Properties

Label 675.2.k.b
Level $675$
Weight $2$
Character orbit 675.k
Analytic conductor $5.390$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + ( - \beta_{9} + 2 \beta_{6}) q^{4} + (\beta_{8} + \beta_{7} - \beta_{3}) q^{7} + (\beta_{10} + \beta_{8} - \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + ( - \beta_{9} + 2 \beta_{6}) q^{4} + (\beta_{8} + \beta_{7} - \beta_{3}) q^{7} + (\beta_{10} + \beta_{8} - \beta_{4}) q^{8} + ( - \beta_{9} + \beta_{6} - \beta_{2} - 1) q^{11} + (\beta_{5} - \beta_{3}) q^{13} + ( - \beta_{11} - \beta_{9} + 3 \beta_{6}) q^{14} + (\beta_{11} + 2 \beta_{6} + \beta_1 - 2) q^{16} + \beta_{10} q^{17} + (\beta_{2} - 1) q^{19} + ( - \beta_{5} - 2 \beta_{4}) q^{22} + (2 \beta_{5} - \beta_{4}) q^{23} + (\beta_{2} + 1) q^{26} + ( - \beta_{10} + 3 \beta_{8} + \beta_{7} - 3 \beta_{4}) q^{28} + ( - \beta_{11} - 2 \beta_{9} - 2 \beta_{6} - 2 \beta_{2} - \beta_1 + 2) q^{29} + ( - 2 \beta_{11} - \beta_{9} - 3 \beta_{6}) q^{31} + (\beta_{4} - 3 \beta_{3}) q^{32} + (\beta_{11} - \beta_{9} + 2 \beta_{6} - \beta_{2} + \beta_1 - 2) q^{34} + (\beta_{10} - 2 \beta_{8} - \beta_{7} + 2 \beta_{4}) q^{37} + (\beta_{10} + \beta_{5}) q^{38} + ( - \beta_{11} + \beta_{9} - 5 \beta_{6}) q^{41} + (\beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{3}) q^{43} + ( - \beta_{2} + \beta_1 - 8) q^{44} + (\beta_{2} - 2 \beta_1) q^{46} + ( - \beta_{10} - 3 \beta_{8} - 3 \beta_{7} - \beta_{5} + 3 \beta_{3}) q^{47} + ( - 2 \beta_{11} - \beta_{9} - \beta_{6}) q^{49} + ( - \beta_{10} + 2 \beta_{8} - 2 \beta_{7} - \beta_{5} + 2 \beta_{3}) q^{52} + ( - 2 \beta_{8} + 2 \beta_{4}) q^{53} + (3 \beta_{6} - 3) q^{56} + ( - \beta_{4} + 3 \beta_{3}) q^{58} + (\beta_{11} + \beta_{6}) q^{59} + (\beta_{11} - \beta_{9} + \beta_{6} - \beta_{2} + \beta_1 - 1) q^{61} + ( - 3 \beta_{10} + 6 \beta_{7}) q^{62} + (\beta_{2} + \beta_1 + 5) q^{64} + (3 \beta_{5} - \beta_{4} + \beta_{3}) q^{67} + ( - \beta_{5} - 2 \beta_{4} - 3 \beta_{3}) q^{68} + ( - 3 \beta_{2} + \beta_1 + 2) q^{71} + ( - 4 \beta_{8} - 2 \beta_{7} + 4 \beta_{4}) q^{73} + (2 \beta_{11} + \beta_{9} - 5 \beta_{6} + \beta_{2} + 2 \beta_1 + 5) q^{74} + (\beta_{11} + \beta_{9} + 4 \beta_{6}) q^{76} + (\beta_{5} - 2 \beta_{4}) q^{77} + (2 \beta_{11} + 2 \beta_1) q^{79} + ( - 3 \beta_{10} - 5 \beta_{8} + 3 \beta_{7} + 5 \beta_{4}) q^{82} + ( - 3 \beta_{8} - 3 \beta_{7} + 3 \beta_{3}) q^{83} + (2 \beta_{11} - 3 \beta_{9} + 11 \beta_{6}) q^{86} + ( - \beta_{10} - 4 \beta_{8} + 3 \beta_{7} - \beta_{5} - 3 \beta_{3}) q^{88} - 3 q^{89} + ( - \beta_{2} + 3) q^{91} + (\beta_{10} + \beta_{8} - 6 \beta_{7} + \beta_{5} + 6 \beta_{3}) q^{92} + (2 \beta_{11} + 4 \beta_{9} - 11 \beta_{6}) q^{94} + (4 \beta_{10} - 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{5} + 2 \beta_{3}) q^{97} + ( - 3 \beta_{10} + 2 \beta_{8} + 6 \beta_{7} - 2 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{4} - 4 q^{11} + 18 q^{14} - 10 q^{16} - 16 q^{19} + 8 q^{26} + 14 q^{29} - 16 q^{31} - 8 q^{34} - 26 q^{41} - 88 q^{44} - 12 q^{46} - 4 q^{49} - 18 q^{56} + 4 q^{59} - 2 q^{61} + 60 q^{64} + 40 q^{71} + 32 q^{74} + 24 q^{76} + 4 q^{79} + 56 q^{86} - 36 q^{89} + 40 q^{91} - 62 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 473 \nu^{11} - 516 \nu^{10} + 1118 \nu^{9} - 3014 \nu^{8} - 4564 \nu^{7} - 5676 \nu^{6} - 2470 \nu^{5} + 66640 \nu^{4} + 113100 \nu^{3} + 96048 \nu^{2} + 54216 \nu - 194196 ) / 51972 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 748 \nu^{11} + 816 \nu^{10} - 1768 \nu^{9} + 4162 \nu^{8} + 6311 \nu^{7} + 8976 \nu^{6} + 7532 \nu^{5} - 93902 \nu^{4} - 164352 \nu^{3} - 141012 \nu^{2} - 80298 \nu + 98004 ) / 51972 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 962 \nu^{11} + 2392 \nu^{10} - 2887 \nu^{9} + 2220 \nu^{8} + 13990 \nu^{7} - 14442 \nu^{6} - 11380 \nu^{5} - 57708 \nu^{4} - 98118 \nu^{3} + 132080 \nu^{2} + \cdots - 15984 ) / 51972 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 962 \nu^{11} + 2392 \nu^{10} - 2887 \nu^{9} + 2220 \nu^{8} + 13990 \nu^{7} - 14442 \nu^{6} - 11380 \nu^{5} - 57708 \nu^{4} - 98118 \nu^{3} + 97432 \nu^{2} - 49284 \nu - 15984 ) / 34648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2327 \nu^{11} + 5669 \nu^{10} - 7159 \nu^{9} + 5370 \nu^{8} + 34543 \nu^{7} - 32710 \nu^{6} - 24718 \nu^{5} - 137484 \nu^{4} - 236286 \nu^{3} + 157488 \nu^{2} + \cdots - 38664 ) / 51972 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 118 \nu^{11} - 90 \nu^{10} + 53 \nu^{9} - 32 \nu^{8} - 1866 \nu^{7} - 1416 \nu^{6} + 1364 \nu^{5} + 10408 \nu^{4} + 27758 \nu^{3} + 22776 \nu^{2} + 12468 \nu + 5172 ) / 1704 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1688 \nu^{11} + 1054 \nu^{10} - 840 \nu^{9} + 684 \nu^{8} + 26378 \nu^{7} + 24587 \nu^{6} - 12648 \nu^{5} - 151968 \nu^{4} - 420176 \nu^{3} - 400856 \nu^{2} + \cdots - 105120 ) / 12993 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14137 \nu^{11} - 3520 \nu^{10} + 77 \nu^{9} - 84 \nu^{8} - 226010 \nu^{7} - 282250 \nu^{6} + 85550 \nu^{5} + 1356228 \nu^{4} + 3952074 \nu^{3} + 4346412 \nu^{2} + \cdots + 1015848 ) / 103944 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17172 \nu^{11} + 12498 \nu^{10} - 4217 \nu^{9} - 3016 \nu^{8} + 280340 \nu^{7} + 206064 \nu^{6} - 219272 \nu^{5} - 1528612 \nu^{4} - 3992950 \nu^{3} + \cdots - 737076 ) / 103944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11258 \nu^{11} - 7163 \nu^{10} + 6136 \nu^{9} - 4254 \nu^{8} - 179857 \nu^{7} - 152420 \nu^{6} + 68632 \nu^{5} + 1007136 \nu^{4} + 2820480 \nu^{3} + 2696676 \nu^{2} + \cdots + 708480 ) / 51972 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29270 \nu^{11} - 21978 \nu^{10} + 11125 \nu^{9} - 2248 \nu^{8} - 467946 \nu^{7} - 351240 \nu^{6} + 350356 \nu^{5} + 2589800 \nu^{4} + 6858478 \nu^{3} + \cdots + 1273908 ) / 103944 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{8} + \beta_{5} - 4\beta_{4} + \beta_{2} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{11} + 4 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 3 \beta_{7} - 12 \beta_{6} + 8 \beta_{5} - 8 \beta_{4} + 6 \beta_{3} + 4 \beta_{2} + 4 \beta _1 + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} + \beta_{9} - 10\beta_{6} + \beta_{2} + 3\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19 \beta_{11} - 32 \beta_{10} + 16 \beta_{9} + 76 \beta_{8} + 42 \beta_{7} - 39 \beta_{6} - 16 \beta_{5} - 38 \beta_{4} - 21 \beta_{3} + 32 \beta_{2} + 38 \beta _1 + 78 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{10} + 32\beta_{8} + 27\beta_{7} - 32\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 47 \beta_{11} - 35 \beta_{10} + 35 \beta_{9} + 94 \beta_{8} + 60 \beta_{7} - 108 \beta_{6} + 35 \beta_{5} - 188 \beta_{4} + 60 \beta_{3} - 35 \beta_{2} - 47 \beta _1 - 108 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 83\beta_{11} + 48\beta_{9} - 223\beta_{6} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 472 \beta_{11} - 164 \beta_{10} + 328 \beta_{9} + 472 \beta_{8} + 321 \beta_{7} - 1140 \beta_{6} - 328 \beta_{5} + 472 \beta_{4} - 642 \beta_{3} + 164 \beta_{2} + 236 \beta _1 + 570 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -262\beta_{10} + 852\beta_{8} + 636\beta_{7} - 262\beta_{5} - 636\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1193 \beta_{11} - 1600 \beta_{10} - 800 \beta_{9} + 4772 \beta_{8} + 3342 \beta_{7} + 2949 \beta_{6} - 800 \beta_{5} - 2386 \beta_{4} - 1671 \beta_{3} - 1600 \beta_{2} - 2386 \beta _1 - 5898 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-\beta_{6}\) \(-1\)

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
−0.180407 + 0.673288i
2.17840 + 0.583700i
−0.403293 + 1.50511i
−1.50511 0.403293i
0.583700 2.17840i
−0.673288 0.180407i
−0.180407 0.673288i
2.17840 0.583700i
−0.403293 1.50511i
−1.50511 + 0.403293i
0.583700 + 2.17840i
−0.673288 + 0.180407i
−2.17731 + 1.25707i 0 2.16044 3.74200i 0 0 −0.445256 + 0.257068i 5.83502i 0 0
199.2 −1.80664 + 1.04307i 0 1.17597 2.03684i 0 0 −3.53869 + 2.04307i 0.734191i 0 0
199.3 −0.495361 + 0.285997i 0 −0.836412 + 1.44871i 0 0 1.23669 0.714003i 2.10083i 0 0
199.4 0.495361 0.285997i 0 −0.836412 + 1.44871i 0 0 −1.23669 + 0.714003i 2.10083i 0 0
199.5 1.80664 1.04307i 0 1.17597 2.03684i 0 0 3.53869 2.04307i 0.734191i 0 0
199.6 2.17731 1.25707i 0 2.16044 3.74200i 0 0 0.445256 0.257068i 5.83502i 0 0
424.1 −2.17731 1.25707i 0 2.16044 + 3.74200i 0 0 −0.445256 0.257068i 5.83502i 0 0
424.2 −1.80664 1.04307i 0 1.17597 + 2.03684i 0 0 −3.53869 2.04307i 0.734191i 0 0
424.3 −0.495361 0.285997i 0 −0.836412 1.44871i 0 0 1.23669 + 0.714003i 2.10083i 0 0
424.4 0.495361 + 0.285997i 0 −0.836412 1.44871i 0 0 −1.23669 0.714003i 2.10083i 0 0
424.5 1.80664 + 1.04307i 0 1.17597 + 2.03684i 0 0 3.53869 + 2.04307i 0.734191i 0 0
424.6 2.17731 + 1.25707i 0 2.16044 + 3.74200i 0 0 0.445256 + 0.257068i 5.83502i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.k.b 12
3.b odd 2 1 225.2.k.b 12
5.b even 2 1 inner 675.2.k.b 12
5.c odd 4 1 135.2.e.b 6
5.c odd 4 1 675.2.e.b 6
9.c even 3 1 inner 675.2.k.b 12
9.c even 3 1 2025.2.b.m 6
9.d odd 6 1 225.2.k.b 12
9.d odd 6 1 2025.2.b.l 6
15.d odd 2 1 225.2.k.b 12
15.e even 4 1 45.2.e.b 6
15.e even 4 1 225.2.e.b 6
20.e even 4 1 2160.2.q.k 6
45.h odd 6 1 225.2.k.b 12
45.h odd 6 1 2025.2.b.l 6
45.j even 6 1 inner 675.2.k.b 12
45.j even 6 1 2025.2.b.m 6
45.k odd 12 1 135.2.e.b 6
45.k odd 12 1 405.2.a.i 3
45.k odd 12 1 675.2.e.b 6
45.k odd 12 1 2025.2.a.o 3
45.l even 12 1 45.2.e.b 6
45.l even 12 1 225.2.e.b 6
45.l even 12 1 405.2.a.j 3
45.l even 12 1 2025.2.a.n 3
60.l odd 4 1 720.2.q.i 6
180.v odd 12 1 720.2.q.i 6
180.v odd 12 1 6480.2.a.bv 3
180.x even 12 1 2160.2.q.k 6
180.x even 12 1 6480.2.a.bs 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 15.e even 4 1
45.2.e.b 6 45.l even 12 1
135.2.e.b 6 5.c odd 4 1
135.2.e.b 6 45.k odd 12 1
225.2.e.b 6 15.e even 4 1
225.2.e.b 6 45.l even 12 1
225.2.k.b 12 3.b odd 2 1
225.2.k.b 12 9.d odd 6 1
225.2.k.b 12 15.d odd 2 1
225.2.k.b 12 45.h odd 6 1
405.2.a.i 3 45.k odd 12 1
405.2.a.j 3 45.l even 12 1
675.2.e.b 6 5.c odd 4 1
675.2.e.b 6 45.k odd 12 1
675.2.k.b 12 1.a even 1 1 trivial
675.2.k.b 12 5.b even 2 1 inner
675.2.k.b 12 9.c even 3 1 inner
675.2.k.b 12 45.j even 6 1 inner
720.2.q.i 6 60.l odd 4 1
720.2.q.i 6 180.v odd 12 1
2025.2.a.n 3 45.l even 12 1
2025.2.a.o 3 45.k odd 12 1
2025.2.b.l 6 9.d odd 6 1
2025.2.b.l 6 45.h odd 6 1
2025.2.b.m 6 9.c even 3 1
2025.2.b.m 6 45.j even 6 1
2160.2.q.k 6 20.e even 4 1
2160.2.q.k 6 180.x even 12 1
6480.2.a.bs 3 180.x even 12 1
6480.2.a.bv 3 180.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 11T_{2}^{10} + 90T_{2}^{8} - 323T_{2}^{6} + 862T_{2}^{4} - 279T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 11 T^{10} + 90 T^{8} - 323 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 19 T^{10} + 322 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( (T^{6} + 2 T^{5} + 12 T^{4} + 8 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 24 T^{10} + 528 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{6} + 20 T^{4} + 112 T^{2} + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 4 T^{2} - 4 T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} - 75 T^{10} + \cdots + 187388721 \) Copy content Toggle raw display
$29$ \( (T^{6} - 7 T^{5} + 78 T^{4} + 101 T^{3} + \cdots + 2601)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + 124 T^{4} + \cdots + 219024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 60 T^{4} + 192 T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 13 T^{5} + 150 T^{4} + 241 T^{3} + \cdots + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 108 T^{10} + 11568 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{12} - 191 T^{10} + \cdots + 18539817921 \) Copy content Toggle raw display
$53$ \( (T^{6} + 44 T^{4} + 496 T^{2} + 576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 2 T^{5} + 24 T^{4} - 8 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + T^{5} + 38 T^{4} - 179 T^{3} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 199 T^{10} + \cdots + 66074188401 \) Copy content Toggle raw display
$71$ \( (T^{3} - 10 T^{2} - 92 T + 708)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 192 T^{4} + 6144 T^{2} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 2 T^{5} + 88 T^{4} + 216 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 171 T^{10} + \cdots + 43046721 \) Copy content Toggle raw display
$89$ \( (T + 3)^{12} \) Copy content Toggle raw display
$97$ \( T^{12} - 396 T^{10} + \cdots + 2891414573056 \) Copy content Toggle raw display
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