Properties

Label 900.3.f.i
Level $900$
Weight $3$
Character orbit 900.f
Analytic conductor $24.523$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 12 x^{14} + 138 x^{12} - 1000 x^{10} + 5291 x^{8} - 17800 x^{6} + 39458 x^{4} - 53588 x^{2} + 32761\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + ( 2 - \beta_{5} ) q^{4} -\beta_{8} q^{7} + ( 2 \beta_{10} - \beta_{12} - \beta_{14} ) q^{8} +O(q^{10})\) \( q + \beta_{10} q^{2} + ( 2 - \beta_{5} ) q^{4} -\beta_{8} q^{7} + ( 2 \beta_{10} - \beta_{12} - \beta_{14} ) q^{8} -\beta_{6} q^{11} + ( -\beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{15} ) q^{13} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{14} + ( -1 - \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{16} + ( -4 \beta_{10} - 2 \beta_{13} + 3 \beta_{14} ) q^{17} + ( -1 - 2 \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{19} + ( -\beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{15} ) q^{22} + ( -5 \beta_{10} - \beta_{13} - \beta_{14} ) q^{23} + ( 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} ) q^{26} + ( -4 \beta_{8} + \beta_{9} + 6 \beta_{11} ) q^{28} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( 1 + 4 \beta_{4} - \beta_{5} - \beta_{7} ) q^{31} + ( -4 \beta_{10} - 3 \beta_{12} + 6 \beta_{13} - \beta_{14} ) q^{32} + ( 14 + 5 \beta_{4} + 7 \beta_{5} ) q^{34} + ( -\beta_{8} + 2 \beta_{9} + 17 \beta_{11} - \beta_{15} ) q^{37} + ( -4 \beta_{10} + 4 \beta_{12} + 8 \beta_{13} + 4 \beta_{14} ) q^{38} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{41} + ( 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{11} - 2 \beta_{15} ) q^{43} + ( 10 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{44} + ( -24 + 4 \beta_{5} ) q^{46} + ( 5 \beta_{10} - 8 \beta_{12} + \beta_{13} - 3 \beta_{14} ) q^{47} + ( 23 - 12 \beta_{5} + 4 \beta_{7} ) q^{49} + ( -2 \beta_{9} + 12 \beta_{11} - 4 \beta_{15} ) q^{52} + ( 15 \beta_{10} - 7 \beta_{13} - 4 \beta_{14} ) q^{53} + ( -8 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} - \beta_{6} ) q^{56} + ( 3 \beta_{8} + \beta_{9} - 5 \beta_{11} ) q^{58} + ( -6 \beta_{1} - 6 \beta_{2} + 18 \beta_{3} - 3 \beta_{6} ) q^{59} + ( 6 \beta_{5} - 2 \beta_{7} ) q^{61} + ( 4 \beta_{10} - 4 \beta_{12} - 12 \beta_{13} ) q^{62} + ( 21 - 7 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} ) q^{64} + ( -6 \beta_{8} - 8 \beta_{9} - 4 \beta_{11} - 4 \beta_{15} ) q^{67} + ( 14 \beta_{10} + 2 \beta_{12} - 10 \beta_{13} + 12 \beta_{14} ) q^{68} + ( -2 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{6} ) q^{71} + ( 2 \beta_{8} - 4 \beta_{9} + 15 \beta_{11} + 2 \beta_{15} ) q^{73} + ( 4 \beta_{1} + 2 \beta_{2} - 18 \beta_{3} - 2 \beta_{6} ) q^{74} + ( 44 - 4 \beta_{4} + 8 \beta_{5} - 4 \beta_{7} ) q^{76} + ( -4 \beta_{10} + 16 \beta_{13} - 6 \beta_{14} ) q^{77} + ( -3 + 4 \beta_{4} + 19 \beta_{5} + 3 \beta_{7} ) q^{79} + ( -4 \beta_{8} - 4 \beta_{9} + 12 \beta_{11} ) q^{82} + ( -40 \beta_{10} + 4 \beta_{12} - 8 \beta_{13} - 6 \beta_{14} ) q^{83} + ( 16 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{86} + ( 4 \beta_{8} - 7 \beta_{9} + 38 \beta_{11} - 2 \beta_{15} ) q^{88} + ( -26 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -2 - 16 \beta_{4} - 6 \beta_{5} + 2 \beta_{7} ) q^{91} + ( -24 \beta_{10} + 4 \beta_{12} + 4 \beta_{14} ) q^{92} + ( 24 - 4 \beta_{4} - 8 \beta_{5} + 8 \beta_{7} ) q^{94} + ( -6 \beta_{8} + 12 \beta_{9} + 15 \beta_{11} - 6 \beta_{15} ) q^{97} + ( 11 \beta_{10} - 16 \beta_{12} + 16 \beta_{13} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 28q^{4} + O(q^{10}) \) \( 16q + 28q^{4} - 28q^{16} + 232q^{34} - 368q^{46} + 304q^{49} + 32q^{61} + 364q^{64} + 768q^{76} + 336q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 12 x^{14} + 138 x^{12} - 1000 x^{10} + 5291 x^{8} - 17800 x^{6} + 39458 x^{4} - 53588 x^{2} + 32761\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 1564 \nu^{14} - 26035 \nu^{12} + 272717 \nu^{10} - 2294806 \nu^{8} + 12106617 \nu^{6} - 45019766 \nu^{4} + 87520785 \nu^{2} - 92455767 \)\()/6203680\)
\(\beta_{2}\)\(=\)\((\)\( -1517 \nu^{14} + 11182 \nu^{12} - 145387 \nu^{10} + 769817 \nu^{8} - 3409951 \nu^{6} + 6222189 \nu^{4} - 11683666 \nu^{2} + 7138059 \)\()/1772480\)
\(\beta_{3}\)\(=\)\((\)\( 12301 \nu^{14} - 134882 \nu^{12} + 1585239 \nu^{10} - 10860033 \nu^{8} + 56040923 \nu^{6} - 173399285 \nu^{4} + 344417086 \nu^{2} - 337443471 \)\()/12407360\)
\(\beta_{4}\)\(=\)\((\)\( 159 \nu^{14} - 1702 \nu^{12} + 19133 \nu^{10} - 128595 \nu^{8} + 614137 \nu^{6} - 1661727 \nu^{4} + 2826986 \nu^{2} - 2555965 \)\()/129920\)
\(\beta_{5}\)\(=\)\((\)\( -279 \nu^{14} + 2818 \nu^{12} - 32761 \nu^{10} + 213667 \nu^{8} - 1032517 \nu^{6} + 2797615 \nu^{4} - 4798574 \nu^{2} + 4194529 \)\()/129920\)
\(\beta_{6}\)\(=\)\((\)\( -4499 \nu^{14} + 45515 \nu^{12} - 538412 \nu^{10} + 3521601 \nu^{8} - 17610972 \nu^{6} + 49688181 \nu^{4} - 97604685 \nu^{2} + 90006842 \)\()/886240\)
\(\beta_{7}\)\(=\)\((\)\( -753 \nu^{14} + 7754 \nu^{12} - 90611 \nu^{10} + 594605 \nu^{8} - 2934199 \nu^{6} + 8031169 \nu^{4} - 13936662 \nu^{2} + 10110515 \)\()/129920\)
\(\beta_{8}\)\(=\)\((\)\( -76 \nu^{15} + 1455 \nu^{13} - 15013 \nu^{11} + 125594 \nu^{9} - 702033 \nu^{7} + 2408754 \nu^{5} - 4496945 \nu^{3} + 4520663 \nu \)\()/839840\)
\(\beta_{9}\)\(=\)\((\)\( 5664 \nu^{15} - 34121 \nu^{13} + 492575 \nu^{11} - 2166718 \nu^{9} + 9649707 \nu^{7} - 8403134 \nu^{5} + 34028543 \nu^{3} - 1684841 \nu \)\()/11757760\)
\(\beta_{10}\)\(=\)\((\)\(2233184 \nu^{15} - 21899443 \nu^{13} + 256907341 \nu^{11} - 1641213002 \nu^{9} + 7874858657 \nu^{7} - 20447352810 \nu^{5} + 33883360909 \nu^{3} - 22596309219 \nu\)\()/ 4491464320 \)
\(\beta_{11}\)\(=\)\((\)\( -2981 \nu^{15} + 30342 \nu^{13} - 360879 \nu^{11} + 2364333 \nu^{9} - 11922963 \nu^{7} + 34130105 \nu^{5} - 66225366 \nu^{3} + 58773711 \nu \)\()/5878880\)
\(\beta_{12}\)\(=\)\((\)\( 85762 \nu^{15} - 867511 \nu^{13} + 10192219 \nu^{11} - 65146100 \nu^{9} + 319614791 \nu^{7} - 796307516 \nu^{5} + 1201016993 \nu^{3} - 195043225 \nu \)\()/ 154878080 \)
\(\beta_{13}\)\(=\)\((\)\(-4949768 \nu^{15} + 50039697 \nu^{13} - 580754295 \nu^{11} + 3781602326 \nu^{9} - 18269059059 \nu^{7} + 48923742998 \nu^{5} - 85332242031 \nu^{3} + 74953045257 \nu\)\()/ 4491464320 \)
\(\beta_{14}\)\(=\)\((\)\(-3371926 \nu^{15} + 33490087 \nu^{13} - 394763119 \nu^{11} + 2544388208 \nu^{9} - 12429430523 \nu^{7} + 33485052920 \nu^{5} - 59181450721 \nu^{3} + 54030795461 \nu\)\()/ 2245732160 \)
\(\beta_{15}\)\(=\)\((\)\( -9430 \nu^{15} + 107187 \nu^{13} - 1220071 \nu^{11} + 8527534 \nu^{9} - 42962735 \nu^{7} + 130623386 \nu^{5} - 242923901 \nu^{3} + 226052383 \nu \)\()/2939440\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14} - \beta_{11} + 2 \beta_{10}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} - 5 \beta_{11} - 8 \beta_{10} + 3 \beta_{8}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{7} - 2 \beta_{6} + 16 \beta_{5} + 4 \beta_{4} - 13 \beta_{3} - 7 \beta_{2} + 12 \beta_{1} - 70\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-10 \beta_{15} - 15 \beta_{14} + 4 \beta_{13} + 18 \beta_{12} + 74 \beta_{11} - 64 \beta_{10} + 20 \beta_{9} + 5 \beta_{8}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-\beta_{7} - 16 \beta_{6} - 19 \beta_{5} - 38 \beta_{4} - 61 \beta_{3} + 27 \beta_{2} + 9 \beta_{1} - 65\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-42 \beta_{15} - 119 \beta_{14} + 258 \beta_{13} - 84 \beta_{12} + 477 \beta_{11} + 340 \beta_{10} + 140 \beta_{9} - 252 \beta_{8}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(192 \beta_{7} + 14 \beta_{6} - 632 \beta_{5} - 56 \beta_{4} + 597 \beta_{3} + 543 \beta_{2} - 920 \beta_{1} + 3278\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(570 \beta_{15} + 966 \beta_{14} - 772 \beta_{13} - 810 \beta_{12} - 3781 \beta_{11} + 2550 \beta_{10} - 804 \beta_{9} - 873 \beta_{8}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-266 \beta_{7} + 1896 \beta_{6} + 3192 \beta_{5} + 4602 \beta_{4} + 8005 \beta_{3} - 2441 \beta_{2} - 2854 \beta_{1} + 632\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(3190 \beta_{15} + 5751 \beta_{14} - 15672 \beta_{13} + 6458 \beta_{12} - 31096 \beta_{11} - 24816 \beta_{10} - 8756 \beta_{9} + 13145 \beta_{8}\)\()/4\)
\(\nu^{12}\)\(=\)\(-3165 \beta_{7} + 244 \beta_{6} + 9437 \beta_{5} - 122 \beta_{4} - 6613 \beta_{3} - 7989 \beta_{2} + 12013 \beta_{1} - 54202\)
\(\nu^{13}\)\(=\)\((\)\(-29380 \beta_{15} - 67131 \beta_{14} + 57268 \beta_{13} + 51256 \beta_{12} + 191671 \beta_{11} - 154818 \beta_{10} + 39000 \beta_{9} + 54600 \beta_{8}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(17696 \beta_{7} - 103684 \beta_{6} - 212746 \beta_{5} - 301226 \beta_{4} - 446003 \beta_{3} + 126875 \beta_{2} + 169616 \beta_{1} - 17522\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-179644 \beta_{15} - 378264 \beta_{14} + 1015030 \beta_{13} - 404910 \beta_{12} + 1739431 \beta_{11} + 1575428 \beta_{10} + 488312 \beta_{9} - 729481 \beta_{8}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−1.97048 + 1.96040i
−1.97048 + 0.960405i
−1.97048 1.96040i
−1.97048 0.960405i
−1.36646 + 0.842371i
−1.36646 0.157629i
−1.36646 0.842371i
−1.36646 + 0.157629i
1.36646 0.157629i
1.36646 + 0.842371i
1.36646 + 0.157629i
1.36646 0.842371i
1.97048 + 0.960405i
1.97048 + 1.96040i
1.97048 0.960405i
1.97048 1.96040i
−1.97048 0.342371i 0 3.76556 + 1.34927i 0 0 −11.5108 −6.95801 3.94792i 0 0
199.2 −1.97048 0.342371i 0 3.76556 + 1.34927i 0 0 11.5108 −6.95801 3.94792i 0 0
199.3 −1.97048 + 0.342371i 0 3.76556 1.34927i 0 0 −11.5108 −6.95801 + 3.94792i 0 0
199.4 −1.97048 + 0.342371i 0 3.76556 1.34927i 0 0 11.5108 −6.95801 + 3.94792i 0 0
199.5 −1.36646 1.46040i 0 −0.265564 + 3.99117i 0 0 −1.87135 6.19161 5.06596i 0 0
199.6 −1.36646 1.46040i 0 −0.265564 + 3.99117i 0 0 1.87135 6.19161 5.06596i 0 0
199.7 −1.36646 + 1.46040i 0 −0.265564 3.99117i 0 0 −1.87135 6.19161 + 5.06596i 0 0
199.8 −1.36646 + 1.46040i 0 −0.265564 3.99117i 0 0 1.87135 6.19161 + 5.06596i 0 0
199.9 1.36646 1.46040i 0 −0.265564 3.99117i 0 0 −1.87135 −6.19161 5.06596i 0 0
199.10 1.36646 1.46040i 0 −0.265564 3.99117i 0 0 1.87135 −6.19161 5.06596i 0 0
199.11 1.36646 + 1.46040i 0 −0.265564 + 3.99117i 0 0 −1.87135 −6.19161 + 5.06596i 0 0
199.12 1.36646 + 1.46040i 0 −0.265564 + 3.99117i 0 0 1.87135 −6.19161 + 5.06596i 0 0
199.13 1.97048 0.342371i 0 3.76556 1.34927i 0 0 −11.5108 6.95801 3.94792i 0 0
199.14 1.97048 0.342371i 0 3.76556 1.34927i 0 0 11.5108 6.95801 3.94792i 0 0
199.15 1.97048 + 0.342371i 0 3.76556 + 1.34927i 0 0 −11.5108 6.95801 + 3.94792i 0 0
199.16 1.97048 + 0.342371i 0 3.76556 + 1.34927i 0 0 11.5108 6.95801 + 3.94792i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.f.i 16
3.b odd 2 1 inner 900.3.f.i 16
4.b odd 2 1 inner 900.3.f.i 16
5.b even 2 1 inner 900.3.f.i 16
5.c odd 4 1 180.3.c.c 8
5.c odd 4 1 900.3.c.o 8
12.b even 2 1 inner 900.3.f.i 16
15.d odd 2 1 inner 900.3.f.i 16
15.e even 4 1 180.3.c.c 8
15.e even 4 1 900.3.c.o 8
20.d odd 2 1 inner 900.3.f.i 16
20.e even 4 1 180.3.c.c 8
20.e even 4 1 900.3.c.o 8
40.i odd 4 1 2880.3.e.i 8
40.k even 4 1 2880.3.e.i 8
60.h even 2 1 inner 900.3.f.i 16
60.l odd 4 1 180.3.c.c 8
60.l odd 4 1 900.3.c.o 8
120.q odd 4 1 2880.3.e.i 8
120.w even 4 1 2880.3.e.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.c.c 8 5.c odd 4 1
180.3.c.c 8 15.e even 4 1
180.3.c.c 8 20.e even 4 1
180.3.c.c 8 60.l odd 4 1
900.3.c.o 8 5.c odd 4 1
900.3.c.o 8 15.e even 4 1
900.3.c.o 8 20.e even 4 1
900.3.c.o 8 60.l odd 4 1
900.3.f.i 16 1.a even 1 1 trivial
900.3.f.i 16 3.b odd 2 1 inner
900.3.f.i 16 4.b odd 2 1 inner
900.3.f.i 16 5.b even 2 1 inner
900.3.f.i 16 12.b even 2 1 inner
900.3.f.i 16 15.d odd 2 1 inner
900.3.f.i 16 20.d odd 2 1 inner
900.3.f.i 16 60.h even 2 1 inner
2880.3.e.i 8 40.i odd 4 1
2880.3.e.i 8 40.k even 4 1
2880.3.e.i 8 120.q odd 4 1
2880.3.e.i 8 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{4} - 136 T_{7}^{2} + 464 \)
\( T_{29}^{4} - 456 T_{29}^{2} + 35344 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 256 - 112 T^{2} + 28 T^{4} - 7 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( 464 - 136 T^{2} + T^{4} )^{4} \)
$11$ \( ( 22736 + 328 T^{2} + T^{4} )^{4} \)
$13$ \( ( 65536 + 528 T^{2} + T^{4} )^{4} \)
$17$ \( ( 150544 + 1096 T^{2} + T^{4} )^{4} \)
$19$ \( ( 118784 + 944 T^{2} + T^{4} )^{4} \)
$23$ \( ( 29696 - 368 T^{2} + T^{4} )^{4} \)
$29$ \( ( 35344 - 456 T^{2} + T^{4} )^{4} \)
$31$ \( ( 742400 + 1840 T^{2} + T^{4} )^{4} \)
$37$ \( ( 802816 + 2832 T^{2} + T^{4} )^{4} \)
$41$ \( ( -832 + T^{2} )^{8} \)
$43$ \( ( 1900544 - 3776 T^{2} + T^{4} )^{4} \)
$47$ \( ( 7602176 - 5552 T^{2} + T^{4} )^{4} \)
$53$ \( ( 21904 + 3624 T^{2} + T^{4} )^{4} \)
$59$ \( ( 51452496 + 16488 T^{2} + T^{4} )^{4} \)
$61$ \( ( -1036 - 4 T + T^{2} )^{8} \)
$67$ \( ( 75732224 - 21664 T^{2} + T^{4} )^{4} \)
$71$ \( ( 363776 + 2848 T^{2} + T^{4} )^{4} \)
$73$ \( ( 19600 + 3880 T^{2} + T^{4} )^{4} \)
$79$ \( ( 65598464 + 16816 T^{2} + T^{4} )^{4} \)
$83$ \( ( 69852416 - 24608 T^{2} + T^{4} )^{4} \)
$89$ \( ( 160985344 - 28704 T^{2} + T^{4} )^{4} \)
$97$ \( ( 71571600 + 20520 T^{2} + T^{4} )^{4} \)
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