Properties

Label 845.2.f.e
Level $845$
Weight $2$
Character orbit 845.f
Analytic conductor $6.747$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(408,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.408");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.f (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: \(6.74735897080\)
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: \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{10} q^{3} + ( - \beta_{8} - \beta_{4} + \beta_{3}) q^{4} + (\beta_{19} + \beta_{13} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{8} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{10} q^{3} + ( - \beta_{8} - \beta_{4} + \beta_{3}) q^{4} + (\beta_{19} + \beta_{13} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{19} - 2 \beta_{18} - 3 \beta_{17} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{3} - 12 q^{4} + 4 q^{6} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{3} - 12 q^{4} + 4 q^{6} + 4 q^{7} - 8 q^{10} + 8 q^{11} - 24 q^{12} + 28 q^{15} + 4 q^{16} - 14 q^{17} + 4 q^{19} - 12 q^{20} + 4 q^{21} - 32 q^{22} + 8 q^{23} - 4 q^{24} + 18 q^{25} + 4 q^{27} - 36 q^{28} + 40 q^{30} + 2 q^{34} - 20 q^{35} + 8 q^{37} - 8 q^{38} - 16 q^{40} - 38 q^{41} + 16 q^{42} - 32 q^{43} - 36 q^{44} - 6 q^{45} + 4 q^{46} - 40 q^{47} + 28 q^{48} - 36 q^{49} + 42 q^{50} - 10 q^{53} + 36 q^{54} - 16 q^{55} + 8 q^{59} + 28 q^{60} + 32 q^{61} + 4 q^{62} + 20 q^{64} - 32 q^{66} - 50 q^{68} + 32 q^{69} - 12 q^{70} - 40 q^{71} - 8 q^{72} + 4 q^{75} - 16 q^{76} - 28 q^{77} + 112 q^{80} + 28 q^{81} - 34 q^{82} + 48 q^{83} + 8 q^{84} - 2 q^{85} + 60 q^{86} - 28 q^{87} - 32 q^{88} + 12 q^{89} + 46 q^{90} - 8 q^{92} - 64 q^{93} + 40 q^{95} + 56 q^{96} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 69 \nu^{18} + 1647 \nu^{16} + 15798 \nu^{14} + 78322 \nu^{12} + 214723 \nu^{10} + 324081 \nu^{8} + \cdots + 539 ) / 1712 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2051 \nu^{19} - 1625 \nu^{18} - 52213 \nu^{17} - 40407 \nu^{16} - 546574 \nu^{15} - 409402 \nu^{14} + \cdots - 11323 ) / 23968 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1625 \nu^{18} - 40407 \nu^{16} - 409402 \nu^{14} - 2186822 \nu^{12} - 6647231 \nu^{10} + \cdots - 35291 ) / 11984 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2137 \nu^{19} + 321 \nu^{18} + 55159 \nu^{17} + 9951 \nu^{16} + 586250 \nu^{15} + 127330 \nu^{14} + \cdots + 27499 ) / 23968 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 908 \nu^{19} - 41 \nu^{18} - 22604 \nu^{17} - 11 \nu^{16} - 229460 \nu^{15} + 13762 \nu^{14} + \cdots + 12681 ) / 11984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2137 \nu^{19} + 321 \nu^{18} - 55159 \nu^{17} + 9951 \nu^{16} - 586250 \nu^{15} + 127330 \nu^{14} + \cdots + 51467 ) / 23968 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2051 \nu^{19} + 1625 \nu^{18} - 52213 \nu^{17} + 40407 \nu^{16} - 546574 \nu^{15} + 409402 \nu^{14} + \cdots + 11323 ) / 23968 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} + \cdots - 15221 ) / 11984 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2753 \nu^{19} + 867 \nu^{18} - 71035 \nu^{17} + 22593 \nu^{16} - 754642 \nu^{15} + 243222 \nu^{14} + \cdots + 15221 ) / 11984 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6161 \nu^{19} + 1625 \nu^{18} - 162487 \nu^{17} + 40407 \nu^{16} - 1774122 \nu^{15} + \cdots + 11323 ) / 23968 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5315 \nu^{19} + 3731 \nu^{18} - 137269 \nu^{17} + 93877 \nu^{16} - 1459766 \nu^{15} + \cdots + 89369 ) / 23968 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 539 \nu^{19} + 13945 \nu^{17} + 148734 \nu^{15} + 848758 \nu^{13} + 2806945 \nu^{11} + \cdots + 22081 \nu ) / 1712 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10681 \nu^{19} - 2701 \nu^{18} - 278863 \nu^{17} - 65979 \nu^{16} - 3007858 \nu^{15} + \cdots - 76919 ) / 23968 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11655 \nu^{19} - 191 \nu^{18} + 301777 \nu^{17} - 4801 \nu^{16} + 3219622 \nu^{15} - 49518 \nu^{14} + \cdots + 54179 ) / 23968 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 5223 \nu^{19} + 1330 \nu^{18} - 135929 \nu^{17} + 34482 \nu^{16} - 1460102 \nu^{15} + \cdots + 34314 ) / 11984 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 1767 \nu^{19} + 203 \nu^{18} + 46090 \nu^{17} + 5334 \nu^{16} + 496321 \nu^{15} + 57876 \nu^{14} + \cdots - 455 ) / 2996 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 14121 \nu^{19} + 1997 \nu^{18} - 366743 \nu^{17} + 48691 \nu^{16} - 3929786 \nu^{15} + \cdots + 34959 ) / 23968 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 17385 \nu^{19} - 3359 \nu^{18} - 451799 \nu^{17} - 85593 \nu^{16} - 4842978 \nu^{15} + \cdots - 41765 ) / 23968 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - \beta_{4} + \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} + \beta_{17} - \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \cdots - 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{19} - \beta_{18} + 2 \beta_{15} + 2 \beta_{13} - \beta_{12} + 2 \beta_{10} - 2 \beta_{9} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{19} - 9 \beta_{18} - 8 \beta_{17} + \beta_{15} + 8 \beta_{14} - 15 \beta_{13} + 9 \beta_{12} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 32 \beta_{19} + 13 \beta_{18} + 2 \beta_{17} + 2 \beta_{16} - 19 \beta_{15} - 19 \beta_{13} + \cdots - 50 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 51 \beta_{19} + 68 \beta_{18} + 54 \beta_{17} - 15 \beta_{15} - 54 \beta_{14} + 99 \beta_{13} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 267 \beta_{19} - 120 \beta_{18} - 28 \beta_{17} - 24 \beta_{16} + 147 \beta_{15} - 4 \beta_{14} + \cdots + 304 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 308 \beta_{19} - 488 \beta_{18} - 352 \beta_{17} + 154 \beta_{15} + 352 \beta_{14} - 648 \beta_{13} + \cdots - 323 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2044 \beta_{19} + 974 \beta_{18} + 274 \beta_{17} + 214 \beta_{16} - 1070 \beta_{15} + 60 \beta_{14} + \cdots - 1940 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1854 \beta_{19} + 3434 \beta_{18} + 2299 \beta_{17} - 1356 \beta_{15} - 2299 \beta_{14} + 4298 \beta_{13} + \cdots + 2590 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 15044 \beta_{19} - 7438 \beta_{18} - 2330 \beta_{17} - 1724 \beta_{16} + 7606 \beta_{15} - 606 \beta_{14} + \cdots + 12755 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11342 \beta_{19} - 23984 \beta_{18} - 15204 \beta_{17} + 11014 \beta_{15} + 15204 \beta_{14} + \cdots - 19642 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 108504 \beta_{19} + 54982 \beta_{18} + 18452 \beta_{17} + 13248 \beta_{16} - 53522 \beta_{15} + \cdots - 85534 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 71003 \beta_{19} + 167135 \beta_{18} + 101945 \beta_{17} - 85268 \beta_{15} - 101945 \beta_{14} + \cdots + 144450 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 773909 \beta_{19} - 398897 \beta_{18} - 140250 \beta_{17} - 99100 \beta_{16} + 375012 \beta_{15} + \cdots + 581415 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 455154 \beta_{19} - 1164737 \beta_{18} - 691774 \beta_{17} + 640259 \beta_{15} + 691774 \beta_{14} + \cdots - 1043445 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 5484892 \beta_{19} + 2861437 \beta_{18} + 1039156 \beta_{17} + 728470 \beta_{16} - 2623455 \beta_{15} + \cdots - 3989746 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 2980455 \beta_{19} + 8124372 \beta_{18} + 4738502 \beta_{17} - 4711075 \beta_{15} - 4738502 \beta_{14} + \cdots + 7456686 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-\beta_{13}\) \(-\beta_{13}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
408.1
2.64975i
2.08794i
1.83163i
1.58474i
0.493902i
0.274809i
0.131303i
1.02262i
1.51805i
2.25081i
2.25081i
1.51805i
1.02262i
0.131303i
0.274809i
0.493902i
1.58474i
1.83163i
2.08794i
2.64975i
2.64975i 0.917096 + 0.917096i −5.02120 1.81654 + 1.30391i 2.43008 2.43008i −0.112348 8.00544i 1.31787i 3.45504 4.81339i
408.2 2.08794i 1.94842 + 1.94842i −2.35949 2.22760 + 0.194361i 4.06818 4.06818i 2.91126 0.750585i 4.59268i 0.405815 4.65110i
408.3 1.83163i −1.40138 1.40138i −1.35488 −1.69810 1.45480i −2.56682 + 2.56682i 3.53890 1.18163i 0.927746i −2.66466 + 3.11030i
408.4 1.58474i −0.139520 0.139520i −0.511395 −2.23506 0.0672627i −0.221103 + 0.221103i −0.548328 2.35905i 2.96107i −0.106594 + 3.54198i
408.5 0.493902i −0.664960 0.664960i 1.75606 −0.284413 + 2.21791i −0.328425 + 0.328425i −3.67549 1.85513i 2.11566i 1.09543 + 0.140472i
408.6 0.274809i 1.67095 + 1.67095i 1.92448 1.45395 1.69883i 0.459191 0.459191i 0.386104 1.07848i 2.58414i −0.466854 0.399558i
408.7 0.131303i −0.243172 0.243172i 1.98276 2.08297 0.813169i 0.0319291 0.0319291i −2.78137 0.522947i 2.88174i 0.106771 + 0.273499i
408.8 1.02262i −1.97063 1.97063i 0.954253 −1.45744 + 1.69584i 2.01520 2.01520i 0.963574 3.02107i 4.76674i −1.73420 1.49040i
408.9 1.51805i 0.478298 + 0.478298i −0.304465 −2.15400 0.600231i −0.726078 + 0.726078i 2.59488 2.57390i 2.54246i 0.911178 3.26987i
408.10 2.25081i 1.40490 + 1.40490i −3.06613 0.247944 + 2.22228i −3.16216 + 3.16216i −1.27718 2.39966i 0.947480i −5.00192 + 0.558075i
437.1 2.25081i 1.40490 1.40490i −3.06613 0.247944 2.22228i −3.16216 3.16216i −1.27718 2.39966i 0.947480i −5.00192 0.558075i
437.2 1.51805i 0.478298 0.478298i −0.304465 −2.15400 + 0.600231i −0.726078 0.726078i 2.59488 2.57390i 2.54246i 0.911178 + 3.26987i
437.3 1.02262i −1.97063 + 1.97063i 0.954253 −1.45744 1.69584i 2.01520 + 2.01520i 0.963574 3.02107i 4.76674i −1.73420 + 1.49040i
437.4 0.131303i −0.243172 + 0.243172i 1.98276 2.08297 + 0.813169i 0.0319291 + 0.0319291i −2.78137 0.522947i 2.88174i 0.106771 0.273499i
437.5 0.274809i 1.67095 1.67095i 1.92448 1.45395 + 1.69883i 0.459191 + 0.459191i 0.386104 1.07848i 2.58414i −0.466854 + 0.399558i
437.6 0.493902i −0.664960 + 0.664960i 1.75606 −0.284413 2.21791i −0.328425 0.328425i −3.67549 1.85513i 2.11566i 1.09543 0.140472i
437.7 1.58474i −0.139520 + 0.139520i −0.511395 −2.23506 + 0.0672627i −0.221103 0.221103i −0.548328 2.35905i 2.96107i −0.106594 3.54198i
437.8 1.83163i −1.40138 + 1.40138i −1.35488 −1.69810 + 1.45480i −2.56682 2.56682i 3.53890 1.18163i 0.927746i −2.66466 3.11030i
437.9 2.08794i 1.94842 1.94842i −2.35949 2.22760 0.194361i 4.06818 + 4.06818i 2.91126 0.750585i 4.59268i 0.405815 + 4.65110i
437.10 2.64975i 0.917096 0.917096i −5.02120 1.81654 1.30391i 2.43008 + 2.43008i −0.112348 8.00544i 1.31787i 3.45504 + 4.81339i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 408.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.f.e 20
5.c odd 4 1 845.2.k.e 20
13.b even 2 1 845.2.f.d 20
13.c even 3 1 65.2.t.a yes 20
13.c even 3 1 845.2.t.f 20
13.d odd 4 1 845.2.k.d 20
13.d odd 4 1 845.2.k.e 20
13.e even 6 1 845.2.t.e 20
13.e even 6 1 845.2.t.g 20
13.f odd 12 1 65.2.o.a 20
13.f odd 12 1 845.2.o.e 20
13.f odd 12 1 845.2.o.f 20
13.f odd 12 1 845.2.o.g 20
39.i odd 6 1 585.2.dp.a 20
39.k even 12 1 585.2.cf.a 20
65.f even 4 1 inner 845.2.f.e 20
65.h odd 4 1 845.2.k.d 20
65.k even 4 1 845.2.f.d 20
65.n even 6 1 325.2.x.b 20
65.o even 12 1 325.2.x.b 20
65.o even 12 1 845.2.t.e 20
65.o even 12 1 845.2.t.g 20
65.q odd 12 1 65.2.o.a 20
65.q odd 12 1 325.2.s.b 20
65.q odd 12 1 845.2.o.e 20
65.r odd 12 1 845.2.o.f 20
65.r odd 12 1 845.2.o.g 20
65.s odd 12 1 325.2.s.b 20
65.t even 12 1 65.2.t.a yes 20
65.t even 12 1 845.2.t.f 20
195.bc odd 12 1 585.2.dp.a 20
195.bl even 12 1 585.2.cf.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.o.a 20 13.f odd 12 1
65.2.o.a 20 65.q odd 12 1
65.2.t.a yes 20 13.c even 3 1
65.2.t.a yes 20 65.t even 12 1
325.2.s.b 20 65.q odd 12 1
325.2.s.b 20 65.s odd 12 1
325.2.x.b 20 65.n even 6 1
325.2.x.b 20 65.o even 12 1
585.2.cf.a 20 39.k even 12 1
585.2.cf.a 20 195.bl even 12 1
585.2.dp.a 20 39.i odd 6 1
585.2.dp.a 20 195.bc odd 12 1
845.2.f.d 20 13.b even 2 1
845.2.f.d 20 65.k even 4 1
845.2.f.e 20 1.a even 1 1 trivial
845.2.f.e 20 65.f even 4 1 inner
845.2.k.d 20 13.d odd 4 1
845.2.k.d 20 65.h odd 4 1
845.2.k.e 20 5.c odd 4 1
845.2.k.e 20 13.d odd 4 1
845.2.o.e 20 13.f odd 12 1
845.2.o.e 20 65.q odd 12 1
845.2.o.f 20 13.f odd 12 1
845.2.o.f 20 65.r odd 12 1
845.2.o.g 20 13.f odd 12 1
845.2.o.g 20 65.r odd 12 1
845.2.t.e 20 13.e even 6 1
845.2.t.e 20 65.o even 12 1
845.2.t.f 20 13.c even 3 1
845.2.t.f 20 65.t even 12 1
845.2.t.g 20 13.e even 6 1
845.2.t.g 20 65.o even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{20} + 26 T_{2}^{18} + 279 T_{2}^{16} + 1604 T_{2}^{14} + 5353 T_{2}^{12} + 10466 T_{2}^{10} + \cdots + 1 \) Copy content Toggle raw display
\( T_{7}^{10} - 2 T_{7}^{9} - 24 T_{7}^{8} + 46 T_{7}^{7} + 166 T_{7}^{6} - 278 T_{7}^{5} - 328 T_{7}^{4} + \cdots - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 26 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} - 4 T^{19} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{20} - 9 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{10} - 2 T^{9} - 24 T^{8} + \cdots - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} - 8 T^{19} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( T^{20} + 14 T^{19} + \cdots + 1168561 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 1583721616 \) Copy content Toggle raw display
$23$ \( T^{20} - 8 T^{19} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 206213167449 \) Copy content Toggle raw display
$31$ \( T^{20} - 104 T^{17} + \cdots + 2166784 \) Copy content Toggle raw display
$37$ \( (T^{10} - 4 T^{9} + \cdots - 67143)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 3748255729 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 1370772640000 \) Copy content Toggle raw display
$47$ \( (T^{10} + 20 T^{9} + \cdots - 28416)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 2978634160384 \) Copy content Toggle raw display
$59$ \( T^{20} - 8 T^{19} + \cdots + 33856 \) Copy content Toggle raw display
$61$ \( (T^{10} - 16 T^{9} + \cdots + 909097)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 15478905336976 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 64\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{10} - 24 T^{9} + \cdots + 3393024)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 329648222500 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
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