Newspace parameters
Level: | \( N \) | \(=\) | \( 1089 = 3^{2} \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1089.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(29.6731007888\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
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: | \( x^{16} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{12}\cdot 11^{4} \) |
Twist minimal: | no (minimal twist has level 99) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{16} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641 \) :
\(\beta_{1}\) | \(=\) | \( ( 45667752444 \nu^{14} - 597804978248 \nu^{12} + 3528390713769 \nu^{10} - 621669021475 \nu^{8} + 721124010273630 \nu^{6} + \cdots + 332158447600837 ) / 39\!\cdots\!45 \) |
\(\beta_{2}\) | \(=\) | \( ( 5084684619 \nu^{14} - 102502036623 \nu^{12} + 1068420907817 \nu^{10} - 6488335560210 \nu^{8} + 114684022273170 \nu^{6} + \cdots + 22\!\cdots\!21 ) / 355575559329595 \) |
\(\beta_{3}\) | \(=\) | \( ( 7530891822 \nu^{15} - 161922628410 \nu^{13} + 1773071437239 \nu^{11} - 11760003820535 \nu^{9} + 185084618505135 \nu^{7} + \cdots + 243765923221832 \nu ) / 355575559329595 \) |
\(\beta_{4}\) | \(=\) | \( ( - 6262229615 \nu^{14} + 123470462925 \nu^{12} - 1253712964279 \nu^{10} + 7604818484385 \nu^{8} - 143729875320340 \nu^{6} + \cdots + 159603221202733 ) / 355575559329595 \) |
\(\beta_{5}\) | \(=\) | \( ( - 226078922543 \nu^{14} + 5080279956993 \nu^{12} - 57615885824889 \nu^{10} + 397910034267221 \nu^{8} + \cdots + 619468120074969 ) / 39\!\cdots\!45 \) |
\(\beta_{6}\) | \(=\) | \( ( 445551515192 \nu^{15} - 9307380902916 \nu^{13} + 100246591559245 \nu^{11} - 655811636243113 \nu^{9} + \cdots - 11\!\cdots\!28 \nu ) / 39\!\cdots\!45 \) |
\(\beta_{7}\) | \(=\) | \( ( 373009540956 \nu^{14} - 8272667397206 \nu^{12} + 93216689770491 \nu^{10} - 641417043065372 \nu^{8} + \cdots - 908176758810299 ) / 39\!\cdots\!45 \) |
\(\beta_{8}\) | \(=\) | \( ( - 598241835319 \nu^{15} + 13289199325505 \nu^{13} - 150985563543754 \nu^{11} + \cdots + 26\!\cdots\!82 \nu ) / 39\!\cdots\!45 \) |
\(\beta_{9}\) | \(=\) | \( ( - 675341589602 \nu^{15} + 13868921723197 \nu^{13} - 147009169987501 \nu^{11} + 940994096476483 \nu^{9} + \cdots + 13\!\cdots\!50 \nu ) / 39\!\cdots\!45 \) |
\(\beta_{10}\) | \(=\) | \( ( 87882372397 \nu^{15} - 1883160071640 \nu^{13} + 20577181071988 \nu^{11} - 136163991479220 \nu^{9} + \cdots + 28\!\cdots\!24 \nu ) / 355575559329595 \) |
\(\beta_{11}\) | \(=\) | \( ( 76482029149 \nu^{14} - 1530855816173 \nu^{12} + 15742685210216 \nu^{10} - 96069570646780 \nu^{8} + \cdots + 40\!\cdots\!68 ) / 355575559329595 \) |
\(\beta_{12}\) | \(=\) | \( ( 914548814197 \nu^{14} - 18896431939389 \nu^{12} + 202148975636349 \nu^{10} + \cdots - 757739325471612 ) / 39\!\cdots\!45 \) |
\(\beta_{13}\) | \(=\) | \( ( - 125817353665 \nu^{15} + 2586721454850 \nu^{13} - 27473712050138 \nu^{11} + 176068897871470 \nu^{9} + \cdots - 39\!\cdots\!64 \nu ) / 355575559329595 \) |
\(\beta_{14}\) | \(=\) | \( ( 1590497768908 \nu^{15} - 33743007445253 \nu^{13} + 368452116183311 \nu^{11} + \cdots - 47\!\cdots\!68 \nu ) / 39\!\cdots\!45 \) |
\(\beta_{15}\) | \(=\) | \( ( 26847260804 \nu^{15} - 566324648655 \nu^{13} + 6124941929985 \nu^{11} - 40072005667775 \nu^{9} + 649352121603960 \nu^{7} + \cdots + 860414508504650 \nu ) / 32325050848145 \) |
\(\nu\) | \(=\) | \( ( -\beta_{15} - \beta_{14} - 2\beta_{13} + 2\beta_{10} + 4\beta_{9} + 6\beta_{6} + 2\beta_{3} ) / 22 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{2} + \beta _1 + 6 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{15} + 2\beta_{14} - \beta_{13} + 3\beta_{10} + 3\beta_{9} + 2\beta_{8} - 12\beta_{3} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( -15\beta_{12} + \beta_{11} + 27\beta_{7} - 6\beta_{5} + 11\beta_{4} - 3\beta_{2} + 52\beta _1 + 3 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( -2\beta_{15} - \beta_{13} + 19\beta_{10} - 160\beta_{3} \) |
\(\nu^{6}\) | \(=\) | \( ( -215\beta_{12} - 15\beta_{11} + 354\beta_{7} - 137\beta_{5} - 127\beta_{4} + 70\beta_{2} + 735\beta _1 - 225 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( - 179 \beta_{15} - 272 \beta_{14} - 146 \beta_{13} + 640 \beta_{10} - 589 \beta_{9} - 354 \beta_{8} - 397 \beta_{6} - 2889 \beta_{3} ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( - 1420 \beta_{12} - 494 \beta_{11} + 807 \beta_{7} - 3356 \beta_{5} - 2533 \beta_{4} + 4305 \beta_{2} + 5239 \beta _1 - 21042 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( - 1537 \beta_{15} - 2486 \beta_{14} - 1325 \beta_{13} + 4352 \beta_{10} - 13120 \beta_{9} - 5583 \beta_{8} - 18507 \beta_{6} - 12658 \beta_{3} ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( -4305\beta_{11} - 17010\beta_{4} + 43805\beta_{2} - 226193 \) |
\(\nu^{11}\) | \(=\) | \( ( 19463 \beta_{15} - 41352 \beta_{14} + 16436 \beta_{13} - 60241 \beta_{10} - 196922 \beta_{9} - 86435 \beta_{8} - 266948 \beta_{6} + 214479 \beta_{3} ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( ( 362330 \beta_{12} - 108925 \beta_{11} - 379152 \beta_{7} + 578566 \beta_{5} - 531443 \beta_{4} + 980208 \beta_{2} - 1290483 \beta _1 - 4851218 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( 531443 \beta_{15} - 578566 \beta_{14} + 422518 \beta_{13} - 2043094 \beta_{10} - 1869049 \beta_{9} - 940896 \beta_{8} - 2031965 \beta_{6} + 10065851 \beta_{3} ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( ( 9766705 \beta_{12} - 640368 \beta_{11} - 13570866 \beta_{7} + 10225888 \beta_{5} - 3503633 \beta_{4} + 5282864 \beta_{2} - 33907236 \beta _1 - 25255599 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( 4302656\beta_{15} + 3322448\beta_{13} - 18032177\beta_{10} + 97256460\beta_{3} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).
\(n\) | \(244\) | \(848\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
604.1 |
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− | 3.86322i | 0 | −10.9245 | −4.15040 | 0 | − | 7.44465i | 26.7509i | 0 | 16.0339i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.2 | − | 3.86322i | 0 | −10.9245 | 4.15040 | 0 | 7.44465i | 26.7509i | 0 | − | 16.0339i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.3 | − | 3.11662i | 0 | −5.71334 | −1.42146 | 0 | 3.80859i | 5.33982i | 0 | 4.43016i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.4 | − | 3.11662i | 0 | −5.71334 | 1.42146 | 0 | − | 3.80859i | 5.33982i | 0 | − | 4.43016i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.5 | − | 1.38910i | 0 | 2.07040 | −4.37284 | 0 | 0.612830i | − | 8.43240i | 0 | 6.07432i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.6 | − | 1.38910i | 0 | 2.07040 | 4.37284 | 0 | − | 0.612830i | − | 8.43240i | 0 | − | 6.07432i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.7 | − | 0.657695i | 0 | 3.56744 | −8.10135 | 0 | 4.08611i | − | 4.97707i | 0 | 5.32822i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.8 | − | 0.657695i | 0 | 3.56744 | 8.10135 | 0 | − | 4.08611i | − | 4.97707i | 0 | − | 5.32822i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.9 | 0.657695i | 0 | 3.56744 | −8.10135 | 0 | − | 4.08611i | 4.97707i | 0 | − | 5.32822i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.10 | 0.657695i | 0 | 3.56744 | 8.10135 | 0 | 4.08611i | 4.97707i | 0 | 5.32822i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.11 | 1.38910i | 0 | 2.07040 | −4.37284 | 0 | − | 0.612830i | 8.43240i | 0 | − | 6.07432i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.12 | 1.38910i | 0 | 2.07040 | 4.37284 | 0 | 0.612830i | 8.43240i | 0 | 6.07432i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.13 | 3.11662i | 0 | −5.71334 | −1.42146 | 0 | − | 3.80859i | − | 5.33982i | 0 | − | 4.43016i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.14 | 3.11662i | 0 | −5.71334 | 1.42146 | 0 | 3.80859i | − | 5.33982i | 0 | 4.43016i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.15 | 3.86322i | 0 | −10.9245 | −4.15040 | 0 | 7.44465i | − | 26.7509i | 0 | − | 16.0339i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
604.16 | 3.86322i | 0 | −10.9245 | 4.15040 | 0 | − | 7.44465i | − | 26.7509i | 0 | 16.0339i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.b | odd | 2 | 1 | inner |
33.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1089.3.c.l | 16 | |
3.b | odd | 2 | 1 | inner | 1089.3.c.l | 16 | |
11.b | odd | 2 | 1 | inner | 1089.3.c.l | 16 | |
11.c | even | 5 | 1 | 99.3.k.b | ✓ | 16 | |
11.d | odd | 10 | 1 | 99.3.k.b | ✓ | 16 | |
33.d | even | 2 | 1 | inner | 1089.3.c.l | 16 | |
33.f | even | 10 | 1 | 99.3.k.b | ✓ | 16 | |
33.h | odd | 10 | 1 | 99.3.k.b | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.3.k.b | ✓ | 16 | 11.c | even | 5 | 1 | |
99.3.k.b | ✓ | 16 | 11.d | odd | 10 | 1 | |
99.3.k.b | ✓ | 16 | 33.f | even | 10 | 1 | |
99.3.k.b | ✓ | 16 | 33.h | odd | 10 | 1 | |
1089.3.c.l | 16 | 1.a | even | 1 | 1 | trivial | |
1089.3.c.l | 16 | 3.b | odd | 2 | 1 | inner | |
1089.3.c.l | 16 | 11.b | odd | 2 | 1 | inner | |
1089.3.c.l | 16 | 33.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 27T_{2}^{6} + 204T_{2}^{4} + 363T_{2}^{2} + 121 \)
acting on \(S_{3}^{\mathrm{new}}(1089, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} + 27 T^{6} + 204 T^{4} + 363 T^{2} + \cdots + 121)^{2} \)
$3$
\( T^{16} \)
$5$
\( (T^{8} - 104 T^{6} + 2921 T^{4} + \cdots + 43681)^{2} \)
$7$
\( (T^{8} + 87 T^{6} + 2004 T^{4} + \cdots + 5041)^{2} \)
$11$
\( T^{16} \)
$13$
\( (T^{8} + 822 T^{6} + 199029 T^{4} + \cdots + 441168016)^{2} \)
$17$
\( (T^{8} + 1177 T^{6} + 275249 T^{4} + \cdots + 28344976)^{2} \)
$19$
\( (T^{8} + 1545 T^{6} + \cdots + 5628000400)^{2} \)
$23$
\( (T^{8} - 1430 T^{6} + \cdots + 1301766400)^{2} \)
$29$
\( (T^{8} + 3795 T^{6} + \cdots + 136766832400)^{2} \)
$31$
\( (T^{4} + 20 T^{3} - 2045 T^{2} + \cdots + 592295)^{4} \)
$37$
\( (T^{4} + 70 T^{3} + 1535 T^{2} + \cdots - 21780)^{4} \)
$41$
\( (T^{8} + 5410 T^{6} + \cdots + 328443610000)^{2} \)
$43$
\( (T^{8} + 5102 T^{6} + \cdots + 431696305296)^{2} \)
$47$
\( (T^{8} - 7755 T^{6} + \cdots + 1065065280400)^{2} \)
$53$
\( (T^{8} - 15780 T^{6} + \cdots + 130872341603025)^{2} \)
$59$
\( (T^{8} - 22220 T^{6} + \cdots + 640510301973025)^{2} \)
$61$
\( (T^{8} + 14865 T^{6} + \cdots + 3114589632400)^{2} \)
$67$
\( (T^{4} - 75 T^{3} - 3655 T^{2} + \cdots - 3482380)^{4} \)
$71$
\( (T^{8} - 21555 T^{6} + \cdots + 128324037120400)^{2} \)
$73$
\( (T^{8} + 30217 T^{6} + \cdots + 494466201667216)^{2} \)
$79$
\( (T^{8} + 39735 T^{6} + \cdots + 436852010010025)^{2} \)
$83$
\( (T^{8} + 35437 T^{6} + \cdots + 14\!\cdots\!01)^{2} \)
$89$
\( (T^{8} - 29246 T^{6} + \cdots + 33083847444736)^{2} \)
$97$
\( (T^{4} + 180 T^{3} - 7305 T^{2} + \cdots - 45053405)^{4} \)
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