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Properties

Label 400.2.q.h
Level $400$
Weight $2$
Character orbit 400.q
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(149,400)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.q (of order \(4\), 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: \(3.19401608085\)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{11} + \beta_{6}) q^{3} + ( - \beta_{14} - \beta_{12} - \beta_{2}) q^{4} + (\beta_{12} + \beta_{11} - \beta_{8} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{15} - 2 \beta_{13} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{11} + \beta_{6}) q^{3} + ( - \beta_{14} - \beta_{12} - \beta_{2}) q^{4} + (\beta_{12} + \beta_{11} - \beta_{8} + \cdots - 1) q^{6}+ \cdots + (2 \beta_{15} + 3 \beta_{13} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 4 q^{4} - 12 q^{6} + 8 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 4 q^{4} - 12 q^{6} + 8 q^{7} - 8 q^{8} - 8 q^{11} + 20 q^{12} - 4 q^{14} + 16 q^{16} + 12 q^{18} + 8 q^{19} - 20 q^{22} + 24 q^{23} - 8 q^{24} - 16 q^{26} + 24 q^{27} + 20 q^{28} + 16 q^{29} + 24 q^{32} - 16 q^{34} - 4 q^{36} - 16 q^{37} - 20 q^{38} - 20 q^{42} - 8 q^{43} - 40 q^{44} - 4 q^{46} + 16 q^{49} - 32 q^{51} - 16 q^{52} - 16 q^{53} - 32 q^{54} + 16 q^{56} - 28 q^{58} + 8 q^{59} + 16 q^{61} + 48 q^{62} + 16 q^{64} - 40 q^{67} - 8 q^{68} - 16 q^{69} - 96 q^{72} + 72 q^{74} - 16 q^{77} + 24 q^{78} - 16 q^{79} - 16 q^{81} + 44 q^{82} + 40 q^{83} + 64 q^{84} + 28 q^{86} - 16 q^{88} + 32 q^{91} + 20 q^{92} - 48 q^{93} + 36 q^{94} + 8 q^{96} - 32 q^{98} + 8 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 39 \nu^{15} - 74 \nu^{14} - 66 \nu^{13} - 16 \nu^{12} + 337 \nu^{11} + 454 \nu^{10} - 654 \nu^{9} + \cdots + 256 ) / 448 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 255 \nu^{15} - 724 \nu^{14} + 52 \nu^{13} + 184 \nu^{12} + 2281 \nu^{11} + 960 \nu^{10} + \cdots - 45056 ) / 2688 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 174 \nu^{15} + 937 \nu^{14} - 622 \nu^{13} - 928 \nu^{12} - 2266 \nu^{11} + 2343 \nu^{10} + \cdots + 83840 ) / 1344 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 139 \nu^{15} - 300 \nu^{14} - 92 \nu^{13} - 24 \nu^{12} + 1069 \nu^{11} + 968 \nu^{10} - 2668 \nu^{9} + \cdots - 13056 ) / 896 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20 \nu^{15} - 20 \nu^{14} - 41 \nu^{13} - 44 \nu^{12} + 108 \nu^{11} + 276 \nu^{10} - 95 \nu^{9} + \cdots + 576 ) / 96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 215 \nu^{15} + 572 \nu^{14} - 40 \nu^{13} - 176 \nu^{12} - 1809 \nu^{11} - 648 \nu^{10} + \cdots + 34176 ) / 896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 703 \nu^{15} + 1200 \nu^{14} + 956 \nu^{13} + 656 \nu^{12} - 5081 \nu^{11} - 7260 \nu^{10} + \cdots + 24448 ) / 2688 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} + \cdots - 83072 ) / 1344 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 536 \nu^{15} - 1145 \nu^{14} - 374 \nu^{13} - 128 \nu^{12} + 4152 \nu^{11} + 4017 \nu^{10} + \cdots - 45888 ) / 1344 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 99 \nu^{15} + 299 \nu^{14} - 68 \nu^{13} - 116 \nu^{12} - 869 \nu^{11} - 111 \nu^{10} + \cdots + 20992 ) / 192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 251 \nu^{15} - 706 \nu^{14} + 84 \nu^{13} + 216 \nu^{12} + 2141 \nu^{11} + 654 \nu^{10} - 6740 \nu^{9} + \cdots - 45952 ) / 384 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 257 \nu^{15} - 786 \nu^{14} + 176 \nu^{13} + 344 \nu^{12} + 2311 \nu^{11} + 246 \nu^{10} + \cdots - 53120 ) / 384 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1791 \nu^{15} + 4570 \nu^{14} + 260 \nu^{13} - 760 \nu^{12} - 15097 \nu^{11} - 8598 \nu^{10} + \cdots + 255872 ) / 2688 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 964 \nu^{15} + 2301 \nu^{14} + 170 \nu^{13} - 148 \nu^{12} - 7556 \nu^{11} - 4869 \nu^{10} + \cdots + 127360 ) / 1344 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 999 \nu^{15} - 2518 \nu^{14} - 128 \nu^{13} + 400 \nu^{12} + 8305 \nu^{11} + 4722 \nu^{10} + \cdots - 137216 ) / 1344 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{10} - \beta_{8} + \beta_{5} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{14} + \beta_{13} - \beta_{12} - 2\beta_{10} + \beta_{9} - 2\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{15} + \beta_{14} + 2\beta_{12} - \beta_{9} - 4\beta_{8} + 3\beta_{6} + \beta_{5} + \beta_{4} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{14} + 3 \beta_{13} - \beta_{12} - \beta_{11} - 3 \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + \beta_{10} + 4 \beta_{9} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6 \beta_{15} + 8 \beta_{14} + 2 \beta_{13} + 4 \beta_{12} + \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + \cdots - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{14} + 3 \beta_{13} + 4 \beta_{12} + 7 \beta_{11} + 13 \beta_{10} - \beta_{9} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 14 \beta_{15} + 5 \beta_{14} + 7 \beta_{13} - 11 \beta_{12} + 6 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + \cdots - 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 14 \beta_{15} + 15 \beta_{14} - 28 \beta_{13} + 26 \beta_{12} + 2 \beta_{11} + 26 \beta_{10} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2 \beta_{15} - 9 \beta_{14} - 15 \beta_{13} - 3 \beta_{12} - 3 \beta_{11} + 9 \beta_{10} - 39 \beta_{9} + \cdots + 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 27 \beta_{15} - 41 \beta_{13} + 26 \beta_{12} - 5 \beta_{11} - 7 \beta_{10} + 12 \beta_{9} - 104 \beta_{8} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 66 \beta_{15} - 6 \beta_{14} - 64 \beta_{13} + 26 \beta_{12} - 41 \beta_{11} - 51 \beta_{10} + \cdots + 53 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 99 \beta_{15} - 23 \beta_{14} - 111 \beta_{13} + 44 \beta_{12} - 51 \beta_{11} + 9 \beta_{10} + \cdots + 67 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 48 \beta_{15} - 69 \beta_{14} - 45 \beta_{13} + 53 \beta_{12} - 56 \beta_{11} - 110 \beta_{10} + \cdots - 26 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(-\beta_{8}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.21331 + 0.726558i
−0.966675 + 1.03225i
1.38652 + 0.278517i
1.26868 + 0.624862i
−0.530822 − 1.31081i
−0.296075 + 1.38287i
1.32070 − 0.505727i
−1.39563 − 0.228522i
1.21331 − 0.726558i
−0.966675 − 1.03225i
1.38652 − 0.278517i
1.26868 − 0.624862i
−0.530822 + 1.31081i
−0.296075 − 1.38287i
1.32070 + 0.505727i
−1.39563 + 0.228522i
−1.36306 + 0.376912i 1.82762 + 1.82762i 1.71587 − 1.02751i 0 −3.18001 − 1.80230i 4.50961 −1.95156 + 2.04729i 3.68037i 0
149.2 −1.29751 − 0.562546i 0.209571 + 0.209571i 1.36708 + 1.45982i 0 −0.154028 − 0.389815i −1.73696 −0.952595 − 2.66319i − 2.91216i 0
149.3 −0.139945 − 1.40727i −2.32624 − 2.32624i −1.96083 + 0.393883i 0 −2.94811 + 3.59920i −0.982011 0.828709 + 2.70430i 7.82281i 0
149.4 0.320793 + 1.37735i −0.720673 − 0.720673i −1.79418 + 0.883688i 0 0.761432 − 1.22381i 4.02840 −1.79271 − 2.18774i − 1.96126i 0
149.5 0.790349 + 1.17275i 1.37027 + 1.37027i −0.750696 + 1.85377i 0 −0.523995 + 2.68998i −2.73482 −2.76732 + 0.584744i 0.755274i 0
149.6 0.889181 − 1.09971i −0.120009 − 0.120009i −0.418713 − 1.95568i 0 −0.238684 + 0.0252650i −2.66881 −2.52299 − 1.27849i − 2.97120i 0
149.7 1.39064 + 0.257150i −1.66366 − 1.66366i 1.86775 + 0.715205i 0 −1.88574 − 2.74137i 2.89402 2.41345 + 1.47488i 2.53555i 0
149.8 1.40956 − 0.114638i 1.42313 + 1.42313i 1.97372 − 0.323179i 0 2.16913 + 1.84284i 0.690576 2.74502 − 0.681804i 1.05061i 0
349.1 −1.36306 − 0.376912i 1.82762 − 1.82762i 1.71587 + 1.02751i 0 −3.18001 + 1.80230i 4.50961 −1.95156 − 2.04729i − 3.68037i 0
349.2 −1.29751 + 0.562546i 0.209571 − 0.209571i 1.36708 − 1.45982i 0 −0.154028 + 0.389815i −1.73696 −0.952595 + 2.66319i 2.91216i 0
349.3 −0.139945 + 1.40727i −2.32624 + 2.32624i −1.96083 − 0.393883i 0 −2.94811 − 3.59920i −0.982011 0.828709 − 2.70430i − 7.82281i 0
349.4 0.320793 − 1.37735i −0.720673 + 0.720673i −1.79418 − 0.883688i 0 0.761432 + 1.22381i 4.02840 −1.79271 + 2.18774i 1.96126i 0
349.5 0.790349 − 1.17275i 1.37027 − 1.37027i −0.750696 − 1.85377i 0 −0.523995 − 2.68998i −2.73482 −2.76732 − 0.584744i − 0.755274i 0
349.6 0.889181 + 1.09971i −0.120009 + 0.120009i −0.418713 + 1.95568i 0 −0.238684 − 0.0252650i −2.66881 −2.52299 + 1.27849i 2.97120i 0
349.7 1.39064 − 0.257150i −1.66366 + 1.66366i 1.86775 − 0.715205i 0 −1.88574 + 2.74137i 2.89402 2.41345 − 1.47488i − 2.53555i 0
349.8 1.40956 + 0.114638i 1.42313 − 1.42313i 1.97372 + 0.323179i 0 2.16913 − 1.84284i 0.690576 2.74502 + 0.681804i − 1.05061i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.q 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.q.h 16
4.b odd 2 1 1600.2.q.g 16
5.b even 2 1 400.2.q.g 16
5.c odd 4 1 80.2.l.a ✓ 16
5.c odd 4 1 400.2.l.h 16
15.e even 4 1 720.2.t.c 16
16.e even 4 1 400.2.q.g 16
16.f odd 4 1 1600.2.q.h 16
20.d odd 2 1 1600.2.q.h 16
20.e even 4 1 320.2.l.a 16
20.e even 4 1 1600.2.l.i 16
40.i odd 4 1 640.2.l.b 16
40.k even 4 1 640.2.l.a 16
60.l odd 4 1 2880.2.t.c 16
80.i odd 4 1 400.2.l.h 16
80.i odd 4 1 640.2.l.b 16
80.j even 4 1 320.2.l.a 16
80.k odd 4 1 1600.2.q.g 16
80.q even 4 1 inner 400.2.q.h 16
80.s even 4 1 640.2.l.a 16
80.s even 4 1 1600.2.l.i 16
80.t odd 4 1 80.2.l.a ✓ 16
160.u even 8 1 5120.2.a.t 8
160.u even 8 1 5120.2.a.u 8
160.bb odd 8 1 5120.2.a.s 8
160.bb odd 8 1 5120.2.a.v 8
240.bd odd 4 1 2880.2.t.c 16
240.bf even 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.c odd 4 1
80.2.l.a ✓ 16 80.t odd 4 1
320.2.l.a 16 20.e even 4 1
320.2.l.a 16 80.j even 4 1
400.2.l.h 16 5.c odd 4 1
400.2.l.h 16 80.i odd 4 1
400.2.q.g 16 5.b even 2 1
400.2.q.g 16 16.e even 4 1
400.2.q.h 16 1.a even 1 1 trivial
400.2.q.h 16 80.q even 4 1 inner
640.2.l.a 16 40.k even 4 1
640.2.l.a 16 80.s even 4 1
640.2.l.b 16 40.i odd 4 1
640.2.l.b 16 80.i odd 4 1
720.2.t.c 16 15.e even 4 1
720.2.t.c 16 240.bf even 4 1
1600.2.l.i 16 20.e even 4 1
1600.2.l.i 16 80.s even 4 1
1600.2.q.g 16 4.b odd 2 1
1600.2.q.g 16 80.k odd 4 1
1600.2.q.h 16 16.f odd 4 1
1600.2.q.h 16 20.d odd 2 1
2880.2.t.c 16 60.l odd 4 1
2880.2.t.c 16 240.bd odd 4 1
5120.2.a.s 8 160.bb odd 8 1
5120.2.a.t 8 160.u even 8 1
5120.2.a.u 8 160.u even 8 1
5120.2.a.v 8 160.bb odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 8 T_{3}^{13} + 112 T_{3}^{12} - 80 T_{3}^{11} + 32 T_{3}^{10} - 176 T_{3}^{9} + 2632 T_{3}^{8} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4 T^{15} + \cdots + 256 \) 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
$7$ \( (T^{8} - 4 T^{7} + \cdots + 452)^{2} \) 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
$17$ \( T^{16} + \cdots + 192876544 \) Copy content Toggle raw display
$19$ \( T^{16} - 8 T^{15} + \cdots + 614656 \) Copy content Toggle raw display
$23$ \( (T^{8} - 12 T^{7} + \cdots + 1316)^{2} \) 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
$47$ \( T^{16} + \cdots + 330675601936 \) 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
$73$ \( (T^{8} - 280 T^{6} + \cdots - 125888)^{2} \) 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
$97$ \( T^{16} + \cdots + 73090735673344 \) Copy content Toggle raw display
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