Newspace parameters
Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 693.bu (of order \(10\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.53363286007\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{10})\) |
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: | \( x^{16} + 20x^{14} + 260x^{12} + 2030x^{10} + 11605x^{8} + 42100x^{6} + 106925x^{4} + 113575x^{2} + 87025 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 77) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \( x^{16} + 20x^{14} + 260x^{12} + 2030x^{10} + 11605x^{8} + 42100x^{6} + 106925x^{4} + 113575x^{2} + 87025 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 473414 \nu^{15} - 121962046 \nu^{13} - 1914724533 \nu^{11} - 19540502651 \nu^{9} - 110976178125 \nu^{7} - 427139590595 \nu^{5} + \cdots - 700136714820 \nu ) / 141563707035 \) |
\(\beta_{3}\) | \(=\) | \( ( - 1553143 \nu^{15} - 83115610 \nu^{13} - 738702950 \nu^{11} - 5169909042 \nu^{9} - 7773695495 \nu^{7} + 12526757215 \nu^{5} + \cdots + 389963137595 \nu ) / 141563707035 \) |
\(\beta_{4}\) | \(=\) | \( ( 2026557 \nu^{14} + 205077656 \nu^{12} + 2653427483 \nu^{10} + 24710411693 \nu^{8} + 118749873620 \nu^{6} + 414612833380 \nu^{4} + \cdots + 310173577225 ) / 141563707035 \) |
\(\beta_{5}\) | \(=\) | \( ( - 2307412 \nu^{14} - 42954924 \nu^{12} - 546435360 \nu^{10} - 4022687807 \nu^{8} - 22443730260 \nu^{6} - 74163373020 \nu^{4} + \cdots - 70674046800 ) / 141563707035 \) |
\(\beta_{6}\) | \(=\) | \( ( 2307412 \nu^{15} + 42954924 \nu^{13} + 546435360 \nu^{11} + 4022687807 \nu^{9} + 22443730260 \nu^{7} + 74163373020 \nu^{5} + \cdots + 70674046800 \nu ) / 141563707035 \) |
\(\beta_{7}\) | \(=\) | \( ( - 8126617 \nu^{15} + 107242387 \nu^{13} + 1727473773 \nu^{11} + 21511248812 \nu^{9} + 103285230840 \nu^{7} + 401725517900 \nu^{5} + \cdots + 534354646485 \nu ) / 141563707035 \) |
\(\beta_{8}\) | \(=\) | \( ( 9624825 \nu^{14} + 326813127 \nu^{12} + 4112826637 \nu^{10} + 33537013348 \nu^{8} + 159160366925 \nu^{6} + 503717613335 \nu^{4} + \cdots + 564950170875 ) / 141563707035 \) |
\(\beta_{9}\) | \(=\) | \( ( - 85328 \nu^{14} - 3129227 \nu^{12} - 37034577 \nu^{10} - 295472690 \nu^{8} - 1274305820 \nu^{6} - 3749548520 \nu^{4} - 3135087840 \nu^{2} + \cdots - 2416417865 ) / 1080638985 \) |
\(\beta_{10}\) | \(=\) | \( ( - 85328 \nu^{15} - 3129227 \nu^{13} - 37034577 \nu^{11} - 295472690 \nu^{9} - 1274305820 \nu^{7} - 3749548520 \nu^{5} - 3135087840 \nu^{3} + \cdots - 2416417865 \nu ) / 1080638985 \) |
\(\beta_{11}\) | \(=\) | \( ( 19576647 \nu^{14} + 220271693 \nu^{12} + 1820750374 \nu^{10} + 4608222307 \nu^{8} + 1502139270 \nu^{6} - 102149491525 \nu^{4} + \cdots - 312330974765 ) / 141563707035 \) |
\(\beta_{12}\) | \(=\) | \( ( - 23957031 \nu^{14} - 304074935 \nu^{12} - 2842215611 \nu^{10} - 12038362652 \nu^{8} - 33691071920 \nu^{6} + 17998043615 \nu^{4} + \cdots + 147206723390 ) / 141563707035 \) |
\(\beta_{13}\) | \(=\) | \( ( - 198348 \nu^{15} - 3886661 \nu^{13} - 41951474 \nu^{11} - 280524995 \nu^{9} - 1163671340 \nu^{7} - 3027593815 \nu^{5} - 2666658230 \nu^{3} + \cdots - 1244021785 \nu ) / 1080638985 \) |
\(\beta_{14}\) | \(=\) | \( ( - 26536505 \nu^{14} - 264615202 \nu^{12} - 2085345531 \nu^{10} - 3473599188 \nu^{8} + 6011890130 \nu^{6} + 135449500340 \nu^{4} + \cdots + 29496038150 ) / 141563707035 \) |
\(\beta_{15}\) | \(=\) | \( ( - 38187887 \nu^{15} - 796505985 \nu^{13} - 8851599651 \nu^{11} - 61721024229 \nu^{9} - 271898350415 \nu^{7} - 782880946375 \nu^{5} + \cdots - 987191416985 \nu ) / 141563707035 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{14} - 2\beta_{11} - 2\beta_{9} - 2\beta_{8} - 5\beta_{5} - \beta_{4} - 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{15} - 2\beta_{13} + \beta_{10} + 2\beta_{7} + 7\beta_{6} + \beta_{3} + 2\beta_{2} + 3\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( -7\beta_{14} + 8\beta_{12} + \beta_{11} + 7\beta_{9} + 16\beta_{8} + 37\beta_{5} - 2\beta_{4} - 30 \) |
\(\nu^{5}\) | \(=\) | \( -8\beta_{15} + 9\beta_{13} - 2\beta_{10} - \beta_{7} - 38\beta_{6} + 9\beta_{3} + \beta_{2} - 47\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( 84\beta_{14} - 29\beta_{12} + 98\beta_{11} + 42\beta_{9} - 13\beta_{8} - 42\beta_{5} + 56\beta_{4} + 231 \) |
\(\nu^{7}\) | \(=\) | \( -14\beta_{15} + 69\beta_{13} - 29\beta_{10} - 98\beta_{7} - 56\beta_{6} - 138\beta_{3} - 111\beta_{2} + 146\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( -245\beta_{14} - 260\beta_{12} - 750\beta_{11} - 490\beta_{9} - 630\beta_{8} - 1210\beta_{5} - 125\beta_{4} - 245 \) |
\(\nu^{9}\) | \(=\) | \( 505 \beta_{15} - 1010 \beta_{13} + 385 \beta_{10} + 750 \beta_{7} + 1960 \beta_{6} + 505 \beta_{3} + 630 \beta_{2} + 1135 \beta_1 \) |
\(\nu^{10}\) | \(=\) | \( - 1420 \beta_{14} + 2645 \beta_{12} + 965 \beta_{11} + 1420 \beta_{9} + 5030 \beta_{8} + 9235 \beta_{5} - 2190 \beta_{4} - 7815 \) |
\(\nu^{11}\) | \(=\) | \( - 2385 \beta_{15} + 3610 \beta_{13} - 2190 \beta_{10} - 965 \beta_{7} - 10200 \beta_{6} + 3610 \beta_{3} + 965 \beta_{2} - 13810 \beta_1 \) |
\(\nu^{12}\) | \(=\) | \( 16410 \beta_{14} - 995 \beta_{12} + 26405 \beta_{11} + 8205 \beta_{9} - 7210 \beta_{8} - 8205 \beta_{5} + 18200 \beta_{4} + 59020 \) |
\(\nu^{13}\) | \(=\) | \( - 9995 \beta_{15} + 25410 \beta_{13} - 995 \beta_{10} - 26405 \beta_{7} - 18200 \beta_{6} - 50820 \beta_{3} - 33615 \beta_{2} + 39825 \beta_1 \) |
\(\nu^{14}\) | \(=\) | \( - 47235 \beta_{14} - 129635 \beta_{12} - 224105 \beta_{11} - 94470 \beta_{9} - 172640 \beta_{8} - 332295 \beta_{5} + 4230 \beta_{4} - 47235 \) |
\(\nu^{15}\) | \(=\) | \( 176870 \beta_{15} - 353740 \beta_{13} + 125405 \beta_{10} + 224105 \beta_{7} + 556400 \beta_{6} + 176870 \beta_{3} + 172640 \beta_{2} + 349510 \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).
\(n\) | \(155\) | \(199\) | \(442\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-\beta_{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
118.1 |
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0.395472 | + | 0.544320i | 0 | 0.478148 | − | 1.47159i | −2.08654 | + | 2.87188i | 0 | −1.94632 | − | 1.79216i | 2.26988 | − | 0.737529i | 0 | −2.38839 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.2 | 0.395472 | + | 0.544320i | 0 | 0.478148 | − | 1.47159i | 2.08654 | − | 2.87188i | 0 | −2.62801 | + | 0.305873i | 2.26988 | − | 0.737529i | 0 | 2.38839 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.3 | 1.41355 | + | 1.94558i | 0 | −1.16913 | + | 3.59821i | −1.97962 | + | 2.72471i | 0 | −1.43059 | + | 2.22563i | −4.07890 | + | 1.32531i | 0 | −8.09942 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
118.4 | 1.41355 | + | 1.94558i | 0 | −1.16913 | + | 3.59821i | 1.97962 | − | 2.72471i | 0 | 0.150818 | − | 2.64145i | −4.07890 | + | 1.32531i | 0 | 8.09942 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
244.1 | −0.478148 | − | 0.155360i | 0 | −1.41355 | − | 1.02700i | −0.572621 | + | 0.186056i | 0 | 2.61795 | + | 0.382556i | 1.10735 | + | 1.52414i | 0 | 0.302703 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
244.2 | −0.478148 | − | 0.155360i | 0 | −1.41355 | − | 1.02700i | 0.572621 | − | 0.186056i | 0 | −1.17282 | − | 2.37160i | 1.10735 | + | 1.52414i | 0 | −0.302703 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
244.3 | 1.16913 | + | 0.379874i | 0 | −0.395472 | − | 0.287327i | −2.26926 | + | 0.737329i | 0 | 1.80595 | + | 1.93354i | −1.79833 | − | 2.47520i | 0 | −2.93316 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
244.4 | 1.16913 | + | 0.379874i | 0 | −0.395472 | − | 0.287327i | 2.26926 | − | 0.737329i | 0 | −2.39697 | − | 1.12006i | −1.79833 | − | 2.47520i | 0 | 2.93316 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
370.1 | 0.395472 | − | 0.544320i | 0 | 0.478148 | + | 1.47159i | −2.08654 | − | 2.87188i | 0 | −1.94632 | + | 1.79216i | 2.26988 | + | 0.737529i | 0 | −2.38839 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
370.2 | 0.395472 | − | 0.544320i | 0 | 0.478148 | + | 1.47159i | 2.08654 | + | 2.87188i | 0 | −2.62801 | − | 0.305873i | 2.26988 | + | 0.737529i | 0 | 2.38839 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
370.3 | 1.41355 | − | 1.94558i | 0 | −1.16913 | − | 3.59821i | −1.97962 | − | 2.72471i | 0 | −1.43059 | − | 2.22563i | −4.07890 | − | 1.32531i | 0 | −8.09942 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
370.4 | 1.41355 | − | 1.94558i | 0 | −1.16913 | − | 3.59821i | 1.97962 | + | 2.72471i | 0 | 0.150818 | + | 2.64145i | −4.07890 | − | 1.32531i | 0 | 8.09942 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
622.1 | −0.478148 | + | 0.155360i | 0 | −1.41355 | + | 1.02700i | −0.572621 | − | 0.186056i | 0 | 2.61795 | − | 0.382556i | 1.10735 | − | 1.52414i | 0 | 0.302703 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
622.2 | −0.478148 | + | 0.155360i | 0 | −1.41355 | + | 1.02700i | 0.572621 | + | 0.186056i | 0 | −1.17282 | + | 2.37160i | 1.10735 | − | 1.52414i | 0 | −0.302703 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
622.3 | 1.16913 | − | 0.379874i | 0 | −0.395472 | + | 0.287327i | −2.26926 | − | 0.737329i | 0 | 1.80595 | − | 1.93354i | −1.79833 | + | 2.47520i | 0 | −2.93316 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
622.4 | 1.16913 | − | 0.379874i | 0 | −0.395472 | + | 0.287327i | 2.26926 | + | 0.737329i | 0 | −2.39697 | + | 1.12006i | −1.79833 | + | 2.47520i | 0 | 2.93316 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
11.d | odd | 10 | 1 | inner |
77.l | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 693.2.bu.d | 16 | |
3.b | odd | 2 | 1 | 77.2.l.b | ✓ | 16 | |
7.b | odd | 2 | 1 | inner | 693.2.bu.d | 16 | |
11.d | odd | 10 | 1 | inner | 693.2.bu.d | 16 | |
21.c | even | 2 | 1 | 77.2.l.b | ✓ | 16 | |
21.g | even | 6 | 1 | 539.2.s.b | 16 | ||
21.g | even | 6 | 1 | 539.2.s.c | 16 | ||
21.h | odd | 6 | 1 | 539.2.s.b | 16 | ||
21.h | odd | 6 | 1 | 539.2.s.c | 16 | ||
33.d | even | 2 | 1 | 847.2.l.i | 16 | ||
33.f | even | 10 | 1 | 77.2.l.b | ✓ | 16 | |
33.f | even | 10 | 1 | 847.2.b.f | 16 | ||
33.f | even | 10 | 1 | 847.2.l.e | 16 | ||
33.f | even | 10 | 1 | 847.2.l.j | 16 | ||
33.h | odd | 10 | 1 | 847.2.b.f | 16 | ||
33.h | odd | 10 | 1 | 847.2.l.e | 16 | ||
33.h | odd | 10 | 1 | 847.2.l.i | 16 | ||
33.h | odd | 10 | 1 | 847.2.l.j | 16 | ||
77.l | even | 10 | 1 | inner | 693.2.bu.d | 16 | |
231.h | odd | 2 | 1 | 847.2.l.i | 16 | ||
231.r | odd | 10 | 1 | 77.2.l.b | ✓ | 16 | |
231.r | odd | 10 | 1 | 847.2.b.f | 16 | ||
231.r | odd | 10 | 1 | 847.2.l.e | 16 | ||
231.r | odd | 10 | 1 | 847.2.l.j | 16 | ||
231.u | even | 10 | 1 | 847.2.b.f | 16 | ||
231.u | even | 10 | 1 | 847.2.l.e | 16 | ||
231.u | even | 10 | 1 | 847.2.l.i | 16 | ||
231.u | even | 10 | 1 | 847.2.l.j | 16 | ||
231.be | even | 30 | 1 | 539.2.s.b | 16 | ||
231.be | even | 30 | 1 | 539.2.s.c | 16 | ||
231.bf | odd | 30 | 1 | 539.2.s.b | 16 | ||
231.bf | odd | 30 | 1 | 539.2.s.c | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.2.l.b | ✓ | 16 | 3.b | odd | 2 | 1 | |
77.2.l.b | ✓ | 16 | 21.c | even | 2 | 1 | |
77.2.l.b | ✓ | 16 | 33.f | even | 10 | 1 | |
77.2.l.b | ✓ | 16 | 231.r | odd | 10 | 1 | |
539.2.s.b | 16 | 21.g | even | 6 | 1 | ||
539.2.s.b | 16 | 21.h | odd | 6 | 1 | ||
539.2.s.b | 16 | 231.be | even | 30 | 1 | ||
539.2.s.b | 16 | 231.bf | odd | 30 | 1 | ||
539.2.s.c | 16 | 21.g | even | 6 | 1 | ||
539.2.s.c | 16 | 21.h | odd | 6 | 1 | ||
539.2.s.c | 16 | 231.be | even | 30 | 1 | ||
539.2.s.c | 16 | 231.bf | odd | 30 | 1 | ||
693.2.bu.d | 16 | 1.a | even | 1 | 1 | trivial | |
693.2.bu.d | 16 | 7.b | odd | 2 | 1 | inner | |
693.2.bu.d | 16 | 11.d | odd | 10 | 1 | inner | |
693.2.bu.d | 16 | 77.l | even | 10 | 1 | inner | |
847.2.b.f | 16 | 33.f | even | 10 | 1 | ||
847.2.b.f | 16 | 33.h | odd | 10 | 1 | ||
847.2.b.f | 16 | 231.r | odd | 10 | 1 | ||
847.2.b.f | 16 | 231.u | even | 10 | 1 | ||
847.2.l.e | 16 | 33.f | even | 10 | 1 | ||
847.2.l.e | 16 | 33.h | odd | 10 | 1 | ||
847.2.l.e | 16 | 231.r | odd | 10 | 1 | ||
847.2.l.e | 16 | 231.u | even | 10 | 1 | ||
847.2.l.i | 16 | 33.d | even | 2 | 1 | ||
847.2.l.i | 16 | 33.h | odd | 10 | 1 | ||
847.2.l.i | 16 | 231.h | odd | 2 | 1 | ||
847.2.l.i | 16 | 231.u | even | 10 | 1 | ||
847.2.l.j | 16 | 33.f | even | 10 | 1 | ||
847.2.l.j | 16 | 33.h | odd | 10 | 1 | ||
847.2.l.j | 16 | 231.r | odd | 10 | 1 | ||
847.2.l.j | 16 | 231.u | even | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 5T_{2}^{7} + 13T_{2}^{6} - 15T_{2}^{5} + 4T_{2}^{4} + 5T_{2}^{3} - 3T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} - 5 T^{7} + 13 T^{6} - 15 T^{5} + \cdots + 1)^{2} \)
$3$
\( T^{16} \)
$5$
\( T^{16} + 5 T^{14} + 235 T^{12} + \cdots + 87025 \)
$7$
\( T^{16} + 10 T^{15} + 46 T^{14} + \cdots + 5764801 \)
$11$
\( (T^{8} + T^{7} - 20 T^{6} - T^{5} + 309 T^{4} + \cdots + 14641)^{2} \)
$13$
\( T^{16} + 15 T^{14} + 125 T^{12} + \cdots + 87025 \)
$17$
\( T^{16} + 15 T^{14} + 425 T^{12} + \cdots + 54390625 \)
$19$
\( T^{16} - 55 T^{14} + 5745 T^{12} + \cdots + 87025 \)
$23$
\( (T^{2} - T - 11)^{8} \)
$29$
\( (T^{8} + 5 T^{7} + 8 T^{6} - 25 T^{5} + \cdots + 961)^{2} \)
$31$
\( T^{16} - 85 T^{14} + 3120 T^{12} + \cdots + 87025 \)
$37$
\( (T^{8} - 2 T^{7} + 43 T^{6} - 319 T^{5} + \cdots + 477481)^{2} \)
$41$
\( T^{16} + 110 T^{14} + \cdots + 61551129025 \)
$43$
\( (T^{2} + 3)^{8} \)
$47$
\( T^{16} - 15 T^{14} + \cdots + 570971025 \)
$53$
\( (T^{8} + 35 T^{6} - 115 T^{5} + 390 T^{4} + \cdots + 25)^{2} \)
$59$
\( T^{16} - 55 T^{14} + \cdots + 1204934313025 \)
$61$
\( T^{16} + \cdots + 283945921983025 \)
$67$
\( (T^{4} + T^{3} - 109 T^{2} - 439 T + 241)^{4} \)
$71$
\( (T^{8} - 28 T^{7} + 453 T^{6} + \cdots + 8288641)^{2} \)
$73$
\( T^{16} + 315 T^{14} + \cdots + 1204934313025 \)
$79$
\( (T^{8} - 25 T^{7} + 343 T^{6} + \cdots + 44521)^{2} \)
$83$
\( T^{16} + 235 T^{14} + \cdots + 1204934313025 \)
$89$
\( (T^{8} + 165 T^{6} + 3150 T^{4} + \cdots + 23895)^{2} \)
$97$
\( T^{16} - 10 T^{14} + \cdots + 11\!\cdots\!25 \)
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