Properties

Label 123.4.a.d
Level $123$
Weight $4$
Character orbit 123.a
Self dual yes
Analytic conductor $7.257$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [123,4,Mod(1,123)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(123, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("123.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 123 = 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 123.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: \(7.25723493071\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 34x^{4} - 26x^{3} + 269x^{2} + 258x - 272 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{4} + 6) q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 + 6) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + ( - \beta_{4} + 6) q^{5} + ( - 3 \beta_1 + 3) q^{6} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 + 6) q^{8} + 9 q^{9} + ( - 2 \beta_{5} - 3 \beta_{3} - \beta_{2} - 8 \beta_1 + 10) q^{10} + (2 \beta_{5} + 3 \beta_{4} - \beta_{3} + 3 \beta_1 + 9) q^{11} + (3 \beta_{2} - 3 \beta_1 + 12) q^{12} + ( - 2 \beta_{5} - \beta_{4} - 4 \beta_{2} - 2 \beta_1 + 6) q^{13} + (3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{14} + ( - 3 \beta_{4} + 18) q^{15} + ( - \beta_{5} - \beta_{4} + 4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 10) q^{16} + ( - \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} + 12 \beta_1 + 5) q^{17} + ( - 9 \beta_1 + 9) q^{18} + (2 \beta_{5} - \beta_{4} - 4 \beta_{3} + 2 \beta_1 + 20) q^{19} + ( - 3 \beta_{5} - 3 \beta_{4} - 7 \beta_{3} + 9 \beta_{2} + 5 \beta_1 + 46) q^{20} + (3 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 6) q^{21} + (10 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - 35) q^{22} + ( - 3 \beta_{5} + 4 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} + 15 \beta_1 + 52) q^{23} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2} - 6 \beta_1 + 18) q^{24} + ( - 7 \beta_{5} - 7 \beta_{4} - 7 \beta_{3} + 7 \beta_{2} + \beta_1 + 41) q^{25} + ( - 2 \beta_{5} - 8 \beta_{4} - \beta_{3} + 5 \beta_{2} + 16 \beta_1 + 34) q^{26} + 27 q^{27} + (6 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 10 \beta_1 - 38) q^{28} + ( - 12 \beta_{5} - 5 \beta_{4} + \beta_{3} - 4 \beta_{2} + 19 \beta_1 + 41) q^{29} + ( - 6 \beta_{5} - 9 \beta_{3} - 3 \beta_{2} - 24 \beta_1 + 30) q^{30} + (2 \beta_{5} + 6 \beta_{4} + 7 \beta_{3} + 6 \beta_{2} + 17 \beta_1 - 25) q^{31} + (7 \beta_{5} - 5 \beta_{4} + 10 \beta_{3} - \beta_{2} + 30 \beta_1 - 26) q^{32} + (6 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} + 9 \beta_1 + 27) q^{33} + ( - 3 \beta_{5} + 7 \beta_{4} + 13 \beta_{3} - 6 \beta_{2} - 9 \beta_1 - 111) q^{34} + (12 \beta_{5} + 6 \beta_{4} + 12 \beta_{3} - 14 \beta_{2} + 42 \beta_1 - 30) q^{35} + (9 \beta_{2} - 9 \beta_1 + 36) q^{36} + (\beta_{5} + 5 \beta_{4} - 12 \beta_{3} + 15 \beta_{2} - 6 \beta_1 - 3) q^{37} + (2 \beta_{5} - 4 \beta_{4} - 17 \beta_{3} - 11 \beta_{2} - 10 \beta_1) q^{38} + ( - 6 \beta_{5} - 3 \beta_{4} - 12 \beta_{2} - 6 \beta_1 + 18) q^{39} + ( - 5 \beta_{5} - 11 \beta_{4} - 3 \beta_{3} - \beta_{2} - 21 \beta_1 - 96) q^{40} + 41 q^{41} + (9 \beta_{5} + 9 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 18) q^{42} + ( - 13 \beta_{5} + 7 \beta_{4} + 13 \beta_{2} - 22 \beta_1 - 83) q^{43} + (13 \beta_{5} - \beta_{4} + 16 \beta_{3} - 14 \beta_{2} + 33 \beta_1 - 96) q^{44} + ( - 9 \beta_{4} + 54) q^{45} + ( - 3 \beta_{5} - 15 \beta_{4} - 6 \beta_{3} - 12 \beta_{2} - 53 \beta_1 - 144) q^{46} + (16 \beta_{5} + 23 \beta_{4} - 5 \beta_{3} - 12 \beta_{2} - 11 \beta_1 - 39) q^{47} + ( - 3 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} - 9 \beta_{2} - 6 \beta_1 - 30) q^{48} + ( - 6 \beta_{5} - 12 \beta_{4} - 10 \beta_{3} - 4 \beta_{2} - 40 \beta_1 - 93) q^{49} + ( - 35 \beta_{5} - 21 \beta_{4} - 35 \beta_{3} - \beta_{2} - 76 \beta_1 + 37) q^{50} + ( - 3 \beta_{5} - 6 \beta_{4} + 18 \beta_{3} - 9 \beta_{2} + 36 \beta_1 + 15) q^{51} + ( - 9 \beta_{5} + 7 \beta_{4} - 25 \beta_{3} + 11 \beta_{2} - 61 \beta_1 - 166) q^{52} + (20 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 26 \beta_{2} + 10 \beta_1 + 64) q^{53} + ( - 27 \beta_1 + 27) q^{54} + (39 \beta_{5} + 6 \beta_{4} + 25 \beta_{3} - 7 \beta_{2} + 25 \beta_1 - 244) q^{55} + ( - 2 \beta_{5} + 2 \beta_{4} + 14 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 128) q^{56} + (6 \beta_{5} - 3 \beta_{4} - 12 \beta_{3} + 6 \beta_1 + 60) q^{57} + ( - 30 \beta_{5} - 26 \beta_{4} + \beta_{2} - 30 \beta_1 - 155) q^{58} + ( - 33 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 23 \beta_{2} - 63 \beta_1 + 96) q^{59} + ( - 9 \beta_{5} - 9 \beta_{4} - 21 \beta_{3} + 27 \beta_{2} + 15 \beta_1 + 138) q^{60} + ( - 3 \beta_{5} + 13 \beta_{4} + 53 \beta_{2} + 28 \beta_1 - 93) q^{61} + (10 \beta_{5} + 24 \beta_{4} + 37 \beta_{3} - 8 \beta_{2} - 20 \beta_1 - 233) q^{62} + (9 \beta_{5} + 9 \beta_{3} - 9 \beta_{2} + 9 \beta_1 + 18) q^{63} + (13 \beta_{5} + 41 \beta_{4} - 24 \beta_{3} - 15 \beta_{2} + 8 \beta_1 - 238) q^{64} + ( - 21 \beta_{5} - 21 \beta_{4} + 3 \beta_{3} - 39 \beta_{2} - 101 \beta_1 + 46) q^{65} + (30 \beta_{5} + 6 \beta_{4} + 12 \beta_{3} - 15 \beta_{2} - 105) q^{66} + (7 \beta_{5} - 45 \beta_{4} + 17 \beta_{3} + 15 \beta_{2} + 59 \beta_1 - 78) q^{67} + (22 \beta_{5} + 30 \beta_{4} + 15 \beta_{3} + 59 \beta_{2} + 26 \beta_1 - 64) q^{68} + ( - 9 \beta_{5} + 12 \beta_{4} - 21 \beta_{3} + 15 \beta_{2} + 45 \beta_1 + 156) q^{69} + (50 \beta_{5} + 34 \beta_{4} + 42 \beta_{3} - 48 \beta_{2} + 90 \beta_1 - 478) q^{70} + (\beta_{5} + 34 \beta_{4} - 30 \beta_{3} + 9 \beta_{2} - 110 \beta_1 + 143) q^{71} + ( - 9 \beta_{5} + 9 \beta_{4} + 9 \beta_{2} - 18 \beta_1 + 54) q^{72} + ( - 10 \beta_{5} - 16 \beta_{4} + 25 \beta_{3} - 26 \beta_{2} + 35 \beta_1 - 201) q^{73} + ( - 3 \beta_{5} - 7 \beta_{4} - 22 \beta_{3} - 3 \beta_{2} - 41 \beta_1 + 17) q^{74} + ( - 21 \beta_{5} - 21 \beta_{4} - 21 \beta_{3} + 21 \beta_{2} + 3 \beta_1 + 123) q^{75} + ( - 9 \beta_{5} - 33 \beta_{4} - 33 \beta_{3} - 15 \beta_{2} + 93 \beta_1 - 38) q^{76} + (9 \beta_{5} - 14 \beta_{4} - 39 \beta_{3} + 25 \beta_{2} - 129 \beta_1 + 352) q^{77} + ( - 6 \beta_{5} - 24 \beta_{4} - 3 \beta_{3} + 15 \beta_{2} + 48 \beta_1 + 102) q^{78} + ( - 14 \beta_{5} + 14 \beta_{4} - 24 \beta_{3} - 46 \beta_{2} + 36 \beta_1 - 70) q^{79} + ( - 7 \beta_{5} + 7 \beta_{4} + 19 \beta_{3} - 55 \beta_{2} + 49 \beta_1 - 196) q^{80} + 81 q^{81} + ( - 41 \beta_1 + 41) q^{82} + ( - 28 \beta_{5} + 47 \beta_{4} - 38 \beta_{3} + 4 \beta_{2} - 66 \beta_1 + 350) q^{83} + (18 \beta_{5} + 24 \beta_{4} + 12 \beta_{3} + 18 \beta_{2} + 30 \beta_1 - 114) q^{84} + (18 \beta_{5} - 21 \beta_{4} + 64 \beta_{3} - 48 \beta_{2} + 222 \beta_1 - 128) q^{85} + ( - 25 \beta_{5} - 13 \beta_{4} + 34 \beta_{3} + 55 \beta_{2} + 19 \beta_1 + 105) q^{86} + ( - 36 \beta_{5} - 15 \beta_{4} + 3 \beta_{3} - 12 \beta_{2} + 57 \beta_1 + 123) q^{87} + ( - 42 \beta_{5} + 28 \beta_{4} - 4 \beta_{2} + 130 \beta_1 - 132) q^{88} + (61 \beta_{5} - 37 \beta_{4} + 33 \beta_{3} - 23 \beta_{2} + 67 \beta_1 + 418) q^{89} + ( - 18 \beta_{5} - 27 \beta_{3} - 9 \beta_{2} - 72 \beta_1 + 90) q^{90} + (4 \beta_{5} - 14 \beta_{4} + 44 \beta_{3} - 70 \beta_{2} + 62 \beta_1 - 150) q^{91} + ( - 62 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 84 \beta_1 + 86) q^{92} + (6 \beta_{5} + 18 \beta_{4} + 21 \beta_{3} + 18 \beta_{2} + 51 \beta_1 - 75) q^{93} + (90 \beta_{5} + 10 \beta_{4} + 38 \beta_{3} - 3 \beta_{2} + 172 \beta_1 + 13) q^{94} + (3 \beta_{5} - 11 \beta_{4} - 21 \beta_{3} + 41 \beta_{2} - 69 \beta_1 + 490) q^{95} + (21 \beta_{5} - 15 \beta_{4} + 30 \beta_{3} - 3 \beta_{2} + 90 \beta_1 - 78) q^{96} + ( - 27 \beta_{5} + 62 \beta_{4} - 95 \beta_{3} + 67 \beta_{2} - 59 \beta_1 + 304) q^{97} + ( - 32 \beta_{5} - 36 \beta_{4} - 60 \beta_{3} + 30 \beta_{2} + 123 \beta_1 + 383) q^{98} + (18 \beta_{5} + 27 \beta_{4} - 9 \beta_{3} + 27 \beta_1 + 81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 18 q^{3} + 26 q^{4} + 37 q^{5} + 18 q^{6} + 14 q^{7} + 36 q^{8} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 18 q^{3} + 26 q^{4} + 37 q^{5} + 18 q^{6} + 14 q^{7} + 36 q^{8} + 54 q^{9} + 47 q^{10} + 50 q^{11} + 78 q^{12} + 27 q^{13} - 34 q^{14} + 111 q^{15} - 54 q^{16} + 43 q^{17} + 54 q^{18} + 111 q^{19} + 273 q^{20} + 42 q^{21} - 200 q^{22} + 294 q^{23} + 108 q^{24} + 239 q^{25} + 217 q^{26} + 162 q^{27} - 206 q^{28} + 234 q^{29} + 141 q^{30} - 121 q^{31} - 116 q^{32} + 150 q^{33} - 649 q^{34} - 166 q^{35} + 234 q^{36} - 28 q^{37} - 67 q^{38} + 81 q^{39} - 581 q^{40} + 246 q^{41} - 102 q^{42} - 492 q^{43} - 542 q^{44} + 333 q^{45} - 894 q^{46} - 280 q^{47} - 162 q^{48} - 590 q^{49} + 101 q^{50} + 129 q^{51} - 1065 q^{52} + 472 q^{53} + 162 q^{54} - 1370 q^{55} - 718 q^{56} + 333 q^{57} - 932 q^{58} + 595 q^{59} + 819 q^{60} - 468 q^{61} - 1317 q^{62} + 126 q^{63} - 1558 q^{64} + 207 q^{65} - 600 q^{66} - 335 q^{67} - 229 q^{68} + 882 q^{69} - 2822 q^{70} + 753 q^{71} + 324 q^{72} - 1177 q^{73} + 34 q^{74} + 717 q^{75} - 333 q^{76} + 2068 q^{77} + 651 q^{78} - 612 q^{79} - 1243 q^{80} + 486 q^{81} + 246 q^{82} + 1919 q^{83} - 618 q^{84} - 633 q^{85} + 830 q^{86} + 702 q^{87} - 870 q^{88} + 2659 q^{89} + 423 q^{90} - 890 q^{91} + 562 q^{92} - 363 q^{93} + 266 q^{94} + 2973 q^{95} - 348 q^{96} + 1584 q^{97} + 2182 q^{98} + 450 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 34x^{4} - 26x^{3} + 269x^{2} + 258x - 272 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 6\nu^{4} + 12\nu^{3} - 88\nu^{2} + 7\nu + 134 ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} + 45\nu^{3} - 57\nu^{2} - 175\nu + 72 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} + 59\nu^{3} - 85\nu^{2} - 399\nu + 212 ) / 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} + 2\beta_{2} + 18\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{5} - 5\beta_{4} + 4\beta_{3} + 23\beta_{2} + 44\beta _1 + 195 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30\beta_{5} - 42\beta_{4} + 10\beta_{3} + 74\beta_{2} + 399\beta _1 + 480 \) 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
5.27916
3.17178
0.657015
−1.93430
−3.29831
−3.87534
−4.27916 3.00000 10.3112 15.7767 −12.8375 5.96642 −9.89000 9.00000 −67.5112
1.2 −2.17178 3.00000 −3.28338 −11.3872 −6.51533 −6.57196 24.5050 9.00000 24.7304
1.3 0.342985 3.00000 −7.88236 9.86667 1.02895 16.4245 −5.44740 9.00000 3.38412
1.4 2.93430 3.00000 0.610133 6.30631 8.80291 18.1286 −21.6841 9.00000 18.5046
1.5 4.29831 3.00000 10.4755 21.2228 12.8949 −27.1762 10.6404 9.00000 91.2221
1.6 4.87534 3.00000 15.7689 −4.78533 14.6260 7.22865 37.8761 9.00000 −23.3301
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(41\) \(-1\)

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 123.4.a.d 6
3.b odd 2 1 369.4.a.f 6
4.b odd 2 1 1968.4.a.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.4.a.d 6 1.a even 1 1 trivial
369.4.a.f 6 3.b odd 2 1
1968.4.a.s 6 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 6T_{2}^{5} - 19T_{2}^{4} + 142T_{2}^{3} + 2T_{2}^{2} - 588T_{2} + 196 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(123))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 6 T^{5} - 19 T^{4} + 142 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$3$ \( (T - 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 37 T^{5} + 190 T^{4} + \cdots + 1135256 \) Copy content Toggle raw display
$7$ \( T^{6} - 14 T^{5} - 636 T^{4} + \cdots + 2293568 \) Copy content Toggle raw display
$11$ \( T^{6} - 50 T^{5} - 4014 T^{4} + \cdots - 81911864 \) Copy content Toggle raw display
$13$ \( T^{6} - 27 T^{5} + \cdots - 978161912 \) Copy content Toggle raw display
$17$ \( T^{6} - 43 T^{5} + \cdots + 19797717802 \) Copy content Toggle raw display
$19$ \( T^{6} - 111 T^{5} + \cdots + 1289742832 \) Copy content Toggle raw display
$23$ \( T^{6} - 294 T^{5} + \cdots + 50890186048 \) Copy content Toggle raw display
$29$ \( T^{6} - 234 T^{5} + \cdots + 2531498501216 \) Copy content Toggle raw display
$31$ \( T^{6} + 121 T^{5} + \cdots - 2218959458272 \) Copy content Toggle raw display
$37$ \( T^{6} + 28 T^{5} + \cdots + 884867511276 \) Copy content Toggle raw display
$41$ \( (T - 41)^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 492 T^{5} + \cdots - 17814848169328 \) Copy content Toggle raw display
$47$ \( T^{6} + 280 T^{5} + \cdots + 96987904976768 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 270637221082112 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 586668983577344 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 499608027642676 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 869522980253504 \) Copy content Toggle raw display
$71$ \( T^{6} - 753 T^{5} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 420232761125654 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 822986137536512 \) Copy content Toggle raw display
$83$ \( T^{6} - 1919 T^{5} + \cdots + 41\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{6} - 2659 T^{5} + \cdots - 65\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{6} - 1584 T^{5} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
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