Properties

Label 8022.2.a.z
Level $8022$
Weight $2$
Character orbit 8022.a
Self dual yes
Analytic conductor $64.056$
Analytic rank $0$
Dimension $15$
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] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, 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("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.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: \(64.0559925015\)
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: \( x^{15} - 53 x^{13} - x^{12} + 1068 x^{11} + 45 x^{10} - 10139 x^{9} - 615 x^{8} + 45390 x^{7} + 2130 x^{6} - 84842 x^{5} + 7822 x^{4} + 62828 x^{3} - 16144 x^{2} + \cdots + 4704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - \beta_1 q^{5} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - \beta_1 q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + \beta_1 q^{10} - \beta_{5} q^{11} + q^{12} + (\beta_{10} + 1) q^{13} - q^{14} - \beta_1 q^{15} + q^{16} + \beta_{7} q^{17} - q^{18} + ( - \beta_{14} + 1) q^{19} - \beta_1 q^{20} + q^{21} + \beta_{5} q^{22} + \beta_{9} q^{23} - q^{24} + (\beta_{2} + 2) q^{25} + ( - \beta_{10} - 1) q^{26} + q^{27} + q^{28} - \beta_{12} q^{29} + \beta_1 q^{30} + ( - \beta_{11} + \beta_{8} + 1) q^{31} - q^{32} - \beta_{5} q^{33} - \beta_{7} q^{34} - \beta_1 q^{35} + q^{36} + ( - \beta_{12} - \beta_{9} - \beta_{6} + \beta_{3} + 2) q^{37} + (\beta_{14} - 1) q^{38} + (\beta_{10} + 1) q^{39} + \beta_1 q^{40} + ( - \beta_{13} + \beta_{11} - \beta_1) q^{41} - q^{42} + (\beta_{9} + \beta_{5} - \beta_{4} + 1) q^{43} - \beta_{5} q^{44} - \beta_1 q^{45} - \beta_{9} q^{46} + (\beta_{14} + \beta_{12} + \beta_{6}) q^{47} + q^{48} + q^{49} + ( - \beta_{2} - 2) q^{50} + \beta_{7} q^{51} + (\beta_{10} + 1) q^{52} + ( - \beta_{14} - \beta_{6} - \beta_{5} + \beta_{4} + 1) q^{53} - q^{54} + ( - \beta_{14} + \beta_{12} - \beta_{8} + \beta_{6} - 2 \beta_{3} + 1) q^{55} - q^{56} + ( - \beta_{14} + 1) q^{57} + \beta_{12} q^{58} + (\beta_{13} - \beta_{11} - \beta_{8} + 1) q^{59} - \beta_1 q^{60} + (\beta_{14} + \beta_{13} + \beta_{10} - \beta_{5} + \beta_{4} + 3) q^{61} + (\beta_{11} - \beta_{8} - 1) q^{62} + q^{63} + q^{64} + (\beta_{13} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 2 \beta_1 - 1) q^{65} + \beta_{5} q^{66} + (\beta_{12} + \beta_{11} - \beta_{10} - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{2} + 2) q^{67} + \beta_{7} q^{68} + \beta_{9} q^{69} + \beta_1 q^{70} + (\beta_{12} + \beta_{10} - \beta_{6} + \beta_{5} - \beta_{2} - 1) q^{71} - q^{72} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{6} + \beta_{3} - \beta_{2} + 1) q^{73} + (\beta_{12} + \beta_{9} + \beta_{6} - \beta_{3} - 2) q^{74} + (\beta_{2} + 2) q^{75} + ( - \beta_{14} + 1) q^{76} - \beta_{5} q^{77} + ( - \beta_{10} - 1) q^{78} + (\beta_{12} + \beta_{9} - \beta_{8} + \beta_{6} - \beta_{4} - \beta_{3} + 2) q^{79} - \beta_1 q^{80} + q^{81} + (\beta_{13} - \beta_{11} + \beta_1) q^{82} + (\beta_{14} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_1 + 1) q^{83} + q^{84} + (2 \beta_{14} - 2 \beta_{13} + 2 \beta_{10} - 2 \beta_{9} + \beta_{5} + \beta_{4} + 1) q^{85} + ( - \beta_{9} - \beta_{5} + \beta_{4} - 1) q^{86} - \beta_{12} q^{87} + \beta_{5} q^{88} + ( - 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{6} - \beta_{4} + \beta_1 - 1) q^{89} + \beta_1 q^{90} + (\beta_{10} + 1) q^{91} + \beta_{9} q^{92} + ( - \beta_{11} + \beta_{8} + 1) q^{93} + ( - \beta_{14} - \beta_{12} - \beta_{6}) q^{94} + (\beta_{13} + \beta_{11} - 2 \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 - 2) q^{95} - q^{96} + ( - \beta_{12} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{97} - q^{98} - \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 15 q^{2} + 15 q^{3} + 15 q^{4} - 15 q^{6} + 15 q^{7} - 15 q^{8} + 15 q^{9} - 7 q^{11} + 15 q^{12} + 10 q^{13} - 15 q^{14} + 15 q^{16} - 3 q^{17} - 15 q^{18} + 12 q^{19} + 15 q^{21} + 7 q^{22} - q^{23} - 15 q^{24} + 31 q^{25} - 10 q^{26} + 15 q^{27} + 15 q^{28} - 3 q^{29} + 19 q^{31} - 15 q^{32} - 7 q^{33} + 3 q^{34} + 15 q^{36} + 25 q^{37} - 12 q^{38} + 10 q^{39} + 8 q^{41} - 15 q^{42} + 25 q^{43} - 7 q^{44} + q^{46} + 11 q^{47} + 15 q^{48} + 15 q^{49} - 31 q^{50} - 3 q^{51} + 10 q^{52} - 4 q^{53} - 15 q^{54} + 9 q^{55} - 15 q^{56} + 12 q^{57} + 3 q^{58} + 27 q^{61} - 19 q^{62} + 15 q^{63} + 15 q^{64} - 2 q^{65} + 7 q^{66} + 31 q^{67} - 3 q^{68} - q^{69} - 16 q^{71} - 15 q^{72} + 26 q^{73} - 25 q^{74} + 31 q^{75} + 12 q^{76} - 7 q^{77} - 10 q^{78} + 32 q^{79} + 15 q^{81} - 8 q^{82} + 7 q^{83} + 15 q^{84} + 26 q^{85} - 25 q^{86} - 3 q^{87} + 7 q^{88} - 11 q^{89} + 10 q^{91} - q^{92} + 19 q^{93} - 11 q^{94} - 8 q^{95} - 15 q^{96} + 30 q^{97} - 15 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 53 x^{13} - x^{12} + 1068 x^{11} + 45 x^{10} - 10139 x^{9} - 615 x^{8} + 45390 x^{7} + 2130 x^{6} - 84842 x^{5} + 7822 x^{4} + 62828 x^{3} - 16144 x^{2} + \cdots + 4704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 47\!\cdots\!99 \nu^{14} + \cdots + 11\!\cdots\!84 ) / 71\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!23 \nu^{14} + \cdots + 97\!\cdots\!64 ) / 71\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\!\cdots\!59 \nu^{14} + \cdots - 20\!\cdots\!96 ) / 71\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!43 \nu^{14} + \cdots + 12\!\cdots\!04 ) / 35\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 43\!\cdots\!85 \nu^{14} + \cdots + 45\!\cdots\!32 ) / 71\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!41 \nu^{14} + \cdots + 80\!\cdots\!56 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!44 \nu^{14} + \cdots - 89\!\cdots\!72 ) / 17\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27\!\cdots\!25 \nu^{14} + \cdots - 19\!\cdots\!60 ) / 35\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!18 \nu^{14} + \cdots - 11\!\cdots\!56 ) / 11\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!21 \nu^{14} + \cdots - 87\!\cdots\!56 ) / 71\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 69\!\cdots\!51 \nu^{14} + \cdots - 52\!\cdots\!28 ) / 35\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!51 \nu^{14} + \cdots - 10\!\cdots\!20 ) / 71\!\cdots\!84 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{11} + 2\beta_{10} - \beta_{8} - \beta_{5} + 2\beta_{4} + 13\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{6} - 2 \beta_{4} + 15 \beta_{2} + 87 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 24 \beta_{14} - 27 \beta_{13} - 2 \beta_{12} - 20 \beta_{11} + 49 \beta_{10} - 7 \beta_{9} - 22 \beta_{8} + \beta_{7} - 8 \beta_{6} - 11 \beta_{5} + 37 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 188 \beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 49 \beta_{14} - 44 \beta_{13} + 48 \beta_{12} + 22 \beta_{11} + 8 \beta_{10} + 36 \beta_{9} - 4 \beta_{8} - 58 \beta_{7} + 39 \beta_{6} - 7 \beta_{5} - 54 \beta_{4} + 6 \beta_{3} + 225 \beta_{2} - 6 \beta _1 + 1201 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 478 \beta_{14} - 541 \beta_{13} - 57 \beta_{12} - 355 \beta_{11} + 968 \beta_{10} - 233 \beta_{9} - 396 \beta_{8} + 41 \beta_{7} - 241 \beta_{6} - 126 \beta_{5} + 621 \beta_{4} + 81 \beta_{3} - 60 \beta_{2} + 2820 \beta _1 + 591 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 968 \beta_{14} - 753 \beta_{13} + 891 \beta_{12} + 457 \beta_{11} - 170 \beta_{10} + 902 \beta_{9} - 70 \beta_{8} - 1252 \beta_{7} + 938 \beta_{6} - 225 \beta_{5} - 1132 \beta_{4} + 262 \beta_{3} + 3441 \beta_{2} + \cdots + 17301 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8912 \beta_{14} - 9849 \beta_{13} - 1287 \beta_{12} - 6009 \beta_{11} + 17662 \beta_{10} - 5529 \beta_{9} - 6731 \beta_{8} + 1117 \beta_{7} - 5307 \beta_{6} - 1675 \beta_{5} + 10188 \beta_{4} + 1597 \beta_{3} + \cdots + 9179 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 17802 \beta_{14} - 11609 \beta_{13} + 15244 \beta_{12} + 9225 \beta_{11} - 8674 \beta_{10} + 19286 \beta_{9} - 310 \beta_{8} - 24290 \beta_{7} + 19432 \beta_{6} - 5136 \beta_{5} - 21634 \beta_{4} + \cdots + 255902 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 160537 \beta_{14} - 171975 \beta_{13} - 26460 \beta_{12} - 99073 \beta_{11} + 309620 \beta_{10} - 114845 \beta_{9} - 112359 \beta_{8} + 25479 \beta_{7} - 104040 \beta_{6} - 25580 \beta_{5} + \cdots + 140454 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 316885 \beta_{14} - 167892 \beta_{13} + 252984 \beta_{12} + 180405 \beta_{11} - 242203 \beta_{10} + 379427 \beta_{9} + 18451 \beta_{8} - 447357 \beta_{7} + 377629 \beta_{6} + \cdots + 3856280 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2832236 \beta_{14} - 2940922 \beta_{13} - 515670 \beta_{12} - 1608888 \beta_{11} + 5308664 \beta_{10} - 2227972 \beta_{9} - 1867397 \beta_{8} + 528420 \beta_{7} - 1927402 \beta_{6} + \cdots + 2111793 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5547762 \beta_{14} - 2303000 \beta_{13} + 4153305 \beta_{12} + 3431803 \beta_{11} - 5584747 \beta_{10} + 7113938 \beta_{9} + 782491 \beta_{8} - 8001130 \beta_{7} + \cdots + 58932157 \) 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
4.01349
3.88066
3.31212
2.90136
1.18405
0.695582
0.638936
0.434267
−0.600780
−1.10784
−1.55328
−2.53934
−3.41684
−3.71340
−4.12898
−1.00000 1.00000 1.00000 −4.01349 −1.00000 1.00000 −1.00000 1.00000 4.01349
1.2 −1.00000 1.00000 1.00000 −3.88066 −1.00000 1.00000 −1.00000 1.00000 3.88066
1.3 −1.00000 1.00000 1.00000 −3.31212 −1.00000 1.00000 −1.00000 1.00000 3.31212
1.4 −1.00000 1.00000 1.00000 −2.90136 −1.00000 1.00000 −1.00000 1.00000 2.90136
1.5 −1.00000 1.00000 1.00000 −1.18405 −1.00000 1.00000 −1.00000 1.00000 1.18405
1.6 −1.00000 1.00000 1.00000 −0.695582 −1.00000 1.00000 −1.00000 1.00000 0.695582
1.7 −1.00000 1.00000 1.00000 −0.638936 −1.00000 1.00000 −1.00000 1.00000 0.638936
1.8 −1.00000 1.00000 1.00000 −0.434267 −1.00000 1.00000 −1.00000 1.00000 0.434267
1.9 −1.00000 1.00000 1.00000 0.600780 −1.00000 1.00000 −1.00000 1.00000 −0.600780
1.10 −1.00000 1.00000 1.00000 1.10784 −1.00000 1.00000 −1.00000 1.00000 −1.10784
1.11 −1.00000 1.00000 1.00000 1.55328 −1.00000 1.00000 −1.00000 1.00000 −1.55328
1.12 −1.00000 1.00000 1.00000 2.53934 −1.00000 1.00000 −1.00000 1.00000 −2.53934
1.13 −1.00000 1.00000 1.00000 3.41684 −1.00000 1.00000 −1.00000 1.00000 −3.41684
1.14 −1.00000 1.00000 1.00000 3.71340 −1.00000 1.00000 −1.00000 1.00000 −3.71340
1.15 −1.00000 1.00000 1.00000 4.12898 −1.00000 1.00000 −1.00000 1.00000 −4.12898
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(191\) \(-1\)

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 8022.2.a.z 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8022.2.a.z 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\( T_{5}^{15} - 53 T_{5}^{13} + T_{5}^{12} + 1068 T_{5}^{11} - 45 T_{5}^{10} - 10139 T_{5}^{9} + 615 T_{5}^{8} + 45390 T_{5}^{7} - 2130 T_{5}^{6} - 84842 T_{5}^{5} - 7822 T_{5}^{4} + 62828 T_{5}^{3} + 16144 T_{5}^{2} + \cdots - 4704 \) Copy content Toggle raw display
\( T_{11}^{15} + 7 T_{11}^{14} - 88 T_{11}^{13} - 632 T_{11}^{12} + 3135 T_{11}^{11} + 23255 T_{11}^{10} - 56697 T_{11}^{9} - 444934 T_{11}^{8} + 513990 T_{11}^{7} + 4648764 T_{11}^{6} - 1535720 T_{11}^{5} - 25004496 T_{11}^{4} + \cdots + 7362048 \) Copy content Toggle raw display
\( T_{13}^{15} - 10 T_{13}^{14} - 60 T_{13}^{13} + 845 T_{13}^{12} + 421 T_{13}^{11} - 24556 T_{13}^{10} + 26843 T_{13}^{9} + 302354 T_{13}^{8} - 476405 T_{13}^{7} - 1675638 T_{13}^{6} + 2307434 T_{13}^{5} + 4567808 T_{13}^{4} + \cdots - 125200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{15} \) Copy content Toggle raw display
$3$ \( (T - 1)^{15} \) Copy content Toggle raw display
$5$ \( T^{15} - 53 T^{13} + T^{12} + 1068 T^{11} + \cdots - 4704 \) Copy content Toggle raw display
$7$ \( (T - 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + 7 T^{14} - 88 T^{13} + \cdots + 7362048 \) Copy content Toggle raw display
$13$ \( T^{15} - 10 T^{14} - 60 T^{13} + \cdots - 125200 \) Copy content Toggle raw display
$17$ \( T^{15} + 3 T^{14} - 187 T^{13} + \cdots - 20952960 \) Copy content Toggle raw display
$19$ \( T^{15} - 12 T^{14} - 62 T^{13} + \cdots + 1165600 \) Copy content Toggle raw display
$23$ \( T^{15} + T^{14} - 133 T^{13} + \cdots + 8372736 \) Copy content Toggle raw display
$29$ \( T^{15} + 3 T^{14} + \cdots - 1972901760 \) Copy content Toggle raw display
$31$ \( T^{15} - 19 T^{14} + \cdots - 3677891584 \) Copy content Toggle raw display
$37$ \( T^{15} - 25 T^{14} + \cdots - 182055680000 \) Copy content Toggle raw display
$41$ \( T^{15} - 8 T^{14} - 222 T^{13} + \cdots + 81371520 \) Copy content Toggle raw display
$43$ \( T^{15} - 25 T^{14} + \cdots - 16833744896 \) Copy content Toggle raw display
$47$ \( T^{15} - 11 T^{14} + \cdots - 193373184 \) Copy content Toggle raw display
$53$ \( T^{15} + 4 T^{14} - 340 T^{13} + \cdots + 298284000 \) Copy content Toggle raw display
$59$ \( T^{15} - 307 T^{13} - 748 T^{12} + \cdots - 442368 \) Copy content Toggle raw display
$61$ \( T^{15} - 27 T^{14} - 137 T^{13} + \cdots - 142976 \) Copy content Toggle raw display
$67$ \( T^{15} - 31 T^{14} + \cdots + 996928004096 \) Copy content Toggle raw display
$71$ \( T^{15} + 16 T^{14} + \cdots + 380643501120 \) Copy content Toggle raw display
$73$ \( T^{15} - 26 T^{14} + \cdots + 1027026776 \) Copy content Toggle raw display
$79$ \( T^{15} - 32 T^{14} + \cdots + 134938205440 \) Copy content Toggle raw display
$83$ \( T^{15} - 7 T^{14} + \cdots - 584237888868 \) Copy content Toggle raw display
$89$ \( T^{15} + 11 T^{14} + \cdots - 5016899430528 \) Copy content Toggle raw display
$97$ \( T^{15} - 30 T^{14} + \cdots + 93789721600 \) Copy content Toggle raw display
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