Properties

Label 35.4.j.a
Level $35$
Weight $4$
Character orbit 35.j
Analytic conductor $2.065$
Analytic rank $0$
Dimension $20$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,4,Mod(4,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 35.j (of order \(6\), 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: \(2.06506685020\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 55 x^{18} + 2018 x^{16} - 42095 x^{14} + 639938 x^{12} - 5744691 x^{10} + 35287093 x^{8} + \cdots + 9834496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{14} q^{3} + (\beta_{4} - 3 \beta_{3} - \beta_{2}) q^{4} + \beta_{10} q^{5} + (\beta_{7} + \beta_{2} - 5) q^{6} + (\beta_{18} - \beta_{17} + \cdots + \beta_{15}) q^{7}+ \cdots + (\beta_{8} - \beta_{5} + 8 \beta_{3} + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{14} q^{3} + (\beta_{4} - 3 \beta_{3} - \beta_{2}) q^{4} + \beta_{10} q^{5} + (\beta_{7} + \beta_{2} - 5) q^{6} + (\beta_{18} - \beta_{17} + \cdots + \beta_{15}) q^{7}+ \cdots + (62 \beta_{13} + 62 \beta_{12} + \cdots + 740) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 30 q^{4} + 3 q^{5} - 96 q^{6} + 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 30 q^{4} + 3 q^{5} - 96 q^{6} + 82 q^{9} - 32 q^{10} + 36 q^{11} + 26 q^{14} - 146 q^{15} - 22 q^{16} - 192 q^{19} + 584 q^{20} - 404 q^{21} - 444 q^{24} - 187 q^{25} + 434 q^{26} + 260 q^{29} - 658 q^{30} + 834 q^{31} - 160 q^{34} + 661 q^{35} + 516 q^{36} + 868 q^{39} + 674 q^{40} - 1224 q^{41} + 542 q^{44} + 60 q^{45} - 1274 q^{46} - 326 q^{49} - 2556 q^{50} + 986 q^{51} - 2808 q^{54} - 742 q^{55} - 36 q^{56} - 2514 q^{59} - 204 q^{60} + 512 q^{61} + 6900 q^{64} - 946 q^{65} + 1396 q^{66} + 3064 q^{69} + 5190 q^{70} - 2944 q^{71} + 1590 q^{74} + 3003 q^{75} + 44 q^{76} + 46 q^{79} + 2304 q^{80} - 130 q^{81} - 12952 q^{84} - 5082 q^{85} - 1592 q^{86} - 5876 q^{89} - 3316 q^{90} + 4348 q^{91} - 3314 q^{94} - 2155 q^{95} + 3756 q^{96} + 13860 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 55 x^{18} + 2018 x^{16} - 42095 x^{14} + 639938 x^{12} - 5744691 x^{10} + 35287093 x^{8} + \cdots + 9834496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 77\!\cdots\!35 \nu^{18} + \cdots + 31\!\cdots\!44 ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!41 \nu^{18} + \cdots + 20\!\cdots\!32 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 94\!\cdots\!97 \nu^{18} + \cdots + 24\!\cdots\!52 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 99\!\cdots\!73 \nu^{18} + \cdots + 33\!\cdots\!52 ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26\!\cdots\!41 \nu^{19} + \cdots - 20\!\cdots\!32 \nu ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 50\!\cdots\!73 \nu^{18} + \cdots + 27\!\cdots\!00 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 61\!\cdots\!51 \nu^{18} + \cdots - 19\!\cdots\!20 ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 70\!\cdots\!21 \nu^{18} + \cdots - 58\!\cdots\!32 ) / 59\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46\!\cdots\!39 \nu^{19} + \cdots + 43\!\cdots\!12 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 46\!\cdots\!39 \nu^{19} + \cdots - 43\!\cdots\!12 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 67\!\cdots\!89 \nu^{19} + \cdots + 36\!\cdots\!08 ) / 66\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 67\!\cdots\!89 \nu^{19} + \cdots + 36\!\cdots\!08 ) / 66\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 26\!\cdots\!97 \nu^{19} + \cdots + 84\!\cdots\!36 \nu ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 50\!\cdots\!83 \nu^{19} + \cdots + 39\!\cdots\!68 \nu ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 33\!\cdots\!57 \nu^{19} + \cdots - 10\!\cdots\!56 \nu ) / 66\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 75\!\cdots\!49 \nu^{19} + \cdots - 23\!\cdots\!20 \nu ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 11\!\cdots\!13 \nu^{19} + \cdots - 12\!\cdots\!16 \nu ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 61\!\cdots\!19 \nu^{19} + \cdots + 46\!\cdots\!58 \nu ) / 74\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 11\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{16} - 2\beta_{15} + 2\beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + 17\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} - 2\beta_{10} - \beta_{8} + 2\beta_{5} + 21\beta_{4} - 201\beta_{3} - 201 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{17} + 30\beta_{16} + 80\beta_{14} - 28\beta_{13} + 28\beta_{12} + 317\beta_{6} - 317\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 72\beta_{13} + 72\beta_{12} + 72\beta_{11} - 72\beta_{10} - 34\beta_{9} + 84\beta_{7} + 449\beta_{2} - 3991 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -811\beta_{19} + 296\beta_{18} + 2390\beta_{15} - 695\beta_{11} - 695\beta_{10} - 6257\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1982 \beta_{13} + 1982 \beta_{12} - 875 \beta_{9} + 875 \beta_{8} + 2622 \beta_{7} + \cdots + 9973 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 20928 \beta_{19} + 8336 \beta_{18} - 8336 \beta_{17} - 20928 \beta_{16} + 64180 \beta_{15} + \cdots - 129581 \beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 49796 \beta_{11} + 49796 \beta_{10} + 20532 \beta_{8} - 72912 \beta_{5} - 227281 \beta_{4} + \cdots + 1804203 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 213568 \beta_{17} - 525077 \beta_{16} - 1637498 \beta_{14} + 388469 \beta_{13} - 388469 \beta_{12} + \cdots + 2788945 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1204074 \beta_{13} - 1204074 \beta_{12} - 1204074 \beta_{11} + 1204074 \beta_{10} + 465429 \beta_{9} + \cdots + 40206193 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12937890 \beta_{19} - 5249720 \beta_{18} - 40630072 \beta_{15} + 9067464 \beta_{11} + \cdots + 61797373 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 28634368 \beta_{13} - 28634368 \beta_{12} + 10446758 \beta_{9} - 10446758 \beta_{8} + \cdots - 123053505 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 314988255 \beta_{19} - 126533176 \beta_{18} + 126533176 \beta_{17} + 314988255 \beta_{16} + \cdots + 1398412305 \beta_{6} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 676391382 \beta_{11} - 676391382 \beta_{10} - 234869951 \beta_{8} + 1198431894 \beta_{5} + \cdots - 21061554309 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 3020954560 \beta_{17} + 7606844548 \beta_{16} + 23967851036 \beta_{14} - 4951028670 \beta_{13} + \cdots - 32115822125 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 15943966460 \beta_{13} + 15943966460 \beta_{12} + 15943966460 \beta_{11} - 15943966460 \beta_{10} + \cdots - 489896458675 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 182686621641 \beta_{19} + 71799750416 \beta_{18} + 575555594274 \beta_{15} + \cdots - 745133606321 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
4.21755 2.43500i
3.52149 2.03314i
3.14484 1.81568i
0.793194 0.457951i
0.736336 0.425124i
−0.736336 + 0.425124i
−0.793194 + 0.457951i
−3.14484 + 1.81568i
−3.52149 + 2.03314i
−4.21755 + 2.43500i
4.21755 + 2.43500i
3.52149 + 2.03314i
3.14484 + 1.81568i
0.793194 + 0.457951i
0.736336 + 0.425124i
−0.736336 0.425124i
−0.793194 0.457951i
−3.14484 1.81568i
−3.52149 2.03314i
−4.21755 2.43500i
−4.21755 + 2.43500i 6.94145 + 4.00765i 7.85846 13.6113i 7.35984 + 8.41622i −39.0345 −17.6364 + 5.65324i 37.5814i 18.6225 + 32.2551i −51.5340 17.5746i
4.2 −3.52149 + 2.03314i −3.93660 2.27280i 4.26728 7.39115i 9.62212 5.69340i 18.4836 17.0662 + 7.19331i 2.17369i −3.16878 5.48848i −22.3088 + 39.6124i
4.3 −3.14484 + 1.81568i −0.336480 0.194267i 2.59336 4.49183i −10.9591 2.21318i 1.41090 −2.86629 18.2971i 10.2160i −13.4245 23.2520i 38.4831 12.9381i
4.4 −0.793194 + 0.457951i 7.59866 + 4.38709i −3.58056 + 6.20172i −8.53056 7.22700i −8.03628 13.8805 + 12.2610i 13.8861i 24.9931 + 43.2892i 10.0760 + 1.82583i
4.5 −0.736336 + 0.425124i −3.23521 1.86785i −3.63854 + 6.30214i −1.93130 + 11.0123i 3.17627 −3.68367 + 18.1502i 12.9893i −6.52228 11.2969i −3.25949 8.92977i
4.6 0.736336 0.425124i 3.23521 + 1.86785i −3.63854 + 6.30214i 10.5026 + 3.83358i 3.17627 3.68367 18.1502i 12.9893i −6.52228 11.2969i 9.36316 1.64208i
4.7 0.793194 0.457951i −7.59866 4.38709i −3.58056 + 6.20172i −1.99348 11.0012i −8.03628 −13.8805 12.2610i 13.8861i 24.9931 + 43.2892i −6.61922 7.81315i
4.8 3.14484 1.81568i 0.336480 + 0.194267i 2.59336 4.49183i 3.56288 10.5974i 1.41090 2.86629 + 18.2971i 10.2160i −13.4245 23.2520i −8.03683 39.7963i
4.9 3.52149 2.03314i 3.93660 + 2.27280i 4.26728 7.39115i −9.74169 + 5.48630i 18.4836 −17.0662 7.19331i 2.17369i −3.16878 5.48848i −23.1509 + 39.1262i
4.10 4.21755 2.43500i −6.94145 4.00765i 7.85846 13.6113i 3.60874 + 10.5819i −39.0345 17.6364 5.65324i 37.5814i 18.6225 + 32.2551i 40.9870 + 35.8425i
9.1 −4.21755 2.43500i 6.94145 4.00765i 7.85846 + 13.6113i 7.35984 8.41622i −39.0345 −17.6364 5.65324i 37.5814i 18.6225 32.2551i −51.5340 + 17.5746i
9.2 −3.52149 2.03314i −3.93660 + 2.27280i 4.26728 + 7.39115i 9.62212 + 5.69340i 18.4836 17.0662 7.19331i 2.17369i −3.16878 + 5.48848i −22.3088 39.6124i
9.3 −3.14484 1.81568i −0.336480 + 0.194267i 2.59336 + 4.49183i −10.9591 + 2.21318i 1.41090 −2.86629 + 18.2971i 10.2160i −13.4245 + 23.2520i 38.4831 + 12.9381i
9.4 −0.793194 0.457951i 7.59866 4.38709i −3.58056 6.20172i −8.53056 + 7.22700i −8.03628 13.8805 12.2610i 13.8861i 24.9931 43.2892i 10.0760 1.82583i
9.5 −0.736336 0.425124i −3.23521 + 1.86785i −3.63854 6.30214i −1.93130 11.0123i 3.17627 −3.68367 18.1502i 12.9893i −6.52228 + 11.2969i −3.25949 + 8.92977i
9.6 0.736336 + 0.425124i 3.23521 1.86785i −3.63854 6.30214i 10.5026 3.83358i 3.17627 3.68367 + 18.1502i 12.9893i −6.52228 + 11.2969i 9.36316 + 1.64208i
9.7 0.793194 + 0.457951i −7.59866 + 4.38709i −3.58056 6.20172i −1.99348 + 11.0012i −8.03628 −13.8805 + 12.2610i 13.8861i 24.9931 43.2892i −6.61922 + 7.81315i
9.8 3.14484 + 1.81568i 0.336480 0.194267i 2.59336 + 4.49183i 3.56288 + 10.5974i 1.41090 2.86629 18.2971i 10.2160i −13.4245 + 23.2520i −8.03683 + 39.7963i
9.9 3.52149 + 2.03314i 3.93660 2.27280i 4.26728 + 7.39115i −9.74169 5.48630i 18.4836 −17.0662 + 7.19331i 2.17369i −3.16878 + 5.48848i −23.1509 39.1262i
9.10 4.21755 + 2.43500i −6.94145 + 4.00765i 7.85846 + 13.6113i 3.60874 10.5819i −39.0345 17.6364 + 5.65324i 37.5814i 18.6225 32.2551i 40.9870 35.8425i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.4.j.a 20
5.b even 2 1 inner 35.4.j.a 20
5.c odd 4 2 175.4.e.g 20
7.b odd 2 1 245.4.j.d 20
7.c even 3 1 inner 35.4.j.a 20
7.c even 3 1 245.4.b.e 10
7.d odd 6 1 245.4.b.f 10
7.d odd 6 1 245.4.j.d 20
35.c odd 2 1 245.4.j.d 20
35.i odd 6 1 245.4.b.f 10
35.i odd 6 1 245.4.j.d 20
35.j even 6 1 inner 35.4.j.a 20
35.j even 6 1 245.4.b.e 10
35.k even 12 2 1225.4.a.bq 10
35.l odd 12 2 175.4.e.g 20
35.l odd 12 2 1225.4.a.bp 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.j.a 20 1.a even 1 1 trivial
35.4.j.a 20 5.b even 2 1 inner
35.4.j.a 20 7.c even 3 1 inner
35.4.j.a 20 35.j even 6 1 inner
175.4.e.g 20 5.c odd 4 2
175.4.e.g 20 35.l odd 12 2
245.4.b.e 10 7.c even 3 1
245.4.b.e 10 35.j even 6 1
245.4.b.f 10 7.d odd 6 1
245.4.b.f 10 35.i odd 6 1
245.4.j.d 20 7.b odd 2 1
245.4.j.d 20 7.d odd 6 1
245.4.j.d 20 35.c odd 2 1
245.4.j.d 20 35.i odd 6 1
1225.4.a.bp 10 35.l odd 12 2
1225.4.a.bq 10 35.k even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(35, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 55 T^{18} + \cdots + 9834496 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 46352367616 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 68542503321600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 3445 T^{8} + \cdots + 3071819776)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 80\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 94\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{5} - 65 T^{4} + \cdots + 1210995800)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + 306 T^{4} + \cdots + 30026757530)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 26\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 31\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + 736 T^{4} + \cdots - 445136828288)^{4} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 38\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 22\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 57\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 56\!\cdots\!24)^{2} \) Copy content Toggle raw display
show more
show less