Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2025,2,Mod(649,2025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2025.649");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2025 = 3^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2025.b (of order \(2\), degree \(1\), 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: | \(16.1697064093\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.5089536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 45) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{5} + 48\nu^{4} - 12\nu^{3} - 2\nu^{2} + 12\nu + 701 ) / 131 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} + 226\nu - 138 ) / 131 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 46\nu^{5} - 56\nu^{4} + 14\nu^{3} + 308\nu^{2} + 772\nu - 534 ) / 393 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 23\nu^{2} + 386\nu - 267 ) / 131 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} + 3\beta_{4} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 4\beta_{3} + 4\beta_{2} + \beta _1 - 5 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 3\beta _1 - 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14\beta_{5} - 3\beta_{4} + 18\beta_{3} - 18\beta_{2} + 7\beta _1 - 31 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(1702\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
− | 2.51414i | 0 | −4.32088 | 0 | 0 | − | 0.514137i | 5.83502i | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 649.2 | − | 2.08613i | 0 | −2.35194 | 0 | 0 | − | 4.08613i | 0.734191i | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 649.3 | − | 0.571993i | 0 | 1.67282 | 0 | 0 | 1.42801i | − | 2.10083i | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 649.4 | 0.571993i | 0 | 1.67282 | 0 | 0 | − | 1.42801i | 2.10083i | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 649.5 | 2.08613i | 0 | −2.35194 | 0 | 0 | 4.08613i | − | 0.734191i | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 649.6 | 2.51414i | 0 | −4.32088 | 0 | 0 | 0.514137i | − | 5.83502i | 0 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2025.2.b.m | 6 | |
| 3.b | odd | 2 | 1 | 2025.2.b.l | 6 | ||
| 5.b | even | 2 | 1 | inner | 2025.2.b.m | 6 | |
| 5.c | odd | 4 | 1 | 405.2.a.i | 3 | ||
| 5.c | odd | 4 | 1 | 2025.2.a.o | 3 | ||
| 9.c | even | 3 | 2 | 675.2.k.b | 12 | ||
| 9.d | odd | 6 | 2 | 225.2.k.b | 12 | ||
| 15.d | odd | 2 | 1 | 2025.2.b.l | 6 | ||
| 15.e | even | 4 | 1 | 405.2.a.j | 3 | ||
| 15.e | even | 4 | 1 | 2025.2.a.n | 3 | ||
| 20.e | even | 4 | 1 | 6480.2.a.bs | 3 | ||
| 45.h | odd | 6 | 2 | 225.2.k.b | 12 | ||
| 45.j | even | 6 | 2 | 675.2.k.b | 12 | ||
| 45.k | odd | 12 | 2 | 135.2.e.b | 6 | ||
| 45.k | odd | 12 | 2 | 675.2.e.b | 6 | ||
| 45.l | even | 12 | 2 | 45.2.e.b | ✓ | 6 | |
| 45.l | even | 12 | 2 | 225.2.e.b | 6 | ||
| 60.l | odd | 4 | 1 | 6480.2.a.bv | 3 | ||
| 180.v | odd | 12 | 2 | 720.2.q.i | 6 | ||
| 180.x | even | 12 | 2 | 2160.2.q.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.2.e.b | ✓ | 6 | 45.l | even | 12 | 2 | |
| 135.2.e.b | 6 | 45.k | odd | 12 | 2 | ||
| 225.2.e.b | 6 | 45.l | even | 12 | 2 | ||
| 225.2.k.b | 12 | 9.d | odd | 6 | 2 | ||
| 225.2.k.b | 12 | 45.h | odd | 6 | 2 | ||
| 405.2.a.i | 3 | 5.c | odd | 4 | 1 | ||
| 405.2.a.j | 3 | 15.e | even | 4 | 1 | ||
| 675.2.e.b | 6 | 45.k | odd | 12 | 2 | ||
| 675.2.k.b | 12 | 9.c | even | 3 | 2 | ||
| 675.2.k.b | 12 | 45.j | even | 6 | 2 | ||
| 720.2.q.i | 6 | 180.v | odd | 12 | 2 | ||
| 2025.2.a.n | 3 | 15.e | even | 4 | 1 | ||
| 2025.2.a.o | 3 | 5.c | odd | 4 | 1 | ||
| 2025.2.b.l | 6 | 3.b | odd | 2 | 1 | ||
| 2025.2.b.l | 6 | 15.d | odd | 2 | 1 | ||
| 2025.2.b.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 2025.2.b.m | 6 | 5.b | even | 2 | 1 | inner | |
| 2160.2.q.k | 6 | 180.x | even | 12 | 2 | ||
| 6480.2.a.bs | 3 | 20.e | even | 4 | 1 | ||
| 6480.2.a.bv | 3 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2025, [\chi])\):
|
\( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 9 \)
|
|
\( T_{11}^{3} - 2T_{11}^{2} - 8T_{11} + 12 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 11 T^{4} + \cdots + 9 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 19 T^{4} + \cdots + 9 \)
|
| $11$ |
\( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \)
|
| $13$ |
\( T^{6} + 24 T^{4} + \cdots + 16 \)
|
| $17$ |
\( T^{6} + 20 T^{4} + \cdots + 144 \)
|
| $19$ |
\( (T^{3} + 4 T^{2} - 4 T - 4)^{2} \)
|
| $23$ |
\( T^{6} + 75 T^{4} + \cdots + 13689 \)
|
| $29$ |
\( (T^{3} + 7 T^{2} - 29 T - 51)^{2} \)
|
| $31$ |
\( (T^{3} - 8 T^{2} + \cdots + 468)^{2} \)
|
| $37$ |
\( T^{6} + 60 T^{4} + \cdots + 16 \)
|
| $41$ |
\( (T^{3} - 13 T^{2} + 19 T - 3)^{2} \)
|
| $43$ |
\( T^{6} + 108 T^{4} + \cdots + 16 \)
|
| $47$ |
\( T^{6} + 191 T^{4} + \cdots + 136161 \)
|
| $53$ |
\( T^{6} + 44 T^{4} + \cdots + 576 \)
|
| $59$ |
\( (T^{3} + 2 T^{2} - 20 T - 24)^{2} \)
|
| $61$ |
\( (T^{3} - T^{2} - 37 T - 71)^{2} \)
|
| $67$ |
\( T^{6} + 199 T^{4} + \cdots + 257049 \)
|
| $71$ |
\( (T^{3} - 10 T^{2} + \cdots + 708)^{2} \)
|
| $73$ |
\( T^{6} + 192 T^{4} + \cdots + 16384 \)
|
| $79$ |
\( (T^{3} + 2 T^{2} - 84 T + 24)^{2} \)
|
| $83$ |
\( T^{6} + 171 T^{4} + \cdots + 6561 \)
|
| $89$ |
\( (T + 3)^{6} \)
|
| $97$ |
\( T^{6} + 396 T^{4} + \cdots + 1700416 \)
|
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