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Properties

Label 273.2.bd.a
Level $273$
Weight $2$
Character orbit 273.bd
Analytic conductor $2.180$
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] = [273,2,Mod(43,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("273.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bd (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: \(2.17991597518\)
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: \( x^{16} + 22x^{14} + 187x^{12} + 774x^{10} + 1619x^{8} + 1618x^{6} + 690x^{4} + 96x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 22x^{14} + 187x^{12} + 774x^{10} + 1619x^{8} + 1618x^{6} + 690x^{4} + 96x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 20\nu^{12} + 147\nu^{10} + 480\nu^{8} + 646\nu^{6} + 183\nu^{4} - 105\nu^{2} + 26\nu - 6 ) / 52 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - 20\nu^{12} - 147\nu^{10} - 480\nu^{8} - 646\nu^{6} - 183\nu^{4} + 105\nu^{2} + 26\nu + 6 ) / 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{14} - 107\nu^{12} - 870\nu^{10} - 3340\nu^{8} - 6095\nu^{6} - 4633\nu^{4} - 1107\nu^{2} - 23 ) / 104 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{14} - 107\nu^{12} - 870\nu^{10} - 3340\nu^{8} - 6095\nu^{6} - 4633\nu^{4} - 1003\nu^{2} + 289 ) / 104 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7 \nu^{15} + 5 \nu^{14} + 149 \nu^{13} + 107 \nu^{12} + 1210 \nu^{11} + 870 \nu^{10} + 4680 \nu^{9} + 3340 \nu^{8} + 8733 \nu^{7} + 6095 \nu^{6} + 6819 \nu^{5} + 4633 \nu^{4} + 1237 \nu^{3} + \cdots + 23 ) / 208 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5 \nu^{15} - 13 \nu^{14} + 115 \nu^{13} - 283 \nu^{12} + 1054 \nu^{11} - 2362 \nu^{10} + 4964 \nu^{9} - 9440 \nu^{8} + 12935 \nu^{7} - 18339 \nu^{6} + 18465 \nu^{5} - 15457 \nu^{4} + \cdots - 215 ) / 208 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15 \nu^{14} + 323 \nu^{12} + 2656 \nu^{10} + 10400 \nu^{8} + 19631 \nu^{6} + 15823 \nu^{4} + 4285 \nu^{2} - 52 \nu + 203 ) / 104 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9 \nu^{15} + 197 \nu^{13} + 2 \nu^{12} + 1662 \nu^{11} + 46 \nu^{10} + 6796 \nu^{9} + 406 \nu^{8} + 13901 \nu^{7} + 1684 \nu^{6} + 13243 \nu^{5} + 3146 \nu^{4} + 5117 \nu^{3} + 1952 \nu^{2} + \cdots + 160 ) / 104 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{15} + 197 \nu^{13} - 2 \nu^{12} + 1662 \nu^{11} - 46 \nu^{10} + 6796 \nu^{9} - 406 \nu^{8} + 13901 \nu^{7} - 1684 \nu^{6} + 13243 \nu^{5} - 3146 \nu^{4} + 5117 \nu^{3} - 1952 \nu^{2} + \cdots - 160 ) / 104 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23 \nu^{15} + 7 \nu^{14} - 501 \nu^{13} + 149 \nu^{12} - 4194 \nu^{11} + 1210 \nu^{10} - 16932 \nu^{9} + 4680 \nu^{8} - 33897 \nu^{7} + 8733 \nu^{6} - 31119 \nu^{5} + 6819 \nu^{4} + \cdots - 167 ) / 208 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23 \nu^{15} - 7 \nu^{14} - 501 \nu^{13} - 149 \nu^{12} - 4194 \nu^{11} - 1210 \nu^{10} - 16932 \nu^{9} - 4680 \nu^{8} - 33897 \nu^{7} - 8733 \nu^{6} - 31119 \nu^{5} - 6819 \nu^{4} + \cdots + 167 ) / 208 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6\nu^{15} + 133\nu^{13} + 1142\nu^{11} + 4791\nu^{9} + 10194\nu^{7} + 10354\nu^{5} + 4323\nu^{3} + 471\nu + 26 ) / 52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 41 \nu^{15} + 5 \nu^{14} - 895 \nu^{13} + 119 \nu^{12} - 7518 \nu^{11} + 1094 \nu^{10} - 30524 \nu^{9} + 4840 \nu^{8} - 61699 \nu^{7} + 10375 \nu^{6} - 57605 \nu^{5} + 9313 \nu^{4} + \cdots - 317 ) / 208 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41 \nu^{15} - 5 \nu^{14} - 895 \nu^{13} - 119 \nu^{12} - 7518 \nu^{11} - 1094 \nu^{10} - 30524 \nu^{9} - 4840 \nu^{8} - 61699 \nu^{7} - 10375 \nu^{6} - 57605 \nu^{5} - 9313 \nu^{4} + \cdots + 317 ) / 208 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 59 \nu^{15} + 5 \nu^{14} + 1293 \nu^{13} + 107 \nu^{12} + 10934 \nu^{11} + 870 \nu^{10} + 44928 \nu^{9} + 3340 \nu^{8} + 92869 \nu^{7} + 6095 \nu^{6} + 90383 \nu^{5} + 4633 \nu^{4} + \cdots - 185 ) / 208 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 5\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{7} - 6\beta_{4} + 8\beta_{3} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{14} - 9 \beta_{13} - 2 \beta_{12} + 10 \beta_{11} + 10 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} - 2 \beta_{5} - \beta_{3} + 29 \beta_{2} + 29 \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{14} - \beta_{13} - 8 \beta_{11} + 8 \beta_{10} - 9 \beta_{7} + 35 \beta_{4} - 57 \beta_{3} + 2 \beta_{2} - 11 \beta _1 - 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{15} + 66 \beta_{14} + 66 \beta_{13} + 24 \beta_{12} - 79 \beta_{11} - 79 \beta_{10} + 35 \beta_{9} + 35 \beta_{8} + \beta_{7} + 2 \beta_{6} + 24 \beta_{5} - 2 \beta_{4} + 12 \beta_{3} - 176 \beta_{2} - 177 \beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 13 \beta_{14} + 13 \beta_{13} + 57 \beta_{11} - 57 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 66 \beta_{7} - 211 \beta_{4} + 391 \beta_{3} - 28 \beta_{2} + 94 \beta _1 + 500 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 60 \beta_{15} - 453 \beta_{14} - 453 \beta_{13} - 218 \beta_{12} + 569 \beta_{11} + 569 \beta_{10} - 211 \beta_{9} - 211 \beta_{8} - 13 \beta_{7} - 26 \beta_{6} - 222 \beta_{5} + 30 \beta_{4} - 111 \beta_{3} + 1098 \beta_{2} + 1111 \beta _1 + 79 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 124 \beta_{14} - 124 \beta_{13} - 403 \beta_{11} + 403 \beta_{10} + 30 \beta_{9} - 30 \beta_{8} - 453 \beta_{7} + 1309 \beta_{4} - 2633 \beta_{3} + 280 \beta_{2} - 733 \beta _1 - 3091 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 620 \beta_{15} + 3026 \beta_{14} + 3026 \beta_{13} + 1778 \beta_{12} - 3927 \beta_{11} - 3927 \beta_{10} + 1309 \beta_{9} + 1309 \beta_{8} + 124 \beta_{7} + 248 \beta_{6} + 1862 \beta_{5} - 310 \beta_{4} + \cdots - 579 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1055 \beta_{14} + 1055 \beta_{13} + 2861 \beta_{11} - 2861 \beta_{10} - 310 \beta_{9} + 310 \beta_{8} + 3026 \beta_{7} - 8282 \beta_{4} + 17572 \beta_{3} - 2440 \beta_{2} + 5466 \beta _1 + 19574 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5500 \beta_{15} - 19978 \beta_{14} - 19978 \beta_{13} - 13702 \beta_{12} + 26552 \beta_{11} + 26552 \beta_{10} - 8282 \beta_{9} - 8282 \beta_{8} - 1055 \beta_{7} - 2110 \beta_{6} - 14822 \beta_{5} + \cdots + 4101 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8466 \beta_{14} - 8466 \beta_{13} - 20354 \beta_{11} + 20354 \beta_{10} + 2750 \beta_{9} - 2750 \beta_{8} - 19978 \beta_{7} + 53090 \beta_{4} - 116816 \beta_{3} + 19762 \beta_{2} - 39740 \beta _1 - 125893 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 45024 \beta_{15} + 131294 \beta_{14} + 131294 \beta_{13} + 102072 \beta_{12} - 177792 \beta_{11} - 177792 \beta_{10} + 53090 \beta_{9} + 53090 \beta_{8} + 8466 \beta_{7} + 16932 \beta_{6} + \cdots - 28524 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(\beta_{12}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
− 2.60802i
− 1.77930i
− 0.775848i
− 0.485989i
0.106359i
1.10207i
1.98765i
2.45308i
2.60802i
1.77930i
0.775848i
0.485989i
− 0.106359i
− 1.10207i
− 1.98765i
− 2.45308i
−2.25861 − 1.30401i −0.500000 + 0.866025i 2.40088 + 4.15844i − 1.50528i 2.25861 − 1.30401i −0.866025 + 0.500000i − 7.30704i −0.500000 − 0.866025i −1.96290 + 3.39983i
43.2 −1.54092 − 0.889651i −0.500000 + 0.866025i 0.582956 + 1.00971i − 0.681820i 1.54092 − 0.889651i 0.866025 − 0.500000i 1.48409i −0.500000 − 0.866025i −0.606581 + 1.05063i
43.3 −0.671904 − 0.387924i −0.500000 + 0.866025i −0.699030 − 1.21076i 3.03444i 0.671904 − 0.387924i 0.866025 − 0.500000i 2.63638i −0.500000 − 0.866025i 1.17713 − 2.03885i
43.4 −0.420879 − 0.242995i −0.500000 + 0.866025i −0.881907 − 1.52751i 1.06536i 0.420879 − 0.242995i −0.866025 + 0.500000i 1.82917i −0.500000 − 0.866025i 0.258876 − 0.448387i
43.5 0.0921099 + 0.0531797i −0.500000 + 0.866025i −0.994344 − 1.72225i − 1.41292i −0.0921099 + 0.0531797i −0.866025 + 0.500000i − 0.424234i −0.500000 − 0.866025i 0.0751388 − 0.130144i
43.6 0.954423 + 0.551037i −0.500000 + 0.866025i −0.392717 − 0.680206i − 3.28432i −0.954423 + 0.551037i 0.866025 − 0.500000i − 3.06975i −0.500000 − 0.866025i 1.80978 − 3.13463i
43.7 1.72135 + 0.993824i −0.500000 + 0.866025i 0.975372 + 1.68939i 2.85284i −1.72135 + 0.993824i −0.866025 + 0.500000i − 0.0979034i −0.500000 − 0.866025i −2.83522 + 4.91075i
43.8 2.12443 + 1.22654i −0.500000 + 0.866025i 2.00879 + 3.47933i − 0.0682999i −2.12443 + 1.22654i 0.866025 − 0.500000i 4.94928i −0.500000 − 0.866025i 0.0837724 − 0.145098i
127.1 −2.25861 + 1.30401i −0.500000 − 0.866025i 2.40088 − 4.15844i 1.50528i 2.25861 + 1.30401i −0.866025 − 0.500000i 7.30704i −0.500000 + 0.866025i −1.96290 − 3.39983i
127.2 −1.54092 + 0.889651i −0.500000 − 0.866025i 0.582956 − 1.00971i 0.681820i 1.54092 + 0.889651i 0.866025 + 0.500000i − 1.48409i −0.500000 + 0.866025i −0.606581 − 1.05063i
127.3 −0.671904 + 0.387924i −0.500000 − 0.866025i −0.699030 + 1.21076i − 3.03444i 0.671904 + 0.387924i 0.866025 + 0.500000i − 2.63638i −0.500000 + 0.866025i 1.17713 + 2.03885i
127.4 −0.420879 + 0.242995i −0.500000 − 0.866025i −0.881907 + 1.52751i − 1.06536i 0.420879 + 0.242995i −0.866025 − 0.500000i − 1.82917i −0.500000 + 0.866025i 0.258876 + 0.448387i
127.5 0.0921099 − 0.0531797i −0.500000 − 0.866025i −0.994344 + 1.72225i 1.41292i −0.0921099 − 0.0531797i −0.866025 − 0.500000i 0.424234i −0.500000 + 0.866025i 0.0751388 + 0.130144i
127.6 0.954423 − 0.551037i −0.500000 − 0.866025i −0.392717 + 0.680206i 3.28432i −0.954423 − 0.551037i 0.866025 + 0.500000i 3.06975i −0.500000 + 0.866025i 1.80978 + 3.13463i
127.7 1.72135 − 0.993824i −0.500000 − 0.866025i 0.975372 − 1.68939i − 2.85284i −1.72135 − 0.993824i −0.866025 − 0.500000i 0.0979034i −0.500000 + 0.866025i −2.83522 − 4.91075i
127.8 2.12443 − 1.22654i −0.500000 − 0.866025i 2.00879 − 3.47933i 0.0682999i −2.12443 − 1.22654i 0.866025 + 0.500000i − 4.94928i −0.500000 + 0.866025i 0.0837724 + 0.145098i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
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 273.2.bd.a ✓ 16
3.b odd 2 1 819.2.ct.b 16
13.e even 6 1 inner 273.2.bd.a ✓ 16
13.f odd 12 1 3549.2.a.bb 8
13.f odd 12 1 3549.2.a.bd 8
39.h odd 6 1 819.2.ct.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.a ✓ 16 1.a even 1 1 trivial
273.2.bd.a ✓ 16 13.e even 6 1 inner
819.2.ct.b 16 3.b odd 2 1
819.2.ct.b 16 39.h odd 6 1
3549.2.a.bb 8 13.f odd 12 1
3549.2.a.bd 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 11 T_{2}^{14} + 88 T_{2}^{12} - 6 T_{2}^{11} - 315 T_{2}^{10} + 12 T_{2}^{9} + 824 T_{2}^{8} + 66 T_{2}^{7} - 758 T_{2}^{6} - 144 T_{2}^{5} + 519 T_{2}^{4} + 288 T_{2}^{3} + 24 T_{2}^{2} - 12 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 11 T^{14} + 88 T^{12} - 6 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 34 T^{14} + 439 T^{12} + 2690 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + 12 T^{15} + 28 T^{14} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{16} - 4 T^{15} - 25 T^{14} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 10 T^{15} + \cdots + 2883582601 \) Copy content Toggle raw display
$19$ \( T^{16} - 95 T^{14} + \cdots + 25335725584 \) Copy content Toggle raw display
$23$ \( T^{16} + 2 T^{15} + \cdots + 1446433024 \) Copy content Toggle raw display
$29$ \( T^{16} - 12 T^{15} + \cdots + 2538849769 \) Copy content Toggle raw display
$31$ \( T^{16} + 222 T^{14} + 17145 T^{12} + \cdots + 7929856 \) Copy content Toggle raw display
$37$ \( T^{16} - 18 T^{15} + \cdots + 16292735449 \) Copy content Toggle raw display
$41$ \( T^{16} - 18 T^{15} + 71 T^{14} + \cdots + 692224 \) Copy content Toggle raw display
$43$ \( T^{16} + 10 T^{15} + \cdots + 15764309136 \) Copy content Toggle raw display
$47$ \( T^{16} + 222 T^{14} + 15381 T^{12} + \cdots + 1336336 \) Copy content Toggle raw display
$53$ \( (T^{8} - 6 T^{7} - 168 T^{6} + 1026 T^{5} + \cdots - 76707)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 60 T^{15} + \cdots + 3887273104 \) Copy content Toggle raw display
$61$ \( T^{16} + 6 T^{15} + \cdots + 103866332089 \) Copy content Toggle raw display
$67$ \( T^{16} + 18 T^{15} + \cdots + 5866334464 \) Copy content Toggle raw display
$71$ \( T^{16} - 6 T^{15} + \cdots + 84\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{16} + 646 T^{14} + \cdots + 17309352523401 \) Copy content Toggle raw display
$79$ \( (T^{8} + 2 T^{7} - 563 T^{6} + \cdots + 75240852)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 684 T^{14} + \cdots + 2140872301584 \) Copy content Toggle raw display
$89$ \( T^{16} - 78 T^{15} + \cdots + 72\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{16} + 54 T^{15} + \cdots + 941955655936 \) Copy content Toggle raw display
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