Properties

Label 912.3.be.i
Level $912$
Weight $3$
Character orbit 912.be
Analytic conductor $24.850$
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] = [912,3,Mod(145,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.145");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.be (of order \(6\), degree \(2\), not 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: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 51 x^{18} + 314 x^{17} + 631 x^{16} - 7264 x^{15} + 8030 x^{14} + 12664 x^{13} + 393231 x^{12} - 447686 x^{11} - 9929427 x^{10} + 44358366 x^{9} + \cdots + 26753228352 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{3} + ( - \beta_{13} - \beta_{2}) q^{5} + \beta_{5} q^{7} + 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{3} + ( - \beta_{13} - \beta_{2}) q^{5} + \beta_{5} q^{7} + 3 \beta_{3} q^{9} + ( - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{8} - \beta_{2}) q^{11} + (\beta_{18} - \beta_{14} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} - \beta_{3}) q^{13} + (\beta_{13} + 2 \beta_{2}) q^{15} + (\beta_{17} - \beta_{16} - \beta_{15} - \beta_{13} + \beta_{9} + \beta_{8} + \beta_{5}) q^{17} + ( - \beta_{19} - \beta_{18} - \beta_{16} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{19}+ \cdots + ( - 3 \beta_{14} - 3 \beta_{13} - 3 \beta_{12} - 3 \beta_{8} + 3 \beta_{4} - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 30 q^{3} + 4 q^{7} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 30 q^{3} + 4 q^{7} + 30 q^{9} + 8 q^{11} - 6 q^{13} + 8 q^{17} + 20 q^{19} - 6 q^{21} + 56 q^{23} - 58 q^{25} - 204 q^{29} - 12 q^{33} - 20 q^{35} + 12 q^{39} + 12 q^{41} - 34 q^{43} - 24 q^{47} + 392 q^{49} - 24 q^{51} + 24 q^{55} - 54 q^{57} - 168 q^{59} + 142 q^{61} + 6 q^{63} - 246 q^{67} - 276 q^{71} - 118 q^{73} - 152 q^{77} + 210 q^{79} - 90 q^{81} + 112 q^{83} - 208 q^{85} + 408 q^{87} - 42 q^{91} - 102 q^{93} - 100 q^{95} - 540 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} - 51 x^{18} + 314 x^{17} + 631 x^{16} - 7264 x^{15} + 8030 x^{14} + 12664 x^{13} + 393231 x^{12} - 447686 x^{11} - 9929427 x^{10} + 44358366 x^{9} + \cdots + 26753228352 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 14\!\cdots\!99 \nu^{19} + \cdots - 13\!\cdots\!52 ) / 40\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 41\!\cdots\!41 \nu^{19} + \cdots - 99\!\cdots\!00 ) / 11\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65\!\cdots\!77 \nu^{19} + \cdots - 22\!\cdots\!28 ) / 15\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!75 \nu^{19} + \cdots - 17\!\cdots\!40 ) / 22\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!03 \nu^{19} + \cdots - 24\!\cdots\!28 ) / 38\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\!\cdots\!31 \nu^{19} + \cdots - 22\!\cdots\!12 ) / 19\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!92 \nu^{19} + \cdots + 16\!\cdots\!28 ) / 10\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!35 \nu^{19} + \cdots - 50\!\cdots\!08 ) / 38\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 77\!\cdots\!09 \nu^{19} + \cdots - 18\!\cdots\!56 ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!54 \nu^{19} + \cdots + 30\!\cdots\!48 ) / 38\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 31\!\cdots\!97 \nu^{19} + \cdots - 15\!\cdots\!48 ) / 77\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!04 \nu^{19} + \cdots + 21\!\cdots\!40 ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!63 \nu^{19} + \cdots - 54\!\cdots\!24 ) / 38\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!80 \nu^{19} + \cdots + 50\!\cdots\!16 ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!92 \nu^{19} + \cdots - 42\!\cdots\!40 ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 31\!\cdots\!08 \nu^{19} + \cdots - 89\!\cdots\!68 ) / 57\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 63\!\cdots\!30 \nu^{19} + \cdots + 22\!\cdots\!08 ) / 10\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 18\!\cdots\!83 \nu^{19} + \cdots - 56\!\cdots\!40 ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 19\!\cdots\!33 \nu^{19} + \cdots - 84\!\cdots\!32 ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{19} - \beta_{18} - 2 \beta_{17} - 2 \beta_{16} + 5 \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{7} + 4 \beta_{5} + \beta_{4} + \beta_{2} - \beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{19} + 7 \beta_{18} + \beta_{17} + 3 \beta_{16} - 7 \beta_{14} - 8 \beta_{13} + \beta_{12} - 6 \beta_{11} + 5 \beta_{10} - 7 \beta_{9} - 4 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} - 8 \beta_{5} - \beta_{4} + 8 \beta_{3} - 8 \beta_{2} + 2 \beta _1 + 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 18 \beta_{19} - 38 \beta_{18} - 31 \beta_{17} - 31 \beta_{16} + 69 \beta_{14} - 22 \beta_{13} - 25 \beta_{12} + 26 \beta_{11} - 63 \beta_{10} + 27 \beta_{9} - 10 \beta_{8} + 43 \beta_{7} - 28 \beta_{6} + 59 \beta_{5} + 20 \beta_{4} + 281 \beta_{3} + \cdots - 366 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 36 \beta_{19} + 179 \beta_{18} + 157 \beta_{17} + 83 \beta_{16} - 20 \beta_{15} - 213 \beta_{14} - 26 \beta_{13} - 2 \beta_{12} - 139 \beta_{11} + 153 \beta_{10} - 143 \beta_{9} - 46 \beta_{8} + 52 \beta_{7} + 160 \beta_{6} - 260 \beta_{5} + \cdots + 390 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 541 \beta_{19} - 960 \beta_{18} - 297 \beta_{17} - 373 \beta_{16} + 456 \beta_{15} + 998 \beta_{14} + 1003 \beta_{13} - 11 \beta_{12} + 755 \beta_{11} - 869 \beta_{10} + 663 \beta_{9} + 311 \beta_{8} + 159 \beta_{7} + \cdots - 7053 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 590 \beta_{19} + 3320 \beta_{18} + 3889 \beta_{17} + 2441 \beta_{16} + 1044 \beta_{15} - 7971 \beta_{14} - 1456 \beta_{13} - 1265 \beta_{12} - 4548 \beta_{11} + 4641 \beta_{10} - 2583 \beta_{9} - 788 \beta_{8} + \cdots + 21588 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 29927 \beta_{19} - 65825 \beta_{18} - 30438 \beta_{17} - 26574 \beta_{16} + 3784 \beta_{15} + 91713 \beta_{14} + 82397 \beta_{13} + 22402 \beta_{12} + 62881 \beta_{11} - 57146 \beta_{10} + \cdots - 314554 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 30204 \beta_{19} + 163819 \beta_{18} + 90799 \beta_{17} + 83417 \beta_{16} + 12676 \beta_{15} - 237725 \beta_{14} - 156634 \beta_{13} - 33145 \beta_{12} - 199060 \beta_{11} + 176925 \beta_{10} + \cdots + 833601 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 525726 \beta_{19} - 1918258 \beta_{18} - 1455091 \beta_{17} - 611251 \beta_{16} + 35992 \beta_{15} + 2884105 \beta_{14} + 1707118 \beta_{13} + 453395 \beta_{12} + 2166246 \beta_{11} + \cdots - 10506962 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2684807 \beta_{19} + 6643786 \beta_{18} + 3496553 \beta_{17} + 1950591 \beta_{16} - 535192 \beta_{15} - 8030656 \beta_{14} - 7766013 \beta_{13} - 2890198 \beta_{12} - 7494630 \beta_{11} + \cdots + 31249518 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9387929 \beta_{19} - 29314236 \beta_{18} - 21172975 \beta_{17} - 9770103 \beta_{16} - 1825340 \beta_{15} + 44897532 \beta_{14} + 37477305 \beta_{13} + 13595831 \beta_{12} + \cdots - 191642005 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 41568578 \beta_{19} + 102772458 \beta_{18} + 65733630 \beta_{17} + 23101882 \beta_{16} - 184824 \beta_{15} - 140345868 \beta_{14} - 141778218 \beta_{13} - 66132084 \beta_{12} + \cdots + 567034878 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 907325867 \beta_{19} - 2167240253 \beta_{18} - 1328334970 \beta_{17} - 514663466 \beta_{16} - 174844432 \beta_{15} + 2916945769 \beta_{14} + 3273121493 \beta_{13} + \cdots - 12344459022 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2625023368 \beta_{19} + 6265865035 \beta_{18} + 4186833057 \beta_{17} + 1032295747 \beta_{16} + 152058544 \beta_{15} - 8422384595 \beta_{14} - 9900914476 \beta_{13} + \cdots + 38024697089 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 35513525462 \beta_{19} - 72935842334 \beta_{18} - 47555850979 \beta_{17} - 5604902419 \beta_{16} - 3653727136 \beta_{15} + 89510172033 \beta_{14} + \cdots - 431944382294 ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 100658320226 \beta_{19} + 190007465425 \beta_{18} + 125339351993 \beta_{17} + 7884664847 \beta_{16} + 7403975196 \beta_{15} - 234678301551 \beta_{14} + \cdots + 1241020515666 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 610479075153 \beta_{19} - 1074764815796 \beta_{18} - 733390851401 \beta_{17} + 68217152379 \beta_{16} - 64032962120 \beta_{15} + 1265256831162 \beta_{14} + \cdots - 7203618568453 ) / 4 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 3675611760874 \beta_{19} + 5499374943296 \beta_{18} + 3772770614951 \beta_{17} - 1014915382745 \beta_{16} + 477392803268 \beta_{15} + \cdots + 39857589938568 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 42296618092011 \beta_{19} - 58408413733317 \beta_{18} - 39989644748166 \beta_{17} + 17304330944594 \beta_{16} - 4798399376792 \beta_{15} + \cdots - 434732456642466 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
3.07263 3.70568i
−4.15322 0.777292i
−2.00924 + 3.91523i
3.19798 2.28035i
3.61172 + 2.91538i
−0.106113 + 3.45557i
−1.45625 + 0.503882i
4.25934 1.63984i
−6.06670 0.463279i
0.649869 0.191583i
3.07263 + 3.70568i
−4.15322 + 0.777292i
−2.00924 3.91523i
3.19798 + 2.28035i
3.61172 2.91538i
−0.106113 3.45557i
−1.45625 0.503882i
4.25934 + 1.63984i
−6.06670 + 0.463279i
0.649869 + 0.191583i
0 −1.50000 + 0.866025i 0 −4.28133 7.41548i 0 10.9641 0 1.50000 2.59808i 0
145.2 0 −1.50000 + 0.866025i 0 −3.37418 5.84424i 0 −11.7715 0 1.50000 2.59808i 0
145.3 0 −1.50000 + 0.866025i 0 −2.33973 4.05253i 0 −3.94782 0 1.50000 2.59808i 0
145.4 0 −1.50000 + 0.866025i 0 −2.12913 3.68775i 0 −3.44542 0 1.50000 2.59808i 0
145.5 0 −1.50000 + 0.866025i 0 0.531831 + 0.921158i 0 2.38472 0 1.50000 2.59808i 0
145.6 0 −1.50000 + 0.866025i 0 0.900429 + 1.55959i 0 12.2498 0 1.50000 2.59808i 0
145.7 0 −1.50000 + 0.866025i 0 1.07526 + 1.86241i 0 5.32572 0 1.50000 2.59808i 0
145.8 0 −1.50000 + 0.866025i 0 2.02727 + 3.51133i 0 −9.21530 0 1.50000 2.59808i 0
145.9 0 −1.50000 + 0.866025i 0 2.76590 + 4.79069i 0 7.80823 0 1.50000 2.59808i 0
145.10 0 −1.50000 + 0.866025i 0 4.82367 + 8.35484i 0 −8.35261 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −4.28133 + 7.41548i 0 10.9641 0 1.50000 + 2.59808i 0
673.2 0 −1.50000 0.866025i 0 −3.37418 + 5.84424i 0 −11.7715 0 1.50000 + 2.59808i 0
673.3 0 −1.50000 0.866025i 0 −2.33973 + 4.05253i 0 −3.94782 0 1.50000 + 2.59808i 0
673.4 0 −1.50000 0.866025i 0 −2.12913 + 3.68775i 0 −3.44542 0 1.50000 + 2.59808i 0
673.5 0 −1.50000 0.866025i 0 0.531831 0.921158i 0 2.38472 0 1.50000 + 2.59808i 0
673.6 0 −1.50000 0.866025i 0 0.900429 1.55959i 0 12.2498 0 1.50000 + 2.59808i 0
673.7 0 −1.50000 0.866025i 0 1.07526 1.86241i 0 5.32572 0 1.50000 + 2.59808i 0
673.8 0 −1.50000 0.866025i 0 2.02727 3.51133i 0 −9.21530 0 1.50000 + 2.59808i 0
673.9 0 −1.50000 0.866025i 0 2.76590 4.79069i 0 7.80823 0 1.50000 + 2.59808i 0
673.10 0 −1.50000 0.866025i 0 4.82367 8.35484i 0 −8.35261 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.be.i 20
4.b odd 2 1 456.3.w.b 20
12.b even 2 1 1368.3.bv.b 20
19.d odd 6 1 inner 912.3.be.i 20
76.f even 6 1 456.3.w.b 20
228.n odd 6 1 1368.3.bv.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.3.w.b 20 4.b odd 2 1
456.3.w.b 20 76.f even 6 1
912.3.be.i 20 1.a even 1 1 trivial
912.3.be.i 20 19.d odd 6 1 inner
1368.3.bv.b 20 12.b even 2 1
1368.3.bv.b 20 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{20} + 154 T_{5}^{18} + 24 T_{5}^{17} + 16398 T_{5}^{16} - 1552 T_{5}^{15} + 859996 T_{5}^{14} - 967176 T_{5}^{13} + 32168116 T_{5}^{12} - 50284464 T_{5}^{11} + 750941712 T_{5}^{10} + \cdots + 1053299900416 \) Copy content Toggle raw display
\( T_{7}^{10} - 2 T_{7}^{9} - 341 T_{7}^{8} + 452 T_{7}^{7} + 40308 T_{7}^{6} - 27876 T_{7}^{5} - 1945372 T_{7}^{4} + 376868 T_{7}^{3} + 34600875 T_{7}^{2} + 8417726 T_{7} - 164147183 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + 154 T^{18} + \cdots + 1053299900416 \) Copy content Toggle raw display
$7$ \( (T^{10} - 2 T^{9} - 341 T^{8} + \cdots - 164147183)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 4 T^{9} - 614 T^{8} + \cdots + 340118400)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 6 T^{19} + \cdots + 48\!\cdots\!61 \) Copy content Toggle raw display
$17$ \( T^{20} - 8 T^{19} + \cdots + 53\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{20} - 20 T^{19} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} - 56 T^{19} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{20} + 204 T^{19} + \cdots + 54\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{20} + 7162 T^{18} + \cdots + 30\!\cdots\!89 \) Copy content Toggle raw display
$37$ \( T^{20} + 7422 T^{18} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( T^{20} - 12 T^{19} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{20} + 34 T^{19} + \cdots + 29\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( T^{20} + 24 T^{19} + \cdots + 41\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{20} - 15750 T^{18} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + 168 T^{19} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{20} - 142 T^{19} + \cdots + 61\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{20} + 246 T^{19} + \cdots + 33\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{20} + 276 T^{19} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{20} + 118 T^{19} + \cdots + 34\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{20} - 210 T^{19} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{10} - 56 T^{9} + \cdots - 10\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} - 22094 T^{18} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{20} + 540 T^{19} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
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