Newspace parameters
Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 720.cu (of order \(12\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.74922894553\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 45) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + 1720995499366 \nu^{11} + \cdots + 33432180594 ) / 3707507912227 \) |
\(\beta_{3}\) | \(=\) | \( ( 94386116202 \nu^{15} - 619740656932 \nu^{14} + 2033008548312 \nu^{13} - 4441986378774 \nu^{12} + 5386888154268 \nu^{11} + \cdots - 80029143512 ) / 3707507912227 \) |
\(\beta_{4}\) | \(=\) | \( ( 140020985001 \nu^{15} - 843322184609 \nu^{14} + 2535236829090 \nu^{13} - 5074237635644 \nu^{12} + 4819083120574 \nu^{11} + \cdots - 5622211270526 ) / 3707507912227 \) |
\(\beta_{5}\) | \(=\) | \( ( - 140094553942 \nu^{15} + 840493754711 \nu^{14} - 2524530400854 \nu^{13} + 5054110370148 \nu^{12} - 4783342099524 \nu^{11} + \cdots - 1652709999986 ) / 3707507912227 \) |
\(\beta_{6}\) | \(=\) | \( ( 190424165289 \nu^{15} - 1059314239720 \nu^{14} + 2941649532255 \nu^{13} - 5449473018186 \nu^{12} + 3809266070290 \nu^{11} + \cdots + 7166890770427 ) / 3707507912227 \) |
\(\beta_{7}\) | \(=\) | \( ( - 250381984552 \nu^{15} + 1889212496921 \nu^{14} - 6927613311171 \nu^{13} + 16582911879478 \nu^{12} - 24294968776849 \nu^{11} + \cdots + 5660844178894 ) / 3707507912227 \) |
\(\beta_{8}\) | \(=\) | \( ( - 254444403226 \nu^{15} + 1391318146672 \nu^{14} - 3674576400880 \nu^{13} + 6138067615014 \nu^{12} - 1944897155536 \nu^{11} + \cdots + 146893504700 ) / 3707507912227 \) |
\(\beta_{9}\) | \(=\) | \( ( 196858530 \nu^{15} - 1212076232 \nu^{14} + 3734556994 \nu^{13} - 7676123862 \nu^{12} + 7906540260 \nu^{11} + 2288455604 \nu^{10} + \cdots + 785074438 ) / 1973128213 \) |
\(\beta_{10}\) | \(=\) | \( ( - 677106342206 \nu^{15} + 3890533925017 \nu^{14} - 11114551602021 \nu^{13} + 21019011866538 \nu^{12} + \cdots - 11893459432082 ) / 3707507912227 \) |
\(\beta_{11}\) | \(=\) | \( ( - 785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} - 34368654754 \nu^{11} - 6224799624 \nu^{10} + \cdots - 3343669588 ) / 1973128213 \) |
\(\beta_{12}\) | \(=\) | \( ( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} - 64714538406 \nu^{11} - 18641341380 \nu^{10} + \cdots - 10233974547 ) / 3810388399 \) |
\(\beta_{13}\) | \(=\) | \( ( - 2029780580195 \nu^{15} + 12360646755715 \nu^{14} - 37626995630777 \nu^{13} + 76333636000087 \nu^{12} + \cdots - 14351633608601 ) / 3707507912227 \) |
\(\beta_{14}\) | \(=\) | \( ( 2802573231023 \nu^{15} - 17485025240258 \nu^{14} + 54572756591878 \nu^{13} - 113605741903926 \nu^{12} + \cdots + 12411058785072 ) / 3707507912227 \) |
\(\beta_{15}\) | \(=\) | \( ( - 2852772870096 \nu^{15} + 17521813816267 \nu^{14} - 53872302373806 \nu^{13} + 110572117714578 \nu^{12} + \cdots - 17972736434118 ) / 3707507912227 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{8} + \beta_{3} - 2\beta_{2} + \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 4\beta_{3} - \beta_{2} + 1 \) |
\(\nu^{4}\) | \(=\) | \( - \beta_{15} + 5 \beta_{14} + 2 \beta_{13} + \beta_{12} + 8 \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} + 5 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} + 8 \) |
\(\nu^{5}\) | \(=\) | \( - 16 \beta_{15} + 8 \beta_{13} + 14 \beta_{12} + 7 \beta_{11} + 3 \beta_{9} - 8 \beta_{8} - 8 \beta_{6} - 8 \beta_{5} + 8 \beta_{3} + 7 \beta_{2} + 11 \beta _1 + 14 \) |
\(\nu^{6}\) | \(=\) | \( - 35 \beta_{15} - 16 \beta_{14} + 8 \beta_{13} + 30 \beta_{12} - 8 \beta_{11} + 16 \beta_{10} - \beta_{9} - 16 \beta_{8} + 8 \beta_{7} - 16 \beta_{6} - 57 \beta_{5} - 8 \beta_{4} + 34 \beta_{3} + 16 \beta_{2} + 25 \beta _1 + 16 \) |
\(\nu^{7}\) | \(=\) | \( \beta_{15} - \beta_{14} - 10 \beta_{12} - 10 \beta_{11} + 51 \beta_{10} - \beta_{8} + 51 \beta_{7} - 98 \beta_{5} + \beta_{4} + 97 \beta_{3} + 10 \beta_{2} + 10 \) |
\(\nu^{8}\) | \(=\) | \( 50 \beta_{15} + 148 \beta_{14} + 50 \beta_{13} - 50 \beta_{12} + 145 \beta_{11} + 50 \beta_{10} - 10 \beta_{9} + 100 \beta_{7} + 50 \beta_{6} + 10 \beta_{5} + 50 \beta_{4} + 128 \beta_{3} - 138 \beta _1 + 50 \) |
\(\nu^{9}\) | \(=\) | \( - 288 \beta_{15} + 278 \beta_{14} + 298 \beta_{13} + 228 \beta_{12} + 456 \beta_{11} - 55 \beta_{9} - 278 \beta_{8} - 10 \beta_{4} - 10 \beta_{3} + 158 \beta_{2} - 278 \beta _1 + 228 \) |
\(\nu^{10}\) | \(=\) | \( - 1385 \beta_{15} - 288 \beta_{14} + 576 \beta_{13} + 1032 \beta_{12} + 288 \beta_{11} + 288 \beta_{10} - 140 \beta_{9} - 1097 \beta_{8} - 288 \beta_{7} - 288 \beta_{6} - 1315 \beta_{5} - 576 \beta_{4} - 218 \beta_{3} + \cdots + 288 \) |
\(\nu^{11}\) | \(=\) | \( - 1533 \beta_{15} - 1603 \beta_{14} + 817 \beta_{12} - 1245 \beta_{11} + 1673 \beta_{10} - 1603 \beta_{8} - 4331 \beta_{5} - 1533 \beta_{4} + 1245 \beta_{2} - 817 \) |
\(\nu^{12}\) | \(=\) | \( 3206 \beta_{15} - 1603 \beta_{13} - 3206 \beta_{12} - 1603 \beta_{11} + 3206 \beta_{10} + 428 \beta_{9} + 1603 \beta_{7} + 3206 \beta_{6} - 3545 \beta_{5} - 1195 \beta_{4} - 428 \beta_{3} - 3545 \beta _1 - 3923 \) |
\(\nu^{13}\) | \(=\) | \( 8782 \beta_{15} + 8782 \beta_{14} - 6751 \beta_{12} + 6751 \beta_{11} + 428 \beta_{8} + 9210 \beta_{6} + 9210 \beta_{5} + 428 \beta_{4} - 8782 \beta_{3} - 2459 \beta_{2} - 14219 \beta _1 - 6751 \) |
\(\nu^{14}\) | \(=\) | \( - 8782 \beta_{15} + 8782 \beta_{14} + 8782 \beta_{13} + 8782 \beta_{12} + 17564 \beta_{11} - 8782 \beta_{10} - 2459 \beta_{9} - 15075 \beta_{8} - 17564 \beta_{7} + 8782 \beta_{6} + 15105 \beta_{5} + \cdots - 8782 \) |
\(\nu^{15}\) | \(=\) | \( - 47744 \beta_{15} - 45285 \beta_{14} + 36503 \beta_{12} - 22803 \beta_{11} - 45285 \beta_{8} - 50203 \beta_{7} - 45285 \beta_{5} - 47744 \beta_{4} - 78841 \beta_{3} + 22803 \beta_{2} + \cdots - 36503 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).
\(n\) | \(181\) | \(271\) | \(577\) | \(641\) |
\(\chi(n)\) | \(1\) | \(1\) | \(\beta_{12}\) | \(-\beta_{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
113.1 |
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0 | −1.35314 | + | 1.08121i | 0 | −2.23073 | − | 0.154373i | 0 | 1.73749 | + | 0.465559i | 0 | 0.661975 | − | 2.92605i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
113.2 | 0 | 0.806271 | − | 1.53295i | 0 | −0.250705 | − | 2.22197i | 0 | 2.35868 | + | 0.632007i | 0 | −1.69985 | − | 2.47194i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
113.3 | 0 | 1.18953 | + | 1.25897i | 0 | −1.59371 | + | 1.56847i | 0 | −1.97869 | − | 0.530190i | 0 | −0.170031 | + | 2.99518i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
113.4 | 0 | 1.72336 | − | 0.173261i | 0 | 1.70912 | + | 1.44185i | 0 | −0.751454 | − | 0.201351i | 0 | 2.93996 | − | 0.597183i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.1 | 0 | −1.53295 | − | 0.806271i | 0 | −2.04963 | + | 0.893868i | 0 | −0.632007 | + | 2.35868i | 0 | 1.69985 | + | 2.47194i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.2 | 0 | −0.173261 | − | 1.72336i | 0 | 2.10323 | + | 0.759216i | 0 | 0.201351 | − | 0.751454i | 0 | −2.93996 | + | 0.597183i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.3 | 0 | 1.08121 | + | 1.35314i | 0 | −1.24906 | − | 1.85468i | 0 | −0.465559 | + | 1.73749i | 0 | −0.661975 | + | 2.92605i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
257.4 | 0 | 1.25897 | − | 1.18953i | 0 | 0.561484 | − | 2.16443i | 0 | 0.530190 | − | 1.97869i | 0 | 0.170031 | − | 2.99518i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.1 | 0 | −1.53295 | + | 0.806271i | 0 | −2.04963 | − | 0.893868i | 0 | −0.632007 | − | 2.35868i | 0 | 1.69985 | − | 2.47194i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.2 | 0 | −0.173261 | + | 1.72336i | 0 | 2.10323 | − | 0.759216i | 0 | 0.201351 | + | 0.751454i | 0 | −2.93996 | − | 0.597183i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.3 | 0 | 1.08121 | − | 1.35314i | 0 | −1.24906 | + | 1.85468i | 0 | −0.465559 | − | 1.73749i | 0 | −0.661975 | − | 2.92605i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
353.4 | 0 | 1.25897 | + | 1.18953i | 0 | 0.561484 | + | 2.16443i | 0 | 0.530190 | + | 1.97869i | 0 | 0.170031 | + | 2.99518i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
497.1 | 0 | −1.35314 | − | 1.08121i | 0 | −2.23073 | + | 0.154373i | 0 | 1.73749 | − | 0.465559i | 0 | 0.661975 | + | 2.92605i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
497.2 | 0 | 0.806271 | + | 1.53295i | 0 | −0.250705 | + | 2.22197i | 0 | 2.35868 | − | 0.632007i | 0 | −1.69985 | + | 2.47194i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
497.3 | 0 | 1.18953 | − | 1.25897i | 0 | −1.59371 | − | 1.56847i | 0 | −1.97869 | + | 0.530190i | 0 | −0.170031 | − | 2.99518i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
497.4 | 0 | 1.72336 | + | 0.173261i | 0 | 1.70912 | − | 1.44185i | 0 | −0.751454 | + | 0.201351i | 0 | 2.93996 | + | 0.597183i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 720.2.cu.c | 16 | |
4.b | odd | 2 | 1 | 45.2.l.a | ✓ | 16 | |
5.c | odd | 4 | 1 | inner | 720.2.cu.c | 16 | |
9.d | odd | 6 | 1 | inner | 720.2.cu.c | 16 | |
12.b | even | 2 | 1 | 135.2.m.a | 16 | ||
20.d | odd | 2 | 1 | 225.2.p.b | 16 | ||
20.e | even | 4 | 1 | 45.2.l.a | ✓ | 16 | |
20.e | even | 4 | 1 | 225.2.p.b | 16 | ||
36.f | odd | 6 | 1 | 135.2.m.a | 16 | ||
36.f | odd | 6 | 1 | 405.2.f.a | 16 | ||
36.h | even | 6 | 1 | 45.2.l.a | ✓ | 16 | |
36.h | even | 6 | 1 | 405.2.f.a | 16 | ||
45.l | even | 12 | 1 | inner | 720.2.cu.c | 16 | |
60.h | even | 2 | 1 | 675.2.q.a | 16 | ||
60.l | odd | 4 | 1 | 135.2.m.a | 16 | ||
60.l | odd | 4 | 1 | 675.2.q.a | 16 | ||
180.n | even | 6 | 1 | 225.2.p.b | 16 | ||
180.p | odd | 6 | 1 | 675.2.q.a | 16 | ||
180.v | odd | 12 | 1 | 45.2.l.a | ✓ | 16 | |
180.v | odd | 12 | 1 | 225.2.p.b | 16 | ||
180.v | odd | 12 | 1 | 405.2.f.a | 16 | ||
180.x | even | 12 | 1 | 135.2.m.a | 16 | ||
180.x | even | 12 | 1 | 405.2.f.a | 16 | ||
180.x | even | 12 | 1 | 675.2.q.a | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
45.2.l.a | ✓ | 16 | 4.b | odd | 2 | 1 | |
45.2.l.a | ✓ | 16 | 20.e | even | 4 | 1 | |
45.2.l.a | ✓ | 16 | 36.h | even | 6 | 1 | |
45.2.l.a | ✓ | 16 | 180.v | odd | 12 | 1 | |
135.2.m.a | 16 | 12.b | even | 2 | 1 | ||
135.2.m.a | 16 | 36.f | odd | 6 | 1 | ||
135.2.m.a | 16 | 60.l | odd | 4 | 1 | ||
135.2.m.a | 16 | 180.x | even | 12 | 1 | ||
225.2.p.b | 16 | 20.d | odd | 2 | 1 | ||
225.2.p.b | 16 | 20.e | even | 4 | 1 | ||
225.2.p.b | 16 | 180.n | even | 6 | 1 | ||
225.2.p.b | 16 | 180.v | odd | 12 | 1 | ||
405.2.f.a | 16 | 36.f | odd | 6 | 1 | ||
405.2.f.a | 16 | 36.h | even | 6 | 1 | ||
405.2.f.a | 16 | 180.v | odd | 12 | 1 | ||
405.2.f.a | 16 | 180.x | even | 12 | 1 | ||
675.2.q.a | 16 | 60.h | even | 2 | 1 | ||
675.2.q.a | 16 | 60.l | odd | 4 | 1 | ||
675.2.q.a | 16 | 180.p | odd | 6 | 1 | ||
675.2.q.a | 16 | 180.x | even | 12 | 1 | ||
720.2.cu.c | 16 | 1.a | even | 1 | 1 | trivial | |
720.2.cu.c | 16 | 5.c | odd | 4 | 1 | inner | |
720.2.cu.c | 16 | 9.d | odd | 6 | 1 | inner | |
720.2.cu.c | 16 | 45.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} - 2 T_{7}^{15} + 2 T_{7}^{14} - 12 T_{7}^{13} - 10 T_{7}^{12} + 62 T_{7}^{11} - 32 T_{7}^{10} + 216 T_{7}^{9} + 283 T_{7}^{8} - 990 T_{7}^{7} + 424 T_{7}^{6} - 3356 T_{7}^{5} + 1850 T_{7}^{4} + 5124 T_{7}^{3} + 1568 T_{7}^{2} + \cdots + 2401 \)
acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} - 6 T^{15} + 18 T^{14} + \cdots + 6561 \)
$5$
\( T^{16} + 6 T^{15} + 16 T^{14} + \cdots + 390625 \)
$7$
\( T^{16} - 2 T^{15} + 2 T^{14} - 12 T^{13} + \cdots + 2401 \)
$11$
\( (T^{8} - 10 T^{6} + 102 T^{4} - 180 T^{3} + \cdots + 4)^{2} \)
$13$
\( T^{16} + 2 T^{15} + 2 T^{14} + \cdots + 4477456 \)
$17$
\( T^{16} + 964 T^{12} + 6504 T^{8} + \cdots + 16 \)
$19$
\( (T^{8} + 60 T^{6} + 864 T^{4} + 1836 T^{2} + \cdots + 324)^{2} \)
$23$
\( T^{16} + 18 T^{15} + 162 T^{14} + \cdots + 62742241 \)
$29$
\( T^{16} + 84 T^{14} + \cdots + 981506241 \)
$31$
\( (T^{8} - 2 T^{7} + 46 T^{6} - 200 T^{5} + \cdots + 676)^{2} \)
$37$
\( (T^{8} - 2 T^{7} + 2 T^{6} + 124 T^{5} + \cdots + 4)^{2} \)
$41$
\( (T^{8} + 12 T^{7} + 26 T^{6} - 264 T^{5} + \cdots + 32761)^{2} \)
$43$
\( T^{16} - 2 T^{15} + 2 T^{14} + \cdots + 11316496 \)
$47$
\( T^{16} - 12 T^{15} + \cdots + 33243864241 \)
$53$
\( T^{16} + 20032 T^{12} + \cdots + 409600000000 \)
$59$
\( T^{16} + 144 T^{14} + \cdots + 592240896 \)
$61$
\( (T^{8} - 4 T^{7} + 118 T^{6} + 956 T^{5} + \cdots + 11449)^{2} \)
$67$
\( T^{16} + 4 T^{15} + \cdots + 539415333601 \)
$71$
\( (T^{8} + 116 T^{6} + 3972 T^{4} + \cdots + 128164)^{2} \)
$73$
\( (T^{8} + 4 T^{7} + 8 T^{6} - 584 T^{5} + \cdots + 270400)^{2} \)
$79$
\( T^{16} - 372 T^{14} + \cdots + 17\!\cdots\!96 \)
$83$
\( T^{16} - 66 T^{15} + \cdots + 13841287201 \)
$89$
\( (T^{8} - 300 T^{6} + 8262 T^{4} + \cdots + 3969)^{2} \)
$97$
\( T^{16} - 28 T^{15} + \cdots + 6146560000 \)
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