Properties

Label 4016.2.a.k
Level $4016$
Weight $2$
Character orbit 4016.a
Self dual yes
Analytic conductor $32.068$
Analytic rank $0$
Dimension $17$
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] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, 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("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.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: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + 6186 x^{9} - 9216 x^{8} - 11921 x^{7} + 13680 x^{6} + 13752 x^{5} - 9400 x^{4} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 251)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{6} q^{5} + \beta_{13} q^{7} + (\beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + \beta_{6} q^{5} + \beta_{13} q^{7} + (\beta_{3} + 1) q^{9} + ( - \beta_{16} - \beta_{11} + \beta_{6} + \beta_1 + 1) q^{11} + ( - \beta_{10} - \beta_{6} + \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{15} + \beta_{11} - \beta_{10} - \beta_{4} + \beta_{3}) q^{15} + ( - \beta_{15} + \beta_{6} + \beta_{5} - \beta_1 - 1) q^{17} + (\beta_{16} + \beta_{12} + \beta_{11} - \beta_{6} + \beta_{2} - \beta_1 - 2) q^{19} + ( - \beta_{16} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{6} - \beta_{5} + \beta_{3} + \cdots + 3) q^{21}+ \cdots + ( - \beta_{16} - \beta_{14} + 2 \beta_{13} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + 6186 x^{9} - 9216 x^{8} - 11921 x^{7} + 13680 x^{6} + 13752 x^{5} - 9400 x^{4} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22 \nu^{16} + 15 \nu^{15} + 647 \nu^{14} - 374 \nu^{13} - 7676 \nu^{12} + 3513 \nu^{11} + 46913 \nu^{10} - 14575 \nu^{9} - 156321 \nu^{8} + 19540 \nu^{7} + 276564 \nu^{6} + \cdots + 4304 ) / 304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 85 \nu^{16} + 178 \nu^{15} + 2264 \nu^{14} - 4694 \nu^{13} - 23617 \nu^{12} + 48558 \nu^{11} + 120979 \nu^{10} - 246626 \nu^{9} - 312422 \nu^{8} + 626264 \nu^{7} + 384541 \nu^{6} + \cdots - 3200 ) / 1216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69 \nu^{16} + 212 \nu^{15} + 1752 \nu^{14} - 5638 \nu^{13} - 17157 \nu^{12} + 59424 \nu^{11} + 79979 \nu^{10} - 313888 \nu^{9} - 174414 \nu^{8} + 866052 \nu^{7} + 146661 \nu^{6} + \cdots - 27776 ) / 1216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{16} - \nu^{15} + 31 \nu^{14} + 30 \nu^{13} - 391 \nu^{12} - 363 \nu^{11} + 2570 \nu^{10} + 2275 \nu^{9} - 9331 \nu^{8} - 7882 \nu^{7} + 18119 \nu^{6} + 14855 \nu^{5} - 16325 \nu^{4} + \cdots + 448 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42 \nu^{16} - 37 \nu^{15} + 1306 \nu^{14} + 1148 \nu^{13} - 16568 \nu^{12} - 14297 \nu^{11} + 109772 \nu^{10} + 91723 \nu^{9} - 401886 \nu^{8} - 323534 \nu^{7} + 783066 \nu^{6} + \cdots + 23776 ) / 608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 235 \nu^{16} + 244 \nu^{15} + 6608 \nu^{14} - 6066 \nu^{13} - 74651 \nu^{12} + 58072 \nu^{11} + 432413 \nu^{10} - 261584 \nu^{9} - 1360330 \nu^{8} + 517908 \nu^{7} + \cdots + 19264 ) / 1216 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55 \nu^{16} + 66 \nu^{15} + 1551 \nu^{14} - 1676 \nu^{13} - 17575 \nu^{12} + 16544 \nu^{11} + 102092 \nu^{10} - 78608 \nu^{9} - 321823 \nu^{8} + 176682 \nu^{7} + 539429 \nu^{6} + \cdots + 1184 ) / 304 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 217 \nu^{16} - 420 \nu^{15} - 5804 \nu^{14} + 10814 \nu^{13} + 61465 \nu^{12} - 109080 \nu^{11} - 326267 \nu^{10} + 539504 \nu^{9} + 908994 \nu^{8} - 1330756 \nu^{7} + \cdots + 8384 ) / 1216 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 273 \nu^{16} - 54 \nu^{15} - 8052 \nu^{14} + 670 \nu^{13} + 96501 \nu^{12} + 3222 \nu^{11} - 601779 \nu^{10} - 92994 \nu^{9} + 2071842 \nu^{8} + 586448 \nu^{7} - 3830305 \nu^{6} + \cdots - 63040 ) / 1216 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 81 \nu^{16} - 120 \nu^{15} - 2231 \nu^{14} + 3068 \nu^{13} + 24529 \nu^{12} - 30574 \nu^{11} - 136930 \nu^{10} + 147646 \nu^{9} + 409343 \nu^{8} - 344458 \nu^{7} - 642219 \nu^{6} + \cdots - 3424 ) / 304 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 81 \nu^{16} + 120 \nu^{15} + 2231 \nu^{14} - 3068 \nu^{13} - 24529 \nu^{12} + 30574 \nu^{11} + 136930 \nu^{10} - 147646 \nu^{9} - 409343 \nu^{8} + 344458 \nu^{7} + 642219 \nu^{6} + \cdots + 3424 ) / 304 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 112 \nu^{16} + 85 \nu^{15} + 3204 \nu^{14} - 2018 \nu^{13} - 36974 \nu^{12} + 17893 \nu^{11} + 219968 \nu^{10} - 68557 \nu^{9} - 715256 \nu^{8} + 72676 \nu^{7} + 1242448 \nu^{6} + \cdots + 17904 ) / 304 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 455 \nu^{16} - 508 \nu^{15} - 12812 \nu^{14} + 12770 \nu^{13} + 144951 \nu^{12} - 124248 \nu^{11} - 840781 \nu^{10} + 576016 \nu^{9} + 2647622 \nu^{8} - 1224636 \nu^{7} + \cdots - 31296 ) / 1216 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 483 \nu^{16} + 268 \nu^{15} + 13936 \nu^{14} - 5874 \nu^{13} - 162659 \nu^{12} + 44784 \nu^{11} + 982565 \nu^{10} - 107976 \nu^{9} - 3259674 \nu^{8} - 264940 \nu^{7} + \cdots + 98624 ) / 1216 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 332 \nu^{16} + 159 \nu^{15} + 9636 \nu^{14} - 3364 \nu^{13} - 113202 \nu^{12} + 23535 \nu^{11} + 688604 \nu^{10} - 33845 \nu^{9} - 2300544 \nu^{8} - 307434 \nu^{7} + \cdots + 80704 ) / 608 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{12} + \beta_{11} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{12} + 3\beta_{11} - 2\beta_{10} + 2\beta_{9} - 2\beta_{8} - 2\beta_{6} - 2\beta_{5} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - \beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + 7\beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{14} - 4 \beta_{13} + 29 \beta_{12} + 11 \beta_{11} - 22 \beta_{10} + 18 \beta_{9} - 20 \beta_{8} + 2 \beta_{7} - 18 \beta_{6} - 22 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 4 \beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{16} + \beta_{15} - 12 \beta_{14} + 2 \beta_{13} - 10 \beta_{11} - 12 \beta_{10} + 12 \beta_{9} - 24 \beta_{8} - 12 \beta_{7} - 12 \beta_{6} - 14 \beta_{5} + \beta_{3} + 48 \beta _1 + 147 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2 \beta_{16} + 6 \beta_{15} - 30 \beta_{14} - 52 \beta_{13} + 177 \beta_{12} + 37 \beta_{11} - 192 \beta_{10} + 144 \beta_{9} - 174 \beta_{8} + 26 \beta_{7} - 144 \beta_{6} - 200 \beta_{5} - 52 \beta_{4} + 50 \beta_{3} + 4 \beta_{2} + \cdots + 162 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 16 \beta_{16} + 18 \beta_{15} - 112 \beta_{14} + 28 \beta_{13} + 2 \beta_{12} - 82 \beta_{11} - 110 \beta_{10} + 114 \beta_{9} - 226 \beta_{8} - 110 \beta_{7} - 108 \beta_{6} - 144 \beta_{5} + 2 \beta_{4} + 14 \beta_{3} + 2 \beta_{2} + \cdots + 986 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 30 \beta_{16} + 102 \beta_{15} - 322 \beta_{14} - 496 \beta_{13} + 1113 \beta_{12} + 67 \beta_{11} - 1546 \beta_{10} + 1126 \beta_{9} - 1448 \beta_{8} + 250 \beta_{7} - 1118 \beta_{6} - 1694 \beta_{5} - 484 \beta_{4} + \cdots + 1436 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 176 \beta_{16} + 222 \beta_{15} - 953 \beta_{14} + 272 \beta_{13} + 34 \beta_{12} - 647 \beta_{11} - 923 \beta_{10} + 1007 \beta_{9} - 1956 \beta_{8} - 911 \beta_{7} - 885 \beta_{6} - 1321 \beta_{5} + 50 \beta_{4} + \cdots + 6825 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 314 \beta_{16} + 1214 \beta_{15} - 3060 \beta_{14} - 4220 \beta_{13} + 7161 \beta_{12} - 557 \beta_{11} - 11994 \beta_{10} + 8786 \beta_{9} - 11814 \beta_{8} + 2120 \beta_{7} - 8586 \beta_{6} - 13870 \beta_{5} + \cdots + 12424 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1665 \beta_{16} + 2343 \beta_{15} - 7772 \beta_{14} + 2270 \beta_{13} + 390 \beta_{12} - 5088 \beta_{11} - 7458 \beta_{10} + 8622 \beta_{9} - 16278 \beta_{8} - 7178 \beta_{7} - 6984 \beta_{6} - 11498 \beta_{5} + \cdots + 48397 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2872 \beta_{16} + 12548 \beta_{15} - 27424 \beta_{14} - 33972 \beta_{13} + 46965 \beta_{12} - 9867 \beta_{11} - 91272 \beta_{10} + 68804 \beta_{9} - 95412 \beta_{8} + 16732 \beta_{7} - 65684 \beta_{6} + \cdots + 105308 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 14572 \beta_{16} + 22760 \beta_{15} - 62088 \beta_{14} + 17404 \beta_{13} + 3808 \beta_{12} - 40160 \beta_{11} - 59188 \beta_{10} + 72596 \beta_{9} - 132704 \beta_{8} - 55032 \beta_{7} + \cdots + 349916 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 24712 \beta_{16} + 120508 \beta_{15} - 237700 \beta_{14} - 265304 \beta_{13} + 313117 \beta_{12} - 106437 \beta_{11} - 687462 \beta_{10} + 541310 \beta_{9} - 766130 \beta_{8} + \cdots + 879042 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 121836 \beta_{16} + 210152 \beta_{15} - 491137 \beta_{14} + 125988 \beta_{13} + 34252 \beta_{12} - 318049 \beta_{11} - 465593 \beta_{10} + 604689 \beta_{9} - 1069530 \beta_{8} + \cdots + 2569939 \) 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
−2.32547
−0.139956
1.37907
1.37191
−0.787554
2.64128
−2.27410
2.18124
−2.51582
1.84638
2.33247
−1.09599
−0.622810
−2.65791
2.82015
−0.932399
0.779516
0 −3.27059 0 −2.03057 0 −1.64874 0 7.69673 0
1.2 0 −2.55238 0 2.96478 0 −0.820250 0 3.51462 0
1.3 0 −2.46273 0 1.31548 0 3.48956 0 3.06502 0
1.4 0 −2.36731 0 −1.32016 0 −4.19189 0 2.60414 0
1.5 0 −1.89108 0 −0.652808 0 −1.12991 0 0.576182 0
1.6 0 −1.66988 0 −3.99830 0 2.26241 0 −0.211508 0
1.7 0 −0.935470 0 3.41593 0 −3.69332 0 −2.12490 0
1.8 0 −0.923498 0 1.20531 0 1.96022 0 −2.14715 0
1.9 0 0.505139 0 −3.97764 0 −1.36760 0 −2.74483 0
1.10 0 0.508487 0 3.51373 0 −0.924114 0 −2.74144 0
1.11 0 0.826533 0 −1.37815 0 −4.67534 0 −2.31684 0
1.12 0 1.16074 0 4.05107 0 4.08218 0 −1.65268 0
1.13 0 1.30185 0 −1.69081 0 −3.93205 0 −1.30520 0
1.14 0 2.62368 0 1.15029 0 2.51426 0 3.88372 0
1.15 0 2.95607 0 2.29008 0 2.82675 0 5.73835 0
1.16 0 3.04458 0 −3.52221 0 4.32039 0 6.26949 0
1.17 0 3.14584 0 1.66398 0 −2.07256 0 6.89631 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(1\)

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 4016.2.a.k 17
4.b odd 2 1 251.2.a.b 17
12.b even 2 1 2259.2.a.k 17
20.d odd 2 1 6275.2.a.e 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
251.2.a.b 17 4.b odd 2 1
2259.2.a.k 17 12.b even 2 1
4016.2.a.k 17 1.a even 1 1 trivial
6275.2.a.e 17 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{17} - 38 T_{3}^{15} - 5 T_{3}^{14} + 582 T_{3}^{13} + 142 T_{3}^{12} - 4602 T_{3}^{11} - 1445 T_{3}^{10} + 20039 T_{3}^{9} + 6280 T_{3}^{8} - 48174 T_{3}^{7} - 10424 T_{3}^{6} + 63091 T_{3}^{5} + 3260 T_{3}^{4} + \cdots - 3164 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} \) Copy content Toggle raw display
$3$ \( T^{17} - 38 T^{15} - 5 T^{14} + \cdots - 3164 \) Copy content Toggle raw display
$5$ \( T^{17} - 3 T^{16} - 54 T^{15} + \cdots - 228857 \) Copy content Toggle raw display
$7$ \( T^{17} + 3 T^{16} - 71 T^{15} + \cdots - 2209789 \) Copy content Toggle raw display
$11$ \( T^{17} - T^{16} - 122 T^{15} + \cdots - 10657792 \) Copy content Toggle raw display
$13$ \( T^{17} - 22 T^{16} + 106 T^{15} + \cdots - 504874 \) Copy content Toggle raw display
$17$ \( T^{17} + T^{16} - 156 T^{15} + \cdots - 54097717 \) Copy content Toggle raw display
$19$ \( T^{17} + 13 T^{16} + \cdots - 130088960 \) Copy content Toggle raw display
$23$ \( T^{17} - 2 T^{16} - 145 T^{15} + \cdots - 201949 \) Copy content Toggle raw display
$29$ \( T^{17} - 28 T^{16} + \cdots + 1937776640 \) Copy content Toggle raw display
$31$ \( T^{17} + 12 T^{16} + \cdots - 10307640389 \) Copy content Toggle raw display
$37$ \( T^{17} - 27 T^{16} + \cdots - 1861132288 \) Copy content Toggle raw display
$41$ \( T^{17} + T^{16} - 327 T^{15} + \cdots - 11114425387 \) Copy content Toggle raw display
$43$ \( T^{17} + 9 T^{16} - 350 T^{15} + \cdots + 11640832 \) Copy content Toggle raw display
$47$ \( T^{17} - 20 T^{16} + \cdots + 62409392128 \) Copy content Toggle raw display
$53$ \( T^{17} - T^{16} + \cdots - 7243329708032 \) Copy content Toggle raw display
$59$ \( T^{17} - 20 T^{16} + \cdots + 139809955840 \) Copy content Toggle raw display
$61$ \( T^{17} - 59 T^{16} + \cdots - 3666674696192 \) Copy content Toggle raw display
$67$ \( T^{17} + 15 T^{16} + \cdots + 1042048845953 \) Copy content Toggle raw display
$71$ \( T^{17} - 26 T^{16} + \cdots + 12296978432 \) Copy content Toggle raw display
$73$ \( T^{17} - 8 T^{16} + \cdots - 371103914897 \) Copy content Toggle raw display
$79$ \( T^{17} + 33 T^{16} + \cdots + 1616495596055 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 295625646813184 \) Copy content Toggle raw display
$89$ \( T^{17} - 11 T^{16} + \cdots - 91153496990 \) Copy content Toggle raw display
$97$ \( T^{17} + 10 T^{16} + \cdots - 41770891288576 \) Copy content Toggle raw display
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