Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,4,Mod(1,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 675.a (trivial) |
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: | yes |
| Analytic conductor: | \(39.8262892539\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 21x^{4} + 5x^{3} + 101x^{2} + 29x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{4}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 135) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - x^{5} - 21x^{4} + 5x^{3} + 101x^{2} + 29x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 6\nu^{5} - 7\nu^{4} - 124\nu^{3} + 54\nu^{2} + 577\nu + 52 ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 6\nu^{5} - 12\nu^{4} - 114\nu^{3} + 114\nu^{2} + 522\nu + 2 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 21\nu^{5} - 27\nu^{4} - 414\nu^{3} + 159\nu^{2} + 1827\nu + 492 ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 18\nu^{5} - 21\nu^{4} - 372\nu^{3} + 147\nu^{2} + 1776\nu + 256 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -33\nu^{5} + 36\nu^{4} + 702\nu^{3} - 252\nu^{2} - 3441\nu - 501 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 5\beta_{4} - 2\beta_{3} - 12\beta _1 + 5 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{5} + 5\beta_{4} - 6\beta_{3} + 24\beta _1 + 215 ) / 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 23\beta_{5} + 75\beta_{4} - 26\beta_{3} - 5\beta_{2} - 96\beta _1 + 245 ) / 30 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 71\beta_{5} + 155\beta_{4} - 102\beta_{3} - 40\beta_{2} + 288\beta _1 + 2715 ) / 30 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 87\beta_{5} + 241\beta_{4} - 82\beta_{3} - 30\beta_{2} - 132\beta _1 + 1111 ) / 6 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.92742 | 0 | 16.2795 | 0 | 0 | 34.7816 | −40.7965 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.62699 | 0 | 5.15507 | 0 | 0 | −22.4519 | 10.3185 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.25118 | 0 | −6.43456 | 0 | 0 | 5.67029 | 18.0602 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.25118 | 0 | −6.43456 | 0 | 0 | 5.67029 | −18.0602 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.62699 | 0 | 5.15507 | 0 | 0 | −22.4519 | −10.3185 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.92742 | 0 | 16.2795 | 0 | 0 | 34.7816 | 40.7965 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 675.4.a.bc | 6 | |
| 3.b | odd | 2 | 1 | inner | 675.4.a.bc | 6 | |
| 5.b | even | 2 | 1 | 675.4.a.bb | 6 | ||
| 5.c | odd | 4 | 2 | 135.4.b.c | ✓ | 12 | |
| 15.d | odd | 2 | 1 | 675.4.a.bb | 6 | ||
| 15.e | even | 4 | 2 | 135.4.b.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.b.c | ✓ | 12 | 5.c | odd | 4 | 2 | |
| 135.4.b.c | ✓ | 12 | 15.e | even | 4 | 2 | |
| 675.4.a.bb | 6 | 5.b | even | 2 | 1 | ||
| 675.4.a.bb | 6 | 15.d | odd | 2 | 1 | ||
| 675.4.a.bc | 6 | 1.a | even | 1 | 1 | trivial | |
| 675.4.a.bc | 6 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):
|
\( T_{2}^{6} - 39T_{2}^{4} + 378T_{2}^{2} - 500 \)
|
|
\( T_{7}^{3} - 18T_{7}^{2} - 711T_{7} + 4428 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 39 T^{4} + \cdots - 500 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{3} - 18 T^{2} + \cdots + 4428)^{2} \)
|
| $11$ |
\( T^{6} - 3357 T^{4} + \cdots - 192820500 \)
|
| $13$ |
\( (T^{3} - 81 T^{2} + \cdots + 128925)^{2} \)
|
| $17$ |
\( T^{6} + \cdots - 76335368000 \)
|
| $19$ |
\( (T^{3} - 5943 T + 119558)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 307148112500 \)
|
| $29$ |
\( T^{6} + \cdots - 4585850680500 \)
|
| $31$ |
\( (T^{3} - 63 T^{2} + \cdots + 186100)^{2} \)
|
| $37$ |
\( (T^{3} - 504 T^{2} + \cdots - 3593970)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 967871758050000 \)
|
| $43$ |
\( (T^{3} - 279 T^{2} + \cdots + 3020436)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 869738053832000 \)
|
| $53$ |
\( T^{6} + \cdots - 108910045472000 \)
|
| $59$ |
\( T^{6} + \cdots - 35\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} - 198 T^{2} + \cdots + 19133944)^{2} \)
|
| $67$ |
\( (T^{3} - 1134 T^{2} + \cdots - 1053540)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 74\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} - 72 T^{2} + \cdots - 51707970)^{2} \)
|
| $79$ |
\( (T^{3} + 549 T^{2} + \cdots - 285462565)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots - 35\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} - 2052 T^{2} + \cdots + 2585429982)^{2} \)
|
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