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Properties

Label 4334.2.a.a
Level $4334$
Weight $2$
Character orbit 4334.a
Self dual yes
Analytic conductor $34.607$
Analytic rank $1$
Dimension $15$
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] = [4334,2,Mod(1,4334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4334 = 2 \cdot 11 \cdot 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4334.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: \(34.6071642360\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 23 x^{13} + 19 x^{12} + 194 x^{11} - 124 x^{10} - 761 x^{9} + 353 x^{8} + 1417 x^{7} - 465 x^{6} - 1128 x^{5} + 288 x^{4} + 316 x^{3} - 79 x^{2} - 20 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} - \beta_{2} q^{5} + \beta_1 q^{6} + \beta_{13} q^{7} - q^{8} + (\beta_{12} - \beta_{11} - 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} - \beta_{2} q^{5} + \beta_1 q^{6} + \beta_{13} q^{7} - q^{8} + (\beta_{12} - \beta_{11} - 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{9} + \beta_{2} q^{10} + q^{11} - \beta_1 q^{12} + (\beta_{12} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{13} - \beta_{13} q^{14} + ( - \beta_{13} + \beta_{11} - \beta_{8} - \beta_{7} - \beta_{4} + \beta_1 - 1) q^{15} + q^{16} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{2}) q^{17} + ( - \beta_{12} + \beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{18} + (\beta_{14} - \beta_{12} + \beta_{10} - \beta_{7} + \beta_{6} - \beta_{4} - 2) q^{19} - \beta_{2} q^{20} + \beta_{3} q^{21} - q^{22} + ( - \beta_{12} + \beta_{11} + \beta_{10} + \beta_{5} + \beta_{2} + \beta_1 - 1) q^{23} + \beta_1 q^{24} + ( - \beta_{14} - \beta_{13} - \beta_{5} + \beta_1) q^{25} + ( - \beta_{12} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_1) q^{26} + ( - \beta_{14} + 2 \beta_{10} + 2 \beta_{8} + \beta_{7} - \beta_{3} - \beta_1) q^{27} + \beta_{13} q^{28} + ( - 2 \beta_{12} + \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots - 2) q^{29}+ \cdots + (\beta_{12} - \beta_{11} - 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9} + 7 q^{10} + 15 q^{11} - q^{12} - q^{13} - q^{14} - 6 q^{15} + 15 q^{16} - 6 q^{17} - 2 q^{18} - 14 q^{19} - 7 q^{20} - 3 q^{21} - 15 q^{22} + 2 q^{23} + q^{24} - 10 q^{25} + q^{26} - 7 q^{27} + q^{28} + 8 q^{29} + 6 q^{30} - 33 q^{31} - 15 q^{32} - q^{33} + 6 q^{34} - 8 q^{35} + 2 q^{36} - 9 q^{37} + 14 q^{38} - 9 q^{39} + 7 q^{40} - 10 q^{41} + 3 q^{42} - 6 q^{43} + 15 q^{44} - 20 q^{45} - 2 q^{46} - q^{47} - q^{48} - 30 q^{49} + 10 q^{50} + 12 q^{51} - q^{52} + 6 q^{53} + 7 q^{54} - 7 q^{55} - q^{56} - 24 q^{57} - 8 q^{58} - 15 q^{59} - 6 q^{60} - 25 q^{61} + 33 q^{62} + 12 q^{63} + 15 q^{64} + 31 q^{65} + q^{66} - 13 q^{67} - 6 q^{68} - 43 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 4 q^{73} + 9 q^{74} - 5 q^{75} - 14 q^{76} + q^{77} + 9 q^{78} - 20 q^{79} - 7 q^{80} + 11 q^{81} + 10 q^{82} + q^{83} - 3 q^{84} - q^{85} + 6 q^{86} + 22 q^{87} - 15 q^{88} - 41 q^{89} + 20 q^{90} - 31 q^{91} + 2 q^{92} + 14 q^{93} + q^{94} + 41 q^{95} + q^{96} - 57 q^{97} + 30 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 23 x^{13} + 19 x^{12} + 194 x^{11} - 124 x^{10} - 761 x^{9} + 353 x^{8} + 1417 x^{7} - 465 x^{6} - 1128 x^{5} + 288 x^{4} + 316 x^{3} - 79 x^{2} - 20 x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 448472053 \nu^{14} - 429772563 \nu^{13} - 10266666737 \nu^{12} + 7711061381 \nu^{11} + 86228693788 \nu^{10} - 44521450596 \nu^{9} + \cdots + 9173520254 ) / 7874747184 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 161438769 \nu^{14} + 64572991 \nu^{13} + 3840262629 \nu^{12} - 994484313 \nu^{11} - 33531038748 \nu^{10} + 4048417900 \nu^{9} + \cdots + 3103607882 ) / 1312457864 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1075926553 \nu^{14} + 1972947231 \nu^{13} + 23883502997 \nu^{12} - 40695091289 \nu^{11} - 192855632140 \nu^{10} + 299079459828 \nu^{9} + \cdots - 5475167462 ) / 7874747184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 494359543 \nu^{14} - 219123313 \nu^{13} - 11510694283 \nu^{12} + 2974994567 \nu^{11} + 98138892948 \nu^{10} - 6660902252 \nu^{9} + \cdots + 2824180666 ) / 2624915728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1646115737 \nu^{14} - 882807951 \nu^{13} - 37887294613 \nu^{12} + 13164700057 \nu^{11} + 317452879004 \nu^{10} - 47189809620 \nu^{9} + \cdots + 15387470326 ) / 7874747184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 617146855 \nu^{14} - 414537745 \nu^{13} - 14272457147 \nu^{12} + 7008393303 \nu^{11} + 120570189908 \nu^{10} - 36285675884 \nu^{9} + \cdots + 1116761994 ) / 2624915728 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 733283991 \nu^{14} + 378020609 \nu^{13} + 17374142651 \nu^{12} - 6186850999 \nu^{11} - 151732628516 \nu^{10} + 29987243308 \nu^{9} + \cdots + 9093329046 ) / 2624915728 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1073285971 \nu^{14} - 1027424005 \nu^{13} - 24473840551 \nu^{12} + 19054790339 \nu^{11} + 203330746100 \nu^{10} - 118912987804 \nu^{9} + \cdots - 5646716174 ) / 2624915728 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 808016611 \nu^{14} + 591576477 \nu^{13} + 18474709535 \nu^{12} - 9961786835 \nu^{11} - 153807583108 \nu^{10} + 50737695288 \nu^{9} + \cdots + 5301776278 ) / 1968686796 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 135407223 \nu^{14} - 73966371 \nu^{13} - 3128469083 \nu^{12} + 1100245833 \nu^{11} + 26410955648 \nu^{10} - 3832014770 \nu^{9} + \cdots - 935882532 ) / 328114466 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1869007139 \nu^{14} + 1158669429 \nu^{13} + 43098528031 \nu^{12} - 18643573363 \nu^{11} - 362523820556 \nu^{10} + 84699645708 \nu^{9} + \cdots + 3854902022 ) / 3937373592 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1551803941 \nu^{14} - 1228926403 \nu^{13} - 35820636625 \nu^{12} + 21803749621 \nu^{11} + 303038933180 \nu^{10} - 125361611188 \nu^{9} + \cdots - 17899048466 ) / 2624915728 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 8043763655 \nu^{14} + 5369684289 \nu^{13} + 186183584971 \nu^{12} - 89823314887 \nu^{11} - 1577959633748 \nu^{10} + \cdots + 81702823190 ) / 7874747184 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{11} - 2\beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - 2\beta_{10} - 2\beta_{8} - \beta_{7} + \beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} - \beta_{13} + 11 \beta_{12} - 12 \beta_{11} - 20 \beta_{10} - 9 \beta_{8} + 12 \beta_{7} - 7 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 9 \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{14} - \beta_{13} - \beta_{12} - 22 \beta_{10} + 4 \beta_{9} - 25 \beta_{8} - 16 \beta_{7} + \beta_{5} + 2 \beta_{4} + 13 \beta_{3} + 3 \beta_{2} + 59 \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 16 \beta_{14} - 20 \beta_{13} + 113 \beta_{12} - 121 \beta_{11} - 195 \beta_{10} + 2 \beta_{9} - 88 \beta_{8} + 119 \beta_{7} + \beta_{6} - 57 \beta_{5} + 94 \beta_{4} + 100 \beta_{3} + 23 \beta_{2} + 85 \beta _1 + 219 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 119 \beta_{14} - 20 \beta_{13} - 10 \beta_{12} - 13 \beta_{11} - 230 \beta_{10} + 59 \beta_{9} - 270 \beta_{8} - 181 \beta_{7} + 5 \beta_{6} + 9 \beta_{5} + 35 \beta_{4} + 148 \beta_{3} + 50 \beta_{2} + 541 \beta _1 - 43 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 181 \beta_{14} - 260 \beta_{13} + 1126 \beta_{12} - 1172 \beta_{11} - 1898 \beta_{10} + 38 \beta_{9} - 883 \beta_{8} + 1125 \beta_{7} + 27 \beta_{6} - 506 \beta_{5} + 882 \beta_{4} + 991 \beta_{3} + 234 \beta_{2} + \cdots + 1922 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1125 \beta_{14} - 281 \beta_{13} - 29 \beta_{12} - 276 \beta_{11} - 2403 \beta_{10} + 666 \beta_{9} - 2783 \beta_{8} - 1825 \beta_{7} + 98 \beta_{6} + 53 \beta_{5} + 456 \beta_{4} + 1586 \beta_{3} + 621 \beta_{2} + 5132 \beta _1 - 431 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1825 \beta_{14} - 2907 \beta_{13} + 11053 \beta_{12} - 11276 \beta_{11} - 18502 \beta_{10} + 514 \beta_{9} - 8880 \beta_{8} + 10512 \beta_{7} + 418 \beta_{6} - 4676 \beta_{5} + 8355 \beta_{4} + \cdots + 17551 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10512 \beta_{14} - 3426 \beta_{13} + 743 \beta_{12} - 4101 \beta_{11} - 25098 \beta_{10} + 6911 \beta_{9} - 28175 \beta_{8} - 17493 \beta_{7} + 1353 \beta_{6} + 103 \beta_{5} + 5385 \beta_{4} + 16555 \beta_{3} + \cdots - 3546 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 17493 \beta_{14} - 30438 \beta_{13} + 107692 \beta_{12} - 108635 \beta_{11} - 180705 \beta_{10} + 6146 \beta_{9} - 89119 \beta_{8} + 98124 \beta_{7} + 5257 \beta_{6} - 44062 \beta_{5} + \cdots + 163776 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 98124 \beta_{14} - 38942 \beta_{13} + 18777 \beta_{12} - 52963 \beta_{11} - 261593 \beta_{10} + 69368 \beta_{9} - 283145 \beta_{8} - 163305 \beta_{7} + 16212 \beta_{6} - 3290 \beta_{5} + 60888 \beta_{4} + \cdots - 23608 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 163305 \beta_{14} - 308859 \beta_{13} + 1045620 \beta_{12} - 1049397 \beta_{11} - 1767987 \beta_{10} + 69402 \beta_{9} - 892773 \beta_{8} + 917749 \beta_{7} + 59841 \beta_{6} + \cdots + 1547591 \) Copy content Toggle raw display

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

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
3.16186
2.47732
2.20706
1.67591
1.03566
0.516772
0.415881
0.167063
−0.275495
−0.670430
−0.995042
−1.70683
−1.77498
−2.19035
−3.04439
−1.00000 −3.16186 1.00000 −1.61035 3.16186 0.0905305 −1.00000 6.99733 1.61035
1.2 −1.00000 −2.47732 1.00000 1.17448 2.47732 0.539926 −1.00000 3.13710 −1.17448
1.3 −1.00000 −2.20706 1.00000 2.02512 2.20706 2.12256 −1.00000 1.87111 −2.02512
1.4 −1.00000 −1.67591 1.00000 −3.90939 1.67591 −0.186287 −1.00000 −0.191317 3.90939
1.5 −1.00000 −1.03566 1.00000 2.52120 1.03566 −1.73580 −1.00000 −1.92741 −2.52120
1.6 −1.00000 −0.516772 1.00000 −1.53195 0.516772 −0.576902 −1.00000 −2.73295 1.53195
1.7 −1.00000 −0.415881 1.00000 −2.24456 0.415881 1.94686 −1.00000 −2.82704 2.24456
1.8 −1.00000 −0.167063 1.00000 −1.86132 0.167063 −4.61549 −1.00000 −2.97209 1.86132
1.9 −1.00000 0.275495 1.00000 −0.829257 −0.275495 4.83738 −1.00000 −2.92410 0.829257
1.10 −1.00000 0.670430 1.00000 1.45227 −0.670430 0.581735 −1.00000 −2.55052 −1.45227
1.11 −1.00000 0.995042 1.00000 2.56257 −0.995042 −3.12926 −1.00000 −2.00989 −2.56257
1.12 −1.00000 1.70683 1.00000 1.14577 −1.70683 −1.19968 −1.00000 −0.0867259 −1.14577
1.13 −1.00000 1.77498 1.00000 −0.781935 −1.77498 2.50677 −1.00000 0.150553 0.781935
1.14 −1.00000 2.19035 1.00000 −2.35673 −2.19035 −0.565985 −1.00000 1.79762 2.35673
1.15 −1.00000 3.04439 1.00000 −2.75592 −3.04439 0.383632 −1.00000 6.26834 2.75592
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)
\(197\) \(-1\)

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 4334.2.a.a ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4334.2.a.a ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} + T_{3}^{14} - 23 T_{3}^{13} - 19 T_{3}^{12} + 194 T_{3}^{11} + 124 T_{3}^{10} - 761 T_{3}^{9} - 353 T_{3}^{8} + 1417 T_{3}^{7} + 465 T_{3}^{6} - 1128 T_{3}^{5} - 288 T_{3}^{4} + 316 T_{3}^{3} + 79 T_{3}^{2} - 20 T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4334))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + T^{14} - 23 T^{13} - 19 T^{12} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{15} + 7 T^{14} - 8 T^{13} - 146 T^{12} + \cdots + 4339 \) Copy content Toggle raw display
$7$ \( T^{15} - T^{14} - 37 T^{13} + 34 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} + T^{14} - 81 T^{13} - 15 T^{12} + \cdots - 14284 \) Copy content Toggle raw display
$17$ \( T^{15} + 6 T^{14} - 68 T^{13} + \cdots - 26093 \) Copy content Toggle raw display
$19$ \( T^{15} + 14 T^{14} + 15 T^{13} - 393 T^{12} + \cdots + 48 \) Copy content Toggle raw display
$23$ \( T^{15} - 2 T^{14} - 145 T^{13} + \cdots - 9538764 \) Copy content Toggle raw display
$29$ \( T^{15} - 8 T^{14} - 238 T^{13} + \cdots - 686143609 \) Copy content Toggle raw display
$31$ \( T^{15} + 33 T^{14} + 302 T^{13} + \cdots + 11274477 \) Copy content Toggle raw display
$37$ \( T^{15} + 9 T^{14} + \cdots - 2749788772 \) Copy content Toggle raw display
$41$ \( T^{15} + 10 T^{14} - 120 T^{13} + \cdots - 1025428 \) Copy content Toggle raw display
$43$ \( T^{15} + 6 T^{14} + \cdots - 34861703012 \) Copy content Toggle raw display
$47$ \( T^{15} + T^{14} - 214 T^{13} + \cdots - 306948 \) Copy content Toggle raw display
$53$ \( T^{15} - 6 T^{14} + \cdots + 244181352116 \) Copy content Toggle raw display
$59$ \( T^{15} + 15 T^{14} + \cdots - 735327198093 \) Copy content Toggle raw display
$61$ \( T^{15} + 25 T^{14} + \cdots + 448552684675 \) Copy content Toggle raw display
$67$ \( T^{15} + 13 T^{14} + \cdots + 55312586988 \) Copy content Toggle raw display
$71$ \( T^{15} + 4 T^{14} + \cdots + 75278625949 \) Copy content Toggle raw display
$73$ \( T^{15} + 4 T^{14} + \cdots - 2700639899 \) Copy content Toggle raw display
$79$ \( T^{15} + 20 T^{14} + \cdots - 5866847775988 \) Copy content Toggle raw display
$83$ \( T^{15} - T^{14} - 807 T^{13} + \cdots - 15317447248 \) Copy content Toggle raw display
$89$ \( T^{15} + 41 T^{14} + \cdots - 20211044090796 \) Copy content Toggle raw display
$97$ \( T^{15} + 57 T^{14} + \cdots + 69305861104657 \) Copy content Toggle raw display
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