Properties

Label 546.2.bd.a
Level $546$
Weight $2$
Character orbit 546.bd
Analytic conductor $4.360$
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] = [546,2,Mod(121,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.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: \(4.35983195036\)
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: \( x^{16} + 26x^{14} + 249x^{12} + 1144x^{10} + 2766x^{8} + 3554x^{6} + 2260x^{4} + 564x^{2} + 9 \) 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.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} - q^{3} + \beta_{3} q^{4} + (\beta_{15} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{2}) q^{5} - \beta_{13} q^{6} + (\beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{11} - \beta_{8} - \beta_{5} - \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{7}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{2} - q^{3} + \beta_{3} q^{4} + (\beta_{15} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{9} + \beta_{6} + \beta_{5} - \beta_{2}) q^{5} - \beta_{13} q^{6} + (\beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{11} - \beta_{8} - \beta_{5} - \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{7}+ \cdots + ( - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} - \beta_{4} - \beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} + 8 q^{4} + 8 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{3} + 8 q^{4} + 8 q^{7} + 16 q^{9} - 8 q^{10} - 8 q^{12} - 10 q^{13} + 4 q^{14} - 8 q^{16} - 8 q^{21} + 6 q^{22} - 16 q^{23} + 2 q^{26} - 16 q^{27} + 10 q^{28} - 4 q^{29} + 8 q^{30} + 12 q^{31} + 16 q^{35} + 8 q^{36} + 30 q^{37} - 2 q^{38} + 10 q^{39} - 4 q^{40} - 18 q^{41} - 4 q^{42} - 32 q^{43} + 6 q^{44} + 12 q^{46} + 66 q^{47} + 8 q^{48} - 2 q^{49} + 36 q^{50} + 4 q^{52} + 2 q^{53} + 16 q^{55} + 2 q^{56} - 36 q^{59} - 8 q^{61} + 4 q^{62} + 8 q^{63} - 16 q^{64} - 28 q^{65} - 6 q^{66} + 16 q^{69} - 6 q^{70} - 30 q^{71} - 18 q^{73} + 6 q^{74} - 18 q^{76} - 34 q^{77} - 2 q^{78} - 24 q^{79} + 16 q^{81} - 12 q^{82} - 10 q^{84} + 72 q^{85} + 4 q^{87} + 12 q^{88} - 42 q^{89} - 8 q^{90} - 18 q^{91} - 32 q^{92} - 12 q^{93} + 48 q^{94} - 40 q^{95} - 6 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 26x^{14} + 249x^{12} + 1144x^{10} + 2766x^{8} + 3554x^{6} + 2260x^{4} + 564x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 36 \nu^{14} + 811 \nu^{12} + 6086 \nu^{10} + 18652 \nu^{8} + 24037 \nu^{6} + 8001 \nu^{4} - 9025 \nu^{2} + 3190 \nu - 879 ) / 6380 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 36 \nu^{14} - 811 \nu^{12} - 6086 \nu^{10} - 18652 \nu^{8} - 24037 \nu^{6} - 8001 \nu^{4} + 9025 \nu^{2} + 3190 \nu + 879 ) / 6380 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 293 \nu^{15} + 7726 \nu^{13} + 75390 \nu^{11} + 353450 \nu^{9} + 866394 \nu^{7} + 1113433 \nu^{5} + 686183 \nu^{3} + 138177 \nu + 9570 ) / 19140 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1810 \nu^{15} - 1059 \nu^{14} + 37922 \nu^{13} - 24681 \nu^{12} + 233613 \nu^{11} - 196947 \nu^{10} + 284221 \nu^{9} - 675429 \nu^{8} - 1397898 \nu^{7} - 1070775 \nu^{6} + \cdots + 143010 ) / 57420 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1160 \nu^{15} - 5139 \nu^{14} + 28507 \nu^{13} - 122868 \nu^{12} + 247428 \nu^{11} - 1020780 \nu^{10} + 954071 \nu^{9} - 3698685 \nu^{8} + 1657872 \nu^{7} - 6096447 \nu^{6} + \cdots + 46764 ) / 57420 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2366 \nu^{15} - 1386 \nu^{14} - 64129 \nu^{13} - 37125 \nu^{12} - 646218 \nu^{11} - 369567 \nu^{10} - 3102776 \nu^{9} - 1763784 \nu^{8} - 7486245 \nu^{7} + \cdots - 227898 ) / 57420 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56\nu^{14} + 1363\nu^{12} + 11694\nu^{10} + 44963\nu^{8} + 82794\nu^{6} + 70141\nu^{4} + 21947\nu^{2} + 312 ) / 330 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3283 \nu^{15} + 1326 \nu^{14} + 77831 \nu^{13} + 28968 \nu^{12} + 635805 \nu^{11} + 200349 \nu^{10} + 2219830 \nu^{9} + 462333 \nu^{8} + 3323844 \nu^{7} - 35856 \nu^{6} + \cdots - 73368 ) / 57420 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3002 \nu^{15} + 3813 \nu^{14} + 72502 \nu^{13} + 93900 \nu^{12} + 614547 \nu^{11} + 820431 \nu^{10} + 2326109 \nu^{9} + 3236352 \nu^{8} + 4265154 \nu^{7} + 6132303 \nu^{6} + \cdots + 55314 ) / 57420 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3002 \nu^{15} + 3813 \nu^{14} - 72502 \nu^{13} + 93900 \nu^{12} - 614547 \nu^{11} + 820431 \nu^{10} - 2326109 \nu^{9} + 3236352 \nu^{8} - 4265154 \nu^{7} + \cdots + 55314 ) / 57420 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 967 \nu^{15} - 2236 \nu^{14} - 22892 \nu^{13} - 55547 \nu^{12} - 188022 \nu^{11} - 492990 \nu^{10} - 682489 \nu^{9} - 1999345 \nu^{8} - 1226019 \nu^{7} - 3960288 \nu^{6} + \cdots + 8766 ) / 19140 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3016 \nu^{15} + 7260 \nu^{14} + 73544 \nu^{13} + 177375 \nu^{12} + 632403 \nu^{11} + 1532520 \nu^{10} + 2432926 \nu^{9} + 5970030 \nu^{8} + 4430475 \nu^{7} + \cdots + 103455 ) / 57420 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3016 \nu^{15} - 7260 \nu^{14} + 73544 \nu^{13} - 177375 \nu^{12} + 632403 \nu^{11} - 1532520 \nu^{10} + 2432926 \nu^{9} - 5970030 \nu^{8} + 4430475 \nu^{7} + \cdots - 103455 ) / 57420 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2127 \nu^{15} - 1624 \nu^{14} + 51399 \nu^{13} - 39527 \nu^{12} + 435450 \nu^{11} - 339126 \nu^{10} + 1636560 \nu^{9} - 1303927 \nu^{8} + 2883891 \nu^{7} - 2401026 \nu^{6} + \cdots - 18618 ) / 19140 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2761 \nu^{15} - 170 \nu^{14} + 68783 \nu^{13} - 6364 \nu^{12} + 614229 \nu^{11} - 87081 \nu^{10} + 2527723 \nu^{9} - 541307 \nu^{8} + 5198259 \nu^{7} - 1571214 \nu^{6} + \cdots - 35352 ) / 19140 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + 2 \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} - 2 \beta_{8} + \beta_{7} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 7 \beta_{2} - 7 \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 3 \beta_{13} + 10 \beta_{12} + \beta_{11} + 12 \beta_{10} + 13 \beta_{9} - \beta_{8} - 14 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} - 8 \beta_{4} - \beta_{3} - 5 \beta_{2} + 12 \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16 \beta_{15} - 30 \beta_{14} - 11 \beta_{13} - 14 \beta_{12} - 16 \beta_{11} + 25 \beta_{10} + 25 \beta_{8} - 15 \beta_{7} - 3 \beta_{6} + 38 \beta_{5} + 13 \beta_{4} + 7 \beta_{3} + 60 \beta_{2} + 57 \beta _1 - 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{15} + 34 \beta_{13} - 94 \beta_{12} - 13 \beta_{11} - 129 \beta_{10} - 144 \beta_{9} + 15 \beta_{8} + 147 \beta_{7} - 86 \beta_{6} - 88 \beta_{5} + 73 \beta_{4} + 15 \beta_{3} + 78 \beta_{2} - 138 \beta _1 - 133 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 203 \beta_{15} + 376 \beta_{14} + 94 \beta_{13} + 146 \beta_{12} + 203 \beta_{11} - 274 \beta_{10} + 9 \beta_{9} - 265 \beta_{8} + 188 \beta_{7} + 52 \beta_{6} - 416 \beta_{5} - 151 \beta_{4} + 53 \beta_{3} - 564 \beta_{2} + \cdots + 445 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 151 \beta_{15} - 342 \beta_{13} + 906 \beta_{12} + 151 \beta_{11} + 1371 \beta_{10} + 1568 \beta_{9} - 197 \beta_{8} - 1499 \beta_{7} + 866 \beta_{6} + 912 \beta_{5} - 715 \beta_{4} - 197 \beta_{3} - 984 \beta_{2} + \cdots + 1218 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2381 \beta_{15} - 4420 \beta_{14} - 774 \beta_{13} - 1456 \beta_{12} - 2381 \beta_{11} + 2942 \beta_{10} - 177 \beta_{9} + 2765 \beta_{8} - 2210 \beta_{7} - 682 \beta_{6} + 4464 \beta_{5} + 1699 \beta_{4} - 1457 \beta_{3} + \cdots - 4563 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1699 \beta_{15} + 3442 \beta_{13} - 9029 \beta_{12} - 1699 \beta_{11} - 14642 \beta_{10} - 17029 \beta_{9} + 2387 \beta_{8} + 15466 \beta_{7} - 8985 \beta_{6} - 9673 \beta_{5} + 7286 \beta_{4} + \cdots - 12021 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 26928 \beta_{15} + 50214 \beta_{14} + 6547 \beta_{13} + 14669 \beta_{12} + 26928 \beta_{11} - 31562 \beta_{10} + 2465 \beta_{9} - 29097 \beta_{8} + 25107 \beta_{7} + 8122 \beta_{6} - 47903 \beta_{5} + \cdots + 47677 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18806 \beta_{15} - 35301 \beta_{13} + 92503 \beta_{12} + 18806 \beta_{11} + 157140 \beta_{10} + 184712 \beta_{9} - 27572 \beta_{8} - 162020 \beta_{7} + 94814 \beta_{6} + 103580 \beta_{5} + \cdots + 123298 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 298798 \beta_{15} - 559224 \beta_{14} - 58196 \beta_{13} - 150785 \beta_{12} - 298798 \beta_{11} + 339383 \beta_{10} - 30263 \beta_{9} + 309120 \beta_{8} - 279612 \beta_{7} - 92589 \beta_{6} + \cdots - 504823 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 206209 \beta_{15} + 368523 \beta_{13} - 966603 \beta_{12} - 206209 \beta_{11} - 1691819 \beta_{10} - 2001694 \beta_{9} + 309875 \beta_{8} + 1717197 \beta_{7} - 1010498 \beta_{6} + \cdots - 1292175 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3280637 \beta_{15} + 6155108 \beta_{14} + 545170 \beta_{13} + 1577135 \beta_{12} + 3280637 \beta_{11} - 3657117 \beta_{10} + 350644 \beta_{9} - 3306473 \beta_{8} + 3077554 \beta_{7} + \cdots + 5390366 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(1 - \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.130758i
0.960282i
2.54804i
1.45057i
1.38609i
0.809195i
3.28902i
1.75225i
0.130758i
0.960282i
2.54804i
1.45057i
1.38609i
0.809195i
3.28902i
1.75225i
−0.866025 + 0.500000i −1.00000 0.500000 0.866025i −0.813575 0.469718i 0.866025 0.500000i 2.13374 1.56433i 1.00000i 1.00000 0.939436
121.2 −0.866025 + 0.500000i −1.00000 0.500000 0.866025i −0.620092 0.358010i 0.866025 0.500000i −2.52418 + 0.792781i 1.00000i 1.00000 0.716021
121.3 −0.866025 + 0.500000i −1.00000 0.500000 0.866025i 0.825077 + 0.476358i 0.866025 0.500000i −1.08980 + 2.41088i 1.00000i 1.00000 −0.952717
121.4 −0.866025 + 0.500000i −1.00000 0.500000 0.866025i 2.34064 + 1.35137i 0.866025 0.500000i 2.61422 0.407273i 1.00000i 1.00000 −2.70274
121.5 0.866025 0.500000i −1.00000 0.500000 0.866025i −3.80406 2.19627i −0.866025 + 0.500000i −0.227820 + 2.63592i 1.00000i 1.00000 −4.39255
121.6 0.866025 0.500000i −1.00000 0.500000 0.866025i −0.594123 0.343017i −0.866025 + 0.500000i 2.63424 0.246484i 1.00000i 1.00000 −0.686034
121.7 0.866025 0.500000i −1.00000 0.500000 0.866025i −0.152918 0.0882870i −0.866025 + 0.500000i −0.623746 2.57117i 1.00000i 1.00000 −0.176574
121.8 0.866025 0.500000i −1.00000 0.500000 0.866025i 2.81905 + 1.62758i −0.866025 + 0.500000i 1.08335 + 2.41379i 1.00000i 1.00000 3.25515
361.1 −0.866025 0.500000i −1.00000 0.500000 + 0.866025i −0.813575 + 0.469718i 0.866025 + 0.500000i 2.13374 + 1.56433i 1.00000i 1.00000 0.939436
361.2 −0.866025 0.500000i −1.00000 0.500000 + 0.866025i −0.620092 + 0.358010i 0.866025 + 0.500000i −2.52418 0.792781i 1.00000i 1.00000 0.716021
361.3 −0.866025 0.500000i −1.00000 0.500000 + 0.866025i 0.825077 0.476358i 0.866025 + 0.500000i −1.08980 2.41088i 1.00000i 1.00000 −0.952717
361.4 −0.866025 0.500000i −1.00000 0.500000 + 0.866025i 2.34064 1.35137i 0.866025 + 0.500000i 2.61422 + 0.407273i 1.00000i 1.00000 −2.70274
361.5 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −3.80406 + 2.19627i −0.866025 0.500000i −0.227820 2.63592i 1.00000i 1.00000 −4.39255
361.6 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −0.594123 + 0.343017i −0.866025 0.500000i 2.63424 + 0.246484i 1.00000i 1.00000 −0.686034
361.7 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i −0.152918 + 0.0882870i −0.866025 0.500000i −0.623746 + 2.57117i 1.00000i 1.00000 −0.176574
361.8 0.866025 + 0.500000i −1.00000 0.500000 + 0.866025i 2.81905 1.62758i −0.866025 0.500000i 1.08335 2.41379i 1.00000i 1.00000 3.25515
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bd.a 16
3.b odd 2 1 1638.2.cr.a 16
7.c even 3 1 546.2.bm.a yes 16
13.e even 6 1 546.2.bm.a yes 16
21.h odd 6 1 1638.2.dt.a 16
39.h odd 6 1 1638.2.dt.a 16
91.u even 6 1 inner 546.2.bd.a 16
273.x odd 6 1 1638.2.cr.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bd.a 16 1.a even 1 1 trivial
546.2.bd.a 16 91.u even 6 1 inner
546.2.bm.a yes 16 7.c even 3 1
546.2.bm.a yes 16 13.e even 6 1
1638.2.cr.a 16 3.b odd 2 1
1638.2.cr.a 16 273.x odd 6 1
1638.2.dt.a 16 21.h odd 6 1
1638.2.dt.a 16 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 20 T_{5}^{14} + 335 T_{5}^{12} - 390 T_{5}^{11} - 1112 T_{5}^{10} + 1212 T_{5}^{9} + 3608 T_{5}^{8} + 1110 T_{5}^{7} - 2168 T_{5}^{6} - 1146 T_{5}^{5} + 1375 T_{5}^{4} + 1680 T_{5}^{3} + 708 T_{5}^{2} + 126 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 20 T^{14} + 335 T^{12} - 390 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{15} + 33 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 68 T^{14} + 1716 T^{12} + \cdots + 1089 \) Copy content Toggle raw display
$13$ \( T^{16} + 10 T^{15} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 54 T^{14} - 4 T^{13} + 2160 T^{12} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{16} + 176 T^{14} + \cdots + 50936769 \) Copy content Toggle raw display
$23$ \( T^{16} + 16 T^{15} + 250 T^{14} + \cdots + 55935441 \) Copy content Toggle raw display
$29$ \( T^{16} + 4 T^{15} + 146 T^{14} + \cdots + 845588241 \) Copy content Toggle raw display
$31$ \( T^{16} - 12 T^{15} + \cdots + 5963391729 \) Copy content Toggle raw display
$37$ \( T^{16} - 30 T^{15} + \cdots + 18014203089 \) Copy content Toggle raw display
$41$ \( T^{16} + 18 T^{15} + \cdots + 24389381241 \) Copy content Toggle raw display
$43$ \( T^{16} + 32 T^{15} + \cdots + 3870808348969 \) Copy content Toggle raw display
$47$ \( T^{16} - 66 T^{15} + \cdots + 24307197434289 \) Copy content Toggle raw display
$53$ \( T^{16} - 2 T^{15} + \cdots + 689748521121 \) Copy content Toggle raw display
$59$ \( T^{16} + 36 T^{15} + \cdots + 63426911409 \) Copy content Toggle raw display
$61$ \( (T^{8} + 4 T^{7} - 156 T^{6} - 1200 T^{5} + \cdots - 2816)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 698 T^{14} + \cdots + 59619415692321 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 610341027216561 \) Copy content Toggle raw display
$73$ \( T^{16} + 18 T^{15} + \cdots + 42\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{16} + 24 T^{15} + \cdots + 1257624369 \) Copy content Toggle raw display
$83$ \( T^{16} + 844 T^{14} + \cdots + 40\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( T^{16} + 42 T^{15} + \cdots + 25608042872721 \) Copy content Toggle raw display
$97$ \( T^{16} + 6 T^{15} + \cdots + 11594767041 \) Copy content Toggle raw display
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