Properties

Label 4017.2.a.e
Level $4017$
Weight $2$
Character orbit 4017.a
Self dual yes
Analytic conductor $32.076$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4017,2,Mod(1,4017)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4017, 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("4017.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4017 = 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4017.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: \(32.0759064919\)
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: \( x^{16} - 21 x^{14} - 3 x^{13} + 177 x^{12} + 45 x^{11} - 763 x^{10} - 251 x^{9} + 1771 x^{8} + 639 x^{7} - 2118 x^{6} - 710 x^{5} + 1113 x^{4} + 243 x^{3} - 183 x^{2} - 10 x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{12} q^{5} - \beta_1 q^{6} + (\beta_{7} - 1) q^{7} + ( - \beta_{3} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{12} q^{5} - \beta_1 q^{6} + (\beta_{7} - 1) q^{7} + ( - \beta_{3} - 1) q^{8} + q^{9} + ( - \beta_{13} - \beta_{7} + \beta_{6} - \beta_{2} + \beta_1 - 1) q^{10} + (\beta_{13} + \beta_{12} + \beta_{3}) q^{11} + (\beta_{2} + 1) q^{12} - q^{13} + (\beta_{15} - \beta_{13} + \beta_{12} + \beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_1 - 1) q^{14} - \beta_{12} q^{15} + (\beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} + \cdots + \beta_1) q^{16}+ \cdots + (\beta_{13} + \beta_{12} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{3} + 10 q^{4} - 6 q^{5} - 13 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{3} + 10 q^{4} - 6 q^{5} - 13 q^{7} - 9 q^{8} + 16 q^{9} - 8 q^{10} - 5 q^{11} + 10 q^{12} - 16 q^{13} - 8 q^{14} - 6 q^{15} - 14 q^{16} - q^{17} + 6 q^{19} - 4 q^{20} - 13 q^{21} - 11 q^{22} - 21 q^{23} - 9 q^{24} - 10 q^{25} + 16 q^{27} - 10 q^{28} - 17 q^{29} - 8 q^{30} - 33 q^{31} - 18 q^{32} - 5 q^{33} - 5 q^{34} - 4 q^{35} + 10 q^{36} - 23 q^{37} - 28 q^{38} - 16 q^{39} - 12 q^{40} + 7 q^{41} - 8 q^{42} - 33 q^{43} + 11 q^{44} - 6 q^{45} - 15 q^{46} - 13 q^{47} - 14 q^{48} - 17 q^{49} + 35 q^{50} - q^{51} - 10 q^{52} - 20 q^{53} - 54 q^{55} + 12 q^{56} + 6 q^{57} - 33 q^{58} + 6 q^{59} - 4 q^{60} - 49 q^{61} - 13 q^{62} - 13 q^{63} - 35 q^{64} + 6 q^{65} - 11 q^{66} - 4 q^{67} - 14 q^{68} - 21 q^{69} - 33 q^{70} - 29 q^{71} - 9 q^{72} - 21 q^{73} + 22 q^{74} - 10 q^{75} + 10 q^{76} - 21 q^{77} - 70 q^{79} - 8 q^{80} + 16 q^{81} - 10 q^{82} + 5 q^{83} - 10 q^{84} + 14 q^{85} + 29 q^{86} - 17 q^{87} - 45 q^{88} - 8 q^{89} - 8 q^{90} + 13 q^{91} - 29 q^{92} - 33 q^{93} + 12 q^{94} - 45 q^{95} - 18 q^{96} - 30 q^{97} + 15 q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 21 x^{14} - 3 x^{13} + 177 x^{12} + 45 x^{11} - 763 x^{10} - 251 x^{9} + 1771 x^{8} + 639 x^{7} - 2118 x^{6} - 710 x^{5} + 1113 x^{4} + 243 x^{3} - 183 x^{2} - 10 x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4 \nu^{15} + 8 \nu^{14} + 62 \nu^{13} - 106 \nu^{12} - 382 \nu^{11} + 488 \nu^{10} + 1224 \nu^{9} - 853 \nu^{8} - 2219 \nu^{7} + 127 \nu^{6} + 2215 \nu^{5} + 936 \nu^{4} - 963 \nu^{3} - 513 \nu^{2} + \cdots + 34 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 3 \nu^{15} - 3 \nu^{14} - 59 \nu^{13} + 50 \nu^{12} + 458 \nu^{11} - 318 \nu^{10} - 1773 \nu^{9} + 955 \nu^{8} + 3544 \nu^{7} - 1328 \nu^{6} - 3365 \nu^{5} + 659 \nu^{4} + 1142 \nu^{3} - 10 \nu^{2} - 71 \nu - 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{15} + 19 \nu^{14} + 193 \nu^{13} - 293 \nu^{12} - 1349 \nu^{11} + 1723 \nu^{10} + 4773 \nu^{9} - 4832 \nu^{8} - 8881 \nu^{7} + 6497 \nu^{6} + 7997 \nu^{5} - 3558 \nu^{4} - 2598 \nu^{3} + \cdots - 19 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - 3 \nu^{15} + 2 \nu^{14} + 63 \nu^{13} - 40 \nu^{12} - 512 \nu^{11} + 299 \nu^{10} + 2035 \nu^{9} - 1039 \nu^{8} - 4091 \nu^{7} + 1667 \nu^{6} + 3800 \nu^{5} - 995 \nu^{4} - 1188 \nu^{3} + 66 \nu^{2} + \cdots + 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16 \nu^{15} + 32 \nu^{14} + 272 \nu^{13} - 499 \nu^{12} - 1831 \nu^{11} + 2990 \nu^{10} + 6195 \nu^{9} - 8665 \nu^{8} - 10901 \nu^{7} + 12397 \nu^{6} + 9055 \nu^{5} - 7776 \nu^{4} - 2484 \nu^{3} + \cdots - 59 ) / 3 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16 \nu^{15} - 29 \nu^{14} - 281 \nu^{13} + 457 \nu^{12} + 1960 \nu^{11} - 2765 \nu^{10} - 6882 \nu^{9} + 8056 \nu^{8} + 12578 \nu^{7} - 11446 \nu^{6} - 10882 \nu^{5} + 6885 \nu^{4} + 3162 \nu^{3} + \cdots + 32 ) / 3 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14 \nu^{15} - 25 \nu^{14} - 253 \nu^{13} + 416 \nu^{12} + 1802 \nu^{11} - 2689 \nu^{10} - 6369 \nu^{9} + 8492 \nu^{8} + 11431 \nu^{7} - 13367 \nu^{6} - 9239 \nu^{5} + 9309 \nu^{4} + 2103 \nu^{3} + \cdots + 82 ) / 3 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23 \nu^{15} + 46 \nu^{14} + 388 \nu^{13} - 704 \nu^{12} - 2600 \nu^{11} + 4108 \nu^{10} + 8817 \nu^{9} - 11453 \nu^{8} - 15745 \nu^{7} + 15428 \nu^{6} + 13583 \nu^{5} - 8709 \nu^{4} + \cdots - 43 ) / 3 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20 \nu^{15} + 34 \nu^{14} + 367 \nu^{13} - 572 \nu^{12} - 2654 \nu^{11} + 3739 \nu^{10} + 9525 \nu^{9} - 11933 \nu^{8} - 17398 \nu^{7} + 18947 \nu^{6} + 14477 \nu^{5} - 13257 \nu^{4} + \cdots - 115 ) / 3 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( - 12 \nu^{15} + 22 \nu^{14} + 212 \nu^{13} - 353 \nu^{12} - 1484 \nu^{11} + 2187 \nu^{10} + 5207 \nu^{9} - 6579 \nu^{8} - 9445 \nu^{7} + 9793 \nu^{6} + 8018 \nu^{5} - 6387 \nu^{4} - 2233 \nu^{3} + \cdots - 54 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41 \nu^{15} + 73 \nu^{14} + 724 \nu^{13} - 1154 \nu^{12} - 5078 \nu^{11} + 7009 \nu^{10} + 17925 \nu^{9} - 20516 \nu^{8} - 32929 \nu^{7} + 29312 \nu^{6} + 28664 \nu^{5} - 17766 \nu^{4} + \cdots - 100 ) / 3 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 49 \nu^{15} + 77 \nu^{14} + 902 \nu^{13} - 1264 \nu^{12} - 6571 \nu^{11} + 8015 \nu^{10} + 23913 \nu^{9} - 24610 \nu^{8} - 44789 \nu^{7} + 37054 \nu^{6} + 39136 \nu^{5} - 23808 \nu^{4} + \cdots - 155 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} + 5 \beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 9 \beta_{3} + 18 \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{13} - 10 \beta_{12} - 9 \beta_{11} + 9 \beta_{10} - 8 \beta_{9} + 9 \beta_{8} + 10 \beta_{7} - 9 \beta_{6} + 10 \beta_{5} + 19 \beta_{3} + 26 \beta_{2} + 9 \beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{15} - 8 \beta_{14} + 11 \beta_{13} - 13 \beta_{12} - 3 \beta_{11} + 11 \beta_{10} - 10 \beta_{9} + 11 \beta_{8} + 20 \beta_{7} + 8 \beta_{6} + 22 \beta_{5} - 10 \beta_{4} + 64 \beta_{3} + 2 \beta_{2} + 89 \beta _1 + 78 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{15} + \beta_{14} + 74 \beta_{13} - 76 \beta_{12} - 64 \beta_{11} + 65 \beta_{10} - 51 \beta_{9} + 65 \beta_{8} + 76 \beta_{7} - 62 \beta_{6} + 78 \beta_{5} - 2 \beta_{4} + 141 \beta_{3} + 141 \beta_{2} + 65 \beta _1 + 418 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 15 \beta_{15} - 47 \beta_{14} + 89 \beta_{13} - 116 \beta_{12} - 42 \beta_{11} + 92 \beta_{10} - 74 \beta_{9} + 90 \beta_{8} + 150 \beta_{7} + 45 \beta_{6} + 179 \beta_{5} - 74 \beta_{4} + 423 \beta_{3} + 30 \beta_{2} + 471 \beta _1 + 555 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 21 \beta_{15} + 18 \beta_{14} + 491 \beta_{13} - 528 \beta_{12} - 425 \beta_{11} + 438 \beta_{10} - 302 \beta_{9} + 437 \beta_{8} + 517 \beta_{7} - 394 \beta_{6} + 555 \beta_{5} - 31 \beta_{4} + 962 \beta_{3} + 789 \beta_{2} + \cdots + 2455 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 155 \beta_{15} - 237 \beta_{14} + 642 \beta_{13} - 900 \beta_{12} - 404 \beta_{11} + 694 \beta_{10} - 486 \beta_{9} + 664 \beta_{8} + 1016 \beta_{7} + 198 \beta_{6} + 1304 \beta_{5} - 492 \beta_{4} + 2716 \beta_{3} + \cdots + 3786 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 264 \beta_{15} + 206 \beta_{14} + 3101 \beta_{13} - 3532 \beta_{12} - 2754 \beta_{11} + 2870 \beta_{10} - 1734 \beta_{9} + 2847 \beta_{8} + 3328 \beta_{7} - 2437 \beta_{6} + 3781 \beta_{5} - 317 \beta_{4} + \cdots + 14762 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1365 \beta_{15} - 1031 \beta_{14} + 4386 \beta_{13} - 6535 \beta_{12} - 3335 \beta_{11} + 4976 \beta_{10} - 3003 \beta_{9} + 4677 \beta_{8} + 6580 \beta_{7} + 564 \beta_{6} + 9025 \beta_{5} - 3130 \beta_{4} + \cdots + 25272 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2630 \beta_{15} + 1935 \beta_{14} + 19131 \beta_{13} - 23214 \beta_{12} - 17691 \beta_{11} + 18583 \beta_{10} - 9815 \beta_{9} + 18272 \beta_{8} + 20812 \beta_{7} - 14978 \beta_{6} + 25184 \beta_{5} + \cdots + 90106 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 11003 \beta_{15} - 3451 \beta_{14} + 29116 \beta_{13} - 45771 \beta_{12} - 25421 \beta_{11} + 34660 \beta_{10} - 17919 \beta_{9} + 32144 \beta_{8} + 41725 \beta_{7} - 1023 \beta_{6} + 60832 \beta_{5} + \cdots + 166649 \) 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
2.54124
2.37644
1.91188
1.71723
1.56390
0.820556
0.291985
0.256478
−0.228474
−0.549353
−1.25986
−1.67713
−1.68093
−1.89635
−1.89812
−2.28950
−2.54124 1.00000 4.45789 −0.349136 −2.54124 −1.96840 −6.24610 1.00000 0.887239
1.2 −2.37644 1.00000 3.64749 1.54914 −2.37644 1.30434 −3.91517 1.00000 −3.68145
1.3 −1.91188 1.00000 1.65528 −0.339131 −1.91188 −3.26096 0.659059 1.00000 0.648377
1.4 −1.71723 1.00000 0.948894 −2.53727 −1.71723 −0.257645 1.80499 1.00000 4.35709
1.5 −1.56390 1.00000 0.445771 0.640434 −1.56390 3.30522 2.43065 1.00000 −1.00157
1.6 −0.820556 1.00000 −1.32669 3.24973 −0.820556 −1.69765 2.72973 1.00000 −2.66659
1.7 −0.291985 1.00000 −1.91474 −3.42004 −0.291985 −3.90865 1.14305 1.00000 0.998600
1.8 −0.256478 1.00000 −1.93422 −0.330746 −0.256478 1.25226 1.00904 1.00000 0.0848290
1.9 0.228474 1.00000 −1.94780 −1.93998 0.228474 1.64440 −0.901970 1.00000 −0.443234
1.10 0.549353 1.00000 −1.69821 1.27882 0.549353 −2.63834 −2.03162 1.00000 0.702521
1.11 1.25986 1.00000 −0.412758 −1.76649 1.25986 −1.24457 −3.03973 1.00000 −2.22553
1.12 1.67713 1.00000 0.812756 −1.82601 1.67713 2.67229 −1.99116 1.00000 −3.06245
1.13 1.68093 1.00000 0.825530 3.10294 1.68093 −2.88861 −1.97420 1.00000 5.21583
1.14 1.89635 1.00000 1.59615 1.10443 1.89635 −4.39855 −0.765840 1.00000 2.09438
1.15 1.89812 1.00000 1.60286 −0.521152 1.89812 0.438711 −0.753828 1.00000 −0.989209
1.16 2.28950 1.00000 3.24180 −3.89554 2.28950 −1.35384 2.84309 1.00000 −8.91883
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(-1\)

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 4017.2.a.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4017.2.a.e 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):

\( T_{2}^{16} - 21 T_{2}^{14} + 3 T_{2}^{13} + 177 T_{2}^{12} - 45 T_{2}^{11} - 763 T_{2}^{10} + 251 T_{2}^{9} + 1771 T_{2}^{8} - 639 T_{2}^{7} - 2118 T_{2}^{6} + 710 T_{2}^{5} + 1113 T_{2}^{4} - 243 T_{2}^{3} - 183 T_{2}^{2} + 10 T_{2} + 7 \) Copy content Toggle raw display
\( T_{23}^{16} + 21 T_{23}^{15} + 52 T_{23}^{14} - 1657 T_{23}^{13} - 11244 T_{23}^{12} + 28996 T_{23}^{11} + 425898 T_{23}^{10} + 361375 T_{23}^{9} - 6162022 T_{23}^{8} - 13651920 T_{23}^{7} + 35744003 T_{23}^{6} + \cdots - 42466673 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 21 T^{14} + 3 T^{13} + 177 T^{12} + \cdots + 7 \) Copy content Toggle raw display
$3$ \( (T - 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 6 T^{15} - 17 T^{14} - 153 T^{13} + \cdots + 61 \) Copy content Toggle raw display
$7$ \( T^{16} + 13 T^{15} + 37 T^{14} + \cdots + 6451 \) Copy content Toggle raw display
$11$ \( T^{16} + 5 T^{15} - 73 T^{14} + \cdots + 231903 \) Copy content Toggle raw display
$13$ \( (T + 1)^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + T^{15} - 87 T^{14} - 127 T^{13} + \cdots + 213 \) Copy content Toggle raw display
$19$ \( T^{16} - 6 T^{15} - 115 T^{14} + \cdots - 55495529 \) Copy content Toggle raw display
$23$ \( T^{16} + 21 T^{15} + 52 T^{14} + \cdots - 42466673 \) Copy content Toggle raw display
$29$ \( T^{16} + 17 T^{15} + \cdots - 15473827063 \) Copy content Toggle raw display
$31$ \( T^{16} + 33 T^{15} + \cdots + 5447822383 \) Copy content Toggle raw display
$37$ \( T^{16} + 23 T^{15} + \cdots + 91853108587 \) Copy content Toggle raw display
$41$ \( T^{16} - 7 T^{15} - 223 T^{14} + \cdots - 82242539 \) Copy content Toggle raw display
$43$ \( T^{16} + 33 T^{15} + \cdots + 43830282901 \) Copy content Toggle raw display
$47$ \( T^{16} + 13 T^{15} + \cdots - 478150441 \) Copy content Toggle raw display
$53$ \( T^{16} + 20 T^{15} - 191 T^{14} + \cdots + 20777267 \) Copy content Toggle raw display
$59$ \( T^{16} - 6 T^{15} + \cdots + 2950602733183 \) Copy content Toggle raw display
$61$ \( T^{16} + 49 T^{15} + \cdots + 2619166748621 \) Copy content Toggle raw display
$67$ \( T^{16} + 4 T^{15} + \cdots + 1461258372841 \) Copy content Toggle raw display
$71$ \( T^{16} + 29 T^{15} + \cdots - 318165392053 \) Copy content Toggle raw display
$73$ \( T^{16} + 21 T^{15} + \cdots - 1058152570629 \) Copy content Toggle raw display
$79$ \( T^{16} + 70 T^{15} + \cdots + 6443862537 \) Copy content Toggle raw display
$83$ \( T^{16} - 5 T^{15} + \cdots + 20518043017729 \) Copy content Toggle raw display
$89$ \( T^{16} + 8 T^{15} + \cdots + 587841659751 \) Copy content Toggle raw display
$97$ \( T^{16} + 30 T^{15} + \cdots + 8166340825801 \) Copy content Toggle raw display
show more
show less