Properties

Label 3009.2.a.f
Level $3009$
Weight $2$
Character orbit 3009.a
Self dual yes
Analytic conductor $24.027$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3009,2,Mod(1,3009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3009 = 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3009.a (trivial)

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: yes
Analytic conductor: \(24.0269859682\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 13 x^{14} + 65 x^{13} + 49 x^{12} - 403 x^{11} + 11 x^{10} + 1205 x^{9} - 452 x^{8} - 1783 x^{7} + 946 x^{6} + 1190 x^{5} - 645 x^{4} - 299 x^{3} + 119 x^{2} + \cdots - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{9} q^{5} + \beta_1 q^{6} - \beta_{14} q^{7} + ( - \beta_{3} - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{9} q^{5} + \beta_1 q^{6} - \beta_{14} q^{7} + ( - \beta_{3} - \beta_{2} - 1) q^{8} + q^{9} + ( - \beta_{8} - \beta_{7} + \beta_1 - 1) q^{10} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - \beta_{4} - \beta_1 - 1) q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{13} - \beta_{10} - \beta_{7} + 1) q^{13} + ( - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{3} + \beta_{2}) q^{14} + \beta_{9} q^{15} + ( - \beta_{15} + \beta_{13} - \beta_{12} - \beta_{9} + \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{16} + q^{17} - \beta_1 q^{18} + (\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{4} + \beta_1 - 2) q^{19} + (\beta_{14} + \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{20} + \beta_{14} q^{21} + (\beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \cdots + 2) q^{22}+ \cdots + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - \beta_{4} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} - 16 q^{3} + 10 q^{4} - 3 q^{5} + 4 q^{6} - 3 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} - 16 q^{3} + 10 q^{4} - 3 q^{5} + 4 q^{6} - 3 q^{7} - 9 q^{8} + 16 q^{9} - 7 q^{11} - 10 q^{12} + 7 q^{13} - 13 q^{14} + 3 q^{15} + 2 q^{16} + 16 q^{17} - 4 q^{18} - 19 q^{19} - 17 q^{20} + 3 q^{21} + 23 q^{22} - 16 q^{23} + 9 q^{24} + 5 q^{25} - 11 q^{26} - 16 q^{27} + 8 q^{28} + 4 q^{29} - 15 q^{31} - 22 q^{32} + 7 q^{33} - 4 q^{34} - 23 q^{35} + 10 q^{36} + 20 q^{37} - 17 q^{38} - 7 q^{39} + 21 q^{40} - 4 q^{41} + 13 q^{42} - 12 q^{43} - 19 q^{44} - 3 q^{45} + 24 q^{46} - 36 q^{47} - 2 q^{48} - 11 q^{49} + 9 q^{50} - 16 q^{51} + 6 q^{52} - 32 q^{53} + 4 q^{54} - 29 q^{55} - 11 q^{56} + 19 q^{57} - 33 q^{58} + 16 q^{59} + 17 q^{60} + 11 q^{62} - 3 q^{63} + 9 q^{64} - 5 q^{65} - 23 q^{66} - 30 q^{67} + 10 q^{68} + 16 q^{69} + 16 q^{70} - 70 q^{71} - 9 q^{72} + 21 q^{73} - 29 q^{74} - 5 q^{75} - 8 q^{76} - 13 q^{77} + 11 q^{78} - 25 q^{79} - 9 q^{80} + 16 q^{81} + 22 q^{82} - 23 q^{83} - 8 q^{84} - 3 q^{85} - 58 q^{86} - 4 q^{87} + 41 q^{88} - 31 q^{89} - 36 q^{91} - 26 q^{92} + 15 q^{93} + 9 q^{94} - 10 q^{95} + 22 q^{96} + 52 q^{97} - 36 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 13 x^{14} + 65 x^{13} + 49 x^{12} - 403 x^{11} + 11 x^{10} + 1205 x^{9} - 452 x^{8} - 1783 x^{7} + 946 x^{6} + 1190 x^{5} - 645 x^{4} - 299 x^{3} + 119 x^{2} + \cdots - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 147 \nu^{15} + 1940 \nu^{14} - 11921 \nu^{13} - 20889 \nu^{12} + 156987 \nu^{11} + 46557 \nu^{10} - 831455 \nu^{9} + 165835 \nu^{8} + 2105046 \nu^{7} - 823267 \nu^{6} + \cdots + 46042 ) / 8770 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 186 \nu^{15} - 230 \nu^{14} - 3808 \nu^{13} + 4443 \nu^{12} + 28606 \nu^{11} - 30044 \nu^{10} - 97010 \nu^{9} + 83025 \nu^{8} + 142958 \nu^{7} - 70631 \nu^{6} - 46293 \nu^{5} + \cdots - 9934 ) / 4385 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 148 \nu^{15} + 117 \nu^{14} + 2898 \nu^{13} - 1753 \nu^{12} - 21583 \nu^{11} + 9129 \nu^{10} + 76078 \nu^{9} - 17545 \nu^{8} - 128934 \nu^{7} - 2624 \nu^{6} + 94689 \nu^{5} + \cdots + 1492 ) / 877 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2539 \nu^{15} + 5450 \nu^{14} + 43447 \nu^{13} - 85757 \nu^{12} - 287039 \nu^{11} + 506351 \nu^{10} + 924685 \nu^{9} - 1404145 \nu^{8} - 1494612 \nu^{7} + 1823729 \nu^{6} + \cdots - 3914 ) / 8770 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2933 \nu^{15} + 5560 \nu^{14} + 50759 \nu^{13} - 89369 \nu^{12} - 329623 \nu^{11} + 544437 \nu^{10} + 980995 \nu^{9} - 1587795 \nu^{8} - 1270294 \nu^{7} + 2266763 \nu^{6} + \cdots - 17008 ) / 8770 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1519 \nu^{15} - 3340 \nu^{14} - 25252 \nu^{13} + 51632 \nu^{12} + 161994 \nu^{11} - 303826 \nu^{10} - 508685 \nu^{9} + 877555 \nu^{8} + 809382 \nu^{7} - 1325924 \nu^{6} + \cdots + 25574 ) / 4385 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1904 \nu^{15} + 5985 \nu^{14} + 28702 \nu^{13} - 95272 \nu^{12} - 160334 \nu^{11} + 576541 \nu^{10} + 414040 \nu^{9} - 1681060 \nu^{8} - 492427 \nu^{7} + 2432414 \nu^{6} + \cdots - 19204 ) / 4385 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5019 \nu^{15} + 11440 \nu^{14} + 82527 \nu^{13} - 180077 \nu^{12} - 507669 \nu^{11} + 1067721 \nu^{10} + 1423005 \nu^{9} - 3002265 \nu^{8} - 1667182 \nu^{7} + \cdots - 5934 ) / 8770 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 502 \nu^{15} - 1262 \nu^{14} - 8052 \nu^{13} + 19658 \nu^{12} + 49066 \nu^{11} - 115761 \nu^{10} - 142711 \nu^{9} + 327055 \nu^{8} + 199758 \nu^{7} - 456123 \nu^{6} - 115945 \nu^{5} + \cdots + 225 ) / 877 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5097 \nu^{15} - 7010 \nu^{14} - 92611 \nu^{13} + 99191 \nu^{12} + 663097 \nu^{11} - 501783 \nu^{10} - 2383405 \nu^{9} + 1108955 \nu^{8} + 4513446 \nu^{7} - 982487 \nu^{6} + \cdots + 7992 ) / 8770 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 582 \nu^{15} + 946 \nu^{14} + 10614 \nu^{13} - 14751 \nu^{12} - 75760 \nu^{11} + 86540 \nu^{10} + 268524 \nu^{9} - 241088 \nu^{8} - 494367 \nu^{7} + 322965 \nu^{6} + \cdots + 700 ) / 877 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2936 \nu^{15} + 9100 \nu^{14} + 44738 \nu^{13} - 147648 \nu^{12} - 247901 \nu^{11} + 914959 \nu^{10} + 589890 \nu^{9} - 2743450 \nu^{8} - 441713 \nu^{7} + 4097791 \nu^{6} + \cdots - 36651 ) / 4385 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{15} + \beta_{13} - \beta_{12} - \beta_{9} + \beta_{7} - \beta_{5} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} + 10 \beta_{3} + 10 \beta_{2} + 19 \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9 \beta_{15} - 2 \beta_{14} + 8 \beta_{13} - 11 \beta_{12} - \beta_{11} + \beta_{10} - 9 \beta_{9} + 6 \beta_{7} + 3 \beta_{6} - 11 \beta_{5} - \beta_{4} + 13 \beta_{3} + 49 \beta_{2} + 13 \beta _1 + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 22 \beta_{14} - 11 \beta_{13} - 24 \beta_{12} - 12 \beta_{11} + \beta_{10} + 10 \beta_{9} + 10 \beta_{8} - 6 \beta_{7} + 14 \beta_{6} - 5 \beta_{5} - 11 \beta_{4} + 81 \beta_{3} + 84 \beta_{2} + 107 \beta _1 + 101 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 63 \beta_{15} - 27 \beta_{14} + 49 \beta_{13} - 95 \beta_{12} - 15 \beta_{11} + 13 \beta_{10} - 60 \beta_{9} + \beta_{8} + 17 \beta_{7} + 41 \beta_{6} - 93 \beta_{5} - 14 \beta_{4} + 125 \beta_{3} + 347 \beta_{2} + 123 \beta _1 + 465 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4 \beta_{15} - 185 \beta_{14} - 91 \beta_{13} - 220 \beta_{12} - 105 \beta_{11} + 19 \beta_{10} + 77 \beta_{9} + 76 \beta_{8} - 100 \beta_{7} + 140 \beta_{6} - 82 \beta_{5} - 94 \beta_{4} + 615 \beta_{3} + 668 \beta_{2} + 677 \beta _1 + 763 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 409 \beta_{15} - 268 \beta_{14} + 264 \beta_{13} - 760 \beta_{12} - 154 \beta_{11} + 125 \beta_{10} - 356 \beta_{9} + 16 \beta_{8} - 90 \beta_{7} + 403 \beta_{6} - 729 \beta_{5} - 144 \beta_{4} + 1076 \beta_{3} + 2479 \beta_{2} + \cdots + 2964 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 68 \beta_{15} - 1430 \beta_{14} - 692 \beta_{13} - 1836 \beta_{12} - 821 \beta_{11} + 227 \beta_{10} + 552 \beta_{9} + 522 \beta_{8} - 1141 \beta_{7} + 1234 \beta_{6} - 918 \beta_{5} - 744 \beta_{4} + 4555 \beta_{3} + \cdots + 5533 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2589 \beta_{15} - 2373 \beta_{14} + 1253 \beta_{13} - 5891 \beta_{12} - 1363 \beta_{11} + 1082 \beta_{10} - 1961 \beta_{9} + 170 \beta_{8} - 2112 \beta_{7} + 3507 \beta_{6} - 5566 \beta_{5} - 1314 \beta_{4} + \cdots + 19501 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 762 \beta_{15} - 10713 \beta_{14} - 5132 \beta_{13} - 14667 \beta_{12} - 6123 \beta_{11} + 2243 \beta_{10} + 3900 \beta_{9} + 3420 \beta_{8} - 11106 \beta_{7} + 10231 \beta_{6} - 8765 \beta_{5} + \cdots + 39491 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 16283 \beta_{15} - 19824 \beta_{14} + 4743 \beta_{13} - 44954 \beta_{12} - 11222 \beta_{11} + 8913 \beta_{10} - 10028 \beta_{9} + 1520 \beta_{8} - 24796 \beta_{7} + 28793 \beta_{6} - 42056 \beta_{5} + \cdots + 130850 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 7148 \beta_{15} - 79304 \beta_{14} - 37887 \beta_{13} - 114374 \beta_{12} - 44758 \beta_{11} + 20094 \beta_{10} + 27709 \beta_{9} + 21895 \beta_{8} - 99230 \beta_{7} + 81949 \beta_{6} + \cdots + 280336 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.70554
2.49904
1.89445
1.87903
1.60855
1.18288
1.03277
0.368531
0.156223
−0.232820
−0.534018
−0.936663
−1.65208
−1.73071
−1.83015
−2.41057
−2.70554 −1.00000 5.31993 −1.56121 2.70554 −0.550914 −8.98218 1.00000 4.22392
1.2 −2.49904 −1.00000 4.24519 0.245151 2.49904 2.59581 −5.61082 1.00000 −0.612642
1.3 −1.89445 −1.00000 1.58894 −3.78565 1.89445 4.81439 0.778724 1.00000 7.17172
1.4 −1.87903 −1.00000 1.53076 0.397990 1.87903 −3.59277 0.881714 1.00000 −0.747835
1.5 −1.60855 −1.00000 0.587427 2.01171 1.60855 −2.60607 2.27219 1.00000 −3.23594
1.6 −1.18288 −1.00000 −0.600803 −0.208439 1.18288 1.11694 3.07643 1.00000 0.246558
1.7 −1.03277 −1.00000 −0.933390 2.84929 1.03277 2.89611 3.02951 1.00000 −2.94266
1.8 −0.368531 −1.00000 −1.86418 2.00812 0.368531 −0.0810239 1.42407 1.00000 −0.740054
1.9 −0.156223 −1.00000 −1.97559 −3.39057 0.156223 −3.12691 0.621081 1.00000 0.529687
1.10 0.232820 −1.00000 −1.94579 −2.72584 −0.232820 0.153308 −0.918659 1.00000 −0.634630
1.11 0.534018 −1.00000 −1.71483 3.93217 −0.534018 −3.40919 −1.98378 1.00000 2.09985
1.12 0.936663 −1.00000 −1.12266 0.344092 −0.936663 1.01488 −2.92488 1.00000 0.322298
1.13 1.65208 −1.00000 0.729355 −0.704608 −1.65208 −2.68183 −2.09920 1.00000 −1.16407
1.14 1.73071 −1.00000 0.995353 1.38078 −1.73071 1.03497 −1.73875 1.00000 2.38974
1.15 1.83015 −1.00000 1.34945 −3.85463 −1.83015 2.00843 −1.19060 1.00000 −7.05456
1.16 2.41057 −1.00000 3.81084 0.0616516 −2.41057 −2.58613 4.36516 1.00000 0.148615
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(17\) \(-1\)
\(59\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3009.2.a.f 16
3.b odd 2 1 9027.2.a.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3009.2.a.f 16 1.a even 1 1 trivial
9027.2.a.m 16 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3009))\):

\( T_{2}^{16} + 4 T_{2}^{15} - 13 T_{2}^{14} - 65 T_{2}^{13} + 49 T_{2}^{12} + 403 T_{2}^{11} + 11 T_{2}^{10} - 1205 T_{2}^{9} - 452 T_{2}^{8} + 1783 T_{2}^{7} + 946 T_{2}^{6} - 1190 T_{2}^{5} - 645 T_{2}^{4} + 299 T_{2}^{3} + 119 T_{2}^{2} + \cdots - 4 \) Copy content Toggle raw display
\( T_{5}^{16} + 3 T_{5}^{15} - 38 T_{5}^{14} - 99 T_{5}^{13} + 547 T_{5}^{12} + 1095 T_{5}^{11} - 3874 T_{5}^{10} - 4784 T_{5}^{9} + 13689 T_{5}^{8} + 6627 T_{5}^{7} - 19664 T_{5}^{6} + 520 T_{5}^{5} + 6720 T_{5}^{4} - 1843 T_{5}^{3} + \cdots - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{15} - 13 T^{14} - 65 T^{13} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 3 T^{15} - 38 T^{14} - 99 T^{13} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{16} + 3 T^{15} - 46 T^{14} - 145 T^{13} + \cdots + 404 \) Copy content Toggle raw display
$11$ \( T^{16} + 7 T^{15} - 66 T^{14} + \cdots + 15641 \) Copy content Toggle raw display
$13$ \( T^{16} - 7 T^{15} - 64 T^{14} + \cdots - 2768 \) Copy content Toggle raw display
$17$ \( (T - 1)^{16} \) Copy content Toggle raw display
$19$ \( T^{16} + 19 T^{15} + 52 T^{14} + \cdots - 22460513 \) Copy content Toggle raw display
$23$ \( T^{16} + 16 T^{15} - 88 T^{14} + \cdots - 34223767 \) Copy content Toggle raw display
$29$ \( T^{16} - 4 T^{15} - 229 T^{14} + \cdots + 758228 \) Copy content Toggle raw display
$31$ \( T^{16} + 15 T^{15} - 115 T^{14} + \cdots - 29771776 \) Copy content Toggle raw display
$37$ \( T^{16} - 20 T^{15} + \cdots + 1170071132 \) Copy content Toggle raw display
$41$ \( T^{16} + 4 T^{15} + \cdots + 3046517648 \) Copy content Toggle raw display
$43$ \( T^{16} + 12 T^{15} + \cdots + 175234414144 \) Copy content Toggle raw display
$47$ \( T^{16} + 36 T^{15} + \cdots + 15877187632 \) Copy content Toggle raw display
$53$ \( T^{16} + 32 T^{15} + \cdots + 2497286843 \) Copy content Toggle raw display
$59$ \( (T - 1)^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 418 T^{14} + \cdots - 4848879517 \) Copy content Toggle raw display
$67$ \( T^{16} + 30 T^{15} + \cdots - 34183161436 \) Copy content Toggle raw display
$71$ \( T^{16} + 70 T^{15} + \cdots - 112146919424 \) Copy content Toggle raw display
$73$ \( T^{16} - 21 T^{15} + \cdots - 11168929997 \) Copy content Toggle raw display
$79$ \( T^{16} + 25 T^{15} + \cdots - 393864146276 \) Copy content Toggle raw display
$83$ \( T^{16} + 23 T^{15} + \cdots - 4577729041364 \) Copy content Toggle raw display
$89$ \( T^{16} + 31 T^{15} + \cdots + 76837230044 \) Copy content Toggle raw display
$97$ \( T^{16} - 52 T^{15} + \cdots + 8197045361 \) Copy content Toggle raw display
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