{17} - 144 \beta_{16} - 83 \beta_{15} - 1703 \beta_{14} + \cdots + 488 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 2604 \beta_{19} + 4155 \beta_{18} - 3617 \beta_{17} - 332 \beta_{16} - 2664 \beta_{15} - 4900 \beta_{14} + \cdots - 5798 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 3900 \beta_{19} + 4770 \beta_{18} + 8186 \beta_{17} - 5238 \beta_{16} - 2133 \beta_{15} + 3841 \beta_{14} + \cdots - 7068 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(-1\) \(\beta_{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.47681 0.905005i
−0.866241 1.49988i
−0.807075 + 1.53252i
0.756150 + 1.55828i
1.68687 0.393031i
−1.45930 + 0.932975i
−0.554989 + 1.64073i
−0.247823 1.71423i
1.26911 + 1.17872i
1.70011 0.331082i
−1.47681 + 0.905005i
−0.866241 + 1.49988i
−0.807075 1.53252i
0.756150 1.55828i
1.68687 + 0.393031i
−1.45930 0.932975i
−0.554989 1.64073i
−0.247823 + 1.71423i
1.26911 1.17872i
1.70011 + 0.331082i
−0.707107 0.707107i −1.47681 0.905005i 1.00000i 2.21547 0.302779i 0.404327 + 1.68420i 2.13164 2.13164i 0.707107 0.707107i 1.36193 + 2.67304i −1.78067 1.35248i
53.2 −0.707107 0.707107i −0.866241 1.49988i 1.00000i −1.91458 1.15515i −0.448047 + 1.67310i −1.34012 + 1.34012i 0.707107 0.707107i −1.49925 + 2.59851i 0.536996 + 2.17063i
53.3 −0.707107 0.707107i −0.807075 + 1.53252i 1.00000i −2.02552 + 0.947239i 1.65435 0.512970i −1.59729 + 1.59729i 0.707107 0.707107i −1.69726 2.47372i 2.10206 + 0.762461i
53.4 −0.707107 0.707107i 0.756150 + 1.55828i 1.00000i 2.23169 0.139915i 0.567192 1.63655i −3.31973 + 3.31973i 0.707107 0.707107i −1.85647 + 2.35659i −1.67698 1.47911i
53.5 −0.707107 0.707107i 1.68687 0.393031i 1.00000i −0.507058 2.17782i −1.47071 0.914881i 0.297077 0.297077i 0.707107 0.707107i 2.69105 1.32598i −1.18141 + 1.89849i
53.6 0.707107 + 0.707107i −1.45930 + 0.932975i 1.00000i 2.00768 0.984487i −1.69159 0.372168i 3.42552 3.42552i −0.707107 + 0.707107i 1.25911 2.72298i 2.11578 + 0.723507i
53.7 0.707107 + 0.707107i −0.554989 + 1.64073i 1.00000i −2.07006 0.845499i −1.55261 + 0.767733i −1.12188 + 1.12188i −0.707107 + 0.707107i −2.38397 1.82117i −0.865892 2.06161i
53.8 0.707107 + 0.707107i −0.247823 1.71423i 1.00000i −1.39944 + 1.74401i 1.03691 1.38738i −1.92224 + 1.92224i −0.707107 + 0.707107i −2.87717 + 0.849651i −2.22275 + 0.243653i
53.9 0.707107 + 0.707107i 1.26911 + 1.17872i 1.00000i 2.09893 + 0.771022i 0.0639155 + 1.73087i −0.340740 + 0.340740i −0.707107 + 0.707107i 0.221259 + 2.99183i 0.938975 + 2.02937i
53.10 0.707107 + 0.707107i 1.70011 0.331082i 1.00000i −0.637125 + 2.14338i 1.43627 + 0.968051i 1.78776 1.78776i −0.707107 + 0.707107i 2.78077 1.12575i −1.96611 + 1.06508i
287.1 −0.707107 + 0.707107i −1.47681 + 0.905005i 1.00000i 2.21547 + 0.302779i 0.404327 1.68420i 2.13164 + 2.13164i 0.707107 + 0.707107i 1.36193 2.67304i −1.78067 + 1.35248i
287.2 −0.707107 + 0.707107i −0.866241 + 1.49988i 1.00000i −1.91458 + 1.15515i −0.448047 1.67310i −1.34012 1.34012i 0.707107 + 0.707107i −1.49925 2.59851i 0.536996 2.17063i
287.3 −0.707107 + 0.707107i −0.807075 1.53252i 1.00000i −2.02552 0.947239i 1.65435 + 0.512970i −1.59729 1.59729i 0.707107 + 0.707107i −1.69726 + 2.47372i 2.10206 0.762461i
287.4 −0.707107 + 0.707107i 0.756150 1.55828i 1.00000i 2.23169 + 0.139915i 0.567192 + 1.63655i −3.31973 3.31973i 0.707107 + 0.707107i −1.85647 2.35659i −1.67698 + 1.47911i
287.5 −0.707107 + 0.707107i 1.68687 + 0.393031i 1.00000i −0.507058 + 2.17782i −1.47071 + 0.914881i 0.297077 + 0.297077i 0.707107 + 0.707107i 2.69105 + 1.32598i −1.18141 1.89849i
287.6 0.707107 0.707107i −1.45930 0.932975i 1.00000i 2.00768 + 0.984487i −1.69159 + 0.372168i 3.42552 + 3.42552i −0.707107 0.707107i 1.25911 + 2.72298i 2.11578 0.723507i
287.7 0.707107 0.707107i −0.554989 1.64073i 1.00000i −2.07006 + 0.845499i −1.55261 0.767733i −1.12188 1.12188i −0.707107 0.707107i −2.38397 + 1.82117i −0.865892 + 2.06161i
287.8 0.707107 0.707107i −0.247823 + 1.71423i 1.00000i −1.39944 1.74401i 1.03691 + 1.38738i −1.92224 1.92224i −0.707107 0.707107i −2.87717 0.849651i −2.22275 0.243653i
287.9 0.707107 0.707107i 1.26911 1.17872i 1.00000i 2.09893 0.771022i 0.0639155 1.73087i −0.340740 0.340740i −0.707107 0.707107i 0.221259 2.99183i 0.938975 2.02937i
287.10 0.707107 0.707107i 1.70011 + 0.331082i 1.00000i −0.637125 2.14338i 1.43627 0.968051i 1.78776 + 1.78776i −0.707107 0.707107i 2.78077 + 1.12575i −1.96611 1.06508i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.l.c 20
3.b odd 2 1 390.2.l.d yes 20
5.c odd 4 1 390.2.l.d yes 20
15.e even 4 1 inner 390.2.l.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.l.c 20 1.a even 1 1 trivial
390.2.l.c 20 15.e even 4 1 inner
390.2.l.d yes 20 3.b odd 2 1
390.2.l.d yes 20 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\):

\( T_{7}^{20} + 4 T_{7}^{19} + 8 T_{7}^{18} + 16 T_{7}^{17} + 662 T_{7}^{16} + 2712 T_{7}^{15} + \cdots + 419904 \) Copy content Toggle raw display
\( T_{17}^{20} + 12 T_{17}^{19} + 72 T_{17}^{18} + 76 T_{17}^{17} + 630 T_{17}^{16} + 8812 T_{17}^{15} + \cdots + 118548544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + 2 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} - 16 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} + 4 T^{19} + \cdots + 419904 \) Copy content Toggle raw display
$11$ \( T^{20} + 108 T^{18} + \cdots + 802816 \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 118548544 \) Copy content Toggle raw display
$19$ \( T^{20} + 184 T^{18} + \cdots + 802816 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 40675414441984 \) Copy content Toggle raw display
$29$ \( (T^{10} - 12 T^{9} + \cdots + 153728)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 4 T^{9} + \cdots - 8661248)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 46049726464 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 2657578123264 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 2429209022464 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 304770032336896 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 1504920469504 \) Copy content Toggle raw display
$59$ \( (T^{10} - 174 T^{8} + \cdots + 1068928)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 12 T^{9} + \cdots - 48140288)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 37\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 6298574012416 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 1755773403136 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{10} - 44 T^{9} + \cdots - 761340928)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 152961091649536 \) Copy content Toggle raw display
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