↑

Properties

Label 574.2.e.h
Level $574$
Weight $2$
Character orbit 574.e
Analytic conductor $4.583$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [574,2,Mod(165,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(6))
 
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("574.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.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: \(4.58341307602\)
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: \( x^{20} - x^{19} + 24 x^{18} - 15 x^{17} + 367 x^{16} - 184 x^{15} + 3257 x^{14} - 765 x^{13} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{10} - 1) q^{4} - \beta_{3} q^{5} - \beta_{2} q^{6} + ( - \beta_{16} - \beta_{7}) q^{7} + q^{8} + (\beta_{19} + 2 \beta_{10} + \cdots - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{10} - 1) q^{4} - \beta_{3} q^{5} - \beta_{2} q^{6} + ( - \beta_{16} - \beta_{7}) q^{7} + q^{8} + (\beta_{19} + 2 \beta_{10} + \cdots - \beta_{3}) q^{9}+ \cdots + ( - 2 \beta_{17} - 2 \beta_{16} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{2} + q^{3} - 10 q^{4} - 2 q^{5} - 2 q^{6} + q^{7} + 20 q^{8} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 10 q^{2} + q^{3} - 10 q^{4} - 2 q^{5} - 2 q^{6} + q^{7} + 20 q^{8} - 17 q^{9} - 2 q^{10} - 11 q^{11} + q^{12} - 8 q^{13} + q^{14} + 8 q^{15} - 10 q^{16} + 5 q^{17} - 17 q^{18} - q^{19} + 4 q^{20} + 10 q^{21} + 22 q^{22} - 9 q^{23} + q^{24} - 24 q^{25} + 4 q^{26} - 14 q^{27} - 2 q^{28} + 46 q^{29} - 4 q^{30} - 5 q^{31} - 10 q^{32} - 5 q^{33} - 10 q^{34} + 31 q^{35} + 34 q^{36} - 16 q^{37} - q^{38} + 7 q^{39} - 2 q^{40} - 20 q^{41} - 2 q^{42} + 40 q^{43} - 11 q^{44} - 42 q^{45} - 9 q^{46} - 16 q^{47} - 2 q^{48} + 13 q^{49} + 48 q^{50} - 13 q^{51} + 4 q^{52} - 26 q^{53} + 7 q^{54} - 14 q^{55} + q^{56} + 74 q^{57} - 23 q^{58} - 10 q^{59} - 4 q^{60} + 10 q^{62} + 21 q^{63} + 20 q^{64} - 18 q^{65} - 5 q^{66} - 7 q^{67} + 5 q^{68} - 78 q^{69} - 20 q^{70} + 10 q^{71} - 17 q^{72} - 13 q^{73} - 16 q^{74} + 19 q^{75} + 2 q^{76} + 25 q^{77} - 14 q^{78} + q^{79} - 2 q^{80} - 18 q^{81} + 10 q^{82} - 42 q^{83} - 8 q^{84} + 68 q^{85} - 20 q^{86} + 2 q^{87} - 11 q^{88} + 6 q^{89} + 84 q^{90} - 23 q^{91} + 18 q^{92} + 5 q^{93} - 16 q^{94} - 24 q^{95} + q^{96} - 58 q^{97} - 26 q^{98} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 24 x^{18} - 15 x^{17} + 367 x^{16} - 184 x^{15} + 3257 x^{14} - 765 x^{13} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 30\!\cdots\!29 \nu^{19} + \cdots - 80\!\cdots\!16 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 32\!\cdots\!70 \nu^{19} + \cdots + 17\!\cdots\!56 ) / 31\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 80\!\cdots\!39 \nu^{19} + \cdots + 59\!\cdots\!64 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 53\!\cdots\!22 \nu^{19} + \cdots - 18\!\cdots\!64 ) / 31\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!05 \nu^{19} + \cdots - 37\!\cdots\!00 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!97 \nu^{19} + \cdots - 12\!\cdots\!36 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21\!\cdots\!47 \nu^{19} + \cdots - 57\!\cdots\!52 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 73\!\cdots\!85 \nu^{19} + \cdots + 15\!\cdots\!68 ) / 21\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 55\!\cdots\!64 \nu^{19} + \cdots - 23\!\cdots\!32 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25\!\cdots\!45 \nu^{19} + \cdots - 17\!\cdots\!56 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 36\!\cdots\!00 \nu^{19} + \cdots - 89\!\cdots\!28 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 45\!\cdots\!37 \nu^{19} + \cdots - 35\!\cdots\!92 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 56\!\cdots\!93 \nu^{19} + \cdots + 13\!\cdots\!04 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 29\!\cdots\!52 \nu^{19} + \cdots - 48\!\cdots\!32 ) / 31\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 71\!\cdots\!33 \nu^{19} + \cdots - 43\!\cdots\!32 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 10\!\cdots\!27 \nu^{19} + \cdots + 11\!\cdots\!76 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 12\!\cdots\!49 \nu^{19} + \cdots - 53\!\cdots\!76 ) / 63\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 13\!\cdots\!27 \nu^{19} + \cdots - 32\!\cdots\!32 ) / 63\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} - 5\beta_{10} + \beta_{8} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} - 8\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{19} + \beta_{18} + \beta_{17} - \beta_{15} - 2 \beta_{14} - \beta_{11} + 37 \beta_{10} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} + 14 \beta_{18} - 13 \beta_{17} + 16 \beta_{16} + 16 \beta_{15} - 13 \beta_{13} + 4 \beta_{12} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 18 \beta_{17} - 18 \beta_{16} + 6 \beta_{15} + 34 \beta_{14} + 34 \beta_{13} - 12 \beta_{12} + \cdots + 313 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 24 \beta_{19} - 163 \beta_{18} + 145 \beta_{17} - 127 \beta_{16} - 68 \beta_{15} - 143 \beta_{14} + \cdots + 547 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 717 \beta_{19} - 167 \beta_{18} + 48 \beta_{17} + 123 \beta_{16} + 123 \beta_{15} - 430 \beta_{13} + \cdots - 2861 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 58 \beta_{17} - 844 \beta_{16} - 1332 \beta_{15} + 1507 \beta_{14} + 1507 \beta_{13} - 2176 \beta_{12} + \cdots + 1168 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6751 \beta_{19} + 1878 \beta_{18} + 2080 \beta_{17} + 1532 \beta_{16} - 2784 \beta_{15} + \cdots - 170 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5364 \beta_{19} + 19257 \beta_{18} - 17705 \beta_{17} + 23417 \beta_{16} + 23417 \beta_{15} + \cdots - 16432 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 31945 \beta_{17} - 31089 \beta_{16} + 18052 \beta_{15} + 53894 \beta_{14} + 53894 \beta_{13} + \cdots + 270477 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 66961 \beta_{19} - 203938 \beta_{18} + 177257 \beta_{17} - 147348 \beta_{16} - 100488 \beta_{15} + \cdots + 540479 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 648475 \beta_{19} - 215658 \beta_{18} + 119530 \beta_{17} + 138856 \beta_{16} + 138856 \beta_{15} + \cdots - 2718809 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 168302 \beta_{17} - 1055472 \beta_{16} - 1546159 \beta_{15} + 1704687 \beta_{14} + 1704687 \beta_{13} + \cdots + 2447698 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6524169 \beta_{19} + 2244043 \beta_{18} + 2402767 \beta_{17} + 2136094 \beta_{16} - 3638029 \beta_{15} + \cdots + 1105937 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 8991055 \beta_{19} + 22390854 \beta_{18} - 21637503 \beta_{17} + 27196388 \beta_{16} + \cdots - 27935344 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 39799006 \beta_{17} - 38777968 \beta_{16} + 22391816 \beta_{15} + 65026990 \beta_{14} + \cdots + 283623605 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 99994236 \beta_{19} - 233222813 \beta_{18} + 207238389 \beta_{17} - 168649209 \beta_{16} + \cdots + 587626291 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(493\)
\(\chi(n)\) \(1\) \(\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−1.62046 − 2.80671i
−1.09739 − 1.90073i
−0.988577 − 1.71227i
−0.782860 − 1.35595i
−0.0255930 − 0.0443284i
0.193097 + 0.334454i
0.657140 + 1.13820i
1.20842 + 2.09305i
1.35554 + 2.34787i
1.60067 + 2.77245i
−1.62046 + 2.80671i
−1.09739 + 1.90073i
−0.988577 + 1.71227i
−0.782860 + 1.35595i
−0.0255930 + 0.0443284i
0.193097 − 0.334454i
0.657140 − 1.13820i
1.20842 − 2.09305i
1.35554 − 2.34787i
1.60067 − 2.77245i
−0.500000 − 0.866025i −1.62046 + 2.80671i −0.500000 + 0.866025i −0.726764 − 1.25879i 3.24091 2.51271 − 0.828435i 1.00000 −3.75175 − 6.49823i −0.726764 + 1.25879i
165.2 −0.500000 − 0.866025i −1.09739 + 1.90073i −0.500000 + 0.866025i −1.37708 − 2.38517i 2.19478 −2.11248 − 1.59293i 1.00000 −0.908521 − 1.57360i −1.37708 + 2.38517i
165.3 −0.500000 − 0.866025i −0.988577 + 1.71227i −0.500000 + 0.866025i 1.68499 + 2.91848i 1.97715 −1.52175 + 2.16432i 1.00000 −0.454570 − 0.787339i 1.68499 − 2.91848i
165.4 −0.500000 − 0.866025i −0.782860 + 1.35595i −0.500000 + 0.866025i −1.49773 − 2.59414i 1.56572 0.322893 + 2.62597i 1.00000 0.274261 + 0.475034i −1.49773 + 2.59414i
165.5 −0.500000 − 0.866025i −0.0255930 + 0.0443284i −0.500000 + 0.866025i −0.474716 − 0.822233i 0.0511860 2.05775 − 1.66303i 1.00000 1.49869 + 2.59581i −0.474716 + 0.822233i
165.6 −0.500000 − 0.866025i 0.193097 − 0.334454i −0.500000 + 0.866025i 2.06965 + 3.58475i −0.386194 −0.873293 − 2.49747i 1.00000 1.42543 + 2.46891i 2.06965 − 3.58475i
165.7 −0.500000 − 0.866025i 0.657140 − 1.13820i −0.500000 + 0.866025i −0.0533723 − 0.0924435i −1.31428 −2.64041 + 0.168037i 1.00000 0.636335 + 1.10216i −0.0533723 + 0.0924435i
165.8 −0.500000 − 0.866025i 1.20842 − 2.09305i −0.500000 + 0.866025i 1.32303 + 2.29155i −2.41684 2.39117 − 1.13238i 1.00000 −1.42056 − 2.46049i 1.32303 − 2.29155i
165.9 −0.500000 − 0.866025i 1.35554 − 2.34787i −0.500000 + 0.866025i 0.216302 + 0.374647i −2.71108 2.09237 + 1.61926i 1.00000 −2.17499 − 3.76719i 0.216302 − 0.374647i
165.10 −0.500000 − 0.866025i 1.60067 − 2.77245i −0.500000 + 0.866025i −2.16431 − 3.74869i −3.20135 −1.72895 + 2.00268i 1.00000 −3.62432 − 6.27750i −2.16431 + 3.74869i
247.1 −0.500000 + 0.866025i −1.62046 − 2.80671i −0.500000 − 0.866025i −0.726764 + 1.25879i 3.24091 2.51271 + 0.828435i 1.00000 −3.75175 + 6.49823i −0.726764 − 1.25879i
247.2 −0.500000 + 0.866025i −1.09739 − 1.90073i −0.500000 − 0.866025i −1.37708 + 2.38517i 2.19478 −2.11248 + 1.59293i 1.00000 −0.908521 + 1.57360i −1.37708 − 2.38517i
247.3 −0.500000 + 0.866025i −0.988577 − 1.71227i −0.500000 − 0.866025i 1.68499 − 2.91848i 1.97715 −1.52175 − 2.16432i 1.00000 −0.454570 + 0.787339i 1.68499 + 2.91848i
247.4 −0.500000 + 0.866025i −0.782860 − 1.35595i −0.500000 − 0.866025i −1.49773 + 2.59414i 1.56572 0.322893 − 2.62597i 1.00000 0.274261 − 0.475034i −1.49773 − 2.59414i
247.5 −0.500000 + 0.866025i −0.0255930 − 0.0443284i −0.500000 − 0.866025i −0.474716 + 0.822233i 0.0511860 2.05775 + 1.66303i 1.00000 1.49869 − 2.59581i −0.474716 − 0.822233i
247.6 −0.500000 + 0.866025i 0.193097 + 0.334454i −0.500000 − 0.866025i 2.06965 − 3.58475i −0.386194 −0.873293 + 2.49747i 1.00000 1.42543 − 2.46891i 2.06965 + 3.58475i
247.7 −0.500000 + 0.866025i 0.657140 + 1.13820i −0.500000 − 0.866025i −0.0533723 + 0.0924435i −1.31428 −2.64041 − 0.168037i 1.00000 0.636335 − 1.10216i −0.0533723 − 0.0924435i
247.8 −0.500000 + 0.866025i 1.20842 + 2.09305i −0.500000 − 0.866025i 1.32303 − 2.29155i −2.41684 2.39117 + 1.13238i 1.00000 −1.42056 + 2.46049i 1.32303 + 2.29155i
247.9 −0.500000 + 0.866025i 1.35554 + 2.34787i −0.500000 − 0.866025i 0.216302 − 0.374647i −2.71108 2.09237 − 1.61926i 1.00000 −2.17499 + 3.76719i 0.216302 + 0.374647i
247.10 −0.500000 + 0.866025i 1.60067 + 2.77245i −0.500000 − 0.866025i −2.16431 + 3.74869i −3.20135 −1.72895 − 2.00268i 1.00000 −3.62432 + 6.27750i −2.16431 − 3.74869i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.10
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 574.2.e.h ✓ 20
7.c even 3 1 inner 574.2.e.h ✓ 20
7.c even 3 1 4018.2.a.bt 10
7.d odd 6 1 4018.2.a.bu 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.h ✓ 20 1.a even 1 1 trivial
574.2.e.h ✓ 20 7.c even 3 1 inner
4018.2.a.bt 10 7.c even 3 1
4018.2.a.bu 10 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\):

\( T_{3}^{20} - T_{3}^{19} + 24 T_{3}^{18} - 15 T_{3}^{17} + 367 T_{3}^{16} - 184 T_{3}^{15} + 3257 T_{3}^{14} + \cdots + 144 \) Copy content Toggle raw display
\( T_{5}^{20} + 2 T_{5}^{19} + 39 T_{5}^{18} + 72 T_{5}^{17} + 994 T_{5}^{16} + 1801 T_{5}^{15} + \cdots + 7056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} - T^{19} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( T^{20} + 2 T^{19} + \cdots + 7056 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} + 11 T^{19} + \cdots + 5143824 \) Copy content Toggle raw display
$13$ \( (T^{10} + 4 T^{9} + \cdots - 2041200)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 260112384 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 982947904 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 12572688384 \) Copy content Toggle raw display
$29$ \( (T^{10} - 23 T^{9} + \cdots - 20160)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + 5 T^{19} + \cdots + 589824 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 3213099270144 \) Copy content Toggle raw display
$41$ \( (T + 1)^{20} \) Copy content Toggle raw display
$43$ \( (T^{10} - 20 T^{9} + \cdots - 64196608)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 55493679550464 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 1966232146176 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 892405688385600 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 215267584 \) Copy content Toggle raw display
$71$ \( (T^{10} - 5 T^{9} + \cdots + 1145259045)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 42\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{10} + 21 T^{9} + \cdots - 309985872)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{10} + 29 T^{9} + \cdots - 1048305664)^{2} \) Copy content Toggle raw display
show more
show less