Properties

Label 1331.1.d.a
Level $1331$
Weight $1$
Character orbit 1331.d
Analytic conductor $0.664$
Analytic rank $0$
Dimension $20$
Projective image $D_{11}$
CM discriminant -11
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1331,1,Mod(161,1331)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1331, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1331.161");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1331 = 11^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1331.d (of order \(10\), degree \(4\), 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: \(0.664255531815\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{10})\)
Coefficient field: 20.0.1402274470934209014892578125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 5 x^{18} - 6 x^{17} + 20 x^{16} - 11 x^{15} + 59 x^{14} - 30 x^{13} + 179 x^{12} - 109 x^{11} + 260 x^{10} - 128 x^{9} + 334 x^{8} + 82 x^{7} + 199 x^{6} + 22 x^{5} + 146 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{11}\)
Projective field: Galois closure of 11.1.4177248169415651.1

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{13} q^{3} + \beta_{9} q^{4} + ( - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{8}) q^{5} + (\beta_{19} + \beta_{17}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{3} + \beta_{9} q^{4} + ( - \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{8}) q^{5} + (\beta_{19} + \beta_{17}) q^{9} + \beta_{3} q^{12} + ( - \beta_{7} - \beta_{5}) q^{15} - \beta_{11} q^{16} + ( - \beta_{19} - \beta_{18} - \beta_{17} - \beta_{16} - \beta_{13} - \beta_{8} + \beta_{3} + \beta_1) q^{20} + ( - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{23} + ( - \beta_{17} - \beta_{16} + \beta_{12} + \beta_{11} - \beta_{9} - \beta_{6} + \beta_{5} - 1) q^{25} + ( - \beta_{12} - \beta_{8}) q^{27} + \beta_{18} q^{31} + ( - \beta_{19} - \beta_{17} - \beta_{15} + \beta_{11} - \beta_{10} - \beta_{7} - \beta_{5} - \beta_{2} + \beta_1 - 1) q^{36} + (\beta_{10} - \beta_{9} + \beta_{7} + \beta_{5} - \beta_1) q^{37} + ( - \beta_{6} - \beta_{3} - 1) q^{45} + ( - \beta_{19} - \beta_{15} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{2} + \beta_1) q^{47} + \beta_1 q^{48} - \beta_{11} q^{49} + \beta_{16} q^{53} - \beta_{5} q^{59} + ( - \beta_{14} - \beta_{12}) q^{60} + \beta_{17} q^{64} + \beta_{4} q^{67} + (\beta_{19} + \beta_{17} + \beta_{16} + \beta_{15} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + \cdots + 1) q^{69}+ \cdots + \beta_{18} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{3} - 5 q^{4} + q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{3} - 5 q^{4} + q^{5} - 4 q^{9} - 4 q^{12} + 2 q^{15} - 5 q^{16} + q^{20} - 4 q^{23} - 4 q^{25} + 2 q^{27} + q^{31} - 4 q^{36} + q^{37} - 12 q^{45} + q^{47} + q^{48} - 5 q^{49} + q^{53} + q^{59} + 2 q^{60} - 5 q^{64} - 4 q^{67} + 2 q^{69} + q^{71} + 3 q^{75} + q^{80} - 3 q^{81} - 4 q^{89} + q^{92} + 2 q^{93} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 5 x^{18} - 6 x^{17} + 20 x^{16} - 11 x^{15} + 59 x^{14} - 30 x^{13} + 179 x^{12} - 109 x^{11} + 260 x^{10} - 128 x^{9} + 334 x^{8} + 82 x^{7} + 199 x^{6} + 22 x^{5} + 146 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 126239 \nu^{19} + 631195 \nu^{18} - 757434 \nu^{17} + 2524780 \nu^{16} - 3335154 \nu^{15} + 7448101 \nu^{14} - 3787170 \nu^{13} + 22596781 \nu^{12} + \cdots - 189837059 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 171700 \nu^{19} + 858500 \nu^{18} - 1030200 \nu^{17} + 3434000 \nu^{16} - 4744285 \nu^{15} + 10130300 \nu^{14} - 5151000 \nu^{13} + 30734300 \nu^{12} + \cdots - 27559952 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 387795 \nu^{19} - 1938975 \nu^{18} + 2326770 \nu^{17} - 7755900 \nu^{16} + 10354441 \nu^{15} - 22879905 \nu^{14} + 11633850 \nu^{13} - 69415305 \nu^{12} + \cdots + 165561080 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 504956 \nu^{19} + 126239 \nu^{18} - 1767346 \nu^{17} + 810374 \nu^{16} - 6059472 \nu^{15} - 3660931 \nu^{14} - 18809611 \nu^{13} - 8836730 \nu^{12} + \cdots - 126239 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 451714 \nu^{19} - 2258570 \nu^{18} + 2710284 \nu^{17} - 9034280 \nu^{16} + 12429463 \nu^{15} - 26651126 \nu^{14} + 13551420 \nu^{13} - 80856806 \nu^{12} + \cdots + 80721890 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2056136 \nu^{19} - 514034 \nu^{18} + 7196476 \nu^{17} - 3408915 \nu^{16} + 24673632 \nu^{15} + 14906986 \nu^{14} + 76591066 \nu^{13} + 35982380 \nu^{12} + \cdots + 514034 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2518527 \nu^{19} + 5037054 \nu^{18} - 14154492 \nu^{17} + 26864288 \nu^{16} - 61284157 \nu^{15} + 73037283 \nu^{14} - 159506710 \nu^{13} + \cdots + 3358036 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3358036 \nu^{19} - 839509 \nu^{18} + 11753126 \nu^{17} - 5993724 \nu^{16} + 40296432 \nu^{15} + 24345761 \nu^{14} + 125086841 \nu^{13} + 58765630 \nu^{12} + \cdots + 839509 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4222416 \nu^{19} + 1055604 \nu^{18} - 14778456 \nu^{17} + 7281980 \nu^{16} - 50668992 \nu^{15} - 30612516 \nu^{14} - 157284996 \nu^{13} + \cdots - 1055604 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6064056 \nu^{19} - 12128112 \nu^{18} + 34426370 \nu^{17} - 64683264 \nu^{16} + 147558696 \nu^{15} - 175857624 \nu^{14} + 384056880 \nu^{13} + \cdots - 8085408 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7040481 \nu^{19} - 14080962 \nu^{18} + 39652961 \nu^{17} - 75098464 \nu^{16} + 171318371 \nu^{15} - 204173949 \nu^{14} + 445897130 \nu^{13} + \cdots - 9387308 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 8410883 \nu^{19} - 8284644 \nu^{18} + 41423220 \nu^{17} - 49707864 \nu^{16} + 165692880 \nu^{15} - 89184559 \nu^{14} + 488793996 \nu^{13} + \cdots - 24853932 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10207293 \nu^{19} + 20414586 \nu^{18} - 57705477 \nu^{17} + 108877792 \nu^{16} - 248377463 \nu^{15} + 296011497 \nu^{14} - 646461890 \nu^{13} + \cdots + 13609724 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12128112 \nu^{19} - 24256224 \nu^{18} + 68852740 \nu^{17} - 129366528 \nu^{16} + 295117392 \nu^{15} - 351715248 \nu^{14} + 768113760 \nu^{13} + \cdots - 16170816 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 18192168 \nu^{19} + 12318270 \nu^{18} - 84896784 \nu^{17} + 78832728 \nu^{16} - 327459024 \nu^{15} + 78832728 \nu^{14} - 1003314858 \nu^{13} + \cdots - 18192168 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 24853932 \nu^{19} + 16443049 \nu^{18} - 115985016 \nu^{17} + 107700372 \nu^{16} - 447370776 \nu^{15} + 107700372 \nu^{14} - 1377197429 \nu^{13} + \cdots - 24853932 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 29941563 \nu^{19} - 20204673 \nu^{18} + 139727294 \nu^{17} - 129746773 \nu^{16} + 538948134 \nu^{15} - 129746773 \nu^{14} + 1652821401 \nu^{13} + \cdots + 29941563 ) / 99123971 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 47189337 \nu^{19} - 31324241 \nu^{18} + 220216906 \nu^{17} - 204487127 \nu^{16} + 849408066 \nu^{15} - 204487127 \nu^{14} + 2613338960 \nu^{13} + \cdots + 47189337 ) / 99123971 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - 2\beta_{11} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} + 3\beta_{13} + 3\beta_{8} - 3\beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{19} + \beta_{18} - 5 \beta_{17} + \beta_{16} - 3 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + 5 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} - 4 \beta_{7} - 4 \beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_{2} + 3 \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{6} - \beta_{4} + 9\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{10} + 5\beta_{9} + 14\beta_{7} + 15\beta_{5} - 15\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{15} - 6\beta_{14} - 20\beta_{12} + \beta_{11} - 34\beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 28\beta_{19} - 20\beta_{18} + 42\beta_{17} - 27\beta_{16} - 28\beta_{13} - 28\beta_{8} + 28\beta_{3} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8 \beta_{19} - 27 \beta_{18} + 9 \beta_{17} - 75 \beta_{16} + 8 \beta_{15} + 27 \beta_{14} - 117 \beta_{13} + 75 \beta_{12} - 9 \beta_{11} + 8 \beta_{10} + \beta_{9} + 35 \beta_{7} - 48 \beta_{6} + 83 \beta_{5} + 27 \beta_{4} + 27 \beta_{3} + 35 \beta_{2} - 8 \beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -35\beta_{6} + 75\beta_{4} - 44\beta_{3} + 165\beta_{2} + 207 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -44\beta_{10} - 9\beta_{9} - 154\beta_{7} - 319\beta_{5} + 451\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 297\beta_{15} + 275\beta_{14} + 429\beta_{12} - 429\beta_{11} + 482\beta_{8} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 207 \beta_{19} + 429 \beta_{18} - 260 \beta_{17} + 1001 \beta_{16} + 1430 \beta_{13} + 1430 \beta_{8} - 1430 \beta_{3} - 1430 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1001 \beta_{19} + 1001 \beta_{18} - 1430 \beta_{17} + 1637 \beta_{16} - 1001 \beta_{15} - 1001 \beta_{14} + 1897 \beta_{13} - 1637 \beta_{12} + 1430 \beta_{11} - 1001 \beta_{10} - 429 \beta_{9} - 2002 \beta_{7} + \cdots - 2431 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2002\beta_{6} - 1637\beta_{4} + 3432\beta_{3} - 2533\beta_{2} - 2793 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 3432\beta_{10} + 1430\beta_{9} + 7071\beta_{7} + 9604\beta_{5} - 10760\beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( -3689\beta_{15} - 6172\beta_{14} - 13243\beta_{12} + 4845\beta_{11} - 18105\beta_{8} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 11933 \beta_{19} - 13243 \beta_{18} + 16795 \beta_{17} - 23104 \beta_{16} - 27949 \beta_{13} - 27949 \beta_{8} + 27949 \beta_{3} + 27949 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 14706 \beta_{19} - 23104 \beta_{18} + 19551 \beta_{17} - 48280 \beta_{16} + 14706 \beta_{15} + 23104 \beta_{14} - 65075 \beta_{13} + 48280 \beta_{12} - 19551 \beta_{11} + 14706 \beta_{10} + 4845 \beta_{9} + \cdots + 42655 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1331\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.592999 1.82506i
−0.404726 1.24562i
−0.0879554 0.270699i
0.256741 + 0.790166i
0.519923 + 1.60016i
−1.36118 0.988953i
−0.672156 0.488350i
0.230270 + 0.167301i
1.05959 + 0.769835i
1.55249 + 1.12795i
−1.36118 + 0.988953i
−0.672156 + 0.488350i
0.230270 0.167301i
1.05959 0.769835i
1.55249 1.12795i
−0.592999 + 1.82506i
−0.404726 + 1.24562i
−0.0879554 + 0.270699i
0.256741 0.790166i
0.519923 1.60016i
0 −0.592999 + 1.82506i 0.309017 + 0.951057i −0.672156 + 0.488350i 0 0 0 −2.17019 1.57674i 0
161.2 0 −0.404726 + 1.24562i 0.309017 + 0.951057i 1.55249 1.12795i 0 0 0 −0.578747 0.420484i 0
161.3 0 −0.0879554 + 0.270699i 0.309017 + 0.951057i −1.36118 + 0.988953i 0 0 0 0.743475 + 0.540166i 0
161.4 0 0.256741 0.790166i 0.309017 + 0.951057i 0.230270 0.167301i 0 0 0 0.250570 + 0.182050i 0
161.5 0 0.519923 1.60016i 0.309017 + 0.951057i 1.05959 0.769835i 0 0 0 −1.48117 1.07613i 0
596.1 0 −1.36118 + 0.988953i −0.809017 0.587785i −0.404726 1.24562i 0 0 0 0.565758 1.74122i 0
596.2 0 −0.672156 + 0.488350i −0.809017 0.587785i −0.0879554 0.270699i 0 0 0 −0.0957092 + 0.294563i 0
596.3 0 0.230270 0.167301i −0.809017 0.587785i 0.519923 + 1.60016i 0 0 0 −0.283982 + 0.874008i 0
596.4 0 1.05959 0.769835i −0.809017 0.587785i −0.592999 1.82506i 0 0 0 0.221062 0.680358i 0
596.5 0 1.55249 1.12795i −0.809017 0.587785i 0.256741 + 0.790166i 0 0 0 0.828940 2.55122i 0
699.1 0 −1.36118 0.988953i −0.809017 + 0.587785i −0.404726 + 1.24562i 0 0 0 0.565758 + 1.74122i 0
699.2 0 −0.672156 0.488350i −0.809017 + 0.587785i −0.0879554 + 0.270699i 0 0 0 −0.0957092 0.294563i 0
699.3 0 0.230270 + 0.167301i −0.809017 + 0.587785i 0.519923 1.60016i 0 0 0 −0.283982 0.874008i 0
699.4 0 1.05959 + 0.769835i −0.809017 + 0.587785i −0.592999 + 1.82506i 0 0 0 0.221062 + 0.680358i 0
699.5 0 1.55249 + 1.12795i −0.809017 + 0.587785i 0.256741 0.790166i 0 0 0 0.828940 + 2.55122i 0
1207.1 0 −0.592999 1.82506i 0.309017 0.951057i −0.672156 0.488350i 0 0 0 −2.17019 + 1.57674i 0
1207.2 0 −0.404726 1.24562i 0.309017 0.951057i 1.55249 + 1.12795i 0 0 0 −0.578747 + 0.420484i 0
1207.3 0 −0.0879554 0.270699i 0.309017 0.951057i −1.36118 0.988953i 0 0 0 0.743475 0.540166i 0
1207.4 0 0.256741 + 0.790166i 0.309017 0.951057i 0.230270 + 0.167301i 0 0 0 0.250570 0.182050i 0
1207.5 0 0.519923 + 1.60016i 0.309017 0.951057i 1.05959 + 0.769835i 0 0 0 −1.48117 + 1.07613i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
11.c even 5 3 inner
11.d odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1331.1.d.a 20
11.b odd 2 1 CM 1331.1.d.a 20
11.c even 5 1 1331.1.b.a 5
11.c even 5 3 inner 1331.1.d.a 20
11.d odd 10 1 1331.1.b.a 5
11.d odd 10 3 inner 1331.1.d.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1331.1.b.a 5 11.c even 5 1
1331.1.b.a 5 11.d odd 10 1
1331.1.d.a 20 1.a even 1 1 trivial
1331.1.d.a 20 11.b odd 2 1 CM
1331.1.d.a 20 11.c even 5 3 inner
1331.1.d.a 20 11.d odd 10 3 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1331, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - T^{19} + 5 T^{18} - 6 T^{17} + 20 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} - T^{19} + 5 T^{18} - 6 T^{17} + 20 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} \) Copy content Toggle raw display
$23$ \( (T^{5} + T^{4} - 4 T^{3} - 3 T^{2} + 3 T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} - T^{19} + 5 T^{18} - 6 T^{17} + 20 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{20} - T^{19} + 5 T^{18} - 6 T^{17} + 20 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( T^{20} - T^{19} + 5 T^{18} - 6 T^{17} + 20 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{20} - T^{19} + 5 T^{18} - 6 T^{17} + 20 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{20} - T^{19} + 5 T^{18} - 6 T^{17} + 20 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{20} \) Copy content Toggle raw display
$67$ \( (T^{5} + T^{4} - 4 T^{3} - 3 T^{2} + 3 T + 1)^{4} \) Copy content Toggle raw display
$71$ \( T^{20} - T^{19} + 5 T^{18} - 6 T^{17} + 20 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{20} \) Copy content Toggle raw display
$79$ \( T^{20} \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( (T^{5} + T^{4} - 4 T^{3} - 3 T^{2} + 3 T + 1)^{4} \) Copy content Toggle raw display
$97$ \( T^{20} - T^{19} + 5 T^{18} - 6 T^{17} + 20 T^{16} + \cdots + 1 \) Copy content Toggle raw display
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