Properties

Label 6045.2.a.bh
Level $6045$
Weight $2$
Character orbit 6045.a
Self dual yes
Analytic conductor $48.270$
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] = [6045,2,Mod(1,6045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.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: \(48.2695680219\)
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: \( x^{17} - 2 x^{16} - 25 x^{15} + 47 x^{14} + 252 x^{13} - 437 x^{12} - 1319 x^{11} + 2056 x^{10} + 3854 x^{9} - 5201 x^{8} - 6304 x^{7} + 6915 x^{6} + 5469 x^{5} - 4238 x^{4} + \cdots + 8 \) 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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} - \beta_1 q^{6} + (\beta_{11} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} - \beta_1 q^{6} + (\beta_{11} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + q^{9} + \beta_1 q^{10} - \beta_{12} q^{11} + ( - \beta_{2} - 1) q^{12} + q^{13} + (\beta_{14} + \beta_1) q^{14} - q^{15} + (\beta_{14} + \beta_{10} + \beta_{9} + \beta_{2} + 1) q^{16} + ( - \beta_{10} - \beta_{4}) q^{17} + \beta_1 q^{18} + ( - \beta_{6} + 1) q^{19} + (\beta_{2} + 1) q^{20} + ( - \beta_{11} - 1) q^{21} + (\beta_{16} - \beta_{13} - \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{9} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{22}+ \cdots - \beta_{12} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 2 q^{2} - 17 q^{3} + 20 q^{4} + 17 q^{5} - 2 q^{6} + 18 q^{7} + 9 q^{8} + 17 q^{9} + 2 q^{10} + 3 q^{11} - 20 q^{12} + 17 q^{13} + q^{14} - 17 q^{15} + 26 q^{16} + 2 q^{18} + 10 q^{19} + 20 q^{20} - 18 q^{21} + 5 q^{22} + 16 q^{23} - 9 q^{24} + 17 q^{25} + 2 q^{26} - 17 q^{27} + 36 q^{28} - 3 q^{29} - 2 q^{30} + 17 q^{31} + 20 q^{32} - 3 q^{33} + q^{34} + 18 q^{35} + 20 q^{36} + 14 q^{37} + 22 q^{38} - 17 q^{39} + 9 q^{40} - 6 q^{41} - q^{42} + 24 q^{43} - 15 q^{44} + 17 q^{45} + 6 q^{46} + 25 q^{47} - 26 q^{48} + 31 q^{49} + 2 q^{50} + 20 q^{52} - 15 q^{53} - 2 q^{54} + 3 q^{55} + 31 q^{56} - 10 q^{57} + 44 q^{58} + 16 q^{59} - 20 q^{60} - 5 q^{61} + 2 q^{62} + 18 q^{63} + 35 q^{64} + 17 q^{65} - 5 q^{66} + 50 q^{67} + 13 q^{68} - 16 q^{69} + q^{70} + 16 q^{71} + 9 q^{72} + 33 q^{73} + 2 q^{74} - 17 q^{75} + 9 q^{77} - 2 q^{78} - 10 q^{79} + 26 q^{80} + 17 q^{81} + 61 q^{82} + 27 q^{83} - 36 q^{84} - 12 q^{86} + 3 q^{87} + 23 q^{88} - 24 q^{89} + 2 q^{90} + 18 q^{91} - 21 q^{92} - 17 q^{93} + 6 q^{94} + 10 q^{95} - 20 q^{96} + 48 q^{97} + 14 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 2 x^{16} - 25 x^{15} + 47 x^{14} + 252 x^{13} - 437 x^{12} - 1319 x^{11} + 2056 x^{10} + 3854 x^{9} - 5201 x^{8} - 6304 x^{7} + 6915 x^{6} + 5469 x^{5} - 4238 x^{4} + \cdots + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 113557 \nu^{16} - 530768 \nu^{15} + 5270067 \nu^{14} + 10382037 \nu^{13} - 83048126 \nu^{12} - 70627697 \nu^{11} + 620969061 \nu^{10} + \cdots - 133114498 ) / 14023831 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 312195 \nu^{16} - 3509363 \nu^{15} + 13349902 \nu^{14} + 87408009 \nu^{13} - 196134497 \nu^{12} - 869010094 \nu^{11} + 1349656637 \nu^{10} + \cdots + 160614960 ) / 28047662 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1500797 \nu^{16} - 2833680 \nu^{15} + 44873866 \nu^{14} + 73542695 \nu^{13} - 532269412 \nu^{12} - 761838312 \nu^{11} + 3189061955 \nu^{10} + \cdots + 64950898 ) / 28047662 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1604091 \nu^{16} - 384325 \nu^{15} - 41605574 \nu^{14} + 2341871 \nu^{13} + 429886899 \nu^{12} + 53714652 \nu^{11} - 2247822761 \nu^{10} + \cdots - 18135454 ) / 28047662 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 823629 \nu^{16} - 1818804 \nu^{15} - 21236683 \nu^{14} + 44900786 \nu^{13} + 221201292 \nu^{12} - 442474917 \nu^{11} - 1194843988 \nu^{10} + \cdots + 154119101 ) / 14023831 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1839987 \nu^{16} - 267610 \nu^{15} - 47761999 \nu^{14} - 450507 \nu^{13} + 494622034 \nu^{12} + 81684169 \nu^{11} - 2598840645 \nu^{10} - 820692828 \nu^{9} + \cdots + 88911066 ) / 28047662 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2263647 \nu^{16} - 1882134 \nu^{15} - 57777861 \nu^{14} + 38112665 \nu^{13} + 589221704 \nu^{12} - 284971261 \nu^{11} - 3057841575 \nu^{10} + \cdots + 58213626 ) / 28047662 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1322580 \nu^{16} + 593343 \nu^{15} + 34139372 \nu^{14} - 9391330 \nu^{13} - 352121239 \nu^{12} + 36045591 \nu^{11} + 1846126566 \nu^{10} + \cdots + 37102250 ) / 14023831 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3448071 \nu^{16} - 591283 \nu^{15} - 91652439 \nu^{14} + 1155319 \nu^{13} + 973882157 \nu^{12} + 133423725 \nu^{11} - 5254617039 \nu^{10} + \cdots - 30204858 ) / 28047662 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3616735 \nu^{16} - 5281842 \nu^{15} - 89703162 \nu^{14} + 117460815 \nu^{13} + 886622524 \nu^{12} - 1014336732 \nu^{11} - 4451885089 \nu^{10} + \cdots + 117339748 ) / 28047662 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2051817 \nu^{16} + 1074872 \nu^{15} + 52769930 \nu^{14} - 18831079 \nu^{13} - 541921869 \nu^{12} + 101643546 \nu^{11} + 2828341110 \nu^{10} + \cdots + 10580640 ) / 14023831 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2558766 \nu^{16} + 1224081 \nu^{15} + 67288514 \nu^{14} - 21959060 \nu^{13} - 709445968 \nu^{12} + 126249210 \nu^{11} + 3820473522 \nu^{10} + \cdots + 1853237 ) / 14023831 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 3116457 \nu^{16} + 2351568 \nu^{15} + 79728849 \nu^{14} - 45045593 \nu^{13} - 816697554 \nu^{12} + 300324502 \nu^{11} + 4272901584 \nu^{10} + \cdots - 88198496 ) / 14023831 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{10} + \beta_{9} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{13} - \beta_{6} + 9\beta_{3} + \beta_{2} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + 12 \beta_{10} + 9 \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} + 46 \beta_{2} + \beta _1 + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{15} + 2 \beta_{14} + 13 \beta_{13} - \beta_{12} - 2 \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} - 14 \beta_{6} + \beta_{4} + 68 \beta_{3} + 13 \beta_{2} + 165 \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{15} + 81 \beta_{14} - 16 \beta_{13} - 14 \beta_{12} - 13 \beta_{11} + 111 \beta_{10} + 67 \beta_{9} + 14 \beta_{8} - \beta_{7} - 2 \beta_{6} - 11 \beta_{5} + 15 \beta_{4} + 14 \beta_{3} + 301 \beta_{2} + 15 \beta _1 + 541 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{16} + 118 \beta_{15} + 33 \beta_{14} + 121 \beta_{13} - 17 \beta_{12} - 27 \beta_{11} + 6 \beta_{10} + 15 \beta_{9} + 13 \beta_{8} + 17 \beta_{7} - 140 \beta_{6} + 4 \beta_{5} + 16 \beta_{4} + 490 \beta_{3} + 121 \beta_{2} + 1007 \beta _1 + 137 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2 \beta_{16} - 23 \beta_{15} + 620 \beta_{14} - 174 \beta_{13} - 140 \beta_{12} - 122 \beta_{11} + 933 \beta_{10} + 477 \beta_{9} + 136 \beta_{8} - 20 \beta_{7} - 38 \beta_{6} - 82 \beta_{5} + 158 \beta_{4} + 134 \beta_{3} + 1982 \beta_{2} + \cdots + 3495 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 19 \beta_{16} + 932 \beta_{15} + 371 \beta_{14} + 991 \beta_{13} - 198 \beta_{12} - 250 \beta_{11} + 120 \beta_{10} + 158 \beta_{9} + 120 \beta_{8} + 192 \beta_{7} - 1229 \beta_{6} + 76 \beta_{5} + 181 \beta_{4} + 3470 \beta_{3} + \cdots + 1167 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 38 \beta_{16} - 314 \beta_{15} + 4651 \beta_{14} - 1613 \beta_{13} - 1235 \beta_{12} - 1018 \beta_{11} + 7471 \beta_{10} + 3356 \beta_{9} + 1145 \beta_{8} - 253 \beta_{7} - 480 \beta_{6} - 498 \beta_{5} + 1442 \beta_{4} + \cdots + 23119 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 240 \beta_{16} + 6878 \beta_{15} + 3556 \beta_{14} + 7618 \beta_{13} - 1968 \beta_{12} - 2001 \beta_{11} + 1571 \beta_{10} + 1447 \beta_{9} + 981 \beta_{8} + 1819 \beta_{7} - 10110 \beta_{6} + 945 \beta_{5} + \cdots + 9262 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 475 \beta_{16} - 3400 \beta_{15} + 34571 \beta_{14} - 13739 \beta_{13} - 10259 \beta_{12} - 8052 \beta_{11} + 58138 \beta_{10} + 23565 \beta_{9} + 8987 \beta_{8} - 2611 \beta_{7} - 5074 \beta_{6} + \cdots + 155389 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2537 \beta_{16} + 48912 \beta_{15} + 31314 \beta_{14} + 56495 \beta_{13} - 17944 \beta_{12} - 14982 \beta_{11} + 17061 \beta_{10} + 12339 \beta_{9} + 7606 \beta_{8} + 15657 \beta_{7} - 80141 \beta_{6} + \cdots + 70526 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 4970 \beta_{16} - 32423 \beta_{15} + 255583 \beta_{14} - 111214 \beta_{13} - 82428 \beta_{12} - 61937 \beta_{11} + 444161 \beta_{10} + 165707 \beta_{9} + 67887 \beta_{8} - 24073 \beta_{7} + \cdots + 1056297 \) 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
−2.65328
−2.37579
−2.18223
−1.58644
−1.45848
−1.21506
−0.565863
−0.287376
−0.0282840
0.551239
1.15560
1.39113
1.42963
2.00742
2.54626
2.59255
2.67896
−2.65328 −1.00000 5.03989 1.00000 2.65328 1.22871 −8.06567 1.00000 −2.65328
1.2 −2.37579 −1.00000 3.64437 1.00000 2.37579 2.23288 −3.90666 1.00000 −2.37579
1.3 −2.18223 −1.00000 2.76211 1.00000 2.18223 −1.44741 −1.66310 1.00000 −2.18223
1.4 −1.58644 −1.00000 0.516801 1.00000 1.58644 5.23184 2.35301 1.00000 −1.58644
1.5 −1.45848 −1.00000 0.127166 1.00000 1.45848 3.69177 2.73149 1.00000 −1.45848
1.6 −1.21506 −1.00000 −0.523635 1.00000 1.21506 −0.863273 3.06636 1.00000 −1.21506
1.7 −0.565863 −1.00000 −1.67980 1.00000 0.565863 −3.78718 2.08226 1.00000 −0.565863
1.8 −0.287376 −1.00000 −1.91742 1.00000 0.287376 −1.07674 1.12577 1.00000 −0.287376
1.9 −0.0282840 −1.00000 −1.99920 1.00000 0.0282840 2.73754 0.113114 1.00000 −0.0282840
1.10 0.551239 −1.00000 −1.69614 1.00000 −0.551239 4.16860 −2.03745 1.00000 0.551239
1.11 1.15560 −1.00000 −0.664588 1.00000 −1.15560 3.45180 −3.07920 1.00000 1.15560
1.12 1.39113 −1.00000 −0.0647466 1.00000 −1.39113 1.38286 −2.87234 1.00000 1.39113
1.13 1.42963 −1.00000 0.0438374 1.00000 −1.42963 −4.47441 −2.79659 1.00000 1.42963
1.14 2.00742 −1.00000 2.02973 1.00000 −2.00742 −0.373125 0.0596763 1.00000 2.00742
1.15 2.54626 −1.00000 4.48346 1.00000 −2.54626 −1.02990 6.32355 1.00000 2.54626
1.16 2.59255 −1.00000 4.72133 1.00000 −2.59255 3.82140 7.05518 1.00000 2.59255
1.17 2.67896 −1.00000 5.17683 1.00000 −2.67896 3.10465 8.51061 1.00000 2.67896
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(-1\)
\(31\) \(-1\)

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 6045.2.a.bh 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.bh 17 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\( T_{2}^{17} - 2 T_{2}^{16} - 25 T_{2}^{15} + 47 T_{2}^{14} + 252 T_{2}^{13} - 437 T_{2}^{12} - 1319 T_{2}^{11} + 2056 T_{2}^{10} + 3854 T_{2}^{9} - 5201 T_{2}^{8} - 6304 T_{2}^{7} + 6915 T_{2}^{6} + 5469 T_{2}^{5} - 4238 T_{2}^{4} + \cdots + 8 \) Copy content Toggle raw display
\( T_{7}^{17} - 18 T_{7}^{16} + 87 T_{7}^{15} + 285 T_{7}^{14} - 3886 T_{7}^{13} + 8156 T_{7}^{12} + 32523 T_{7}^{11} - 163078 T_{7}^{10} + 89191 T_{7}^{9} + 650335 T_{7}^{8} - 936806 T_{7}^{7} - 1037415 T_{7}^{6} + \cdots + 300032 \) Copy content Toggle raw display
\( T_{11}^{17} - 3 T_{11}^{16} - 118 T_{11}^{15} + 332 T_{11}^{14} + 5573 T_{11}^{13} - 13939 T_{11}^{12} - 138926 T_{11}^{11} + 289292 T_{11}^{10} + 2006215 T_{11}^{9} - 3185249 T_{11}^{8} - 16985131 T_{11}^{7} + \cdots + 4723456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 2 T^{16} - 25 T^{15} + 47 T^{14} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( (T + 1)^{17} \) Copy content Toggle raw display
$5$ \( (T - 1)^{17} \) Copy content Toggle raw display
$7$ \( T^{17} - 18 T^{16} + 87 T^{15} + \cdots + 300032 \) Copy content Toggle raw display
$11$ \( T^{17} - 3 T^{16} - 118 T^{15} + \cdots + 4723456 \) Copy content Toggle raw display
$13$ \( (T - 1)^{17} \) Copy content Toggle raw display
$17$ \( T^{17} - 194 T^{15} + 116 T^{14} + \cdots + 3557248 \) Copy content Toggle raw display
$19$ \( T^{17} - 10 T^{16} - 89 T^{15} + \cdots - 39512384 \) Copy content Toggle raw display
$23$ \( T^{17} - 16 T^{16} - 43 T^{15} + \cdots + 41598976 \) Copy content Toggle raw display
$29$ \( T^{17} + 3 T^{16} + \cdots + 10158432256 \) Copy content Toggle raw display
$31$ \( (T - 1)^{17} \) Copy content Toggle raw display
$37$ \( T^{17} - 14 T^{16} - 126 T^{15} + \cdots + 47612288 \) Copy content Toggle raw display
$41$ \( T^{17} + 6 T^{16} + \cdots - 1836031564 \) Copy content Toggle raw display
$43$ \( T^{17} - 24 T^{16} + \cdots - 63721621376 \) Copy content Toggle raw display
$47$ \( T^{17} - 25 T^{16} + \cdots - 39938914816 \) Copy content Toggle raw display
$53$ \( T^{17} + 15 T^{16} + \cdots - 145888741792 \) Copy content Toggle raw display
$59$ \( T^{17} - 16 T^{16} + \cdots + 406374236928 \) Copy content Toggle raw display
$61$ \( T^{17} + 5 T^{16} - 430 T^{15} + \cdots + 197654464 \) Copy content Toggle raw display
$67$ \( T^{17} - 50 T^{16} + \cdots - 2542811148544 \) Copy content Toggle raw display
$71$ \( T^{17} - 16 T^{16} + \cdots + 30247986528256 \) Copy content Toggle raw display
$73$ \( T^{17} - 33 T^{16} + \cdots + 2671259776 \) Copy content Toggle raw display
$79$ \( T^{17} + 10 T^{16} + \cdots - 2222561675264 \) Copy content Toggle raw display
$83$ \( T^{17} - 27 T^{16} + \cdots + 57\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{17} + 24 T^{16} + \cdots - 1997470044544 \) Copy content Toggle raw display
$97$ \( T^{17} - 48 T^{16} + \cdots - 17240741888 \) Copy content Toggle raw display
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