Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [560,2,Mod(17,560)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(560, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("560.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 560 = 2^{4} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 560.ci (of order \(12\), degree \(4\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(4.47162251319\) |
| 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
|
|
| Defining polynomial: |
\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + \cdots + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 140) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
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
\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + \cdots + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 25 \nu^{14} - 175 \nu^{13} + 964 \nu^{12} - 3509 \nu^{11} + 10031 \nu^{10} - 22160 \nu^{9} + \cdots - 4676 ) / 2156 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 79 \nu^{14} + 553 \nu^{13} - 3422 \nu^{12} + 13343 \nu^{11} - 44289 \nu^{10} + 112314 \nu^{9} + \cdots - 11564 ) / 308 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4834 \nu^{15} - 101205 \nu^{14} + 704128 \nu^{13} - 3926292 \nu^{12} + 15242328 \nu^{11} + \cdots - 14479640 ) / 466774 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4834 \nu^{15} + 28695 \nu^{14} - 205172 \nu^{13} + 1782813 \nu^{12} - 7191402 \nu^{11} + \cdots + 8834812 ) / 466774 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 17435 \nu^{15} + 211950 \nu^{14} - 1405979 \nu^{13} + 7058965 \nu^{12} - 26329116 \nu^{11} + \cdots + 18788784 ) / 933548 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 17435 \nu^{15} + 35286 \nu^{14} - 169331 \nu^{13} - 612063 \nu^{12} + 3620628 \nu^{11} + \cdots + 8216656 ) / 933548 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2249 \nu^{15} - 27909 \nu^{14} + 180284 \nu^{13} - 887769 \nu^{12} + 3199847 \nu^{11} + \cdots - 796740 ) / 84868 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 24739 \nu^{15} - 207842 \nu^{14} + 1289025 \nu^{13} - 5512533 \nu^{12} + 18704048 \nu^{11} + \cdots - 1768592 ) / 933548 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 24739 \nu^{15} - 163243 \nu^{14} + 976832 \nu^{13} - 3587415 \nu^{12} + 11211849 \nu^{11} + \cdots + 9046016 ) / 933548 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 34102 \nu^{15} + 371809 \nu^{14} - 2442791 \nu^{13} + 11725816 \nu^{12} - 42750227 \nu^{11} + \cdots + 21672924 ) / 933548 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 34102 \nu^{15} + 139721 \nu^{14} - 818175 \nu^{13} + 1712258 \nu^{12} - 3788887 \nu^{11} + \cdots - 8891680 ) / 933548 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 25839 \nu^{15} + 131224 \nu^{14} - 795654 \nu^{13} + 2382058 \nu^{12} - 7119904 \nu^{11} + \cdots - 2395204 ) / 466774 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 6947 \nu^{15} - 52752 \nu^{14} + 331837 \nu^{13} - 1358817 \nu^{12} + 4632502 \nu^{11} + \cdots - 431788 ) / 84868 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 76661 \nu^{15} + 455233 \nu^{14} - 2856600 \nu^{13} + 10109135 \nu^{12} - 33483101 \nu^{11} + \cdots + 10482976 ) / 933548 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 852 \nu^{15} - 6390 \nu^{14} + 40676 \nu^{13} - 167479 \nu^{12} + 580296 \nu^{11} - 1562572 \nu^{10} + \cdots - 175506 ) / 9526 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - 2\beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{13} - 3\beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 4\beta _1 - 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} + \beta_{12} - 9 \beta_{11} + 7 \beta_{10} - 7 \beta_{9} + \cdots - 9 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 4 \beta_{15} - 4 \beta_{14} + 27 \beta_{13} + 23 \beta_{12} - 21 \beta_{11} + 17 \beta_{10} + \cdots + 17 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 20 \beta_{15} + 14 \beta_{14} + 5 \beta_{13} + 25 \beta_{12} + 49 \beta_{11} - 35 \beta_{10} + 31 \beta_{9} + \cdots + 71 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 70 \beta_{15} + 52 \beta_{14} - 207 \beta_{13} - 137 \beta_{12} + 221 \beta_{11} - 169 \beta_{10} + \cdots - 33 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 116 \beta_{15} - 60 \beta_{14} - 247 \beta_{13} - 351 \beta_{12} - 157 \beta_{11} + 83 \beta_{10} + \cdots - 519 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 800 \beta_{15} - 492 \beta_{14} + 1383 \beta_{13} + 631 \beta_{12} - 1893 \beta_{11} + 1345 \beta_{10} + \cdots - 331 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 156 \beta_{15} - 10 \beta_{14} + 3339 \beta_{13} + 3381 \beta_{12} - 619 \beta_{11} + 705 \beta_{10} + \cdots + 3427 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 7290 \beta_{15} + 4024 \beta_{14} - 7549 \beta_{13} - 1189 \beta_{12} + 14353 \beta_{11} - 9429 \beta_{10} + \cdots + 6173 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 6610 \beta_{15} + 4150 \beta_{14} - 33959 \beta_{13} - 27511 \beta_{12} + 18835 \beta_{11} + \cdots - 19277 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 55336 \beta_{15} - 28692 \beta_{14} + 25577 \beta_{13} - 19709 \beta_{12} - 95753 \beta_{11} + \cdots - 67705 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 115110 \beta_{15} - 62912 \beta_{14} + 294485 \beta_{13} + 195181 \beta_{12} - 243247 \beta_{11} + \cdots + 77217 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 341246 \beta_{15} + 169732 \beta_{14} + 91907 \beta_{13} + 356907 \beta_{12} + 526209 \beta_{11} + \cdots + 606325 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 1314998 \beta_{15} + 682938 \beta_{14} - 2246473 \beta_{13} - 1174327 \beta_{12} + 2458635 \beta_{11} + \cdots + 26483 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(337\) | \(351\) | \(421\) |
| \(\chi(n)\) | \(1 + \beta_{15}\) | \(-\beta_{8} - \beta_{9}\) | \(1\) | \(1\) |
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −2.42519 | + | 0.649827i | 0 | −2.19862 | − | 0.407542i | 0 | −2.59537 | + | 0.513853i | 0 | 2.86119 | − | 1.65191i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | −1.52691 | + | 0.409133i | 0 | 2.14461 | − | 0.632955i | 0 | 2.59572 | + | 0.512081i | 0 | −0.434025 | + | 0.250584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | 0.827625 | − | 0.221762i | 0 | 0.543268 | + | 2.16907i | 0 | −2.22829 | − | 1.42643i | 0 | −1.96229 | + | 1.13293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | 3.12447 | − | 0.837199i | 0 | 1.01073 | − | 1.99460i | 0 | −0.870132 | + | 2.49857i | 0 | 6.46333 | − | 3.73161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.1 | 0 | −2.42519 | − | 0.649827i | 0 | −2.19862 | + | 0.407542i | 0 | −2.59537 | − | 0.513853i | 0 | 2.86119 | + | 1.65191i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | −1.52691 | − | 0.409133i | 0 | 2.14461 | + | 0.632955i | 0 | 2.59572 | − | 0.512081i | 0 | −0.434025 | − | 0.250584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | 0.827625 | + | 0.221762i | 0 | 0.543268 | − | 2.16907i | 0 | −2.22829 | + | 1.42643i | 0 | −1.96229 | − | 1.13293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | 3.12447 | + | 0.837199i | 0 | 1.01073 | + | 1.99460i | 0 | −0.870132 | − | 2.49857i | 0 | 6.46333 | + | 3.73161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.1 | 0 | −0.649827 | + | 2.42519i | 0 | −0.746366 | − | 2.10783i | 0 | 0.513853 | − | 2.59537i | 0 | −2.86119 | − | 1.65191i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.2 | 0 | −0.409133 | + | 1.52691i | 0 | 1.62046 | + | 1.54081i | 0 | 0.512081 | + | 2.59572i | 0 | 0.434025 | + | 0.250584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.3 | 0 | 0.221762 | − | 0.827625i | 0 | −1.60683 | + | 1.55502i | 0 | −1.42643 | − | 2.22829i | 0 | 1.96229 | + | 1.13293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 257.4 | 0 | 0.837199 | − | 3.12447i | 0 | 2.23274 | − | 0.121977i | 0 | 2.49857 | − | 0.870132i | 0 | −6.46333 | − | 3.73161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.1 | 0 | −0.649827 | − | 2.42519i | 0 | −0.746366 | + | 2.10783i | 0 | 0.513853 | + | 2.59537i | 0 | −2.86119 | + | 1.65191i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.2 | 0 | −0.409133 | − | 1.52691i | 0 | 1.62046 | − | 1.54081i | 0 | 0.512081 | − | 2.59572i | 0 | 0.434025 | − | 0.250584i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.3 | 0 | 0.221762 | + | 0.827625i | 0 | −1.60683 | − | 1.55502i | 0 | −1.42643 | + | 2.22829i | 0 | 1.96229 | − | 1.13293i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.4 | 0 | 0.837199 | + | 3.12447i | 0 | 2.23274 | + | 0.121977i | 0 | 2.49857 | + | 0.870132i | 0 | −6.46333 | + | 3.73161i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 35.k | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 560.2.ci.d | 16 | |
| 4.b | odd | 2 | 1 | 140.2.u.a | ✓ | 16 | |
| 5.c | odd | 4 | 1 | inner | 560.2.ci.d | 16 | |
| 7.d | odd | 6 | 1 | inner | 560.2.ci.d | 16 | |
| 12.b | even | 2 | 1 | 1260.2.dq.a | 16 | ||
| 20.d | odd | 2 | 1 | 700.2.bc.b | 16 | ||
| 20.e | even | 4 | 1 | 140.2.u.a | ✓ | 16 | |
| 20.e | even | 4 | 1 | 700.2.bc.b | 16 | ||
| 28.d | even | 2 | 1 | 980.2.v.a | 16 | ||
| 28.f | even | 6 | 1 | 140.2.u.a | ✓ | 16 | |
| 28.f | even | 6 | 1 | 980.2.m.a | 16 | ||
| 28.g | odd | 6 | 1 | 980.2.m.a | 16 | ||
| 28.g | odd | 6 | 1 | 980.2.v.a | 16 | ||
| 35.k | even | 12 | 1 | inner | 560.2.ci.d | 16 | |
| 60.l | odd | 4 | 1 | 1260.2.dq.a | 16 | ||
| 84.j | odd | 6 | 1 | 1260.2.dq.a | 16 | ||
| 140.j | odd | 4 | 1 | 980.2.v.a | 16 | ||
| 140.s | even | 6 | 1 | 700.2.bc.b | 16 | ||
| 140.w | even | 12 | 1 | 980.2.m.a | 16 | ||
| 140.w | even | 12 | 1 | 980.2.v.a | 16 | ||
| 140.x | odd | 12 | 1 | 140.2.u.a | ✓ | 16 | |
| 140.x | odd | 12 | 1 | 700.2.bc.b | 16 | ||
| 140.x | odd | 12 | 1 | 980.2.m.a | 16 | ||
| 420.br | even | 12 | 1 | 1260.2.dq.a | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.2.u.a | ✓ | 16 | 4.b | odd | 2 | 1 | |
| 140.2.u.a | ✓ | 16 | 20.e | even | 4 | 1 | |
| 140.2.u.a | ✓ | 16 | 28.f | even | 6 | 1 | |
| 140.2.u.a | ✓ | 16 | 140.x | odd | 12 | 1 | |
| 560.2.ci.d | 16 | 1.a | even | 1 | 1 | trivial | |
| 560.2.ci.d | 16 | 5.c | odd | 4 | 1 | inner | |
| 560.2.ci.d | 16 | 7.d | odd | 6 | 1 | inner | |
| 560.2.ci.d | 16 | 35.k | even | 12 | 1 | inner | |
| 700.2.bc.b | 16 | 20.d | odd | 2 | 1 | ||
| 700.2.bc.b | 16 | 20.e | even | 4 | 1 | ||
| 700.2.bc.b | 16 | 140.s | even | 6 | 1 | ||
| 700.2.bc.b | 16 | 140.x | odd | 12 | 1 | ||
| 980.2.m.a | 16 | 28.f | even | 6 | 1 | ||
| 980.2.m.a | 16 | 28.g | odd | 6 | 1 | ||
| 980.2.m.a | 16 | 140.w | even | 12 | 1 | ||
| 980.2.m.a | 16 | 140.x | odd | 12 | 1 | ||
| 980.2.v.a | 16 | 28.d | even | 2 | 1 | ||
| 980.2.v.a | 16 | 28.g | odd | 6 | 1 | ||
| 980.2.v.a | 16 | 140.j | odd | 4 | 1 | ||
| 980.2.v.a | 16 | 140.w | even | 12 | 1 | ||
| 1260.2.dq.a | 16 | 12.b | even | 2 | 1 | ||
| 1260.2.dq.a | 16 | 60.l | odd | 4 | 1 | ||
| 1260.2.dq.a | 16 | 84.j | odd | 6 | 1 | ||
| 1260.2.dq.a | 16 | 420.br | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 78 T_{3}^{12} - 120 T_{3}^{11} + 1068 T_{3}^{9} + 6443 T_{3}^{8} + 9360 T_{3}^{7} + \cdots + 14641 \)
acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} \)
|
| $3$ |
\( T^{16} - 78 T^{12} + \cdots + 14641 \)
|
| $5$ |
\( T^{16} - 6 T^{15} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} + 2 T^{15} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{8} + 13 T^{6} + \cdots + 100)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 268435456 \)
|
| $17$ |
\( T^{16} - 18 T^{15} + \cdots + 256 \)
|
| $19$ |
\( T^{16} + \cdots + 4711998736 \)
|
| $23$ |
\( T^{16} - 16 T^{15} + \cdots + 7890481 \)
|
| $29$ |
\( (T^{8} + 162 T^{6} + \cdots + 16384)^{2} \)
|
| $31$ |
\( (T^{8} - 6 T^{7} + \cdots + 414736)^{2} \)
|
| $37$ |
\( T^{16} + 14 T^{15} + \cdots + 4096 \)
|
| $41$ |
\( (T^{8} + 170 T^{6} + \cdots + 12544)^{2} \)
|
| $43$ |
\( (T^{8} + 14 T^{7} + \cdots + 10432900)^{2} \)
|
| $47$ |
\( T^{16} - 6 T^{15} + \cdots + 10000 \)
|
| $53$ |
\( T^{16} + \cdots + 35477982736 \)
|
| $59$ |
\( T^{16} + \cdots + 6263627420176 \)
|
| $61$ |
\( (T^{8} - 30 T^{7} + \cdots + 4092529)^{2} \)
|
| $67$ |
\( T^{16} + 8 T^{15} + \cdots + 2401 \)
|
| $71$ |
\( (T^{4} - 2 T^{3} + \cdots + 448)^{4} \)
|
| $73$ |
\( T^{16} + \cdots + 3841600000000 \)
|
| $79$ |
\( T^{16} + \cdots + 3647809685776 \)
|
| $83$ |
\( T^{16} + \cdots + 5006411536 \)
|
| $89$ |
\( T^{16} + \cdots + 33243864241 \)
|
| $97$ |
\( T^{16} + \cdots + 47698139955456 \)
|
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