Newspace parameters
Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 504.by (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.7330053238\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
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: | \( x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + 623931 x^{8} + 289492 x^{7} + 241286 x^{6} - 4777608 x^{5} + \cdots + 1148023744 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{18} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + 623931 x^{8} + 289492 x^{7} + 241286 x^{6} - 4777608 x^{5} + \cdots + 1148023744 \) :
\(\beta_{1}\) | \(=\) | \( ( - 10\!\cdots\!45 \nu^{15} + \cdots + 21\!\cdots\!24 ) / 13\!\cdots\!04 \) |
\(\beta_{2}\) | \(=\) | \( ( - 11\!\cdots\!90 \nu^{15} + \cdots + 62\!\cdots\!96 ) / 12\!\cdots\!92 \) |
\(\beta_{3}\) | \(=\) | \( ( - 11\!\cdots\!90 \nu^{15} + \cdots + 62\!\cdots\!96 ) / 12\!\cdots\!92 \) |
\(\beta_{4}\) | \(=\) | \( ( 32\!\cdots\!85 \nu^{15} + \cdots - 16\!\cdots\!08 ) / 12\!\cdots\!92 \) |
\(\beta_{5}\) | \(=\) | \( ( 40\!\cdots\!73 \nu^{15} + \cdots - 21\!\cdots\!08 ) / 12\!\cdots\!28 \) |
\(\beta_{6}\) | \(=\) | \( ( - 16\!\cdots\!11 \nu^{15} + \cdots - 42\!\cdots\!00 ) / 25\!\cdots\!84 \) |
\(\beta_{7}\) | \(=\) | \( ( 12\!\cdots\!21 \nu^{15} + \cdots + 91\!\cdots\!12 ) / 18\!\cdots\!56 \) |
\(\beta_{8}\) | \(=\) | \( ( 38\!\cdots\!09 \nu^{15} + \cdots - 20\!\cdots\!64 ) / 46\!\cdots\!64 \) |
\(\beta_{9}\) | \(=\) | \( ( - 10\!\cdots\!46 \nu^{15} + \cdots - 10\!\cdots\!40 ) / 12\!\cdots\!92 \) |
\(\beta_{10}\) | \(=\) | \( ( - 83\!\cdots\!77 \nu^{15} + \cdots + 12\!\cdots\!16 ) / 64\!\cdots\!96 \) |
\(\beta_{11}\) | \(=\) | \( ( - 25\!\cdots\!43 \nu^{15} + \cdots + 12\!\cdots\!92 ) / 18\!\cdots\!56 \) |
\(\beta_{12}\) | \(=\) | \( ( 96\!\cdots\!19 \nu^{15} + \cdots - 13\!\cdots\!76 ) / 64\!\cdots\!96 \) |
\(\beta_{13}\) | \(=\) | \( ( 26\!\cdots\!95 \nu^{15} + \cdots - 11\!\cdots\!92 ) / 12\!\cdots\!92 \) |
\(\beta_{14}\) | \(=\) | \( ( - 27\!\cdots\!00 \nu^{15} + \cdots - 18\!\cdots\!16 ) / 12\!\cdots\!92 \) |
\(\beta_{15}\) | \(=\) | \( ( - 30\!\cdots\!35 \nu^{15} + \cdots - 18\!\cdots\!96 ) / 12\!\cdots\!92 \) |
\(\nu\) | \(=\) | \( ( -\beta_{3} + \beta_{2} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( 2\beta_{15} + \beta_{13} + \beta_{10} - 3\beta_{9} - \beta_{8} + 4\beta_{7} - \beta_{4} + \beta _1 - 18 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( - \beta_{15} + 2 \beta_{14} - 19 \beta_{13} + 4 \beta_{12} + \beta_{11} + 5 \beta_{10} + 15 \beta_{9} + 25 \beta_{8} - 6 \beta_{7} + 22 \beta_{6} + 176 \beta_{5} - 8 \beta_{4} + 41 \beta_{3} - 45 \beta_{2} + 2 \beta _1 + 47 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( - 109 \beta_{15} - 4 \beta_{14} + 16 \beta_{13} + 59 \beta_{12} + 32 \beta_{11} - 50 \beta_{10} + 105 \beta_{9} - 305 \beta_{7} + 27 \beta_{6} - 317 \beta_{5} + 41 \beta_{4} + 135 \beta_{3} - 9 \beta_{2} + 29 \beta _1 + 332 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 279 \beta_{15} - 468 \beta_{14} + 871 \beta_{13} - 442 \beta_{12} + 198 \beta_{11} - 563 \beta_{10} - 1557 \beta_{9} - 941 \beta_{8} + 1315 \beta_{7} - 1338 \beta_{6} - 14216 \beta_{5} + 428 \beta_{4} - 3351 \beta_{3} + \cdots - 6081 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( 5209 \beta_{15} - 24 \beta_{14} - 1485 \beta_{13} - 5865 \beta_{12} - 3017 \beta_{11} + 1797 \beta_{10} - 2202 \beta_{9} + 707 \beta_{8} + 16955 \beta_{7} - 1410 \beta_{6} + 25140 \beta_{5} - 2985 \beta_{4} + \cdots - 22583 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( - 24675 \beta_{15} + 32956 \beta_{14} - 45339 \beta_{13} + 43890 \beta_{12} - 8589 \beta_{11} + 27685 \beta_{10} + 79415 \beta_{9} + 39298 \beta_{8} - 94080 \beta_{7} + 82082 \beta_{6} + \cdots + 456701 ) / 4 \) |
\(\nu^{8}\) | \(=\) | \( ( - 281111 \beta_{15} + 7840 \beta_{14} + 70358 \beta_{13} + 314552 \beta_{12} + 189210 \beta_{11} - 87940 \beta_{10} + 59028 \beta_{9} - 58808 \beta_{8} - 929365 \beta_{7} + 47320 \beta_{6} + \cdots + 1501223 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( 1823989 \beta_{15} - 1925334 \beta_{14} + 2500899 \beta_{13} - 2474216 \beta_{12} + 189113 \beta_{11} - 1228677 \beta_{10} - 4490331 \beta_{9} - 2225317 \beta_{8} + 6002452 \beta_{7} + \cdots - 31098363 ) / 4 \) |
\(\nu^{10}\) | \(=\) | \( ( 15761336 \beta_{15} - 406340 \beta_{14} - 4194951 \beta_{13} - 16548277 \beta_{12} - 10315623 \beta_{11} + 5308169 \beta_{10} - 1948744 \beta_{9} + 4285624 \beta_{8} + \cdots - 80877241 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( - 127051415 \beta_{15} + 107202502 \beta_{14} - 138044043 \beta_{13} + 143357780 \beta_{12} - 3359816 \beta_{11} + 55725989 \beta_{10} + 271490321 \beta_{9} + \cdots + 1912446727 ) / 4 \) |
\(\nu^{12}\) | \(=\) | \( ( - 869585982 \beta_{15} - 516652 \beta_{14} + 290868584 \beta_{13} + 894758565 \beta_{12} + 576135224 \beta_{11} - 329410426 \beta_{10} + 28956455 \beta_{9} + \cdots + 3998642990 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( 8475803371 \beta_{15} - 5956688682 \beta_{14} + 7417316179 \beta_{13} - 9265432204 \beta_{12} - 419542935 \beta_{11} - 2598095535 \beta_{10} - 16031100855 \beta_{9} + \cdots - 112654415615 ) / 4 \) |
\(\nu^{14}\) | \(=\) | \( ( 46926047268 \beta_{15} + 2026278086 \beta_{14} - 19502362417 \beta_{13} - 47573182678 \beta_{12} - 32663193668 \beta_{11} + 20102185643 \beta_{10} + \cdots - 192474230932 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( - 546882905353 \beta_{15} + 333119813700 \beta_{14} - 393537231435 \beta_{13} + 600336340426 \beta_{12} + 70136917089 \beta_{11} + 120588039003 \beta_{10} + \cdots + 6633855205101 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).
\(n\) | \(73\) | \(127\) | \(253\) | \(281\) |
\(\chi(n)\) | \(1 + \beta_{5}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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73.1 |
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0 | 0 | 0 | −7.85541 | + | 4.53532i | 0 | 1.17763 | + | 6.90023i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
73.2 | 0 | 0 | 0 | −4.51611 | + | 2.60738i | 0 | 6.91807 | − | 1.06788i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
73.3 | 0 | 0 | 0 | −2.18217 | + | 1.25988i | 0 | −3.00972 | − | 6.31993i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
73.4 | 0 | 0 | 0 | −1.27883 | + | 0.738332i | 0 | −4.08597 | + | 5.68374i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
73.5 | 0 | 0 | 0 | 1.27883 | − | 0.738332i | 0 | −4.08597 | + | 5.68374i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
73.6 | 0 | 0 | 0 | 2.18217 | − | 1.25988i | 0 | −3.00972 | − | 6.31993i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
73.7 | 0 | 0 | 0 | 4.51611 | − | 2.60738i | 0 | 6.91807 | − | 1.06788i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
73.8 | 0 | 0 | 0 | 7.85541 | − | 4.53532i | 0 | 1.17763 | + | 6.90023i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.1 | 0 | 0 | 0 | −7.85541 | − | 4.53532i | 0 | 1.17763 | − | 6.90023i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.2 | 0 | 0 | 0 | −4.51611 | − | 2.60738i | 0 | 6.91807 | + | 1.06788i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.3 | 0 | 0 | 0 | −2.18217 | − | 1.25988i | 0 | −3.00972 | + | 6.31993i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.4 | 0 | 0 | 0 | −1.27883 | − | 0.738332i | 0 | −4.08597 | − | 5.68374i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.5 | 0 | 0 | 0 | 1.27883 | + | 0.738332i | 0 | −4.08597 | − | 5.68374i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.6 | 0 | 0 | 0 | 2.18217 | + | 1.25988i | 0 | −3.00972 | + | 6.31993i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.7 | 0 | 0 | 0 | 4.51611 | + | 2.60738i | 0 | 6.91807 | + | 1.06788i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
145.8 | 0 | 0 | 0 | 7.85541 | + | 4.53532i | 0 | 1.17763 | − | 6.90023i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 504.3.by.d | ✓ | 16 |
3.b | odd | 2 | 1 | inner | 504.3.by.d | ✓ | 16 |
4.b | odd | 2 | 1 | 1008.3.cg.q | 16 | ||
7.c | even | 3 | 1 | 3528.3.f.i | 16 | ||
7.d | odd | 6 | 1 | inner | 504.3.by.d | ✓ | 16 |
7.d | odd | 6 | 1 | 3528.3.f.i | 16 | ||
12.b | even | 2 | 1 | 1008.3.cg.q | 16 | ||
21.g | even | 6 | 1 | inner | 504.3.by.d | ✓ | 16 |
21.g | even | 6 | 1 | 3528.3.f.i | 16 | ||
21.h | odd | 6 | 1 | 3528.3.f.i | 16 | ||
28.f | even | 6 | 1 | 1008.3.cg.q | 16 | ||
84.j | odd | 6 | 1 | 1008.3.cg.q | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.3.by.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
504.3.by.d | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
504.3.by.d | ✓ | 16 | 7.d | odd | 6 | 1 | inner |
504.3.by.d | ✓ | 16 | 21.g | even | 6 | 1 | inner |
1008.3.cg.q | 16 | 4.b | odd | 2 | 1 | ||
1008.3.cg.q | 16 | 12.b | even | 2 | 1 | ||
1008.3.cg.q | 16 | 28.f | even | 6 | 1 | ||
1008.3.cg.q | 16 | 84.j | odd | 6 | 1 | ||
3528.3.f.i | 16 | 7.c | even | 3 | 1 | ||
3528.3.f.i | 16 | 7.d | odd | 6 | 1 | ||
3528.3.f.i | 16 | 21.g | even | 6 | 1 | ||
3528.3.f.i | 16 | 21.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} - 118 T_{5}^{14} + 10739 T_{5}^{12} - 334630 T_{5}^{10} + 7682449 T_{5}^{8} - 58300664 T_{5}^{6} + 325701440 T_{5}^{4} - 638105600 T_{5}^{2} + 959512576 \)
acting on \(S_{3}^{\mathrm{new}}(504, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} \)
$5$
\( T^{16} - 118 T^{14} + \cdots + 959512576 \)
$7$
\( (T^{8} - 2 T^{7} + 48 T^{6} + \cdots + 5764801)^{2} \)
$11$
\( T^{16} + 666 T^{14} + \cdots + 37\!\cdots\!36 \)
$13$
\( (T^{8} + 766 T^{6} + 165641 T^{4} + \cdots + 63043600)^{2} \)
$17$
\( T^{16} - 920 T^{14} + \cdots + 44\!\cdots\!96 \)
$19$
\( (T^{8} + 12 T^{7} - 95 T^{6} + \cdots + 2458624)^{2} \)
$23$
\( T^{16} + 3152 T^{14} + \cdots + 36\!\cdots\!56 \)
$29$
\( (T^{8} - 4554 T^{6} + \cdots + 142968684544)^{2} \)
$31$
\( (T^{8} + 42 T^{7} - 1544 T^{6} + \cdots + 74287318249)^{2} \)
$37$
\( (T^{8} + 34 T^{7} + \cdots + 212012360704)^{2} \)
$41$
\( (T^{8} + 10520 T^{6} + \cdots + 12478867111936)^{2} \)
$43$
\( (T^{4} - 20 T^{3} - 5901 T^{2} + \cdots + 3352240)^{4} \)
$47$
\( T^{16} - 12976 T^{14} + \cdots + 47\!\cdots\!36 \)
$53$
\( T^{16} + 7314 T^{14} + \cdots + 74\!\cdots\!76 \)
$59$
\( T^{16} - 10038 T^{14} + \cdots + 13\!\cdots\!76 \)
$61$
\( (T^{8} - 108 T^{7} + \cdots + 2040555110400)^{2} \)
$67$
\( (T^{8} - 28 T^{7} + 6317 T^{6} + \cdots + 2997781504)^{2} \)
$71$
\( (T^{8} - 6544 T^{6} + \cdots + 21724401664)^{2} \)
$73$
\( (T^{8} - 78 T^{7} + \cdots + 999648030976)^{2} \)
$79$
\( (T^{8} - 14 T^{7} + \cdots + 3096660231289)^{2} \)
$83$
\( (T^{8} + 5366 T^{6} + \cdots + 1728025927936)^{2} \)
$89$
\( T^{16} - 46136 T^{14} + \cdots + 46\!\cdots\!76 \)
$97$
\( (T^{8} + 48470 T^{6} + \cdots + 14\!\cdots\!00)^{2} \)
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