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Properties

Label 209.2.g.b
Level $209$
Weight $2$
Character orbit 209.g
Analytic conductor $1.669$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [209,2,Mod(65,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.g (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: \(1.66887340224\)
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: \( x^{20} + 14 x^{18} + 125 x^{16} + 674 x^{14} + 2641 x^{12} + 6937 x^{10} + 13442 x^{8} + 17506 x^{6} + \cdots + 3249 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_1) q^{2} - \beta_{12} q^{3} + ( - \beta_{13} + \beta_{12} - \beta_{11} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{8} - \beta_{7} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_1) q^{2} - \beta_{12} q^{3} + ( - \beta_{13} + \beta_{12} - \beta_{11} + \cdots - 1) q^{4}+ \cdots + (\beta_{15} + 2 \beta_{10} - 2 \beta_{9} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 8 q^{4} - 12 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 8 q^{4} - 12 q^{5} - 6 q^{9} - 24 q^{11} - 6 q^{14} - 6 q^{15} + 20 q^{16} + 12 q^{20} + 30 q^{22} + 14 q^{23} - 30 q^{25} - 28 q^{26} + 12 q^{33} + 18 q^{34} - 4 q^{36} - 18 q^{38} + 2 q^{42} + 18 q^{44} - 28 q^{45} + 4 q^{47} + 24 q^{48} - 48 q^{49} + 48 q^{53} - 2 q^{55} + 28 q^{58} + 54 q^{59} + 36 q^{60} - 8 q^{64} - 28 q^{66} + 36 q^{67} + 72 q^{70} - 96 q^{71} + 24 q^{77} - 48 q^{78} - 4 q^{80} + 14 q^{81} - 8 q^{82} - 96 q^{86} - 54 q^{89} + 48 q^{91} + 32 q^{92} + 40 q^{93} - 96 q^{97} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 14 x^{18} + 125 x^{16} + 674 x^{14} + 2641 x^{12} + 6937 x^{10} + 13442 x^{8} + 17506 x^{6} + \cdots + 3249 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2556111024 \nu^{18} - 79933025646 \nu^{16} - 1188089865341 \nu^{14} + \cdots - 426134371434612 ) / 124858643331687 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 36525880578 \nu^{18} - 484605931967 \nu^{16} - 4225841138650 \nu^{14} + \cdots - 109878683491143 ) / 124858643331687 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 61890752758 \nu^{19} + 1005214016597 \nu^{17} + 9433019518079 \nu^{15} + \cdots + 20\!\cdots\!43 \nu ) / 374575929995061 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 63535317119 \nu^{19} + 26923748039 \nu^{17} + 3650126001773 \nu^{15} + \cdots + 12\!\cdots\!09 \nu ) / 374575929995061 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 36525880578 \nu^{19} - 484605931967 \nu^{17} - 4225841138650 \nu^{15} + \cdots - 109878683491143 \nu ) / 124858643331687 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 113564708159 \nu^{18} - 1605280249410 \nu^{16} - 14308771411765 \nu^{14} + \cdots - 354134221818267 ) / 124858643331687 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6623303400 \nu^{18} + 84542092960 \nu^{16} + 724176187629 \nu^{14} + 3572352185214 \nu^{12} + \cdots - 6528835526985 ) / 6571507543773 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8309577374 \nu^{18} + 94088659161 \nu^{16} + 756453856772 \nu^{14} + 3231440713878 \nu^{12} + \cdots - 11709615762501 ) / 6571507543773 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 167031723789 \nu^{18} - 2040637245138 \nu^{16} - 16673129264638 \nu^{14} + \cdots + 285043082162142 ) / 124858643331687 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 174737725901 \nu^{18} + 2393619213966 \nu^{16} + 21006814296734 \nu^{14} + \cdots + 624388295333103 ) / 124858643331687 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 193788119914 \nu^{18} - 2380531178738 \nu^{16} - 19584614944903 \nu^{14} + \cdots + 540946426159281 ) / 124858643331687 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 252276162129 \nu^{18} - 3666052252048 \nu^{16} - 33071061998967 \nu^{14} + \cdots - 10\!\cdots\!36 ) / 124858643331687 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 419318748491 \nu^{19} - 6867660313666 \nu^{17} - 64743916795942 \nu^{15} + \cdots - 31\!\cdots\!75 \nu ) / 374575929995061 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 24842176103 \nu^{19} - 336589017727 \nu^{17} - 2957948176624 \nu^{15} + \cdots - 114153921052086 \nu ) / 19714522631319 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 579719795381 \nu^{19} + 8173731300850 \nu^{17} + 71836990354561 \nu^{15} + \cdots + 15\!\cdots\!09 \nu ) / 374575929995061 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 622723471835 \nu^{19} + 7481714612485 \nu^{17} + 60680496930742 \nu^{15} + \cdots - 20\!\cdots\!61 \nu ) / 374575929995061 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 286245931683 \nu^{19} + 4230591209661 \nu^{17} + 38484993002958 \nu^{15} + \cdots + 17\!\cdots\!78 \nu ) / 124858643331687 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1061034038771 \nu^{19} + 13296985796089 \nu^{17} + 111390590594542 \nu^{15} + \cdots - 732317892792045 \nu ) / 374575929995061 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{11} - \beta_{9} + \beta_{8} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} + \beta_{17} - 4\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{13} + 6 \beta_{12} - 7 \beta_{11} - 6 \beta_{10} + 6 \beta_{9} - 6 \beta_{8} - \beta_{7} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{19} + 7\beta_{18} - 7\beta_{17} + 8\beta_{15} + 7\beta_{14} + 25\beta_{6} + 6\beta_{4} - 26\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -31\beta_{12} + \beta_{11} + 31\beta_{10} - 8\beta_{8} + \beta_{2} + 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{19} - 39 \beta_{18} + \beta_{17} - 2 \beta_{16} - 30 \beta_{15} - 40 \beta_{14} + \cdots + 125 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 155\beta_{13} - \beta_{12} + 204\beta_{11} - 157\beta_{9} + 216\beta_{8} + 73\beta_{7} - 255\beta_{3} - 61\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 216 \beta_{19} - 12 \beta_{18} + 204 \beta_{17} + 13 \beta_{16} - 132 \beta_{15} + 12 \beta_{14} + \cdots + 59 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 783 \beta_{13} + 783 \beta_{12} - 1144 \beta_{11} - 797 \beta_{10} + 797 \beta_{9} - 894 \beta_{8} + \cdots - 1233 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1047 \beta_{19} + 1144 \beta_{18} - 1144 \beta_{17} + 111 \beta_{16} + 1505 \beta_{15} + \cdots - 3521 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 125 \beta_{13} - 3818 \beta_{12} + 666 \beta_{11} + 4068 \beta_{10} - 1283 \beta_{8} - 125 \beta_{7} + \cdots + 6048 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 666 \beta_{19} - 5351 \beta_{18} + 666 \beta_{17} - 1582 \beta_{16} - 3277 \beta_{15} + \cdots + 16133 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 19035 \beta_{13} - 916 \beta_{12} + 27376 \beta_{11} - 20867 \beta_{9} + 31573 \beta_{8} + \cdots - 12538 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 31573 \beta_{19} - 4197 \beta_{18} + 27376 \beta_{17} + 5113 \beta_{16} - 27441 \beta_{15} + \cdots + 10706 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 101448 \beta_{13} + 101448 \beta_{12} - 165575 \beta_{11} - 107477 \beta_{10} + 107477 \beta_{9} + \cdots - 149772 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 140446 \beta_{19} + 165575 \beta_{18} - 165575 \beta_{17} + 31158 \beta_{16} + 229702 \beta_{15} + \cdots - 480922 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 37187 \beta_{13} - 481056 \beta_{12} + 145543 \beta_{11} + 555430 \beta_{10} - 167386 \beta_{8} + \cdots + 753660 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 145543 \beta_{19} - 722816 \beta_{18} + 145543 \beta_{17} - 365460 \beta_{16} - 372700 \beta_{15} + \cdots + 2177449 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).

\(n\) \(78\) \(134\)
\(\chi(n)\) \(1 - \beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.14653 + 1.98584i
1.04471 + 1.80950i
0.737456 + 1.27731i
0.583939 + 1.01141i
0.457406 + 0.792250i
−0.457406 − 0.792250i
−0.583939 − 1.01141i
−0.737456 − 1.27731i
−1.04471 − 1.80950i
−1.14653 − 1.98584i
1.14653 − 1.98584i
1.04471 − 1.80950i
0.737456 − 1.27731i
0.583939 − 1.01141i
0.457406 − 0.792250i
−0.457406 + 0.792250i
−0.583939 + 1.01141i
−0.737456 + 1.27731i
−1.04471 + 1.80950i
−1.14653 + 1.98584i
−1.14653 + 1.98584i −0.623450 − 0.359949i −1.62905 − 2.82160i −0.846537 + 1.46624i 1.42961 − 0.825383i 4.75542i 2.88491 −1.24087 − 2.14926i −1.94116 − 3.36218i
65.2 −1.04471 + 1.80950i 2.10634 + 1.21610i −1.18285 − 2.04876i −0.788647 + 1.36598i −4.40105 + 2.54095i − 0.569411i 0.764121 1.45779 + 2.52496i −1.64782 − 2.85411i
65.3 −0.737456 + 1.27731i −0.443483 − 0.256045i −0.0876839 − 0.151873i 1.85115 − 3.20628i 0.654099 − 0.377644i − 2.80482i −2.69117 −1.36888 − 2.37097i 2.73028 + 4.72899i
65.4 −0.583939 + 1.01141i 0.791358 + 0.456891i 0.318031 + 0.550846i −1.78609 + 3.09359i −0.924209 + 0.533592i − 0.142706i −3.07860 −1.08250 − 1.87495i −2.08593 − 3.61294i
65.5 −0.457406 + 0.792250i −1.83077 − 1.05699i 0.581560 + 1.00729i −1.42988 + 2.47662i 1.67481 − 0.966950i − 4.02173i −2.89366 0.734472 + 1.27214i −1.30807 − 2.26564i
65.6 0.457406 − 0.792250i −1.83077 − 1.05699i 0.581560 + 1.00729i −1.42988 + 2.47662i −1.67481 + 0.966950i 4.02173i 2.89366 0.734472 + 1.27214i 1.30807 + 2.26564i
65.7 0.583939 − 1.01141i 0.791358 + 0.456891i 0.318031 + 0.550846i −1.78609 + 3.09359i 0.924209 − 0.533592i 0.142706i 3.07860 −1.08250 − 1.87495i 2.08593 + 3.61294i
65.8 0.737456 − 1.27731i −0.443483 − 0.256045i −0.0876839 − 0.151873i 1.85115 − 3.20628i −0.654099 + 0.377644i 2.80482i 2.69117 −1.36888 − 2.37097i −2.73028 − 4.72899i
65.9 1.04471 − 1.80950i 2.10634 + 1.21610i −1.18285 − 2.04876i −0.788647 + 1.36598i 4.40105 − 2.54095i 0.569411i −0.764121 1.45779 + 2.52496i 1.64782 + 2.85411i
65.10 1.14653 − 1.98584i −0.623450 − 0.359949i −1.62905 − 2.82160i −0.846537 + 1.46624i −1.42961 + 0.825383i − 4.75542i −2.88491 −1.24087 − 2.14926i 1.94116 + 3.36218i
164.1 −1.14653 − 1.98584i −0.623450 + 0.359949i −1.62905 + 2.82160i −0.846537 − 1.46624i 1.42961 + 0.825383i − 4.75542i 2.88491 −1.24087 + 2.14926i −1.94116 + 3.36218i
164.2 −1.04471 − 1.80950i 2.10634 − 1.21610i −1.18285 + 2.04876i −0.788647 − 1.36598i −4.40105 − 2.54095i 0.569411i 0.764121 1.45779 − 2.52496i −1.64782 + 2.85411i
164.3 −0.737456 − 1.27731i −0.443483 + 0.256045i −0.0876839 + 0.151873i 1.85115 + 3.20628i 0.654099 + 0.377644i 2.80482i −2.69117 −1.36888 + 2.37097i 2.73028 − 4.72899i
164.4 −0.583939 − 1.01141i 0.791358 − 0.456891i 0.318031 − 0.550846i −1.78609 − 3.09359i −0.924209 − 0.533592i 0.142706i −3.07860 −1.08250 + 1.87495i −2.08593 + 3.61294i
164.5 −0.457406 − 0.792250i −1.83077 + 1.05699i 0.581560 − 1.00729i −1.42988 − 2.47662i 1.67481 + 0.966950i 4.02173i −2.89366 0.734472 − 1.27214i −1.30807 + 2.26564i
164.6 0.457406 + 0.792250i −1.83077 + 1.05699i 0.581560 − 1.00729i −1.42988 − 2.47662i −1.67481 − 0.966950i − 4.02173i 2.89366 0.734472 − 1.27214i 1.30807 − 2.26564i
164.7 0.583939 + 1.01141i 0.791358 − 0.456891i 0.318031 − 0.550846i −1.78609 − 3.09359i 0.924209 + 0.533592i − 0.142706i 3.07860 −1.08250 + 1.87495i 2.08593 − 3.61294i
164.8 0.737456 + 1.27731i −0.443483 + 0.256045i −0.0876839 + 0.151873i 1.85115 + 3.20628i −0.654099 − 0.377644i − 2.80482i 2.69117 −1.36888 + 2.37097i −2.73028 + 4.72899i
164.9 1.04471 + 1.80950i 2.10634 − 1.21610i −1.18285 + 2.04876i −0.788647 − 1.36598i 4.40105 + 2.54095i − 0.569411i −0.764121 1.45779 − 2.52496i 1.64782 − 2.85411i
164.10 1.14653 + 1.98584i −0.623450 + 0.359949i −1.62905 + 2.82160i −0.846537 − 1.46624i −1.42961 − 0.825383i 4.75542i −2.88491 −1.24087 + 2.14926i 1.94116 − 3.36218i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.d odd 6 1 inner
209.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.g.b ✓ 20
11.b odd 2 1 inner 209.2.g.b ✓ 20
19.d odd 6 1 inner 209.2.g.b ✓ 20
209.g even 6 1 inner 209.2.g.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.g.b ✓ 20 1.a even 1 1 trivial
209.2.g.b ✓ 20 11.b odd 2 1 inner
209.2.g.b ✓ 20 19.d odd 6 1 inner
209.2.g.b ✓ 20 209.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 14 T_{2}^{18} + 125 T_{2}^{16} + 674 T_{2}^{14} + 2641 T_{2}^{12} + 6937 T_{2}^{10} + \cdots + 3249 \) acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 14 T^{18} + \cdots + 3249 \) Copy content Toggle raw display
$3$ \( (T^{10} - 6 T^{8} + 32 T^{6} + \cdots + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + 6 T^{9} + \cdots + 10201)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 47 T^{8} + \cdots + 19)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 12 T^{9} + \cdots + 161051)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 49 T^{18} + \cdots + 3249 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( (T^{10} - 7 T^{9} + \cdots + 68121)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 100139588678529 \) Copy content Toggle raw display
$31$ \( (T^{10} + 94 T^{8} + \cdots + 1302843)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 122 T^{8} + \cdots + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 275952697344 \) Copy content Toggle raw display
$43$ \( T^{20} - 83 T^{18} + \cdots + 29241 \) Copy content Toggle raw display
$47$ \( (T^{10} - 2 T^{9} + \cdots + 91030681)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 24 T^{9} + \cdots + 6456267)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} - 27 T^{9} + \cdots + 133563)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 515214230185561 \) Copy content Toggle raw display
$67$ \( (T^{10} - 18 T^{9} + \cdots + 243)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 48 T^{9} + \cdots + 6360978627)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 34\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 47105065335969 \) Copy content Toggle raw display
$83$ \( (T^{10} + 458 T^{8} + \cdots + 3335659)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 27 T^{9} + \cdots + 4191120387)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 48 T^{9} + \cdots + 13318347)^{2} \) Copy content Toggle raw display
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