Properties

Label 1110.2.x.d
Level $1110$
Weight $2$
Character orbit 1110.x
Analytic conductor $8.863$
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] = [1110,2,Mod(751,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1110.751");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.x (of order \(6\), degree \(2\), 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: \(8.86339462436\)
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: \( x^{16} + 60x^{14} + 1362x^{12} + 15028x^{10} + 86441x^{8} + 260376x^{6} + 382684x^{4} + 224224x^{2} + 38416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 60x^{14} + 1362x^{12} + 15028x^{10} + 86441x^{8} + 260376x^{6} + 382684x^{4} + 224224x^{2} + 38416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11 \nu^{14} + 613 \nu^{12} + 12389 \nu^{10} + 113739 \nu^{8} + 490228 \nu^{6} + 982004 \nu^{4} + 849840 \nu^{2} - 34944 \nu + 206192 ) / 69888 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16872 \nu^{15} - 85673 \nu^{14} - 845916 \nu^{13} - 4862599 \nu^{12} - 13686912 \nu^{11} - 100872023 \nu^{10} - 65555292 \nu^{9} + \cdots - 1892498384 ) / 1061598720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1843 \nu^{15} + 87690 \nu^{14} - 55469 \nu^{13} + 5110410 \nu^{12} + 441587 \nu^{11} + 110660190 \nu^{10} + 28000317 \nu^{9} + 1128794190 \nu^{8} + \cdots + 3013123680 ) / 303313920 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1406 \nu^{15} + 70493 \nu^{13} + 1140576 \nu^{11} + 5462941 \nu^{9} - 21516416 \nu^{7} - 230818172 \nu^{5} - 501688272 \nu^{3} - 208036752 \nu ) / 44233280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1843 \nu^{15} - 87690 \nu^{14} - 55469 \nu^{13} - 5110410 \nu^{12} + 441587 \nu^{11} - 110660190 \nu^{10} + 28000317 \nu^{9} - 1128794190 \nu^{8} + \cdots - 3013123680 ) / 303313920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 52443 \nu^{15} - 10192 \nu^{14} - 3509229 \nu^{13} - 343196 \nu^{12} - 91328373 \nu^{11} + 666008 \nu^{10} - 1178556723 \nu^{9} + 128790228 \nu^{8} + \cdots + 9551468864 ) / 1061598720 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12239 \nu^{15} - 23772 \nu^{14} + 694657 \nu^{13} - 1327536 \nu^{12} + 14410289 \nu^{11} - 26856732 \nu^{10} + 136836879 \nu^{9} - 245232792 \nu^{8} + \cdots - 92593536 ) / 151656960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 263 \nu^{15} + 15241 \nu^{13} + 328169 \nu^{11} + 3345303 \nu^{9} + 17160772 \nu^{7} + 44457716 \nu^{5} + 52527696 \nu^{3} + 17328752 \nu + 1712256 ) / 3424512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12239 \nu^{15} + 23772 \nu^{14} + 694657 \nu^{13} + 1327536 \nu^{12} + 14410289 \nu^{11} + 26856732 \nu^{10} + 136836879 \nu^{9} + 245232792 \nu^{8} + \cdots + 92593536 ) / 151656960 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17481 \nu^{15} - 1169743 \nu^{13} - 30442791 \nu^{11} - 392852241 \nu^{9} - 2634019244 \nu^{7} - 8742547388 \nu^{5} - 12512903888 \nu^{3} + \cdots - 5058208848 \nu ) / 176933120 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 407712 \nu^{15} + 191303 \nu^{14} - 23678916 \nu^{13} + 10175809 \nu^{12} - 509843112 \nu^{11} + 188699273 \nu^{10} - 5151114132 \nu^{9} + \cdots - 2614850896 ) / 2123197440 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 407712 \nu^{15} + 191303 \nu^{14} + 23678916 \nu^{13} + 10175809 \nu^{12} + 509843112 \nu^{11} + 188699273 \nu^{10} + 5151114132 \nu^{9} + \cdots - 2614850896 ) / 2123197440 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 152793 \nu^{15} + 28329 \nu^{14} - 8812379 \nu^{13} + 1614627 \nu^{12} - 188046063 \nu^{11} + 33560079 \nu^{10} - 1883030733 \nu^{9} + \cdots - 1258733168 ) / 353866240 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 539909 \nu^{15} - 84987 \nu^{14} - 31161847 \nu^{13} - 4843881 \nu^{12} - 665870579 \nu^{11} - 100680237 \nu^{10} - 6686136129 \nu^{9} + \cdots + 4306998864 ) / 1061598720 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{5} - 2\beta_{3} - 2\beta_{2} - \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{14} - 2 \beta_{11} - 2 \beta_{10} - 13 \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 12 \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 18 \beta_{15} + 18 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \beta_{11} - 28 \beta_{10} - 18 \beta_{9} + 28 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 48 \beta_{2} + 24 \beta _1 + 95 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 27 \beta_{15} + 27 \beta_{14} + 44 \beta_{11} + 70 \beta_{10} + 303 \beta_{9} + 70 \beta_{8} + 34 \beta_{6} - 139 \beta_{5} + 34 \beta_{4} + 216 \beta _1 - 165 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 328 \beta_{15} - 328 \beta_{14} + 54 \beta_{13} + 54 \beta_{12} - 63 \beta_{11} + 690 \beta_{10} + 328 \beta_{9} - 690 \beta_{8} + 126 \beta_{7} - 54 \beta_{6} + 243 \beta_{5} + 54 \beta_{4} + 486 \beta_{3} - 1044 \beta_{2} + \cdots - 1631 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 639 \beta_{15} - 639 \beta_{14} - 72 \beta_{13} + 72 \beta_{12} - 964 \beta_{11} - 1846 \beta_{10} - 6395 \beta_{9} - 1846 \beta_{8} - 458 \beta_{6} + 3619 \beta_{5} - 458 \beta_{4} - 4404 \beta _1 + 3517 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6356 \beta_{15} + 6356 \beta_{14} - 1086 \beta_{13} - 1086 \beta_{12} + 2027 \beta_{11} - 16442 \beta_{10} - 6356 \beta_{9} + 16442 \beta_{8} - 4054 \beta_{7} + 1206 \beta_{6} - 8747 \beta_{5} + \cdots + 31135 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14483 \beta_{15} + 14483 \beta_{14} + 2848 \beta_{13} - 2848 \beta_{12} + 21832 \beta_{11} + 44942 \beta_{10} + 135095 \beta_{9} + 44942 \beta_{8} + 5690 \beta_{6} - 90523 \beta_{5} + 5690 \beta_{4} + \cdots - 74789 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 130184 \beta_{15} - 130184 \beta_{14} + 19958 \beta_{13} + 19958 \beta_{12} - 53735 \beta_{11} + 384754 \beta_{10} + 130184 \beta_{9} - 384754 \beta_{8} + 107470 \beta_{7} - 26118 \beta_{6} + \cdots - 632007 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 323863 \beta_{15} - 323863 \beta_{14} - 81352 \beta_{13} + 81352 \beta_{12} - 501140 \beta_{11} - 1058558 \beta_{10} - 2896595 \beta_{9} - 1058558 \beta_{8} - 65386 \beta_{6} + \cdots + 1610229 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2771924 \beta_{15} + 2771924 \beta_{14} - 358558 \beta_{13} - 358558 \beta_{12} + 1317035 \beta_{11} - 8899826 \beta_{10} - 2771924 \beta_{9} + 8899826 \beta_{8} - 2634070 \beta_{7} + \cdots + 13344047 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7228043 \beta_{15} + 7228043 \beta_{14} + 2067696 \beta_{13} - 2067696 \beta_{12} + 11521008 \beta_{11} + 24521326 \beta_{10} + 63035871 \beta_{9} + 24521326 \beta_{8} + \cdots - 35131957 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 60481488 \beta_{15} - 60481488 \beta_{14} + 6504422 \beta_{13} + 6504422 \beta_{12} - 31115103 \beta_{11} + 204305602 \beta_{10} + 60481488 \beta_{9} - 204305602 \beta_{8} + \cdots - 289019607 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 161646111 \beta_{15} - 161646111 \beta_{14} - 49841816 \beta_{13} + 49841816 \beta_{12} - 264166636 \beta_{11} - 562800062 \beta_{10} - 1388334603 \beta_{9} + \cdots + 774990357 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
4.75759i
0.535537i
0.871333i
1.68974i
3.25684i
1.95985i
2.16125i
3.78749i
4.75759i
0.535537i
0.871333i
1.68974i
3.25684i
1.95985i
2.16125i
3.78749i
−0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i −2.37879 + 4.12019i 1.00000i −0.500000 0.866025i 1.00000
751.2 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i −0.267768 + 0.463788i 1.00000i −0.500000 0.866025i 1.00000
751.3 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 0.435667 0.754597i 1.00000i −0.500000 0.866025i 1.00000
751.4 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 0.844871 1.46336i 1.00000i −0.500000 0.866025i 1.00000
751.5 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i −1.62842 + 2.82051i 1.00000i −0.500000 0.866025i 1.00000
751.6 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i −0.979923 + 1.69728i 1.00000i −0.500000 0.866025i 1.00000
751.7 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 1.08063 1.87170i 1.00000i −0.500000 0.866025i 1.00000
751.8 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 1.89374 3.28006i 1.00000i −0.500000 0.866025i 1.00000
841.1 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −2.37879 4.12019i 1.00000i −0.500000 + 0.866025i 1.00000
841.2 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −0.267768 0.463788i 1.00000i −0.500000 + 0.866025i 1.00000
841.3 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0.435667 + 0.754597i 1.00000i −0.500000 + 0.866025i 1.00000
841.4 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0.844871 + 1.46336i 1.00000i −0.500000 + 0.866025i 1.00000
841.5 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i −1.62842 2.82051i 1.00000i −0.500000 + 0.866025i 1.00000
841.6 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i −0.979923 1.69728i 1.00000i −0.500000 + 0.866025i 1.00000
841.7 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 1.08063 + 1.87170i 1.00000i −0.500000 + 0.866025i 1.00000
841.8 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 1.89374 + 3.28006i 1.00000i −0.500000 + 0.866025i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.x.d 16
37.e even 6 1 inner 1110.2.x.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.x.d 16 1.a even 1 1 trivial
1110.2.x.d 16 37.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 2 T_{7}^{15} + 32 T_{7}^{14} + 8 T_{7}^{13} + 623 T_{7}^{12} + 70 T_{7}^{11} + 6068 T_{7}^{10} - 4080 T_{7}^{9} + 37621 T_{7}^{8} - 20766 T_{7}^{7} + 121838 T_{7}^{6} - 85068 T_{7}^{5} + 256600 T_{7}^{4} + \cdots + 38416 \) acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + 32 T^{14} + \cdots + 38416 \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} - 62 T^{6} - 232 T^{5} + \cdots + 24336)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 6 T^{15} - 56 T^{14} + \cdots + 37417689 \) Copy content Toggle raw display
$17$ \( T^{16} - 6 T^{15} - 14 T^{14} + \cdots + 82944 \) Copy content Toggle raw display
$19$ \( T^{16} + 12 T^{15} - 21 T^{14} + \cdots + 26873856 \) Copy content Toggle raw display
$23$ \( T^{16} + 160 T^{14} + 9172 T^{12} + \cdots + 5308416 \) Copy content Toggle raw display
$29$ \( T^{16} + 222 T^{14} + \cdots + 10585940544 \) Copy content Toggle raw display
$31$ \( T^{16} + 198 T^{14} + \cdots + 201867264 \) Copy content Toggle raw display
$37$ \( T^{16} - 12 T^{15} + \cdots + 3512479453921 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 215838879399936 \) Copy content Toggle raw display
$43$ \( T^{16} + 142 T^{14} + 5869 T^{12} + \cdots + 9216 \) Copy content Toggle raw display
$47$ \( (T^{8} - 34 T^{7} + 344 T^{6} + \cdots - 178848)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 12 T^{15} + \cdots + 20517855952896 \) Copy content Toggle raw display
$59$ \( T^{16} - 6 T^{15} + \cdots + 14648745024 \) Copy content Toggle raw display
$61$ \( T^{16} - 12 T^{15} + \cdots + 940868960256 \) Copy content Toggle raw display
$67$ \( T^{16} + 36 T^{15} + \cdots + 1874890000 \) Copy content Toggle raw display
$71$ \( T^{16} - 6 T^{15} + \cdots + 2190240000 \) Copy content Toggle raw display
$73$ \( (T^{8} + 8 T^{7} - 253 T^{6} + \cdots - 914048)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 24 T^{15} + \cdots + 239802172416 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 180864626782464 \) Copy content Toggle raw display
$89$ \( T^{16} - 24 T^{15} + \cdots + 7020380160000 \) Copy content Toggle raw display
$97$ \( T^{16} + 270 T^{14} + 20893 T^{12} + \cdots + 5760000 \) Copy content Toggle raw display
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