Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [57,4,Mod(7,57)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(57, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("57.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 57.e (of order \(3\), degree \(2\), 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: | \(3.36310887033\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + 30x^{8} - 25x^{7} + 712x^{6} - 447x^{5} + 5277x^{4} + 5604x^{3} + 23688x^{2} + 3744x + 576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} + 30x^{8} - 25x^{7} + 712x^{6} - 447x^{5} + 5277x^{4} + 5604x^{3} + 23688x^{2} + 3744x + 576 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 35591 \nu^{9} - 4293851 \nu^{8} + 23229030 \nu^{7} - 120778775 \nu^{6} + 559467668 \nu^{5} + \cdots + 13212052704 ) / 86043890808 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 354855 \nu^{9} + 1846775 \nu^{8} - 9990750 \nu^{7} + 44510573 \nu^{6} - 240625700 \nu^{5} + \cdots + 86042182440 ) / 7170324234 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1526193 \nu^{9} + 20268653 \nu^{8} - 109650090 \nu^{7} + 540378617 \nu^{6} + \cdots + 261510626244 ) / 14340648468 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 45875183 \nu^{9} - 45946365 \nu^{8} + 1384843192 \nu^{7} - 1193337635 \nu^{6} + \cdots + 4737055536 ) / 172087781616 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 63412483 \nu^{9} - 325909519 \nu^{8} + 1763117070 \nu^{7} - 10705841371 \nu^{6} + \cdots - 3007233996624 ) / 172087781616 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 45875183 \nu^{9} - 45946365 \nu^{8} + 1384843192 \nu^{7} - 1193337635 \nu^{6} + \cdots + 176824837152 ) / 14340648468 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 887204611 \nu^{9} + 1238616705 \nu^{8} - 26487673640 \nu^{7} + 31359031471 \nu^{6} + \cdots - 83413210608 ) / 172087781616 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 308763829 \nu^{9} + 275337543 \nu^{8} - 9130040400 \nu^{7} + 7000266625 \nu^{6} + \cdots - 1077594055776 ) / 57362593872 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - 12\beta_{5} - 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} + 18\beta_{2} + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} - 2\beta_{8} - 23\beta_{7} + 221\beta_{5} - \beta_{4} + 23\beta_{3} + 11\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 28\beta_{9} - 2\beta_{8} + 62\beta_{7} - 2\beta_{6} - 214\beta_{5} - 355\beta_{2} - 355\beta _1 - 214 \)
|
| \(\nu^{6}\) | \(=\) |
\( -60\beta_{6} + 28\beta_{4} - 503\beta_{3} + 474\beta_{2} + 4404 \)
|
| \(\nu^{7}\) | \(=\) |
\( -655\beta_{9} + 64\beta_{8} - 1628\beta_{7} + 7589\beta_{5} - 655\beta_{4} + 1628\beta_{3} + 7310\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 781 \beta_{9} + 1438 \beta_{8} + 11093 \beta_{7} + 1438 \beta_{6} - 91501 \beta_{5} - 14737 \beta_{2} + \cdots - 91501 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1314\beta_{6} + 14626\beta_{4} - 40580\beta_{3} + 154935\beta_{2} + 219408 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).
| \(n\) | \(20\) | \(40\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−2.08662 | − | 3.61412i | 1.50000 | + | 2.59808i | −4.70793 | + | 8.15437i | −4.26171 | − | 7.38149i | 6.25985 | − | 10.8424i | −34.6578 | 5.90869 | −4.50000 | + | 7.79423i | −17.7851 | + | 30.8047i | ||||||||||||||||||||||||||||||||||
| 7.2 | −1.89477 | − | 3.28184i | 1.50000 | + | 2.59808i | −3.18033 | + | 5.50849i | 7.86295 | + | 13.6190i | 5.68432 | − | 9.84553i | 25.2871 | −6.21237 | −4.50000 | + | 7.79423i | 29.7970 | − | 51.6099i | |||||||||||||||||||||||||||||||||||
| 7.3 | 0.0794808 | + | 0.137665i | 1.50000 | + | 2.59808i | 3.98737 | − | 6.90632i | −9.14468 | − | 15.8390i | −0.238442 | + | 0.412994i | 23.2129 | 2.53937 | −4.50000 | + | 7.79423i | 1.45365 | − | 2.51780i | |||||||||||||||||||||||||||||||||||
| 7.4 | 0.989242 | + | 1.71342i | 1.50000 | + | 2.59808i | 2.04280 | − | 3.53824i | 6.19376 | + | 10.7279i | −2.96773 | + | 5.14025i | −8.04938 | 23.9112 | −4.50000 | + | 7.79423i | −12.2542 | + | 21.2250i | |||||||||||||||||||||||||||||||||||
| 7.5 | 2.41267 | + | 4.17886i | 1.50000 | + | 2.59808i | −7.64191 | + | 13.2362i | −2.65032 | − | 4.59049i | −7.23800 | + | 12.5366i | 9.20712 | −35.1469 | −4.50000 | + | 7.79423i | 12.7887 | − | 22.1506i | |||||||||||||||||||||||||||||||||||
| 49.1 | −2.08662 | + | 3.61412i | 1.50000 | − | 2.59808i | −4.70793 | − | 8.15437i | −4.26171 | + | 7.38149i | 6.25985 | + | 10.8424i | −34.6578 | 5.90869 | −4.50000 | − | 7.79423i | −17.7851 | − | 30.8047i | |||||||||||||||||||||||||||||||||||
| 49.2 | −1.89477 | + | 3.28184i | 1.50000 | − | 2.59808i | −3.18033 | − | 5.50849i | 7.86295 | − | 13.6190i | 5.68432 | + | 9.84553i | 25.2871 | −6.21237 | −4.50000 | − | 7.79423i | 29.7970 | + | 51.6099i | |||||||||||||||||||||||||||||||||||
| 49.3 | 0.0794808 | − | 0.137665i | 1.50000 | − | 2.59808i | 3.98737 | + | 6.90632i | −9.14468 | + | 15.8390i | −0.238442 | − | 0.412994i | 23.2129 | 2.53937 | −4.50000 | − | 7.79423i | 1.45365 | + | 2.51780i | |||||||||||||||||||||||||||||||||||
| 49.4 | 0.989242 | − | 1.71342i | 1.50000 | − | 2.59808i | 2.04280 | + | 3.53824i | 6.19376 | − | 10.7279i | −2.96773 | − | 5.14025i | −8.04938 | 23.9112 | −4.50000 | − | 7.79423i | −12.2542 | − | 21.2250i | |||||||||||||||||||||||||||||||||||
| 49.5 | 2.41267 | − | 4.17886i | 1.50000 | − | 2.59808i | −7.64191 | − | 13.2362i | −2.65032 | + | 4.59049i | −7.23800 | − | 12.5366i | 9.20712 | −35.1469 | −4.50000 | − | 7.79423i | 12.7887 | + | 22.1506i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 57.4.e.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 171.4.f.g | 10 | ||
| 19.c | even | 3 | 1 | inner | 57.4.e.a | ✓ | 10 |
| 19.c | even | 3 | 1 | 1083.4.a.i | 5 | ||
| 19.d | odd | 6 | 1 | 1083.4.a.g | 5 | ||
| 57.h | odd | 6 | 1 | 171.4.f.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.4.e.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 57.4.e.a | ✓ | 10 | 19.c | even | 3 | 1 | inner |
| 171.4.f.g | 10 | 3.b | odd | 2 | 1 | ||
| 171.4.f.g | 10 | 57.h | odd | 6 | 1 | ||
| 1083.4.a.g | 5 | 19.d | odd | 6 | 1 | ||
| 1083.4.a.i | 5 | 19.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + T_{2}^{9} + 30 T_{2}^{8} + 25 T_{2}^{7} + 712 T_{2}^{6} + 447 T_{2}^{5} + 5277 T_{2}^{4} + \cdots + 576 \)
acting on \(S_{4}^{\mathrm{new}}(57, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + T^{9} + \cdots + 576 \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{5} \)
|
| $5$ |
\( T^{10} + \cdots + 25910697024 \)
|
| $7$ |
\( (T^{5} - 15 T^{4} + \cdots - 1507701)^{2} \)
|
| $11$ |
\( (T^{5} - 66 T^{4} + \cdots + 110808)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 46\!\cdots\!41 \)
|
| $17$ |
\( T^{10} + \cdots + 47\!\cdots\!44 \)
|
| $19$ |
\( T^{10} + \cdots + 15\!\cdots\!99 \)
|
| $23$ |
\( T^{10} + \cdots + 12\!\cdots\!56 \)
|
| $29$ |
\( T^{10} + \cdots + 19\!\cdots\!84 \)
|
| $31$ |
\( (T^{5} - 55 T^{4} + \cdots - 364568753753)^{2} \)
|
| $37$ |
\( (T^{5} + 437 T^{4} + \cdots - 60106677263)^{2} \)
|
| $41$ |
\( T^{10} + \cdots + 24\!\cdots\!76 \)
|
| $43$ |
\( T^{10} + \cdots + 64\!\cdots\!01 \)
|
| $47$ |
\( T^{10} + \cdots + 50\!\cdots\!84 \)
|
| $53$ |
\( T^{10} + \cdots + 23\!\cdots\!36 \)
|
| $59$ |
\( T^{10} + \cdots + 78\!\cdots\!64 \)
|
| $61$ |
\( T^{10} + \cdots + 14\!\cdots\!41 \)
|
| $67$ |
\( T^{10} + \cdots + 68\!\cdots\!69 \)
|
| $71$ |
\( T^{10} + \cdots + 18\!\cdots\!16 \)
|
| $73$ |
\( T^{10} + \cdots + 65\!\cdots\!09 \)
|
| $79$ |
\( T^{10} + \cdots + 10\!\cdots\!69 \)
|
| $83$ |
\( (T^{5} + \cdots - 24928873486848)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 23\!\cdots\!04 \)
|
| $97$ |
\( T^{10} + \cdots + 17\!\cdots\!84 \)
|
| show more | |
| show less |