Newspace parameters
Level: | \( N \) | \(=\) | \( 539 = 7^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 539.s (of order \(30\), degree \(8\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.30393666895\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{30})\) |
Coefficient field: | 16.0.9234096523681640625.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + 24 x^{6} + 8 x^{5} + 80 x^{4} - 160 x^{3} + 128 x^{2} - 128 x + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 77) |
Sato-Tate group: | $\mathrm{U}(1)[D_{30}]$ |
$q$-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 \( x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + 24 x^{6} + 8 x^{5} + 80 x^{4} - 160 x^{3} + 128 x^{2} - 128 x + 256 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - \nu^{15} + 3 \nu^{14} + 261 \nu^{12} - 7 \nu^{11} - 11 \nu^{10} + 16 \nu^{9} + 11 \nu^{8} + 55 \nu^{7} - 32 \nu^{6} - 88 \nu^{5} + 80 \nu^{4} + 32 \nu^{3} + 352 \nu^{2} - 128 \nu - 256 ) / 11392 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{15} + \nu^{14} + 128 \nu^{13} - \nu^{12} - 5 \nu^{11} + 11 \nu^{10} + 8 \nu^{9} + 17 \nu^{8} - 11 \nu^{7} - 32 \nu^{6} + 44 \nu^{5} + 56 \nu^{4} + 96 \nu^{3} - 192 \nu + 128 ) / 11392 \) |
\(\beta_{4}\) | \(=\) | \( ( -11\nu^{15} - 89\nu^{10} - 979\nu^{5} - 1024 ) / 2848 \) |
\(\beta_{5}\) | \(=\) | \( ( - 2 \nu^{15} - 2 \nu^{14} + 11 \nu^{13} + 2 \nu^{12} + 10 \nu^{11} - 22 \nu^{10} - 16 \nu^{9} + 55 \nu^{8} + 22 \nu^{7} + 64 \nu^{6} - 88 \nu^{5} - 112 \nu^{4} + 75 \nu^{3} + 384 \nu - 256 ) / 712 \) |
\(\beta_{6}\) | \(=\) | \( ( 45 \nu^{15} - 135 \nu^{14} - 353 \nu^{12} + 315 \nu^{11} + 495 \nu^{10} - 720 \nu^{9} - 495 \nu^{8} - 2475 \nu^{7} + 1440 \nu^{6} + 3960 \nu^{5} - 3600 \nu^{4} - 1440 \nu^{3} - 15840 \nu^{2} + \cdots + 11520 ) / 11392 \) |
\(\beta_{7}\) | \(=\) | \( ( - 91 \nu^{15} - 37 \nu^{14} - 182 \nu^{13} + 455 \nu^{12} - 455 \nu^{11} - 91 \nu^{10} - 546 \nu^{9} - 455 \nu^{8} + 1911 \nu^{7} - 910 \nu^{6} - 2184 \nu^{5} - 728 \nu^{4} - 7280 \nu^{3} + \cdots + 11648 ) / 11392 \) |
\(\beta_{8}\) | \(=\) | \( ( - 2 \nu^{15} - 2 \nu^{14} + 11 \nu^{13} + 2 \nu^{12} + 10 \nu^{11} - 22 \nu^{10} - 16 \nu^{9} + 55 \nu^{8} + 22 \nu^{7} + 64 \nu^{6} - 88 \nu^{5} - 112 \nu^{4} + 431 \nu^{3} + 384 \nu - 256 ) / 356 \) |
\(\beta_{9}\) | \(=\) | \( ( - 91 \nu^{15} - 91 \nu^{14} - 256 \nu^{13} + 91 \nu^{12} + 455 \nu^{11} - 1001 \nu^{10} - 728 \nu^{9} - 1547 \nu^{8} + 1001 \nu^{7} + 2912 \nu^{6} - 4004 \nu^{5} - 5096 \nu^{4} + \cdots - 11648 ) / 11392 \) |
\(\beta_{10}\) | \(=\) | \( ( - 23 \nu^{15} + 46 \nu^{14} - 115 \nu^{13} + 115 \nu^{12} - 422 \nu^{11} + 138 \nu^{10} + 115 \nu^{9} - 483 \nu^{8} + 230 \nu^{7} - 1495 \nu^{6} + 184 \nu^{5} + 1840 \nu^{4} - 3680 \nu^{3} + \cdots + 5888 ) / 5696 \) |
\(\beta_{11}\) | \(=\) | \( ( 33 \nu^{15} - 57 \nu^{14} + 66 \nu^{13} - 165 \nu^{12} + 165 \nu^{11} + 33 \nu^{10} - 514 \nu^{9} + 165 \nu^{8} - 693 \nu^{7} + 330 \nu^{6} + 792 \nu^{5} - 1872 \nu^{4} + 2640 \nu^{3} - 5280 \nu^{2} + \cdots - 4224 ) / 5696 \) |
\(\beta_{12}\) | \(=\) | \( ( -\nu^{15} + 93 ) / 89 \) |
\(\beta_{13}\) | \(=\) | \( ( -17\nu^{15} - 267\nu^{10} - 1513\nu^{5} - 5984 ) / 1424 \) |
\(\beta_{14}\) | \(=\) | \( ( - \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 5 \nu^{12} + \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 21 \nu^{8} + 10 \nu^{7} + 24 \nu^{6} + 8 \nu^{5} + 80 \nu^{4} - 160 \nu^{3} + 128 \nu^{2} - 35 \nu + 256 ) / 178 \) |
\(\beta_{15}\) | \(=\) | \( ( 4 \nu^{15} - 15 \nu^{14} + 8 \nu^{13} - 20 \nu^{12} + 20 \nu^{11} + 4 \nu^{10} - 65 \nu^{9} + 20 \nu^{8} - 84 \nu^{7} + 40 \nu^{6} + 96 \nu^{5} - 235 \nu^{4} + 320 \nu^{3} - 640 \nu^{2} + 512 \nu - 512 ) / 712 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( - \beta_{15} + \beta_{13} - 2 \beta_{10} - \beta_{9} + 2 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{8} - 2\beta_{5} \) |
\(\nu^{4}\) | \(=\) | \( 3\beta_{15} + 2\beta_{14} + 2\beta_{10} - 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + 2 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{13} + \beta_{12} - 6\beta_{4} + 1 \) |
\(\nu^{6}\) | \(=\) | \( 2\beta_{14} - 5\beta_{12} - 5\beta_{11} + 2\beta_{10} + 5\beta_{8} + 5\beta_{6} - 5\beta_{2} - 5 \) |
\(\nu^{7}\) | \(=\) | \( 7 \beta_{15} - 7 \beta_{13} + 10 \beta_{10} + 7 \beta_{9} - 10 \beta_{7} - 10 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} - 3 \beta_{3} - 7 \beta_1 \) |
\(\nu^{8}\) | \(=\) | \( -3\beta_{9} - 3\beta_{8} + 14\beta_{5} - 17\beta_{3} \) |
\(\nu^{9}\) | \(=\) | \( -6\beta_{14} - 17\beta_{11} - 6\beta_{10} + 6\beta_{6} - 6\beta_{5} - 6 \) |
\(\nu^{10}\) | \(=\) | \( -11\beta_{13} + 34\beta_{4} - 34 \) |
\(\nu^{11}\) | \(=\) | \( 23\beta_{12} + 23\beta_{11} - 22\beta_{10} - 23\beta_{8} - 23\beta_{6} + 23\beta_{2} - 23\beta _1 + 23 \) |
\(\nu^{12}\) | \(=\) | \( \beta_{6} + 45\beta_{2} \) |
\(\nu^{13}\) | \(=\) | \( \beta_{9} + 91\beta_{3} \) |
\(\nu^{14}\) | \(=\) | \( -91\beta_{15} + 91\beta_{11} + 2\beta_{7} \) |
\(\nu^{15}\) | \(=\) | \( -89\beta_{12} + 93 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).
\(n\) | \(199\) | \(442\) |
\(\chi(n)\) | \(1 - \beta_{4}\) | \(-\beta_{10} - \beta_{14}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
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−1.75895 | + | 0.184873i | 0 | 1.10343 | − | 0.234542i | 0 | 0 | 0 | 1.46663 | − | 0.476537i | −0.313585 | − | 2.98357i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.2 | −0.132743 | + | 0.0139519i | 0 | −1.93887 | + | 0.412119i | 0 | 0 | 0 | 0.505505 | − | 0.164249i | −0.313585 | − | 2.98357i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
68.1 | −1.15386 | + | 1.03894i | 0 | 0.0429395 | − | 0.408542i | 0 | 0 | 0 | −1.45037 | − | 1.99627i | 2.00739 | + | 2.22943i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
68.2 | 2.02748 | − | 1.82555i | 0 | 0.568981 | − | 5.41349i | 0 | 0 | 0 | −5.52176 | − | 7.60006i | 2.00739 | + | 2.22943i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
117.1 | 0.0542889 | + | 0.121935i | 0 | 1.32634 | − | 1.47305i | 0 | 0 | 0 | 0.505505 | + | 0.164249i | 2.74064 | − | 1.22021i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
117.2 | 0.719370 | + | 1.61573i | 0 | −0.754835 | + | 0.838329i | 0 | 0 | 0 | 1.46663 | + | 0.476537i | 2.74064 | − | 1.22021i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
129.1 | 0.0542889 | − | 0.121935i | 0 | 1.32634 | + | 1.47305i | 0 | 0 | 0 | 0.505505 | − | 0.164249i | 2.74064 | + | 1.22021i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
129.2 | 0.719370 | − | 1.61573i | 0 | −0.754835 | − | 0.838329i | 0 | 0 | 0 | 1.46663 | − | 0.476537i | 2.74064 | + | 1.22021i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
178.1 | −0.322819 | − | 1.51874i | 0 | −0.375277 | + | 0.167084i | 0 | 0 | 0 | −1.45037 | − | 1.99627i | −2.93444 | + | 0.623735i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
178.2 | 0.567234 | + | 2.66862i | 0 | −4.97271 | + | 2.21399i | 0 | 0 | 0 | −5.52176 | − | 7.60006i | −2.93444 | + | 0.623735i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
215.1 | −0.322819 | + | 1.51874i | 0 | −0.375277 | − | 0.167084i | 0 | 0 | 0 | −1.45037 | + | 1.99627i | −2.93444 | − | 0.623735i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
215.2 | 0.567234 | − | 2.66862i | 0 | −4.97271 | − | 2.21399i | 0 | 0 | 0 | −5.52176 | + | 7.60006i | −2.93444 | − | 0.623735i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
227.1 | −1.75895 | − | 0.184873i | 0 | 1.10343 | + | 0.234542i | 0 | 0 | 0 | 1.46663 | + | 0.476537i | −0.313585 | + | 2.98357i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
227.2 | −0.132743 | − | 0.0139519i | 0 | −1.93887 | − | 0.412119i | 0 | 0 | 0 | 0.505505 | + | 0.164249i | −0.313585 | + | 2.98357i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
325.1 | −1.15386 | − | 1.03894i | 0 | 0.0429395 | + | 0.408542i | 0 | 0 | 0 | −1.45037 | + | 1.99627i | 2.00739 | − | 2.22943i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
325.2 | 2.02748 | + | 1.82555i | 0 | 0.568981 | + | 5.41349i | 0 | 0 | 0 | −5.52176 | + | 7.60006i | 2.00739 | − | 2.22943i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
11.d | odd | 10 | 1 | inner |
77.l | even | 10 | 1 | inner |
77.n | even | 30 | 1 | inner |
77.o | odd | 30 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 539.2.s.a | 16 | |
7.b | odd | 2 | 1 | CM | 539.2.s.a | 16 | |
7.c | even | 3 | 1 | 77.2.l.a | ✓ | 8 | |
7.c | even | 3 | 1 | inner | 539.2.s.a | 16 | |
7.d | odd | 6 | 1 | 77.2.l.a | ✓ | 8 | |
7.d | odd | 6 | 1 | inner | 539.2.s.a | 16 | |
11.d | odd | 10 | 1 | inner | 539.2.s.a | 16 | |
21.g | even | 6 | 1 | 693.2.bu.a | 8 | ||
21.h | odd | 6 | 1 | 693.2.bu.a | 8 | ||
77.h | odd | 6 | 1 | 847.2.l.c | 8 | ||
77.i | even | 6 | 1 | 847.2.l.c | 8 | ||
77.l | even | 10 | 1 | inner | 539.2.s.a | 16 | |
77.m | even | 15 | 1 | 847.2.b.b | 8 | ||
77.m | even | 15 | 1 | 847.2.l.a | 8 | ||
77.m | even | 15 | 1 | 847.2.l.c | 8 | ||
77.m | even | 15 | 1 | 847.2.l.d | 8 | ||
77.n | even | 30 | 1 | 77.2.l.a | ✓ | 8 | |
77.n | even | 30 | 1 | inner | 539.2.s.a | 16 | |
77.n | even | 30 | 1 | 847.2.b.b | 8 | ||
77.n | even | 30 | 1 | 847.2.l.a | 8 | ||
77.n | even | 30 | 1 | 847.2.l.d | 8 | ||
77.o | odd | 30 | 1 | 77.2.l.a | ✓ | 8 | |
77.o | odd | 30 | 1 | inner | 539.2.s.a | 16 | |
77.o | odd | 30 | 1 | 847.2.b.b | 8 | ||
77.o | odd | 30 | 1 | 847.2.l.a | 8 | ||
77.o | odd | 30 | 1 | 847.2.l.d | 8 | ||
77.p | odd | 30 | 1 | 847.2.b.b | 8 | ||
77.p | odd | 30 | 1 | 847.2.l.a | 8 | ||
77.p | odd | 30 | 1 | 847.2.l.c | 8 | ||
77.p | odd | 30 | 1 | 847.2.l.d | 8 | ||
231.be | even | 30 | 1 | 693.2.bu.a | 8 | ||
231.bf | odd | 30 | 1 | 693.2.bu.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
77.2.l.a | ✓ | 8 | 7.c | even | 3 | 1 | |
77.2.l.a | ✓ | 8 | 7.d | odd | 6 | 1 | |
77.2.l.a | ✓ | 8 | 77.n | even | 30 | 1 | |
77.2.l.a | ✓ | 8 | 77.o | odd | 30 | 1 | |
539.2.s.a | 16 | 1.a | even | 1 | 1 | trivial | |
539.2.s.a | 16 | 7.b | odd | 2 | 1 | CM | |
539.2.s.a | 16 | 7.c | even | 3 | 1 | inner | |
539.2.s.a | 16 | 7.d | odd | 6 | 1 | inner | |
539.2.s.a | 16 | 11.d | odd | 10 | 1 | inner | |
539.2.s.a | 16 | 77.l | even | 10 | 1 | inner | |
539.2.s.a | 16 | 77.n | even | 30 | 1 | inner | |
539.2.s.a | 16 | 77.o | odd | 30 | 1 | inner | |
693.2.bu.a | 8 | 21.g | even | 6 | 1 | ||
693.2.bu.a | 8 | 21.h | odd | 6 | 1 | ||
693.2.bu.a | 8 | 231.be | even | 30 | 1 | ||
693.2.bu.a | 8 | 231.bf | odd | 30 | 1 | ||
847.2.b.b | 8 | 77.m | even | 15 | 1 | ||
847.2.b.b | 8 | 77.n | even | 30 | 1 | ||
847.2.b.b | 8 | 77.o | odd | 30 | 1 | ||
847.2.b.b | 8 | 77.p | odd | 30 | 1 | ||
847.2.l.a | 8 | 77.m | even | 15 | 1 | ||
847.2.l.a | 8 | 77.n | even | 30 | 1 | ||
847.2.l.a | 8 | 77.o | odd | 30 | 1 | ||
847.2.l.a | 8 | 77.p | odd | 30 | 1 | ||
847.2.l.c | 8 | 77.h | odd | 6 | 1 | ||
847.2.l.c | 8 | 77.i | even | 6 | 1 | ||
847.2.l.c | 8 | 77.m | even | 15 | 1 | ||
847.2.l.c | 8 | 77.p | odd | 30 | 1 | ||
847.2.l.d | 8 | 77.m | even | 15 | 1 | ||
847.2.l.d | 8 | 77.n | even | 30 | 1 | ||
847.2.l.d | 8 | 77.o | odd | 30 | 1 | ||
847.2.l.d | 8 | 77.p | odd | 30 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 7 T_{2}^{14} + 20 T_{2}^{13} + 30 T_{2}^{12} + 140 T_{2}^{11} + 367 T_{2}^{10} + 800 T_{2}^{9} + 1529 T_{2}^{8} + 2600 T_{2}^{7} + 3713 T_{2}^{6} + 4300 T_{2}^{5} + 3770 T_{2}^{4} + 530 T_{2}^{3} + 33 T_{2}^{2} + 10 T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} + 7 T^{14} + 20 T^{13} + 30 T^{12} + \cdots + 1 \)
$3$
\( T^{16} \)
$5$
\( T^{16} \)
$7$
\( T^{16} \)
$11$
\( T^{16} - 4 T^{15} + 11 T^{14} + \cdots + 214358881 \)
$13$
\( T^{16} \)
$17$
\( T^{16} \)
$19$
\( T^{16} \)
$23$
\( (T^{8} + 8 T^{7} + 115 T^{6} + \cdots + 383161)^{2} \)
$29$
\( (T^{8} - 112 T^{6} - 290 T^{5} + \cdots + 259081)^{2} \)
$31$
\( T^{16} \)
$37$
\( T^{16} - 18 T^{15} + \cdots + 4810832476321 \)
$41$
\( T^{16} \)
$43$
\( (T^{8} + 402 T^{6} + 53459 T^{4} + \cdots + 16072081)^{2} \)
$47$
\( T^{16} \)
$53$
\( T^{16} + 30 T^{15} + \cdots + 36325030350625 \)
$59$
\( T^{16} \)
$61$
\( T^{16} \)
$67$
\( (T^{8} + 4 T^{7} + 335 T^{6} + \cdots + 300710281)^{2} \)
$71$
\( (T^{8} + 48 T^{7} + 1123 T^{6} + \cdots + 19321)^{2} \)
$73$
\( T^{16} \)
$79$
\( T^{16} + \cdots + 544688429550721 \)
$83$
\( T^{16} \)
$89$
\( T^{16} \)
$97$
\( T^{16} \)
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