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Properties

Label 400.2.l.h
Level $400$
Weight $2$
Character orbit 400.l
Analytic conductor $3.194$
Analytic rank $0$
Dimension $16$
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] = [400,2,Mod(101,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.l (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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 80)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + (\beta_{11} + \beta_{4}) q^{3} + ( - \beta_{14} + \beta_{3} + \beta_1) q^{4} + ( - \beta_{13} - \beta_{12} + \beta_{7} + \cdots - 1) q^{6}+ \cdots + (\beta_{15} - \beta_{12} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + (\beta_{11} + \beta_{4}) q^{3} + ( - \beta_{14} + \beta_{3} + \beta_1) q^{4} + ( - \beta_{13} - \beta_{12} + \beta_{7} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{15} + \beta_{14} + \beta_{11} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} - 12 q^{6} - 8 q^{11} + 12 q^{12} + 4 q^{14} + 16 q^{16} - 8 q^{19} + 20 q^{22} + 8 q^{24} - 16 q^{26} - 24 q^{27} + 4 q^{28} - 16 q^{29} + 16 q^{34} - 4 q^{36} + 16 q^{37} - 20 q^{38} - 60 q^{42} - 8 q^{43} + 40 q^{44} - 4 q^{46} + 40 q^{47} + 40 q^{48} - 16 q^{49} - 32 q^{51} - 56 q^{52} - 16 q^{53} + 32 q^{54} + 16 q^{56} + 12 q^{58} - 8 q^{59} + 16 q^{61} + 8 q^{62} - 40 q^{63} - 16 q^{64} - 40 q^{67} + 48 q^{68} + 16 q^{69} + 40 q^{72} - 72 q^{74} - 16 q^{77} + 16 q^{78} + 16 q^{79} - 16 q^{81} + 76 q^{82} - 40 q^{83} - 64 q^{84} + 28 q^{86} + 32 q^{91} + 52 q^{92} + 48 q^{93} - 36 q^{94} + 8 q^{96} - 60 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{15} - 70 \nu^{14} + 120 \nu^{13} + 144 \nu^{12} + 89 \nu^{11} - 606 \nu^{10} - 968 \nu^{9} + \cdots - 9472 ) / 384 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 39 \nu^{15} - 74 \nu^{14} - 66 \nu^{13} - 16 \nu^{12} + 337 \nu^{11} + 454 \nu^{10} - 654 \nu^{9} + \cdots + 256 ) / 448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 121 \nu^{15} + 65 \nu^{14} + 264 \nu^{13} + 372 \nu^{12} - 487 \nu^{11} - 1725 \nu^{10} + \cdots - 5056 ) / 1344 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 285 \nu^{15} + 446 \nu^{14} + 436 \nu^{13} + 496 \nu^{12} - 1739 \nu^{11} - 3378 \nu^{10} + \cdots + 9088 ) / 2688 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 61 \nu^{15} - 82 \nu^{14} - 124 \nu^{13} - 112 \nu^{12} + 411 \nu^{11} + 870 \nu^{10} - 532 \nu^{9} + \cdots + 1536 ) / 384 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 177 \nu^{15} - 604 \nu^{14} + 240 \nu^{13} + 328 \nu^{12} + 1607 \nu^{11} - 368 \nu^{10} + \cdots - 48256 ) / 896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 289 \nu^{15} - 1206 \nu^{14} + 604 \nu^{13} + 832 \nu^{12} + 3119 \nu^{11} - 1446 \nu^{10} + \cdots - 99328 ) / 1344 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 81 \nu^{15} - 184 \nu^{14} - 56 \nu^{13} - 8 \nu^{12} + 679 \nu^{11} + 588 \nu^{10} - 1680 \nu^{9} + \cdots - 7808 ) / 384 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20 \nu^{15} - 20 \nu^{14} - 41 \nu^{13} - 44 \nu^{12} + 108 \nu^{11} + 276 \nu^{10} - 95 \nu^{9} + \cdots + 576 ) / 96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} + \cdots - 83072 ) / 1344 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 827 \nu^{15} + 2780 \nu^{14} - 904 \nu^{13} - 1480 \nu^{12} - 7821 \nu^{11} + 744 \nu^{10} + \cdots + 210048 ) / 2688 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 127 \nu^{15} - 336 \nu^{14} - 32 \nu^{13} + 88 \nu^{12} + 1145 \nu^{11} + 636 \nu^{10} - 3256 \nu^{9} + \cdots - 16384 ) / 384 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 206 \nu^{15} - 633 \nu^{14} + 178 \nu^{13} + 268 \nu^{12} + 1770 \nu^{11} + 57 \nu^{10} - 6150 \nu^{9} + \cdots - 46400 ) / 448 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1431 \nu^{15} - 3748 \nu^{14} + 136 \nu^{13} + 856 \nu^{12} + 11857 \nu^{11} + 5328 \nu^{10} + \cdots - 219776 ) / 2688 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2295 \nu^{15} - 6418 \nu^{14} + 580 \nu^{13} + 1936 \nu^{12} + 19969 \nu^{11} + 7086 \nu^{10} + \cdots - 392960 ) / 2688 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} + \beta_{13} + \beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{6} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{15} + 3 \beta_{14} + \beta_{13} - 4 \beta_{10} + 3 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \cdots + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{15} + 5 \beta_{13} + \beta_{12} + 4 \beta_{11} - 3 \beta_{10} - 4 \beta_{9} + 4 \beta_{8} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{14} + 2 \beta_{13} - \beta_{12} - 2 \beta_{11} - 3 \beta_{10} - 4 \beta_{9} + 3 \beta_{8} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} - 7 \beta_{12} + \beta_{11} + 5 \beta_{10} - \beta_{9} + \cdots - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - \beta_{15} + 10 \beta_{14} + 5 \beta_{13} - \beta_{12} - 4 \beta_{11} - \beta_{10} - 24 \beta_{9} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} - 2 \beta_{12} + \beta_{11} + 8 \beta_{10} - 3 \beta_{9} + \cdots - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9 \beta_{15} - 3 \beta_{14} - 11 \beta_{13} - 22 \beta_{12} - 30 \beta_{11} + 26 \beta_{10} - 4 \beta_{9} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19 \beta_{15} + 4 \beta_{14} - 5 \beta_{13} - 19 \beta_{12} - 12 \beta_{11} + 3 \beta_{10} - 22 \beta_{9} + \cdots + 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 22 \beta_{15} - \beta_{14} - 8 \beta_{13} + 19 \beta_{12} - 36 \beta_{11} - 41 \beta_{10} + 16 \beta_{9} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 58 \beta_{15} - 34 \beta_{14} - 26 \beta_{13} - 75 \beta_{12} - 23 \beta_{11} + 19 \beta_{10} + \cdots + 53 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 101 \beta_{15} - 32 \beta_{14} - 97 \beta_{13} - 29 \beta_{12} - 60 \beta_{11} - 99 \beta_{10} + \cdots + 67 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 69 \beta_{15} + 74 \beta_{14} + 51 \beta_{13} + 16 \beta_{12} + 11 \beta_{11} - 190 \beta_{10} + \cdots - 26 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(\beta_{10}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
1.38652 − 0.278517i
−0.296075 − 1.38287i
−0.966675 − 1.03225i
−1.39563 + 0.228522i
1.32070 + 0.505727i
1.21331 − 0.726558i
−0.530822 + 1.31081i
1.26868 − 0.624862i
1.38652 + 0.278517i
−0.296075 + 1.38287i
−0.966675 + 1.03225i
−1.39563 − 0.228522i
1.32070 − 0.505727i
1.21331 + 0.726558i
−0.530822 − 1.31081i
1.26868 + 0.624862i
−1.40727 + 0.139945i 2.32624 − 2.32624i 1.96083 − 0.393883i 0 −2.94811 + 3.59920i 0.982011i −2.70430 + 0.828709i − 7.82281i 0
101.2 −1.09971 − 0.889181i 0.120009 − 0.120009i 0.418713 + 1.95568i 0 −0.238684 + 0.0252650i 2.66881i 1.27849 − 2.52299i 2.97120i 0
101.3 −0.562546 + 1.29751i −0.209571 + 0.209571i −1.36708 − 1.45982i 0 −0.154028 − 0.389815i 1.73696i 2.66319 − 0.952595i 2.91216i 0
101.4 −0.114638 − 1.40956i −1.42313 + 1.42313i −1.97372 + 0.323179i 0 2.16913 + 1.84284i − 0.690576i 0.681804 + 2.74502i − 1.05061i 0
101.5 0.257150 − 1.39064i 1.66366 − 1.66366i −1.86775 − 0.715205i 0 −1.88574 − 2.74137i − 2.89402i −1.47488 + 2.41345i − 2.53555i 0
101.6 0.376912 + 1.36306i −1.82762 + 1.82762i −1.71587 + 1.02751i 0 −3.18001 − 1.80230i − 4.50961i −2.04729 − 1.95156i − 3.68037i 0
101.7 1.17275 − 0.790349i −1.37027 + 1.37027i 0.750696 − 1.85377i 0 −0.523995 + 2.68998i 2.73482i −0.584744 − 2.76732i − 0.755274i 0
101.8 1.37735 − 0.320793i 0.720673 − 0.720673i 1.79418 − 0.883688i 0 0.761432 − 1.22381i − 4.02840i 2.18774 − 1.79271i 1.96126i 0
301.1 −1.40727 − 0.139945i 2.32624 + 2.32624i 1.96083 + 0.393883i 0 −2.94811 − 3.59920i − 0.982011i −2.70430 − 0.828709i 7.82281i 0
301.2 −1.09971 + 0.889181i 0.120009 + 0.120009i 0.418713 − 1.95568i 0 −0.238684 − 0.0252650i − 2.66881i 1.27849 + 2.52299i − 2.97120i 0
301.3 −0.562546 − 1.29751i −0.209571 − 0.209571i −1.36708 + 1.45982i 0 −0.154028 + 0.389815i − 1.73696i 2.66319 + 0.952595i − 2.91216i 0
301.4 −0.114638 + 1.40956i −1.42313 − 1.42313i −1.97372 − 0.323179i 0 2.16913 − 1.84284i 0.690576i 0.681804 − 2.74502i 1.05061i 0
301.5 0.257150 + 1.39064i 1.66366 + 1.66366i −1.86775 + 0.715205i 0 −1.88574 + 2.74137i 2.89402i −1.47488 − 2.41345i 2.53555i 0
301.6 0.376912 − 1.36306i −1.82762 − 1.82762i −1.71587 − 1.02751i 0 −3.18001 + 1.80230i 4.50961i −2.04729 + 1.95156i 3.68037i 0
301.7 1.17275 + 0.790349i −1.37027 − 1.37027i 0.750696 + 1.85377i 0 −0.523995 − 2.68998i − 2.73482i −0.584744 + 2.76732i 0.755274i 0
301.8 1.37735 + 0.320793i 0.720673 + 0.720673i 1.79418 + 0.883688i 0 0.761432 + 1.22381i 4.02840i 2.18774 + 1.79271i − 1.96126i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.l.h 16
4.b odd 2 1 1600.2.l.i 16
5.b even 2 1 80.2.l.a ✓ 16
5.c odd 4 1 400.2.q.g 16
5.c odd 4 1 400.2.q.h 16
15.d odd 2 1 720.2.t.c 16
16.e even 4 1 inner 400.2.l.h 16
16.f odd 4 1 1600.2.l.i 16
20.d odd 2 1 320.2.l.a 16
20.e even 4 1 1600.2.q.g 16
20.e even 4 1 1600.2.q.h 16
40.e odd 2 1 640.2.l.a 16
40.f even 2 1 640.2.l.b 16
60.h even 2 1 2880.2.t.c 16
80.i odd 4 1 400.2.q.h 16
80.j even 4 1 1600.2.q.h 16
80.k odd 4 1 320.2.l.a 16
80.k odd 4 1 640.2.l.a 16
80.q even 4 1 80.2.l.a ✓ 16
80.q even 4 1 640.2.l.b 16
80.s even 4 1 1600.2.q.g 16
80.t odd 4 1 400.2.q.g 16
160.y odd 8 1 5120.2.a.t 8
160.y odd 8 1 5120.2.a.u 8
160.z even 8 1 5120.2.a.s 8
160.z even 8 1 5120.2.a.v 8
240.t even 4 1 2880.2.t.c 16
240.bm odd 4 1 720.2.t.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.l.a ✓ 16 5.b even 2 1
80.2.l.a ✓ 16 80.q even 4 1
320.2.l.a 16 20.d odd 2 1
320.2.l.a 16 80.k odd 4 1
400.2.l.h 16 1.a even 1 1 trivial
400.2.l.h 16 16.e even 4 1 inner
400.2.q.g 16 5.c odd 4 1
400.2.q.g 16 80.t odd 4 1
400.2.q.h 16 5.c odd 4 1
400.2.q.h 16 80.i odd 4 1
640.2.l.a 16 40.e odd 2 1
640.2.l.a 16 80.k odd 4 1
640.2.l.b 16 40.f even 2 1
640.2.l.b 16 80.q even 4 1
720.2.t.c 16 15.d odd 2 1
720.2.t.c 16 240.bm odd 4 1
1600.2.l.i 16 4.b odd 2 1
1600.2.l.i 16 16.f odd 4 1
1600.2.q.g 16 20.e even 4 1
1600.2.q.g 16 80.s even 4 1
1600.2.q.h 16 20.e even 4 1
1600.2.q.h 16 80.j even 4 1
2880.2.t.c 16 60.h even 2 1
2880.2.t.c 16 240.t even 4 1
5120.2.a.s 8 160.z even 8 1
5120.2.a.t 8 160.y odd 8 1
5120.2.a.u 8 160.y odd 8 1
5120.2.a.v 8 160.z even 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{16} + 8 T_{3}^{13} + 112 T_{3}^{12} + 80 T_{3}^{11} + 32 T_{3}^{10} + 176 T_{3}^{9} + 2632 T_{3}^{8} + \cdots + 16 \) Copy content Toggle raw display
\( T_{7}^{16} + 64 T_{7}^{14} + 1616 T_{7}^{12} + 20736 T_{7}^{10} + 145224 T_{7}^{8} + 549632 T_{7}^{6} + \cdots + 204304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{13} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 64 T^{14} + \cdots + 204304 \) Copy content Toggle raw display
$11$ \( T^{16} + 8 T^{15} + \cdots + 1290496 \) Copy content Toggle raw display
$13$ \( T^{16} - 128 T^{13} + \cdots + 20647936 \) Copy content Toggle raw display
$17$ \( (T^{8} - 72 T^{6} + \cdots + 13888)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 8 T^{15} + \cdots + 614656 \) Copy content Toggle raw display
$23$ \( T^{16} + 128 T^{14} + \cdots + 1731856 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 3017085184 \) Copy content Toggle raw display
$31$ \( (T^{8} - 96 T^{6} + \cdots - 20224)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 16 T^{15} + \cdots + 18939904 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 110660014336 \) Copy content Toggle raw display
$43$ \( T^{16} + 8 T^{15} + \cdots + 53640976 \) Copy content Toggle raw display
$47$ \( (T^{8} - 20 T^{7} + \cdots + 575044)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 383725735936 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 12227051776 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1393986371584 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 46120451769616 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 3333516427264 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 15847788544 \) Copy content Toggle raw display
$79$ \( (T^{8} - 8 T^{7} + \cdots + 4352)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 2050640656 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 684153962496 \) Copy content Toggle raw display
$97$ \( (T^{8} - 440 T^{6} + \cdots - 8549312)^{2} \) Copy content Toggle raw display
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