Properties

Label 6030.2.d.j
Level $6030$
Weight $2$
Character orbit 6030.d
Analytic conductor $48.150$
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] = [6030,2,Mod(2411,6030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6030.2411");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.d (of order \(2\), degree \(1\), 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: \(48.1497924188\)
Analytic rank: \(0\)
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: \( x^{16} + 36x^{14} + 519x^{12} + 3876x^{10} + 16111x^{8} + 36772x^{6} + 41293x^{4} + 16036x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{67}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + \beta_{12} q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - q^{5} + \beta_{12} q^{7} + q^{8} - q^{10} + (\beta_{11} - 1) q^{11} - \beta_{14} q^{13} + \beta_{12} q^{14} + q^{16} + (\beta_{8} + \beta_{7} - \beta_{4} - \beta_1) q^{17} + (\beta_{13} + \beta_{11} - \beta_{10} - \beta_{3}) q^{19} - q^{20} + (\beta_{11} - 1) q^{22} + (\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - 1) q^{23} + q^{25} - \beta_{14} q^{26} + \beta_{12} q^{28} + ( - \beta_{12} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 + 1) q^{29} + ( - \beta_{4} - \beta_1) q^{31} + q^{32} + (\beta_{8} + \beta_{7} - \beta_{4} - \beta_1) q^{34} - \beta_{12} q^{35} + (\beta_{15} - \beta_{13} - \beta_{10} + \beta_{6} - \beta_{5} + 2) q^{37} + (\beta_{13} + \beta_{11} - \beta_{10} - \beta_{3}) q^{38} - q^{40} + ( - \beta_{15} + \beta_{13} + \beta_{10} + \beta_{9} - \beta_{6} + \beta_{5} - 2 \beta_{3}) q^{41} + (3 \beta_{14} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - 1) q^{43} + (\beta_{11} - 1) q^{44} + (\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - 1) q^{46} + (\beta_{14} + \beta_{12} - \beta_{8} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 2) q^{47} + (\beta_{15} + \beta_{13} + \beta_{11} + 2 \beta_{10} - \beta_{6} + \beta_{5} - \beta_{3} - 5) q^{49} + q^{50} - \beta_{14} q^{52} + (\beta_{13} - \beta_{11} + \beta_{10} + \beta_{3}) q^{53} + ( - \beta_{11} + 1) q^{55} + \beta_{12} q^{56} + ( - \beta_{12} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 + 1) q^{58} + (\beta_{12} + \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 2) q^{59} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 2) q^{61} + ( - \beta_{4} - \beta_1) q^{62} + q^{64} + \beta_{14} q^{65} + (\beta_{15} + \beta_{14} - \beta_{12} - \beta_{10} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{67} + (\beta_{8} + \beta_{7} - \beta_{4} - \beta_1) q^{68} - \beta_{12} q^{70} + ( - 2 \beta_{14} + \beta_{12} - \beta_{7} - \beta_{4} - 3 \beta_{2}) q^{71} + ( - \beta_{15} + \beta_{13} - \beta_{11} - \beta_{9} - \beta_{3} + 1) q^{73} + (\beta_{15} - \beta_{13} - \beta_{10} + \beta_{6} - \beta_{5} + 2) q^{74} + (\beta_{13} + \beta_{11} - \beta_{10} - \beta_{3}) q^{76} + ( - \beta_{14} + 2 \beta_{8} - \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - \beta_{4} + 2 \beta_{2} + \cdots - 3) q^{77}+ \cdots + (\beta_{15} + \beta_{13} + \beta_{11} + 2 \beta_{10} - \beta_{6} + \beta_{5} - \beta_{3} - 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8} - 16 q^{10} - 20 q^{11} + 16 q^{16} - 8 q^{19} - 16 q^{20} - 20 q^{22} + 16 q^{25} + 16 q^{32} + 32 q^{37} - 8 q^{38} - 16 q^{40} - 8 q^{41} - 20 q^{44} - 88 q^{49} + 16 q^{50} + 8 q^{53} + 20 q^{55} + 16 q^{64} + 4 q^{67} + 16 q^{73} + 32 q^{74} - 8 q^{76} - 16 q^{80} - 8 q^{82} - 20 q^{88} + 40 q^{91} + 8 q^{95} - 88 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 36x^{14} + 519x^{12} + 3876x^{10} + 16111x^{8} + 36772x^{6} + 41293x^{4} + 16036x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 57 \nu^{15} + 5016 \nu^{13} + 123508 \nu^{11} + 1353680 \nu^{9} + 7499857 \nu^{7} + 21144180 \nu^{5} + 27059854 \nu^{3} + 10321564 \nu ) / 102712 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 516 \nu^{15} - 19730 \nu^{13} - 305161 \nu^{11} - 2461588 \nu^{9} - 11058568 \nu^{7} - 27049648 \nu^{5} - 31840219 \nu^{3} - 12228614 \nu ) / 102712 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1977 \nu^{14} - 58425 \nu^{12} - 636150 \nu^{10} - 3175738 \nu^{8} - 7255081 \nu^{6} - 5874285 \nu^{4} + 677236 \nu^{2} + 329028 ) / 205424 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1977 \nu^{15} - 58425 \nu^{13} - 610472 \nu^{11} - 2456754 \nu^{9} - 65241 \nu^{7} + 25401519 \nu^{5} + 57554006 \nu^{3} + 30680424 \nu ) / 205424 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 740 \nu^{15} - 447 \nu^{14} - 26603 \nu^{13} - 13658 \nu^{12} - 380804 \nu^{11} - 159030 \nu^{10} - 2795276 \nu^{9} - 912120 \nu^{8} - 11245706 \nu^{7} + \cdots - 230080 ) / 102712 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 740 \nu^{15} + 447 \nu^{14} - 26603 \nu^{13} + 13658 \nu^{12} - 380804 \nu^{11} + 159030 \nu^{10} - 2795276 \nu^{9} + 912120 \nu^{8} - 11245706 \nu^{7} + \cdots + 332792 ) / 102712 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2585 \nu^{15} - 86251 \nu^{13} - 1114754 \nu^{11} - 7104130 \nu^{9} - 23349573 \nu^{7} - 37106779 \nu^{5} - 23101196 \nu^{3} - 4830228 \nu ) / 205424 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2463 \nu^{15} + 101193 \nu^{13} + 1674352 \nu^{11} + 14314902 \nu^{9} + 67408995 \nu^{7} + 171560325 \nu^{5} + 210318866 \nu^{3} + 85883832 \nu ) / 205424 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2117 \nu^{14} + 70745 \nu^{12} + 912924 \nu^{10} + 5759508 \nu^{8} + 18326919 \nu^{6} + 26438303 \nu^{4} + 11363520 \nu^{2} - 151268 ) / 102712 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4343 \nu^{14} - 138243 \nu^{12} - 1675058 \nu^{10} - 9775258 \nu^{8} - 28440475 \nu^{6} - 37492323 \nu^{4} - 15315920 \nu^{2} + 611668 ) / 205424 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5311 \nu^{14} + 172071 \nu^{12} + 2139046 \nu^{10} + 12952366 \nu^{8} + 39719383 \nu^{6} + 56465971 \nu^{4} + 26403696 \nu^{2} + 881532 ) / 205424 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2562 \nu^{15} - 84227 \nu^{13} - 1074603 \nu^{11} - 6827302 \nu^{9} - 22989322 \nu^{7} - 39944865 \nu^{5} - 31892425 \nu^{3} - 9441406 \nu ) / 102712 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6677 \nu^{14} - 215245 \nu^{12} - 2653638 \nu^{10} - 15835558 \nu^{8} - 47211081 \nu^{6} - 62897213 \nu^{4} - 23205608 \nu^{2} + 1155036 ) / 205424 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3147 \nu^{15} + 110029 \nu^{13} + 1525895 \nu^{11} + 10838358 \nu^{9} + 42318483 \nu^{7} + 89653347 \nu^{5} + 92797759 \nu^{3} + 33771266 \nu ) / 102712 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12673 \nu^{14} - 409079 \nu^{12} - 5060170 \nu^{10} - 30433338 \nu^{8} - 92396805 \nu^{6} - 128998247 \nu^{4} - 57051452 \nu^{2} + 203436 ) / 205424 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{8} - 3\beta_{6} - 3\beta_{5} + 2\beta_{4} + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{15} + 6\beta_{11} + 2\beta_{10} - \beta_{6} + \beta_{5} - 2\beta_{3} - 25 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{14} - 6 \beta_{12} + 16 \beta_{8} - 6 \beta_{7} + 21 \beta_{6} + 21 \beta_{5} - 10 \beta_{4} + 6 \beta_{2} + 6 \beta _1 - 21 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 14 \beta_{15} + 18 \beta_{13} - 48 \beta_{11} - 26 \beta_{10} + 18 \beta_{9} + 7 \beta_{6} - 7 \beta_{5} + 2 \beta_{3} + 181 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 78 \beta_{14} + 84 \beta_{12} - 146 \beta_{8} + 60 \beta_{7} - 171 \beta_{6} - 171 \beta_{5} + 74 \beta_{4} - 102 \beta_{2} - 96 \beta _1 + 171 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 22 \beta_{15} - 80 \beta_{13} + 142 \beta_{11} + 114 \beta_{10} - 100 \beta_{9} - 31 \beta_{6} + 31 \beta_{5} + 18 \beta_{3} - 527 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 906 \beta_{14} - 1002 \beta_{12} + 1492 \beta_{8} - 486 \beta_{7} + 1527 \beta_{6} + 1527 \beta_{5} - 718 \beta_{4} + 1350 \beta_{2} + 1146 \beta _1 - 1527 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 38 \beta_{15} + 2634 \beta_{13} - 4056 \beta_{11} - 4378 \beta_{10} + 3918 \beta_{9} + 1337 \beta_{6} - 1337 \beta_{5} - 818 \beta_{3} + 15347 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 10422 \beta_{14} + 11532 \beta_{12} - 16130 \beta_{8} + 3552 \beta_{7} - 14409 \beta_{6} - 14409 \beta_{5} + 7910 \beta_{4} - 16314 \beta_{2} - 12672 \beta _1 + 14409 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 6842 \beta_{15} - 27756 \beta_{13} + 40350 \beta_{11} + 54190 \beta_{10} - 47388 \beta_{9} - 17939 \beta_{6} + 17939 \beta_{5} + 9242 \beta_{3} - 157955 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 119850 \beta_{14} - 131406 \beta_{12} + 178600 \beta_{8} - 23118 \beta_{7} + 141177 \beta_{6} + 141177 \beta_{5} - 90706 \beta_{4} + 189690 \beta_{2} + 137526 \beta _1 - 141177 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 43054 \beta_{15} + 97350 \beta_{13} - 137920 \beta_{11} - 217582 \beta_{10} + 185118 \beta_{9} + 75609 \beta_{6} - 75609 \beta_{5} - 32466 \beta_{3} + 560003 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1375422 \beta_{14} + 1492020 \beta_{12} - 1996922 \beta_{8} + 118692 \beta_{7} - 1424307 \beta_{6} - 1424307 \beta_{5} + 1047146 \beta_{4} - 2168478 \beta_{2} - 1492704 \beta _1 + 1424307 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1876414 \beta_{15} - 3103416 \beta_{13} + 4340634 \beta_{11} + 7708622 \beta_{10} - 6409932 \beta_{9} - 2758477 \beta_{6} + 2758477 \beta_{5} + 1013062 \beta_{3} + \cdots - 18218221 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 15732498 \beta_{14} - 16909002 \beta_{12} + 22418236 \beta_{8} - 170766 \beta_{7} + 14720895 \beta_{6} + 14720895 \beta_{5} - 12052510 \beta_{4} + 24602262 \beta_{2} + \cdots - 14720895 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times\).

\(n\) \(1207\) \(3151\) \(4691\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2411.1
3.35739i
2.10268i
1.72684i
2.22811i
2.83449i
0.870493i
0.0157987i
1.88888i
1.88888i
0.0157987i
0.870493i
2.83449i
2.22811i
1.72684i
2.10268i
3.35739i
1.00000 0 1.00000 −1.00000 0 5.18179i 1.00000 0 −1.00000
2411.2 1.00000 0 1.00000 −1.00000 0 5.11107i 1.00000 0 −1.00000
2411.3 1.00000 0 1.00000 −1.00000 0 4.32270i 1.00000 0 −1.00000
2411.4 1.00000 0 1.00000 −1.00000 0 3.75929i 1.00000 0 −1.00000
2411.5 1.00000 0 1.00000 −1.00000 0 2.81381i 1.00000 0 −1.00000
2411.6 1.00000 0 1.00000 −1.00000 0 1.74812i 1.00000 0 −1.00000
2411.7 1.00000 0 1.00000 −1.00000 0 1.45101i 1.00000 0 −1.00000
2411.8 1.00000 0 1.00000 −1.00000 0 1.06257i 1.00000 0 −1.00000
2411.9 1.00000 0 1.00000 −1.00000 0 1.06257i 1.00000 0 −1.00000
2411.10 1.00000 0 1.00000 −1.00000 0 1.45101i 1.00000 0 −1.00000
2411.11 1.00000 0 1.00000 −1.00000 0 1.74812i 1.00000 0 −1.00000
2411.12 1.00000 0 1.00000 −1.00000 0 2.81381i 1.00000 0 −1.00000
2411.13 1.00000 0 1.00000 −1.00000 0 3.75929i 1.00000 0 −1.00000
2411.14 1.00000 0 1.00000 −1.00000 0 4.32270i 1.00000 0 −1.00000
2411.15 1.00000 0 1.00000 −1.00000 0 5.11107i 1.00000 0 −1.00000
2411.16 1.00000 0 1.00000 −1.00000 0 5.18179i 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2411.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
201.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6030.2.d.j yes 16
3.b odd 2 1 6030.2.d.i 16
67.b odd 2 1 6030.2.d.i 16
201.d even 2 1 inner 6030.2.d.j yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6030.2.d.i 16 3.b odd 2 1
6030.2.d.i 16 67.b odd 2 1
6030.2.d.j yes 16 1.a even 1 1 trivial
6030.2.d.j yes 16 201.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6030, [\chi])\):

\( T_{7}^{16} + 100 T_{7}^{14} + 3985 T_{7}^{12} + 80856 T_{7}^{10} + 887880 T_{7}^{8} + 5215776 T_{7}^{6} + 15513616 T_{7}^{4} + 21456384 T_{7}^{2} + 10653696 \) Copy content Toggle raw display
\( T_{11}^{8} + 10T_{11}^{7} + T_{11}^{6} - 204T_{11}^{5} - 230T_{11}^{4} + 1256T_{11}^{3} + 1238T_{11}^{2} - 2280T_{11} - 1224 \) Copy content Toggle raw display
\( T_{41}^{8} + 4 T_{41}^{7} - 240 T_{41}^{6} - 956 T_{41}^{5} + 15702 T_{41}^{4} + 66784 T_{41}^{3} - 213392 T_{41}^{2} - 1005888 T_{41} - 760608 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 100 T^{14} + \cdots + 10653696 \) Copy content Toggle raw display
$11$ \( (T^{8} + 10 T^{7} + T^{6} - 204 T^{5} + \cdots - 1224)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 108 T^{14} + \cdots + 10653696 \) Copy content Toggle raw display
$17$ \( T^{16} + 200 T^{14} + \cdots + 3121680384 \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} - 100 T^{6} + \cdots - 165376)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 132 T^{14} + 5844 T^{12} + \cdots + 589824 \) Copy content Toggle raw display
$29$ \( T^{16} + 348 T^{14} + \cdots + 107495424 \) Copy content Toggle raw display
$31$ \( T^{16} + 140 T^{14} + \cdots + 80568576 \) Copy content Toggle raw display
$37$ \( (T^{8} - 16 T^{7} - 27 T^{6} + 1664 T^{5} + \cdots - 39304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 4 T^{7} - 240 T^{6} + \cdots - 760608)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 600 T^{14} + \cdots + 25817455263744 \) Copy content Toggle raw display
$47$ \( T^{16} + 516 T^{14} + \cdots + 2848997376 \) Copy content Toggle raw display
$53$ \( (T^{8} - 4 T^{7} - 252 T^{6} + 1360 T^{5} + \cdots - 82944)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 360 T^{14} + \cdots + 122645643264 \) Copy content Toggle raw display
$61$ \( T^{16} + 812 T^{14} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 406067677556641 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 248181962379264 \) Copy content Toggle raw display
$73$ \( (T^{8} - 8 T^{7} - 136 T^{6} + \cdots - 490624)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 572 T^{14} + \cdots + 20571377082624 \) Copy content Toggle raw display
$83$ \( T^{16} + 856 T^{14} + \cdots + 54047375262864 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 517575968087616 \) Copy content Toggle raw display
$97$ \( T^{16} + 720 T^{14} + \cdots + 11098812294144 \) Copy content Toggle raw display
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