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Properties

Label 418.2.f.h
Level $418$
Weight $2$
Character orbit 418.f
Analytic conductor $3.338$
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] = [418,2,Mod(115,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.115");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.f (of order \(5\), degree \(4\), 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: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 11 x^{18} - 3 x^{17} + 103 x^{16} + 50 x^{15} + 1002 x^{14} + 1120 x^{13} + \cdots + 1936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} - \beta_{4} q^{3} + ( - \beta_{11} + \beta_{9} + \beta_{8} - 1) q^{4} + \beta_{10} q^{5} + \beta_1 q^{6} + (\beta_{15} - \beta_{12} - \beta_{9} + \cdots + 1) q^{7}+ \cdots + (\beta_{19} + \beta_{15} + \cdots + \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} - \beta_{4} q^{3} + ( - \beta_{11} + \beta_{9} + \beta_{8} - 1) q^{4} + \beta_{10} q^{5} + \beta_1 q^{6} + (\beta_{15} - \beta_{12} - \beta_{9} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{18} + 2 \beta_{17} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{2} - q^{3} - 5 q^{4} - q^{5} + q^{6} + 13 q^{7} + 5 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 5 q^{2} - q^{3} - 5 q^{4} - q^{5} + q^{6} + 13 q^{7} + 5 q^{8} - 6 q^{9} + 6 q^{10} + q^{11} + 4 q^{12} - 2 q^{13} - 13 q^{14} - 8 q^{15} - 5 q^{16} + 11 q^{17} + 6 q^{18} + 5 q^{19} - q^{20} + 2 q^{21} + 4 q^{22} + 28 q^{23} + q^{24} - 30 q^{25} - 13 q^{26} - 31 q^{27} - 2 q^{28} + 28 q^{29} + 8 q^{30} - q^{31} - 20 q^{32} + 9 q^{33} + 24 q^{34} - 11 q^{35} - 6 q^{36} + 8 q^{37} - 5 q^{38} + 18 q^{39} - 4 q^{40} - 5 q^{41} - 22 q^{42} - 44 q^{43} + 11 q^{44} - 4 q^{45} + 7 q^{46} - 39 q^{47} - q^{48} + 4 q^{49} - 25 q^{50} - 11 q^{51} + 13 q^{52} - q^{53} - 4 q^{54} + 8 q^{55} + 22 q^{56} + q^{57} - 28 q^{58} + 6 q^{59} + 7 q^{60} + 10 q^{61} + 11 q^{62} + 34 q^{63} - 5 q^{64} - 8 q^{65} + 41 q^{66} + 18 q^{67} + 11 q^{68} - 63 q^{69} + q^{70} - 3 q^{71} + 6 q^{72} + 5 q^{73} - 8 q^{74} + 5 q^{75} - 20 q^{76} + 36 q^{77} + 22 q^{78} + 19 q^{79} + 4 q^{80} + 63 q^{81} - 9 q^{83} - 23 q^{84} + 30 q^{85} - 26 q^{86} - 16 q^{87} - q^{88} + 44 q^{89} + 14 q^{90} - 68 q^{91} - 7 q^{92} + 27 q^{93} - 31 q^{94} - 4 q^{95} + q^{96} - 71 q^{97} + 6 q^{98} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 11 x^{18} - 3 x^{17} + 103 x^{16} + 50 x^{15} + 1002 x^{14} + 1120 x^{13} + \cdots + 1936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\!\cdots\!69 \nu^{19} + \cdots - 22\!\cdots\!20 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\!\cdots\!02 \nu^{19} + \cdots + 13\!\cdots\!40 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 76\!\cdots\!55 \nu^{19} + \cdots + 99\!\cdots\!12 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37\!\cdots\!86 \nu^{19} + \cdots + 58\!\cdots\!16 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 47\!\cdots\!19 \nu^{19} + \cdots + 38\!\cdots\!52 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22\!\cdots\!84 \nu^{19} + \cdots - 18\!\cdots\!96 ) / 49\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51\!\cdots\!17 \nu^{19} + \cdots + 10\!\cdots\!52 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 67\!\cdots\!40 \nu^{19} + \cdots + 16\!\cdots\!52 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 88\!\cdots\!19 \nu^{19} + \cdots + 18\!\cdots\!84 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 99\!\cdots\!22 \nu^{19} + \cdots - 91\!\cdots\!84 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!01 \nu^{19} + \cdots - 28\!\cdots\!56 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 22\!\cdots\!04 \nu^{19} + \cdots + 42\!\cdots\!20 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 23\!\cdots\!66 \nu^{19} + \cdots - 43\!\cdots\!40 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 25\!\cdots\!26 \nu^{19} + \cdots - 60\!\cdots\!88 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 25\!\cdots\!07 \nu^{19} + \cdots - 58\!\cdots\!48 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 17\!\cdots\!48 \nu^{19} + \cdots + 44\!\cdots\!40 ) / 54\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 53\!\cdots\!10 \nu^{19} + \cdots - 48\!\cdots\!44 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 50\!\cdots\!05 \nu^{19} + \cdots + 77\!\cdots\!36 ) / 99\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{18} + \beta_{17} - \beta_{14} - \beta_{12} + 4\beta_{11} + \beta_{8} + \beta_{7} - \beta_{5} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - \beta_{14} + 2 \beta_{11} + 2 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10 \beta_{19} + \beta_{17} + \beta_{16} + 11 \beta_{15} - \beta_{12} + 4 \beta_{11} + 2 \beta_{10} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{19} + 14 \beta_{18} + 2 \beta_{16} + 15 \beta_{15} + 15 \beta_{14} - 2 \beta_{13} - 45 \beta_{9} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 21 \beta_{19} + 117 \beta_{18} - 21 \beta_{17} + 103 \beta_{14} - 19 \beta_{13} + 17 \beta_{12} + \cdots - 66 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 349 \beta_{18} - 174 \beta_{17} - 38 \beta_{16} - 173 \beta_{15} + 101 \beta_{14} - 199 \beta_{13} + \cdots - 434 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 301 \beta_{19} + 950 \beta_{18} - 950 \beta_{17} - 275 \beta_{16} - 977 \beta_{15} + 235 \beta_{14} + \cdots - 1158 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2074 \beta_{19} - 1901 \beta_{17} - 1927 \beta_{16} - 1465 \beta_{15} + 1855 \beta_{12} - 4060 \beta_{11} + \cdots + 1855 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 9698 \beta_{19} - 9698 \beta_{18} - 3756 \beta_{16} - 2999 \beta_{15} - 2999 \beta_{14} + \cdots + 19223 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 20217 \beta_{19} - 44283 \beta_{18} + 20217 \beta_{17} - 18709 \beta_{14} + 19295 \beta_{13} + \cdots + 44244 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 146655 \beta_{18} + 101250 \beta_{17} + 39688 \beta_{16} + 36599 \beta_{15} - 97225 \beta_{14} + \cdots + 54138 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 214205 \beta_{19} - 274134 \beta_{18} + 274134 \beta_{17} + 198475 \beta_{16} + 224465 \beta_{15} + \cdots - 35194 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1073122 \beta_{19} + 526805 \beta_{17} + 498599 \beta_{16} + 1007961 \beta_{15} - 433775 \beta_{12} + \cdots - 433775 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3083534 \beta_{19} + 3083534 \beta_{18} + 960580 \beta_{16} + 2138707 \beta_{15} + 2138707 \beta_{14} + \cdots - 2588903 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6006461 \beta_{19} + 17494399 \beta_{18} - 6006461 \beta_{17} + 10622249 \beta_{14} - 5685575 \beta_{13} + \cdots - 14569156 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 58672939 \beta_{18} - 34389982 \beta_{17} - 11042764 \beta_{16} - 22586203 \beta_{15} + 29557777 \beta_{14} + \cdots - 48348578 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 67705117 \beta_{19} + 123804714 \beta_{18} - 123804714 \beta_{17} - 63947759 \beta_{16} + \cdots - 67949562 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 381334070 \beta_{19} - 260381725 \beta_{17} - 237026387 \beta_{16} - 331581857 \beta_{15} + \cdots + 240024067 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/418\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(343\)
\(\chi(n)\) \(1\) \(-1 + \beta_{8} + \beta_{9} - \beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
−0.657798 + 2.02449i
−0.649220 + 1.99809i
−0.370901 + 1.14152i
0.488718 − 1.50412i
0.880184 − 2.70893i
−2.05454 + 1.49271i
−1.03876 + 0.754706i
0.0978632 − 0.0711018i
1.12827 − 0.819735i
2.67618 − 1.94436i
−0.657798 − 2.02449i
−0.649220 − 1.99809i
−0.370901 − 1.14152i
0.488718 + 1.50412i
0.880184 + 2.70893i
−2.05454 − 1.49271i
−1.03876 − 0.754706i
0.0978632 + 0.0711018i
1.12827 + 0.819735i
2.67618 + 1.94436i
−0.309017 − 0.951057i −1.72214 − 1.25121i −0.809017 + 0.587785i −0.137434 + 0.422980i −0.657798 + 2.02449i −0.658492 + 0.478423i 0.809017 + 0.587785i 0.473189 + 1.45633i 0.444747
115.2 −0.309017 − 0.951057i −1.69968 − 1.23489i −0.809017 + 0.587785i 0.760196 − 2.33964i −0.649220 + 1.99809i 2.50791 − 1.82211i 0.809017 + 0.587785i 0.436908 + 1.34467i −2.46005
115.3 −0.309017 − 0.951057i −0.971032 − 0.705496i −0.809017 + 0.587785i −1.28699 + 3.96096i −0.370901 + 1.14152i 0.280040 − 0.203461i 0.809017 + 0.587785i −0.481873 − 1.48305i 4.16480
115.4 −0.309017 − 0.951057i 1.27948 + 0.929596i −0.809017 + 0.587785i 0.804188 − 2.47504i 0.488718 − 1.50412i 1.25444 − 0.911401i 0.809017 + 0.587785i −0.154132 − 0.474370i −2.60241
115.5 −0.309017 − 0.951057i 2.30435 + 1.67421i −0.809017 + 0.587785i −0.948974 + 2.92064i 0.880184 − 2.70893i 3.77922 − 2.74577i 0.809017 + 0.587785i 1.58001 + 4.86277i 3.07094
191.1 0.809017 + 0.587785i −0.784763 + 2.41525i 0.309017 + 0.951057i −1.46809 + 1.06663i −2.05454 + 1.49271i −0.0865782 − 0.266460i −0.309017 + 0.951057i −2.79053 − 2.02744i −1.81466
191.2 0.809017 + 0.587785i −0.396773 + 1.22114i 0.309017 + 0.951057i 1.09974 − 0.799009i −1.03876 + 0.754706i 0.979051 + 3.01321i −0.309017 + 0.951057i 1.09330 + 0.794325i 1.35936
191.3 0.809017 + 0.587785i 0.0373804 − 0.115045i 0.309017 + 0.951057i 1.75481 − 1.27494i 0.0978632 − 0.0711018i −1.23224 − 3.79244i −0.309017 + 0.951057i 2.41521 + 1.75475i 2.16907
191.4 0.809017 + 0.587785i 0.430960 − 1.32636i 0.309017 + 0.951057i −1.74807 + 1.27005i 1.12827 − 0.819735i 0.365332 + 1.12438i −0.309017 + 0.951057i 0.853548 + 0.620139i −2.16074
191.5 0.809017 + 0.587785i 1.02221 − 3.14604i 0.309017 + 0.951057i 0.670629 − 0.487241i 2.67618 − 1.94436i −0.688687 − 2.11956i −0.309017 + 0.951057i −6.42562 − 4.66849i 0.828943
229.1 −0.309017 + 0.951057i −1.72214 + 1.25121i −0.809017 − 0.587785i −0.137434 − 0.422980i −0.657798 − 2.02449i −0.658492 − 0.478423i 0.809017 − 0.587785i 0.473189 − 1.45633i 0.444747
229.2 −0.309017 + 0.951057i −1.69968 + 1.23489i −0.809017 − 0.587785i 0.760196 + 2.33964i −0.649220 − 1.99809i 2.50791 + 1.82211i 0.809017 − 0.587785i 0.436908 − 1.34467i −2.46005
229.3 −0.309017 + 0.951057i −0.971032 + 0.705496i −0.809017 − 0.587785i −1.28699 − 3.96096i −0.370901 − 1.14152i 0.280040 + 0.203461i 0.809017 − 0.587785i −0.481873 + 1.48305i 4.16480
229.4 −0.309017 + 0.951057i 1.27948 − 0.929596i −0.809017 − 0.587785i 0.804188 + 2.47504i 0.488718 + 1.50412i 1.25444 + 0.911401i 0.809017 − 0.587785i −0.154132 + 0.474370i −2.60241
229.5 −0.309017 + 0.951057i 2.30435 − 1.67421i −0.809017 − 0.587785i −0.948974 − 2.92064i 0.880184 + 2.70893i 3.77922 + 2.74577i 0.809017 − 0.587785i 1.58001 − 4.86277i 3.07094
267.1 0.809017 − 0.587785i −0.784763 − 2.41525i 0.309017 − 0.951057i −1.46809 − 1.06663i −2.05454 − 1.49271i −0.0865782 + 0.266460i −0.309017 − 0.951057i −2.79053 + 2.02744i −1.81466
267.2 0.809017 − 0.587785i −0.396773 − 1.22114i 0.309017 − 0.951057i 1.09974 + 0.799009i −1.03876 − 0.754706i 0.979051 − 3.01321i −0.309017 − 0.951057i 1.09330 − 0.794325i 1.35936
267.3 0.809017 − 0.587785i 0.0373804 + 0.115045i 0.309017 − 0.951057i 1.75481 + 1.27494i 0.0978632 + 0.0711018i −1.23224 + 3.79244i −0.309017 − 0.951057i 2.41521 − 1.75475i 2.16907
267.4 0.809017 − 0.587785i 0.430960 + 1.32636i 0.309017 − 0.951057i −1.74807 − 1.27005i 1.12827 + 0.819735i 0.365332 − 1.12438i −0.309017 − 0.951057i 0.853548 − 0.620139i −2.16074
267.5 0.809017 − 0.587785i 1.02221 + 3.14604i 0.309017 − 0.951057i 0.670629 + 0.487241i 2.67618 + 1.94436i −0.688687 + 2.11956i −0.309017 − 0.951057i −6.42562 + 4.66849i 0.828943
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.f.h ✓ 20
11.c even 5 1 inner 418.2.f.h ✓ 20
11.c even 5 1 4598.2.a.cc 10
11.d odd 10 1 4598.2.a.cd 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.h ✓ 20 1.a even 1 1 trivial
418.2.f.h ✓ 20 11.c even 5 1 inner
4598.2.a.cc 10 11.c even 5 1
4598.2.a.cd 10 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + T_{3}^{19} + 11 T_{3}^{18} + 23 T_{3}^{17} + 93 T_{3}^{16} + 140 T_{3}^{15} + 612 T_{3}^{14} + \cdots + 1936 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + T^{19} + \cdots + 1936 \) Copy content Toggle raw display
$5$ \( T^{20} + T^{19} + \cdots + 121801 \) Copy content Toggle raw display
$7$ \( T^{20} - 13 T^{19} + \cdots + 3481 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( T^{20} + 2 T^{19} + \cdots + 3936256 \) Copy content Toggle raw display
$17$ \( T^{20} - 11 T^{19} + \cdots + 1907161 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$23$ \( (T^{10} - 14 T^{9} + \cdots - 377581)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 376648106315776 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 31086021834256 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 67868586256 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{10} + 22 T^{9} + \cdots + 350900)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 120944668441 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 533794816 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 117416560921 \) Copy content Toggle raw display
$67$ \( (T^{10} - 9 T^{9} + \cdots + 6908404)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 555678974248336 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 71861495677456 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{10} - 22 T^{9} + \cdots - 64913216)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 74\!\cdots\!16 \) Copy content Toggle raw display
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