↑

Properties

Label 2070.2.j.j
Level $2070$
Weight $2$
Character orbit 2070.j
Analytic conductor $16.529$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2070,2,Mod(323,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2070.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.j (of order \(4\), 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: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 20 x^{18} + 187 x^{16} - 1012 x^{14} + 3533 x^{12} - 7896 x^{10} + 10837 x^{8} - 5668 x^{6} + \cdots + 3721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} + \beta_{5} q^{4} + ( - \beta_{16} - \beta_{12}) q^{5} + (\beta_{7} + \beta_{5} - \beta_{2} - 1) q^{7} + \beta_{12} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{2} + \beta_{5} q^{4} + ( - \beta_{16} - \beta_{12}) q^{5} + (\beta_{7} + \beta_{5} - \beta_{2} - 1) q^{7} + \beta_{12} q^{8} + ( - \beta_{6} + 1) q^{10} + \beta_{11} q^{11} + (\beta_{8} - \beta_{5} - \beta_{3} + \cdots - 1) q^{13}+ \cdots + (2 \beta_{18} - 2 \beta_{17} + \cdots - 9 \beta_{12}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 16 q^{7} + 12 q^{10} - 12 q^{13} - 20 q^{16} - 24 q^{25} - 16 q^{28} - 48 q^{31} - 60 q^{37} - 16 q^{43} + 20 q^{46} + 12 q^{52} - 32 q^{55} + 4 q^{58} + 104 q^{61} - 56 q^{67} - 8 q^{70} - 20 q^{73} + 40 q^{76} - 28 q^{82} - 40 q^{85} - 32 q^{91} - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 20 x^{18} + 187 x^{16} - 1012 x^{14} + 3533 x^{12} - 7896 x^{10} + 10837 x^{8} - 5668 x^{6} + \cdots + 3721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9967066941 \nu^{18} - 272372915591 \nu^{16} + 3594422491973 \nu^{14} + \cdots + 625072837249106 ) / 38037780473340 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6165537229 \nu^{18} - 94241982629 \nu^{16} + 720988671062 \nu^{14} - 3542160728360 \nu^{12} + \cdots + 103334075474974 ) / 19018890236670 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4617044613 \nu^{18} - 92866849757 \nu^{16} + 870804191465 \nu^{14} - 4699135846682 \nu^{12} + \cdots + 12014477992988 ) / 7607556094668 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33919961647 \nu^{18} - 712776569037 \nu^{16} + 6893779213971 \nu^{14} + \cdots + 149202740481302 ) / 38037780473340 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 87789 \nu^{18} + 1831526 \nu^{16} - 17645580 \nu^{14} + 97833022 \nu^{12} - 344620552 \nu^{10} + \cdots - 342161314 ) / 64306234 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12005736535 \nu^{18} - 228088789659 \nu^{16} + 2070898908699 \nu^{14} + \cdots + 132987487324604 ) / 7607556094668 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42271474243 \nu^{18} + 731020269433 \nu^{16} - 5698634980594 \nu^{14} + \cdots - 9119894905473 ) / 19018890236670 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 46760965092 \nu^{18} - 965951829062 \nu^{16} + 9311370725561 \nu^{14} + \cdots + 256611179940557 ) / 19018890236670 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49299521753 \nu^{18} - 1012824265843 \nu^{16} + 9716502298309 \nu^{14} + \cdots + 336128386688288 ) / 19018890236670 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 470000606992 \nu^{19} + 10698216321017 \nu^{17} - 98459993774411 \nu^{15} + \cdots + 11\!\cdots\!73 \nu ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 611242032316 \nu^{19} - 11998105260136 \nu^{17} + 109968894006568 \nu^{15} + \cdots + 57\!\cdots\!21 \nu ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1418558761258 \nu^{19} + 31636712638763 \nu^{17} - 327648534796289 \nu^{15} + \cdots - 12\!\cdots\!53 \nu ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1929288087016 \nu^{19} - 41541595241281 \nu^{17} + 417436481425753 \nu^{15} + \cdots + 16\!\cdots\!21 \nu ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1098396297641 \nu^{19} - 20764662706796 \nu^{17} + 186012498972938 \nu^{15} + \cdots + 73\!\cdots\!91 \nu ) / 773434869624580 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 4817799225356 \nu^{19} + 93207753144371 \nu^{17} - 830093277457583 \nu^{15} + \cdots - 66\!\cdots\!01 \nu ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 5044530402157 \nu^{19} + 98543070400002 \nu^{17} - 900595897065486 \nu^{15} + \cdots - 32\!\cdots\!47 \nu ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 1741588298547 \nu^{19} - 35212018560062 \nu^{17} + 336330384904576 \nu^{15} + \cdots + 15\!\cdots\!47 \nu ) / 773434869624580 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 6973530573471 \nu^{19} + 138284193322766 \nu^{17} + \cdots - 36\!\cdots\!41 \nu ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 4964843374903 \nu^{19} - 103200967017608 \nu^{17} + \cdots + 37\!\cdots\!98 \nu ) / 11\!\cdots\!70 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{18} - \beta_{17} - \beta_{16} + \beta_{14} + 2\beta_{13} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{19} + 2 \beta_{18} - 2 \beta_{17} - 2 \beta_{16} - \beta_{15} + 2 \beta_{14} + \cdots + \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{8} - 2\beta_{7} - 3\beta_{6} + 20\beta_{5} + 13\beta_{4} + 11\beta_{3} - 4\beta_{2} + 3\beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4 \beta_{19} + 8 \beta_{18} + 12 \beta_{17} + 6 \beta_{16} - 7 \beta_{15} + 4 \beta_{14} + \cdots - 3 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 15\beta_{9} + 7\beta_{8} - \beta_{7} - 3\beta_{6} + 82\beta_{5} + 42\beta_{4} + 43\beta_{3} - 16\beta_{2} - 17 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24 \beta_{19} + 23 \beta_{18} + 131 \beta_{17} + 103 \beta_{16} - 18 \beta_{15} - 5 \beta_{14} + \cdots - 66 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 267 \beta_{9} - 11 \beta_{8} + 90 \beta_{7} + 106 \beta_{6} + 907 \beta_{5} + 433 \beta_{4} + 509 \beta_{3} + \cdots - 612 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 453 \beta_{19} - 122 \beta_{18} + 758 \beta_{17} + 760 \beta_{16} + 111 \beta_{15} + \cdots - 477 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1466 \beta_{9} - 764 \beta_{8} + 1010 \beta_{7} + 1421 \beta_{6} + 2936 \beta_{5} + 1501 \beta_{4} + \cdots - 5250 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3708 \beta_{19} - 2632 \beta_{18} + 2830 \beta_{17} + 3772 \beta_{16} + 1963 \beta_{15} + \cdots - 2139 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2413 \beta_{9} - 4091 \beta_{8} + 3334 \beta_{7} + 5225 \beta_{6} - 2477 \beta_{5} - 302 \beta_{4} + \cdots - 16023 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 20102 \beta_{19} - 24665 \beta_{18} + 1525 \beta_{17} + 10017 \beta_{16} + 16692 \beta_{15} + \cdots - 3796 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 5131 \beta_{9} - 57793 \beta_{8} + 28724 \beta_{7} + 51326 \beta_{6} - 165593 \beta_{5} + \cdots - 133568 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 61893 \beta_{19} - 160408 \beta_{18} - 87448 \beta_{17} - 38304 \beta_{16} + 99543 \beta_{15} + \cdots + 40637 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 233154 \beta_{9} - 288398 \beta_{8} + 36954 \beta_{7} + 119719 \beta_{6} - 1533138 \beta_{5} + \cdots - 154520 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 142578 \beta_{19} - 721246 \beta_{18} - 957402 \beta_{17} - 771376 \beta_{16} + \cdots + 560993 \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 1109236 \beta_{9} - 377474 \beta_{8} - 366357 \beta_{7} - 382672 \beta_{6} - 4789871 \beta_{5} + \cdots + 1794207 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 3892810 \beta_{19} - 1258539 \beta_{18} - 6614415 \beta_{17} - 6533395 \beta_{16} + \cdots + 4272186 \beta_{10} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(1891\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
0.0229150 − 0.707107i
−2.39613 − 0.707107i
1.75570 − 0.707107i
−1.39971 − 0.707107i
1.31011 − 0.707107i
−1.31011 + 0.707107i
1.39971 + 0.707107i
−1.75570 + 0.707107i
2.39613 + 0.707107i
−0.0229150 + 0.707107i
0.0229150 + 0.707107i
−2.39613 + 0.707107i
1.75570 + 0.707107i
−1.39971 + 0.707107i
1.31011 + 0.707107i
−1.31011 − 0.707107i
1.39971 − 0.707107i
−1.75570 − 0.707107i
2.39613 − 0.707107i
−0.0229150 − 0.707107i
−0.707107 − 0.707107i 0 1.00000i −1.88578 − 1.20159i 0 −1.61288 + 1.61288i 0.707107 − 0.707107i 0 0.483797 + 2.18310i
323.2 −0.707107 − 0.707107i 0 1.00000i −1.24750 + 1.85573i 0 −3.35392 + 3.35392i 0.707107 − 0.707107i 0 2.19432 − 0.430087i
323.3 −0.707107 − 0.707107i 0 1.00000i −0.967383 − 2.01598i 0 0.545849 − 0.545849i 0.707107 − 0.707107i 0 −0.741469 + 2.10956i
323.4 −0.707107 − 0.707107i 0 1.00000i 0.125717 + 2.23253i 0 3.50921 − 3.50921i 0.707107 − 0.707107i 0 1.48974 − 1.66753i
323.5 −0.707107 − 0.707107i 0 1.00000i 1.85363 + 1.25063i 0 −3.08825 + 3.08825i 0.707107 − 0.707107i 0 −0.426386 − 2.19504i
323.6 0.707107 + 0.707107i 0 1.00000i −1.85363 − 1.25063i 0 −3.08825 + 3.08825i −0.707107 + 0.707107i 0 −0.426386 − 2.19504i
323.7 0.707107 + 0.707107i 0 1.00000i −0.125717 − 2.23253i 0 3.50921 − 3.50921i −0.707107 + 0.707107i 0 1.48974 − 1.66753i
323.8 0.707107 + 0.707107i 0 1.00000i 0.967383 + 2.01598i 0 0.545849 − 0.545849i −0.707107 + 0.707107i 0 −0.741469 + 2.10956i
323.9 0.707107 + 0.707107i 0 1.00000i 1.24750 − 1.85573i 0 −3.35392 + 3.35392i −0.707107 + 0.707107i 0 2.19432 − 0.430087i
323.10 0.707107 + 0.707107i 0 1.00000i 1.88578 + 1.20159i 0 −1.61288 + 1.61288i −0.707107 + 0.707107i 0 0.483797 + 2.18310i
737.1 −0.707107 + 0.707107i 0 − 1.00000i −1.88578 + 1.20159i 0 −1.61288 − 1.61288i 0.707107 + 0.707107i 0 0.483797 − 2.18310i
737.2 −0.707107 + 0.707107i 0 − 1.00000i −1.24750 − 1.85573i 0 −3.35392 − 3.35392i 0.707107 + 0.707107i 0 2.19432 + 0.430087i
737.3 −0.707107 + 0.707107i 0 − 1.00000i −0.967383 + 2.01598i 0 0.545849 + 0.545849i 0.707107 + 0.707107i 0 −0.741469 − 2.10956i
737.4 −0.707107 + 0.707107i 0 − 1.00000i 0.125717 − 2.23253i 0 3.50921 + 3.50921i 0.707107 + 0.707107i 0 1.48974 + 1.66753i
737.5 −0.707107 + 0.707107i 0 − 1.00000i 1.85363 − 1.25063i 0 −3.08825 − 3.08825i 0.707107 + 0.707107i 0 −0.426386 + 2.19504i
737.6 0.707107 − 0.707107i 0 − 1.00000i −1.85363 + 1.25063i 0 −3.08825 − 3.08825i −0.707107 − 0.707107i 0 −0.426386 + 2.19504i
737.7 0.707107 − 0.707107i 0 − 1.00000i −0.125717 + 2.23253i 0 3.50921 + 3.50921i −0.707107 − 0.707107i 0 1.48974 + 1.66753i
737.8 0.707107 − 0.707107i 0 − 1.00000i 0.967383 − 2.01598i 0 0.545849 + 0.545849i −0.707107 − 0.707107i 0 −0.741469 − 2.10956i
737.9 0.707107 − 0.707107i 0 − 1.00000i 1.24750 + 1.85573i 0 −3.35392 − 3.35392i −0.707107 − 0.707107i 0 2.19432 + 0.430087i
737.10 0.707107 − 0.707107i 0 − 1.00000i 1.88578 − 1.20159i 0 −1.61288 − 1.61288i −0.707107 − 0.707107i 0 0.483797 − 2.18310i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2070.2.j.j ✓ 20
3.b odd 2 1 inner 2070.2.j.j ✓ 20
5.c odd 4 1 inner 2070.2.j.j ✓ 20
15.e even 4 1 inner 2070.2.j.j ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2070.2.j.j ✓ 20 1.a even 1 1 trivial
2070.2.j.j ✓ 20 3.b odd 2 1 inner
2070.2.j.j ✓ 20 5.c odd 4 1 inner
2070.2.j.j ✓ 20 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2070, [\chi])\):

\( T_{7}^{10} + 8 T_{7}^{9} + 32 T_{7}^{8} + 48 T_{7}^{7} + 624 T_{7}^{6} + 4800 T_{7}^{5} + 19584 T_{7}^{4} + \cdots + 32768 \) Copy content Toggle raw display
\( T_{11}^{10} + 36T_{11}^{8} + 348T_{11}^{6} + 720T_{11}^{4} + 528T_{11}^{2} + 128 \) Copy content Toggle raw display
\( T_{17}^{20} + 2400T_{17}^{16} + 1125120T_{17}^{12} + 10248192T_{17}^{8} + 24182784T_{17}^{4} + 16777216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 12 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{10} + 8 T^{9} + \cdots + 32768)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 36 T^{8} + \cdots + 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 6 T^{9} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + 2400 T^{16} + \cdots + 16777216 \) Copy content Toggle raw display
$19$ \( (T^{10} + 136 T^{8} + \cdots + 2715904)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$29$ \( (T^{10} - 266 T^{8} + \cdots - 39854592)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 12 T^{4} + \cdots + 3712)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 30 T^{9} + \cdots + 101274912)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 138 T^{8} + \cdots + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 8 T^{9} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 232294665158656 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{10} - 616 T^{8} + \cdots - 25690112)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 26 T^{4} + \cdots - 61456)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + 28 T^{9} + \cdots + 476288)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 176 T^{8} + \cdots + 123008)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 10 T^{9} + \cdots + 112740128)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 232 T^{8} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 106970533134336 \) Copy content Toggle raw display
$89$ \( (T^{10} - 146 T^{8} + \cdots - 373248)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 22 T^{9} + \cdots + 896168448)^{2} \) Copy content Toggle raw display
show more
show less