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Properties

Label 165.2.k.c
Level $165$
Weight $2$
Character orbit 165.k
Analytic conductor $1.318$
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] = [165,2,Mod(23,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.k (of order \(4\), 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: \(1.31753163335\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} + 19 x^{12} - 80 x^{11} + 168 x^{10} + 28 x^{9} + 119 x^{8} - 432 x^{7} + 784 x^{6} + 84 x^{5} + 169 x^{4} - 420 x^{3} + 392 x^{2} - 112 x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} + 19 x^{12} - 80 x^{11} + 168 x^{10} + 28 x^{9} + 119 x^{8} - 432 x^{7} + 784 x^{6} + 84 x^{5} + 169 x^{4} - 420 x^{3} + 392 x^{2} - 112 x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4128071754399 \nu^{15} + 11898447714838 \nu^{14} - 15277599976658 \nu^{13} - 34812594791870 \nu^{12} + \cdots - 80\!\cdots\!20 ) / 27\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21143084361889 \nu^{15} - 76316193938758 \nu^{14} + 145347779465436 \nu^{13} + 30555199953316 \nu^{12} + \cdots - 19\!\cdots\!44 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 38450704565045 \nu^{15} - 59618622420401 \nu^{14} - 46616013291644 \nu^{13} + 663284102485220 \nu^{12} + \cdots + 14\!\cdots\!96 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 106357174118413 \nu^{15} - 451134746013694 \nu^{14} + 911994273881980 \nu^{13} - 31886019956504 \nu^{12} + \cdots - 12\!\cdots\!60 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 157213841010915 \nu^{15} - 509728662885302 \nu^{14} + 841883624893756 \nu^{13} + 708288711249504 \nu^{12} + \cdots + 16\!\cdots\!72 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 207551406788003 \nu^{15} - 560257236979352 \nu^{14} + 726910331340020 \nu^{13} + \cdots + 29\!\cdots\!80 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 247784897113907 \nu^{15} - 933231595130810 \nu^{14} + \cdots - 40\!\cdots\!32 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 249727967307343 \nu^{15} - 956625700505594 \nu^{14} + \cdots - 14\!\cdots\!96 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 83688943960991 \nu^{15} - 319218427676056 \nu^{14} + 612574342939380 \nu^{13} + 102474554907584 \nu^{12} + \cdots - 12\!\cdots\!88 ) / 27\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 426823538486029 \nu^{15} + \cdots + 58\!\cdots\!96 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 233776061987612 \nu^{15} - 998483250651247 \nu^{14} + \cdots - 28\!\cdots\!44 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 67717762917308 \nu^{15} + 258235473611088 \nu^{14} - 497634782511666 \nu^{13} - 80312689721047 \nu^{12} + \cdots + 41\!\cdots\!60 ) / 13\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 626947109265783 \nu^{15} + \cdots + 25\!\cdots\!04 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 627014941431923 \nu^{15} + \cdots - 41\!\cdots\!28 ) / 10\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + 2\beta_{9} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{9} + \beta_{5} + 4\beta_{3} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{8} + \beta_{5} + \beta_{3} - 6\beta_{2} - \beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{14} - 8 \beta_{13} + \beta_{12} - \beta_{11} - 10 \beta_{9} - 8 \beta_{8} + \beta_{6} + 2 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} - 21 \beta _1 - 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 37 \beta_{13} + 4 \beta_{12} - 9 \beta_{11} - \beta_{10} - 52 \beta_{9} - 11 \beta_{8} + \beta_{7} - 11 \beta_{5} + 12 \beta_{4} - 47 \beta_{3} - 47 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{15} - 9 \beta_{14} - 57 \beta_{13} + 15 \beta_{12} - 13 \beta_{11} - 77 \beta_{9} + 4 \beta_{7} - 15 \beta_{6} - 57 \beta_{5} + 26 \beta_{4} - 126 \beta_{3} + 57 \beta_{2} - 57 \beta _1 + 134 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 53 \beta_{15} - 53 \beta_{14} - 11 \beta_{11} + 15 \beta_{10} + 92 \beta_{8} + 15 \beta_{7} - 60 \beta_{6} - 92 \beta_{5} + 4 \beta_{4} - 81 \beta_{3} + 236 \beta_{2} + 81 \beta _1 + 567 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 62 \beta_{15} - 92 \beta_{14} + 394 \beta_{13} - 154 \beta_{12} + 92 \beta_{11} + 60 \beta_{10} + 554 \beta_{9} + 402 \beta_{8} - 154 \beta_{6} - 216 \beta_{4} + 394 \beta_{3} + 394 \beta_{2} + 807 \beta _1 + 948 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 60 \beta_{15} - 60 \beta_{14} + 1543 \beta_{13} - 612 \beta_{12} + 496 \beta_{11} + 154 \beta_{10} + 2200 \beta_{9} + 702 \beta_{8} - 154 \beta_{7} + 702 \beta_{5} - 954 \beta_{4} + 2149 \beta_{3} + 2149 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 702 \beta_{15} + 394 \beta_{14} + 2697 \beta_{13} - 1356 \beta_{12} + 1006 \beta_{11} + 3893 \beta_{9} - 612 \beta_{7} + 1356 \beta_{6} + 2839 \beta_{5} - 2058 \beta_{4} + 5344 \beta_{3} - 2697 \beta_{2} + 2697 \beta _1 - 6590 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2227 \beta_{15} + 2227 \beta_{14} + 744 \beta_{11} - 1356 \beta_{10} - 5149 \beta_{8} - 1356 \beta_{7} + 5336 \beta_{6} + 5149 \beta_{5} - 612 \beta_{4} + 4363 \beta_{3} - 10268 \beta_{2} - 4363 \beta _1 - 25195 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2437 \beta_{15} + 5149 \beta_{14} - 18424 \beta_{13} + 11019 \beta_{12} - 5149 \beta_{11} - 5336 \beta_{10} - 27114 \beta_{9} - 20058 \beta_{8} + 11019 \beta_{6} + 13456 \beta_{4} - 18424 \beta_{3} + \cdots - 45538 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5336 \beta_{15} + 5336 \beta_{14} - 69179 \beta_{13} + 42920 \beta_{12} - 25741 \beta_{11} - 11019 \beta_{10} - 102334 \beta_{9} - 37029 \beta_{8} + 11019 \beta_{7} - 37029 \beta_{5} + \cdots - 99995 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 37029 \beta_{15} - 14991 \beta_{14} - 126005 \beta_{13} + 85363 \beta_{12} - 57911 \beta_{11} - 188245 \beta_{9} + 42920 \beta_{7} - 85363 \beta_{6} - 141543 \beta_{5} + 122392 \beta_{4} + \cdots + 314250 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
1.86535 − 1.86535i
1.63522 − 1.63522i
1.05652 − 1.05652i
0.416745 − 0.416745i
0.178913 − 0.178913i
−0.688858 + 0.688858i
−1.14633 + 1.14633i
−1.31757 + 1.31757i
1.86535 + 1.86535i
1.63522 + 1.63522i
1.05652 + 1.05652i
0.416745 + 0.416745i
0.178913 + 0.178913i
−0.688858 − 0.688858i
−1.14633 − 1.14633i
−1.31757 − 1.31757i
−1.86535 − 1.86535i 1.54580 − 0.781338i 4.95908i 0.0840146 + 2.23449i −4.34094 − 1.42600i 2.42600 − 2.42600i 5.51973 − 5.51973i 1.77902 − 2.41559i 4.01139 − 4.32483i
23.2 −1.63522 − 1.63522i −1.37033 + 1.05933i 3.34791i 1.69456 − 1.45893i 3.97305 + 0.508557i 0.491443 − 0.491443i 2.20413 − 2.20413i 0.755630 − 2.90328i −5.15665 − 0.385296i
23.3 −1.05652 − 1.05652i −1.71860 − 0.215468i 0.232479i −0.158945 + 2.23041i 1.58809 + 2.04338i −1.04338 + 1.04338i −1.86743 + 1.86743i 2.90715 + 0.740604i 2.52441 − 2.18855i
23.4 −0.416745 − 0.416745i 0.554354 − 1.64094i − 1.65265i −2.22420 + 0.230100i −0.914879 + 0.452830i 0.547170 − 0.547170i −1.52222 + 1.52222i −2.38538 − 1.81933i 1.02282 + 0.831030i
23.5 −0.178913 − 0.178913i 0.720382 + 1.57513i − 1.93598i 0.754048 − 2.10509i 0.152926 − 0.410697i 1.41070 − 1.41070i −0.704196 + 0.704196i −1.96210 + 2.26940i −0.511536 + 0.241719i
23.6 0.688858 + 0.688858i −1.67171 − 0.453204i − 1.05095i −2.14206 − 0.641537i −0.839376 − 1.46376i 2.46376 − 2.46376i 2.10167 − 2.10167i 2.58921 + 1.51525i −1.03365 − 1.91750i
23.7 1.14633 + 1.14633i −0.582649 + 1.63111i 0.628149i −0.515221 + 2.17590i −2.53770 + 1.20188i −0.201884 + 0.201884i 1.57260 − 1.57260i −2.32104 − 1.90073i −3.08492 + 1.90369i
23.8 1.31757 + 1.31757i 1.52275 + 0.825374i 1.47196i −1.49219 − 1.66534i 0.918834 + 3.09381i −2.09381 + 2.09381i 0.695724 − 0.695724i 1.63751 + 2.51367i 0.228136 − 4.16026i
122.1 −1.86535 + 1.86535i 1.54580 + 0.781338i − 4.95908i 0.0840146 − 2.23449i −4.34094 + 1.42600i 2.42600 + 2.42600i 5.51973 + 5.51973i 1.77902 + 2.41559i 4.01139 + 4.32483i
122.2 −1.63522 + 1.63522i −1.37033 − 1.05933i − 3.34791i 1.69456 + 1.45893i 3.97305 − 0.508557i 0.491443 + 0.491443i 2.20413 + 2.20413i 0.755630 + 2.90328i −5.15665 + 0.385296i
122.3 −1.05652 + 1.05652i −1.71860 + 0.215468i − 0.232479i −0.158945 − 2.23041i 1.58809 − 2.04338i −1.04338 − 1.04338i −1.86743 − 1.86743i 2.90715 − 0.740604i 2.52441 + 2.18855i
122.4 −0.416745 + 0.416745i 0.554354 + 1.64094i 1.65265i −2.22420 − 0.230100i −0.914879 − 0.452830i 0.547170 + 0.547170i −1.52222 − 1.52222i −2.38538 + 1.81933i 1.02282 − 0.831030i
122.5 −0.178913 + 0.178913i 0.720382 − 1.57513i 1.93598i 0.754048 + 2.10509i 0.152926 + 0.410697i 1.41070 + 1.41070i −0.704196 − 0.704196i −1.96210 − 2.26940i −0.511536 − 0.241719i
122.6 0.688858 − 0.688858i −1.67171 + 0.453204i 1.05095i −2.14206 + 0.641537i −0.839376 + 1.46376i 2.46376 + 2.46376i 2.10167 + 2.10167i 2.58921 − 1.51525i −1.03365 + 1.91750i
122.7 1.14633 − 1.14633i −0.582649 − 1.63111i − 0.628149i −0.515221 − 2.17590i −2.53770 − 1.20188i −0.201884 − 0.201884i 1.57260 + 1.57260i −2.32104 + 1.90073i −3.08492 − 1.90369i
122.8 1.31757 − 1.31757i 1.52275 − 0.825374i − 1.47196i −1.49219 + 1.66534i 0.918834 − 3.09381i −2.09381 − 2.09381i 0.695724 + 0.695724i 1.63751 − 2.51367i 0.228136 + 4.16026i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.k.c ✓ 16
3.b odd 2 1 165.2.k.d yes 16
5.b even 2 1 825.2.k.j 16
5.c odd 4 1 165.2.k.d yes 16
5.c odd 4 1 825.2.k.i 16
15.d odd 2 1 825.2.k.i 16
15.e even 4 1 inner 165.2.k.c ✓ 16
15.e even 4 1 825.2.k.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.k.c ✓ 16 1.a even 1 1 trivial
165.2.k.c ✓ 16 15.e even 4 1 inner
165.2.k.d yes 16 3.b odd 2 1
165.2.k.d yes 16 5.c odd 4 1
825.2.k.i 16 5.c odd 4 1
825.2.k.i 16 15.d odd 2 1
825.2.k.j 16 5.b even 2 1
825.2.k.j 16 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 4 T_{2}^{15} + 8 T_{2}^{14} + 19 T_{2}^{12} + 80 T_{2}^{11} + 168 T_{2}^{10} - 28 T_{2}^{9} + 119 T_{2}^{8} + 432 T_{2}^{7} + 784 T_{2}^{6} - 84 T_{2}^{5} + 169 T_{2}^{4} + 420 T_{2}^{3} + 392 T_{2}^{2} + 112 T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{15} + 8 T^{14} + 19 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} - T^{14} + 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 8 T^{15} + 41 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{15} + 32 T^{14} - 56 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{16} - 40 T^{13} + 1304 T^{12} + \cdots + 9216 \) Copy content Toggle raw display
$17$ \( T^{16} - 8 T^{15} + 32 T^{14} + 88 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{16} + 228 T^{14} + \cdots + 35426304 \) Copy content Toggle raw display
$23$ \( T^{16} - 14 T^{15} + \cdots + 1615396864 \) Copy content Toggle raw display
$29$ \( (T^{8} + 8 T^{7} - 70 T^{6} - 504 T^{5} + \cdots - 3392)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{7} - 111 T^{6} + \cdots + 218512)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 6 T^{15} + \cdots + 214369000000 \) Copy content Toggle raw display
$41$ \( T^{16} + 380 T^{14} + \cdots + 514444693504 \) Copy content Toggle raw display
$43$ \( T^{16} - 16 T^{15} + \cdots + 1923348736 \) Copy content Toggle raw display
$47$ \( T^{16} - 48 T^{15} + \cdots + 171409248256 \) Copy content Toggle raw display
$53$ \( T^{16} + 4 T^{15} + 8 T^{14} + \cdots + 51609856 \) Copy content Toggle raw display
$59$ \( (T^{8} + 34 T^{7} + 313 T^{6} + \cdots + 1225520)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 4 T^{7} - 188 T^{6} + \cdots - 441344)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 6 T^{15} + \cdots + 3634642944 \) Copy content Toggle raw display
$71$ \( T^{16} + 702 T^{14} + \cdots + 210027890944 \) Copy content Toggle raw display
$73$ \( T^{16} - 736 T^{13} + \cdots + 15170339487744 \) Copy content Toggle raw display
$79$ \( T^{16} + 764 T^{14} + \cdots + 14903526400 \) Copy content Toggle raw display
$83$ \( T^{16} + 8 T^{15} + \cdots + 4917936384 \) Copy content Toggle raw display
$89$ \( (T^{8} - 16 T^{7} - 179 T^{6} + \cdots + 112304)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 629744570362944 \) Copy content Toggle raw display
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