Properties

Label 6010.2.a.c
Level $6010$
Weight $2$
Character orbit 6010.a
Self dual yes
Analytic conductor $47.990$
Analytic rank $1$
Dimension $16$
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] = [6010,2,Mod(1,6010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6010, 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("6010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6010 = 2 \cdot 5 \cdot 601 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6010.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: \(47.9900916148\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 9 x^{14} + 75 x^{13} - 178 x^{12} - 232 x^{11} + 872 x^{10} + 228 x^{9} - 1986 x^{8} + 164 x^{7} + 2332 x^{6} - 440 x^{5} - 1344 x^{4} + 244 x^{3} + 295 x^{2} + \cdots - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 9 x^{14} + 75 x^{13} - 178 x^{12} - 232 x^{11} + 872 x^{10} + 228 x^{9} - 1986 x^{8} + 164 x^{7} + 2332 x^{6} - 440 x^{5} - 1344 x^{4} + 244 x^{3} + 295 x^{2} + \cdots - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25986 \nu^{15} - 3039938 \nu^{14} + 22373644 \nu^{13} - 24822547 \nu^{12} - 180887926 \nu^{11} + 419917416 \nu^{10} + 422463844 \nu^{9} - 1628277055 \nu^{8} + \cdots - 17499053 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 75052 \nu^{15} - 416320 \nu^{14} + 7114944 \nu^{13} - 14421854 \nu^{12} - 48394711 \nu^{11} + 157537117 \nu^{10} + 79579696 \nu^{9} - 536624577 \nu^{8} + \cdots - 2133268 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 180376 \nu^{15} - 2419781 \nu^{14} + 9497505 \nu^{13} + 2661981 \nu^{12} - 91910225 \nu^{11} + 118706510 \nu^{10} + 274101206 \nu^{9} - 560118954 \nu^{8} + \cdots - 20383548 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 284786 \nu^{15} - 1819192 \nu^{14} - 441924 \nu^{13} + 20718600 \nu^{12} - 13794772 \nu^{11} - 100540960 \nu^{10} + 74146068 \nu^{9} + 279407436 \nu^{8} + \cdots + 706474 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 362678 \nu^{15} - 2892660 \nu^{14} + 4353048 \nu^{13} + 18928481 \nu^{12} - 57805768 \nu^{11} - 10924214 \nu^{10} + 182827079 \nu^{9} - 139100342 \nu^{8} + \cdots - 6358506 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 384044 \nu^{15} + 3421318 \nu^{14} - 6474594 \nu^{13} - 22475282 \nu^{12} + 82865319 \nu^{11} + 16650207 \nu^{10} - 294055924 \nu^{9} + 129035112 \nu^{8} + \cdots - 714592 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 39704 \nu^{15} + 366746 \nu^{14} - 785482 \nu^{13} - 2133030 \nu^{12} + 9493309 \nu^{11} - 971276 \nu^{10} - 32730627 \nu^{9} + 23795599 \nu^{8} + \cdots - 43793 ) / 163957 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 800612 \nu^{15} + 5318060 \nu^{14} - 244087 \nu^{13} - 58445943 \nu^{12} + 60827688 \nu^{11} + 255794130 \nu^{10} - 323031488 \nu^{9} - 600603130 \nu^{8} + \cdots - 7297457 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 972224 \nu^{15} + 8826787 \nu^{14} - 18013961 \nu^{13} - 54357008 \nu^{12} + 225954995 \nu^{11} + 3659483 \nu^{10} - 818923620 \nu^{9} + \cdots + 4252139 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1048336 \nu^{15} + 8878763 \nu^{14} - 14466751 \nu^{13} - 64724264 \nu^{12} + 206543323 \nu^{11} + 92532283 \nu^{10} - 804630339 \nu^{9} + \cdots + 5288765 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1124047 \nu^{15} - 8847625 \nu^{14} + 11444848 \nu^{13} + 67320977 \nu^{12} - 173501774 \nu^{11} - 128681409 \nu^{10} + 645278405 \nu^{9} + \cdots - 1358606 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1618763 \nu^{15} + 13952863 \nu^{14} - 25482926 \nu^{13} - 88167362 \nu^{12} + 323120564 \nu^{11} + 34111348 \nu^{10} - 1108457254 \nu^{9} + \cdots - 7526881 ) / 3115183 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1717836 \nu^{15} - 13768724 \nu^{14} + 19404207 \nu^{13} + 100594394 \nu^{12} - 280173236 \nu^{11} - 160180219 \nu^{10} + 1017753578 \nu^{9} + \cdots - 10162035 ) / 3115183 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{15} + 6 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + \beta_{7} - 2 \beta_{6} + 7 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + 10 \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{15} + 28 \beta_{13} - 12 \beta_{12} + 13 \beta_{11} + 13 \beta_{10} - 12 \beta_{9} + 9 \beta_{8} - 6 \beta_{6} + \beta_{5} + 31 \beta_{4} - 16 \beta_{3} + 28 \beta_{2} + 34 \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 40 \beta_{15} - 3 \beta_{14} + 87 \beta_{13} - 33 \beta_{12} + 36 \beta_{11} + 42 \beta_{10} - 32 \beta_{9} + 19 \beta_{8} + 2 \beta_{7} - 31 \beta_{6} + 3 \beta_{5} + 103 \beta_{4} - 56 \beta_{3} + 109 \beta_{2} + 84 \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 140 \beta_{15} - 13 \beta_{14} + 319 \beta_{13} - 132 \beta_{12} + 149 \beta_{11} + 148 \beta_{10} - 100 \beta_{9} + 60 \beta_{8} - 21 \beta_{7} - 102 \beta_{6} + 15 \beta_{5} + 368 \beta_{4} - 191 \beta_{3} + 344 \beta_{2} + \cdots - 49 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 443 \beta_{15} - 71 \beta_{14} + 1025 \beta_{13} - 403 \beta_{12} + 461 \beta_{11} + 487 \beta_{10} - 262 \beta_{9} + 121 \beta_{8} - 84 \beta_{7} - 396 \beta_{6} + 50 \beta_{5} + 1209 \beta_{4} - 642 \beta_{3} + 1221 \beta_{2} + \cdots - 107 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1476 \beta_{15} - 276 \beta_{14} + 3512 \beta_{13} - 1416 \beta_{12} + 1652 \beta_{11} + 1638 \beta_{10} - 748 \beta_{9} + 301 \beta_{8} - 467 \beta_{7} - 1330 \beta_{6} + 192 \beta_{5} + 4075 \beta_{4} - 2123 \beta_{3} + \cdots - 629 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4758 \beta_{15} - 1126 \beta_{14} + 11468 \beta_{13} - 4492 \beta_{12} + 5313 \beta_{11} + 5404 \beta_{10} - 1938 \beta_{9} + 438 \beta_{8} - 1772 \beta_{7} - 4716 \beta_{6} + 659 \beta_{5} + 13342 \beta_{4} + \cdots - 1896 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 15673 \beta_{15} - 4126 \beta_{14} + 38386 \beta_{13} - 15058 \beta_{12} + 18058 \beta_{11} + 17929 \beta_{10} - 5242 \beta_{9} + 396 \beta_{8} - 7120 \beta_{7} - 15845 \beta_{6} + 2354 \beta_{5} + \cdots - 7473 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 51058 \beta_{15} - 15215 \beta_{14} + 126139 \beta_{13} - 48520 \beta_{12} + 58871 \beta_{11} + 59106 \beta_{10} - 13264 \beta_{9} - 2769 \beta_{8} - 25673 \beta_{7} - 54176 \beta_{6} + \cdots - 24198 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 167860 \beta_{15} - 53769 \beta_{14} + 418395 \beta_{13} - 159874 \beta_{12} + 196018 \beta_{11} + 195119 \beta_{10} - 33961 \beta_{9} - 18334 \beta_{8} - 93348 \beta_{7} - 181236 \beta_{6} + \cdots - 85607 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 549791 \beta_{15} - 189315 \beta_{14} + 1377611 \beta_{13} - 518871 \beta_{12} + 642097 \beta_{11} + 642570 \beta_{10} - 81445 \beta_{9} - 90804 \beta_{8} - 325797 \beta_{7} + \cdots - 282370 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1807643 \beta_{15} - 653596 \beta_{14} + 4551335 \beta_{13} - 1699915 \beta_{12} + 2120493 \beta_{11} + 2116648 \beta_{10} - 189849 \beta_{9} - 374428 \beta_{8} - 1134451 \beta_{7} + \cdots - 960385 \) Copy content Toggle raw display

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
−2.06059
−1.45194
−1.28136
−1.23485
−0.981279
−0.624714
−0.136067
0.293003
0.630852
1.27770
1.52769
1.74152
1.84416
2.41044
2.75851
3.28691
1.00000 −3.06059 1.00000 1.00000 −3.06059 0.302669 1.00000 6.36721 1.00000
1.2 1.00000 −2.45194 1.00000 1.00000 −2.45194 1.53125 1.00000 3.01199 1.00000
1.3 1.00000 −2.28136 1.00000 1.00000 −2.28136 1.99437 1.00000 2.20462 1.00000
1.4 1.00000 −2.23485 1.00000 1.00000 −2.23485 −0.971978 1.00000 1.99454 1.00000
1.5 1.00000 −1.98128 1.00000 1.00000 −1.98128 −4.22080 1.00000 0.925466 1.00000
1.6 1.00000 −1.62471 1.00000 1.00000 −1.62471 −0.333284 1.00000 −0.360304 1.00000
1.7 1.00000 −1.13607 1.00000 1.00000 −1.13607 −1.31630 1.00000 −1.70935 1.00000
1.8 1.00000 −0.706997 1.00000 1.00000 −0.706997 −2.05365 1.00000 −2.50015 1.00000
1.9 1.00000 −0.369148 1.00000 1.00000 −0.369148 4.16284 1.00000 −2.86373 1.00000
1.10 1.00000 0.277700 1.00000 1.00000 0.277700 2.55233 1.00000 −2.92288 1.00000
1.11 1.00000 0.527686 1.00000 1.00000 0.527686 −1.45681 1.00000 −2.72155 1.00000
1.12 1.00000 0.741521 1.00000 1.00000 0.741521 −2.92320 1.00000 −2.45015 1.00000
1.13 1.00000 0.844163 1.00000 1.00000 0.844163 −0.532701 1.00000 −2.28739 1.00000
1.14 1.00000 1.41044 1.00000 1.00000 1.41044 −3.15104 1.00000 −1.01066 1.00000
1.15 1.00000 1.75851 1.00000 1.00000 1.75851 −0.115571 1.00000 0.0923708 1.00000
1.16 1.00000 2.28691 1.00000 1.00000 2.28691 −3.46813 1.00000 2.22998 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(601\) \(1\)

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 6010.2.a.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6010.2.a.c 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 8 T_{3}^{15} + 9 T_{3}^{14} - 79 T_{3}^{13} - 204 T_{3}^{12} + 206 T_{3}^{11} + 1015 T_{3}^{10} + 71 T_{3}^{9} - 2142 T_{3}^{8} - 864 T_{3}^{7} + 2183 T_{3}^{6} + 979 T_{3}^{5} - 1132 T_{3}^{4} - 370 T_{3}^{3} + 268 T_{3}^{2} + \cdots - 19 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{15} + 9 T^{14} - 79 T^{13} + \cdots - 19 \) Copy content Toggle raw display
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 10 T^{15} + 6 T^{14} - 244 T^{13} + \cdots - 104 \) Copy content Toggle raw display
$11$ \( T^{16} + 14 T^{15} + 30 T^{14} + \cdots - 1810 \) Copy content Toggle raw display
$13$ \( T^{16} + 20 T^{15} + 105 T^{14} + \cdots - 131590 \) Copy content Toggle raw display
$17$ \( T^{16} + 27 T^{15} + \cdots - 123709702 \) Copy content Toggle raw display
$19$ \( T^{16} + 17 T^{15} + 23 T^{14} + \cdots + 152506 \) Copy content Toggle raw display
$23$ \( T^{16} + 9 T^{15} - 131 T^{14} + \cdots + 1723900 \) Copy content Toggle raw display
$29$ \( T^{16} + 23 T^{15} + 25 T^{14} + \cdots + 19604501 \) Copy content Toggle raw display
$31$ \( T^{16} + 21 T^{15} + \cdots - 331425593 \) Copy content Toggle raw display
$37$ \( T^{16} + 16 T^{15} + \cdots + 400525795984 \) Copy content Toggle raw display
$41$ \( T^{16} + 35 T^{15} + \cdots + 1159200242 \) Copy content Toggle raw display
$43$ \( T^{16} - 3 T^{15} + \cdots + 12020031088 \) Copy content Toggle raw display
$47$ \( T^{16} + 25 T^{15} + \cdots - 233890112 \) Copy content Toggle raw display
$53$ \( T^{16} + 39 T^{15} + \cdots - 527055664361 \) Copy content Toggle raw display
$59$ \( T^{16} + 32 T^{15} + \cdots + 1424918763584 \) Copy content Toggle raw display
$61$ \( T^{16} + 38 T^{15} + \cdots - 19460885020 \) Copy content Toggle raw display
$67$ \( T^{16} - 5 T^{15} + \cdots + 5064340186319 \) Copy content Toggle raw display
$71$ \( T^{16} + 16 T^{15} + \cdots + 14387146030835 \) Copy content Toggle raw display
$73$ \( T^{16} + 17 T^{15} + \cdots - 98804247364918 \) Copy content Toggle raw display
$79$ \( T^{16} + 40 T^{15} + \cdots + 1084834349107 \) Copy content Toggle raw display
$83$ \( T^{16} + 22 T^{15} + \cdots - 4218579966400 \) Copy content Toggle raw display
$89$ \( T^{16} + 46 T^{15} + \cdots - 1877257307863 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 239291495389240 \) Copy content Toggle raw display
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