Properties

Label 45.3.i.a
Level $45$
Weight $3$
Character orbit 45.i
Analytic conductor $1.226$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,3,Mod(11,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 45.i (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: \(1.22616118962\)
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: \( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{11} - \beta_{7}) q^{3} + ( - \beta_{14} + 2 \beta_{7} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{4} + \beta_{9} q^{5} + (\beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} - \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} + \cdots - 2) q^{6}+ \cdots + (\beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{4} - \beta_{2} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{11} - \beta_{7}) q^{3} + ( - \beta_{14} + 2 \beta_{7} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{4} + \beta_{9} q^{5} + (\beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} - \beta_{8} - \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} + \cdots - 2) q^{6}+ \cdots + (14 \beta_{15} - 13 \beta_{14} - 22 \beta_{13} - 12 \beta_{12} + 14 \beta_{11} + \cdots - 51) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} + 16 q^{4} - 22 q^{6} + 2 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{3} + 16 q^{4} - 22 q^{6} + 2 q^{7} + 8 q^{9} - 18 q^{11} - 22 q^{12} - 10 q^{13} - 54 q^{14} + 10 q^{15} - 32 q^{16} - 8 q^{18} - 52 q^{19} + 72 q^{21} - 24 q^{22} - 54 q^{23} + 108 q^{24} + 40 q^{25} + 34 q^{27} + 32 q^{28} - 54 q^{29} - 100 q^{30} + 32 q^{31} + 216 q^{32} + 62 q^{33} + 54 q^{34} - 86 q^{36} + 44 q^{37} + 252 q^{38} + 160 q^{39} - 30 q^{40} + 144 q^{41} - 270 q^{42} - 124 q^{43} + 140 q^{45} - 108 q^{46} - 216 q^{47} - 172 q^{48} - 54 q^{49} - 106 q^{51} + 62 q^{52} - 316 q^{54} - 18 q^{56} - 236 q^{57} + 90 q^{58} - 486 q^{59} - 10 q^{60} + 62 q^{61} - 132 q^{63} + 256 q^{64} - 90 q^{65} + 208 q^{66} + 14 q^{67} - 288 q^{68} + 90 q^{69} - 60 q^{70} + 804 q^{72} - 268 q^{73} + 540 q^{74} - 20 q^{75} - 106 q^{76} + 702 q^{77} + 290 q^{78} - 40 q^{79} - 112 q^{81} - 204 q^{82} + 522 q^{83} + 714 q^{84} + 30 q^{85} + 54 q^{86} + 106 q^{87} + 144 q^{88} + 250 q^{90} + 136 q^{91} - 1332 q^{92} + 90 q^{93} - 150 q^{94} + 180 q^{95} + 166 q^{96} - 142 q^{97} - 824 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 43 \nu^{12} + 715 \nu^{10} + 5777 \nu^{8} + 23051 \nu^{6} + 39821 \nu^{4} + 25605 \nu^{2} - 2376 \nu + 28755 ) / 4752 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{14} - 43 \nu^{12} - 715 \nu^{10} - 5777 \nu^{8} - 23051 \nu^{6} - 39821 \nu^{4} - 20853 \nu^{2} + 2376 \nu - 243 ) / 4752 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{15} - 68 \nu^{14} + 369 \nu^{13} - 3056 \nu^{12} + 7299 \nu^{11} - 53504 \nu^{10} + 65653 \nu^{9} - 460024 \nu^{8} + 241317 \nu^{7} - 2004520 \nu^{6} + 46971 \nu^{5} + \cdots - 532116 ) / 175824 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53 \nu^{15} + 155 \nu^{14} + 2741 \nu^{13} + 6659 \nu^{12} + 54989 \nu^{11} + 108827 \nu^{10} + 534013 \nu^{9} + 837547 \nu^{8} + 2524279 \nu^{7} + 2999593 \nu^{6} + \cdots + 1364877 ) / 351648 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 215 \nu^{15} + 195 \nu^{14} + 10139 \nu^{13} + 9423 \nu^{12} + 187691 \nu^{11} + 177387 \nu^{10} + 1720855 \nu^{9} + 1631991 \nu^{8} + 8064517 \nu^{7} + 7502193 \nu^{6} + \cdots + 1868049 ) / 351648 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3 \nu^{15} - 143 \nu^{13} - 2693 \nu^{11} - 25397 \nu^{9} - 125029 \nu^{7} - 304045 \nu^{5} - 313711 \nu^{3} - 87147 \nu - 2376 ) / 4752 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 261 \nu^{15} + 145 \nu^{14} - 12289 \nu^{13} + 6301 \nu^{12} - 227389 \nu^{11} + 106117 \nu^{10} - 2091313 \nu^{9} + 878585 \nu^{8} - 9940775 \nu^{7} + \cdots + 1084239 ) / 351648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 247 \nu^{15} - 77 \nu^{14} + 11847 \nu^{13} - 3245 \nu^{12} + 225075 \nu^{11} - 52613 \nu^{10} + 2150779 \nu^{9} - 413677 \nu^{8} + 10811157 \nu^{7} - 1622335 \nu^{6} + \cdots + 678645 ) / 351648 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 247 \nu^{15} - 77 \nu^{14} - 11847 \nu^{13} - 3245 \nu^{12} - 225075 \nu^{11} - 52613 \nu^{10} - 2150779 \nu^{9} - 413677 \nu^{8} - 10811157 \nu^{7} - 1622335 \nu^{6} + \cdots + 678645 ) / 351648 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 67 \nu^{15} - 5 \nu^{14} + 3257 \nu^{13} - 290 \nu^{12} + 62816 \nu^{11} - 6572 \nu^{10} + 610152 \nu^{9} - 73609 \nu^{8} + 3120355 \nu^{7} - 424519 \nu^{6} + 7992463 \nu^{5} + \cdots - 202257 ) / 87912 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 245 \nu^{15} + 19 \nu^{14} - 11805 \nu^{13} + 769 \nu^{12} - 225273 \nu^{11} + 12253 \nu^{10} - 2159993 \nu^{9} + 105755 \nu^{8} - 10866783 \nu^{7} + 598721 \nu^{6} + \cdots + 336609 ) / 175824 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 161 \nu^{15} - 16 \nu^{14} + 7747 \nu^{13} - 632 \nu^{12} + 147749 \nu^{11} - 9072 \nu^{10} + 1418111 \nu^{9} - 55196 \nu^{8} + 7164083 \nu^{7} - 102148 \nu^{6} + 18140737 \nu^{5} + \cdots + 92304 ) / 58608 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17 \nu^{15} - 815 \nu^{13} - 15443 \nu^{11} - 146605 \nu^{9} - 727123 \nu^{7} - 1784449 \nu^{5} - 1861413 \nu^{3} - 2376 \nu^{2} - 527391 \nu - 14256 ) / 4752 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1361 \nu^{15} - 13 \nu^{14} - 65433 \nu^{13} - 421 \nu^{12} - 1245657 \nu^{11} - 2413 \nu^{10} - 11920553 \nu^{9} + 54859 \nu^{8} - 59989011 \nu^{7} + 842569 \nu^{6} + \cdots + 1470933 ) / 351648 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + 2\beta_{11} - \beta_{8} + 3\beta_{7} - 2\beta_{4} - \beta_{2} - 10\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} + 3\beta_{10} + 3\beta_{9} - \beta_{7} - 18\beta_{3} - 12\beta_{2} + 3\beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 18 \beta_{15} + 26 \beta_{14} + 6 \beta_{13} + 4 \beta_{12} - 34 \beta_{11} - 6 \beta_{10} - \beta_{9} + 10 \beta_{8} - 63 \beta_{7} + 7 \beta_{5} + 36 \beta_{4} + 4 \beta_{3} + 22 \beta_{2} + 116 \beta _1 - 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - 23 \beta_{12} - 27 \beta_{11} - 57 \beta_{10} - 59 \beta_{9} - 4 \beta_{8} + 21 \beta_{7} - 8 \beta_{6} + 10 \beta_{5} - 18 \beta_{4} + 277 \beta_{3} + 139 \beta_{2} - 67 \beta _1 - 681 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 267 \beta_{15} - 471 \beta_{14} - 144 \beta_{13} - 96 \beta_{12} + 504 \beta_{11} + 186 \beta_{10} - 9 \beta_{9} - 75 \beta_{8} + 1020 \beta_{7} - 177 \beta_{5} - 552 \beta_{4} - 102 \beta_{3} - 369 \beta_{2} - 1421 \beta _1 + 684 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 108 \beta_{15} + 108 \beta_{14} - 48 \beta_{13} + 432 \beta_{12} + 528 \beta_{11} + 900 \beta_{10} + 942 \beta_{9} + 108 \beta_{8} - 378 \beta_{7} + 216 \beta_{6} - 258 \beta_{5} + 468 \beta_{4} - 4022 \beta_{3} + \cdots + 8187 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3806 \beta_{15} + 7550 \beta_{14} + 2568 \beta_{13} + 1746 \beta_{12} - 7258 \beta_{11} - 3834 \beta_{10} + 570 \beta_{9} + 314 \beta_{8} - 15072 \beta_{7} + 3264 \beta_{5} + 8080 \beta_{4} + 1872 \beta_{3} + \cdots - 10198 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2088 \beta_{15} - 2088 \beta_{14} + 924 \beta_{13} - 7298 \beta_{12} - 9146 \beta_{11} - 13506 \beta_{10} - 14112 \beta_{9} - 2088 \beta_{8} + 6452 \beta_{7} - 4176 \beta_{6} + 4782 \beta_{5} - 8640 \beta_{4} + \cdots - 102042 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 53703 \beta_{15} - 114319 \beta_{14} - 41040 \beta_{13} - 28598 \beta_{12} + 103604 \beta_{11} + 67170 \beta_{10} - 13732 \beta_{9} + 3493 \beta_{8} + 213993 \beta_{7} - 53438 \beta_{5} + \cdots + 146268 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 35528 \beta_{15} + 35528 \beta_{14} - 16156 \beta_{13} + 115609 \beta_{12} + 147921 \beta_{11} + 198303 \beta_{10} + 205525 \beta_{9} + 35528 \beta_{8} - 105171 \beta_{7} + 71056 \beta_{6} + \cdots + 1306266 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 755544 \beta_{15} + 1678848 \beta_{14} + 622506 \beta_{13} + 443874 \beta_{12} - 1472856 \beta_{11} - 1083540 \beta_{10} + 258213 \beta_{9} - 132204 \beta_{8} - 2981067 \beta_{7} + \cdots - 2059032 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 566496 \beta_{15} - 566496 \beta_{14} + 265242 \beta_{13} - 1758429 \beta_{12} - 2288913 \beta_{11} - 2875095 \beta_{10} - 2948883 \beta_{9} - 566496 \beta_{8} + 1648629 \beta_{7} + \cdots - 17072361 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 10619317 \beta_{15} - 24221449 \beta_{14} - 9179076 \beta_{13} - 6672042 \beta_{12} + 20879918 \beta_{11} + 16666128 \beta_{10} - 4330389 \beta_{9} + 2724767 \beta_{8} + \cdots + 28720238 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1 + \beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
3.73655i
3.27064i
1.39204i
0.0476108i
0.692902i
1.83391i
2.82877i
3.09125i
3.73655i
3.27064i
1.39204i
0.0476108i
0.692902i
1.83391i
2.82877i
3.09125i
−3.23594 1.86827i 2.62117 + 1.45927i 4.98088 + 8.62715i 1.93649 1.11803i −5.75562 9.61919i 3.63061 6.28840i 22.2764i 4.74103 + 7.65001i −8.35517
11.2 −2.83245 1.63532i −2.26486 1.96733i 3.34853 + 5.79983i −1.93649 + 1.11803i 3.19790 + 9.27615i −3.16931 + 5.48940i 8.82112i 1.25919 + 8.91148i 7.31337
11.3 −1.20554 0.696021i 2.33148 1.88791i −1.03111 1.78593i −1.93649 + 1.11803i −4.12473 + 0.653207i 4.41004 7.63842i 8.43886i 1.87156 8.80325i 3.11270
11.4 −0.0412321 0.0238054i −1.76580 2.42528i −1.99887 3.46214i 1.93649 1.11803i 0.0150729 + 0.142035i 1.90756 3.30399i 0.380778i −2.76393 + 8.56508i −0.106461
11.5 0.600071 + 0.346451i 2.96601 + 0.450312i −1.75994 3.04831i 1.93649 1.11803i 1.62381 + 1.29780i −6.14112 + 10.6367i 5.21055i 8.59444 + 2.67126i 1.54938
11.6 1.58822 + 0.916957i 0.822398 + 2.88508i −0.318381 0.551452i −1.93649 + 1.11803i −1.33934 + 5.33623i 3.23398 5.60142i 8.50342i −7.64732 + 4.74536i −4.10076
11.7 2.44978 + 1.41438i 0.110987 2.99795i 2.00096 + 3.46576i −1.93649 + 1.11803i 4.51214 7.18734i −3.97472 + 6.88441i 0.00541780i −8.97536 0.665467i −6.32531
11.8 2.67710 + 1.54563i −2.82138 + 1.01971i 2.77793 + 4.81151i 1.93649 1.11803i −9.12922 1.63094i 1.10296 1.91037i 4.80953i 6.92040 5.75396i 6.91225
41.1 −3.23594 + 1.86827i 2.62117 1.45927i 4.98088 8.62715i 1.93649 + 1.11803i −5.75562 + 9.61919i 3.63061 + 6.28840i 22.2764i 4.74103 7.65001i −8.35517
41.2 −2.83245 + 1.63532i −2.26486 + 1.96733i 3.34853 5.79983i −1.93649 1.11803i 3.19790 9.27615i −3.16931 5.48940i 8.82112i 1.25919 8.91148i 7.31337
41.3 −1.20554 + 0.696021i 2.33148 + 1.88791i −1.03111 + 1.78593i −1.93649 1.11803i −4.12473 0.653207i 4.41004 + 7.63842i 8.43886i 1.87156 + 8.80325i 3.11270
41.4 −0.0412321 + 0.0238054i −1.76580 + 2.42528i −1.99887 + 3.46214i 1.93649 + 1.11803i 0.0150729 0.142035i 1.90756 + 3.30399i 0.380778i −2.76393 8.56508i −0.106461
41.5 0.600071 0.346451i 2.96601 0.450312i −1.75994 + 3.04831i 1.93649 + 1.11803i 1.62381 1.29780i −6.14112 10.6367i 5.21055i 8.59444 2.67126i 1.54938
41.6 1.58822 0.916957i 0.822398 2.88508i −0.318381 + 0.551452i −1.93649 1.11803i −1.33934 5.33623i 3.23398 + 5.60142i 8.50342i −7.64732 4.74536i −4.10076
41.7 2.44978 1.41438i 0.110987 + 2.99795i 2.00096 3.46576i −1.93649 1.11803i 4.51214 + 7.18734i −3.97472 6.88441i 0.00541780i −8.97536 + 0.665467i −6.32531
41.8 2.67710 1.54563i −2.82138 1.01971i 2.77793 4.81151i 1.93649 + 1.11803i −9.12922 + 1.63094i 1.10296 + 1.91037i 4.80953i 6.92040 + 5.75396i 6.91225
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.3.i.a 16
3.b odd 2 1 135.3.i.a 16
4.b odd 2 1 720.3.bs.c 16
5.b even 2 1 225.3.j.b 16
5.c odd 4 2 225.3.i.b 32
9.c even 3 1 135.3.i.a 16
9.c even 3 1 405.3.c.a 16
9.d odd 6 1 inner 45.3.i.a 16
9.d odd 6 1 405.3.c.a 16
12.b even 2 1 2160.3.bs.c 16
15.d odd 2 1 675.3.j.b 16
15.e even 4 2 675.3.i.c 32
36.f odd 6 1 2160.3.bs.c 16
36.h even 6 1 720.3.bs.c 16
45.h odd 6 1 225.3.j.b 16
45.j even 6 1 675.3.j.b 16
45.k odd 12 2 675.3.i.c 32
45.l even 12 2 225.3.i.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.i.a 16 1.a even 1 1 trivial
45.3.i.a 16 9.d odd 6 1 inner
135.3.i.a 16 3.b odd 2 1
135.3.i.a 16 9.c even 3 1
225.3.i.b 32 5.c odd 4 2
225.3.i.b 32 45.l even 12 2
225.3.j.b 16 5.b even 2 1
225.3.j.b 16 45.h odd 6 1
405.3.c.a 16 9.c even 3 1
405.3.c.a 16 9.d odd 6 1
675.3.i.c 32 15.e even 4 2
675.3.i.c 32 45.k odd 12 2
675.3.j.b 16 15.d odd 2 1
675.3.j.b 16 45.j even 6 1
720.3.bs.c 16 4.b odd 2 1
720.3.bs.c 16 36.h even 6 1
2160.3.bs.c 16 12.b even 2 1
2160.3.bs.c 16 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 24 T^{14} + 408 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{15} + 4 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + \cdots + 4654875290256 \) Copy content Toggle raw display
$11$ \( T^{16} + 18 T^{15} + \cdots + 112356358416 \) Copy content Toggle raw display
$13$ \( T^{16} + 10 T^{15} + \cdots + 69380636886016 \) Copy content Toggle raw display
$17$ \( T^{16} + 2790 T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + 26 T^{7} - 1415 T^{6} + \cdots - 3133657244)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 54 T^{15} + \cdots + 64\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} + 54 T^{15} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{16} - 32 T^{15} + \cdots + 396718580736 \) Copy content Toggle raw display
$37$ \( (T^{8} - 22 T^{7} + \cdots - 143779124336)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 144 T^{15} + \cdots + 44\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{16} + 124 T^{15} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + 216 T^{15} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + 30792 T^{14} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + 486 T^{15} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{16} - 62 T^{15} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{16} - 14 T^{15} + \cdots + 39\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{16} + 24048 T^{14} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + 134 T^{7} + \cdots - 3873480104384)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 40 T^{15} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{16} - 522 T^{15} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + 18666 T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + 142 T^{15} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
show more
show less