Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2548,2,Mod(589,2548)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2548, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2548.589");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2548.u (of order \(6\), 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: | \(20.3458824350\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(9\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - x^{17} + 17 x^{16} - 6 x^{15} + 188 x^{14} - 49 x^{13} + 1116 x^{12} - x^{11} + 4649 x^{10} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 364) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - x^{17} + 17 x^{16} - 6 x^{15} + 188 x^{14} - 49 x^{13} + 1116 x^{12} - x^{11} + 4649 x^{10} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 51\!\cdots\!75 \nu^{17} + \cdots + 19\!\cdots\!64 ) / 47\!\cdots\!71 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\!\cdots\!80 \nu^{17} + \cdots + 78\!\cdots\!24 ) / 47\!\cdots\!71 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 19\!\cdots\!81 \nu^{17} + \cdots + 18\!\cdots\!68 ) / 18\!\cdots\!84 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 16\!\cdots\!85 \nu^{17} + \cdots - 11\!\cdots\!92 ) / 14\!\cdots\!13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 20\!\cdots\!81 \nu^{17} + \cdots - 22\!\cdots\!88 ) / 14\!\cdots\!13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 80\!\cdots\!19 \nu^{17} + \cdots - 28\!\cdots\!54 ) / 28\!\cdots\!26 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 19\!\cdots\!81 \nu^{17} + \cdots + 18\!\cdots\!68 ) / 47\!\cdots\!71 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 12\!\cdots\!15 \nu^{17} + \cdots + 31\!\cdots\!14 ) / 28\!\cdots\!26 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 27\!\cdots\!49 \nu^{17} + \cdots + 22\!\cdots\!44 ) / 56\!\cdots\!52 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 28\!\cdots\!73 \nu^{17} + \cdots - 42\!\cdots\!72 ) / 56\!\cdots\!52 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 13\!\cdots\!18 \nu^{17} + \cdots + 36\!\cdots\!22 ) / 14\!\cdots\!13 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 34\!\cdots\!17 \nu^{17} + \cdots - 15\!\cdots\!64 ) / 18\!\cdots\!84 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 26\!\cdots\!98 \nu^{17} + \cdots - 43\!\cdots\!55 ) / 14\!\cdots\!13 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 10\!\cdots\!55 \nu^{17} + \cdots + 52\!\cdots\!40 ) / 56\!\cdots\!52 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 58\!\cdots\!98 \nu^{17} + \cdots - 28\!\cdots\!11 ) / 14\!\cdots\!13 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 14\!\cdots\!53 \nu^{17} + \cdots + 17\!\cdots\!40 ) / 28\!\cdots\!26 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} - 4\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{17} - \beta_{16} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - 5\beta_{3} + \beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{17} - 2 \beta_{16} - 2 \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} + \cdots - 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{17} - 2 \beta_{16} - 2 \beta_{15} - 12 \beta_{14} - 6 \beta_{13} - 10 \beta_{12} + \cdots - 32 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 29 \beta_{17} + 15 \beta_{16} + 12 \beta_{15} - 14 \beta_{14} + 26 \beta_{13} + 14 \beta_{12} + \cdots + 207 \)
|
| \(\nu^{7}\) | \(=\) |
\( 89 \beta_{17} + 122 \beta_{16} + 63 \beta_{15} + 89 \beta_{14} + 89 \beta_{13} + 122 \beta_{12} + \cdots + 294 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 157 \beta_{17} + 157 \beta_{16} + 173 \beta_{15} + 327 \beta_{14} - 125 \beta_{13} + 170 \beta_{12} + \cdots + 209 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1193 \beta_{17} - 789 \beta_{16} - 343 \beta_{15} + 404 \beta_{14} - 747 \beta_{13} - 404 \beta_{12} + \cdots - 3143 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1771 \beta_{17} - 3399 \beta_{16} - 3008 \beta_{15} - 1771 \beta_{14} - 1589 \beta_{13} + \cdots - 14725 \)
|
| \(\nu^{11}\) | \(=\) |
\( 4427 \beta_{17} - 4427 \beta_{16} - 4544 \beta_{15} - 11550 \beta_{14} + 1002 \beta_{13} + \cdots - 14581 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 34133 \beta_{17} + 17845 \beta_{16} + 11667 \beta_{15} - 16288 \beta_{14} + 27955 \beta_{13} + \cdots + 132000 \)
|
| \(\nu^{13}\) | \(=\) |
\( 65591 \beta_{17} + 111597 \beta_{16} + 83765 \beta_{15} + 65591 \beta_{14} + 63794 \beta_{13} + \cdots + 327660 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 160099 \beta_{17} + 160099 \beta_{16} + 177531 \beta_{15} + 337065 \beta_{14} - 113959 \beta_{13} + \cdots + 274743 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 1078369 \beta_{17} - 613705 \beta_{16} - 354497 \beta_{15} + 464664 \beta_{14} - 819161 \beta_{13} + \cdots - 3260085 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 1739546 \beta_{17} - 3299254 \beta_{16} - 2823663 \beta_{15} - 1739546 \beta_{14} - 1576975 \beta_{13} + \cdots - 11361824 \)
|
| \(\nu^{17}\) | \(=\) |
\( 4616962 \beta_{17} - 4616962 \beta_{16} - 4884083 \beta_{15} - 10425525 \beta_{14} + \cdots - 10358511 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(885\) | \(1275\) |
| \(\chi(n)\) | \(\beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 589.1 |
|
0 | −1.55551 | + | 2.69422i | 0 | 1.19594i | 0 | 0 | 0 | −3.33921 | − | 5.78368i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 589.2 | 0 | −1.19055 | + | 2.06209i | 0 | − | 1.91620i | 0 | 0 | 0 | −1.33482 | − | 2.31198i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 589.3 | 0 | −0.893365 | + | 1.54735i | 0 | 1.41239i | 0 | 0 | 0 | −0.0962010 | − | 0.166625i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 589.4 | 0 | −0.141496 | + | 0.245078i | 0 | − | 0.818894i | 0 | 0 | 0 | 1.45996 | + | 2.52872i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 589.5 | 0 | −0.118765 | + | 0.205706i | 0 | − | 2.57768i | 0 | 0 | 0 | 1.47179 | + | 2.54921i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 589.6 | 0 | 0.302937 | − | 0.524702i | 0 | 4.02670i | 0 | 0 | 0 | 1.31646 | + | 2.28017i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 589.7 | 0 | 0.591009 | − | 1.02366i | 0 | − | 3.88123i | 0 | 0 | 0 | 0.801416 | + | 1.38809i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 589.8 | 0 | 1.23985 | − | 2.14749i | 0 | 0.559885i | 0 | 0 | 0 | −1.57447 | − | 2.72707i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 589.9 | 0 | 1.26588 | − | 2.19257i | 0 | 1.99909i | 0 | 0 | 0 | −1.70492 | − | 2.95301i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1765.1 | 0 | −1.55551 | − | 2.69422i | 0 | − | 1.19594i | 0 | 0 | 0 | −3.33921 | + | 5.78368i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1765.2 | 0 | −1.19055 | − | 2.06209i | 0 | 1.91620i | 0 | 0 | 0 | −1.33482 | + | 2.31198i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1765.3 | 0 | −0.893365 | − | 1.54735i | 0 | − | 1.41239i | 0 | 0 | 0 | −0.0962010 | + | 0.166625i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1765.4 | 0 | −0.141496 | − | 0.245078i | 0 | 0.818894i | 0 | 0 | 0 | 1.45996 | − | 2.52872i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1765.5 | 0 | −0.118765 | − | 0.205706i | 0 | 2.57768i | 0 | 0 | 0 | 1.47179 | − | 2.54921i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1765.6 | 0 | 0.302937 | + | 0.524702i | 0 | − | 4.02670i | 0 | 0 | 0 | 1.31646 | − | 2.28017i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1765.7 | 0 | 0.591009 | + | 1.02366i | 0 | 3.88123i | 0 | 0 | 0 | 0.801416 | − | 1.38809i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1765.8 | 0 | 1.23985 | + | 2.14749i | 0 | − | 0.559885i | 0 | 0 | 0 | −1.57447 | + | 2.72707i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1765.9 | 0 | 1.26588 | + | 2.19257i | 0 | − | 1.99909i | 0 | 0 | 0 | −1.70492 | + | 2.95301i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.e | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2548.2.u.d | 18 | |
| 7.b | odd | 2 | 1 | 2548.2.u.e | 18 | ||
| 7.c | even | 3 | 1 | 2548.2.bb.f | 18 | ||
| 7.c | even | 3 | 1 | 2548.2.bq.f | 18 | ||
| 7.d | odd | 6 | 1 | 364.2.bb.a | ✓ | 18 | |
| 7.d | odd | 6 | 1 | 364.2.bq.a | yes | 18 | |
| 13.e | even | 6 | 1 | inner | 2548.2.u.d | 18 | |
| 21.g | even | 6 | 1 | 3276.2.fe.i | 18 | ||
| 21.g | even | 6 | 1 | 3276.2.hi.i | 18 | ||
| 91.k | even | 6 | 1 | 2548.2.bq.f | 18 | ||
| 91.l | odd | 6 | 1 | 364.2.bq.a | yes | 18 | |
| 91.p | odd | 6 | 1 | 364.2.bb.a | ✓ | 18 | |
| 91.t | odd | 6 | 1 | 2548.2.u.e | 18 | ||
| 91.u | even | 6 | 1 | 2548.2.bb.f | 18 | ||
| 273.y | even | 6 | 1 | 3276.2.hi.i | 18 | ||
| 273.br | even | 6 | 1 | 3276.2.fe.i | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 364.2.bb.a | ✓ | 18 | 7.d | odd | 6 | 1 | |
| 364.2.bb.a | ✓ | 18 | 91.p | odd | 6 | 1 | |
| 364.2.bq.a | yes | 18 | 7.d | odd | 6 | 1 | |
| 364.2.bq.a | yes | 18 | 91.l | odd | 6 | 1 | |
| 2548.2.u.d | 18 | 1.a | even | 1 | 1 | trivial | |
| 2548.2.u.d | 18 | 13.e | even | 6 | 1 | inner | |
| 2548.2.u.e | 18 | 7.b | odd | 2 | 1 | ||
| 2548.2.u.e | 18 | 91.t | odd | 6 | 1 | ||
| 2548.2.bb.f | 18 | 7.c | even | 3 | 1 | ||
| 2548.2.bb.f | 18 | 91.u | even | 6 | 1 | ||
| 2548.2.bq.f | 18 | 7.c | even | 3 | 1 | ||
| 2548.2.bq.f | 18 | 91.k | even | 6 | 1 | ||
| 3276.2.fe.i | 18 | 21.g | even | 6 | 1 | ||
| 3276.2.fe.i | 18 | 273.br | even | 6 | 1 | ||
| 3276.2.hi.i | 18 | 21.g | even | 6 | 1 | ||
| 3276.2.hi.i | 18 | 273.y | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{18} + T_{3}^{17} + 17 T_{3}^{16} + 6 T_{3}^{15} + 188 T_{3}^{14} + 49 T_{3}^{13} + 1116 T_{3}^{12} + \cdots + 16 \)
acting on \(S_{2}^{\mathrm{new}}(2548, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} \)
|
| $3$ |
\( T^{18} + T^{17} + \cdots + 16 \)
|
| $5$ |
\( T^{18} + 50 T^{16} + \cdots + 14283 \)
|
| $7$ |
\( T^{18} \)
|
| $11$ |
\( T^{18} - 6 T^{17} + \cdots + 1581228 \)
|
| $13$ |
\( T^{18} + \cdots + 10604499373 \)
|
| $17$ |
\( T^{18} + \cdots + 14846935104 \)
|
| $19$ |
\( T^{18} - 21 T^{17} + \cdots + 2988012 \)
|
| $23$ |
\( T^{18} + \cdots + 2496801024 \)
|
| $29$ |
\( T^{18} - 2 T^{17} + \cdots + 4100625 \)
|
| $31$ |
\( T^{18} + \cdots + 10703377083 \)
|
| $37$ |
\( T^{18} + \cdots + 1104538032 \)
|
| $41$ |
\( T^{18} + \cdots + 3793327443 \)
|
| $43$ |
\( T^{18} + \cdots + 1883413140625 \)
|
| $47$ |
\( T^{18} + \cdots + 14333519021787 \)
|
| $53$ |
\( (T^{9} - 13 T^{8} + \cdots - 1053081)^{2} \)
|
| $59$ |
\( T^{18} + \cdots + 501089328 \)
|
| $61$ |
\( T^{18} + \cdots + 1673300836 \)
|
| $67$ |
\( T^{18} + \cdots + 26266034700 \)
|
| $71$ |
\( T^{18} + \cdots + 44122876875 \)
|
| $73$ |
\( T^{18} + \cdots + 3223781883 \)
|
| $79$ |
\( (T^{9} - 14 T^{8} + \cdots + 2307735)^{2} \)
|
| $83$ |
\( T^{18} + \cdots + 93\!\cdots\!00 \)
|
| $89$ |
\( T^{18} + \cdots + 465414910128 \)
|
| $97$ |
\( T^{18} + \cdots + 938674322067 \)
|
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