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Properties

Label 1040.2.da.f
Level $1040$
Weight $2$
Character orbit 1040.da
Analytic conductor $8.304$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1040,2,Mod(641,1040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1040, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1040.641");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.da (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 22x^{14} + 183x^{12} + 730x^{10} + 1485x^{8} + 1552x^{6} + 812x^{4} + 192x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 520)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{10} + \beta_{9}) q^{5} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{5} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - \beta_{15} - \beta_{13} + \beta_{12} + 2 \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{10} + \beta_{9}) q^{5} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{5} + \cdots - \beta_{2}) q^{7}+ \cdots + (\beta_{15} + 2 \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{10} + 2 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} - 6 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{3} - 6 q^{7} - 16 q^{9} + 6 q^{11} - 2 q^{13} + 4 q^{17} - 30 q^{19} - 6 q^{23} - 16 q^{25} - 44 q^{27} - 16 q^{29} + 24 q^{33} + 6 q^{35} - 24 q^{37} + 8 q^{39} - 24 q^{41} - 6 q^{43} + 12 q^{45} - 4 q^{49} + 40 q^{51} + 4 q^{53} + 6 q^{55} - 12 q^{59} - 2 q^{61} + 60 q^{63} - 10 q^{65} + 6 q^{67} + 52 q^{69} - 72 q^{71} - 4 q^{75} + 32 q^{77} - 36 q^{79} - 28 q^{81} + 22 q^{87} + 24 q^{89} + 22 q^{91} - 96 q^{93} - 10 q^{95} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 22x^{14} + 183x^{12} + 730x^{10} + 1485x^{8} + 1552x^{6} + 812x^{4} + 192x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 18 \nu^{15} - 385 \nu^{13} - 3058 \nu^{11} - 11251 \nu^{9} - 19650 \nu^{7} - 15009 \nu^{5} - 3626 \nu^{3} + 148 \nu + 104 ) / 208 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4 \nu^{15} + 45 \nu^{14} + 113 \nu^{13} + 956 \nu^{12} + 1240 \nu^{11} + 7515 \nu^{10} + 6627 \nu^{9} + 27224 \nu^{8} + 17644 \nu^{7} + 46681 \nu^{6} + 21141 \nu^{5} + 36086 \nu^{4} + \cdots + 904 ) / 208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{15} + 12 \nu^{14} - 288 \nu^{13} + 248 \nu^{12} - 2981 \nu^{11} + 1848 \nu^{10} - 15504 \nu^{9} + 5932 \nu^{8} - 42411 \nu^{7} + 7120 \nu^{6} - 57946 \nu^{5} - 472 \nu^{4} + \cdots - 1312 ) / 208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11 \nu^{15} + 12 \nu^{14} + 288 \nu^{13} + 248 \nu^{12} + 2981 \nu^{11} + 1848 \nu^{10} + 15504 \nu^{9} + 5932 \nu^{8} + 42411 \nu^{7} + 7120 \nu^{6} + 57946 \nu^{5} - 472 \nu^{4} + \cdots - 1312 ) / 208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45 \nu^{14} + 956 \nu^{12} + 7515 \nu^{10} + 27224 \nu^{8} + 46681 \nu^{6} + 36086 \nu^{4} + 11548 \nu^{2} + 104 \nu + 1216 ) / 104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 45\nu^{14} + 956\nu^{12} + 7515\nu^{10} + 27224\nu^{8} + 46681\nu^{6} + 36086\nu^{4} + 11444\nu^{2} + 904 ) / 104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{15} - 22\nu^{13} - 183\nu^{11} - 730\nu^{9} - 1485\nu^{7} - 1552\nu^{5} - 812\nu^{3} - 192\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 39 \nu^{15} - 34 \nu^{14} + 806 \nu^{13} - 720 \nu^{12} + 6045 \nu^{11} - 5626 \nu^{10} + 20046 \nu^{9} - 20144 \nu^{8} + 28379 \nu^{7} - 33754 \nu^{6} + 12896 \nu^{5} - 25096 \nu^{4} + \cdots - 928 ) / 208 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76 \nu^{15} + 39 \nu^{14} - 1627 \nu^{13} + 806 \nu^{12} - 12952 \nu^{11} + 6045 \nu^{10} - 47965 \nu^{9} + 20046 \nu^{8} - 85636 \nu^{7} + 28379 \nu^{6} - 71271 \nu^{5} + 12896 \nu^{4} + \cdots - 624 ) / 208 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 76 \nu^{15} + 39 \nu^{14} + 1627 \nu^{13} + 806 \nu^{12} + 12952 \nu^{11} + 6045 \nu^{10} + 47965 \nu^{9} + 20046 \nu^{8} + 85636 \nu^{7} + 28379 \nu^{6} + 71271 \nu^{5} + 12896 \nu^{4} + \cdots - 624 ) / 208 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -76\nu^{14} - 1627\nu^{12} - 12952\nu^{10} - 47965\nu^{8} - 85636\nu^{6} - 71271\nu^{4} - 25626\nu^{2} - 3044 ) / 52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 107 \nu^{15} - 15 \nu^{14} - 2298 \nu^{13} - 284 \nu^{12} - 18389 \nu^{11} - 1777 \nu^{10} - 68706 \nu^{9} - 3476 \nu^{8} - 124591 \nu^{7} + 3801 \nu^{6} - 106456 \nu^{5} + \cdots + 2160 ) / 208 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 81 \nu^{15} + 119 \nu^{14} - 1778 \nu^{13} + 2546 \nu^{12} - 14723 \nu^{11} + 20237 \nu^{10} - 58150 \nu^{9} + 74638 \nu^{8} - 115517 \nu^{7} + 131711 \nu^{6} - 113736 \nu^{5} + \cdots + 3768 ) / 208 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 122 \nu^{15} - 45 \nu^{14} + 2569 \nu^{13} - 956 \nu^{12} + 19906 \nu^{11} - 7515 \nu^{10} + 70323 \nu^{9} - 27224 \nu^{8} + 115018 \nu^{7} - 46681 \nu^{6} + 81049 \nu^{5} - 36086 \nu^{4} + \cdots - 904 ) / 208 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 183 \nu^{15} - 178 \nu^{14} + 3925 \nu^{13} - 3774 \nu^{12} + 31341 \nu^{11} - 29570 \nu^{10} + 116671 \nu^{9} - 106486 \nu^{8} + 210227 \nu^{7} - 180398 \nu^{6} + 177727 \nu^{5} + \cdots - 4320 ) / 208 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{14} + \beta_{10} - \beta_{9} + \beta_{7} - 2\beta_{6} + 2\beta_{5} - \beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - 4 \beta_{14} + \beta_{12} + \beta_{11} + 5 \beta_{10} - 6 \beta_{9} + 3 \beta_{7} + 2 \beta_{5} - 4 \beta_{2} - 2 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{15} + 13 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 11 \beta_{10} + 17 \beta_{9} - 4 \beta_{8} - 15 \beta_{7} + 14 \beta_{6} - 26 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 13 \beta_{2} + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 24 \beta_{15} + 59 \beta_{14} + 8 \beta_{13} - 24 \beta_{12} - 20 \beta_{11} - 83 \beta_{10} + 107 \beta_{9} - 51 \beta_{7} + 2 \beta_{6} - 48 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 63 \beta_{2} + 36 \beta _1 - 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{15} - 59 \beta_{14} - 24 \beta_{13} + 9 \beta_{12} + 12 \beta_{11} + 46 \beta_{10} - 87 \beta_{9} + 24 \beta_{8} + 71 \beta_{7} - 50 \beta_{6} + 118 \beta_{5} + 11 \beta_{4} + 11 \beta_{3} - 59 \beta_{2} - 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 222 \beta_{15} - 435 \beta_{14} - 120 \beta_{13} + 222 \beta_{12} + 162 \beta_{11} + 649 \beta_{10} - 871 \beta_{9} + 425 \beta_{7} - 30 \beta_{6} + 444 \beta_{5} + 56 \beta_{4} + 64 \beta_{3} - 495 \beta_{2} - 296 \beta _1 + 310 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 324 \beta_{15} + 991 \beta_{14} + 468 \beta_{13} - 144 \beta_{12} - 222 \beta_{11} - 739 \beta_{10} + 1567 \beta_{9} - 468 \beta_{8} - 1225 \beta_{7} + 750 \beta_{6} - 1982 \beta_{5} - 208 \beta_{4} - 208 \beta_{3} + \cdots + 1178 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 941 \beta_{15} + 1615 \beta_{14} + 640 \beta_{13} - 941 \beta_{12} - 621 \beta_{11} - 2486 \beta_{10} + 3427 \beta_{9} - 1722 \beta_{7} + 158 \beta_{6} - 1882 \beta_{5} - 288 \beta_{4} - 352 \beta_{3} + 1939 \beta_{2} + \cdots - 1230 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3082 \beta_{15} - 8089 \beta_{14} - 4236 \beta_{13} + 1154 \beta_{12} + 1882 \beta_{11} + 5891 \beta_{10} - 13369 \beta_{9} + 4236 \beta_{8} + 10207 \beta_{7} - 5778 \beta_{6} + 16178 \beta_{5} + \cdots - 8118 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 15364 \beta_{15} - 24169 \beta_{14} - 11968 \beta_{13} + 15364 \beta_{12} + 9380 \beta_{11} + 37877 \beta_{10} - 53241 \beta_{9} + 27457 \beta_{7} - 2902 \beta_{6} + 30728 \beta_{5} + 5280 \beta_{4} + \cdots + 19394 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 13721 \beta_{15} + 32540 \beta_{14} + 18376 \beta_{13} - 4655 \beta_{12} - 7682 \beta_{11} - 23407 \beta_{10} + 55394 \beta_{9} - 18376 \beta_{8} - 41728 \beta_{7} + 22526 \beta_{6} - 65080 \beta_{5} + \cdots + 28932 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 123118 \beta_{15} + 182167 \beta_{14} + 104792 \beta_{13} - 123118 \beta_{12} - 70722 \beta_{11} - 288649 \beta_{10} + 411767 \beta_{9} - 216953 \beta_{7} + 24954 \beta_{6} - 246236 \beta_{5} + \cdots - 152526 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 234792 \beta_{15} - 519021 \beta_{14} - 309908 \beta_{13} + 75116 \beta_{12} + 123118 \beta_{11} + 370949 \beta_{10} - 901885 \beta_{9} + 309908 \beta_{8} + 673975 \beta_{7} + \cdots - 422094 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 488399 \beta_{15} - 690968 \beta_{14} - 441948 \beta_{13} + 488399 \beta_{12} + 267425 \beta_{11} + 1102735 \beta_{10} - 1591134 \beta_{9} + 852893 \beta_{7} - 103578 \beta_{6} + \cdots + 598824 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
− 1.97402i
− 2.79253i
2.44974i
1.44614i
0.897932i
− 0.432338i
0.956612i
− 0.551543i
1.97402i
2.79253i
− 2.44974i
− 1.44614i
− 0.897932i
0.432338i
− 0.956612i
0.551543i
0 −1.43538 + 2.48615i 0 1.00000i 0 −0.0113994 + 0.00658143i 0 −2.62062 − 4.53905i 0
641.2 0 −1.00284 + 1.73697i 0 − 1.00000i 0 −1.48627 + 0.858099i 0 −0.511370 − 0.885719i 0
641.3 0 −0.275748 + 0.477609i 0 1.00000i 0 1.57306 − 0.908206i 0 1.34793 + 2.33468i 0
641.4 0 −0.268727 + 0.465448i 0 − 1.00000i 0 0.331682 − 0.191497i 0 1.35557 + 2.34792i 0
641.5 0 0.647893 − 1.12218i 0 − 1.00000i 0 1.06291 − 0.613670i 0 0.660468 + 1.14396i 0
641.6 0 1.19037 − 2.06179i 0 1.00000i 0 −3.14022 + 1.81301i 0 −1.33398 − 2.31052i 0
641.7 0 1.52075 − 2.63402i 0 1.00000i 0 2.67664 − 1.54536i 0 −3.12538 − 5.41331i 0
641.8 0 1.62367 − 2.81228i 0 − 1.00000i 0 −4.00640 + 2.31309i 0 −3.77262 − 6.53437i 0
881.1 0 −1.43538 − 2.48615i 0 − 1.00000i 0 −0.0113994 − 0.00658143i 0 −2.62062 + 4.53905i 0
881.2 0 −1.00284 − 1.73697i 0 1.00000i 0 −1.48627 − 0.858099i 0 −0.511370 + 0.885719i 0
881.3 0 −0.275748 − 0.477609i 0 − 1.00000i 0 1.57306 + 0.908206i 0 1.34793 − 2.33468i 0
881.4 0 −0.268727 − 0.465448i 0 1.00000i 0 0.331682 + 0.191497i 0 1.35557 − 2.34792i 0
881.5 0 0.647893 + 1.12218i 0 1.00000i 0 1.06291 + 0.613670i 0 0.660468 − 1.14396i 0
881.6 0 1.19037 + 2.06179i 0 − 1.00000i 0 −3.14022 − 1.81301i 0 −1.33398 + 2.31052i 0
881.7 0 1.52075 + 2.63402i 0 − 1.00000i 0 2.67664 + 1.54536i 0 −3.12538 + 5.41331i 0
881.8 0 1.62367 + 2.81228i 0 1.00000i 0 −4.00640 − 2.31309i 0 −3.77262 + 6.53437i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.da.f 16
4.b odd 2 1 520.2.bu.b ✓ 16
13.e even 6 1 inner 1040.2.da.f 16
52.i odd 6 1 520.2.bu.b ✓ 16
52.l even 12 1 6760.2.a.bk 8
52.l even 12 1 6760.2.a.bl 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.bu.b ✓ 16 4.b odd 2 1
520.2.bu.b ✓ 16 52.i odd 6 1
1040.2.da.f 16 1.a even 1 1 trivial
1040.2.da.f 16 13.e even 6 1 inner
6760.2.a.bk 8 52.l even 12 1
6760.2.a.bl 8 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 4 T_{3}^{15} + 28 T_{3}^{14} - 60 T_{3}^{13} + 327 T_{3}^{12} - 574 T_{3}^{11} + 2508 T_{3}^{10} - 2550 T_{3}^{9} + 9913 T_{3}^{8} - 5782 T_{3}^{7} + 28018 T_{3}^{6} - 2252 T_{3}^{5} + 25368 T_{3}^{4} + 16680 T_{3}^{3} + \cdots + 2704 \) acting on \(S_{2}^{\mathrm{new}}(1040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{15} + 28 T^{14} + \cdots + 2704 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 6 T^{15} - 8 T^{14} - 120 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} - 6 T^{15} - 64 T^{14} + \cdots + 38800441 \) Copy content Toggle raw display
$13$ \( T^{16} + 2 T^{15} + 38 T^{14} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 4 T^{15} + 104 T^{14} + \cdots + 63425296 \) Copy content Toggle raw display
$19$ \( T^{16} + 30 T^{15} + 388 T^{14} + \cdots + 6235009 \) Copy content Toggle raw display
$23$ \( T^{16} + 6 T^{15} + \cdots + 1235663104 \) Copy content Toggle raw display
$29$ \( T^{16} + 16 T^{15} + 202 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$31$ \( T^{16} + 288 T^{14} + \cdots + 38554107904 \) Copy content Toggle raw display
$37$ \( T^{16} + 24 T^{15} + 164 T^{14} + \cdots + 21557449 \) Copy content Toggle raw display
$41$ \( T^{16} + 24 T^{15} + \cdots + 1386221824 \) Copy content Toggle raw display
$43$ \( T^{16} + 6 T^{15} + \cdots + 143344017664 \) Copy content Toggle raw display
$47$ \( T^{16} + 388 T^{14} + \cdots + 1307674583296 \) Copy content Toggle raw display
$53$ \( (T^{8} - 2 T^{7} - 230 T^{6} + \cdots - 663344)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 12 T^{15} + \cdots + 3760434585856 \) Copy content Toggle raw display
$61$ \( T^{16} + 2 T^{15} + \cdots + 65554433296 \) Copy content Toggle raw display
$67$ \( T^{16} - 6 T^{15} + \cdots + 676131241984 \) Copy content Toggle raw display
$71$ \( T^{16} + 72 T^{15} + \cdots + 667763540224 \) Copy content Toggle raw display
$73$ \( T^{16} + 544 T^{14} + \cdots + 129784385536 \) Copy content Toggle raw display
$79$ \( (T^{8} + 18 T^{7} - 98 T^{6} + \cdots + 659776)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 176 T^{14} + \cdots + 89718784 \) Copy content Toggle raw display
$89$ \( T^{16} - 24 T^{15} + \cdots + 5504781135529 \) Copy content Toggle raw display
$97$ \( T^{16} - 60 T^{15} + \cdots + 2668426795024 \) Copy content Toggle raw display
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