Properties

Label 1.32.a.a
Level 1
Weight 32
Character orbit 1.a
Self dual yes
Analytic conductor 6.088
Analytic rank 0
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.08771328190\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{18295489}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 19980 - \beta ) q^{2} + ( 8681580 - 432 \beta ) q^{3} + ( 886267168 - 39960 \beta ) q^{4} + ( -9695609010 + 1418560 \beta ) q^{5} + ( 1311583748112 - 17312940 \beta ) q^{6} + ( 15128763788600 + 71928864 \beta ) q^{7} + ( 80077529352960 + 462815680 \beta ) q^{8} + ( -50633228151963 - 7500885120 \beta ) q^{9} +O(q^{10})\) \( q +(19980 - \beta) q^{2} +(8681580 - 432 \beta) q^{3} +(886267168 - 39960 \beta) q^{4} +(-9695609010 + 1418560 \beta) q^{5} +(1311583748112 - 17312940 \beta) q^{6} +(15128763788600 + 71928864 \beta) q^{7} +(80077529352960 + 462815680 \beta) q^{8} +(-50633228151963 - 7500885120 \beta) q^{9} +(-3930986106140760 + 38038437810 \beta) q^{10} +(-3891176872559388 - 29000909200 \beta) q^{11} +(53173705477656960 - 729783353376 \beta) q^{12} +(37354476525130310 + 4661016429888 \beta) q^{13} +(112772481922620576 - 13691625085880 \beta) q^{14} +(-1698672911337290520 + 16503845217120 \beta) q^{15} +(-1522606456842450944 + 14982974507520 \beta) q^{16} +(8612303914493544690 - 45085348093056 \beta) q^{17} +(18749808114787989180 - 99234456545637 \beta) q^{18} +(-6185281664011082020 + 157805792764560 \beta) q^{19} +(-\)\(15\!\cdots\!80\)\( + 1644659689877680 \beta) q^{20} +(49477478708035580832 - 5911169769550080 \beta) q^{21} +(-1341356516498345040 + 3311738706743388 \beta) q^{22} +(\)\(94\!\cdots\!40\)\( + 18557618179251808 \beta) q^{23} +(\)\(16\!\cdots\!40\)\( - 30575521329304320 \beta) q^{24} +(\)\(73\!\cdots\!75\)\( - 27507606234451200 \beta) q^{25} +(-\)\(11\!\cdots\!08\)\( + 55772631744031930 \beta) q^{26} +(\)\(27\!\cdots\!40\)\( + 223588927516223520 \beta) q^{27} +(\)\(58\!\cdots\!60\)\( - 540797210397718848 \beta) q^{28} +(\)\(64\!\cdots\!70\)\( - 234192357384448960 \beta) q^{29} +(-\)\(77\!\cdots\!20\)\( + 2028419738775348120 \beta) q^{30} +(\)\(62\!\cdots\!32\)\( - 2412621171020361600 \beta) q^{31} +(-\)\(24\!\cdots\!20\)\( + 828077182664699904 \beta) q^{32} +(-\)\(77\!\cdots\!40\)\( + 1429214695653119616 \beta) q^{33} +(\)\(29\!\cdots\!96\)\( - 9513109169392803570 \beta) q^{34} +(\)\(12\!\cdots\!40\)\( + 20763665018078951360 \beta) q^{35} +(\)\(74\!\cdots\!16\)\( - 4624484415843298680 \beta) q^{36} +(-\)\(41\!\cdots\!30\)\( - 32224244113578511296 \beta) q^{37} +(-\)\(53\!\cdots\!60\)\( + 9338241403446990820 \beta) q^{38} +(-\)\(49\!\cdots\!56\)\( + 24327853158530769120 \beta) q^{39} +(\)\(95\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( \beta) q^{40} +(\)\(43\!\cdots\!42\)\( - \)\(18\!\cdots\!00\)\( \beta) q^{41} +(\)\(16\!\cdots\!40\)\( - \)\(16\!\cdots\!32\)\( \beta) q^{42} +(-\)\(91\!\cdots\!00\)\( + \)\(34\!\cdots\!48\)\( \beta) q^{43} +(-\)\(39\!\cdots\!84\)\( + \)\(12\!\cdots\!80\)\( \beta) q^{44} +(-\)\(27\!\cdots\!70\)\( + 899377225198297920 \beta) q^{45} +(-\)\(29\!\cdots\!28\)\( - \)\(57\!\cdots\!00\)\( \beta) q^{46} +(\)\(47\!\cdots\!60\)\( - \)\(51\!\cdots\!16\)\( \beta) q^{47} +(-\)\(30\!\cdots\!60\)\( + \)\(78\!\cdots\!08\)\( \beta) q^{48} +(\)\(84\!\cdots\!93\)\( + \)\(21\!\cdots\!00\)\( \beta) q^{49} +(\)\(87\!\cdots\!00\)\( - \)\(12\!\cdots\!75\)\( \beta) q^{50} +(\)\(12\!\cdots\!72\)\( - \)\(41\!\cdots\!60\)\( \beta) q^{51} +(-\)\(45\!\cdots\!00\)\( + \)\(26\!\cdots\!84\)\( \beta) q^{52} +(\)\(97\!\cdots\!30\)\( + \)\(12\!\cdots\!68\)\( \beta) q^{53} +(-\)\(53\!\cdots\!20\)\( + \)\(17\!\cdots\!60\)\( \beta) q^{54} +(-\)\(70\!\cdots\!20\)\( - \)\(52\!\cdots\!80\)\( \beta) q^{55} +(\)\(12\!\cdots\!20\)\( + \)\(12\!\cdots\!40\)\( \beta) q^{56} +(-\)\(23\!\cdots\!20\)\( + \)\(40\!\cdots\!40\)\( \beta) q^{57} +(\)\(19\!\cdots\!60\)\( - \)\(68\!\cdots\!70\)\( \beta) q^{58} +(-\)\(99\!\cdots\!60\)\( + \)\(44\!\cdots\!80\)\( \beta) q^{59} +(-\)\(32\!\cdots\!60\)\( + \)\(82\!\cdots\!60\)\( \beta) q^{60} +(-\)\(60\!\cdots\!38\)\( - \)\(11\!\cdots\!00\)\( \beta) q^{61} +(\)\(76\!\cdots\!60\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{62} +(-\)\(21\!\cdots\!80\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{63} +(-\)\(37\!\cdots\!52\)\( + \)\(22\!\cdots\!80\)\( \beta) q^{64} +(\)\(17\!\cdots\!80\)\( + \)\(77\!\cdots\!20\)\( \beta) q^{65} +(-\)\(37\!\cdots\!56\)\( + \)\(29\!\cdots\!20\)\( \beta) q^{66} +(-\)\(48\!\cdots\!60\)\( + \)\(88\!\cdots\!44\)\( \beta) q^{67} +(\)\(12\!\cdots\!80\)\( - \)\(38\!\cdots\!08\)\( \beta) q^{68} +(-\)\(12\!\cdots\!96\)\( - \)\(24\!\cdots\!40\)\( \beta) q^{69} +(-\)\(52\!\cdots\!60\)\( + \)\(29\!\cdots\!60\)\( \beta) q^{70} +(\)\(27\!\cdots\!72\)\( + \)\(95\!\cdots\!00\)\( \beta) q^{71} +(-\)\(13\!\cdots\!80\)\( - \)\(62\!\cdots\!40\)\( \beta) q^{72} +(\)\(31\!\cdots\!90\)\( - \)\(15\!\cdots\!92\)\( \beta) q^{73} +(\)\(76\!\cdots\!36\)\( - \)\(22\!\cdots\!50\)\( \beta) q^{74} +(\)\(37\!\cdots\!00\)\( - \)\(55\!\cdots\!00\)\( \beta) q^{75} +(-\)\(22\!\cdots\!60\)\( + \)\(38\!\cdots\!80\)\( \beta) q^{76} +(-\)\(64\!\cdots\!00\)\( - \)\(71\!\cdots\!32\)\( \beta) q^{77} +(-\)\(16\!\cdots\!00\)\( + \)\(54\!\cdots\!56\)\( \beta) q^{78} +(-\)\(59\!\cdots\!80\)\( - \)\(46\!\cdots\!60\)\( \beta) q^{79} +(\)\(70\!\cdots\!40\)\( - \)\(23\!\cdots\!40\)\( \beta) q^{80} +(-\)\(19\!\cdots\!79\)\( + \)\(53\!\cdots\!60\)\( \beta) q^{81} +(\)\(57\!\cdots\!60\)\( - \)\(80\!\cdots\!42\)\( \beta) q^{82} +(-\)\(13\!\cdots\!80\)\( + \)\(62\!\cdots\!28\)\( \beta) q^{83} +(\)\(66\!\cdots\!76\)\( - \)\(72\!\cdots\!60\)\( \beta) q^{84} +(-\)\(25\!\cdots\!60\)\( + \)\(12\!\cdots\!60\)\( \beta) q^{85} +(-\)\(10\!\cdots\!68\)\( + \)\(16\!\cdots\!40\)\( \beta) q^{86} +(\)\(82\!\cdots\!20\)\( - \)\(29\!\cdots\!40\)\( \beta) q^{87} +(-\)\(34\!\cdots\!80\)\( - \)\(41\!\cdots\!40\)\( \beta) q^{88} +(-\)\(10\!\cdots\!90\)\( - \)\(26\!\cdots\!80\)\( \beta) q^{89} +(-\)\(55\!\cdots\!20\)\( + \)\(27\!\cdots\!70\)\( \beta) q^{90} +(\)\(14\!\cdots\!12\)\( + \)\(73\!\cdots\!40\)\( \beta) q^{91} +(-\)\(11\!\cdots\!60\)\( - \)\(21\!\cdots\!56\)\( \beta) q^{92} +(\)\(32\!\cdots\!60\)\( - \)\(48\!\cdots\!24\)\( \beta) q^{93} +(\)\(23\!\cdots\!56\)\( - \)\(57\!\cdots\!40\)\( \beta) q^{94} +(\)\(64\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( \beta) q^{95} +(-\)\(30\!\cdots\!48\)\( + \)\(11\!\cdots\!60\)\( \beta) q^{96} +(-\)\(45\!\cdots\!90\)\( + \)\(16\!\cdots\!84\)\( \beta) q^{97} +(-\)\(40\!\cdots\!60\)\( - \)\(41\!\cdots\!93\)\( \beta) q^{98} +(\)\(77\!\cdots\!44\)\( + \)\(30\!\cdots\!60\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 39960q^{2} + 17363160q^{3} + 1772534336q^{4} - 19391218020q^{5} + 2623167496224q^{6} + 30257527577200q^{7} + 160155058705920q^{8} - 101266456303926q^{9} + O(q^{10}) \) \( 2q + 39960q^{2} + 17363160q^{3} + 1772534336q^{4} - 19391218020q^{5} + 2623167496224q^{6} + 30257527577200q^{7} + 160155058705920q^{8} - 101266456303926q^{9} - 7861972212281520q^{10} - 7782353745118776q^{11} + 106347410955313920q^{12} + 74708953050260620q^{13} + 225544963845241152q^{14} - 3397345822674581040q^{15} - 3045212913684901888q^{16} + 17224607828987089380q^{17} + 37499616229575978360q^{18} - 12370563328022164040q^{19} - \)\(31\!\cdots\!60\)\(q^{20} + 98954957416071161664q^{21} - 2682713032996690080q^{22} + \)\(18\!\cdots\!80\)\(q^{23} + \)\(33\!\cdots\!80\)\(q^{24} + \)\(14\!\cdots\!50\)\(q^{25} - \)\(23\!\cdots\!16\)\(q^{26} + \)\(54\!\cdots\!80\)\(q^{27} + \)\(11\!\cdots\!20\)\(q^{28} + \)\(12\!\cdots\!40\)\(q^{29} - \)\(15\!\cdots\!40\)\(q^{30} + \)\(12\!\cdots\!64\)\(q^{31} - \)\(48\!\cdots\!40\)\(q^{32} - \)\(15\!\cdots\!80\)\(q^{33} + \)\(58\!\cdots\!92\)\(q^{34} + \)\(24\!\cdots\!80\)\(q^{35} + \)\(14\!\cdots\!32\)\(q^{36} - \)\(83\!\cdots\!60\)\(q^{37} - \)\(10\!\cdots\!20\)\(q^{38} - \)\(99\!\cdots\!12\)\(q^{39} + \)\(19\!\cdots\!00\)\(q^{40} + \)\(87\!\cdots\!84\)\(q^{41} + \)\(33\!\cdots\!80\)\(q^{42} - \)\(18\!\cdots\!00\)\(q^{43} - \)\(79\!\cdots\!68\)\(q^{44} - \)\(55\!\cdots\!40\)\(q^{45} - \)\(59\!\cdots\!56\)\(q^{46} + \)\(95\!\cdots\!20\)\(q^{47} - \)\(60\!\cdots\!20\)\(q^{48} + \)\(16\!\cdots\!86\)\(q^{49} + \)\(17\!\cdots\!00\)\(q^{50} + \)\(25\!\cdots\!44\)\(q^{51} - \)\(91\!\cdots\!00\)\(q^{52} + \)\(19\!\cdots\!60\)\(q^{53} - \)\(10\!\cdots\!40\)\(q^{54} - \)\(14\!\cdots\!40\)\(q^{55} + \)\(25\!\cdots\!40\)\(q^{56} - \)\(46\!\cdots\!40\)\(q^{57} + \)\(38\!\cdots\!20\)\(q^{58} - \)\(19\!\cdots\!20\)\(q^{59} - \)\(64\!\cdots\!20\)\(q^{60} - \)\(12\!\cdots\!76\)\(q^{61} + \)\(15\!\cdots\!20\)\(q^{62} - \)\(43\!\cdots\!60\)\(q^{63} - \)\(74\!\cdots\!04\)\(q^{64} + \)\(34\!\cdots\!60\)\(q^{65} - \)\(75\!\cdots\!12\)\(q^{66} - \)\(96\!\cdots\!20\)\(q^{67} + \)\(24\!\cdots\!60\)\(q^{68} - \)\(25\!\cdots\!92\)\(q^{69} - \)\(10\!\cdots\!20\)\(q^{70} + \)\(55\!\cdots\!44\)\(q^{71} - \)\(26\!\cdots\!60\)\(q^{72} + \)\(62\!\cdots\!80\)\(q^{73} + \)\(15\!\cdots\!72\)\(q^{74} + \)\(75\!\cdots\!00\)\(q^{75} - \)\(44\!\cdots\!20\)\(q^{76} - \)\(12\!\cdots\!00\)\(q^{77} - \)\(32\!\cdots\!00\)\(q^{78} - \)\(11\!\cdots\!60\)\(q^{79} + \)\(14\!\cdots\!80\)\(q^{80} - \)\(39\!\cdots\!58\)\(q^{81} + \)\(11\!\cdots\!20\)\(q^{82} - \)\(26\!\cdots\!60\)\(q^{83} + \)\(13\!\cdots\!52\)\(q^{84} - \)\(50\!\cdots\!20\)\(q^{85} - \)\(21\!\cdots\!36\)\(q^{86} + \)\(16\!\cdots\!40\)\(q^{87} - \)\(69\!\cdots\!60\)\(q^{88} - \)\(21\!\cdots\!80\)\(q^{89} - \)\(11\!\cdots\!40\)\(q^{90} + \)\(28\!\cdots\!24\)\(q^{91} - \)\(22\!\cdots\!20\)\(q^{92} + \)\(65\!\cdots\!20\)\(q^{93} + \)\(46\!\cdots\!12\)\(q^{94} + \)\(12\!\cdots\!00\)\(q^{95} - \)\(60\!\cdots\!96\)\(q^{96} - \)\(90\!\cdots\!80\)\(q^{97} - \)\(80\!\cdots\!20\)\(q^{98} + \)\(15\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2139.16
−2138.16
−31347.9 −1.34921e7 −1.16479e9 6.31161e10 4.22947e11 1.88207e13 1.03833e14 −4.35638e14 −1.97855e15
1.2 71307.9 3.08552e7 2.93733e9 −8.25073e10 2.20022e12 1.14368e13 5.63222e13 3.34371e14 −5.88342e15
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.32.a.a 2
3.b odd 2 1 9.32.a.a 2
4.b odd 2 1 16.32.a.b 2
5.b even 2 1 25.32.a.a 2
5.c odd 4 2 25.32.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.32.a.a 2 1.a even 1 1 trivial
9.32.a.a 2 3.b odd 2 1
16.32.a.b 2 4.b odd 2 1
25.32.a.a 2 5.b even 2 1
25.32.b.a 4 5.c odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{32}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 39960 T + 2059617280 T^{2} - 85813446574080 T^{3} + 4611686018427387904 T^{4} \)
$3$ \( 1 - 17363160 T + 819046287028710 T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(38\!\cdots\!09\)\( T^{4} \)
$5$ \( 1 + 19391218020 T + \)\(41\!\cdots\!50\)\( T^{2} + \)\(90\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 30257527577200 T + \)\(53\!\cdots\!50\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + 7782353745118776 T + \)\(39\!\cdots\!66\)\( T^{2} + \)\(14\!\cdots\!36\)\( T^{3} + \)\(36\!\cdots\!21\)\( T^{4} \)
$13$ \( 1 - 74708953050260620 T + \)\(12\!\cdots\!70\)\( T^{2} - \)\(25\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!69\)\( T^{4} \)
$17$ \( 1 - 17224607828987089380 T + \)\(34\!\cdots\!90\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!89\)\( T^{4} \)
$19$ \( 1 + 12370563328022164040 T + \)\(87\!\cdots\!38\)\( T^{2} + \)\(54\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!61\)\( T^{4} \)
$23$ \( 1 - \)\(18\!\cdots\!80\)\( T + \)\(32\!\cdots\!30\)\( T^{2} - \)\(31\!\cdots\!60\)\( T^{3} + \)\(26\!\cdots\!29\)\( T^{4} \)
$29$ \( 1 - \)\(12\!\cdots\!40\)\( T + \)\(83\!\cdots\!58\)\( T^{2} - \)\(27\!\cdots\!60\)\( T^{3} + \)\(46\!\cdots\!41\)\( T^{4} \)
$31$ \( 1 - \)\(12\!\cdots\!64\)\( T + \)\(22\!\cdots\!86\)\( T^{2} - \)\(21\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!61\)\( T^{4} \)
$37$ \( 1 + \)\(83\!\cdots\!60\)\( T + \)\(56\!\cdots\!70\)\( T^{2} + \)\(34\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!69\)\( T^{4} \)
$41$ \( 1 - \)\(87\!\cdots\!84\)\( T + \)\(12\!\cdots\!46\)\( T^{2} - \)\(86\!\cdots\!44\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \)
$43$ \( 1 + \)\(18\!\cdots\!00\)\( T + \)\(64\!\cdots\!50\)\( T^{2} + \)\(79\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - \)\(95\!\cdots\!20\)\( T + \)\(15\!\cdots\!10\)\( T^{2} - \)\(65\!\cdots\!60\)\( T^{3} + \)\(46\!\cdots\!09\)\( T^{4} \)
$53$ \( 1 - \)\(19\!\cdots\!60\)\( T + \)\(57\!\cdots\!10\)\( T^{2} - \)\(55\!\cdots\!20\)\( T^{3} + \)\(80\!\cdots\!09\)\( T^{4} \)
$59$ \( 1 + \)\(19\!\cdots\!20\)\( T + \)\(10\!\cdots\!18\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(62\!\cdots\!81\)\( T^{4} \)
$61$ \( 1 + \)\(12\!\cdots\!76\)\( T + \)\(80\!\cdots\!66\)\( T^{2} + \)\(26\!\cdots\!36\)\( T^{3} + \)\(49\!\cdots\!21\)\( T^{4} \)
$67$ \( 1 + \)\(96\!\cdots\!20\)\( T + \)\(81\!\cdots\!90\)\( T^{2} + \)\(39\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \)
$71$ \( 1 - \)\(55\!\cdots\!44\)\( T + \)\(32\!\cdots\!26\)\( T^{2} - \)\(13\!\cdots\!24\)\( T^{3} + \)\(59\!\cdots\!41\)\( T^{4} \)
$73$ \( 1 - \)\(62\!\cdots\!80\)\( T + \)\(12\!\cdots\!30\)\( T^{2} - \)\(36\!\cdots\!60\)\( T^{3} + \)\(33\!\cdots\!29\)\( T^{4} \)
$79$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(80\!\cdots\!58\)\( T^{2} + \)\(79\!\cdots\!40\)\( T^{3} + \)\(44\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 + \)\(26\!\cdots\!60\)\( T + \)\(53\!\cdots\!90\)\( T^{2} + \)\(82\!\cdots\!20\)\( T^{3} + \)\(96\!\cdots\!89\)\( T^{4} \)
$89$ \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(47\!\cdots\!78\)\( T^{2} + \)\(58\!\cdots\!20\)\( T^{3} + \)\(72\!\cdots\!21\)\( T^{4} \)
$97$ \( 1 + \)\(90\!\cdots\!80\)\( T + \)\(97\!\cdots\!10\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!09\)\( T^{4} \)
show more
show less