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Properties

Label 847.2.e.i
Level $847$
Weight $2$
Character orbit 847.e
Analytic conductor $6.763$
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] = [847,2,Mod(485,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.485");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.e (of order \(3\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 15 x^{18} - 14 x^{17} + 149 x^{16} - 131 x^{15} + 825 x^{14} - 595 x^{13} + 3197 x^{12} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{17} q^{3} + (\beta_{17} - \beta_{14} - \beta_{12} + \cdots - 1) q^{4}+ \cdots + (\beta_{16} + \beta_{14} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{17} q^{3} + (\beta_{17} - \beta_{14} - \beta_{12} + \cdots - 1) q^{4}+ \cdots + (\beta_{19} + \beta_{18} + \beta_{16} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{2} + 3 q^{3} - 9 q^{4} + 2 q^{5} - 18 q^{6} - 11 q^{7} + 6 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{2} + 3 q^{3} - 9 q^{4} + 2 q^{5} - 18 q^{6} - 11 q^{7} + 6 q^{8} - 9 q^{9} + 7 q^{10} - 9 q^{12} - 2 q^{13} - 19 q^{14} - 2 q^{15} - 15 q^{16} + 19 q^{17} + 17 q^{18} + 28 q^{19} - 10 q^{20} - q^{21} - 7 q^{23} + 19 q^{24} + 8 q^{25} + 5 q^{26} + 12 q^{27} + 8 q^{28} + 30 q^{29} - 22 q^{30} + 14 q^{31} - 15 q^{32} + 24 q^{34} + 8 q^{35} + 32 q^{36} - 13 q^{37} - 24 q^{38} + 4 q^{39} + 10 q^{40} - 70 q^{41} + 25 q^{42} - 36 q^{43} - 8 q^{45} + 9 q^{46} - 16 q^{47} - 66 q^{48} - 25 q^{49} - 12 q^{50} - 21 q^{51} - 4 q^{52} + 9 q^{53} + 17 q^{54} + 12 q^{56} - 8 q^{57} + 9 q^{58} - 12 q^{59} + 21 q^{60} + 20 q^{61} - 76 q^{62} + 12 q^{63} - 58 q^{64} - 20 q^{65} - 19 q^{67} + 56 q^{68} + 18 q^{69} + 21 q^{70} + 30 q^{71} - 4 q^{72} + 3 q^{73} + 42 q^{74} + 27 q^{75} - 48 q^{76} - 50 q^{78} - 32 q^{79} + 6 q^{80} + 46 q^{81} + 18 q^{82} - 58 q^{83} + 73 q^{84} + 46 q^{85} - 9 q^{86} + 24 q^{87} - 5 q^{89} - 24 q^{90} + 28 q^{91} + 30 q^{92} + q^{93} - 19 q^{94} + q^{95} + 46 q^{96} + 8 q^{97} + 58 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 15 x^{18} - 14 x^{17} + 149 x^{16} - 131 x^{15} + 825 x^{14} - 595 x^{13} + 3197 x^{12} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\!\cdots\!76 \nu^{19} + \cdots - 86\!\cdots\!08 ) / 12\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 57\!\cdots\!09 \nu^{19} + \cdots - 19\!\cdots\!81 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34\!\cdots\!23 \nu^{19} + \cdots - 44\!\cdots\!75 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37\!\cdots\!46 \nu^{19} + \cdots + 53\!\cdots\!24 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 45\!\cdots\!67 \nu^{19} + \cdots - 12\!\cdots\!47 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 67\!\cdots\!46 \nu^{19} + \cdots + 13\!\cdots\!40 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 60\!\cdots\!69 \nu^{19} + \cdots - 29\!\cdots\!82 ) / 42\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 87\!\cdots\!97 \nu^{19} + \cdots - 60\!\cdots\!69 ) / 42\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 34\!\cdots\!72 \nu^{19} + \cdots + 23\!\cdots\!47 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 36\!\cdots\!67 \nu^{19} + \cdots + 34\!\cdots\!19 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 39\!\cdots\!58 \nu^{19} + \cdots - 53\!\cdots\!87 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 40\!\cdots\!51 \nu^{19} + \cdots - 35\!\cdots\!02 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 86\!\cdots\!08 \nu^{19} + \cdots + 13\!\cdots\!43 ) / 12\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 63\!\cdots\!31 \nu^{19} + \cdots - 66\!\cdots\!90 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 79\!\cdots\!98 \nu^{19} + \cdots + 14\!\cdots\!04 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 12\!\cdots\!37 \nu^{19} + \cdots + 21\!\cdots\!10 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 18\!\cdots\!14 \nu^{19} + \cdots - 25\!\cdots\!82 ) / 84\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 27\!\cdots\!21 \nu^{19} + \cdots + 38\!\cdots\!81 ) / 84\!\cdots\!85 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} - 3\beta_{14} - \beta_{12} - \beta_{9} - \beta_{6} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - 2\beta_{8} + \beta_{7} + \beta_{4} + 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{17} + \beta_{16} - \beta_{15} + 16\beta_{14} + 6\beta_{12} + 8\beta_{9} + \beta_{8} - \beta_{7} - 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{19} - 6 \beta_{16} + 18 \beta_{15} - 18 \beta_{14} - \beta_{13} - 9 \beta_{12} - \beta_{9} + \cdots - 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{11} + \beta_{10} - 12 \beta_{8} + 8 \beta_{7} + 55 \beta_{6} + 3 \beta_{5} + 35 \beta_{4} + \cdots + 93 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 58 \beta_{19} - 3 \beta_{18} + 32 \beta_{16} - 134 \beta_{15} + 140 \beta_{14} + 14 \beta_{13} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 28 \beta_{19} + 14 \beta_{18} + 219 \beta_{17} - 53 \beta_{16} + 114 \beta_{15} - 607 \beta_{14} + \cdots - 607 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 407 \beta_{11} + 43 \beta_{10} - 944 \beta_{8} + 169 \beta_{7} + 148 \beta_{6} + 134 \beta_{5} + \cdots + 897 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 282 \beta_{19} - 134 \beta_{18} - 1349 \beta_{17} + 338 \beta_{16} - 980 \beta_{15} + 3954 \beta_{14} + \cdots + 424 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2808 \beta_{19} + 424 \beta_{18} + 98 \beta_{17} - 909 \beta_{16} + 6509 \beta_{15} - 7401 \beta_{14} + \cdots - 7401 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2481 \beta_{11} + 1113 \beta_{10} - 7927 \beta_{8} + 2147 \beta_{7} + 15600 \beta_{6} + 3593 \beta_{5} + \cdots + 22634 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 19193 \beta_{19} - 3593 \beta_{18} - 1246 \beta_{17} + 5037 \beta_{16} - 44472 \beta_{15} + \cdots + 8645 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 20298 \beta_{19} + 8645 \beta_{18} + 52850 \beta_{17} - 13713 \beta_{16} + 61551 \beta_{15} + \cdots - 175974 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 130530 \beta_{11} + 28156 \beta_{10} - 302726 \beta_{8} + 28917 \beta_{7} + 94090 \beta_{6} + \cdots + 304624 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 158844 \beta_{19} - 64754 \beta_{18} - 335423 \beta_{17} + 88294 \beta_{16} - 464526 \beta_{15} + \cdots + 210831 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 885578 \beta_{19} + 210831 \beta_{18} + 121783 \beta_{17} - 172333 \beta_{16} + 2058493 \beta_{15} + \cdots - 2591178 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1207014 \beta_{11} + 474628 \beta_{10} - 3435581 \beta_{8} + 573286 \beta_{7} + 4468617 \beta_{6} + \cdots + 6551299 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 6003296 \beta_{19} - 1534679 \beta_{18} - 1059189 \beta_{17} + 1064864 \beta_{16} - 14001217 \beta_{15} + \cdots + 3430646 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(\beta_{14}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
−1.31579 + 2.27901i
−1.11855 + 1.93739i
−0.637536 + 1.10424i
−0.292639 + 0.506866i
0.0380109 − 0.0658368i
0.218609 − 0.378642i
0.336982 − 0.583670i
0.924415 − 1.60113i
1.12270 − 1.94457i
1.22380 − 2.11968i
−1.31579 − 2.27901i
−1.11855 − 1.93739i
−0.637536 − 1.10424i
−0.292639 − 0.506866i
0.0380109 + 0.0658368i
0.218609 + 0.378642i
0.336982 + 0.583670i
0.924415 + 1.60113i
1.12270 + 1.94457i
1.22380 + 2.11968i
−1.31579 + 2.27901i 1.10673 + 1.91691i −2.46258 − 4.26532i 0.588178 − 1.01875i −5.82486 1.11286 + 2.40032i 7.69779 −0.949686 + 1.64490i 1.54783 + 2.68093i
485.2 −1.11855 + 1.93739i −0.667467 − 1.15609i −1.50231 − 2.60208i 0.433167 − 0.750267i 2.98638 −0.983749 + 2.45606i 2.24744 0.608975 − 1.05478i 0.969038 + 1.67842i
485.3 −0.637536 + 1.10424i 0.0696550 + 0.120646i 0.187097 + 0.324061i −0.592535 + 1.02630i −0.177630 −0.573698 − 2.58280i −3.02727 1.49030 − 2.58127i −0.755525 − 1.30861i
485.4 −0.292639 + 0.506866i 0.691326 + 1.19741i 0.828725 + 1.43539i 1.98209 − 3.43308i −0.809236 −0.0768837 − 2.64463i −2.14063 0.544137 − 0.942473i 1.16007 + 2.00931i
485.5 0.0380109 − 0.0658368i 1.17753 + 2.03955i 0.997110 + 1.72705i −1.28904 + 2.23267i 0.179037 −1.24572 + 2.33413i 0.303648 −1.27318 + 2.20520i 0.0979948 + 0.169732i
485.6 0.218609 − 0.378642i 1.29176 + 2.23740i 0.904420 + 1.56650i 0.607894 − 1.05290i 1.12957 −2.54037 + 0.739259i 1.66530 −1.83731 + 3.18231i −0.265782 − 0.460348i
485.7 0.336982 − 0.583670i −1.07557 − 1.86294i 0.772886 + 1.33868i −0.844546 + 1.46280i −1.44979 −2.51615 − 0.817927i 2.38972 −0.813692 + 1.40936i 0.569194 + 0.985873i
485.8 0.924415 − 1.60113i −0.591002 − 1.02365i −0.709086 − 1.22817i 1.18135 − 2.04616i −2.18532 1.65295 + 2.06586i 1.07570 0.801433 − 1.38812i −2.18411 − 3.78299i
485.9 1.12270 − 1.94457i 0.954542 + 1.65332i −1.52089 − 2.63427i −1.23252 + 2.13479i 4.28664 1.96682 − 1.76964i −2.33923 −0.322301 + 0.558242i 2.76749 + 4.79344i
485.10 1.22380 − 2.11968i −1.45751 − 2.52448i −1.99536 − 3.45607i 0.165961 − 0.287453i −7.13479 −2.29605 − 1.31460i −4.87248 −2.74868 + 4.76085i −0.406206 − 0.703570i
606.1 −1.31579 − 2.27901i 1.10673 − 1.91691i −2.46258 + 4.26532i 0.588178 + 1.01875i −5.82486 1.11286 − 2.40032i 7.69779 −0.949686 − 1.64490i 1.54783 − 2.68093i
606.2 −1.11855 − 1.93739i −0.667467 + 1.15609i −1.50231 + 2.60208i 0.433167 + 0.750267i 2.98638 −0.983749 − 2.45606i 2.24744 0.608975 + 1.05478i 0.969038 − 1.67842i
606.3 −0.637536 − 1.10424i 0.0696550 − 0.120646i 0.187097 − 0.324061i −0.592535 − 1.02630i −0.177630 −0.573698 + 2.58280i −3.02727 1.49030 + 2.58127i −0.755525 + 1.30861i
606.4 −0.292639 − 0.506866i 0.691326 − 1.19741i 0.828725 − 1.43539i 1.98209 + 3.43308i −0.809236 −0.0768837 + 2.64463i −2.14063 0.544137 + 0.942473i 1.16007 − 2.00931i
606.5 0.0380109 + 0.0658368i 1.17753 − 2.03955i 0.997110 − 1.72705i −1.28904 − 2.23267i 0.179037 −1.24572 − 2.33413i 0.303648 −1.27318 − 2.20520i 0.0979948 − 0.169732i
606.6 0.218609 + 0.378642i 1.29176 − 2.23740i 0.904420 − 1.56650i 0.607894 + 1.05290i 1.12957 −2.54037 − 0.739259i 1.66530 −1.83731 − 3.18231i −0.265782 + 0.460348i
606.7 0.336982 + 0.583670i −1.07557 + 1.86294i 0.772886 − 1.33868i −0.844546 − 1.46280i −1.44979 −2.51615 + 0.817927i 2.38972 −0.813692 − 1.40936i 0.569194 − 0.985873i
606.8 0.924415 + 1.60113i −0.591002 + 1.02365i −0.709086 + 1.22817i 1.18135 + 2.04616i −2.18532 1.65295 − 2.06586i 1.07570 0.801433 + 1.38812i −2.18411 + 3.78299i
606.9 1.12270 + 1.94457i 0.954542 − 1.65332i −1.52089 + 2.63427i −1.23252 − 2.13479i 4.28664 1.96682 + 1.76964i −2.33923 −0.322301 − 0.558242i 2.76749 − 4.79344i
606.10 1.22380 + 2.11968i −1.45751 + 2.52448i −1.99536 + 3.45607i 0.165961 + 0.287453i −7.13479 −2.29605 + 1.31460i −4.87248 −2.74868 − 4.76085i −0.406206 + 0.703570i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.e.i 20
7.c even 3 1 inner 847.2.e.i 20
7.c even 3 1 5929.2.a.bw 10
7.d odd 6 1 5929.2.a.bx 10
11.b odd 2 1 847.2.e.h 20
11.c even 5 2 77.2.m.b ✓ 40
11.c even 5 2 847.2.n.i 40
11.d odd 10 2 847.2.n.h 40
11.d odd 10 2 847.2.n.j 40
33.h odd 10 2 693.2.by.b 40
77.h odd 6 1 847.2.e.h 20
77.h odd 6 1 5929.2.a.by 10
77.i even 6 1 5929.2.a.bz 10
77.j odd 10 2 539.2.q.h 40
77.m even 15 2 77.2.m.b ✓ 40
77.m even 15 2 539.2.f.h 20
77.m even 15 2 847.2.n.i 40
77.o odd 30 2 847.2.n.h 40
77.o odd 30 2 847.2.n.j 40
77.p odd 30 2 539.2.f.g 20
77.p odd 30 2 539.2.q.h 40
231.z odd 30 2 693.2.by.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.b ✓ 40 11.c even 5 2
77.2.m.b ✓ 40 77.m even 15 2
539.2.f.g 20 77.p odd 30 2
539.2.f.h 20 77.m even 15 2
539.2.q.h 40 77.j odd 10 2
539.2.q.h 40 77.p odd 30 2
693.2.by.b 40 33.h odd 10 2
693.2.by.b 40 231.z odd 30 2
847.2.e.h 20 11.b odd 2 1
847.2.e.h 20 77.h odd 6 1
847.2.e.i 20 1.a even 1 1 trivial
847.2.e.i 20 7.c even 3 1 inner
847.2.n.h 40 11.d odd 10 2
847.2.n.h 40 77.o odd 30 2
847.2.n.i 40 11.c even 5 2
847.2.n.i 40 77.m even 15 2
847.2.n.j 40 11.d odd 10 2
847.2.n.j 40 77.o odd 30 2
5929.2.a.bw 10 7.c even 3 1
5929.2.a.bx 10 7.d odd 6 1
5929.2.a.by 10 77.h odd 6 1
5929.2.a.bz 10 77.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - T_{2}^{19} + 15 T_{2}^{18} - 14 T_{2}^{17} + 149 T_{2}^{16} - 131 T_{2}^{15} + 825 T_{2}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - T^{19} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} - 3 T^{19} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{20} - 2 T^{19} + \cdots + 2401 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( (T^{10} + T^{9} - 47 T^{8} + \cdots - 49)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 12800885881 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 244328161 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 187991521 \) Copy content Toggle raw display
$29$ \( (T^{10} - 15 T^{9} + \cdots - 33129)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} - 14 T^{19} + \cdots + 2019241 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 5012937403681 \) Copy content Toggle raw display
$41$ \( (T^{10} + 35 T^{9} + \cdots - 49)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 18 T^{9} + \cdots + 181456)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 350145268684561 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 2159345019841 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 1241223361 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 12137217790201 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 5650457030761 \) Copy content Toggle raw display
$71$ \( (T^{10} - 15 T^{9} + \cdots - 1017431)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( (T^{10} + 29 T^{9} + \cdots - 326647279)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{10} - 4 T^{9} + \cdots + 3426031)^{2} \) Copy content Toggle raw display
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