Properties

Label 750.2.h.d
Level $750$
Weight $2$
Character orbit 750.h
Analytic conductor $5.989$
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] = [750,2,Mod(49,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("750.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.h (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: \(5.98878015160\)
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: \( x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + 6259 x^{8} - 11958 x^{7} - 15752 x^{6} + 14670 x^{5} + 18271 x^{4} + \cdots + 11105 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 150)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{3} q^{3} + \beta_{8} q^{4} + \beta_{2} q^{6} + ( - \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} - \beta_{11} q^{8} + ( - \beta_{8} + \beta_{7} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + \beta_{3} q^{3} + \beta_{8} q^{4} + \beta_{2} q^{6} + ( - \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} - \beta_{11} q^{8} + ( - \beta_{8} + \beta_{7} + \beta_{2} + 1) q^{9} + (\beta_{15} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{11} - \beta_1 q^{12} + ( - \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{5} - \beta_{3} + \cdots - 2) q^{13}+ \cdots + ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 4 q^{6} + 4 q^{9} + 2 q^{11} - 20 q^{13} + 2 q^{14} - 4 q^{16} + 30 q^{17} - 2 q^{21} + 20 q^{22} + 10 q^{23} - 16 q^{24} + 4 q^{26} - 10 q^{29} - 18 q^{31} + 20 q^{33} + 12 q^{34} - 4 q^{36} - 20 q^{37} - 10 q^{38} - 4 q^{39} + 22 q^{41} + 8 q^{44} - 6 q^{46} + 50 q^{47} - 52 q^{49} + 28 q^{51} - 20 q^{52} - 30 q^{53} + 4 q^{54} - 2 q^{56} + 30 q^{58} + 20 q^{59} + 12 q^{61} - 50 q^{62} - 10 q^{63} + 4 q^{64} + 2 q^{66} + 50 q^{67} + 6 q^{69} - 28 q^{71} - 20 q^{73} + 12 q^{74} + 20 q^{76} - 100 q^{77} - 20 q^{79} - 4 q^{81} + 30 q^{83} + 2 q^{84} - 6 q^{86} - 10 q^{87} + 70 q^{89} + 12 q^{91} + 30 q^{92} + 2 q^{94} - 4 q^{96} + 10 q^{97} - 60 q^{98} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + 6259 x^{8} - 11958 x^{7} - 15752 x^{6} + 14670 x^{5} + 18271 x^{4} + \cdots + 11105 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 21\!\cdots\!17 \nu^{15} + \cdots - 36\!\cdots\!85 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!46 \nu^{15} + \cdots - 21\!\cdots\!00 ) / 12\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\!\cdots\!03 \nu^{15} + \cdots + 54\!\cdots\!40 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34\!\cdots\!64 \nu^{15} + \cdots - 82\!\cdots\!95 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38\!\cdots\!68 \nu^{15} + \cdots - 73\!\cdots\!90 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\!\cdots\!47 \nu^{15} + \cdots + 35\!\cdots\!10 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!10 \nu^{15} + \cdots + 26\!\cdots\!20 ) / 12\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!10 \nu^{15} + \cdots - 10\!\cdots\!70 ) / 12\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 52\!\cdots\!43 \nu^{15} + \cdots + 85\!\cdots\!40 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 33\!\cdots\!69 \nu^{15} + \cdots + 23\!\cdots\!95 ) / 15\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 65\!\cdots\!33 \nu^{15} + \cdots - 61\!\cdots\!15 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66\!\cdots\!11 \nu^{15} + \cdots - 11\!\cdots\!05 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 86\!\cdots\!62 \nu^{15} + \cdots + 27\!\cdots\!60 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!63 \nu^{15} + \cdots + 11\!\cdots\!40 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12\!\cdots\!49 \nu^{15} + \cdots + 20\!\cdots\!20 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} + 4 \beta_{12} - \beta_{11} + 3 \beta_{10} + 3 \beta_{8} + 2 \beta_{7} - 2 \beta_{4} - \beta_{3} - \beta _1 - 1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 5 \beta_{15} + 2 \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 4 \beta_{10} - \beta_{9} + 7 \beta_{8} - \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + 7 \beta _1 + 16 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{15} + \beta_{14} - 8 \beta_{13} + 25 \beta_{12} - 26 \beta_{11} + 17 \beta_{10} - 12 \beta_{9} + 53 \beta_{8} + 14 \beta_{7} + 23 \beta_{6} + 6 \beta_{5} - 8 \beta_{4} - 12 \beta_{3} - 3 \beta_{2} - 14 \beta _1 + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 29 \beta_{15} + 18 \beta_{14} - 23 \beta_{13} + 17 \beta_{12} - 89 \beta_{11} + 36 \beta_{10} - 27 \beta_{9} + 107 \beta_{8} - 50 \beta_{7} + 8 \beta_{6} - 4 \beta_{5} - 29 \beta_{4} + 35 \beta_{3} - 83 \beta_{2} + 93 \beta _1 + 66 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 81 \beta_{15} - 6 \beta_{14} - 140 \beta_{13} + 179 \beta_{12} - 356 \beta_{11} + 173 \beta_{10} - 140 \beta_{9} + 593 \beta_{8} + 7 \beta_{7} + 285 \beta_{6} + 80 \beta_{5} - 77 \beta_{4} - 26 \beta_{3} - 55 \beta_{2} - 16 \beta _1 + 69 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 25 \beta_{15} + 57 \beta_{14} - 384 \beta_{13} + 339 \beta_{12} - 1679 \beta_{11} + 434 \beta_{10} - 456 \beta_{9} + 1647 \beta_{8} - 996 \beta_{7} - 176 \beta_{6} + 63 \beta_{5} - 406 \beta_{4} + 398 \beta_{3} - 1254 \beta_{2} + \cdots - 104 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 853 \beta_{15} - 209 \beta_{14} - 1803 \beta_{13} + 1400 \beta_{12} - 4661 \beta_{11} + 1877 \beta_{10} - 1652 \beta_{9} + 6368 \beta_{8} - 1981 \beta_{7} + 2448 \beta_{6} + 1001 \beta_{5} - 1078 \beta_{4} + 1483 \beta_{3} + \cdots + 659 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2371 \beta_{15} - 1092 \beta_{14} - 4833 \beta_{13} + 4697 \beta_{12} - 22869 \beta_{11} + 5536 \beta_{10} - 6217 \beta_{9} + 22292 \beta_{8} - 14235 \beta_{7} - 3012 \beta_{6} + 1906 \beta_{5} - 5419 \beta_{4} + \cdots - 6464 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8816 \beta_{15} - 4051 \beta_{14} - 19930 \beta_{13} + 10989 \beta_{12} - 61441 \beta_{11} + 20443 \beta_{10} - 19660 \beta_{9} + 69933 \beta_{8} - 43763 \beta_{7} + 13790 \beta_{6} + 11550 \beta_{5} + \cdots + 1354 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 44065 \beta_{15} - 27243 \beta_{14} - 55419 \beta_{13} + 51794 \beta_{12} - 277069 \beta_{11} + 67399 \beta_{10} - 76546 \beta_{9} + 273867 \beta_{8} - 186976 \beta_{7} - 34036 \beta_{6} + \cdots - 101659 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 94563 \beta_{15} - 70294 \beta_{14} - 201808 \beta_{13} + 80655 \beta_{12} - 797386 \beta_{11} + 220777 \beta_{10} - 232477 \beta_{9} + 792838 \beta_{8} - 692681 \beta_{7} - 8057 \beta_{6} + \cdots - 106706 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 564651 \beta_{15} - 418752 \beta_{14} - 593638 \beta_{13} + 457702 \beta_{12} - 3193009 \beta_{11} + 769981 \beta_{10} - 885537 \beta_{9} + 3144842 \beta_{8} - 2414770 \beta_{7} + \cdots - 1309799 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1081916 \beta_{15} - 1118506 \beta_{14} - 1915745 \beta_{13} + 467779 \beta_{12} - 9996001 \beta_{11} + 2345053 \beta_{10} - 2702835 \beta_{9} + 9072163 \beta_{8} - 9531453 \beta_{7} + \cdots - 2885261 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 6140735 \beta_{15} - 5535813 \beta_{14} - 5892454 \beta_{13} + 2882489 \beta_{12} - 35942739 \beta_{11} + 8257539 \beta_{10} - 9796001 \beta_{9} + 34294982 \beta_{8} + \cdots - 16212534 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 12746988 \beta_{15} - 16418039 \beta_{14} - 16898518 \beta_{13} + 61645 \beta_{12} - 119837181 \beta_{11} + 24216397 \beta_{10} - 30549527 \beta_{9} + 101919263 \beta_{8} + \cdots - 51277636 ) / 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(\beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.543374 + 0.809017i
−1.16141 + 0.809017i
2.17199 0.809017i
−2.79002 0.809017i
−1.16141 0.809017i
0.543374 0.809017i
−2.79002 + 0.809017i
2.17199 + 0.809017i
2.32349 + 0.309017i
−0.705457 + 0.309017i
3.42137 0.309017i
−1.80334 0.309017i
−0.705457 0.309017i
2.32349 0.309017i
−1.80334 + 0.309017i
3.42137 + 0.309017i
−0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 2.70913i 0.951057 + 0.309017i 0.809017 0.587785i 0
49.2 −0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 0.533559i 0.951057 + 0.309017i 0.809017 0.587785i 0
49.3 0.587785 0.809017i 0.951057 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 4.80694i −0.951057 0.309017i 0.809017 0.587785i 0
49.4 0.587785 0.809017i 0.951057 0.309017i −0.309017 0.951057i 0 0.309017 0.951057i 4.63137i −0.951057 0.309017i 0.809017 0.587785i 0
199.1 −0.587785 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 0.533559i 0.951057 0.309017i 0.809017 + 0.587785i 0
199.2 −0.587785 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 2.70913i 0.951057 0.309017i 0.809017 + 0.587785i 0
199.3 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 4.63137i −0.951057 + 0.309017i 0.809017 + 0.587785i 0
199.4 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0 0.309017 + 0.951057i 4.80694i −0.951057 + 0.309017i 0.809017 + 0.587785i 0
349.1 −0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 0.329315i −0.587785 + 0.809017i −0.309017 + 0.951057i 0
349.2 −0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 3.23143i −0.587785 + 0.809017i −0.309017 + 0.951057i 0
349.3 0.951057 0.309017i −0.587785 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 2.61995i 0.587785 0.809017i −0.309017 + 0.951057i 0
349.4 0.951057 0.309017i −0.587785 0.809017i 0.809017 0.587785i 0 −0.809017 0.587785i 3.52206i 0.587785 0.809017i −0.309017 + 0.951057i 0
649.1 −0.951057 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 3.23143i −0.587785 0.809017i −0.309017 0.951057i 0
649.2 −0.951057 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 0.329315i −0.587785 0.809017i −0.309017 0.951057i 0
649.3 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 3.52206i 0.587785 + 0.809017i −0.309017 0.951057i 0
649.4 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0 −0.809017 + 0.587785i 2.61995i 0.587785 + 0.809017i −0.309017 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.h.d 16
5.b even 2 1 150.2.h.b 16
5.c odd 4 1 750.2.g.f 16
5.c odd 4 1 750.2.g.g 16
15.d odd 2 1 450.2.l.c 16
25.d even 5 1 150.2.h.b 16
25.d even 5 1 3750.2.c.k 16
25.e even 10 1 inner 750.2.h.d 16
25.e even 10 1 3750.2.c.k 16
25.f odd 20 1 750.2.g.f 16
25.f odd 20 1 750.2.g.g 16
25.f odd 20 1 3750.2.a.u 8
25.f odd 20 1 3750.2.a.v 8
75.j odd 10 1 450.2.l.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.b 16 5.b even 2 1
150.2.h.b 16 25.d even 5 1
450.2.l.c 16 15.d odd 2 1
450.2.l.c 16 75.j odd 10 1
750.2.g.f 16 5.c odd 4 1
750.2.g.f 16 25.f odd 20 1
750.2.g.g 16 5.c odd 4 1
750.2.g.g 16 25.f odd 20 1
750.2.h.d 16 1.a even 1 1 trivial
750.2.h.d 16 25.e even 10 1 inner
3750.2.a.u 8 25.f odd 20 1
3750.2.a.v 8 25.f odd 20 1
3750.2.c.k 16 25.d even 5 1
3750.2.c.k 16 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 82 T_{7}^{14} + 2683 T_{7}^{12} + 44874 T_{7}^{10} + 407105 T_{7}^{8} + 1927704 T_{7}^{6} + 3943448 T_{7}^{4} + 1326272 T_{7}^{2} + 99856 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 82 T^{14} + 2683 T^{12} + \cdots + 99856 \) Copy content Toggle raw display
$11$ \( T^{16} - 2 T^{15} + 55 T^{14} + \cdots + 524176 \) Copy content Toggle raw display
$13$ \( T^{16} + 20 T^{15} + 188 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{16} - 30 T^{15} + 432 T^{14} + \cdots + 3748096 \) Copy content Toggle raw display
$19$ \( T^{16} + 100 T^{14} - 220 T^{13} + \cdots + 2560000 \) Copy content Toggle raw display
$23$ \( T^{16} - 10 T^{15} + \cdots + 20533743616 \) Copy content Toggle raw display
$29$ \( T^{16} + 10 T^{15} + 80 T^{14} + \cdots + 40960000 \) Copy content Toggle raw display
$31$ \( T^{16} + 18 T^{15} + \cdots + 205176976 \) Copy content Toggle raw display
$37$ \( T^{16} + 20 T^{15} + 117 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{16} - 22 T^{15} + \cdots + 78050949376 \) Copy content Toggle raw display
$43$ \( T^{16} + 328 T^{14} + \cdots + 15083769856 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 172199901270016 \) Copy content Toggle raw display
$53$ \( T^{16} + 30 T^{15} + \cdots + 111534721 \) Copy content Toggle raw display
$59$ \( T^{16} - 20 T^{15} + \cdots + 1600000000 \) Copy content Toggle raw display
$61$ \( T^{16} - 12 T^{15} + 140 T^{14} + \cdots + 10137856 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 201983672713216 \) Copy content Toggle raw display
$71$ \( T^{16} + 28 T^{15} + \cdots + 3398330023936 \) Copy content Toggle raw display
$73$ \( T^{16} + 20 T^{15} + \cdots + 4778526048256 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 110872476160000 \) Copy content Toggle raw display
$83$ \( T^{16} - 30 T^{15} + \cdots + 25573127056 \) Copy content Toggle raw display
$89$ \( T^{16} - 70 T^{15} + \cdots + 36100000000 \) Copy content Toggle raw display
$97$ \( T^{16} - 10 T^{15} + \cdots + 91700905971481 \) Copy content Toggle raw display
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