Properties

Label 261.3.b.a
Level $261$
Weight $3$
Character orbit 261.b
Analytic conductor $7.112$
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] = [261,3,Mod(233,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.233");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.b (of order \(2\), degree \(1\), 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: \(7.11173489980\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 60 x^{18} + 1502 x^{16} + 20402 x^{14} + 163953 x^{12} + 798708 x^{10} + 2330953 x^{8} + 3915186 x^{6} + 3550046 x^{4} + 1539500 x^{2} + 233289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{12} q^{5} + ( - \beta_{7} + 1) q^{7} + ( - \beta_{14} + \beta_{13} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{12} q^{5} + ( - \beta_{7} + 1) q^{7} + ( - \beta_{14} + \beta_{13} - 2 \beta_1) q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} - 1) q^{10} + ( - \beta_{16} + \beta_{12} + \beta_{11} - \beta_1) q^{11} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} - 2) q^{13} + ( - \beta_{18} - \beta_{14} + \beta_{12} + \beta_1) q^{14} + (\beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{2} + 4) q^{16} + (\beta_{19} + \beta_{18} + 2 \beta_{17} - \beta_{15} - \beta_{12} + \beta_{11} + \beta_1) q^{17} + (\beta_{10} + \beta_{9} + \beta_{6} + \beta_{4} + \beta_{3} + 3) q^{19} + (\beta_{18} + \beta_{17} - \beta_{15} - 2 \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_1) q^{20} + (\beta_{10} + 3 \beta_{7} + \beta_{6} + \beta_{4} - 3 \beta_{3} + 3) q^{22} + ( - 2 \beta_{17} + \beta_{16} + \beta_{15} + 2 \beta_{12} - 2 \beta_1) q^{23} + ( - 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} - 4) q^{25} + ( - \beta_{16} - 2 \beta_{15} - 5 \beta_{14} + \beta_{13} - 3 \beta_{12} - \beta_{11} - 4 \beta_1) q^{26} + ( - 2 \beta_{9} + \beta_{7} - 2 \beta_{3} + \beta_{2} - 8) q^{28} - \beta_{14} q^{29} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{31} + ( - \beta_{19} - 2 \beta_{17} + 3 \beta_{16} + 3 \beta_{15} - \beta_{13} - \beta_{12} - 2 \beta_{11} + \cdots + 3 \beta_1) q^{32}+ \cdots + ( - \beta_{19} - \beta_{18} + 2 \beta_{17} - 6 \beta_{16} - \beta_{15} + \beta_{14} + 5 \beta_{13} + \cdots - 13 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 40 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 40 q^{4} + 16 q^{7} - 24 q^{10} - 24 q^{13} + 72 q^{16} + 40 q^{19} + 92 q^{22} - 76 q^{25} - 124 q^{28} - 56 q^{31} + 4 q^{34} - 80 q^{40} + 64 q^{43} + 120 q^{46} + 28 q^{49} + 452 q^{52} - 280 q^{55} - 16 q^{61} - 92 q^{64} - 88 q^{67} - 576 q^{70} + 320 q^{73} - 344 q^{76} + 328 q^{79} + 208 q^{82} + 328 q^{85} - 528 q^{88} + 248 q^{91} + 636 q^{94} - 464 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 60 x^{18} + 1502 x^{16} + 20402 x^{14} + 163953 x^{12} + 798708 x^{10} + 2330953 x^{8} + 3915186 x^{6} + 3550046 x^{4} + 1539500 x^{2} + 233289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1186 \nu^{18} + 69859 \nu^{16} + 1700104 \nu^{14} + 22081995 \nu^{12} + 164745778 \nu^{10} + 703909200 \nu^{8} + 1597631676 \nu^{6} + 1564676419 \nu^{4} + \cdots - 50934177 ) / 8426880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 561 \nu^{18} + 35240 \nu^{16} + 929061 \nu^{14} + 13354940 \nu^{12} + 113729488 \nu^{10} + 582818878 \nu^{8} + 1740559925 \nu^{6} + 2775084232 \nu^{4} + \cdots + 439805142 ) / 4213440 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21 \nu^{18} - 1283 \nu^{16} - 32607 \nu^{14} - 446255 \nu^{12} - 3555538 \nu^{10} - 16635322 \nu^{8} - 43858427 \nu^{6} - 59271811 \nu^{4} - 35545567 \nu^{2} + \cdots - 6782769 ) / 126720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45 \nu^{18} + 2533 \nu^{16} + 58443 \nu^{14} + 713809 \nu^{12} + 4978142 \nu^{10} + 19958786 \nu^{8} + 44053951 \nu^{6} + 48537893 \nu^{4} + 22987019 \nu^{2} + \cdots + 3124611 ) / 177408 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 764 \nu^{18} - 44515 \nu^{16} - 1072694 \nu^{14} - 13855895 \nu^{12} - 104061522 \nu^{10} - 461671012 \nu^{8} - 1177013270 \nu^{6} - 1603681923 \nu^{4} + \cdots - 209659023 ) / 2106720 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3275 \nu^{18} - 194129 \nu^{16} - 4774553 \nu^{14} - 63148725 \nu^{12} - 486598310 \nu^{10} - 2210070102 \nu^{8} - 5690111421 \nu^{6} + \cdots - 544849011 ) / 8426880 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6176 \nu^{18} - 365341 \nu^{16} - 8964578 \nu^{14} - 118243805 \nu^{12} - 908277198 \nu^{10} - 4112949676 \nu^{8} - 10592724014 \nu^{6} + \cdots - 1714471101 ) / 8426880 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13211 \nu^{18} + 774057 \nu^{16} + 18751145 \nu^{14} + 243076765 \nu^{12} + 1823657398 \nu^{10} + 7994730614 \nu^{8} + 19667260093 \nu^{6} + \cdots + 2314348155 ) / 16853760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11581 \nu^{19} - 733224 \nu^{17} - 19541689 \nu^{15} - 285447860 \nu^{13} - 2493952208 \nu^{11} - 13354360630 \nu^{9} - 43128840641 \nu^{7} + \cdots - 21113434278 \nu ) / 193818240 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35683 \nu^{19} - 2131619 \nu^{17} - 53009941 \nu^{15} - 712208695 \nu^{13} - 5611492434 \nu^{11} - 26307264446 \nu^{9} - 70893577921 \nu^{7} + \cdots - 9577416357 \nu ) / 387636480 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 413 \nu^{19} - 23745 \nu^{17} - 562067 \nu^{15} - 7081837 \nu^{13} - 51294982 \nu^{11} - 215369138 \nu^{9} - 503631511 \nu^{7} - 603730945 \nu^{5} + \cdots - 62227839 \nu ) / 4080384 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 413 \nu^{19} - 23745 \nu^{17} - 562067 \nu^{15} - 7081837 \nu^{13} - 51294982 \nu^{11} - 215369138 \nu^{9} - 503631511 \nu^{7} - 603730945 \nu^{5} + \cdots - 103031679 \nu ) / 4080384 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4893 \nu^{19} + 290866 \nu^{17} + 7187205 \nu^{15} + 95815510 \nu^{13} + 748847884 \nu^{11} + 3493745342 \nu^{9} + 9502862029 \nu^{7} + \cdots + 3406938280 \nu ) / 32303040 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 4681 \nu^{19} - 277479 \nu^{17} - 6824299 \nu^{15} - 90252035 \nu^{13} - 695616938 \nu^{11} - 3166310530 \nu^{9} - 8232904151 \nu^{7} + \cdots - 1483563213 \nu ) / 20401920 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 63351 \nu^{19} - 3707059 \nu^{17} - 89608569 \nu^{15} - 1157014135 \nu^{13} - 8611513778 \nu^{11} - 37111183190 \nu^{9} - 87712061701 \nu^{7} + \cdots + 2117289747 \nu ) / 193818240 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 34256 \nu^{19} - 2016651 \nu^{17} - 49247318 \nu^{15} - 647229715 \nu^{13} - 4971447298 \nu^{11} - 22698665636 \nu^{9} - 59904756514 \nu^{7} + \cdots - 9591666771 \nu ) / 96909120 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 94659 \nu^{19} + 5601133 \nu^{17} + 137587225 \nu^{15} + 1819941185 \nu^{13} + 14071579342 \nu^{11} + 64641546646 \nu^{9} + 171643225757 \nu^{7} + \cdots + 37372224175 \nu ) / 129212160 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + \beta_{13} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - 14\beta_{2} + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} - 2 \beta_{17} + 3 \beta_{16} + 3 \beta_{15} + 16 \beta_{14} - 17 \beta_{13} - \beta_{12} - 2 \beta_{11} + 115 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -22\beta_{10} - 18\beta_{8} - 24\beta_{7} - 22\beta_{5} - 20\beta_{4} + 4\beta_{3} + 182\beta_{2} - 689 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 18 \beta_{19} - 2 \beta_{18} + 52 \beta_{17} - 86 \beta_{16} - 68 \beta_{15} - 232 \beta_{14} + 248 \beta_{13} + 20 \beta_{12} + 48 \beta_{11} - 1407 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 364 \beta_{10} - 20 \beta_{9} + 264 \beta_{8} + 424 \beta_{7} - 14 \beta_{6} + 386 \beta_{5} + 300 \beta_{4} - 114 \beta_{3} - 2381 \beta_{2} + 8402 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 222 \beta_{19} + 80 \beta_{18} - 978 \beta_{17} + 1746 \beta_{16} + 1178 \beta_{15} + 3311 \beta_{14} - 3473 \beta_{13} - 294 \beta_{12} - 836 \beta_{11} + 17840 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5487 \beta_{10} + 694 \beta_{9} - 3673 \beta_{8} - 6737 \beta_{7} + 500 \beta_{6} - 6295 \beta_{5} - 4109 \beta_{4} + 2246 \beta_{3} + 31588 \beta_{2} - 106074 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2171 \beta_{19} - 1944 \beta_{18} + 16342 \beta_{17} - 30981 \beta_{16} - 18705 \beta_{15} - 47184 \beta_{14} + 47981 \beta_{13} + 3869 \beta_{12} + 12958 \beta_{11} - 231437 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 79644 \beta_{10} - 15888 \beta_{9} + 50344 \beta_{8} + 101884 \beta_{7} - 11868 \beta_{6} + 99072 \beta_{5} + 54440 \beta_{4} - 38228 \beta_{3} - 424388 \beta_{2} + 1369993 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 15028 \beta_{19} + 38128 \beta_{18} - 258112 \beta_{17} + 512896 \beta_{16} + 286192 \beta_{15} + 673448 \beta_{14} - 660380 \beta_{13} - 47648 \beta_{12} - 190556 \beta_{11} + \cdots + 3050493 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1137128 \beta_{10} + 304312 \beta_{9} - 688460 \beta_{8} - 1499764 \beta_{7} + 236584 \beta_{6} - 1525048 \beta_{5} - 713008 \beta_{4} + 604816 \beta_{3} + 5760613 \beta_{2} + \cdots - 17984778 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3772 \beta_{19} - 666948 \beta_{18} + 3952244 \beta_{17} - 8146376 \beta_{16} - 4297020 \beta_{15} - 9629521 \beta_{14} + 9091561 \beta_{13} + 550396 \beta_{12} + 2730908 \beta_{11} + \cdots - 40687314 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 16119489 \beta_{10} - 5293684 \beta_{9} + 9436337 \beta_{8} + 21723809 \beta_{7} - 4288356 \beta_{6} + 23108613 \beta_{5} + 9316013 \beta_{4} - 9185616 \beta_{3} + \cdots + 239070180 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2846471 \beta_{19} + 10898004 \beta_{18} - 59385226 \beta_{17} + 125922679 \beta_{16} + 63803979 \beta_{15} + 137887432 \beta_{14} - 125415577 \beta_{13} - 5839157 \beta_{12} + \cdots + 547811247 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 227852930 \beta_{10} + 86874176 \beta_{9} - 129834330 \beta_{8} - 311498724 \beta_{7} + 73283148 \beta_{6} - 346005782 \beta_{5} - 121929012 \beta_{4} + \cdots - 3210229065 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 75192698 \beta_{19} - 170519970 \beta_{18} + 881284308 \beta_{17} - 1910010470 \beta_{16} - 940372620 \beta_{15} - 1976286464 \beta_{14} + 1734710596 \beta_{13} + \cdots - 7433242107 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
233.1
3.77386i
3.50376i
3.33061i
2.73126i
2.44669i
2.16402i
1.39441i
1.10795i
0.882022i
0.556561i
0.556561i
0.882022i
1.10795i
1.39441i
2.16402i
2.44669i
2.73126i
3.33061i
3.50376i
3.77386i
3.77386i 0 −10.2420 0.989973i 0 −2.12908 23.5565i 0 −3.73602
233.2 3.50376i 0 −8.27632 6.57696i 0 0.415091 14.9832i 0 23.0441
233.3 3.33061i 0 −7.09298 6.39075i 0 9.86432 10.3015i 0 −21.2851
233.4 2.73126i 0 −3.45979 3.43176i 0 4.03252 1.47545i 0 9.37304
233.5 2.44669i 0 −1.98629 3.05895i 0 −8.99498 4.92692i 0 −7.48431
233.6 2.16402i 0 −0.682976 9.41763i 0 9.78219 7.17810i 0 −20.3799
233.7 1.39441i 0 2.05563 7.37680i 0 −3.80285 8.44401i 0 10.2863
233.8 1.10795i 0 2.77246 3.27042i 0 0.561257 7.50351i 0 3.62345
233.9 0.882022i 0 3.22204 4.87805i 0 −10.6840 6.37000i 0 −4.30255
233.10 0.556561i 0 3.69024 2.04634i 0 8.95559 4.28009i 0 −1.13891
233.11 0.556561i 0 3.69024 2.04634i 0 8.95559 4.28009i 0 −1.13891
233.12 0.882022i 0 3.22204 4.87805i 0 −10.6840 6.37000i 0 −4.30255
233.13 1.10795i 0 2.77246 3.27042i 0 0.561257 7.50351i 0 3.62345
233.14 1.39441i 0 2.05563 7.37680i 0 −3.80285 8.44401i 0 10.2863
233.15 2.16402i 0 −0.682976 9.41763i 0 9.78219 7.17810i 0 −20.3799
233.16 2.44669i 0 −1.98629 3.05895i 0 −8.99498 4.92692i 0 −7.48431
233.17 2.73126i 0 −3.45979 3.43176i 0 4.03252 1.47545i 0 9.37304
233.18 3.33061i 0 −7.09298 6.39075i 0 9.86432 10.3015i 0 −21.2851
233.19 3.50376i 0 −8.27632 6.57696i 0 0.415091 14.9832i 0 23.0441
233.20 3.77386i 0 −10.2420 0.989973i 0 −2.12908 23.5565i 0 −3.73602
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 233.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.b.a 20
3.b odd 2 1 inner 261.3.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.3.b.a 20 1.a even 1 1 trivial
261.3.b.a 20 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(261, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 60 T^{18} + 1502 T^{16} + \cdots + 233289 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 288 T^{18} + \cdots + 981431011584 \) Copy content Toggle raw display
$7$ \( (T^{10} - 8 T^{9} - 220 T^{8} + \cdots + 631708)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 1796 T^{18} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{10} + 12 T^{9} - 962 T^{8} + \cdots + 2126522608)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + 3344 T^{18} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{10} - 20 T^{9} - 1828 T^{8} + \cdots - 16689921696)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 4436 T^{18} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{2} + 29)^{10} \) Copy content Toggle raw display
$31$ \( (T^{10} + 28 T^{9} + \cdots - 1599317862912)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} - 3422 T^{8} + \cdots + 1381667331176)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + 20988 T^{18} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{10} - 32 T^{9} + \cdots + 95883094711136)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + 17472 T^{18} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{20} + 29252 T^{18} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{20} + 49504 T^{18} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{10} + 8 T^{9} + \cdots - 30\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 44 T^{9} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + 66836 T^{18} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{10} - 160 T^{9} + \cdots + 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} - 164 T^{9} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + 71348 T^{18} + \cdots + 50\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{20} + 93148 T^{18} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{10} + 232 T^{9} + \cdots + 44\!\cdots\!04)^{2} \) Copy content Toggle raw display
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