Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2254,4,Mod(1,2254)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2254, 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("2254.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2254 = 2 \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2254.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: | \(132.990305153\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - 38x^{4} + 203x^{2} - 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 38x^{4} + 203x^{2} - 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 18\nu^{3} - 517\nu ) / 90 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{4} - 54\nu^{2} - 8 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} + 81\nu^{3} - 451\nu ) / 45 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 36\nu^{2} + 113 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 36\nu^{3} - 149\nu ) / 18 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{3} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{4} + \beta_{2} + 26 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -33\beta_{5} + 37\beta_{3} - 17\beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -27\beta_{4} + 18\beta_{2} + 355 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -1111\beta_{5} + 1183\beta_{3} - 463\beta_1 ) / 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
2.00000 | −5.32887 | 4.00000 | 6.69892 | −10.6577 | 0 | 8.00000 | 1.39685 | 13.3978 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.92350 | 4.00000 | −13.4068 | −7.84699 | 0 | 8.00000 | −11.6062 | −26.8137 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.34803 | 4.00000 | −2.71681 | −6.69606 | 0 | 8.00000 | −15.7907 | −5.43361 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 3.34803 | 4.00000 | 2.71681 | 6.69606 | 0 | 8.00000 | −15.7907 | 5.43361 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 3.92350 | 4.00000 | 13.4068 | 7.84699 | 0 | 8.00000 | −11.6062 | 26.8137 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 5.32887 | 4.00000 | −6.69892 | 10.6577 | 0 | 8.00000 | 1.39685 | −13.3978 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2254.4.a.n | ✓ | 6 |
| 7.b | odd | 2 | 1 | inner | 2254.4.a.n | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2254.4.a.n | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2254.4.a.n | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 55T_{3}^{4} + 928T_{3}^{2} - 4900 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{6} \)
|
| $3$ |
\( T^{6} - 55 T^{4} + \cdots - 4900 \)
|
| $5$ |
\( T^{6} - 232 T^{4} + \cdots - 59536 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T^{3} + 42 T^{2} + \cdots - 6424)^{2} \)
|
| $13$ |
\( T^{6} + \cdots - 1826365696 \)
|
| $17$ |
\( T^{6} + \cdots - 167032420416 \)
|
| $19$ |
\( T^{6} + \cdots - 13946665216 \)
|
| $23$ |
\( (T - 23)^{6} \)
|
| $29$ |
\( (T^{3} + 11 T^{2} + \cdots - 729928)^{2} \)
|
| $31$ |
\( T^{6} + \cdots - 1234556543236 \)
|
| $37$ |
\( (T^{3} + 90 T^{2} + \cdots + 6509592)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 30316916966400 \)
|
| $43$ |
\( (T^{3} + 300 T^{2} + \cdots - 5836864)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 423968113716196 \)
|
| $53$ |
\( (T^{3} + 438 T^{2} + \cdots + 10077416)^{2} \)
|
| $59$ |
\( T^{6} + \cdots - 10\!\cdots\!24 \)
|
| $61$ |
\( T^{6} + \cdots - 135511297828624 \)
|
| $67$ |
\( (T^{3} + 46 T^{2} + \cdots - 309969128)^{2} \)
|
| $71$ |
\( (T^{3} + 717 T^{2} + \cdots - 85056944)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 12\!\cdots\!56 \)
|
| $79$ |
\( (T^{3} + 1410 T^{2} + \cdots - 1011139928)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 49182056792064 \)
|
| $89$ |
\( T^{6} + \cdots - 52\!\cdots\!84 \)
|
| $97$ |
\( T^{6} + \cdots - 26\!\cdots\!00 \)
|
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