Properties

Label 9295.2.a.bd
Level $9295$
Weight $2$
Character orbit 9295.a
Self dual yes
Analytic conductor $74.221$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9295,2,Mod(1,9295)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9295, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9295.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9295 = 5 \cdot 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9295.a (trivial)

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: yes
Analytic conductor: \(74.2209486788\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 24 x^{14} + 232 x^{12} - 4 x^{11} - 1154 x^{10} + 64 x^{9} + 3124 x^{8} - 368 x^{7} - 4455 x^{6} + 900 x^{5} + 2915 x^{4} - 826 x^{3} - 551 x^{2} + 150 x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 715)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{11} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{11} - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{6} + (\beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{9} - \beta_{8} - \beta_1) q^{8} + (\beta_{15} - \beta_{12} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{11} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{11} - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{6} + (\beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{9} - \beta_{8} - \beta_1) q^{8} + (\beta_{15} - \beta_{12} - \beta_{2}) q^{9} + \beta_1 q^{10} + q^{11} + (2 \beta_{15} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{3} - \beta_1) q^{12} + ( - 2 \beta_{15} + \beta_{13} + \beta_{12} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{3} - 1) q^{14} + \beta_{11} q^{15} + ( - 2 \beta_{15} + \beta_{13} + \beta_{12} + 2 \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{16}+ \cdots + (\beta_{15} - \beta_{12} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 16 q^{5} + 10 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 16 q^{5} + 10 q^{7} + 8 q^{9} + 16 q^{11} + 8 q^{12} - 26 q^{14} - 24 q^{17} - 8 q^{18} - 12 q^{19} - 16 q^{20} + 4 q^{21} + 6 q^{23} + 20 q^{24} + 16 q^{25} - 6 q^{27} + 12 q^{28} - 20 q^{29} - 20 q^{32} + 36 q^{34} - 10 q^{35} - 42 q^{36} + 6 q^{37} - 32 q^{38} - 12 q^{41} - 38 q^{42} - 28 q^{43} + 16 q^{44} - 8 q^{45} + 20 q^{46} - 28 q^{47} + 10 q^{48} - 22 q^{49} - 52 q^{51} - 10 q^{53} - 36 q^{54} - 16 q^{55} + 16 q^{56} + 4 q^{57} - 30 q^{58} - 12 q^{59} - 8 q^{60} - 44 q^{61} - 6 q^{62} - 6 q^{63} - 52 q^{64} + 30 q^{67} - 48 q^{68} - 20 q^{69} + 26 q^{70} - 12 q^{71} + 20 q^{72} + 82 q^{73} - 24 q^{74} - 26 q^{76} + 10 q^{77} - 10 q^{79} - 12 q^{81} + 34 q^{82} + 36 q^{83} + 14 q^{84} + 24 q^{85} - 32 q^{86} - 28 q^{87} - 36 q^{89} + 8 q^{90} - 14 q^{92} + 12 q^{93} - 22 q^{94} + 12 q^{95} - 12 q^{96} + 10 q^{97} - 14 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 24 x^{14} + 232 x^{12} - 4 x^{11} - 1154 x^{10} + 64 x^{9} + 3124 x^{8} - 368 x^{7} - 4455 x^{6} + 900 x^{5} + 2915 x^{4} - 826 x^{3} - 551 x^{2} + 150 x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6 \nu^{15} + 250 \nu^{14} - 202 \nu^{13} - 5112 \nu^{12} + 5285 \nu^{11} + 40145 \nu^{10} - 42249 \nu^{9} - 150151 \nu^{8} + 153331 \nu^{7} + 265525 \nu^{6} - 262793 \nu^{5} + \cdots - 1096 ) / 1573 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 685 \nu^{15} - 1068 \nu^{14} + 16194 \nu^{13} + 24780 \nu^{12} - 147910 \nu^{11} - 226010 \nu^{10} + 653843 \nu^{9} + 1007024 \nu^{8} - 1430500 \nu^{7} - 2199730 \nu^{6} + \cdots - 513 ) / 51909 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 278 \nu^{15} - 87 \nu^{14} - 6492 \nu^{13} + 2258 \nu^{12} + 59987 \nu^{11} - 21622 \nu^{10} - 279711 \nu^{9} + 94175 \nu^{8} + 699880 \nu^{7} - 177645 \nu^{6} - 937774 \nu^{5} + \cdots - 11532 ) / 17303 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 281 \nu^{15} - 1322 \nu^{14} + 7561 \nu^{13} + 28440 \nu^{12} - 80374 \nu^{11} - 237630 \nu^{10} + 428126 \nu^{9} + 968113 \nu^{8} - 1195039 \nu^{7} - 1967888 \nu^{6} + \cdots + 46032 ) / 17303 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1201 \nu^{15} - 2604 \nu^{14} + 28176 \nu^{13} + 57594 \nu^{12} - 261799 \nu^{11} - 500081 \nu^{10} + 1226150 \nu^{9} + 2145023 \nu^{8} - 3044950 \nu^{7} + \cdots + 65196 ) / 51909 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 525 \nu^{15} + 815 \nu^{14} + 11185 \nu^{13} - 16090 \nu^{12} - 93826 \nu^{11} + 122346 \nu^{10} + 392642 \nu^{9} - 446475 \nu^{8} - 860261 \nu^{7} + 781313 \nu^{6} + \cdots + 41367 ) / 17303 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 525 \nu^{15} - 815 \nu^{14} - 11185 \nu^{13} + 16090 \nu^{12} + 93826 \nu^{11} - 122346 \nu^{10} - 392642 \nu^{9} + 446475 \nu^{8} + 860261 \nu^{7} - 781313 \nu^{6} + \cdots - 41367 ) / 17303 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 124 \nu^{15} - 153 \nu^{14} - 2433 \nu^{13} + 2841 \nu^{12} + 17824 \nu^{11} - 20170 \nu^{10} - 57863 \nu^{9} + 69832 \nu^{8} + 67171 \nu^{7} - 129575 \nu^{6} + 34098 \nu^{5} + \cdots + 8499 ) / 3993 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1594 \nu^{15} - 711 \nu^{14} + 30363 \nu^{13} + 20835 \nu^{12} - 205057 \nu^{11} - 225176 \nu^{10} + 515525 \nu^{9} + 1153883 \nu^{8} + 122186 \nu^{7} - 2920645 \nu^{6} + \cdots + 133392 ) / 51909 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1588 \nu^{15} + 3756 \nu^{14} - 39522 \nu^{13} - 84564 \nu^{12} + 397945 \nu^{11} + 744566 \nu^{10} - 2076551 \nu^{9} - 3226214 \nu^{8} + 5956987 \nu^{7} + 7076017 \nu^{6} + \cdots - 202959 ) / 51909 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 380 \nu^{15} + 18 \nu^{14} - 8544 \nu^{13} - 798 \nu^{12} + 76118 \nu^{11} + 9574 \nu^{10} - 341140 \nu^{9} - 46975 \nu^{8} + 805898 \nu^{7} + 94250 \nu^{6} - 957750 \nu^{5} + \cdots + 16941 ) / 4719 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 4450 \nu^{15} + 3885 \nu^{14} - 103965 \nu^{13} - 87345 \nu^{12} + 959989 \nu^{11} + 759767 \nu^{10} - 4426934 \nu^{9} - 3197813 \nu^{8} + 10576378 \nu^{7} + \cdots - 273006 ) / 51909 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 5438 \nu^{15} + 1545 \nu^{14} - 122880 \nu^{13} - 40311 \nu^{12} + 1103639 \nu^{11} + 389761 \nu^{10} - 5016367 \nu^{9} - 1747000 \nu^{8} + 12168995 \nu^{7} + \cdots - 62874 ) / 51909 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} + \beta_{13} + \beta_{12} + 2 \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} + 8 \beta_{9} + 9 \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 27 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 19 \beta_{15} + \beta_{14} + 8 \beta_{13} + 10 \beta_{12} + 20 \beta_{10} + 11 \beta_{9} + \beta_{8} - 10 \beta_{7} + 11 \beta_{6} + 10 \beta_{5} + 11 \beta_{4} + 10 \beta_{3} + 34 \beta_{2} + 10 \beta _1 + 73 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{15} + 10 \beta_{14} - 10 \beta_{13} + 12 \beta_{12} + 15 \beta_{10} + 55 \beta_{9} + 63 \beta_{8} + 9 \beta_{7} + 11 \beta_{6} + 14 \beta_{5} + 12 \beta_{4} + 2 \beta_{3} - 10 \beta_{2} + 153 \beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 140 \beta_{15} + 12 \beta_{14} + 51 \beta_{13} + 76 \beta_{12} + 2 \beta_{11} + 156 \beta_{10} + 88 \beta_{9} + 14 \beta_{8} - 78 \beta_{7} + 88 \beta_{6} + 78 \beta_{5} + 88 \beta_{4} + 76 \beta_{3} + 194 \beta_{2} + 77 \beta _1 + 399 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 45 \beta_{15} + 76 \beta_{14} - 74 \beta_{13} + 102 \beta_{12} + 3 \beta_{11} + 150 \beta_{10} + 361 \beta_{9} + 408 \beta_{8} + 55 \beta_{7} + 88 \beta_{6} + 131 \beta_{5} + 106 \beta_{4} + 28 \beta_{3} - 70 \beta_{2} + 892 \beta _1 + 165 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 944 \beta_{15} + 102 \beta_{14} + 302 \beta_{13} + 522 \beta_{12} + 34 \beta_{11} + 1110 \beta_{10} + 622 \beta_{9} + 133 \beta_{8} - 559 \beta_{7} + 621 \beta_{6} + 559 \beta_{5} + 625 \beta_{4} + 520 \beta_{3} + \cdots + 2238 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 453 \beta_{15} + 522 \beta_{14} - 492 \beta_{13} + 759 \beta_{12} + 55 \beta_{11} + 1270 \beta_{10} + 2320 \beta_{9} + 2559 \beta_{8} + 262 \beta_{7} + 624 \beta_{6} + 1047 \beta_{5} + 833 \beta_{4} + 269 \beta_{3} + \cdots + 1171 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6108 \beta_{15} + 759 \beta_{14} + 1726 \beta_{13} + 3411 \beta_{12} + 376 \beta_{11} + 7552 \beta_{10} + 4142 \beta_{9} + 1086 \beta_{8} - 3851 \beta_{7} + 4119 \beta_{6} + 3857 \beta_{5} + 4206 \beta_{4} + \cdots + 12767 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3854 \beta_{15} + 3411 \beta_{14} - 3120 \beta_{13} + 5289 \beta_{12} + 639 \beta_{11} + 9858 \beta_{10} + 14743 \beta_{9} + 15819 \beta_{8} + 851 \beta_{7} + 4177 \beta_{6} + 7740 \beta_{5} + 6174 \beta_{4} + \cdots + 7937 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 38677 \beta_{15} + 5289 \beta_{14} + 9645 \beta_{13} + 21681 \beta_{12} + 3436 \beta_{11} + 50136 \beta_{10} + 26739 \beta_{9} + 8231 \beta_{8} - 25967 \beta_{7} + 26407 \beta_{6} + 26089 \beta_{5} + \cdots + 73734 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 29963 \beta_{15} + 21681 \beta_{14} - 19347 \beta_{13} + 35527 \beta_{12} + 6061 \beta_{11} + 72675 \beta_{10} + 93095 \beta_{9} + 97169 \beta_{8} - 578 \beta_{7} + 27123 \beta_{6} + 54782 \beta_{5} + \cdots + 52540 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.54349
2.40504
2.14365
1.92682
1.21972
1.10218
0.614596
0.343293
−0.0512307
−0.504538
−1.26917
−1.61252
−2.00728
−2.05251
−2.36230
−2.43924
−2.54349 −0.175846 4.46935 −1.00000 0.447263 1.86809 −6.28078 −2.96908 2.54349
1.2 −2.40504 −0.463938 3.78420 −1.00000 1.11579 0.175077 −4.29106 −2.78476 2.40504
1.3 −2.14365 2.37286 2.59524 −1.00000 −5.08658 5.00848 −1.27598 2.63047 2.14365
1.4 −1.92682 −2.20067 1.71262 −1.00000 4.24028 −1.16702 0.553725 1.84293 1.92682
1.5 −1.21972 3.02373 −0.512279 −1.00000 −3.68810 0.593036 3.06428 6.14291 1.21972
1.6 −1.10218 −2.03352 −0.785210 −1.00000 2.24129 3.61660 3.06979 1.13519 1.10218
1.7 −0.614596 0.192134 −1.62227 −1.00000 −0.118085 2.67113 2.22623 −2.96308 0.614596
1.8 −0.343293 1.76650 −1.88215 −1.00000 −0.606428 1.83793 1.33272 0.120523 0.343293
1.9 0.0512307 −3.00165 −1.99738 −1.00000 −0.153777 0.637306 −0.204788 6.00993 −0.0512307
1.10 0.504538 1.71117 −1.74544 −1.00000 0.863352 −2.14540 −1.88972 −0.0718874 −0.504538
1.11 1.26917 −2.84577 −0.389215 −1.00000 −3.61176 −1.19503 −3.03231 5.09843 −1.26917
1.12 1.61252 −1.08841 0.600219 −1.00000 −1.75507 −1.83425 −2.25717 −1.81537 −1.61252
1.13 2.00728 2.00590 2.02916 −1.00000 4.02640 −1.12224 0.0585271 1.02365 −2.00728
1.14 2.05251 −0.675239 2.21280 −1.00000 −1.38593 −0.426557 0.436774 −2.54405 −2.05251
1.15 2.36230 −0.329556 3.58045 −1.00000 −0.778510 4.18714 3.73350 −2.89139 −2.36230
1.16 2.43924 1.74230 3.94990 −1.00000 4.24989 −2.70431 4.75628 0.0356015 −2.43924
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9295.2.a.bd 16
13.b even 2 1 9295.2.a.be 16
13.f odd 12 2 715.2.z.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
715.2.z.b 32 13.f odd 12 2
9295.2.a.bd 16 1.a even 1 1 trivial
9295.2.a.be 16 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9295))\):

\( T_{2}^{16} - 24 T_{2}^{14} + 232 T_{2}^{12} + 4 T_{2}^{11} - 1154 T_{2}^{10} - 64 T_{2}^{9} + 3124 T_{2}^{8} + 368 T_{2}^{7} - 4455 T_{2}^{6} - 900 T_{2}^{5} + 2915 T_{2}^{4} + 826 T_{2}^{3} - 551 T_{2}^{2} - 150 T_{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{16} - 28 T_{3}^{14} + 2 T_{3}^{13} + 305 T_{3}^{12} - 34 T_{3}^{11} - 1644 T_{3}^{10} + 150 T_{3}^{9} + 4602 T_{3}^{8} + 90 T_{3}^{7} - 6358 T_{3}^{6} - 1606 T_{3}^{5} + 3477 T_{3}^{4} + 2038 T_{3}^{3} + 183 T_{3}^{2} - 74 T_{3} - 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 24 T^{14} + 232 T^{12} + 4 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{16} - 28 T^{14} + 2 T^{13} + 305 T^{12} + \cdots - 11 \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 10 T^{15} + 5 T^{14} + 204 T^{13} + \cdots - 327 \) Copy content Toggle raw display
$11$ \( (T - 1)^{16} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + 24 T^{15} + 160 T^{14} + \cdots - 13387251 \) Copy content Toggle raw display
$19$ \( T^{16} + 12 T^{15} - 97 T^{14} + \cdots + 1700637 \) Copy content Toggle raw display
$23$ \( T^{16} - 6 T^{15} - 78 T^{14} + \cdots + 4033221 \) Copy content Toggle raw display
$29$ \( T^{16} + 20 T^{15} + \cdots - 251555691 \) Copy content Toggle raw display
$31$ \( T^{16} - 329 T^{14} + \cdots + 199075848873 \) Copy content Toggle raw display
$37$ \( T^{16} - 6 T^{15} - 263 T^{14} + \cdots + 21820416 \) Copy content Toggle raw display
$41$ \( T^{16} + 12 T^{15} + \cdots - 1117367775 \) Copy content Toggle raw display
$43$ \( T^{16} + 28 T^{15} + \cdots - 6012295156583 \) Copy content Toggle raw display
$47$ \( T^{16} + 28 T^{15} + \cdots - 148228211871 \) Copy content Toggle raw display
$53$ \( T^{16} + 10 T^{15} + \cdots - 130808389647 \) Copy content Toggle raw display
$59$ \( T^{16} + 12 T^{15} + \cdots + 278166537 \) Copy content Toggle raw display
$61$ \( T^{16} + 44 T^{15} + \cdots - 15053321408 \) Copy content Toggle raw display
$67$ \( T^{16} - 30 T^{15} + \cdots + 23221658133 \) Copy content Toggle raw display
$71$ \( T^{16} + 12 T^{15} + \cdots - 1034199923427 \) Copy content Toggle raw display
$73$ \( T^{16} - 82 T^{15} + \cdots + 3730202068725 \) Copy content Toggle raw display
$79$ \( T^{16} + 10 T^{15} + \cdots - 137337441743 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 302412218481297 \) Copy content Toggle raw display
$89$ \( T^{16} + 36 T^{15} + 208 T^{14} + \cdots - 7322967 \) Copy content Toggle raw display
$97$ \( T^{16} - 10 T^{15} + \cdots - 85868304867 \) Copy content Toggle raw display
show more
show less