Properties

Label 6027.2.a.bm
Level $6027$
Weight $2$
Character orbit 6027.a
Self dual yes
Analytic conductor $48.126$
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] = [6027,2,Mod(1,6027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, 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("6027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6027.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: \(48.1258372982\)
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: \( x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 64 x^{12} - 456 x^{11} - 54 x^{10} + 1532 x^{9} - 400 x^{8} - 2708 x^{7} + 1218 x^{6} + 2424 x^{5} - 1276 x^{4} - 960 x^{3} + 500 x^{2} + \cdots - 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{5} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{5} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2} - 1) q^{8} + q^{9} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{10} + (\beta_{15} - \beta_{14} - \beta_{13} - \beta_{11} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{11} + (\beta_{2} + 1) q^{12} + ( - \beta_{15} - \beta_{5}) q^{13} + (\beta_{5} - 1) q^{15} + (\beta_{10} - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{2} + \beta_1) q^{16} + (\beta_{15} - \beta_{14} - \beta_{11} - \beta_{8} + \beta_{4} - 1) q^{17} - \beta_1 q^{18} + (\beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{8} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{19} + (\beta_{14} + \beta_{13} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{2} - \beta_1 - 2) q^{20} + ( - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{11} + 2 \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} + \beta_{3} + \cdots - 1) q^{22}+ \cdots + (\beta_{15} - \beta_{14} - \beta_{13} - \beta_{11} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 64 x^{12} - 456 x^{11} - 54 x^{10} + 1532 x^{9} - 400 x^{8} - 2708 x^{7} + 1218 x^{6} + 2424 x^{5} - 1276 x^{4} - 960 x^{3} + 500 x^{2} + \cdots - 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{15} + 316 \nu^{14} - 525 \nu^{13} - 5554 \nu^{12} + 7439 \nu^{11} + 39832 \nu^{10} - 38854 \nu^{9} - 146235 \nu^{8} + 97946 \nu^{7} + 285731 \nu^{6} - 129437 \nu^{5} + \cdots - 16891 ) / 1659 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 344 \nu^{15} - 711 \nu^{14} - 6475 \nu^{13} + 11870 \nu^{12} + 48798 \nu^{11} - 77407 \nu^{10} - 186478 \nu^{9} + 251383 \nu^{8} + 380344 \nu^{7} - 426432 \nu^{6} - 399882 \nu^{5} + \cdots + 13062 ) / 553 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 615 \nu^{15} - 1501 \nu^{14} - 10822 \nu^{13} + 24523 \nu^{12} + 75753 \nu^{11} - 155889 \nu^{10} - 266506 \nu^{9} + 489196 \nu^{8} + 494579 \nu^{7} - 790497 \nu^{6} + \cdots + 17073 ) / 553 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 615 \nu^{15} - 1501 \nu^{14} - 10822 \nu^{13} + 24523 \nu^{12} + 75753 \nu^{11} - 155889 \nu^{10} - 266506 \nu^{9} + 489196 \nu^{8} + 494579 \nu^{7} - 790497 \nu^{6} + \cdots + 18179 ) / 553 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 137 \nu^{15} + 316 \nu^{14} + 2471 \nu^{13} - 5199 \nu^{12} - 17793 \nu^{11} + 33328 \nu^{10} + 64712 \nu^{9} - 105797 \nu^{8} - 124989 \nu^{7} + 173926 \nu^{6} + 123884 \nu^{5} + \cdots - 4621 ) / 79 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 230 \nu^{15} + 553 \nu^{14} + 4089 \nu^{13} - 9080 \nu^{12} - 28999 \nu^{11} + 58088 \nu^{10} + 103836 \nu^{9} - 183812 \nu^{8} - 197609 \nu^{7} + 300321 \nu^{6} + 193949 \nu^{5} + \cdots - 7202 ) / 79 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5042 \nu^{15} + 11929 \nu^{14} + 89943 \nu^{13} - 195436 \nu^{12} - 639991 \nu^{11} + 1246429 \nu^{10} + 2297459 \nu^{9} - 3928944 \nu^{8} - 4373500 \nu^{7} + \cdots - 153286 ) / 1659 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5042 \nu^{15} + 11929 \nu^{14} + 89943 \nu^{13} - 195436 \nu^{12} - 639991 \nu^{11} + 1246429 \nu^{10} + 2297459 \nu^{9} - 3928944 \nu^{8} - 4373500 \nu^{7} + \cdots - 144991 ) / 1659 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2533 \nu^{15} - 5925 \nu^{14} - 45416 \nu^{13} + 97309 \nu^{12} + 324855 \nu^{11} - 622481 \nu^{10} - 1172474 \nu^{9} + 1970354 \nu^{8} + 2243872 \nu^{7} - 3224355 \nu^{6} + \cdots + 78449 ) / 553 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2547 \nu^{15} + 5925 \nu^{14} + 45612 \nu^{13} - 96924 \nu^{12} - 325870 \nu^{11} + 617063 \nu^{10} + 1174231 \nu^{9} - 1941640 \nu^{8} - 2241205 \nu^{7} + 3152381 \nu^{6} + \cdots - 72814 ) / 553 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1177 \nu^{15} + 2765 \nu^{14} + 21060 \nu^{13} - 45329 \nu^{12} - 150389 \nu^{11} + 289313 \nu^{10} + 542200 \nu^{9} - 912900 \nu^{8} - 1037609 \nu^{7} + 1487065 \nu^{6} + \cdots - 35294 ) / 237 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 9529 \nu^{15} - 22436 \nu^{14} - 170457 \nu^{13} + 367622 \nu^{12} + 1218296 \nu^{11} - 2345744 \nu^{10} - 4404277 \nu^{9} + 7401420 \nu^{8} + 8477201 \nu^{7} + \cdots + 286916 ) / 1659 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5834 \nu^{15} - 13746 \nu^{14} - 104349 \nu^{13} + 225401 \nu^{12} + 745048 \nu^{11} - 1438573 \nu^{10} - 2687259 \nu^{9} + 4536144 \nu^{8} + 5150163 \nu^{7} + \cdots + 170730 ) / 553 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{6} - \beta_{5} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} + 2\beta_{10} + 8\beta_{6} - 7\beta_{5} + 11\beta_{2} + 19\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{13} + 10 \beta_{10} - 8 \beta_{9} + \beta_{7} + 12 \beta_{6} - 10 \beta_{5} + \beta_{4} + 48 \beta_{2} + 13 \beta _1 + 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10 \beta_{13} - \beta_{11} + 23 \beta_{10} - 3 \beta_{9} + 58 \beta_{6} - 45 \beta_{5} + 2 \beta_{4} + 96 \beta_{2} + 103 \beta _1 + 109 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{14} - 12 \beta_{13} + \beta_{12} + 80 \beta_{10} - 54 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} + 110 \beta_{6} - 81 \beta_{5} + 12 \beta_{4} + 337 \beta_{2} + 123 \beta _1 + 495 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{15} + \beta_{14} - 73 \beta_{13} - 12 \beta_{11} + 199 \beta_{10} - 44 \beta_{9} + 3 \beta_{8} - \beta_{7} + 419 \beta_{6} - 296 \beta_{5} + 28 \beta_{4} - \beta_{3} + 770 \beta_{2} + 619 \beta _1 + 894 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3 \beta_{15} + 15 \beta_{14} - 99 \beta_{13} + 12 \beta_{12} - 3 \beta_{11} + 605 \beta_{10} - 355 \beta_{9} + 30 \beta_{8} + 53 \beta_{7} + 916 \beta_{6} - 618 \beta_{5} + 108 \beta_{4} - \beta_{3} + 2413 \beta_{2} + 1028 \beta _1 + 3297 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 19 \beta_{15} + 24 \beta_{14} - 469 \beta_{13} + \beta_{12} - 98 \beta_{11} + 1571 \beta_{10} - 441 \beta_{9} + 55 \beta_{8} - 25 \beta_{7} + 3055 \beta_{6} - 2006 \beta_{5} + 274 \beta_{4} - 15 \beta_{3} + 5940 \beta_{2} + 4001 \beta _1 + 6978 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 58 \beta_{15} + 161 \beta_{14} - 689 \beta_{13} + 94 \beta_{12} - 50 \beta_{11} + 4503 \beta_{10} - 2348 \beta_{9} + 310 \beta_{8} + 224 \beta_{7} + 7288 \beta_{6} - 4598 \beta_{5} + 888 \beta_{4} - 20 \beta_{3} + 17500 \beta_{2} + \cdots + 22773 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 239 \beta_{15} + 339 \beta_{14} - 2779 \beta_{13} + 12 \beta_{12} - 686 \beta_{11} + 11965 \beta_{10} - 3795 \beta_{9} + 661 \beta_{8} - 385 \beta_{7} + 22482 \beta_{6} - 13893 \beta_{5} + 2351 \beta_{4} - 152 \beta_{3} + \cdots + 53012 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 730 \beta_{15} + 1547 \beta_{14} - 4238 \beta_{13} + 599 \beta_{12} - 527 \beta_{11} + 33376 \beta_{10} - 15762 \beta_{9} + 2807 \beta_{8} + 313 \beta_{7} + 56597 \beta_{6} - 33753 \beta_{5} + 7042 \beta_{4} + \cdots + 160772 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2528 \beta_{15} + 3805 \beta_{14} - 15189 \beta_{13} + 48 \beta_{12} - 4449 \beta_{11} + 89759 \beta_{10} - 30258 \beta_{9} + 6660 \beta_{8} - 4736 \beta_{7} + 166624 \beta_{6} - 97558 \beta_{5} + \cdots + 396492 \) 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
2.72726
2.69180
1.83908
1.64639
1.63508
1.44011
1.04648
0.631061
0.304034
−0.404293
−0.819942
−1.10456
−1.49558
−1.87592
−2.11474
−2.14627
−2.72726 1.00000 5.43794 −0.416230 −2.72726 0 −9.37616 1.00000 1.13517
1.2 −2.69180 1.00000 5.24578 −3.62847 −2.69180 0 −8.73699 1.00000 9.76712
1.3 −1.83908 1.00000 1.38223 −2.07594 −1.83908 0 1.13613 1.00000 3.81783
1.4 −1.64639 1.00000 0.710595 0.0457395 −1.64639 0 2.12286 1.00000 −0.0753050
1.5 −1.63508 1.00000 0.673497 1.18651 −1.63508 0 2.16894 1.00000 −1.94005
1.6 −1.44011 1.00000 0.0739309 3.20961 −1.44011 0 2.77376 1.00000 −4.62221
1.7 −1.04648 1.00000 −0.904885 1.81328 −1.04648 0 3.03990 1.00000 −1.89755
1.8 −0.631061 1.00000 −1.60176 −1.65938 −0.631061 0 2.27293 1.00000 1.04717
1.9 −0.304034 1.00000 −1.90756 −0.824488 −0.304034 0 1.18803 1.00000 0.250672
1.10 0.404293 1.00000 −1.83655 −3.62653 0.404293 0 −1.55109 1.00000 −1.46618
1.11 0.819942 1.00000 −1.32769 0.332064 0.819942 0 −2.72852 1.00000 0.272273
1.12 1.10456 1.00000 −0.779950 −3.67747 1.10456 0 −3.07062 1.00000 −4.06198
1.13 1.49558 1.00000 0.236756 1.30108 1.49558 0 −2.63707 1.00000 1.94586
1.14 1.87592 1.00000 1.51907 −1.33107 1.87592 0 −0.902182 1.00000 −2.49697
1.15 2.11474 1.00000 2.47212 −0.285010 2.11474 0 0.998417 1.00000 −0.602721
1.16 2.14627 1.00000 2.60647 −2.36369 2.14627 0 1.30166 1.00000 −5.07312
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(41\) \(-1\)

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 6027.2.a.bm yes 16
7.b odd 2 1 6027.2.a.bl 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6027.2.a.bl 16 7.b odd 2 1
6027.2.a.bm yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{16} + 4 T_{2}^{15} - 14 T_{2}^{14} - 68 T_{2}^{13} + 64 T_{2}^{12} + 456 T_{2}^{11} - 54 T_{2}^{10} - 1532 T_{2}^{9} - 400 T_{2}^{8} + 2708 T_{2}^{7} + 1218 T_{2}^{6} - 2424 T_{2}^{5} - 1276 T_{2}^{4} + 960 T_{2}^{3} + 500 T_{2}^{2} + \cdots - 49 \) Copy content Toggle raw display
\( T_{5}^{16} + 12 T_{5}^{15} + 36 T_{5}^{14} - 100 T_{5}^{13} - 730 T_{5}^{12} - 732 T_{5}^{11} + 2800 T_{5}^{10} + 6192 T_{5}^{9} - 1464 T_{5}^{8} - 12676 T_{5}^{7} - 6132 T_{5}^{6} + 7724 T_{5}^{5} + 6906 T_{5}^{4} + 404 T_{5}^{3} + \cdots + 7 \) Copy content Toggle raw display
\( T_{13}^{16} - 84 T_{13}^{14} + 88 T_{13}^{13} + 2588 T_{13}^{12} - 4612 T_{13}^{11} - 36046 T_{13}^{10} + 87176 T_{13}^{9} + 212080 T_{13}^{8} - 706696 T_{13}^{7} - 208090 T_{13}^{6} + 2155768 T_{13}^{5} - 1691066 T_{13}^{4} + \cdots + 6489 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{15} - 14 T^{14} - 68 T^{13} + \cdots - 49 \) Copy content Toggle raw display
$3$ \( (T - 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 12 T^{15} + 36 T^{14} - 100 T^{13} + \cdots + 7 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 4 T^{15} - 80 T^{14} + \cdots + 1136 \) Copy content Toggle raw display
$13$ \( T^{16} - 84 T^{14} + 88 T^{13} + \cdots + 6489 \) Copy content Toggle raw display
$17$ \( T^{16} + 8 T^{15} - 86 T^{14} + \cdots - 12012100 \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} - 162 T^{14} + \cdots + 34097252 \) Copy content Toggle raw display
$23$ \( T^{16} + 12 T^{15} - 68 T^{14} + \cdots - 38586529 \) Copy content Toggle raw display
$29$ \( T^{16} + 16 T^{15} - 58 T^{14} + \cdots + 11398527 \) Copy content Toggle raw display
$31$ \( T^{16} + 4 T^{15} - 264 T^{14} + \cdots + 227939012 \) Copy content Toggle raw display
$37$ \( T^{16} + 48 T^{15} + 824 T^{14} + \cdots - 29586151 \) Copy content Toggle raw display
$41$ \( (T - 1)^{16} \) Copy content Toggle raw display
$43$ \( T^{16} + 16 T^{15} + \cdots - 277170012 \) Copy content Toggle raw display
$47$ \( T^{16} + 36 T^{15} + \cdots + 387817355639 \) Copy content Toggle raw display
$53$ \( T^{16} + 60 T^{15} + \cdots - 5526668001953 \) Copy content Toggle raw display
$59$ \( T^{16} + 36 T^{15} + \cdots + 129945682972 \) Copy content Toggle raw display
$61$ \( T^{16} + 4 T^{15} + \cdots - 150423854076 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 111899955411809 \) Copy content Toggle raw display
$71$ \( T^{16} + 12 T^{15} + \cdots - 1629727719300 \) Copy content Toggle raw display
$73$ \( T^{16} + 16 T^{15} + \cdots - 2108630966236 \) Copy content Toggle raw display
$79$ \( T^{16} + 36 T^{15} + 36 T^{14} + \cdots - 56486767 \) Copy content Toggle raw display
$83$ \( T^{16} + 32 T^{15} + \cdots + 70542671296 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots - 227280314382608 \) Copy content Toggle raw display
$97$ \( T^{16} - 48 T^{15} + \cdots - 30092709652439 \) Copy content Toggle raw display
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