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Properties

Label 77.4.e.c
Level $77$
Weight $4$
Character orbit 77.e
Analytic conductor $4.543$
Analytic rank $0$
Dimension $20$
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] = [77,4,Mod(23,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.e (of order \(3\), 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: \(4.54314707044\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 67 x^{18} - 2 x^{17} + 2960 x^{16} - 261 x^{15} + 74338 x^{14} - 19762 x^{13} + \cdots + 649230400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{6} q^{3} + ( - \beta_{14} + 5 \beta_{5} + \cdots - 5) q^{4}+ \cdots + (\beta_{19} - \beta_{18} + 2 \beta_{17} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{6} q^{3} + ( - \beta_{14} + 5 \beta_{5} + \cdots - 5) q^{4}+ \cdots + ( - 11 \beta_{18} + 11 \beta_{17} + \cdots - 77) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 54 q^{4} - 10 q^{5} + 106 q^{6} - 30 q^{7} + 6 q^{8} - 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 54 q^{4} - 10 q^{5} + 106 q^{6} - 30 q^{7} + 6 q^{8} - 76 q^{9} - 63 q^{10} + 110 q^{11} - 55 q^{12} + 324 q^{13} + 58 q^{14} + 60 q^{15} - 286 q^{16} - 200 q^{17} + 252 q^{18} - 252 q^{19} + 192 q^{20} + 458 q^{21} - 134 q^{23} - 786 q^{24} - 86 q^{25} - 363 q^{26} + 348 q^{27} - 1057 q^{28} + 296 q^{29} + 316 q^{30} - 530 q^{31} + 731 q^{32} + 204 q^{34} - 288 q^{35} + 2574 q^{36} + 902 q^{37} - 66 q^{38} + 208 q^{39} - 2163 q^{40} + 336 q^{41} - 1482 q^{42} + 236 q^{43} + 594 q^{44} - 58 q^{45} + 210 q^{46} - 288 q^{47} - 1700 q^{48} - 1072 q^{49} + 4650 q^{50} + 1022 q^{51} - 1663 q^{52} + 608 q^{53} - 2312 q^{54} - 220 q^{55} - 4905 q^{56} + 1656 q^{57} + 1951 q^{58} + 464 q^{59} + 818 q^{60} - 3484 q^{61} - 1618 q^{62} - 3948 q^{63} + 6090 q^{64} + 1560 q^{65} + 583 q^{66} - 142 q^{67} - 1145 q^{68} + 3432 q^{69} - 4133 q^{70} + 668 q^{71} + 1176 q^{72} - 1466 q^{73} + 3460 q^{74} - 2982 q^{75} + 6774 q^{76} - 330 q^{77} + 10840 q^{78} - 578 q^{79} - 2911 q^{80} - 118 q^{81} + 307 q^{82} + 1092 q^{83} - 6786 q^{84} + 5164 q^{85} - 2597 q^{86} - 1516 q^{87} + 33 q^{88} - 3150 q^{89} - 3672 q^{90} - 3638 q^{91} + 4326 q^{92} + 1484 q^{93} - 5700 q^{94} + 1338 q^{95} - 5429 q^{96} + 3308 q^{97} - 3186 q^{98} - 1672 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 67 x^{18} - 2 x^{17} + 2960 x^{16} - 261 x^{15} + 74338 x^{14} - 19762 x^{13} + \cdots + 649230400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38\!\cdots\!27 \nu^{19} + \cdots - 12\!\cdots\!08 ) / 17\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53\!\cdots\!53 \nu^{19} + \cdots + 15\!\cdots\!44 ) / 12\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\!\cdots\!55 \nu^{19} + \cdots + 15\!\cdots\!60 ) / 34\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!42 \nu^{19} + \cdots + 44\!\cdots\!20 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\!\cdots\!06 \nu^{19} + \cdots + 55\!\cdots\!00 ) / 88\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!49 \nu^{19} + \cdots + 12\!\cdots\!80 ) / 88\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 46\!\cdots\!51 \nu^{19} + \cdots + 46\!\cdots\!28 ) / 17\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21\!\cdots\!71 \nu^{19} + \cdots - 17\!\cdots\!60 ) / 58\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!51 \nu^{19} + \cdots - 55\!\cdots\!60 ) / 17\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 71\!\cdots\!89 \nu^{19} + \cdots + 26\!\cdots\!00 ) / 88\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 18\!\cdots\!69 \nu^{19} + \cdots + 25\!\cdots\!40 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!17 \nu^{19} + \cdots + 49\!\cdots\!00 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10\!\cdots\!42 \nu^{19} + \cdots + 44\!\cdots\!20 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 13\!\cdots\!64 \nu^{19} + \cdots + 15\!\cdots\!80 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14\!\cdots\!29 \nu^{19} + \cdots - 10\!\cdots\!20 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 10\!\cdots\!89 \nu^{19} + \cdots + 87\!\cdots\!60 ) / 88\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 32\!\cdots\!09 \nu^{19} + \cdots + 10\!\cdots\!08 ) / 27\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 35\!\cdots\!51 \nu^{19} + \cdots - 22\!\cdots\!00 ) / 17\!\cdots\!60 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - 13\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{18} - \beta_{17} + \beta_{16} + \beta_{13} + \beta_{12} - \beta_{10} + 2 \beta_{9} + \cdots - 20 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{19} + \beta_{17} + 4 \beta_{16} + \beta_{15} - 33 \beta_{14} - 3 \beta_{11} - 2 \beta_{9} + \cdots - 275 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{19} + 38 \beta_{18} + 23 \beta_{17} - 4 \beta_{15} - 44 \beta_{14} - 40 \beta_{12} + 38 \beta_{11} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 148 \beta_{19} - 131 \beta_{18} + 217 \beta_{17} - 191 \beta_{16} - 148 \beta_{15} - 17 \beta_{13} + \cdots + 6789 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 278 \beta_{19} + 753 \beta_{17} - 1939 \beta_{16} - 191 \beta_{15} + 1693 \beta_{14} - 1277 \beta_{13} + \cdots + 3546 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2885 \beta_{19} + 4462 \beta_{18} - 8022 \beta_{17} + 2622 \beta_{15} + 28244 \beta_{14} + 1083 \beta_{12} + \cdots + 451 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 19793 \beta_{19} - 40332 \beta_{18} - 37051 \beta_{17} + 66516 \beta_{16} + 19793 \beta_{15} + \cdots - 173931 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 81958 \beta_{19} + 2553 \beta_{17} + 244463 \beta_{16} + 103961 \beta_{15} - 824249 \beta_{14} + \cdots - 4981387 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 252108 \beta_{19} + 1236735 \beta_{18} + 132516 \beta_{17} - 478436 \beta_{15} - 2092943 \beta_{14} + \cdots + 1019672 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6006373 \beta_{19} - 4163266 \beta_{18} + 7828784 \beta_{17} - 8080479 \beta_{16} - 6006373 \beta_{15} + \cdots + 140524516 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 16615462 \beta_{19} + 33190853 \beta_{17} - 68987615 \beta_{16} - 8537997 \beta_{15} + 69734656 \beta_{14} + \cdots + 190009976 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 111404064 \beta_{19} + 120721790 \beta_{18} - 227229173 \beta_{17} + 78250132 \beta_{15} + \cdots - 17154508 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 832115238 \beta_{19} - 1095971516 \beta_{18} - 1111583106 \beta_{17} + 2155445106 \beta_{16} + \cdots - 7586835942 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2421778760 \beta_{19} - 794580116 \beta_{17} + 8356465144 \beta_{16} + 3489892556 \beta_{15} + \cdots - 116217559461 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 9293514288 \beta_{19} + 32056344517 \beta_{18} + 2335120613 \beta_{17} - 17571013440 \beta_{15} + \cdots + 32162479224 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 182873480959 \beta_{19} - 95664470631 \beta_{18} + 232737367585 \beta_{17} - 264736024076 \beta_{16} + \cdots + 3428403000718 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 552373617412 \beta_{19} + 978549201086 \beta_{17} - 2048468084291 \beta_{16} - 301640123788 \beta_{15} + \cdots + 7330358559258 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−2.63576 − 4.56527i
−2.31251 − 4.00539i
−1.64894 − 2.85605i
−1.37828 − 2.38724i
−0.268161 − 0.464468i
0.674308 + 1.16794i
1.11799 + 1.93642i
1.41608 + 2.45272i
2.26498 + 3.92307i
2.77029 + 4.79828i
−2.63576 + 4.56527i
−2.31251 + 4.00539i
−1.64894 + 2.85605i
−1.37828 + 2.38724i
−0.268161 + 0.464468i
0.674308 − 1.16794i
1.11799 − 1.93642i
1.41608 − 2.45272i
2.26498 − 3.92307i
2.77029 − 4.79828i
−2.63576 + 4.56527i −0.604092 − 1.04632i −9.89443 − 17.1377i −9.10933 + 15.7778i 6.36896 −7.73915 − 16.8257i 62.1452 12.7701 − 22.1185i −48.0200 − 83.1730i
23.2 −2.31251 + 4.00539i −4.69688 − 8.13523i −6.69544 − 11.5968i 2.00754 − 3.47716i 43.4464 −3.31004 + 18.2221i 24.9329 −30.6213 + 53.0377i 9.28493 + 16.0820i
23.3 −1.64894 + 2.85605i 2.68131 + 4.64416i −1.43800 − 2.49069i −4.02020 + 6.96320i −17.6853 15.2781 + 10.4681i −16.8983 −0.878841 + 1.52220i −13.2581 − 22.9638i
23.4 −1.37828 + 2.38724i −0.285547 − 0.494582i 0.200708 + 0.347637i 10.1080 − 17.5075i 1.57425 −12.2187 − 13.9178i −23.1589 13.3369 − 23.1002i 27.8631 + 48.2604i
23.5 −0.268161 + 0.464468i 1.97250 + 3.41648i 3.85618 + 6.67910i 1.47169 − 2.54903i −2.11579 −8.16569 + 16.6229i −8.42687 5.71846 − 9.90466i 0.789296 + 1.36710i
23.6 0.674308 − 1.16794i −2.90728 − 5.03555i 3.09062 + 5.35311i −10.5235 + 18.2273i −7.84160 −12.9848 + 13.2058i 19.1250 −3.40453 + 5.89681i 14.1922 + 24.5816i
23.7 1.11799 − 1.93642i −0.448504 − 0.776832i 1.50018 + 2.59839i 2.13369 − 3.69565i −2.00570 4.50836 − 17.9631i 24.5967 13.0977 − 22.6859i −4.77090 − 8.26344i
23.8 1.41608 − 2.45272i 4.34207 + 7.52069i −0.0105427 − 0.0182606i −1.36796 + 2.36938i 24.5948 −18.4977 − 0.914217i 22.5975 −24.2072 + 41.9281i 3.87427 + 6.71043i
23.9 2.26498 − 3.92307i −3.59080 − 6.21944i −6.26030 − 10.8432i 2.34561 − 4.06272i −32.5324 14.1667 + 11.9292i −20.4782 −12.2876 + 21.2828i −10.6256 − 18.4040i
23.10 2.77029 − 4.79828i 3.53721 + 6.12662i −11.3490 − 19.6570i 1.95454 − 3.38537i 39.1963 13.9630 − 12.1670i −81.4350 −11.5237 + 19.9596i −10.8293 − 18.7569i
67.1 −2.63576 − 4.56527i −0.604092 + 1.04632i −9.89443 + 17.1377i −9.10933 − 15.7778i 6.36896 −7.73915 + 16.8257i 62.1452 12.7701 + 22.1185i −48.0200 + 83.1730i
67.2 −2.31251 − 4.00539i −4.69688 + 8.13523i −6.69544 + 11.5968i 2.00754 + 3.47716i 43.4464 −3.31004 − 18.2221i 24.9329 −30.6213 − 53.0377i 9.28493 − 16.0820i
67.3 −1.64894 − 2.85605i 2.68131 − 4.64416i −1.43800 + 2.49069i −4.02020 − 6.96320i −17.6853 15.2781 − 10.4681i −16.8983 −0.878841 − 1.52220i −13.2581 + 22.9638i
67.4 −1.37828 − 2.38724i −0.285547 + 0.494582i 0.200708 − 0.347637i 10.1080 + 17.5075i 1.57425 −12.2187 + 13.9178i −23.1589 13.3369 + 23.1002i 27.8631 − 48.2604i
67.5 −0.268161 − 0.464468i 1.97250 − 3.41648i 3.85618 − 6.67910i 1.47169 + 2.54903i −2.11579 −8.16569 − 16.6229i −8.42687 5.71846 + 9.90466i 0.789296 − 1.36710i
67.6 0.674308 + 1.16794i −2.90728 + 5.03555i 3.09062 − 5.35311i −10.5235 − 18.2273i −7.84160 −12.9848 − 13.2058i 19.1250 −3.40453 − 5.89681i 14.1922 − 24.5816i
67.7 1.11799 + 1.93642i −0.448504 + 0.776832i 1.50018 − 2.59839i 2.13369 + 3.69565i −2.00570 4.50836 + 17.9631i 24.5967 13.0977 + 22.6859i −4.77090 + 8.26344i
67.8 1.41608 + 2.45272i 4.34207 − 7.52069i −0.0105427 + 0.0182606i −1.36796 − 2.36938i 24.5948 −18.4977 + 0.914217i 22.5975 −24.2072 − 41.9281i 3.87427 − 6.71043i
67.9 2.26498 + 3.92307i −3.59080 + 6.21944i −6.26030 + 10.8432i 2.34561 + 4.06272i −32.5324 14.1667 − 11.9292i −20.4782 −12.2876 − 21.2828i −10.6256 + 18.4040i
67.10 2.77029 + 4.79828i 3.53721 − 6.12662i −11.3490 + 19.6570i 1.95454 + 3.38537i 39.1963 13.9630 + 12.1670i −81.4350 −11.5237 − 19.9596i −10.8293 + 18.7569i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.4.e.c ✓ 20
7.c even 3 1 inner 77.4.e.c ✓ 20
7.c even 3 1 539.4.a.n 10
7.d odd 6 1 539.4.a.m 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.e.c ✓ 20 1.a even 1 1 trivial
77.4.e.c ✓ 20 7.c even 3 1 inner
539.4.a.m 10 7.d odd 6 1
539.4.a.n 10 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 67 T_{2}^{18} - 2 T_{2}^{17} + 2960 T_{2}^{16} - 261 T_{2}^{15} + 74338 T_{2}^{14} + \cdots + 649230400 \) acting on \(S_{4}^{\mathrm{new}}(77, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 649230400 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 99566122681 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 11 T + 121)^{10} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 38048696721917)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 15\!\cdots\!25)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 35\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 91\!\cdots\!20)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 78\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 45\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 68\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 30\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 47\!\cdots\!13)^{2} \) Copy content Toggle raw display
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