Properties

Label 889.2.a.b
Level $889$
Weight $2$
Character orbit 889.a
Self dual yes
Analytic conductor $7.099$
Analytic rank $0$
Dimension $15$
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] = [889,2,Mod(1,889)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(889, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("889.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 889 = 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 889.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: \(7.09870073969\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + 726 x^{3} + 145 x^{2} - 83 x - 13 \) 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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{6} - q^{7} + ( - \beta_{14} - \beta_{12} + \beta_{9} - \beta_{6} + \beta_{4}) q^{8} + ( - \beta_{13} + \beta_{9}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{8} q^{5} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{6} - q^{7} + ( - \beta_{14} - \beta_{12} + \beta_{9} - \beta_{6} + \beta_{4}) q^{8} + ( - \beta_{13} + \beta_{9}) q^{9} + (\beta_{7} - \beta_{5} - \beta_{3} - \beta_1 + 1) q^{10} + (\beta_{14} + \beta_{12} + \beta_{8} - \beta_{4}) q^{11} + ( - \beta_{14} + 2 \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} + 1) q^{12} + ( - \beta_{4} - \beta_1) q^{13} + \beta_1 q^{14} + ( - \beta_{10} - \beta_{2}) q^{15} + (\beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + 2 \beta_{2} + 2) q^{16} + (\beta_{12} + \beta_{6} + 1) q^{17} + (\beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{18} + (\beta_{14} + \beta_{11} - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - \beta_{2} + 1) q^{19} + (\beta_{14} + \beta_{13} - \beta_{12} - 3 \beta_{9} - \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{20} - \beta_{9} q^{21} + (\beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{2} - 2 \beta_1 - 1) q^{22} + (\beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{6} - \beta_{4} + 1) q^{23} + ( - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{24} + ( - \beta_{14} - \beta_{10} + \beta_{8} - 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{25} + ( - \beta_{11} - 2 \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_1 + 3) q^{26} + ( - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{8} - \beta_1 + 2) q^{27} + ( - \beta_{2} - 1) q^{28} + (\beta_{14} + \beta_{13} - \beta_{12} + 2 \beta_{10} + \beta_{3} + 2) q^{29} + (\beta_{14} + \beta_{13} + \beta_{12} - \beta_{9} - 2 \beta_{4} - \beta_{3} + \beta_{2}) q^{30} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + 2) q^{31} + ( - 2 \beta_{14} - \beta_{12} - \beta_{11} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 3 \beta_{5} + \cdots + 2) q^{32}+ \cdots + (3 \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - 3 \beta_{4} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 4 q^{3} + 14 q^{4} + 7 q^{5} + 8 q^{6} - 15 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 4 q^{3} + 14 q^{4} + 7 q^{5} + 8 q^{6} - 15 q^{7} + 9 q^{9} + 10 q^{10} + 14 q^{11} + 10 q^{12} + 6 q^{13} + 6 q^{15} + 20 q^{16} + 10 q^{17} + q^{18} + 13 q^{19} + 8 q^{20} - 4 q^{21} - 11 q^{22} + 15 q^{23} + 34 q^{24} + 22 q^{26} + 22 q^{27} - 14 q^{28} + 16 q^{29} + 7 q^{30} + 22 q^{31} + 14 q^{33} - 15 q^{34} - 7 q^{35} + 20 q^{36} - 14 q^{37} - 6 q^{38} + 29 q^{39} + 22 q^{40} + 19 q^{41} - 8 q^{42} - q^{43} + 25 q^{44} - 8 q^{45} - 28 q^{46} + 49 q^{47} - 14 q^{48} + 15 q^{49} + 24 q^{50} - 8 q^{51} - 17 q^{52} - 28 q^{53} + 13 q^{54} + 39 q^{55} - 12 q^{57} - 10 q^{58} + 43 q^{59} - 60 q^{60} + 27 q^{61} + 14 q^{62} - 9 q^{63} + 18 q^{64} - 8 q^{65} - 36 q^{66} + 3 q^{67} + 13 q^{68} - 17 q^{69} - 10 q^{70} + 55 q^{71} - 21 q^{72} - 3 q^{73} - 12 q^{74} + 8 q^{75} - 20 q^{76} - 14 q^{77} - 6 q^{78} + 18 q^{79} + 29 q^{80} - 17 q^{81} + 14 q^{82} + 17 q^{83} - 10 q^{84} + 7 q^{85} + 4 q^{86} + 35 q^{87} - 114 q^{88} + 36 q^{89} - 39 q^{90} - 6 q^{91} + 45 q^{92} + 15 q^{93} - 15 q^{94} + 59 q^{95} + 85 q^{96} - 2 q^{97} + 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + 726 x^{3} + 145 x^{2} - 83 x - 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5995 \nu^{14} - 50387 \nu^{13} - 70689 \nu^{12} + 1136059 \nu^{11} - 104425 \nu^{10} - 9736369 \nu^{9} + 4215784 \nu^{8} + 39113971 \nu^{7} - 17615205 \nu^{6} + \cdots - 1488901 ) / 624662 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13615 \nu^{14} - 201176 \nu^{13} - 414259 \nu^{12} + 4188080 \nu^{11} + 4802080 \nu^{10} - 32567535 \nu^{9} - 26615464 \nu^{8} + 116731019 \nu^{7} + \cdots - 2677270 ) / 312331 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 46461 \nu^{14} + 346281 \nu^{13} - 856573 \nu^{12} - 7272379 \nu^{11} + 5275825 \nu^{10} + 57143503 \nu^{9} - 10437030 \nu^{8} - 207601489 \nu^{7} + \cdots + 3303757 ) / 624662 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 61025 \nu^{14} + 251381 \nu^{13} - 1185327 \nu^{12} - 5203625 \nu^{11} + 8156911 \nu^{10} + 39981133 \nu^{9} - 22898694 \nu^{8} - 140135849 \nu^{7} + \cdots + 1161803 ) / 624662 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 89367 \nu^{14} + 251991 \nu^{13} - 1775425 \nu^{12} - 5236867 \nu^{11} + 12760035 \nu^{10} + 40507123 \nu^{9} - 39479230 \nu^{8} - 144087755 \nu^{7} + \cdots - 424393 ) / 624662 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 124199 \nu^{14} - 36911 \nu^{13} - 2688475 \nu^{12} + 848163 \nu^{11} + 22201561 \nu^{10} - 7588635 \nu^{9} - 87863826 \nu^{8} + 32388591 \nu^{7} + \cdots - 4275743 ) / 624662 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 71314 \nu^{14} + 21453 \nu^{13} + 1521763 \nu^{12} - 459426 \nu^{11} - 12246648 \nu^{10} + 3742105 \nu^{9} + 46094392 \nu^{8} - 14021902 \nu^{7} + \cdots + 295618 ) / 312331 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 124624 \nu^{14} + 196305 \nu^{13} - 2558030 \nu^{12} - 4051663 \nu^{11} + 19403758 \nu^{10} + 31050350 \nu^{9} - 66378289 \nu^{8} - 109257358 \nu^{7} + \cdots + 2534792 ) / 312331 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 165788 \nu^{14} + 202150 \nu^{13} - 3394240 \nu^{12} - 4150019 \nu^{11} + 25626527 \nu^{10} + 31492447 \nu^{9} - 86766924 \nu^{8} - 108999529 \nu^{7} + \cdots + 2727176 ) / 312331 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 457793 \nu^{14} - 140549 \nu^{13} - 9783541 \nu^{12} + 3108413 \nu^{11} + 79045069 \nu^{10} - 26346187 \nu^{9} - 300616102 \nu^{8} + 104110845 \nu^{7} + \cdots - 6116335 ) / 624662 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 243430 \nu^{14} + 83019 \nu^{13} - 5093930 \nu^{12} - 1622009 \nu^{11} + 39878673 \nu^{10} + 11264670 \nu^{9} - 144341909 \nu^{8} - 33752528 \nu^{7} + \cdots - 283714 ) / 312331 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 317108 \nu^{14} - 235139 \nu^{13} + 6591938 \nu^{12} + 4776260 \nu^{11} - 51045558 \nu^{10} - 35642903 \nu^{9} + 181236326 \nu^{8} + 120721619 \nu^{7} + \cdots + 95614 ) / 312331 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{12} - \beta_{9} + \beta_{6} - \beta_{4} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{14} + 9 \beta_{12} + \beta_{11} - 10 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 8 \beta_{6} + 3 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + \beta_{2} + 22 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{14} + 9 \beta_{13} - 11 \beta_{12} - 10 \beta_{11} + 9 \beta_{10} - 13 \beta_{9} + 2 \beta_{8} - 11 \beta_{7} - \beta_{4} + 57 \beta_{2} + \beta _1 + 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 79 \beta_{14} - \beta_{13} + 68 \beta_{12} + 13 \beta_{11} - 2 \beta_{10} - 81 \beta_{9} + 26 \beta_{8} - 28 \beta_{7} + 56 \beta_{6} + 37 \beta_{5} - 70 \beta_{4} + 23 \beta_{3} + 14 \beta_{2} + 137 \beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 8 \beta_{14} + 66 \beta_{13} - 91 \beta_{12} - 77 \beta_{11} + 67 \beta_{10} - 128 \beta_{9} + 32 \beta_{8} - 97 \beta_{7} + \beta_{6} + 3 \beta_{5} - 18 \beta_{4} + 2 \beta_{3} + 398 \beta_{2} + 16 \beta _1 + 601 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 580 \beta_{14} - 13 \beta_{13} + 486 \beta_{12} + 117 \beta_{11} - 25 \beta_{10} - 616 \beta_{9} + 245 \beta_{8} - 280 \beta_{7} + 380 \beta_{6} + 337 \beta_{5} - 521 \beta_{4} + 200 \beta_{3} + 150 \beta_{2} + 901 \beta _1 - 208 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 24 \beta_{14} + 461 \beta_{13} - 673 \beta_{12} - 545 \beta_{11} + 473 \beta_{10} - 1123 \beta_{9} + 352 \beta_{8} - 792 \beta_{7} + 21 \beta_{6} + 58 \beta_{5} - 222 \beta_{4} + 34 \beta_{3} + 2775 \beta_{2} + 180 \beta _1 + 3848 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4133 \beta_{14} - 121 \beta_{13} + 3390 \beta_{12} + 919 \beta_{11} - 212 \beta_{10} - 4567 \beta_{9} + 2051 \beta_{8} - 2454 \beta_{7} + 2562 \beta_{6} + 2738 \beta_{5} - 3801 \beta_{4} + 1579 \beta_{3} + 1430 \beta_{2} + \cdots - 1566 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 235 \beta_{14} + 3186 \beta_{13} - 4698 \beta_{12} - 3722 \beta_{11} + 3269 \beta_{10} - 9269 \beta_{9} + 3314 \beta_{8} - 6231 \beta_{7} + 282 \beta_{6} + 738 \beta_{5} - 2302 \beta_{4} + 395 \beta_{3} + 19409 \beta_{2} + \cdots + 25023 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 29066 \beta_{14} - 987 \beta_{13} + 23369 \beta_{12} + 6765 \beta_{11} - 1512 \beta_{10} - 33474 \beta_{9} + 16229 \beta_{8} - 20119 \beta_{7} + 17306 \beta_{6} + 21050 \beta_{5} - 27447 \beta_{4} + \cdots - 10818 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5213 \beta_{14} + 22030 \beta_{13} - 31683 \beta_{12} - 24988 \beta_{11} + 22382 \beta_{10} - 73792 \beta_{9} + 28733 \beta_{8} - 48003 \beta_{7} + 3120 \beta_{6} + 7809 \beta_{5} - 21594 \beta_{4} + \cdots + 164749 \) 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
2.71493
2.30266
2.16969
1.32572
1.22206
1.15799
0.371530
−0.146355
−0.601235
−0.735585
−0.955258
−1.64397
−2.07483
−2.51564
−2.59170
−2.71493 −2.30660 5.37083 1.33828 6.26225 −1.00000 −9.15156 2.32040 −3.63333
1.2 −2.30266 0.689513 3.30224 −1.35662 −1.58771 −1.00000 −2.99860 −2.52457 3.12384
1.3 −2.16969 2.40688 2.70756 −2.09371 −5.22219 −1.00000 −1.53520 2.79307 4.54272
1.4 −1.32572 −2.26843 −0.242472 −1.07970 3.00730 −1.00000 2.97289 2.14577 1.43138
1.5 −1.22206 0.302349 −0.506578 1.90746 −0.369487 −1.00000 3.06318 −2.90859 −2.33103
1.6 −1.15799 2.52691 −0.659052 3.10556 −2.92615 −1.00000 3.07916 3.38529 −3.59621
1.7 −0.371530 −1.28062 −1.86197 0.962757 0.475791 −1.00000 1.43484 −1.36000 −0.357693
1.8 0.146355 −0.729258 −1.97858 −2.93285 −0.106730 −1.00000 −0.582285 −2.46818 −0.429238
1.9 0.601235 0.551784 −1.63852 4.40463 0.331752 −1.00000 −2.18760 −2.69553 2.64822
1.10 0.735585 3.18186 −1.45891 0.430466 2.34053 −1.00000 −2.54433 7.12424 0.316644
1.11 0.955258 −1.40478 −1.08748 −2.56419 −1.34193 −1.00000 −2.94934 −1.02659 −2.44946
1.12 1.64397 −2.36628 0.702628 1.50234 −3.89009 −1.00000 −2.13284 2.59930 2.46980
1.13 2.07483 1.81987 2.30492 1.11193 3.77592 −1.00000 0.632654 0.311930 2.30706
1.14 2.51564 2.69647 4.32847 −1.19101 6.78336 −1.00000 5.85760 4.27094 −2.99615
1.15 2.59170 0.180340 4.71692 3.45466 0.467388 −1.00000 7.04143 −2.96748 8.95345
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(127\) \(-1\)

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 889.2.a.b 15
3.b odd 2 1 8001.2.a.q 15
7.b odd 2 1 6223.2.a.j 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
889.2.a.b 15 1.a even 1 1 trivial
6223.2.a.j 15 7.b odd 2 1
8001.2.a.q 15 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 22 T_{2}^{13} + 186 T_{2}^{11} - 763 T_{2}^{9} + 7 T_{2}^{8} + 1588 T_{2}^{7} - 64 T_{2}^{6} - 1625 T_{2}^{5} + 185 T_{2}^{4} + 726 T_{2}^{3} - 145 T_{2}^{2} - 83 T_{2} + 13 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(889))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 22 T^{13} + 186 T^{11} + \cdots + 13 \) Copy content Toggle raw display
$3$ \( T^{15} - 4 T^{14} - 19 T^{13} + 82 T^{12} + \cdots - 32 \) Copy content Toggle raw display
$5$ \( T^{15} - 7 T^{14} - 13 T^{13} + \cdots - 2294 \) Copy content Toggle raw display
$7$ \( (T + 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} - 14 T^{14} + 2 T^{13} + \cdots - 302112 \) Copy content Toggle raw display
$13$ \( T^{15} - 6 T^{14} - 62 T^{13} + 318 T^{12} + \cdots - 296 \) Copy content Toggle raw display
$17$ \( T^{15} - 10 T^{14} - 48 T^{13} + \cdots + 13032 \) Copy content Toggle raw display
$19$ \( T^{15} - 13 T^{14} - 43 T^{13} + \cdots - 2261282 \) Copy content Toggle raw display
$23$ \( T^{15} - 15 T^{14} - 26 T^{13} + \cdots + 159348 \) Copy content Toggle raw display
$29$ \( T^{15} - 16 T^{14} - 56 T^{13} + \cdots - 24350536 \) Copy content Toggle raw display
$31$ \( T^{15} - 22 T^{14} + \cdots - 508362642 \) Copy content Toggle raw display
$37$ \( T^{15} + 14 T^{14} + \cdots - 40940005196 \) Copy content Toggle raw display
$41$ \( T^{15} - 19 T^{14} - 83 T^{13} + \cdots + 14583848 \) Copy content Toggle raw display
$43$ \( T^{15} + T^{14} - 315 T^{13} + \cdots + 2881799908 \) Copy content Toggle raw display
$47$ \( T^{15} - 49 T^{14} + \cdots - 127047598350 \) Copy content Toggle raw display
$53$ \( T^{15} + 28 T^{14} + \cdots + 33403676712 \) Copy content Toggle raw display
$59$ \( T^{15} - 43 T^{14} + \cdots - 925256454912 \) Copy content Toggle raw display
$61$ \( T^{15} - 27 T^{14} + \cdots - 26580006408 \) Copy content Toggle raw display
$67$ \( T^{15} - 3 T^{14} + \cdots - 38013704592 \) Copy content Toggle raw display
$71$ \( T^{15} - 55 T^{14} + \cdots - 23410866384 \) Copy content Toggle raw display
$73$ \( T^{15} + 3 T^{14} + \cdots - 1903703128 \) Copy content Toggle raw display
$79$ \( T^{15} - 18 T^{14} + \cdots + 1041479699184 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 328479194135568 \) Copy content Toggle raw display
$89$ \( T^{15} - 36 T^{14} + \cdots + 217040265526 \) Copy content Toggle raw display
$97$ \( T^{15} + 2 T^{14} + \cdots - 1094692262 \) Copy content Toggle raw display
show more
show less