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: |
\( x^{10} - x^{9} + 34 x^{8} - 57 x^{7} + 966 x^{6} - 1413 x^{5} + 7305 x^{4} - 9864 x^{3} + 40104 x^{2} + \cdots + 69696 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| 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
\( x^{10} - x^{9} + 34 x^{8} - 57 x^{7} + 966 x^{6} - 1413 x^{5} + 7305 x^{4} - 9864 x^{3} + 40104 x^{2} + \cdots + 69696 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2032295 \nu^{9} - 28338727 \nu^{8} + 190787486 \nu^{7} - 935102223 \nu^{6} + \cdots - 966135893184 ) / 619115892888 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 12390055 \nu^{9} - 418166390 \nu^{8} + 1725268279 \nu^{7} - 13745054347 \nu^{6} + \cdots - 14902591002816 ) / 619115892888 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 152483569 \nu^{9} - 130128324 \nu^{8} + 4872715349 \nu^{7} - 6592901087 \nu^{6} + \cdots - 4518945562560 ) / 6810274821768 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 24274137 \nu^{9} - 93350729 \nu^{8} + 628473922 \nu^{7} - 2965023337 \nu^{6} + \cdots - 7559843774928 ) / 619115892888 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 57100545 \nu^{9} + 205285604 \nu^{8} - 2472056413 \nu^{7} + 6827194559 \nu^{6} + \cdots - 457876174536 ) / 619115892888 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 2112414721 \nu^{9} + 1510070539 \nu^{8} - 66119352540 \nu^{7} + 82014490765 \nu^{6} + \cdots + 52637743050816 ) / 6810274821768 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 610410073 \nu^{9} + 2448250897 \nu^{8} - 16482590546 \nu^{7} + 96143017369 \nu^{6} + \cdots + 55498476420240 ) / 1238231785776 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 5206545835 \nu^{9} - 11616089500 \nu^{8} - 173584234171 \nu^{7} - 221958421055 \nu^{6} + \cdots - 119284965490680 ) / 6810274821768 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + 14\beta_{4} - \beta_{2} - \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{3} + 22\beta_{2} + 11 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} - 31\beta_{7} - \beta_{6} - 31\beta_{5} - 311\beta_{4} + 36\beta _1 - 311 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} + 2\beta_{8} + 14\beta_{7} + 418\beta_{4} + 32\beta_{3} - 567\beta_{2} - 567\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 68\beta_{8} + 46\beta_{6} + 841\beta_{5} - 46\beta_{3} + 1179\beta_{2} + 7988 \)
|
| \(\nu^{7}\) | \(=\) |
\( -44\beta_{9} - 752\beta_{7} - 887\beta_{6} - 752\beta_{5} - 14257\beta_{4} + 15192\beta _1 - 14257 \)
|
| \(\nu^{8}\) | \(=\) |
\( -1862\beta_{9} - 1862\beta_{8} + 22423\beta_{7} + 213005\beta_{4} + 1639\beta_{3} - 37374\beta_{2} - 37374\beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( -446\beta_{8} + 24062\beta_{6} + 29702\beta_{5} - 24062\beta_{3} + 412257\beta_{2} + 464608 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).
| \(n\) | \(20\) | \(40\) |
| \(\chi(n)\) | \(1\) | \(\beta_{4}\) |
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 |
|
−2.66732 | − | 4.61994i | −1.50000 | − | 2.59808i | −10.2292 | + | 17.7176i | 8.59290 | + | 14.8833i | −8.00197 | + | 13.8598i | −21.2765 | 66.4617 | −4.50000 | + | 7.79423i | 45.8401 | − | 79.3974i | ||||||||||||||||||||||||||||||||||
| 7.2 | −1.27816 | − | 2.21384i | −1.50000 | − | 2.59808i | 0.732617 | − | 1.26893i | −2.45209 | − | 4.24715i | −3.83448 | + | 6.64151i | 0.148382 | −24.1962 | −4.50000 | + | 7.79423i | −6.26833 | + | 10.8571i | |||||||||||||||||||||||||||||||||||
| 7.3 | 0.898167 | + | 1.55567i | −1.50000 | − | 2.59808i | 2.38659 | − | 4.13370i | −8.79657 | − | 15.2361i | 2.69450 | − | 4.66701i | −24.6867 | 22.9449 | −4.50000 | + | 7.79423i | 15.8016 | − | 27.3691i | |||||||||||||||||||||||||||||||||||
| 7.4 | 1.10163 | + | 1.90807i | −1.50000 | − | 2.59808i | 1.57283 | − | 2.72423i | 1.84710 | + | 3.19927i | 3.30488 | − | 5.72422i | 32.6653 | 24.5567 | −4.50000 | + | 7.79423i | −4.06963 | + | 7.04881i | |||||||||||||||||||||||||||||||||||
| 7.5 | 2.44569 | + | 4.23606i | −1.50000 | − | 2.59808i | −7.96280 | + | 13.7920i | 6.80866 | + | 11.7929i | 7.33707 | − | 12.7082i | −13.8505 | −38.7671 | −4.50000 | + | 7.79423i | −33.3037 | + | 57.6838i | |||||||||||||||||||||||||||||||||||
| 49.1 | −2.66732 | + | 4.61994i | −1.50000 | + | 2.59808i | −10.2292 | − | 17.7176i | 8.59290 | − | 14.8833i | −8.00197 | − | 13.8598i | −21.2765 | 66.4617 | −4.50000 | − | 7.79423i | 45.8401 | + | 79.3974i | |||||||||||||||||||||||||||||||||||
| 49.2 | −1.27816 | + | 2.21384i | −1.50000 | + | 2.59808i | 0.732617 | + | 1.26893i | −2.45209 | + | 4.24715i | −3.83448 | − | 6.64151i | 0.148382 | −24.1962 | −4.50000 | − | 7.79423i | −6.26833 | − | 10.8571i | |||||||||||||||||||||||||||||||||||
| 49.3 | 0.898167 | − | 1.55567i | −1.50000 | + | 2.59808i | 2.38659 | + | 4.13370i | −8.79657 | + | 15.2361i | 2.69450 | + | 4.66701i | −24.6867 | 22.9449 | −4.50000 | − | 7.79423i | 15.8016 | + | 27.3691i | |||||||||||||||||||||||||||||||||||
| 49.4 | 1.10163 | − | 1.90807i | −1.50000 | + | 2.59808i | 1.57283 | + | 2.72423i | 1.84710 | − | 3.19927i | 3.30488 | + | 5.72422i | 32.6653 | 24.5567 | −4.50000 | − | 7.79423i | −4.06963 | − | 7.04881i | |||||||||||||||||||||||||||||||||||
| 49.5 | 2.44569 | − | 4.23606i | −1.50000 | + | 2.59808i | −7.96280 | − | 13.7920i | 6.80866 | − | 11.7929i | 7.33707 | + | 12.7082i | −13.8505 | −38.7671 | −4.50000 | − | 7.79423i | −33.3037 | − | 57.6838i | |||||||||||||||||||||||||||||||||||
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.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 171.4.f.f | 10 | ||
| 19.c | even | 3 | 1 | inner | 57.4.e.b | ✓ | 10 |
| 19.c | even | 3 | 1 | 1083.4.a.h | 5 | ||
| 19.d | odd | 6 | 1 | 1083.4.a.j | 5 | ||
| 57.h | odd | 6 | 1 | 171.4.f.f | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.4.e.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 57.4.e.b | ✓ | 10 | 19.c | even | 3 | 1 | inner |
| 171.4.f.f | 10 | 3.b | odd | 2 | 1 | ||
| 171.4.f.f | 10 | 57.h | odd | 6 | 1 | ||
| 1083.4.a.h | 5 | 19.c | even | 3 | 1 | ||
| 1083.4.a.j | 5 | 19.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - T_{2}^{9} + 34 T_{2}^{8} - 57 T_{2}^{7} + 966 T_{2}^{6} - 1413 T_{2}^{5} + 7305 T_{2}^{4} + \cdots + 69696 \)
acting on \(S_{4}^{\mathrm{new}}(57, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - T^{9} + \cdots + 69696 \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{5} \)
|
| $5$ |
\( T^{10} + \cdots + 5563966464 \)
|
| $7$ |
\( (T^{5} + 27 T^{4} + \cdots + 35261)^{2} \)
|
| $11$ |
\( (T^{5} + 10 T^{4} + \cdots + 79362288)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 1376564839441 \)
|
| $17$ |
\( T^{10} + \cdots + 33\!\cdots\!24 \)
|
| $19$ |
\( T^{10} + \cdots + 15\!\cdots\!99 \)
|
| $23$ |
\( T^{10} + \cdots + 31\!\cdots\!44 \)
|
| $29$ |
\( T^{10} + \cdots + 30\!\cdots\!96 \)
|
| $31$ |
\( (T^{5} + 571 T^{4} + \cdots + 20911632441)^{2} \)
|
| $37$ |
\( (T^{5} + 229 T^{4} + \cdots - 43430079)^{2} \)
|
| $41$ |
\( T^{10} + \cdots + 87\!\cdots\!16 \)
|
| $43$ |
\( T^{10} + \cdots + 10\!\cdots\!61 \)
|
| $47$ |
\( T^{10} + \cdots + 18\!\cdots\!64 \)
|
| $53$ |
\( T^{10} + \cdots + 66\!\cdots\!24 \)
|
| $59$ |
\( T^{10} + \cdots + 19\!\cdots\!24 \)
|
| $61$ |
\( T^{10} + \cdots + 94\!\cdots\!89 \)
|
| $67$ |
\( T^{10} + \cdots + 80\!\cdots\!09 \)
|
| $71$ |
\( T^{10} + \cdots + 33\!\cdots\!76 \)
|
| $73$ |
\( T^{10} + \cdots + 14\!\cdots\!41 \)
|
| $79$ |
\( T^{10} + \cdots + 16\!\cdots\!81 \)
|
| $83$ |
\( (T^{5} + \cdots + 6461469294336)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 79\!\cdots\!24 \)
|
| $97$ |
\( T^{10} + \cdots + 30\!\cdots\!76 \)
|
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