Properties

Label 1792.2.m.g
Level $1792$
Weight $2$
Character orbit 1792.m
Analytic conductor $14.309$
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] = [1792,2,Mod(449,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.m (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: \(14.3091920422\)
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: \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{3} + \beta_{10} q^{5} - \beta_{5} q^{7} + (\beta_{15} + \beta_{11} - \beta_{8} - \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{3} + \beta_{10} q^{5} - \beta_{5} q^{7} + (\beta_{15} + \beta_{11} - \beta_{8} - \beta_{5}) q^{9} + (\beta_{15} - \beta_{13} - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{2} + \beta_1) q^{11} + (\beta_{12} - \beta_{11} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{13} + ( - \beta_{14} + \beta_{12} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + 2 \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{15} + (\beta_{14} - \beta_{9} + \beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_1 - 1) q^{17} + (\beta_{15} - \beta_{13} - \beta_{10} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{19} - \beta_{8} q^{21} + (3 \beta_{15} - \beta_{14} - \beta_{13} - \beta_{10} - \beta_{9} + \beta_{6} - \beta_{3} + 2 \beta_{2}) q^{23} + ( - \beta_{15} + \beta_{12} - \beta_{10} + \beta_{5} - \beta_{2}) q^{25} + (\beta_{15} - \beta_{13} + \beta_{10} - \beta_{8} - \beta_{6} - 3 \beta_{5} + \beta_{4} + \beta_1 - 2) q^{27} + (\beta_{15} + \beta_{14} - \beta_{7} + \beta_{4} + \beta_{2}) q^{29} + ( - \beta_{14} - \beta_{12} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} - \beta_1 - 1) q^{31} + (\beta_{14} + \beta_{12} + \beta_{10} - \beta_{9} - \beta_{7} + 3 \beta_{4} + 3 \beta_1 + 3) q^{33} + \beta_{12} q^{35} + ( - 2 \beta_{15} + 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + \cdots - 2 \beta_{2}) q^{37}+ \cdots + ( - 3 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{3} - 4 q^{5} - 8 q^{11} + 12 q^{13} - 8 q^{17} + 4 q^{19} - 4 q^{21} - 56 q^{27} + 8 q^{31} + 16 q^{33} - 4 q^{35} - 8 q^{37} - 24 q^{43} - 36 q^{45} + 40 q^{47} - 16 q^{49} + 24 q^{51} - 32 q^{53} - 4 q^{59} - 20 q^{61} - 24 q^{63} + 72 q^{65} + 32 q^{67} + 56 q^{69} - 28 q^{75} - 8 q^{77} - 40 q^{81} + 36 q^{83} - 12 q^{91} + 8 q^{93} + 80 q^{95} - 72 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 951071408409672 \nu^{15} + \cdots - 29\!\cdots\!59 ) / 25\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 837450978918 \nu^{15} + 3268059368601 \nu^{14} - 9774129145049 \nu^{13} + 39378227440746 \nu^{12} - 52647402989314 \nu^{11} + \cdots + 22416575382206 ) / 1646719437545 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27\!\cdots\!92 \nu^{15} + \cdots + 63\!\cdots\!84 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\!\cdots\!76 \nu^{15} + \cdots - 23\!\cdots\!62 ) / 25\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19482487322 \nu^{15} - 71829768307 \nu^{14} + 211530176114 \nu^{13} - 869648666130 \nu^{12} + 1035106924390 \nu^{11} + \cdots - 258618081490 ) / 18265555126 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 74\!\cdots\!34 \nu^{15} + \cdots - 12\!\cdots\!68 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37\!\cdots\!11 \nu^{15} + \cdots - 67\!\cdots\!12 ) / 25\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41\!\cdots\!91 \nu^{15} + \cdots + 40\!\cdots\!92 ) / 25\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 51700125227286 \nu^{15} + 181410266124727 \nu^{14} - 529691956965078 \nu^{13} + \cdots + 339029565430912 ) / 28585593772190 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 51\!\cdots\!39 \nu^{15} + \cdots - 58\!\cdots\!28 ) / 25\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 53\!\cdots\!28 \nu^{15} + \cdots - 69\!\cdots\!76 ) / 25\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 55\!\cdots\!52 \nu^{15} + \cdots + 76\!\cdots\!04 ) / 25\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!28 \nu^{15} + \cdots + 13\!\cdots\!34 ) / 517399247276639 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 96105262004818 \nu^{15} + 355577051279181 \nu^{14} + \cdots + 11\!\cdots\!96 ) / 28585593772190 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12\!\cdots\!07 \nu^{15} + \cdots + 12\!\cdots\!04 ) / 25\!\cdots\!95 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} + \beta_{12} + 2\beta_{10} - \beta_{9} + 2\beta_{5} + 2\beta_{3} - \beta_{2} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} + 4 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - \beta_{2} + 8 \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - \beta_{9} + 5 \beta_{8} + 2 \beta_{6} - 11 \beta_{5} + 4 \beta_{4} - 5 \beta_{3} + 4 \beta_{2} + 4 \beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18 \beta_{15} + 10 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 10 \beta_{10} - 32 \beta_{9} - 8 \beta_{8} + 30 \beta_{7} - 26 \beta_{6} + 48 \beta_{5} + 7 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 2 \beta _1 + 78 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 74 \beta_{15} - 56 \beta_{14} + 24 \beta_{13} + 124 \beta_{12} + 88 \beta_{11} + 82 \beta_{10} - 46 \beta_{9} + 32 \beta_{8} - 118 \beta_{7} - 4 \beta_{6} + 226 \beta_{5} + 81 \beta_{4} + 98 \beta_{3} - 25 \beta_{2} + 146 \beta _1 + 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 21 \beta_{15} - 222 \beta_{13} + 68 \beta_{10} + 44 \beta_{9} + 190 \beta_{8} - 201 \beta_{7} + 178 \beta_{6} - 268 \beta_{5} + 111 \beta_{4} + 111 \beta_{2} + 222 \beta _1 - 46 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 142 \beta_{15} + 592 \beta_{14} - 1292 \beta_{13} - 1282 \beta_{12} - 890 \beta_{11} - 904 \beta_{10} - 366 \beta_{9} + 212 \beta_{8} + 1092 \beta_{7} - 70 \beta_{6} - 1480 \beta_{5} - 99 \beta_{4} - 1058 \beta_{3} + \cdots + 1868 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2330 \beta_{15} - 738 \beta_{14} + 1498 \beta_{13} + 636 \beta_{12} + 484 \beta_{11} - 880 \beta_{10} - 2740 \beta_{9} - 1810 \beta_{8} + 2206 \beta_{7} - 3154 \beta_{6} + 8300 \beta_{5} - 143 \beta_{4} + 842 \beta_{3} + \cdots + 3942 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2531 \beta_{15} - 3492 \beta_{14} - 210 \beta_{13} + 5181 \beta_{12} + 3691 \beta_{11} + 4159 \beta_{10} + 298 \beta_{9} + 2848 \beta_{8} - 7989 \beta_{7} + 2392 \beta_{6} + 6387 \beta_{5} + 3015 \beta_{4} + \cdots - 6421 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3944 \beta_{15} + 7978 \beta_{14} - 52842 \beta_{13} - 26600 \beta_{12} - 17948 \beta_{11} - 6776 \beta_{10} + 13828 \beta_{9} + 31098 \beta_{8} - 15224 \beta_{7} + 32910 \beta_{6} - 86808 \beta_{5} + \cdots - 9098 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 21134 \beta_{15} + 46534 \beta_{14} - 57378 \beta_{13} - 126974 \beta_{12} - 86542 \beta_{11} - 116430 \beta_{10} - 55584 \beta_{9} - 43698 \beta_{8} + 171842 \beta_{7} - 69398 \beta_{6} + \cdots + 184392 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 111990 \beta_{15} - 96551 \beta_{14} + 154426 \beta_{13} + 127911 \beta_{12} + 92227 \beta_{11} + 26515 \beta_{10} - 96551 \beta_{9} - 92227 \beta_{8} - 133222 \beta_{6} + 518948 \beta_{5} + \cdots - 101791 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 137632 \beta_{15} - 548602 \beta_{14} - 478874 \beta_{13} + 701572 \beta_{12} + 507786 \beta_{11} + 806284 \beta_{10} + 522960 \beta_{9} + 881276 \beta_{8} - 1824304 \beta_{7} + \cdots - 1929410 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 994874 \beta_{15} + 1834282 \beta_{14} - 5092398 \beta_{13} - 4608284 \beta_{12} - 3155984 \beta_{11} - 2431792 \beta_{10} + 1059000 \beta_{9} + 1822086 \beta_{8} + 1631110 \beta_{7} + \cdots + 1312114 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1679682 \beta_{15} + 1116243 \beta_{14} + 1730018 \beta_{13} - 3703000 \beta_{12} - 2499720 \beta_{11} - 5022618 \beta_{10} - 3827007 \beta_{9} - 4740003 \beta_{8} + 9202782 \beta_{7} + \cdots + 8689045 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.339278 + 0.0446668i
−1.09227 0.838128i
0.792206 1.03242i
0.277956 + 0.213283i
−0.424637 + 3.22544i
2.69978 + 0.355433i
−0.709944 + 0.925217i
0.117630 0.893490i
0.339278 0.0446668i
−1.09227 + 0.838128i
0.792206 + 1.03242i
0.277956 0.213283i
−0.424637 3.22544i
2.69978 0.355433i
−0.709944 0.925217i
0.117630 + 0.893490i
0 −1.48474 + 1.48474i 0 1.83598 + 1.83598i 0 1.00000i 0 1.40890i 0
449.2 0 −1.26274 + 1.26274i 0 −2.95746 2.95746i 0 1.00000i 0 0.189043i 0
449.3 0 −1.18265 + 1.18265i 0 −1.87820 1.87820i 0 1.00000i 0 0.202696i 0
449.4 0 −0.328027 + 0.328027i 0 1.40197 + 1.40197i 0 1.00000i 0 2.78480i 0
449.5 0 0.171192 0.171192i 0 0.268425 + 0.268425i 0 1.00000i 0 2.94139i 0
449.6 0 1.62602 1.62602i 0 1.16900 + 1.16900i 0 1.00000i 0 2.28788i 0
449.7 0 2.04137 2.04137i 0 0.701647 + 0.701647i 0 1.00000i 0 5.33435i 0
449.8 0 2.41958 2.41958i 0 −2.54136 2.54136i 0 1.00000i 0 8.70871i 0
1345.1 0 −1.48474 1.48474i 0 1.83598 1.83598i 0 1.00000i 0 1.40890i 0
1345.2 0 −1.26274 1.26274i 0 −2.95746 + 2.95746i 0 1.00000i 0 0.189043i 0
1345.3 0 −1.18265 1.18265i 0 −1.87820 + 1.87820i 0 1.00000i 0 0.202696i 0
1345.4 0 −0.328027 0.328027i 0 1.40197 1.40197i 0 1.00000i 0 2.78480i 0
1345.5 0 0.171192 + 0.171192i 0 0.268425 0.268425i 0 1.00000i 0 2.94139i 0
1345.6 0 1.62602 + 1.62602i 0 1.16900 1.16900i 0 1.00000i 0 2.28788i 0
1345.7 0 2.04137 + 2.04137i 0 0.701647 0.701647i 0 1.00000i 0 5.33435i 0
1345.8 0 2.41958 + 2.41958i 0 −2.54136 + 2.54136i 0 1.00000i 0 8.70871i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.m.g yes 16
4.b odd 2 1 1792.2.m.e 16
8.b even 2 1 1792.2.m.f yes 16
8.d odd 2 1 1792.2.m.h yes 16
16.e even 4 1 1792.2.m.f yes 16
16.e even 4 1 inner 1792.2.m.g yes 16
16.f odd 4 1 1792.2.m.e 16
16.f odd 4 1 1792.2.m.h yes 16
32.g even 8 1 7168.2.a.ba 8
32.g even 8 1 7168.2.a.be 8
32.h odd 8 1 7168.2.a.bb 8
32.h odd 8 1 7168.2.a.bf 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1792.2.m.e 16 4.b odd 2 1
1792.2.m.e 16 16.f odd 4 1
1792.2.m.f yes 16 8.b even 2 1
1792.2.m.f yes 16 16.e even 4 1
1792.2.m.g yes 16 1.a even 1 1 trivial
1792.2.m.g yes 16 16.e even 4 1 inner
1792.2.m.h yes 16 8.d odd 2 1
1792.2.m.h yes 16 16.f odd 4 1
7168.2.a.ba 8 32.g even 8 1
7168.2.a.bb 8 32.h odd 8 1
7168.2.a.be 8 32.g even 8 1
7168.2.a.bf 8 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1792, [\chi])\):

\( T_{3}^{16} - 4 T_{3}^{15} + 8 T_{3}^{14} + 16 T_{3}^{13} + 40 T_{3}^{12} - 152 T_{3}^{11} + 416 T_{3}^{10} + 1072 T_{3}^{9} + 1412 T_{3}^{8} - 1280 T_{3}^{7} + 5888 T_{3}^{6} + 17984 T_{3}^{5} + 25728 T_{3}^{4} + 6912 T_{3}^{3} + \cdots + 256 \) Copy content Toggle raw display
\( T_{5}^{16} + 4 T_{5}^{15} + 8 T_{5}^{14} - 40 T_{5}^{13} + 104 T_{5}^{12} + 264 T_{5}^{11} + 1024 T_{5}^{10} - 5344 T_{5}^{9} + 10500 T_{5}^{8} - 5184 T_{5}^{7} + 37376 T_{5}^{6} - 170496 T_{5}^{5} + 380416 T_{5}^{4} + \cdots + 16384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{15} + 8 T^{14} + 16 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{15} + 8 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + 8 T^{15} + 32 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{16} - 12 T^{15} + 72 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$17$ \( (T^{8} + 4 T^{7} - 100 T^{6} + \cdots + 116608)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} + 8 T^{14} + 40 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{16} + 232 T^{14} + \cdots + 4983230464 \) Copy content Toggle raw display
$29$ \( T^{16} + 64 T^{13} + \cdots + 8425771264 \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{7} - 132 T^{6} + \cdots - 251648)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 8 T^{15} + \cdots + 483102843136 \) Copy content Toggle raw display
$41$ \( T^{16} + 296 T^{14} + \cdots + 87310336 \) Copy content Toggle raw display
$43$ \( T^{16} + 24 T^{15} + \cdots + 2111586304 \) Copy content Toggle raw display
$47$ \( (T^{8} - 20 T^{7} + 68 T^{6} + 784 T^{5} + \cdots - 18176)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 32 T^{15} + \cdots + 133651661056 \) Copy content Toggle raw display
$59$ \( T^{16} + 4 T^{15} + \cdots + 166320414976 \) Copy content Toggle raw display
$61$ \( T^{16} + 20 T^{15} + \cdots + 9688858624 \) Copy content Toggle raw display
$67$ \( T^{16} - 32 T^{15} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{16} + 288 T^{14} + \cdots + 67108864 \) Copy content Toggle raw display
$73$ \( T^{16} + 592 T^{14} + \cdots + 58046357241856 \) Copy content Toggle raw display
$79$ \( (T^{8} - 288 T^{6} - 704 T^{5} + \cdots - 4822784)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 401670219890944 \) Copy content Toggle raw display
$89$ \( (T^{8} + 96 T^{6} + 800 T^{4} + 1536 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 36 T^{7} + 116 T^{6} + \cdots - 2571392)^{2} \) Copy content Toggle raw display
show more
show less