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Properties

Label 55.2.j.a
Level $55$
Weight $2$
Character orbit 55.j
Analytic conductor $0.439$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,2,Mod(4,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 55.j (of order \(10\), degree \(4\), 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: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{14} + 25x^{12} - 57x^{10} + 194x^{8} - 303x^{6} + 235x^{4} - 33x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{13} - \beta_{12} + \beta_1) q^{2} + ( - \beta_{15} + \beta_{13} + \beta_{5}) q^{3} + (\beta_{8} + \beta_{6} - \beta_{4}) q^{4} + (\beta_{10} + \beta_{6} - \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{9} - 2 \beta_{8} + \cdots - 2) q^{6}+ \cdots + (\beta_{11} + \beta_{9} - \beta_{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} - \beta_{12} + \beta_1) q^{2} + ( - \beta_{15} + \beta_{13} + \beta_{5}) q^{3} + (\beta_{8} + \beta_{6} - \beta_{4}) q^{4} + (\beta_{10} + \beta_{6} - \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{9} - 2 \beta_{8} + \cdots - 2) q^{6}+ \cdots + (\beta_{11} - 5 \beta_{9} - \beta_{8} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{9} - 6 q^{11} - 12 q^{14} - 16 q^{15} + 16 q^{16} + 6 q^{19} - 8 q^{20} + 8 q^{21} + 6 q^{24} - 16 q^{25} + 40 q^{26} + 2 q^{29} + 26 q^{30} + 8 q^{31} - 16 q^{34} + 22 q^{35} + 10 q^{36} + 30 q^{39} + 12 q^{40} - 52 q^{41} + 4 q^{44} + 12 q^{45} - 62 q^{46} - 10 q^{49} + 28 q^{50} - 42 q^{51} - 40 q^{54} - 8 q^{55} - 20 q^{56} + 2 q^{59} - 32 q^{60} - 40 q^{61} - 8 q^{64} - 40 q^{65} + 58 q^{66} + 26 q^{69} - 34 q^{70} + 36 q^{71} + 48 q^{74} - 20 q^{75} + 56 q^{76} + 38 q^{79} + 34 q^{80} + 68 q^{81} + 12 q^{84} + 58 q^{85} + 22 q^{86} + 24 q^{89} + 78 q^{90} - 20 q^{91} + 14 q^{94} + 48 q^{95} - 86 q^{96} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{14} + 25x^{12} - 57x^{10} + 194x^{8} - 303x^{6} + 235x^{4} - 33x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1829973 \nu^{15} - 24476532 \nu^{13} + 152757677 \nu^{11} - 557238290 \nu^{9} + \cdots - 7498263355 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 608673 \nu^{14} + 4779625 \nu^{12} - 9344761 \nu^{10} - 2560494 \nu^{8} - 9430964 \nu^{6} + \cdots + 31775447 ) / 609644662 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2744937 \nu^{14} - 22433610 \nu^{12} + 99837091 \nu^{10} - 276654262 \nu^{8} + 814713604 \nu^{6} + \cdots - 771926881 ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2744937 \nu^{15} - 22433610 \nu^{13} + 99837091 \nu^{11} - 276654262 \nu^{9} + \cdots - 771926881 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2924455 \nu^{14} + 43736988 \nu^{12} - 193036323 \nu^{10} + 488836470 \nu^{8} + \cdots - 236512683 ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1351905 \nu^{14} + 5558819 \nu^{12} - 7678521 \nu^{10} - 13223046 \nu^{8} - 42401020 \nu^{6} + \cdots - 629543387 ) / 609644662 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6960689 \nu^{14} + 30587176 \nu^{12} - 61040185 \nu^{10} + 38446538 \nu^{8} + \cdots - 1738523017 ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3962283 \nu^{14} + 31992860 \nu^{12} - 118526613 \nu^{10} + 271533274 \nu^{8} + \cdots - 383811549 ) / 1219289324 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8234811 \nu^{15} - 67300830 \nu^{13} + 299511273 \nu^{11} - 829962786 \nu^{9} + \cdots - 2315780643 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 770947 \nu^{14} - 4704418 \nu^{12} + 13758165 \nu^{10} - 28071690 \nu^{8} + 126275508 \nu^{6} + \cdots - 243303327 ) / 221688968 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10669503 \nu^{15} - 86419330 \nu^{13} + 336890317 \nu^{11} - 819720810 \nu^{9} + \cdots - 4303783 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8815096 \nu^{15} + 58503253 \nu^{13} - 198965251 \nu^{11} + 429083674 \nu^{9} + \cdots + 352179707 \nu ) / 1219289324 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19639683 \nu^{15} + 162786416 \nu^{13} - 643887411 \nu^{11} + 1627587846 \nu^{9} + \cdots + 4755604557 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1496498 \nu^{15} + 9707543 \nu^{13} - 31032777 \nu^{11} + 60570826 \nu^{9} + \cdots - 172220447 \nu ) / 110844484 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 2\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + 3\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{9} - 4\beta_{8} + \beta_{7} - 4\beta_{6} + 4\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 5\beta_{14} - \beta_{13} - 10\beta_{12} - \beta_{10} + 15\beta_{5} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{11} - 22\beta_{8} + 21\beta_{7} + \beta_{4} - 7\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 21\beta_{15} - 21\beta_{14} - 36\beta_{13} - 36\beta_{12} + 29\beta_{5} - 28\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -85\beta_{11} - 78\beta_{9} - 87\beta_{8} + 85\beta_{7} + 49\beta_{6} - 7\beta_{4} - 49\beta_{3} - 87 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 85\beta_{15} - 36\beta_{14} - 136\beta_{13} - 45\beta_{12} + 36\beta_{10} + 36\beta_{5} - 121\beta_{2} - 85\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -342\beta_{11} - 396\beta_{9} - 230\beta_{8} + 166\beta_{7} + 166\beta_{6} - 166\beta_{4} - 529 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 166\beta_{15} - 220\beta_{13} + 342\beta_{10} - 220\beta_{5} - 342\beta_{2} - 475\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -728\beta_{11} - 1653\beta_{9} + 728\beta_{6} - 1170\beta_{4} + 651\beta_{3} - 1653 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 728\beta_{14} + 1002\beta_{12} + 1379\beta_{10} - 2823\beta_{5} - 1002\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -4644\beta_{9} + 3748\beta_{8} - 3109\beta_{7} + 2472\beta_{6} - 3748\beta_{4} + 3109\beta_{3} - 3109 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3109 \beta_{15} + 5581 \beta_{14} + 4385 \beta_{13} + 8392 \beta_{12} + 3109 \beta_{10} + \cdots - 1276 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-1\) \(\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
1.92464 + 0.625353i
1.17360 + 0.381325i
−1.17360 − 0.381325i
−1.92464 − 0.625353i
−0.972539 − 1.33858i
−0.471815 − 0.649397i
0.471815 + 0.649397i
0.972539 + 1.33858i
1.92464 − 0.625353i
1.17360 − 0.381325i
−1.17360 + 0.381325i
−1.92464 + 0.625353i
−0.972539 + 1.33858i
−0.471815 + 0.649397i
0.471815 − 0.649397i
0.972539 − 1.33858i
−1.18949 + 1.63719i 2.49233 + 0.809808i −0.647481 − 1.99274i −1.06421 − 1.96658i −4.29042 + 3.11717i −0.918552 + 0.298456i 0.183406 + 0.0595923i 3.12889 + 2.27327i 4.48555 + 0.596911i
4.2 −0.725323 + 0.998322i −0.346168 − 0.112477i 0.147481 + 0.453901i 0.0238439 + 2.23594i 0.363371 − 0.264005i 2.45903 − 0.798988i −2.90731 − 0.944641i −2.31987 − 1.68548i −2.24948 − 1.59798i
4.3 0.725323 − 0.998322i 0.346168 + 0.112477i 0.147481 + 0.453901i −2.11914 − 0.713621i 0.363371 − 0.264005i −2.45903 + 0.798988i 2.90731 + 0.944641i −2.31987 − 1.68548i −2.24948 + 1.59798i
4.4 1.18949 − 1.63719i −2.49233 − 0.809808i −0.647481 − 1.99274i 1.54147 + 1.61983i −4.29042 + 3.11717i 0.918552 − 0.298456i −0.183406 − 0.0595923i 3.12889 + 2.27327i 4.48555 − 0.596911i
9.1 −1.57360 − 0.511294i 1.16075 + 1.59764i 0.596764 + 0.433574i 1.09069 + 1.95203i −1.00970 − 3.10753i 1.31845 − 1.81468i 1.22769 + 1.68978i −0.278050 + 0.855749i −0.718246 − 3.62937i
9.2 −0.763412 − 0.248048i −1.03494 − 1.42447i −1.09676 − 0.796845i 1.42953 − 1.71943i 0.436748 + 1.34417i −0.348029 + 0.479022i 1.58326 + 2.17917i −0.0309674 + 0.0953077i −1.51782 + 0.958043i
9.3 0.763412 + 0.248048i 1.03494 + 1.42447i −1.09676 − 0.796845i −2.16717 − 0.550792i 0.436748 + 1.34417i 0.348029 − 0.479022i −1.58326 − 2.17917i −0.0309674 + 0.0953077i −1.51782 − 0.958043i
9.4 1.57360 + 0.511294i −1.16075 − 1.59764i 0.596764 + 0.433574i 0.264988 + 2.22031i −1.00970 − 3.10753i −1.31845 + 1.81468i −1.22769 − 1.68978i −0.278050 + 0.855749i −0.718246 + 3.62937i
14.1 −1.18949 − 1.63719i 2.49233 − 0.809808i −0.647481 + 1.99274i −1.06421 + 1.96658i −4.29042 − 3.11717i −0.918552 − 0.298456i 0.183406 − 0.0595923i 3.12889 − 2.27327i 4.48555 − 0.596911i
14.2 −0.725323 − 0.998322i −0.346168 + 0.112477i 0.147481 − 0.453901i 0.0238439 − 2.23594i 0.363371 + 0.264005i 2.45903 + 0.798988i −2.90731 + 0.944641i −2.31987 + 1.68548i −2.24948 + 1.59798i
14.3 0.725323 + 0.998322i 0.346168 − 0.112477i 0.147481 − 0.453901i −2.11914 + 0.713621i 0.363371 + 0.264005i −2.45903 − 0.798988i 2.90731 − 0.944641i −2.31987 + 1.68548i −2.24948 − 1.59798i
14.4 1.18949 + 1.63719i −2.49233 + 0.809808i −0.647481 + 1.99274i 1.54147 − 1.61983i −4.29042 − 3.11717i 0.918552 + 0.298456i −0.183406 + 0.0595923i 3.12889 − 2.27327i 4.48555 + 0.596911i
49.1 −1.57360 + 0.511294i 1.16075 − 1.59764i 0.596764 − 0.433574i 1.09069 − 1.95203i −1.00970 + 3.10753i 1.31845 + 1.81468i 1.22769 − 1.68978i −0.278050 − 0.855749i −0.718246 + 3.62937i
49.2 −0.763412 + 0.248048i −1.03494 + 1.42447i −1.09676 + 0.796845i 1.42953 + 1.71943i 0.436748 − 1.34417i −0.348029 − 0.479022i 1.58326 − 2.17917i −0.0309674 − 0.0953077i −1.51782 − 0.958043i
49.3 0.763412 − 0.248048i 1.03494 − 1.42447i −1.09676 + 0.796845i −2.16717 + 0.550792i 0.436748 − 1.34417i 0.348029 + 0.479022i −1.58326 + 2.17917i −0.0309674 − 0.0953077i −1.51782 + 0.958043i
49.4 1.57360 − 0.511294i −1.16075 + 1.59764i 0.596764 − 0.433574i 0.264988 − 2.22031i −1.00970 + 3.10753i −1.31845 − 1.81468i −1.22769 + 1.68978i −0.278050 − 0.855749i −0.718246 − 3.62937i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.2.j.a ✓ 16
3.b odd 2 1 495.2.ba.a 16
4.b odd 2 1 880.2.cd.c 16
5.b even 2 1 inner 55.2.j.a ✓ 16
5.c odd 4 2 275.2.h.d 16
11.b odd 2 1 605.2.j.d 16
11.c even 5 1 inner 55.2.j.a ✓ 16
11.c even 5 1 605.2.b.g 8
11.c even 5 2 605.2.j.h 16
11.d odd 10 1 605.2.b.f 8
11.d odd 10 1 605.2.j.d 16
11.d odd 10 2 605.2.j.g 16
15.d odd 2 1 495.2.ba.a 16
20.d odd 2 1 880.2.cd.c 16
33.h odd 10 1 495.2.ba.a 16
44.h odd 10 1 880.2.cd.c 16
55.d odd 2 1 605.2.j.d 16
55.h odd 10 1 605.2.b.f 8
55.h odd 10 1 605.2.j.d 16
55.h odd 10 2 605.2.j.g 16
55.j even 10 1 inner 55.2.j.a ✓ 16
55.j even 10 1 605.2.b.g 8
55.j even 10 2 605.2.j.h 16
55.k odd 20 2 275.2.h.d 16
55.k odd 20 2 3025.2.a.bl 8
55.l even 20 2 3025.2.a.bk 8
165.o odd 10 1 495.2.ba.a 16
220.n odd 10 1 880.2.cd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a ✓ 16 1.a even 1 1 trivial
55.2.j.a ✓ 16 5.b even 2 1 inner
55.2.j.a ✓ 16 11.c even 5 1 inner
55.2.j.a ✓ 16 55.j even 10 1 inner
275.2.h.d 16 5.c odd 4 2
275.2.h.d 16 55.k odd 20 2
495.2.ba.a 16 3.b odd 2 1
495.2.ba.a 16 15.d odd 2 1
495.2.ba.a 16 33.h odd 10 1
495.2.ba.a 16 165.o odd 10 1
605.2.b.f 8 11.d odd 10 1
605.2.b.f 8 55.h odd 10 1
605.2.b.g 8 11.c even 5 1
605.2.b.g 8 55.j even 10 1
605.2.j.d 16 11.b odd 2 1
605.2.j.d 16 11.d odd 10 1
605.2.j.d 16 55.d odd 2 1
605.2.j.d 16 55.h odd 10 1
605.2.j.g 16 11.d odd 10 2
605.2.j.g 16 55.h odd 10 2
605.2.j.h 16 11.c even 5 2
605.2.j.h 16 55.j even 10 2
880.2.cd.c 16 4.b odd 2 1
880.2.cd.c 16 20.d odd 2 1
880.2.cd.c 16 44.h odd 10 1
880.2.cd.c 16 220.n odd 10 1
3025.2.a.bk 8 55.l even 20 2
3025.2.a.bl 8 55.k odd 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(55, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{16} - 7 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 9 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( (T^{8} + 3 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 10 T^{14} + \cdots + 47265625 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 1675346761 \) Copy content Toggle raw display
$19$ \( (T^{8} - 3 T^{7} + \cdots + 121)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 99 T^{6} + \cdots + 26411)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - T^{7} + 25 T^{6} + \cdots + 121)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{7} + 39 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 28 T^{14} + \cdots + 15768841 \) Copy content Toggle raw display
$41$ \( (T^{8} + 26 T^{7} + \cdots + 3876961)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 173 T^{6} + \cdots + 212531)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 59639012521 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 239836452361 \) Copy content Toggle raw display
$59$ \( (T^{8} - T^{7} + 129 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 20 T^{7} + \cdots + 43681)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 149 T^{6} + \cdots + 18491)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 18 T^{7} + \cdots + 6305121)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1636073786281 \) Copy content Toggle raw display
$79$ \( (T^{8} - 19 T^{7} + \cdots + 101761)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45169425961 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} + \cdots + 1871)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 25937424601 \) Copy content Toggle raw display
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