Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,4,Mod(1,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.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: | \(2.89109359028\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{65})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 35x^{2} + 36x + 194 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 7 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 35x^{2} + 36x + 194 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 100\nu - 79 ) / 57 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 11\nu^{2} + 12\nu - 182 ) / 19 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\nu^{3} + 12\nu^{2} - 265\nu - 392 ) / 57 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 7\beta _1 + 7 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + 10\beta_{2} + 7\beta _1 + 133 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8\beta_{3} - 5\beta_{2} + 23\beta _1 + 39 \)
|
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 |
|
−3.53113 | −7.82220 | 4.46887 | −2.07730 | 27.6212 | 0 | 12.4689 | 34.1868 | 7.33521 | ||||||||||||||||||||||||||||||
| 1.2 | −3.53113 | 7.82220 | 4.46887 | 2.07730 | −27.6212 | 0 | 12.4689 | 34.1868 | −7.33521 | |||||||||||||||||||||||||||||||
| 1.3 | 4.53113 | −3.57956 | 12.5311 | 13.4791 | −16.2194 | 0 | 20.5311 | −14.1868 | 61.0753 | |||||||||||||||||||||||||||||||
| 1.4 | 4.53113 | 3.57956 | 12.5311 | −13.4791 | 16.2194 | 0 | 20.5311 | −14.1868 | −61.0753 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
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 | 49.4.a.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 441.4.a.u | 4 | ||
| 4.b | odd | 2 | 1 | 784.4.a.bf | 4 | ||
| 5.b | even | 2 | 1 | 1225.4.a.bb | 4 | ||
| 7.b | odd | 2 | 1 | inner | 49.4.a.e | ✓ | 4 |
| 7.c | even | 3 | 2 | 49.4.c.e | 8 | ||
| 7.d | odd | 6 | 2 | 49.4.c.e | 8 | ||
| 21.c | even | 2 | 1 | 441.4.a.u | 4 | ||
| 21.g | even | 6 | 2 | 441.4.e.y | 8 | ||
| 21.h | odd | 6 | 2 | 441.4.e.y | 8 | ||
| 28.d | even | 2 | 1 | 784.4.a.bf | 4 | ||
| 35.c | odd | 2 | 1 | 1225.4.a.bb | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 49.4.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 49.4.a.e | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 49.4.c.e | 8 | 7.c | even | 3 | 2 | ||
| 49.4.c.e | 8 | 7.d | odd | 6 | 2 | ||
| 441.4.a.u | 4 | 3.b | odd | 2 | 1 | ||
| 441.4.a.u | 4 | 21.c | even | 2 | 1 | ||
| 441.4.e.y | 8 | 21.g | even | 6 | 2 | ||
| 441.4.e.y | 8 | 21.h | odd | 6 | 2 | ||
| 784.4.a.bf | 4 | 4.b | odd | 2 | 1 | ||
| 784.4.a.bf | 4 | 28.d | even | 2 | 1 | ||
| 1225.4.a.bb | 4 | 5.b | even | 2 | 1 | ||
| 1225.4.a.bb | 4 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(49))\):
|
\( T_{2}^{2} - T_{2} - 16 \)
|
|
\( T_{3}^{4} - 74T_{3}^{2} + 784 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T - 16)^{2} \)
|
| $3$ |
\( T^{4} - 74T^{2} + 784 \)
|
| $5$ |
\( T^{4} - 186T^{2} + 784 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 50 T + 40)^{2} \)
|
| $13$ |
\( T^{4} - 3234 T^{2} + \cdots + 2458624 \)
|
| $17$ |
\( T^{4} - 14564 T^{2} + \cdots + 9746884 \)
|
| $19$ |
\( T^{4} - 14746 T^{2} + \cdots + 52591504 \)
|
| $23$ |
\( (T^{2} - 176 T + 6704)^{2} \)
|
| $29$ |
\( (T^{2} - 130 T + 1040)^{2} \)
|
| $31$ |
\( T^{4} - 100104 T^{2} + \cdots + 629407744 \)
|
| $37$ |
\( (T^{2} - 106 T + 2224)^{2} \)
|
| $41$ |
\( T^{4} - 66836 T^{2} + \cdots + 307721764 \)
|
| $43$ |
\( (T^{2} - 270 T - 61400)^{2} \)
|
| $47$ |
\( T^{4} - 187240 T^{2} + \cdots + 8332038400 \)
|
| $53$ |
\( (T^{2} - 8 T - 21044)^{2} \)
|
| $59$ |
\( T^{4} - 189354 T^{2} + \cdots + 1600960144 \)
|
| $61$ |
\( T^{4} - 360266 T^{2} + \cdots + 5022273424 \)
|
| $67$ |
\( (T^{2} + 972 T + 142336)^{2} \)
|
| $71$ |
\( (T^{2} - 1124 T + 303104)^{2} \)
|
| $73$ |
\( T^{4} - 276756 T^{2} + \cdots + 12428236324 \)
|
| $79$ |
\( (T^{2} + 524 T - 505696)^{2} \)
|
| $83$ |
\( T^{4} - 11466 T^{2} + \cdots + 6492304 \)
|
| $89$ |
\( T^{4} - 3623876 T^{2} + \cdots + 2844693770884 \)
|
| $97$ |
\( T^{4} - 3082884 T^{2} + \cdots + 841222821124 \)
|
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