Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2548,2,Mod(373,2548)]
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, 2, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2548.373");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2548.l (of order \(3\), degree \(2\), 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: | \(20.3458824350\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(9\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - x^{17} + 19 x^{16} - 16 x^{15} + 244 x^{14} - 181 x^{13} + 1600 x^{12} - 607 x^{11} + \cdots + 1296 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | no (minimal twist has level 364) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - x^{17} + 19 x^{16} - 16 x^{15} + 244 x^{14} - 181 x^{13} + 1600 x^{12} - 607 x^{11} + \cdots + 1296 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 22\!\cdots\!28 \nu^{17} + \cdots - 92\!\cdots\!92 ) / 34\!\cdots\!13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 18\!\cdots\!85 \nu^{17} + \cdots - 13\!\cdots\!40 ) / 34\!\cdots\!13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 70\!\cdots\!47 \nu^{17} + \cdots + 48\!\cdots\!71 ) / 52\!\cdots\!95 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 28\!\cdots\!33 \nu^{17} + \cdots + 16\!\cdots\!76 ) / 13\!\cdots\!52 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11\!\cdots\!57 \nu^{17} + \cdots + 42\!\cdots\!16 ) / 52\!\cdots\!95 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 39\!\cdots\!91 \nu^{17} + \cdots + 48\!\cdots\!68 ) / 52\!\cdots\!95 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 28\!\cdots\!33 \nu^{17} + \cdots - 16\!\cdots\!76 ) / 34\!\cdots\!13 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 19\!\cdots\!35 \nu^{17} + \cdots - 57\!\cdots\!44 ) / 20\!\cdots\!80 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 11\!\cdots\!64 \nu^{17} + \cdots + 15\!\cdots\!28 ) / 10\!\cdots\!39 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 23\!\cdots\!02 \nu^{17} + \cdots + 22\!\cdots\!43 ) / 17\!\cdots\!65 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 30\!\cdots\!71 \nu^{17} + \cdots - 13\!\cdots\!59 ) / 17\!\cdots\!65 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 20\!\cdots\!63 \nu^{17} + \cdots - 47\!\cdots\!40 ) / 52\!\cdots\!95 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 22\!\cdots\!90 \nu^{17} + \cdots + 44\!\cdots\!92 ) / 52\!\cdots\!95 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 33\!\cdots\!51 \nu^{17} + \cdots + 70\!\cdots\!16 ) / 69\!\cdots\!60 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 52\!\cdots\!39 \nu^{17} + \cdots + 16\!\cdots\!56 ) / 10\!\cdots\!39 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 36\!\cdots\!27 \nu^{17} + \cdots + 36\!\cdots\!08 ) / 69\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{8} - 4\beta_{5} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{17} - \beta_{15} - \beta_{10} + 2\beta_{7} - 6\beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{17} + \beta_{15} + \beta_{14} + 2 \beta_{13} - \beta_{11} - 2 \beta_{9} + 10 \beta_{8} + \cdots - 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{17} + 12 \beta_{16} + 11 \beta_{15} - 25 \beta_{14} - 5 \beta_{13} - 11 \beta_{12} + \cdots - 43 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 14 \beta_{12} + 14 \beta_{11} - 6 \beta_{10} + 21 \beta_{7} - 27 \beta_{6} - 30 \beta_{4} + \cdots + 230 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 106 \beta_{17} - 123 \beta_{16} + 35 \beta_{15} + 263 \beta_{14} + 85 \beta_{13} + 141 \beta_{12} + \cdots + 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 157 \beta_{17} + 114 \beta_{16} - 146 \beta_{15} - 302 \beta_{14} - 298 \beta_{13} + 146 \beta_{12} + \cdots - 254 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1449 \beta_{17} - 1449 \beta_{15} - 448 \beta_{12} + 448 \beta_{11} - 1221 \beta_{10} + 2661 \beta_{7} + \cdots + 277 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1366 \beta_{17} - 1557 \beta_{16} + 1660 \beta_{15} + 3757 \beta_{14} + 3143 \beta_{13} + 294 \beta_{12} + \cdots - 18622 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 5123 \beta_{17} + 12081 \beta_{16} + 9425 \beta_{15} - 26677 \beta_{14} - 12044 \beta_{13} + \cdots - 27139 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 5202 \beta_{17} - 5202 \beta_{15} - 17252 \beta_{12} + 17252 \beta_{11} - 18831 \beta_{10} + \cdots + 175508 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 88768 \beta_{17} - 120024 \beta_{16} + 55706 \beta_{15} + 267329 \beta_{14} + 129439 \beta_{13} + \cdots - 108695 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 178561 \beta_{17} + 215088 \beta_{16} - 101756 \beta_{15} - 489242 \beta_{14} - 342616 \beta_{13} + \cdots - 437948 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( 1428192 \beta_{17} - 1428192 \beta_{15} - 590998 \beta_{12} + 590998 \beta_{11} - 1199103 \beta_{10} + \cdots + 1448905 \)
|
| \(\nu^{16}\) | \(=\) |
\( 823504 \beta_{17} - 2382291 \beta_{16} + 1848118 \beta_{15} + 5368078 \beta_{14} + 3581528 \beta_{13} + \cdots - 16360906 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 6191582 \beta_{17} + 12043875 \beta_{16} + 7913018 \beta_{15} - 27060034 \beta_{14} + \cdots - 24751717 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(885\) | \(1275\) |
| \(\chi(n)\) | \(-\beta_{5}\) | \(-1 + \beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 373.1 |
|
0 | −2.98937 | 0 | 2.11635 | − | 3.66562i | 0 | 0 | 0 | 5.93631 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.2 | 0 | −2.52589 | 0 | −1.33439 | + | 2.31123i | 0 | 0 | 0 | 3.38014 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.3 | 0 | −1.57983 | 0 | −0.985205 | + | 1.70642i | 0 | 0 | 0 | −0.504145 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.4 | 0 | −1.46962 | 0 | 0.879363 | − | 1.52310i | 0 | 0 | 0 | −0.840227 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.5 | 0 | 0.421254 | 0 | 0.466726 | − | 0.808393i | 0 | 0 | 0 | −2.82254 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.6 | 0 | 0.652064 | 0 | 1.14588 | − | 1.98472i | 0 | 0 | 0 | −2.57481 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.7 | 0 | 1.04054 | 0 | −1.59004 | + | 2.75404i | 0 | 0 | 0 | −1.91727 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.8 | 0 | 2.23212 | 0 | −0.660291 | + | 1.14366i | 0 | 0 | 0 | 1.98235 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.9 | 0 | 3.21873 | 0 | −0.0383870 | + | 0.0664882i | 0 | 0 | 0 | 7.36021 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.1 | 0 | −2.98937 | 0 | 2.11635 | + | 3.66562i | 0 | 0 | 0 | 5.93631 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.2 | 0 | −2.52589 | 0 | −1.33439 | − | 2.31123i | 0 | 0 | 0 | 3.38014 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.3 | 0 | −1.57983 | 0 | −0.985205 | − | 1.70642i | 0 | 0 | 0 | −0.504145 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.4 | 0 | −1.46962 | 0 | 0.879363 | + | 1.52310i | 0 | 0 | 0 | −0.840227 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.5 | 0 | 0.421254 | 0 | 0.466726 | + | 0.808393i | 0 | 0 | 0 | −2.82254 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.6 | 0 | 0.652064 | 0 | 1.14588 | + | 1.98472i | 0 | 0 | 0 | −2.57481 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.7 | 0 | 1.04054 | 0 | −1.59004 | − | 2.75404i | 0 | 0 | 0 | −1.91727 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.8 | 0 | 2.23212 | 0 | −0.660291 | − | 1.14366i | 0 | 0 | 0 | 1.98235 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1537.9 | 0 | 3.21873 | 0 | −0.0383870 | − | 0.0664882i | 0 | 0 | 0 | 7.36021 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 91.g | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2548.2.l.n | 18 | |
| 7.b | odd | 2 | 1 | 364.2.l.a | yes | 18 | |
| 7.c | even | 3 | 1 | 2548.2.i.n | 18 | ||
| 7.c | even | 3 | 1 | 2548.2.k.i | 18 | ||
| 7.d | odd | 6 | 1 | 364.2.i.a | ✓ | 18 | |
| 7.d | odd | 6 | 1 | 2548.2.k.h | 18 | ||
| 13.c | even | 3 | 1 | 2548.2.i.n | 18 | ||
| 21.c | even | 2 | 1 | 3276.2.x.k | 18 | ||
| 21.g | even | 6 | 1 | 3276.2.u.k | 18 | ||
| 91.g | even | 3 | 1 | inner | 2548.2.l.n | 18 | |
| 91.h | even | 3 | 1 | 2548.2.k.i | 18 | ||
| 91.m | odd | 6 | 1 | 364.2.l.a | yes | 18 | |
| 91.n | odd | 6 | 1 | 364.2.i.a | ✓ | 18 | |
| 91.v | odd | 6 | 1 | 2548.2.k.h | 18 | ||
| 273.bf | even | 6 | 1 | 3276.2.x.k | 18 | ||
| 273.bn | even | 6 | 1 | 3276.2.u.k | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 364.2.i.a | ✓ | 18 | 7.d | odd | 6 | 1 | |
| 364.2.i.a | ✓ | 18 | 91.n | odd | 6 | 1 | |
| 364.2.l.a | yes | 18 | 7.b | odd | 2 | 1 | |
| 364.2.l.a | yes | 18 | 91.m | odd | 6 | 1 | |
| 2548.2.i.n | 18 | 7.c | even | 3 | 1 | ||
| 2548.2.i.n | 18 | 13.c | even | 3 | 1 | ||
| 2548.2.k.h | 18 | 7.d | odd | 6 | 1 | ||
| 2548.2.k.h | 18 | 91.v | odd | 6 | 1 | ||
| 2548.2.k.i | 18 | 7.c | even | 3 | 1 | ||
| 2548.2.k.i | 18 | 91.h | even | 3 | 1 | ||
| 2548.2.l.n | 18 | 1.a | even | 1 | 1 | trivial | |
| 2548.2.l.n | 18 | 91.g | even | 3 | 1 | inner | |
| 3276.2.u.k | 18 | 21.g | even | 6 | 1 | ||
| 3276.2.u.k | 18 | 273.bn | even | 6 | 1 | ||
| 3276.2.x.k | 18 | 21.c | even | 2 | 1 | ||
| 3276.2.x.k | 18 | 273.bf | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2548, [\chi])\):
|
\( T_{3}^{9} + T_{3}^{8} - 18T_{3}^{7} - 17T_{3}^{6} + 97T_{3}^{5} + 69T_{3}^{4} - 183T_{3}^{3} - 45T_{3}^{2} + 129T_{3} - 36 \)
|
|
\( T_{5}^{18} + 25 T_{5}^{16} + 22 T_{5}^{15} + 445 T_{5}^{14} + 361 T_{5}^{13} + 3697 T_{5}^{12} + \cdots + 729 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} \)
|
| $3$ |
\( (T^{9} + T^{8} - 18 T^{7} + \cdots - 36)^{2} \)
|
| $5$ |
\( T^{18} + 25 T^{16} + \cdots + 729 \)
|
| $7$ |
\( T^{18} \)
|
| $11$ |
\( (T^{9} + 2 T^{8} + \cdots - 468)^{2} \)
|
| $13$ |
\( T^{18} + \cdots + 10604499373 \)
|
| $17$ |
\( T^{18} + \cdots + 38805848064 \)
|
| $19$ |
\( (T^{9} + T^{8} - 77 T^{7} + \cdots + 1896)^{2} \)
|
| $23$ |
\( T^{18} + \cdots + 141580617984 \)
|
| $29$ |
\( T^{18} - 2 T^{17} + \cdots + 927369 \)
|
| $31$ |
\( T^{18} + 7 T^{17} + \cdots + 2064969 \)
|
| $37$ |
\( T^{18} + \cdots + 287364612096 \)
|
| $41$ |
\( T^{18} + \cdots + 291104490681 \)
|
| $43$ |
\( T^{18} + \cdots + 256754010681 \)
|
| $47$ |
\( T^{18} + \cdots + 215391594609 \)
|
| $53$ |
\( T^{18} + \cdots + 1455498801 \)
|
| $59$ |
\( T^{18} + \cdots + 6347497291776 \)
|
| $61$ |
\( (T^{9} + T^{8} + \cdots + 10140682)^{2} \)
|
| $67$ |
\( (T^{9} + 29 T^{8} + \cdots + 7838036)^{2} \)
|
| $71$ |
\( T^{18} + \cdots + 3994485901641 \)
|
| $73$ |
\( T^{18} + \cdots + 123708070942569 \)
|
| $79$ |
\( T^{18} + \cdots + 55833279675889 \)
|
| $83$ |
\( (T^{9} - 15 T^{8} + \cdots - 874800)^{2} \)
|
| $89$ |
\( T^{18} + \cdots + 1557623810304 \)
|
| $97$ |
\( T^{18} + \cdots + 54\!\cdots\!09 \)
|
| show more | |
| show less |